Principia Mathematica (Motte Translation, 1848) — part 44 of 45

move i]j conic sections that have one focus in the centre of the sun, and, by radii drawn to the sun, to describe areas proportional to the times ; for that force is propagated to an immense distance, and will govern the motions of bodies far beyond the orbit of Saturn. ^

There are three hypotheses about comets (p. 466) ; for some will have it that they are generated and perish as often as they appear and vanish j others^ that they come from the regions of the fixed stars, and are seen by us in their passage through the system of our planets ; and, lastly, others, that they are bodies perpetually revolving about the sun in very eccentric orbits. In the first case, the comets, according to their different velocities, will move in conic sections of all sorts ; in the second, they will describe hyperbolas, and in either of the two will frequent indifferently all quar- ters of the heavens, as well those about the poles as those towards the ecliptic ; in the third, their motions will be performed in ellipses very ec- centric, and very nearly approaching to parabolas. But (if thei law of the planets is observed) their ^ orbits will not much decline from the plane of the ecliptic ; and, so far as I could hitherto observe, the third case obtains ; for the comets do, indeed, chiefly frequent the zodiac, and scarcely ever attain to a heliocentric latitude of 40°. And that they move in orbits very nearly parabolical, I infer from their velocity ; for the velocity with which a parabola is described is every where to the velocity with which a comet or planet may be revolved about the sun in a circle at the same dis- tance in the subduplicate ratio of 2 to 1 (by Cor. YII, Prop. XYI) ; and, by my computation, the velocity of comets is found to be much about the same. I examined the thing by inferring nearly the velocities from the distances, and the distances both from the parallaxes and the ph^eno- mena of the tails, and never found the errors of excess or defect in the ve- locities greater than what might have arose from the errors in the dis- tances collected after that manner. But I likewise made use of the reason- ing that follows.

Supposing the radius of the orbis magniis to be divided into 1000 parts : let the numbers in the first column of the following table represent the distance of the vertex of the parabola from the sun's centre, expressed by those parts ; and a comet in the times expressed in col. 2, will pass from its perihelion to the surface of the sphere which is described about the sun as a centre with the radius of the orUs magnus ; and in the times expressed in col. 3, 4, and 5, it will double; triple, and quadruple, that its distance from the sun.

THE SYSTEM OF THE WORLD.

563

TABLE

T.

The dis- tance of a

comet's nerihelion from the Sun's cen- tre.

The time of a comet's pa'?f5age from its perihelion to a distance from the sun equal to

The radii of \he orbis rna^nus.

To its double.

Toi

ts triple

To its Quad- ruple.

0

d. h. '

27 11 12

d. h. '

77 16 28

d.

142

h. '

17 14

d. h. ' 219 17 30

5 10

27 16 07 27 21 00

77 23 14

78 06 24

20

28 06 40

78 20 13

144

03 19

221 08 54

40

29 01 32

79 23 34

80

30 13 25

82 04 56

160 320

33 05 29 37 13 46

86 10 26 93 23 38

153

16^08

232 12 20

640

37 09 49

105 01 28

1280

106 06 35

200

06 43

297 03 46

2560

147

22 31

300 06 03

[This table, here corrected, is made on the supposition that the earth's diurnal motion is just 59', and the measure of one minute loosely 0,2909, in respect of the radius 1000. If those measures are taken true, the true numbers of the table will all come out less. But the difference, even when greatest, and to the quadruple of the earth's distance from the sun, amounts only to 16^. 55'.] The time of a comet's ingress into the sphere of the orbis magnus, or

of its egress from the same, may be inferred nearly from its parallax, but

with more expedition by the following

TABLE IL

Its distance from

The apparent

Its apparent diur-

the earth m parts

elongation of

nal motion in its

whereof the radiut;

a comet from

own orbit.

of the orbis magnus

the sun.

contains 1000.

Direct.

Retrog.

60«

2° 18'

00° 20'

1000

65

2 33

00 35

845

70

2 55

00 57

684

72

3 07

01 09

618

74

3 23

01 25

551

76

3 43

01 45

484

78

4 10

02 12

416

80

4 57

02 49

347

. 82

5 45

03 47

278

84

7 18

05 20

209

86

10 27

08 19

140

88

18 37

16 39

70

90

Infi'ite

Infi'ite

00

564

THE SYSTEM OF THE WORLD.

