the obscure line Ei parallel to AC. Join the obscure line Si, cutting AC
in A, and complete the parallelogram il AJU. Take \o equal to 3IA ; and
through the sun S draw the obscure line<0 equal to 3So -f 3 fa. Then,
cancelling the letters A, E, C, I, from the point B towards the point ,
draw the new obscure line BE, which may be to the former BE in the
duplicate proportion of the distance BS to the quantity Sju + 1 fa. And
through the point E draw again the right line AEC by the same rule as
before ; that is, so as its parts AE and EC may be one to the other as the
times V and W between the observations. Thus A and C will be the
places of the comet more accurately.
Upon AC, bisected in I, erect the perpendiculars AM, CN, IO, of which
AM and CN may be the tangents of the latitudes in the first and third ob
servations, to the radii TA and TC. Join MN, cutting IO in O. Draw the
rectangular parallelogram zlAjt/, as before. In IA produced take ID equal to
Sfi + f fa. Then in MN, towards N, take MP, which may be to the
above found length X in the subduplicate proportion of the mean distance
of the earth from the sun (or of the semi-diameter of the orbis tnagnus]
to the distance OD. If the point P fall upon the point N; A, B, and C,
<vill be three places of the comet, through which its orbit is to be described
in the plane of the ecliptic. But if the point P falls not upon the point
N, in the right line AC take CG equal to NP, so as the points G and P
may lie on the same side of the line NC.
By the same method as the points E, A, C, G, were found from the as
sumed point B, from other points 6 and j3 assumed at pleasure, find out the
new points e, a, c, g ; and e, a, , y. Then through G, g-, and y, draw the
circumference of a circle G^y, cutting the right line rC in Z : and Z will
be one place of the comet in the plane of the ecliptic. And in AC, ac, OK,
making AF, a/, a</>, equal respectively to CG, eg, KJ ; through the points P,
f, and 0, draw the circumference of a circle
Vf<t>, cutting the right line AT
in X ; and the point X will be another place of the comet in the plane of
BOOK III.] OF NATURAL PHILOSOPHY. 4~3
the ecliptic. And at the points X and Z, erecting the tangents of the
latitudes of the comet to the radii TX and rZ, two places of the comet in
its own orbit will be determined. Lastly, if (by Prop. XIX., Book 1) to
the focus S a parabola is described passing through those two places, this
parabola will be the orbit of the comet. Q.E.L
The demonstration of this construction follows from the preceding Lem
mas, because the right line AC is cut in E in the proportion of the times,
by Lem. VI L, as it ought to be, by Lem. VIII.
; and BE, by Lem. XL, is a
portion of the right line BS or B in the plane of the ecliptic, intercepted
between the arc ABC and the chord AEC ; and MP (by Cor. Lem. X.) is
the length of the chord of that arc, which the comet should describe in its
proper orbit between the firs : and third observation, and therefore is equal
to MN, providing B is a true place of the comet in the plane of the
ecliptic.
But it will be convenient to assume the points B, b, (3, not at random,
but nearly true. If the angle AQ/, at which the projection of the orbit in
the plane of the ecliptic cuts the right line B, is rudely known, at that
angle with Bt draw the obscure line AC, which may be to -fTT in the subduplicate
proportion of SQ, to S/ ; and, drawing the right line SEB so as
its part EB may be equal to the length
, the point B will be determined,
which we are to use for the first time. Then, cancelling the right line
AC, and drawing anew AC according to the preceding construction, and,
aioreover, finding the length MP, in tB take the point b, by this rule, that,
if TA and rC intersect each other in Y, the distance Y6 may be to the
distance YB in a proportion compounded of the proportion of MP to MN,
and the subduplicate proportion of SB to Sb. And by the same method
you may find the third point 18, if you please to repeat the operation the
third time ; but if this method is followed, two operations generally will be
sufficient ; for if the distance Bb happens to be very small, after the points
F,/, and G, , are found, draw the right lines F/and G^-, and they will
cut TA and rC in the points required, X and Z.
EXAMPLE.
Let the comet of the year 1680 be proposed. The following table shews
the motion thereof, as observed by Flamsted, and calculated afterwards by
him from his observations, and corrected by Dr. Halley from the same ob
servations.
