INDIVISIBLES

, are those indefinitely small elements, or principles, into which any body or figure may ultimately be divided.

A line is said to consist of points, a surface of parallel lines, and a solid of parallel surfaces: and because each of these elements is supposed Indivisible, if in any figure a line be drawn perpendicularly through all the elements, the number of points in that line, will be the same as the number of the elements.

Whence it appears, that a parallelogram, or a prism, or a cylinder, is resolvable into elements, as Indivisibles, all equal to each other, parallel, and like or similar to the base; for which reason, one of these elements multiplied by the number of them, that is the base of the figure multiplied by its height, gives the area or content. And a triangle is resolvable into lines parallel to the base, but decreasing in arithmetical progression; so also do the circles, which constitute the parabolic conoid, as well as those which constitute the plane of a circle, or the surface of a cone. In all which cases, as the last or least term of the arithmetic progression is 0, and the length of the figure the same thing as the number of the terms, therefore the greatest term, or base, being multiplied by the length of the figure, half the product is the sum of the whole, or the content of the figure.

And in any other figure or solid, if the law of the decrease of the elements be known, and thence the relation of the sum to the greatest term, which is the base, the whole number of them being the altitude of the sigure, then the said sum of the elements is always the content of the figure.

A cylinder may also be resolved into cylindrical curve surfaces, having all the same height, and continually decreasing inwards, as the circles of the base do, on which they insist.

This way of considering magnitudes, is called the Method of Indivisibles, which is only the ancient method of exhaustions, a little disguised and contracted. And it is found of good use, both in computing the contents of figures in a very short and easy way, as above instanced, and in shortening other demonstrations in mathematics; an instance of which may here be given in that celebrated proposition of Archimedes, that a sphere is two-thirds of its circumscribed cylinder. Thus,

Suppose a cylinder, a hemisphere, and an inverted cone, having all the same base and altitude, and cut by an infinite number of planes all parallel to the base, of which EFGH is one; it is evident that the square of EI, the radius of the cylinder, is every where equal| to the square of SF, the radius of the sphere; and also that the square of EI, or of SF, is equal to the sum of the squares of IF and IS, or of IF and IK, because IK = IS; that is, , in every position; but IE is the radius of the cylinder, IF the corresponding radius of the sphere, and IK that of the cone; and the circular sections of these bodies, are as the squares of their radii; therefore the section of the cylinder is every where equal to the sum of the sections of the hemisphere and cone; and, as the number of all those sections, which is the common height of the figures, is the same, therefore all the sections, or elements, of the cylinder, will be equal to the sum of all those of the hemisphere and cone taken together; that is, the cylinder is equal to both the hemisphere and cone: but as the cone itself is equal to one-third part of the cylinder; therefore the hemisphere is equal to the other two-thirds of it.

The Method of Indivisibles was introduced by Cavalerius, in 1635, in his Geometria Indivisibilium. The same was also pursued by Torricelli in his works, printed 1644: and again by Cavalerius himself in another treatise, published in 1647.

INERTIÆ Vis. See Vis Inertiæ.