SECTION SECOND.

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.

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.

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,

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.

Direction of the Meridian at Black Down. On the 18th of April, in the forenoon, the angle between the Pole Star, when at its greatest apparent elongation from the meridian, was observed, and found to be And on the 19th, in the afternoon Half their sum is the angle between the meridian and Abbotsbury staff

104° 19′ 19′′,25

98 42 47

On the 20th of April, in the forenoon, the angle between the Pole Star, when at its greatest apparent elongation from the meridian, was observed, and found to be

And on the 19th, in the afternoon

Half their sum is the angle between the meridian and Abbotsbury staff

101 31 3

104 19 25,25 98 42 35,5

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:

DDC84° 54′ 52,5"

In the triangle BDC CDB=91 2-44,75

CB

CBD 4 2 22,75

ABD 85 57 36,75

And in the triangle ABD BAD=88 57 16,25

Now the distance between

BAD ==

BDA 5 5 7

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

BC=

313128

{CD=31146,9 } feet.

And in the triangle ABD {AB=318681,2) feet.

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