400 Dr. hutton on Cubic Equations 
3 1 . From the bare infpeclion of this table feveral ufe- 
fnl and curious obfervations may be made. And firft it 
appears, that when q is pofitive, as in all the forms after 
the i ath, r is the greateft root; but when q is negative, 
or in all the cafes to the 1 2th, r is one of the lefs roots. 
32. In all cafes before the 4th form r is the lead: root, 
becaufe or &c, is always greater than 1 ; 
and in albfuch forms \q is lefs than \ but the for- 
mer approaches nearer and nearer to an equality with the 
latter till the 4th form, where ftp is become = fp\ and 
r is then equal to one of the other roots, becaufe 
^ii = 2 -= 1. 
2 2 
33. From hence r becomes the middle root, and con- 
tinues foto the 1 ath form, where it becomes equal to what 
has hitherto been the greateft root, and the other root 
becomes at this place = o ; and fp has decreafed from 
the 4th form all the way more and more in refpedt of 
till at this 1 2th form it has become = o,or infinitely 
lefs than 
34. From this place r becomes the greateft root, the 
fign of q changes to + , and fp again increafes in refpedt 
of fp , till at the 13th cafe it becomes again equal to it, 
and the two lefs roots equal to each other, like as at the 
4th form. 
3 
35. From 
