## Mathematics and astronomy

From Euclid and Pythagoras to Archimedes, the Ancient Greeks were the masters of mathematics, geometry, and astronomy. Explore their accomplishments here! Concerned in this article with mathematics, not on its own account, but as it was related to Greek philosophy - a relation which, especially in Plato, was very close.

### Beginning of Greek mathematics

# Introduction Greek mathematician geometry

#### Greek mathematics contributions

** ****who is Eudoxus**

**Eudoxus**(ca. 408 - ca. 355 B.C.). Before him, there was only the arithmetical theory of proportion.

**Euclid **geometry

**Euclid**, who was still, lived at Alexandria, about 300 B.C., a few years after the death of Alexander and Aristotle. Euclid of Alexandria was another important mathematician and philosopher during the sixth century BC in Greece. He is best known for writing the Elements, which is often called the most influential mathematical work of all time. It included definitions, postulates, propositions, and proofs that form the basis of modern mathematics. In addition to developing basic geometrical theory such as points, lines, circles, angles, and theorems such as circle area calculation he also wrote extensively on number theory. He showed how to construct regular polygons using a ruler and compass and compared incommensurable magnitudes. His works have proved essential for many developments in science and mathematics.

#### Euclid elementary

**exhaustion**," which was subsequently used with great success by Archimedes.

## Archimedes contribution in mathematics

This method is an anticipation of the integral calculus. Take, for example, the question of the area of a circle. You can inscribe in a circle a regular hexagon, or a regular dodecagon, or a regular polygon of a thousand or a million sides. The area of such a polygon, however many sides it has, is proportional to the square on the diameter of the circle. The more sides the polygon has, the more nearly it becomes equal to the circle. You can prove that, if you give the polygon enough sides, its area can be got to differ from that of the circle by less than any previously assigned area, however small. For this purpose, the "axiom of Archimedes" is used. This states (when somewhat simplified) that if the greater of two quantities is halved, and then the half is halved, and so on, a quantity will be reached, at last, which is less than the smaller of the original two quantities. In other words, if a is greater than b, there is some whole number n such that 2n times b is greater than a.

### Archimedes method

The method of exhaustion sometimes leads to an exact result, as in squaring the parabola, which was done by Archimedes; sometimes, as in the attempt to square the circle, it can only lead to successive approximations. The problem of squaring the circle is the problem of determining the ratio of the circumference of a circle to the diameter, which is called Ï€. Archimedes used the approximation 22/7 in calculations; by inscribing and circumscribing a regular polygon of 96 sides, he proved that Ï€ is less than 3 1/7 and greater than 3 10/71. The method could be carried to any required degree of approximation, and that is all that any method can do in this problem. The use of inscribed and circumscribed polygons for approximations to Ï€ goes back to Antiphon, who was a contemporary of Socrates.

#### Greek astronomy

### Greek astronomy history

**astronomy**, where Greek achievements were as remarkable as in geometry.The Ancient Greeks made major strides in the development of astronomy. Euclid, who is often referred to as the ‘Founder of Geometry’, developed a treatise for astronomy which was used for generations. Hipparchus is known for creating a star catalog and calculating the length of a year with remarkable accuracy at that time. Also, Eratosthenes calculated the circumference of the Earth.

**ancient Greek astronomy discoveries**

Let us begin with some of the earliest discoveries and correct hypotheses. Anaximander thought that the earth floats freely, and is not supported on anything. Aristotle, who often rejected the best hypotheses of his time, objected to the theory of Anaximander, that the earth, being at the centre, remained immovable because there was no reason for moving in one direction rather than another. If this were valid, he said, a man placed the centre of a circle with food at various points of the circumference would starve to death for lack of reason to choose one portion of food rather than another. This argument reappears in scholastic philosophy, not in connection with astronomy, but with free will. It reappears in the form of "Buridan's ass," which was unable to choose between two bundles of hay placed at equal distances to right and left, and therefore died of hunger.

