**XVII**

Now that we have made the acquaintance of Plato the man, the philosopher, the politician, the moralist, it is time to ask ourselves what kind of man of science he was.

There is a great contrast between his manner of thinking and that of such men as Hippocrates and Thucydides, and even Herodotos. We already realize that Plato is the typical and “ideal” philosopher, whose knowledge or wisdom is supposed to come from above, and to stoop like an eagle on the objects below. The knowledge of a worthy metaphysician is complete to begin with and proceeds from heaven downward; the knowledge of the man of science, on the contrary, begins with homely things on the face of the earth, then soars slowly heavenward. The two points of view are fundamentally different. Indeed, Plato would have gone so far as to say that men of science have only opinions, no substantial knowledge, for knowledge can be derived only from absolute ideals while material objects can yield nothing more valuable than doubtful and precarious opinions.

His philosophy was colored with mathematical ideas, which he obtained from his Pythagorean friends, and especially from Theodoros of Cyrene and from Archytas of Tarentum. We have already spoken of Theodoros, who was a much older man (p. 282), and we shall come back to Archytas presently. We may assume that Plato had received a good mathematical training; though Socrates did not care for mathematics, he was fond of using forms of argument that could be easily applied to mathematical questions. Hence, the paradox that Plato had received an essential part of his mathematical training from Socrates, who was definitely not a mathematician.

**MATHEMATICS**

Plato’s general attitude to mathematics is well explained in the *Republic:*

“It is befitting, then, Glaucon, that this branch of learning should be prescribed by our law and that we should induce those who are to share the highest functions of state to enter upon that study of calculation and take hold of it, not as amateurs, but to follow it up until they attain to the contemplation of the nature of number, by pure thought, not for the purpose of buying and selling, as if they were preparing to be merchants or hucksters, but for the uses of war and for facilitating the conversion of the soul itself from the world of generation to essence and truth.” “Excellently said,” he replied. “And, further,” I said, “it occurs to me, now that the study of reckoning has been mentioned, that there is something fine in it, and that it is useful for our purpose in many ways, provided it is pursued for the sake of knowledge and not for huckstering.” “In what respect?” he said. “Why, in respect of the very point of which we were speaking, that it strongly directs the soul upward and compels it to discourse about pure numbers, never acquiescing if anyone proffers to it in the discussion numbers attached to visible and tangible bodies.”__¹¹⁴¹__

Irrespective of its mathematical interest, this extract is typically Platonic because of its juristic slant. Mathematics in Plato’s eyes is very important, so much so that “there ought to be a law” making the study of it obligatory for would-be statesmen (I wonder how our own statesmen would take to that).

When he speaks of mathematics, Plato is thinking, of course, of pure mathematics which gives us a vision of eternal truth and affords the best means of raising one’s soul to the Idea of Good, and to God. Plato carried his dislike of “applied mathematics” to the extreme of deprecating the use of instruments, except perhaps the ruler and the compass.__¹¹⁴²__

His general point of view is beautifully expressed in the statement that “God is always geometrizing” (God is primarily a mathematician!).__¹¹⁴³__ It is illustrated by the traditional inscription upon the door of the Academy: “Nobody should enter who is not a mathematician.”__¹¹⁴⁴__

The Platonic Idea is perfectly understood in the mathematical field, and it was possibly from his conception of it in that field that he ventured to extend it to the whole universe of thought. If we define the circle as a closed plane curve every point of which is equidistant from a point within, we create an Idea, the ideal or essential circle *(autos ho cyclos),* which no drawn circle could possibly emulate. The same applies to every mathematical definition; we can define a tangent, but it is impossible even with the finest instruments to draw a line and a circle having only one point in common. The ideal circle makes sense, while the ideal horse does not. And yet, according to Aristotle, Plato placed things mathematical *(ta mathematica*) somewhat below the pure Ideas, and considered them intermediate between the latter and tangible things, because the Idea of triangle is one, while there are many “ideal triangles.”__¹¹⁴⁵__ This seems farfetched. In spite of that quibbling, we may safely assume that the Platonic theory of Ideas had a mathematical origin, and take its formulation as one proof among others of Plato’s immoderate and irrational mathematization of everything.

Plato’s contributions to mathematical knowledge were chiefly of the philosophic kind; he improved the definitions and increased the logical tightness of the elements. It is not possible to measure the extent of those contributions and their originality. The Academy attached much importance to mathematical discussions; an increase in mathematical rigor was the main result, which cannot be ascribed definitely to the master or to any other member of the school but was to some extent a collective achievement.

*Fig. 83. The locus of points equidistant from two intersecting lines*

Did Plato invent geometric analysis? It is highly probable that the invention was made by Hippocrates of Chios (p. 277). Yet Plato may have improved it, or explained it more clearly (classroom discussion would easily lead to that), or he may have been the first to realize the need of completing the analysis with a synthesis.

*Example* of *analysis.* Suppose we have to prove that *A* is *B.* Let us assume that *A* is *B,* then *B* is *C, C* is *D, D* is *E;* therefore *A* is *E.* If that is untrue, then the theorem is disproved by *reductio ad absurdum.*

But if *A* is *E,* the theorem is not yet proved, and the analysis must be completed by the reverse process, called synthesis.

*Synthesis. If A* is *E, E* is *D, D* is *C, C* is *B;* therefore *A* is *B.*

It is also possible that Plato was the inventor (or developer) of problematic analysis.

Suppose one has to find the locus of all points at equal distances from two intersecting lines. Consider the two lines AB and *CD* crossing in *O* (__Fig. 83__), and suppose we have found one point *M* at equal distances from both lines. This means that if we draw perpendiculars from *M* to both lines, the segments *MN* and *MP* are equal. Let us draw the line *OM* and compare the triangles *OMN* and *OMP;* these triangles are equal; hence, the angles *NOM* and *MOP* are equal. Hence, *OM* is the bisector of the acute angle. A similar result would be obtained if one considered a point *M′*in the obtuse angle.

The next step is the construction of the locus, that is, the drawing of the two bisectors.

The last step is the synthesis, which consists in proving (1) that any point on the bisectors is at equal distances from both lines, (2) that any other point is not at equal distances from both lines.

Or suppose we are asked to draw a tangent from a point *A* to a circle *C* (the circle and point being in the same plane) (__Fig. 84__). Let us assume that the tangent is *AT;* then the radius *CT* is the shortest distance from *C* to *AT* and the angle *ATC* is a right angle. The locus of the vertexes of right angles subtended by *AC* is the circle of which *AC* is a diameter. Let us construct that circle. It cuts the circle *C* in two points, *T* and *T′,* and therefore we can draw two tangents, *AT* and *AT′.* Synthesis: We must now prove that *AT* and *AT′* are really tangents, and that there are no others.

Did Plato develop these methods? Or were they developed by his disciples, with him or without him, in the Academy or outside? It is impossible to tell, but the Platonic or Academic invention or rigorous formulation of the invention is very plausible.

