enlarged but little by the followers of Euclid at Alexandria. In plane geometry, the subject of isoperimeters was further developed, and solid geometry received practically its present form. But above all, the important work of this period was the development of the theory of the conic sections and of the geometry of measurement. The first of these has never been unified with elementary geometry; the second has, although in varying degree in the different countries and institutions.

THE ORIENTALS

Little was added to geometry by the Hindus. Just how much was original with them it is hard to say. We find Brahmagupta1 (b. 598 A.D.) using Heron's formula for finding the area. of a triangle. He was also familiar with the Pythagorean proposition (Euclid I, 47). One feature of his work was his distinguishing between approximate answers and exact answers. He called the first gross answers. Brahmagupta thus takes = 3, thereby giving the gross value of the circumference of a circle, and in a problem following he takes π = V 10, giving what he calls an

exact value for circumference and area."

The method of proof used by the Hindus was probably characterized by its brevity. In fact, Aryabhatta (b. 476 A.D.) would only state the theorem, add the figure, and then write "behold!"'3 We know that Aryabhatta, among other Hindu writers, wrote on mathematics in verse. Whether geometry was taught by means of rhymes, we do not know, but the study of geometry would have offered splendid opportunity for such a practice wherever learning by heart was encouraged.

They

The Arabs contributed even less than the Hindus. were influenced on the one side by the Hindus, on the other by

1 Colebrook, trans. of Algebra with Arithmetic and Mensuration, from the Sanscrit of Brahmagupta and Bhascara, pp. 295-318. Also see Cantor, I, pp. 605-614.

The area of a triangle whose sides are 13, 14, 15 is found by getting the half sum of 13 and 15 and multiplying by of 14 or 7. In the case of the isosceles triangle whose sides are 10, 13, 13, Brahmagupta multiplies 13 by of 10. The exact values are also given by using Heron's formula. 3 Fink, A Brief History of Mathematics, trans. by Beman and Smith, p. 215. See Rodet, Leçons calcul d'Aryabhata

3

the classical learning of the Greeks. The Arabs have already been mentioned' as the preservers of the mathematical learning of Alexandria. We recall that among other works Euclid's "Elements" was translated into the Arabic at Bagdad and was introduced into Europe by way of Spain.

THE ROMANS

The Roman mind was concerned with the practical. The youth was trained in oratory that he might make use of it for practical ends. So it was with mathematics, the end was practical. The geometry of the Romans was associated with surveying and the engineering of warfare. It is known that Julius Cæsar caused a survey to be made of the Roman Empire. For our own system of land surveying we are indebted to the Romans. According to Cantor,3 the temple-fields of the Etruscans were truly orientated. How this was done is not known, but the Romans later knew how to lay out meridians. Thus Vitruvius, an architect of the time of Augustus (cir. 15 B.C.), and Hyginus, a surveyor of the time of Trajan (cir. 100 A.D.), knew two methods of doing this. On account of these practical interests of the Romans, their mathematical writings were chiefly on mensuration and surveying. The writings of some of them are collected in a work known as the Codex Arcerianus, the contributions being fragments of the works of Frontinus, Hyginus, Balbus, Nipsus, Epaphroditus, and Vitruvius Rufus, all of whom lived during the first two centuries of the Christian era.5

1 See above, p. 32.

2 Cantor, Die römischen Agrimensoren und ihre Stellung in der Geschichte der Feldmesskunst, p. 75.

3 Ibid., pp. 65-66.

* One of these methods was as follows: Let AC be a stake set upright in the ground. At a certain time in the forenoon the shadow will be represented by a line such as BC. With C as a center and BC as a radius draw a circle. Let CD be the position of the shadow in the afternoon when its extremity just touches the circumference. BC = CD. Join B, D and draw the perpendicular bisector of BD. This is the required meridian line.

Cantor I, p. 513ff; Günther, Geschichte des mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525, p. 115. Hereafter referred to as Günther. The "Codex" was discovered in 980 by Gerbert, who became Pope Silvester II. See Gow, p. 206.

The impulse to develop this practical geometry came certainly from the nature of their own needs, but the Romans were undoubtedly influenced by the practical geometers of the Alexandrian school. The work of Heron, who developed surveying at Alexandria, was known to them. Archimedes (b. 287 B.C.), we recall, passed the greater part of his life at Syracuse in Sicily. We know that his writings were made generally known to Latin Europe from the Arab translations carried into Spain, but the Roman architect Vitruvius Pollio knew of his work, and it is safe to suppose that use was made of it.

One might expect that Euclid would have found its way into Italy by way of Sicily. The practical geometry of Alexandria did, why not the theoretical? Because the Romans were not interested in that side. But we have evidence that knowledge of the subject-matter and method of Euclid was not unknown to the Romans during the first century of the Christian era. The following passage from Quintilian3 will show the truth of this: "Order, in the first place, is necessary in geometry; and is it not also necessary in eloquence? Geometry proves what follows from what precedes, what is unknown from what is known; and do we not draw similar conclusions in speaking? Does not the well known mode of deduction from a number of proposed questions consist almost wholly in syllogisms? Accordingly you may find more persons to say that geometry is allied to logic, than that it is allied to rhetoric. . . Besides of all proofs, the strongest are what are called geometrical demonstrations; and what does oratory make its object more indisputable than proof? Geometry, often, moreover, by demonstration, proves what is apparently true to be false. . . Who would not believe the asserter of the following proposition: 'Of whatever places the boundary lines measure the same length, of those places the areas also, which are contained by those lines, must necessarily be equal?' But this proposition is fallacious; for it would make a vast difference what figure the boundary lines may form; and historians, who have thought that the dimensions of islands are

1 Cantor, I, p. 515.

? Vitruvius relates how Archimedes came to consider the laws of floating bodies. Vitruvius, Architecture, ed. Newton, IX, 3.

Institutes of Oratory, Book I; ch. X, 37-41, trans. Watson.

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

prung

with

anity

Already before the light of Greek learning he been extinguished at Alexandria a new sort of educatio up there. The doctrines of Christianity came Greek thought, and Christian leaders saw that 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.

'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.