Early Mathematics and Astronomy

chapter B2 Early Mat h e mat i c s and Ast ronomy Leonid Zhmud There is no generally agreed starting point for the history of Greek mathematics and astronomy. hose scholars who prefer dealing with the fully preserved works begin with Euclid’s Elements and Autolycus’ of Pitane On the Moving Sphere, written ca 300 bce, when Greek mathemata—geometry, arithmetic, astronomy. and harmonics—were already fully formed. An awareness that scientiic methods and theories known from the works of Euclid and Autolycus are not exactly their own methods and theories, but very oten originate from the 4th, the 5th, and even the 6th centuries, leads other scholars to search for the earliest written text in mathemata. Such a text is represented by a long fragment from the writing of Hippocrates of Chios (ca 440/30 bce) on the squaring of lunes (moon-shaped areas between circular arcs). his takes us almost 150 years back, to a period when Greek mathematicians and astronomers systematically started to reveal their theories in writing and arrange previous discoveries. Indeed, Hippocrates was the author of the irst Elements (Euclid’s Elements were the fourth such work), where geometrical theorems were systematically expounded in a deductive though not yet entirely axiomatic way. he irst systematic work in astronomy was written most probably by Hippocrates’ compatriot Oenopides of Chios (ca 450 bce). Archytas of Tarentum, a generation younger than Hippocrates, was the author of the irst writings speciically on arithmetic and harmonics known to us. 1. Eudemus: The Milesians of the 6th century bce Hippocrates’ fragment came to us as a quotation from the History of Geometry by Eudemus of Rhodes (ca 330 bce), a student of Aristotle and the author of the irst

histories of science. As well as the History of Geometry, he wrote the History of Astronomy, beginning these sciences with Thales of Miletus (fl. ca 585 bce)— the famous Sage, whom Aristotle regarded as the founder of natural philosophy. hales wrote nothing; the other famous mathematician of the 6th century bce, Pythagoras of Samos, also let nothing in writing. One has to concede that to write the history of preEuclidean mathematics on the basis of contemporaneous texts is impossible: there are no such texts for the time from hales to Hippocrates (and almost none for that from Hippocrates to Euclid). What is known about the earliest period of Greek mathemata amounts very oten to the fragmentary evidence provided by Eudemus; in some cases it can be augmented by the independent testimonia but never by a preserved though fragmentary text. Nevertheless, if one does not want to overlook the century and a half preceding Oenopides and Hippocrates, a period in which geometry and astronomy came into being and took shape, the best thing to do is to follow Eudemus’ reports, while subjecting them to critical scrutiny (Zhmud 2006; cf. Netz 2004). Most modern histories of Greek science bear some important features inherent in Eudemus’ histories. One of them consists in regarding the ancient Orient as a source of Greek mathematics and astronomy. Geometry, says Eudemus, was discovered by the Egyptians as a result of the practical needs of land surveying, and arithmetic, in turn, was discovered by the Phoenicians, who were employed in trade. hales, irst having traveled to Egypt, brought geometry to Greece; he discovered much himself and instructed his successors in the principles of the other things (fr. 133 W.). he Egyptian origin of geometry is already attested in Herodotus (2.109), and ater him in Aristotle (Metaphysics A 1.981b23), meaning that Eudemus simply relected the widespread egyptophilia of the Greeks, especially in their approach to the past. he prestige of Egyptian geometry was so great that the gited mathematician Democritus boasted that nobody excelled him in the construction of lines with proofs, even the Egyptian “rope stretchers,” that is, land surveyors (DK 68 B 299). Ater more than a century’s investigation of Egyptian mathematics, however, there is no basis to assume the presence in it of anything resembling theory or proof. It is more probable that in the Archaic period the Greeks borrowed from Egypt practical knowledge needed for land surveying, building, and the like, the more so as early Greek architecture and sculpture bear obvious traces of Egyptian inluence. All available evidence on Egyptian borrowings relates to practical mathematics, moreover to arithmetic rather than geometry. hus, late scholia to Plato’s Charmides (163e) refer to Egyptian methods of multiplication and division and also to operations with fractions (Heath 1921, 14, 41, 52). As the most conventional histories of science still do, Eudemus focused in his works on speciic discoveries in mathemata and their authors, the “irst discoverers.” His list of hales’ discoveries runs as follows. hales: (1) was the irst to prove that the diameter divides the circle into two equal parts (Euclid 1.def.17); (2) was the irst to learn and state that the angles at the base of any isosceles triangle are equal (1.5), calling them, in the archaic manner, similar, not equal; (3) was the irst to discover that if two straight lines intersect, the vertical angles are equal (1.15); and (4) knew the theorem about the equality of the triangles that have one side and two angles equal (1.26), which he must have used

to determine the distances of ships from the shore. Obviously, hales’ theorems of angles and triangles cannot have originated in Egyptian geometry, since the Egyptians neither engaged in comparing the size of angles nor the similarity of triangles. In Egyptian and Babylonian mathematics there was no notion of the angle as a measurable magnitude. As Kurt von Fritz (1971, 568 n. 79) observed, “All theorems ascribed to hales are either directly related to the problems of symmetry and can be ‘demonstrated’ by the method of superposition, or such that the irst step of the demonstration is evidently based on considerations of symmetry while the second, which brings the argument to conclusion, is simply an addition or subtraction.” Indeed, hales’ propositions can be reduced to the symmetries of the so-called halesian basic igure (Becker 1966, 37), that is, a rectangle with the diagonals inscribed in a circle, the center of which is on the intersection of the diagonals (see igure B2.1). hales appealed in his demonstrations to the visualizability of the geometrical drawing but certainly went beyond this. Aristotle (Analytica Priora 41b13–22) refers to an archaic-looking proof of a theorem that the angles at the base of any isosceles triangle are equal (Euclid, 1.5), which might well go back to hales (Heath 1926, 1:252–253; Becker 1966, 38–39). It is based on the equality of mixed angles, in particular angles in a semicircle and angles of a segment of a circle, which could be proved by using only the superposition method. he proof in Aristotle can be re-established in igure B2.2. ABC is an isosceles triangle with its vertex in the center of the circle. Prove that its base angles are equal. ∠ 1 is equal to ∠ 2, since they are angles of a semicircle; ∠ 3 is equal to ∠ 4, since they are angles of a segment of a circle. Taking equal angles from equal an-gles, we obtain that angles CAB and ACB are equal. Thus, the proof demonstrates the normal procedure of deductive reasoning. he idea of proof is vital for the history of Greek mathematics, for this is what both distinguishes it from the earlier mathematical cultures, like Egypt and Babylon, and makes it akin to modern mathematics. (Recent history of mathematical proof, Chemla 2012, sheds new light on the methods of proving the correctness of algorithms and computations in the East Asian cultures but does not change the traditional view on the Greek origin of deductive proof.) he systematic application of deductive proof was Figure B2.1 halesian basic igure. Drawing by W. Sinelnikow based on O. Becker, Das mathematische Denken in der Antike. Göttingen, 1966.

