自然哲学的数学原理-52

the obscure line Ei parallel to AC. Join the obscure line Si, cutting ACin A, and complete the parallelogram il AJU. Take 。o equal to 3IA ; andthrough the sun S draw the obscure line<0 equal to 3So -f 3 fa. Then,cancelling the letters A, E, C, I, from the point B towards the point ,draw the new obscure line BE, which may be to the former BE in theduplicate proportion of the distance BS to the quantity Sju + 1 fa. Andthrough the point E draw again the right line AEC by the same rule asbefore ; that is, so as its parts AE and EC may be one to the other as thetimes V and W between the observations. Thus A and C will be theplaces of the comet more accurately.Upon AC, bisected in I, erect the perpendiculars AM, CN, IO, of whichAM and CN may be the tangents of the latitudes in the first and third observations, to the radii TA and TC. Join MN, cutting IO in O. Draw therectangular parallelogram zlAjt/, as before. In IA produced take ID equal toSfi + f fa. Then in MN, towards N, take MP, which may be to theabove found length X in the subduplicate proportion of the mean distanceof the earth from the sun (or of the semi-diameter of the orbis tnagnus]to the distance OD. If the point P fall upon the point N; A, B, and C,<vill be three places of the comet, through which its orbit is to be describedin the plane of the ecliptic. But if the point P falls not upon the pointN, in the right line AC take CG equal to NP, so as the points G and Pmay 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 6 and j3 assumed at pleasure, find out thenew points e, a, c, g ; and e, a, , y. Then through G, g-, and y, draw thecircumference of a circle G^y, cutting the right line rC in Z : and Z willbe one place of the comet in the plane of the ecliptic. And in AC, ac, OK,making AF, a/, a</>, equal respectively to CG, eg, KJ ; through the points P,f, and 0, draw the circumference of a circleVf<t>, cutting the right line ATin X ; and the point X will be another place of the comet in the plane ofBOOK III.] OF NATURAL PHILOSOPHY. 4~3the ecliptic. And at the points X and Z, erecting the tangents of thelatitudes of the comet to the radii TX and rZ, two places of the comet inits own orbit will be determined. Lastly, if (by Prop. XIX., Book 1) tothe focus S a parabola is described passing through those two places, thisparabola will be the orbit of the comet. Q.E.LThe 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. VI L, as it ought to be, by Lem. VIII.; and BE, by Lem. XL, is aportion of the right line BS or B in the plane of the ecliptic, interceptedbetween the arc ABC and the chord AEC ; and MP (by Cor. Lem. X.) isthe length of the chord of that arc, which the comet should describe in itsproper orbit between the firs : and third observation, and therefore is equalto MN, providing B is a true place of the comet in the plane of theecliptic.But it will be convenient to assume the points B, b, (3, not at random,but nearly true. If the angle AQ/, at which the projection of the orbit inthe plane of the ecliptic cuts the right line B, is rudely known, at thatangle with Bt draw the obscure line AC, which may be to -fTT in the subduplicateproportion of SQ, to S/ ; and, drawing the right line SEB so asits part EB may be equal to the length , the point B will be determined,which we are to use for the first time. Then, cancelling the right lineAC, and drawing anew AC according to the preceding construction, and,aioreover, finding the length MP, in tB take the point b, by this rule, that,if TA and rC intersect each other in Y, the distance Y6 may be to thedistance YB in a proportion compounded of the proportion of MP to MN,and the subduplicate proportion of SB to Sb. And by the same methodyou may find the third point 18, if you please to repeat the operation thethird time ; but if this method is followed, two operations generally will besufficient ; for if the distance Bb happens to be very small, after the pointsF,/, and G, , are found, draw the right lines F/and G^-, and they willcut TA and rC in the points required, X and Z.EXAMPLE.Let the comet of the