47.1 THE MATHEMATICAL PRINCIPLES FBooK III.
To these you may add some observations of mine.
These observations were made by a telescope of 7 feet, with a microme
ter and threads placed in the focus of the telescope; by Avhich instruments
we determined the positions both of the fixed stars among themselves, and
of the comet in respect of the fixed stars. Let A represent the star of the
fourth magnitude in the left heel of Perseus (Bayer s o), B the following
star of the third magnitude in the left foot (Bayer s s), C a star of the
sixth magnitude (Bayer s 11} in the heel of the same foot, and 1). E, F, G,
H, I, K. L, M, N, O, Z, a, j3, y, S, other smaller stars in the same foot;
and let p, P, Q, R, S, T, V, X, represent the places of the comet in the
observations above set down ; and, reckoning the distance AB of 80 r\ parts,
AC was 52i of those parts; BC, 5Sf ; AD, 57T\ ; BD, S2 T
"
T ; CD, 23f :
AE, 29i ; CE, 57i ; DE, 49J4 ; AI, 27 T\ ; BI, 52} ; OF, 36 rV ; Dl, 53/r ;
AK, 38| ; BK, 43; OK, 31$; FK, 29; FB, 23; FC, 36i ; AH, 1S| ;
DH, 50J; BN, 46 T\ ; ON, 31 1; BL, 45T\; NL, 31f HO was to HI
as 7 to 6, and. produced, did pass between the stars D and E, so as the
distance of the star D from this right line was jCD. LM was to LN as
2 to 9, and, produced, did pass through the star H. Thus were the posi
tions of the fixed stars determined in respect of one another.
3,K)K HI.] OF NATURAL PHILOSOPHY. 475
*2
Mr. Pound has since observed a second time the positions of thcst fixed
stars amongst themselves, and collected their longitudes and lat" /udes ac
cording to the following table
4^6 THE MATHEMATICAL PRINCIPLES [BOOK III.
The positions of the comet to these fixed stars were observed to be as
follow :
Friday, February 25, O.S. at 8ih
. P. M. the distance of the comet in p
from the star E wai less than T\AE, and greater than }AE, and therefore
nearly equal to T
3
S AE; and the angle AjoE was a little obtuse, but almost
right. For from A, letting fall a perpendicular on pE, the distance of the
comet from that perpendicular was j/E.
The same night, at 9|h
., the distance of the comet in P from the star E
was greater than AE, and less than AE, and therefore nearly equal
to
j^-
of AE, or /^ AE. But the distance of the comet from the perpen-
^8"
dicular let fall from the star A upon the right line PE was jPE.
Sunday, February 27, 8|
h
. P. M. the distance of the comet in Q, from
the star O was equal to the distance of the stars O and H and the risjht
line QO produced passed between the stars K and B. I could not, by
reason of intervening clouds, determine the position of the star to greater
accuracy.
Tuesday, March 1, ] l h
. P. M. the comet in R lay exactly in a line be
tween the stars K and C, so as the part CR of the right line CRK was a
little greater than CK, and a little less than JCK + jCR, and therefore = iCK + A CR, or ifCK.
Wednesday, March 2, S1
. P. M. the distance of the comet in S from the
star C was nearly FC ; the distance of the star F from the right line OS
produced was g^FC ; and the distance of the star B from the same right
line was five times greater than the distance of the star F ; and the right
line NS produced passed between the stars H and I five or six times nearer
to the star H than to the star I.
Saturday, March 5. lHh
. P. M. when the comet was in T, the right line
MT was equal to ^ML, and the right line LT produced passed between B
and F four or five times nearer to F than to B, cutting off from BF a fifth
or sixth part thereof towards F : and MT produced passed on the outside
of the space BF towards the star B four times nearer to the star B than
to the star F. M was a very small star, scarcely to be seen by the tele
scope; but the star L was greater, and of about the eighth magnitude.
Monday, March 7, Qih
. P. M. the comet being in V, the right line Va
produced did pass between B and F, cutting off, from BF towards F, T\ of
BF, and was to the right line Yj3 as 5 to 4. And the distance of the comet
from the right line a(3 was |V/3.
Wednesday, March 9, S|-
h
. P. M. the comet being in X, the right line
yX was equal to
jy<? ; and the perpendicular let fall from the star 6 upon
the right yX was f of yd.
