## Site.OperatingInstructionsForAPascaline History

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18 May 2017
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Changed line 58 from:

* ~~All~~ of the surviving Pascalines (including the ~~replica~~ in Calculant) have dots engraved on two of the spokes of each input wheel which can be used to input 9s complements. To do that the stylus is placed in the marked segment and rotated to the number for which a complement is sought. The complement then shows correctly in the upper (complement) row of the results window.

to:

* Most of the surviving Pascalines (including the exemplar in Calculant, but not the replica of the "Queen of Poland" Pascaline) have dots engraved on two of the spokes of each input wheel which can be used to input 9s complements. To do that the stylus is placed in the marked segment and rotated to the number for which a complement is sought. The complement then shows correctly in the upper (complement) row of the results window.

26 November 2014
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Changed line 56 from:

* The results 'top window' of the Pascaline has a cover bar which can be slid up or down to reveal either a lower row of numbers or an upper row of their 9s complements. The lower row is often rendered in black, whilst the upper complements (subtracted from 9) are often rendered in red (at ~~lest~~ in replicas). It is therefore possible to dial in a number, lower the cover bar, and read off the complement.

to:

* The results 'top window' of the Pascaline has a cover bar which can be slid up or down to reveal either a lower row of numbers or an upper row of their 9s complements. The lower row is often rendered in black, whilst the upper complements (subtracted from 9) are often rendered in red (at least in replicas). It is therefore possible to dial in a number, lower the cover bar, and read off the complement.

18 July 2013
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Changed line 137 from:

|see also [[Site.PascalineMechanism|Pascaline mechanism]]>>

to:

| see also [[Site.PascalineMechanism|Pascaline mechanism]]>>

10 July 2013
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Changed line 57 from:

* All of the surviving Pascalines (including the replica in Calculant) have dots engraved on two of the spokes of each input wheel which can be used to input 9s complements. To do that the stylus is placed in the marked segment and rotated to the number for which a complement is sought. The complement then shows correctly in the upper complement row of the results window.

to:

* All of the surviving Pascalines (including the replica in Calculant) have dots engraved on two of the spokes of each input wheel which can be used to input 9s complements. To do that the stylus is placed in the marked segment and rotated to the number for which a complement is sought. The complement then shows correctly in the upper (complement) row of the results window.

22 June 2013
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Changed line 3 from:

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate. Blame for the explanations of the different possible modes of subtraction explained below is however accepted by the site author.

to:

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate. Blame for the explanations of the different possible modes of subtraction explained below is however accepted by the site author and have been verified on the [[Site.Pascaline1652|Jan Meyer replica in this collection]].

16 June 2013
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Changed lines 88-89 from:

2) Set the subtrahend **s** as its 9s complement~~ (using one of the methods above)~~.

to:

2) Set the subtrahend **s** as its 9s complement.

Changed line 109 from:

2) Set the minuend **m** (say 21) as its 9s complement~~ (using one of the methods above)~~.

to:

2) Set the minuend **m** (say 21) as its 9s complement.

16 June 2013
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Changed line 129 from:

->In order to display 890 £ 9 s 7 d in the windows, I don't put the stylus in the same digits I want to display as we do for the addition, but I take the complement to 9. That is to say~~ ~~: I put the stylus at the digit whose~~ the~~ sum is 9 when we add it to the one we want. In the example, I put the stylus at 1 and I turn the wheel until the needle stops me. At this time I see in the display window the digit 8 that I want. I then go to the next wheel to introduce the second digit who is 9, since 9 is already displayed, I go the the next wheel.[^Quoted by François Babillot from "Usage de la machine"[[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_addition_soustraction.php?lang=eng]] viewed 15 June 2013^]

to:

->In order to display 890 £ 9 s 7 d in the windows, I don't put the stylus in the same digits I want to display as we do for the addition, but I take the complement to 9. That is to say: I put the stylus at the digit whose sum is 9 when we add it to the one we want. In the example, I put the stylus at 1 and I turn the wheel until the needle stops me. At this time I see in the display window the digit 8 that I want. I then go to the next wheel to introduce the second digit who is 9, since 9 is already displayed, I go the the next wheel.[^Quoted by François Babillot from "Usage de la machine"[[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_addition_soustraction.php?lang=eng]] viewed 15 June 2013^]

16 June 2013
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Changed line 129 from:

->In order to display 890 £ 9 s 7 d in the windows, I don't put the stylus in the same digits I want to display as we do for the addition, but I take the complement to 9. That is to say : I put the stylus at the digit whose the sum is 9 when we add it to the one we want. In the example, I put the stylus at 1 and I turn the wheel until the needle stops me. At this time I see in the display window the digit 8 that I want. I then go to the next wheel to introduce the second digit who is 9, since 9 is already displayed, I go the the next wheel.[^Quoted ~~from "Usage de la machine" by François Babillot~~ [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_addition_soustraction.php?lang=eng]] viewed 15 June 2013^]

to:

->In order to display 890 £ 9 s 7 d in the windows, I don't put the stylus in the same digits I want to display as we do for the addition, but I take the complement to 9. That is to say : I put the stylus at the digit whose the sum is 9 when we add it to the one we want. In the example, I put the stylus at 1 and I turn the wheel until the needle stops me. At this time I see in the display window the digit 8 that I want. I then go to the next wheel to introduce the second digit who is 9, since 9 is already displayed, I go the the next wheel.[^Quoted by François Babillot from "Usage de la machine"[[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_addition_soustraction.php?lang=eng]] viewed 15 June 2013^]

16 June 2013
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Changed line 3 from:

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate. Blame for the explanations of the different possible modes of subtraction explained below ~~are~~ however accepted by the site author.

to:

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate. Blame for the explanations of the different possible modes of subtraction explained below is however accepted by the site author.

16 June 2013
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Changed line 3 from:

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate. Blame for the explanations of the different possible modes of subtraction are however accepted by the site author.

to:

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate. Blame for the explanations of the different possible modes of subtraction explained below are however accepted by the site author.

16 June 2013
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Changed line 8 from:

The Pascaline adds by summing the rotations of the input dials through 1 to 9 units. When a dial moves from 9 to 0, 1 is automatically carried to the next dial ~~rotating that~~ dial by one unit. The process of addition is thus simple:

to:

The Pascaline adds by summing the rotations of the input dials through 1 to 9 units. When a dial moves from 9 to 0, 1 is automatically carried to the next higher dial (eg from units to tens) rotating that higher dial by one unit. The process of addition is thus simple:

16 June 2013
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16 June 2013
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Changed lines 3-5 from:

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful.~~ ~~ This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate. Blame for the explanations of the different possible modes of subtraction are however accepted by the site author.

to:

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate. Blame for the explanations of the different possible modes of subtraction are however accepted by the site author.

Changed lines 55-56 from:

* The results 'top window' of the Pascaline has a cover bar which can be slid up or down to reveal either a lower row of numbers or an upper row of their 9s complements. The lower row is often rendered in black, whilst the upper complements (subtracted from 9) are often rendered in red. It is therefore possible to dial in a number, lower the cover bar, and read off the complement.

to:

* The results 'top window' of the Pascaline has a cover bar which can be slid up or down to reveal either a lower row of numbers or an upper row of their 9s complements. The lower row is often rendered in black, whilst the upper complements (subtracted from 9) are often rendered in red (at lest in replicas). It is therefore possible to dial in a number, lower the cover bar, and read off the complement.