The ingress ot a comet into the sphere of the orbis magnus, or its e^ess from the same, happens at the time of its elongation from the sun, expressed in col. 1, against its diurnal motion. So in the comet of 1681, Jan. 4, O.S. the apparent diurnal motion in its orbit was about 3° 5', and the corresponding elongation 71|° ; and the comet had acquired this elon- gation from the sun Ja?i. 4, about six in the evening. Again, in the year 1680, Nov. 11, the diurnal motion of the comet that then appeared was about 4|° ; and the corresponding elongation 79f happened Nov. 10, a little before midnight. Now at the times named these comets had arrived at an equal distance from the sun with the earth, and the earth was then almost in its perihelion. But the first table is fitted to the earth's mean distance from the sun assumed of 1000 parts ; and this distance is greater by such an excess of space as the earth might describe by its annual motion in one day's time, or the comet by its motion in 16 hours. To reduce the comet to this mean distance of 1000 parts, we add those 16 hours to the former time, and subduct them from the latter : and thus the former be- comes Ja?i. 4*^. 10*". afternoon ; the latter Nov. 10, about six in the morn- ing. But from the tenor and progress of the diurnal motions it appears that both comets were in conjunction with the sun between Dec. 7 and Dec, 8 ; and from thence to Jan. 4'^ 10'\ afternoon on one side, and to Nov. 10'^. 6^^. of the morning on the other, there are about 28 days. And so many days (by Table 1) the motions in parabolic trajectories do require.

But though we have hitherto considered those comets as two, yet, from the coincidence of their perihelions and agreement of their velocities, it is probable that in effect they were but one and the same ; and if so, the orbit of this comet must have either been a parabola, or at least a conic section very little differing from a parabola, and at its vertex almost in contact with the surface of the sun. For (by Tab. 2) the distance of the comet from the earth, Nov. 10, was about 360 parts, and Jan. 4, about 630. From which distances, together with its longitudes and latitudes, we infer the distance of the places in which the comet was at those times to have been about 280 : the half of which, viz., 140, is an ordinate to the comet's orbit, cutting off a portion of its axis nearly equal to the radius of the orbis magnus, that is, to 1000 parts. And, therefore, dividing the square of the ordinate 140 by 1000, the segment of the axis, we find the latus rectum 19, 16, or in a round number 20 ; the fourth part whereof, 5, is the distance of the vertex of the orbit from the sun's centre. But the time corresponding to the distance of 5 parts in Tab. 1 is 27^. 16^. 7'. In which time, if the comet moved in a parabolic orbit, it would have been carried from its perihelion to the surface of the sphere of the orbis mag- nus described with the radius 1000, and would have spent the double of that time, viz., hb'^. 8i'\ in the whole course of its motion within that sphere : and so in fact it did \ for from Nov. 10*^. 6^. of the morning, the

THE SYSTEM OF THE WORLD. 565

time of the comet's ingress into the sphere of the orbis magnus, to Jan. 4:\ lO"". afternoon, the time of its egress from the same, there are 55"^. 16^ The small difference of 7f '\ in this rude way of computing is to be neg- lected, and perhaps may arise from the comet's motion being some small matter slower, as it must have been if the true orbit in which it was car- ried was an ellipsis. The middle time between its ingress and egress was Decemher 8^. 2^. of the morning ; and therefore at this time the comet ought to have been in its perihelion. And accordingly that very day, just before sunrising, Dr. Halley (as we said) saw the tail short and broad, but very bright, rising perpendicularly from the horizon. From the position of the tail it is certain that the comet had then crossed over the ecliptic, and got into north latitude, and therefore had passed by its perihelion, which lay on the other side of the ecliptic, though it had not yet come into conjunction with the sun ; and the comet [see more of this famous comet, p. 475 to 486] being at this time between its perihelion and its conjunc- tion with the sun, must have been in its perihelion a few hours before ; for in so near a distance from the sun it must have been carried with great velocity, and have apparently described almost half a degree every hour.

