20 
ARCHIMEDES 
birth to the calculus of the infinite conceived and brought to 
perfection successively by Kepler, Cavalieri, Fermat, Leibniz 
and Newton ’. He performed in fact what is equivalent to 
integration in finding the area of a parabolic segment, and of 
a spiral, the surface and volume of a sphere and a segment of 
a sphere, and the volumes of any segments of the solids of 
revolution of the second degree. In arithmetic he calculated 
approximations to the value of tt, in the course of which cal 
culation he shows that he could approximate to the value of 
square roots of large or small non-square numbers ; he further 
invented a system of arithmetical terminology by which he 
could express in language any number up to that which we 
should write down with 1 followed by 80,000 million million 
ciphers. In mechanics he not only worked out the principles of 
the subject but advanced so far as to find the centre of gravity 
of a segment of a parabola, a semicircle, a cone, a hemisphere, 
a segment of a sphere, a right segment of a paraboloid and 
a spheroid of revolution. His mechanics, as we shall see, has 
become more important in relation to his geometry since the 
discovery of the treatise called The Method which was formerly 
supposed to be lost. Lastly, he invented the whole science of 
hydrostatics, which again he carried so far as to give a most 
complete investigation of the positions of rest and stability of 
a right segment of a paraboloid of revolution floating in a 
fluid with its base either upwards or downwards, but so that 
the base is either wholly above or wholly below the surface of 
the fluid. This represents a sum of mathematical achieve 
ment unsurpassed by any one man in the world’s history. 
Character of treatises. 
The treatises are, without exception, monuments of mathe 
matical exposition ; the gradual revelation of the plan of 
attack, the masterly ordering of the propositions, the stern 
elimination of everything not immediately relevant to the 
purpose, the finish of the whole, are so impressive in their 
perfection as to create a feeling akin to awe in the mind of 
the reader. As Plutarch said, £ It is not possible to find in 
geometry more difficult and troublesome questions or proofs 
set out in simpler and clearer propositions There is at the 
1 Plutarch, Marcellus, c. 17. 
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