It is convenient to divide this chapter into three sections — Mathematics, Astronomy, Technology — in spite of the fact that this may oblige us to come back twice or thrice to the same personalities.



Students of early Greek mathematics are kept in a state of continual amazement by two complementary (or contradictory) facts: the neglect of plain arithmetic, and the singular depth of mathematical thought. The early Pythagoreans did not pay attention to ordinary methods of reckoning, yet their geometric ideas were largely based on numbers. A point for them is simply a unit having position; any geometric figure, beginning with the straight line, can be conceived and represented as a number of points. This raises the problems of continuity and infinite divisibility, or, more exactly, it raised those problems in the Greek minds, because those minds were ready for philosophical discussion. We have many evidences of Greek genius, but none is stronger and more startling than the mathematical thinking of this time, incited by logical difficulties that the average man of today (twenty-five centuries later) would hardly notice. As a first approximation, one is tempted to say that the more intelligent people are the quicker they are in their understanding; but one is very soon obliged to abandon that statement and almost to reverse it. Foolish people understand quickly, or believe they do, because they are not capable of imagining the difficulties and hence have no hurdles to leap over. The immense difference between Egyptian and Babylonian mathematics, on the one hand, and Greek mathematics, on the other, consists in that the former did not even conceive of some of the difficulties with which the Greeks were now beginning to struggle.

We may recall that Zenon visited Athens with his master Parmenides about the middle of the century. It was perhaps in Athens that he came across mathematicians like Hippocrates who were then trying to reduce geometric knowledge to a closely knit system. Being primarily a philosopher and a logician, Zenon perceived conceptual difficulties that would never have occurred to practical mathematicians (even Greek ones!). These would consider a straight line as made up of points. How can we reconcile that notion with the line’s continuity? The line is not a series of points or, to put it otherwise, a series of holes; it is a continuous whole. The practical mathematician would say: The points can be brought as near to one another, and the holes be as small, as you please; if the distance between two points is too large to satisfy you, well, divide it into a thousand or a million parts and imagine additional points in all those parts. The logician would demur and answer: The actual distance between any two points does not affect the argument; however small that distance may be, the two points remain separate and are different from the line or space joining them. Similar difficulties concerned the divisions of time (should we conceive it as continuous or discontinuous?) and motion (the passing of a body from one place to another in a given time). The paradoxical results of Zenon’s meditations on these riddles are known to us through the Physics of Aristotle,⁶⁵⁸ who called them fallacies, yet was unable to refute them, partly through the commentary on Aristotle by Simplicios (VI–I), and they cut so deep that they have exercised the minds of philosophers and mathematicians to our own day. Those questions are so subtle that to give a complete and accurate account of them would take considerable space. It must suffice here to indicate their general nature. Following Cajori’s model, we shall call Zenon’s four arguments against motion by the names “dichotomy,” “Achilles,” “arrow,” and “stade,” and summarize them as he does:

1. Dichotomy. You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you traverse the whole, and the half of that again before you can traverse it. This goes on ad infinitum, so that (if space is made up of points) there are an infinite number in any given space, and it cannot be traversed in a finite time.

2. Achilles, the second argument, is the famous puzzle of Achilles and the tortoise. Achilles must first reach the place from which the tortoise started. By that time the tortoise will have got on a little way. Achilles must then traverse that, and still the tortoise will be ahead. He is always nearer but he never makes up to it.

3. Arrow. The third argument against the possibility of motion through a space made up of points is that, on this hypothesis, an arrow in any given moment of its flight must be at rest in some particular point.

4. Stade. Suppose three parallel rows of points in juxtaposition:

Fig. 1

Fig. 2

One of these (B) is immovable; while A and C move in opposite directions with equal velocity so as to come into position represented in Fig. 2. The movement of C relatively to A will be double its movement relatively to B, or, in other words, any given point in C has passed twice as many points in A it has in B. It cannot, therefore, be the case that an instant of time corresponds to the passage from one point to another.⁶⁵⁹

The four arguments seem to be directed against the belief held by most people at that time (including the Pythagoreans and Empedocles), and still held by the majority of people of our own time, that space is the sum of points and time the sum of instants. Zenon argued that motion was not conceivable on a pluralistic basis.


Democritos was born about thirty years later than Zenon. The dates of his birth and death are uncertain, but we shall not be far wrong if we write c. 460, c. 370. It does not follow that the mathematical speculations of Democritos were later than those of Zenon and that Democritos was acquainted with Zenon’s perplexities. At any rate, those perplexities, or others of the same kind, were unavoidable as soon as one began to think rigorously on continuity and infinity, and the Greeks — not one of them, but many — were doing just that. In the catalogue of Democritos’ works published by Diogenes Laërtios (III-1), five mathematical works are listed: (1) on the contact of a circle and a sphere, (2 and 3) on geometry, (4) on numbers, (5) on irrationals. We shall come back to the last one when we discuss that topic presently. The titles of items 2 to 4 are too vague to be useful. As to the first item, if we assume the title to mean the contact between a sphere and a tangent plane, we are drawn to the consideration of an infinitesimal angle. If we consider the simpler case (as Democritos probably did) of the angle between a circle and a tangent, the inherent difficulties obtrude rapidly. First, it was necessary to define the tangent; Democritos’ mind was sharp enough to realize that the tangent and circle had only one point in common, though this could not be illustrated by any drawing. Then the angle must be considered. It had to be exceedingly small, because if the tangent were turned ever so little around its point of contact it would include a second point of the circle, and it would cease to be tangent.

Plato ignored Democritos, but Aristotle spoke very warmly of his ideas on change and growth. A century later Archimedes referred to Democritos’ greatest mathematical discovery, that the volumes of a cone and a pyramid are one-third of the volumes of the cylinder and prism having the same base and height, adding that the proofs of Democritos’ theorems were given not by him, but later by Eudoxos.⁶⁶⁰ How did Democritos discover them? It is probable that he used a crude and intuitive method of integration, dividing the pyramid (or cone) into a large number of parallel slices. We shall come back to that when we discuss Eudoxos’ discovery and use of the method of exhaustion.

The beginnings of perspective as applied to the designing of stage scenery were ascribed by Vitruvius to Democritos as well as to Agatharchos and Anaxagoras. These ascriptions are plausible but unproved. It is certain that problems of perspective had to be solved by the designers of scenery, but good solutions could be found in an empirical way.


