Takleton — The Relation of Mathematics to Physical Science. 165 



truth of the theory of parallel lines as expounded by Euclid, on which rests 

 the fundamental theorem that the three angles of a triangle are equal to 

 two right angles. 



There is no doubt that the truth of Euclid's axiom, on which the theory 

 of parallels depends, is by no means obvious intuitively ; but Legendre showed 

 that it could be deduced from another assertion which would, I think, be 

 generally admitted. 



It is easy to see, by superposition, that triangles having two angles 

 and the intervening side equal are equal in every respect. Hence the 

 vertical angle of a triangle must be a function of the base and base-angles. 

 One quantity cannot be a function of another of a different kind from itself 

 unless something else enters into the equation to reduce the heterogeneous 

 quantity to the proper nature. Hence the only way in which one angle 

 could be a function of other angles, and of the length of a line, would be 

 through the entrance of another line into the equation. If, then, there, be 

 an absolute standard of linear magnitude, depending on the nature of space, 

 and not on any arbitrary unit, as there is an absolute standard of angular 

 magnitude, viz. a right angle, it is possible, but otherwise impossible, that 

 the length of the base should affect the magnitude of the ^^ertical angle when 

 the base-angles are given. Everything turns, then, on the admission or denial 

 of a standard length dependent on the nature of space. ]\Iost people will, I 

 think, admit that there is no absolute standard of length. If this be 

 admitted, it follows that one angle of a triangle is a function of the other two, 

 and independent of the length of the sides. From this the Euclidean theory 

 of parallels can be deduced. 



It may, however, be said that if our space were a space of three dimen- 

 sions having a curvature in a space of four dimensions, as the surface of a 

 sphere is a space of two dimensions having a curvature in a space of three 

 dimensions, the curvature of space would supply an absolute standard of 

 magnitude. Under these circumstances, also, we might have two non- 

 coincident straight lines having two points in common. This mode of 

 regarding the matter is indeed the simplest way of construing the systems of 

 Lobatschewsky, Bolyai, Kiemann, and Helmboltz. 



In a sense these systems are all correct and consistent ; but if we ask are 

 they true, the correct answer seems to be that they are not true for our 

 minds. 



The great mathematician and physicist M. Poincare holds a somewhat 

 peculiar theory. According to him : — 



The axioms of Geometry are neither synthetic a ]_)noTi intuitions nor 

 experimental facts, but only conventions. Geometrical space differs in its 



