Ancient Philosophy

Socrates, Plato, and Aristotle

Early Greek Mathematics and Astronomy

In this chapter, I shall address mathematics not as an isolated discipline but in its intricate connection with Greek philosophy—a connection particularly pronounced in the works of Plato. The Greeks displayed their superiority in mathematics and astronomy more distinctly than in any other field. While their achievements in art, literature, and philosophy may be appreciated to varying degrees based on individual taste, their contributions to geometry stand beyond dispute. Some elements were inherited from Egypt, and a lesser amount from Babylon; in mathematics, they primarily acquired simple techniques from these sources, while their astronomical knowledge was grounded in long-standing observational records. The art of mathematical proof is almost entirely of Greek origin.

Many intriguing accounts (likely apocryphal) exist regarding the practical problems that spurred mathematical inquiries. The earliest and simplest tale pertains to Thales, who was once asked by the king of Egypt to calculate the height of a pyramid. Thales waited for a moment in the day when his shadow matched his height, then measured the shadow of the pyramid, which, of course, equated to its height. It is said that the laws of perspective were first studied by the geometer Agatharchus, who sought to create backdrops for Aeschylus's plays. The task of determining the distance to a ship at sea, which Thales reportedly examined, was already solved in ancient times. One of the significant challenges engaging Greek geometers was the problem of doubling the volume of a cube. This issue allegedly arose among the priests of a temple, who were informed by an oracle that their god desired a statue twice the size of the one they possessed. Initially, they considered simply doubling the dimensions of the statue, but soon realized that this would yield a statue eight times larger than the original, incurring expenses far exceeding the divine request. They then dispatched a delegation to Plato, hoping that someone from the Academy might resolve their conundrum. Geometers engaged with the problem, laboring for centuries and producing numerous remarkable works along the way. This challenge ultimately boils down to extracting the cube root of 2.

The square root of 2—the first of the discovered irrational numbers—was known to early Pythagoreans, who devised clever methods for approximating its value. The best methods involved creating two columns of numbers, which we shall denote as a and b; each column begins with one. Each subsequent a is formed by summing the last a and b obtained. The following b is derived by adding twice the previous a to the preceding b. This generates the first six pairs: (1, 1), (2, 3), (5, 7), (12, 17), (29, 41), (70, 99). For each pair, the expression 2 * a² — b² yields either 1 or —1. Thus, b/a approximates the square root of 2, approaching √2 with each new step. For instance, the reader may find that (99/70)² is nearly equal to 2.

Pythagoras, whose persona remains somewhat enigmatic, was named by Proclus as the first to integrate geometry into general education. Many authorities, including Thomas Heath, believe that Pythagoras may indeed have discovered the theorem bearing his name; this theorem states that in a right triangle, the square of the side opposite the right angle is equal to the sum of the squares of the other two sides. In any case, this theorem was known to the Pythagoreans long ago. They also understood that the sum of the angles of a triangle equals two right angles.

Irrational numbers, apart from the square root of 2, were examined in isolated instances by Theodorus, a contemporary of Socrates, and more broadly by Theaetetus, who lived around Plato's time or perhaps slightly earlier. Democritus wrote a treatise on irrational numbers, but little is known about its content. Plato expressed a deep interest in this issue; he references the works of Theodorus and Theaetetus in a dialogue named after the latter. In the Laws, he states that general ignorance in this domain is shameful and implies that he himself learned of it at a relatively late age. The discovery of irrational numbers undoubtedly held great significance for Pythagorean philosophy.

One of the most important consequences of the discovery of irrational numbers was the creation of a geometric theory of proportion by Eudoxus (circa 408—355 BCE). Prior to him, only an arithmetic theory of proportion existed. According to this theory, the ratio of a to b equals the ratio of c to d if a, taken d times, equals b, taken c times. This definition, in the absence of an arithmetic theory for irrational numbers, could only apply to rational numbers. However, Eudoxus provided a new definition that transcended this limitation, resembling the methods of modern mathematical analysis. This theory was further developed by Euclid and exhibits considerable logical elegance.

Eudoxus also invented or refined the "method of exhaustion," which was later successfully employed by Archimedes. This method foreshadows integral calculus. Consider, for example, the question of the area of a circle. You may inscribe a regular hexagon or a regular dodecagon, or a regular polygon with thousands or millions of sides within the circle. The area of such a polygon, regardless of the number of sides it possesses, is proportional to the square of the diameter of the circle. The more sides the polygon has, the closer it approaches the circle. It can be proven that if the polygon has a sufficiently large number of sides, the difference between its area and that of the circle will be less than any predetermined quantity, no matter how small. This requires the use of Archimedes' axiom, which states (if somewhat simplified) that if the greater of two magnitudes is halved, and then that half is halved again, and so forth, after a finite number of steps, a magnitude will be reached that is less than the lesser of the two initial magnitudes. In other words, if a is greater than b, there exists a whole number n such that 2ⁿ * b will be greater than a.

