Notes to accompany Dr. Webb's Lecture on the Hellenistic Period I

Plato's school, under the name of the Academy, persisted for many centuries, but was chiefly occupied with philosophical discussion. One of his first disciples to distinguish himself in science was Eudoxus (409-356 B.C.) of Cnidus, the founder of observational cosmology. Eudoxus had also studied with the Pythagoreans. Under the stimulus of Plato he made advances in mathematical theory, but occupied himself chiefly with examining the heavens. Among his achievements is his remarkably accurate estimate of the solar year as 365 days and 6 hours. His most influential contribution was his view that the heavenly bodies move on a series of concentric spheres, of which the center is Earth, itself a sphere. Eudoxus had observed the irregularities in the movements of the planets. To explain these he supposed each planet to occupy its own sphere. The poles of each planetary sphere were supposed to be attached to a larger sphere rotating round other poles. The secondary spheres could be succeeded by tertiary or quaternary spheres according to mathematical and observational needs. For Sun and Moon Eudoxus found three spheres each sufficient. In the explanation of the movements of the other planets, four spheres each were demanded. For the fixed stars one sphere sufficed. Thus twenty seven spheres in all were demanded. These spheres-save that of the fixed stars-were treated by Eudoxus not as material but in the manner of mathematical constructions.

Heracleides of Pontus (c. 388-315 B.C.), a pupil of Plato, contributed to astronomy a suggestion that the Earth rotates on its own axis once in twenty four hours, and that Mercury and Venus circle round the Sun like satellites. His teaching led on to that of Aristarchus. Many others of Plato's followers made contributions to pure mathematics, and, in the sense all subsequent mathematicians are Plato's spiritual heirs. They is also evidence of a certain amount of botanical activity in the Academy, and some physiological theories which became popular in later centuries may be traced to Plato. Platonism passed into Christianity early, mainly through St. Augustine, so that the Christian Middle Ages, until the twelfth century, were main Platonic. The later school of philosophy known as 'Neoplatonisn' also profoundly influenced Christianity.

Aristarchus of Samos (c. 310-230 B.C.) taught at Alexandria soon after Euclid. He was himself the pupil of a disciple of Strato. The peculiar views of Aristarchus on the position of the Earth among the heavenly bodies have earned him the title of the 'Copernicus of Antiquity'. He extended the view of an earlier philosopher that the Earth rotates about its own axis by maintaining that the Sun itself is at rest, and that not only Mercury and Venus but also all the other planets, of which the Earth is one, revolve in circles about the Sun. It is interesting to observe that this view of Aristarchus brought on him the same charge of impiety as had descended on the head of Anaxagoras two centuries earlier.

We owe to Aristarchus the first scientific attempt to measure the distances of the Sun and Moon from the Earth, and their sizes relative to each other.

He knew that the light of the Moon is reflected from the Sun. When the Moon is exactly at the half, the line of vision from the observer on the Earth to the center of the Moon's disk M must be at right angles to the line of light passing from the center of the Sun's disk S to the center of the Moon's disk M. Now the observer can measure the angle that the Sun and Moon form at his own eye O. With a knowledge of the two angles at M and O the relative lengths of the sides OS and OM can be determined. This gives the relative distances of Sun and Moon from the observer. The difficulty lay in determining exactly the angle at O. A very small error here makes a very great difference in the result. Aristarchus estimated this angle as 87 degrees when the reality is 89 degrees 52 minutes. In the resulting calculation he estimated the Sun as 18 times more distant than the Moon, instead of over 346 times more distant!

If we have the relative distances of Sun and Moon from the observer, the relative sizes of these bodies can be estimated, provided that we know the relative sizes of their disks, as they appear to an observer on the Earth. On this basis Aristarchus calculated that the Sun was seven thousand times larger than the Moon. Here further observational errors were introduced, and the ratio is very far from the truth. Nevertheless Aristarchus per­ceived that while the Moon is smaller than the Earth, the Sun is enormously greater. This fundamental relationship may well have affected his thought, for it seems inherently improbable that an enormously large body would revolve round a relatively minute one.

Contemporary with Aristarchus at Alexandria were other astronomers who recorded the positions of stars by measurements of their distances from fixed positions in the sky. Thus they defined the position of the more important stars in the signs of the zodiac, near to which all the planets in their orbits pass. They thereby facilitated accurate observations and record of the move­ments of the planets. Their observations were used by later astronomers, notably by Hipparchus.

