Ancient knowledge of the sciences was often wrong and wholly unsatisfactory by modern standards. However not all of the knowledge of the more learned peoples of the past was false. In fact without people like Euclid or Plato we may not have been as advanced in this age as we are. Mathematics is an adventure in ideas. Within the history of mathematics, one finds the ideas and lives of some of the most brilliant people in the history of mankind’s' populace upon Earth. The period of Greek Mathematics in which some very significant discoveries were made was from 600 B.C.E to about 300 B.C.E. (Allen, 1997, para. 2). For the earliest period of Greek mathematics, there are few primary sources of information. We are therefore forced to rely extensively on the Euclidean Summary of Proclus. This contains an outline of the development of Greek geometry from early times until Euclid (Allen, 1997). Thales is often considered to be one of the first Greek mathematicians. The proposition known as the “Theorem of Thales” states, “The diameter of a circle always subtends a right angle to any point on the circle” (Thomas, 1991, p. 119). There are few primary sources that are able to describe early Greek Mathematics, hence the reliance on Proclus, but it is Thales who was accredited with the first proofs of this and therefore he is still very significant. These discoveries, along with 4 others that have been rumored to be proven by Thales, were the first rationally organized proofs in Geometry (Allen, 1997, para. 5). Whether it was Thales or not who put this in a logical structure or if it was added by other Greeks up to two centuries later is still up for debate. This is still however a significant start to the development of Greek Mathematics and Geometry. Pythagoras was perhaps the most significant early mathematician in the development of Greek mathematics (O’Connor, Robertson, n.d., para. 3). It is however impossible to distinguish between the works of Pythagoras and his followers the Pythagoreans. The Pythagoreans worked on many properties of numbers and also looked into the realms of philosophy but perhaps their most significant discovery was the Pythagorean Theorem. This theorem states, “in any right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the two sides” (O’Connor, Robertson, n.d., para. 4). This discovery is probably the most significant in this early period of Greek mathematics and is now universally used theoretically and also in many practical realms of engineering and science. This is where the first criticism can be leveled against the significance of both Pythagoras and Thales. Both of these were travelers and these two previously mentioned significant discoveries have been thought to have been known by the Babylonians some 1000 years earlier. This however is a very limited criticism, of Pythagoras in particular, since it is likely that despite all of the information being known to the Babylonians, Pythagoras and Thales were most likely to have been the first to have written down the most general proofs (O’Connor, Robertson, n.d., para. 6).
The next significant period of Greek mathematics is called the Classical period. In this period, the majority of the writing from the period came from two men, Euclid and Apollonius (Thomas, 1991, p. 126). Euclid in particular is considered to one of the most significant contributors to Geometry. His main, and perhaps most known, work is called the Elements. This consisted of thirteen books and contains 467 propositions (Thomas, 1991, p. 127). It is not thought that Euclid necessarily discovered all of the proofs and propositions but it is the logical presentation that really set it apart. The first influence can be found in the elements laid out as an axiomatic system. An axiomatic system being a logical system which possesses and explicitly stated set of axioms from which...