We now come to the greatest mathematician of the century, the first man to illustrate the name of Hippocrates. Almost every educated person is familiar with that name, but it evokes in his mind the memory of another man, the father of medicine, Hippocrates of Cos. The name Hippocrates is not uncommon in Greece,⁶⁶¹ but it is remarkable that the two most illustrious bearers of it were contemporaries and came from the same group of islands, the Sporades off the coast of Asia Minor. The mathematician, who was the older man, was born in Chios and flourished in Athens during the third quarter of the fifth century. The physician belonged to the following generation; he was still a child when the mathematician was in his maturity, and he was active at the turn of the century. He came from the island of Cos (one of the Southern Sporades, the group also called Dodecanese islands).⁶⁶² We shall give him all the space that he so fully deserves in another chapter, but it was necessary to introduce him here and to place him for a moment near his older contemporary. I much hope that the readers of this book will remember that there were two Hippocrates’, whose achievements were equally outstanding but so different that no comparison between them is possible. One could certainly not say that the second was a greater man than the first, yet he alone is remembered by most people while the older man is almost forgotten. Well, never mind.

The reason for Hippocrates’ arrival in Athens about the middle of the century is traditionally believed to have been the loss of his possessions and his attempt to recover them. According to one story, he was a merchant whose ship had been captured by pirates; according to another story (told by Aristotle),⁶⁶³ he was a geometer robbed of much money by the customs.collector at Byzantium “owing to his silliness.” Of course, mathematicians (from Thales down to Poincaré) have often been accused of being incompetent in ordinary life, but those stories are interesting in other respects; they help us to realize the existence of other sides of Greek life: the merchants, the pirates, and the wicked customhouse officers. Apparently, Hippocrates had started to be a merchant as well as a mathematician; the combination was not incongruous in the Greek community. Having lost his belongings, he applied himself to mathematics and was one of the first to teach for money; why should he not have been remunerated as well as the sophists; he might be called a sophist himself, though he specialized in the mathematical field.

Before explaining his work, we must recall another story which is very typical of the intellectual climate of that time. Three famous problems were then exercising the minds of the Athenian mathematicians: (1) the quadrature of the circle, (2) the trisection of the angle, (3) the duplication of the cube. How did these three problems emerge? The first is very ancient, and it was as yet impossible to know that an exact solution of it cannot be obtained. The two others are less natural. At least two legends, both reported by Eratosthenes, were circulated to account for the third problem. It will suffice to tell one of them. The people of Delos, suffering from a pestilence, were ordered by the oracle to double a certain altar which had the form of a cube; therefore, that problem was called the Delian problem. The legend has all the earmarks of an invented post factum,and, as far as I know, there never was a cubic altar in Delos or anywhere else.⁶⁶⁴ A simpler explanation is that some mathematicians may have wished to generalize a problem of plane geometry. To double a square it suffices to draw a new square on its diagonal. Could not a similar rule be found for the cube? It was not as easy as it looked. The emergence of these three problems, out of an infinity of others, is a new proof of the Greek genius, for they all combine apparent simplicity with inherent difficulties of a very high order.⁶⁶⁵ They are insoluble except approximately; the second and third cannot be solved by simple geometric methods (that is, with ruler and compass); yet they were solved theoretically by the Greek mathematicians of the fifth century.

Fig. 63. The lunes of Hippocrates of Chios.

Hippocrates did not deal with the second problem, but we owe to him incomplete solutions of the two others. His attempts to square the circle led him to the discovery of lunes which could be squared; strangely enough, he discovered three out of the five species of lunes that can be squared in a simple way. This must have been very exciting, for it proved that at least some curvilinear figures were susceptible of quadrature.

Here is the simplest example of Hippocrates’ lunes. Consider the half square ABC inscribed in the semicircle with center O (Fig. 63). Let us draw another semicircle on AB as diameter. The two semicircles are to each other as the squares of their diameters: AC² = 2AB². Hence, half of the larger semicircle is equal to the smaller one. Take out the segment common to both areas, and the remaining areas, to wit, the lune and the triangle ABO, are equal.

This is simple enough, yet it implies knowledge of the proposition that circles are to one another as the squares of their diameters.⁶⁶⁶ If Hippocrates found the area of that lune, we must assume that he knew that proposition. His knowledge of it might have been intuitive; according to Eudemos, he was able to prove it, but if so, we do not know how he did it.

Hippocrates’ work on the squaring of the lunes is very important in another way: it is the only fragment of Hellenic (pre-Alexandrian) mathematics that has been transmitted to us in its integrity, but the transmission was very indirect and slow.⁶⁶⁷

This illustrates once more how hard it is to know the facts of early Greek mathematics and how prudent the historian must be.

His solution of the third problem, the duplication of the cube, is equally interesting in its implication, for it shows that he had a clear understanding of compound ratios. That knowledge was derived from numbers and applied intuitively to lines.

If the side of the given cube is a, the problem is to determine x, such that = 2a³. It is solved by finding two mean proportionals in continued proportion between a and 2a: a/x = x/y = y/2a; for then = ay, y² = 2ax; hence x⁴ = 2a³x or = 2a³.

By the middle of the fifth century, so many geometric theorems had already been established and so many problems solved that it became increasingly necessary to put all these data in good logical order. This implied not only the classification of the results already obtained but, what is more important, the strengthening of the proofs. In many cases (as illustrated above about the proposition in Euclid) knowledge was intuitive, or the proof, if it had been found, had failed to be transmitted. If every item were put in its logical place, the gaps would be detected. The geometric edifice as far as it could be built would be stronger, and one would know more definitely what to do in order to bring it nearer to completion and logical perfection. It would seem that Hippocrates was one of the first to attempt that task, that is, he was the first forerunner of Euclid, not only as a discoverer of individual propositions, but as a builder of the geometric monument that was to be called later the Elements.

If the text of Hippocrates relative to the quadrature of lunes transmitted to us by Simplicios is really as he wrote it, then he is the first mathematician known to us who used letters in geometric figures and thus made possible the unambiguous description of such figures.⁶⁶⁸ The manuscript tradition was thus greatly facilitated, for the figures, which may be difficult to draw neatly, can be omitted. They are not indispensable, for the reader can easily reconstruct them on the basis of the text. We are not surprised to find that Hippocrates’ use of letters was not yet quite as simple and clear as Euclid’s, but it was a very important beginning, almost necessary for the future progress of mathematics.

