Principia Mathematica (Motte Translation, 1848) — part 32 of 45

other side of the equator ; L the place of the moon three hours before ; H the place of the earth directly under it ; h the opposite place ; K, k the places at 90 degrees distance ; CH, C/i, the greatest heights of the sea from the centre of the earth ; and CK, Ck^ its least heights : and if with the axes Hh, Kk, an ellipsis is described, and by the revolution of that ellipsis about its longer axis H/i a spheroid HFKhpk is formed, this sphe- roid will nearly represent the figure of the sea ; and CF, C/, CD, Cd, will represent the heights of the sea in the places F/, Dd. But far- ther ; in the said revolution of the ellipsis any point N describes the circle NM cutting the parallels F/, Dd, in any places RT, and the equator AE in S ; CN will represent the height of the sea in all those places R, S, T, situated in this circle. Wherefore, in the diurnal revolution of any place F. the greatest flood will be in F, at the third hour after the appulse of the moon to the meridian above the horizon ; and afterw^ards the great- est ebb in Q, at the third hour after the setting of the moon ; and then the greatest flood in/, at the third hour after the appulse of the moon to the meridian under the horizon : and, lastly, the greatest ebb in Q,, at the third hour after the rising of the moon ; and the latter flood in / will be less than the preceding flood in F. For the whole sea is divided into two hemisphericai floods, one in the hemisphere KHA; on the north side, the other in the opposite hemisphere Khk, which we may therefore call the northern and the southern floods. These floods, being always opposite the one to the other, come by turns to the meridians of all places, after an interval of 12 lunar hours. And seeing the northern countries partake more of the northern flood, and the southern countries more of the southern flood, thence arise tides, alternately greater and less in all places without the equator, in which the luminaries rise and set. But the greatest tide will happen when the moon declines towards the vertex of the place, about the third hour after the appulse of the moon to the meridian above the hori- zon : and when the moon changes its declination to the other side of the equator^ that which was the greater tide will be changed into a lesser. And the greatest difference of the floods will fall out about the times of the solstices ; especially if the ascending node of the moon is about the first of Aries. So it is found by experience that the morning tides in winter exceed those of the evening, and the evening tides in summer ex- ceed those of the morning ; at Plymouth by the height of one foot, but at Bristol by the height of 15 inches, according to the observations of Cole- press and Sturmy.

But the motions which we have been describing sufifer some alteration from that force of reciprocation, which the waters, being once moved, retain a little while hy their vis insita. Whence it comes to pass that the tides may continue for some time, though the actions of the luminaries should

■ 27

418 THE MATHEMATICAL PRINCIPLES [BoOK III.

cease. This power of retaining the impressed motion lessens the difference of the alternate tides, and makes those tides which immediately succeed after the syzygies greater, and those which follow next after the quadra- tures less. And hence it is that the alternate tides at Plymouth and Brisiiol do not differ much more one from the other than by the height of a foot or 15 inches, and that the greatest tides of all at those ports are not the first but the third after the syzygies. And, besides, all the motions are retarded in their passage through shallow channels, so that the greatest tides of all, in some straits and mouths of rivers, are the fourth or even the fifth after the syzygies.

Farther, it may happen that the tide may be propagated from the ocean through different channels towards the same port, and may pass quicker through some channels than through others ; in which case the same tide, divided into two or more succeeding one another, may compound new mo- tions of different kinds. Let us suppose two equal tides flowing towards the same port from different places, the one preceding the other by 6 hours ; and suppose the first tide to happen at the third hour of the appulse of the moon to the meridian of the port. If the moon at the time of the appulse to the meridian was in the equator, every 6 hours alternately there would arise equal floods, which, meeting with as many equal ebbs, would so bal- ance one the other, that for that day, the water would stagnate and remain quiet. If the moon then declined from the. equator, the tides in the ocean would be alternately greater and less, as was said ; and from thence two greater and two lesser tides would be alternately propagated towards that port. But the two greater floods would make the greatest height of the waters to fall out in the middle time betwixt both ; and the greater and lesser floods would make the waters to rise to a mean height in the middle time between them, and in the middle time between the two lesser floods the waters would rise to their least height. Thus in the space of 24 hours the waters would come, not twice, as commonl}'', but once only to their great- est, and once only to their least height ; and their greatest height, if the moon declined towards the elevated pole, would happen at the 6th or 30th hour after the appulse of the moon to the meridian ; and when the moon changed its declination, this flood would be changed into an ebb. An ex- ample of all which Dr. Halley has given us, from the observations of sea- men in the port of Batsham, in the kingdom of Tunquin, in the latitude of 20° 50' north. In that port, on the day which follows after the passage of the moon over the equator, the waters stagnate : when the moon declines to the north, they begin to flow and ebb, not twice, as in other ports, but once only every day ; and the flood happens at the setting, and the greatest ebb at the rising of the moon. This tide increases with the declination of the moon till the 7th or 8th day ; then for the 7 or 8 days following it

