of many days, and partaking of the motion of the heads which
they had from the beginning, continue to go along together with them
through the heavens. From again we have another argument
proving the celestial spaces to be free, and without resistance, since in
them not only the solid bodies of the planets and comets, but also the ex
tremely rare vapours of comets1
tails, maintain their rapid motions with
great freedom, and for an exceeding long time.
Kepler ascribes the ascent of the tails of the comets to the atmospheres
of their heads : and their direction towards the parts opposite to the sun to
the action of the rays of light carrying along with them the matter of the
comets tails
; and without any great incongruity we may suppose, that, in
so free spaces, so fine a matter as that of the aether may yield to the action
of the rays of the sun s light, though those rays are not able sensibly to
move the gross substances in our parts, which are clogged with so palpable
a resistance. Another author thinks that there may be a sort of particles
of matter endowed with a principle of levity, as well as others are with a
power of gravity ;
that the matter of the tails of comets may be of the
former sort, and that its ascent from the sun may be owing to its levity ;
but, considering that the gravity of terrestrial bodies is as the matter of
the bodies, and therefore can be neither more nor less in the same quantity
of matter, I am inclined to believe that this ascent may rather proceed from
the rarefaction of the matter of the comets tails. The ascent of smoke in
a chimney is owing to the impulse of the air with which it is entangled.
The air rarefied by heat ascends, because its specific gravity is diminished,
and in its ascent carries along with it the smoke with which it is engaged ;
ind why may not the tail of a comet rise from the sun after the same man
492 THE MATHEMATICAL PHINCIPLES [BOOK III.
ner ? For the sun s rays do not act upon the mediums which they per
vade otherwise than by reflection and refraction ; and those reflecting par
ticles heated by this action, heat the matter of the aether which is involved
with them. That matter is rarefied by the heat which it acquires, and beoause,
by this rarefaction, the specific gravity with which it tended towards
the sun before is diminished, it will ascend therefrom, and carry along with
it the reflecting particles of which the tail of the comet is composed. But
the ascent of the vapours is further promoted by their circumgyration
about the sun, in consequence whereof they endeavour to recede from the
sun, while the sun s atmosphere and the other matter of the heavens are
either altogether quiescent, or are only moved with a slower circumgyra
tion derived from the rotation of the sun. And these are the causes of the
ascent of the tails of the comets in the neighbourhood of the sun, where
their orbits are bent into a greater curvature, and the comets themselves
are plunged into the denser and therefore heavier parts of the sun s atmos
phere : upon which account they do then emit tails of an huge length ;
for
the tails which then arise, retaining their own proper motion, and in the
mean time gravitating towards the sun, must be revolved in ellipses about
the sun in like manner as the heads are, and by that motion must always
accompany the heads, and freely adhere to them. For the gravitation ot
the vapours towards the sun can no more force the tails to abandon the
heads, and descend to the sun,*than the gravitation of the heads can oblige
them to fall from the tails. They must by their common gravity either
fall together towards the sun, or be retarded together in their comii>ori as
cent therefrom ; and, therefore (whether from the causes already described,
or from any others), the tails and heads of comets may easily acquire and
freely retain any position one to the other, without disturbance or impedi
ment from that common gravitation.
The tails, therefore, that rise in the perihelion positions of the comets
will go along with their heads into far remote parts, and together with
the heads will either return again from thence to us, after a long course of
years, or rather will be there rarefied, and by degrees quite vanish away ;
for afterwards, in the descent of the heads towards the sun, new short tails
will be emitted from the heads with a slow motion; and those tails by de
grees will be augmented immensely, especially in such comets as in their
perihelion distances descend as low as the sun s atmosphere ;
for all vapour
in those free spaces is in a perpetual state of rarefaction and dilatation ;
and from hence it is that the tails of all comets are broader at their upper
extremity than near their heads. And it is not unlikely but that the va
pour, thus perpetually rarefied and dilated, may be at last dissipated and
scattered through the whole heavens, and by little and little be attracted
towards the planets by its gravity, and mixed with their atmosphere; for
as the seas are absolutely necessary to the constitution of our earth,
BOOK III.] OF NATURAL PHILOSOPHY. 493
from them, the sun, by its heat, may exhale a sufficient quantity of vapours,
which, being gathered together into clouds, may drop down in rain, for
watering of the earth, and for the production and nourishment of vegeta
bles; or, being condensed with cold on the tops of mountains (as some phi
losophers with reason judge), may run down in springs and rivers; so for
the conservation of the seas, and fluids of the planets, comets seem to be
required, that, from their exhalations and vapours condensed, the wastes of
the planetary fluids spent upon vegetation and putrefaction, and converted
into dry earth, may be continually supplied and made up; for all vegeta
bles entirely derive their growths from fluids, and afterwards, in great
measure, are turned into dry earth by putrefaction : and a sort of slime is
always found to settle at the bottom of putrefied fluids; and hence it is
that the bulk of the solid earth is continually increased; and the fluids, if
they are not supplied from without, must be in a continual decrease,
and quite fail at last. I suspect, moreover, that it is chiefly from the
comets that spirit comes, which is indeed the smalles; but the most subtle
and useful part of our air, arid so much required to sustain the life of all
things with us.
