Volume 1

The elements of that mathematical art commonly called algebra, expounded in four books / By John Kersey.

John Kersey the elder

Date:: 1673

Credit: The elements of that mathematical art commonly called algebra, expounded in four books / By John Kersey. Source: Wellcome Collection.

270/350 (page 250)

$. / . 13. The fquare Root of the faW half Remainder is? -_ the leiTer part of the Root fought, to wit, . 'V*V'5 — VS* 14. I fay, the two parts in the eighth and thirteenth^ Reps, (the former of Which pahs is a Binomial, andC /, ~T~. 7-. j / —-7— the latter a Refidual) being fconnefted by fliill^ V• v5 v 3 • v • v 5 — v 3 i be the fquare Root fought j to vVit, . . . . j Which Root'anfwers to the irrational line which (in Prop.^2^ & 60. Uh.\o.Eltm End.) is called j a line containing in Power two Medial ReHangle i: And, if the Icfler part of the faid Root be fubtraaed from the greater by the interpolition of the fign_, it gives W) +^3: — V-V5 : which is the Root of the fixth Refidual and anfwers to the irrational line which i^y.lih. 10. Euclid. ) is called a line making with a Medial Re^angle a Whole S^ace Medial, The Proof of the Root above extraHed m of the fixth BinOmial.' The Squares of the parts of Wf +V3^ ‘“h VW5—V3 :? the Root fought, are . . . *.^ V5 2nd y J—>/3 Therefore the fumm of the faid Squares of the parts is > that is, The Produif of the parts multiplied one into the other is > 5_^: that is Jz The double of the faid Produft is ... , . * . . i . / y 8 Therefore the fumm of the faid futtfm of* the Squares of? , , the parts and double Product is Whence it is manifeft that -j- >^^8 is the Square of VV 5 + v 3 : + VWS~V3- therefore this is that fquare Root of the fixth Binomial: which Was to be proved. More¬ over, if the faid double ProduCt be fubtraaed from the faid fumm of the Squares of ihe parts, the Remainder —VS is the Square of V:V5 +V3: — V^ V5 ~V3 : ibcre- fore this is the fquare Root of that fixth Rclidual. Note, In every Binomial and Refidual conftituted according to the preceding i c* the fquare Root of the difference of the Squares of the Names or parts is efaual to the dif¬ ference of the Square of the parts of the Root of the Binomial or Refidual. ^ As in the firft Binomial V704> whofe (quare Root hath before been fount) 4-hV‘M the Square of 27, towif, 729, exceeds 704, the Square of V704i by 2 j * whofe fquare Root 5 is equal to the difference of the Squares of the parts of the Root of the Binomial propofed, to wit, the difference between 16 and II. This property may be demonfirated thus, let ^-f-W reprefent a Binomial Root whofe greater partis b^ then the Square of that Root is bbM2b^d-{-dy this divided into its Narap or parts makes the Binomial bb^\^d more aiv^. then the Squares of the parts of this Binomial are bbbb -I-' 2bbd~-\- dd and ^bbd^ and the difference of thefe Squares is bbbb — 2bbd-\-dd, whofe fquare Root hb — d is manifefily the difference of the Squares of the parts of the Root b<~\-^d firft propofed: which was to be (hewn. The like property may be demonftrated in a Refidual. How to extra& the Square Root out of a Binomial defignd by Letters^ if it hath a Binomial Root, By the fame general Rule which hath before been exercis’d in extra^lingthe fqaslre Root out or Binomials expreft by Numbers, we may extfad the fquare Root out of a Binomial defign d by Letters, when it hath a binomial Root^ as will be evident by the following - more apparent diftinftion of the parts of the given Binomial, initead of I fet the word [more} between the parts , .and inftead of — 1 fet the word [lejs] between the parts of a given Refidual^ Example ii Let it be required to extraft the fquare Root out of, this Binomial, . , * , . ..i > bb^d more 2b^d. The Operation^ . hb^\-d,') viz. from.. ^ ti Subtraa the Square of the leffer part i l^d, to wit ,* ! ■ ti From the Square of the greateC parr* (which fuppofe to be hbbh “|— 2 dbh “-I— dd „ , ^dbb$