heads. In the former case, we must shorten the distance of the comets,
lest we be obliged to allow that the smoke arising from their heads is
propagated through such a vast extent of space, and with such a velocity
and expansion as will seem altogether incredible ; in the latter case, the
whole light of both head arid tail is to be ascribed to the central nucleus.
But. then, if we suppose all this light to be united and condensed within
the disk of the nucleus, certainly the nucleus will by far exceed Jupiter
itself in splendor, especially when it emits a very large and lucid tail If.
therefore, under a less apparent diameter, it reflects more light, it must be
much more illuminated by the sun, and therefore much nearer to it; and
the same argument will bring down the heads of comets sometimes within
the orb of Venus, viz., when, being hid under the sun s rays, they emit such
huge and splendid tails, like beams of fire, as sometimes they do ; for if all
that light was supposed to be gathered together into one star, it would
sometimes exceed not one Venus only, but a great many such united
into one.
Lastly ; the same thing is inferred from the light of the heads, which
increases in the recess of the cornets from the earth towards the sun, and
decreases in their return from the sun towards the earth ; for so the comet
of the year 1665 (by the observations of Hevelius], from the time that it
was first seen, was always losing of its apparent motion, and therefore had
already passed its perigee ; but yet the splendor of its head was daily in
creasing, till, being hid under the sun s rays, the comet ceased to appear.
The comet of the year 1683 (by the observations of the same LJevelius),
about the end of July, when it first appeared, moved at a very slow rate,
advancing only about 40 or 45 minutes in its orb in a day s time ; but
from that time its diurnal motion was continually upon the increase, till
September 4, when it arose to about 5 degrees ; and therefore, in all this
interval of time, the comet was approaching to the earth. Which is like
wise proved from the diameter of its head, measured with a micrometer ;
for, August 6, Hevelius found it only 6 05", including the coma, which,
September 2 he observed to be 9 07", and therefore its head appeared far
less about, the beginning than towards the end of the motion ; though
about the beginning, because nearer to the sun, it appeared far more lucid
than towards the end, as the same Hevelius declares. Wherefore in all
this interval of time, on account of its recess from the sun, it decreases
464 THE MATHEMATICAL PRINCIPLES [BOOK III.
in splendor, notwithstanding its access towards the earth. The comet of
the year 1618, about the middle of December, and that of the year 1680,
about the end of the same month, did both move with their greatest velo
city, and were therefore then in their perigees : but the greatest splendor
of their heads was seen two weeks before, when they had just got clear of
the sun s rays ; and the greatest splendor of their tails a little more early,
when yet nearer to the sun. The head of the former comet (according to
the observations of Cysdtus], Dece/itber 1, appeared greater than the stars
of
^
the first magnitude: and, December 16 (then in the perigee), it was
but little diminished in magnitude, but in the splendor and brightness of
its light a great deal. January 7, Kepler, being uncertain about the
head, left oif observing. December 12, the head of the latter comet was
seen and observed by Mr. Flamsted, when but 9 degrees distant from the
sun ; which is scarcely to be done in a star of the third magnitude. De
cember 15 and 17, it appeared as a star of the third magnitude, its lustre
being diminished by the brightness of the clouds near the setting sun.
December 26, when it moved with the greatest velocity, being almost in
its perigee, it was less than the mouth of Pegasus, a star of the third
magnitude. January 3, it appeared as a star of the fourth. January 9,
as one of the fifth. January 13, it was hid by the splendor of the moon,
then in her increase. January 25, it was scarcely equal to the stars of
the seventh magnitude. If we compare equal intervals of time on one
side and on the other from the perigee, we shall find that the head of the
comet, which at both intervals of time was far, but yet equally, removed
from the earth, and should have therefore shone with equal splendor, ap
peared brightest on the side of the perigee towards the sun, and disap
peared on the other. Therefore, from the great difference of light in the
one situation and in the other, we conclude the great vicinity of the sun
and comet in the former ; for the light of comets uses to be regular, and
to appear greatest when the heads move fastest, and are therefore in their
perigees ; excepting in so far as it is increased by their nearness to the
sun. -
COR. 1. Therefore the comets shine by the sun s light, which they reflect.
