Part 7
140. The _Ptolemean System_ § 96, which asserts the Earth to be at rest in the Center of the Universe, and all the Planets with the Sun and Stars to move round it, is evidently false and absurd. For if this hypothesis were true, Mercury and Venus could never be hid behind the Sun, as their Orbits are included within the Sun’s: and again, these two Planets would always move direct, and be as often in Opposition to the Sun as in Conjunction with him. But the contrary of all this is true: for they are just as often behind the Sun as before him, appear as often to move backwards as forwards, and are so far from being seen at any time in the side of the Heavens opposite to the Sun, that they were never seen a quarter of a circle in the Heavens distant from him.
[Sidenote: Appearances of Mercury and Venus.]
141. These two Planets, when viewed with a good telescope, appear in all the various shapes of the Moon; which is a plain proof that they are enlightened by the Sun, and shine not by any light of their own: for if they did, they would constantly appear round as the Sun does; and could never be seen like dark spots upon the Sun when they pass directly between him and us. Their regular Phases demonstrate them to be Spherical bodies; as may be shewn by the following experiment.
[Sidenote: Experiment to prove they are round.]
Hang an ivory ball by a thread, and let any Person move it round the flame of a candle, at two or three yards distance from your Eye: when the ball is beyond the candle, so as to be almost hid by the flame, it’s enlightened side will be towards you, and appear round like the Full Moon: When the ball is between you and the candle, it’s enlightened side will disappear, as the Moon does at the Change: When it is half way between these two positions, it will appear half illuminated, like the Moon in her Quarters: But in every other place between these positions, it will appear more or less horned or gibbous. If this experiment be made with a circular plate which has a flat surface, you may make it appear fully enlightened, or not enlightened at all; but can never make it seem either horned or gibbous.
[Sidenote: PLATE II.
Experiment to represent the motions of Mercury and Venus.]
142. If you remove about six or seven yards from the candle, and place yourself so that it’s flame may be just about the height of your eye, and then desire the other person to move the ball slowly round the candle as before, keeping it as near of an equal height with the flame as he possibly can, the ball will appear to you not to move in a circle, but rather to vibrate backward and forward like a pendulum; moving quickest when it is directly between you and the candle, and when directly beyond it; and gradually slower as it goes farther to the right or left side of the flame, until it appears at the greatest distance from the flame; and then, though it continues to move with the same velocity, it will seem to stand still for a moment. In every Revolution it will shew all the above Phases § 141; and if two balls, a smaller and a greater, be moved in this manner round the candle, the smaller ball being kept nearest the flame, and carried round almost three times as often as the greater, you will have a tolerably good representation of the apparent Motions of Mercury and Venus; especially, if the bigger ball describes a circle almost twice as large in diameter as the circle described by the lesser.
[Sidenote: Fig. III.
The elongations or digressions of Mercury from the Sun.
PLATE II.]
143. Let _ABCDE_ be a part or segment of the visible Heavens, in which the Sun, Moon, Planets, and Stars appear to move at the same distance from the Earth _E_. For there are certain limits, beyond which the eye cannot judge of different distances; as is plain from the Moon’s appearing to be no nearer to us than the Sun and Stars are. Let the circle _fghiklmno_ be the Orbit in which Mercury _m_ moves round the Sun _S_, according to the order of the letters. When Mercury is at _f_, he disappears to the Earth at _E_, because his enlightened side is turned from it; unless he be then in one of his Nodes § 20, 25; in which case, he will appear like a dark spot upon the Sun. When he is at _g_ in his Orbit, he appears at _B_ in the Heavens, westward of the Sun _S_, which is seen at _C_: when at _h_, he appears at _A_, at his greatest western elongation or distance from the Sun; and then seems to stand still. But, as he moves from _h_ to _i_, he appears to go from _A_ to _B_; and seems to be in the same place when at _i_ as when he was at _g_, only not near so big: at _k_ he is hid from the Earth _E_ by the Sun _S_; being then in his superiour Conjunction. In going from _k_ to _l_, he appears to move from _C_ to _D_; and when he is at _n_, he appears stationary at _E_; being seen as far east from the Sun then, as he was west from him at _A_. In going from _n_ to _o_ in his Orbit, he seems to go back again in the Heavens, from _E_ to _D_; and is seen in the same place (with respect to the Sun) at _o_ as when he was at _l_; but of a larger diameter at _o_, because he is then nearer the Earth _E_: and when he comes to _f_, he again passes by the Sun, and disappears as before. In going from _n_ to _h_ in his Orbit, he seems to go backward in the Heavens from _E_ to _A_; and in going from _h_ to _n_, he seems to go forward from _A_ to _E_. As he goes on from _f_ a little of his enlightened side at _g_ is seen from _E_; at _h_ he appears half full, because half of his enlightened side is seen; at _i_, gibbous, or more than half full; and at _k_ he would appear quite full, were he not hid from the Earth _E_ by the Sun _S_. At _l_ he appears gibbous again; at _n_ half decreased, at _o_ horned, and at _f_ new like the Moon at her Change. He goes sooner from his eastern station at _n_ to his western station at _h_ than from _h_ to _n_ again; because he goes through less than half his Orbit in the former case, and more in the latter.
