The Elements: Books I-XIII (Barnes & Noble Library of Essential Reading) by Euclid, Sir Thomas L. Heath (Translator), Andrew Aberdein (Introduction)

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Euclid's Elements is a fundamental landmark of mathematical achievement. Firstly, it is a compendium of the principal mathematical work undertaken in classical Greece, for which in many cases no other source survives. Secondly, it is a model of organizational clarity which has had a deep influence on the way almost all subsequent mathematical research has been conducted. Thirdly, it is the most successful textbook ever written, only seriously challenged as an account of elementary geometry in the nineteenth century, more than two thousand years after its first publication.

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• Forewords & Introductions

Forewords & Introductions

Euclid's Elements is without question a true masterpiece of Western civilization. It is one of the most widely disseminated and most influential books of all time. A fundamental landmark of mathematical achievement, the Elements is profoundly important for several distinct reasons. Firstly, it is a compendium of the principal mathematical work undertaken in classical Greece, for which in many cases no other source survives. Secondly, it is a model of organizational clarity which has had a deep influence on the way almost all subsequent mathematical research has been conducted. Thirdly, it is the most successful textbook ever written, only seriously challenged as an account of elementary geometry in the nineteenth century, more than two thousand years after its first publication.

Homer, the greatest Greek poet, became so obscure that skeptics were able to speculate that he never existed; Euclid, the greatest Greek scientist, is comparably enigmatic. We do not know where Euclid was born. Some medieval writers referred to him as Euclid of Megara, but it is now apparent that this is a confusion with a different Euclid, a student of Socrates (469-399 BC) who was a century older than the author of the Elements. We do not know with any certainty when Euclid was born. Our best source, Proclus (AD 410-485), argues that Euclid lived some time between the death of Plato (427-347 BC) and the birth of Archimedes (287-212 BC). However, it is clear that even this is conjectural. Euclid probably learned mathematics at Plato's Academy in Athens. Some commentators have detected platonist echoes in Euclid's writing, but it is a fairly safe bet that he studied in Athens, since we know of nowhere else he could have pursued his subject to comparable depth. We can be reasonably confident that Euclid taught at Alexandria in Egypt, most likely during the reign of Ptolemy I (306-283 BC). One of Ptolemy's first acts as ruler had been to establish a library, the celebrated Great Library of Alexandria, and a school of advanced study, patterned after those in Athens, and known as the Museum. Euclid appears to have been hired as one of the original faculty and to have remained there for the rest of his career. Little else is known, although a few anecdotes have been preserved. One has Euclid responding to a student who had asked what he would gain by the geometry he had learnt by directing his slave to "Give him threepence, since he must profit by what he learns." Another has him reproaching an impatient Ptolemy that there is "no royal road in geometry." However, both stories are of much later date and the latter at least is also told of other ancient mathematicians.

The Greeks were not the first civilization to study geometry. They themselves attributed the invention of the discipline to Egyptian surveyors seeking to compensate landowners for the annual inundation of the Nile. Whatever the status of this story, it is true that the Egyptians did have practical surveying techniques at a very early date, as did ancient Babylonian, Chinese, and Indian civilizations. What influence these civilizations had on each other, or on the Greeks, is much harder to determine. The Babylonians left behind a rich legacy of mathematical cuneiform tablets, but they were far more interested in algebra than geometry. Documents from other early civilizations are either fragmentary or frustratingly hard to date. Despite some tantalizing passages, particularly in the Indian S´ulvasu¯tra tradition, all these sources lack any concept of rigorous proof. This specifically mathematical practice of deriving results with certainty from undisputed axioms, rather than by generalization from practical cases, appears to be a Greek innovation. Rigorous proof may have antedated Euclid by little more than a century: Traditional attributions of geometrical proofs to the early Greek philosophers Thales (c. 624-548 BC) and Pythagoras (585-497 BC) are hard to justify. However, both men left behind disciples, and it is amongst these less well-known figures that the concept of proof is likely to have originated. The next decisive step in mathematical history was the establishment of Plato's Academy in about 387 BC. Plato was fascinated by mathematics: Although he was not himself a mathematician, he made geometry central to the curriculum of his prototype university and recruited the greatest mathematicians of his day, including such luminaries as Eudoxus (408-355 BC) and Theaetetus (415-369 BC), much of whose work is preserved in Euclid's Elements.

