This number then is the proportion existing for the teeth and pitch  diameters of the 4th wheel and escape pinion. We must now find a suitable number of teeth for this wheel and pinion. Of available pinions for a watch, the only one which would answer would be one of 8 leaves, as any other number would give a fractional number of teeth for the 4th wheel, therefore 9.375 × 8 = 75 teeth in 4th wheel. Now as to the proof: as is well known, if we multiply the number of teeth contained in 4th and escape wheels also by 2, for the reason previously given, and divide by the leaves in the escape pinion, we get the number of beats made per minute; therefore ^{(75 × 16 × 2)}⁄₈ = 300 beats per minute.

Pallets can be made to embrace more than three teeth, but would be much heavier and therefore the mechanical action would suffer. They can also be made to embrace fewer teeth, but the necessary side shake in the pivot holes would prove very detrimental to a total lifting angle of 10°, which represents the angle of movement in modern watches. Some of the finest ones only make 8 or 9° of a movement; the smaller the angle the greater will the effects of defective workmanship be; 10° is a common-sense angle and gives a safe escapement capable of fine results. Theoretically, if a timepiece could be produced in which the balance would vibrate without being connected with an escapement, we would have reached a step nearer the goal. Practice has shown this to be the proper theory to work on. Hence, the smaller the pallet and impulse angles the less will the balance and escapement be connected. The chronometer is still more highly detached than the lever.

The pallet embracing three teeth is sound and practical, and when applied to a 15 tooth wheel, this arrangement offers certain geometrical and mechanical advantages in its construction, which we will notice in due time. 15 teeth divide evenly into 360° leaving an interval of 24° from tooth to tooth, which is also the angle at which the locking faces of the teeth are inclined from the center, which fact will be found convenient when we come to cut our wheel.

 From locking to locking on the pallet scaping over three teeth, the angle is 60°, which is equal to 2½ spaces of the wheel. Fig. 1 illustrates the lockings, spanning this arc. If the pallets embraced 4 teeth, the angle would be 84°; or in case of a 16 tooth wheel scaping over three teeth, the angle would be 360 × ^2.5⁄₁₆ = 56¼°.

Fig. 1.

Pallets may be divided into two kinds, namely: equidistant and circular. The equidistant pallet is so-called because the lockings are an equal distance from the center; sometimes it is also called the tangential escapement, on account of the unlocking taking place on the intersection of tangent AC with EB, and FB with AD, the tangents, which is the valuable feature of this form of escapement.

Fig. 2.

AC and AD, Fig. 2, are tangents to the primitive circle GH. ABE and ABF are angles of 30° each, together  therefore forming the angle FBE of 60°. The locking circle MN is struck from the pallet center A; the interangles being equal, consequently the pallets must be equidistant.

The weak point of this pallet is that the lifting is not performed so favorably; by examining the lifting planes MO and NP, we see that the discharging edge, O, is closer to the center, A, than the discharging edge, P; consequently the lifting on the engaging pallet is performed on a shorter lever arm than on the disengaging pallet, also any inequality in workmanship would prove more detrimental on the engaging than on the disengaging pallet. The equidistant pallet requires fine workmanship throughout. We have purposely shown it of a width of 10°, which is the widest we can employ in a 15 tooth wheel, and shows the defects of this escapement more readily than if we had used a narrow pallet. A narrower pallet is advisable, as the difference in the discharging edges will be less, and the lifting arms would, therefore, not show so much difference in leverage.

Fig. 3.

The circular pallet is sometimes appropriately called “the pallet with equal lifts,” as the lever arms AMO and ANP, Fig. 3, are equal lengths. It will be noticed by examining the diagram, that the pallets are bisected by the 30° lines EB and FB, one-half their width being placed on each side of these lines. In this pallet we have two locking circles, MP  for the engaging pallet, and NO for the disengaging pallet. The weak points in this escapement are that the unlocking resistance is greater on the engaging than on the disengaging pallet, and that neither of them lock on the tangents AC and AD, at the points of intersection with EB and FB. The narrower the circular pallet is made, the nearer to the tangent will the unlocking be performed. In neither the equidistant or circular pallets can the unlocking resistance be exactly the same on each pallet, as in the engaging pallet the friction takes place before AB, the line of centers, which is more severe than when this line has been passed, as is the case with the disengaging pallet; this fact proportionately increases the existing defects of the circular over the equidistant pallet, and vice versa, but for the same reason, the lifting in the equidistant is proportionately accompanied by more friction than in the circular.

Both equidistant and circular pallets have their adherents; the