The atmospheres of comets, in their descent towards the sun, by running
out into the tails, are spent and diminished, and become narrower, at least
on that side which regards the sun ; and in receding from the sun, when
they less run out into the tails, they are again enlarged, if Hevelins has
justly marked their appearances. But they are seen least of all just after
they have been most heated by the sun, and on that account then emit the
longest and most resplendent tails; and, perhaps, at the same time, the
nuclei are environed with a denser and blacker smoke in the lowermost
parts of their atmosphere ;
for smoke that is raised by a great and intense
heat is commonly the denser and blacker. Thus the head of that cornet
which we have been describing, at equal distances both from the sun and
from the earth, appeared darker after it had passed by its perihelion than
it did before ;
for in the month of December it was commonly compared
with the stars of the third magnitude, but in November with those of the
first or second ; and such as saw both appearances have described the first
as of another and greater comet than the second. For, November 19. this
comet appeared to a young man at Cambridge, though with a pale and
dull light, yet equal to Spica Virg-inis ; and at that time it shone with
greater brightness than it did afterwards. And Montenari, November 20,
et. vet. observed it larger than the stars of the first magnitude, its tail
being then 2 degrees long. And Mr. Storer (by letters which have come
into my hands) writes, that in the month of December, when the tail ap
peared of the greatest bulk and splendor, the head was but small, and far
less than that which was seen in the month of November before sun- rising;
and, conjecturing at the cause of the appearance, he judged it to proceed
494 THE MATHEMATICAL PRINCIPLES [BOOK J II
from there being a greater quantity of matter in the head at first, which
was afterwards gradually spent.
And, which farther makes for the same purpose, I find, that the heads of
other cornets, which did put forth tails of the greatest bulk and splendor,
have appeared but obscure and small. For in Brazil, March 5, 1 668, 7h
.
P. M.; St. N. P. Valentin its Esta. tcws saw a comet near the horizon, and
towards the south west, with a head so small as scarcely to be discerned,
but with a tail above measure splendid, so that the reflection thereof from
the sea was easily seen by those who stood upon the shore ; and it looked
like a fiery beam extended 23 in length from the west to south, almost
parallel to the horizon. But this excessive splendor continued only three
days, decreasing apace afterwards ; arid while the splendor was decreasing,
the bulk of the tail increased : whence in Portugal it is said to have taken
ap one quarter of the heavens, that is, 45 degrees, extending from west to
3ast with a very notable splendor, though the whole tail was not seen in
those parts, the head was always hid under the horizon : and from
the increase of the hulk arid decrease of the splendor of the tail, it appears
that the head vis then in its recess from the sun, and had been very near
to it in its perihelion, as the comet of 1680 was. And we read, in the
Saxon Chronicle, of a like comet appearing in the year 1 106, the star
whereof was small and obscure (as that of 1680), but the splendour of its
tail w^s very bright, and like a hugefiery beam stretched out in a direc
tion fetween the east and north, as Hevelius has it also from Simeon, the
monk of Durham. This comet appeared in the beginning of February.