year 1680 be proposed. The following table shewsthe motion thereof, as observed by Flamsted, and calculated afterwards byhim from his observations, and corrected by Dr. Halley from the same observations.47.1 THE MATHEMATICAL PRINCIPLES FBooK III.To these you may add some observations of mine.These observations were made by a telescope of 7 feet, with a micrometer and threads placed in the focus of the telescope; by Avhich instrumentswe determined the positions both of the fixed stars among themselves, andof the comet in respect of the fixed stars. Let A represent the star of thefourth magnitude in the left heel of Perseus (Bayer s o), B the followingstar of the third magnitude in the left foot (Bayer s s), C a star of thesixth magnitude (Bayer s 11} in the heel of the same foot, and 1). E, F, G,H, I, K. L, M, N, O, Z, a, j3, y, S, other smaller stars in the same foot;and let p, P, Q, R, S, T, V, X, represent the places of the comet in theobservations above set down ; and, reckoning the distance AB of 80 r。 parts,AC was 52i of those parts; BC, 5Sf ; AD, 57T。 ; BD, S2 T"T ; CD, 23f :AE, 29i ; CE, 57i ; DE, 49J4 ; AI, 27 T。 ; BI, 52} ; OF, 36 rV ; Dl, 53/r ;AK, 38| ; BK, 43; OK, 31$; FK, 29; FB, 23; FC, 36i ; AH, 1S| ;DH, 50J; BN, 46 T。 ; ON, 31 1; BL, 45T。; NL, 31f HO was to HIas 7 to 6, and. produced, did pass between the stars D and E, so as thedistance of the star D from this right line was jCD. LM was to LN as2 to 9, and, produced, did pass through the star H. Thus were the positions of the fixed stars determined in respect of one another.3,K)K HI.] OF NATURAL PHILOSOPHY. 475*2Mr. Pound has since observed a second time the positions of thcst fixedstars amongst themselves, and collected their longitudes and lat" /udes according to the following table4^6 THE MATHEMATICAL PRINCIPLES [BOOK III.The positions of the comet to these fixed stars were observed to be asfollow :Friday, February 25, O.S. at 8ih. P. M. the distance of the comet in pfrom the star E wai less than T。AE, and greater than }AE, and thereforenearly equal to T3S AE; and the angle AjoE was a little obtuse, but almostright. For from A, letting fall a perpendicular on pE, the distance of thecomet from that perpendicular was j/E.The same night, at 9|h., the distance of the comet in P from the star Ewas greater than AE, and less than AE, and therefore nearly equaltoj^-of AE, or /^ AE. But the distance of the comet from the perpen-^8"dicular let fall from the star A upon the right line PE was jPE.Sunday, February 27, 8|h. P. M. the distance of the comet in Q, fromthe star O was equal to the distance of the stars O and H and the risjhtline QO produced passed between the stars K and B. I could not, byreason of intervening clouds, determine the position of the star to greateraccuracy.Tuesday, March 1, ] l 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 alittle greater than CK, and a little less than JCK + jCR, and therefore = iCK + A CR, or ifCK.Wednesday, March 2, S1. P. M. the distance of the comet in S from thestar C was nearly FC ; the distance of the star F from the right line OSproduced was g^FC ; and the distance of the star B from the same rightline was five times greater than the distance of the star F ; and the rightline NS produced passed between the stars H and I five or six times nearerto the star H than to the star I.Saturday, March 5. lHh. P. M. when the comet was in T, the right lineMT was equal to ^ML, and the right line LT produced passed between Band F four or five times nearer to F than to B, cutting off from BF a fifthor sixth part thereof towards F : and MT produced passed on the outsideof the space BF towards the star B four times nearer to the star B thanto 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, Qih. P. M. the comet being in V, the right line Vaproduced did pass between B and F, cutting off, from BF towards F, T。 ofBF, and was to the right line Yj3 as 5 to 4. And the distance of the cometfrom the right line a(3 was |V/3.Wednesday, March 9, S|-h. P. M. the comet being in X, the right lineyX was equal tojy<? ; and the perpendicular let fall from the star 6 uponthe right yX was f of yd.The same night, at 12h. the comet being in Y, the right line yY wasBOOK III.] OF NATURAL PHILOSOPHY. 