Ancient Greek Astronomy and Mathematics

Ancient Greek Astronomy and Mathematics In this chapter, my focus is not on mathematics per se, but rather on mathematics in connection to Greek philosophy, a relationship that was particularly intimate in Plato. More than anything else, mathematics and astronomy demonstrate the Greeks' supremacy. What they accomplished in geometry is indisputable, but what they produced in art, literature, and philosophy may be rated as better or worse depending on personal preference. They learned a little from Egypt and a little less from Babylonia; but, the majority of what they learned from these sources was in the fields of mathematics and astronomy, where they found long-term records of observations. Nearly all mathematical demonstration techniques have Greek roots. Numerous amusing anecdotes exist, most of which are probably not historical, that illustrate what real-world issues sparked mathematical research. The oldest and most basic one concerns Thales, who was requested to determine the height of a pyramid by the pharaoh when he was in Egypt. He measured the pyramid's shadow, which naturally matched its height, after waiting for the moment of day when his shadow was as long as he was tall. It is supposed that the geometer Agatharcus first studied the principles of perspective in order to paint the landscape for Aeschylus' plays. It is reported that Thales researched the challenge of determining a ship's distance at sea, which was successfully resolved early on. The story goes that the priests of a certain temple were instructed by an oracle that the deity desired a statue twice as huge as the one they possessed, which led to the beginning of one of the major issues that plagued Greek geometers: the duplication of the cube. Initially, they considered just doubling the statue's overall proportions, but they soon discovered that this would result in an eightfold increase in size, exceeding the god's requested budget. at order to find out if anyone at the Academy might help them with their issue, they dispatched a deputation to Plato. The geometers took up the task and labored on it for centuries, resulting in a great deal of noteworthy work. Naturally, finding the cube root of two is the issue. The early Pythagoreans were aware of the square root of 2, the first irrational to be found, and they devised clever ways to approximate its value. The most effective method was to create two numerical columns, referred to as the a's and b's, with a starting value of 1. At every step, the next an is created by adding the last a and b that have previously been acquired; the next b is created by adding the previous a and b twice. These six pairings are (1,1), (2, 3), (5, 7), (12, 17), (29, 41), and (70, 99) like the first six pairs. 2a 2 -b 2 is either 1 or -1 in each combination. As a result, b/a approaches the square root of two with every new step. The reader can assure himself, for example, that the square of 99/70 is extremely close to 2. Proclus characterizes Pythagoras—always a rather enigmatic figure—as the pioneer who introduced mathematics into the liberal arts curriculum. Numerous scholars, Sir Thomas Heath among them, * think that he most likely developed the theorem that bears his name, which states that the square on the side opposite the right angle of a right-angled triangle equals the sum of the squares on the other two sides. Regardless, the Pythagoreans were aware of this theorem from a very early time. They were also aware that a triangle's total angles equals its two right angles. Theodorus, a Socrates contemporary, examined irrationals other than the square root of two in specific instances, while Theaetetus, a considerably older but roughly contemporary of Plato, investigated irrationals more broadly. Though not much is known about it, Democritus authored a dissertation on irrationals. The work of Theodorus and Theaetetus is mentioned by Plato in the discourse named after the latter, demonstrating his deep interest in the topic. In the Laws (819–820), he remarks that it is shameful that there is so widespread misunderstanding on the issue and suggests that he himself only learned about it much later in life. Naturally, it had a significant impact on the Pythagorean school of thought. The development of Eudoxus' geometrical theory of proportion (c. 408–355) was one of the most significant outcomes of the identification of irrationals. There was just the mathematical theory of proportion available to him. This theory states that if a times d equals b times c, then the ratio of a to b is equal to the ratio of c to d. This definition only applies to rationals in the absence of an arithmetical theory of irrationals. Eudoxus, on the other hand, provided a new definition that was unaffected by this limitation and was structured in a way that alluded to contemporary analytical techniques. The hypothesis has a great deal of logical elegance and is explored in Euclid. Additionally, Eudoxus developed or devised the "method of exhaustion," which Archimedes later applied to great effect. This approach foreshadows the integral calculus. Consider the topic of what a circle's area is. A regular polygon with a thousand or a million sides, a regular hexagon, or a regular dodecagon can all be inscribed on a circle. No matter how many sides a polygon has, its area is always proportionate