646 PHILOSOPHICAL TRANSACTIONS. [aNNO 17g5. 



consequence on wliat hypothesis thai it be obtained; but if 6ll73 fathoms be 

 assumed for the length of a degree of a great circle peq)endicular to the meridian 

 at M, then mb = 9' 37". 1 9, and the latitude of b, or Beachy Head, will be 

 found = 50°44'23".71. 



Again; in fig. 18, let wb be the arc of a great circle perpendicular to the 

 meridian of Beachy Head at b, meeting that of Dunnose in w ; and let dr be 

 another arc of a great circle perpendicular to the meridian of Dunnose in d, 

 meeting that of Beachy Head in r ; then we shall have 2 small spheroidical 

 triangles wbd and rdb. having in each 2 angles given, namely wdb = 81° 56' 

 53", and WBD = 6° 55' 58" in the triangle wbd ; and dbr = 83° 4' 2", with 

 BDR = 8° 3' 7" in the triangle dbr ; and these reduced to the angles formed by 

 the chords, give the following triangles for computation, namely, 



fwBD= 6° 55' 57".2 .. j-EDR = 8° 3' 6" 



In the triangle wbd< wdb = 81 56 52 A .. And in the triangle bdr.^ dbr = 83 4 1 



fDWB=91 7 10.4.. Ldrb = 88 52 53 



In which it must be noted, that the reduced angles are given to the nearest y. 



Now the chord of the arc bd, or the distance between Beachy Head and Dun- 

 nose, is 339397.6 feet, which used in the 



Triangle WBD f ew = 336X15.6 feet 7 and the triangle f dr = 336980 feet 

 gives 1 Dw = 40973.4 feet j bdr 1 br = 47547.1 feet. 



Again ; let bl and de be the parallels of latitude of Beachy Head and Dun- 

 nose, meeting the meridians in l and e : then, to find lw and er we have two 

 small triangles which may be considered as plane ones, namely, lew and edr, in 

 which the angles at w and r are given, nearly. Now the excess of the 3 angles 

 above 180° in the triangle dbw, considered as a spherical one, is 3" nearly; 

 therefore the angle dwb will be 91° 7' 12" nearly; hence bwl = 88° 52' 48'': 

 consequently the angle blw = 90° 33' 36", and lbw = 0° 33' 36". Therefore 

 with the chord of the arc wb = 336l 15.6 feet, we get wl = 3285.2 feet, which 

 added to wd, as found above, gives 44258.6 feet, for the distance between the 

 parallels of Beachy Head and Dunnose. 



Again ; in the triangle bdr, considered as a spherical one, the excess is about 

 3"-^ ; hence, from the 2 observed angles at d and b, namely, 8° 3' 7", and 83 

 4' 2", we get the 3d angle brd = 88° 52' 54".5 ; and taking the triangle erd as 

 a plane one, the other angles will be 0° 33' 32". 76 (edr), and 90° 33' 32".75 

 (der) ; therefore, with the chord of the arc dr = 33698O feet, we get re = 

 3288.2 feet, which taken from br, as found above, leaves 44258.9 feet for the 

 meridional arc, or the distance between the parallels of Beachy Head and Dun- 

 nose ; which is nearly the same as before. This method of determining the dis- 

 tance between the parallels is sufficiently correct ; but the same conclusion may be 

 deduced from a different principle, thus : Let the difference of longitude, or the 

 angle at p, be found, on any hypothesis of the earth's figure, and also the lati- 



o 