We have already explained that Plato was profoundly impressed by the mathematical regularities that the Pythagoreans had discovered in the musical intervals. Thus, mathematics was connected with music on the one hand and with astronomy on the other. Might one not conclude then that there was music in astronomy? This was an intoxicating thought, which led Plato to his conception of the harmony of heavens, or the harmony of the world soul.__¹¹⁴⁶__

__Fig. 84__. Construction of a tangent to a circle from a point.

The reader is familiar with the medieval conception of the seven liberal arts, which is generally traced back to Boetius (VI—1) but can already be found in St. Augustine (V—1).__¹¹⁴⁷__ In reality, the idea is more ancient (as far as the *quadrivium* is concerned). The liberal arts constituted (and still do) a kind of general education *(encyclios paideia).*__¹¹⁴⁸__ In the course of time their number and contents varied. According to the medieval combination with which we are most familiar, the seven arts were divided into two groups, the *trivium* (grammar, logic, rhetoric) and the *quadrivium*(arithmetic, geometry, music, astronomy). This means that the second or higher level of general education was wholly mathematical.__¹¹⁴⁹__ That idea is often ascribed to Plato, but it is more correct to call it Pythagorean, though we cannot trace it further back than Plato’s time. Plato conceived a kind of mathematical *quadrivium,* but curiously enough it did not include music. It included arithmetic, geometry, stereometry, astronomy; the distinction between planimetry and stereometry or between plane and solid geometry betrays the immaturity of contemporary mathematics. The familiar division of the *quadrivium* (with music and without stereometry) was adumbrated by Archytas (text quoted below), but then disappeared; it reappeared only in the first century of our era, in the *Pinax* of pseudo-Cebes and in Seneca (I—2), then in Sextos Empiricos (II—2) and in Porphyry (III—2), later in St. Augustine (V—1), Martianus Capella (V—2), Boetius (VI—1), Cassiodorus (VI—1), Isidore of Seville (VII—1), and others. Plato did not introduce the medieval *quadrivium,* but it was he who caused the higher general studies to be mathematical.

The discovery of the regular solids has sometimes been ascribed to Plato. What does that mean? The regular solids were certainly known before him; the simplest had been known from time immemorial. The most difficult to recognize, the dodecahedron, was already known to Hippasos of Metapontum (p. 283) or to other Pythagoreans who liked to play with pentagrams and pentagons. We may thus assume that the Pythagoreans knew the five regular solids. They could construct them by putting together 4, 8, or 20 equilateral triangles, 6 squares, or 12 pentagons. That was not very difficult. But did they realize that there could not be more than five regular solids? That realization was the crux of the discovery, which was probably made by Theaitetos, and communicated to his friend Plato. The latter’s original contribution to the theory is of very doubtful quality. Considering on the one hand the four elements and on the other the five solids, these two groups must be somehow related, must they not? Plato associated the tetrahedron (pyramid) with fire, the hexahedron (cube) with earth, the octahedron with air, the icosahedron with water. What then shall be done with the fifth solid? That is easy. Plato associated it with the whole universe.__¹¹⁵⁰__

It has been argued that because Plato assumed that the particles of earth were cubic, the particles of fire pyramidal, and so on, he was an atomist. That is quibbling. Plato was definitely on the side of the antiatomists with Anaxagoras and with Aristotle. He rejected the possibility of a vacuum.__¹¹⁵¹__ He was not interested in the regular solids as “atoms” but simply as means for cosmologic analogies. The theory of the four elements was nonsensical and the attempt to dovetail the four elements with the five solids, doubly so.

Another mystical fancy to which innumerable commentaries have been devoted is the geometric or nuptial number of the Republic.__¹¹⁵²__ The number is called “nuptial” because it is related by Plato, in a language that is suitably obscure, to the time required for the breeding of perfect rulers. “Now for divine begettings there is a period comprehended by a perfect number” *(arithmos teleios)* and the perfect number is determined in such a sibylline manner that the interpretations of it vary considerably. There are really two numbers to be determined, not one, and Hultsch and Adam have arrived at the same two numbers in different ways. We quote their solution for the sake of example, and without attaching too much importance to it, for whether we know these numbers or do not is immaterial. Their two numbers are 216 = 2’ + 3° + 4³ = 2³ × 3³, and 12,960,000 = 60⁴ = 3600² = 4800 × 2700.

The first number, 216, may express the shortest period of human gestation in days. As to the larger one, 12,960,000, it “represents two current aeons in the life of the universe, in which the world waxes and wanes alternately, the harmony 3600’ meaning the cycle of uniformity and the harmony 4800 × 2700 the cycle of dissimilarity described by Plato in the *Statesman.”*__¹¹⁵³__

Let us approach the matter from another angle; the number 3600, being one of the units of the sexagesimal system, suggests a Babylonian origin: 12,960,000 = 3600 × 3600 = 360 × 36,000, that is, 36,000 years of 360 days; __¹¹⁵⁴__ according to Berossos (III—1 B.C.), the period of 36,000 years was the duration of a Babylonian cycle, it was called later *magnus platonicus annus.* Moreover, all the multiplication and division tables from the temple libraries of Nippur and Sippar and from the library of Ashurbâna–pal are based upon 12,960,000. This coincidence can scarcely be accidental. We must necessarily come to the conclusion that Plato, or rather Pythagoras, whom he closely followed, borrowed his famous number and the whole idea of a decisive influence exercised by it upon the life of man directly from Babylonia.__¹¹⁵⁵__

This much is clear: the geometric number is almost certainly of Babylonian origin, but we need not worry concerning Plato’s interpretation of it or modern interpretations of Plato’s. It is typical of the harm done by the *Timaios* that many scholars have racked their poor brains and perhaps been driven to distraction and insanity by the puzzle offered to them, in such solemn terms, by the divine Plato. We should not imitate them, and abondon the solution of Platonic riddles to wits, or better still, to half–wits.__¹¹⁵⁶__

Even if Plato made no mathematical discoveries (and there is no evidence that he made any), he was probably an up–to–date mathematician, but very definitely an amateur.__¹¹⁵⁷__ Yet his influence upon the progress of mathematics was immense. This was very neatly expressed by Proclos (V—2) in his commentary on the first book of Euclid:

He caused mathematics in general and geometry in particular to make a very great advance by reason of his enthusiasm for them, which of course is obvious from the way in which he filled his books with mathematical illustrations and everywhere tries to kindle admiration for these subjects in those who make a pursuit of philosophy.__¹¹⁵⁸__

One could not put it better. It was because of Plato that the higher level of the liberal arts was mathematical. His mathematical enthusiasm was contagious. One must love mathematics before one knows it; otherwise, one would never learn it; it is a kind of faith which Plato communicated to others. He did not create mathematics, but he created mathematicians.

He intimated again and again that a gentleman should know mathematics and for that reason mathematics was an essential part of the classical tradition nourished by the public schools of England. Most of the boys took heir mathematics as they took cod–liver oil; it was a painful process but one had to submit to it; some of them, however, took to it in great earnest. Plato was their initiator and their leader — in this respect, at least, a good leader.

Unfortunately, Plato was not unlike other amateurs, even amateurs of genius, in that his enthusiasm betrayed him and caused him to misuse mathematics grievously. We have given enough examples of his misapplications in this chapter and in a previous one. He was a mathematician with a vengeance.

The mathematical tradition that Plato initiated in the Academy was continued by his successors, and the Academy remained through the centuries a cradle of mathematicians. Let us speak now of those who were his contemporaries, were influenced by him and influenced him in return. It is a curious situation: they were genuine mathematicians, which he was not, yet they owed perhaps their vocation to him, and in any case he fostered it.

