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. 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) 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 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. 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: 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. 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”. 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." Euclid of Alexandria (about 325 BC - about 265 BC) Some quotes by Euclid In reply to King Ptolemy Euclid said 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 • “There is no royal road to geometry”. 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 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. 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. 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 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. 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.) 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. 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. 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 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. 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. 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: The Area S The Riemann Integral S b a f ( x ) dx 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 G.B Riemann 1826 – 1866 G.B Riemann Riemann Integration 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) 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. The Prime number theorem 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, The Prime number theorem with B=-1.08366 (where B is sometimes called Legendre’s constant). See also http://mathworld.wolfram.com/PrimeNumberTheorem.html 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. 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:- Proclus Diadochus 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. 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. 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”. Euclid The geometry Applet http://aleph0.clarku.edu/~djoyce/java/ele ments/usingApplet.html 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 Kampyle of Eudoxus a2x4 = b4(x2 + y2) 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. 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. Archimedes Archimedes Burning mirrors 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 midwinter. 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 HISTORY Translated by Earnest Cary, Loeb Classical Library, Harvard University Press, Cambridge, 1914, Volume II, Page 171 Mirrors 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 WORKS Translated by Ivor Thomas, Loeb Classical Library, Harvard University Press, Cambridge, 1941, Volume II, Page 19 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. Archimedes and pi Pi: continued fractions What about this constant π What is π ? It is the ratio of the circumference of a circle to its diameter, the Greeks were aware that C/D was the same for every circle Is this ratio the same as A/r2? • • How did Archimedes obtain his estimate? By a form of integral calculus, known as the method of exhaustion Euclid The second type of information is that Euclid was born at Megara. This is due to an error on the part of the authors who first gave this information. In fact there was a Euclid of Megara, who was a philosopher who lived about 100 years before the mathematician Euclid of Alexandria. It is not quite the coincidence that it might seem that there were two learned men called Euclid. Euclid In fact Euclid was a very common name around this period and this is one further complication that makes it difficult to discover information concerning Euclid of Alexandria since there are references to numerous men called Euclid in the literature of this period. Euclid Returning to the quotation from Proclus given above, the first point to make is that there is nothing inconsistent in the dating given. However, although we do not know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring to, in what has come down to us there is only one reference to Euclid and this occurs in On the sphere and the cylinder. The obvious conclusion, therefore, is that all is well with the argument of Proclus and this was assumed until challenged by Hjelmslev. Euclid He argued that the reference to Euclid was added to Archimedes book at a later stage, and indeed it is a rather surprising reference. It was not the tradition of the time to give such references, moreover there are many other places in Archimedes where it would be appropriate to refer to Euclid and there is no such reference. Despite Hjelmslev's claims that the passage has been added later, Bulmer-Thomas writes in:- Euclid Although it is no longer possible to rely on this reference, a general consideration of Euclid's works ... still shows that he must have written after such pupils of Plato, Eudoxus and before Archimedes. Plato (427-347BCE) The son of wealthy and influential Athenian parents, Plato began his philosophical career as a student of Socrates. When the master died, Plato traveled to Egypt and Italy, studied with students of Pythagoras, and spent several