IRREDUCIBLE CASE. 



foii: 



■r 1 r 1 "^^ ^ ' ^ 2 \^ ? 



lie ; and, conlcqucntly, — < — , or c < — ^— ^, or 

 4 27 9 



c < ..?849i ; aflTuining this, thcivfore, as the maximum va- 

 lue of e, we find that the greatell value of y can never ex- 

 ceed 1.1549, nor can the Icall be kis than unity ; fo that all 

 pofTible values x>{ y lie between the limits 1 and 1.IJ49 ; and 

 if, therefore, we have a table containing all the values of c 

 to tbofe of y, between the above limits, wc (hall have by 

 infpeftion the folution of every equation of the irreducible 

 cafe, when converted into the form y' — y =z c \ and hence 

 alfo of every irreducible equation of the form .v ' — ax = h, 

 becaiile x ~ y ^ a. 



The following is a table of this defcription, the value of j: 

 being arranged in the leading column, except the lalt digit, 

 which is found in the upper horizontal line, and the refpcftivc 

 values of c are found in the other columns, being exactly 

 the form that is ufually given to the common tables of loga- 

 rithms ; and the finding of any value of _)•, correfponding to 

 a given value of c, is performed cxaClly the fame as thit of 

 finding the natural number to a given logarithm. Thus, for 

 example, giving _)• ' — y =l .3S37 to find^y : in the table the 

 number correfponding to .38^69912 is 1.1543 ; tiiat h,y =. 

 I.I 543. It is obvious, therefore, that, from bare infpeCtion, 

 the val'ie of _y may be found true to five places of figures, 

 but thefe are extended to 8 or 9 places true, by taking pro- 

 portional parts as in logarillims ; thus, find the next greater 

 and lefs tabular number to that propofed ; and then fay, 

 as the difference of thefe is to .0001 ; fo is the difference 

 between the lead of them and the number propofed, to the 

 part which is to be added unto the leall value ij{ y ; and this 

 proportion may be fafely carried to four places at leaft, and 

 thus the value of y will be known to eight places. This 

 may be demonllrated as fi)llows : the proportional part that 

 is thus added unto y, is always lefs than .0001, and, there- 

 fore, lefs than the ^-i-^dth part of ji, and hence it may be 

 faid to be very fmall with regard to_)'. Now, I fay, that if 

 a and b be both very fmall with regard to a third quantity _y, 

 that the following proportion is very nearly true ; ix's. 



(7T^' - y -^r a) - {y' - y) : (/T7)' - yTl>) 



- (v' - y\ :: a : I ; 

 for, by rejefting all thofe powers of a and h higher than the 

 firll, this proportion becomes precifely 



ily'- - i)" : iy^ - ^) b :: a : h ; 

 and fincc, in this cafe, <? and b are lefs than .0001, (or at leaft 

 neither of them ever exceed this,) by rejefting the fquares 

 and cubes of thefe quantities, we cannot have affefted the 

 fourth place of fij^uies in the rclult, and therefore the {)rij- 



portioi) may always be depended upon to four places. An J 

 thus, tile following rule may be deduced for every irreducible 

 cubic equation, x- - nv ~ h ; viz.. 



Find 



b ^/ a 



the following table, and take out the cor- 



refponding value of y ; then will y ,/ a =: r be the root 

 fought. Let us, for inftance, take the example propof-d 

 by Cardan to Tartalea ; namely, .v' — 9* := lo, to find x. 



Here -— = ~| = .37037O.37- 



Tab. 11. .37057828. correfp. n. i . 1 499'37037037 given n. , 

 ne.xtlefs .3702S164 n. i.i498i37028i"64 lefstab.n. 



As 29664 ; 



Therefore J' — 1.14982991 



08873 • 2991 



= ,/ a 



r = 3.44948973 as required. 

 -Required the value of v in the equation 



,32396954; 



Tabula n. .32398835, correfp. n. 

 Next lefs .323-0267, n. 



and 



'•-3.^9 .3239^9^4 



As 28568 : I ;: 266.';7 : 9342 



Therefore the value of _v is 1.I3389J42 ; 

 y s' 1 = i-i.^3''9342 X 2-6457JI3 = 



2.999999999, as will be found by the operation ; which 

 anlweristrue to the 10th place, the real root being 3, as ii 

 obvious from the equation. It will be unneceflary to give 

 any other examples, as the fame method is purfued in a!} 

 caies, and the fame accuracy may be depended upon in the 

 refults. The reader who will trouble himfelf to coinpare 

 the rule here laid down, with any other that has been before 

 given by other writers, will foon be convinced of the im.- 

 menfe labour that is faved by the following table ; at the 

 fame time, that the number of figures in the refult are nearly- 

 double thofe that can be fafely obtained by the tables of 

 fines and tangents. 



For the method of finding the other two roots after one 

 is obtained, fee E(^UATioxs ; for the folution of cubic equa- 

 tions by the tiifettion of an angle, fee Bonnycaftle's Trigo- 

 nometry ; and for the method by infinite fcries, fee Philol.i- 

 phical Tranfaiftioiis, vols. Ixviii. and Ixx. 



S^: 



TaUe 



