自然哲学的数学原理-62

which, in Tab. I, belong 33^ days. Lastly ; that memorable comet ofRegiomontanus, which in 1472 was carried through the circum-polarparts of our northern hemisphere with such rapidity as to describe 40566 THE SYSTEM OF THE WORLD.degrees in one day, entered the sphere of the orbis magnus Jan 21, abonlthe time that it was passing by the pole, and, hastening from them*towards the sun, was hid under the sun s rays about the end of Feb. ,whence it is probable that 30 days, or a few more, were spent between itsingress into the sphere of the orbis magnus and its perihelion. Nor didthis comet truly move with more velocity than other comets, but owed thegreatness of its apparent velocity to its passing by the earth at a neardistance.It appears, then, that the velocity of comets (p. 471), so far as it can bedetermined by these rude ways of computing, is that very velocity withwhich parabolas, or ellipses near to parabolas, ought to be described; andtherefore the distance between a comet and the sun being given, the velocityof the comet is nearly given. And hence arises this problem.4 PROBLEM.The relation betwixt the velocity of a comet and its distance from thesun s centre being given, the comet s trajectory is required.If this problem was resolved, we should thence have a method of determining the trajectories of comets to the greatest accuracy : for if that relation be twice assumed, and from thence the trajectory be twice computed,and the error of each trajectory be found from observations, the assumptionmay be corrected by the Rule of False, and a third trajectory may thencebe found that will exactly agree with the observations. And bv determining the trajectories of comets after this method, we may come" at last,to a more exact knowledge of the parts through which those bodies travel,of the velocities with which they are carried, what sort of trajectories theydescribe, and what are the true magnitudes and forms of their tails according to the various distances of their heads from the sun ; whether, aftercertain intervals of time, the same comets do return again, and in whatperiods they complete their several revolutions. But hhe problem ma? beresolved by determining, first, the hourly motion of a comet to a ffiven timefrom three or more observations, and then deriving the trajectory from thismotion. And thus the invention of the trajectory, depending on one observation, and its hourly motion at the time of this observation, will eitherconfirm or disprove itself; for the conclusion that is drawn from the motion only of an hour or two and a false hypothesis, will never agree withthe motions of the comets from beginning to end. The method of fh*whole computation is this.THE SYSTEM OF THE WORLD. 567LEMMA I.To cut two right lines OR, TP, given in, position, by a third right lineRP, so as TRP may be a right angle ; and, if another right line SPis drawn to any given point S, the solid contained under this line SP5and the square of the right line OR terminated at a given point O,may be of a given magnitude.It is done by linear description thus. Let the given magnitude of thesolid be M2 x N : from any point r of the right line OR erect the perpendicular rp meeting TP in p. Then through the point Sp draw theM2 X Nline Sq equal to ^ 2. In like manner draw three or more right linesS2q, S3<7, &c. ; and a regular line q2q3q, drawn through all the pointsy2q3q, &c., will cut the right line TP in the point P, from which the perpendicular PR is to be let fall. Q.E.F.By trigonometry thus. Assuming the right line TP as found by thepreceding method, the perpendiculars TR, SB, in the triangles TPR, TPS,will be thence given ; and the side SP in the triangle SBP, as well as theM2 X Nerror ^r^ SP. Let this error, suppose D, be to a new error, suppose E, as the error 2p2q + 3p3q to the error 2p3p ; or as the error 2p2qH- D to the error 2pP ; and this new error added to or subducted from thelength TP, will give the correct length TP + E. The inspection of thefigure will shew whether we are to add to or subtract ; and if at any timethere should be use for a farther correction, the operation may be repeated668 THE SYSTEM OF THE WORLD.By arithmetic thus. Let us suppose the thing done, and let TP -f- e be thecorrect length of the right line TP as found out by delineation : and thenceTRthe correct lengths of the lines OR. BP, and SP, will be OR ^^e.BP + e, and ^/SP 2 + 2BPe + ee = M2NQRa 20RX iTPop SR 2Whence, by the method of converging series, we have SP -f- -p6 + op~jM2 N 2TR M2 N 3TR 2 M2Nee, <fcc., = 2 + X 3e + x l66 ^ tor the givenM2 N ^ 2TR M2N BP 3TR 2 M2 N SB 2co-efficients ^-2 SP, Tp X-^-3gp> Tppl"xQR4~2SP~JF F Fputting F, , ppj,and carefully