Principia Mathematica (Motte Translation, 1848) — part 38 of 45

in the extremity of the tail, had begun to rise from the head before De- cember U, and therefore had spent in its whole ascent 45 days ; but that the whole tail which appeared on December 10 had finished its ascent in the space of the two days then elapsed from the time of the comet's being in its perihelion. The vapour, therefore, about the beginning and in the neighbourhood of the sun rose with the greatest velocity, and afterwards continued to ascend with a motion constantly retarded by its own gravity ; and the higher it ascended, the more it added to the length of the tail ; and while the tail continued to be seen, it was made up of almost all that vapour which had risen since the time of the comet's being in its perihe- lion ; nor did that part of the vapour which had risen first, and which formed the extremity of the tail, cease to appear, till its too great dis- tance, as well from the sun, from which it received its light, as from our eyes, rendered it invisible. Whence also it is that the tails of other comets which are short do not rise from their heads with a swift and continued motion, and soon after disappear, but are permanent and lasting columns of vapours and exhalations, which, ascending from the heads with a slow motion 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. Prom whence 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 comets' tails, maintain their rapid motions with great freedom, and for an exceeding long time.

Keyler 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 ; and why may not the tail of a comet rise from the sun after the same man-

492 THE MATHEMATICAL PRINCIPLES [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 be- cause, 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 of 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 common 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, that

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 smallest but the most subtle and useful part of our air, and 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 Hevelms 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 comet 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 Virginis ; and at that time it shone with greater brightness than it did afterwards. And Montenari, November 20, St. 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 ^efore sun-rising j and, conjecturing at the cause of the appearance, he judged it to proceed

494 THE MATHEMATICAL PRINCIPLES [BoOK III.

from there being a greater quantity of matter in the head at first, which was afterwards gradually spent.

And, whicb farther makes for the same purpose, I find, that the heads of other comets, which did put forth tails of the greatest bulk and splendor, have appeared but obscure and small. For in Brazil, March 5, ] 668, 7^ P. M., >S'^. N. P. Vale?itinus Estancius 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 ; and while the splendor was decreasing, the bulk of the tail increased : whence in Portugal it is said to have taken up one quarter of the heavens, that is, 45 degrees, extending from west to east with a very notable splendor, though the whole tail was not seen in those parts, because the head was always hid under the horizon : and from the increase of the bulk and decrease of the splendor of the tail, it appears that the head was 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 1106, the star whereof was small and obscure (as that of 1680), hut the splendour of its tail luas very bright, and like a huge fiery beam stretched out in a direc- tion betioeen 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; from whence, and from the position of the tail, we infer that the head was near the sun. Mattheio Paris says, It 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 loas 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) loas now diminished (the head of), the comet recovered its native brightness ; arid the splendour (of its tail) reached noio to a third part of the heavens (that is, to 60°). This appearance was in the ivinter season (an. 4, Olymp. 101), and, rising to 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 ih^ 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 Yenus, 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 are 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, I leave them to be determined by comparing comets together which 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 comefs 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 second 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, D 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. XVL, 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.

Oper. 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 thenij and the two areas of the same described between the observation, which call 6 and £ ; and let r be the whole time in which the whole area (J + e 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 (5 to e ; 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, = mG — mg

• nG — ny ; and 2T — 2S = mT — mi + nT — 7ir. 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 + nOi will be the true inclination of the plane of the trajectory to the plane of the ecliptic, and K -f- 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 y"? y T'

the transverse diameters of the same, then R + mr — mR + wp — liK

will be the true parameter, and ^ — ■ -, r — ; -. =r- will be the

^ Li -- 'ml — mJL + wA — wL

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 fhrnnornena^ 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 Hevelkis^ 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° 18' 40" ; the dis- tance of its perihelion from the node, estimated in the comet's orbit, 49° 27' 30", its perihelion in ^ 8° 40' 30", with heliocentric latitude south

OF NATURAL PHILOSOPHY.

497

Book III.]

16° 01' 45" ; the comet to have been in its perihelion November 24^. llu. 52' P.M. equal time at London, or 13^ 8' at Dantzick, O. S. ; and that the latus rectum of the parabola was 410286 such parts as the sun's mean distance from the earth is supposed to contain 100000. 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,

Appar. Time at Dantzick.

