heads. In the former case, we must shorten the distance of the comets,lest we be obliged to allow that the smoke arising from their heads ispropagated through such a vast extent of space, and with such a velocityand expansion as will seem altogether incredible ; in the latter case, thewhole light of both head arid tail is to be ascribed to the central nucleus.But. then, if we suppose all this light to be united and condensed withinthe disk of the nucleus, certainly the nucleus will by far exceed Jupiteritself in splendor, especially when it emits a very large and lucid tail If.therefore, under a less apparent diameter, it reflects more light, it must bemuch more illuminated by the sun, and therefore much nearer to it; andthe same argument will bring down the heads of comets sometimes withinthe orb of Venus, viz., when, being hid under the sun s rays, they emit suchhuge and splendid tails, like beams of fire, as sometimes they do ; for if allthat light was supposed to be gathered together into one star, it wouldsometimes exceed not one Venus only, but a great many such unitedinto one.Lastly ; the same thing is inferred from the light of the heads, whichincreases in the recess of the cornets from the earth towards the sun, anddecreases in their return from the sun towards the earth ; for so the cometof the year 1665 (by the observations of Hevelius], from the time that itwas first seen, was always losing of its apparent motion, and therefore hadalready passed its perigee ; but yet the splendor of its head was daily increasing, till, being hid under the sun s rays, the comet ceased to appear.The comet of the year 1683 (by the observations of the same LJevelius),about the end of July, when it first appeared, moved at a very slow rate,advancing only about 40 or 45 minutes in its orb in a day s time ; butfrom that time its diurnal motion was continually upon the increase, tillSeptember 4, when it arose to about 5 degrees ; and therefore, in all thisinterval of time, the comet was approaching to the earth. Which is likewise proved from the diameter of its head, measured with a micrometer ;for, August 6, Hevelius found it only 6 05", including the coma, which,September 2 he observed to be 9 07", and therefore its head appeared farless about, the beginning than towards the end of the motion ; thoughabout the beginning, because nearer to the sun, it appeared far more lucidthan towards the end, as the same Hevelius declares. Wherefore in allthis interval of time, on account of its recess from the sun, it decreases464 THE MATHEMATICAL PRINCIPLES [BOOK III.in splendor, notwithstanding its access towards the earth. The comet ofthe year 1618, about the middle of December, and that of the year 1680,about the end of the same month, did both move with their greatest velocity, and were therefore then in their perigees : but the greatest splendorof their heads was seen two weeks before, when they had just got clear ofthe sun s rays ; and the greatest splendor of their tails a little more early,when yet nearer to the sun. The head of the former comet (according tothe observations of Cysdtus], Dece/itber 1, appeared greater than the starsof^the first magnitude: and, December 16 (then in the perigee), it wasbut little diminished in magnitude, but in the splendor and brightness ofits light a great deal. January 7, Kepler, being uncertain about thehead, left oif observing. December 12, the head of the latter comet wasseen and observed by Mr. Flamsted, when but 9 degrees distant from thesun ; which is scarcely to be done in a star of the third magnitude. December 15 and 17, it appeared as a star of the third magnitude, its lustrebeing diminished by the brightness of the clouds near the setting sun.December 26, when it moved with the greatest velocity, being almost inits perigee, it was less than the mouth of Pegasus, a star of the thirdmagnitude. January 3, it appeared as a star of the fourth. January 9,as one of the fifth. January 13, it was hid by the splendor of the moon,then in her increase. January 25, it was scarcely equal to the stars ofthe seventh magnitude. If we compare equal intervals of time on oneside and on the other from the perigee, we shall find that the head of thecomet, which at both intervals of