Changed line 80 from:

Since the difference, d, is negative we need to compute -d = s - ~~m~~

to:

Since the difference, d, is negative we need to compute -d = s - m\\

Changed lines 100-102 from:

For 9s complementation of a six digit number A=999999.

if s = 000021

then s* = 999999 - 000021 =~~999978~~

if s = 000021

then s* = 999999 - 000021 =

to:

For 9s complementation of a six digit number A=999999.\\

if s = 000021\\

then s* = 999999 - 000021 = 999978\\

if s = 000021\\

then s* = 999999 - 000021 = 999978\\

15 June 2013
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Changed lines 125-127 from:

Kistermann believes that this approach is more consistent with actual historical practice. However, Kistermann does not mention the marked spokes of the input wheels on ~~at least several of the surviving Pascalines which could be used directly to input 9s complements. This suggests that arithmetic may well have been performed using~~ the ~~more obvious 9s complements methods along ~~the ~~lines discussed earlier here.~~

In 1982, a manuscript from the 18th century, entitled "Usage de la machine" (usage of the machine),[^Courrier du centre international Blaise Pascal, (Clermont-Ferrand), "Usage de la machine", Vol 8, 1986.^] was bought by the CIBP (Blaise Pascal International Center) in Clermont Ferrand. In this paper it is clear that the 9s complementation method is to be used:

In

to:

Kistermann believes that this approach is more consistent with actual historical practice. However, Kistermann does not mention the marked spokes of the input wheels on the surviving Pascalines which could be used directly to input 9s complements. This suggests that arithmetic may well have been performed using the 9s complements methods along the lines discussed earlier here.

Consistent with the above, in 1982, a manuscript from the 18th century, entitled "Usage de la machine" (usage of the machine),[^Courrier du centre international Blaise Pascal, (Clermont-Ferrand), "Usage de la machine", Vol 8, 1986.^] was bought by the CIBP (Blaise Pascal International Center) in Clermont Ferrand. In this paper it is clear that the 9s complementation method is to be used:

Consistent with the above, in 1982, a manuscript from the 18th century, entitled "Usage de la machine" (usage of the machine),[^Courrier du centre international Blaise Pascal, (Clermont-Ferrand), "Usage de la machine", Vol 8, 1986.^] was bought by the CIBP (Blaise Pascal International Center) in Clermont Ferrand. In this paper it is clear that the 9s complementation method is to be used:

15 June 2013
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Changed lines 57-59 from:

* ~~Some Pascalines (including ~~the ~~replica in Calculant) have dots engraved on two of the spokes~~ of ~~each input wheel which can be used to input 9s complements. To do that the stylus is placed in the marked segment and rotated to the number for which a complement is sought. The ~~complement ~~then shows correctly in the upper complement row of ~~the ~~results window.~~

* Some Pascalines had an inner disk set into each input wheel showing the 9s complements to assist their input.

* Some Pascalines had an inner disk set into each input wheel showing the 9s complements to assist their

to:

* All of the surviving Pascalines (including the replica in Calculant) have dots engraved on two of the spokes of each input wheel which can be used to input 9s complements. To do that the stylus is placed in the marked segment and rotated to the number for which a complement is sought. The complement then shows correctly in the upper complement row of the results window.

* Some Pascalines had an inner disk set into each input wheel showing digits which may have assisted the input of the 9s complements.

* Some Pascalines had an inner disk set into each input wheel showing digits which may have assisted the input of the 9s complements.

15 June 2013
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Changed line 127 from:

In 1982, a manuscript from the 18th century, entitled "Usage de la machine" (usage of the machine),[^Courrier du centre international Blaise Pascal (~~in fr~~) ~~(Clermont-Ferrand), ~~"Usage de la machine", Vol 8, 1986. was bought by the CIBP (Blaise Pascal International Center) in Clermont Ferrand. In this paper it is clear that the 9s complementation method is to be used:

to:

In 1982, a manuscript from the 18th century, entitled "Usage de la machine" (usage of the machine),[^Courrier du centre international Blaise Pascal, (Clermont-Ferrand), "Usage de la machine", Vol 8, 1986.^] was bought by the CIBP (Blaise Pascal International Center) in Clermont Ferrand. In this paper it is clear that the 9s complementation method is to be used:

15 June 2013
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Changed line 127 from:

In 1982, a manuscript from the 18th century, entitled "Usage de la machine" (usage of the machine), was bought by the CIBP (Blaise Pascal International Center) in Clermont Ferrand. In this paper it is clear that the 9s complementation method is to be used:

to:

In 1982, a manuscript from the 18th century, entitled "Usage de la machine" (usage of the machine),[^Courrier du centre international Blaise Pascal (in fr) (Clermont-Ferrand), "Usage de la machine", Vol 8, 1986. was bought by the CIBP (Blaise Pascal International Center) in Clermont Ferrand. In this paper it is clear that the 9s complementation method is to be used:

15 June 2013
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Added lines 126-130:

In 1982, a manuscript from the 18th century, entitled "Usage de la machine" (usage of the machine), was bought by the CIBP (Blaise Pascal International Center) in Clermont Ferrand. In this paper it is clear that the 9s complementation method is to be used:

->In order to display 890 £ 9 s 7 d in the windows, I don't put the stylus in the same digits I want to display as we do for the addition, but I take the complement to 9. That is to say : I put the stylus at the digit whose the sum is 9 when we add it to the one we want. In the example, I put the stylus at 1 and I turn the wheel until the needle stops me. At this time I see in the display window the digit 8 that I want. I then go to the next wheel to introduce the second digit who is 9, since 9 is already displayed, I go the the next wheel.[^Quoted from "Usage de la machine" by François Babillot [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_addition_soustraction.php?lang=eng]] viewed 15 June 2013^]

15 June 2013
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Changed line 123 from:

**NB. 1** An alternative procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) by Kistermann[^From 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. 63.^] Kistermann prefers this approach because if the 10s complements are formed across the entire set of places of the machine, and if the highest place is reserved, it may act as an indicator of whether the result of a subtraction is positive or negative. If it is positive a zero appears in the highest place, and if it is negative a 9 appears in the highest place, in which case the cover is lowered to read off the result from the upper (compliment) window. ~~Analogous to reasons already discussed in Method 2, above, 1 must be added to~~ the ~~units position and this can only be done outside~~ the ~~machine. Once that has occurred, as in the above methods,~~ further calculation without resetting the machine is impractical.

to:

**NB. 1** An alternative procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) by Kistermann.[^From 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. 63.^] Kistermann prefers this approach because if the 10s complements are formed across the entire set of places of the machine, and if the highest place is reserved, it may act as an indicator of whether the result of a subtraction is positive or negative. If it is positive a zero appears in the highest place, and if it is negative a 9 appears in the highest place, in which case the cover is lowered to read off the result from the upper (compliment) window. An attractive feature of this approach is that positive outcomes are always read from the lower row, whilst negative outcomes are always read from the upper (complements) row of the result window. However, analogous to reasons already discussed in Method 2, above, for negative outcomes 1 must be added to the units position of the result in the complements row, and this can only be done outside the machine. Once that has occurred, as in the methods described earlier, further calculation without resetting the machine is impractical.

15 June 2013
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Changed lines 123-125 from:

**NB.** An alternative procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) by Kistermann[^From 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. 63.^]

The Pascaline may be used to assist multiplication or division, but for practical purposes multiplication tables either written down or remembered must be used, whilst the Pascaline sums up the intermediate results ('partial products'). For that see the [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|explanation of multiplication and division using the Pascaline]] by François Babillot.