A 1 3 4 2 C Figure B2.2 Aristotle’s proof of equality of base angles in isosceles triangle. Drawing by W. Sinelnikow based on O. Becker, Das mathematische Denken in der Antike. Göttingen, 1966. the most important factor in the formation of theoretical mathematics on. an axiomatic basis; this led to the formulation of theorems valid for any numbers, and consequently ousted the empirical, computational methods from mathematical science. Further, it stimulated the search for the axiomatic bases of mathematical theory, since deductive constructions in order to be true and noncontradictory must of necessity rest on initial propositions accepted without proof. Some scholars believe that deductive proof appeared at the very beginning of Greek geometry; others insist that mathematics developed empirically until the early 5th century bce, whereas deductive proof was borrowed from the Eleatic philosophy or was gradually developed in geometry itself. he problem with the extra- mathematical origin of the deductive proof is that in philosophy it does not possess the logical cogency and irrefutability that it does in mathematics (cf. section 2). Yet the intra-mathematical origin of the deductive method is also not without problems, insofar as this method is not something inherent in dealing with numbers and igures: for thousands of years mathematics developed without it in the ancient Orient, including India and China. Could mathematics of the practical and computational kind, as it existed in archaic Greece, give rise of itself to a striving for strict proof? Hardly: hales in geometry and Pythagoras in arithmetic began by proving things of no practical use that were also too simple to be demonstrations of technical virtuosity. (Høyrup [1994] regards the demonstration of technical virtuosity as one of the chief stimuli in the development by Babylonian scribes of increasingly complex types of calculation.) If mathematics did not of itself give rise to deductive proof, or adopt it from outside, then, most probably, it came into being in mathematics under the inluence of external impulses. As distinct from Babylonian and Egyptian scribes versed in computation, hales was not a professional: he was a wealthy and politically inluential aristocrat. Why did he decide to prove that the angles at the base of an isosceles triangle are equal? And why did he achieve public recognition in this pursuit? Two centuries ater hales’ birth an Athenian audience knew him as a famous geometer (Aristophanes, Birds, 1009; Clouds, 180), which would be impossible if their attitudes to fame and geometry did not

partially overlap. he problem is more general than hales’ geometry, it relates to how Greek science was born and what distinguishes it from similar pursuits in other ancient cultures. A comparison of Greek and Chinese intellectual traditions, ofered by G. E. R. Lloyd, emphasizes a conspicuous feature of Greek science: its highly competitive character, which relects, in turn, an agonistic character of Greek society and culture revealed by Jacob Burckhardt (1898–1902). “he competitiveness of Greek intellectual life” was the decisive factor in the formation of Greek science and, in particular, axiomaticodeductive mathematics (Lloyd 2004, 133, 140, 144). his spirit of pure competition arose in Greek agonistics and then spread to areas of intellectual creativity, multiplying tenfold the force of those striving for truth (Zaicev 1994). A second important factor was that, in the Greece of the 8th to 5th centuries, for the irst time in human history, all aspects of productive cultural activity, including those lacking a direct utilitarian purpose, gained public approval. he social climate of the time encouraged any and all creative achievements, independent of the extent of their practical value, thus establishing the most powerful stimuli for new investigations. Once set on the path of free research, unconstrained by narrow practicality and corporative ethos, the mathematicians quickly realized that to apply strict, logical proof makes it possible in this pursuit to achieve irrefutable and hence universally recognized results (Zaicev 1994, 167). “hales seems by some accounts to have been the irst to study astronomy, the irst to predict eclipses of the sun, so (says) Eudemus in his History of Astronomy” (fr. 144). he prediction of the solar eclipse was the most famous “discovery” made by hales, and it was relected in many early sources, among them in his younger contemporary Xenophanes (DK 21 B 19). Successful prediction captured the imagination of the Greeks and made hales the “father of astronomy,” but what is meant by “prediction,” and how can it be explained? hales could not have had a theory ofering a correct explanation of solar eclipses—such a theory only appeared in the mid-5th century bce. Since Greek tradition before hales does not know of any predictions of eclipses, the very idea could only have been of Babylonian origin. In the early 6th century bce, Babylonian astronomy was the only one capable of making predictions that concerned all potential lunar and solar eclipses for a given year, without trying to explain them. Until the mid20th century, the predominant opinion was that hales’ prediction could have relied on the so-called Saros, a period of 223 synodic months (≈18 years), used by the Babylonians to predict lunar and solar eclipses. Later it became known that the Babylonians were unable to reliably predict solar eclipses for a given point either in the 6th century or later (Neugebauer 1957, 142–143). It is quite probable, however, that hales having known about one of the Babylonian schemes, boldly used it to ix the date of the next solar eclipse and thus by lucky coincidence “predicted” the eclipse of May 25, 585 bce, almost full in Miletus. hales’ prediction, no matter how famous it was, let no traces in Greek astronomy, which later on was concerned with explanations of the celestial phenomena, including eclipses, not their predictions. Another discovery of hales, “that the sun’s period with respect to the solstices is not always the same” ( Eud. fr. 145 W.), had a more durable efect. It seems that hales tried to estimate the solstices’ dates and, hence, the length of the solar

year more accurately than known before. Such an activity became a part of the “calendaric astronomy” that tried to ind the best ratio between the solar year and the lunar month for the luni-solar calendar. he results of these investigations, however, were never applied to the civic calendar, remaining a matter for astronomers and philosophers. In the 6th to early 5th centuries among them were the astronomers Cleostratus of Tenedos, Harpalus and Matricetas, and the philosophers Xenophanes and Heraclitus. Cleostratus, in particular, suggested the irst intercalation period for a luni-solar calendar, octaëteris (eight years), which presumed the year to be 365¼ days long; it was further improved by Harpalus and others engaged in astronomical observations (DK 6 A 1, B 4). In contrast to Babylonian astronomy that was in principle ageometric, the most important stream of Greek astronomy was the creation of geometrical models representing and explaining the apparent motion of the heavenly bodies. It started with Anaximander of Miletus (l. ca 570 bce), the irst Greek thinker who revealed his theories in writing; he also created the irst geographical map of the earth. he system of Anaximander was a peculiar combination of bold speculations, geometrical and spatial imagination, and astronomical observations (he used the gnomon to determine the solstices and equinoxes and set up a sundial in Sparta: DK 12 B 1). he earth in this system has the shape of a column’s drum, and its depth is a third of its width. he earth is freely suspended, supported by nothing, and is aloat in the center of the cosmos. It is this counterintuitive idea, unprecedented in the preceding astronomy, that became a cornerstone of the speciically Greek conception of the universe (Couprie 2011, 99). Anaximander imagined further that the earth is enclosed by three wheels, which consist of thick air (and thus are invisible) and are full of ire: the sun, the moon, and the stars being the holes in the wheels. he sun is the same size as the earth (another revolutionary insight!), and the wheel of sun is the highest of all, followed by the moon and stars; distances from the earth to the stars, the moon, and the sun are equal to 9, 18, and 27 radii of the earth. Solar and lunar eclipses occur when the openings in the rim of the wheel are stopped, which means the moon shines with its own light. hough Anaximander’s model of the cosmos was not yet purely geometrical, but also physical, with time this physical component receded into the background, whereas geometry became the basis of what Aristotle called “mathematical astronomy.” he other conspicuous features of this system, instrumental in shaping Greek astronomy, are that it is devoid of any divine presence and inluence, which are typical, for example, of Babylonian and Chinese astronomy, and that it rests on the assumption of a concealed order of the world that can be revealed both geometrically and numerically. Pythagoras and his school shared this assumption. In a new epistemological situation in the 5th century, the idea of the invisible things and/or regularities in nature was expressed in the pregnant dictum of Anaxagoras of Clazomenae (ca 500‒ca 428 bce): “appearances are a sight of the unseen” (DK 59 B 21a). Developing this line of thought, Eudoxus of Cnidus (ca 390‒ca 337 bce) put forward the principle of “saving (preserving) the appearances” that was to underlie the whole subsequent history of Greek astronomy: to explain the apparently irregular movement of the sun, moon, and planets along the ecliptic by attributing uniform circular movement to them.

