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.

289/350 (page 269)

$But that the Learner may the more eafily perceive my meaning, J fliill premife a few Definitions in three Sedions next following. 11. When the known Abfdlute number in an Equation folely pofielTeth one part thereof, let it be transferr’d to the other part by the fign —, and then there will be an Equation which hath o or nothing for one part, and the other part is by Cartefim called the fiumm of the Equation pf opofed. As, for example, if this Equation be propofed, viz,. aaa — 94a-\-^z6a ^ 24, by tranfpofition of 24 it makes aaa — 9aa-\-26a — = o , whofe firft part is called the Sumra of the Equation propofed. I I f, In the Equations handled in this Chapter, 1 put a, e or j/ tofignifie an unknown Quantity j and by the firft Term of an Equation is meant the higheft unknown Power to wir, that which hath itioft Dimenfions or Degrees of 4 ; by the fccond Term that which hath fewer Dimenfions by one than the firft , and fo downwards. As in this Equation; aaa — ^aa-j^zCa—24 = o , the firft Term is 444 , whofe Index is 5 j the feccnd Term is ^9aaj where the Index of 44 is 2 5 the third Term is 4 2^^,’ where the Index of 4 is i j and the laft Term is —24, the known Abfolute number whofe Index is o. IV. Equation are of thfee kinds, vtz,. either Affirmative, or Negative, or Impoffible: an affirmative Koot is a quantity greater than nothing , as A- 5 or ^1-^2 0 : a negative Root (which Caru/iHs calls a falfe Root ) expreflethVquantity whofe Denomination is oppolite to an affirmative • as _ 5 , or — 20 . the former of which wants $ , and the latter 20 of being equal to nothing: laftly, impoffible Roots are fuch whofe values cannot be conceived or comprehended either Arithmetically or Geo¬ metrically : As in this Equation , 4 — 2 — ^ _ i, where V — i, that is, the fquare Root of — I is no manner of way intelligible j for no number can be imagined , which being multiplied by it felf according to any Rule of Multiplication, will produce — i. V. Thefe things premifed, I ftiall proceed to the forming of Equations which lhall have many Roots. ♦ ^ PROP, I. To form an Equation which fjall have two /Iffi) maiive Root si — 2 that is, 4—2 = that is, 4 — 3 = o o xi Suppofe ; . . . . . . . ^ ~ 2. Then by multiplying the faid 4 — 2 = o by 7 4 — 5 = o, this Equation is produced, ^ ~ o .3. That is, by tranfpofition j.^.^$a~-aa ^ 6 Which laft Equation falls under the laft of the three Forms in SeU:. Chap. i 5. and may be expounded by either of two Roots or values of a, which by the Canon in' SeU.io. of the fame Chap, will be found 2 and 3 , to wit, thofe from Which the faid Equation was produced by Multiplication, as above. Again, if this Equation 44 64 — 5 5 = o, ( that is, 44 + ^4 ='55,) which hath one affirmative Root, to wit, 5, be multiplied by 4—-5 = o , there will be produced 444 — 9i4*-[-'33o =: o, (that is, ^i4 — 44^ — 330,) which hath two affirmative Roots or values of 4, to wit, 5 and 6 , which may be found out by the Rule hereafter deli¬ vered in Sed:, 9. of this Chapt. PROP. 11/ To form an Equation which frail have one Afirmaiive^ and one Negative Root,, 1. Suppofe . . . . . . . ; . ^ — ^5 that is; 4—3 = d C a — —2y that is, 4-1-2 = o 2, Then by multiplying the faid 4 - 3 r= o by 7 4-1- 2 =r o, thisEquation is produced, 3* That IS, . 44 Which laft Equation falls under the fecond of the three Forms in Sed. i. Chap i 3 . Book_ I. and may be expounded by either of tvvo Roots or values of a, whercol one is Affirmative, and the other Negative ; which ,' after the manner of rcfolving i; in Sdd.']. of the fame Chapi. will be found -|- 3 and — 2 , to wit, thofe from which the faid Equation was produced by Multiplication; as beffire. PROP. 4 — 6 — 0 a — 6$