Euclid: The Creation of Mathematics

Forty years ago, Jean Dieudonné declared "Euclid must go!" Our school system, remarkably, seems to have taken up this slogan, though it is hard to imagine that Dieudonné would have been pleased by what the schools have put in Euclid's place.

But of course, Euclid has not gone. There seems to be an endless fascination with Euclid's Elements, the greatest textbook ever written. Since Dieudonné's call for his ouster, a great deal of work has been done which deepens our understanding of Euclid ---notably, in recent years, the work of the late Wilbur Knorr and of David Fowler.

"Euclid", in Benno Artmann's new book Euclid: The Creation of Mathematics, refers exclusively to the Euclid of the Elements; none of Euclid's other works is discussed. There is nothing wrong with this, of course, though it might be a little misleading in suggesting to the reader that the Elements is the only work of Euclid which we possess. There is a tendency among modern authors to take the style of the Elements to be typical of "the Greeks", though in fact even the other works of Euclid himself are rather different from the Elements.

Artmann's book is not a thorough and systematic exposition of the Elements, like Ian Mueller's Philosophy of Mathematics and Deductive Structure in Euclid's Elements. Addressed to "all lovers of mathematics", not just "the narrow circle of specialists in the history of mathematics", it gives an introductory overview of the Elements. Artmann outlines the structure of each of the thirteen books of the Elements, usually singling out what he sees as the key propositions, and giving a sketch of their proofs.

Interspersed among the sections dealing with the contents of the Elements are sections which use Euclid's text as a starting-point to investigate "the origins of mathematics": the axiomatic method, the role of definitions, the importance of generalization, the nature of infinity, and incommensurability.

Artmann also tries to connect Euclid with other parts of mathematics, as well as with culture in general. For example, after looking at Euclid's constructions for certain regular polygons in Book IV, Artmann goes on to discuss Archimedes's "neusis" constructions of regular 7- and 9-gons. (Roughly speaking, a "neusis" construction involves marking a certain length on a ruler, which is then slid into place in the diagram.) Artmann then describes Gauss's results in the Disquisitiones Arithmeticae concerning the construction of regular polygons with straight-edge and compasses. Artmann compares Euclid's number theory with that of Nicomachus and that of Diophantus. And there are quotations from the philosophers Plato, Aristotle, and Plotinus, from the Greek dramatists Aeschylus and Aristophanes, from Dante, from the eighteenth-century German poet Adelbert von Chamisso, from Marcel Proust, from Robert Musil (author of The Man without Qualities), from Lewis Carroll, and from the mathematicians G. H. Hardy and the late G.-C. Rota.

In recent years, the study of Euclid has become something of a mine-field, various authors having staked out contentious positions with regard to certain points of interpretation. In 1975, the historian Sabetai Unguru published a paper, "On the need to rewrite the history of Greek mathematics", in which he attacked the doctrine, which goes back to the nineteenth-century mathematical historians Paul Tannery and H. G. Zeuthen, that parts of the Elements, particularly Books II and VI, could best be understood as "geometric algebra", later identified by Otto Neugebauer as a development of the "algebra" of the Babylonians. Unguru particularly criticized the presentation of this doctrine by the mathematician cum historian B. L. van der Waerden in his book Science Awakening. Van der Waerden replied with a vigorous defense of his position, in which he was later joined by the mathematicians Hans Freudenthal and André Weil.

With the hindsight of two decades, it appears (to me at least) that this debate was largely a dispute about the meaning of the word "algebra." In particular, Unguru claimed that it was not legitimate to restate Euclid's arguments using modern algebraic notation (as T. L. Heath does in the commentary which accompanies his translation of the Elements), since Euclid himself did not use "algebraic symbols". In his rejoinder, Weil asked, with some justice: what are the Greek capital letters which Euclid uses to stand for numbers, for example in the proof of the arithmetical proposition IX.8, if not "symbols"?

One aspect of Euclid, however, which seems to have escaped both Unguru and his mathematical opponents, has recently been emphasized by David Fowler. Euclid's geometry, unlike ours, is not "arithmetized". When we think of line segments, or of regions of the plane, or of space, we naturally think of their lengths, areas, or volumes as numbers. We attach numbers, as it were, to the geometric objects. For Euclid, on the other hand, only the positive integers are "numbers", whereas line segments, rectangles, and so forth are themselves geometrical "magnitudes", independent of any numerical interpretation.

Suppose, then, that we have, say, two line segments. Euclid might have denoted them by "AB" and "CD". I cannot see that it would distort Euclid's thinking at all if we were to use instead "algebraic" symbols such as "x" and "y". But suppose we then wrote down an expression such as "xy" (or, for that matter, "AB.CD"). What kind of object would such an expression represent? We, today, would naturally think of x and y as numbers, so that their product, xy, is also a number. Not so for Euclid! He could, and frequently did, form the rectangle having AB and CD as sides, but this rectangle, a geometrical region or area, was not a number, and was a "magnitude" of a different kind from the segments AB and CD. (Thus, for Euclid, an expression such as "xy + x" could have no meaning.) We could, if we liked, denote the rectangle formed by AB and CD by "AB.CD", as long as we were clear what the symbolism stood for. However, many readers would have an impulse to read into this symbolism the modern idea of multiplication of {it numbers}, and to think of the product as a number representing the area.

