# The Theory of Gravitation

## Isaac Newton

## PROPOSITION I. THEOREM I.

*That the areas, which bodies, when moving in curves, cut off by radii drawn to a fixed center of force, are in one fixed plane and are proportional to the times.*

Let the time be divided into equal parts, and in the first period of time let the body driven by one force describe the line AB. In the second period, it would, if nothing hindered it, go on to c, describing the line Be equal to AB. Then by the radii AS, BS, cS to the center S would be cut off the equal areas ASB, BSc [the bases being equal and the altitude the same]. Now when the body comes to B, a centripetal force [in the direction BS] acts upon it with uniform impulse, and makes it leave the line of direction Bc and pass along the line BC. Let cC be drawn parallel to the direction of the force BS, meeting BC in C. Then at the end of the second (equal) period the body will be found at C, in the same plane with the triangle ASB. Draw SC. Then the triangle SBC, on account of the parallels SB and cC, will be equal to the triangle SBc and therefore to the triangle SAB, etc.—Therefore in equal times equal areas will be described in the same plane.—Let the number of the triangles be increased and their altitude diminished to infinity: their ultimate perimeter will be a curve (Cor. iv. Lem. iii.). And therefore a centripetal force, by which a body is continually drawn from a course tangent to this curve, will act along this radius and whatever areas have been described proportional to the times, will remain proportional to the same times when curvilinear. [p.136]

## PROPOSITION II. THEOREM II.

*Every body, which is moved in any curve described in a plane, and cuts off, by radii drawn to a center, that is stationary or moving in a straight line with uniform motion, areas about the center proportional to the times, is drawn by a centripetal force urging it toward the center.*

For every body that is moved in a curved line, is turned from its course by some force acting upon it. And that force by which a body is turned from a straight line, and is made to describe the supposed equal triangles SAB, SBC, etc. about the fixed center S in equal times must act at the point B in a line parallel to Cc.

[For extend AB to c making AB=Bc. Then c is where the body would have been had it not been drawn by the new force at B. Hence at B the force acts in the direction Cc.]

But cC is parallel to BS. [For since the triangle SCB=triangle SAB by hypothesis, and triangle SAB=SBc (equal altitude and bases) then triangles SBC and SBc must be equal and Cc and SB must be parallel, in order to have the altitude equal.] Therefore at B the force acts along the line 115 toward the center S. Therefore the force always acts toward the immovable center S.

[It will be remembered that Kepler had already shown that the planets move in ellipses, and do cut off areas proportioned to the times. Hence they act as if drawn by a centripetal force. Then what is this force? The next great step was to prove it identical with weight.]

## PROPOSITION IV. THEOREM IV.

*That the moon is drawn by gravity [weight] toward the earth, and is deflected by the force of gravity from a straight line [tangent], and thus held in her orbit.*

The mean distance of the moon from the earth in terms of [p.137] semi-diameters of the earth is, according to Ptolemy and many astronomers, 59; according to Vendelius and Huyghens, 60; according to Copernicus, 60 1–5; according to Streetus, 60 2–5, and according to Tycho, 56 1–2. (But Tycho has erred . . .) Let us assume that the mean distance is 60 semi-diameters of the earth. The moon completes her full periodic times (goes round the earth) in 27 days, 7 hours, 43 minutes, as is determined by astronomers. The circumference of the earth is 123,249,600 Paris feet, as has been calculated by the French measurements. If the moon should be deprived of every other motion, and drawn by that one alone by which she is held in her orbit, she would fall to the earth. The distance she would fall in the first minute would be 15 1–12 Paris feet. This follows from calculation or from Proposition xxvi., Bk. I., or (what amounts to the same thing) from Cor. ix., Prop. iv., the same Book. For the versed sine (distance along the radius from the chord to the circumference) of that are which the moon describes in one minute at her mean motion and at a distance of 60 semi-diameters of the earth from the earth is about 15 1–2 Paris feet, or, more accurately, 15 feet, 1 inch, and 1 4–9 lines. [This is found as follows: The distance of the moon from the earth is 60 radii of the earth. Hence the orbit of the moon equals 60 times the circumference of the earth. Divide this result by the number of minutes (39,343) in the moon’s periodic time, and the quotient is the arc passed over by the moon in one minute (about 187,964 Paris feet). In the diagram Mm is the arc passed over by the moon in one minute, Mx is the distance the moon has been deflected from a tangent in one minute and the distance she would fall toward the earth in this time if acted on by gravity alone. Arc Mm squared equals Mx times MA (diameter moon’s orbit), or Mx=Mm2 divided by MA, or 35,330,465,296 feet divided by 2,353,893,976 or 15 1–12 feet.]