[Sidenote: Fig. III.
The Elongations and Phases of Venus.
The greatest Elongations of Mercury and Venus.]
144. In the same Figure, let _FGHIKLMN_ be the Orbit in which Venus _v_ moves round the Sun _S_, according to the order of the letters: and let _E_ be the Earth as before. When Venus is at _F_ she is in her inferiour Conjunction; and disappears like the New Moon because her dark side is toward the Earth. At _G_ she appears half enlightened to the Earth, like the Moon in her first quarter: at _h_ she appears gibbous; at _I_, almost full; her enlightened side being then nearly towards the Earth: at _K_, she would appear quite full to the Earth _E_; but is hid from it by the Sun _S_: at _L_, she appears upon the decrease, or gibbous; at _M_, more so; at _N_, only half enlightened; and at _F_ she disappears again. In moving from _N_ to _G_, she seems to go backward in the Heavens; and from _G_ to _N_, forward: but, as she describes a much greater portion of her Orbit in going from _G_ to _N_ than from _N_ to _G_, she appears much longer direct than retrograde in her motion. At _N_ and _G_ she appears stationary; as Mercury does at _n_ and _h_. Mercury, when stationary seems to be only 28 degrees from the Sun; and Venus when so, 47; which is a demonstration that Mercury’s Orbit is included within Venus’s, and Venus’s within the Earth’s.
[Sidenote: Morning and Evening Star, what.]
145. Venus, from her superiour Conjunction at _K_ to her inferiour Conjunction at _F_ is seen on the east side of the Sun _S_ from the Earth. _E_; and therefore she shines in the Evening after the Sun sets, and is called _the Evening Star_: for, the Sun being then to the westward of Venus, he must set first. From her inferiour Conjunction to her superiour, she appears on the west side of the Sun; and therefore rises before him, for which reason she is called _the Morning Star_. When she is about _N_ or _G_, she shines so bright, that bodies cast shadows in the night-time.
[Sidenote: PLATE II.
The stationary places of the Planets variable.]
146. If the Earth kept always at _E_, it is evident that the Stationary places of Mercury and Venus would always be in the same points of the Heavens where they were before. For example; whilst Mercury _m_ goes from _h_ to _n_, according to the order of the letters, he appears to describe the arc _ABCDE_ in the Heavens, direct: and whilst he goes from _n_ to _h_, he seems to describe the same arc back again, from _E_ to _A_, retrograde: always at _n_ and _h_ he appears stationary at the same points _E_ and _A_ as before. But Mercury goes round his Orbit, from _f_ to _f_ again, in 88 days; and yet there are 116 days from any one of his Conjunctions, or apparent Stations, to the same again: and the places of these Conjunctions and Stations are found to be about 114 degrees eastward from the points of the Heavens where they were last before; which proves, that the Earth has not kept all that time at _E_, but has had a progressive motion in it’s Orbit from _E_ to _t_. Venus also differs every time in the places of her Conjunctions and Stations; but much more than Mercury; because, as Venus describes a much larger Orbit than Mercury does, the Earth advances so much the farther in it’s annual path before Venus comes round again.
[Sidenote: The Elongations of all Saturn’s inferiour Planets as seen from him.]
