sign in sign up
<<
Home , Method of exhaustion, Rotating spheres

Study Guide for the Final

Math 330: History of

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 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. (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. .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 . In it, he showed that a certain lune is equal in to a certain triangle. Thus he showed that some curved regions could be squared. He was apparently trying to square the , 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 , 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 ’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 .) Eudoxus used the method of exhaustion to prove that the volume of a 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. later applied the method of exhaustion to figure out the volume of a sphere, and to study other geometric volumes, , 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 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 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 (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 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. . 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 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. 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. 5This fact seems to have been known, but probably not proved, before Archimedes.

3 commentary on earlier mathematicians. At this time, scholars were trying to keep alive the earlier scientific and mathematical culture and so would write commentaries to make the earlier works accessible. Of course, they would sometimes add their own contributions to these earlier subjects. Hypatia was the last major intellectual figure of Alexandria and the Roman Empire. (Even after the western Roman Empire fell, Byzantine mathematicians wrote commentaries to keep the tradition alive.) She was killed by a fanatical mob of Christians (in 415). (At that time, pagan scholars were not welcome in Alexandria, and many left. This marked the beginning of the end of Alexandria as the major center of learning.)

11. and Nine Chapters on the Mathematical Art. Liu Hui wrote (in 263 AD) commentary and additions to the Chinese classic: Nine Chapters on the Mathematical Arts. (He lived after the Han dynasty.) The Nine Chapters on the Mathematical Arts (Jiuzhang suanshu) was a practical textbook of mathematics (dating from the Han dynasty or earlier) used to train civil servants who had to take tests to get their positions. The Nine Chapters is the most important early work of . Its influence was so strong that it might be called the “Euclid of China”, but without proofs. It consists of word problems (246 problems, surveying, trade, taxation, distribution problems, construction of canals, and other problems you might see in as word problems in a textbook today.) Topics: Arithmetic including rules for common fractions. Areas. Proportion, percentages and exchange rates. Square and cube roots. Volumes (but not the volume of a sphere). Linear and quadratic equations. Pythagorean theorem (which the Chinese called the Gougu rule), and examples of Pythagorean triples. Similar triangles. Some interesting features; use of matrices to solve systems of linear equations (up to six equa- tions and six unknowns). The use of negative numbers, especially for matrix manipulations. Correct rules for adding and subtracting negative numbers. Liu Hui supplied explanations, some of which can be regarded as proofs, for much of the Nine Chapters. He also pointed out that 3, the value of π used in the book, was not a good approximation and gave an excellent approximation (3.14159) using n-gons (in fact a 3072-gon) in a manner similar to Archimedes.

12. Brahmagupta (died 670). Indian mathematician and astronomer (head of an astronomical observatory). He came up with an interesting formula for cyclic quadrilaterals. He was an early user of negative numbers and zero, and gave rules for them. In modern notation: a + 0 = a, a − 0 = a, a0 = 0, 0 − 0 = 0, product or quotient of two negatives is a positive, product or quotient of a positive with a negative is a negative, 0/a = 0. He was the first that I know of to give rules for multiplying and dividing negative numbers. He even has rules that we don’t quite agree with today: 0/0 = 0. Interestingly enough, later mathematicians did not always accept negative numbers. For example, Arabic mathematicians avoided negative numbers, and treated zero as just a place holder. (His works discuss geometry, algebra, Pythagorean triples, indeterminate equations including 2 2 2 2 8x + 1 = y and 61x + 1 = y with solution x = 226153980√ and y = 1766319049, sine functions. Even has the formula for the first n squares. He used 10 for π.)

4 13. Al-Khwarizmi (c. 780 - c. 850). One of the earliest and most important Arabic mathe- maticians. Worked at the famous House of Wisdom which was located in Baghdad. At the time, Baghdad was a new city, and was the capital of the Islamic world. The word “algebra” comes from the title of his most famous work (Hisab al-jabr w’al-muqabala concerning calculation using the operations of al-jabr and al-muqabala. The term al-jabr means “completion” or “restoration”. It refers to adding to both sides of an equation to compensate for a subtraction. For example, going from x2 − 5 = x to x2 = x + 5. The term al-muqabala means “balancing”. It refers to the operation of cancelling from both sides of the equation. For example, going from x2 + 3x = 5x + 1 to x2 = 2x + 1. With these two operations every quadratic equation can be reduced to one of al-Khwarizmi’s standard forms.) This book discusses how to solve quadratic equations, but does not use symbolism: everything is written out in words. It justifies the procedures using geometric arguments: when al-Khwarizmi completes the square, he really completes the square. Of course, quadratic problems had been solved long before al-Khwarizmi, but he lays things out in a very organized, practical, and accessible way. Because of this his book was very influential. This book was very influential in Europe as well.6 (This book also discusses linear equations, areas and volumes, and algebraic problems arising from dividing inheritances.) Another important book, which we only know about through its Latin translation, is Algoritmi de numero Indorum. The word Algoritmi is a Latinized version of Al-Khwarizmi’s name. This translated book was influential in Europe in spreading Hindu-Arabic numbers. It discusses how to calculate with these numbers, which were new at the time. It is from this book that we derive our word algorithm. (Al-Khwarizmi also wrote on astronomy and trigonometry based on Indian sources, geography, the sundial, the astrolabe, the Jewish calendar, and history. His geography improves on Ptolomy’s).

14. Omar Khayyam (1048-1122). Famous Persian poet, mathematician, scientist, and philoso- pher. (His Arabic name was al-Khayyami). Wrote mathematics in Arabic but poetry in Persian. He also wrote a book on al-jabr, the first to tackle general cubic equations. His solutions were geometric: involving the intersection of curves such as conics and . (He wrote this book in Samarkand, a famous ancient city now in Uzbekistan). He wrote a critique of Euclid (discussing the fifth postulate and the theory of proportion). (He also reformed the calendar based on his accurate estimates for the length of a solar year. His idea was to have 8 leap years every 33 years. He is famous worldwide outside of mathematics for writing the poem called the Rubaiyat.)

15. Bhaskara (1114-1185) Indian mathematician and astronomer. He wrote the Lilavati which was dedicated to his daughter Lilavati to consol her. Why did she need consoling? Thought that a/0 = ∞. He represents the high point of Indian mathematics

16. Fibonacci. (1170-1250) His real name was Leonardo of Pisa. An Italian mathematician, the most famous and important medieval European mathematician. From a merchant family, he travelled widely in the European and Arabic worlds, and learned some Arabic mathe- matics. He advocated for the Hindu-Arabic numerals instead of the then current Roman 6It was translated by Robert of Chester into Latin in 1145.

5 numerals. Despite the efforts of Fibonacci and others, Roman numerals remained popular for several more centuries. He is most famous for his Fibonacci sequence motivated by a problem involving the population of rabbits. What are the Fibonacci numbers? How are they formed? His performance at a mathematics contest made him famous. (For example, he showed that the cubic x3 +2x2 +10x = 20 cannot be solved with square roots, but he gave a very accurate approximate solution 1.3688081075, good to 9 places. He also found a rational square such that if you add or subtract five, you still have a rational square: (41/12)2. He realized, as did others, that if p and q are the sum of two squares, then so is pq.)

17. Thomas Harriot (1560-1621). English mathematician of the late Renaissance. Science ad- visor of Sir Walter Raleigh (where his duties included ship design, developing and teaching navigational techniques, surveying, keeping the financial accounts, and solving various mathe- matical problems put to him.) He was one of the first Englishmen to travel to North America. In fact, he was on Raleigh’s expedition to Virginia (1585-86, perhaps also to Roanok Island 1584. He died of cancer of the nose caused by his habit of inhaling tobacco smoke, a practice he learned in Virginia.) In algebra he introduced the symbols > and <. Harriot advocated moving all terms of an algebraic equation to one side of the equation and setting this equal to zero (he was very comfortable with zero and negative numbers). He made several other advances in algebra as well. He came up with an interesting formula for the area of spherical triangle. Let H be the area of a hemisphere, and T the area of a triangle on a sphere.Then T E = H 360 where E is the excess angle sum: the sum of the three minus 180 degrees. (He made observations with telescopes about the same time as Galileo: mountains on the moon, moons of Jupiter, and sunspots — he even found the period of rotation of the sun based on the sunspots. His interests included algebra, astronomy, optics, chemistry, trajectories of projectiles, the rainbow.)

18. Cardano (1501-1576). Famous Renaissance Italian physician, mathematician, notorious gambler, and astrologer (at one time he was an astrologer to the Pope). He convinced another mathematician (the famous Tartaglia) into telling him his method for cubic equations (1539). Cardano swore he would not tell anyone. Yet in his influential algebra book Ars magna “the great art” (1545) he explained how to solve cubics. (Cardano’s student, Ferrari, claimed that Cardano also learned it from another source, and then extended it in original ways so he could legitimately write about it.) The Ars magna also has a solution to the fourth degree polynomial (discovered by his student Ferrari). It was only after 1800 that mathematicians, including Abel, were able to show that general polynomials of degree five or more cannot be solved with these types of algebraic methods. (He also was a forerunner to probability, due partly to his gambling.)

19. Cavalieri. (Bonaventura Cavalieri, 1598-1647). An Italian Monk who became a disciple of Galileo. A key predecessor for integral calculus: he developed his method of indivisibles for finding areas and volumes which is related to the method of exhaustion of Archimedes

6 and Eudoxus. His method was not rigorous, but it produced correct answers fairly easily. So some people criticized his method, while others including Galileo were very impressed. Cavalieri thought of a volume as an infinite stack of areas, and an area as a stack of lines. A simple version of Cavalieri’s Principle: if two solids have cross sections at each height with equal area, then they have the same volume. A simple proof for the volume of the sphere can be given using his principle. This proof takes as given the more elementary formulas for and cylinders. (A version of Cavalieri’s Principle holds for areas with equal length cross sections. In both versions, you can replace ‘equal’ with ‘in a fixed proportion’.)

20. Descartes. (Ren´eDescartes. 1596–1650). French philosopher and mathematician. One of the most famous philosophers of all times. He wrote the La g´eom´etrie as one of the appendices to his most important philosophical work (Discours de la m´ethode). La g´eom´etrie introduced coordinate geometry. His notation was almost modern: used x, y, z for variables, and a, b, c for constants. He used xn for the nth power. He did not really introduce the so called “Cartesian” coordinate system, but did take the first important steps. In fact, he did not consider points with negative coordinates. He thought of x and y as lengths, and the point determined by x and y was drawn by drawing x horizontal from then origin, then a segment of length y was drawn at an angle (usually 90 degrees) from the endpoint of the x length. When the angle is 90, this gives the same point that we get in modern coordinate geometry, but the thought process is somewhat different. He interpreted multiplication in a more modern way where a length A times another length B is a third length C. Before his time, most mathematicians regarded AB as a rectangle, A3 as a cube, and An for n > 3 was considered nonsensical. But from Descartes’ time, they are all lengths, so one should not be afraid to take large powers. While in school he was allowed to stay in bed thinking until 11 in the morning. He kept this habit for much of his life (even when he was in the military). He spent much of his productive years in Holland. He moved to Sweden to work for the Queen (Queen Christina). She wanted him to get up at 5AM to give her lessons. That and the cold climate was hard on him: he died in a few months.

21. Fermat. (Pierre de Fermat. 1601–1665) French Lawyer working in the city parliament. Did he get paid for his mathematical research? Answer: no. Did he correspond with other mathematicians? Answer: yes, that is the main way in which his discoveries spread. Fermat and Descartes are considered the inventors of coordinate or . Fer- mat’s approach was more understandable than Descartes: he started with linear equations and lines, then built up to quadratic equations and conics. (Descartes basically skipped lines and their linear equations). Fermat was a key predecessor to calculus. He used infinitesimal methods to find tangents to curves and areas under curves. However, he did not realize that tangents problem and the area problem (the problem) were connected: the Fundamental Theorem of Calculus had to wait until later. Fermat read a translation of Diophantus, and so became interested in number theory. In some ways, Fermat can be regarded as the inventor of number theory. What primes are the sum of two squares? Answer: a prime p > 2 is the sum of two squares if and only if p is one more than a multiple of four.

7 22. Isaac 17th century England (specifically 1643-1727, or by the old calendar: born on Christmas 1642). One of the two founders of calculus. His theory was built around “fluons” and “fluxions”. It was not considered as notationally satisfying as Leibniz’s system. He knew the Fundamental Theorem of Calculus, a relationship between rates of change (derivatives) and areas under curves (). Newton also discovered the binomial , and many other things. Although he is remembered today primarily for his physics, his mathematical accomplishments are enough to make him one of the greatest mathematicians of all times. His theory of gravity is explained in his book, the Principia (1687) which is probably the most important scientific book ever published. In it he formulated his inverse square law. He showed that all three of Kepler’s laws follow from the inverse square law. The thing people found strange about gravity is the idea of “force at a distance”. He also formulated laws of motion, and made contributions to the theory of light and optics, especially the idea that white light is a mix of colors (and that different colors are refracted at different angles). In 1665, while he was a student at Trinity College in Cambridge University, the University closed due to the plague. It did not open until 1667. Newton went home and had a few years of time to reflect on his interests. In that period he developed the seeds for his theory of gravity and for calculus. (He wrote up his ideas about calculus in 1671, but it wasn’t until 1736 that they were published. They were spread long before 1736, however, through letters, lectures, word of mouth, etc.)

23. Leibniz. 17th century Germany. (More specifically, Gottfried Leibniz, 1646-1716). Like Descartes, he is famous today both as a philosopher and a mathematician. One of the two founders of calculus (the other is Newton). He invented the integral sign R as a kind of letter S for a kind of infinite sum of infinitesimals (1675), and dx and dy for infinitesimal differences. Infinitesimals are infinitely small quantities. According to him, the ratio of two infinitesimals is often not infinitely small, but is often an ordinary quantity. What does the ratio dy/dx mean in that case? answer: slope of the tangent line. Leibniz did not think in terms of limits. So his conception of derivative is different than ours. He gave the basic rules of derivatives (1675-76) and integrals, and knew the inverse relationship between them: the Fundamental Theorem of Calculus. Finding good notation was very important to Leibniz. In philosophy, Leibniz is responsible for the idea that we are living in the best possible (but far from perfect) universe. He was more than just a philosopher and a mathematician. His interests and contributions were very broad (including law, logic, physics, Latin poetry, religion, theology, diplomacy, calculating machines, geology, languages including Latin, Greek, and Sanskrit.) He built a calculating machine: the first that could execute all four arithmetic operations (1694). This machine used binary. (Pascal’s early machine could only add and subtract, and didn’t use binary). There was a bitter dispute between Newton and Leibniz on who invented the main ideas of calculus. (Leibniz first published his ideas in 1684. Newton’s Principia was published 1687, but Newton had actually written on calculus in 1671 before Leibniz started working on calculus in 1673. Newton’s 1671 paper was not published until 1736, but Newton’s ideas were spread through letters, lectures, etc.)

