The Moon and the Planets on Superphysics

1: Rules of Reasoning

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Book 1 laid down the principles of mathematical philosophy. This will be the basis of our reasonings on philosophical inquiries.

These mathematical principles are the laws and conditions of certain motions and forces. I use them to build a system of the universe.

2: Phenomena

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The circumjovial planets, by radii drawn to Jupiter’s centre, describe areas proportional to the times of description; and that their periodic times, the fixed stars being at rest, are in the sesquiplicate proportion of their distances from, its centre.

The Attraction of Bodies

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Proposition 1 Theorem 1

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

The Moon's Gravitation

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Proposition 4 Theorem 4

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

The mean distance of the moon from the earth in the syzygies in semi-diameters of the earth is:

The Moon's Gravitation

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Proposition 5 Theorem 5

The circumjovial planets gravitate towards Jupiter. The circnntsaturnal gravitate towards Saturn. The circumsolar gravitate towards the sun.

By their gravity, they are maintained in curvilinear orbits.

The Moon's Gravitation

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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.

Propositions 9-10

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Proposition 9 Theorem 9

Gravity, as a downward pull of a planet, 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 Prop. LXXIII. Book I).

Propositions 11-12

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Proposition 11 Theorem 11

The common centre of gravity of the earth, the sun, and all the planets, is immovable.

For (by Cor. 4 of the Laws) that centre either is at rest, or moves uniformly forward in a right line; but if that centre moved, the centre of the world would move also, against the Hypothesis.

The Planets Move in Ellipses

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Proposition 13 Theorem 13

The planets move in ellipses which have their common focus in the sun’s centre. By radii drawn to that centre, they describe areas proportional to the times of description.

The Orbits of Planets

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Proposition 14 Theorm 14: The aphelions and nodes of the orbits of the planets are fixed.

The aphelions are immovable by Prop. XI, Book I; and so are the planes of the orbits, by Prop. I of the same Book. And if the planes are fixed, the nodes must be so too. It is true, that some inequalities may arise from the mutual actions of the planets and comets in their revolutions; but these will be so small, that they may be here passed by.

The Orbits of Planets

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Proposition 15 Problem 1

Find the principal diameters of the orbits of the planets.

They are in the sub-sesquiplicate proportion of the periodic times, by Prop. 15, Book 1. Then they are augmented in the proportion of the sum of the masses of matter in the sun and each planet to the first of two mean proportionals betwixt that sum and the quantity of matter in the sun, by Prop. 60, Book 1.

Propositions 18-19

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Proposition 18 Theorem 16: The axes of planets

The axes of the planets are less than the diameters drawn perpendicularly to the axes.

The equal gravitation would give a spherical shape to the planets.

The orbits of Planets

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Proposition 19 Problem 3: Find the proportion of the axis of a planet to the diameter, perpendicular thereto

Norwood measured a distance of 905751 feet of London measure between London and York.

Weights on Earth

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Proposition 20 Problem 4: Find and compare together the weights of bodies in the different regions of our earth.

Because the weights of the unequal legs of the canal of water ACQqca are equal; and the weights of the parts proportional to the whole legs, and alike situated in them, are one to another as the weights of the wholes, and therefore equal betwixt themselves; the weights of equal parts, and alike situated in the legs, will be reciprocally as the legs, that is, reciprocally as 230 to 229.

Weights on Earth

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Proposition 21 Theorem 17: The equinoctial points go backward

The axis of the earth, by a nutation in every annual revolution, twice vibrates towards the ecliptic, and as often returns to its former position.

The Moons of Jupiter and Saturn

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Proposition 23. Problem 5: Derive the unequal motions of the satellites of Jupiter and Saturn from the motions of our moon.

From the motions of our moon we deduce the corresponding motions of the moons or satellites of Jupiter in this manner, by Cor. 16, Prop. LXVI, Book I.

The Motions of the Moon

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Proposition 25 Problem 6: Find the forces with which the sun disturbs the motions of the moon

Let S represent the sun, T the earth, P the moon, CADB the moon’s orbit.

