This course combines Greek and Geometry to show how to read Euclid's Elements in the original language

*"I would make them all learn English; and then I would let the clever ones learn Latin as an honour, and Greek as a treat"*

--Sir Winston Churchill

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*Eu)klei/dou Stoixei~a*, Euclid's Elements, the classical textbook in geometry, is easy to read in the original ancient Greek, but its grammar and vocabulary are not those familiar from the usual course in elementary Greek, with peculiarities that make it difficult for the beginner. The text of the Elements that we have is written in the literary koinh/ typical of the 1st century AD. This course concentrates on exactly what is necessary to read Euclid, both in vocabulary and grammar. Its sole aim is to teach how to read this work, and similar texts in Greek mathematics, and not to compose Greek sentences, nor to read the Iliad or Plato. All necessary information is included in the course. A great amount of scholarship has been devoted to Euclid, mainly in Latin or German, and this course may expose some of it to a larger audience, to whom it has been largely inaccessible. For authoritative details, reference must be made to these sources, since the present one claims no expertise. There are many websites with information on Euclid and geometry. For example, look at the link to Euclid in the Seven Wonders website that is referenced in the Classics Index page, under the heading Pharos of Alexandria. As is typical of education on the Internet, many sites are poor, repetitive or childish, however.

If you already have some knowledge of Greek, as well as of the rudiments of Geometry, this course will be easy and entertaining. If you have no Geometry, this course may introduce you to its beauties. This is certainly a proper place to begin, as the Elements were intended as an introduction, and have been used as such for two millennia. Mathematics, as what is taught in schools is wrongly called, is not calculation but an exercise in reason, and Euclid does it with words. Literacy is the foundation of knowledge, including mathematics. There are not two kinds of knowledge, as commonly supposed. It is simply easier to acquire verbal knowledge, since it is built in, but mathematics requires intelligence and effort. If you have Geometry but no Greek, the road is harder. It will take a great deal of work to become accustomed to the alphabet, and then to the grammar and vocabulary. You should consider it as an exercise in decoding. It may awaken a desire to learn Greek properly. If you have neither Greek nor Geometry, you have the hardest task of all, but one which will be valuable to you in opening doors whose existence was hardly suspected. Above all, what is essential is a fascination with both Greek and Geometry.

Greek geometry traditionally begins with Thales of Miletus (624-547 BC), one of the Seven Wise Men of the ancient world (to people of the classical world), who is said to have brought the rudiments from Egypt. It is more probable that the roots are to be found in Mesopotamia, however. Pythagoras, born in Samos in 582 BC, founded his famous colony in Crotona in 529 BC. Plato used mathematical illustrations in much of his work, making it popular among philosophers. Eudoxus (408-355), a great mathematician, developed geometry further, as did Aristotle and Menaechmus, discoverer of the conic sections. It was on this basis that The Elements was built. Later workers, such as Archimedes (287-212 BC), Eratosthenes (b. 284 BC), Apollonius of Perga (fl. 220 BC), and Ptolemy (fl. 150) advanced geometry and astronomy greatly. Geometry was still actively pursued in the time of Proclus (410-485) of Athens, and Pappus (fl. 320) of Alexandria, but ceased in the disorders and intellectual collapse as the Western World turned to superstition. Much of the work, especially the more advanced parts, were lost, and we know little of the lives of the geometers. Although they added very little, the Arabs preserved much what has survived, and deserve great credit for this.

The best modern translation and commentary on Euclid is the work by T. L. Heath, *Euclid's Elements*, 2nd ed., 3 vols., (New York: Dover Publications, 1956). An excellent small volume containing I. i's edition of 1862, based on Simson, was published in the Everyman's Library (London: Dent and New York: Dutton), the last printing being, apparently, in 1967. This series seems no longer to be published, which is a pity, and it is a greater pity that Euclid is not easily available to the intelligent. If you want to learn about Geometry, a 20th-century school textbook (from when there were such things) is a bad place to begin. Mathematical knowledge among the public is declining to practically nothing, except for mere calculation.

The definitive text is *Euclidis Elementa*, by I. L. Heiberg (Leipzig: B. G. Teubner, 1883), which is the source of all the text in these pages. This work is currently quite difficult to obtain, although a good library may possess a copy. In it, Greek text is on the left-hand pages, with a Latin translation on the right, with footnotes explaining the sources and variant readings. The best and earliest sources for Euclid seem to be a Vatican parchment of the 10th century, discovered by Peyrard in 1808, and some earlier palimpsests in the British Museum from the 7th and 8th centuries. The thought of ignorant, quasi-literate monks scraping off Euclid to replace it by credulous fables still elicits disgust, and shows how contrary superstition is to reason. No Euclid was preserved in Western Europe. Adelard of Bath (fl. 1116-1142) made the first Latin translation from the Arabic of al-Hajjaj (fl. 786-833), who had translated it from Greek at the court of Caliph Harun-ar-Rashid, whence it came to Spain, thence to England.