By like computations I find that the comet of 1618 entered the sphere of the orbis magnus Dece?nber 7, towards sun-setting ; but its conjunc- tion with the sun was Nov. 9, or 10, about 28 days intervening, as in the preceding comet ; for from the size of the tail of this, in which it was equal to the preceding, it is probable that this comet likewise did come almost into a contact with the sun. Four comets were seen that year of which this was the last. The second, which made its first appearance October 31, in the neighbourhood of the rising sun, and was soon after hid under the sun's rays, I suspect to have been the same with the fourth, which emerged out of the sun's rays about Nov. 9. To these we may add the comet of 1607, which entered the sphere of the orbis magnus ^ept. 14, O.S. and arrived at its perihelion distance from the sun about October 19; 35 days intervening. Its perihelion distance subtended an apparent angle at the earth of about 23 degrees, and was therefore of 390 parts. And to this number of parts about 34 days correspond in Tab. 1. Far- ther ; the comet of 1665 entered the sphere of the orbis magnus about March 17, and came to its perihelion about April 16, 30 days intervening. Its perihelion distance subtended an angle at the earth of about seven degrees, and therefore was of 122 parts : and corresponding to this number of parts, in Tab. 1, we find 30 days. Again ; the comet of 1682 entered the sphere of the orbis magnus about Aug. 11, and arrived at its perihe- lion about Sep. 16, being then distant from the sun by about 350 parts, to which, in Tab. 1, belong 33 1 days. Lastly ; that memorable comet of Regiomontanus, which in 1472 was carried through the circum-polar parts of our northern hemisphere with such rapidity as to describe 40

566 THE SYSTEM OF THE WORLD.

degrees in one day, entered tlie sphere of the orbis magnus Jan. 21, about the time that it was passing by the pole, and, hastening from thence towards the sun, was hid under the sun's rays about the end of Feb, ; whence it is probable that 30 days, or a few more, were spent between its ingress into the sphere of the orbis ^magnus and its perihelion. Nor did this comet truly move with more velocity than other comets, but owed the greatness of its apparent velocity to its passing by the earth at a near distance.

It appears, then, that the velocity of comets (p. 471)^ so far as it can be determined by these rude ways of computing, is that very velocity with which parabolas, or ellipses near to parabolas, ought to be described ; and therefore the distance between a comet and the sun being given, the velocity of the comet is nearly given. And hence arises this problem.

PROBLEM.

The relation betwixt the velocity of a comet and its distance from the sun^s centre being given, the comefs trajectory is required.

If this problem was resolved, we should thence have a method of deter- mining the trajectories of comets to the greatest accuracy ; for if that re- lation be twice assumed, and from thence the trajectory be twice computed, and the error of each trajectory be found from observations, the assumption may be corrected by the Rule of False, and a third trajectory may thence be found that will exactly agree with the observations. And by deter- mining the trajectories of comets after this method, we may come, at last, to a more exact knowledge of the parts through which those bodies travel, of the velocities with which they are carried, what sort of trajectories they describe, and what are the true magnitudes and forms of their tails accord- ing to the various distances of their heads from the sun : whether, after certain intervals of time, the same comets do return again, and in what periods they complete their several revolutions. But the problem may be resolved by determining, first, the hourly motion of a comet to a given time from three or more observations, and then deriving the trajectory from this motion. And thus the invention of the trajectory, depending on one ob- servation, and its hourly motion at the time of this observation, will either confirm or disprove itself; for the conclusion that is drawn from the mo- tion only of an hour or two and a false hypothesis, will never agree with the motions of the comets from beginning to end. The method of the whole computation is this.

THE SYSTEM OF THE WORLD.

567

LEMMA I.

To cut two right lines OR, TP, given in position, by a third right line RP, so as TRP may be a right angle ; and, if another right line SP is drawn to any given point S, the solid contained imder this line SP, and the square of the right line OR termijiated at a given point O, may be of a given magnitude. ' It is done by linear description tlius. Let the given magnitude of the

solid be M^ x N; from any point r of the right line OR erect the per-

pendicular rp meeting TP in p. Then through the point Sp draw the

M^ X N line 8q equal to — tt-^ — . In like manner draw three or more right lines

82^', 83q, &c. ; and a regular line q2q3q, drawn through all the points q2q3q, <kc., will cut the right line TP in the point P, from which the per- pendicular PR is to be let fall. Gt.E.F.

