The sphere in softwood is little required for other than ornamental purposes, and for these it can readily be turned with the gouge and chisel; while the material is hardly suitable for spheres of greater accuracy, such as may be made in hardwood or ivory, by the method described in later pages. This figure has been considered last in the series of elementary planes and solids, because facility of hand acquired on the cylinder, surface and cone, is necessary to correct and easy guidance of the chisel on the sphere; the path of the tool being a combination of the motions separately used for all three.

The sphere has been variously defined, but the path necessarily followed by the chisel in its production may perhaps be allowed to give a practical or turner's definition, viz., that the contour of the sphere is formed by a series of minute portions of cones, extending each way equally from its circumference or diameter to its axis; the examination of which definition may afford some assistance in the manipulation of the tool, during the successive stages of the development of the softwood sphere from the cylinder.

In the sphere fig. 373, revolving upon a. b. the axis of the lathe mandrel, it will be seen that there is one plane c. d., where the chisel can only cut when it lies on its face, as in turning the cylinder; this place in the traverse of the tool may therefore be considered as a minute portion of a cylinder. There is another plane a. b., where at the point b., the chisel can only cut when supported on its edge, as in turning the surface, this is therefore as a minute portion of a surface. In traversing the successive portions of the curve d. b. the cutting position of the edge of the chisel is gradually transferred from that for the cylinder at d., to that for the surface at b. During which progress the edge passes through a series of positions, in every respect equivalent to those required for a series of cones, each of minute length; while from the continuous motion of the tool these minute portions of cones are continuously produced and glide one into the other, forming the curve. The chisel travels round the sphere by a triple impulse or change; it slides laterally along the rest, the distance from d. to b.; it rotates one quarter turn, in the course of its traverse between these two points, for the blade which lies flat at d., is on its edge at b.; and thirdly, the cutting edge is continuously depressed from d., when it cuts by a portion of the edge as on a cylinder, until it arrives at b., where the cut is terminated at the center by the extreme point or angle, as on the surface. The rest stands parallel with the axis of the mandrel, necessary to the commencement, progress, and termination of the cut; and the tool having traversed the curve d. b. is turned over to proceed from d. to a.

Fig. 373. Fig. 374. Fig. 375.

The chisel travels in either direction around the curve, with the obtuse angle leading, but it is absolutely essential to successful cutting on the soft wood sphere, that its straight cutting edge be always maintained fairly parallel with the mandrel axis a. b. The stem of the chisel, stands therefore nearly at right angles to a. b.; while should it at any time wander from this position, so as to acquire more horizontal angle, the cutting edge ceasing to be parallel to a. b., immediately catches in and damages the surface of the sphere.

Small spheres not requiring particular accuracy, may be turned at once by the gouge and chisel; first taking the precaution to turn the cylinder true, and to mark off upon it as a guide, its diameter, and the central line c. d., fig. 374, the base of the two hemispheres. As the sphere then depends entirely on the path given to the tools, their use, especially that of the chisel, requires a little practice. If the chisel be twisted too rapidly, it will produce two rounded cones base to base, instead of two hemispheres, as shown by the dotted lines fig. 378; but if not sufficiently so, it will leave the material in excess, the lesser evil, as admitting correction.

Fig. 376. Fig. 377. Fig. 378.

A guide for the path of the tool for larger or more perfect spheres, may be obtained by the polygonal system, figs. 374 to 378. The material is first turned to a true cylinder, a little larger than the diameter of the intended sphere, and the end surface turned true. The exact diameter is then taken with a pair of callipers, and marked off upon the cylinder, measured exactly from the end or surface fig. 374; a line to mark it being slightly cut into the cylinder by the edge of the chisel, presented vertically. The diameter thus marked as the length of the cylinder is then divided by a pair of spring dividers, and a second or center line c. d., the base on which the two hemispheres are to be turned, is marked in the same manner. The portion of the cylinder next the chuck beyond the first line, is then reduced somewhat to the shape required to fix the sphere to further portions of the work; the back surface is turned true, without encroaching upon the length of the block left for the sphere, and the distances a. - c. and c. - b., bisected and marked with the lines, fig. 375.

The corners formed by the cylinder and its surfaces, which seen in section are right angles, are turned away so far as the lines last marked, to an exact angle of 45°, indicated by the dotted lines fig. 375, under the guidance of a bevil, applied on the cylinder or surface. When, provided that in the first instance, the length of the block exactly equalled its diameter, the length of the cylindrical portion and the diameter of the surface remaining, will be equal. These two bevils or cones turned upon the cylinder are then still further reduced with the chisel, and still strictly maintained at the angle of 45°. As they are turned, they become wider and gradually encroach upon the portions of the cylinder and surface; until as tested by measurement, the cylindrical portion, the cones, and the diameter of the surface, are all of equal width or length, fig. 376. Carrying this a step further with the same means, the eight angles of 135° thus formed, are reduced or divided to sixteen planes, and angles of 157½°, fig. 377, and if the sphere be large enough to require it, then to thirty-two; thus actually in practice continually reducing the cylinder to the series of short cones, which were said theoretically to compose the path of the tool. The series of rings or facets reduces the block almost to the circular line, and forms an efficient guide for the traverse of the tool, around the contour of the sphere. The final process, is to make the chisel traverse this path from d. to b. and from d. to a., its stem always at about right angles to the axis a. - b., aiming to remove an equal thickness of shaving throughout, and cutting no more than is sufficient to obliterate the angles and surfaces of the short pieces of cones to melt them into one circular line. When the sphere has to be separated from the neck, the principal work is done and the sphere completed up to the neck while that is still strong; the small portion remaining, is then finished simultaneously with the reduction of the neck to separation, as already described for the back surface.

Too much material left, or too deep a cut of the chisel at any portion of the spherical line, shows either as a swell, or as a flat or groove around the sphere. These errors have to be corrected by further traverses of the tool, either to remove a partial excess, or to reduce the whole sphere uniformly, to the depth to which the tool at any part may have accidentally entered upon it. A ring turned perfectly true, with its internal diameter somewhat less than that of the sphere, is employed as a gage to detect such errors. It is applied at different angles all over the sphere, fig. 878, which when true, exactly fills or agrees with the true circle of the ring; any excess or depressions are readily appreciated, and their situation marked with a pencil for subsequent correction.