to the square of the circle's diameter. The polygon approaches circular equality more closely the more sides it has. You may demonstrate that, given enough sides in the polygon, its area can be made to deviate from the circle's area by less than any previously specified area, no matter how little. The “axiom of Archimedes” is applied in this context. This says, to put it simply, that if you first halve the larger of two amounts, and then you halve the half, and so on, until you reach a quantity that is less than the smaller of the initial two values. Put otherwise, if an is greater than b, then 2n times b is more than a for some whole number n. Sometimes, as in Archimedes' case of squaring the parabola, the exhaustion approach yields an accurate solution; other times, like in the case of the endeavor to square the circle, it can only result in consecutive approximations. Finding the ratio, or π, between a circle's diameter and circumference is known as the "squaring the circle" issue. Using the approximation 22/7, Archimedes showed that π is higher than 3 10/71 and less than 3 1/7 by circumscribing and inscribing a regular polygon of 96 sides. That's all any approach can accomplish in this problem—it may be used to whatever necessary degree of approximation. Approximations to π using inscribed and circumscribed polygons date back to Antiphon, a Socrates contemporary. During my childhood, Euclid was the only recognized geometry textbook for boys. He lived in Alexandria circa 300 B.C., following the demise of Aristotle and Alexander. Although the majority of his Elements were not original, he was mostly responsible for the logical framework and propositional order. These are viewed as more commendable the more geometry is studied. Utilizing the well-known premise of parallels to analyze parallels has the dual benefits of rigorous deduction and exposing the questionable nature of the initial assumption. The theory of proportion, which is based on Eudoxus, overcomes all the problems associated with irrationals by using techniques that are basically the same as those that Weierstrass brought to nineteenth-century analysis. After that, Euclid moves on to a form of geometric algebra and discusses irrationals in Book X. He then moves on to solid geometry, concluding with the creation of regular solids, which Theaetetus had mastered and which Plato had assumed in his Timaeus. One of the best books ever written and one of the most exquisite examples of Greek intellectualism is unquestionably Euclid's Elements. Naturally, it has the usual constraints of the Greek language: the approach is entirely deductive and does not allow for the verification of the underlying hypotheses. Although these presumptions were assumed to be indisputable, non-Euclidean geometry of the nineteenth century demonstrated that they may be partially incorrect and that the only way to know for sure would be via observation. Euclid possesses the disdain for usefulness that Plato had ingrained in him. After seeing a demonstration, a student reportedly asked Euclid what he would gain from learning geometry. Euclid responded, calling the student a slave, "Give the young man threepence, since he must needs make a gain out of what he learns." Despite this, his disdain for practice was rationalized. In the Greek era, conic sections were not seen to be useful at all. It wasn't until the seventeenth century that Galileo and Kepler discovered that projectiles move in parabolas and ellipses, respectively. All of a sudden, the work that the Greeks had done out of a pure love of theory was crucial to astronomy and combat. The Romans were too pragmatic to value Euclid; Cicero was the first to acknowledge him, and it's likely that no Latin translation existed during his time. In fact, no Latin translation exists before Boethius (c. A.D. 480). The Arabs were more grateful: in the year or so A.D., the Byzantine emperor gave a copy to the caliph. 760, and about the year 760 A.D., Harun al Rashid translated it into Arabic. 800. Athelhard of Bath translated from the Arabic into the earliest Latin text that is still in existence in the year AD. I 120. After then, geometry was gradually revitalized in the West, though significant progress was not achieved until the late Renaissance. I now turn to astronomy, where Greek contributions were just as noteworthy as those in geometry. Centuries of observation among the Egyptians and Babylonians before them had created the groundwork. The planets' apparent movements were known, but it was unknown if the morning and evening stars were the same. A pattern of eclipses had been found, most likely in Egypt and Babylonia, which made it possible to anticipate lunar eclipses with some degree of accuracy, but not solar eclipses, which were not always seen at a certain location. The Babylonians are responsible for the division of the right angle into ninety degrees and the degree into sixty minutes. They also developed a numeration system based on the number sixty. The Greeks like to credit their ancestors' knowledge to their journeys in Egypt, although prior to the Greeks, not much had been accomplished. Nonetheless, Thales' eclipse forecast was an example of foreign influence; there is little reason to believe he added anything to what he already knew from Babylonian or Egyptian sources, and the fact that his prediction came true was merely