Volume 1

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

  • John Kersey the elder
Date:
1673
Catalogue details

Licence: Public Domain Mark

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

    264/350 (page 244)
    Now if any Binomial or Refidual be given, we may eafify find out another of the fame kind in this manner, viz. For the firfi: and fourth Binomials, if it t?e made as the greater Name or Part to the leller, fo any Rational number affumed for the greater Part of a new firfi; or fourth Binomial, to a fourth Proportional number, this number Iball be the lefier Part of the new firfi or fourth Binomial. But for the fecond and fifth, if it be made as the lefler Part to the greater, fo any Rational number taken for the lefier Part of a new fecond or fifth Binomial to a fourth Proportional, the number fo produced fhall be the greater Part of the new fecond or fifth Binomial. And laftly, for the third and lixth Bi¬ nomials, if it be made as the greater Part to the lefier, (each of which is a Surd fquarc Root,) fo any Surd fquare Root aflumed for the greater Part of a new third or fixth Binomial, to a fourtfi Proportional, there will, come forth the lefier Part of a new third or fixth Binomial. (The reafon of this Operation is manifeft , per Prep, i Elem.io, EhcIU, ) And , after a new Binomial is found put, its correfpondent Refidual is alfo made, by changing theTign into —, as before hath been faid. As, for example, if a firfi Binomial 3 J be propofed, to find another like to it ♦ I take a R.ational number at pleafure,as,8,for the greater Parc of the Binomial fought; then by the RuU' of Three, as 5 h(to v'5, fo 8 to a fourth Proportional, to wit, > for foe lefier Part fought, therefore 8 fiiall be a new firfi Binomial, and b .— a new firfi Refidual j and fo of the reft. Sc (ft. XVI. Concerning the extra&ion of the Square Root out of Binomials afzd Rejiduals confhtuted in fetch manner as hath been fietpn in the ceding Se(ft. 15. Every one of the Binomials and Refiduals whofe Conftr-uflion hath been (hewn in foe preceding Sed:. hath a fquare Root, that is, fuch a Binomial or Refidual that if it be mnltipli^ into it felf will produce the given Binomial or Refidual j as may be evidently pollened out ol Prop. 55,56,5-7,58,5-9, and 60, Alfo out of Prop. and-py. of* the Tenth Book of Emli^% Elements. As, for example, a Binomial of the firfi kind, fuppofe 74-^4^ hath a fquare Root, to wit, 2 Vs ; •for this being fquared (or multiplied into it felf) produceth that Binomial 74^48; whofe greater Part 7 is compofed of 4 and 3 the Squares of the Parts of the Root ; and the lefier Part V4S is the double of the Produft made by the multi¬ plication of z into v'3, the Parts of the Root 2 V? = all which is evident by the mul¬ tiplication of 1 -j-V? into it felf. The likeeffedi will be found in every one of the reft of the Binomials confiituted in the preceding SdP. 1 Therefore if a Binomial be propofed, and its fquare Root defired, there is given the fumm of the Squares.of the Parts of the Root ,♦ ( which fumm is the greater Part of foe Binomial propofed;) and the double of foe Pro- dudl of the Parts ot the Root, ( which double Produdf is the lefier Parc of the Binomial propofed,) to find out the two Parts of the Root feverally. And therefore in order to the Extraftion of the fquare Root of a Binomial, it will be requifite to feaich out a Canon for the folving of this following XP £ ST. The fumm ( ^ ) of the Squared of two numbers being given ] as alfo ( c ) the double Produdl of the multiplication of the fame two numbers 5 to find the numbers feverally. RESOLVTION. z c c 3. .Which Produdl divided by the firfi number a gives the other number. 0 9 4i Therefore the Square of foe firfi number is . . ... . j> • • cc 5. And the Square of the other number is. 6. Therefore the fumm of the vSquares of foe two numbers is . 7. Whifo' #