There are several tempting misconceptions about the nature of the Elements: that it was the first such work of its kind; that the results, or at least the proofs, were largely Euclid's own work; that it is exhaustive of Greek mathematics. None of these claims are true. Several earlier authors are credited with having written Elements before Euclid, the earliest known being Hippocrates of Chios (fl. c. 430 BC). The success of Euclid's Elements obliterated all these earlier works, which have been lost since antiquity: Then as now, obsolete textbooks fell swiftly from favor. However, it would appear that all such works sought to assemble the core subject matter of Greek mathematics into a logical sequence, beginning with first principles which have to be accepted without proof, and then deriving some of the most useful and generally applicable results in the field from these principles. Euclid's genius was not as an original mathematician, but as a brilliant expositor. His work brings together many of the most important mathematical results then known. Euclid never attributes these results to their original discoverers, although we are sometimes able to identify these mathematicians through discussions in earlier works, such as those of Aristotle (384-322 BC), and references in ancient commentaries on Euclid. In this manner much of Books V and VI may be attributed to Eudoxus and much of X and XIII to Theaetetus. However, Euclid was not attempting to anthologize all prior mathematics, only the 'elementary' results, those which were most important to the subject and essential to a thorough mathematical education. Material that was more advanced, such as the theory of conic sections, or more rudimentary, such as everyday methods of calculation, is excluded from the Elements.

It is also somewhat misleading to say that the Elements is only concerned with geometry. Although the entire content of the book is set out geometrically, much of it is concerned with subject matter that we would now expect to be articulated by other means. Specifically, Books I through IV are concerned with plane geometry. Book I sets out the underlying principles of this discipline: It begins with definitions of the principal terms used in geometry, thereby setting boundaries to the subject. Next Euclid states five postulates and five common notions. The latter are to be understood as subject neutral claims, common to any discipline, whereas the former constitute axioms from which the rest of geometry may proceed. The remainder of the book sets out the principal results in the geometry of triangles. In Book II Euclid extends his treatment to rectangles, in Book III circles, and in Book IV polygons. Book V introduces a theory of proportion, which we would find more familiar in an algebraic format. Book VI applies this theory to specifically geometrical questions. Books VII, VIII, and IX are given over to results in arithmetic, that is elementary number theory. Book X confronts what had been for the Greeks the vexing topic of irrational numbers and incommensurable magnitudes. Pythagoras had taught that all magnitude could be expressed as ratios of whole numbers; the discovery by later members of his school that this was not so had been profoundly unsettling to the foundations of their mathematics and the quasi-religious beliefs from which they were derived. Finally, Books XI to XIII are concerned with solid geometry and culminate in a demonstration that the 'Platonic' solids, regular polyhedra having regular polygons for faces, are exactly five in number.

Euclid is known to have written several other works, four of which survive: the Data, the Division of Figures, the Phaenomena, and the Optics. The Data forms a sort of companion to the first six books of the Elements, consisting of worked problems in which various magnitudes are given and others are to be found. The Division of Figures exists only in an Arabic translation. It shows how to divide figures of plane geometry, such as triangles, circles, and quadrilaterals, into parts whose areas stand in a given ratio to each other. The Phaenomena is a text on astronomy, which includes some work on spherical geometry. The Optics is mostly concerned with perspective: It shows Euclid's allegiance to Plato's theory of vision, in which we see by way of rays emitted from our eyes, rather than incident upon them. Euclid is still able to derive sound conclusions, since he grasps the underlying point that light travels in straight lines. Lost works attributed to Euclid include the Conics, the Porisms, and the Pseudaria. The Conics appears to have been an anthology of prior work on conic sections, which was subsumed into the first three books of the Conic Sections of Apollonius (c. 262-190 BC), a work which survives in a mixture of Greek and Arabic editions. The Porisms was a treatment of propositions of a somewhat elusive nature, midway between theorems and problems, in the sense that they require a construction, but of something that is already known to exist. The Pseudaria was an anthology of fallacies in elementary geometry, intended as a teaching aid to the Elements.

As we do with many ancient works, we rely on a precarious process of transmission for the text of the Elements that we read today. Paradoxically, the much earlier Babylonian mathematical sources survive in original editions, sometimes the authors' own manuscripts. However, the Babylonians wrote on clay tablets, which in hot, dry conditions are virtually indestructible, whereas the Greeks wrote on much less durable papyrus or parchment. A few scraps of Euclid on papyrus, written perhaps four hundred years after the book's original publication, have been recovered from the Egyptian desert, but the earliest complete manuscript, now in the Bodleian Library at Oxford, dates from AD 888. This is closer to our own time than it is to Euclid's.

