After Aristotle, the overwhelming consensus among astronomers and philosophers (aside from a few like Lactantius) was that the Earth is a sphere. With the mind’s eye, Archimedes even saw the spherical shape of the Earth in a glass of water; in Proposition 2 of On Floating Bodies, he demonstrates, “The surface of any fluid at rest is the surface of a sphere whose center is the Earth.”6 (This would be true only in the absence of surface tension, which Archimedes neglected.)
Now I come to what in some respects is the most impressive example of the application of mathematics to natural science in the ancient world: the work of Aristarchus of Samos. Aristarchus was born around 310 BC on the Ionian island of Samos; studied as a pupil of Strato of Lampsacus, the third head of the Lyceum in Athens; and then worked at Alexandria until his death around 230 BC. Fortunately, his masterwork On the Sizes and Distances of the Sun and Moon has survived.7 In it, Aristarchus takes four astronomical observations as postulates:
1. “At the time of Half Moon, the Moon’s distance from the Sun is less than a quadrant by one-thirtieth of a quadrant.” (That is, when the Moon is just half full, the angle between the lines of sight to the Moon and to the Sun is less than 90° by 3°, so it is 87°.)
2. The Moon just covers the visible disk of the Sun during a solar eclipse.
3. “The breadth of the Earth’s shadow is that of two Moons.” (The simplest interpretation is that at the position of the Moon, a sphere with twice the diameter of the Moon would just fill the Earth’s shadow during a lunar eclipse. This was presumably found by measuring the time from when one edge of the Moon began to be obscured by the Earth’s shadow to when it became entirely obscured, the time during which it was entirely obscured, and the time from then until the eclipse was completely over.)
4. “The Moon subtends one fifteenth part of the zodiac.” (The complete zodiac is a full 360° circle, but Aristarchus here evidently means one sign of the zodiac; the zodiac consists of 12 constellations, so one sign occupies an angle of 360°/12 = 30°, and one fifteenth part of that is 2°.)
From these assumptions, Aristarchus deduced in turn that:
1. The distance from the Earth to the Sun is between 19 and 20 times larger than the distance of the Earth to the Moon.
2. The diameter of the Sun is between 19 and 20 times larger than the diameter of the Moon.
3. The diameter of the Earth is between 108/43 and 60/19 times larger than the diameter of the Moon.
4. The distance from the Earth to the Moon is between 30 and 45/2 times larger than the diameter of the Moon.
At the time of his work, trigonometry was not known, so Aristarchus had to go through elaborate geometric constructions to get these upper and lower limits. Today, using trigonometry, we would get more precise results; for instance, we would conclude from point 1 that the distance from the Earth to the Sun is larger than the distance from the Earth to the Moon by the secant (the reciprocal of the cosine) of 87°, or 19.1, which is indeed between 19 and 20. (This and the other conclusions of Aristarchus are re-derived in modern terms in Technical Note 11.)
From these conclusions, Aristarchus could work out the sizes of the Sun and Moon and their distances from the Earth, all in terms of the diameter of the Earth. In particular, by combining points 2 and 3, Aristarchus could conclude that the diameter of the Sun is between 361/60 and 215/27 times larger than the diameter of the Earth.
The reasoning of Aristarchus was mathematically impeccable, but his results were quantitatively way off, because points 1 and 4 in the data he used as a starting point were badly in error. When the Moon is half full, the actual angle between the lines of sight to the Sun and to the Moon is not 87° but 89.853°, which makes the Sun 390 times farther away from the Earth than the Moon, and hence much larger than Aristarchus thought. This measurement could not possibly have been made by naked-eye astronomy, though Aristarchus could have correctly reported that when the Moon is half full the angle between the lines of sight to the Sun and Moon is not less than 87°. Also, the visible disk of the Moon subtends an angle of 0.519°, not 2°, which makes the distance of the Earth to the Moon more like 111 times the diameter the Moon. Aristarchus certainly could have done better than this, and there is a hint in Archimedes’ The Sand Reckoner that in later work he did so.*
It is not the errors in his observations that mark the distance between the science of Aristarchus and our own. Occasional serious errors continue to plague observational astronomy and experimental physics. For instance, in the 1930s the rate at which the universe is expanding was thought to be about seven times faster than we now know it actually is. The real difference between Aristarchus and today’s astronomers and physicists is not that his observational data were in error, but that he never tried to judge the uncertainty in them, or even acknowledged that they might be imperfect.