The

to:

**NB. 1** An alternative procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) by Kistermann[^From 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. 63.^] Kistermann prefers this approach because if the 10s complements are formed across the entire set of places of the machine, and if the highest place is reserved, it may act as an indicator of whether the result of a subtraction is positive or negative. If it is positive a zero appears in the highest place, and if it is negative a 9 appears in the highest place, in which case the cover is lowered to read off the result from the upper (compliment) window. Analogous to reasons already discussed in Method 2, above, 1 must be added to the units position and this can only be done outside the machine. Once that has occurred, as in the above methods, further calculation without resetting the machine is impractical.

Kistermann believes that this approach is more consistent with actual historical practice. However, Kistermann does not mention the marked spokes of the input wheels on at least several of the surviving Pascalines which could be used directly to input 9s complements. This suggests that arithmetic may well have been performed using the more obvious 9s complements methods along the lines discussed earlier here.

**NB. 2.** The Pascaline may be used to assist multiplication or division, but for practical purposes multiplication tables either written down or remembered must be used, whilst the Pascaline sums up the intermediate results ('partial products'). For that see the [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|explanation of multiplication and division using the Pascaline]] by François Babillot.

Kistermann believes that this approach is more consistent with actual historical practice. However, Kistermann does not mention the marked spokes of the input wheels on at least several of the surviving Pascalines which could be used directly to input 9s complements. This suggests that arithmetic may well have been performed using the more obvious 9s complements methods along the lines discussed earlier here.

**NB. 2.** The Pascaline may be used to assist multiplication or division, but for practical purposes multiplication tables either written down or remembered must be used, whilst the Pascaline sums up the intermediate results ('partial products'). For that see the [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|explanation of multiplication and division using the Pascaline]] by François Babillot.

15 June 2013
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Added line 77:

Changed lines 91-94 from:

4) The result ~~will show in the numbers of ~~the **upper** (complement) row of the result window.

5) Further ~~numbers may be subtracted by dialling them in directly as true numbers, in the usual way~~.~~ Each time the result will show in the upper (complement) row of the result window. ~~

5)

to:

4) The result for (-d) will show in the numbers of the **upper** (complement) row of the result window.

5) Further calculation without resetting the machine is impractical.

5) Further calculation without resetting the machine is impractical.

Changed lines 120-139 from:

Yet another approach is to just add a number B to the minuend to shift the calculation back into a subtraction with a positive outcome, and then take off the amount at the end of the calculation.

Again d* = m* + s\\

Let B be a number big enough so that m+B>s. (e.g., for d=21-38 let B be 100.)\\

∴ A-d = A-m-B + s+B\\

∴ d* = A-(m+B) + (s+B)\\

∴ d* = (m+B)* + (s+B)\\

\\

1) Reset the machine

2) Add B to the minuend and set the result **(m+B)** as its 9s complement (eg the complement of 138).

3) Add B to the subtrahend and set the result **(s+B)** in as a true number, in the usual way by rotating the wheel from the number to the stop bar.

4) The result will show in the numbers of the **upper** (complement) row of the result window.

5) Further numbers may be subtracted by dialling them in directly as true numbers, in the usual way. Each time the result will show in the upper (complement) row of the result window.

to:

5) Further calculation without resetting the machine is impractical.

15 June 2013
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Changed line 123 from:

Let B be a number big enough so that ~~d~~>~~0~~. (e.g., for 21-38 let B be 100.)\\

to:

Let B be a number big enough so that m+B>s. (e.g., for d=21-38 let B be 100.)\\

15 June 2013
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Added lines 117-137:

!!!!Method 3. Same order - add a number to the minuend to make the calculation positive.

Yet another approach is to just add a number B to the minuend to shift the calculation back into a subtraction with a positive outcome, and then take off the amount at the end of the calculation.

Again d* = m* + s\\

Let B be a number big enough so that d>0. (e.g., for 21-38 let B be 100.)\\

∴ A-d = A-m-B + s+B\\

∴ d* = A-(m+B) + (s+B)\\

∴ d* = (m+B)* + (s+B)\\

1) Reset the machine

2) Add B to the minuend and set the result **(m+B)** as its 9s complement (eg the complement of 138).

3) Add B to the subtrahend and set the result **(s+B)** in as a true number, in the usual way by rotating the wheel from the number to the stop bar.

4) The result will show in the numbers of the **upper** (complement) row of the result window.

5) Further numbers may be subtracted by dialling them in directly as true numbers, in the usual way. Each time the result will show in the upper (complement) row of the result window.

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The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate.

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!! ~~Operating Instructions for the Pascaline~~

This is a brief introduction to operating the ~~Pascaline.~~ François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate.

This is a brief introduction to operating

to:

!!! Introduction

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate.

The Pascaline was probably mostly used for addition or simple subtraction where the result was still positive. Using it for other purposes (when the outcome is negative, or for multiplication or division) is possible, but progressively more complicated and therefore less helpful. This is a brief introduction to operating the Pascaline in its simpler modes of addition and subtraction. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate.

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!!!!Method 2. Same order - read from lower row

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!!!!Method 2. Same order - add overflow, read from lower row

15 June 2013
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!!!!Method 1. Reverse order - read result in upper row as usual

Changed line 94 from:

to:

!!!!Method 2. Same order - read from lower row

15 June 2013
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4) Recalling that d is negative, -d is precisely the number we can compute. Note that d is a true number, not a complement, so the result will show in the numbers of the **lower** row of the result window. However, as seen above, the 1 that overflowed and was ignored must now be added back to the result to give 000017. Of course we must remember we have calculated -d and the actual ~~result of the subtraction is ~~-000017~~ ~~.

to:

4) Recalling that d is negative, -d is precisely the number we can compute. Note that d is a true number, not a complement, so the result will show in the numbers of the **lower** row of the result window. However, as seen above, the 1 that overflowed and was ignored must now be added back to the result to give 000017. Of course we must remember we have calculated -d and the actual difference d, is -000017.

15 June 2013
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4) Recalling that d is negative, -d is precisely the number we can compute. ~~This result will show in the numbers of the **lower** row of ~~the result ~~window. However, as seen above,~~ the ~~1 that overflowed and was ignored must now be added back to the result to give 000017. Of course we must remember we have calculated -d and the actual result of the subtraction is thus~~ -000017 .

to:

4) Recalling that d is negative, -d is precisely the number we can compute. Note that d is a true number, not a complement, so the result will show in the numbers of the **lower** row of the result window. However, as seen above, the 1 that overflowed and was ignored must now be added back to the result to give 000017. Of course we must remember we have calculated -d and the actual result of the subtraction is -000017 .

15 June 2013
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4) Recalling that d is negative, -d is precisely the number we ~~need. ~~ This result will show in the numbers of the **lower** row of the result window. However, as seen above, the 1 that overflowed and was ignored must now be added back to the result to give 000017 ~~which of ~~course must ~~now be treated as a negative number.~~

**NB.** An alternative and in some ways preferable procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) by Kistermann[^From 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. 63.^]

**NB.** An alternative and in some ways preferable

to:

4) Recalling that d is negative, -d is precisely the number we can compute. This result will show in the numbers of the **lower** row of the result window. However, as seen above, the 1 that overflowed and was ignored must now be added back to the result to give 000017. Of course we must remember we have calculated -d and the actual result of the subtraction is thus -000017 .

