CHAPTER V. THE SECOND ALEXANDRIAN SCHOOL.1 30 B.C.-641 A.D. I CONCLUDED the last chapter by stating that the first school of Alexandria may be said to have come to an end at about the same time as the country lost its nominal independence. But, although the schools at Alexandria suffered from the disturbances which affected the whole Roman world in the transition, in fact if not in name, from a republic to an empire, there was no break of continuity; the teaching in the university was never abandoned; and as soon as order was again established, students began once more to flock to Alexandria. This time of confusion was, however, contemporaneous with a change in the prevalent views of philosophy which thenceforward were mostly neo-platonic or neo-pythagorean, and it therefore fitly marks the commencement of a new period. These mystical opinions reacted on the mathematical school, and this may partially account for the paucity of good work. Though Greek influence was still predominant and the Greek language always used, Alexandria now became the intellectual centre for most of the Mediterranean nations which were subject to Rome. It should be added, however, that the direct connection with it of many of the mathematicians 1 For authorities, see footnote above on p. 50. All dates given hereafter are to be taken as anno domini unless the contrary is expressly stated. of this time is at least doubtful, but their knowledge was ultimately obtained from the Alexandrian teachers, and they are usually described as of the second Alexandrian school. Such mathematics as were taught at Rome were derived from Greek sources, and we may therefore conveniently consider their extent in connection with this chapter. The first century after Christ. There is no doubt that throughout the first century after Christ geometry continued to be that subject in science to which most attention was devoted. But by this time it was evident that the geometry of Archimedes and Apollonius was not capable of much further extension; and such geometrical treatises as were produced consisted mostly of commentaries on the writings of the great mathematicians of a preceding age. In this century the only original works of any ability of which we know anything were two by Serenus and one by Menelaus. Serenus. Menelaus. Those by Serenus of Antissa or of Antinoe, circ. 70, are on the plane sections of the cone and cylinder,1 in the course of which he lays down the fundamental proposition of transversals. That by Menelaus of Alexandria, circ. 98, is on spherical trigonometry, investigated in the Euclidean method. The fundamental theorem on which the subject is based is the relation between the six segments of the sides of a spherical triangle, formed by the arc of a great circle which cuts them [book III, prop. 1]. Menelaus also wrote on the calculation of chords, that is, on plane trigonometry; this is lost. Nicomachus. Towards the close of this century, circ. 100, a Jew, Nicomachus, of Gerasa, published an Arithmetic,3 which (or rather the Latin translation of it) remained for a 1 These have been edited by J. L. Heiberg, Leipzig, 1896; and by E. Halley, Oxford, 1710. 2 This was translated by E. Halley, Oxford, 1758. 3 The work has been edited by R. Hoche, Leipzig, 1866. thousand years a standard authority on the subject. Geometrical demonstrations are here abandoned, and the work is a mere classification of the results then known, with numerical illustrations: the evidence for the truth of the propositions enunciated, for I cannot call them proofs, being in general an induction from numerical instances. The object of the book is the study of the properties of numbers, and particularly of their ratios. Nicomachus commences with the usual distinctions between even, odd, prime, and perfect numbers; he next discusses fractions in a somewhat clumsy manner; he then turns to polygonal and to solid numbers; and finally treats of ratio, proportion, and the progressions. Arithmetic of this kind is usually termed Boethian, and the work of Boethius on it was a recognised text-book in the middle ages. The second century after Christ. Theon. Another text-book on arithmetic on much the same lines as that of Nicomachus was produced by Theon of Smyrna, circ. 130. It formed the first book of his work1 on mathematics, written with the view of facilitating the study of Plato's writings. Thymaridas. Another mathematician, reckoned by some writers as of about the same date as Theon, was Thymaridas, who is worthy of notice from the fact that he is the earliest known writer who explicitly enunciates an algebraical theorem. He states that, if the sum of any number of quantities be given, and also the sum