Blake on the Teeth of Cog- Wheels. 101 



Prop. 12. — If the velocity of the circles be uniform at 

 the point of contact, and the velocity of the describing point 

 be also uniform, the curves described are Logistic Spirals. 



For since the velocity of the circle a, Fig. 9, is uniform at 

 p, the lines pe, di, &o. described in successive equal por- 

 tions of time, -are equal to each other. And for the same 

 reasons, since the velocity of the describing point p is uni- 

 form, the successive lines ed, ih^ &c. are equal. Therefore, 

 in the two right angled triangles hid, dep, the sides which 

 inclose the right angles, are respectively equal to each other. 

 Wherefore the angles ihd, edp are equal to each other. 

 Hence all lines drawn from the point a to the curve pA make 

 the same angles with the curve. Let a and b, Fig. 10, b» 

 the centres of the circles, and ce the curve described on the 

 plane of a, during the uniform motion of the circle at the 

 point p, and of the point j?. Divide the arc of a circle gf, 

 into indefinitely small equal parts, and through the points of 

 division draw the lines ad, ap, &c. Now in the triangles 

 Medf adp, ape, the angles at a are equal, because they sub- 

 tend equal arcs, and the remaining angles have already been 

 shown to be respectively equal. Therefore the triangles 

 are similar each to each. Hence ac I ap'.'.ap I ad'.'.ad : ae, 

 &c. That is, the radii ac, ap, ad, ae, &c. are continued 

 proportionals ; which is the fundamental characteristic of the 

 Logistic Spiral. The same might be shown of the curve 

 hi on the plane of the circle b. 



ScHOL. — The Logistic Spiral is perhaps the most simple 

 form for atripsic teeth of wheels. 



Prop. 13. — Any curve on the plane of a circle may be 

 driven by some other curve without friction, when every 

 successive part of it, commencing with that extreme which 

 is nearest the centre, recedes continually both from the cen- 

 tre and from the radius in which it commences. 



Let a. Fig. 11, be the centre of the circle, to the plane of 

 which is attached the curve pe answering the condition of 

 the proposition. Through p, the nea'rest extremity of the 

 curve pe, draw the line ab, and through the other extremity 

 e, draw the arc ec of a circle whose centre is a. Now if 

 the circle a revolve on its centre till the point e comes into 

 the line of centres at c, the point of intersection between the 

 Qurve pe and the line ab will advance from p to c. If 



