Mr. J. Walsh on parallel straight Lines. 101 



to demonstrate, by the theory of functions, the property of 

 plane triangles, depending on the preceding proposition, that 

 the sums of the three angles are equal to two right angles. 

 M. Legendre, though vanquished, continues to argue still, and 

 is supported in his defence by a foreign geometer of eminence, 

 M. Maurice, as well as by the celebrated British geometers 

 Dr. Brewster and Professor Playfair. When such authorities 

 contend, it were perhaps presumptuous to interfere. It ap- 

 pears to me extraordinary that M. Legendre, a geometer so 

 deeply acquainted with theproperties of numbers, should really 

 mistake the nature of number itself. This circumstance is, 

 I believe, common to most geometers. Number expresses the 

 relation between two homogeneous magnitudes. It cannot 

 represent magnitude. Numbers are therefore abstract things. 

 To say " abstract numbers," is to say "abstract abstract things !" 

 It is as correct language to say " straight right line." 



At the origin of grammar, men, then not acquainted with 

 the real nature of numbers, arranged them very improperly 

 under the head of adjectives. We say, "The money is one 

 pound sterling;" "The distance is three metres;" "The 

 right angle is one," &c. We ought to say, " The money has 

 the relation one to the pound sterling;" " The distance has 

 the relation three to the metre ;" " The right angle has the 

 relation one to the right angle," &c. : — the pound sterling, the 

 metre, the right angle, &c, being the bases of comparison. 



Let A B C be the angles of any plane triangle, and c the 

 side adjacent to the angles A B. It is required to determine 

 the angle c. For this we have C = 0(ABc). Now M. Le- 

 gendre°says, in a note to the tenth edition of his Elements of 

 Geometry, that " the side c is of a nature heterogeneous to the 

 ano-les A B, and cannot coalesce with them in the equation 

 C = 9(ABc). The right angle being the natural unity of an- 

 gles, it is therefore a number. The angles A and B are 

 therefore numbers. They cannot then coalesce with the side 

 c, which is a straight line ; then C is entirely determined by 

 the angles A and B alone ; therefore, when two angles of one 

 triangle are equal to two angles of another, the third angle of 

 the one is equal to the third angle of the other." 



Now I shall demonstrate that the preceding reasoning fails 

 in three different ways. 1st. It is said the right angle is the 

 natural, that is to say, the necessary unity of angles. ^ I have 

 shown that a number cannot be put for magnitude. The right 

 angle is not the necessary, but is made the arbitrary base of 

 comparison in respect to angles. In respect to the sides, I 

 shall make the metre the base of comparison; then instead of 

 the side c I shall substitute its relation to the metre, and sub- 

 stituting 



