Page:Scientific Memoirs, Vol. 3 (1843).djvu/720

710 of quantities, is brought into use during a calculation. We easily ascertain this, from the inspection of any vertical column or columns of the diagram in which that quantity may appear. Thus, in the present case, we see that all the data, and all the intermediate results likewise, are used twice, excepting $\scriptstyle{(mn'-m'n)}$, which is used three times.

The order in which it is possible to perform the operations for the present example, enables us to effect all the eleven operations of which it consists, with only three Operation-cards; because the problem is of such a nature that it admits of each class of operations being performed in a group together; all the multiplications one after another, all the subtractions one after another, &amp;c. The operations are $\scriptstyle{\left\{6(\times),\;3(-),\; 2(\div)\right\}}$.|undefined

Since the very definition of an operation implies that there must be two numbers to act upon, there are of course two Supplying Variable-cards necessarily brought into action for every operation, in order to furnish the two proper numbers. (See Note B.) Also, since every operation must produce a result, which must be placed somewhere, each operation entails the action of a Receiving Variable-card, to indicate the proper locality for the result. Therefore, at least three times as many Variable-cards as there are operations (not Operation-cards, for these, as we have just seen, are by no means always as numerous as the operations) are brought into use in every calculation. Indeed, under certain contingencies, a still larger proportion is requisite; such, for example, would probably be the case when the same result has to appear on more than one Variable simultaneously (which is not unfrequently a provision necessary for subsequent purposes in a calculation), and in some other cases which we shall not here specify. We see therefore that a great disproportion exists between the amount of Variable and of Operation-cards requisite for the working of even the simplest calculation.

All calculations do not admit, like this one, of the operations of the same nature being performed in groups together. Probably very few do so without exceptions occurring in one or other stage of the progress; and some would not admit it at all. The order in which the operations shall be performed in every particular case, is a very interesting and curious question, on which our space does not permit us fully to enter. In almost every computation a great variety of arrangements for the succession of the processes is possible, and various considerations must influence the selection amongst them for the purposes of a Calculating Engine. One essential object is to choose that arrangement which shall tend to reduce to a minimum the time necessary for completing the calculation.

It must be evident how multifarious and how mutually complicated are the considerations which the workings of such an engine involve. There are frequently several distinct sets of effects going on simultaneously; all in a manner independent of each other, and yet to a greater or less degree exercising a mutual influence. To adjust each to every other, and indeed even to perceive and trace them out with perfect correctness and success, entails difficulties whose nature partakes to a certain extent of those involved in every question where conditions are very numerous and inter-complicated; such as for instance the estimation of the mutual relations amongst statistical phænomena, and of those involved in many other classes of facts.