692 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR 
These cards, however, have nothing to do with the regulation of the 
particular numerical data. They merely determine the operations* to 
be effected, which operations may of course be performed on an infi- 
nite 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 x, and for 
the other primitive quantities which enter into the function. 
Again, let us suppose that for F we put two complete equations of 
the fourth degree between # andy. 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 accord- 
ingly. The following is one mode that might be adopted. The engine 
is able to multiply together any two functions of the form 
a+ba+e2u?+.,...pa". 
This granted, the two equations may be arranged according to the 
powers of y, and the coefficients of the powers of y may be arranged 
according to powers of 2. The elimination of y will result from the 
successive multiplications and subtractions of several such functions. 
In this, and in all other instances, as was explained above, the parti- 
cular numerical data and the numerical results are determined by 
means and by portions of the mechanism which act quite indepen- 
dently 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 appre- 
ciation is essential to the attainment of any very just and adequate gene- 
ral 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 ope- 
rations 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 di- 
stinct sets of objects, viz. Cards of the Operations, Cards of the Variabies, and 
certain Cards of Numbers. (See pages 678 and 687.) ; 
