The Shape of the Moon

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.

Corollary

Hence it is that the same face of the moon always respects the earth; nor can the body of the moon possibly rest in any other position, but would return always by a libratory motion to this situation; but those librations, however, must be exceedingly slow, because of the weakness of the forces which excite them; so that the face of the moon, which should be always obverted to the earth, may, for the reason assigned in Prop. XVII be turned towards the other focus of the moon’s orbit, without being immediately drawn back, and converted again towards the earth.

Lemma 1

The Earth is represented by APEp with the centre C.

• Its poles P, p, and the equator is AE

The sphere Pape is around centre C with the radius CP.

QR denotes the plane on which a right line, drawn from the centre of the sun to the centre of the earth, insists at right angles.

The several particles of the whole exterior earth PapAPepE, without the height of the said sphere, endeavour to recede towards this side and that side from the plane QR, every particle by a force proportional to its distance from that plane; I say, in the first place, that the whole force and efficacy of all the particles that are situate in AE, the circle of the equator, and disposed uniformly without the globe, encompassing the same after the manner of a ring, to wheel the earth about its centre, is to the whole force and efficacy of as many particles in that point A of the equator which is at the greatest distance from the plane QR, to wheel the earth about its centre with a like circular motion, as 1 to 2. And that circular motion will be performed about an axis lying in the common section of the equator and the plane QR.

For let there be described from the centre K, with the diameter IL, the semi-circle INL. Suppose the semi-circumference INL to be divided into innumerable equal parts, and from the several parts N to the diameter

IL let fall the sines NM. Then the sums of the squares of all the sines NM will be equal to the sums of the squares of the sines KM, and both sums together will be equal to the sums of the squares of as many semi-diameters KN; and therefore the sum of the squares of all the sines NM will be but half so great as the sum of the squares of as many semi-diameters KN.

Suppose now the circumference of the circle AE to be divided into the like number of little equal parts, and from every such part F a perpendicular FG to be let fall upon the plane QR, as well as the perpendicular AH from the point A. Then the force by which the particle F recedes from the plane QR will (by supposition) be as that perpendicular FG; and this force multiplied by the distance CG will represent the power of the particle F to turn the earth round its centre. And, therefore, the power of a particle in the place F will be to the power of a particle in the place A as FG × GC to AH × HC; that is, as FC² to AC²: and therefore the whole power of all the particles F, in their proper places F, will be to the power of the like number of particles in the place A as the sum of all the FC² to the sum of all the AC², that is (by what we have demonstrated before), as 1 to 2. Q.E.D.

And because the action of those particles is exerted in the direction of lines perpendicularly receding from the plane QR, and that equally from each side of this plane, they will wheel about the circumference of the circle of the equator, together with the adherent body of the earth, round an axis which lies as well in the plane QR as in that of the equator.

LEMMA 2

The same things still supposed, I say, in the second place, that the total force or power of all the particles situated every where about the sphere to turn the earth about the said axis is to the whole force of the like number of particles, uniformly disposed round the whole circumference of the equator AE in the fashion of a ring, to turn the whole earth about with the like circular motion, as 2 to 5.