of every pair which contains one of them, then this quantity is equal to one (n − 2)th part of the difference between the sum of these pairs and the first given sum. Thus, if and x1+xn=8n, + Sn−S)/(n − 2). - 1 The Greek text of those parts which are now extant, with a French translation, was issued by J. Dupuis, Paris, 1892. He does not seem to have used a symbol to denote the unknown quantity, but he always represents it by the same word, which is an approximation to symbolism. Ptolemy.1 About the same time as these writers Ptolemy of Alexandria, who died in 168, produced his great work on astronomy, which will preserve his name as long as the history of science endures. This treatise is usually known as the Almagest the name is derived from the Arabic title al midschisti, which is said to be a corruption of μεγίστη [μαθηματική] σúvragis. The work is founded on the writings of Hipparchus, and, though it did not sensibly advance the theory of the subject, it presents the views of the older writer with a completeness and elegance which will always make it a standard treatise. We gather from it that Ptolemy made observations at Alexandria from the years 125 to 150; he, however, was but an indifferent practical astronomer, and the observations of Hipparchus are generally more accurate than those of his expounder. The work is divided into thirteen books. In the first book Ptolemy discusses various preliminary matters; treats of trigonometry, plane or spherical; gives a table of chords, that is, of natural sines (which is substantially correct and is probably taken from the lost work of Hipparchus); and explains the obliquity of the ecliptic; in this book he uses degrees, minutes, and seconds as measures of angles. The second book is devoted chiefly to phenomena depending on the spherical form of the earth he remarks that the explanations would be much simplified if the earth were supposed to rotate on its axis once a day, but states that this hypothesis is inconsistent with known facts. In the third book he explains the motion of the 1 See the article Ptolemaeus Claudius, by A. De Morgan in Smith's Dictionary of Greek and Roman Biography, London, 1849; S. P, Tannery, Recherches sur l'histoire de l'astronomie ancienne, Paris, 1893; and J. B. J. Delambre, Histoire de l'astronomie ancienne, Paris, 1817, vol. ii, An edition of all the works of Ptolemy which are now extant was published at Bâle in 1551. The Almagest with various minor works was edited by M. Halma, 12 vols. Paris, 1813-28, and a new edition, in two volumes, by J. L. Heiberg, Leipzig, 1898, 1903, 1907. sun round the earth by means of excentrics and epicycles: and in the fourth and fifth books he treats the motion of the moon in a similar way. The sixth book is devoted to the theory of eclipses; and in it he gives 3° 8' 30", that is 31, as the approximate value of 7, which is equivalent to taking it equal to 3.1416. The seventh and eighth books contain a catalogue (probably copied from Hipparchus) of 1028 fixed stars determined by indicating those, three or more, that appear to be in a plane passing through the observer's eye and in another work Ptolemy added a list of annual sidereal phenomena. The remaining books are given up to the theory of the planets. This work is a splendid testimony to the ability of its author. It became at once the standard authority on astronomy, and remained so till Copernicus and Kepler shewed that the sun and not the earth must be regarded as the centre of the solar system. The idea of excentrics and epicycles on which the theories of Hipparchus and Ptolemy are based has been often ridiculed in modern times. No doubt at a later time, when more accurate observations had been made, the necessity of introducing epicycle on epicycle in order to bring the theory into accordance with the facts made it very complicated. But De Morgan has acutely observed that in so far as the ancient astronomers supposed that it was necessary to resolve every celestial motion into a series of uniform circular motions they erred greatly, but that, if the hypothesis be regarded as a convenient way of expressing known facts, it is not only legitimate but convenient. The theory suffices to describe either the angular motion of the heavenly bodies or their change in distance. The ancient astronomers were concerned only with the former question, and it fairly met their needs; for the latter question it is less convenient. In fact it was as good a theory as for their purposes and with their instruments and knowledge it was possible to frame, and corresponds to the expression of a given function as a sum of sines or cosines, a method which is of frequent use in modern analysis. |