he circular motion of the sun and moon around the earth, most probably by analogy with the visible circular motion of the stars around the North Pole, was postulated already by Anaximander’s system, though his wheels related rather to the diurnal motion of the two luminaries than to their motion along the ecliptic. Unlike Anaximander, his student Anaximenes of Miletus (l. ca 550 bce) introduced no new geometrical concepts; he only “moved” the stars beyond the moon, sun, and planets, to the outer place, where they have a ixed position on the celestial vault (DK 13 A 14). As a separate group, distinct from the ixed stars, the planets (“wandering stars”) appeared for the irst time in Anaximenes, but he did not say anything speciic about them. On the whole, the Greeks learned about planets rather late and slowly. In the 6th century, they had no ixed names for them, with the exception of Venus, which was called the Evening and Morning Star, depending on the time of its appearance. hat it is the same planet was irst attested, according to heophrastus, in the learned poem of Parmenides (ca 475 bce). he word “planet” and the ixed number of the planets appear in the late 5th century, and still later, in the mid-4th century, their names, borrowed from Babylon: the stars of Hermes, Aphrodite, Ares, Zeus, and Kronos. he sun and moon were also regarded as planets, for they, too, have independent movement along the ecliptic. 2. The Contributions of the Pythagoreans In the late 6th century the Ionian tradition of geometry and astronomy was transferred to Magna Graecia by Pythagoras, who circa 530 bce let his native Samos because of Polycrates’ tyranny and moved to Croton. Pythagoras taught metempsychosis, and many of his ethical rules were supported by belief in his god-like nature. he dual nature of this igure was attested by Aristotle: “Pythagoras, the son of Mnesarchus, irst dedicated himself to the study of mathemata, especially numbers, but later could not refrain from the wonder-working of Pherecydes” (Arist. fr. 191). his combination of the rational and the religious is not unique among the pre-Socratics: the natural philosopher Empedocles pretended to be a wonder-worker and was a proponent of metempsychosis. However peculiar Pythagoras’ personality was, in mathemata he continued the work of hales and Anaximander, and none of the Pythagoreans known to us by name are linked with anything remotely supernatural or miraculous. he Pythagorean school existed until the mid- 4th century bce, having in almost every generation signif- icant mathematicians and astronomers: Hippasus of Metapontum (fl. ca 500/ 490 bce), Theodorus of Cyrene (active ca 440‒ca 400 bce), Philolaus of Croton (active ca 440‒ca 400 bce), Archytas (active ca 410˗360 bce), Ecphantus of Syracuse (first part of the 4th century bce). The names of the other Pythagorean mathematical scientists re- main unknown.

Pythagoras’ contributions to astronomy are hard to discern, partly because the late antique tradition ascribes too much to him: discovering the sphericity of the earth, the obliquity of the ecliptic, the planets’ motion along the ecliptic, dividing the celestial and terrestrial spheres into zones, and so on. Early sources are much more reticent. Even if they do connect Pythagoras with astronomy, as with Aristotle’s Protrepticus (fr. 18, 20 Düring), they do not refer to any speciic discoveries. Eudemus, in particular, mentions his followers, rather than Pythagoras: “Anaximander was the irst to ind an account of the sizes and distances (of the planets), as Eudemus says, adding that the Pythagoreans were the irst who found the order of their position” (fr. 146 W). Although Eudemus’ fragment does not indicate the number and order of the heavenly bodies, he clearly had in mind their “correct” arrangement, which was accepted in the astronomy of his time: moon—sun—Venus—Mercury—Mars—Jupiter—Saturn—celestial sphere. It was established that, relative to the stars, Mercury and Venus moved the fastest (their sidereal period was equated to that of the sun), Mars more slowly, Jupiter more slowly still, and Saturn extremely slowly. hese observations, together with data on the relative brightness of some of the planets (Venus being brighter than Mercury), formed the basis of their order. Which Pythagoreans did Eudemus mean? In Philolaus’ system, ive planets were located between the moon and the sun on one side and the stars on the other. Philolaus, however, radically transformed the order of the planets by introducing the Central Fire (Hestia), which he situated in the center of the universe, and around which he made the earth, the invisible counter-earth and all other celestial bodies revolve (DK 47 A 16–17). We should look, then, at the earlier stage of Pythagorean astronomy. Alcmaeon of Croton (ca 500/490 bce), the Pythagorean natural philosopher, thought, according to the late doxographer Aëtius, that the planets move from west to east in a direction opposite to the movement of the ixed stars (DK 24 A 4). If we believe this evidence, Alcmaeon was aware that the planets, sun and moon, apart from their diurnal movement, also have an annual movement along the ecliptic from west to east, which is to say that they rise each day further to the east in the zodiacal constellations. hough Alcmaeon was not an astronomer (his other astronomical views look rather naive), he might have gained this knowledge from the other Pythagoreans. he evidence of Aëtius implies that the motion of the planets along the ecliptic is circular, as we see later in Oenopides, Hippocrates, and Philolaus. Aristotle says that Alcmaeon taught that the soul was immortal because, like all divine celestial bodies—the sun, moon, planets, and the whole heaven—it is in constant motion (DK 24 A 12). his kind of motion also had to be circular. Transferring the circular motion from Anaximander’s model to the motion of the sun, moon, and planets along the ecliptic, the Pythagoreans must have proceeded both from observations and from considerations of symmetry as they attempted to regularize the motion of all the celestial bodies following a single principle. Since a circle was at that time the only possible method of geometrical presentation of planetary motion (only a circular motion is continuous, says Aristotle, Physics 264b9–28), the planets’ numerous deviations from circular orbits were simply ignored.

