Proposition 8 Theorem 8

The Moon's Gravitation

The forces that keeps the planets in their orbits comes from Jupiter's center.

Newton Newton
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Proposition 8 Theorem 8

In 2 spheres gravitating towards the other, if the matter in places on all sides round about and equi-distant 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 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 Prop. LXXV and LXXVI, Book I, and their Corollaries, I was at last satisfied of the truth of the Proposition, as it now lies before us.

Corollary 1

We can use this to find the weights of planets. The weights of bodies orbiting planets are (by Cor. 2, Prop. 4, Book I) as the diameters of the circles directly and the squares of their periodic times reciprocally.

Their weights at the surfaces of the planets, or at any other distances from their centres, are (by this Prop.) greater or less in the reciprocal duplicate proportion of the distances.

Thus, from the orbital period of:

  • Venus is 224d 16 3/4h

  • outermost moon of Jupiter is 16d 16 h

  • of the Huygenian satellite about Saturn in 15d.22⅔h.; 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 Huygenian 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

This is also the way to find the quantity of matter [mass] in the planets. This is because the quantities of matter are as the forces of gravity at equal distances from their centres.

In the sun, Jupiter, Saturn, and the earth respectively are: 1, 1/xxxx, 1/xxxx, and 1/xxxxx

If the parallax of the sun is 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

This is how we find the densities of the planets. According to Book 1 Prop 72, the weights of equal bodies Spheres are, at the surfaces of those spheres, as the diameters of the spheres 5 and therefore the densities of dissimilar spheres are as those weights applied.

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 defined.

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 rarefied state. The moon is denser than the earth, as shall appear afterward.

Corollary 4

The smaller the planet, and the nearer to the sun, the more dense it is.

The gravity on the surface come nearer to equality.

  • Jupiter is denser than Saturn
  • Earth is denser than Jupiter

placed at different distances from the sun, that, according to their degrees of density, they might enjoy a greater or less proportion to the sun s heat.

Earth’s water would be converted to:

  • ice if it were transferred to Saturn.
  • vapor if it were transferred to Mercury from boiling

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