SUMMARY 
217 
the main varieties of irrationals distinguished, though their 
classification was not carried so far as in Euclid. 
The substance of Book XL 1—19 must already have been in 
cluded in the Elements (e.g. Eucl. XL 19 is assumed in Archytas’s 
construction for the two mean proportionals), and the whole 
theory of the section of Book XI in question would be required 
for Theaetetus’s work on the five regular solids: XI. 21 must 
have been known to the Pythagoreans: while there is nothing 
in the latter portion of the book about parallelepipedal solids 
which (subject to the want of a rigorous theory of proportion) 
was not within the powers of those who were familiar with 
the theory of plane and solid numbers. 
Book XII employs throughout the method of exhaustion, 
the orthodox form of which is attributed to Eudoxus, who 
grounded it upon a lemma known as Archimedes’s Axiom or 
its equivalent (Eucl. X. 1). Yet even XII. 2, to the effect that 
circles are to one another as the square of their diameters, had 
already been anticipated by Hippocrates of Chios, while 
Democritus had discovered the truth of the theorems of 
XII. 7, For., about the volume of a pyramid, and XII. 10, 
about the volume of a cone. 
As in the case of Book X, it would appear that Euclid was 
indebted to Theaetetus for much of the substance of Book XIII, 
the latter part of which (Props. 12-18) is devoted to the 
construction of the five regular solids, and the inscribing of 
them in spheres. 
There is therefore probably little in the whole compass of 
the Elements of Euclid, except the new theory of proportion due 
to Eudoxus and its consequences, which was not in substance 
included in the recognized content of geometry and arithmetic 
by Plato’s time, although the form and arrangement of the 
subject-matter and the methods employed in particular cases 
were different from what we find in Euclid.