## Site.Introduction History

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17 May 2017
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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]]\\([[Site.Pascaline1652|Pascaline "1652" - working exemplar~~ - ~~]] - [[Site.ObjectsGallery|collection Calculant]])

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]]\\([[Site.Pascaline1652|Pascaline "1652" - working exemplar]] - [[Site.ObjectsGallery|collection Calculant]])

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]]\\([[Site.Pascaline1652|~~Replica ~~Pascaline "1652"]] - [[Site.ObjectsGallery|collection Calculant]])

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]]\\([[Site.Pascaline1652|Pascaline "1652" - working exemplar - ]] - [[Site.ObjectsGallery|collection Calculant]])

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%center% http://metastudies.net/pmwiki/uploads/PA_1.jpg\\(~~Replica ~~Pascaline "1652" - collection Calculant)

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%center% http://metastudies.net/pmwiki/uploads/PA_1.jpg\\(Pascaline "1652" - working exemplar - collection Calculant)

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But the same is true of invention. At different times and in different cultures there have been quite different views taken on the value of change, and thus invention. At some points in history the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). ~~A~~ other times or places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do it. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. This is as true in mathematics as in other areas of human activity.

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But the same is true of invention. At different times and in different cultures there have been quite different views taken on the value of change, and thus invention. At some points in history the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). At other times or places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do it. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. This is as true in mathematics as in other areas of human activity.

14 April 2014
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**Welcome to [[http://things-that-count.com|things-that-count.~~com~~]].** This website describes a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) and uses it to help develop a historical account of the way humans developed the need and capacity to calculate, the things they used to help them, and how human societies (and even human brains) evolved with those developments.

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**Welcome to [[http://things-that-count.com|things-that-count.net]].** This website describes a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) and uses it to help develop a historical account of the way humans developed the need and capacity to calculate, the things they used to help them, and how human societies (and even human brains) evolved with those developments.

12 April 2014
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(:description A site dealing with the history of ~~calculators and~~ calculating technology:)

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(:description A site dealing with the history of counting, history of calculators, and history of calculating technology:)

12 April 2014
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12 April 2014
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08 April 2014
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(:description A site dealing with the history of calculators and calculating technology:)

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The objects (:if equal {Site.PrintBook$:PSW} "False":)in [[[[Objects gallery|"collection Calculant"]](:ifend:) described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (which are described in more detail in an accompanying web site - "things-that-count.com")[^The full address is http://things-that-count.com. The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:)~~.~~ The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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The objects (:if equal {Site.PrintBook$:PSW} "False":)in [[[[Objects gallery|"collection Calculant"]](:ifend:) described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (which are described in more detail in an accompanying web site - "things-that-count.com")[^The full address is http://things-that-count.com. The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:) The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

02 April 2014
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(:keywords history, calculators, rechemaschine, rechenschieber, slide rule, Babylonia, Summeria, Ancient Rome, Ancient Greece, Ancient China, France, Germany, renaissance, Caculatrice, Pascal, Leibnitz, mathematics, arithmetic, Napier, Neper, Briggs, computer, social history, calculator, adding machine, mathematical instruments, Hewlett Packard, MADAS, Thomas de Colmar, arithmometer, pin wheel, calculator, logarithms, Gunter scale, History of Science, History of Technology, Science and Technology Studies, STS, Engineering, Astronomy, Ptolemy, Horology, anthropology, Ancient Egypt :)

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(:keywords history, calculators, rechemaschine, rechenschieber, slide rule, Babylonia, Summeria, Ancient Rome, Ancient Greece, Ancient China, France, Germany, renaissance, Caculatrice, Pascal, Leibnitz, mathematics, arithmetic, Napier, Neper, Briggs, computer, social history, calculator, adding machine, mathematical instruments, Hewlett Packard, MADAS, Thomas de Colmar, arithmometer, pin wheel, calculator, logarithms, Gunter scale, History of Science, History of Technology, Science and Technology Studies, STS, Engineering, Astronomy, Ptolemy, Horology, anthropology, Ancient Egypt, things that count, things-that-count, Jim Falk, University of Melbourne :)

02 April 2014
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(:keywords history calculators rechemaschine rechenschieber slide ~~rules~~ :)

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(:keywords history, calculators, rechemaschine, rechenschieber, slide rule, Babylonia, Summeria, Ancient Rome, Ancient Greece, Ancient China, France, Germany, renaissance, Caculatrice, Pascal, Leibnitz, mathematics, arithmetic, Napier, Neper, Briggs, computer, social history, calculator, adding machine, mathematical instruments, Hewlett Packard, MADAS, Thomas de Colmar, arithmometer, pin wheel, calculator, logarithms, Gunter scale, History of Science, History of Technology, Science and Technology Studies, STS, Engineering, Astronomy, Ptolemy, Horology, anthropology, Ancient Egypt :)

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(:keywords history calculators rechemaschine rechenschieber slide rules :)

31 March 2014
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30 March 2014
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This page has been visited {$PageCount} times.

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This page has been visited {$PageCount} times since 31 March 2014.

24 March 2014
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23 March 2014
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The calculational devices that were developed show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But, as already noted, it is also important to understand /~~/~~why/~~/~~ they were invented and used.

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The calculational devices that were developed show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But, as already noted, it is also important to understand '/why/' they were invented and used.

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Even keeping our attention restricted to the basic arithmetic operations, it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, /~~/~~The Oxford handbook of the history of mathematics/~~/~~, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, /~~/~~Mathematics in Society and History: Sociological Inquiries/~~/~~, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" of mathematics it will still be useful to take some account of this literature and its insights. In particular, whether concerning ourselves with the evolution of the simple areas of mathematics or the more obstruse areas, one question is always raised: what led to this particular development happening as and when it did?

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Even keeping our attention restricted to the basic arithmetic operations, it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, '/The Oxford handbook of the history of mathematics/', Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, '/Mathematics in Society and History: Sociological Inquiries/', Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" of mathematics it will still be useful to take some account of this literature and its insights. In particular, whether concerning ourselves with the evolution of the simple areas of mathematics or the more obstruse areas, one question is always raised: what led to this particular development happening as and when it did?

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In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom. As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, /~~/~~The Mathematical Practitioners of Tudor & Stuart England 1485-1714/~~/~~, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

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In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom. As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, '/The Mathematical Practitioners of Tudor & Stuart England 1485-1714/', Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

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/~~/~~In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting)./~~/~~ Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book~~"~~The Calculating Machines (Die Rechenmaschinen)~~"~~ is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, /~~/~~The Calculating Machines (Die Rechenmaschinen)/~~/~~, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which would widely be understood to be a machine. But with that caveat, the term "calculator" will be used here very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

to:

'/In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting)./' Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book '/The Calculating Machines (Die Rechenmaschinen)/' is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, '/The Calculating Machines (Die Rechenmaschinen)/', 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which would widely be understood to be a machine. But with that caveat, the term "calculator" will be used here very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), '/Science in Context/', The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book '/The Calculating Machines (Die Rechenmaschinen)/' is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, '/The Calculating Machines (Die Rechenmaschinen)/', 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which would widely be understood to be a machine. But with that caveat, the term "calculator" will be used here very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), '/Science in Context/', The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

22 March 2014
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In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom. As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

22 March 2014
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Thus the historical account is broken into three parts. The first part looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop but only ~~refers to one object in the collection.~~ Apart from that object (which is some 4,000 years old) the objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800) when mechanical calculation began to gain greater use in the broader society.

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Thus the historical account is broken into three parts. The first part looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop but only one object in the collection is of an appropriate age. Apart from that object (which is some 4,000 years old) the objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800) when mechanical calculation began to gain greater use in the broader society.

22 March 2014
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22 March 2014
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The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what became increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning, the aides~~ were~~ were very simple - for example, marks scribed on bones and patterns arranged with pebbles. Later, primitive devices and tables were developed and sold. Over time, much more elaborate mechanical devices were developed to help in this task. Many of these devices, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

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The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what became increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning, the aides were very simple - for example, marks scribed on bones and patterns arranged with pebbles. Later, primitive devices and tables were developed and sold. Over time, much more elaborate mechanical devices were developed to help in this task. Many of these devices, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

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Early "calculators" were not things~~, but rather~~ people who were employed to calculate. Over time these people were first aided, ~~and then~~ replaced by calculating devices. These devices became very widely used across many countries. ~~But we are now passing the heyday of such stand-alone calculators. Increasingly, since the advent of electronic computing~~, ~~the aides to calculation appear in virtual form in~~ apps in phones, tablets and laptops. The end of calculators, seen as devices, in this sense is looming.

to:

Early "calculators" were not things. Rather they were people who were employed to calculate. Over time these people were first aided, but later replaced by calculating devices. These devices became very widely used across many countries. There is evidence we may now be passing the heyday of such stand-alone calculators. This is because, increasingly since the advent of electronic computing, the aides to calculation have begun to appear in virtual form as apps in phones, tablets and laptops. The end of calculators, seen as devices, in this sense is looming.

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The objects (:if equal {Site.PrintBook$:PSW} "False":)in [[[[Objects gallery|"collection Calculant"]](:ifend:) described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (~~the name also of an accompanying web~~ site)[^~~see~~ http://things-that-count.com. The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The objects (:if equal {Site.PrintBook$:PSW} "False":)in [[[[Objects gallery|"collection Calculant"]](:ifend:) described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (which are described in more detail in an accompanying web site - "things-that-count.com")[^The full address is http://things-that-count.com. The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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As most people know, the spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction - from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

to:

As most people know, the spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn, ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction - from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

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Even keeping our attention restricted to the basic arithmetic operations, it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" ~~to~~ mathematics it will still be useful to take some account of this literature and its insights. In particular, whether concerning ourselves with the evolution of the simple areas of mathematics or the more obstruse areas, one question is always raised: what led to this particular development happening ~~how~~ and when it did?

to:

Even keeping our attention restricted to the basic arithmetic operations, it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" of mathematics it will still be useful to take some account of this literature and its insights. In particular, whether concerning ourselves with the evolution of the simple areas of mathematics or the more obstruse areas, one question is always raised: what led to this particular development happening as and when it did?

Changed lines 44-51 from:

It might be assumed that arithmetic, and more broadly, mathematics, developed through a process that was entirely internal to itself. For example, this development might have been propelled forward because people could ask questions which arise within mathematics, but require the invention of new mathematics to answer them. ~~For example if you know about addition and that 2+2 =4 then it is possible to ask what number added to 4 gives ~~2. Answering that involves ~~some~~ idea of a negative number. This ~~progress through 'completing mathematics' is certainly part of the story. Yet~~ the ~~literature on history of~~ mathematics ~~tells us this cannot ~~be ~~all~~.

The idea of 'mathematics', and doing it, are themselves inventions. The question of what sort of problems mathematical thinking should be applied to will have different answers in different cultures. In different societies different sorts of issues will be seen as interesting or important (and only some of these will be usefully tackled with mathematics). Also different groups of people will be educated to different degrees (if at all) in what is known inmathematics. ~~ Finally, different groups may also have influence in framing the questions mathematicians are encouraged to explore~~.

Similarly at different ~~times and in different cultures there~~ have ~~been very different views taken on the value of invention. At some moments~~ the ~~mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it ~~is ~~to be used for, and by whom.~~

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

The idea of 'mathematics', and doing it, are themselves inventions. The question of what sort of problems mathematical thinking should be applied to will have different answers in different cultures. In different societies different sorts of issues will be seen as interesting or important (and only some of these will be usefully tackled with mathematics). Also different groups of people will be educated to different degrees (if at all) in what is known in

Similarly at

As

to:

It might be assumed that arithmetic, and more broadly, mathematics, developed through a process that was entirely internal to itself. For example, this development might have been propelled forward because people could ask questions which arise within mathematics, but require the invention of new mathematics to answer them.

Suppose we know about addition and that 2+2 =4. Then it is possible to ask what number added to 4 would give 2. Answering that involves inventing the idea of a negative number. This leads to progress through 'completing mathematics' (i.e. seeking to answer all the questions that arise in mathematics which cannot yet be answered.) That must be part of the story of how mathematics develops. Yet the literature on the history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The question of when mathematics might be useful will have different answers in different cultures. Different societies may identify different sorts of issues as interesting or important (and only some of these will be usefully tackled with mathematics). Also different groups of people in those societies will be educated in what is known in mathematics. Finally, different groups of people, or organisations, may have influence in framing the questions that mathematicians are encouraged (and resourced) to explore.

But the same is true of invention. At different times and in different cultures there have been quite different views taken on the value of change, and thus invention. At some points in history the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). A other times or places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do it. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. This is as true in mathematics as in other areas of human activity.

In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom. As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

Suppose we know about addition and that 2+2 =4. Then it is possible to ask what number added to 4 would give 2. Answering that involves inventing the idea of a negative number. This leads to progress through 'completing mathematics' (i.e. seeking to answer all the questions that arise in mathematics which cannot yet be answered.) That must be part of the story of how mathematics develops. Yet the literature on the history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The question of when mathematics might be useful will have different answers in different cultures. Different societies may identify different sorts of issues as interesting or important (and only some of these will be usefully tackled with mathematics). Also different groups of people in those societies will be educated in what is known in mathematics. Finally, different groups of people, or organisations, may have influence in framing the questions that mathematicians are encouraged (and resourced) to explore.

But the same is true of invention. At different times and in different cultures there have been quite different views taken on the value of change, and thus invention. At some points in history the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). A other times or places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do it. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. This is as true in mathematics as in other areas of human activity.

In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom. As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

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This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which would widely be understood to be a machine. But with that caveat, ~~we will be using the term "calculator"~~ very broadly.

to:

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which would widely be understood to be a machine. But with that caveat, the term "calculator" will be used here very broadly.

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Thus the historical account is broken into three parts. The first part looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop but ~~does not refer at all to specific objects in the collection. The~~ objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800) when mechanical calculation began to gain greater use in the broader society.

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Thus the historical account is broken into three parts. The first part looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop but only refers to one object in the collection. Apart from that object (which is some 4,000 years old) the objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800) when mechanical calculation began to gain greater use in the broader society.

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**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website describes a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) and uses it to help develop a historical account of the way humans developed the need and capacity calculate, the things they used to help them, and how human societies (and even human brains) evolved with those developments.

to:

**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website describes a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) and uses it to help develop a historical account of the way humans developed the need and capacity to calculate, the things they used to help them, and how human societies (and even human brains) evolved with those developments.

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**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website describes a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) and uses it to help develop a historical account of the way humans~~, over millennia, have developed the need and capacity to count and calculate,~~ and how human societies (and even human brains)~~ have~~ evolved with those developments.

to:

**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website describes a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) and uses it to help develop a historical account of the way humans developed the need and capacity calculate, the things they used to help them, and how human societies (and even human brains) evolved with those developments.

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**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website~~ both~~ describes a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) and uses it to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

to:

**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website describes a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) and uses it to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

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Fortunately in order to understand what has shaped the development of these calculational aids we can largely avoid talking much about mathematics. This is lucky because mathematics is by now a huge field of knowledge. So ~~in this site we will avoid~~ calculus, set and group theory, the mathematics of infinite dimensional vector spaces that make the modern formulation of quantum mechanics possible, and tensors which Einstein used to express his wonderfully neat equations for the shape of space-time.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It will be sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations which in the end are constituted out of additions and subtractions (and multiplications and divisions) and can only be carried out in workable times with the use of ever faster calculating devices.

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Fortunately in order to understand what has shaped the development of these calculational aids we can largely avoid talking much about mathematics. This is lucky because mathematics is by now a huge field of knowledge. So you are entitled to relief that in this site we will avoid much of mathematics. We need not touch, for example, on calculus, set and group theory, the mathematics of infinite dimensional vector spaces that make the modern formulation of quantum mechanics possible, and tensors which Einstein used to express his wonderfully neat equations for the shape of space-time.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It will be sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations which in the end are constituted out of additions and subtractions (and multiplications and divisions) and can only be carried out in workable times with the use of ever faster calculating devices.

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It might be assumed that ~~mathematics developed through a process that was entirely internal to itself. For example, its development might have been propelled forward because people could ask questions which arise within mathematics, but require the invention of new mathematics to answer them. This~~ is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

to:

It might be assumed that arithmetic, and more broadly, mathematics, developed through a process that was entirely internal to itself. For example, this development might have been propelled forward because people could ask questions which arise within mathematics, but require the invention of new mathematics to answer them. For example if you know about addition and that 2+2 =4 then it is possible to ask what number added to 4 gives 2. Answering that involves some idea of a negative number. This progress through 'completing mathematics' is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

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The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what ~~were becoming ~~increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning the aides were were very simple - for example, marks scribed on bones and patterns arranged with pebbles. Later primitive devices and tables were developed and sold. Over time much more elaborate mechanical devices were developed to help in this task. Many of these ~~things~~, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

The need for calculation, however has prospered. As societies have become more complex, transactions in them depending on arithmetic~~- ~~the familiar tasks of counting, adding, subtracting, multiplying and dividing ~~- and~~ more complex mathematics have intensified. Yet over much of this period, for many people in these societies, doing even the ~~simple~~ arithmetic tasks has been neither easy nor ~~even accessible~~. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to ~~these simple goals (rather than the development of complex~~ mathematics) which is the primary focus of this website.

Early "calculators" were not things, but rather people who were employed to calculate. Over time these people were first aided, and then replaced by calculating devices. These devices became very widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Increasingly since the advent of electronic computing, the aides to calculation appear in virtual form in apps in phones, tablets and laptops. The end of calculators~~in this sense is looming. ~~

One might imagine that a history of calculators would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this and the minutae of mechanical invention. But to focus simply on that is to ~~oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg~~ a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

The need for calculation, however has prospered. As societies have become more complex, transactions in them depending on arithmetic

Early "calculators" were not things, but rather people who were employed to calculate. Over time these people were first aided, and then replaced by calculating devices. These devices became very widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Increasingly since the advent of electronic computing, the aides to calculation appear in virtual form in apps in phones, tablets and laptops. The end of calculators

One might imagine that a history of calculators would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this and the minutae of mechanical invention. But to focus simply on that is

to:

The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what became increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning, the aides were were very simple - for example, marks scribed on bones and patterns arranged with pebbles. Later, primitive devices and tables were developed and sold. Over time, much more elaborate mechanical devices were developed to help in this task. Many of these devices, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

The need for calculation, however has prospered. As societies have become more complex, transactions in them depending on arithmetic (the familiar tasks of counting, adding, subtracting, multiplying and dividing), as well as more complex mathematics, have intensified. Yet over much of this period, for many people in these societies, doing even the simplest arithmetic tasks has been neither easy nor, for some, comprehensible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to aid achieving these simple goals (rather than the development of complex mathematics) which is the primary focus of this website.

Early "calculators" were not things, but rather people who were employed to calculate. Over time these people were first aided, and then replaced by calculating devices. These devices became very widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Increasingly, since the advent of electronic computing, the aides to calculation appear in virtual form in apps in phones, tablets and laptops. The end of calculators, seen as devices, in this sense is looming.

One might imagine that a history of calculators would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this with loving details of the minutae of mechanical invention. But to focus simply on that is to oversimplify and lose much of what is potentially interesting. Across human history many weird things were indeed devised for doing simple calculations. But the development of these begs a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

The need for calculation, however has prospered. As societies have become more complex, transactions in them depending on arithmetic (the familiar tasks of counting, adding, subtracting, multiplying and dividing), as well as more complex mathematics, have intensified. Yet over much of this period, for many people in these societies, doing even the simplest arithmetic tasks has been neither easy nor, for some, comprehensible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to aid achieving these simple goals (rather than the development of complex mathematics) which is the primary focus of this website.

Early "calculators" were not things, but rather people who were employed to calculate. Over time these people were first aided, and then replaced by calculating devices. These devices became very widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Increasingly, since the advent of electronic computing, the aides to calculation appear in virtual form in apps in phones, tablets and laptops. The end of calculators, seen as devices, in this sense is looming.

