Principia

BOOK III. (part 2)

The effect of either luminary doth likewise depend upon its declination or distance from the equator; for if the luminary was placed at the pole, it would constantly attract all the parts of the waters without any intension or remission of its action, and could cause no reciprocation of motion. And, therefore, as the luminaries decline from the equator towards either pole, they will, by degrees, lose their force, and on this account will excite lesser tides in the solstitial than in the equinoctial syzygies. But in the solstitial quadratures they will raise greater tides than in the quadratures about the equinoxes; because the force of the moon, then situated in the equator, most exceeds the force of the sun. Therefore the greatest tides fall out in those syzygies, and the least in those quadratures, which happen about the time of both equinoxes: and the greatest tide in the syzygies is always succeeded by the least tide in the quadratures, as we find by experience. But, because the sun is less distant from the earth in winter than in summer, it comes to pass that the greatest and least tides more frequently appear before than after the vernal equinox, and more frequently after than before the autumnal.

Moreover, the effects of the luminaries depend upon the latitudes of places. Let ApEP represent the earth covered with deep waters; C its centre; P, p its poles; AE the equator; F any place without the equator; Ff the parallel of the place; Dd the correspondent parallel on the[Pg 417] other side of the equator; L the place of the moon three hours before; H the place of the earth directly under it; h the opposite place; K, k the places at 90 degrees distance; CH, Ch, the greatest heights of the sea from the centre of the earth; and CK, Ck, its least heights: and if with the axes Hh, Kk, an ellipsis is described, and by the revolution of that ellipsis about its longer axis Hh a spheroid HPKhpk is formed, this spheroid will nearly represent the figure of the sea; and CF, Cf, CD, Cd, will represent the heights of the sea in the places Ff, Dd. But farther; in the said revolution of the ellipsis any point N describes the circle NM cutting the parallels Ff, Dd, in any places RT, and the equator AE in S; CN will represent the height of the sea in all those places R, S, T, situated in this circle. Wherefore, in the diurnal revolution of any place F, the greatest flood will be in F, at the third hour after the appulse of the moon to the meridian above the horizon; and afterwards the greatest ebb in Q, at the third hour after the setting of the moon; and then the greatest flood in f, at the third hour after the appulse of the moon to the meridian under the horizon; and, lastly, the greatest ebb in Q, at the third hour after the rising of the moon; and the latter flood in f will be less than the preceding flood in F. For the whole sea is divided into two hemispherical floods, one in the hemisphere KHk on the north side, the other in the opposite hemisphere Khk, which we may therefore call the northern and the southern floods. These floods, being always opposite the one to the other, come by turns to the meridians of all places, after an interval of 12 lunar hours. And seeing the northern countries partake more of the northern flood, and the southern countries more of the southern flood, thence arise tides, alternately greater and less in all places without the equator, in which the luminaries rise and set. But the greatest tide will happen when the moon declines towards the vertex of the place, about the third hour after the appulse of the moon to the meridian above the horizon; and when the moon changes its declination to the other side of the equator, that which was the greater tide will be changed into a lesser. And the greatest difference of the floods will fall out about the times of the solstices; especially if the ascending node of the moon is about the first of Aries. So it is found by experience that the morning tides in winter exceed those of the evening, and the evening tides in summer exceed those of the morning; at Plymouth by the height of one foot, but at Bristol by the height of 15 inches, according to the observations of Colepress and Sturmy.

But the motions which we have been describing suffer some alteration from that force of reciprocation, which the waters, being once moved, retain a little while by their vis insita. Whence it comes to pass that the tides may continue for some time, though the actions of the luminaries should[Pg 418] cease. This power of retaining the impressed motion lessens the difference of the alternate tides, and makes those tides which immediately succeed after the syzygies greater, and those which follow next after the quadratures less. And hence it is that the alternate tides at Plymouth and Bristol do not differ much more one from the other than by the height of a foot or 15 inches, and that the greatest tides of all at those ports are not the first but the third after the syzygies. And, besides, all the motions are retarded in their passage through shallow channels, so that the greatest tides of all, in some straits and mouths of rivers, are the fourth or even the fifth after the syzygies.

Farther, it may happen that the tide may be propagated from the ocean through different channels towards the same port, and may pass quicker through some channels than through others; in which case the same tide, divided into two or more succeeding one another, may compound new motions of different kinds. Let us suppose two equal tides flowing towards the same port from different places, the one preceding the other by 6 hours; and suppose the first tide to happen at the third hour of the appulse of the moon to the meridian of the port. If the moon at the time of the appulse to the meridian was in the equator, every 6 hours alternately there would arise equal floods, which, meeting with as many equal ebbs, would so balance one the other, that for that day, the water would stagnate and remain quiet. If the moon then declined from the equator, the tides in the ocean would be alternately greater and less, as was said; and from thence two greater and two lesser tides would be alternately propagated towards that port. But the two greater floods would make the greatest height of the waters to fall out in the middle time betwixt both; and the greater and lesser floods would make the waters to rise to a mean height in the middle time between them, and in the middle time between the two lesser floods the waters would rise to their least height. Thus in the space of 24 hours the waters would come, not twice, as commonly, but once only to their greatest, and once only to their least height; and their greatest height, if the moon declined towards the elevated pole, would happen at the 6th or 30th hour after the appulse of the moon to the meridian; and when the moon changed its declination, this flood would be changed into an ebb. An example of all which Dr. Halley has given us, from the observations of seamen in the port of Batsham, in the kingdom of Tunquin, in the latitude of 20° 50' north. In that port, on the day which follows after the passage of the moon over the equator, the waters stagnate: when the moon declines to the north, they begin to flow and ebb, not twice, as in other ports, but once only every day: and the flood happens at the setting, and the greatest ebb at the rising of the moon. This tide increases with the declination of the moon till the 7th or 8th day; then for the 7 or 8 days following it[Pg 419] decreases at the same rate as it had increased before, and ceases when the moon changes its declination, crossing over the equator to the south. After which the flood is immediately changed into an ebb; and thenceforth the ebb happens at the setting and the flood at the rising of the moon; till the moon, again passing the equator, changes its declination. There are two inlets to this port and the neighboring channels, one from the seas of China, between the continent and the island of Leuconia; the other from the Indian sea, between the continent and the island of Borneo. But whether there be really two tides propagated through the said channels, one from the Indian sea in the space of 12 hours, and one from the sea of China in the space of 6 hours, which therefore happening at the 3d and 9th lunar hours, by being compounded together, produce those motions; or whether there be any other circumstances in the state of those seas, I leave to be determined by observations on the neighbouring shores.

Thus I have explained the causes of the motions of the moon and of the sea. Now it is fit to subjoin something concerning the quantity of those motions.

PROPOSITION XXV. PROBLEM VI.

To find the forces with which the sun disturbs the motions of the moon.

Let S represent the sun, T the earth, P the moon, CADB the moon's orbit. In SP take SK equal to ST; and let SL be to SK in the duplicate proportion of SK to SP: draw LM parallel to PT; and if ST or SK is supposed to represent the accelerated force of gravity of the earth towards the sun, SL will represent the accelerative force of gravity of the moon towards the sun. But that force is compounded of the parts SM and LM, of which the force LM, and that part of SM which is represented by TM, disturb the motion of the moon, as we have shewn in Prop. LXVI, Book I, and its Corollaries. Forasmuch as the earth and moon are revolved about their common centre of gravity, the motion of the earth about that centre will be also disturbed by the like forces; but we may consider the sums both of the forces and of the motions as in the moon, and represent the sum of the forces by the lines TM and ML, which are analogous to them both. The force ML (in its mean quantity) is to the centripetal force by which the moon may be retained in its orbit revolving about the earth at rest, at the distance PT, in the duplicate proportion of the periodic time of the moon about the earth to the periodic time of the earth about the sun (by Cor. 17, Prop. LXVI, Book I); that is, in the duplicate proportion of 27d. 7h. 43' to 365d. 6h. 9'; or as 1000 to 178725; or as 1 to . But in the[Pg 420] 4th Prop. of this Book we found, that, if both earth and moon were revolved about their common centre of gravity, the mean distance of the one from the other would be nearly mean semi-diameters of the earth; and the force by which the moon may be kept revolving in its orbit about the earth in rest at the distance PT of semi-diameters of the earth, is to the force by which it may be revolved in the same time, at the distance of 60 semi-diameters, as to 60: and this force is to the force of gravity with us very nearly as 1 to 60 × 60. Therefore the mean force ML is to the force of gravity on the surface of our earth as to , or as 1 to 638092,6; whence by the proportion of the lines TM, ML, the force TM is also given; and these are the forces with which the sun disturbs the motions of the moon. Q.E.I.

PROPOSITION XXVI. PROBLEM VII.

To find the horary increment of the area which the moon, by a radius drawn to the earth, describes in a circular orbit.

We have above shown that the area which the moon describes by a radius drawn to the earth is proportional to the time of description, excepting in so far as the moon's motion is disturbed by the action of the sun; and here we propose to investigate the inequality of the moment, or horary increment of that area or motion so disturbed. To render the calculus more easy, we shall suppose the orbit of the moon to be circular, and neglect all inequalities but that only which is now under consideration; and, because of the immense distance of the sun, we shall farther suppose that the lines SP and ST are parallel. By this means, the force LM will be always reduced to its mean quantity TP, as well as the force TM to its mean quantity 3PK. These forces (by Cor. 2 of the Laws of Motion) compose the force TL; and this force, by letting fall the perpendicular LE upon the radius TP, is resolved into the forces TE, EL; of which the force TE, acting constantly in the direction of the radius TP, neither accelerates nor retards the description of the area TPC made by that radius TP; but EL, acting on the radius TP in a perpendicular direction, accelerates or retards the description of the area in proportion as it accelerates or retards the moon.[Pg 421] That acceleration of the moon, in its passage from the quadrature C to the conjunction A, is in every moment of time as the generating accelerative force EL, that is, as . Let the time be represented by the mean motion of the moon, or (which comes to the same thing) by the angle CTP, or even by the arc CP. At right angles upon CT erect CG equal to CT; and, supposing the quadrantal arc AC to be divided into an infinite number of equal parts Pp, &c., these parts may represent the like infinite number of the equal parts of time. Let fall pk perpendicular on CT, and draw TG meeting with KP, kp produced in F and f; then will FK be equal to TK, and Kk be to PK as Pp to Tp, that is, in a given proportion; and therefore FK × Kk, or the area FKkf, will be as , that is, as EL; and compounding, the whole area GCKF will be as the sum of all the forces EL impressed upon the moon in the whole time CP; and therefore also as the velocity generated by that sum, that is, as the acceleration of the description of the area CTP, or as the increment of the moment thereof. The force by which the moon may in its periodic time CADB of 27d. 7h. 43' be retained revolving about the earth in rest at the distance TP, would cause a body falling in the time CT to describe the length , and at the same time to acquire a velocity equal to that with which the moon is moved in its orbit. This appears from Cor. 9, Prop. IV., Book I. But since Kd, drawn perpendicular on TP, is but a third part of EL, and equal to the half of TP, or ML, in the octants, the force EL in the octants, where it is greatest, will exceed the force ML in the proportion of 3 to 2; and therefore will be to that force by which the moon in its periodic time may be retained revolving about the earth at rest as 100 to , or 11915; and in the time CT will generate a velocity equal to parts of the velocity of the moon; but in the time CPA will generate a greater velocity in the proportion of CA to CT or TP. Let the greatest force EL in the octants be represented by the area FK × Kk, or by the rectangle , which is equal thereto; and the velocity which that greatest force can generate in any time CP will be to the velocity which any other lesser force EL can generate in the same time as the rectangle to the area KCGF; but the velocities generated in the whole time CPA will be one to the other as the rectangle to the triangle TCG, or as the quadrantal arc CA to the radius TP; and therefore the latter velocity generated in the whole time will be parts of the velocity of the moon. To this velocity of the moon, which is proportional to the mean moment of the area (supposing this mean moment to be represented by the number 11915), we add and subtract the half of the other velocity; the sum 11915 + 50, or 11965, will represent the greatest moment of the area in the syzygy A; and the[Pg 422] difference 11915 - 50, or 11865, the least moment thereof in the quadratures. Therefore the areas which in equal times are described in the syzygies and quadratures are one to the other as 11965 to 11865. And if to the least moment 11865 we add a moment which shall be to 100, the difference of the two former moments, as the trapezium FKCG to the triangle TCG, or, which comes to the same thing, as the square of the sine PK to the square of the radius TP (that is, as Pd to TP), the sum will represent the moment of the area when the moon is in any intermediate place P.

But these things take place only in the hypothesis that the sun and the earth are at rest, and that the synodical revolution of the moon is finished in 27d. 7h. 43'. But since the moon's synodical period is really 29d. 12h. 44', the increments of the moments must be enlarged in the same proportion as the time is, that is, in the proportion of 1080853 to 1000000. Upon which account, the whole increment, which was parts of the mean moment, will now become parts thereof; and therefore the moment of the area in the quadrature of the moon will be to the moment thereof in the syzygy as 11023 - 50 to 11023 + 50; or as 10973 to 11073; and to the moment thereof, when the moon is in any intermediate place P, as 10973 to 10973 + Pd; that is, supposing TP = 100.

The area, therefore, which the moon, by a radius drawn to the earth, describes in the several little equal parts of time, is nearly as the sum of the number 219,46, and the versed sine of the double distance of the moon from the nearest quadrature, considered in a circle which hath unity for its radius. Thus it is when the variation in the octants is in its mean quantity. But if the variation there is greater or less, that versed sine must be augmented or diminished in the same proportion.

PROPOSITION XXVII. PROBLEM VIII.

From the horary motion of the moon to find its distance from the earth.

The area which the moon, by a radius drawn to the earth, describes in every moment of time, is as the horary motion of the moon and the square of the distance of the moon from the earth conjunctly. And therefore the distance of the moon from the earth is in a proportion compounded of the subduplicate proportion of the area directly, and the subduplicate proportion of the horary motion inversely. Q.E.I.

COR. 1. Hence the apparent diameter of the moon is given; for it is reciprocally as the distance of the moon from the earth. Let astronomers try how accurately this rule agrees with the phænomena.

[Pg 423]

COR. 2. Hence also the orbit of the moon may be more exactly defined from the phænomena than hitherto could be done.

PROPOSITION XXVIII. PROBLEM IX.

To find the diameters of the orbit, in which, without eccentricity, the moon would move.

The curvature of the orbit which a body describes, if attracted in lines perpendicular to the orbit, is as the force of attraction directly, and the square of the velocity inversely. I estimate the curvatures of lines compared one with another according to the evanescent proportion of the sines or tangents of their angles of contact to equal radii, supposing those radii to be infinitely diminished. But the attraction of the moon towards the earth in the syzygies is the excess of its gravity towards the earth above the force of the sun 2PK (see Fig. Prop. XXV), by which force the accelerative gravity of the moon towards the sun exceeds the accelerative gravity of the earth towards the sun, or is exceeded by it. But in the quadratures that attraction is the sum of the gravity of the moon towards the earth, and the sun's force KT, by which the moon is attracted towards the earth. And these attractions, putting N for , are nearly as

and , or as 178725N × CT2 - 2000AT2 × CT, and 178725N × AT2 + 1000CT2 × AT. For if the accelerative gravity of the moon towards the earth be represented by the number 178725, the mean force ML, which in the quadratures is PT or TK, and draws the moon towards the earth, will be 1000, and the mean force TM in the syzygies will be 3000; from which, if we subtract the mean force ML, there will remain 2000, the force by which the moon in the syzygies is drawn from the earth, and which we above called 2PK. But the velocity of the moon in the syzygies A and B is to its velocity in the quadratures C and D as CT to AT, and the moment of the area, which the moon by a radius drawn to the earth describes in the syzygies, to the moment of that area described in the quadratures conjunctly; that is, as 11073CT to 10973AT. Take this ratio twice inversely, and the former ratio once directly, and the curvature of the orb of the moon in the syzygies will be to the curvature thereof in the quadratures as 120406729 × 178725AT2 × CT2 × N - 120406729 × 2000AT4 × CT to 122611329 × 178725AT2 × CT2 × N + 122611329 × 1000CT4 × AT, that is, as 2151969AT × CT × N - 24081AT3 to 2191371AT × CT × N + 12261CT3.

Because the figure of the moon's orbit is unknown, let us, in its stead, assume the ellipsis DBCA, in the centre of which we suppose the earth to be situated, and the greater axis DC to lie between the quadratures as the lesser AB between the syzygies. But since the plane of this ellipsis is revolved about the earth by an angular motion, and the orbit, whose curvature we now examine, should be described in a plane void of such motion[Pg 424] we are to consider the figure which the moon, while it is revolved in that ellipsis, describes in this plane, that is to say, the figure Cpa, the several points p of which are found by assuming any point P in the ellipsis, which may represent the place of the moon, and drawing Tp equal to TP in such manner that the angle PTp may be equal to the apparent motion of the sun from the time of the last quadrature in C; or (which comes to the same thing) that the angle CTp may be to the angle CTP as the time of the synodic revolution of the moon to the time of the periodic revolution thereof, or as 29d. 12h. 44' to 27d. 7h. 43'. If, therefore, in this proportion we take the angle CTa to the right angle CTA, and make Ta of equal length with TA, we shall have a the lower and C the upper apsis of this orbit Cpa. But, by computation, I find that the difference betwixt the curvature of this orbit Cpa at the vertex a, and the curvature of a circle described about the centre T with the interval TA, is to the difference between the curvature of the ellipsis at the vertex A, and the curvature of the same circle, in the duplicate proportion of the angle CTP to the angle CTp; and that the curvature of the ellipsis in A is to the curvature of that circle in the duplicate proportion of TA to TC; and the curvature of that circle to the curvature of a circle described about the centre T with the interval TC as TC to TA; but that the curvature of this last arch is to the curvature of the ellipsis in C in the duplicate proportion of TA to TC; and that the difference betwixt the curvature of the ellipsis in the vertex C, and the curvature of this last circle, is to the difference betwixt the curvature of the figure Cpa, at the vertex C, and the curvature of this same last circle, in the duplicate proportion of the angle CTp to the angle CTP; all which proportions are easily drawn from the sines of the angles of contact, and of the differences of those angles. But, by comparing those proportions together, we find the curvature of the figure Cpa at a to be to its curvature at C as

to ; where the number represents the difference of the squares of the angles CTP and CTp, applied to the square of the lesser angle CTP; or (which is all one) the difference of the squares of the times 27d. 7h. 43', and 29d. 12h. 44', applied to the square of the time 27d. 7h. 43'.

Since, therefore, a represents the syzygy of the moon, and C its quadrature, the proportion now found must be the same with that proportion of the curvature of the moon's orb in the syzygies to the curvature thereof in the quadratures, which we found above. Therefore, in order to find the[Pg 425] proportion of CT to AT, let us multiply the extremes and the means, and the terms which come out, applied to AT × CT, become 2062,79CT4 - 2151969N × CT3 + 368676N × AT × CT2 + 36342AT2 × CT2 - 362047N × AT2 × CT + 2191371N × AT3 + 4051,4AT4 = 0. Now if for the half sum N of the terms AT and CT we put 1, and x for their half difference, then CT will be = 1 + x, and AT = 1 - x. And substituting those values in the equation, after resolving thereof, we shall find x = 0,00719; and from thence the semi-diameter CT = 1,00719, and the semi-diameter AT = 0,99281, which numbers are nearly as , and . Therefore the moon's distance from the earth in the syzygies is to its distance in the quadratures (setting aside the consideration of eccentricity) as to ; in round numbers, as 69 to 70.

PROPOSITION XXIX. PROBLEM X.

To find the variation of the moon.

This inequality is owing partly to the elliptic figure of the moon's orbit, partly to the inequality of the moments of the area which the moon by a radius drawn to the earth describes. If the moon P revolved in the ellipsis DBCA about the earth quiescent in the centre of the ellipsis, and by the radius TP, drawn to the earth, described the area CTP, proportional to the time of description; and the greatest semi-diameter CT of the ellipsis was to the least TA as 70 to 69; the tangent of the angle CTP would be to the tangent of the angle of the mean motion, computed from the quadrature C, as the semi-diameter TA of the ellipsis to its semi-diameter TC, or as 69 to 70. But the description of the area CTP, as the moon advances from the quadrature to the syzygy, ought to be in such manner accelerated, that the moment of the area in the moon's syzygy may be to the moment thereof in its quadrature as 11073 to 10973; and that the excess of the moment in any intermediate place P above the moment in the quadrature may be as the square of the sine of the angle CTP; which we may effect with accuracy enough, if we diminish the tangent of the angle CTP in the subduplicate proportion of the number 10973 to the number 11073, that is, in proportion of the number 68,6877 to the number 69. Upon which account the tangent of the angle CTP will now be to the tangent of the mean motion as 68,6877 to 70; and the angle CTP in the octants, where the mean motion is 45°, will be found 44° 27' 28'', which subtracted from 45°, the angle of the mean motion, leaves the greatest variation 32' 32''. Thus it would be, if the moon, in passing from the quadrature to the syzygy, described an angle CTA of 90 degrees only. But because of the motion of the earth, by which the sun is apparently transferred in consequentia, the moon, before it overtakes the sun, describes an angle CTa, greater than a right angle, in the proportion of the time of the synodic revolution of the moon to the [Pg 426]time of its periodic revolution, that is, in the proportion of 29d. 12h. 44' to 27d. 7h. 43'. Whence it comes to pass that all the angles about the centre T are dilated in the same proportion; and the greatest variation, which otherwise would be but 32' 32'', now augmented in the said proportion, becomes 35' 10''.

And this is its magnitude in the mean distance of the sun from the earth, neglecting the differences which may arise from the curvature of the orbis magnus, and the stronger action of the sun upon the moon when horned and new, than when gibbous and full. In other distances of the sun from the earth, the greatest variation is in a proportion compounded of the duplicate proportion of the time of the synodic revolution of the moon (the time of the year being given) directly, and the triplicate proportion of the distance of the sun from the earth inversely. And, therefore, in the apogee of the sun, the greatest variation is 33' 14'', and in its perigee 37' 11'', if the eccentricity of the sun is to the transverse semi-diameter of the orbis magnus as to 1000.

Hitherto we have investigated the variation in an orb not eccentric, in which, to wit, the moon in its octants is always in its mean distance from the earth. If the moon, on account of its eccentricity, is more or less removed from the earth than if placed in this orb, the variation may be something greater, or something less, than according to this rule. But I leave the excess or defect to the determination of astronomers from the phænomena.

PROPOSITION XXX. PROBLEM XI.

To find the horary motion of the nodes of the moon in a circular orbit.

Let S represent the sun, T the earth, P the moon, NPn the orbit of the moon, Npn the orthographic projection of the orbit upon the plane of the ecliptic; N, n the nodes, nTNm the line of the nodes produced indefinitely;[Pg 427] PI, PK perpendiculars upon the lines ST, Qq; Pp a perpendicular upon the plane of the ecliptic; A, B the moon's syzygies in the plane of the ecliptic; AZ a perpendicular let fall upon Nn, the line of the nodes; Q, q the quadratures of the moon in the plane of the ecliptic, and pK a perpendicular on the line Qq lying between the quadratures. The force of the sun to disturb the motion of the moon (by Prop. XXV) is twofold, one proportional to the line LM, the other to the line MT, in the scheme of that Proposition; and the moon by the former force is drawn towards the earth, by the latter towards the sun, in a direction parallel to the right line ST joining the earth and the sun. The former force LM acts in the direction of the plane of the moon's orbit, and therefore makes no change upon the situation thereof, and is upon that account to be neglected; the latter force MT, by which the plane of the moon's orbit is disturbed, is the same with the force 3PK or 3IT. And this force (by Prop. XXV) is to the force by which the moon may, in its periodic time, be uniformly revolved in a circle about the earth at rest, as 3IT to the radius of the circle multiplied by the number 178,725, or as IT to the radius thereof multiplied by 59,575. But in this calculus, and all that follows, I consider all the lines drawn from the moon to the sun as parallel to the line which joins the earth and the sun; because what inclination there is almost as much diminishes all effects in some cases as it augments them in others; and we are now inquiring after the mean motions of the nodes, neglecting such niceties as are of no moment, and would only serve to render the calculus more perplexed.

Now suppose PM to represent an arc which the moon describes in the least moment of time, and ML a little line, the half of which the moon, by the impulse of the said force 3IT, would describe in the same time; and joining PL, MP, let them be produced to m and l, where they cut the plane of the ecliptic, and upon Tm let fall the perpendicular PH. Now, since the right line ML is parallel to the plane of the ecliptic, and therefore can never meet with the right line ml which lies in that plane, and yet both those right lines lie in one common plane LMPml, they will be parallel, and upon that account the triangles LMP, lmP will be similar. And seeing MPm lies in the plane of the orbit, in which the moon did move while in the place P, the point m will fall upon the line Nn, which passes through the nodes N, n, of that orbit. And because the force by which the half of the little line LM is generated, if the whole had been together, and it once impressed in the point P, would have generated that whole line, and caused the moon to move in the arc whose chord is LP; that is to say, would have transferred the moon from the plane MPmT into the plane LPlT; therefore the angular motion of the nodes generated by that force will be equal to the angle mTl. But ml is to mP as ML to MP; and since MP, because of the time given, is also given, ml will be as the rectangle[Pg 428] ML × mP, that is, as the rectangle IT × mP. And if Tml is a right angle, the angle mTl will be as , and therefore as , that is (because Tm and mP, TP and PH are proportional), as ; and, therefore, because TP is given, as IT × PH. But if the angle Tml or STN is oblique, the angle mTl will be yet less, in proportion of the sine of the angle STN to the radius, or AZ to AT. And therefore the velocity of the nodes is as IT × PH × AZ, or as the solid content of the sines of the three angles TPI, PTN, and STN.

If these are right angles, as happens when the nodes are in the quadratures, and the moon in the syzygy, the little line ml will be removed to an infinite distance, and the angle mTl will become equal to the angle mPl. But in this case the angle mPl is to the angle PTM, which the moon in the same time by its apparent motion describes about the earth, as 1 to 59,575. For the angle mPl is equal to the angle LPM, that is, to the angle of the moon's deflexion from a rectilinear path; which angle, if the gravity of the moon should have then ceased, the said force of the sun 3IT would by itself have generated in that given time; and the angle PTM is equal to the angle of the moon's deflexion from a rectilinear path; which angle, if the force of the sun 3IT should have then ceased, the force alone by which the moon is retained in its orbit would have generated in the same time. And these forces (as we have above shewn) are the one to the other as 1 to 59,575. Since, therefore, the mean horary motion of the moon (in respect of the fixed stars) is 32' 56'' 27''' . the horary motion of the node in this case will be 33'' 10''' 33iv. 12v. But in other cases the horary motion will be to 33'' 10''' 33iv. 12v. as the solid content of the sines of the three angles TPI, PTN, and STN (or of the distances of the moon from the quadrature, of the moon from the node, and of the node from the sun) to the cube of the radius. And as often as the sine of any angle is changed from positive to negative, and from negative to positive, so often must the regressive be changed into a progressive, and the progressive into a regressive motion. Whence it comes to pass that the nodes are progressive as often as the moon happens to be placed between either quadrature, and the node nearest to that quadrature. In other cases they are regressive, and by the excess of the regress above the progress, they are monthly transferred in antecedentia.

COR. 1. Hence if from P and M, the extreme points of a least arc PM, on the line Qq joining the quadratures we let fall the perpendiculars PK Mk, and produce the same till they cut the line of the nodes Nn in D and d, the horary motion of the nodes will be as the area MPDd, and the square of the line AZ conjunctly. For let PK, PH, and AZ, be the three said sines, viz., PK the sine of the distance of the moon from the quadrature,[Pg 429] PH the sine of the distance of the moon from the node, and AZ the sine of the distance of the node from the sun; and the velocity of the node will be as the solid content of PK × PH × AZ. But PT is to PK as PM to Kk; and, therefore, because PT and PM are given, Kk will be as PK. Likewise AT is to PD as AZ to PH, and therefore PH is as the rectangle PD × AZ; and, by compounding those proportions, PK × PH is as the solid content Kk × PD × AZ, and PK × PH × AZ as Kk × PD × AZ2; that is, as the area PDdM and AZ2 conjunctly. Q.E.D.

COR. 2. In any given position of the nodes their mean horary motion is half their horary motion in the moon's syzygies; and therefore is to 16'' 35''' 16iv. 36v. as the square of the sine of the distance of the nodes from the syzygies to the square of the radius, or as AZ2 to AT2. For if the moon, by an uniform motion, describes the semi-circle QAq, the sum of all the areas PDdM, during the time of the moon's passage from Q to M, will make up the area QMdE, terminating at the tangent QE of the circle; and by the time that the moon has arrived at the point n, that sum will make up the whole area EQAn described by the line PD: but when the moon proceeds from n to q, the line PD will fall without the circle, and describe the area nqe, terminating at the tangent qe of the circle, which area, because the nodes were before regressive, but are now progressive, must be subducted from the former area, and, being itself equal to the area QEN, will leave the semi-circle NQAn. While, therefore, the moon describes a semi-circle, the sum of all the areas PDdM will be the area of that semi-circle; and while the moon describes a complete circle, the sum of those areas will be the area of the whole circle. But the area PDdM, when the moon is in the syzygies, is the rectangle of the arc PM into the radius PT; and the sum of all the areas, every one equal to this area, in the time that the moon describes a complete circle, is the rectangle of the whole circumference into the radius of the circle; and this rectangle, being double the area of the circle, will be double the quantity of the former sum.[Pg 430] If, therefore, the nodes went on with that velocity uniformly continued which they acquire in the moon's syzygies, they would describe a space double of that which they describe in fact; and, therefore, the mean motion, by which, if uniformly continued, they would describe the same space with that which they do in fact describe by an unequal motion, is but one-half of that motion which they are possessed of in the moon's syzygies. Wherefore since their greatest horary motion, if the nodes are in the quadratures, is 33'' 10''' 33iv. 12v. their mean horary motion in this case will be 16'' 35''' 16iv. 36v. And seeing the horary motion of the nodes is every where as AZ2 and the area PDdM conjunctly, and, therefore, in the moon's syzygies, the horary motion of the nodes is as AZ2 and the area PDdM conjunctly, that is (because the area PDdM described in the syzygies is given), as AZ2, therefore the mean motion also will be as AZ2; and, therefore, when the nodes are without the quadratures, this motion will be to 16'' 35''' 16iv. 36v. as AZ2 to AT2. Q.E.D.

PROPOSITION XXXI. PROBLEM XII.

To find the horary motion of the nodes of the moon in an elliptic orbit.

Let Qpmaq represent an ellipsis described with the greater axis Qq, and the lesser axis ab; QAqB a circle circumscribed; T the earth in the common centre of both; S the sun; p the moon moving in this ellipsis; and[Pg 431] pm an arc which it describes in the least moment of time; N and n the nodes joined by the line Nn; pK and mk perpendiculars upon the axis Qq, produced both ways till they meet the circle in P and M, and the line of the nodes in D and d. And if the moon, by a radius drawn to the earth, describes an area proportional to the time of description, the horary motion of the node in the ellipsis will be as the area pDdm and AZ2 conjunctly.

For let PF touch the circle in P, and produced meet TN in F; and pf touch the ellipsis in p, and produced meet the same TN in f, and both tangents concur in the axis TQ at Y. And let ML represent the space which the moon, by the impulse of the above-mentioned force 3IT or 3PK, would describe with a transverse motion, in the meantime while revolving in the circle it describes the arc PM; and ml denote the space which the moon revolving in the ellipsis would describe in the same time by the impulse of the same force 3IT or 3PK; and let LP and lp be produced till they meet the plane of the ecliptic in G and g, and FG and fg be joined, of which FG produced may cut pf, pg, and TQ, in c, e, and R respectively; and fg produced may cut TQ in r. Because the force 3IT or 3PK in the circle is to the force 3IT or 3pK in the ellipsis as PK to pK, or as AT to aT, the space ML generated by the former force will be to the space ml generated by the latter as PK to pK; that is, because of the similar figures PYKp and FYRc, as FR to cR. But (because of the similar triangles PLM, PGF) ML is to FG as PL to PG, that is (on account of the parallels Lk, PK, GR), as pl to pe, that is (because of the similar triangles plm, cpe), as lm to ce; and inversely as LM is to lm, or as FR is to cR, so is FG to ce. And therefore if fg was to ce as fy to cY, that is, as fr to cR (that is, as fr to FR and FR to cR conjunctly, that is, as fT to FT, and FG to ce conjunctly), because the ratio of FG to ce, expunged on both sides, leaves the ratios fg to FG and fT to FT, fg would be to FG as fT to FT; and, therefore, the angles which FG and fg would subtend at the earth T would be equal to each other. But these angles (by what we have shewn in the preceding Proposition) are the motions of the nodes, while the moon describes in the circle the arc PM, in the ellipsis the arc pm; and therefore the motions of the nodes in the circle and in the ellipsis would be equal to each other. Thus, I say, it would be, if fg was to ce as fY to cY, that is, if fg was equal to . But because of the similar triangles fgp, cep, fg is to ce as fp to cp; and therefore fg is equal to ; and therefore the angle which fg subtends in fact is to the former angle which FG subtends, that is to say, the motion of the nodes in the ellipsis is to the motion of the same in the circle as this fg or to the former fg or , that is, as fp ×[Pg 432] cY to fY × cp, or as fp to fY, and cY to cp; that is, if ph parallel to TN meet FP in h, as Fh to FY and FY to FP; that is, as Fh to FP or Dp to DP, and therefore as the area Dpmd to the area DPMd. And, therefore, seeing (by Corol. 1, Prop. XXX) the latter area and AZ2 conjunctly are proportional to the horary motion of the nodes in the circle, the former area and AZ2 conjunctly will be proportional to the horary motion of the nodes in the ellipsis. Q.E.D.

COR. Since, therefore, in any given position of the nodes, the sum of all the areas pDdm, in the time while the moon is carried from the quadrature to any place m, is the area mpQEd terminated at the tangent of the ellipsis QE; and the sum of all those areas, in one entire revolution, is the area of the whole ellipsis; the mean motion of the nodes in the ellipsis will be to the mean motion of the nodes in the circle as the ellipsis to the circle; that is, as Ta to TA, or 69 to 70. And, therefore, since (by Corol. 2, Prop. XXX) the mean horary motion of the nodes in the circle is to 16'' 35''' 16iv. 36v. as AZ2 to AT2, if we take the angle 16'' 21''' 3iv. 30v. to the angle 16'' 35''' 16iv. 36v. as 69 to 70, the mean horary motion of the nodes in the ellipsis will be to 16'' 21''' 3iv. 30v. as AZ2 to AT2; that is, as the square of the sine of the distance of the node from the sun to the square of the radius.

But the moon, by a radius drawn to the earth, describes the area in the syzygies with a greater velocity than it does that in the quadratures, and upon that account the time is contracted in the syzygies, and prolonged in the quadratures; and together with the time the motion of the nodes is likewise augmented or diminished. But the moment of the area in the quadrature of the moon was to the moment thereof in the syzygies as 10973 to 11073; and therefore the mean moment in the octants is to the excess in the syzygies, and to the defect in the quadratures, as 11023, the half sum of those numbers, to their half difference 50. Wherefore since the time of the moon in the several little equal parts of its orbit is reciprocally as its velocity, the mean time in the octants will be to the excess of the time in the quadratures, and to the defect of the time in the syzygies arising from this cause, nearly as 11023 to 50. But, reckoning from the quadratures to the syzygies, I find that the excess of the moments of the area, in the several places above the least moment in the quadratures, is nearly as the square of the sine of the moon's distance from the quadratures; and therefore the difference betwixt the moment in any place, and the mean moment in the octants, is as the difference betwixt the square of the sine of the moon's distance from the quadratures, and the square of the sine of 45 degrees, or half the square of the radius; and the increment of the time in the several places between the octants and quadratures, and the decrement thereof between the octants and syzygies, is in the same proportion. But the motion of the nodes, while the moon describes the several little equal parts of its orbit, is accelerated or retarded[Pg 433] in the duplicate proportion of the time; for that motion, while the moon describes PM, is (cæteris paribus) as ML, and ML is in the duplicate proportion of the time. Wherefore the motion of the nodes in the syzygies, in the time while the moon describes given little parts of its orbit, is diminished in the duplicate proportion of the number 11073 to the number 11023; and the decrement is to the remaining motion as 100 to 10973; but to the whole motion as 100 to 11073 nearly. But the decrement in the places between the octants and syzygies, and the increment in the places between the octants and quadratures, is to this decrement nearly as the whole motion in these places to the whole motion in the syzygies, and the difference betwixt the square of the sine of the moon's distance from the quadrature, and the half square of the radius, to the half square of the radius conjunctly. Wherefore, if the nodes are in the quadratures, and we take two places, one on one side, one on the other, equally distant from the octant and other two distant by the same interval, one from the syzygy, the other from the quadrature, and from the decrements of the motions in the two places between the syzygy and octant we subtract the increments of the motions in the two other places between the octant and the quadrature, the remaining decrement will be equal to the decrement in the syzygy, as will easily appear by computation; and therefore the mean decrement, which ought to be subducted from the mean motion of the nodes, is the fourth part of the decrement in the syzygy. The whole horary motion of the nodes in the syzygies (when the moon by a radius drawn to the earth was supposed to describe an area proportional to the time) was 32'' 42'' 7iv. And we have shewn that the decrement of the motion of the nodes, in the time while the moon, now moving with greater velocity, describes the same space, was to this motion as 100 to 11073; and therefore this decrement is 17''' 43iv. 11v. The fourth part of which 4''' 25iv. 48v. subtracted from the mean horary motion above found, 16'' 21'' 3iv. 30v. leaves 16'' 16''' 37iv. 42v. their correct mean horary motion.

[Pg 434]

If the nodes are without the quadratures, and two places are considered, one on one side, one on the other, equally distant from the syzygies, the sum of the motions of the nodes, when the moon is in those places, will be to the sum of their motions, when the moon is in the same places and the nodes in the quadratures, as AZ2 to AT2. And the decrements of the motions arising from the causes but now explained will be mutually as the motions themselves, and therefore the remaining motions will be mutually betwixt themselves as AZ2 to AT2; and the mean motions will be as the remaining motions. And, therefore, in any given position of the nodes, their correct mean horary motion is to 16'' 16''' 37iv. 42v. as AZ2 to AT2; that is, as the square of the sine of the distance of the nodes from the syzygies to the square of the radius.

PROPOSITION XXXII. PROBLEM XIII.

To find the mean motion of the nodes of the moon.

The yearly mean motion is the sum of all the mean horary motions throughout the course of the year. Suppose that the node is in N, and that, after every hour is elapsed, it is drawn back again to its former place; so that, notwithstanding its proper motion, it may constantly remain in the same situation with respect to the fixed stars; while in the mean time the sun S, by the motion of the earth, is seen to leave the node, and to proceed till it completes its apparent annual course by an uniform motion. Let Aa represent a given least arc, which the right line TS always drawn to the sun, by its intersection with the circle NAn, describes in the least given moment of time; and the mean horary motion (from what we have above shewn) will be as AZ2, that is (because AZ and ZY are proportional), as the rectangle of AZ into ZY, that is, as the area AZYa; and the sum of all the mean horary motions from the beginning will be as the sum of all the areas aYZA, that is, as the area NAZ. But the greatest AZYa is equal to the rectangle of the arc Aa into the radius of the circle; and therefore the sum of all these rectangles in the whole circle will be to the like sum of all the greatest rectangles as the area of the whole circle to the rectangle of the whole circumference into the radius, that is, as 1 to 2. But the horary motion corresponding to that greatest rectangle was 16'' 16''' 37iv. 42v. and this motion in the complete course of the sidereal year, 365d. 6h. 9', amounts to 39° 38' 7'' 50''', and therefore the half thereof, 19° 49' 3'' 55''', is the mean motion of the nodes corresponding to the whole circle. And the motion of the nodes, in the time while the sun is carried from N to A, is to 19° 49' 3'' 55'' as the area NAZ to the whole circle.

Thus it would be if the node was after every hour drawn back again to its former place, that so, after a complete revolution, the sun at the year's end would be found again in the same node which it had left when the year begun. But, because of the motion of the node in the mean time, the sun must needs meet the node sooner; and now it remains that we compute the abbreviation of the time. Since, then, the sun, in the course of the year, travels 360 degrees, and the node in the same time by its greatest motion would be carried 39° 38' 7'' 50''', or 39,6355 degrees; and the mean motion of the node in any place N is to its mean motion in its quadratures as AZ2 to AT2; the motion of the sun will [Pg 435]be to the motion of the node in N as 360AT2 to 39·6355AZ2; that is, as 9,0827646AT2 to AZ2. Wherefore if we suppose the circumference NAn of the whole circle to be divided into little equal parts, such as Aa, the time in which the sun would describe the little arc Aa, if the circle was quiescent, will be to the time of which it would describe the same arc, supposing the circle together with the nodes to be revolved about the centre T, reciprocally as 9,0827646AT2 to 9,0827646AT2 + AZ2; for the time is reciprocally as the velocity with which the little arc is described, and this velocity is the sum of the velocities of both sun and node. If, therefore, the sector NTA represent the time in which the sun by itself, without the motion of the node, would describe the arc NA, and the indefinitely small part ATa of the sector represent the little moment of the time in which it would describe the least arc Aa; and (letting fall aY perpendicular upon Nn) if in AZ we take dZ of such length that the rectangle of dZ into ZY may be to the least part ATa of the sector as AZ2 to 9,0827646AT2 + AZ2, that is to say, that dZ may be to as AT2 to 9,0827646AT2 + AZ2; the rectangle of dZ into ZY will represent the decrement of the time arising from the motion of the node, while the arc Aa is described; and if the curve NdGn is the locus where the point d is always found, the curvilinear area NdZ will be as the whole decrement of time while the whole arc NA is described; and, therefore, the excess of the sector NAT above the area NdZ will be as the whole time. But because the motion of the node in a less time is less in proportion of the time, the area AaYZ must also be diminished in the same proportion; which may be done by taking in AZ the line eZ of such length, that it may be to the length of AZ as AZ2 to 9,0827646AT2 + AZ2; for so the rectangle of eZ into ZY will be to the area AZYa as the decrement of the time in which the arc Aa is described to the whole time in which it would have been described, if the node had been quiescent; and, therefore, that rectangle will be as the decrement of the motion of the node. And if the curve NeFn is the locus of the point e, the whole area NeZ, which is the sum of all the decrements of that motion, will be as the whole decrement thereof during the time in which the arc AN is described; and the remaining area NAe will be as the remaining motion, which is the true motion of the node, during the time in which the whole arc NA is described by the joint motions of both sun and node. Now the area of the semi-circle is to the area of the figure NeFn found by the method of infinite series nearly as 793 to 60. But the motion corresponding or proportional to the whole circle was 19° 49' 3'' 55'''; and therefore the motion corresponding to double the figure NeFn is 1° 29' 58'' 2''', which taken from the former motion leaves 18° 19' 5'' 53''', the whole motion of the node with respect to the fixed stars in the interval between two of its conjunctions with the sun; and this motion subducted from the annual motion of the sun 360°, leaves 341° 40' 54'' 7''',[Pg 436] the motion of the sun in the interval between the same conjunctions. But as this motion is to the annual motion 360°, so is the motion of the node but just now found 18° 19' 5'' 53''' to its annual motion, which will therefore be 19° 18' 1'' 23'''; and this is the mean motion of the nodes in the sidereal year. By astronomical tables, it is 19° 21' 21'' 50'''. The difference is less than part of the whole motion, and seems to arise from the eccentricity of the moon's orbit, and its inclination to the plane of the ecliptic. By the eccentricity of this orbit the motion of the nodes is too much accelerated; and, on the other hand, by the inclination of the orbit, the motion of the nodes is something retarded, and reduced to its just velocity.

PROPOSITION XXXIII. PROBLEM XIV.

To find the true motion of the nodes of the moon.

