Chapter XII: Introduction (8)

30. THUS the moon in the case before us, while it passes from the plane of the earth’s motion in the quarter, till it comes again into the same plane, has the nodes of its orbit continually moved backward, and the inclination of its orbit is at first diminished, viz. till it comes to G in fig. 100, which is near to its conjunction with the sun, but afterwards is increased again almost by the same degrees, till upon the moon’s arrival again to the plane of the earth’s motion, the inclination of the orbit is restored to something more than its first magnitude, though the difference is not very great, because the points I and C are not far distant from each other[190].

31. AFTER the same manner, if the moon had departed from the quarter in C, it should have described the curve line C X W (in fig. 98.) between the planes A F C and A D C, which would be convex to the former of those planes, and concave to the latter; so that, here also, the nodes should continually recede, and the inclination of the orbit gradually diminish more and more, till the moon arrived near its opposition to the sun in X; but from that time the inclination should again increase, till it became a little greater than at first. This will easily appear, by considering, that as the action of the sun upon the moon, by exceeding its action upon the earth, drew it out of the plane A E C towards the sun, while the moon passed from A to I; so, during its passage from C to W, the moon being all that time farther from the sun than the earth, it will be attracted less; and the earth, together with the plane A E C F, will as it were be drawn from the moon, in such sort, that the path the moon describes shall appear from the earth, as it did in the former case by the moon’s being drawn away.

32. THESE are the changes, which the nodes and the inclination of the moon’s orbit undergo, when the nodes are in the quarters; but when the nodes by their motion, and the motion of the sun together, come to be situated between the quarter and conjunction or opposition, their motion and the change made in the inclination of the orbit are somewhat different.

33. LET A G C H (in fig. 103.) be a circle described in the plane of the earth’s motion, having the earth in T for its center. Let the point opposite to the sun be A, and the point G a fourth part of the circle distant from A. Let the nodes of the moon’s orbit be situated in the line B T D, and B the node, falling between A, the place where the moon would be in the full, and G the place where the moon would be in the quarter. Suppose B E D F to be the plane, in which the moon essays to move, when it proceeds from the point B. Because the moon in B is more distant from the sun than the earth, it shall be less attracted by the sun, and shall not descend towards the sun so fast as the earth: consequently it shall quit the plane B E D F, which we suppose to accompany the earth, and describe the line B I K convex thereto, till such time as it comes to the point K, where it will be in the quarter: but from thenceforth being more attracted than the earth, the moon shall change its course, and the following part of the path it describes shall be concave to the plane B E D or B G D, and shall continue concave to the plane B G D, till it crosses that plane in L, just as in the preceding case. Now I say, while the moon is passing from B to K, the nodes, contrary to what was found in the foregoing case, will proceed forward, or move the same way with the moon[191]; and at the same time the inclination of the orbit will increase[192].

34. WHEN the moon is in the point I, let the plane M I N pass through the earth T, and touch the path of the moon in I, cutting the plane of the earth’s motion, in the line M T N, and the plane B E D in the line T O. Because the line B I K is convex to the plane B E D, which touches it in B, the plane N I M must cross the plane D E B, before it meets the plane C G B; and therefore the point M will fall from B towards G, and the node of the moon’s orbit being translated from B to M is moved forward.

35. I SAY farther, the angle under O M G, which the plane M O N makes with the plane B G C, is greater than the angle under O B G, which the plane B O D makes with the same. This appears from what has been already explained; because the arches B O, O M are each less than the quarter of a circle, and therefore taken both together are less than a semicircle.

36. AGAIN, when the moon is come to the point K in its quarter, the nodes will be advanced yet farther forward, and the inclination of the orbit also more augmented. Hitherto the moon’s motion has been referred to the plane, which passing through the earth touches the path of the moon in the point, where the moon is, according to what was asserted at the beginning of this discourse upon the nodes, that it is the custom of astronomers so to do. But here in the point K no such plane can be found; on the contrary, seeing the line of the moon’s motion on one side the point K is convex to the plane B E D, and on the other side concave to the same, no plane can pass through the points T and K but will cut the line B K L in that point. Therefore instead of such a touching plane, we must here make use of what is equivalent, the plane P K Q, with which the line B K L shall make a less angle than with any other plane; for this plane does as it were touch the line B K in the point K, since it so cuts it, that no other plane can be drawn so, as to pass between the line B K and the plane P K Q. But now it is evident, that the point P, or the node, is removed from M towards G, that is, has moved yet farther forward; and it is likewise as manifest, that the angle under K P G, or the inclination of the moon’s orbit in the point K, is greater than the angle under I M G, for the reason so often assigned.

