roots, and giving them an univerfal C«Je root* it is 
3 3 
Vf48i fvi79f6f fv i4S^—vi78^<Ji9to 9 the root fought. 
In the former Scheme ^B. and .^P. may fignifie the roots 
of Cardans Binomials that run infinitely Upward and terminate 
at Q^as is mentioned in SetUonthe ^th. And il they can be 
continued downwards^ probably they will terminate at O. 
and ft'. The t'^uck line in Sediou 2.I may here be reprefented 
by the line 9 S. and the Chord hne between 9 and t by 7. from 
whence tis plain that any root between 9 and 8 found near, 
may be limited by Approximations of Majus and Mwm, 
As to CARDANS RTJLZS 
1 The defcription of the Loa/j is before liandled. 
2 The tcf4ch line afFordidg approaches by an ^Equation de- 
rived out of that propofed is before defcribed, and the me- 
thod of drawing is mentioned by IdvJ'^aihs in theTranfadiions, 
3. The Limits are of two kinds {viz.) either xht Baje li- 
mits when the refolvend is o. and the ^equation falls a degree 
lower : or the diorifiick limits whereby a pair of roots gain or 
loofe their poffibility, as is before defcribed 
4 Cardans canons are but the fum of the roots of a folid 
quadratick asquation arifing out of half the dionfttck limit as 
the V of the rectangle, and the refolvend as the fumm 
5' If the roots of thofe bimmtaU are feparately pricl^ down 
as ordinates on their re/o/w«r/j,they beget cur'ie^ infinitely con- 
tinued upward, and meeting in a point bifedting the root 
that is equal to a pair of equal roots, when the sequation is 
juil limited, or dioriftick as aforefaid in the Figure at jS^ 
6 If ^tk, binomials are prickt down as ordinates to their re- 
folvends, Mr. Ntimon upon fudden thoughts, fuppofed they 
may defcribe both fides of an Hyperbole, 
7 If fo they cannot be continued downwards, but by the 
method in Mtrcators Lcgartthmotechnta : moft numbers of a con- 
ftant habitude belonging to any arithmetical progreffion, 
may by aid of the differences, and a Table of Figurative 
numbers (yea, and I add otherwife) be continued upward 
or downward, and if thefe run downward they will proba- 
bly end both in the baje hmits at O and R. \ , 
8 If thefe binomial c^rw be continued downward, and fe- 
parately found Ihould always added make the root of a cu- 
bick -Equation capable of 5 roots : then Cardans impoflSble or 
negative roots are prov'd poffil3le, and we only in ignorance 
how to extradt them. 
9 Afltime any root within the limits of 3 poifible roots, and 
raife a refolvend to it, and when you have done, by Cm-dan's 
llules improved, you may find that root; and, with a httle va- 
I'y ng 
