176 THE ELEMENTS DOWN TO PLATO’S TIME 
lay down the restriction of the means permissible in construc 
tions to the ruler and compasses which became a canon of 
Greek geometry for all ‘ plane ’ constructions, i.e. for all 
problems involving the equivalent of the solution of algebraical 
equations of degree not higher than the second. 
Democritus, as mathematician, may be said to have at last 
come into his own. In the Method of Archimedes, happily 
discovered in 1906, we are told that Democritus w T as the first 
to state the important propositions that the volume of a cone 
is one third of that of a cylinder having the same base and 
equal height, and that the volume of a pyramid is one third of 
that of a prism having the same base and equal height; that is 
to say, Democritus enunciated these propositions some fifty 
years or more before they were first scientifically proved by 
Eudoxus. 
Democritus came from Abdera, and, according to his own 
account, w T as young w r hen Anaxagoras was old. Apollodorus 
placed his birth in 01. 80 (■= 460-457 B.C.), while according 
to Thrasyllus he was born in 01. 77. 3 (= 470/69 B.C.), being 
one year older than Socrates. He lived to a great age, 90 
according to Diodorus, 104, 108, 109 according to other 
authorities. He was indeed, as Thrasyllus called him, 
irevTaOXos in philosophy 1 ; there was no subject to which he 
did not notably contribute, from mathematics and physics on 
the one hand to ethics and poetics on the other; he even went 
by the name of ‘ Wisdom ’ {Zocpia). 2 Plato, of course, ignores 
him throughout his dialogues, and is said to have wished to 
burn all his works; Aristotle, on the other hand, pays 
handsome tribute to his genius, observing, e.g., that on the 
subject of change and growth no one save Democritus had 
observed anything except superficially; whereas Democritus 
seemed to have thought of everything. 3 He could say 
of himself (the fragment is, it is true, considered by Diels 
to be spurious, while Gomperz held it to be genuine), ‘ Of 
all my contemporaries I have covered the most ground in 
my travels, making the most exhaustive inquiries the while ; 
I have seen the most climates and countries and listened to 
1 Diog. L. ix. 37 (Vors. ii 3 , p. 11. 24-30). 
2 Clem. Strom, vi. 32 (Vor<. ii 3 , p. 16. 28). 
3 Arist. De yen. et corr. i. 2, 315 a 35.