37. AFTER the moon has passed the quarter, the path of the moon being concave to the plane A G C H, the nodes, as in the preceding case, shall recede, till the moon arrives at the point L; which shews, that considering the whole time of the moon’s passing from B to L, at the end of that time the nodes shall be found to have receded, or to be placed backwarder, when the moon is in L, than when it was in B. For the moon takes a longer time in passing from K to L, than in passing from B to K; and therefore the nodes continue to recede a longer time, than they moved forwards; so that their recess must surmount their advance.

38. IN the same manner, while the moon is in its passage from K to L, the inclination of the orbit shall diminish, till the moon comes to the point, in which it is one quarter part of a circle distant from its node; suppose in the point R; and from that time the inclination shall again increase. Since therefore the inclination of the orbit increases, while the moon is passing from B to K, and diminishes itself again only, while the moon is passing from K to R, and then augments again, till the moon arrive in L; while the moon is passing from B to L, the inclination of the orbit is much more increased than diminished, and will be distinguishably greater, when the moon is come to L, than when it set out from B.

39. IN like manner, while the moon is passing from L on the other side the plane A G C H, the node shall advance forward, as long as the moon is between the point L and the next quarter; but afterwards it shall recede, till the moon come to pass the plane A G C H again in the point V, between B and A: and because the time between the moon’s passing from L to the next quarter is less, than the time between that quarter and the moon’s coming to the point V, the node shall have more receded than advanced; so that the point V will be nearer to A, than L is to C. So also the inclination of the orbit, when the moon is in V, will be greater, than when the moon was at L; for this inclination increases all the time the moon is between L and the next quarter; it decreases only while the moon is passing from this quarter to the mid way between the two nodes, and from thence increases again during the whole passage through the other half of the way to the next node.

40. THUS we have traced the moon from her node in the quarter, and shewn, that at every period of the moon the nodes will have receded, and thereby will have approached toward a conjunction with the sun. But this conjunction will be much forwarded by the visible motion of the sun itself. In the last scheme the sun will appear to move from S toward W. Suppose it appeared to have moved from S to W, while the moon’s node has receded from B to V, then drawing the line W T X, the arch V X will represent the distance of the line drawn between the nodes from the sun, when the moon is in V; whereas the arch B A represented that distance, when the moon was in B. This visible motion of the sun is much greater, than that of the node; for the sun appears to revolve quite round each year, and the node is near 19 years in making one revolution. We have also seen, that when the node was in the quadrature, the inclination of the moon’s orbit decreased, till the moon came to the conjunction, or opposition, according to which node it set out from; but that afterwards it again increased, till it became at the next node rather greater than at the former. When the node is once removed from the quarter nearer to a conjunction with the sun, the inclination of the moon’s orbit, when the moon comes into the node, is more sensibly greater, than it was in the node preceding; the inclination of the orbit by this means more and more increasing till the node comes into conjunction with the sun; at which time it has been shewn above, that the sun has no power to change the plane of the moon’s motion; and consequently has no effect either on the nodes, or on the inclination of the orbit.

41. AS soon as the nodes, by the action of the sun, are got out of conjunction toward the other quarters, they begin again to recede as before; but the inclination of the orbit in the appulse of the moon to each succeeding node is less than at the preceding, till the nodes come again into the quarters. This will appear as follows. Let A (in fig. 104.) represent one of the moon’s nodes placed between the point of opposition B and the quarter C. Let the plane A D E pass through the earth T, and touch the path of the moon in A. Let the line A F G H be the path of the moon in her passage from A to H, where she crosses again the plane of the earth’s motion. This line will be convex toward the plane A D E, till the moon comes to G, where she is in the quarter; and after this, between G and H, the same line will be concave toward this plane. All the time this line is convex toward the plane A D E, the nodes will recede; and on the contrary proceed, while it is concave to that plane. All this will easily be conceived from what has been before so largely explained. But the moon is longer in passing from A to G, than from G to H; therefore the nodes recede a longer time, than they proceed; consequently upon the whole, when the moon is arrived at H, the nodes will have receded, that is, the point H will fall between B and E. The inclination of the orbit will decrease, till the moon is arrived to the point F, in the middle between A and H. Through the passage between F and G the inclination will increase, but decrease again in the remaining part of the passage from G to H, and consequently at H must be less than at A. The like effects, both in respect to the nodes and inclination of the orbit, will take place in the following passage of the moon on the other side of the plane A B E C, from H, till it comes over that plane again in I.

