That the moon gravitates toward the earth, and by the force of gravity is continually drawn off from a rectilinear motion and retained in its orbit.


The demonstration of this proposition may be more diffusely explained after the following manner. Suppose several moons to revolve about the earth, as in the system of Jupiter or Saturn; the periodic times of these moons (by the argument of induction) would observe the same law which Kepler found to obtain among the planets, and therefore their centripetal forces would be inversely as the squares of the distances from the center of the earth, by Prop. 1 of this book. Now if the lowest of these were very small and were so near the earth as almost to touch the tops of the highest mountains, the centripetal force thereof, retaining in its orbit, would be nearly equal to the weights of any terrestrial bodies that should be found upon the tops of those mountains, as may be known by the foregoing computation. Therefore, if the same little moon should be deserted by its centrifugal force that carries it through its orbit, and be disabled from going onward therein, it would descend to the earth; and that with the same velocity with which heavy bodies actually fall upon the tops of those very mountains, because of the equality of the forces that oblige them both to descend. And if the force by which the lowest moon would descend were different from gravity, and if the moon were to gravitate toward the earth, as we find terrestrial bodies do upon the tops of mountains, it would then descend with twice the velocity, as being impelled by both these forces conspiring together. Therefore, since both these forces, that is, the gravity of heavy bodies and the centripetal forces of the moons, are directed to the center of the earth and are similar and equal between themselves, they will (by Rules 1 and 2) have one and the same cause. And therefore the force which retains the moon in its orbit is that very force which we commonly call gravity, because otherwise this little moon at the top of a mountain must either be without gravity, or fall twice as swiftly as heavy bodies are wont to do.


That the circumjovial planets gravitate towards Jupiter; the circimsaturnal towards Saturn; the circumsolar towards the sun; and by the forces of their gravity are drawn off from rectilinear motions, and retained in curvilinear orbits.


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, by Rules 1, 2, and 4.


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

It has been now for 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 experiments with 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 center 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 forward and backward for a long time with equal vibrations. And therefore the quantity of matter in the gold (by Cor. I and VI, Prop. 24, Book II) was to the quantity of matter in the wood as the action of the motive force (or vis motrix) upon all the gold to the action of the same upon all the wood; that is, as the weight of the one to the weight of the other: 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 toward the planets is the same as toward the earth. For, should we imagine our terrestrial bodies taken to the orbit of the moon and there, together with the moon, deprived of all motion, to be let go, so as to fall together toward the earth, it is certain, from what we have demonstrated before, that in equal times they would describe equal spaces with the moon, and of consequence are to the moon, in quantity of matter, as their weights to its weight. Moreover, since the satellites of Jupiter perform their revolutions in times which observe the 3/2th power of the proportion of their distances from Jupiter’s center, their accelerative gravities toward Jupiter will be inversely as the squares of their distances from Jupiter’s center; that is, equal, at equal distances. And therefore these satellites, if supposed to fall toward Jupiter from equal heights, would describe equal spaces in equal times, in like manner as heavy bodies do on our earth. And, by the same argument, if the circumsolar planets were supposed to be let fall at equal distances from the sun, they would, in their descent toward the sun, describe equal spaces in equal times. But forces which equally accelerate unequal bodies must be as those bodies: that is to say, the weights of the planets toward the sun must be as their quantities of matter. Further, that the weights of Jupiter and of his satellites toward the sun are proportional to the several quantities of their matter, appears from the exceedingly regular motions of the satellites (by Cor. III, Prop. 65, Book 1). For if some of those bodies were more strongly attracted to the sun in proportion to their quantity of matter than others, the motions of the satellites would be disturbed by that inequality of attraction (by Cor. II, Prop. 65, Book 1). If, at equal distances from the sun, any satellite, in proportion to the quantity of its matter, did gravitate toward 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 centers of the sun and of the satellite’s orbit would be always greater than the distance between the centers of the sun and of Jupiter, nearly as the square root of that proportion: as by some computations I have found. And if the satellite did gravitate toward the sun with a force less in the proportion of e to d, the distance of the center of the satellite’s orbit from the sun would be less than the distance of the center of Jupiter from the sun as the square root of the same proportion. Therefore if, at equal distances from the sun, the accelerative gravity of any satellite toward the sun were greater or less than the accelerative gravity of Jupiter toward the sun but by one 1/1,000 part of the whole gravity, the distance of the center of the satellite’s orbit from the sun would be greater or less than the distance of Jupiter from the sun by one 1/2,000 part of the whole distance; that is, by a fifth part of the distance of the utmost satellite from the center of Jupiter; an eccentricity of the orbit which would be very sensible. But the orbits of the satellites are concentric to Jupiter, and therefore the accelerative gravities of Jupiter and of all its satellites toward the sun are equal among themselves. And by the same argument the weights of Saturn and of his satellites toward the sun, at equal distances from the sun, are as their several quantities of matter; and the weights of the moon and of the earth toward the sun are either none, or accurately proportional to the masses of matter which they contain. But some weight they have, by Cor. I and III, Proposition 5.
But further; the weights of all the parts of every planet toward 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 to the orbit 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 shown above.

COR. I. 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. II. Universally, all bodies about the earth gravitate toward the earth; and the weights of all, at equal distances from the earth’s center, 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. If the ether, or any other body, were either altogether void of gravity or were to gravitate less in proportion to its quantity of matter, then, because (according to Aristotle, Descartes, and others) there is no difference between that and other bodies but in mere form of matter, by a successive change from form to form, it might be changed at last into a body of the same condition with those which gravitate most in proportion to their quantity of matter; and, on the other hand, the heaviest bodies, acquiring the first form of that body, might by degrees quite lose their gravity. And therefore the weights would depend upon the forms of bodies and, with those forms, might be changed, contrary to what was proved in the preceding Corollary.

COR. III. All spaces are not equally full; for if all spaces were equally full, then the specific gravity of the fluid which fills the region of the air, on account of the extreme density of the matter, would fall nothing short of the specific gravity of quicksilver or gold or any other the most dense body, and therefore neither gold nor any other body could descend in air, for bodies do not descend in fluids unless they are specifically heavier than the fluids. And if the quantity of matter in a given space can, by any rarefaction, be diminished, what should hinder a diminution to infinity?

COR. IV. If all the solid particles of all bodies are of the same density and cannot be rarefied without pores, then a void space or vacuum must be granted. By bodies of the same density I mean those whose inertias are in the proportion of their bulks.

COR. V. 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 as the square, but almost as the cube of the distance, as nearly as I could judge from some rude observations.

[End of Book 3. Rules of Reasoning in Philosophy. On Gravity]