394 
EUCLID 
Therefore, ex aequcdi (V. 22), 
{ A BCD) : iCEFG) = K : M. 
The important Proposition 25 (to construct a rectilineal figure 
similar to one, and equal to another, given rectilineal figure) is 
one of the famous problems alternatively associated with the 
story of Pythagoras’s sacrifice 1 ; it is doubtless Pythagorean. 
The given figure (P, say) to which the required figure is to be 
similar is transformed (I. 44) into a parallelogram on the same 
base BC. Then the other figure (Q, say) to which the required 
figure is to be equal is (I. 45) transformed into a parallelo 
gram on the base CF (in a straight line with BC) and of equal 
height with the other parallelogram. Then (P): (Q) = BC: CF 
(1). It is then only necessary to take a straight line GH 
a mean proportional between BC and CF, and to describe on 
GH as base a rectilineal figure similar to P which has BC as 
base (VI. 18). The proof of the correctness of the construction 
follows from VI. 19 Por. 
In 27, 28, 29 we reach the final problems in the Pythagorean 
application of areas, which are the geometrical equivalent of 
the algebraical solution of the most general form of quadratic 
equation where that equation has a real and positive root. 
Detailed notice of these propositions is necessary because of 
their exceptional historic importance, which arises from the 
fact that the method of these propositions was constantly used 
by the Greeks in the solution of problems. They constitute, 
for example, the foundation of Book X of the Elements and of 
1 Plutarch, Non posse suavitervivi secundum Epicurum, c, 11.