One might imagine that a history of calculators would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this with loving details of the minutae of mechanical invention. But to focus simply on that is to oversimplify and lose much of what is potentially interesting. Across human history many weird things were indeed devised for doing simple calculations. But the development of these begs a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

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The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what were becoming increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning the aides were were very simple - for example, marks scribed on bones and patterns arranged with pebbles. Later devices and tables were developed and sold~~, and over~~ time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

to:

The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what were becoming increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning the aides were were very simple - for example, marks scribed on bones and patterns arranged with pebbles. Later primitive devices and tables were developed and sold. Over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

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The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what were becoming increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning the aides were were very simple - for example, ~~utilising ~~marks on bones and patterns ~~of~~ pebbles. Later devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

to:

The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what were becoming increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning the aides were were very simple - for example, marks scribed on bones and patterns arranged with pebbles. Later devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

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The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over ~~this ~~time these capabilities became essential ingredients in ~~the skills which enabled them to build increasingly complex societies. The operation of these societies were increasingly dependent on more of their citizens being able to manipulate numbers in their daily lives. As the need for this ability spread a wide range of aides of various sorts were developed to help~~. At the beginning the aides were were very simple - for example, utilising marks on bones and patterns of pebbles. Later devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

to:

The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what were becoming increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning the aides were were very simple - for example, utilising marks on bones and patterns of pebbles. Later devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

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The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over this time these became essential ingredients in the skills which enabled them to build increasingly complex societies. ~~These societies depended on more of their citizens being able to manipulate numbers in their daily lives. As this requirement~~ spread a wide range of aides of various sorts were developed to help. At the beginning the aides were were very simple - for example, utilising marks on bones and patterns of pebbles. Later devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

to:

The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over this time these capabilities became essential ingredients in the skills which enabled them to build increasingly complex societies. The operation of these societies were increasingly dependent on more of their citizens being able to manipulate numbers in their daily lives. As the need for this ability spread a wide range of aides of various sorts were developed to help. At the beginning the aides were were very simple - for example, utilising marks on bones and patterns of pebbles. Later devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

19 March 2014
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The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and ~~then develop aides to help them do simple arithmetic sums. Over this time these techniques~~ enabled them to build increasingly complex societies. These ~~required more of their citizens to be~~ able to manipulate numbers in their daily lives. As this requirement spread a wide range of aides of various sorts were developed to help. At the beginning the aides were were very simple - for example, utilising marks on bones and patterns of pebbles. Later devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

to:

The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over this time these became essential ingredients in the skills which enabled them to build increasingly complex societies. These societies depended on more of their citizens being able to manipulate numbers in their daily lives. As this requirement spread a wide range of aides of various sorts were developed to help. At the beginning the aides were were very simple - for example, utilising marks on bones and patterns of pebbles. Later devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

19 March 2014
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The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story ~~- during~~ which humans came to record their counting and then ~~developed~~ aides to help them do simple arithmetic sums. Over this time these techniques enabled them to build ~~societies that increasingly~~ required ~~their citizens to rely on numbers and then~~ manipulate ~~them simply to succeed in their daily lives. As this became more central in living a wide range of aides of various sorts were developed to help. At the beginning they were implemented in simple ways - for example, as marks on bones~~ and ~~patterns of pebbles. Later through simple devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive,~~ represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

to:

The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story, one in which humans came to count, record their counting and then develop aides to help them do simple arithmetic sums. Over this time these techniques enabled them to build increasingly complex societies. These required more of their citizens to be able to manipulate numbers in their daily lives. As this requirement spread a wide range of aides of various sorts were developed to help. At the beginning the aides were were very simple - for example, utilising marks on bones and patterns of pebbles. Later devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

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The account ~~in this website outlines~~ (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story - ~~of how~~ humans came to ~~count, then developed aides to help them do simple sums. Over this time they built societies that increasingly required their citizens~~ to ~~rely on numbers and then manipulate them simply ~~to ~~succeed in their daily lives. To help people do this a wide range of aides of various sorts were developed. At the beginning they were as simple as marks on bones and patterns of pebbles, and later simple devices and tables. But over time much more elaborate mechanical devices were developed to help in this task. Many of these things are now little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.~~

Increasingly, as societies have become more complex, transactions in them have depended on mathematics, and in particular on simple arithmetic-~~ the familiar tasks of counting, adding, subtracting~~, ~~multiplying and dividing~~. ~~Yet over much of this period, for many people ~~in ~~these societies, doing such ~~arithmetic ~~tasks has been neither easy nor even accessible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people~~, ~~has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals, rather than the development of complex mathematics~~, ~~which is~~ the ~~focus of this website.~~

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly since the advent of electronic computing, the aides to calculation appear in virtual form in apps in phones, tablets and laptops. The end of calculators in this sense is looming.

Increasingly, as societies have become more complex, transactions in them have depended on mathematics, and in particular on simple arithmetic

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly

to:

The account given here is of (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story - during which humans came to record their counting and then developed aides to help them do simple arithmetic sums. Over this time these techniques enabled them to build societies that increasingly required their citizens to rely on numbers and then manipulate them simply to succeed in their daily lives. As this became more central in living a wide range of aides of various sorts were developed to help. At the beginning they were implemented in simple ways - for example, as marks on bones and patterns of pebbles. Later through simple devices and tables were developed and sold, and over time much more elaborate mechanical devices were developed to help in this task. Many of these things, where they survive, represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

The need for calculation, however has prospered. As societies have become more complex, transactions in them depending on arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing - and more complex mathematics have intensified. Yet over much of this period, for many people in these societies, doing even the simple arithmetic tasks has been neither easy nor even accessible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals (rather than the development of complex mathematics) which is the primary focus of this website.

Early "calculators" were not things, but rather people who were employed to calculate. Over time these people were first aided, and then replaced by calculating devices. These devices became very widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Increasingly since the advent of electronic computing, the aides to calculation appear in virtual form in apps in phones, tablets and laptops. The end of calculators in this sense is looming.

The need for calculation, however has prospered. As societies have become more complex, transactions in them depending on arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing - and more complex mathematics have intensified. Yet over much of this period, for many people in these societies, doing even the simple arithmetic tasks has been neither easy nor even accessible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals (rather than the development of complex mathematics) which is the primary focus of this website.

Early "calculators" were not things, but rather people who were employed to calculate. Over time these people were first aided, and then replaced by calculating devices. These devices became very widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Increasingly since the advent of electronic computing, the aides to calculation appear in virtual form in apps in phones, tablets and laptops. The end of calculators in this sense is looming.

Changed lines 31-41 from:

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal.

The calculational devices

Fortunately in order

The focus here will thus be confined to a tiny simple bit of mathematics - the basic numerical calculations required to finally evaluate practical applications of mathematical formulae. Yet even keeping our attention as restricted as this

to:

As most people know, the spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction - from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. As already noted, it is that which is dealt with here.

The calculational devices that were developed show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But, as already noted, it is also important to understand //why// they were invented and used.

Fortunately in order to understand what has shaped the development of these calculational aids we can largely avoid talking much about mathematics. This is lucky because mathematics is by now a huge field of knowledge. So in this site we will avoid calculus, set and group theory, the mathematics of infinite dimensional vector spaces that make the modern formulation of quantum mechanics possible, and tensors which Einstein used to express his wonderfully neat equations for the shape of space-time.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It will be sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations which in the end are constituted out of additions and subtractions (and multiplications and divisions) and can only be carried out in workable times with the use of ever faster calculating devices.

Even keeping our attention restricted to the basic arithmetic operations, it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will still be useful to take some account of this literature and its insights. In particular, whether concerning ourselves with the evolution of the simple areas of mathematics or the more obstruse areas, one question is always raised: what led to this particular development happening how and when it did?

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. As already noted, it is that which is dealt with here.

The calculational devices that were developed show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But, as already noted, it is also important to understand //why// they were invented and used.

Fortunately in order to understand what has shaped the development of these calculational aids we can largely avoid talking much about mathematics. This is lucky because mathematics is by now a huge field of knowledge. So in this site we will avoid calculus, set and group theory, the mathematics of infinite dimensional vector spaces that make the modern formulation of quantum mechanics possible, and tensors which Einstein used to express his wonderfully neat equations for the shape of space-time.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It will be sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations which in the end are constituted out of additions and subtractions (and multiplications and divisions) and can only be carried out in workable times with the use of ever faster calculating devices.

Even keeping our attention restricted to the basic arithmetic operations, it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will still be useful to take some account of this literature and its insights. In particular, whether concerning ourselves with the evolution of the simple areas of mathematics or the more obstruse areas, one question is always raised: what led to this particular development happening how and when it did?

18 March 2014
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The account in this website outlines (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story - of how humans came to count, then developed aides to help them do simple sums. Over this time they built societies that increasingly required their citizens to rely on numbers and ~~the~~ manipulate them simply to succeed in their daily lives. To help people do this a wide range of aides of various sorts were developed. At the beginning they were as simple as marks on bones and patterns of pebbles, and later simple devices and tables. But over time much more elaborate mechanical devices were developed to help in this task. Many of these things are now little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

to:

The account in this website outlines (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story - of how humans came to count, then developed aides to help them do simple sums. Over this time they built societies that increasingly required their citizens to rely on numbers and then manipulate them simply to succeed in their daily lives. To help people do this a wide range of aides of various sorts were developed. At the beginning they were as simple as marks on bones and patterns of pebbles, and later simple devices and tables. But over time much more elaborate mechanical devices were developed to help in this task. Many of these things are now little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

18 March 2014
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Nevertheless, in much of daily life, despite all the power of modern computers, much of what we transact depends on simple arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing. Yet through much of history for many people these sorts of tasks have been neither seen as simple nor even accessible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals, rather than the development of complex mathematics, which is the focus of this website.

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly, they appear in virtual form as

to:

The account in this website outlines (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story - of how humans came to count, then developed aides to help them do simple sums. Over this time they built societies that increasingly required their citizens to rely on numbers and the manipulate them simply to succeed in their daily lives. To help people do this a wide range of aides of various sorts were developed. At the beginning they were as simple as marks on bones and patterns of pebbles, and later simple devices and tables. But over time much more elaborate mechanical devices were developed to help in this task. Many of these things are now little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

Increasingly, as societies have become more complex, transactions in them have depended on mathematics, and in particular on simple arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing. Yet over much of this period, for many people in these societies, doing such arithmetic tasks has been neither easy nor even accessible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals, rather than the development of complex mathematics, which is the focus of this website.

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly since the advent of electronic computing, the aides to calculation appear in virtual form in apps in phones, tablets and laptops. The end of calculators in this sense is looming.

Increasingly, as societies have become more complex, transactions in them have depended on mathematics, and in particular on simple arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing. Yet over much of this period, for many people in these societies, doing such arithmetic tasks has been neither easy nor even accessible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals, rather than the development of complex mathematics, which is the focus of this website.

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly since the advent of electronic computing, the aides to calculation appear in virtual form in apps in phones, tablets and laptops. The end of calculators in this sense is looming.

18 March 2014
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It is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story - of how humans ~~invented ways ~~to count, then ~~help them do simple sums, and to depend on numbers and the manipulation of them in so much of their daily lives. To help them~~ in ~~this task they developed a wide range of physical aides - ranging from marks on bones to elaborate mechanical devices. The remains of those can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become more subtle, and so much part of our lives that they are often almost invisible.~~

As with much of daily life, despite all the power of modern computers, much of ~~what we transact depends on simple arithmetic - the familiar tasks~~ of ~~counting, adding, subtracting, multiplying and dividing. Yet through~~ much ~~of history these sorts ~~of ~~tasks have been neither seen as simple nor even available to many people. And finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to ~~these ~~simple goals, rather than the development of complex mathematics, which is the focus of~~ this ~~website.~~

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly, they appear in virtual form as apps in phones, tablets and laptops. The end of calculators in this sense is looming, and ~~forms an end point to this account. ~~

One might imagine that this history would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this and the minutae of mechanical invention. But to focus simply on that is to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

As with much of daily life, despite all the power of modern computers, much

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly, they appear in virtual form as apps in phones, tablets and laptops. The end of calculators in this sense is looming,

One might imagine that this history

to:

It is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story - of how humans came to count, then developed aides to help them do simple sums, and the built societies that required their citizens to rely on numbers and the manipulate them simply to succeed in their daily lives. To help people do this they developed a wide range of physical aides. At the beginning they were as simple as marks on bones and patterns of pebbles. But later much more elaborate mechanical devices were developed to help in this task. Many of these are now essentially mechanical fossils. Now unused their remains can be found scattered across history from earliest human pre-history to our present moment. The arrival of electronic computing heralded the arrival of a much more subtle way of assisting people in their use of numbers, so much so that much of the time these are by now often almost invisible.

Nevertheless, in much of daily life, despite all the power of modern computers, much of what we transact depends on simple arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing. Yet through much of history for many people these sorts of tasks have been neither seen as simple nor even accessible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals, rather than the development of complex mathematics, which is the focus of this website.

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly, they appear in virtual form as apps in phones, tablets and laptops. The end of calculators in this sense is looming.

One might imagine that a history of calculators would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this and the minutae of mechanical invention. But to focus simply on that is to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

Nevertheless, in much of daily life, despite all the power of modern computers, much of what we transact depends on simple arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing. Yet through much of history for many people these sorts of tasks have been neither seen as simple nor even accessible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals, rather than the development of complex mathematics, which is the focus of this website.

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly, they appear in virtual form as apps in phones, tablets and laptops. The end of calculators in this sense is looming.

One might imagine that a history of calculators would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this and the minutae of mechanical invention. But to focus simply on that is to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

18 March 2014
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Changed lines 17-23 from:

As with

The history developed here stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, and forward to the subsequent progressive disappearance

Such a history could be simply an account of the progressive discovery

to:

It is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story - of how humans invented ways to count, then help them do simple sums, and to depend on numbers and the manipulation of them in so much of their daily lives. To help them in this task they developed a wide range of physical aides - ranging from marks on bones to elaborate mechanical devices. The remains of those can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become more subtle, and so much part of our lives that they are often almost invisible.

As with much of daily life, despite all the power of modern computers, much of what we transact depends on simple arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing. Yet through much of history these sorts of tasks have been neither seen as simple nor even available to many people. And finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals, rather than the development of complex mathematics, which is the focus of this website.

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly, they appear in virtual form as apps in phones, tablets and laptops. The end of calculators in this sense is looming, and forms an end point to this account.

One might imagine that this history would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this and the minutae of mechanical invention. But to focus simply on that is to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

As with much of daily life, despite all the power of modern computers, much of what we transact depends on simple arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing. Yet through much of history these sorts of tasks have been neither seen as simple nor even available to many people. And finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to these simple goals, rather than the development of complex mathematics, which is the focus of this website.

The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. Now, increasingly, they appear in virtual form as apps in phones, tablets and laptops. The end of calculators in this sense is looming, and forms an end point to this account.

One might imagine that this history would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this and the minutae of mechanical invention. But to focus simply on that is to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

18 March 2014
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18 March 2014
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As with much of daily life, despite all the power of modern computers, much of what is transacted depends on simple arithmetic - the familiar tasks of counting, adding, subtracting, multiplying and dividing. Yet through much of history none of these have been either seen as simple or even available to many people. And finding ways to do these faster and more accurately has been a preoccupation during millennia. This, rather than the development of complex mathematics, is the focus of this account.

The history developed here stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, and forward to the subsequent progressive disappearance of calculating devices into phones, tablets and laptops. The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. The end of calculators in this sense is part of this account.

The history developed here stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, and forward to the subsequent progressive disappearance of calculating devices into phones, tablets and laptops. The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. The end of calculators in this sense is part of this account.

18 March 2014
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This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year ~~account~~ - ranging across things which humans invented to help them do simple sums as they came to count and then to depend on counting in so much of their daily lives. Examples of things devised to assist counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

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This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend on counting in so much of their daily lives. Examples of things devised to assist counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

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This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a ~~10~~,000 year ~~story~~ - ranging across things which humans invented to help them do simple sums as they came to count and then to depend on counting in so much of their daily lives. Examples of things devised to assist counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

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This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 37,000 year account - ranging across things which humans invented to help them do simple sums as they came to count and then to depend on counting in so much of their daily lives. Examples of things devised to assist counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

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(:if equal {Site.PrintBook$:PSW} "True":)

%center% http://metastudies.net/pmwiki/uploads/PA_1.jpg\\(Replica Pascaline "1652" - collection Calculant)

(:ifend:)

(:if equal {Site.PrintBook$:PSW} "False":)

%center% http://metastudies.net/pmwiki/uploads/PA_1.jpg\\(Replica Pascaline "1652" - collection Calculant)

(:ifend:)

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(:if equal {Site.PrintBook$:PSW} "False":)

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**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website ~~uses~~ a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

to:

**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website both describes a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) and uses it to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

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**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website ~~() utilises~~ a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

to:

**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website uses a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

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**Welcome to ~~"Things ~~that~~ Count"~~.** This website (~~[[http://things-that-count.com|things-that-count.com]]~~) utilises a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

to:

**Welcome to [[http://things-that-count.com|things-that-count.com]].** This website () utilises a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

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This history stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, and forward to the subsequent progressive disappearance of calculating devices into phones, tablets and laptops. The first calculators were people who calculated. ~~They were replaced by calculating devices. We~~ are now passing the heyday of such stand-alone calculators. The end of calculators in this sense is part of this account.

to:

This history stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, and forward to the subsequent progressive disappearance of calculating devices into phones, tablets and laptops. The first calculators were people who calculated. Over time those people were first aided, and then replaced by calculating devices. These devices became incredibly widely used across many countries. But we are now passing the heyday of such stand-alone calculators. The end of calculators in this sense is part of this account.

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The objects ~~in [[[[Objects gallery|~~"~~collection Calculant~~"~~]]~~ described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (the name also of an accompanying web site)[^see http://things-that-count.com. The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The objects (:if equal {Site.PrintBook$:PSW} "False":)in [[[[Objects gallery|"collection Calculant"]](:ifend:) described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (the name also of an accompanying web site)[^see http://things-that-count.com. The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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The objects in [[[[Objects gallery|"collection Calculant"]] described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The objects in [[[[Objects gallery|"collection Calculant"]] described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (the name also of an accompanying web site)[^see http://things-that-count.com. The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

16 March 2014
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The objects in [[[[Objects gallery|"collection Calculant"]] described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"~~)~~[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The objects in [[[[Objects gallery|"collection Calculant"]] described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate"[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

16 March 2014
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16 March 2014
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The objects in [[[[Objects gallery|"collection Calculant"]] described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. ~~For convenience, I will refer to this set of artifacts as "collection Calculant~~"~~(:if equal {Site.PrintBook$:PSW} ~~"~~True":)~~ (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The objects in [[[[Objects gallery|"collection Calculant"]] described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. (:if equal {Site.PrintBook$:PSW} "True":)For convenience, I will refer to this set of artifacts as "collection Calculant" (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

16 March 2014
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The objects in [[[[Objects gallery|"collection Calculant"]] described here, which ~~is~~ used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The objects in [[[[Objects gallery|"collection Calculant"]] described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

16 March 2014
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The [[[[Objects gallery|"~~Collection~~ Calculant"]] described here, which is used to help illustrate answers to these sort of questions are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The objects in [[[[Objects gallery|"collection Calculant"]] described here, which is used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

16 March 2014
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The [[[[Objects gallery|"Collection Calculant"]] described here, which is used to help illustrate answers to these sort of questions are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:)~~. So part of the story here also is that of a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating~~. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The [[[[Objects gallery|"Collection Calculant"]] described here, which is used to help illustrate answers to these sort of questions are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

16 March 2014
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The [[[[Objects gallery|"Collection Calculant"]] ~~which will be~~ used ~~as guides to answering some of ~~these questions are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is that of a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The [[[[Objects gallery|"Collection Calculant"]] described here, which is used to help illustrate answers to these sort of questions are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is that of a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 10,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend ~~(or 'count') on counting in so much of their daily ~~lives. Examples of things~~ humans have~~ devised to assist counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

to:

This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 10,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend on counting in so much of their daily lives. Examples of things devised to assist counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

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**Welcome to "Things that Count".** This website ([[http://things-that-count.com|things-that-count.com]]) utilises a collection of antique calculators ~~-~~[[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]~~ -~~ to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

to:

**Welcome to "Things that Count".** This website ([[http://things-that-count.com|things-that-count.com]]) utilises a collection of antique calculators ([[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^]) to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

16 March 2014
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This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 10,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of things ~~for counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-~~history ~~to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.~~

This history of calculation stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, ~~to the subsequent progressive disappearance of calculating devices into~~ phones, tablets and laptops. The first calculators were people who calculated. They were replaced by calculating devices. We are now passing the heyday of such stand-alone calculators. The end of calculators in this sense is part of this account.

