1842. On the Theory of Angular Geometry. 231 



An advanced student, who has acquired the idea in question, will find 

 nothing very objectionable in thus expressing himself, for he knows 

 what is to be described, and mentally assigns a due scientific meaning 

 to the general term of common parlance. But to the beginner, there 

 appears something as vague in the word " inclination," as in the term 

 " direction," when applied to a straight line. It even appears more vague, 

 for the genus of a straight line is given — it is the line of direction : but 

 the angle, is it then inclination itself ? The student is apprised, that 

 his attention is to be confined to points, lines, surfaces, and solids, things 

 of which he has definite conceptions : but here at the very outset is a 

 subject introduced, which appears to be distinct from all, and to be a 

 quality of figure rather than an existence. It afterwards turns out that 

 the only practically useful explanation relative to an angle requires 

 merely, that it should measure this quality of position. 



Considerations of this nature have induced some distinguishedly suc- 

 cessful elementary writers, to deviate from the usual custom in seeking 

 for such a definition of an angle as should appear to be a natural 

 description of it, to be free from metaphysical objections, and to permit 

 of immediate use in the investigation of the properties of angles, or 

 failing that, through medium of such simple considerations as may 

 appear almost axiomatically deducible from the definition. 



Of this class is Bossat's statement, that the angle is the opening be- 

 tween two lines, with an explanation impressing the definiteness of the 

 conception, and the mode of comparison naturally resulting from it. 

 This was followed by Professor Young in England. But the nature of 

 the idea thus attempted to be expressed by the word opening, did not 

 seem to be yet satisfactorily developed, and Legendre, accordingly ven- 

 tured to substitute " quantity" for " opening." The American edition 

 of Brewster's translation calls an angle, " the quantity by which two 

 intersecting lines are separated from each other;" and Francceur, I pre- 

 sume after Legendre, adopts a similar definition in his admirable course. 

 It is, however, easily seen, that very little is gained by this step, on the 

 score of clearness or precision, as the kind of quantity is not speci- 

 fied. 



Leslie attempted quite another path, suggested by the cerelations of 

 angles and arcs ; viz., that angular magnitude is generated by the revo- 

 lution of a line round a fixed point : but we are not told what angular 