477equal to ^ of yd, or a little less, as perhaps T5g of yd ; and a perpendicularlet 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 discernible, and therefore its place could not be determined with that certainty asin the foregoing observations.Prom these observations, by constructions of figures and calculations, Ideduced the longitudes and latitudes of the comet ; and Mr. Pound, bycorrecting the places of the fixed stars, hath determined more correctly theplaces of the comet, which correct places are set down above. Though mymicrometer was none of the best, yet the errors in longitude and latitude(as derived from my observations) scarcely exceed one minute. The comet(according to my observations), about the end of its motion. besraD **> J;;oiinesensibly towards the north, from the parallel which it described about theend of February.Now, in order to determine the orbit of the comet out of the observationsabove described, I selected those three which Flamsted made, Dec. 21, Jan.5, and Jan. 25; from which I found S^ of 9842,1 parts, and V of 455such as the semi-diameter of the orbis magnus contains 10000. Then forthe first observation, assuming tE cf 5657 of those parts, 1 found SB 9747,BE for the first time 412, Sji 9503, U 413, BE for the second time 421,OD 10186, X 8528,4, PM 8450, MN 8475, NP 25; from whence, by thesecond operation. I collected the distance tb 5640 ; and by this operation 1at last deduced the distances TX 4775 and rZ 11322. From which, limiting the orbit, I found its descending node in 25, and ascending node in V?1 53 ; the inclination of its plane to the plane of the ecliptic 61 20^ ,the vertex thereof (or the perihelion of the comet) distant from the node8 38 , and in t 27 43 , with latitude 7 34 south; its lotus return236.8; and the diurnal area described by a radius drawn to the sun 93585,supposing the square of the semi-diameter of the orbis magnus lOUOOOOOO ;that the comet in this orbit moved directly according to the order of thesigns, and on DM. 8(1. OO1. 04 P. M was in the vertex or perihelion of itsorbit. All which I determined by scale and compass, and the chords ofangles, taken from the table of natural sines, in a pretty large figure, inwhich, to wit, the radius of the orbis magnus (consisting of 10000 parts)was equal to 16^ inches of an English foot.Lastly, in order to discover whether the comet did truly move in theorbit so determined, I investigated its places in this orbit partly by arithmetical operations, and partly by scale and compass, to the times of gomeof the observations, as may be seen in the following table :478 THE MATHEMATICAL PRINCIPLES [BOOK III,IBut 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 s and ^ 1 53 , and the inclination of the plane of the orbit to the ecliptic 61 20| , as well as the timeof the comet s being in perihelio, Dec. 8(i. OUh. 04 , he found the distanceof the perihelion from the ascending node measured in the comet s orbit9 20 , and the Ititus rectum of the parabola 2430 parts, supposing themean distance of the sun from the earth to be 100000 parts ;arid fromthese data, by an accurate arithmetical calculus, he computed the placesof the comet to the times of the observations as follows :This comet also appeared in the November before, and at Coburg, inSaxony, was observed by Mr. Gottfried Kirch, on the 4th of that month, onthe 6th and llth O. S.; from its positions to the nearest fixed stars observedwith sufficient accuracy, sometimes with a two feet, and sometimes with aten 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 :BOOK III.] OF NATURAL PHILOSOPHY. 479Nov. 3, 17h. 2 , apparent time fit London, the comet was in 71 29 deg.51 , with 1 deg. 17 45" latitude north.November 5. 