XXVIII. On the Singular Points of Curves, and on Newton's Method of Co- 

 ordinated Exponents. By Augustus De Morgan, of Trinity College, Vice- 

 President of the Royal Astronomical Society, and Professor of Mathematics 

 in University College, London. 



[Read May 21, 1855.] 



Many of the critical points of Algebra have first been brought to notice in the treatment 

 of curves : and all have been usefully illustrated. The subject also has its controversies. 

 Some of these might have been avoided, if writers had distinctly stated what they meant by a 

 curve, and by a singular point. In algebraic Geometry, or geometry investigated algebraically, 

 a curve should be either the intersection of two surfaces, or a continuous line described by a 

 point moving under one permanent law. But in geometrical Algebra, or algebra represented 

 in space, a curve ought to be defined as the collection of all points whatsoever, and howsoever 

 connected, whose co-ordinates satisfy a given equation. The present paper is on algebra, and 

 on real values only, so far as interpretation is concerned. 



According to Cramer, singular points are those " qui ont quelque chose qui les distingue 

 des autres." This is somewhat too general : the point at which x = 3, y = 4^, has something 

 which distinguishes it from all others ; namely, that x = 3, y = 4j. Dr Peacock, in the heading 

 of a chapter, speaks of "singular or remarkable" points, in which the disjunctive particle 

 appears to have a defining force. According to M. Cauchy, singular points are those which 

 present some remarkable peculiarity inherent in the curve, and independent of the position of 

 the axes. I should prefer to divide singular points with reference to position and to curvature. 

 A point of singular position is one which has a notable property, such as no continuous arc of 

 the curve *, however small, can have at all its points, and which has reference to the axes : 

 a point of singular curvature has such a property independently of all reference to the axes. 

 But since the point is especially taken into consideration as a point in a curve, we must under- 

 stand that reference to adjacent points is intended in both cases : that is, the singular property 

 must either involve some use of differential coefficients, or some other reference to adjacent 

 points. In truth, we are rather concerned with singular elements, infinitely small arcs, than 

 with singular points. 



* Of the curve : thus y'= gives a point of singular position 

 on all curves except y = const. I cannot find a definition free 

 from the idea of notability, which contains matter of opinion, 

 varying according to circumstances. When we think of a curve 

 as the tabulation of y in terms of x, maxima, contrary flexures, 

 and points of regression, are single and notable. But let a 



curve be used to supply directing data for the formation of 

 other curves, and a point of the most ordinary character may 

 be perfectly single, and especially notable. Use a curve y = <j>x 

 for the determination of <jxpx, rfxfxpx, &c, and the point at 

 which y = x is the most notable of any. 