The observed Distances of the Comet from

1 The Places The observed Places. 1 computed in \ the Orb.

December d. h. ' 3.18.29ii

O 1 II

The Lion's heart 46.24.20 The Virgin's spike 22.52.10

O 1 II

Long. ^ 7.01.00 Lat. S. 21.39. 0

O 1 II

r- 7. 1.29 21.38.50

4.18. \h

The Lion's heart 46. 2.45 The Virgin's spike 23.52.40

Long. :i^ 6.15. 0 Lat. S. 22.24. 0

^ 6.16. 5 22.24. 0

7.17.48

The Lion's heart 44.48. 0 The Virgin's spike 27.56.40

Long. -- 3. 6. 0 Lat. S. 25.22. 0

-^ 3. 7.33 25.21.40

17.14.43

The Lion's heart 53.15.15 Orion's right shoulder 45.43.30

Long, a 2.56. 0 Lat. S. 49.25. 0

a 2.56. 0 49.25. 0

19. 9.25

Procyon 35.13.50 Bright star of Whale's jaw 52.56. 0

Long, n 28.40.30 Lat. S. 45.48. 0

n 28.43. 0 45.46. 0

20. 9.53i^

Procyon 40.49. 0 Bright star of Whale's jaw 40.04. 0

Long, n 13.03. 0 Lat. S. 39.54. 0

n 13. 5. 0 39.53. 0

21. 9. H

. Orion's right shoulder 26.21.25 Brightstarof Whale's jaw 29.28. 0

Long, n 2.16. 0 Lat. S. 33.41. 0

n 2.18.30 33.39.40

22. 9. 0

Orion's right shoulder 29.47. 0 Bright star of Whale's jaw 20.29.30

Long. fc( 24.24. 0 Lat. S. 27.45. 0

b 24.27. 0 27.46. 0

26. 7.58

The bright star of Aries 23.20.0 Aldebaran 26.44. 0

Long, b 9. 0. 0 La^ S. 12.36. 0

y 9. 2.28 12.34.13

27. 6.45

The bright star of Aries 20.45. 0 Aldebaran 28.10. 0

Long. « 7. 5.40 Lat. S. 10.23. 0

b 7. 8.45 10.23.13

28. 7.39

The bright star of Aries 18.29.0 Palilicium 29.37. 0

Long. « 5.24.45 , Lat. S. 8.22.50

b 5.27.52 8.23.37

31. 6.45

Andromeda's girdle 30.48.10 Palilicium 32.53.30

Long, b 2. 7.40 Lat. S. 4.13. 0

« 2. 8.20 4.16.25

Jan. 1665 7. 7.37^

Andromeda's girdle 25.11.0 Palilicium 37.12.25

Long. T 28.24.47 Lat. N. 0.54. 0

T 28.24. 0 0.53. 0

13. 7. 0

Andromeda's head 28. 7.10 Palilicium 38.55.20

Long. T 27. 6.54 Lat. N. 3. 6.50

T 27. 6.39 3. 7.40

24. 7.29

Andromeda's girdle 20.32.15 Palilicium 40. 5. 0

Long. T 26.29.15 Lat. N. 5.25.50

T 26.28.50 5.26. 0

Feb. 7. 8.37

Long. T 27. 4.46 Lat. N. 7. 3.29

T 27.24.55 7. 3.15

22. 8.46

Long. T 28.29.46 Lat. N. 8.12.36

T 28.29.58 8.10.25

March.

1. 8.16

Long. T 29.18 15 Lat. N. 8.36.26

T 29.18.20 8.36.12

7. 8.37

Long, b 0. 2.48 Lat. N. 8.56.30

b 0. 2.42 8.56.56

In February/, the beginning of the year which I shall hereafter call y, was in T 28°

32

1665, the first star of Aries, 30' 15", with 7° 8' 58" north

498 THE MATHEMATICAL PRINCIPLES [BoOK IIL

lat. ; the second star of Aries was in T 29° 17' 18", with 8° 28' 16" north lat. ; and another star of the seventh magnitude, which I call A, was in ^ 28° 24' 45", with 8° 28' 83" north lat. The comet Feb, 7\ 7\ 30' at Paris (that is, Feb. 7^. 8^ 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 ^ 27° 0', nearly ; and, by the scheme in which Dr. Hooke delineated its motion, it was then in T 26° 59' 24". I place it in T 27° 4' 46", taking the middle between the two extremes.

From the same observations, M. Auzout made the latitude of the comet 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.

February 22^ 7^ 30' a-t London, that is, February 22^. 8^ 46' at Dantzick, the distance of the comet from the star A, according to Dr. Hooke' s observation, as was delineated by himself in a scheme, and also by the observations of M. Auzout, delineated in like manner by M. Petitj 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 fifth part, that is, 4' ; and therefore the comet was in T 28° 29' 46", with 8° 12' 36" north lat.