he revolution of the celestial bodies around the earth is attested in the other early Pythagorean theory (prior to Philolaus), the famous “harmony of the spheres” that was borrowed by Plato in the Republic (616b–617d) and acknowledged but refuted by Aristotle: he theory that music is produced by their (sc. planets and stars) movement, because the sounds they make are harmonious, although ingeniously and brilliantly formulated by its authors, does not contain the truth. It seems to some thinkers that bodies so great must inevitably produce a sound by their movement: even bodies on earth do so, although they are neither so great in bulk nor moving at so high a speed, and as for the sun and moon, and the stars, it is incredible that they should fail to produce a noise of surpassing loudness. Taking this as their hypothesis, and also that the speeds of the stars, judged by their distances, are in the ratio of the musical consonances, they airm that the sound of the stars as they revolve is concordant. (De caelo 290b, tr. W. Guthrie) Like Anaximander’s model, this theory has a physical component, lacking in Philolaus. here is no sound without movement, said Archytas’ Pythagorean predecessors in harmonics (DK 47 B 1); consequently, there can be no movement without sound, even though we do not hear the celestial harmony. he speed of rotation of the celestial bodies in this system is directly proportional to their distances from the earth, which, according to the late commentator Alexander (Aristotle, fr. 13 Ross), make up the arithmetical progression 1, 2, 3, 4. . . (n + 1). hus, the ratios of the distances correspond to the ratios of the basic concords: the octave (2:1), the ith (3:2), the fourth (4:3), and so on. he doctrine of heavenly harmony does not lend itself to detailed reconstruction, especially in its musical part. What is important for us is to state that it is based on Pythagoras’ discovery of a link between music and number, which led to the inclusion of harmonics in the mathemata. Late antique tradition about how Pythagoras discovered the ratios of concords, such as Nicomachus’ story about an experiment with the hammers (Harmonics, 6), is unreliable, but the discovery itself is attested by Plato’s student Xenocrates (ca 395–313 bce), who let behind numerous works on mathematical sciences: “Pythagoras discovered also that the intervals in music do not come into being apart from number, for they are an interrelation of quantity with quantity” (fr. 87 Isnardi Parente). hat Pythagoras found the numerical expressions of the octave, the ith, and the fourth is indirectly conirmed by the evidence of the famous musicologist Aristoxenus (active ca 340—ca 300 bce), a student of the last Pythagoreans and then of Aristotle. He says that Hippasus fashioned four bronze discs of the same diameter, with thickness in the ratios 2:1, 3:2 and 4:3; when struck they produced harmonic concordance (Aristox. fr. 90 W.). Hippasus of Metapontum was a student of Pythagoras, and his experiment was conducted to conirm what Pythagoras had already discovered, most likely by observations and experiments with a stringed instrument. (hough the Greeks knew no regular practice of experimentation, sporadic experiments were performed.) he ratios of the basic concords are closely bound up with arithmetic b = (a + c ) / 2

and harmonic b = 2ac/(a+c) means,, which, according to information that goes back to Eudemus, were known to Pythagoras and Hippasus (Zhmud 2006, 173– 175). Thus, the fifth (3:2) is the arithmetic mean between the terms of the octave (2:1), and the fourth (4:3) is the harmonic mean between them; taken together, they form a “musical” proportion (12:9 = 8:6). The only preserved fragment of Eudemus’ History of Arithmetic deals with the Pythagorean ratios of the three concords (fr. 142 W.). During the 5th century bce, arithmetic and harmonics as related sciences remained a monopoly of the Pythagorean school: whereas the Ionias Oenopides and Hippocrates studied only geometry and astronomy, Hippasus (DK 18 A 12– 15), Theodorus (DK 43 A 4), Philolaus (DK 44 Α 26, B 5–6), and Archytas (DK 47 A 16– 19, B 1– 2) were engaged also in two other sciences of the quadrivium. “Pythagoras more than anybody else seems to have valued the science (or theory) of numbers and to have advanced it, separating it from the merchants’ business and likening all things to numbers,” says Aristoxenus in his On Arithmetic (fr. 23). his is close to what Aristotle noted about Pythagoras’ study of numbers (fr. 191), but is more speciically related to the origin of arithmetic as a theoretical science, distinct from the art of calculation. he arithmetic known to us from the three books of Euclid’s Elements (books 7–9) is the theory of arithmoi, which is to say whole numbers greater than one, and their properties. “A unit is a beginning of a number” (and thus not a number), and “a number is a multitude consisting of units”—these deinitions from the same fragment of Aristoxenus are likely to have opened an early Pythagorean arithmetical treatise. he next deinitions introduce two basic kinds of number: even numbers are divisible into equal parts, odd number are divisible into unequal parts and have a middle. (Philolaus, following the arithmetic of his time, also mentions the division of numbers into even, odd, and even-odd: DK 44 B 5). he latter assertion indicates that the early Pythagoreans represented numbers not by line segments, as Archytas (DK 47 A 19) and later Euclid did, but by psephoi, counting stones. (Hence there is no “middle” in Euclid’s deinition of the odd number: 7.def.7). If you add or subtract a psephos to or from an even number, you get an odd number (DK 24 B 4), says a character from the comedy of the Sicilian writer Epicharmus (ca 480 bce), alluding most probably to Pythagorean arithmetic. (Practical arithmetic does not need and, thus, does not know odd and even numbers. It is Epicharmus’ fragment, where “even” and “odd” in their mathematical meaning irst occur in Greek literature, whereas the practical and computational mathematics of Mesopotamia and Egypt did not have special terms for odd and even numbers.) he simplest example of this arithmetic is a summation of odd and even numbers, represented by pebbles; such arithmetical series produce the so-called igurate numbers (igure B2.3). he added number, called the gnomon, preserves the form of that to which it is added. square number 1 + 3 + 5 + ... + (2n − 1) = n2 ; oblong number 2 + 4 + 6 + ... + 2n = n (n + 1). he presence of deinitions in the early Pythagorean arithmetic implies that it contained some deductively proved propositions. he high standard of Archytas’

Figure B2.3 Gnomon for square and oblong numbers. Drawing by W. Sinelnikow based on T.L. Heath, A History of Greek Mathematics. Oxford, 1921. arithmetical proofs (DK 47 A 19) shows that by the late 5th century bce arithmetic was established as a demonstrative science. In Archytas’ opinion, it even surpassed geometry in clarity and exactness, accomplishing proofs where geometry failed (DK 47 B 4). An early specimen of the axiomatic-deductive method in arithmetic is the theory of even and odd numbers, preserved at the very end of the last arithmetical book of the Elements (Becker 1966, 44–49). his theory, consisting of propositions 9.21–34, based only on deinitions of even and odd numbers (7.def.6–11), is of an elementary character and lacks any intrinsic connection with the material of other arithmetical books. Here are its irst ive propositions in abridged form: 21. 22. 23. 24. 25. he sum of even numbers is even. he sum of an even number of odd numbers is even. he sum of an odd number of odd numbers is odd. An even number minus an even number is even. An even number minus an odd number is odd. Becker showed that both the propositions and their proofs retained by Euclid are easily illustrated through the use of psephoi. Meanwhile, four of these propositions (9.30–31, 33–34) are proved by reductio ad absurdum, one of the powerful tools of Greek mathematics, which allows the establishment of a proposition by showing that its contradictory involves impossible consequences, for example that the same number is both even and odd. We see again how very simple mathematical problems lead to nontrivial results. It is hard to establish whether indirect proof originated in arithmetic or earlier in geometry (proposition I, 26, attributed by Eudemus to hales, is proved indirectly). Judging by the preponderance of reductio ad absurdum in the theory of even and odd, one can reasonably infer that deduction, which is to say a formal proof technique, was shaped by the early Pythagorean psephoi-arithmetic, which appealed not to the (then nonexistent) lettered diagram (cf. Netz 1999) but to pebbles arranged in such a way as to give an ocular demonstration. Further nontrivial results of Pythagorean arithmetic appeared rather quickly. First was the discovery of the irrationality of √2, the classic example of which is the incommensurability of the diagonal of a square with its side. he probable context of the discovery was the search for the ratios of the sides in the right-angled triangle