15h. 58 the comet was in ^ 3 23 , with 1 6 nortl lat.November 10, 16h. 31 , the comet was equally distant from two stars in1, which are <r and T in Bayer ; but it had not quite touched the rightline that joins them, but was very little distant from it. In Flamstecfscatalogue this star o was then in ^ 14 15 , with 1 deg. 41 lat. northnearly, and r in W 17 3^ with deg. 34 lat. south; and the middlepoint between those stars was lrJZ 15 39} , with 33i lat. north. Letthe distance of the cornet from that right line be about 10 or 12 : andthe difference of the longitude of the comet and that middle point will be7 ;arid the difference of the latitude nearly 7。 ; and thence it followsthat the comet was in T02 15 32 , with about 26 lat. north.The first observation from the position of the comet with respect trcertain small fixed stars had all the exactness that could be desired ; UKsecond also was accurate enough. In the third observation, which was theleast accurate, there might be an error of 6 or 7 minutes, but hardlygreater. The longitude of the comet, as found in the first and mostaccurate observation, being computed in the aforesaid parabolic orbit,comes out U 29 30 22", its latitude north 1 25 7", and its distancefrom the sun 115546.Moreover, Dr. Halley, observing that a remarkable comet had appearedfour times at equal intervals of 575 years (that is, in the month of September after Julius Ccesar was killed ; An. Chr. 531, in the consulate ofLainpadins and Orestes; An. Chr. 1106, in the month of February ;and at the end of the year 16SO; and that with a long and remarkabletail, except when it was seen after C(BsaiJs death, at which time, by reasonof the inconvenient, situation of the earth, the tail was not so conspicuous),set himself to find out an elliptic orbit whose greater axis should be1382957 parts, the mean distance of the earth from the sun containing10000 such ;in which orbit a comet might revolve in 575 years ; and,placing the ascending node in 25 2 2 , the inclination of the plane of theorbit to the plane of the ecliptic in an angle of 61 6 48", the perihelionof the comet in this plane in t 22 44 25", the equal time of the perihelion December 7 1. 23h. 9 , the distance of the perihelion from the ascending node in the plane of the ecliptic 9^ 17 35", and its conjugate axis18481,2, he computed the motions of the comet in this elliptic orbit. Theplaces of the comet, as deduced from the observations, and as arising fromcomputation made in this orbit, may be seen in the following table.480 THE MATHEMATICAL PRINCIPLES [BOOK 111The observations of this comet from the beginning to the end agree atporfectly with the motion of the comet in the orbit just now described asthe motions of the planets do with the theories from whence they are calculated ; and by this agreement plainly evince that it was one and thesame comet that appeared all that time, and also that the orbit of thatcomet 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. Ponthczns and his companions, at 6h. in the morning at Rome (that is, 5h. 10 at London], by threadsdirected to the fixed stars, observed the comet in === 8 30 , with latitude40 south. Their observations may be seen in a treatise which Ponthc&uspublished concerning this comet. Celliits, who was present, and communicated his observations in a letter to Cassitn} saw the comet at the samehour in ^= 8 30 , with latitude 30 south. It was likewise seen byGalletius at the same hour at Avignon (that is, at 5h. 42 morning atLondon] in ^= 8 without latitude. But by the theory the comet was atthat time in ^ 8 16 45", and its latitude was 53 7" south.Nov. 18, at 6h. 30 in the morning at Rome (that is, at 5h. 40 at London), PonthcEns observed the comet in ^ 13 30 , with latitude 1 20BOOK III.] OF NATURAL PHII OSOPHY. 48lsouth ; and Cellius in ^ 13 30 , with latitude 1 00 south. But at 5b.30 in the morning at Aviation, Galletius saw it in ^ 13 00 , with latitude 1 00 south. In the University of La Fleche, in Prance, at 5h. inthe morning (that is. at 5h. 9 at London.