In order to avoid this misinterpretation, E. J. Dijksterhuis, in his editions of Euclid and Archimedes, which were published long before Unguru's broadside, devised special symbols for the rectangle formed from two line segments ("O(AB, CD)") and for the square whose side is a given line segment ("T(AB)"). Artmann, in the book under review, uses, in the early chapters, a notation similar to that of Dijksterhuis. Instead of "O," Artmann uses a little rectangle, and, instead of "T," a little square. He does not, however, discuss the non-arithmetized nature of Euclid's geometry, nor does he take up the issue of representing Euclid's arguments algebraically, except, very briefly, on p. 42. There he makes the contradictory remarks, "Because the geometric version is quite sufficient for the understanding of Euclid's text, we will leave the formulas aside. Occasionally we will use them in order to facilitate understanding for the modern reader." Indeed, in the later parts of his book he switches to modern algebraic symbolism. Of course, even the most purist authors (Unguru included) find it necessary to use some kind of symbolism in presenting Euclid's proofs, because a purely verbal account is too hard to follow. I suspect that this was just as true for Euclid's own students.

Another disputed point in the interpretation of Euclid concerns whether it is legitimate to use concepts from modern mathematics to describe Euclid's work. Thus, in another article, Weil remarked, "Anyway, it is impossible for us to analyze properly the contents of Books V and VII of Euclid without the concept of group and even that of groups with operators, since the ratios of magnitudes are treated as a multiplicative group operating on the additive group of the magnitudes themselves." For this remark, Weil was severely taken to task by Joseph Dauben. Of course, it is a purely factual question whether Euclid's "magnitudes" have the structure of a group (Weil stated specifically that he was not attributing the abstract concept of a group to Euclid); but, for Dauben, such an observation is not "history".

At any rate, Artmann, p. 129, lists the statements of the first six propositions of Book V (in algebraic symbolism). They are essentially the defining properties of a group with operators. (The "operators" here are the positive integers, not the ratios of magnitudes, as in Weil's remark.) Artmann even goes on to note the similarity with the modern concept of vector space. (The parallel is not quite exact, since the positive integers do not form a field. I suppose that Artmann assumed that his readers would be more likely to be acquainted with the concept of a vector space than with the more general concept of a group with operators.)

In other passages, too, Artmann does not shy away from interpreting Euclid's ideas in terms of modern mathematical concepts. Thus, he notes (p. 168) that Euclid's theory of "the least numbers of those which have the same ratio with them", Book VII, Props. 20--33, amounts to our concept of the uniqueness of the prime factorization.

A well-known topic in Euclid is that of a "construction" with straightedge and compasses. There is a common impression, indeed, that "the Greeks" had a rule that those instruments were the only ones which were permitted to be used in geometrical constructions. For example, W. W. Rouse Ball writes, in A Short Account of the History of Mathematics, "Thus the objection that he [Plato] expressed to the use in the construction of curves of any instruments other than rulers and compasses was at once accepted as a canon which must be observed in such problems." Clearly, this cannot be correct, as Archimedes and Apollonius frequently perform constructions which are not possible with straightedge and compasses; for example, as we mentioned above, Archimedes shows how to construct a regular 7-gon by means of a "neusis" construction. After giving an analogous construction for a regular pentagon by means of a "neusis", Artmann remarks (p. 104): "It seems that at some time and for some unknown reason neusis constructions were decided to be unacceptable. Only ruler and compass were permitted for geometrical constructions. It is sometimes said that this goes back to Plato, but an adequate reference to Plato's writings has not been supplied." The only citation Artmann gives for this claim is a 1936 article by A. D. Steele, which, far from showing that "neusis" constructions became unacceptable, investigates the whole issue of the supposed restriction to straightedge and compasses, and in particular Plato's alleged role in it, and finds the evidence wanting.

On p. 300, Artmann speculates that the Pythagorean discovery of the regular dodecahedron might have been inspired by the occurrence of the mineral pyrite in crystals which are "almost regular dodecahedra". It would have been instructive to point out that a regular dodecahedral crystal is impossible because of the "crystallographic restriction." Ordinary crystals are thought to be formed from regular lattices of atoms, and it is a theorem of geometry that such a lattice cannot have any 5-fold symmetries. I do not know who first proved this theorem, but it must have been known to the French scientist Bravais by 1850. Thus the excitement when "quasicrystals", which can have 5-fold symmetry, were discovered in 1984. For an account of this development, see Marjorie Senechal's book, Quasicrystals and Symmetry.

