A History of Western Philosophy

by Bertrand Russell

  Copernicus was not in a position to give any conclusive evidence in favour of his hypothesis, and for a long time astronomers rejected it. The next astronomer of importance was Tycho Brahe (1546-1601), who adopted an intermediate position: he held that the sun and moon go round the earth, but the planets go round the sun. As regards theory he was not very original. He gave, however, two good reasons against Aristotle’s view that everything above the moon is unchanging. One of these was the appearance of a new star in 1572, which was found to have no daily parallax, and must therefore be more distant than the moon. The other reason was derived from observation of comets, which were also found to be distant. The reader will remember Aristotle’s doctrine that change and decay are confined to the sublunary sphere; this, like everything else that Aristotle said on scientific subjects, proved an obstacle to progress.

  The importance of Tycho Brahe was not as a theorist, but as an observer, first under the patronage of the king of Denmark, then under the Emperor Rudolf II. He made a star catalogue, and noted the positions of the planets throughout many years. Towards the end of his life Kepler, then a young man, became his assistant. To Kepler his observations were invaluable.

  Kepler (1571-1630) is one of the most notable examples of what can be achieved by patience without much in the way of genius. He was the first important astronomer after Copernicus to adopt the heliocentric theory, but Tycho Brahe’s data showed that it could not be quite right in the form given to it by Copernicus. He was influenced by Pythagoreanism, and more or less fancifully inclined to sun-worship, though a good Protestant. These motives no doubt gave him a bias in favour of the heliocentric hypothesis. His Pythagoreanism also inclined him to follow Plato’s Timaeus in supposing that cosmic significance must attach to the five regular solids. He used them to suggest hypotheses to his mind; at last, by good luck, one of these worked.

  Kepler’s great achievement was the discovery of his three laws of planetary motion. Two of these he published in 1609, and the third in 1619. His first law states: The planets describe elliptic orbits, of which the sun occupies one focus. His second law states: The line joining a planet to the sun sweeps out equal areas in equal times. His third law states: The square of the period of revolution of a planet is proportioned to the cube of its average distance from the sun.

  Something must be said in explanation of the importance of these laws.

  The first two laws, in Kepler’s time, could only be proved in the case of Mars; as regards the other planets, the observations were compatible with them, but not such as to establish them definitely. It was not long, however, before decisive confirmation was found.

  The discovery of the first law, that the planets move in ellipses, required a greater effort of emancipation from tradition than a modern man can easily realize. The one thing upon which all astronomers, without exception, had been agreed, was that all celestial motions are circular, or compounded of circular motions. Where circles were found inadequate to explain planetary motions, epicycles were used. An epicycle is the curve traced by a point on a circle which rolls on another circle. For example: take a big wheel and fasten it flat on the ground; take a smaller wheel which has a nail through it, and roll the smaller wheel (also flat on the ground) round the big wheel, with the point of the nail touching the ground. Then the mark of the nail in the ground will trace out an epicycle. The orbit of the moon, in relation to the sun, is roughly of this kind: approximately, the earth describes a circle round the sun, and the moon meanwhile describes a circle round the earth. But this is only an approximation. As observation grew more exact, it was found that no system of epicycles would exactly fit the facts. Kepler’s hypothesis, he found, was far more closely in accord with the recorded positions of Mars than was that of Ptolemy, or even that of Copernicus.

  The substitution of ellipses for circles involved the abandonment of the aesthetic bias which had governed astronomy ever since Pythagoras. The circle was a perfect figure, and the celestial orbs were perfect bodies—originally gods, and even in Plato and Aristotle closely related to gods. It seemed obvious that a perfect body must move in a perfect figure. Moreover, since the heavenly bodies move freely, without being pushed or pulled, their motion must be “natural.” Now it was easy to suppose that there is something “natural” about a circle, but not about an ellipse. Thus many deep-seated prejudices had to be discarded before Kepler’s first law could be accepted. No ancient, not even Aristarchus of Samos, had anticipated such an hypothesis.

  The second law deals with the varying velocity of the planet at different points of its orbit. If S is the sun, and P1, P2, P3, P4, P5 are successive positions of the planet at equal intervals of time—say at intervals of a month—then Kepler’s law states that the areas P1SP2, P2SP3, P3SP4, P4SP5 are all equal. The planet therefore moves fastest when it is nearest to the sun, and slowest when it is farthest from it. This, again, was shocking; a planet ought to be too stately to hurry at one time and dawdle at another.

  The third law was important because it compared the movements of different planets, whereas the first two laws dealt with the several planets singly. The third law says: If r is the average distance of a planet from the sun, and T is the length of its year, then r3 divided by T2 is the same for all the different planets. This law afforded the proof (as far as the solar system is concerned) of Newton’s law of the inverse square for gravitation. But of this we shall speak later.

  Galileo (1564-1642) is the greatest of the founders of modern science, with the possible exception of Newton. He was born on about the day on which Michelangelo died, and he died in the year in which Newton was born. I commend these facts to those (if any) who still believe in metempsychosis. He is important as an astronomer, but perhaps even more as the founder of dynamics.