8 Leibniz independently discovered the following formula (1674?): π 1 1 1 1 1 = 1 − + − + − + ... 4 3 5 7 9 11 which when multiplied by 4, gives a formula for π. (Other work in mathematics by Leibniz includes sums of infinite series, the multinomial theorem, theory of determinates, osculating circles and envelopes. His idea of an algebra or calculus of rational thought was an important predecessor of modern mathematical logic.) 24. Euler. (1707-1783). He was from Switzerland, but spent his career in St Petersburg in Russia and Berlin in Germany. Euler was the most important mathematician between Leibniz and Gauss, and the most productive mathematician of all time (his life work fills almost 80 volumes. You can go to the mathematical library at UCSD and see how much library he fills compared to the collected works of other notable mathematicians). Even when he went blind, later in life, he continued his research (with the help of his sons and several younger mathematicians).7 He emphasized the function approach to mathematics. For example, he treated sines and cosines as functions of angles instead of as just lengths. Much of precalculus notation is due to his textbooks including f(x) for a function. He was one of the first people to use π as the symbol for the ratio of the circumference to the diameter, and was the first to use e for the base of the natural logarithm. He introduced i for the square root of −1. (Of course, these numbers were used before Euler, but under other names). He also introduced Σ for summation. (Euler was important in the development of virtually all branches of mathematics which existed in his day. He made notable contributions in calculus, differential equations, con- vergence of series, calculus of variations, complex analysis including Euler’s formula, special functions, number theory, continued fractions, algebra, analytic geometry, trigonometry, ge- ometry including early work in differential geometry and topology, logarithms. In applied mathematics: acoustics, cartography, magnetism, ship building, navigation, tides, mechanics, artillery and ballistics, optics, insurance, planetary motion including the three-body problem and the motion of the moon, elasticity, wave theory of light, music, hydraulics.) (He is also responsible for reviving number theory. He studied Fermat’s collected works with the goal of proving or disproving all Fermat’s unproved claims about number theory. After a while he proved or disproved all of them except Fermat’s Last Theorem which he could only prove for n = 3 and n = 4.) 25. Sophie Germain (1776-1831). French mathematician who grew up during the French Rev- olution. While in her teens, during the turbulent French Revolution, she was inspired by the story of the death of Archimedes to take solace in mathematical pursuit during violent times. As a woman, she was not allowed to attend the university (Ecole´ Polytechnique in Paris), but she managed to obtain the lecture notes from courses there. She wrote mathematical papers under a pseudonym (M. LeBlanc). She also corresponded with the leading mathematicians including Gauss under her pseudonym. She made significant progress in Fermat’s Last The- orem in number theory (1820’s), and the theory of elastic surfaces in applied mathematics (she won a prize for it in 1815 or 1816. She also wrote on differential geometry in 1831).

7He was also prolific in his personal life: he had 13 children.

9 26. Abel. From Norway in the early 1800’s (1802-1829). When he was young he thought he figured out a way to solve the general fifth degree equation (“the quintic”) with radicals, but when asked to give a specific numerical example, he detected a mistake. He later showed that √ general polynomials of degree 5 or more are not solvable using radicals n . This showed that Cardan’s techniques cannot be extended beyond degree 4. He died very young. (He lived a life plagued by poverty. He had trouble getting teaching job in mathematics. Finally, a few days after he died, a letter came informing him that he was being awarded a teaching position. Starting in 2002, in honor of Abel’s 200th birthday, there is a new prize in mathematics called the Abel prize which is personally awarded by the king of Norway: yes, Norway has a king. This prize is annual, and is being promoted as the Nobel prize of mathematics. It is worth almost a million dollars.)

27. Galois. France, early 1800’s. (1811-1832, born during the rule of Napoleon). Built on Abel’s result to invent a new theory in algebra called Galois theory based on the concept of a group. It tells you, among other things, whether or not a given polynomial equation is solvable using radicals, or whether you can construct a certain length with a straightedge and compass. In fact, Galois gave a solution to the following problem: given a polynomial of prime degree, when can it be solved by radicals. This paper was published over ten years after his death. Even though he was clearly passionate about mathematics, he was bad at communicating his ideas and intelligence. Because of that he was denied admission to the leading French university (where the notable mathematicians of his time were located). He was kicked out of college, and later arrested, for being a political rebel. He died when he was twenty years old in a mysterious duel (probably over the love of a woman). The night before he was shot, he fervently wrote some of his very original ideas and gave it to a friend who was instructed to send it to the leading mathematicians. He became famous only after his demise.

28. Gauss. Germany, 1800’s (more specifically, 1777 - 1855). Some consider him the best math- ematician of all times. He contributed enormously to applied mathematics and pure mathe- matics. In his teen years, he discovered how to construct the 17-gon with a straightedge and compass, something the ancient Greeks were never able to do. Because of this he decided to become a mathematician (instead of a philologist). Also, when he was a in his teens he revolu- tionized number theory by proving his famous reciprocity law, and introducing congruences. Based on his ideas in number theory, it is possible to prove that it is impossible to trisect an angle, or duplicate the cube with only a straightedge and a compass.8 In his doctoral dissertation, he proved the fundamental theorem of algebra which says that every polynomial factors as the product of linear polynomials (where one must use complex numbers). If you do not want to use complex numbers, then this theorem says that every polynomial factors as the product of linear and quadratic factors. Unlike Euler, he was very picky about publishing work before it was made almost perfect. (His motto was Pauca sed matura “few, but ripe”.) Many mathematicians who were excited by their own great discoveries would often be disappointed to learn that Gauss had discovered the same results decades before, but had not gotten around to publish them.

