The process however which was actually followed was this : forms similar to (A) having been observed, in which, whatever might be thought of them as they stood, were found ready means of returning to well-known truths, it was natural to ask whether an application of algebra to the form (A), producing of course a transformation of both sides, would lead to a result from which, by the same method of returning, another known truth might be produced. For example, assume that n and A are to be treated as quantities : then A= es —1 gives 1 + A= sy, log. (1+ A)= D, or 1 1 D A — + A• — Now restore ler to every term, and let n and A reassume their opera tive meanings, so that nox is the differential coefficient, and a0x, &c., are the successive DIFFERENCES of 0x, x being changed into x + 1 at each step. We have then 1 1 D = A ithr — (P.0 + 5 (px — . .
a theorem which must be true if the preceding method be legitimate. This theorem is found to be true by other and certain modes of de monstration : consequently the legitimacy of the preceding method has some presumption in its favour, arising from its leading to an otherwise known truth.
In this way many theorems were investigated, upon the following plan of proceeding :-1. Throw away symbols of quantity from a known theorem, and proceed in any manner which may appear eligible with the symbols of operation, treating them as if they were quan tities. 2. When a result has been obtained, restore the symbols of quantity to their old places, and those of operation to their old mean ings. 3. The result as thus viewed has all the presumption in its favour which arises from its being the legitimate consequence of a method which, whether accurate or not, has never been found to lead to anything but what could otherwise be satisfactorily shown to be true. And though Lagrange himself, Arbogast, the English translators of Lacroix, Brinkley, &c., may have used language in reference to this method which would seem to imply that they considered it as one of demonstration, yet it is obvious, from the care taken by them to have abundant external verification in every case, that their results were considered by themselves as resting on such a presumption as that above noted ; though it is also evident that they considered the pre sumption as amounting to moral certainty, which indeed they were justified in doing.
A student who reads on this subject for the first time will be apt to let his ideas run farther than they should, and to imagine that this treatment of operations may be made universal. For instance, if 0x= s and tpx= xi, and if 0+4' be taken as representing + X2, he might suppose that 0+ 40 performed twice, or (xl + + + represented by should be the same thing as or + + (xs) .
This however will be found not to bo the case, and thus it appears that a line is to be drawn, distinguishing operations which may be used independently of quantities from those which may not. Until this line can be properly drawn, nothing like demonstration of the method, when true, can be given ; or rather perhaps the converse, that is to say, a method of demonstration of such cases as give truths will draw the line which separates these from the rest. We proceed to give some account of this method of demonstration.
We do not know how far those who used the separation of the symbols of operation and quantity (as it was called) had before their minds a view that would have made their method intelligible in a rational point of view, which was all it trAthematical exactness. But, looking only at their modes of expression, we atnnot find anything of the kind. Lagrange (' 516m. Acad. Berlin,' 1772) gave only theorems without any mode of deducing them. Arbogast assumes the " Mpa ration des eehelles" without remark. Laplace, by the aid of his theory of generating functions, must be held to have strictly demonstrated some isolated classes of the theorems which this method gives. But nothing like a general account of the reason why this separation of the symbols of operation end quantity leads to truth in certain cases and not in others, ever Appeared, to our knowledge, before the publication of a memoir by 11i. Servois in the 5th volume of the Annales de Mathematiques' (Lacroix, voL iii., p. 726). The author exhibits those properties of the separable operations on which the legitimacy of the separation depends ; and making a separate species of calculus of functions out of those properties, fully succeeds in demonstrating that differences, differentiations, and multiplications by any factors which are independent of the variables, may be used as if their symbols of operation were common algebraical quantities.
The last step towards the full conception of a calculus of operations was virtually made by Dr. Peacock, in his Algebra ' (first edition, 1830); for though this work does not mention the subject, yet it is the first which lays down the principles on which the theory of separation is neither a resemblance of algebra, nor a calculus of functions founded on algebra, but an algebra, or if the reader pleases, algebra itself ; so that its conclusions rest upon the same foundation as those of ordinary algebra.