Aristotle, having spent twenty years of his life in or near the Academy, was necessarily a mathematician. He was not a professional mathematician like Eudoxos, Menaichmos, or Theudios, but he was less of an amateur than Plato. This is proved positively by the mass of his mathematical disquisitions¹³²⁶ and negatively by his lack of interest in the mathematical occultism and nonsense that disgraced Platonic thought. He was well trained but not quite up–to–date, and inclined to avoid technical difficulties. He was probably well acquainted with Eudoxos’ ideas, but not so well with those of other contemporaries like Menaichmos. His references to incommensurable quantities are frequent, but the only example quoted by him is the simplest of all, the irrationality of the diagonal of a square in relation to its side. He was primarily a philosopher, and his mathematical knowledge was sufficient for his purpose. All considered, he is one of the greatest mathematicians among philosophers, being surpassed in this respect only by Descartes and Leibniz. Most of his examples of scientific method were taken from his mathematical experience.

In his classification of sciences he considered most exact those that are most concerned with first principles. On that basis, mathematics came first, arithmetic being ahead of geometry.¹³²⁷ Like Plato he was interested in knowledge for its own sake, for the contemplation of truth, rather than for its applications. Moreover, he was more interested in generalities than in particularities, and more interested in the determination of general causes than in the multiplicity of consequences.

He made a distinction between axioms (common to all sciences) and postulates (relative to each science). Examples of axioms or common notions (coinai en–noiai) are the “law of excluded middle” (everything must be either affirmed or denied), the “law of contradiction” (a thing cannot at the same time both be and not be), and “if equals are subtracted from equals the remainders are equal.” As to definitions, they must be understood; they do not necessarily assert the existence or nonexistence of the object defined. We must assume in arithmetic the existence of the unit or monad, and in geometry of points and lines. More complex things, like triangles or tangents, must be proved to exist, and the best proof is the actual construction.

Aristotle’s greatest service to mathematics lies in his cautious discussion of continuity and infinity. The latter, he remarked, exists only potentially, not in actuality. His views on those fundamental questions, as developed and illustrated by Archimedes and Apollonios, were the basis of the calculus invented in the seventeenth century by Fermat, John Wallis, Leibniz, and the two Isaacs, Barrow and Newton (as opposed to the lax handling of pseudo infinitesimals by Kepler and Cavalieri).¹³²⁸ This statement, which cannot be amplified in a book meant for nonmathematical readers, is very high praise indeed, but justice obliged us to make it, the more so because Plato is more famous as a mathematician than Aristotle, and that is exceedingly unfair. Aristotle was sound but dull; Plato was more attractive but as unsound as could be. Aristotle and his contemporaries built the best foundation for the magnificent achievements of Euclid, Archimedes, and Apollonios, while Plato’s seductive example encouraged all the follies of arithmology and gematria and induced other superstitions. Aristotle was the honest teacher, Plato the magician, the Pied Piper; it is not surprising that the followers of the latter were far more numerous than those of the former. But we should always remember with gratitude that many great mathematicians owed their vocation to Plato; they obtained from him the love of mathematics, but they did not otherwise follow him and their own genius was their salvation.


Let us now leave Aristotle and the Lyceum and return to the Academy. We should always bear in mind that mathematical studies were then fashionable in Athens and were conducted in both schools, probably in friendly emulation. Most of the mathematical work was probably done in the Academy; Speusippos and Xenocrates were Plato’s successors at the head of it; the brothers Menaichmos and Deinostratos were both mentioned by Proclos ¹³²⁹ as friends of Plato and pupils of Eudoxos; Theudios of Magnesia wrote the textbook of the Academy; on the other hand, Eudemos of Rhodes, quoted as a pupil of Aristotle and Theophrastos, must be assigned to the Lyceum. These matters cannot be settled with any certainty, for we know the headmasters of both schools (some of them at least), but there never were any lists of students, and it is possible that attendance was informal. So–and–so are named disciples of Plato or of Aristotle, not members of the Academy or the Lyceum.

Speusippos, nephew of Plato, succeeded him in 348/47 as master of the Academy. Judging from the fragments, his lost work “On the Pythagorean numbers” was derived from Philolaos and dealt with polygonal numbers, primes versus composite numbers, and the five regular solids.


At the time of Speusippos’s death there was an election for a new master and the votes were almost equally divided between Heracleides of Pontos and Xenocrates of Chalcedon, but the latter won and was the head of the Academy for twenty–five years (339–315). Note that Aristotle, Heracleides, Xenocrates were all “northerners,” and that the new master was an old friend of Aristotle (who referred many times to him in his writings). Hence, we must assume that Xenocrates was as familiar with Aristotle’s mathematical views as with Plato’s. He continued Plato’s policy of excluding from the Academy the applicants who lacked geometric knowledge and said to one of them, “Go thy way for thou hast not the means of getting a grip of philosophy.”¹³³¹ The story is plausible.

Xenocrates wrote a great many treatises, all of which are lost, but judging from the titles¹³³² some of them dealt with numbers and with geometry. The perennial controversy on geometric continuity which had been dramatized by Zeno’s paradoxes led him to the conception of indivisible lines. He calculated the number of syllables that could be formed with the letters of the alphabet (according to Plutarch that number was 1,002,000,000,000); this is the earliest problem of combinatorial analysis on record.¹³³³Unfortunately, we know nothing about his activities but the meager information just given.


Menaichmos and Deinostratos were two brothers, about whose circumstances we know only what Proclos told us in a short paragraph of his commentary on Book I of the Elements of Euclid: “Amyclas of Heraclea, one of Plato’s friends, Menaichmos, a pupil of Eudoxos who had also studied with Plato, and Deinostratos his brother made the whole of geometry more nearly perfect.” ¹³³⁴

We do not know when and where these brothers were born, but they lived in Athens, attended the Academy, and sat at the feet of Plato and later of Eudoxos. We may conclude that they flourished about the middle of the century.

