CHAPTER III

THE TEACHING OF GEOMETRY FROM EUCLID TO THE RISE

OF THE CHRISTIAN SCHOOLS

AT ALEXANDRIA

The Elements" of Euclid marked the culmination of the development of elementary geometry. From a logical standpoint, the system was now complete, so no new developments could be expected in this direction, but there was a wide field for the applications of geometry. Solid geometry was not yet fully developed, and in the field of deductive mathematics there was yet the development of the geometry of the conic sections. To these other fields the later Alexandrian mathematicians turned their attention. In treating that which is contribution to subjectmatter, we shall discover some facts as to method that have a particular bearing on our subject.

The first geometer after Euclid was Archimedes of Syracuse (b. 287 B.C.). Although he probably did not live at Alexandria, his writings show a thorough acquaintance with the mathematical knowledge of the earlier Alexandrian school. Archimedes was famous for his applications of geometry to science. The nature of his contributions is best seen from the titles of some of his works.1 "Equiponderance of Planes," "The Quadrature of the Parabola," "On the Sphere and the Cylinder," "On The Measurement of the Circle," "On Spirals," "On Conoids and Spheroids," "On Floating Bodies." He also wrote a treatise on the half-regular polyhedra and his addition to the geometry of the three round bodies was considerable. Two of his propositions, at least, are well known. He proved that the area of a spherical surface equals four times the area of a great circle, and secondly, that the volume and surface of a sphere equal two

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1 Archimedes, Opera omnia, ed. Heiberg.

thirds the volume and total surface of the cylinder in which it is inscribed.1 The value of π obtained by Archimedes is so close an approximation that even to-day it is used in ordinary work. He expressed it in the form, "A circle has to the square on its diameter the ratio 11 : 14 very nearly."

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In methods of attack, Archimedes continued the use of analysis and exhaustions as begun at Athens. The process of exhaustion as applied to the quadrature of the parabola deserves special mention, as it laid the foundations of the integral calculus.3 Archimedes frequently gave mechanical proofs for some of his propositions. For example, in effecting the quadrature of the parabola, he gave both a geometric and a mechanical proof.* This is of interest pedagogically, for in recent years we hear of efforts being made to make the teaching of geometry more experimental. In the above case we see some historic basis for modern "laboratory methods" in the teaching of geometry.

In his division of subject-matter, Archimedes drew no hard and fast line between the geometric, the arithmetical, and the mechanical. For example, the above proposition occurs in his "Quadrature of the Parabola," but the physical principles on which it is based are found in his Book I of "Equiponderance of Planes or Centers of Plane Gravities." So we see here a recognition of the unity of the mathematical sciences, a principle that we are too far from fully recognizing to-day.

Concerning the work of Archimedes, then, we can say that he added to the subject-matter of solid geometry, he placed physical 1 According to the wish of Archimedes, a cylinder with its inscribed sphere was engraved on his tomb. Plutarch, Marcellus, 17.

3 For an outline of the plan, see Gow, pp. 226-227. Archimedes' method as explained by him in a letter to Eratosthenes is found in the newly discovered MS. mentioned above. See note, p. 16.

4 In his Quadrature of the Parabola (prop. 6), a proposition is proved mechanically as follows: AT is a lever with B its mid-point. A rightangled triangle BAT is suspended from B T, having B г equal to one-half the length of the lever, the right angle being at B. An area Z is suspended from A, and balances the triangle. It is proved that the area of Z equals one-third the area of the triangle. The center of gravity of the triangle has already been determined in the Equiponderance of Planes (I, 14). Other proportions are proved according to the same methods. Archimedes, Opera omnia, Heiberg ed.

science on a mathematical basis, and in turn made mathematics practical. He employed the geometric methods of attack already existing. His method in the large showed that he recognized the unity of the mathematical subjects.

It has already been stated that the conic sections were excluded by the Greeks from the domain of elementary geometry. Only those constructions were allowed in the "Elements" which could be effected by means of compasses and the straight edge. Menæchmus (b. cir. 375 B.C.) invented the geometry of the conic sections, but it was Apollonius of Perga (b. 260 B.C.) who systematized the work and put it on a scientific basis. While the invention of this new geometry has certainly had an influence on the teaching of elementary geometry, the significant thing for us here is that the geometry as defined by Euclid and that by Apollonius were not shaped into one coherent system. A complete treatise on the conic sections would include the geometry of the straight line and the circle, two special kinds of conics. But this generalizing treatment was not made then, nor since, in the realm of synthetic geometry. It has been only since the invention of analytic geometry by Descartes that this treatment has found recognition.