Hippocrates writes “the line on which is AB,” or “the point upon which is K,” while Euclid and we ourselves simply write “the line AB,” “the point K.” Such differences occur repeatedly in the history of mathematical notations, and, we might say more generally, in the history of science. The inventor is seldom able to express his invention in the simplest manner, and it takes another man, or many men, less intelligent but more practical than himself, to complete the invention. Hippocrates’ invention might have been perfected, for example, by other teachers or even by students who would use the shorter phrase “the line AB” out of sheer laziness.

If Hippocrates actually wrote the first textbook of geometry, which is not only possible but plausible, he was obliged to tighten the proofs, and we may believe Proclos’ statement that he invented the method of geometric reduction (apagõg ), that is, the passage from one problem or theorem to another, the solution of which entails the solution of the former. We shall discuss that later on.

The achievements of Hippocrates of Chios were considerable, so great indeed that he might be called the father of geometry with as much justice as Hippocrates of Cos was called the father of medicine. It is better to avoid such metaphors, however, for there are no absolute fathers except Our Father in Heaven.


According to Proclos (V-2), Oinopides was a little younger than Anaxagoras; he places him ahead of Hippocrates and Theodoros. We may assume that Oinopides flourished in the third quarter of the century. It is interesting to note that he was not only the contemporary of Hippocrates, but also his fellow citizen. They must have known each other either in Chios or in Athens. Whether he was a little younger than Hippocrates or not hardly concerns us, for what matters is the chronologic order of discoveries, which is different from the chronologic order of birth dates; some men do their best work when they are young, others in old age.

Oinopides is more important as an astronomer than as a mathematician, and we shall devote more space to him in the second section of this chapter. His mathematical contributions are modest, yet significant. He was the first to solve the following problems: (1) to draw a perpendicular to a given straight line from a given point; (2) at a given point of a straight line to construct an angle equal to a given angle.

As everybody could solve these problems roughly, the ascription of their solution to Oinopides means that he was the first to show how to solve them rigorously with ruler and compass. Such problems had to be solved to make possible the writing of the Elements, yet Proclos says that Oinopides solved the first problem for astronomic reasons; he also says that Oinopides used the old name for perpendicular (cata gn mona instead of orthios). All of which illustrates the transitional nature of this period: geometric knowledge is gradually ordered and crystallized, the Elementsare in the making.


Hippias came from Elis,⁶⁷⁰ a small country in the northwest corner of the Peloponnesos, renowned for horse breeding and almost sacred to the Greek people on account of the Olympic games which took place every fourth year in the plains of Olympia. He was born c. 460 and is much better known than his seniors Hippocrates and Oinopides, because he traveled considerably all over Greece giving public lectures and teaching; he was a kind of wandering sophist whose activities were dominated by the love of fame and of money. He was ready to discuss any subject but was especially interested in mathematics and science. When he reached Sparta he was disappointed, because the Spartans did not care enough about science to reward scientific lectures. He is immortalized by two Platonic dialogues, the Hippias major and the Hippias minor, wherein he appears as a sophist, vain and arrogant. This is not attractive, yet his mathematical fame is secure because of a single discovery, which is truly astonishing.

Fig. 64. The quadratrix of Hippias of Elis.

In order to solve the problem of the trisection of the angle, Hippias invented a new curve, the first example in history of a higher curve, a curve that could not be drawn by any instrument but only point by point. That is, at the very time when the best mathematicians are working hard to consolidate geometric knowledge into a well-ordered edifice, he is bold enough to jump out of it and begin exploring the mysterious unknown outside of it.

The curve discovered by Hippias was called the quadratrix (its name will be justified later on) and generated as follows (Fig. 64). Suppose we have the square ABCD (side a) and within it the quarter of a circle of radius a with center at A. Let us assume that the radius turns with constant speed from the position AB to AD, and that in the same time the side BC moves down to AD with constant speed, remaining parallel to itself. The locus of the points of intersection (such as F and L) of the two lines is the quadratrix. Now < BAD: < EAD = arc BD: arc ED = BA:FH. Consider the vector AF connecting the center A with a point F of the curve; let its length be ρ and let it make an angle φ with AD; then a/ (ρ sin φ) = (π/2)/φ.

The curve can be used to trisect any angle such as φ. Let us divide the line FH into two parts in the ratio 2:1, so that FF1 = 2F1H. Then draw the line B″C″ which cuts FH in F1 and the curve in L, and join AL. The angle NAD will be one-third of φ.

The curve could be used equally to divide any angle in any ratio; it would suffice (in our example) to divide the line FH in that ratio and continue the construction as before.

The same curve was used a century later by Deinostratos (IV–2 B.C.) and others for the quadrature of the circle, and it is for that reason that it received the name quadratrix (tetrag nizusa).


The mathematician Theodoros of Cyrene ⁶⁷¹ is best known to us because of the beginning of Plato’s Theaitetos, in which he is introduced as a famous master. He was then (399) ⁶⁷² an old man, hence we might say that he was born c. 470. It is said that Plato had visited him in Cyrene; at any rate, Theodoros was in Athens about the end of the century; he belonged to the Socratic group and was (or may have been) the mathematics teacher of Plato. A single mathematical discovery is ascribed to him, but it is a startling one. He is said to have proved the irrationality of the square roots of 3, 5, 7, ..., 17.

It is significant that the discovery of the irrationality of V2 is not ascribed to him, which can only mean that it was known before him. Indeed, such knowledge is ascribed to the early Pythagoreans. The discovery of the irrationality of √2 was a shocking surprise, and the Pythagoreans seem to have thought for a time that it was an exception.

The square root of 2 appears very naturally and simply, because it is the diagonal of the unit square (side and area equal 1). How did the early Pythagoreans discover the irrationality of √2?

We must first introduce another man, Hippasos of Metapontum,⁶⁷³ an early Pythagorean about whom some extraordinary stories circulated. They said that he had been expelled from the Pythagorean school for having revealed mathematical secrets. According to one tradition, he had revealed the construction of a dodecahedron in a sphere and claimed it as his own; according to another, he had revealed the discovery of irrational quantities — and this would very probably refer to √2 or to √5. Before we abandon Hippasos, one more mathematical thing may be said of him. The early Pythagoreans distinguished three kinds of mean — arithmetic, geometric, and subcontrary.⁶⁷⁴ Hippasos suggested that the name harmonic be given to the third, a name well applied because of the importance of the harmonic means in musical theory, and he defined three other medieties (means). Let us now return to the discovery of the existence of irrational quantities, which was for the mathematicians of the sixth and fifth centuries a kind of logical scandal.