a matter of chance. Let's start with a few of the most accurate theories and early discoveries. According to Anaximander, the earth is free to float and needs no external support. Anaximander's hypothesis that the earth, being in its center, remained immobile because there was no reason for it to move in one direction over another was met with opposition by Aristotle, * who was known for rejecting the greatest theories of his day. He asserted that if this were true, a man positioned in the middle of a circle with food arranged at different locations around it would starve to death as he would have no motivation to select one dish over another. This argument recurs in Scholastic philosophy, this time about free will rather than astronomy. It resurfaces as "Buridan's ass," which perished from starvation after being unable to decide between two hay bundles positioned equally to the right and left. Most likely, Pythagoras was the first to propose that the earth is spherical, but his reasoning was probably more artistic than scientific. But shortly, scientific explanations were discovered. Anaxagoras provided the correct explanation for eclipses after discovering that the moon glows due to reflected light. Although he continued to believe that the globe was flat, the Pythagoreans' persuasive arguments for its spherical shape came from the shape of the earth's shadow during lunar eclipses. They even went so far as to consider Earth to be a planet. They had discovered that the moon always turned the same face toward the earth, and they believed that the earth always turned the same face toward the "central fire." The Mediterranean regions were on the side turned away from the central fire, which was always invisible. They also believed that all the planets, including the earth, moved in circles, not around the sun but round the "central fire." They are said to have learned this information from Pythagoras himself. The sun was meant to shine through light reflected from the center fire, which was referred to as "the house of Zeus," or "the Mother of the gods." At the same distance from the center fire as the earth, there existed another body—the counter-earth. They had two explanations for this: one was scientific, and the other came from their mathematical mysticism. The accurate observation that a lunar eclipse may occasionally happen while the sun and moon are both above the horizon provided the scientific justification. They were unaware of the phenomenon known as refraction, and they assumed that in these kinds of situations, the eclipse was caused by the shadow of a body other than the earth. The other reason was that ten was the mystic number of the Pythagoreans, and the sun, moon, five planets, earth, counterearth, and center fire made up the 10 heavenly bodies. The person who is credited with this Pythagorean notion is Philolaus, a Theban who lived towards the close of the fifth century B.C. It is highly significant despite being fantastical and somewhat nonscientific since it represents the majority of the creative work needed to develop the Copernican idea. It demonstrated a remarkable liberation from anthropocentric thought to think of the earth not as the center of the cosmos but rather as one of the planets, traveling through space rather than being permanently fixed. Once men's innate understanding of the cosmos had been shaken, it was not too difficult to be persuaded to adopt a more correct hypothesis by scientific reasoning. Several observations contributed to this. Slightly after Anaxagoras, Oenopides discovered the obliquity of the ecliptic. Those who denied that the earth is the center of the universe were reinforced by the fact that it was quickly evident that the sun must be far larger than the earth. Shortly after Plato's time, the Pythagoreans abandoned the center fire and the counter-earth. After seeing that Venus and Mercury revolved around the sun, Heraclides of Pontus— whose dates range from around 388 to 315 B.C., making him a contemporary of Aristotle—adopted the theory that the earth revolves on its own axis once every twenty-four hours. This final action, which no one before had done, was crucial. Heraclides, who attended Plato's school, must have been a wonderful man, yet he did not receive the respect one would have thought because he was portrayed as a chubby dandy. The most fascinating of all ancient astronomers is Aristarchus of Samos, who lived roughly from 310 to 230 B.C. and was thus roughly 25 years older than Archimedes. He advanced the entire Copemiccn hypothesis, which holds that all planets, including the earth, revolve in circles around the sun and that the earth rotates once every twenty-four hours on its axis. The fact that Aristarchus's sole surviving work, On the Sizes and Distances of the Sun and the Moon, supports the geocentric theory is a bit disheartening. Although it is true that the theories that are adopted have no bearing on the problems this book addresses, he might have believed it foolish to burden his calculations with needless opposition to the consensus of astronomers, or he might have discovered the Copernican hypothesis only after writing this book. In his work on Aristarchus, * which includes a translation of the book's text, Sir Thomas Heath leans toward the latter opinion. In any event, the evidence supporting Aristarchus's suggestion of the Copernican