The philosophy which was the parent of science among the Greeks concerned itself in three main aspects of the material world: (a) number and form and their relation to each other and to material objects, (b) the form and workings of the universe, and (c) the nature of man. In Alexandria, where science had freed itself from philosophy, it was thus to be expected that the systematization of mathematics and astronomy would be accom­panied by a similar development in the basic studies by which alone medicine can continue its progressive, scientific tradition.

The greatest astronomer of antiquity was Hipparchus of Nicaea (c. 190-120 B.C.). He worked at Rhodes, where he erected an observatory and made most important researches. He developed trigonometry by which numerical calculations can be applied to figures drawn on either plane or spherical surfaces. Trigonometry, it turns out, is of great value to astronomy.

Hipparchus made numerous accurate astronomical observa­tions. He also collected and collated the records of previous observers to see if astronomical changes had taken place in the course of the ages. There were available to him records of his Alexandrian and earlier Greek predecessors, and also those of the yet more ancient Babylonian astronomers. As a result of these comparisons he gave to the world two brilliant astronomical con­ceptions. (a) One of these, the precession of the equinoxes, was of permanent value. (b) The other, his theory of the movements of the planets and notably of the Sun and Moon, was of value to subsequent generations for the calculation of eclipses.

(a) Precession of the equinoxes. In 134 B.C. Hipparchus observed a new star in the constellation Scorpio. This suggested to him that he should prepare a catalogue of star positions. He therefore drew up a list of upwards of a thousand stars, each of which was given its celestial latitude and longitude. The constellations to which Hipparchus referred these stars are those which are to-day gener­ally accepted. He showed great foresight in recording a number of cases in which three or more stars were in a line, so that astronomers of subsequent ages might detect changes. in their relative positions.

Hipparchus proceeded to compare his observations with others of about 150 years earlier. He found that in this lapse of time there had been changes in the distance of the stars from certain fixed points in the heavens. The changes were of a kind that could only be explained by a rotation of the axis of the earth in the direction of the apparent daily motion of the stars. This causes the equi­noxes to fall a little earlier each year. The knowledge of this precession of the equinoxes and of the rate at which it takes place was necessary for the progress of accurate astronomical observa­tion. The complete cycle of precession takes 26,000 years.

(b) Theory of motion of the Planets. When Hipparchus came to examine the apparent movement of the planets he had before him two theories, namely, that of `epicyclical motion' and that of 'eccentric motion'. Certain of his predecessors, notably Apol­lonius of Perga, had suggested the epicyclical view. According to this each planet moves on a circle the center of which moves on another circle, the center of which is the center of the Earth. Others of his predecessors had set forth the view of eccentric motion. According to this the planet moves around the Earth but in a circle whose center is not at the center of the Earth. This secondary center may also be represented as moving on a circle (see class handout on Ptolemaic Astronomy.)  Hipparchus explained the behavior of the sun by fixed and the moon by a moving eccentric.  (The geometric results of moving eccentric and epicycle are identical.)

The epicyclical view finally prevailed through the mediation of the astronomer Ptolemy.  The theory of the eccentric motion of the Moon and to a less extent of the Sun, as enunciated by Hipparchus, was, however, of great service in that calculations based on it accorded much more closely with actual observations than did calculations based on any older doctrine of their movements. From the time of Hipparchus onward eclipses of the Moon could be predicted within an hour or two. Eclipses of the Sun could be predicted less accurately.

Ptolemy of Alexandria (flourished A.D. 170), who provided the final astronomical and geographical syntheses of antiquity, con­tributed also to the knowledge of optics. He not only knew that luminous rays in passing from one medium to another are deflected, but he actually measured the angle of deflection. Applying the known principle of the refraction of light, Ptolemy points out that the light of a star on entering the earthly atmosphere and on penetrating to the lower and denser parts must at each stage be gradually bent or refracted. Thus it will appear to be nearer the zenith than is actually the case.

The great work of Ptolemy known as the Almagest has proved one of the most influential of all scientific writings. The very name has a history. The Greeks called the work the megale syntaxis, i.e. `great composition'. The later translators from the Greek into Arabic, either from admiration or carelessness, converted the positive megale into the superlative megiste. Thus it became in Arabic Almagisti, from the Latin Almagestum and colloquial Almagest.

