§4 iSlake on the Teeth of Cog-Wheels. 



circumference of the first concentric circle as a centre, with 

 a radius equal to dm, one of the divisions of the arc, inter- 

 sect the next concentric circle at t'; and from the point i^ 

 with the same radius intersect the next, and so on, till all 

 be intersected. Then, if a curve be traced through these 

 several points of intersection, and applied at the point p to 

 the point d of the circle, and rolled round upon it to e, the 

 point in the plane of this curve \m'\\\ trace the curve he. 



Take or equal to op and join p, r ; and take np equal 

 to db^ and join d. n. \{dni is infinitely small, pi is also in- 

 finitely small, and they may both be taken as straight lines. 

 The angle por is also infinitely small. But since the three 

 angles of the isosceles triangle por are equal to two right • 

 angles, when the angle por is infinitely small, or vanishes, 

 the remaining angles are equal, each to one right angle- 

 Since therefore pro is a right angle, pri is also a right 

 angle. Again ; when dm vanishes, mh coincides with 

 db. Therefore, when dm is infinitely small, mh may be 

 taken as parallel to dh, and bh will be infinitely smalL and 

 may be taken as a straight line. But nh is equal to dh, 

 therefore dn, bh, are parallel. Consequently the angle 

 dnm is a right angle. Since by the construction po is 

 equal to dh, and mh to io, and since nh equals dh, and or 

 equals op, nh is equal to ro, and the remaining parts mn, ir, 

 are equal. In the two triangles dmn pir, dm is equal to 

 pi by the construction, and it has been shown that ir is 

 equal to mn and that the angles at n and r are right an- 

 gles. Hence the angle mdri is equal to the angle ipr. To 

 the angle mdn add the right angle 7idb, and to the angle ipr 

 add the right angle rpo. Then the angle mdb is equal to 

 the angle ipo. If therefore, the point p of the curve be . 

 made to coincide with or touch the circle at the point d, po 

 Coincides with db and o is in b. If the curve and circle 

 toucheach other in any other corresponding points, it may 

 be shown in the same manner that the point o is in a corres- 

 ponding point of the curve be. Now let the point p of the 

 curve touch the circle at the point d, and let the curve be 

 rolled round upon the circle to e. Since tire small arcs of 

 the curve pi he. are by the construction equal in length 

 and number to the small arcs dm he. of the circle, the cor- 

 responding points p, d, i, m, &tc. will successively coincide. 

 But it has been shown that when these points are in contact. 



