640 PHILOSOPHICAL TRANSACTIONS. [aNNO 1795. 



The length of a degree on the meridian in these latitudes being about 60874 

 fathoms, and that of a degree perpendicular to the meridian, about 6ll83; it 

 follows, that the values of all the oblique arcs are between these extremes: now 

 having obtained the sides of the triangles within a few feet by a rough compu- 

 tation, we take their values in parts of a degree, nearly as their inclinations to 

 the meridian; this proportion, though not found on an ellipsoid, is sufficiently 

 true for finding the values of the sides of the triangles; for in this case great 

 accuracy is not necessary. With the sides thus determined, we compute the 3 

 angles of each triangle by spherical trigonometry; and taking twice the natural 

 sines of half the arcs, we get, by plane trigonometry, the angles formed by the 

 chords; then, from the difFerences of these angles we infer the corrections to be 

 applied to the observed angles, to reduce them for computation: an example 

 however will make this matter much plainer; for which purpose we shall take 

 the very oblique triangle formed by the stations Beachy Head, Chanctonbury 

 Ring, and Rook's Hill. 



^Rook's Hill and B. Head 39' 7" -> 113785156 



, ^'"'^ J Ch. Ring and B. Head 25 47 chords > 75000501 



iRook's Hill and Ch. Ring 1-J. J 40724320 



Hence the angles by spherical trigonometry will be o , „ 



At Chanctonbury Ring 157 59 36.29 



Rook's Hill 14 17 58.32 



Beachy Head 7 42 26.56 



And the angles formed by the chords 157 59 27.44 



14 18 3.44 

 7 42 29.12 



We have given the results to the 2d place in decimals, though perhaps they 

 are true only to the nearest 1 0th of a second. In finding the angles formed by 

 the chords, we have used Rheticiis's large triangular canon, where the natural 

 sines are given to every 10" of the quadrant, and computed to the radius 

 10000000000. It is remarked, that great accuracy in the values of the sides in 

 degrees, &c. is not necessary, and that this is true will be found on examination; 

 for in the foregoing example, if the sides of the triangle be varied, so that the 

 resulting angles are several minutes different from those found above, still the 

 differences between the spherical and plane triangles will be very nearly the same. 



When the 3 angles of any triangle appear to have been observed correctly, 

 by their sum being equal to 180 degrees plus the computed excess, the cor- 

 rections for the chord angles have been added to, or taken from them, as that 

 correction has been negative or affirmative, and the triangle rendered fit for 

 computation. Also, if any triangle, where the sum has either fallen short of, 

 or exceeded 1 80 degrees plus the computed excess, one or two of the observed 

 angles have appeared to have been determined with sufficient accuracy, as sh6wn 

 by the agreement of the angles obtained on different parts of the arch; the cor- 



