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.

HOW THE BASIS FOR CLOSE MEASUREMENTS IS OBTAINED.