The Tides of the Sea

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Proposition 24. Theorem 19: The Flux And Reflux Of The Sea Arise From The Actions Of The Sun And Moon.

By Cor. 19 and 20, Prop. LXVI, Book 1:

the waters of the sea should twice rise and twice to fall every day, as well lunar as solar.
the greatest height of the waters in the open and deep seas should follow the appulse of the luminaries to the meridian of the place by a less interval than 6 hours.

This happens in all that eastern tract of the Atlantic and Æthiopic seas between France and the Cape of Good Hope; and on the coasts of Chili and Peru, in the South Sea; in all which shores the flood falls out about the second, third, or fourth hour, unless where the motion propagated from the deep ocean is by the shallowness of the channels, through which it passes to some particular places, retarded to the fifth, sixth, or seventh hour, and even later. The hours I reckon from the appulse of each luminary to the meridian of the place; as well under as above the horizon; and by the hours of the lunar day I understand the 24th parts of that time which the moon, by its apparent diurnal motion, employs to come about again to the meridian of the place which it left the day before. The force of the sun or moon in raising the sea is greatest in the appulse of the luminary to the meridian of the place; but the force impressed upon the sea at that time continues a little while after the impression, and is afterwards increased by a new though less force still acting upon it. This makes the sea rise higher and higher, till this new force becoming too weak to raise it any more, the sea rises to its greatest height. And this will come to pass, perhaps, in one or two hours, but more frequently near the shores in about three hours, or even more, where the sea is shallow.

The Orbit of the Moon

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Proposition 26 Problem 7: Find the horary increment of the area which the moon, by a radius drawn to the earth, describes in a circular orbit

The area which the moon describes by a radius drawn to the earth is proportional to the time of description, excepting in so far as the moon’s motion is disturbed by the action of the sun.

Projectivle

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Proposition 27 Problem 8: From the horary motion of the moon to find its distance from the earth.

The area which the moon, by a radius drawn to the earth, describes in every, moment of time, is as the horary motion of the moon and the square of the distance of the moon from the earth conjunctly. And therefore the distance of the moon from the earth is in a proportion compounded of the subduplicate proportion of the area directly, and the subduplicate proportion of the horary motion inversely. Q.E.I.

Find the variation of the moon

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Proposition 29 Problem 10: Find the variation of the moon.

This inequality is owing partly to the elliptic figure of the moon’s orbit, partly to the inequality of the moments of the area which the moon by a radius drawn to the earth describes.

Find the variation of the moon

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PROPOSITION 30 PROBLEM 11: Find the horary motion of the nodes of the moon, in a circular orbit.

Let S represent the sun, T the earth, P the moon, NPn the orbit of the moon, Npn the orthographic projection of the orbit upon the plane of the ecliptic; N, n the nodes, nTNm the line of the nodes produced indefinitely;

Projectivle

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Proposition 31 Problem 12

Find the horary motion of the nodes of the moon in an elliptic orbit

Let:

Qjpmaq represent an ellipsis with the greater axis Qq, and the lesser axis ab QAqB a circle circumscribed
T is the earth in the common centre of both
S the sun
p is the moon moving in this ellipsis
pm is an arc which it describes in the least moment of time
N and n are the nodes joined by the line Nn
pK and mk are perpendiculars upon the axis Qq produced both ways till they meet the circle in P and M, and the line of the nodes in D and d

If the moon, by a radius drawn to the earth, describes an area proportional to the time of description, the horary motion of the node in the ellipsis will be as the area pDdm and AZ2 conjunctly.

Projectivle

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Proposition 32 Problem 13: Find the mean motion of the nodes of the moon

The yearly mean motion is the sum of all the mean horary motions throughout the course of the year. Suppose that the node is in N, and that, after every hour is elapsed, it is drawn back again to its former place; so that, notwithstanding its proper motion, it may constantly remain in the same situation with respect to the fixed stars; while in the mean time the sun S, by the motion of the earth, is seen to leave the node, and to proceed till it completes its apparent annual course by an uniform motion. Let Aa represent a given least arc, which the right line TS always drawn to the sun, by its intersection with the circle NAn, describes in the least given moment of time.