viewpoint is quite convincing. The earliest and most reliable evidence comes from Archimedes, who was Aristarchus's younger contemporary, as we have shown. In a letter to Gelon, the King of Syracuse, he describes how Aristarchus published "a book consisting of certain hypotheses," adding, "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." Cleanthes, who was Aristarchus' contemporary, believed that it was the Greeks' duty to indict Aristarchus of Samos for allegedly putting the Hearth of the Universe (i.e., the earth), as a result of his attempt to preserve the phenomena by supposing that the heavens remain at rest and the earth revolves in an oblique circle, while it rotates, at the same time, about its own axis. Plutarch states in a different text that Aristarchus only put forward this theory as a hypothesis, but Seleucus, his successor, upheld it as a firm belief. Selucas reached its peak about 150 B.C. The heliocentric theory was also pushed by Aristarchus, according to Aerius and Sextus Empiricus, although they do not state that he only proposed it as a hypothesis. Even if he did, it seems likely that he was swayed by the fear of upsetting religious preconceptions, just as Galileo two millennia later. This worry appears to have been well-founded, based on Cleanthes's (above) attitude. After been put out by Aristarchus, whether positively or tentatively, Seleucus unquestionably accepted the Copernican idea, but no other ancient astronomer did. The main cause of this widespread rejection was Hipparchus, who lived from 161 to 126 B.C. * 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. Heath refers to him as "the greatest astronomer of antiquity." In contrast to Aristarchus' heliocentric theory, he adopted and refined Apollonius' theory of epicycles, which dates back to around 220 B.C. This theory's evolution became known as the Ptolemaic system, named for the astronomer Ptolemy, who lived in the middle of the second century A.D. Copernicus was delighted to discover historical support for his innovation and learned a little something about Aristarchus's nearly forgotten idea. Otherwise, this concept had virtually no impact on later astronomy. Although the absence of precise tools hindered ancient astronomers, their techniques for determining the sizes of the earth, moon, and sun, as well as their distances from them, were theoretically sound. Considering this deficiency, several of their outcomes were unexpectedly strong. The earth's diameter is only around fifty miles smaller than Eratosthenes' estimate of 7850 miles. Ptolemy calculated the moon's mean distance to be 29.2 times the diameter of the earth; however, the actual value is closer to 30.2. They all miscalculated the sun's size and distance, and none of them came close. Aristarchus estimated the earth's diameter to be 180, Hipparchus, 1245, and Posidonius, 6545. The number 11,726 is accurate. As we shall see, these approximations improved steadily with time (Ptolemy's estimate, however, displayed a regression); Posidonius*'s estimate is around half off. Overall, their depiction of the solar system was not too off from reality. Greek astronomy was not dynamic, but geometrical. The celestial bodies' movements were perceived by the ancients as being round and consistent, or as a combination of circular motions. They did not understand the concept of force. The several celestial bodies were fastened on spheres that moved collectively. Newton and gravity introduced a new, less geometrical perspective. The idea of force in the Newtonian sense has been expelled from Einstein's General Theory of Relativity, and it is interesting to note that this viewpoint has returned. The challenge facing astronomers is to hypothesise the introduction of a third coordinate, depth, in order to simplify the explanation of the phenomena in light of the apparent movements of the celestial bodies on the celestial sphere. Because of the relativity of motion, the Copernican hypothesis is superior to other theories in terms of simplicity rather than truth. Though not directly, the Greeks were approaching the issue in the most scientifically sound manner as they looked for theories that would "save the phenomena." Every learner must be convinced of their genuinely astounding talent by comparing them to their predecessors and then to their successors up to Copernicus. The elite group of Greek mathematicians is rounded out by two very notable individuals from the third century B.C., Archimedes and Apollonius. Archimedes was slain in 212 B.C. when the Romans took control of Syracuse, where he was a friend and perhaps a relative of the king. Apollonius had resided in Alexandria since his early years. In addition to being a mathematician, Archimedes also studied hydrostatics and physics. The main contribution of Apollonius is the study of conic sections. I won't talk about them any more because they arrived too late to have an impact on philosophy. The magnificent age came to an end with these two individuals, despite the fact that Alexandria continued to produce excellent art. Under Roman rule, the Greeks lost the self-assurance that comes with political freedom and, as a result, developed a crippling reverence for their forebears. The Roman soldier who murdered Archimedes served as a metaphor for Rome's contribution to the decline of creative ideas in the Hellenic world.