SOURCE: "The History of Geometry to the Time of Euclid: Commentary on Euclid's 'Elements I,'" in A Source Book In Greek Science, edited by Morris R. Cohen and I. E. Drabkin, translated by I. E. Drabkin, McGraw-Hill Book Company, Inc., 1948, pp. 33-85.

[In the following excerpt—with translation by Drabkin and notes by Cohen and Drabkin, Proclus offers a brief overview of geometry, from that of the ancient Egyptians up to that of Euclid's Elements.]

We must next speak of the origin of geometry in the present world cycle. For, as the remarkable Aristotle tells us, the same ideas have repeatedly come to men at various periods of the universe. It is not, he goes on to say, in our time or in the time of those known to us that the sciences have first arisen, but they have appeared and again disappeared, and will continue to appear and to disappear, in various cycles, of which the number both past and future is countless. But since we must speak of the origin of the arts and sciences with reference to the present world cycle, it was, we say, among the Egyptians that geometry is generally held to have been discovered. It owed its discovery to the practice of land measurement. For the Egyptians had to perform such measurements because the overflow of the Nile would cause the boundary of each person's land to disappear.¹ Furthermore, it should occasion no surprise that the discovery both of this science and of the other sciences proceeded from utility, since everything that is in the process of becoming advances from the imperfect to the perfect. The progress, then, from sense perception to reason and from reason to understanding is a natural one. And so, just as the accurate knowledge of numbers originated with the Phoenicians through their commerce and their business transactions, so geometry was discovered by the Egyptians for the reason we have indicated.

It was Thales,² who, after a visit to Egypt, first brought this study to Greece. Not only did he make numerous discoveries himself, but laid the foundations for many other discoveries on the part of his successors, attacking some problems with greater generality and others more empirically. After him Mamercus,³ the brother of the poet Stesichorus, is said to have embraced the study of geometry, and in fact Hippias of Elis⁴ writes that he achieved fame in that study.

After these Pythagoras⁵ changed the study of geometry, giving it the form of a liberal discipline, seeking its first principles in ultimate ideas, and investigating its theorems abstractly and in a purely intellectual way. It was he who discovered the subject of proportions⁶ and the construction of the cosmic figures.⁷ After him Anaxagoras⁸ of Clazomenae devoted himself to many of the problems of geometry, as did Oenopides⁹ of Chios, who was a little younger than Anaxagoras. Plato in The Rivals'¹⁰ mentions them both as having achieved repute in mathematics. After them Hippocrates of Chios¹¹ the discoverer of the quadrature of the lune, and Theodorus of Cyrene¹² gained fame in geometry. For Hippocrates is the first man on record who also composed Elements.

Plato, who lived after Hippocrates and Theodorus, stimulated to a very high degree the study of mathematics and of geometry in particular because of his zealous interest in these subjects. For he filled his works with mathematical discussions, as is well known, and everywhere sought to awaken admiration for mathematics in students of philosophy.

At this time lived also Leodamas of Thasos, Archytas of Tarentum,¹³ and Theaetetus of Athens,¹⁴ all of whom increased the number of theorems and made progress towards a more scientific ordering thereof. Neoclides and his pupil, Leon, who were younger than Leodamas, made many additions to the work of their predecessors. As a result the Elements composed by Leon are more carefully worked out both in the number and in the usefulness of the propositions proved. Leon also first discussed the diorismi (distinctions), that is the determination of the conditions under which the problem posed is capable of solution, and the conditions under which it is not.¹⁵

A little younger than Leon was Eudoxus of Cnidus,¹⁶ an associate of Plato's school, who first increased the number of the general theorems, as they are called, and added three new proportions to the three then known. Using the method of analysis he greatly extended the theory of the section, a subject which had originated with Plato. Amyclas of Heraclea, a friend of Plato, Menaechmus,¹⁷ a pupil of Eudoxus who had also been associated with Plato, and Dinostratus,¹⁸ a brother of Menaechmus, brought the whole of geometry to an even higher degree of perfection. Theudius of Magnesia achieved a reputation for excellence both in mathematics and in the other parts of philosophy, for his Elements were excellently arranged and many of the (theretofore) limited¹⁹ propositions were put in more general form. Furthermore, Athenaeus of Cyzicus, a contemporary, distinguished himself in mathematics generally and in geometry in particular. These men spent their time together in the Academy and collaborated in their investigations. Hermotimus of Colophon extended the work that had been done by Eudoxus and Theaetetus, made many discoveries in the Elements, and put together some material on the subject of Loci. Philippus of Mende²⁰ who, as a pupil of Plato, was inspired by him to the study of mathematics, conducted investigations along lines suggested by Plato, and in particular set before himself researches which he thought would contribute to the philosophy of Plato. Now those who have written histories trace the devel opment of the science of geometry down to Philippus.²¹

