The desire to cast history as a succession of heroes is deep in many cultures, and many things (including national pride) can hang on who the heroes are finally taken to be. Yet the idea that there is a ‘father’ of the modern calculator, or that one particular person should be allocated the role of its inventor, has many problems associated with it. Rather than a train line from the first to the last, the process of innovation can proceed on many tracks in a direction which has no single end. These problems can be found in the frequent somewhat crude debate over whether the claim to ‘inventor of the modern mechanical calculator’ or some similar title, should be conferred on Schickard on the one hand, or Pascal on the other.1

The debate is often not very edifying as, especially in various locations in the Wikipedia, the whole history of calculational technology has, at times, been “cherry picked” to try to advance the case for one side of this debate. The background to this debate (including sources not explicitly provided here) is recounted elsewhere in this site (Part 2: the Modern Era). What is not in doubt is that whatever role one wishes to ascribe to it, Schickard’s invention (reported in 1623) preceded Pascal’s by 19 years (Schickard having died when Pascal was 13 years old). Beyond that the debate must hinge not around timing but the “success” or “significance” of each innovation.

The judgement on this of course becomes rather more subjective (and to an extent, aesthetic). Collection Calculant - which is associated with this site - is unusual in that it contains both a replica of Schickard’s ‘Calculating Clock” and of a Pascaline. It has been possible to experiment with these to assist in framing this judgement.

First, what is required for an invention? It could be argued that someone is not the inventor of a machine until we have historical proof it has been demonstrated in practice, better still the evidence supported with still surviving examples. This is not the position taken in the legal process of recognition of invention in intellectual property - that is when a patent is granted. Here, what is required is that the invention must be characterised (in writing) in a way that shows the basis of it, and what is new, useful, and non-obvious in its conception (‘the inventive step’). No working embodiment of the inventive idea is required to be produced or, although it must in principle be possible to be constructed, it need not even exist.

The above applies to the criteria for a patentable invention. But if we interrogate the word ‘invention’ we find that it has an even broader usage. The word comes from the Middle English word for ‘finding out, discovery’ (or in French, invenire “to discover”).2 (As opposed to “invention” it is the process of “innovation” where inventions are taken up and adopted in practice.) Consistent with this, for example, the Macmillan Dictionary stresses that an invention is “a machine, tool, or system that someone has made, designed, or thought of for the first time” (my emphasis).3 So from this point of view, both Pascal and Schickard invented (thought of and discovered) and elucidated a new mechanism for achieving a useful purpose. From this point of view both Pascal and Schickard’s work qualifies as an ‘invention’. To go beyond that requires us to explore the other questions which rest upon the significance of the two inventions, which includes having regard to the purpose for which they were invented.


The success of a machine must be partly judged in terms of the objectives for which it was invented. A fork may be a good invention for eating peas, but not so successful for drinking soup. It is important that inventions are assessed, at least in part, in the context of the aims of their inventors. So for a start it is important to note that Schickard and Pascal were not bent on the same tasks.

Schickard’s aims.

Schickard was inspired by conversations in a fairly small international network of natural philosophers, theologians, and associated scholars, with sufficient time and resources to follow increasing interest in matters of mathematical, astronomical and philosophical inquiry. In particular, Schickard was in correspondence with Kepler and Napier. Kepler, who had a very high assessment of Schickard’s capability as an intellectual who was both mathematically and mechanically well informed, made clear to Schickard the tedious nature of the massive calculations that needed to be performed in order to understand the motions of the heavenly bodies.

Whilst there were well established methods of adding and subtracting (for example, with abacus, counting board and calculi, or plume et jeton) the processes of multiplication and division of multi-digit numbers was very challenging, even for that compact elite who had been trained to memorise and use multiplication tables). Gallileo and Hood had tackled this problem several decades before with the invention of the sector. However, helpful though this was for rough calculation one could not use it with facility to calculate the product or quotient of large numbers.

Napier first tackled the question of how to facilitate multiplication and division by a process he named Rabdology (Rabdologiae) which provided a means through the use of a box of appropriately numbered “Napier’s bones” to break down the calculation of one number by another into a series of partial products which could then be added up. The bones effectively replaced memorised multiplication tables, whilst the sets of resulting partial products mimicked the process of long multiplication with paper and pen. Napier’s later invention, of logarithmic tables, whilst less general than rabdology, could produce the outcome of a multiplication (provided the numbers to be multiplied were not too long) by looking up the logarithmic equivalents of the two numbers, adding them (or subtracting them in the case of division), and then looking up the number corresponding to the resulting logarithm.

Schickard’s machine was described by him as an “arithmeticum organum” (or arithmetic calculating instrument) in his letter to Kepler. For the purposes of reference the reader may consult extracts of the key paragraphs of Kepler’s letters here.4 However, in his note to artisans constructing his finalised version of the instrument he referred to it in German as a Rechenuhr (calculating clock).

The over-arching purpose of this instrument was designed to implement the insights of Rabdology in a mechanised fashion. That is, Schickard set himself the task of tackling the problem of mechanising long multiplication and long division. To this end he constructed a machine which had two highly insightful inventive leaps. The first was to turn the independent rectangular Napier’s bones, into a machine form which could be utilised to multiply a multi-digit number automatically by any unit from 1 to 9, and read-out the partial products. He achieved this by imprinting the numbers of Napier’s bones onto vertical rotatable cylinders which could be uncovered by windows. Then, in the base, he did something equally clever. He built an adding machine worked by interlocking gears.

The adding machine of Schickard has only come to us through the comments in letters by him to Kepler and some notes to artisans on building the machine. However, none of these notes give the full mechanism of the machine, and from this point of view what cannot be found on the historical record leaves room for interpolation. Such interpolation however, should be treated with caution.

Help is provided at least by Schickard’s note to Kepler, since he describes a functioning adding device, when on 20 September 1623 he wrote to Kepler to tell him :

What you have done in a logistical way (i.e. by calculation) I have just tried to do by mechanics. I have constructed a machine consisting of eleven complete and six incomplete (“mutliated”) sprocket wheels which can calculate. You would burst out laughing if you were present to see how it carries by itself from one column of tens to the next or borrows from them during subtraction.5

Finally, in the base of the machine he provided another set of windows in which digits could be set with knobs to store intermediate results.

Thus the aim of Schickard’s machine, to build a fully fledged “arithmeticum organum” was to be served with the triple innovation of rotatable multiplying cylinders with a setting mechanism, an adding machine for adding up the partial products so displayed, and what in modern terms would be called a ‘memory register’ to hold intermediate results.

An important motivation appears to have been “by mechanics” to assist the process of doing the sorts of four function calculations that Kepler was finding so tedious. Unfortunately, the machine that apparently could be seen to work, was destroyed by a fire in 1624, along with other papers, leaving Schickard both devastated and very short of time. No record of further work on this particular project has been recovered, and in 1635 he and his entire family died as a result of the bubonic plague brought by troops near the end of the 30 year war.

Pascal’s aims

Pascal, at 19 years of age, was deeply involved in his father’s important work as “Commissioner deputed by His Majesty for the Imposition and Collection of Taxes in Upper Normandy” (to which he was appointed in 1640 by Cardinal de Richilieu). His background was was thus, at least superficially, more tinged with commercial acumen and interest than that of Schickard (although he was soon to also demonstrate his extraordinary talents as a brilliant mathematician). However the task at hand, according to Pascal, was to assist in the seemingly endless and tedious task of adding and subtracting (and of course also doing more complex calculations) as vast numbers of taxes on different items from different citizens across 1,800 communities in Normandy had to be assessed, collected, paid, accounted for, and audited, with his father seldom going to sleep before 2.00 AM.6 Blaise focussed on the most repetitious of these tasks: that of adding and subtracting sums of money and built what would now be understood as an adding machine.

There is no evidence that Pascal had any knowledge of Schickard’s invention. Nevertheless it was clear that he was infected with the growing understanding that his invention could now be turned to reducing labour in new ways. His Pascaline turned out to be an adding machine, rather like that in the base of Schickard’s clock, but with two important improvements (and one significant deficiency).

As with the Schickard’s machine, numbers were added by turning disks with a stylus which in turn turned interlinked gears. Pascal however experimented with ways to improve the practical functioning of this.

First, drawing firmly on the history of clock design, Pascal introduced gears which were minaturised versions of the “lantern gears” used in tower clocks and mills. These could withstand very great stresses and still operate smoothly. Second, he sought to achieve a carry mechanism which could operate even if many numbers needed to be carried at once (e.g. for 9999999+1 to give 10000000). He soon realised that a single tooth gear when turned as an input dial was turned from 9 to 0 (through 36 degrees) could turn a small number of interconnected gears in a row, but the force would dissipate due to friction for each successive gear that had to be turned. To overcome this problem he had the smart idea of storing the energy of each dial as it turned from 0 to 9, and then releasing that stored energy whenever it was needed to achieve the carry. To do this, as each gear passed from 0 to 9 a fork-shaped weighted arm (the “sautoir”) on a pivot was slowly rotated upward. As the gear passed from 9 to 0 this weighted arm was then released to drop thrusting an attached lever forward to rotate the gear ahead of it by one unit.

Pascal’s innovations, the conception of an adding machine (almost certainly without knowledge of the work of Schickard), its implementation using the addition of rotations, the refinement to the use of lantern gears, the crowning achievement of the introduction of a sautoir, and the use of complementary arithmetic (somewhat less elegantly) for subtraction (since only clockwise rotation was permitted by his mechanism), certainly testified to his mathematical and mechanical skill. But beyond that was something quite different - a tenacity in engaging with the process, and artisans, to produce more and more improved designs (some 50 in all). The purpose of this was not just scientific interest, or indeed not simply to help his father, but to achieve commercial success. In the search of this Pascal even went so far as to obtain, in 1649, a royal privilege (the then equivalent of a Patent) to the exclusive right to develop and sell his invention.

As notes Jean-François Gauvin, in his fascinating doctoral thesis:

In Pascal’s dedicatory letter to the chancelier Séguier, the young Archimedes—as he was sometimes called by Marin Mersenne—mentioned the difficulty of using the common method of calculations to perform lengthy arithmetical operations such as the determination of taxes. Only after an extensive period of meditation, Pascal wrote, did he determine that a simpler and faster approach could actually be discovered. It is interesting to note that Pascal’s moyens ordinaires to perform arithmetical operations was the plume and jetons, nothing else. It was the only method referred to in Séguier’s dedication, in the Avis nécessaire cited earlier in the chapter, in the privilège and in Gilberte’s Vie de Monsieur Pascal. No mention whatsoever of rabdology and logarithms, as we saw above in relation to the English context, though Pascal père and fils were most likely aware of these two, by then well-known, mathematical practices as regular members of Mersenne’s Parisian circle of mathematicians.7

Gauvin points out there was a reason for this:

First, rabdology and logarithms had a marginal impact on the French milieu of reckonmasters—and by ricochet on merchants, financiers and bankers, despite efforts by individuals such as Henrion, Petit and Buot. Pascal, therefore, did not find his primary insight from mathematical practitioners in developing the arithmetical machine… Second, the pascaline’s rhetoric of effortless calculations was meant to convince and meet the needs of a specially targeted clientèle, one which owned money and, contrary to Pascal’s father, did not necessarily want to learn complex mathematical procedures.

In short, Pascal aimed first to alleviate the immediately most tedious work of addition and subtraction suffered by his father (and himself in support of his father’s work) and more broadly, over time, to make a commercial success of it. In the end, for a number of good reasons (including not only the high cost of the machine, but also the quite severe task of learning to use it efficiently for anything other than the most simple addition), it proved to be more a curiosity joining the other curiosities that were collected by persons of elevated status.

Issues of mechanism.

The mechanism of Schickard’s Calculating Clock

For Schickard’s Calculating Clock two major mechanical deficiencies came to light in the effort to make a working replica from the surviving notes. Both of these relate to the adding machine component of the calculator and were noted by Baron Bruno von Freytag Löringhoff, a mathematician who built the first working replicas.

The first was the absence of a mechanism to hold in place a number wheel as it rotated from a digit to the next successive digit (eg. from 7 to 8). Something was needed to lock it into place. The simple method adopted by v. Freytag Löringhoff was the use of a spring ‘detente’ which clicked into place on the corresponding gear wheel. The fact that this was not included in the notes by Schickard does not mean there was no such mechanism. In his letters Schickard had made clear that he would “describe the computer more precisely some other time, now I don’t have enough time.” Schickard in any case had consigned the final production of the machine to a clock maker, and the principles of the detente catchment were well understood within this guild. If, as is likely, it had been required, it would have been added.

The second deficiency was more serious. When attempts were made to build a working replica of Schickard’s Calculating Clock the information above, combined with his sketches, was taken to mean that the machine consisted of 6 rotating numbered output drums on axles connected to external “input dials” which could be turned by means of a stylus. On all but the left-most of these axles, inside the machine were additional cogs which were turned by one unit when the next right drum turned from “9” to “0”. To achieve this, on each of the left-most five axles a further “mutilated” gear was fitted which had a single spoke that would engage with its next left counterpart once in each revolution to achieve the “carry” by adding one unit of rotation.

This method runs into trouble where a large number of numbers are to “carry” simultaneously - most notably when 1 is added to 999999 - which in Schickard’s 6 dial machine should rotate all dials to give 000000. The problem is that the energy required to drive this across all dials derives from the advance of a single dial from 9 to 0. The more dials that have to be moved by this single advance of a single dial through 36 degrees, the more friction will tend to overpower the force of the input. As a result, for some number of dials, the carry will not be achieved.

If the input force is too high it could damage the mechanism. If it is too low then the machine can jam. Experiments with the replica in this collection show that propagation of a carry beyond three dials causes the machine to jam. In the circumstances that this is encountered the fact that there is too much resistance will be obvious to the operator. Then with a little assistance from the operator the recalcitrant wheels can be turned and the addition or subtraction can be achieved satisfactorily.

There is however more to be said on this point:

  • The Schickard clock is primarily intended as a multiplying aid. To the extent that this is its purpose the adding machine is provided to add partial products. These are rendered as two digit numbers which must be added one at a time so when used for this purpose the carry problem will only be encountered if the number to which the partial product is to be added already has three or more nines where the partial product is to be added.
  • When used as an adding machine the carry problem can almost entirely be avoided if the number to be added is entered from the left rather than is conventional from the right. (i.e. if 134876 is to be added to 787345, 1 is first added to 7, then 3 to 8, then 4 to 7, etc.) The only time that a carry problem will be created is if at some point there happen to be three or more nines to the left of the number being added.
  • On the rare occasions a carry problem remains, the operator, sensing the resistance, can manually intervene to help the recalcitrant wheels turn. However, a simple alternative approach is to set each of the 9s to zero and then complete the addition, and finally subtract the carried 1 which appears to the left of the left-most digit added. (Eg. for 212131 + 787891 = (1) 000022, starting from left to right we get 9999 and when we try to add 9 to 3 hit resistance. Setting the left numbers to 0000 (0000=9999+1), we may then add 9+3=12 and 1+2=2 giving 000122, and finally subtract the 1 from the place it was added to give 000022).
  • The above approach might be summed up for the student Schickard operator with a little jingle:
Always add from left to right
But if 9s on the left make turning tight
Zero each 9, turn with effort now light
Subtract the 1 and the answer is right

Although intervention (either arithmetic, or ‘brute’ manual assistance) will be required most infrequently (on average 0.0009% of the time or less than once in every 5 months for an operator working 40 hour weeks, doing an addition every 30 seconds),8 the less than consistently smooth operation in the carry mechanism and requirement for sporadic operator intervention is a deficiency in the design. However, because unlike with the Pascaline, subtraction can be directly performed by turning the wheels backwards, with a simply learned technique the inadequacies of the Schickard carry mechanism can be overcome. The need to apply such an approach is less than ideal, but like the requirement to use complementary numbers for subtraction on the Pascaline, it is hardly a fatal flaw.

Putting aside these available approaches to the carry problem, the way to get the most out of such a design is to make the bearings of the machine as friction free as possible. (We see this approach in the C20 SEE demonstration adder which can carry across 4 wheels). How Schickard might have approached it, beyond providing operator ‘assistance’ when it occurred is not known. We do know that Schickard reported that he had already had this system carrying numbers from one column of tens to the next on addition, and borrowing back on subtraction. And beyond that, the multiplication and lower ‘memory register’ part of the machine seems very easy to implement, and works well in replicas.

The mechanism of the Pascaline

Clearly Pascal managed to overcome the problem of friction across the carry mechanism by his most ingenious innovation in the development of the weighted sautoir. The effect of this was to store energy for each wheel as it turned from 0 to 9 releasing it to help turn the next wheel when the time came for a carry.

But beyond that, how well did the mechanism of the Pascaline work? One might think that because we have a number of surviving Pascalines this would be easy to answer. But, as Kisterman points out, “Unfortunately, little is known about the condition of these original machines. Indeed, a detailed and systematic inspection or analysis report about the existing PAMs is not available.”9 He goes on to note that Dorr Eugene Felt (inventor of the Comptometer) reported:

He didn’t make one that would calculate accurately, even if you handled it with the greatest care and took hold of the wheels and cogs after taking the top off the machine, trying to help them along I have tried them myself on several of his machines which are preserved, and making due allowance for age they never could have been in any sense accurate mechanical calculators.10

Similarly, Ernst Martin in his seminal book on the evolution of calculating machines in 1925 wrote of the Pascaline (albeit without citing his source): “whereby it is worth mentioning that the carry does not propagate throughout the machine, it runs merely across a few positions.”11

Of course if one was to try out a bicycle or clock after 400 years one would be surprised and mightily pleased if it worked with the same efficiency it might have when it was brand new. So the above merely indicate that despite the fact that surviving machines exist it is still not really possible to judge the effectiveness of the mechanism from them. Therefore, as with Schickard’s machine, additional information can only be extracted by the attempt to use and build modern replicas of the Pascaline.

The replica in this collection was built by a master craftsman (Jan Meyer) with loving care over several years, entirely in brass, and using modern tools, but similar methods to the construction of the original Pascalines.

As for Schickard’s machine, we cannot know precisely how well problems encountered now and solved with modern tools would have been solved 400 years ago with the tools that then existed. So, as Jan Meyer, who constructed the Pascaline in this collection points out, the answer to how well the Pascaline worked may never be finally settled but he reports there is reason to maintain doubts over its absolute reliability.12

First a positive observation. It has been said that the Pascaline would have been sensitive to position. However, the replica Pascaline in this collection works very reliably even when tilted through 25 degrees in any direction.13 The replica machine requires a certain determined gesture with the stylus to ensure that a number is correctly input with the number fully displaying in its correct position in the window. However, provided the appropriate gesture is learned and applied in its close to new state the machine operates close to faultlessly (both for input and thus addition, and carry). Beyond that, there are some potential issues.

Prior to constructing the Pascaline, Jan Meyer famously constructed a replica Hahn Calculator - a rare and complex machine (from 1773) of which two originals are known to survive (together with with two similar machines made later around 1822 by Hahn’s brother in law Johann Schuster). Meyer’s Hahn replica worked perfectly from the day it was completed. But despite now having 3D-CAD and simulation to assist him, there was a long litany of challenges in getting the replica Pascaline to work reliably. Why was this so?

