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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 (1,11); 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."

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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

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1 Chasles, Aperçu historique, p. 26.

2 Theodosius, Sphærica, ed. Barrow.

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

* La géométrie grecque, p. 166ff 5 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.3

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.

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.

'Gow, p. 285.

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

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enlarged but little by the followers of Euclid at Alexandria. 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 π = 3, thereby giving the gross value of the circumference of a circle, and in a problem following he takes = 10, giving what he calls an exact value for circumference and area.2

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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!" 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'Âryabhata

the classical learning of the Greeks. The Arabs have already been mentioned1 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.2 For our own system of land surveying we are indebted to the Romans. According to Cantor, 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.

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* 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. 5 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.

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

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