Book IIL] of natural philosophy. 419

decreases at the same rate as it had increased before, and ceases when the moon changes its declination, crossing over the equator to the south. Af- ter which the flood is immediately changed into an ebb ; and thenceforth the ebb happens at the setting and the flood at the rising of the moon ; till the moon, again passing the equator, changes its declination. There are two inlets to this port and the neighboring channels, one from the seas of China, between the continent and the island of Leuconia ; tb^ other from the Indian sea, between the continent and the island of Borneo. But whether there be really two tides propagated through the said channels, one from the India7i sea in the space of 12 hours, and one from the sea of China in the space of 6 hours, which therefore happening at the 3d and 9th lunar hours, by being compounded together, produce those motions ; or whether there be any other circumstances in the state of those seas, I leave to be determined by observations on the neighbouring shores.

Thus I have explained the causes of the motions of the moon and of the sea. Now it is fit to subjoin something concerning the quantity of those motions.

PROPOSITION XXV. PROBLEM VI.

To find the forces ivith 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. In SP take SK equal to ST ; and let SL be to g SK in the duplicate proportion of SK to SP : draw LM parallel

to PT ; and if ST or SK is sup- ^

posed to represent the accelerated force of gravity of the earth towards the sun, SL will represent the accelerative force of gravity of the moon towards the sun. But that force is compounded of the parts SM and LM, of which the force LM, and that part of SM which is represented by TM, disturb the motion of the moon, as we 'have shewn in Prop. LXVI, Book I, and its Corollaries. Forasmuch as the earth and moon are revolved about their common centre of gravity, the motion of the earth about that centre will be also disturbed by the like forces ; but we may consider the sums both of the forces and of the motions as in the m.oon, and represent the sum of the forces by the lines TM and ML, which are analogous to them both. The force ML (in its mean quantity) is to the centripetal force by which the moon may be retained in its orbit revolving about the earth at rest, at the distance PT, in the duplicate proportion of the periodic time of the moon about the earth to the periodic time of the earth about the sun (by Cor. 17, Prop. LXVI, Book I) ; that is, in the duplicate proportion of 27^. 7\ 43' to 365^^. 6\ 9' ; or as 1000 to 178725 ; or as 1 to 178| f . But in the

420

THE MATHEMATICAL PRINCIPLES

[Book III.

4th Prop, of this Book we found, that, if both earth and moon were revolved about their common centre of gravity, the mean distance of the one from the other would be nearly 60|- mean semi-diameters of the earth; and the force by which the moon may be kept revolving in its orbit about the earth in rest at the distance PT of 60|- semi-diameters of the earth, is to the force by whi^ it may be revolved in the same time, at the distance of 60 semi-diameters, as 60|- to 60 : and this force is to the force of gravity with us very nearly as 1 to 60 X 60. Therefore the mean force ML is to the force of gravity on the surface of our earth as 1 X 60-} to 60 X 60 X 60 X 178|f, or as 1 to 638092,6 ; whence by the proportion of the lines TM, ML, the force TM is also given ; and these are the forces with which the sun disturbs the motions of the moon. Q.E.L

PROPOSITION XXYL PROBLEM YIl.

To find the horary increment of the area which the moon, by a radius drawn to the earth, describes in a circular orbit.