years advising the ruling family of Syracuse. Eventually, he returned to Athens and established his own school of philosophy at the Academy. Plato Plato (429–347 B.C.E.) is, by any reckoning, one of the most dazzling writers in the Western literary tradition and one of the most penetrating, wideranging, and influential authors in the history of philosophy. Plato An Athenian citizen of high status, he displays in his works his absorption in the political events and intellectual movements of his time, but the questions he raises are so profound and the strategies he uses for tackling them so richly suggestive and provocative that educated readers of nearly every period have in some way been influenced by him, and in practically every age there have been philosophers who count themselves Platonists in some important respects. Plato He was not the first thinker or writer to whom the word “philosopher” should be applied. But he was so self-conscious about how philosophy should be conceived, and what its scope and ambitions properly are, and he so transformed the intellectual currents with which he grappled, that the subject of philosophy, as it is often conceived—a rigorous and systematic examination of ethical, political, metaphysical, and epistemological issues, armed with a distinctive method—can be called his invention. Few other authors in the history of Western philosophy approximate him in depth and range: perhaps only Aristotle (who studied with him), Aquinas, and Kant would be generally agreed to be of the same rank. Plato For students enrolled there, Plato tried both to pass on the heritage of a Socratic style of thinking and to guide their progress through mathematical learning to the achievement of abstract philosophical truth. The written dialogues on which his enduring reputation rests also serve both of these aims. Plato In his earliest literary efforts, Plato tried to convey the spirit of Socrates's teaching by presenting accurate reports of the master’s conversational interactions, for which these dialogues are our primary source of information. Early dialogues are typically devoted to investigation of a single issue, about which a conclusive result is rarely achieved. Pythagoras of Samos about 569 BC - about 475 BC Euclid This is far from an end to the arguments about Euclid the mathematician. The situation is best summed up by Itard who gives three possible hypotheses. Euclid (i) Euclid was an historical character who wrote the Elements and the other works attributed to him. (ii) Euclid was the leader of a team of mathematicians working at Alexandria. They all contributed to writing the 'complete works of Euclid', even continuing to write books under Euclid's name after his death. (iii) Euclid was not an historical character. The 'complete works of Euclid' were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about 100 years earlier. Euclid It is worth remarking that Itard, who accepts Hjelmslev's claims that the passage about Euclid was added to Archimedes, favors the second of the three possibilities that was listed on the last slide (i.e. of a team) We should, however, make some comments on the three possibilities which, it is fair to say, sum up pretty well all possible current theories. Euclid There is some strong evidence to accept (i). (Euclid was the sole author). It was accepted without question by everyone for over 2000 years and there is little evidence which is inconsistent with this hypothesis. It is true that there are differences in style between some of the books of the Elements yet many authors vary their style. Again the fact that Euclid undoubtedly based the Elements on previous works means that it would be rather remarkable if no trace of the style of the original author(s) remained. Euclid Even if we accept (i) then there is little doubt that Euclid built up a vigorous school of mathematics at Alexandria. He therefore would have had some able pupils who may have helped out in writing the books. However hypothesis (ii) (team) goes much further than this and would suggest that different books were written by different mathematicians. Euclid Other than the differences in style referred to above, there is little direct evidence of this. Although on the face of it (iii) might seem the most fanciful of the three suggestions, nevertheless the 20th century example of Bourbaki shows that it is far from impossible. Henri Cartan, Andre Weil, Jean Dieudonne, Claude Chevalley, and Alexander Grothendieck wrote collectively under the name of Bourbaki and Boubaki's Eléments de mathématiques contains more than 30 volumes. Euclid Of course if (iii) were the correct hypothesis then Apollonius, who studied with the pupils of Euclid in Alexandria, must have known there was no person 'Euclid' but the fact that he wrote:- Euclid .... Euclid did not work out the syntheses of the locus with respect to three and four lines, but only a chance portion of it ... certainly does not prove that Euclid was an historical character since there are many similar references to Bourbaki by mathematicians who knew perfectly well that Bourbaki was fictitious. Euclid Nevertheless the mathematicians who made up the Bourbaki team are all well known in their own right and this may be the greatest