IM6 philosophical transactions. [anno] 737. 



other times geometrical, according as he finds the one to be plainer and shorter 

 than the other. 



The second book treats of the direct method of fluxions. And here he 

 hopes the first principles of this method are laid down, not only in a new, but 

 very plain and concise manner. He proceeds to show the use effluxions in the 

 solution of the common problems of finding the maxima and minima of quan- 

 tities, the radii of the evolution of curves, and the radii of refraction and re- 

 flection. Under the first of these heads he says, particular care has been taken 

 to distinguish the maximums from the minimums, a thing which has not been 

 noticed so much as it ought to have been. And whereas some mathematicians, 

 having made use of what they call infinitely small quantities, are forced to re- 

 ject something out of the equation, for finding the fluxion of a rectangle, whose 

 sides are varying quantities, Mr. MuUer uses only finite quantities ; and finds 

 the fluxion of such a rectangle after a new manner, without rejecting any 

 quantity for its smallness. He does the same in finding the fluxion of a power. 

 And to avoid the use of infinitely small quantities, introduces a new principle, 

 viz. that a curve line may be considered as generated by the motion of a point 

 carried along by two forces or motions, one in a direction always parallel to the 

 absciss, and the other in a direction always parallel to the ordinate. Hence he 

 infers, that the fluxion of the ordinates is to the fluxion of the absciss, as the 

 ordinate is to the subtangent of the curve. 



Having likewise proved from the first supposition, that if the describing point, 

 when arrived at any place given, should continue to move onwards, with the 

 velocity it has there, it would proceed in a right line, which would touch the 

 curve in that point ; he concludes that the direction of the force in that place, 

 is in the tangent to the curve : consequently, the 3 directions being known in 

 each place, the proportion between the velocities of the urging forces will be 

 likewise known. So that the nature of the curve being given, the law observed 

 by these velocities may be found ; and if the law of the velocities be given, the 

 nature of the curve may likewise be given. 



In the third and last book, we have the inverse method of fluxions, with its 

 application to the several problems solvable by it ; such as the superficial and 

 solid contents of curvilineal figures, the rectification of curve lines, centres of 

 gravity, oscillation and percussion. Here also Mr. Cotes's tables of fluents are 

 explained and illustrated by examples. 



He finishes this book with a great variety of problems, of a physico-mathe- 

 matical nature, several of which are new, and were proposed to him by Mr. 

 Belidor. Some indeed are not so, having been solved by Messieurs Varignon 

 and Parent ; but then he has solved them after a different, and, as he hopes, a 