We have above shown that the area which the moon de- scribes by a radius drawn to the earth is proportional to M the time of descrip- tion, excepting in so far as the moon's motion is disturbed by the action of the sun; and here we propose to investi- gate the inequality of the moment, or horary increment of that area or fiiotion so disturbed. To render the calculus more easy, we shall suppose the orbit of the moon to be circular, and neglect all inequalities but that only which is now under consideration ; and, because of the immense dis- tance of the sun, we shall farther suppose that the lines SP and ST are parallel. By this means, the force LM will be always reduced to its mean quantity TP, as well as the force TM to its mean quantity 3PK. These forces (by Cor. 2 of the Laws of Motion) compose the force TL ; and this force, by letting fall the perpendicular LE upon the radius TP, is resolved into the forces TE, EL ; of which the force TE, acting constantly in the direction of the radius TP, neither accelerates nor retards the de- scription of the area TPC made by that radius TP ; but EL, acti7ig on the radius TP in a perpendicular direction, accelerates or retards the de- scription of the area in proportion as it accelerates or retards the moon.

^''^

<^^..

/^ X T.

/ ^^

\ /

T

D

Book III.] of natural philosophy. 421

That acceleration of the moon, in its passage from the quadrature C to the

conjunction A, is in every moment of time as the generating accelerative

3PK X TK force EL, that is, as ^^ . Let the time be represented by the

mean motion of the moon, or (which comes to the same thing) by the angle CTP, or even by the arc CP. At right angles upon CT erect CG equal to CT ; and, supposing the quadrantal arc AC to be divided into an infinite number of equal parts P/?, &c., these parts may represent the like infinite number of the equal parts of time. Let fall pk perpendicular on CT, and draw TG meeting with KP, kp produced in F and /; then will FK be equal to TK, and K^^ be to PK as P/? to Tp, that is, in a given propor-

3PK X TK

tion ; and therefore FK X K^^, or the area FK^/, will be as ^f 5 y

that is, as EL; and compounding, the whole area GCKF will be as the sum of all the forces EL impressed upon the moon in the whole time CP ; and therefore also as the velocity generated by that sum, that is, as the ac- celeration of the description of the area CTP, or as the increment of the moment thereof. The force by which the moon may in its periodic time CADB of 27'^. T\ 43' be retained revolving about the earth in rest at the distance TP, would cause a body falling in the time CT to describe the length ^CT, and at the same time to acquire a velocity equal to that with which the moon is moved in its orbit. This appears from Cor. 9, Prop. IV., Book I. But since Kd, drawn perpendicular on TP, is birt a third part of EL, and equal to the half of TP, or ML, in the octants, the force EL in the octants, where it is greatest, will exceed the force ML in the proportion of 3 to 2 ; and therefore will be to that force by which the moon in its periodic time may be retained revolving about the earth at rest as 100 to f X 17872I-, or 11915; and in the time CT will generate a ve- locity equal to yylfs- parts of the velocity of the moon; but in the time CPA will generate a greater velocity in the proportion of CA to CT or TP. Let the greatest force EL in the octants be represented by the area FK X Kk, or by the rectangle iTP X Pjo, which is equal thereto ; and the velocity which that greatest force can generate in any time CP will be to the velocity which any other lesser force EL can generate in the same time as the rectangle |TP X CP to the area KCGF ; but the velocities generated in the whole time CPA will be one to the other as the rectangle ^TP X CA to the triangle TCG, or as the quadrantal arc CA to the radius TP ; and therefore the latter velocity generated in the whole time will be ttIts P^^^s of the velocity of the moon. To this velocity of the moon, which is proportional to the mean moment of the area (mipposing this mean moment to be represented by the number 11915), we add and subtract the half of the other velocity ; the sum 11915 + 50, or 11965, will represent the greatest moment of the area in the syzygy A ; and the

422 THE MATHEMATICAL PRINCIPLES [BoOK III.

difference 11915 — 50^ or 11865^ the least moment thereof in the quadra- tures. Therefore the areas which in equal times are described in the syzy- gies and quadratures are one to the other as 11965 to 11865. And if to the least moment 11865 we add a moment which shall be to 100, the dif- ference of the two former moments, as the trapezium FKCGto the triangle TCG, or, which comes to the same thing, as the square of the sine PK to the square of the radius IT (that is, as Vd to TP), the sum will represent the moment of the area when the moon is in any intermediate place P.