Physicists and astronomers today are trained to take experimental uncertainty very seriously. Even though as an undergraduate I knew that I wanted to be a theoretical physicist who would never do experiments, I was required along with all other physics students at Cornell to take a laboratory course. Most of our time in the course was spent estimating the uncertainty in the measurements we made. But historically, this attention to uncertainty was a long time in coming. As far as I know, no one in ancient or medieval times ever tried seriously to estimate the uncertainty in a measurement, and as we will see in Chapter 14, even Newton could be cavalier about experimental uncertainties.
We see in Aristarchus a pernicious effect of the prestige of mathematics. His book reads like Euclid’s Elements: the data in points 1 through 4 are taken as postulates, from which his results are deduced with mathematical rigor. The observational error in his results was very much greater than the narrow ranges that he rigorously demonstrated for the various sizes and distances. Perhaps Aristarchus did not mean to say that the angle between the lines of sight to the Sun and the Moon when half full is really 87°, but only took that as an example, to illustrate what could be deduced. Not for nothing was Aristarchus known to his contemporaries as “the Mathematician,” in contrast to his teacher Strato, who was known as “the Physicist.”
But Aristarchus did get one important point qualitatively correct: the Sun is much bigger than the Earth. To emphasize the point, Aristarchus noted that the volume of the Sun is at least (361/60)3 (about 218) times larger than the volume of the Earth. Of course, we now know that it is much bigger than that.
There are tantalizing statements by both Archimedes and Plutarch that Aristarchus had concluded from the great size of the Sun that it is not the Sun that goes around the Earth, but the Earth that goes around the Sun. According to Archimedes in The Sand Reckoner,8 Aristarchus had concluded not only that the Earth goes around the Sun, but also that the Earth’s orbit is tiny compared with the distance to the fixed stars. It is likely that Aristarchus was dealing with a problem raised by any theory of the Earth’s motion. Just as objects on the ground seem to be moving back and forth when viewed from a carousel, so the stars ought to seem to move back and forth during the year when viewed from the moving Earth. Aristotle had seemed to realize this, when he commented9 that if the Earth moved, then “there would have to be passings and turnings of the fixed stars. Yet no such thing is observed. The same stars always rise and set in the same parts of the Earth.” To be specific, if the Earth goes around the Sun, then each star should seem to trace out in the sky a closed curve, whose size would depend on the ratio of the diameter of the Earth’s orbit around the Sun to the distance to the star.
So if the Earth goes around the Sun, why didn’t ancient astronomers see this apparent annual motion of the stars, known as annual parallax? To make the parallax small enough to have escaped observation, it was necessary to assume that the stars are at least a certain distance away. Unfortunately, Archimedes in The Sand Reckoner made no explicit mention of parallax, and we don’t know if anyone in the ancient world used this argument to put a lower bound on the distance to the stars.
; Aristotle had given other arguments against a moving Earth. Some were based on his theory of natural motion toward the center of the universe, mentioned in Chapter 3, but one other argument was based on observation. Aristotle reasoned that if the Earth were moving, then bodies thrown straight upward would be left behind by the moving Earth, and hence would fall to a place different from where they were thrown. Instead, as he remarks,10 “heavy bodies forcibly thrown quite straight upward return to the point from which they started, even if they are thrown to an unlimited distance.” This argument was repeated many times, for instance by Claudius Ptolemy (whom we met in Chapter 4) around AD 150, and by Jean Buridan in the Middle Ages, until (as we will see in Chapter 10) an answer to this argument was given by Nicole Oresme.
