Euclid of Alexandria

1. Euclid of Alexandria Euclid was a Greek mathematician best known for his treatise on geometry: The Elements. This book influenced the development of Western mathematics for more than 2000 years.

2. History of Mathematics The lecturer of this course Dr Vasos Pavlika Vasos.Pavlika@conted.ox.ac.uk vp4@soas.ac.uk vpavlika@lse.ac.uk VPavlika@sgul.ac.uk (Vas for short)

3. Course Lecturer • Dr Vasos Pavlika, Subject Lecturer at SOAS, University of London. • Subject Lecturer and online Tutor in Mathematical Economics at SOAS, University of London. • Senior Teaching Fellow, SOAS, University of London • Lecturer for the Department for Continuing Education, University of Oxford. • Associate Lecturer: New College, Oxford • Saturday School Lecturer: The London School of Economics and Political Science. • Associate Tutor: St George’s Medical School, University of London. • Consultant Mathematician. • Previously Senior Lecturer at the University of Westminster. • Field Chair at the University of Gloucestershire

4. Portfolio Exercises • There will be a portfolio exercise for the course. • This will be an extended essay of a topic discussed on the course. • This essay will count towards to 10 CATS points.

5. Euclid • Little is known about Euclid's actual life. • He lived in Alexandria about 300 B.C.E. based on a passage in Proclus' Commentary on the First Book of Euclid's Elements. • Indeed, much of what is known or conjectured is based on what Proclus says. • After mentioning two students of Plato, Proclus writes:

6. Euclid • All those who have written histories bring to this point their account of the development of this science. • Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors.

7. Euclid • He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. • It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that • “There is no royal road to geometry”.

8. Euclid • It is apparent that Proclus had no direct evidence for when Euclid lived, but managed to place him between Plato's students and Archimedes, putting him, very roughly, about 300 B.C.E. • Proclus lived about 800 years later, in the fifth century C.E. • There are a few other historical comments about Euclid. • The most important being Pappus' (fourth century C.E.) comment that Apollonius (third century B.C.E.) studied "with the students of Euclid at Alexandria."

9. Euclid of Alexandria (about 325 BC - about 265 BC)

10. Some quotes by Euclid • In reply to King Ptolemy Euclid said • “There is no royal road to geometry”. • A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, "What do I get by learning these things?" So Euclid called a slave and said: • "Give him threepence, since he must make a gain out of what he learns."Stobaeus, Extracts

11. The parallel postulate • That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. (see next slide)[the 5th postulate] • This is the famous parallel postulate, which caused so many problems for mathematicians in the late 17th and 18th century. • Led to the development of so-called non-Euclidean geometries. • This is a consistent geometry that contradicts (or does away with) the 5th or parallel postulate

12. The Parallel Postulate If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

13. What are postulates • Results that can be taken as being true without proof. • It has been said (E.T. Bell and Heath) that Euclid’s genius lay in the fact that he was aware that the five postulates that he chose were the essential ones required to derive his theorems.

14. The parallel postulate • This postulate influenced the creation of non-Euclidean geometry at the hands of • Gauss (the Prince of Mathematics) • Riemann (Ph.D., examined by Gauss) • Bolyai (friend of Gauss) • Lobachevsky

15. Euclid’s postulates • A straight line segment can be drawn joining any two points. • 2. Any straight line segment can be extended indefinitely in a straight line. • 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre. • 4. All right angles are congruent. • 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

16. Euclid’s Postulates • Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. • Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. • In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. • (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

17. J.C.F. Gauss • Johann Carl Friedrich Gauss • 1777 - 1855 Known as the Prince of Mathematics. Quotes by Gauss Pauca sed matura (few but ripe) I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.

18. Sums an arithmetic progression • Gauss at 7 sums the first 100 integers in class. • His teacher immediately realised his potential. • Gauss was supported by the Prince of Brunswick, enabled him to study • At 17 proves the fundamental theorem of algebra, bit too advanced to show.

19. Gauss • Gauss worked in a wide variety of fields in both mathematics and physics including: • Number Theory: Higher Arithmetic, his favorite past time • Analysis • Differential geometry: geodesics • Geodesy: map making (similar to Leibniz) • Magnetism • Astronomy and optics. • Ordinary Least squares. Show this • His work has had an immense influence in many areas. There are few areas of mathematics where his name does not appear

20. Georg Friedrich Bernhard Riemann • 1826 – 1866 (student of Gauss) (died young) • Riemann's ideas concerning geometry of space had a profound effect on the development of modern theoretical physics. • General Relativity • He clarified the notion of the integral by defining what we now call the Riemann integral.

21. Riemann integral • Riemann integrals • The Riemann hypothesis • The holy grail of mathematics now that Fermat has been dispensed with • Goldbach conjecture also very old conjecture • Known more formally as the Euler-Goldbach conjecture • Any even integer can be expressed as the sum of two primes.