about the evening, and towards the south west part of heaven ;
frc-in
whence, and from the position of the tail, we infer that the head was near
the sun. Matthew Paris says, // was distant from the sun by about a
cubit, from, three of the clock (rather six) till nine, putting forth a long
tail. Such also was that most resplendent comet described by Aristotle,
lib. 1, Meteor. 6. The head whereof could not be seen, because it had set
before the sun, or at least was hid under the sun s rays ; but next day
it was seen, as well as might be ; for, having left the sun but a very lit
tle way, it set immediately after it. And the scattered light of the head,
obscured by the too great splendour (of the tail) did not yet appear. But
afterwards (as Aristotle says) when the splendour (of the tail) was now
diminished (the head of), the comet recovered its native brightness ; and
the splendour (of its tail) reached now to a third part of the heavens (that
is, to 60). This appearance was in the winter season (an. 4, Olymp.
101), and, rising tit Orion s girdle, it there vanished away. It is true
that the comet of 1618, which came out directly from under the sun s rays
with a very large tail, seemed to equal, if not to exceed, the stars of the
first magnitude: but, then, abundance of other comets have appeared yet
greater than this, that put forth shorter tails; some of which are said
BOOK III.] OF NATURAL PHILOSOPHY. 495
to have appeared as big as Jupiter, others as big as Venus, or even as
the moon.
We have said, that comets are a sort of planets revolved in very eccen
tric orbits about the sun : and as, in the planets which are without tails,
those are commonly less which are revolved in lesser orbits, and nearer to
the sun, so in comets it is probable that those which in their perihelion ap
proach nearer to the sun ate generally of less magnitude, that they may
not agitate the sun too much by their attractions. But, as to the trans
verse diameters of their orbits, and the periodic times of their revolutions,
1 leave them to be determined by comparing comets together ^hich after
long intervals of time return again in the same orbit. In the mean time,
the following Proposition may give some light in that inquiry.
PROPOSITION XLIL PROBLEM XXII.
To correct a cornet s trajectory found as above.
OPERATION 1. Assume that position of the plane of the trajectory which
was determined according to the preceding proposition; and select three
places of the comet, deduced from very accurate observations, and at great
distances one from the other. Then suppose A to represent the time be
tween the first observation and the second, and B the time between the
secoi.d and the third
; but it will be convenient that in one of those times
the comet be in its perigeon, or at least not far from it. From those ap
parent places find, by trigonometric operations, the three true places of the
comet in that assumed plane of the trajectory ;
then through the places
found, and about the centre of the sun as the focus, describe a conic section
by arithmetical operations, according to Prop. XXL, Book 1. Let the
areas of this figure which are terminated by radii drawn from the sun to
the places found be D and E; to wit, I) the area between the first observa
tion and the second, and E the area between the second and third ; and let
T represent the whole time in which the whole area D + E should be de
scribed with the velocity of the comet found by Prop. XVI., Book 1.
OPER. 2. Retaining the inclination of the plane of the trajectory to the
plane of the ecliptic, let the longitude of the nodes of the plane of the tra
jectory be increased by the addition of 20 or 30 minutes, which call P.
Then from the aforesaid three observed places of the comet let the three
true places be found (as before) in this new plane; as also the orbit passing
through those places, and the two areas of the same described between the
two observations, which call d and e ; and let t be the whole time in which
the whole area d + e should be described.
OrER. 3. Retaining the longitude of the nodes in the first operation, let
the inclination of the plane of the trajectory to the plane of the ecliptic be
increased by adding thereto 20 or 30 , which call Q,. Then from the
496 THE MATHEMATICAL PRINCIPLES [BOOK 111
aforesaid three observed apparent places of the comet let the three true
places be found in this new plane, as well as the orbit passing through
them, and the two areas of the same described between the observation,
which call d and ; and let r be the whole time in which the whole area
(5 -4- should be described.