But these things take place only in the hypothesis that the sun and the earth are at rest, and that the synodical revolution of the moon is finished in 27'\ 7\ 43'. But since the moon's synodical period is really 29"^. 12^ 44', the increments of the moments must be enlarged in the same propor- tion as the time is, that is, in the proportion of 1080853 to 1000000. Upon which account, the whole increment, which was ji-lfs parts of the mean moment, will now become Trfls- P^^rts thereof; and therefore the moment of the area in the quadrature of the moon will be to the moment thereof in. the syzygy as 11023 — 50 to 11023 + 50; or as 10973 to 11073; and to the moment thereof, when the moon is in any intermediate place P, as 10973 to 10973 + Fd ; that is, supposing TP = 100.

The area, therefore, which the moon, by a radius drawn to the earth, describes in the several little equal parts of time, is nearly as the sum of the number 219,46, and the versed sine of the double distance of the moon from the nearest quadrature, considered in a circle which hath unity for its radius. Thus it is when the variation in the octants is in its mean quantity. But if the variation there is greater or less, that versed sine must be aug- mented or diminished in the same proportion.

PROPOSITION XXVII. PROBLEM YIII.

From the horary 'jnotion of the mooji 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 subduplioate propor- tion of the horary motion inversely. Q.E.I.

Cor. 1 . Hence the apparent diameter of the moon is given ; for it is re- ciprocally as the distance of the moon from the earth. Let astronomers try how accurately this rule agrees with the phsenomena.

Cor. 2. Hence also the orbit of the moon may be more exactly defined from the phsenomena than hitherto could be done.

Book III] of natural philosophy. 423

PROPOSITION XXYIIt. PROBLEM IX.

To find the diameters of the orbit, in which, loithout eccentricity, the

moon woidd move. The curvature of the orbit which a body describes, if attracted in lines perpendicular to the orbit, is as the force of attraction directly, and the square of the velocity inversely. I estimate the curvatures of lines com- pared one with another according to the evanescent proportion of the sines or tangents of their angles of contact to equal radii, supposing those radii to be infinitely diminished. But the attraction of the moon towards the earth in the syzygies is the excess of its gravity towards the earth above the force of the sun 2PK (see Fig. Prop. XXV), by which force the accel- erative gravity of the moon towards the sun exceeds the accelerative gravity of the earth towards the sun, or is exceeded by it. But in the quadratures that attraction is the sum of the gravity of the moon towards the earth, and the sun's force KT, by which the moon is attracted towards the earth.

. . , . . ^, , AT + CT , 178725

And these attractions, putting N for ^- , are nearly as ^^a —

2000 ^„, 1 rSTp ^ 1000 ^^ ^ ^^g^3^^ ^ ^^^ _ ^^^^^^^

CT X N — CT^ ' AT X N^ X CT, and 178725N X AT^ + lOOOCT^ x AT. For if the accelera- tive gravity of the moon towards the earth be represented by the number 178725; the mean force ML, which in the quadratures is PT or TK, and draws the moon towards the earth, will be 1000, and the mean force TM in the syzygies will be 3000 ; from which, if we subtract the mean force ML, there will remain 2000, the force by which the moon in the syzygies is drawn from the earth, and which we above called 2PK. But the velocity of the moon in the syzygies A and B is to its velocity in the quadratures C and D as CT to AT, and the moment of the area, which the moon by a radius drawn to the earth describes in the syzygies, to the moment of that area described in the quadratures conjunctly; that is, as 11073CT to 10973AT. Take this ratio twice inversely, and the former ratio once di- rectly, and the curvature of the orb of the moon in the syzygies will be to the curvature thereof in the quadratures as 120406729 X 178725AT2 x CT^ X N— 120406729 X 2000AT4 x CTto 122611329 X 178725AT2 X CT2 X N + 122611329 X lOOOCT* X AT, that is, as 2151969AT X CT X N ~ 24081xiT^ to 2191371AT X CT x N + 12261CT^

