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In Fig. 5. of Plate XIV let B M A be the Earth, V Venus, and the Sun. The Earth's motion on its axis from west to east, or in the direction AMB, carries an observer on that side contrary to the motion of Venus in her orbit, which is in the direction UVW; and will therefore cause her motion to appear quicker on the Sun's disc, than it would appear to an observer placed at the Earth's centre C, or at either of its poles. For, if Venus were to stand still in her orbit at V for twelve hours, the observer on the Earth's surface would in that time be carried from A to B, through the arc AMB. When he was at A, he would see Venus on the Sun at R; when at M, he would see her at S; and when he was at B, he would see her at T: so that his own motion would cause the planet to appear in motion on the Sun through the line RST'; which being in the direction of her apparent motion on the Sun as she moves in her orbit UW, her motion will be accelerated on the Sun to this observer, just as much as his own motion would shift her apparent place on the Sun, if she were at rest in her orbit at V.
But as the whole duration of the transit, from first to last internal contact, will not be quite six hours; an observer, who has the Sun on his meridian at the middle of the transit, will be carried only from a to b during the whole time thereof. And therefore, the
. duration will be much less contracted by his own motion, than if the planet were to be twelve hours in passing over the Sun, as seen from the Earth's centre.
25. The nearer Venus is to the Earth, the greater is her parallax, and the more will the true duration of her transit be contracted thereby; the farther she is from the Earth, the contrary: so that the contraction will be in direct proportion to the parallax. Therefore, by observing, at proper places, how much the duration of the transit is less than its true duration at the Earth's centre, where it is 5 hours 58 minutes, as given by the astronomical tables, the parallax of Venus will be ascertained.
26. The above method ($ 17, & seq.) is much the same as was prescribed long ago by Doctor Halley; but the calculations differ considerably from his; as will appear in the next article, which contains a translation of the Doctor's whole dissertation on that subject.--He had not computed his own tables when he wrote it, nor had he time before-hand to make a suffi. cient number of observations on the motion of Venus, so as to determine whether the nodes of her orbit are at rest or not; and was therefore obliged to trust to other tables, which are now found to be erroneous.
Containing Doctor Halley's Dissertation on the
method of finding the Sun's parallax and distance from the Earth, by the transit of Venus over the Sun's disc, June the 6th, 1761. Translated from the Latin in Mottee's Abridgment of the Philosophical Transactions, Vol. I. page 243; with additional notes.
There are many things exceedingly paradoxical, and that seem quite incredible to the illiterate, which yet by means of mathematical principles may be easily solved. Scarce any problem will appear more hard and difficult, than that of determining the distance of the Sun from the Earth very near the truth: but even this, when we are made acquainted with some exact observations, taken at places fixed upon, and chosen before-hand, will without much labour be effected. And this is what I am now desirous to lay before this illustrious Society* (which I foretel will continue for ages), that I may explain before-hand to young astronomers, who may perhaps live to observe these things,
• The Royal Society.
a method by which the immense distance of the Sun may be truly obtained, to within a five-hundredth part
a of what it really is.
It is well known that the distance of the Sun from the Earth is by different astronomers supposed different, according to what was judged most probable from the best conjecture that each could form. Pto. lemy and his followers, as also Copernicus and Tycho Brahe, thought it to be 1200 semidiameters of the Earth; Kepler, 3500 nearly: Ricciolus doubles the distance mentioned by Kepler; and Hevelius only increases it by one half. But the planets Venus and Mercury having, by the assistance of the telescope, been seen on the disc of the Sun, deprived of their borrowed brightness, it is at length found that the apparent diameters of the planets are much less than they were formerly supposed; and that the semidiameter of Venus seen from the Sun subtends an angle of no more than a fourth part of a minute, or 15 seconds, while the semidiameter of Mercury, at its mean distance from the Sun, is seen under an angle only of ten seconds; that the semidiameter of Saturn seen from the Sun appears under the same angle; and that the semidiameter of Jupiter, the largest of all the planets, subtends an angle of no more than a third part of a minute at the Sun. Whence, keeping the proportion, some modern astronomers have thought, that the semidia. meter of the Earth, seen from the Sun, would sub. tend a mean angle between that larger one subtended by Jupiter, and that smaller one subtended by Saturn and Mercury; and equal to that subtended by Venus (namely, fifteen seconds): and have thence concluded, that the Sun is distant from the Earth almost 14000 of the Earth's semidiameters. But the same authors have on another account somewhat increased this distance: for inasmuch as the Moon's diameter is a little more than a fourth part of the diameter of the Earth, if the Sun's parallax should be supposed fifteen seconds, it would follow that the body of the Moon is larger than that of Mercury; that is, that a secondary planet would be greater than a primary; which would seem inconsistent with the uniformity of the mundane system. And on the contrary, the same regularity and uniformity seems scarcely to admit that Venus, an inferior planet, that has no satellite, should be greater than our Earth, which stands higher in the system, and has such a splendid attendant. There. fore, to observe a mean, let us suppose the semidiameter of the Earth seen from the Sun, or, which is the same thing, the Sun's horizontal parallax, to be twelve seconds and a half; according to which, the Moon will be less than Mercury, and the Earth larger than Venus; and the Sun's distance from the Earth will come out nearly 16,500 of the Earth's semidiameters. This distance I assent to at present, as the true one, till it shall become certain what it is, by the experiment which I propose. . Nor am I induced to alter my opinion by the authority of those (however weighty it may be) who are for placing the Sun at an immense distance beyond the bounds here assigned, relying on observations made upon the vibrations of a pendulum, in order to determine those exceeding small angles; but which, as it seems, are not sufficient to be depended upon; at least, by this method of investigating the parallax, it will sometimes come out nothing, or even negative; that is, the distance would either become infinite, or greater than infinite ; which is absurd. And indeed, to confess the truth, it is hardly possible for a man to distinguish, with any degree of certainty, seconds, or even ten seconds, with instruments, let them be ever so skilfully made: therefore, it is not at all to be wondered at, that the excessive nicety of this matter has eluded the many and ingenious endeavours of such skilful operators.
pole, I had an opportunity of observing, with the greatest diligence, Mercury passing over the disc of the Sun; and (which succeeded better than I could have hoped for) I observed, with the greatest degree of accuracy, by means of a telescope 24 feet long, the very moment when Mercury entering upon the Sun seemed to touch its limb within, and also the moment when going off it struck the limb of the Sun's disc, forming the angle of interior contact: whence I found the interval of time, during which Mercury then appeared within the Sun's disc, even without an error of one second of time. For the lucid line intercepted between the dark limb of the planet and the bright limb of the Sun, although exceeding fine, is seen by the eye;
and the little dent made in the Sun's limb, by Mercury's entering the disc, appears to vanish in a moment; and also that made by Mercury, when leaving the disc, seems to begin in an instant.—When I perceived this, it immediately came into my mind, that the Sun's parallax might be accurately determined by such kind of observations as these; provided Mercury were but nearer to the Earth, and had a greater parallax from the Sun; but the difference of these parallaxes is so little, as always to be less than the solar parallax which we see; and therefore Mercury, though frequently to be seen on the Sun, is not to be looked upon as fit for our purpose.
There remains then the transit of Venus over the Sun's disc; whose parallax, being almost four times as great as the solar parallax, will cause very sensible differences between the times in which Venus will seem to be passing over the Sun at different parts of the Earth. And from these differences, if they be observed as they ought, the Sun's parallax may be determined even to a small part of a second. Nor do we require any other instruments for this purpose, than common telescopes and clocks, only good of their kind : and in the observers, nothing more is needful