CHAPTER VI THE DECLINE OF ALEXANDRIAN SCIENCE The century which produced Euclid, Archimedes and Apollonius was... the time at which Greek mathematical genius attained its highest development. For many centuries afterwards geometry remained a favorite study, but no substantive work fit to be compared with the Sphere and Cylinder or the Conics was ever produced. One great invention, trigonometry, remains to be completed, but trigonometry with the Greeks remained always the instrument of astronomy and was not used in any other branch of mathematics, pure or applied. The geometers who succeed to Apollonius are professors who signalised themselves by this or that pretty little discovery or by some commentary on the classical treatises. The force of nature could go no further in the same direction than the ingenious applications of exhaustion by Archimedes and the portentous sentences in which Apollonius enunciates a proposition in conics. A briefer symbolism, an analytical geometry, an infinitesimal calculus were wanted, but against these there stood the tremendous authority of the Platonic and Euclidean tradition, and no discoveries were made in physics or astronomy which rendered them imperatively necessary. It remained only for mathematicians, as Cantor says, to descend from the height which they had reached and "in the descent to pause here and there and look around at details which had been passed by in the hasty ascent." The elements of planimetry were exhausted, and the theory of conic sections. In stereometry something still remained to be done, and new curves, suggested by the spiral of Archimedes, could still be investigated. Finally, the arithmetical determination of geometrical ratios, in the style of the Measurement of the Circle, offered a considerable field of research, and to these subjects mathematicians now devoted themselves. - Gow. In the second century B.C. Hypsicles developed the theory of arithmetical progression and added two books of elements to Euclid's thirteen, but the chief mathematical work of this cen tury was due to Hipparchus, a great astronomer, and Hero, an engineer. ORBITAL MOTION OF THE EARTH. ARISTARCHUS. - Before dealing with Hipparchus and Hero, however, we have to consider the highly interesting and significant astronomical theories of Aristarchus of Samos (270 B.C.-?), who was the author of a treatise On the Dimensions and Distances of the Sun and Moon. He endeavored to determine these distances relatively by ascertaining or estimating the angular distance between the two bodies when the moon is just half illuminated, that is, when the lines joining sun, earth, and moon form a right angle at the moon — a method which may have been due to Eudoxus. The difficulties of this determination are so serious, however, that no high degree of accuracy could be attained, the actual result of Aristarchus of a right angle — against the true § corresponding to a ratio of about 1 to 19 of the two distances. Aristarchus had no trigonometry, and no other method of attacking this problem seems to have been known to the Greeks. In his Sand Counting already mentioned, Archimedes says of Aristarchus, He supposes that the fixed stars and the sun are immovable, but that the earth is carried round the sun in a circle which is in the middle of the course; but the sphere of the fixed stars, lying with the sun round the same centre, is of such a size that the circle, in which he supposes the earth to move, has the same ratio to the distance of the fixed stars as the centre of the sphere has to the surface. But this is evidently impossible, for as the centre of the sphere has no magnitude, it follows that it has no ratio to the surface. It is therefore to be supposed that Aristarchus meant that as we consider the earth as the centre of the world, then the earth has the same ratio to that which we call the world, as the sphere in which is the circle, described by the earth according to him, has to the sphere of the fixed stars. Aristarchus thus meets the objection that motion of the earth would cause changes in the apparent positions of the stars by assuming that their distances are so great as to render the motion of the earth a negligible factor. Another reference to Aristarchus, in Plutarch, mentions an opinion that he ought to be accused of impiety for moving the hearth of the world, as the man in order to save the phenomena supposed that the heavens stand still and the earth moves in an oblique circle at the same time as it turns round its axis. How far this remarkable anticipation of the Copernican theory was a conviction rather than a mere fortunate speculation cannot be known, but at any rate it failed of that acceptance necessary to its permanence. In the next century the rotation of the earth on its axis was indeed taught by Seleucus, an Asiatic astronomer, but it was 1700 years before these daring theories were again advanced. Seleucus also observed the tides, saying "that the revolution of the moon is opposed to the earth's rotation, but the air between the two bodies being drawn forward falls upon the Atlantic Ocean, and the sea is disturbed in proportion." PLANETARY IRREGULARITIES. — The earlier theory of homocentric spheres, while accounting more or less successfully for the apparent motions of the heavenly bodies, had