1     Introduction

Astronomy is often considered the zenith of ‘the exact sciences’ in antiquity. Highly mathematical, it is quite distinct from cosmology, and was more concerned with modelling celestial phenomena than with speculation about why the heavens appeared as they did. In modelling the heavens astronomers sought to ‘save the phenomena’, that is, to ‘explain’ observations by means of a mathematical construction which located certain celestial bodies at certain places and certain times. Each construction aimed to be systematic – even axiomatic in the Euclidean style – and leave as few as possible ‘anomalies’, i.e., unexplained phenomena. After centuries of effort by mathematicians and astronomers, Claudius Ptolemy synthesized and developed their achievements to produce the Almagest. This is the Arabic name by which his Mathematical Syntax, the rules of the motions of the heavens, is better known. It became the bible of astronomy in the West and the Islamic world for the next 1500 years, until displaced in the Copernican Revolution.

Throughout antiquity astronomy was not clearly distinguished from astrology, which grew out of it. Indeed, the same Ptolemy went on to write the bible of astrology, the Four Books, or Tetrabiblos. Astrology aimed to explain events in a ‘rational’ ‘natural’ way, just like other sciences: celestial bodies, principally the sun, moon, and five visible planets, were thought to be the causes of natural phenomena, like the seasons and the tides, and exerted their influence over all natural things: plants, minerals, regions of the earth, the people who live there, and parts of the body, for example. It was thought possible to explain and predict astral influences, as eclipses could be explained and predicted.¹ And because of widespread belief in astral influences, astronomical knowledge had far wider application than it has today – in biology for example.² The most obvious application of astronomy is of course to time-keeping.

Before Sosigenes created a purely solar calendar for Julius Caesar’s administrative convenience (and that of others wishing to run an empire), knowing exactly whereabouts in the seasonal cycle one was – as opposed to the official date on one or more of the local calendars – required an astronomer. The natural clock offered by the moon does not synchronize with the natural clock offered by the sun, and purely lunar calendars lose coherence with the seasons unless corrected. I am not suggesting that anyone other than astronomers wanted to know the time to within, say, a quarter of an hour.³ Most ancient time-telling devices⁴ were probably used as event markers rather than clocks as such: it is time to do X (meet, for example) when the shadow strikes Y.⁵ But periodically all local calendars were corrected to correspond with the seasons, for the very good reason that religious sacrifices needed to be offered at the appropriate time of year.⁶ Assuming that the Greeks involved in making these adjustments in democracies behaved like democratic Greeks deciding any other matter, then they would have resolved the matter by argument and evidence, i.e. by application of astronomical theories as understood by ho boulomenos.⁷ The archon’s calendar ‘was controlled throughout by the politicians’, as Pritchett said forcibly (Pritchett 1963 p. 339), and in Athens, the politicians were the politai, the citizens.

One brilliant politician – Augustus – used astronomy as the basis for a piece of propaganda in an ideological campaign on those he sought to rule: the Horologium Augusti .⁸ His association with the heavens, the gods, and time itself, was cemented when the eighth month was named after him, as the seventh had been named after Julius (Caesar). Coming down to earth again, anyone wishing to find their orientation with respect to the compass points, because they were lost, or laying out a new colony,⁹ a farmhouse,¹⁰ a temple, a body, or anything else that they wanted to align in a particular direction, could use a fragment of astronomical theory to settle the matter accurately, simply by pushing a stick in the ground and noting the position of its shadow on two moments either side of midday. And on the edges of the known world, explorers with a little astronomical knowledge could work out their latitude, so know at least how far north round the globe of the earth they were.

Modern works in this area tend to be focused on either astronomy or astrology, and there is very little modern literature on astronomy and other subjects, with the obvious exception of its application to calendar-making (e.g. Samuel 1972). These studies, like works on astronomy generally, tend to be highly technical works demanding fluent numeracy on the part of their readers, and they sometimes lose sight of the everyday life aspects, even of time-keeping.

I shall divide up the subject into mapping the heavens and modelling the heavens, though there is some overlap between them. The latter includes a look at Aristarchus’ calculation of the distance of the sun from earth – all done with a handful of simple hypotheses and some squares – which shows well what can be done with a little geometry and a lot of imagination. Finally I turn to a different sort of text altogether, Plutarch’s discussion of the face in the moon.

2     Mapping the Heavens

Theorising about the nature and motion of the stars considerably preceded precise data gathering, including developing a system of co-ordinates by which to locate accurately the position of celestial bodies. Eudoxus, for example, proposed his system of concentric spheres and the hippopede to explain planetary motion without having first observed carefully even where the centre of the heavens was.¹¹ Not surprisingly then, radically different explanations were offered by early astronomers for the same phenomena. Even when better data was available, it was sometimes impossible to choose between theories on the basis of observation, as the best of those who argued such cases recognized.

The accumulation of the basic data of astronomy, especially the movement of sun and moon and the location and brightness of the fixed and the wandering stars, took many centuries. The wandering stars were the stars which seemed to move erratically, sometimes stopping and going backwards and then stopping and going forwards again; we call them the planets. Their movements (and the moon’s) were ever a problem to explain, and the challenge of modelling their motions attracted the attention of some of the best mathematical minds, e.g. Eudoxus and Euclid.¹² But even the fixed stars were ‘located’ only in very vague terms (such as ‘near X is Y’) for a long time. Hesiod (Works and Days) and his contemporaries had talked of the risings and settings of the major constellations in the seventh century BC, yet there was still no good system for identifying particular stars and where to find them in the sky five hundred years later. Then in the C2 BC Hipparkhos set about making a star catalogue,¹³ and tried to give with reasonable accuracy some locational co-ordinates – inconsistently using several different reference systems, mostly declinations – for about half of the 850 stars he identified. Ptolemy added about 170 more stars about 200 years later, and used one system of proper co-ordinates (ecliptic longitudes and latitudes) for them all. His astronomy superceded all that had gone before and became the orthodox account of the heavens for the next 1500 years.