March 1, 7^. 0' at London, that is, March 1, 8\ 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. Gottignies. 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. Auzout) ; 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° 18', with 8° 36' 26" north lat.

March 7, 7\ 30' at Paris (that is, March 7, 8^. 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 « 0° 2' 48". From the scheme of the observations of M. Auzout, constructed by M. Petit, Hevelius collected the latitude of the comet 8° 54'. But the engraver did not rightly trace the curvature of the comet's way towards the end of the motion ; and Hevelius^ in the scheme of M. Auzoufs observations which he constructed himself, corrected this irregular curvature, and so made the latitude of the comet 8° 56' 30". And, by farther correcting this irregularity, the lati- tude may become 8° 56', or 8° 67',

This comet was also seen March 9, and at that time its place must have been in « 0° 18', with 9° 3i' 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 towards 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 49° 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 orhis 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 m 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 magnus contains 100000 ; and the time of its perihelion July 2^. 3^ 50'. And the places thereof, computed by Dr. Halley in this orbit, are compared with the places of the same observed by Mr. Flamsted, in the following table : —

500

THE MATHEMATICAL PRINCIPLES

[Book HI.

1683

Comet's

Lat.Nor.

Comet's

Lat.Nor.

Diff.

Diff.

Eq. time.

Sun's place

Long. com.

comput.

Long, obs'd

observ'd

Long.

Lat.

d. h. '

o / //

f 1 II

Q / //

O / II

O f II

» II

July 13.12.55

^ 1.02.30

2d 13.05.42

29.28.13

25 13. 6.42

29.28.20

• 1.00

• 0.07

15.11.15

2.53.12

11.37.48

29.34. 0

11.39.43

29.34.50

• 1.55

• 0.50

17.10.20

4.45.45

10. 7. 6

29.33.30

10. 8.40

29.34. 0

• 1.34

• 0.30

23.13.40

10.38.21

5.10.27

28.51.42

5.11.30

28.50.28

• 1.03

— 1.14

25.14. 5

12.35.28

3.27.53

24.24.47

3.27. 0

28.23.40

— 0.53

— 1. 7

31. 9.42

18.09.22

n 27.55. 3

26.22.52

n 27.54.24

26.22.25

— 0.39

— 0.27

31.14.55

18.21.53

27.41. 7

26.16.57

27.41. 8

26.14.50

• 0. 1

— 2. 7

Aug. 2.14.56

20.17.16

25.29.32

25.16.19

25.28.46

25.17.28

— 0.46

• 1. 9

4.10.49

22.02.50

23.18.20

24.10.49

23.16.55

24.12.19

— 1.25

• 1.30

6.10. 9

23.56.45

20.42.23

22.47. 5

20.40.32

22.49. 5

— 1.51

• 2. 0

9.10.26

26.50.52

16. 7.57

17. 6.37

18. 5.55

19. 6.10

— 2. 2

— 0.27

15.14. 1

rrji 2.47.13

3.30.48

11.37.33

3.26.18

11.32. 1

— 4.30

— 5.32

16.15.10

3.48. 2

0.43. 7

9.34.16

0.41.55

9.34.13

— 1.12

— 0. 3

18.15.44

5.45.33

"6 24.52.53

5.11.15 South.

b 24.49. 5

5. 9.11

South

— 3.48

— 2. 4 1

22.14.44

9.35.49

11. 7.14

5.16.58

11.07.12

5.16.58

— 0. 2

— 0. 3

23.15.52

10.36.48

7. 2.18

8.17. 9

7. 1.17

8.16.41

— 1. 1

— 0.28

26.16. 2

13.31.10

T 24.45.31

16.38. 0

T 24.44.00

16.38.20

— 1.31

• 0.20

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. Hallexfs computation) was in "6 2V^ 16' 30" ; the inclination of its orbit to the plane of the ecliptic 17° b^' 00" ; its perihelion in ca 2° 52' 50" ; its perihelion distance from the sun 58328 parts, of which the radius of the orhis magnus contains 100000 ; the equal time of the comet's being in its perihelion Sept, A^. 7^. 39'. And its places, collected from Mr. Flamstedfs observations, are compared with its places computed from our theory in the following table : —

1682

Comet's

Lat.Nor

Com. Long.

Lat.Nor

Diff.

Diff.

App. Time.