~~One might suppose that a history of such devices might be simply served by an account of the progressive discovery and invention of ever more effective and sophisticated calculating devices. But that would be to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history.~~ ~~But they beg a series of questions: When~~ and why ~~were they made, how were they used, and why at times were they forgotten for centuries or even millennia?~~

Supporting this account is [[[[Objects gallery|"Collection Calculant"]] - a collection of 'things' which were constructed at one time or another, over some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is that of a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

This history of calculation stretches from the beginnings of

Supporting this account is [[[[Objects gallery|"Collection Calculant"]] - a collection of 'things' which were constructed at one time or another, over

to:

This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 10,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of things humans have devised to assist counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

This history stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, and forward to the subsequent progressive disappearance of calculating devices into phones, tablets and laptops. The first calculators were people who calculated. They were replaced by calculating devices. We are now passing the heyday of such stand-alone calculators. The end of calculators in this sense is part of this account.

Such a history could be simply an account of the progressive discovery and invention of ever more effective and sophisticated calculating devices. But that would be to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

The [[[[Objects gallery|"Collection Calculant"]] which will be used as guides to answering some of these questions are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is that of a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

This history stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, and forward to the subsequent progressive disappearance of calculating devices into phones, tablets and laptops. The first calculators were people who calculated. They were replaced by calculating devices. We are now passing the heyday of such stand-alone calculators. The end of calculators in this sense is part of this account.

Such a history could be simply an account of the progressive discovery and invention of ever more effective and sophisticated calculating devices. But that would be to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

The [[[[Objects gallery|"Collection Calculant"]] which will be used as guides to answering some of these questions are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is that of a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

16 March 2014
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**Welcome to "Things that Count".** This website ([[http://things-that-count.com|things-that-count.com]]) utilises a collection of antique calculators -[[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^] to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

to:

**Welcome to "Things that Count".** This website ([[http://things-that-count.com|things-that-count.com]]) utilises a collection of antique calculators -[[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^] - to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

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**Welcome to "Things that Count".** This website ([[http://things-that-count.com|things-that-count.com]]) ~~both describes ~~a ~~[[Objects gallery|collection ~~of antique ~~objects~~ - "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^] ~~and also uses that collection to help build an account of the interesting history of~~ the ~~way humans, over millennia, have not only developed devices to assist a developing social need to count and calculate, but indeed how human society~~ (and even human brains) have evolved with those developments.

to:

**Welcome to "Things that Count".** This website ([[http://things-that-count.com|things-that-count.com]]) utilises a collection of antique calculators -[[Objects gallery| "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^] to help develop a historical account of the way humans, over millennia, have developed the need and capacity to count and calculate, and how human societies (and even human brains) have evolved with those developments.

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**Welcome to "Things that Count".** This website ([[http://things-that-count.com|things-that-count.com]]) both describes ~~an intriguing ~~[[Objects gallery|collection of antique objects - "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^] ~~from can be drawn~~ an account of the interesting history of the way humans, over millennia, have not only developed devices to assist a developing social need to count and calculate, but indeed how human society (and even human brains) have evolved with those developments.

to:

**Welcome to "Things that Count".** This website ([[http://things-that-count.com|things-that-count.com]]) both describes a [[Objects gallery|collection of antique objects - "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^] and also uses that collection to help build an account of the interesting history of the way humans, over millennia, have not only developed devices to assist a developing social need to count and calculate, but indeed how human society (and even human brains) have evolved with those developments.

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**Welcome to "Things that Count".** This ~~is first a website ~~([[http://things-that-count.com|things-that-count.com]])~~. Its purpose is to describe an intriguing [[Objects gallery|collection of antique objects - ~~"~~collection Calculant"~~]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^] from can be drawn an account of the interesting history of the way humans, over millennia, have not only developed devices to assist a developing social need to count and calculate, but indeed how human society (and even human brains) have evolved with those developments.

This is~~(:fiend:)~~(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 10,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of things for counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

This is

to:

**Welcome to "Things that Count".** This website ([[http://things-that-count.com|things-that-count.com]]) both describes an intriguing [[Objects gallery|collection of antique objects - "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^] from can be drawn an account of the interesting history of the way humans, over millennia, have not only developed devices to assist a developing social need to count and calculate, but indeed how human society (and even human brains) have evolved with those developments.

This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 10,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of things for counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

This is (:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 10,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of things for counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

16 March 2014
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**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com|things-that-count.com]]). Its purpose is to describe ~~a rather ~~intriguing [[Objects gallery|collection of antique objects - "collection Calculant"]] ~~which are used to build an evocative picture of how humans have come to use things to help them calculate.~~

Whilst you are currently reading "~~Things that Count" on this website it will soon also be able to be down-loaded as a short book. It's topic is the fascinating ways in which ways of using things~~ to ~~add things up and human society (and even the~~ human ~~brain~~) have evolved ~~together.~~

The collection name - "calculant" is from the Latin for"~~they calculate~~"~~.[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that~~ count~~" spans some 10,000 years, and so this explanation is a bit extended(:fiend:)~~

(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of ~~their daily ~~lives~~. Examples of these things can be found scattered across history ~~from ~~earliest human pre-history~~ to ~~our present moment. Now, of course, with electronic computers, the aides they provide to doing everyday calculations have become so much part of our lives that they are often almost invisible.~~

This history of calculation stretches from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. ~~ The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday~~ of ~~the stand-alone device known as a calculator~~. ~~The end of calculators in this sense is part~~ of ~~this story. So also are the ends that calculators have at different times been created to meet~~.

One might suppose that for the purposes of such a history a story of progressive discovery and invention would suffice. But that would be to oversimplify and lose much of what is potentially interesting. The appearance of what were often weird things devised for doing simple calculations across human history begs a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

Whilst you are currently reading

The collection name - "calculant" is from the Latin for

(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in

This history of calculation stretches from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops

One might suppose that for the purposes of such a history a story of progressive discovery and invention would suffice. But that would be to oversimplify and lose much of what is potentially interesting. The appearance of what were often weird things devised for doing simple calculations across human history begs

to:

**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com|things-that-count.com]]). Its purpose is to describe an intriguing [[Objects gallery|collection of antique objects - "collection Calculant"]][^"Calculant" in Latin means literally "they calculate". (More precisely it is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute").^] from can be drawn an account of the interesting history of the way humans, over millennia, have not only developed devices to assist a developing social need to count and calculate, but indeed how human society (and even human brains) have evolved with those developments.

This is (:fiend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 10,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of things for counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

This history of calculation stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, to the subsequent progressive disappearance of calculating devices into phones, tablets and laptops. The first calculators were people who calculated. They were replaced by calculating devices. We are now passing the heyday of such stand-alone calculators. The end of calculators in this sense is part of this account.

One might suppose that a history of such devices might be simply served by an account of the progressive discovery and invention of ever more effective and sophisticated calculating devices. But that would be to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

This is (:fiend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines(:ifend:) a 10,000 year story - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of things for counting - from marks on bones to elaborate mechanical devices - can be found scattered across history from earliest human pre-history to our present moment. With the arrival of electronic computing, such aides have become so much part of our lives that they are often almost invisible.

This history of calculation stretches from the beginnings of counting, to the invention of tools to help manipulate numbers, to the subsequent progressive disappearance of calculating devices into phones, tablets and laptops. The first calculators were people who calculated. They were replaced by calculating devices. We are now passing the heyday of such stand-alone calculators. The end of calculators in this sense is part of this account.

One might suppose that a history of such devices might be simply served by an account of the progressive discovery and invention of ever more effective and sophisticated calculating devices. But that would be to oversimplify and lose much of what is potentially interesting. Many weird things were indeed devised for doing simple calculations across human history. But they beg a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

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**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com|things-that-count.com]]). Its purpose is to describe a rather intriguing [[Objects gallery|collection of antique objects~~]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing a quite wide-ranging essay on this~~.

The collection is ~~called "Collection Calculant", whilst this essay is titled "Things that Count". You are currently reading this essay on ~~the ~~website. But it will soon also be able to be down-loaded as a short book. It's topic ~~is the ~~fascinating ways in which ways~~ of ~~using things to add things up and human society (and even the human brain) have evolved together.~~

The collection name -"~~calculant" is from the Latin for "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ~~(~~ calculare, calculavi~~) ~~meaning~~ "~~they calculate, they compute".^] The history of human calculation using "~~things ~~that count" spans some 10,000 years, and so this explanation is a bit extended~~(~~:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and ~~things ~~which humans invented as they came to count and then~~ to ~~depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are~~ almost invisible.

The collection

The collection name -

to:

**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com|things-that-count.com]]). Its purpose is to describe a rather intriguing [[Objects gallery|collection of antique objects - "collection Calculant"]] which are used to build an evocative picture of how humans have come to use things to help them calculate.

Whilst you are currently reading "Things that Count" on this website it will soon also be able to be down-loaded as a short book. It's topic is the fascinating ways in which ways of using things to add things up and human society (and even the human brain) have evolved together.

The collection name - "calculant" is from the Latin for "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:fiend:)

(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these things can be found scattered across history from earliest human pre-history to our present moment. Now, of course, with electronic computers, the aides they provide to doing everyday calculations have become so much part of our lives that they are often almost invisible.

Whilst you are currently reading "Things that Count" on this website it will soon also be able to be down-loaded as a short book. It's topic is the fascinating ways in which ways of using things to add things up and human society (and even the human brain) have evolved together.

The collection name - "calculant" is from the Latin for "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:fiend:)

(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across things which humans invented to help them do simple sums as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these things can be found scattered across history from earliest human pre-history to our present moment. Now, of course, with electronic computers, the aides they provide to doing everyday calculations have become so much part of our lives that they are often almost invisible.

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This ~~perspective on the long history of calculation thus stretches from~~ the ~~beginnings~~ of ~~counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated~~. ~~That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in~~ this ~~sense is part of this story, but so also are the ends that calculators have at different times been created to meet.~~

One might suppose that this would amount to little more than a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. Across human history can be found an often weird variety of things for calculating. When and why they were made, how ~~they ~~were used, and why at times ~~they ~~were forgotten for centuries or even millennia~~ is also a poignant part of the picture. So several stories are tangled together here, each informing the other.~~

One might suppose that this would amount to little more than a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. Across human history can be found an often weird variety of things for calculating. When and why they

to:

This history of calculation stretches from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in this sense is part of this story. So also are the ends that calculators have at different times been created to meet.

One might suppose that for the purposes of such a history a story of progressive discovery and invention would suffice. But that would be to oversimplify and lose much of what is potentially interesting. The appearance of what were often weird things devised for doing simple calculations across human history begs a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

One might suppose that for the purposes of such a history a story of progressive discovery and invention would suffice. But that would be to oversimplify and lose much of what is potentially interesting. The appearance of what were often weird things devised for doing simple calculations across human history begs a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

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(:if equal {Site.PrintBook$:PSW} "False":)

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**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com|things-that-count.com]]). Its purpose is to describe a rather intriguing [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing a quite wide-ranging essay on this.

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(:if equal {Site.PrintBook$:PSW} "False":)**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com|things-that-count.com]]). Its purpose is to describe a rather ~~extraordinary~~ [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing a quite wide-ranging essay on this.

to:

(:if equal {Site.PrintBook$:PSW} "False":)**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com|things-that-count.com]]). Its purpose is to describe a rather intriguing [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing a quite wide-ranging essay on this.

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]]\\(Replica Pascaline - collection Calculant)

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]]\\([[Site.Pascaline1652|Replica Pascaline "1652"]] - [[Site.ObjectsGallery|collection Calculant]])

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]] ~~(Replica ~~Pascaline - collection ~~Calculant~~

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]]\\(Replica Pascaline - collection Calculant)

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg (Replica Pascaline - collection ~~Calculant]]~~

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]] (Replica Pascaline - collection Calculant

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg (Replica Pascaline - collection Calculant]]

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%center% http://metastudies.net/pmwiki/uploads/PA_1.~~jpg~~

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%center% [[Site.Pascaline1652|http://metastudies.net/pmwiki/uploads/PA_1.jpg]]

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Supporting this account is [[[[Objects gallery|"Collection Calculant"]] - a collection of 'things' which were constructed at one time or another, over some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^~~http://calculant~~.~~metastudies.net~~ The author, Jim Falk, may be contacted through this website.^]. The name ~~is taken from the Latin meaning simply "they calculate~~")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

Supporting this account is [[[[Objects gallery|"Collection Calculant"]] - a collection of 'things' which were constructed at one time or another, over some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^things-that-count.com The author, Jim Falk, may be contacted through this website.^]. The name of the collection is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is that of a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

16 March 2014
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(:if equal {Site.PrintBook$:PSW} "False":)**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com]]). Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing a quite wide-ranging essay on this.

to:

(:if equal {Site.PrintBook$:PSW} "False":)**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com|things-that-count.com]]). Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing a quite wide-ranging essay on this.

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The ~~associated ~~collection is called "Collection Calculant", whilst this essay is titled "Things that Count". You are reading ~~the~~ essay ~~now here.~~ But it will soon also be able to be down-loaded ~~from here as a short ~~book. It is ~~about ~~the fascinating ways in which ~~calculation technology~~ and human society (and even the human brain) have evolved together.

to:

The collection is called "Collection Calculant", whilst this essay is titled "Things that Count". You are currently reading this essay on the website. But it will soon also be able to be down-loaded as a short book. It's topic is the fascinating ways in which ways of using things to add things up and human society (and even the human brain) have evolved together.

12 March 2014
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The associated collection is called "Collection Calculant", whilst this essay is titled "Things that Count". You are reading the essay now here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

12 March 2014
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**The collection is called "Collection Calculant", whilst this essay is titled "Things that Count".** You are reading the essay now here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

to:

**The associated collection is called "Collection Calculant", whilst this essay is titled "Things that Count".** You are reading the essay now here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

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12 March 2014
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The collection name - "calculant" is from the Latin for "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

12 March 2014
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**The collection name - "calculant" is from the Latin for "they calculate".**[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

to:

**The collection name - "calculant" is from the Latin for "they calculate".** [^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

12 March 2014
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The ~~[[Objects gallery|collection]] ~~name - ~~**~~"calculant" is from the Latin for "they calculate".**[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

to:

**The collection name - "calculant" is from the Latin for "they calculate".**[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

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Supporting this account is ~~"Collection Calculant~~" - a collection of 'things' which were constructed at one time or another, over some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net The author, Jim Falk, may be contacted through this website.^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

Supporting this account is [[[[Objects gallery|"Collection Calculant"]] - a collection of 'things' which were constructed at one time or another, over some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net The author, Jim Falk, may be contacted through this website.^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

12 March 2014
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The [[Objects gallery|collection]] name - **"calculant" is from the Latin for "they calculate".**[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

12 March 2014
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The [[Objects gallery|collection]] name - "calculant" is from the Latin for "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

to:

**The [[Objects gallery|collection]] name - "calculant" is from the Latin for "they calculate".**[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

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(:if equal {Site.PrintBook$:PSW} "False":)**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com]]). Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing ~~quite ~~a wide-ranging essay on this.

**The collection is called "Collection Calculant", whilst~~the~~ essay is titled "Things that Count".** You are reading the essay now here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

**The collection is called "Collection Calculant", whilst

to:

(:if equal {Site.PrintBook$:PSW} "False":)**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com]]). Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing a quite wide-ranging essay on this.

**The collection is called "Collection Calculant", whilst this essay is titled "Things that Count".** You are reading the essay now here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

**The collection is called "Collection Calculant", whilst this essay is titled "Things that Count".** You are reading the essay now here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

12 March 2014
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**The collection is called "Collection Calculant", whilst the essay is titled "Things that Count".** You are reading ~~it~~ now here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

to:

**The collection is called "Collection Calculant", whilst the essay is titled "Things that Count".** You are reading the essay now here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

12 March 2014
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**The collection is called "Collection Calculant", whilst the essay is titled "Things that Count".** You ~~can read~~ it here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

to:

**The collection is called "Collection Calculant", whilst the essay is titled "Things that Count".** You are reading it now here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

12 March 2014
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The collection is called "Collection Calculant"~~. The~~ essay is titled "Things that Count". You can read it here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

to:

**The collection is called "Collection Calculant", whilst the essay is titled "Things that Count".** You can read it here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

12 March 2014
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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Things that Count". This is first a website ([[http://things-that-count.com]]). Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing quite a wide-ranging essay on this.

to:

(:if equal {Site.PrintBook$:PSW} "False":)**Welcome to "Things that Count".** This is first a website ([[http://things-that-count.com]]). Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing quite a wide-ranging essay on this.

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Supporting this account is ~~something else~~ - a collection of 'things' ~~constructed at one time or~~ another, over some ~~500~~ years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net The author, Jim Falk, may be contacted through this website.^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

Supporting this account is "Collection Calculant" - a collection of 'things' which were constructed at one time or another, over some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net The author, Jim Falk, may be contacted through this website.^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

12 March 2014
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The collection is called "Collection Calculant". The ~~Essay~~ is titled "Things that Count". ~~It is being prepared as a short book - on the interesting way in which calculation technology and human society (and even~~ the ~~human brain) have evolved together. In due course you will be able to download this book from this website~~.

to:

The collection is called "Collection Calculant". The essay is titled "Things that Count". You can read it here. But it will soon also be able to be down-loaded from here as a short book. It is about the fascinating ways in which calculation technology and human society (and even the human brain) have evolved together.

12 March 2014
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12 March 2014
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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Things that Count". This is first a website. Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing quite a wide-ranging essay on this.

to:

(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Things that Count". This is first a website ([[http://things-that-count.com]]). Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing quite a wide-ranging essay on this.

12 March 2014
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The [[Objects gallery|collection]] name - ~~calculate~~ is from the Latin for "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

to:

The [[Objects gallery|collection]] name - "calculant" is from the Latin for "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

12 March 2014
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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Things that Count". This is first a website. Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which ~~were acquired to illustrate the history~~ of ~~calculation~~. ~~ The collection is used to provide the 'framework' for a quite wide-ranging essay on the same subject.~~

The collection is called "Collection Calculant", whilst the Essay and website are called "Things that Count". The essay ~~is also being prepared so it can be downloaded as a short book - on the interesting way in which calculation technology and human society (and even ~~the ~~human brain) have evolved together.~~

The collection name - calculant - in Latin means "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

The collection is called "Collection Calculant", whilst the Essay and website are called "Things that Count". The

The collection name - calculant - in Latin means

to:

(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Things that Count". This is first a website. Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which have been brought together over a number of years. They provide an evocative picture of their role in the evolution of how humans have come to use things to help them calculate. That is used as an excuse for writing quite a wide-ranging essay on this.

The collection is called "Collection Calculant". The Essay is titled "Things that Count". It is being prepared as a short book - on the interesting way in which calculation technology and human society (and even the human brain) have evolved together. In due course you will be able to download this book from this website.

The [[Objects gallery|collection]] name - calculate is from the Latin for "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

The collection is called "Collection Calculant". The Essay is titled "Things that Count". It is being prepared as a short book - on the interesting way in which calculation technology and human society (and even the human brain) have evolved together. In due course you will be able to download this book from this website.

The [[Objects gallery|collection]] name - calculate is from the Latin for "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

12 March 2014
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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Things that Count". This is a website ~~containing the description of~~ a ~~[[Objects gallery|collection of antique objects to do with the history of~~ calculation~~]], and an~~ essay on the same subject.

to:

(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Things that Count". This is first a website. Its purpose is to describe a rather extraordinary [[Objects gallery|collection of antique objects]] which were acquired to illustrate the history of calculation. The collection is used to provide the 'framework' for a quite wide-ranging essay on the same subject.

12 March 2014
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The collection name - calculant - in Latin means "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ~~an account of the~~ ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

to:

The collection name - calculant - in Latin means "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - ranging across ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

12 March 2014
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The collection name - calculant - in Latin means "they calculate"~~)~~.[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

to:

The collection name - calculant - in Latin means "they calculate".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

12 March 2014
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The collection is called "Collection Calculant", whilst the Essay and website are called "Things that Count". The essay is also being prepared so it can be downloaded as a short book - on the way calculation technology and human society (and even the human brain) have evolved together.

to:

The collection is called "Collection Calculant", whilst the Essay and website are called "Things that Count". The essay is also being prepared so it can be downloaded as a short book - on the interesting way in which calculation technology and human society (and even the human brain) have evolved together.