An irrational number (alogos) is one that cannot be expressed as a quotient of two integers; the discovery was made geometrically when it was realized that the diagonal of a unit square could not be measured in terms of the side or in terms of any of the equal parts, however small, into which that side can be subdivided.

How could one prove that irrationality? The traditional proof is referred to by Aristotle; ⁶⁷⁵ it is a reductio ad absurdum. The argument is so short and easy that we reproduce it.

Consider the square whose side is a and diagonal c. We must prove that a and c are incommensurable. Let us assume that they are not, and that their ratio c/a is expressed in the simplest manner by γ/α. Then c²/a² = γ²/α²; but c² = 2a²; thus γ² = 2α². Hence γ² is even and γ is even, and a must be odd. If γ is even, we may write γ = 2β; then γ² = 4β² = 2α², so that, α² = 2β². Hence α² is even, and α is even. We have found that α is both even and odd, which is impossible; hence a and c are incommensurable.

Fig. 65. Pentagons and pentagrams.

It is quite possible that the first irrational quantity was discovered by Hippasos (if not before), but one cannot prove that. One is tempted to assume it because of the tradition reported above, and because this leaves a little time for the theory of irrationality to develop. Yet the proof of the irrationality of √2 that has just been given, simple as it is, implies a degree of abstraction that can hardly have obtained as early as Hippasos’ time, say the beginning of the fifth century. But another tradition ascribed to Hippasos some knowledge of the dodecahedron, a regular solid whose twelve faces are regular pentagons. An interest in the pentagon was natural enough to a Pythagorean, whose symbol was a pentagram (a regular pentagon whose sides have been extended to their points of intersection).

Now Kurt von Fritz has made the very interesting suggestion ⁶⁷⁶ that Hippasos’ interest in pentagrams and pentagons and the numbers and ratios incorporated in them would have led him to the notion of incommensurability. How would a craftsman try to find the common measure of two lines a and b? He would try to measure the longer a in terms of the shorter b, and if that failed, he would try to measure it in terms of fractions of b. Now, such a method could not be applied in this case, because of the coarseness of physical measurements. Yet, if Hippasos had considered the pentagon with all its diagonals, he would have seen that the diagonals constitute a pentagram and enclose a smaller pentagon (Fig. 65). The same procedure might be continued, and this would be tempting enough. In practice, one could not continue it very long, but it is obvious that in theory it might be continued indefinitely, and this means that the diagonals and the side are irreducible to a common measure, are incommensurable.

The discovery of incommensurable quantities might have been made intuitively by Hippasos before their existence had been completely proved. It is possible even that Greek mathematicians had begun before the end of the century to consider more complicated cases. In Hippias major(303 B.C.) occurs the remark that just as an even number may be the sum of two even or two odd numbers, even so the sum of two irrationals may be either rational or irrational. A good example is that of the rational line cut in extreme and mean ratio; the three ratios of those segments and of the whole line are irrational.

Fig. 66. Simple construction of various incommensurable quantities.

Assuming that Hippasos had discovered the irrationality of √2 and √5, how did Theodoros find the other surds up to ? He might have constructed many of them in an easy way, as shown in Fig. 66. Once the possibility of irrational quantities was realized and admitted, it was not difficult to find new ones. The main difficulties were of another kind: if there were numbers that could not be represented by any ratio such as n/m, then the Pythagorean similitude between numbers and lines or between arithmetic and geometry could not be maintained any longer — or could it? We have no reason to suppose that these deep difficulties were solved before the fourth century, but a long period of ferment of ideas represented by Hippasos and Theodoros⁶⁷⁷ prepared the age of Theaitetos and Eudoxos. We shall continue our discussions of this topic when we reach that age.

The Greek genius had intuitions of mathematical truth just as it had intuitions of beauty. It seems to have understood, if not at the very beginning, at any rate very early, that mathematics cannot be built up with logical rigor without solving many problems implying infinity. The uncanny depth of that genius may be better appreciated if we bear in mind that there are many educated people, well-educated even, say physicians or grammarians, who would hardly understand such things today, not to mention discover them. This chapter has already given many examples of Greek intuitions concerning infinity, to wit, the views explained by Zenon, Democritos, Hippasos, Theodoros, and now we add two more, Antiphon and Bryson.


Antiphon was an Athenian, who flourished at the same time as Socrates, and was to some extent the latter’s rival as an educator of youth. He was a sophist, interested in many sciences but also in divination and the interpretation of dreams. We should never forget that divination and particularly oneiromancy⁶⁷⁹ were then legitimate parts of science, attracting the intelligent curiosity of some of the best minds, for the limitations of knowledge could not be as keenly realized then as they are now. He deserves our attention, however, because he invented a new method for the solution of the old problem, the quadrature of the circle.

Antiphon suggested that a simple regular polygon, say a square, be inscribed in the given circle. Then an isosceles triangle could be built on each side, its vertex being on the circumference. A regular octagon would thus be constructed, and continuing in the same manner one would easily construct regular polygons of 16, 32, 64, ... sides. Now it is obvious that the area of each of those successive polygons comes closer to the area of the circle than the preceding one, or, to put it otherwise, as more complex polygons are inscribed in the same circle, the area of that circle is gradually exhausted. Now the areas of these polygons can be exactly computed, or the polygons can be “squared”; they increase gradually though they cannot possibly increase beyond a given limit, the area of the circle itself.

This method was criticized by Aristotle, his commentators, and others on the ground that no matter how many times the number of sides of each polygon is doubled, the area of the circle can never be completely used up.


Bryson was the son of the logographer or mythologist Herodoros of Heraclea Pontica.⁶⁸¹ He was a pupil of Socrates and also of Socrates’ disciple, Euclid of Megara. Bryson thus belongs to a later generation than Antiphon and must have flourished in the first half of the fourth century, but we must speak of him here, because his work completes so well that of Antiphon.

While Antiphon’s method consisted in drawing in the circle a series of polygons with 4, 8, 16, 32, . . . sides, Bryson proposed to construct a series of polygons circumscribed about the same circle. The areas of the circumscribed polygons diminish gradually. The area of the circle is the upper limit of the inscribed polygons and the lower limit of the circumscribed ones. Of course, Bryson was subject to the same criticisms as Antiphon and he was duly criticized by Aristotle, Simplicios, and many historians of mathematics.