42. THUS the inclination of the orbit is greatest, when the line drawn between the moon’s nodes will pass through the sun; and least, when this line lies in the quarters, especially if the moon at the same time be in conjunction with the sun, or in the opposition. In the first of these cases the nodes have no motion, in all others, the nodes will each month have receded: and this regressive motion will be greatest, when the nodes are in the quarters; for in that case the nodes have no progressive motion during the whole month, but in all other cases the nodes do at some times proceed forward, viz. whenever the moon is between either quarter, and the node which is less distant from that quarter than a fourth part of a circle.

43. IT now remains only to explain the irregularities in the moon’s motion, which follow from the elliptical figure of the orbit. By what has been said at the beginning of this chapter it appears, that the power of the earth on the moon acts in the reciprocal duplicate proportion of the distance: therefore the moon, if undisturbed by the sun, would move round the earth in a true ellipsis, and the line drawn from the earth to the moon would pass over equal spaces in equal portions of time. That this description of the spaces is altered by the sun, has been already declared. It has also been shown, that the figure of the orbit is changed each month; that the moon is nearer the earth at the new and full, and more remote in the quarters, than it would be without the sun. Now we must pass by these monthly changes, and consider the effect, which the sun will have in the different situations of the axis of the orbit in respect of that luminary.

44. THE action of the sun varies the force, wherewith the moon is drawn toward the earth; in the quarters the force of the earth is directly increased by the sun; at the new and full the same is diminished; and in the intermediate places the influence of the earth is sometimes aided, and sometimes lessened by the sun. In these intermediate places between the quarters and the conjunction or opposition, the sun’s action is so oblique to the action of the earth on the moon, as to produce that alternate acceleration and retardment of the moon’s motion, which I observed above to be stiled the variation. But besides this effect, the power, by which the earth attracts the moon toward itself, will not be at full liberty to act with the same force, as if the sun acted not at all on the moon. And this effect of the sun’s action, whereby it corroborates or weakens the action of the earth, is here only to be considered. And by this influence of the sun it comes to pass, that the power, by which the moon is impelled toward the earth, is not perfectly in the reciprocal duplicate proportion of the distance. Consequently the moon will not describe a perfect ellipsis. One particular, wherein the moon’s orbit will differ from an ellipsis, consists in the places, where the motion of the moon is perpendicular to the line drawn from itself to the earth. In an ellipsis, after the moon should have set out in the direction perpendicular to this line drawn from itself to the earth, and at its greatest distance from the earth, its motion would again become perpendicular to this line drawn between itself and the earth, and the moon be at its nearest distance from the earth, when it should have performed half its period; after performing the other half of its period its motion would again become perpendicular to the forementioned line, and the moon return into the place whence it set out, and have recovered again its greatest distance. But the moon in its real motion, after setting out as before, sometimes makes more than half a revolution, before its motion comes again to be perpendicular to the line drawn from itself to the earth, and the moon is at its nearest distance; and then performs more than another half of an intire revolution before its motion can a second time recover its perpendicular direction to the line drawn from the moon to the earth, and the moon arrive again to its greatest distance from the earth. At other times the moon will descend to its nearest distance, before it has made half a revolution, and recover again its greatest distance, before it has made an intire revolution. The place, where the moon is at its greatest distance from the earth, is called the moon’s apogeon, and the place of the least distance the perigeon. This change of the place, where the moon successively comes to its greatest distance from the earth, is called the motion of the apogeon. In what manner the sun causes the apogeon to move, I shall now endeavour to explain.

45. OUR author shews, that if the moon were attracted toward the earth by a composition of two powers, one of which were reciprocally in the duplicate proportion of the distance from the earth, and the other reciprocally in the triplicate proportion of the same distance; then, though the line described by the moon would not be in reality an ellipsis, yet the moon’s motion might be perfectly explained by an ellipsis, whose axis should be made to move round the earth; this motion being in consequence, as astronomers express themselves, that is, the same way as the moon itself moves, if the moon be attracted by the sum of the two powers; but the axis must move in antecedence, or the contrary way, if the moon be acted on by the difference of these powers. What is meant by duplicate proportion has been often explained; namely, that if three magnitudes, as A, B, and C, are so related, that the second B bears the same proportion to the third C, as the first A bears to the second B, then the proportion of the first A to the third C, is the duplicate of the proportion of the first A to the second B. Now if a fourth magnitude, as D, be assumed, to which C shall bear the same proportion as A bears to B, and B to C, then the proportion of A to D is the triplicate of the proportion of A to B.