observing the signs, wo find F + ^ e -fF eei = 0, and e + YT= G. Whence, neglecting the very small He 2 e 2term ^, e comes out equal to G. If the error ^ is not despicable, takeGjj= e.And it is to be observed that here a general method is hinted at forsolving the more intricate sort of problems, as well by trigonometry as byarithmetic, without those perplexed computations and resolutions of affectedequations which hitherto have been in use.LEMMA II.To cut three right lines given in position by a fourth right line thatshall pass through a point assigned in any of the three, and so as itsintercepted parts shall be in a given ratio one to the other.Let AB, AC, BC, be the right lines given in position, and suppose D tobe the given point in the line AC. Parallel to AB draw DG meeting BCin G ; and, taking GF to BG in the given ratio, draw FDE ; and FDwill be to 1)E as FG to BG. Q.E.F.THE SYSTEM OF THE WORLD. 569By trigonometry thus. In the triangle CGD all the angles and the sideCD are given, and from thence its remaining sides are found ; and fromthe given ratios the lines GF and BE are also given.LEMMA III.Tofind and represent hy a linear description the hourly motion of a cometto any given time.From observations of the best credit, let three longitudes of the cometbe given, and, supposing ATR, RTB, to be their differences, let the hourlymotion be required to the time of the middle observation TR. By LemII. draw the right line ARB, so as its intercepted parts AR, RB, may b<as the times between the observations ; and if we suppose a body in thewhole time to describe the whole line AB with an equal motion, and to bein the mean time viewed from the place T, the apparent motion of thatbody about the point R will be nearly the same with that of the comet atthe time of the observation TR.The same more accurately.Let Ta, T6, be two longitudes given at a greater distance on one sftleand on the other ; and by Lem,. II draw the right line aRb so as its intercepted parts aR, Rft may be as the times between the observations aTR, RTA.Suppose this to cut the lines TA, TB, in D and E ; and because the errorof the inclination TRa increases nearly in the duplicate ratio of the timebetween the observations, draw FRG, so as either the angle DRF may beto the angle ARF, or the line DF to the line AF, in the duplicate ratioof the whole time between the observations aTB to the whole time betweenthe observations A IB, and use the line thus found FG in place of theline AB found above.It will be convenient that the angles ATR, RTB, aTA, BT6, be ncless than of ten or fifteen degrees, the times corresponding no greater than5~0 THE SYSTEM OF THE WORLD.of eight or twelve days, and the longitude^ taken when the comet jnoveswith the greatest velocity for thus the errors of the observation 。s willbear a less proportion to the differences of the longitudes.LEMMA IV.Tofind the longitudes of a comet to any given times.It is done by taking in the line FG the distances Rr, Rp, proportionalto the times, and drawing the lines Tr, Tp. The way of working bythgonometry is manifest.LEMMA V.To find the latitudes.On TF, TR, TG, as radiuses, at right angles erect F/, RP, Gg-, tangents of the observed latitudes ; and parallel to fg draw PH. The perpendiculars rp, pw, meeting PH, will be the tangents of the sought latitudesto Tr and Tp as radiuses.PROBLEM I.Prow, the assumed ratio of the velocity to determine the trajectory oj acomet.Let S represent the sun ; /, T, r} three places of the earth in its orbitat e^ual distances ; p, P, o5? as many corresponding places of the comet inits trajectory, so as the distances interposed betwixt place and place mayanswer to the motion of one hour ; pr, PR, wp, perpendiculars let fall onthe plane of the ecliptic, and rRp the vestige of the trajectory in thisplane. Join S/?, SP, Sc5, SR, ST, tr, TR, rp, TP , and let tr, -p, meet inO, TR will nearly converge to the same point O, or the error will be inconsiderable. By the premised lemmas the angles rOR, ROp, are given,as well as the ratios pr to //;, PR to TR, and wp to rp.rrlie figure TrOTHE SYSTEM OF THE WORLD. 