Sun's place

Lon. comp.

comp.

observed.

observ.

O 1 II

Long.

Lat. 1 II

d. h. '

O 1 II

O / II

o / II

9 1 II

/ II

^wg 19.16.38

W 7. 0. 7

^l 18.14.28

25.50. 7

a 18.14.40

25.49.55

— 0.12

• 0.12

20.15.38

7.55.52

24.46.23

26.14.42

24.46.22

26.12.52

• 0. 1

• 1.50

21. 8.21

8.36.14

29.37.15

26.20. 3

29.38.02

26.17.37

j— 0.47

• 2.26

22. 8. 8

9.33.55

Tlj? 6.29.53

26. 8.42

TTg 6.30. 3

26. 7.12

— 0.10

• 1.30

29.08.20

16.22.40

^ 12.37.54

18.37.47

:^ 12.37.49

18.34. 5

• 0. 5

• 3.42

30. 7.45

17.19.41

15.36. 1

17.26.43

15.35.18

17.27.17

• 0.43

— 0.34

Sept. 1. 7.33

19.16. 9

20.30.53

15.13. 0

20.27. 4

15. 9.49

• 3.49

• 3.11

4. 7.22

22.11.28

25.42. 0

12.23.48

25.40.58

12.22. 0

• 1. 2

• 1.48

5. 7.32

23.10.29

27. 0.46

11.33.08

26.59.24

11.33.51

• 1.22

— 0.43

8. 7.16

9. 5.58

29.58.44

9.26.46

29.58.45

9.26.43

_0. 1

• 0. 3

9. 7.26

10. 5. 9

TU 0.44.10

8.49.10

TU 0.44. 4

8.48.25

• 0. 6

• 0.45

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 to the computation of Mr. Bradley, Savilian Professor of Astronomy at Oxford) Avas in T 14° 16'. The inclination of the orbit to the plane of the ecliptic 49° 59'. Its perihelion was in « 12° 15' 20". Its perihelion distance from the sun 998651 parts, of which the radius of the orbis mag' nus contains 1000000, and the equal time of its perihelion September 16^.

OF NATURAL PHILOSOPHY.

501

Book III.]

16i». 10'. The places of this comet computed in this orbit by Mr. Bradley, and compared with the places observed by himself, his uncle Mr. Pound, and Dr. Halley, may be seen in the following table.

''

1723

Comet's

Lat.Nor

Comet's

Lat.Nor

Diff.

DifF.

Eq

. time.

Long. obs.

obs.

Lon. com.

comp.

Lon.

Lat.

d. h. '

o , ,/

~d } IT

O t II Q 1 II

"

Oct.

9.8. 5

OCC 7.22.15

5. 2. 0

OX 7.21.26

5. 2.47

• 49

— 47

10.6.21

6.41.12

7.44.13

6.41.42

7.43.18

— 50

^55

12.7.22

5 39.58

11.55. 0

5.40.19

11.54.55

— 21

• 5

14.8.57

4.59.49

14.43.50

5. 0.37

14.44. 1

— 48

— 11

15.6.35

4.47.41

15.40.51

4.47.45

15.40.55

— 4

— 4

21.6.22

4. 2.32

19.41.49

4. 2.21

19.42. 3

• 11

— 14

22.6.24

3.59. 2

20. 8.12

3.59.10

20. 8.17

— 8

__ 5

24.8. 2

3.55.29

20.55.18

3.55.11

20.55. 9

• 18

• 9

29.8.56

3.56.17

22.20.27

3.56.42

22.20.10

— 25

• 17

30.6.20

3.58. 9

22.32.28

3.58.17

22.32.12

— 8

• 16

Nov

. 5.5.53

4.16.30

23.38.33

4.16.23

23.38. 7

• 7

• 26

8.7. 6

4.29.36

24. 4.30

4.29.54

24. 4.40

— 18

— 10

14.6.20

5. 2.16

24.48.46

5. 2.51

24.48.16

— 35

• 30

20.7.45

5.42.20

25.24.45

5.43.13

25.25.17

— 53

— 32

Dec.

7.6.45

8. 4.13

26.54.18

8. 3.55

26.53.42

• 18

• 36

From these examples it is abundantly evident that the motions of com- ets are no less accurately represented by our theory than the motions of the planets commonly are by the theories of them ; and, therefore, by means of this theory, we may enumerate the orbits of comets, and so discover the periodic time of a comet's revolution in any orbit ; whence, at last, we shall have the transverse diameters of their elliptic orbits and their aphe- lion distances.