denotes that the cube root of 27 is to be extracted; and 81=4th root of 81.

When the power is expressed by several numbers, with the signs + or between them, a line or vinculum is drawn from the top of the sign over all the parts of it. Thus, the square root of 30-5 is expressed √30—5.

A TABLE OF THE SQUARES AND CUBES OF THE NINE DIGITS.

EXTRACTION OF THE SQUARE ROOT.

To extract the Square Root of any number, is, to find a number which, being multiplied into itself, shall produce the given number.

RULE.

1. Point off the given number into periods of two figures each, by putting a dot over the units, another over the place of hundreds, and so on; and if there be decimals, point them in the same manner from units towards the right hand, which dots will show the number of figures the root will consist of.

2. Find the greatest square number in the first, or left hand period; place its root as a quotient in division, and place the square number under the period and subtract it therefrom; and to the remainder, bring down the next period for a dividend.

3. Double the root already found, and place it at the left hand of the dividend for a divisor.

4. Place such a figure at the right hand of the divisor, and also the same figure in the root, as, when multiplied into the divisor thus increased, the product shall be equal to, or next less than the dividend; this will be the second figure in the root.

5. Multiply the whole increased divisor by the last figure of the root; place the product under the dividend;

subtract it therefrom, and to the remainder bring down the next period for a new dividend.

6. Double the figures already found in the root for a new divisor; and from these find the next figure in the root as last directed; and continue the operation in the same manner until all the periods are brought down.

Note. When there is a deficiency in any period of decimals, you may annex a cipher; or, when there is a remainder, you may continue the operation to decimals, by annexing periods of ciphers.

EXAMPLES.

1. What is the length of one side of a square field which contains 1225 square rods? or what is the square root of 1225 ?

Operation

1225(35

9

65)325
325

Illustration. By the Rule, we point the given number into periods of two figures each, by putting a dot over the unit's place and another over the place of hundreds, making 2 periods, which show that the root will consist of 2 places of figures, viz.: a ten and a unit. This Rule for determining the number of figures of which the root will consist is founded on 00 the known principle that the places of figures in the product of any two factors cannot exceed the number of places contained in both of those factors, nor can they be but one less than the places in both factors: and as the square root of any power is a factor which, being multiplied into itself, exactly produces that power, consequently, any square number contains just twice as many places of figures as its root, or at least, but one less than twice that number. Then the first period being 12 (hundreds,) we seek for the greatest square number that is contained therein, which we find to be 9 (hundreds,) the root of which is 3 (tens 30.) We therefore place 3 (tens) in the root, and its square 9 (hundreds,) under the period 12 (hundreds,) which being deducted, the remainder is 3 (hundreds,) to which we join the next period, 25, making the dividend 325. These 3 (tens-30) in the quotient, or root, it will be recollected, are the length of one side of a square field which contains 9 (hundred) square rods, which are contained in the square figure A;

30 rods.

5 rods.

30 rods.

A

30

30

5 150

30
900 sq. rods.

5 rods.

D

25:

30

5

5 rods.

(30x30 900,) and 900 square rods deducted from 1225 square rods leaves 325 square rods to be added to the square figure A.

Now to dispose of the remaining 325 rods so as to retain the square form of the figure A, it is evident that we must make the addition on two sides the length of each is 30 rods, and 30+30 60 rods, or which is the same thing, double the root already found: then, 3 (tens) makes 6 (tens,) or 60 for a divisor; then if we divide 325 by 60, or, which is the same thing, neglect the cipher in the divisor and divide 6 150 (tens) into 32 (tens,) it shows that the breadth of the addition must be 5 rods, which is the next figure in the root; then if we examine the figure, we shall find that it is not yet complete, for there yet remains a small square in the corner D, each side of which is 5 rods, to the last quotient figure. This quotient figure, 5, we must add to the divisor 60, (by the Rule) making 65 the whole divisor. Or, which is the same thing, we place the figure 5 at the right

30 rods.

5 rods.

30 rods.

hand of the 6 (tens,) making 65 the whole divisor, which multiplied by the quotient figure 5 gives the whole number of square rods in the addition around the sides of the square, A; 65 x 5=325 square rods, which being deducted from the dividend, 325, leaves 00.

Hence we find that the square root of 1225 is 35, which is the length of one side of the field.

2

Proof. This question may be proved by Involution; thus, 35, or 35 × 35=1225: or by adding the several parts of the figure together;

Thus, A contains 900 square rods.

Ans. 372.

Ans. 40,6.

Ans. 2574.

4. What is the square root of 138384 ? 5. What is the square root of 1648,36? 6. What is the square root of 6625476 ? 7. What is the square root of 488,631025? Ans. 22,105. 8. What is the square root of ,002304? 9. What is the square root of 30138696025?

Ans. ,048.

Ans. 173605.

10. What is the square root of 1355 ?-In this example, after bringing down all the figures there is a remainder, to which we annex a period of ciphers and continue the operation to decimals; and there is still a remainder, to

1. Let 6498 mén be so formed that the number in rank may be double the number in file.

6498÷2=3249, and √/3249-57 in file; and
57x2=114 in rank, Ans.

2. A man wishes to plant 1875 trees in an orchard which is 3 times as long as it is broad, so that they may stand in rows in equal distances apart; how many rows must he make, and how many trees must he plant in each row? Ans. 25 rows of 75 trees.

3. There is a certain lot of land containing 6 acres 2 roods and 18 rods in the form of a long square, the length of which is twice as much as the breadth; required the length and the breadth.

Ans. The width is 23 rods, and the length 46 rods. PROB. V.-The diameter of a circle being given, to make another circle which shall be proportionably greater or smaller than the given circle.

Note. The areas, or contents of circles, are in proportion to the squares of their diameters or circumferences.

RULE.

Square the given diameter; then multiply by the given proportion, if greater, (but divide if smaller,) and extract the square root of the product, (or quotient,) which will give the required diameter.

1. Suppose there is a certain garden whose diameter is 9 rods, and it is required to lay out another which shall contain 3 times as much; what must be its diameter ?

9x9x3=15,58+rods, Ans. 2. There is a circular field whose area, or contents, is 64 rods, the diameter of which is 9 rods; required the diameter of another which shall contain just one-fourth as much. Ans. 4,5rds.

3. The quantity of water discharged through a certain pipe, which is 2,5 inches in diameter, will fill a certain cistern in one hour; what is the diameter of another pipe which will fill another cistern four times as large, in the same time? √/2,5X2,5X4-5 inches diameter, Ans. PROB. VI. The sum of two numbers and their product given, to find those numbers,

RULE.

From the square of their sum subtract 4 times their product, and the square root of the remainder will be their difference; half the said difference added to the half sum will be the greater of the two numbers; and half the said difference subtracted from the half, will be the lesser number.

1. The sum of two numbers is 58, and their product is 792; what are those numbers?

58×58=3364=square of their sum.

792 × 4=3168=4 times their product.

PROB. VII.-The difference and the product of two numbers given, to find those numbers.

RULE.

Add the square of half the difference of the numbers to their product, and the square root of that amount will be half the sum of the two numbers. Then to the half sum add half the difference, gives the greater number; and from the half sum, subtract half the difference, gives the lesser number.

EXAMPLE.

1. The difference of two numbers is 9, and their product is 442; what are those numbers?

Difference of the numbers 9 Product 442