Both brothers were concerned with the building up of a geometric synthesis. Menaichmos was especially interested in the old problem of the duplication of the cube. That problem had been reduced by Hippocrates of Chios (V B.C.) to the finding of two mean proportionals between one straight line and another twice as long. In modern language we would say that Hippocrates had reduced the solution of a cubic equation to that of two quadratic equations. How would these be solved? Menaichmos found two ways of solving them by determining the intersection of two conics — two parabolas in the first case, a parabola and a rectangular hyperbola in the second.

This marks the appearance of conics in world literature, and the discovery of those curves is ascribed to Menaichmos. His construction of them seems very peculiar to us; he imagined that a plane cuts a right circular cone, the plane being always perpendicular to the generating line of that cone. The three different conics (which he seems to have differentiated) were obtained by increasing the cone’s angle;¹³³⁵ as long as the angle is acute, the section is an ellipse; when the angle is right the section is a parabola; when the angle is obtuse one obtains the two branches of a hyperbola. Neugebauer has surmised that Menaichmos may have been led to his discovery by the use of sundials.¹³³⁶ If he is right (and his argument is very plausible to me), it is strange to think that those curves, of astronomic origin, were not introduced into astronomic theory until almost two millennia later. Menaichmos discovered them (c. 350 B.C.) because of his solar observations, but not until Kepler (1609) were they used for the explanation of the solar system.

Alexander the Great asked Menaichmos whether there was not a short cut to geometric knowledge and Menaichmos answered, “O King, for traveling over the country there are royal roads and roads for common citizens, but in geometry there is one road for all.” ¹³³⁷ The story has become a commonplace, and it has been ascribed to Euclid and Ptolemy, as well as to Menaichmos. It fits the last best because he is the most ancient and because Alexander, whose intellectual ambitions had been fanned by Aristotle, might well have asked such a question. The great king was naturally impatient, but he had to find out that it might take longer to acquire sound knowledge than to conquer the world.


We have explained above (p. 278) that geometric thinking was activated in the fifth century by the emergence of three problems: (1) the squaring of the circle, (2) the trisection of the angle, (3) the duplication of the cube. Hippocrates of Chios and Menaichmos were especially interested in the third of those problems; Hippias of Elis found an ingenious solution of the second by means of the curve invented by him, the quadratrix. That name was given to it because Deinostratos, Menaichmos’ brother, applied it to the solution of the first problem. We thus see that the three famous problems were still exercising the minds of the geometers of the Academy in the fourth century and helping them to extend the frontiers of their knowledge.


Said Proclos: “Theudios of Magnesia distinguished himself in mathematics and in other branches of philosophy; he arranged beautifully the Elements (ta stoicheia) and made many partial theorems more general.”¹³³⁸

This statement is very significant in spite of its concision. It reveals the existence of a book which might be called “The geometric textbook” (or the “Elements”) of the Academy. The mathematicians of that time were interested, some in discovery, others in synthesis and logical consistency; the former were like adventurers or conquerors, the latter like colonizers. The two tendencies have always coexisted in times of healthy mathematical development, and they are equally necessary. There must be continual pressure on the frontiers and better organization within. As far as we can guess from Proclos’ laconic account, Theudios’ task was to put the geometric knowledge already obtained by the pioneers into as strong and beautiful a logical order as possible. Theudios was the forerunner of Euclid, and made the latter’s achievement easier.


Eudemos was a pupil of Aristotle and a friend of Theophrastos. We may thus conclude that he flourished in the third quarter of the century and that he was a member of the Lyceum. In fact, Proclos, who quotes him four times in his commentary on Euclid I, calls him Eudemos the Peripatetic.¹³³⁹ Among the writings ascribed to him, but lost, were histories of arithmetic, geometry, and astronomy. He is the first historian of science on record,¹³⁴⁰ and, though only fragments have come to us, we have good reason to assume that his work was the main source out of which whatever knowledge we possess of pre–Euclidean mathematics has trickled down. One of the most important fragments is the one concerning the quadrature of the lunes by Hippocrates of Chios, of which we have already spoken.

The appearance at this time of a historian of mathematics and astronomy is very significant, for it proves that so much work had already been accomplished in these two fields that a historical survey had become necessary. Let us remember with gratitude the name of the first historian of mathematics and consider his presence in Athens around the year 325 as a new illustration of the glory of Hellenism.¹³⁴¹


The last mathematician of this century marks the transition between the age of Aristotle and the age of Euclid. Two treatises of great originality are ascribed to him. One of them was devoted to solid loci connected with conics, that is, it was a treatise on conics regarded as loci, and was prior to Euclid’s book on the same subject.¹³⁴³ He defined the different kinds of conics in the same way as Menaichmos, as sections of cones with acute, right, and obtuse angles. The other book was entitled Comparison of the five figures, meaning the five regular solids, and among other things it proved the remarkable proposition that “the same circle circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron when both solids are inscribed in the same sphere.“¹³⁴⁴

How beautiful a result this was, and how unexpected! For who could have foreseen that the faces of two different regular solids are equally distant from the center of the sphere enveloping them? These two solids, the icosahedron and the dodecahedron, had thus a special relation which the three other solids did not have. How much more beautiful indeed in its truth and honesty than the Platonic illusions on the same “figures.”


The second half of the century did not witness the renewal of revolutionary efforts comparable in their pregnancy to those of Eudoxos of Cnidos, yet the total amount of new mathematics was splendid. The members of the Lyceum headed by Aristotle improved the definitions and axioms and more generally the philosophic substructure; Eudemos facilitated the needed synthesis by his historical surveys. Under the guidance of Speusippos and Xenocrates the Academy continued geometric investigations of various kinds which led to the composition of the “Elements” by Theudios. The brothers Menaichmos and Deinostratos, and Aristaios were creative geometers of the first order. We owe to Menaichmos and to Aristaios the first study of conics.



