74 I'HILOSOPHICAl. TRANSACTIONS. [anNO 1776. 



before, he tells us, " The knowledge of the foregoing particulars is absolutely 

 necessary for setting an undershot wheel to work ; but the advantage to be 

 reaped from it would be still guess-work, and we should be still at a loss to find 

 out tlie utmost it can perform, if we had not an ingenious proposition of that 

 excellent mechanic M. Parent, of the Royal Academy of Sciences, who has 

 given us a maximum in this case, by showing, that an undershot wheel can do 

 the most work, when its velocity is equal to the 3d part of the velocity of the 

 water that drives it, &c. because then f of the water is employed in driving the 

 wheel with a force proportionable to the square of its velocity. If we multiply 

 the surface of the adjutage or opening by the height of the water, we shall have 

 the column of water that moves the wheel. The wheel thus moved will sustain 

 on the opposite side only -S- of that weight, which will keep it in equilibrio ; 

 but what it can move with the velocity it goes with, will be but 4- of that weight 

 of equilibrium ; that is, -^ of the weight of the first column, &c. — This is the 

 utmost that can be expected." 



The same conclusion is likewise adopted by Maclaurin, in art. Q07 , p. 728, 

 of his Fluxions, where, giving the fluxionary deduction of M. Parent's proposition, 

 he says, " that if a represents the weight which would balance the force of the 

 stream, when its velocity is a; and u represents the velocity of that part of the 

 engine, which it strikes when the motion of the machine is uniform, &c. — the 

 machine will have the greatest effect when u is equal to 4 a ; that is, if the 

 weight that is raised by the engine be less than the weight which would balance 

 the power, in the proportion of 4 to Q, and the momentum of the weight 

 is -^V Aa." 



Finding that these conclusions were far from the truth, and seeing, from 

 many other circumstances, that the practical theory of making water and wind- 

 mills was but very imperfectly delivered by any author I had then an opportunity 

 of consulting ;* in the year 1751 I began a course of experiments on this 



* Belidor, Architecture Hydraulique, greatly prefers the application of water to an undershot 

 mill, instead of an overshot ; and attempts to demonstrate, lliat water ajiplied undersliot will do 6" 

 times more execution than the same applied overshot. See vol. I, p. '2S6. Wliile Desaguliers, 

 endeavouring to invalidate what had been advanced by Belidor, and greatly preferring an overshot 

 to an undershot, says, Annotat. on lecture \'Z, vol.2, p. 53 J, tliat from his own experience, "a 

 well made overshot mill ground as much corn in the same time with 10 times less water ;" so that 

 between Belidor and Desaguliers here is a difference of no less than 60 to I. 



Again, Belidor, vol. 2, p. 72, says, that tlie centre of gravity of each sail of a windmill should 

 travel in its own circle with one-third of the velocity of tlie wind ; so that, taking the distance of 

 this centre of gravity from the centre of motion at 20 feet, as he states it p. 38, art. 849, the 

 circumference will be exceeding 1 26 feet English measure ; a wind therefore, to malie the mill go 

 20 turns per minute, which they frequently do witli a fresh wind, and all their cloth spreail, would 

 require the wind to move above 80 miles an hour ; a velocity perhaps liardiy eijualled in the greatest 

 storms we experience in this climate.— Orig. 



