The First Six Books of the Elements of Euclid

by Euclid; John Casey

24. If h be the perpendicular distance between two parallel planes, which cut a sphere in sections whose radii are ρ₁, ρ₂, the volume of the frustum is

{h² + 3(ρ₁² + ρ₂²)}.

25. If δ be the distance of a point P from the centre of a sphere whose radius is R, the sum of the squares of the six segments at three rectangular chords passing through P is = 6R² − 2δ².

26. The volume of a sphere : the volume of an inscribed cube as π : 2.

27. If O–ABC be a tetrahedron whose angles AOB, BOC, COA are right, the square of the area of the triangle ABC is equal to the sum of the squares of the three other triangular faces.

28. In the same case, if p be the perpendicular from O on the face ABC,

29. If h be the height of an æronaut, and R the radius of the earth, the extent of surface visible =