**NB.** An alternative procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) by Kistermann[^From 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. 63.^]

**NB.** An alternative procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) by Kistermann[^From 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. 63.^]

15 June 2013
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For 9s complementation of a six digit number A=999999

if s = 000021

then s* = 999999 - 000021 = 999978

to:

**Method 1. Reverse order - read result in upper row as usual**

Since the difference, d, is negative we need to compute -d = s - m

again -d = s-m\\

∴ A-(-d) = A-(s-m) = (A-s)+m\\

∴ (-d)*= s*+ m\\

Since the difference, d, is negative we need to compute -d = s - m

again -d = s-m\\

∴ A-(-d) = A-(s-m) = (A-s)+m\\

∴ (-d)*= s*+ m\\

Added lines 87-104:

2) Set the subtrahend **s** as its 9s complement (using one of the methods above).

3) Add the minuend **m** in as a true number, in the usual way by rotating the wheel from the number to the stop bar.

4) The result will show in the numbers of the **upper** (complement) row of the result window.

5) Further numbers may be subtracted by dialling them in directly as true numbers, in the usual way. Each time the result will show in the upper (complement) row of the result window.

**Method 2. Same order - read from lower row**

But actually our approach used for positive numbers will work if we treat our complementary numbers correctly. It may seem tricker, because we may have got away in the above by not inputting the full complementary number, but rather just the complements for the number of actual digits we are dealing with. (The effect of that was to leave a string of nines in the places higher than our result, which we may have ignored.) Now it is necessary to use full complementary numbers formed across the whole set of input dials. ie for a six place machine (like the Calculant replica):

For 9s complementation of a six digit number A=999999.

if s = 000021

then s* = 999999 - 000021 = 999978

1) Reset the machine

3) Add the minuend **m** in as a true number, in the usual way by rotating the wheel from the number to the stop bar.

4) The result will show in the numbers of the **upper** (complement) row of the result window.

5) Further numbers may be subtracted by dialling them in directly as true numbers, in the usual way. Each time the result will show in the upper (complement) row of the result window.

**Method 2. Same order - read from lower row**

But actually our approach used for positive numbers will work if we treat our complementary numbers correctly. It may seem tricker, because we may have got away in the above by not inputting the full complementary number, but rather just the complements for the number of actual digits we are dealing with. (The effect of that was to leave a string of nines in the places higher than our result, which we may have ignored.) Now it is necessary to use full complementary numbers formed across the whole set of input dials. ie for a six place machine (like the Calculant replica):

For 9s complementation of a six digit number A=999999.

if s = 000021

then s* = 999999 - 000021 = 999978

1) Reset the machine

Changed lines 109-111 from:

The machine will compute ~~999978+000038~~ = ~~(1)000016.~~ ~~The (1) represents a~~ 1 ~~that has been carried beyond the machine and thus is ignored as an overflow. ~~

4) The result will show in the numbers of the **lower** row of the result window. However, the 1 that overflowed and was ignored must now be added back to the result to give 000017 which of course must now be treated as a negative number.

4) The result will show in the numbers of the **lower** row of the result window. However,

to:

The machine will compute d* = m* + s = 999978+000038 = (1)000016. The (1) represents a 1 that has been carried beyond the machine and thus was ignored as an overflow.

To understand how to treat this note that 1000000 that has been lost = A+1\\

Let the truncated result (0000016) be denoted t\\

∴ Then d* = t + A + 1\\

∴ t+1 = d*-A = -(A-d*) = -d\\

4) Recalling that d is negative, -d is precisely the number we need. This result will show in the numbers of the **lower** row of the result window. However, as seen above, the 1 that overflowed and was ignored must now be added back to the result to give 000017 which of course must now be treated as a negative number.

To understand how to treat this note that 1000000 that has been lost = A+1\\

Let the truncated result (0000016) be denoted t\\

∴ Then d* = t + A + 1\\

∴ t+1 = d*-A = -(A-d*) = -d\\

4) Recalling that d is negative, -d is precisely the number we need. This result will show in the numbers of the **lower** row of the result window. However, as seen above, the 1 that overflowed and was ignored must now be added back to the result to give 000017 which of course must now be treated as a negative number.

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This is a brief introduction to operating the Pascaline. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly ~~fair, ~~well researched~~,~~ and accurate.

to:

This is a brief introduction to operating the Pascaline. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly well researched and accurate.

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The Pascaline may be used to assist multiplication or division, but for practical purposes multiplication tables either written down or remembered must be used, whilst the Pascaline sums up the intermediate results ('partial products'). For ~~more on this see the explanation by ~~[[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|~~François Babillot~~]]

to:

The Pascaline may be used to assist multiplication or division, but for practical purposes multiplication tables either written down or remembered must be used, whilst the Pascaline sums up the intermediate results ('partial products'). For that see the [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|explanation of multiplication and division using the Pascaline]] by François Babillot.

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The Pascaline may be used to assist multiplication or division, but for practical purposes multiplication tables either written down or remembered must be used, whilst the Pascaline sums up the intermediate results ('partial products'). For more on this see the explanation by ~~[^~~[[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|François Babillot]]

to:

The Pascaline may be used to assist multiplication or division, but for practical purposes multiplication tables either written down or remembered must be used, whilst the Pascaline sums up the intermediate results ('partial products'). For more on this see the explanation by [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|François Babillot]]

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The Pascaline may be used to assist multiplication or division, but for practical purposes multiplication tables either written down or remembered must be used, whilst the Pascaline sums up the intermediate results ('partial products'). For more on this see the explanation by [^[[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|François Babillot]]

15 June 2013
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This will seem a bit trickier, because we may have got away in the above by not inputting the full complementary number, but rather just the complements for the number of actual digits we are dealing with. ~~Now it is necessary~~ to ~~use full complementary numbers formed across~~ the ~~whole set of output windows.~~ ie for a six place machine (like the Calculant replica):

to:

This will seem a bit trickier, because we may have got away in the above by not inputting the full complementary number, but rather just the complements for the number of actual digits we are dealing with. (The effect of that was to leave a string of nines in the places higher than our result, which we may have ignored.) Now it is necessary to use full complementary numbers formed across the whole set of input dials. ie for a six place machine (like the Calculant replica):

15 June 2013
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This ~~is~~ a bit trickier, because we have got away in the above by not inputting the full complementary number, but rather just the complements for the number of actual digits we are dealing with. Now it is necessary to use full complementary numbers formed across the whole set of output windows. ie for a six place machine (like the Calculant replica):

to:

This will seem a bit trickier, because we may have got away in the above by not inputting the full complementary number, but rather just the complements for the number of actual digits we are dealing with. Now it is necessary to use full complementary numbers formed across the whole set of output windows. ie for a six place machine (like the Calculant replica):

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4) The result will show in the numbers of the upper (complement) row of the result window.

to:

4) The result will show in the numbers of the **upper** (complement) row of the result window.

Changed line 92 from:

4) The result will show in the numbers of the lower row of the result window. However, the 1 that overflowed and was ignored must now be added back to the result to give 000017 which of course must now be treated as a negative number.

to:

4) The result will show in the numbers of the **lower** row of the result window. However, the 1 that overflowed and was ignored must now be added back to the result to give 000017 which of course must now be treated as a negative number.

15 June 2013
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NB. An alternative and in some ways preferable procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements)~~ as given~~ by Kistermann[^From 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. 63.^]

to:

**NB.** An alternative and in some ways preferable procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) by Kistermann[^From 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. 63.^]

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!!!Simple subtraction (where the outcome will be positive) on the Pascaline (using 9s complements)

Changed lines 68-108 from:

2) Set the minuend ~~'c'~~ as its 9s complement~~.~~ ~~Some Pascalines have ~~the ~~complements marked on an inner wheel. In others (including the Computant replica) two spokes of each input wheel are marked with dots. Move~~ the ~~marked segment to the number c to input its complement. The~~ complement ~~shows in~~ the ~~upper row of the top ~~window ~~(red~~ numbers ~~in the Calculant replica).~~

3) Add the subtrahend 'b' in the usual way.

