the figure may retain its square form, it is evident, the addition must be made on two sides. Now, if the 225 feet be divided by the length of the two sides, (20+ 20 =40,) the quotient will be the breadth of this new addition of 225 feet to the sides ed and c of the square A.

But our root already found, 2 tens, is the length of one side of the figure A; we therefore take double this root, = 4 tens for a DIVISOR.

The divisor, 4, (tens,) is in reality 40, and we are to seek how many times 40 is contained in 225, or, which is the same thing, we may seek how many times 4 (tens,) is contained in 22 tens,) rejecting the right hand figure of the dividend, because we have rejected the cipher in the divisor. We find our quotient, that is, the breadth of the addition, to be 5 feet; but if we look at Fig. II. we shall perceive that this addition of 5 feet to the two sides does not complete the square; for there is still wanting, in the corner D, a small square, each side of which is equal to this last quotient, 5; we must, therefore, add this quotient, 5, to the divisor, 40, that is, place it to the right hand of the 4 (tens,) making it 45; and then the whole divisor, 45, multiplied by the quotient. 5, will give the contents of the whole addition around the sides of the figure A, which, in this case, being 225 feet, the same as our dividend, we have no remainder, and the work is done. Consequently Fig. II. represents a square, one side of which is 25 feet, the answer.

If we would prove the operation, we may do it by adding together the several parts of the figure, thus:

254. From the example and illustration now given, we derive the following

RULE

FOR EXTRACTING THE SQUARE root.

I. POINT OFF the given number, from unit's place, into periods of Two FIGURES each.

II. Find the greatest SQUARE NUMBER in the left hand period, place its root in the quotient; subtract the square number from the left hand period, and to the remainder bring down the next period for a DIVIDEND.

III. DOUBLE the root already found for a DIVISOR; seek how many times the divisor is contained in the dividend, (rejecting the right hand figures,) and place the result in the quotient, and also at the right hand of the divisor; multiply the divisor thus augmented, by the last quotient figure, and subtract the product from the dividend; to the remainder bring down the next period, and proceed as before.

NOTE. As the value of figures, whether integers or decimals, is determined by their distance from the place of units, so we must always begin at unit's place to

point off. Mixed decimals must, therefore, be pointed off both ways from units, and if there be a deficiency in any period of decimals, it must be supplied by a cipher.

1. What is the square root of 207936 ?

256. We have seen, (Art. 243,) that a fraction is involved by involving the numerator and denominator. Hence, it is evident that the ROOT of a fraction may be obtained by extracting the root of the NUMERATOв and of the DENOMINATOR. Ans. to the above, 3.

Or, 24 = 2.25, & √2.25 = 1.5, as before.

257. When the numerator and denominator are not exact powers

the fraction may be reduced to a decimal, and the approximate rool found as above.

16. What is the square root of ?

Ans. .89442719

THEORETICAL QUESTIONS.

the 1st power?

What is involution? What are powers? the 2nd power? the 3d power? What is the index, or exponent? How is a frac tion involved? — a mixed number? What is evolution? What is a root?

-the square root? — the cube root? What does the character ✔, or an inverted r, express? How is the same character made to express any other root? Of how many figures at most, can the square of any root consist? Why? How, then, can we tell now many figures there will be in the square root of any number? Of how many figures can the cube of any root consist? How, then, can the num ber of figures in the cube root of any number be ascertained? Can the exact root of all numbers be found? What are surd numbers? — rational numbers ? What is it to extract the square root? In the operation, why do we double the root for a divisor? Why do we, in dividing, reject the right hand figure of the dividend? Why do we place the quotient figure to the right hand of the divisor?

EXAMPLES FOR PRACTICE.

1. If 1369 fruit trees be planted in a square orchard, how many must be in a row each way 2. If a square field contain 2025 square rods, how many rods will it measure on each side?

Ans. 37.
Ans. 45.

3. There is an army of 4096 men; how many must be placed in rank and file to form them into a square? Ans. 64. 4. There is a circle, whose area, or superficial contents, is 5625 feet; what will be the length of one side of an equal square?

✔5625=75, Ans.

5. A man has two fields, one containing 40 acres, and the other 50; B offers him, in exchange, a square field containing the same number of acres; how many rods must each side of this field measure?

Ans. 120 rods. 6. Suppose I have an eliptical fish pond, containing 9A. 2R. 15 poles, and would have a square one of the same superficies, what will be the length of each side? Ans. 215.484918 yards. 7. There is a square field, which measures 20 rods on each side; what will be the side of a square, which contains 4 times as much?

9 times as large?

√20 × 20 × 4 = 40 rods, Ans. S. If the side of a square be 3 feet, what will be the side of one 3 times as large? 25 times as large? 36 times as large ? Ans. to last, 18 feet. 9. It is required to lay out 338 rods of land in the form of a parallelogram, so that it shall be twice as many rods in length as it is in breadth?

If the field be divided in the middle, it will form two equal squares; hence 338÷ 2= 169, then ✓169 = 13 rods, the width; and 13 × 2=26 rods, the length. 10. I would set out at equal distances 1444 apple trees, so that my orchard may be 4 times as long as it is wide; how many rows of trees must I have, and how many trees in each row?