Pythagoras, in all probability, was the first to think the earth spherical, but his reasons were (one must suppose) aesthetic rather than scientific. Scientific reasons, however, were soon found. Anaxagoras discovered that the moon shines by reflected light, and gave the right theory of eclipses. He himself still thought the earth flat, but the shape of the earth's shadow in lunar eclipses gave the Pythagoreans conclusive arguments in favour of its being spherical. They went further, and regarded the earth as one of the planets. They knew-from Pythagoras himself, it is said-that the morning star and the evening star are identical, and they thought that all the planets, including the earth, move in circles, not round the sun, but round the "central fire."

They had discovered that the moon always turns the same face to the earth, and they thought that the earth always turns the same face to the "central fire." The Mediterranean regions were on the side turned away from the central fire, which was therefore always invisible. The central fire was called "the house of Zeus," or "the Mother of the gods." The sun was supposed to shine by light reflected from the central fire. In addition to the earth, there was another body, the counter-earth, at the same distance from the central fire. For this, they had two reasons, one scientific, one derived from their arithmetical mysticism.

The scientific reason was the correct observation that an eclipse of the moon sometimes occurs when both sun and moon are above the horizon. Refraction, which is the cause of this phenomenon, was unknown to them, and they thought that, in such cases, the eclipse must be due to the shadow of a body other than the earth. The other reason was that the sun and moon, the five planets, the earth and counter-earth, and the central fire, made ten heavenly bodies, and ten was the mystic number of the Pythagoreans.

This Pythagorean theory is attributed to Philolaus, a Theban, who lived at the end of the fifth century B.C. Although it is fanciful and in part quite unscientific, it is very important, since it involves the greater part of the imaginative effort required for conceiving the Copernican hypothesis. To conceive of the earth, not as the centre of the universe, but as one among the planets, not as eternally fixed, but as wandering through space, an extraordinary emancipation from anthropocentric thinking. When once this jolt had given to men's natural picture of the universe, it was not so very difficult to be led by scientific arguments to a more accurate theory.

To this various observations contributed. Oenopides, who was slightly later than Anaxagoras, discovered the obliquity of the ecliptic. It soon became clear that the sun must be much larger than the earth, which fact supported those who denied that the earth is the centre of the universe. The central fire and the counter-earth were dropped by the Pythagoreans soon after the time of Plato. Heraclides of Pontus (whose dates are about 388 to 315 B.C., contemporary with Aristotle) discovered that Venus and Mercury revolve about the sun, and adopted the view that earth rotates on its own axis once every twenty-four hours. This last was a very important step, which no predecessor had taken. Heraclides was of Plato's school, and must have been a great man, but was not as much respected as one would expect; he is described as a fat dandy.

#### famous Greek astronomers

**Aristarchus of Samos**, who lived approximately from 310 to 230 B.C., and was thus about twenty-five years older than Archimedes, is the most interesting of all ancient astronomers, because he advanced the complete Copernican hypothesis, that all the planets, including the , revolve in circles round the sun, and that the earth rotates on its axis once in twenty-four hours. It is a little disappointing to find that the only extant work of Aristarchus, On the Sizes and Distances of the Sun and the Moon, adheres to the geocentric view. It is true that, for the problems with which this book deals, it makes no difference which theory is adopted, and he may therefore have thought it unwise to burden his calculations with an unnecessary opposition to the general opinion of astronomers; or he may have only arrived at the Copernican hypothesis after writing this book. Sir Thomas Heath, in his work on Aristarchus, which contains the text of this book with a translation, inclines to the latter view. The evidence that Aristarchus suggested the Copernican view is, in any case, quite conclusive.

The first and best evidence is that of Archimedes, who, as we have seen, was a younger contemporary of Aristarchus. Writing to Gelon, King of Syracuse, he says that Aristarchus brought out "a book consisting of certain hypotheses," and continues: "His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit." There is a passage in Plutarch saying that Cleanthes "thought it was the duty of the Greeks to indict Aristarchus of Samos on the charge of impiety for putting in motion the Hearth of the Universe (i.e. the earth), this being the effect of his attempt to save the phenomena by supposing the heaven to remain at rest and the earth to revolve in an oblique circle, while it rotates, at the same time, about its own axis." Cleanthes was a contemporary of Aristarchus, and died about 232 B.C. In another passage, Plutarch says that Aristarchus advanced this view only as a hypothesis, but that his successor Seleucus maintained it as a definite opinion. ( Seleucus flourished about 150 B.C.). AÃ«tius and Sextus Empiricus also assert that Aristarchus advanced the heliocentric hypothesis, but do not say that it was set forth by him only as a hypothesis. Even if he did so, it seems not unlikely that he, like Galileo two thousand years later, was influenced by the fear of offending religious prejudices, a fear which the attitude of Cleanthes (mentioned above) shows to have been well grounded.