571is likewise given both in magnitude and position, together with the distance ST, and the angles STR, PTR, STP. Let us assume the velocityof the comet in the place P to be to the velocity of a planet revolvedabout the sun in a circle, at the same distance SP, as V to 1 ; and we shallhave a line pP& to be determined, of this condition, that the space /?w,described by the comet in two hours, may be to the space V X tr (that is.to the space which the earth describes in the same time multiplied by thenumber V) in the subduplicate ratio of ST, the distance of the earth fromthe sun, to SP, the distance of the comet from the sun ; and that the spacepP, described by the comet in the first hour, may be to the space Pw, described by the comet in the second hour, as the velocity in p to the velocityin P ; that is, in the subduplicate ratio of the distance SP to the distanceS/7, or in the ratio of 2Sp to SP + Sp ; for in this whole work I neglectsmall fractions that can produce no sensible error.In the first place, then, as mathematicians, in the resolution of affectedequations, are wont, for the first essay, to assume the root by conjecture,so, in this analytical operation, I judge of the sought distance TR as Ibest can by conjecture. Then, by Lem. II. I draw rp, first supposing / Requal to Rp, and again (after the ratio of SP to Sp is discovered) so asrR may be to Rp as 2SP to SP + Sp, and I find the ratios of the linespw, rp, and OR, one to the other. Let M be to V X tr as OR to pi** ; andbecause the square ofp<*>is to the square of V X tr as ST to SP, weshall have, ex aquo, OR2 to M2 as ST to SP, and therefore the solidOR2 X SP equal to the given solid M2 X ST; whence (supposing thetriangles STP, PTR, to be now placed in the same plane) TR, TP, SP,PR, will be given, by Lem. I. All this I do, first by delineation in a rudeand hasty way ; then by a new delineation with greater care ; and, lastly,by an arithmetical computation. Then I proceed to determine the positionof the lines rp, pti, with the greatest accuracy, together with the nodes andinclination of the plane Spti to the plane of the ecliptic ; and in thatplane Spti I describe the trajectory in which a body let go from the placeP in the direction of the given right line jf?c5 would be carried with i velocity that is to the velocity of the earth as pti to V X tr. Q.E.F.PROBLEM II.To correct the assumed ratio of the velocity and the trajectory thencefound.Take an observation of the comet about the end of its appearance, orany other observation at a very great distance from the observations usedbefore, and find the intersection of a right line drawn to the comet, in thatobservation with the plane Sjow, as well as the comet s place in its trajectory to the time of the observation. If that intersection happens in thisplace, it is a proof that the trajectory was rightly determined ; if other572 THE SYSTEM OF THE WORLD.wise, a new number V is to be assumed, and a new trajectory to be found ;Z.L.。。 then tlu place of tke comet in this trajectory to the time of that probatoryobservation, and the intersection of a right line drawn to the cometwith the plane of the trajectory, are to be determined as before; and bycomparing the variation of the error with the variation of the other quantities, we may conclude, by the Rule of Three, how far those otherquantities ought to be varied or corrected, so as the error may become assmall as possible. And by means of these corrections we may have thetrajectory exactly, providing the observations upon which the computationwas founded were exact, and that we did not err much in the assumptionof the quantity V : for if we did, the operation is to be repeated till thetrajectory is exactly enough determined. Q,.E.F.CNJ) OF THE SYSTEM OF THE WORLD.CONTENTSOFTHE SYSTEM OF THE WORLD.That the matter of the heavens is fluid, 51 jThe principle of circular motion in free spaces, ........ . 5I jThe ettects of centripetal forces, ... .. 512The certainty of the argument, 514Wh.t follows from the .-upposed diurnal motion of the stars, 514The incongruous consequences of this supposition. 514That there is a centripetal force really directed to the centre of every planet, . . . 515<.-; Centripetal forces decrease in duplicate proportion of distances from the centre of every planet, 5i6That the superior ets are revolved about the sun, and by radii drawn to the sun describeareas proportional to the times, 517That the force which governs the superior planets is directed not to the earth, but to the sun, . 51t>That the ci; cuin-solar force throughout all the regions of the planets decreaseth in the duplicateproportion of the distances from the sun, 519That the circum-terrestrial force decreases in the duplicate proportion of the distances from theearth proved in the hypothesis of the earth s being at rest, 519The same proved in the hypothesis of the earth s motion 520The decrement of the forces in the duplicate proportion of the distances from the earth and planets, proved from the eccentricity of the planets, and the very slow motion of their apses, . 520The quantity of the forces tending towards the several planets : the circuni-solar very great, . 