46. THE way of representing the moon’s motion in this case is thus. T denoting the earth (in fig. 105, 106.) suppose the moon in the point A, its apogeon, or greatest distance from the earth, moving in the direction A F perpendicular to A B, and acted upon from the earth by two such forces as have been named. By that power alone, which is reciprocally in the duplicate proportion of the distance, if the moon let out from the point A with a proper degree of velocity, the ellipsis A M B may be described. But if the moon be acted upon by the sum of the forementioned powers, and the velocity of the moon in the point A be augmented in a certain proportion[193]; or if that velocity be diminished in a certain proportion, and the moon be acted upon by the difference of those powers; in both these cases the line A E, which shall be described by the moon, is thus to be determined. Let the point M be that, into which the moon would have arrived in any given space of time, had it moved in the ellipsis A M B. Draw M T, and likewise C T D in such sort, that the angle under A T M shall bear the same proportion to the angle under A T C, as the velocity, with which the ellipsis A M B must have been described, bears to the difference between this velocity, and the velocity, with which the moon must set out from the point A in order to describe the path A E. Let the angle A T C be taken toward the moon (as in fig. 105.) if the moon be attracted by the sum of the powers; but the contrary way (as in fig. 106.) if by their difference. Then let the line A B be moved into the position C D, and the ellipsis A M B into the situation C N D, so that the point M be translated to L: then the point L shall fall upon the path of the moon A E.

47. THE angular motion of the line A T, wereby it is removed into the situation C T, represents the motion of the apogeon; by the means of which the motion of the moon might be fully explicated by the ellipsis A M B, if the action of the sun upon it was directed to the center of the earth, and reciprocally in the triplicate proportion of the moon’s distance from it. But that not being so, the apogeon will not move in the regular manner now described. However, it is to be observed here, that in the first of the two preceding cases, where the apogeon moves forward, the whole centripetal power increases faster, with the decrease of distance, than if the intire power were reciprocally in the duplicate proportion of the distance; because one part only is in that proportion, and the other part, which is added to this to make up the whole power, increases faster with the decrease of distance. On the other hand, when the centripetal power is the difference between these two, it increases less with the decrease of the distance, than if it were simply in the reciprocal duplicate proportion of the distance. Therefore if we chuse to explain the moon’s motion by an ellipsis (as is most convenient for astronomical uses to be done, and by reason of the small effect of the sun’s power, the doing so will not be attended with any sensible error;) we may collect in general, that when the power, by which the moon is attracted to the earth, by varying the distance, increases in a greater than in the duplicate proportion of the distance diminished, a motion in consequence must be ascribed to the apogeon; but that when the attraction increases in a less proportion than that named, the apogeon must have given to it a motion in antecedence[194]. It is then observed by Sir IS. NEWTON, that the first of these cases obtains, when the moon is in the conjunction and opposition; and the latter, when the moon is in the quarters: so that in the first the apogeon moves according to the order of the signs; in the other, the contrary way[195]. But, as was said before, the disturbance given to the action of the earth by the sun in the conjunction and opposition being near twice as great as in the quarters[196], the apogeon will advance with a greater velocity than recede, and in the compass of a whole revolution of the moon will be carried in consequence[197].

48. IT is shewn in the next place by our author, that when the line A B coincides with that, which joins the earth and the sun, the progressive motion of the apogeon, when the moon is in the conjunction or opposition, exceeds the regressive in the quadratures more than in any other situation of the line A B[198]. On the contrary, when the line A B makes right angles with that, which joins the earth and sun, the retrograde motion will be more considerable[199], nay is found so great as to exceed the progressive; so that in this case the apogeon in the compass of an intire revolution of the moon is carried in antecedence. Yet from the considerations in the last paragraph the progressive motion exceeds the other; so that in the whole the mean motion of the apogeon is in consequence, according as astronomers find. Moreover, the line A B changes its situation with that, which joins the earth and sun, by such slow degrees, that the inequalities in the motion of the apogeon arising from this last consideration, are much greater than what arises from the other[200].