time was far, but yet equally, removedfrom the earth, and should have therefore shone with equal splendor, appeared brightest on the side of the perigee towards the sun, and disappeared on the other. Therefore, from the great difference of light in theone situation and in the other, we conclude the great vicinity of the sunand comet in the former ; for the light of comets uses to be regular, andto appear greatest when the heads move fastest, and are therefore in theirperigees ; excepting in so far as it is increased by their nearness to thesun. -COR. 1. Therefore the comets shine by the sun s light, which they reflect.COR. 2. From what has been said, we may likewise understand whycomets are so frequently seen in that hemisphere in which the sun is, andso seldom in the other. If they were visible in the regions far aboveSaturn, they would appear more frequently in the parts opposite to thesun ;for such as were in those parts would be nearer to the earth, whereasthe presence of the sun must obscure and hide those that appear in thehemisphere in which he is. Yet, looking over the history of comets, Ifind that four or five times more have been seen in the hemisphere towardsthe sun than in the opposite hemisphere ; besides, without doubt, not afew, which have been hid by the light of the sun : for comets descendingBOOK III.]0* NATURAL PHILOSOPHY. !fi5into our parts neither emit tails, nor are so well illuminated by the sun,as to discover themselves to our naked eyes, until they are come nearer tous than Jupiter. But the far greater part of that spherical space, whichis described about the sun with so small an interval, lies on that side ofthe earth which regards the sun ; and the comets in that greater part arecommonly more strongly illuminated, as being for the most part nearer tothe sun.COR. 3. Hence also it is evident that the celestial spaces are void ofresistance ;for though the comets are carried in oblique paths, and sometimes contrary to the course of the planets, yet they move every way withthe greatest freedom, and preserve their motions for an exceeding longtime, even where contrary to the course of the planets. I am out in myjudgment if they are not a sort of planets revolving in orbits returninginto themselves with a perpetual motion ; for, as to what some writerscontend, that they are no other than meteors, led into this opinion by theperpetual changes that happen to their heads, it seems to have no foundation ;for the heads of comets are encompassed with huge atmospheres,and the lowermost parts of these atmospheres must be the densest; andtherefore it is in the clouds only, not in the bodies of the comets themselves, that these changes are seen. Thus the earth, if it was viewed fromthe planets, would, without all doubt, shine by the light of its clouds, andthe solid body would scarcely appear through the surrounding clouds.Thus also the belts of Jupiter are formed in the clouds of that planet,for they change their position one to another, and the solid body of Jupiteris hardly to be seen through them ; and much more must the bodies ofcomets be hid under their atmospheres, which are both deeper and thicker.PROPOSITION XL. THEOREM XX.That the comets mnve in some of the conic sections, having their fociin the centre of the sun ; and by radii drawn to the sun describeareas proportional to the times.This proposition appears from Cor. 1, Prop. XIII, Book 1, comparedvith Prop. VIII, XII, and XIII, Book HI.COR. 1. Hence if comets are revolved in orbits returning into themselves, those orbits will be ellipses ; and their periodic times be to theperiodic times of the planets in the sesquiplicate proportion of their principal axes. And therefore the comets, which for the most part of theircourse are higher than the planets, and upon that account describe orbitswith greater axes, will require a longer time to finish their revolutions.Thus if the axis of a comet s orbit was four times greater than the axisof the orbit of Saturn, the time of the revolution of the comet would beto the time of the revolution of Saturn, that is, to 30 years, as 4 ^/ 4(or 8) to 1, and would therefore be 240 years.30THE MATHEMATICAL PRINCIPLES [BOOK III.COR. 2. But their orbits will be so near to parabolas, that parabolasmay