521The circum-terrestrial force very small, 521The apparent diameters of the planets, 5^1The correction of the apparent diameters, 522Why the density is greater in some of the planets and less in others; but the forces in all are astheir quantities of matter, 524Another analogy between the forces and bodies, proved in the celestial bodies, .... 525Proved in terrestrial bodies, 525The affinity of those analogies, 526And coincidence, . ... 526That the forces of small bodies are insensible, 527Which, notwithstanding, there are forces tending towards all terrestrial bodies proportional totheir quantities of matter, 528L roved that the same forces tend towards the celestial bodies, 528That from the surfaces of the planets, reckoning outward, their forces decrease in thj duplicate ;but, reckoning inward, in the simple proportion of the distances from their centres, . 52rThe quantities of the forces and of the motions arising in the several cases, .... 52V.That all the planets revolve about the sun, 529That the commun centre of gravity of all the planets is quiescent. That the sun is agitatedwith a very slow motion. This motion defined, 531That the planets, nevertheless, are revolved in ellipses having their foci in the sun; and by radiidrawn to the sun describe areas proportional to the times, 531Jf the dimensions of the orbits, and of the motions of their aphelions and nodes, . . . 532All the motions of the moon that have hitherto been observed by astronomers derived from theforegoing principles, 532As also some other unequable motions that hitherto have not been observed, .... 533And the distance of the moon from the earth to any given time, 533The motions of the satellites of Jupiter and Saturn derived from the motions of our moon, . 534That the planets, in respect of the fixed stars, are revolved by equable motions about theirproper axes. And that (perhaps) those motions are the most fit for the equation of time, 534The moon likewise is revolved by a diurnal motion about its axis, and its libration thence arises, 535That the sea ought twice to flow, and twice to ebb, every day ; that the highest water must fallout in the chird hour after the appulse of the luminaries to the meridian of the place, . 333674 CONTENTS OF THE SYSTEM OF THE WORLD.fke precession of the equinoxes, and the libratory motion of the axes of the earth and planet , 535? That the greatest tides happen in the syzygies of the luminaries, the least in their quadratures;and that at the third hour after the appulse of the moon to the meridian of the place. Batthat out of the syzygies and quadratures those greatest and least tides deviate a little fromthat third hour towards the third hour after the appulse of the sun to the meridian, . 536That the tides are greatest when the luminaries are in their perigees, 536That the tides are greatest about the equinoxes, 536That out of the equator the tides are greater and less alternately, . .... 537That, by the conservation of the impressed motion, the difference of the tides is diminished ; andthat hence it may happen that the greatest inensti ual tide will be the third after the syzygy, 5:38Thit the motio is of the sea may be retarded by impediments in its channels, .... 538That from the impediments of channels and shores various phenomena do arite, as that the seamay flow but once every day, 539That the times of the tides within the channels of rivers are more unequal than in the ocean, . 5401 hat the tides are greater in greater and deeper seas; greater on the shores of continents than(1 islands in the middle of the sea; and yet greater in shallow bays that open with wideinlets to the sea,.*..... 540The force of the sun to disturb the motions of the moon, computed from the foregoing principks, 542The force of the sun to move the sea computed, 543The height of the tide under the equator arising from the force of the sun computed, . . 543The height of the tides under the parallels arising from the sun s force computed, . . . 544The proportion of the tides under the equator, in the syzygies and quadratures, arising from thejoint forces of both sun and moon, ........... 545The force of the moon to excite tides, and the height of the water thence arising, computed, . 545That those forces of the sun and moon are scarcely ?en?ible by any other effect beside the tideswhich they raise in the sea, 546That the body of the moon is about six times more dense than the body of the sun, . . . 547That the moon is more dense than the earth in a ratio of about three to two, .... 547Of the distance ot the fixed stars, 547That the comets, as o ten as they become visible to us, are nearer than Jupiter, proved fromtheir parallax in longitude, . 548The same proved from their parallax in latitude, ......... 549The same proved otherwise by the parallax, .......... 