10 On the Theory of 



and in a note, p. 44, he thus expresses himself, " Fig. 3, Plate III., 

 (Cumming's Work) exhibits at one view the length of the pallet, 

 and the distances of the center of the verge from that of the swing 

 wheel, according to the number of the teeth of the wheel which the 

 pallet takes in from two to twelve*, (the wheel is supposed a wheel 

 of thirty teeth), by which it appears that the distance of those 

 centers is the secant, and the length of the pallets the tangent of 

 half the angle subtended at the center of the swing wheel by such 

 number of teeth." To illustrate Mr. Cumming's observation re- 

 specting the secant and tangent of half the angle of the number of 

 teeth taken over ; suppose the case of the pallets taking over twelve 

 teeth, (see Fig. 4, Plate IV.,) DC the distance between the centers 

 will be the secant, and D B the length of the supposed pallets the 

 tangent to the angle A C B, which angle is half the angle sub- 

 tended at the center of the wheel by such number of teeth. Here 

 Mr. dimming determines the center of action of the pallets by 

 tangents drawn from the points of the teeth of the wheel ; which, 

 as will shortly be shewn, is not the most correct method. 



I. A. Lepantre, in his Traitc d'Horlogerie, 4to, Paris, 1767, page 

 188, No. 58, states, that Mr. Graham made his dead escapement 

 for clocks with the rests at equal distance from the center of action 

 of the pallets as represented Fig. 2, Plate II., and has so repre- 

 sented them in Plate XIII., Fig. 9, of his work. 



Whatever advantage may be supposed to result from the rests 

 being at equal distance from the center, it is more than counter- 

 balanced by the inequality of the distance from the center of the 

 pallets, at which the impulse is given by the wheel. In the first 

 place this difference of distance causes the lengths of the inclined 



* In the plate of Mr. Cumming's work, the number of teeth taken in by the 

 pallets is from three to thirteen, not from two to twelve, as stated in the text, 

 and as marked in figures in the plate. Fig. 4, Plate IV. represents the tan- 

 gents drawn, taking in from two to twelve teeth, as quoted in the text, Fig. 5, 

 taking in from three to thirteen teeth, as represented in Mr. Cumming's figure. 



It is worthy of notice, that in the case of taking in thirteen teeth the distance 

 between the centres, supposing the tangents drawn from the points of the teeth, 

 (see Fig. 5, Plate IV.,) is not exactly 1 and } diameter of the wheel, the dis- 

 tance determined by the supposed rule of Graham's, but a little more. 