8The proof that it was impossible to square the circle would have to wait until the proof that π was transcendental.

10 3 Historical Dates, Events, and People

1. The first Olympic games in Greece: 776 BC. Also, approximately the time of the inven- tion of the Greek alphabet. In our course we take this to be the official start of ancient Greek culture that soon gave rise to Thales and the other early Greek mathematicians.

2. Plato. Follower of Socrates. Founded the Academy in Athens. Not a great mathematician himself, but advocated the philosophic value of mathematical study. According to legend he had the following motto written above his Academy: let no one ignorant of geometry enter.

3. Alexander the Great from Macedon. Conquered Greece, the Persian Empire, Egypt, and more. He died in 323 BC. His death is the beginning of the Hellenistic Period. During this period the center of culture was the city of Alexandria in Egypt. Hellenistic culture was really a mix of Greek, Persian, Egyptian, Babylonian, Hebrew, and other cultures. Aristotle was Alexander’s personal tutor. Alexander died young in 323 BC. His generals divided his empire into several smaller kingdoms. One of his generals was King Ptolomy of Egypt.

4. Alexandria. A city in Northern Egypt founded by Alexander the Great. It had a famous Museum (a kind of university) and Library. It became the pre-eminant cosmopolitan hel- lenistic city. It replaced Athens as the center of culture and learning.

5. The Roman Empire. The major cities included Rome, of course, but also Constantinople (or Byzantine, today Istanbul) in Asia Minor (today Turkey). However, Alexandria remained a center of mathematics and culture. In the western half of the Roman Empire, people spoke Latin. (In fact, Spanish, French, Italian, Portuguese and even Romanian all descend directly from Latin. These languages are called Romance Languages.) In the eastern half of the Roman Empire, including Alexandria, people continued to write in Greek.

6. 476: The fall of the (western) Roman Empire. Rome was conquered by Germanic tribes.9 The eastern Roman Empire lasted for almost 1000 years after this, and evolved into the Byzantine Empire.

7. The development of the Hindu-Arabic number system and the positional decimal system. From about 600-800. Until Al-Khwarizmi, our information is very sketchy.

8. The growth of Islamic culture. During the so called dark ages in Europe, Islamic culture prospered. The center of culture was Baghdad (although the religious center was Mecca). The House of Wisdom was in Baghdad. Baghdad was founded in 766. The house of Wisdom was set up in Baghdad about 50 years later.

9. Renaissance in Europe. Know the date 1450: the approximate date of the fall of Con- stantinople (Eastern Roman Empire) and the beginning of printing in Europe. It marks the height of the Renaissance (in Italy and Central Europe. It would wait to come to England until the late 1500’s).

9England was also conquered by Germanic tribes, the Angles and the Saxons. The English language is a Germanic language, but it has a huge amount of French and Latin influences as well. In fact, English draws from many languages for its vocabulary. We have seen the importance of Greek words to our English vocabulary.

11 10. 1600. Approximate end of the Renaissance, and the start of the modern period. Some people date the start of modern period with Descartes. 11. 1665-66. The plague years when Newton discovered calculus and the laws of gravity.

4 Summary of Dates

Know the following dates and their significance. • 776 BC • 323 BC • 476 AD • 766 • 600-800 (Development of the Hindu-Arabic numbers) • 1450 • 1600 • 1665-66

5 Languages

An important part of this course was language and its relationship with mathematics. Be able to answer the following: 1. What language did the Greek and Roman era mathematicians use? Answer: usually Greek. 2. List the Greek alphabet: upper case, lower case, name, Latin equivalent. (See handout). 3. What language did the Indian (Hindu) mathematicians use? Answer: Sanskrit. 4. What language did the Islamic mathematicians use? Answer: Arabic. 5. What language did the Western European Mathematicians use before Descartes? Answer: Usually Latin.

