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ANALYTICAL GEOMETRY 51 

Zeno showed that we meet difficulties if we hold that time 
and space are infinitely divisible. Of the arguments which 
he invented to show this, the best known is the puzzle of 
Achilles and the Tortoise. Zeno argued that, if Achilles ran 
ten times as fast as a tortoise, yet, if the tortoise has (say) 
1000 yards start, it could never be overtaken. For, when 
Achilles had gone the 1000 yards, the tortoise would still be 
100 yards in front of him ; by the time he had covered these 
100 yards, it would still be 10 yards in front of him ; and 
so on for ever : thus Achilles would get nearer and nearer to 
the tortoise, but never overtake it. Zeno invented some other 
subtle puzzles for much the same purpose, and they could only 
be discussed really satisfactorily by quite modern mathematics. 

To avoid the use of infinitesimals, Eudoxus (408-355 B.C.) 
devised a method, exposed by Euclid in the Twelfth Book of 
his Elements and used by Archimedes to demonstrate many of 
his great discoveries, of verifying results found by the doubtful 
infinitesimal considerations. When the Greeks wished to dis- 
cover the properties of a curve, they regarded it as the fixed 
boundary to which the inscribed and circumscribed polygons 
approach continually, and as much as they pleased, according 
as they increased the number of their sides. Thus they ex- 
hausted in some measure the space comprised between these 
polygons and the curve, and doubtless this gave to this operation 
the name of "the method of exhaustion." As these polygons 
terminated by straight lines were known figures, their con- 
tinual approach to the curve gave an idea of it more and more 
precise, and, the law of continuity serving as a guide, the 
Greeks could eventually arrive at the exact knowledge of its 
properties. But it was not sufficient for geometricians to have 
observed, and, as it were, guessed at these properties ; it was 
necessary to verify them in an unexceptionable way ; and this 
they did by proving that every supposition contrary to the 
existence of these properties would necessarily lead to some 
contradiction : thus, after, by infinitesimal considerations, they 
had found the area (say) of a curvilinear figure to be a, they 
verified it by proving that, if it is not a, it would yet be 
greater than the area of some polygon inscribed in the curvi- 
linear figure whose area is palpably greater than that of the 
polygon. 

In the seventeenth century we have a complete contrast