60 A History ofMathematics
including Archimedes, Heron of Alexandria, and Ptolemy. All were tremendously influential in the
later development of mathematics, and all raise interesting questions about the varieties of Greek
mathematics which were (one might guess) competing for influence over the long historical period
which concerns us.
At the same time, because of the length of the period, and the variety of the work produced, it
would be impractical in a book of this kind to try to cover everything. In particular for the important
work of Apollonius, Diophantus, and Pappus, you will have to look elsewhere.
The remarks on sources made in the previous chapter apply on the whole. The major historian
who has recently concentrated attention on the late period is Cuomo, to whose works (2000) and
(2001) we shall return in due course.Exercise 1. Check that Menaechmus’s construction does give a line of length a^3√
2. How would you
generalize it to solve the problem of increasing the volume of the cube by a factor m?Exercise2.Hippocratesof Chios(fifthcenturybce)showedthatthegeneralproblem(multiplyingacube
by m) can be solved if, between two given linesA,B, with the ratioB:A=m, one can construct two ‘mean
proportionals’C,D; that is so that the ratiosA:C,C:D,D:Bare equal.Why is this true?2. Archimedes
Archimedes is one of the most heroized figures in the history of science; but unlike Galileo and
Newton, whose lives are available in minute detail, we know rather little about him. There is a
growing literature on him; not so much ‘biographical’ as an attempt to understand him from his
works. True, his life is better documented than that of any other Greek mathematician (with the
possible exception of Hypatia), but that is not saying much. The chief sources tend to concentrate
on a few memorable events—the ‘Eureka story’, his role in the siege of Syracuse, and his death at
the hands of a Roman soldier. His works have always been seen as uniquely brilliant and difficult,
and perhaps his portrait has been constructed to fit them; though unusually, there are letters
introducing several of the writings which are ‘personal’ as not much else is in Greek mathematics.
A late portrait of Archimedes as the absent-minded pure researcher is given in Plutarch’sLife of
Marcellus, and for whatever reason it has become influential. In line with a Platonic propagandist
viewpoint, Plutarch (while crediting Archimedes with major military inventions), claims that such
practical considerations were unimportant to him.Yet Archimedes possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that though
these inventions had now obtained him the renown of more than human sagacity, he yet would not deign to leave
behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of
engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in
those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which
to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects
examined, or the precision and cogency of the methods and means of proof, most deserve our admiration. It is not
possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some
ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearances,
easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once
seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the
conclusion required. (Plutarch, in Fauvel and Gray 4.B.1)