Principia Mathematica (Motte Translation, 1848) — part 37 of 45

Latit'de

Longitude

North

Longitude

Latitude

Errors in

True time.

observed.

obs.

comp.

computed.

Long.

Lat.

(1. h. '

p / //

O / «

i> 1 It

o ; II

/ //

1 II

Nov. 3.16.47

^ 29.51. 0

1.17.45

a 29.51.22

1.17.32 N

_i.0.22

— 0.13

5.15.37

rrg 3.23. 0

1. 6. 0

nj^ 3.24.32

1. 6. 9

• 1.32

• 0. 9

10.16.18

15.32. 0

0.27. 0

15.33. 2

0.25. 7

• 1. 2

— 1.53

16.17.00

-= 8.16.45

0.53. 7 S

18.21.34

18.52.15

1.26.54

20.17. 0

28.10.36

1.53.35

23.17. 5

\ 13.22.42

2.29. 0

Dec. 12. 4.46

\S 6.32.30

8.28. 0

V^' 6.31.20

8.29. 6 N

— 1.10

• 1. 6

21. 6.37

^ 5. 8.12

21.42.13

-)X 5. 6.14

21.44.42

— 1.58

• 2.29

24. 6.18

18.49.23

25.23. 5

18.47.30

25.23.35

— 1.53

• 0.30

26. 5.21

28.24.13

27. 0.52

28.21.42

27. 2. 1

— 2.31

• 1. 9

29. 8. 3

X 13.10.41

28. 9.58

X 13.11.14

28.10.38

• 0.33

• 0.40

30. 8.10

17.38. 0

28.11.53

17.38.27

28.11.37

• 0. 7

— 0.16

Jan. 5. 6. U

T 8.48.53

26.15. 7

q^ 8.48.51

26.14.57

— 0. 2

— 0.10

9. 7. 1

18.44. 4

24.11.56

18.43.51

24.12.17

— 0.13

• 0.21

10. 6. 6

20.40.50

23.43.32

20.40.23

23.43.25

— 0.27

— 0. 7

13. 7. 9

25.59.48

22.17.28

26. 0. 8

22.16.32

• 0.20

— 0.56

25. 7.59

« 9.35. 0

17.56.30

y 9.34.11

17.56. 6

— 0.49

— 0.24

30. 8.22

13.19.51

16.42.18

13.18.28

16.40. 5

— 1.23

— 2.13

Feb. 2. 6.35

15.13.53

16. 4. 1

15.11.59

16. 2.17

— 1.54

— 1.54

5. 7. 4^

16.59. 6

15.27. 3

16.59.17

15.27. 0

• 0.11

— 0. 3

25. 8.41

26.18.35

12.46.46

26.16.59

12.45.22

— 1.36

— 1.24

Mar. 1.11.10

27.52.42

12.23.40

27.51.47

12.22.28

— 0.55

— 1.12

5.11.39

29.18. 0

12. 3.16

29.20.11

12. 2.50

• 2.11

— 0.26

9. 8.38

n 0.43. 4

11.45.52

n 0.42.43

11.45.35

— 0.21

-0.17

The observations of this comet from the beginning to the end agree as perfectly 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. PonthcBus and his compan- ions, at 6^ in the morning at Rome (that is, 5^. 10' at London), by threads directed to the fixed stars, observed the comet in ^ 8° 30', with latitude 0° 40' south. Their observations may be seen in a treatise which PontkcBus published concerning this comet. Cellius, who was present, and commu- nicated his observations in a letter to Cassini^ saw the comet at the same hour in ^ 8° 30', with latitude 0° 30' south. It was likewise seen by Galletius at the same hour at Avignon (that is, at 5\ 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 0° 53' 7" south.

