- ARCHIMEDES 299
cones and cylinders are proportional to the cubes of their linear dimensions, end-
ing with the proof that spheres are proportional to the cubes on their diameters
(Proposition 18). As we noted above, Archimedes (or someone who edited his
works) credited these theorems to Eudoxus.
Book 13, the last book of the Elements, is devoted to the construction of the
regular solids and the relation between their dimensions and the dimensions of
the sphere in which they are inscribed. The last proposition (Proposition 18) sets
out the sides of these regular solids and their ratios to one another. An informal
discussion following this proposition concludes that there can be only five regular
solids.
2.2. The Data. Euclid's Elements assume a certain familiarity with the principles
of geometric reasoning, principles that are explained in more detail in the Data. The
Greek name of this work (Dedomena) means [Things That Are] Given, just as Data
does in Latin. The propositions in this book can be interpreted in various ways.
Some can be looked at as uniqueness theorems. For example (Proposition 53), if
the shapes—that is, the angles and ratios of the sides—are given for two polygons,
and the ratio of the areas of the polygons is given, then the ratio of any side of
one to any side of the other is given. Here, being given means being uniquely
determined. Uniqueness is needed in proofs and constructions so that one can be
sure that the result will be the same no matter what choices are made. It is an
issue that arises frequently in modern mathematics, where operations on sets are
defined by choosing representatives of the sets; when that is done, it is necessary to
verify that the operation is well defined, that is, independent of the choice made. In
geometry we frequently say, "Let ABC be a triangle having the given properties and
having such-and-such a property," such as being located in a particular position.
In such cases we need to be sure that the additional condition does not restrict
the generality of the argument. In another sense, this same proposition reassures
the reader that an explicit construction is possible, and removes the necessity of
including it in the exposition of a theorem.
Other propositions assert that certain properties are invariant. For example
(Proposition 81), when four lines Α, Β, Ã, and Ä are given, and the line Ç is such
that Ä : Ε = A : Ç, where Ε is the fourth proportional to A, B, and Ã, then
Ä : Ã = Β : Ç. This last proposition is a lemma that can be useful in working out
locus problems, which require finding a curve on which a point must, lie if it satisfies
certain prescribed conditions. Finally, a modern mathematician might interpret the
assertion that an object is "given" as saying that the object "exists" and can be
meaningfully talked about. To Euclid, that existence would mean that the object
was explicitly constructible.
3. Archimedes
Archimedes is one of a small number of mathematicians of antiquity of whose
works we know more than a few fragments and of whose life we know more than
the approximate time and place. The man indirectly responsible for his death, the
Roman general Marcellus, is also indirectly responsible for the preservation of some
of what we know about him. Archimedes lived in the Greek city of Syracuse on
the island of Sicily during the third century BCE and is said by Plutarch to have
been "a relative and a friend" of King Hieron II. Since Sicily lies nearly on a direct
line between Carthage and Rome, it became embroiled in the Second Punic War.