# The Mathematical Principles of Natural Philosophy (1846)/BookIII-Prop5

LEMMA I.
    If APEp represent the earth uniformly dense, marked with the centre C, the poles P, p, and the equator AE; and if about the centre C, with the radius CP, we suppose the sphere Pape to be described, and QR to denote the plane on which a right line, drawn from the centre of the sun to the centre of the earth, insists at right angles; and further suppose that 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 $\scriptstyle \times$ GC to AH $\scriptstyle \times$ 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 II.
    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 $\scriptstyle \times$ MC, and lm $\scriptstyle \times$ mC; that is, as LN $\scriptstyle \times$ MC + NM $\scriptstyle \times$ MC, and ln $\scriptstyle \times$ mC - nm $\scriptstyle \times$ mC; or LN $\scriptstyle \times$ MC + NM $\scriptstyle \times$ MC, and LN $\scriptstyle \times$ mC - NM $\scriptstyle \times$ mC, and LN $\scriptstyle \times$ Mm - NM $\scriptstyle \times {\overline {MC+mC}}$, the difference of the two, is the force of both taken together to turn the earth round. The affirmative part of this difference LN $\scriptstyle \times$ Mm, or 2LN $\scriptstyle \times$ NX, is to 2AH $\scriptstyle \times$ HC, the force of two particles of the same size situated in A, as LX² to AC²; and the negative part NM ​$\scriptstyle \times {\overline {MC+mC}}$, or 2XY $\scriptstyle \times$ CY, is to 2AH $\scriptstyle \times$ 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.
Now 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 AC$^4$ - 4AC² $\scriptstyle \times$ CX² + 3CX$^4$ to as many AC$^4$ - AC² $\scriptstyle \times$ CX²; that is, as the whole fluent quantity, whose fluxion is AC$^4$ - 4AC² $\scriptstyle \times$ CX² + 3CX$^4$, to the whole fluent quantity, whose fluxion is AC$^4$ - AC² $\scriptstyle \times$ CX²; and, therefore, by the method of fluxions, as AC$^4$ $\scriptstyle \times$ CX - $^4$⁄$_3$AC² $\scriptstyle \times$ CX³ + $^3$⁄$_5$CX$^5$ to AC$^4$ $\scriptstyle \times$ CX - ⅓AC² $\scriptstyle \times$ CX³; that is, if for CX we write the whole Cp, or AC, as $^4$⁄$_{15}$ AC$^5$ to ⅔AC$^5$; that is, as 2 to 5. Q.E.D.
LEMMA III.
    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 II.
    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 23½ 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.
PROPOSITION XXXIX. PROBLEM XX.
To find the precession of the equinoxes.
The middle horary motion of the moon's nodes in a circular orbit, when the nodes are in the quadratures, was 16″ 35‴ 16$^{iv}$.36$^v$.; the half of which, 8″ 17‴ 38$^{iv}$.18$^v$. (for the reasons above explained) is the mean horary motion of the nodes in such an orbit, which motion in a whole sidereal year becomes 20° 11′ 46″. Because, therefore, the nodes of the moon in such an orbit would be yearly transferred 20° 11′ 46″ in antecedentia; and, if there were more moons, the motion of the nodes of every one (by Cor. 16, Prop. LXVI. Book 1) would be as its periodic time; if upon the surface of the earth a moon was revolved in the time of a sidereal day, the annual motion of the nodes of this moon would be to 20° 11′ 46″ as 23$^h$.56′, the sidereal day, to 27$^d$.7$^h$.43′, the periodic time of our moon, that is, as 1436 to 39343. And the same thing would happen to the nodes of a ring of moons encompassing the earth, whether these moons did not mutually touch each the other, or whether they were molten, and formed into a continued ring, or whether that ring should become rigid and inflexible.
Let us, then, suppose that this ring is in quantity of matter equal to the whole exterior earth PapAPepE, which lies without the sphere Pape (see fig. Lem. II); and because this sphere is to that exterior earth as aC² to AC² - aC², that is (seeing PC or aC the least semi-diameter of the earth is to AC the greatest semi-diameter of the same as 229 to 230), as 52441 to 459; if this ring encompassed the earth round the equator, and both together were revolved about the diameter of the ring, the motion of ​the ring (by Lem. III) would be to the motion of the inner sphere as 459 to 52441 and 1000000 to 925275 conjunctly, that is, as 4590 to 485223; and therefore the motion of the ring would be to the sum of the motions of both ring and sphere as 4590 to 489813. Wherefore if the ring adheres to the sphere, and communicates its motion to the sphere, by which its nodes or equinoctial points recede, the motion remaining in the ring will be to its former motion as 4590 to 489813; upon which account the motion of the equinoctial points will be diminished in the same proportion. Wherefore the annual motion of the equinoctial points of the body, composed of both ring and sphere, will be to the motion 20° 11′ 46″ as 1436 to 39343 and 4590 to 489813 conjunctly, that is, as 100 to 292369. But the forces by which the nodes of a number of moons (as we explained above), and therefore by which the equinoctial points of the ring recede (that is, the forces 3IT, in fig. Prop. XXX), are in the several particles as the distances of those particles from the plane QR; and by these forces the particles recede from that plane: and therefore (by Lem. II) if the matter of the ring was spread all over the surface of the sphere, after the fashion of the figure PapAPepE, in order to make up that exterior part of the earth, the total force or power of all the particles to wheel about the earth round any diameter of the equator, and therefore to move the equinoctial points, would become less than before in the proportion of 2 to 5. Wherefore the annual regress of the equinoxes now would be to 20° 11′ 46″ as 10 to 73092; that is, would be 9″ 56‴ 50$^{iv}$.
But because the plane of the equator is inclined to that of the ecliptic, this motion is to be diminished in the proportion of the sine 91706 (which is the co-sine of 23½ deg.) to the radius 100000; and the remaining motion will now be 9″ 7‴ 20$^{iv}$. which is the annual precession of the equinoxes arising from the force of the sun.
But the force of the moon to move the sea was to the force of the sun nearly as 4,4815 to 1; and the force of the moon to move the equinoxes is to that of the sun in the same proportion. Whence the annual precession of the equinoxes proceeding from the force of the moon comes out 40″ 52‴ 52$^{iv}$. and the total annual precession arising from the united forces of both will be 50″ 00‴ 12$^{iv}$. the quantity of which motion agrees with the phaenomena; for the precession of the equinoxes, by astronomical observations, is about 50″ yearly.
If the height of the earth at the equator exceeds its height at the poles by more than 17$^1$⁄$_6$ miles, the matter thereof will be more rare near the surface than at the centre; and the precession of the equinoxes will be augmented by the excess of height, and diminished by the greater rarity.
And now we have described the system of the sun, the earth, moon, and planets, it remains that we add something about the comets.