There is scarcely a mechanical combination of any kind into which toothed or gear whecls do not more or less enter into the arrangement. Their advantages for conveying motion are so obvious and well known that it is almost an unnecessary task to recapitulate them ; they are more certain than belts or straps ; their motion is easy and regular, and they can be made of any size or strength. A tooth wheel is essentially a wheel having on its periphery a nnmber of projections at equal distances apart, with corresponding indentations or spaces into which the teeth of the correspouding teeth can fit with ease. The distance from the center of one of these teeth to the center of the next is called thepitch of the teeth; or, in other words, the pitch is the space occupied by one tooth and one space. When the motion is not intended to be conveyed parallel to the motor or shaft which gives motion to the first, but at some angle with it, then the teeth are placed at a suitable angle with the shaft, and are called bevel wheels. The simplest form of a toothed wheel is evidently a wheel set on an. axle, a number of pegs being inserted in the periphery of the wheel, at right angles with the axle, and these fitting into the spaces between pegs placed in the rim of another wheel parallel to the shaft or axle. In constructing these teeth there are certain principles to be remembered and attended to, which we will state as briefly as we can. The first is, that ge"ring wheels act by direct pressure, tooth against tooth, and consequently the teeth must not be too long, or they will snap off; and, secondly, they must be of such a shape that they will meet, fit into one another and separate with ease, and yet remain in contact from the moment they meet to that in which they part, so that no pressure is lost, but a pressure is always excrted in proportion to the amount of surface in contact. On this account the shape of these teeth is of the highest importance, and to determine the proper form, many mathematicians have spent much time and labor, among whom we may honorably mention the names of Camus, Emerson, Young, and Willis. They have determined that the tooth must be thicker in the center than at either end, for there it receives the greatest amount of pressure, this through all the teeth is called the line of centers, and that if the wheel is to gear into another wheel, the sides of them shonld be a segment of a cycloid ; this is the curve formed by any point of a wheel rotating along a level plane ; and if the wheel is intended to gear into a rack, the epicycloid is the best form ; an epicycloidal curve is formed by a point in a small wheel rotating aronnd the periphery of another circle. Another good and useful curve of which to form them is the involute, which is constructed by fastening a pencil to a piece of string and winding it round a cylinder, then holding the point of the pencil on a piece of paper, and pulling it round so as to nnwind the string. Millwrights and machinists sre generally in too great a hurry to attend to such minutir and they adopt a much more simple method, by drawing a number of circles until they obtain such a curve as will fit, or else swing the patterns, and cut two teeth that will gear, gradually cutting away until they obtain the desired form ; and from those teeth make the remainder. Cog, gear or toothed wheels are now made on a large scale and kept in sizes, so that you have only to send the pitch and size of the wheel you want, and you will receive from the foundry or millwright's shop a wheel that will gear into the one you require.