- GREEK AND ROMAN MATHEMATICS 45
himself. Lasserre (1964, P- 17) believes that the most important mathematical work
at the Academy was done between 375 and 350 BCE.
Socrates explained that arithmetic was needed both to serve the eye of the soul
and as a practical instrument in planning civic projects and military campaigns:
The kind of knowledge we are seeking seems to be as follows. It is
necessary for a military officer to learn (mathein) these things for
the purpose of proper troop deployment, and the philosopher must
have risen above change, in order to grasp the essence of things, or
else never become skilled in calculation (logistiko).
iPlato, through Socrates, complains of the lack of a government subsidy for
geometry. In his day solid geometry was underdeveloped in comparison with plane
geometry, and Socrates gave what he thought were the reasons for its backwardness:
First, no government holds [the unsolved problems in solid geom-
etry] in honor; and they are researched in a desultory way, being
difficult. Second, those who are doing the research need a mentor,
without which they will never discover anything. But in the first
place, to become a mentor is difficult; and in the second place,
after one became a mentor, as things are just now, the arrogant
people doing this research would never listen to him. But if the
entire state were to act in concert in conducting this research with
respect, the researchers would pay heed, and by their combined
intensive work the answers would become clear.Plato himself was among that group of people mentioned in Chapter 2, for
whom the "eye of the soul" was sufficient justification for intellectual activity. He
seems to have had a rather dim view of the second group, the practical-minded
people. In his long dialogue The Laws, one of the speakers, an Athenian, rants
about the shameful Greek ignorance of incommensurables, surely a topic of limited
application in the lives of most people.
1.3. Works and authors. Books on mathematics written in Greek begin appear-
ing early in Hellenistic times (third century BCE) and continue in a steady stream
for hundreds of years. We list here only a few of the most outstanding authors.Euclid. This author lived and worked in Alexandria, having been invited by Ptolemy
Soter (Ptolemy I) shortly after the city was founded. Essentially nothing is known
of his life beyond that fact, but his famous treatise on the basics of geometry (the
Elements) has become a classic known all over the world. Several of his minor
works—the Optics, the Data, and the Phenomena—also have been preserved. Un-
like Aryabhata and Bhaskara, Euclid did not provide any preface to tell us why he
wrote his treatise. We do, however, know enough of the Pythagorean philosophy
to understand why they developed geometry and number theory to the extent that
they did, and it is safe to conclude that this kind of work was considered valuable
because it appealed to the intellect of those who could understand it.