The Copernican hypothesis, after being advanced, whether positively or tentatively, by Aristarchus, was definitely adopted by Seleucus, but by no other ancient astronomer. This general rejection was mainly due to Hipparchus, who flourished from 161 to 126 B.C. He is described by Heath as "the greatest astronomer of antiquity." He was the first to write systematically on trigonometry; he discovered the precession of the equinoxes; he estimated the length of the lunar month with an error of less than one second; he improved Aristarchus's estimates of the sizes and distances of the sun and moon; he made a catalogue of eight hundred and fifty fixed stars, giving their latitude and longitude. As against the heliocentric hypothesis of Aristarchus, he adopted and improved the theory of epicycles which had been invented by Apollonius, who flourished about 220 B.C.; it was a development of this theory that came to be known, later, as the Ptolemaic system, after the astronomer Ptolemy, who flourished in the middle of the second century A.D. Copernicus came to know something, though not much, of the almost forgotten hypothesis of Aristarchus, and was encouraged by finding ancient authority for his innovation. Otherwise, the effect of this hypothesis on subsequent astronomy was practically nil.

Ancient astronomers, in estimating the sizes of the earth, moon, and sun, and the distances of the moon and sun, used methods which were theoretically valid, but they were hampered by the lack of instruments of precision.

Many of their results, in view of this lack, were surprisingly good. Eratosthenes estimated the earth's diameter at 7850 miles, which is only about fifty miles short of the truth. Ptolemy estimated the mean distance of the moon at 29 Â½ times the earth's diameter; the correct figure is about 30.7. None of them got anywhere near the size and distance of the sun, which all underestimated. Their estimates, in terms of the earth's diameter, were: Aristarchus, 180; Hipparchus, 1245; Posidonius, 6545.

The correct figure is 11,726. It will be seen that these estimates continually improved (that of Ptolemy, however, showed a retrogression); that of Posidonius (Posidonius was Cicero's teacher. He flourished in the latter half of the second century. B.C. ) is about half the correct figure. On the whole, their picture of the solar system was not so very far from the truth.

## Greek astronomy facts

Greek astronomy was geometrical, not dynamic. The ancients thought of the motions of the heavenly bodies as uniform and circular, or compounded of circular motions. They had not the conception of force. There were spheres which moved as a whole, and on which the various heavenly bodies were fixed. With Newton and gravitation a new point of view, less geometrical, was introduced. It is curious to observe that there is a reversion to the geometrical point of view in Einstein's General Theory of Relativity, from which the conception of force, in the Newtonian sense, has been banished.

The problem for the astronomer is this: given the apparent motions of the heavenly bodies on the celestial sphere, to introduce, by hypothesis, a third co-ordinate, depth, in such a way as to make the description of the phenomena as simple as possible. The merit of the Copernican hypothesis is not truth, but simplicity; in view of the relativity of motion, no question of truth is involved. The Greeks, in their search for hypotheses which would "save the phenomena," were in effect, though not altogether in intention, tackling the problem in the scientifically correct way. A comparison with their predecessors, and with their successors until Copernicus, must convince every student of their truly astonishing genius.

Two very great men, Archimedes and Apollonius, in the third century B.C., complete the list of first-class Greek mathematicians. Archimedes was a friend, probably a cousin, of the king of Syracuse, and was killed when that city was captured by the Romans in 212 B.C. Apollonius, from his youth, lived at Alexandria. Archimedes was not only a mathematician, but also a physicist and student of hydrostatics. Apollonius is chiefly noted for his work on conic sections. I shall say no more about them, as they came too late to influence philosophy.

After these two men, though respectable work continued to be done in Alexandria, the great age was ended. Under the Roman domination, the Greeks lost the self-confidence that belongs to political liberty, and in losing it acquired a paralysing respect for their predecessors. The Roman soldier who killed Archimedes was a symbol of the death of original thought that Rome caused throughout the Hellenic world.

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