Let us, confining ourselves to the main subject of pure 
geometry by way of example, anticipate so far as to mark 
certain definite stages in its development, with the intervals 
separating them. In Thales’s time (about 600 b. c.) we find 
the first glimmerings of a theory of geometry, in the theorems 
that a circle is bisected by any diameter, that an isosceles 
triangle has the angles opposite to the equal sides equal, and 
(if Thales really discovered this) that the angle in a semicircle 
is a right angle. Rather more than half a century later 
Pythagoras was taking the first steps towards the theory of 
numbers and continuing the work of making geometry a 
theoretical science; he it was who first made geometry one of 
the subjects of a liberal education. The Pythagoreans, before 
the next century was out (i, e. before, say, 450 B. c.), had practi 
cally completed the subject-matter of Books I—II, IY, VI (and 
perhaps III) of Euclid’s Elements, including all the essentials 
of the ‘geometrical algebra’ which remained fundamental in 
Greek geometry; the only drawback was that their theory of 
proportion was not applicable to incommensurable but only 
to commensurable magnitudes, so that it proved inadequate 
as soon as the incommensurable came to be discovered. 
In the same fifth century the difficult problems of doubling 
the cube and trisecting any angle, which are beyond the 
geometry of the straight line and circle, were not only mooted 
but solved theoretically, the former problem having been first 
reduced to that of finding two mean proportionals in continued 
proportion (Hippocrates of Chios) and then solved by a 
remarkable construction in three dimensions (Archytas), while 
the latter was solved by means of the curve of Hippias of 
Elis known as the quadratrix; the problem of squaring the 
circle was also attempted, and Hippocrates, as a contribution 
to it, discovered and squared three out of the five lunes which 
can be squared by means of the straight line and circle. In 
the fourth century Eudoxus discovered the great theory of 
proportion expounded in Euclid, Book Y, and laid down the 
principles of the method of exhaustion for measuring areas and 
volumes ; the conic sections and their fundamental properties 
were discovered by Menaechmus; the theory of irrationals 
(probably discovered, so far as V 2 is concerned, by the 
early Pythagoreans) was generalized by Theaetetus; and the 
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