Because the figure of the moon's orbit is unknown, let us, in its stead, assume the ellipsis DBCA, in the centre of which we suppose the earth to be situated, and the greater axis DC to lie between the quadratures as the lesser AB between the syzygies. But since the plane of this ellipsis is re- volved about the earth by an angular motion, and the orbit, whose curva- ture we now examine, should be described in a plane void of such motion,

424

THE MATHEMATICAL PRINCIPLES

[Book III.

we are to consider the figure which the mooiij while it is revolved in that ellipsis, describes in this plane, that is to say, the figure Cpa, the several points p of which are found by assuming any point P in the ellipsis, which may represent the place of the moon, and drawing Tp equal to TP in such manner that the angle PT/? may be equal to the apparent motion of the sun from the time of the last quadrature in C ; or (which comes to the same thing) that the angle CTp may be to the angle CTP as the time of the synodic revolution of the moon to the time of the periodic revolution thereof, or as 29^. 12^. 44' to 27'^. 7\ 43'. If, there- fore, in this proportion we take the angle CTa to the right angle CTA, and make T« of equal length with TA, we shall have a the lower and 0 the upper apsis of this orbit Cpa. But, by computation, I find that the difference betwixt the curvature of this orbit Cpa at the vertex a, and the curvature of a circle described about the centre T with the interval TA, is to the difference between the curvature of the ellipsis at the vertex A, and the curvature of the same circle, in the duplicate proportion of the angle CTP to the angle CTp ; and that the curvature of the ellipsis in A is to the curvature of that circle in the duplicate proportion of TA to TC ; and the curvature of that circle to the curvature of a circle described about the centre T with the interval TC as TC to TA ; but that the curvature of this last arch is to the curvature of the ellipsis in C in the duplicate pro- portion of T A to TC ; and that the difference betwixt the curvature of the ellipsis in the vertex C, and the curvature of this last circle, is to the dif- ference betwixt the curvature of the figure Cpa, at the vertex C, and the curvature of this same last circle, in the duplicate proportion of the angle CTp to the angle CTP ; all which proportions are easily drawn from the sines of the angles of contact, and of the differences of those angles. But, by comparing those proportions together, we find the curvature of the figure Cpa at a to be to its curvature at C as AT^— ^VoWoCT^ AT to CT=^ -l- j_6_8_2_4_AT2 X CT ,^ where the number yVVoVo represents the difference of the squares of the angles CIT and CTp, applied to the square of the lesser angle CTP ; or (which is all one) the difference of the squares of the times 27^ 7^ 43', and 29'^. 12^. 44', applied to the square of the time 27^.

7\ 43'.

Since, therefore, a represents the syzygy of the moon, and C its quadra- ture, the proportion now found must be the same with that proportion of the curvature of the moon's orb in the syzygies to the curvature thereof in ihe quadratures, which we found above. Therefore, in order to find the

Book III.] of natural philosophy. 425

proportion of CT to AT, let us multiply the extremes and the means, and the terms which come out, applied to AT X CT, become 2062,79CT4 — 2151969N X CT^ + 36S676N X AT X CT^ + 36342 AT=^ X CT^ — 362047N X AT^ x CT + 21913nN X AT^ -f 4051,4AT^ = 0.

Now if for the half sum N of the terms AT and CT we put 1, and x for their half difference, then CT will be = 1 + ^, and AT = 1 — x. And substituting those values in the equation, after resolving thereof, we shall find X == 0,00719 ; and from thence the semi-diameter CT = 1,00719, and the semi-diameter AT = 0,99281, which numbers are nearly as 70 ^V? ^i^cl 692V- Therefore the moon's distance from the earth in the syzygies is to its distance in the quadratures (setting aside the consideration of eccentrici- ty) as 692V to 70^'^ ; or, in round numbers, as 69 to 70.

PROPOSITION XXIX. PROBLEM X.