The relationship between these early texts and a modern edition such as this one is somewhat circuitous. It is believed that a Latin edition of the Elements may have circulated in the later Roman Empire, but all traces of it have long since disappeared. The first language into which we know Euclid was translated is Arabic. The earliest such translation appeared in the reign of Ha¯ru¯n al-Rashi¯d (786-809), the Caliph of Baghdad familiar from so many of the tales of The Thousand and One Nights. Although this edition is lost, several subsequent Arabic translations survive, which have been of some use in standardizing the text, on the hypothesis that the translators were working from good quality Greek manuscripts. All three of the earliest surviving Latin editions, by Athelhard of Bath (fl. c. 1120), Gherard of Cremona (1114-1187), and Johannes Campanus (fl. 1261-1281), are primarily translations from the Arabic. The third of these became the first edition of Euclid to appear in print, in Venice in 1482. Meanwhile, Greek manuscripts had been preserved, mostly in the monastic libraries of the Byzantine Empire, and began to trickle into western Europe, perhaps as early as the fourteenth century.

The search for a sound Greek text is further complicated by the poor quality of some of these early manuscripts, which after all are at the very least copies of copies of copies. Moreover, the copyists may have had a poor grasp of the subject matter, or have been working from inferior originals. In particular, the earliest printed Greek edition of 1533, the so-called editio princeps, upon which many subsequent translations were based, was derived from especially weak manuscripts. More importantly, this and all editions of the Elements until the nineteenth century were ultimately derived from a version prepared by Theon of Alexandria in about AD 400. This edition contained many emendations, most not clearly marked, where Theon had sought to improve upon Euclid, sometimes successfully, sometimes not. Only in the early nineteenth century did the French scholar François Peyrard (1760-1822) identify a tenth-century manuscript, now known as P, from the Vatican Library as a copy of an earlier edition from which Theon's changes were absent. The precise connection between P and the so-called Theonine editions is complicated by textual evidence that suggests that Theon was working from an even earlier and more accurate edition. (After all, he had at his disposal the resources of the Library of Alexandria, where the finest editions still extant in the fourth century were likely to be located.) Some early nineteenth century editions of the Elements, including Peyrard's own, made minor use of P, but modern Euclid scholarship really began in 1883-8 with the publication of a Greek text prepared by the Danish philologist Johan Ludvig Heiberg (1854-1928). He based his text primarily on P and, by comparing it with many other manuscripts, was able to reconstruct an edition similar to that which circulated prior to Theon's editorial intervention. All important modern editions of the Elements are derived from Heiberg's text.

The first English edition of the Elements appeared in 1570 in a translation by Sir Henry Billingsley (c. 1545-1606), who subsequently became Lord Mayor of London. This substantial volume contains numerous annotations, and an extensive preface by John Dee (1527-1608), better known for his interest in magic, who may have had a hand in the translation as well. Of the many subsequent English editions, the most influential was first published in 1756 by Robert Simson (1687-1768). However, the 1908 translation by Thomas Heath (1861-1940) contained in this book was the first to employ Heiberg's text and eclipses all its predecessors. Heath was one of the last great academic amateurs: Despite being a world class expert in the history of mathematics, he spent his entire working life in the British Civil Service, rising to the senior rank of Permanent Secretary to the Treasury, a post he held throughout the First World War. He was knighted twice, in 1909 and 1916, for his administrative work, and awarded fellowships of the Royal Society and the British Academy for his academic achievements. Besides his edition of Euclid's Elements, his publications included editions of the surviving works of Archimedes and Apollonius, and influential histories of Greek mathematics and astronomy. In 1925 Heath produced a second edition, 'revised with additions', of the Elements. Most of the text was reproduced photographically and the changes are comparatively minor: The 'Addenda and Corrigenda' were incorporated into the main text, and two short essays were added on Pythagoras and on the traditional names associated with the proposition I.5 ('Pons Asinorum' or 'Asses' Bridge' and 'Elefuga' or 'the flight of the miserable') and Pythagoras's Theorem I.47 ('the Franciscan's cowl', 'Dulcarnon', 'the bride's chair', and 'the theorem of the bride'). However, the most conspicuous asset of Heath's edition, the enormous volume of annotation explaining Euclid's work and linking it to that of subsequent mathematicians, was largely unchanged. Heath's methods have been brought up to date in the most recent new edition, Bernard Vitrac's extensively annotated French translation, published in four volumes between 1990 and 2001. Unfortunately for the non-francophone, this work is unavailable in English.

