CHAP. VIII.] OF REDUCTION. 125 



the chapter referred to, is of this kind. Such equations, how- 

 ever, as it has been seen, have a meaning. Should it, for cu- 

 riosity, or for any other motive, be determined to employ them, 

 it will be best to reduce them by the Rule (VI. 5). 



10. Ex. 2. Let us take the following Propositions of Ele- 

 mentary Geometry : 



1st. Similar figures consist of all whose corresponding angles 

 are equal, and whose corresponding sides are proportional. 



2nd. Triangles whose corresponding angles are equal have 

 their corresponding sides proportional, and vice versa. 

 To represent these premises, let us make 

 s = similar. 

 t = triangles. \ 



q = having corresponding angles equal. 

 r = having corresponding sides proportional. 



Then the premises are expressed by the following equations : 



s = qr, (1) 



tq=tr. (2) 



Reducing by the Rule, or, which amounts to the same thing, 

 bringing the terms of these equations to the first side, squaring 

 each equation, and then adding, we have 



s + qr - 2qrs + tq + ir - 2tqr = 0. (3) 



Let it be required to deduce a description of dissimilar figures 

 formed out of the elements expressed by the terms, triangles, 

 having corresponding angles equal, having corresponding sides 

 proportional. 



We have from (3), 



_ tq + qr + rt - 2tqr 



0) 



And fully developing the second member, we find 



1 - s = Qtqr + 2tq (l-r) + 2*r (1 - q) + t (1 - q) (1 - r) 