Pride of place in our astronomic section must be given to Heracleides not only because of his age but also because of his singular greatness. He was born in Heracleia Pontica ¹³⁴⁵ c. 388, before Aristotle, and he lived until the ninth decade of the century (c. 315–310). His singularity was such that he has been called “the Paracelsus of antiquity,” a silly nickname, yet meaningful, whether it is taken as praise or blame. To compare him with a man who appeared nineteen centuries later is to invite unnecessary trouble; it is more helpful to compare him with his predecessor, Empedocles, a man whom he greatly admired and tried to emulate.

We know little of his life except that he was wealthy, emigrated to Athens, and was a pupil of Plato and Speusippos, perhaps also of Aristotle. When Speusippos died in 339 and was replaced by Xenocrates (Aristotle’s friend), Heracleides returned to his country. He wrote many books on philosophy and mythology which obtained some popularity not only among the Greeks but also among the Romans of the last century B.C. For example, Cicero admired him and one can detect traces of Heracleides’ influence in “Scipio’s dream.”¹³⁴⁶ Even as Plato had written a revelation of other–world mysteries in his myth of Er, Heracleides wrote a similar revelation in his myth of Empedotimos: ¹³⁴⁷ his Hades where the disin–carnated souls found their last refuge was located in the Milky Way; the souls were illuminated!

Such poetic fancies explain his popularity but would not justify our own praise in this volume. Yet to be a spiritual descendant of Empedocles was a remarkable thing and we must pause a moment to consider it: there was an irrational trend in Greek thought cutting through the centuries via the Pythagoreans, Empedocles, Plato, Heracleides, and their epigoni. Heracleides, however, combined his apocalyptic with scientific tendencies, and we must speak of him at greater length because of his astronomic theories, which make him one of the forerunners of modern science.

One more word, however, concerning his relation with Empedocles. The latter’s view of the universe included the four elements and the two antagonistic forces (love and strife). Heracleides conceived the world as made up of jointless particles (anarmoi oncoi),as opposed probably to the Democritean atoms, which had various shapes and could cling to one another. The Heracleidean particles might hold together by some kind of Empedoclean attraction.¹³⁴⁸

Heracleides’ astronomy was more rational, as we would expect, than his cosmology. He had probably heard of the views expressed by Hicetas and Ecphantos and agreed with them. On the basis of those views and of other Pythagorean–Platonic ideas he explained his own theory, which can be summarized as follows. The universe is infinite. The Earth is in the center of the solar system; the Sun, Moon, and superior planets revolve around the Earth; Venus and Mercury (the inferior planets) revolve around the Sun; the Earth rotates daily on its own axis (this rotation replaces the daily rotation of all the stars around the Earth).¹³⁴⁹ This geoheliocentric system had an astounding fortune. It was not sufficiently bolstered up with observations to deserve the acceptance of the practical astronomers of Heracleides’ time; yet the hypotheses that it included were never forgotten. They reappeared in Chalcidius (IV–1), Macrobius (V–1), Martianus Capella (V–2), John Scotus Erigena (IX–2), William of Conches (XII–1).¹³⁵⁰

Looked at from the modern point of view, Heracleides’ system is a compromise between the Ptolemaic (centered upon the Earth) and the Copernican (centered upon the Sun), but this should not be exaggerated as is done by the historians who call Heracleides the Greek Tycho!¹³⁵¹ The compromise suggested by Tycho Brahe (1588; regular publication, 1603) and by Nicholas Reymers (1588) was deeper: all the planets, not two only, were supposed to revolve around the Sun. Strangely enough, the Jesuit, Giovanni Battista Riccioli, in his Almagestum novum published half a century later (Bologna, 1651), came back somewhat closer to Heracleides, for he accepted the rotation of three planets around the Sun, the two most remote ones (Jupiter and Saturn ) moving around the Earth.¹³⁵²

Heracleides was not a Copernicus, nor even a Brahe, yet his conception of the solar system, imperfect as it was, was astoundingly good for its time.


In the meanwhile, the work of Eudoxos was being continued by Aristotle and Callippos. They worked together at the Lyceum; though Callippos was somewhat younger than his chief, he seems to have been the originator in astronomic research. That would be natural enough, for Aristotle was obliged to busy himself with the whole institution and with the logical and philosophic teaching. If he had been tempted to make special investigations on his own account, he would probably have made them in the field of zoölogy, or he would have devoted more time to zoology than he was able to do.

After his return from Egypt, Eudoxos had spent some time in Cyzicos (Sea of Marmara), where he started a school of his own. Now, Callippos was born in that very place c. 370 and he may have known Eudoxos in his youth. In any case, he must have heard of Eudoxos’ mathematical and astronomic teaching, either directly or from a disciple such as his countryman, Polemarchos of Cyzicos, who is quoted as one of the first critics of the theory of homocentric spheres.¹³⁵³ Indeed, he was Polemarchos’ pupil and followed him to Athens, where “he stayed with Aristotle helping the latter to correct and complete the discoveries of Eudoxos.”¹³⁵⁴ The date of Callippos’ arrival in Athens was probably after the beginning of Alexander’s rule (336), and before the beginning of Callippos’ cycle (330). According to Aristotle,¹³⁵⁵ Callippos realized the imperfections of Eudoxos’ system and tried to remove them by adding seven more spheres, that is, two each for the Sun and the Moon, and one more for each of the other planets, except Jupiter and Saturn. The theory as improved by Callippos thus required a total of 33 concentric spheres rotating simultaneously each on its own axis and with its own speed.

