CHAP. VIII.] OF REDUCTION. 127 



which, on interpretation, would give for the first term of the de- 

 scription of dissimilar figures, " Triangles whose corresponding 

 sides are not proportional," instead of the fuller description origi- 

 nally obtained. A regard to convenience must always determine 

 the propriety of such reduction. 



12. A reduction which is always advantageous (VII. 15) con- 

 sists in collecting the terms of the immediate description sought, 

 as of the second member of (5) or (6), into as few groups as 

 possible. Thus the third and fourth terms of the second mem- 

 ber of (6) produce by addition the single term (1 - t) (1 - q). 

 If this reduction be combined with the last, we have 



1 - . = /(I - r) + (1 - t)q (1 -r) + (I-*) (1 - q), 

 the interpretation of which is 



Dissimilar figures consist of all triangles whose corresponding 

 sides are not proportional, and all figures not being triangles which 

 have either their corresponding angles unequal, or their corresponding 

 angles equal, but sides not proportional. 



The fulness of the general solution is therefore not a super- 

 fluity. While it gives us all the information that we seek, it 

 provides us also with the means of expressing that information 

 in the mode that is most advantageous. 



13. Another observation, illustrative of a principle which has 

 already been stated, remains to be made. Two of the terms in 

 the full development of 1 - s in (5) have 2 for their coefficients, 



instead of -. It will hereafter be shown that this circumstance 



indicates that the two premises were not independent. To verify 

 this, let us resume the equations of the premises in their reduced 

 forms, viz., ( 



s(l - qr) -t qr(l - s) = 0, 



Now if the first members of these equations have any common 

 constituents, they will appear on multiplying the equations to- 

 gether. If we do this we obtain 



stq(l - r) + str(l - q) = 0. 



