B00B7H7M2E EBOK, page 3
There is still one circumstance to be mentioned – a simple, trivial matter, yet Eratosthenes’s successful measurement of the circumference of the Earth would not have taken place without it. A happenstance, perhaps, that such a small gem of information reached the ears of this man who realized what it meant and what could be done with it. It is true that the fact that this snippet of news reached him did have something to do with the broadened mental horizons of the world, with improved communications from remote areas, with Eratosthenes’s own world centring on northern Africa, and with his habit of keeping his ears and eyes open and wanting to know everything and anything. He was indeed the right man in the right time and place. Perhaps there was no other so likely to run across this back-page news and recognize its worth:
In a well located at Syene (near modern Aswan), on the day of the summer solstice, a shaft of sunlight penetrated all the way to the bottom of the well.
To Eratosthenes there was nothing trivial about this information. It meant that the Sun was shining directly down at Syene, not at an angle, and he knew this showed that Syene was on the tropic. A stick set up at noon at Syene on the day of the summer solstice would not cast a shadow. A stick set up at Alexandria (which he thought was the same longitude as Syene) would. Accordingly, Eratosthenes set up a stick at Alexandria on the day of the summer solstice and measured the angle of its shadow when that shadow was at its shortest.
Figure 1.1 below shows the stick at Alexandria and its shadow, and what is meant by ‘the angle of the shadow’. The figure illustrates how that must also be the angle ‘subtended’ by the arc Syene–Alexandria at the centre of the Earth. To put that a bit more simply: if we draw a straight line from the point marked Alexandria (where the stick is casting a shadow) to the centre of the Earth, and a second straight line from the point marked Syene (where the stick casts no shadow) to the centre of the Earth, those lines will of course meet at the centre of the Earth. We want to know the angle between those two lines where they meet. Geometry tells us, as it told Eratosthenes, that the angle at the centre of the Earth and the angle of the shadow at Alexandria will be the same angle.
Figure 1.1
Because the Sun’s rays are running parallel as they strike the Earth, if a line is drawn from Alexandria (a) where the stick casts a shadow, to the centre of the Earth, and a second line from Syene (s) where there is no shadow, to the centre of the Earth, the angle where those two lines meet will be the same as the angle of the shadow at Alexandria.
Figure 1.2 illustrates Eratosthenes’s measurement.
Figure 1.2
Because the sunlight shone all the way to the bottom of the well at Syene (s), Eratosthenes knew that the Sun was shining straight down on the Earth there. He set up a stick at Alexandria (a), where the Sun wasn’t shining straight down, and he measured the angle (x) of the shadow cast by the stick. He knew that because the Sun’s rays all run parallel as they strike the Earth, the angle (y) where a line drawn straight down from Alexandria and a line drawn straight down from Syene would meet at the centre of the Earth would be the same angle as the angle of the shadow cast by the stick (x). If Syene is due south of Alexandria, then the distance between Syene and Alexandria must be the same fraction of the Earth’s total circumference as the angle at x or y is of 360°
Eratosthenes found that the shadow angle at Alexandria was 7°, and so he knew that the angle between the ‘Syene–Alexandria lines’ (meeting at the centre of the Earth) was also 7°. A circle has 360°, and it is a simple process to find out how many of the Syene–Alexandria angles (7°) it will take to make 360°. Think of the cross-section of the Earth as a pie and the two lines coming from Syene and Alexandria as cutting out a wedge of pie. How many wedges of that size can you cut from the whole pie? Divide 360 by 7, and it comes out to 50 wedges. If we say (as Eratosthenes did) that the distance between Syene and Alexandria at the surface of the Earth (at the pie-crust edge of the pie) is ‘5,000 stades’, then we can multiply 5,000 by 50 and conclude that the distance all the way around the Earth – the circumference of the Earth – is 250,000 stades. Eratosthenes later fine-tuned this to 252,000 stades.
