Page:Scientific Memoirs, Vol. 3 (1843).djvu/723

Rh separate. The engine must in such a case appropriate as many columns to results as there are terms to compute.

Thirdly. It may be desired to compute the numerical value of various subdivisions of each term, and to keep all these results separate. It may be required, for instance, to compute each coefficient separately from its variable, in which particular case the engine must appropriate two result-columns to every term that contains both a variable and coefficient.

There are many ways in which it may be desired in special cases to distribute and keep separate the numerical values of different parts of an algebraical formula; and the power of effecting such distributions to any extent is essential to the algebraical character of the Analytical Engine. Many persons who are not conversant with mathematical studies, imagine that because the business of the engine is to give its results in numerical notation, the nature of its processes must consequently be arithmetical and numerical, rather than algebraical and analytical. This is an error. The engine can arrange and combine its numerical quantities exactly as if they were letters or any other general symbols; and in fact it might bring out its results in algebraical notation, were provisions made accordingly. It might develope three sets of results simultaneously, viz. symbolic results (as already alluded to in Notes A. and B.); numerical results (its chief and primary object); and algebraical results in literal notation. This latter however has not been deemed a necessary or desirable addition to its powers, partly because the necessary arrangements for effecting it would increase the complexity and extent of the mechanism to a degree that would not be commensurate with the advantages, where the main object of the invention is to translate into numerical language general formulæ of analysis already known to us, or whose laws of formation are known to us. But it would be a mistake to suppose that because its results are given in the notation of a more restricted science, its processes are therefore restricted to those of that science. The object of the engine is in fact to give the utmost practical efficiency to the resources of numerical interpretations of the higher science of analysis, while it uses the processes and combinations of this latter.

To return to the trigonometrical series. We shall only consider the four first terms of the factor ($\scriptstyle{A+A_1\cos\theta+}$&c., since this will be sufficient to show the method. We propose to obtain separately the numerical value of each coefficient $\scriptstyle{C_0}$, $\scriptstyle{C_1}$, &amp;c. of (1.). The direct multiplication of the two factors gives {{centre|$$\scriptstyle{\left.\begin{align}\scriptstyle{BA}&\scriptstyle{+BA_1\cos\theta}&\scriptstyle{+BA_2\qquad\cos2\theta}&\scriptstyle{+BA_3\qquad\cos3\theta}&\scriptstyle{+~.~.~.~.~.~.~.~.~.~.}\\&\scriptstyle{+B_1A\cos\theta}&\scriptstyle{+B_1A_1\cos\theta.\cos\theta}&\scriptstyle{+B_1A_2\cos2\theta.\cos2\theta}&\scriptstyle{+B_1A_3\cos3\theta.\cos\theta}\end{align}\right\}}$$(2.)}} a result which would stand thus on the engine:—