Euclid—The Creation of Mathematics

The description “Ancient Greece” refers to the period roughly from 800 b.c.e. to 150 b.c.e., from Homer to the time when Rome established political hegemony over the Greek world. The first Olympic games took place in 776 b.c.e.; democracy was gradually introduced in the political life of the city-states from 600 b.c.e. onwards. The Greeks defended their freedom against the Persians in the “Persian wars” (500-480), after which the great classical period of Greece under the cultural leadership of Athens lasted until the Macedonian Kings Philip and Alexander the Great established monarchic rules around 330 and spread Greek culture over the whole ancient world in Hellenistic times, 300-50 b.c.e. Science and philosophy remained the domain of Greeks until the end of the Roman empire. Boetius, the “last Roman “ was the first writer to translate mathematical texts from Greek into Latin, about 500 c.e. The Romans ran their Imperium without any mathematics.

Traditionally, the Elements have been divided into three main parts: 1.Plane geometry, Books I-VI;2.Arithmetic, Books VII-X;3.Solid geometry, Books XI-XIII.

Our most important source about the history of Greek mathematics before Euclid originates from Eudemus of Rhodes, a student of Aristotle, who lived from about 350 to 300 b.c.e. He wrote a book on the history of mathematics, which has, however, been lost except for a few passages quoted by other authors. The following one was preserved in Proclus’s Commentary on Euclid. Proclus (410-485 c.e.) is writing about the origin and development of geometry, and the Eudemus passage starts with the second paragraph:

Euclid created the model of a mathematical text: Start with explicitly formulated definitions and axioms, then proceed with theorems and proofs. Unlike modern authors, who do not pretend to know what a set is, Euclid wants to say what he is talking about, or to give some sort of description of the objects of geometry. He does this in the first group of definitions, 1-9.

The discussion of the parallel axiom has been the driving force behind the axiomatization of mathematics. We will sketch the development in the history of mathematics. (For more detailed information see the Notes.)

Pythagoras lived about 570-490 B.C.E. The only roughly determined date in his life is ≈ 530, when he left Samos to settle in Crotona, in southern Italy. At Crotona he founded a religious and philosophical society that soon came to exert considerable political influence in the Greek cities of southern Italy. He was forced to leave Crotona about 500 and retired to Metapontum, where he died (see Fig. 6.1).

Book II is short and homogeneous, with only 14 propositions and two definitions at the beginning. For the most part it is about various combinations of rectangles and squares of equal content. At the end we find the generalization of the theorem of Pythagoras to what today is called the law of cosines and, as the last proposition, the squaring of a rectangle. There are reasons to assume a Pythagorean origin for the main part of Book II, but some historians have different opinions.

Theorem II.14 solves an important problem: Every rectilinear figure can be squared. As usual in mathematics, a problem is solved only to beget another one. The next most prominent figure is the circle. How to square it? Proclus observes in his comment on Prop. I.45, which is the last step before II.14:

Equal circles are those the diameters of which are equal, or the radii of which are equal.

In section C of Book III Euclid presents the prototype of a mathematical theory. He has a clear sense of its architecture. Let us recapitulate the main steps: III.20 is the preparing lemma with specific information about the size of angles.III.21 is the main theorem about an invariant.III.23-29 are technical expansions of the main theorem.III.31/32 deal with two remaining extreme cases.III.33/34 modifies the extreme case 32 to get a tool for applications.

We will use the standard term “regular polygon” (or n-gon) for what Euclid calls in particular cases an “equilateral and equiangular polygon.” Convexity is always tacitly assumed. Book IV follows a tight plan and has none of the subdivisions of some other books. We repeat what has been said in the section about the contents of the Elements. Four problems are treated systematically: (i) to inscribe a rectilinear figure in or(ii) to circumscribe it about a given circle;(ii) to inscribe a circle in or(iv) circumscribe it about a given rectilinear figure.