Callippos concerned himself also with the reform of the calendar, the last establishment of which had been made in Athens in 432 by Meton and Euctemon. Better solstitial and equinoctial observations enabled him to determine more exactly the lengths of the seasons (beginning with the spring, 94, 92, 89, 90 days, the errors ranging from 0.08 to 0.44 day). He improved the Metonic cycle of 19 years by dropping 1 day out of each period of [19 × 4 = ] 76 years. The epoch of the new era was possibly 29 June 330.¹³⁵⁶The comparison of Callippos’ calendar with Meton’s gives us a measure of the progress in astronomic observation that had been achieved in a century.


Aristotle’s views on astronomy are explained in Metaphysics lambda, in Physics, in De caelo,¹³⁵⁷ and in Simplicios’ Commentary. He was not satisfied with the theory of homocentric spheres, even as perfected by Callippos. As Heath puts it,

In his matter–of–fact way, he thought it necessary to transform the system into a mechanical one, with material spherical shells one inside the other and mechanically acting on one another. The object was to substitute one system of spheres for the Sun, Moon, and planets together, instead of a separate system for each heavenly body. For this purpose he assumed sets of reacting spheres between successive sets of the original spheres. Saturn being, for instance, moved by a set of four spheres, he had three reacting spheres to neutralize the last three, in order to restore the outermost sphere to act as the first of the four spheres producing the motion of the next lower planet, Jupiter, and so on. In Callippos’ system there were thirty–three spheres in all; Aristotle added twenty–two reacting spheres making fifty–five. The change was not an improvement.¹³⁵⁸

This is typical of Aristotle’s mind; in his anxiety to give a mechanical and tangible explanation of planetary movements, he introduced unnecessary complications. Did Aristotle believe in the physical reality of the homocentric spheres? We cannot be sure; yet his transformation of the geometric concept into a mechanical one suggests such a belief. It is a good example of the eternal conflict between the explanation that satisfies the mathematician and the one that the practical man requires. The practical man is often defeated by his very practicality, and so was Aristotle in this case.

We cannot dissociate his astronomic views from the physical ones. Let us describe them rapidly together. There are three kinds of motion in space: (1) rectilinear, (2) circular, (3) mixed. The bodies of the sublunar world are made out of the four elements. These elements tend to move along straight lines, earth downward, fire upward; water and air, being relatively heavy and relatively light, fall in between. Hence, the natural order of the elements, starting from the Earth, is: earth, water, air, fire. Celestial bodies are made out of another substance, not earthly, but divine or transcendent, the fifth element or aether, whose natural motion is circular, changeless, and eternal.

The universe is spherical and finite; it is spherical, because the sphere is the most perfect shape; it is finite, because it has a center, the center of the earth, and an infinite body cannot have a center.¹³⁵⁹ There is but one universe and that universe is complete; there can be nothing (not even space) outside of it.

Is there a transcendent mover of the spheres (that is, a superior and unmoved mover of the spheres and of everything else)? Aristotle could not reach a certain answer on that fundamental question.¹³⁶⁰ His final conclusion in De caelo was that the sphere of the fixed stars was the prime mover (though itself moving) and hence the foremost and highest god; ¹³⁶¹ but in the Metaphysics lambda, his conclusion is that there is behind the fixed stars an unmoved mover influencing all the celestial motions as the Beloved influences the Lover. This implies that the celestial bodies are not only divine but alive, sensitive, and makes us realize once more, and more deeply, that ancient physics and ancient astronomy were very close to metaphysics, so close that one could not know any more where one was. Is this astronomy or metaphysics or theology?

We come closer to reality in Aristotle’s discussion of the shape of the Earth and estimate of its size. The Earth must be spherical for reasons of symmetry and equilibrium; the elements that fall upon it fall from every direction and the final result of all the deposits can only be a sphere. Moreover, during lunar eclipses the edge of the shadow is always circular, and when one travels northward (or southward) the general layout of the starry heavens changes; one sees new stars or ceases to see familiar ones. The fact that a small change in our position (along a meridian) makes so much difference is a proof that the Earth is relatively small. Here is the relevant text:

There is much change, I mean, in the stars which are overhead, and the stars seen are different, as one moves northward or southward. Indeed there are some stars seen in Egypt and in the neighborhood of Cyprus which are not seen in the northerly regions; and stars, which in the north are never beyond the range of observation, in those regions rise and set. All of which goes to show not only that the earth is circular in shape, but also that it is a sphere of no great size: for otherwise the effect of so slight a change of place would not be so quickly apparent. Hence one should not be too sure of the incredibility of the view of those who conceive that there is continuity between the parts about the Pillars of Hercules and the parts about India, and that in this way the ocean is one. As further evidence in favor of this they quote the case of elephants, a species occurring in each of these extreme regions, suggesting that the common characteristic of these extremes is explained by their continuity. Also, those mathematicians who try to calculate the size of the earth’s circumference arrive at the figure 400,000 stades. This indicates not only that the earth’s mass is spherical in shape, but also that as compared with the stars it is not of great size.¹³⁶²

The mathematicians referred to are probably Eudoxos and Callippos. Their estimate of the size of the Earth as quoted by Aristotle is the earliest of its kind; it was too large yet very remarkable.¹³⁶³ This fragment of Aristotle was the first seed out of which grew eventually in 1492 the heroic experiments of Christopher Columbus.

The main achievement of the astronomers of this period, if not of Aristotle himself, was the completion of the theory of homocentric spheres. This achievement implied the availability of a fairly large number of solar, lunar, and planetary observations. Where did Eudoxos, Callippos, and Aristotle obtain them? In Egypt and Babylonia.

According to Simplicios’ commentary on the De caelo, the Egyptians possessed a treasure of observations extending over 630,000 years, and the Babylonians had accumulated observations for 1,440,000 years.¹³⁶⁴ A more modest estimate was quoted by Simplicios from Porphyry, according to which the observations sent from Babylon by Callisthenes, at Aristotle’s request, covered a period of 31,000 years. All that is fantastic, but Oriental observations covering many centuries were actually available to the Greek theorists and were sufficient for their purpose. The Greeks obtained them in Egypt and Babylonia; they could not have obtained them in Greece, where men of science had preferred to philosophize each in his own way and where no institution had ever been ready to continue astronomic observations throughout the centuries. Simplicios’ exaggerations are simply a tribute to the antiquity and the admirable continuity of Oriental astronomy.

