Principia Mathematica (Motte Translation, 1848) — part 36 of 45

Cor. 1. Hence if comets are revolved in orbits retm-ning 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

466

THE MATHEMATICAL PrvINCIPLES

[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. XYI, 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 sub- duplicate 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 magmis, or the greatest semi- diameter 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 71675-^- 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 ^/ 2 to 1, would by its diurnal motion describe 2432747 parts, and 101364| 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 subdu- plicate proportion of the distances, and is therefore given.

Cor. 4. Wherefore if the latus recAiim of the parabola is quadruple of the radius of the orbis ??iagmis, 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 1216373|- parts, and the horary area will be 50682^ parts. But, if the latiis 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.

To find a curve line of the parabolic kind ivIiicJi s kail pass through any given number of points. Let those points be A, B, C, D, E, F, &c., and from the same to any right line HN, given in position, let fall as many perpendiculars AH, BI, CK, DL, EM, FN, c^c.

b 2b 3b 4:b 5b

JB H

c 2c 3c 4c d 2d 3d

e 2e

f

Case 1. If HI, IK, KL, &c., the intervals of the points H, I, K, L, M, N, &c., are equal, take b, 2b, 3b, 46, 5b, &c., the first diiferences of the per- pendiculars AH, BI, CK, <fcc. ; their second differences c, 2c, 3c, 4.c, (fee. ; their third, d, 2d, 3d, &c., that is to say. so as AH — BI may be = b, BI

Book III.] of natural philosophy. 467

— . CK = 2b, CK — DL = 3b, DL + EM = U, — EM + FN = 55,

&c. ; then b — 2b = c, &.G., 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 ordinate, suppose the intervals HI, IK, KL, LM, 6cc., to be units, and let AH = a, — HS == J), Ijo into — IS = q, ^q into + SK = r, jr into + SL == 5, Is into + SM = t ; proceeding, to wit, to ME, the last perpendicular but one, and prefixing negative signs before the terms HS, IS, (fee, which lie from S towards A ; and affirmative signs before the terms SK, SL, &c., which lie on the other side of the point S ; and, observing well the signs, RS will he = a + bp + cq + dr -- es + ft, + (fee.

Case 2. But if HI, IK, ifec, the intervals of the points H, I, K, L, «fee, are unequal, take b, 2b, 3b, Ab, 56, (fee, the first differences of the perpen- diculars AH, BI, CK, (fee, divided by the intervals between those perpen- diculars ; c, 2c, 3c, 4c, (fee, their second differences, divided by the intervals between every two ; d, 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 = jjj — j ^^ = fir — ^ ^" "^ tFi -' ^^' *"^^

b-~2b ^ 2b-~3b ^ 35-46 '^ - 2c

c = -jj^, 2c == -^j--, 3c = -^- (fee, then d == ~^—, 2d

2c 3c

= — yTz — , iScc. 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 -h SM = t ; proceeding, to wit, to ME, the last perpendicular but one ; and the ordinate RS will he = a + bp --'cq -- dr -- es + ft, + (fee

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 will 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 YL

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 ABCDE 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 BO, whose parts PB, PC, cut off hy two right lines AB, AC, given in position, may he one to the other in a given proportion.

E

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. I^et EC be parallel to AD. Draw CPB, and PC will be to PB as PE to PD. Q.E.F,

• LEMMA VIIL

Let ABC be a parabola, having its focus in S. By the chord AC bi- sected in I cut off the segment ABCI, lohose diameter is Ijtx and vertex pt. In I// produced take /iO equal to one half of Ifi. Join OS, and produce it to ^, so as S^ may be equal to 2S0. Noio, supposing a comet to revolve in the arc OBA, draw ^B, cutting AC in E ; I say, the point E will cut off from 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 ^i, and meeting EO in X, the curvi- linear area AEXjwA 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 ASEX//A will be to the whole area ASC Y^A as

Book III.] of natural philosophy. 469

AE to AC. But, because |0 is to SO as 3 to 1, and EO to XO 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 ASEXuA we add the triangle EXB, and from the sum subduct the triangle SEB, there will remain the area ASBX^wA, equal to the area ASEXjuA, and therefore in proportion to the area ASCY//A as AE to AC. But the area ASBYftA is nearly equal to the area ASBXftA; and this area ASBY/iA is to the area ASCYf^A 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 jW of the parabola, AE is to AC accurately in the proportion of the times.

SCHOLIUM.

If we join jw^ cutting AC in ^, and in it take ^n in proportion to ^B as 27MI to WMfi, and draw Bn, this Bn will cut the chord AC, in the pro- portion of the times, more accurately than before; but the point ?i 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 jf^.

LEMMA IX.

4S/^-

AP

The right lines Ifi and ftM, and the length j^, are equal a^nong them-

selves. For 4S|U is the latus rectum of the pa^rabola belonging to the vertex ^.

