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there will remain 13 minutes 21 seconds (or 801 se. conds) for their difference or elapse, in absolute time, between the beginning of egress, as seen from these two places.
Divide 801 seconds by the Sun's parallax 12}", and the quotient will be 64 seconds and a small fraction. So that for each second of a degree in the Sun's horizontal parallax (supposing it to be 12}") there will be a difference or elapse of 64 seconds of absolute time between the beginning of egress as seen from London, and as seen from St. Helena ; and consequently 32 seconds of time for every half second of the Sun's parallax ; 16 seconds of time for every fourth part of a second of the Sun's parallax ; 8 seconds of time for the eighth part of a second of the Sun's parallax; and full 4 seconds for a sixteenth part of the Sun's parallax. For in so small an angle as that of the Sun's parallax, the arc is not sensibly different from either its sine or its tangent: and therefore the quantity of this parallax is in direct proportion to the absolute difference in the time of egress arising from it at different parts of the Earth.
67. Therefore, when this difference is ascertained by good observations, made at different places, and compared together, the true quantity of the Sun's parallax will be very nearly determined. For, since it may be presumed that the beginning of egress can be observed within 2 seconds of its real time, the Sun's parallax may then be found within the 32d part of a second of its true quantity; and consequently, his distance may be found within a 400th part of the whole, provided his parallax be not less than 12}"; for 32 times 12; is 400.
68. But since Dr. HALLEY has assured us, that he had observed the two internal contacts of the planet Mercury with the Sun's edge so exactly as not to err one second in the time, we may well imagine that the internal contacts of Venus with the Sun may be observed with as great accuracy. So that we may hope to have the absolute interval between the moments of her beginning of egress, as seen from London, and from St. Helena, true to a second of time; and if so, the Sun's parallax may be determined to the 64th part of a second, provided it be not less than 12" and consequently his distance may be found, within its 800th part; for 64 times 121 is 800: which is still nearer the truth than Dr. HalLEY expected it might be found by observing the whole duration of the transit in the East-Indies and at Port-Nelson. So that our present astronomers have judiciously resolved to improve the Doctor's method, by taking only the interval between the absolute times of its ending at different places. If the Sun's parallax be greater or less than 121", the elapse or difference of absolute time between the beginning of egress at London and at St. Helena, will be found by observation to be greater or less than 801 seconds accordingly.
69. There will also be a great difference between the absolute times of egress at St. Helena and the northern parts of Russia, which would make these places very proper for observation. The difference between them at Tobolsk in Siberia, and at St. Helena, will be 11 minutes, according to De L'Isle's map: at Archangel it will be but about 40 seconds less than at Tobolsk; and only a minute and a quarter less at Petersburgh, even if the Sun's parallax be no more than 103". At Wardhus the same advantage would nearly be gained as at Tobolsk ; but if the observers could go still farther to the east, as to Yakoutsk in Siberia, the advantage would be still greater : for, as M. DE L'Isle very justly observes, in a memoir presented to the French king with his map of the transit, the difference of time between Venus's egress from the Sun at Yakoutsk and at the Cape of Good Hope will be 134 minutes.
70. This method requires that the longitude each place of observation be ascertained t
greatest degree of nicety, and that cach observer's clock be exactly regulated to the equal time at his place : for without these particulars it would be im. possible for the observers to reduce the times to those which are reckoned under any given meridian; and without reducing the observed times of egress at different places to the time at some given place, the absolute time that elapses between the egress at one place and at another could not be found. But the longitudes may be found by observing the eclipses of Jupiter's satellites; and a true meridian, for regulating the clock, to the time at any place, may be had by observing when any given star within 20 or 30 degrees of the pole, is stationary with regard to its azimuth on the east and west sides of the pole; the pole itself being the middle point between these two stationary positions of the star. And it is not mate. rial for the observers to know exactly either the true angular measure of the Sun's diameter, or of Venus's, in this case; for whatever their diameters be, it will make no sensible difference in the observed interval between the same contact, as seen from different places.
71. In the geometrical construction of transits, the scale AB (Fig. 3. of Plate XVI) may be divided into any given number of equal parts, answering to any assumed quantity of Venus's horizontal parallax from the Sun (which is always the difference between the horizontal parallax of Venus and that of the Sun), provided the whole length of the scale be equal to the semidiameter of the Earth's disc in Fig. 4.-Thus if we suppose Venus's hori. zontal parallax from the Sun to be only 26" (in. stead of 31") in which case the Sun's horizontal parallax must be 10".3493, as in 20, the rest of the projection will answer to that scale: as CD, which contains only 26 equal parts, is the same length as AB, which contains 31. And by working in all other respects as taught from § 45 to § 62, you will find the times of total ingress and be. ginning of egress; and consequently the duration of the transit at any given place, which must result from such a parallax.
72. In projections of this kind, it may be easily conceived, that a right line passing continually through the centre of Venus, and a given point of the Earth, and produced to the Sun's disc, will mark the path of Venus on the Sun, as seen from the given point of the Earth : and in this there are three cases.
1. When the given point is the Earth's centre, at which there is no parallax, either in longłtude or latitude. 2. When the given point is one of the poles, where there is no parallax of longitude ; but a parallax of latitude, whose quantity is easily determined, by letting fall a perpendicular from the pole upon the plane of the ecliptic, and setting off the parallax of latitude on this perpendicular: and here the polar transit-lines will be parallel to the central, as the poles have no motion arising from the Earth's diurnal rotation. 3. The last case is, when the given point of the Earth is any point of its surface, whose latitude is less than 90 degrees: then there is a parallax in latitude proportional to the perpendicular let fall upon the abovesaid plane, from the given point; and a parallax in longitude proportional to the perpendicular let fall upon the axis of that plane, from the said given point. And the effect of this last will be to alter the transit-line, both in position and length; and will prevent its being parallel to the central transit-line, unless when its axis and the axis of the Earth coincide, as seen from the Sun; which is a thing that may not happen in many ages.
Concerning the map of the transit. Plate XVII.
73. The title of this map, and the lines drawn upon it, together with the words annexed to these lines, and the numbers (hours and minutes) on the dotted lines, explain the whole of it so well, that no farther description seems requisite.
74. So far as I can examine the map by a good globe, the black curve-lines are in general pretty well laid down, for shewing at what places the transit will begin, or end, at sun-rising or sun-setting, to all those places through which they are drawn, according to the times mentioned in the map. Only I question much whether the transit will begin at sunrise to any place in Africa, that is west of the RedSea; and am pretty certain that the Sun will not be risen to the northernmost part of Madagascar when the transit begins, as M. De L'Isle reckons the first contact of Venus with the Sun to be the beginning of the transit. So that the line which shews the entrance of Venus on the Sun's disc at sun-rising, seems to be a little too far west in the map, at all places which are south of Asia Minor: but in Europe, I think it is very well.
75. In delineating this map, I had M. DE L'Isle's map of the transit before me. And the only difference between his map and this, is, 1.
That in his map, the times are computed to the meridian of Paris; in this they are reduced to the meridian of London. 2. I have changed his meridional projection into that of the equatorial; by which, I apprehend that the black curve.lines, shewing at what places the transit begins, or ends, with the rising or setting Sun, appear more natural to the eye, and are more fully seen at once, than in the map from which I copied; for in that map the lines are interrupted and broken in the meridian