Euclid's Elements are just that, the fundamentals of plane and solid geometry that form a basis for advanced work. They do not represent the limits of Greek mathematics, only its basis. They were assembled from existing knowledge around 300 BC, shortly after the foundation of the library at Alexandria by the Macedonian general Ptolemy I *So/ter* (367-283 BC), the "Saviour", in 322, a year after the death of Alexander, or soon afterwards. Euclid is the name given to the man supposed to have supervised this collection. Nothing is known of the life of Euclid, except that he worked in Alexandria, and is sometimes erroneously called Euclid of Megara, who was another philosopher entirely. There are websites that presume to give exact dates, and even a "biography," but this is nonsense. The text of the Elements is the result of a millennium of editing and recopying, of additions and deletions, and "improvements" by commentators. Heiberg's text was the first to be independent of Theon's influential edition of the 4th century, which was the basis of most translations until recently. Theon's recension had considerable merit, but differed from the earlier editions in important respects.

The word stoixei~on was used for the shadow of the gnomon in the sun, but more commonly for one of a row, or more abstractly, a letter of the alphabet, and from that--in the plural--to the elements, or basic components, of something. It is the neuter diminutive of a word meaning a line of poles, perhaps remotely cognate to the English word *stockade*. Under Euclid's name, works on Optics (theory of perspective), Catoptrics (mirrors), Phenomena (spherical astronomy), Data and Music have also been preserved or reconstructed. Some works have been lost, and are known only by references to them. Greek mathematics was further developed by Archimedes, who discussed spirals, among much else, Apollonius of Perga (conic sections), and others. It is by no means a small or easily-understood body of work. It is, incidentally, also correct.

In the rise of modern physical science in the 17th century, Euclid was considered part of the new science, not part of the discredited scholastic Aristotelian science that had proved erroneous and infertile. Newton used the style of Euclid, that of formal propositions and proofs, in the *Principia*, a method which still has much to recommend it, and which is still observed in principle in mathematics. Euclid was regarded as fundamental in the training of a scientist, its mood and rigour lying at the root of the new scientific method, in which experimental facts provided the postulates and also guaranteed their validity. Indeed, Euclid was the school text for geometry until the late 19th century. It has now vanished from schools, even in simplified editions, as being too difficult for the modern pupil (not to mention the teachers). Most aspects of reasoning have been purged from education, especially in the United States, replaced by a cult of manipulation and following rules and prescriptions. Euclid firmly connects modern science with ancient, Newton with Archimedes. Algebra is a modern addition; geometry with algebra is a very powerful scientific tool reaching its highest development in the calculus, which, in fact, was invented by Newton and Leibniz.

The Elements consists of 13 Books (two more were added later, apparently the work of Hypsicles, and generally numbered as books 14 and 15), containing about 465 propositions in all. A majority of the propositions consist of geometrical calculation with magnitudes, involving such things as proportion, commensurability, prime numbers and other such matters. Much of this is now done with algebra. However, there is still a great deal of material on the kind of geometry with which the reader is familiar. Book I on triangles, Book III on circles, Book IV on regular polygons, Book VI on similar plane figures, and Books XI-XIII on solids contain this material, and are those that were generally presented to students in modern times. Book II was also usually included, since it included the solution of certain numerical problems of general utility. Book V, on proportion, was also often studied. Most of the examples in this course are taken from Books I and III, with a few from Books II, IV and VI, and from other works under Euclid's name.

The propositions in the early books are often obvious, but the emphasis in the Elements is on rigour, and teaching the methods and forms of proof. There is good reason for this: "common sense" led to the scientific errors in Aristotle, and only rigour leads to truth. Some important propositions are not at all obvious, however, and understanding their solutions is a delight. When considering a proposition, it is good first to form a mental sketch of a possible proof, before getting lost in the words of the text, which must be rigorously chosen and carefully made to depend only on previously demonstrated propositions. Sometimes it helps to start from the conclusion, and work towards the beginning, in devising a method of proof. The Greeks were proud of this process, called *analysis*. In modern mathematics, "analysis" has a different meaning. Also note that the converse of every important proposition is also proved in Euclid. The converse interchanges the statement and conclusion of a proposition. For example, if the proposition is that equal angles are opposite equal sides (Prop. I-5), the converse is that equal sides are opposite equal angles (Prop. I-6).