In the time which is as the area NTA - NdZ (in the preceding Fig.) that motion is as the area NAe, and is thence given; but because the calculus is too difficult, it will be better to use the following construction of the Problem. About the centre C, with any interval CD, describe the circle BEFD; produce DC to A so as AB may be to AC as the mean motion to half the mean true motion when the nodes are in their quadratures (that is, as 19° 18' 1'' 23''' to 19° 49' 3'' 55'''; and therefore BC to AC as the difference of those motions 0° 31' 2'' 32''' to the latter motion 19° 49' 3'' 55'', that is, as 1 to ). Then through the point D draw the indefinite line Gg, touching the circle in D; and if we take the angle BCE, or BCF, equal to the double distance of the sun from the place of the node, as found by the mean motion, and drawing AE or AF cutting the perpendicular DG in G, we take another angle which shall be to the whole motion of the node in the interval between its syzygies (that is, to 9° 11'' 3''') as the tangent DG to the whole circumference of the circle BED, and add this last angle (for which the angle DAG may be used) to the mean motion of the nodes, while they are passing from the quadratures to the syzygies, and subtract it from their mean motion while they are passing from the syzygies to the quadratures, we shall have their true motion; for the true motion so found will nearly agree with the true motion which comes out from assuming the times as the area NTA - NdZ, and the motion of the node as the area NAe; as whoever will please to examine and make the computations will find: and this is the semi-menstrual equation of the motion of the nodes. But there is also a menstrual equation, but which is by no means necessary for finding[Pg 437] of the moon's latitude; for since the variation of the inclination of the moon's orbit to the plane of the ecliptic is liable to a twofold inequality, the one semi-menstrual, the other menstrual, the menstrual inequality of this variation, and the menstrual equation of the nodes, so moderate and correct each other, that in computing the latitude of the moon both may be neglected.

COR. From this and the preceding Prop. it appears that the nodes are quiescent in their syzygies, but regressive in their quadratures, by an hourly motion of 16'' 19''' 26iv.; and that the equation of the motion of the nodes in the octants is 1° 30'; all which exactly agree with the phænomena of the heavens.

SCHOLIUM.

Mr. Machin, Astron., Prof. Gresh., and Dr. Henry Pemberton, separately found out the motion of the nodes by a different method. Mention has been made of this method in another place. Their several papers, both of which I have seen, contained two Propositions, and exactly agreed with each other in both of them. Mr. Machin's paper coming first to my hands, I shall here insert it.

OF THE MOTION OF THE MOON'S NODES.

PROPOSITION I.

"The mean motion of the sun from the node is defined by a geometric mean proportional between the mean motion of the sun and that mean motion with which the sun recedes with the greatest swiftness from the node in the quadratures.

"Let T be the earth's place, Nn the line of the moon's nodes at any given time, KTM a perpendicular thereto, TA a right line revolving about the centre with the same angular velocity with which the sun and the node recede from one another, in such sort that the angle between the quiescent right line Nn and the revolving line TA may be always equal to the distance of the places of the sun and node. Now if any right line TK be divided into parts TS and SK, and those parts be taken as the mean horary motion of the sun to the mean horary motion of the node in the quadratures, and there be taken the right line TH, a mean proportional between the part TS and the whole TK, this right line will be proportional to the sun's mean motion from the node.

"For let there be described the circle NKnM from the centre T and with the radius TK, and about the same centre, with the semi-axis TH[Pg 438] and TN, let there be described an ellipsis NHnL; and in the time in which the sun recedes from the node through the arc Na, if there be drawn the right line Tba, the area of the sector NTa will be the exponent of the sum of the motions of the sun and node in the same time. Let, therefore, the extremely small arc aA be that which the right line Tba, revolving according to the aforesaid law, will uniformly describe in a given particle of time, and the extremely small sector TAa will be as the sum of the velocities with which the sun and node are carried two different ways in that time. Now the sun's velocity is almost uniform, its inequality being so small as scarcely to produce the least inequality in the mean motion of the nodes. The other part of this sum, namely, the mean quantity of the velocity of the node, is increased in the recess from the syzygies in a duplicate ratio of the sine of its distance from the sun (by Cor. Prop. XXXI, of this Book), and, being greatest in its quadratures with the sun in K, is in the same ratio to the sun's velocity as SK to TS, that is, as (the difference of the squares of TK and TH, or) the rectangle KHM to TH2. But the ellipsis NBH divides the sector ATa, the exponent of the sum of these two velocities, into two parts ABba and BTb, proportional to the velocities. For produce BT to the circle in , and from the point B let fall upon the greater axis the perpendicular BG, which being produced both ways may meet the circle in the points F and f; and because the space ABba is to the sector TBb as the rectangle to BT2 (that rectangle being equal to the difference of the squares of TA and TB, because the right line is equally cut in T, and unequally in B), therefore when the space ABba is the greatest of all in K, this ratio will be the same as the ratio of the rectangle KHM to HT2. But the greatest mean velocity of the node was shewn above to be in that very[Pg 439] ratio to the velocity of the sun; and therefore in the quadratures the sector ATa is divided into parts proportional to the velocities. And because the rectangle KHM is to HT2 as FBf to BG2, and the rectangle is equal to the rectangle FBf, therefore the little area ABba, where it is greatest, is to the remaining sector TBb as the rectangle to BG2 But the ratio of these little areas always was as the rectangle to BT2; and therefore the little area ABba in the place A is less than its correspondent little area in the quadratures in the duplicate ratio of BG to BT, that is, in the duplicate ratio of the sine of the sun's distance from the node. And therefore the sum of all the little areas ABba, to wit, the space ABN, will be as the motion of the node in the time in which the sun hath been going over the arc NA since he left the node; and the remaining space, namely, the elliptic sector NTB, will be as the sun's mean motion in the same time. And because the mean annual motion of the node is that motion which it performs in the time that the sun completes one period of its course, the mean motion of the node from the sun will be to the mean motion of the sun itself as the area of the circle to the area of the ellipsis; that is, as the right line TK to the right line TH, which is a mean proportional between TK and TS; or, which comes to the same as the mean proportional TH to the right line TS.

PROPOSITION II.

"The mean motion of the moon's nodes being given, to find their true motion.

"Let the angle A be the distance of the sun from the mean place of the node, or the sun's mean motion from the node. Then if we take the angle B, whose tangent is to the tangent of the angle A as TH to TK, that is,[Pg 440] in the sub-duplicate ratio of the mean horary motion of the sun to the mean horary motion of the sun from the node, when the node is in the quadrature, that angle B will be the distance of the sun from the node's true place. For join FT, and, by the demonstration of the last Proposition, the angle FTN will be the distance of the sun from the mean place of the node, and the angle ATN the distance from the true place, and the tangents of these angles are between themselves as TK to TH.

"COR. Hence the angle FTA is the equation of the moon's nodes; and the sine of this angle, where it is greatest in the octants, is to the radius as KH to TK + TH. But the sine of this equation in any other place A is to the greatest sine as the sine of the sums of the angles FTN + ATN to the radius; that is, nearly as the sine of double the distance of the sun from the mean place of the node (namely, 2FTN) to the radius.

"SCHOLIUM.

"If the mean horary motion of the nodes in the quadratures be 16'' 16''' 37iv. 42v. that is, in a whole sidereal year, 39° 38' 7'' 50''', TH will be to TK in the subduplicate ratio of the number 9,0827646 to the number 10,0827646, that is, as 18,6524761 to 19,6524761. And, therefore, TH is to HK as 18,6524761 to 1; that is, as the motion of the sun in a sidereal year to the mean motion of the node 19° 18' 1'' .

"But if the mean motion of the moon's nodes in 20 Julian years is 386° 50' 15'', as is collected from the observations made use of in the theory of the moon, the mean motion of the nodes in one sidereal year will be 19° 20' 31'' 58'''. and TH will be to HK as 360° to 19° 20' 31'' 58'''; that is, as 18,61214 to 1: and from hence the mean horary motion of the nodes in the quadratures will come out 16'' 18''' 48iv. And the greatest equation of the nodes in the octants will be 1° 29' 57''."

PROPOSITION XXXIV. PROBLEM XV.

To find the horary variation of the inclination of the moon's orbit to the plane of the ecliptic.

Let A and a represent the syzygies; Q, and q the quadratures; N and n the nodes; P the place of the moon in its orbit; p the orthographic projection of that place upon the plane of the ecliptic; and mTl the momentaneous motion of the nodes as above. If upon Tm we let fall the perpendicular PG, and joining pG we produce it till it meet Tl in g, and join also Pg, the angle PGp will be the inclination of the moon's orbit to the plane of the ecliptic when the moon is in P; and the angle Pgp will be the inclination of the same after a small moment of time is elapsed; and therefore the angle GPg will be the momentaneous variation of the inclination. But this angle GPg is to the angle GTg as TG to PG and Pp to PG conjunctly. And, therefore, if for the moment of time we assume[Pg 441] an hour, since the angle GTg (by Prop. XXX) is to the angle 33'' 10''' 33iv. as IT × PG × AZ to AT3, the angle GPg (or the horary variation of the inclination) will be to the angle 33'' 10''' 33iv. as

to AT3. Q.E.I.

And thus it would be if the moon was uniformly revolved in a circular orbit. But if the orbit is elliptical, the mean motion of the nodes will be diminished in proportion of the lesser axis to the greater, as we have shewn above; and the variation of the inclination will be also diminished in the same proportion.

COR. 1. Upon Nn erect the perpendicular TF, and let pM be the horary motion of the moon in the plane of the ecliptic; upon QT let fall the perpendiculars pK, Mk, and produce them till they meet TF in H and h; then IT will be to AT as Kk to Mp; and TG to Hp as TZ to AT; and, therefore, IT × TG will be equal to , that is, equal to the area HpMh multiplied into the ratio : and therefore the horary variation of the inclination will be to 33'' 10''' 33iv. as the area HpMh multiplied into

to AT3.

COR. 2. And, therefore, if the earth and nodes were after every hour drawn back from their new and instantly restored to their old places, so as their situation might continue given for a whole periodic month together, the whole variation of the inclination during that [Pg 442]month would be to 33'' 10''' 33iv. as the aggregate of all the areas HpMh, generated in the time of one revolution of the point p (with due regard in summing to their proper signs + -), multiplied into

to Mp × AT3; that is, as the whole circle QAqa multiplied into to Mp × AT3, that is, as the circumference QAqa multiplied into to 2Mp × AT2.

COR. 3. And, therefore, in a given position of the nodes, the mean horary variation, from which, if uniformly continued through the whole month, that menstrual variation might be generated, is to 33'' 10''' 33iv. as to 2AT2, or as

to PG × 4AT; that is (because Pp is to PG as the sine of the aforesaid inclination to the radius, and

to 4AT as the sine of double the angle ATn to four times the radius), as the sine of the same inclination multiplied into the sine of double the distance of the nodes from the sun to four times the square of the radius.

[Pg 443]

COR. 4. Seeing the horary variation of the inclination, when the nodes are in the quadratures, is (by this Prop.) to the angle 33'' 10''' 33iv. as to AT3, that is, as

to 2AT, that is, as the sine of double the distance of the moon from the quadratures multiplied into to twice the radius, the sum of all the horary variations during the time that the moon, in this situation of the nodes, passes from the quadrature to the syzygy (that is, in the space of hours) will be to the sum of as many angles 33'' 10''' 33iv. or 5878'', as the sum of all the sines of double the distance of the moon from the quadratures multiplied into to the sum of as many diameters; that is, as the diameter multiplied into to the circumference; that is, if the inclination be 5° 1', as to 22, or as 278 to 10000. And, therefore, the whole variation, composed out of the sum of all the horary variations in the aforesaid time, is 163'', or 2' 43''.

PROPOSITION XXXV. PROBLEM XVI.

To a given time to find the inclination of the moon's orbit to the plane of the ecliptic.

Let AD be the sine of the greatest inclination, and AB the sine of the least. Bisect BD in C; and round the centre C, with the interval BC, describe the circle BGD. In AC take CE in the same proportion to EB as EB to twice BA. And if to the time given we set off the angle AEG equal to double the distance of the nodes from the quadratures, and upon AD let fall the perpendicular GH, AH will be the sine of the inclination required.

For GE2 is equal to GH2 + HE2 = BHD + HE2 = HBD + HE2 - BH2 = HBD + BE2 - 2BH × BE = BE2 + 2EC × BH = 2EC × AB + 2EC × BH = 2EC × AH; wherefore since 2EC is given. GE2 will be as AH. Now let AEg represent double the distance of the nodes from the quadratures, in a given moment of time after, and the arc Gg, on account of the given angle GEg, will be as the distance GE. But Hh is to Gg as GH to GC, and, therefore, Hh is as the rectangle GH × Gg, or GH × GE, that is, as , or ; that is, as AH and the sine of the angle AEG conjunctly. If, therefore, in any one case, AH be the sine of inclination, it will increase by the same increments as the sine of inclination doth, by Cor. 3 of the preceding Prop. and therefore will always continue equal to that sine. But when the point G falls upon either point B or D, AH is equal to this sine, and therefore remains always equal thereto. Q.E.D.

In this demonstration I have supposed that the angle BEG, representing double the distance of the nodes from the quadratures, increaseth uniformly; for I cannot descend to every minute circumstance of inequality. Now suppose that BEG is a right angle, and that Gg is in this case the horary increment of double the distance of the nodes from the sun; then, by Cor. 3 of the last Prop. the horary variation of the inclination in the same case will be to 33'' 10''' 33iv. as the rectangle of AH, the sine of the inclination, into the sine of the right angle BEG, double the distance of the nodes from the sun, to four times the square of the radius; that is, as AH,[Pg 444] the sine of the mean inclination, to four times the radius; that is, seeing the mean inclination is about 5° , as its sine 896 to 40000, the quadruple of the radius, or as 224 to 10000. But the whole variation corresponding to BD, the difference of the sines, is to this horary variation as the diameter BD to the arc Gg, that is, conjunctly as the diameter BD to the semi-circumference BGD, and as the time of hours, in which the node proceeds from the quadratures to the syzygies, to one hour, that is as 7 to 11, and to 1. Wherefore, compounding all these proportions, we shall have the whole variation BD to 33'' 10''' 33iv. as to 110000, that is, as 29645 to 1000; and from thence that variation BD will come out 16' .

And this is the greatest variation of the inclination, abstracting from the situation of the moon in its orbit; for if the nodes are in the syzygies, the inclination suffers no change from the various positions of the moon. But if the nodes are in the quadratures, the inclination is less when the moon is in the syzygies than when it is in the quadratures by a difference of 2' 43'', as we shewed in Cor. 4 of the preceding Prop.; and the whole mean variation BD, diminished by 1' , the half of this excess, becomes 15' 2'', when the moon is in the quadratures; and increased by the same, becomes 17' 45'' when the moon is in the syzygies. If, therefore, the moon be in the syzygies, the whole variation in the passage of the nodes from the quadratures to the syzygies will be 17' 45''; and, therefore, if the inclination be 5° 17' 20'', when the nodes are in the syzygies, it will be 4° 59' 35'' when the nodes are in the quadratures and the moon in the syzygies. The truth of all which is confirmed by observations.

Now if the inclination of the orbit should be required when the moon is in the syzygies, and the nodes any where between them and the quadratures, let AB be to AD as the sine of 4° 59' 35" to the sine of 5° 17' 20'', and take the angle AEG equal to double the distance of the nodes from the quadratures; and AH will be the sine of the inclination desired. To this inclination of the orbit the inclination of the same is equal, when the moon is 90° distant from the nodes. In other situations of the moon, this menstrual inequality, to which the variation of the inclination is obnoxious in the calculus of the moon's latitude, is balanced, and in a manner took off, by the menstrual inequality of the motion of the nodes (as we said before), and therefore may be neglected in the computation of the said latitude.

SCHOLIUM.

By these computations of the lunar motions I was willing to shew that by the theory of gravity the motions of the moon could be calculated from their physical causes. By the same theory I moreover found that the annual equation of the mean motion of the moon arises from the various[Pg 445] dilatation which the orbit of the moon suffers from the action of the sun according to Cor. 6, Prop. LXVI. Book I. The force of this action is greater in the perigeon sun, and dilates the moon's orbit; in the apogeon sun it is less, and permits the orbit to be again contracted. The moon moves slower in the dilated and faster in the contracted orbit; and the annual equation, by which this inequality is regulated, vanishes in the apogee and perigee of the sun. In the mean distance of the sun from the earth it arises to about 11' 50''; in other distances of the sun it is proportional to the equation of the sun's centre, and is added to the mean motion of the moon, while the earth is passing from its aphelion to its perihelion, and subducted while the earth is in the opposite semi-circle. Taking for the radius of the orbis magnus 1000, and for the earth's eccentricity, this equation, when of the greatest magnitude, by the theory of gravity comes out 11' 49''. But the eccentricity of the earth seems to be something greater, and with the eccentricity this equation will be augmented in the same proportion. Suppose the eccentricity and the greatest equation will be 11' 51''.

Farther; I found that the apogee and nodes of the moon move faster in the perihelion of the earth, where the force of the sun's action is greater, than in the aphelion thereof, and that in the reciprocal triplicate proportion of the earth's distance from the sun; and hence arise annual equations of those motions proportional to the equation of the sun's centre. Now the motion of the sun is in the reciprocal duplicate proportion of the earth's distance from the sun; and the greatest equation of the centre which this inequality generates is 1° 56' 20'', corresponding to the above-mentioned eccentricity of the sun, . But if the motion of the sun had been in the reciprocal triplicate proportion of the distance, this inequality would have generated the greatest equation 2° 54' 30''; and therefore the greatest equations which the inequalities of the motions of the moon's apogee and nodes do generate are to 2° 54' 30'' as the mean diurnal motion of the moon's apogee and the mean diurnal motion of its nodes are to the mean diurnal motion of the sun. Whence the greatest equation of the mean motion of the apogee comes out 19' 43'', and the greatest equation of the mean motion of the nodes 9' 24''. The former equation is added, and the latter subducted, while the earth is passing from its perihelion to its aphelion, and contrariwise when the earth is in the opposite semi-circle.

By the theory of gravity I likewise found that the action of the sun upon the moon is something greater when the transverse diameter of the moon's orbit passeth through the sun than when the same is perpendicular upon the line which joins the earth and the sun; and therefore the moon's orbit is something larger in the former than in the latter case. And hence arises another equation of the moon's mean motion, depending[Pg 446] upon the situation of the moon's apogee in respect of the sun, which is in its greatest quantity when the moon's apogee is in the octants of the sun, and vanishes when the apogee arrives at the quadratures or syzygies; and it is added to the mean motion while the moon's apogee is passing from the quadrature of the sun to the syzygy, and subducted while the apogee is passing from the syzygy to the quadrature. This equation, which I shall call the semi-annual, when greatest in the octants of the apogee, arises to about 3' 45'', so far as I could collect from the phænomena: and this is its quantity in the mean distance of the sun from the earth. But it is increased and diminished in the reciprocal triplicate proportion of the sun's distance, and therefore is nearly 3' 34'' when that distance is greatest, and 3' 56'' when least. But when the moon's apogee is without the octants, it becomes less, and is to its greatest quantity as the sine of double the distance of the moon's apogee from the nearest syzygy or quadrature to the radius.

By the same theory of gravity, the action of the sun upon the moon is something greater when the line of the moon's nodes passes through the sun than when it is at right angles with the line which joins the sun and the earth; and hence arises another equation of the moon's mean motion, which I shall call the second semi-annual; and this is greatest when the nodes are in the octants of the sun, and vanishes when they are in the syzygies or quadratures; and in other positions of the nodes is proportional to the sine of double the distance of either node from the nearest syzygy or quadrature. And it is added to the mean motion of the moon, if the sun is in antecedentia, to the node which is nearest to him, and subducted if in consequentia; and in the octants, where it is of the greatest magnitude, it arises to 47'' in the mean distance of the sun from the earth, as I find from the theory of gravity. In other distances of the sun, this equation, greatest in the octants of the nodes, is reciprocally as the cube of the sun's distance from the earth; and therefore in the sun's perigee it comes to about 49'', and in its apogee to about 45''.

By the same theory of gravity, the moon's apogee goes forward at the greatest rate when it is either in conjunction with or in opposition to the sun, but in its quadratures with the sun it goes backward; and the eccentricity comes, in the former case, to its greatest quantity; in the latter to its least, by Cor. 7, 8, and 9, Prop. LXVI, Book I. And those inequalities, by the Corollaries we have named, are very great, and generate the principal which I call the semi-annual equation of the apogee; and this semi-annual equation in its greatest quantity comes to about 12° 18', as nearly as I could collect from the phænomena. Our countryman, Horrox, was the first who advanced the theory of the moon's moving in an ellipsis about the earth placed in its lower focus. Dr. Halley improved the notion, by putting the centre of the ellipsis in an epicycle whose centre[Pg 447] is uniformly revolved about the earth; and from the motion in this epicycle the mentioned inequalities in the progress and regress of the apogee, and in the quantity of eccentricity, do arise.

Suppose the mean distance of the moon from the earth to be divided into 100000 parts, and let T represent the earth, and TC the moon's mean eccentricity of 5505 such parts. Produce TC to B, so as CB may be the sine of the greatest semi-annual equation 12° 18' to the radius TC; and the circle BDA described about the centre C, with the interval CB, will be the epicycle spoken of, in which the centre of the moon's orbit is placed, and revolved according to the order of the letters BDA. Set off the angle BCD equal to twice the annual argument, or twice the distance of the sun's true place from the place of the moon's apogee once equated, and CTD will be the semi-annual equation of the moon's apogee, and TD the eccentricity of its orbit, tending to the place of the apogee now twice equated. But, having the moon's mean motion, the place of its apogee, and its eccentricity, as well as the longer axis of its orbit 200000, from these data the true place of the moon in its orbit, together with its distance from the earth, may be determined by the methods commonly known.

In the perihelion of the earth, where the force of the sun is greatest, the centre of the moon's orbit moves faster about the centre C than in the aphelion, and that in the reciprocal triplicate proportion of the sun's distance from the earth. But, because the equation of the sun's centre is included in the annual argument, the centre of the moon's orbit moves faster in its epicycle BDA, in the reciprocal duplicate proportion of the sun's distance from the earth. Therefore, that it may move yet faster in the reciprocal simple proportion of the distance, suppose that from D, the centre of the orbit, a right line DE is drawn, tending towards the moon's apogee once equated, that is, parallel to TC; and set off the angle EDF equal to the excess of the aforesaid annual argument above the distance of the moon's apogee from the sun's perigee in consequentia; or, which comes to the same thing, take the angle CDF equal to the complement of the sun's true anomaly to 360°; and let DF be to DC as twice the eccentricity of the orbis magnus to the sun's mean distance from the earth, and the sun's mean diurnal motion from the moon's apogee to the sun's mean diurnal motion from its own apogee conjunctly, that is, as to 1000, and 52' 27'' 16''' to 59' 8'' 10''' conjunctly, or as 3 to 100; and imagine the centre of the moon's orbit placed in the point F to be revolved in an epicycle whose centre is D, and radius DF, while the point D moves in the circumference of the circle DABD: for by this means the centre of[Pg 448] the moon's orbit comes to describe a certain curve line about the centre C with a velocity which will be almost reciprocally as the cube of the sun's distance from the earth, as it ought to be.

The calculus of this motion is difficult, but may be rendered more easy by the following approximation. Assuming, as above, the moon's mean distance from the earth of 100000 parts, and the eccentricity TC of 5505 such parts, the line CB or CD will be found , and DF of those parts; and this line DF at the distance TC subtends the angle at the earth, which the removal of the centre of the orbit from the place D to the place F generates in the motion of this centre; and double this line DF in a parallel position, at the distance of the upper focus of the moon's orbit from the earth, subtends at the earth the same angle as DF did before, which that removal generates in the motion of this upper focus; but at the distance of the moon from the earth this double line 2DF at the upper focus, in a parallel position to the first line DF, subtends an angle at the moon, which the said removal generates in the motion of the moon, which angle may be therefore called the second equation of the moon's centre; and this equation, in the mean distance of the moon from the earth, is nearly as the sine of the angle which that line DF contains with the line drawn from the point F to the moon, and when in its greatest quantity amounts to 2' 25". But the angle which the line DF contains with the line drawn from the point F to the moon is found either by subtracting the angle EDF from the mean anomaly of the moon, or by adding the distance of the moon from the sun to the distance of the moon's apogee from the apogee of the sun; and as the radius to the sine of the angle thus found, so is 2' 25'' to the second equation of the centre: to be added, if the forementioned sum be less than a semi-circle; to be subducted, if greater. And from the moon's place in its orbit thus corrected, its longitude may be found in the syzygies of the luminaries.

The atmosphere of the earth to the height of 35 or 40 miles refracts the sun's light. This refraction scatters and spreads the light over the earth's shadow; and the dissipated light near the limits of the shadow dilates the shadow. Upon which account, to the diameter of the shadow, as it comes out by the parallax, I add 1 or minute in lunar eclipses.

But the theory of the moon ought to be examined and proved from the phænomena, first in the syzygies, then in the quadratures, and last of all in the octants; and whoever pleases to undertake the work will find it not amiss to assume the following mean motions of the sun and moon at the Royal Observatory of Greenwich, to the last day of December at noon, anno 1700, O.S. viz. The mean motion of the sun ♑ 20° 43' 40'', and of its apogee ♋ 7° 44' 30''; the mean motion of the moon ♒ 15° 21' 00''; of its apogee, ♓ 8° 20' 00''; and of its ascending node ♌ 27° 24' 20''; and the difference of meridians betwixt the Observatory at Greenwich and[Pg 449] the Royal Observatory at Paris, 0h. 9' 20'': but the mean motion of the moon and of its apogee are not yet obtained with sufficient accuracy.

PROPOSITION XXXVI. PROBLEM XVII.

To find the force of the sun to move the sea.

The sun's force ML or PT to disturb the motions of the moon, was (by Prop. XXV.) in the moon's quadratures, to the force of gravity with us, as 1 to 638092,6; and the force TM - LM or 2PK in the moon's syzygies is double that quantity. But, descending to the surface of the earth, these forces are diminished in proportion of the distances from the centre of the earth, that is, in the proportion of to 1; and therefore the former force on the earth's surface is to the force of gravity as 1 to 38604600; and by this force the sea is depressed in such places as are 90 degrees distant from the sun. But by the other force, which is twice as great, the sea is raised not only in the places directly under the sun, but in those also which are directly opposed to it; and the sum of these forces is to the force of gravity as 1 to 12868200. And because the same force excites the same motion, whether it depresses the waters in those places which are 90 degrees distant from the sun, or raises them in the places which are directly under and directly opposed to the sun, the aforesaid sum will be the total force of the sun to disturb the sea, and will have the same effect as if the whole was employed in raising the sea in the places directly under and directly opposed to the sun, and did not act at all in the places which are 90 degrees removed from the sun.

And this is the force of the sun to disturb the sea in any given place, where the sun is at the same time both vertical, and in its mean distance from the earth. In other positions of the sun, its force to raise the sea is as the versed sine of double its altitude above the horizon of the place directly, and the cube of the distance from the earth reciprocally.

COR. Since the centrifugal force of the parts of the earth, arising from the earth's diurnal motion, which is to the force of gravity as 1 to 289, raises the waters under the equator to a height exceeding that under the poles by 85472 Paris feet, as above, in Prop. XIX., the force of the sun, which we have now shewed to be to the force of gravity as 1 to 12868200, and therefore is to that centrifugal force as 289 to 12868200, or as 1 to 44527, will be able to raise the waters in the places directly under and directly opposed to the sun to a height exceeding that in the places which are 90 degrees removed from the sun only by one Paris foot and inches; for this measure is to the measure of 85472 feet as 1 to 44527.

PROPOSITION XXXVII. PROBLEM XVIII.

To find the force of the moon to move the sea.

The force of the moon to move the sea is to be deduced from its proportion[Pg 450] to the force of the sun, and this proportion is to be collected from the proportion of the motions of the sea, which are the effects of those forces. Before the mouth of the river Avon, three miles below Bristol, the height of the ascent of the water in the vernal and autumnal syzygies of the luminaries (by the observations of Samuel Sturmy) amounts to about 45 feet, but in the quadratures to 25 only. The former of those heights arises from the sum of the aforesaid forces, the latter from their difference. If, therefore, S and L are supposed to represent respectively the forces of the sun and moon while they are in the equator, as well as in their mean distances from the earth, we shall have L + S to L - S as 45 to 25, or as 9 to 5.

At Plymouth (by the observations of Samuel Colepress) the tide in its mean height rises to about 16 feet, and in the spring and autumn the height thereof in the syzygies may exceed that in the quadratures by more than 7 or 8 feet. Suppose the greatest difference of those heights to be 9 feet, and L + S will be to L - S as to , or as 41 to 23; a proportion that agrees well enough with the former. But because of the great tide at Bristol, we are rather to depend upon the observations of Sturmy; and, therefore, till we procure something that is more certain, we shall use the proportion of 9 to 5.

But because of the reciprocal motions of the waters, the greatest tides do not happen at the times of the syzygies of the luminaries, but, as we have said before, are the third in order after the syzygies; or (reckoning from the syzygies) follow next after the third appulse of the moon to the meridian of the place after the syzygies; or rather (as Sturmy observes) are the third after the day of the new or full moon, or rather nearly after the twelfth hour from the new or full moon, and therefore fall nearly upon the forty-third hour after the new or full of the moon. But in this port they fall out about the seventh hour after the appulse of the moon to the meridian of the place; and therefore follow next after the appulse of the moon to the meridian, when the moon is distant from the sun, or from opposition with the sun by about 18 or 19 degrees in consequentia. So the summer and winter seasons come not to their height in the solstices themselves, but when the sun is advanced beyond the solstices by about a tenth part of its whole course, that is, by about 36 or 37 degrees. In like manner, the greatest tide is raised after the appulse of the moon to the meridian of the place, when the moon has passed by the sun, or the opposition thereof, by about a tenth part of the whole motion from one greatest tide to the next following greatest tide. Suppose that distance about degrees; and the sun's force in this distance of the moon from the syzygies and quadratures will be of less moment to augment and diminish that part of the motion of the sea which proceeds from the motion of the moon than in the syzygies and quadratures themselves in the proportion of the radius to[Pg 451] the co-sine of double this distance, or of an angle of 37 degrees; that is in proportion of 10000000 to 7986355; and, therefore, in the preceding analogy, in place of S we must put 0,7986355S.

But farther; the force of the moon in the quadratures must be diminished, on account of its declination from the equator; for the moon in those quadratures, or rather in degrees past the quadratures, declines from the equator by about 23° 13'; and the force of either luminary to move the sea is diminished as it declines from the equator nearly in the duplicate proportion of the co-sine of the declination; and therefore the force of the moon in those quadratures is only 0,8570327L; whence we have L + 0,7986355S to 0,8570327L - 0,7986355S as 9 to 5.

BOOK III. (part 3)

Farther yet; the diameters of the orbit in which the moon should move, setting aside the consideration of eccentricity, are one to the other as 69 to 70; and therefore the moon's distance from the earth in the syzygies is to its distance in the quadratures, cæteris paribus, as 69 to 70; and its distances, when degrees advanced beyond the syzygies, where the greatest tide was excited, and when degrees passed by the quadratures, where the least tide was produced, are to its mean distance as 69,098747 and 69,897345 to . But the force of the moon to move the sea is in the reciprocal triplicate proportion of its distance; and therefore its forces, in the greatest and least of those distances, are to its force in its mean distance as 0,9830427 and 1,017522 to 1. From whence we have 1,017522L × 0,7986355S to 0,9830427 × 0,8570327L - 0,7986355S as 9 to 5; and S to L as 1 to 4,4815. Wherefore since the force of the sun is to the force of gravity as 1 to 12868200, the moon's force will be to the force of gravity as 1 to 2871400.

COR. 1. Since the waters excited by the sun's force rise to the height of a foot and inches, the moon's force will raise the same to the height of 8 feet and inches; and the joint forces of both will raise the same to the height of feet; and when the moon is in its perigee to the height of feet, and more, especially when the wind sets the same way as the tide. And a force of that quantity is abundantly sufficient to excite all the motions of the sea, and agrees well with the proportion of those motions; for in such seas as lie free and open from east to west, as in the Pacific sea, and in those tracts of the Atlantic and Æthiopic seas which lie without the tropics, the waters commonly rise to 6, 9, 12, or 15 feet; but in the Pacific sea, which is of a greater depth, as well as of a larger extent, the tides are said to be greater than in the Atlantic and Æthiopic seas; for to have a full tide raised, an extent of sea from east to west is required of no less than 90 degrees. In the Æthiopic sea, the waters rise to a less height within the tropics than in the temperate zones, because of the narrowness of the sea between Africa and the southern parts of America. In the middle of the open sea the waters cannot rise without[Pg 452] falling together, and at the same time, upon both the eastern and western shores, when, notwithstanding, in our narrow seas, they ought to fall on those shores by alternate turns; upon which account there is commonly but a small flood and ebb in such islands as lie far distant from the continent. On the contrary, in some ports, where to fill and empty the bays alternately the waters are with great violence forced in and out through shallow channels, the flood and ebb must be greater than ordinary; as at Plymouth and Chepstow Bridge in England, at the mountains of St. Michael, and the town of Avranches, in Normandy, and at Cambaia and Pegu in the East Indies. In these places the sea is hurried in and out with such violence, as sometimes to lay the shores under water, sometimes to leave them dry for many miles. Nor is this force of the influx and efflux to be broke till it has raised and depressed the waters to 30, 40, or 50 feet and above. And a like account is to be given of long and shallow channels or straits, such as the Magellanic straits, and those channels which environ England. The tide in such ports and straits, by the violence of the influx and efflux, is augmented above measure. But on such shores as lie towards the deep and open sea with a steep descent, where the waters may freely rise and fall without that precipitation of influx and efflux, the proportion of the tides agrees with the forces of the sun and moon.

COR. 2. Since the moon's force to move the sea is to the force of gravity as 1 to 2871400, it is evident that this force is far less than to appear sensibly in statical or hydrostatical experiments, or even in those of pendulums. It is in the tides only that this force shews itself by any sensible effect.

COR. 3. Because the force of the moon to move the sea is to the like force of the sun as 4,4815 to 1, and those forces (by Cor. 14, Prop. LXVI, Book I) are as the densities of the bodies of the sun and moon and the cubes of their apparent diameters conjunctly, the density of the moon will be to the density of the sun as 4,4815 to 1 directly, and the cube of the moon's diameter to the cube of the sun's diameter inversely; that is (seeing the mean apparent diameters of the moon and sun are 31' , and 32' 12''), as 4891 to 1000. But the density of the sun was to the density of the earth as 1000 to 4000; and therefore the density of the moon is to the density of the earth as 4891 to 4000, or as 11 to 9. Therefore the body of the moon is more dense and more earthly than the earth itself.

COR. 4. And since the true diameter of the moon (from the observations of astronomers) is to the true diameter of the earth as 100 to 365, the mass of matter in the moon will be to the mass of matter in the earth as 1 to 39,788.

COR. 5. And the accelerative gravity on the surface of the moon will be[Pg 453] about three times less than the accelerative gravity on the surface of the earth.

COR. 6. And the distance of the moon's centre from the centre of the earth will be to the distance of the moon's centre from the common centre of gravity of the earth and moon as 40,788 to 39,788.

COR. 7. And the mean distance of the centre of the moon from the centre of the earth will be (in the moon's octants) nearly of the greatest semi-diameters of the earth; for the greatest semi-diameter of the earth was 19658600 Paris feet, and the mean distance of the centres of the earth and moon, consisting of 60 such semi-diameters, is equal to 1187379440 feet. And this distance (by the preceding Cor.) is to the distance of the moon's centre from the common centre of gravity of the earth and moon as 40,788 to 39,788; which latter distance, therefore, is 1158268534 feet. And since the moon, in respect of the fixed stars, performs its revolution in 27d. 7h. , the versed sine of that angle which the moon in a minute of time describes is 12752341 to the radius 1000,000000,000000; and as the radius is to this versed sine, so are 1158268534 feet to 14,7706353 feet. The moon, therefore, falling towards the earth by that force which retains it in its orbit, would in one minute of time describe 14,7706353 feet; and if we augment this force in the proportion of to , we shall have the total force of gravity at the orbit of the moon, by Cor. Prop. III; and the moon falling by this force, in one minute of time would describe 14,8538067 feet. And at the 60th part of the distance of the moon from the earth's centre, that is, at the distance of 197896573 feet from the centre of the earth, a body falling by its weight, would, in one second of time, likewise describe 14,8538067 feet. And, therefore, at the distance of 19615800, which compose one mean semi-diameter of the earth, a heavy body would describe in falling 15,11175, or 15 feet, 1 inch, and lines, in the same time. This will be the descent of bodies in the latitude of 45 degrees. And by the foregoing table, to be found under Prop. XX, the descent in the latitude of Paris will be a little greater by an excess of about parts of a line. Therefore, by this computation, heavy bodies in the latitude of Paris falling in vacuo will describe 15 Paris feet, 1 inch, lines, very nearly, in one second of time. And if the gravity be diminished by taking away a quantity equal to the centrifugal force arising in that latitude from the earth's diurnal motion, heavy bodies falling there will describe in one second of time 15 feet, 1 inch, and line. And with this velocity heavy bodies do really fall in the latitude of Paris, as we have shewn above in Prop. IV and XIX.

COR. 8. The mean distance of the centres of the earth and moon in the syzygies of the moon is equal to 60 of the greatest semi-diameters of the earth, subducting only about one 30th part of a semi-diameter: and in the[Pg 454] moon's quadratures the mean distance of the same centres is such semi-diameters of the earth; for these two distances are to the mean distance of the moon in the octants as 69 and 70 to , by Prop. XXVIII.

COR. 9. The mean distance of the centres of the earth and moon in the syzygies of the moon is 60 mean semi-diameters of the earth, and a 10th part of one semi-diameter; and in the moon's quadratures the mean distance of the same centres is 61 mean semi-diameters of the earth, subducting one 30th part of one semi-diameter.

COR. 10. In the moon's syzygies its mean horizontal parallax in the latitudes of 0, 30, 38, 45, 52, 60, 90 degrees is 57' 20'', 57' 16'', 57' 14'', 57' 12'', 57' 10'', 57' 8'', 57' 4'', respectively.

In these computations I do not consider the magnetic attraction of the earth, whose quantity is very small and unknown: if this quantity should ever be found out, and the measures of degrees upon the meridian, the lengths of isochronous pendulums in different parallels, the laws of the motions of the sea, and the moon's parallax, with the apparent diameters of the sun and moon, should be more exactly determined from phænomena: we should then be enabled to bring this calculation to a greater accuracy.

PROPOSITION XXXVIII. PROBLEM XIX.

To find the figure of the moon's body.

If the moon's body were fluid like our sea, the force of the earth to raise that fluid in the nearest and remotest parts would be to the force of the moon by which our sea is raised in the places under and opposite to the moon as the accelerative gravity of the moon towards the earth to the accelerative gravity of the earth towards the moon, and the diameter of the moon to the diameter of the earth conjunctly; that is, as 39,788 to 1, and 100 to 365 conjunctly, or as 1081 to 100. Wherefore, since our sea, by the force of the moon, is raised to feet, the lunar fluid would be raised by the force of the earth to 93 feet; and upon this account the figure of the moon would be a spheroid, whose greatest diameter produced would pass through the centre of the earth, and exceed the diameters perpendicular thereto by 186 feet. Such a figure, therefore, the moon affects, and must have put on from the beginning. Q.E.I.

[Pg 455]

COR. Hence it is that the same face of the moon always respects the earth; nor can the body of the moon possibly rest in any other position, but would return always by a libratory motion to this situation; but those librations, however, must be exceedingly slow, because of the weakness of the forces which excite them; so that the face of the moon, which should be always obverted to the earth, may, for the reason assigned in Prop. XVII. be turned towards the other focus of the moon's orbit, without being immediately drawn back, and converted again towards the earth.

LEMMA I.

If APEp represent the earth uniformly dense, marked with the centre C, the poles P, p, and the equator AE; and if about the centre C, with the radius CP, we suppose the sphere Pape to be described, and QR to denote the plane on which a right line, drawn from the centre of the sun to the centre of the earth, insists at right angles; and further suppose that the several particles of the whole exterior earth PapAPepE, without the height of the said sphere, endeavour to recede towards this side and that side from the plane QR, every particle by a force proportional to its distance from that plane; I say, in the first place, that the whole force and efficacy of all the particles that are situate in AE, the circle of the equator, and disposed uniformly without the globe, encompassing the same after the manner of a ring, to wheel the earth about its centre, is to the whole force and efficacy of as many particles in that point A of the equator which is at the greatest distance from the plane QR, to wheel the earth about its centre with a like circular motion, as 1 to 2. And that circular motion will be performed about an axis lying in the common section of the equator and the plane QR.

For let there be described from the centre K, with the diameter IL, the semi-circle INL. Suppose the semi-circumference INL to be divided into innumerable equal parts, and from the several parts N to the diameter IL let fall the sines NM. Then the sums of the squares of all the sines NM will be equal to the sums of the squares of the sines KM, and both sums together will be equal to the sums of the squares of as many semi-diameters KN; and therefore the sum of the squares of all the sines NM will be but half so great as the sum of the squares of as many semi-diameters KN.

Suppose now the circumference of the circle AE to be divided into the like number of little equal parts, and from every such part F a perpendicular FG to be let fall upon the plane QR, as well as the perpendicular AH from the point A. Then the force by which the particle F recedes[Pg 456] from the plane QR will (by supposition) be as that perpendicular FG; and this force multiplied by the distance CG will represent the power of the particle F to turn the earth round its centre. And, therefore, the power of a particle in the place F will be to the power of a particle in the place A as FG × GC to AH × HC; that is, as FC2 to AC2: and therefore the whole power of all the particles F, in their proper places F, will be to the power of the like number of particles in the place A as the sum of all the FC2 to the sum of all the AC2, that is (by what we have demonstrated before) as 1 to 2. Q.E.D.

And because the action of those particles is exerted in the direction of lines perpendicularly receding from the plane QR, and that equally from each side of this plane, they will wheel about the circumference of the circle of the equator, together with the adherent body of the earth, round an axis which lies as well in the plane QR as in that of the equator.

LEMMA II.

The same things still supposed, I say, in the second place, that the total force or power of all the particles situated every where about the sphere to turn the earth about the said axis is to the whole force of the like number of particles, uniformly disposed round the whole circumference of the equator AE in the fashion of a ring, to turn the whole earth about with the like circular motion, as 2 to 5.

For let IK be any lesser circle parallel to the equator AE, and let Ll be any two equal particles in this circle, situated without the sphere Pape; and if upon the plane QR, which is at right angles with a radius drawn to the sun, we let fall the perpendiculars LM, lm, the total forces by which these particles recede from the plane QR will be proportional to the perpendiculars LM, lm. Let the right line Ll be drawn parallel to the plane Pape, and bisect the same in X; and through the point X draw Nn parallel to the plane QR, and meeting the perpendiculars LM, lm, in N and n; and upon the plane QR let fall the perpendicular XY. And the contrary forces of the particles L and l to wheel about the earth contrariwise are as LM × MC, and lm × mC; that is, as LN × MC + NM × MC, and ln × mC - nm × mC; or LN × MC + NM × MC, and LN × mC - NM × mC, and , the difference of the two, is the force of both taken together to turn the earth round. The affirmative part of this difference LN × Mm, or 2LN × NX, is to 2AH × HC, the force of two particles of the same size [Pg 457]situated in A, as LX2 to AC2; and the negative part , or 2XY × CY, is to 2AH × HC, the force of the same two particles situated in A, as CX2 to AC2. And therefore the difference of the parts, that is, the force of the two particles L and l, taken together, to wheel the earth about, is to the force of two particles, equal to the former and situated in the place A, to turn in like manner the earth round, as LX2 - CX2 to AC2. But if the circumference IK of the circle IK is supposed to be divided into an infinite number of little equal parts L, all the LX2 will be to the like number of IX2 as 1 to 2 (by Lem. 1); and to the same number of AC2 as IX2 to 2AC2; and the same number of CX2 to as many AC2 as 2CX2 to 2AC2. Wherefore the united forces of all the particles in the circumference of the circle IK are to the joint forces of as many particles in the place A as IX2 - 2CX2 to 2AC2; and therefore (by Lem. 1) to the united forces of as many particles in the circumference of the circle AE as IX2 - 2CX2 to AC2.

Now if Pp, the diameter of the sphere, is conceived to be divided into an infinite number of equal parts, upon which a like number of circles IK are supposed to insist, the matter in the circumference of every circle IK will be as IX2; and therefore the force of that matter to turn the earth about will be as IX2 into IX2 - 2CX2; and the force of the same matter, if it was situated in the circumference of the circle AE, would be as IX2 into AC2. And therefore the force of all the particles of the whole matter situated without the sphere in the circumferences of all the circles is to the force of the like number of particles situated in the circumference of the greatest circle AE as all the IX2 into IX2 - 2CX2 to as many IX2 into AC2; that is, as all the AC2 - CX2 into AC2 - 3CX2 to as many AC2 - CX2 into AC2; that is, as all the AC4 - 4AC2 × CX2 + 3CX4 to as many AC4 - AC2 × CX2; that is, as the whole fluent quantity, whose fluxion is AC4 - 4AC2 × CX2 + 3CX4, to the whole fluent quantity, whose fluxion is AC4 - AC2 × CX2; and, therefore, by the method of fluxions, as

to ; that is, if for CX we write the whole Cp, or AC, as to ; that is, as 2 to 5. Q.E.D.

