622 



Mr DE MORGAN, ON THE SINGULAR POINTS OF CURVES, 



great peculiarity. He himself feels that it is his main point. I almost suspect* him to have 

 imagined that his analytical triangle and ruler have a power to which algebraical process 

 cannot supply an equivalent : I can in no other way explain the question asked in the passage 

 cited below, as coming from an algebraist accustomed to the ignotum per ignotum. 



e-i 



In the diagram, the points PQRSTUVWX are the exponent points in the equation 

 previously discussed : the essential points are PQSUVWX. In the vertical side PQ, P counts 

 on the lower contour, and Q on the higher. 



Given the lower or higher solutions of an equation <p (x, y) = 0, as to the character of 

 their leading terms ; that is, given r in y =x r (u + U), where U vanishes with x, or with x~ l ; 

 or given both ; — required all conditions of compatibility, and the essential terms of the 

 equation. A side of the polygon drawn from (n, m) to (n, m) denotes solutions of the 

 degree (m' -»»):(»- »'), higher or lower, according as the side is on the higher or lower 

 contour : this may be denoted by — Am : Aw. Where integers are in question, the number 

 of solutions is n—ri or ri — n, whichever is positive. Hence sides which when produced, 

 would cut a right-angled triangle out of the region in which m and n are positive, denote 

 solutions of positive degree ; other sides, negative. The degree must be given so that the 

 denominator expresses the number of roots wanted : thus 12 roots of the degree -J must be 

 demanded as of the degree -f-^. 



This being premised, let it be required that when x is infinitely small, we shall have 2, 4, 

 2, 1, 2, 1 branches of the dimensions 1, ^, 0, — 1, — |, — 2. Write the dimensions down in 

 descending order of magnitude, with the proper denominators : 



2 20-1 -3-2 

 2 4 2 1 2 1 



the varieties of ory*. I suspect that this conversion of the 

 plane of co-ordinates into a sort of abacus was one great cause 

 of the gradual dismissal of the method. The last notice that 

 I can find of it is in the Encyclopedic Mithodique, in which 

 D'Alembert, who is of all writers the one to be relied upon for 

 separating the kernel from the shell, requires the term x m y" to 

 be imagined at the centre of each square, and preserves the 

 squares in which the terms are supposed to be written. Under 

 such an arrangement, it would of course not be very easy to say, 



in all cases, whether the ruler would pass on one side or the 

 other of the centre of a square. 



* Quel moyen y aura-t-il done pour discemer les plus 

 grands termes d'une Equation, puisqu'il semble qu'on ne les 

 peut reconnoitre sans savoirde quel Ordreest^ par rapport a jr, 

 et qu'on nepeut de"couvrir ce rapport dcya i sans avoir sdpare' 

 des autres les plus grands termes de lVquation ? Quel fil nous 

 conduira dans ce Labyrinthe ? (p 152). 



