Part 38

SIR ISAAC NEWTON

Principia

Sir Isaac Newton was born at Woolsthorpe, Lincolnshire, England, Dec. 25, 1642, the son of a small landed proprietor. For the famous episode of the falling apple, Voltaire, who admirably explained his system for his countrymen, is responsible. It was in 1680 that Newton discovered how to calculate the orbit of a body moving under a central force, and showed that if the force varied as the inverse square of the distance, the orbit would be an ellipse with the centre of force in one focus. The great discovery, which made the writing of his "Philosophiæ Naturalis Principia Mathematica" possible, was that the attraction between two spheres is the same as it would be if we supposed each sphere condensed to a point at its centre. The book was published as a whole in 1687. Of its author it was said by Lagrange that not only was he the greatest genius that ever existed, but also the most fortunate, "for we cannot find more than once a system of the world to establish." Newton died on March 20, 1727.

Our design (writes Newton in his preface) not respecting arts but philosophy, and our subject not manual but natural powers, we consider those things which relate to gravity, levity, elastic force, the resistance of fluids and the like forces, whether attractive or impulsive; and, therefore, we offer this work as the mathematical principles of philosophy, for all the difficulty of philosophy seems to consist in this--from the phenomena of motions to investigate the forces of nature, and from these forces to demonstrate the other phenomena, and to this end the general propositions in the first and second book are directed. In the third book, we give an example of this in the explication of the system of the world; for by the propositions mathematically demonstrated in the former books, we in the third derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the motions of the planets, the comets, the moon, and the sea.

Upon this subject I had (he says) composed the third book in a popular method, that it might be read by many, but afterward, considering that such as had not sufficiently entered into the principles could not easily discern the strength of the consequences, nor lay aside the prejudices to which they had been many years accustomed, therefore, to prevent the disputes which might be raised upon such accounts, I chose to reduce the substance of this book into the form of Propositions (in the mathematical way). So that this third book is composed both "in popular method" and in the form of mathematical propositions.

Books I and II

The principle of universal gravitation, namely, "That every particle of matter is attracted by or gravitates to every other particle of matter with a force inversely proportional to the squares of their distances," is the discovery which characterises the "Principia." This principle the author deduced from the motion of the moon and the three laws of Kepler; and these laws in turn Newton, by his greater law, demonstrated to be true.

From the first law of Kepler, namely, the proportionality of the areas to the times of their description, Newton inferred that the force which retained the planet in its orbit was always directed to the sun. From the second, namely, that every planet moves in an ellipse with the sun as one of foci, he drew the more general inference that the force by which the planet moves round that focus varies inversely as the square of its distance therefrom. He demonstrated that a planet acted upon by such a force could not move in any other curve than a conic section; and he showed when the moving body would describe a circular, an elliptical, a parabolic, or hyperbolic orbit. He demonstrated, too, that this force or attracting, gravitating power resided in even the least particle; but that in spherical masses it operates as if confined to their centres, so that one sphere or body will act upon another sphere or body with a force directly proportional to the quantity of matter and inversely as the square of the distance between their centres, and that their velocities of mutual approach will be in the inverse ratio of their quantities of matter. Thus he outlined the universal law.

The System of the World

It was the ancient opinion of not a few (writes Newton in Book III.) in the earliest ages of philosophy that the fixed stars stood immovable in the highest parts of the world; that under the fixed stars the planets were carried about the sun; that the earth, as one of the planets, described an annual course about the sun, while, by a diurnal motion, it was in the meantime revolved about its own axis; and that the sun, as the common fire which served to warm the whole, was fixed in the centre of the universe. It was from the Egyptians that the Greeks derived their first, as well as their soundest notions of philosophy. It is not to be denied that Anaxagoras, Democritus and others would have it that the earth possessed the centre of the world, but it was agreed on both sides that the motions of the celestial bodies were performed in spaces altogether free and void of resistance. The whim of solid orbs was of later date, introduced by Endoxus, Calippus and Aristotle, when the ancient philosophy began to decline.

As it was the unavoidable consequence of the hypothesis of solid orbs while it prevailed that the comets must be thrust down below the moon, so no sooner had the late observations of astronomers restored the comets to their ancient places in the higher heavens than these celestial spaces were at once cleared of the encumbrance of solid orbs, which by these observations were broken to pieces and discarded for ever.

Whence it was that the planets came to be retained within any certain bounds in these free spaces, and to be drawn off from the rectilinear courses, which, left to themselves, they should have pursued, into regular revolutions in curvilinear orbits, are questions which we do not know how the ancients explained; and probably it was to give some sort of satisfaction to this difficulty that solid orbs were introduced.