The method of exhaustion sometimes yields an exact result, as in the problem of squaring the parabola, which Archimedes solved; at other times, as in the attempt to compute the squaring of the circle, it may lead only to successive approximations. The problem of squaring the circle involves determining the ratio of the circumference of a circle to its diameter, known as "π." Archimedes, in his calculations, used the approximation 22/7; by inscribing and circumscribing a regular polygon with 96 sides, he demonstrated that "π" is less than 3 1/7 and greater than 3 10/71. This method can achieve any required degree of approximation, and this is all that any method can accomplish in resolving this problem. The use of inscribed and circumscribed polygons to approximate "π" dates back to Antiphon, a contemporary of Socrates.

Euclid, whose works were still the sole recognized geometry textbook for students during my youth, lived in Alexandria around 300 BCE, shortly after the deaths of Alexander the Great and Aristotle. Much of his Elements was not original, but the order of theorems and the logical structure were largely his own. The more one studies geometry, the more remarkable they seem. The interpretation of parallels through the famous parallel postulate possesses a dual merit: the deduction is rigorous and yet the questionable nature of the initial assumption is not concealed. The theory of proportion (the triple rule) followed by Eudoxus circumvents all the difficulties associated with irrational numbers through methods fundamentally akin to those introduced in mathematical analysis by Weierstrass in the 19th century. Euclid then transitions to a kind of geometric algebra and treats irrational numbers in Book X. Subsequently, he considers spatial geometry, concluding with the construction of regular polyhedra, which was refined by Theaetetus and embraced in Plato's Timaeus.

The "Elements" of Euclid is undoubtedly one of the greatest works ever written, a consummate testament to the intellect of ancient Greece. Of course, this book also bears the hallmarks of typically Greek limitations: its method is purely deductive, devoid of any means to verify its foundational assumptions. These assumptions were regarded as incontrovertible; however, the advent of non-Euclidean geometry in the nineteenth century revealed that some of them might have been erroneous, and only observation could determine their validity.

Euclid held the practical utility espoused by Plato in disdain. It is said that when a student, after hearing a proof, inquired about the benefits of studying geometry, Euclid summoned a slave and declared, "Give the young man a penny, for he must certainly profit from what he studies." Yet, this contempt for practicality was pragmatically justified. In the days of the Greeks, no one envisioned that the study of conic sections would yield any benefit; yet, by the seventeenth century, Galileo discovered that projectiles travel along parabolic paths, and Kepler established that planets move in elliptical orbits. The work undertaken by the Greeks out of pure love for theory unexpectedly became the key to waging war and advancing astronomy.

The Romans were far too pragmatic to properly appreciate Euclid; the first among them to mention him was Cicero, during a time when a Latin translation of Euclid’s works likely did not exist. In fact, there is no written evidence of a Latin translation prior to Boethius (480 AD). The Arabs, however, valued him more highly: a copy of Euclid’s works was gifted to the caliph by the Byzantine emperor around 760 AD, and during the reign of Harun al-Rashid, around 800 AD, a translation into Arabic was made. The first surviving Latin translation from Arabic was completed by Adelard of Bath in 1120 AD. From that point on, the study of geometry gradually revived in the West; yet it was only in the late Renaissance that significant progress was made in this field.

Now I turn to astronomy, where the Greeks achieved remarkable advancements akin to those in geometry. Long before them, the Babylonians and Egyptians laid the foundations of astronomy through centuries of observation. They recorded the visible movements of the planets, yet were unaware that the morning star and the evening star were one and the same. In Babylon, and perhaps in Egypt, the cycle of eclipses was discerned, which enabled reasonably accurate predictions of lunar eclipses (though not solar ones, as they were not always visible from any given location). The Babylonians gifted us the division of a right angle into ninety degrees and a degree into sixty minutes; they were fond of the number sixty, which they even used as a basis for their counting system. The Greeks enjoyed attributing the wisdom of their early pioneers to journeys in Egypt, but in truth, very little was achieved before their time. However, the prediction of a solar eclipse by Thales serves as an example of foreign influence; there is no reason to believe that he added anything to what he learned from Egyptian and Babylonian sources, and it was sheer luck that his prediction came to pass.

Let us begin with some of the earliest discoveries and valid hypotheses. Anaximander proposed that the Earth floats freely, unsupported by anything. Aristotle, who frequently dismissed the best hypotheses of his time, countered Anaximander's theory, which posited that the Earth, being at the center, remains stationary because it has no reason to move in one direction rather than another. If this were correct, he argued, a person placed at the center of a circle, with food located at various points on its circumference, would starve due to the lack of a reason to choose one food over another. This argument resurfaces in scholastic philosophy, though not in connection with astronomy, but rather in discussions about free will. It takes the form of the tale of "Buridan's Ass," who, unable to choose between two equal bundles of hay placed equidistantly to his left and right, perished of hunger.