The high points of Books I-IV are worth a review. The high point of Book I is, of course, the Pythagorean theorem and its converse, which are the last two propositions, towards which the whole book progresses. The most interesting proposition of Book II is the division of a line by the Golden Section, Proposition 11, the way to which is prepared by Proposition 6. Book III culminates with its final two propositions, that the product of the distances from an external point to the two points where a secant cuts the circle is equal to the square of the tangent from the external point, and its converse. On the way, it is proved that the rectangles enclosed by the two segments of each of two intersecting chords of a circle are equal, that the opposite angles of a quadrilateral inscribed in a circle are supplementary, that the central angle is twice the angle from the circumference standing on a given chord, and the very old observation that the angle in a semicircle is a right angle. On the way to these results, one learns the properties of triangles, parallels and circles, and how to carry out constructions with compass and straightedge, all of which is still of practical value.

Mathematics has been cultivated only by advanced cultures. Most people have been, and still are, content with counting their days, possessions and money, for which the elementary processes of what was known as *logistic*, and now generally *arithmetic*, are sufficient. Mathematics, on the other hand, was studied for more intellectual purposes. One purpose was practical, for land surveying , construction or astronomy. The other was recreational, the devising of ingenious and entertaining problems and puzzles. Egypt, Babylonia, China and India all were advanced in these pursuits, and their ancient achievements are impressive, some of which have left traces in modern times. Empirical knowledge of the Pythagorean Theorem (which, of course, did not originate with Pythagoras), and of integer solutions such as the 3,4,5 triangle, was widespread. The Greek mathematics displayed in the Elements is completely different, going far beyond these mathematical recreations into another world, that appears nowhere else, in any equivalent form. Many commentators simply put all ancient mathematics in a single basket, and, in their ignorance of mathematics, assume that knowledge of the Pythagorean theorem or similar notions is similar whether Greek or Chinese. This, I believe, is not the case. In the Elements, the Pythagorean theorem is embedded in a stream of logic starting from simple beginnings and extending to complex and unexpected conclusions, and this is found nowhere else. The one unique factor in Western thought and science was its foundations in the Elements of Euclid, and social anthropologists need look no further for the cause of the universal spread of European thought. Incidentally, Euclid shows that the Pythagorean Theorem holds not just for squares, but for any similar figures (such as, the triangle on the hypotenuse is the sum of the triangles on the legs) in Book VI.

It is well-known that all the constructions in Euclidean geometry must be capable of being done with a straigtedge and a compass that cannot be lifted from the paper. This is not just some silly arbitrary rule, but embodies the postulates on which the logic is based. If other means of construction were admitted, the postulates would become more complicated and less elegant. Once a construction is demonstrated, it can then be carried out by any practical means available. The use of dividers and triangles is one step, and the use of graduated scales and protractors a further step into numerical geometry.

The arrangement of the course is a challenge. The Elements were not written as a beginning Greek reader, so a fairly wide knowledge is necessary to begin. The early part of the course is, therefore, mainly grammar, with words from Euclid introduced as examples. This material cannot be fully grasped at a first reading, and should be regarded as reference material. These early lessons are, however, full of information, and should be read through. More and more examples from the Elements are introduced as the course progresses, and when the necessary grammar has been explained, the amount of Geometry in the course increases. Space constraints preclude a complete presentation of even Book I, but the flavour of Euclid will be evident.

For those with little Latin or Greek, the grammar will probably be difficult. Conjugated verbs will probably not be too unfamiliar, but the cases of nouns will give some problems. This is, however, an essential matter, and a sensitivity to the use of the cases should be cultivated. This is easier than it might appear, and facilitates understanding. In Euclid, case usage is simple and straightforward, adding greatly to the clarity of the text. Cases free word order to express the matter at hand as clearly as possible. The reader may find Peter Jones' popular Greek course, *Learning Ancient Greek*, valuable and entertaining as an introduction.

