circle whose radius is equal to Z A, and whose arc is equal to the circumference of the circle A B. The length of Z A, corresponding to any given point on the sphere, is easily obtained. The diameter A B being known, the radius A C is known, the square of which being taken from the square of A O the radius of the sphere, the remainder will give the square of CO, which will therefore be known; but the ratio of CO to CA will be the same as that of O A to AZ. The length of A Z will therefore be determined. The angle Z' (fig. 181.), which, with a radius equal to ZA (fig.180.), will give an arc ab equal to the circumference of the parallel A B, may be easily determined; for this angle, expressed in degrees, will bear to 360 degrees the same proportion as the circumference of the circle A B bears to the circumference of a circle whose diameter is twice Z A. The angle Z' (fig. 181.) will therefore be found by multiplying 360 degrees by CA, and dividing the product by Z A. In this manner a series of narrow zones may be formed, which, when laid upon the sphere, will very nearly cover it, - the edges uniting without perceptible folds or wrinkles; and the more narrow such zones are formed the more nearly will they cover the sphere. fig. 182. (515.) Another method of covering a spherical surface consists in dividing it by a number of meridians, as represented in fig. 182., forming with each other angles so small that the arcs of parallel circles intercepted between them may be considered as straight lines. If these meridians be themselves divided into small and equal arcs by parallel circles intersecting the axis of the sphere at right angles, the whole spherical surface will be divided into small quadrilateral figures bounded by the parts of the meridians and parallels, which may be considered as plane trapeziums: the form and magnitude of each series of these being determined, and the substance intended to cover the sphere being resolved into corresponding pieces, the object of covering the sphere by plane figures will be attained; and the precision with which this will be accomplished will be proportional to the smallness of the pieces into which the sphere is divided. (516.) The sphere is not the only surface which can be formed by the revolution of a circle round a straight line: we have seen that a semicircle revolving on its diameter will generate a sphere; but other segments revolving on their chords will generate solids of other forms. Thus a segment less than a semicircle revolving on its chord P P' (fig. 183.) will generate a solid, such as there represented, having pointed ends. A segment of a circle greater than a semicircle, as represented in fig. 184., revolving on its chord as an axis, will generate a figure such as represented in fig. 184., having a hollow at top and bottom resembling that of the end of an apple from which the stalk proceeds. If a circle A B (fig. 185.) revolve round a line such as P P', drawn in its plane, but outside it as an axis, it will generate an annulus, the centre C of the circle describing a circle round P P', which will be the axis of the annulus. Such a solid is represented in perspective in fig. 186. If an arc of a circle such as A Brevolve round a line PP' drawn on the convex side of it and in its plane, as an axis, it will generate a figure with concave cylindrical sides, such as is represented in fig. 187. (517.) Almost all the variety of vases of metal and porcelain used in domestic economy, ancient and modern, and adopted for ornamental purposes in fig. 188. the arts, are surfaces produced by the revolution of the arcs of curves round lines drawn in their planes, within or without them, in the manner above described, combined with cylindrical surfaces, and those of truncated cones; all the surfaces of revolution composing the same vessel having a common axis, as represented in fig. 188. (518.) A circle is not the only line by the revolution of which round a fixed axis a surface may be generated; on the contrary, this method of producing a surface is general, and has given rise to a class of surfaces called surfaces of revolution, and which are the class of geometrical forms of the most frequent occurrence both in natural and artificial productions. Any line whatever, whether straight or curved, may revolve round another line as an axis, and by such revolution it will generate a surface of revolution, the form and properties of which will depend on the species of line which revolves, and its position with respect to the axis of revolution. The right circular cylinder and cone, as has been already observed, belong to the family of surfaces of revolution. If one of two parallel right lines revolve round the other as an axis, it will produce the surface of a right circular cylinder; and if one side of a plane rectilinear angle revolve round its other side as an axis, it will produce the surface of a right circular cone. (519.) From the mode in which they are generated, it follows, that the sections of all surfaces of revolution made by planes at right angles to the axis of revolution, are circles having their centres in the axis of revolution: this is a characteristic property of such surfaces; and, as it belongs to none other whatever, it may be, and sometimes is, taken as the basis of their definition. It is evident, that, in the production of a surface of revolution, all the points of the revolving line move in parallel planes, and, as they preserve their distances from the axis of revolution, each must describe a circle whose centre is in that axis. (520.) Surfaces of revolution are infinitely various, not only in consequence of the great variety of lines by the revolution of which they may be produced, but by reason of the variety of surfaces which may be produced by the same line revolving under different circumstances. When a right line is in the same plane with the axis round which it revolves, it will produce, as has been shown, either a cylindrical or conical surface, according as it is parallel or not to the axis of revolution; but if it be not in the same plane with the axis of revolution, it will produce a curved surface (fig. 189.) whose cross section shall be a circle whose radius is the least distance of the revolving line from the axis of re (521.) Among the productions of nature, the great bodies of the universe - the sun, planets, and satellitesare surfaces produced by the revolution of an oval or ellipse round its lesser axis (fig. 190.). Fruit of almost every kind are surfaces of revolution produced by the segment of a circle revolving round a chord. fig. 190. A lemon affords an example of a surface of revolution (fig. 191.) formed by a segment less than fig. 191. a semicircle revolving on its chord. An apple (fig. 184.), of a surface formed by a segment greater than a semicircle revolving on its chord. An orange is an example of a surface of revolution (fig. 190.) formed by an oval revolving on its shorter axis. A plum (fig. 192.), of a surface of revolution formed by an oval revolving on its longer axis. fig. 192. (522.) Every species of dome in architecture is a surface of revolution. A hemispherical dome is formed by a semicircle revolving round the radius which is perpendicular to its diameter (fig. 193.). fig. 193. 4 An oblate elliptical dome (fig. 194.) is a surface of revolution produced by the revolution of a semi-ellipse round its lesser semi-axis. fig. 194. fig. 195. |