49. FARTHER, this unsteady motion in the apogeon is attended with another inequality in the motion of the moon, that it cannot be explained at all times by the same ellipsis. The ellipsis in general is called by astronomers an eccentric orbit. The point, in which the two axis’s cross, is called the center of the figure; because all lines drawn through this point within the ellipsis, from side to side, are divided in the middle by this point. But the center, about which the heavenly bodies revolve, lying out of this center of the figure in one focus, these orbits are said to be eccentric; and where the distance of the focus from this center bears the greatest proportion to the whole axis, that orbit is called the most eccentric: and in such an orbit the distance from the focus to the remoter extremity of the axis bears the greatest proportion to the distance of the nearer extremity. Now whenever the apogeon of the moon moves in consequence, the moon’s motion must be referred to an orbit more eccentric, than what the moon would describe, if the whole power, by which the moon was acted on in its passing from the apogeon, changed according to the reciprocal duplicate proportion of the distance from the earth, and by that means the moon did describe an immoveable ellipsis; and when the apogeon moves in antecedence, the moon’s motion must be referred to an orbit less eccentric. In the first of the two figures last referred to, the true place of the moon L falls without the orbit A M B, to which its motion is referred: whence the orbit A L E, truly described by the moon, is less incurvated in the point A, than is the orbit A M B; therefore the orbit A M B is more oblong, and differs farther from a circle, than the ellipsis would, whose curvature in A were equal to that of the line A L B, that is, the proportion of the distance of the earth T from the center of the ellipsis to its axis will be greater in the ellipsis A M B, than in the other; but that other is the ellipsis, which the moon would describe, if the power acting upon it in the point A were altered in the reciprocal duplicate proportion of the distance. In the second figure, when the apogeon recedes, the place of the moon L falls within the orbit A M B, and therefore that orbit is less eccentric, than the immoveable orbit which the moon should describe. The truth of this is evident; for, when the apogeon moves forward, the power, by which the moon is influenced in its descent from the apogeon, increases faster with the decrease of distance, than in the duplicate proportion of the distance; and consequently the moon being drawn more forcibly toward the earth, it will descend nearer to it. On the other hand, when the apogeon recedes, the power acting on the moon increases with the decrease of distance in less than the duplicate proportion of the distance; and therefore the moon is less impelled toward the earth, and will not descend so low.

50. NOW suppose in the first of these figures, that the apogeon A is in the situation, where it is approaching toward the conjunction or opposition of the sun. In this case the progressive motion of the apogeon is more and more accelerated. Here suppose that the moon, after having descended from A through the orbit A E as far as F, where it is come to its nearest distance from the earth, ascends again up the line F G. Because the motion of the apogeon is here continually more and more accelerating, the cause of its motion is constantly upon the increase; that is, the power, whereby the moon is drawn to the earth, will decrease with the increase of distance, in the moon’s ascent from F, in a greater proportion than that wherewith it increased with the decrease of distance in the moon’s descent to F. Consequently the moon will ascend higher than to the distance A T, from whence it descended; therefore the proportion of the greatest distance of the moon to the least is increased. And when the moon descends again, the power will yet more increase with the decrease of distance, than in the last ascent it decreased with the augmentation of distance; the moon therefore must descend nearer to the earth than it did before, and the proportion of the greatest distance to the least yet be more increased. Thus as long as the apogeon is advancing toward the conjunction or opposition, the proportion of the greatest distance of the moon from the earth to the least will continually increase; and the elliptical orbit, to which the moon’s motion is referred, will be rendered more and more eccentric.

51. AS soon as the apogeon is passed the conjunction with the sun or the opposition, the progressive motion thereof abates, and with it the proportion of the greatest distance of the moon from the earth to the least distance will also diminish; and when the apogeon becomes regressive, the diminution of this proportion will be still farther continued on, till the apogeon comes into the quarter; from thence this proportion, and the eccentricity of the orbit will increase again. Thus the orbit of the moon is most eccentric, when the apogeon is in conjunction with the sun, or in opposition to it, and least of all when the apogeon is in the quarters.

52. THESE changes in the nodes, in the inclination of the orbit to the plane of the earth’s motion, in the apogeon, and in the eccentricity, are varied like the other inequalities in the motion of the moon, by the different distance of the earth from the sun; being greatest, when their cause is greatest, that is, when the earth is nearest to the sun.

53. I SAID at the beginning of this chapter, that Sir ISAAC NEWTON has computed the very quantity of many of the moon’s inequalities. That acceleration of the moon’s motion, which is called the variation, when greatest, removes the moon out of the place, in which it would otherwise be found, something more than half a degree[201]. In the phrase of astronomers, a degree is 1/360 part of the whole circuit of the moon or any planet. If the moon, without disturbance from the sun, would have described a circle concentrical to the earth, the sun will cause the moon to approach nearer to the earth in the conjunction and opposition, than in the quarters, nearly in the proportion of 69 to 70[202]. We had occasion to mention above, that the nodes perform their period in almost 19 years. This the astronomers found by observation; and our author’s computations assign to them the same period[203]. The inclination of the moon’s orbit when least, is an angle about 1/18 part of that angle, which constitutes a perpendicular; and the difference between the greatest and least inclination of the orbit is determined by our author’s computation to be about 1/18 of the least inclination[204]. And this also is agreeable to the observations of astronomers. The motion of the apogeon, and the changes in the eccentricity, Sir ~ISAAC NEWTON~ has not computed. The apogeon performs its revolution in about eight years and ten months. When the moon’s orbit is most eccentric, the greatest distance of the moon from the earth bears to the least distance nearly the proportion of 8 to 7; when the orbit is least eccentric, this proportion is hardly so great as that of 12 to 11.

