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point P; DFH the evolute of the curve AEP ; EF 
the radius of curvature at the point E ; (which we will 
fuppofe to have the fame latitude with the point E in 
Fig. 7.) and let EF be produced till it cuts the axis 
PH in G : then with the radius EG and centre G, 
defcribethe arc IEK, which will be the leaH circle, 
that can touch the curve AE P at the point E, with- 
out cutting it. Let now the curve PEA, the line 
EG, and the arc IEK revolve about PH as an axis, 
and, P E being equivalent to P E in the former figure, 
the point E in the latter figure will delcribe the paral- 
lel DEF in the former; AEP at the fame time 
defcribing the furface of the earth, and IK defcribing 
a portion of a fphere, which will be every where a 
tangent to the parallel DEF, and whofe centre will 
be G. The curvature therefore of this fphere will be 
lefs than the curvature of the earth, in the dire&ion of 
the meridian, at the point E, as the radius GE is 
.greater than the radius FE; but this, in moderate 
difiances, can caufe no fenfible error. The difference 
between AD the radius of curvature, at the point A, 
on the earth’s furface, and the line AC, according to 
that hypothefis, which makes it the greateft, does not 
exceed one lixtieth part of the whole, and upon the 
lame hypothefis, the part FG of the line EG, fup- 
pofing E to be in the latitude of 45 0 , would not 
exceed part of the whole. If then we take any 
other point upon the furface of the earth as M, at a 
Email diHance from E, the diHance between that point 
and the fphere defcribed by the arc IK, will be 
only -j-'.-g- part of the verfed fine of the arc EIVT, and 
the perpendicular Handing upon the furface of the 
earth at M, will be inclined to the perpendicular 
Handing 
