Expression of the sides of Right-angled Triangles, ^c. 69 



Let m, and n, be any rational numbers whatever ; then m^+^S 

 m'-* — w% and 2mn, will be sides of a right-angled triangle. For, 



If m remains constant for any set of examples, and n be succes- 

 sively increased by unity ; the hypothenuse will increase by the fol- 

 lowing series, 3, 5, 7, Stc. the base will decrease by the same series, 

 and the perpendicular will increase constantly by 2m. 



In the following examples m remains constantly equal to 10, and n 

 increases by successive units. The required triangular sides may 

 be extended ad infinitum by the process above explained, without 

 changing the value of m. If m be changed, other series equally un- 

 limited will arise. 



From this set of examples, it will be seen, 



1. That when m=-n, then the hypothenuse and perpendicular are 

 equal, and the base vanishes. 



2. That as n increases above m, the bases are negative, that is, 

 they lie on the contrary side of the perpendicular. 



