VOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 455 



given rise to a general relaxation with regard to the true principles, and to the 

 adoption of rules and measures arbitrary, vague, and which produce very con- 

 siderable errors: such, for instance, is the error of taking a little more than 3 

 points of a tooth, in order to determine the size of the pinions of 6, without 

 regard to the revolutions which the different numbers of the teeth of the wheels 

 produce. The consequence is, that an equal measure being taken on a wheel 

 of 18, and on one of 7*2 teeth, produces a variation of 1 entire revolution, as 

 will be shown hereafter. For instance, the pinion is the divisor of the wheel as 

 well as that of the circle, each tooth of which is to raise its required proportion of 

 degrees. A pinion of 6 is to raise 6o° per tooth, because 6 x 6o = 360; one of 7 

 is to raise 51° 25-|- and some seconds; one of 8 is to raise 45°; one of 10 is to 

 raise 36°: finally, one of 12 must incontestibly raise 30° per tooth, since 

 12 X 30 = 360. 



Among various methods that have been used for the solution of this im- 

 portant problem, I shall mention only the most simple one, easily understood. 

 I set in motion two wheels, each of 12 teeth. By making one wheel turn the 

 other, it is evident that their two diameters must be perfectly equal, allow- 

 ing for the necessary shake between the teeth.* It will hence follow that these 

 two movers, which we will suppose wheel and pinion, must reciprocally raise 

 the 30° required; but if we increase one of these so as to make a revolution 

 more than the other, which is fixed at J 2 teeth, and at 1 line diameter, this last 

 will indeed have 24 teeth ; but instead of 2 lines of diameter, it will have only 

 23 and — relatively to that of the pinion ; or if we give it exactly the double 

 diameter, it will have 25 teeth instead of 24 ; consequently, on 3 revolutions, or 

 36 teeth, we must subtract from the wheel 2 and Vt oi a line; and so on as far 

 as 12, or rather 11, effective revolutions; for the first, being supposed to be in 

 equilibrio with its pinion, ought not to be reckoned. The wheel will then 

 indeed have 144 teeth, or 12 X 12; but i*t will only have 11 times as much in 

 diameter, that is, 11 lines instead of 12, in order that the angles of these two 

 movers may always be in the same perfect proportion to each other, of which 

 the precise raising of 30° per tooth is one of the most convincing proofs. 



To make this more intelligible, as the primitive radii ought to be equal 

 betwen these movers, a pinion of 6 wings, being a line in diameter, requires the 

 taking off of i- to take from it what is useless in its catch ; this is what forms 

 the apparent diameter. This deduction will reduce the primitive radius of this 

 pinion to -f- of a line; for as its revolutions about a wheel of 12 teeth are as 2 to 



* This shake is very inconsiderable, especially when the pinion and the teeth of the wheel are 

 properly opened ; for, according to my experiments, the deduction to be made on this account is 

 reduced to the 96th part of the circumference of a pinion of 1 2, which produces the sum of the 8th 

 part of a tooth on the wheel, whatever its number may be. — Orig. 