Important developments in geometry since Euclid's day include the splitting off of trigonometry, algebra and number theory as separate disciplines, the invention of analytic or coordinate geometry and the eventual heresy of non-Euclidean geometry. As we observed above, all of Euclid's results are arrived at geometrically, although many of them would seem more natural to us in a different mathematical idiom. For example, Propositions II.12 and II.13 are more familiar as the laws of cosines for obtuse and acute angles. However, the systematic study of trigonometry did not begin until more than a century after the Elements, with the work of Hipparchus of Nicaea (c. 180-125 BC). Similarly, Greek interest in algebraic and number theoretic problems remained essentially wedded to geometry until the publication of the Arithmetica of Diophantus (fl. c. AD 250), which may have been influenced by much earlier Babylonian work, probably unknown to Euclid. Euclid's practice of solving algebraic problems with geometry was eventually inverted, with the application of a much more sophisticated algebra to geometry, most notably in the work of René Descartes (1596-1650), who pioneered the use of algebraic equations as representations of geometrical curves.

The story of non-Euclidean geometry is one of frequent ineffectual anticipation: Numerous mathematicians stumbled on the idea, but failed either to realize its importance or to attract any attention to their discovery. The earliest was Geralmo Saccheri (1667-1733) in his 1733 book Euclides ab Omni Naevo Vindicatus: 'Euclid Cleared of Every Flaw'-an ironic choice of title given the nature of the discoveries it contains. He had intended to resolve the perceived awkwardness of the fifth or parallel postulate, by deriving it from the others, an exercise attempted since antiquity. His chosen method was innovative: An application of the rule of logical inference known to the medievals as consequentia mirabilis: that any proposition entailed by its own negation must be true. Hence he sought to show that, even if we assume the parallel postulate is false, it will still be possible to derive the postulate, thereby demonstrating that the postulate must be true after all. Saccheri's demonstration doesn't work, although he seems not to have realized this, precisely because the systems arising from the denial of the parallel postulate are internally consistent alternative geometries. Saccheri considered three possibilities, that the angles in a triangle add to two right angles (180°), to more than that, or to less than that, which give rise respectively to Euclidean geometry, elliptic or Riemannian geometry, and hyperbolic or Lobachevskian geometry. Saccheri's accomplishment was repeated by Johann Heinrich Lambert (1728-1777) in his Die Theorie der Parallellinein, posthumously published in 1786. Lambert came closer than Saccheri to realizing that he had found a new form of geometry, but not quite close enough.

The first mathematician who did realize the magnitude of this discovery was Carl Friedrich Gauss (1777-1855), perhaps the greatest mathematician of his generation. However, Gauss did not publish his results on non-Euclidean geometry, which only became widely known after his death. In the meantime, hyperbolic geometry had been independently and simultaneously rediscovered by two comparatively obscure figures, the Russian Nicolai Ivanovich Lobachevsky (1793-1856) and the Hungarian Janos Bolyai (1802-1860). Both recognized their discovery for what it was in papers coincidentally each published in 1829. However, the material remained obscure and little read; Gauss praised both works privately, but declined to do so in print, perhaps conscious of how controversial such a departure from Euclid would appear. Georg Riemann (1826-1866) developed an elliptic geometry in his seminal habilitation lecture of 1854, which was, however, not published until after his death. Finally, Hermann von Helmholtz (1821-1894) independently arrived at and extended Riemann's ideas in papers published from 1868 onwards. Hence it was only in the 1870s, nearly a century and a half after the earliest work in the field that non-Euclidean geometry became widely known amongst mathematicians. Riemann had observed that since Euclidean geometry is not unique, whether we live in a Euclidean or non-Euclidean world is a question for physicists, not mathematicians. Famously, it was settled in favor of non-Euclidean geometry with the confirmation of Albert Einstein's (1879-1955) theory of relativity.

Non-Euclidean geometry establishes the logical independence of Euclid's controversial fifth postulate. A more modest arbitrariness attaches to his first three postulates, which constrain the admissible methods of proof to straight edge and compass constructions. Crucially, neither device can be used to transfer a magnitude from one part of a construction to another: The straight edge is not a ruler, and the compass collapses when lifted from the page. These constraints make it impossible to solve three famous ancient geometrical puzzles: The trisection of an arbitrary angle, the doubling of a cube, and the squaring of a circle. This impossibility may have been suspected by the Greeks, but it only received a rigorous demonstration in the work of Gauss and Evariste Galois (1812-1832).

