VOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 457 



proper space between the teetli of each wheel, whence follows the curve, equally 

 constant and advantageous in every respect, both with regard to the execution and 

 to the uniformity in its lead on a pinion such as this, the thickness of the teeth of 

 which is a 6th part of its diameter, just as the excess of the primitive radius or 

 true diameter is the 6th part of its apparent diameter. 



According to this exposition, founded on experiments, it is easy to see what 

 errors must have crept into clock-making by the old method of taking, for the 

 pinion of 6, 3 or a little more than 3 points of teeth; for a pinion of 7, the 3 

 full teeth when finished , for a pinion of 8, the 3 full teeth and the void space as 

 far as the 4th ; for a pinion of 10, 4 full teeth, so as the wheel comes out of the 

 engine, (the making of which has hitherto been entrusted to women, children, 

 and servants); finally, for a pinion of 12, 5 points of rather strong teeth, with 

 the same arbitrary formality. These methods then, are only fit to perpetuate 

 misunderstanding and division on the manner of fixing and ascertaining the first 

 principles of watch-making. Nothing therefore ought to make us more sensible 

 of the want there is of our compasses of proportion, than the consideration of 

 measures as absurd in themselves as they are impossible to fix; there is no pro- 

 portion which can be fixed (except it be by mere chance) between these movers, 

 not even within so much as 8 or 10 degrees of rise per tooth; whereas by our 

 machines, whose utility and the manner of making use of them is com- 

 prehended at first sight, we may come within a quarter or even an 8th of a rise 

 per tooth, or nearly so. It is not so easy to adapt these principles to our 

 machines for all sorts of pinions, numbers of teeth, and sizes of wheels, and 

 generally to whatever acts by lever, or wheel and pinion, or whatever is suscep- 

 tible of relation, dimension, proportion, and compensation, from the first to 

 the last movers of a watch or clock ; for the least fault, either in the systems or 

 geometrical principles, or the mechanical execution, is as fatal as would be the 

 setting down a cypher too much or too little in the solution of a problem; and 

 the more fatal as the machine is more complicated. 



As to the true point of the opening of the teeth, this is my rule. I take 

 rather a soft file, in thickness 6 and a quarter of its diameter for a pinion of 6, 

 7 and a quarter for one of 7, 8 and a quarter for one of 8, 10 and a quarter for 

 one of 10, 12 and a quarter for one of 12, and so on; and I believe that this rule 

 may be the easiest for the preservation both of the best figure of the curves and 

 the necessary force of their teeth. Just as the thickness of the teeth of a 

 pinion of 6 must be the 6th part of its diameter; that of the pinion of 7, a 

 7th; one of 8, an 8th; and so on, for any number of pinions whatever. Just 

 in the same manner as the length of the teeth of the wheel, which shall be 

 exactly the aperture of their angles, or (which comes to the same) half the 



VOL. XIV. 



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