VOL. LXXXV.J PHILOSOPHICAL TRANSACTIttNS 647 



tudes of Beachy Head and Dunnose ; with these compute the latitudes of the 

 points R and w ; then it will be found, that the arc re is -y-i-^" greater than lw ; 

 and since -^-^ of a second on the meridian is nearly a foot, re is 5 feet more 



than LW ; hence — —-^ '- = 44257.B feet is the distance between 



the parallels, and which is very nearly the same as found by the other method. 



It seems therefore, that whatever be the value of the arch between those 

 parallels in parts of a degree, the distance between them is obtained sufficiently 

 near the truth ; therefore, taking 6085l fathoms for the length of a degree on the 

 meridian, we get the arch subtended by 44258.7 feet = 7' l6".4, which sub- 

 tracted from the latitude of Beachy Head, namely, 50° 44' 23".71, leaves 50" 

 37' 7".31 for the latitude of Dunnose, We have therefore, for finding the 

 length of the degree of a great circle perpendicular to the meridian at Beachy 

 Head, or Dunnose, the latitudes of the 2 stations, and the angles which those 

 stations make with each other and the pole. 



Now it is proved in the Philos. Trans, vol. 80, that the sum of the horizontal 

 angles (such as pdb, pbd, fig. 16) on a spheroid, is nearly the same as the sum 

 of those which would be observed on a sphere, the latitudes and also the differ- 

 ence of longitude being the same on both figures. We therefore shall have re- 

 course to that determination, and apply it to the present question. The co- 

 latitudes of D and B, or the arches dp and bp, are 39° 22' 52'''.69, and 39° 15' 

 36''.29, therefore half their sum is 39° 19' 14".49, and half their difference 3' 

 38'''.2. Half the sum of the angles pdb and pbd is 89° 26' 25".5 ; therefore, as 

 tang. 39° 19' 14'''.49 : tang. 3' 38'''.2 :: tang. 89° 26' 25".5 : tang. 7° 3l' 57".7I, 

 or half the difference of the angles : hence the angles for computation are 81° 

 54' 17". 79, and 96° 58' 23".21, which, with the co-latitudes of d and b, give 

 the difference of longitude between Beachy Head and Dunnose, or the angle 

 DPB = 1° 26' 47".93. We have now 2 right-angled triangles, which may be 

 considered spherical, namely, pbw and pdr, in which the angle at the pole p is 

 given, and the sides pb and pd ; therefore, using these data, we find the arc 

 BW = 54' 56".21, and the arc dr = 55' 4".74. 



The chords of the two perpendicular arcs are about 3-|- feet less than the arcs 

 themselves; therefore bw = 336119.1 feet, and dr = 336983.5 feet; and by 

 proportioning these arcs to their respective values in fathoms, we get the length 

 of the degree of the great circle perpendicular to the meridian in the middle 

 point between w and b = 61182.8 fathoms, and in the middle point between r 

 and D = 6 1 1 8 1 .8 fathoms. Therefore 61182.3 fathoms is the length of a degree 

 of the great circle perpendicular to the meridian, in latitude 50° 41', which is 

 nearly that of the middle point between Beachy Head and Dunnose. 



If the horizontal angles, or the directions of the meridians, have been ob- 



