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] 12th century, says that he was born at Ptolemais Hermii, a Grecian city of the Thebaid. It is certain that he observed at Alexandria during the reigns of Hadrian and Antoninus Pius, and that he survived Antoninus. Olympiodorus, a philosopher of the Neoplatonic school who lived in the reign of the emperor Justinian, relates in his scholia on the Phaedo of Plato that Ptolemy devoted his life to astronomy and lived for forty years in the so-called undefined, probably elevated terraces of the temple of Serapis at Canopus near Alexandria, where they raised pillars with the results of his astronomical discoveries engraved upon them. This statement is probably correct; we have indeed the direct evidence of Ptolemy himself that he made astronomical observations during a long series of years; his first recorded observation was made in the eleventh year of Hadrian, 127, and his last in the fourteenth year of Antoninus, 151 Ptolemy, moreover, says, “We make our observations in the parallel of Alexandria.” St Isidore of Seville asserts that he was of the royal race of the Ptolemies, and even calls him king of Alexandria; this assertion has been followed by others, but there is no ground for their opinion. Indeed Fabricius shows by numerous instances that the name Ptolemy was common in Egypt. Weidler, from whom this is taken, also tells us that according to Arabian tradition Ptolemy lived to the age of seventy-eight years; from the same source some description of his personal appearance has been handed down, which is generally considered as not trustworthy, but which may be seen in Weidler, Historia astronomiae, p. 177, or in the preface to Halma’s edition of the Almagest, p. 61.

Mathematics. Ptolemy’s work as a geographer is discussed below, and an account of the discoveries in astronomy of Hipparchus and Ptolemy is given in the article : History. Their contributions to pure mathematics, however, require to be noticed here. Of these the chief is the foundation of trigonometry, plane and spherical, including the formation of a table of chords, which served the same purpose as our table of sines. This branch of mathematics was created by Hipparchus for the use of astronomers, and its exposition was given by Ptolemy in a form so perfect that for 1400 years it was not surpassed. In this respect it may be compared with the doctrine as to the motion of the heavenly bodies so well known as the Ptolemaic system, which was paramount for about the same period of time. There is, however, this difference, that, whereas the Ptolemaic system was then overthrown, the theorems of Hipparchus and Ptolemy, on the other hand, will be, as Delambre says, for ever the basis of trigonometry. The astronomical and trigonometrical systems are contained in the great work of Ptolemy, undefined, or, as Fabricius after Syncellus writes it, undefined; and in like manner Suidas says undefined. The Syntaxis of Ptolemy was called undefined to distinguish it from another collection called undefined, also highly esteemed by the Alexandrian school, which contained some works of Autolycus, Euclid, Aristarchus, Theodosius of Tripolis, Hypsicles and Menelaus. To designate the great work of Ptolemy the Arabs used the superlative undefined, from which, the article al being prefixed, the hybrid name Almagest, by which it is now universally known, is derived.

We proceed now to consider the trigonometrical work of Hipparchus and Ptolemy. In the ninth chapter of the first book of the Almagest Ptolemy shows how to form a table of chords. He supposes the circumference divided into 360 equal parts (undefined), and then bisects each of these parts. Further, he divides the diameter into 120 equal parts, and then for the subdivisions of these he employs the sexagesimal method as most convenient in practice, i.e. he divides each of the sixty parts of the radius into sixty equal parts, and each of these parts he further subdivides into sixty equal parts. In the Latin translation these subdivisions become “partes minutae primae” and “partes minutae secundae” whence our “minutes” and “seconds” have arisen. It must not be supposed, however, that these sexagesimal divisions are due to Ptolemy; they must have been familiar to his predecessors, and were handed down from the Chaldaeans. Nor did the formation of the table of chords originate with Ptolemy; indeed, Theon of Alexandria, the father of Hypatia, who lived in the reign of Theodosius, in his commentary on the Almagest says expressly that Hipparchus had already given the doctrine of chords inscribed in a circle in twelve books, and that Menelaus had done the same in six books, but, he continues, every one must be astonished at the ease with which Ptolemy, by means of a few simple theorems, has found their values; hence it is inferred that the method of calculation in the Almagest is Ptolemy’s own.