The Moon's Nodes

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Proposition 33 Problem 14: Find the true motion of the moon’s nodes

In the time which is as the area NTA-NdZ that motions is as the area NAe, and is thence given but because the calculus is too difficult, it will be better to use the following construction of the Problem.

Syzygies

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Proposition 34 Problem 15

Find the horary variation of the inclination of the moon’s orbit to the plane of the ecliptic.

Let:

A and a are syzygies
Q and q are the quadratures
N and n are the nodes
P is the place of the moon in its orbit
p is the orthographic projection of that place upon the plane of the ecliptic
mTl is the momentaneous motion of the nodes as above.

If upon Tm we let fall the perpendicular PG, and joining pG we produce it till it meet Tl in g, and join also Pg, the angle PGp will be the inclination of the moon’s orbit to the plane of the ecliptic when the moon is in P.

The Moon's Orbit

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Proposition 35 Problem 16

With a given time, find the inclination of the moom’s orbit to the plane of the eclipltic.

Let:

AD is the sine of the greatest inclination
AB is the sine of the least inclination

Bisect BD inC and around the centre C, with the interval BC, describe the circle BGD.

The Force of the Sun to Move the Sea

Mon, 01 Jan 0001 00:00:00 +0000

Proposition 36 Problem 17: Find the force of the sun to move the sea

The sun s force ML or PT to disturb the motions of the moon, was (by Prop 25) in the moon’s quadratures, to the force of gravity with us, as 1 to 638092.6

The Force of the moon to move the sea

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Proposition 37 Problem 18: Find the force of the moon to move the sea

The force of the moon to move the sea is to be deduced from its proportion to the sun’s force. This proportion is to be collected from the proportion of the motions of the sea, which are the effects of those forces.

The Shape of the Moon

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Proposition 38 Problem 19: Find the shape of the moon’s body.

If the moon’s body were fluid like our sea, the force of the earth to raise that fluid in the nearest and remotest parts would be to the force of the moon by which our sea is raised in the places under and opposite to the moon as the accelerative gravity of the moon towards the earth to the accelerative gravity of the earth towards the moon, and the diameter of the moon to the diameter of the earth conjunctly; that is, as 39,788 to 1, and 100 to 365 conjunctly, or as 1081 to 100. Wherefore, since our sea, by the force of the moon, is raised to 83⁄5 feet, the lunar fluid would be raised by the force of the earth to 93 feet; and upon this account the figure of the moon would be a spheroid, whose greatest diameter produced would pass through the centre of the earth, and exceed the diameters perpendicular thereto by 186 feet. Such a figure, therefore, the moon affects, and must have put on from the beginning. Q.E.I.

The Precession of the Equinoxes

Mon, 01 Jan 0001 00:00:00 +0000

Proposition 39 Problem 20: Find the precession of the equinoxes.

The middle horary motion of the moon’s nodes in a circular orbit when the nodes are in quadratures was 16" 35’" 16 36

The Comets

Mon, 01 Jan 0001 00:00:00 +0000

Lemma 4: The comets are higher than the moon and in the regions of the planets.

Astronomers put the comets beyond the moon because they were found to have no diurnal parallax.

The Movement of Comets

Mon, 01 Jan 0001 00:00:00 +0000

Proposition 40 Theorem 20: The comets move in some of the conic sections, having their foci as the sun. They trace an orbit to the sun by radii drawn to the sun’s center.

centre of the in the a r eas proportional to the times.

General Scholium

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The hypothesis of vortices is problematic.

It says that:

each planet by a radius drawn to the sun may describe areas proportional to the times of description
the periodic times of the several parts of the vortices should observe the square of their distances from the sun
the periodic times of the planets may obtain the 3/2th power of their distances from the sun
the periodic times of the parts of the vortex should be as the 3/2th power of their distances

the smaller vortices may maintain their lesser revolutions around Saturn, Jupiter, and other planets, and swim quietly and undisturbed in the greater vortex of the sun
the periodic times of the parts of the sun’s vortex should be equal
the rotation of the sun and planets around their axes should correspond with the motions of their vortices, but recede far from all these proportions.

I have explained that the motions of the comets are very regular*. These: