The Mathematical Principles of Natural Philosophy (1729)/Proposition 7: Gravity tending to all Bodies
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 prov'd before; as well as that the force of gravity towards every one of them, consider'd apart, is reciprocally as the square of the distance of places from the centre of the planet. And thence (by proposition 69. book. 1. 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 5.) to every action corresponds an equal re-action: 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.
Corollary 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 form'd 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, whereas no such gravitation any where appears: 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.
Corollary 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 corrolary 3. proposition 74. book 1.
Proposition VIII. Theorem VIII. In two spheres mutually gravitating each towards the other, if the matter in places on all sides round about and equidistant from the centres, is similar; the weight of either sphere towards the other, will be reciprocally as the square of the distance between their centres.
After I had found that the force of gravity towards a whole planet did arise from, and was compounded of the the forces of gravity towards all its parts; and towards every one part, was in the reciprocal proportion of the squares of the distances from the part: I was yet in doubt, whether that reciprocal duplicate proportion did accurately hold, or but nearly so, in the total force compounded of so many partial ones. For it might be that the proportion which accurately enough took place in greater distances, should be wide of the truth near the surface of the planet, where the distances of the particles are unequal, and their situation dissimilar. But by the help of proposition 75. and 76. book I. and their corollaries, I was at last satisfy'd of the truth of the proposition, as it now lies before us.
Corollary 1. Hence we may find and compare together the weights of bodies towards different planets. For the weights of bodies revolving in circles about planets, are (by corrolary 2. proposition 4. book I.) as the diameters of the circles directly, and the squares of their periodic times reciprocally; and their weights at the surfaces of the planets, or at any other distances from their centres, are (by this proposition) greater or less, in the reciprocal duplicate proportion of the distances. Thus from the periodic times of Venus, revolving about the Sun, in 224d. 16¼h. of the utmost circumjovial satellite revolving about Jupiter, in 16d. 161/15h; of the Hugenian satellite about Saturn in 15d. 222/3h; and of the Moon about the Earth in 27d. 7h. 43'; compared with the mean distance of Venus from the Sun, and with the greatest heliocentric elongations of the outmost circumjovial satellite from Jupiter's centre, 8'.16" of the Hugenian satellite from the centre of Saturn, 3'.4", and of the Moon from the Earth, 10'.33"; by computation I found, that the weight of equal bodies, at equal distances from the centres of the Sun, of Jupiter, of Saturn, and of the Earth, towards the Sun, Jupiter, Saturn, and the Earth, were one to another, as 1, 1/1067, 1/3021, and 1/169282 respectively. Then because as the distances are increased or diminished, the weights are diminished or increased in a duplicate ratio; the weights of equal bodies towards the Sun, Jupiter, Saturn, and the Earth, at the distances 10000, 997, 791 and 109 from their centres, that is, at their very superficies, will be as 10000, 943, 529 and 435 respectively. How much the weights of bodies are at the superficies of the Moon, will be'shewn hereafter.
Corollary 2. Hence likewise vwe discoVeiTthe quantity of matter in the several Planets. For their quantities of matter are as the forces of gravity at equal distances from their centres, that is, in the Sun, Jupiter, Saturn, and the Earth, as 1, 1/1067, 1/3021, and 1/169282 respectively. If the parallax of the Sun be taken greater or less than 10", 30"', the quantity of matter in the Earth must be augmented or diminished in the triplicate of that proportion.
Corollary 3. Hence also we find the densities of the Planets. For (by proposition 72. book I.) the weights of equal and similar bodies towards similar spheres, are, at the surfaces of those spheres, as the diameters of the spheres. And therefore the densities of dissimilar spheres are as those weights applied to the diameters of the spheres. But the true diameters of the Sun, Jupiter, Saturn, and the Earth, were one to another as 10000, 997, 791 and 109; and the weights towards the same, as 10000, 943, 529, and 435 respectively; and therefore their densities are as 100, 94½, 67 and 400. The density of the Earth, which comes out by this computation, does not depend upon the parallax of the Sun, but is determined by the parallax of the Moon, and therefore is here truly defin'd. The Sun therefore is a little denser than Jupiter, and Jupiter than Saturn, and the Earth four times denser than the Sun; for the Sun, by its great heat, is kept in a sort of a rarefy'd state. The Moon is denser than the Earth, as shall appear afterwards.
Corollary 4. The smaller the Planets are, they are, ceteris paribus, of so much the greater density. For so the powers of gravity on their several surfaces, come nearer to equality. They are likewise, ceteris paribus, of the greater density, as they are nearer to the Sun. So Jupiter is more dense than Saturn, and the Earth than Jupiter. For the Planets were to be placed at different distances from the Sun, that according to their degrees of density, they might enjoy a greater or less proportion of the Sun's heat. Our water, if it were remov'd as far as the orb of Saturn, would be converted into ice, and the orb of Mercury would quickly fly away in vapour. For the light of the Sun, to which its heat is proportional, is seven times denser in the orb of the Mercury than with us: and by the thermometer I have found, that a sevenfold heat of our summer-sun will make water boil. Nor are we to doubt, that the matter of Mercury is adapted to its heat, and is therefore more dense than the matter of our Earth; since, in a denser matter, the operations of nature require a stronger heat.
Proposition IX. Theorem IX. That the force of gravity, consider'd downwards from the surface of the planets, decreases nearly in the proportion of the distances from their centres.
If the matter of the planet were of an uniform density, this proposition would be accurately true, (by proposition 75. book I.) The error therefore can be no greater than what may arise from the inequality of the density.