In the Hahn mechanism, the whole force of the turning crank is brought directly into the mechanism. If some component has a little resistance in the axles and gear wheels it does not matter. A little extra force on the crank will cause everything to rotate as it should. And each gear has a latching function with a spring so each gear is always exactly in the right position - there is no possibility of a gear assuming an undefined position in between.

The Pascaline concept however is very different. The whole force is brought by a small stylus only applied to just one of the input dials of the Pascaline mechanism. That dial is rotated by the amount (in increments of 36 degrees) where that is required to input a number from 1 to 9.14 The addition occurs as a result of this, and if the dial is rotated beyond 9 then a “carry” (of 10) is required to the next dial to the left. When that carry is triggered, what happens to the dials to the left is totally dependent on the stored energy in the sautoir ‘ carry weights’ which fall to rotate these dials as necessary, by one increment. And that is what causes the problems. If the carry weights are too heavy and all bearings run too smoothly then as the carry weights fall down, the next display digit can be increased by two or three or something in between. One might call this “overshoot”. To deal with this the maker will try to reduce the carry weights. But it is not easy to find the correct value. If it is too heavy we get overshoot. If it is too light the tens do not carry and the next dial remains unmoved.

One might make a further observation following from the above. The correct ‘tuning’ of the sautoir will be established against the resistive force created by friction in the bearings of the axles. When all is adjusted at delivery the new mechanism will presumably work as intended. However, it will only take so much wear, corrosion, or the accumulation of grease or grime in the bearings (let alone variations in alignment caused by transport and use) to change the balance between sautoir weight and friction. This opens an explanation for the experience described above that when used, the propogation of the carry mechanism was not reliable.

Ensuring that the numbers display consistently in the centre of each window presents another challenge. Jan Meyer found that this works fine for the upper digits (6- 9) but not with the lower digits (1–5). The reason traces back once more to the carry weights. For the higher digits the weight has been lifted some distance and it places a restoring torque on the wheel lifting it. The wheel is however held in place by the ratchet which is in place for that purpose. At the lower positions there is a tendency for the weight to not properly engage so the wheel can slip back against the rachet. To fix this problem Jan Meyer introduced a small bow in the lever to ensure that the weight is always engaged (except, of course at the highest left-most position where the carry is to occur). This tendency to mis-position the output numbers can be seen in other replica Pascalines.

It has been argued that the fact that Pascal chose to zero his machine by first setting it as 9s across all the dials, and then to add one so that a carry propagated across the machine, was a testament to the smooth working of the carry mechanism. But, it is equally possible that he chose to do this as a check that the machine was in adjustment before use precisely because it had a tendency to go out of adjustment.

Indeed, experimenting with precision made replicas indicates that it is difficult to avoid occasional under or overshoot and quite easy for the machine to go out of adjustment. Thus it seems reasonable to conclude that the machine Pascal designed can work well after special adjustment and carefully operation. But it was not a reliable tool for daily calculation in the tax office. And, indeed, despite the energetic promotional efforts of Pascal, there is no surviving evidence that it was ever used effectively for this purpose, or proved to be superior to the methods of jeton and pen, or jeton and counting board, that were in common use at the time and in the hands of a skilled operator, could be very efficient indeed.

Other considerations

In the various discussions that can be found about the two inventions the fact is unchallenged that the Schickard was invented well before Pascal turned his attention to the invention of the Pascaline. There is no suggestion that the documentation that has survived on the Schickard is anything but genuine, nor that the statements therein should not be accepted as the literal truth as understood by the correspondents. Schickard had a long standing friendship with Kepler, each had expressed admiration of the other, and this was just one of a number of topics on which they corresponded.

Given this, any claim of precedence for the Pascaline as an invention tends to be that the Pascaline should be accepted as the first successful modern innovation in the long line of evolution of the modern calculating machine. How fair is this as a characterisation?

(i) Was the Pascaline, as opposed, to the Schickard machine, the fitter candidate as the first modern “calculating machine”? This takes us into the pedantic question of what should be taken to be a “calculating machine”. One person’s idea of a machine may not be another’s. The Oxford Dictionary defines a machine as “An apparatus using mechanical power and having several parts, each with a definite function and together performing a particular task”.15 Both inventions satisfy this definition. However, a calculating machine is something more specialised and when there is further debate Martin’s seminal work on the history of calculators in 1925 is often taken as good as any authority for arbitration.

