Sphere
Collier's New Encyclopedia (1921)
in astronomy, a term formerly applied to any one of the concentric and eccentric revolving transparent shells in which the heavenly bodies were supposed to be fixed, and by which they were carried so as to produce their apparent motions. The word now signifies the vault of heaven, which to the eye seems the concave side of a hollow sphere, and on which the imaginary circles marking the positions of the equator, the ecliptic, etc., are supposed to be drawn. It is that po portion of limitless space which the eye is powerful enough to penetrate, and appears a hol hollow sphere because the capacity of the eye for distant vision is equal in every direction. In geometry, a solid or volume bounded by a surface, every point of which is equally distant from a point within, called the center. Or it is a volume that may be generated by revolving a semicircle about its diameter as an axis. The distance from any part of the surface to the center is called a radius of the sphere. Every section of a sphere made by a plane is a circle, and all sections made by planes equally dis-sect, completely paralyzed, but alive, and tant from the center are equal. A circle of the sphere whose plane passes through the center is a great circle; all other circles are small circles. All great circles are equal, and their radii are equal to the radii of the sphere. The surface of a sphere is equal to the product of the diameter by the circumference of a great circle; or it is equivalent to the area of four great circles. Denoting the radius of the sphere by r, and its diameter by d, we have the foltowing formula for the surface: s=4pr² = pd² = 3.14159 d². The volume of a sphere is equal to the product of its surface by one-third of its radius. It is also equivalent to twothirds of the volume of its circumscribing cylinder. The following formula gives the value of the volume of any sphere, whose radius is r and diameter is d: v= pr. Spheres are to one another as the cubes of their diameters. In logic, the extension of a general. conception; the individuals and species comprised in any general conception. The doctrine of the sphere is the application of geometrical principles to geography and astronomy. An oblique sphere, or spherical projection, is the projection made on the plane of the horizon of any place not on the equator or at the poles. SPHERICAL TRIGONOMETRY. See TRIGONOMETRY.
What Has Changed in the Last Century
Every formula Collier's gave in 1921 still holds: the surface is 4πr², the volume two-thirds of the circumscribing cylinder, a result Archimedes had already proved and which no century can undo (the printed volume formula, "v = pr", is simply a typesetting casualty of 4/3πr³). What has moved is the mathematics built around the sphere rather than the sphere itself: Grigori Perelman's work of 2002–03 proved the Poincaré conjecture, settling how a three-dimensional sphere can be recognised from its topology alone, and Thomas Hales proved the Kepler sphere-packing conjecture in 1998. Two other usages here have quietly aged — the "sphere" of a general conception in logic is now called its extension, and the crystalline shells of the old astronomy are mentioned today only as history, though the celestial sphere survives exactly as Collier's describes it, as a convenient imaginary vault for mapping positions. Physically, near-perfect silicon spheres were machined in the 2000s for the Avogadro project, part of the work that led to the kilogram being redefined in 2019.
Written for this edition, 2026.