argument against hypothesis (iii) in that the 'Euclid team' would have had to had consisted of outstanding mathematicians. So who were they? Euclid We shall assume that hypothesis (i) is true but, having no knowledge of Euclid, we must concentrate on his works after making a few comments on possible historical events. Euclid must have studied in Plato’s Academy in Athens to have learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar. Euclid None of Euclid's works have a preface, at least none has come down to us so it is highly unlikely that any ever existed, so we cannot see any of his character, as we can of some other Greek mathematicians, from the nature of their prefaces. Pappus writes that Euclid was:- Euclid ... most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself. Euclid Some claim these words have been added to Pappus, and certainly the point of the passage (in a continuation which we have not quoted) is to speak harshly (and almost certainly unfairly) of Apollonius. The picture of Euclid drawn by Pappus is, however, certainly in line with the evidence from his mathematical texts. Another story told by Stobaeus is the following:- Euclid Euclid's most famous work is his treatise on mathematics • The Elements. The book was a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid but the organisation of the material and its exposition are certainly due to him. Euclid In fact there is ample evidence that Euclid is using earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used such as that of an oblong, a rhombus, and a rhomboid. Euclid’ postulates again The fourth and fifth postulates are of a different nature. Postulate four states that all right angles are equal. This may seem "obvious" but it actually assumes that space in homogeneous - by this we mean that a figure will be independent of the position in space in which it is placed. The famous fifth, or parallel, postulate states that one and only one line can be drawn through a point parallel to a given line. Euclid's decision to make this a postulate led to Euclidean geometry. It was not until the 19th century that this postulate was dropped and non-Euclidean geometries were studied. Euclid There are also axioms which Euclid calls 'common notions'. These are not specific geometrical properties but rather general assumptions which allow mathematics to proceed as a deductive science. For example:Things which are equal to the same thing are equal to each other. Zeno Zeno of Sidon, about 250 years after Euclid wrote the Elements, seems to have been the first to show that Euclid's propositions were not deduced from the postulates and axioms alone, and Euclid does make other subtle assumptions. The Elements is divided into 13 books. Books one to six deal with plane geometry. In particular books one and two set out basic properties of triangles, parallels, parallelograms, rectangles and squares. Achilles and the tortoise “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.” —Aristotle, Physics, VI:9, 239b15 Achilles and the tortoise In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise's starting point. Achilles and the tortoise During this time, the tortoise has run a much shorter distance, for example 10 feet. It will then take Achilles some further time to run that distance, in which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been-he can never overtake the tortoise Euclid Book three studies properties of the circle while book four deals with problems about circles and is thought largely to set out work of the followers of Pythagoras. Book five lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes. Heath says: Euclid Greek mathematics can boast no finer discovery than this theory, which put on a sound footing so much of geometry as depended on the use of proportion. Book six looks at applications of the results of book five to plane geometry. Books seven to nine deal with number theory. In particular book seven is a self-contained introduction to number theory and contains the Euclidean Algorithm for finding the greatest common divisor of two numbers. Book eight looks at numbers in geometrical progression but van der Waerden writes that it contains:- Euclid ... cumbersome enunciations, needless repetitions, and even logical fallacies. Apparently Euclid's exposition excelled only in those parts in which he had excellent sources at his disposal. Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus. Euclid Books eleven to thirteen deal with three-dimensional geometry. In book eleven the basic definitions needed for the three books together are given. The theorems then follow a fairly similar pattern to the two-dimensional analogues previously given in books one and four. The main results of book twelve are that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters. These results are certainly due to Eudoxus. Euclid Euclid proves these theorems using the “method of exhaustion" as invented by Eudoxus. The Elements ends with book thirteen which discusses the properties of the five regular polyhedra and gives a proof that there are precisely five. This book appears to be based largely on an earlier treatise by Theaetutus. Euclid Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. The standard of rigor