For much of its history Euclid's Elements was a paragon of mathematical rigor. All other fields of mathematics lagged behind until modern times: This includes fields such as logic and arithmetic, which would now be considered more foundational than geometry. As recently as the eighteenth century, many mathematicians sought to ground all mathematics in geometry. However by the nineteenth century it was clear that Euclidean geometry could be given a more rigorous foundation than that provided by Euclid. This work was pioneered by Moritz Pasch (1843-1930) and perfected by David Hilbert (1862-1943) in his 1899 Grundlagen der Geometrie. Hilbert's axiomatization achieves a far higher standard of rigor than Euclid's, with the mutual consistency and independency of the axioms explicitly proven, and the empirical specifics of the subject matter wholly abstracted away. As Hilbert is said to have remarked, "One must at all times be able to replace 'points, lines, planes' by 'tables, chairs, beer mugs'."

Hilbert's work may be understood as the elimination of diagrams from geometry: An important criticism of the Elements was that the diagrams often form essential parts of the proofs. However, diagrams remain essential for the effective study of geometry. Greek geometers are often represented scratching diagrams in sand with a stick, and they also used them freely in their texts, as had the Egyptians before them. Over the centuries a variety of proposals have been made to facilitate understanding by improving geometrical diagrams. Components that are variously given, unknown, or constructed have been distinguished in various ways, often by the thickness of the lines, but in at least one edition by color. One of the most helpful innovations of all is a product of the Internet. David Joyce of Clark University, Massachusetts, has made interactive versions of all Euclid's diagrams available online in a form that allows one to manipulate the variable parts and instantly see the changes to the whole.

The formal methods of proof characteristic of geometry and logic both originated in ancient Greece at much the same time, although probably independently of each other. (Euclid often provides separate proofs for propositions which in logic would follow immediately from propositions he has already established.) Hilbert's work finally demonstrated the superiority of logic to geometry as a foundation for mathematics, but the disciplines also rivaled each other as models of reasoning. In particular, the account of geometrical 'analysis and synthesis' provides a valuable insight into the heuristics, or problem-solving techniques, in use amongst the Greeks. Analysis describes the method of working backwards from the sought conclusion to known propositions, whereas synthesis proceeds in the opposite direction. This has been a direct inspiration for modern studies of mathematical problem solving, such as that of George Pólya (1887-1985). This work in turn has been a source for research into automated theorem proving: the programming of computers to prove mathematical theorems. One of the earliest successes of this approach was the independent discovery by a computer program called the Geometry Machine, developed in the 1950s by Herbert Gelernter of IBM, of a more natural proof of Euclid's I.4. Unbeknownst to Gelernter (or his program), this proof was first discovered by Pappus (fl. c. 300 AD), to whom we owe the most detailed account of analysis and synthesis to have survived.

In 1879 the Oxford mathematician Charles Dodgson (1832-1898), better known as Lewis Carroll, published a book with the tantalizing title Euclid and his Modern Rivals. However, the rivals he had in mind were not the pioneers of non-Euclidean geometry, a subject of which English mathematicians were only just beginning to take notice, but rather the authors of competing geometry textbooks. Sadly we can only speculate what the author of Through the Looking Glass might have done with such rich material. However, Carroll's determination to defend the superiority of Euclid for teaching purposes was not quite the quixotic or reactionary enterprise that it might at first appear. Indeed he makes short work of the vast majority of the books he considers, showing up fallacies and confusions in their attempts to improve upon Euclid, not much different from those in Theon's work 1500 years earlier. Two crucial exceptions to Carroll's critique are the books by the American Benjamin Peirce (1809-1880) and the Frenchman Adrien Marie Legendre (1752-1833), both of which are dismissed not for their errors, but for their supposed unsuitability for beginning students. There is something in this: As the introduction to the modern edition of Carroll's book observes, "there is much to be said for his standpoint that the degree of rigor in Euclid's Elements is just right for high school." However, Legendre's 1794 book Éléments de Géométrie represents an important redevelopment of the teaching of geometry, simplifying Euclid's exposition by the use of trigonometrical and algebraic techniques. This work ran into many editions, and became particular influential in American geometry teaching. In the twentieth century, geometry teaching has been largely in retreat, occupying less and less of the mathematics syllabus in both schools and universities, while what is taught is more in the spirit of Legendre than of Euclid. However, there are some signs of a recent return to Euclid. J. L. Heilbron's richly illustrated Geometry Civilized "follows more or less in order the material in Books I-IV, and some of that in Book VI of the Elements." While this work is intended to be accessible to the high school student, as well as the general reader, Robin Hartshorne's Geometry: Euclid and Beyond offers a similarly structured approach to the subject for the undergraduate mathematician. Euclid's place in the present century seems as assured as it has been for the previous twenty-three centuries.