The second law deals with the varying velocity of the planet at different points of its orbit. If S is the sun, and P 1 , P 2 , P 3 , P 4 , P 5 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 P 1 SP 2 , P 2 SP 3 , P 3 SP 4 , P 4 SP 5 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
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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 acceleration, 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,
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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
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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