The same night, at 12h
. the comet being in Y, the right line yY was
BOOK III.] OF NATURAL PHILOSOPHY. 477
equal to ^ of yd, or a little less, as perhaps T
5
g of yd ; and a perpendicular
let fall from the star 6 on the right line yY was equal to about or | yd.
But the comet being then extremely near the horizon, was scarcely discern
ible, and therefore its place could not be determined with that certainty as
in the foregoing observations.
Prom these observations, by constructions of figures and calculations, I
deduced the longitudes and latitudes of the comet ; and Mr. Pound, by
correcting the places of the fixed stars, hath determined more correctly the
places of the comet, which correct places are set down above. Though my
micrometer was none of the best, yet the errors in longitude and latitude
(as derived from my observations) scarcely exceed one minute. The comet
(according to my observations), about the end of its motion. besraD **> J;;oiine
sensibly towards the north, from the parallel which it described about the
end of February.
Now, in order to determine the orbit of the comet out of the observations
above described, I selected those three which Flamsted made, Dec. 21, Jan.
5, and Jan. 25; from which I found S^ of 9842,1 parts, and V of 455
such as the semi-diameter of the orbis magnus contains 10000. Then for
the first observation, assuming tE cf 5657 of those parts, 1 found SB 9747,
BE for the first time 412, Sji 9503, U 413, BE for the second time 421,
OD 10186, X 8528,4, PM 8450, MN 8475, NP 25; from whence, by the
second operation. I collected the distance tb 5640 ; and by this operation 1
at last deduced the distances TX 4775 and rZ 11322. From which, lim
iting the orbit, I found its descending node in 25, and ascending node in V?
1 53 ; the inclination of its plane to the plane of the ecliptic 61 20^ ,
the vertex thereof (or the perihelion of the comet) distant from the node
8 38 , and in t 27 43 , with latitude 7 34 south; its lotus return
236.8; and the diurnal area described by a radius drawn to the sun 93585,
supposing the square of the semi-diameter of the orbis magnus lOUOOOOOO ;
that the comet in this orbit moved directly according to the order of the
signs, and on DM. 8(1
. OO1
. 04 P. M was in the vertex or perihelion of its
orbit. All which I determined by scale and compass, and the chords of
angles, taken from the table of natural sines, in a pretty large figure, in
which, to wit, the radius of the orbis magnus (consisting of 10000 parts)
was equal to 16^ inches of an English foot.
Lastly, in order to discover whether the comet did truly move in the
orbit so determined, I investigated its places in this orbit partly by arith
metical operations, and partly by scale and compass, to the times of gome
of the observations, as may be seen in the following table :
478 THE MATHEMATICAL PRINCIPLES [BOOK III,
I
But afterwards Dr. Halley did determine the orbit to a greater accu
racy by an arithmetical calculus than could be done by linear descriptions :
and, retaining the place of the nodes in s and ^ 1 53 , and the inclina
tion of the plane of the orbit to the ecliptic 61 20| , as well as the time
of the comet s being in perihelio, Dec. 8(i
. OUh
. 04 , he found the distance
of the perihelion from the ascending node measured in the comet s orbit
9 20 , and the Ititus rectum of the parabola 2430 parts, supposing the
mean distance of the sun from the earth to be 100000 parts ;
arid from
these data, by an accurate arithmetical calculus, he computed the places
of the comet to the times of the observations as follows :
This comet also appeared in the November before, and at Coburg, in
Saxony, was observed by Mr. Gottfried Kirch, on the 4th of that month, on
the 6th and llth O. S.
; from its positions to the nearest fixed stars observed
with sufficient accuracy, sometimes with a two feet, and sometimes with a
ten feet telescope; from the difference of longitudes of Coburg and Lon
don, 11; and from the places of the fixed stars observed by Mr. Pound,
Dr. Halley has determined the places of the comet as follows :
BOOK III.] OF NATURAL PHILOSOPHY. 479
Nov. 3, 17h
. 2 , apparent time fit London, the comet was in 71 29 deg.
51 , with 1 deg. 17 45" latitude north.
November 5. 15h
. 58 the comet was in ^ 3 23 , with 1 6 nortl lat.
November 10, 16h
. 31 , the comet was equally distant from two stars in
1, which are <r and T in Bayer ; but it had not quite touched the right
line that joins them, but was very little distant from it. In Flamstecfs
catalogue this star o was then in ^ 14 15 , with 1 deg. 41 lat. north
nearly, and r in W 17 3^ with deg. 34 lat. south; and the middle
point between those stars was lr
JZ 15 39} , with 33i lat. north. Let
the distance of the cornet from that right line be about 10 or 12 : and
the difference of the longitude of the comet and that middle point will be
7 ;
arid the difference of the latitude nearly 7\ ; and thence it follows
that the comet was in T
02 15 32 , with about 26 lat. north.
The first observation from the position of the comet with respect tr
certain small fixed stars had all the exactness that could be desired ; UK
second also was accurate enough. In the third observation, which was the
least accurate, there might be an error of 6 or 7 minutes, but hardly
greater. The longitude of the comet, as found in the first and most
accurate observation, being computed in the aforesaid parabolic orbit,
comes out U 29 30 22", its latitude north 1 25 7", and its distance
from the sun 115546.
Moreover, Dr. Halley, observing that a remarkable comet had appeared
four times at equal intervals of 575 years (that is, in the month of Sep
tember after Julius Ccesar was killed ; An. Chr. 531, in the consulate of
Lainpadins and Orestes; An. Chr. 1106, in the month of February ;
and at the end of the year 16SO; and that with a long and remarkable
tail, except when it was seen after C(BsaiJs death, at which time, by reason
of the inconvenient, situation of the earth, the tail was not so conspicuous),
set himself to find out an elliptic orbit whose greater axis should be
1382957 parts, the mean distance of the earth from the sun containing
10000 such ;
in which orbit a comet might revolve in 575 years ; and,
placing the ascending node in 25 2 2 , the inclination of the plane of the
orbit to the plane of the ecliptic in an angle of 61 6 48", the perihelion
of the comet in this plane in t 22 44 25", the equal time of the perihe
lion December 7 1
. 23h
. 9 , the distance of the perihelion from the ascend
ing node in the plane of the ecliptic 9^ 17 35", and its conjugate axis
18481,2, he computed the motions of the comet in this elliptic orbit. The
places of the comet, as deduced from the observations, and as arising from
computation made in this orbit, may be seen in the following table.
480 THE MATHEMATICAL PRINCIPLES [BOOK 111
The observations of this comet from the beginning to the end agree at
porfectly with the motion of the comet in the orbit just now described as
the motions of the planets do with the theories from whence they are cal
culated ; and by this agreement plainly evince that it was one and the
same comet that appeared all that time, and also that the orbit of that
comet is here rightly defined.
In the foregoing table we have omitted the observations of Nov. 16,
18, 20. and 23, as not sufficiently accurate, for at those times several per
sons had observed the comet. Nov. 17, O. S. Ponthczns and his compan
ions, at 6h
. in the morning at Rome (that is, 5h
. 10 at London], by threads
directed to the fixed stars, observed the comet in === 8 30 , with latitude
40 south. Their observations may be seen in a treatise which Ponthc&us
published concerning this comet. Celliits, who was present, and commu
nicated his observations in a letter to Cassitn} saw the comet at the same
hour in ^= 8 30 , with latitude 30 south. It was likewise seen by
Galletius at the same hour at Avignon (that is, at 5h
. 42 morning at
London] in ^= 8 without latitude. But by the theory the comet was at
that time in ^ 8 16 45", and its latitude was 53 7" south.
Nov. 18, at 6h
. 30 in the morning at Rome (that is, at 5h
. 40 at Lon
don), PonthcEns observed the comet in ^ 13 30 , with latitude 1 20
BOOK III.] OF NATURAL PHII OSOPHY. 48l
south ; and Cellius in ^ 13 30 , with latitude 1 00 south. But at 5b
.
30 in the morning at Aviation, Galletius saw it in ^ 13 00 , with lati
tude 1 00 south. In the University of La Fleche, in Prance, at 5h
. in
the morning (that is. at 5h
. 9 at London.}, it was seen by P. Ango, in the
middle between two small stars, one of which is the middle of the three
which lie in a right line in the southern hand of Virgo, Bayers i/> ; and
the other is the outmost of the wing, Bayer s 0. Whence the comet was
then in ^ 12 46 with latitude 50 south. And I was informed by Dr.
ffalley, that on the same day at Boston in New England, in the latitude
of 42| deg. at 5h
. in the morning (that is, at 9h
. 44 in the morning at
London), the comet was seen near === 14, with latitude 1 30 south.