LEMMA III.

The same things still supposed, I say, in the third place, that the motion of the whole earth about the axis above-named arising from the motions of all the particles, will be to the motion of the aforesaid ring about the same axis in a proportion compounded of the proportion of the matter in the earth to the matter in the ring; and the proportion of three squares of the quadrantal arc of any circle to two squares of its diameter, that is, in the proportion of the matter to the matter, and of the number 925275 to the number 1000000.

For the motion of a cylinder revolved about its quiescent axis is to the[Pg 458] motion of the inscribed sphere revolved together with it as any four equal squares to three circles inscribed in three of those squares; and the motion of this cylinder is to the motion of an exceedingly thin ring surrounding both sphere and cylinder in their common contact as double the matter in the cylinder to triple the matter in the ring; and this motion of the ring, uniformly continued about the axis of the cylinder, is to the uniform motion of the same about its own diameter performed in the same periodic time as the circumference of a circle to double its diameter.

HYPOTHESIS II.

If the other parts of the earth were taken away, and the remaining ring was carried alone about the sun in the orbit of the earth by the annual motion, while by the diurnal motion it was in the mean time revolved about its own axis inclined to the plane of the ecliptic by an angle of degrees, the motion of the equinoctial points would be the same, whether the ring were fluid, or whether it consisted of a hard and rigid matter.

PROPOSITION XXXIX. PROBLEM XX.

To find the precession of the equinoxes.

The middle horary motion of the moon's nodes in a circular orbit, when the nodes are in the quadratures, was 16'' 35''' 16iv. 36v.; the half of which, 8'' 17''' 38iv. 18v. (for the reasons above explained) is the mean horary motion of the nodes in such an orbit, which motion in a whole sidereal year becomes 20° 11' 46''. Because, therefore, the nodes of the moon in such an orbit would be yearly transferred 20° 11' 46'' in antecedentia; and, if there were more moons, the motion of the nodes of every one (by Cor. 16, Prop. LXVI, Book I) would be as its periodic time; if upon the surface of the earth a moon was revolved in the time of a sidereal day, the annual motion of the nodes of this moon would be to 20° 11' 46'' as 23h. 56', the sidereal day, to 27d. 7h. 43', the periodic time of our moon, that is, as 1436 to 39343. And the same thing would happen to the nodes of a ring of moons encompassing the earth, whether these moons did not mutually touch each the other, or whether they were molten, and formed into a continued ring, or whether that ring should become rigid and inflexible.

Let us, then, suppose that this ring is in quantity of matter equal to the whole exterior earth PapAPepE, which lies without the sphere Pape (see fig. Lem. II); and because this sphere is to that exterior earth as aC2 to AC2 - aC2, that is (seeing PC or aC the least semi-diameter of the earth is to AC the greatest semi-diameter of the same as 229 to 230), as 52441 to 459; if this ring encompassed the earth round the equator, and both together were revolved about the diameter of the ring, the motion of[Pg 459] the ring (by Lem. III) would be to the motion of the inner sphere as 459 to 52441 and 1000000 to 925275 conjunctly, that is, as 4590 to 485223; and therefore the motion of the ring would be to the sum of the motions of both ring and sphere as 4590 to 489813. Wherefore if the ring adheres to the sphere, and communicates its motion to the sphere, by which its nodes or equinoctial points recede, the motion remaining in the ring will be to its former motion as 4590 to 489813; upon which account the motion of the equinoctial points will be diminished in the same proportion. Wherefore the annual motion of the equinoctial points of the body, composed of both ring and sphere, will be to the motion 20° 11' 46'' as 1436 to 39343 and 4590 to 489813 conjunctly, that is, as 100 to 292369. But the forces by which the nodes of a number of moons (as we explained above), and therefore by which the equinoctial points of the ring recede (that is, the forces 3IT, in fig. Prop. XXX), are in the several particles as the distances of those particles from the plane QR; and by these forces the particles recede from that plane: and therefore (by Lem. II) if the matter of the ring was spread all over the surface of the sphere, after the fashion of the figure PapAPepE, in order to make up that exterior part of the earth, the total force or power of all the particles to wheel about the earth round any diameter of the equator, and therefore to move the equinoctial points, would become less than before in the proportion of 2 to 5. Wherefore the annual regress of the equinoxes now would be to 20° 11' 46'' as 10 to 73092; that is, would be 9'' 56''' 50iv.

But because the plane of the equator is inclined to that of the ecliptic, this motion is to be diminished in the proportion of the sine 91706 (which is the co-sine of deg.) to the radius 100000; and the remaining motion will now be 9'' 7''' 20iv. which is the annual precession of the equinoxes arising from the force of the sun.

But the force of the moon to move the sea was to the force of the sun nearly as 4,4815 to 1; and the force of the moon to move the equinoxes is to that of the sun in the same proportion. Whence the annual precession of the equinoxes proceeding from the force of the moon comes out 40'' 52''' 52iv. and the total annual precession arising from the united forces of both will be 50'' 00''' 12iv. the quantity of which motion agrees with the phænomena; for the precession of the equinoxes, by astronomical observations, is about 50'' yearly.

If the height of the earth at the equator exceeds its height at the poles by more than miles, the matter thereof will be more rare near the surface than at the centre; and the precession of the equinoxes will be augmented by the excess of height, and diminished by the greater rarity.

And now we have described the system of the sun, the earth, moon, and planets, it remains that we add something about the comets.

[Pg 460]

LEMMA IV.

That the comets are higher than the moon, and in the regions of the planets.

As the comets were placed by astronomers above the moon, because they were found to have no diurnal parallax, so their annual parallax is a convincing proof of their descending into the regions of the planets; for all the comets which move in a direct course according to the order of the signs, about the end of their appearance become more than ordinarily slow or retrograde, if the earth is between them and the sun; and more than ordinarily swift, if the earth is approaching to a heliocentric opposition with them; whereas, on the other hand, those which move against the order of the signs, towards the end of their appearance appear swifter than they ought to be, if the earth is between them and the sun; and slower, and perhaps retrograde, if the earth is in the other side of its orbit. And these appearances proceed chiefly from the diverse situations which the earth acquires in the course of its motion, after the same manner as it happens to the planets, which appear sometimes retrograde, sometimes more slowly, and sometimes more swiftly, progressive, according as the motion of the earth falls in with that of the planet, or is directed the contrary way. If the earth move the same way with the comet, but, by an angular motion about the sun, so much swifter that right lines drawn from the earth to the comet converge towards the parts beyond the comet, the comet seen from the earth, because of its slower motion, will appear retrograde; and even if the earth is slower than the comet, the motion of the earth being subducted, the motion of the comet will at least appear retarded; but if the earth tends the contrary way to that of the comet, the motion of the comet will from thence appear accelerated; and from this apparent acceleration, or retardation, or regressive motion, the distance of the comet may be inferred in this manner. Let , , , be three observed longitudes of the comet about the time of its first appearing, and its last observed longitude before its disappearing. Draw the right line ABC, whose parts AB, BC, intercepted between the right lines QA and QB, QB and QC, may be one to the other as the two times between the three first observations. Produce AC to G, so as AG may be to AB as the time between the first and last observation to the time between the first and second; and join QG. Now if the comet did move uniformly in a right line, and the earth either stood still, or was likewise carried forwards in a right line by an uniform motion, the angle would be the[Pg 461] longitude of the comet at the time of the last observation. The angle, therefore, FQG, which is the difference of the longitude, proceeds from the inequality of the motions of the comet and the earth; and this angle, if the earth and comet move contrary ways, is added to the angle , and accelerates the apparent motion of the comet; but if the comet move; the same way with the earth, it is subtracted, and either retards the motion of the comet, or perhaps renders it retrograde, as we have but now explained. This angle, therefore, proceeding chiefly from the motion of the earth, is justly to be esteemed the parallax of the comet; neglecting, to wit, some little increment or decrement that may arise from the unequal motion of the comet in its orbit; and from this parallax we thus deduce the distance of the comet.

Let S represent the sun, acT the orbis magnus, a the earth's place in the first observation, c the place of the earth in the third observation, T the place of the earth in the last observation, and a right line drawn to the beginning of Aries. Set off the angle equal to the angle , that is, equal to the longitude of the comet at the time when the earth is in T; join ac, and produce it to g, so as ag may be to ac as AG to AC; and g will be the place at which the earth would have arrived in the time of the last observation, if it had continued to move uniformly in the right line ac. Wherefore, if we draw parallel to , and make the angle equal to the angle , this angle will be equal to the longitude of the comet seen from the place g, and the angle TVg will be the parallax which arises from the earth's being transferred from the place g into the place T; and therefore V will be the place of the comet in the plane of the ecliptic. And this place V is commonly lower than the orb of Jupiter.

[Pg 462]

The same thing may be deduced from the incurvation of the way of the comets; for these bodies move almost in great circles, while their velocity is great; but about the end of their course, when that part of their apparent motion which arises from the parallax bears a greater proportion to their whole apparent motion, they commonly deviate from those circles, and when the earth goes to one side, they deviate to the other; and this deflexion, because of its corresponding with the motion of the earth, must arise chiefly from the parallax; and the quantity thereof is so considerable, as, by my computation, to place the disappearing comets a good deal lower than Jupiter. Whence it follows that when they approach nearer to us in their perigees and perihelions they often descend below the orbs of Mars and the inferior planets.

The near approach of the comets is farther confirmed from the light of their heads; for the light of a celestial body, illuminated by the sun, and receding to remote parts, is diminished in the quadruplicate proportion of the distance; to wit, in one duplicate proportion, on account of the increase of the distance from the sun, and in another duplicate proportion, on account of the decrease of the apparent diameter. Wherefore if both the quantity of light and the apparent diameter of a comet are given, its distance will be also given, by taking the distance of the comet to the distance of a planet in the direct proportion of their diameters and the reciprocal subduplicate proportion of their lights. Thus, in the comet of the year 1682, Mr. Flamsted observed with a telescope of 16 feet, and measured with a micrometer, the least diameter of its head, 2' 00; but the nucleus or star in the middle of the head scarcely amounted to the tenth part of this measure; and therefore its diameter was only 11'' or 12''; but in the light and splendor of its head it surpassed that of the comet in the year 1680, and might be compared with the stars of the first or second magnitude. Let us suppose that Saturn with its ring was about four times more lucid; and because the light of the ring was almost equal to the light of the globe within, and the apparent diameter of the globe is about 21'', and therefore the united light of both globe and ring would be equal to the light of a globe whose diameter is 30'', it follows that the distance of the comet was to the distance of Saturn as 1 to inversely, and 12'' to 30'' directly; that is, as 24 to 30, or 4 to 5. Again; the comet in the month of April 1665, as Hevelius informs us, excelled almost all the fixed stars in splendor, and even Saturn itself, as being of a much more vivid colour; for this comet was more lucid than that other which had appeared about the end of the preceding year, and had been compared to the stars of the first magnitude. The diameter of its head was about 6'; but the nucleus, compared with the planets by means of a telescope, was plainly less than Jupiter; and sometimes judged less, sometimes judged equal, to the globe of Saturn within the ring. Since, then, the diameters of the heads of the comets seldom exceed 8' or 12', and the diameter of the nucleus or central star is but about a tenth or perhaps fifteenth part of the diameter of the head, it appears that these stars are generally of about the same apparent magnitude with the planets. But in regard that their light may be often compared with the light of Saturn, yea, and sometimes exceeds it, it is evident that all comets in their perihelions must either be placed below or not far above Saturn; and they are much mistaken who remove them almost as far as the fixed stars; for if it was so, the comets could receive no more light from our sun than our planets do from the fixed stars.

So far we have gone, without considering the obscuration which comets suffer from that plenty of thick smoke which encompasseth their heads, and through which the heads always shew dull, as through a cloud; for by[Pg 463] how much the more a body is obscured by this smoke, by so much the more near it must be allowed to come to the sun, that it may vie with the planets in the quantity of light which it reflects. Whence it is probable that the comets descend far below the orb of Saturn, as we proved before from their parallax. But, above all, the thing is evinced from their tails, which must be owing either to the sun's light reflected by a smoke arising from them, and dispersing itself through the æther, or to the light of their own 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 is propagated through such a vast extent of space, and with such a velocity and expansion as will seem altogether incredible; in the latter case, the whole light of both head and tail is to be ascribed to the central nucleus. But, then, if we suppose all this light to be united and condensed within the disk of the nucleus, certainly the nucleus will by far exceed Jupiter itself 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 be much more illuminated by the sun, and therefore much nearer to it; and the same argument will bring down the heads of comets sometimes within the orb of Venus, viz., when, being hid under the sun's rays, they emit such huge and splendid tails, like beams of fire, as sometimes they do; for if all that light was supposed to be gathered together into one star, it would sometimes exceed not one Venus only, but a great many such united into one.

Lastly; the same thing is inferred from the light of the heads, which increases in the recess of the comets from the earth towards the sun, and decreases in their return from the sun towards the earth; for so the comet of the year 1665 (by the observations of Hevelius), from the time that it was first seen, was always losing of its apparent motion, and therefore had already 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 Hevelius), 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; but from that time its diurnal motion was continually upon the increase, till September 4, when it arose to about 5 degrees; and therefore, in all this interval 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 far less about the beginning than towards the end of the motion; though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the same Hevelius declares. Wherefore in all this interval of time, on account of its recess from the sun, it decreased[Pg 464] in splendor, notwithstanding its access towards the earth. The comet of the 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 splendor of their heads was seen two weeks before, when they had just got clear of the 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 to the observations of Cysatus), December 1, appeared greater than the stars of the first magnitude; and, December 16 (then in the perigee), it was but little diminished in magnitude, but in the splendor and brightness of its light a great deal. January 7, Kepler, being uncertain about the head, left off observing. December 12, the head of the latter comet was seen and observed by Mr. Flamsted, when but 9 degrees distant from the sun; 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 lustre being diminished by the brightness of the clouds near the setting sun. December 26, when it moved with the greatest velocity, being almost in its perigee, it was less than the mouth of Pegasus, a star of the third magnitude. 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 of the seventh magnitude. If we compare equal intervals of time on one side and on the other from the perigee, we shall find that the head of the comet, which at both intervals of time was far, but yet equally, removed from 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 the one situation and in the other, we conclude the great vicinity of the sun and comet in the former; for the light of comets uses to be regular, and to appear greatest when the heads move fastest, and are therefore in their perigees; excepting in so far as it is increased by their nearness to the sun.

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 why comets are so frequently seen in that hemisphere in which the sun is, and so seldom in the other. If they were visible in the regions far above Saturn, they would appear more frequently in the parts opposite to the sun; for such as were in those parts would be nearer to the earth, whereas the presence of the sun must obscure and hide those that appear in the hemisphere in which he is. Yet, looking over the history of comets, I find that four or five times more have been seen in the hemisphere towards the sun than in the opposite hemisphere; besides, without doubt, not a few, which have been hid by the light of the sun: for comets descending[Pg 465] into 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 to us than Jupiter. But the far greater part of that spherical space, which is described about the sun with so small an interval, lies on that side of the earth which regards the sun; and the comets in that greater part are commonly more strongly illuminated, as being for the most part nearer to the sun.

COR. 3. Hence also it is evident that the celestial spaces are void of resistance; for though the comets are carried in oblique paths, and sometimes contrary to the course of the planets, yet they move every way with the greatest freedom, and preserve their motions for an exceeding long time, even where contrary to the course of the planets. I am out in my judgment if they are not a sort of planets revolving in orbits returning into themselves with a perpetual motion; for, as to what some writers contend, that they are no other than meteors, led into this opinion by the perpetual 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; and therefore 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 from the planets, would, without all doubt, shine by the light of its clouds, and the 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 Jupiter is hardly to be seen through them; and much more must the bodies of comets be hid under their atmospheres, which are both deeper and thicker.

PROPOSITION XL. THEOREM XX.

That the comets move in some of the conic sections, having their foci in the centre of the sun; and by radii drawn to the sun describe areas proportional to the times.

This proposition appears from Cor. 1, Prop. XIII, Book I, compared with Prop. VIII, XII, and XIII, Book III.

COR. 1. Hence if comets are revolved in orbits returning into themselves, those orbits will be ellipses; and their periodic times be to the periodic times of the planets in the sesquiplicate proportion of their principal axes. And therefore the comets, which for the most part of their course are higher than the planets, and upon that account describe orbits with 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 axis of the orbit of Saturn, the time of the revolution of the comet would be to the time of the revolution of Saturn, that is, to 30 years, as (or 8) to 1, and would therefore be 240 years.

[Pg 466]

COR. 2. But their orbits will be so near to parabolas, that parabolas may be used for them without sensible error.

COR. 3. And, therefore, by Cor. 7, Prop. XVI, Book I, the velocity of every comet will always be to the velocity of any planet, supposed to be revolved at the same distance in a circle about the sun, nearly in the subduplicate proportion of double the distance of the planet from the centre of the sun to the distance of the comet from the sun's centre, very nearly. Let us suppose the radius of the orbis magnus, or the greatest semi-diameter of the ellipsis which the earth describes, to consist of 100000000 parts; and then the earth by its mean diurnal motion will describe 1720212 of those parts, and by its horary motion. And therefore the comet, at the same mean distance of the earth from the sun, with a velocity which is to the velocity of the earth as to 1, would by its diurnal motion describe 2432747 parts, and parts by its horary motion. But at greater or less distances both the diurnal and horary motion will be to this diurnal and horary motion in the reciprocal subduplicate proportion of the distances, and is therefore given.

COR. 4. Wherefore if the latus rectum of the parabola is quadruple of the radius of the orbis magnus, and the square of that radius is supposed to consist of 100000000 parts, the area which the comet will daily describe by a radius drawn to the sun will be parts, and the horary area will be parts. But, if the latus rectum is greater or less in any proportion, the diurnal and horary area will be less or greater in the subduplicate of the same proportion reciprocally.

LEMMA V.

To find a curve line of the parabolic kind which shall pass through any given number of points.

Let those points be A, B, C, D, E, F, &c., and from the same to any right line HN, given in position, let fall as many perpendiculars AH, BI, CK, DL, EM, FN, &c.

CASE 1. If HI, IK, KL, &c., the intervals of the points H, I, K, L, M, N, &c., are equal, take b, 2b, 3b, 4b, 5b, &c., the first differences of the perpendiculars AH, BI, CK, &c.; their second differences c, 2c, 3c, 4c, &c.; their third, d, 2d, 3d, &c., that is to say, so as AH - BI may be = b, BI[Pg 467] - CK = 2b, CK - DL = 3b, DL + EM = 4b, - EM + FN = 5b, &c.; then b - 2b = c, &c., and so on to the last difference, which is here f. Then, erecting any perpendicular RS, which may be considered as an ordinate of the curve required, in order to find the length of this ordinate, suppose the intervals HI, IK, KL, LM, &c., to be units, and let , , into , into , into + SL = s, into + SM = t; proceeding, to wit, to ME, the last perpendicular but one, and prefixing negative signs before the terms HS, IS, &c., which lie from S towards A; and affirmative signs before the terms SK, SL, &c., which lie on the other side of the point S; and, observing well the signs, RS will be = a + bp + cq + dr + es + ft, + &c.

CASE 2. But if HI, IK, &c., the intervals of the points H, I, K, L, &c., are unequal, take b, 2b, 3b, 4b, 5b, &c., the first differences of the perpendiculars AH, BI, CK, &c., divided by the intervals between those perpendiculars; c, 2c, 3c, 4c, &c., their second differences, divided by the intervals between every two; d, 2d, 3d, &c., their third differences, divided by the intervals between every three; e, 2e, &c., their fourth differences, divided by the intervals between every four; and so forth; that is, in such manner, that b may be , 2b , 3b , &c., then c , 2c , 3c , &c., then d , 2d , &c. 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 + SM = t; proceeding, to wit, to ME, the last perpendicular but one: and the ordinate RS will be = a + bp + cq + dr + es + ft, + &c.

COR. Hence the areas of all curves may be nearly found; for if some number of points of the curve to be squared are found, and a parabola be supposed to be drawn through those points, the area of this parabola will be nearly the same with the area of the curvilinear figure proposed to be squared: but the parabola can be always squared geometrically by methods vulgarly known.

LEMMA VI.

Certain observed places of a comet being given, to find the place of the same to any intermediate given time.

Let HI, IK, KL, LM (in the preceding Fig.), represent the times between the observations; HA, IB, KC, LD, ME, five observed longitudes of the comet; and HS the given time between the first observation and the longitude required. Then if a regular curve ABCDE is supposed to be drawn through the points A, B, C, D, E, and the ordinate RS is found out by the preceding lemma, RS will be the longitude required.

[Pg 468]

After the same method, from five observed latitudes, we may find the latitude to a given time.

If the differences of the observed longitudes are small, suppose of 4 or 5 degrees, three or four observations will be sufficient to find a new longitude and latitude; but if the differences are greater, as of 10 or 20 degrees, five observations ought to be used.

LEMMA VII.

Through a given point P to draw a right line BC, whose parts PB, PC, cut off by two right lines AB, AC, given in position, may be one to the other in a given proportion.

From the given point P suppose any right line PD to be drawn to either of the right lines given, as AB; and produce the same towards AC, the other given right line, as far as E, so as PE may be to PD in the given proportion. Let EC be parallel to AD. Draw CPB, and PC will be to PB as 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, whose diameter is and vertex . In produced take equal to one half of . Join OS, and produce it to as may be equal to 2SO. Now, supposing a comet to revolve in the arc CBA, draw , cutting AC in E; I say, the point E will cut off from the chord AC the segment AE, nearly proportional to the time.

For if we join EO, cutting the parabolic arc ABC in Y, and draw touching the same arc in the vertex , and meeting EO in X, the curvilinear area will be to the curvilinear area as AE to AC; and, therefore, since the triangle ASE is to the triangle ASC in the same proportion, the whole area will be to the whole area as[Pg 469] AE to AC. But, because is to SO as 3 to 1, and EO to XO in the same proportion, SX will be parallel to EB; and, therefore, joining BX, the triangle SEB will be equal to the triangle XEB. Wherefore if to the area we add the triangle EXB, and from the sum subduct the triangle SEB, there will remain the area , equal to the area ; and therefore in proportion to the area as AE to AC. But the area is nearly equal to the area ; and this area is to the area as the time of description of the arc AB to the time of description of the whole arc AC; and, therefore, AE is to AC nearly in the proportion of the times. Q.E.D.

COR. When the point B falls upon the vertex of the parabola, AE is to AC accurately in the proportion of the times.

SCHOLIUM.

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

LEMMA IX.

The right lines and , and the length , are equal among themselves.

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

LEMMA X.

Produce to N and P, so as may be one third of , and SP may be to SN as SN to ; and in the time that a comet would describe the arc , if it was supposed to move always forwards with the velocity which it hath in a height equal to SP, it would describe a length equal to the chord AC.

For if the comet with the velocity which it hath in was in the said time supposed to move uniformly forward in the right line which touches the parabola in , the area which it would describe by a radius drawn to the point S would be equal to the parabolic area ; and therefore the space contained under the length described in the tangent and the length would be to the space contained under the lengths AC and SM as the[Pg 470] area to the triangle ASC, that is, as SN to SM. Wherefore AC is to the length described in the tangent as to SN. But since the velocity of the comet in the height SP (by Cor. 6, Prop. XVI., Book I) is to the velocity of the same in the height in the reciprocal subduplicate proportion of SP to , that is, in the proportion of to SN, the length described with this velocity will be to the length in the same time described in the tangent as to SN. Wherefore since AC, and the length described with this new velocity, are in the same proportion to the length described in the tangent, they must be equal betwixt themselves. Q.E.D.

COR. Therefore a comet, with that velocity which it hath in the height , would in the same time describe the chord AC nearly.

LEMMA XI.

If a comet void of all motion was let fall from the height SN, or , towards the sun, and was still impelled to the sun by the same force uniformly continued by which it was impelled at first, the same, in one half of that time in which it might describe the arc AC in its own orbit, would, in descending describe a space equal to the length .

For in the same time that the comet would require to describe the parabolic arc AC, it would (by the last Lemma), with that velocity which it hath in the height SP, describe the chord AC: and, therefore (by Cor. 7, Prop. XVI, Book I), if it was in the same time supposed to revolve by the force of its own gravity in a circle whose semi-diameter was SP, it would describe an arc of that circle, the length of which would be to the chord of the parabolic arc AC in the subduplicate proportion of 1 to 2. Wherefore if with that weight, which in the height SP it hath towards the sun, it should fall from that height towards the sun, it would (by Cor. 9, Prop. XVI, Book I) in half the said time describe a space equal to the square of half the said chord applied to quadruple the height SP, that is, it would describe the space . But since the weight of the comet towards the sun in the height SN is to the weight of the same towards the sun in the height SP as SP to , the comet, by the weight which it hath in the height SN, in falling from that height towards the sun, would in the same time describe the space ; that is, a space equal to the length or . Q.E.D.

[Pg 471]

PROPOSITION XLI. PROBLEM XXI.

From three observations given to determine the orbit of a comet moving in a parabola.

This being a Problem of very great difficulty, I tried many methods of resolving it; and several of these Problems, the composition whereof I have given in the first Book, tended to this purpose. But afterwards I contrived the following solution, which is something more simple.

Select three observations distant one from another by intervals of time nearly equal; but let that interval of time in which the comet moves more slowly be somewhat greater than the other; so, to wit, that the difference of the times may be to the sum of the times as the sum of the times to about 600 days; or that the point E may fall upon M nearly, and may err therefrom rather towards I than towards A. If such direct observations are not at hand, a new place of the comet must be found, by Lem. VI.

Let S represent the sun; T, t, , three places of the earth in the orbis magnus; TA, tB, , three observed longitudes of the comet; V the time between the first observation and the second; W the time between the second and the third; X the length which in the whole time V + W the comet might describe with that velocity which it hath in the mean distance of the earth from the sun, which length is to be found by Cor. 3,[Pg 472] Prop. XL, Book III; and tV a perpendicular upon the chord . In the mean observed longitude tB take at pleasure the point B, for the place of the comet in the plane of the ecliptic; and from thence, towards the sun S, draw the line BE, which may be to the perpendicular tV as the content under SB and St2 to the cube of the hypothenuse of the right angled triangle, whose sides are SB, and the tangent of the latitude of the comet in the second observation to the radius tB. And through the point E (by Lemma VII) draw the right line AEC, whose parts AE and EC, terminating in the right lines TA and , may be one to the other as the times V and W: then A and C will be nearly the places of the comet in the plane of the ecliptic in the first and third observations, if B was its place rightly assumed in the second.

Upon AC, bisected in I, erect the perpendicular Ii. Through B draw the obscure line Bi parallel to AC. Join the obscure line Si, cutting AC in , and complete the parallelogram iI . Take equal to ; and through the sun S draw the obscure line equal to . Then, cancelling the letters A, E, C, I, from the point B towards the point , draw the new obscure line BE, which may be to the former BE in the duplicate proportion of the distance BS to the quantity . And through the point E draw again the right line AEC by the same rule as before; that is, so as its parts AE and EC may be one to the other as the times V and W between the observations. Thus A and C will be the places of the comet more accurately.

Upon AC, bisected in I, erect the perpendiculars AM, CN, IO, of which AM and CN may be the tangents of the latitudes in the first and third observations, to the radii TA and . Join MN, cutting IO in O. Draw the rectangular parallelogram , as before. In IA produced take ID equal to . Then in MN, towards N, take MP, which may be to the above found length X in the subduplicate proportion of the mean distance of the earth from the sun (or of the semi-diameter of the orbis magnus) to the distance OD. If the point P fall upon the point N; A, B, and C, will be three places of the comet, through which its orbit is to be described in the plane of the ecliptic. But if the point P falls not upon the point N, in the right line AC take CG equal to NP, so as the points G and P may lie on the same side of the line NC.

By the same method as the points E, A, C, G, were found from the assumed point B, from other points b and assumed at pleasure, find out the new points e, a, c, g; and , , , . Then through G, g, and , draw the circumference of a circle , cutting the right line in Z: and Z will be one place of the comet in the plane of the ecliptic. And in AC, ac, , making AF, af, , equal respectively to CG, cg, ; through the points F, f, and , draw the circumference of a circle , cutting the right line AT in X; and the point X will be another place of the comet in the plane of[Pg 473] the ecliptic. And at the points X and Z, erecting the tangents of the latitudes of the comet to the radii TX and , two places of the comet in its own orbit will be determined. Lastly, if (by Prop. XIX., Book I) to the focus S a parabola is described passing through those two places, this parabola will be the orbit of the comet. Q.E.I.

The demonstration of this construction follows from the preceding Lemmas, because the right line AC is cut in E in the proportion of the times, by Lem. VII, as it ought to be, by Lem. VIII.; and BE, by Lem. XI., is a portion of the right line BS or in the plane of the ecliptic, intercepted between the arc ABC and the chord AEC; and MP (by Cor. Lem. X.) is the length of the chord of that arc, which the comet should describe in its proper orbit between the first and third observation, and therefore is equal to MN, providing B is a true place of the comet in the plane of the ecliptic.

But it will be convenient to assume the points B, b, , not at random, but nearly true. If the angle AQt, at which the projection of the orbit in the plane of the ecliptic cuts the right line tB, is rudely known, at that angle with Bt draw the obscure line AC, which may be to in the subduplicate proportion of SQ to St; and, drawing the right line SEB so as its part EB may be equal to the length Vt, the point B will be determined, which we are to use for the first time. Then, cancelling the right line AC, and drawing anew AC according to the preceding construction, and, moreover, finding the length MP, in tB take the point b, by this rule, that, if TA and intersect each other in Y, the distance Yb may be to the distance YB in a proportion compounded of the proportion of MP to MN, and the subduplicate proportion of SB to Sb. And by the same method you may find the third point , if you please to repeat the operation the third time; but if this method is followed, two operations generally will be sufficient; for if the distance Bb happens to be very small, after the points F, f, and G, g, are found, draw the right lines Ff and Gg, and they will cut TA and in the points required, X and Z.

EXAMPLE.

[Pg 474]

Let the comet of the year 1680 be proposed. The following table shews the motion thereof, as observed by Flamsted, and calculated afterwards by him from his observations, and corrected by Dr. Halley from the same observations.

Time. Sun's

Longitude. Comet's

Appar. True.

Longitude. Lat. N.

h. '' h. ' '' ° ' '' ° ' '' ° ' ''

1680, Dec. 12 4.46 4.46. 0 ♑ 1.51.23 ♑ 6.32.30 8.28. 0

21

6.36.59 11.06.44 ♒ 5.08.12 21.42.13

24 6.12 6.17.52 14.09.26 18.49.23 25.23. 5

26 5.14 5.20.44 16.09.22 28.24.13 27.00.52

29 7.55 8.03.02 19.19.43 ♓ 13.10.41 28.09.58

30 8.02 8.10.26 20.21.09 17.38.20 28.11.53

1681, Jan. 5 5.51 6.01.38 26.22.18 ♈ 8.48.53 26.15. 7

9 6.49 7.00.53 ♒ 0.29.02 18.44.04 24.11.56

10 5.54 6.06.10 1.27.43 20.40.50 23.43.52

13 6.56 7.08.55 4.33.20 25.59.48 22.17.28

25 7.44 7.58.42 16.45.36 ♉ 9.35. 0 17.56.30

30 8.07 8.21.53 21.49.58 13.19.51 16.42.18

Feb. 2 6.20 6.34.51 24.46.59 15.13.53 16.04. 1

5 6.50 7.04.41 27.49.51 16.59.06 15.27. 3

To these you may add some observations of mine.

Ap.

Time. Comet's

Longitude. Lat. N.

h. ' ° ' '' ° ' ''

1681, Feb. 25 8.30 ♉︎ 26.18.35 12.46.46

27 8.15 27.04.30 12.36.12

Mar. 1 11. 0 27.52.42 12.23.40

2 8. 0 28.12.48 12.19.38

5 11.30 29.18. 0 12.03.16

7 9.30 ♊︎ 0. 4. 0 11.57. 0

9 8.30 0.43. 4 11.45.52

These observations were made by a telescope of 7 feet, with a micrometer and threads placed in the focus of the telescope; by which instruments we determined the positions both of the fixed stars among themselves, and of the comet in respect of the fixed stars. Let A represent the star of the fourth magnitude in the left heel of Perseus (Bayer's ο), B the following star of the third magnitude in the left foot (Bayer's ), C a star of the sixth magnitude (Bayer's ) in the heel of the same foot, and D, E, F, G, H, I, K, L, M, N, O, Z, , , , , other smaller stars in the same foot; and let p, P, Q, R, S, T, V, X, represent the places of the comet in the observations above set down; and, reckoning the distance AB of parts, AC was of those parts; BC, ; AD, ; BD, ; CD, ; AE, ; CE, ; DE, ; AI, ; BI, ; CI, ; DI, ; AK, ; BK, 43; CK, ; FK, 29; FB, 23; FC, ; AH, ; DH, ; BN, ; CN, ; BL, ; NL, . HO was to HI as 7 to 6, and, produced, did pass between the stars D and E, so as the distance of the star D from this right line was . LM was to LN as 2 to 9, and, produced, did pass through the star H. Thus were the positions of the fixed stars determined in respect of one another.

[Pg 475]

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

The fixed

stars. Their

Longitudes. Latitute

North. The fixed

stars. Their

Longitudes. Latitute

North.

° ' '' ° ' ''

° ' '' ° ' ''

A ♉︎ 26.41.50 12. 8.36 L ♉︎ 29.33.34 12. 7.48

B 28.40.23 11.17.54 M 9.18.54 2. 7.20

C 27.58.30 12.40.25 N 28.48.29 2.31. 9

E 26.27.17 12.52. 7 Z 29.44.48 11.57.13

F 28.28.37 11.52.22

29.52. 3 11.55.48

G 26.56. 8 12. 4.58

♊︎ 0. 8.23 11.48.56

H 27.11.45 12. 2. 1

0.40.10 11.55.18

I 27.25. 2 11.53.11

1. 3.20 11.30.42

K 27.42. 7 11.53.26

[Pg 476]

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

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

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

Sunday, February 27, P. M. the distance of the comet in Q, from the star O was equal to the distance of the stars O and H; and the right line QO produced passed between the stars K and B. I could not, by reason of intervening clouds, determine the position of the star to greater accuracy.

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

Wednesday, March 2, 8h. P. M. the distance of the comet in S from the star C was nearly ; the distance of the star F from the right line CS produced was ; and the distance of the star B from the same right line was five times greater than the distance of the star F; and the right line NS produced passed between the stars H and I five or six times nearer to the star H than to the star I.

Saturday, March 5, P. M. when the comet was in T, the right line MT was equal to , and the right line LT produced passed between B and F four or five times nearer to F than to B, cutting off from BF a fifth or sixth part thereof towards F: and MT produced passed on the outside of the space BF towards the star B four times nearer to the star B than to the star F. M was a very small star, scarcely to be seen by the telescope; but the star L was greater, and of about the eighth magnitude.

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

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

The same night, at 12h. the comet being in Y, the right line was[Pg 477] equal to of , or a little less, as perhaps of ; and a perpendicular let fall from the star on the right line was equal to about or . But the comet being then extremely near the horizon, was scarcely discernible, and therefore its place could not be determined with that certainty as in the foregoing observations.

From these observations, by constructions of figures and calculations, I deduced the longitudes and latitudes of the comet; and Mr. Pound, by correcting the places of the fixed stars, hath determined more correctly the places of the comet, which correct places are set down above. Though my micrometer was none of the best, yet the errors in longitude and latitude (as derived from my observations) scarcely exceed one minute. The comet (according to my observations), about the end of its motion, began to decline sensibly towards the north, from the parallel which it described about the end of February.

Now, in order to determine the orbit of the comet out of the observations above described, I selected those three which Flamsted made, Dec. 21, Jan. 5, and Jan. 25; from which I found St of 9842,1 parts, and Vt of 455, such as the semi-diameter of the orbis magnus contains 10000. Then for the first observation, assuming tB of 5657 of those parts, I found SB 9747, BE for the first time 412, 9503, 413, BE for the second time 421, OD 10186, X 8528,4, PM 8450, MN 8475, NP 25; from whence, by the second operation, I collected the distance tb 5640; and by this operation I at last deduced the distances TX 4775 and 11322. From which, limiting the orbit, I found its descending node in ♋, and ascending node in ♑ 1° 53'; the inclination of its plane to the plane of the ecliptic 61° 20, the vertex thereof (or the perihelion of the comet) distant from the node 8° 38', and in ♐ 27° 43', with latitude 7° 34' south; its latus rectum 236,8; and the diurnal area described by a radius drawn to the sun 93585, supposing the square of the semi-diameter of the orbis magnus 100000000; that the comet in this orbit moved directly according to the order of the signs, and on Dec. 8d. 00h. 04' P. M. was in the vertex or perihelion of its orbit. All which I determined by scale and compass, and the chords of angles, taken from the table of natural sines, in a pretty large figure, in which, to wit, the radius of the orbis magnus (consisting of 10000 parts) was equal to inches of an English foot.

[Pg 478]

Lastly, in order to discover whether the comet did truly move in the orbit so determined, I investigated its places in this orbit partly by arithmetical operations, and partly by scale and compass, to the times of some of the observations, as may be seen in the following table:—

The Comet's

Dist.

from

sun. Longitude

computed. Latitud.

computed. Longitude

observed. Latitute

observed. Dif.

Lo. Dif.

Lat.

Dec. 12 2792 ♑ 6°.32' 8°.18 ♑ 6°.31 8°.26 +1 - 7

29 8403 ♓ 13 .13 28. 00 ♓ 13.11 28 .10 +2 -10

Feb. 5 16669 ♉ 17 .00 15. 29 ♉ 16 .59 15 .27 +0 + 2

Mar. 5 21737 29 .19 12. 4 29 .20 12 . 3 -1 +

But afterwards Dr. Halley did determine the orbit to a greater accuracy by an arithmetical calculus than could be done by linear descriptions; and, retaining the place of the nodes in ♋ and ♑ 1° 53' and the inclination of the plane of the orbit to the ecliptic 61° , as well as the time of the comet's being in perihelion, Dec. 8d. 00h. 04' he found the distance of the perihelion from the ascending node measured in the comet's orbit 9° 20', and the latus rectum of the parabola 2430 parts, supposing the mean distance of the sun from the earth to be 100000 parts; and from these data, by an accurate arithmetical calculus, he computed the places of the comet to the times of the observations as follows:—

The Comet's

True time. Dist. from

the sun. Longitude

computed. Latitude

computed. Errors in

Long.

Lo. Lat.

Lat.

d. h. ' ''

° ' '' ° ' '' ' '' ' ''

Dec. 12. 4.46. 28028 ♑︎ 6.29.25 8.26. 0 bor. -3. 5 -2. 0

21. 6.37. 61076 ♒︎ 5. 6.30 21.43.20 -1.42 +1. 7

24. 6.18. 70008 18.48.20 25.22.40 -1. 3 -0.25

26. 5.20. 75576 28.22.45 27. 1.36 -1.28 +0.44

29. 8. 3. 84021 ♓︎ 13.12.40 28.10.10 +1.59 +0.12

30. 8.10. 86661 17.40. 5 28.11.20 +1.45 -0.33

Jan. 5. 6. 1. 101440 ♈︎ 8.49.49 26.15.15 +0.56 +0. 8

9. 7. 0. 110959 18.44.36 24.12.54 +0.32 +0.58

10. 6. 6. 113162 20.41. 0 23.44.10 +0.10 +0.18

13. 7. 9. 120000 26. 0.21 22.17.30 +0.33 +0. 2

25. 7.59. 145370 ♉︎ 9.33.40 17.57.55 -1.20 +1.25

30. 8.22. 155303 13.17.41 16.42. 7 -2.10 -0.11

Feb. 2. 6.35. 160951 15.11.11 16. 4.15 -2.42 +0.14

5. 7. 4. 166686 16.58.55 15.29.13 -0.41 +2. 0

25. 8.41. 202570 26.15.46 12.48. 0 -2.49 +1.10

Mar. 5.11.39. 216205 29.18.35 12. 5.40 +0.35 +2.14

This comet also appeared in the November before, and at Coburg, in Saxony, was observed by Mr. Gottfried Kirch, on the 4th of that month, on the 6th and 11th O. S.; from its positions to the nearest fixed stars observed with sufficient accuracy, sometimes with a two feet, and sometimes with a ten feet telescope; from the difference of longitudes of Coburg and London; 11°; and from the places of the fixed stars observed by Mr. Pound, Dr. Halley has determined the places of the comet as follows:—

[Pg 479]

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

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

November 10, 16h. 31', the comet was equally distant from two stars in ♌, which are and in Bayer; but it had not quite touched the right line that joins them, but was very little distant from it. In Flamsted's catalogue this star was then in ♍︎ 14° 15', with 1 deg. 41' lat. north nearly, and in ♍ 17° with 0 deg. 34' lat. south; and the middle point between those stars was ♍ 15° , with 0° lat. north. Let the distance of the comet from that right line be about 10' or 12'; and the difference of the longitude of the comet and that middle point will be 7'; and the difference of the latitude nearly ; and thence it follows that the comet was in ♍ 15° 32', with about 26' lat. north.

The first observation from the position of the comet with respect to certain small fixed stars had all the exactness that could be desired; the second also was accurate enough. In the third observation, which was the least accurate, there might be an error of 6 or 7 minutes, but hardly greater. The longitude of the comet, as found in the first and most accurate observation, being computed in the aforesaid parabolic orbit, comes out ♌ 29° 30' 22'', its latitude north 1° 25' 7'', and its distance from the sun 115546.

[Pg 480]

Moreover, Dr. Halley, observing that a remarkable comet had appeared four times at equal intervals of 575 years (that is, in the month of September after Julius Cæsar was killed; An. Chr. 531, in the consulate of Lampadius and Orestes; An. Chr. 1106, in the month of February; and at the end of the year 1680; and that with a long and remarkable tail, except when it was seen after Cæsar's death, at which time, by reason of the inconvenient situation of the earth, the tail was not so conspicuous), set himself to find out an elliptic orbit whose greater axis should be 1382957 parts, the mean distance of the earth from the sun containing 10000 such; in which orbit a comet might revolve in 575 years; and, placing the ascending node in ♋ 2° 2', the inclination of the plane of the orbit to the plane of the ecliptic in an angle of 61° 6' 48'', the perihelion of the comet in this plane in ♐ 22° 44' 25'', the equal time of the perihelion December 7d. 23h. 9', the distance of the perihelion from the ascending node in the plane of the ecliptic 9° 17' 35'', and its conjugate axis 18481,2, he computed the motions of the comet in this elliptic orbit. The places of the comet, as deduced from the observations, and as arising from computation made in this orbit, may be seen in the following table.

True time. Longitude

observed. Latit'de

North

obs. Longitude

comp. Latitude

computed. Errors in

Long.

Lo. Lat.

