The History of Mathematics: A Brief Course

300 10. EUCLIDEAN GEOMETRY

Marcellus took the city of Syracuse after a long siege, and Archimedes was killed
by a Roman soldier in the chaos of the final fall of the city. In the course of writing
a biography of Marcellus, the polymath Plutarch included some information on
mathematics and philosophy in general.
According to Plutarch's biography of Marcellus, the general was very upset
that Archimedes had been killed and had his body buried in a suitably imposing
tomb. According to Eutocius, a biography of Archimedes was written by a certain
Heracleides, who is mentioned in some of Archimedes' letters. However, no copy of
this biography is known to exist today.
There are many legends connected with Archimedes, scattered among the var-
ious sources. Plutarch, for instance, says that Archimedes made many mechanical
contrivances but generally despised such work in comparison with pure mathemati-
cal thought. Plutarch also reports three different stories of the death of Archimedes
and tells us that Archimedes wished to have a sphere inscribed in a cylinder carved
on his tombstone. The famous story that Archimedes ran naked through the streets
shouting "Eureka!" ("I've got it!") when he discovered the principle of specific grav-
ity in the baths is reported by the Roman architect Vitruvius. Proclus gives another
well-known anecdote: that Archimedes built a system of pulleys that enabled him
(or King Hieron) single-handedly to pull a ship through the water. Finally, Plutarch
and Pappus both quote Archimedes as saying in connection with his discovery of
the principle of the lever that if there were another Earth, he could move this one
by standing on it.
With Archimedes we encounter the first author of a considerable body of orig-
inal mathematical research that has been preserved to the present day. He was
one of the most versatile, profound, creative, imaginative, rigorous, and influential
mathematicians who ever lived. Ten of Archimedes' treatises have come down to
the present, along with a Book of Lemmas that seems to be Archimedean. Some of
these works are prefaced by a "cover letter" intended to explain their contents to the
person to whom Archimedes sent them. These correspondents of Archimedes were:
Gelon, son of Hieron II and one of the kings of Syracuse during Archimedes' life;
Dositheus, a student of Archimedes' student and close friend Conon; and Eratos-
thenes, an astronomer who worked in Alexandria. Like the manuscripts of Euclid,
all of the Archimedean manuscripts date from the ninth century or later. These
manuscripts have been translated into English and published by various authors.
A complete set of Medieval manuscripts of Archimedes' work has been published
by Marshall Clagett in the University of Wisconsin series on Medieval Science.
The 10 treatises referred to above are the following.

1. On the Equilibrium of Planes, Part I


2. Quadrature of the Parabola


3. On the Equilibrium of Planes, Part II


4. On the Sphere and the Cylinder, Parts I and II


5. On Spirals


6. On Conoids and Spheroids


7. On Floating Bodies


8. Measurement of a Circle


9. The Sand-reckoner


10. The Method