Our historical reconstructions about the pentagon maybe hypothetical. Nevertheless, we can use them as an example for some remarks on rigor in mathematics. What is meant by saying that an argument is rigorous and not just intuitively right?

In Prop. IV. 16 Euclid constructs a regular 15-gon by superimposing an equilateral triangle on a regular pentagon (Fig. 13.1).

One of the main discoveries of Pythagoras was the relation between musical harmonics and the ratios of segments on a monochord, the simplest one being 2 : 1 for the octave. Much has been written about this. We will complement it by a short look at architecture and the arts.

The basic theorem of Book VI looks innocent enough, but it is the foundation of Euclid’s similarity geometry.

One of the most essential features of a modern mathematical theory is its generality. Mathematicians try to present their results as generally as possible. Uniting seemingly disparate phenomena in a new theory earns great praise from fellow researchers. In this section we will study, in a few cases, the movement from a particular to a more general statement.

In Prometheus Bound, Aeschylus, the first of the three great classical writers, tells us how Prometheus taught humans not just the use of fire but also so many other worthwhile things. In particular, Prometheus relates, “And numbers, too, the chiefest of sciences, I invented for them” (lines 459/460). These lines, written about 465 b.c.e., bear witness to the high esteem the early Greeks had for arithmetic.

Aside from Euclid, there are two mathematicians from antiquity whose books about arithmetic have survived.

There is no break in the contents between Books VIII and IX. Book IX just continues to treat the problems from the end of Book VIII. For that reason we will show the internal subdivisions of the two books in one picture.

The Greek word theorema is related to theater and means “something seen,” or, in mathematics, an insight, understanding, or knowledge. A mathematical theorem should answer a question or solve a problem that posed itself during the discussion of a mathematical subject. Very often, a theorem, once proved, is used in various ways to develop a mathematical theory further. In this process it changes its character. The original questions recede, and the theorem becomes a basic tool for the practitioners.

The first two propositions of Book IX have already been dealt with in the section on similar plane numbers. Props. IX. 3-6 are concerned with cube numbers, for instance Prop. IX. 3. x is a cube ⇒ x2 is a cube. Prop. IX. 6. x2 is a cube ⇒ x is a cube.

When mathematicians praise the aesthetic qualities of their subject, they usually meet raised eyebrows from sceptical nonbelievers. What will be said here will probably not convince outsiders, but it may clarify the situation somewhat for ourselves. Using examples as well as comments by some prominent authors, we will try to be more specific than just stating an emotional opinion. Proclus will again be our main witness from antiquity. He discusses the applications of mathematics and then goes on to describe its superior beauty and its value for the study of philosophy. In part he elaborates a short passage from Aristotle, who says what beauty is: “The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree” (Aristotle, Metaphysics 1078 a34-b2). (Here, as in the quotation from Proclus below and in many other places, the translator has used the modern “symmetry” for the Greek “symmetria “ which, at least in a mathematical context and by its literal translation, should mean “of common measure.” In most cases the literal translation makes much more sense than the modern word “symmetry,” which has taken on quite a different meaning outside of mathematics.)

Book X comprises fully one quarter of the Elements. Not only is it the most voluminous book, but also the most difficult to read. Setting aside the bulk of the material of Book X, we concentrate our attention on the introductory part of the book, which is of general interest. Some recent studies have done much to elucidate the tedious and long-winded considerations of Book X. The interested reader should look up the papers by Taisbak [1982], Knorr [1985], Fowler [1992], as well as the respective chapters in Mueller [1981] and van der Waerden [1954]. The original motivations for the classification of incommensurable lines apparently can be found in the material presented in Book XIII. During the discussion of the relevant propositions of Book XIII we will return to the problems of Book X.

