Mathematics in Ancient Greece

For school purposes only.

1. MATHEMATICS IN ANCIENT GREECE

2. T he Greek mathematician, Pythagoras, used small rocks to signify numbers while working on mathematical equations. Trivia

3. Thales of Miletus Pythagoras of Samos Pythagorean School Plato and his influence on Mathematics and Science Aristotle’s Logic Eudoxus of Cnidus Euclid and his “Elements” Archimedes of Syracuse Method of Exhaustion Discussions about…

4. Often thought of as the first western scientist. Olive Trader based in Miletus, a city-state in Anatolia Contact with Babylonian traders/scholars probably led to his learning some geometry and an attempt to organize his discoveries. Stated some of the first abstract propositions.(Angles at the base of an isosceles triangle are equal, any circle is bisected by it’s diameter, and a triangle inscribed in a semi-circle is right angled or a.k.a. “Thales’ Theorem” ) THALES OF MILETUS (624-546 BC)

5. He is one of the earliest Greek mathematicians and many historians regard him as the first true mathematician. Very little is known about Thales; including when he was born d when he died. He used geometry to solve problems including determining the height of ancient Egyptian pyramids based on the lengths of its shadows. The “Theorem of Thales” and the “Intercept of Theorem”, an important theory in geometry, are attributed to him. THALES OF MILETUS (624-546 BC)

6. More of mystic/philosopher than a mathematician. Core belief that number is fundamental to nature. Motto: “All is number”. Emphasized form, pattern, proportion. Pythagoreans essentially practiced a mini-religion (they were vegetarians, believed in the transmigration of souls, etc.). Using such intervals to tune ,musical instruments(in particular pianos) is still known as the Pythagorean tuning . -Identical strings whose lengths are in the ratio 2:1 vibrate an octave part. - A perfect fifth corresponds to the ratio 3:2 -A perfect fourth corresponds to the ratio 4:3 Pythagoras of samos (572-497 BC)

7. THEOREMS 21-34 in Book IX of Euclid’s Elements are Pythagorean in origin: Theorem(IX.21). A sum of even numbers is even. (2+2= 4 ) Theorem(IX.27). Odd less odd is even (5 - 3 = 2 ) T he Pythagoreans studied perfect numbers: equal to the sum of their proper divisors (e.g. 6 = 1+2+3). Theorem(IX.36). If is prime then s perfect   Pythagoras of samos (572-497 BC)

8. The school of Pythagoras was every bit as much a religion as a school of mathematics. For example, here are some rules: To abstain from beans. Not to pick up what has fallen. Not to touch a white cock. Not to stir the fire with iron. Do not look in a mirror beside the light. Vegetarianism was strictly practiced probably because Pythagoras preached the transmigration of souls. Pythagorean school

9. Life in the Pythagorean society was more –or-less egalitarian. The Pythagorean school regarded men and women equally. They enjoyed a common way of life. Property was communal. Even mathematical discoveries were communal and by association attributed to Pythagoras himself – even from the grave. Hence, exactly what Pythagoras discovered is difficult to ascertain. Pythagorean school

10. Plato was also one of the ancient Greece’s most important patrons of mathematics. He was convinced that geometry was the key to unlocking the secrets of the universe. There was sign above the entrance of his Academy “Let no-one ignorant of geometry enter here”. The first 10 years of the 15 year course at the Academy involved the study of mathematics and science, including plane, solid geometry, astronomy, and harmonics. Plato became known as the “marker of mathematicians”. Platonic Solids ( Tetrahedron, octahedron, cube, icosahedron, dodecahedron) Plato and his influence on mathematics and science

11. ARISTOTLE’S LOGICAL WORKS -Contain the earliest formal study of logic that we have. It is therefore all the more remarkable that together they comprise a highly developed logical theory, one that was able to command immense respect for many centuries. In the last century, Aristotle’s reputation as a logician has undergone two remarkable reversals. Aristotle’s logic; today, very few would try to maintain that it is adequate as a basis for understanding science, mathematics, or even everyday reasoning. Aristotle’s logic

