INDIVISIBLE% in geometry, the ele ments or principles into which any body or figure may be ultimately resolved which elements are supposed infinitely small : thus a line may be said to consist of points, a surface of parallel lines, and a solid of parallel and similar surfaces ; and then, because each of these elements is supposed indivisible, if in any figure a line be drawn through the elements per pendicularly, the number of points in that line will be the same as the number of the elements; whence we may see, that a parallelogram, prism, or cylinder, is re solvable into elements or indivisibles, all equal to each other, parallel and like to the base ; a triangle into lines parallel to the base, but decreasing in arithmetical proportion ; and so are the circles which constitute the parabolic conoid, and those which constitute the plane of a circle, or surface of an isosceles cone. See INFINITESIMALS.
A cylinder may be resolved into cylin drical curve surfaces, having all the same height, and continually decreasing in wards, as the circles of the base do on which they insist.
The method of indivisibles is only the ancient method of exhaustions, a little dis guised and contracted. It is found of great
use in shortening mathematical demon strations, of which take the following in stance, in the famous proposition of Ar chimedes, viz. that a sphere is two thirds of a cylinder circumscribing it.
Suppose a cylinder, an hemisphere, and an inverted cone (Plate Miscel. VI. fig. 13) to have the same base and altitude, and to be cut by infinite planes, all paral lel to the base, of which dg is one. It is plain the square of d h will be every where equal to the square of k c (the ra dius of the sphere) the square h c = e h square ; and consequently, since circles are to one another as the squares of the radii, all the circles of the hemisphere will be equal to all those of the cylinder, deducting thence all those of the cone ; wherefore, the cylinder, deducting the cone, is equal to the hemisphere ; but it is known that the cone is one third of the cylinder, and, consequently, the sphere must be two.thirds of it. Q. E. D.