For if we join EO, cutting the parabolic arc ABC in Y, and draw touching the same arc in the vertex , and meeting EO in X, the curvilinear area will be to the curvilinear area as AE to AC; and, therefore, since the triangle ASE is to the triangle ASC in the same proportion, the whole area will be to the whole area as AE to AC. But, because 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 triangle SEB will be equal to the triangle XEB. Wherefore if to the area we add the triangle EXB, and from the sum subduct the triangle SEB, there will remain the area , equal to the area ; and therefore in proportion to the area as AE to AC. But the area is nearly equal to the area ; and this area is to the area 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 of the parabola, AE is to AC accurately in the proportion of the times.

SCHOLIUM.

If we join cutting AC in ; and in it take in proportion to as 27MI to , and draw Bn, this Bn will cut the chord AC, in the proportion of the times, more accurately than before; but the point n is to be taken beyond or on this side the point , according as the point B is more or less distant from the principal vertex of the parabola than the point .

LEMMA IX.

For is the latus rectum of the parabola belonging to the vertex .

LEMMA X.

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 , the area which it would describe by a radius drawn to the point S would be equal to the parabolic area ; and therefore the space contained under the length described in the tangent and the length would be to the space contained under the lengths AC and SM as the area to the triangle ASC, that is, as SN to SM. Wherefore AC is to the length described in the tangent as to SN. But since the velocity of the comet in the height SP (by Cor. 6, Prop. XVI., Book I) is to the velocity of the same in the height in the reciprocal subduplicate proportion of SP to , that is, in the proportion of to SN, the length described with this velocity will be to the length in the same time described in the tangent as 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 , would in the same time describe the chord AC nearly.

LEMMA XI.

For in the same time that the comet would require to describe the parabolic 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 I), 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. Wherefore 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 I) 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, it would describe the space . But since the weight of the comet towards the sun in the height SN is to the weight of the same towards the sun in the height SP as SP to , the comet, by the weight which it hath in the height SN, in falling from that height towards the sun, would in the same time describe the space ; that is, a space equal to the length or . Q.E.D.

PROPOSITION XLI. PROBLEM XXI.

This being a Problem of very great difficulty, I tried many methods of resolving it; and several of these Problems, the composition whereof I have 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 difference 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, , three places of the earth in the orbis magnus; TA, tB, , three observed longitudes of the comet; V the time between the first observation and the second; W the time between the second and the third; X the length which in the whole time V + W the comet might describe with that velocity which it hath in the mean distance of the earth from the sun, which length is to be found by Cor. 3, Prop. XL, Book III; and tV a perpendicular upon the chord . In the mean observed longitude tB 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 tV as the content under SB and St² to the cube of the hypothenuse of the right angled triangle, whose sides are SB, and the tangent of the latitude of the comet in the second observation to the radius tB. And through the point E (by Lemma VII) draw the right line AEC, whose parts AE and EC, terminating in the right lines TA and , 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 Ii. Through B draw the obscure line Bi parallel to AC. Join the obscure line Si, cutting AC in , and complete the parallelogram iI . Take equal to ; and through the sun S draw the obscure line equal to . Then, cancelling the letters A, E, C, I, from the point B towards the point , draw the new obscure line BE, which may be to the former BE in the duplicate proportion of the distance BS to the quantity . And through the point E draw again the right line AEC by the same rule as before; that is, so as its parts AE and EC may be one to the other as the times V and W between the observations. Thus A and C will be the places of the comet more accurately.

Upon AC, bisected in I, erect the perpendiculars AM, CN, IO, of which AM and CN may be the tangents of the latitudes in the first and third observations, to the radii TA and . Join MN, cutting IO in O. Draw the rectangular parallelogram , as before. In IA produced take ID equal to . 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 magnus) to the distance OD. If 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 assumed point B, from other points b and assumed at pleasure, find out the new points e, a, c, g; and , , , . Then through G, g, and , draw the circumference of a circle , cutting the right line in Z: and Z will be one place of the comet in the plane of the ecliptic. And in AC, ac, , making AF, af, , equal respectively to CG, cg, ; through the points F, f, and , draw the circumference of a circle , cutting the right line AT in X; and the point X will be another place of the comet in the plane of the ecliptic. And at the points X and Z, erecting the tangents of the latitudes of the comet to the radii TX and , two places of the comet in its own orbit will be determined. Lastly, if (by Prop. XIX., Book I) 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 Lemmas, because the right line AC is cut in E in the proportion of the times, by Lem. VII, as it ought to be, by Lem. VIII.; and BE, by Lem. XI., is a portion of the right line BS or 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 first and third observation, and therefore is equal to MN, providing B is a true place of the comet in the plane of the ecliptic.

