Chapter 27 Chapter 24 Mathematics and Astronomy in Early Greece

What I am going to discuss in this chapter is mathematics, not for its own sake, but because of its relationship to Greek philosophy—a very close relationship (especially in Plato's thought).The excellence of the Greeks is more evident in mathematics and astronomy than in anything else.The achievements of the Greeks in art, literature, and philosophy may be judged according to personal taste; but their achievements in geometry are beyond doubt.They got something from Egypt, and very little from Babylon; and what they got from these sources was mainly crude experience in mathematics, and a very long record of observations in astronomy.Mathematical proof methods are almost entirely Greek in origin.

There are many very interesting stories—perhaps not historically true—that show which practical problems stimulated the study of mathematics.The earliest and simplest story is about Thales. It is said that when he was in Egypt, the king asked him to find out the height of a pyramid.He waited until the sun showed that the length of his own shadow was equal to his height, and then he measured the shadow of the pyramid; this shadow was of course equal to the height of the pyramid.It is said that the laws of perspective were first studied by the geometer Agatacus in order to set the scene for the drama of Ischylus.It is said that the problem of finding the distance of a ship at sea was studied by Thales, and it was solved correctly at a very early stage.One of the great concerns of Greek geometry, that of doubling a cube, is said to have originated with the priests in a certain temple; That one is twice as big.At first they only thought of doubling the size of the original image, but then they realized that the result would be eight times larger, which would be far more expensive than God required.So they sent a messenger to Plato to ask if there was anyone in his academy who could solve the problem.Geometers have taken up this problem, worked on it for many centuries, and produced a lot of amazing results incidentally.This problem is of course also the problem of finding the cube root of 2.

The square root of 2, the first irrational number yet to be discovered, was known to the early Pythagoreans and discovered ingenious ways to approximate it.The best method is as follows: Suppose there are two columns of numbers, which we call column a and column b; each column starts from 1, and a in each step is formed by adding the last a and b that have been obtained; The next b is made up of twice the previous a plus the previous b.The initial 6 pairs obtained in this way are (1, 1), (2, 3), (5, 7), (12, 17), (29, 41), (70, 99).In each pair of numbers, 2a2-b2 is either 1 or -1.Then b/a is roughly the square root of 2, and gets closer and closer with each step.For example, the reader will be satisfied to find that the square of 99B70 is very nearly equal to 2.

Procrusus described Pythagoras - always a rather shadowy figure - as the first to treat geometry as an art.Many authorities, including Sir Thomas Heath, believe that Watagoras may have discovered the theorem that bears his name; the theorem that in a right triangle the square of the chord is equal to two The sum of the squares of the sides.In any case, this theorem was known to the Pythagoreans at a very early time.They also knew that the sum of the interior angles of a triangle equals two right angles. In addition to the square root of 2, other irrational numbers have also been studied in special cases by Diodorus, a contemporary of Socrates, and in a more general way by Tyodorus, a contemporary of Plato and a little earlier. Ated studied.Democritus wrote a treatise on irrational numbers, but we don't know much about its contents.Plato was deeply interested in this subject; he mentioned the works of Diodorus and Theaetetus in the dialogue entitled "Theaetetus".In the Laws he says that the common ignorance of the subject is disgraceful, and implies that he himself came to know it very late.It certainly has an important relation to Pythagorean philosophy.

One of the most important consequences of the discovery of irrational numbers was the invention of the geometric theory of proportion by Eudoxo (c. 408-355 BC).Before him, there was only an arithmetic theory of proportions.According to this theory, if a times d is equal to b times c, then a is to b as c is to d.This definition, when there is no geometric theory of irrational numbers, can only be applied to rational numbers.Eudoxo, however, proposes a new definition free from such limitations, constructed in a way that suggests modern methods of analysis.This theory is developed in Euclid's book and has great logical beauty.

Eudoxo also invented or completed the "method of exhaustion", which was later used very successfully by Archimedes.This method is a kind of anticipation of calculus.For example, we can take the area of ​​a circle as an example.You can inscribe a circle to make a regular hexagon, or a regular dodecagon, or a regular polygon of a thousand or a million sides.Such a polygon, no matter how many sides it has, has an area proportional to the square of the diameter of the circle.The more sides this polygon has, the closer it is to being equal to a circle.You can prove that if you can make the polygon have enough sides, the difference between its area and the area of ​​the circle can be made smaller than any prespecified area, no matter how small.For this purpose, the "Axioms of Archimedes" are invoked.This axiom (after somewhat simplification) says: Suppose there are two quantities, and if you divide the larger one in half, divide the half in half, and so on, you will end up with a quantity that is smaller than the original quantity. The smaller of the two quantities of .In other words, if a is greater than b, there must be some integer n that makes 2n times b greater than a.

