GEOMETRY 
would be, in the language of application of areas, ‘ To a given 
straight line (a) to apply a rectangle which shall be equal 
to a given square (b-) and shall fall short by a square figure.’ 
As the Pythagoreans solved the somewhat similar equation 
in II. 11, they cannot have failed to solve this one, as well as 
the equations corresponding to II. 6. For in the present case 
it is only necessary to draw CQ at right angles to MR from 
its middle point C, to make CQ equal to h, and then, with 
centre Q and radius equal to GB, or \a, to draw a circle 
cutting QC produced in R and^ CB in D [h 2 must be not 
greater than \a 2 ; otherwise a solution is impossible). 
Then the determination of the point D constitutes the 
solution of the quadratic. 
For, by the proposition II. 6, 
AD.DB + CD 2 = GB 2 
= QD 2 = QC 2 + GB 2 ; 
therefore AD. DB = QG 2 , 
or ax — x 2 — h 2 . 
Similarly II. 6 enables us to solve the equations 
ax + a; 2 = b 2 , 
and x 2 — ax = b 2 ; 
the first equation corresponding to AB — a, BD — x and the 
second to AB = a, AD — x, in the figure of the proposition. 
The application of the theory to conics by Apollonius will 
be described when we come to deal with his treatise. 
One great feature of Book II of Euclid’s Elements is the 
use of the gnomon (Props. 5 to 8), which is undoubtedly 
Pythagorean and is connected, as we have seen, with the