6 Numerals

1. Know Ionic numerals (I will help you by partially filling up the table include the 3 archaic characters: digamma, koppa, and san). Write numbers up to 9999. (Numbers larger than that, and fractions, could be written in a variety of ways depending on the particular author). 2. Be able to write Chinese rod numerals. For the Chinese rod numerals see the Chinese numerals entry at Mac-Tutor. www-groups.dcs.st-and.ac.uk/∼history/HistTopics/Chinese numerals.html

3. Be able to write Roman numerals up to 6000. See page 68–69 of Math through the Ages.

12 7 Mathematical Topics

1. Pythagorean Number Theory. Illustrate with a picture the fact that two times the nth trianglular number Tn is an n by n + 1 rectangular number (called an “oblong number”). Conclude that Tn = n(n + 1)/2. (Hint: just illustrate the case n = 3 or n = 4. The general case obviously works in the same way.)

2. Pythagorean Number Theory. Illustrate with a picture how the nth square number is the sum of the first n odd numbers.

3. Thales’ Theorem. State and prove this theorem.

4. The Pythagorean Theorem. Give a proof of the Pythagorean Theorem involving similar triangles. Also draw a picture of the Pythagorean Theorem involving three squares attached to the triangle. State the theorem in terms of adding areas of squares. What is the generalization of the Pythagorean theorem used by Hippocrates? The Pythagorean theorem was known to other civilizations before Pythagoras.

5. Geometric Mean. What is the definition of the geometric mean of two lengths? How can it be constructed? Prove that the construction works. How can you use this construction to square a rectangle? How can you use this construction to square a triangle?

6. The Lune of Hippocrates. Explain how the lune is defined on a 45-45-90 right triangle. Sketch it. Why is it called a lune? Prove that the area of the lune is one half the area of the triangle. (This implies that the lune can be squared). This was the first curved region that we know of that was successfully squared, and with only a straightedge and compass . It can be thought of as a step on the way to .

7. Rectification. Given a curve, if you construct a straight line segment of the same length as the given curve then the curve is said to be rectified. Rectifying a circle is an important step in squaring the circle. Describe how to rectify one quarter of a circle using the spiral of Archimedes. Prove that it works using polar coordinates. Hint r = aθ. Thus x = aθ cos(θ) and y = aθ sin(θ). Then dy/dx = (dy/dθ)/(dx/dθ). Evaluate dy/dx at θ = π/2 to find the slope. Use the slope to show the result.

8. The Method of Exhaustion. In elementary geometry, two areas or volumes are shown to be equal by cutting up the first region and reassembling it to be equal to the second region. However, curved areas and general solids cannot always be studied in this way. The method of exhaustion is a more sophisticated method based on lower and upper approximations of the region, and using these approximations to show that the area or volume cannot be bigger, nor smaller, than the proposed answer. Eudoxus initiated this method, and it was used by Euclid. Archimedes also used this method.10

9. The Elements. The most famous mathematics book of all time. Written by Euclid (around 300 BC). About half of it is devoted to plane geometry including the Pythagorean theorem and the construction of the pentagon. Some of it deals with algebra, in a geometric style

10It is called “exhaustion”, not (only?) because it is tiring, but because the inscribed approximations take away or “exhaust” as much of the areas or volume as you need for the proof.

13 (including methods for solving quadratic equations). Some of it deals with number theory (where it proves that there are an infinite number of primes, gives the , and discusses even perfect numbers). Some of it deals with solid geometry. It gives Eudoxus’ theory of proportion and uses his method of exhaustion. (It gives Theaetetus’ theory of incommensurability.) The Elements culminate in the constructions of the Platonic solids. Are conics covered in the Elements? No, but Euclid wrote another book about conics (which is now lost).

10. Volumes of Cones and Pyramids. The volume of such solids is the height times 1/3 the area of the base. Some civilizations knew this, at least for pyramids, before Eudoxus’ time (but some got it wrong). However, Eudoxus is the first to prove it (using the method of exhaustion). A proof can be found in Euclid’s Elements.

11. Area of Circles. Archimedes wrote a treatise on the circle. In it he (i) approximated π as 22/7 by approximating a circle with inscribed and circumscribed 96-gons, and (ii) used the method of exhaustion to prove that the area of a circle is equal to a triangle whose base is the circumference of the circle, and whose height is the radius of the circle. Derive the modern formula A = πr2 from this theorem. Hint: Use the fact that c = 2πr.