For let IK be any lesser circle parallel to the equator AE, and let Ll be any two equal particles in this circle, situated without the sphere Pape; and if upon the plane QR, which is at right angles with a radius drawn to the sun, we let fall the perpendiculars LM, lm, the total forces by which these particles recede from the plane QR will be proportional to the perpendiculars LM, lm. Let the right line Ll be drawn parallel to the plane Pape, and bisect the same in X; and through the point X draw Nn parallel to the plane QR, and meeting the perpendiculars LM, lm, in N and n; and upon the plane QR let full the perpendicular XY. And the contrary forces of the particles L and l to wheel about the earth contrariwise are as LM × MC, and lm × mC; that is, as LN × MC + NM × MC, and ln × mC - nm × mC; or LN × MC + NM × MC, and LN × mC - NM × mC, and LN × Mm - NM × M C + m C ¯ {\displaystyle \scriptstyle \times {\overline {MC+mC}}}, the difference of the two, is the force of both taken together to turn the earth round. The affirmative part of this difference LN × Mm, or 2LN × NX, is to 2AH × HC, the force of two particles of the same size situated in A, as LX² to AC²; and the negative part NM × M C + m C ¯ {\displaystyle \scriptstyle \times {\overline {MC+mC}}}, or 2XY × CY, is to 2AH × HC, the force of the same two particles situated in A, as CX² to AC². And therefore the difference of the parts, that is, the force of the two particles L and l, taken together, to wheel the earth about, is to the force of two particles, equal to the former and situated in the place A, to turn in like manner the earth round, as LX² - CX² to AC². But if the circumference IK of the circle IK is supposed to be divided into an infinite number of little equal parts L, all the LX² will be to the like number of IX² as 1 to 2 (by Lem. 1); and to the same number of AC² as IX² to 2AC²; and the same number of CX² to as many AC² as 2CX² to 2AC². Wherefore the united forces of all the particles in the circumference of the circle IK are to the joint forces of as many particles in the place A as IX² - 2CX² to 2AC²; and therefore (by Lem. 1) to the united forces of as many particles in the circumference of the circle AE as IX² - 2CX² to AC².

If Pp, the diameter of the sphere, is conceived to be divided into an infinite number of equal parts, upon which a like number of circles IK are supposed to insist, the matter in the circumference of every circle IK will be as IX²; and therefore the force of that matter to turn the earth about will be as IX² into IX² - 2CX²; and the force of the same matter, if it was situated in the circumference of the circle AE, would be as IX² into AC². And therefore the force of all the particles of the whole matter situated without the sphere in the circumferences of all the circles is to the force of the like number of particles situated in the circumference of the greatest circle AE as all the IX² into IX² - 2CX² to as many IX² into AC²; that is, as all the AC² - CX² into AC² - 3CX² to as many AC² - CX² into AC²; that is, as all the AC4 - 4AC² × CX² + 3CX4 to as many AC4 - AC² × CX²; that is, as the whole fluent quantity, whose fluxion is AC4 - 4AC² × CX² + 3CX4, to the whole fluent quantity, whose fluxion is AC4 - AC² × CX²; and, therefore, by the method of fluxions, as AC4 × CX - 4⁄3AC² × CX³ + 3⁄5CX5 to AC4 × CX - ⅓AC² × CX³; that is, if for CX we write the whole Cp, or AC, as 4⁄15 AC5 to ⅔AC5; that is, as 2 to 5. Q.E.D.

LEMMA 3

The same things still supposed, I say, in the third place, that the motion of the whole earth about the axis above-named arising from the motions of all the particles, will be to the motion of the aforesaid ring about the same axis in a proportion compounded of the proportion of the matter in the earth to the matter in the ring; and the proportion of three squares of the quadrantal arc of any circle to two squares of its diameter, that is, in the proportion of the matter to the matter, and of the number 925275 to the number 1000000.

For the motion of a cylinder revolved about its quiescent axis is to the motion of the inscribed sphere revolved together with it as any four equal squares to three circles inscribed in three of those squares; and the motion of this cylinder is to the motion of an exceedingly thin ring surrounding both sphere and cylinder in their common contact as double the matter in the cylinder to triple the matter in the ring; and this motion of the ring, uniformly continued about the axis of the cylinder, is to the uniform motion of the same about its own diameter performed in the same periodic time as the circumference of a circle to double its diameter.

HYPOTHESIS 2

If the other parts of the earth were taken away, and the remaining ring was carried alone about the sun in the orbit of the earth by the annual motion, while by the diurnal motion it was in the mean time revolved about its own axis inclined to the plane of the ecliptic by an angle of 23 1/2 degrees, the motion of the equinoctial points would be the same, whether the ring were fluid, or whether it consisted of a hard and rigid matter.