Pythagoras is likely the first to have contemplated the spherical nature of the Earth, but his reasoning presumably belonged more to the realm of aesthetics than to science. Nevertheless, scientific arguments soon emerged. Anaxagoras discovered that the Moon shines with reflected light and formulated a correct theory of eclipses. Although he himself still believed the Earth to be flat, the shape of the Earth's shadow during lunar eclipses provided the Pythagoreans with conclusive evidence that the Earth is spherical. They advanced further, considering the Earth as one of the planets. They recognized (reportedly from Pythagoras himself) that the morning star and the evening star are the same, and believed that all planets, including the Earth, move in circles—not around the Sun, but around a "central fire." They observed that the Moon always presents the same face to the Earth, and posited that the Earth is always oriented with one face toward the "central fire." The Mediterranean regions are constantly turned away from the "central fire," making it eternally invisible to them. The "central fire" was referred to as "the home of Zeus" or "the Mother of the Gods." It was assumed that the Sun shines with light reflected from the "central fire." In addition to the Earth, there was another body, the counter-Earth, situated at the same distance from the "central fire." They based this on two grounds: one scientific and the other stemming from their arithmetic mysticism. The scientific basis was the correct observation that a lunar eclipse sometimes occurs when both the Sun and the Moon are above the horizon. They were unaware of the refraction of light, which causes this phenomenon, and believed that in such cases, the eclipse must be caused by the shadow of some other body, not the Earth. The second basis was that the Sun, the Moon, the five planets, the Earth, the counter-Earth, and the "central fire" comprised ten celestial bodies, and ten was a mystical number for the Pythagoreans.

This Pythagorean theory is attributed to Philolaus, a Theban who lived at the end of the fifth century BC. Although it is unrealistic and to some extent entirely unscientific, it is significant because it encompasses much of the imaginative effort required to give birth to Copernican hypothesis. To begin to conceive of the Earth not as the center of the universe, but as one of the planets—not as permanently fixed in one place, but as wandering through space—is evidence of an extraordinary liberation from anthropocentric thinking. When the established conceptions of humanity's place in the universe were challenged, it became less difficult to arrive at a more accurate theory through scientific arguments.

Various observations facilitated this shift. Eudoxus, who lived shortly after Anaxagoras, discovered the inclination of the ecliptic. It soon became apparent that the Sun must be many times larger than the Earth; this fact reinforced the views of those who denied that the Earth is the center of the universe. The theories of the "central fire" and the counter-Earth were discarded by the Pythagoreans shortly after Plato's time. Heraclides of Pontus (who lived approximately from 388 to 315 BC, a contemporary of Aristotle) discovered that Venus and Mercury revolve around the Sun and adopted the view that the Earth completes a full rotation on its axis every twenty-four hours. This discovery was a significant step forward, one that none of his predecessors had made. Heraclides was a follower of Plato's school and must have been a remarkable individual, yet he did not receive the respect one might have expected; he is described as a portly dandy.

Aristarchus of Samos, who lived from around 310 to 230 B.C.E., thus being roughly twenty-five years older than Archimedes, stands as the most intriguing of all ancient astronomers for having posited a hypothesis entirely akin to Copernicus's: that all planets, including the Earth, revolve in circular orbits around the Sun, while the Earth completes a rotation on its axis every twenty-four hours. It is somewhat disappointing that the only surviving work of Aristarchus, On the Distances of the Sun and Moon, is framed from a geocentric perspective. It is true that for the problems discussed in this book, the specific theory adopted is of little consequence; perhaps he thought it unwise to enter into unnecessary conflicts with the prevailing astronomical views in his calculations, or he may have arrived at a hypothesis similar to Copernicus's only after the writing of this book. Thomas Heath, in his study of Aristarchus, which includes the text of this work with translation, leans toward the latter conjecture. In any case, the evidence that Aristarchus proposed a viewpoint akin to Copernicus's is quite persuasive.