COR. 2. From what has been said, we may likewise understand why
comets are so frequently seen in that hemisphere in which the sun is, and
so seldom in the other. If they were visible in the regions far above
Saturn, they would appear more frequently in the parts opposite to the
sun ;
for such as were in those parts would be nearer to the earth, whereas
the presence of the sun must obscure and hide those that appear in the
hemisphere in which he is. Yet, looking over the history of comets, I
find that four or five times more have been seen in the hemisphere towards
the sun than in the opposite hemisphere ; besides, without doubt, not a
few, which have been hid by the light of the sun : for comets descending
BOOK III.]
0* NATURAL PHILOSOPHY. !fi5
into our parts neither emit tails, nor are so well illuminated by the sun,
as to discover themselves to our naked eyes, until they are come nearer to
us than Jupiter. But the far greater part of that spherical space, which
is described about the sun with so small an interval, lies on that side of
the earth which regards the sun ; and the comets in that greater part are
commonly more strongly illuminated, as being for the most part nearer to
the sun.
COR. 3. Hence also it is evident that the celestial spaces are void of
resistance ;
for though the comets are carried in oblique paths, and some
times contrary to the course of the planets, yet they move every way with
the greatest freedom, and preserve their motions for an exceeding long
time, even where contrary to the course of the planets. I am out in my
judgment if they are not a sort of planets revolving in orbits returning
into themselves with a perpetual motion ; for, as to what some writers
contend, that they are no other than meteors, led into this opinion by the
perpetual changes that happen to their heads, it seems to have no founda
tion ;
for the heads of comets are encompassed with huge atmospheres,
and the lowermost parts of these atmospheres must be the densest
; and
therefore it is in the clouds only, not in the bodies of the comets them
selves, that these changes are seen. Thus the earth, if it was viewed from
the planets, would, without all doubt, shine by the light of its clouds, and
the solid body would scarcely appear through the surrounding clouds.
Thus also the belts of Jupiter are formed in the clouds of that planet,
for they change their position one to another, and the solid body of Jupiter
is hardly to be seen through them ; and much more must the bodies of
comets be hid under their atmospheres, which are both deeper and thicker.
PROPOSITION XL. THEOREM XX.
That the comets mnve in some of the conic sections, having their foci
in the centre of the sun ; and by radii drawn to the sun describe
areas proportional to the times.
This proposition appears from Cor. 1, Prop. XIII, Book 1, compared
vith Prop. VIII, XII, and XIII, Book HI.
COR. 1. Hence if comets are revolved in orbits returning into them
selves, those orbits will be ellipses ; and their periodic times be to the
periodic times of the planets in the sesquiplicate proportion of their prin
cipal axes. And therefore the comets, which for the most part of their
course are higher than the planets, and upon that account describe orbits
with greater axes, will require a longer time to finish their revolutions.
Thus if the axis of a comet s orbit was four times greater than the axis
of the orbit of Saturn, the time of the revolution of the comet would be
to the time of the revolution of Saturn, that is, to 30 years, as 4 ^/ 4
(or 8) to 1, and would therefore be 240 years.
30
THE MATHEMATICAL PRINCIPLES [BOOK III.
COR. 2. But their orbits will be so near to parabolas, that parabolas
may be used for them without sensible error.
COR. 3. And, therefore, by Cor. 7, Prop. XVI, Book 1, the velocity of
every comet will always be to the velocity of any planet, supposed to be
revolved at the same distance in a circle about the sun, nearly in the subduplicate
proportion of double the distance of the planet from the centre
of the sun to the distance of the comet from the sun s centre, very nearly.
Let us suppose the radius of the orbis wagmis, or the greatest semidiameter
of the ellipsis which the earth describes, to consist of 100000000
parts ; and then the earth by its mean diurnal motion will describe
1720212 of those parts, and 716751 by its horary motion. And there
fore the comet, at the same mean distance of the earth from the sun, with
a velocity which is to the velocity of the earth as v/ 2 to I, would by its
diurnal motion describe 2432747 parts, and 101.3641 parts by its horary
motion. But at greater or less distances both the diurnal and horary
motion will be to this diurnal and horary motion in the reciprocal subduplicate
proportion of the distances, and is therefore given.
COR. 4. Wherefore if the lattis rectum of the parabola is quadruple of
the radius of the orbis maginis, and the square of that radius is sup
posed to consist of 100000000 parts, the area which the comet will daily
describe by a radius drawn to the sun will be 12163731 parts, and the
horary area will be 506821 parts. But, if the latus rectum is greater
or less in any proportion, the diurnal and horary area will be less or
greater in the subduplicate of the same proportion reciprocally.
LEMMA V.
Tofind a curve line of the parabolic kind which shall pass through any
given number of points.
Let those points be A, B, C, D, E, F, (fee., and from the same to any
right line HN, given in position, let fall as many perpendiculars AH, BI,
CK, DL, EM, FN, tfec.
b 2b 3b 45 5b
c 2c 3c 4c
d 2d 3d
H
e 2e
f
CASE 1. If HI, IK, KL, &c., the intervals of the points H, I, K, L, M
N, (fee., are equal, take b, 2b, 3b, 46, 56, (fee., the first differences of the per
pendiculars AH. BI, CK, (fee.
;
their second differences c, 2c, 3c, 4r, <fec. :
their third, d, 2d, 3d, (fee., that is to say, so as AH BI may be== b, 01
BOOK III.] OF NATURAL PHILOSOPHY. 467
CK = 2b, CK DL = 36, DL + EM = 46, EM + FN = 56,
&c. ; then 6 2b == c, &c., and so on to the last difference, which is here
/*. Then, erecting any perpendicular RS, which may be considered as an
ordinate of the curve required, in order to find the length of this ordinatc,
suppose the intervals HI. IK, KL, LM, (fee., to be units, and let AH = a.
-KS=f>, \p into IS = q, q into + SK = r,
into + SL = s,
\s into 4- SM = t ; proceeding, to wit, to ME, the last perpendicular but
one, and prefixing negative signs before the terms HS, IS, &c., which lie
from S towards A; and affirmative signs before the terms SK, SL, (fee..
which lie on the other side of the point S ; and, observing well the signs,
RS will be = a + bp + cq + dr + es + ft, + (fee.
CASE 2. But if HI, IK, (fee., the intervals of the points H, I, K, L, <fcc.,
are unequal, take 6, 26, 36, 46, 56, (fee., the first differences of the perpen
diculars AH, BI, CK, cfec., divided by the intervals between those perpen
diculars ; c, 2Cj 3c, 4c, (fee., their second differences, divided by the intervals
between every two ; c/, 2d, 3d, (fee., their third differences, divided by the
intervals between every three; e, 2e, (fee., their fourth differences, divided
by the intervals between every four ; and so forth
;
that is, in such manner,
AH BI
*
BI CK , CK DL
that b may be = ---^ , 2b = --.-==
, 6b = --==- -----
, (fee., then
2b 2b 3b 36 46
(
c 2c
&c then rf - "
2</
2c 3c
-=
j-T7
, (fee. And those differences being found, let AH be = a,
HS = p, p into IS = q, q into + SK = r, r into + SL =. s, s into
-f- SM = t
; proceeding, to wit, to ME, the last perpendicular but one : .
and the ordinate RS will be = a -f- bp + cq + dr + es -f //, + tfec.
COR. Hence the areas of all curves may be nearly found ; for if some
number of points of the curve to be squared are found, and a parabola be
supposed to be drawn through those points, the area of this parabola willi
be nearly the same with the area of the curvilinear figure proposed to be
squared : but the parabola can be always squared geometrically by methods
vulgarly known.
LEMMA VI.
Certain observed places of a comet being" given, to find the place of the
same, to any intermediate given time.
Let HI, IK, KL, LM (in the preceding Fig.), represent the times between
the observations ; HA, IB, KC, LD, ME, five observed longitudes of the
comet ; and HS the given time between the first observation and the longi
tude required. Then if a regular curve ABODE is supposed to be drawn
through the points A, B, C, D, E, and the ordinate RS is found out by the
preceding lemma, RS will be the longitude required.
46S THE MATHEMATICAL PRINCIPLES [BooK III.
After the same method, from five observed latitudes, we may find the
latitude to a given time.
If the differences of the observed longitudes are small, suppose of 4 or 5
degrees, three or four observations will be sufficient to find a new longitude
and latitude : but if the differences are greater, as of 10 or 20 degrees, five
observations ought to be used.
LEMMA VII.
Through a given point P to draw a right line BC, rvhose parts PB, PC,
cut off by two right lines AB, AC, given in position, may be one to the
other in. a given proportion.
P
i\ From the given point P suppose any right line
PD to be drawn to either of the right lines given,
as AB; and produce the same towards AC, the
other given right line, as far as E; so as PE may
be to PD in the given proportion. Let EC be
parallel to A D. Draw CPB, and PC will be to PB
as PE to PD. Q.E.F.
LEMMA VIII.
Let ABC be a parabola, having its focus in S. By the chord AC bi
sected in I cut off the segment ABCI, ivhose diameter is Ip and vertex
I . In I/i produced take pO equal to one half of I//. Join OS, and
produce it to so as S may be equal to 2SO. Now, supposing a comet
to revolve in the arc CBA, draw B, cutting AC in E ; I say, the point
E will cut offfrom the chord AC the segment AE, nearly proportional
to the time.
For if we join EO, cutting the parabolic arc ABC in Y, and draw //X
touching the same arc in the vertex //, and meeting EO in X, the curvi
linear area AEXjuA will be to the curvilinear area ACY//A as AE to AC ;
and. therefore, since the triangle ASE is to the triangle ASC in the same
proportion, the whole area ASEXjuA will be to the whole area ASCY/^A as
BOOK II Lj OF NATURAL PHILOSOPHY. 469
AE to AC. But, because O is to SO as 3 to 1, and EG to XC in the same
proportion, SX will be parallel to EB ; and, therefore, joining BX, the tri
angle SEB will be equal to the
triangle XEB. Wherefore if to the area
ASEX.uA we add the triangle EXB, and from the sum subduct the triangle
SEB, there will remain the area ASBX,wA, equal to the area ASEX/^A. and
therefore in proportion to the area ASCY//A as AE to AC. But the area
ASBYwA is nearly equal to the area ASBX//A; and this area ASBY/zA
is to the area ASCYwA as the time of description of the arc AB to the
time of description of the whole arc AC ; and, therefore, AE is to AC
nearly in the proportion of the times. Q.E.D.
COR. When the point B falls upon the vertex \i of the parabola, AE is
to AC accurately in the proportion of the times.
SCHOLIUM.
If we join // cutting AC in d, and in it take //, in proportion to ^B as
27MI to 16Mf/, and draw B/?, this Bu will cut the chord AC, in the pro
portion of the times, more accurately than before; but the point n is to be
taken beyond or on this side the point , according as the point B is
more or less distant from the principal vertex of the parabola than the
point p.
LEMMA IX.
AI 2
The right lines Ip and /zM, and the length j~-, are equal among them
selves.
For 4.S/Z is the latus rectum of the parabola belonging to the vertex ft.
LEMMA X.
Produce Su to N and P, so as ^N may be one third of //I, and SP may
be to SN as SN to S" ; and in the time that a comet would describe
the arc AjuC. if it was supposed to move always forwards with the ve
locity which it hath in a height equal to SP, it would describe a length
equal to the chord AC.
For if the comet with the velocity
which it hath in \i was in the said time
supposed to move uniformly forward in
the right line which touches the parabola
in p, the area which it would describe by
a radius drawn to the point S would be
equal to the parabolic area ASC/zA ; and
therefore the space contained under the
length described in the tangent and the
length Su would be to the space contained under the lengths AC and SM as the
4/0 THE MATHEMATICAL PRINCIPLES [BOOK 111
area ASC//A to the triangle A SO, that is, as SN to SM. Wherefore AC
is to the length described in the tangent as Sf* to SN. But since the ve
locity of the comet in the height SP (by Cor. 6, Prop. XVI., Book I ) is to
the velocity of the same in the height Sfi in the reciprocal subduplicate
proportion of SP to Sft, that is, in the proportion of S/^ to SN, the length
described with this velocity will be to the length in the same time described
in the tangent as Su to SN. Wherefore since AC, and the length described
with this new velocity, are in the same proportion to the length described
in the tangent, they must be equal betwixt themselves. Q.E.D.
COR. Therefore a comet, with that velocity which it hath in the height
S/x + fI,, would in the same time describe the chord AC nearly.
LEMMA XI.
If a comet void of all motion was let fallfrom the heigJit SN, or $n +
J Ift, towards the sun, and was still impelled to the sun by the same
force uniformly continued by ivhich it was impelled at first, the same,
in, one half of that time in which it might describe the arc AC in its
own orbit, would in. descending describe a space equal to the fen gift
fa
For in the same time that the comet would require to describe the para
bolic arc AC, it would (by the last Lemma), with that velocity which it
hath in the height SP, describe the chord AC: and, therefore (by Cor. 7,
Prop. XVI, Book 1), if it was in the same time supposed to revolve by the
force of its own gravity in a circle whose semi- diameter was SP. it would
describe an arc of that circle, the length of which would be to the chord
of the parabolic arc AC in the subduplicate proportion of 1 to 2. Where
fore if with that weight, which in the height SP it hath towards the sun,
it should fall from that height towards the sun, it would (by Cor. 9,
Prop. XVI, Book 1) in half the said time describe a space equal to the
square of half the said chord applied to quadruple the height SP, that is,
AI 2
it would describe the space ,^p.
But since the weight of the comet
towards the sun in the height SN is to
the weight of the same towards the
sun in the height SP as SP to S^, the
comet, by the weight which it hath in
the height SN. in falling from that
height towards the sun, would in tin:
AI 2
same time describe the space 7^-; that
4S^
is, a space equa] to the length I// OT
wM. Q.E.D
BOOK III.] OF NATURAL PHILOSOPHY. 47
PROPOSITION XLL PROBLEM XXI.
Prom three observations given to determine the orbit of a comet moving
in a parabola.
This being a Problem of very great difficulty, I tried many methods of
resolving it
; and several of these Problems, the composition whereof I
have triven in the first Book, tended to this purpose. But afterwards I
contrived the following solution, which is something more simple.
Select three observations distant one from another by intervals of time
nearly equal ; but let that interval of time in which the comet moves
more slowly be somewhat greater than the other
; so, to wit, that the dif
ference of the times may be to the sum of the times as the sum of the
At
imes to about 600 days ; or that the point E may fall upon M nearly,
and may err therefrom rather towards 1 than towards A. If such direct
observations are not at hand, a new place of the comet must be found, by
Lem. VI.
Let S represent the sun ; T, t,
r
} three places of the earth in the orbis
mag-mis; TA, /B, rC, three observed longitudes of the comet; V the
time between the first observation and the second ; W the time between
the second and the third
; X the length which in the whole time V + W
the comet might describe with that velocity which it hath in the mean
distance of the earth from the sun, which length is to be found by Cor. 3,
THE MATHEMATICAL PRINCIPLES [BOOK III.
Prop. XL, Book III
;
and tV a perpendicular upon the chord TT. In the
mean observed longitude tfB take at pleasure the point B, for the place of
the comet in the plane of the ecliptic ; and from thence, towards the sun
S, draw the line BE, which may be to the perpendicular /V as the content
under SB and St 2 to the cube of the hypothenuse of the right angled tri
angle, whose sides are SB, and the tangent of the latitude of the comet in
the second observation to the radius ^B. And through the point E (by
Lemma VII) draw the right line AEC, whose parts AE and EC, terminat
ing in the right lines TA and rC. may be one to the other as the times V
and \V : then A and C will be nearly the places of the comet in the plane
of the ecliptic in the first and third observations, if B was its place
rightly assumed in the second.
Upon AC, bisected in I, erect the perpendicular li. Through B draw