For the practical student of the history of mathematics, it is a great relief to pass from Plato to the real mathematicians, from hot air to golden fruits. We shall restrict ourselves to Theaitetos, Leodamas, Neocleides, Leon, Archytas, and the greatest of them all, Eudoxos.

**THEAITETOS**

We do not know much about the life of Theaitetos (c. 415—c. 369), not even his father’s name, but we know that he was an Athenian, a disciple of Socrates and Theodoros of Cyrene and a contemporary of Plato and Archytas.

One of the best of Plato’s dialogues, entitled Theaitetos, is a conversation between the mathematician in his youth, Theodoros of Cyrene, and Socrates, shortly before the latter’s death. Their conversation is not reported directly. A prelude describes another conversation, occurring in 369 in Megara, between Euclid and Terpsion in front of the former’s house. Euclid tells Terpsion__¹¹⁵⁹__ that as he was going to the harbor he met Theaitetos, who had been wounded in a battle fought for the Athenians near Corinth, and was carried back to Athens, suffering from his wounds and from dysentery, more dead than alive. They praise him for his courage and genius and Euclid recalls the dialogue proper which he has written down and which is now being read to them by his servant. The *Theaitetos* is thus a dialogue within a dialogue. As Plato knew the man, we may trust the physical portrait that he gives of him. Theodoros introduces him to Socrates with the following words,

Yes, Socrates, I have become acquainted with one very remarkable Athenian youth, whom I commend to you as well worthy of your attention. If he had been a beauty I should have been afraid to praise him, lest you should suppose that I was in love with him; but he is no beauty, and you must not be offended if I say that he is very like you; for he has a snub nose and projecting eyes, although these features are less marked in him than in you.__¹¹⁶⁰__

And at the end of the dialogue, Socrates remarks to Theaitetos that his “snub–nosedness” is characteristic of him. Thus, if we do not know Theaitetos very well, yet we can see him with the eyes of our imagination.

From that dialogue we also gather that Theaitetos was not only a mathematician but also a philosopher who distinguished between numbers perceived by the senses and numbers conceived by the mind. This is not very surprising, because every mathematician of that time was a philosopher.

Moreover, we can be sure that he was a Pythagorean, because the two subjects to which he owes his fame are both Pythagorean — the theory of irrationals and the theory of the regular solids.

The early history of irrationals has been told above apropos of Theaitetos’s master, Theodoros of Cyrene (pp. 282–285). Theaitetos continued the elaboration of that theory; he introduced distinctions between various kinds of irrationals (medial, binomial, apotome) which are described in Book x of the *Elements.*__¹¹⁶¹__ In particular, proposition 9 of that book is definitely ascribed to him (that the sides of squares that have not to one another the ratio of a square number to a square number are incommensurable). In short, he laid the foundation of the knowledge included in Euclid, x.__¹¹⁶²__

As to the theory of regular solids, it was said that Theaitetos discovered the octahedron and the icosahedron, and was the first to write on the five regular solids. The first part of that statement cannot be correct as it stands. Earlier Pythagoreans knew these two solids and were probably able to build them by means of 8 or 20 equilateral triangles (cut in leather, wood, or stone). That is, by holding 3, 4, or 5 equilateral triangles (of equal size) around a common vertex, they could form solid angles, and by combining 4, 6, or 12 such solid angles they could construct regular solids of 4, 8, and 20 faces. That was one thing, but the geometric construction was quite another. Still another was the realization that there were five regular solids and that there could not be more.

Theaitetos was the first to write on the five regular solids,__¹¹⁶³__ How much did he write? In the case of the irrationals, we ascribed to him an indefinite part of Book x of the *Elements*; in the regular solids we can ascribe to him in the same imprecise way a part of Book XIII. It was natural for him to study the regular solids, the mathematical construction of which implied irrationals. If he wrote of the five solids, this implies that he knew that there could not be more. Could he know that? Why not? After all, the proof as given in Euclid __¹¹⁶⁴__ is simple enough, so simple indeed that we can afford to insert it here (but I give it in my own way, for greater clearness).

There can be only five regular convex solids.

1. In every solid angle the sum of the plane angles is smaller than four right angles. The maximum (four right angles) would be attained only if the solid angle were completely flattened out around its vertex; that is, the solid angle would cease to exist.

2. If the faces are triangles, there can be around a point:

(i) three triangles, and the solid will be a tetrahedron or pyramid (4 faces);

(ii) four triangles, and the solid will be an octahedron (8 faces);

(iii) five triangles, and the solid will be an icosahedron (20 faces).

(There cannot be six, for six such angles would equal 4 right angles.)

3. If the faces are squares, there can be only three around a point, and the resulting solid will be a hexahedron (cube, 6 faces).

4. If the faces are pentagons, there can be only three around a point (for the angle of a pentagon equals 6/5 of a right angle) and this leads to a dodecahedron (12 faces).

5. No others are possible, for the angle of the hexagon equals 4/3 of a right angle, and three of these equal 4 right angles.

6. Thus there can be only five regular solids, having respectively 4, 6, 8, 12, 20 equal faces.

*Example,* The two segments of the “golden section” are apotomes (Euclid, *Elements,* Book XIII, prop. 6).

It was necessary to add the word “convex” at the head of the demonstration, because it was discovered much later that there were other regular solids which were not convex; these are called stellated polyhedrons and are to the convex polyhedrons somewhat as pentagrams are to pentagons. In 1810, Louis Poinsot (1777— 1859) discovered four stellated polyhedrons, to wit, three dodecahedrons and one icosahedron; in 1813, Augustin Cauchy (1789—1857) proved that these nine solids exhaust the series of regular solids; his proof was rigorous but difficult. It was simplified by Joseph Bertrand (1822—1900), who showed that the vertexes of every stellated polyhedron must be the vertexes of a convex concentric one. It suffices then to take the five Pythagorean solids and to examine which other regular solids could be obtained by grouping their vertexes differently.__¹¹⁶⁵__

To return to the five convex solids, the discovery that there could only be five of them, whether it was made by Theaitetos or not, must have been a great surprise and a shock. The investigation of polygons had not prepared one for that, because the regular polygons are infinite in number. If there is a regular polygon with n sides, one can easily construct others with 2n sides, 4n sides, etc. It is strange to pass from an infinity of polygons to the very small group of five polyhedrons. That extraordinary and sudden restriction appeared to Plato a mathematical mystery, requiring some kind of philosophic interpretation. If the regular solids are restricted to five, those five bodies (later called the Platonic bodies) must each have some definite meaning. They could not be connected with the planets, of which there were seven. Plato bethought himself of the four elements; the fifth solid would then represent the whole universe. This patching up of the theory, plus the finding of a meaning for the superfluous solid, is typical of the analogies invented by numerologists and other mystical mathematicians; they change the rules of their game as often as necessary and thus prove what they want to prove. In his interpretation of the regular solids, Plato stooped to the level of the Chinese cosmologists.

**LEODAMAS, NEOCLEIDES, AND LEON**

The progress of geometric discovery and organization under the influence of the Academy is symbolized by these three mathematicians, about whom we know only what Proclos tells us in his commentary on Euclid, Book 1, and that is more tantalizing than sufficient.