  Galileo first discovered the importance of acceleration in dynamics. “Acceleration” means change of velocity, whether in magnitude or direction; thus a body moving uniformly in a circle has at all times an acceleration towards the centre of the circle. In the language that had been customary before his time, we might say that he treated uniform motion in a straight line as alone “natural,” whether on earth or in the heavens. It had been thought “natural” for heavenly bodies to move in circles, and for terrestrial bodies to move in straight lines: but moving terrestrial bodies, it was thought, would gradually cease to move if they were let alone. Galileo held, as against this view, that every body, if let alone, will continue to move in a straight line with uniform velocity; any change, either in the rapidity or the direction of motion, requires to be explained as due to the action of some “force.” This principle was enunciated by Newton as the “first law of motion.” It is also called the law of inertia. I shall return to its purport later, but first something must be said as to the detail of Galileo’s discoveries.

  Galileo was the first to establish the law of falling bodies. This law, given the concept of “acceleration,” is of the utmost simplicity. It says that, when a body is falling freely, its acceleration is constant, except in so far as the resistance of the air may interfere; further, the acceleration is the same for all bodies, heavy or light, great or small. The complete proof of this law was not possible until the air pump had been invented, which was about 1654. After this, it was possible to observe bodies falling in what was practically a vacuum, and it was found that feathers fell as fast as lead. What Galileo proved was that there is no measurable difference between large and small lumps of the same substance. Until his time it had been supposed that a large lump of lead would fall much quicker than a small one, but Galileo proved by experiment that this is not the case. Measurement, in his day, was not such an accurate business as it has since become; nevertheless he arrived at the true law of falling bodies. If a body is falling freely in a vacuum, its velocity increases at a constant rate. At the end of the first second, its velocity will be 32 feet per second; at the end of another second, 64 feet per second; at the end of the third, 96 feet per second; and so on. The a
cceleration, i.e., the rate at which the velocity increases, is always the same; in each second, the increase of velocity is (approximately) 32 feet per second.

  Galileo also studied projectiles, a subject of importance to his employer, the duke of Tuscany. It had been thought that a projectile fired horizontally will move horizontally for a while, and then suddenly begin to fall vertically. Galileo showed that, apart from the resistance of the air, the horizontal velocity would remain constant, in accordance with the law of inertia, but a vertical velocity would be added, which would grow according to the law of falling bodies. To find out how the projectile will move during some short time, say a second, after it has been in flight for some time, we proceed as follows: First, if it were not falling, it would cover a certain horizontal distance, equal to that which it covered in the first second of its flight. Second, if it were not moving horizontally, but merely falling, it would fall vertically with a velocity proportional to the time since the flight began. In fact, its change of place is what it would be if it first moved horizontally for a second with the initial velocity, and then fell vertically for a second with a velocity proportional to the time during which it has been in flight. A simple calculation shows that its consequent course is a parabola, and this is confirmed by observation except in so far as the resistance of the air interferes.

  The above gives a simple instance of a principle which proved immensely fruitful in dynamics, the principle that, when several forces act simultaneously, the effect is as if each acted in turn. This is part of a more general principle called the parallelogram law. Suppose, for example, that you are on the deck of a moving ship, and you walk across the deck. While you are walking the ship has moved on, so that, in relation to the water, you have moved both forward and across the direction of the ship’s motion. If you want to know where you will have got to in relation to the water, you may suppose that first you stood still while the ship moved, and then, for an equal time, the ship stood still while you walked across it. The same principle applies to forces. This makes it possible to work out the total effect of a number of forces, and makes it feasible to analyse physical phenomena, discovering the separate laws of the several forces to which moving bodies are subject. It was Galileo who introduced this immensely fruitful method.

  In what I have been saying, I have tried to speak, as nearly as possible, in the language of the seventeenth century. Modern language is different in important respects, but to explain what the seventeenth century achieved it is desirable to adopt its modes of expression for the time being.

  The law of inertia explained a puzzle which, before Galileo, the Copernican system had been unable to explain. As observed above, if you drop a stone from the top of a tower, it will fall at the foot of the tower, not somewhat to the west of it; yet, if the earth is rotating, it ought to have slipped away a certain distance during the fall of the stone. The reason this does not happen is that the stone retains the velocity of rotation which, before being dropped, it shared with everything else on the earth’s surface. In fact, if the tower were high enough, there would be the opposite effect to that expected by the opponents of Copernicus. The top of the tower, being further from the centre of the earth than the bottom, is moving faster, and therefore the stone should fall slightly to the east of the foot of the tower. This effect, however, would be too slight to be measurable.