12 March 2014
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Changed line 11 from:

The collection is called "Collection Calculant", whilst the Essay and website are called "Things that Count". ~~"Things that Count" ~~is also being prepared so it can be downloaded as a short book -~~ an essay~~ on the way calculation technology and human society (and even the human brain) have evolved together.

to:

The collection is called "Collection Calculant", whilst the Essay and website are called "Things that Count". The essay is also being prepared so it can be downloaded as a short book - on the way calculation technology and human society (and even the human brain) have evolved together.

12 March 2014
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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to ~~both ~~"Things that Count"~~, the title of this explanation for what can be found in "Calculant". (Calculant is thus both the name of this web site~~ and ~~also a collection of antique related objects~~. ~~In Latin it~~ means "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

to:

(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Things that Count". This is a website containing the description of a [[Objects gallery|collection of antique objects to do with the history of calculation]], and an essay on the same subject.

The collection is called "Collection Calculant", whilst the Essay and website are called "Things that Count". "Things that Count" is also being prepared so it can be downloaded as a short book - an essay on the way calculation technology and human society (and even the human brain) have evolved together.

The collection name - calculant - in Latin means "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

The collection is called "Collection Calculant", whilst the Essay and website are called "Things that Count". "Things that Count" is also being prepared so it can be downloaded as a short book - an essay on the way calculation technology and human society (and even the human brain) have evolved together.

The collection name - calculant - in Latin means "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

27 January 2014
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This perspective on the long history of calculation thus stretches from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in this sense is part of this story, but so also are the ends that calculators have at different times been created to meet.

27 January 2014
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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to both "Things that Count", the title of this explanation for what can be found in "Calculant" (~~the name of this web site and a corresponding collection of antique objects. In Latin it means "they calculate~~").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The ~~explanation is a little extended as it~~ spans some 10,000 years ~~of human history~~(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

In brief~~the account here presents a~~ perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in this sense is part of this story, but so also are the ends that calculators have at different times been created to meet.

In brief

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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to both "Things that Count", the title of this explanation for what can be found in "Calculant". (Calculant is thus both the name of this web site and also a collection of antique related objects. In Latin it means "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The history of human calculation using "things that count" spans some 10,000 years, and so this explanation is a bit extended(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

In brief this is one perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in this sense is part of this story, but so also are the ends that calculators have at different times been created to meet.

In brief this is one perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in this sense is part of this story, but so also are the ends that calculators have at different times been created to meet.

22 January 2014
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22 January 2014
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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to both "Things that Count", the title of this ~~account, and "Calculant" (the name of this web site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi~~) ~~meaning "they calculate, they compute".^] This includes a 10,000 year story~~(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to both "Things that Count", the title of this explanation for what can be found in "Calculant" (the name of this web site and a corresponding collection of antique objects. In Latin it means "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] The explanation is a little extended as it spans some 10,000 years of human history(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

22 January 2014
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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "~~Calculant" (the name of this~~ site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to both "Things that Count", the title of this account, and "Calculant" (the name of this web site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

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(:title ~~Stepping stones: toward the end of calculators~~ :)

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(:title Things that Count :)

15 August 2013
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In brief the account here presents a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in this sense is part of this story, but ~~sp~~ also ~~is~~ the ends that calculators have at different times been created to meet.

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In brief the account here presents a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in this sense is part of this story, but so also are the ends that calculators have at different times been created to meet.

15 August 2013
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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. ~~These "stepping stones" stretch forward from earliest human pre-~~history ~~to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.~~

In brief the account here presents a perspective on the long history of calculation - from the beginnings of counting, tothe ~~invention~~ of ~~calculational tools~~, ~~and ~~to ~~their subsequent progressive disappearance into phones~~, ~~tablets ~~and ~~laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (potentially final disappearance) of calculators.~~

One might suppose that this would amount to little more than a ~~story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. These stepping stones are scattered across human history.~~ When and why they were made, how they were used, and why at times they were forgotten for centuries or even millennia is also a poignant part of the picture. So several stories are tangled together here, each informing the other.

In brief the account here presents a perspective on the long history of calculation - from the beginnings of counting, to

One might suppose that this would amount to little more than

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(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. Examples of these can be found scattered across history from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

In brief the account here presents a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in this sense is part of this story, but sp also is the ends that calculators have at different times been created to meet.

One might suppose that this would amount to little more than a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. Across human history can be found an often weird variety of things for calculating. When and why they were made, how they were used, and why at times they were forgotten for centuries or even millennia is also a poignant part of the picture. So several stories are tangled together here, each informing the other.

In brief the account here presents a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The first calculators were people who calculated. That role has essentially vanished. But we are also passing the heyday of the stand-alone device known as a calculator. The end of calculators in this sense is part of this story, but sp also is the ends that calculators have at different times been created to meet.

One might suppose that this would amount to little more than a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. Across human history can be found an often weird variety of things for calculating. When and why they were made, how they were used, and why at times they were forgotten for centuries or even millennia is also a poignant part of the picture. So several stories are tangled together here, each informing the other.

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It will be absolutely familiar to most people that the spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction - from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

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The ~~"stepping stones"~~ considered here~~ as laying out the path of this development~~ show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this construction strips away much that may be important in why they were invented and used.

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The calculational devices considered here show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this construction strips away much that may be important in why they were invented and used.

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This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which ~~we ~~would ~~comfortably refer~~ to ~~as~~ a machine. But with that caveat, we will be using the term "calculator" very broadly.

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This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which would widely be understood to be a machine. But with that caveat, we will be using the term "calculator" very broadly.

15 August 2013
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15 August 2013
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Supporting this account is something else - a collection of 'things' constructed at one time or another, over ~~more than~~ 500 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net The author, Jim Falk, may be contacted through this website.^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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Supporting this account is something else - a collection of 'things' constructed at one time or another, over some 500 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net The author, Jim Falk, may be contacted through this website.^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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Thus the historical account is broken into three parts. The first part~~,~~ looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop~~,~~ but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800) when mechanical calculation began to gain greater use in the broader society.

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Thus the historical account is broken into three parts. The first part looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800) when mechanical calculation began to gain greater use in the broader society.

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In ~~short~~ the account here ~~is devoted to developing~~ a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (potentially final disappearance) of calculators.

to:

In brief the account here presents a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (potentially final disappearance) of calculators.

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Supporting this account is something else - a collection of 'things' constructed at one time or another, over more than 500 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net The author, Jim Falk, may be contacted through this website.^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

15 August 2013
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Underlying this account is something else - a collection of 'things' constructed at one time or another, over more than 500 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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Underlying this account is something else - a collection of 'things' constructed at one time or another, over more than 500 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net The author, Jim Falk, may be contacted through this website.^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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"Calculator" could be taken to mean a variety of things. It could be calculation 'app.' on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

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"Calculator" could be taken to mean a variety of things. It could be calculation 'app.' on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. In earlier times it could simply mean someone who calculates. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

15 August 2013
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One might suppose that this would ~~really ~~amount to ~~just~~ a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. These stepping stones are scattered across human history. When and why they were made, how they were used, and why at times they were forgotten for centuries or even millennia is also a poignant part of the picture. So several stories are tangled together here, each informing the other.

to:

One might suppose that this would amount to little more than a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. These stepping stones are scattered across human history. When and why they were made, how they were used, and why at times they were forgotten for centuries or even millennia is also a poignant part of the picture. So several stories are tangled together here, each informing the other.

15 August 2013
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One might ~~think~~ this ~~is~~ really just a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. These stepping stones are scattered across human history. When and why they were made, how they were used, and why at times they were forgotten for centuries or even millennia is also a poignant part of the picture. So several stories are tangled together here, each informing the other.

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One might suppose that this would really amount to just a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. These stepping stones are scattered across human history. When and why they were made, how they were used, and why at times they were forgotten for centuries or even millennia is also a poignant part of the picture. So several stories are tangled together here, each informing the other.

15 August 2013
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Underlying this account is something else - a collection of 'things' constructed at one time or another, over more than 500 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site[^http://calculant.metastudies.net^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

Underlying this account is something else - a collection of 'things' constructed at one time or another, over more than 500 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site)[^http://calculant.metastudies.net^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

15 August 2013
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Underlying this account is something else - a collection of 'things' constructed at one time or another, over more than 500 years of history. Each of them was ~~supposed to aid the acts of thinking about~~ numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site[^http://calculant.metastudies.net^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

Underlying this account is something else - a collection of 'things' constructed at one time or another, over more than 500 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site[^http://calculant.metastudies.net^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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(:if equal {Site.PrintBook$:PSW} "False":)

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story~~- an account of the ideas and things which humans invented as they came to count~~ and ~~then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from earliest human pre-history to our present moment.~~ ~~We are living in a time when electronic computers, and the counting which forms a central part of how they work~~, ~~are becoming so much part of our lives that~~ they are ~~almost invisible.~~

(:ifend:)

(:if equal {Site.PrintBook$:PSW} "True":)

This short book outlines a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives.~~ These "stepping stones" stretch forward from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.~~

(:ifend:)

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story

(:ifend:)

(:if equal {Site.PrintBook$:PSW} "True":)

This short book outlines a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives

(:ifend:)

to:

(:if equal {Site.PrintBook$:PSW} "False":)Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)This short book outlines a 10,000 year story(:ifend:) - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. These "stepping stones" stretch forward from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

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This short book outlines a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

to:

This short book outlines a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to depend (or 'count') on counting in so much of their daily lives. These "stepping stones" stretch forward from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

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Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from ~~the ~~earliest ~~moments of human ~~pre-history ~~stretch ~~to our present~~. ~~ We are living a ~~moment where~~ electronic computers, and the counting which forms a central part of ~~their~~ work, are becoming so much part of our lives that they are almost invisible.

to:

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

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This short book outlines a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from ~~the ~~earliest ~~moments of human ~~pre-history ~~stretch ~~to our present~~. ~~ We are living a ~~moment where~~ electronic computers, and the counting which forms a central part of ~~their~~ work, are becoming so much part of our lives that they are almost invisible.

to:

This short book outlines a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from earliest human pre-history to our present moment. We are living in a time when electronic computers, and the counting which forms a central part of how they work, are becoming so much part of our lives that they are almost invisible.

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Thus the historical account is broken into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800)~~,~~ when mechanical calculation began to gain greater use in the broader society.

to:

Thus the historical account is broken into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800) when mechanical calculation began to gain greater use in the broader society.

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A word also about the way I have constructed the historical account. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]]^]), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon~~,~~ the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^ibid, pp. 132-45.^] (but there is not much need to focus on that here)

to:

A word also about the way I have constructed the historical account. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]]^]), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^ibid, pp. 132-45.^] (but there is not much need to focus on that here).

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This book ~~tells~~ a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from the earliest moments of human pre-history stretch to our present. We are living a moment where electronic computers, and the counting which forms a central part of their work, are becoming so much part of our lives that they are almost invisible.

to:

This short book outlines a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from the earliest moments of human pre-history stretch to our present. We are living a moment where electronic computers, and the counting which forms a central part of their work, are becoming so much part of our lives that they are almost invisible.

14 August 2013
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The "stepping stones" considered here as laying out the path of this development show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this construction strips away much that may be important in why they were invented and used.

14 August 2013
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The story here is based on a series

to:

(:if equal {Site.PrintBook$:PSW} "False":)

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from the earliest moments of human pre-history stretch to our present. We are living a moment where electronic computers, and the counting which forms a central part of their work, are becoming so much part of our lives that they are almost invisible.

(:ifend:)

(:if equal {Site.PrintBook$:PSW} "True":)

This book tells a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from the earliest moments of human pre-history stretch to our present. We are living a moment where electronic computers, and the counting which forms a central part of their work, are becoming so much part of our lives that they are almost invisible.

(:ifend:)

In short the account here is devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (potentially final disappearance) of calculators.

One might think this is really just a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. These stepping stones are scattered across human history. When and why they were made, how they were used, and why at times they were forgotten for centuries or even millennia is also a poignant part of the picture. So several stories are tangled together here, each informing the other.

Underlying this account is something else - a collection of 'things' constructed at one time or another, over more than 500 years of history. Each of them was supposed to aid the acts of thinking about numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site[^http://calculant.metastudies.net^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This includes a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from the earliest moments of human pre-history stretch to our present. We are living a moment where electronic computers, and the counting which forms a central part of their work, are becoming so much part of our lives that they are almost invisible.

(:ifend:)

(:if equal {Site.PrintBook$:PSW} "True":)

This book tells a 10,000 year story - an account of the ideas and things which humans invented as they came to count and then to 'count' on counting in so much of their daily lives. These "stepping stones" stretch forward from the earliest moments of human pre-history stretch to our present. We are living a moment where electronic computers, and the counting which forms a central part of their work, are becoming so much part of our lives that they are almost invisible.

(:ifend:)

In short the account here is devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (potentially final disappearance) of calculators.

One might think this is really just a story of progressive discovery and invention. But that would be to oversimplify and lose much of what is potentially really interesting. These stepping stones are scattered across human history. When and why they were made, how they were used, and why at times they were forgotten for centuries or even millennia is also a poignant part of the picture. So several stories are tangled together here, each informing the other.

Underlying this account is something else - a collection of 'things' constructed at one time or another, over more than 500 years of history. Each of them was supposed to aid the acts of thinking about numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of 'vanishing point' for all that had come before. For convenience, I will refer to this set of artifacts as "collection Calculant"(:if equal {Site.PrintBook$:PSW} "True":) (the name also of an accompanying web site[^http://calculant.metastudies.net^]. The name is taken from the Latin meaning simply "they calculate")[^"Calculant" is the third-person plural present active of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^](:ifend:). So part of the story here also is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

14 August 2013
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14 August 2013
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A word also about the way I have constructed the historical account. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself~~), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth ~~of ~~the first modern homo-sapiens to the beginning of the ~~"Modern ~~Period~~"~~. This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of ~~the ~~sixteenth century//, with ~~the "~~Early ~~Modern Period" ~~continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into ~~the ~~twentieth~~ century, ~~and //terminating around the two world wars~~//~~. From thereon, the world is regarded by Joseph Camilleri and myself as entering a ~~//~~period of transition~~//~~[^Joseph Camilleri and Jim Falk~~, ~~[[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45.~~/^~~] (but there is not much need to focus on that~~ here)~~.~~

to:

A word also about the way I have constructed the historical account. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]]^]), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^ibid, pp. 132-45.^] (but there is not much need to focus on that here)

14 August 2013
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A word also about the way I have ~~approached constructing~~ the historical account. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

to:

A word also about the way I have constructed the historical account. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

14 August 2013
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Thus the historical account is broken into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

14 August 2013
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(:if equal {Site.PrintBook$:PSW} "False":)~~A word on the state and structure of this essay. First~~ it is ~~a work in progress~~. ~~That is why I have chosen to publish it as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much ~~appreciated~~.~~

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here)

to:

(:if equal {Site.PrintBook$:PSW} "False":)

This is a work in progress, which in part is why it is formed as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

This is a work in progress, which in part is why it is formed as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

Changed lines 60-62 from:

(:if equal {Site.PrintBook$:PSW} "True":)This book should be regarded as a work in progress. Corrections, additional insights, or links to other resources I should know about will be much appreciated~~.(:ifend:)~~

A word also about the way the book is structured. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

A word also about the way the book is structured. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here)

to:

(:if equal {Site.PrintBook$:PSW} "True":)

This book should be regarded as a work in progress. Corrections, additional insights, or links to other resources I should know about will be much appreciated.

This book should be regarded as a work in progress. Corrections, additional insights, or links to other resources I should know about will be much appreciated.

Changed lines 64-66 from:

to:

A word also about the way I have approached constructing the historical account. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

Similarly the historical account breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

Similarly the historical account breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

14 August 2013
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(:if equal {Site.PrintBook$:PSW} "False":)A word on the state and structure of this essay. First it is a work in progress. That is why I have chosen to publish it as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

Changed lines 59-60 from:

to:

(:ifend:)

(:if equal {Site.PrintBook$:PSW} "True":)This book should be regarded as a work in progress. Corrections, additional insights, or links to other resources I should know about will be much appreciated.(:ifend:)

A word also about the way the book is structured. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

(:ifend:)

Similarly the main body of the book breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

(:if equal {Site.PrintBook$:PSW} "True":)This book should be regarded as a work in progress. Corrections, additional insights, or links to other resources I should know about will be much appreciated.(:ifend:)

A word also about the way the book is structured. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

(:ifend:)

Similarly the main body of the book breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

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It might be assumed that mathematics developed through a process that was entirely internal to itself. For example, its development might have been propelled forward because people could ask questions which arise within mathematics, but require the invention of new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

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The focus here will thus be confined to a tiny simple bit of mathematics - the basic numerical calculations required to finally evaluate practical applications of mathematical formulae. Yet even keeping our attention as restricted as this it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will still be useful to take some account of this literature and its insights.

to:

The focus here will thus be confined to a tiny simple bit of mathematics - the basic numerical calculations required to finally evaluate practical applications of mathematical formulae. Yet even keeping our attention as restricted as this it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will still be useful to take some account of this literature and its insights. In particular, whether concerning ourselves with the evolution of the simple areas of mathematics or the more obstruse areas, one question is always raised: what led to this particular development happening how and when it did?

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The focus here will thus be confined to a tiny simple bit of mathematics - the basic numerical calculations required in practical applications of mathematical analysis

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Fortunately in order to understand what has shaped the development of these calculational aids we can largely avoid talking much about mathematics - that huge field of knowledge that has been developed over millennia. For example, we may avoid calculus, set and group theory, the mathematics of infinite dimensional vector spaces that make the modern formulation of quantum mechanics possible, and tensors which Einstein used to express his wonderfully neat equations for the shape of space-time.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It will be sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of additions and subtractions (and multiplications and divisions) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus here will thus be confined to a tiny simple bit of mathematics - the basic numerical calculations required to finally evaluate practical applications of mathematical formulae. Yet even keeping our attention as restricted as this it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will still be useful to take some account of this literature and its insights.

The focus here will thus be confined to a tiny simple bit of mathematics - the basic numerical calculations required to finally evaluate practical applications of mathematical formulae. Yet even keeping our attention as restricted as this it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will still be useful to take some account of this literature and its insights.

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We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

~~The invention and design of technologies has been shaped in major part by what~~ they ~~were~~ to be used for. ~~This raises a wider set of questions for this discussion~~. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

to:

We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this construction strips away much that may be important in why they were invented and used.

Clearly part of what shapes the invention and design of technologies is the purpose which they are to be used for. For calculating technologies this raises a wider set of questions. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

Clearly part of what shapes the invention and design of technologies is the purpose which they are to be used for. For calculating technologies this raises a wider set of questions. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

Changed lines 22-26 from:

As already noted, it will not be necessary to consider much of the huge corpus of extraordinary ~~development~~ in mathematics that have occurred over millennia. For example~~, important though they are we will not need to talk about the development of set and group theory, nor~~ of the ~~development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations ~~for ~~the shape of space time of this universe.~~[~~^See for example ~~[~~[~~http://~~mathworld~~.~~wolfram~~.~~com~~/~~EinsteinFieldEquations~~.~~html~~]] ~~or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf~~]~~] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from~~ the ~~prediction~~ of ~~climate under~~ the ~~stress~~ of ~~global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.~~

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems, in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from manyperspectives. These include those from the mainstream of ~~philosophy~~ and ~~history~~ of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be ~~important~~ to~~ at least~~ take some account of this literature and its insights.

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems, in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many

to:

As already noted, it will not be necessary to consider much of the huge corpus of extraordinary developments in mathematics that have occurred over millennia. For example we will not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we study the creation of the tensor theory which enabled Einstein to write his wonderfully neat equations for the shape of space-time in this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of additions and subtractions (and multiplications and divisions) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus here will thus be confined to a tiny simple bit of mathematics - the basic numerical calculations required in practical applications of mathematical analysis. Yet even keeping our attention as restricted as this it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will still be useful to take some account of this literature and its insights.

The focus here will thus be confined to a tiny simple bit of mathematics - the basic numerical calculations required in practical applications of mathematical analysis. Yet even keeping our attention as restricted as this it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will still be useful to take some account of this literature and its insights.