So the two stars forming the back end of the little dipper could be used as a pointer to the celestial north pole, just as the back end of the big dipper is now used to point to Polaris. For this reason these end stars in Ursa Minor were known for centuries as ‘the guardians of the pole’¹⁵; they circled it fairly tightly (about two thumbspans). According to Diogenes Laertius (1.23), Thales first drew the Greeks’ attention to the fact that while Greek sailors navigated by Ursa Major, the Phoenicians used Ursa Minor. Aratus confirms the distinction for his own times (C3 BC):

‘Encompassing it [the north celestial pole] and together circling round it are the two Bears, which are called Wagons¹⁶…Now the one men call by the name of Cynosura,¹⁷ the other they call Helike.¹⁸ It is by Helike that the Achaians on the sea judge where to direct the course of their ships, while the Phoenicians put their trust in the other as they cross the sea.¹⁹ Now Helike is clear and easy to recognize, being large and visible from earliest nightfall; the other is smaller and yet better for sailors, for the whole of it turns in a lesser circuit, and by it the men of Sidon steer the straightest course’ (Phainomena 26-44 Mair trans. slightly modified).

The anonymous Aetna suggests that the educated Roman layman of the C1 BC still used Ursa Major as a guide to the pole:

‘Not cattle-like to gaze on the world’s marvels merely with the eye, not to lie outstretched upon the ground feeding a weight of flesh, but to grasp the proof of things and search into doubtful causes, to hallow genius, to raise the head to the sky, to know the number and character of natal elements in the mighty universe…to know the axle of Helice… [to know] under what constellation the sailor must furl or spread his sails, to know the paths of the sea and learn betimes the course of the heavens…’ (224-9, 242, 245-6, Duff trans).

The apparent centre of the heavens has moved, and it was Hipparkhos who first realized that it was moving, everso slowly, at about 1° every 100 years he thought. This phenomenon is called the precession of the equinoxes, and it is caused by the earth’s polar axis (which tilts) rotating about a true vertical axis very slowly.²⁰ According to modern computations, a complete revolution takes about 25,800 years. Hipparkhos did not observe this movement: it could only be detected by comparison of then-current star positions with old records of star locations. The records used by Hipparkhos had been made by Timokharis of Alexandria about 170 years earlier, in 294 and 282 BC. Precession determines that the circumpolar stars have changed, and some stars which were always visible from Greece are no longer so, but dip beneath the horizon for part of the year, while others which were never visible are so now.

A less obvious but equally important difference between ancient observations of the sky and those made in the present is that all the Greeks could see was what they could see with the naked, unassisted, eye: there were no optical lenses in antiquity, to enhance the sight of someone with 20/20 vision, or to correct defective eyesight. Lentoids, sometimes called lenses, are more convincingly explained as eyes for sculptures and other decorative pieces than as optical quality lenses (see Plantzos 1997). Burning glasses, on the other hand, which are essentially the same shape, did exist, at least from Aristophanes’ time (Clouds 771-3), to start fires by concentrating the rays of the sun. Images, including ‘tiny and obscure writing’ could be magnified by viewing them though a glass ball filled with water, and magnifying mirrors existed (Seneca Nat Quest. 1.6.5 and 15.7 respectively). The absence of light pollution may have compensated for unassisted observation in terms of the average person’s perception of the fainter and transient lights in the sky, but atmospheric pollution, at least in some areas, could have been worse than in modern Britain.²¹

Measurements could be made by very rough and ready methods, such as using body parts for estimates, or by very sophisticated methods, such as constructing a celestial sphere following Ptolemy’s text.²² The adult’s hand is an adequate tool for many purposes. Held up to the sky at arm’s length, the tip of a finger (at the nail) covers about 1°; the fist about 10°. Observations of eclipses without damaging the eyes were made possible by viewing the event in a basin of pitch or thick oil (Seneca Nat Quest. 1.12.1). Ptolemy’s instruments for accurate observations, on the other hand, need expert machining to work properly, even the ‘simple’ pieces of apparatus (see e.g. Almagest 1.12, 5.12), never mind the astrolabe (5.1) or globe (8.3).²³ The famous Antikythera mechanism was a C1 BC ‘calendar-computer’, with fantastically complicated interlocking bronze cog wheels designed to show on the display plate the positions of the sun and moon, and the risings and settings of major constellations, set to the Metonic cycle of 19 years.²⁴

Numerous small errors in ancient observations of the night sky (not just those made by poets like Hesiod; also people like Euclid) must sometimes derive from the fact that observations were made with the naked eye and roughly ‘measured’ angles. The ancient themselves were aware of this: Hipparkhos, for example, commented on the unreliability of the data.²⁵ His discovery of precession was based on a shift of a mere 2° in the position of Spica relative to the autumnal equinox, and he was rightly cautious about the accuracy of the old records he depended upon. It would have been easy for him to dismiss Timokharis’ figures as everso slightly inaccurate.

