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.

318/350 (page 298)

$From the three laft Equations tis evident, that the three defired numbers a, y may be either 9, 15 , or gf-, 8|-, and 13^ : For firft, 5, 9, 13 are in Arithmetical Pro- greffion; and if 5 be multiplied by 1, 9 by 2, and 13 by 3 ^ the fumra of the three Produds is 62 . moreoveri the fumm of the Squares of 9,13 makes 2 75;j as was required. The like may be prov^ by 3 f-, 8and 13 J^BST. zo. To find three fuch numbers, that the Square of the firft being added to the Product of the firft multiplied into the fecond may make the fumm 48; alfo, that the Square of the firft being fubtrafted from the Product of the firft multiplied into the third, the Remainder may be 3 2 ^ and that the fumm of the Squares of the firft and third may have the fame proportion to the Square of the fccofid as 5 to 2. Let the three numbers fought be reprefented by 4 j e, 7, and then thcQueftion may be ftated thus, ' I. If . i . : : : . V . : ; 1. And . .. 3* Aiud • . • .(•' What arc the numbers 4, e, 7 ? RESO LVTIO N. 4^ From the firft Equation by tranfpofition ofj y this arifeih, . 5* And by dividing each part of the laft Equa-i tion by it gives . . .. 6. And by tranfpofition of — aa in the fecond? Equation , it makes.' • o 7. And by dividing the fixth Equation by , > there arifeth . ...0 8. From the Analogy in the third ftep, by com¬ paring the Produd of the extremes to the Pro-, dud of the means, this Equation arifeth, 9. The Square of the feventh Equation is . . )> » _ • At rrr 48 ;• . ay — aa z=z 2^ • ce 5 , l at — ^8-~^aa ,-Al aa ay — aar\* 2^ ^ aa -1— 2 2 ; =-!- a ^ee ■=. iaa-\-^2yy jy rr: 6^aa-\-a‘^ 10. The double of the ninth Equation is . ^ J> 11. If inftead of lyy in the latter part of the eighth Equation there be taken the latter part( of the tenth, the eighth will be converted into( this, 'vvzjt . ...... 12. The Square of the fifth Equation is . . .9 . / 13. The twelfth Equation multiplied by 5 gives {>.1. 14. From the eleventh and thirteenth Equations, by comparing their latter parts one to the other,' and reducing the Equation thereby refulting, this Equation arifeih, viz,. 3 5. Which Equation in the ftep being refol- ved by the Canon in SeU.io. Chap.i j. £00^1. will difeover two values of a, viz, . . . 3 5. But the leffer of thofe two values of a, to^ wit, 4, is the firft number fought by the Que- ftion, for the Square of the greater value exceeds 48 , but according to the fuppofition 9 in the firft ftep it ought to be Icfs than 48 • ] 1 fuppofing then ^ = 4, it follows from the fifth 11 ftep, that.. . J 37. Laftly, from the 1 •^th and yth Equations, 9 So three numbers are found out, to wit, 4, 8 and as may ealily be proved. V) ^ee _ 'i-oa8-\^\2 8aaJ^7af^ aa _ 2 04 8 12 8aa-\^^a'^ aa ee _ 2304 — g6aa a* aa. ^ee 115-20—48 aa 6o8aa—a'^— 9472 4 = \^S9z or 4 c = 8 12 7—12 which will fatisfie the Queftion,$