was to become a goal for the inventors of the calculus centuries later. As Heath writes: This wonderful book, with all its imperfections, which are indeed slight enough when account is taken of the date it appeared, is and will doubtless remain the greatest mathematical textbook of all time. ... Even in Greek times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, altering the language here and there, mostly with a view to greater clearness and consistency... Euclid It is a fascinating story how the Elements has survived from Euclid's time and this is told well by Fowler. He describes the earliest material relating to the Elements which has survived:Our earliest glimpse of Euclidean material will be the most remarkable for a thousand years, six fragmentary ostraca containing text and a figure ... found on Elephantine Island in 1906/07 and 1907/08... Euclid These texts are early, though still more than 100 years after the death of Plato (they are dated on palaeographic grounds to the third quarter of the third century BC); advanced (they deal with the results found in the "Elements" [book thirteen] ... on the pentagon, hexagon, decagon, and icosahedron); and they do not follow the text of the Elements. ... So they give evidence of someone in the third century BC, located more than 500 miles south of Alexandria, working through this difficult material... this may be an attempt to understand the mathematics, and not a slavish copying ... Euclid The next fragment that we have dates from 75 - 125 AD and again appears to be notes by someone trying to understand the material of the Elements. More than one thousand editions of The Elements have been published since it was first printed in 1482. Heath discusses many of the editions and describes the likely changes to the text over the years. B L van der Waerden assesses the importance of the Elements:- Euclid Almost from the time of its writing and lasting almost to the present, the Elements has exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century. It is sometimes said that, next to the Bible, the "Elements" may be the most translated, published, and studied of all the books produced in the Western world. Euclid Euclid also wrote the following books which have survived: Data (with 94 propositions), which looks at what properties of figures can be deduced when other properties are given; On Divisions which looks at constructions to divide a figure into two parts with areas of given ratio; Optics which is the first Greek work on perspective; and Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set. Euclid Euclid's following books have all been lost: • Surface Loci (two books), Porisms Conics (four books), Book of Fallacies and Elements of Music. The Book of Fallacies is described by Proclus:- Euclid Since many things seem to conform with the truth and to follow from scientific principles, but lead astray from the principles and deceive the more superficial, [Euclid] has handed down methods for the clear-sighted understanding of these matters also ... The treatise in which he gave this machinery to us is entitled Fallacies, enumerating in order the various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of the error with practical illustration. Euclid Elements of Music is a work which is attributed to Euclid by Proclus. We have two treatises on music which have survived, and have by some authors attributed to Euclid, but it is now thought that they are not the work on music referred to by Proclus. Euclid Euclid may not have been a first class mathematician but the long lasting nature of The Elements must make him the leading mathematics teacher of antiquity or perhaps of all time. As a final personal note let me add that my own introduction to mathematics at school did not include in the late 1970s-80s any edition or part of Euclid's Elements and thus the concept of proof was lacking as it is in the mathematics teaching in schools today. Euclid Book 1 of The Elements begins with numerous definitions followed by the famous five postulates. Then, before Euclid starts to prove theorems, he gives a list of common notions. The first few definitions are: Def. 1.1. A point is that which has no part. Def. 1.2. A line is a breadthless length. Def. 1.3. The extremities of lines are points. Def. 1.4. A straight line lies equally with respect to the points on itself. Euclid The postulates are ones of construction such as: One can draw a straight line from any point to any point. The common notions are axioms such as: Things equal to the same thing are also equal to one another. • If a=b and b=c then a=c We should note certain things. Euclid Euclid seems to define a point twice (definitions 1 and 3) and a line twice (definitions 2 and 4). This is rather strange. Euclid never makes use of the definitions and never refers to them in the rest of the text. Some concepts are never defined. For example there is no notion of ordering the points on a line, so the idea that one point is between two others is never defined, but of course it is used. Euclid As we noted in the real numbers, Pythagoras to Stevin, Book V of The Elements considers magnitudes and the theory of proportion of magnitudes. However Euclid leaves the concept of magnitude undefined and this