A P P

A P P

penommator or Divisor is to be 22 (the doubleRoot of the fubstracted Square) or 224 1. (thatdouble Root increased by one) the true value fal-ling between these two; sometime the one.somc-time the other, being nearest to the true value.Rut (lor avoiding ot negative Numbers,) the lat-ter is commonly directed.

The true ground of the Ru’c is this; 22 being(by Construction) the greatest Integer Squarecontained in n, his evident that e must be lessthan t, (otherwise not aa, but the Square of a -j- 1,or some greater than it, would be the greatestInteger Square contained in n). Now if the re-mainder b—2ae-{- cc be divided by 22, the resultw iil be too great for e, (the Divisor being toolittle, for it should be 22-f -e to make the Quoti-ent e:j Rut if (to rectisie thisj we diminish theQuotient by increasing the Divisor, adding 1 coit, it becomes too litue, because the Divifor isnow too big. i< or ( e being less than 1) -f-1 is

more than za st- c ; and therefore too big.

As tor instance ; If the Non-quadrat proposedbe »= 5 , the greatest Integer Square thereincontained is <0152:4 (the Square of 2=2.) whichbeing fubstracted, leaves n- —22=5*—,40=15 szbzxtl*c~\~cc ■ which being divided by 1-255:4, givesv ; bur divided by 224 1=4+ 1 = 5 . gives ( :That too great, and this too little for c. Andtherefore the true Root (,i+ee=\ln) is less than24=2,25, but greater than 25 = 1, 2: And thisWas anciently thought an approach near enough.

If this Approach be not now thought near e.nough, the lame Process may be again repeated;and that as oft as is thought necessary.

Take now for a, 2(=2, 1, whose Square is4, 8455522 (now considered as an Integer in thesecond place of Decimal Parts) this fubstractedfrom 5,00. (or, which is the fame, 0.84, the ex-cess ot this Square above the former, front 1,which was then the remainder,) leaves a new re-mainder s-=o,,L ; which divided by 22=4.4,S lv « 1 : 44 =^5=0.03636-(-,too much.- But divi-ded by i.-i-f-1=4.5 it gives■ ? l /i'=,'h= 0 , 0 3555t-, too little. The true value (between theselWo ) being 2.226 proxime, whole Square is4 - 999696 . r

If this be not thought near enough, fubstractthis Square from 5.000000: The remainder £=°-oqo 3 o 4, divided by 225-4.472,, or by 2,2+ 1 =4 ' 473 , gives (either way ) o.000068—; whichoddest to 2=2.236, makes 2 236068—, lotne-what too big ■ but 2,2160674- would be much^ore too little.

Which gives us the Square Root of 5, adjusted£ o the sixth pi ace of Decimal Parts, at three steps.And by the fame method, if it be thought need-u > we may proceed further.

It were easie to compound the Process of twoth 'T 6 Ste P s inco one, and give (for the Rule)me Result of such Composition, which would' a ke it seem more intricate and mysterious, toamuse the Reader.

In the Cubicle Root, (consonant to the Quadra-llc jp the Rule is this .-

i n u° m lhe blck. proposed, (suppose n)

ltra * the greatest Cube in Integers thereinntamd, ( foppofe 222) the Remainder ( sup-er e ..— 3 'Utc-j- 2tec~\-occ) is to be the Numera-/• u ° a Inaction for designing the value of e,‘ e remaining part of the Root sought, a-{-c~

j/n.) To this Numerator, if (for the Denomi-nator or Divisor) we lubjoyn 322, the Resultwill certainly be too great for c, because the De-visor is too little : (For it should be 3224-3224cc, to give the true value of c.) It for the Di-visor, we take 3224-324-1, it wiil certainly betoo little, because the Divisor is too great. (Fore by construction is lei’s than 1.) It must there-fore (between these Limits) be more than thislatter ; and therefore this latter Result being ad-ded to 2, will give a Root whole Cube may befubstracted from the Non-Cubick proposed inorder to another step.

But if for the Diviibr, we take 3224-32, (oreven less than so) the Result may be tco great;or (in case b be small) it may be too little, andoft is lo.)

Which comes to pass from hence, because c(by Construction) is less than 1 ; and therefore 32cless than 32 ; and perhaps lo much, as that theAddition of ce will not redress it. And when itIb happens, 3224-32 is a better Divifor than3224-324-1, (or even somewhat less than ci-ther.) But because it doth not always lo hap-pen, (though for the most part it doth) theRule doth rather direct the other ; as whichdoth certainly give a Root Jess than the true va-lue, whole Cube may always be fubstracted fromthe Non-Cubick proposed. T he design being tohave such a Cube as (being fubstracted) mayleave another b to be ordered in like manner fora new Approach. But for the most part 322 may-be safely taken for the DivisorFor though theResult will then be somewhat too big, yet theexcess may be so small as to be neglected ; or arleast, we may thence easily judge what Number(somewhat less than it) may be lately taken;and if we chance to take it somewhat too big,the Inconvenience will be but this, that b for thenext step will be a Negative; of which Cafe weshall speak anon;

Thus for instance ; if the Non Cube proposedbe 9=w ; the greatest Integer Cube therein con-tained is 8=222 (whole Cubick Root is 2=2)which Cube fubstracted, leaves 9—8=i=H=3222-)-32-tc-I-ere. , This divided by 322=12,gives -14=0.083334~» too big for e; but thefame divided by 3224324 < = u 464 1 = 19,gives 4=0,052634, too little; or if bur by3224 32=12.4 6=18,it gives-(=-4=0,0 5 5 5 5 -st,yet too little. For the Cube of 24 o,c6,=2,06,is but 8,742—, which is short of 9; And somuch short ot it, that we may safely take 2,07,as not too big ; or perhaps 2 08, (which if icchance to be too big, it will not be much too big,(as shall be farther (liewn;) And upon tryal itwill be found not too big ; for the Cube of 2,08,is 8.998912.

If this Step.bc not near enough, this Cubefubstracted from 9,000000, leaves a new />=0,001085, which divided by 322=12,9796;gives 0,000084.—, which will be somewhat toobig, but not too much (for e is now so small, asthat 32c may be safely neglected (and ce muchmore;) so that it to 2.08, we add 0.000084—,the Result 2,080084 will be too big,but 2.080083will be more too little ; (as will appear if wetake the Cube of each,) 16 that either of themac the second Step, gives the true Root withinan Unite in the sixth place of Decimal Parts. .