Nov. 19, at 4|
h
. at Cambridge, the comet (by the observation of a
young man) was distant from Spica $ about 2 towards the north west.
Now the spike was at that time in ^ 19 23 47", with latitude 2 1 59"
south. The same day, at 5h
. in the morning, at Boston in New England,
the comet wTas distant from Spica nj? 1, with the difference of 40 in lati
tude. The same day, in the island of Jamaica, it was about 1 distant
from Spica W. The same day, Mr. Arthur Storer, at the river Patuxent,
near Hunting Creek, in Maryland, in the confines of Virginia, in lat.
38i, at 5 in the morning (that is, at 10h
. at London), saw the comet
above Spica W, and very nearly joined with it, the distance between them
being about of one deg. And from these observations compared, I con
clude, that at 9h
. 44 at London, the comet was in === 18 50 , with about
1 25 latitude south. Now by the theory the comet was at that time in
^ 18 52 15", with 1 26 54" lat. south.
Nov. 20, Montenari, professor of astronomy at Padua, at 6h
. in the
morning at Venice (that is, 5h
. 10 at London), saw the comet in === 23,
with latitude 1 30 south. The same day, at Boston, it was distant from
Spica W by about 4 of longitude east, and therefore was in ^ 23 24
nearly.
Nov. 21, Ponthceus and his companions, at 7|
h
. in the morning, ob
served the comet in == 27 50 , with latitude 1 16 south ; Cellius, in ^=
28 ; P. Ango at 5h
. in the morning, in === 27 45 ; Montenari in ^
27 51 . The same day, in the island of Jamaica, it was seen near the
beginning of ^1, and of about the same latitude with Spica u%, that is, 2
2 . The same day, at 5h
. morning, at Ballasore, in the East Indies (that
is, at ll h
. 20 of the night preceding at London), the distance of the
comet from Spica W was taken 7 35 to the east. It was in a right line
between the spike and the balance, and therefore was then in == 26 58 ,
with about 1 11 lat. south; and after 5h
. 40 (that is. at 5h
. morning at
London), it was in === 28 12 . with 1 16 lat. south. Now by the theory
the comet was then in *= 28 10 36", with 1 53 35" lat. south.
Nov. 22, the comet was seen by Montenari in ^ 2 33 : hut at Boston
31
482 THE MATHEMATICAL PRINCIPLES [BOOK 111.
in New England, it was found in about ^l 3, and with almost the same
latitude as before, that is, 1 30 . The same day, at 5h
. morning at
Ballasore, the comet was observed in ^l 1 50 ; and therefore at 5h
. morn
ing at London, the comet was in iU 3 5 nearly. The same day, at 6^h
.
in the morning at London, Dr. Hook observed it in about nt 3 30 , and
that in the right line which passeth through Spica ^ and Cor Leonis ;
not, indeed, exactly, but deviating a little from that line towards the
north. Montenari likewise observed, that this day, and some days after,
a right line drawn from the comet through Spica passed by the south
side of Cor Lt>oi\is at a very small distance therefrom. The right line
through Cor Leonis and Spica ^ did cut the ecliptic in ^ 3 46 at an
ano-le of 2 51 ; and if the comet had been in this line and in W. 3, its
latitude would have been 2 26 ; but since Hook and Montenari agree
that the comet was at some small distance from this line towards the
north, its latitude must have been something less. On the 20th, by the
observation of Montenari, its latitude was almost the same with that of
Spica ^l7
, that is, about 1 30 . But by the agreement of Hook, Montenari,
and Align, the latitude was continually increasing, and therefore
must now, on the 22ci be sensibly greater than t 30 : and, taking a
mean between the extreme limits but now stated. 2 26 and 1 30 , the
latitude will be about 1 58 . Hook and Montenari agree that the tail
of the comet was "directed towards Spica W, declining a little from that
star towards the south according to Hook, but towards the north according
to Montenari ; and, therefore, that declination was scarcely sensible
; and