On this matter, Martin writes:

In the calculating machine trade, particularly in European countries and in the United States, it has become customary in recent years to call only those products that possess automatic tens-carry by the term calculating machine: if this feature is missing, the term calculating aid is more appropriate…
It is necessary to discuss the term calculating machine still further in order to avoid misunderstandings and errors. In the broadest sense, calculating machines are understood to be machines that are equally suited for the performance of multiplication, division, addition, and subtraction. The term also applies to machines that are normally intended for addition and subtraction only and do not permit the operation of multiplication, or if they do, only as a makeshift or less advantageous operation. In a narrower sense, however, we term only the former ones calculating machines. With the latter ones forming a separate class under the term adding machines. Adding machines are primarily intended for adding and usually, although not always, for subtracting. Such machines may permit the operations of multiplication and division, but in this respect they cannot compete with calculating machines proper. Calculating machines are able to carry out all four fundamental operations with great rapidity.16

The above of course is a C20 analysis. Three hundred years prior the only way to carry out these calculations with great rapidity was by a highly skilled rechenmeister/calculator utilising the tools of his trade (whether it be abacus, calculi and counting board, or pen and paper). Neither Schickard’s nor Pascal’s devices fit Martin’s description of a C20 calculating machine. But, this was the beginning. Did one conform so much closer to that description as to wipe out the candidacy of the other for being an early point in the evolution of the C20 calculating machine?

(ii) How well did each meet their objectives? Of the two it is clear that the primary purpose of the Pascaline was as an adding machine, in the terms defined by Martin, above. It had its flaws but it was an extraordinarily good adding machine for such an early step forward, albeit carrying the rather considerable blemish of requiring an understanding of complementary arithmetic in order to perform subtractions. It is possible to perform a multiplication between two multi-digit numbers upon it, but the process is so confusing as to be of no advantage in relation to simply doing this by pen and paper, a methodology which in any case would have to be understood in order to reliably perform such an operation.

Schickard’s machine was actually intended to facilitate all four operations of arithmetic. In that sense it was a “calculating machine” in Martin’s terms. Some work and skill was still needed to multiply two multi-digit numbers, or divide them, on it, but that was what it was designed to do. From the partial information we have, the adding machine in the base had more flaws than the Pascaline, probably the most serious being that a carry would have only been possible across all six dials simultaneously with some judicious help from the operator. But, from modern replicas, we know that with some learned skill this is possible.

To make this clear let us look at multiplication on each machine. Consider the multiplication of 654 x 432. In each case you do this by long multiplication which amounts to 654×400 + 654 x 30 + 654 x 2. On the Schickard you set 654 with the napier rods. Pull out the 2 slider revealing the results of multiplying 654×2 = 1308. (Actually it appears as 12,10,08 which can be entered into the adding machine in three parts shifting one to the left each time to give 1308). Once entered into the adding machine, pull out the 3 slider (654 x 30 = 19620). Shifting one place to your left add the number (1912) shown. Pull out the 4 slider (654 x 400 = 261600). Shifting one place again to the left add the number 2616 shown. The result (282528) will be shown by the adding machine. The process is easily learned. No memorised tables, paper and pen or counters are required (and, as is usually the case, the adding machine does the job smoothly without any special operator intervention).
Compare the same process on the Pascaline. It follows the same way except you have no way of multiplying other than repetitive adding. So, for example, to multiply 654 x 2 we must dial in 654 twice. Then for 654 x 30 you shift one wheel to your left and then dial in 654 three times. And then finally to multiply by 400 we shift one wheel to the left again and dial in 654 four times. This is clearly quite a bit more tedious - probably too tedious to be worthwhile, especially if larger numbers (for example 987 x 987) are to be multiplied.

Having said that, it must also be conceded that using Napier’s rods, even in this way, is not very easy, and certainly no easier than long multiplication and long division with pen and paper, for someone (such as Kepler) adept at addition and multiplication in their head. However, the rods were intended for those who had not acquired that skill. But acquiring a rapid skill with the rods, memory store, and adding machine of the Schickard calculator might well be quite daunting - although perhaps not quite as daunting as learning the multiplication tables let alone tackling the taxing task of trying to do multiplication or division on a Pascaline.

In the above sense each machine met its objectives as a machine (with some flaws) and neither was able to become a success in the sense it was taken up and widely used (a more overt objective for the Pascal than for Schickard). However, all this tells us is that these were early in the innovation process. Full scale success would require a closer convergence of need, design, and social and economic organisation in the society in which the innovation might be taken up.

(iii) Which invention was the more influential? From one point of view so little is known of the Schickard invention other than the response of Kepler, the unfortunate fire, the devastating effect on Schickard, and his and his family’s subsequent death of plague. Despite the fact that Schickard widely corresponded and mixed with influential thinkers of his day (including, for example, Bernegger, Diodati, Peiresc, Gassendi, Bouilliau, Morin, Luillier, Rivet, Golius, Grotius, and Hortensius),17 and news of the invention was published in every subsequent century, we do not have evidence of the extent to which news of it might have influenced subsequent thinking. On the other hand, we even have surviving examples of the multiple products, and publicity penned in support of Pascal’s energetic and entrepreneurial efforts to commercialise his invention.

