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

10 Spherical Geometry.
and whose dihedral angles (Eue. XI., Def. Il.) are equal to
the angles of the spherical triangle (~ 8). There is then a
correspondence between the spherical triangle ABC and the
solid angle O-ABC: every property of one gives a property
of the other.
DEF. XI.-The portion of a sphere comprised between two halves
of great circles is called a LUNE.
Three great circles intersect in six points A, A'; B, B'; C, C'.
These are two by two diametrically opposite, and divide the
sphere into eight triangles.
DEF. XII. —Two triangles BCA, B'CA, which have a common
side CA, and whose other sides belong to the same great circles,
are called COLUNAR TRIANGLES. The triangle ABC has three
colunar triangles, viz. A'BC, B'CA, C'AB.
DEF. XIII.-Two triangles ABC, A'B'C', whose corresponding vertices are diametrically opposite, are called ANTIPODAL
TRIANGLES.
12. Any two sides of a spherical triangle are together greater
than the third, and the sum of the three sides is less than a
great circle.
DEaM.-Let ABC be the spherical triangle (see fig., ~ 11),
O the centre of the sphere. Join OA, OB, OC; then (Euc.
XI., xx.) any two of the plane angles forming the trihedral
angle O - ABC are together greater than the third; but the
arcs AB, BC, CA are the measures of the plane angles A OB,
BOC, COA. ience the sum of any two of the arcs AB, BC, CA
is greater than the third.
Again, the sum of the three plane angles AOB, BOC, COA
is (Ecu. XI., xxi.) less than four right angles. ifence the sum
of the three arcs AB, B C CCA is less than a great circle.
In the same manner it follows that the sum of the sides of any
convex spherical polygon is less than a great circle.