147. As Mercury and Venus, seen from the Earth, have their respective Elongations from the Sun, and Stationary places; so has the Earth, seen from Mars; and Mars, seen from Jupiter; and Jupiter, seen from Saturn. That is, to every superiour Planet, all the inferiour ones have their Stations and Elongations; as Venus and Mercury have to the Earth. As seen from Saturn, Mercury never goes above 2-1/2 degrees from the Sun; Venus 4-1/3; the Earth 6; Mars 9-1/2; and Jupiter 33-1/4: so that Mercury, as seen from the Earth, has almost as great a Digression or Elongation from the Sun, as Jupiter seen from Saturn.
[Sidenote: A proof of the Earth’s annual motion.]
148. Because the Earth’s Orbit is included within the Orbits of Mars, Jupiter, and Saturn, they are seen on all sides of the Heavens; and are as often in Opposition to the Sun as in Conjunction with him. If the Earth stood still, they would always appear direct in their motions, never retrograde nor stationary. But they seem to go just as often backward as forward; which, if gravity be allowed to exist, affords a sufficient proof of the Earth’s annual motion.
[Sidenote: Fig. III.
PLATE II.
General Phenomena of a superiour Planet to an inferiour.]
149. As Venus and the Earth are superiour Planets to Mercury, they shew much the same Appearances to him that Mars and Jupiter do to us. Let Mercury _m_ be at _f_, Venus _v_ at _F_, and the Earth at _E_; in which situation Venus hides the Earth from Mercury; but, being in opposition to the Sun, she shines on Mercury with a full illumined Orb; though, with respect to the Earth, she is in conjunction with the Sun and invisible. When Mercury is at _f_, and Venus at _G_, her enlightened side not being directly towards him, she appears a little gibbous; as Mars does in a like situation to us: but, when Venus is at _I_, her enlightened side is so much towards Mercury at _f_, that she appears to him almost of a round figure. At _K_, Venus disappears to Mercury at _f_, being then hid by the Sun; as all our superiour Planets are to us, when in conjunction with the Sun. When Venus has, as it were, emerged out of the Sun beams, as at _L_, she appears almost full to Mercury at _f_; at _M_ and _N_, a little gibbous; quite full at _F_, and largest of all; being then in opposition to the Sun, and consequently nearest to Mercury at _f_; shining strongly on him in the night, because her distance from him then is somewhat less than a fifth part of her distance from the Earth, when she appears roundest to it between _I_ and _K_, or between _K_ and _L_, as seen from the Earth _E_. Consequently, when Venus is opposite to the Sun as seen from Mercury, she appears more than 25 times as large to him as she does to us when at the fullest. Our case is almost similar with respect to Mars, when he is opposite to the Sun; because he is then so near the Earth, and has his whole enlightened side towards it. But, because the Orbits of Jupiter and Saturn are very large in proportion to the Earth’s, these two Planets appear much less magnified at their Oppositions or diminished at their Conjunctions than Mars does, in proportion to their mean apparent Diameters.
CHAP. VII.
_The physical Causes of the Motions of the Planets. The Excentricities of their Orbits. The Times in which the Action of Gravity would bring them to the Sun._ ARCHIMEDES_’s ideal Problem for moving the Earth. The World not eternal._
[Sidenote: Gravitation and Projection.
Fig. IV.
PLATE II.
Circular Orbits.
Fig. IV.]
150. From the uniform projectile motion of bodies in straight lines, and the universal power of attraction, arises the curvilineal motions of all the Heavenly bodies. If the body _A_ be projected along the right line _ABX_, in open Space, where it meets with no resistance, and is not drawn aside by any other power, it will for ever go on with the same velocity, and in the same direction. For, the force which moves it from _A_ to _B_ in any given time, will carry it from _B_ to _X_ in as much more time; and so on, there being nothing to obstruct or alter it’s motion. But if, when this projectile force has carried it, suppose to _B_, the body _S_ begins to attract it, with a power duly adjusted, and perpendicular to it’s motion at _B_, it will then be drawn from the straight line _ABX_, and forced to revolve about _S_ in the Circle _BYTU_. When the body _A_ comes to _U_, or any other part of it’s Orbit, if the small body _u_, within the sphere of _U_’s attraction, be projected as in the right line _Z_, with a force perpendicular to the attraction of _U_, then _u_ will go round _U_ in the Orbit _W_, and accompany it in it’s whole course round the body _S_. Here, _S_ may represent the Sun, _U_ the Earth, and _u_ the Moon.
