COMPARISON OF VARIABLE QUANTITIES. 13 



(hU)-=-,(50); and zzrz. (hi zyzzs-. (y. hi n)T:r. C— -f {\x\x).y\ 



Z X 



(48, 50)rzyxy-^x-\-{h\ x) zy. 



52. Theorem. When a variable quantity is greatest 

 or least, its fluxion vanishes. 



For a quantity is greatest when it ceases to increase, and before it 

 begins to decrease ; that is, when it has neither increment nor decre- 

 ment ; and it is least when it has ceased to have a decrement and 

 has not yet an increment. 



53. Problem. To solve a numerical equation by 

 approximation. 



The most general and useful mode of solving all numerical equa- 

 tions is by approximation. Substitute for the unknown quantity a 

 number, found by trial, which nearly answers to tlie conditions ; 

 then the error will be a finite difference of the whole equation ; which, 

 when small, will be to the error of the quantity substituted, nearly in 

 the ratio of the evanescent differences, or of the fluxions ; and this 

 ratio may be easily determined. 



Thus, ir x^—6x^-{-4xzz6699, call 6699, y, then 3x^x—l2xx-{-4xzzy, 



and;eT= — , and x'zz — nearly ; now assume xiz. 



3x2— 12a:+4 Sx^^i2x+4 



20, then yzi5680, and jzi 1019, whence x' 1.05, and x corrected is 



21.05 ; by repeating the operation we may approach still nearer to 



the true value 21. 



If x^y, xz=. -2. — , whence the common rule for the extraction of 



roots is derived. In order to find the nearest integer root, the digits 

 must be divided, beginning with the units, into parcels of as many as 

 there are units in the index, and the nearest root of the last or high- 

 est parcel being found, and its power subtracted, the remainder must 

 be divided by its next inferior power multiplied by the given index, 

 in order to find the next figure, adding the next parcel to the re- 

 mainder before the division. There are also, in particular cases, 

 other more compendious methods. 



It is, however, often more convenient to solve an equation by the 

 rule of double position, taking two approximate values of the root, 

 and finding a third which diflers from one of them by a quantity bear- 

 ing the same proportion to their difference as the error of that one 

 bears to the difference of the two errors. 