But it will be convenient to assume the points B, b, , not at random, but nearly true. If the angle AQt, at which the projection of the orbit in the plane of the ecliptic cuts the right line tB, is rudely known, at that angle with Bt draw the obscure line AC, which may be to in the subduplicate proportion of SQ to St; and, drawing the right line SEB so as its part EB may be equal to the length Vt, 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 tB take the point b, by this rule, that, if TA and intersect each other in Y, the distance Yb 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 , 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, f, and G, g, are found, draw the right lines Ff and Gg, and they will cut TA and 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 observations.

To these you may add some observations of mine.

These observations were made by a telescope of 7 feet, with a micrometer 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 (Bayer's ο), B the following star of the third magnitude in the left foot (Bayer's ), C a star of the sixth magnitude (Bayer's ) in the heel of the same foot, and D, E, F, G, H, I, K, L, M, N, O, Z, , , , , 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 parts, AC was of those parts; BC, ; AD, ; BD, ; CD, ; AE, ; CE, ; DE, ; AI, ; BI, ; CI, ; DI, ; AK, ; BK, 43; CK, ; FK, 29; FB, 23; FC, ; AH, ; DH, ; BN, ; CN, ; BL, ; NL, . HO was to HI as 7 to 6, and, produced, did pass between the stars D and E, so as the distance of the star D from this right line was . LM was to LN as 2 to 9, and, produced, did pass through the star H. Thus were the positions of the fixed stars determined in respect of one another.

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

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

Friday, February 25, O.S. at P.M. the distance of the comet in p from the star E was less than , and greater than , and therefore nearly equal to ; and the angle ApE was a little obtuse, but almost right. For from A, letting fall a perpendicular on pE, the distance of the comet from that perpendicular was .

The same night, at , the distance of the comet in P from the star E was greater than , and less than , and therefore nearly equal to of AE, or . But the distance of the comet from the perpendicular let fall from the star A upon the right line PE was .

Sunday, February 27, 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 QO produced passed between the stars K and B. I could not, by reason of intervening clouds, determine the position of the star to greater accuracy.

Tuesday, March 1, 11^h. P. M. the comet in R lay exactly in a line between the stars K and C, so as the part CR of the right line CRK was a little greater than , and a little less than , and therefore = , or .

Wednesday, March 2, 8^h. P. M. the distance of the comet in S from the star C was nearly ; the distance of the star F from the right line CS produced was ; 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, P. M. when the comet was in T, the right line MT was equal to , 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 telescope; but the star L was greater, and of about the eighth magnitude.

Monday, March 7, P. M. the comet being in V, the right line produced did pass between B and F, cutting off, from BF towards F, of BF, and was to the right line as 5 to 4. And the distance of the comet from the right line was .

Wednesday, March 9, P. M. the comet being in X, the right line was equal to ; and the perpendicular let fall from the star upon the right was of .

The same night, at 12^h. the comet being in Y, the right line was equal to of , or a little less, as perhaps of ; and a perpendicular let fall from the star on the right line was equal to about or . But the comet being then extremely near the horizon, was scarcely discernible, and therefore its place could not be determined with that certainty as in the foregoing 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 places of the comet, which correct places are set down above. Though my micrometer was none of the best, yet the errors in longitude and latitude (as derived from my observations) scarcely exceed one minute. The comet (according to my observations), about the end of its motion, began to decline sensibly towards the north, from the parallel which it described about the end of February.

Now, in order to determine the orbit of the comet out of the observations above described, I selected those three which Flamsted made, Dec. 21, Jan. 5, and Jan. 25; from which I found St of 9842,1 parts, and Vt of 455, such as the semi-diameter of the orbis magnus contains 10000. Then for the first observation, assuming tB of 5657 of those parts, I found SB 9747, BE for the first time 412, 9503, 413, BE for the second time 421, OD 10186, X 8528,4, PM 8450, MN 8475, NP 25; from whence, by the second operation, I collected the distance tb 5640; and by this operation I at last deduced the distances TX 4775 and 11322. From which, limiting the orbit, I found its descending node in ♋, and ascending node in ♑ 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 ♐ 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^d. 00^h. 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 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 arithmetical operations, and partly by scale and compass, to the times of some of the observations, as may be seen in the following table:—

But afterwards Dr. Halley did determine the orbit to a greater accuracy by an arithmetical calculus than could be done by linear descriptions; and, retaining the place of the nodes in ♋ and ♑ 1° 53' and the inclination of the plane of the orbit to the ecliptic 61° , as well as the time of the comet's being in perihelion, Dec. 8^d. 00^h. 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:—

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 London; 11°; and from the places of the fixed stars observed by Mr. Pound, Dr. Halley has determined the places of the comet as follows:—

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

November 5, 15^h. 58' the comet was in ♍ 3° 23', with 1° 6' lat. north.

