THE METHOD OF FINDING THE DISTANCES OF THE PLANETS FROM THE SUN. CHAPTER XXIII. ARTICLE I. TS Concerning parallaxes, and their use in general. HE* approaching transit of Venus over the 1. Sun has justly engaged the attention of as. tronomers, as it is a phenomenon seldom seen, and as the parallaxes of the Sun and planets, and their distances from one another, may be found with greater accuracy by it, than by any other method yet known. 2. The parallax of the Sun, Moon, or any planet, is the distance between its true and apparent place in the heavens. The true place of any celestial ob. ject, referred to the starry heaven, is that in which it would appear if seen from the centre of the Earth; the apparent place is that in which it appears as seen from the Earth's surface. To explain this, let ABDH be the Earth (Fig. I. of Plate XIV.), C its centre, M the Moon, and ZXR an arc.of the starry heaven. To an observer at C (supposing the Earth to be transparent) the Moon M will appear at U, which is her true place, The whole of this Dissertation was published in the beginning of the year 1761, before the time of the transit, except the 7th and 8th articles, which are added since that time. referred to the starry firmament: but at the same instant, to an observer at A, she will appear at ui, below her true place among the stars. The angle AMC is called the Moon's parallax, and is equal to the opposite angle UMu, whose measure is the celestial arc Uu. The whole earth is but a point if compared with its distance from the fixed stars, and therefore we consider the stars as having no parallax at all. 3. The nearer the object is to the horizon, the greater is its parallax ; the nearer it is to the zenith, the less. In the horizon it is greatest of all; in the zenith it is nothing.–Thus let ALt be the sensible horizon of an observer at A; to him the Moon at L is in the horizon, and her parallax is the angle ALC, under which the Earth's semidiameter AC appears as seen from her. This angle is called the Moon's horizontal parallax, and is equal to the opposite angle TLt, whose measure is the arc Tt in the starry heaven. As the Moon rises higher and higher to the points M, N, O, P, in her diurnal course, the parallactic angles UMu, XNx, Yoy diminish, and so do the arcs Uu, Xx, Yy, which are their measures, until the Moon comes to P; and then she appears in the zenith Z without any parallax, her place being the same whether it be seen from A on the Earth's surface, or from C its centre. 4. If the observer at A could take the true measure or quantity of the parallactic angle ALC, he might by that means find the Moon's distance from the centre of the Earth. For, in the plane triangle LAC, the side AC, which is the Earth's semidiameter, the angle ALC, which is the Moon's horizontal parallax, and the right angle CAL, would be given. Therefore, by trigonometry, as the tangent of the parallactic angle ALC is to radius, so is the Earth's semidiameter AC to the Moon's distance CL from the Earth's centre C.But because we consider the Earth's semidiameter as unity, and the logarithm of unity is nothing, sub. tract the logarithmic tangent of the angle ALC from radius, and the remainder will be the logarithm of CL, and its corresponding number is the number of semi-diameters of the Earth which the Moon is distant from the Earth's centre. Thus, supposing the angle ALC of the Moon's horizontal parallax to be 57' 18', From the radius 10.0000000 Subtract the tangent of 57' 18" 8.2219207 And there will remain 1.7780793 which is the logarithm of 59.99, the number of semidiameters of the Earth which are equal to the Moon's distance from the Earth's centre. Then, 59.99 being multiplied by 3985, the number of miles contained in the Earth's semidiameter, will give 239060 miles for the Moon's distance from the centre of the Earth, by this parallax. 5. But the true quantity of the Moon's horizontal parallax cannot be accurately determined by observing the Moon in the horizon, on account of the inconstancy of the horizontal refractions, which always vary according to the state of the atmosphere; and at a mean rate, elevate the Moon's apparent place near the horizon half as much as her parallax depresses it. And therefore to have her parallax more accurate, astronomers have thought of the following method, which seems to be a very good one, but hath not yet been put in practice. Let two observers be placed under the same meridian, one in the northern hemisphere, and the other in the southern, at such a distance from each other, that the arc of the celestial meridian included between their two zeniths may be at least 80 or 90 degrees. Let each observer take the distance of the Moon's centre from his zenith, by means of an exceeding good instrument, at the moment of her passing the meridian: add these two zenith-distan. ces of the Moon together, and their excess above the distance between the two zeniths will be the distance between the two apparent places of the Moon. Then, as the sum of the natural sines of the two ze. nith-distances of the Moon is to radius, so is the distance between her two apparent places to her hori. zontal parallax : which being found, her distance from the Earth's centre may be found by the analogy mentioned in $ 4. Thus, in Fig. 2. let VECQ be the Earth, M the Moon, and Zbaz an arc of the celestial meridian. Let V be Vienna, whose latitude EV is 48° 20' north; and C the Cape of Good Hope, whose latitude EC is 34° 30' south: both which latitudes we suppose to be accurately determined before-hand by the ob. servers. As these two places are on the sanie meridian nVECs, and in different hemispheres, the Sum of their latitudes 82° 50' is their distance from each other. Z is the zenith of Vienna, and z the ze. nith of the Cape of Good Hope; which two zeniths are also 82° 50' distant from each other, in the common celestial meridian Zz. To the observer at Vienna, the Moon's centre will appear at a in the celestial meridian; and at the same instant, to the observer at the Cape it will appear at b. Now suppose the Moon's distance Za from the zenith of Vi- . enna to be 38° 1' 53"; and her distance zb from the zenith of the Cape of Good Hope to be 46° 4' 41" ; the sum of these two zenith-distances / Za+zb) is 84° 6' 34', from which subtract 82° 50', the distance Zz between the zeniths of these two places, and there will remain 1° 16' 34" for the arc ba, or distance between the two apparent places of the Moon's centre as seen from Vand from C. Then, supposing the tabular radius to be 10000000, the natural sine of 38° 1' 53" (the arc Za) is 6160816, and the natural sine of 46° 4' 41" (the arc 2b) is 7202821; the sum of both these sines is 13363637. Say, therefore, As 13363637 |