88 Blake on the Teeth ofCog-PVheek. 



cycloidal teeth; and in this view of the subject, the suc- 

 ceeding writers have hastily concurred with him. That 

 they are entirely distinct from that principle may be seen 

 by referring to the subsequent remarks on involutes of cir- 

 cles, taken in connexion with the rest of the subject. 



These discoveries of Roemer, De la Hire and Robison 

 constitute the whole extent to which the science has hith- 

 erto been carried. 



I apprehend that the subject has not yet been placed on 

 its true and proper basis, and that those who have written up- 

 on it have commenced their labours at one of the branches, 

 and not at the root of the principle ; in consequence of 

 which they have not only failed to circumscribe it with 

 those clear and distinct outlines which should always define 

 the extent of mathematical truth, but, as the following trea- 

 tise will show, have embraced in their views only an infi- 

 nitely small part of the subject. It is true, as shown by 

 Roemer and De la Hire, that epicycloidal curves possess 

 the property of transmitting a uniform force and velocity ; 

 but this property does not, as they have supposed, depend 

 at all on the circular form of the generating curve ; for it 

 will be shown that all curves possess the same property, 

 which are generated, or are capable of being generated by 

 any curve or line whatever, rolling on the pitch circles. — 

 Nor is it necessary, as in epicycloidal curves, that the tra- 

 cer or describing point should be situated in the generating 

 curve, for the curve generated will possess the same prop- 

 erly if the describing point is any where else in the plane 

 of the generating curve. 



I propose in the first part of the following treatise to set 

 forth in a few concise propositions the true principle in its 

 whole extent ; embracing all possible curves which possess 

 the property of transmitting a uniform force and velocity; 

 to which will be added such remarks and illustrations as 

 readily flow from the subject, and are thought of practical 

 utility. 



Since the principle thus exhibited will be found to em- 

 brace an infinite variety of curves, most of which are un- 

 known to mathematicians by any appropriate name, yet 

 all possessing one common property; to avoid circumlocu- 

 tion we shall call them Isosagistic curves, a term sig- 

 nificant of their characteristic property. It is proper to re- 