Nov. 18, at 6^ 30' in the morning at Rome (that is, at 5'\ 40' at Lon- don)^ Ponthmis observed the comet in ^ 13° 30', with latitude 1° 20'

Book III] of natural philosophy. 481

south ; and Cellius in ^ 13° 30', with latitude 1° 00' south. But at 5\ 30' in the morning at Avignon, Galletius saw it in ^ 13° 00', with lati- tude 1° 00' south. In the University of La Fleche, in France, at 5^ in the morning (that is, at 5''. 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 Yirgo, Bayers ip ; and the other is the outmost of the wing, Bayefs 6. Whence the comet was then in ^ 12° 46' with latitude 50' south. And I was informed by Dr. Halley, that on the same day at Boston in New England, in the latitude of 42| deg. at 5'\ in the morning (that is, at 9'". 44' in the morning at London), the comet was seen near ^ 14°, with latitude 1° 30' south.

Nov. 19, at 4|^ at Cambridge, the comet (by the observation of a young man) was distant from iSpica W about 2° towards the north west. Now the spike was at that time in ^ 19° 23' 47", with latitude 2° V 59" south. The same day, at 5K in the morning, at Boston in Neiv England, the comet was distant from Spica W 1°, with the diiFerence 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. 38|-°, at 5 in the morning (that is, at 10^ at London), saw the comet above JSpica 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 9^ 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 6^ in the morning at Venice (that is, 5^ 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, PonthcBus and his companions, at 7l\ in the morning, ob- served the comet in =^ 27° 50', with latitude 1° 16' south ; Cellius, in ^ 28° : P. Ango at 5^ 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 ^11^ and of about the same latitude with Spica W, that is, 2° 2'. The same day, at 5*\ morning, at Ballasore, in the East Indies (that is, at 11^. 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 5\ 40' (that is, at 5\ 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 ^U 2° 33' ; but at Boston,

31

482 THE MATHEMATICAL PRINCIPLES [BoOK IIL

in New England, it was found in about ^. 3°, and with almost tLe same latitude as before, tliat is, 1° 30'. The same day, at 5^ morning at Ballasore, the comet was observed in ^ 1° 50' ; and therefore at b\ morn- ing at London, the comet was in ^ 3° 5' nearly. The same day, at 6|^. in the morning at London^ Dr. Hook observed it in about R 3° 30', and that in the right line which passeth through Spica W 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 Leonis at a very small distance therefrom. The right line through Cor Leonis. and Spica W did cut the ecliptic in W 3° 46' at an anoie of 2^ 51' ; and if the comet had been in this line and in '^ 3°, its latitude would have been 2° 26' ; but since Hook and Mojitenari 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 W, that is, about 1° 30'. But by the agreement of Hook, Monte- nari, and Ango, the latitude was continually increasing, and therefore must now, on the 22d, be sensibly greater than 1° 30' ; and, taking a mean between the extreme limits but now stated, 2° 26' and V 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 the tail, lying nearly parallel to the equator, deviated a little from the op- position of the sun towards the north.

Nov. 23, O. S. at 5^. morning, at Nuremberg (that is, at 4|^ at Lon- don), Mr. Zimmerman saw the comet in '^\ 8° 8', with 2° 31' south lat. its place being collected by taking its distances from fixed stars.

Nov. 24, before sun-rising, the comet was seen by Montenari in ^ 12° 52' on the north side of the right line through Cor Leonis and ^pica W, and therefore its latitude was something less than 2° 38' ; and since the latitude, as we said, by the concurring observations of Montenari, Ango, and Hook, was continually increasing, therefore, it was now, on the 24th, something greater than 1° 58' ; and, taking the mean quantity, may be reckoned 2° 18', without any considerable error. PonthcBus and Galletiiis will have it that the latitude was now decreasing ; and Cellius, and the observer in New England, that it continued the same, viz., of about 1°, or H°. The observations of Ponthmns and Cellins are more rude, espe- cially those which were made by taking the azimuths and altitudes ; as are also the observations of Galletius. Those are better which were made by taking the position of the comet to the fixed stars by Montenari^ Hook, Ango, and the observer in New England, and sometimes by

Book III] of natural philosophy. 4S3

PonthcBus and Cellms. The same day, at 5''. morning, at Ballasore, the comet was observed in ^U 11° 45' ; and, therefore, at 5^. morning at Lon- don, was in ^l 13° nearly. And, by the theory, the comet was at that time in lU 13° 22' 42".

