sufficiently indicated by the space traversed in sailing round them have been justly censured by geometricians. For the nearer to perfection any figure is, the greater is its capacity; and if the boundary line, accordingly shall form a circle, which of all plane figures is the most perfect, it will embrace a larger area than if it shall form a square of equal circumference. Squares, again, contain more than triangles of equal circuit, and triangles themselves contain more when their sides are equal than when they are unequal. .". The method of geometry was, for Quintilian, of practical value in making more perfect the art of oratory.

We thus see that the subject-matter of geometry under the Romans was of a practical nature. The logic of geometry was of value inasmuch as it was an aid in oratory. Only in one way has the work of the Romans influenced the later teaching of geometry. The later Italian practical geometries under the influence of both the practical work of the Romans and that of Archimedes and Heron, kept alive the interest in applied geometry. As a rule, owing to the standard set by Euclid, it was not combined with the theoretical.

CHAPTER IV

THE TEACHING OF GEOMETRY FROM THE RISE OF THE

CHRISTIAN SCHOOLS TO THE YEAR 1525

THE CHRISTIAN SCHOOLS OF THE MIDDLE AGES

Already before the light of Greek learning had been extinguished at Alexandria a new sort of education had sprung The doctrines of Christianity came in conflict with Greek thought, and Christian leaders saw that if Christianity was to attain any success in this competition, its teachers must be trained in the Greek learning.1 So there arose at Alexandria the catechetical schools, and out of these grew the episcopal schools of later times. All these schools had as their chief function the training of their members for the priesthood. So it is not surprising that the study of mathematics was neglected.

Although Christian education began at Alexandria, the little geometry taught in the schools was derived, not from the Alexandrians, but from the later Roman writers, Martianus Capella (cir. 420 A.D.), Boethius (cir. 480-524), and Isidore of Seville (cir. 570-636). The books written by these men were the great text-books of the Middle Ages up to the thirteenth century.' They contained, however, but little geometry. It will be shown

1 Monroe, Text-book in the History of Education, p. 233.

2 Laurie, The Rise and Early Constitution of Universities, pp. 37-38: Günther, pp. 1-2.

About the beginning of the first century B.C., Varro, a Roman, wrote on grammar, rhetoric, dialectic, arithmetic, geometry, astronomy, music, philosophy, and other branches. See Davidson, The Seven Liberal Arts, in

the Educational Review, 2, p. 469. This also appears in his Aristotle and Ancient Educational Ideals, appendix. See also Parker, The Seven Liberal Arts, in The English Historical Review, vol. V, p. 431.

43

later that the geometry of Boethius, as a text, was lost to Europe until a copy was found by the great Gerbert (cir. 980). The work of Capella1 is embodied in an encyclopædia of nine books, the sixth book treating of geometry, which is more properly geography. This part of the work was followed by definitions of lines, figures, and solids, then the most necessary "demands," all according to Euclid and using the Greek terminology. The geometry of Boethius2 (cir. 480-524) in like manner is part of a larger work. It begins with definitions, postulates, and axioms. After these follow Book I of Euclid and selected propositions from Books II, III, and IV, but the propositions are merely stated. It is only at the end that any proofs are given, and then only for the first three propositions of Book I. After further definitions and a discussion on the operations with numbers, there is inserted the much discussed chapter on the Gobar numerals.3 The rest of the geometry is concerned with the mensuration of various geometric figures, most emphasis being laid on the simplest plane figures.1 As in the geometry of Capella there is no surveying. The geometry of Boethius shows on the whole a tendency to follow the practical geometry of the Romans, but the attention given to Euclid marks it as different from any European geometry up to the time of the introduction of Euclid into Latin Europe by way of Spain.

Two other names are associated with the early Christian education of the Middle Ages, those of Cassiodorus (cir. 480-575) and Isidore of Seville (cir. 570-636). Both of these recognized the order of the trivium and quadrivium in the arrangement of the subject-matter in their writings.5 Under the trivium were grammar, rhetoric, and dialectics; under the quadrivium, arithmetic, music, geometry, and astronomy. These constituted the seven liberal arts of the Middle Ages. But the geometry 1 De nuptiis philologia et septem artibus liberalibus. Libri novem. Edition 1539. See Cantor, I, pp. 527-528; Kästner, I, p. 251. 'De Institutione arithmetica, musica, geometria, ed. Friedlein, p. 373ff. Some claim that this treatment of the Gobar (dust) numerals is not authentic. See Weissenborn, Die Boetius Frage.

