250 Further Remarks on the Theory of parallel Lines. 



magnitude to which it has a relation. The transcendental 



c c 



quantities — and -^- find their way into the equation by 



means of , which is the ratio of two straight lines. All 



the quantities -— - , — , are functions of no dimensions 



derived from the same angle ; and we know that any one of 

 such quantities may be substituted for any other of them, 

 in a proposed function. This proves that Professor Leslie's 

 reasoning fails ; but it does not appear that the fallacy of it 

 could be deduced merely from Legendre's definition, which 

 determines all angles by their proportions to a right angle. 

 It follows from what we have shown that the angles of the 

 triangle may be considered as ratios, or as numbers indepen- 

 dent of arbitrary measurement, whether they be estimated in 

 parts of a right angle, or of any other determinate angle. 

 But Legendre's procedure is defective inasmuch as he infers 

 inconclusively from a definition, what can be proved only by 

 the principles we have explained. 



When the understanding is made fully master of the case 

 by acquiring distinct ideas on the disputed points, it will 

 hardly be allowed that the functional equations are so simple 

 as they appear to be. To reason about them with intelligence, 

 many notions seem to be necessary that are far removed from 

 the first principles of geometry. They cannot, with much 

 propriety, be considered as propositions merely elementary. 

 We are almost inclined to think that the geometer must have 

 plodded on to the end of his science, before he has acquired 

 knowledge enough to judge critically of the functional demon- 

 strations of the first principles. 



In this Journal for February, we observe that Mr. Walsh 

 is a strenuous advocate for Professor Leslie's opinions. His 

 communication is remarkable for being wrong in every point 

 relating to geometry. What can be a greater mistake than 

 to suppose that the angles of a triangle must be evanescent at 

 the same time with the base ? Every one knows that the an- 

 gles may remain the same, while the sides increase to be in- 

 finitely great on one hand, or decrease to zei*o on the other. 

 But Mr. Walsh makes ample amends by the display of his 

 recondite researches. We are carried back to the principles 

 of philosophical grammar; we are taught to speak accurately 

 in the language of the schools; and the poor geometers, made 

 the sport of every petty whipster, are utterly condemned as 

 ignorant of the philosophy of number. If there be any doubt 



whether 