550From the light of the comets heads it is proved that they descend to the orbit of Saturn, . 550And also below the orb of Jupiter, and sometimes below the orb of the earth, . . . 551The same proved from the extraordinary splendor of their tails when they are near the sun, . 551The same proved from the light of their heads, as being greater, c&teris paribus, when theycome near to the sun, 553The same confirmed by the great number of comets seen in the region of the sun, . . . 555This also confirmed by the greater magnitude and splendor of the tails after the conjunction ofthe heads with the sun than before, 555That the tails arise from the atmospheres of the comets, 556That the air and vapour in the celestial spaces is of an immense rarity ; and that a small quantity of vapour may be sufficient to explain all the phtcnomena of the tails of comets, . . 558After what manner the tails of comets may arise from the atmospheres of their heads. . . 559That the tails do indeed arise from those atmospheres, proved from several of their pheenomena, 559That comets do sometimes descend below the orbit of Mercury, proved from their tails, . 560That the comets move in conic sections, having one focus in the centre of the sun, and by radiiIrawn to that centre do describe areas proportional to the times, 561That those conic sections are near to parabolas, proved from the velocity of the comets, . 561*"~*" In what space of time cornets describing parabolic trajectories pass through the sphere of theorbis magnus, ...... 562At what time comets enter into and pass out of the sphere of the vrbis magnus, . . . 563With what velocity the comets of 1680 passed through the sphere of the orbis magnus, . . 564 ~i*That these were not two, but one and the same comet. In what orbit and with what velocitythis comet was carried through the heavens described more exactly, .... 564With what velocity corsets are carried, shewed by more examples, . . ... 565The investigation of the trajectory of comets proposed, .... ... 566"Lemmas premised to the solution of the problem, ..... ... 567The problem resolved, . ..... ... 57CINDEX TO THE PRINC1PIA.j their prsecession the cause of that motion shewn, 413" the quantity of that motion computed from the causes, 4oJA.IR, its density at any height, collected by Prop. XXII, Book II, and its density at the heightof one semi-diameter of the earth, shewn, 489its elastic force, what cause it may be attributed to, 302its gravity compared with that of water, -l^t" its resistance, collected by experiments of pendulums, 315" the same more accurately by experiments of falling bodies, and a theory, .... 353ANGLE S of contact not all of the same kind, but some infinitely less than others, . . . 101APSIDES, their motion shewn, 172, 173AREAS which revolving bodies, by radii drawn to the centre of force describe, compared with thetimes of description, 103, 105, 106, 195, 2(!<>As, the mathematical signification of this word defined, . . .100ATTRACTION of all bodies demonstrated, 3 >7" the certainty of this demonstration shewn, 384the cause or manner thereof no where defined by the author, .... 507the common centre of gravity of the earth, sun, and all the planets, is at rest, confirmed by Cor. 2, Prop. XIV, Book HI, 401" the common centre of gravity of the earth and moon goes round the orbis magnus, 402" its distance from the earth and from the moon, 452CENTRE, the common centre of gravity of many bodies does not alter its state of motion or restby the actions of the bodies among themselves, 87" of the forces by which revolving bodies are retained in their orbits, how indicated bythe description of areas, 107" how found by the given velocities of the revolving bodies, . ..... 110CIRCLE, by what law of centripetal force tending to any given point its circumference may bedescribed, . 108,111,114COMETS, a sort of planets, not meteors, 465,486" higher than the moon, and ir. the- planetary regions, 460" their distance how collected very nearly by observations, 401" more of them observed in the hemisphere towards the sun than in the opposite hemisphere; and how this comes to pa?s, 464" shine by the sun s light reflected from them, 464" surrounded with vast atmospheres, 463, 465" those which come nearest to the sun probably the least, ... . . 4P5" why they are not comprehended within a zodia , like the planets, but move differentlyinto all parts of the heavens, ... 502" may sometimes fall into the sun, and afford a new supply of fire, 502the use of them hinted, 492" move in conic sections, having their foci in the sun s centre, and by radii drawn to thesun describe areas proportional to the times. Move in ellipses if they come round again