402 - ' PHILOSOPHICAL TRANSACTIONS. [aNNO idSJ, 



And this may suffice for cubic equations ; but because of the excellent use of 

 the method, in which, by means of a table of sines, the roots of these equations 

 are found ; it was thought proper to add an example or two, by which the com- 

 pendium of that practice may appear. Let then the equation z^ — 3g z^ + 

 479 z — 1881 = O be proposed, to find the roots z, ^ -^bb — -yj = ^Q±^ = 

 V d, whose double v'37-j- is the radius of the circle ; and 



that is, making a division by means of the logarithms, log. Q.Q'ZSisSo, to 

 which answers the angle of 57° 19' \l\" % the third part of this is, 19° 6' 24"; 

 and ^ of its supplement 40° 53' 36*'; the sines give the logarithms 9:514983, 

 and9.8l6oi], which multiplied into the radius v'37-^, produce Y & and Y 8c, 

 log. 0.301030 = 2, and log. O.601059 = 4> but the third Y& is equal to their 

 sum, or 6. Therefore the roots are 13 —'4 = 9, 13 — 2 = 11, and 13 + 

 6 = 19, of which several quantities the above-mentioned equation is composed. 

 Where it is to be observed, that the two lesser roots do not exceed -J-d, or 1 3, 

 because the centre R in the construction falls on the right hand side of the axis ; 

 that is, ^bp is less than -^^b^ + 4?- 



For another example, let x'' — \bx^ — 229^ — 525 = O be an equation, 

 whose roots are sought. Here ^ ^b^ + -t/* = v' 101-^ =. »/ d, and the radius 



<- ,. . , ,.^-, i_ aV + i*;* + -i? 125 + 5724 + 2624^ 960 



of the circle = /405i; also JL—^—±^ _____ = __^_ 



= the tabular sine of an arch, whose log. is g.9736426, and the arch itself 70° 

 14' 22"; the third part of it is 23° 24' 47''4., and of its supplement, is 36° 35' 

 12*4; whose log. sines are 9.599183, and g.775275, to which adding the log. 

 of ^4054-, we have the log. O.903O89 = 8, and log. I.079I8I = 12, whose 

 sum is = 20. Hence we conclude that 20 + \b or 25 is equal to the affirma- 

 tive root, and 8 and 12 — \h, that is 8 and 7> are equal to the negative roots. 

 But if the equation had been c^ -\- ISo?^ — 229a; + 525 = 0, then 8 and ^ 

 would have been the affirmative roots, and 25 the negative : as for the other 

 cubic equations, which are explicable by one root only, they are to be resolved 

 by Cardan's rules, after the second term is taken away, nor do I see that the 

 thing can be done with less calculation. But if this root be desired to be ex- 

 pressed by the quantities by p, q ; in the first formula it is, ^^ + or — the 

 sum or difference of the cubic roots of 



^igq- T^fb^ + -^b'q - ibpq + ^p^ ± ^\b' + ^q - ^bp (viz. -f , 

 if -jV^* + \q be greater than -^bp, otherwise — ) the sum, when ^bb is greater 

 than p, the difference when less. And in the other formulas, the root always 

 consists of the same parts, changing the signs + ^"d — , as will be easily seen 

 upon trial. 