* Some of the examples are not worked correctly. For example, Boethius finds the area of an equilateral triangle whose side is 30 to be 390, while for one whose side is only 28 the area is 406.

5 Cantor, I, pp. 529-532; 772-775.

in these writings was, like that of Capella, closely related to geography.

2

Up to the time of Gerbert (d. 1003), we do not find that instruction in geometry went beyond the learning of definitions and rules, and the making of a few simple constructions. The subject was generally taught in the church schools, but it was given stepmotherly care because geometry was of no practical significance for those studying for the clergy. There seems to have been no use of the Roman practical geometry of the time of Vitruvius. The study of geometry received no greater attention in the Palace School of Charles the Great, where, under the direction of Alcuin (735-804), the subject was taught as one of the seven liberal arts.1

After Gerbert, the teaching in the church schools was still based on the medieval writers already mentioned, but there was added a practical phase which carries us back to the geometry of Heron and that of the Roman surveyors (gromatici). Gerbert, who was the head of the cathedral school at Rheims and later became Pope Silvester II, made two discoveries of mathematical treatises that were of importance. The first was the finding of the Codex Arcerianus, which contained the works of the Roman surveyors. The second was the finding of a copy of the geometry of Boethius at Mantua. These discoveries have a double significance for us. The fact that Gerbert discovered them means that those works as such were unknown in the medieval schools up to that time, and the nature of the subject-matter of the geometry then taught bears out this fact. We recall here that the Codex treated geometry with reference to its application in surveying and the measuring of heights and distances, and the geometry of Boethius the mensuration of ordinary geometric figures only, this being prefaced by some propositions from Euclid. Since the discoveries of Gerbert are considered genuine, one must conclude that the church

1 Cantor I, p. 113.

2 Specht, Geschichte des Unterrichtswesens in Deutschland von den altesten Zeiten bis zur Mitte des dreizehnten Jahrhunderts, p. 144.

3 Ibid., p. 143.

'Günther, pp. 25-26.

5 Cantor, I, pp. 797-824

schools were not in possession of the geometry of Boethius, although they were acquainted with his other writings.1 2

5

1

2

1

The geometry of Gerbert (cir. 980) shows that he was influenced by the practical work of the Codex, but not by the Euclidean feature of the geometry of Boethius. The book begins with definitions, then comes a discussion of units of measure, after which follows a great deal of work on mensuration, more than is given by Boethius. Gerbert also gives problems on finding heights and distances, using the astrolabe and the mirror." Gerbert used some inexact methods in finding areas. Thus in an equilateral triangle whose side is a he finds the area by the formula, Area —a (a ——a). He takes a = 7 and thus finds the area to equal 21. In another place the area of an isosceles trapezoid is found by multiplying one-half the sum of the parallel bases by one of the equal legs. Brahmagupta, we recall, was more particular, for he gave both "gross and "exact" answers. If we are to accept the opinion of Gow,' this tendency to inaccuracy can be traced back to Heron of Alexandria. The particular formula used by Gerbert for the area of an isosceles trapezoid can be traced back even to the ancient Egyptians, for we remember that Ahmes employed this incorrect rule. Although Gerbert's book was decidedly practical, we catch a glimpse of a logical proof where he shows that the angle-sum of a triangle equals two right angles, using the Euclidean method of proof by drawing a line through a vertex parallel to the opposite

1 Günther in a recent article says that the teaching of geometry in the early Middle Ages was based on the geometry incorporated in the treatises of Capella and Isidore of Seville. No reference is made to the geometry of Boethius. See Günther, Le développement historique de l'enseignement mathématique en Allemagne, trans. by Bernoud in Enseignement Mathématique, 1900, pp. 237-264. Hereafter referred to as Günther, Ens. Math.

2 Weissenborn does not credit Boethius with having written the geometry ascribed to him. See his Boetius Frage

3 For an edition of Gerbert's geometry, see Opera mathematica, edited by Bubnov. Also Migne, Patrologic cursus completus, v. 139, pp. 84-134.