The Odontography invented in 1838 by Professor Willis, has been used to determine the radii of arcs of circles that shall approximate to the epicycloidal and hypocycloidal curves which should be used if perfect forms are wanted for the teeth of wheels. The instrument consists of a scale and a table. The first may be set out as follows on a piece of cardboard about 14- in. high by 71/2 in. broad. At the right-hand edge, and about 21/2 in. from the base, take a point. From this point divide the edge into lengths of 1/2 in. and number the divisions 10, 20, 30, etc., both above and below the point first marked, which should be numbered 0. Then subdivide each 1/2-in. division into ten equal parts, and from the point first marked (0) set off a line towards the base at an angle of 750 with the vertical. The tables on the instrument show the place of the centres of the arcs of thinks and faces upon the scales for wheels with teeth numbering 12 to 150, and for racks, the pitches varying from 1 in. to 3 in. Other pitches may be found proportionately; thus, for a,;-in. pitch, take out half the table value for a pitch of l 1/2 in. To use the instrument, one half the pitch is marked along the pitch circle of the wheel to be set out at each side of a radial line.

From the two points thus found radial lines are set off. Then the sloping line of the Instrument is placed so as to coincide with one radial line, with the edge of the scale over the point on the pitch circle. Then consult the table of centres for flanks of teeth; the number in the table, which varies with the pitch and the number of teeth. shows the point on the scale line above 0 at which the centre of the curve for the flank of the tooth is situate 1. Similarly for the centre of the face of the tooth set the sloping line on the other radial line with 0 on the pitch circle. Then the table shows the position of the centre on the scale measured downward from 0.