620 



Ma DE MORGAN, ON THE SINGULAR POINTS OF CURVES, 



couple of diagrams, side by side, the one containing terms of the form x m y n arranged in 

 squares placed in rank and file, the other exhibiting stars which, had there been but one more, 

 might have passed for a representation of the Great Bear, very badly figured. Had this 

 epistle ever been read by any one of our time whose mind was not full of the controversy on 

 fluxions, the method which I shall proceed to develope would have been seen, briefly but 

 clearly indicated in its leading step. The cognate application to expansion from differential 

 equations appears in those letters to Wallis of 1692, which Wallis incorporated in his second 

 volume of works : and also, but much less clearly, in the Quadratura Curvarum. Stirling, in 

 his Linece Tertii Ordinis, &c. (1617) has given a good and full account of the method, "qua?, 

 quamvis a plurimis qui earn haud intellexerunt, ut Methodus Mechanica diu neglecta jacuit, 

 est omnium quam quis excogitare potest generalissima et elegantissima." In this opinion I 

 fully coincide, even up to potest. Taylor, in his Methodus Incrementorum (1615) had given 

 a very sufficient account of the application to differential equations. De Gua (as I learn from 

 Cramer and Lacroix) adopted this method in his Usage de V Analyse de Descartes. John 

 Stewart explained it fully* in his commentary on the Analysis per JEquationes, &c. And 

 lastly, Cramer has made it the leading method of his well-known work on curve lines, in 

 which those who have searched for examples or classifications have probably passed it over as 

 some fanciful form of a tentative method. But all, from Newton to Cramer, have rather 

 dwelt upon the meaning of the separate sides of the polygon than taken the polygon, as a 

 whole, in its relation to the equation, as the containing form of all its solutions : to this point 

 Stewart comes far the nearest. I cannot doubt that both Newton's method, and Lagrange's 

 arithmetical translation of it, will henceforward find that place in elementary systems which 

 they ought long ago to have occupied. 



The method of co-ordinated exponents connects every equation of the form ~2Ax m y n = 0, or 

 which takes that form, quam proxime, when w is infinitely small, or infinitely great, with 

 a convex rectilinear polygon, from one part of the contour of which we read the ultimate 

 relations between y and x when x is infinitely small, and from the other part the ultimate 

 relations when x is infinitely great. 



Let the exponent of y in any one term be made an abscissa, and that of x an ordinate, so 

 that every possible case of x m y" has what we may call its exponent point (w, m). When none 

 but integer exponents occur, the exponent points will all be found at the corners of squares of 

 integer sides. Lay down all the exponent points belonging to the terms of an equation, and 



" 'Sir Isaac Newton's two Treatises.. .explained.. ..By John 

 Stewart, A.M. Professor of Mathematicks in the Marischal 

 College and University of Aberdeen.' London, 1745, 4to. 

 This was an invasion from Scotland of a very serious character : 

 it contains the Quadratura Curvarum and Analysis per Mqua- 

 liones, &c. in English ; 55 pages of Newton, and 424 pages of 

 explanation. It was published by a ' Society for the encourage- 

 ment of Learning' (on which see Nichols's Anecdotes, Vol. 11. 

 p. 90), to which we owe both Tanner's Nolitia Monastica and 

 his Bibliotheca. In the Analysis, &c. Newton shews a decided 

 attempt at what he afterwards succeeded in by his parallelo- 

 gram: of which attempt Stewart takes advantage to introduce 

 a very full account of the subsequent methods. He has given 



the theorem, which I call Lagrange's, so nearly, that nothing 

 but a repetition is wanted : he obtains only the first of the 

 maxima from which dimensions are derived. For the lower 

 dimension, the equation being 3C + 'S,x r y' = 0, he directs to 

 "run over the several terms, and observe where the value of 



<^is greatest,' but he lays it down that on,y one dimension 



of solution of each kind can be found in this way ; and when 

 all are wanted, he has recourse to the polygon (pp. 428-436), 

 under express declaration of the necessity of such a step. And 

 from this, looking at the fact that Stewart was well acquainted 

 with the earlier Newtonian school, I derive my final assurance 

 that the complete theorem belongs to Lagrange. 