Lat.

d. h. ' '' ° ' '' ° ' '' ° ' '' ' '' ' '' ' ''

Nov. 3.16.47 ♌ 29.51. 0 1.17.45 ♌ 29.51.22 1.17.32 N +0.22 -0.13

5.15.37 ♍ 3.23. 0 1. 6. 0 ♍ 3.24.32 1. 6. 9 +1.32 +0. 9

10.16.18 15.32. 0 0.27. 0 15.33. 2 0.25. 7 +1. 2 -1.53

16.17.00

♎ 8.16.45 0.53. 7 S

18.21.34

18.52.15 1.26.54

20.17. 0

28.10.36 1.53.35

23.17. 5

♏ 13.22.42 2.29. 0

Dec. 12. 4.46 ♑ 6.32.30 8.28. 0 ♑ 6.31.20 8.29. 6 N -1.10 +1. 6

21. 6.37 ♒ 5. 8.12 21.42.13 ♒ 5. 6.14 21.44.42 -1.58 +2.29

24. 6.18 18.49.23 25.23. 5 18.47.30 25.23.35 -1.53 +0.30

26. 5.21 28.24.13 27. 0.52 28.21.42 27. 2. 1 -2.31 +1. 9

29. 8. 3 ♓ 13.10.41 28. 9.58 ♓ 13.11.14 28.10.38 +0.33 +0.40

30. 8.10 17.38. 0 28.11.53 17.38.27 28.11.37 +0. 7 -0.16

Jan. 5. 6. 1 ♈ 8.48.53 26.15. 7 ♈ 8.48.51 26.14.57 -0. 2 -0.10

9. 7. 1 18.44. 4 24.11.56 18.43.51 24.12.17 -0.13 +0.21

10. 6. 6 20.40.50 23.43.32 20.40.23 23.43.25 -0.27 -0. 7

13. 7. 9 25.59.48 22.17.28 26. 0. 8 22.16.32 +0.20 -0.56

25. 7.59 ♉ 9.35. 0 17.56.30 ♉ 9.34.11 17.56. 6 -0.49 -0.24

30. 8.22 13.19.51 16.42.18 13.18.28 16.40. 5 -1.23 -2.13

Feb. 2. 6.35 15.13.53 16. 4. 1 15.11.59 16. 2.17 -1.54 -1.54

5. 7. 4 16.59. 6 15.27. 3 16.59.17 15.27. 0 +0.11 -0. 3

25. 8.41 26.18.35 12.46.46 26.16.59 12.45.22 -1.36 -1.24

Mar. 1.11.10 27.52.42 12.23.40 27.51.47 12.22.28 -0.55 -1.12

5.11.39 29.18. 0 12. 3.16 29.20.11 12. 2.50 +2.11 -0.26

9. 8.38 ♊ 0.43. 4 ♊.45.52 ♊ 0.42.43 11.45.35 -0.21 -0.17

The observations of this comet from the beginning to the end agree as perfectly with the motion of the comet in the orbit just now described as the motions of the planets do with the theories from whence they are calculated; and by this agreement plainly evince that it was one and the same comet that appeared all that time, and also that the orbit of that comet is here rightly defined.

In the foregoing table we have omitted the observations of Nov. 16, 18, 20, and 23, as not sufficiently accurate, for at those times several persons had observed the comet. Nov. 17, O. S. Ponthæus and his companions, at 6h. in the morning at Rome (that is, 5h. 10' at London), by threads directed to the fixed stars, observed the comet in ♎ 8° 30', with latitude 0° 40' south. Their observations may be seen in a treatise which Ponthæus published concerning this comet. Cellius, who was present, and communicated his observations in a letter to Cassini, saw the comet at the same hour in ♎ 8° 30', with latitude 0° 30' south. It was likewise seen by Galletius at the same hour at Avignon (that is, at 5h. 42' morning at London) in ♎ 8° without latitude. But by the theory the comet was at that time in ♎ 8° 16' 45'', and its latitude was 0° 53' 7'' south.

Nov. 18, at 6h. 30' in the morning at Rome (that is, at 5h. 40' at London), Ponthæus observed the comet in ♎ 13° 30', with latitude 1° 20'[Pg 481] south; and Cellius in ♎ 13° 30', with latitude 1° 00' south. But at 5h. 30' in the morning at Avignon, Galletius saw it in ♎ 13° 00' with latitude 1° 00' south. In the University of La Fleche, in France, at 5h. in the morning (that is, at 5h. 9' at London), it was seen by P. Ango, in the middle between two small stars, one of which is the middle of the three which lie in a right line in the southern hand of Virgo, Bayers ; and the other is the outmost of the wing, Bayer's . Whence the comet was then in ♎ 12° 46' with latitude 50' south. And I was informed by Dr. Halley, that on the same day at Boston in New England, in the latitude of deg. at 5h. in the morning (that is, at 9h. 44' in the morning at London), the comet was seen near ♎ 14°, with latitude 1° 30' south.

Nov. 19, at at Cambridge, the comet (by the observation of a young man) was distant from Spica ♍ about 2° towards the north west. Now the spike was at that time in ♎ 19° 23' 47'', with latitude 2° 1' 59'' south. The same day, at 5h. in the morning, at Boston in New England, the comet was distant from Spica ♍ 1°, with the difference of 40' in latitude. The same day, in the island of Jamaica, it was about 1° distant from Spica ♍. The same day, Mr. Arthur Storer, at the river Patuxent, near Hunting Creek, in Maryland, in the confines of Virginia, in lat. °, at 5 in the morning (that is, at 10h. at London), saw the comet above Spica ♍, and very nearly joined with it, the distance between them being about of one deg. And from these observations compared, I conclude, that at 9h. 44' at London, the comet was in ♎ 18° 50', with about 1° 25' latitude south. Now by the theory the comet was at that time in ♎ 18° 52' 15'', with 1° 26' 54'' lat. south.

Nov. 20, Montenari, professor of astronomy at Padua, at 6h. in the morning at Venice (that is, 5h. 10' at London), saw the comet in ♎ 23°, with latitude 1° 30' south. The same day, at Boston, it was distant from Spica ♍ by about 4° of longitude east, and therefore was in ♎ 23° 24' nearly.

Nov. 21, Ponthæus and his companions, at in the morning, observed the comet in ♎ 27° 50', with latitude 1° 16' south; Cellius, in ♎ 28°; P. Ango at 5h. in the morning, in ♎ 27° 45'; Montenari in ♎ 27° 51'. The same day, in the island of Jamaica, it was seen near the beginning of ♏, and of about the same latitude with Spica ♍, that is, 2° 2'. The same day, at 5h. morning, at Ballasore, in the East Indies (that is, at 11h. 20' of the night preceding at London), the distance of the comet from Spica ♍ was taken 7° 35' to the east. It was in a right line between the spike and the balance, and therefore was then in ♎ 26° 58', with about 1° 11' lat. south; and after 5h. 40' (that is, at 5h. morning at London), it was in ♎ 28° 12', with 1° 16' lat. south. Now by the theory the comet was then in ♎ 28° 10' 36'', with 1° 53' 35'' lat. south.

Nov. 22, the comet was seen by Montenari in ♏ 2° 33'; but at Boston[Pg 482] in New England, it was found in about ♏ 3°, and with almost the same latitude as before, that is, 1° 30'. The same day, at 5h. morning at Ballasore, the comet was observed in ♏ 1° 50'; and therefore at 5h. morning at London, the comet was in ♏ 3° 5' nearly. The same day, at in the morning at London, Dr. Hook observed it in about ♏ 3° 30', and that in the right line which passeth through Spica ♍ and Cor Leonis; not, indeed, exactly, but deviating a little from that line towards the north. Montenari likewise observed, that this day, and some days after, a right line drawn from the comet through Spica passed by the south side of Cor Leonis at a very small distance therefrom. The right line through Cor Leonis and Spica ♍ did cut the ecliptic in ♍ 3° 46' at an angle of 2° 51'; and if the comet had been in this line and in ♏ 3°, its latitude would have been 2° 26'; but since Hook and Montenari agree that the comet was at some small distance from this line towards the north, its latitude must have been something less. On the 20th, by the observation of Montenari, its latitude was almost the same with that of Spica ♍, that is, about 1° 30'. But by the agreement of Hook, Montenari, and Ango, the latitude was continually increasing, and therefore must now, on the 22d, be sensibly greater than 1° 30'; and, taking a mean between the extreme limits but now stated, 2° 26' and 1° 30', the latitude will be about 1° 58'. Hook and Montenari agree that the tail of the comet was directed towards Spica ♍, declining a little from that star towards the south according to Hook, but towards the north according to Montenari; and, therefore, that declination was scarcely sensible; and the tail, lying nearly parallel to the equator, deviated a little from the opposition of the sun towards the north.

Nov. 23, O. S. at 5h. morning, at Nuremberg (that is, at at London), Mr. Zimmerman saw the comet in ♏ 8° 8', with 2° 31' south lat. its place being collected by taking its distances from fixed stars.

Nov. 24, before sun-rising, the comet was seen by Montenari in ♏ 12° 52' on the north side of the right line through Cor Leonis and Spica ♍, and therefore its latitude was something less than 2° 38'; and since the latitude, as we said, by the concurring observations of Montenari, Ango, and Hook, was continually increasing, therefore, it was now, on the 24th, something greater than 1° 58'; and, taking the mean quantity, may be reckoned 2° 18', without any considerable error. Ponthæus and Galletius will have it that the latitude was now decreasing; and Cellius, and the observer in New England, that it continued the same, viz., of about 1°, or °. The observations of Ponthæus and Cellius are more rude, especially those which were made by taking the azimuths and altitudes; as are also the observations of Galletius. Those are better which were made by taking the position of the comet to the fixed stars by Montenari, Hook, Ango, and the observer in New England, and sometimes by[Pg 483] Ponthæus and Cellius. The same day, at 5h. morning, at Ballasore, the comet was observed in ♏ 11° 45'; and, therefore, at 5h. morning at London, was in ♏ 13° nearly. And, by the theory, the comet was at that time in ♏ 13° 22' 42''.

Nov. 25, before sunrise, Montenari observed the comet in ♏ ° nearly; and Cellius observed at the same time that the comet was in a right line between the bright star in the right thigh of Virgo and the southern scale of Libra; and this right line cuts the comet's way in ♏ 18° 36'. And, by the theory, the comet was in ♏ ° nearly.

From all this it is plain that these observations agree with the theory, so far as they agree with one another; and by this agreement it is made clear that it was one and the same comet that appeared all the time from Nov. 4 to Mar. 9. The path of this comet did twice cut the plane of the ecliptic, and therefore was not a right line. It did cut the ecliptic not in opposite parts of the heavens, but in the end of Virgo and beginning of Capricorn, including an arc of about 98°; and therefore the way of the comet did very much deviate from the path of a great circle; for in the month of Nov. it declined at least 3° from the ecliptic towards the south; and in the month of Dec. following it declined 29° from the ecliptic towards the north; the two parts of the orbit in which the comet descended towards the sun, and ascended again from the sun, declining one from the other by an apparent angle of above 30°, as observed by Montenari. This comet travelled over 9 signs, to wit, from the last deg. of ♌ to the beginning of ♊, beside the sign of ♌, through which it passed before it began to be seen; and there is no other theory by which a comet can go over so great a part of the heavens with a regular motion. The motion of this comet was very unequable; for about the 20th of Nov. it described about 5° a day. Then its motion being retarded between Nov. 26 and Dec. 12, to wit, in the space of days, it described only 40°. But the motion thereof being afterwards accelerated, it described near 5° a day, till its motion began to be again retarded. And the theory which justly corresponds with a motion so unequable, and through so great a part of the heavens, which observes the same laws with the theory of the planets, and which accurately agrees with accurate astronomical observations, cannot be otherwise than true.

And, thinking it would not be improper, I have given a true representation of the orbit which this comet described, and of the tail which it emitted in several places, in the annexed figure; protracted in the plane of the trajectory. In this scheme ABC represents the trajectory of the comet, D the sun, DE the axis of the trajectory, DF the line of the nodes, GH the intersection of the sphere of the orbis magnus with the plane of the trajectory, I the place of the comet Nov. 4, Ann. 1680; K the place of the same Nov. 11; L the place of the same Nov. 19; M its place Dec. 12; N[Pg 484] its place Dec. 21; O its place Dec. 29; P its place Jan. 5 following; Q its place Jan. 25; R its place Feb. 5; S its place Feb. 25; T its place March 5; and V its place March 9. In determining the length of the tail, I made the following observations.

Nov. 4 and 6, the tail did not appear; Nov. 11, the tail just begun to shew itself, but did not appear above deg. long through a 10 feet telescope; Nov. 17, the tail was seen by Ponthæus more than 15° long; Nov. 18, in New-England, the tail appeared 30° long, and directly opposite to the sun, extending itself to the planet Mars, which was then in ♍, 9° 54'; Nov. 19. in Maryland, the tail was found 15° or 20° long; Dec. 10 (by[Pg 485] the observation of Mr. Flamsted), the tail passed through the middle of the distance intercepted between the tail of the Serpent of Ophiuchus and the star in the south wing of Aquila, and did terminate near the stars A, , b, in Bayer's tables. Therefore the end of the tail was in ♑ °, with latitude about ° north; Dec. 11, it ascended to the head of Sagitta (Bayer's , ), terminating in ♑ 26° 43' with latitude 38° 34' north; Dec. 12, it passed through the middle of Sagitta, nor did it reach much farther; terminating in ♒ 4°, with latitude ° north nearly. But these things are to be understood of the length of the brighter part of the tail; for with a more faint light, observed, too, perhaps, in a serener sky, at Rome, Dec. 12, 5h. 40', by the observation of Ponthæus, the tail arose to 10° above the rump of the Swan, and the side thereof towards the west and towards the north was 45' distant from this star. But about that time the tail was 3° broad towards the upper end; and therefore the middle thereof was 2° 15' distant from that star towards the south, and the upper end was ♓ in 22°, with latitude 61° north; and thence the tail was about 70° long; Dec. 21, it extended almost to Cassiopeia's chair, equally distant from and from Schedir, so as its distance from either of the two was equal to the distance of the one from the other, and therefore did terminate in ♈ 24°, with latitude °; Dec. 29, it reached to a contact with Scheat on its left, and exactly filled up the space between the two stars in the northern foot of Andromeda, being 54° in length; and therefore terminated in ♉ 19°, with 35° of latitude; Jan. 5, it touched the star in the breast of Andromeda on its right side, and the star of the girdle on its left; and, according to our observations, was 40° long; but it was curved, and the convex side thereof lay to the south; and near the head of the comet it made an angle of 4° with the circle which passed through the sun and the comet's head; but towards the other end it was inclined to that circle in an angle of about 10° or 11°; and the chord of the tail contained with that circle an angle of 8°. Jan. 13, the tail terminated between Alamech and Algol, with a light that was sensible enough; but with a faint light it ended over against the star in Perseus's side. The distance of the end of the tail from the circle passing through the sun and the comet was 3° 50'; and the inclination of the chord of the tail to that circle was °. Jan. 25 and 26, it shone with a faint light to the length of 6° or 7°; and for a night or two after, when there was a very clear sky, it extended to the length of 12°, or something more, with a light that was very faint and very hardly to be seen; but the axis thereof was exactly directed to the bright star in the eastern shoulder of Auriga, and therefore deviated from the opposition of the sun towards the north by an angle of 10°. Lastly, Feb. 10, with a telescope I observed the tail 2° long; for that fainter light which I spoke of did not appear through the glasses. But Ponthæus writes, that, on Feb. 7, he saw the tail 12° long. Feb. 25, the comet was without a tail, and so continued till it disappeared.

[Pg 486]

Now if one reflects upon the orbit described, and duly considers the other appearances of this comet, he will be easily satisfied that the bodies of comets are solid, compact, fixed, and durable, like the bodies of the planets; for if they were nothing else but the vapours or exhalations of the earth, of the sun, and other planets, this comet, in its passage by the neighbourhood of the sun, would have been immediately dissipated; for the heat of the sun is as the density of its rays, that is, reciprocally as the square of the distance of the places from the sun. Therefore, since on Dec. 8, when the comet was in its perihelion, the distance thereof from the centre of the sun was to the distance of the earth from the same as about 6 to 1000, the sun's heat on the comet was at that time to the heat of the summer-sun with us as 1000000 to 36, or as 28000 to 1. But the heat of boiling water is about 3 times greater than the heat which dry earth acquires from the summer-sun, as I have tried; and the heat of red-hot iron (if my conjecture is right) is about three or four times greater than the heat of boiling water. And therefore the heat which dry earth on the comet, while in its perihelion, might have conceived from the rays of the sun, was about 2000 times greater than the heat of red-hot iron. But by so fierce a heat, vapours and exhalations, and every volatile matter, must have been immediately consumed and dissipated.

This comet, therefore, must have conceived an immense heat from the sun, and retained that heat for an exceeding long time; for a globe of iron of an inch in diameter, exposed red-hot to the open air, will scarcely lose all its heat in an hour's time; but a greater globe would retain its heat longer in the proportion of its diameter, because the surface (in proportion to which it is cooled by the contact of the ambient air) is in that proportion less in respect of the quantity of the included hot matter; and therefore a globe of red hot iron equal to our earth, that is, about 40000000 feet in diameter, would scarcely cool in an equal number of days, or in above 50000 years. But I suspect that the duration of heat may, on account of some latent causes, increase in a yet less proportion than that of the diameter; and I should be glad that the true proportion was investigated by experiments.

BOOK III. (part 4)

It is farther to be observed, that the comet in the month of December, just after it had been heated by the sun, did emit a much longer tail, and much more splendid, than in the month of November before, when it had not yet arrived at its perihelion; and, universally, the greatest and most fulgent tails always arise from comets immediately after their passing by the neighbourhood of the sun. Therefore the heat received by the comet conduces to the greatness of the tail: from whence, I think I may infer, that the tail is nothing else but a very fine vapour, which the head or nucleus of the comet emits by its heat.

But we have had three several opinions about the tails of comets; for[Pg 487] some will have it that they are nothing else but the beams of the sun's light transmitted through the comets' heads, which they suppose to be transparent; others, that they proceed from the refraction which light suffers in passing from the comet's head to the earth; and, lastly, others, that they are a sort of clouds or vapour constantly rising from the comets' heads, and tending towards the parts opposite to the sun. The first is the opinion of such as are yet unacquainted with optics; for the beams of the sun are seen in a darkened room only in consequence of the light that is reflected from them by the little particles of dust and smoke which are always flying about in the air; and, for that reason, in air impregnated with thick smoke, those beams appear with great brightness, and move the sense vigorously; in a yet finer air they appear more faint, and are less easily discerned; but in the heavens, where there is no matter to reflect the light they can never be seen at all. Light is not seen as it is in the beam, but as it is thence reflected to our eyes; for vision can be no otherwise produced than by rays falling upon the eyes; and therefore, there must be some reflecting matter in those parts where the tails of the comets are seen: for otherwise, since all the celestial spaces are equally illuminated by the sun's light, no part of the heavens could appear with more splendor than another. The second opinion is liable to many difficulties. The tails of comets are never seen variegated with those colours which commonly are inseparable from refraction; and the distinct transmission of the light of the fixed stars and planets to us is a demonstration that the æther or celestial medium is not endowed with any refractive power: for as to what is alleged, that the fixed stars have been sometimes seen by the Egyptians environed with a Coma or Capillitium, because that has but rarely happened, it is rather to be ascribed to a casual refraction of clouds; and so the radiation and scintillation of the fixed stars to the refractions both of the eyes and air; for upon laying a telescope to the eye, those radiations and scintillations immediately disappear. By the tremulous agitation of the air and ascending vapours, it happens that the rays of light are alternately turned aside from the narrow space of the pupil of the eye; but no such thing can have place in the much wider aperture of the object-glass of a telescope; and hence it is that a scintillation is occasioned in the former case, which ceases in the latter; and this cessation in the latter case is a demonstration of the regular transmission of light through the heavens, without any sensible refraction. But, to obviate an objection that may be made from the appearing of no tail in such comets as shine but with a faint light, as if the secondary rays were then too weak to affect the eyes, and for that reason it is that the tails of the fixed stars do not appear, we are to consider, that by the means of telescopes the light of the fixed stars may be augmented above an hundred fold, and yet no tails are seen; that the light of the planets is yet more copious without any[Pg 488] tail; but that comets are seen sometimes with huge tails, when the light of their heads is but faint and dull. For so it happened in the comet of the year 1680, when in the month of December it was scarcely equal in light to the stars of the second magnitude, and yet emitted a notable tail, extending to the length of 40°, 50°, 60°, or 70°, and upwards; and afterwards, on the 27th and 28th of January, when the head appeared but as a star of the 7th magnitude, yet the tail (as we said above), with a light that was sensible enough, though faint, was stretched out to 6 or 7 degrees in length, and with a languishing light that was more difficultly seen, even to 12°, and upwards. But on the 9th and 10th of February, when to the naked eye the head appeared no more, through a telescope I viewed the tail of 2° in length. But farther; if the tail was owing to the refraction of the celestial matter, and did deviate from the opposition of the sun, according to the figure of the heavens, that deviation in the same places of the heavens should be always directed towards the same parts. But the comet of the year 1680, December 28d. . P. M. at London, was seen in ♓ 8° 41', with latitude north 28° 6'; while the sun was in ♑ 18° 26'. And the comet of the year 1577, December 29d. was in ♓ 8° 41', with latitude north 28° 40', and the sun, as before, in about ♑ 18° 26'. In both cases the situation of the earth was the same, and the comet appeared in the same place of the heavens; yet in the former case the tail of the comet (as well by my observations as by the observations of others) deviated from the opposition of the sun towards the north by an angle of degrees; whereas in the latter there was (according to the observations of Tycho) a deviation of 21 degrees towards the south. The refraction, therefore, of the heavens being thus disproved, it remains that the phænomena of the tails of comets must be derived from some reflecting matter.

And that the tails of comets do arise from their heads, and tend towards the parts opposite to the sun, is farther confirmed from the laws which the tails observe. As that, lying in the planes of the comets' orbits which pass through the sun, they constantly deviate from the opposition of the sun towards the parts which the comets' heads in their progress along these orbits have left. That to a spectator, placed in those planes, they appear in the parts directly opposite to the sun; but, as the spectator recedes from those planes, their deviation begins to appear, and daily becomes greater. That the deviation, cæteris paribus, appears less when the tail is more oblique to the orbit of the comet, as well as when the head of the comet approaches nearer to the sun, especially if the angle of deviation is estimated near the head of the comet. That the tails which have no deviation appear straight, but the tails which deviate are likewise bended into a certain curvature. That this curvature is greater when the deviation is greater; and is more sensible when the tail, cæteris paribus is longer; for in the shorter tails the curvature is hardly to be perceived.[Pg 489] That the angle of deviation is less near the comet's head, but greater towards the other end of the tail; and that because the convex side of the tail regards the parts from which the deviation is made, and which lie in a right line drawn out infinitely from the sun through the comet's head. And that the tails that are long and broad, and shine with a stronger light, appear more resplendent and more exactly defined on the convex than on the concave side. Upon which accounts it is plain that the phænomena of the tails of comets depend upon the motions of their heads, and by no means upon the places of the heavens in which their heads are seen; and that, therefore, the tails of comets do not proceed from the refraction of the heavens, but from their own heads, which furnish the matter that forms the tail. For, as in our air, the smoke of a heated body ascends either perpendicularly if the body is at rest, or obliquely if the body is moved obliquely, so in the heavens, where all bodies gravitate towards the sun, smoke and vapour must (as we have already said) ascend from the sun, and either rise perpendicularly if the smoking body is at rest, or obliquely if the body, in all the progress of its motion, is always leaving those places from which the upper or higher parts of the vapour had risen before; and that obliquity will be least where the vapour ascends with most velocity, to wit, near the smoking body, when that is near the sun. But, because the obliquity varies, the column of vapour will be incurvated; and because the vapour in the preceding sides is something more recent, that is, has ascended something more late from the body, it will therefore be something more dense on that side, and must on that account reflect more light, as well as be better defined. I add nothing concerning the sudden uncertain agitation of the tails of comets, and their irregular figures, which authors sometimes describe, because they may arise from the mutations of our air, and the motions of our clouds, in part obscuring those tails; or, perhaps, from parts of the Via Lactea, which might have been confounded with and mistaken for parts of the tails of the comets as they passed by.

But that the atmospheres of comets may furnish a supply of vapour great enough to fill so immense spaces, we may easily understand from the rarity of our own air; for the air near the surface of our earth possesses a space 850 times greater than water of the same weight; and therefore a cylinder of air 850 feet high is of equal weight with a cylinder of water of the same breadth, and but one foot high. But a cylinder of air reaching to the top of the atmosphere is of equal weight with a cylinder of water about 33 feet high: and, therefore, if from the whole cylinder of air the lower part of 850 feet high is taken away, the remaining upper part will be of equal weight with a cylinder of water 32 feet high: and from thence (and by the hypothesis, confirmed by many experiments, that the compression of air is as the weight of the incumbent atmosphere, and[Pg 490] that the force of gravity is reciprocally as the square of the distance from the centre of the earth) raising a calculus, by Cor. Prop. XXII, Book II, I found, that, at the height of one semi-diameter of the earth, reckoned from the earth's surface, the air is more rare than with us in a far greater proportion than of the whole space within the orb of Saturn to a spherical space of one inch in diameter; and therefore if a sphere of our air of but one inch in thickness was equally rarefied with the air at the height of one semi-diameter of the earth from the earth's surface, it would fill all the regions of the planets to the orb of Saturn, and far beyond it. Wherefore since the air at greater distances is immensely rarefied, and the coma or atmosphere of comets is ordinarily about ten times higher, reckoning from their centres, than the surface of the nucleus, and the tails rise yet higher, they must therefore be exceedingly rare; and though, on account of the much thicker atmospheres of comets, and the great gravitation of their bodies towards the sun, as well as of the particles of their air and vapours mutually one towards another, it may happen that the air in the celestial spaces and in the tails of comets is not so vastly rarefied, yet from this computation it is plain that a very small quantity of air and vapour is abundantly sufficient to produce all the appearances of the tails of comets: for that they are, indeed, of a very notable rarity appears from the shining of the stars through them. The atmosphere of the earth, illuminated by the sun's light, though but of a few miles in thickness, quite obscures and extinguishes the light not only of all the stars, but even of the moon itself; whereas the smallest stars are seen to shine through the immense thickness of the tails of comets, likewise illuminated by the sun, without the least diminution of their splendor. Nor is the brightness of the tails of most comets ordinarily greater than that of our air, an inch or two in thickness, reflecting in a darkened room the light of the sun-beams let in by a hole of the window-shutter.

And we may pretty nearly determine the time spent during the ascent of the vapour from the comet's head to the extremity of the tail, by drawing a right line from the extremity of the tail to the sun, and marking the place where that right line intersects the comet's orbit: for the vapour that is now in the extremity of the tail, if it has ascended in a right line from the sun, must have begun to rise from the head at the time when the head was in the point of intersection. It is true, the vapour does not rise in a right line from the sun, but, retaining the motion which it had from the comet before its ascent, and compounding that motion with its motion of ascent, arises obliquely; and, therefore, the solution of the Problem will be more exact, if we draw the line which intersects the orbit parallel to the length of the tail; or rather (because of the curvilinear motion of the comet) diverging a little from the line or length of the tail. And by means of this principle I found that the vapour which, January 25, was[Pg 491] in the extremity of the tail, had begun to rise from the head before December 11, and therefore had spent in its whole ascent 45 days; but that the whole tail which appeared on December 10 had finished its ascent in the space of the two days then elapsed from the time of the comet's being in its perihelion. The vapour, therefore, about the beginning and in the neighbourhood of the sun rose with the greatest velocity, and afterwards continued to ascend with a motion constantly retarded by its own gravity; and the higher it ascended, the more it added to the length of the tail; and while the tail continued to be seen, it was made up of almost all that vapour which had risen since the time of the comet's being in its perihelion; nor did that part of the vapour which had risen first, and which formed the extremity of the tail, cease to appear, till its too great distance, as well from the sun, from which it received its light, as from our eyes, rendered it invisible. Whence also it is that the tails of other comets which are short do not rise from their heads with a swift and continued motion, and soon after disappear, but are permanent and lasting columns of vapours and exhalations, which, ascending from the heads with a slow motion of many days, and partaking of the motion of the heads which they had from the beginning, continue to go along together with them through the heavens. From whence again we have another argument proving the celestial spaces to be free, and without resistance, since in them not only the solid bodies of the planets and comets, but also the extremely rare vapours of comets' tails, maintain their rapid motions with great freedom, and for an exceeding long time.

Kepler ascribes the ascent of the tails of the comets to the atmospheres of their heads; and their direction towards the parts opposite to the sun to the action of the rays of light carrying along with them the matter of the comets' tails; and without any great incongruity we may suppose, that, in so free spaces, so fine a matter as that of the æther may yield to the action of the rays of the sun's light, though those rays are not able sensibly to move the gross substances in our parts, which are clogged with so palpable a resistance. Another author thinks that there may be a sort of particles of matter endowed with a principle of levity, as well as others are with a power of gravity; that the matter of the tails of comets may be of the former sort, and that its ascent from the sun may be owing to its levity; but, considering that the gravity of terrestrial bodies is as the matter of the bodies, and therefore can be neither more nor less in the same quantity of matter, I am inclined to believe that this ascent may rather proceed from the rarefaction of the matter of the comets' tails. The ascent of smoke in a chimney is owing to the impulse of the air with which it is entangled. The air rarefied by heat ascends, because its specific gravity is diminished, and in its ascent carries along with it the smoke with which it is engaged; and why may not the tail of a comet rise from the sun after the same manner?[Pg 492] For the sun's rays do not act upon the mediums which they pervade otherwise than by reflection and refraction; and those reflecting particles heated by this action, heat the matter of the æther which is involved with them. That matter is rarefied by the heat which it acquires, and because, by this rarefaction, the specific gravity with which it tended towards the sun before is diminished, it will ascend therefrom, and carry along with it the reflecting particles of which the tail of the comet is composed. But the ascent of the vapours is further promoted by their circumgyration about the sun, in consequence whereof they endeavour to recede from the sun, while the sun's atmosphere and the other matter of the heavens are either altogether quiescent, or are only moved with a slower circumgyration derived from the rotation of the sun. And these are the causes of the ascent of the tails of the comets in the neighbourhood of the sun, where their orbits are bent into a greater curvature, and the comets themselves are plunged into the denser and therefore heavier parts of the sun's atmosphere: upon which account they do then emit tails of an huge length; for the tails which then arise, retaining their own proper motion, and in the mean time gravitating towards the sun, must be revolved in ellipses about the sun in like manner as the heads are, and by that motion must always accompany the heads, and freely adhere to them. For the gravitation of the vapours towards the sun can no more force the tails to abandon the heads, and descend to the sun, than the gravitation of the heads can oblige them to fall from the tails. They must by their common gravity either fall together towards the sun, or be retarded together in their common ascent therefrom; and, therefore (whether from the causes already described, or from any others), the tails and heads of comets may easily acquire and freely retain any position one to the other, without disturbance or impediment from that common gravitation.

The tails, therefore, that rise in the perihelion positions of the comets will go along with their heads into far remote parts, and together with the heads will either return again from thence to us, after a long course of years, or rather will be there rarefied, and by degrees quite vanish away; for afterwards, in the descent of the heads towards the sun, new short tails will be emitted from the heads with a slow motion; and those tails by degrees will be augmented immensely, especially in such comets as in their perihelion distances descend as low as the sun's atmosphere; for all vapour in those free spaces is in a perpetual state of rarefaction and dilatation; and from hence it is that the tails of all comets are broader at their upper extremity than near their heads. And it is not unlikely but that the vapour, thus perpetually rarefied and dilated, may be at last dissipated and scattered through the whole heavens, and by little and little be attracted towards the planets by its gravity, and mixed with their atmosphere; for as the seas are absolutely necessary to the constitution of our earth, that[Pg 493] from them, the sun, by its heat, may exhale a sufficient quantity of vapours, which, being gathered together into clouds, may drop down in rain, for watering of the earth, and for the production and nourishment of vegetables; or, being condensed with cold on the tops of mountains (as some philosophers with reason judge), may run down in springs and rivers; so for the conservation of the seas, and fluids of the planets, comets seem to be required, that, from their exhalations and vapours condensed, the wastes of the planetary fluids spent upon vegetation and putrefaction, and converted into dry earth, may be continually supplied and made up; for all vegetables entirely derive their growths from fluids, and afterwards, in great measure, are turned into dry earth by putrefaction; and a sort of slime is always found to settle at the bottom of putrefied fluids; and hence it is that the bulk of the solid earth is continually increased; and the fluids, if they are not supplied from without, must be in a continual decrease, and quite fail at last. I suspect, moreover, that it is chiefly from the comets that spirit comes, which is indeed the smallest but the most subtle and useful part of our air, and so much required to sustain the life of all things with us.

The atmospheres of comets, in their descent towards the sun, by running out into the tails, are spent and diminished, and become narrower, at least on that side which regards the sun; and in receding from the sun, when they less run out into the tails, they are again enlarged, if Hevelius has justly marked their appearances. But they are seen least of all just after they have been most heated by the sun, and on that account then emit the longest and most resplendent tails; and, perhaps, at the same time, the nuclei are environed with a denser and blacker smoke in the lowermost parts of their atmosphere; for smoke that is raised by a great and intense heat is commonly the denser and blacker. Thus the head of that comet which we have been describing, at equal distances both from the sun and from the earth, appeared darker after it had passed by its perihelion than it did before; for in the month of December it was commonly compared with the stars of the third magnitude, but in November with those of the first or second; and such as saw both appearances have described the first as of another and greater comet than the second. For, November 19, this comet appeared to a young man at Cambridge, though with a pale and dull light, yet equal to Spica Virginis; and at that time it shone with greater brightness than it did afterwards. And Montenari, November 20, st. vet. observed it larger than the stars of the first magnitude, its tail being then 2 degrees long. And Mr. Storer (by letters which have come into my hands) writes, that in the month of December, when the tail appeared of the greatest bulk and splendor, the head was but small, and far less than that which was seen in the month of November before sun-rising; and, conjecturing at the cause of the appearance, he judged it to proceed[Pg 494] from there being a greater quantity of matter in the head at first, which was afterwards gradually spent.

And, which farther makes for the same purpose, I find, that the heads of other comets, which did put forth tails of the greatest bulk and splendor, have appeared but obscure and small. For in Brazil, March 5, 1668, 7h. P. M., St. N. P. Valentinus Estancius saw a comet near the horizon, and towards the south west, with a head so small as scarcely to be discerned, but with a tail above measure splendid, so that the reflection thereof from the sea was easily seen by those who stood upon the shore; and it looked like a fiery beam extended 23° in length from the west to south, almost parallel to the horizon. But this excessive splendor continued only three days, decreasing apace afterwards: and while the splendor was decreasing, the bulk of the tail increased: whence in Portugal it is said to have taken up one quarter of the heavens, that is, 45 degrees, extending from west to east with a very notable splendor, though the whole tail was not seen in those parts, because the head was always hid under the horizon: and from the increase of the bulk and decrease of the splendor of the tail, it appears that the head was then in its recess from the sun, and had been very near to it in its perihelion, as the comet of 1680 was. And we read, in the Saxon Chronicle, of a like comet appearing in the year 1106, the star whereof was small and obscure (as that of 1680), but the splendour of its tail was very bright, and like a huge fiery beam stretched out in a direction between the east and north, as Hevelius has it also from Simeon, the monk of Durham. This comet appeared in the beginning of February, about the evening, and towards the south west part of heaven; from whence, and from the position of the tail, we infer that the head was near the sun. Matthew Paris says, It was distant from the sun by about a cubit, from three of the clock (rather six) till nine, putting forth a long tail. Such also was that most resplendent comet described by Aristotle, lib. 1, Meteor. 6. The head whereof could not be seen, because it had set before the sun, or at least was hid under the sun's rays; but next day it was seen as well as might be; for, having left the sun but a very Little way, it set immediately after it. And the scattered light of the head, obscured by the too great splendour (of the tail) did not yet appear. But afterwards (as Aristotle says) when the splendour (of the tail) was now diminished (the head of), the comet recovered its native brightness; and the splendour (of its tail) reached now to a third part of the heavens (that is, to 60°). This appearance was in the winter season (an. 4, Olymp. 101), and, rising to Orion's girdle, it there vanished away. It is true that the comet of 1618, which came out directly from under the sun's rays with a very large tail, seemed to equal, if not to exceed, the stars of the first magnitude; but, then, abundance of other comets have appeared yet greater than this, that put forth shorter tails; some of which are said[Pg 495] to have appeared as big as Jupiter, others as big as Venus, or even as the moon.

We have said, that comets are a sort of planets revolved in very eccentric orbits about the sun; and as, in the planets which are without tails, those are commonly less which are revolved in lesser orbits, and nearer to the sun, so in comets it is probable that those which in their perihelion approach nearer to the sun are generally of less magnitude, that they may not agitate the sun too much by their attractions. But as to the transverse diameters of their orbits, and the periodic times of their revolutions, I leave them to be determined by comparing comets together which after long intervals of time return again in the same orbit. In the mean time, the following Proposition may give some light in that inquiry.

PROPOSITION XLII. PROBLEM XXII.

To correct a comet's trajectory found as above.

OPERATION 1. Assume that position of the plane of the trajectory which was determined according to the preceding proposition; and select three places of the comet, deduced from very accurate observations, and at great distances one from the other. Then suppose A to represent the time between the first observation and the second, and B the time between the second and the third; but it will be convenient that in one of those times the comet be in its perigeon, or at least not far from it. From those apparent places find, by trigonometric operations, the three true places of the comet in that assumed plane of the trajectory; then through the places found, and about the centre of the sun as the focus, describe a conic section by arithmetical operations, according to Prop. XXI., Book I. Let the areas of this figure which are terminated by radii drawn from the sun to the places found be D and E; to wit, D the area between the first observation and the second, and E the area between the second and third; and let T represent the whole time in which the whole area D + E should be described with the velocity of the comet found by Prop. XVI., Book I.

OPER. 2. Retaining the inclination of the plane of the trajectory to the plane of the ecliptic, let the longitude of the nodes of the plane of the trajectory be increased by the addition of 20 or 30 minutes, which call P. Then from the aforesaid three observed places of the comet let the three true places be found (as before) in this new plane; as also the orbit passing through those places, and the two areas of the same described between the two observations, which call d and e; and let t be the whole time in which the whole area d + e should be described.

OPER. 3. Retaining the longitude of the nodes in the first operation, let the inclination of the plane of the trajectory to the plane of the ecliptic be increased by adding thereto 20' or 30', which call Q. Then from the[Pg 496] aforesaid three observed apparent places of the comet let the three true places be found in this new plane, as well as the orbit passing through them, and the two areas of the same described between the observation, which call and ; and let be the whole time in which the whole area should be described.

Then taking C to 1 as A to B; and G to 1 as D to E; and g to 1 as d to e; and to 1 as to ; let S be the true time between the first observation and the third; and, observing well the signs + and -, let such numbers m and n be found out as will make ; and . And if, in the first operation, I represents the inclination of the plane of the trajectory to the plane of the ecliptic, and K the longitude of either node, then I + nQ will be the true inclination of the plane of the trajectory to the plane of the ecliptic, and K + mP the true longitude of the node. And, lastly, if in the first, second, and third operations, the quantities R, r, and , represent the parameters of the trajectory, and the quantities , , , the transverse diameters of the same, then will be the true parameter, and will be the true transverse diameter of the trajectory which the comet describes; and from the transverse diameter given the periodic time of the comet is also given. Q.E.I. But the periodic times of the revolutions of comets, and the transverse diameters of their orbits, cannot be accurately enough determined but by comparing comets together which appear at different times. If, after equal intervals of time, several comets are found to have described the same orbit, we may thence conclude that they are all but one and the same comet revolved in the same orbit; and then from the times of their revolutions the transverse diameters of their orbits will be given, and from those diameters the elliptic orbits themselves will be determined.

To this purpose the trajectories of many comets ought to be computed, supposing those trajectories to be parabolic; for such trajectories will always nearly agree with the phænomena, as appears not only from the parabolic trajectory of the comet of the year 1680, which I compared above with the observations, but likewise from that of the notable comet which appeared in the year 1664 and 1665, and was observed by Hevelius, who, from his own observations, calculated the longitudes and latitudes thereof, though with little accuracy. But from the same observations Dr. Halley did again compute its places; and from those new places determined its trajectory, finding its ascending node in ♊ 21° 13' 55''; the inclination of the orbit to the plane of the ecliptic 21° 18' 40''; the distance of its perihelion from the node, estimated in the comets orbit, 49° 27' 30'', its perihelion in ♌ 8° 40' 30'', with heliocentric latitude south[Pg 497] 16° 01' 45''; the comet to have been in its perihelion November 24d. 11h. 52' P.M. equal time at London, or 13h. 8' at Dantzick, O. S.; and that the latus rectum of the parabola was 410286 such parts as the sun's mean distance from the earth is supposed to contain 100000. And how nearly the places of the comet computed in this orbit agree with the observations, will appear from the annexed table, calculated by Dr. Halley.

Appar. Time

The observed Distances of the Comet from Observed Places. The Places

computed

in the Orb.

December ° ' '' ' '' ° ' ''

d. h. ' The Lion's heart 46.24.20 Long. ♎ 7.01.00 ♎ 7. 1.29

3.18.29 The Virgin's spike 22.52.10 Lat. S. 21.39. 0 21.38.50

4.18. 1 The Lion's heart 46. 2.45 Long. ♎ 6.15. 0 ♎ 6.16. 5

The Virgin's spike 23.52.40 Lat. S. 22.24. 0 22.24. 0

7.17.48 The Lion's heart 44.48. 0 Long. ♎ 3. 6. 0 ♎ 3. 7.33

The Virgin's spike 27.56.40 Lat. S. 25.22. 0 25.21.40

17.14.43 The Lion's heart 53.15.15 Long. ♌ 2.56. 0 ♌ 2.56. 0

Orion's right shoulder 45.43.30 Lat. S. 49.25. 0 49.25. 0

19. 9.25 Procyon 35.13.50 Long. ︎ 28.40.30 ♊ 28.43. 0

Bright star Whale's jaw 52.56. 0 Lat. S. 45.48. 0 45.46. 0

20. 9.53 Procyon 40.49. 0 Long. ♊ 13.03. 0 ♊ 13. 5. 0

Bright star Whale's jaw 40.04. 0 Lat. S. 39.54. 0 39.53. 0

21. 9. 9 Orion's right shoulder 26.21.25 Long. ♊ 2.16. 0 ♊ 2.18.30

Bright star Whale's jaw 29.28. 0 Lat. S. 33.41. 0 33.39.40

22. 9. 0 Orion's right shoulder 29.47. 0 Long. ♉ 24.24. 0 ♉ 24.27. 0

Bright star Whale's jaw 20.29.30 Lat. S. 27.45. 0 27.46. 0

26. 7.58 Bright star of Aries 23.20. 0 Long. ♉ 9. 0. 0 ♉. . 2.28

Aldebaran 26.44. 0 Lat. S. 12.36. 0 12.34.13

27. 6.45 Bright star of Aries 20.45. 0 Long. ♉ 7. 5.40 ♉. . 7. 8.45

Aldebaran 28.10. 0 Lat. S. 10.23. 0 10.23.13

28. 7.39 Bright star of Aries 18.29. 0 Long. ♉ 5.24.45 ♉. . 5.27.52

Palilicium 29.37. 0 Lat. S. 8.22.50 8.23.37

31. 6.45 Andromeda's girdle 30.48.10 Long. ♉ 2. 7.40 ♉. . 2. 8.20

Palilicium 32.53.30 Lat. S. 4.13. 0 4.16.25

Jan. 1665 Andromeda's girdle 25.11. 0 Long. ♈ 28.24.47 ︎ 28.24. 0

7. 7.37 Palilicium 37.12.25 Lat. N. 0.54. 0 0.53. 0

13. 7. 0 Andromeda's head 28. 7.10 Long. ♈ 27. 6.54 ♈ 27. 6.39

Palilicium 38.55.20 Lat. N. 3. 6.50 3. 7.40

24. 7.29 Andromeda's girdle 20.32.15 Long. ♈ 26.29.15 ♈ 26.28.50

Palilicium 40. 5. 0 Lat. N. 5.25.50 5.26. 0

Feb.

Long. ♈ 27. 4.46 ♈ 27.24.55

7. 8.37

Lat. N. 7. 3.29 7. 3.15

22. 8.46

Long. ♈ 28.29.46 ♈ 28.29.58

Lat. N. 8.12.36 8.10.25

March

Long. ♈. 29.18.15 ♈. 29.18.20

1.18.16

Lat. N. 8.36.26 8.36.12

7. 8.37

Long. ♈. . 0. 2.48 ♈ 0. 2.42

Lat. N. 8.56.30 8.56.56

In February, the beginning of the year 1665, the first star of Aries, which I shall hereafter call , was in ♈ 28° 30' 15'', with 7° 8' 58'' north[Pg 498] lat.; the second star of Aries was in ♈ 29° 17' 18'', with 8° 28' 16'' north lat.; and another star of the seventh magnitude, which I call A, was in ♈ 28° 24' 45'', with 8° 28' 33'' north lat. The comet Feb. 7d. 7h. 30' at Paris (that is, Feb. 7d. 8h. 37' at Dantzick) O. S. made a triangle with those stars and A, which was right-angled in ; and the distance of the comet from the star was equal to the distance of the stars and A, that is, 1° 19' 46'' of a great circle; and therefore in the parallel of the latitude of the star it was 1° 20' 26''. Therefore if from the longitude of the star there be subducted the longitude 1° 20' 26'', there will remain the longitude of the comet ♈ 27° 9' 49''. M. Auzout, from this observation of his, placed the comet in ♈ 27° 0', nearly; and, by the scheme in which Dr. Hooke delineated its motion, it was then in ♈ 26° 59' 24''. I place it in ♈ 27° 4' 46'', taking the middle between the two extremes.

