46 3. MATHEMATICAL CULTURES IIArchimedes. Much more is known of Archimedes (ca. 287-212 BCE). About 10 of
his works have been preserved, including the prefaces that he wrote in the form
of "cover letters" to the people who received the works. Here is one such letter,
which accompanied a report of what may well be regarded as his most profound
achievement—proving that the area of a sphere is four times the area of its equa-
torial circle.On a former occasion I sent you the investigations which I had
up to that time completed, including the proofs, showing that any
segment bounded by a straight line and a section of a right-angled
cone [parabola] is four-thirds of the triangle which has the same
base with the segment and equal height. Since then certain theo-
rems not hitherto demonstrated have occurred to me, and I have
worked out the proofs of them. They are these: first, that the
surface of any sphere is four times its greatest circle... For, though
these properties also were naturally inherent in the figures all along,
yet they were in fact unknown to all the many able geometers who
lived before Eudoxus, and had not been observed by anyone. Now,
however, it will be open to those who possess the requisite ability
to examine these discoveries of mine. [Heath, 1897, Dover edition,
pp. 1-2]As this letter shows, mathematics was a "going concern" by Archimedes' time,
and a community of mathematicians existed. Archimedes is known to have studied
in Alexandria. He perished when his native city of Syracuse was taken by the
Romans during the Second Punic War. Some of Archimedes' letters, like the one
quoted above, give us a glimpse of mathematical life during his time. Despite being
widely separated, the mathematicians of the time sent one another challenges and
communicated their achievements.
Apollonius. Apollonius, about one generation younger than Archimedes, was a na-
tive of what is now Turkey. He studied in Alexandria somewhat after the time of
Euclid and is also said to have taught there. He eventually settled in Pergamum
(now Bergama in Turkey). He is the author of eight books on conic sections, four
of which survive in Greek and three others in an Arabic translation. We know
that there were originally eight books because commentators, especially Pappus,
described the work and told how many propositions were in each book.
In his prefaces Apollonius implies that geometry was simply part of what an
educated person would know, and that such people were as fascinated with it in
his time as they are today about the latest scientific achievements. Among other
things, he said the following.
During the time I spent with you at Pergamum I observed your
eagerness to become aquainted with my work in conies. [Book I]
I undertook the investigation of this subject at the request of Nau-
crates the geometer, at the time when he came to Alexandria and
stayed with me, and, when I had worked it out in eight books, I
gave them to him at once, too hurriedly, because he was on the
point of sailing; they had therefore not been thoroughly revised,