Changed lines 29-33 from:

The question is simple enough, but the answer is less so. First it raises the ~~question~~ of what caused the developments in mathematics. It might be assumed that mathematics developed through a process that was entirely internal to itself. ~~The development of mathematics~~ might~~, for example, have occurred~~ because people could ask questions which ~~arose~~ within ~~what is known in mathematics, but required ~~new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The~~decision about the~~ sort of problems mathematical thinking ~~might~~ be applied to ~~depends also on the culture of the society in which that thinking is developing and what is ~~seen as interesting or important~~. Beyond that there has historically been a shifting story about who may be taught mathematical thinking and who should frame the questions mathematicians explore. ~~

Similarly at different times and in ~~different cultures have been very differing ideas about~~ the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

The idea of 'mathematics', and doing it, are themselves inventions. The

Similarly at different times and

to:

The question is simple enough, but the answer is less so. First it raises the issue of what caused the developments in mathematics. It might be assumed that mathematics developed through a process that was entirely internal to itself. For example, its development might have been propelled forward because people could ask questions which arise within mathematics, but require the invention of new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The question of what sort of problems mathematical thinking should be applied to will have different answers in different cultures. In different societies different sorts of issues will be seen as interesting or important (and only some of these will be usefully tackled with mathematics). Also different groups of people will be educated to different degrees (if at all) in what is known in mathematics. Finally, different groups may also have influence in framing the questions mathematicians are encouraged to explore.

Similarly at different times and in different cultures there have been very different views taken on the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

The idea of 'mathematics', and doing it, are themselves inventions. The question of what sort of problems mathematical thinking should be applied to will have different answers in different cultures. In different societies different sorts of issues will be seen as interesting or important (and only some of these will be usefully tackled with mathematics). Also different groups of people will be educated to different degrees (if at all) in what is known in mathematics. Finally, different groups may also have influence in framing the questions mathematicians are encouraged to explore.

Similarly at different times and in different cultures there have been very different views taken on the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

30 July 2013
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Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

to:

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (potentially final disappearance) of calculators.

30 July 2013
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Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (~~potentially ~~final disappearance) of calculators.

to:

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

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Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. It is that which is dealt with here.

We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies has been shaped in major part by what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

As already noted, it will not be necessary to consider much of the huge corpus of extraordinary development in mathematics that have occurred over millennia. For example, important though they are we will not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems, in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

!!![[#did]]Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

The question is simple enough, but the answer is less so. First it raises the question of what caused the developments in mathematics. It might be assumed that mathematics developed through a process that was entirely internal to itself. The development of mathematics might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but required new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on the culture of the society in which that thinking is developing and what is seen as interesting or important. Beyond that there has historically been a shifting story about who may be taught mathematical thinking and who should frame the questions mathematicians explore.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. For this reason, amongst others already mentioned, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

History of course relies on evidence. We can only know where and when innovations have occurred when evidence of them can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

!!![[#what]]What is a calculator?

"Calculator" could be taken to mean a variety of things. It could be calculation 'app.' on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which we would comfortably refer to as a machine. But with that caveat, we will be using the term "calculator" very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

!!![[#discussion]]A discussion in three parts.

A final word on the state and structure of this essay. First it is a work in progress. That is why I have chosen to publish it as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

Similarly the history here breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

<<|[[HistoryTrail|History Contents]]|>>

[^#^]

<<|[[HistoryTrail|History Contents]]|>>

We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies has been shaped in major part by what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

As already noted, it will not be necessary to consider much of the huge corpus of extraordinary development in mathematics that have occurred over millennia. For example, important though they are we will not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems, in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

!!![[#did]]Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

The question is simple enough, but the answer is less so. First it raises the question of what caused the developments in mathematics. It might be assumed that mathematics developed through a process that was entirely internal to itself. The development of mathematics might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but required new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on the culture of the society in which that thinking is developing and what is seen as interesting or important. Beyond that there has historically been a shifting story about who may be taught mathematical thinking and who should frame the questions mathematicians explore.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. For this reason, amongst others already mentioned, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

History of course relies on evidence. We can only know where and when innovations have occurred when evidence of them can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

!!![[#what]]What is a calculator?

"Calculator" could be taken to mean a variety of things. It could be calculation 'app.' on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which we would comfortably refer to as a machine. But with that caveat, we will be using the term "calculator" very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

!!![[#discussion]]A discussion in three parts.

A final word on the state and structure of this essay. First it is a work in progress. That is why I have chosen to publish it as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

Similarly the history here breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

<<|[[HistoryTrail|History Contents]]|>>

[^#^]

<<|[[HistoryTrail|History Contents]]|>>

30 July 2013
by -

Changed lines 7-8 from:

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

to:

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (potentially final disappearance) of calculators.

Changed lines 16-65 from:

We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies has been shaped in major part by what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

As already noted, it will not be necessary to consider much of the huge corpus of extraordinary development in mathematics that have occurred over millennia. For example, important though they are we will not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems, in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

!!![[#did]]Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

The question is simple enough, but the answer is less so. First it raises the question of what caused the developments in mathematics. It might be assumed that mathematics developed through a process that was entirely internal to itself. The development of mathematics might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but required new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on the culture of the society in which that thinking is developing and what is seen as interesting or important. Beyond that there has historically been a shifting story about who may be taught mathematical thinking and who should frame the questions mathematicians explore.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. For this reason, amongst others already mentioned, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

History of course relies on evidence. We can only know where and when innovations have occurred when evidence of them can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

!!![[#what]]What is a calculator?

"Calculator" could be taken to mean a variety of things. It could be calculation 'app.' on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which we would comfortably refer to as a machine. But with that caveat, we will be using the term "calculator" very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

!!![[#discussion]]A discussion in three parts.

A final word on the state and structure of this essay. First it is a work in progress. That is why I have chosen to publish it as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

Similarly the history here breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

<<|[[HistoryTrail|History Contents]]|>>

[^#^]

<<|[[HistoryTrail|History Contents]]|>>

to:

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they

30 July 2013
by -

Changed lines 7-9 from:

The story here is based on a series of stepping stones which stretch back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). This is a collection of

to:

Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

The story here is based on a series of stepping stones which stretch back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). This is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

!!![[#initial]]Initial observations.

The spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction - from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. It is that which is dealt with here.

We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies has been shaped in major part by what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

As already noted, it will not be necessary to consider much of the huge corpus of extraordinary development in mathematics that have occurred over millennia. For example, important though they are we will not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems, in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

!!![[#did]]Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

The question is simple enough, but the answer is less so. First it raises the question of what caused the developments in mathematics. It might be assumed that mathematics developed through a process that was entirely internal to itself. The development of mathematics might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but required new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on the culture of the society in which that thinking is developing and what is seen as interesting or important. Beyond that there has historically been a shifting story about who may be taught mathematical thinking and who should frame the questions mathematicians explore.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. For this reason, amongst others already mentioned, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

History of course relies on evidence. We can only know where and when innovations have occurred when evidence of them can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

!!![[#what]]What is a calculator?

"Calculator" could be taken to mean a variety of things. It could be calculation 'app.' on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which we would comfortably refer to as a machine. But with that caveat, we will be using the term "calculator" very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

!!![[#discussion]]A discussion in three parts.

A final word on the state and structure of this essay. First it is a work in progress. That is why I have chosen to publish it as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

Similarly the history here breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

<<|[[HistoryTrail|History Contents]]|>>

[^#^]

<<|[[HistoryTrail|History Contents]]|>>

The story here is based on a series of stepping stones which stretch back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). This is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

!!![[#initial]]Initial observations.

The spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction - from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. It is that which is dealt with here.

We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies has been shaped in major part by what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

As already noted, it will not be necessary to consider much of the huge corpus of extraordinary development in mathematics that have occurred over millennia. For example, important though they are we will not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems, in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

!!![[#did]]Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

The question is simple enough, but the answer is less so. First it raises the question of what caused the developments in mathematics. It might be assumed that mathematics developed through a process that was entirely internal to itself. The development of mathematics might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but required new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on the culture of the society in which that thinking is developing and what is seen as interesting or important. Beyond that there has historically been a shifting story about who may be taught mathematical thinking and who should frame the questions mathematicians explore.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. For this reason, amongst others already mentioned, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

History of course relies on evidence. We can only know where and when innovations have occurred when evidence of them can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

!!![[#what]]What is a calculator?

"Calculator" could be taken to mean a variety of things. It could be calculation 'app.' on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which we would comfortably refer to as a machine. But with that caveat, we will be using the term "calculator" very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

!!![[#discussion]]A discussion in three parts.

A final word on the state and structure of this essay. First it is a work in progress. That is why I have chosen to publish it as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

Similarly the history here breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

<<|[[HistoryTrail|History Contents]]|>>

[^#^]

<<|[[HistoryTrail|History Contents]]|>>

30 July 2013
by -

Changed lines 7-65 from:

The story here is based on a series of stepping stones which stretch back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). This is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

!!![[#initial]]Initial observations.

The spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction - from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. It is that which is dealt with here.

We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies has been shaped in major part by what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

As already noted, it will not be necessary to consider much of the huge corpus of extraordinary development in mathematics that have occurred over millennia. For example, important though they are we will not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems, in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

!!![[#did]]Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

The question is simple enough, but the answer is less so. First it raises the question of what caused the developments in mathematics. It might be assumed that mathematics developed through a process that was entirely internal to itself. The development of mathematics might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but required new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on the culture of the society in which that thinking is developing and what is seen as interesting or important. Beyond that there has historically been a shifting story about who may be taught mathematical thinking and who should frame the questions mathematicians explore.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. For this reason, amongst others already mentioned, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

History of course relies on evidence. We can only know where and when innovations have occurred when evidence of them can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

!!![[#what]]What is a calculator?

"Calculator" could be taken to mean a variety of things. It could be calculation 'app.' on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which we would comfortably refer to as a machine. But with that caveat, we will be using the term "calculator" very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

!!![[#discussion]]A discussion in three parts.

A final word on the state and structure of this essay. First it is a work in progress. That is why I have chosen to publish it as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

Similarly the history here breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

<<|[[HistoryTrail|History Contents]]|>>

[^#^]

<<|[[HistoryTrail|History Contents]]|>>

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Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo ( calculare, calculavi) meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (potentially final disappearance) of calculators.

The story here is based on a series of stepping stones which stretch back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). This is a collection of

The story here is based on a series of stepping stones which stretch back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). This is a collection of

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Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

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Welcome to "Calculant" (the name of this site~~, and~~ in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

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Welcome to "Calculant" (the name of this site - in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

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Welcome to "Calculant" (~~which in case you wonder~~, ~~is~~ Latin ~~for~~ "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

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Welcome to "Calculant" (the name of this site, and in Latin meaning "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

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Welcome to "Calculant" (which in case you wonder, is Latin for "they calculate).[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

to:

Welcome to "Calculant" (which in case you wonder, is Latin for "they calculate").[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

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Welcome to "Calculant".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

to:

Welcome to "Calculant" (which in case you wonder, is Latin for "they calculate).[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

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This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with~~ a~~ more than just a basic mechanism which we would comfortably refer to as a machine. But with that caveat, we will be using the term "calculator" very broadly.

to:

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which we would comfortably refer to as a machine. But with that caveat, we will be using the term "calculator" very broadly.

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History of course relies on evidence. We can only know where and when innovations occurred evidence of ~~it~~ can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

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History of course relies on evidence. We can only know where and when innovations have occurred when evidence of them can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

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The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

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The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems, in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

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The invention and design of technologies has been shaped in major part by what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning~~,~~ which in various ways they have been made to assist.

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The invention and design of technologies has been shaped in major part by what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning which in various ways they have been made to assist.

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The invention and design of technologies has shaped in major part ~~on~~ what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist.

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The invention and design of technologies has been shaped in major part by what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist.

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The spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

Prior calculating technologies~~,~~ had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. It is that which is dealt with here.

Prior calculating technologies

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The spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction - from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. It is that which is dealt with here.

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. It is that which is dealt with here.

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The question is simple enough, but the answer is less so. First it raises the question of what caused the developments in mathematics~~?~~ It might be assumed that mathematics developed through a process that was entirely internal to itself. The development of mathematics might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but required new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on~~social considerations. Beyond that there is a shifting story about who might legitimately be taught what is known about mathematics and who should address themselves to such questions. ~~

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the ~~crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation, or as stated in a holy book). But at other times and place much greater value has been placed on inventing new knowledge. But even~~ when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that

to:

The question is simple enough, but the answer is less so. First it raises the question of what caused the developments in mathematics. It might be assumed that mathematics developed through a process that was entirely internal to itself. The development of mathematics might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but required new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on the culture of the society in which that thinking is developing and what is seen as interesting or important. Beyond that there has historically been a shifting story about who may be taught mathematical thinking and who should frame the questions mathematicians explore.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on the culture of the society in which that thinking is developing and what is seen as interesting or important. Beyond that there has historically been a shifting story about who may be taught mathematical thinking and who should frame the questions mathematicians explore.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). But at other times and places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

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One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. ~~So, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading. ~~

We can only know where development occurred from where there is anyevidence~~ remaining~~. ~~Even this reveals a patchwork of developments in different directions. No doubt this is but a shadow of ~~the ~~totality constituting a complex pattern of discovery, invention, forgetting, and re-discovery all according to the particular needs and constraints of different cultures, values, political structures~~, ~~religions~~, and ~~practices. In short, understanding the evolution of calculating machines is will be seen to be illuminated by investigating it in the context of the evolution of mathematical thinking. But that~~ is ~~no simple picture. The history~~ of ~~developments in calculators and mathematics has been embroidered and shaped by the~~ ~~the social, political and economic circumstances in which they emerged. And at times, mathematical developments have shaped developments in calculators, and times, vice-versa~~.

We can only know where development occurred from where there is any

to:

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. For this reason, amongst others already mentioned, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

History of course relies on evidence. We can only know where and when innovations occurred evidence of it can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

History of course relies on evidence. We can only know where and when innovations occurred evidence of it can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

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"Calculator" could be taken to mean a variety of things. It could be calculation 'app.' on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

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This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices")~~,~~ ~~rather than "calculating machines~~"~~.~~

to:

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"). Where the phrase "calculating machine" is used it will be in the sense used by Martin, referring to something with a more than just a basic mechanism which we would comfortably refer to as a machine. But with that caveat, we will be using the term "calculator" very broadly.

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In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

to:

A final word on the state and structure of this essay. First it is a work in progress. That is why I have chosen to publish it as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

Second, a word about the different parts of the essay. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

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Welcome to "Calculant".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the ~~development~~ of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

The story here is based on a series of stepping stones~~stretching~~ back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). ~~ It ~~is ~~thus ~~a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. ~~In this way~~ the ~~collection and the history it illustrates~~ form a duet - ~~where ~~the two voices each ~~tell~~ part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

The story here is based on a series of stepping stones

to:

Welcome to "Calculant".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the beginnings of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

The story here is based on a series of stepping stones which stretch back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). This is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

The story here is based on a series of stepping stones which stretch back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). This is a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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The ~~early personal calculators of the 1980s presaged ~~the ~~birth of the personal computer. Before long, with every more sophisticated miniaturisation~~, ~~ these~~ began to be embodied in an ever expanding array of converging devices~~, which in ~~turn ~~diffused ~~ever greater computing power across the planet. However~~, however ~~sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and" ~~and ~~"~~not~~" and ~~arithmetic operations of addition, subtraction, multiplication~~ and ~~division ). On top of this were layers of sophisticated programming, memory and input and output.~~

Prior calculating technologies, had to rely on slower mechanical processes. ~~ This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures in order to gain much from looking at how the technologies of mathematics developed.~~

The stepping stones along the path of the development of ~~human aids to calculation show an obvious progression in complexity, sophistication and style from~~ the ~~earliest to~~ the ~~latest. Corresponding ~~to ~~this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented~~ and ~~used.~~

The invention and design of technologies depends in major part on what they were to ~~be used for. There are a number~~ of ~~aspects to this and a range of them will be dealt with in this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist "perhaps"!~~

Prior calculating technologies, had to rely on slower mechanical processes

The stepping stones along the path of the development

The invention and design of technologies depends in major part on what they were

to:

The spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and", "or" and "not", and the arithmetic operations of addition and subtraction from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

Prior calculating technologies, had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. It is that which is dealt with here.

We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies has shaped in major part on what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist.

Prior calculating technologies, had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. It is that which is dealt with here.

We may consider a range of examples of these human aids to calculation as "stepping stones" laying out the path of their development. These show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies has shaped in major part on what they were to be used for. This raises a wider set of questions for this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist.

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The focus

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As already noted, it will not be necessary to consider much of the huge corpus of extraordinary development in mathematics that have occurred over millennia. For example, important though they are we will not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

The focus here will thus be confined to a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

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The question is ~~important,~~ but the answer is ~~a bit more complex. First there is the question of what caused the developments in ~~mathematics~~!~~ ~~ ~~It might be assumed that mathematics developed through a process that was entirely internal to itself. ~~It might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but needed to develop~~ new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

to:

The question is simple enough, but the answer is less so. First it raises the question of what caused the developments in mathematics? It might be assumed that mathematics developed through a process that was entirely internal to itself. The development of mathematics might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but required new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

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Similarly the history here breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to greater ~~purchase~~ in the broader society.

to:

Similarly the history here breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to gain greater use in the broader society.

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Similarly the history here breaks into three parts. The first part, looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop, but does not refer at all to specific objects in the collection. The objects in this "collection Calculant" are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800), when mechanical calculation began to greater purchase in the broader society.

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The story here is based on a series of stepping stones stretching back millenia. Some of the more recent of ~~those~~ (~~over~~ the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). It is thus a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In this way the collection and the history it illustrates form a duet - where the two voices each tell part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The story here is based on a series of stepping stones stretching back millenia. Some of the more recent of these (from the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). It is thus a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In this way the collection and the history it illustrates form a duet - where the two voices each tell part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

24 July 2013
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Welcome to "Calculant".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the development of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "~~stepping stones: toward~~ the~~ end of calculators"~~ title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

to:

Welcome to "Calculant".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the development of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "end of calculators" in the title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

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Welcome to "Calculant".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the development of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops.

The "stepping stones: toward the end of calculators" title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

The "stepping stones: toward the end of calculators" title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

to:

Welcome to "Calculant".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the development of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops. The "stepping stones: toward the end of calculators" title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

24 July 2013
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Welcome to "Calculant"[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^]~~.~~ This is a site devoted to developing a perspective on the long history of calculation - from the development of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops.

to:

Welcome to "Calculant".[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] This is a site devoted to developing a perspective on the long history of calculation - from the development of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops.

24 July 2013
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The story here is based on a series of stepping stones stretching back millenia. Some of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is~~ often~~ called "collection Calculant"). It is thus a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In this way the collection and the history it illustrates form a duet - where the two voices each tell part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

to:

The story here is based on a series of stepping stones stretching back millenia. Some of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is called "collection Calculant"). It is thus a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In this way the collection and the history it illustrates form a duet - where the two voices each tell part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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The ~~title ~~"stepping stones: toward the end of calculators" contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

to:

The "stepping stones: toward the end of calculators" title hence contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

24 July 2013
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Welcome to "Calculant"[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] ~~-~~ a site devoted to ~~providing one~~ perspective on the long history ~~from the development of~~ the ~~human ability~~ to ~~count, to the invention of calculational ~~tools, and to their subsequent progressive disappearance into ~~the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end ~~(objective) and end (final disappearance) of calculators. ~~ The story ~~is ~~of~~ a series of stepping stones stretching back millenia. ~~In this site many~~ of ~~the more recent of those (over the last 400 years) are drawn from a ~~[[Site.ObjectsInTheCollection|personal collection]] ~~- a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way~~ the ~~collection and the history form a duet, where each tells part of~~ the ~~story with the history informing in part what is to be collected, and the collection in part shaping~~ how the story is told.

to:

Welcome to "Calculant"[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^]. This is a site devoted to developing a perspective on the long history of calculation - from the development of counting, to the invention of calculational tools, and to their subsequent progressive disappearance into phones, tablets and laptops.

The title "stepping stones: toward the end of calculators" contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

The story here is based on a series of stepping stones stretching back millenia. Some of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is often called "collection Calculant"). It is thus a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In this way the collection and the history it illustrates form a duet - where the two voices each tell part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

The title "stepping stones: toward the end of calculators" contains a deliberate ambiguity. It refers both to the end (objective) and end (final disappearance) of calculators.

