41 



OPERATION. 



OPERATION. 



may represent the result of so doing : similarly 3 may be the direction 

 to proceed as in I + 1 + 1, and 3 1 its result. And 3x2 would be a 

 direction to perform 3 upon the result of 2, or to take 2l + 2i + 2l, or 

 (i + i) + (i + i) + (l + i), or 6L If then we say 3x (2i) = 6i, we have an 

 equation between magnitudes ; but if we throw away I, as we just now 

 did Q.r, we have 3 x 2 = 6, an expression of equivalence of operations. 

 Now it might very reasonably have been asked whether these operations 

 must be the only ones which will admit of being treated by themselves 

 and viewed independently of the subjects of operation ; and a direct 

 assumption of such modes of notation as that marked (A), even when 

 A and D were considered independently, though it might not have 

 been fully explicable, would have appeared to be nothing but a natural 

 extension of views which had already been taken to a limited extent. 



The process however which was actually followed was this : forma 

 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 D and A are t<> be treated as quantities : then A=c" 1 

 gives 1 + A = f D , log. (l-rA) = D, or 



Now restore <f>x to every term, and let D and A reassume their opera- 

 tive meanings, so that D<p.r is the differential coefficient, and A<f>.<-, 

 A-<px, &c., are the successive DIFFERENCES of <fae, x being changed into 

 x + 1 at each step. We have then 



A' <t>x- ... 



a theorem which must be true if the preceding method be legitimate. 

 This theorem ia 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 .1 

 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 thuir 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 KnglUh translators 

 of Lacroix, liriukley, &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 <f>.c= 

 x 1 and <fix=jp, and if Q + ty be taken as representing a? + x 1 , he might 

 suppose that ^ + if* performed twice, or 



represented by (<p + 1^)', should be the same thing as 

 , or (**) + 2(i 



This however will be found not to be 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 

 of this method of demonstration. 



We do not know how far those who vised the neparativn of the lymbol* 

 i if 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 wanted of nftthetn.itic.tl exactness. But, 

 looking only at their modes cf expression, we cannot find anything of 

 the kind. Lagrange ('Mem. A cad. Berlin,' 1772) gave only theorems 

 without any mode of deducing them. Arbogast assumes the " sepa- 

 ration des cchelles " 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 

 ymbols of operation and quantity leads to truth in certain cases and 

 not in other", ever appeared, to our knowledge, before the publication 



f a memoir by M. Servois in the 5th volume of the ' Annales de 

 Math^matiques ' (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 hia ' 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. 



We have [ALGEBRA] pointed out what is meant by symbolical 

 algebra, as distinguished from explained or interpreted algebra. Granting 

 a certain number of fundamental relations, which are to be true, the 

 logical consequences of combining those relations must be true also : 

 thus, if it be universally true that a + b =b + a, and that xy=yx, it 

 follows, even before a, b, + , x, xy, &c., have any meaning assigned, 

 that (o + 6)z=j(a + 6)=2(A + a). If, as in the article cited, we select all 

 the primary relations on which algebraical transformations depend, and 

 then bear in mind that the truth of all then- consequences depends on 

 the truth of those relations only, not on the relations being true for 

 one meaning or another meaning of the symbols, but on the truth only 

 of the relations, come how it may we shall then see that all formula; 

 of algebra may be used as expressions of truths, whatever may be the 

 meaning of the symbols employed, provided only that, such meanings 

 being given, the fundamental relations are true. We have already 

 seen that this may be carried the length of extending the meanings of 

 all the symbols of algebra, in such manner that a science is created 

 with definitions wide enough to include among its rational objects not 

 only the negative quantity, but also its square root. This was extension 

 only ; we shall now show a process which, though it be still extension, 

 is of another character. 



In our present inquiry, we need not trouble ourselves to make any 

 particular consideration of the signs + and. They retain their 

 algebraical meaning, so that whatever A and B may stand for, + ( + A) 

 = + A, ( A) = A, &c. 



If we now ask, what are the fundamental symbolical relations of 

 algebra which remain, after those which depend on + and are taken 

 away, we shall find them to be as follows : 1. The distributive character, 

 as it is called, of the operation ab, with respect to + and , as shown 

 in a(t> + c e) = ab + ac at. 2. The commutative or convertible character 

 of the same operation with respect to others of the same kind and 

 itself, as shown in abc=acb = bca, &c., and ab-ba. 3. The drprarible 

 character of operations of the species a", when performed upon other 



operations of the same kind, as shown in a*a"=o"+", (o m ) = a"". 

 These laws of operation being granted, no matter what the nature of 

 the interpretation under which it is found possible to grant them, all 

 that is necessary to the mechanism of algebra will be found to have 

 been granted. It will be remembered that we speak of l-=-o under the 

 symbol o~'. 



In arithmetic, as already seen, we may, if we please, consider the 

 symbols 2, 3, &e., as indicative of operations performed upon the unit. 

 Let us extend this notion, and, instead of the unit, make <px, any 

 function of a variable x, the subject of operation ; this function being 

 always understood, if not expressed. Thus any symbol E has an 

 operative meaning in itself, but when written in an equation stands 

 for the result obtained by performing that operation upon tpx, which 

 may also be signified by E (^x). Also let E + F and E F stand for the 

 algebraical sum and difference of the results of the operations E and F 

 performed upon ^r. Let us now appropriate E to stand for the simple 

 operation of changing x into x + h, or x + any quantity independent of 

 x in value : and to distinguish the different increments, let E V E t , &c., 

 denote the operations of changing x into x + h, x + k, &c. It is then 

 very easily shown that E possesses the distributive, convertible, and 

 depressible characters essential to its being logically the object of 

 algebraical transformation. Take two functions, <t>x and i^.r, either 

 assumed independently or resulting from preceding operations : it 

 follows then that E* (<px + ifix) is <f> (x + h) + <fy (x + A), which is 

 Z 4 <t>x + E A ifix ; or the distributive character is established. Again, 

 r K 4 (F t <ps) and E t (E 4 <(>x) : first, K k <f>x means < (x + i'), on 

 which perform E t , or change x into x + h, giving <p (.c + h + k) ; 

 next, E, ipx is (x + A), on which E t being performed, gives 

 9 (x + k + h), the same as <f> (x + h + t). Consequently E t E A <f>x 

 = K t E 4 (/>.r, or the convertible character of K is established. Thirdly, 

 consider Ej 'E* , meaning that the operation E 4 having been twice per- 

 formed, E 4 is to be three times performed upon the result : we have 

 evidently <f> (x + 5h), or Ej <f>,t ; and if Ej were to be performed four 

 times running, we should have E J J . Hence the depreasible character of 

 the successive operations is established : and though it be a wide step 

 or the beginner to make, the applicability of all the formula! of 



