Figure 73 Eratosthenes’ method of measuring the Earth’s circumference.
Until the nineteenth century the name of one man so dominated the field of geometry that, when it was
shown at that time that new geometries could be seriously pursued, these new fields came to be referred
to as “non-Euclidean” geometry, in reference to the Alexandrian mathematician Euclid, who was active
some time around 300 BC. Almost nothing is known about Euclid, but his work, The Elements in 13
books, served as the fundamental textbook of geometry until as recently as two hundred years ago. The
Greek title of Euclid’s work is the same as the word for “letters” of the alphabet (p. 39); just as the few
letters of the alphabet are the building blocks of words and sentences and dissertations, so Euclid begins
with a handful of definitions, postulates, and self-evident notions (for example, “The whole is greater than
the part.”) and uses those elements and no others to construct incontrovertible proofs of hundreds of
propositions, including the Pythagorean Theorem (p. 216). It is not clear how much of The Elements is
original with Euclid and how much draws upon the thinking of other brilliant scholars whose works have
been lost, some of whom we know to have been associated with Plato’s Academy. What is remarkable
about The Elements is its clear-headed, rational structure, which successfully reduces the whole of
geometry as it was then understood to a well-organized, self-contained system. As was the case with the
alphabet, the roots of Euclid’s work stretch back, ultimately, to the non-Greek people of western Asia,
specifically the accomplished mathematicians of Babylon, as well as to the Egyptians, whom the Greeks
considered to be the inventors of geometry.
For all his considerable merits as an expounder of geometry, however, Euclid fell far short of the
accomplishments of the most brilliant of ancient mathematicians. The level of sophistication of
Hellenistic science and mathematics was made apparent by a dramatic discovery in 1900. At that time
sponge divers discovered the wreck of a Roman ship that had gone down in a storm some time in the first