tract 28.

and infinite series.

91

s 4- d~r the 1st root,

— \f — 3 = the 2d root,

— —- — - */ — 3 = the 3d root.

Z I i

45. The first of these roots, ^ or r, — s q- d —l/\b + \/(l> z + a 3 )] -\-%/[b — V(£* + « s )]> is that which iscalled Cardan’s rule, by whom it was first published, butinvented by Ferreus. And this is always a real root,though it is not always the greatest root, as it has beencommonly thought to be.

46. The first root r = s + d = 3 /[b + V (b z -f- a 3 )]+ l/[b — x / (b l + a 3 )], though it be always a real quantity,yet often assumes an imaginary form, when particular num-bers are substituted instead of the letters a and b, or p andq. And this it is evident will happen whenever a is nega-tive, and a 3 greater than b r . or (ip) 3 greater than (^q ) 1 ; forthen ^/(b z + a 3 ) becomes */{b l — a 3 ), the square root of anegative quantity', which is imaginary. And this will evi-dently happen whenever the equation has three real roots,but at no time else, that is in all the first 13 cases of theforegoing table, where (|jo) 3 is greater than (~qY, and p ne-gative ; the 4th and 13th only excepted, when (yp ) 3 is —(iqY, and therefore */[b 3 - a») = 0, and two of the rootsbecome equal, but with contrary signs. -.This root cannever assume au -imaginary' form when a or p is positive,nor y'et when p is negative and (ig) z greater than (yp ) 3 ; forin both these cases the quantity */(b z + as) is real, or thesquare root of a positive quantity. And these take placeafter the first 13 cases of the tabl£ of forms, that is, in allthe cases which have only one real root. So that this ruleof Cardan’s always gives the root in an imaginary formwhen the equation has no imaginary roots, but in the formof a real quantity when it has imaginary roots.

47. It may perhaps seem wonderful, that Cardan’s theo-rem should thus exhibit the root of an equation under theibrm of an imaginary or impossible quantity, always when