It would seem to me that the modern historians (such as Rudio⁶⁸² and Heiberg) have been unduly severe to Antiphon and Bryson. The procedure of the latter lacked rigor but it derived from a sound intuition, and it led eventually to the method of exhaustion (formulated by Eudoxos) and to the integral calculus.

One cannot deny Bryson a definite discovery: that the area of the circle is a limit to the increasing areas of inscribed polygons and to the decreasing areas of the circumscribed polygons, and as the number of sides of those two series of polygons are increased, their areas approach closer to the area of the circle on both sides of it. The method was actually applied by Archimedes (III–2 B.C.), who measured the areas of two inscribed and circumscribed polygons of 96 sides each and reached the conclusion that < π < (3.141 < π < 3.142).

Before concluding this section it is worth while to observe that the men whose mathematical ideas have been reviewed (excepting perhaps Hippocrates) were not mathematicians in the restricted sense of today; they were philosophers and sophists who realized the fundamental importance of mathematics and tried to understand it as well as possible. Note that they came from many parts of the Greek world. Zenon hailed from Magna Grecia, Hippocrates and Oinopides from Ionia, Democritos from Thrace, Hippias from the Peloponnesos, Theodoros from Cyrenaica, Bryson from the Black Sea; Antiphon was, as far as we know, an Athenian, the only one among them. If we had spoken of Archytas, who is astride of both centuries⁶⁸³ and will be dealt with later, another country would have to be added to the list, Sicily. This shows that the mathematical genius was as widely distributed across Hellas as was the artistic or literary genius. That genius was not Athenian, or restricted to any locality; it was the genius of Greece.


In our review of astronomic ideas in the fifth century we may leave out those of the philosophers like Heraclitos, Empedocles, Anaxagoras and restrict ourselves largely to the Pythagoreans. Indeed, the Pythagorean school was the leading school of astronomy in that century and the most progressive. Their mathematical mysticism had its useful side, for it helped to assume regularities in the celestial motions and to discover planetary laws. As Plato put it,⁶⁸⁴ “as the eyes are designed to look up at the stars, so are the ears to hear harmonious motions, and these are sister sciences as the Pythagoreans say.” This expresses beautifully the Pythagorean concept of the unity of mathematics, music, and astronomy, which influenced astronomic thinking down to the time of Kepler.

When we speak of Pythagorean astronomers, we do not mean only those who were fully initiated in all the Pythagorean mysteries, but also those who accepted, if only in part, the Pythagorean views on the system of the world. Thus, we shall begin our account with Parmenides (who was not a Pythagorean but the founder of the Eleatic school), and then deal with Philolaos, Hicetas, and a few others.

The Pythagoreans were the first to call the world cosmos (implying that it is a well-ordered and harmonious system) and to say that the earth is round. These designations are ascribed to Pythagoras himself and also to Parmenides; it is not easy to separate Parmenidean inventions from older Pythagorean doctrines, but we need not worry too much about that. The first section of our account may be understood as representing not only the views of Parmenides but also those of the Pythagoreans about the middle of the century. By that time, some points of Pythagorean cosmology were already determined; the universe is a well-ordered system; the most perfect shape is the sphere and the earth is round;⁶⁸⁵ the planets are not “errant” bodies but have regular motions; those motions are uniform. It is possible that other ideas were already accepted, such as the divinity of the stars and planets, and the essential duality of the world — superlunar (perfect) and sublunar (imperfect).⁶⁸⁶ Such ideas take us away from astronomy into mythology and religion. Their coexistence with the other, more scientific, ideas explains the paradox that the Pythagorean school was at one and the same time the cradle of mathematical astronomy and the cradle of astrology. These two aspects are apparently incompatible, yet they reappear repeatedly throughout the history of science (at least down to the seventeenth century). One cannot understand the development of ancient and medieval astronomy unless one bears that essential polarity constantly in mind.


Parmenides came to Athens about the middle of the century, but he was then in his fifties and it is possible that his astronomic views were already crystallized. He was the first to assume that the spherical earth was divided into five zones, but those zones were not well defined, and he conceived the central zone, torrid and uninhabited, to be twice as broad as it really is. We cannot attach much importance to those zones, which were too speculative. As to the spherical shape of the earth, we do not know how the early Pythagoreans, Parmenides among them, reached that conclusion. It is probable that it was at first an a priori conception but that it was promptly and repeatedly confirmed by observation of the stars. The world known to the Greeks extended from at least latitude 45° N (top of the Black Sea) to the Tropic of Cancer or even farther — a belt that was 20° to 25° of latitude wide. Now, this was more than enough to observe considerable changes in the starry heavens. As one traveled northward certain stars became circumpolar; on the other hand, a very bright star (Canopus), invisible in Greece proper, was just visible above the horizon in Crete and it rose higher as one proceeded to Egypt and sailed up the Nile. Moreover, travelers must have noticed the lengthening days as one went northward; this was enough to lead to the idea of zones. Parmenides was the first to conceive the universe as a continuous series of spheres or crowns (stephanai) concentric with the earth, which is at rest at the center. We need not recall his other astronomic views, some of which were not new (for example, that the Moon derives its light from the Sun) or were mere figments (for example, that the Moon and Sun are fragments from the Milky Way). It is remarkable, however, that a pure metaphysician, as he was, could guess so much of the truth, his vague anticipation of geographic zones is almost as astonishing as Democritos’ anticipation of atoms.


Philolaos came from Croton or from Tarentum (both places are in the same region, the Gulf of Tarantum). It was in Croton that Pythagoras had established his school; it is not surprising then that Philolaos is quoted as a Pythagorean. He was a contemporary of Socrates, but so was Parmenides; hence, we cannot conclude that he was much younger than the latter. He was probably born after Parmenides and before Socrates, for he was the teacher of Simmias and Cebes in Thebes, and these two were among the last disciples of Socrates.⁶⁸⁸

His astronomy was Pythagorean, and he is often described as the first exponent of Pythagorean astronomy, a statement that must be qualified in two ways. First, Parmenides, who was not a full Pythagorean, however, was possibly older than he. Second, he represents a second (or third) and more sophisticated stage in the evolution of Pythagorean astronomy. His writings are unfortunately lost, except for a very few fragments.

