94 Blake 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 7; and from the point 7, 
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 oin the plane of this curve will trace the curve bc. 
Take or equal to op and join p,r; and take np equal 
to db, and joind.n. fdm is infinitely small, pi is also in- 
nitely 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 isa 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 parailel to db, and bh will be infinitely small. and 
may be taken as a straight line. But nh is equal to db, 
therefore dn, 6h, are parallel. Consequently the angle 
dnm is a right angle. Since by the construction po is 
equal to db, and mh to to, and since nk equals db, 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 andr are right an- 
gles. Hence the angle mdn is equal to the angle ipr. To 
the angle mdn add the right angle ndb, 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 
eoincides with db and o isin 6. If the curve and circle 
the curve pi Xe. are by the construction equal in length 
and number to the small arcs dm Wc. of the circle, the cor- 
responding points p, d, i, m, &c. will successively coincide. 
But it has been shown that when these points are in contact, 
