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Determination of the Latitudes and Longitudes of the Stations on Black Down, in Dorsetshire, Butterton, in Devonshire, and St. Agnes Beacon, in Cornwall.
ART. XVI.-Calculation of the Distance between Black Down and Dunnose in the Isle of Wight.
To complete this distance, I shall have recourse to the xxvith and xxviith triangles, published in the Philosophical Transactions of 1795, and LIIId and LIVth of the Trans. for 1797, together with the observations made at Black Down, in the latter year. (See also Pl. XXX. Fig. 1.).
The most eligible method of calculating with these data, seems to be that of first finding the cross-distance between Black Down and Dean Hill. To do this, we have the angle at Nine Barrow Down, between Black Down and Dean Hill, and the respective distances from the first to the latter stations, together with the newly observed angle between Dunnose and Nine Barrow Down; from which we obtain the angles of a triangle, constituted by Dunnose, Nine Barrow Down, and Black Down.
The distance from Nine Barrow Down to Dean Hill is 166497 feet, and, from the same station to Black Down, the distance is 126782 feet, (see Phil. Trans. for 1795, p. 502, and for 1797, p. 455,) and the angle comprehended by those distances
110° 30′13′′,25. The difference between the horizontal angle and that formed by the chords is 3",25, which, substracted from 110° 30′ 13′′,25, leaves 110° 30' 10": computing with this
angle and the sides spoken of, there results the following triangle, viz.
Nine Barrow Down
29 22 55,75
This, using the side Nine Barrow and Dean Hill, (166497 feet,) gives 240236,7 feet, for the distance between Black Down
and Dean Hill.
The angle at Dean Hill, between Nine Barrow Down and Dunnose, is 64° 50' 19", (see Phil. Trans. for 1795. p. 501,) and the angle between Black Down and Nine Barrow, as just found, is 29° 22′ 55′′,75, which, increased by the proper correction for the difference between the chord and horizontal angles, becomes 29° 22′ 57′′,5. The sum of these angles,94° 13′ 16′′,5, is the horizontal angle between Black Down and Dunnose.
The angle at Black Down, between Dunnose and Nine Barrow Down, deduced from observations made in 1797, is found to be 4° 30' 25",75: this, subtracted from the angle between Dean Hill and Dunnose, leaves 35° 36′ 29′′, for the angle at Black Down; which, corrected for the purpose of reduction to their respective chord angles, become 94° 13′ 11′′,5, and 35° 36' 25",75, from whence we get the angle at Dunnose = = 50° 10' 22",75. We have, therefore, the following triangle, viz.
110° 30′ 10′′
50 10 22,75
The distance between Dean Hill and Dunnose is 183496,2 feet, (Phil. Trans. for 1795, p. 501,) and that between Black Down and Dean Hill, according to the foregoing computation, is 240236,7 feet: these, applied to the angles of the above triangle,,
94° 13' 11,5"
35 36 25,75
give 314309,6, and 314305,4 feet, respectively, for the distance between Black Down and Dunnose: wherefore, the mean 314307,5 feet, 59,528 miles, may be considered as the true distance between those stations.
104 19 25,25 98.42 35,5
And on the 19th, in the afternoon :
Half their sum is the angle between the meridian and Abbotsbury staff
101 31 0,5
Therefore, 101° 31′ 2′′ may be taken for the angle between the meridian and Abbotsbury staff.
ART. XVII. Latitude and Longitude of Black Down.
The angle between Dunnose and the Abbotsbury Staff was observed, and found = 164° 26′ 35′′25; and the angle between the meridian and the same staff, by double azimuths of the Pole Star, 101° 31' 2". Wherefore their sum, subtracted from 360°, leaves 94° 2′ 22",75, the angle which Dunnose makes with the meridian.
In Fig. 4. Plate XXX. let Z be the zenith, B the station on Black Down, and ZBA its meridian; also, let D be Dunnose, and ZD its meridian; likewise, suppose BC to be an arc of a great circle, perpendicular to the meridian at B, and DA another arc of a great circle, perpendicular to the meridian at D, BF and ED being the parallels of latitude at Black Down and Dunnose.
In the spherical triangle BZD, the angles at B and D are given, the first being 94° 2′-22",75, and the second 84° 54′ 53′′; therefore, in the triangle ABD the angle at B is 85° 57' 36",75, and, in the triangle BDC, the angle at D=84° 54′ 53′′: hence, the angles of these triangles, when reduced to those formed by the chords, are as follows:
DDC=84° 54′ 52,5"
In the triangle BDC CDB =91 2.44,75
CBD 4 ABD = 85 57 36,75 And in the triangle ABD BAD=88 57 16,25 = - 5 5 7
Now the distance between 'Black Down and Dunnose, BD, has been already found to be 314307,5 feet; therefore, using the above angles with that distance, (after the proper corrections are applied for reducing the horizontal angles to those formed by the chords,) we get,
In the triangle BCD
And in the triangle ABD
Again, in the two small triangles formed by the parallels BF and ED, the perpendiculars BC and DA, and the small arcs CF and AE, we have the angles at C and A given, the
first being 91° 2′ 45′′,75, and the last 88° 57′ 15′′; which angles, however, are augmented by the addition of the differences between the horizontal angles and those formed by the chords, We have therefore,
91° 2′ 45,75′′ In the triangle BCF BFC 88 25 51,5 FBC 0 31 22,75
EAD 88 57 17
And in the triangle AED AED=90 31 21,5
And, using BC and AD, as found above, we get
Therefore FD=DC+ CF = 22146,9 +2859,1=25006
feet. And BE BAEA = 27864,5-2859,8=25004,7 feet. The mean, 25005,3 feet, may be considered as very nearly the true distance between the parallels of Black Down and Dunnose. This method is the same as that made use of in the Phil. Trans. for 1795, p. 521, and affords the means of very accurately determining the distance between the parallels of latitude of the two stations, when the angles were observed with precision, and the direction in which the stations lie, is not much removed from east and west.
This small space, 25004,7 feet, corresponds to 4′ 6′′,5, in which I use 60851 fathoms for the length of a degree of the meridian in 50° 41'. See Phil. Trans. for 1795, p. 537.
Now the latitude of Dunnose is 50° 37' 7",3, and its longitude 1° 11′ 36′′; (Phil. Trans. for 1795, p. 536;) therefore, 50° 37′7′′3 +4' 6",5=50° 41′ 13′′,8, is the latitude of Black Down.
This method of finding the latitude seems to be more correct than by spherical computation; yet, by this latter, nearly the