151. If a Planet at _B_ gravitates, or is attracted, toward the Sun, so as to fall from _B_ to _y_ in the time that the projectile force would have carried it from _B_ to _X_, it will describe the curve _BY_ by the combined action of these two forces, in the same time that the projectile force singly would have carried it from _B_ to _X_, or the gravitating power singly have caused it to descend from _B_ to _y_; and these two forces being duly proportioned, and perpendicular to one another, the Planet obeying them both, will move in the circle _BYTU_[30].
[Sidenote: Elliptical Orbits.
PLATE II.]
152. But if, whilst the projectile force carries the Planet from _B_ to _b_, the Sun’s attraction (which constitutes the Planet’s gravitation) should bring it down from _B_ to I, the gravitating power would then be too strong for the projectile force; and would cause the Planet to describe the curve _BC_. When the Planet comes to _C_, the gravitating power (which always increases as the square of the distance from the Sun _S_ diminishes) will be yet stronger for the projectile force; and by conspiring in some degree therewith, will accelerate the Planet’s motion all the way from _C_ to _K_; causing it to describe the arcs _BC_, _CD_, _DE_, _EF_, &c. all in equal times. Having it’s motion thus accelerated, it gains so much centrifugal force, or tendency to fly off at _K_ in the line _Kk_, as overcomes the Sun’s attraction: and the centrifugal force being too great to allow the Planet to be brought nearer the Sun, or even to move round him in the Circle _Klmn_, &c. it goes off, and ascends in the curve _KLMN_, &c. it’s motion decreasing as gradually from _K_ to _B_ as it increased from _B_ to _K_, because the Sun’s attraction acts now against the Planet’s projectile motion just as much as it acted with it before. When the Planet has got round to _B_, it’s projectile force is as much diminished from it’s mean state about _G_ or _N_, as it was augmented at _K_; and so, the Sun’s attraction being more than sufficient to keep the Planet from going off at _B_, it describes the same Orbit over again, by virtue of the same forces or laws.
[Sidenote: Fig. IV.
The Planets describe equal Areas in equal times.]
153. A double projectile force will always balance a quadruple power of gravity. Let the Planet at _B_ have twice as great an impulse from thence towards _X_, as it had before: that is, in the same length of time that it was projected from _B_ to _b_, as in the last example, let it now be projected from _B_ to _c_; and it will require four times as much gravity to retain it in it’s Orbit: that is, it must fall as far as from _B_ to 4 in the time that the projectile force would carry it from _B_ to _c_; otherwise it could not describe the curve _BD_, as is evident by the Figure. But, in as much time as the Planet moves from _B_ to _C_ in the higher part of it’s Orbit, it moves from _I_ to _K_ or from _K_ to _L_ in the lower part thereof; because, from the joint action of these two forces, it must always describe equal areas in equal times, throughout it’s annual course. These Areas are represented by the triangles _BSC_, _CSD_, _DSE_, _ESF_, &c. whose contents are equal to one another, quite round the Figure.
[Sidenote: A difficulty removed.]
154. As the Planets approach nearer the Sun, and recede farther from him, in every Revolution; there may be some difficulty in conceiving the reason why the power of gravity, when it once gets the better of the projectile force, does not bring the Planets nearer and nearer the Sun in every Revolution, till they fall upon and unite with him. Or why the projectile force, when it once gets the better of gravity, does not carry the Planets farther and farther from the Sun, till it removes them quite out of the sphere of his attraction, and causes them to go on in straight lines for ever afterward. But by considering the effects of these powers as described in the two last Articles, this difficulty will be removed. Suppose a Planet at _B_ to be carried by the projectile force as far as from _B_ to _b_, in the time that gravity would have brought it down from _B_ to 1: by these two forces it will describe the curve _BC_. When the Planet comes down to _K_, it will be but half as far from the Sun _S_ as it was at _B_; and therefore, by gravitating four times as strongly towards him, it would fall from _K_ to _V_ in the same length of time that it would have fallen from _B_ to 1 in the higher part of it’s Orbit, that is, through four times as much space; but it’s projectile force is then so much increased at _K_, as would carry it from _K_ to _k_ in the same time; being double of what it was at _B_, and is therefore too strong for the tendency of the gravitating power, either to draw the Planet to the Sun, or cause it to go round him in the circle _Klmn_, &c. which would require it’s falling from _K_ to _w_, through a greater space than gravity can draw it whilst the projectile force is such as would carry it from _K_ to _k_: and therefore the Planet ascends in it’s Orbit _KLMN_, decreasing in it’s velocity for the cause already assigned in § 152.