For further work, a Greek grammar, such as Abbot and Mansfield's *A Primer of Greek Grammar* (London: Duckworth, 1977), a Greek introductory text, such as H. L. Crosby and J. N. Schaeffer, *Introduction to Greek*, (Boston: Allyn and Bacon, 1928), and a lexicon, such as Liddell and Scott's, which is available in several sizes, are recommended. The site Thrasymachus offers a Greek primer. My best conjectures as to the grammar are given, but I probably suffer from many misconceptions, and will greatly appreciate any illumination. Greek is a living language with a continuous tradition of three millennia, which has been subject to constant evolution in that time, so there cannot be a single 'correct' authority. There has been a huge amount of tillage in this field by scholars, who assert many facts that could not possibly be known, and these scholars are influenced by every sort of prejudice. All explanations given here are offered for the purpose of interpreting the mathematical texts, and are not warranted to be either general or historically accurate, or in agreement with any particular academic fashion. I am motivated only by *studium*.

The SPIonic font is used for Greek in the lesson text, with the Symbol font as a backup. To get SPIonic on your computer, see below. Unlike Jones, I hold accents to be very important, since they help you in internal vocalization, as well as reflecting the grammar. Since there are differences between SPIonic and Symbol, you may note improper spellings with Symbol, as well as diacritical marks rendered by such as /, \, (, ) and ~. These correspond to acute and grave accents, rough and smooth breathing, and the circumflex accent. Not all the examples have been adjusted for SPIonic as yet. Proper text with accents will be given in boxes rendered as GIF graphics until these, too, are rendered in SPIonic. Until things settle down, expect errors. Translations into English are rather literal, so that the correspondence with the Greek is clear. The student should recast them into better English after they are understood. Every effort should be made to go directly from Greek to meaning without passing through English, but this takes considerable practice.

The font SPIonic, a property of Scholars' Press, can be downloaded from Download SPIonic. When I downloaded the font by clicking on the link given there, the file wound up in Temporary Internet Files as spionic.ttf. Use Cut/Paste to transfer this file to the Fonts folder in your Settings folder. The directions given in Thrasymachus were not valid for my system, but the process is a very simple one, and completely safe. Windows should recognize the new font automatically. You can verify its presence with any of the font selection boxes in, for example, Word. Having this font will allow you to type ancient Greek in your word processor and e-mails as well.

Another change is the addition of explanatory pop-ups to serve as footnotes and additional explanations. They will appear in a distinctive colour (at present, green) but will not be underlined as a link is. When you click on one, or sometimes just move the mouse cursor over one, an "alert" box will appear with the information. Close the box by clicking on the OK button. Try it now with quadrilateral. These, also, will appear gradually.

I gratefully acknowledge the collaboration of Mr Wayne Collins in the proofreading and general improvement of this site. He is responsible for encouraging me to do this work, and has made many valuable additions and suggestions. He is preparing a paper copy of the course for those who may want a hard copy, or who do not have Internet access. At this time, our work has just begun.

- Sir Thomas Heath,
*A History of Greek Mathematics*(New York: Dover, 1981), Reprint of the Clarendon Press 1921 edition. 2 vols. - I. Thomas,
*Selections Illustrating the History of Greek Mathematics*(Cambridge, MA: Harvard Univ. Press, 1939), 2 vols.

- Lesson α': The Alphabet
- Lesson β': Cases and the o-Declension
- Lesson γ': The α-Declension and Adjectives
- Lesson δ': Verbs
- Lesson ε': Little Words
- Lesson ': To Be; Pronouns
- Lesson ζ': Participles
- Lesson η': More Declensions
- Lesson θ': The Euclidean Proposition
- Lesson ι': Adjectives and Comparisons
- Lesson ια': Prepositions
- Lesson ιβ': Adverbs
- Lesson ιγ': Third-Person Imperatives
- Lesson ιδ': Analysis of a Sentence
- Lesson ιε': A Remarkable Proposition
- Lesson ι': Definitions
- Lesson ιζ': Postulates
- Lesson ιη': Axioms
- Lesson ιθ': Book I, Proposition 5, Angles at Base of Isosceles Triangle Equal
- Lesson κ': Book I, Proposition 47, The Pythagorean Theorem
- Lesson κα': Book II, Proposition 11, The Golden Section
- Lesson κβ': Book III, Proposition 17, Drawing a Tangent to a Circle
- Lesson κγ': Book III, Proposition 35, Intersecting Chords in a Circle
- Lesson κδ': A Theorem From The Optics, Viewing Angle Not Proportional to Distance
- Lesson κε': A Theorem From The Phaenomena, Fixed Stars Rise and Set at the Same Points
- Appendix α': Declension
- Index: Grammar and Geometry

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Composed by J. B. Calvert

Created 6 August 2000

Last revised 16 June 2002