The exhaustive method can sometimes give exact results, such as Archimedes did to find the area of ​​a parabola; sometimes it can only get continuous approximations, such as when we try to find the area of ​​a circle.The problem of finding the area of ​​a circle is also a problem of determining the ratio of the circumference to the diameter, this ratio is called pi;.Archimedes used the approximate value of 22/7 in the calculation. He made the inscribed and circumscribed regular 96 polygons, thus proving that pi is less than 3 and 1/7 and greater than 3 and 10/71.The method can be continued to any desired degree of approximation, and that is as far as any method can do on this problem.The use of inscribed and circumscribed polygons to find an approximation of pi should go back to Antiphon, a contemporary of Socrates.

Euclid - when I was young it was the only recognized textbook of geometry for schoolchildren - lived in Alexandria around 300 BC, a few years after the deaths of Alexander and Aristotle port.Most of his "Elementary Geometry" is not his original idea, but the order of propositions and logical structure are mostly his.The more one studies geometry, the more admirable they are.His treatment of the character line with the famous character theorem has the double advantage; the deduction is powerful without hiding the dubiousness of the original assumption.The theory of proportion was inherited from Eudoxo, using methods essentially similar to those of analytical mathematics introduced by Weierstrass to the nineteenth century, thus avoiding the difficulties concerning irrational numbers.Euclid then transitioned to a kind of geometric algebra, and in Book Ten he dealt with the subject of irrational numbers.After this he proceeds to solid geometry, concluding with the problem of constructing regular polyhedra, which was solved by Theaetetus and mentioned in Plato's Timaeus.

Euclid's Elements is undoubtedly one of the greatest works of all time, one of the most perfect monuments of Greek reason.Of course he also had typical Greek limitations: his method was purely deductive, and there was nothing in it to test the underlying assumptions.These assumptions were considered by him to be unquestionable, but in the nineteenth century, non-Euclidean geometry indicated that some of them could be wrong, and only observation could determine whether they were wrong. Euclidean geometry despises practical value, which has long been taught by Plato.It is said that after hearing a proof, a student asked what good geometry could do, so Euclid called in a slave and said, "Go and give this young man three cents, because he must learn from what he has learned." get the good out of things.” And yet the contempt for the practical is pragmatically justified.No one in Greek times could have imagined that conics were of any use; it was not until the seventeenth century that Galileo discovered that projectiles move along parabolas, and Kepler that planets move in ellipses.Thus, the work done by the Greeks out of pure love for theory suddenly became a key to solving tactics and astronomy.

The Roman mind was too practical to appreciate Euclid; the first Roman to mention Euclid was Cicero, in whose time there was probably no Latin translation of Euclid; and in There is indeed no record of any Latin translation before Boiseus (c. 480 AD).Arabs can appreciate Euclid better; about 760 AD, the Byzantine emperor gave a copy of Euclid to the Muslim Caliph; about 800 AD, when Harun al-Rashid reigned At that time, Euclid had an Arabic translation.The earliest known Latin translation is Adelard of Bath's translation from Arabic in AD 1120.From this time onwards the study of geometry was gradually revived in the West; but it was not until the late Renaissance that important advances were made.

