Page:Philosophical Transactions - Volume 014.djvu/183

Rh roots, and giving them an universal Cube root, it is. 3 3

vr48% 'l'V179§ 5f 'l' v I4-8%-~v1r8s619to 9 theroot fought. In the former Scheme, QB and QP. may signifie the roots of Cardam Binomial; that run infinitely upward, and terminate at Q, as is mentioned in Sectim fbe pb. And if they can be continued downwards, probably they will terminate at 0, and H. The $0146/E line in Section 2.-1. may here be represented by the line 9 S. and the Chord line between 9 and ¥> by if from whence tis plain that any root between 9 and 8 found near, may be limited by Approximations of Majm and Minus. As ru CAZ{DAlK§ RULES”

1 The description of the Low: is before handled. 2 The tout/5 line affording approaches by an [Equation derived ourof that proposed is before described, and the method of drawing is mentioned by DMWHM in theTranfactions-3-The Limits are of two kinds (fain) either the in/e limits when the resolvent is O~ and the equation falls a dcg1'€€ lower: or the dim/hc/1 limits whereby a pair of roots gain or loose their poffibility, as is before described. 4. Car/lon; canons are but the sum of the roots of a solid quadratic equation arising out of half the diar;{li¢l= limit as the v of the rectangle, and the resolvent as the fumm 5' If the roots of those lfimmmlf are separately Erickt down as ordinates on their rejolizfenigtlqey begat cur-réf infinitely continued upward, and meeting in a point bisecting the root that is equal to a 'pair of equal roots, when the equation is Juli: limited, or dioristick as aforesaid in the Figure at, Q 6 If these /fimwwlr are prickt down as ordinates to their rcfolvends, Mr. 1Qu/ran upon sudden thoughts, supposed they may describe both sides of an H)']'CTb0!¢. 7 If so they cannot be continued downwards, but by the method in Mercator: Logaritbmotcchnia: most: numbers of a conilant habitude belonging to any arithmetical progression, may by aid of the differences, and a Table of Figurative numbers (yea, and I add otherwise) be continued upward or downward, and if these run downward they will probably end both in the bafé limits at'O and R.

8 If these binomial Cf/5Y'U€5 be continued downward, and separately found should always added make the root of a cubick Equation capable of 3 roots: then Cardam impossible or negative ro0ts are prov'd possible, and we only in ignorance how to extract them.

9 Assume any root within the limits of 3 possible roots, and raise a resolvent to it, and when you have done, by Cnrclanu Rules improved; you may find that root, and, with a little