54. SIR ~ISAAC NEWTON~ shews farther, how, by comparing the periods of the motion of the satellites, which revolve round Jupiter and Saturn, with the period of our moon round the earth, and the periods of those planets round the sun with the period of our earth’s motion, the inequalities in the motion of those satellites may be derived from the inequalities in the moon’s motion; excepting only in regard to that motion of the axis of the orbit, which in the moon makes the motion of the apogeon; for the orbits of those satellites, as far as can be discerned by us at this distance, appearing little or nothing eccentric, this motion, as deduced from the moon, must be diminished.

~CHAP. IV.~

Of ~Comets~.

IN the former of the two preceding chapters the powers have been explained, which keep in motion those celestial bodies, whose courses had been well determined by the astronomers. In the last chapter we have shewn, how those powers have been applied by our author to the making a more perfect discovery of the motion of those bodies, the courses of which were but imperfectly understood; for some of the inequalities, which we have been describing in the moon’s motion, were unknown to the astronomers. In this chapter we are to treat of a third species of the heavenly bodies, the true motion of which was not at all apprehended before our author writ; in so much, that here Sir ~ISAAC NEWTON~ has not only explained the causes of the motion of these bodies, but has performed also the part of an astronomer, by discovering what their motions are.

2. THAT these bodies are not meteors in our air, is manifest; because they rise and set in the same manner, as the sun and stars. The astronomers had gone so far in their inquiries concerning them, as to prove by their observations, that they moved in the etherial spaces far beyond the moon; but they had no true notion at all of the path, which they described. The most prevailing opinion before our author was, that they moved in straight lines; but in what part of the heavens was not determined. DESCARTES[205] removed them far beyond the sphere of Saturn, as finding the straight motion attributed to them, inconsistent with the vortical fluid, by which he explains the motions of the planets, as we have above related[206]. But Sir ISAAC NEWTON distinctly proves from astronomical observation, that the comets pass through the region of the planets, and are mostly invisible at a less distance, than that of Jupiter[207].

3. AND from hence finding the comets to be evidently within the sphere of the sun’s action, he concludes they must, necessarily move about the sun, as the planets do[208]. The planets move in ellipsis’s; but it is not necessary that every body, which is influenced by the sun, should move in that particular kind of line. However our author proves, that the power of the sun being reciprocally in the duplicate proportion of the distance, every body acted on by the sun must either fall directly down, or move in some conic section; of which lines I have above observed, that there are three species, the ellipsis, parabola, and hyperbola[209]. If a body, which descends toward the sun as low as the orbit of any planet, move with a swifter motion than the planet does, that body will describe an orbit of a more oblong figure, than that of the planet, and have a longer axis at least. The velocity of the body may be so great, that it shall move in a parabola, and having once passed about the sun, shall ascend for ever without returning any more: but the sun will be placed in the focus of this parabola. With a velocity still greater the body will move in an hyperbola. But it is most probable, that the comets move in elliptical orbits, though of a very oblong, or in the phrase of astronomers, of a very eccentric form, such as is represented in fig. 107, where S is the sun, C the comet, and A B D E its orbit, wherein the distance of S and D far exceeds that of S and A. Whence it is, that they sometimes are found at a moderate distance from the sun, and appear within the planetary regions; at other times they ascend to vast distances, far beyond the very orbit of Saturn, and so become invisible. That the comets do move in this manner is proved by our author, from computations built upon the observations, which astronomers had made on many comets. These computations were performed by Sir ~ISAAC NEWTON~ himself upon the comet, which appeared toward the latter end of the year 1680, and at the beginning of the year following[210]; but the learned Dr. HALLEY prosecuted the like computations more at large in this, and also in many other comets[211]. Which computations are made upon propositions highly worthy of our author’s unparallel’d genius, such as could scarce have been discovered by any one not possessed of the utmost force of invention;

4. THOSE computations depend upon this principle, that the eccentricity of the orbits of the comets is so great, that if they are really elliptical, yet they approach so near to parabolas in that part of them, where they come under our view, that they may be taken for such without sensible error[212]: as in the preceding figure the parabola F A G differs in the lower part of it about A very little from the ellipsis D E A B. Upon which ground our great author teaches a method of finding by three observations made upon any comet the parabola, which nearest agrees with its orbit[213].

5. NOW what confirms this whole theory beyond the least room for doubt is, that the places of the comets computed in the orbits, which the method here mentioned assigns them, agree to the observations of astronomers with the same degree of exactness, as the computations of the primary planets places usually do; and this in comets, whose motions are very extraordinary[214].

