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Raschid (786-809), and completely so under Al Mamun.1 The Arabs are to be considered the preservers of Euclid. They contributed nothing to the science of geometry, their achievements being in arithmetic, trigonometry, and algebra. When they conquered Spain (747), they brought their Euclid with them, but it was nearly 400 years later that this learning was given to Christian Europe.

The Moors guarded their learning very jealously, but in 1120 an English monk, Adelard of Bath, while studying in Spain, succeeded in getting a copy of the "Elements" and translated it into Latin. Another translation was made by Gherardo of Cremona in about 1186, and in 1260 Johannus Campanus made a copy of Adelard's translation and gave it out as his own. These translations had a stimulating effect on the study of geometry. Leonardo of Pisa, in 1220, wrote the first original work on mathematics in Europe that was based on Euclid, Archimedes, and Ptolemy.2

In the curricula of the universities, which began their existence at this time, Euclid and Aristotle found a place, but it was only in the German institutions that any great attention was given to the study of the "Elements." In Italy especially the study of Euclid was associated with astrology.3 We thus see that up to the time of the invention of printing, the study of Euclid was a rather formal affair. The influence of the "Elements" was not far reaching. But with the invention of printing it became possible for the work to be read by a greater number of people, and its use became more common in the higher institutions of learning.

Some of the important editions through which the "Elements" passed should be mentioned. The first edition to be printed was in Latin from the Adelard-Campanus translation Hankel, pp. 231-234; Sédillot, Matériaux pour servir à l'histoire comparée des sciences mathématiques chez les grecs et les orientaux, Tome I, p. 377. 2 Hankel, pp. 342-348. See below, p. 48

1

3 For a more detailed account of the teaching of Euclid in the early universities, see below, pp. 51-53.

For a fairly complete list, see De Morgan's article, Eucleides, in Smith's Dictionary of Greek and Roman Biography and Mythology. Also see Heiberg, Litterargeschichtliche Studien über Euklid; and Kästner, Geschichte der Mathematik, I, pp. 279-398. The latter work is hereafter referred to as Kästner.

from the Arabic. It comprised the full fifteen books, and was printed by Ernest Ratdolt at Venice in 1482. Other editions followed, until in 1509 appeared the one by Paciuolo, who was the first to print a work on algebra. The fifth edition was printed at Paris in 1516, under the title "Contenta." De Morgan says, "From it we date the time when a list of enunciations merely was universally called the complete work of Euclid." This idea was quite common during the sixteenth century. The above edition contained, besides the enunciations of the theorems, the commentaries of Campanus, Theon, and Hypsicles. The idea was prevalent that Theon had supplied the proofs, which Euclid had failed to insert.1 This is shown by a statement in Xylander's German translation (Basel, 1562), where the reader is warned that the demonstrations were added "nit von jme dem Euclid selbs," but by other learned men, Theon, Hypsicles, Campanus, etc. These five editions appeared within thirtyfour years, which shows a revival of interest in the "Elements." Then began the period of printed translations into Greek. The first was by Simon Grynæus at Basel, in 1533. In 1551, Robert Recorde published his "Pathway of Knowledge," which contained the enunciations of the first four books of Euclid, but not in Euclid's order. The first translation of the complete "Elements" into English was by Henry Billingsley, in 1570. The next important English edition was by Robert Simson in 1756, In 1795 appeared Playfair's Euclid, which was a departure, as solid geometry was added from other sources. But the first radical departure from Euclid that was generally adopted was the text of Legendre, which appeared in 1794.

The extent to which the various countries have adhered to the geometry of Euclid will be considered later.3 This preliminary survey suffices to show us that the influence of the "Elements" has been most enduring since its introduction into the curriculum of the early universities.

1 Gow, p. 200.

2 Heiberg, Litterargeschichtliche Studien über Euklid, p. 175.

3 See especially Chapters V and VI.

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

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."

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.1 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.

2 From this π=3/, approximately.

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 There is some doubt regarding the dates to assign to Heron.

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