[Sidenote: The Planetary Orbits elliptical.
Their Excentricities.]
155. The Orbits of all the Planets are Ellipses, very little different from Circles: but the Orbits of the Comets are very long Ellipses; the lower focus of them all being in the Sun. If we suppose the mean distance (or middle between the greatest and least) of every Planet and Comet from the Sun to be divided into 1000 equal parts, the Excentricities of their Orbits, both in such parts and in _English_ miles, will be as follows. Mercury’s, 210 parts, or 6,720,000 miles; Venus’s, 7 parts, or 413,000 miles; the Earth’s, 17 parts, or 1,377,000 miles; Mars’s, 93 parts, or 11,439,000 miles; Jupiter’s, 48 parts, or 20,352,000 miles; Saturn’s, 55 parts, or 42,735,000 miles. Of the nearest of the three forementioned Comets, 1,458,000 miles; of the middlemost, 2,025,000,000 miles; and of the outermost, 6,600,000,000.
[Sidenote: The above laws sufficient for motions both in circular and elliptic Orbits.]
156. By the above-mentioned laws § 150 _& seq._ bodies will move in all kinds of Ellipses, whether long or short, if the spaces they move in be void of resistance. Only, those which move in the longer Ellipses, have so much the less projectile force impressed upon them in the higher parts of their Orbits; and their velocities, in coming down towards the Sun, are so prodigiously increased by his attraction, that their centrifugal forces in the lower parts of their Orbits are so great as to overcome the Sun’s attraction there, and cause them to ascend again towards the higher parts of their Orbits; during which time, the Sun’s attraction acting so contrary to the motions of those bodies, causes them to move slower and slower, until their projectile forces are diminished almost to nothing; and then they are brought back again by the Sun’s attraction, as before.
[Sidenote: In what times the Planets would fall to the Sun by the power of gravity.]
157. If the projectile forces of all the Planets and Comets were destroyed at their mean distances from the Sun, their gravities would bring them down so, as that Mercury would fall to the Sun in 15 days 13 hours; Venus in 39 days 17 hours; the Earth or Moon in 64 days 10 hours; Mars in 121 days; Jupiter in 290; and Saturn in 767. The nearest Comet in 13 thousand days; the middlemost in 23 thousand days; and the outermost in 66 thousand days. The Moon would fall to the Earth in 4 days 20 hours; Jupiter’s first Moon would fall to him in 7 hours, his second in 15, his third in 30, and his fourth in 71 hours. Saturn’s first Moon would fall to him in 8 hours; his second in 12, his third in 19, his fourth in 68 hours, and the fifth in 336. A stone would fall to the Earth’s center, if there were an hollow passage, in 21 minutes 9 seconds. Mr. WHISTON gives the following Rule for such Computations. “[31]It is demonstrable, that half the Period of any Planet, when it is diminished in the sesquialteral proportion of the number 1 to the number 2, or nearly in the proportion of 1000 to 2828, is the time that it would fall to the Center of it’s Orbit.” This proportion is, when a quantity or number contains another once and a half as much more.
[Sidenote: The prodigious attraction of the Sun and Planets.]
158. The quick motions of the Moons of Jupiter and Saturn round their Primaries, demonstrate that these two Planets have stronger attractive powers than the Earth has. For, the stronger that one body attracts another, the greater must be the projectile force, and consequently the quicker must be the motion of that other body, to keep it from falling to it’s primary or central Planet. Jupiter’s second Moon is 124 thousand miles farther from Jupiter than our Moon is from us; and yet this second Moon goes almost eight times round Jupiter whilst our Moon goes only once round the Earth. What a prodigious attractive power must the Sun then have, to draw all the Planets and Satellites of the System towards him; and what an amazing power must it have required to put all these Planets and Moons into such rapid motions at first! Amazing indeed to us, because impossible to be effected by the strength of all the living Creatures in an unlimited number of Worlds, but no ways hard for the Almighty, whose Planetarium takes in the whole Universe!
[Sidenote: ARCHIMEDES’s Problem for raising the Earth.]