As starting-point the values of certain chords in terms of the diameter were already known, or could be easily found by means of the Elements of Euclid. Thus the side of the hexagon, or the chord of 60°, is equal to the radius, and therefore contains sixty parts. The side of the decagon, or the chord of 36°, is the greater segment of the radius cut in extreme and mean ratio, and therefore contains approximately 37p 4′ 55″ parts, of which the diameter contains 120 parts. Further, the square on the side of the regular pentagon is equal to the sum of the squares on the sides of the regular hexagon and of the regular decagon, all being inscribed in the same circle (Eucl. XIII. 10); the chord of 72° can therefore be calculated, and contains approximately 70p 32′ 3″. In like manner, the square on the chord of 90°, which is the side of the inscribed square, is twice the square on the radius; and the square on the chord of 120°, or the side of the equilateral triangle, is three times the square on the radius; these chords can thus be calculated approximately. Further, from the values of all these chords we can calculate at once the chords of the arcs which are their supplements.

This being laid down, we now proceed to give Ptolemy’s exposition of the mode of obtaining his table of chords, which is a piece of geometry of great elegance, and is indeed, as De Morgan says, “one of the most beautiful in the Greek writers.”

He takes as basis and sets forth as a lemma the well-known theorem, which is called after him, concerning a quadrilateral inscribed in a circle: The rectangle under the diagonals is equal to the sum of the rectangles under the opposite sides. By means of this theorem the chord of the sum or the difference of two arcs whose chords are given can be easily found, for we have only to draw a diameter from the common vertex of the two arcs the chord of whose sum or difference is required, and complete the quadrilateral; in one case a diagonal, in the other one of the sides is a diameter of the circle. The relations thus obtained are equivalent to the fundamental formulae of our trigonometry—

sin (A+B)＝sin A cos B+cos A sin B, sin (A−B)＝sin A cos B−cos A sin B,

which can therefore be established in this simple way.

Ptolemy then gives a geometrical construction for finding the chord of half an arc from the chord of the arc itself. By means of the foregoing theorems, since we know the chords of 72° and of 60°, we can find the chord of 12°; we can then find the chords of 6°, 3°, 1° and three-fourths of 1°, and lastly, the chords of 4°, 7°, 9°, 10°, &c.—all those arcs, namely, as Ptolemy says, which being doubled are divisible by 3. Performing the calculations, he finds that the chord of 1° contains approximately 1p 34′ 55″, and the chord of three-fourths of 1° contains 0p 47′ 8″. A table of chords of arcs increasing by 1° can thus be formed; but this is not sufficient for Pto1emy’s purpose, which was to frame a table of chords increasing by half a degree. This could be effected if he knew the chord of one-half of 1°; but, since this chord cannot be found geometrically from the chord of 1°, inasmuch as that would come to the trisection of an angle, he proceeds to seek in the first place the chord of 1°, which he finds approximately by means of a lemma of great elegance, due probably to Apollonius. It is as follows: If two unequal chords be inscribed in a circle, the greater will be to the less in a less ratio than the arc described on the greater will be to the arc described on the less. Having proved this theorem, he proceeds to employ it in order to find approximately the chord of 1°, which he does in the following manner—

$chord 60′⁄chord 45′$ < $60⁄45$, i.e. < $4⁄3$, ∴ chord 1° <$4⁄3$ chord 45′;

again— $chord 90′⁄chord 60′$ < $90⁄60$, i.e. < $3⁄2$, ∴ chord 1° >$2⁄3$ chord 90′.

For brevity we use a modern notation. It has been shown that the chord of 45′ is 0p 47′ 8″ q.p., and the chord of 90′ is 1p 34′ 15″ q.p.; hence it follows that approximately

chord 1° < 1p 2′ 50″ 40′″ and > 1p 2′ 50″.

Since these values agree as far as the seconds, Ptolemy takes 1p 2′ 50″ as the approximate value of the chord of 1°. The chord of 1° being thus known, he finds the chord of one-half of a degree, the approximate value of which is of 31′ 25″, and he is at once in a position to complete his table of chords for arcs increasing by half a degree. Ptolemy then gives his table of chords, which is arranged in three columns; in the first he has entered the arcs, increasing by half degrees, from 0° to 180°; in the second he gives the values of the