By trigonometry thus. Assuming the right line TP as found by the preceding method, the perpendiculars TR, SB, in the triangles TPR, TPS, will be thence given ; and the side SP in the triangle SBP, as well as the

error

M_2_X_N 0R=^

— SP. Let this error, suppose D, be to a new error, sup-

pose E, as the error 2p2q + 3p3q to the error 2p3p ; or as the error 2p2q

• D to the error 2pV ; and this new error added to or subducted from the length TP, will give the correct length TP + E. The inspection of the figure will shew whether we are to add to or subtract ; and if at any time there should be use for a farther correction, the operation may be repeated.

568 THE SYSTEM OF THE WORLD.

By arithmetic thus. Let us suppose the thing done, and let TP + e be the correct length of the right line TP as found out by delineation ; and thence

rpr»

the correct lengths of the lines OR. BP, and SP, will be OR — FF^^t

BP + e, and ^SF^ + 2BPe + ee =

^^ ^ , 20R X TR , TR= OR^ + ^p e -f ^j^ee.

BP SB 2

Whence, by the method of converging series, we have SP + -^^e

SP ' 2SP3

M^N 2TR M^N STR^ M^N , ^ , . ee,&c.; = oR^"^ TP" ^ OR^^ "^ "TP^ ^ OR^ '^^' For the given

M^N 2TR M^^ BP 3TR^ M^N SB^

co-eflicients ^^^^ _ SP, ^^^ X ^j^3 — gp, ^p^ X ^^- — ggps'

F F P

putting F, kj pTTj and carefully observing the signs, we find F + p e +

F ee

t;ppj ee = 0, and 6 + ^5:= — G. Whence, neglecting the very small

term ^tj ^ comes out equal to — G. If the error ^t is not despicable, take

And it is to be observed that here a general method is hinted at for solving the more intricate sort of problems, as well by trigonometry as by arithmetic, without those perplexed computations and resolutions of affected equations which hitherto have been in use.

LEMMA IL

To cut three right lines given in position by a fourth right line that shall pass through a point assigned in any of the three, and so as its intercepted parts shall be in a given ratio one to the other. Let AB, AC, BC, be the right lines given in position, and suppose D to

be the given point in the line AC. Parallel to AB draw DG meeting BC

in G ; and, taking GF to BG in the given ratio, draw FDE ; and FD wiU be to DE as FG to BG. Q.E.F.

THE SYSTEM OF THE WORLD.

569

By trigonometry thus. In the triangle CGD all the angles and the side CD are given, and from thence its remaining sides are found ; and from the given ratios the lines GF and BE are also given.

LEMMA III. f

To find and Tepresent by a linear description the hourly motion of a comet

to any given time.

From observations of the best credit, let three longitudes of the comet be given, and, supposing ATR, RTB, to be their differences, let the hourly motion be required to the time of the middle observation TR. By Lem. II, draw the right line ARB, so as its intercepted parts AR, RB, may be

P ClJ H

as the times between the observations ; and if we suppose a body in the whole time to describe the whole line AB with an equal motion, and to be in the mean time viewed from the place T, the apparent motion of that body about the point R will be nearly the same with that of the comet at the time of the observation TR.

The same m^ore accurately.

Let Ta, T^, be two longitudes given at a greater distance on one side and on the other ; and by Lem. II draw the right line aR6 so as its inter- cepted parts aR, R6 may be as the times between the observations «TR, RT6. Suppose this to cut the lines TA, TB, in D and E ; and because the error of the inclination TRa increases nearly in the duplicate ratio of the time between the observations, draw FRG, so as either the angle DRF may be to the angle ARF, or the line DF to the line AF, in the duplicate ratio of the whole time between the observations «TB to the whole time between the observations ATB, and use the line thus found FG in place of the line AB found above.

It will be convenient that the angles ATR, RTB, «TA, WTb, be no less than of ten or fifteen degrees, the times corresponding no greater than

570

THE SYSTEM OF THE WORLD.

of eight or twelve days, and the longitudes? taken when the comet moves with the greatest velocity ; for thus the errors of the observations will bear a less proportion to the differences of the longitudes.