4) The result will show in the numbers of the upper row of the ~~top~~ window.

!!!~~ If~~ the ~~result of the above is ~~negative

1) Observe an 8 showing to the left of the position of the left-most digit of the ~~minuend~~

2) Move the bar to reveal the lower row in the ~~top window~~

3) The highest position will show a 1. Delete that and add 1 to the remaining number to give the negative difference.

!!! Operating Instructions for subtraction on the Pascaline (using 10s complements) as given by Kistermann[^From 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. 63.^]~~

1) Resetthe ~~machine.~~

2) Enter (~~dial) the minuend ~~as ~~a true number.~~

3) Enter the 10s complement of the subtrahend.

5) Ignore any overflow carry.

//[6) According to Kistermann If the result is positive, the ~~highest position will show a zero. If ~~the ~~result is negative, ~~the ~~highest position~~ will ~~show a nine. - This doesn't seem to work!// (To be confirmed)~~

7) If the result is positive, the result (ignoring any overflow carry forward) is shown in the lower row of the ~~top window.~~

8) Ifthe ~~result is negative, lower ~~the ~~cover bar to obtain ~~the ~~(nines) recomplement, then add a one to the units position (outside~~ the ~~machine) and write down this negative result.~~

**Notes**

1) The highest position of the machine must not be used for subtraction, because it acts as the sign indicator forthe ~~result (9 is ~~negative~~, 0 positive).~~

2) The cover bar is not used for the actual subtract operations, it is used merely for the complementation of the result.

3) The red numerals indicate a negative result if the cover bar is down. This alerts the operator to write down a minus sign for the result.~~ ~~

4) Multiple sequential additions or multiple sequential subtractions, and especially mixed additions and subtractions should not incur any complications provided that the highest position is not involved in the arithmetical operations.

3) Add the subtrahend 'b' in the usual

4) The result will show in the numbers of

1) Observe an 8 showing to the left of the position of the left-most digit of

2) Move the bar to reveal the lower row in

3) The highest position will show a 1. Delete that and add 1 to the remaining number to give the negative difference.

!!! Operating Instructions for subtraction on the Pascaline (using 10s complements) as given by Kistermann[^From Friedrich W. Kistermann, "Blaise Pascal’s Adding Machine: New Findings and Conclusions", //IEEE Annals of the History of Computing//, vol. 20, no.

1) Reset

2) Enter

3) Enter the 10s complement of the subtrahend.

5) Ignore any overflow carry.

//[6) According to Kistermann If the result is positive

7) If the result is positive, the result (ignoring any overflow carry forward) is shown in the lower row of

8) If

**Notes**

1) The highest position of the machine must not be used for subtraction, because it acts as the sign indicator for

2) The cover bar is not used for the actual subtract operations, it is used merely for the complementation of the result.

3) The red numerals indicate a negative result if the cover bar is down. This alerts the operator to write down a minus sign for the result

4) Multiple sequential additions or multiple sequential subtractions, and especially mixed additions and subtractions should not incur any complications provided that the highest position is not involved in the arithmetical operations.

to:

2) Set the minuend **m** as its 9s complement (using one of the methods above).

3) Add the subtrahend **s** in as a true number, in the usual way by rotating the wheel from the number to the stop bar.

4) The result will show in the numbers of the upper (complement) row of the result window.

5) Further numbers may be subtracted by dialling them in directly as true numbers, in the usual way. Each time the result will show in the upper (complement) row of the result window.

!!!Simple subtraction (where the outcome will be negative) on the Pascaline (using 9s complements)

This is a bit trickier, because we have got away in the above by not inputting the full complementary number, but rather just the complements for the number of actual digits we are dealing with. Now it is necessary to use full complementary numbers formed across the whole set of output windows. ie for a six place machine (like the Calculant replica):

For 9s complementation of a six digit number A=999999

if s = 000021

then s* = 999999 - 000021 = 999978

1) Reset the machine

2) Set the minuend **m** (say 21) as its 9s complement (using one of the methods above).

3) Add the subtrahend **s** (38) in as a true number, in the usual way by rotating the wheel from the number to the stop bar.

The machine will compute 999978+000038 = (1)000016. The (1) represents a 1 that has been carried beyond the machine and thus is ignored as an overflow.

4) The result will show in the numbers of the lower row of the result window. However, the 1 that overflowed and was ignored must now be added back to the result to give 000017 which of course must now be treated as a negative number.

NB. An alternative and in some ways preferable procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) as given by Kistermann[^From 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. 63.^]

3) Add the subtrahend **s** in as a true number, in the usual way by rotating the wheel from the number to the stop bar.

4) The result will show in the numbers of the upper (complement) row of the result window.

5) Further numbers may be subtracted by dialling them in directly as true numbers, in the usual way. Each time the result will show in the upper (complement) row of the result window.

!!!Simple subtraction (where the outcome will be negative) on the Pascaline (using 9s complements)

This is a bit trickier, because we have got away in the above by not inputting the full complementary number, but rather just the complements for the number of actual digits we are dealing with. Now it is necessary to use full complementary numbers formed across the whole set of output windows. ie for a six place machine (like the Calculant replica):

For 9s complementation of a six digit number A=999999

if s = 000021

then s* = 999999 - 000021 = 999978

1) Reset the machine

2) Set the minuend **m** (say 21) as its 9s complement (using one of the methods above).

3) Add the subtrahend **s** (38) in as a true number, in the usual way by rotating the wheel from the number to the stop bar.

The machine will compute 999978+000038 = (1)000016. The (1) represents a 1 that has been carried beyond the machine and thus is ignored as an overflow.

4) The result will show in the numbers of the lower row of the result window. However, the 1 that overflowed and was ignored must now be added back to the result to give 000017 which of course must now be treated as a negative number.

NB. An alternative and in some ways preferable procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) as given by Kistermann[^From 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. 63.^]

15 June 2013
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Because A can be any number we have a variety of possible approaches to implementing this on the Pascaline. We can simply chose A and then do the "complementation" (calculating x*=A-x) and "re-complementation" (calculating x=A+x*) in our heads. But that somewhat takes away the point of having a machine to do the 'grunt' of addition! ~~ The various surviving Pascalines show attempts to make this task ~~easier.

to:

Because A can be any number we have a variety of possible approaches to implementing this on the Pascaline.

* We can simply chose A and then do the "complementation" (calculating x*=A-x) and "re-complementation" (calculating x=A+x*) in our heads. But that somewhat takes away the point of having a machine to do the 'grunt' of addition!

The various surviving Pascalines show attempts to make this task easier.

* The results 'top window' of the Pascaline has a cover bar which can be slid up or down to reveal either a lower row of numbers or an upper row of their 9s complements. The lower row is often rendered in black, whilst the upper complements (subtracted from 9) are often rendered in red. It is therefore possible to dial in a number, lower the cover bar, and read off the complement.

* Some Pascalines (including the replica in Calculant) have dots engraved on two of the spokes of each input wheel which can be used to input 9s complements. To do that the stylus is placed in the marked segment and rotated to the number for which a complement is sought. The complement then shows correctly in the upper complement row of the results window.

* Some Pascalines had an inner disk set into each input wheel showing the 9s complements to assist their input.

Whichever way is chosen or available the process of subtraction can be done in a variety of ways. The simplest is given below.

* We can simply chose A and then do the "complementation" (calculating x*=A-x) and "re-complementation" (calculating x=A+x*) in our heads. But that somewhat takes away the point of having a machine to do the 'grunt' of addition!