22. The Riemann Integral • Let f(x) be a non-negative real-valued function of the interval [a,b], and let • S = {(x,y) | 0 < y < f(x)} be the region of the plane under the function f(x) and above the interval [a,b] (see the figure on the next slide). • We are interested in measuring the area of S. • Once we have measured it, we will denote the area by:

23. The Area S

24. The Riemann Integral

25. The Riemann Integral • The basic idea of the Riemann integral is to use very simple approximations for the area of S. • By taking better and better approximations, we can say that "in the limit" we get exactly the area of S under the curve. • Note that where f can be both positive and negative, the integral corresponds to a signed area; that is, the area above the x-axis minus the area below the x-axis. • http://en.wikipedia.org/wiki/Riemann_integral

26. G.B Riemann 1826 – 1866 • G.B Riemann • Riemann Integration

27. The Riemann Hypothesis • When studying the distribution of prime numbers Riemann extended Euler's zeta function (defined just for s with real part greater than one)

28. The Riemann Hypothesis • to the entire complex plane (with simple pole at s = 1).  • Riemann noted that his zeta function had trivial zeros at -2, -4, -6, ... and that all nontrivial zeros that he could calculate were symmetric about the line Re(s) = ½   • The Riemann hypothesis is that all nontrivial zeros are on this line. • Proving the Riemann Hypothesis would allow us to greatly sharpen many number theoretical results. 

29. The Prime number theorem

30. The Prime number theorem • The prime number theorem gives an asymptotic form for the prime counting function, which counts the number of primes less than some integer. • Legendre (1808) suggested that for large n,

31. The Prime number theorem • with B=-1.08366 (where B is sometimes called Legendre’s constant). • See also • http://mathworld.wolfram.com/PrimeNumberTheorem.html

32. Nikolai Ivanovich Lobachevsky 1792 - 1856 • In 1829 Lobachevsky published his non-Euclidean geometry, the first account of the subject to appear in print, • contradicting the 5th • postulate of Euclid.

33. Euclid of Alexandria • Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics • The Elements • The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. • However little is known of Euclid's life except that he taught at Alexandria in Egypt. • Proclus, the last major Greek philosopher, who lived around 450 AD wrote about Euclid:-

34. Proclus Diadochus

35. Proclus Diadochus • Proclus was a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians. • He was more of a promoter of Greek thought.

36. Euclid according to Proclus • Not much younger than these [pupils of Plato] is Euclid, who put together the "Elements", arranging in order many of Eudoxus’s theorems, perfecting many of Theaetetus’s, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors.

37. Euclid • This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied • “that there was no royal road to geometry”.

38. Euclid • The geometry Applet • http://aleph0.clarku.edu/~djoyce/java/elements/usingApplet.html

39. Eudoxus of Cnidus (410 or 408 BC – 355 or 347 BC) • An algebraic curve (the Kampyle of Eudoxus) is named after him • a2x4 = b4(x2 + y2). • Eudoxus was a Greek mathematician and astronomer who contributed to Euclid's Elements. • He mapped the stars and compiled a map of the known world. • His philosophy influenced Aristotle. • Influenced Alexander the Great

40. Kampyle of Eudoxus a2x4 = b4(x2 + y2)

41. Euclid • There is other information about Euclid given by certain authors but it is not thought to be reliable. • Two different types of this extra information exists. • The first type of extra information is that given by Arabian authors who state that Euclid was the son of Naucrates (not much literature is available on him) and that he was born in Tyre (Lebanon). • It is believed by historians of mathematics that this is entirely fictitious and was merely invented by the authors.

42. Archimedes • Born about 287 BC in Syracuse, Sicily. • At the time Syracuse was an independent Greek city-state with a 500-year history. • Died 212 or 211 BC in Syracuse when it was being sacked/attacked by a Roman army. • He was killed by a Roman soldier who did not know who he was. • Archimedes was in the middle of doing some work when the soldier asked him to leave the building, Archimedes asked for more time to complete his work and as result of this suggestion he was killed.

43. Archimedes

44. Archimedes

45. Burning mirrors

46. Mirrors • When Marcellus withdrew them [his ships] a bow-shot, the old man [Archimedes] constructed a kind of hexagonal mirror, and at an interval proportionate to the size of the mirror he set similar small mirrors with four edges, moved by links and by a form of hinge, and made it the centre of the sun's beams--its noon-tide beam, whether in summer or in mid-winter.

47. Mirrors • At last in an incredible manner he [Archimedes] burned up the whole Roman fleet. • For by tilting a kind of mirror toward the sun he concentrated the sun's beam upon it; and owing to the thickness and smoothness of the mirror he ignited the air from this beam and kindled a great flame, the whole of which he directed upon the ships that lay at anchor in the path of the fire, until he consumed them all. • The above passage is from • DIO'S ROMAN HISTORYTranslated by Earnest Cary,Loeb Classical Library, Harvard University Press, Cambridge, 1914, Volume II, Page 171

48. Mirrors

49. Archimedes • Afterwards, when the beams were reflected in the mirror, a fearful kindling of fire was raised in the ships, and at the distance of a bow-shot he turned them into ashes. • In this way did the old man prevail over Marcellus with his weapons. • This is most likely apocryphal • The previous passage is from • GREEK MATHEMATICAL WORKSTranslated by Ivor Thomas, Loeb Classical Library, Harvard University Press, Cambridge, 1941, Volume II, Page 19

50. Archimedes and Pi • Archimedes calculated that • 220/71<π<22/7 • 3.14< π <3.157 • A little bit of trivia 14th March is Einstein’s birthday • A better approximation was not obtained for another 2000 years.