LEMMA IV.

To find the longitudes of a comet to any given times.

It is done by taking in the line FG the distances Rr, R/o, proportional to the times, and drawing the lines Tr, Tp. The way of working by trigonometry is manifest.

LEMMA y.

To find the latitudes. On TF, TR, TG, as radiuses, at right angles erect F/", RP, Gg, tan- gents of the observed latitudes : and parallel to fg draw PH. The per- pendiculars rp, pw, meeting PH, will be the tangents of the sought latitudes to Tr and Tp as radiuses.

PROBLEM L

From, the assumed ratio of the velocity to determine the trajectory of a

comet.

Let S represent the sun ; t, T, r, three places of the earth in its orbit at equal distances ; p, P, a>, as many corresponding places of the comet in

its trajectory, so as the distances interposed betwixt place and place may answer to the motion of one hour ; jor, PR, wp, perpendiculars let fall on the plane of the ecliptic, and ?'Rp the vestige of the trajectory in this plane. Join Sjo, SP, Sw, SR, ST, tr, TR, rp, TP , and let tr, rp, meet in O, TR will nearly converge to the same point O, or the error will be in- considerable. By the premised lemmas the angles ?'0R, ROp, are given, as well as the ratios j9r to ^r, PR to TR^ and ojp to rp. The figure ^TrO

THE SYSTEM OF THE WORLD. 571

is likewise given both in magnitude and position, together with the dis- tance ST, and the angles STR, PTR, STP. Let us assume the velocity of the comet in the place P to be to the velocity of a planet revolved about the sun in a circle, at the same distance SP, as Y to 1 ; and we shall have a line pFc^ to be determined, of this condition, that the space jpw, described by the comet in two hours, may be to the space V X /^ (that is, to the space which the earth describes in the same time multiplied by the number V) in the subduplicate ratio of ST, the distance of the earth from the sun, to SP, the distance of the comet from the sun ; and that the space pP, described by the comet in the first hour, may be to the space Pw, de- scribed by the comet in the second hour, as the velocity in p to the velocity in P ; that is, in the subduplicate ratio of the distance SP to the distance 8p, or in the ratio of 2Sp to SP + Sp ; for in this whole work I neglect small fractions that can produce no sensible error.

In the first place, then, as mathematicians, in the resolution of affected equations, are wont, for the first essay, to assume the root by conjecture, so, in this analytical operation, I judge of the sought distance TR as I best can by conjecture. Then, by Le?n. 11. I draw rp, first supposing rR equal to Rp, and again (after the ratio of SP to Sp is discovered) so as rR may be to Rp as 2SP to SP + Sp, and I find the ratios of the lines poj, rp, and OR, one to the other. Let M be to V X if^ as OR to pcj ; and because the square of jow is to the square of Y X ^t as ST to SP, we shall have, ex tzquo, 0R=^ to M^ as ST to SP, and therefore the solid 0R2 X SP equal to the given solid M^ x ST; whence (supposing the triangles STP, PTR, to be now placed in the same plane) TR, TP, SP, PR, will be given, by Lent. L All this I do, first by delineation in a rude and hasty way ; then by a new delineation with greater care ; and, lastly, by an arithmetical computation. Then I proceed to determine the position of the lines rp, poj, with the greatest accuracy, together with the nodes and inclination of the plane Spw to the plane of the ecliptic ; and in that plane Spw I describe the trajectory in which a body let go from the place P in the direction of the given right line poj would be carried with a velo- city that is to the velocity of the earth as jow to Y X tr. Q.E.F.

PROBLEM II.

To correct the ossimied ratio of the velocity and the trajectory thence

found. Take an observation of the comet about the end of its appearance, or any other observation at a very great distance from the observations used before, and find the intersection of a right line drawn to the comet, in that observation with the plane Sjt?w, as well as the comet's place in its trajec- tory to the time of the observation. If that intersection happens in this place, it is a proof that the trajectory was rightly determined ,• if other-

572 THE SYSTEM OF THE WORLD.

wise, a new number V is to be assumed, and a new trajectory to be found ; and then the place of the comet in this trajectory to the time of that pro- batory observation, and the intersection of a right line drawn to the comet with the plane of the trajectory, are to be determined as before ; and by comparing the variation of the error with the variation of the other quan- tities, we may conclude, by the Rule of Three, how far those other quantities ought to be varied or corrected, so as the error may become as small as possible. And by means of these corrections we may have the trajectory exactly, providing the observations upon which the computation was founded were exact, and that we did not err much in the assumption of the quantity V ; for if we did, the operation is to be repeated till the trajectory is exactly enough determined. Q.EJ'.

