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planets. But, on the Ptolemaic system, they are inexplicable, without calling in the aid of a very complicated hypothesis.

When the inferior planets are passing from their greatest elongation, on one side of the sun, through their superior conjunction, to their greatest elongation on the other side, their motion, as viewed from the earth, is direct. In order to explain this proposition, we shall first suppose the earth to be at rest at A, plate 6, fig. 2; and correct this supposition afterwards, by shewing that the apparent motion of Venus, or Mercury, seen from the earth, is the same in this respect, whether the earth moves in its orbit, or rests at A.

The proposition to be explained is this : that as Venus, for instance, moves from x, its greatest elongation on one side of the sun, through G, its superior conjunction, to H, its greatest elongation on the other side, it will appear to a spectator upon the earth, to move from west to east, according to the order of the signs; that is, its geocentric motion will be direct.

The planets move round the sun from west to east; and, consequently, if there was a spectator at the sun, they would appear to him to move through the zodiac, according to the order of the signs; or, in other words, the heliocentric motion of Venus is direct. Now if the sun S, and the earth A, are both on the same side of the plaret, a spectator at the earth is in the same situation, with respect to the planet and its motion, as if he had been at the sun : for

whilst the planet is moving from X, through G, to H, a spectator, either at A or S, is on the concave side of the planet's orbit; and, consequently, the planet will appear to move in the same manner from either ; but the apparent motion of the planet, when seen from the sun, is direct; and, consequently, its motion, when seen from the earth, will also be direct.

For, when Venus is at x, it appears to a spectator on the earth at A, to be in the line AxV, or is seen among the stars at V; when Venus has moved to K, it is seen among the fixed stars at P; when it has moved to G, it is in its superior conjunction ; when it has moved to C, it appears among the fixed stars at R; and when it is come to H, it appears among the fixed stars at T. Thus, whilst Venus has moved in its orbit from x, its greatest elongation on one side of the sun, through G, its superior conjunction, to H, its greatest elongation on the other side, it appears to have described the arc VPQRT, in the concave sphere of the heavens ; ' but the letters «K, GCH, lie from west to east, because they lie in the same direction that the planet moves round the sun; and the letters VPQRT lie in the same direction with «KGCH. Therefore, as the planet seems to a spectator on the earth, to describe the arc VPQRT, its apparent motion, seen from the earth, is direct, or from west to east,

The second proposition is this : that while the inferior planets move from their greatest elongation on one side of the sun, through their inferior conjunc

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tion, to their greatest elongation on the other side, their geocentric motion is retrograde.

In other words, whilst Venus is moving from its greatest elongation H, plate 6, fig. 2, through its inferior conjunction E, to its other greatest elonga

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appears to a spectator upon the earth at A, to move backwards, or from east to west, contrary to the order of the signs.

A spectator at the sun is on the concave side of the planet's orbit, viewing it from withinside. But, whilst Venus is moving from its greatest elongation H, on one side, through E, its inferior conjunction, to x, its greatest elongation on the other side, a spectator upon the earth is on the convex side of its orbit, viewing it from without.

Therefore, if a spectator at the sun S would see the planet move one way, a spectator at the earth A will see it move the contrary way; or

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geocentric motion will be contrary to its heliocentric motion, and therefore retrograde; for, as seen from the sun, its motion is always direct.

That two spectators, one at the earth, the other at the sun, as they are on the contrary sides of the arc HEX, will see the planet apparently move contrary ways, may be rendered more plain by the following familiar consideration. If two men stand with their faces towards each other, and a ball is rolled along upon the ground, this ball will move from the right hand of one of the men towards his left, and from the left hand of the other towards his right. In like manner, if one man is at the earth

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A, and the other at the sun S, then whilst the planet is describing the arc HEX, which is between them, it will appear to move from the right hand of the man at S, towards his left ; and from the left hand of the man at A, towards his right.

Whilst the motion of Venus is direct, or while it is describing the are xGH, it appears to move from V to T, among the fixed stars. But, after it has been carried in its orbit from H to %; it the line AzR, and is seen among the fixed stars at R. When it comes to E, it appears at Q; and when at y, its apparent place in the heavens is at P. Thus, as the planet passes from its greatest elongation H, on one side of the sun, through its inferior conjunction E, to its greatest elongation t, on the other side, it apparently runs back from T to V, or its motion is retrograde.

Our third proposition is, that Venus is stationary, or has no apparent motion for some time, when it is at its two greatest elongations; that is, when it is at H or x, and its apparent place is either at Tor V.

When either of the inferior planets, Venus for instance, is at its greatest elongation H or x, a line drawn from the earth through the planet, as AHT, or AxV, is a tangent to the orbit. Now, though a right line touches a circle but in one point, yet some part of the circle greater than a point is so near to the tangent, as not to be distinguished from it. Thus the arc bd, so nearly coincides with the tangent AHT, that a spectator's eye, placed at A, could not distinguish the tangent from this part of the curve ;

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consequently, while the planet is describing this arc, no other change will be made in its geocentric place, than if it was to move in the tangent.

But the geocentric place of the planet would not be altered, if the planet was to move in the tangent. For if it was to move from T towards A, or from A to V, the apparent place of it in the heavens would, in one case, be at T, in the other case at V. Therefore, while the planet is at its greatest elongation, and is describing a small arc in its orbit, that nearly coincides with the tangent, its geocentric place does not alter, but it appears to continue for some time in the same part of the heavens, or is stationary.

I have hitherto supposed the earth to be at rest ; and, upon that supposition, have explained the progress and

regress, the conjuctions and stations of the inferior planets. If this supposition was true, VT, or the arc which the planet at any time describes in its progress, and TV, the arc which it describes in its regress, would always be in the same part of the heavens. The planet, when in conjunction, would always appear at Q among the same fixed stars; and at its elongation, or when it is stationary, it would always appear among the same fixed stars at T, on one side of the sun, and at V on the other.

But this supposition is not true; for the earth revolves in its orbit ABO round the sun. Now, if the earth is at A, at the time of either conjunction, the planet at this conjunction would appear among the fixed stars at Q; and the arcs of the greatest elongation QV and QT, would be on each side of those

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