The various surviving Pascalines show attempts to make this task easier.

* The results 'top window' of the Pascaline has a cover bar which can be slid up or down to reveal either a lower row of numbers or an upper row of their 9s complements. The lower row is often rendered in black, whilst the upper complements (subtracted from 9) are often rendered in red. It is therefore possible to dial in a number, lower the cover bar, and read off the complement.

* Some Pascalines (including the replica in Calculant) have dots engraved on two of the spokes of each input wheel which can be used to input 9s complements. To do that the stylus is placed in the marked segment and rotated to the number for which a complement is sought. The complement then shows correctly in the upper complement row of the results window.

* Some Pascalines had an inner disk set into each input wheel showing the 9s complements to assist their input.

Whichever way is chosen or available the process of subtraction can be done in a variety of ways. The simplest is given below.

Changed lines 64-65 from:

to:

!!!Operating Instructions for simple subtraction (where outcome is positive) on the Pascaline (using 9s complements)

15 June 2013
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Changed lines 8-9 from:

The Pascaline adds by summing the rotations of the input dials through 1 to 9 units. When a dial moves from 9 to 0, ~~one~~ is automatically carried to the next dial rotating that dial by one unit. The process of addition is thus simple:

to:

The Pascaline adds by summing the rotations of the input dials through 1 to 9 units. When a dial moves from 9 to 0, 1 is automatically carried to the next dial rotating that dial by one unit. The process of addition is thus simple:

Added lines 46-47:

!!!!Making the complement in practice.

15 June 2013
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Changed line 3 from:

This is a brief introduction to operating the Pascaline. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly fair and accurate.

to:

This is a brief introduction to operating the Pascaline. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly fair, well researched, and accurate.

15 June 2013
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Changed line 3 from:

This is a brief introduction to operating the Pascaline. François Babillot[^François Babillot teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly fair and accurate.

to:

This is a brief introduction to operating the Pascaline. François Babillot[^[[http://calmeca.free.fr/calculmecanique_php/index_calmeca.php?lang=eng|François Babillot]] teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly fair and accurate.

15 June 2013
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Added lines 2-4:

This is a brief introduction to operating the Pascaline. François Babillot[^François Babillot teaches Physics at the Blaise Pascal University of Clermont-Ferrand in France^] has prepared a [[http://calmeca.free.fr/calculmecanique_php/rubriques/Fichiers_Blaise_Pascal/Fichiers_mode_emploi/Pascaline_mode_emploi.php?lang=eng|more complete and very nicely illustrated explanation of how the Pascaline can be used]] which seems particularly fair and accurate.

15 June 2013
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Changed line 1 from:

!! Operating Instructions for~~ addition on~~ the Pascaline

to:

!! Operating Instructions for the Pascaline

15 June 2013
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Added lines 43-45:

Because A can be any number we have a variety of possible approaches to implementing this on the Pascaline. We can simply chose A and then do the "complementation" (calculating x*=A-x) and "re-complementation" (calculating x=A+x*) in our heads. But that somewhat takes away the point of having a machine to do the 'grunt' of addition! The various surviving Pascalines show attempts to make this task easier.

15 June 2013
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Changed lines 5-7 from:

to:

The Pascaline adds by summing the rotations of the input dials through 1 to 9 units. When a dial moves from 9 to 0, one is automatically carried to the next dial rotating that dial by one unit. The process of addition is thus simple:

1) Reset the machine (by setting 9 in each window and then adding 1 to the right-most input dial. As it turns to 0, 1 is carried to the next dial causing it also to turn to 0, and so on along each dial until they are all set at zero. No carry is possible from the left-most 'highest' dial and so it simply turns to 0 the carry being ignored as 'an overflow').

1) Reset the machine (by setting 9 in each window and then adding 1 to the right-most input dial. As it turns to 0, 1 is carried to the next dial causing it also to turn to 0, and so on along each dial until they are all set at zero. No carry is possible from the left-most 'highest' dial and so it simply turns to 0 the carry being ignored as 'an overflow').

15 June 2013
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Changed lines 36-39 from:

As a specific example,

put A=99, m=21 and s=38

then since d* = m* + s, d* = 78+38 = 116

and since d*=A-d, -d=d*-A, so -d = 116-99 = 17 which is correct.

put A=99, m=21 and s=38

then since d* = m* + s, d* = 78+38 = 116

and since d*=A-d, -d=d*-A, so -d = 116-99 = 17 which is correct.

to:

As a specific example,\\

put A=99, m=21 and s=38\\

then since d* = m* + s, d* = 78+38 = 116\\

and since d*=A-d, -d=d*-A, so -d = 116-99 = 17 which is correct.\\

put A=99, m=21 and s=38\\

then since d* = m* + s, d* = 78+38 = 116\\

and since d*=A-d, -d=d*-A, so -d = 116-99 = 17 which is correct.\\

15 June 2013
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Changed lines 22-24 from:

Consider (the 'difference') d resulting from the subtraction of (the 'subtrahend') s from (the 'minuend') m.

let d = m - s\\

let d = m - s\\

to:

Consider (the 'difference') **d** resulting from the subtraction of (the 'subtrahend') **s** from (the 'minuend') **m**.

let **d = m - s**\\

let **d = m - s**\\

15 June 2013
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Changed lines 20-21 from:

In the Pascaline, since the mechanism could only rotate in the addition direction, subtraction could only be achieved by a method that transformed addition to subtraction. To do this "complementary numbers" were utilised.

to:

In the Pascaline, since the mechanism could only rotate in the addition direction, subtraction could only be achieved by a method that transformed addition to subtraction. To do this "complementary numbers" were utilised. The abstract idea is shown below:

Changed lines 32-36 from:

d*=A-d as the complement of d, and\\

->m*=A-m as the complement of m

->m*=A-m as the complement of m

to:

d*=A-d as the 'complement of d', and\\

->m*=A-m as the 'complement of m'

As a specific example,

put A=99, m=21 and s=38

then since d* = m* + s, d* = 78+38 = 116

and since d*=A-d, -d=d*-A, so -d = 116-99 = 17 which is correct.

->m*=A-m as the 'complement of m'

As a specific example,

put A=99, m=21 and s=38

then since d* = m* + s, d* = 78+38 = 116

and since d*=A-d, -d=d*-A, so -d = 116-99 = 17 which is correct.

15 June 2013
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Changed line 32 from:

to:

d*=A-d as the complement of d, and\\

15 June 2013
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Changed lines 26-27 from:

∴ ~~A-d =~~ A ~~+ s - m, where A is any number~~\\

to:

∴ for any number A

A-d = A + s - m\\

A-d = A + s - m\\

Changed lines 32-33 from:

d*=A-d as the complement of d, and\\

m*=A-m as the complement of m

m*=A-m as the complement of m

to:

->d*=A-d as the complement of d, and\\

->m*=A-m as the complement of m

->m*=A-m as the complement of m

15 June 2013
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Changed lines 24-32 from:

let d = m - s

∴ -d = s - m

∴ A-d = A + s - m, where A is any number

∴ (A-d)=(A-m) + s

∴ d* = m* + s

->where we have defined~~ ~~

d*=A-d as the complement of d, and

m*=A-m as the complement of~~m.~~

d*=A-d as the complement of d, and

m*=A-m as the complement of

to:

let d = m - s\\

∴ -d = s - m\\

∴ A-d = A + s - m, where A is any number\\

∴ (A-d)=(A-m) + s\\

∴ d* = m* + s\\

->where we have defined\\

d*=A-d as the complement of d, and\\

m*=A-m as the complement of m

∴ -d = s - m\\

∴ A-d = A + s - m, where A is any number\\

∴ (A-d)=(A-m) + s\\

∴ d* = m* + s\\

->where we have defined\\

d*=A-d as the complement of d, and\\

m*=A-m as the complement of m

15 June 2013
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Changed lines 22-24 from:

ie we seek d = m - s

to:

Consider (the 'difference') d resulting from the subtraction of (the 'subtrahend') s from (the 'minuend') m.

let d = m - s

let d = m - s

Changed lines 28-29 from:

to:

∴ d* = m* + s

->where we have defined

d*=A-d as the complement of d, and

m*=A-m as the complement of m.

