The principles of action of all cutting tools, and of some others, whether guided by hand or by machinery, resolve themselves into the simple condition, that the work is the combined copy of the form of the tool, and of the motion employed. Or in other words, that we exactly put into practice the geometrical definitions employed to convey to us the primary ideas of lines, superficies, and solids; namely, that the line results from the lotion of a point, the superficies from the motion of a line, and the solid from the motion of a superficies.

I therefore follows, as will be shown, that when the tool is a point having no measurable magnitude, that two motions must be impressed upon it, one equivalent to the breadth, and another equivalent to the length of the superficies. When the tool is wide, so as to represent the one dimension of the superficies say its breadth, then only one motion is to be impressed, say a motion equivalent to the length of the superficies; and these two are cither rectilinear or curvilinear, accordingly as straight or curved superficies are to be produced.

To illustrate this in a more familiar way than by the ideal mathematical conceptions, that a point is without magnitude, a line is without breadth, and a superficies without thickness; we will suppose these to be materialised, and to become pieces of wood, and that the several results are formed through their agency on soft clay.

Fig. 317.

Thus supposing g g, to be two boards, the edges of which are parallel and exactly in one plane, and that the interval between them is filled with clay; by sliding the board p, along the edges of g g, the point in p, would produce a line, and if so many lines were ploughed, that every part of the clay were acted upon by the point, a level surface would at length result. The line l, such as a string or wire, carried along g g, Mould at one process reduce the clay to the level of the edges of the box.

Either the point or the line, might be applied in any direction whatever, and still they would equally produce the plane, provided that every part of the material were acted upon; and this, because the section of a plane is everywhere a right line, and which conditions are fulfilled in the elementary apparatus, as the edges of g g are straight and give in every case the longitudinal guide; and with f, the second line is formed at once, either with a string, a wire, or a straight board; but in p, the point requires a second or transverse guide, and which is furnished by the straight parts of the board p, rubbing on the edges of g g, and therefore the point obtains both a longitudinal and a transverse guide, which were stated to be essential.

The board c, with a circular edge, and m, with a moulding, would respectively produce circular and moulded pieces, which would be straight in point of length in virtue of gg, the line of and curved in width in virtue of cor m, the lines of the Hut now c, and m, must always advance parallel with their starting positions, or the width of the moulding would vary; and this is true, whenever curved guides or curved tools arc employed, as the angular relation of the tool must be then constantly maintained, which it is supposed to be by the external piece or guide attached to m.

Supposing g g, each to have circular edges, as represented by the dotted arc a a, or to be curved into any arbitrary mould-ing, the same boards p, l, c, m, would produce results of the former transverse sections, but the clay would in each case present, longitudinally, the curved figure of the curved longitudinal boards a a; here also the line of the tool and the line of tin-motion would obtain in the result.

If, to carry out the supposition, we conceive the board a a, to be continued until it produced the entire circle, we should obtain a cylinder at one single sweep, if the wire /, were carried round right angles to a a. But to produce the same result with the point p, it must be done either by sweeping it round to make circular furrows very near together, or by traversing the point from side to side, to make a multitude of contiguous lines, parallel with the axis of the cylinder. In either case we should apply the point to every part of the surface of the cylinder, which is the object to be obtained, as we copy the circle of a a, (which is supposed to be complete,) and the line l; or the trans-verse motion of p, which is equivalent to a line.

But it is obvious that, in every case referred to, there is the choice of moving either the clay or the tool, without variation in the effect. If in respect to the circular guide a a, we set the clay to rotate upon its center, we should produce all the results without the necessity for the guide boards a a, as the axis being fixed, and the tool also fixed, the distance from the circumference to the center would be everywhere alike, and we should obtain the condition of the circle by motion alone, instead of by guide; and such in effect, is turning.

An every day example of this identical supposition is scon in the potter's wheel; and the potter also, instead of always describing the lines of his works with his hands, as in sketching, occasionally resorts to curved boards or templets, as for making the mouldings for the base of a column, or any other circular ornament. But here, as also in ordinary turning, we have choice, either to employ a figured tool, or to impress on a pointed tool a path identical with the one section; for example, the sphere is turned either by a semicircular tool applied parallel with the axis, or else by sweeping a narrow or pointed tool around the sphere, in the same semicircular path.

Having shown that in every case, the superficies is a copy of the tool and of the one motion, or of the point and the two motions, it will be easily conceived that the numerous superficies and solids, emanating from the diagonal, spiral, oval, cycloid, epicycloid, and other acknowledged lines, which are mostly themselves the compositions of right lines and of circles, may be often mechanically produced in three different ways.