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It would have been fortunate, had the difference of latitude between Black Down and Butterton, and Butterton and St. Agnes Beacon, been determined in the same manner, since the latitudes of all these important stations would, in that case, have been found with evident accuracy; but, whoever has leisure and inclination to go through these calculations, will find that, by means of the directions of meridians at Butterton and St. Agnes Beacon, the latitudes of those stations may be found to within half a second. By this I mean, that, allowing the latitude of Black Down to be 50° 41′ 13′′,8, the latitude of Butterton, 50° 24′ 46′′,3, will not deviate more than half a second from the truth; and the same may be said with respect to the latitude of St. Agnes, that of Butterton being admitted as correct. Supposing, therefore, the latitude of Greenwich to be 51° 28' 40", we may rely on the assurance of the latitude of St. Agnes Beacon being determined within 14" of the truth.
With respect to the longitudes of these stations, their accuracy entirely depends on the observations made at Dunnose and Beachy Head, for determining the length of a degree of a great circle perpendicular to the meridian. The truth of the deduction drawn from those observations rests on their accuracy; and it can scarcely be deemed presumptuous to assert, that an error of more than 1" cannot have existed in either of the angles. On this account, therefore, I should suppose, that the difference of longitude between those stations, has been found so nearly as to leave no greater error than 1". The whole of the operation to which I now allude, was performed with great care; the directions of the meridians having been determined by means of double azimuths of the Pole Star, confirmed by computed azimuths. In returning to the consideration of this sub
ject, I do not perceive any source of error likely to affect the conclusions, unless it be that to which all astronomical observations, made with instruments adjusted by plumb-lines or levels, are liable. In determining differences of longitude through these means, the direction in which any lateral attraction must act, to produce a maximum of error, is at right angles to the meridian. If the attraction be in the plane of it, it is obvious the double azimuth, although the telescope of the theodolite does not move in a vertical, will nevertheless give, almost exactly, the true direction of the meridian.
The high lands about St. Catherine's Light-House, in the Isle of Wight, are about six miles from Dunnose, and nearly west of it; but it does not appear that the effect of their lateral attraction can have produced any sensible error; since it may be shewn, that the plumb-line of the sector at Schehallien would have deviated only a small part of a second from the true vertical, had the sector itself been placed at that distance from the hill. Beachy Head is situated at the eastern extremity of the South Downs; a defect of matter towards the east immediately taking place. This circumstance renders the observations liable to some small errors, on account of the superior lateral attraction in the opposite direction; but, notwithstanding it is very probable that an error induced by either of these attractions, is so very small as to render the subject scarcely worth consideration, yet, as both lie the same way, it is satisfactory to consider that they mutually tend to correct the errors which may result from either; we may, therefore, safely conclude, that 1° 11′ 36′′ is very nearly the true longitude between the station on Beachy Head and that on Dunnose. Under this persuasion, I consider it probable that the longitude of Black
Down cannot err in excess or defect more than 3"; that of Butterton 5"; and that of St. Agnes Beacon 6′′.
The latitudes and longitudes of these important stations, brought under one point of view, will be as follows:
Note. It may probably be expected, that I should determine the directions of the meridians at Black Down, Butterton Hill, and St. Agnes Beacon, by calculation, and afterwards compare them with the observed ones. I have desisted from the measure in the body of the work, and reserved the little I have to say for this note.
If the earth were a perfect sphere, or an ellipsoid of known diameters, the direction of the meridian, at any station not very remotely situated from the parallel of another, might be determined, provided the direction of the meridian at that station were observed, and the value of the arc subtended by the space between them pretty accurately ascertained, and also the latitude of the station, at which the angle is given, nearly obtained.
Thus, if it be required to find the angle at Dunnose, between Beachy Head and the meridian, from the observed angle at the latter station, and the arc between them, we shall have 39° 15' 36",3, the co-latitude of Beachy Head, and 55′ 28",7 for the oblique arc. These data (two sides and an included angle) give 1° 26′ 48′′,4, for the difference of longitude between Beachy Head and Dunnose, and 81° 56′52′′,6, for the angle which the meridian at the latter makes with the former station. The difference of longitude found in a rather more correct way, has been heretofore shewn to be 1o 26′ 47′′,93, (see Philos. Trans. 1795. p. 523,) and the angle at Dunnose was also shewn to be 81° 56′ 53′′, from observation, which may be considered the same with that found by this mode of computation. In all cases in which the data were equally correct, no doubt the direction of meridians might be computed, without fear of the results deviating much from the truth; but, if it be required to find the angle at Black Down, from the observed direction of the meridian at Dunnose, a different method must be used. It is, however, less accurate than the former one, and it has been expressly for this reason, that I have not introduced this subject into the account.
In the adjoining diagram, suppose B, Black Down; D, Dunnose; and, N, Nine Barrow Down: also, let PB, the meridian of Black Down, be prolonged to M, and DM be drawn, PM being PD. Then we shall have three spherical triangles BPD, END, and BMD. Now, the angle NBD was found from observations to be 4° 30′ 28′′, and BND 172° 27' 33",5; these give the angle BDN 3° 1′ 59′′,5, nearly, because the excess of the three angles above 180° is 1". The observed angle at D, Dunnose, between Nine Barrow Down and the meridian DP, or PDN, was 87° 56' 53"; therefore, 87° 56′ 53′′-3° 1′ 59′′,5 = 84° 54′ 53,5, is the angle at D, between the meridian and the station on Black Down. Now, the difference of longitude between B and D, or the angle at P, has been already found 1° 20′ 46′′,4; and, since BP is very nearly PD, and BD is small, we shall have rad. : tang. P :: cosine DP: cosine BMD = 89° 28′ 47". But the angle PDB has been found 84° 54' 53".5; therefore, 89° 28′ 47" - 84° 54′53′′,5 4° 33′ 53,5, the angle BDM; hence, 180° o' 2" — 94° 2′ 40′′,5 = 85° 57′ 21,"5, or MBD; therefore, 94° 2′ 38,"5, or DBP, is the angle at Black Down obtained in this way, which differs nearly 16" from the observed one, viz. 94° 2′ 22′′,75. It is probable, some portion of this arises from defects in the observation made at Dunnose, on the lights fired at Nine Barrow Down: only two lights were seen; and, as the ob servations differed 5" from each other, some degree of doubt exists, as to the accuracy of the angle. The angle at Nine Barrow Down, between Black Down and Dunnose, is not absolutely to be depended on for purposes of this kind, although there can be no doubt of its being sufficiently near the truth, for that to which it has been before applied. In the correction of the angles at that station, in our former accounts, we proceeded on the supposition of their being less satisfactory than the other angles of the triangles to which Nine Barrow Down is a common station. For these reasons, I am of opinion the computed angle cannot be applied as a test to the observed one; and it also appears to me, that greater objections lie against similar comparisons between the computed and observed angles at Butterton and St. Agnes; as those stations could not be seen from each other, nor the latter from Black Down. Although the computed directions of the meridians differ some seconds from the observed ones, I am by no means doubtful of the truth of the latter; as the double azimuths of the Pole Star, found from computation, agree very satisfactorily with those which have been used in obtaining the directions of the several meridians.In finding the value of the oblique arc, or the line which joins Black Down and Dunnose, as used in the first method of computation, I have had recourse to the following correct expression, viz. d= ; where d is the length of the required degree, p that of the great Þ + m - p. s2 circle perpendicular to the meridian, m that of a degree of the meridian itself, and s the sine of the angle constituted by the oblique arc and the meridian.