12. Eudoxus was arguably the most prolific of the pre- Euclidean mathematicians. He is famous for explaining how to calculate with ratios of lengths . e.g. A : B is greater than C : D if there are positive integers m , n, such that mA nB and mC nD .   Eudoxus of cnidus (390-337BC )

13. Euclid worked in the library of Alexandria, now on the north coast of Egypt. The library is thought to have been constructed around 320BC as a means of organizing the knowledge of the world and for the demonstration of G reek power. Although the library was destroyed, the knowledge was preserved. Euclid’s Elements was the most influential mathematics text in history. it is better thought of as a compilation of the work of earlier mathematicians rather than an original work. Partly due to its fame it was edited and added to over the centuries, eclipsing and subsuming other important works. Theon of Alexandria and his daughter Hypatia famously edited the elements in an attempt to improve the work and make it easier to follow. Euclid and his “elements”

14. Euclid and his “elements” The earliest (almost) complete copies of Euclid date from the ninth century. The pictured version, written in greek , is at the Vatican, and does not contain some of Theon’s edits, thus showing that multiple versions of the text were circulating during the first millennium AD.

15. EDITIONS AND VARIATIONS THAT HAVE BEEN PRODUCED

16. BOOK I: Consists of 48 theorems, the final two being Pythagoras’ and its converse. Euclid organized this book with the goal of proving this important result in a thorough manner. It begins with a list of definitions, axioms, and postulate. 1. Given any two points, a straight line can be drawn between them 2. Any line may be indefinitely extended 3. Given a center and a radius, a circle may be drawn 4. All right angles are equal to each other 5.If a straight line crosses two others so that the angles on the same side make less than two right angles, then the two lines meet on that side of the original. Euclid and his “elements”

17. BOOK II: Geometric solutions to problems: constructing a length, even if incommensurate with the unit BOOK III: Mostly theorems regarding circles(e.g. Thales’ theorem) and tangency Probably most of the material came from Hippocrates BOOK IV: Construction of regular 3, 4, 5, 6, 15-sided polygons in and around a circle Also from Hippocrates BOOK V: Ratios/ magnitudes á la Eudoxus If a : b = c : d and c : d = e : f then a : b = e : f Euclid and his “elements”

18. BOOK VI: Ratios of magnitudes applied to geometry Triangles with equal angles have corresponding sides proportional BOOK VII: Divisibility and the Euclidean algorithm BOOK VIII: Number and progressions, geometric sequences BOOK IX: Numbers: even/odd + perfect numbers BOOK X: Discussion of commensurable and incommensurable lines Long and difficult, possibly derived from Theaetetus Euclid and his “elements”

19. BOOK X I: Solid geometry (lines/planes in 3D) BOOK X II: Ratios of areas volumes ( Eudoxus ) BOOK XIII: Construction of regular polyhedra inside a sphere and their classification. Euclid and his “elements”

20. Archimedes is arguably the greatest of the ancient Greek mathematicians. His place of birth and residence (Syracuse) is in modern-day Sicily,the large island at the foot of the Italian peninsula. He famously helped defend Syracuse against the Romans using catapults and he died at the hands of the Romans after they captured the city. He travelled to Alexandria in his youth and perhaps met and studied with eminences such as Eratosthenes. He is rare among ancient Greek mathematicians for being highly practical. He is credited with a large number of inventions and technical innovations, including Archimedes’ screw. Archimedes of syracuse (287-212 BC )

21. Is a technique that the classical Greek mathematicians used to prove results that would now be dealt with by means of limits. 1. Archimedes ‘ formula for the area of a circle -the way Archimedes formulated his Proposition about the area of a circle is that it is equal to the area of a triangle whose height is equal to its radius and whose base is equal to its circumference: (1/2)(r x 2 π r)= 2. Convexity -   The method of exhaustion

22. 3. Archimedes’ Axioms -Axioms for Archimedes are assumptions, not to be proven. 4. The proof Archimedes formula -The basic idea is to approximate the area of a circle from the above and below by circumscribing and inscribing regular polygons of a larger and number of sides. The method of exhaustion

23. Luke Mastin ,(2010);The story of mathematics; https ://www.storyofmathematics.com / https://www.math.uci.edu/~ndonalds/math184/greece.pdf References: