VOL. LXXXV,] PHILOSOPHICAL TRANSACTIONS. 649 



the same conclusions may be otherwise deduced; for the chords of the parallels 

 may be found from the small triangles bwl and der, (fig. 18) and these, when 

 augmented by the differences between them and the arcs, give the length of the 

 degree of longitude at Beachy Head 38719 fathoms, and Dunnose 38819 fathoms. 

 Problem. — Having given the length of a degree of a great circle perpendi- 

 cular to the meridian, in the latitude whose tangent is t, and cosine s, and the 

 length of the degree on the meridian ; to find the diameters of the earth, sup- 

 posing it an ellipsoid. 



In fig. 20, let APAP be the elliptical meridian, passing through the point b, 

 the tangent of its latitude being t, and cosine s; and put ac = x, cp = c, d 

 = the length of the degree of the great circle, d = that of the degree on the 

 meridian, and r = 57°.2Q &c. the degrees in radius. Then if bf, and af be the 

 ordinate and abscissa to the point b ; 



And 



FC = 



rD = 



rd = 





= BR, the radius of curvature of the great circle, 

 the radius of curvature of the meridional degree. 



there- 



These equations give dc^ = ds^ (t- -j- i^c") ; hence c = ^t v^ ^ _ ^^,y , 

 fore c : T :: i/ d : \/ (d -\- (d — d) t'), which call as 1 : m; then ro 



and c = 



srD 1^ {m} + t'-) 



therefore t may readily be found. 



The account next gives the following table, containing a comparison between 

 the degrees on the meridian, which have been measured in different latitudes, 

 with those computed on 3 ellipsoids whose magnitudes have been determined by 

 data applied to the conclusions derived from the foregoing problem. 



Deg. on meridian in lat. 50° 41' 

 Deg. perpendicular to meridian. 



Lat. 



0° 0' 

 39 12 

 43 

 45 

 48 43 

 51 41 

 6o 20 



Measured 

 Fath. 

 e0482 

 60628 

 60725 

 60778 

 60839 

 60851 

 61194 



1st Ellipsoid. 

 60851 fath. 

 61182 



Com- 

 puted. 

 60122 

 60607 

 60687 

 60730 

 608(!6 

 60851 

 61148 



Dili'. 



-360 



— 21 



- 38 



- 30 

 



— 4() 



2d Ellipsoid. 



60870 



61182 



Com- 

 puted. 

 60183 

 60640 

 60716 

 60756 

 60831 

 60870 

 61150 



Diff. 



-299 

 + 12 



- 9 



- 22 



+ 19 



— 44 



3d Ellipsoid. 



60851 



61191 



Com- 

 puted. 

 60103 

 60600 

 6o683 

 60727 

 6O8O8 

 6o851 

 61156 



DiiF. 



-379 



- 28 



- 42 



- 51 



- 31 

 



- 38 



Bouguer, &c 



Mason and Dixon .... 



Boscovich, &c 



Cassini 



Leisganig 



Betw. Greenwich and Paris 

 Maupertuis, &c 



The contents of the above table are computed from the data expressed in the 



different columns at top. In the 3d column, 60851 fathoms is nearly the length 



of the degree on the meridian, as derived by the application of the measured arc 



between Greenwich and Paris to the difference of latitude, namely, 2° 38' 26". 



The 5th, contains the degrees on an ellipsoid, computed from a different length 



vol. XVII. 4 O 



