which, in Tab. I, belong 33^ 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 the sphere of the orbis magnus Jan 21, abonl
the time that it was passing by the pole, and, hastening from them*
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.
4 PROBLEM.
The relation betwixt the velocity of a comet and its distance from the
sun s centre being given, the comet s 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 bv 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 hhe problem ma? be
resolved by determining, first, the hourly motion of a comet to a ffiven 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 fh*
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 under this line SP5
and the square of the right line OR terminated at a given point O,
may be of a given magnitude.
It is done by linear description thus. Let the given magnitude of the
solid be M2 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
M2 X N
line Sq equal to ^ 2
. In like manner draw three or more right lines
S2q, S3<7, &c. ; and a regular line q2q3q, drawn through all the points
y2q3q, &c., will cut the right line TP in the point P, from which the per
pendicular PR is to be let fall. Q.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
M2 X N
error ^r^ 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
H- D to the error 2pP ; 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
668 THE SYSTEM OF THE WORLD.
By arithmetic thus. Let us suppose the thing done, and let TP -f- e be the
correct length of the right line TP as found out by delineation : and thence
TR
the correct lengths of the lines OR. BP, and SP, will be OR ^^e.
BP + e, and ^/SP 2 + 2BPe + ee = M2N
QRa 20RX i
TP
op SR 2
Whence, by the method of converging series, we have SP -f- -p6 + op~j
M2 N 2TR M2 N 3TR 2 M2N
ee, <fcc., = 2 + X 3
e + x l
66 ^ tor the given
M2 N ^ 2TR M2N BP 3TR 2 M2 N SB 2
co-efficients ^-2 SP, Tp X
-^-3
gp> Tppl"
x
QR4
~
2SP~J
F F F
putting F, , ppj,
and carefully observing the signs, wo find F + ^ e -f
F ee
i = 0, and e + YT= G. Whence, neglecting the very small H
e 2 e 2
term ^, e comes out equal to G. If the error ^ is not despicable, take
G
jj
= e.
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 II.
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
will be to 1)E 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.
Tofind and represent hy 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 b<
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 more accurately.
Let Ta, T6, be two longitudes given at a greater distance on one sftle
and on the other ; and by Lem,. II draw the right line aRb so as its inter
cepted parts aR, Rft may be as the times between the observations aTR, RTA.
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 aTB to the whole time between
the observations A IB, and use the line thus found FG in place of the
line AB found above.
It will be convenient that the angles ATR, RTB, aTA, BT6, be nc
less than of ten or fifteen degrees, the times corresponding no greater than
5~0 THE SYSTEM OF THE WORLD.
of eight or twelve days, and the longitude^ taken when the comet jnoves
with the greatest velocity for thus the errors of the observation \s will
bear a less proportion to the differences of the longitudes.
LEMMA IV.
Tofind the longitudes of a comet to any given times.
It is done by taking in the line FG the distances Rr, Rp, proportional
to the times, and drawing the lines Tr, Tp. The way of working by
thgonometry is manifest.
LEMMA V.
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 I.
Prow, the assumed ratio of the velocity to determine the trajectory oj a
comet.
Let S represent the sun ; /, T, r
} three places of the earth in its orbit
at e^ual distances ; p, P, o5
? 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 ; pr, PR, wp, perpendiculars let fall on
the plane of the ecliptic, and rRp the vestige of the trajectory in this
plane. Join S/?, SP, Sc5, SR, ST, tr, TR, rp, TP , and let tr, -p, 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 rOR, ROp, are given,
as well as the ratios pr to //;, PR to TR, and wp to rp.
rr
lie 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 V to 1 ; and we shall
have a line pP& to be determined, of this condition, that the space /?w,
described by the comet in two hours, may be to the space V X tr (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
S/7, 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 Lem. II. I draw rp, first supposing / R
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
pw, rp, and OR, one to the other. Let M be to V X tr as OR to pi** ; and
because the square of
p<*>
is to the square of V X tr as ST to SP, we
shall have, ex aquo, OR2 to M2 as ST to SP, and therefore the solid
OR2 X SP equal to the given solid M2 X ST; whence (supposing the
triangles STP, PTR, to be now placed in the same plane) TR, TP, SP,
PR, will be given, by Lem. I. 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, pti, with the greatest accuracy, together with the nodes and
inclination of the plane Spti to the plane of the ecliptic ; and in that
plane Spti I describe the trajectory in which a body let go from the place
P in the direction of the given right line jf?c5 would be carried with i velo
city that is to the velocity of the earth as pti to V X tr. Q.E.F.
