84 PHILOSOPHICAL TRANSACTIONS. [anNO 177(5. 



let me expend at what rate I will, if when this is expended, I must wait another 

 hour before it be renewed, by the natural flow of a river, or otherwise, I can 

 then only expend 12 such quantities of power in 24 hours ; but if, while I am 

 expending 1000 tuns in 1 hour, the stream renews me the same quantity, then 

 I can expend 24 such quantities of power in 24 hours ; that is, I can go on con- 

 tinually at that rate, and the product or effect will be in proportion to time, 

 which is the common measure ; but the quantity of mechanic power arising 

 from the flow of the two rivers, compared by taking an equal portion of time, 

 is double in the one to the other, though each has a mill, that, when going, 

 will grind an equal quantity of corn in an hour.* 



XXVIII. A New and General Method of finding Simple and Quickly-converging 

 Series; by which the Proportion of the Diameter of a Circle to its Circumfer- 

 ence may easily be computed to a great Number of Places of Figures. By 

 Charles Hutton, Esq. F. R. S. p. 476. -^ 



' ' • -. f: r: ;. : i-,.(r ; 



In a late examination of the methods of Mr. Machin and others, for computing 

 the proportion of the diameter of a circle to its circumference, I discovered the 

 method explained in this paper. This method is very general, and discovers 

 many series which are very fit for the abovementioned purpose. The advantage 

 of this method is primarily owing to the simplicity of the series by which an arc 

 is found from its tangent. For if t denote the tangent of an arc a, the radius 

 being i, then it is well known, that the arc a will be equal to the infinite series, 

 < — - t^ + r ^ ' — i i'^ + g t'^ — &c. where the form is as simple as can be de- 

 sired. And it is evident that nothing further is required, than to contrive 

 matters so as that the value of the quantity /, in this series, may be both a 

 small and a very simple number. Small, that the series may be made to con- 

 verge sufHciently fast ; and simple, that the several powers oi t may be raised 

 by easy multiplications, or easy divisions. 



Since the first discovery of the above series, many have used it, and that after 



* In this paper, Mr. Smeaton does not seem clearly to distinguish between what he calls Mecha- 

 nical Power, and die Newtonian term Momentum, or Quantity of Motion. These two powers are, 

 from their very definition, as well as from their nature, of ditferent kinds. The one is measured or 

 estimated by its momentary or instantaneous action ; the other by its action during some certain time. 

 The one, by its definition, is in tlie compound ratio of the mass of a body and its velocity, or as the 

 product of the body and its velocity, and therefore simply as tlie velocity in a given body. Whereas 

 the other, by its definition, is estimated by the mass or weight compounded witJi the space it has 

 fallen or described in acquiring its velocity : and since, as is well known, tlie space faUeii by a body, 

 is as the square of the velocity acquired ; it follows that tliis force must needs be as die square of die 

 velocity in a given body. The Newtonian momentum or force, therefore, and Mr. Smeaton's me- 

 chanical force, are two things that are quite ditferent in their measure, and in their mode of action : 

 tliough botli may produce true results when applied to their proper objects. 