Said Proclos:

At about the same time Leodamas of Thasos, Archytas of Tarentum and Theaitetos of Athens increased the number of theorems and inserted them in a more scientific context; then Neocleides who was younger than Leodamas and his disciple Leon [IV–1 B.C.] produced many more things than their forerunners, so much so that Leon was able to assemble *Elements (ta stoicheia)* which are of great interest because of their number and their usefulness, and to invent distinctions *(diorismoi)* which show when a problem is solvable and when it is not.__¹¹⁶⁶__

That is all he has to say about Neoclides and Leon, but he adds this about Leodamas: “Plato had explained to him the analytical method which enabled him (Leodamas), it is reported, to invent many things in geometry.” That information is meager and vague, but helps us to realize that much geometric research was done by younger contemporaries of Plato. There was an emulation among them for the discovery of new theorems and, what is even more important, for a better dovetailing of all of them in a single synthesis. Proclos has nothing more to say about Archytas but happily we know much about the latter from various other sources.

**ARCHYTAS OF TARENTUM**

At the time of Plato’s first visit to Sicily in 388 he made the acquaintance of the Pythagorean Archytas, who was a very important man in Tarentum, not simply as a philosopher and mathematician, but also as a politician (or statesman) and as a general. He is said to have saved Plato’s life, through his influence with Dionysios. At the time of Plato’s last visit to Sicily (in 361–60) Archytas was still alive.

As far as one can judge from the fragments of his lost writings, his was a rich and complex personality. One of them shows that the classification of mathematical subjects that was later crystallized in the *quadrivium* existed already in the minds of the early Pythagoreans, or at least in his own.

The mathematicians *(hoi peri ta math mata)* seem to me to have arrived at correct conclusions, and it is not therefore surprising that they have a true conception of the nature of each individual thing: for, having reached such correct conclusions regarding the nature of the universe, they were bound to see in its true light the nature of particular things as well. Thus, they have handed down to us clear knowledge about the speed of the stars, their risings and settings, and about geometry, arithmetic, and sphae–ric, and last, not least, about music; for these branches of knowledge*(math mata)* seem to be sisters.__¹¹⁶⁷__

Archytas was an astronomer whose memory as such was still alive in the time of the poet Horace (65–8 B.C.), who celebrated it in one of his odes.__¹¹⁶⁸__ He speculated concerning the finitude or infinity of the universe and concluded that it must be unlimited. The most astonishing of his mathematical achievements is his solution of the famous problem of the duplication of the cube. That problem had been reduced by Hippocrates of Chios to the finding of two mean proportionals in continued proportion between two given straight lines. Archytas determined these two mean proportionals by means of the intersection of three surfaces of revolution. The intersection of two of these, a cylinder and a tore (or anchor ring) with inner diameter zero, is a curve of double curvature. The point where that curve pierces the third surface, a right cone, gives the solution. This is the first example in history of the use of a curve of double curvature. Archytas’ boldness is amazing.

His was an inventive mind, mechanical. It is said that he had devised a flying toy, a wooden dove, which did not continue its flight, however, after it had settled down. In Aristotle’s *Politics* we find an amusing reference to another toy: Children should have something to do, and the rattle of Archytas, which people give to their children in order to amuse them and prevent them from breaking anything in the house, was a capital invention, for a young thing cannot be quiet.__¹¹⁶⁹__

The anecdote is charming, but assuming that it refers to our Archytas, it does not help us to appreciate his mechanical ingenuity. The creation of a flying dove would have been a very remarkable achievement, but no mechanical genius was needed to make a good rattle.

Did Archytas write a book on mechanics, which would be, of course, the first of its kind? We do not know. Was he the founder of theoretical mechanics? __¹¹⁷⁰__ We have no right to make such a statement. All we can say is that he was interested in mechanics (in the crude sense of that word); he may have seen the possibility of relations between mechanics and mathematics even as he improved the mathematical study of music;__¹¹⁷¹__ he had found a mechanical solution of a mathematical problem,__¹¹⁷²__ and he may have thought of applying mathematics to mechanics. We cannot say more than that. At any rate, that Sicilian philosopher and mathematician is a kind of prototype of a greater countryman of his, Archimedes of Syracuse (III–2 B.C.).

**EUDOXOS OF CNIDOS**

As far as we can trust Diogenes Laërtios (III–1), and there is no reason to distrust the substance of his account, Eudoxos’ life is fairly well known and of unusual interest to the student of international relations. The dates of his birth and death are uncertain, but we may assume them to be c. 408 and 355.__¹¹⁷³__ Eudoxos, son of Aischines, was born in Cnidos; he learned geometry from Archytas and medicine from Philistion of Locroi. At the age of 23 (c. 385) he traveled to Athens and became a pupil of Plato (the Academy had been opened in 387); his traveling expenses were defrayed by Theomedon the physician. He was so poor that he remained in the Peiraieus, where he had disembarked, and walked to Athens every day. After spending two months in that way he returned to Cnidos; later, he traveled to Egypt with the physician Chrysippos of Cnidos, bearing a letter of introduction from Agesilaos to Nectanabis,__¹¹⁷⁴__ who recommended him to the priests (learned men). He remained in Egypt sixteen months, conforming to the customs of his hosts (he had his beard and eyebrows shaved), and there he wrote his *Octaët ris*. From Egypt he went to Cyzicos, on the southern shore of the Propontis (Sea of Marmara) and to other places in that neighborhood, earning his living as a teacher *(sophisteuonta),* then returned to his native region and attended the court of Mausolos in Halicarnassos.__¹¹⁷⁵__ He then visited Athens, but this time he came not as a poor young student, but as a master accompanied by his own disciples. Plato gave a banquet in his honor. After his return to Cnidos he helped to write laws for his fellow citizens and was very much honored by them.

Apollodoros of Athens (II–2 B.C.) said that he died in his fifty–third year (which fixes 355 if we accept 408). According to Favorinos of Arles (fl. under Hadrian, emperor 117–138), when Eudoxos was in Egypt with Chonuphis of Heliopolis, the sacred bull Apis licked his cloak and the priests augured that he would become famous but would not live very long. (The statements of Apollodoros and Favorinos are reported by Diogenes.)

The prediction of the Egyptian priests was confirmed, imperfectly with regard to his age (for to attain his fifty–third year was not so bad), completely with regard to his fame. He is considered to be the greatest mathematician and astronomer of his age, and one would have to speak of him even in the briefest outline of the whole history of science. Plato is known to a much larger public, but from the point of view of science, the time of Plato should be called the time of Eudoxos.

His well–deserved mathematical fame rests on three grounds — his general theory of proportion, the golden section, and the method of exhaustion. On that triple basis he deserves to be called one of the greatest mathematicians of all time.

