Blake on the Teeth of Cog-Wheeh. 03 



and the curve ph on the plane of the circle 6. These 

 curves are isosagistic. 



For the line pd is constantly the describing radius to 

 the successive points of both curves pi, ph. These curves 

 are therefore both constantly perpendicular to the line pd. 

 Hence the curves pi, ph are constantly coincident at the 

 point/). Consequently the point p is constantly the point 

 ©faction. Therefore the curves are constantly perpen- 

 dicular at their point of action to the line pd. Wherefore 

 (Prop. 2.) they are isosagistic. 



CoR.l. — Any curve on the plane of a circle which is sus- 

 ceptible of being generated by another curve rolling on the 

 circle is isosagistic. 



Cor. 2. — The base circles being considered as the pitch 

 circles of two wheels, the curves generated are sections of 

 the acting faces of fellow isosagistic teeth. 



Cor. 3. — When the base circles and generating curve si- 

 multaneously roll together, the describing point and point 

 of action constantly coincide. 



Scholium. — When the generating curve is a circle, and 

 the describing point its centre, the curves generated are cir- 

 cles concentric with their bases. These are manifestly in- 

 capable of acting on each other, and therefore, though they 

 are rather in than beyond the limits of the principle, they 

 will not be considered as included in the general term, iso- 

 sagistic curves. 



Prop. 5. — Any curve on the plane of a circle is suscepti- 

 ble of being generated by a curve rolling on the circle, when 

 perpendiculars from every successive point of it, com- 

 mencing with one extreme, fall on corresponding successive 

 points of the circle; and its generating curve may be found. 



Let a, Fig. 4, be the circle, and he any curve on its plane 

 answering the condition of the proposition. It is required 

 lo find the curve which rolling on the circumference of the 

 circle a, will generate the curve he. 



From the points h and c, draw lines perpendicular to ic, 

 meeting the circle at d and e. Divide the arc de into inde- 

 finitely small equal parts, and suppose lines to be di'awn 

 from each point of division perpendicular to he. From any 

 point 0, with each of the perpendiculars snccessively as ra- 

 dius, describe concentric circles. From any point J5, in the 



