Page:Scientific Memoirs, Vol. 3 (1843).djvu/730

applied to the engine) that a certain prism which is a part of the mechanism (see Note C), turns a new face, and thus presents a new card to act on the bundles of levers of the engine; the new card being perforated with holes, which are arranged according to the peculiarities of the operation of addition, or of multiplication, &c. Again, the numbers in the preceding formula (8.), each of them really represents one of these very pieces of card that are hung over the prism.

Now in the use made in the formulæ (7.), (8.) and (10.), of the notation of the integral calculus, we have glimpses of a similar new application of the language of the higher mathematics. $\scriptstyle {\Sigma }$, in reality, here indicates that when a certain number of cards have acted in succession, the prism over which they revolve must rotate backwards, so as to bring those cards into their former position; and the limits 1 to $\scriptstyle {n}$, 1 to $\scriptstyle {p}$, &c., regulate how often this backward rotation is to be repeated.A. A. L.

 

Note F.—Page 688.

There is in existence a beautiful woven portrait of Jacquard, in the fabrication of which 24,000 cards were required.

The power of repeating the cards, alluded to by M. Menabrea in page 680, and more fully explained in Note C, reduces to an immense extent the number of cards required. It is obvious that this mechanical improvement is especially applicable wherever cycles occur in the mathematical operations, and that, in preparing data for calculations by the engine, it is desirable to arrange the order and combination of the processes with a view to obtain them as much as possible symmetrically and in cycles, in order that the mechanical advantages of the backing system may be applied to the utmost. It is here interesting to observe the manner in which the value of an analytical resource is met and enhanced by an ingenious mechanical contrivance. We see in it an instance of one of those mutual adjustments between the purely mathematical and the mechanical departments, mentioned in Note A. as being a main and essential condition of success in the invention of a calculating engine. The nature of the resources afforded by such adjustments would be of two principal kinds. In some cases, a difficulty (perhaps in itself insurmountable) in the one department, would be overcome by facilities in the other; and sometimes (as in the present case) a strong point in the one, would be rendered still stronger and more available, by combination with a corresponding strong point in the other.

As a mere example of the degree to which the combined systems of cycles and of backing can diminish the number of cards requisite, we shall choose a case which places it in strong evidence, and which has likewise the advantage of being a perfectly different kind of problem from those that are mentioned in any of the other Notes. Suppose it be required to eliminate nine variables from ten simple equations of the form—

$\scriptstyle {ax_{0~}+bx_{1~}+cx_{2~}+dx_{3~}+.\ldots .=p}$	(1.)
$\scriptstyle {a^{1}x_{0}+b^{1}x_{1}+c^{1}x_{2}+d^{1}x_{3}+.\ldots .=p^{1}}$	(2.)
&c.&c.&c.&c.

We should explain, before proceeding, that it is not our object to con-