LEMMA X.

Produce Sa to N ajid P, so as |uN maij be one third of fjj, ajid SP may be to SN as SN to S/^ ; and in the time that a comet ivoidd describe the arc AjwC, if it was supposed to move alwaijs forwards loith 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 /^ was in the said time supposed to move uniformly forward in the right line which touches the parabola in fi, the area which it would describe by a radius drawn to the point S would be equal to the parabolic area ASCf^A ; and therefore the space contained under the lengtb described in the tangent and the length Sju would be to the space contained under the lengths AC and SM as the

470 THE MATHEMATICAL PRINCIPLES [BoOK III.

area ASCjuA to the triangle ASC, that is, as SN to SM. Wherefore AC is to the length described in the tangent as Sjt^ 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 S/z in the reciprocal subduplicate proportion of SP to Sfi, 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/i + flju, would in the same time describe the chord AC nearly.

LEMMA XL

If a comet void of all motion was let fall froin the height SN, or Sfi -f

llfi, towards the sun, and was still impelled to the sun by the same

force uniformity continued by lohich it was impelled at first, the same^

in one hxdf of that time in which it m^ight describe the arc AC in its

own orbit, %ooidd in descending describe a space equal to the length

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,

AP

it would describe the space -t^. But since the weight of the comet

towards the sun in the height SN is to the weigrht of the same towards the sun in the height SP as SP to S/i, the comet, by thei^eight which it hath in the height SN, in falling from that height towards the sun, would in the

AI^

same time describe the space j^ ; that

is, a space equal to the length Ijw or /iM. aE.D.

Book III.

OF NATURAL PHILOSOPHY.

471

PROPOSITION XLI. PROBLEM XXL

Prom three observations given to determine the orbit of a comet movitig

in a parabola.

This being a Problem of very great difficulty, I tried many methods of resolving it ; and several of those Problems, the composition whereof I have given 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

times to about 600 days ; or that the point E may fall upon M nearly, and may err therefrom rather towards I 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 magnus ; TA, i^B, rC, three observed longitudes of the comet; Y 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,

472 THE MATHEMATICAL PRINCIPLES [BoOK III.

Prop. XL, Book III ; and iY a perpendicular upon the chord Tr. In the mean observed longitude ^B 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 S^^ 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 Txl and tC; may be one to the other as the times V and W : 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 the obscure line Bi parallel to AC. Join the obscure line Si, cutting AC in Aj and complete the parallelogram il A^. Take \a equal to 31/1 ; and through the sun S draw the obscure line o^ equal to 3Sc7 + 3 ih Then, cancelling the letters A, E, C, I, from the point B towards the point Ij draw the new obscure line BE, which may be to the former BE in the duplicate proportion of the distance BS to the quantity Sjfx 4- ^ i^- 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 Y 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, 10, of which AM and CN may be the tangents of the latitudes in the first and third ob- servations, to the radii TA and rC. Join MN, cutting 10 in O. Draw the rectangular parallelogram iIA,a, as before. In lA produced take ID equal to S/^ + f a. 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 magmis) to the distance OD. K the point P fall upon the point N; A, B, and C, will 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 b and f3 assumed at pleasure, find out the new points e, a, c, g / and e, a, k, y. Then through G, g, and y, draw the circumference of a circle Ogy, 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, uk, taking AF, af, a(p, equal respectively to CG, eg, Ky ; through the points F, /, and (p, draw the circumference of a circle Ff(f>, cutting the right line AT in X ; and the point X will be another plaoe of the comet in the plane of

Book III.] of natural philosophy. 473

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.I.

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. VIC, as it ought to be, by Lem. VIII. ; and BE, by Lem. XL, is a portion of the right line BS or Bs 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 firsb 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, h, fi, not at random, but nearly true. If the angle AQ^t, 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 B^ draw the obscure line AC, which may be to |Tt in the sub- duplicate proportion of SQ, to S^ ; and, drawing the right line SEB so as its part EB may be equal to the length V^, 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, moreover, finding the length MP, in ^B take the point b, by this rule, that, if TA and tC 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 13, 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, g, are found, draw the right lines Ff and Gg, and they will cut TA and tC 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.

474

THE MATHEMATICAL PRINCIPLES

[Book III.

Time.

Comet's

Sun's

Appar.

True.

Longitude.

Longitude.

Lat. N.

h. "

h. ' "

O 1 II

o / //

o / //

1680,

Dec. 12

4.46

4.46. 0

V9 1.51.23

V?

