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The distance from Hensbarrow to Carraton Hill, is 100416 feet, and from Butterton to that station 131576 feet. (See Phil. Trans. for 1797, p. 458, 460.) These data give the following triangle, viz.
136° 52′ 39′′
24 35 57,5
18 31 23,5, which gives 21602 feet, for the distance between Hensbarrow and Butterton Hill.
The angle between Carraton Hill and Rippin Tor was observed in 1796, and found — 101° 3′ 44′′,25. (See Phil. Trans. 1797.) The angle between Hensbarrow and Rippin Tor is 119° 35′ 3′′,25; therefore, 18° 31′ 19′′ is the angle between Hensbarrow and Carraton. The difference between the horizontal and chord angle is o",25 nearly; this, added to 18° 31′ 23′′,5, gives 18° 31′ 23",75, which is nearly the same as the observed angle. This agreement proves, that the angles of the triangles connecting Butterton and Hensbarrow have been observed correctly.
Latitude and Longitude of Hensbarrow.
The angle between Hensbarrow and Hemmerdon, (see Observations made at Butterton,) was 1° 52′ 4′′,5; therefore, as the angle between the latter and the meridian = 94° 16′ 44′′, we get 92° 24′ 39′′,5, for the angle which Hensbarrow makes with the same meridian. The distance from Hensbarrow to Butterton, as found above, is 21602 feet; this, with the angle 92° 24′ 39′′,5, gives the distance of Hensbarrow from the meridian = 215871 feet, and from the perpendicular 9089 feet; these, converted into parts of degrees, become 35' 17",1, and 1' 29",62. There
fore, the latitude of Hensbarrow is 50° 23′ 3′′,3, and its longitude, west of Butterton, 55′ 20′′,2; consequently, its longitude, west of Greenwich, is 3° 52′ 47′′,5 + 55′ 20′′,2 = 4° 48′ 7′′‚7.
Direction of the Meridian at St. Agnes Beacon. On the 22d of May, in the forenoon, the angle between the Pole Star, when at its greatest elongation from the meridian, and the staff near Peranzabulo, was observed, and found to be
And on the 22d, in the afternoon
Half their sum is the angle between the meridian and staff
41 13 17,5
The angle between the staff at Peranzabulo and the station Hensbarrow, was also observed at the same station, and found to be 31° 50′ 55′′,5; wherefore, 41° 13′ 17′′,5 + 31° 50′ 55′′,5 73° 4′ 13′′, is the angle between Hensbarrow and St. Agnes Beacon.
38° 26' 1",5 44
ART. XXIII. To find the Latitude and Longitude of St. Agnes
In Plate XXX. Fig. 3. Let A be the station at St. Agnes, P the pole, H Hensbarrow, and B the point where the parallel to the meridian of St. Agnes cuts that meridian, BHP being a right angled spherical triangle on the earth's surface.
PH has been already found = 39° 36′ 56′′,7; and, as BH, the distance of Hensbarrow from the meridian, 92878, and AB, the distance from the perpendicular, 28271, we get BH =15′ 10′′,9, and AB=4′ 38′′,8; which arcs are found by using 61182 and 60845 fathoms, for the length of their respective
degrees. From these data, the latitude of the point B is easily derived; for cosine 15' 10",9: radius:: cosine 39° 36′ 56′′,7: cosine 39° 36′ 54′′,2, the co-latitude of B; hence 39° 36′ 54′′,2 +4′38′′,839° 41′ 33′′,0 the co-latitude of A; hence 50° 18′ 27" is the latitude of St. Agnes. Its longitude, west from Hensbarrow, is also found by a simple proportion; sine 39° 36′ 54′′,2: radius :: sine 15′ 10′′,9: sine o° 23′ 48′′; therefore, 4° 48′ 7′′,7 + 0° 23′ 48′′ = 5° 11′ 55′′,7, is the longitude of St. Agnes, west of Greenwich.
I have shewn, with attention to minuteness, the manner in which the latitudes and longitudes of the stations on which directions of meridians have been observed are determined. It now remains to be considered, how far the uncertain state in which we remain, with respect to the figure of the earth, may affect the accuracy of those conclusions.
If the earth were homogeneous, it would necessarily be an ellipsoid; and, were its diameters known, the longitudes and latitudes of places on its surface might be accurately computed, provided their geodetical situations were correctly ascertained, and the latitude of one station in the series of triangles truly determined.
As there is, however, great reason to suppose that the earth is not any regular geometrical figure, from the impossibility of reconciling the results of the various measurements for ascertaining the lengths of degrees of latitude, some uncertainty must remain with respect to our deductions; but there seems to be reasons for supposing the errors, thence resulting, are confined within moderate limits.
In making computations on a given hypothesis of the earth's figure, the truth of the conclusions, as well as the ease with which they are found, materially depends on the distances of the objects from their respective fixed meridians.
If the difference of longitude approaches nearly to, or exceeds 3o, to compute that longitude, and also the latitude, it is necessary the precise figure should be understood; because the analogy does not hold good, in that case, between the equality of the sums of the angles of spherical and spheroidical triangles on the earth's surface. With regard to latitudes, more particularly when the distances are diminished by means of frequent new directions of meridians, a knowledge of the exact length of a degree of a great circle is not necessary; because the determination of those latitudes, by means of spherical computation, being true as to sense, the cosines of those small arcs will remain the same.
As there cannot be a doubt justly entertained of the latitude of Greenwich being very accurately determined, as particularly set forth by the Astronomer Royal in his reply to M. CASSINI, it is reasonable to suppose, that if any errors do exist in the latitudes of those stations, they can only have arisen from the computations being made with erroneous lengths of degrees on the meridian.
In our former Papers on this subject, we have taken it for granted, that the length of a degree of the meridian at the middle point between Greenwich and Paris, (50° 10′,) is 60842 fathoms, (which supposition may be considered just, provided the latitude of Paris, 48° 50′ 14′′, be as near the truth as 51° 28′ 40" is to that of Greenwich,) and afterwards added 9 fathoms, O
making it 60851, in order to get the length of the degree in 50° 41'; (see Phil. Trans. 1795, p. 537;) these 9 fathoms, however, were not arbitrarily assumed, but computed. If the latitude of Paris be 48° 50' 15", (Conn. des Tems, 1797-98, p. 373,) the length of the degree will be about 7 fathoms greater, which will make the degree in 50° 41′, 60849 instead of 60842 fathoms.
The latitude of the station on Beachy Head, 50° 44′ 23′′,7, was found by using 60861 fathoms for the length of a degree on the meridian in 51° 6'; but, if it be true that 48° 50' 15" is the latitude of Paris, the latitude of Beachy Head will be about one-third of a second greater. This seems to be the limit of the probable error in the computed latitude of this station; since its proximity to the meridian of Greenwich, obviates any doubt of the conclusions being affected by any uncertainty respecting the length of the degree of the great circle perpendicular to the meridian.
The latitude of Dunnose was determined by computing the distance between the parallels of that station and Beachy Head; (see Phil. Trans. for 1795, p. 522;) which method is very exact, and preferable to any other, since the small space between the parallels was determined with great accuracy, leaving not a doubt of a greater error than 3 feet, a quantity corresponding to about 3d part of a second. And, since the same method has been adopted to find the difference of latitude between Black Down and Dunnose, it is highly probable that the latitude of the former station is not removed more than 3ths of a second from the true one, that of Beachy Head being supposed =50° 44′ 23",7