A new theory of proportion had become necessary because of the revolutionary discovery of irrationals by Theodoros of Cyrene and Theaitetos of Athens. The Pythagoreans had observed parallelisms between numbers and lines (such as triangular or square numbers, and the theorem of Pythagoras). A ratio between lines might be represented by the ratio between two integers m and n, and conversely *m/n* might represent the ratio of two lines *m* and n units long. Now, the newly discovered lines or numbers, the irrationals *(alogos)*__¹¹⁷⁶__ were not integers, and could not be represented by any ratio of integers. Thus the structure of Pythagorean mathematics was breaking down. There were only two ways out, either to reject the parallelism between geometry and arithmetic, or to recognize a new kind of number, the irrationals. The second alternative was more complex than the nonmathematician would imagine, for it implied not only the definition of those numbers and the proof of their existence but also the proof that they could be handled in the same way as other numbers and the validation of geometric demonstrations that included or might include irrational elements. To put it otherwise, it was necessary to extend the idea of number, so that irrationals would be included, and to extend the idea of length so that theorems concerning any lines would still be correct if some of the lines were irrational. This extension was accomplished by Eudoxos in his general theory of proportions, which was to be developed later in Books V and VI of Euclid’s *Elements.* How much of that was done by Theaitetos and how much by Eudoxos it is impossible to state with precision, but it is traditionally assumed that the latter’s contribution was decisive.

What is the golden section? According to Proclus,__¹¹⁷⁷__ the theorems concerning “the section” *(ta peri ten tom n)* originated with Plato, and Theaitetos applied to them the method of analysis. It is more likely that the theorems were discovered by Theaitetos or other mathematicians and that Plato applied them to his own fancy. The curious use of the definite article “the section” (*h tom* ) must refer to a very exceptional section, almost certainly to the division of a straight line in extreme and mean ratio,__¹¹⁷⁸__ which obtruded itself in the construction of the pentagon and of the dodecahedron. At a later time, that remarkable section was called divine (by Luca Pacioli, 1509), and much later still, golden.__¹¹⁷⁹__ The term “golden section” was enormously successful, and a number of artists and mystics played with the idea that that particular section was one of the secrets of beauty.__¹¹⁸⁰__

Eudoxos’ share in the theory of the golden section gives him a modicum of popularity and glamour, but his outstanding mathematical achievements are the general theory of proportion and the method of exhaustion.

The method of exhaustion is a true infinitesimal method, the first of its kind, and it was based upon a rigorous notion of limit. By his invention of it Eudoxos became one of the distant ancestors of the integral calculus. Integration of simple areas had been made before him, and such results as that circles are to one another as the squares of their diameters __¹¹⁸¹__ had no doubt been obtained. Indeed, Hippocrates is said to have proved that theorem. How did he do it?

The demonstration given by Euclid is based on the use of the method of exhaustion invented by Eudoxos, and hence we may assume that it is essentially Eudoxos’ demonstration.

Given two circles, of areas *A* and *B and* radii *a* and *b;* we claim that *A*/*B* = *a*²/*b*²*.*

It has been proved before that similar polygons inscribed in circles are to one another as the squares of the diameters.__¹¹⁸²__ That was easy; the difficulty was to pass to the limit.

(1) Let us inscribe in the circles *A* and *B* regular polygons of areas *A′* and *B′,* having so many sides that the differences *A*–*A′* and *B*–*B′* are arbitrarily small.

(2) We have to prove that

*a*²/*b*² = *A*/*B*.

Let us suppose that that is not true, and that

*a*²/*b*² = *A*/*C*.

Can *C* be smaller than *B*?

We can reduce the difference *B*–*B*′ so much that

*B*–*B*′ < *B*–*C* or *B*′ > *C*.

The equalities

*a*²/*b*² = *A*/*C* = *A*′/*B*′

are irreconcilable because

*A* > *A*′, *C* < *B*′.

One shows in the same way that *C* cannot be greater than *B*. If *C* can be neither smaller nor greater than *B*, then *C* = *B* and the theorem is proved.

*Fig. 85*

This might be generalized, but the ancients failed to do so. The method of exhaustion was rigorous but particular; a special demonstration had to be elaborated in each case. Its use enabled Eudoxos to prove rigorously the formulas concerning the volumes of the pyramid and the cone that Democritos had discovered.__¹¹⁸³__

By the middle of the fourth century, thanks chiefly to the efforts of Theaitetos and Eudoxos, geometry had been raised to a much higher plane, approaching the Euclidean one. We are now past the stage of intuitive discovery, and mathematicians well trained in logic are no longer satisfied with partial results; they need rigor. What had been the share of Plato in that development? It is impossible to say. He may have insisted on clearness and good logic, but the main achievements, the purely mathematical ones, were not his. He may have helped the mathematicians; they could have done without him; he could not have done without them.

**ASTRONOMY**

The astronomic deeds of the Platonic age are as brilliant as the mathematical ones, and they were accomplished chiefly by the same man, Eudoxos of Cnidos. The history that we have to tell is rather complex. We shall deal first with the achievements ascribed to Babylonian astronomers. The Greek story must be divided into three sections: The Precursors; Eudoxos; Plato and Philip of Opus.

**KIDINNU**

In order to explain the part that Babylonian astronomers may have played in the development of Greek astronomy, we must anticipate a little. According to Ptolemy (II–1),__¹¹⁸⁴__ Hipparchos of Nicaea (II–2 B.C.), comparing his own observations of the fixed stars with other observations made a century earlier in Alexandria by Aristyllos and Timocharis (III–1 B.C.), concluded that all the stars had moved a little toward the east, that is, he discovered the precession of the equinoxes. Hipparchos assumed that the displacement of the stars in longitude, that is, the precession, amounted to 45” or 46” a year, which would be only 1°10’ in a century. (Ptolemy corrected that to 36” a year or exactly 1° in a century, but Hipparchos was nearer to the truth, the real value being 50”.26.) Could Hipparchos have noticed a difference of the order of 1°? Yes, that was not impossible, but it would have been easier for him to discover the precession if older observations had been available to him,__¹¹⁸⁵__ and accurate observations made by Babylonians may have been. Ptolemy refers to Chaldean observations made in 244, 236, 229 B.C.__¹¹⁸⁶__ The theory has been advanced that Hipparchos not only had earlier Oriental observations at his disposal (which is very likely), but that the precession had been discovered as early as 379 by the Babylonian astronomer Kidinnu.__¹¹⁸⁷__

It is certain that Chaldean astronomers had gathered a large number of observations of astounding precision. The earliest of them known to us by name were Naburianos (Naburimannu son of Balatu), who flourished at Babylon in 491, and Kidinnu, who flourished c. 379; they devised lunar tables according to two different systems; then came the astronomers responsible for the Chaldean observations recorded in the *Almagest*. It is almost certain that Hipparchos was acquainted with these observations, which facilitated his own work, in particular his discovery of the precession.__¹¹⁸⁸__

The discovery, it should be noted, was unavoidable just as soon as lists of stars sufficiently distant in time were compared. The astronomers making those comparisons could not fail to notice that all the longitudes were increased by the same quantity; that quantity was very small, about 1°24′ in a century, 4°12′ in three centuries, 5°36′ in four centuries. No matter how gross the observations, a time must come when the precession is noticed (I do not say explained; that is another story).

We cannot abandon this subject without introducing another remark even if this obliges us to make more anticipations. The precession being finally recognized, as it had been by Hipparchos, and published by Ptolemy,__¹¹⁸⁹__ additional observations of stellar longitudes would confirm it, and therefore one would expect such a fundamental discovery to be firmly established. Not at all! Most of the followers of Ptolemy let it drop; the only ones who refer to it are Theon of Alexandria (IV–2) and Proclos (V–2), but the latter denied it and Theon, while accepting the Ptolemaic value (1° per century), suggests that it is restricted to an oscillation along an arc of 8°, which means that the precession would accumulate for some eight centuries and then be reversed. Proclos made a similar admission; the tropical points do not move in a whole circle but some degrees to and fro.

