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 ARITHME TICS. 421 the right of the firft part; and you have the divifor com- Examp. III. Required the fquare root of .2916. plete. IV. Multiply the divifor thus compleated by the fi.2.916(.54 root. gure put in the quot, fubtratt the product from the re25 folvend,' and to the remainder bring down the following 104) 416 period for a new refolvend, and then proceed as before. 416 Note 1. If the firft part of the divifor, with unity fuppofed to be annexed to it, happen to be greater than the refolvend, in this cafe place o in the quot, and alfo If the fquare root of a vulgar fraction be required, on the fight of the partial divifor; to- the refolvend find the root of the given numerator for a new numebring down another period; and proceed to divide as rator, and find the root of the given denominator foran(a new denominator.$ Thus, before. an< the fquare root of 4 is 4> I Note a. If the product, of the quotient-figure into the the root of i r > I thus the root of y (=65) is divifor happen to be greater than the refolvend, youmuft t— 2-j-. But if the root of either the numerator or denominago back, and give a lefler figure to the quot. Note 3. If, after every period of the given number is tor cannot be extracted without a remainder, reduce the brought down, there happen at laft to be a remainder, vulgar fraction to a decimal, and then extraCt the root, you may continue the operation, by annexing periods or as in Example HI. above. pairs of ciphers, till there be no remainder, or till the II. Extraction of the Cube Root. decimal part of the quot repeat or circulate, or till you Rule I. Divide the given number into periods of think proper to limit it. Examp. I. Required thefquare root of 133225. three figures, beginning at the right hand in integers, and pointing toward the left. But in decimals, begin at the place of thoufands, and point toward the right. The Square number 133225(365 root 365 number of periods ftiews the number of figures in the 9 365 root. II. Find by the table of powers, or by trial, the 1 div. 66) 432 refolvend. 1825 neareft lefler root of the left-hand period; place the fi396^produ6t. 2190 gure fo found in the quot; fubtrad its cube from the 1095 faid period; and to the remainder bring down the next 2 div. 725) 3625 refolvend. 3625 produ6t. 133225 proof. period for a dividual or refolvend. Examp. II. Required the fquare root of 72, to The divifor confifts of three parts which may be found as follows. eight decimal places. III. The firft part of the divifor is found thus: Multiply the fquare of the quot by 3, and to the pro72.00000000(8.48528137 root. duct annex two ciphers; then inquire how often this firft 64 part of the divifor is contained in the refolvend, and place the figure denoting the anfwer in the quot. 164)800 IV. Multiply the former quot by 3, and the product 656 by the figure now put in the quot; to this laft product annex a cipher; and you have the fecond part of the di1688)14400 13504 After getting half of, the de- vifor. Again, fquare the figure now put in the quot for cimal places, work by contract- the third part of the divifor; place thefe three parts uned divifion for the other half; der one another, as in addition; and itheir fum will be 16965)89600 848 and obtain them with the fame the divifor complete. accuracy as if the work had V. Multiply the divifor, thus completed, -by the figure laft pur in the quot, fubtraCt the product from the 169702)477500 been at large. refolvend, and to the remainder bring down the follow3394°4 ing period for a new refolvend, and then proceed as before. 169704)138096 •••• 135763 Note 1. If the firft part of the divifor happen to be equal to or greater than the refolvend, in this cafe place o in the quot, annex two ciphers to the faid firft part of the divifor, to the refolvend bring down another period, and proceed to divide as before. 636 Note 2. If the product of the quotient-figure into the 509 divifor happen to be greater than the refolvend, you muft go back, and give a lefler figure to the quot. 127 Note 3. If, after every period of the given number is 118 brought down, there happen at laft to be a remainder, you may continue the operation by annexing periods of three ciphers till there be no remainder, or till you have Vol. I. No. (9)18. 5O as 3