Mr. Walsh on Parallel Straight Lines. 273 



distorted by the false language of the infinitesimal calculi, 

 will have assumed its proper empire in mathematical and phy- 

 sical investigations. 



I cannot discover in the second note, tenth edition, of his 

 Geometry, where he applies functional equations to the theory 

 of parallel lines, that M. Legendre assumes, that if at the ends 

 of any straight lines cf, c" &c. angles be formed equal each 

 to each to the angles at the base c of any given plane triangle, 

 triangles will be formed on each of those lines. In fact, his 

 reasoning is similar to the following : " Having measured 

 two angles of any plane triangle, it is required from these 

 to determine the third. 



" The vertical angle C is determined by the angles A, B, 

 and the base c. For when A = 0, 13 = 0, c=0, then will 

 C = 0. But C, a number, being heterogeneous to the line c, 

 then any particular magnitude of C cannot depend on any 

 particular magnitude of c, then the magnitude of C can only 

 depend on the magnitudes of A, B, and be derived from 

 them. Therefore when any two plane triangles have two an- 

 gles of the one equal each to each to two angles of the other, 

 the third angle of the one will be equal to the third angle of 

 the other. It follows from this, that the sum of the three 

 angles of any plane triangle is equal to two right angles. 



" The base and its adjacent angles being given, to construct 

 the triangle ? 



" With the base make an angle equal to the given acute 

 angle ; then at any point in the base, or the second line, ac- 

 cording as the other given angle is acute or obtuse, make an 

 angle equal to the other acute angle ; then a triangle will be 

 formed, having its angles equal each to each to the angles of 

 the triangle to be constructed. Then at the other end of the 

 base make an angle equal to the other given angle, and the 

 thing required is done. For if the lines forming the angles 

 at the base, do not meet when produced, then the vertical 

 angle will vary for particular values of the base, which is 

 shown to be impossible." 



Now I shall object to the preceding reasoning, that the arc 

 C and the base c may both be compared to the straight line 

 radius, and may be expressed by equations of equal dimen- 

 sions, as I have demonstrated in my Essay on the Binomial 

 Calculus. The reasoning of M. Legendre therefore fails al- 

 together, as must all attempts at obviating the difficulty of 

 the axiom of the illustrious Greek geometer. 



Cork, April 1.5, 1824, John WALSH. 



Vol.63. No. 312 April 1824. Mm XLVII. Zoo- 



