Greek Mathematicians: Ancient Influencers in Mathematical Sciences, Study notes of Engineering

Download Greek Mathematicians: Ancient Influencers in Mathematical Sciences and more Study notes Engineering in PDF only on Docsity! Greek (Polymaths) Mathematicians Panos M. Pardalos Center for Applied Optimization, ISE Department 303 Weil Hall, University of Florida, Gainesville, FL 32611 August 28, 2006 http://www.ise.ufl.edu/pardalos 1 1 From ancient to moderm times It is well acknowledged that the ancient Greek mathematicians had a great influence in the establishment of foundations and development of mathematical sciences. In the past century, influential scientistists like C. Caratheodory and C. Papakyriakopoulos re- established the presense of Greeks in the inter- national mathematical community. 2 4 Heraclitus - the fire priest (540 BC - 480 BC) This cosmos was not made by gods or men, but always was, and is, and ever shall be ever- living fire. You could not step twice into the same river; for other waters are ever flowing on to you. If you do not expect the unexpected, you will not find it, for it is trackless and unexplored. The road up and the road down is one and the same. Lambek, J., 1982, The Influence of Heraclitus on Modern Mathematics, Scientific Philosophy Today, J. Agassi and R.S. Cohen, eds., Dordrecht, Reidel, 111?”122. 5 5 Archimedes What else did he do besides cry Eureka? His research into the equilibriun of floating bodies is the first recorded work in hydrostat- ics and naval architecture. π : Defined as the proportion “circumference: diameter”. Archimedes found the first decimal digits of π. Archimedes used two regular 96-sided polygons, one cir- cumscribing the circle, one inscribing. With the aid of some elegant geometry and arithmetic, he estimated the perimeters of these polygons and therefore the circumfer- ence of the circle between them. 6 6 Anaxagoras Anaxagoras of Clazomenae (499BC - 428BC) He was imprisoned for claiming that the Sun was not a god and that the Moon reflected the Sun’s light. There is no smallest among the small and no largest among the large, But always something still smaller and something still larger. Anaxagoras was the he first mathematician who is on record as having attempted to square the circle. Plutarch, in his work On Exile which was written in the first cen- tury AD, says: There is no place that can take away the happiness of a man, nor yet his virtue or wisdom. Anaxagoras wrote on the squaring of the circle while in prison. 7 8 Democritus of Abdera (460BC - 370BC) Democritus is best known for his atomic theory but he was also an excellent geometer. Quotations by Democritus: I would rather discover one scientific fact than become King of Persia. Everything existing in the Universe is the fruit of chance and necessity. Nothing exists except atoms and empty space; everything else is opinion. He wrote On numbers, On geometry, On tangencies, On mappings, On irrationals but none of these works survive. ... Democritus was the first to state the important propo- sitions that the volume of a cone is one third of that of a cylinder having the same base and equal height, and that the volume of a pyramid is one third of that of a prism having the same base and equal height; that is to say, Democritus enunciated these propositions some fifty years or more before they were first scientifically proved by Eudoxus. 10 9 Plato (427BC - 347BC) Plato is one of the most important Greek philosophers who founded the Academy in Athens, an institution devoted to research and instruction in philosophy and the sciences. His works on philosophy, politics and math- ematics were very influencial and laid the foundations for Euclid’s systematic approach to mathematics. Plato’s Academy flourished until 529 AD when it was closed down by the Christian Emperor Justinian who claimed it was a pagan establishment. Having sur- vived for 900 years the Plato’s Academy of Athens is the longest surviving university known. Let no one ignorant of Mathematics enter here. [Said to have been above the doorway of his Academy.] 