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same conclusion is derived; for, the bearing of Black Down west of Dunnose being 84° 54' 52",5, we get the distance of that station from the meridian of the latter 313072 feet, and from the perpendicular, 27861 feet; which, converted into parts of an arch, according to the lengths of their respective degrees, gives 50° 41′ 14′′ for the latitude, and 1° 20′ 46′′,4 for the longitude west of Dunnose. According to the troublesome yet ingenious method recommended by M. SEJOUR, in his Traité Analytique des Mouvemens apparens des Corps Célestes, the latitude of Black Down comes out 50° 41′ 13′′,9, and the longitude 1° 20′ 45′′,75. We may, therefore, admitting the supposition of Dunnose being situated in 50° 37′ 7′′,3, safely take 50° 41′ 13′′,8 for the latitude, and 2° 32′ 22′′,4 for the longitude, of Black Down; that of Dunnose being 1° 11′ 36′′ west of the meridian of Greenwich.
ART. XVIII. Calculation of the Distance between the Stations on Black Down, in Dorsetshire, and Rippin Tor, in Devonshire.
For the calculation of this distance, we must have recourse to the XLVIIth, XLVIIIth, XLIXth, and Lth triangles. (See Philosophical Transactions for 1797, and Plate XXX, Fig. 1 of this Volume.) In the two first, we have the whole angle at Pilsden, between Dumpdon and Black Down 152° 37′ 27′′,25, which, reduced to the angle formed by the chords, becomes 152° 37′24′′,25. The sides forming this angle, are Dumpdon and Pilsden, Pilsden and Black Down: the distance between the two first stations being 78459,3 feet, and between the two last 79110,7 feet. From these data, the distance between Dumpdon and Black Down is found to be 153095,7 feet, the triangle for computation being,
152° 37′ 24′′,25
13 37 50 5
13 44 45,25
But this side may be also found, by computing with the whole angle at Charton Common, which angle, when reduced to the plane of the chords, becomes 141° 33′ 53′′,75. The two sides are 581012,5 feet, and 103345 feet; which data give the following triangle:
141° 33′ 53′′,5
24 48 39,25
• 13 37 27,25; from whence we find the distance from Dumpdon to Black Down = 153094,6 feet. Wherefore, the mean, 153095,2 feet, may be considered to be very nearly the true distance.
In the Lth triangle, (Cawsand Beacon, Dumpdon, and Little Haldon) the angle at Cawsand Beacon is 43° 14′ 21′′,25; and in the List, (Rippin Tor, Cawsand Beacon, and Little Haldon) the angle at the same station is 25° 30′ 39′′,75; their sum is 68° 45′ 1′′, and, adding 1" for the necessary correction, it becomes 68° 45′ 2′′. Computing with this angle, and the including sides, (64020,5 and 18334 feet,) we obtain the following triangle:
90° 34′ 35′′
68 45 2
20 40 23, which gives the distance from Dumpdon to Cawsand Beacon = 169014 feet. In the XLIXth triangle, the observed angle at Dumpdon is found to be 86° 39′ 8′′5, and, by adding to it the horizontal angle at Dumpdon, between Rippin Tor and Little Haldon, and also that between Black Down and Charton Common, we get 125° 54′ 30′′,5, for the horizontal angle between Rippin
Tor and Cawsand Beacon. To reduce this angle to that formed by the chords, 6" must be subtracted; therefore, 125° 54′ 24′′,5 is the angle for computation. The sides Dumpdon and Rippin Tor, Dumpdon and Black Down, (169014 and 153095,2 feet,) with this angle, give the following triangle:
25° 36′ 4′′,5
125 54 24,5
28 29 31, which gives the distance from Rippin Tor to Black Down = 286973,3 feet. On referring to the observations made in 1797, on Black Down, it will be seen that the angle between Rippin Tor and the staff erected near Abbotsbury, was g°8′52′′,5, and the angle between Pilsden and the same staff 45° 16′ 13′′; their difference, 42° 7' 20",5, is the angle between Rippin Tor and Pilsden. Now, if the angles of the triangles, five in number, used in finding the distance between Rippin Tor and Black Down have been observed correctly, and the calculations properly made, the computed angle at Blackdown, between those stations, should be, of course, the same; but the angle formed by the chords of the arcs between Blackdown and Pilsden and Dumpdon, has been found = 13° 37′ 50′′,5, (which is very nearly the same as the horizontal one,) and the angle between Dumpdon and-Rippin Tor= 28° 29′ 31", which it is also unnecessary to correct: their sum is 42° 7′ 21′′,5, the very angle observed. It is not, perhaps, proper to dismiss this consideration, without observing that this agreement affords a strong proof of the excellence of our instrument, as the triangles, from their magnitude and nature, are not so disposed as to favour the comparison.
