once that this condition could be obtained in a perfect convex surface
fitting a perfect concave of the same radius. Fortunately we have a check
to guard against this error. To produce a perfect plane, _three surfaces
must_ be worked together, unless we have a true plane to commence with;
but to make this true plane by this method we _must_ work three together,
and if each one comes up to the demands of this most rigorous test, we may
rest assured that we have attained a degree of accuracy almost beyond
human conception. Let me illustrate. Suppose we have plates 1, 2, and 3,
Fig. 11. Suppose 1 and 2 to be accurately convex and 3 accurately concave,
of the same radius. Now it is evident that 3 will exactly fit 1 and 2, and
that 1 and 2 will separately fit No. 3, _but_ when 1 and 2 are placed
together, they will only touch in the center, and there is no possible
way to make three plates coincide when they are alternately tested upon
one another than to make _perfect planes_ out of them. As it is difficult
to see the colors well on metal surfaces, a one-colored light is used,
such as the sodium flame, which gives to the eye in our test, dark and
bright bands instead of colored ones. When these plates are worked and
tested upon one another until they all present the same appearance, one
may be reserved for a test plate for future use. Here is a small test
plate made by the celebrated Steinheil, and here two made by myself, and I
may be pardoned in saying that I was much gratified to find the
coincidence so nearly perfect that the limiting error is much less than
0.00001 of an inch. My assistant, with but a few months' experience, has
made quite as accurate plates. It is necessary of course to have a glass
plate to test the metal plates, as the upper plate _must_ be transparent.
So far we have been dealing with perfect surfaces. Let us now see what
shall occur in surfaces that are not plane. Suppose we now have our
perfect test plate, and it is laid on a plate that has a compound error,
say depressed at center and edge and high between these points. If this
error is regular, the central bands arrange themselves as in Fig. 9. You