324 Astronomical and Nautical Collections, 



mur, illorum signis longe anteferenda videntur.' In fact, however, 

 the English do not call the evanescent increments fluxions, any more 

 than a mile is an evanescent quantity, when we speak of a velocity 

 of a mile an hour. There are certainly some cases in which the 

 fluxional notation is inconvenient; thus, when we have occasion 

 to write d$x = $dx, it would be impossible to express this equation 

 without deviating from that method ; we might, indeed, write 

 Qx)' zr &c, but we should still introduce one heterogeneous cha- 

 racter " £, in order to denote the fluxion of the quantity x, upon a 

 different supposition respecting the generation of the quantity to 

 be compared, which is called, for distinction, the variation of x ; 

 as if we supposed the ordinate x to be moved laterally along the 

 absciss, and to assume the place of its neighbouring ordinates in 

 succession on the one hand, or to be simply prolonged or dimi- 

 nished on the other, the two elementary changes of its magnitude 

 being expressed by dx and ix respectively, or by $x and dx, at the 

 option of the writer. 



" It is surely a great inelegance, to say the least, not to distin- 

 guish a characteristic mark from a multiplying quantity by a dif- 

 ference of type ; for dx must mean, according to all analogy, the 

 product of d and x : and it is much more intelligible to write d<r, 

 as Lacroix, and many others, have done, instead of dx, as it is ge- 

 nerally printed in the works of Laplace. 



" It must be understood, then, that dx, as well as dc, denotes a 

 finite quantity proportional to an evanescent increment ; but when 

 we use other characteristics of variation, such as £ or A , it is not 

 always necessary to limit their signification so precisely : it will, 

 however, sometimes be convenient to employ the mark d for an 

 element of matter, considered as evanescent, and A x for an evane- 

 scent increment of x, corresponding to the fluxion dx, while the 

 larger A is employed to denote a finite difference, whether greater 

 or smaller. ,, 



*P. 93. " The initial value of any variable quantity u has some- 

 times been distinguished by a capital letter u ; sometimes by a 

 point, u, especially where it is supposed to begin from nothing; but 

 we must not altogether forget that this is the Newtonian character 