The story here is based on a series of stepping stones stretching back millenia. Some of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] (which for shortness is often called "collection Calculant"). It is thus a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In this way the collection and the history it illustrates form a duet - where the two voices each tell part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

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Welcome to "Calculant"[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute"~~ or more figuratively "they esteem"~~^] - a site devoted to providing one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

to:

Welcome to "Calculant"[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute".^] - a site devoted to providing one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

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Welcome to "Calculant" ~~-~~ the ~~name given to this site[^"Calculant" is~~ the ~~third-person plural present active indicative of the Latin verb calculo meaning~~ "they ~~calculate, they compute~~" ~~or more figuratively "they esteem"^]. The site is~~ devoted to providing one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

to:

Welcome to "Calculant"[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute" or more figuratively "they esteem"^] - a site devoted to providing one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

24 July 2013
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Welcome to "Calculant" - the name given to this site[^Calculant is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute" or more figuratively "they esteem"^]. The site is devoted to providing one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

to:

Welcome to "Calculant" - the name given to this site[^"Calculant" is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute" or more figuratively "they esteem"^]. The site is devoted to providing one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

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Welcome to Calculant~~.~~ ~~This site provides~~ one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

to:

Welcome to "Calculant" - the name given to this site[^Calculant is the third-person plural present active indicative of the Latin verb calculo meaning "they calculate, they compute" or more figuratively "they esteem"^]. The site is devoted to providing one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

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Welcome to Calculant. This site provides one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a personal collection - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

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Welcome to Calculant. This site provides one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a [[Site.ObjectsInTheCollection|personal collection]] - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

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(:title Stepping stones toward the end of calculators :)

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(:title Stepping stones: toward the end of calculators :)

24 July 2013
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Welcome to Calculant. This site provides one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (~~object~~) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a personal collection - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

to:

Welcome to Calculant. This site provides one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (objective) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a personal collection - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

24 July 2013
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Welcome to Calculant. This site provides one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a personal collection - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

to:

Welcome to Calculant. This site provides one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The title thus contains a deliberate ambiguity, referring both to the end (object) and end (final disappearance) of calculators. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a personal collection - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

24 July 2013
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(:title Stepping stones ~~towards~~ the end of calculators :)

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(:title Stepping stones toward the end of calculators :)

24 July 2013
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24 July 2013
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(:title Stepping stones ~~to~~ the end of calculators :)

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(:title Stepping stones towards the end of calculators :)

24 July 2013
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Welcome to Calculant. This site provides one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent disappearance into the converging world of electronic computerised phones, tablets and laptops. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a personal collection - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

to:

Welcome to Calculant. This site provides one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent progressive disappearance into the converging world of electronic computerised phones, tablets and laptops. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a personal collection - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

24 July 2013
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The early personal calculators the 1980s ~~gave~~ birth ~~to~~ the personal computer. Before long, with every more sophisticated miniaturisation, these began to be embodied in an ever expanding array of converging devices, which in turn diffused ever greater computing power across the planet. However, however sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). On top of this were layers of sophisticated programming, memory and input and output.

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The early personal calculators of the 1980s presaged the birth of the personal computer. Before long, with every more sophisticated miniaturisation, these began to be embodied in an ever expanding array of converging devices, which in turn diffused ever greater computing power across the planet. However, however sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). On top of this were layers of sophisticated programming, memory and input and output.

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The stepping stones along the path of the development of human aids to calculation show an obvious progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

24 July 2013
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Welcome to Calculant. This site provides one perspective on the long history from the development of the human ability to count, to the invention of calculational tools, and to their subsequent disappearance into the converging world of electronic computerised phones, tablets and laptops. The story is of a series of stepping stones stretching back millenia. In this site many of the more recent of those (over the last 400 years) are drawn from a personal collection - a collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. In a way the collection and the history form a duet, where each tells part of the story with the history informing in part what is to be collected, and the collection in part shaping how the story is told.

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The early personal calculators the 1980s gave birth to the personal computer. Before long, with every more sophisticated miniaturisation, these began to be embodied in an ever expanding array of converging devices, which in turn diffused ever greater computing power across the planet. However, however sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). On top of this were layers of sophisticated programming, memory and input and output.

24 July 2013
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(:title ~~A collection & history: ~~the ~~development~~ of ~~mechanical calculation~~ :)

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(:title Stepping stones to the end of calculators :)

24 July 2013
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(:title A collection & history: the development of mechanical calculation :)

16 July 2013
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In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

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In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

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The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

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This site is built around a personal collection~~ -~~ of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. To understand the collection this essay sets those historical items against the history of calculation (starting before even the invention of writing, and stretching forward to the first personal electronic calculators). In fact, the devices that have been collected have very much been chosen to illustrate that history. That creates an interesting dynamic which gives both the collection and the history its own peculiar shape.

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This site is built around a personal collection of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. To understand the collection this essay sets those historical items against the history of calculation (starting before even the invention of writing, and stretching forward to the first personal electronic calculators). In fact, the devices that have been collected have very much been chosen to illustrate that history. That creates an interesting dynamic which gives both the collection and the history its own peculiar shape.

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**[[Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|**Click here For Part 2. The Modern Era**]]

[^#^]

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[^#^]

<<|[[HistoryTrail|History Contents]]|>>

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15 July 2013
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<<[[HistoryTrail|History Contents~~|~~]]>>

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<<|[[HistoryTrail|History Contents]]|>>

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<<[[HistoryTrail|History Contents]]>>

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<<[[HistoryTrail|History Contents|]]>>

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<<~~|~~[[HistoryTrail|History Contents]]~~|~~>>

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<<[[HistoryTrail|History Contents]]>>

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<<|[[HistoryTrail|History ~~Contentsl~~]]|>>

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<<|[[HistoryTrail|History Contents]]|>>

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<<|[[HistoryTrail|~~HistoryTrail~~]]|>>

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<<|[[HistoryTrail|History Contentsl]]|>>

12 July 2013
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This site is built around a personal collection - of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. To understand the collection ~~there is also an essay setting~~ those historical items against the history of calculation (starting before even the invention of writing, and stretching forward to the first personal electronic calculators). In fact, the devices that have been collected have very much been chosen to illustrate that history. That creates an interesting dynamic which gives both the collection and the history its own peculiar shape.

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This site is built around a personal collection - of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. To understand the collection this essay sets those historical items against the history of calculation (starting before even the invention of writing, and stretching forward to the first personal electronic calculators). In fact, the devices that have been collected have very much been chosen to illustrate that history. That creates an interesting dynamic which gives both the collection and the history its own peculiar shape.

12 July 2013
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Suppose you find yourself in a new land with which you are not yet very familiar. Trying to find your way forward you stumble across a strange dark place. There is a door which you open, and you enter

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch which, with some hesitation, you flick on. Now next to each case you see a softly illuminated picture. Each depicts the associated objects in use by one or more humans, although the use is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics - some crude, others progressively more recognisably contemporary. You can also make out branching passage ways. Each is characterised by different figures, clothing and, not infrequently, ethnic features. Yet it is still really not clear what all this is about.

Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator. If this is about calculating then, perhaps this exhibition is intended as a history - perhaps of the developments in attempts to build devices to aid calculation - an effort brought effectively to a close during the abrupt transition to personal electronic mathematical calculators in the early 1970s. This seems a plausible explanation of the purpose of the display, but why these particular choices? Perhaps somewhere there is guide or catalogue that might provide more context and explanation?

What follows is one such guide for the collection assembled in this website. It is an attempt to provide some useful context and at least partial explanation of why the objects that are presented might be of interest.

to:

This site is built around a personal collection - of the ever more peculiar machines and devices invented in the past to assist the very human task of calculating. To understand the collection there is also an essay setting those historical items against the history of calculation (starting before even the invention of writing, and stretching forward to the first personal electronic calculators). In fact, the devices that have been collected have very much been chosen to illustrate that history. That creates an interesting dynamic which gives both the collection and the history its own peculiar shape.

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As most people know following the early personal calculators the 1980s gave birth to the personal computer. Before long, with every more sophisticated miniaturisation, these began to be embodied in an ever expanding array of converging devices, which in turn diffused ever greater computing power across the planet. However, however sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this were layers of sophisticated programming, memory and input and output.

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To begin: As most people know following the early personal calculators the 1980s gave birth to the personal computer. Before long, with every more sophisticated miniaturisation, these began to be embodied in an ever expanding array of converging devices, which in turn diffused ever greater computing power across the planet. However, however sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this were layers of sophisticated programming, memory and input and output.

01 July 2013
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!!![[#discussion]]A discussion in ~~two~~ parts.

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!!![[#discussion]]A discussion in three parts.

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In relation to [[Site.ObjectsInTheCollection|the collection of objects]], for which this discussion forms a context, the content similarly breaks effectively into ~~two~~ major parts. The first part, which looks at the relationship between the evolution of calculating and calculators in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects in the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century).

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In relation to [[Site.ObjectsInTheCollection|the collection of objects]], for which this discussion forms a context, the content similarly breaks effectively into three major parts. The first part, which looks at the relationship between the evolution of calculating and calculators in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects in the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century), and the Late Modern Period (from 1800), when mechanical calculation began to be adopted in the broader society.

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<<|[[HistoryTrail|~~Part 1 Origins~~]]|>>

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<<|[[HistoryTrail|HistoryTrail]]|>>

11 November 2012
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A collection is like a puzzle. It ~~raises~~ many questions without the need for a single word. ~~This is ~~the opposite~~. Here some words are required to set the scene~~ to ~~enable key questions to be articulated.~~ Bear with me, and I will try to illustrate what I mean with a metaphor.

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A collection is like a puzzle. It can raise many questions without the need for a single word. But the opposite is also true. Framing interesting questions in relation to the objects in the collection may work best after some words of explanation. Bear with me, and I will try to illustrate what I mean with a metaphor.

11 November 2012
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A collection is like a puzzle. It raises many questions without the need for a single word. This is the opposite. Here some words are required to set the scene to enable key questions to be articulated. ~~It may be helpful to think of this through~~ a metaphor.

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A collection is like a puzzle. It raises many questions without the need for a single word. This is the opposite. Here some words are required to set the scene to enable key questions to be articulated. Bear with me, and I will try to illustrate what I mean with a metaphor.

10 November 2012
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As most people know following the early personal calculators ~~came ~~the ~~personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their~~ ever ~~more functional surface) simply did a few things extremely fast (logic operations such as "if", "and"~~ and ~~"not" and arithmetic operations ~~of ~~addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility ~~and ~~adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For~~ this ~~reason, whilst mathematics encompasses much more than arithmetic it is not necessary~~ to ~~consider all the historical development of its more elaborate analytic structures.~~

It is also clear that this collection is in some sense a history. Laid out, as it is, along a time-line there is an obvious progression in complexity, ~~sophistication and style from the earliest to the later devices. It is possible to construct histories ~~of ~~technical devices such as calculators as some sort of evolution based on solving technical problems with consequent improvements in design~~. ~~ But this strips away much that may be important in why they were invented and used. The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and~~ some ~~others will be dealt with elsewhere in this site. But clearly~~ one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist "perhaps"!

It is also clear that this collection is in some sense a history. Laid out, as it is, along a time-line there is an obvious progression in complexity

to:

As most people know following the early personal calculators the 1980s gave birth to the personal computer. Before long, with every more sophisticated miniaturisation, these began to be embodied in an ever expanding array of converging devices, which in turn diffused ever greater computing power across the planet. However, however sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this were layers of sophisticated programming, memory and input and output.

Prior calculating technologies, had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures in order to gain much from looking at how the technologies of mathematics developed.

One thing is clear from this collection. Laid out, as it is, along a time-line there is an obvious progression in complexity, sophistication and style from the earliest to the later devices. Corresponding to this it is possible to construct histories of such instruments as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and a range of them will be dealt with in this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist "perhaps"!

Prior calculating technologies, had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures in order to gain much from looking at how the technologies of mathematics developed.

One thing is clear from this collection. Laid out, as it is, along a time-line there is an obvious progression in complexity, sophistication and style from the earliest to the later devices. Corresponding to this it is possible to construct histories of such instruments as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But this strips away much that may be important in why they were invented and used.

The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and a range of them will be dealt with in this discussion. Clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist "perhaps"!

10 November 2012
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You are puzzled. Retracing your steps in the gloom you find a dimly lit switch which, with some hesitation, you flick on. Now next to each case you see a softly illuminated picture. ~~These depict~~ the associated objects in use by one or more humans, although the use is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics - some crude, others progressively more recognisably contemporary. You can also make out branching passage ways. Each is characterised by different figures, clothing and, not infrequently, ethnic features. Yet it is still really not clear what all this is about.

to:

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch which, with some hesitation, you flick on. Now next to each case you see a softly illuminated picture. Each depicts the associated objects in use by one or more humans, although the use is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics - some crude, others progressively more recognisably contemporary. You can also make out branching passage ways. Each is characterised by different figures, clothing and, not infrequently, ethnic features. Yet it is still really not clear what all this is about.

10 November 2012
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A collection is like a puzzle. It raises many questions without the need for a single word. This is the opposite. Here some words are ~~needed first ~~to set the scene ~~before~~ key questions ~~can~~ be ~~usefully posed~~. It may be helpful to think of ~~the role of this collection through a metaphor. ~~

Suppose you find yourself in a new land with which you are not yet very familiar. Trying to find your way forward you stumble across a strange dark place. There is a door whichyou ~~open, and you enter. Inside it is so dark you cannot make out more than the beginning of a path. As you take a few more steps on your left a light soundlessly turns on illuminating a glass case. In it is a pile of sand, but~~ you ~~cannot yet understand its purpose. As you walk on the light extinguishes and a new light illuminates the next case~~. In ~~it is~~ a ~~small group of stones~~. ~~ ~~In the next~~,~~ a ~~rope~~ with ~~knots. In the next a clay object with symbols on it. In the next a set of beads~~ on a ~~string. In the next strange symbols scribed on a decaying yellowing material. Soon an assortment of devices with gears, dials, and various displays emerge.~~

You are puzzled. Retracing your steps in the gloom you ~~find a dimly lit switch. Hesitatingly~~ you ~~decide to flick it to the other position. Now next to each case you see a softly illuminated ageing picture. These are there to show the objects in use by humans. Each picture shows a human figure doing something with the objects. But what that something is is still not clear. One or two figures are naked. Most are dressed in clothing of different fabrics - some crude, others progressively more recognisably of your time. In this half light you see branching passage ways. Each is characterised by different figures, clothing and, not infrequently, ethnic features. Yet it is still really not clear what all this is about.~~

Getting to what seems to be the end of the display you see in ~~the last case something you recognise. It is a pocket calculator.~~

If this is about calculating then, perhaps it represents a history - and probably that is of the developments in calculation technology which ended with the ~~abrupt transition to personal electronic mathematical calculators in the early 1970s. This seems a plausible explanation of the purpose of the display, but why these particular objects in this arrangement? Some more explanation would be helpful. Perhaps somewhere there is guide or catalogue that might provide more context and explanation? What follows is one such guide. It is an attempt to provide some useful context and at least partial explanation of why the objects in this collection have been chosen~~.

Suppose you find yourself in a new land with which you are not yet very familiar. Trying to find your way forward you stumble across a strange dark place. There is a door which

You are puzzled. Retracing your steps in the gloom

Getting to what seems to be the end of the display you see

If this is about calculating then, perhaps it represents a history - and probably that is of the developments in calculation technology which ended with

to:

A collection is like a puzzle. It raises many questions without the need for a single word. This is the opposite. Here some words are required to set the scene to enable key questions to be articulated. It may be helpful to think of this through a metaphor.

Suppose you find yourself in a new land with which you are not yet very familiar. Trying to find your way forward you stumble across a strange dark place. There is a door which you open, and you enter. Inside it is so dark you cannot make out more than the beginning of a path. You take a few steps forward and on your left a light clicks on illuminating a glass case. In it is a pile of sand, but its purpose is unclear. As you walk on that light extinguishes and a new light illuminates another case. In it is a small group of stones. In the next, a rope with knots. In the next a clay object with symbols on it. In the next a set of beads on a string. In the next strange symbols scribed on a decaying yellowing material. Soon an assortment of devices with gears, dials, and various displays emerge.

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch which, with some hesitation, you flick on. Now next to each case you see a softly illuminated picture. These depict the associated objects in use by one or more humans, although the use is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics - some crude, others progressively more recognisably contemporary. You can also make out branching passage ways. Each is characterised by different figures, clothing and, not infrequently, ethnic features. Yet it is still really not clear what all this is about.

Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator. If this is about calculating then, perhaps this exhibition is intended as a history - perhaps of the developments in attempts to build devices to aid calculation - an effort brought effectively to a close during the abrupt transition to personal electronic mathematical calculators in the early 1970s. This seems a plausible explanation of the purpose of the display, but why these particular choices? Perhaps somewhere there is guide or catalogue that might provide more context and explanation?

What follows is one such guide for the collection assembled in this website. It is an attempt to provide some useful context and at least partial explanation of why the objects that are presented might be of interest.

Suppose you find yourself in a new land with which you are not yet very familiar. Trying to find your way forward you stumble across a strange dark place. There is a door which you open, and you enter. Inside it is so dark you cannot make out more than the beginning of a path. You take a few steps forward and on your left a light clicks on illuminating a glass case. In it is a pile of sand, but its purpose is unclear. As you walk on that light extinguishes and a new light illuminates another case. In it is a small group of stones. In the next, a rope with knots. In the next a clay object with symbols on it. In the next a set of beads on a string. In the next strange symbols scribed on a decaying yellowing material. Soon an assortment of devices with gears, dials, and various displays emerge.

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch which, with some hesitation, you flick on. Now next to each case you see a softly illuminated picture. These depict the associated objects in use by one or more humans, although the use is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics - some crude, others progressively more recognisably contemporary. You can also make out branching passage ways. Each is characterised by different figures, clothing and, not infrequently, ethnic features. Yet it is still really not clear what all this is about.

Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator. If this is about calculating then, perhaps this exhibition is intended as a history - perhaps of the developments in attempts to build devices to aid calculation - an effort brought effectively to a close during the abrupt transition to personal electronic mathematical calculators in the early 1970s. This seems a plausible explanation of the purpose of the display, but why these particular choices? Perhaps somewhere there is guide or catalogue that might provide more context and explanation?

What follows is one such guide for the collection assembled in this website. It is an attempt to provide some useful context and at least partial explanation of why the objects that are presented might be of interest.

10 November 2012
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A collection is like a puzzle. It raises many questions without the need for a single word. This is the opposite. Here ~~are posed quite a few words even before ~~the ~~questions are fully articulated.~~

Consider this. You are in a new land with which you are not yet very familiar. You stumble acrossa ~~strange dark place ~~you ~~have never visited. There is a door. You open it and enter~~.

Inside it is so dark you cannot make out more than the beginning of a path. ~~As you take ~~a ~~few more steps on your left a light soundlessly turns on illuminating a glass case. In it is a pile~~ of ~~sand. You ponder why it is there, but cannot make it out. As you walk~~ on ~~the light extinguishes and a new light illuminates the next case. In it is a small group of stones~~. ~~ In the next, a rope with knots. In the next a clay object with symbols on ~~it~~. In the next a set of beads on~~ a ~~string. In the next strange symbols scribed on a decaying yellowing material. Soon an assortment of devices with gears, dials, and various displays emerge~~.~~..~~

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. ~~Now next to each case~~ you ~~see~~ a ~~softly illuminated ageing picture~~. ~~Each picture shows a human figure doing something with the objects - although what the purpose is is not clear. One or two figures ~~are ~~naked. Most are dressed~~ in ~~clothing of different fabrics, perhaps from different times. In this half light you see branching passage ways with different figures, clothing, and seemingly ethnic features. You~~ are ~~seeing more and yet it is not clear what it really all means.~~

Getting to what seems to be the end of the display you see ~~in the last case something you recognise. It is a pocket calculator. So, you surmise, perhaps this may seek to illustrate the developments in calculation technology which ended with~~ the ~~abrupt transition to personal electronic mathematical calculators ~~in the ~~early 1970s.~~

This seems a plausible explanation of the purpose of the display, but why these particular objects, ~~and what does their arrangement signify? Is there a card (or catalogue essay) somewhere to provide more context and explanation? This is one such essay.~~

The above story is of course a ~~metaphor for~~ the ~~personal collection ~~of ~~objects which is the subject of this website. This essay seeks to provide a context and at least partial explanation of the choice and meaning of them~~.

Consider this. You are in a new land with which you are not yet very familiar. You stumble across

Inside it is so dark you cannot make out more than the beginning of a path

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position

Getting to what seems to be the end of the display

This seems a plausible explanation of the purpose of the display, but why these particular objects

The above story is of course

to:

A collection is like a puzzle. It raises many questions without the need for a single word. This is the opposite. Here some words are needed first to set the scene before key questions can be usefully posed. It may be helpful to think of the role of this collection through a metaphor.