that corresponded to Pythagoras’ theorem (see the end of this section). It was found then that the side and diagonal of a square cannot be expressed as a ratio of two numbers. his t h eorem w as o ne o f A ristotle’s f avorite m athematical e xamples: referring to it more than 15 times, he twice alludes to the fact that its indirect proof relies on the theory of odd and even numbers (Analytica priora 1.23, 41а24–27, and 1.44, 50а37). It might have been that Archytas had this very proof in mind, saying that arithmetic accomplishes proofs where geometry fails (DK 47 A 4). “he analysis of certain classes of problems in geometry, e.g. the construction of irrational lines, can only be completed by means of arithmetical principles” (Knorr 1975, 311). Plato ascribes to heodorus a proof of irrationality of the magnitudes between √3 to √17 (DK 43 A 4), which means that the proof of the irrationality of √2 was found earlier. Ancient tradition, probably going back to Eudemus, attributes the discovery of irrationality to the Pythagoreans; the name of Hippasus is mentioned or implied in the legendary stories surrounding it (von Fritz 1974, 545–575; Zhmud 2012, 274–275). he ancient (though not the original) arithmetical proof of the proposition that the diagonal and side of a square are incommensurable in length is preserved at the end of book 10 of the Elements (app. 27); it makes use of the Pythagoras’ theorem, the theory of even and odd numbers, the method of reductio ad absurdum, and the least numbers in a given ratio. his all points to its Pythagorean origin. As we know from Archytas (DK 47 A 17) and Eudemus (fr. 142 W.), the early Pythagoreans took the ratios of the concords in lowest terms (2:1, 3:2, 4:3), which they called “irst numbers,” or pythmenes (base numbers). Archytas’ proof that a superparticular ratio (n + 1): n, and so the concordant intervals represented by it, for example the ith and the fourth, cannot be divided into equal parts (DK 47 A 19) and have no mean proportional (or geometric mean), also contains reductio ad absurdum and the least numbers in the same ratio. he problems evoked by the discovery of irrationality provided the impulse for the research of heodorus and his student heaetetus (discussed later), the author of the general theory of irrational magnitudes (book 10 of Euclid’s Elements) and led to the development of Eudoxus’ theory of proportions, which was applicable to commensurable and incommensurable magnitudes (book 5). In the modern literature, the impact of Hippasus’ discovery has oten been overrated. hus, it was widely believed that it was originally motivated by Pythagoras’ dogma “all is number” and then had dealt a “fatal blow” to this dogma by demonstrating the existence of incommensurable magnitudes in geometry, which in turn led to the “foundation crisis” in Greek mathematics. All three assumptions are not borne out by the reliable sources. he “foundation crisis” of the 5th century bce is a retrospective projection of what happened in mathematics at the turn of the 20th century (Knorr 2001). he motto “all is number” is unattested in ancient Pythagoreanism; it was irst ascribed to the unnamed Pythagoreans by Aristotle, who mistakenly regarded them as the predecessors of the Platonic number doctrine (Zhmud 2012, 433–452). As for general interaction between mathematics and philosophy, Greek mathematics appeared to have been independent of contemporary philosophy, whereas the latter was frequently inluenced by mathematical ideas (Knorr 1981). One of the earliest examples of such an inluence was systematic deductive reasoning, including

indirect proofs, employed by Parmenides (DK 28 B 8) and his student Zeno (DK 29 A 15, B 1–2) in attempting to prove their bold theses that contradicted all experience, for example, that there is no movement or plurality. “Parmenides’ reasoning is the extension of the Pythagorean proof . . . . Not only in mathematics, where the Pythagoreans had already developed reductio ad absurdum proofs in their exploration of quantities, but throughout nature—in philosophy, physics, everywhere—it became possible to show simply by examining their logical consequences that some generalizations cannot be true” (Brumbaugh 1981, 54–55). he Eleatics put deductive proof in a much wider context, but, in contrast to the Pythagorean mathematicians, they succeeded neither in proving any of their basic theses nor even in formulating their indirect proofs in a rigorous form. heir reductio ad absurdum proofs are formally incomplete. Two pieces of early Greek geometry—the theorem of Pythagoras and the theory of the application of areas that Eudemus deemed “ancient” and attributed to the “Pythagorean muse” (fr. 137 W.)—were from the 1930s considered derived from Babylonian mathematics. One of its rediscoverers, O. Neugebauer (1957, 40), believed to ind on the tablet Plimpton 322 (18th century bce) “the fundamental formula for the construction of triples of Pythagorean numbers,ˮ that is, positive integers (a, b, c) for which a2 + b2 = c2. he much-repeated idea that the Babylonians knew the Pythagorean theorem became a cliché, and Pythagoras was regarded as the transmitter of Babylonian knowledge (van der Waerden 1961, 92–93). Over recent decades,, the leading students of Babylonian mathematics have changed this trend. First, the Babylonians knew not the theorem, but the rule for determining the values numerically, which they did not prove or even formulate explicitly (Høyrup 1998). Secondly, a detailed examination of the tablet has shown that it has nothing to do with number-theoretical problems in general, nor with Pythagorean numbers in particular, but contains a school problem using a list of reciprocal pairs (Robson 2001). As for the Greeks, Proclus (5th century AD) in his commentary on the irst book of Euclid (In Euclid, 428.7–21) ascribes to Pythagoras the method of deining Pythagorean triples, starting from the odd number, which is based on igurate numbers (Heath 1926, 1:356). he irst author to claim that Pythagoras proved the theorem named ater him was a certain Apollodorus the Arithmetician (Diogenes Laertius 8.12), who may be identical with the Democritean Apollodorus of Cyzicus (second half of the 4th century bce); he was followed by virtually all the Greek writers who wrote about it. his evidence, though not irrefutable, is conirmed by the fact that the proof of irrationality of √2, associated with Hippasus, is based on Pythagoras’ theorem. Hippocrates already knew the generalized Pythagorean theorem for acute- and obtuse-angled triangles (2.12–13); it comes from book 2 of the Elements, which belongs to the Pythagoreans. he application of areas with excess or defect, “one of the most powerful methods on which Greek geometry reliedˮ (Heath 1926, 1:343), relates to the transformation of areas into equivalent areas of diferent shape. he propositions of this theory, comprising theorems 1.44–45, the entire book 2 of the Elements, and theorems 6.27–29, can be reformulated into algebraic identities and quadratic equations. hus, the application of areas with defect means the construction on a given line a of the rectangle ax,

