Credited with formulating the method of exhaustion for approximating a circle by polygons

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1 MATH 300 History of Mathematics Figures in Greek Mathematics Sixth Century BCE Thales of Miletus May have formulated earliest theorems in geometry (e.g., ASA) Predicted an eclipse in 585 BCE Pythagoras of Samos Established a brotherhood of disciples (Mathematikoi) in Crotona, which lasted 600 years Developed a mathematical theory of musical harmony based on small number ratios Credited with proofs of a number of theorems (incl. the one that bears his name) Held up number as the unifying principle of all reality Fifth Century BCE Zeno of Elea Monist philosopher: all is one Wrote a collection of Paradoxes concerning the infinite and the infinitesimal Antiphon of Athens Credited with formulating the method of exhaustion for approximating a circle by polygons Hippocrates of Chios Discovered quadrature of lunes Hippasus of Metapontum Credited with the discovery of the irrationality of 2 Theodorus of Cyrene Discovered that n is irrational for integer n not a perfect square (n 17) Hippias of Elis Invented the quadratrix curve for trisection of angles Bryson of Heraclea Showed how inscribed and circumscribed polygons are used to approximately square the circle

2 Fourth Century BCE Archytas of Tarentum Named and studied the harmonic mean Solved the duplication problem by intersecting a cylinder sphere and cone Plato of Athens Established a school of philosophy, The Academy, in Athens (Let no one ignorant of geometry enter here) Recognized that mathematical thought expresses the most definite and precise truths; emphasized value of proof Associated with identifying the five regular polyhedra Thaetetus of Athens Inspiration for Books X, XIII of Euclid s Elements on incommensurable ratios, solid geometry Eudoxus of Cnidos Mathematical astronomer: compiled a comprehensive star map Inspiration for Book V of Euclid s Elements on comparison of incommensurables Aristotle of Athens Wrote influential works on logic, rhetoric Read and commented on earlier mathematical philosophers Menaechmus of Alopeconnesus First to identify the conic sections, used to solve the duplication problem Said to have been tutor to Alexander the Great Aristaeus Wrote a five-book work on conic sections Eudemus of Rhodes First historian of mathematics, wrote histories of arithmetic, geometry, and astronomy

3 Third Century BCE Euclid of Alexandria Thought to have been one of the first scholars to work at the Library in Alexandria Wrote the most influential work in all of mathematics, the Elements (of Geometry) in 13 books The Elements superseded all previous work and set standard for mathematical writing Organized mathematics axiomatically, providing proofs for all propositions Also wrote other works that survive (Data, On Divisions, Optics, Phenomena) Aristarchus of Samos Noted mathematician and astronomer, who formulated a heliocentric model of the universe The Sun was the center of the circular orbits of the six known planets Archimedes of Syracuse Greatest mathematician of antiquity Famous for applying mathematical principles in the defense of Syracuse Author of 9 mathematical works: On Sphere and Cylinder Measurement of the Circle On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Method Quadrature of the Parabola The Sand Reckoner On Floating Bodies Nicomedes Invented the conchoid curve to solve the trisection and duplication problems Used the quadratrix curve of Hippias to square the circle Eratosthenes of Cyrene Worked at the Alexandrian Library Developed the sieve method for identifying prime numbers Measured the circumference of the Earth Made contributions to astronomy and geography Apollonius of Perga The Great Geometer His Conics set a unified theory of these curves; introduced terms parabola, ellipse, hyperbola Also author of six other extant works (incl. Cutting off a Ratio) and a number of lost works

4 Second Century BCE Zenodorus of Athens Studied the isoperimetric problem in 2 and 3 dimensions Studied the reflective properties of curved surfaces Hipparchus of Rhodes Mathematical astronomer considered the Father of Trigonometry Compiled the first table of chords Studied the fundamental relations of spherical triangles First Century BCE First Century CE Heron of Alexandria Applied geometer, wrote many works, e.g, on surveying, engineering, optics Developed a method for computing the area of a triangle from the three side lengths Nicomachus of Gerasa Pythagorean author of an Introduction to Arithmetic which was influential for centuries Second Century CE Menelaus of Alexandria Developed spherical geometry and its applications to astronomy Claudius Ptolemy of Alexandria Most influential Greek astronomer Almagest: developed a geocentric model of the universe which became widely accepted until 17th c.; 7 planets (incl. Sun and Moon) move on epicyclical orbits around the Earth; theory compared with observations via extensive use of spherical trigonometry; prepared an expanded table of chords Geographica: comprehensive treatise on cartography and world geography Tetrabiblos: authoritative work on astrology and preparation of horoscopes

5 Third Century CE Diophantus of Alexandria Author of Arithmetica, compilation of 130 numerical problems, mostly of linear and quadratic type Used a syncopated notation for abbreviating mathematical expressions Only integer or rational solutions are provided, all via ad hoc methods Fourth Century BCE Pappus of Alexandria Author of the Mathematical Collection, a handbook reviewing many results in geometry Theon of Alexandria Wrote important commentaries on Euclid s Elements, Ptolemy s Almagest Father of Hypatia Hypatia of Alexandria Female philosopher who worked at the Alexandrian Library Contributed to Theon s commentaries Assisted Theon in preparing an edition of Euclid s Elements One of the last generation of philosophers at the Library Fifth Century CE Proclus of Constantinople Wrote a Commentary on Euclid that provides much of the history of Greek mathematics Sixth Century CE Eutocius of Ascalon Wrote commentaries of works of Archimedes and Apollonius Simplicius of Cilicia Wrote commentaries on works of Aristotle and Euclid

Preparation suggestions for the second examination

Math 153 Spring 2012 R. Schultz Preparation suggestions for the second examination The second examination will be about 75 per cent problems and 25 per cent historical or short answer with extra credit