Since M. Camus has treated of no other curve than the epicycloid, it would appear that he considered it to supersede all others for the figure of the teeth of wheels and pinions. And the editor must candidly acknowledge that he entertained the same opinion until after the greater part of the foregoing sheets were printed off; but on critically examining the properties of the involute with a view to the better explaining of its application to the formation of the teeth of wheels and pinions, the editor has discovered advantages which had before escaped his notice, owing, perhaps, to his prejudice in favor of the epicycloid, from having, during a long life, heard it extolled above all other curves; a prejudice strengthened too by the supremacy given to it by De la Hire, Doctor Robison, Sir David Brewster, Dr. Thomas Young, Mr. Thomas Reid, Mr. Buchannan, and many others, who have, indeed, described the involute as a curve by which equable motion _might_ be communicated from wheel to wheel, but scarce any of whom have held it up as equally eligible with the epicycloid; and owing also to his perfect conviction, resulting from strict research, that a wheel and pinion, or two wheels, accurately formed according to the epicycloidal curve, would work with the least possible degree of friction, and with the greatest durability. But the editor had not sufficiently adverted to the case where one wheel or pinion drives, at the same time, two or more wheels or pinions of different diameters, for which purpose the epicycloid is not perfectly applicable, because the form of the tooth of the driving wheel cannot be generated by a circle equal to the _radius_ of more than one of the driven wheels or pinions. In considering this case, he found that the involute satisfies all the conditions of perfect figure, for wheels of any sizes, to work smoothly in wheels of any other sizes; although, perhaps, not equal to the epicycloid for pinions of few leaves. With Joseph Clement, he experimented somewhat to determine the relative end-thrust of involute and cycloidal teeth, deciding that the advantage, if any, lay with the former. He details methods of laying out involute teeth and concludes: Before dismissing the involute it may be well to remark that what has been said respecting that curve should be considered as a mere sketch, there appearing to be many very interesting points in regard to its application in the formation of the teeth of wheels which require strict investigation and experiment.