appears to modern readers as though Euclid has failed to set up magnitudes with the rigor for which he is famed. Euclid When Euclid introduces magnitudes and numbers he gives some definitions but no postulates or common notions. For example one might expect Euclid to postulate a + b = b + a, (a + b) + c = a + (b + c), etc., but he does not. When Euclid introduces numbers in Book VII he does make a definition rather similar to the basic ones at the beginning of Book I: A unit is that by virtue of which each of the things that exist are called one. Euclid Some historians have suggested that the difference between the way that basic definitions occur at the beginning of Book I and of Book V is not because Euclid was less rigorous in Book V, rather they suggest that Euclid always left his basic concepts undefined and the definitions at the beginning of Book I are later additions. What is the evidence for this? The first comment would be that this would explain why Euclid never refers to the basic definitions. If they were not in the text that Euclid wrote then of course he could not refer to them. Euclid The next point to note is that they are very similar to the work which is ascribed to Heron called Definitions of terms in geometry. This contains 133 definitions of geometrical terms beginning with points, lines etc. which are very close to those given by Euclid. Knorr argues convincingly that this work is in fact due to Diophantus. The point here is the following. Is Definitions of terms in geometry based on Euclid's Elements or have the basic definitions from this work been inserted into later versions of The Elements? Diophantus of Alexandria about 200 - about 284 Diophantus was a Greek mathematician sometimes known as 'the father of algebra' who is best known for his Arithmetica. This had an enormous influence on the development of number theory. • Pierre de Fermat Diophantus: Fermat’s Last theorem "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" Heron of Alexandria about 10 - about 75 Heron or Hero of Alexandria was an important geometer and worker in mechanics who invented many machines including 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. Euclid We have to consider what Sextus Empiricus says about definitions. First note that Sextus wrote about 200 AD and it was believed until comparatively recently that Heron lived later than this. Were this the case, then of course Sextus could not have referred to anything written by Heron. However more recently Heron has been dated to the first century AD and this tells us that Sextus wrote after Heron. The other part of the puzzle we have to consider here is the earliest version of Euclid’d Elements to be found. When Vesuvius erupted in 79 AD, Euclid Herculaneum together with Pompeii and Stabiae, were destroyed. Herculaneum was buried by a compact mass of material about 16 m deep which preserved the city until excavations began in the 18th century. Euclid Special conditions of humidity of the ground conserved wood, cloth, food, and in particular papyri which give us important information. One papyrus found there contains fragments of The Elements and was clearly written before 79 AD. Since Philodemus, a student of Zeno of Sidon, took his library of papyri there some time soon after 75 BC the version of The Elements is likely to be of around that date. Zeno of Sidon about 150 BC - about 70 BC Zeno Zeno of Sidon was a Greek philosopher who became head of the Epicurean school. He criticised some of the axioms that Euclid set out in The Elements. Euclid Let us go back to Sextus who writes about "mathematicians describing geometrical entities" and it is interesting that the word "describing" is not used in The Elements but is used by Heron in Definitions of terms in geometry. Again the descriptions he gives are closer to the exact words appearing in Heron than those of Euclid. When Sextus gives "the definition of a circle" he uses the word "definition" which is that of Euclid. Sextus quotes the precise definition of a circle which appears in the Herculaneum fragment. Euclid This does not include a definition of "circumference" although Euclid does use the notion of circumference of a circle. The later versions of The Elements which have come down to us include a definition of "circumference" within the definition of a circle. Euclid None of the above proves whether the basic definitions of geometric objects have been added to The Elements later. They do show fairly convincingly that the definition of a circle has been extended to include the definition of circumference in later editions of the book. The hypothesis is that Sextus has The Elements and Definitions of terms in geometry in front of him when he is writing and he uses the word "describe" when he refers to Heron and "define" when he refers to Euclid. Euclid Even it this is correct it still doesn't prove that the version of The Elements sitting in front of Sextus does not contain basic definitions of geometric objects but it does make such a possibility at least worth debating. What do you think? Euclid One last point to think about. We quoted above: Def. 1.4. A straight line lies equally with respect to the points on itself. What does this mean? It does seem a strange description