From the same observations, M. Auzout made the latitude of the comet at that time 7° and 4' or 5' to the north; but he had done better to have made it 7° 3' 29'', the difference of the latitudes of the comet and the star being equal to the difference of the longitude of the stars and A.

February 22d. 7h. 30' at London, that is, February 22d. 8h. 46' at Dantzick, the distance of the comet from the star A, according to Dr. Hooke's observation, as was delineated by himself in a scheme, and also by the observations of M. Auzout, delineated in like manner by M. Petit, was a fifth part of the distance between the star A and the first star of Aries, or 15' 57''; and the distance of the comet from a right line joining the star A and the first of Aries was a fourth part of the same fifth part, that is, 4'; and therefore the comet was in ♈ 28° 29' 46'', with 8° 12' 36'' north lat.

March 1, 7h. 0' at London, that is, March 1, 8h. 16' at Dantzick, the comet was observed near the second star in Aries, the distance between them being to the distance between the first and second stars in Aries, that is, to 1° 33', as 4 to 45 according to Dr. Hooke, or as 2 to 23 according to M. Gottignies. And, therefore, the distance of the comet from the second star in Aries was 8' 16'' according to Dr. Hooke, or 8' 5'' according to M. Gottignies; or, taking a mean between both, 8' 10''. But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth part of the space that it commonly went over in a day, to wit, about 1' 35'' (in which he agrees very well with M. Auzout); or, according to Dr. Hooke, not quite so much, as perhaps only 1'. Wherefore if to the longitude of the first star in Aries we add 1', and 8' 10'' to its latitude, we shall have the longitude of the comet ♈ 29° 18', with 8° 36' 26'' north lat.

March 7, 7h. 30' at Paris (that is, March 7, 8h. 37' at Dantzick), from the observations of M. Auzout, the distance of the comet from the second star in Aries was equal to the distance of that star from the star[Pg 499] A, that is, 52' 29''; and the difference of the longitude of the comet and the second star in Aries was 45' or 46', or, taking a mean quantity, 45' 30''; and therefore the comet was in ♉ 0° 2' 48''. From the scheme of the observations of M. Auzout, constructed by M. Petit, Hevelius collected the latitude of the comet 8° 54'. But the engraver did not rightly trace the curvature of the comets way towards the end of the motion; and Hevelius, in the scheme of M. Auzout's observations which he constructed himself, corrected this irregular curvature, and so made the latitude of the comet 8° 55' 30''. And, by farther correcting this irregularity, the latitude may become 8° 56', or 8° 57'.

This comet was also seen March 9, and at that time its place must have been in ♉ 0° 18' with 9° north lat. nearly.

This comet appeared three months together, in which space of time it travelled over almost six signs, and in one of the days thereof described almost 20 deg. Its course did very much deviate from a great circle, bending towards the north, and its motion towards the end from retrograde became direct; and, notwithstanding its course was so uncommon, yet by the table it appears that the theory, from beginning to end, agrees with the observations no less accurately than the theories of the planets usually do with the observations of them; but we are to subduct about 2' when the comet was swiftest, which we may effect by taking off 12'' from the angle between the ascending node and the perihelion, or by making that angle 49° 27' 18''. The annual parallax of both these comets (this and the preceding) was very conspicuous, and by its quantity demonstrates the annual motion of the earth in the orbis magnus.

[Pg 500]

This theory is likewise confirmed by the motion of that comet, which in the year 1683 appeared retrograde, in an orbit whose plane contained almost a right angle with the plane of the ecliptic, and whose ascending node (by the computation of Dr. Halley) was in ♍ 23° 23'; the inclination of its orbit to the ecliptic 83° 11'; its perihelion in ♊ 25° 29' 30''; its perihelion distance from the sun 56020 of such parts as the radius of the orbis magnus contains 100000; and the time of its perihelion July 2d. 3h. 50'. And the places thereof, computed by Dr. Halley in this orbit, are compared with the places of the same observed by Mr. Flamsted, in the following table:—

1683

Eq. time Sun's place Comet's

Lon. com. Lat. Nor.

comput. Comet's

Long. obs'd. Lat. Nor.

observ'd. Diff.

Long. Diff.

Lat.

d. h. ' ° ' '' ° ' '' ° ' '' ° ' '' ° ' '' ' '' ' ''

July 13.12.55 ♌ 1.02.30 ♋ 13.05.42 29.28.13 ♋ 13. 6.42 29.28.20 +1.00 +0.07

15.11.15 2.53.12 11.37.48 29.34. 0 11.39.43 29.34.50 +1.55 +0.50

17.10.20 4.45.45 10. 7. 6 29.33.30 10. 8.40 29.34. 0 +1.34 +0.30

23.13.40 10.38.21 5.10.27 28.51.42 5.11.30 28.50.28 +1.03 -1.14

25.14. 5 12.35.28 3.27.53 24.24.47 3.27. 0 28.23.40 -0.53 -1.7

31. 9.42 18.09.22 ♊ 27.55. 3 26.22.52 ♊ 27.54.24 26.22.25 -0.39 -0.27

31.14.55 18.21.53 27.41. 7 26.16.57 27.41. 8 26.14.50 +0.1 -2.7

Aug. 2.14.56 20.17.16 25.29.32 25.16.19 25.28.46 25.17.28 -0.46 +1.9

4.10.49 22.02.50 23.18.20 24.10.49 23.16.55 24.12.19 -1.25 +1.30

6.10. 9 23.56.45 20.42.23 22.47. 5 20.40.32 22.49. 5 -1.51 +2.0

9.10.26 26.50.52 16. 7.57 20. 6.37 16. 5.55 20. 6.10 -2.2 -0.27

15.14. 1 ♍ 2.47.13 3.30.48 11.37.33 3.26.18 11.32. 1 -4.30 -5.32

16.15.10 3.48. 2 0.43. 7 9.34.16 0.41.55 9.34.13 -1.12 -0.3

18.15.44 5.45.33 ♉ 24.52.53 5.11.15 ♉ 24.49. 5 5. 9.11 -3.48 -2.4

South.

South.

22.14.44 9.35.49 11. 7.14 5.16.58 11.07.12 5.16.58 -0.2 -0.3

23.15.52 10.36.48 7. 2.18 8.17. 9 7. 1.17 8.16.41 -1.1 -0.28

26.16. 2 13.31.10 ♈ 24.45.31 16.38. 0 ♈ 24.44.00 16.38.20 -1.31 +0.20

This theory is yet farther confirmed by the motion of that retrograde comet which appeared in the year 1682. The ascending node of this (by Dr. Halley's computation) was in ♉ 21° 16' 30''; the inclination of its orbit to the plane of the ecliptic 17° 56' 00''; its perihelion in ♒ 2° 52' 50''; its perihelion distance from the sun 58328 parts, of which the radius of the orbis magnus contains 100000; the equal time of the comet's being in its perihelion Sept. 4d. 7h. 39'. And its places, collected from Mr. Flamsted's observations, are compared with its places computed from our theory in the following table:—

1682

App. time Sun's place Comet's

Lon. comp. Lat. Nor.

comp. Com. Long.

observed. Lat. Nor.

observ'd. Diff.

Long. Diff.

Lat.

d. h. ' ° ' '' ° ' '' ° ' '' ° ' '' ° ' '' ' '' ' ''

Aug 19.16.38 ♍︎ 7. 0. 7 ♌︎ 18.14.28 25.50. 7 ♌︎ 18.14.40 25.49.55 -0.12 +0.12

20.15.38 7.55.52 24.46.23 26.14.42 24.46.22 26.12.52 +0.1 +1.50

21. 8.21 8.36.14 29.37.15 26.20. 3 29.38.02 26.17.37 -0.47 +2.26

22. 8. 8 9.33.55 ♍︎ 6.29.53 26. 8.42 ♍︎ 6.30. 3 26. 7.12 -0.10 +1.30

29.08.20 16.22.40 ♎︎ 12.37.54 18.37.47 ♎︎ 12.37.49 18.34. 5 +0.5 +3.42

30. 7.45 17.19.41 15.36. 1 17.26.43 15.35.18 17.27.17 +0.43 -0.34

Sept. 1. 7.33 19.16. 9 20.30.53 15.13. 0 20.27. 4 15. 9.49 +3.49 +3.11

4. 7.22 22.11.28 25.42.0 12.23.48 25.40 58 12.22. 0 +1.2 +1.48

5. 7.32 23.10.29 27. 0.46 11.33.08 26.59.24 11.33.51 +1.22 -0.43

8. 7.16 26. 5.58 29.58.44 9.26.46 29.58.45 9.26.43 -0.1 +0.3

9. 7.26 27. 5. 9 ♏︎ 0.44. 10 8.49.10 ♏︎ 0. 44. 4 8.48.25 +0.6 +0.45

This theory is also confirmed by the retrograde motion of the comet that appeared in the year 1723. The ascending node of this comet (according to the computation of Mr. Bradley, Savilian Professor of Astronomy at Oxford) was in ♈ 14° 16'. The inclination of the orbit to the plane of the ecliptic 49° 59'. Its perihelion was in ♉ 12° 15' 20''. Its perihelion distance from the sun 998651 parts, of which the radius of the orbis magnus contains 1000000, and the equal [Pg 501]time of its perihelion September 16d 16h. 10'. The places of this comet computed in this orbit by Mr. Bradley, and compared with the places observed by himself, his uncle Mr. Pound, and Dr. Halley, may be seen in the following table.

1723

Eq. time Comet's

Long. obs. Lat. Nor.

obs. Comet's

Lon. com. Lat. Nor.

comp. Diff.

Long. Diff.

Lat.

d. h. ' ° ' '' ° ' '' ° ' '' ° ' '' '' ''

Oct. 9. 8. 5 ♒︎ 7.22.15 5. 2. 0 ♒︎ 7.21.26 5. 2.47 +49 -47

10. 6.21 6.41.12 7.44.13 6.41.42 7.43.18 -50 +55

12. 7.22 5.39.58 11.55. 0 5.40.19 11.54.55 -21 + 5

14. 8.57 4.59.49 14.43.50 5. 0.37 14.44. 1 -48 -11

15. 6.35 4.47.41 15.40.51 4.47.45 15.40.55 - 4 - 4

21. 6.22 4. 2.32 19.41.49 4. 2.21 19.42. 3 +11 -14

22. 6.24 3.59. 2 20. 8.12 3.59.10 20. 8.17 - 8 - 5

24. 8. 2 3.55.29 20.55.18 3.55.11 20.55. 9 +18 + 9

29. 8.56 3.56.17 22.20.27 3.56.42 22.20.10 -25 +17

30. 6.20 3.58. 9 22.32.28 3.58.17 22.32.12 - 8 +16

Nov. 5. 5.53 4.16.30 23.38.33 4.16.23 23.38. 7 + 7 +26

8. 7. 6 4.29.36 24. 4.30 4.29.54 24. 4.40 -18 -10

14. 6.20 5. 2.16 24.48.46 5. 2.51 24.48.16 -35 +30

20. 7.45 5.42.20 25.24.45 5.43.13 25.25.17 -53 -32

Dec. 7. 6.45 8. 4.13 26.54.18 8. 3.55 26.53.42 +18 +36

From these examples it is abundantly evident that the motions of comets are no less accurately represented by our theory than the motions of the planets commonly are by the theories of them; and, therefore, by means of this theory, we may enumerate the orbits of comets, and so discover the periodic time of a comet's revolution in any orbit; whence, at last, we shall have the transverse diameters of their elliptic orbits and their aphelion distances.

That retrograde comet which appeared in the year 1607 described an orbit whose ascending node (according to Dr. Halley's computation) was in ♉ 20° 21'; and the inclination of the plane of the orbit to the plane of the ecliptic 17° 2'; whose perihelion was in ♒ 2° 16'; and its perihelion distance from the sun 58680 of such parts as the radius of the orbis magnus contains 100000; and the comet was in its perihelion October 16d. 3h. 50'; which orbit agrees very nearly with the orbit of the comet which was seen in 1682. If these were not two different comets, but one and the same, that comet will finish one revolution in the space of 75 years; and the greater axis of its orbit will be to the greater axis of the orbis magnus as to 1, or as 1778 to 100, nearly. And the aphelion distance of this comet from the sun will be to the mean distance of the earth from the sun as about 35 to 1; from which data it will be no hard matter to determine the elliptic orbit of this comet. But these things are to be supposed on condition, that, after the space of 75 years, the same comet shall return again in the same orbit. The other comets seem to ascend to greater heights, and to require a longer time to perform their revolutions.

But, because of the great number of comets, of the great distance of their[Pg 502] aphelions from the sun, and of the slowness of their motions in the aphelions, they will, by their mutual gravitations, disturb each other; so that their eccentricities and the times of their revolutions will be sometimes a little increased, and sometimes diminished. Therefore we are not to expect that the same comet will return exactly in the same orbit, and in the same periodic times: it will be sufficient if we find the changes no greater than may arise from the causes just spoken of.

And hence a reason may be assigned why comets are not comprehended within the limits of a zodiac, as the planets are; but, being confined to no bounds, are with various motions dispersed all over the heavens; namely, to this purpose, that in their aphelions, where their motions are exceedingly slow, receding to greater distances one from another, they may suffer less disturbance from their mutual gravitations: and hence it is that the comets which descend the lowest, and therefore move the slowest in their aphelions, ought also to ascend the highest.

The comet which appeared in the year 1680 was in its perihelion less distant from the sun than by a sixth part of the sun's diameter; and because of its extreme velocity in that proximity to the sun, and some density of the sun's atmosphere, it must have suffered some resistance and retardation; and therefore, being attracted something nearer to the sun in every revolution, will at last fall down upon the body of the sun. Nay, in its aphelion, where it moves the slowest, it may sometimes happen to be yet farther retarded by the attractions of other comets, and in consequence of this retardation descend to the sun. So fixed stars, that have been gradually wasted by the light and vapours emitted from them for a long time, may be recruited by comets that fall upon them; and from this fresh supply of new fuel those old stars, acquiring new splendor, may pass for new stars. Of this kind are such fixed stars as appear on a sudden, and shine with a wonderful brightness at first, and afterwards vanish by little and little. Such was that star which appeared in Cassiopeia's chair; which Cornelius Gemma did not see upon the 8th of November, 1572, though he was observing that part of the heavens upon that very night, and the sky was perfectly serene; but the next night (November 9) he saw it shining much brighter than any of the fixed stars, and scarcely inferior to Venus in splendor. Tycho Brahe saw it upon the 11th of the same month, when it shone with the greatest lustre; and from that time he observed it to decay by little and little; and in 16 months' time it entirely disappeared. In the month of November, when it first appeared, its light was equal to that of Venus. In the month of December its light was a little diminished, and was now become equal to that of Jupiter. In January 1573 it was less than Jupiter, and greater than Sirius; and about the end of February and the beginning of March became equal to that star. In the months of April and May it was equal to a star of the second magnitude;[Pg 503] in June, July, and August, to a star of the third magnitude; in September, October, and November, to those of the fourth magnitude; in December and January 1574 to those of the fifth; in February to those of the sixth magnitude; and in March it entirely vanished. Its colour at the beginning was clear, bright, and inclining to white; afterwards it turned a little yellow; and in March 1573 it became ruddy, like Mars or Aldebaran: in May it turned to a kind of dusky whiteness, like that we observe in Saturn; and that colour it retained ever after, but growing always more and more obscure. Such also was the star in the right foot of Serpentarius, which Kepler's scholars first observed September 30, O.S. 1604, with a light exceeding that of Jupiter, though the night before it was not to be seen; and from that time it decreased by little and little, and in 15 or 16 months entirely disappeared. Such a new star appearing with an unusual splendor is said to have moved Hipparchus to observe, and make a catalogue of, the fixed stars. As to those fixed stars that appear and disappear by turns, and increase slowly and by degrees, and scarcely ever exceed the stars of the third magnitude, they seem to be of another kind, which revolve about their axes, and, having a light and a dark side, shew those two different sides by turns. The vapours which arise from the sun, the fixed stars, and the tails of the comets, may meet at last with, and fall into, the atmospheres of the planets by their gravity, and there be condensed and turned into water and humid spirits; and from thence, by a slow heat, pass gradually into the form of salts, and sulphurs, and tinctures, and mud, and clay, and sand, and stones, and coral, and other terrestrial substances.

GENERAL SCHOLIUM.

The hypothesis of vortices is pressed with many difficulties. That every planet by a radius drawn to the sun may describe areas proportional to the times of description, the periodic times of the several parts of the vortices should observe the duplicate proportion of their distances from the sun; but that the periodic times of the planets may obtain the sesquiplicate proportion of their distances from the sun, the periodic times of the parts of the vortex ought to be in the sesquiplicate proportion of their distances. That the smaller vortices may maintain their lesser revolutions about Saturn, Jupiter, and other planets, and swim quietly and undisturbed in the greater vortex of the sun, the periodic times of the parts of the sun's vortex should be equal; but the rotation of the sun and planets about their axes, which ought to correspond with the motions of their vortices, recede far from all these proportions. The motions of the comets are exceedingly regular, are governed by the same laws with the motions of the planets, and can by no means be accounted for by the hypothesis of vortices; for comets are carried with very eccentric motions through all parts of the[Pg 504] heavens indifferently, with a freedom that is incompatible with the notion of a vortex.

Bodies projected in our air suffer no resistance but from the air. Withdraw the air, as is done in Mr. Boyle's vacuum, and the resistance ceases; for in this void a bit of fine down and a piece of solid gold descend with equal velocity. And the parity of reason must take place in the celestial spaces above the earth's atmosphere; in which spaces, where there is no air to resist their motions, all bodies will move with the greatest freedom; and the planets and comets will constantly pursue their revolutions in orbits given in kind and position, according to the laws above explained; but though these bodies may, indeed, persevere in their orbits by the mere laws of gravity, yet they could by no means have at first derived the regular position of the orbits themselves from those laws.

The six primary planets are revolved about the sun in circles concentric with the sun, and with motions directed towards the same parts, and almost in the same plane. Ten moons are revolved about the earth, Jupiter and Saturn, in circles concentric with them, with the same direction of motion, and nearly in the planes of the orbits of those planets; but it is not to be conceived that mere mechanical causes could give birth to so many regular motions, since the comets range over all parts of the heavens in very eccentric orbits; for by that kind of motion they pass easily through the orbs of the planets, and with great rapidity; and in their aphelions, where they move the slowest, and are detained the longest, they recede to the greatest distances from each other, and thence suffer the least disturbance from their mutual attractions. This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One; especially since the light of the fixed stars is of the same nature with the light of the sun, and from every system light passes into all the other systems: and lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed those systems at immense distances one from another.

This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God παντοκράτωρ, or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or[Pg 505] my Perfect: these are titles which have no respect to servants. The word God[1] usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done. He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. He endures for ever, and is every where present; and by existing always and every where, he constitutes duration and space. Since every particle of space is always, and every indivisible moment of duration is every where, certainly the Maker and Lord of all things cannot be never and no where. Every soul that has perception is, though in different times and in different organs of sense and motion, still the same indivisible person. There are given successive parts in duration, co-existent parts in space, but neither the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole life, in all and each of his organs of sense. God is the same God, always and every where. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him[2] are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God exists necessarily; and by the same necessity he exists always and every where. Whence also he is all similar, all eye, all ear, all brain, all arm, all power to perceive, to understand, and to act; but in a manner not at all human, in a manner not at all corporeal, in a manner utterly unknown to us. As a blind man has no idea of colours, so have we no idea of the manner by[Pg 506] which the all-wise God perceives and understands all things. He is utterly void of all body and bodily figure, and can therefore neither be seen, nor heard, nor touched; nor ought he to be worshipped under the representation of any corporeal thing. We have ideas of his attributes, but what the real substance of any thing is we know not. In bodies, we see only their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savours; but their inward substances are not to be known either by our senses, or by any reflex act of our minds: much less, then, have we any idea of the substance of God. We know him only by his most wise and excellent contrivances of things, and final causes; we admire him for his perfections; but we reverence and adore him on account of his dominion: for we adore him as his servants; and a god without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind metaphysical necessity, which is certainly the same always and every where, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing. But, by way of allegory, God is said to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build; for all our notions of God are taken from the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to Natural Philosophy.

Hitherto we have explained the phænomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, without suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes use to do), but according to the quantity of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always in the duplicate proportion of the distances. Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed; and in receding from the sun decreases accurately in the duplicate proportion of the distances as far as the orb of Saturn, as evidently appears from the quiescence of the aphelions of the planets; nay, and even to the remotest aphelions of the comets, if those aphelions are also quiescent. But hitherto I have not been able to discover the cause of those properties of gravity from phænomena, and I frame no hypotheses; for whatever is not deduced from the phænomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental[Pg 507] philosophy. In this philosophy particular propositions are inferred from the phænomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, and the impulsive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.

And now we might add something concerning a certain most subtle Spirit which pervades and lies hid in all gross bodies; by the force and action of which Spirit the particles of bodies mutually attract one another at near distances, and cohere, if contiguous; and electric bodies operate to greater distances, as well repelling as attracting the neighbouring corpuscles; and light is emitted, reflected, refracted, inflected, and heats bodies; and all sensation is excited, and the members of animal bodies move at the command of the will, namely, by the vibrations of this Spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But these are things that cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elastic Spirit operates.

FOOTNOTES:

[1] Dr. Pocock derives the Latin word Deus from the Arabic du (in the oblique case di), which signifies Lord. And in this sense princes are called gods, Psal. lxxxii. ver. 6; and John x. ver. 35. And Moses is called a god to his brother Aaron, and a god to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the same sense the souls of dead princes were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

[2] This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i. Thales, Anaxagoras, Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses, in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters supposed the sun, moon, and stars, the souls of men, and other parts of the world, to be parts of the Supreme God, and therefore to be worshipped; but erroneously.

END OF THE MATHEMATICAL PRINCIPLES.

[Pg 508]

[Pg 509]

THE SYSTEM OF THE WORLD.

[Pg 511]

THE SYSTEM OF THE WORLD.

It was the ancient opinion of not a few, in the earliest ages of philosophy, that the fixed stars stood immoveable in the highest parts of the world; that under the fixed stars the planets were carried about the sun; that the earth, as one of the planets, described an annual course about the sun, while by a diurnal motion it was in the mean time revolved about its own axis; and that the sun, as the common fire which served to warm the whole, was fixed in the centre of the universe.

This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it.

The Egyptians were early observers of the heavens; and from them, probably, this philosophy was spread abroad among other nations; for from them it was, and the nations about them, that the Greeks, a people of themselves more addicted to the study of philology than of nature, derived their first, as well as soundest, notions of philosophy; and in the vestal ceremonies we may yet trace the ancient spirit of the Egyptians; for it was their way to deliver their mysteries, that is, their philosophy of things above the vulgar way of thinking, under the veil of religious rites and hieroglyphic symbols.

It is not to be denied but that Anaxagoras, Democritus, and others, did now and then start up, who would have it that the earth possessed the centre of the world, and that the stars of all sorts were revolved towards the west about the earth quiescent in the centre, some at a swifter, others at a slower rate.

However, it was agreed on both sides that the motions of the celestial bodies were performed in spaces altogether free and void of resistance. The whim of solid orbs was of a later date, introduced by Eudoxus, Calippus, and Aristotle; when the ancient philosophy began to decline, and to give place to the new prevailing fictions of the Greeks.

But, above all things, the phænomena of comets can by no means consist[Pg 512] with the notion of solid orbs. The Chaldeans, the most learned astronomers of their time, looked upon the comets (which of ancient times before had been numbered among the celestial bodies) as a particular sort of planets, which, describing very eccentric orbits, presented themselves to our view only by turns, viz., once in a revolution, when they descended into the lower parts of their orbits.

And as it was the unavoidable consequence of the hypothesis of solid orbs, while it prevailed, that the comets should be thrust down below the moon, so no sooner had the late observations of astronomers restored the comets to their ancient places in the higher heavens, but these celestial spaces were at once cleared of the incumbrance of solid orbs, which by these observations were broke into pieces, and discarded for ever.

Whence it was that the planets came to be retained within any certain bounds in these free spaces, and to be drawn off from the rectilinear courses, which, left to themselves, they should have pursued, into regular revolutions in curvilinear orbits, are questions which we do not know how the ancients explained; and probably it was to give some sort of satisfaction to this difficulty that solid orbs were introduced.

The later philosophers pretend to account for it either by the action of certain vortices, as Kepler and Des Cartes; or by some other principle of impulse or attraction, as Borelli, Hooke, and others of our nation; for, from the laws of motion, it is most certain that these effects must proceed from the action of some force or other.

But our purpose is only to trace out the quantity and properties of this force from the phænomena (p. 218), and to apply what we discover in some simple cases as principles, by which, in a mathematical way, we may estimate the effects thereof in more involved cases; for it would be endless and impossible to bring every particular to direct and immediate observation.

We said, in a mathematical way, to avoid all questions about the nature or quality of this force, which we would not be understood to determine by any hypothesis; and therefore call it by the general name of a centripetal force, as it is a force which is directed towards some centre; and as it regards more particularly a body in that centre, we call it circum-solar, circum-terrestrial, circum-jovial; and in like manner in respect of other central bodies.

That by means of centripetal forces the planets may be retained in certain orbits, we may easily understand, if we consider the motions of projectiles (p. 75, 76, 77); for a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground; and the greater the velocity is with which it is projected, the farther it goes before it falls to the earth. We may therefore suppose the velocity to be so increased,[Pg 513] that it would describe an arc of 1, 2, 5,10, 100, 1000 miles before it arrived at the earth, till at last, exceeding the limits of the earth, it should pass quite by without touching it.

Let AFB represent the surface of the earth, C its centre, VD, VE, VF, the curve lines which a body would describe, if projected in an horizontal direction from the top of an high mountain successively with more and more velocity (p. 400); and, because the celestial motions are scarcely retarded by the little or no resistance of the spaces in which they are performed, to keep up the parity of cases, let us suppose either that there is no air about the earth, or at least that it is endowed with little or no power of resisting; and for the same reason that the body projected with a less velocity describes the lesser arc VD, and with a greater velocity the greater arc VE, and, augmenting the velocity, it goes farther and farther to F and G, if the velocity was still more and more augmented, it would reach at last quite beyond the circumference of the earth, and return to the mountain from which it was projected.

And since the areas which by this motion it describes by a radius drawn to the centre of the earth are (by Prop. 1, Book I, Princip. Math.) proportional to the times in which they are described, its velocity, when it returns to the mountain, will be no less than it was at first; and, retaining the same velocity, it will describe the same curve over and over, by the same law.

[Pg 514]

But if we now imagine bodies to be projected in the directions of lines parallel to the horizon from greater heights, as of 5, 10, 100, 1000, or more miles, or rather as many semi-diameters of the earth, those bodies, according to their different velocity, and the different force of gravity in different heights, will describe arcs either concentric with the earth, or variously eccentric, and go on revolving through the heavens in those trajectories, just as the planets do in their orbs.

As when a stone is projected obliquely, that is, any way but in the perpendicular direction, the perpetual deflection thereof towards the earth from the right line in which it was projected is a proof of its gravitation to the earth, no less certain than its direct descent when only suffered to fall freely from rest; so the deviation of bodies moving in free spaces from rectilinear paths, and perpetual deflection therefrom towards any place, is a sure indication of the existence of some force which from all quarters impels those bodies towards that place.

And as, from the supposed existence of gravity, it necessarily follows that all bodies about the earth must press downwards, and therefore must either descend directly to the earth, if they are let fall from rest, or at least perpetually deviate from right lines towards the earth if they are projected obliquely; so from the supposed existence of a force directed to any centre, it will follow, by the like necessity, that all bodies upon which this force acts must either descend directly to that centre, or at least deviate perpetually towards it from right lines, if otherwise they should have moved obliquely in these right lines.

And how from the motions given we may infer the forces, or from the forces given we may determine the motions, is shewn in the two first Books of our Principles of Philosophy.

If the earth is supposed to stand still, and the fixed stars to be revolved in free spaces in the space of 24 hours, it is certain the forces by which the fixed stars are retained in their orbs are not directed to the earth, but to the centres of the several orbs, that is, of the several parallel circles, which the fixed stars, declining to one side and the other from the equator, describe daily; also that by radii drawn to the centres of those orbs the fixed stars describe areas exactly proportional to the times of description. Then, because the periodic times are equal (by Cor. III, Prop. IV, Book I), it follows that the centripetal forces are as the radii of the several orbs, and that they will perpetually revolve in the same orbs. And the like consequences may be drawn from the supposed diurnal motion of the planets.

That forces should be directed to no body on which they physically depend, but to innumerable imaginary points in the axis of the earth, is an hypothesis too incongruous. It is more incongruous still that those forces should increase exactly in proportion of the distances from this axis; for[Pg 515] this is an indication of an increase to immensity, or rather to infinity; whereas the forces of natural things commonly decrease in receding from the fountain from which they flow. But, what is yet more absurd, neither are the areas described by the same star proportional to the times, nor are its revolutions performed in the same orb; for as the star recedes from the neighbouring pole, both areas and orb increase; and from the increase of the area it is demonstrated that the forces are not directed to the axis of the earth. And this difficulty (Cor. 1, Prop. II) arises from the twofold motion that is observed in the fixed stars, one diurnal round the axis of the earth, the other exceedingly slow round the axis of the ecliptic. And the explication thereof requires a composition of forces so perplexed and so variable, that it is hardly to be reconciled with any physical theory.

That there are centripetal forces actually directed to the bodies of the sun, of the earth, and other planets, I thus infer.

The moon revolves about our earth, and by radii drawn to its centre (p. 390) describes areas nearly proportional to the times in which they are described, as is evident from its velocity compared with its apparent diameter; for its motion is slower when its diameter is less (and therefore its distance greater), and its motion is swifter when its diameter is greater.

The revolutions of the satellites of Jupiter about that planet are more regular (p. 386); for they describe circles concentric with Jupiter by equable motions, as exactly as our senses can distinguish.

And so the satellites of Saturn are revolved about this planet with motions nearly (p. 387) circular and equable, scarcely disturbed by any eccentricity hitherto observed.

That Venus and Mercury are revolved about the sun, is demonstrable from their moon-like appearances (p. 388); when they shine with a full face, they are in those parts of their orbs which in respect of the earth lie beyond the sun; when they appear half full, they are in those parts which lie over against the sun; when horned, in those parts which lie between the earth and the sun; and sometimes they pass over the sun's disk, when directly interposed between the earth and the sun.

And Venus, with a motion almost uniform, describes an orb nearly circular and concentric with the sun.

But Mercury, with a more eccentric motion, makes remarkable approaches to the sun, and goes off again by turns; but it is always swifter as it is near to the sun, and therefore by a radius drawn to the sun still describes areas proportional to the times.

Lastly, that the earth describes about the sun, or the sun about the earth, by a radius from the one to the other, areas exactly proportional to the times, is demonstrable from the apparent diameter of the sun compared with its apparent motion.

These are astronomical experiments; from which it follows, by Prop. I,[Pg 516] II, III, in the first Book of our Principles, and their Corollaries (p. 212, 213, 214), that there are centripetal forces actually directed (either accurately or without considerable error) to the centres of the earth, of Jupiter, of Saturn, and of the sun. In Mercury, Venus, Mars, and the lesser planets, where experiments are wanting, the arguments from analogy must be allowed in their place.

That those forces (p. 212, 213, 214) decrease in the duplicate proportion of the distances from the centre of every planet, appears by Cor. VI, Prop. IV, Book I; for the periodic times of the satellites of Jupiter are one to another (p. 386, 387) in the sesquiplicate proportion of their distances from the centre of this planet.

This proportion has been long ago observed in those satellites; and Mr. Flamsted, who had often measured their distances from Jupiter by the micrometer, and by the eclipses of the satellites, wrote to me, that it holds to all the accuracy that possibly can be discerned by our senses. And he sent me the dimensions of their orbits taken by the micrometer, and reduced to the mean distance of Jupiter from the earth, or from the sun, together with the times of their revolutions, as follows:—

The greatest elongation

of the satellites

from the centre

of Jupiter as

seen from the sun. The periodic

times of their

revolutions.

' '' '' d h ' ''

1st 1 48 or 108 1 18 28 36

2d 3 01 or 181 3 13 17 54

3d 4 46 or 286 7 03 59 36

4th 8 13 or 493 16 18 5 13

Whence the sesquiplicate proportion may be easily seen. For example; the 16d. 18h. 05' 13'' is to the time 1d. 18h. 28' 36'' as to , neglecting those small fractions which, in observing, cannot be certainly determined.

Before the invention of the micrometer, the same distances were determined in semi-diameters of Jupiter thus:—

Distance of the 1st 2d 3d 4th

By Galileo, 6 10 16 28

" Simon Marius 6 10 16 26

" Cassini 5 8 13 23

" Borelli, more exactly 5 8 14 24

After the invention of the micrometer:—

By Townley 5,51 8,78 13,47 24,72

" Flamsted 5,31 8,85 13,98 24,23

More accurately by

the eclipses 5,578 8,876 14,159 24,903

[Pg 517]

And the periodic times of those satellites, by the observations of Mr. Flamsted, are 1d. 18h. 28' 36'' | 3d. 13". 17' 54'' | 7d. 3h. 59' 36'' | 16d. 18h. 5' 13'', as above.

And the distances thence computed are 5,578 | 8,878 | 14,168 | 24,968, accurately agreeing with the distances by observation.

Cassini assures us (p. 388, 389) that the same proportion is observed in the circum-saturnal planets. But a longer course of observations is required before we can have a certain and accurate theory of those planets.

In the circum-solar planets, Mercury and Venus, the same proportion holds with great accuracy, according to the dimensions of their orbs, as determined by the observations of the best astronomers.

That Mars is revolved about the sun is demonstrated from the phases which it shews, and the proportion of its apparent diameters (p. 388, 389, and 390); for from its appearing full near conjunction with the sun, and gibbous in its quadratures, it is certain that it surrounds the sun.

And since its diameter appears about five times greater when in opposition to the sun than when in conjunction therewith, and its distance from the earth is reciprocally as its apparent diameter, that distance will be about five times less when in opposition to than when in conjunction with the sun; but in both cases its distance from the sun will be nearly about the same with the distance which is inferred from its gibbous appearance in the quadratures. And as it encompasses the sun at almost equal distances, but in respect of the earth is very unequally distant, so by radii drawn to the sun it describes areas nearly uniform; but by radii drawn to the earth, it is sometimes swift, sometimes stationary, and sometimes retrograde.

That Jupiter, in a higher orb than Mars, is likewise revolved about the sun, with a motion nearly equable, as well in distance as in the areas described, I infer thus.

Mr. Flamsted assured me, by letters, that all the eclipses of the innermost satellite which hitherto have been well observed do agree with his theory so nearly, as never to differ therefrom by two minutes of time; that in the outmost the error is little greater; in the outmost but one, scarcely three times greater; that in the innermost but one the difference is indeed much greater, yet so as to agree as nearly with his computations as the moon does with the common tables; and that he computes those eclipses only from the mean motions corrected by the equation of light discovered and introduced by Mr. Romer. Supposing, then, that the theory differs by a less error than that of 2' from the motion of the outmost satellite as hitherto described, and taking as the periodic time 16d. 18h. 5' 13'' to 2' in time, so is the whole circle or 360° to the arc 1' 48'', the error of Mr. Flamsted's computation, reduced to the satellite's orbit, will be less than 1' 48''; that is, the longitude of the satellite, as seen from the centre of Jupiter, will be determined with a less error than 1' 48''. But when[Pg 518] the satellite is in the middle of the shadow, that longitude is the same with the heliocentric longitude of Jupiter; and, therefore, the hypothesis which Mr. Flamsted follows, viz., the Copernican, as improved by Kepler, and (as to the motion of Jupiter) lately corrected by himself, rightly represents that longitude within a less error than 1' 48''; but by this longitude, together with the geocentric longitude, which is always easily found, the distance of Jupiter from the sun is determined; which must, therefore, be the very same with that which the hypothesis exhibits. For that greatest error of 1' 48'' that can happen in the heliocentric longitude is almost insensible, and quite to be neglected, and perhaps may arise from some yet undiscovered eccentricity of the satellite; but since both longitude and distance are rightly determined, it follows of necessity that Jupiter, by radii drawn to the sun, describes areas so conditioned as the hypothesis requires, that is, proportional to the times.

And the same thing may be concluded of Saturn from his satellite, by the observations of Mr. Huygens and Dr. Halley; though a longer series of observations is yet wanting to confirm the thing, and to bring it under a sufficiently exact computation.

For if Jupiter was viewed from the sun, it would never appear retrograde nor stationary, as it is seen sometimes from the earth, but always to go forward with a motion nearly uniform (p. 389). And from the very great inequality of its apparent geocentric motion, we infer (by Prop. III Cor. IV) that the force by which Jupiter is turned out of a rectilinear course, and made to revolve in an orb, is not directed to the centre of the earth. And the same argument holds good in Mars and in Saturn. Another centre of these forces is therefore to be looked for (by Prop. II and III, and the Corollaries of the latter), about which the areas described by radii intervening may be equable; and that this is the sun, we have proved already in Mars and Saturn nearly, but accurately enough in Jupiter. It may be alledged that the sun and planets are impelled by some other force equally and in the direction of parallel lines; but by such a force (by Cor. VI of the Laws of Motion) no change would happen in the situation of the planets one to another, nor any sensible effect follow: but our business is with the causes of sensible effects. Let us, therefore, neglect every such force as imaginary and precarious, and of no use in the phænomena of the heavens; and the whole remaining force by which Jupiter is impelled will be directed (by Prop. III, Cor. I) to the centre of the sun.

The distances of the planets from the sun come out the same, whether, with Tycho, we place the earth in the centre of the system, or the sun with Copernicus: and we have already proved that these distances are true in Jupiter.

Kepler and Bullialdus have, with great care (p. 388), determined the distances of the planets from the sun; and hence it is that their tables[Pg 519] agree best with the heavens. And in all the planets, in Jupiter and Mars, in Saturn and the earth, as well as in Venus and Mercury, the cubes of their distances are as the squares of their periodic times; and therefore (by Cor. VI, Prop. IV) the centripetal circum-solar force throughout all the planetary regions decreases in the duplicate proportion of the distances from the sun. In examining this proportion, we are to use the mean distances, or the transverse semi-axes of the orbits (by Prop. XV), and to neglect those little fractions, which, in defining the orbits, may have arisen from the insensible errors of observation, or may be ascribed to other causes which we shall afterwards explain. And thus we shall always find the said proportion to hold exactly; for the distances of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury, from the sun, drawn from the observations of astronomers, are, according to the computation of Kepler, as the numbers 951000, 519650, 152350, 100000, 72400, 38806; by the computation of Bullialdus, as the numbers 954198, 522520, 152350, 100000, 72398, 38585; and from the periodic times they come out 953806, 520116, 152399, 100000, 72333, 38710. Their distances, according to Kepler and Bullialdus, scarcely differ by any sensible quantity, and where they differ most the distances drawn from the periodic times, fall in between them.

That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, ; according to Tycho, ; to Vendelin, 60; to Copernicus, ; to Kircher, (p. 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed those of the fixed stars, and that by about four or five minutes in the horizon, did thereby augment the horizontal parallax of the moon by about the like number of minutes; that is, by about the 12th or 15th part of the whole parallax. Correct this error, and the distance will become 60 or 61 semi-diameters of the earth, nearly agreeing with what others have determined.

Let us, then, assume the mean distance of the moon 60 semi-diameters of the earth, and its periodic time in respect of the fixed stars 27d. 7h. 43', as astronomers have determined it. And (by Cor. VI, Prop. IV) a body revolved in our air, near the surface of the earth supposed at rest, by means of a centripetal force which should be to the same force at the distance of the moon in the reciprocal duplicate proportion of the distances from the centre of the earth, that is, as 3600 to 1, would (secluding the resistance of the air) complete a revolution in 1h. 24' 27''.

Suppose the circumference of the earth to be 123249600 Paris feet, as[Pg 520] has been determined by the late mensuration of the French (vide p. 406); then the same body, deprived of its circular motion, and falling by the impulse of the same centripetal force as before, would, in one second of time, describe Paris feet.

This we infer by a calculus formed upon Prop. XXXVI, and it agrees with what we observe in all bodies about the earth. For by the experiments of pendulums, and a computation raised thereon, Mr. Huygens has demonstrated that bodies falling by all that centripetal force with which (of whatever nature it is) they are impelled near the surface of the earth, do, in one second of time, describe Paris feet.

But if the earth is supposed to move, the earth and moon together (by Cor. IV of the Laws of Motion, and Prop. LVII) will be revolved about their common centre of gravity. And the moon (by Prop. LX) will in the same periodic time, 27d. 7h. 43', with the same circum-terrestrial force diminished in the duplicate proportion of the distance, describe an orbit whose semi-diameter is to the semi-diameter of the former orbit, that is, to 60 semi-diameters of the earth, as the sum of both the bodies of the earth and moon to the first of two mean proportionals between this sum and the body of the earth; that is, if we suppose the moon (on account of its mean apparent diameter ) to be about of the earth, as 43 to , or as about 128 to 127. And therefore the semi-diameter of the orbit, that is, the distance between the centres of the moon and earth, will in this case be semi-diameters of the earth, almost the same with that assigned by Copernicus, which the Tychonic observations by no means disprove; and, therefore, the duplicate proportion of the decrement of the force holds good in this distance. I have neglected the increment of the orbit which arises from the action of the sun as inconsiderable; but if that is subducted, the true distance will remain about semi-diameters of the earth.

But farther (p. 390); this proportion of the decrement of the forces is confirmed from the eccentricity of the planets, and the very slow motion of their apses; for (by the Corollaries of Prop. XLV) in no other proportion could the circum-solar planets once in every revolution descend to their least and once ascend to their greatest distance from the sun, and the places of those distances remain immoveable. A small error from the duplicate proportion would produce a motion of the apses considerable in every revolution, but in many enormous.

BOOK III. (part 5)

But now, after innumerable revolutions, hardly any such motion has been perceived in the orbs of the circum-solar planets. Some astronomers affirm that there is no such motion; others reckon it no greater than what may easily arise from the causes hereafter to be assigned, and is of no moment in the present question.

[Pg 521]

We may even neglect the motion of the moon's apsis (p. 390, 391), which is far greater than in the circum-solar planets, amounting in every revolution to three degrees; and from this motion it is demonstrable that the circum-terrestrial force decreases in no less than the duplicate, but far less than the triplicate proportion of the distance; for if the duplicate proportion was gradually changed into the triplicate, the motion of the apsis would thereby increase to infinity; and, therefore, by a very small mutation, would exceed the motion of the moon's apsis. This slow motion arises from the action of the circum-solar force, as we shall afterwards explain. But, secluding this cause, the apsis or apogeon of the moon will be fixed, and the duplicate proportion of the decrease of the circum-terrestrial force in different distances from the earth will accurately take place.

Now that this proportion has been established, we may compare the forces of the several planets among themselves (p. 391).

In the mean distance of Jupiter from the earth, the greatest elongation of the outmost satellite from Jupiter's centre (by the observations of Mr. Flamsted) is 8' 13''; and therefore the distance of the satellite from the centre of Jupiter is to the mean distance of Jupiter from the centre of the sun as 124 to 52012, but to the mean distance of Venus from the centre of the sun as 124 to 7234; and their periodic times are and ; and from hence (according to Cor. II, Prop. IV), dividing the distances by the squares of the times, we infer that the force by which the satellite is impelled towards Jupiter is to the force by which Venus is impelled towards the sun as 442 to 143; and if we diminish the force by which the satellite is impelled in the duplicate proportion of the distance 124 to 7234, we shall have the circum-jovial force in the distance of Venus from the sun to the circum-solar force by which Venus is impelled as to 143, or as 1 to 1100; wherefore at equal distances the circum-solar force is 1100 times greater than the circum-jovial.

And, by the like computation, from the periodic time of the satellite of Saturn 15d. 22h. and its greatest elongation from Saturn, while that planet is in its mean distance from us, 3' 20'', it follows that the distance of this satellite from Saturn's centre is to the distance of Venus from the sun as to 7234; and from thence that the absolute circum-solar force is 2360 times greater than the absolute circum-saturnal.

From the regularity of the heliocentric and irregularity of the geocentric motions of Venus, of Jupiter, and the other planets, it is evident (by Cor. IV, Prop. III) that the circum-terrestrial force, compared with the circum-solar, is very small.