Theon was thus the originator of the theory of the “trepidation of the equinoxes,” which enjoyed a long popularity in spite of its wrongness. The theory of continuous precession as discovered by Hipparchos and explained by Ptolemy and the theory of trepidation were contradictory, though many astronomers tried to find a compromise. Trepidation was accepted by the Hindu astronomer ryabha a (V–2), who was perhaps the link between Theon and Proclos on one hand and Th bit ibn Qurra (IX–2), the first Arabic writer to speak of it, on the other. It must be said to the credit of the Muslim astronomers that most of them rejected the idea of trepidation; that is the case for al–Fargh n (IX–1), al–Batt n (IX—2), ‘Abd al–Rahm n al– f (X—2), and Ibn Y nus (XI—1). But unfortunately al–Zarq l (IX–2) al–Bi r j (XII–2) sponsored that wrong idea, and as their influence was considerable, they were largely responsible for its diffusion among the Muslim, Jewish, and Christian astronomers, so much so that Johann Werner (1522) and Copernicus himself (1543) were still accepting it; Tycho Brahe and Kepler had doubts concerning the continuity and regularity of the precession, but they finally rejected the trepidation.__¹¹⁹⁰__ The matter could not be completely elucidated until the precession of the equinoxes had been explained in Newton’s *Principia* (1687).

The persistence of the false theory of trepidation is difficult to understand. At the very beginning of our era, the time span of the observations was still too small to measure the precession with precision and without ambiguity, but as the centuries passed there could not remain any ambiguity. Between the stellar observations registered in the Almagest__¹¹⁹¹__ and those that could be made by Copernicus almost fifteen centuries had elapsed, and the difference of longitudes would amount to 21°.__¹¹⁹²__ How could such a difference be explained in terms of trepidation? How could it be explained otherwise than by a steady accumulation of differences of the same sign?__¹¹⁹³__

The vicissitudes of the theories of precession and trepidation, the true versus the false, is one of the best examples of human inertia. It helps us not to be too optimistic, and to remain modest. If it is so difficult to establish scientific facts, which are relatively tangible and unambiguous, we should not expect much progress in other fields and be very humble and very patient.

**THE PRECURSORS OF SCIENTIFIC ASTRONOMY: PHILOLAOS, HICETAS, AND ECPHANTOS**

Philolaos was a contemporary of Socrates; Hicetas and Ecphantos, both of Syracuse, were younger; the first flourished perhaps and the second certainly in the fourth century. Their views were explained in a previous chapter (pp. 288–291), as it was more convenient not to separate Hicetas and Ecphantos from Philolaos, but we should remember that the fruition of their ideas occurred definitely in Plato’s age. These ideas may be summarized as follows. The universe is spherical and limited; the earth is not necessarily in the center, it is a planet like the others, and turns eastward around its axis.__¹¹⁹⁴__ Did Plato know of them? He mentions Philolaos in *Phaidon*;__¹¹⁹⁵__ considering his Pythagorean and Sicilian connections, he probably heard of the other men, yet he does not refer to them.

**THE FOUNDER OF SCIENTIFIC ASTRONOMY, EUDOXOS OF CNIDOS, AND HIS THEORY OF HOMOCENTRIC SPHERES**

We have given above an outline of Eudoxos’ biography and told that he spent sixteen months in Egypt (sometime in the period 378—364) and was admitted to the company of learned priests. Before that he had studied at the Academy and had become familiar with Pythagorean astronomy. Yet that did not suffice him and, being of a rigorous cast of mind, he may have been especially dissatisfied with the lack of observations. Not only did he obtain communication of Egyptian observations, but he made new ones and the observatory used by him, between Heliopolis and Cercesura,__¹¹⁹⁶__ was still pointed out in the time of Augustus (emperor, 27 B.C.–A.D. 14). Later, he built another observatory in his native land, Cnidos, and there he observed the star Canopus, which was not then visible in higher latitudes.

Eudoxos’ relatively long stay in Egypt accounts for his knowledge of Egyptian astronomy, but was he familiar with Babylonian astronomy, which was so much richer? There is no evidence of his having traveled in Mesopotamia and Persia, but he had a deep knowledge of the ancient world and wrote an elaborate description of it (*Periodos g s*) which was the first of its kind and scope. As far as can be judged from the fragments that have come down to us, Eudoxos’ geography contained a vast body of geodesic and topographic data, plus information on natural history, medicine, ethnology, religion. For example, he had noted the importance of Zoroastrianism, and Plutarch’s knowledge of Isis and Osiris was partly derived from his.__¹¹⁹⁷__ Far superior to the geographers of the fifth century, he is in this respect the forerunner of Eratosthenes of Cyrene (III–2 B.C.).

Even if Eudoxos did not travel to Mesopotamia, his residence in Cnidos enabled him to drink from Asiatic springs, whether Persian or Chaldean, for Cnidos (as well as its neighbors, Halicarnassos and Cos) was a cosmopolitan center of the first order. There is some probability that he is the author of bad–weather predictions (*cheim nos progn stica*),__¹¹⁹⁸__ which are definitely of Babylonian origin. He had identified the twelve main gods of the Greeks with the zodiacal signs. This is interesting, but we shall not insist upon it, for no matter how well he was acquainted with Egyptian and Babylonian astronomy his merit lies in another direction. No doubt he had profited by his study of Oriental methods of observation and he had played with Chaldean astrology, but no Oriental astronomer could have suggested to him his main achievement: the theory of homocentric spheres.__¹¹⁹⁹__

The purpose of that theory was to give a mathematical account of the positions of the heavenly bodies at any time, or, if we may use the strong Greek phrase, “to save the phenomena” (*s zein ta phainomena*). That was easy enough as far as the stars were concerned, but how could one account for the planets, the trajectories of which were extremely puzzling? At times they seem to stop, to retrograde, and to trace a singular curve such as the one that Eudoxos investigated and that he called the *hippoped* or horsefetter, a spherical lemniscate, looking like a figure eight. This was a difficult geometric or kinematic problem. He had to discover a combination of movements, circular or spherical, of such a nature that a single planet, say Mercury or Venus, would seem to describe in the heavens a *hippoped .*

Eudoxos’ solution of that problem was typical of the Greek mathematical genius and of his own. Mercury is supposed to be placed on the equator of a sphere centered upon the earth and revolving with constant speed around one of its diameters (on account of an old Pythagorean prejudice all the movements must be circular and uniform). Let us call that diameter after its two poles *AA′.* If that diameter did not change its position, Mercury (*M*) would describe a circle around the earth, but let us suppose that the diameter *AA′,* instead of being fixed, is carried by another concentric sphere turning with constant speed around the diameter *BB′*; the apparent motion of *M* will then be the combination of the two rotations, of speed *ω* around *AA′* and *ω′* around *BB′.* If that is not enough to “save the phenomena” we may suppose that the diameter *BB′* is not fixed but is carried by a third concentric sphere rotating with speed *ω′′* around the axis *CC′;* the apparent motion of *M* will then be the kinematic resultant of the three rotations of speeds *ω, ω′, ω′′* around the axis *AA′, BB′, CC′.* There is no need to stop at this third sphere; once the principle is accepted one may use as many auxiliary or starless (*anastroi)* spheres as are needed. And the problem can thus be stated in these terms: given the apparent trajectory of any celestial body, to find enough spheres concentric with the earth of speeds *ω, ω′, ω′′,* ... and axes *AA′, BB′, CC′,* ... to account for it. When a solution has been found, it can be checked as frequently as one pleases and in fact it is checked every time one compares the calculated position of that body with the observed one. If these two positions do not coincide, the solution may be improved either by changing the speeds and axes of the auxiliary spheres or by adding a new one.