11 Perhaps the best overview of Plato’s views can be gained from examining what he thought that a proper course of education should con- sist. Here is his course of study: ... the exact sciences - arithmetic, plane and solid geometry, astronomy, and harmonics - would first be studied for ten years to familiarise the mind with relations that can only be apprehended by thought. Five years would then be given to the still severer study of dialectic. Dialectic is the art of conversation, of ques- tion and answer; and according to Plato, dialectical skill is the ability to pose and answer questions about the essences of things. The dialectician replaces hy- potheses with secure knowledge, and his aim is to ground all science, all knowledge, on some unhypothetical first principle. 12 12 Eratosthenes of Cyrene (276BC - 194BC) Eratosthenes was a Greek mathematician who is famous for his work on prime numbers (Sieve of Eratosthenes) and for measuring the diameter of the earth. Eratosthenes also measured the distance to the sun as 804,000,000 stadia and the distance to the Moon as 780,000 stadia. He computed these distances using data obtained during lunar eclipses. Ptolemy tells us that Eratosthenes measured the tilt of the Earth’s axis with great accuracy obtaining the value of 11/83 of 180degrees, namely 23degrees 51’ 15”. Eratosthenes made many other major contributions to the progress of science. He worked out a calendar that included leap years, and he laid the foundations of a systematic chronography of the world when he tried to give the dates of literary and political events from the time of the siege of Troy. He is also said to have compiled a star catalogue containing 675 stars. 15 13 Apollonius of Perga (262BC - 190BC) Apollonius was known as ”The Great Geometer”. His works had a very great influence on the development of mathematics and his famous 8 books Conics introduced the terms parabola, ellipse and hyperbola. A conic or conic section is one of the three curves: a parabola, hyperbola or ellipse which one can obtain by intersecting a plane with a (double sided) cone. Other books by Apollonius: Cutting of a ratio (in two books) Cutting an area (in two books) On determinate section (in two books) Tangencies (in two books) Plane loci (in two books) Quick Delivery On verging constructions (in two books). 16 For example, in Tangencies, Apollonius shows how to construct the circle which is tangent to three given circles. More generally he shows how to con- struct the circle which is tangent to any three objects, where the objects are points or lines or circles. In the book Quick Delivery, Apollonius obtained an ap- proximation for π better than the 223/71 < π < 22/7 known to Archimedes. Problem: Prove that the locus of points whose distance from a afixed point A is a multiple of its distance from another fixed point B is a circle (one of Apollonius circles). 17 16 Heron of Alexandria (10 - 75) Heron or Hero of Alexandria was an important geometer and worker in mechanics who invented many machines ncluding a steam turbine. His best known mathematical work is the formula for the area of a triangle in terms of the lengths of its sides: If A is the area of a triangle with sides a, b and c and s = (a + b + c)/2 then A2 = s(s − a)(s − b)(s − c). 20 17 Hypatia of Alexandria (370 - 415) Hypatia was the first woman to make a substantial con- tribution to the development of mathematics. She was killed by a fanatical Christian sect. This event marked the beginning of the de- cline of Alexandria as a major center of ancient learning. M Dzielska, Hypatia of Alexandria (Harvard, 1995). 21 18 Kallinikos (7th-century) In order to maintain all the lands that its precessor once controlled, Byzantium had to put all its forces in one direction : to defend. Roman army’s only goal was to conquer. On the contrary, Byzantine army was in a decisive position. Barbarians from all over saw a great opportunity in the Byzantium, due to its richness and fertility. Greek, or liquid fire: The Byzantines’ most impor- tant weapon, which played a decisive role in all naval operations. It consisted of a developed recipe of older chemical compounds attributed to Kallinikos, a Greek engineer from Helioupolis in Syria. Liquid fire contained sulphur, nitrogen, and naphtha (petrol) and other sub- stances and put fear into the enemy fleet, since it burnt even on water. In 678 The Byzantines utterly destroyed a Muslim fleet (it is believed over 30,000 men were lost) and also in 717-718, when Caliph Suleiman attacked Constantinople. Most of the Muslim fleet was once again destroyed by Greek Fire, and the Caliph was ultimately forced to flee. 22