Nov. 25, before sunrise, Montenari observed the comet in ^U 17^ nearly ; and Cellms observed at the same time that the comet was in a right line between the bright star in the right thigh of Virgo and the southern scale of Libra ; and this right line cuts the comet's way in ^. 18° 36'. And, by the theory, the comet was in ^. 18^° nearly.

From all this it is plain that these observations agree with the theory, so far as they agree with one another ; and by this agreement it is made clear that it was one and. the same comet that appeared all the time from Nov. 4 to Mar. 9. The path of this comet did twice cut the plane of the ecliptic, and therefore was not a right line. It did cut the ecliptic not in opposite parts of the heavens, but in the end of Virgo and beginning of Capricorn, including an arc of about 98° ; and therefore the way of the comet did very much deviate from the path of a great circle ; for in the month of Nov. it declined at least 3° from the ecliptic towards the south ; and in the month of Dec. following it declined 29° from the ecliptic to- wards the north ; the two parts of the orbit in which the comet descended towards the sun, and ascended again from the sun, declining one from the other by an apparent angle of above 30°, as observed by Montenari, This comet travelled over 9 signs, to wit, from the last deg, of ^ to the heginr ning of n, beside the sign of U, through which it passed before it began to be seen ; and there is no other theory by which a comet can go over so great a part of the heavens with a regular motion. The motion of this comet was very unequable ; for about the 20th of Nov. it described about 5° a day. Then its motion keing retarded between Nov. 26 and Dec 12, to wit, in the space of 15i days, it described only 40°- But the mo- tion thereof being afterwards accelerated, it described near 5° a day, till its motion began to be again retarded. And the theory which justly cor- responds with a motion so unequable, and through so great a part of the heavens, which observes the same laws with the theory of the planets, and which accurately agrees with accurate astronomical observations, cannot be otherwise than true.

And, thinking it would not be improper, I have given a true representar tion of the orbit which this comet described, and of the tail which it emitted in several places, in the annexed figure ; protracted in the plane of the trajectory. In this scheme ABC represents the trajectory of the comet, D the sun DE the axis of the trajectory, DF the line of the nodes, GH the intersection of the sphere of the orbis magnus with the plane of the trajectory, I the place of the comet -Nov. 4, Ann. 1680 ; K the place of the same Nov. 11 ; L the place of the same Nov. 19 ; M its place Dec. 12; N

484

THE MATHEMATICAL PRINCIPLES

[Book III.

its place Dec. 21 ; O its place Dec. 29 ; P its place Jan. 5 following ; Q, its place Jan. 25 ; R its place Feb. 5 ; S its place Feb. 25 ; T its place March 5 ; and Y its place March 9. In determining the length of the tail, I made the following observations.

Nov. 4 and 6, the tail did not appear; Nov. 11, the tail just begun to shew itself, but did not appear above | deg. long through a 10 feet tele- scope ; Nov. 17, the tail was seen by PonthcBus more than 15° long ; Nov. 18, in New-England^ the tail appeared 30° long, and directly opposite to the sun, extending itself to the planet -Mars, which was then in ttj?, 9° 54 ; Nov. 19, in Maryland^ the tail was found 15° or 20° long ; Dec. 10 (by