In order to explain the motions of all the celestial bodies, Eudoxos was forced to postulate the existence of no fewer than 27 concentric spheres,__¹²⁰⁰__ each of which turned around a definite axis with a definite speed. The boldness of that conception is staggering. It was the first attempt to explain astronomic phenomena in mathematical terms; the explanation was very complicated (inasmuch as it required the simultaneous motion of 27 spheres rotating with different speeds around different axes), but it was adequate and elegant. It did “save the phenomena” with sufficient approximation. The working out of that solution implied an advanced knowledge of spherical geometry; it is probable that Eudoxos himself contributed to its advancement, for he needed it badly.

The theory of homocentric spheres is a magnificent example of Greek rationalism. Eudoxos introduced as many spheres as were needed for his kinematic purpose; he did not speculate on the real existence of those spheres or on the cause of their motions. It matters not, he might have said, whether the spheres exist or not, or why they move as they do; the only thing that matters is that their imaginary functioning together “saves the phenomena.” The theory provides a kinematic re–creation and verification of the observations.

Remarkable as it was, that theory was unavoidably imperfect; the observations available to Eudoxos were themselves insufficient in number and precision. His ideas on the sizes and distances of the celestial bodies were very crude. For example, we know from Aristarchos of Samos (III—1 B.C.) that he believed the diameter of the Sun to be nine times that of the Moon.

Eudoxos wrote two astronomic books, entitled the *Mirror* (*Enoptron*) and the *Phainomena,* a description of the heavens, which was the source of a famous astronomic poem by Aratos of Soli (III–1 B.C.).__¹²⁰¹__ The *Phainomena* of Eudoxos and of Aratos were commented upon by Hipparchos (II–2 B.C.) in his youth, and this commentary is, strangely enough, the only text of Hipparchos that has been transmitted to us in its integrity. Hipparchos corrected some of Eudoxos’ errors, for example, the latter’s belief that the north pole was occupied by a particular star; the north pole, said Hipparchos, is empty but there are three stars close to it [*α* and *κ* of Draco and *β* of Ursa minor] with which the point at the pole forms a square.

According to Diogenes Laërtios,__¹²⁰²__ Eudoxos wrote the *Octaët ris* while he was in Egypt. This may refer to an attempt made by Eudoxos to discuss or correct the eight–year cycle introduced by Cleostratos (p. 179), but we do not know the nature of his correction.

These are secondary matters, however. The fame of Eudoxos rests upon his invention and development of the theory of homocentric spheres, thanks to which he must be considered the founder of scientific astronomy and one of the greatest astronomers of all ages.

**THE ASTRONOMIC FANCIES OF PLATO AND PHILIP OF OPUS THE INTRODUCTION OF SIDEREAL RELIGION INTO THE WESTERN WORLD**

After having breathed the pure air of Eudoxean rationalism, it is a terrible shock to come down again to the low level of Platonic sublimities. Plato insists __¹²⁰³__ that “each planet moves in the same path, not in many paths, but in one only, which is circular, and the varieties are only apparent. Nor are we right in supposing that the swiftest of them is the slowest, nor conversely, that the slowest is the quickest,” and also __¹²⁰⁴__ that “the movements can be apprehended only by reason and thought, not by sight.” That is, he realized that the universe is a cosmos, but that its order and regularity could not be immediately deduced from the appearances. __¹²⁰⁵__ Eudoxos had proved that, for if it were possible to represent the motions of the celestial bodies by a kinematic system, these motions were well ordered; one might not know their causes or the rules producing them, but one could be certain that there were rules (natural laws).

The relation between Plato and Eudoxos is not clear. The latter was a younger contemporary of the former, and was for a time his pupil but left, whether rejected by the master or disgusted with his philosophy. There were certainly exchanges of influence between Eudoxos and the Academy. Eudoxos is nowhere mentioned by Plato. I suspect that they could not understand each other; they spoke different languages.

The astronomic views of Eudoxos have been explained in the previous section; these views were scientific views of the highest order. The observations at Eudoxos’ disposal were insufficient in number and precision, but his method was excellent. On the other hand, the Platonic views expressed in *Timaios* (and also in *Phaidon, Republic,* and *Laws)* are unscientific; Plato asserts this and that but he proves nothing, and his language is often as unclear as that of any soothsayer. His astronomic knowledge was of Pythagorean origin; it was far from up–to–date, being inferior not only to that of Eudoxos, but also to that of the later Pythagoreans like Philolaos and Hicetas. Let us give a brief outline of it.

The universe is spherical; in the center of it lies the Earth, which is also spherical and immovable, and stays there (in the center) for reasons of symmetry. The axis of the universe and of the Earth passes through their common center. The outer sphere of the universe rotates around that axis with constant speed in 24 hours, as is witnessed by the motion of the fixed stars. The Sun, Moon, and other planets are also carried round by the motion of the outer sphere, but they have other circular motions of their own. On account of these independent motions the planet’s real trajectory is a spiral in the zodiacal zone. The angular speeds of planets decrease in the following order: Moon, Sun, Venus and Mercury traveling with the Sun, Mars, Jupiter, Saturn. The order of distances from the Earth is the same, and the distances are deduced from two geometric progressions: 1, 2, 4, 8; 1, 3, 9, 12, the distances being: Moon, 1; Sun, 2; Venus, 3; Mercury, 4; Mars, 8; Jupiter, 9; Saturn, 12.

In the *Timaios*__¹²⁰⁶__ it is suggested that Venus and Mercury move in a direction opposite to that of the Sun.__¹²⁰⁷__ Plato knew the periods of Moon, Sun, Venus, and Mercury (the periods of the three last named he believed to be the same, 1 year),__¹²⁰⁸__ but not those of the other planets, yet he speaks of the Great Year __¹²⁰⁹__ when the eight revolutions (the seven planetary ones plus that of the outer sphere) are brought back to their starting point. That Great Year is equal to 36,000 years.__¹²¹⁰__ How did he measure it? He measured nothing, but copied a Babylonian tradition (see p. 71).

We leave out other fancies connecting the planets with the regular solids or with musical notes, the harmony of the spheres. The music of the heavens referred to in *Timaios* is not one audible to human ears, however. It may be caused by the relative speeds of the planets, yet it exists only in the world soul. Do not expect me to explain these mysteries.

According to Aristotle, Plato believed that the Earth rotates around its axis; according to Theophrastos, “in his old age Plato repented of having given the Earth the central place in the universe, to which it had no right.” These two sayings have caused polemics, but we are justified in rejecting them, because they contradict Plato’s own writings, all of which have come down to us.

