304 10. EUCLIDEAN GEOMETRY
a cone, the rectangle LGFE generates a cylinder, and each horizontal line such as
Ì Í generates a disk. The point A is the midpoint of CH. Archimedes shows that
the area of the disk generated by revolving QR plus the area of the disk generated
by revolving OP has the same ratio to the area of the disk generated by revolving
Ì Í that AS has to AH. It follows from his work on the equilibrium of planes that
if the first two of these disks are hung at H, they will balance the third disk about
A as a fulcrum. Archimedes concluded that the sphere and cone together placed
with their centers of gravity at Ç would balance (about the point A) the cylinder,
whose center of gravity is at K.
Therefore,
HA : AK = (cylinder) : (sphere + cone).
But HA = 2AK. Therefore, the cylinder equals twice the sum of the sphere and
the cone AEF. And since it is known that the cylinder is three times the cone
AEF, it follows that the cone AEF is twice the sphere. But since EF = 2BD,
cone AEF is eight times cone ABD, and the sphere is four times the cone ABD.
From this fact Archimedes easily deduces the famous result allegedly depicted
on his tombstone: The cylinder circumscribed about a sphere equals the volume of
the sphere plus the volume of a right circular cone inscribed in the cylinder.
Having concluded the demonstration, Archimedes reveals that this method en-
abled him to discover the area of a sphere. He writes
For I realized that just as every circle equals a triangle having as
its base the circumference of the circle and altitude equal to the
[distance] from the center to the circle [that is, the radius], in the
same way every sphere is equal to a cone having as its base the
surface [area] of the sphere and altitude equal to the [distance]
from the center to the sphere.
The Method gives an inside view of the route by which Archimedes discovered
his results. The method of exhaustion is convincing as a method of proving a
theorem, but useless as a way of discovering it. The Method shows us Archimedes'
route to discovery.
4. Apollonius
From what we have already seen of Greek geometry we can understand how the
study of the conic sections came to seem important. From commentators like Pap-
pus we know of treatises on the subject by Aristaeus, a contemporary of Euclid who
is said to have written a book on Solid Loci, and by Euclid himself. We have also
just seen that Archimedes devoted a great deal of attention to the conic sections.
The only treatise on the subject that has survived, however, is that of Apollonius,
and even for this work, unfortunately, no faithful translation into English exists.
The version most accessible is that of Heath, who says in his preface that writing his
translation involved "the substitution of a new and uniform notation, the conden-
sation of some propositions, the combination of two or more into one, some slight
re-arrangements of order for the purpose of bringing together kindred propositions
in cases where their separation was rather a matter of accident than indicative of
design, and so on." He might also have mentioned that he supplemented Apol-
lonius' purely synthetic methods with analytic arguments, based on the algebraic
notation we are familiar with. All this labor has no doubt made Apollonius more