3     Modelling the Heavens

Herakleides of Pontus²⁶ was the first person we know to take the amazing anti-intuitive leap and think the unthinkable: day and night are caused not by the rotation of sun and stars around a stationary us, but by earth rotating on its axis, to face alternately towards and away from the stationary sun. Most people thought he was crazy because they realized that the speeds we would have to be travelling at, just from axial rotation, would be very high, and that didn’t fit with anyone’s experience of watching the world go by.²⁷ As Ptolemy states it (in Toomer’s translation):

‘Although there is perhaps nothing in the celestial phenomena which would count against that hypothesis, at least from simpler considerations, nevertheless from what would occur here on earth and in the air, one can see that such a notion is quite ridiculous. Let us concede to them [for the sake of argument] that such an unnatural thing could happen as that the most rare and light of matter [stars etc.] should either not move at all or should move in a way no different from that of matter with the opposite nature (although things in the air, which are less rare [than the heavens] obviously move with a more rapid motion than any earthly object)²⁸; [let us also concede that] the densest and heaviest objects have a proper motion of the quick and uniform kind which they suppose (although again, as all agree, earthy objects are sometimes not readily moved even by an external force). Nevertheless, they would have to admit that the revolving motion of the earth must be the most violent of all motions associated with it, seeing that it makes one revolution in such a short time; the result would be that all objects not actually standing on the earth would appear to have the same motion, opposite to that of the earth: neither clouds nor other flying or thrown objects would ever be seen moving towards the east, since the earth’s motion towards the east would always outrun and overtake them, so that all other objects would seem to move in the direction of the west and the rear. But if they said that the air is carried round in the same direction and with the same speed as the earth, the compound objects in the air would none the less always seem to be left behind by the motion of both [the earth and the air]; or if those objects too were carried round, fused, as it were, to the air, then they would never appear to have any motion either in advance or rearwards: they would always appear still, neither wandering nor changing position, whether they were flying or thrown objects. Yet we quite plainly see that they do undergo all these kinds of motion, in such a way that they are not even slowed down or speeded up at all by any motion of the earth.’ (1.7)

But certain people,²⁹ propounding what they consider to be a more persuasive view, suppose that nothing is stationary and that there is no middle place in the universe. They suppose that the earth revolves around the sun, making one revolution in a year, that the sun revolves around the Milky Way, taking 200 million years to complete one revolution, and that the Milky Way is flying through space towards the Centaurus constellation. But if this were the case, then the earth would be travelling around the sun at about 67,000 mph, and the solar system would be travelling around the Milky Way at about 515,000 mph, and the Milky Way would be moving across space at about 1,342,000 mph.³⁰ ‘But such things are utterly ridiculous even to think of’, Ptolemy would have said.

Such things run counter to our intuitions, our common-sense, and our experience of the cosmos and our place in it. Look out of the window: are we really travelling at nearly 2,000,000 miles per hour? Look up into a clear night sky: Cassiopeia is there, a big W, as always, like clockwork, slowly circling the pole around which the heavens rotate each year, every year. If we are moving so far and so fast, why doesn’t the night sky change, you might ask, quite reasonably expecting the appearance of something to change as we change our viewing position (a phenomenon called parallax). This is the same question that people asked of Aristarchus’ radical idea (heliocentric theory) that we go round the sun, and not the sun round us. And the answer then and now is essentially the same: the appearance of the heavens doesn’t change because the stars are so far away that the earth has to them the same relation that a point has to the sphere in which the moon or stars move. In other words, the universe is so big that the movement of the earth is insignificant in observational terms.

Ptolemy, and almost everyone else³¹ in antiquity and the Middle Ages, thought that this answer was clever but insensible.³² Most of us, I think, would side with them if we relied on our senses and were not so well trained as to accept, without too much fuss, what the experts say. And so, if we consider the implications of what our experts tell us is the case, then we accept that we are on a small planet, one of nine, going round a star, which is one of millions of stars, in a galaxy, which is one of millions of galaxies, in a universe which is vastly bigger than the one that Aristarkhos had to suppose, Archimedes could count grains of sand in, and almost everybody else rejected as ridiculous. It is by thinking about parallax (which is observable, by the way, but not with the naked eye) or rather its implications³³ that we can perhaps best comprehend why astronomical theories sometimes impinge on people’s beliefs, and attitudes to life in general, to such an extent that astronomers have been lampooned, as Aristophanes lampooned Sokrates in the Clouds, or prosecuted, as Cleanthes thought Aristarkhos should have been but evidently wasn’t.

Who am I? Why am I here? Who cares? Is there a god? The answers to these sorts of questions tend to be very different depending on whether I think I am on the only inhabited place, sitting pretty in the centre of a bounded universe; whether I think I am too ignoble to occupy such a place (as the Pythagoreans did), so must be in orbit around the most noble thing in the cosmos (an invisible central hearth in Pythagoreanism); or whether I think I am on a fleck of stellar dust, somewhere (relatively speaking) in a boundless vastness, which will end in freezing darkness with the last visible star fading from sight as the universe flies apart. The former view puts humans at the centre of everything. The middle view puts humans in what the Pythagoreans considered to be an appropriate place in the cosmos. The latter view renders humans insignificant.

Some ancients were persuaded by Herakleides’ arguments. One amongst them was Aristarkhos, who is sometimes known as the Copernicus of antiquity because he suggested that the earth orbits the sun, and not vice versa. Unfortunately, we do not even know the name of the book in which he hypothesised this, never mind have a text surviving. What we do have is his treatise On the sizes and distances of the sun and the moon. This was not perhaps the first attempt at such a calculation, and it was certainly not the best in terms of agreement with modern figures – for the size and distance of the sun, that honour goes to Poseidonius; for the size and distance of the moon, to Ptolemy. But Aristarkhos’s method is an ingenious early attempt to quantify celestial phenomena, and, if one changes Hypothesis 4 from 87° to 89° 50’, then one gets (as near as matters) the right answer! It also has the advantage of being easily reproducible: one can contemplate the method by observing any full moon or half moon, both of which occur once a month – or indeed the moon in any state when it is visible during daylight, so that one can see both the moon and sun simultaneously, as happens not infrequently. As the Greeks themselves observed,³⁴ the atmosphere around the earth distorts light rays, so whilst one may thus contemplate the method, achieving accuracy is not quite as straightforward as it might appear.

