APPLICATION OF AREAS 
153 
application of areas. The whole of Book II, with the latter 
section of Book I from Prop. 42 onwards, may he said to deal 
with the transformation of areas into equivalent areas of 
different shape or composition by means of £ application ’ 
and the use of the theorem of I. 47. Eucl. II. 9 and 10 are 
special cases which are very useful in geometry generally, but 
were also employed by the Pythagoreans for the specific purpose 
of proving the property of ‘ side- ’ and ‘ diameter- ’ numbers, 
the object of which was clearly to develop a series of closer 
and closer approximations to the value of V2 (see p. 93 ante). 
The geometrical algebra, therefore, as we find it in Euclid, 
Books I and II, was Pythagorean. It was of course confined 
to problems not involving expressions above the second degree. 
Subject to this, it was an effective substitute for modern 
algebra. The product of two linear factors was a rect 
angle, and Book II of Euclid made it possible to multiply 
two factors with any number of linear terms in each; the 
compression of the result into a single product (rectangle) 
followed by means of the application-theorem (Eucl. I. 44). 
That theorem itself corresponds to dividing the product of 
any two linear factors by a third linear expression. To trans 
form any area into a square, we have only to turn the area 
into a rectangle (as in Eucl. I. 45), and then find a square 
equal to that rectangle by the method of Eucl. II. 14; the 
latter problem then is equivalent to the extraction of the square 
root. And we have seen that the theorems of Eucl. II. 5, 6 
enable mixed quadratic equations of certain types to be solved 
so far as their roots are real. In cases where a quadratic 
equation has one or both roots negative, the Greeks would 
transform it into one having a positive root or roots (by the 
equivalent of substituting — x for x) ; thus, where one root is 
positive and one negative, they would solve the problem in 
two parts by taking two cases. 
The other great engine of the Greek geometrical algebra, 
namely the method of proportions, was not in its full extent 
available to the Pythagoreans because their theory of pro 
portion was only applicable to commensurable magnitudes 
(Eudoxus was the first to establish the general theory, applic 
able to commensurables and incommensurables alike, which 
we find in Eucl. Y, YI). Yet it cannot be doubted that they