But in terms of influence, the task of comparison remains one of comparing unlikes, with too many centuries between to distinguish rhetoric from reality. We do not, for example, know who precisely knew or was inspired by Schickard’s invention. Taton has argued that Schickard’s work had no impact on the development of mechanical calculators.18 On the other hand, Adam citing a similar view by Prager offers his judgement in favour of the position “that the idea of the Kepler-Schickart calculating-machine found its way to France and thus into Pascal’s circle, so that he was able to adopt this salient principle. Technically speaking, Pascal’s adding and subtracting machine (1641–45) shows an unmistakable relation to the gear mechanism of Schickart’s construction.”19 In my view, none of these statements are conclusive. We simply do not have the necessary evidence.

Certainly, there were devices after Schickard and Pascal, such as Moreland’s multiplying and adding instruments when used together, Caspar Schott’s Cistula, René Grillet’s machine arithmétique, and Claude Perrault’s rhabdologique at the end of the century, and later, the Bamberger Omega developed in the early 20th Century, amongst others, which followed the same path pioneered by Schickard with his ground breaking combination of a form of Napier’s rods and adding machine designed to assist multiplication.

It is hard to find an example of the very clever development of Pascal’s sautoir being taken up in any subsequent machine. Certainly the problem he sought to address so successfully was now recognised and a range of approaches (from the use of springs to store the necessary energy, to systems which could harness more mechanical energy at input from a crank) were developed. The beautiful brass-work in Pascal’s can definitely be found reflected in subsequent devices, but that is probably as much a reflection of the style of the clock makers at the time, and the wealthy and influential market that they were intending to please.

Attempts to draw a direct line from Pascal’s invention to subsequent devices however are somewhat artificial. Later machines tended to draw on mechanism that promised greater capability or reduced cost, or both. The most important step towards commercialisation of a full-functioned calculating machine came with Leibniz’s invention, and his clever introduction of a stepped drum and moveable carriage (albeit combined with an imperfect carry mechanism). Leibniz certainly knew of Pascal’s invention, although he probably had not seen its mechanism nor understood its limitations (notably in only doing subtraction by complementary arithmetic) when he considered using it as a basis for a four function calculating machine, an idea he abandoned. And just to demonstrate how easy it is to assume that once Leibniz had invented the stepped drum that meant that Thomas de Colmar utilised that invention in his commercially successful arithmometer, Kisterman mounts a compelling case that Thomas de Colmar could not have known of Leibniz’s invention, and certainly not the detail of its mechanism, until 1892, decades after he first adopted the stepped drum to his invention.20

Enough has been said to indicate that influence is hard to prove, and even if not able to be tracked, may still have been historically present. However, beyond that, it must be said that the fact that a working adding machine (in the case of Pascal), and an ingenious calculating machine incorporating an adding machine, had been developed and produced would, where known, have provided inspiration to others to try to improve on that. Beyond that, the effort to determine which was “more influential” in some abstract way, when the evolution of calculators followed so many different weird and wonderful paths, is probably unlikely to provide compelling closure.


The two machines were essentially different in that Pascal’s machine was designed primarily for addition (and with the use of complementary numbers) for subtraction whilst Schickard’s was closer to a full-function calculating machine.The adding machine in Schickard’s design may have jammed in the unusual case of a carry being required across too many dials. On the other hand, Schickard’s invention could smoothly subtract by reversing the motion of the input dials, in a way that was not possible in the Pascaline. (The Schickard adding machine also had provision for an audible warning when an output was too large for the available dials, a feature that was not provided for in the Pascaline.) In this sense both machines have a firm place in the evolution of the modern calculating machine.

It is difficult to make the case that the Pascaline was so “successful” that the Schickard machine should be discounted especially as the adding machine in Schickard’s design appears to have been introduced to assist the broader objective of multiplication (through the calculation of partial products using a clever mechanised embodiment of Napier’s rods, a mechanism that can also be used to assist division). In this sense, an attempt to compare the “success” of the two machines is as subjective as comparing the success of apples versus oranges.

Even if we, somewhat arbitrarily, restrict our attention to the adding machine mechanisms in each device, experiments with building modern replicas of either machine suggest that there were things that in practice could interfere with the smooth performance of both the Pascaline and the adding machine in Schickard’s design. Modern replicas of the Pascaline (when constructed with similar materials and technique to the originals) demonstrate that with some additional tweaks it can work perfectly for addition with careful adjustment and operation. But it does not take much to throw it into less than perfect performance. The Schickard could resist carry-over to too many output wheels at once, and on the occasions this occurred would need to be assisted by the operator to complete the carry. So it depends on what is considered as important as to whether either machine could be seen as a success or failure.

In short, neither invention was a success in the sense that it was taken up and used widely in practice (which appears to have been more the objective for the Pascal than for Schickard). From what is known neither can be shown to be a failure in the sense that it could not be used by a careful operator to calculate in one way or another.