Euclid,²² who was not much younger than Hermotimus and Philippus, composed Elements, putting in order many of the theorems of Eudoxus, perfecting many that had been worked on by Theaetetus, and furnishing with rigorous proofs propositions that had been demonstrated less rigorously by his predecessors. Euclid lived in the time of the first Ptolemy, for Archimedes, whose life overlapped the reign of this Ptolemy too,²³ mentions Euclid. Furthermore, there is a story that Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by the study of the Elements. Whereupon Euclid answered that there was no royal road to geometry. He is, then, younger than Plato's pupils and older than Eratosthenes and Archimedes, who, as Eratosthenes somewhere remarks, were contemporaries.

By choice Euclid was a follower of Plato and connected with this school of philosophy. In fact he set up as the goal of the Elements as a whole the construction of the so-called Platonic figures.²⁴

There are, in addition, many other mathematical works by Euclid, written with remarkable accuracy and scientific insight, such as the Optics, the Catoptrics, works on the Elements of Music, and the book On Divisions.²⁵

But he is most to be admired for his Elements of Geometry because of the choice and arrangement of the theorems and problems made with regard to the elements. For he did not include all that he might have included, but only those theorems and problems which could fulfill the functions of elements…. If you seek to add or subtract anything, are you not unwittingly cast adrift from science and carried away toward falsehood and ignorance?

Now there are many things which seem to be grounded in truth and to follow from scientific principles but actually are at variance with these principles and deceive the more superficial. It was for this reason that Euclid set forth methods for intelligent discrimination in such matters, too. With these methods not only shall we be able to train beginners in this study to detect fallacies, but we shall be able to escape deception ourselves. The work in which he gives us this preparation he entitled Pseudaria.²⁶ In it he recounts in order various types of fallacies, training us to understand each type by all sorts of theorems, setting the true alongside the false, and linking the refutation of the fallacy with empirical examples. This book, therefore, is for the purification and training of the understanding, while the Elements contain the complete and irrefutable guide to the scientific study of the subject of geometry.

Notes

¹ Cf. Herodotus II. 109: "This king [Sesostris, ca. 1300 B.C.] moreover (so they said) divided the country among all the Egyptians by giving each an equal square parcel of land, and made this his source of revenue, appointing the payment of a yearly tax. And any man who was robbed by the river of a part of his land would come to Sesostris and declare what had befallen him; then the king would send men to look into it and measure the space by which the land was diminished, so that thereafter it should pay the appointed tax in proportion to the loss. From this, to my thinking, the Greeks learnt the art of measuring land; the sunclock and the sundial, and the twelve divisions of the day, came to Hellas not from Egypt but from Babylonia." (Translation of A. D. Godley, London, 1921.)

Note the different emphasis in Aristotle, Metaphysics 981b20-25 : "Hence when all such inventions were already established, the sciences which do not aim at giving pleasure or at the necessities of life were discovered, and first in the places where men first began to have leisure. That is why the mathematical arts were founded in Egypt; for there the priestly caste was allowed to be at leisure." (Translation of W. D. Ross, Oxford, 1928.)

The debt of Greek geometry to Egypt is implied also in passages of Plato, Hero of Alexandria, Diodorus Siculus, Strabo, and others.

² See p. 92 [A Source Book in Greek Science, 1948. All following page references refer to this text]. Several geometric propositions are associated with the name of Thales in Proclus's commentary, e.g., that a diameter bisects a circle, that the base angles of an isosceles triangle are equal, that when two lines intersect the vertical angles are equal, that two triangles are congruent when two angles and a side of one are equal, respectively, to two angles and the side of the other. Proclus tells us, on the authority of Eudemus, that Thales found the distance of ships at sea by using this last theorem. Diogenes Laertius preserves a statement of Pamphila that Thales first inscribed a right triangle in a circle, though Diogenes notes another tradition, that the discoverer was Pythagoras. How much Thales had to do with these propositions is the subject of controversy.

In this connection it must be remembered that what constitutes an acceptable demonstration is not the same in every period. Just as early attempts at demonstrations must have differed considerably from the later canonical proofs, so the modern mathematician cannot in every case be satisfied with Euclid's proofs.

³ Other readings are Mamertius and Ameristus.

⁴ Hippias flourished in the latter half of the fifth cen tury B.C. He may have discovered the quadratrix, a curve that may be used for squaring the circle or trisecting an angle (see p. 57).

⁵ See the special note on Pythagorean geometry, p. 41.