Suppose you find yourself in a new land with which you are not yet very familiar. Trying to find your way forward you stumble across a strange dark place. There is a door which you open, and you enter. Inside it is so dark you cannot make out more than the beginning of a path. As you take a few more steps on your left a light soundlessly turns on illuminating a glass case. In it is a pile of sand, but you cannot yet understand its purpose. As you walk on the light extinguishes and a new light illuminates the next case. In it is a small group of stones. In the next, a rope with knots. In the next a clay object with symbols on it. In the next a set of beads on a string. In the next strange symbols scribed on a decaying yellowing material. Soon an assortment of devices with gears, dials, and various displays emerge.

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. Now next to each case you see a softly illuminated ageing picture. These are there to show the objects in use by humans. Each picture shows a human figure doing something with the objects. But what that something is is still not clear. One or two figures are naked. Most are dressed in clothing of different fabrics - some crude, others progressively more recognisably of your time. In this half light you see branching passage ways. Each is characterised by different figures, clothing and, not infrequently, ethnic features. Yet it is still really not clear what all this is about.

Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator.

If this is about calculating then, perhaps it represents a history - and probably that is of the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s. This seems a plausible explanation of the purpose of the display, but why these particular objects in this arrangement? Some more explanation would be helpful. Perhaps somewhere there is guide or catalogue that might provide more context and explanation? What follows is one such guide. It is an attempt to provide some useful context and at least partial explanation of why the objects in this collection have been chosen.

Suppose you find yourself in a new land with which you are not yet very familiar. Trying to find your way forward you stumble across a strange dark place. There is a door which you open, and you enter. Inside it is so dark you cannot make out more than the beginning of a path. As you take a few more steps on your left a light soundlessly turns on illuminating a glass case. In it is a pile of sand, but you cannot yet understand its purpose. As you walk on the light extinguishes and a new light illuminates the next case. In it is a small group of stones. In the next, a rope with knots. In the next a clay object with symbols on it. In the next a set of beads on a string. In the next strange symbols scribed on a decaying yellowing material. Soon an assortment of devices with gears, dials, and various displays emerge.

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. Now next to each case you see a softly illuminated ageing picture. These are there to show the objects in use by humans. Each picture shows a human figure doing something with the objects. But what that something is is still not clear. One or two figures are naked. Most are dressed in clothing of different fabrics - some crude, others progressively more recognisably of your time. In this half light you see branching passage ways. Each is characterised by different figures, clothing and, not infrequently, ethnic features. Yet it is still really not clear what all this is about.

Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator.

If this is about calculating then, perhaps it represents a history - and probably that is of the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s. This seems a plausible explanation of the purpose of the display, but why these particular objects in this arrangement? Some more explanation would be helpful. Perhaps somewhere there is guide or catalogue that might provide more context and explanation? What follows is one such guide. It is an attempt to provide some useful context and at least partial explanation of why the objects in this collection have been chosen.

Changed lines 16-18 from:

As ~~you already will ~~know following ~~those~~ early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

It is also clear that this collection is in some sense a history.~~It is laid ~~out ~~along a~~ time-line~~. There~~ is an obvious progression in complexity, sophistication and style from the earliest to the later devices. It is possible to construct histories of technical devices such as calculators as some sort of evolution based on solving technical problems with consequent improvements in design. But this strips away much that may be important in why they were invented and used. The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and some others will be dealt with elsewhere in this site. But clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist "perhaps"!

It is also clear that this collection is in some sense a history.

to:

As most people know following the early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

It is also clear that this collection is in some sense a history. Laid out, as it is, along a time-line there is an obvious progression in complexity, sophistication and style from the earliest to the later devices. It is possible to construct histories of technical devices such as calculators as some sort of evolution based on solving technical problems with consequent improvements in design. But this strips away much that may be important in why they were invented and used. The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and some others will be dealt with elsewhere in this site. But clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist "perhaps"!

It is also clear that this collection is in some sense a history. Laid out, as it is, along a time-line there is an obvious progression in complexity, sophistication and style from the earliest to the later devices. It is possible to construct histories of technical devices such as calculators as some sort of evolution based on solving technical problems with consequent improvements in design. But this strips away much that may be important in why they were invented and used. The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and some others will be dealt with elsewhere in this site. But clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist "perhaps"!

23 July 2012
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23 July 2012
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23 July 2012
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<|[[Site.TrailIndex1| ~~+~~ ]]|>

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<|[[Site.TrailIndex1| Historical Context ]]|>

23 July 2012
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30 June 2012
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So what is this rather long and overly footnoted essay doing in a website about a calculator collection?

30 June 2012
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**[[#did|Did increases in the power of mathematics lead the development of calculators? Was it the other way round?]]

**[[#what|What is a calculator?]]

**[[#discussion|A discussion in two parts]]

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(:toc:)

16 June 2012
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16 June 2012
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**[[Part 1 BeforeTheModernEpoch|**Click here For Part 1. Origins - Before the Modern ~~Era~~**]]

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**[[Part 1 BeforeTheModernEpoch|**Click here For Part 1. Origins - Before the Modern Epoch**]]

16 June 2012
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**[[Part 1 BeforeTheModernEpoch|**Click here For Part 1. Origins**]]

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**[[Part 1 BeforeTheModernEpoch|**Click here For Part 1. Origins - Before the Modern Era**]]

16 June 2012
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**[[~~Site.Part1BeforeTheModernEpoch~~|**Click here For Part 1. Origins**]]

**[[~~Site.~~Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|**Click here For Part 2. The Modern Era**]]

**[[

to:

**[[Part 1 BeforeTheModernEpoch|**Click here For Part 1. Origins**]]

**[[Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|**Click here For Part 2. The Modern Era**]]

**[[Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|**Click here For Part 2. The Modern Era**]]

16 June 2012
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**~~*~~[[Site.Part1BeforeTheModernEpoch|Click here For Part 1. Origins~~]]~~**

**~~*~~[[Site.Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|Click here For Part 2. The Modern Era~~]]~~**

to:

**[[Site.Part1BeforeTheModernEpoch|**Click here For Part 1. Origins**]]

**[[Site.Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|**Click here For Part 2. The Modern Era**]]

**[[Site.Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|**Click here For Part 2. The Modern Era**]]

16 June 2012
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*[Site.Part1BeforeTheModernEpoch|Click here For Part 1. Origins]]

*[Site.Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|Click here For Part 2. The Modern Era]]

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***[[Site.Part1BeforeTheModernEpoch|Click here For Part 1. Origins]]**

***[[Site.Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|Click here For Part 2. The Modern Era]]**

***[[Site.Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|Click here For Part 2. The Modern Era]]**

16 June 2012
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*[Site.Part1BeforeTheModernEpoch|Click here For Part 1. Origins]]

*[Site.Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|Click here For Part 2. The Modern Era]]

*[Site.Part2.TheModernEpochAndTheEmergenceOfTheModernCalculator|Click here For Part 2. The Modern Era]]

16 June 2012
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!!Introduction

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**[[#did|~~!!!~~Did increases in the power of mathematics lead the development of calculators? Was it the other way round?]]

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**[[#did|Did increases in the power of mathematics lead the development of calculators? Was it the other way round?]]

16 June 2012
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!! [[#introduction]]Introduction

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!! [[#introduction]]Introduction

16 June 2012
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*[[#introduction|Introduction]]

**[[#initial|Initial observations]]

**[[#did|!!!Did increases in the power of mathematics lead the development of calculators? Was it the other way round?]]

**[[#what|What is a calculator?]]

**[[#discussion|A discussion in two parts]]

!! [[#introduction]]Introduction

**[[#initial|Initial observations]]

**[[#did|!!!Did increases in the power of mathematics lead the development of calculators? Was it the other way round?]]

**[[#what|What is a calculator?]]

**[[#discussion|A discussion in two parts]]

!! [[#introduction]]Introduction

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!!!Initial observations.

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!!![[#initial]]Initial observations.

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!!!Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

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!!![[#did]]Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

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!!!What is a calculator?

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!!![[#what]]What is a calculator?

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!!!A discussion in two parts.

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!!![[#discussion]]A discussion in two parts.

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!!![[Site.Part1BeforeTheModernEpoch|Click here For Part 1. ~~Beginnings: Before the Modern Epoch ~~]]

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!!![[Site.Part1BeforeTheModernEpoch|Click here For Part 1. Origins]]

18 February 2012
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This seems a plausible explanation of the purpose of the display, but why these particular objects, and what does their arrangement signify? Is there a card (or catalogue essay) somewhere to provide more context and explanation? This is one such essay.

18 February 2012
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Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator. So, you surmise, perhaps this ~~is a collection of objects relating to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early ~~1970s.

So we have an explanation of the purpose of the display, but why these objects, and what does their arrangement signify? Is there a card (or catalogue essay) somewhere to provide more context and explanation? This is one such essay.

The above story is of course a metaphor for the personal collection of~~calculation devices ~~which is the subject of this website. This essay seeks to provide a context and at least partial explanation of the choice and meaning of ~~those objects~~.

So we have an explanation of the purpose of the display, but why these objects, and what does their arrangement signify? Is there a card (or catalogue essay) somewhere to provide more context and explanation? This is one such essay.

The above story is of course a metaphor for the personal collection of

to:

Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator. So, you surmise, perhaps this may seek to illustrate the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

So we have an explanation of the purpose of the display, but why these particular objects, and what does their arrangement signify? Is there a card (or catalogue essay) somewhere to provide more context and explanation? This is one such essay.

The above story is of course a metaphor for the personal collection of objects which is the subject of this website. This essay seeks to provide a context and at least partial explanation of the choice and meaning of them.

So we have an explanation of the purpose of the display, but why these particular objects, and what does their arrangement signify? Is there a card (or catalogue essay) somewhere to provide more context and explanation? This is one such essay.

The above story is of course a metaphor for the personal collection of objects which is the subject of this website. This essay seeks to provide a context and at least partial explanation of the choice and meaning of them.

18 February 2012
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Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator. So, you ~~propose~~, perhaps this is a collection of objects relating to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

to:

Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator. So, you surmise, perhaps this is a collection of objects relating to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

18 February 2012
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18 February 2012
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18 February 2012
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!!!Did increases in the power of mathematics ~~led~~ the development of calculators? Was it the other way round?

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!!!Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

18 February 2012
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You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. Now next to each case you see a softly illuminated ageing picture. Each picture shows a human figure doing something with the objects - although what the purpose is is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics, perhaps from different times. In this half light you see branching passage ways with different figures, clothing, and seemingly ethnic features. You are seeing more and yet it is not clear what it really all means~~. Is there a card (or catalogue essay) somewhere to provide more context? This is one such essay~~.

to:

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. Now next to each case you see a softly illuminated ageing picture. Each picture shows a human figure doing something with the objects - although what the purpose is is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics, perhaps from different times. In this half light you see branching passage ways with different figures, clothing, and seemingly ethnic features. You are seeing more and yet it is not clear what it really all means.

Changed lines 15-17 from:

to:

So we have an explanation of the purpose of the display, but why these objects, and what does their arrangement signify? Is there a card (or catalogue essay) somewhere to provide more context and explanation? This is one such essay.

The above story is of course a metaphor for the personal collection of calculation devices which is the subject of this website. This essay seeks to provide a context and at least partial explanation of the choice and meaning of those objects.

The above story is of course a metaphor for the personal collection of calculation devices which is the subject of this website. This essay seeks to provide a context and at least partial explanation of the choice and meaning of those objects.

18 February 2012
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Getting to the end of the display you see in the last case something you recognise. It is a pocket calculator. So, you propose, perhaps this is a collection of objects relating to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

to:

Getting to what seems to be the end of the display you see in the last case something you recognise. It is a pocket calculator. So, you propose, perhaps this is a collection of objects relating to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

18 February 2012
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!!!Initial observations.

As you already will know following those early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

As you already will know following those early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

18 February 2012
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You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. Now next to each case you see ~~an~~ softly illuminated ageing picture. Each picture shows a human figure doing something with the objects - although what the purpose is is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics, perhaps from different times. In this half light you see branching passage ways with different figures, clothing, and seemingly ethnic features. You are ~~getting much ~~more~~ information~~ and yet it is not clear what it really all means. Is there a card (or catalogue essay) somewhere to provide more context? This is one such essay.

to:

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. Now next to each case you see a softly illuminated ageing picture. Each picture shows a human figure doing something with the objects - although what the purpose is is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics, perhaps from different times. In this half light you see branching passage ways with different figures, clothing, and seemingly ethnic features. You are seeing more and yet it is not clear what it really all means. Is there a card (or catalogue essay) somewhere to provide more context? This is one such essay.

Changed lines 15-17 from:

Getting to the ~~last display one thing is obvious enough. This collection relates to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s. ~~

Perhaps we already know that following those early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

Perhaps we

to:

Getting to the end of the display you see in the last case something you recognise. It is a pocket calculator. So, you propose, perhaps this is a collection of objects relating to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

The above story is of course a metaphor for the personal collection of calculation devices which is the subject of this website.

We already know that following those early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

The above story is of course a metaphor for the personal collection of calculation devices which is the subject of this website.

We already know that following those early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

13 February 2012
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//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however ~~primitive~~, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a ~~primitive~~ calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

to:

//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

Changed line 55 from:

In relation to [[Site.ObjectsInTheCollection|the collection of objects]], for which this discussion forms a context, the content similarly breaks effectively into two major parts. The first part, which looks at the relationship between the evolution of calculating and ~~calculating~~ in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects in the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century).

to:

In relation to [[Site.ObjectsInTheCollection|the collection of objects]], for which this discussion forms a context, the content similarly breaks effectively into two major parts. The first part, which looks at the relationship between the evolution of calculating and calculators in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects in the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century).

12 February 2012
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Changed lines 9-11 from:

Inside it is so dark you cannot make out more than ~~a path. As you walk down it on your left a light soundlessly turns on illuminating ~~a ~~glass case. In it is ~~a ~~pile of sand~~. ~~You ponder why it is there, but cannot make it out. As you walk on the light extinguishes and a new light illuminates~~ the ~~next case. In it is a small group of pebbles~~. ~~ ~~In ~~the next,~~ a ~~knotted rope. In the next a clay envelope with symbols on it~~. ~~ ~~In the next a ~~set of beads~~ on ~~a string~~. ~~Then various machines emerge...~~

It is all meaningless until, retracing your steps in the ~~gloom you find a dimly lit lever with a large knob. You pull the lever towards you. Now next to each case you see an illuminated picture. Each picture shows ~~a ~~human figure doing something with the sand, pebbles, rope. The figures are dressed in period clothing. A possibility occurs to you. You return to the lever and turn the knob. On each case a date flashes on. You see branching passage ways with different figures, clothing, and~~ ethnic features. You are getting much more information and yet it is not clear what it really all means. Is there a card (or catalogue essay) somewhere to provide more context? This is one such essay.

It is all meaningless until, retracing your steps in

to:

Inside it is so dark you cannot make out more than the beginning of a path. As you take a few more steps on your left a light soundlessly turns on illuminating a glass case. In it is a pile of sand. You ponder why it is there, but cannot make it out. As you walk on the light extinguishes and a new light illuminates the next case. In it is a small group of stones. In the next, a rope with knots. In the next a clay object with symbols on it. In the next a set of beads on a string. In the next strange symbols scribed on a decaying yellowing material. Soon an assortment of devices with gears, dials, and various displays emerge...

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. Now next to each case you see an softly illuminated ageing picture. Each picture shows a human figure doing something with the objects - although what the purpose is is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics, perhaps from different times. In this half light you see branching passage ways with different figures, clothing, and seemingly ethnic features. You are getting much more information and yet it is not clear what it really all means. Is there a card (or catalogue essay) somewhere to provide more context? This is one such essay.

You are puzzled. Retracing your steps in the gloom you find a dimly lit switch. Hesitatingly you decide to flick it to the other position. Now next to each case you see an softly illuminated ageing picture. Each picture shows a human figure doing something with the objects - although what the purpose is is not clear. One or two figures are naked. Most are dressed in clothing of different fabrics, perhaps from different times. In this half light you see branching passage ways with different figures, clothing, and seemingly ethnic features. You are getting much more information and yet it is not clear what it really all means. Is there a card (or catalogue essay) somewhere to provide more context? This is one such essay.

05 February 2012
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Changed line 5 from:

A collection is like a puzzle. It raises many questions without ~~a single word. This is the opposite. Here are posed quite a few words even before the questions are fully formed~~.

to:

A collection is like a puzzle. It raises many questions without the need for a single word. This is the opposite. Here are posed quite a few words even before the questions are fully articulated.

05 February 2012
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Changed lines 9-12 from:

Inside it is so dark you cannot make out more than a path. As you walk down it on your left a light ~~snaps~~ on illuminating a glass case. In it is a pile of sand. You ponder why it is there, but cannot make it out. As you walk on the light extinguishes and a new light illuminates ~~a new~~ case. In it is a small group of pebbles. In the next, a knotted rope. In the next a clay envelope with symbols on it. In the next a set of beads on a string. Then various machines emerge...

It is all meaningless until, retracing your steps~~you find a lever with~~ a ~~large knob. You pull the lever towards you. Now next to each case ~~you ~~see an illuminated picture. Each picture shows a human figure doing something with the sand, pebbles, rope. The figures are dressed in period clothing~~. ~~A possibility occurs to you. You return to the lever and turn the knob~~. ~~ With each case a date flashes on along~~. ~~You see branching passage ways with different figures, clothing, and ethnic features. You are getting much more information and yet it is not clear what it really all means. Is there a card (or cards) somewhere to provide more context? This is one such card~~.

It is all meaningless until, retracing your steps

to:

Inside it is so dark you cannot make out more than a path. As you walk down it on your left a light soundlessly turns on illuminating a glass case. In it is a pile of sand. You ponder why it is there, but cannot make it out. As you walk on the light extinguishes and a new light illuminates the next case. In it is a small group of pebbles. In the next, a knotted rope. In the next a clay envelope with symbols on it. In the next a set of beads on a string. Then various machines emerge...

It is all meaningless until, retracing your steps in the gloom you find a dimly lit lever with a large knob. You pull the lever towards you. Now next to each case you see an illuminated picture. Each picture shows a human figure doing something with the sand, pebbles, rope. The figures are dressed in period clothing. A possibility occurs to you. You return to the lever and turn the knob. On each case a date flashes on. You see branching passage ways with different figures, clothing, and ethnic features. You are getting much more information and yet it is not clear what it really all means. Is there a card (or catalogue essay) somewhere to provide more context? This is one such essay.

It is all meaningless until, retracing your steps in the gloom you find a dimly lit lever with a large knob. You pull the lever towards you. Now next to each case you see an illuminated picture. Each picture shows a human figure doing something with the sand, pebbles, rope. The figures are dressed in period clothing. A possibility occurs to you. You return to the lever and turn the knob. On each case a date flashes on. You see branching passage ways with different figures, clothing, and ethnic features. You are getting much more information and yet it is not clear what it really all means. Is there a card (or catalogue essay) somewhere to provide more context? This is one such essay.

Changed lines 15-17 from:

We know that following that came the personal computer which in various converging guises (including even phones) diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply do

to:

Getting to the last display one thing is obvious enough. This collection relates to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

Perhaps we already know that following those early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

Perhaps we already know that following those early personal calculators came the personal computer. In various converging guises (including even phones) these diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply did a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

05 February 2012
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Changed lines 25-28 from:

!!!Did ~~mathematics lead to ~~the ~~need for calculators?~~

For a start, it might be tempting to see the ~~developments as being created through some process which~~ is ~~entirely internal to mathematical thinking~~. ~~For example, development might be seen to occur because people can ask questions which arise within what is known in~~ mathematics~~, but need to develop new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all. The idea of '~~mathematics~~', and doing it, are themselves inventions. The decision about~~ the ~~sort of problems mathematical thinking might be applied to is a social choice. Beyond that there is a shifting story about who might legitimately be taught what is known about mathematics and who should address themselves~~ to ~~such questions.~~ Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation, or as stated in a holy book). But at other times and place much greater value has been placed on inventing new knowledge. But even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

For a start, it might be tempting to see

to:

!!!Did increases in the power of mathematics led the development of calculators? Was it the other way round?

The question is important, but the answer is a bit more complex. First there is the question of what caused the developments in mathematics! It might be assumed that mathematics developed through a process that was entirely internal to itself. It might, for example, have occurred because people could ask questions which arose within what is known in mathematics, but needed to develop new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all.

