the bifnittfunal Calculus. iy$ 



'Hence it evidently follows, that into all unafligned quan- 

 tities there neceflarilv enters fomething arbitrary ; for, if this 

 were not the cafe, their value would then be afligned, by the 

 veryrconditions of the problem, and confequently they would 

 wholly depend on propofed quantities; which is contrary to 

 the fuppofition. 



t6. Since in the mathematics, two lines, two furfaccs, 

 two folids, in fhort two quantities of any kind whatever, arc 

 fuppofed continually to approach each other by infenfible 

 degrees; fo that their proportion, or the quotient reprefent- 

 ing that proportion, differs lefs and lefs, and as little as we 

 pleafe, from unity, we fay, that the laft, or ultimate ratio of 

 thofe two quantities is a ratio of equality. 



ly. If one of the quantities be afligned, and the other 

 auxiliary, the firft is called the limit, or ultimate value of the 

 fecond. Thus a limit is nothing elfe than an afligned quan- 

 tity, to which an auxiliary quantity is fuppofed continually 

 to approach, in fuch a manner that their difference may be 

 rendered as fmall as we pleafe, and that their ultimate ratio 

 mav 'be a ratio of equality. 



Thus auxiliary quantities alone can be properly laid to 

 have limits; for afligned quantities, being fuppofed conftant 

 and unchangeable, and to be fherhfelves the terms, or ulti- 

 mate values of auxiliary quantities, cannot, in ftri&nefs, be 

 faid to have anv limits ; unlefs we choofe to fay, that every 

 afligned quantity is its own limit, a liberty of fpeech which 

 cannot be refufed us; fince the ultimate value of any de- 

 terminate quantity whatever can be nothing e'fe than that 

 quantity itfelf. 



18. Henee, in general, we give the names of ultimate 

 values and ultimate ratios of quantities, to the values and 

 ratios which are, in fact, the laft, or ultimate, ones afligned 

 to thofe quantities and their relations, by the law of conti- 

 nuity, when each of them is fuppofed to approach, by con- 

 tinued and infenlible degrees, to its correfponding afligned 

 quantity. 



19. We generally give the name of an Infinitely fmall 

 quantity, to the difference between any auxiliary quantity 

 whatever and its limit. Thus, for iiiitance, RZ, which is 



Hlu the 



