(Continued from page 147.) No branch of machinery, probably, has received more valuable assistance from mathematical science than that which formerly was known more especially as " Mill-work," but which is now generally designated by the ti-tile that forms the heading of this article. What were the uncouth and almost ludicrous-shaped wheels of the past race of millwrights may be conceived on inspecting the mechanical works ot the last century. While the beautiful symmetry of their construction as at present made, is well known to all who are in any way employed about machinery. Not that the machinists of past times were less ingenious than their successors, but thay worked mostly at random, unaided by the light ot science, whose followers, at that period, spurned for the most part, the researches of any knowledge that could not, strictly, be classed under pure mathematics. A more liberal and enlightened spirit, however, has at length prevailed, and many of the most illustrious disciples of Newton have since, like him, been practical philosophers. More especially with regard to geared wheels have their studies been found of inestimable advantage to mechanics, as all can testify who have heard of Professor Willis, or who have availed themselves of his theory for the construction of toothed wheels. But, as the study of theories is often neglected, and the theory itself sometimes too intricate for the hasty seeker of information, we will here mention that the practical application of the above is to be found in a scale termed the " Odontograph," and which is extensively employed by machinists. Before entering upon the shape of the teeth, it is worth while to enquire what are the mechanical laws affecting systems of geared wheels, which, if traced to their simple origin, are found in reality to be only a form of the compound lever, and that the conditions of equilibrium are the same. From the fact that the arms of wheels are as levers fixed at one end, and loaded at the other, and that, consequently, the greatest strain is upon that part of the arm next the axle, is derived the mode forming the arms strongest at the axle and tapering towards the rim. In order that the power applied through the intervention of gearing may be used with the greatest effect, it is necessary that the wheel-work be properly designed and executed, otherwise power is expended to no purpose, and it should be especially noted that the primary object aimed at in the construction ef toothed gear is the uniform transmission of the power, supposing that to be constant and equal. This implies that the one wheel ought to conduct the other, as if they simply touch ed in the plane passing through both their centres,—these considerations will show the importance of a right form ol tooth for the wheels. Of the various methods which have been employed to determine the forms of teeth, that which is termed the epicycloidal curve, has been an especial favorite. This shape is produced by rolling a circle equal in diameter to the radius of the pinion upon another circle equal in diameter to the radius of the wheel, the diameters being taken at the pitch lines, which are the circles described by the wheel and pinion at their point of contact, the curves so struck, commencing at the pitch lines, torm the points of the teeth. They are struck in opposite directions, the space between their starting points being the thickness of the tooth ; and from these two points radial lines are drawn to the centres of the wheel and pinion, which forms the sides of the teeth included between them, within the pitch line. This form, it will be observed, made the tooth smallest at the root by the convergence of the radial lines, and conser quently tended to weaken it j this was reme- died in the pinion by casting a plate upon the teeth, which, forming part of them, served not only to bind, as it were, all the teeth together, but to strengthen the body ol the pinion, perforated and weakened by the axle passing through it. "The roots of the teeth" ura the wheel were strengthened by small angle p:ei"es, for which space was found without the curved line described by the teeth of the pinion. Such teeth worked freely and equably together. But it will be observed that the side of each tooth of the wheel consisted partly of a radial line, partly of an epicycloidal curve, and partly of such a concave angle piece as might be found to clear the pinion : and it will also be observed that the wheel and pinion were adapted to each other ; consequently another pinion, differing much in diameter from the first, would not act well with the same wheel. A mode ot forming the teeth of wheels, by which this inconvenience is obviated, has been proposed by Pro-lessor Willis, and the form of tooth thus produced is much superior to the old-fashioned plan. If tor a set of wheels of the same pitch a constant-describing circle be taken to trace those parts of the teeth which project beyond each pitch line, by rolling on the exterior circumference, and those parts which be within it, by rolling on the interior circumference, then any two wheels of the set will work correctly together. The describing or "Pitch Circle" should be equal in diameter to the radius of the smallest pinion, which, in this case should not have less than twelve teeth. When rolled upon the interior circumference of a circle equal in diameter to the pinion, a point upon the periphery of the pitch circle will describe radial lines through the centre of the larger circle representing the pinion, which is twice the diameter, so that the form of the pinion teeth within the pitch line may be at once drawn in straight lines from the centre. When rolled on the exterior circumference, epicyloidal curves, forming the teeth of the pinion beyond the pitch line are described by the tracing point. But when these operations are performed by rolling the pitch circle upon another of much larger diameter, representing the wheel, the interior and exterior epicycloids form a tooth of very different shape; it is no longer contained within radial lines, but spreads out at the root, giving great strength and firmness at the point, where they are most needed. The exterior epicycloid forms the point ol the tooth in a manner similar to that already described j but any wheel or pinion having teeth described by a common pitch circle will work together; even the teeth of a rack, which, being placed upon a straight line, may be regarded as the segment of a wheel of infinite radius can be formed in the same manner, and will work equally well with the wheels. The principles above discussed are applicable to both spur and bevel wheels; there is, however, another lorm in which teeth are shaped when the wheel and tangent screw principle is employed, and tha thread of a cylindrical screw gives motion to a wheel, a plan which is often employed to diminish a high velocity. (To be Continued.)