History of Mathematics Week 3 Notes Ionia and the Pythagoreans • Mystery surrounds Thales (6th century BC). To him are attributed many things, like predicting a solar eclipse, that seem historically implausi- ble. • Thales traveled to Egypt, Chaldea, and possibly Babylon to exchange ideas • Theorem of Thales - A triangle inscribed in a circle is a right triangle • other theorems attributed to Thales - (1) a diameter bisects a circle (2) the base angles of an isosceles triangle are equal (3) the pairs of vertical angles formed by intersecting lines are equal (4) AAS congruence • No evidence of deductive approach to mathematics • Diogenes Laertius reports Thales computed heights of pyramids using shadows. • Pythagoras traveled to Egypt and Babylon, maybe even India, and may have studied under Thales • Head of a communal group of mathematicians known as the Pythagore- ans. They were vegetarians due to a belief in transmigration of souls and did not eat lentils. • Pythagoras is rumored to have coined the terms Philosophy, love of wisdom, and Mathematics, that which is learned • Pythagoras created the liberal education • Motto \All is number", logo pentagram • Perhaps the Pythagoreans were the first to prove the Pythagorean the- orem, hence earned naming rights. They also knew the rule of (m2 − 1), 2m, and (m2 + 1 for Pythagorean triples, even though this may have been known to Babylonians. • Dividing a line into extreme and mean ratio or \section" • Numerology- 1 is the generator and number of reason, 2 is female and number of opinion, 3 first male number and number of harmony, 4 justice and retribution, 5 marriage as union of male and female, 6 creation, 10 is the 1 sum of generators of dimensions (not anatomical) • Developed astronomical model involving earth rotating around a cen- tral fire with a counterearth added to make ten heavenly bodies (in order to be numerologically correct) • Philolaus - All things which can be known have number; for it is not possible that without number anything can be conceived or known. • Quantitative musical theory with relative lengths of vibrating strings • Pythagorean science was an odd congeries of sober thought and fanciful speculation. • Greeks used two systems of numeration: Attic and Ionian. Attic re- sembled Roman numerals, with a combination of base 5 and 10. Ionian had different symbols for 1-9, 10-90, and 100-900 (alphabetic characters), but was nearly positional, with extra ticks and dots added to larger places for clarity. The Heroic Age • Anaxagoras imprisoned for impiety on suggesting the moon was an in- habited planet which took light from the sun, which he claimed was not a diety. • Anaxagoras had a book On Nature which could be bought in Athens for a drachma. • Anaxagoras worked on squaring the circle while in prison, which is the oldest historical record of the problem. • Doubling the cube is reported to have roots in the plague. A delegation went to consult the Oracle at Delos, but it demanded the altar to Apollo be doubled. • In Athens the problem of trisecting an angle was developed. • Hippocrates was a geometer who wrote a book on its elements, to whom Eudemus attributed the theorem that similar segments of circles are in the same ratio as the squares on their bases. • Hippocrates used this theorem to discover areas of lunes, and so gave the first measurement of curves. • Hippias invented a curve that could be used to trisect an angle and square the circle 2 • Archytas the Pythagorean designated the quadrivium • Archytas showed how to double the cube using essentially analytical geometry • Incommensurable magnitudes were discovered, likelyp around 410 BC (Hindu claims are unsupported). Typical accounts are that 2 was the first one found, although it is suggested that the golden section may have been the first. • Zeno's Paradoxes - Achilles and the Hare; an arrow is at each time at rest, so motion is an illusion; there cannot be a minimum time interval • Greeks had a geometric algebra, whereby equations like ax = bc are statements about areas • Democritus of Abdera wrote On Numbers, On Geometry, On Tan- gencies, On Mappings, On Irrationals, On the Pythagoreans, On the World Order, On Ethics. • Democritus was primarily a chemist, but viewed solids and shapes with some sort of \geometric atomism", which enabled infinitesimal techniques. He is credited with the theorem that the volume of a pyramid is Bh=3, even though he was not the first to state or prove it. 3.