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