40 3 Dr. button on Cubic Equations 
manner from the.ta.ble of forms, which contains all the 
roots of every equation, thus: by a few trials I find 
. /3L = V_ji.’ , ihearly, and therefore iEEil = 1*854 is 
V 20-05., -V 10-05 A’ )6‘95/> 
the. le aft root, becaufe here n - 17*95 which lies far 
n&fl! ’ 7 4 < •# oj */ S * r : i.vil) 1 . : r • - 
above thp limit, for the leaft roots, which is at the fourth 
form, where n is = 9. And laftly, = \f~~- 
nearly, and therefore, l°± 5 2 S = 4-854 is the greateft 
9 * 43 /* 
root, becaufe 3 ~— S7 7 is found between the 1 2th and 1 3 th 
V 43 ? . 5 
forms, which are the limits between which lies the 
greateft root of every equation that has all its roots real. 
?f (\ 1 i | gff jj . ■ V 
39. Again, let the equation be A 3 + ix — 12. Here 
p - 2, and q - 125 hence - \/ %p - s/ and 
=-^/ 8 = 2 alfo; therefore /z - 3 = 2, or 
*#>+ 1 - 6, either of which gives zz - .5. Confequently, 
r= - ---- ? = '2, and the other two roots are 
* • ‘ - 
n+ i . p 
ft £ 
6 /. 2 p 
t — - 
6 
- -f r . 1 ± \/-n = - 1 . 1 ± n/~c = - i+\/-c. 
/ ( fjR if ;■ . . ; ' ? ,7 3 
40. But it is only by, trials that we find out a proper 
! • :f : • ii I ■ m 1 01 u» . < 
value for n in fuch cafes as thefe; and this is perhaps at- 
* ' ■ ' ‘ * > ' : : ' ' . j * ■ , ' ; f V* , 1 ■' ' '• .. ~ 
tended with no lefs trouble than the fearching out one of 
the roots by trials from the original cubic equation itfelf. 
This method of finding the roots would indeed be ef- 
j 1 . ■ ! : if.'. . !■ i 
feftual and fatisfaftory if we had a direct method of 
deter- 