Book III.] of natural philosophy. 485

the observation of Mr. Flamsted), the tail passed through the middle of the distance intercepted between the tail of the Serpent of Ophiuchus and the star 6 in the south wing of Aquila, and did terminate near the stars A, 0), b, in Bayer^s tables. Therefore the end of the tail was in \3 19^°, with latitude about 34^° north; Dec. 11, it ascended to the head of Sagit- ta {Bayer's a, iS), terminating in \3 2Q'' 43', with latitude 38° 34' north ; Dec. 12j it passed through the middle of Sagitta, nor did it reach much farther ; terminating in cci 4°, with latitude 42|-° north nearly. But these things are to be understood of the length of the brighter part of the tail ; for with a more faint light, observed, too, perhaps, in a serener sky, at Rome, Dec. 12, 6\ 40', by the observation of Ponthceus, the tail arose to 10° above the rump of the Swan, and the side thereof towards the west and towards the north was 45' distant from this star. But about that time the tail was 3° broad towards the upper end ; and therefore the middle thereof was 2° 15' distant from that star towards the south, and the upper end was X in 22°, with latitude 61° north ; and thence the tail was about 70° long ; Dec. 21, it extended almost to Cassiopeia's chair, equally dis- tant from |3 and from Schedir, so as its distance from either of the two was equal to the distance of the one from the other, and therefore did ter- minate in T 24°, with latitude 47|° ; Dec. 29, it reached to a contact with iScheat on its left, and exactly filled up the space between the tw^o stars in the northern foot of Andro'/neda, being 54° in length ; and therefore ter- minated in ^ 19°, with 35° of latitude; Jan. 5, it touched the star tt in the breast of Andromeda on its right side, and the star // of the girdle on its left; and, according to our observations, was 40° long; but it was curved, and the convex side thereof lay to the south ; and near the head of the comet it made an angle of 4° with the circle which passed through the sun and the comet's head ; but towards the other end it was inclined to that circle in an angle of about 10° or 11° ; and the chord of the tail con- tained with that circle an angle of 8°. Jan. 13, the tail terminated be- tween Alamech and Algol, with a light that was sensible enough ; but with a faint light it ended over against the star k in Perseus^s side. The distance of the end of the tail from the circle passing through the sun and the comet was 3° 50' ; and the inclination of the chord of the tail to that circle was 8|°. Jan. 25 and 26, it shone with a faint light to the length of 6° or 7° ; and for a night or two after, when there was a very clear sky, it extended to the length of 12°, or something more, with a light that was very faint and very hardly to be seen; but the axis thereof was exactly di- rected to the bright star in the eastern shoulder of Auriga, and therefore deviated from the opposition of the sun towards the north by an angle of 10°. Lastly, Feb. 10, with a telescope I observed the tail 2° long ; for that fainter light which I spoke of did not appear through the glasses. But Ponthmis writes, that, on Feb. 7, he saw the tail 12° long. Feb. 25, the comet was without a tail, and so continued till it disappeared.

486 THE MATHEMATICAL PRINCIPLES [BoOK III.

Now if one reflects upon the orbit described, and duly considers the other appearances of this comet, he will be easily satisfied that the bodies of comets are solid, compact, fixed, and durable, like the bodies of the planets ; for if they were nothing else but the vapours or exhalations of the earth, of the smi; and other planets, this comet, in its passage by the neighbourhood of the sun, would have been immediately dissipated ; for the heat of the sun is as the density of its rays, that is, reciprocally as the square of the distance of the places from the sun. Therefore, since on Dec. 8, when the comet was in its perihelion, the distance thereof from the centre of the sun was to the distance of the earth from the same as about 6 to 1000, the sun's heat on the comet was at that time to the heat of the summer-sun with us as 1000000 to 36, or as 28000 to 1. But the heat of boiling water is about 3 times greater than the heat which dry earth acquires from the summer-sun, as I have tried ; and the heat of red-hot iron (if my con- jecture is right) is about three or four times greater than the heat of boil- ing water. And therefore the heat which dry earth on the comet, while in its perihelion, might have conceived from the rays of the sun, was about 2000 times greater than the heat of red-hot iron. But by so fierce a heat, Tapours and exhalations, and every volatile matter, must have been imme- diately consumed and dissipated.