How sophisticated his views were will appear in a moment. They illustrate once more the theoretical boldness of the early Greek men of science, unfettered as they were by religious prejudices or by common-sense restrictions. The whole question for them is to give a consistent explanation of reality, and no hypothesis is too daring if it provides such an explanation. Philolaos did not hesitate to reject the geocentric idea which earlier Pythagoreans had taken for granted. The universe is spherical and limited. In the very center is the central fire (Hearth of the Universe, Watchtower of Zeus, etc.) which is also the central force or the central motor. Around it rotate ten bodies: first, the counterearth (antichth n) which always accompanies the Earth and shades the fire from it, second, the Earth itself, then the Moon, the Sun, and the five planets, finally the fixed stars. We do not see the counterearth because the Earth is always turning its back to it, that is, to the center of the universe. This implies that the Earth rotates around its own axis, while it turns around the center of the world.⁶⁸⁹

The boldness of that conception is staggering. Not only did Philolaos reject the geocentric hypothesis, but he did not hesitate to consider the Earth a planet like the others, and to assume that it rotates around the center of the world and also (perhaps) around its own axis. Moreover, he postulated the existence of another planet which remains always invisible! This seems exceedingly artificial. Why did Philolaos introduce the counterearth? According to Aristotle, he did so to account for eclipses and particularly for the greater frequency of lunar eclipses as compared with solar ones.³²a

If the Earth turns around the center of the world, then the apparent movements of the stars might be accounted for by the rotation of the Earth in an opposite direction. In spite of that, Philolaos assumed that the sphere of fixed stars was rotating like the other spheres. This is a good example of great boldness combined with timidity (a phenomenon which is so common in the history of science that we might consider it the rule rather than the exception). It would have been much simpler, indeed, to assume that the outside sphere did not move. Philolaos could not bring himself to do that — because all the spheres move . . . The gratuitous extra complication that he thus introduced did not necessarily conflict with reality. As the radii of the spheres increased, their angular speed decreased, and one might always determine the angular speeds of both the Earth and the fixed stars in such a manner that the apparent motion of the stars was exactly compensated. The very slow motion ascribed to the outside sphere might have been introduced in order to account for the precessiou of the equinoxes, but in spite of long centuries of Egyptian and Babylonian observations that phenomenon was then unknown; it remained unknown until the time of Hipparchos (11-2 B.C.).⁶⁹⁰


The world system that has just been described was ascribed to Philolaos by Aëtios,⁶⁹¹ but Diogenes Laërtios ascribed it to Hicetas, and Aristotle, to the Pythagoreans in general.

Even if the system was invented by Philolaos, it is possible that Hicetas improved it. For example, Hicetas may have drawn out the implication that the Earth turns around its own axis, and may have abandoned the fantastic and gratuitous conception of central fire and counterearth. This is substantiated by Cicero (1–1 B.C.); he is a late witness, but he was using a text of Theophrastos (IV–2 B.C.), who was much closer to the event. The date of Hicetas is unknown; we may assume that he was a younger contemporary of Philolaos. “Hicetas the Syracusan, as Theophrastos says, believes that sky, Sun, Moon, stars, and in fine all heavenly bodies are at a standstill, and that no body in the universe except the Earth is in motion; and that as this turns and revolves round its axis with great velocity, all the phenomena come into view which would be produced if the Earth were at rest and the heavens in motion.”⁶⁹²

Cicero’s statement that nothing in the universe is moving except the Earth is of course wrong in any case, but the exaggeration is understandable in the mouth of a man who was not an astronomer and who overemphasized the thought expressed by Hicetas and Theophrastos: it is the Earth that turns around its axis every day, not the starry heavens.

On the strength of that tradition, it is permissible to ascribe to Philolaos the system making the Earth a planet like the others turning around the central fire at the same speed as the counterearth, and to Hicetas the system replacing the Earth in the center and explaining the apparent rotation of the stars by a real rotation of the Earth around its own axis.


In order to complete this story, we must say a few words of Ecphantos, though he belongs probably to the following century. As he was a Syracusan and a Pythagorean like Hicetas, we may assume that he was a direct or indirect disciple of the latter, According to the Placita of Aëtios,⁶⁹³ “Heraclides of Pontos and Ecphantos the Pythagorean move the earth, not however in the sense of translation but in the sense of rotation, like a wheel fixed on its axis, from west to east, about its own center.” Thus, Ecphantos at least (if not Hicetas before him), affirmed unambiguously the daily rotation of the Earth. The fact that Aëtios associates him with Heraclides and even names the latter before him suggests that they were contemporaries (Heraclides of Pontos was born c. 388 and died c. 315–310).⁶⁹⁴ Ecphantos is said to have combined Pythagorean with atomistic doctrines; this also would place him in the fourth century, even in Heracleides’ time.


The founders of the atomic theory were great cosmologists but poor astronomers. To consider Democritos alone,

He said that the ordered worlds are boundless and differ in size, and that in some there is neither Sun nor Moon, but that in others both are greater than with us, and in yet others more in number. And that the intervals between the ordered worlds are unequal, here more and there less, and that some increase, others flourish and others decay, and here they come into being and there they are eclipsed. But that they are destroyed by colliding with one another. And that some ordered worlds are bare of animals and plants and of all water. And that in our cosmos the Earth came into being first of the stars and that the Moon is the lowest of the stars, and then comes the Sun and then the fixed stars; but that the planets are not all at the same height. And he laughed at everything, as if all things among men deserved laughter.⁶⁹⁵

These statements are made by St. Hippolytos (111-1); assuming that they represent Democritos’ thoughts, they are remarkable because of their boldness and their gratuitousness. It is clear that Democritos could base them on nothing, and yet his intuitions have been confirmed by modern science. For example, we now know that the number of universes is, if not infinite, at least so large as to stagger the imagination; we also know that the stars are of many different kinds and are in different stages of evolution, some going up, others down. That is not science, of course, but poetic fancy. Some of his cosmologic views, however, were as prophetic as his atomic theory. How could he make such guesses? One cannot help wondering; and why in his abysmal ignorance did he venture to speculate on such matters?