be used for them without sensible error.COR. 3. And, therefore, by Cor. 7, Prop. XVI, Book 1, the velocity ofevery comet will always be to the velocity of any planet, supposed to berevolved at the same distance in a circle about the sun, nearly in the subduplicateproportion of double the distance of the planet from the centreof the sun to the distance of the comet from the sun s centre, very nearly.Let us suppose the radius of the orbis wagmis, or the greatest semidiameterof the ellipsis which the earth describes, to consist of 100000000parts ; and then the earth by its mean diurnal motion will describe1720212 of those parts, and 716751 by its horary motion. And therefore the comet, at the same mean distance of the earth from the sun, witha velocity which is to the velocity of the earth as v/ 2 to I, would by itsdiurnal motion describe 2432747 parts, and 101.3641 parts by its horarymotion. But at greater or less distances both the diurnal and horarymotion will be to this diurnal and horary motion in the reciprocal subduplicateproportion of the distances, and is therefore given.COR. 4. Wherefore if the lattis rectum of the parabola is quadruple ofthe radius of the orbis maginis, and the square of that radius is supposed to consist of 100000000 parts, the area which the comet will dailydescribe by a radius drawn to the sun will be 12163731 parts, and thehorary area will be 506821 parts. But, if the latus rectum is greateror less in any proportion, the diurnal and horary area will be less orgreater in the subduplicate of the same proportion reciprocally.LEMMA V.Tofind a curve line of the parabolic kind which shall pass through anygiven number of points.Let those points be A, B, C, D, E, F, (fee., and from the same to anyright line HN, given in position, let fall as many perpendiculars AH, BI,CK, DL, EM, FN, tfec.b 2b 3b 45 5bc 2c 3c 4cd 2d 3dHe 2efCASE 1. If HI, IK, KL, &c., the intervals of the points H, I, K, L, MN, (fee., are equal, take b, 2b, 3b, 46, 56, (fee., the first differences of the perpendiculars AH. BI, CK, (fee.;their second differences c, 2c, 3c, 4r, <fec. :their third, d, 2d, 3d, (fee., that is to say, so as AH BI may be== b, 01BOOK III.] OF NATURAL PHILOSOPHY. 467CK = 2b, CK DL = 36, DL + EM = 46, EM + FN = 56,&c. ; then 6 2b == c, &c., and so on to the last difference, which is here/*. Then, erecting any perpendicular RS, which may be considered as anordinate of the curve required, in order to find the length of this ordinatc,suppose the intervals HI. IK, KL, LM, (fee., to be units, and let AH = a.-KS=f>, 。p into IS = q, q into + SK = r, into + SL = s,。s into 4- SM = t ; proceeding, to wit, to ME, the last perpendicular butone, and prefixing negative signs before the terms HS, IS, &c., which liefrom S towards A; and affirmative signs before the terms SK, SL, (fee..which lie on the other side of the point S ; and, observing well the signs,RS will be = a + bp + cq + dr + es + ft, + (fee.CASE 2. But if HI, IK, (fee., the intervals of the points H, I, K, L, <fcc.,are unequal, take 6, 26, 36, 46, 56, (fee., the first differences of the perpendiculars AH, BI, CK, cfec., divided by the intervals between those perpendiculars ; c, 2Cj 3c, 4c, (fee., their second differences, divided by the intervalsbetween every two ; c/, 2d, 3d, (fee., their third differences, divided by theintervals between every three; e, 2e, (fee., their fourth differences, dividedby the intervals between every four ; and so forth;that is, in such manner,AH BI*BI CK , CK DLthat b may be = ---^ , 2b = --.-==, 6b = --==- -----, (fee., then2b 2b 3b 36 46(c 2c&c then rf - "2</2c 3c-=j-T7, (fee. And those differences being found, let AH be = a,HS = p, p into IS = q, q into + SK = r, r into + SL =. s, s into-f- SM = t; proceeding, to wit, to ME, the last perpendicular but one : .and the ordinate RS will be = a -f- bp + cq + dr + es -f //, + tfec.COR. Hence the areas of all curves may be nearly found ; for if somenumber of points of the curve to be squared are found, and a parabola besupposed to be drawn through those points, the area of this parabola willibe nearly the same with the area of the curvilinear figure proposed to besquared : but the parabola can be always squared geometrically by methodsvulgarly known.LEMMA