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 88 algebra. In extraS ir,? roots it will often happen that the exa& ,f found lefs than the two firft periods of the given numroot cannot he 1found in finite terms; thus the fquare “ her, the fecond figure of the root is right. But, if “ it be found greater, you muft diminifh the fecond firoot of <* * ~j- * is found to be “ gure of the root till that power be found equal to or 7. &c. 5^ a + 2ti 8a} “ lefs than thofe periods of the given number. Sub-< i6a* 128a * “ traft it, and to the remainder annex the next period; The operation is thus; “ and proceed till you have gone through the whole ginumber, finding the third figure by means of the <bc. ““ ven (a+— — 8a * two firft, as you found the fecond by the firft; and “ afterwards finding, the fourth figure (if therf be a “ fourth period) after the fame manner from the three “ firft.” Thus to find the cube root of 13824, point it 13824; find the greateft cube in 13, viz. 8, whofe cube foot 2 is the firft figure of the root required. Subtrad 8 from 13, and to the remainder 5 annex 8, the firft figure of the fecond period; divide 58 by triple the fquare of 2, viz. 12, and the quotient is 4, which is the fecond figure of 'a 8a') the root required, fince the cube of 24 gives 13824, the number propofed. After the fame manner the cube root 8a') 4a* 8<8a4 ^ 64ae of 13312053 is found to be 237. 13824 (24 Subtr. 8 = 2x2x3 £C The general theorem which we gave for the invq3x4=12) 58 (4 “ lution of binomials will ferve alfo for their evo}uSubtr. 24x24x24=13824 “ tion;” becaufe to extrad any root of a given quantity is the fame thing as to raife that quantity to a ... o. „ power whofe exponent is a fradion that has its denomi-. In extrading ofRem. after you have gone through nator equal to the number that exprefles what kind of the number propofed,roots, if there remainder, you may root is to be extraded. Thus, to extrad the fquare continue the operation by addingis aperiods of cyphers to root of a b is to raife a b to a power whofe expo- that remainder*, and find the true root in decimals to any nent is The roots of numbers are to be extraded as thofe of degree of exadnefs. algebraic quantities. “ Place a point over the units, “ and then place points over every third, fourth, or Chap. VIII. Cy Proportion. “ fifth figure towards the left hand, according as it is “ the root of the cube, of the 4th or jth power that is “ required; and, if there be any decimals annexed to When quantities of the fame kind are compared, it “ the number, point them after the fame manner, pro- may be confidered, either how much the one is greater “ ceeding from the place of units towards the right- than the other, and what is their difference; or, it may “ hand. By this means the number will be divided in- be confidered how many times the one is contained in “ to fo many periods as there are figures in the root re- the other; or, more generally, what is their quotient. “ quired. Then inquire which is the greateft cube, The firft relation of quantities is expreffed by their a“ biquadrate, or yth power in the firft period, and the rithmetical ratio ; the fecond by their geometrical ra“ root of that power will give the firft figure of the root tio. That term whofe ratio is inquired into is called the “ required. Subtrad the greateft cube, biquadrate, or antecedent, and that with which it is compared is called “ 5th power, from the firft period, and to the remainder the confequent. “ annex the firft figure of your fecond period, which When of four quantities the difference betwixt the “ fhall give your dividend. firft and fecond is equal to the difference betwixt the “ Raife the firft figure already found to a power lefs third and fourth, thofe quantities are called arithmetical “ by unit than the power whofe root is fought, that is, proportionals as the numbers 3, 7, 12, 16. And the “c to the 2d, 3d, or 4th power, according as it is the quantities a, a--b, e, effb. But quantities form a fe‘ cube root, the root of the 4th, or the root of the ries in arithmetical proportion, when they “ increafe or “ jth power that is required, and multiply that power “ decreafe by the fame conftant difference.” As thefe, “ by the index of the cube, 4th, or 5th power, and di- a, a+b, a--2b, a+33, rf+4£, 'be. x, x—b, x—2b, be. “ vide the dividend by this produd, fo ftiall the quo- or the numbers 1, 2, 3, 4, J, be. and 10, 7, 4, 1, —2, “ tient be the fecond figure of the root required. —y, —8, be. “ Raife the part already found of the root, to the In four quantities arithmetically proportional, “ the “ power whofe root is required, and if that power be “ fum of the extremes is equal to the fum of the mean “ terms.”
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