The argument is not difficult to follow if you take your time and keep relating the text to the figure. Readers without trigonometry will find that it is easier to follow Aristarkhos than it is to follow some modern discussions of this computation.³⁵

‘[Hypotheses]

1. That the moon receives its light from the sun.

2. That the earth is in the relation of a point and centre to the sphere in which the moon moves.

3. That, when the moon appears to us halved, the great circle which divides the dark and the bright portions of the moon is in the direction of our eye.

4. That, when the moon appears to us halved, its distance from the sun is then less than a quadrant by one thirtieth of a quadrant.³⁶

5. That the breadth of the [earth’s] shadow is [that] of two moons.

6. That the moon subtends one fifteenth part of a sign of the zodiac.³⁷

We are now in a position to prove the following propositions:

1. The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon [from earth]; this follows from the hypothesis about the halved moon.

2. The diameter of the sun has the same ratio [as aforesaid] to the diameter of the moon.

3. The diameter of the sun has to the diameter of the earth a ratio greater than that which 19 has to 3, but less than that which 43 has to 6; this follows from the ratio thus discovered between the distances, the hypothesis about the shadow, and the hypothesis that the moon subtends one fifteenth part of a sign of the zodiac.

…

FIGURE to follow

[Proposition] 7

The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon from the earth.

For let A be the centre of the sun, B that of the earth. Let AB be joined and produced. Let C be the centre of the moon when halved; let a plane be carried through AB and C, and let the section made by it in the sphere on which the centre of the sun moves³⁸ be the great circle ADE. Let AC,CB be joined, and let BC be produced to D. Then, because the point C is the centre of the moon when halved, the angle ACB will be right. Let BE be drawn from B at right angles to BA; then the circumference ED will be one thirtieth of the circumference EDA; for, by hypothesis [4], when the moon appears to us halved, its distance from the sun is less than a quadrant by one thirtieth of a quadrant. Thus the angle EBC is also one thirtieth of a right angle. Let the parallelogram AE be completed, and let BF be joined. Then the angle FBE will be half a right angle. Let the angle FBE be bisected by the straight line BG; therefore the angle GBE is one fourth part of a right angle. But the angle DBE is also one thirtieth part of a right angle; therefore the ratio of the angle GBE to the angle DBE is that which 15 has to 2; for, if a right angle be regarded as divided into 60 equal parts, the angle GBE contains 15 of such parts, and the angle DBE contains 2. Now, since GE has to EH³⁹ a ratio greater than that which the angle GBE has to the angle DBE, therefore GE has to EH a ratio greater than that which 15 has to 2. Next, since BE is equal to EF, and the angle at E is right, therefore the square on FB is double of the square on BE.⁴⁰ But, as the square on FB is to the square on BE, so is the square on FG to the square on GE;⁴¹ therefore the square on FG is double of the square on GE. Now 49 is less than double of 25, so that the square on FG has to the square on GE a ratio greater than that which 49 has to 25; therefore FG also has to GE a ratio greater than that which 7 has to 5.⁴² Therefore [adding the 5 of GE to the 7 of FG to produce the total for FE⁴³], FE has to EG a ratio greater than that which 12 has to 5, that is, than that which 36 has to 15.⁴⁴ But it was also proved that GE has to EH a ratio greater than that which 15 has to 2; therefore, equally, FE has to EH a ratio greater than that which 36 has to 2, that is,⁴⁵ than that which 18 has to 1; therefore FE is greater than 18 times EH. And FE is equal to BE; therefore BE is also greater than 18 times EH; therefore BH is much greater than 18 times HE. But, as BH is to HE, so is AB to BC, because of the similarity of the triangles; therefore AB is also greater than 18 times BC. And AB is the distance of the sun from the earth, while CB is the distance of the moon from the earth; therefore the distance of the sun from the earth is greater than 18 times the distance of the moon from the earth.

Again, I say that it is also less than 20 times that distance.’ [Heath trans. The demonstration follows.]

It is worth noticing that the essential geometry for this tour de force – that the moon receives its light from the sun (hypothesis 1) and that when the moon appears halved the great circle which divides the dark and bright portions is in the direction of our eye (hypothesis 3) – is given in a more clumsy form in [Aristotle] Problems 15.7.

4     Aperitifs and astrophysics

Plutarch’s long book On the Face in the orb of the Moon is an extraordinary piece of literature. The scene is a dinner party, and the text pretends to be a discussion between the diners. The intended audience, one assumes, is the educated elite of the Roman empire. One of the speakers, appropriately called Aristotle, represents the Peripatetic view of things; another, Pharnaces, represents Stoicism; another, Lucius, brings in Pythagoreanism but generally takes a Peripatetic stance; two others, Apollonides and Menelaus, represent expert mathematical astronomers; another, Theon, represents literary critics; another, Sulla, stands for the mythographers; and finally Lamprias, the main speaker and the ‘I’ of the dialogue, is Plutarch’s brother, and perhaps stands for the educated layman. What is extraordinary about this text from the point of view of Greek science is not merely what it states, which is surprising enough, but also what it doesn’t say explicitly: what knowledge it assumes of its audience.

The text is mentioned most frequently in the modern literature for a paragraph in which one speaker comments that Cleanthes the Stoic philosopher thought that Aristarkhos should have been prosecuted for impiety for moving the earth from the centre of the heavens. Since references to Aristarkhos (by name) in ancient literature are few and far between, and references to clashes between science and religion are fewer still, it is understandable that these few sentences get highlighted, even if they are very unrepresentative of ancient concerns. But they are not even representative of this text, or of the participants’ opinions on Aristarkhos’s bold suggestion that the sun, not the earth, is at the centre of the cosmos. What this discussion really shows, and should be emphasized, is that what then passed for astrophysics is a suitable subject of conversation at a dinner party, and that everyone present knows a fair amount about it, and other topics in natural history.