6.32.30

8.28. 0

21

6.32i

6.36.59

11.06.44

^

5.08.12

21.42.13

24

6.12

6.17.52

14.09.26

18.49.23

25.23. 5

26

5.14

5 20.44

16.09.22

28.24.13

27.00.52

29

7.55

8.03.02

19.19.43

X

13.10.41

28.09.58

30

8.02

8.10.26

20.21.09

17.38.20

28.11.53

1681,

Jan. 5

5.51

6.01.38

26.22.18

T

8.48.53

26.15. 7

9

6.49

7.00.53

^ 0.29.02

18.44.04

24.11.56

10

5.54

6.06.10

1.27.43

20.40.50

23.43.52

13

6.56

7.08.55

4.33.20

25.59.48

22.17.28

25

7.44

7.58.42

16.45.36

«

9.35. 0

17.56.30

30

8.07

8.21.53

21.49.58

13.19.51

16.42.18

Feb. 2

6.20

6.34.51

24.46.59

15.13.53

16.04. 1

5

6.50

7.04.41

27.49.51

16.59.06

15.27. 3

To these you may add some observations of mine.

Ap. Time

Comet's 1

i

Longitude.

Lat. N.

h. '

o / /;

o / n

1681, Feb. 25

8.30

y 26.18.35

12.46.46

27

8.15

27.04.30

12.36.12

Mar. 1

11. 0

27.52.42

12.23.40

2

8. 0

28.12.48

12.19.38

5

11.30

29.18. 0

12.03.16

7

9.30

n 0. 4. 0

11.57. 0

9

8.30

0.43. 4

11.45.52

These observations were made by a telescope of 7 feet, with a microme- ter and threads placed in the focus of the telescope ; by which 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 [Bayefs o), B the following star of the third magnitude in the left foot [Bayer^s ^), C a star of the sixth magnitude [Bayefs n) in the heel of the same foot, and D, E, F, G, H, I, K, L, M, N, O, Z, a, 3, y, 6^ 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 /^ parts, AC was 521 of those parts; BC, 58| ; AD, B7^] BD, 82yV; CD, 23|; AE, 291 ; CE, 57i ; DE, 49ii ; AI, 27 ^\ ; BI, 52i ; CT, 36/^ ; DI, 53^ ; AK, 38f ; BK, 43; CK, 3H; FK, 29; FB, 23; FC, 36i ; AH, 18|;

BL, 45tV; NL, 31f HO was to HI

DH, 50|; BN, 46/^;

CN,

3U;

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 |CD. 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.

Book III.]

or NATURAL PHILOSOPHY.

475

^

l<*^

N»

Mr. Pound has since observed a second time the positions of these fixed stars amongst themselves, and collected their longitudes and latitudes ac- cording to the following table.

The

The

fixed

Their

Latitude

fixed

Their

Latitude

stars.

Longiludes

North.

stars.

Longitudes

North.

o 1 II

O / II

O f //

o , r

A

b 26.41.50

12. 8.36

L

y 29.33.34

12. 7.48

B

28.40.23

11.17.54

M

29.18.54

12. 7.20

C

27.58.30

12.40.25

N

28.48.29

12.31. 9

E

26.27.17

12.52. 7

Z

29.44.48

11.57.13

F

28.28.37

11.52.22

a

29.52. 3

11.55.48

G

26.56. 8

12. 4.58

a

n 0. 8.23

11.48.56

H

27.11.45

12. 2. 1

y

0.40.10

11.55.18

I

27.25. 2

11.53.11

6

1. 3.20

11.30.42

K

27.42. 7

11.53.26

4T6 THE MATHEMAtlCAL PRINCIPLES [BoOK III.

The positions of the comet to these fixed stars were observed to be as follow :

Friday, February 25, O.S. at S^\ P. M. the distance of the comet in p from the star E was less than j^AE, and greater than |AE, and therefore nearly equal to f^AE ; and the angle Ajt?E was a little obtuse, but almost right. For from A, letting fall a perpendicular on jdE, the distance of the comet from that perpendicular was |joE.

The same night, at 9^, the distance of the comet in P from the star E

was greater than jy AE, and less than — AE, and therefore nearly equal

to 27 of AE, or /g-AE. But the distance of the comet from the perpen-

dicular let fall from the star A upon the right line PE was |PE.

Sunday, February 27, S\ 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 right line 0.0 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, UK 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 ^CK + |CR, and therefore

CK + j\ CR, or i|CK.

Wednesday, March 2, S\ 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 CS produced was aV^'C ; 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, ll|h. 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, 9^K P. M. the comet being in Y, the right line Ya produced did pass between B and F, cutting off, from BF towards F, y*^ of BF, and was to the right line Y(3 as 5 to 4. And the distance of the comet from the right line aj3 was |-V/3.

Wednesday, March 9, 8^i\ P. M. the comet being in X, the right line yX was equal to lyd ; and the perpendicular let fall from the star d upon the right yX was | of yd.

The same night, at 12'^ the comet being in Y, the right line yY was

Book III.] of natural philosophy. 47'?'

equal to i of yd, or a little less, as perhaps y\ 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 foreo-oino- observations.