PROBLEM II.
To correct the assumed 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 Sjow, 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 ;
Z.L.\\ then tlu place of tke comet in this trajectory to the time of that probatory
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,.E.F.
CNJ) OF THE SYSTEM OF THE WORLD.
CONTENTS
OF
THE SYSTEM OF THE WORLD.
That the matter of the heavens is fluid, 51 j
The principle of circular motion in free spaces, ........ . 5I j
The ettects of centripetal forces, ... .. 512
The certainty of the argument, 514
Wh.t follows from the .-upposed 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, 5i6
That the superior ets 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, . 51t>
That the ci; cuin-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 being at rest, 519
The same proved in the hypothesis of the earth s motion 520
The decrement of the forces in the duplicate 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 circuni-solar very great, . 521
The circum-terrestrial force very small, 521
The apparent diameters of the planets, 5^1
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
L roved that the same forces tend towards the celestial bodies, 528
That from the surfaces of the planets, reckoning outward, their forces decrease in thj duplicate ;
but, reckoning inward, in the simple proportion of the distances from their centres, . 52r
The quantities of the forces and of the motions arising in the several cases, .... 52V.
That all the planets revolve about the sun, 529
That the commun 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
Jf 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 chird hour after the appulse of the luminaries to the meridian of the place, . 333
674 CONTENTS OF THE SYSTEM OF THE WORLD.
fke 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 in their quadratures;
and that at the third hour after the appulse of the moon to the meridian of the place. Bat
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 inensti ual tide will be the third after the syzygy, 5:38
Thit the motio is of the sea may be retarded by impediments in its channels, .... 538
That from the impediments of channels and shores various phenomena do arite, 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
1 hat the tides are greater in greater and deeper seas; greater on the shores of continents than
(1 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 principks, 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 ?en?ible 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 ot 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, ......... 549
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, c&teris paribus, when they
come near to the sun, 553
The same confirmed by the great number 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 phtcnomena 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 pheenomena, 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
Irawn 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 vrbis magnus, . . . 563
With what velocity the comets of 1680 passed through the sphere of the orbis magnus, . . 564 ~i
*
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 corsets are carried, shewed by more examples, . . ... 565
The investigation of the trajectory of comets proposed, .... ... 566"
Lemmas premised to the solution of the problem, ..... ... 567
The problem resolved, . ..... ... 57C
INDEX TO THE PRINC1PIA.
j their prsecession the cause of that motion shewn, 413
" the quantity of that motion computed from the causes, 4oJ
A.IR, its density at any height, collected by Prop. XXII, Book II, and its density at the height
of one semi-diameter of the earth, shewn, 489
its elastic force, what cause it may be attributed to, 302
its gravity compared with that of water, -l^t
" its resistance, collected by experiments of pendulums, 315
" the same more accurately by experiments of falling bodies, and a theory, .... 353
ANGLE S 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, 2(!<>
As, the mathematical signification of this word defined, . . .100
ATTRACTION of all bodies demonstrated, 3 >7
" the certainty of this demonstration shewn, 384
the cause or manner thereof no where defined by the author, .... 507
the common centre of gravity of the earth, sun, and all the planets, is at rest, con
firmed by Cor. 2, Prop. XIV, Book HI, 401
" the common centre of gravity of the earth and moon goes round the orbis magnus, 402
" its distance from the earth and from the moon, 452
CENTRE, the common centre of gravity of many bodies does not alter its state of motion or rest
by the actions of the bodies among themselves, 87
" of the forces by which revolving bodies are retained in their orbits, how indicated by
the description of areas, 107
" how found by the given velocities of the revolving bodies, . ..... 110
CIRCLE, by what law of centripetal force tending to any given point its circumference may be
described, . 108,111,114
COMETS, a sort of planets, not meteors, 465,486
" higher than the moon, and ir. the- planetary regions, 460
" their distance how collected very nearly by observations, 401
" more of them observed in the hemisphere towards the sun than in the opposite hemis
phere; and how this comes to pa?s, 464
" shine by the sun s light reflected from them, 464
" surrounded with vast atmospheres, 463, 465
" those which come nearest to the sun probably the least, ... . . 4P5
" why they are not comprehended within a zodia , like the planets, but move differently
into all parts of the heavens, ... 502
" may sometimes fall into the sun, and afford a new supply of fire, 502
the use of them hinted, 492
" move in conic sections, having their foci in the sun s centre, and by radii drawn to the
sun describe areas proportional to the times. Move in ellipses if they come round again