The typography and editing of Artmann's book are not up to the usual standard of the Springer-Verlag. There are a large number of typographical errors and incorrect references to figures. The figures have been drawn by Artmann himself; some of the labels are missing, and in some cases the hand-lettered Greek letters used as labels for angles are too small to read clearly.

The reader of Artmann's book will need to have the Elements itself at hand; Artmann frequently refers to Euclid's propositions by book and proposition number. And this is as it should be. Indeed, the only way to come to know Euclid is to read Euclid himself. Fortunately, Heath's magnificent edition of the Elements is still readily available. (The mathematical community owes a debt of gratitude to the Dover Publishing Company for keeping this great work in print, at a low price.) Artmann's book, read just by itself, is sometimes rather thin and unsatisfying, just a selection of highlights, with proofs only sketched, or omitted altogether. Notoriously difficult parts of Euclid (such as Book X) are skated over rather quickly, though references are provided for the reader who wishes to go further. But the reader who uses Artmann as a guide to the actual reading of Euclid will find a helpful introduction to Euclid's often unfamiliar ways of thought, and an orientation to the broader context of the Elements.

References:

Dieudonné pronounced his dictum "Euclid must go!" in his talk to the Royaumont Seminar, 1959. See New Thinking in School Mathematics, Organisation for European Economic Co-operation, 1961, p. 35. For a review by David Joyce of a standard current high-school geometry textbook, see http://aleph0.clarku.edu/~djoyce/java/elements/geotfacw.html.

For the work of Knorr and Fowler:

• Wilbur Knorr, The Evolution of the Euclidean Elements, Dordrecht, Reidel, 1975.
• Wilbur Knorr, The Ancient Tradition of Geometric Problems, Birkhäuser Boston, 1986, Dover reprint, 1993.
• Wilbur Knorr, Textual Studies in Ancient and Medieval Geometry, Birkhäuser Boston, 1989.
• David Fowler, The Mathematics of Plato's Academy, 2d edition, Oxford, 1999. (See the MAA review.)

   

Mueller's and van der Waerden's books:

• Ian Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements, MIT Press, 1981, Dover 2006.
• B. L. van der Waerden, Science Awakening, 4th edition, Noordhoff, 1975.

   

Neugebauer's views on "geometric algebra" are given in:

• Otto Neugebauer, The Exact Sciences in Antiquity, 2d edition, Brown University Press, 1957, Dover reprint, 1969.

   

Unguru and his critics:

• Sabetai Unguru, "On the need to rewrite the history of Greek mathematics", Archive for History of Exact Sciences, 15, 1975, 67--114.
• B. L. van der Waerden, "Defense of a `shocking' point of view", Archive for History of Exact Sciences, 15, 1976, 199--210.
• Hans Freudenthal, "What is algebra and what has been its history?", Archive for History of Exact Sciences, 16, 1977, 189--200.
• André Weil, "Who betrayed Euclid?", Archive for History of Exact Sciences, 19, 1978, 91--93.
• Sabetai Unguru, "History of Ancient Mathematics: Some reflections on the state of the art", Isis, 70, 1979, 555--565.

   

Dijksterhuis's notation:

• E. J. Dijksterhuis, Archimedes, Ejnar Munksgaard, 1956; Princeton paperback, 1987.

   

Weil's remark on groups with operators and Dauben's reply:

• André Weil, "History of Mathematics: Why and how", Collected Papers, vol. III, Springer-Verlag, 1979, 434--442.
• Joseph Dauben, "Mathematics: an Historian's Perspective", in The Intersection of History and Mathematics, Birkhäuser, 1994, 1--13.

Rouse Ball's quotation:

• W. W. Rouse Ball, A Short Account of the History of Mathematics, 4th edition, Dover reprint, 1960, p. 40.

   

Steele's article:

• Arthur Donald Steele, "Über die Rolle von Zirkel und Lineal in der griechischen Mathematik", Quellen und Studien zur Geschichte der Mathematik, Astronomie, und Physik, Abteilung B, 3, 1936, 287--369.

   

For the history of crystallography, see the remarks in

• R. L. E. Schwarzenberger, N-dimensional crystallography, Pitman, 1980, pp. 132--135.

   

Senechal's book:

• Marjorie Senechal, Quasicrystals and geometry, Cambridge, 1995.

   

Heath's edition of Euclid:

• T. L. Heath, The Thirteen Books of Euclid's Elements, 3 volumes, Cambridge University Press, 1926, Dover reprint, 1956.Euclid on the web:

• Euclid's Elements, by David Joyner. This web edition uses Java applets to illuminate the propositions.
• The Perseus Project at Tufts University also has made Heath's translation of the Elements available online.

Another work of Euclid:

• J. L. Berggren and R. S. D. Thomas, Euclid's Phaenomena: A Translation and Study of a Hellenistic Treatise in Spherical Astronomy, Garland, 1996.

Stacy G. Langton (langton@acusd.edu) is Professor of Mathematics and Computer Science at the University of San Diego. During his recent sabbatical, he translated three articles by Euler.