We measure the angle *i n B*, Fig. 92, and find
it to be seventy-four degrees; we draw the line *v
t* to the same angle with *v B*, and we define
the inner face of our pallet in the new position.
We draw a line parallel with *v t* from the intersection
of the line *v y* with the arc *p*, and
we define our locking face. If now we revolve
the lines we have just drawn on the center *B*
until the line *l B* coincides with the line
*f B*, we will find the line *y y* to coincide
with *h h*, and the line *v v’* with
*n i*.

HIGHER MATHEMATICS APPLIED TO THE LEVER ESCAPEMENT.

We have now instructed the reader how to delineate either tooth or pallet in any conceivable position in which they can be related to each other. Probably nothing has afforded more efficient aid to practical mechanics than has been afforded by the graphic solution of abstruce mathematical problems; and if we add to this the means of correction by mathematical calculations which do not involve the highest mathematical acquirements, we have approached pretty close to the actual requirements of the practical watchmaker.

[Illustration: Fig. 93]

To better explain what we mean, we refer the reader
to Fig. 93, where we show preliminary drawings for
delineating a lever escapement. We wish to ascertain
by the graphic method the distance between the centers
of action of the escape wheel and the pallet staff.
We make our drawing very carefully to a given scale,
as, for instance, the radius of the arc *a* is
5”. After the drawing is in the condition
shown at Fig. 93 we measure the distance on the line
*b* between the points (centers) *A B*,
and we thus by graphic means obtain a measure of the
distance between *A B*. Now, by the use
of trigonometry, we have the length of the line *A
f* (radius of the arc *a*) and all the angles
given, to find the length of *f B*, or *A B*,
or both *f B* and *A B*. By adopting
this policy we can verify the measurements taken from
our drawings. Suppose we find by the graphic
method that the distance between the points *A B*
is 5.78”, and by trigonometrical computation
find the distance to be 5.7762”. We know
from this that there is .0038” to be accounted
for somewhere; but for all practical purposes either
measurement should be satisfactory, because our drawing
is about thirty-eight times the actual size of the
escape wheel of an eighteen-size movement.