ART. XIX. Latitude and Longitude of Rippin Tor.
The angle at Blackdown, between the staff at Abbotsbury and the meridian, has been found 101° 31′ 1′′,5, nearly, and that between Rippin Tor and the same staff=3° 8′52′′,5; therefore, 98° 22′ 8′′ is the angle which Rippin Tor makes with the meridian, and this, taken from 180°, leaves 81° 37′ 52′′, the bearing of Rippin Tor SW from Black Down.
This angle, with the distance found above, gives 28585,3 feet, for the distance of Rippin Tor from the meridian of Black Down, and 56086,0 feet, for that from its perpendicular; therefore, the latitude is 50° 33′ 59′′,1, and the longitude west from Black Down, 1° 13′ 3′′,8; consequently, its longitude west of Greenwich is 3° 45′ 26′′2.
Direction of the Meridian at Butterton Hill.
On the 6th of May, in the afternoon, the angle between the Pole Star, when at its greatest apparent elongation from the meridian, and the staff on Hemmerdon Ball was observed, and found to be
And on the 7th, in the afternoon
Half their sum is the angle between the meridian and the staff on Hemmerdon Ball
Again, on the 7th, in the afternoon, the angle between the Pole Star, when at its greatest apparent elongation from the meridian, and the staff on Hemmerdon Ball was observed, and found to be
Half the sum of this, and the angle observed
91° 29′ 13′′75
97 4 14
94 16 44
91 29 12
in the forenoon of the same day, (97° 4′ 14′′)
94° 16' 43"
Hence, 94° 16′ 44′′ may be considered as the true angle be-tween the meridian and the staff on Hemmerdon Ball.
The angle between the station on Rippin Tor and Hemmerdon Ball, is 121° 17′ 7',75; therefore, 121° 17′ 7′′,75 — 94° 16′ 44′′ = 27° 0′ 23′′,75, is the bearing of Rippin Tor, north-east of Butterton. This angle, with 62951 feet, gives 28585,2 feet, and 56086,6 feet, for the distance of Rippin Tor from the meridian and perpendicular; which, using 61182 and 60847 fathoms, for the lengths of degrees on the meridian and perpendicular, respectively become 4′ 40′′,3, and 9′ 13′′. Therefore, in the right angled spherical triangle BPT, (Plate XXX, Fig. 2,) in which B is Butterton, P the pole, T Rippin Tor, and R the point where the parallel to the perpendicular cuts the meridian, we have the co-latitude of T, or Rippin Tor, = 39o 26′ 0′′,9, and RT = 4′ 40",3, We have, consequently, cosine 4′ 40′′,3 : radius : : cosine 39° 26' 0,′′9: cosine 39° 26′ 0,′′7, the co-latitude of the point R. So PB PR + RT 39° 26' 0",7 + 9' 13" 39° 35' 13",7; therefore, the latitude of Butterton is 50° 24′ 46′′,3, and its longitude west from Greenwich, 3° 52′ 47′′,5.
ART. XX. Calculation of the Distance between Hensbarrow and Butterton.
The most convenient, as well as the most accurate means of computing this distance, will be by referring to the Lvith, LVIIth, and LXIVth triangles, in the series of 1796, where the sum of the observed angles at Carraton Hill is 136° 52′ 43′′. The correction for reducing this angle to that formed by the chords, is 4′′; therefore, 136° 52′ 39′′ is the proper angle for computation.