636 PHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3, 



preceding that term, by axiom 2, and hence it is demonstrated, that the fluxion 

 of the triangle is accurately measured by the rectangle contained by the corres- 

 ponding ordinate of the triangle, and the right line which measures the fluxion 

 of the base. The increment which the triangle acquires in any time, is re- 

 solved into two parts ; that which is generated in consequence of the motion 

 with which the triangle flows at the beginning of the time, and that which is 

 generated in consequence of the acceleration of this motion for the same time. 

 The latter is justly neglected in measuring that motion, or the fluxion of the 

 triangle at that term, but may serve for measuring its acceleration, of the 2d 

 fluxion of the triangle. The motion with which the triangle flows, is similar 

 to that of a body descending in free spaces by a uniform gravity, the velocity of 

 which, at any term of the time, is not to be measured by the space described 

 by the body in a given time, either before or after that term, because the mo- 

 tion continually increases, but by a mean between these spaces. 



When the sides of a rectangle increase or decrease with uniform motions, it 

 may be always considered as the sum or difference of a triangle and trapezium ; 

 and its fluxion is derived from the last proposition. If the sides increase with 

 uniform motions, the rectangle increases with an accelerated motion ; and in 

 measuring this motion at any term of the time, a part of the increment of the 

 rectangle, that is here determined, is rejected, as generated in consequence of 

 the acceleration of that motion. 



The fluxions of a curvilineal area (whether it be generated by an ordinate 

 moving parallel to itself, or by a ray revolving about a given centre) and of the 

 solid, generated by the area revolving about the base, are determined by de- 

 monstrations of the same kind; and when the ordinates of the figure increase, 

 the increment of the area is resolved in like manner into two parts, one of 

 which is only to be retained in measuring the fluxion of the area, the other be- 

 ing rejected as generated in consequence of the acceleration of the motion with 

 which the figure flows. An illustration of 2d and 3d fluxions is given by re- 

 solving the increment of a pyramid or cone into the several respective parts that 

 are conceived to be generated in consequence of the 1st, 2d, and 3d fluxions 

 of the solid, when the axis is supposed to flow uniformly. 



In chap. 5, a series of lines in geometrical progression are represented by an 

 easy construction. The first term being supposed invariable, and the second to 

 increase uniformly, all the subsequent terms increase with accelerated motions. 

 The velocities of the points that describe those lines being compared, it is de- 

 monstrated, from the axioms by common geometry, that the fluxions of any 

 two terms are in a ratio compounded of the ratio of the terms, and of the ratio 

 of the numbers that express how many terms precede them in the progression. 



