These cards, however, have nothing to do with the regulation of the particular *numerical* data. They merely determine the *operations*^{[1]} to be effected, which operations may of course be performed on an infinite variety of particular numerical values, and do not bring out any definite numerical results unless the numerical data of the problem have been impressed on the requisite portions of the train of mechanism. In the above example, the first essential step towards an arithmetical result, would be the substitution of specific numbers for , and for the other primitive quantities which enter into the function.

Again, let us suppose that for we put two complete equations of the fourth degree between and . We must then express on the cards the law of elimination for such equations. The engine would follow out those laws, and would ultimately give the equation of one variable which results from such elimination. Various *modes* of elimination might be selected; and of course the cards must be made out accordingly. The following is one mode that might be adopted. The engine is able to multiply together any two functions of the form

.

This granted, the two equations may be arranged according to the powers of , and the coefficients of the powers of may be arranged according to powers of . The elimination of will result from the successive multiplications and subtractions of several such functions. In this, and in all other instances, as was explained above, the particular *numerical* data and the *numerical* results are determined by means and by portions of the mechanism which act quite independently of those that regulate the *operations*.

In studying the action of the Analytical Engine, we find that the peculiar and independent nature of the considerations which in all mathematical analysis belong to *operations*, as distinguished from *the objects operated upon* and from the *results* of the operations performed upon those objects, is very strikingly defined and separated.

It is well to draw attention to this point, not only because its full appreciation is essential to the attainment of any very just and adequate general comprehension of the powers and mode of action of the Analytical Engine, but also because it is one which is perhaps too little kept in view in the study of mathematical science in general. It is, however, impossible to confound it with other considerations, either when we trace the manner in which that engine attains its results, or when we prepare the data for its attainment of those results. It were much to be desired, that when mathematical processes pass through the human brain instead of through the medium of inanimate mechanism, it were equally a necessity of things that the reasonings connected with *operations* should hold the same just place as a clear and well-defined branch of the subject of analysis, a fundamental but yet independent

- ↑ We do not mean to imply that the
*only*use made of the Jacquard cards is that of regulating the algebraical*operations*. But we mean to explain that*those*cards and portions of mechanism which regulate these*operations*, are wholly independent of those which are used for other purposes. M. Menabrea explains that there are*three*classes of cards used in the engine for three distinct sets of objects, viz.*Cards of the Operations*,*Cards of the Variables*, and certain*Cards of Numbers*. (See pages 678 and 687.)