The success of Plato’s astronomy, like that of his mathematics, was due to a series of misunderstandings: the philosophers believed that he had obtained his results by the aid of his mathematical genius; the mathematicians did not like to discuss the same results because they ascribed them to his metaphysical genius. He was speaking in riddles, and nobody dared to admit that he did not understand him for fear of being considered a poor mathematician or a poor metaphysician. Almost everybody was deceived, either by his own ignorance and conceit or by his subservience to fatuous authorities. The Platonic tradition is very largely a chain of prevarications.

**THE EPINOMIS**

We must still speak of a short dialogue called *Epinomis* (or *Nocturnal council,* or *Philosopher*)*.* As its main title indicates, it is an appendix to the *Laws*.__¹²¹¹__ The Nocturnal Council named in the second title was a secret body of inquisitors who had to make sure that the laws were obeyed. The *Epinomis* might be described as a discussion concerning the education of the members of that Council; but because this purpose appears only in the first and final paragraphs, the reader is likely to forget it. According to Diogenes Laërtios and Suidas, the *Epinomis* was written or published posthumously by one of Plato’s disciples, Philip of Opus.__¹²¹²__ Philip had been Plato’s secretary (*anagrapheus*) in the latter’s old age; he edited the *Laws,* divided them into twelve books, and added the *Epinomis.* Many other writings are ascribed to him, dealing with mathematics (for example, on polygonal numbers and on means), astronomy (on the distances of the planets, *parapegma,* which are astronomic tables or almanacs), optics, meteorology, ethics. Was he the author or simply the editor of the *Epinomis*? And if he was the editor, how far did his editorship extend? It is impossible to answer such questions. We must take the *Epinomis* as we find it (the text itself gives no clue as to authorship or editorship). It is Platonic in form and contents, though more Pythagorean than the other Platonic writings. The astronomy of *Epinomis* is essentially the same as that of *Timaios,* except for the stronger Pythagorean note, which concerns the metaphysics of astronomy rather than astronomy proper.

The main purpose of the *Epinomis* is to emphasize the importance of astronomy for the attainment of true wisdom. As expressed by a master of the history of ancient religion, the late Franz Cumont, the *Epinomis* was “the first Gospel preached to the Hellenes of the stellar religion of Asia.”__¹²¹³__ That religion was born in Babylonia, where priests were astronomers and where an amazingly clear sky invited observations. At the beginning of the Achaimenian empire (Cyros the Great, ruling 559–529), which included Babylonia, these ideas were diffused by Persian priests called magi and by native ones called Chaldeans. It reached Greek–speaking people from both sources (Persian and Chaldean) and the *Epinomis* was its first evangel in the Greek language.

The argument of the *Epinomis* is far from clear, but we shall indicate some of the salient ideas. Number is extremely important, and nowhere more obviously so than in the regular motions of celestial bodies, the stars, Sun and Moon, and the planets. The five regular solids are equated to the five elements, but the fifth element is called aether.__¹²¹⁴__ The soul is older and more divine than the body. Order is equated to intelligence, and disorder to nonintelligence; the supreme order of the celestial motions represents supreme intelligence. There are eight “powers in heaven” (the seven planets and the eighth sphere), which are equally divine. The planets must be gods. That is what the Egyptians and Syrians (meaning Babylonians) have known for thousands of years; we should accept their knowledge and their religion, after having improved it, as the Hellenes always do when they borrow anything from the barbarians. While paying due reverence to the ancient gods, according to venerable traditions, the cult of the visible gods, the celestial bodies, should become a state religion. That religion would give the Hellenes a vision of divine unity, and would provide for them a bond (*desmos*) universal and immaterial. Note that many of the astrologic fancies were already expressed in other Platonic writings (*Phaidros, Timaios, Laws*). The novelty lies in the religious accent, the equation of astronomy and piety, the idea of an astrologic state religion.

The aim of wisdom is to contemplate numbers, and especially celestial numbers. The most beautiful things are those revealed to our understanding by our own souls, or the cosmic soul, and the heavenly regularities.__¹²¹⁵__ The cult of the stars must be introduced into the laws.

Astronomy is not only the climax of scientific knowledge, it is the rational theology. Members of the Nocturnal Council should receive a mathematical education, leading them to astronomy and religion. The supreme magistrates of the city are not to be philosophers, but rather astronomers, that is, theologians.

There are so many irrational statements in *Epinomis* (parading under the garb of supreme rationalism) that discussion of them would be as endless as it would be futile. There is one point, however, that I would like to touch, because it has puzzled me more than the others and has received but little attention. The author criticizes foolish people for associating initiative (freedom) with intelligence;__¹²¹⁶__ real intelligence, however, is characterized by repetitious order; the planets reveal supreme divine intelligence by the eternal accuracy of their motion. Now we might admit with Plato that the planetary motions reveal God, but not that the planets themselves are gods. Think of the popular argument of the clock. Its mechanism and regular motions reveal the existence of a clockmaker. Nobody ever said that the clockmaker was in the clock, or that the clock was itself the clockmaker. Yet, according to the new astrologic religion, the planets did not simply reveal God, they were themselves gods, each planet regulating its own motion with divine intelligence and repeating it eternally to evidence its own wisdom. Does that make sense? Yet the argument was accepted by the New Academy and by the Stoics and we find it very clearly stated by Cicero.__¹²¹⁷__ The confusion of thought was probably caused by a wrong generalization: the soul or intelligence of an animal is within itself; we may say that the animal has intelligence or that he is an intelligent being; his intelligence, however, is revealed not by the regularity and precision of his motions, but rather by their unexpectedness.

It is significant that the *Epinomis,* first Greek gospel of the astrologic religion, does not include any astrology in the vulgar sense. There is a passing reference __¹²¹⁸__ to the divinity of birth (*to theion t s genese s*), but it is not clear and does not prove that the author had accepted the fundamental assumption of practical astrology: that a man’s fate is determined at the time of his birth (or conception?) and can be deduced from the study of his horoscope.__¹²¹⁹__ Yet judicial or mundane astrology had been practiced in Babylonia from time immemorial, and as to the Greeks, if they took astronomy and divination with equal seriousness, the step to astrologic divination was unavoidable.

If we believe that the stars or planets are gods and that there are communications between them and us, we cannot but conclude that they must influence us. The communication is sufficiently proved by the fact that we see them, for this implies that something passes from them to us.__¹²²⁰__ Astrologic divination becomes possible only after certain assumptions had been made, such as the one referred to above, and after the establishment of “scientific” horoscopes had been determined by the acceptance of a series of conventions.__¹²²¹__

In any case, the astrologic religion first explained in the *Epinomis* became gradually the highest religion of the pagan world, Greek and Latin. The old gods were still worshiped and the old myths were celebrated by the poets and the artists, but thinking people could no longer acquiesce in them except in a traditional mood and with poetic ambiguities. As compared with the mythologic childishness and immorality, the cult of the stars seemed highly rational. Not only the *Epinomis,* but Pythagorean and Platonic ideas in general, provided a philosophic substructure upon which the new religion could be established so solidly that most members of the intellectual elite took it as a kind of science. The influence of that “Pagan science” upon the best minds of the Roman empire was so deep that Christianity itself could not completely erase it. Indeed, it is represented to this day by one of our oldest and most ubiquitous institutions, one that controls the labor and rest of every man, the week. The number of days in the week is of astrologic origin, and their names in most European languages are planetary names.__¹²²²__