VOL. LXXXI.] PHILOSOPHICAL TRANSACTIONS. 6Q 



fore, if each of the former degrees was about 140 fathoms less, the computed 

 and measured arcs in latitude 50° Q'i would be nearly the same. But, that they 

 also may nearly agree in latitude 45°, let the degrees at the equator, and in lati- 

 titude 50° g'-^, be taken 60344 and 60844; then, from these two degrees, the 

 ratio of the axes will be found as the tangents of the arcs 50° p'-i- and 50° l' 35%; 

 and the semi-axes 3489932 and 3473656 fathoms*. 



The length of the whole meridional arc between Greenwich and Paris on this 

 ellipsoid is 6 fathoms greater than the measured arc; the degree in latitude 48° 

 43', l6 fathoms less; in latitude 45°, 10 fathoms less; in 43°, 13 fathoms 

 greater; and that in latitude 39° 12', 54 fathoms greater. The degrees at the 

 equator and polar circle are considerably less than the measured ones, conformable 

 to the hypothesis. 



Suppose CE, cp (fig. 13, pi. 1,) are the greater and less semi-axes of the el 

 lipsoid; g Greenwich; pge its meridian; pd the meridian of Dunkirk; and let 

 GBA be perpendicular to the curve of the meridian at g; then ga will be the 

 shorter axis of the elliptical section which is the perpendicular to the meridian 

 at Greenwich, and the angle ebg will be the latitude of Greenwich, or 5 1° 28' 

 40". Let HO (parallel to ga) be the section of the parallel to that perpendicular, 

 passing through Dunkirk. Then the arc gh is 152549 feet, but this arc ex- 

 ceeds the real distance of the parallels ga, ho, not more than a fathom; there- 

 fore this distance may be taken = 25424 fathoms. Now the sections ga, ho, 

 of the ellipsoid being similar, from the known properties of the figure, we shall 

 get ho the shorter axis of the section of the parallel = 6959396, its longer axis 



* Determined thus : If right lines be drawn perpendicular to the curve of a conic section to meet 

 the axis, it is known, that the radii of curvature at the points in the curve from whence these lines 

 «re drawn, will be as the cubes of these lines. Hence, if pc, gb, ec, pi. 1, fig. 13, are perpen- 



pc* 

 dicular to the curve, the radii of curvature at p, g, e, will be as pc', gb', and ( — )' because at 



PC* 



the point e, or equator, the line so drawn will become the radius of curvature itself, or — . There- 



PC* 



fore gb' : ( — )' :: rad. curv. at g : rad. curv. at e :: length of a deg. in the lat. of G : length of a 



deg. at E, the equator. Let the arc erl be described witli the radius ce; draw cr parallel to gb, 

 RS parallel to pc, and join ck; then, by the nature of the ellipse, or (ce) : ck :: gb : half the 



CP* CP* 



parameter, or — ; therefore ce' : ck' :: gb' : ( — )' -.: 60844 : 60344 (supposing the lat. of the 



'^ CE CE J 



point G to be 50° 9'|,) or ce (ck) : ck :: (60844)3 : (60344) '; but cr : CK :: sine skc : sine 



60844 , 

 KBC (colat.); therefore, ("60344^^ ^ cosine lat. = sine skc ; hence tlie angle sck is given (50° 



1' 35"|) ; therefore, as tang, sck : tang. lat. (scr) :: sk : sr :: lesser semi-axis cp : greater cE. 

 And putting d = 57.295779, &c. the degrees in the circular arc which is equal to the radius) we have 



tang, lat. ^ 60344 d = 3480032 fathoms the longer semi-axis ; and -^'- — '- x 603Hd = 

 ^tang. SCK tang, sck 



3473656, the shorter. 