It might be possible to judge how far the idea of a moving Earth spread in the ancient world if we had a good description of an ancient orrery, a mechanical model of the solar system.* Cicero in On the Republic tells of a conversation about an orrery in 129 BC, twenty-three years before he himself was born. In this conversation, one Lucius Furius Philus was supposed to have told about an orrery made by Archimedes that had been taken after the fall of Syracuse by its conqueror Marcellus, and that was later seen in the house of Marcellus’ grandson. It is not easy to tell from this thirdhand account how the orrery worked (and some pages from this part of De Re Publica are missing), but at one point in the story Cicero quotes Philus as saying that on this orrery “were delineated the motion of the Sun and Moon and of those five stars that are called wanderers [planets],” which certainly suggests that the orrery had a moving Sun, rather than a moving Earth.11
As we will see in Chapter 8, long before Aristarchus the Pythagoreans had the idea that both the Earth and the Sun move around a central fire. For this they had no evidence, but somehow their speculations were remembered, while that of Aristarchus was almost forgotten. Just one ancient astronomer is known to have adopted the heliocentric ideas of Aristarchus: the obscure Seleucus of Seleucia, who flourished around 150 BC. In the time of Copernicus and Galileo, when astronomers and churchmen wanted to refer to the idea that the Earth moves, they called it Pythagorean, not Aristarchean. When I visited the island of Samos in 2005, I found plenty of bars and restaurants named for Pythagoras, but none for Aristarchus of Samos.
It is easy to see why the idea of the Earth’s motion did not take hold in the ancient world. We do not feel this motion, and no one before the fourteenth century understood that there is no reason why we should feel it. Also, neither Archimedes nor anyone else gave any indication that Aristarchus had worked out how the motion of the planets would appear from a moving Earth.
The measurement of the distance from the Earth to the Moon was much improved by Hipparchus, generally regarded as the greatest astronomical observer of the ancient world.12 Hipparchus made astronomical observations in Alexandria from 161 BC to 146 BC, and then continued until 127 BC, perhaps on the island of Rhodes. Almost all his writings have been lost; we know about his astronomical work chiefly from the testimony of Claudius Ptolemy, three centuries later. One of his calculations was based on the observation of an eclipse of the Sun, now known to have occurred on March 14, 189 BC. In this eclipse the disk of the Sun was totally hidden at Alexandria, but only four-fifths hidden on the Hellespont (the modern Dardanelles, between Asia and Europe). Since the apparent diameters of the Moon and Sun are very nearly equal, and were measured by Hipparchus to be about 33' (minutes of arc) or 0.55°, Hipparchus could conclude that the direction to the Moon as seen from the Hellespont and from Alexandria differed by one-fifth of 0.55°, or 0.11°. From observations of the Sun Hipparchus knew the latitudes of the Hellespont and Alexandria, and he knew the location of the Moon in the sky at the time of the eclipse, so he was able to work out the distance to the Moon as a multiple of the radius of the Earth. Considering the changes during a lunar month of the apparent size of the Moon, Hipparchus concluded that the distance from the Earth to the Moon varies from 71 to 83 Earth radii. The average distance is actually about 60 Earth radii.
I should pause to say something about another great achievement of Hipparchus, even though it is not directly relevant to the measurement of sizes and distances. Hipparchus prepared a star catalog, a list of about 800 stars, with the celestial position given for each star. It is fitting that our best modern star catalog, which gives the positions of 118,000 stars, was made by observations from an artificial satellite named in honor of Hipparchus.
The measurements of star positions by Hipparchus led him to the discovery of a remarkable phenomenon, which was not understood until the work of Newton. To explain this discovery, it is necessary to say something about how celestial positions are described. The catalog of Hipparchus has not survived, and we don’t know just how he described these positions. There are two possibilities commonly used from Roman times on. One method, used later in the star catalog of Ptolemy,13 pictures the fixed stars as points on a sphere, whose equator is the ecliptic, the path through the stars apparently traced in a year by the Sun. Celestial latitude and longitude locate stars on this sphere in the same way that ordinary latitude and longitude give the location of points on the Earth’s surface.* In a different method, which may have been used by Hipparchus,14 the stars are again taken as points on a sphere, but this sphere is oriented with the Earth’s axis rather than the ecliptic; the north pole of this sphere is the north celestial pole, about which the stars seem to revolve every night. Instead of latitude and longitude, the coordinates on this sphere are known as declination and right ascension.