Then taking C to 1 as A to B ; and G to 1 as D to E ; and g to 1 as
d to e; and y to 1 as cJ to c; let S be the true time between the first ob
servation and the third ; and, observing well the signs + and , let such
numbers m and n be found out as will make 2G 2C, = raG m
+ nG uy ; and 2T 28 = mT - nil + nT nr. And if, in
the first operation, I represents the inclination of the plane of the trajec
tory to the plane of the ecliptic, and K the longitude of either node, then
I + 7/Q will be the true inclination of the plane of the trajectory to the
plane of the ecliptic, and K + mP the true longitude of the node. And.
lastly, if in the first, second, and third operations, the quantities R, r, and
p, represent the parameters of the trajectory, and the quantities -7", -7, -,
LA I A.
the transverse diameters of the same, then R -f mr mR + up /?R
will be the true parameter, and =
;
- : .- =- will be the L + inl mL + nh w.L
true transverse diameter of the trajectory which the comet describes
; and
from the transverse diameter given the periodic time of the comet is also
given. Q.E.I. But the periodic times of the revolutions of comets, and
the transverse diameters of their orbits, cannot be accurately enough de
termined but by comparing comets together which appear at different
times. If, after equal intervals of time, several comets are found to have
described the same orbit, we may thence conclude that they are all but one
and the same comet revolved in the same orbit ; and then from the times
of their revolutions the transverse diameters of their orbits will be given,
and from those diameters the elliptic orbits themselves will be determined.
To this purpose the trajectories of many comets ought to be computed,
supposing those trajectories to be parabolic; for such trajectories will
always nearly agree with the ph&nomena, as appears not only from the
parabolic trajectory of the comet of the year 1680, which I compared
above with the observations, but likewise from that of the notable comet
which appeared in the year 1664 and 1665, and was observed by Hevelins,
who, from his own observations, calculated the longitudes and latitudes
thereof, though with little accuracy. But from the same observations Dr.
Halley did again compute its places; and from those new places deter
mined its trajectory, finding its ascending node in n 21 13 55"
; the in
clination of the orbit to the plane of the ecliptic 21 IS 40"
; the dis
tance of its perihelion from the node, estimated in the comet s orbit, 49
27 30",- its perihelion in P, 8 40 30", with heliocentric latitude south
BOOK IIL1 OF NATURAL PHILOSOPHY. 497
16 UT 45"
;
the comet to have been in its perihelion November 21(l
. Hi,.
52 P.M. equal time at London, or 13h
. 8 at Duiitzick, O. S.; and that
the latus rectum of the parabola was 4102S6 such parts as the sun s mean
distance from the earth is supposed to contain 100UOO. And how nearly
the places of the comet computed in this orbit agree with the observations,
will appear from the annexed table, calculated by Dr. Halley.
In February, the beginning of the year
which I shall hereafter call y, was in HP 28
1665, the first star of Aries,
30 15", with 7 8 58" north
498 THE MATHEMATICAL PRINCIPLES [BOOK III.
lat.; the second star of Aries was in w 29 IT IS , with 8 28 16" north
lat.; and another star of the seventh magnitude, which I call A, was in
v 28 24 45", with 8 28 33" north lat,
~
The comet Feb. 7(1
. 7h
. 30 tt
Paris (that is. Feb. 7 1
. 8h
. 37 at Dantzick] O. S. made a triangle with
those stars y and A. which was right-angled in y; and the distance of the
comet from the star y was equal to the distance of the stars y and A, that
is, 1 19 46 of a great circle ; and therefore in the parallel of the lati
tude of the star y it was 1 20 26". Therefore if from the longitude of
the star y there be subducted the longitude 1 20 26". there will remain
the longitude of the comet T 27 9 49". M. Auzout, from this observa
tion of his, placed the comet in 1P 27 , nearly ; and, by the scheme in
which Dr. Hooke delineated its motion, it was then in T 26 59 24 . 1
place it in CP 27 4 46 , taking the middle between the two extremes.
From the same observations, M. Anzont made the latitude of the cornet
at that time 7 and 4 or 5 to the north ; but he had done better to have
made it 7 3 29", the difference of the latitudes of the comet and the star
y being equal to the difference of the longitude of the stars y and A.
Ftbmury 22(i
. 71
. 30 at London, that is, February 22 . 8h
. 46 at
Dantzick, the distance of the comet from the star A, according to Dr.
JJooke s observation, as was delineated by himself in a scheme, and also
by the observations of M. Auzout, delineated in like manner by M. Petit,
was a fifth part of the distance between the star A and the first star of
Aries, or 15 57"
; and the distance of the comet from a right line joining
the star A and the first of Aries was a fourth part of the same iifth part,
that is, 4 ; and therefore the comet was in T 28 29 46", with 8 12
36" north lat.
