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.
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]].
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.
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.