A Theory of Fluxions. 349 



cond fluxion ocx' 2 , by a parallelopipedon, which is double the 

 prism EIHei'N ; the third fluxion x' 3 , by a cube, which is 

 made up of six prisms, similar and equal to eNih. Because 

 a;* 3 is a constant quantity, the fourth fluxion is equal to 0. 

 From the section it is evident, that the increment DEeGHA 

 is composed of three parts ; the prism EDeIG/, which is the 

 first fluxion ; the prism EIHeiN, which is one half the second 

 fluxion ; and the pyramid eNih, which is one sixth part of the 

 third fluxion. 



For the benefit of the Mathematical students, I have pre- 

 sented to Yale College, a set of models designed to give a 

 geometrical illustration of the higher orders of fluxions by 

 the sections of the Pyramid and Cube. No. I. represents 

 the several orders of fluxions arising from the Pyramid, and 

 explains by solid figures the diagram in Maclaurin's fluxions 

 illustrating his celebrated theorem, which is, that the first 

 fluxion, half the second fluxion, and one sixth of the third 

 fluxion, are equal to the corresponding increment in the third 

 power. The fluent is represented by a Pyramid of a white 

 color, whose base is half a square of two inches. The first 

 fluxion by a short prism of a red color, one inch in length. 

 The second fluxion by a parallelopipedon of a yellow color 

 two inches in length, having its ends a square of one inch, 

 formed by the union of two equal prisms. The third fluxion 

 by a small cube of a blue color, having its sides one inch, 

 formed by the union of the six little pyramids. No. II. is all 

 of a white color, and represents merely the increment 3x a x'-{- 

 3xx' 2 -+-x' 3 of the cube. 



An actual inspection of these sections, as all must be sensi- 

 ble, will give the clearest idea of the nature of these quanti- 

 ties, but for the sake of those who may not have the oppor- 

 tunity of viewing the models, I have attempted to explain 

 No. III. by drawings. Fig. 1. is one of the three pyramids, 

 of which the fluent, (fig. 2.) which is a cube, is composed. 

 The apex of the first may be conceived to be situated at the 

 nearest left hand corner at the top, having its base at the 

 bottom. The second has its apex at the same corner, and 

 base at the side opposite : and the third pyramid has its apex 

 at the same corner, and its base at the opposite end. Fig. 

 3. 4. and 5., are kathetic views of the several orders of flux- 

 ions arising from the three pyramids just mentioned ; in 

 which the lines representing their thickness in the projection, 

 fall behind and are hidden. When the first fluxion is, produ- 



