Operating Instructions For A Pascaline

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 Babillot¹ has prepared a 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 Jan Meyer replica in this collection.

Addition

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:

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’).

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 black numbers in the lower row of the top window (where the cover bar is up).

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. The abstract idea is shown below:

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

let d = m - s
∴ -d = s - m

∴ for any number A
A-d = A + s - m
∴ (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’

As a specific example,
let 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.

Making the complement in practice.

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 (at least in replicas). It is therefore possible to dial in a number, lower the cover bar, and read off the complement.
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.
Some Pascalines had an inner disk set into each input wheel showing digits which may have assisted the input of the 9s complements.

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

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

1) Reset the machine

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)

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

1) Reset the machine

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

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 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.

Method 2. Same order - add overflow, 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

Bearing this in mind the method is:

1) Reset the machine

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

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 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 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.

5) Further calculation without resetting the machine is impractical.

NB. 1 An alternative procedure for subtraction on the Pascaline, where negative outcomes may arise, is given (using 10s complements) by Kistermann.² 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.

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),³ 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 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.⁴

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 explanation of multiplication and division using the Pascaline by François Babillot.