6. OUR author afterwards shews how to make use of any small deviation from the parabola, that shall be observed, to determine whether the orbits of the comets are elliptical or not, and so to discover if the same comet returns at certain periods[215]. And upon examining the comet in 1680, by the rule laid down for this purpose, he finds its orbit to agree more exactly to an ellipsis than to a parabola, though the ellipsis be so very eccentric, that the comet cannot perform its period through it in the space of 500 years[216]. Upon this Dr. HALLEY observed, that mention is made in history of a comet, with the like eminent tail as this, having appeared three several times before; the first of which appearances was at the death of JULIUS CESAR, and each appearance was at the distance of 575 years from the next preceding. He therefore computed the motion of this comet in such an elliptic orbit, as would require this number of years for the body to revolve through it; and these computations agree yet more perfectly with the observations made on this comet, than any parabolical orbit will do[217].

7. THE comparing together different appearances of the same comet, is the only way to discover certainly the true form of the orbit: for it is impossible to determine with exactness the figure of an orbit so exceedingly eccentric, from single observations taken in one part of it; and therefore Sir ~ISAAC NEWTON~[218] proposes to compare the orbits, upon the supposition that they are parabolical, of such comets as appear at different times; for if the same orbit be found to be described by a comet at different times, in all probability it will be the same comet which describes it. And here he remarks from Dr. HALLEY, that the same orbit very nearly agrees to two appearances of a comet about the space of 75 years distance[219]; so that if those two appearances were really of the same comet, the transverse axis of the orbit of the comet would be near 18 times the axis of the earth’s orbit; and the comet, when at its greatest distance from the sun, will be removed not less than 35 times as far as the middle distance of the earth.

8. AND this seems to be the shortest period of any of the comets. But it will be farther confirmed, if the same comet should return a third time after another period of 75 years. However it is not to be expected, that comets should preserve the same regularity in their periods, as the planets; because the great eccentricity of their orbits makes them liable to suffer very considerable alterations from the action of the planets, and other comets, upon them.

9. IT is therefore to prevent too great disturbances in their motions from these causes, as our author observes, that while the planets revolve all of them nearly in the same plane, the comets are disposed in very different ones; and distributed over all parts of the heavens; that, when in their greatest distance from the sun, and moving slowest, they might be removed as far as possible out of the reach of each other’s action[220]. The same end is likewise farther answered in those comets, which by moving slowest in the aphelion, or remotest distance from the sun, descend nearest to it, by placing the aphelion of these at the greatest height from the sun[221].

10. OUR philosopher being led by his principles to explain the motions of the comets, in the manner now related, takes occasion from thence to give us his thoughts upon their nature and use. For which end he proves in the first place, that they must necessarily be solid and compact bodies, and by no means any sort of vapour or light substance exhaled from the planets or stars: because at the near distance, to which some comets approach the sun, it could not be, but the immense heat, to which they are exposed, should instantaneously disperse and scatter any such light volatile substance[222]. In particular the forementioned comet of 1680 descended so near the sun, as to come within a sixth part of the sun’s diameter from the surface of it. In which situation it must have been exposed, as appears by computation, to a degree of heat exceeding the heat of the sun upon our earth no less than 28000 times; and therefore might have contracted a degree of heat 2000 times greater, than that of red hot iron[223]. Now a substance, which could endure so intense a heat, without being dispersed in vapor, must needs be firm and solid.

11. IT is shewn likewise, that the comets are opake substances, shining by a reflected light, borrowed from the sun[224]. This is proved from the observation, that comets, though they are approaching the earth, yet diminish in lustre, if at the same time they recede from the sun; and on the contrary, are found to encrease daily in brightness, when they advance towards the sun, though at the same time they move from the earth[225].

12. THE comets therefore in these respects resemble the planets; that both are durable opake bodies, and both revolve about the sun in conic sections. But farther the comets, like our earth, are surrounded by an atmosphere. The air we breath is called the earth’s atmosphere; and it is most probable, that all the other planets are invested with the like fluid. Indeed here a difference is found between the planets and comets. The atmospheres of the planets are of so fine and subtile a substance, as hardly to be discerned at any distance, by reason of the small quantity of light which they reflect, except only in the planet Mars. In him there is some little appearance of such a substance surrounding him, as stars which have been covered by him are said to look somewhat dim a small space before his body comes under them, as if their light, when he is near, were obstructed by his atmosphere. But the atmospheres which surround the comets are so gross and thick, as to reflect light very copiously. They are also much greater in proportion to the body they surround, than those of the planets, if we may judge of the rest from our air; for it has been observed of comets, that the bright light appearing in the middle of them, which is reflected from the solid body, is scarce a ninth or tenth part of the whole comet,