On the other hand, Democritos did not believe that the Earth was round (the conception of a spherical Earth was apparently a Pythagorean monopoly, one that the people of other sects did not care to infringe). He had spent some time in the East and his astronomic ideas were definitely Babylonian. One of his tetralogies deals with uranography, geography, polography, and meteorology. The first part can be reconstructed from Vitruvius; ⁶⁹⁶ it was possibly accompanied with plani-spheres “on which were pictured in imitation of the Babylonian terms, the human and animal figures which have come to represent the constellations.”⁶⁹⁷ In spite of his idea that the earth is flat, “disk-like laterally, but hollowed out in the middle,” ⁶⁹⁸; ⁶⁹⁹ he accepted the possibility of “zones,” but in the Babylonian manner. The Babylonians divided the celestial sphere into three concentric zones: The Way of Anu, above the pole, circumpolar stars; the Way of Enlil, the middle part, or zodiac; the Way of Ea, god of the deep, far down. Democritos abandoned that division into three and replaced it by a division into two hemispheres, northern and southern. The hypothesis that there existed southern constellations different from the northern ones was plausible enough, for when one traveled southward, across the Mediterranean and up the Nile, new constellations gradually appeared. But how could he reconcile these views with the flatness of the earth? The earth is flat but not perpendicular to the axis of the celestial sphere. This was not promising, though Democritos’ descriptions prepared those of Eudoxos (IV–1 B.C.), and later of Aratos of Soli (III–1 B.C.), which enjoyed a long popularity.”

Democritos was also acquainted with the Greek astronomic views, especially those of Anaxagoras, whom he followed. There is a curious difference between them, however, concerning the order of the planets. Anaxagoras had put them in this order: Moon, Sun, five planets, stars; Democritos put Venus between the Moon and the Sun. That is, he recognized Venus as an “inferior” planet, though not Mercury, and to that extent he paved the way for Heracleides of Pontos.


The mathematician Oinopides, who was a younger contemporary of Anaxagoras, is credited with two astronomic discoveries. The first is that of the obliquity of the ecliptic. That idea had been adumbrated by Anaximander of Miletos; indeed, it was possible not only to deduce the idea from the observations that he made with a gnomon (the simplest of astronomic instruments), but also to measure the obliquity. Yet, even if Anaximander measured the obliquity, one could hardly say that he understood it. On the other hand, if Oinopides was acquainted with Pythagorean astronomy (as he very probably was), it became possible for him really to understand the obliquity of the ecliptic, that is, to discover it.

The early measurement of the obliquity known to Euclid (24°; real value, 23° 27’) was not made by Oinopides but by other astronomers following him. It has been suggested that Euclid took an interest in certain mathematical problems because of their astronomic applications, and Proclos gives as an example Euclid’s construction of the regular polygon of fifteen sides.⁷⁰⁰ “For when we have inscribed the fifteen-angled figure in the circle through the poles, we have the distance from the poles both of the equator and the zodiac, since they are distant from one another by the side of the fifteen-angled figure.” ⁷⁰¹

His second discovery is that of the “great year” (megas eniautos) of 59 years, or he introduced it from Babylonia. Assuming the lengths of the year and the month to be 365 and 29½ days, 59 is the smallest integral number of years containing an integral number (730) of months.⁷⁰² This is very puzzling, for if it is true that the Egyptians had known that length of the year (365 days) since the Third Dynasty (thirtieth century), the Babylonians had known a 19-year cycle since 747. That cycle included hollow and full months of 29 and 30 days alternately, plus 7 intercalary months; that was better than the Egyptian year.⁷⁰³ The 8-year cycle (octaë-teris) of Cleostratos of Tenedos implied a year of 365¼ or days. How did Oinopides come to insist on 365 days? According to Censorinus (III–I) Oinopides made the length of the year to be . Tannery’s explanation of that contradiction is as follows: Having found the number of months in the great year, 730 [= 365 × 2], he had now to determine the number of days and did so on the basis of the Athenian calendar, recording the exact lengths of the synodic months (from full moon to full moon, or from first moonlight to first moonlight). This number was 21,557 days, which divided by 59 gives days as the length of the year. It should be noted that Oinopides as well as Philolaos had a pretty accurate knowledge (correct within 1 percent) of the periods of revolution of Saturn, Jupiter, and Mars; such knowledge might have been obtained from Babylonia.⁷⁰⁴

Oinopides had traveled to Egypt soon after 459, and his calendar reform reestablishing the Pythagorean great year of 59 years was published on a large bronze tablet exhibited in Olympia in 456. Thus, all the visitors to the Olympic games could know of Oinopides’ reform if they cared enough about it. Judging from the results, they did not care very much.


The first accurate solstitial observations were made by Meton and Euctemon in Athens in 432. These observations enabled them to determine the length of the seasons with greater precision. They introduced in that same year a new cycle, called the Metonic cycle, a period of 19 solar years, equivalent to 235 lunar months; this implied a year of c. days, that is, 30 min 10 sec too long, a much better approximation than those of Cleostrotos and Oinopides, as the following table shows:

Length of the year

Our knowledge of the observations made by Meton and Euctemon is obtained from a papyrus (now in the Louvre) called “The art of Eudoxos” (or the papyrus of Eudoxos). It is probably the notebook of a student who flourished in Alexandria c. 193–190.

We must not continue this story because we cannot devote too much space to the calendar, which takes us away from the history of astronomy, to a mixed field wherein astronomic knowledge is dominated by religious and political needs.⁷⁰⁵


The history of the arts and crafts and of various forms of engineering and architecture is almost endless, and we must restrict ourselves to a few significant examples.


One of the outstanding engineering undertakings of the century was the digging of a canal across the Athos peninsula ⁷⁰⁶ by order of Xerxes (king of Persia, 485— 465). Navigation is so dangerous around that mountainous peninsula that the great king ordered the digging of the canal in order to insure the safety of his fleet. Details are given by Herodotos.⁷⁰⁷ Two Persians, Bubares son of Megabazos and Artachaies son of Artaios, were in charge (epestasan tu ergu). Artechaies was high in the king’s favor and he stood very high in his own sandals, for he was the tallest man in Persia (8 ft high!). He died during the work or soon after; he was mourned by the king and the army and given a funeral and burial of great pomp. The isthmus is 2500 yd across, and traces of the digging can still be seen, or could be seen a century agu. ”The canal forms a line of ponds from 2 to 8 ft deep and from 60 to 90 broad. It was cut through beds of tertiary sands and marls, being probably where it was deepest not more than 60 ft below the natural surface of the ground, which at its highest point rises only 51 ft above the sea level (Rawlinson).” ⁷⁰⁸


It was said of Anaxagoras (p. 243) that he had written a book on scenography. Agatharchos, who was born c. 490 and flourished in Athens from 460 to 417, was a painter who actually practiced that new art and produced scenery or stage settings for Aischylos. He is the earliest painter known to us who used perspective on a large scale (that is, in wall painting or scenery, as opposed to vase painting) . He may have done that before Anaxagoras wrote his book and rationalized the art, for Anaxagoras is associated with Euripides. Agatharchos did not simply practice the arts but he also wrote a technical memoir (hypomn mata) concerning it. How his writing would compare with that of Anaxagoras or Democritos we have no means of judging, as all have vanished. At any rate, it is significant that three men of this time, Agatharchos, Anaxagoras, and Democritos, are associated with scenography and hence we may safely assume that the art began in this time, which was natural enough, considering that it was the golden age of tragedy.


