ron or lead on the lathe mandrel or in
the chuck, and set the slide rest so that it is free to slide up or
down the lathe bed. Take a bar of tool steel and cut it a little
longer than the radius of curvature required. Forge and finish one
end of the bar into a pointed turning tool of the ordinary kind.
Measure the radius of curvature from the point of the tool along the
bar, and bore a hole, whose centre is at this point, through the bar
from the upper to the lower face. I regard the upper face as the one
whose horizontal plane contains the cutting point when the tool is in
use. Clamp a temporary back centre to the lathe bed, and let it carry
a pin in the vertical plane through the lathe centres, and let this
pin exactly fit the hole in the bar.

Fig. 48.

Place the "radius" tool in position for cutting, and let it be lightly
held in the slide rest nearly at the cutting point, the centre of
rotation of the pedestal (or its equivalent) passing through the
central line of the bar. Then adjust the temporary back rest, so that
the tool will take a cut. In the sketch the tool is shown swinging
about the back centre instead of about a pin--there is little to
choose between the methods unless economy of tool steel is an object.
The tool must now be fed across the work. The pedestal must of course
be free to rotate, and the slide rest to slip up and down the bed. In
this way a better concave grinding tool can be made than would be made
by a beginner by turning to a template--though an expert turner would
probably carry out the latter operation so as to obtain an' accuracy
of the same order, and would certainly do it in much less time than
would be required in setting up the special arrangements here
described.

On the other hand, if several surfaces have to be prepared, as in the
making of an achromatic lens, the quickest way would be by the use of
the radius tool, bored of course to work at the several radii
required. I have tried both methods, and my choice would depend
partly on the lathe at my disposal, and partly on the number of
grinding tools that had to be prepared.

Having obtained a concave tool of any given radius, it is easily
copied--negatively, so as to make a convex tool in the following
manner. Adjust the concave tool already made on the back rest, so
that if it rotated about the line of centres, it would rotate about
its axis of figure.

Arrangements for this can easily be made, but of course t