so that by subtracting from it the square x2, the given square b2 is obtained (ax – x2= b2). Proposition 2.3 can be presented as the identity (a + b)a = ab + a2 and 2.4 as (a + b)2 = a2 + 2ab + b2. Since the late 19th century, these propositions have come to be known as geometric algebra and seen as a geometric reformulation of algebraic problems. When Neugebauer found in Babylonian mathematics corresponding identities and equations, he concluded that the algebra reformulated by the Greeks was Babylonian. hat he regarded his interpretation as a working hypothesis, unconirmed by documentary evidence (Neugebauer 1957, 147), did not prevent it from soon becoming the dominant theory. his theory came under attack from S. Unguru (1975), who claimed that the application of areas was not a reformulation of Babylonian algebra, but arose on Greek soil in the course of solving purely geometric problems. Ater a lengthy discussion, most historians of Greek mathematics accepted his view. “We have no good reason to believe,ˮ noted Taisbak (2003, 306), “that the Greeks were thinking of quadratic equations in any form when working with the diferent types of application of areas.ˮ Revealingly, there is no evidence of the practice of mathematics analogous to geometric algebra in Mesopotamia in the 6th‒5th centuries: all extant texts relate to the Old Babylonian period. “Old Babylonian mathematics cannot have inluenced early Greek developments: it was a part of a scribal culture that all but died out nearly a millennium before the earliest Greek literate culture, 1200 miles away” (Robson 2005, 13). Real or assumed isomorphism between two mathematical theories, formulas, or methods oten gives rise to common-origin hypotheses, but only the theories placed in a speciic historical setting with identiiable ways of transmission survive the tests. 3. The Milesians, Pythagoreans, and Athenians: Productive Interactions In the mid-5th century bce, studies of geometry and astronomy were revived in Ionia by two natives of Chios, Oenopides and Hippocrates. Before them we know only Anaxagoras, who taught that the moon received its light from the sun and ofered correct explanations for both lunar and solar eclipses (DK 59 B 8, A 76–77). On the whole, however, his astronomy was physical rather than mathematical. Oenopides, mentioned by Eudemus in both the History of Astronomy and the History of Geometry, attempted to establish closer connections between these two mathemata. According to the late evidence, he “was the irst among the Greeks who wrote down the methods of (mathematical) astronomyˮ (Boll 1894, 53–55), which is essentially conirmed by the early sources. Eudemus attributes to Oenopides two elementary geometrical constructions that later entered Euclid’s book 1: to draw a perpendicular to a given straight line from a point outside it (1.12); at a point on a given straight line, to construct a rectilinear angle equal to a given rectilinear angle (1.23). Oenopides considered problem 1.12 useful for astronomy. Proclus says the same about proposition 4.16 (this is the last proposition of book 4, which

the scholia to Euclid 273.3–13, probably on the authority of Eudemus, ascribe to the Pythagoreans), on a regular pentadecagon inscribed in the circle: its side is equal to the angle between the celestial equator and the zodiacal circle, that is, 24° (In Euclid, 283.7–10, 269.8–18). heon of Smyrna’s excerpt from Eudemus clariies the way in which it may be related to Oenopides’ astronomy: he “was the irst to discover the obliquity of the zodiacal circle” (Eud. fr. 145 W.). his can mean either that Oenopides discovered that the annual path of the sun is inclined to the celestial equator or that he irst measured the angle of the obliquity of the ecliptic (Bodnar 2006, 4–6). he latter variant seems more plausible in view of Aëtius’ evidence about Alcmaeon and the zodiacal motion of the planets (DK 24 A 4). he ecliptical motion of the sun, moon, and planets against the background of the celestial sphere is attested both in Philolaus (DK 44 A 21) and in Hippocrates (DK 42 A 5), which is hard to explain if Oenopides shortly before them discovered that the annual path of the sun is oblique. Von Fritz (1937, 2258–2259) argued convincingly that the end of heon’s excerpt from Eudemus was originally related to Oenopides: “And others discovered in addition to this that the ixed stars move round the immobile axis that passes through the poles, whereas the planets move round the axis perpendicular to the zodiac and that the axis of the ixed stars and that of the planets are separated from one another by the side of a (regular) pentadecagon” (fr. 145 W.). hough Oenopides’ astronomical system deies reconstruction, we can surmise that his work, irstly, incorporated geometrical notions of the structure of the universe developed by the Greeks from Anaximander to Anaxagoras, removing them from the cosmological context to which they belonged in the works of natural philosophers, and secondly, expounded them in conformity with the requirements of the deductive geometry of the mid-5th century. here is reciprocal inluence between the Pythagoreans and the Chians: Oenopides held the same theory of the Milky Way, as being the former course of the sun, as did the Pythagoreans (DK 41 A 10); Philolaus borrowed from him the 59-year luni-solar cycle (Eudemus fr. 145 W.; DK 44 A 22). Hippocrates shared the view of some Pythagoreans that a comet is one of the planets, visible at long intervals and rising low over the horizon (DK 42 A 5). Hippocrates’ theory as set out by Aristotle is more complex than the Pythagorean, demonstrating advanced concepts of the geometry of the universe: the celestial sphere is divided into zones by a celestial equator and two tropic circles crossed by the oblique circle of the zodiac; the planets move in circular orbits along the ecliptic; the horizon divides these circular orbits into unequal segments; and the earth, to all appearances, is spherical (Wilson 2008). he sphericity of the earth, safely attested for Philolaus, is related in the Greek tradition alternatively to Pythagoras and Parmenides (Diogenes Laertius 8.48). From what we know about their astronomy, neither appears to be a suitable candidate for this discovery; it is safer to attribute it to the Pythagorean tradition of the 5th century, though certainty is impossible. he discovery of the earth’s spherical shape led to the formation of the main astronomical model of antiquity, which consisted of two concentric spheres, the celestial and the terrestrial, divided into zones. In the generation of Philolaus and Hippocrates, this two-sphere model of the cosmos was still in the making (Philolaus’ spherical earth was not the center of the cosmos), in a more developed form we ind it in Plato’s Republic and later in his Timaeus.