Ricciolus and Vendelin have severally tried to determine the sun's parallax from the moon's dichotomies observed by the telescope, and they agree that it does not exceed half a minute.

Kepler, from Tycho's observations and his own, found the parallax of[Pg 522] Mars insensible, even in opposition to the sun, when that parallax is something greater than the sun's.

Flamsted attempted the same parallax with the micrometer in the perigeon position of Mars, but never found it above 25''; and thence concluded the sun's parallax at most 10''.

Whence it follows that the distance of the moon from the earth bears no greater proportion to the distance of the earth from the sun than 29 to 10000; nor to the distance of Venus from the sun than 29 to 7233.

From which distances, together with the periodic times, by the method above explained, it is easy to infer that the absolute circum-solar force is greater than the absolute circum-terrestrial force at least 229400 times.

And though we were only certain, from the observations of Ricciolus and Vendelin, that the sun's parallax was less than half a minute, yet from this it will follow that the absolute circum-solar force exceeds the absolute circum-terrestrial force 8500 times.

By the like computations I happened to discover an analogy, that is observed between the forces and the bodies of the planets; but, before I explain this analogy, the apparent diameters of the planets in their mean distances from the earth must be first determined.

Mr. Flamsted (p. 387), by the micrometer, measured the diameter of Jupiter 40'' or 41''; the diameter of Saturn's ring 50''; and the diameter of the sun about 32' 13'' (p. 387).

But the diameter of Saturn is to the diameter of the ring, according to Mr. Huygens and Dr. Halley, as 4 to 9; according to Galletius, as 4 to 10; and according to Hooke (by a telescope of 60 feet), as 5 to 12. And from the mean proportion, 5 to 12, the diameter of Saturn's body is inferred about 21''.

Such as we have said are the apparent magnitudes; but, because of the unequal refrangibility of light, all lucid points are dilated by the telescope, and in the focus of the object-glass possess a circular space whose breadth is about the 50th part of the aperture of the glass.

It is true, that towards the circumference the light is so rare as hardly to move the sense; but towards the middle, where it is of greater density, and is sensible enough, it makes a small lucid circle, whose breadth varies according to the splendor of the lucid point, but is generally about the 3d, or 4th, or 5th part of the breadth of the whole.

Let ABD represent the circle of the whole light; PQ the small circle of the denser and clearer light; C the centre of both; CA, CB, semi-diameters of the greater circle containing a right angle at C; ACBE the square comprehended under these semi-diameters; AB the diagonal of that square; EGH an hyperbola with the centre C and asymptotes CA, CB; PG a perpendicular erected from any point P of the line BC, and meeting the hyperbola in G, and the right lines AB, AE, in K and F: and the[Pg 523] density of the light in any place P, will, by my computation, be as the line FG, and therefore at the centre infinite, but near the circumference very small. And the whole light within the small circle PQ, is to the whole without as the area of the quadrilateral figure CAKP to the triangle PKB. And we are to understand the small circle PQ to be there terminated, where FG, the density of the light, begins to be less than what is required to move the sense.

Hence it was, that, at the distance of 191382 feet, a fire of 3 feet in diameter, through a telescope of 3 feet, appeared to Mr. Picart of 8'' in breadth, when it should have appeared only of 3'' 14'''; and hence it is that the brighter fixed stars appear through the telescope as of 5'' or 6'' in diameter, and that with a good full light; but with a fainter light they appear to run out to a greater breadth. Hence, likewise, it was that Hevelius, by diminishing the aperture of the telescope, did cut off a great part of the light towards the circumference, and brought the disk of the star to be more distinctly defined, which, though hereby diminished, did yet appear as of 5'' or 6'' in diameter. But Mr. Huygens, only by clouding the eye-glass with a little smoke, did so effectually extinguish this scattered light, that the fixed stars appeared as mere points, void of all sensible breadth. Hence also it was that Mr. Huygens, from the breadth of bodies interposed to intercept the whole light of the planets, reckoned their diameters greater than others have measured them by the micrometer; for the[Pg 524] scattered light, which could not be seen before for the stronger light of the planet, when the planet is hid, appears every way farther spread. Lastly, from hence it is that the planets appear so small in the disk of the sun being lessened by the dilated light. For to Hevelius, Galletius, and Dr. Halley, Mercury did not seem to exceed 12'' or 15''; and Venus appeared to Mr. Crabtrie only 1' 3''; to Horrox but 1' 12''; though by the mensurations of Hevelius and Hugenius without the sun's disk, it ought to have been seen at least 1' 24''. Thus the apparent diameter of the moon, which in 1684, a few days both before and after the sun's eclipse, was measured at the observatory of Paris 31' 30'', in the eclipse itself did not seem to exceed 30' or 30' 05''; and therefore the diameters of the planets are to be diminished when without the sun, and to be augmented when within it, by some seconds. But the errors seem to be less than usual in the mensurations that are made by the micrometer. So from the diameter of the shadow, determined by the eclipses of the satellites, Mr. Flamsted found that the semi-diameter of Jupiter was to the greatest elongation of the outmost satellite as 1 to 24,903. Wherefore since that elongation is 8' 13'', the diameter of Jupiter will be ; and, rejecting the scattered light, the diameter found by the micrometer 40'' or 41'' will be reduced to ; and the diameter of Saturn 21'' is to be diminished by the like correction, and to be reckoned 20'', or something less. But (if I am not mistaken) the diameter of the sun, because of its stronger light, is to be diminished something more, and to be reckoned about 32', or 32' 6''.

That bodies so different in magnitude should come so near to an analogy with their forces, is not without some mystery (p. 400).

It may be that the remoter planets, for want of heat, have not those metallic substances and ponderous minerals with which our earth abounds; and that the bodies of Venus and Mercury, as they are more exposed to the sun's heat, are also harder baked, and more compact.

For, from the experiment of the burning-glass, we see that the heat increases with the density of light; and this density increases in the reciprocal duplicate proportion of the distance from the sun; from whence the sun's heat in Mercury is proved to be sevenfold its heat in our summer seasons. But with this heat our water boils; and those heavy fluids, quicksilver and the spirit of vitriol, gently evaporate, as I have tried by the thermometer; and therefore there can be no fluids in Mercury but what are heavy, and able to bear a great heat, and from which substances of great density may be nourished.

And why not, if God has placed different bodies at different distances from the sun, so as the denser bodies always possess the nearer places, and each body enjoys a degree of heat suitable to its condition, and proper for its nourishment? From this consideration it will best appear that the weights of all the planets are one to another as their forces.

[Pg 525]

But I should be glad the diameters of the planets were more accurately measured; and that may be done, if a lamp, set at a great distance, is made to shine through a circular hole, and both the hole and the light of the lamp are so diminished that the spectrum may appear through the telescope just like the planet, and may be defined by the same measure: then the diameter of the hole will be to its distance from the objective glass as the true diameter of the planet to its distance from us. The light of the lamp may be diminished by the interposition either of pieces of cloth, or of smoked glass.

Of kin to the analogy we have been describing, there is another observed between the forces and the bodies attracted (p. 395, 396, p. 397). Since the action of the centripetal force upon the planets decreases in the duplicate proportion of the distance, and the periodic time increases in the sesquiplicate thereof, it is evident that the actions of the centripetal force, and therefore the periodic times, would be equal in equal planets at equal distances from the sun; and in equal distances of unequal planets the total actions of the centripetal force would be as the bodies of the planets; for if the actions were not proportional to the bodies to be moved, they could not equally retract these bodies from the tangents of their orbs in equal times: nor could the motions of the satellites of Jupiter be so regular, if it was not that the circum-solar force was equally exerted upon Jupiter and all its satellites in proportion of their several weights. And the same thing is to be said of Saturn in respect of its satellites, and of our earth in respect of the moon, as appears from Cor. II and III, Prop. LXV. And, therefore, at equal distances, the actions of the centripetal force are equal upon all the planets in proportion of their bodies, or of the quantities of matter in their several bodies; and for the same reason must be the same upon all the particles of the same size of which the planet is composed; for if the action was greater upon some sort of particles than upon others than in proportion to their quantity of matter, it would be also greater or less upon the whole planets not in proportion to the quantity only, but likewise of the sort of the matter more copiously found in one and more sparingly in another.

In such bodies as are found on our earth of very different sorts, I examined this analogy with great accuracy (p. 343, 344).

If the action of the circum-terrestrial force is proportional to the bodies to be moved, it will (by the Second Law of Motion) move them with equal velocity in equal times, and will make all bodies let fall to descend through equal spaces in equal times, and all bodies hung by equal threads to vibrate in equal times. If the action of the force was greater, the times would be less; if that was less, these would be greater.

But it has been long ago observed by others, that (allowance being made for the small resistance of the air) all bodies descend through equal spaces[Pg 526] in equal times; and, by the help of pendulums, that equality of times may be distinguished to great exactness.

I tried the thing in gold, silver, lead, glass, sand, common salt wood, water, and wheat. I provided two equal wooden boxes. I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes, hung by equal threads of 11 feet, made a couple of pendulums perfectly equal in weight and figure, and equally exposed to the resistance of the air: and, placing the one by the other, I observed them to play together forwards and backwards for a long while, with equal vibrations. And therefore (by Cor. I and VI, Prop. XXIV, Book II) the quantity of matter in the gold was to the quantity of matter in the wood as the action of the motive force upon all the gold to the action of the same upon all the wood; that is, as the weight of the one to the weight of the other.

And by these experiments, in bodies of the same weight, could have discovered a difference of matter less than the thousandth part of the whole.

Since the action of the centripetal force upon the bodies attracted is, at equal distances, proportional to the quantities of matter in those bodies, reason requires that it should be also proportional to the quantity of matter in the body attracting.

For all action is mutual, and (p. 83, 93, by the Third Law of Motion) makes the bodies mutually to approach one to the other, and therefore must be the same in both bodies. It is true that we may consider one body as attracting, another as attracted; but this distinction is more mathematical than natural. The attraction is really common of either to other, and therefore of the same kind in both.

And hence it is that the attractive force is found in both. The sun attracts Jupiter and the other planets; Jupiter attracts its satellites; and, for the same reason, the satellites act as well one upon another as upon Jupiter, and all the planets mutually one upon another.

And though the mutual actions of two planets may be distinguished and considered as two, by which each attracts the other, yet, as those actions are intermediate, they do not make two but one operation between two terms. Two bodies may be mutually attracted each to the other by the contraction of a cord interposed. There is a double cause of action, to wit, the disposition of both bodies, as well as a double action in so far as the action is considered as upon two bodies; but as betwixt two bodies it is but one single one. It is not one action by which the sun attracts Jupiter, and another by which Jupiter attracts the sun; but it is one action by which the sun and Jupiter mutually endeavour to approach each the other. By the action with which the sun attracts Jupiter, Jupiter and the sun endeavours to come nearer together (by the Third Law of Motion); and by the action with which Jupiter attracts the sun, likewise Jupiter[Pg 527] and the sun endeavor to come nearer together. But the sun is not attracted towards Jupiter by a twofold action, nor Jupiter by a twofold action towards the sun; but it is one single intermediate action, by which both approach nearer together.

Thus iron draws the load-stone (p. 93), as well as the load-stone draws the iron; for all iron in the neighbourhood of the load-stone draws other iron. But the action betwixt the load-stone and iron is single, and is considered as single by the philosophers. The action of iron upon the load-stone, is, indeed, the action of the load-stone betwixt itself and the iron, by which both endeavour to come nearer together: and so it manifestly appears; for if you remove the load-stone, the whole force of the iron almost ceases.

In this sense it is that we are to conceive one single action to be exerted betwixt two planets, arising from the conspiring natures of both: and this action standing in the same relation to both, if it is proportional to the quantity of matter in the one, it will be also proportional to the quantity of matter in the other.

Perhaps it may be objected, that, according to this philosophy (p. 398), all bodies should mutually attract one another, contrary to the evidence of experiments in terrestrial bodies; but I answer, that the experiments in terrestrial bodies come to no account; for the attraction of homogeneous spheres near their surfaces are (by Prop. LXXII) as their diameters. Whence a sphere of one foot in diameter, and of a like nature to the earth, would attract a small body placed near its surface with a force 20000000 times less than the earth would do if placed near its surface; but so small a force could produce no sensible effect. If two such spheres were distant but by of an inch, they would not, even in spaces void of[Pg 528] resistance, come together by the force of their mutual attraction in less than a month's time; and less spheres will come together at a rate yet slower, viz., in the proportion of their diameters. Nay, whole mountains will not be sufficient to produce any sensible effect. A mountain of an hemispherical figure, three miles high, and six broad, will not, by its attraction, draw the pendulum two minutes out of the true perpendicular; and it is only in the great bodies of the planets that these forces are to be perceived, unless we may reason about smaller bodies in manner following.

Let ABCD (p. 93) represent the globe of the earth cut by any plane AC into two parts ACB, and ACD. The part ACB bearing upon the part ACD presses it with its whole weight; nor can the part ACD sustain this pressure and continue unmoved, if it is not opposed by an equal contrary pressure. And therefore the parts equally press each other by their weights, that is, equally attract each other, according to the third Law of Motion; and, if separated and let go, would fall towards each other with velocities reciprocally as the bodies. All which we may try and see in the load-stone, whose attracted part does not propel the part attracting, but is only stopped and sustained thereby.

Suppose now that ACB represents some small body on the earth's surface; then, because the mutual attractions of this particle, and of the remaining part ACD of the earth towards each other, are equal, but the attraction of the particle towards the earth (or its weight) is as the matter of the particle (as we have proved by the experiment of the pendulums), the attraction of the earth towards the particle will likewise be as the matter of the particle; and therefore the attractive forces of all terrestrial bodies will be as their several quantities of matter.

The forces (p. 396), which are as the matter in terrestrial bodies of all forms, and therefore are not mutable with the forms, must be found in all sorts of bodies whatsoever, celestial as well as terrestrial, and be in all proportional to their quantities of matter, because among all there is no difference of substance, but of modes and forms only. But in the celestial bodies the same thing is likewise proved thus. We have shewn that the action of the circum-solar force upon all the planets (reduced to equal distances) is as the matter of the planets; that the action of the circumjovial force upon the satellites of Jupiter observes the same law; and the same thing is to be said of the attraction of all the planets towards every planet: but thence it follows (by Prop. LXIX) that their attractive forces are as their several quantities of matter.

As the parts of the earth mutually attract one another, so do those of all the planets. If Jupiter and its satellites were brought together, and formed into one globe, without doubt they would continue mutually to attract one another as before. And, on the other hand, if the body of Jupiter was broke into more globes, to be sure, these would no less attract[Pg 529] one another than they do the satellites now. From these attractions it is that the bodies of the earth and all the planets effect a spherical figure, and their parts cohere, and are not dispersed through the æther. But we have before proved that these forces arise from the universal nature of matter (p. 398), and that, therefore, the force of any whole globe is made up of the several forces of all its parts. And from thence it follows (by Cor. III, Prop. LXXIV) that the force of every particle decreases in the duplicate proportion of the distance from that particle; and (by Prop. LXXIII and LXXV) that the force of an entire globe, reckoning from the surface outwards, decreases in the duplicate, but, reckoning inwards, in the simple proportion of the distances from the centres, if the matter of the globe be uniform. And though the matter of the globe, reckoning from the centre towards the surface, is not uniform (p. 398, 399), yet the decrease in the duplicate proportion of the distance outwards would (by Prop. LXXVI) take place, provided that difformity is similar in places round about at equal distances from the centre. And two such globes will (by the same Proposition) attract one the other with a force decreasing in the duplicate proportion of the distance between their centres.

Wherefore the absolute force of every globe is as the quantity of matter which the globe contains; but the motive force by which every globe is attracted towards another, and which, in terrestrial bodies, we commonly call their weight, is as the content under the quantities of matter in both globes applied to the square of the distance between their centres (by Cor. IV, Prop. LXXVI), to which force the quantity of motion, by which each globe in a given time will be carried towards the other, is proportional. And the accelerative force, by which every globe according to its quantity of matter is attracted towards another, is as the quantity of matter in that other globe applied to the square of the distance between the centres of the two (by Cor. II, Prop. LXXVI); to which force, the velocity by which the attracted globe will, in a given time, be carried towards the other is proportional. And from these principles well understood, it will be now easy to determine the motions of the celestial bodies among themselves.

From comparing the forces of the planets one with another, we have above seen that the circum-solar does more than a thousand times exceed all the rest; but by the action of a force so great it is unavoidable but that all bodies within, nay, and far beyond, the bounds of the planetary system must descend directly to the sun, unless by other motions they are impelled towards other parts: nor is our earth to be excluded from the number of such bodies; for certainly the moon is a body of the same nature with the planets, and subject to the same attractions with the other planets, seeing it is by the circum-terrestrial force that it is retained in its orbit. But that the earth and moon are equally attracted towards the sun, we have above proved; we have likewise before proved that all bodies are subject to[Pg 530] the said common laws of attraction. Nay, supposing any of those bodies to be deprived of its circular motion about the sun, by having its distance from the sun, we may find (by Prop. XXXVI) in what space of time it would in its descent arrive at the sun; to wit, in half that periodic time in which the body might be revolved at one half of its former distance; or in a space of time that is to the periodic time of the planet as 1 to ; as that Venus in its descent would arrive at the sun in the space of 40 days, Jupiter in the space of two years and one month, and the earth and moon together in the space of 66 days and 19 hours. But, since no such thing happens, it must needs be, that those bodies are moved towards other parts (p. 75), nor is every motion sufficient for this purpose. To hinder such a descent, a due proportion of velocity is required. And hence depends the force of the argument drawn from the retardation of the motions of the planets. Unless the circum-solar force decreased in the duplicate ratio of their increasing slowness, the excess thereof would force those bodies to descend to the sun; for instance, if the motion (cæteris paribus) was retarded by one half, the planet would be retained in its orb by one fourth of the former circum-solar force, and by the excess of the other three fourths would descend to the sun. And therefore the planets (Saturn, Jupiter, Mars, Venus, and Mercury) are not really retarded in their perigees, nor become really stationary, or regressive with slow motions. All these are but apparent, and the absolute motions, by which the planets continue to revolve in their orbits, are always direct, and nearly equable. But that such motions are performed about the sun, we have already proved; and therefore the sun, as the centre of the absolute motions, is quiescent. For we can by no means allow quiescence to the earth, lest the planets in their perigees should indeed be truly retarded, and become truly stationary and regressive, and so for want of motion should descend to the sun. But farther; since the planets (Venus, Mars, Jupiter, and the rest) by radii drawn to the sun describe regular orbits, and areas (as we have shewn) nearly and to sense proportional to the times, it follows (by Prop. III, and Cor. III, Prop. LXV) that the sun is moved with no notable force, unless perhaps with such as all the planets are equally moved with, according to their several quantities of matter, in parallel lines, and so the whole system is transferred in right lines. Reject that translation of the whole system, and the sun will be almost quiescent in the centre thereof. If the sun was revolved about the earth, and carried the other planets round about itself, the earth ought to attract the sun with a great force, but the circum-solar planets with no force producing any sensible effect, which is contrary to Cor. III, Prop. LXV. Add to this, that if hitherto the earth, because of the gravitation of its parts, has been placed by most authors in the lowermost region of the universe; now, for better reason, the sun possessed of a centripetal force exceeding our terrestrial gravitation a thousand[Pg 531] times and more, ought to be depressed into the lowermost place, and to be held for the centre of the system. And thus the true disposition of the whole system will be more fully and more exactly understood.

Because the fixed stars are quiescent one in respect of another (p. 401, 402), we may consider the sun, earth, and planets, as one system of bodies carried hither and thither by various motions among themselves; and the common centre of gravity of all (by Cor. IV of the Laws of Motion) will either be quiescent, or move uniformly forward in a right line: in which case the whole system will likewise move uniformly forward in right lines. But this is an hypothesis hardly to be admitted; and, therefore, setting it aside, that common centre will be quiescent: and from it the sun is never far removed. The common centre of gravity of the sun and Jupiter falls on the surface of the sun; and though all the planets were placed towards the same parts from the sun with Jupiter the common centre of the sun and all of them would scarcely recede twice as far from the sun's centre; and, therefore, though the sun, according to the various situation of the planets, is variously agitated, and always wandering to and fro with a slow motion of libration, yet it never recedes one entire diameter of its own body from the quiescent centre of the whole system. But from the weights of the sun and planets above determined, and the situation of all among themselves, their common centre of gravity may be found; and, this being given, the sun's place to any supposed time may be obtained.

About the sun thus librated the other planets are revolved in elliptic orbits (p. 403), and, by radii drawn to the sun, describe areas nearly proportional to the times, as is explained in Prop. LXV. If the sun was quiescent, and the other planets did not act mutually one upon another, their orbits would be elliptic, and the areas exactly proportional to the times (by Prop. XI, and Cor. 1, Prop. XIII). But the actions of the planets among themselves, compared with the actions of the sun on the planets, are of no moment, and produce no sensible errors. And those errors are less in revolutions about the sun agitated in the manner but now described than if those revolutions were made about the sun quiescent (by Prop. LXVI, and Cor. Prop. LXVIII), especially if the focus of every orbit is placed in the common centre of gravity of all the lower included planets; viz., the focus of the orbit of Mercury in the centre of the sun; the focus of the orbit of Venus in the common centre of gravity of Mercury and the sun; the focus of the orbit of the earth in the common centre of gravity of Venus, Mercury, and the sun; and so of the rest. And by this means the foci of the orbits of all the planets, except Saturn, will not be sensibly removed from the centre of the sun, nor will the focus of the orbit of Saturn recede sensibly from the common centre of gravity of Jupiter and the sun. And therefore astronomers are not far from the truth, when they reckon the sun's centre the common focus of all the planetary orbits. In Saturn itself[Pg 532] the error thence arising does not exceed 1' 45''. And if its orbit, by placing the focus thereof in the common centre of gravity of Jupiter and the sun, shall happen to agree better with the phænomena, from thence all that we have said will be farther confirmed.

If the sun was quiescent, and the planets did not act one on another, the aphelions and nodes of their orbits would likewise (by Prop. 1, XI, and Cor. Prop. XIII) be quiescent. And the longer axes of their elliptic orbits would (by Prop. XV) be as the cubic roots of the squares of their periodic times: and therefore from the given periodic times would be also given. But those times are to be measured not from the equinoctial points, which are moveable, but from the first star of Aries. Put the semi-axis of the earth's orbit 100000, and the semi-axes of the orbits of Saturn, Jupiter, Mars, Venus, and Mercury, from their periodic times, will come out 953806, 520116, 152399, 72333, 38710 respectively. But from the sun's motion every semi-axis is increased (by Prop. LX) by about one third of the distance of the sun's centre from the common centre of gravity of the sun and planet (p. 405, 406.) And from the actions of the exterior planets on the interior, the periodic times of the interior are something protracted, though scarcely by any sensible quantity; and their aphelions are transferred (by Cor. VI and VII, Prop. LXVI) by very slow motions in consequentia. And on the like account the periodic times of all, especially of the exterior planets, will be prolonged by the actions of the comets, if any such there are, without the orb of Saturn, and the aphelions of all will be thereby carried forwards in consequentia. But from the progress of the aphelions the regress of the nodes follows (by Cor. XI, XIII, Prop. LXVI). And if the plane of the ecliptic is quiescent, the regress of the nodes (by Cor. XVI, Prop. LXVI) will be to the progress of the aphelion in every orbit as the regress of the nodes of the moon's orbit to the progress of its apogeon nearly, that is, as about 10 to 21. But astronomical observations seem to confirm a very slow progress of the aphelions, and a regress of the nodes in respect of the fixed stars. And hence it is probable that there are comets in the regions beyond the planets, which, revolving in very eccentric orbs, quickly fly through their perihelion parts, and, by an exceedingly slow motion in their aphelions, spend almost their whole time in the regions beyond the planets; as we shall afterwards explain more at large.

The planets thus revolved about the sun (p. 413, 414, 415) may at the same time carry others revolving about themselves as satellites or moons, as appears by Prop. LXVI. But from the action of the sun our moon must move with greater velocity, and, by a radius drawn to the earth, describe an area greater for the time; it must have its orbit less curve, and therefore approach nearer to the earth in the syzygies than in the quadratures, except in so far as the motion of eccentricity hinders those effects.[Pg 533] For the eccentricity is greatest when the moon's apogeon is in the syzygies, and least when the same is in the quadratures; and hence it is that the perigeon moon is swifter and nearer to us, but the apogeon moon slower and farther from us, in the syzygies than in the quadratures. But farther; the apogeon has a progressive and the nodes a regressive motion, both unequable. For the apogeon is more swiftly progressive in its syzygies, more slowly regressive in its quadratures, and by the excess of its progress above its regress is yearly transferred in consequentia; but the nodes are quiescent in their syzygies, and most swiftly regressive in their quadratures. But farther, still, the greatest latitude of the moon is greater in its quadratures than in its syzygies; and the mean motion swifter in the aphelion of the earth than in its perihelion. More inequalities in the moon's motion have not hitherto been taken notice of by astronomers: but all these follow from our principles in Cor. II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, Prop. LXVI, and are known really to exist in the heavens. And this may seen in that most ingenious, and if I mistake not, of all, the most accurate, hypothesis of Mr. Horrox, which Mr. Flamsted has fitted to the heavens; but the astronomical hypotheses are to be corrected in the motion of the nodes; for the nodes admit the greatest equation or prosthaphæresis in their octants, and this inequality is most conspicuous when the moon is in the nodes, and therefore also in the octants; and hence it was that Tycho, and others after him, referred this inequality to the octants of the moon, and made it menstrual; but the reasons by us adduced prove that it ought to be referred to the octants of the nodes, and to be made annual.

Beside those inequalities taken notice of by astronomers (p. 414, 445, 447,) there are yet some others, by which the moon's motions are so disturbed, that hitherto by no law could they be reduced to any certain regulation. For the velocities or horary motions of the apogee and nodes of the moon, and their equations, as well as the difference betwixt the greatest eccentricity in the syzygies and the least in the quadratures, and that inequality which we call the variation, in the progress of the year are augmented and diminished (by Cor. XIV, Prop. LXVI) in the triplicate ratio of the sun's apparent diameter. Beside that, the variation is mutable nearly in the duplicate ratio of the time between the quadratures (by Cor. I and II, Lem. X, and Cor. XVI, Prop. LXVI); and all those inequalities are something greater in that part of the orbit which respects the sun than in the opposite part, but by a difference that is scarcely or not at all perceptible.

By a computation (p. 412), which for brevity's sake I do not describe, I also find that the area which the moon by a radius drawn to the earth describes in the several equal moments of time is nearly as the sum of the number , and versed sine of the double distance of the moon from[Pg 534] the nearest quadrature in a circle whose radius is unity; and therefore that the square of the moon's distance from the earth is as that sum divided by the horary motion of the moon. Thus it is when the variation in the octants is in its mean quantity; but if the variation is greater or less, that versed sine must be augmented or diminished in the same ratio. Let astronomers try how exactly the distances thus found will agree with the moon's apparent diameters.

From the motions of our moon we may derive the motions of the moons or satellites of Jupiter and Saturn (p. 413); for the mean motion of the nodes of the outmost satellite of Jupiter is to the mean motion of the nodes of our moon in a proportion compounded of the duplicate proportion of the periodic time of the earth about the sun to the periodic time of Jupiter about the sun, and the simple proportion of the periodic time of the satellite about Jupiter to the periodic time of our moon about the earth (by Cor. XVI, Prop. LXVI): and therefore those nodes, in the space of a hundred years, are carried 8° 24' backwards, or in antecedentia. The mean motions of the nodes of the inner satellites are to the (mean) motion of (the nodes of) the outmost as their periodic times to the periodic time of this, by the same corollary, and are thence given. And the motion of the apsis of every satellite in consequentia is to the motion of its nodes in antecedentia, as the motion of the apogee of our moon to the motion of its nodes (by the same Corollary), and is thence given. The greatest equations of the nodes and line of the apses of each satellite are to the greatest equations of the nodes and the line of the apses of the moon respectively as the motion of the nodes and line of the apses of the satellites in the time of one revolution of the first equations to the motion of the nodes and apogeon of the moon in the time of one revolution of the last equations. The variation of a satellite seen from Jupiter is to the variation of our moon in the same proportion as the whole motions of their nodes respectively, during the times in which the satellite and our moon (after parting from) are revolved (again) to the sun, by the same Corollary; and therefore in the outmost satellite the variation does not exceed 5'' 12'''. From the small quantity of those inequalities, and the slowness of the motions, it happens that the motions of the satellites are found to be so regular, that the more modern astronomers either deny all motion to the nodes, or affirm them to be very slowly regressive.

(P. 404). While the planets are thus revolved in orbits about remote centres, in the mean time they make their several rotations about their proper axes; the sun in 26 days; Jupiter in 9h. 56'; Mars in ; Venus in 23h.; and that in planes not much inclined to the plane of the ecliptic, and according to the order of the signs, as astronomers determine from the spots or maculæ that by turns present themselves to our sight in their bodies; and there is a [Pg 535]like revolution of our earth performed in 24h.; and those motions are neither accelerated nor retarded by the actions of the centripetal forces, as appears by Cor. XXII, Prop. LXVI; and therefore of all others they are the most equable and most fit for the mensuration of time; but those revolutions are to be reckoned equable not from their return to the sun, but to some fixed star: for as the position of the planets to the sun is unequably varied, the revolutions of those planets from sun to sun are rendered unequable.

In like manner is the moon revolved about its axis by a motion most equable in respect of the fixed stars, viz., in 27d. 7h. 43', that is, in the space of a sidereal month; so that this diurnal motion is equal to the mean motion of the moon in its orbit; upon which account the same face of the moon always respects the centre about which this mean motion is performed, that is, the exterior focus of the moon's orbit nearly; and hence arises a deflection of the moon's face from the earth, sometimes towards the east, and other times towards the west, according to the position of the focus which it respects; and this deflection is equal to the equation of the moon's orbit, or to the difference betwixt its mean and true motions; and this is the moon's libration in longitude: but it is likewise affected with a libration in latitude arising from the inclination of the moon's axis to the plane of the orbit in which the moon is revolved about the earth; for that axis retains the same position to the fixed stars nearly, and hence the poles present themselves to our view by turns, as we may understand from the example of the motion of the earth, whose poles, by reason of the inclination of its axis to the plane of the ecliptic, are by turns illuminated by the sun. To determine exactly the position of the moon's axis to the fixed stars, and the variation of this position, is a problem worthy of an astronomer.

By reason of the diurnal revolutions of the planets, the matter which they contain endeavours to recede from the axis of this motion; and hence the fluid parts rising higher towards the equator than about the poles (p. 405), would lay the solid parts about the equator under water, if those parts did not rise also (p. 405, 409): upon which account the planets are something thicker about the equator than about the poles; and their equinoctial points (p. 413) thence become regressive; and their axes, by a motion of nutation, twice in every revolution, librate towards their ecliptics, and twice return again to their former inclination, as is explained in Cor. XVIII, Prop. LXVI; and hence it is that Jupiter, viewed through very long telescopes, does not appear altogether round (p. 409), but having its diameter that lies parallel to the ecliptic something longer than that which is drawn from north to south.

And from the diurnal motion and the attractions (p. 415, 418) of the sun and moon our sea ought twice to rise and twice to fall every day, as well lunar as solar (by Cor. XIX, XX, Prop. LXVI), and the greatest[Pg 536] height of the water to happen before the sixth hour of either day and after the twelfth hour preceding. By the slowness of the diurnal motion the flood is retracted to the twelfth hour; and by the force of the motion of reciprocation it is protracted and deferred till a time nearer to the sixth hour. But till that time is more certainly determined by the phænomena, choosing the middle between those extremes, why may we not conjecture the greatest height of the water to happen at the third hour? for thus the water will rise all that time in which the force of the luminaries to raise it is greater, and will fall all that time in which their force is less; viz., from the ninth to the third hour when that force is greater, and from the third to the ninth when it is less. The hours I reckon from the appulse of each luminary to the meridian of the place, as well under as above the horizon; and by the hours of the lunar day I understand the twenty-fourth parts of that time which the moon spends before it comes about again by its apparent diurnal motion to the meridian of the place which it left the day before.

But the two motions which the two luminaries raise will not appear distinguished, but will make a certain mixed motion. In the conjunction or opposition of the luminaries their forces will be conjoined, and bring on the greatest flood and ebb. In the quadratures the sun will raise the waters which the moon depresseth, and depress the waters which the moon raiseth; and from the difference of their forces the smallest of all tides will follow. And because (as experience tells us) the force of the moon is greater than that of the sun, the greatest height of the water will happen about the third lunar hour. Out of the syzygies and quadratures the greatest tide which by the single force of the moon ought to fall out at the third lunar hour, and by the single force of the sun at the third solar hour, by the compounded forces of both must fall out in an intermediate time that approaches nearer to the third hour of the moon than to that of the sun; and, therefore, while the moon is passing from the syzygies to the quadratures, during which time the third hour of the sun precedes the third of the moon, the greatest tide will precede the third lunar hour, and that by the greatest interval a little after the octants of the moon; and by like intervals the greatest tide will follow the third lunar hour, while the moon is passing from the quadratures to the syzygies.

But the effects of the luminaries depend upon their distances from the earth; for when they are less distant their effects are greater, and when more distant their effects are less, and that in the triplicate proportion of their apparent diameters. Therefore it is that the sun in the winter time, being then in its perigee, has a greater effect, and makes the tides in the syzygies something greater, and those in the quadratures something less, cæteris paribus, than in the summer season; and every month the moon, while in the perigee, raiseth greater tides than at the distance of 15 days[Pg 537] before or after, when it is in its apogee. Whence it comes to pass that two highest tides do not follow one the other in two immediately succeeding syzygies.

The effect of either luminary doth likewise depend upon its declination or distance from the equator; for if the luminary was placed at the pole, it would constantly attract all the parts of the waters, without any intension or remission of its action, and could cause no reciprocation of motion; and, therefore, as the luminaries decline from the equator towards either pole, they will by degrees lose their force, and on this account will excite lesser tides in the solstitial than in the equinoctial syzygies. But in the solstitial quadratures they will raise greater tides than in the quadratures about the equinoxes; because the effect of the moon, then situated in the equator, most exceeds the effect of the sun; therefore the greatest tides fall out in those syzygies, and the least in those quadratures, which happen about the time of both equinoxes; and the greatest tide in the syzygies is always succeeded by the least tide in the quadratures, as we find by experience. But because the sun is less distant from the earth in winter than in summer, it comes to pass that the greatest and least tides more frequently appear before than after the vernal equinox, and more frequently after than before the autumnal.

Moreover, the effects of the luminaries depend upon the latitudes of places. Let ApEP represent the earth on all sides covered with deep waters; C its centre; P, p, its poles; AE the equator; F any place without the equator; Ff the parallel of the place; Dd the correspondent parallel on the other side of the equator; L the place which the moon possessed three hours before; H the place of the earth directly under it; h the opposite place; K, k, the places at 90 degrees distance; CH, Ch, the greatest heights of the sea from the centre of the earth; and CK, Ck, the least heights; and if with the axes Hh, Kk, an ellipsis is described, and by the revolution of that ellipsis about its longer axis Hh a spheroid HPKhpk is formed, this spheroid will nearly represent the figure of the sea; and CF, Cf, CD, Cd, will represent the sea in the places F, f, D, d. But farther; if in the said revolution of the ellipsis any point N describes the circle NM, cutting the parallels Ff, Dd, in any places R, T, and the equator AE in S, CN will represent the height of the sea in all those places R, S, T, situated in this circle. Wherefore in the diurnal revolution of any place F the greatest flood will be in F, at the third hour after the appulse of the moon to the meridian above the horizon; and afterwards the greatest ebb in Q, at the third hour after the setting of the moon; and then the greatest flood in f,[Pg 538] at the third hour after the appulse of the moon to the meridian under the horizon, and, lastly, the greatest ebb in Q, at the third hour after the rising of the moon; and the latter flood in f will be less than the preceding flood in F. For the whole sea is divided into two huge and hemispherical floods, one in the hemisphere KHkC on the north side, the other in the opposite hemisphere KHkC, which we may therefore call the northern and the southern floods: these floods being always opposite the one to the other, come by turns to the meridians of all places after the interval of twelve lunar hours; and, seeing the northern countries partake more of the northern flood, and the southern countries more of the southern flood, thence arise tides alternately greater and less in all places without the equator in which the luminaries rise and set. But the greater tide will happen when the moon declines towards the vertex of the place, about the third hour after the appulse of the moon to the meridian above the horizon; and when the moon changes its declination, that which was the greater tide will be changed into a lesser; and the greatest difference of the floods will fall out about the times of the solstices, especially if the ascending node of the moon is about the first of Aries. So the morning tides in winter exceed those of the evening, and the evening tides exceed those of the morning in summer; at Plymouth by the height of one foot, but at Bristol by the height of 15 inches, according to the observations of Colepress and Sturmy.

But the motions which we have been describing suffer some alteration from that force of reciprocation which the waters [having once received] retain a little while by their vis insita; whence it comes to pass that the tides may continue for some time, though the actions of the luminaries should cease. This power of retaining the impressed motion lessens the difference of the alternate tides, and makes those tides which immediately succeed after the syzygies greater, and those which follow next after the quadratures less. And hence it is that the alternate tides at Plymouth and Bristol do not differ much more one from the other than by the height of a foot, or of 15 inches; and that the greatest tides of all at those ports are not the first but the third after the syzygies.

And, besides, all the motions are retarded in their passage through shallow channels, so that the greatest tides of all, in some straits and mouths of rivers, are the fourth, or even the fifth, after the syzygies.

It may also happen that the greatest tide may be the fourth or fifth after the syzygies, or fall out yet later, because the motions of the sea are retarded in passing through shallow places towards the shores; for so the tide arrives at the western coast of Ireland at the third lunar hour, and an hour or two after at the ports in the southern coast of the same island; as also at the islands Cassiterides, commonly Sorlings; then successively at Falmouth, Plymouth, Portland, the isle of Wight, Winchester, Dover,[Pg 539] the mouth of the Thames, and London Bridge, spending twelve hours in this passage. But farther; the propagation of the tides may be obstructed even by the channels of the ocean itself, when they are not of depth enough, for the flood happens at the third lunar hour in the Canary islands; and at all those western coasts that lie towards the Atlantic ocean, as of Ireland, France, Spain, and all Africa, to the Cape of Good Hope, except in some shallow places, where it is impeded, and falls out later; and in the straits of Gibraltar, where, by reason of a motion propagated from the Mediterranean sea, it flows sooner. But, passing from those coasts over the breadth of the ocean to the coasts of America, the flood arrives first at the most eastern shores of Brazil, about the fourth or fifth lunar hour; then at the mouth of the river of the Amazons at the sixth hour, but at the neighbouring islands at the fourth hour; afterwards at the islands of Bermudas at the seventh hour, and at port St. Augustin in Florida at seven and a half. And therefore the tide is propagated through the ocean with a slower motion than it should be according to the course of the moon; and this retardation is very necessary, that the sea at the same time may fall between Brazil and New France, and rise at the Canary islands, and on the coasts of Europe and Africa, and vice versa: for the sea cannot rise in one place but by falling in another. And it is probable that the Pacific sea is agitated by the same laws; for in the coasts of Chili and Peru the highest flood is said to happen at the third lunar hour. But with what velocity it is thence propagated to the eastern coasts of Japan, the Philippine and other islands adjacent to China, I have not yet learned.

Farther; it may happen (p. 418) that the tide may be propagated from the ocean through different channels towards the same port, and may pass quicker through some channels than through others, in which case the same tide, divided into two or more succeeding one another, may compound new motions of different kinds. Let us suppose one tide to be divided into two equal tides, the former whereof precedes the other by the space of six hours, and happens at the third or twenty-seventh hour from the appulse of the moon to the meridian of the port. If the moon at the time of this appulse to the meridian was in the equator, every six hours alternately there would arise equal floods, which, meeting with as many equal ebbs, would so balance one the other, that, for that day, the water would stagnate, and remain quiet. If the moon then declined from the equator, the tides in the ocean would be alternately greater and less, as was said; and from hence two greater and two lesser tides would be alternately propagated towards that port. But the two greater floods would make the greatest height of the waters to fall out in the middle time betwixt both, and the greater and lesser floods would make the waters to rise to a mean height in the middle time between them; and in the middle time between[Pg 540] the two lesser floods the waters would rise to their least height. Thus in the space of twenty-four hours the waters would come, not twice, but once only to their greatest, and once only to their least height; and their greatest height, if the moon declined towards the elevated pole, would happen at the sixth or thirtieth hour after the appulse of the moon to the meridian; and when the moon changed its declination, this flood would be changed into an ebb.

Of all which we have an example in the port of Batsham, in the kingdom of Tunquin, in the latitude of 20° 50' north. In that port, on the day which follows after the passage of the moon over the equator, the waters stagnate; when the moon declines to the north, they begin to flow and ebb, not twice, as in other ports, but once only every day; and the flood happens at the setting, and the greatest ebb at the rising of the moon. This tide increaseth with the declination of the moon till the seventh or eighth day; then for the seventh or eighth day following it decreaseth at the same rate as it had increased before, and ceaseth when the moon changeth its declination. After which the flood is immediately changed into an ebb; and thenceforth the ebb happens at the setting and the flood at the rising of the moon, till the moon again changes its declination. There are two inlets from the ocean to this port; one more direct and short between the island Hainan and the coast of Quantung, a province of China; the other round about between the same island and the coast of Cochim; and through the shorter passage the tide is sooner propagated to Batsham.

In the channels of rivers the influx and reflux depends upon the current of the rivers, which obstructs the ingress of the waters from the sea, and promotes their egress to the sea, making the ingress later and slower, and the egress sooner and faster; and hence it is that the reflux is of longer duration that the influx, especially far up the rivers, where the force of the sea is less. So Sturmy tells us, that in the river Avon, three miles below Bristol, the water flows only five hours, but ebbs seven; and without doubt the difference is yet greater above Bristol, as at Caresham or the Bath. This difference does likewise depend upon the quantity of the flux and reflux; for the more vehement motion of the sea near the syzygies of the luminaries more easily overcoming the resistance of the rivers, will make the ingress of the water to happen sooner and to continue longer, and will therefore diminish this difference. But while the moon is approaching to the syzygies, the rivers will be more plentifully filled, their currents being obstructed by the greatness of the tides, and therefore will something more retard the reflux of the sea a little after than a little before the syzygies. Upon which account the slowest tides of all will not happen in the syzygies, but precede them a little; and I observed above that the tides before the syzygies were also retarded by the force of the sun; and from both[Pg 541] causes conjoined the retardation of the tides will be both greater and sooner before the syzygies. All which I find to be so, by the tide-tables which Flamsted has composed from a great many observations.

