To Explain the World: The Discovery of Modern Science (12 page)

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 Cyrene, 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.

8

The Problem of the Planets

The Sun and Moon are not alone in moving from west to east through the zodiac while they share the quicker daily revolution of the stars from east to west around the north celestial pole. In several ancient civilizations it was noticed that over many days five “stars” travel from west to east through the fixed stars along pretty much the same path as the Sun and Moon. The Greeks called them wandering stars, or planets, and gave them the names of gods: Hermes, Aphrodite, Ares, Zeus, and Cronos, translated by the Romans into Mercury, Venus, Mars, Jupiter, and Saturn. Following the lead of the Babylonians, they also included the Sun and Moon as planets,
*
making seven in all, and on this based the week of seven days.
*

The planets move through the sky at different speeds: Mercury and Venus take 1 year to complete one circuit of the zodiac; Mars takes 1 year and 322 days; Jupiter 11 years and 315 days;
and Saturn 29 years and 166 days. All these are average periods, because the planets do not move at constant speed through the zodiac—they even occasionally reverse the direction of their motion for a while, before resuming their eastward motion. Much of the story of the emergence of modern science deals with the effort, extending over two millennia, to explain the peculiar motions of the planets.

An early attempt at a theory of the planets and Sun and Moon was made by the Pythagoreans. They imagined that the five planets, together with the Sun and Moon and also the Earth, all revolve around a central fire. To explain why we on Earth do not see the central fire, the Pythagoreans supposed that we live on the side of the Earth that faces outward, away from the fire. (Like almost all the pre-Socratics, the Pythagoreans believed the Earth to be flat; they thought of it as a disk always presenting the same side to the central fire, with us on the other side. The daily motion of the Earth around the central fire was supposed to explain the apparent daily motion of the more slowly moving Sun, Moon, planets, and stars around the Earth.)
¹
According to Aristotle and Aëtius, the Pythagorean Philolaus of the fifth century BC invented a counter-Earth, orbiting where on our side of the Earth we can’t see it, either between the Earth and the central fire or on the other side of the central fire from the Earth. Aristotle explained the introduction of the counter-Earth as a result of the Pythagoreans’ obsession with numbers. The Earth, Sun, Moon, and five planets together with the sphere of the fixed stars made nine objects about the central fire, but the Pythagoreans supposed that the number of these objects must be 10, a perfect number in the sense that 10 = 1 + 2 + 3 + 4. As described somewhat scornfully by Aristotle,
²
the Pythagoreans

supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they collected and fitted into their
scheme, and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. For example, as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth—the “counter-Earth.”

Apparently the Pythagoreans never tried to show that their theory explained in detail the apparent motions in the sky of the Sun, Moon, and planets against the background of fixed stars. The explanation of these apparent motions was a task for the following centuries, not completed until the time of Kepler.

This work was aided by the introduction of devices like the gnomon, for studying the motions of the Sun, and other instruments that allowed the measurement of angles between the lines of sight to various stars and planets, or between such astronomical objects and the horizon. Of course, all this was naked-eye astronomy. It is ironic that Claudius Ptolemy, who had deeply studied the phenomena of refraction and reflection (including the effects of refraction in the atmosphere on the apparent positions of stars) and who as we will see played a crucial role in the history of astronomy, never realized that lenses and curved mirrors could be used to magnify the images of astronomical bodies, as in Galileo Galilei’s refracting telescope and the reflecting telescope invented by Isaac Newton.

It was not just physical instruments that furthered the great advances of scientific astronomy among the Greeks. These advances were made possible also by improvements in the discipline of mathematics. As matters worked out, the great debate in ancient and medieval astronomy was not between those who thought that the Earth or the Sun was in motion, but between two different conceptions of how the Sun and Moon and planets revolve around a stationary Earth. As we will see, much of this debate had to do with different conceptions of the role of mathematics in the natural sciences.

This story begins with what I like to call Plato’s homework
problem. According to the Neoplatonist Simplicius, writing around AD 530 in his commentary on Aristotle’s
On the Heavens
,

Plato lays down the principle that the heavenly bodies’ motion is circular, uniform, and constantly regular. Therefore he sets the mathematicians the following problem: What circular motions, uniform and perfectly regular, are to be admitted as hypotheses so that it might be possible to save the appearances presented by the planets?
³

“Save (or preserve) the appearances” is the traditional translation; Plato is asking what combinations of motion of the planets (here including the Sun and Moon) in circles at constant speed, always in the same direction, would present an appearance just like what we actually observe.

This question was first addressed by Plato’s contemporary, the mathematician Eudoxus of Cnidus.
⁴
He constructed a mathematical model, described in a lost book,
On Speeds
, whose contents are known to us from descriptions by Aristotle
⁵
and Simplicius.
⁶
According to this model, the stars are carried around the Earth on a sphere that revolves once a day from east to west, while the Sun and Moon and planets are carried around the Earth on spheres that are themselves carried by other spheres. The simplest model would have two spheres for the Sun. The outer sphere revolves around the Earth once a day from east to west, with the same axis and speed of rotation as the sphere of the stars; but the Sun is on the equator of an inner sphere, which shares the rotation of the outer sphere as if it were attached to it, but that also revolves around its own axis from west to east once a year. The axis of the inner sphere is tilted by 23½° to the axis of the outer sphere. This would account both for the Sun’s daily apparent motion, and for its annual apparent motion through the zodiac. Likewise the Moon could be supposed to be carried around the Earth by two other counter-rotating spheres, with the difference that the inner sphere on which the Moon rides makes a full
rotation from west to east once a month, rather than once a year. For reasons that are not clear, Eudoxus is supposed to have added a third sphere each for the Sun and Moon. Such theories are called “homocentric,” because the spheres associated with the planets as well as the Sun and the Moon all have the same center, the center of the Earth.

The irregular motions of the planets posed a more difficult problem. Eudoxus gave each planet four spheres: the outer sphere rotating once a day around the Earth from east to west, with the same axis of rotation as the sphere of the fixed stars and the outer spheres of the Sun and Moon; the next sphere like the inner spheres of the Sun and Moon revolving more slowly at various speeds from west to east around an axis tilted by about 23½° to the axis of the outer sphere; and the two innermost spheres rotating, at exactly the same rates, in opposite directions around two nearly parallel axes tilted at large angles to the axes of the two outer spheres. The planet is attached to the innermost sphere. The two outer spheres give each planet its daily revolution following the stars around the Earth and its
average
motion over longer periods through the zodiac. The effects of the two oppositely rotating inner spheres would cancel if their axes were precisely parallel, but because these axes are supposed to be not quite parallel, they superimpose a figure eight motion on the average motion of each planet through the zodiac, accounting for the occasional reversals of direction of the planet. The Greeks called this path a
hippopede
because it resembled the tethers used to keep horses from straying.

The model of Eudoxus did not quite agree with observations of the Sun, Moon, and planets. For instance, its picture of the Sun’s motion did not account for the differences in the lengths of the seasons that, as we saw in
Chapter 6
, had been found with the use of the gnomon by Euctemon. It quite failed for Mercury, and did not do well for Venus or Mars. To improve things, a new model was proposed by Callippus of Cyzicus. He added two more spheres to the Sun and Moon, and one more each to
Mercury, Venus, and Mars. The model of Callippus generally worked better than that of Eudoxus, though it introduced some new fictitious peculiarities to the apparent motions of the planets.