An Account of the Measurement 
We have then found the total length of the whole meridional 
arc, and also the distances of two intermediate points from 
either, or from both, of its extremities. And, to bring the whole 
The demonstration of this rule I did not insert, on account of the previous length of 
the note alluded to ; but I take this opportunity of supplying the omission, in the 
words of Mr. Dalby. 
Having the length of the degree on the meridian, and 
also that of the degree perpendicular to it, at the same 
point ; to find the length of a degree in any other given di- 
rection, supposing the earth to be an ellipsoid. 
Let EP be one-fourth of the elliptic meridian ; C the 
centre of the earth; CE, CP, the equatorial and polar 
semiaxes ; G a given point on the meridian EP. Draw GR 
perpendicular to the meridian at G, meeting the axis PR in 
R ; then RG is the radius of curvature of the ellipse, at the point G, which is per- 
pendicular to the meridian at G. 
Conceive another ellipsoid FGSO to touch the given one in the point G. Then, it 
is evident, that if the curvature be respectively the same in the direction of the meri- 
dian and the perpendicular, on both ellipsoids at the point G, the curvature will also 
be equal on both figures, in any other direction at that point. And the like is manifest 
in spheroids of any other kind. 
Let M be the radius of curvature of the meridian at the point G ; then, because RG 
is the radius of curvature in the perpendicular direction, if we take FS (at right angles 
toRG) =r 2 V'RGxM, and about FS, the axis to the semidiameter RG, describe the 
ellipsoid FGSO, it will be that having the curvature of G the same as on the other 
ellipsoid at that point. 
Let OGR be the plane of an ellipse, inclined to the meridian EGP, or to the plane 
FGS, in a given angle FRO, whose sine and cosine are s and c ■ Then, since RG, or rather 
its equal, is a semitransverse, in the plane FOSR, (which is perpendicular to RG,) to the 
semiconjugate RF, we shall have w Wch, divided by RG, (RG 
being the semitransverse toRO in the perpendicular plane ROG,) gives R( ^^, * 
for the radius of curvature of the inclined ellipse OG at the point G. But, because the 
lengths of the degrees are proportional to their radii of curvature, if we put m andp 
for the meridional and perpendicular degrees, then RF or V RG a M and RG may be 
expounded by v' pm, and p ; hence, the expression will become 7T» f° r l en gfk 
