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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