I come now to astronomy, in which the achievements of the Greeks are as remarkable as in geometry.Before Greece, the observations of the Babylonians and Egyptians over many centuries had laid a foundation.They recorded the apparent motion of the planets, but they didn't know that the morning and evening stars were one.Babylonia, and probably Egypt, had discovered the cycles of eclipses, which made it possible to predict lunar eclipses with considerable reliability, but not solar eclipses; for solar eclipses were not always visible from the same spot .Dividing a right angle into ninety degrees, and one degree into sixty points, we also got it from the Babylonians; the Babylonians liked the number sixty, and even had a numbering system based on sexages.The Greeks always liked to attribute the wisdom of their ancestors to the travels in Egypt, but before the Greeks, very little had been achieved.Yet Thales' prediction of a lunar eclipse is an example of foreign influence; we have no reason to suppose that he added anything new to what he learned from Egypt and Babylon, and that his predictions were confirmed, It's also a purely lucky coincidence. Let's look first at some of the earliest discoveries and correct hypotheses of the Greeks.Anaximander believed that the earth was floating and had nothing to support it.Aristotle was always against the best hypotheses of his day, so he refuted Anaximander's theory that the earth was at the center and would never move because it had no reason to move in one direction. No movement in the other direction.Aristotle said that if this were valid, a man who stood at the center of a circle and was filled with food at every point on the circumference would starve to death, since there would be no reason to choose which part food without choosing another part of the food.This argument reappears in scholasticism, but not in connection with astronomy, but with free will.It reappears in the form of "Bridden's Donkey", who starved to death because he couldn't choose between two bales of hay that were equally spaced to the left and right. Pythagoras was most likely the first to think that the earth was spherical, but his reasons (we must assume) were aesthetic rather than scientific.However, a scientific reason was discovered before long.Anaxagoras discovered that the moon shines by reflecting light, and made a correct theory of lunar eclipses.He himself still believed that the earth was flat, but the shape of the earth's shadow during a lunar eclipse gave the Pythagoreans the final conclusive argument in favor of a spherical earth.They went a step further and saw the earth as one of the planets.They knew—from Pythagoras himself, it is said—that the morning and evening stars were one and the same, and they thought that all stars, including the earth, moved in circles, but not around Instead, the sun surrounds a "central fire".They had discovered that the moon was always facing the earth with the same face, and they thought the earth was always facing the "central fire" with the same face.The Mediterranean area is on the opposite side from the central fire, so the central fire is never seen.The fire in the center is called the "House of Zeus" or "Mother of the Gods".The sun shines by reflecting the fire in the center.Besides the Earth there is another body, called the anti-Earth, at an equal distance from the central fire.For this they have two reasons; one scientific, the other derived from their mathematical mysticism.The scientific reason is that they correctly observed that lunar eclipses sometimes occur when the sun and moon are above the horizon.The cause of this phenomenon was refraction, which they did not know, and they thought that in this case the eclipse must be due to the shadow of another body other than the earth.Another reason is that the sun, the moon, the five stars, the earth and the anti-earth, and the fire in the center constitute ten celestial bodies, and ten is a mysterious number of Pythagoreans.This Pythagorean doctrine is attributed to Philaulos, a Theban who lived towards the end of the fifth century BC.Although this theory is fantastic and in some parts very unscientific, it is very important because it contains most of the imagination necessary to conceive the Copernican hypothesis.Thinking of the earth not as the center of the universe but as one of the planets, not as eternally fixed but as traveling in space, this shows a remarkable liberation from anthropocentrism.Once the natural image of man in the universe has been shaken in this way, it will not be difficult to lead it to a more correct theory through scientific arguments. There are many observations that contribute to this point.The inclination of the ecliptic was discovered by Emobides a little later than Anaxagoras.It soon became clear that the sun was much larger than the earth, and this fact supported those who denied that the earth was the center of the universe.Fire and Anti-Earth at the center, abandoned by the Pythagoreans shortly after Plato's time.Heraclides of Pontus (who lived around 388-315 BC, contemporaneous with Aristotle) ​​discovered that both Venus and Mercury revolved around the sun, and took The insight that one's own axis turns around.This insight is a very important step no one has taken before.Heraclidus belonged to the Platonic school, and must have been a great figure, but he was not respected as much as we might expect; he is described as a fat dandy. Aristotle of Samoa lived about 310-230 B.C., and was therefore about twenty-five years older than Archimedes; he is the most interesting of all the ancient astronomers, because he presented a complete The Copernican hypothesis that all planets, including the Earth, orbit the Sun in circles, and that the Earth rotates once on its own axis every 24 hours.But the only surviving work of Aristotle, "On the Size and Distance of the Sun and Moon", still sticks to the view of the center of the earth, which is a bit disappointing.Indeed, as far as the book is concerned, it does not make any difference whichever theory he adopts; so he may have thought that by throwing an unnecessary objection to the general opinion of astronomers, It would have been unwise to burden his