This comet, therefore, must have conceived an immense heat from the snn, and retained that heat for an exceeding long time ; for a globe of iron of an inch in diameter, exposed red-hot to the open air, will scarcely lose all its heat in an hour's time ; but a greater globe would retain its heat longer in the proportion of its diameter, because the surface (in proportion to which it is cooled by the contact of the ambient air) is in that proportion less in respect of the quantity of the included hot matter; and therefore a globe of red hot iron equal to our earth, that is, about 40000000 feet in diameter, would scarcely cool in an equal number of days, or in above 50000 years. But I suspect that the duration of heat may, on account of some latent causes, increase in a yet less proportion than that of the diameter ; and I should be glad that the true proportion was investigated by experiments.

It is farther to be observed, that the comet in the month of December. just after it had been heated by the sun, did emit a much longer tail, and much more splendid, than in the month of November before, when it had not yet arrived at its perihelion ; and, universally, the greatest and most fulgent tails always arise from comets immediately after their passing by the neighbourhood of the sun. Therefore the heat received by the comet conduces to the greatness of the tail: from whence, I thmk. I may infer, that the tail is nothing else but a very fine vapour, which the head or nucleus of the comet emits by its heat.

But we have' had three several opinions about the tails of comets; for

Book III.] of natural philosophy. 4S7

some will have it that they are nothing else but the beams of the sun's light transmitted through the comets' heads, which they suppose to be transparent ; others, that they proceed from the refraction which light suf- fers in passing from the comet's head to the earth ; and, lastly, others, that they are a sort of clouds or vapour constantly rising from the comets^ heads, and tending towards the parts opposite to the sun. The first is the opin> ion of such as are yet unacquainted with optics ; for the beams of the sun are seen in a darkened room only in consequence of the light that is re- flected from them by the little particles of dust and smoke Avhich are always flying about in the air ; and, for that reason, in air impregnated with thick smoke, those beams appear with great brightness, and move the sense vigorously ; in a yet finer air they appear more faint, and are less easily discerned ; but in the heavens, where there is no matter to reflect the light, they can never be seen at all. Light is not seen as it is in the beam, but as it is thence reflected to our eyes ; for vision can be no other- wise produced than by rays falling upon the eyes ; and, therefore, there must be some reflecting matter in those parts where the tails of the comets are seen : for otherwise, since all the celestial spaces are equally illumin- ated by the sun's light, no part of the heavens could appear with more splendor than another. The second opinion is liable to many difficulties. The tails of comets are never seen variegated with those colours which commonly are inseparable from refraction ; and the distinct transmission of the light of the fixed stars and planets to us is a demonstration that the sether or celestial medium is not endowed with any refractive power : for as to what is alleged, that the fixed stars have been sometimes seen by the Egyptians environed with a Coma or Capitlitium, because that has but rarely happened, it is rather to be ascribed to a casual refraction of clouds ; and so the radiation and scintillation of the fixed stars to the refractions both of the eyes and air ; for upon laying a telescope to the eye, those radiations and scintillations immediately disappear. By the trem- ulous agitation of the air and ascending vapours, it happens that the rays of light are alternately turned aside from the narrow space of the pupil of the eye ; but no such thing can have place in the much wider aperture of the ob- ject-glass of a telescope ; and hence it is that a scintillation is occasioned in the former case, which ceases in the latter ; and this cessation in the latter case is a demonstration of the reo^ular transmission of lis-ht throuo:h the heavens, without any sensible refraction. But, to obviate an objection that may be made from the appearing of no tail in such comets as shine but with a faint light, as if the secondary rays were then too weak to af- fect the eyes, and for that reason it is that the tails of the fixed stars do not appear, we are to consider, that by the means of telescopes the light of the fixed stars may be augmented above an hundred fold, and yet no tails are seen ; that the light of the planets is yet more copious without any