March 1, 7h
. at Londou, that is, March 1, 8h
. 16 at Dantzick, the
comet was observed near the second star in Aries, the distance between
them being to the distance between the first and second stars in Aries, that
is, to 1 33 , as 4 to 45 according to Dr. Hooke, or as 2 to 23 according
to M. Gottiguies. And, therefore, the distance of the comet from the
second star in Aries was 8 16" according to Dr. Hooke, or 8 5" according
to M. Gottignies ; or, taking a mean between both, 8 10". But, accord
ing to M. Gottignies, the comet had gone beyond the second star of Aries
about a fourth or a fifth part of the space that it commonly went over in
a day, to wit, about 1 35" (in which he agrees very well with M. Auzo-nf] ;
or, according to Dr. Hooke, not quite so much, as perhaps only .1 . Where
fore if to the longitude of the first star in Aries we add 1
, and 8 10" to
its latitude, we shall have the longitude of the comet T 29 IS , with S
36 26" north lat.
March 7, 7h
. 30 at Paris (that is, March 7, 8h
. 37 at Dantzick),
from the observations of M. Auzout, the distance of the comet from the
second star in Aries was equal to the distance of that star from the star
BOOK III.] OF NATURAL PHILOSOPHY. 499
A, that is, 52/29" ; and the difference of the longitude of the comet and
the second star in Aries was 45 or 46 , or, taking a mean quantity, 45
30"
;
and therefore the comet was in tf 2 48". From the scheme of
the observations of M. Auzout, constructed by M. Petit, Hevelius collected
the latitude of the comet 8D 54 . But the engraver did not rightly trace
the curvature of the comet s way towards the end of the motion ; and
Hevdius, in the scheme of M. Auzoiifs observations which he constructed
himself, corrected this irregular curvature, and so made the latitude of the
comet 8 55 30". And, by farther correcting this irregularity, the lati
tude may become 8 56 , or 8 57 .
This comet was also seen March 9, and at that time its place must have
been in 8 18 . with 9 3f north lat. nearly.
This comet appeared three months together, in which space of time it
travelled over almost six signs, and in one of the days thereof described
almost 20 deg. Its course did very much deviate from a great circle, bend
ing towatds the north, and its motion towards the end from retrograde be
came direct ; and, notwithstanding its course was so uncommon, yet by the
table it appears that the theory, from beginning to end, agrees with the
observations no less accurately than the theories of the planets usually do
with the observations of them : but we are to subduct about 2 when the
comet was swiftest, which we may effect by taking off 12" from the angle
between the ascending node and the perihelion, or by making that angle
493 27 18". The annual parallax of both these comets (this and the
preceding) was very conspicuous, and by its quantity demonstrates the an
nual motion of the earth in the orbis magnus.
This theory is likewise confirmed by the motion of that comet, which
in the year 1683 appeared retrograde, in an orbit whose plane contained
almost a right angle with the plane of the ecliptic, and whose ascending
node (by the computation of Dr. Halley) was in ng 23 23 ; the inclina
tion of its orbit to the ecliptic 83 11 ;
its perihelion in. n 25 29 30"
its perihelion distance from the sun 56020 of such parts as the radius of
the orbis maguiis contains 100000 ; and the time of its perihelion July
2 1
. 3h
. 50 . And the places thereof, computed by Dr. Halley in this orbit,
are compared with the places of the same observed by Mr. Flamsted. iD
the following table :
500 THE MATHEMATICAL PRINCIPLES [BOOK III.
This theory is yet farther confirmed by the motion of that retrograde
comet which appeared in the year 1682. The ascending node of this (by
Dr. Halleifs computation) was in & 21 16 30"
;
the inclination of its
orbit to the plane of the ecliptic 17 56 00"
; its perihelion in z, 2 52
50 ; its perihelion distance from the sun 5S32S parts, of which the radius
of the orbift matrnus contains 100000 ; the equal time of the comet s
being in its perihelion Sept. 4 1
. 7h
. 39 . And its places, collected from
Mr. Flamsted s observations, are compared with its places computed from
our theory in the following table :
This theory is also confirmed by the retrograde motion of the comet that
appeared in the year 1723. The ascending node of this comet (according