Ans. 19 rows, and 76 trees in each row. 11. There is an oblong piece of land, containing 192 square rods, of which the breadth is as much as the length; required its length and

th.

aken from the length, it will form a square, of which the length on each

8

side is equal to the width of the oblong: hence the area of this square will be less than that of the oblong; 192 193 144, and ✔/14412, the width; then 12+

12 16, the length.

12. Suppose 450 men be drawn up in oblong battle, so that the number in rank be 12 times as many as in file; required the number in rank and and file?

Ans. 6 and 75.

13. There is a parallelogram, containing 2744 square rods, the breadth of which is as much as the length; required its dimensions.

Ans. 56 by 49. 14. There is a circle whose diameter is 3 inches; what is the diameter of one 4 times as large?

258. The AREAS of circles are in proportion to the SQUARES of their DIAMETERS.

Hence, if the diameter of any circle be given, to find the diameter of one, 2, 3, 3, &c., times as large,

RULE. Multiply the square of the diameter by the GIVEN RATIO; and the square root of the product will be the answer. Applying the rule, . √3×3×4=6, Ans, to the above.

15. If the diameter of a circle be 5 feet, what will be the diameter of one 4 times as large? 9 times as large ? 16 times as large? 25 times as large? Ans. to last, 25 feet. 16. If the diameter of a circle be 12 feet, what will be the diameter of one as large? Ans. 6 feet.

17. There are two circular ponds in a gentleman's pleasure ground; the diameter of the less is 48 feet, and the greater is 3 times as large; what is its diameter ? Ans. 83.13+. 259. 18. A carpenter has a large wooden square; one part of it is 4 feet, and the other part 3; what length of cord will reach from one end to the other?

C

RULE 1. The SQUARE ROOT of the SUM OF The squares of the two shortest sides, is the length of the HYPOTENUSE.

RULE 2. The SQUARE ROOT of the DIFFERENCE OF THE SQUARES of the HYPOTENUSE and EITHER of the other sides, is the length of the REMAINING side.

Applying the rule to example 18, √42 +32= 5 feet, Ans.

19. If the two legs, that is, the two shortest sides, of a triangle be 12 inches each, what is the length of the hypotenuse?

Ans. 16.97. But the carpenters use 17 for their square rule. 20. The wall of a town is 25 feet high, and is surrounded by a moat of 30 feet in breadth; I desire to know the length of a ladder that will reach from the outside of the moat to the top of the wall. Ans. 39.05 feet.

EXTRACTION OF THE CUBE ROOT.

260. A CUBE is a solid body having six equal sides, and each of the sides an EXACT SQUARE.

261. The ROOT of a CUBE is, therefore, the measure in LENGTH of ONE of its SIDES.

For the length, breadth, and thickness, of such a body are ALL EQUAL.

Consequently, the length of one side of a cube, raised to the 3d power, gives the solid contents.

262. EXTRACTING THE CUBE ROOT of any number is therefore finding the length of ONE SIDE of a cube whose SOLIDITY is equal to THAT NUMBER.

1. Required the solid contents of a cubic block, one side of which measures 5 feet. 53125 feet, Ans.

2. Required the length of one side of a cubic block, that shall contain 125 solid feet. 1255 feet, Ans.

3. A man has 13824 feet of timber, in separate blocks of 1 cubic foot each, and wishes to put them up in a cubic pile, what will be the length of one side of such a pile?

The answer to this question is evidently found by extracting the cube root of the whole number of feet, (13824.)

To do which we must

1st. Ascertain the number of figures, of which the root will consist; This we do by pointing off the given number into periods of THREE FIGURES each. Thus we find, that the root will consist of two figures, a ten and a unit.

'Let us, then, seek for the first figure, or tens of the root, which must evidently be extracted from the left hand period, 13 (thousand). The greatest cube in 13 (thousand) we find by trial to be 8 (thousand) the root of which is 2 (tens ;) we therefore place 2 (tens) in the root. The root, it will be recollected, is one side of a cube. Let us, then, form a cube, (Fig. I,) each side of which shall be supposed 20 feet, expressed by the root now obtained. The contents of this cube are 20 X 20 X 20 8000 solid feet, which are now disposed of, and which, consequently, are to be deducted from the whole number of feet, 13824. 8000 feet taken from 13824 leaves 5824 feet. This deduction is most readily performed by subtracting the cubic number, 8, or the cube of 2, (the figure of the root already found,) from the period 13 (thousand) and bringing down the next period by the side of the remainder, making 5824, as before.

2nd. The cubic pile, A D, is now to be enlarged by the addition of 5824 solid feet, and in order to preserve the cubic form of the pile, the addition must be made on one half of its sides, that is, on 3 sides, a, b, and c. Now, if the 5824 solid feet be divided by the square contents of these 3 equal sides, that is, by 3 times, (20 X 20 =400) 1200, the quotient will be the thickness of the addition to be made to

=