126 OF REDUCTION. [CHAP. VIII. 



In the above development two of the terms have the coefficient 

 2, these must be equated to by the Rule, then those terms 

 whose coefficients are being rejected, we have 



1 -,=(! -q) (l-r) + (l -t)q(l -r) + (l-t)r (1 - j) 



+ (l-f)(l-y)(l-r); (6) 



<y(l-r) = 0; (7) 



<r(l- 3 )-0; (8) 



the direct interpretation of which is 



1st. Dissimilar figures consist of all triangles which have not their 

 corresponding angles equal and sides proportional, and of all figures 

 not being triangles which have either their angles equal, and sides not 

 proportional, or their corresponding sides proportional, and angles 

 not equal, or neither their corresponding angles equal nor corres- 

 ponding sides proportional. 



2nd. There are no triangles whose corresponding angles are equal, 

 and sides not proportional. 



3rd. There are no triangles whose corresponding sides are pro- 

 portional and angles not equal. 



\ 1 . Such are the immediate interpretations of the final equa- 

 tion. It is seen, in accordance with the general theory, that in 

 deducing a description of a particular class of objects, viz., dis- 

 similar figures, in terms of certain other elements of the original 

 premises, we obtain also the independent relations which exist 

 among those elements in virtue of the same premises. And that 

 this is not superfluous information, even as respects the imme- 

 diate object of inquiry, may easily be shown. For example, the 

 independent relations may always be made use of to reduce, if it 

 be thought desirable, to a briefer form, the expression of that re- 

 lation which is directly sought. Thus if we write (7) in the 

 form 



and add it to (6), we get, since 



(!-*)(! -?)(i -r), 



