Blake on the Teeth of Cog- Wheels. 9S 



the point o is in a corresponding point of the curve be. 

 And since the equal arcs are by the construction indefinite- 

 ly small, the point o is constantly in the curve be;- first at 

 b, and lastly at c. Therefore the point o describes the 

 ©urve be. 



CoK. 1. — Curves are susceptible of being generated by 

 two different curves, rolling on different arcs of the circle as 

 the base of generation, when their perpendiculars produced 

 fall on another part of the circle, in the same order of suc- 

 cession. 



CoR. 2. — Any curve being given on the plane of a circle, 

 it may be ascertained whether it can be generated, and if 

 it can be generated, its generating curve may be found. 



Prop. 6. — All isosagistic curves are susceptible of being 

 generated by curves rolling on their pitch circles. 



Let the point g, (Fig. 1.) be the extremity of the isosa- 

 gistic curve gi, in action at the pointy, and let the curve gi 

 drive the curve gh toward e, and let ei be the arc which 

 rolls on the fellow-circle during the action of the curve gi. 

 By prop. 3. Cor. the perpendiculars from every part of gi 

 fall on ei. Now, from the nature of curves the perpendic- 

 ulars from two adjacent points make an infinitely small an- 

 gle with each other, and consequently can intersect but an 

 infinitely small portion of an arc between them. Therefore 

 the perpendicular from the point h (not given in the figure) 

 adjacent to the point g, in the curve gi, either falls on the 

 point e, or on a point x, (not shown in the figure) adjacent 

 to e, between e and i. Now when the point n is in action, 

 the point x is in the line ab. But the point x is the next 

 point of the arc ei, which in the progress of the action comes 

 into the line ab. Therefore the point n is the next point 

 of the curve gi which is in action. In the same manner it 

 may be shown with regard to every successive point of the 

 curve, that it is in action when the next successive point of 

 the circle is in contact. But the perpendicular from the 

 po.nt of action falls on the point of contact. Therefore, 

 since the successive points of the circle come successively 

 in contact, perpendiculars from the successive points of the 

 curve fall on successive points of the circle. Therefore 

 (prop. 5) the curve gi is susceptible of being generated by 

 a curve rolling on its circle. The same may be shown of 

 any other isosagistic curve. 