maintained each of them at a constant distance from the earth, and thus quite failed to explain the differences of brightness which were soon discovered, as well as the variations in the apparent size of the moon. The conception of motion in neither a straight line nor a circle was repugnant to the Greek philosophers, and the difficulty was therefore met, first by supposing the earth not to be exactly at the centre of the circular orbits about it, second by introducing subsidiary circles or epicycles. EXCENTRIC CIRCULAR ORBITS. The complete planetary system according to the excentric circle theory was therefore as follows. In the centre of the universe the earth, round which moved the moon in 27 days, and the sun in a year, probably in concentric circles. Mercury and Venus moved on circles, the centres of which were always on the straight line from the earth to the sun, so that the earth was always outside these circles, for which reason the two planets are always within a certain limited angular distance of the sun, from tury was due to Hipparchus, a great astronomer, and Hero, an engineer. ORBITAL MOTION OF THE EARTH. ARISTARCHUS. - Before dealing with Hipparchus and Hero, however, we have to consider the highly interesting and significant astronomical theories of Aristarchus of Samos (270 B.C.-?), who was the author of a treatise On the Dimensions and Distances of the Sun and Moon. He endeavored to determine these distances relatively by ascertaining or estimating the angular distance between the two bodies when the moon is just half illuminated, that is, when the lines joining sun, earth, and moon form a right angle at the moon — a method which may have been due to Eudoxus. The difficulties of this determination are so serious, however, that no high degree of accuracy could be attained, the actual result of Aristarchus of a right angle - against the true corresponding to a ratio of about 1 to 19 of the two distances. Aristarchus had no trigonometry, and no other method of attacking this problem seems to have been known to the Greeks. In his Sand Counting already mentioned, Archimedes says of Aristarchus, He supposes that the fixed stars and the sun are immovable, but that the earth is carried round the sun in a circle which is in the middle of the course; but the sphere of the fixed stars, lying with the sun round the same centre, is of such a size that the circle, in which he supposes the earth to move, has the same ratio to the distance of the fixed stars as the centre of the sphere has to the surface. But this is evidently impossible, for as the centre of the sphere has no magnitude, it follows that it has no ratio to the surface. It is therefore to be supposed that Aristarchus meant that as we consider the earth as the centre of the world, then the earth has the same ratio to that which we call the world, as the sphere in which is the circle, described by the earth according to him, has to the sphere of the fixed stars. Aristarchus thus meets the objection that motion of the earth would cause changes in the apparent positions of the stars by assuming that their distances are so great as to render the motion of the earth a negligible factor. Another reference to Aristarchus, in Plutarch, mentions an opinion that he ought to be accused of impiety for moving the hearth of the world, as the man in order to save the phenomena supposed that the heavens stand still and the earth moves in an oblique circle at the same time as it turns round its axis. How far this remarkable anticipation of the Copernican theory was a conviction rather than a mere fortunate speculation cannot be known, but at any rate it failed of that acceptance necessary to its permanence. In the next century the rotation of the earth on its axis was indeed taught by Seleucus, an Asiatic astronomer, but it was 1700 years before these daring theories were again advanced. Seleucus also observed the tides, saying "that the revolution of the moon is opposed to the earth's rotation, but the air between the two bodies being drawn forward falls upon the Atlantic Ocean, and the sea is disturbed in proportion.' PLANETARY IRREGULARITIES. -The earlier theory of homocentric spheres, while accounting more or less successfully for the apparent motions of the heavenly bodies, had maintained each of them at a constant distance from the earth, and thus quite failed to explain the differences of brightness which were soon discovered, as well as the variations in the apparent size of the moon. The conception of motion in neither a straight line nor a circle was repugnant to the Greek philosophers, and the difficulty was therefore met, first by supposing the earth not to be exactly at the centre of the circular orbits about it, second by introducing subsidiary circles or epicycles. EXCENTRIC CIRCULAR ORBITS. The complete planetary system according to the excentric circle theory was therefore as follows. In the centre of the universe the earth, round which moved the moon in 27 days, and the sun in a year, probably in concentric circles. Mercury and Venus moved on circles, the centres of which were always on the straight line from the earth to the sun, so that the earth was always outside these circles, for which reason the two planets are always within a certain limited angular distance of the sun, from |