By Frater Apollonius 4°=7□

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Of Geometry

The story that Ptolemy, the king of Alexandria, asked Euclid if there was a royal road to geometry, to which he replied there was none; is both problematic and emblematic. While Proclus reported this about 450 AD, over six hundred years later, a markedly different version of this story comes from Stobaeus. Stobaeus (late 5th century AD) states that it was Alexander the Great (356-323 BC) who asked this question of another mathematician by the name of Menaechmus. But, the second version of this story is even less reliable than the first. Consider that Alexander was the private pupil of Aristotle, who had been Plato's student. Plato's Academy, in which Aristotle studied for over twenty years, reportedly proclaimed the following over its entrance: "Let no one untrained in Geometry enter here." Now, this tale, too, may be spurious. But one also should note that Plato started a work Epinomis, posthumously finished by Philippus of Mende, which laid out the appropriate mathematical curriculum for an ideal head of state.

The point here is that Alexander, with the benefit of Aristotle's teaching, would have at least received a primary introduction to the elements of geometry (It is said by Proclus that Hippocrates was the first to compile Elements, thus preceding Euclid), and in view of his training, would have never have asked such a question in the first place. This point is further supported by the following passage from Plutarch's life of Alexander The Great (1st Century AD) that goes on to quote from a letter actually authored by Alexander:

"It seems clear that Alexander was instructed by his teacher not only in the principles and ethics of politics, but also in those secret and more esoteric studies which philosophers do not impart to the general run of students, but only by word of mouth to a select circle of the initiated. Some years later,..., he learned that Aristotle had published some treatises dealing with these esoteric matters, and he wrote to him in blunt language and took him to task..."

The introductory quote is also highly emblematic of the role that mathematics, and specifically geometry, would come to play in the development of Greek civilization. While the Greeks did not have a monopoly on the subject, they were able to take mathematical thinking to an entirely new level and into an unique direction to which much is owed. But, the history of Greek mathematics, which stretches back to 600 BC, abounds with such problematic anecdotes, which is why during the past hundred years, it has become fashionable to dismiss much of it out of hand. But the naysayers miss the mark in two key respects.

The first is that the history of mathematics, apocryphal tales and all, was to have a profound effect on the Middle Ages and the Renaissance, a time not just of discovering the new but of recovering ancient learning and lore. Those who rediscovered the beginnings of mathematics through their study of Plato, Plutarch, Pliny, Iamblichus and Proclus, drank in every word.

The second brings us back to Plato. Plato, who taught Aristotle, was himself taught by Socrates. Plato's dialogues were, in fact, Socratic dialogues that Plato formalized through transcription into philosophical arguments. What the reader witnesses though these dialogues is a great oral tradition that in antiquity was a primary means of imparting instruction and knowledge. Texts were used and available, but there was also an esoterical oral tradition central to teaching mathematics and philosophy which is clearly indicated by Plutarch's reference. This tradition, which as will see, dated back to the time of Pythagoras, must have continued well after the transmission of Socratic ideas through Plato's written dialogues and lasted at least until Justinian closed Plato's Academy in 529 AD. Elsewhere, such as Alexandria, it may have continued even longer. Hence, an oral tradition of teaching mathematics may have extended an additional eight hundred years after Plato's death. This being so, some of the documentation that comes to us from much later sources such as Nicomachus, Iamblichus and Proclus in the late first century, and from the third and fifth centuries, may well contain ideas culled from this oral teaching as well as preceding texts that are long lost, a tradition that was possibly diminished yet still extant. This oral tradition would eventually dim with the advent of a larger historical decline. But one can well imagine that at one point, a standard work such as Euclid, would have been taught in conjunction with a parallel verbal curriculum.

An inherent problem obtains with any oral tradition. That problem is entropy. With time, distortion and loss of energy alters the message being transmitted. However, such effects were not lost on the Greeks or other ancient cultures. Precautions were taken, two of which were initiation and secrecy. Mnemonic devices (poems) and riddles were also used. Furthermore, over time the encoding of the oral message could be changed to preserve the value of the information. In the two versions of the tale cited earlier, similar information is conveyed through two different contexts. The first had the conversation occurring between Euclid and Ptolemy Soter, the second, between Alexander and Meneachmus. However, the exchange in each case is identical: Question: "Is there a royal road to Geometry?" Answer: "No. There is no short cut!"

The history of Greek mathematics can be divided into roughly four periods. (fig. 1)

(fig. 1)

The first is the age of Thales and Pythagoras (600 BC to 400 BC). The second is known as the Platonic Age (400 BC to 300 BC); the third, the Alexandrian Age (300 BC to 1 AD) and the fourth, the Ptolemaic Age or post-Euclidean Age (1 AD to 300 AD).

Thales of Milete (585 BC) was said to be the first of the seven sages to arise as the Greek polis emerged from a former feudal system. He was a statesman, philosopher, mathematician and astronomer who at some point traveled the Mediterranean basin in search of knowledge and who returned with extensive empirical observations from the civilizations of Babylon and Egypt. Thales then organized some of this information into the beginnings of a formal mathematical system. He is, as a result, regarded as the father of Greek mathematics.

Thales was famous in antiquity for predicting a total solar eclipse that occurred on May 28, 585 BC. Herodotus reports that this occurred during the battle of the Halys between the Medes and Lydians. At the time the Ionians, Thales was Ionian, were allied with the Lydians. The battle came to a sudden stop, as day turned to night, and the cessation of hostilities led to a lasting peace between both sides. This story is also reported by Diogenes Laertius, who states that Xenophanes (philosopher and poet, c. 570-490 BC) was greatly impressed by this feat. Thales, who had spent time studying astronomy in Babylon, may have returned with intimate knowledge of astronomical observations collected over many centuries by the Babylonians. They may have included certain patterns in solar and lunar eclipses, one being that a solar eclipse will occur 23 1/2 synodic months after a total lunar eclipse. Thales may have predicted the eclipse by counting off the months between events.

Thales is credited with at least four principal geometrical discoveries. He was the first to observe that a circle is divided into two equal parts by its diameter. He also recognized the equality of base angles in the isosceles triangle and observed that when two straight lines intersect each other, the opposing angles are also equal. He provided the definition of congruence which states that any two triangles with one similar side and two similar angles are equal. (fig. 2)

(fig. 2)

There may be other achievements that can be attributed to him, but certainly one of the most important is that he was Pythagoras' teacher.

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