By the laws we have been describing, the times of the tides are governed; but the greatness of the tides depends upon the greatness of the seas. Let C represent the centre of the earth, EADB the oval figure of the seas, CA the longer semi-axis of this oval, CB the shorter insisting at right angles upon the former, D the middle point between A and B, and ECF or eCf the angle at the centre of the earth, subtended by the breadth of the sea that terminates in the shores E, F, or e, f. Now, supposing that the point A is in the middle between the points E, F, and the point D in the middle between the points e, f, if the difference of the heights CA, CB, represent the quantity of the tide in a very deep sea surrounding the whole earth, the excess of the height CA above the height CE or CF will represent the quantity of the tide in the middle of the sea EF, terminated by the shores E, F; and the excess of the height Ce above the height Cf will nearly represent the quantity of the tide on the shores f of the same sea. Whence it appears that the tides are far less in the middle of the sea than at the shores; and that the tides at the shores are nearly as EF (p. 451, 452), the breadth of the sea not exceeding a quadrantal arc. And hence it is that near the equator, where the sea between Africa and America is narrow, the tides are far less than towards either side in the temperate zones, where the seas are extended wider; or on almost all the shores of the Pacific sea, as well towards America as towards China, and within as well as without the tropics; and that in islands in the middle of the sea they scarcely rise higher than two or three feet, but on the shores of great continents are three or four times greater, and above, especially if the motions propagated from the ocean are by degrees contracted into a narrow space, and the water, to fill and empty the bays alternately, is forced to flow and ebb with great violence through shallow places; as Plymouth and Chepstow Bridge in England, at the mount of St. Michael and town of Avranches in Normandy, and at Cambaia and Pegu in the East Indies. In which places,[Pg 542] the sea, hurried in and out with great violence, sometimes lays the shores under water, sometimes leaves them dry, for many miles. Nor is the force of the influx and efflux to be broke till it has raised or depressed the water to forty or fifty feet and more. Thus also long and shallow straits that open to the sea with mouths wider and deeper than the rest of their channel (such as those about Britain and the Magellanic Straits at the eastern entry) will have a greater flood and ebb, or will more intend and remit their course, and therefore will rise higher and be depressed lower. On the coast of South America it is said that the Pacific sea in its reflux sometimes retreats two miles, and gets out of sight of those that stand on shore. Whence in these places the floods will be also higher; but in deeper waters the velocity of influx and efflux is always less, and therefore the ascent and descent is so too. Nor in such places is the ocean known to ascend to more than six, eight, or ten feet. The quantity of the ascent I compute in the following manner:

Let S represent the sun, T the earth (419, 420), P the moon, PAGB the moon's orbit. In SP take SK equal to ST and SL to SK in the duplicate ratio of SK to SP. Parallel to PT draw LM; and, supposing the mean quantity of the circum-solar force directed towards the earth to be represented by the distance ST or SK, SL will represent the quantity thereof directed towards the moon. But that force is compounded of the parts SM, LM; of which the force LM and that part of SM which is represented by TM, do disturb the motion of the moon (as appears from Prop. LXVI, and its Corollaries). In so far as the earth and moon are revolved about their common centre of gravity, the earth will be liable to the action of the like forces. But we may refer the sums as well of the forces as of the motions to the moon, and represent the sums of the forces by the lines TM and ML, which are proportional to them. The force LM, in its mean quantity, is to the force by which the moon may be revolved in an orbit, about the earth quiescent, at the distance PT in the duplicate ratio of the moon's periodic time about the earth to the earth's periodic time about the sun (by Cor. XVII, Prop. LXVI); that is, in the duplicate ratio of 27d. 7h. 43' to 365d. 6h. 9'; or as 1000 to 178725, or 1 to . The force by which the moon may be revolved in its orb about the earth in rest, at the distance PT of semi-diameters of the earth, is to the force by which it may revolve in the same time at the distance of 60 semi-diameters as to 60; and this force is to the force of gravity with us as 1 to 60 × 60 nearly; and therefore the mean force ML is to the force of gravity at the surface of the earth as to , or 1 to[Pg 543] 638092,6. Whence the force TM will be also given from the proportion of the lines TM, ML. And these are the forces of the sun, by which the moon's motions are disturbed.

If from the moon's orbit (p. 449) we descend to the earth's surface, those forces will be diminished in the ratio of the distances and 1; and therefore the force LM will then become 38604600 times less than the force of gravity. But this force acting equally every where upon the earth, will scarcely effect any change on the motion of the sea, and therefore may be neglected in the explication of that motion. The other force TM, in places where the sun is vertical, or in their nadir, is triple the quantity of the force ML, and therefore but 12868200 times less than the force of gravity.

Suppose now ADBE to represent the spherical surface of the earth, aDbE the surface of the water overspreading it, C the centre of both, A the place to which the sun is vertical, B the place opposite; D, E, places at 90 degrees distance from the former; ACEmlk a right angled cylindric canal passing through the earth's centre. The force TM in any place is as the distance of the place from the plane DE, on which a line from A to C insists at right angles, and therefore in the part of the canal which is represented by EClm is of no quantity, but in the other part AClk is as the gravity at the several heights; for in descending towards the centre of the earth, gravity is (by Prop. LXXIII) every where as the height; and therefore the force TM drawing the water upwards will diminish its gravity in the leg AClk of the canal in a given ratio: upon which account the water will ascend in this leg, till its defect of gravity is supplied by its greater height; nor will it rest in an equilibrium till its total gravity becomes equal to the total gravity in EClm, the other leg of the canal. Because the gravity of every particle is as its distance from the earth's centre, the weight of the whole water in either leg will increase in the duplicate ratio of the height; and therefore the height of the water in the leg AClk will be to the height thereof in the leg ClmE in the subduplicate ratio of the number 12868201 to 12868200, or in the ratio of the number 25623053 to the number 25623052, and the height of the water in the leg EClm to the difference of the heights, as 25623052 to 1. But the height in the leg EClm is of 19615800 Paris feet, as has been lately[Pg 544] found by the mensuration of the French; and, therefore, by the preceding analogy, the difference of the heights comes out inches of the Paris foot; and the sun's force will make the height of the sea at A to exceed the height of the same at E by 9 inches. And though the water of the canal ACEmlk be supposed to be frozen into a hard and solid consistence, yet the heights thereof at A and E, and all other intermediate places, would still remain the same.

Let Aa (in the following figure) represent that excess of height of 9 inches at A, and hf the excess of height at any other place h; and upon DC let fall the perpendicular fG, meeting the globe of the earth in F; and because the distance of the sun is so great that all the right lines drawn thereto may be considered as parallel, the force TM in any place f will be to the same force in the place A as the sine FG to the radius AC. And, therefore, since those forces tend to the sun in the direction of parallel lines, they will generate the parallel heights Ff, Aa, in the same ratio; and therefore the figure of the water Dfaeb will be a spheroid made by the revolution of an ellipsis about its longer axis ab. And the perpendicular height fh will be to the oblique height Ff as fG to fC, or as FG to AC: and therefore the height fh is to the height Aa in the duplicate ratio of FG to AC, that is, in the ratio of the versed sine of double the angle DCf to double the radius, and is thence given. And hence to the several moments of the apparent revolution of the sun about the earth we may infer the proportion of the ascent and descent of the waters at any given place under the equator, as well as of the diminution of that ascent and descent, whether arising from the latitude of places or from the sun's declination; viz., that on account of the latitude of places, the ascent and descent of the sea is in all places diminished in the duplicate ratio of the co-sines of latitude; and on account of the sun's declination, the ascent and descent under the equator is diminished in the duplicate ratio of the co-sine of declination. And in places without the equator the half sum of the morning and evening ascents (that is, the mean ascent) is diminished nearly in the same ratio.

Let S and L respectively represent the forces of the sun and moon placed in the equator, and at their mean distances from the earth; R the radius; T and V the versed sines of double the complements of the sun[Pg 545] and moon's declinations to any given time; D and E the mean apparent diameters of the sun and moon: and, supposing F and G to be their apparent diameters to that given time, their forces to raise the tides under the equator will be, in the syzygies ; in the quadratures, . And if the same ratio is likewise observed under the parallels, from observations accurately made in our northern climates we may determine the proportion of the forces L and S; and then by means of this rule predict the quantities of the tides to every syzygy and quadrature.

At the mouth of the river Avon, three miles below Bristol (p. 450 to 453), in spring and autumn, the whole ascent of the water in the conjunction or opposition of the luminaries (by the observation of Sturmy) is about 45 feet, but in the quadratures only 25. Because the apparent diameters of the luminaries are not here determined, let us assume them in their mean quantities, as well as the moon's declination in the equinoctial quadratures in its mean quantity, that is, °; and the versed sine of double its complement will be 1682, supposing the radius to be 1000. But the declinations of the sun in the equinoxes and of the moon in the syzygies are of no quantity, and the versed sines of double the complements are each 2000. Whence those forces become L + S in the syzygies, and in the quadratures; respectively proportional to the heights of the tides of 45 and 25 feet, or of 9 and 5 paces. And, therefore, multiplying the extremes and the means, we have , or .

But farther; I remember to have been told that in summer the ascent of the sea in the syzygies is to the ascent thereof in the quadratures as about 5 to 4. In the solstices themselves it is probable that the proportion may be something less, as about 6 to 5; whence it would follow that L is = [for then the proportion is ]. Till we can more certainly determine the proportion from observation, let us assume ; and since the heights of the tides are as the forces which excite them, and the force of the sun is able to raise the tides to the height of nine inches, the moon's force will be sufficient to raise the same to the height of four feet. And if we allow that this height may be doubled, or perhaps tripled, by that force of reciprocation which we observe in the motion of the waters, and by which their motion once begun is kept[Pg 546] up for some time, there will be force enough to generate all that quantity of tides which we really find in the ocean.

Thus we have seen that these forces are sufficient to move the sea. But, so far as I can observe, they will not be able to produce any other effect sensible on our earth; for since the weight of one grain in 4000 is not sensible in the nicest balance; and the sun's force to move the tides is 12868200 less than the force of gravity; and the sum of the forces of both moon and sun, exceeding the sun's force only in the ratio of to 1, is still 2032890 times less than the force of gravity; it is evident that both forces together are 500 times less than what is required sensibly to increase or diminish the weight of any body in a balance. And, therefore, they will not sensibly move any suspended body; nor will they produce any sensible effect on pendulums, barometers, bodies swimming in stagnant water, or in the like statical experiments. In the atmosphere, indeed, they will excite such a flux and reflux as they do in the sea, but with so small a motion that no sensible wind will be thence produced.

If the effects of both moon and sun in raising the tides (p. 454), as well as their apparent diameters, were equal among themselves, their absolute forces would (by Cor. XIV, Prop. LXVI) be as their magnitudes. But the effect of the moon is to the effect of the sun as about to 1; and the moon's diameter less than the sun's in the ratio of to , or of 45 to 16. Now the force of the moon is to be increased in the ratio of the effect directly, and in the triplicate ratio of the diameter inversely. Whence the force of the moon compared with its magnitude will be to the force of the sun compared with its magnitude in the ratio compounded of to 1, and the triplicate of 45 to 46 inversely, that is, in the ratio of about to 1. And therefore the moon, in respect of the magnitude of its body, has an absolute centripetal force greater than the sun in respect of the magnitude of its body in the ratio to to 1, and is therefore more dense in the same ratio.

In the time of 27d. 7h. 43', in which the moon makes its revolution about the earth, a planet may be revolved about the sun at the distance of 18,954 diameters of the sun from the sun's centre, supposing the mean apparent diameter of the sun to be ; and in the same time the moon may be revolved about the earth at rest, at the distance of 30 of the earth's diameters. If in both cases the number of diameters was the same, the absolute circum-terrestrial force would (by Cor. II, Prop. LXXII) be to the absolute circum-solar force as the magnitude of the earth to the magnitude of the sun. Because the number of the earth's diameters is greater in the ratio of 30 to 18,954, the body of the earth will be less in the triplicate of that ratio, that is, in the ratio of to 1. Wherefore the earth's force, for the magnitude of its body, is to the sun's force, for the magnitude of its body, as to 1; and consequently the earth's density to the sun's will be in[Pg 547] the same ratio. Since, then, the moon's density is to the sun's density as to 1, the moon's density will be to the earth's density as to , or as 23 to 16. Wherefore since the moon's magnitude is to the earth's magnitude as about 1 to , the moon's absolute centripetal force will be to the earth's absolute centripetal force as about 1 to 29, and the quantity of matter in the moon to the quantity of matter in the earth in the same ratio. And hence the common centre of gravity of the earth and moon is more exactly determined than hitherto has been done; from the knowledge of which we may now infer the moon's distance from the earth with greater accuracy. But I would rather wait till the proportion of the bodies of the moon and earth one to the other is more exactly defined from the phænomena of the tides, hoping that in the mean time the circumference of the earth may be measured from more distant stations than any body has yet employed for this purpose.

Thus I have given an account of the system of the planets. As to the fixed stars, the smallness of their annual parallax proves them to be removed to immense distances from the system of the planets: that this parallax is less than one minute is most certain; and from thence it follows that the distance of the fixed stars is above 360 times greater than the distance of Saturn from the sun. Such as reckon the earth one of the planets, and the sun one of the fixed stars, may remove the fixed stars to yet greater distances by the following arguments: from the annual motion of the earth there would happen an apparent transposition of the fixed stars, one in respect of another, almost equal to their double parallax: but the greater and nearer stars, in respect of the more remote, which are only seen by the telescope, have not hitherto been observed to have the least motion. If we should suppose that motion to be but less than 20'', the distance of the nearer fixed stars would exceed the mean distance of Saturn by above 2000 times. Again; the disk of Saturn, which is only 17'' or 18'' in diameter, receives but about of the sun's light; for so much less is that disk than the whole spherical surface of the orb of Saturn. Now if we suppose Saturn to reflect about of this light, the whole light reflected from its illuminated hemisphere will be about of the whole light emitted from the sun's hemisphere; and, therefore, since light is rarefied in the duplicate ratio of the distance from the luminous body, if the sun was times more distant than Saturn, it would yet appear as lucid as Saturn now does without its ring, that is, something more lucid than a fixed star of the first magnitude. Let us, therefore, suppose that the distance from which the sun would shine as a fixed star exceeds that of Saturn by about 100,000 times, and its apparent diameter will be 7v. 16vi. and its parallax arising from the annual motion of the earth 13'''': and so great will be the distance, the apparent diameter, and the parallax of the fixed stars of the first magnitude, in bulk and light equal to our sun.[Pg 548] Some may, perhaps, imagine that a great part of the light of the fixed stars is intercepted and lost in its passage through so vast spaces, and upon that account pretend to place the fixed stars at nearer distances; but at this rate the remoter stars could be scarcely seen. Suppose, for example, that of the light perish in its passage from the nearest fixed stars to us; then will twice perish in its passage through a double space, thrice through a triple, and so forth. And, therefore, the fixed stars that are at a double distance will be 16 times more obscure, viz., 4 times more obscure on account of the diminished apparent diameter; and, again, 4 times more on account of the lost light. And, by the same argument, the fixed stars at a triple distance will be 9 × 4 × 4, or 144 times more obscure; and those at a quadruple distance will be 16 × 4 × 4 × 4, or 1024 times more obscure; but so great a diminution of light is no ways consistent with the phænomena and with that hypothesis which places the fixed stars at different distances.

The fixed stars being, therefore, at such vast distances from one another (p. 460, 461), can neither attract each other sensibly, nor be attracted by our sun. But the comets must unavoidably be acted on by the circum-solar force; for as the comets were placed by astronomers above the moon, because they were found to have no diurnal parallax, so their annual parallax is a convincing proof of their descending into the regions of the planets. For all the comets which move in a direct course, according to the order of the signs, about the end of their appearance become more than ordinarily slow, or retrograde, if the earth is between them and the sun; and more than ordinarily swift if the earth is approaching to a heliocentric opposition with them. Whereas, on the other hand, those which move against the order of the signs, towards the end of their appearance, appear swifter than they ought to be if the earth is between them and the sun; and slower, and perhaps retrograde, if the earth is in the other side of its orbit. This is occasioned by the motion of the earth in different situations. If the earth go the same way with the comet, with a swifter motion, the comet becomes retrograde; if with a slower motion, the comet becomes slower, however; and if the earth move the contrary way, it becomes swifter; and by collecting the differences between the slower and swifter motions, and the sums of the more swift and retrograde motions, and comparing them with the situation and motion of the earth from whence they arise, I found, by means of this parallax, that the distances of the comets at the time they cease to be visible to the naked eye are always less than the distance of Saturn, and generally even less than the distance of Jupiter.

The same thing may be collected from the curvature of the way of the comets (p. 462). These bodies go on nearly in great circles while their motion continues swift; but about the end of their course, when that part[Pg 549] of their apparent motion which arises from the parallax bears a greater proportion to their whole apparent motion, they commonly deviate from those circles; and when the earth goes to one side, they deviate to the other; and this deflection, because of its corresponding with the motion of the earth, must arise chiefly from the parallax; and the quantity thereof is so considerable, as, by my computation, to place the disappearing comets a good deal lower than Jupiter. Whence it follows, that, when they approach nearer to us in their perigees and perihelions, they often descend below the orbits of Mars and the inferior planets.

Moreover, this nearness of the comets is confirmed by the annual parallax of the orbit, in so far as the same is pretty nearly collected by the supposition that the comets move uniformly in right lines. The method of collecting the distance of a comet according to this hypothesis from four observations (first attempted by Kepler, and perfected by Dr. Wallis and Sir Christopher Wren) is well known; and the comets reduced to this regularity generally pass through the middle of the planetary region. So the comets of the year 1607 and 1618, as their motions are defined by Kepler, passed between the sun and the earth; that of the year 1664 below the orbit of Mars; and that in 1680 below the orbit of Mercury, as its motion was defined by Sir Christopher Wren and others. By a like rectilinear hypothesis, Hevelius placed all the comets about which we have any observations below the orbit of Jupiter. It is a false notion, therefore, and contrary to astronomical calculation, which some have entertained, who, from the regular motion of the comets, either remove them into the regions of the fixed stars, or deny the motion of the earth; whereas their motions cannot be reduced to perfect regularity, unless we suppose them to pass through the regions near the earth in motion; and these are the arguments drawn from the parallax, so far as it can be determined without an exact knowledge of the orbits and motions of the comets.

The near approach of the comets is farther confirmed from the light of their heads (p. 463, 465); for the light of a celestial body, illuminated by the sun, and receding to remote parts, is diminished in the quadruplicate proportion of the distance; to wit, in one duplicate proportion on account of the increase of the distance from the sun; and in another duplicate proportion on account of the decrease of the apparent diameter. Hence it may be inferred, that Saturn being at a double distance, and having its apparent diameter nearly half of that of Jupiter, must appear about 16 times more obscure; and that, if its distance were 4 times greater, its light would be 256 times less; and therefore would be hardly perceivable to the naked eye. But now the comets often equal Saturn's light, without exceeding him in their apparent diameters. So the comet of the year 1668, according to Dr. Hooke's observations, equalled in brightness the light of a fixed star of the first magnitude; and its head, or the star in[Pg 550] the middle of the coma, appeared, through a telescope of 15 feet, as lucid as Saturn near the horizon; but the diameter of the head was only 25'' that is, almost the same with the diameter of a circle equal to Saturn and his ring. The coma or hair surrounding the head was about ten times as broad; namely, min. Again; the least diameter of the hair of the comet of the year 1682, observed by Mr. Flamsted with a tube of 16 feet and measured with the micrometer, was 2' 0''; but the nucleus, or star in the middle, scarcely possessed the tenth part of this breadth, and was therefore only 11'' or 12'' broad; but the light and clearness of its head exceeded that of the year 1680, and was equal to that of the stars of the first or second magnitude. Moreover, the comet of the year 1665, in April, as Hevelius informs us, exceeded almost all the fixed stars in splendor, and even Saturn itself, as being of a much more vivid colour; for this comet was more lucid than that which appeared at the end of the foregoing year and was compared to the stars of the first magnitude. The diameter of the coma was about 6'; but the nucleus, compared with the planets by means of a telescope, was plainly less than Jupiter, and was sometimes thought less, sometimes equal to the body of Saturn within the ring. To this breadth add that of the ring, and the whole face of Saturn will be twice as great as that of the comet, with a light not at all more intense; and therefore the comet was nearer to the sun than Saturn. From the proportion of the nucleus to the whole head found by these observations, and from its breadth, which seldom exceeds 8' or 12', it appears that the stars of the comets are most commonly of the same apparent magnitude as the planets; but that their light may be compared oftentimes with that of Saturn, and sometimes exceeds it. And hence it is certain that in their perihelia their distances can scarcely be greater than that of Saturn. At twice that distance, the light would be four times less, which besides by its dim paleness would be as much inferior to the light of Saturn as the light of Saturn is to the splendor of Jupiter: but this difference would be easily observed. At a distance ten times greater, their bodies must be greater than that of the sun; but their light would be 100 times fainter than that of Saturn. And at distances still greater, their bodies would far exceed the sun; but, being in such dark regions, they must be no longer visible. So impossible is it to place the comets in the middle regions between the sun and fixed stars, accounting the sun as one of the fixed stars; for certainly they would receive no more light there from the sun than we do from the greatest of the fixed stars.

So far we have gone without considering that obscuration which comets suffer from that plenty of thick smoke which encompasseth their heads, and through which the heads always shew dull as through a cloud; for by how much the more a body is obscured by this smoke, by so much the more near it must be allowed to come to the sun, that it may vie with the[Pg 551] planets in the quantity of light which it reflects; whence it is probable that the comets descend far below the orbit of Saturn, as we proved before from their parallax. But, above all, the thing is evinced from their tails, which must be owing either to the sun's light reflected from a smoke arising from them, and dispersing itself through the æther, or to the light of their own 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 is propagated through such a vast extent of space, and with such a velocity of expansion, as will seem altogether incredible; in the latter case the whole light of both head and tail must be ascribed to the central nucleus. But, then, if we suppose all this light to be united and condensed within the disk of the nucleus, certainly the nucleus will by far exceed Jupiter itself 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 be much more illuminated by the sun, and therefore much nearer to it. So the comet that appeared Dec. 12 and 15, O.S. Anno 1679, at the time it emitted a very shining tail, whose splendor was equal to that of many stars like Jupiter, if their light were dilated and spread through so great a space, was, as to the magnitude of its nucleus, less than Jupiter (as Mr. Flamsted observed), and therefore was much nearer to the sun: nay, it was even less than Mercury. For on the 17th of that month, when it was nearer to the earth, it appeared to Cassini through a telescope of 35 feet a little less than the globe of Saturn. On the 8th of this month, in the morning, Dr. Halley saw the tail, appearing broad and very short, and as if it rose from the body of the sun itself, at that time very near its rising. Its form was like that of an extraordinary bright cloud; nor did it disappear till the sun itself began to be seen above the horizon. Its splendor, therefore, exceeded the light of the clouds till the sun rose, and far surpassed that of all the stars together, as yielding only to the immediate brightness of the sun itself. Neither Mercury, nor Venus, nor the moon itself, are seen so near the rising sun. Imagine all this dilated light collected together, and to be crowded into the orbit of the comet's nucleus which was less than Mercury; by its splendor, thus increased, becoming so much more conspicuous, it will vastly exceed Mercury, and therefore must be nearer to the sun. On the 12th and 15th of the same month, this tail, extending itself over a much greater space, appeared more rare; but its light was still so vigorous as to become visible when the fixed stars were hardly to be seen, and soon after to appear like a fiery beam shining in a wonderful manner. From its length, which was 40 or 50 degrees, and its breadth of 2 degrees, we may compute what the light of the whole must be.

[Pg 553]

This near approach of the comets to the sun is confirmed from the situation they are seen in when their tails appear most resplendent; for when the head passes by the sun, and lies hid under the solar rays, very bright and shining tails, like fiery beams, are said to issue from the horizon; but afterwards, when the head begins to appear, and is got farther from the sun, that splendor always decreases, and turns by degrees into a paleness like to that of the milky way, but much more sensible at first; after that vanishing gradually. Such was that most resplendent comet described by Aristotle, Lib. 1, Meteor. 6. "The head thereof could not be seen, because it set before the sun, or at least was hid under the sun's rays; but the next day it was seen as well as might be; for, having left the sun but a very little way, it set immediately after it; and the scattered light of the head obscured by the too great splendour (of the tail) did not yet appear. But afterwards (says Aristotle), when the splendour of the tail was now diminished (the head of), the comet recovered its native brightness. And the splendour of its tail reached now to a third part of the heavens (that is, to 60°). It appeared in the winter season, and, rising to Orion's girdle, there vanished away." Two comets of the same kind are described by Justin, Lib. 37, which, according to his account, "shined so bright, that the whole heaven seemed to be on fire; and by their greatness filled up a fourth part of the heavens, and by their splendour exceeded that of the sun." By which last words a near position of these bright comets and the rising or setting sun is intimated (p. 494, 495). We may add to these the comet of the year 1101 or 1106, "the star of which was small and obscure (like that of 1680); but the splendour arising from it extremely bright, reaching like a fiery beam to the east and north," as Hevelius has it from Simeon, the monk of Durham. It appeared at the beginning of February about the evening in the south-west. From this and from the situation of the tail we may infer that the head was near the sun. Matthew Paris says, "it was about one cubit from the sun; from the third [or rather the sixth] to the ninth hour sending out a long stream of light." The comet of 1264, in July, or about the solstice, preceded the rising sun, sending out its beams with a great light towards the west as far as the middle of the heavens; and at the beginning it ascended a little above the horizon: but as the sun went forwards it retired every day farther from the horizon, till it passed by the very middle of the heavens. It is said to have been at the beginning large and bright, having a large coma, which decayed from day to day. It is described in Append. Matth. Paris, Hist. Ang. after this manner: "An. Christi 1265, there appeared a comet so wonderful, that none then living had ever seen the like; for, rising from the east with a great brightness, it extended itself with a great light as far as the middle of the hemisphere towards the west." The Latin original being somewhat barbarous and obscure, it is here subjoined. Ab oriente enim cum magno fulgore surgens, usque ad medium hemisphærii versus occidentem, omnia perlucide pertrahebat.

BOOK III. (part 6)

"In the year 1401 or 1402, the sun being got below the horizon, there appeared in the west a bright and shining comet, sending out a tail upwards, in splendor like a flame of fire, and in form like a spear, darting its rays from west to east. When the sun was sunk below the horizon, by the lustre of its own rays it enlightened all the borders of the earth, not permitting the other stars to shew their light, or the shades of night to darken the air, because its light exceeded that of the others, and extended itself to the upper part of the heavens, flaming," &c., Hist. Byzant. Duc. Mich. Nepot. From the situation of the tail of this comet, and the time of its first appearance, we may infer that the head was then near the sun, and went farther from him every day; for that comet continued three months. In the year 1527, Aug. 11, about four in the morning, there was seen almost throughout Europe a terrible comet in Leo, which continued flaming an hour and a quarter every day. It rose from the east, and ascended to the south and west to a prodigious length. It was most conspicuous to the north, and its cloud (that is, its tail) was very terrible; having, according to the fancies of the vulgar, the form of an arm a little bent holding a sword of a vast magnitude. In the year 1618, in the end of November, there began a rumour, that there appeared about sun-rising a bright beam, which was the tail of a comet whose head was yet concealed within the brightness of the solar rays. On Nov. 24, and from that time, the comet itself appeared with a bright light, its head and tail being extremely resplendent. The length of the tail, which was at first 20 or 30 deg., increased till December 9, when it arose to 75 deg., but with a light much more faint and dilute than at the beginning. In the year 1668, March 5, N. S., about 7 in the evening, P. Valent. Estancius, being in Brazil, saw a comet near the horizon in the south-west. Its head was small, and scarcely discernible, but its tail extremely bright and refulgent, so that the reflection of it from the sea was easily seen by those who stood upon the shore. This great splendor lasted but three days, decreasing very remarkably from that time. The tail at the beginning extended itself from west to south, and in a situation almost parallel to the horizon, appearing like a shining beam 23 deg. in length. Afterwards, the light decreasing, its magnitude increased till the comet ceased to be visible; so that Cassini, at Bologna, saw it (Mar. 10, 11, 12) rising from the horizon 32 deg. in length. In Portugal it is said to have taken up a fourth part of the heavens (that is, 45 deg.), extending itself from west to east with a notable brightness; though the whole of it was not seen, because the head in this part of the world always lay hid below the horizon. From the increase of the tail it is plain that the head receded from the sun, and was nearest to it at the beginning, when the tail appeared brightest.

To all these we may add the comet of 1680, whose wonderful splendor at the conjunction of the head with the sun was above described. But so[Pg 554] great a splendor argues the comets of this kind to have really passed near the fountain of light, especially since the tails never shine so much in their opposition to the sun; nor do we read that fiery beams have ever appeared there.

Lastly, the same thing is inferred (p. 466, 467) from the light of the heads increasing in the recess of the comets from the earth towards the sun, and decreasing in their return from the sun towards the earth; for so the last comet of the year 1665 (by the observation of Hevelius), from the time that it was first seen, was always losing of its apparent motion, and therefore had already passed its perigee; yet the splendor of its head was daily increasing, till, being hid by the sun's rays, the comet ceased to appear. The comet of the year 1683 (by the observation of the same Hevelius), about the end of July, when it first appeared, moved at a very slow rate, advancing only about 40 or 45 minutes in its orbit in a day's time. But from that time its diurnal motion was continually upon the increase till September 4, when it arose to about 5 degrees; and therefore in all this interval 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 the 6th, Hevelius found it only 6' 5'', including the coma; which, September 2, he observed 9' 7''. And therefore its head appeared far less about the beginning than towards the end of its motion, though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the same Hevelius declares. Wherefore in all this interval of time, on account of its recess from the sun, it decreased in splendor, notwithstanding its access towards the earth. The comet of the 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 splendor of their heads was seen two weeks before, when they had just got clear of the 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 to the observations of Cysatus, Dec. 1, appeared greater than the stars of the first magnitude; and, Dec. 16 (being then in its perigee), of a small magnitude, and the splendor or clearness was much diminished. Jan. 7, Kepler, being uncertain about the head, left off observing. Dec. 12, the head of the last comet was seen and observed by Flamsted at the distance of 9 degrees from the sun, which a star of the third magnitude could hardly have been. December 15 and 17, the same appeared like a star of the third magnitude, its splendor being diminished by the bright clouds near the setting sun. Dec. 26, when it moved with the greatest swiftness, and was almost in its perigee, it was inferior to Os Pegasi, a star of the third magnitude. Jan. 3, it appeared like a star of the fourth; Jan. 9, like a star of the fifth. Jan. 13, it disappeared, by reason of the[Pg 555] brightness of the moon, which was then in its increase. Jan. 25, it was scarcely equal to the stars of the seventh magnitude. If we take equal times on each hand of the perigee, the heads placed at remote distances would have shined equally before and after, because of their equal distances from the earth. That in one case they shined very bright, and in the other vanished, is to be ascribed to the nearness of the sun in the first case, and his distance in the other; and from the great difference of the light in these two cases we infer its great nearness in the first of them; for the light of the comets uses to be regular, and to appear greatest when their heads move the swiftest, and are therefore in their perigees, excepting in so far as it is increased by their nearness to the sun.

From these things I at last discovered why the comets frequent so much the region of the sun. If they were to be seen in the regions a great way beyond Saturn, they must appear oftener in these parts of the heavens that are opposite to the sun; for those which are in that situation would be nearer to the earth, and the interposition of the sun would obscure the others: but, looking over the history of comets, I find that four or five times more have been seen in the hemisphere towards the sun than in the opposite hemisphere; besides, without doubt, not a few which have been hid by the light of the sun; for comets descending into our parts neither emit tails, nor are so well illuminated by the sun, as to discover themselves to our naked eyes, till they are come nearer to us than Jupiter. But the far greater part of that spherical space, which is described about the sun with so small an interval, lies on that side of the earth which regards the sun, and the comets in that greater part are more strongly illuminated, as being for the most part nearer to the sun: besides, from the remarkable eccentricity of their orbits, it comes to pass that their lower apsides are much nearer to the sun than if their revolutions were performed in circles concentric to the sun.

Hence also we understand why the tails of the comets, while their heads are descending towards the sun, always appear short and rare, and are seldom said to have exceeded 15 or 20 deg. in length; but in the recess of the heads from the sun often shine like fiery beams, and soon after reach to 40, 50, 60, 70 deg. in length, or more. This great splendor and length of the tails arises from the heat which the sun communicates to the comet as it passes near it. And thence, I think, it may be concluded, that all the comets that have had such tails have passed very near the sun.

Hence also we may collect that the tails arise from the atmospheres of the heads (p. 487 to 488): but we have had three several opinions about the tails of comets; for some will have it that they are nothing else but the beams of the sun's light transmitted through the comets' heads, which they suppose to be transparent; others, that they proceed from the refraction which light suffers in passing from the comet's head to the earth;[Pg 556] and, lastly, others, that they are a sort of clouds or vapour constantly rising from the comets' heads, and tending towards the parts opposite to the sun. The first is the opinion of such as are yet unacquainted with optics; for the beams of the sun are not seen in a darkened room, but in consequence of the light that is reflected from them by the little particles of dust and smoke which are always flying about in the air; and hence it is that in air impregnated with thick smoke they appear with greater brightness, and are more faintly and more difficultly seen in a finer air; but in the heavens, where there is no matter to reflect the light, they are not to be seen at all. Light is not seen as it is in the beams, but as it is thence reflected to our eyes; for vision is not made but by rays falling upon the eyes, and therefore there must be some reflecting matter in those parts where the tails of comets are seen; and so the argument turns upon the third opinion; for that reflecting matter can be no where found but in the place of the tail, because otherwise, since all the celestial spaces are equally illuminated by the sun's light, no part of the heavens could appear with more splendor than another. The second opinion is liable to many difficulties. The tails of comets are never seen variegated with those colours which ever use to be inseparable from refraction; and the distinct transmission of the light of the fixed stars and planets to us is a demonstration that the æther or celestial medium is not endowed with any refractive power. For as to what is alledged that the fixed stars have been sometimes seen by the Egyptians environed with a coma or capillitium because that has but rarely happened, it is rather to be ascribed to a casual refraction of clouds, as well as the radiation and scintillation of the fixed stars to the refractions both of the eyes and air; for upon applying a telescope to the eye, those radiations and scintillations immediately disappear. By the tremulous agitation of the air and ascending vapours, it happens that the rays of light are alternately turned aside from the narrow space of the pupil of the eye; but no such thing can have place in the much wider aperture of the object-glass of a telescope; and hence it is that a scintillation is occasioned in the former case which ceases in the latter; and this cessation in the latter case is a demonstration of the regular transmission of light through the heavens without any sensible refraction. But, to obviate an objection that may be made from the appearing of no tail in such comets as shine but with a faint light, as if the secondary rays were then too weak to affect the eyes, and for this reason it is that the tails of the fixed stars do not appear, we are to consider that by the means of telescopes the light of the fixed stars may be augmented above an hundred fold and yet no tails are seen; that the light of the planets is yet more copious without any tail, but that comets are seen sometimes with huge tails when the light of their heads is but faint and dull; for so it happened in the comet of the year 1680, when in the month of December[Pg 557] it was scarcely equal in light to the stars of the second magnitude, and yet emitted a notable tail, extending to the length of 40°, 50°, 60°, or 70°, and upwards; and afterwards, on the 27th and 28th of January, the head appeared but as a star of the seventh magnitude; but the tail (as was said above), with a light that was sensible enough, though faint, was stretched out to 6 or 7 degrees in length, and with a languishing light that was more difficultly seen, even to 12° and upwards. But on the 9th and 10th of February, when to the naked eye the head appeared no more, I saw through a telescope the tail of 2° in length. But farther; if the tail was owing to the refraction of the celestial matter, and did deviate from the opposition of the sun, according as the figure of the heavens requires, that deviation, in the same places of the heavens, should be always directed towards the same parts: but the comet of the year 1680, December 28d. P. M. at London, was seen in Pisces, 8° 41', with latitude north 28° 6', while the sun was in Capricorn 18° 26'. And the comet of the year 1577, December 29, was in Pisces 8° 41', with latitude north 28° 40'; and the sun, as before, in about Capricorn 18° 26'. In both cases the situation of the earth was the same, and the comet appeared in the same place of the heavens; yet in the former case the tail of the comet (as well by my observations as by the observations of others) deviated from the opposition of the sun towards the north by an angle of degrees, whereas in the latter there was (according to the observation of Tycho) a deviation of 21 degrees towards the south. The refraction, therefore, of the heavens being thus disproved, it remains that the phænomena of the tails of comets must be derived from some reflecting matter. That vapours sufficient to fill such immense spaces may arise from the comet's atmospheres, may be easily understood from what follows.

It is well known that the air near the surface of our earth possesses a space about 1200 times greater than water of the same weight; and therefore a cylindric column of air 1200 feet high is of equal weight with a cylinder of water of the same breadth, and but one foot high. But a cylinder of air reaching to the top of the atmosphere is of equal weight with a cylinder of water about 33 feet high; and therefore if from the whole cylinder of air the lower part of 1200 feet high is taken away, the remaining upper part will be of equal weight with a cylinder of water 32 feet high. Wherefore at the height of 1200 feet, or two furlongs, the weight of the incumbent air is less, and consequently the rarity of the compressed air greater, than near the surface of the earth in the ratio of 33 to 32. And, having this ratio, we may compute the rarity of the air in all places whatsoever (by the help of Cor. Prop. XXII, Book II), supposing the expansion thereof to be reciprocally proportional to its compression; and this proportion has been proved by the experiments of Hooke and others. The result of the computation I have set down in the following[Pg 558] table, in the first column of which you have the height of the air in miles, whereof 4000 make a semi-diameter of the earth; in the second the compression of the air, or the incumbent weight; in the third its rarity or expansion, supposing gravity to decrease in the duplicate ratio of the distances from the earth's centre. And the Latin numeral characters are here used for certain numbers of ciphers, as 0,xvii 1224 for 0,000000000000000001224, and 26956 xv for 26956000000000000000.

AIR's

Height. Compression. Expansion.

0 33 1

5 17,8515 1,8486

10 9,6717 3,4151

20 2,852 11,571

40 0,2525 136,83

400 0,xvii 1224 26956 xv

4000 0,cv. 4465 73907 cii

40000 0,cxcii 1628 20263 clxxxix

400000 0,ccx 7895 41798 ccvii

4000000 0,ccxii 9878 33414 ccix

Infinite. 0,ccxii 6041 54622 ccix

But from this table it appears that the air, in proceeding upwards, is rarefied in such manner, that a sphere of that air which is nearest to the earth, of but one inch in diameter, if dilated with that rarefaction which it would have at the height of one semi-diameter of the earth, would fill all the planetary regions as far as the sphere of Saturn, and a great, way beyond; and at the height of ten semi-diameters of the earth would fill up more space than is contained in the whole heavens on this side the fixed stars, according to the preceding computation of their distance. And though, by reason of the far greater thickness of the atmospheres of comets, and the great quantity of the circum-solar centripetal force, it may happen that the air in the celestial spaces, and in the tails of comets, is not so vastly rarefied, yet from this computation it is plain that a very small quantity of air and vapour is abundantly sufficient to produce all the appearances of the tails of comets; for that they are indeed of a very notable rarity appears from the shining of the stars through them. The atmosphere of the earth, illuminated by the sun's light, though but of a few miles in thickness, obscures and extinguishes the light not only of all the stars, but even of the moon itself; whereas the smallest stars are seen to shine through the immense thickness of the tails of comets, likewise illuminated by the sun, without the least diminution of their splendor.

Kepler ascribes the ascent of the tails of comets to the atmospheres of their heads, and their direction towards the parts opposite to the sun to the action of the rays of light carrying along with them the matter of the comets' tails; and without any great incongruity we may suppose that, in so free spaces, so fine a matter as that of the æther may yield to the action[Pg 559] of the rays of the sun's light, though those rays are not able sensibly to move the gross substances in our parts, which are clogged with so palpable a resistance. Another author thinks that there may be a sort of particles of matter endowed with a principle of levity as well as others are with a power of gravity; that the matter of the tails of comets may be of the former sort, and that its ascent from the sun may be owing to its levity; but, considering the gravity of terrestrial bodies is as the matter of the bodies, and therefore can be neither more nor less in the same quantity of matter, I am inclined to believe that this ascent may rather proceed from the rarefaction of the matter of the comets' tails. The ascent of smoke in a chimney is owing to the impulse of the air with which it is entangled. The air rarefied by heat ascends, because its specific gravity is diminished, and in its ascent carries along with it the smoke with which it is engaged. And why may not the tail of a comet rise from the sun after the same manner? for the sun's rays do not act any way upon the mediums which they pervade but by reflection and refraction; and those reflecting particles heated by this action, heat the matter of the æther which is involved with them. That matter is rarefied by the heat which it acquires, and because by this rarefaction the specific gravity, with which it tended towards the sun before, is diminished, it will ascend therefrom like a stream, and carry along with it the reflecting particles of which the tail of the comet is composed; the impulse of the sun's light, as we have said, promoting the ascent.

But that the tails of comets do arise from their heads (p. 488), and tend towards the parts opposite to the sun, is farther confirmed from the laws which the tails observe; for, lying in the planes of the comets' orbits which pass through the sun, they constantly deviate from the opposition of the sun towards the parts which the comets' heads in their progress along those orbits have left; and to a spectator placed in those planes they appear in the parts directly opposite to the sun; but as the spectator recedes from those planes, their deviation begins to appear, and daily becomes greater. And the deviation, cæteris paribus, appears less when the tail is more oblique to the orbit of the comet, as well as when the head of the comet approaches nearer to the sun; especially if the angle of deviation is estimated near the head of the comet. Farther; the tails which have no deviation appear straight, but the tails which deviate are likewise bended into a certain curvature; and this curvature is greater when the deviation is greater, and is more sensible when the tail, cæteris paribus, is longer; for in the shorter tails the curvature is hardly to be perceived. And the angle of deviation is less near the comet's head, but greater towards the other end of the tail, and that because the lower side of the tail regards the parts from which the deviation is made, and which lie in a right line drawn out infinitely from the sun through the comet's head. And the tails that are[Pg 560] longer and broader, and shine with a stronger light, appear more resplendent and more exactly defined on the convex than on the concave side. Upon which accounts it is plain that the phænomena of the tails of comets depend upon the motions of their heads, and by no means upon the places of the heavens in which their heads are seen; and that, therefore, the tails of the comets do not proceed from the refraction of the heavens, but from their own heads, which furnish the matter that forms the tail; for as in our air the smoke of a heated body ascends either perpendicularly, if the body is at rest, or obliquely if the body is moved obliquely, so in the heavens, where all the bodies gravitate towards the sun, smoke and vapour must (as we have already said) ascend from the sun, and either rise perpendicularly, if the smoking body is at rest, or obliquely, if the body, in the progress of its motion, is always leaving those places from which the upper or higher parts of the vapours had risen before. And that obliquity will be less where the vapour ascends with more velocity, to wit, near the smoking body, when that is near the sun; for there the force of the sun by which the vapour ascends is stronger. But because the obliquity is varied, the column of vapour will be incurvated; and because the vapour in the preceding side is something more recent, that is, has ascended something more lately from the body, it will therefore be something more dense on that side, and must on that account reflect more light, as well as be better defined; the vapour on the other side languishing by degrees, and vanishing out of sight.

But it is none of our present business to explain the causes of the appearances of nature. Let those things which we have last said be true or false, we have at least made out, in the preceding discourse, that the rays of light are directly propagated from the tails of comets in right lines through the heavens, in which those tails appear to the spectators wherever placed; and consequently the tails must ascend from the heads of the comets towards the parts opposite to the sun. And from this principle we may determine anew the limits of their distances in manner following. Let S represent the sun, T the earth, STA the elongation of a comet from the sun, and ATB the apparent length of its tail; and because the light is propagated from the extremity of the tail in the direction of the right, line TB, that extremity must lie somewhere in the line TB. Suppose it in D, and join DS cutting TA in C. Then, because the tail is always stretched out towards the parts nearly opposite to the sun, and therefore[Pg 561] the sun, the head of the comet, and the extremity of the tail, lie in a right line, the comet's head will be found in C. Parallel to TB draw SA, meeting the line TA in A, and the comet's head C must necessarily be found between T and A, because the extremity of the tail lies somewhere in the infinite line TB; and all the lines SD which can possibly be drawn from the point S to the line TB must cut the line TA somewhere between T and A. Wherefore the distance of the comet from the earth cannot exceed the interval TA, nor its distance from the sun the interval SA beyond, or ST on this side the sun. For instance: the elongation of the comet of 1680 from the sun, Dec. 12, was 9°, and the length of its tail 35° at least. If, therefore, a triangle TSA is made, whose angle T is equal to the elongation 9°, and angle A equal to ATB, or to the length of the tail, viz., 35°, then SA will be to ST, that is, the limit of the greatest possible distance of the comet from the sun to the semi-diameter of the orbis magnus, as the sine of the angle T to the sine of the angle A, that is, as about 3 to 11. And therefore the comet at that time was less distant from the sun than by of the earth's distance from the sun, and consequently either was within the orb of Mercury, or between that orb and the earth. Again, Dec. 21, the elongation of the comet from the sun was °, and the length of its tail 70°. Wherefore as the sine of ° to the sine of 70°, that is, as 4 to 7, so was the limit of the comet's distance from the sun to the distance of the earth from the sun, and consequently the comet had not then got without the orb of Venus. Dec. 28, the elongation of the comet from the sun was 55°, and the length of its tail 56°; and therefore the limit of the comet's distance from the sun was not yet equal to the distance of the earth from the same, and consequently the comet had not then got without the earth's orbit. But from its parallax we find that its egress from the orbit happened about Jan. 5, as well as that it had descended far within the orbit of Mercury. Let us suppose it to have been in its perihelion Dec. the 8th, when it was in conjunction with the sun; and it will follow that in the journey from its perihelion to its exit out of the earth's orbit it had spent 28 days; and consequently that in the 26 or 27 days following, in which it ceased to be farther seen by the naked eye, it had scarcely doubled its distance from the sun; and by limiting the distances of other comets by the like arguments, we come at last to this conclusion,—that all comets, during the time in which they are visible by us, are within the compass of a spherical space described about the sun as a centre, with a radius double, or at most triple, of the distance of the earth from the sun.