An Athenian astronomer Meton (ca 430 bce) belonged probably to the same generation as Philolaus and Hippocrates. Meton and his colleague Euctemon made systematic observations in diferent regions of Greece; created the irst astronomical calendars, the so-called parapegmata; suggested a new 19-year calendar cycle; and determined the inequality of the four astronomical seasons (according to their calculations, the seasons are 90, 90, 92, and 93 days, starting with the summer solstice). Meton and Euctemon were the earliest of the Greek astronomers whose dated observations are cited by Ptolemy. By the time of Hippocrates several geometrical problems, such as squaring the circle and doubling the cube, became famous, attracting the attention of audiences far beyond a narrow circle of specialists. Aristophanes ridicules Meton for promising to square the circle (Birds 1004–1009); Plutarch describes Anaxagoras as busy in prison squaring the circle (DK 59 A 38); Aristotle and Eudemus record unsuccessful attempts by the Sophists Antiphon of Athens and Bryson of Heraclea to solve the same problem. he agonistic spirit that surrounded the problem of doubling the cube led Greek geometers to continually search for new solutions to the problem long ater it had been solved, irst by Archytas, and then by his student Eudoxus and by Eudoxus’ student Menaechmus (Knorr 1986). Eratosthenes’ dialogue Platonicus, relying on the Academic legend of Plato as the architect of mathēmata, ascribes to the latter an instrumental role in doubling the cube, but this tradition is unreliable (Zhmud 2006: 84– 86; Kouremenos 2011). he way to Archytas’ solution was paved by Hippocrates, who was the irst to reduce the problem of doubling the cube to inding two mean proportionals x and y in continuous proportion between two lines, a (side of the cube) and 2a, that is, if a: x = x: y = y: 2a, then x 3 = 2a3, x = a 3 2 . It was suggested long ago that Hippocrates came to this idea by analogy with the planimetric problem, solved by the Pythagoreans, of doubling the square, which is equivalent to the problem of inding the mean proportional x between two lines, a and 2a, x 2 = 2a2, x = a 2 (Heath 1921, 201). In turn, Archytas found a brilliant solution to the problem formulated by Hippocrates, which was reported by Eudemus (fr. 141 W.). Archytas constructed a series of similar right triangles AMI, AIK, AKD and then showed that their sides are in continued proportion, so that AM: AI = AI: AK = AK: AD, where AM was equal to the side of the original cube and AD = 2AM (igure B2.4). To prove this, he employed a remarkable stereometric construction, which for the irst time introduced movement into geometry (note that the moving point D appears twice). Point K, the key point for the construction of similar triangles, was determined as the intersection of three surfaces of revolution: the right cone, the torus, and the half-cylinder (Knorr 1986, 50–52; Hufman 2005, 342–346). he problem of squaring the circle arose in the irst part of the 5th century, ater the Pythagoreans had found how to square a rectangle (Euclid 2.14). Being equivalent to constructing a line segment whose length is √π times the radius of the circle, the problem is unsolvable using compass-and-straightedge techniques, or even algebraic equations, as was established in the late 19th century. (It does not seem, however, that in the preEuclidean period Greek mathematicians consciously restricted the means allowable to

K K M M P B A D D T I I T A AM : AI :: AI : AK :: AK : AD D E Z O Figure B2.4 Archytas’ stereometric construction, and the similar triangles whose sides are in continued proportion. Drawing by W. Sinelnikow based on C. Huffman, Archytas of Tarentum. Cambridge, 2005. P1 P2 P3 Figure B2.5 Squaring the circle by inscribed polygons. Drawing by W. Sinelnikow based on W. Knorr. The ancient tradition of geometric problems. Boston: Birkhäuser, 1986. . their constructions to compass and straightedge; see Knorr 1986, 40–41). It is unknown, whether Anaxagoras came up with a solution of the problem. he solutions of Antiphon and Bryson, says Aristotle, were “eristic” (Sophistic Refutations sec. 11, 171b16–18, 172a2– 7; Physics 1.2, 185a14–17), which is to say unscientiic, since they proceeded not from geometrical principles. Eudemus passes over Bryson in silence but speciies Antiphon’s procedure (fr. 140 W.): the latter started by inscribing a regular polygon in a circle; then, by doubling the number of its sides repeatedly, he obtained an inscribed polygon whose sides coincided with the circumference (see igure B2.5). hus, concludes Eudemus, Antiphon did not admit the basic principles of geometry, in particular, that geometrical magnitudes are ininitely divisible. his criticism, which relected a position of the mathematicians, applies to Bryson as well. We know from late sources that, squaring the circle, he added circumscribed polygons to the inscribed ones and claimed that by multiplying their sides he could obtain an intermediate polygon equal to the circle. T. L. Heath, the author of the still-standard history of Greek mathematics, believed that Antiphon’s and Bryson’s procedures anticipated the famous method of exhaustion, discovered by Eudoxus (Heath 1921, 222), but this idea did not ind much support.

Whereas Aristotle believed that Hippocrates pretended to have solved the problem of squaring the circle, but had committed a logical mistake (Sophistic Refutations sec. 11, 171b12–16; Physics 1.2, 185a14–17), Eudemus disagreed with his teacher: he quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation (of lunes) to the circle, were irst investigated by Hippocrates, and his exposition was thought to be in correct form. (Fr. 140 W., tr. T. Heath) he opinion of specialists, to which Eudemus refers, implies that though originally squaring the lunes was most probably intended to lead to squaring the circle, Hippocrates did not claim to have solved the last problem, so Aristotle’s interpretation was incorrect (Lloyd 1987). But Hippocrates succeeded in squaring three out of the ive lunes that are possible in plane geometry (two others were found in the 18th century), namely, with the outer circumference equal to a semicircle (see igure B2.6a), greater than a semicircle (see igure B2.6b), and smaller than a semicircle, the most elaborate case. He also squared a igure that consisted of a lune and a circle. In his problem-solving attempts, Hippocrates did not proceed axiomatically. hus, he started his quadrature of the lunes not from deinitions or unproved principles, but by proving two theorems: irst, similar segments of circles have the same ratio as the squares on their bases (12.2), which he then reduced to the second theorem, that the squares on the diameters have the same ratio as the circles. But Hippocrates’ Elements, (a) Γ b b’ ∆ a A B (b) b’ ∆ c’’ c’ b b’’ a c A Γ a’ E c’‘’ Figure B2.6 Two of the lunes of Hippocrates. Drawing by Paul A. Whyman.

whose goal was to organize interrelated mathematical propositions in their logical sequence, must have built them on the explicitly formulated deinitions and axioms. A papyrus text, most probably dating back to a Platonist of the 4th century bce, asserts that in Plato’s time “the theory of proportions (μετρολογία) and research on deinitions reached their peak, as Eudoxus and his students completely revised the old theory of Hippocrates” (Zhmud 2006, 87–89). Whereas Eudoxus created a new theory of proportions applicable to commensurable and incommensurable magnitudes, Hippocrates, working around 75 years before him, applied Pythagorean theory of proportions to a new ield—solid geometry—and worked out the axiomatic basis for his Elements. It is generally believed that his compendium contained much of books 1–4 and 6 of the Euclidean Elements and that most propositions of book 3 belonged to Hippocrates himself. he authorship of arithmetical books 7–9 is a tricky question. Eudemus’ History of Geometry did not touch on this, and from his History of Arithmetic only one fragment is preserved. Many scholars believed that an arithmetical compendium analogous to Hippocrates’ Elements in geometry existed before Archytas, but what did it comprise? Archytas obviously relied on the basis of book 7, which may have then belonged to heodorus, a contemporary of Hippocrates, though this is no more than conjecture; Knorr (1979, 244) attributed book 7 to heaetetus. Book 8 is usually related to Archytas; the end of book 9 to the early Pythagoreans. What is certain is that heaetetus’ theory of irrational magnitudes is based on these arithmetical books. he irst signiicant geometer who was born in Athens, heaetetus was, as mentioned, a student of heodorus and belonged, according to Eudemus (fr. 133 W.), to the generation of Archytas and Plato. his places his birth around 435/425 bce, but since Plato depicts him in the heaetetus, whose dramatic date is 399 bce, as an adolescent, his birth date is usually given as 415/413 bce. It is known, however, that Plato sometimes changed the age of his personages depending on the dramatic situation in the dialogue, so that it may be safer to stick to the dating provided by Eudemus, who was particular about chronology. heaetetus’ main achievements in mathematics, the theory of irrational lines (book 10), and the theory of the regular solids (book 13) show him as a successor of the Pythagoreans. He proved that there is an ininite number of straight lines, which are incommensurable in length or both in length and in square; and introduced three particular kinds of such lines, medial, binomial, and apotome, associating them with three known means, the geometric, the arithmetic, and the harmonic (Eudemus, fr. 141-I W.). According to a scholion on book 13 (Scholia in Euclid, 654.3), which very likely derives from Eudemus, the Pythagoreans constructed three regular solids, pyramid, cube, and dodecahedron, to which Theaetetus added the octahedron and icosahedron (see figure B2.7). hough the construction of the octahedron, a combination of two pyramids on a square base, is much simpler than that of the dodecahedron, they are ascribed respectively to heaetetus and Hippasus, who lived a century before him (von Fritz 1945;