They discuss, for example, the theoretical implications and consequences of various theories on planetary motion and speed of travel, on optics, on possible inhabitants of the moon, on the nature of the stars (where Plutarch scores points off Aristotle with his notions of natural place and levity), and the nature and behaviour of matter more generally. In this dinner party piece – not a scientific treatise – we find the nearest thing in all surviving ancient literature to the concept of gravity. All of them are also sufficiently familiar with literature to be able to quote from memory e.g. lines from the Attic tragedians or Pindar’s odes, the sort of literature we more commonly associate with sympotic discussions. Although they each have special interests, they are all polymaths by today’s standards. Let us let them speak for themselves, in an excerpt which gives some of the flavour of this long text (Moralia 922f-925d, Cherniss trans.).

‘Pharnaces spoke: "Here we are faced again with that stock manoeuvre of the Akademy: on each occasion that they engage in discourse with others they will not offer any accounting of their own assertions but must keep their interlocutors on the defensive lest they become prosecutors. Well, me you will not today entice into defending the Stoics against your charges until I have called you people to account for turning the world upside down." Thereupon Lucius laughed and said: "Oh, sir, just don’t bring suit against us for impiety as Kleanthes thought that the Greeks ought to lay an action for impiety against Aristarkhos the Samian on the ground that he was disturbing the hearth of the universe because he sought to save [the] phenomena by assuming that the heaven is at rest while the earth is revolving along the ecliptic and at the same time is rotating about its own axis. We express no opinion of our own now; but those who suppose that the moon is earth, why do they, my dear sir, turn things upside down any more than you do who station the earth here suspended in the air? Yet the earth is a great deal larger than the moon according to the mathematicians⁴⁶ who during the occurrences of eclipses and the transits of the moon through the shadow calculate her magnitude by the length of time that she is obscured. For the shadow of the earth grows smaller the further it extends, because the body that casts the light is larger than the earth; and that the upper part of the shadow itself is taper and narrow was recognized, as they say, even by Homer, who called night ‘nimble’ because of the ‘sharpness’ of the shadow. Yet captured by this part in eclipses the moon barely escapes from it in a space thrice her own magnitude. Consider then how many times as large as the moon the earth is, if the earth casts a shadow which at its narrowest is thrice as broad as the moon. All the same, you fear for the moon lest it fall; whereas concerning the earth perhaps Aiskhulos has persuaded you that Atlas

Stands, staying on his back the prop of earth

And sky, no tender burden to embrace.

Or, while under the moon there stretches air unsubstantial and incapable of supporting a solid mass, the earth, as Pindar says, is encompassed by ‘steel-shod pillars’; and therefore Pharnaces is himself without any fear that the earth may fall but is sorry for the Ethiopians or Taprobanians,⁴⁷ who are situated under the circuit of the moon, lest such a great weight fall upon them. Yet the moon is saved from falling by its very motion and the rapidity of its revolution, just as missiles placed in slings are kept from falling by being whirled around in a circle. For each thing is governed by its natural motion unless it be diverted by something else. That is why the moon is not governed by its weight: the weight has its influence frustrated by the rotary motion. Nay, there would be more reason perhaps to wonder if she were absolutely unmoved and stationary like the earth. As it is, while [the] moon has good cause for not moving in this direction, the influence of weight alone might reasonably move the earth, since it has no part in any other motion; and the earth is heavier than the moon not merely in proportion to its greater size but still more, inasmuch as the moon has, of course, become light through the action of heat and fire. In short, your own statements seem to make the moon, if it is fire, stand in greater need of earth, that is of matter to serve it as a foundation, as something to which to adhere, as something to lend it coherence, and as something that can be ignited by it, for it is impossible to imagine fire being maintained without fuel, but you people say that earth does abide without roof or foundation." "Certainly it does", said Pharnaces, "in occupying the proper and natural place that belongs to it, the middle, for this is the place about which all weights in their natural inclination press against one another and towards which they move and converge from every direction, whereas all the upper space, even if it receive something earthy which has been forcibly hurled up into it, straightway extrudes it into our region or rather lets it go where its proper inclination causes it naturally to descend."

At this–for I wished Lucius to have time to collect his thoughts–I called to Theon. "Which of the tragic poets was it, Theon," I asked, "who said that physicians

With bitter drugs the bitter bile purge?"

Theon replied that it was Sophokles. "Yes", I said, "and we have of necessity to allow them this procedure; but to philosophers one should not listen if they desire to repulse paradoxes with paradoxes and in struggling against opinions that are amazing fabricate others that are more amazing and outlandish, as these people do in introducing their ‘motion to the centre’. What paradox is not involved in this doctrine? …[many follow]… Nevertheless, though of tall tales of such a kind and number they have shouldered and lugged in – not a wallet-full, by Heaven, but some juggler’s pack and hotchpotch – still they say that others are playing the buffoon by placing the moon, though it is earth, on high and not where the centre is. Yet if all heavy body converges to the same point and is compressed in all its parts upon its own centre, it is no more as centre of the sum of things than as a whole that the earth would appropriate to herself the heavy bodies that are parts of herself; and [the downward tendency] of falling bodies proves not that the [earth] is in the centre of the cosmos but that those bodies which when thrust away from the earth fall back to her again have some affinity and cohesion with her. For as the sun attracts to itself the parts of which it consists so the earth too accepts as [her] own the stone that has properly a downward tendency, and consequently every such thing ultimately unites and coheres with her. If there is a body, however, that was not originally allotted to the earth or detached from it but has somewhere independently a constitution and nature of its own, as those men would say of the moon, what is to hinder it from being permanently separate in its own place, compressed and bound together by its own parts? For it has not been proved that the earth is the centre of the sum of things, and the way in which things in our region press together and concentrate upon the earth suggests how in all probability things in that region converge upon the moon and remain there. The man who drives together into a single region all earthy and heavy things and makes them part of a single body–I do not see for what reason he does not apply the same compulsion to light objects in their turn but allows so many separate concentrations of fire and, since he does not collect all the stars together, clearly does not think that there must also be a body common to all things that are fiery and have an upward tendency.