488 THE MATHEMATICAL PRINCIPLES [BoOK III.

tail ; but that comets are seen sometimes with huge tails, when the light of their heads is but faint and dull. For so it happened in the comet of the year 1680, when in the month of December it was scarcely equal in light to the stars of the second magnitude, and yet emitted a notable tail, extending to the length of 40°, 50°, 60°, or 70°, and upwards ; and after- wards, on the 27th and 28th of January, when the head appeared but as a star of the 7th magnitude, yet the tail (as we said above), with a light that was sensible enough, though faint, was stretched out to 6 or 7 degrees in length, and with a languishing light that was more difficultly seen, even to 12°, and upwards. But on the 9th and 10th of February, when to the naked eye the head appeared no more, through a telescope I viewed the tail of 2° in length. But farther ; if the tail was owing to the refrac- tion of the celestial matter, and did deviate from the opposition of the sun, according to the figure of the heavens, that deviation in the same places of the heavens should be always directed towards the same parts. But the comet of the year 1680, December 2S^. 8l^ P. M. at Lo?idon, was seen in X 8° 41', with latitude north 28° 6' ; while the sun was in ^9 18° 26'. And the comet of the year 1577, Dece?nber 29"^. was in X 8° 41', with latitude north 28' 40', and the sun, as before, in about V? 18° 26'. In both cases the situation of the earth was the same, and the comet ap- peared in the same place of the heavens ; yet in the former case the tail of the comet (as well by my observations as by the observations of others) deviated from the opposition 'of the sun towards the north by an angle of 4|- degrees ; whereas in the latter there was (according to the observations of Tycho) a deviation of 21 degrees towards the south. The refraction, therefore, of the heavens being thus disproved, it remains that thejt?/i«- nomena of the tails of comets must be derived from some reflecting matter. And that the tails of comets do arise from their heads, and tend towards the parts opposite to the sun, is farther confirmed from the laws which the tails observe. As that, lying in the planes of the comets' orbits which pass tlirough the sun, they constantly deviate from the opposition of the sun towards the parts which the comets' heads in their progress along these orbits have left. That to a spectator, placed in those planes, they appear in the parts directly opposite to the sun ; but, as the spectator recedes from those planes, their deviation begins to appear, and daily be- comes greater. That the deviation, ccEteris paribus, appears less when the tail is more oblique to the orbit of the comet, as well as when the head of the comet approaches nearer to the sun. especially if the angle of deviation is estimated near the head of the comet. That the tails which have no deviation appear straight, but the tails which deviate are like- wise bended into a certain curvature. That this curvature is greater when the deviation is greater ; and is more sensible when the tail, cceteris pari- bus, is longer ; for in the shorter tails the curvature is hardly to be per-

Book III.] of natural philosophy. 489

ceived. That the angle of deviation is less near the comet's head, but greater towards the other end of the tail ; and that because the convex side of the tail regards the parts from which the deviation is made, and which lie in a right line drawn out infinitely from the sun through the comet's head. And that the tails that are long and broad, and shine with a stronger light, appear more resplendent and more exactly defined on the convex than on the concave side. Upon which accounts it is plain that the phcBuomena of the tails of comets depend upon the motions of their heads, and by no means upon the places of the heavens in which their heads are seen ; and that, therefore, the tails of comets do not proceed from the refraction of the heavens, but from their own heads, which furnish the matter that forms the tail. For, as in our air, the smoke of a heated body ascends either perpendicularly if the body is at rest, or obliquely if the body is moved obliquely, so in the heavens, where all bodies gravitate to- wards the sun, smoke and vapour must (as we have already said) ascend from the sun, and either rise perpendicularly if the smoking body is at rest, or obliquely if the body, in all the progress of its motion, is always leaving those places from which the upper or higher parts of the vapour had risen before ; and that obliquity will be least where the vapour ascends with most velocity, to wit, near the smoking body, when that is near the sun. But, because the obliquity varies, the column of vapour will be in- curvated; and because the vapour in the preceding sides is something more recent, that is, has ascended something more late from the body, it will therefore be something more dense on that side, and must on that account reflect more light, as well as be better defined. I add nothing concerning the sudden uncertain agitation of the tails of comets, and their irregular figures, which authors sometimes describe, because they may arise from the mutations of our air, and the motions of our clouds, in part obscuring those tails ; or, perhaps, from parts of the Via Lactea, which might have been confounded with and mistaken for parts of the tails of the comets as they passed by.