Now since this force in approaching the earth increases, in inverse ratio with the square of the distance, therefore at the surface of the earth it will be greater by 60x60 than at the moon [the distance being 60 radii of the earth]. Then a body driven by this force in falling in our locality ought to pass over in the first minute 60x60x15 1–12 Paris feet, and in the space of one second 15 1–12 Paris feet or, more accurately 15 feet, 1 inch, and 1 4–9 lines. But heavy bodies do actually fall at this rate on the earth. For the length of a pendulum, oscillating each Second in the latitude of Lutetia, Paris, is three Paris feet and 8 1–2 lines, as Huyghens has observed, and the distance which a body falls in a second when pulled by gravity is to the length of such a pendulum as the square of the circumference of a circle to its diameter, as Huyghens has also observed; and this is 15 Paris feet, 1 inch, 1 7–9 lines. Hence the force by which the moon is held in its orbit, if it were brought down upon earth, would be equal to the force of gravity among us, and hence is that very force which we are wont to call (weight or) gravity.

## BOOK III. PROPOSITION V. THEOREM V. SCHOLIUM

The force which retains the celestial bodies in their orbits has been hitherto called centripetal force; but it being now made plain that it Can be no other than a gravitating force, we shall hereafter call it gravity. For the cause of that centripetal force which retains the moon in its orbit will extend itself to all the planets:

## BOOK III. PROPOSITION VI. THEOREM VI.

*That all bodies gravitate towards every planet; and that the weights of bodies towards any the same planet, at equal distances from the centre of the planet, are proportional to the quantities of matter which they severally contain.*

It has been, now of a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to the earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums. I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal; I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forwards and backwards, for a long time, with equal vibrations. . . . and the like happened in the other bodies. By these experiments, in bodies of the same weight, I could manifestly have discovered a difference of matter less than the thousandth part of the whole, had any such been. But, without all doubt, the nature of gravity towards the planets is the same as towards the earth . . . Moreover, since the satellites of Jupiter perform their revolutions in times which observe the sesquiplicate proportion of their distances from Jupiter’s centre—that is, equal at equal distances. And, therefore, these [p.139] satellites, if supposed to fall towards Jupiter from equal heights, would describe equal spaces in equal times, in like manner as heavy bodies do on our earth . . . If, at equal distances from the sun, any satellite, in proportion to the quantity of its matter, did gravitate towards the sun with a force greater than Jupiter in proportion to his, according to any given proportion, suppose of d to e; then the distance between the centres of the sun and of the satellite’s orbit would be always greater than the distance between the centres of the sun and of Jupiter nearly in the sub-duplicate of that proportion; as by some computations I have found. And if the satellite did gravitate towards the sun with a force, lesser in the proportion of e to d, the distance of the centre of the satellite’s orbit from the sun would be less than the distance of the centre of Jupiter from the sun in the sub-duplicate of the same proportion. Therefore if, at equal distances from the sun, the accelerative gravity of any satellite towards the sun were greater or less than the accelerative gravity of Jupiter towards the sun but one 1–1000 part of the whole gravity, the distance of the centre of the satellite’s orbit from the sun would be greater or less than the distance of Jupiter from the sun by one 1–2000 part of the whole distance—that is, by a fifth part of the distance of the utmost satellite from the centre of Jupiter; an eccentricity of the orbit which would be very sensible. But the orbits of the satellite are concentric to Jupiter, and therefore the accelerative gravities of Jupiter, and of all its satellites towards the sun, are equal among themselves . . .

