240 On the Theory of Angular Geometry. [No. 123. 



" to the modes of thinking which involve the idea of infinite magnitude 

 " under any form whatsoever, it may be reasonably suspected that he 

 " would admit the following axiom, Magnitudes which can be made to 

 " coincide in all their parts are equal, as applicable to infiniteas well as to 

 " finite spaces. Not having done so, the adherence to his standard has 

 " to this day excluded the only proof of the theory of parallels, which 

 " does not assume the axiom of Euclid, or an equivalent." 



Remarks on the Essay " on the Theory of Angular Geometry." By Capt. 



Shortrede. 



A definition is perfect, when it includes all that has the property 

 intended to be defined, while it excludes all that has it not. 



If we would have a true definition of angle, or of any thing else, it is 

 of the utmost importance that we have a clear idea of the thing, and 

 then use such words as plainly to convey the idea. If there be any 

 neglect in either of these, our definition must necessarily be imperfect. 



Geometry as commonly defined, treats of figured space. If this defini- 

 tion be correct, (and I find no fault with it), then it is plainly improper 

 to introduce indefiniteness, or boundlessness, or infinity, as part and 

 parcel of the definition of a thing or idea, of which the property signified 

 by these terms, is not necessarily a part. I can conceive of an angle 

 formed by finite lines : unboundedness is therefore not necessary to 

 the idea of angle, and therefore ought not to form a part of the 

 definition. 



Since the idea of angle is somehow sooner or later convertible to, and 

 commensurable by that of circular arc, every attempt at defining angle 

 should be made with this in view, otherwise the definer will discover, 

 (or some one will discover it for him), that his definition is not perfect. 



As the author of this Essay introduces unlimitedness in the containing 

 lines as part of his definition of angle, I do not see why the plane sur- 

 face of a hyperbola between its assymptotes may not be angle, as well 

 as the thing intended by him. If it be said, the meaning is the whole 

 plane surface between the lines, I rejoin that the whole plane surface 

 being unlimited, I cannot form an idea of how much it is. 