To fijid 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. If the moon P revolved in the ellipsis DBCA about the earth quiescent in the centre of the ellipsis, and by the radius TP, drawn to the earth, described the area CTP, proportional to the time of description ; and the greatest semi-diameter CT of the ellipsis was to the least TA as 70 to 69 ; the tangent of the angle CIT would be to the tangent of the angle of the mean motion, computed from the quad- rature C, as the semi-diameter TA of the ellipsis to its semi-diameter TC, or as 69 to 70. But the description of the area CTP, as the moon advan- ces from the quadrature to the syzygy, ought to be in such manner accel- erated, that the moment of the area in the moon's syzygy may be to the moment thereof in its quadrature as 11073 to 10973; and that the excess of the moment in any intermediate place P above the moment in the quad- rature may be as the square of the sine of the angle CTP ; which we may effect with accuracy enough, if we diminish the tangent of the angle CTP in the subduplicate proportion of the number 10973 to the number 11073, that is, in proportion of the number 68,6877 to the number 69. Upon which account the tangent of the angle CTP will now be to the tangent of the mean motion as 68,6877 to 70 ; and the angle CTP in the octants, where the mean motion is 45°, will be found 44° 27' 28", which sub- tracted from 45°, the angle of the mean motion, leaves the greatest varia- tion 32' 32". Thus it would be, if the moon, in passing from the quad- rature to the syzygy, described an angle CTA of 90 degrees only. But because of the motion of the earth, by which the sun is apparently trans- ferred in consequential the moon, before it overtakes the sun, describes an angle CTcf, greater than a right angle, in the proportion of the time of the synodic revolution of the moon to the time of its periodic revolution, that

426

THE MATHEMATICAL PRINCIPLES

[Book III.

is, in the proportion of 29'^. 12^. 44' to 27\ 7^. 43'. Whence it comes to pass that all the angles about the centre T are dilated in the same pro- portion ; and the greatest variation, which otherwise would be but 32' 32", now augmented in the said proportion, becomes 35' 10".

And this is its magnitude in the mean distance of the sun from the earth, neglecting the differences which may arise from the curvature of the 07'bis magmis, and the stronger action of the sun upon the moon when horned and new, than when gibbous and full. In other distances of the sun from the earth, the greatest variation is in a proportion compounded of the duplicate proportion of the time of the synodic revolution of the moon (the time of the year being given) directly, and the triplicate pro- portion of the distance of the sun from the earth inversely. And, there- fore, in the apogee of the sun, the greatest variation is 33' 14", and in its perigee 37' 11", if the eccentricity of the sun is to the transverse semi-di- ameter of the orbis magnus as 16|| to 1000.

Hitherto we have investigated the variation in an orb not eccentric, in which, to wit, the moon in its octants is always in its mean distance from the earth. If the moon, on account of its eccentricity, is more or less re- moved from the earth than if placed in this orb, the variation may be something greater, or something less, than according to this rule. But I leave the excess or defect to the determination of astronomers from the phsenomena.

PROPOSITION XXX. PROBLEM XL To find the horary motion of the nodes of the moon in a circidar orbit.

Let S represent the sun, T the earth, P the moon, NPy? the orbit of the moon, Nj!9?i the orthographic projection of the orbit upon the plane of the ecliptic ; N, n the nodes, ?iTNm the line of the nodes produced indefi-

M-

K

11 \ ^^'

//

V\

Aj

1 I

/

w'T ■■

V

/

T

J

B

Book III] of natural philosophy. 427

nitely ; PI, PK perpendiculars upon the lines ST, Qq ; Vp a perpendicu- lar upon the plane of the ecliptic ; A, B the moon^s syzygies in the plane of the ecliptic; AZ a perpendicular let fall upon N??., the line of the nodes ; Q,, q the quadratures of the moon in the plane of the ecliptic, and pK a perpendicular on the line Q,q lying between the quadratures. The force of the sun to disturb the motion of the moon (by Prop. XXV) is twofold, one proportional to the line LM, the other to the line MT, in the scheme of that Proposition ; and the moon by the former force is drawn towards the earth, by the latter towards the sun, in a direction parallel to the right line ST joining the earth and the sun. The former force LM acts in the direction of the plane of the moon's orbit, and therefore makes no change upon the situation thereof, and is upon that account to be neg- lected ; the latter force MT, by which the plane of the moon's orbit is dis- turbed, is the same with the force 3PK or 3IT. And this force (by Prop. XXY) is to the force by which the moon may, in its periodic time, be uni- formly revolved in a circle about the earth at rest, as 3IT to the radius of the circle multiplied by the number 178,725, or as IT to the radius there- of multiplied by 59,575. But in this calculus, and all that follows, I consider all the lines drawn from the moon to the sun as parallel to the line which joins the earth and the sun ; because what inclination there is almost as much diminishes all effects in some cases as it augments them in others : and we are now inquiring after the mean motions of the nodes, neglecting such niceties as are of no moment, and would only serve to ren- der the calculus more perplexed.

