120 PHILOSOPHICAL TRANSACTIONS. [aNNO 1714. 



and that the right line bk, 1, is the moment by which the area ak, a:, gradually 

 increases, and the right line bd, y, the moment by which the curvilinear area 

 ABD gradually increases; and that from the moment. bd continually given, you 

 may by the three preceding rules investigate the area abd, described bv it, or 

 compare it with the area ak, or, described by the moment 1." This is Mr. 

 Newton's idea of the work in squaring of curves, and how he applies this to 

 other problems, he expresses in the next words. " Now, says he, by the same 

 method that the superficies abd is found, by the three foregoing rules, from its 

 moment being continually given, by the very same any other quantity may be 

 found from its moment in like manner given." And after some examples he 

 adds his method of regression from the area, arc, or solid content, to the 

 abscissa; and shows how the same method extends to mechanical curves, for 

 determining their ordinates, tangents, areas, lengths, &c. And that by assum- 

 ing any equation, expressing the relation between the area and abscissa of a 

 curve, you may find the ordinate by this method. And this is the foundation 

 of the method of fluxions and moments, which Mr. Newton in his letter, 

 dated Oct. 24, 1676, comprehended in this sentence; data aequatione quot- 

 cunque fluentes quantitates involvente, invenire fluxiones; et vice versa; that 

 is, from a given equation involving any number of fluents, or flowing quan- 

 tities, to find the fluxions, and vice versa. 



In this compendium, Mr. Newton represents the uniform fluxion of time, 

 or of any exponent of time, by an unit; the moment of time, or of its expo- 

 nent by the letter 0; the fluxions of other quantities by any other symbols; the 

 moments of those quantities by the rectangles under those symbols and the 

 letter 0; and the area of a curve by the ordinate inclosed in a square, the area 

 being put for a fluent, and the ordinate for its fluxion. When he is demon- 

 strating any proposition he uses the letter for the finite moment of time, or 

 of its exponent, or of any quantity flowing uniformly, and performs the whole 

 calculation by the geometry of the ancients, in finite figures or schemes, without 

 any approximation ; and as soon as the calculation is at an end, and the equation 

 is reduced, he supposes that the moment decreases in infinitum and vanishes. 

 But when he is not demonstrating, but only investigating a proposition, for 

 making dispatch he supposes the moment to be infinitely little, and forbears 

 to write it down, using all manner of approximations, which he conceives will 

 produce no error in the conclusion. An example of the first kind you have in 

 the end of this compendium, in demonstrating the first of the three rules laid 

 down in the beginning of the book. Examples of the second kind you have in 

 the same compendium, in finding the length of curve lines, p. 15, and in find- 

 ing the ordinates, areas, and lengtl>s of mechanical curves, p. 18, 19. And he 



