1 90 On prima and ultimate Ratios, 



creased, and the fluxion of the area is (#+ i) 2 — # 4 ; that 

 is = c jxx-\-x 2 . Now in order that we may neglect i ,; 

 without affecting the result of the operation, we must sup- 

 pose that x is a quantity less than any assignable, or that 

 it is only in a nascent state : according to this supposition, 

 the error, by neglecting x l , will be extremely small, and 

 will no way affect the fluxionary increase of the square ; 

 but, except x vanishes, and that would annihilate the 

 fluxional increase altogether, we are obliged to acknowledge 

 that the result is not strictly and logically true. 



Again, let x and y denote the sides of a rectangle, and, 

 by the motion of those sides, let them become aj + iand 

 y+y) then, the fluxion of the area of this rectangle will 

 be (x-\-x) X (y+y) — XV, or = xy+iy + xy. Here, also, that 

 the rectangle iy .may be neglected, i and z/ must be inde- 

 finitely small, or in a nascent state; but even then an error 

 is committed, and, however trifling it may be, the result 

 will not be strictly and geometrically true. Fluxions, then, 

 do not produce results which are exactly true; but, as was 

 observed above, they give us approximations differing from 

 the truth by less than any assignable quantity, however 

 small, and, therefore, may be esteemed as true with respect 

 to their practical conclusion. To proceed further would 

 be of no use: the application of those principles to curvi- 

 linear spaces is given in every book of fluxions. What has 

 been given above may probably be of some use to students, 

 as it may possibly serve to elucidate the principles of a 

 science, which has been the instrument by which almost all 

 the improvements in philosophy have been brought to 

 light. The principle of such a science ought to be esta- 

 blished upon a sure foundation ; and should what has been 

 said be of any use in removing the cavils that have been 

 made against the fluxionary calculus, a service will be done 

 to philosophy, and the writer of this essay mav at least 

 hope to be excused for endeavouring to contribute some-' 

 thing towards elucidating the elements of those very use- 

 ful but too much neglected studies. 



I remain, sir, 



Your very humble servant, 



Boston, Sept. 10, 1810. W. MARRAT. 



XXXV. Com' 