But that the atmospheres of comets may furnish a supply of vapour great enough to fill so immense spaces, we may easily understand from the rarity of our own air; for the air near the surface of our earth possesses a space 850 times greater than water of the same weight ; and therefore a cylinder of air 850 feet high is of equal weight with a cylinder of water of the same breadth, and but one foot high. But a cylinder of air reach- ing to the top of the atmosphere is of equal weight with a cylinder of water about 33 feet high : and, therefore, if from the whole cylinder of air the lower part of 850 feet high is taken away, the remaining upper part will be of equal weight with a cylinder of water 32 feet high : and from thence (and by the hypothesis, confirmed by many experiments, that the compression of air is as the weight of the incumbent atmosphere, and

490 THE MATHEMATICAL PRINCIPLES [BoOK III.

that tlie force of gravity is reciprocally as the square of the distance from the centre of the earth) raising a calculuSj by Cor. Prop. X.XII, Book II, I found, that, at the height of one semi-diameter of the earth, reckoned from the earth's surface, the air is more rare than with us in a far grpater proportion than of the whole space within the orb of Saturn to a spherical space of one inch in diameter ; and therefore if a sphere of our air of but one inch in thickness was equally rarefied with the air at the height of one semi-diameter of the earth from the earth's surface, it would fill all the regions of the planets to the orb of Saturn, and far beyond it. Where- fore since the air at greater distances is immensely rarefied, and the coma or atmosphere of comets is ordinarily about ten times higher, reckoning from their centres, than the surface of the nucleus, and the tails rise yet higher, they must therefore be exceedingly rare ; and though, on account of the much thicker atmospheres' of comets, and the great gravitation of their bodies towards the sun, as well as of the particles of their air and vapours mutually one towards another, it may happen that the air in the celestial spaces and in the tails of comets is not so vastly rarefied, yet from this computation it is plain that a very small quantity of air and vapour is abundantly sufficient to produce all the appearances of the tails of comets ; for that they are, indeed, of a very notable rarity appears from the shining of the stars through them. The atmosphere of the earth, illuminated by the sun's light, though but of a few miles in thickness, quite obscures and extinguishes the light not only of all the stars, but even of the moon itself; whereas the smallest stars are seen to shine through the immense thickness of the tails of comets, likewise illuminated by the sun, without the least diminution of their splendor. Nor is the brightness of "the tails of most comets ordinarily greater than that of our air, an inch or two in thickness, reflecting in a darkened room the light of the sun-beams let in by a hole of the window-shutter.

And we may pretty nearly determine the time spent during the ascent of the vapour from the comet's head to the extremity of the tail, by draw- ing a right line from the extremity of the tail to the sun, and marking the place where that right line intersects the comet's orbit ; for the vapour that is now in the extremity of the tail, if it has ascended in a right line from the sun, must have begun to rise from the head at the time when the head was in the point of intersection. It is true, the vapour does not rise in a right line from the sun, but, retaining the motion which it had from the comet before its ascent, and compounding that motion with its motion of ascent, arises obliquely ; and, therefore, the solution of the Problem will be more exact, if we draw the line which intersects the orbit parallel to the length of the tail ; or rather (because of the curvilinear motion of the comet) diverging a little from the line or length of the tail. And by means of this principle I found that the vapour which, January 25, was

Doom 111.1 of natural philosophy. 491