The later philosophers pretend to account for it either by the action of certain vortices, as Kepler and Descartes, or by some other principle of impulse or attraction, for it is most certain that these effects must proceed from the action of some force or other. This we will call by the general name of a centripetal force, as it is a force which is directed to some centre; and, as it regards more particularly a body in that centre, we call it circum-solar, circum-terrestrial, circum-jovial.

Centre-Seeking Forces

That by means of centripetal forces the planets may be retained in certain orbits we may easily understand if we consider the motions of projectiles, for a stone projected is by the pressure of its own weight forced out of the rectilinear path, which, by the projection alone, it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground, and the greater the velocity is with which it is projected the further it goes before it falls to earth. We can, therefore, suppose the velocity to be so increased that it would describe an arc of 1, 2, 5, 10, 100, 1,000 miles before it arrived at the earth, till, at last, exceeding the limits of the earth, it should pass quite by it without touching it.

And because the celestial motions are scarcely retarded by the little or no resistance of the spaces in which they are performed, to keep up the parity of cases, let us suppose either that there is no air about the earth or, at least, that it is endowed with little or no power of resisting.

And since the areas which by this motion it describes by a radius drawn to the centre of the earth have previously been shown to be proportional to the times in which they are described, its velocity when it returns to the point from which it started will be no less than at first; and, retaining the same velocity, it will describe the same curve over and over by the same law.

But if we now imagine bodies to be projected in the directions of lines parallel to the horizon from greater heights, as from 5, 10, 100, 1,000 or more miles, or, rather, as many semi-diameters of the earth, those bodies, according to their different velocity and the different force of gravity in different heights, will describe arcs either concentric with the earth or variously eccentric, and go on revolving through the heavens in those trajectories just as the planets do in their orbs.

As when a stone is projected obliquely, the perpetual deflection thereof towards the earth is a proof of its gravitation to the earth no less certain than its direct descent when suffered to fall freely from rest, so the deviation of bodies moving in free spaces from rectilinear paths and perpetual deflection therefrom towards any place, is a sure indication of the existence of some force which from all quarters impels those bodies towards that place.

That there are centripetal forces actually directed to the bodies of the sun, of the earth, and other planets, I thus infer.

The moon revolves about our earth, and by radii drawn to its centre describes areas nearly proportional to the times in which they are described, as is evident from its velocity compared with its apparent diameter; for its motion is slower when its diameter is less (and therefore its distance greater), and its motion is swifter when its diameter is greater.

The revolutions of the satellites of Jupiter about the planet are more regular; for they describe circles concentric with Jupiter by equable motions, as exactly as our senses can distinguish.

And so the satellites of Saturn are revolved about this planet with motions nearly circular and equable, scarcely disturbed by any eccentricity hitherto observed.

That Venus and Mercury are revolved about the sun is demonstrable from their moon-like appearances. And Venus, with a motion almost uniform, describes an orb nearly circular and concentric with the sun. But Mercury, with a more eccentric motion, makes remarkable approaches to the sun and goes off again by turns; but it is always swifter as it is near to the sun, and therefore by a radius drawn to the sun still describes areas proportional to the times.

Lastly, that the earth describes about the sun, or the sun about the earth, by a radius from one to the other, areas exactly proportional to the times is demonstrable from the apparent diameter of the sun compared with its apparent motion.

These are astronomical experiments; from which it follows that there are centripetal forces actually directed to the centres of the earth, of Jupiter, of Saturn, and of the sun.

That these forces decrease in the duplicate proportion of the distances from the centre of every planet appears by Cor. vi., Prop. iv., Book I. for the periodic times of the satellites of Jupiter are one to another in the sesquiplicate proportion of their distances from the centre of this planet. Cassini assures us that the same proportion is observed in the circum-Saturnal planets. In the circum-solar planets Mercury and Venus, the same proportional holds with great accuracy.

That Mars is revolved about the sun is demonstrated from the phases which it shows and the proportion of its apparent diameters; for from its appearing full near conjunction with the sun and gibbous in its quadratures, it is certain that it travels round the sun. And since its diameter appears about five times greater when in opposition to the sun than when in conjunction therewith, and its distance from the earth is reciprocally as its apparent diameter, that distance will be about five times less when in opposition to than when in conjunction with the sun; but in both cases its distance from the sun will be nearly about the same with the distance which is inferred from its gibbous appearance in the quadratures. And as it encompasses the sun at almost equal distances, but in respect of the earth is very unequally distant, so by radii drawn to the sun it describes areas nearly uniform; but by radii drawn to the earth it is sometimes swift, sometimes stationary, and sometimes retrograde.