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.

343/350 page 323

$19. Whence by tranrpofuion of 3«, this Equation? __ atifeth,..0 ^ 20. Then by fetting the latter part of the Equation \t\ ? the nineteenth ftep in the place of j in the thirteenth, > = 8 this arifeth,.. . .3 21. Whence, after due Redudlion, this Equation arifeth, > e — 7h — 2 22. From the latter part of the nineteenth Equation, it? _ t may be inferrd that.^ ^ 23. 'And from the latter part of the twenty-firft Equa-5 ..s ^ tion tr* I Now fince by the twenty-fecond and twenty-third Reps, « ( or the number of Girls) IS found to fall between 1 and 3| , let 2 be taken for the value of «, viz. 24. Suppofe..)> « = 2 2^, Then from the nineteenth and twenty-fourth fteps, J> jy = 4 ( — 10 — 3») 2 5. And from the twenty-firft and twenty-fourth fteps, <? = 2(r=2/# — 2) Thus by the eleventh , twenty-fixth , twenty-fifth and twenty-fourth fteps , four whole numbers are difeovered, to wit, 12, 2,4 and 2, for the values of and «. Again, by taking 3 for the value of « , ( which is within the limits before difeovered) the nineteenth and twenty-firft fteps will difeover i and 4 for the values of y and e,{a being 12 , as before. Wherefore two Anfwers to the Queftion are found out 5 for the number of Men being put 12 , the number of Women will be 2, the number 01 Boys 4, and the number of Girls 2 • or the number of Men being 12 as before , there will be four Women, i Boy and 3 Girls. Again, if 11 be put equal to A, (or the number of Men,) and the procefs be repeated from the eleventh ftep to the end of the Refolution, there will be found two Anfwers more in whole numbers. In like manner, if 9^ o and 13 be fcverally put equal to a, three Anfwers rao^jwill be difeovered; But if 8 and 14 be fcverally put eqilal to a, although they be within the limits in the eighth and tenth fteps, yet the work being repeated as before will not fucceed to find e, j and h in whole numbers • fo that there arc only feven Anfwers, to wit, thofeinferted in the Table • but that every one of them will folve the Queftion may eafily be proved. If a Queftion of this nature be delired that hath butoneAnfwer in whole numbers^ let the number of perfons be 60, and 100 the number of (hillings fpent ; alfo let every Man fpend 2 (hillings, every Woman f of a (hilling, every Boy j- of a (hilling, and every Girl t of a (hilling; then by forming the Refolution as before, the number of Men will be found 46 , the number of Women 3 , the number of Boys y, and the number of Girls 6. a e JL. u 9 9 1 I 10 6 3 I 11 S 2 2 II 3 5 I 12 z 4 2 i 2 4 I 3 1 1 3 3 J^EST. 23. . . To divide 200 into five fuch whole numbers, that if the firft be multiplied by 12, the fecond by 3, the third by i, the fourth by ^nd the fifth by j, the fumm of the Pro- duds may alfo make 200. This Queftion may be refolved like the foregoing twentyeth and twenty-fecond, but I (hall leave it as an exercife to the induftrious Analyft , who , (if he thinks it to be worth his pains,) may find out 6639 Anfwers to it in whole numbers, '(as Sachet y in the two laft pages of his little Book before cited in Se^. i. of this Chapter, doth affirm. .Nicholas Tartaglia handling this very Queftion, ( which is the la(f of the feventcenth Book of the firft Part of his Arithmetick,) thought it a great matter that he had found out one fingle Anfwer to it in thefe five whole numbers, to wit, 6, 12, 34, 72, 95, and alTerted, That Queftions of this fort could not be perfedly folved, either by the v^lgebrakal Art, or any certain Rule ; but the contents of this Chapter do manifeftly (liew, that the Imperfedion was in the Artift, and not in the Art. ihc End of the Second BOOK* , ,$