308 Cambridge Philosophical Society : — 



April 23. — A paper was read by the Master of Trinity, on Plato's 

 Survey of the Sciences, contained in the seventh book of the Republic. 



Plato, like Francis Bacon, took a review of the sciences of his time ; 

 and like him, complained how little attention was given to the phi- 

 losophy which they involved. The sciences which Plato enumerates 

 are arithmetic and plane geometry, treated as collections of a1)stract 

 and permanent truths; solid geometrj', which he "notes as deficient" 

 in his time, although, in fact, he and his school were in possession 

 of the doctrine of the " five regular solids ; " astronomy, in which he 

 demands a science which should be elevated above the mere know- 

 ledge of phsenomena. I'he visible appearances of the heavens only 

 suggest the problems with which true astronomy deals ; as beautiful 

 geometrical diagrams do not prove, but only suggest geometrical 

 propositions. Finally, Plato notices the subject of harmonics, in 

 which he requires a science which shall deal with truths more exact 

 than the ear can establish, as in astronomy he requires truths more 

 exact than the eye can assure us of. It was remarked also, that 

 such requirements had led to the progress of science in general, and 

 to such inquiries and discoveries as those of Kepler in particular. 



May 21. — A paper was read " On the singular Points of Curves." 

 By Professor De Morgan. 



Mr. De Morgan defines a curve as the collection of all points 

 whose co-ordinates satisfy a given equation ; and contends for this 

 definition as necessary in geometrical algebra, whatever limitation 

 may be imposed in algebraic geometry. He divides singular points 

 into points of singular position and points of singular curvature ; the 

 character of the former depending on the axes, but not that of the 

 latter. Both .species are defined as possessing a notable property, 

 and such as no arc of the curve, however small, can have at all its 

 points. 



The form first considered is that of which the case usually taken 

 is an algebraic curve. Let (j>{x, y) be a function which for all real 

 and finite values of x and y is real, finite, and univocal ; let the curve 

 be ^(j:-, y) =0, considered as an individual of the family (p{x, y) =const. 

 The two curves c?^ : dx=^0, d(p : c?y = 0, or 0^=0, 0^ = 0, are the sub- 

 ordinates of this system, on which the singular points of all depend. 



When <j> is not reducible to another function of the same kind by 

 extraction of a root, it divides the plane of co-ordinates into regions 

 in which, severally, it is always positive or always negative. By 

 this consideration it is easily shown (independently of y', y", &c.), that 

 if (x -}- dx, y + dy) be a point on the tangent at (x, y), (j){x + dx, y -f- dy) 

 has the sign oi ^xxdx"-\-2(i),„jdxdy -'r<Py,,dy'^. Hence, immediately after 

 leaving the curve, (p agrees with or differs from —(p^y" at the point 

 left, according as the curve is left on the convex or the concave side. 

 Hence easily follow the criteria of flexure, and also the following 

 relation between any two points whatsoever of the curve. 



Let two points be called sitnilar when a line drawn from one 

 to the other cuts the curve an even number of times (0 included) 

 with the same abutments (on convexity or on concavity), or an odd 

 number of times with different abutments. Let other points be 



