Slake on the Teeth of Cog-W heels. St 



ihat they may act with a constant and equable force, and 

 produce a uniform velocity. Several of the most eminent 

 mathematicians of the last century gave their attention to 

 this branch of mechanical science. The first who came to 

 any practical result was Olaus Roemer, the celebrated as- 

 tronomer and mechanist of Denmark. He discovered that 

 wheels possess the property of transmitting a uniform force 

 and velocity, when sections of the acting faces of the teeth 

 of one of them are the incipient portions ol exterior Epicy- 

 cloids, generated on the pitch circle by a circle whose di- 

 ameter is equal to half the pitch diameter of the other 

 wheel ; and the teeth of the other, the incipient portions of 

 interior Epicycloids, generated on its pitch circle by the 

 same generating circle. 



M, De la Hire soon afterward took up the subject, and 

 proved that it is not necessary, as Roemer had supposed, 

 that the diameter of the generating circle should be half 

 that of the circle on which the interior epicycloid is descri- 

 bed, since the same result will follow if the generating cir- 

 cle be of any other diameter whatever. He also proved 

 that it is not necessary in order that the teeth may possess 

 the property of transmitting a uniform force and velocity, 

 " that they should be exact epicycloids in the sense to 

 which geometers usually restrict that term, since they will 

 possess this property if the teeth of one wheel be triangular, 

 circular, or of any other regular figure, provided the 

 teeth of the other are of a figure compounded of that fig- 

 ure, and of an epicycloid." 



The subject was afterward investigated by Camus, Eu- 

 ler, Varignon, and others on the Continent. But their la- 

 bours, though they afforded some valuable elucidations of 

 the subject, contributed nothing to extend it beyond the dis- 

 coveries of De la Hire. Nor has any advancement been 

 since made in the science, except by Professor Robison, 

 ©f Edinburgh, who pointed out a single species of teeth 

 possessing this property, which are not embraced in the 

 epicycloidal principle of De la Hire. These teeth are in 

 the form of the involutes of circles, smaller than the pitch 

 circles, and concentric with them, whose diameters are to 

 each other as the diameters of the pitch circles. Mr. 

 Brewster has erroneously remarked that " the principle of 

 these teeth is not new;" and classed them among the epi- 