  Galileo ardently adopted the heliocentric system; he corresponded with Kepler, and accepted his discoveries. Having heard that a Dutchman had lately invented a telescope, Galileo made one himself, and very quickly discovered a number of important things. He found that the Milky Way consists of a multitude of separate stars. He observed the phases of Venus, which Copernicus knew to be implied by his theory, but which the naked eye was unable to perceive. He discovered the satellites of Jupiter, which, in honour of his employer, he called “sidera medicea.” It was found that these satellites obey Kepler’s laws. There was, however, a difficulty. There had always been seven heavenly bodies, the five planets and the sun and moon; now seven is a sacred number. Is not the Sabbath the seventh day? Were there not the seven-branched candlesticks and the seven churches of Asia? What, then, could be more appropriate than that there should be seven heavenly bodies? But if we have to add Jupiter’s four moons, that makes eleven—a number which has no mystic properties. On this ground the traditionalists denounced the telescope, refused to look through it, and maintained that it revealed only delusions. Galileo wrote to Kepler wishing they could have a good laugh together at the stupidity of “the mob”; the rest of his letter makes it plain that “the mob” consisted of the professors of philosophy, who tried to conjure away Jupiter’s moons, using “logic-chopping arguments as though they were magical incantations.”

  Galileo, as every one knows, was condemned by the Inquisition, first privately in 1616, and then publicly in 1633, on which latter occasion he recanted, and promised never again to maintain that the earth rotates or revolves. The Inquisition was successful in putting an end to science in Italy, which did not revive there for centuries. But it failed to prevent men of science from adopting the heliocentric theory, and did considerable damage to the Church by its stupidity. Fortunately there were Protestant countries, where the clergy, however anxious to do harm to science, were unable to gain control of the State.

  Newton (1642-1727) achieved the final and complete triumph for which Copernicus, Kepler, and Galileo had prepared the way. Starting from his three laws of motion—of which the first two are due to Galileo—he proved that Kepler’s three laws are equivalent to the proposition that every planet, at every moment, has an acceleration towards the sun which varies inversely as the square of the distance from the sun. He showed that accelerations towards the earth and the sun, following the same formula, explain the moon’s motion, and that the acceleration of falling bodies on the earth’s surface is again related to that of the moon according to the inverse square law. He defined “force” as the cause of change of motion, i.e., of acceleration. He was thus able to enunciate his law of universal gravitation: “Every body attracts every other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.” From this formula he was able to deduce everything in planetary theory: the motions of the planets and their satellites, the orbits of comets, the tides. It appeared later that even the minute departures from elliptical orbits on the part of the planets were deducible from Newton’s law. The triumph was so complete that Newton was in danger of becoming another Aristotle, and imposing an insuperable barrier to progress. In England, it was not till a century after his death that men freed themselves from his authority sufficiently to do important original work in the subjects of which he had treated.

  The seventeenth century was remarkable, not only in astronomy and dynamics, but in many other ways connected with science.

  Take first the question of scientific instruments.* The compound microscope was invented just before the seventeenth century, about 1590. The telescope was invented in 1608, by a Dutchman named Lippershey, though it was Galileo who first made serious use of it for scientific purposes. Galileo also invented the thermometer—at least, this seems most probable. His pupil Torricelli invented the barometer. Guericke (1602-86) invented the air pump. Clocks, though not new, were greatly improved in the seventeenth century, largely by the work of Galileo. Owing to these inventions, scientific observation became immensely more exact and more extensive than it had been at any former time.

  Next, there was important work in other sciences than astronomy and dynamics. Gilbert (1540-1603) published his great book on the magnet in 1600. Harvey (1578-1657) discovered the circulation of the blood, and published his discovery in 1628. Leeuwenhoek (1632-1723) discovered spermatozoa, though another man, Stephen Hamm, had discovered them, apparently, a few months earlier; Leeuwenhoek also discovered protozoa or unicellular organisms, and even bacteria. Robert Boyle (1627-91) was, as children were taught when
I was young, “the father of chemistry and son of the Earl of Cork”; he is now chiefly remembered on account of “Boyle’s Law,” that in a given quantity of gas at a given temperature, pressure is inversely proportional to volume.

  I have hitherto said nothing of the advances in pure mathematics, but these were very great indeed, and were indispensable to much of the work in the physical sciences. Napier published his invention of logarithms in 1614. Co-ordinate geometry resulted from the work of several seventeenth-century mathematicians, among whom the greatest contribution was made by Descartes. The differential and integral calculus was invented independently by Newton and Leibniz; it is the instrument for almost all higher mathematics. These are only the most outstanding achievements in pure mathematics; there were innumerable others of great importance.

  The consequence of the scientific work we have been considering was that the outlook of educated men was completely transformed. At the beginning of the century, Sir Thomas Browne took part in trials for witchcraft; at the end, such a thing would have been impossible. In Shakespeare’s time, comets were still portents; after the publication of Newton’s Principia in 1687, it was known that he and Halley had calculated the orbits of certain comets, and that they were as obedient as the planets to the law of gravitation. The reign of law had established its hold on men’s imaginations, making such things as magic and sorcery incredible. In 1700 the mental outlook of educated men was completely modern; in 1600, except among a very few, it was still largely medieval.

  In the remainder of this chapter I shall try to state briefly the philosophical beliefs which appeared to follow from seventeenth-century science, and some of the respects in which modern science differs from that of Newton.