for Euclid to give, for it appears to be meaningless. Compare it with the definition of a straight line in Definitions of terms in geometry: A straight line is a line that equally with respect to all points on itself lies straight and maximally taut between its extremities. Euclid Again we ask: do you think that the definition appearing in The Elements is a corruption of Heron’s definition and so was added later, or do you think that Euclid gave a rather poor definition which was improved by Heron? Why do neither use the definition of a straight line as the shortest distance between two points? Euclid He is therefore younger than Platro’s circle, but older than Erarosthenes and Archimedes; for these were contemporaries, as Erarosthenes somewhere says. In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole "Elements" the construction of the so-called Platonic figures. Bertrand Russell on Euclid Bertrand Russell wrote an article The Teaching of Euclid in which he was highly critical of the Euclid's axiomatic approach. Although this article is very interesting, it seems extremely harsh to criticize Euclid in the way that Russell does. As someone once said, Euclid's main fault in Russell's eyes is that he hadn't read the work of Russell. The article appeared in The Mathematical Gazette in 1902. Its full reference is B Russell, The Teaching of Euclid, The Mathematical Gazette 2 (33) (1902), 165-167. We give below our version of Russell's article. The Teaching of Euclid It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. The Teaching of Euclid A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid's earlier proofs fail before this test. The first proposition assumes that the circles used in the construction intersect - an assumption not noticed by Euclid because of the dangerous habit of using a figure. We require as a lemma, before the construction can be known to succeed, the following: Euclid If A and B be any two given points, there is at least one point C whose distances from A and B are both equal to AB. Euclid This lemma may be derived from an axiom of continuity. The fact that in elliptic space it is not always possible to construct an equilateral triangle on a given base, shows also that Euclid has assumed the straight line to be not a closed curve - an assumption which certainly is not made explicit. When these facts are taken account of, it will be found that the first proposition has a rather long proof, and presupposes the fourth. We require the axiom: on any straight line there is at least one point whose distance from a given point on or off the line exceeds a given distance. Euclid Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics. It has influenced all branches of science but none so much as mathematics and the exact sciences. The Elements have been studied 24 centuries in many languages starting, of course, in the original Greek, then in Arabic, Latin, and many modern languages. Euclid I am discussing Euclid's Elements for a couple of reasons. The main one is to rekindle an interest in the Elements, and the web is a great way to do that. Using Java applets we can illustrate geometry. That also helps to bring the Elements alive. The text of all 13 Books is complete, and all of the figures are illustrated using the Geometry Applet, even those in the last three books on solid geometry that are three-dimensional. Euclid The Geometry Applet is used to illustrate the figures in the Elements. With the help of this applet, you can manipulate the figures by dragging points. In order to take advantage of this applet, be sure that you have enabled Java on your browser. If you disable Java, or if your browser is not Java-capable, then the illustrations in the elements will still appear, but as plain images. If you click on a point in the figure, you can usually move it in some way. The free points, usually colored red, can be freely dragged about, and as they move, the rest of the diagram (except the other free points) will adjust appropriately. Sliding points, usually colored orange, can be dragged about like the free points, except their motion is limited to either a straight line, a circle, a plane, or a sphere, depending on the point. Other points can be dragged to translate the entire diagram. But if a pivot point appears, usually colored green, then the diagram will be rotated and scaled around that pivot point. Euclid Take, for example, the figure below showing the relation between a tetrahedron and a cube inscribed in a sphere. The diameter of the sphere has length AB, and you can drag the endpoints A and B to change the size of the sphere. The side of the cube has length BD, and the side of the tetrahedron has length AD. The cube is drawn with red edges while the tetrahedron is shaded light blue and drawn with blue edges. The center of the sphere is the red dot, and you can drag it to move the sphere around. The point E can be dragged anywhere on the surface of the sphere. The point F has to be at length BD from E on the surface of the sphere, and so it drags along a certain circle on the sphere. The rest of the cube and tetrahedron are then determined. (See proposition XIII.15 for background on the mathematics.)

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