END OF THE SYSTEM OF THE WORLD.

CONTENTS

OF

THE SYSTEM OF THE WORLD,

That the matter of the heavens is fluid, 511

The principle of circular motion in free spaces, . 512

The effects of centripetal forces, , 512

The certainty of the argument, 514

What follows from the supposed diurnal motion of the stars, ....... 514

The incongruous consequences of this supposition, 514

That there is a centripetal force really directed to the centre of every planet, . . . 515

Centripetal forces decrease in duplicate proportion of distances from the centre of every planet, 516 That the superior planets are revolved about the sun, and by radii drawn to the sun describe

areas proportional to the times, 517

That the force which governs the superior planets is directed not to the earth, but to the sun, . 518 That the circum-solar force throughout all the regions of the planets decreaseth in the duplicate

proportion of the distances from the sun, 519

That the circum -terrestrial force decreases in the duplicate proportion of the distances from the

earth proved in the hypothesis of the earth's bemg at rest, 519

The same proved in the hypothesis of the earth's motion, 520

The decrement of the forces in the duplicatie proportion of the distances from the earth and plan- ets, proved from the eccentricity of the planets, and the very slow motion of their apses, . 520 The quantity of the forces tending towards the several planets : the circum-solar very great, . 521

The circum-terrestrial force very small, . . t 521

The apparent diameters of the planets, 521

The correction of the apparent diameters, 522

Why the density is greater in some of the planets and less in others; but the forces in all are as

their quantities of matter, 524

Another analogy between the forces and bodies, proved in the celestial bodies, .... 525

Proved in terrestrial bodies, 525

The affinity of those analogies, 526

And coincidence, 526

That the forces of small bodies are insensible, 527

Which, notwithstanding, there are forces tending towards all terrestrial bodies proportional to

their quantities of matter, 528

Proved that the same forces tend towards the celestial bodies, 528

That from the surfaces of the planets, reckoning outward, their forces decrease in the duplicate ;

but, reckoning inward, in the simple proportion of the distances from their centres, . 529

The quantities of the forces and of the motions arising in the several cases, .... 529

That all the planets revolve about the sun, 529

That the common centre of gravity of all the planets is quiescent. That the sun is agitated

with a very slow motion. This motion defined, 531

That the planets, nevertheless, are revolved in ellipses having their foci in the sun ; and by radii

drawn to the sun describe areas proportional to the times, 531

Of the dimensions of the orbits, and of the motions of their aphelions and nodes, . . . 532 All the motions of the moon that have hitherto been observed by astronomers derived from the

foregoing principles, 532

As also some other unequable motions that hitherto have not been observed, .... 533

And the distance of the moon from the earth to any given time, 533

The motions of the satellites of Jupiter and Saturn derived from the motions of our moon, . 534 That the planets, in respect of the fixed stars, are revolved by equable motions about their

proper axes. And that (perhaps) those motions are the most fit for the equation of time, 534

The moon likewise is revolved by a diurnal motion about its axis, and its libration thence arises, 535 That the sea ought twice to flow, and twice to ebb, every day ; that the highest water must fall

out in the third hour after the appulse of the luminaries to the meridian of the place, . 535

574 CONTENTS OF THE SYSTEM OF THE WORLD,

The precession of the equinoxes, and the libratory motion of the axes of the earth and planet?, 535 That the greatest tides happen in the syzygies of the luminaries, the least ia their quadratures;

and that at the third hour after the appulse of the moon to the meridian of the place. But

that out of the syzygies and quadratures those greatest and least tides deviate a little from

that third hour towards the third hour after the appulse of the sun to the meridian, . 536

