Spherical Geometry

Excerpts

• Spherical Geometry

  Since there are two poles for each side of a spherical triangle, *eight* triangles can be formed having for their angular points poles of the sides of the given triangle; but there is only one triangle in which these poles $A'$, $B'$, $C'$ lie towards the same parts with the corresponding angles $A$, $B$, $C$; and this is the triangle which is known under the name of the *polar triangle*.

• Spherical Geometry

  The sides and angles of the polar triangle are respectively the supplements of the angles and sides of the primitive triangle.

• Spherical Geometry

  Thus any such theorem will remain true when the angles are changed into the supplements of the corresponding sides and the sides into the supplements of the corresponding angles.

• Spherical Geometry

  Any two sides of a spherical triangle are together greater than the third side.

• Spherical Geometry

  The three angles of a spherical triangle are together greater than two right angles and less than six right angles.

• Spherical Geometry

  This Chapter might be extended; but it is unnecessary to do so because the Trigonometrical formul of the next Chapter supply an easy method of investigating the theorems of Spherical Geometry.

Equations

• Spherical Geometry

  A' &= \pi - a

  The angle A' of the polar triangle is the supplement of the side a of the primitive triangle, in circular measure.

• Spherical Geometry

  B' &= \pi - b

  The angle B' of the polar triangle is the supplement of the side b of the primitive triangle, in circular measure.

• Spherical Geometry

  C' &= \pi - c

  The angle C' of the polar triangle is the supplement of the side c of the primitive triangle, in circular measure.

• Spherical Geometry

  a' &= \pi - A

  The side a' of the polar triangle is the supplement of the angle A of the primitive triangle, in circular measure.

• Spherical Geometry

  b' &= \pi - B

  The side b' of the polar triangle is the supplement of the angle B of the primitive triangle, in circular measure.

• Spherical Geometry

  c' &= \pi - C

  The side c' of the polar triangle is the supplement of the angle C of the primitive triangle, in circular measure.

• Spherical Geometry

  \dfrac{AB}{OA} + \dfrac{BC}{OA} + \dfrac{CA}{OA} \text{ is less than } 2\pi

  Dividing the three sides of a spherical triangle by the sphere's radius, their sum is less than 2π, because the three plane angles at the centre O sum to less than four right angles.

• Spherical Geometry

  AB+BC+CD \text{ is less than } 2\pi\times OA

  The sum of the arcs forming the sides of the spherical figure is less than the circumference of a great circle, 2π times the radius OA. The book writes CD where the sides of the figure are CA; this looks like a typo for CA, flagged for checking.

• Spherical Geometry

  AD+BC\text{ is greater than }AC

  In a four-sided spherical polygon with each angle less than two right angles, the sum of two opposite-ended sides AD and BC exceeds the diagonal AC; obtained by repeated use of the triangle inequality on the sphere.

• Spherical Geometry

  AB+BC+CD\text{ is greater than }AC+CD

  For the four-sided spherical polygon, the sum of the sides AB, BC, CD exceeds the sum AC+CD, and so exceeds AD.

• Spherical Geometry

  A+B+C\text{ is greater than }\pi

  The three angles of a spherical triangle sum to more than π (two right angles), from the polar triangle's side-sum being less than 2π.

Problems

No exercises in this chapter.