The Almagest, a work of utmost skill, was of the highest significance for mathematical development. It has provided the foundations of trigonometry, both plane and spherical. Its basic cosmic conceptions, however, Ptolemy certainly derived from his predecessors. Thus he invoked epicycles to explain the movements and behavior of the planets, employing them to resolve some errors and inconsistencies of Hipparchus. He retained, however, eccentrics to explain the movements of the Sun and Moon.

Among the contents of the Almagest is an account of the construction of the astrolabe, the chief astronomical instrument of ancient and medieval times. It was, in essence, a device for determining the angle of elevation of a heavenly body. Ptolemy used the instrument to obtain the distance of the Moon by parallax. The method is substantially that still in use and is, in principle, very simple. If in one place Z, the Moon is at zenith, then a line passing from the Moon at that place passes also through the center of the Earth C. If an observer 0 takes at the same time the elevation of the center of. the Moon M, then we know the angle at 0 of the triangle MOC. If we know the distance from O to Z we can calculate the angle at C. We thus know the three angles and therefore the relative lengths of the sides of the triangle MOC. Thus we can determine the ratio of CM to CO. Ptolemy thus estimates the Moon's distance to be 59 times the radius of the Earth, which is not very far from the current value. Working on an eclipse method of Hipparchus he estimates the Sun, however, to be only 1,210 Earth radii distant. This number is about one twentieth of the true reckoning. He tells us that he has no means of estimating the distances of the lesser planets, but he follows tradition in accepting rapidity of motion as the main test of nearness. Thus from within outward his universe consists of Earth, Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn. This scheme was passed on to the Middle Ages.

Ptolemy's other great work was his Geographical Outline. This was essentially a product of the knowledge brought by the expansion of the Roman Empire. He studied itineraries of Roman officials and merchants. Thus he may be said to have preserved for us a summary of Roman knowledge of the Earth's surface, presented, however, in a form quite beyond the capacity of any Latin geographical writer. Ptolemy may well have had access to the great map prepared by Vipsanius Agrippa at Rome.

Ptolemy developed his own manner of representing the curved surface of the Earth on a plane surface. In his scheme of projec­tion the parallels of latitude are arcs of concentric circles, the centers of which are at the North Pole. Chief among the parallels are the Equator and the circles passing through Thule, through Rhodes, and through Meroe. The meridians of longitude are represented by straight lines which converge to the Pole.  He delineates in this manner the whole of the then known world. Its boundaries are: on the north, the ocean which surrounds the British Isles, the northern parts of Europe, and the unknown land in the northern region of Asia; on the south, the unknown land which encloses the Indian Sea, and the unknown land to the south of Libya and Ethiopia; on the east, the unknown land which adjoins these eastern nations of Asia, the Sinae (Chinese) and the people of Serica, the silk-producing land; on the west, the great Western Ocean and unknown parts of Libya. The portion of the Earth thus surveyed covers in length a hemisphere and in breadth between 63° north latitude and 16° south latitude.

He has another scheme of projection in which the meridians are also curved.  As originally written Ptolemy's geography was furnished with maps. These have long since disappeared, but as Ptolemy gives the latitude and longitude of the places that he mentions his charts can be reconstructed. A peculiar interest attaches to the map of Britain, which can thus be put together. Scotland is bent eastward with its axis at a right angle to that of England. This is an unusual degree of error for Ptolemy. It is probable that he was here working not on the records of travelers, but on maps of the island, and that he had made the error of fitting the map of Scotland on to that of England on the wrong side!

Ptolemy exhibits the final extension of scientific geography in the Empire. How far the average educated citizen of the Empire was able or willing to appreciate science in general and geography in particular is another matter. It was the attitude of the Romans end especially of the Roman ruling class to things of the mind that determined the fate of science and with it, perhaps, the fate of the Empire. To estimate the attitude of the Roman to science we must turn to geographical works in Latin.

The Almagest of Ptolemy was translated into Latin in the later twelfth century and his Geography in the fifteenth. Thus they could not directly influence the earlier Middle Ages during which a simpler cosmic scheme based on Aristotle prevailed. In the later Middle Ages conflict between the views of Aristotle and those of Ptolemy became of considerable importance for the history of science.