But further; the weights of all the parts of every planet towards any other planet are one to another as the matter in the several parts; for if some parts did gravitate more, others less, than for the quantity of their matter, then the whole planet, according to the sort of parts with which it most abounds, would gravitate more or less than in proportion to the quantity of matter in the whole. Nor is it of any moment whether these parts are external or internal; for if, for example, we should imagine the terrestrial bodies with us to be raised up to the orb of the moon, to be there compared with its body; if the weights of such bodies were to the weights of the external parts of the moon as the quantities of matter in the one and in the other respectively; but to the weights of the internal parts in a greater or less proportion, then likewise the weights of those bodies would be to the weight of the whole moon in a greater or less proportion; against what we have showed above.

Cor. 1. Hence the weights of bodies do not depend upon their forms and textures; for if the weights could be altered with the forms, they would be greater or less, according to the variety of forms, in equal matter; altogether against experience.

Cor. 2. Universally, all bodies about the earth gravitate towards the earth; and the weights of all, at equal distances from the earth’s centre, are as the quantities of matter which they severally contain. This is the quality of all bodies within the reach of our experiments; and therefore (by rule 3) to be affirmed of all bodies whatsoever . . .

Cor. 5. The power of gravity is of a different nature from the power of magnetism; for the magnetic attraction is not as the matter attracted. Some bodies are attracted more by the magnet; others less; most bodies not at all. The power of magnetism in one and the same body may be increased and diminished; and is sometimes far stronger, for the quantity of matter, than the power of gravity; and in receding from the magnet decreases not in the duplicate but almost in the triplicate proportion of the distance, as nearly as I could judge from some rude observations.

## BOOK III. PROPOSITION VII. THEOREM VII.

*That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain.*

That all the planets mutually gravitate one towards another, we have proved before; as well as that the force of gravity towards every one of them, considered apart, is reciprocally as the square of the distance of places from the centre of the planet. And thence (by prop, 69, book I, and its corollaries) it follows, that the gravity tending towards all the planets is proportional to the matter which they contain.

Moreover, since all the parts of any planet A gravitate towards any other planet B; and the gravity of every part is to the gravity of the whole as the matter of the part to the matter of the whole; and (by law 3) to every action corresponds an equal reaction; therefore the planet B will, on the other hand, gravitate towards all the parts of the planet A; and its gravity towards any one part will be to the gravity towards the whole as the matter of the part to the matter of the whole. Q.E.D.

Cor. 1. Therefore the force of gravity towards any whole planet arises from, and is compounded of, the forces of gravity towards all its parts. Magnetic and electric attractions afford us examples of this; for all attraction towards the whole arises from the attractions towards the several parts. The thing may be easily understood in gravity, if we consider a greater planet as formed of a number of lesser planets meeting together in one globe; for hence it would appear that the force of the whole must arise from the forces of the component parts. If it is objected that, according to this law, all bodies with us must mutually gravitate one towards another, I answer, that since the gravitation towards these bodies is to the gravitation towards the whole earth as these bodies are to the whole earth, the gravitation towards them must be far less than to fall under the observation of our senses.

Cor. 2. The force of gravity towards the several equal particles of any body is reciprocally as the square of the distance of places from the, particles; as appears from cor. 3, prop. 74, book 1.

*[Under Proposition X. is the following important passage:]* However the planets have been formed while they were yet in fluid masses, all the heavier matter subsided to the centre. Since, therefore, the common matter of our earth on the surface thereof is about twice as heavy as water, and a little lower, in mines, is found about three, or four, or even five times more heavy, it is probable that the quantity of the whole matter of the earth may be five or six times greater than if it consisted all of water.—Translated from the "Principia."