Now suppose PM to represent an arc which the moon describes in the least moment of time, and ML a little line, the half of which the moon, by the impulse of the said force 3IT, would describe in the same time ; and joining PL, MP, let them be produced to m and /, where they cut the plane of the ecliptic, and upon Tw- let fall the perpendicular PH. Now, since the right line ML is parallel to the plane of the ecliptic, and therefore can never meet with the right line 7nl which lies in that plane, and yet both those right lines lie in one common plane LMPm/, they will be parallel, and upon that account the triangles LMP, ItuP will be similar. And seeing MPm lies in the plane of the orbit, in which the moon did move while in the place P, the point m will fall upon the line N??, which passes through the nodes N, n, of that orbit. And because the force by which the half of the little line LM is generated, if the whole had been together, and at once impressed in the point P, would have generated that whole line, and caused the moon to move in the arc whose chord is LP ; that is to say, would have transferred the moon from the plane MPmT into the plane LPZT ; therefore the angular motion of the nodes generated by that force will be equal to the angle mTl. But ml is to mP as ML to MP ; and since MP, because of the time o^iven, is also o-iven, ml will be as the rectan-

428 THE MATHEMATICAL PRINCIPLES [BoOK III.

gle ML X "niP, that is, as the rectangle IT X mP. And if Tml is a right

ml IT X Pw? angle, the angle mTl will be as 7^—' ^^^ therefore as — ^ that is (be-

IT X PH •

cause Tm and mP, TP and PH are proportional), as — Ffrp~' and, there- fore, because TP is given, as IT X PH. But if the angle Tml or STN is oblique, the angle mTl will be yet less, in proportion of the sine of the angle STN to the radius, or AZ to AT. And therefore the velocity of the nodes is as IT X PH X AZ, or as the solid content of the sines of the three angles TPI, PTN, and STN.

If these are right angles, as happens when the nodes are in the quadra- tures, and the moon in the syzygy, the little line fnl will be removed to an infinite distance, and the angle ?nTl will become equal to the angle mFl. But in this case the angle mFl is to the angle PTM, which the moon in the same time by its apparent motion describes about the earth, as 1 to 59,575. For the angle mVl is equal to the angle LPM, that is, to the angle of the moon's deflexion from a rectilinear path ; which angle, if the gravity of the moon should have then ceased, the said force of the sun 3IT would by itself have generated in that given time ; and the angle PTM is equal to the angle of the moon's deflexion from a rectilinear path ; which angle, if the force of the sun 31T should have then ceased, the force alone by which the moon is retained in its orbit would have generated in the same time. And these forces (as we have above shewn) are the one to the other as I to 59,575. Since, therefore, the mean horary motion of the moon (in respect of the fixed stars) is 32' 56" 27'" 12^^'*'. the horary motion of the node in this case will be 33" 10'" 33^^ 12^ But in other cases the horary motion will be to 33" 10"' 33'^ 12\ as the solid content of the sines of the three angles TPI, PTN, and STN (or of the distances of the moon from the quadrature, of 'the moon from the node, and of the node from the sun) to the cube of the radius. And as often as the sine of any angle is changed from positive to negative, and from negative to positive, so often must the regressive be changed into a progressive, and the progressive into a regressive motion. Whence it comes to pass that the nodes are pro- gressive as often as the moon happens to be placed between either quadra- ture, and the node nearest to that quadrature. In other cases they are regressive, and by the excess of the regress above the progress, they are monthly transferred in mitecedentia.