Another remarkable symbol of Greek maturity is given to us by the appearance of the first town planner. Hippodamos was an architect who planned the construction of Athens’ harbor, Peiraieus (before 466), and of the Athenian colony of Thurii ⁷¹¹ in 443, but was not responsible for the building of Rhodes (in 408). We may thus say that he flourished shortly after the middle of the century.

He was concerned not only with the physical structure of cities (streets, squares, location of public buildings, and so on) but also with their moral structure, and was one of the forerunners of Plato in political thinking. He tried to establish an ideal constitution, which Aristotle criticized without sympathy. But Aristotle’s introduction of him is interesting and picturesque.

Hippodamos, the son of Euryphon, a native of Miletos, the same who invented the art of planning cities, and who also laid out the Peiraieus, — a strange man, whose fondness for distinction led him into a general eccentricity of life, which made some think him affected (for he would wear Howing hair and expensive ornaments; but these were worn on a cheap but warm garment both in winter and summer); he, besides aspiring to be an adept in the knowledge of nature, was the first person not a statesman who made inquiries about the best form of government.

The city of Hippodamos was composed of 10,000 citizens divided into three parts — one of artisans, one of husbandmen, and a third of armed defenders of the state. He also divided the land into three parts, one sacred, one public, the third private: the first was set apart to maintain the customary worship of the gods, the second was to support the warriors, the third was the property of the husbandmen. He also divided laws into three classes, and no more, for he maintained that there are three subjects of lawsuits — insult, injury, and homicide.⁷¹² (Then follows a long description and discussion.)

What shocked Aristotle most was that Hippodamos had no political experience as a statesman or administrator, but was simply a Pythagorean dreamer. Yet some of his dreams were more practical than Aristotle realized. For example, Hippodamos wanted his city to include farmers cultivating their own land for their private benefit. “What use are farmers to the city?” asked Aristotle. Well, Hippodamos must have believed that a “garden city” was a healthier place for every citizen to live in than a city of houses and shops, and was he not right? He was a dreamer indeed but a good dreamer, the distant ancestor of such men as Patrick Geddes (1854–1932) of our own time, who tried to harmonize the physical requirements of town planning with the moral and social aspects.⁷¹³


A little before reaching the Sunion promontory, which is the southern extremity of Attica, one passes through the Laurion region, rich in minerals. That region of about 80 km² has been mined from time immemorial (say from the early Iron Age on). The Greeks were working it mainly to obtain argentiferous galena, an ore that includes 65 percent of lead, and other metals were available, such as zinc and iron; there was even gold, but too little of it to be extracted by the old methods. Thanks to the Laurion mines, Attica was the sole producer of lead in the Greek world. Yet the main purpose of the Athenians was to obtain silver. About the beginning of the fifth century richer bodies of ore were discovered. The state took charge of the exploitation,⁷¹⁵ which was so profitable that about 483 each citizen received a bonus. Themistocles, who sensed the Persian danger ahead of others and realized the need for a strong navy, persuaded the Athenian government to devote the Laurion revenues to that urgent purpose. The victory of Salamis (480) was the fruit of that policy. Later the Laurion silver enabled Pericles to rebuild Athens in a magnificent way. When we admire the Parthenon we should always bear in mind the Laurion mines and the slave labor that made it possible; it is not pleasant to think that genius was not enough but that mining and slavery were needed to create that masterpiece, but we cannot dismiss those painful thoughts without hypocrisy.

The mines were overexploited in the fifth century. By the middle of the following century only old workings were open and there was no more prospecting. Xenophon, who makes that remark,⁷¹⁶ “proposes a socialist system of exploitation and that the state should hire out slaves as required, owing to the scarcity of private capital. But the orators show that there was much money available at Athens for investment in mercantile and other ventures, so either the mines no longer paid well, or the more important deposits had been discovered and there was greater risk of failure in a new prospect.” Efforts were made to revive the mines in the third and second centuries, but those efforts were jeopardized by labor troubles and stopped by a slave revolt in 103. In the time of Strabo (1–2 B.C.), the Athenians were already obliged to work out the stones and slag that had been thrown aside; in the time of Pausanias (11–2) the mines were completely abandoned. Since 1860 better methods and new goals have made it profitable to reëxploit the mines and tailings, not for silver, but for lead, cadmium, and manganese. Remains of the ancient exploitation can still be observed in situ: narrow shafts, galleries, furnaces, cisterns, washing tables, and other equipment.

Of course, mining and metallurgy were not a novelty in the fifth century; they had been practiced for thousands of years by the Egyptians and other peoples. Neither was the state monopoly a novelty, nor the use of ore for military and monumental purposes. It was in the nature of things that rulers finding such riches would use and abuse it for their needs. The fifth-century exploitation of the Laurion mines is the earliest, however, to be known with some archaeologic, political, and economic details. It is very important to remember that the glory of Athens in the fifth century was based not only on the Greek genius but also on the exploitation of silver mines. The human spirit is never separated from bodies, or beauty from hard labor and suffering, nor other spiritual creations from villenage and innumerable agonies.

There were other mines in the Greek world than those of Attica. Herodotos refers to mines near Mount Pangaios (in Macedonia), in Thrace, and in the islands of Siphnos and Thasos.

As to mining in Palestine and Western Asia, we discover a faint echo of them in the Book of Job.

Surely there is a vein for the silver, and a place for gold where they fine it. Iron is taken out of the earth, and brass is molten out of the stone. He setteth an end to darkness, and searcheth out all perfection: the stone of darkness, and the shadow of death.⁷¹⁷

This implies a certain amount of mining, and even metallurgic, experience. Such experience would be available at this time in many countries, all over the world, but the miners and metalworkers were illiterate people who lacked the desire and the power to describe it. More than any other craft, mining has always been combined with an extraordinary amount of ignorance and superstition.⁷¹⁸

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