First, by the employment of tools figured to the various shapes, and used with only one motion or traverse; secondly, by the use of figured guides, cams, or shaper-plates, by which the motion is constrained, just the same as p makes a right or a curved line, in virtue of its straight or curved guide; and thirdly, by the employment of a point actuated by two motions, by the composition of which most geometric lines are expressed.

Thus when uniform motions are employed, two rectilinear motions produce a diagonal to themselves; one circular and one continued right-line motion, give the spiral, the screw, and the cycloid; also if during one circular revolution, either the circle or the point make one oscillation in a right line, we obtain the oval; by two circular movements we obtain the epicycloid, by three motions the compound or double epicycloid, and so on. And when one or both of the rectilinear or circular generating motions, are variable as to velocity, we obtain many different kinds of curves, as the parabola, hyperbola, and others; and thence the solids, arising from the revolutions of some of these curves upon an axis.

We produce the practical composition of any two lines or movements, whether regular or irregular, by impressing these movements on the opposite extremities of an inflexible line or rod; from which rod we obtain a compounded line, if we trace the motion of a point inserted in any part of the rod, and we obtain a compounded superficies, if we copy the motion of the entire line. This may need explanation.

Supposing that in fig.87 the back guide g g, to remain a straight line, the front to become the circular arc a a, the board p, being now traversed in contact both with the straight and curved edges, the point p would describe a line if it were close against the line g g; or an arc if close against the arc a a; midway it would describe an are of about half the original curvature. On the other hand, the line b would cut off the clay in a superficies, possessing at the three parts these same conditions, and merging gradually from the right line to the arc a a.

But a similar composition of the two lines or motions would occur, were the lines g g, a a, to be exchanged for any others, similiar or dissimilar, parallel or oblique, or irregular in two directions; and in mechanical practice we combine, in like manner, two motions to produce a compound line or a com-pound superficies. Indeed in many cases there is no alternative but to impart to two edges g a of a block, the marginal outlines of the superficies, and then, generally by hund-labour, to reduce all the intermediate portions under the guidance of a straight edge applied at short intervals upon the two edges, which thus become compounded or melted together in the superficies. Nurs-bers of irregular surfaces can be produced by this mode alone. In fine, in mechanical processes, we translate the mathematical conceptions of the rectilinear, circular, and mixed motions of points and lines, into the mechanical realities of rectilinear, circular, and mixed motions of pointed or linear tools.

It is not imperative, however, that the tools should have but one fixed point or edge, as without change of principle a succession of similar points may be arranged in a circle, to constitute a revolving cutter, which by its motions will continually present a new point, and multiply the rapidity of the effect. In most cases, the introduction of a tool with a figured outline, cancels the necessity for the means otherwise required to generate such figured line by by the motion of a point; and a tool with a figured superficies, cancels also the remaining motion required to produce the raperficiee, and the tool is simply impressed as a stamp or di

In tracing the method of applying these theoretical views to the explanation of the general employment of cutting tools, or the practice of the workshop, we may safely abandon all apprehension of complexity, notwithstanding the almost boundless variety of the elements of machinery, and other works of cutting tools. For although all the regular figures and solids referred to, are in reality met with, besides a still greater number of others of an irregular or arbitrary character, still by far the greater majority of pieces resolve themselves into very few and simple parts, namely, solids with plane superficies, such as prisms, pyramids, and wedges, and solids with circular superficies, such as cylinders, cones, and spheres. These are frequently as it were strung together in groups, either in their entire or dissected states; but as they are only wrought one surface at a time, the whole inquiry may be considered to resolve itself into the production of superficies.

And it may be further stated that, the difference between the modes of accomplishing the same results, by hand tools or by machinery, bears a very close resemblance to the difference between the practices, of the artist who draws the right line and circle by aid of the unassisted hand, and of the mechanical draftsman, who obtains the same lines with more defined exactness, under the guidance of the rule and compasses.

The guide principle is to be traced in most of our tools. In the joiner's plane it exists in the form of the stock or sole of the plane, which commonly possesses the same superficies as it is desired to produce. For instance, the carpenter's plane used for flat surfaces is itself fiat, both in length and width, and therefore furnishes a double guide. The flat file is somewhat trader the same circumstances, but as it cuts at every part of its surface, from thousands of points being grouped together, it is more treacherous than the plane, as regards the surface from which it derives its guidance, and from this and other reasons, it is far more difficult to manage than the carpenter's plane.

In many other cases the cutting instrument and the guide are entirely detached; this is strictly the case in ordinary turning, in which the circular guide is given by the revolution of the lathe mandrel which carries the work, the surface of which becomes the copy of the tool, or of the motion impressed upon the tool, either by the hand of the workman under the guidance of his eye alone, or by appropriate mechanism.