Figure B2.7 he ive regular solids. Drawing by W. Sinelnikow. Zhmud 2012, 275). To divide the theories of regular polyhedra into two stages—the investigation of individual polyhedra and their general theory—helps clarify why the more complex polyhedron was constructed before the simpler one (Waterhouse 1973). Hippasus studied not the theory of regular solids as such, but the dodecahedron itself. On the other hand, heaetetus, having posed the question of which regular solids could be constructed, easily discovered the octahedron. He wrote a systematic treatise, in which he set forth methods for constructing the ive regular solids and for inscribing them in a sphere; he also described the relations between the edges of the regular solids and the diameter of the sphere. he last book of Euclidean Elements is based on this treatise. he ive regular solids became famous outside of mathematics, ater Plato used them in his Timaeus to impart a geometric structure to the four physical elements traditional for Greek philosophy. Creating the world, the Platonic demiurge makes ire from pyramids, air from octahedra, water from icosahedra, earth from cubes, and he uses the dodecahedron to decorate the whole universe. In the Hellenistic era, the ive regular solids were called “Platonic bodies,” and Proclus even claimed that Euclid belonged to the Platonic school “and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic iguresˮ (In Euclid, 68.20–23). Proclus’ teleological view of the history of mathematics is typically Neoplatonic but is akin to Plato’s own “appropriative” approach to mathematics. Since the geometricians and astronomers do not know how to make use of their discoveries, asserts Plato in his early Euthydemus (290c), those of them who are not utter blockheads must hand these discoveries over to the dialecticians, who will ind proper use for them—just as hunters and fishermen give what they catch to cooks!

4. Mathematics: The Beginning of Self-Reflection he successes of mathemata during the 5th century bce made their methods of attaining true knowledge highly attractive, especially against the background of the endless debates of the natural philosophers about basic principles, as well as doubts and denials that the truth is attainable, expressed by the Sophists. Philolaus became one of the irst pre-Socratics to introduce mathemata into a philosophical work and to make its results and methods an object of discussion and analysis. (Parmenides and Zeno took from mathematics the technique of deductive proof, but in them we ind no relection on the subject of their borrowed methods.) he Pythagoreans, involved in mathemata, were the irst to look at mathematics from an epistemological point of view. In his treatise On Nature, Philolaus declares: “And indeed all the things that are known have number. For without it we can neither understand nor know anythingˮ (DK 44 B 4). his fragment of Philolaus oten has been taken as evidence of the Pythagorean doctrine that “everything is number.” But “to have number” does not mean “to consist of numbers,” it means “to be countable,” since “number or that which has number is countable” (Nussbaum 1979). hus, number in Philolaus makes a knowable thing countable, for example, by representing the octave as a ratio 2:1, the ith as 3:2, and the fourth as 4:3 (fr. 6a Hufman). “Fr. 6a suggests that the whole-number ratios which govern musical scales served as the model of the kind of mathematical account which should be supplied for all phenomenaˮ (Hufman 2012). Archytas started his Harmonics by praising his Pythagorean predecessors, “those concerned with the mathematical sciences” (hoi peri ta mathemata), for their, one might say, great epistemological successes. They showed true insight, and it is not strange that they have a correct understanding of particular things as they really are: For since they exercised good discrimination about the nature of the universe (peri tas tōn holōn phusios), they were likely also to get a good view of the way things really are taken part by part. hey have handed down to us a clear understanding of the speed of the heavenly bodies and their risings and settings, of geometry, of numbers, and not least of music. For these sciences seem to be sisters. (DK 47 B 1, tr. A. D. Barker, slightly modiied) In Archytas, the word mathemata acquires its terminological character and designates a particular group of four mathematical sciences, all of which he regards as related. (his quadrivium soon appears in Plato’s Republic.) It is these sciences, claims Archytas, that give us real understanding of the world and everything in it. his claim is, irstly, very un-Platonic, for Archytas obviously did not need any intermediary to interpret results of scientiic research; and secondly, it is quite unusual, for in antiquity claims to true understanding of reality were usually raised by philosophers rather than by mathematicians.

here were other exceptions, too (Feke 2014). Of all the mathemata, Archytas clearly preferred arithmetic, declaring in particular that it surpassed all other arts, including geometry, in clearness, evidence, and obviousness, which makes it, in comparison, more demonstrative (DK 47 B 4). Apart from the fact that arithmetic is more exact than geometry, it is also socially useful. In the introduction to On Mathematical Sciences, Archytas relates important social changes, such as an increase of concord and an advance toward greater equality, the to the discovery of calculation. Moreover, calculation proves capable of improving people’s moral qualities, keeping them from greed and injustice or, at any rate, exposing these vices (DK 47 B 3). Archytas’ conviction that mathematical knowledge makes a man and, accordingly, the society in which he lives better, was shared by his friend Plato. In the same fragment, Archytas again tackles epistemological issues, presenting diferent ways of acquiring knowledge: To know what was heretofore unknown, one has either to learn it from another, or to discover oneself. What one has learnt, he has learnt from another and with another’s assistance, what one has found, he has found himself and by his own means. Discovery without research is difficult and rare, by research easy and practicable, but without knowing (how) to research it is impossible to research. (DK 47 B 3) To make a discovery, conscious research is needed because one cannot conduct research without knowing how to do it. What, then, must the researcher know? To all appearances, he must know what and how to seek—in other words, he must know the object and method of his research. It follows, then, that the method, which is to say the art of correct research, becomes for Archytas a prerequisite for success in science, although he did not altogether rule out the chance, small as it might appear, of an accidental discovery. hus, by the beginning of the 4th century bce, Greek exact sciences not only succeeded in creating new powerful methods and in solving many diicult problems but began also to look narrowly at themselves: What did they achieve, and why did this become possible? We can only regret that the results of this self-analysis are so seldom available to us. Bibliography Becker, O. Das mathematische Denken der Antike. Göttingen: Vandenhoeck & Ruprecht, 1966. Bodnár, I. Oenopides of Chius. Preprint 327 of the Max Planck Institute for the History of Science, Berlin, 2007. Boll, F. Studien über Claudius Ptolemäus. Leipzig: Teubner, 1894. Brumbaugh, Robert. he Philosophers of Greece. Albany: State University of New York Press, 1981. Burckhardt, Jacob. Griechische Kulturgeschichte. Berlin: Spemann, 1898–1902. Chemla, Karine, ed. he History of Mathematical Proof in Ancient Traditions. Cambridge: Cambridge University Press, 2012.

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