VI.Certain observed places of a comet being" given, to find the place of thesame, to any intermediate given time.Let HI, IK, KL, LM (in the preceding Fig.), represent the times betweenthe observations ; HA, IB, KC, LD, ME, five observed longitudes of thecomet ; and HS the given time between the first observation and the longitude required. Then if a regular curve ABODE is supposed to be drawnthrough the points A, B, C, D, E, and the ordinate RS is found out by thepreceding lemma, RS will be the longitude required.46S THE MATHEMATICAL PRINCIPLES [BooK III.After the same method, from five observed latitudes, we may find thelatitude to a given time.If the differences of the observed longitudes are small, suppose of 4 or 5degrees, three or four observations will be sufficient to find a new longitudeand latitude : but if the differences are greater, as of 10 or 20 degrees, fiveobservations ought to be used.LEMMA VII.Through a given point P to draw a right line BC, rvhose parts PB, PC,cut off by two right lines AB, AC, given in position, may be one to theother in. a given proportion.Pi。 From the given point P suppose any right linePD to be drawn to either of the right lines given,as AB; and produce the same towards AC, theother given right line, as far as E; so as PE maybe to PD in the given proportion. Let EC beparallel to A D. Draw CPB, and PC will be to PBas PE to PD. Q.E.F.LEMMA VIII.Let ABC be a parabola, having its focus in S. By the chord AC bisected in I cut off the segment ABCI, ivhose diameter is Ip and vertexI . In I/i produced take pO equal to one half of I//. Join OS, andproduce it to so as S may be equal to 2SO. Now, supposing a cometto revolve in the arc CBA, draw B, cutting AC in E ; I say, the pointE will cut offfrom the chord AC the segment AE, nearly proportionalto the time.For if we join EO, cutting the parabolic arc ABC in Y, and draw //Xtouching the same arc in the vertex //, and meeting EO in X, the curvilinear area AEXjuA will be to the curvilinear area ACY//A as AE to AC ;and. therefore, since the triangle ASE is to the triangle ASC in the sameproportion, the whole area ASEXjuA will be to the whole area ASCY/^A asBOOK II Lj OF NATURAL PHILOSOPHY. 469AE to AC. But, because O is to SO as 3 to 1, and EG to XC in the sameproportion, SX will be parallel to EB ; and, therefore, joining BX, the triangle SEB will be equal to thetriangle XEB. Wherefore if to the areaASEX.uA we add the triangle EXB, and from the sum subduct the triangleSEB, there will remain the area ASBX,wA, equal to the area ASEX/^A. andtherefore in proportion to the area ASCY//A as AE to AC. But the areaASBYwA is nearly equal to the area ASBX//A; and this area ASBY/zAis to the area ASCYwA as the time of description of the arc AB to thetime of description of the whole arc AC ; and, therefore, AE is to ACnearly in the proportion of the times. Q.E.D.COR. When the point B falls upon the vertex 。i of the parabola, AE isto AC accurately in the proportion of the times.SCHOLIUM.If we join // cutting AC in d, and in it take //, in proportion to ^B as27MI to 16Mf/, and draw B/?, this Bu will cut the chord AC, in the proportion of the times, more accurately than before; but the point n is to betaken beyond or on this side the point , according as the point B ismore or less distant from the principal vertex of the parabola than thepoint p.LEMMA IX.AI 2The right lines Ip and /zM, and the length j~-, are equal among themselves.For 4.S/Z is the latus rectum of the parabola belonging to the vertex ft.LEMMA X.Produce Su to N and P, so as ^N may be one third of //I, and SP maybe to SN as SN to S" ; and in the time that a comet would describethe arc AjuC. if it was supposed to move always forwards with the velocity which it hath in a height equal to SP, it would describe a lengthequal to the chord AC.For if the comet with the velocitywhich it hath in 。