In the figuration of materials by abrasion, the principal dependence for the correctness of form, is generally placed upon the abrasive tool, or grinder, being exactly a counterpart of the form to be produced; thus for plane surfaces a flat grinder is employed, for concave surfaces a convex grinder, and so on. In numerous cases the grinder is made as a revolving wheel, figured to the required counterpart form, either upon the edge, or upon the side, and the work is simply held to the grinder by hand, without the assistance of any mechanical guidance. In other cases the work is traversed on slides beneath revolving or reciprocating grinders; and in some few instances, where great accuracy of form is required, the principal dependence is placed upon the relative motions of the grinder and work, both usually under the control of mechanism.

The natural grindstone is in general only used for the rough preparation of the surfaces, which are afterwards more accurately figured with metal grinders supplied with abrasive powders. Within certain limits, it may be said generally, that the greater the accuracy desired in the surfaces to be produced, the harder should be the material of which the grinder is composed; while, upon the other hand, the finer the surface, or the higher the desired polish, the softer should be the material of the grinder. These opposite qualities required in the grinder, combined with other circumstances, render the attainment of very accurate, and, at the same time, highly polished surfaces, a point of considerable practical difficulty, as will be adverted to hereafter.

The principal contents of the present chapter will be divided into four sections, relating respectively to the methods of grinding and polishing plane surfaces, cylindrical surfaces, conical surfaces, and spherical surfaces. These elementary forms may be considered to include, by their combination, nearly every figure required in the mechanical arts; and the concluding section of the chapter, will be devoted to a brief notice of the practice of glass-cutting, in which all kinds of mixed and arbitrary forms are produced by very simple apparatus, under the guidance of the hand alone.

The present section will refer, first, to the grinding and polishing of flat surfaces in hardened steel, and the metals generally, and this will be followed by a description of the methods of working stone and marble, materials that are almost exclusively wrought by abrasion, and both the manual and machine processes will be noticed, as a general example of the production of form by abrasion. Plate and sheet glass will be next alluded to, and the section will conclude with some account of the methods of grinding the more accurate plane surfaces required for optical purposes.

Revolving laps of metal used upon their flat sides, and supplied with emery and water, are extensively employed by mechanicians for finishing flat surfaces of small and medium size, requiring tolerable accuracy. Sometimes the lap is employed for brass, iron, and soft steel, but more generally the flat surfaces of works in these metals, are wrought by the planing machine or file, and finished in the manner described in the catalogue of grinding processes, pages 1074 to 1076; and the lap is principally employed for correcting works in hardened steel, such as the broad flat surfaces of cutting tools, the faces of dies, hardened steel plates, and numerous other objects. All these works are made nearly flat, either with the grindstone or file, prior to their being hardened, as the general accuracy of the forms may be much quicker produced by these means; and the lap is chiefly resorted to for removing those slight distortions occasioned in hardening, that are beyond the correction of the hack hammer, described at page 247, Vol. I, and also for giving a smooth and finished surface to the work.

Sometimes the laps are made of cast-iron, or copper, because these hard metal laps longer retain their forms uninjured; but, as previously mentioned, lead hardened with a little antimony, is the metal generally used for laps by mechanicians, as the lead being yielding, allows the emery to become embedded in its surface,, and consequently a smooth face can be produced upon the work with an emery, the particles of which are sufficiently large to cut rapidly. Whereas when iron or copper laps are employed, the emery can scarcely penetrate the lap, but is partially lost, and the remainder rolls over, and makes scratches in the work nearly equal in depth to the size of the emery powder.

Laps not exceeding a few inches in diameter, used by mechanicians, are generally mounted vertically, not upon the middle of long spindles, after the method of those used for cutlery, but screwed as chucks upon the mandrel of a lathe, as shown in fig. 1054. This method is adopted in order to avoid the interference of the spindle, and render the entire side of the lap available for works of a moderate size. Larger laps are mounted to revolve horizontally, somewhat after the manner shown in fig. 1039, but in much stronger frames, and generally driven by steam power, as the diameter of these horizontal laps is sometimes as much as five or six feet. The varying velocity of the surface of the lap, which continually decreases from the periphery to the center, is however very objectionable in large laps, as it renders the tool much less effective near the middle, and is besides liable to cause the lap to become conical, from being less worn near the center. To avoid these interferences as much as possible, large laps are in most cases made as annular disks, cast upon iron plates or wheels, so as to leave a central aperture of about one-third the extreme diameter of the lap.

In lapping small works the object, if thin, is held between the thumb and finger nail, and placed fairly in its position on the lap while the latter is at rest; the lap is then put in rotation, and the work is held quite steady to the face of the lap with moderate pressure, and the lap is stopped before the removal of the work, in order to examine its progress. Larger pieces that can be conveniently held in the fingers are applied to the lap while it is in rotation; the work is quickly placed in its position, and the pressure is steadily applied on the back of the work as near as convenient to its center, in order to feel when it bears uniformly upon the lap; the work is retained in its position for a few seconds, and then, in order to examine whether it has been properly placed, the work is lifted at once perpendicularly from the face of the lap, and not gradually drawn off, as the latter course would be liable to round off the edges of the work. Should it appear to have been incorrectly placed on the lap, the work is applied in another position, but the principal dependance is placed upon the sense of feeling, as with a little practice the fingers readily appreciate when the work lies fairly upon the surface of the lap.