382 Mr Taylor, Newton s organic description of curves. [May 17, 



" Ce thdoreme, qui est une generalisation de celui de Newton, 

 n'est lui-meme qu'une maniere particuliere, entre une infinite 

 d'autres semblables, pour former les coniques par I'intersection de 

 deux droites mobiles autour de deux pomts fixes, &c. Ainsi le 

 theoreme de Newton, qui a eu quelque celebrite, et qui a paru 

 capital dans la theorie des coniques, ne se trouve plus qu'un cas 

 tres-particulier d'un mode general de description de ces courbes." 



But in the first place the particular "■ generalisation " of 

 Newton's theorem here referred to, and elsewhere cited as Chasles' 

 extension of Newton's theorem, is itself only " un cas tres-parti- 

 culier" of Newton's Gurvarum Bescrijjtio Organica*. And in the 

 next place this extension would have served no useful purpose 

 in the Principia, where the primary theorem is given -f-. Newton 

 is there dealing with the determination of orbits from given con- 

 ditions, and his construction enables him to determine a conic or 

 orbit of which five points are given. Chasles' extension only 

 enables us to draw a conic by supposing a conic to be already 

 drawn. 



2, Newton's Enumeratio linearum Tertii Ordinis, first pub- 

 lished as an appendix to his Opticks (1704), contains a general 

 statement of his Theoremata de Gurvarum descriptione organica 

 (§ XXXI.). In this tract he enunciates the following proposi- 

 tions : — 



If two angles of given magnitudes PAD and PBD turn about 

 A and B as poles given in position, then if the intersection P of 

 one pair of their arms be made to describe a conic, the inter- 

 section I) of the other pair will in general describe a curve of the 

 third ge7ius [or fourth ordei^'] having double points at A and B and 

 at the limiting position of D where the angles BAP and ABP 

 vanish together : but the locus of D will be of the second genus if 

 the angles BAD and ABD vanish together. If P describes a conic 

 passing through A, then D describes a cubic having a double point 

 at A and passing through B. 



The well-known case in which the locus is a conic is again stated, 

 and the cases in which it degenerates into a straight line are pointed 

 out. It is further evident that when P describes a conic through 

 both A and B, the cubic, the locus of D, has two double points 

 and must therefore degenerate. Thus the so-called extension of 

 Newton's organic description is seen to be a special case of it. 



3. A double point A and six other points BGDEFG of a cubic 

 being given, the curve is organically described as follows : Draw 



* The Apergu historique does contain a reference to this (p. 145) but in the 

 above-mentioned Note xv., which must have misled many readers, it is over- 

 looked. 



+ Principia, Lib. i. sect. 5, lemma 21. See also Aritlivietica Universalis, prob. 53 

 (1707). 