What is this odd unit of measurement, the stade? That question brings up a problem in evaluating Eratosthenes’s result. Whether or not that result matches modern measurements for the circumference of the Earth depends on the length of ‘stade’ he was using, and it isn’t known exactly what that length was. If there are 157.5 metres in a stade, Eratosthenes’s result comes to 39,690 kilometres or 24,608 miles for the circumference of the Earth. That is very near the modern calculation – 24,857 miles (40,009 kilometres) around the poles and 24,900 miles (40,079 kilometres) around the equator. After he had found the circumference, Eratosthenes calculated the diameter of the Earth as 7,850 miles (12,631 kilometres), close to today’s mean value of 7,918 miles (12,740 kilometres).
Another way of figuring a stade was as or of a Roman mile, and that would make Eratosthenes’s result too large by modern standards. There was one additional small difficulty. Eratosthenes assumed that Syene lay on the same line of longitude as Alexandria. Actually, it does not.
But this is nit-picking! No apology need be made for Eratosthenes. First of all, he arguably came astonishingly near to matching the modern measurement. Second, he was probably, for all his curiosity about the world, enough a man of his time to find the puzzle of how to solve this problem by the imaginative use of geometry at least as interesting as the actual measurement. The method is ingenious and it is correct. If the numerical result is a little fuzzy because of a lack of agreement about the length of a stade and the impossibility of determining longitude precisely, that does not prevent our recognizing what a brilliant achievement this was or appreciating the intellectual leap involved in recognizing that it could be done and how it could be done.
Eratosthenes didn’t focus his thoughts only on the Earth. He also raised them above the horizon to consider astronomical questions of his day. When it came to measuring the distances to the Sun and the Moon, he must have realized that he had no tool at his fingertips to equal the news about the well in Syene. Nevertheless, he gave it a try, with far less success than he had in measuring the Earth’s circumference.
Another Hellenistic scholar, Aristarchus of Samos, also tried to measure the distances to the Moon and Sun. Little information exists about him as a person. He lived from about 310 to 230 BC and would already have been a grown man when Eratosthenes was born. The island of Samos was under the rule of the Ptolemys during Aristarchus’s lifetime and it is possible that he worked in Alexandria. Archimedes was certainly aware of his contributions.
The only written work of Aristarchus that has survived is a little book called On the Dimensions and Distances of the Sun and Moon. In it he describes the way he went about trying to determine these dimensions and distances and the results he got.
The book begins with six ‘hypotheses’:
The Moon receives its light from the Sun.
The Moon’s movement describes a sphere and the Earth is at the central point of that sphere.
At the time of ‘half Moon’, the great circle that divides the dark portion of the Moon from the bright portion is in the direction of our eye. (In other words, we are viewing the shadow edge-on.)
At the time of ‘half Moon’, the angle (at the Earth as shown in Figure 1.3) is 87°.
The breadth of the Earth’s shadow (at the distance where the Moon passes through it during an eclipse of the Moon) is the breadth of two Moons.
The portion of the sky that the Moon covers at any one time is equal to of a sign of the zodiac.
Aristarchus’s fourth and sixth assumptions are both far from accurate. The actual angle at the Earth in Aristarchus’s triangle would be 89° 52’, not 87°, and 89° 52’ is very close to 90°. The angle at the Moon in Aristarchus’s triangle is 90°. That makes lines B and C so close to parallel that, on a drawing, the triangle would close up and be no triangle at all. The portion of one sign of the zodiac that the Moon covers is not , and it isn’t clear why Aristarchus, who must have known this from observation, chose that value.
Figure 1.3
Aristarchus’s measurement of the relative distances to the Moon and the Sun: when the Moon is a half Moon, the angle at the Moon (in this triangle) must be 90°. So a measurement of the angle at the Earth determines the ratio of the Earth–Moon line to the Earth–Sun line; in other words, the ratio of the Moon’s distance to the Sun’s distance.