That the tides are greatest when the luminaries are in their perigees, 536

That the tides are greatest about the equinoxes, 536

That out of the equator the tides are greater and less alternately, 537

That, by the conservation of the impressed motion, the difference of the tides is diminished ; and

that hence it may happen that the greatest menstrual tide will be the third after the syzygy, 538

That the motions of the sea may be retarded by impediments in its channels, .... 538 That from the impediments of channels and shores various phsenomena do arise, as that the sea

may flow but once every day, 539

That the times of the tides within the channels of rivers are more unequal than in the ocean. . 540 That the tides are greater in greater and deeper seas; greater on the shores of continents than

of islands in the middle of the sea; and yet greater in shallow bays that open with wide

inlets to the sea, . •. 540

The force of the sun to disturb the motions of the moon, computed from the foregoing principles, 542

The force of the sun to move the sea computed, 543

The height of the tide under the equator arising from the force of the sun computed, . . 543

The height of the tides under the parallels arising from the sun's force computed, . . . 544 The proportion of the tides under the equator, in the syzygies and quadratures, arising from the

joint forces of both sun and moon, 545

The force of the moon to excite tides, and the height of the water thence arising, computed, . 545 That those forces of the sun and moon are scarcely sensible by any other effect beside the tides

which they raise in the sea, 546

That the body of the moon is about six times more dense than the body of the sun, . . . 547

That the moon is more dense than the earth in a ratio of about three to two, .... 547

Of the distance of the fixed stars, 547

That the comets, as o'ten as they become visible to us, are nearer than Jupiter, proved from

their parallax in longitude, 548

The same proved from their parallax in latitude. M9

The same proved otherwise by the parallax, 550

From the light of the comets' heads it is proved that they descend to the orbit of Saturn, . 550

And also below the orb of Jupiter, and sometimes below the orb of the earth, .... 551

The same proved from the extraordinary splendor of their tails when they are near the sun, . 551 The same proved from the light of their heads, as being greater, cceteris paribus, when they

come near to the sun, , 553

The same confirmed by the great niunber of comets seen in the region of the sun, . . . 555

This also confirmed by the greater magnitude and splendor of the tails after the conjunction of «

the heads with the sun than before, 555

That the tails arise from the atmospheres of the comets, 556

That the air and vapour in the celestial spaces is of an immense rarity ; and that a small quan- tity of vapour may be sufficient to explain all the phgenomena of the tails of comets, . . 558 After what manner the tails of comets may arise from the atmospheres of their heads, . . 559 That the tails do indeed arise from those atmospheres, proved from several of their phcenomena, 559 That comets do sometimes descend below the orbit of Mercury, proved from their tails, . 560 That the comets move in conic sections, having one focus in the centre of the sun, and by radii

drawn to that centre do describe areas proportional to the times, 561

That those conic sections are near to parabolas, proved from the velocity of the comets, . 561 In what space of time cornets describing parabolic trajectories pass through the sphere of the

orbis magnus, 562

At what time comets enter into and pass out of the sphere of the orbis magnus, , . . 563

With what velocity the comets of 1680 passed through the sphere of the orbis magnus, . . 564 That these were not two, but one and the same comet. In what orbit and with what velocity

this comet was carried through the heavens described more exactly, .... 564

With what velocity comets are carried, shewed by more examples, 565

The investigation of the trajectory of comets proposed, . . . • • . . • 566

Lemmas premised to the solution of the problem, 667

The problem resolved, • . ... 570

INDEX TO THE PRINCIPIA,

^auiNoxESj their praecession — the cause of that motion shewn, 413

" the quantity of that motion computed from the causes, 458

Air, its density at any height, collected hy Prop. XXII, Book 11, and its density at the height

of one semi-diameter of the earth, shewn, 489

" its elastic force, what cause it may he attributed to, 302

" its gravity compared with that of water, 489

" its resistance, collected by experiments of pendulums, 315

" the same more accurately by experiments of falling bodies, and a theory, .... 355 Angles of contact not all of the same kind, but some infinitely less than others, . . . 101

Apsides, their motion shewn, 172, 173

Areas which revolving bodies, by radii drawn to the centre of force describe, compared with the

times of description, 103, 105, 106, 195, 200

As, the mathematical signification of this word defined, 100

Attraction of all bodies demonstrated, 397