13. I SPEAK only of the heads of the comets, the most lucid part of which is surrounded by a fainter light, the most lucid part being usually not above a ninth or tenth part of the whole in breadth[226]. Their tails are an appearance very peculiar, nothing of the same nature appertaining in the least degree to any other of the celestial bodies. Of that appearance there are several opinions; our author reduces them to three[227]. The two first, which he proposes, are rejected by him; but the third he approves. The first is, that they arise from a beam of light transmitted through the head of the comet, in like manner as a stream of light is discerned, when the sun shines into a darkened room through a small hole. This opinion, as Sir ~ISAAC NEWTON~ observes, implies the authors of it wholly unskilled in the principles of optics; for that stream of light, seen in a darkened room, arises from the reflection of the sun beams by the dust and motes floating in the air: for the rays of light themselves are not seen, but by their being reflected to the eye from some substance, upon which they fall[228]. The next opinion examined by our author is that of the celebrated DESCARTES, who imagins these tails to be the light of the comet refracted in its passage to us, and thence affording an oblong representation; as the light of the sun does, when refracted by the prism in that noted experiment, which will have a great share in the third book of this discourse[229]. But this opinion is at once overturned from this consideration only, that the planets could be no more free from this refraction than the comets; nay ought to have larger or brighter tails, than they, because the light of the planets is strongest. However our author has thought proper to add some farther objections against this opinion: for instance, that these tails are not variegated with colours, as is the image produced by the prism, and which is inseparable from that unequal refraction, which produces that disproportioned length of the image. And besides, when the light in its passage from different comets to the earth describes the same path through the heavens, the refraction of it should of necessity be in all respects the same. But this is contrary to observation; for the comet in 1680, the 28th day of December, and a former comet in the year 1577, the 29th day of December, appear’d in the same place of the heavens, that is, were seen adjacent to the same fixed stars, the earth likewise being in the same place at both times; yet the tail of the latter comet deviated from the opposition to the sun a little to the northward, and the tail of the former comet declined from the opposition of the sun five times as much southward[230].

14. THERE are some other false opinions, though less regarded than these, which have been advanced upon this argument. These our excellent author passes over, hastening to explain, what he takes to be the true cause of this appearance. He thinks it is certainly owing to steams and vapours exhaled from the body, and gross atmosphere of the comets, by the heat of the sun; because all the appearances agree perfectly to this sentiment. The tails are but small, while the comet is descending to the sun, but enlarge themselves to an immense degree, as soon as ever the comet has passed its perihelion; which shews the tail to depend upon the degree of heat, which the comet receives from the sun. And that the intense heat to which comets, when nearest the sun, are exposed, should exhale from them a very copious vapour, is a most reasonable supposition; especially if we consider, that in those free and empty regions steams will more easily ascend, than here upon the surface of the earth, where they are suppressed and hindered from rising by the weight of the incumbent air: as we find by experiments made in vessels exhausted of the air, where upon removal of the air several substances will fume and discharge steams plentifully, which emit none in the open air. The tails of comets, like such a vapour, are always in the plane of the comet’s orbit, and opposite to the sun, except that the upper part thereof inclines towards the parts, which the comet has left by its motion; resembling perfectly the smoak of a burning coal, which, if the coal remain fixed, ascends from it perpendicularly; but, if the coal be in motion, ascends obliquely, inclining from the motion of the coal. And besides, the tails of comets may be compared to this smoak in another respect, that both of them are denser and more compact on the convex side, than on the concave. The different appearance of the head of the comet, after it has past its perihelion, from what it had before, confirms greatly this opinion of their tails: for smoke raised by a strong heat is blacker and grosser, than when raised by a less; and accordingly the heads of comets, at the same distance from the sun, are observed less bright and shining after the perihelion, than before, as if obscured by such a gross smoke.

15. THE observations of HEVELIUS upon the atmospheres of comets still farther illustrate the same; who relates, that the atmospheres, especially that part of them next the sun, are remarkably contracted when near the sun, and dilated again afterwards.

16. TO give a more full idea of these tails, a rule is laid down by our author, whereby to determine at any time, when the vapour in the extremity of the tail first rose from the head of the comet. By this rule it is found, that the tail does not consist of a fleeting vapour, dissipated soon after it is raised, but is of long continuance; that almost all the vapour, which rose about the time of the perihelion from the comet of 1680, continued to accompany it, ascending by degrees, being succeeded constantly by fresh matter, which rendered the tail contiguous to the comet. From this computation the tails are found to participate of another property of ascending vapours, that, when they ascend with the greatest velocity, they are least incurvated.