Aristarchus’s results are not what we now measure these relative distances to be. By his calculation, the distance to the Sun is about 19 times the distance to the Moon and the Sun is 19 times as large as the Moon. The modern ratio between their distances is 400 to one. The measurement Aristarchus was trying to make was extremely difficult with the instruments available to him. It is no simple undertaking to determine the precise centres of the Sun and the Moon or to know when the Moon is exactly a half Moon. Aristarchus chose the smallest angle that would accord with his observations, perhaps to keep the ratio believable. Throughout antiquity and the Middle Ages, estimates of the relative distances to the Sun and Moon would continue to be too small.
Aristarchus didn’t stop with estimating the ratios, but found ways of converting them into actual numerical distances to the Sun and Moon and diameters for both bodies. He could see that the apparent size of the Moon and the Sun (meaning the size they appear to be when viewed from Earth) are about the same. During a solar eclipse, the Moon just about exactly covers the Sun. To put that in more technical language: they both have approximately the same ‘angular size’. Angular size tells how much of the sky a body ‘covers’ and is measured in ‘degrees of arc’. Both the Sun and the Moon have angular sizes of about one half of a ‘degree of arc’. (For a fuller explanation of those terms, see Figure 4.4.) For that to be true, the two bodies don’t actually have to be the same size, for how large they appear when viewed from Earth also depends on how distant they are. (See Figure 1.4a.) Aristarchus assumed that the Sun is much larger than the Earth, and that it was safe to assume also that the shadow cast by the Earth has about the same angular size as the Sun and the Moon (½ a degree of arc). (See Figure 1.4b.)
Figure 1.4 Aristarchus’s calculation of the size and distance of the Moon.
a. Surprisingly, all three of these bodies look the same size when viewed from Earth. We observe the ‘angular size’ of a body like the Moon or Sun, not its true size. It could be small and close or large and far away and still have the same ‘angular size’. Aristarchus saw that the Moon and the Sun have about the same angular size; that is, they look the same size when viewed from Earth, but he knew they are not the same true size.
b. (The angles shown in this drawing are much larger than those that really exist.)
Aristarchus assumed that the Sun is much larger than the Earth. If that is true, then the angle at the point of the Earth’s shadow is about equal to the angular size of the Sun as viewed from Earth.
c. Observing an eclipse, Aristarchus concluded that the breadth of the Earth’s shadow where the Moon crossed it was approximately twice the diameter of the Moon. He knew the angle formed at the point of the Earth’s shadow and also the angular size of the Moon. There was only one distance to put the Moon where it would cover half the area of the shadow.
Note: These drawings are not to scale.
Aristarchus arrived at his fifth ‘hypothesis’ above – the breadth of the Earth’s shadow (at the distance where the Moon passes through it during an eclipse of the Moon) is the breadth of two Moons – by observing a lunar eclipse of maximum duration, which means an eclipse in which the Moon passes through the exact centre of the Earth’s shadow. He measured the time that elapsed between the instant that the Moon first touched the edge of the Earth’s shadow and the instant that it was totally hidden. He then found that that length of time was the same as the length of time during which the Moon was totally hidden. He reasoned that the breadth of the Earth’s shadow where it was crossed by the Moon must therefore be approximately twice the diameter of the Moon itself (Figure 1.4c). If, as he thought, the angle formed at the point of the Earth’s shadow was the same as the angular size of the Moon, that gave him only one distance at which to put the Moon where it would cover half of the area of the shadow.
Aristarchus concluded that the Moon was ¼ the size of the Earth, and that the distance to the Moon was about 60 times the radius of the Earth. Both of those values are close to the modern values. Using Eratosthenes’s calculation of the Earth’s radius, Aristarchus arrived at an actual distance to the Moon in stades. He had less success with the distance to the Sun. His earlier estimate – that the Sun’s distance is about 19 times the Moon’s distance – was in error, and a second approach he tried, though it was ingenious and correct, required timing the phases of the Moon with a precision impossible in his day.
It was another of Aristarchus’s ideas that secured his place much more firmly in the annals of astronomy. Hearing of it, one has a chilling sensation of stumbling into a prophetic vision. For Aristarchus suggested, 17 centuries before Copernicus, that the Earth is not the unmoving centre of everything but instead moves round the Sun, and that the universe is many times larger than anyone in his time thought – perhaps infinitely large.