For this reason, there seems insufficient reason to seek to annul the place of Schickard’s Arithmeticum Organum as being an earlier step in the development of the modern calculating machine than the Pascaline. On the other hand, this is a rather empty observation. The linear model of innovation, where innovation proceeds like a train on a track from a fixed starting point to some final goal, would be widely regarded as misleading and obsolete in the relevant scholarly literature. History is littered with multiple strands of innovation catalysed not so much by a single blinding insight by a genius, but rather by forces which are collectively causing clever inventors, skilled artisans, and commercially astute partners to collaborate together in the development and marketing of new technologies.

Even what success might look like is hard to imagine before it is acclaimed. But if it is to be taken as wide-spread adoption of a technology, then there may be many false steps, and quite a bit of social development, before success was won in that sense. From this point of view Pascal and Schickard can be seen as important early innovators in a complex evolutionary process which soon brought in many other players, and had already been preceded by the inventions of important reckoning devices such as the sector. None of this is to diminish the significance of the elegant insights of either Pascal or Schickard. Both were great figures in the struggle to make more accessible and simple the process of calculation in the seventeenth century, and a debate about which was the greater is probably as empty as it is irrelevant to modern concerns.


1 The most coherent statement of the claim I have found is in M. Rene Taton, “Sur l’invention de la machine arithmétique, in Revue d’histoire des sciences et de leurs applications, vol 16, no 2, 1963, pp. 139–160. The claims there, as well as some cruder restatements, have motivated this review of the issues raised. (↑)

2 See for example, the Oxford English Dictionary, http://www.oxforddictionaries.com/definition/english/invention viewed 18 April 2014 (↑)

3 Macmillan Dictionary http://www.macmillandictionary.com/dictionary/british/invention viewed 18 April 2014 (↑)

4 Taken from the elegant article http://history-computer.com/MechanicalCalculators/Pioneers/Schickard.htmll (↑)

5 Translation as quoted in Michael Williams, A History of Computing Technology, 2nd Edition, IEEE Computer Society and The Institute of Electrical and Electronics Engineers, Inc., USA, 1997, pp. 120–1. (↑)

6 from a postscript in a letter from Blaise Pascal to Gilberte Périer, 31 January 1643, in Pascal, OC, ii:282–283 (↑)

7 Jean-François Gauvin , Habits of Knowledge: Artisans, Savants and Mechanical Devices in Seventeenth-Century French Natural Philosophy, The Department of the History of Science, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of History of Science, Harvard University, Cambridge, Mass., USA, November 2008, pp. 210 (↑)

8 The numbers to be added can occur in pairs of ten million possibilities. Of these a string of 3 or more 9s to the left can occur in (xx999y, x999xy, 999xy, x9999y, 9999xy, 99999y) = 81×2+9×2+1 = 181 ways or on average 0.000905% of the time. So, if (unrealistically) a Schickard calculator operator was to use the above method for an addition every 30 seconds, 8 hours a day, 5 days per week, all year without stopping then he/she would need to zero the problem string of 9s on average a little over once every 5 months. (↑)

9 Friedrich W. Kistermann, “Blaise Pascal’s Adding Machine: New Findings and Conclusions”, IEEE Annals of the History of Computing, Vol. 20, No. 1, 1998, p. 70. (↑)

10 D.E. Felt, Mechanical Arithmetic or the History of the Counting Machine. Racine, Pitman Publishing Co., Wisc., 1916. cited in Kisterman, “Blaise Pascal’s Adding Machine”, p. 70. (↑)

11 cited in Kisterman, “Blaise Pascal’s Adding Machine”, p. 70 (↑)

12 Jan Meyer, Private Communication to Jim Falk, 2013 (↑)

13 François Babillot also observes that his Pierre Charrier replica will work inclined at angles of six degrees without difficulty. see http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_contestation/contestable.php?lang=eng (↑)

14 Note that this discussion is for a Pascaline designed for scientific calculation where all dials run from 1 to 9. Some modification is required for the accounting machines (such as the “Queen of Poland” Pascaline) where some dials have more digits. (↑)

15 Oxford Dictionary. http://www.oxforddictionaries.com/definition/english/machine, viewed 7 April 2014. (↑)

16 Ernest Martin, The Calculating Machines (a translated reprint of Die Rechenmaschinen, 1925), The MIT Press Cambridge, Massachusetts London, England, and Tomash Publishers Los Angeles and San Francisco, 199, pp. 1–2. (↑)

17 Encyclopédie méthodique. Mathématiques, Paris, 1784, pp. 136–142. (↑)

18 M. Rene Taton, “Sur l’invention de la machine arithmétique”, Revue Histoire des Sciences, vol. 16, no. 2, p. 142. (↑)

19 Adolf Adam, “The Kepler-Schickart Calculating Machine”, Vistas in Astronomy, vol. 18, pp. 884. (↑)

20 Friedrich W. Kistermann, “When Could Anyone Have Seen Leibniz’s Stepped Wheel”, IEEE Annals of the History of Computing, Vol. 21, No. 2, 1999, p. 70. (↑)

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