From 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 plaoes 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, began to decline sensibly towards the north, from the parallel which it described about the end of Fehntary.

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 Y^ of 455, such as the semi-diameter of the or bis magnus contains 10000. l^hen for the first observation, assuming fB of 5^67 of those parts, I found SB 9747, BE for the first time 412, S/z 9503, il 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 th 5640 ; and by this operation I 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 \S 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 latus rectum 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 100000000 ; that the comet in this orbit moved directly according to the order of the signs, and on Dec. 8'^. 00\ 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 16i 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 some of the observations, as may be seen in the following table : —

4TS

THE MATHEMATICAL PRINCIPLES

[Book 111.

The Comet's

Dist.

from sun.

jLatitud. Longitude compu- computed.! ted.

Dec. 12 2792

29 8403

Feb. 516669

Mar. 5I21737

V^ 6^32' X 13 .I3i « 17.00"' 29 .19^

8M81 28. 00" 15. 29| 12. 4

Longitude observed.

V5^ 6°.3U

y< 13 .ii|

b 16 .591

29 .20f

Latitude observed

8°.26 28 .10^- 15 .271 12. 3i

Dif Lo.

• 1 +2 +0 -1

Dif.

Lat.

• 7i

• 21

• 4 
  

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 25 and V? 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^. 00^ 04', he found the distance of the perihelion from the ascending node measured in the comet's orbit 9° 20', and the latus rectum of the parabola 2430 parts, supposing the mean distance of the sun from the earth to be 100000 parts ; and from these data^ by an accurate arithmetical calculus, he computed the places of the comet to the times of the observations as follows : —

The Cornel's | I

Dist from Longitude

Latitude

Errors in

True time.

the sun.

computed.

computed.

Long.

Lat.

d.

h. ' "

w / //

0 r II

/ //

r II

Dec.

4.46.

28028

\S 6.29.25

8.26. Obor.

— 3. 5

— 2. 0

6.37.

61076

^ 5. 6.30

21.43.20

— 1.42

• 1. 7

6.18.

70008

18.48.20

25.22.40

— 1. 3

— 0.25

5.20.

75576

28.22.45

27. 1.36

— 1.28

• 0.44

84021

X 13.12.40

28.10.10

• 1.59

• 0.12

8.10.

86661

17.40. 5

28.11.20

• 1.45

— 0.33

Jan.

6. l.h

101440

^ 8.49.49

26.15.15

• 0.56

• 0. 8

110959

18.44.36

24.12.54

• 0.32

• 0.58

113162

20.41. 0

23.44.10

• 0.10

• 0.18

120000

26. 0.21

22.17.30

• 0.33

• 0. 2

7.59.

145370

« 9.33.40

17.57.55

— 1.20

• 1.25

8.22.

155303

13.17.41

16.42. 7

— 2.10

— 0.11

Feb.

6.35.

160951

15.11.11

16. 4.15

— 2.42

• 0.14

6. 4.i

166686

16.58.55

15.29.13

— 0.41

• 2. 0

8.41.

202570

26.15.46

12.48. 0

— 2.49

• 1.10

Mir.

5.11.39.

216205

29.18.35

12. 5.40

• 0.35

• 2.14

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 11th 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. Pounds Dr. Halley has determined the places of the comet as follows : —

Book III.] of natural philosophy. 479

Nov. 3, 17^. 2', apparent time at London^ the comet was in bi 29 deg. 51', with 1 deg. 17' 45" latitude north.

November 5, W\ 58' the comet was in ^ 3° 23', with 1° 6' north lat.

November 10, 16^. 31', the comet was equally distant from two stars in ,U, which are o and t in Bayer ; but it had not quite touched the right line that joins them, but was very little distant from it. In Flamsted's catalogue this star a was then in W 14^ 15', with 1 deg. 41' lat. north nearly, and r in W 17° 3^' with 0 deg. 34' lat. south ; and the middle point between those stars was ^15° 39|', with 0° 33|^' lat. north. Let the distance of the comet 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' ; and the diiFerence of the latitude nearly 7|' ; and thence it follows that the comet was in W 15° 32', with about 26' lat. north.

The first observation from the position of the comet with respect to certain small fixed stars had all the exactness that could be desired ; the 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 ^l 29° 30' 22", its latitude north 1° 25' 7", and its distance from the sun 115546.

Moreover, Dr. Hallei/, observing that a remarkable comet had appeared four times at equal intervals of 575 years (that is, in the month of Sep- tember after Jidiiis Ccesar was killed ; An. Chr. 531, in the consulate of Lampadius and Orestes; An. Chr. 1106, in the month of February ; and at the end of the year 1680 ; and that with a long and remarkable tail, except when it was seen after Ccesafs 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 2b 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^. 23^. 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 III.