For centuries it had been widely assumed that the Earth was the centre of everything. The accepted picture of the cosmos was a series of concentric spheres – spheres embedded one within the other – with the Earth resting motionless at the centre of the system. (See Figure 1.5.)
Plato and Euxodus of Cnidus, a younger contemporary of Plato, had introduced this model, and Aristotle’s model of the universe was a further development of it, though he differed from Euxodus as to the number and nature of the spheres. However, it wouldn’t be correct to think that everyone, without exception, since the dawn of human thought had agreed that the Earth was the centre and didn’t move. Some Pythagorean thinkers had decided in the fifth century BC, largely for symbolic and religious reasons, that the Earth was a planet and that the centre of the universe must be an invisible fire. Heraclides of Pontus, a member of Plato’s Academy under Plato, proposed that the daily rising and setting of all the celestial bodies could be nicely explained if the Earth rotated on its axis once every 24 hours.
Figure 1.5
But Aristarchus went further. Although information about his theory of a Sun-centred cosmos comes second-hand, no one disputes his authorship of the idea because there is plenty of secondary evidence. According to Archimedes:
Aristarchus of Samos brought out a book of certain hypotheses, in which it follows from what is assumed that the universe is many times greater than that now so called. He hypothesizes that the fixed stars and the Sun remain unmoved; that the Earth is borne round the Sun on the circumference of a circle . . . and that the sphere of the fixed stars, situated about [that is, centred on] the same centre as the Sun, is so great that the circle in which he hypothesizes that the Earth revolves bears such a proportion to the distance of the fixed stars as the centre of the sphere does to its surface.
Aristarchus had done no less than move the centre of the cosmos to the Sun. In this astounding turn-about, the Earth moves round the Sun and, rather than the sphere of the fixed stars making a revolution of the heavens once every 24 hours, it is the Earth that turns, rotating on its axis – as Heraclides had suggested. The stars are extremely far away. The implication is, infinitely far.
Did Aristarchus also speculate that the other planets move round the Sun? It would seem a logical next step, but there is no historical evidence that he did. In any case, it’s unlikely that he understood the enormous significance of his model, that it provides, at a sweep, the basis for explaining the planets’ positions and movements far more simply than a model with the Earth as centre. It is impossible to tell from the surviving evidence whether Aristarchus really was personally disposed to thinking that the Earth moved around the Sun or whether he made the suggestion merely for the sake of argument, as in ‘Let’s just suppose for the moment that this is how things work.’ Why did this revolutionary suggestion came at this time and place in history? The simple answer may be that this was an intellectual environment that encouraged one to make suggestions and put forward hypotheses, even hypotheses based on assumptions that were known to be incorrect – in no way claiming they were true – as the starting point for an interesting line of enquiry.
With Aristarchus the question must also be turned on its head to ask not only why this idea emerged but why it died at birth. Seleucus of Seleucia, a Chaldaean or Babylonian astronomer (Seleucia was on the Tigris river) in the second century BC, took Aristarchus’s suggestion seriously – not merely as a hypothesis. Seleucus believed Aristarchus was right. However, no one else for the remainder of antiquity did, and the remainder of antiquity was by no means a dark age when it came to astronomy. It’s only partly correct to blame this resistance on an ideological attachment to having Earth and humanity the centre of everything.
If surviving information can be trusted, public opinion reacted almost not at all. Aristarchus’s idea must have been too far removed from common knowledge and common sense to draw much popular attention. The historian Plutarch reports one comment from the Stoic Cleanthes (the Stoics were reputedly weak in natural science and even ‘anti-scientific’) that Aristarchus of Samos ought to be indicted on a charge of impiety for putting the ‘Hearth of the Universe’ in motion. There is no record of anyone trying to take Cleanthes’s advice. Some philosophers scolded Aristarchus for trespassing in an area of knowledge that was their sole domain; and there were also complaints accusing him of undermining the art of divination.