⁶ Following a conjecture of Diels. The reading of the manuscripts, referring to the subject of irrationals, can hardly stand.

⁷ See p. 43.

⁸ See p. 93. Plutarch says (De Exilio 17.607 E) that Anaxagoras while in prison wrote on the squaring of the circle.

⁹ There is a tradition that Oenopides discovered the obliquity of the ecliptic and calculated a Great Year of 59 years. (According to Heath, History of Greek Mathematics I. 175, Oenopides was concerned with the problem of finding the least integral number of complete years that would contain an exact number of lunar months.) The problems of drawing a perpendicular to a line from a point outside the line and of constructing at a given point in a given line an angle equal to a given angle are attributed to Oenopides by Proclus (Commentary on Euclid's Elements I, pp. 283.7; 333.5).

¹⁰Erastae 132 A-B.

¹¹ See p. 54.

¹² See p. 14.

¹³ The work of Archytas, a contemporary of Plato, is known only through fragments and references in the works of others. He found a remarkable solution of the problem of doubling the cube (p. 63), wrote on arithmetic and musical theory (p. 286), and seems to have invented mechanical contrivances (p. 335). His proof that between two numbers n and n + 1 no rational geometric mean may be inserted is repeated by Euclid, Sectio Canonis 3.

¹⁴ Theaetetus, another contemporary of Plato, made important investigations in connection with irrationals and the geometry of the five regular solids (see also pp. 14-15, 43).

¹⁵ There is a passage in Plato's Meno (86 E-87 B) which indicates that Leon was not the discoverer of diorismi.

¹⁶ The importance of Eudoxus for the history of mathematics can hardly be exaggerated. In perfecting a theory of proportion that included incommensurable as well as commensurable magnitudes he laid the foundation for the method of exhaustion. This enabled the Greek mathematicians to solve problems that are es sentially problems in infinitesimal analysis. Archimedes, whose work represents the highest development in this field among the Greeks, refers to Eudoxus' discoveries (see p. 70).

The precise meaning of Proclus's statements is obscure. Is the "section" that of solids by planes, or the so-called "golden section" (p. 50)? On these and the other questions see T. L. Heath, History of Greek Mathematics I. 323-326. On Eudoxus' theory of concentric spheres see p. 101, below.

¹⁷ Menaechmus used the parabola and the rectangular hyperbola in his solutions of the Delian problem. The tradition that he was the discoverer of the conic sections arose from this fact and the reference of Eratosthenes (?) to the "triads" of Menaechmus (see p. 66).

¹⁸ According to Pappus, Dinostratus made use of the quadratrix in connection with the problem of squaring the circle.

¹⁹ Reading [merikōn].

²⁰ Medma, according to another reading. He is often identified with Philip of Opus, the astronomer and mathematician, whose name is sometimes associated with the publication of Plato's Laws and the authorship of the Epinomis.

²¹ The discussion thus far is probably based, directly or indirectly, on the History of Geometry by Eudemus, a pupil of Aristotle.

²² Euclid, of whose life very little is known other than what is contained in the account given by Proclus, flourished about 300 B.C. and is one of the outstanding figures not only of the golden age of Greek mathematics but of all mathematical history. Though his Elements represent the consummation of three hundred years of Greek mathematics, Euclid's original contribution was by no means insignificant. This work, which in one form or another has served as a textbook for twenty-two hundred years and has passed through countless editions, is in fifteen books, of which all but the last two are probably by Euclid. Besides the Elements Euclid wrote various works in pure and applied mathematics of which the Data, Phaenomena, and Optics are extant in Greek; the book on Divisions of Figures is extant in Arabic and Latin translations. The Porisms, Conics, Pseudaria, and Surface Loci have been lost. An extant Catoptrics, attributed to Euclid, is almost certainly not his. Of the two extant theoretical books on music attributed to Euclid, the Sectio Canonis may be his, but the Introductio Harmonica is almost certainly not. There is, however, no reason to doubt Proclus's statement that Euclid wrote on music. The correctness of the attribution to Euclid of various other works extant in Arabic translations from the Greek is very doubtful.

²³ The reference is to the fact that most of Archimedes' life fell in the succeeding reigns. Ptolemy I ruled from 306 to 283 B.C., Archimedes lived from 287 to 212, and Euclid fluorished about 300.

²⁴ I.e., the five regular polyhedra, the subject of the last book (XIII) of Euclid's Elements (see p. 43 below). One is hardly justified in speaking of this as the goal of the whole work.

²⁵ See pp. 257, 261. For the work On Divisions see T. L. Heath, History of Greek Mathematics I. 425-430.

²⁶ The Pseudaria is not extant.