November 10, 16^h. 31', the comet was equally distant from two stars in ♌, which are and 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 was then in ♍︎ 14° 15', with 1 deg. 41' lat. north nearly, and in ♍ 17° with 0 deg. 34' lat. south; and the middle point between those stars was ♍ 15° , with 0° 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 difference of the latitude nearly ; and thence it follows that the comet was in ♍ 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 ♌ 29° 30' 22'', its latitude north 1° 25' 7'', and its distance from the sun 115546.

Moreover, Dr. Halley, observing that a remarkable comet had appeared four times at equal intervals of 575 years (that is, in the month of September after Julius Cæsar 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 Cæsar's 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 ♋ 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 ♐ 22° 44' 25'', the equal time of the perihelion December 7^d. 23^h. 9', the distance of the perihelion from the ascending 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.

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 calculated; 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 persons had observed the comet. Nov. 17, O. S. Ponthæus and his companions, at 6^h. in the morning at Rome (that is, 5^h. 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 Ponthæus published concerning this comet. Cellius, who was present, and communicated 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^h. 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^h. 30' in the morning at Rome (that is, at 5^h. 40' at London), Ponthæus observed the comet in ♎ 13° 30', with latitude 1° 20' south; and Cellius in ♎ 13° 30', with latitude 1° 00' south. But at 5^h. 30' in the morning at Avignon, Galletius saw it in ♎ 13° 00' with latitude 1° 00' south. In the University of La Fleche, in France, at 5^h. in the morning (that is, at 5^h. 9' at London), it was seen by P. Ango, in the middle between two small stars, one of which is the middle of the three which lie in a right line in the southern hand of Virgo, Bayers ; and the other is the outmost of the wing, Bayer's . 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 deg. at 5^h. in the morning (that is, at 9^h. 44' in the morning at London), the comet was seen near ♎ 14°, with latitude 1° 30' south.

Nov. 19, at at Cambridge, the comet (by the observation of a young man) was distant from Spica ♍ about 2° towards the north west. Now the spike was at that time in ♎ 19° 23' 47'', with latitude 2° 1' 59'' south. The same day, at 5^h. in the morning, at Boston in New England, the comet was distant from Spica ♍ 1°, with the difference of 40' in latitude. The same day, in the island of Jamaica, it was about 1° distant from Spica ♍. The same day, Mr. Arthur Storer, at the river Patuxent, near Hunting Creek, in Maryland, in the confines of Virginia, in lat. ^°, at 5 in the morning (that is, at 10^h. at London), saw the comet above Spica ♍, and very nearly joined with it, the distance between them being about of one deg. And from these observations compared, I conclude, that at 9^h. 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^h. in the morning at Venice (that is, 5^h. 10' at London), saw the comet in ♎ 23°, with latitude 1° 30' south. The same day, at Boston, it was distant from Spica ♍ by about 4° of longitude east, and therefore was in ♎ 23° 24' nearly.

Nov. 21, Ponthæus and his companions, at in the morning, observed the comet in ♎ 27° 50', with latitude 1° 16' south; Cellius, in ♎ 28°; P. Ango at 5^h. 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 ♏, and of about the same latitude with Spica ♍, that is, 2° 2'. The same day, at 5^h. morning, at Ballasore, in the East Indies (that is, at 11^h. 20' of the night preceding at London), the distance of the comet from Spica ♍ 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^h. 40' (that is, at 5^h. morning at London), it was in ♎ 28° 12', with 1° 16' lat. south. Now by the theory the comet was then in ♎ 28° 10' 36'', with 1° 53' 35'' lat. south.

Nov. 22, the comet was seen by Montenari in ♏ 2° 33'; but at Boston in New England, it was found in about ♏ 3°, and with almost the same latitude as before, that is, 1° 30'. The same day, at 5^h. morning at Ballasore, the comet was observed in ♏ 1° 50'; and therefore at 5^h. morning at London, the comet was in ♏ 3° 5' nearly. The same day, at in the morning at London, Dr. Hook observed it in about ♏ 3° 30', and that in the right line which passeth through Spica ♍ and Cor Leonis; not, indeed, exactly, but deviating a little from that line towards the north. Montenari likewise observed, that this day, and some days after, a right line drawn from the comet through Spica passed by the south side of Cor Leonis at a very small distance therefrom. The right line through Cor Leonis and Spica ♍ did cut the ecliptic in ♍ 3° 46' at an angle 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 Montenari agree that the comet was at some small distance from this line towards the north, its latitude must have been something less. On the 20th, by the observation of Montenari, its latitude was almost the same with that of Spica ♍, that is, about 1° 30'. But by the agreement of Hook, Montenari, 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 1° 30', the latitude will be about 1° 58'. Hook and Montenari agree that the tail of the comet was directed towards Spica ♍, 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 opposition of the sun towards the north.