To return to Aristotle, though he was acquainted in a general way with Egyptian and Babylonian astronomy, he did not need their observations as keenly as did professionals like Eudoxos and Callippos. Being primarily a philosopher, he was more interested in questions of such generality that observations were of little help. For example, in the De caelo we find discussions concerning the general shape of the heavens, the shape of the stars, the substance of the stars and planets (which he assumed to be “aether”), the musical harmony caused by their motions. This may seem very foolish, but in justice to Aristotle and his contemporaries we should remember that many irrelevant and futile questions had to be asked and discussed before the pertinent ones were disentangled from the rest. In science immense progress is made whenever the right question is asked, the asking in proper form is almost half of the solution, but we can hardly expect these right questions to be discovered at the beginning.

The fortune of Aristotelian astronomy was singular. The theory of homocentric spheres was eventually displaced by the theories of eccentrics and epicycles, which was eventually crystallized in the Almagest of Ptolemy (II–1). Later, as the weaknesses of the Almagest appeared more clearly, some astronomers went back to Aristotle. The history of medieval astronomy is largely a history of the conflict between Ptolemaic and Aristotelian ideas; the latter were relatively backward and hence the growth of Aristotelianism retarded the progress of astronomy.¹³⁶⁵


In order to complete our survey of mathematics and astronomy in this golden age we must still speak of one great person, whose appearance ends it beautifully. Autolycos was born in Pitane ¹³⁶⁶ in the second half of the century, and he flourished probably in the last decade. He was an older contemporary of Euclid.¹³⁶⁷ Hence, he represents the transition between the great Hellenic school of mathematics and the Alexandrian age.

We know almost nothing about him, not even the place where he flourished. Did he go to Athens? That would have been natural enough. Yet Pitane was a civilized and sophisticated place, a well–located harbor facing Lesbos, not very far from Assos where Aristotle had taught. We know that Autolycos was the teacher of a fellow citizen of his, Arcesilaos of Pitane (315–240), founder of the Middle Academy. This suggests that he resided in Pitane and fixes the date approximately, the turn of the century.

Our ignorance concerning his personality is in paradoxical contrast with the fact that he wrote two important mathematical treatises, which are the earliest Greek books of their kind transmitted to us in their integrity. We know his works exceedingly well, but nothing of himself, except that he was the author of them.

Before speaking of these two books we must refer briefly to a third one which is lost and wherein he criticized the theory of homocentric spheres. He wondered how that theory could be reconciled with the changes of relative size of Sun and Moon and with the variations in the brightness of the planets, especially Mars and Venus. Judging from his controversy with Aristotheros, he could not solve that difficulty.¹³⁶⁸

The two books that have come down to us deal with the geometry of the sphere.¹³⁶⁹ As all the stars were supposed to be on a single sphere (and in any case one might always consider their central projections on that sphere), mathematical problems concerning their relations were problems of spherical geometry. For example, any three stars are the vertexes of a spherical triangle, the sides of which are great circles. When we try to measure the distance between two stars on that sphere (one side of the triangle), what we measure really is the angle which that side subtends at the center of the earth or as seen by a terrestrial observer. All such problems are solved now by means of spherical trigonometry, but trigonometry had not yet been invented in Autolycos’s time and he tried to obtain geometric solutions.

Irrespective of their practical value, which was considerable, these books are of great interest to us because of their Euclidean form, before Euclid. That is, the propositions follow one another in logical order; each proposition is clearly enunciated with reference to lettered figures, then proved. Some propositions, however, are not proved; that is, they are taken for granted, and this suggests that Autolycos’ books were not the first treatises on spherical geometry, but had been preceded by at least one other now lost. The substance of the lost treatise is somewhat preserved in the Sphaerics of Theodosios of Bithynia (I–1 B.C.), which gives the proofs of theorems unproved by Autolycos.

The first of Autolycos’ treatises, entitled On the moving sphere, deals with spherical geometry proper; the second, On risings and settings [of stars], is more astronomic, that is, it implies observations. Both treatises are too technical to be analyzed here.

How did it happen that such books were preserved? Their practical value was immediately realized by mathematical astronomers, who transmitted them from generation to generation with special care. Their preservation was facilitated and insured by the fact that they were eventually included in a collection called “Little astronomy” (in opposition to the “Great collection,” Ptolemy’s Almagest). The “Little astronomy” was transmitted in its integrity to the Arabic astronomers, and became in Arabic translation a substantial part of what they called the “Intermediate books.”¹³⁷⁰ The maxim “l’union fait la force” (part of the heraldic achievement of Belgium) applies to books as well as to men: when books become parts of homogeneous collections, each helps the other to survive.


The main achievement is the completion of the theory of homocentric spheres by Callippos; this may be put to the credit of the Lyceum. The Greeks were theorists rather than observers, but they were fortunate in that a treasure of Egyptian and Babylonian observations was available to them. It is almost impossible to determine their use of it except in a very general way. We can see only the fruits of that use, the main one being the theory of homocentric spheres. Heracleides was the first to propose a kind of geoheliocentric system, that is, to postulate the rotation of some planets around the Sun. He may be called the first Greek forerunner of the Copernican astronomy. At the end of the century Autolycos was building the geometric foundation of astronomy. Aristotle helped to state astronomic problems and to explain their relation to the rest of knowledge.

Note that none of these men was a Greek of Greece proper; their birthplaces were in Macedonia (Stageira) or in Asia Minor (Heracleia Pontica, Cyzicos, and Pitane).