The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to depends also on social considerations. Beyond that there is a shifting story about who might legitimately be taught what is known about mathematics and who should address themselves to such questions.

Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation, or as stated in a holy book). But at other times and place much greater value has been placed on inventing new knowledge. But even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

Changed lines 38-50 from:

We can only know where development occurred from where there is any evidence remaining. Even this reveals a patchwork of developments in different directions. No doubt this is but a shadow of the totality constituting a complex pattern of discovery, invention, forgetting, and re-discovery all according to the particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is ~~likely to ~~be ~~illuminated by seeking to position that within~~ the ~~evolution~~ of ~~mathematical thinking. But that is no simple picture and its history will be embroidered and configured by the the social, political~~ and ~~economic circumstances in which that thinking has~~ emerged.

!!!Relationship to this collection

In keeping with the analysis provided elsewhere (in ~~a book by Joseph Camilleri and myself), human development, but with a firm focus on Europe for these periods, will roughly be divided into a set semi-distinct (but overlapping) epochs in which the "Modern Period" is set as beginning (somewhat earlier than is conventional) in the //middle~~ of ~~the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth~~ century~~, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//.[^Joseph Camilleri and Jim Falk, [[http://worlds-~~in~~-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp~~. ~~132-45.~~/~~^]~~

In~~relation to [[Site.ObjectsInTheCollection|the collection of objects]],~~ for ~~which this discussion forms a context, the content breaks effectively into two major parts. The first part, which looks at the relationship between the evolution of calculating and calculating in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects ~~in ~~the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century).~~

!!!Calculating technologies, ~~"calculator" ~~and ~~"calculating machine"~~

Finally, a note on ~~the terminology used here. "Calculator" could be taken ~~to ~~mean~~ a ~~variety of things~~. ~~ For some, it may conjure up an 'app~~.~~' on an iphone for doing a range of calculations. For others it may evoke the small digital calculating devices (such~~ as ~~the Hewlett Packard HP-35) which became pervasive in the last three decades of the twentieth century. For others~~ it ~~may include ~~the ~~motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to ~~the ~~abstract manipulation ~~of ~~'numbers'. //In this discussion~~, ~~I will take calculator as shorthand for "calculating technology" and in particular to mean any physically embodied methodology, however primitive, used to assist the performance of arithmetic operations (including counting)~~.~~//~~ ~~Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if~~ in ~~three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a primitive calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand~~, ~~clay or papyrus) to achieve a similar result.~~

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)~~" is at pains to argue of the abacus (as well as slide rules~~, ~~and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines ~~(~~Die Rechenmaschinen~~)~~//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or ~~"~~calculating devices~~"~~), rather than "calculating machines". This decision to apparently stretch ~~the ~~concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that~~ in the ~~end, technique and technology, or science and technology, are not completely distinct categories.~~ ~~Technologies embody knowledge, the development of technologies can press~~ forward the ~~boundaries of knowledge~~, and ~~technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr~~, ~~"The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later,~~ the ~~mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically~~, ~~and philosophically~~.

!!!Relationship to this collection

In keeping with the analysis provided elsewhere (

In

!!!Calculating technologies

Finally, a note

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen

to:

We can only know where development occurred from where there is any evidence remaining. Even this reveals a patchwork of developments in different directions. No doubt this is but a shadow of the totality constituting a complex pattern of discovery, invention, forgetting, and re-discovery all according to the particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is will be seen to be illuminated by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. And at times, mathematical developments have shaped developments in calculators, and times, vice-versa.

!!!What is a calculator?

The above raises the question of what is to be meant by a "calculator". "Calculator" could be taken to mean a variety of things. For some, it may conjure up an 'app.' on an iphone for doing a range of calculations. For others it may evoke the small digital calculating devices (such as the Hewlett Packard HP-35) which became pervasive in the last three decades of the twentieth century. For others it may include the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however primitive, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a primitive calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"), rather than "calculating machines".

Thr decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

!!!A discussion in two parts.

In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

In relation to [[Site.ObjectsInTheCollection|the collection of objects]], for which this discussion forms a context, the content similarly breaks effectively into two major parts. The first part, which looks at the relationship between the evolution of calculating and calculating in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects in the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century).

!!!What is a calculator?

The above raises the question of what is to be meant by a "calculator". "Calculator" could be taken to mean a variety of things. For some, it may conjure up an 'app.' on an iphone for doing a range of calculations. For others it may evoke the small digital calculating devices (such as the Hewlett Packard HP-35) which became pervasive in the last three decades of the twentieth century. For others it may include the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'.

//In this discussion, "calculator" is used as shorthand for "calculating technology". In particular it is taken to mean any physically embodied methodology, however primitive, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a primitive calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"), rather than "calculating machines".

Thr decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

!!!A discussion in two parts.

In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself), human development, will roughly be divided into a set semi-distinct (but overlapping) epochs, preceded by a "pre-Modern Era" spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the "Modern Period". This beginning is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^] (but there is not much need to focus on that here).

In relation to [[Site.ObjectsInTheCollection|the collection of objects]], for which this discussion forms a context, the content similarly breaks effectively into two major parts. The first part, which looks at the relationship between the evolution of calculating and calculating in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects in the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century).

05 February 2012
by -

Changed lines 3-9 from:

This site focuses on the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s. Following that came the personal computer which in various converging guises (including even phones) diffused unparalleled computing power across the planet

Since this is about developments in calculating technologies prior to the advent of personal electronic calculators, whilst mathematics forms part of the context for their development,

So the focus is on a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis

to:

So what is this rather long and overly footnoted essay doing in a website about a calculator collection?

A collection is like a puzzle. It raises many questions without a single word. This is the opposite. Here are posed quite a few words even before the questions are fully formed.

Consider this. You are in a new land with which you are not yet very familiar. You stumble across a strange dark place you have never visited. There is a door. You open it and enter.

Inside it is so dark you cannot make out more than a path. As you walk down it on your left a light snaps on illuminating a glass case. In it is a pile of sand. You ponder why it is there, but cannot make it out. As you walk on the light extinguishes and a new light illuminates a new case. In it is a small group of pebbles. In the next, a knotted rope. In the next a clay envelope with symbols on it. In the next a set of beads on a string. Then various machines emerge...

It is all meaningless until, retracing your steps you find a lever with a large knob. You pull the lever towards you. Now next to each case you see an illuminated picture. Each picture shows a human figure doing something with the sand, pebbles, rope. The figures are dressed in period clothing. A possibility occurs to you. You return to the lever and turn the knob. With each case a date flashes on along. You see branching passage ways with different figures, clothing, and ethnic features. You are getting much more information and yet it is not clear what it really all means. Is there a card (or cards) somewhere to provide more context? This is one such card.

!!!Initial observations.

One thing is obvious enough. This collection relates to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

We know that following that came the personal computer which in various converging guises (including even phones) diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply do a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

It is also clear that this collection is in some sense a history. It is laid out along a time-line. There is an obvious progression in complexity, sophistication and style from the earliest to the later devices. It is possible to construct histories of technical devices such as calculators as some sort of evolution based on solving technical problems with consequent improvements in design. But this strips away much that may be important in why they were invented and used. The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and some others will be dealt with elsewhere in this site. But clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist "perhaps"!

So how much of this history of mathematical reasoning is necessary? Thankfully much of the huge corpus of extraordinary development in mathematics need not be considered. For example, important though they are we do not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus need only be on a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

!!!Did mathematics lead to the need for calculators?

A collection is like a puzzle. It raises many questions without a single word. This is the opposite. Here are posed quite a few words even before the questions are fully formed.

Consider this. You are in a new land with which you are not yet very familiar. You stumble across a strange dark place you have never visited. There is a door. You open it and enter.

Inside it is so dark you cannot make out more than a path. As you walk down it on your left a light snaps on illuminating a glass case. In it is a pile of sand. You ponder why it is there, but cannot make it out. As you walk on the light extinguishes and a new light illuminates a new case. In it is a small group of pebbles. In the next, a knotted rope. In the next a clay envelope with symbols on it. In the next a set of beads on a string. Then various machines emerge...

It is all meaningless until, retracing your steps you find a lever with a large knob. You pull the lever towards you. Now next to each case you see an illuminated picture. Each picture shows a human figure doing something with the sand, pebbles, rope. The figures are dressed in period clothing. A possibility occurs to you. You return to the lever and turn the knob. With each case a date flashes on along. You see branching passage ways with different figures, clothing, and ethnic features. You are getting much more information and yet it is not clear what it really all means. Is there a card (or cards) somewhere to provide more context? This is one such card.

!!!Initial observations.

One thing is obvious enough. This collection relates to the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s.

We know that following that came the personal computer which in various converging guises (including even phones) diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply do a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

It is also clear that this collection is in some sense a history. It is laid out along a time-line. There is an obvious progression in complexity, sophistication and style from the earliest to the later devices. It is possible to construct histories of technical devices such as calculators as some sort of evolution based on solving technical problems with consequent improvements in design. But this strips away much that may be important in why they were invented and used. The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and some others will be dealt with elsewhere in this site. But clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning, which in various ways they have been made to assist. In deference to later considerations we might add a qualifier - which they have been made to assist "perhaps"!

So how much of this history of mathematical reasoning is necessary? Thankfully much of the huge corpus of extraordinary development in mathematics need not be considered. For example, important though they are we do not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

The focus need only be on a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

!!!Did mathematics lead to the need for calculators?

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Since this is about developments in calculating technologies prior to the advent of personal electronic calculators, whilst mathematics forms part of the context for their development, much of the huge corpus of extraordinary development in mathematics need not be considered here. For example, important though they are we do not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^~~see~~ for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

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Since this is about developments in calculating technologies prior to the advent of personal electronic calculators, whilst mathematics forms part of the context for their development, much of the huge corpus of extraordinary development in mathematics need not be considered here. For example, important though they are we do not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^See for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

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So the focus is on a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^~~see~~ for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^~~see~~ for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

to:

So the focus is on a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^See for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^See for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

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In keeping with the analysis provided elsewhere (in a book by Joseph Camilleri and myself), human development, but with a firm focus on Europe for these periods, will roughly be divided into a set semi-distinct (but overlapping) epochs in which the "Modern Period" is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//.[^~~Falk and ~~Camilleri~~, ~~[[http://worlds-in-transition.com|//Worlds in Transition~~//]], pp. 132-45~~/^]

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In keeping with the analysis provided elsewhere (in a book by Joseph Camilleri and myself), human development, but with a firm focus on Europe for these periods, will roughly be divided into a set semi-distinct (but overlapping) epochs in which the "Modern Period" is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//.[^Joseph Camilleri and Jim Falk, [[http://worlds-in-transition.com|//Worlds in Transition: Evolving Governance Across a Stressed Planet//, Edward Elgar, UK, 2009]], pp. 132-45./^]

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!! Introduction

It is possible to construct histories of technical devices such as calculators as some sort of evolution based on solving technical problems with consequent improvements in design. But this strips away much that may be important in why they were invented and used. The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and some others will be dealt with elsewhere in this site. But clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning.

This site focuses on the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s. Following that came the personal computer which in various converging guises (including even phones) diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply do a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

Since this is about developments in calculating technologies prior to the advent of personal electronic calculators, whilst mathematics forms part of the context for their development, much of the huge corpus of extraordinary development in mathematics need not be considered here. For example, important though they are we do not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^see for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

So the focus is on a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^see for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^see for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

For a start, it might be tempting to see the developments as being created through some process which is entirely internal to mathematical thinking. For example, development might be seen to occur because people can ask questions which arise within what is known in mathematics, but need to develop new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all. The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to is a social choice. Beyond that there is a shifting story about who might legitimately be taught what is known about mathematics and who should address themselves to such questions. Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation, or as stated in a holy book). But at other times and place much greater value has been placed on inventing new knowledge. But even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. So, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

We can only know where development occurred from where there is any evidence remaining. Even this reveals a patchwork of developments in different directions. No doubt this is but a shadow of the totality constituting a complex pattern of discovery, invention, forgetting, and re-discovery all according to the particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is likely to be illuminated by seeking to position that within the evolution of mathematical thinking. But that is no simple picture and its history will be embroidered and configured by the the social, political and economic circumstances in which that thinking has emerged.

!!!Relationship to this collection

In keeping with the analysis provided elsewhere (in a book by Joseph Camilleri and myself), human development, but with a firm focus on Europe for these periods, will roughly be divided into a set semi-distinct (but overlapping) epochs in which the "Modern Period" is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//.[^Falk and Camilleri, [[http://worlds-in-transition.com|//Worlds in Transition//]], pp. 132-45/^]

In relation to [[Site.ObjectsInTheCollection|the collection of objects]], for which this discussion forms a context, the content breaks effectively into two major parts. The first part, which looks at the relationship between the evolution of calculating and calculating in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects in the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century).

!!!Calculating technologies, "calculator" and "calculating machine"

Finally, a note on the terminology used here. "Calculator" could be taken to mean a variety of things. For some, it may conjure up an 'app.' on an iphone for doing a range of calculations. For others it may evoke the small digital calculating devices (such as the Hewlett Packard HP-35) which became pervasive in the last three decades of the twentieth century. For others it may include the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'. //In this discussion, I will take calculator as shorthand for "calculating technology" and in particular to mean any physically embodied methodology, however primitive, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a primitive calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"), rather than "calculating machines". This decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

[^#^]

It is possible to construct histories of technical devices such as calculators as some sort of evolution based on solving technical problems with consequent improvements in design. But this strips away much that may be important in why they were invented and used. The invention and design of technologies depends in major part on what they were to be used for. There are a number of aspects to this and some others will be dealt with elsewhere in this site. But clearly one important factor shaping the need for, role, and design of calculators has been the parallel developments in mathematical reasoning.

This site focuses on the developments in calculation technology which ended with the abrupt transition to personal electronic mathematical calculators in the early 1970s. Following that came the personal computer which in various converging guises (including even phones) diffused unparalleled computing power across the planet. But even the most sophisticated modern computers at heart (though not on their ever more functional surface) simply do a few things extremely fast (logic operations such as "if", "and" and "not" and arithmetic operations of addition, subtraction, multiplication and division ). Of course on top of this are layers of sophisticated programming, memory and input and output. Prior calculating technologies, whilst much more limited in speed, flexibility and adaptability, were nevertheless similarly restricted to the same simple arithmetic operations. For this reason, whilst mathematics encompasses much more than arithmetic it is not necessary to consider all the historical development of its more elaborate analytic structures.

Since this is about developments in calculating technologies prior to the advent of personal electronic calculators, whilst mathematics forms part of the context for their development, much of the huge corpus of extraordinary development in mathematics need not be considered here. For example, important though they are we do not need to talk about the development of set and group theory, nor of the development by Hilbert of the mathematics of infinite dimensional vector space that made the modern formulation of quantum mechanics possible. Nor need we do we need to study the creation of the tensor theory which enabled Einstein in general relativity to write his wonderfully neat field equations for the shape of space time of this universe.[^see for example [[http://mathworld.wolfram.com/EinsteinFieldEquations.html]] or for more explanation [[http://physics.gmu.edu/~joe/PHYS428/Topic9.pdf]] (both viewed 26 Dec 2011)^] It is sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations of addition and subtraction (multiplication and division) which can only be carried out in workable times with the use of ever faster calculating devices.

So the focus is on a tiny simple bit of mathematics - and then primarily on the numerical calculation required to carry out practical applications of mathematical analysis. Yet oddly, it seems in doing so we come across many of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. The history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of philosophy and history of science[^see for example, Eleanor Robson, Jacqueline A. Stedall, //The Oxford handbook of the history of mathematics//, Oxford University Press, UK, 2009^] through to the sociology of science.[^see for example, Sal Restivo, //Mathematics in Society and History: Sociological Inquiries//, Kluwer Academic Publishers, Netherlands, 1992^] Even though this discussion here focuses on only a tiny "arithmetic core" to mathematics it will be important to at least take some account of this literature and its insights.

For a start, it might be tempting to see the developments as being created through some process which is entirely internal to mathematical thinking. For example, development might be seen to occur because people can ask questions which arise within what is known in mathematics, but need to develop new mathematics to answer them. This is certainly part of the story. Yet the literature on history of mathematics tells us this cannot be all. The idea of 'mathematics', and doing it, are themselves inventions. The decision about the sort of problems mathematical thinking might be applied to is a social choice. Beyond that there is a shifting story about who might legitimately be taught what is known about mathematics and who should address themselves to such questions. Similarly at different times and in different cultures have been very differing ideas about the value of invention. At some moments the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation, or as stated in a holy book). But at other times and place much greater value has been placed on inventing new knowledge. But even when invention is in good standing there can be a big question of who is to be permitted to do that. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. In short, a lot of factors can shape what is seen as "mathematics", what it is to be used for, and by whom.

As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: "The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?"[^E.G.R. Taylor, //The Mathematical Practitioners of Tudor & Stuart England 1485-1714//, Cambridge University Press for the Institute of Navigation, 1970, p. 4.^] Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be 'smutted with the Black Art'.[^John Aubrey quoted in Taylor, ibid, p. 8.^] However, despite these negative connotations, practioners of "the dark arts" played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of "star signs" as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in "mathematics" which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. So, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

We can only know where development occurred from where there is any evidence remaining. Even this reveals a patchwork of developments in different directions. No doubt this is but a shadow of the totality constituting a complex pattern of discovery, invention, forgetting, and re-discovery all according to the particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is likely to be illuminated by seeking to position that within the evolution of mathematical thinking. But that is no simple picture and its history will be embroidered and configured by the the social, political and economic circumstances in which that thinking has emerged.

!!!Relationship to this collection

In keeping with the analysis provided elsewhere (in a book by Joseph Camilleri and myself), human development, but with a firm focus on Europe for these periods, will roughly be divided into a set semi-distinct (but overlapping) epochs in which the "Modern Period" is set as beginning (somewhat earlier than is conventional) in the //middle of the sixteenth century//, with the "Early Modern Period" continuing from the //mid-sixteenth to late eighteenth century//, and the "Late Modern Period" stretching forward into the twentieth century, and //terminating around the two world wars//. From thereon, the world is regarded by Joseph Camilleri and myself as entering a //period of transition//.[^Falk and Camilleri, [[http://worlds-in-transition.com|//Worlds in Transition//]], pp. 132-45/^]

In relation to [[Site.ObjectsInTheCollection|the collection of objects]], for which this discussion forms a context, the content breaks effectively into two major parts. The first part, which looks at the relationship between the evolution of calculating and calculating in the pre-Modern period, forms a backdrop, which important as it may be, does not refer at all to specific objects in the collection. As for the collection, its objects are drawn in their entirety from what, in the above sense, can be considered the Modern Period (the earliest of these objects being from the early seventeenth century).

!!!Calculating technologies, "calculator" and "calculating machine"

Finally, a note on the terminology used here. "Calculator" could be taken to mean a variety of things. For some, it may conjure up an 'app.' on an iphone for doing a range of calculations. For others it may evoke the small digital calculating devices (such as the Hewlett Packard HP-35) which became pervasive in the last three decades of the twentieth century. For others it may include the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of 'numbers'. //In this discussion, I will take calculator as shorthand for "calculating technology" and in particular to mean any physically embodied methodology, however primitive, used to assist the performance of arithmetic operations (including counting).// Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a primitive calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book "The Calculating Machines (Die Rechenmaschinen)" is at pains to argue of the abacus (as well as slide rules, and similar devices), that "it is erroneous to term this instrument a machine because it lacks the characteristics of a machine".[^Ernest Martin, //The Calculating Machines (Die Rechenmaschinen)//, 1925, Translated and reprinted by Peggy Aldrich Kidwell and Michael R. Williams for the Charles Babbage Institute, Reprint Series for the History of Computing, Vol 16, MIT Press, Cambridge, Mass, 1992, p. 1.^] In deference to this what is referred to here is "calculators" (and sometimes "calculating technologies or "calculating devices"), rather than "calculating machines". This decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, "If we can make out boundaries at all between what we call science and technology, they are usually arbitrary."[^Otto Mayr, "The science-technology relationship", in Barry Barnes and David Edge (eds), Science in Context, The MIT Press, Cambridge USA, 1982, p.157.^] Indeed, as will be described later, the mental image that mathematics is the work of mathematicians ('thinkers') whilst calculators are the work of artisans ('practical working people') is an attempt at a distinction that falls over historically, sociologically, and philosophically.

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