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EXAMPLE S.

1. What is the fquare root of 3456?

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Anf.

Anf.

z. I demand the square root of 14? 3. Find the fquare root of 24.

405

-Anf. 790569.
Anf. 645497+.
Anf. 43.
Anf. 3
Anf. 4%.

4. What is the fquare root of 7?-Anf. 72414+. 5. Required the fquare root of 6. Find the fquare root of 14. 7. Extract the fquare root of 1719. 8. What is the fquare root of 101? 9. I demand the fquare root of 201? 10. What is the square root of 7614?

Anf. 8.7649+.

11. Extract the fquare root of 63.-Anf. 2.5298+. 12. Find the fquare root of 9

CASE

Anf. 3.0822+.

III.

To find a mean proportional between two given num

bers.

RULE.

Multiply the two given numbers together, and extract the fquare root of the product, which root will be the mean proportional fought.

EXAMPLE

S.

1. What is the mean proportion between 16 and 36?

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2. Find the mean proportional between 2 and 5.

Anf. 3.162278-.

3. What is the mean proportional between 67 and 124? Anf. 91 14823t.

4. Required the mean proportional between 8 and 17? Anf. 11.6619+,

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To find the fide of a fquare equal in area to any given superficies.

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Extract the square root of the given fuperficies, which root will be the fide of the square fought.

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1. The area of a given triangle is 870.25 yards; I demand the side of a square equal in area thereto ?

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2. If the area of a given circle is 1315.6 feet; what is the fide of a square whose area is equal thereto ? Anf. 36.27+.

3. An oval fish-pond contains 9 as. 3 rs. 21 ps.; required the fide of a fquare fifh-pond that fhall contain the fame quantity of land? Anf. 218.6898+ yds.

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Any two fides of a right-angled triangle being given

to find the remaining fide.

RULE.

R U UL L E.

If the hypothenufe or longeft fide be required; extract the fquare root of the fum of the fquares of the two given fides, which root will be the hypothenufe fought. But, if either of the other two fides be wanted; extract the square root of the difference of the fquares of the given fides for the antwer.

EXAMPLE S.

1. The two fhorteft fides of a right-angled triangle are 23 and 27 yards; required the hypothenufe or longeft fide?

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gds.
1258(35°47-Anf.

27

9

189

65)358

54

325

729

704)3300

529

2816

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2. The hypothenufe of a right-angled triangle is 50, and its bafe 40 feet; required its perpendicular?

Anf. 30 feet.

3. A line of 320 feet will reach from the top of a rock, ftanding clofe by the fide of a brook, to the oppofite bank; required the breadth of the brook; the height of the rock being 103 feet?

Anf. 302-9703-feet. 4. The base of a right-angled triangle is 77, and the perpendicular 36 yards; what is the hypothenufe? Anf. 85 yards. 5. A ladder 40 feet long may be fo planted, that it' fhall reach a window 33 feet from the ground on one

fide of the street; and, without moving it at the foot, will do the fame by a window 21 feet high on the other fide; the breadth of the ftreet is required?

Auf. 56·64+ feet.

REM ARK.

The three laft cafes fhew fome of the many uses to which the fquare root is applicable. It may alfo be applied to many other purposes which are quite foreign to a course of Arithmetic.

EXTRACTION of the CUBE ROOT.

EXtrading the cube root is, the finding out a number, which being multiplied into itself, and then again into the product, will produce the given number.

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To extract the cube root of integers, or decimals, or both mixed together.

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Separate the given number into periods of three figures each, by putting a point over every third figure from the place of units, and make the decimals confift of three, fix, or nine, &c. places, by annexing one or more ciphers if neceffary. Then, feek the greatest cube in the left hand period, write the root in the quotient, and the cube under the period, from which fubtract it. To the remainder annex the next period, and call the whole fum the refolvend. Under this refolvend, write the triple fquare of the root, fo that units in the latter may fland under the place of hundreds in the former; and under the faid triple square write the triple root, removed one place to the right hand, and the fum of these two lines call a divifor.

Seek

Seek how often this divifor may be had in the refolvend, its right hand place excepted, and write the refult in the quotient. Under the divifor write the product of the triple fquare of the root by the laft quotient figure, fetting the units place of this line, under that of tens in the divifor; under this line, write the product of the triple root by the fquare of the last quotient figure, let this line be removed one place beyond the right hand of the former; and under this line, removed one place forward to the right hand, write the cube of the laft quotient figure. Call the fum of these three lines the fubtrahend, which fubtract from the refolvend, and to the remainder bring down the next period for a new refolvend, with which proceed as before, and fo on till the whole is finished.

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