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4. The cube root of 367061696 is required.
Note.-When there is a remainder after the periods of fig: ures are brought down ; annex periods of cyphers (three al once) and point the root for decimals, and proceed as far as you please in decimals; or you may find a denominator to the remainder by the following Rule, viz. Square the root and mul. tiply the square by 3: then multiply the root hy 3; add the ? products together, and their sum is a denominator to the remainder, which fraction must be annexed to the root to make it complete.
When there are decimals annexed to integers.. Rule.- Prepare the decimals that a point may fall upon units in the integers, then proceed as in casc firsti
Ans. 12.12. 2. The cube root of 1815.848 is required.
Ans. 12°2 3. The cube root of 1.442897 is required.
CASE III. To extract the cube root of a vulgar fraction. Rrle.-Extract the root of the numerator, for a new numerator; and the root of the denominator, for a new denominator ; if the fraction be a surd, that is, one whose rout cannot exactly be found, reduce it to a decimal and extract the root.
To extract the cube root of mixed numbers. RULE.-Reduce the vulgar fraction to a decimal, annex it to the integers, and proceed as in case second.
Cube root applied in finding the solidity of globular fig. ures, by having the diameter and solidity
of one given. RULEGlobes are in proportion to one another as the cubes of their diameters; therefore cube the diameter of the given globe, and also of the required globe; and then say as the cube of the diameter of the given globe, is to its solidity ; so is the cube of the diameter of the required globe to its solidity.
Examples. 1. If a cannon ball 6 inches in diameter weighs 25 Ib. ; I demand the weight of another of the like metal, whose diameter inches.
6X6X6 = 216. the cube of the given diameter. 3X3X3 = 27 the cube of the required diameter.
As 216 : 251b. : : 271b. : 3:12tlb, Ans. 2. If a ball of silver 12 inches in diameter is worth $600 ; I demand the value of another ball, whose diameter is 15 inches.
CASE II. Haring one side of a cubical figure given, to find the side of another, that will contain 2, 3, 4, 5 or 6 times as
much as the given one Rule.-Cube the given side, then multiply the cube by the number nientioned in the question, (if three times as large, multiply by 3, &c.) the cube root of the product will be the side required,
NOTE.--If it is required to make one which is 2, 3, 4, 5 or 6, &c. time less than the given one : divide the cube of the given side by the number mentioned in the question, and extract the root of the quotient; the root is the auswer sought.
l. There is a box that is 3 feet high, 3 feet long, and 3 feet wide ; I demand the side of another which shall contain 4 times the quantity.
feet. 3XüX3 = 27X4 = 31084•7 nearly Ansä 2. There is a box that is 4 feet wide, 4 feet high and 4 feet long; I demand the side of another that shall contain of the quantity,
Ans, 2 st.
To find the side of a cubical box that shall contain a quan.
tity equal to any given solidity. RULE.—The cube root of the given solidity is the side of a box that will contain the same quantity.
Examples. 1. There is a cylindrical cistern that contains 3204 solid inches; I demand the side of a cubical box, that shall contain the same quantity.
33204 = 14.741 in. Ans. 2. The side of a cubical cistern, that shall hold just as much liquor as a cask whose solid content is 255.19•1.196 inches, is required.
2.9.444 in. Ans.
REDUCTION OF COINS.
Under this head it may be thought necessary to say something relative to the former currency of the United States, before the FEDERAL CURRENCY was adopted.
In all the states, accompts were kept in pounds, shillings, pence, and farthings; but the value of these sev. eral denominations were different, in different states.
They all reckoned, 4 farthings to 1 penny, 12 pence to 1 shilling, 20 shillings to 1 pound.
For the benefit of those who may have retained the former method of keeping accompts, and wish to know the value of the same in Federal Money, the following rules are inserted,
The former currency of New Hampshire, Vermont, Mas. sachusetts, Connecticut, and Rhode Island reduced
to dollars, cente, &c. When the sum to be reduced is pounds only. Rule-Annex a cypher to the pounds, divide by 3, the quotient is dollars; if there is a remainder, annex three cyphers, continue dividing, the quotient will be dimes, cents, and mills; 6 shillings is equal to , of
Examples. 1. It is required to reduce 621 pounds, into. dollars,
6210 = 3 82070 Ans. 2. It is required to reduce 6.19 into dollars, dimes, cents and mills,
6.190.000 + 3 = 863:3337 Ans. 3. It is required to reduce 6.121 to dols. dimes cents,
Ans. $403.333t. 4. It is required to reduce 4.16 to dols. dimes, cents,
When the sum to be reduced consists of pounds, shillings,
pence and farthings. Rule. To the pounds annex half the number of shillings, and 2 cyphers in decimals, if the shillings are even; if they are odd, annex the greatest even half, and 5 tenths and one cypher in decimals; if there are pence and farthings in the suin, reduce the pence to farthings; observing to increase the sum by 1 if it exceed 12, and by 2 if it exceed 37; add the farthings, thus increased, in the place of tenths and hundreths, divide the whole by 3, the quotient is cents.
sas bit ede
Examples 1. Reduce 63 10s. 11 d. to dols. cents, and mills.
35.48+3=$11.8267 Ans: 2. Reduce 617 198. 11 d. to dols. cents, and mills.
To change dols, cts. &c. to the former currency of NewHampshire, Vermont, Massachusetts, Rhode
Island and Connecticut. RULE.. When the sum to be reduced is dollars; multiply the dollars by 3; double the right hand figure of the product for shillings; the remaining figures are pounds. If the sum to be reduced is dols. cts. and mills, multiply the whole by 3, point off 4 figures from the right for decimals; the rest of the product is pouuds; (note, if mills are not named in the sum, three figures only must be pointed off;) find the value of the decimal part by multiplying by 20 ; 12 and 4 ; observing to point off for decimals each time of multiplying; and the sums standing on the left of the separatrix will be shillings, pence and farthings.
Examples. 1. It is required to reduce $251 to lawful money.
251X33575 68. Ans. 2. It is required to reduce $9529 to lawful money.
9529X3=42858 148. Ads. 3. It is required to reduce $192 to lawful money.
192X3=657 128. Ans. 4. It is required to reduce $999 to lawful money.
6299 146. Ansi