Since the cards do nothing but indicate in what manner and on what columns the machine shall act, it is clear that we must still, in every particular case, introduce the numerical data for the calculation. Thus, in the example we have selected, we must previously inscribe the numerical values of , , , , , , in the order and on the columns indicated, after which the machine when put in action will give the value of the unknown quantity for this particular case. To obtain the value of another series of operations analogous to the preceding must be performed. But we see that they will be only four in number, since the denominator of the expression for , excepting the sign, is the same as that for and equal to . In the preceding table it will be remarked that the column for operations indicates four successive multiplications, two subtractions, and one division. Therefore, if [Pg 15] desired, we need only use three operation cards; to manage which, it is sufficient to introduce into the machine an apparatus which shall, after the first multiplication, for instance, retain the card which relates to this operation, and not allow it to advance so as to be replaced by another one, until after this same operation shall have been four times repeated. In the preceding example we have seen, that to find the value of we must begin by writing the coefficients , , , , , upon eight columns, thus repeating and twice. According to the same method, if it were required to calculate likewise, these coefficients must be written on twelve different columns. But it is possible to simplify this process, and thus to diminish the chances of errors, which chances are greater, the larger the number of the quantities that have to be inscribed previous to setting the machine in action. To understand this simplification, we must remember that every number written on a column must, in order to be arithmetically combined with another number, be effaced from the column on which it is, and transferred to the mill. Thus, in the example we have discussed, we will take the two coefficients and , which are each of them to enter into two different products, that is into and , into and . These coefficients will be inscribed on the columns and . If we commence the series of operations by the product of into , these numbers will be effaced from the columns and , that they may be transferred to the mill, which will multiply them into each other, and will then command the machine to represent the result, say on the column . But as these numbers are each to be used again in another operation, they must again be inscribed somewhere; therefore, while the mill is working out their product, the machine will inscribe them anew on any two columns that may be indicated to it through the cards; and, as in the actual case, there is no reason why they should not resume their former places, we will suppose them again inscribed on and ,