i was in the said timesupposed to move uniformly forward inthe right line which touches the parabolain p, the area which it would describe bya radius drawn to the point S would beequal to the parabolic area ASC/zA ; andtherefore the space contained under thelength described in the tangent and thelength Su would be to the space contained under the lengths AC and SM as the4/0 THE MATHEMATICAL PRINCIPLES [BOOK 111area ASC//A to the triangle A SO, that is, as SN to SM. Wherefore ACis to the length described in the tangent as Sf* to SN. But since the velocity of the comet in the height SP (by Cor. 6, Prop. XVI., Book I ) is tothe velocity of the same in the height Sfi in the reciprocal subduplicateproportion of SP to Sft, that is, in the proportion of S/^ to SN, the lengthdescribed with this velocity will be to the length in the same time describedin the tangent as Su to SN. Wherefore since AC, and the length describedwith this new velocity, are in the same proportion to the length describedin the tangent, they must be equal betwixt themselves. Q.E.D.COR. Therefore a comet, with that velocity which it hath in the heightS/x + fI,, would in the same time describe the chord AC nearly.LEMMA XI.If a comet void of all motion was let fallfrom the heigJit SN, or $n +J Ift, towards the sun, and was still impelled to the sun by the sameforce uniformly continued by ivhich it was impelled at first, the same,in, one half of that time in which it might describe the arc AC in itsown orbit, would in. descending describe a space equal to the fen giftfaFor 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 ithath in the height SP, describe the chord AC: and, therefore (by Cor. 7,Prop. XVI, Book 1), if it was in the same time supposed to revolve by theforce of its own gravity in a circle whose semi- diameter was SP. it woulddescribe an arc of that circle, the length of which would be to the chordof 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 1) in half the said time describe a space equal to thesquare of half the said chord applied to quadruple the height SP, that is,AI 2it would describe the space ,^p.But since the weight of the comettowards the sun in the height SN is tothe weight of the same towards thesun in the height SP as SP to S^, thecomet, by the weight which it hath inthe height SN. in falling from thatheight towards the sun, would in tin:AI 2same time describe the space 7^-; that4S^is, a space equa] to the length I// OTwM. Q.E.DBOOK III.] OF NATURAL PHILOSOPHY. 47PROPOSITION XLL PROBLEM XXI.Prom three observations given to determine the orbit of a comet movingin a parabola.This being a Problem of very great difficulty, I tried many methods ofresolving it; and several of these Problems, the composition whereof Ihave triven in the first Book, tended to this purpose. But afterwards Icontrived the following solution, which is something more simple.Select three observations distant one from another by intervals of timenearly equal ; but let that interval of time in which the comet movesmore 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 theAtimes to about 600 days ; or that the point E may fall upon M nearly,and may err therefrom rather towards 1 than towards A. If such directobservations are not at hand, a new place of the comet must be found, byLem. VI.Let S represent the sun ; T, t,r} three places of the earth in the orbismag-mis; TA, /B, rC, three observed longitudes of the comet; V thetime between the first observation and the second ; W the time betweenthe second and the third; X the length which in the whole time V + Wthe comet might describe with that velocity which it hath in the meandistance of the earth from the sun, which length is to be found by Cor. 3,THE MATHEMATICAL PRINCIPLES [BOOK III.Prop. XL, Book III;and tV a perpendicular upon the chord TT. In themean observed longitude tfB take at pleasure the point B, for the place ofthe comet in the plane of the ecliptic ; and from thence, towards the sunS, draw the line BE, which may be to the perpendicular /V as the contentunder SB and St 2 to the cube of the hypothenuse of the right angled triangle, whose sides are SB, and the tangent of the latitude of the comet inthe second observation to the radius ^B. And through the point E (byLemma VII) draw the right line AEC, whose parts AE and EC, terminating in the right lines TA and rC. may be one to the other as the times Vand 。V : then A and C will be nearly the places of the comet in the planeof the ecliptic in the first and third observations, if B was its placerightly assumed in the second.Upon AC, bisected in I, erect the perpendicular li. Through B draw