And hence it follows that the comets, during the whole time of their appearance unto us, being within the sphere of activity of the circum-solar force, and therefore agitated by the impulse of that force, will (by Cor. 1, Prop. XII, Book I, for the same reason as the planets) be made to[Pg 562] move in conic sections that have one focus in the centre of the sun, and by radii drawn to the sun, to describe areas proportional to the times; for that force is propagated to an immense distance, and will govern the motions of bodies far beyond the orbit of Saturn.

There are three hypotheses about comets (p. 466); for some will have it that they are generated and perish as often as they appear and vanish; others, that they come from the regions of the fixed stars, and are seen by us in their passage through the system of our planets; and, lastly, others, that they are bodies perpetually revolving about the sun in very eccentric orbits. In the first case, the comets, according to their different velocities, will move in conic sections of all sorts; in the second, they will describe hyperbolas, and in either of the two will frequent indifferently all quarters of the heavens, as well those about the poles as those towards the ecliptic; in the third, their motions will be performed in ellipses very eccentric, and very nearly approaching to parabolas. But (if the law of the planets is observed) their orbits will not much decline from the plane of the ecliptic; and, so far as I could hitherto observe, the third case obtains; for the comets do, indeed, chiefly frequent the zodiac, and scarcely ever attain to a heliocentric latitude of 40°. And that they move in orbits very nearly parabolical, I infer from their velocity; for the velocity with which a parabola is described is every where to the velocity with which a comet or planet may be revolved about the sun in a circle at the same distance in the subduplicate ratio of 2 to 1 (by Cor. VII, Prop. XVI); and, by my computation, the velocity of comets is found to be much about the same. I examined the thing by inferring nearly the velocities from the distances, and the distances both from the parallaxes and the phænomena of the tails, and never found the errors of excess or defect in the velocities greater than what might have arose from the errors in the distances collected after that manner. But I likewise made use of the reasoning that follows.

[Pg 563]

Supposing the radius of the orbis magnus to be divided into 1000 parts: let the numbers in the first column of the following table represent the distance of the vertex of the parabola from the sun's centre, expressed by those parts: and a comet in the times expressed in col. 2, will pass from its perihelion to the surface of the sphere which is described about the sun as a centre with the radius of the orbis magnus; and in the times expressed in col. 3, 4, and 5, it will double, triple, and quadruple, that its distance from the sun.

TABLE I.

The distance

of a

comet's

perihelion

from the

Sun's centre. The time of a comet's passage from its perihelion to a

distance from the sun equal to

The radii of

the orbis

magnus. To its

double. To its

triple. To its

Quadruple.

d. h. ' '' d. h. ' '' d. h. ' '' d. h. ' ''

0 27 11 12 77 16 28 142 17 14 219 17 30

5 27 16 07 77 23 14

10 27 21 00 78 06 24

20 28 06 40 78 20 13 144 03 19 221 08 54

40 29 01 32 79 23 34

80 30 13 25 82 04 56

160 33 05 29 86 10 26 153 16 08 232 12 20

320 37 13 46 93 23 38

640 37 09 49 105 01 28

1280

106 06 35 200 06 43 297 03 46

2560

147 22 31 300 06 03

[This table, here corrected, is made on the supposition that the earth's diurnal motion is just 59', and the measure of one minute loosely 0,2909, in respect of the radius 1000. If those measures are taken true, the true numbers of the table will all come out less. But the difference, even when greatest, and to the quadruple of the earth’s distance from the sun, amounts only to 16h. 55'.]

The time of a comet's ingress into the sphere of the orbis magnus, or of its egress from the same, may be inferred nearly from its parallax, but with more expedition by the following

TABLE II.

The apparent

elongation of

a comet from

the sun. Its apparent diurnal

motion in its

own orbit. Its distance from

the earth in parts

whereof the radius

of the orbis magnus

contains 1000.

Direct. Retrog.

60° 2° 18' 00° 20' 1000

65 2 33 00 35 845

70 2 55 00 57 684

72 3 07 01 09 618

74 3 23 01 25 551

76 3 43 01 45 484

78 4 10 02 12 416

80 4 57 02 49 347

82 5 45 03 47 278

84 7 18 05 20 209

86 10 27 08 19 140

88 18 37 16 39 70

90 Infi'ite Infi'ite 00

[Pg 564]

The ingress of a comet into the sphere of the orbis magnus, or its egress from the same, happens at the time of its elongation from the sun, expressed in col. 1, against its diurnal motion. So in the comet of 1681, Jan. 4, O.S. the apparent diurnal motion in its orbit was about 3° 5', and the corresponding elongation °; and the comet had acquired this elongation from the sun Jan. 4, about six in the evening. Again, in the year 1680, Nov. 11, the diurnal motion of the comet that then appeared was about °; and the corresponding elongation happened Nov. 10, a little before midnight. Now at the times named these comets had arrived at an equal distance from the sun with the earth, and the earth was then almost in its perihelion. But the first table is fitted to the earth's mean distance from the sun assumed of 1000 parts; and this distance is greater by such an excess of space as the earth might describe by its annual motion in one day's time, or the comet by its motion in 16 hours. To reduce the comet to this mean distance of 1000 parts, we add those 16 hours to the former time, and subduct them from the latter; and thus the former becomes Jan. 4d. 10h. afternoon; the latter Nov. 10, about six in the morning. But from the tenor and progress of the diurnal motions it appears that both comets were in conjunction with the sun between Dec. 7 and Dec. 8; and from thence to Jan. 4d. 10h. afternoon on one side, and to Nov. 10d. 6h. of the morning on the other, there are about 28 days. And so many days (by Table 1) the motions in parabolic trajectories do require.

But though we have hitherto considered those comets as two, yet, from the coincidence of their perihelions and agreement of their velocities, it is probable that in effect they were but one and the same; and if so, the orbit of this comet must have either been a parabola, or at least a conic section very little differing from a parabola, and at its vertex almost in contact with the surface of the sun. For (by Tab. 2) the distance of the comet from the earth, Nov. 10, was about 360 parts, and Jan. 4, about 630. From which distances, together with its longitudes and latitudes, we infer the distance of the places in which the comet was at those times to have been about 280: the half of which, viz., 140, is an ordinate to the comet's orbit, cutting off a portion of its axis nearly equal to the radius of the orbis magnus, that is, to 1000 parts. And, therefore, dividing the square of the ordinate 140 by 1000, the segment of the axis, we find the latus rectum 19, 16, or in a round number 20; the fourth part whereof, 5, is the distance of the vertex of the orbit from the sun's centre. But the time corresponding to the distance of 5 parts in Tab. 1 is 27d. 16h. 7'. In which time, if the comet moved in a parabolic orbit, it would have been carried from its perihelion to the surface of the sphere of the orbis magnus described with the radius 1000, and would have spent the double of that, time, viz., 55d. in the whole course of its motion within that sphere: and so in fact it did; for from Nov. 10d. 6h. of the morning, the[Pg 565] time of the comet's ingress into the sphere of the orbis magnus, to Jan. 4d. 10h. afternoon, the time of its egress from the same, there are 55d. 16h. The small difference of in this rude way of computing is to be neglected, and perhaps may arise from the comet's motion being some small matter slower, as it must have been if the true orbit in which it was carried was an ellipsis. The middle time between its ingress and egress was December 8d. 2h. of the morning; and therefore at this time the comet ought to have been in its perihelion. And accordingly that very day, just before sunrising, Dr. Halley (as we said) saw the tail short and broad, but very bright, rising perpendicularly from the horizon. From the position of the tail it is certain that the comet had then crossed over the ecliptic, and got into north latitude, and therefore had passed by its perihelion, which lay on the other side of the ecliptic, though it had not yet come into conjunction with the sun; and the comet [see more of this famous comet, p. 475 to 486] being at this time between its perihelion and its conjunction with the sun, must have been in its perihelion a few hours before; for in so near a distance from the sun it must have been carried with great velocity, and have apparently described almost half a degree every hour.

By like computations I find that the comet of 1618 entered the sphere of the orbis magnus December 7, towards sun-setting; but its conjunction with the sun was Nov. 9, or 10, about 28 days intervening, as in the preceding comet; for from the size of the tail of this, in which it was equal to the preceding, it is probable that this comet likewise did come almost into a contact with the sun. Four comets were seen that year of which this was the last. The second, which made its first appearance October 31, in the neighbourhood of the rising sun, and was soon after hid under the sun's rays, I suspect to have been the same with the fourth, which emerged out of the sun's rays about Nov. 9. To these we may add the comet of 1607, which entered the sphere of the orbis magnus Sept. 14, O.S. and arrived at its perihelion distance from the sun about October 19, 35 days intervening. Its perihelion distance subtended an apparent angle at the earth of about 23 degrees, and was therefore of 390 parts. And to this number of parts about 34 days correspond in Tab. 1. Farther; the comet of 1665 entered the sphere of the orbis magnus about March 17, and came to its perihelion about April 16, 30 days intervening. Its perihelion distance subtended an angle at the earth of about seven degrees, and therefore was of 122 parts: and corresponding to this number of parts, in Tab. 1, we find 30 days. Again; the comet of 1682 entered the sphere of the orbis magnus about Aug. 11, and arrived at its perihelion about Sep. 16, being then distant from the sun by about 350 parts, to which, in Tab. 1, belong days. Lastly; that memorable comet of Regiomontanus, which in 1472 was carried through the circum-polar parts of our northern hemisphere with such rapidity as to describe 40[Pg 566] degrees in one day, entered the sphere of the orbis magnus Jan 21, about the time that it was passing by the pole, and, hastening from thence 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 its ingress into the sphere of the orbis magnus and its perihelion. Nor did this comet truly move with more velocity than other comets, but owed the greatness of its apparent velocity to its passing by the earth at a near distance.

It appears, then, that the velocity of comets (p. 471), so far as it can be determined by these rude ways of computing, is that very velocity with which parabolas, or ellipses near to parabolas, ought to be described; and therefore the distance between a comet and the sun being given, the velocity of the comet is nearly given. And hence arises this problem.

PROBLEM.

The relation betwixt the velocity of a comet and its distance from the sun's centre being given, the comet's trajectory is required.

[Pg 567]

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 assumption may be corrected by the Rule of False, and a third trajectory may thence be found that will exactly agree with the observations. And by 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 they describe, and what are the true magnitudes and forms of their tails according to the various distances of their heads from the sun; whether, after certain intervals of time, the same comets do return again, and in what periods they complete their several revolutions. But the problem may be resolved by determining, first, the hourly motion of a comet to a given time from three or more observations, and then deriving the trajectory from this motion. And thus the invention of the trajectory, depending on one observation, and its hourly motion at the time of this observation, will either confirm 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 with the motions of the comets from beginning to end. The method of the whole computation is this.

LEMMA I.

To cut two right lines OR, TP, given in position, by a third right line RP, so as TRP may be a right angle; and, if another right line SP is drawn to any given point S, the solid contained under this line SP, and 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 the solid be M2 × N; from any point r of the right line OR erect the perpendicular rp meeting TP in p. Then through the point Sp draw the line Sq equal to . In like manner draw three or more right lines S2q, S3q, &c.; and a regular line q2q3q, drawn through all the points q2q3q, &c., will cut the right line TP in the point P, from which the perpendicular PR is to be let fall. Q.E.F.

[Pg 568]

By trigonometry thus. Assuming the right line TP as found by the preceding 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 the error . 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 2p2q D to the error 2pP; and this new error added to or subducted from the length TP, will give the correct length TP E. The inspection of the figure will shew whether we are to add to or subtract; and if at any time there should be use for a farther correction, the operation may be repeated.

By arithmetic thus. Let us suppose the thing done, and let TP + e be the correct length of the right line TP as found out by delineation; and thence the correct lengths of the lines OR, BP, and SP, will be , BP + e, and Whence, by the method of converging series, we have , &c., = , &c. For the given co-efficients , ,

putting , , , and carefully observing the signs, we find , and . Whence, neglecting the very small term , e comes out equal to -G. If the error is not despicable, take .

And it is to be observed that here a general method is hinted at for solving the more intricate sort of problems, as well by trigonometry as by arithmetic, without those perplexed computations and resolutions of affected equations which hitherto have been in use.

LEMMA II.

To cut three right lines given in position by a fourth right line that shall pass through a point assigned in any of the three, and so as its intercepted 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 to be the given point in the line AC. Parallel to AB draw DG meeting BC in G; and, taking GF to BG in the given ratio, draw FDE; and FD will be to DE as FG to BG. Q.E.F.

[Pg 569]

By trigonometry thus. In the triangle CGD all the angles and the side CD are given, and from thence its remaining sides are found; and from the given ratios the lines GF and BE are also given.

LEMMA III.

To find and represent by a linear description the hourly motion of a comet to any given time.

From observations of the best credit, let three longitudes of the comet be given, and, supposing ATR, RTB, to be their differences, let the hourly motion be required to the time of the middle observation TR. By Lem. II, draw the right line ARB, so as its intercepted parts AR, RB, may be as the times between the observations; and if we suppose a body in the whole time to describe the whole line AB with an equal motion, and to be in the mean time viewed from the place T, the apparent motion of that body about the point R will be nearly the same with that of the comet at the time of the observation TR.

The same more accurately.

Let Ta, Tb, be two longitudes given at a greater distance on one side and on the other; and by Lem. II draw the right line aRb so as its intercepted parts aR, Rb may be as the times between the observations aTR, RTb. Suppose this to cut the lines TA, TB, in D and E; and because the error of the inclination TRa increases nearly in the duplicate ratio of the time between the observations, draw FRG, so as either the angle DRF may be to the angle ARF, or the line DF to the line AF, in the duplicate ratio of the whole time between the observations aTB to the whole time between the observations ATB, and use the line thus found FG in place of the line AB found above.

It will be convenient that the angles ATR, RTB, aTA, BTb, be no less than of ten or fifteen degrees, the times corresponding no greater than[Pg 570] of eight or twelve days, and the longitudes taken when the comet moves with the greatest velocity; for thus the errors of the observation will bear a less proportion to the differences of the longitudes.

LEMMA IV.

To find the longitudes of a comet to any given times.

It is done by taking in the line FG the distances Rr, , proportional to the times, and drawing the lines Tr, . The way of working by trigonometry is manifest.

LEMMA V.

To find the latitudes.

On TF, TR, TG, as radiuses, at right angles erect Ff, RP, Gg, tangents of the observed latitudes; and parallel to fg draw PH. The perpendiculars rp, , meeting PH, will be the tangents of the sought latitudes to Tr and as radiuses.

PROBLEM I.

From the assumed ratio of the velocity to determine the trajectory of a comet.

Let S represent the sun; t, T, , three places of the earth in its orbit at equal distances; p, P, , as many corresponding places of the comet in its trajectory, so as the distances interposed betwixt place and place may answer to the motion of one hour; pr, PR, , perpendiculars let fall on the plane of the ecliptic, and the vestige of the trajectory in this plane. Join Sp, SP, , SR, ST, tr, TR, , TP, and let tr, , meet in O, TR will nearly converge to the same point O, or the error will be inconsiderable. By the premised lemmas the angles rOR, , are given, as well as the ratios pr to tr, PR to TR, and to . The figure [Pg 571] is likewise given both in magnitude and position, together with the distance ST, and the angles STR, PTR, STP. Let us assume the velocity of the comet in the place P to be to the velocity of a planet revolved about the sun in a circle, at the same distance SP, as V to 1; and we shall have a line to be determined, of this condition, that the space , described by the comet in two hours, may be to the space (that is, to the space which the earth describes in the same time multiplied by the number V) in the subduplicate ratio of ST, the distance of the earth from the sun, to SP, the distance of the comet from the sun; and that the space pP, described by the comet in the first hour, may be to the space , described by the comet in the second hour, as the velocity in p to the velocity in P; that is, in the subduplicate ratio of the distance SP to the distance Sp, or in the ratio of 2Sp to SP + Sp; for in this whole work I neglect small fractions that can produce no sensible error.

In the first place, then, as mathematicians, in the resolution of affected equations, 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 I best can by conjecture. Then, by Lem. II. I draw , first supposing rR equal to , and again (after the ratio of SP to Sp is discovered) so as rR may be to as 2SP to SP + Sp, and I find the ratios of the lines , , and OR, one to the other. Let M be to as OR to ; and because the square of is to the square of as ST to SP, we shall have, ex æquo, OR2 to M2 as ST to SP, and therefore the solid OR2 × SP equal to the given solid M2 × ST; whence (supposing the triangles 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 rude and hasty way; then by a new delineation with greater care; and, lastly, by an arithmetical computation. Then I proceed to determine the position of the lines , , with the greatest accuracy, together with the nodes and inclination of the plane to the plane of the ecliptic; and in that plane I describe the trajectory in which a body let go from the place P in the direction of the given right line would be carried with a velocity that is to the velocity of the earth as to . Q.E.F.

PROBLEM II.

To correct the assumed ratio of the velocity and the trajectory thence found.

Take an observation of the comet about the end of its appearance, or any other observation at a very great distance from the observations used before, and find the intersection of a right line drawn to the comet, in that observation with the plane , as well as the comet's place in its trajectory to the time of the observation. If that intersection happens in this place, it is a proof that the trajectory was rightly determined; if otherwise,[Pg 572] a new number V is to be assumed, and a new trajectory to be found; and then the place of the comet in this trajectory to the time of that probatory observation, and the intersection of a right line drawn to the comet with the plane of the trajectory, are to be determined as before; and by comparing the variation of the error with the variation of the other quantities, we may conclude, by the Rule of Three, how far those other quantities ought to be varied or corrected, so as the error may become as small as possible. And by means of these corrections we may have the trajectory exactly, providing the observations upon which the computation was founded were exact, and that we did not err much in the assumption of the quantity V; for if we did, the operation is to be repeated till the trajectory is exactly enough determined. Q.E.F.

END OF THE SYSTEM OF THE WORLD.

[Pg 573]

CONTENTS

OF

THE SYSTEM OF THE WORLD.

That the matter of the heavens is fluid, 511

The principle of circular motion in free spaces, 512

The effects of centripetal forces, 512

The certainty of the argument, 514

What follows from the supposed diurnal motion of the stars, 514

The incongruous consequences of this supposition, 514

That 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, 516

That the superior planets are revolved about the sun, and by radii

drawn to the sun describe areas proportional to the times, 517

That the force which governs the superior planets is directed not to

the earth, but to the sun, 518

That the circum-solar force throughout all the regions of the planets

decreaseth in the duplicate proportion of the distances from the sun, 519

That the circum-terrestrial force decreases in the duplicate proportion

of the distances from the earth proved in the hypothesis of the earth's

being at rest, 519

The same proved in the hypothesis of the earth's motion, 520

The 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, 520

The quantity of the forces tending towards the several planets: the

circum-solar very great, 521

The circum-terrestrial force very small, 521

The apparent diameters of the planets, 521

The correction of the apparent diameters, 522

Why the density is greater in some of the planets and less in others;

but the forces in all are as their quantities of matter, 524

Another analogy between the forces and bodies, proved in the celestial

bodies, 525

Proved in terrestrial bodies, 525

The affinity of those analogies, 526

And coincidence, 526

That the forces of small bodies are insensible, 527

Which, notwithstanding, there are forces tending towards all

terrestrial bodies proportional to their quantities of matter, 528

Proved that the same forces tend towards the celestial bodies, 528

That from the surfaces of the planets, reckoning outward, their forces

decrease in the duplicate; but, reckoning inward, in the simple

proportion of the distances from their centres, 529

The quantities of the forces and of the motions arising in the several

cases, 529

That all the planets revolve about the sun, 529

That the common centre of gravity of all the planets is quiescent. That

the sun is agitated with a very slow motion. This motion defined, 531

That the planets, nevertheless, are revolved in ellipses having

their foci in the sun; and by radii drawn to the sun describe areas

proportional to the times, 531

Of the dimensions of the orbits, and of the motions of their aphelions

and nodes, 532

All the motions of the moon that have hitherto been observed by

astronomers derived from the foregoing principles, 532

As also some other unequable motions that hitherto have not been

observed, 533

And the distance of the moon from the earth to any given time, 533

The motions of the satellites of Jupiter and Saturn derived from the

motions of our moon, 534

That the planets, in respect of the fixed stars, are revolved by

equable motions about their proper axes. And that (perhaps) those

motions are the most fit for the equation of time, 534

The moon likewise is revolved by a diurnal motion about its axis, and

its libration thence arises, 535

That the sea ought twice to flow, and twice to ebb, every day; that the

highest water must fall out in the third hour after the appulse of the

luminaries to the meridian of the place, 535[Pg 574]

The 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. But that out of the

syzygies and quadratures those greatest and least tides deviate a

little from that third hour towards the third hour after the appulse of

the sun to the meridian, 536

That the tides are greatest when the luminaries are in their perigees, 536

That the tides are greatest about the equinoxes, 536

That out of the equator the tides are greater and less alternately, 537

That, by the conservation of the impressed motion, the difference of

the tides is diminished; and that hence it may happen that the greatest

menstrual tide will be the third after the syzygy, 538

That the motions of the sea may be retarded by impediments in its

channels, 538

That from the impediments of channels and shores various phænomena do

arise, as that the sea may flow but once every day, 539

That the times of the tides within the channels of rivers are more

unequal than in the ocean, 540

That the tides are greater in greater and deeper seas; greater on the

shores of continents than of islands in the middle of the sea; and yet

greater in shallow bays that open with wide inlets to the sea, 540

The force of the sun to disturb the motions of the moon, computed from

the foregoing principles, 542

The force of the sun to move the sea computed, 543

The height of the tide under the equator arising from the force of the

sun computed, 543

The height of the tides under the parallels arising from the sun's

force computed, 544

The proportion of the tides under the equator, in the syzygies and

quadratures, arising from the joint forces of both sun and moon, 545

The force of the moon to excite tides, and the height of the water

thence arising, computed, 545

That those forces of the sun and moon are scarcely sensible by any

other effect beside the tides which they raise in the sea, 546

That the body of the moon is about six times more dense than the body

of the sun, 547

That the moon is more dense than the earth in a ratio of about three to

two, 547

Of the distance of the fixed stars, 547

That the comets, as often as they become visible to us, are nearer than

Jupiter, proved from their parallax in longitude, 548

The same proved from their parallax in latitude, 549

The same proved otherwise by the parallax, 550

From the light of the comets' heads it is proved that they descend to

the orbit of Saturn, 550

And also below the orb of Jupiter, and sometimes below the orb of the

earth, 551

The same proved from the extraordinary splendor of their tails when

they are near the sun, 551

The same proved from the light of their heads, as being greater,

cæteris paribus, when they come near to the sun, 553

The same confirmed by the great number of comets seen in the region of

the sun, 555

This also confirmed by the greater magnitude and splendor of the tails

after the conjunction of the heads with the sun than before, 555

That the tails arise from the atmospheres of the comets, 556

That 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 phænomena of the tails of comets, 558

After what manner the tails of comets may arise from the atmospheres of

their heads, 559

That the tails do indeed arise from those atmospheres, proved from

several of their phænomena, 559

That comets do sometimes descend below the orbit of Mercury, proved

from their tails, 560

That the comets move in conic sections, having one focus in the centre

of the sun, and by radii drawn to that centre do describe areas

proportional to the times, 561

That those conic sections are near to parabolas, proved from the

velocity of the comets, 561

In what space of time comets describing parabolic trajectories pass

through the sphere of the orbis magnus, 562

At what time comets enter into and pass out of the sphere of the

orbis magnus, 563

With what velocity the comets of 1680 passed through the sphere of the

orbis magnus, 564

That these were not two, but one and the same comet. In what orbit and

with what velocity this comet was carried through the heavens described

more exactly, 564

With what velocity comets are carried, shewed by more examples, 565

The investigation of the trajectory of comets proposed, 566

Lemmas premised to the solution of the problem, 567

The problem resolved, 570

[Pg 575]

INDEX TO THE PRINCIPIA.

ÆQUINOXES, their præcession—the cause of that motion shewn, 413

" the quantity of that motion computed from the causes, 458

AIR, its density at any height, collected by Prop. XXII, Book II, and

its density at the height of one semi-diameter of the earth, shewn, 489

" its elastic force, what cause it may be attributed to, 302

" its gravity compared with that of water, 489

" its resistance, collected by experiments of pendulums, 315

" the same more accurately by experiments of falling bodies, and a

theory, 355

ANGLES of contact not all of the same kind, but some infinitely less

than others, 101

APSIDES, their motion shewn, 172, 173

AREAS which revolving bodies, by radii drawn to the centre of force

describe, compared with the times of description, 103, 105, 106, 195,

200

AS, the mathematical signification of this word defined, 100

ATTRACTION of all bodies demonstrated, 397

" the certainty of this demonstration shewn, 384

" the cause or manner thereof no where defined by the author, 507

" the common centre of gravity of the earth, sun, and all the planets,

is at rest, confirmed by Cor. 2, Prop. XIV, Book III, 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, 452

CENTRE, the common centre of gravity of many bodies does not alter its

state of motion or rest by the actions of the bodies among themselves,

87

" of the forces by which revolving bodies are retained in their orbits,

how indicated by the description of areas, 107

" how found by the given velocities of the revolving bodies, 110

CIRCLE, by what law of centripetal force tending to any given point its

circumference may be described, 108, 111, 114

COMETS, a sort of planets, not meteors, 465, 486

" higher than the moon, and in the planetary regions, 460

" their distance how collected very nearly by observations, 461

" more of them observed in the hemisphere towards the sun than in the

opposite hemisphere; and how this comes to pass, 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, 495

" why they are not comprehended within a zodiac, like the planets, but

move differently into all parts of the heavens, 502

" may sometimes fall into the sun, and afford a new supply of fire, 502

" the use of them hinted, 492

" move in conic sections, having their foci in the sun's centre, and

by radii drawn to the sun describe areas proportional to the times.

Move in ellipses if they come round again in their orbits, but these

ellipses will be near to parabolas, 466

COMET'S parabolic trajectory found from three observations given, 472

" corrected when found, 495

" place in a parabola found to a given time, 466

" velocity compared with the velocity of the planets, 466

COMETS' TAILS directed from the sun, 489

" brightest and largest immediately after their passage through the

neighbourhood of the sun, 487

" their wonderful rarity, 490

" their origin and nature, 463

" in what space of time they ascend from their heads, 490[Pg 576]

COMET of the years 1664 and 1665—the observations of its motion

compared with the theory, 496

" of the years 1680 and 1681—observations of its motion, 474

" its motion computed in a parabolic orbit, 478

" in an elliptic orbit, 479

" its trajectory, and its tail in the several parts of its orbit,

delineated, 484

" of the year 1682—its motion compared with the theory, 500

" seems to have appeared in the year 1607, and likely to return again

after a period of 75 years, 501, 502

" of the year 1683—its motion compared with the theory, 499

" of the year 1723—its motion compared with the theory, 501

CONIC SECTIONS, by what law of centripetal force tending to any given

point they may be described by revolving bodies, 125

" the geometrical description of them when the foci are given, 125

" when the foci are not given, 131

" when the centres or asymptotes are given, 147

CURVATURE of figures how estimated, 271, 423

CURVES distinguished into geometrically rational and geometrically

irrational, 157

CYCLOID, or EPICYCLOID, its rectification, 184

" its evoluta, 185

CYLINDER, the attraction of a cylinder composed of attracting

particles, whose forces are reciprocally as the square of the

distances, 239

DESCENT of heavy bodies in vacuo, how much it is, 405

" and ascent of bodies in resisting mediums, 252, 265, 281, 283,

345

DESCENT or ASCENT rectilinear, the spaces described, the times of

description, and the velocities acquired in such ascent or descent,

compared, on the supposition of any kind of centripetal force, 160

EARTH, its dimension by Norwood, by Picart, and by

Cassini, 405

" its figure discovered, with the proportion of its diameters, and the

measure of the degrees upon the meridian, 405, 409

" the excess of its height at the equator above its height at the

poles, 407, 412

" its greatest and least semi-diameter, 407

" its mean semi-diameter, 407

" the globe of the earth more dense than if it was entirely water, 400

" the nutation of its axis, 413

" the annual motion thereof in the orbis magnus demonstrated, 498

" the eccentricity thereof how much, 452

" the motion of its aphelion how much, 404

ELLIPSES, by what law of centripetal force tending to the centre of the

figure it is described by a revolving body, 114

" by what law of centripetal force tending to the focus of the figure

it is described by a revolving body, 116

FLUID, the definition thereof, 108

FLUIDS, the laws of their density and compression shewn, 293

" their motion in running out at a hole in a vessel determined, 331

FORCES, their composition and resolution, 84

" attractive forces of spherical bodies, composed of particles

attracting according to any law, determined, 218

" attractive forces of bodies not spherical, composed of particles

attracting according to any law, determined, 233

" the invention of the centripetal forces, when a body is revolved in a

non-resisting space about an immoveable centre in any orbit, 103, 116

" the centripetal forces tending to any point by which any figure may

be described by a revolving body being given, the centripetal forces

tending to any other point by which the same figure may be described in

the same periodic time are also given, 113

" the centripetal forces by which any figure is described by a

revolving body being given, there are given the forces by which a new

figure may be described, if the ordinates are augmented or diminished

in any given ratio, or the angle of their inclination be any how

changed, the periodic time remaining the same, 116

" centripetal forces decreasing in the duplicate proportion of the

distances, what figures may be described by them, 120, 196[Pg 577]

FORCE, centripetal force defined, 74

" the absolute quantity of centripetal force defined, 75

" the accelerative quantity of the same defined, 76

" the motive quantity of the same defined, 76

" the proportion thereof to any known force how collected, 109

" a centripetal force that is reciprocally as the cube of the ordinate

tending to a vastly remote centre of force will cause a body to move in

any given conic section, 114

" a centripetal force that is as the cube of the ordinate tending

to a vastly remote centre of force will cause a body to move in an

hyperbola, 243

" centrifugal force of bodies on the earth's equator, how great, 405

GOD, his nature, 506

GRAVITY mutual between the earth and its parts, 94

" of a different nature from magnetical force, 397

" the cause of it not assigned, 507

" tends towards all the planets, 393

" from the surfaces of the planets upwards decreases in the duplicate

ratio of the distances from the centre, 400

" from the same downwards decreases nearly in the simple ratio of the

same, 400

" tends towards all bodies, and is proportional to the quantity of

matter in each, 397

" is the force by which the moon is retained in its orbit, 391

" the same proved by an accurate calculus, 453

" is the force by which the primary planets and the satellites of

Jupiter and Saturn are retained in their orbits, 393

HEAT, an iron rod increases in length by heat, 412

" of the sun, how great at different distances from the sun, 486

" how great in Mercury, 400

" how great in the comet of 1680, when in its perihelion, 486

HEAVENS are void of any sensible resistance, 401, 445, 492; and,

therefore, of almost any corporeal fluid whatever, 355, 356

" suffer light to pass through them without any refraction, 485

HYDROSTATICS, the principles thereof delivered, 293

HYPERBOLA, by what law of centrifugal force tending from the centre of

the figure it is described by a revolving body, 116

" by what law of centrifugal force tending from the focus of the figure

it is described by a revolving body, 117

" by what law of centripetal force tending to the focus of the figure

it is described by a revolving body, 118

HYPOTHESES of what kind soever rejected from this philosophy, 506

JUPITER, its periodic time, 388

" its distance from the sun, 388

" its apparent diameter, 386

" its true diameter, 399

" its attractive force, how great, 398

" the weights of bodies on its surface, 399

" its density, 399

" its quantity of matter, 399

" its perturbation by Saturn, how much, 403

" the proportion of its diameters exhibited by computation, 409

" and compared with observations, 409

" its rotation about its axis, in what time performed, 409

" the cause of its belts hinted at, 445

LIGHT, its propagation not instantaneous, 246

" its velocity different in different mediums, 245

" a certain reflection it sometimes suffers explained, 245

" its refraction explained, 243

" refraction is not made in the single point of incidence, 247

" an incurvation of light about the extremities of bodies observed by

experiments, 246

" not caused by the agitation of any ethereal medium, 368

MAGNETIC force, 94, 304, 397,[Pg 578]

454

MARS, its periodic time, 388

" its distance from the sun, 389

" the motion of its aphelion, 405

MATTER, its quantity of matter defined, 73

" its vis insita defined, 74

" its impressed force defined, 74

" its extension, hardness, impenetrability, mobility, vis

inertiæ, gravity, how discovered, 385

" subtle matter of Descartes inquired into, 320

MECHANICAL POWERS explained and demonstrated, 94

MERCURY, its periodic time, 388

" its distance from the sun, 389

" the motion of its aphelion, 405

METHOD of first and last ratios, 95

" of transforming figures into others of the same analytical order, 141

" of fluxions, 261

" differential, 447

" of finding the quadratures of all curves very nearly true, 448

" of converging series applied to the solution of difficult problems,

271, 436

MOON, the inclination of its orbit to the ecliptic greatest in the

syzygies of the node with the sun, and least in the quadratures, 208

" the figure of its body collected by calculation, 454

" its librations explained, 405

" its mean apparent diameter, 453

" its true diameter, 453

" weight of bodies on its surface, 453

" its density, 453

" its quantity of matter, 453

" its mean distance from the earth, how many greatest semi-diameters of

the earth contained therein, 453

" how many mean semi-diameters, 454

" its force to move the sea how great, 449

" not perceptible in experiments of pendulums, or any statical or

hydrostatical observations, 452

" its periodic time, 454

" the time of its synodical revolution, 422

" its motions, and the inequalities of the same derived from their

causes, 413, 444

" revolves more slowly, in a dilated orbit, when the earth is in its

perihelion; and more swiftly in the aphelion the same, its orbit being

contracted, 413, 444, 445

" revolves more slowly, in a dilated orbit, when the apogæon is in the

syzygies with the sun; and more swiftly, in a contracted orbit, when

the apogæon is in the quadratures, 445

" revolves more slowly, in a dilated orbit, when the node is in the

syzygies with the sun; and more swiftly, in a contracted orbit, when

the node is in the quadratures, 446

" moves slower in its quadratures with the sun, swifter in the

syzygies; and by a radius drawn to the earth describes an area, in the

first case less in proportion to the time, in the last case greater, 413

" the inequality of those areas computed, 420

" its orbit is more curve, and goes farther from the earth in the first

case; in the last case its orbit is less curve, and comes nearer to the

earth, 415

" the figure of this orbit, and the proportion of its diameters

collected by computation, 423

" a method of finding the moon's distance from the earth by its horary

motion, 423

" its apogæon moves more slowly when the earth is in its aphelion, more

swiftly in the perihelion, 414, 445

" its apogæon goes forward most swiftly when in the syzygies with the

sun; and goes backward in the quadratures, 414, 446

" its eccentricity greatest when the apogæon is in the syzygies with

the sun; least when the same is in the quadratures, 414, 446

" its nodes move more slowly when the earth is in its aphelion, and

more swiftly in the perihelion, 414, 445

" its nodes are at rest in their syzygies with the sun, and go back

most swiftly in the quadratures, 414[Pg 579]

MOON, the motions of the nodes and the inequalities of its motions

computed from the theory of gravity, 427, 430, 434, 436

" the same from a different principle, 437

" the variations of the inclination computed from the theory of

gravity, 441, 443

" the equations of the moon's motions for astronomical uses, 445

" the annual equation of the moon's mean motion, 445

" the first semi-annual equation of the same, 446

" the second semi-annual equation of the same, 447

" the first equation of the moon's centre, 447

" the second equation of the moon's centre, 448

MOON'S first variation, 425

" the annual equation of the mean motion of its apogee, 445

" the semi-annual equation of the same, 447

" the semi-annual equation of its eccentricity, 447

" the annual equation of the mean motion of its nodes, 445

" the semi-annual equation of the same, 437

" the semi-annual equation of the inclination of the orbit to the

ecliptic, 444

" the method of fixing the theory of the lunar motions from

observations, 464

MOTION, its quantity defined, 73

" absolute and relative, 78

" absolute and relative, the separation of one from the other possible,

demonstrated by an example, 82

" laws thereof, 83

" of concurring bodies after their reflection, by what experiments

collected, 91

" of bodies in eccentric sections, 116

" in moveable orbits, 172

" in given superficies, and of the reciprocal motion of pendulums, 183

" of bodies tending to each other with centripetal forces, 194

" of very small bodies agitated by centripetal forces tending to each

part of some very great body, 233

" of bodies resisted in the ratio of the velocities, 251

" in the duplicate ratio of the velocity, 258

" partly in the simple and partly in the duplicate ratio of the same,

280

" of bodies proceeding by their vis insita alone in resisting

mediums, 251, 258, 259, 280, 281, 330

" of bodies ascending or descending in right lines in resisting

mediums, and acted on by an uniform force of gravity, 252, 265, 281, 283

" of bodies projected in resisting mediums, and acted on by an uniform

force of gravity, 255, 268

" of bodies revolving in resisting mediums, 287

" of funependulous bodies in resisting mediums, 304

" and resistance of fluids, 323

" propagated through fluids, 356

" of fluids after the manner of a vortex, or circular, 370

MOTIONS, composition and resolution of them, 84

OVALS for optic uses, the method of finding them which Cartesius

concealed, 246

" a general solution of Cartesius's problem, 247, 248

ORBITS, the invention of those which are described by bodies going off

from a given place with a given velocity according to a given right

line, when the centripetal force is reciprocally as the square of the

distance, and the absolute quantity of that force is known, 123

" of those which are described by bodies when the centripetal force is

reciprocally as the cube of the distance, 114, 171, 176

" of those which are described by bodies agitated by any centripetal

forces whatever, 168

PARABOLA, by what law of centripetal force tending to the focus of the

figure the same may be described, 120

PENDULUMS, their properties explained, 186, 190, 304

" the diverse lengths of isochronous pendulums in different latitudes

compared among themselves, both by observations and by the theory of

gravity, 409 to 413

PLACE defined, and distinguished into absolute and relative, 78

PLACES of bodies moving in conic sections found to any assigned time,

153

PLANETS not carried about by corporeal vortices, 378[Pg 580]

PLANETS, their periodic times, 388

" their distances from the sun, 389

" the aphelia and nodes of their orbits do almost rest, 405

" their orbits determined, 406

" the way of finding their places in their orbits, 347 to 350

" their density suited to the heat they receive from the sun, 400

" their diurnal revolutions equable, 406

" their axes less than the diameters that stand upon them at right

angles, 406

PLANETS, PRIMARY, surround the sun, 387

" move in ellipses whose focus is in the sun's centre, 403

" by radii drawn to the sun describe areas proportional to the times,

388, 403

" revolve in periodic times that are in the sesquiplicate proportion

of the distances from the sun, 387

" are retained in their orbits by a force of gravity which respects

the sun, and is reciprocally as the square of the distance from the

sun's centre, 389, 393

PLANETS, SECONDARY, move in ellipses having their focus in the centre

of the primary, 413

" by radii drawn to their primary describe areas proportional to the

times, 386, 387, 390

" revolve in periodic times that are in the sesquiplicate proportion

of their distances from the primary, 386, 387

PROBLEM KEPLERIAN, solved by the trochoid and by approximations, 157 to

160

" of the ancients, of four lines, related by Pappus, and attempted

by Cartesius, by an algebraic calculus solved by a geometrical

composition, 135

PROJECTILES move in parabolas when the resistance of the medium is

taken away, 91, 115, 243, 273

" their motions in resisting mediums, 255, 268

PULSES of the air, by which sounds are propagated, their intervals or

breadths determined, 368, 370

" these intervals in sounds made by open pipes probably equal to twice

the length of the pipes, 370

QUADRATURES general of oval figures not to be obtained by finite terms,

153

QUALITIES of bodies how discovered, and when to be supposed universal,

384

RESISTANCE, the quantity thereof in mediums not continued, 329

" in continued mediums, 409

" in mediums of any kind whatever, 338

" of mediums is as their density, cæteris paribus, 320, 321,

324, 329, 344, 355

" is in the duplicate proportion of the velocity of the bodies

resisted, cæteris paribus, 258, 314, 324, 329, 344,

351

" is in the duplicate proportion of the diameters of spherical bodies

resisted, cæteris paribus, 317, 318, 329, 344

" of fluids threefold, arises either from the inactivity of the fluid

matter, or the tenacity of its parts, or friction, 286

" the resistance found in fluids, almost all of the first kind, 321, 354

" cannot be diminished by the subtilty of the parts of the fluid, if

the density remain, 355

" of a globe, what proportion it bears to that of a cylinder, in

mediums not continued, 327

" in compressed mediums, 343

" of a globe in mediums not continued, 329

" in compressed mediums, 344

" how found by experiments, 345 to 355

" to a frustum of a cone, how made the least possible, 328

" what kind of solid it is that meets with the least, 329

RESISTANCES, the theory thereof confirmed by experiments of pendulums,

313 to 321

" by experiments of falling bodies, 345 to 356

REST, true and relative, 78

RULES of philosophy, 384

SATELLITES, the greatest heliocentric elongation of Jupiter's

satellites, 387

" the greatest heliocentric elongation of the Huygenian

satellite from Saturn's centre, 398

" the periodic times of Jupiter's satellites, and their distances from

his centre, 386, 387

" the periodic times of Saturn's satellites, and their distances from

his centre, 387, 388

" the inequalities of the motions of the satellites of Jupiter and

Saturn derived from the motions of the moon, 413

SESQUIPLICATE proportion defined, 101[Pg 581]

SATURN, its periodic time, 388

" its distance from the sun, 388

" its apparent diameter, 388

" its true diameter, 399

" its attractive force, how great, 398

" the weight of bodies on its surface, 399

" its density, 399

" its quantity of matter, 399

" its perturbation by the approach of Jupiter how great, 403

" the apparent diameter of its ring, 388

SHADOW of the earth to be augmented in lunar eclipses, because of the

refraction of the atmosphere, 447

SOUNDS, their nature explained, 360, 363, 365, 366, 367,

368, 369

" not propagated in directum, 359

" caused by the agitation of the air, 368

" their velocity computed, 368, 369

" somewhat swifter by the theory in summer than in winter, 370

" cease immediately, when the motion of the sonorous body ceases, 365

" how augmented in speaking trumpets, 370

SPACE, absolute and relative, 78, 79

" not equally full, 396

SPHEROID, the attraction of the same when the forces of its particles

are reciprocally as the squares of the distances, 239

SPIRAL cutting all its radii in a given angle, by what law of

centripetal force tending to the centre thereof it may be described by

a revolving body, 107, 287, 291

SPIRIT pervading all bodies, and concealed within them, hinted at, as

required to solve a great many phænomena of Nature, 508

STARS, the fixed stars demonstrated to be at rest, 404

" their twinkling what to be ascribed to, 487

" new stars, whence they may arise, 502

SUBSTANCES of all things unknown, 507

SUN, moves round the common centre of gravity of all the planets, 401

" the periodic time of its revolution about its axis, 405

" its mean apparent diameter, 453

" its true diameter, 398

" its horizontal parallax, 398

" has a menstrual parallax, 403

" its attractive force how great, 398

" the weight of bodies on its surface, 399

" its density, 399

" its quantity of matter, 399

" its force to disturb the motions of the moon, 391, 419

" its force to move the sea, 448

TIDES of the sea derived from their cause, 415, 448, 449

TIME, absolute and relative, 78, 79

" the astronomical equation thereof proved by pendulum clocks, and the

eclipses of Jupiter's satellites, 79

A VACUUM proved, or that all spaces (if said to be full) are not

equally full, 396

VELOCITIES of bodies moving in conic sections, where the centripetal

force tends to the focus, 121

VELOCITY, the greatest that a globe falling in a resisting medium can

acquire, 344

VENUS, its periodic time, 388

" its distance from the sun, 388

" the motion of its aphelion, 405

VORTICES, their nature and constitution examined, 504

WAVES, the velocity with which they are propagated on the superficies

of stagnant water, 361

WEIGHTS of bodies towards the sun, the earth, or any planet, are, at

equal distances from the centre, as the quantities of matter in the

bodies, 394

" they do not depend upon the forms and textures of bodies, 395

" of bodies in different regions of the earth found out, and compared

together, 409

TRANSCRIBER’S NOTES

Simple typographical errors have been silently corrected; unbalanced quotation marks were remedied when the change was obvious, and otherwise left unbalanced.

Punctuation and spelling were made consistent when a predominant preference was found in the original book; otherwise they were not changed.

Inconsistent hyphens left as printed.

Due to lack of clarity, the figures on pages 82 and 144 have been replaced by the corresponding figures from the Latin version.

In the table on page 318, in the fifth row, the third column has been repositioned in the fourth column, according to the text that follows on page 319.