Cone

CONE, in its earliest geometric use, denoted the solid space swept over by a right triangle rotating about one side (altitude or axis, a) the other side (base, r) tracing out the circle base of the cone, and the hypotenuse (slant height, s) its curved surface, the vertex V of triangle, and cone being the same (fig. 1).

Cones and Conics.

At first the size of the vertical angle of such a right circular cone (twice that of the triangle) appeared important, and hence such cones were divided into three classes, according to its size, and hence were named "acute-, right-, ob tuse-angled." The natural position of a plane cutting through a cone seemed to be perpendicular to the slant height (i.e., the hypotenuse or generatrix in any position, called also an element, e, of the surface). It is said that Menaechmus (c.

35o B.C.), in striving to construct the so-called double mean pr op o r t i on a 1, a:x=x:y=y:b, discriminated three cor responding types of conic sections, after wards named respectively ellipse, parabola and hyperbola. It was the "great geometer," Apollonius, born at Perga in Pam phylia (fl. c. 225 B.C. ), student in the Euclid school of Alexandria, who perceived and showed that the type of cone was indifferent, any plane section of any circular cone, parallel to an element, yielding a parabola, and other sections yielding either an ellipse or hyperbola.

Volume and Area.—The conception of a cone as solid called for the measurement of its content, a fact enunciated first by the travelling philosopher Democritus (c. 46o–c. 370 B.C.), proved by Eudoxus of Knidos (c. 4o8—c. 355 B.C.), and later completed by the great reckoner, Archimedes (287-212 B.C.), who showed that a cone, hemisphere and cylinder, all of the same base and height (fig. 2) have volumes respectively as 1, 2, 3,—a relation holding for any type of the extremes—the cone and cylinder. If they had equal bases and heights, the volume of the one is one third that of the other, which is the product of base and height. Plainly the curved surface of a right circular cone may be thought as slit all along any element and then flattened out on a plane (tan gent along the opposite element) into a circular sector ; hence the curved surface equals in area the sector area, i.e., the half-product of base-circumference as arc and slant height as radius.

Oblique Cone.—The circular cone is defined by Apollonius more generally as the surface (or its enclosed volume) traced by a right line passing through a fixed point (vertex) and gliding along a fixed circle. The right line through this vertex and the cen tre of this circle is the axis; if this be at right angles to the base (the plane of the circle), the cone is right; otherwise, it is oblique, in which case the vertical angle is not of constant size. Any plane containing the axis cuts the solid in such a vertical angle, and the surface in its sides (elements of the cone). The plane containing the axis and being perpendicular to the base of the cone forms the "principal section"; any plane per pendicular thereto and inclined equally but inversely (with the base) to the gen erator-elements therein is called a "sub contrary section." This cuts the cone in a circle, as do all planes parallel to it or to the base, all which are therefore called "cyclic planes" (fig. 3).

Cone as Surface.

It was natural to gard and define the solid first rather than the bounding curved surface, and hence the early Greek achievements were largely in stereometry. Only very gradually it came to be felt that the surface alone possessed peculiar properties, the enclosed space being indifferent. Hence the very recent change in definition and treatment from three to two dimensions. From the new point of view the cone concept goes a broad generalization, designating any path of (or surface traced by) a right line (the generatrix, g) that passes always through a fixed point V. This path, to be definite, is directed by some curve (the directrix D), along which the line glides always (fig. 4). Thus, in the right circular cone, D is the circle bounding the cone's base, the track of the moving end of the hypotenuse (g). In the oblique circular D is still a circle, but no longer traced by one certain point of the line g. If D be a conic (as an ellipse) the surface is called a "quadric cone." If a fixed direction (or line called the axis) be assumed as passing through the fixed vertex V, the motion of the tracing line g might be directed by ordered vari ation in the vertical angle between the fixed and the moving line. In the most important case, the right circular cone, this variation is the simplest possible; 0, the angle is constant ; the two lines are rigidly joined at V; and the generatrix swings freely round the axis (which may be thought as turning in itself) .

Tangency.—Such seems to be the sim plest and most vivid conception of theright circular cone ; the two halves meeting at V and extending oppositely without end are called nappes or sheets. If slit through out along an element, and rolled out (de veloped) on the plane tangent along the opposite element, the nappes would appear as centrically sym metric sectors of an infinite circle about V as centre. Any plane tangent to a cone passes through V and touches the cone along some element throughout. All such planes would touch the whole cone-surface and would envelop it completely. The tangent plane at any given point of the surface (except V) is quite definite, but at V this definiteness disappears, all tangent-planes pass alike through V,—a matter of importance in treating the umbilical points in Fresnel's wave-surface, in connection with Hamilton's discovery of the conical refraction of light.

Equation.—In analytic geometry the equation of the right circular cone traced by rotating the line y=mx round the y axis is This simply means that, as any point P (of the line) rotates round the y axis, its y remains unchanged and also its distance Vx from the y axis (OY), and the fixed ratio of these two lengths is m.

Intersection.—The doctrine of the intersection of a cone with other surfaces belongs rather to constructive geometry, where it is developed and applied. Its importance may be presumed from the fact that central projection is cone-like, the lines of projec tion issuing like rays of light from a point, the vertex of the cone. An intere.ing special case is the spliero-conic, the intersection of any quadric cone with a sphere about the vertex of the cone as centre.

Truncation.

In more elementary measurement we meet with the frustum (fig. 5), especially of the right circular cone. The word, meaning piece, denotes a truncated cone, or the portion of a cone (viewed as a solid) between the base and a cutting plane generally parallel thereto. The problem of computation was ap parently first proposed with respect to truncated pyramids, and very naturally in Egypt. Indeed, one of the earliest extant com putations of volume is Egyptian and relates to such a figure. Much later, and in the Stereometrica of the encyclopedic Heron "the mechanic" of Alexandria (c. A.D. 5o? or as late as A.D. 200?) the volume is calculated for a pyramid-frustum on a square base (zoo), the top-section a square (4), the oblique edge 9; the height is found to be 7, which multiplied by (I°-{-2)2-{-1CI°-212 2 2 gives 2891, a correct result as seen by the now well known rule or formula, one-third the product of height by the sum of the bases and their geometric mean. But in a second example the sides of the square (base and top) are given as 28 and 4, and the slant edge as 15, which is impossible, since even the projection of any such slant on the base plane would be or 288, which is more than 15 = V 2 2 5. Accordingly the reckoner in applying his method, of which the foregoing rule is the resultant formula, meets with V/ 63 which he treats as V 63. Whether or not this be the work of Heron, it is notable as the oldest known appearance of the so-called "imaginary," the square root of a negative num exclusively residential section,has a large number of summer homes and the handsome club-house of the Atlantic Yacht Club. A shore drive connects it on the east with West Brighton, the most popular amusement centre, to which the name Coney island has come to be more especially applied. Its amusement parks, "side shows," cafes, and dancing halls have made Coney island a well known resort. There are a public bathing beach and bath house and the Coney island boardwalk, extending from Sea Gate east ward for about 13,000 feet. Eastward from West Brighton are the less crowded beaches and residential sections of Brighton Beach and Manhattan Beach.