Aristotle, his colleagues, and his younger disciples must have devoted much time to the discussion of physical questions; it was the old Ionian tradition of research de natura rerum, though already much better focused. A part of that was astronomic, but astronomy was always mixed with physics. The great advantage of astronomy proper, and the main cause of its early progress, was that some problems at least were very definite, and could be isolated with relative ease — such problems as how to account for the regular irregularities of planetary motions, or what are the shapes of the Earth and the planets, their mutual distances, their sizes. Not only was it possible to state these problems, but solutions were offered, some of which were sufficient at least as first approximations.

The universe was divided into two parts, essentially different — the sublunar world and the rest. Physical questions applied mainly to the sublunar world, astronomical ones to the Moon and beyond.

Fig. 93. Beginning of the Aristotelian physics in Latin translation, Physica sive De physico auditu (Padua, 1472–1475; Klebs, 93.1). First edition of the Physics in any language. It contains the double text in Latin with commentary by Ibn Rushd (XII–2). The anonymous printer was Laurenzius Canozius, in Padua. [Courtesy of the Bibliothèque Nationale, Paris.]

Aristotelian physics, or more correctly Peripatetic physics, is found in many books, such as Physica (Fig. 93), Meteorologica, Mechanica, De caelo, De generatione et corruptione, and even in Metaphysica, and the dating of some of these works is very uncertain. For example, theMechanica has been ascribed not only to Aristotle, but also to Straton of Lampsacos (III–1 B.C.), who was Euclid’s contemporary. The fourth book of the Meteorology is also ascribed to Straton. Let us forget for a moment these differences and try to describe the physical ideas that were explained in the Lyceum in the fourth and third centuries.

In order to avoid confusion we must try to forget another thing, our present conception of physics, which is relatively recent. In ancient and medieval times, and even down to the seventeenth century, physics concerned the study of nature in general, inorganic and organic.

The center of Aristotelian¹³⁷¹ physics is the theory of motion or of change. Aristotle distinguished four kinds of motion:

(1) Local motion, that is, our kind, translation of an object from one place to another. Such local motion, Aristotle recognized, is fundamental; it may and does occur in the other kinds.

(2) Creation and destruction; metamorphoses. As such changes are eternal, they imply compensations, or some kind of cyclic return. If they proceeded only in one direction they could not continue eternally. Creation is the passage from a lesser to a higher perfection (say the birth of a living being); destruction is the passage from a higher form to a lower (say the passage from life to death). There is neither absolute creation nor absolute destruction.

(3) Alterations, which do not affect the substance. Objects may receive another shape yet remain substantially alike. A man’s body may be altered by injury or by disease.

(4) Increase and decrease.

Everything that happens, happens because of some kind of motion as defined above. The physicist studies these “motions” for their own sake but also better to understand the substance undergoing them.

It is impossible, however, to explain nature only in terms of “material motions” or mechanism. One has to take into account some general ideas, such as that of universal economy: God (or nature) does nothing in vain. Every motion has a direction and a purpose. The direction is toward something better or more beautiful. The purpose of a being is revealed by the study of its genesis and evolution. We are falling back upon the theory of finalism (or teleology) which has been discussed in the previous chapter.

Everything in nature has a double aspect: material and formal. The form expresses the aim, which cannot be accomplished, however, except through some kind of matter. The weaknesses, imperfections, monstrosities that occur in nature are caused by the blind inertia of matter, defeating the purpose.

Aristotle had inherited and accepted the theory of four elements, at least to account for the changes that occur in the sublunar world. (For the changeless world above the Moon it was necessary to postulate a fifth, incorruptible, element, the aether.) He had also accepted the four qualities; at least, he considered them (wet and dry, hot and cold) the fundamental ones, to which others (for example, soft and hard) could be reduced. Only the necessities are formal; individual objects are contingent. It is the forms that the scientist must try to understand, but he cannot understand them except through individual (accidental) examples. We are thinking of Plato, and in some way, Aristotle is as idealistic as his predecessor, yet with a difference: Plato passes from the Form (the Idea) to the object, Aristotle does the reverse. That difference is simple but immense.

Aristotle made an exception, however, for some fundamental beings, such as the Prime Mover or the Elements, beings whose essence implies existence, and which cannot be known except a priori. All the rest can be known only empirically, by gradual induction, from individual cases to more general ones, and from inferior forms to superior ones. Mechanism alone can never explain the universe, yet analyses, descriptions, and inductions must precede every synthesis. That procedure is essentially the procedure of modern science.

Though he often quoted Democritos and praised him repeatedly, Aristotle rejected the atomic theory and what might be called Democritian materialism. He rejected the concept of vacuum,¹³⁷² because he could not conceive motion except in a definite medium, and was not everything that happened due to a kind of motion? It is possible that Aristotle rejected the atomic theory only because of the wrong use that Democritos (or his disciples) had made of it. It was claimed that Democritos tried to explain everything in mechanical terms, while the Aristotelian explanations were partly material and partly formal.

Celestial bodies move eternally, with constant speed, along circles. Sublunar bodies do not move if they are in their natural places; if they are removed from those places they tend to return to them along a straight line. There are two possible motions along a straight line, upward and downward.¹³⁷³ Heavy bodies like earth move downward; light ones like fire, upward. Between these two elements, which are absolutely heavy and absolutely light, occur the two others, water and air, which are respectively less heavy than earth and less light than fire.

Aristotelian mechanics includes adumbrations of the principle of the lever, of the principle of virtual velocities, of the parallelogram of forces, of the concept of center of gravity, and of the concept of density. Some of these ideas were to be given explicit and quantitative formulation by Archimedes of Syracuse (III–1 B.C.), others would be developed later, but the germs were already in the Aristotelian corpus.