}, it was seen by P. Ango, in themiddle between two small stars, one of which is the middle of the threewhich lie in a right line in the southern hand of Virgo, Bayers i/> ; andthe other is the outmost of the wing, Bayer s 0. Whence the comet wasthen in ^ 12 46 with latitude 50 south. And I was informed by Dr.ffalley, that on the same day at Boston in New England, in the latitudeof 42| deg. at 5h. in the morning (that is, at 9h. 44 in the morning atLondon), the comet was seen near === 14, with latitude 1 30 south.Nov. 19, at 4|h. at Cambridge, the comet (by the observation of ayoung man) was distant from Spica $ about 2 towards the north west.Now the spike was at that time in ^ 19 23 47", with latitude 2 1 59"south. The same day, at 5h. in the morning, at Boston in New England,the comet wTas distant from Spica nj? 1, with the difference of 40 in latitude. The same day, in the island of Jamaica, it was about 1 distantfrom Spica W. The same day, Mr. Arthur Storer, at the river Patuxent,near Hunting Creek, in Maryland, in the confines of Virginia, in lat.38i, at 5 in the morning (that is, at 10h. at London), saw the cometabove Spica W, and very nearly joined with it, the distance between thembeing about of one deg. And from these observations compared, I conclude, that at 9h. 44 at London, the comet was in === 18 50 , with about1 25 latitude south. Now by the theory the comet was at that time in^ 18 52 15", with 1 26 54" lat. south.Nov. 20, Montenari, professor of astronomy at Padua, at 6h. in themorning at Venice (that is, 5h. 10 at London), saw the comet in === 23,with latitude 1 30 south. The same day, at Boston, it was distant fromSpica W by about 4 of longitude east, and therefore was in ^ 23 24nearly.Nov. 21, Ponthceus and his companions, at 7|h. in the morning, observed the comet in == 27 50 , with latitude 1 16 south ; Cellius, in ^=28 ; P. Ango at 5h. in the morning, in === 27 45 ; Montenari in ^27 51 . The same day, in the island of Jamaica, it was seen near thebeginning of ^1, and of about the same latitude with Spica u%, that is, 22 . The same day, at 5h. morning, at Ballasore, in the East Indies (thatis, at ll h. 20 of the night preceding at London), the distance of thecomet from Spica W was taken 7 35 to the east. It was in a right linebetween the spike and the balance, and therefore was then in == 26 58 ,with about 1 11 lat. south; and after 5h. 40 (that is. at 5h. morning atLondon), it was in === 28 12 . with 1 16 lat. south. Now by the theorythe comet was then in *= 28 10 36", with 1 53 35" lat. south.Nov. 22, the comet was seen by Montenari in ^ 2 33 : hut at Boston31482 THE MATHEMATICAL PRINCIPLES [BOOK 111.in New England, it was found in about ^l 3, and with almost the samelatitude as before, that is, 1 30 . The same day, at 5h. morning atBallasore, the comet was observed in ^l 1 50 ; and therefore at 5h. morning at London, the comet was in iU 3 5 nearly. The same day, at 6^h.in the morning at London, Dr. Hook observed it in about nt 3 30 , andthat in the right line which passeth through Spica ^ and Cor Leonis ;not, indeed, exactly, but deviating a little from that line towards thenorth. Montenari likewise observed, that this day, and some days after,a right line drawn from the comet through Spica passed by the southside of Cor Lt>oi。is at a very small distance therefrom. The right linethrough Cor Leonis and Spica ^ did cut the ecliptic in ^ 3 46 at anano-le of 2 51 ; and if the comet had been in this line and in W. 3, itslatitude would have been 2 26 ; but since Hook and Montenari agreethat the comet was at some small distance from this line towards thenorth, its latitude must have been something less. On the 20th, by theobservation of Montenari, its latitude was almost the same with that ofSpica ^l7, that is, about 1 30 . But by the agreement of Hook, Montenari,and Align, the latitude was continually increasing, and thereforemust now, on the 22ci be sensibly greater than t 30 : and, taking amean between the extreme limits but now stated. 2 26 and 1 30 , thelatitude will be about 1 58 . Hook and Montenari agree that the tailof the comet was "directed towards Spica W, declining a little from thatstar towards the south according to Hook, but towards the north accordingto Montenari ; and, therefore, that declination was scarcely sensible; and