RECAPITULATION 
167 
form and, in particular, the whole method of application of 
areas, constituting a geometrical algebra, whereby they effected 
the equivalent of the algebraical processes of addition, sub 
traction, multiplication, division, squaring, extraction of the 
square root, and finally the complete solution of the mixed 
quadratic equation x i ±px + q = 0, so far as its roots are real. 
Expressed in terms of Euclid, this means the whole content of 
Book I. 35-48 and Book II. The method of application of 
areas is one of the most fundamental in the whole of later 
Greek geometry; it takes its place by the side of the powerful 
method of proportions; moreover, it is the starting point of 
Apollonius’s theory of conics, and the three fundamental 
terms, parabole, ellipsis, and hyperbole used to describe the 
three separate problems in ‘application’ were actually em 
ployed by Apollonius to denote the three conics, names 
which, of course, are those which we use to-day. Nor was 
the use of the geometrical algebra for solving numerical 
problems unknown to the Pythagoreans; this is proved by 
the fact that the theorems of Eucl. II. 9, 10 were invented 
for the purpose of finding successive integral solutions of the 
indeterminate equations 
2 x 2 — y 2 = + 1. 
3. They had a theory of proportion pretty fully developed. 
We know nothing of the form in which it was expounded; 
all we know is that it took no account of incommensurable 
magnitudes. Hence we conclude that it was a numerical 
theory, a theory on the same lines as that contained in 
Book VII of Euclid’s Elements. 
They were aware of the properties of similar figures. 
This is clear from the fact that they must be assumed 
to have solved the problem, which was, according to 
Plutarch, attributed to Pythagoras himself, of describing a 
figure which shall be similar to one given figure and equal in 
area to another given figure. This implies a knowledge of 
the proposition that similar figures (triangles or polygons) are 
to one another in the duplicate ratio of corresponding sides 
(Eucl. VI. 19, 20). As the problem is solved in Eucl. VI. 25, 
we assume that, subject to the qualification that their 
theorems about similarity, &c., were only established of figures