298 NOTES. 



The curve considered at length in Tract V. is another 

 instance of the failure of the XXVIIIth Lemma; for that 

 line continues through the cusps and returns into itself, though 

 not, strictly speaking, an oval. The cardioide also. 



The demonstration given, instead of Sir I. Newton's, in 

 Tract I., is not exposed to the other objections which have 

 been made to the Newtonian demonstration ; but it is equally 

 liable to the objection now urged from the consideration of the 

 equation to the class of curves whereof the lemniscata is one, 

 and from the case of the curve described in Tract V., where the 

 rectification is possible, as well as the quadrature. Perhaps we 

 should extend the exception in the Lemma to curves which 

 consist of two or more ovals touching each other, and to curves 

 having cusps though without any infinite branches. 



NOTE II., pp. 23, 74. 



In the Encyc. xiii. p. 126, D'Alembert states Forism to be 

 synonymous with Lemma in the ancient writers, but he adds 

 that lemma is the only word used in modern times. His 

 definition is not inaccurate as applied to lemma, a proposition 

 of which we have need in order to pass to another more im- 

 portant ; and on this he grounds his notion of porism, from 

 Tropof, passage. Under the word Poristique in another part 

 of the Encyc., he gives a different definition of porism. Some 

 authors, he says, call by this name the description " de la 

 " maniere de determiner par quels moyens, et de combien, de 

 " differentes faons un probleme peut 6tre resolu." 



Nothing can be deduced from the Greek for passage, because 

 the word Tropur/ia is plainly not derived from Tropoe, but from 

 7roptw, which rather sanctions the opinion connecting porism 

 with corollary, than the opinion .in the text of a transition 

 from determinate to indeterminate. That the ancients some- 

 times used the word as synonymous with corollary there can 

 be no doubt. 



The subject has been handled incidentally by one of the 

 most eminent geometricians of our day, M. Chasles, with his 