Now", said I, "my dear Apollonides, you mathematicians say that the sun is an immense distance from the upper circumference and that above the sun Venus and Mercury and the other planets revolve lower than the fixed stars and at great intervals from one another; but you think that in the cosmos there is no scope and extension for heavy and earthy objects. You see that it is ridiculous for us to deny that the moon is earth because she stands apart from the nether region and yet to call her a star, although we see her removed so many thousands of miles from the upper circumference as if plunged [into] a pit. So far beneath the stars is she that the distance cannot be expressed, but you mathematicians in trying to calculate it run short of numbers;⁴⁸ she practically grazes the earth and revolving close to it

Whirls like a chariot’s axle-box about,

Empedokles says,

That skims [the post in passing].

Frequently she does not even surmount the earth’s shadow, though it extends but a little way, because the illuminating body is very large; but she seems to revolve so close, almost within arm’s reach of the earth, as to be screened by it from the sun unless she rises above this shadowy, terrestrial, and noctural place which is earth’s estate. Therefore we must boldly declare, I think, that the moon is within the confines of [the] earth inasmuch as she is occulted by its extremities.

Dismiss the fixed stars and the other planets and consider the demonstrations of Aristarkhos in his treatise On Sizes and Distances that ‘the distance of the sun is more than eighteen times and less than twenty times the distance of the moon’, that is its distance from us…"’

This excerpt illustrates well not only the astronomical and physical debates of the time but also many of the points made elsewhere on these pages on the nature and practice of Greek science. For example: the influence of past theories on present ones, and of present ones on each other; the distinction between the reasonably accurate observation or calculation of various phenomena versus the much more speculative and widely divergent explanations for those phenomena; the penetration of scientific ideas into ordinary culture; and the use of rhetoric in argument.

Notes

1. Bowen 1999 argues that predicting eclipses in a precise way – when, where, duration of occultation – was not part of the astronomer’s self-defined tasks until the C1 AD. He sees it growing out of a literary topos. I would look also to the growth of astrology, for this is approximately the same period in which personal horoscopy developed.

2. Thompson made what he called ‘a somewhat startling’ claim that ‘very many of [ancient statements concerning animals] deserve not a zoological but an astronomical interpretation’ 1895 p. xii.

3. Hipparkhos observed that even for professional astronomers large errors could arise from the instruments used to measure it, apud Ptolemy Almagest 3.1: ‘In the case of the solstices, I have to admit that both I and Archimedes may have committed errors of up to a quarter of a day in our observations and calculations [of the time]’, Toomer trans., emphasis added. Ptolemy’s attribution of such errors to the ‘construction and positioning of the instruments’ slightly precedes this quote.

4. Principally gnomons and sundials, which could be flat or hemispherical, fixed horizontally or vertically, or portable.

5. Similarly most water clocks were devices to apportion time, on the basis of the time it took for the water in the clepsydra to run out of the hole at the bottom, rather than devices to tell the time of day. Their use in the courts, for example, regulated the duration of speeches: the clock was started when the speaker started, plugged during the reading out of laws, and the speaker had to finish when all the water had run out. In like fashion, Herophilus’ portable clepsydra ‘measured’ the patient’s pulse rate.

6. See Geminos Elements of astronomy 8. One of Aristophanes’ complaints in the Clouds (production date 423) is that the monthly feasts and festivals are falling on days which do not correspond with the appearance of the moon. He is assuming that the gods of Olympos work with the lunar calendar, which we may thus suppose to be the traditional one. In the Athens of his day this has been interfered with by political adjustments or set aside in favour of a luni-solar or solar calendar, long enough for a very obvious asynchronisation to have developed, and recently enough for Aristophanes to make an issue of it. Meton had offered to sort out the calendar in Athens nine years before Aristophanes produced the Clouds (see Diod. Sic. 12.36.2, recording public honours voted to Meton). He came up with his eponymous Metonic cycle of 19 years, and the first cycle started on 13 Skirophorion (the summer solstice) in 432 BC. This is usually said to have been used only by the astronomers, and did not become the basis for a reformed civil calendar, but it does not follow from this that it was not used for occasional corrections to the civil calendar. Diodorus says that ‘even down to our own day, the majority of the Greeks (hoi pleistoi twn hellhnwn) use the nineteen year cycle’ 12.36.3. Diodorus may not be the best historian from antiquity, but he did his research at the Library at Alexandria and had access to much material which has been lost to us. Corrections to the calendar in our own time may deal with seconds rather than months, but such correction is still necessary and is necessarily sporadic. Meton is actually a character in Aristophanes’ Birds (produced 414), and does geometry at 1000 sqq.

7. Such adjustments could involve dropping or adding months, not a mere 11 days (which caused so much upset for Pope Gregory’s contemporaries). In Athens intercalations were made by the archon, and the calendar so adjusted was the one known as the archon’s calendar. He made adjustments by moving the peg in the official parapegma (labelled board) the required number of slots forward or backward; on parapegmata see Neugebauer HAMA vol 1. 587-9. There were two other calendars in operation in Athens simultaneously: 1. the prytany calendar, of 10 months of 36 or 35 days each (Ath Pol 43.2), by which the ten tribes of Athens took turns to be presidents of the state; 2. the kata theon (‘according to the god’) calendar, the nature of which is disputed. See Pritchett 1963 esp. § 4.