The geometry developed at Alexandria after Apollonius was confined almost entirely to the practical. Under Eratosthenes (b. 276 B.C.), Hipparchus (b. 180 B.C.), and Claudius Ptolemæus (b. cir. 87 A.D.), geometry found an application in astronomy. Surveying was put on a scientific basis by Heron (b. 125 B.C.),1 and extended later by Sextus Julius Africanus (cir. 200 A.D.). By both of these the measurements of heights and distances were emphasized, as was the case in the Italian practical geometries up to the middle of the seventeenth century. The formula for the area of a triangle in terms of its sides is due to Heron. During this period, trigonometry was developed with respect to its applications by Hipparchus and Ptolemy (Claudius Ptolemæus).

Little was added to the subject-matter of geometry during this later period. With the practical completion of solid geometry by Archimedes, there seemed little else to add. We learn that the subject of isoperimeters was studied by Zenodorus (cir. 150 B.C.), who wrote a treatise on this important branch of geometry. The geometry of the sphere was somewhat further 1 1 There is some doubt regarding the dates to assign to Heron.

extended by Menelaus (cir. 100 A.D.). His "Sphæra," in three books, is a treatise on spherical triangles. His treatment corresponds in a sense to Euclid's treatment of plane triangles. For example, he proves: In every spherical triangle the sum of two sides is greater than the third (1,5); The sum of the three angles is greater than two right angles (I,11); Equal sides subtend equal angles and the greatest side the greatest angle (1, 8, 9); The arcs which bisect the angles meet in a point (III, 9). The importance of one of his theorems (III,1) to modern geometry is pointed out by Chasles. He proved the theorem both for plane and spherical geometry. It is, "If the three sides of a triangle be cut by a straght line, the product of three segments which have no common extremity is equal to the product of the other three." Concerning the importance of this theorem, Chasles writes, "The proposition in plane geometry of which we shall speak below in the article on Ptolemy... has acquired a new and great importance in modern geometry, where the illustrious Carnot has introduced it, making it the base of his theory of transversals." A century earlier Theodosius wrote a complete treatise on the sphere in three books, but he added little to what was already known. One of his theorems, (I, 13) was, "If in a sphere a great circle cut another circle at right angles, it bisects it and passes through its poles." The converse was also proved. Another important theorem was added to solid geometry by Pappus, who lived at the end of the third century. He proved that the volume of a solid of revolution equals the product of the area of the generating plane figure by the circumference of the circle generated by the center of gravity of the figure thus revolved."

That interest in the "Elements" of Euclid itself did not die out at Alexandria is shown by some of the commentaries upon it. According to Tannery,* Heron wrote a commentary. Cantor,5 however, doubts this. Pappus in his voluminous writings discussed some of Euclid's propositions. Theon (cir. 370 A.D.), who wrote an edition of the "Elements," added much to it by

3 Pappus, Collectio, ed. Hultsch, p. 682. This is known as Guldin's rule

• La géométrie grecque, p. 166ff 'Cantor, I, p. 354.

way of commentary. Finally, Proclus (b. 412 A.D.) wrote a commentary on Book I.1 We must not forget the work of Hypsicles (cir. 180 B.C.), who wrote the fourteenth book of Euclid, also Geminus (cir. 70 B.C.), who included much valuable historic material in his "Arrangement of Mathematics," and Damascius of Damascus (cir. 490 A.D.), who is thought to have written the fifteenth book.

Claudius Ptolemæus (cir. 139 A.D.) wrote on pure geometry. Proclus2 (pp. 362-368) has preserved extracts from this work in which it is shown that Ptolemy was not satisfied with Euclid's axiom of parallels and so proposed a proof for the same. This inaugurated the long series of futile attempts to prove this axiom of Euclid."

The treatment of subject-matter by those interested in the practical side of geometry is of significance to us on the method side. It has already been shown that Archimedes tended to unify the various branches of applied mathematics. Heron was the first to carry over geometric symbolism into algebraic operations." "He is the first Greek writer who uses a geometrical nomenclature and symbolism, without the geometrical limitations, for algebraical purposes, who adds lines to areas and multiplies squares by squares and finds numerical roots for quadratic equations."5 The fact that Heron placed the exercises on heights and distances in his Stereometry II is of historic interest. In many of the later Italian practical geometries we find just this arrangement, and to-day in our sequence of mathematical subjects, it is common to place trigonometry (which grew out of such mensuration) after solid geometry in the school curriculum. There are certainly strong reasons for thinking that we have been following the example set us by Heron of Alexandria.

Stated briefly, the subject-matter of elementary geometry was

1 Friedlein edited this in 1873. There is an English translation by Thomas Taylor written in 1792. See also Frankland, The First Book of Euclid's Elements with a Commentary, 1905.

2 Cantor, I, pp. 395-396; Gow, pp. 300-301.

3 For a history of the theory of parallel lines, see Stäckel und Engel,

Die Theorie der Parallellinien von Euclid bis auf Gauss, pp. 31-135.

4 Algebra as a science had not yet been developed.

5 Gow, p. 285.

Cantor, I, p. 363; Gow, p. 281.