VOL. LVI.] PHILOSOPHICAL TRANSACTIONS. 293 



Let now the curve pea, the line eg, and the arc iek revolve about ph as an axis; 

 then, PE being equivalent to pe in the former figure, the point e in the latter 

 figure will describe the parallel def in the former ; abp at the same time describ- 

 ing the surface of the earth, and ik describing a portion of a sphere, which will 

 be every where a tangent to the parallel def, and whose centre will be g. The 

 curvature therefore of this sphere will be less than the curvature of the earth, in 

 the direction of the meridian, at the point e, as the radius ge is greater than the 

 radius fe; but this, in moderate distances, can cause no sensible error. The dif- 

 ference between ad, the radius of curvature at the point a, on the earth's surface, 

 and the line ac, according to that hypothesis, which makes it the greatest, does 

 not exceed -bV part of the whole; and on the same hypothesis, the part fg of the 

 line EG, supposing e to be in the latitude of 45°, would not exceed -i4-„ part of the 

 whole. If then we take any other point on the surface of the earth, as m, at a 

 small distance from e, the distance between that point and the sphere described by 

 the arc ik, will be only -j-^ part of the versed sine of the arc em; and the perpen- 

 dicular standing on the surface.of the earth at m, will be inclined to the perpendi- 

 cular standing on the sphere, in an angle, which is equal to -pw P^'^t of the angle 

 subtended by the arc em. And in higher latitudes these quantities will be still less. 

 Let us now return to fig. 1 J , and supposing the point e to be situated in latitude 

 45°, let the arc em, cutting pe at right angles, consist of 2°, near 140 statute 

 miles; then will the side pm, of the triangle pme, consist of 45° 2' 5Y, and conse- 

 quently, if lm in fig. 1 1, be supposed to correspond to em in fig. 12, the distance 

 of these two points e and m, in the latter figure, will be only 2' 5-J-'', the -p^ part 

 of the versed sine of which is a little more than ^ of an inch, to the radius of 

 the earth, which will therefore be the distance of the point m on the earth's sur- 

 face, and the point of the imaginary sphere, described by ik, immediately over 

 it. Hence also the inclination of the real perpendicular at m, and the imaginary 

 one standing on the arc ik, at the same place, to each other, will be something 

 less than a second, a quantity in itself almost too small to be regarded, unless 

 the instruments made use of are both very large and very excellent in their kinds, 

 and which, being wholly in the plane of the meridian, will produce an error, 

 that must be perfectly insensible, with any instruments whatever, in an observa- 

 tion of the angle pme, fig. 1 1 , which will therefore, to all intents and purposes, 

 be the same, as if the curvature" of the earth in the direction of the meridian, 

 and in the direction of me or le were accurately the same. 



I have supposed the arc me to stand at right angles to the meridian pe, which 

 passes through one of the extreme stations; the method here proposed is how- 

 ever liable to the least error, when the meridian cuts the arc to be measured at 

 right angles in the middle of it; but this makes so very small a difference, that 

 it is not worth regarding; nor is it indeed necessary that the arc should not de- 