8. Dedicated in 10/9 BC. Some important remaining parts were found and published by its excavator, Buchner 1982. Besides being emperor, Augustus also had responsibility for keeping the civil calendar in step with the seasons (qua Pontifex Maximus), which at this date meant fine-tuning Caesar’s system. This monumental sundial, next to the Ara Pacis, combined time-keeping and ideological (religious as well as political) functions. The ideological function was obvious to Cicero when Caesar first introduced this calendar: ‘the heavens have to obey the dictator’, Plutarch Caesar 59.

9. Astronomy was part of the formal education of the Roman agrimensores, on which see Dilke 1971 chapter 4.

10. Besides optimising solar gain in winter and cool shade in summer for the residents of the building, the importance of correct aspect for storage rooms of different products is stressed by most of the agricultural writers. See e.g. Columella on the olive-oil store.

11. See n. 14 below. Eudoxus had produced a (lost) description of the heavens, which Aratus versified, but its style and content must have been very similar to Aratus’ surviving Phainomena. Hipparkhos considered them together in his Commentary on the Phainomena of Aratus and Eudoxus, and says explicitly ‘Now Eudoxus gives the same collection of phenomena as Aratus, but has set them forth with greater knowledge…It is, perhaps, not fair to blame Aratus if in some points he is found to be in error; for in writing his Phainomena he has followed the arrangement of Eudoxus, without making any observations on his own account, and without professing to be speaking with the authority of a mathematician when giving details of celestial happenings which afterwards prove to be inaccurate… that Aratus followed Eudoxus’ account of the phenomena may be gathered by comparing, at length, Eudoxus’ text with that of the poem dealing with the same topic in each case…they practically agree in all but a very few details’ [four examples follow] Heath trans. 1932 pp. 117-8.

12. Eudoxus’ work survives only in fragments, bits quoted or paraphrased by later authors. For discussion of his ideas and contribution to Greek astronomy, see Neugebauer HAMA vol. 2 pp. 675-83. For Euclid’s Phainomena see Heath 1921 vol 1 pp. 348-53 and 440-1. Both are excerpted and translated in Heath 1932 pp. 65-7 and 96-100 respectively.

13. On which see Neugebauer HAMA 1 pp. 277-88.

14. Comm on Phainomena of Aratos and Eudoxus 1.4.1: ‘In fact there is no star at the pole but an empty space close to which three stars lie, which together with the point of the pole make a rough quadrangle, as Pytheas of Massalia [better known as an explorer] tells us’, trans. Heath, 1932 p. 119, who thought the stars in question were probably Draco a and k and UMin b; Roseman 1994 pp. 118-9 prefers nearer stars of lesser magnitude. Eudoxus’ error had been repeated by Euclid Phainomena Pref. ‘But a certain star is seen between the Bears which does not change from place to place, but turns about the position where it is’.

15. Jones 1995 p.49. He does not say by whom they were thus known, but it would appear to be the Phoenicians (see the sources about to be cited in text). Nor is a date given for this label, but the stars would have lined up with the pole around the time of Christ, and would have orbited it tightly for centuries either side of that.

16. This name is surely descriptive of their shape. An explanation for how the Greek name Bear could have derived from a Near Eastern word for wagon is given in Kidd 1997 p. 181 on 27.

17. Literally, ‘dog’s tail’. Ursa Minor.

18. Literally, ‘helix’; something with a spiral shape, which goes round, circles, coils. This is the first appearance of this name and surely reflects its circumpolar motion; Homer calls it the Bear. See Kidd p. 188 on 37.

19. When mariners had the courage to cross open waters the position of the sun or the stars were their navigational guides. Deep water archaeology is re-writing the picture of ancient trade in the Mediterranean, which (as is now apparent) did not always tramp round the coasts in shallow waters but sometimes struck out for a direct crossing.

20. Think of an earth globe and the tilt of the poles; they point 23.5° off the vertical. Rotate the globe’s stand round 360° and you have this motion. As it moves round, the pole will point to different parts of the ceiling, describing a circle. So in reality earth’s north pole points toward different parts of the sky, describing a cone from the centre of the earth 47° in diameter over the course of nearly 26,000 years.

21. The range of meteoric phenomena distinguished (e.g. in Aristotle’s Meteorology book 1; ‘burning flames’, ‘shooting stars’, ‘torches’, ‘goats’, ‘chasms’, ‘trenches’, ‘comets’, ‘bearded stars’ and ‘haloes’) illustrates the former point. For the latter, deposits in the polar ice-caps reveal that global atmospheric lead pollution was worse in Roman times than at any other time in history.

22. Archimedes wrote a treatise (lost) On sphere-making, and Cicero mentions both Archimedes’ and Poseidonios’ own spheres in De Natura Deorum 2.88, Rep 1.21 and 1. 28, and Tusc. Disp. 1.63.

23. On such instruments see de Solla Price 1957; also Maddison 1963.

24. On which see de Solla Price 1975.

25. E.g. ‘In On the length of the year [Hipparkhos] assumes only the motion which takes place about the poles of the ecliptic, although he is still dubious, as he himself declares, both because the observations of the school of Timokharis are not trustworthy, having been made very crudely, and because the difference in time between [Timokharis and himself] is not sufficient to provide a secure result’, Ptolemy Almagest 7.3 Toomer trans.