->where we have defined

d*=A-d as the complement of d, and

m*=A-m as the complement of m.

15 June 2013
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Changed lines 1-2 from:

!!~~!~~ Operating Instructions for addition on the Pascaline

to:

!! Operating Instructions for addition on the Pascaline

!!! Addition

!!! Addition

Changed lines 16-19 from:

!!! ~~Operating Instructions for simple subtraction (where outcome is positive) on ~~the ~~Pascaline (using 9s complements)~~

Note: for c − b = a

the traditional names for the terms are: minuend 'c' − subtrahend 'b' = difference 'a').

Note: for c − b = a

the traditional names for the terms are: minuend 'c' − subtrahend 'b' = difference 'a')

to:

!!! Subtraction

!!!! Introduction to terms and methods

In the Pascaline, since the mechanism could only rotate in the addition direction, subtraction could only be achieved by a method that transformed addition to subtraction. To do this "complementary numbers" were utilised.

We seek to find the difference 'd' resulting from the subtraction of (the 'subtrahend') 's' from (the 'minuend') 'm'.

ie we seek d = m - s

∴ -d = s - m

∴ A-d = A + s - m, where A is any number

∴ (A-d)=(A-m) + s

∴ d* = m* + s, where we have defined d*=A-d as the complement of d, and m*=A-m as the complement of m.

Operating Instructions for simple subtraction (where outcome is positive) on the Pascaline (using 9s complements)

!!!! Introduction to terms and methods

In the Pascaline, since the mechanism could only rotate in the addition direction, subtraction could only be achieved by a method that transformed addition to subtraction. To do this "complementary numbers" were utilised.

We seek to find the difference 'd' resulting from the subtraction of (the 'subtrahend') 's' from (the 'minuend') 'm'.

ie we seek d = m - s

∴ -d = s - m

∴ A-d = A + s - m, where A is any number

∴ (A-d)=(A-m) + s

∴ d* = m* + s, where we have defined d*=A-d as the complement of d, and m*=A-m as the complement of m.

Operating Instructions for simple subtraction (where outcome is positive) on the Pascaline (using 9s complements)

14 June 2013
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Changed line 47 from:

//6) ~~If the result is positive, ~~the ~~highest position will show a zero. If the result is negative,~~ the ~~highest position will show a~~ nine.// (To be confirmed)

to:

//[6) According to Kistermann If the result is positive, the highest position will show a zero. If the result is negative, the highest position will show a nine. - This doesn't seem to work!// (To be confirmed)

14 June 2013
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Changed lines 47-49 from:

6) If the result is positive, the highest position will show a zero. If the result is negative, the highest position will show a nine.

7) If the result is positive, lower the bar to obtain the nines recomplement and add 1 to give the ~~result (the tens complement)~~.

7) If the result is positive, lower the bar to obtain the nines recomplement and add 1 to give

to:

//6) If the result is positive, the highest position will show a zero. If the result is negative, the highest position will show a nine.// (To be confirmed)

7) If the result is positive, the result (ignoring any overflow carry forward) is shown in the lower row of the top window.

7) If the result is positive, the result (ignoring any overflow carry forward) is shown in the lower row of the top window.

14 June 2013
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Changed lines 43-46 from:

3) ~~Form~~ the 10s complement of~~ the subtrahend.~~

4) Enter (dial) the subtrahend.

4) Enter (dial)

to:

3) Enter the 10s complement of the subtrahend.

Changed lines 47-49 from:

6) If the result is positive, the highest position will show a zero.

7) If the result is negative, the highest position will show a nine.

7)

to:

6) If the result is positive, the highest position will show a zero. If the result is negative, the highest position will show a nine.

7) If the result is positive, lower the bar to obtain the nines recomplement and add 1 to give the result (the tens complement).

7) If the result is positive, lower the bar to obtain the nines recomplement and add 1 to give the result (the tens complement).

14 June 2013
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Changed lines 21-23 from:

2) Set the minuend ~~(~~c~~)~~ as its 9s complement. Some Pascalines have the complements marked on an inner wheel. In others (including the Computant replica) two spokes of each input wheel are marked with dots. Move the marked segment to the number c to input its complement. The complement shows in the upper row of the top window (red numbers in the Calculant replica).

3) Add the subtrahend~~(~~b~~)~~ in the usual way.

3) Add the subtrahend

to:

2) Set the minuend 'c' as its 9s complement. Some Pascalines have the complements marked on an inner wheel. In others (including the Computant replica) two spokes of each input wheel are marked with dots. Move the marked segment to the number c to input its complement. The complement shows in the upper row of the top window (red numbers in the Calculant replica).

3) Add the subtrahend 'b' in the usual way.

3) Add the subtrahend 'b' in the usual way.

14 June 2013
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Changed line 17 from:

the traditional names for the terms are: minuend ~~[~~c~~]~~ − subtrahend ~~[~~b~~]~~ = difference ~~[~~a~~]~~).

to:

the traditional names for the terms are: minuend 'c' − subtrahend 'b' = difference 'a').

14 June 2013
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Changed line 17 from:

the traditional names for the terms are: minuend ~~(~~c~~)~~ − subtrahend ~~(~~b~~)~~ = difference ~~(~~a).

to:

the traditional names for the terms are: minuend [c] − subtrahend [b] = difference [a]).

14 June 2013
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Changed lines 11-13 from:

5) The result will show in the black numbers in the lower row of the top window.

to:

5) The result will show in the black numbers in the lower row of the top window (where the cover bar is up).

Changed line 61 from:

3) The red numerals indicate a negative result~~ in the PAM~~ if the cover bar is down. This alerts the operator to write down a minus sign for the result.

to:

3) The red numerals indicate a negative result if the cover bar is down. This alerts the operator to write down a minus sign for the result.

14 June 2013
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Changed lines 14-15 from:

!!! Operating Instructions for simple subtraction (where outcome is positive) on the Pascaline

to:

!!! Operating Instructions for simple subtraction (where outcome is positive) on the Pascaline (using 9s complements)

Changed line 37 from:

!!! Operating Instructions for subtraction on the Pascaline as given by Kistermann[^From 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. 63.^]

to:

!!! Operating Instructions for subtraction on the Pascaline (using 10s complements) as given by Kistermann[^From 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. 63.^]

14 June 2013
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Changed lines 11-15 from:

!!! Operating Instructions for subtraction on the Pascaline[^From 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. 63.^]

to:

5) The result will show in the black numbers in the lower row of the top window.

!!! Operating Instructions for simple subtraction (where outcome is positive) on the Pascaline

Note: for c − b = a

the traditional names for the terms are: minuend (c) − subtrahend (b) = difference (a).

1) Reset the machine

2) Set the minuend (c) as its 9s complement. Some Pascalines have the complements marked on an inner wheel. In others (including the Computant replica) two spokes of each input wheel are marked with dots. Move the marked segment to the number c to input its complement. The complement shows in the upper row of the top window (red numbers in the Calculant replica).