According to Ptolemy,15 the measurements of Hipparchus were sufficiently accurate for him to notice that the celestial longitude (or right ascension) of the star Spica had changed by 2° from what had been observed long before at Alexandria by the astronomer Timocharis. It was not that Spica had changed its position relative to the other stars; rather, the location of the Sun on the celestial sphere at the autumnal equinox, the point from which celestial longitude was then measured, had changed.
It is difficult to be precise about how long this change took. Timocharis was born around 320 BC, about 130 years before Hipparchus; but it is believed that he died young around 280 BC, about 160 years before Hipparchus. If we guess that about 150 years separated their observations of Spica, then these observations indicate that the position of the Sun at the autumnal equinox changes by about 1° every 75 years.* At that rate, this equinoctal point would precess through the whole 360° circle of the zodiac in 360 times 75 years, or 27,000 years.
Today we understand that the precession of the equinoxes is caused by a wobble of the Earth’s axis (like the wobble of the axis of a spinning top) around a direction perpendicular to the plane of its orbit, with the angle between this direction and the Earth’s axis remaining nearly fixed at 23.5°. The equinoxes are the dates when the line separating the Earth and the Sun is perpendicular to the Earth’s axis, so a wobble of the Earth’s axis causes the equinoxes to precess. We will see in Chapter 14 that this wobble was first explained by Isaac Newton, as an effect of the gravitational attraction of the Sun and Moon for the equatorial bulge of the Earth. It actually takes 25,727 years for the Earth’s axis to wobble by a full 360°. It is remarkable how accurately the work of Hipparchus predicted this great span of time. (By the way, it is the precession of the equinoxes that explains why ancient navigators had to judge the direction of north from the position in the sky of constellations near the north celestial pole, rather from the position of the North Star, Polaris. Polaris has not moved relative to the other stars, but in ancient times the Earth’s axis did not point at Polaris as it does now, and in the future Polaris will again not be at the north celestial pole.)
Returning now to celestial measurement, all of the estimates by Aristarchus and Hipparchus expressed the size and distances of the Moon and Sun as multiples of the size of the Earth. The size of the Earth was measured a few decades after the work of Aristarchus by Eratosthenes. Eratosthenes was born in 273 BC at Cy
rene, a Greek city on the Mediterranean coast of today’s Libya, founded around 630 BC, that had become part of the kingdom of the Ptolemies. He was educated in Athens, partly at the Lyceum, and then around 245 BC was called by Ptolemy III to Alexandria, where he became a fellow of the Museum and tutor to the future Ptolemy IV. He was made the fifth head of the Library around 234 BC. His main works—On the Measurement of the Earth, Geographic Memoirs, and Hermes—have all unfortunately disappeared, but were widely quoted in antiquity.
The measurement of the size of the Earth by Eratosthenes was described by the Stoic philosopher Cleomedes in On the Heavens,16 sometime after 50 BC. Eratosthenes started with the observations that at noon at the summer solstice the Sun is directly overhead at Syene, an Egyptian city that Eratosthenes supposed to be due south of Alexandria, while measurements with a gnomon at Alexandria showed the noon Sun at the solstice to be one-fiftieth of a full circle, or 7.2°, away from the vertical. From this he could conclude that the Earth’s circumference is 50 times the distance from Alexandria to Syene. (See Technical Note 12.) The distance from Alexandria to Syene had been measured (probably by walkers, trained to make each step the same length) as 5,000 stadia, so the circumference of the Earth must be 250,000 stadia.
How good was this estimate? We don’t know the length of the stadion as used by Eratosthenes, and Cleomedes probably didn’t know it either, since (unlike our mile or kilometer) it had never been given a standard definition. But without knowing the length of the stadion, we can judge the accuracy of Eratosthenes’ use of astronomy. The Earth’s circumference is actually 47.9 times the distance from Alexandria to Syene (modern Aswan), so the conclusion of Eratosthenes that the Earth’s circumference is 50 times the distance from Alexandria to Syene was actually quite accurate, whatever the length of the stadion.* In his use of astronomy, if not of geography, Eratosthenes had done quite well.
To Explain the World: The Discovery of Modern Science Page 8