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The addition and the subtraction with the pascaline

The addition :

Once the pascaline is initialized (see "zeroing the pascaline"), one just has to enter the numbers we want to add. The result displays instantaneously in the windows.

About the carry, the 18th century manuscript says :

"[...] we have to note here that when a digit is entered on a wheel, the figures change on the other dials. This is not a disorder in the machine, the sum is constructed with this alteration of the digits when they are entered on the wheels".

This comment points out that the carry who seems to be evident today was not easy to understand at the Blaise Pascal time.

The subtraction :

The manuscript takes the following example :

890 £ 9 s 7 d - 206 £ 12 s 9 d (this is the french money unit. £ is for livres, s for sols and d for deniers)

a) initialization of the pascaline :

Firstly, the machine is initialized in the same manner as for the addition and the cover slide is set toward the operator in order to show in the display windows the descending figures of the printed drums.

b) entering the first number :

"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."

"For the third digit who is 0, I say 0 and 9 that I enter. I put the stylus at the 9 and I turn the wheel until the needle stops me. The 0 appears in the display window."

"Now I must go to the 'sols' where the digit 9 is required. Instead of 9, I take the complement to 19 who is the largest figure of this wheel, that is to say I put the stylus at 10 and I turn the wheel until the needle stops me. The digit 9 appears in the display window."

"I do the same on the wheel of the 'deniers'. Because the largest number of this wheel is 11, I consider the complement to 11, that is to say the figure who being added to the one I want to display gives 11. Since in the example I wish to display 7 "deniers", I don't put the stylus at 7 but at 4 because 7 added to 4 gives 11. And after turning the wheel until the needle stops me, 7 appears in the display window."

Remarks :

1)There is a mistake in the manuscript. In the explanation, the digit 9 (of the "sols") is replaced by the digit 13. At the end of the explanation it takes again its true value 9 which gives the correct result of the subtraction.

2) It is surprising that a succesion of mental calculations (to find the complement to 9, 19 and 11) is demanded to the operator in order to enter the first number. Indeed, the using of the marked spokes simplifies the work as well as for the zeroing of the pascaline as for the entering of the first number of a subtraction. One can see on the Bellair drawing that the only thing to do is to put the stylus in between the marked spokes and to bring them in regard of the digit we want to display. The following drawing shows that this method works also for an accountant pascaline.

c) subtraction :

At this time, we have to enter the number we want to substract (as we do for an addition). 890 £ 9 s 7 d which is the result is then displayed in the windows.

pascaline division 1

Subtraction on a decimal pascaline replica.

The slide being at the subtract position, “46” is written while placing the marked rays facing the figures of the number.

Then the number to subtract is entered (in the same manner as for an addition). The result is dispayed in the windows.

The manuscript points out an avantage of the pascaline :

"[...] In this way the machine shortens much more than if we work with a pencil. Because in order to accomplish a subtraction in which several numbers are to substract of one number, we must make two operations : one addition of the numbers to substract in order to reduce them to one unique number [...] finaly we must substract this number from the first one. Instead that, on the pascaline, I find with identical operations the remainder after the subtraction of several numbers from one other without doing other thing than entering all the numbers one after each other. "

Another superiority of the calculator is noticed in the manuscript. In case of a schedule between partners, the successive entering of the numbers to substract allows to note intermediates results without any other manipulation, on the contrary without the machine one has to do as much subtractions that the number of required intermediate results.

From my point of view, especially in a time where atithmetics computations was the matter of specialists, the risk of mistakes was less important when the pascaline was used for calculations (particularly in non decimal systems such as monetary base) either for addition or for subtractions. This could have made it possible to entrust calculation work to someone trained to enter numbers in the pascaline, but not overcoming the arithmetic rules.