3) Add the subtrahend (b) in the usual way.

4) The result will show in the numbers of the upper row of the top window.

!!! If the result of the above is negative

1) Observe an 8 showing to the left of the position of the left-most digit of the minuend

2) Move the bar to reveal the lower row in the top window

3) The highest position will show a 1. Delete that and add 1 to the remaining number to give the negative difference.

!!! Operating Instructions for subtraction on the Pascaline as given by Kistermann[^From 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. 63.^]

!!! Operating Instructions for simple subtraction (where outcome is positive) on the Pascaline

Note: for c − b = a

the traditional names for the terms are: minuend (c) − subtrahend (b) = difference (a).

1) Reset the machine

2) Set the minuend (c) as its 9s complement. Some Pascalines have the complements marked on an inner wheel. In others (including the Computant replica) two spokes of each input wheel are marked with dots. Move the marked segment to the number c to input its complement. The complement shows in the upper row of the top window (red numbers in the Calculant replica).

3) Add the subtrahend (b) in the usual way.

4) The result will show in the numbers of the upper row of the top window.

!!! If the result of the above is negative

1) Observe an 8 showing to the left of the position of the left-most digit of the minuend

2) Move the bar to reveal the lower row in the top window

3) The highest position will show a 1. Delete that and add 1 to the remaining number to give the negative difference.

!!! Operating Instructions for subtraction on the Pascaline as given by Kistermann[^From 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. 63.^]

Changed line 61 from:

3) The red numerals indicate a negative result in the PAM if the cover bar is down. This alerts the ~~op- erator~~ to write down a minus sign for the result.

to:

3) The red numerals indicate a negative result in the PAM if the cover bar is down. This alerts the operator to write down a minus sign for the result.

01 June 2013
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Changed line 34 from:

1) The highest position of the machine must not be used for subtraction, because it ~~is~~ the sign indicator for the result.

to:

1) The highest position of the machine must not be used for subtraction, because it acts as the sign indicator for the result (9 is negative, 0 positive).

01 June 2013
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Changed lines 38-40 from:

3) ~~When using subtraction Method 3, red numerals would make sense, because they would indicate a negative result in~~ the ~~PAM if the cover bar is down. This would alert ~~the ~~op- erator to write down a minus sign for the result. However, there is no real need for red numerals, because the nine in the highest position is already the clear indication of a negative result and should cause the operator to lower the cover bar for complementation.~~

4) Multiple sequential additions or multiple sequential sub- tractions, and especially mixed additions and subtractions, would not incur any complications with Method 3, provided that the highest position is not involved in the arithmetical operations.

4) Multiple sequential additions or multiple sequential sub- tractions, and especially mixed additions and subtractions, would not incur any complications with Method 3,

to:

3) The red numerals indicate a negative result in the PAM if the cover bar is down. This alerts the op- erator to write down a minus sign for the result.

4) Multiple sequential additions or multiple sequential subtractions, and especially mixed additions and subtractions should not incur any complications provided that the highest position is not involved in the arithmetical operations.

4) Multiple sequential additions or multiple sequential subtractions, and especially mixed additions and subtractions should not incur any complications provided that the highest position is not involved in the arithmetical operations.

01 June 2013
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Changed line 11 from:

5) The result will show in the top window~~ ~~.

to:

5) The result will show in the top window.

01 June 2013
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Added lines 1-13:

!!! Operating Instructions for addition on the Pascaline

1) Reset the machine (by setting 9 in each window and then adding 1 to the right-most input dial).

2) For the first number to be added (say 23), working from the right-most input dial input the units (3) for the number to be added by inserting the stylus into the corresponding segment (3) of the dial and turning clockwise until the stop catch is encountered. Repeat for tens, hundreds, etc as necessary.

3) Repeat step 2 for the second number to be added.

4) Continue until all numbers are added.

5) The result will show in the top window .

1) Reset the machine (by setting 9 in each window and then adding 1 to the right-most input dial).

2) For the first number to be added (say 23), working from the right-most input dial input the units (3) for the number to be added by inserting the stylus into the corresponding segment (3) of the dial and turning clockwise until the stop catch is encountered. Repeat for tens, hundreds, etc as necessary.

3) Repeat step 2 for the second number to be added.

4) Continue until all numbers are added.

5) The result will show in the top window .

01 June 2013
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Changed line 1 from:

to:

!!! Operating Instructions for subtraction on the Pascaline[^From 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. 63.^]

01 June 2013
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Changed line 1 from:

**Operating Instructions for ~~the PAM~~**[^From 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. 63.^]

to:

**Operating Instructions for subtraction on the Pascaline**[^From 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. 63.^]

01 June 2013
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Added line 4:

Added line 6:

Added line 8:

Added line 10:

Added line 12:

Added line 14:

Changed lines 16-18 from:

8) If the result is negative, lower the cover bar to obtain the

(nines) recomplement, then add a one to the units position (outside the machine) and write down this negative result.

(nines) recomplement, then add a one to the units position (outside the machine) and write down this negative result.

to:

8) If the result is negative, lower the cover bar to obtain the (nines) recomplement, then add a one to the units position (outside the machine) and write down this negative result.

Added line 20:

Added line 22:

Added line 24:

Added line 26:

01 June 2013
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Added lines 1-19:

**Operating Instructions for the PAM**[^From 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. 63.^]

1) Reset the machine.

2) Enter (dial) the minuend as a true number.

3) Form the 10s complement of the subtrahend.

4) Enter (dial) the subtrahend.

5) Ignore any overflow carry.

6) If the result is positive, the highest position will show a zero.

7) If the result is negative, the highest position will show a nine.

8) If the result is negative, lower the cover bar to obtain the

(nines) recomplement, then add a one to the units position (outside the machine) and write down this negative result.

**Notes**

1) The highest position of the machine must not be used for subtraction, because it is the sign indicator for the result.

2) The cover bar is not used for the actual subtract operations, it is used merely for the complementation of the result.

3) When using subtraction Method 3, red numerals would make sense, because they would indicate a negative result in the PAM if the cover bar is down. This would alert the op- erator to write down a minus sign for the result. However, there is no real need for red numerals, because the nine in the highest position is already the clear indication of a negative result and should cause the operator to lower the cover bar for complementation.

4) Multiple sequential additions or multiple sequential sub- tractions, and especially mixed additions and subtractions, would not incur any complications with Method 3, provided that the highest position is not involved in the arithmetical operations.

[^#^]

1) Reset the machine.

2) Enter (dial) the minuend as a true number.

3) Form the 10s complement of the subtrahend.

4) Enter (dial) the subtrahend.

5) Ignore any overflow carry.

6) If the result is positive, the highest position will show a zero.

7) If the result is negative, the highest position will show a nine.

8) If the result is negative, lower the cover bar to obtain the

(nines) recomplement, then add a one to the units position (outside the machine) and write down this negative result.

**Notes**

1) The highest position of the machine must not be used for subtraction, because it is the sign indicator for the result.

2) The cover bar is not used for the actual subtract operations, it is used merely for the complementation of the result.

3) When using subtraction Method 3, red numerals would make sense, because they would indicate a negative result in the PAM if the cover bar is down. This would alert the op- erator to write down a minus sign for the result. However, there is no real need for red numerals, because the nine in the highest position is already the clear indication of a negative result and should cause the operator to lower the cover bar for complementation.

4) Multiple sequential additions or multiple sequential sub- tractions, and especially mixed additions and subtractions, would not incur any complications with Method 3, provided that the highest position is not involved in the arithmetical operations.

[^#^]