calculations; or he may have arrived at the Copernican hypothesis only after writing this book.Sir Thomas Heath, in his book on Aristotle (which includes the original text and translation), is inclined to the latter view.But in either case, the evidence for the fact that Aristotle ever suggested a Copernican point of view is overwhelmingly conclusive. The first and best proof is that of Archimedes, who we have already said was a younger contemporary of Aristarchus.In his letter to Glenn, King of Syracuse, Aristotle wrote "a book containing certain hypotheses"; , the earth revolves around the sun along a circle, with the sun in the middle of the orbit".A passage in Plutarch's book mentions that Creander "thought it was the Greeks' duty to punish Aristarchus for impiety, because he set the furnace of the universe (that is, the earth) in motion. It seems that this is the result of his attempt to simplify the phenomenon by assuming that the sky is stationary and the earth revolves along an oblique circle and rotates around its own axis at the same time."Creander was a contemporary of Aristotle and died about 232 BC.In another passage, Plutarch also said that Aristotle proposed this view only as a hypothesis, but Aristotle's successor Seleucus regarded it as a certainty. Views. (The heyday of Seleucus was about 150 BC.) Isius and Sexto Empiricus also said that Aristarchus proposed the heliocentric theory, but they did not say that he proposed This theory is only a hypothesis.Even if he did make such a formulation, it is likely that he, like Galileo two thousand years later, was caused by the fear of offending the influence of religious prejudice-the attitude of Creander we mentioned above is that There is good reason for this fear. The Copernican hypothesis, advanced (either formally or tentatively) by Aristarchus, was explicitly accepted by Seleucus, but not by any other ancient accepted by astronomers.This general opposition is largely due to Hipparchus, who flourished in 161-126 BC.Heath described Hipparchus as "the greatest astronomer of antiquity".Hipparchus was the first to systematically discuss trigonometry; he discovered the precession; he calculated the length of the lunar moon with an error of no more than one second; Calculations of distances; he cataloged 850 fixed stars, noting their latitude and longitude.In order to oppose Aristotle's sun-centered hypothesis, he adopted and improved the theory of epicycles created by Apollonius (the peak period was about 220 BC); this theory was later developed and named after Ptolemy. It is famous for its system, which is based on the name of the astronomer Ptolemy who flourished in the second century AD. Copernicus happened to know, though not much, of some of the almost forgotten hypotheses of Aristarchus; and he was encouraged that his originality had found an ancient authority.Otherwise, the impact of this hypothesis on future generations of astronomy will actually be zero. The methods ancient astronomers used to calculate the sizes of the Earth, the sun, and the moon, and the distances between them, were theoretically valid, but they were hampered by a lack of precise instruments.When you think about it, many of their results are truly amazing.Eratosthenes calculated the diameter of the earth to be 7,850 miles, which is only fifty miles short of the actual diameter.Ptolemy calculated the average distance of the Moon to be 29 1/2 times the Earth's diameter; the correct figure is about 30.2 times.None of them came anywhere near the size and distance of the Sun, and they all underestimated it too low.Their estimate, expressed in terms of the diameter of the Earth, is Aristocrat is 180 times, Hibagu is 1,245 times, Poseidon is 6,546 times; The correct number is 11,726 times.We can see that these calculations are constantly improving (however, only Ptolemy's calculation shows a regression); Posidonian's calculation is about half of the correct figure.In general their picture of the solar system is not too far from the truth. Greek astronomy was geometric rather than dynamic.The ancients thought of the motion of celestial bodies as constant circular motion, or the compound of circular motion.They have no concept of force.The celestial sphere is in motion as a whole, and all kinds of celestial bodies are fixed on the celestial sphere.By the time of Newton and the theory of gravity, a new, less geometric view was introduced.Oddly enough, we see a return to geometry in Einstein's Universal Theory of Relativity, where the concept of force in the Newtonian sense has been discarded again. The problem for astronomers is: given the apparent motion of celestial bodies on the celestial sphere, how can they use hypotheses to introduce the third coordinate, that is, depth, so as to describe the phenomenon as simply as possible.The advantage of the Copernican hypothesis lies not in its authenticity but in its simplicity; from the perspective of the relativity of motion, no question of authenticity arises.The Greeks were chasing hypotheses that would "simplify phenomena," which in fact approached the problem in a scientifically correct way, though not entirely intentionally.Just a comparison of their predecessors and their descendants (up to Copernicus) is enough to convince everyone of their truly astonishing genius.Two other very great figures, Archimedes and Apollonis in the third century BC, complete this list of the leading Greek mathematicians.Archimedes, a friend and possibly cousin of the king of Syracuse, was killed when the Romans captured the city in 212 BC.Apollonius lived in Alexandria from his youth.Archimedes was not only a mathematician, but also a physicist and hydrostaticist.Apollonius is primarily known for his work on conics.I will say no more about these two men, since they appeared too late to have had any influence on philosophy. After these two men, though admirable work continued at Alexandria, the great age was over.Under the Romans the Greeks lost that self-confidence which came with political liberty, and in losing it they developed a callous respect for their predecessors.The killing of Archimedes by the Roman army is a symbol of Rome's strangulation of creative thought throughout the Hellenistic world.