The earliest and most credible testimony comes from Archimedes, who, as we have seen, was a younger contemporary of Aristarchus. In a letter to the Syracusan king Hieron, he relays that Aristarchus published “a book consisting of certain hypotheses” and goes on to state: “His hypotheses are such that the stars are immovable and the Sun remains fixed, while the Earth revolves around the Sun in a circular path, the Sun being at the center of the orbit.” Cleanthes, it is said in one passage of Plutarch, “believed it was the duty of the Greeks to accuse Aristarchus of impiety for having set in motion the Hearth of the Universe (that is, the Earth), an outcome of his attempt to ’save appearances’ by proposing that the heavens remain at rest while the Earth moves in an inclined circle and simultaneously rotates about its own axis.” Cleanthes was a contemporary of Aristarchus and died around 232 B.C.E. In another excerpt from Plutarch, it is noted that Aristarchus presented this view merely as a hypothesis, while his follower Seleucus endorsed it as a definitive viewpoint (the flourishing of Seleucus's work being around 150 B.C.E.). Aetius and Sextus Empiricus also assert that Aristarchus advanced the heliocentric hypothesis, though they do not specify that it was merely a hypothesis. Yet even if he did so, it seems highly probable that he, like Galileo two millennia later, succumbed to the fear of offending religious prejudices (a fear, as indicated by the position of the aforementioned Cleanthes, that was well-founded).

The hypothesis akin to that of Copernicus, once articulated by Aristarchus—be it positively or as an attempt—was ultimately embraced by Seleucus, but no other ancient astronomer followed suit. This general dismissal was largely attributable to Hipparchus, who lived from 161 to 126 B.C.E. He is characterized by Heath as “the greatest astronomer of antiquity.” He was the first to systematically address trigonometric issues, discovered the precession of the equinoxes, calculated the duration of the lunar month with an error of less than one second, improved Aristarchus's estimates of the sizes of the Moon and the Sun and the distances to them, and compiled a catalog of eight hundred and fifty fixed stars, indicating their latitudes and longitudes. In apparent contradiction to the heliocentric hypothesis of Aristarchus, he adopted and enhanced the theory of epicycles devised by Apollonius, whose work dates to around 220 B.C.E. This theory, in its subsequent development, became known as the Ptolemaic system (named after the astronomer Ptolemy, who lived in the mid-second century C.E.).

Copernicus gleaned some, albeit little, from the nearly forgotten hypothesis of Aristarchus and was pleased to find an ancient authority to support his innovation. Moreover, the impact of this hypothesis on the subsequent development of astronomy was virtually negligible.

Ancient astronomers, in determining the sizes of the Earth, Moon, and Sun, as well as the distances to the Moon and Sun, employed theoretically sound methods, yet they lacked precise measuring instruments. Many of the results they achieved were, considering this shortcoming, remarkably accurate. Eratosthenes estimated the diameter of the Earth at 7,850 miles, yielding an error of approximately only 50 miles. Ptolemy calculated that the average distance to the Moon was 29.5 times the diameter of the Earth (the correct figure being around 30.2). None of them could approach an accurate calculation of the size of the Sun or its distance; all underestimated this distance. According to their calculations, it was as follows:

• Aristarchus: 180
• Hipparchus: 1,245
• Posidonius: 6,545 Earth diameters.

The accurate figure is 11,726 Earth diameters. Subsequently, these calculations were continually revised (although, in Ptolemy’s case, the errors in calculations increased; Posidonius's estimation is about half the correct figure). Overall, however, these astronomers' conceptions of the solar system were not so far from the truth.

Greek astronomy was geometric rather than dynamic. The ancients envisioned the motion of celestial bodies as uniform and circular or composed of circular movements. They had no conception of force. There were spheres moving as a whole, on which various fixed celestial bodies resided. With the advent of Newton and his law of gravitation, a new perspective was introduced, one less geometric in nature. Interestingly, there is a return to the geometric perspective in Einstein's general theory of relativity, which expels the concept of force in the Newtonian sense.

The problem for the astronomer is this: to introduce a third coordinate—depth—based on the visible movements of celestial bodies according to the hypothesis, thus simplifying the description of phenomena as much as possible. The essence of the Copernican hypothesis lies not in its truth but in its simplicity; in connection with the relativity of motion, the question of truth is hardly raised. The Greeks, in their quest for hypotheses that would “save appearances,” were, in fact, albeit not entirely intentionally, grappling with this problem in a scientifically sound manner. A comparison of their work with that of their predecessors and successors prior to Copernicus should convince all researchers of their truly astonishing genius.

Two great figures—Archimedes and Apollonius—in the third century B.C.E. complete the roster of first-rate Greek mathematicians. Archimedes was a friend, and possibly a cousin, of the king of Syracuse, and he met his end when the Romans captured the city in 212 B.C.E. Apollonius lived in Alexandria from his youth. Archimedes was not only a mathematician but also a physicist who studied hydrostatics. Apollonius is chiefly renowned for his work on conic sections. I will confine my examination to these contributions, for they lived in an era too late to significantly influence philosophy.

After these two figures, although significant work continued in Alexandria, the great age came to a close. Under Roman dominion, the Greeks lost that self-confidence inherent to political freedom, and having lost it, they acquired a “paralyzing” respect for their predecessors. The Roman soldier who killed Archimedes symbolized the demise of original thought brought upon the entire Hellenistic world by Roman rule.