26. On whom see Gottschalk 1980.

27. The alternative is of course that the celestial sphere is rotating even faster, but this was acceptable. Indeed, it was even used by Nigidius ‘the potter’ as an argument in defense of astrology to explain why twins can be so different. Augustine City of God 5.3 reports Nigidius’ demonstration of the point using a potter’s wheel.

28. This and the next clause in the argument are based on the notion of natural motion, according to which one would expect light things to move quickly and heavy things to move slowly. Axial rotation theory implies an ‘unnatural’ reversal of this situation, with rapidly moving earth and stationary stars.

29. This paragraph is a paraphrase of Ptolemy’s comments which lead into the section 1.7 quoted above, which I have rewritten to give the modern view of these matters. ‘Certain people’ here means all modern astronomers and astrophysicists. In the original version, Ptolemy presumably meant Herakleides of Pontus, Aristarchus of Samos, and Seleucus of Seleucia at least, but he omits their names and anonymizes them.

30. ≈30 km/s in Kaufmann 1994 p.146, ≈230 km/s op.cit. p.312, and ≈600 km/s op.cit. p.533 respectively.

31. Ecphantus, Hiketas, Aristarchus and Seleucus sided with Herakleides. In addition, before and after Herakleides the Pythagoreans (who numbered many) believed that the earth did not occupy the middle of the universe, but orbited a central fire; therefore they must have had some answer to the sort of objections raised by Ptolemy.

32. See the discussion of this point by Hanson 1973 pp.18-21.

33. Specifically, just how far away the stars must be and just how big the universe must be for our ever-changing position not to have any apparent observational effect.

34. In the context of a lunar eclipse. Since a lunar eclipse is caused by the earth passing between sun and moon, it should not be possible to observe both sun and eclipsed moon simultaneously; the fact that on occasion both were visible simultaneously suggested to some that light rays were bent by earth’s atmosphere. See Kleomedes De motu circulari 2.6, conveniently in translation in Heath 1932 pp. 162-6, and discussed by Pederson and Pihl 1974 p. 135.

35. There was no trigonometry in Aristarkhos’ time. With modern trigonometry (specifically the cosine function) it is quick and easy to prove Aristarkhos’ Proposition 7 from his hypotheses. The origins of trigonometry can be found in the chord functions developed and used by Hipparkhos, Menelaos and later Ptolemy, who drew up tables of their values to enable them to compute astronomical data.

36. A quadrant is a quarter of a circle. If we divide it into 360°, a quadrant is then 90°, and one thirtieth of that is 3°. Therefore Aristarkhos hypothesizes that the angle is 87°. If you substitute ‘less than a quadrant by one five hundred and fortieth of a quadrant’ (89°50’) in the rest of the computation, then you’ll get the modern figure, as near as matters.

37. A sign of the zodiac is one twelfth of 360°, so 30°. One fifteenth of that is 2°.

38. Although Aristarkhos proposed heliocentric theory, he here assumes a geocentric model; the sun A moves round a stationary earth B.

39. Aristarkhos has neglected to tell us where to put H. From the demonstration as a whole it becomes clear that it is the point where an extension of BD cuts FE.

40. By Pythagoras’ Theorem, for which see the Mathematics page.

41. Aristarkhos assumes the reader knows this to be true. It can be demonstrated by adding one point, X, to the diagram, and one line, GX. Since G is found by bisecting FBE, GX and GE are equal. FG is the hypotenuse of the triangle FXG, and X is a right angle. Therefore FG is to XG as FB is to FE. Since XG and GE are equal, so FG is to GE as FB is to FE. Note that Aristarkhos is working with squares on the lines, and only reduces to the roots, the lines, in the next step of the argument.

42. The square roots of 49 and 25 respectively. He needed to choose numbers with integer square roots in the same proportions as his ratios, and he opted for 49 as not quite (less than) 50, which is double 25.

43. Additional to Heath’s translation, which has componendo here.

44. Aristarkhos now multiplies both by 3, to compare with the 15 as a figure for the quarter of a right angle of 60 parts (done above).

45. Dividing both by 2.

46. Mathematikoi usually means more generally ‘learned men’. Intellectual life was not such that learned men pursued one, and only one, subject. Some, however, had greater facility with mathematics than most, and these are clearly the sort of people Lucius has in mind here.

47. Taprobane is Sri Lanka. Eratosthenes had located it on the same parallel as Ethiopia.

48. It is precisely in the context of Aristarkhos’ heliocentric model that Arkhimedes advances his ‘Sand-Reckoner’, as a system by which the huge numbers implied by Aristarkhos’ immense cosmos might be expressed.

Further Reading

J Evans History and practice of ancient astronomy (Oxford 1999) is now the best place to start. O Neugebauer’s History of ancient mathematical astronomy (HAMA), 3 vols (Berlin 1975) is the modern classic on ancient astronomy, but this is not easy reading. It is an excellent example of a modern mathematical commentary on the work of the ancients, going beyond implicit translation to create a modern reconstruction. For those undaunted by mathematical astronomy in principle, but who find HAMA more or less incomprehensible, it may be found helpful to have on the table at the same time a work like C Barlow & G Bryan Elementary Mathematical Astronomy (5th ed. revised by H Spenser Jones). M Crowe’s Theories of the world from antiquity to the Copernican revolution is very good for trying to see the heavens from a pre-modern perspective. F Hodson (ed) The place of astronomy in the ancient world considers the usefulness of astronomy in ancient life, amongst other things. For astrology see T Barton Ancient Astrology (London, 1994).

There is a good selection of astronomical works in translation in T B L Heath Greek astronomy 1932, reprinted by AMS Press (New York, 1969), many of which are also in his Aristarchus of Samos. The mathematical highpoints of Greek astronomy can also be found in I Thomas Greek Mathematical Works Loeb, 2 vols, and Cohen and Drabkin Sourcebook in Greek Science.