As regards the exalted adhesion of the parallel surfaces, we fancy there is more harm feared than really exists, because, to take the worst view of the situation, such parallelism only exists for the briefest duration, in a practical sense, because theoretically these surfaces never slide on each other as parallel planes. Mathematically considered, the theoretical plane represented by the impulse face of the tooth approaches parallelism with the plane represented by the impulse face of the pallet, arrives at parallelism and instantly passes away from such parallelism.
TO DRAW A PALLET IN ANY POSITION.
As delineated in Fig. 92, the impulse planes of the tooth and pallet are in contact; but we have it in our power to delineate the pallet at any point we choose between the arcs p s. To describe and illustrate the above remark, we say the lines B e and B f embrace five degrees of angular motion of the pallet. Now, the impulse plane of the pallet occupies four of these five degrees. We do not draw a radial line from B inside of the line B e to show where the outer angle of the impulse plane commences, but the reader will see that the impulse plane is drawn one degree on the arc p below the line B e. We continue the line h h to represent the impulse face of the tooth, and measure the angle B n h and find it to be twenty-seven degrees. Now suppose we wish to delineate the entrance pallet as if not in contact with the escape-wheel tooth—for illustration, say, we wish the inner angle of the pallet to be at the point v on the arc s. We draw the radial line B l through v; and if we draw another line so it passes through the point v at an angle of twenty-seven degrees to B l, and continue said line so it crosses the arc p, we delineate the impulse face of our pallet.
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.
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.
Let us further suppose the diameter of our actual escape wheel to be .26", and we were constructing a watch after the lines of our drawing. By "lines," in this case, we mean in the same general form and ratio of parts; as, for illustration, if the distance from the intersection of the arc a with the line b to the point B was one-fifteenth of the diameter of the escape wheel, this ratio would hold good in the actual watch, that is, it would be the one-fifteenth part of .26". Again, suppose the diameter of the escape wheel in the large drawing is 10" and the distance between the centers A B is 5.78"; to obtain the actual distance for the watch with the escape wheel .26" diameter, we make a statement in proportion, thus: 10 : 5.78 :: .26 to the actual distance between the pivot holes of the watch. By computation we find the distance to be .15". These proportions will hold good in every part of actual construction.