Study Guide for the Final Math 330: History of Mathematics December 10, 2006. 1 Introduction What follows is a list of topics that might be on the Final. History is full of names of people, place names, and dates. Don’t worry: the only personal names, place names, or dates you need to memorize are given in this study guide. (Footnotes and parenthetical remarks in this study guide are given for information and clarification. You do not need to know them for the exam.) 2 Mathematicians 1. Thales from Miletus in Ionia, a Greek region on the coast of Asia Minor (in modern day Turkey). Said to have predicted an eclipse (of 585 BC) which actually occurred according to modern astronomers. Credited with the first use of proof in mathematics. Said to have brought geometry to Greece from Egypt. He is also credited with initiating the study of natural philosophy among the Greeks. He is given credit for several theorems including Thales’s theorem. 2. Pythagoras (6th century BC) from Samos, a Greek island close to Ionia. Lived later than Thales. He is said to have travelled to Egypt and Babylon, and learned from priests in the Egyptian and Babylonian religions. After coming back to the Greek world he founded a philosophical and religious society in Southern Italy (at Croton). He and his followers, called the Pythagoreans, were fascinated with mathematics, musical harmonies, and the concept of proportions and harmony in all aspects of life, and the universe or kosmos. The Pythagoreans included women as well as men. They discovered incommensurables, and at least some of the regular solids (including the dodecahedron). They were the first Greeks to make mathematics part of higher education. They were a very important influence on the later Greek mathematical tradition. In those days Mathematics, a term invented by Pythagoras, included arithmetic, geometry, music, and astronomy. At first, their knowledge was kept secret from outsiders. Pythagoras said that all is number, and discovered the simple mathematical ratios associated to harmonious notes in music. He is given credit for the first proof of the Pythagorean theorem (the theorem, but perhaps not the proof, was known to the Babylonians). 3. Hippocrates of Chios.1 Started out as a merchant, but during an extended stay in Athens due to a lawsuit, he decided to stay in Athens and do Geometry instead. Active in Athens 1Not to be confused with Hippocrates of Cos, the famous physician from about the same time who came up with the “Hippocratic oath”. 1 about the time of Socrates. Author of the first surviving fragment of Greek mathematics. In it, he showed that a certain lune is equal in area to a certain triangle. Thus he showed that some curved regions could be squared. He was apparently trying to square the circle, and his construction showed that some curved regions could be squared with straightedge and compass. Hippocrates was the first to write an Elements of geometry, now lost. He was also interested in duplicating the cube. He did not solve it completely, but he did reduce the problem to the problem of two mean proportionals: find x and y with a : x = x : y = y : 2a. The reduction of one problem to another is a common theme in mathematical research, and this example is the earliest known example of this phenomenon. (This idea was later used by Archytas, Menaechmus and Eratothenes to find a solution to the Delian problem.) 4. Eudoxus. The most famous mathematician associated with Plato’s Academy. Thought to have come up with some of the ideas incorporated into Euclid’s Elements, especially the theory of proportion and the method of exhaustion. (The theory of proportion is related to the modern idea of Dedekind cuts. The theory of exhaustion is based on a sophisticated use of proof by contradiction, and is an early example of a squeeze theorem. It is a precursor to integral calculus.) Eudoxus used the method of exhaustion to prove that the volume of a cone is one-third the base area times the height. Likewise, he proved that the volume of the pyramid is one-third the base area times the height. This last formula was known to the Egyptians, but Eudoxus seems to be the first to have proved it. Archimedes later applied the method of exhaustion to figure out the volume of a sphere, and to study other geometric volumes, areas, and centers of gravities. Eudoxus was also famous as an astronomer.2 5. Euclid. The first major mathematician who lived in Alexandria in Egypt, but he was likely educated in Athens. He lived after Alexander the Great (about 300 B.C.), and in his time the main center of mathematics shifted from Athens to Alexandria. The most famous mathemat- ics book of all times is Euclid’s Elements. King Ptolomy3, a successor to Alexander the Great who became king of Egypt, is said to have asked Euclid to show him a shorter way to learn geometry than the Elements. Euclid responded that there was no royal road to geometry. 