Most discussions of Aristotelian mechanics center upon his dynamics. The genesis of Aristotle’s ideas on this subject is extremely instructive. We have seen that he did not accept the concept of vacuum.¹³⁷⁴ Motion is inconceivable in emptiness; hence, when he considered the movement of bodies it was always in a resisting medium. On the basis of gross observations he concluded that the speed of a body is proportional to the force pushing (or pulling) it and inversely proportional to the resistance of the medium. Any object moving in a resisting medium is bound to come to a standstill unless a force continues to push it. (In a vacuum, the resistance would be zero and the speed infinite. ) He also remarked that the speed of a falling body would be proportional to its weight, and that it would increase as the body was further removed from its point of release and came closer to its natural place. Hence the velocity would be proportional to the distance fallen.

The discovery of the true laws of motion became possible only when the Aristotelian prejudice against a vacuum was removed. Instead of rejecting motion in a vacuum as absurd, one assumed its possibility and considered what would happen if resistances were eliminated. Thanks to that happy abstraction, Galileo found that the speed was independent of the weight or mass of the falling body. He first thought that the speed would be proportional to the distance fallen but then realized that it was proportional to the time elapsed. The final laws of motion were discovered by Newton, chiefly the one that motive forces are proportional not to the speed of the body moved but to its acceleration. In fairness to Aristotle, however, one must remember that his conclusions were not unreasonable within the frame of his experimental knowledge. Mach was unjust to him and Duhem perhaps too generous. It is just as unfair to condemn Aristotle for not accepting what the invention of the air pump would prove, as for not seeing what could be seen only after the invention of the telescope.

The great difficulty of terrestrial (as compared with celestial) mechanics consisted in the extreme complexity of natural events. These could not become understandable without abstractions of great boldness. Aristotle’s imagination was not equal to that, not because it was inferior to Galileo’s or Newton’s, but because it could not depend upon the same mass of experience and could not soar off from the same altitude.

The Meteorologica ascribed to Aristotle contains meteorology in our sense, plus much else that we would classify under physics, astronomy, geology, even chemistry. ¹³⁷⁵ The astronomic part came in because Aristotle considered such phenomena as comets and the Milky Way as originating below the Moon; these phenomena were thus for him meteorologic rather than astronomic. Such errors were natural and pardonable in his time, and indeed until the end of the sixteenth and the seventeenth centuries. The unpredictable behavior of comets seemed absolutely different from the complex and solemn regularities of planetary motions. The planets suggest eternity and divinity; on the contrary, what better examples of capriciousness and evanescence could one adduce than the comets, which appear in the sky and after a relatively short time dissolve and disappear? Moreover, comets were generally seen outside of the zodiac. That Aristotelian prejudice was not shaken until the publication by Tycho Brahe, in 1588, of his observations of the comet of 1577. Brahe proved that its parallax was so small that the comet could not be sublunar; its orbit exceeded that of Venus.¹³⁷⁶

As to the Milky Way, which divides the heavens as a great circle along the solstitial colure, it also was supposed to be a meteorologic phenomenon, formed by dry and hot exhalations, similar to those that cause the meteors. A better understanding of the Milky Way was hardly possible without a telescope. Aristotle’s views were finally disproved by Kepler, according to whom the Milky Way was concentric with the Sun, on the inner surface of the starry sphere.

A great many other phenomena are described and discussed in the Meteorology, such as meteors, rain, dew, hail, snow, winds, rivers and springs, the saltness of the sea, thunder and lightning, earthquakes. The consideration of each of them would require at least a page, and space is lacking, the patience of our readers limited. Let us restrict ourselves to a few remarks concerning Aristotle’s optical theories. He rejected the view that light is material, being due to corpuscles emitted by the luminous object or emanating from the eye; on the contrary, he suggested that it was a kind of aetherial phenomenon. (Please do not call this an anticipation of the wave theory of light.) He was aware of the repercussions of sound (echo) and of light, and offered a theory of the rainbow, based upon the reflection of light in water drops and thus incomplete, yet very remarkable. His theory of colors has been compared to Goethe’s, a comparison that is not very complimentary to the latter but is very much to Aristotle’s credit.¹³⁷⁷

It is right to marvel at the endless number of physical questions in the Aristotelian corpus, but one should resist the temptation of reading into them too many ideas that are comparable to modern ideas yet could not possibly have had in their author’s mind the meaning and pregnancy that they have in ours. One should never forget that the authority of a statement is a direct function of the knowledge and experience upon which it is based; many Aristotelian statements are brilliant, yet as irresponsible as the queries of an intelligent child.

The fourth book of the Meteorology is probably the work of Straton.¹³⁷⁸ As it has come to us, it might be called the first textbook of chemistry. It discusses the constitution of bodies, the elements and qualities, generation and putrefaction, concoction and inconcoction (indigestion), solidification and solution, properties of composite bodies, what can and what cannot be solidified and melted, homoiomerous bodies.¹³⁷⁹ The final conclusion is that end and function are more evident in nonhomoiomerous bodies than in the homoiomerous bodies that compose them, and in these than in the elements. Aristotle (or Straton) had been thinking hard on the differences that may or may not occur when two different bodies are mixed together; they may remain separate or separable, or they may be combined into something essentially new; their two forms may disappear or exist only in potentia, while a new form is created.¹³⁸⁰

All of which is again very impressive, especially when we bear in mind the impenetrability of the chemical jungle until the end of the eighteenth century. Aristotle and Straton went as far as it was possible to go in their time, or more exactly, their thinking far exceeded their experimental reach, and more than two thousand years would be needed to bring it to maturity and to fruitage.

We have given a few examples of the long acceptance of Aristotelian ideas and prejudices. One might say in a general way that Aristotelian physics dominated European thought until the sixteenth century. Then the revolt that had been gathering strength for centuries became more articulate, more intense, and better organized. In the middle of that century Ramus¹³⁸¹ went to the extreme of proclaiming that everything that Aristotle had said was false. The foundations of Aristotelian physics were undermined in the following century by Gassendi, who revived atomism, and by Descartes,¹³⁸² who accepted some of Aristotle’s prejudices yet built up an entirely new structure. Yet even then the general conception of physics remained as broad as ever. Knowledge was hardly strong and sharp enough in any part of the immense field to separate that part from the rest, or to create physics as we understand it now.¹³⁸³

Aristotle’s views were rejected, but they were not forgotten or overlooked, and there remained an active Scholastic and Peripatetic opposition. Aristotle was still very much alive, though on the defensive, as late as the eighteenth century.