12. Volume and Surface Areas of Spheres. Archimedes showed that the volume of a sphere is 2/3 times the volume of the circumscribed cylinder. Show that this leads to the modern 4 3 formula V = 3 πr . Hint: even before Archimedes, it was known that the volume of the cylinder was given by the base area times the height. He also showed that the surface area of a sphere was 2/3 time the surface area of the circumscribed cylinder.

13. Cavelieri’s method. Know Cavelieri’s method for finding areas and volumes. Be able to answer simple questions about slanted cylinders, etc. Finally, derive the formula for the volume of a sphere using Cavelieri’s method. Is Cavelieri’s method rigorous? Answer: not originally, but later mathematicians made it rigorous.

14. Spherical triangles. Angles do not add up to 180 degrees. In fact, you can measure area by finding the sum of the angles; be able to do this. This is a simple example of non-. Spherical trigonometry was an important topic in astronomy and trigonometry in the . Know how to use the following: Let H be the area of a hemisphere, and T the area of a triangle on a sphere. Then T E = H 360 where E is the excess angle sum: the sum of the three angles minus 180 degrees.

15. Analytic Geometry. Discovered by Fermat and Descartes. Sketch how they would have thought of x and y (as lengths of segments, with the segment for y placed at the end of the horizontal segment for x). Original version: no negative, no y-axis.

16. Calculus. Name the official inventors. Answer: Newton and Leibniz. Name two pioneers who proceeded them. Answer: Fermat and Cavalieri. Who developed the rules and proved the fundamental theorem of calculus? Answer: Newton and Leibniz. Describe some of Leibniz’s notation. Answer: dx, dy stand for infinitesimals. The ratio dx/dy gives a real

14 number: the derivative. The sign R is a fancy S which stands for the continuous summation of infinitesimals.

17. Euler Topic 1: Complex Variables. Source William Dunham, Euler, Chapter 5. What is de Moivre’s theorem? [page 88]. If you accept only real roots, how many cube roots does a real number have? Answer: only one. If you allow complex roots does the answer change? Answer: yes, you get three roots for any non-zero number. If one uses complex numbers then any such (non-zero) quantity has how many square roots? How many cube roots? How many fourth roots? [page 88]. Did Euler’s approach to complex numbers involve trigonometry? What is Euler’s formula for eix? Did Euler allow you to have imaginary powers? Answer: yes, that is what eix represents. If x = π what famous formula do you get? [page 96]. Did Euler study ii? [page 101].

18. Euler Topic 2: Algebra. Source William Dunham, Euler, Chapter 6. Was Euler the first to solve the general fourth-degree equation? Complete this quote: “One aspect of this problem — apart from its length — deserves particular note: solving the quartic rests on the ability to solve a related ”. Was Euler the first mathematician to solve the quintic (fifth-degree)? Before Euler, mathematicians conjectured that every real polynomial factors as the product of what two types of factors? Did Leibniz believe this? Did Nicolaus Bernoulli? Did Euler believe it? What is the fundamental theorem of algebra as now perceived? Euler showed that xn − 1 = 0 has how many complex solutions? What is notable about Nicolaus Bernoulli’s example. How does this conjecture (original version of the fundamental theorem of algebra) help you integrate rational functions? (Hint: partial fractions). Did Euler succeed in proving this factorization theorem for arbitrary polynomials? What about for quartic polynomials? What did he prove about quartic polynomials? What about fifth degree polynomials? answer: by the intermediate value theorem, these always have a root [page 115]. Did his treatment of eighth degree polynomials have holes? [page 118]. Did Abel come before or after Euler? What did Abel show about a solution by radicals of the general fifth-degree equation? Where do abstract concepts such as “group” and “field” come from? [page 120]. Who first proved the Fundamental Theorem of Algebra? How many additional proofs did Gauss give?

19. Extra Credit. Make a list of pages and sketches of Math through the Ages that you have not read yet. Also choose a chapter of William Dunham, Euler. Choose either Chapter 1, 2, 3, 4, 7, or 8. Then read these pages and sketches of Math through the Ages and the selected chapter of Euler by the end of 2006. If you can commit to doing this you will earn 3 points of extra credit on the final.

15