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If there should not be so many figures in the quotient as there should be decimals, prefix ciphers on the left hand to make up the number.
Example. -Divide 1.4850 by 247.5.
Thus, 247.5 = .006. And if there be not as many decimal figures in the dividend as in the divisor, you may annex a sufficient number of ciphers; and if there be not a remainder, you must add ciphers to the right hand of the quotient till you have taken as many in the dividend as will make the decimal figures therein equal to those in the divisor: thus,
A Table of the Fractional parts of an Inch when
divided into Thirty-two Parts; likewise a Foot of Twelve Inches reduced to Decimals.
.46875 11 ☆ & tó=.9375 s & T=.4375 10
.90625 $ & .40626 9
.8125 1 & t=.3125 6
4 s & t=.71875 Š & z' .21875 3
& tó = .6875 ☆ & t=.1875 2
.65625 f & ;= .15625 1
The utility of this table will appear evident by means of the following example :
Suppose a board, or plate, to be 304 inches long; 84 inches broad; and & t of an inch in thickness; required its content in cubic inches.
30.25 X 8.625 = 260.90625 X .4375 = 114,146, &c. cubic inches.
OF THE SQUARE ROOT.
When a number is multiplied by itself, as 6 x 6, or 9 x 9, &c., it produces the square or second power of that number; and the number itself is called the root of that square.
A root consisting of a single figure is found by inspection of the following table :
To Extract or find the Square Root of any Number,
which consists of more figures than one.
Rule.- Make a point or dot over every second figure, commencing at the right hand, by which the given square will be pointed into periods of two figures each, except the first or left hand period, which will sometimes have but one.
The unit figure must always be the latter figure in the perod; for the decimal point must be between the periods, and not in the middle of a period.
Find the greatest root in the first period, which write in the quotient or root, and the square thereof under the same period; subtract therefrom, and to the remainder annex the next period for a dividend.
Double the quotient for a divisor, ask how oft the divisor is contained in the dividend, with this consideration, that the answer must be the unit's figure of the divisor.
Write the answer in the quotient, also in the unit place of the divisor; then multiply the divisor, so completed, by the last quotient figure; write the product under the dividend, and subtract therefrom; to the remainder annex the next period for a new dividend.
Thus proceed with every period; and if there is still a remainder, annex pairs of ciphers for additional periods, till you have a competent number of decimals in the root.
Vulgar fractions, &c., may be reduced to decimals.
The periods which are whole numbers give whole numbers, and decimals periods give decimals in the root.
Ex. 1.—What is the square root of 76176 ?
76176(276. or 276 x 276 = 76176.
Ex. 2.-Required the root of .75.
OF THE CUBE ROOT.
When a square is multiplied again by its root, as 6 x 6 x 6, it produces the cube or third power of that root. · Single cubes are found by inspection of the preceding table. To Extract the Root of any Number that consists of
more than one figure. RULB.-Point the given cube into periods of three figures, and so that the unit figure be the last in its period; then from the first period subtract the greatest cube it contains; put the root as a quotient, and to the remainder bring down the next period for a dividend.
Find a divisor by multiplying the square of the root by 300; see how often it is contained in the dividend; and the answer gives the next figure in the root.
Multiply the divisor by the last figure in the root. Multiply all the figures in the root by 30, except the last; and that product by the square of the last. Cube the last figure in the root; add these three last found numbers together, and subtract this sum from the dividend; to the remainder bring down the next period for a new dividend, and proceed as before. EXAMPLE. -Required the cube root of 444194947.
Involution and Evolution of numbers are very conveniently performed upon the Engineer's Slide Rule; for when the slide is set straight at both ends, C is a line of squares, and D a line of roots; consequently, against any number upon D is its square upon C, and against any number upon C is its root
EXAMPLE 1.-What is the square of 16 ?
EXAMPLE 2.-Required the square root of 625.
Opposite 625 upon C is 25 upon D, the root required.
The cube root is performed by inverting the slide, and setting the number to be cubed upon B to the same number upon D, and against 1 or 10 upon D is the cube number upon B. Also set the cube number upon B to 1 or 10 upon D, and where two numbers of equal value meet upon the lines B and D is the root required.
EXAMPLE 1.-Required the cube of 9.
Set 9 upon B to 9 upon D, and against 10 upon D is 729 upon B.
EXAMPLE 2.-Required the cube root of 343.
Set 343 upon B to 10 upon D; and against 7 upon B is 7 upon D, the root required.
These lines also serve to multiply the square of any number, any number of times; thus,
To find the product of 6 times 6, multiplied
Set 3 upon B to 6 upon D, and against 10 upon D is 108 upon B.