One disciple of Aristotle must still be introduced before we close this chapter, not the least of them, the musician, or rather the theorist of music, Aristoxenos. Aristotle himself was much interested in music, not only in the ethical value of it, somewhat in the Platonic manner,¹³⁸⁴ but also in the more technical sense. He was familiar with the Pythagorean discovery, the numerical aspect of musical harmony. Pythagoras or one of his early disciples had observed that when the vibrating string of a musical instrument was divided in simple ratios (1:¾: :½) one obtained very pleasant accords. Aristotle¹³⁸⁵ extended the same operation to reed pipes.¹³⁸⁶ He realized the importance of frequency of vibration, yet confused it with speed of transmission, and wrongly believed with Archytas that the speed of sound increased with the pitch. He asked the question, Why is the voice higher when it echoes back?¹³⁸⁷ The question was curious and pertinent, but it was not answered until 1873 by Lord Rayleigh’s theory of harmonic echoes.¹³⁸⁸

It is probable that other members of the Lyceum discussed questions concerning acoustics and music, because the books of Aristoxenos, which we shall examine presently, contain a body of knowledge on that subject that is remarkable alike because of its relative depth, extent, and complexity.

Most of what we know concerning Aristoxenos is derived from Suidas (X–2), but Suidas used ancient books that are lost to us, and whatever he tells us is sufficiently confirmed from various other sources to be reliable. Aristoxenos was born in Tarentum, close to the country where Pythagorean fancies had matured; he was educated by his father, Spintharos, who was a musician, by Lampros of Erythrai and Xenophilos the Pythagorean,¹³⁸⁹ finally by Aristotle. After the master’s death the election of Theophrastos instead of himself as head of the Lyceum infuriated him. Suidas says that he flourished in the 111th Olympiad (336–333)¹³⁹⁰ and that he was a contemporary of Dicaiarchos of Messina; he adds that Aristoxenos’ writings dealt with music, philosophy, history, and all the problems of education, and that he wrote altogether 453 books!

The only work of his that has come down to us is his Elements of harmony (Harmonica stoicheia), which is the most significant treatise of its kind in ancient literature. As we have it, it seems to be an artificial recombination of two separate works. It covers (in Macran’s edition ) 70 pages or some 1610 lines.¹³⁹¹ It is a tedious book wherein Aristoxenos applied the logical methods of the Lyceum to the exposition of the knowledge that had been transmitted to him by Spintharos, Lampros, and Xenophilos or that he had obtained by his own experiments. It is divided into three parts, treating (1) generalities, pitch, notes, intervals, scales; (2) idem, plus keys, modulation, melody (the polemical tone of this suggests the existence of other writings now lost ) ; (3) some twenty–six theorems on the combination of intervals and tetrachords in scales.

The most original part of Aristoxenos’ work is the theoretical determinations of the intervals. Starting from the three Pythagorean intervals ( , , ; octave, fifth, and fourth) he takes as unit the difference between the fifth and the fourth (the tone). That unit is too large, however; in order to obtain subunits he divides the interval arithmetically (not by extraction of roots). For example, in the descending fourth lami he inserts two tones, which gives the notes sol, fa. The new interval between fa and mi is the semitone. If this new interval is really a semitone, there are 5 semitones in the fourth, 7 in the fifth, and 12 in the octave. Aristoxenos went even further and considered not only semitones but also thirds, fourths, and even eights of the tone; these smaller divisions fell into abeyance. The empirical confusion between a leimma ¹³⁹² and a semitone led Aristoxenos to a calculus comparable to the calculus by logarithms: the intervals (which are ratios) are calculated by means of additive units. This is extremely interesting, yet it would be foolish to conclude that Aristoxenos was a forerunner of Napier! There’s many a slip ’twixt the cup and the lip, and there are many more between an idea and the theory eventually built upon it.¹³⁹³

The treatise of Aristoxenos is nevertheless highly significant, one of the masterpieces of Hellenic thought. Its influence was considerable, either directly or through the intermediary of the Harmonics of Ptolemy (II–1). The higher learning of late antiquity and of the medieval period included four main subjects (hence the name quadrivium),¹³⁹⁴ and those four subjects were arithmetic, music, geometry, astronomy. Music, not physics! Thanks to Pythagoras and Aristoxenos, music was a mathematical science, while physics remained in a qualitative stage, close to philosophy.

Aristoxenos was less influential in the West, because the first great teacher of music in the Latin language was Boetius (VI–1), whose handbook was based chiefly upon the Pythagorean tradition rather than on the Aristoxenian one. The Byzantine musicologists, on the contrary, followed Aristoxenos. For Manuel Bryennios (XIV–1), who composed the latest Byzantine Harmonics, the history of music was divided into three periods — pre–Pythagorean, Pythagorean, and post–Pythagorean. The third of these periods was the one initiated by Aristoxenos and continued by the other musicologists of classical and Byzantine times; Manuel himself was still in that third and last age, the age of Aristoxenos. Indeed, Greek musical theory never surpassed Aristoxenos’ exposition; nor did the practice of music (composition, playing, singing, teaching) change materially after him.¹³⁹⁵

Ancient music included not only music as we understand it but also metrics, poetry, for Greek poetry was composed to be chanted. Moreover, it had an ethical and cosmologic aspect; the theory of harmony in music was a part of the theory of harmony in the whole cosmos or in the soul of man. Thus music was a branch of philosophy as well as a branch of mathematics. It brought the humanities into the quadrivium.

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