A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

2 Spherical Geometry.
2. Every section of a sphere made by a plane is a circle.
1~. If the plane passes through the centre, such as ABC,
the proposition is evident, since every point in the surface is
equally distant from the centre.
2~. When the plane does not pass through the centre, such as
DEF. Let O be the centre. From O let fall a perpendicular
OIon DEF (Euc. XI. xi.). Take any point F in the section
DEF. Join OF, IF. Then, since OI is normal to the plane,
the angle OIF is right; therefore IF2 = OF2 - 012; but OF is
constant, being the radius of the sphere. Hence IF is constant,
and therefore the section DEF is a circle, whose centre is I
and radius IF.
A
l' 0 /F 1 ig 1\p
B
Fig. 1.
Cor. 1.-If R be the radius of the sphere, r the radius of the
section, d the distance of the plane of section from the centre
of the sphere,
r =R2 - d2. (1)
Cor. 2.-If R = d, r = 0. Hence the section will reduce to
a point, and the plane will touch the sphere.