150 
PYTHAGOREAN GEOMETRY 
(y) 
Application of areas and geometrical algebra. 
We have seen that, in connexion with the story of the 
sacrifice of an ox, Plutarch attributes to Pythagoras himself 
the discovery of the problem of the application of an area 
or, as he says in another place, the problem ‘ Given two 
figures, to “ apply ” a third figure which shall be equal to the 
one, and similar to the other (of the given figures).’ The 
latter problem (= Eucl. VI. 25) is, strictly speaking, not so 
much a case of applying an area as of constructing a figure, 
because the base is not given in length; but it depends 
directly upon the simplest case of ‘ application of areas ’, 
namely the problem, solved in Eucl. I. 44, 45, of applying 
to a given straight line as base a parallelogram containing 
a given angle and equal in area to a given triangle or 
rectilineal figure. The method of application of areas is 
fundamental in Greek geometry and requires detailed notice. 
We shall see that in its general form it is equivalent to the 
geometrical solution of a mixed quadratic equation, and it is 
therefore an essential part of what has been appropriately 
called geometrical algebra. 
It is certain that the theory of application of areas 
originated with the Pythagoreans, if not with Pythagoras 
himself. We have this on the authority of Eudemus, quoted 
in the following passage of Proclus: 
‘ These things, says Eudemus, are ancient, being discoveries 
of the Muse of the Pythagoreans, I mean the application of 
areas {napafioXT] rcor ycopiW), their exceeding (VTrepfioXrj) and 
their falling short (eXXeLyfns). It was from the Pythagoreans 
that later geometers [i. e. Apollonius of Perga] took the 
names, which they then transferred to the so-called conic 
lines (curves), calling one of these a parabola (application), 
another a hyperbola (exceeding), and the third an ellipse 
(falling short), whereas those god-like men of old saw the 
things signified by these names in the construction, in a plane, 
of areas upon a given finite straight line. For, when you 
have a straight line set out, and lay the given area exactly 
alongside the whole of the straight line, they say that you 
apply the said area; when, however, you make the length of 
the area greater than the straight line, it is said to exceed, 
and, when you make it. less, in which case after the area has 
been drawn there is some part of the straight line extending