Aristotle, near the beginning of his Metaphysics, a most prominent place, quotes incommensurability as the prototype of a scientific discovery. After a short general description of philosophy he writes: [The first philosophers] were pursuing science in order to know, and not for any utilitarian end…Evidently then we do not seek it [philosophy] for the sake of any other advantage; but as the man is free, we say, who exists for himself and not for another, so we pursue this as the only free science, for it alone exists for itself [982b 20-27]…All the sciences, indeed, are more necessary than this [philosophy], but none is better. Yet the acquisition of it must in a sense end in something which is the opposite of our original inquiries. For all men begin, as we said, by wondering that the matter is so (as in the case of … the solstices or the incommensurability of the diagonal, for it seems wonderful to all men who have not yet perceived the explanation that there is a thing which cannot be measured even by the smallest unit). But we must end in the contrary and, according to the proverb, the better state, as is the case in these instances when men learn the cause; for there is nothing which would surprise a geometer so much as if the diagonal turned out to be commensurable. [Metaphysics 983a 10-21]

The definitions at the beginning of Book XI are intended to serve for all of the three Books XI-XIII. Accordingly, we find three groups of definitions: Defs. 1-8 determine angles between planes and similar objects. Next it is fixed what is to be understood by similar solid angles (9-11). Pyramids and prisms (12,13) are needed in Book XII, which culminates in the study of cylinders, cones, and spheres.

We have seen how Euclid formulates a definition of the sphere that goes back to the actual process of making one. Plato, in this case more advanced than Euclid, seems to waver between the artisan’s and the mathematician’s definition when he adds the condition of equidistance from a center. The example shows how a particular mathematical concept changes from a practical and realistic origin toward a more abstract notion that fits better into a developed theory. Usually in the Elements we find this mathematically advanced kind of definition. Many of them are explicit descriptions of geometrical objects like “equilateral triangle” or “gnomon,” which simply determine certain notions for later use. However, not everything can be defined. Every mathematical theory starts with some undefined concepts. Today these may be in principle no more than “set” and “function” or one of these two, but one cannot start from nothing. In the Elements a typical example of this sort is “to measure.” We are never told what “measuring” is, in spite of its fundamental importance in Book V for magnitudes and in Book VII for numbers. Somehow one knows what is meant, and it works. We have strong connotations of an everyday procedure that help us to make sense of the concept.

As a preparation for his main theorem on circles Euclid needs some information on polygons.

Using finite means for mastering the infinite is a hallmark of mathematics. Here again Euclid did ground-breaking work. We will list a series of his examples, all of which were discussed in greater detail at their respective places.

Parts A and B are mainly preparatory for part C, but some of the theorems in part B are of independent interest. We will discuss the latter ones in this section. The history of the regular polyhedra from Hippasus to finite simple groups will be the subject of the next section, “Symmetry through the ages.”

From the time of the Greeks, people have been fascinated by the regular polyhedra. They provide us with one of the first complete mathematical theories: a general definition together with a complete classification of all the objects satisfying the definition. In this section the most important steps are presented in the development of a subject that goes back to the very beginnings of mathematics and is still alive today. Hippasus provided the first significant example, and Theaetetus created the mathematical theory. Pacioli revived the subject after it had lain dormant for about a thousand years. Felix Klein replaced the polyhedra by their symmetry groups and opened vast new areas of research. One path leads into function theory and algebraic geometry, while the other starts with the group of rotations of the dodecahedron and goes on to simple groups.

Heiberg, the modern editor of the Greek original of the Elements and the most eminent Euclid scholar of his time, thought in 1904 that it was nearly impossible to reconstruct earlier versions of Greek mathematics from the Elements (Heiberg [1904], p. 4). Mathematical and stylistic analyses over the last 100 years have disproved Heiberg’s claim and revealed a great deal about the prehistory of this monumental compendium of Greek mathematics. Our present, fairly complete, picture of the different contributions of pre-Euclidean mathematicians to the Elements is to a large extent due to the detailed studies begun by Becker in the 1930s, and continued by Neuenschwander on the geometrical books and by Mueller on the Elements as a whole. Heath and the other translators of the Elements provide valuable commentaries, but they are primarily concerned with individual definitions, theorems, and proofs. The global picture emerges from the investigation of the mathematical architecture of the Elements in combination with the study of other ancient sources.