AND ON NEWTON'S METHOD OF CO-ORDINATED EXPONENTS. 



619 



The two curves <p (.r, y) = 0, (p (a? -1 , y' 1 ) = 0, the latter equation being cleared of 

 fractions, have the final branches of either connected with the initial branches of the other, 

 in a manner which the preceding theorem readily points out. This connexion, when its study 

 shall be called for, will elucidate a point to which little attention has been given, the relation 

 of to . y/ - 1, and of co to co y'- I. If a ■» give y = + . y/- 1, we admit the 

 origin as an isolated point. But if w = co give y = co + co -y/- J, we do not admit a real 

 branch at an infinite distance: when an imaginary branch terminates in an isolated point at 

 the origin, we admit the point ; but in the reciprocal curve we do not admit that imaginary 

 branches begin to be real at as =co . For this I hold we have full reason in the true meaning 

 of co , as used, which is the reciprocal of an infinitely small quantity, and not of 0. The 

 infinite among infinites which is properly represented by i is not yet a recognised notion, as 

 distinguished from any one of the successive status through which it is approached. 



This simple, powerful, and fundamental theorem seems hitherto to have been eliminated 

 between the theory of equations and the differential calculus: it belongs to the first, but 

 elementary writers, without whose help no theorem can obtain due notoriety, have referred the 

 points in which it is most useful to the second. That it has fallen quite out of notice might 

 be shewn* in many ways. But, remarkable as this may be, it is still more remarkable that 

 Newton's parallelogram, or method of co-ordinated exponents, as I shall call it, has still more 

 completely fallen into oblivion. This method has not dropped because it was contained in a 

 part of Newton's writings little known or seldom cited ; for it occupies a prominent place in the 

 second epistle to Oldenburgh. Any one who has turned f over the pages of that memorable 

 letter, while passing from one of the anagrams to the other, must have been struck by a 



* For instance, writers developing y in descending powers 

 ofx, from (p(x,y) = 0, an algebraical equation, almost univer- 

 sally obtain only the developments which give the rectilinear 

 asymptotes of the curve. M. Serret himself has got no further 

 in his first edition : and he is a writer who shews abundant 

 acquaintance with his predecessors. Again, D. F. Gregory, a 

 real reader of the older works, and especially of Lagrange's 

 Berlin memoirs, had no knowledge of this theorem, though 

 well prepared to see its use, if it had ever caught his eye : for 

 he abandons the differential formula;, when the point is more 

 than double, as "too complicated to be of much use." But 

 the following is still more remarkable. Gregory prefers Stir- 

 ling's method of finding asymptotes by expanding y in descend- 

 ing powers of x, and cites the Linece, &c. p. 48. The very 

 first example which Stirling gives obliges him to refer back to 

 p. 18, in which the matter of the reference is a use of co- 

 ordinated exponents. But Gregory makes no allusion to any 

 method of expansion as found in Stirling, though he ap- 

 pears, in this and other places, to be well acquainted with the 

 Linete, &c. This is the only proof I have found, in no small 

 amount of search, that any writer of our own day has, while 

 thinking of curves, cast his eye upon any case of co-ordinated 

 exponents. 



+ Hardly anything has been done with these letters for 

 more than a century, except as to the passages which bear on 

 the dispute about fluxions. Montucla, in his rapid enumera- 

 tion of the contents of the second letter, includes " l'usage 

 de son parallelogramme pour la resolution des equations." 



Newton's parallelogram has been mentioned by several writers : 

 but I suspect, with Stirling, that the meaning of it has not 

 been understood. Perhaps many have passed it over with the 

 impression that it was some kind of arithmetical triangle 

 which must needs have been superseded by some kind of 

 binomial theorem. But, as it happens, the algebraical substi- 

 tute commenced by several, and completed by Lagrange, has 

 been equally neglected with its predecessor. 



Lagrange himself did not refer to Newton, but to Taylor 

 and Stirling, of whom he says "...la m<Sthode du premier 

 demande une espece de construction ge'omdtrique, et...celle du 

 second depend du paralldlogramme de Newton, et par conse- 

 quent ne peut etre regardee que comme une me'thode mecanique." 

 This is not a correct criticism : the methods of both, though 

 founded on the geometrical idea, are algebraically applied, 

 insomuch that to Lagrange himself was left only the digestion 

 of the theorem which both Taylor and Stirling had used. I 

 fully admit that the extrication of a theorem which has only 

 existed in combination with other things is a great step, and 

 one for which we have sometimes long to wait : but due ac- 

 knowledgment must be made to those who furnished the 

 combined material. Stirling (p. 22) does with n — 2, re + 1, and 

 1, and Taylor (p. 32) does with 3- + 1, 23+£, &c. precisely 

 what Lagrange does with m +na, m'+n'a, &c. To Lagrange 

 belongs that canonical exhibition, the want of which would 

 have prevented Taylor or Stirling from treating a very com- 

 plicated example without reference to Newton's parallelogram 

 for mechanical help. 