6. Archimedes. Active in Syracuse in Sicily, before Sicily became part of the Roman Empire. (Syracuse was conquered by Rome in 212 BC on the day that Archimedes died.) He was amazingly talented in both pure mathematics and applied mathematics including engineering and early physics. You should know the following things he did: (i) Established the basics of hydrostatics and mechanics including the law of the lever and the law concerning the weight of a submerged object. (ii) Found areas, volumes, and centers of gravity associated with various surfaces and solids and gave proofs using the method of exhaustion. (iii) Approximated π 10 1 as 22/7 using inscribed and circumscribed 96-gons (he actually showed 3 71 < π < 3 7 ) and 2Eudoxus also worked on the Delian problem. In astronomy he developed an ingenious theory of planetary motion based on a complicated arrangement of 27 nested rotating spheres. He also wrote a book on geography. 3This Ptolomy is not the famous astronomer who lived later. 2 demonstrated that with enough skill and patience π could be approximated to arbitrarily high precision by using inscribed and circumscribed n-gons with n sufficiently large.4 (iv) Proved that the area of a circle was equal to the area of a certain triangle; describe this triangle.5 (v) Discovered and proved that the volume of a sphere was 2/3 that of the circumscribed cylinder. (vi) Discovered and proved that the surface area of a sphere was 2/3 that of the circumscribed cylinder. (vii) Showed that his spiral could be used to rectify and square the circle. (viii) Came up with a creative way to trisect an angle using a straightedge with two markings. (ix) Discovered the area of parabolic segments. Be able to answer the following: According to Archimedes’ hydrostatic principle, how much weight does a submerged object lose? What did he say when he discovered this law? What did this have to do with a crown allegedly of gold? How did Archimedes help in the defense of Syracuse? How did Archimedes die? What was inscribed on his tomb? Derive the modern formula for the area of the circle and the volume of the sphere using Archimedes result. 7. Apollonius (of Perga). Active in Alexandria (and Pergamum). A few decades younger than Archimedes. Famous for his work on conic sections. He was also famous for his work in astronomy. 8. Ptolomy. Lived in Alexandria which was then part of the Roman Empire. He was an astronomer, mathematician, and geographer. His most famous book is commonly called the Almagest (original name: Syntaxis Mathematica). This book gives a mathematical treatment of planetary motion, and the motion of the sun and the moon. His geocentric theory did not use simple circular orbits. What did it use? answer: epicyles. His complicated system represents the observations quite well. It needed to be complicated to be somewhat accurate since it was based on a flawed premise: the geocentric model of the solar system. What type of trigonometric function did he use? answer: the chord function. How is it related to sin θ? answer: Chord(θ) = 2 sin(θ/2). Draw a picture illustrating this. (Today we use a unit circle. Ptolomy used a circle of radius of 60 for convenience). He used base sixty and the idea of a circle having 360 degrees. These he inherited from the ancient Babylonians. 17 (He approximated π using an inscribed 360-gon: his value was 3 120 . He also cataloged the position of over a 1000 stars. Ptolomy also listed estimated latitudes and longtitudes of major known locations on the earth.) 9. Diophantus. Lived in Alexandria which was then part of the Roman Empire. First person to use algebraic symbolism instead of just words (syncopated algebra instead of rhetorical algebra). His book, the Arithmetica was quite sophisticated, certainly the most sophisticated algebra in the Greek speaking world. Solving equations for him requires methods quite a bit different than the standard methods taught today in algebra. This is mainly because he didn’t just want real number solutions, but wanted rational or integer solutions. His work later inspired Fermat to invent number theory. In fact, there is a modern branch of number theory called Diophantine equations. 10. Hypatia and the commentators Hypatia was more than just a mathematician. She was also a neo-Platonist philosopher and was known as a charismatic teacher. She wrote 4Archimedes and other ancient mathematicians did not use the letter π, instead they talked directly about the ratio of the circumference to the diameter or they referred to other related ratios.