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ciphers to the last term of the second column, and so on; and use the same trial divisor as before, increased however by the additional ciphers.

NOTE 2. - When the given number does not have an exact root, periods of ciphers may be annexed.

NOTE 3. - When the root is required to many places of decimals, the work may be contracted rejecting one figure at the right from the number in the column next to the last, two from the number in the column next farther to the left, and so on, and otherwise proceeding as directed in the rule, except that the new figure of the root is not added to the first column. As soon as all the figures are rejected in the number of the first column, the remainder of the work may be performed as in contracted division of decimals (Art. 276).

EXAMPLES.

2. Required the fourth root of 1.016397 to eight places of decimals.

Ans. 1.00407427.

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3. What is the cube root of 41673648563 ? Ans. 3467. 4. What is the cube root of 43614208 ?

5. What is the cube root of 1.05 to six places of decimals?

Ans. 1.016397. 6. What is the fifth root of 184528125 ?

Ans. 45. 7. Required the fourth root of 100 to six places of decimals ?

Ans. 3.162278. 8. Required the fifth root of the fourth power of 9 to seven places of decimals?

Ans. 5.7995466.

APPLICATIONS OF POWERS AND ROOTS.

534. A TRIANGLE is a figure having three sides and three angles. When one of the sides of a triangle is perpendicular to another side, the opening between them is called a right angle, and the triangle is called a right-angled triangle.

The lowest side, A B, is called the base of the triangle A B C, the side B C the perpendicular, the longest side, A C, the hypothenuse, and the angle at B is a right angle. Also, the line B C, being perpendicular to the base, is the altitude.

535. The square described upon the hypothenuse of a rightangled triangle is equivalent to the sum of the squares described upon

the other two sides.

u Perpendic.

Hypothenuse.

Base.

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536. To find the hypothenuse, the base and perpendicular being given.

Add together the square of the base and the square of the perpendicular, and extract the square root of the sum.

Thus, if the base be 4 and the perpendicular 3, the hypothenuse will equal „42 + 32 = V25 5.

537. To find the perpendicular, the base and hypothenuse being given.

Subtract the square of the base from the square of the hypothenuse, and extract the square root of the remainder.

Thus, if the base be 4 and the hypothenuse 5, the perpendicular will equal „ 5- 4 = N9= 3.

538. To find the base, the hypothenuse and perpendicular being given.

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A

B

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Subtract the square of the perpendicular from the square of the hypothenuse, and extract the square root of the remainder.

Thus, if the perpendicular be 3 and the hypothenuse 5, the base will equal ✓ 5-3= 16 = 4.

539. All triangles having the same base are to each other as their altitudes.

All similar triangles, and other similar rectilineal figures, are to each other as the squares of their homologous or corresponding sides.

Thus, the triangles A C E and A C D, having the same base, A C, are to each other as the altitude E C of the one is to the altitude D C of the other.

Also, the triangles ACE and B CD, having their corresponding angles the same, and their sides in direct proportion, are said to be similar, and are to each other as the squares of their corresponding sides, or as (AE)' is to (B D), (A C) is to (B C), and (CE) is to (C D)? Likewise the larger square, of which A C is one of the equal sides, is to the smaller square, of which B C is one of the equal sides, as (A C) is to (BC)?

540. All circles (Art. 143) are to each other as the squares of their diameters, semidiameters, or circumferences.

The circumference of a circle is the line which bounds it; and the diameter is a line drawn through the center, and terminated by the circumference; as A B and C D.

Then, the larger circle, of which A B is the diameter, is to the smaller, of which C D is the diameter, as (A B)' is to (CD)', &c.

541. To find the side, diameter, or circumference of any surface, which is similar to a given surface.

State the question as in Proportion, and square the given sides, diameters, or circumferences, and the square root of the fourth term of the proportion will be the answer required.

Thus, if 12 feet be the length of a side of a triangle whose area is 72 square feet, the length of the corresponding side of a

similar triangle whose area is 32 square feet would be found as follows: 72 : 32 :: 122 = 144 : 64; v 64 = 8 feet, length required.

542. To find the area of any surface which is similar to a given surface.

State the question as in Proportion, and square the giren sides, diameters, or circumferences, and the fourth term of the proportion will be the answer required.

Thus, if 72 square feet be the area of a triangle of which 12 feet is one of the sides, the area of a similar triangle of which the corresponding side is 8 feet would be found as follows: 122 = 144:82 64 :: 72 sq. ft. : 32 sq. ft., area required.

543. To find the side of a square equal in area to any given surface.

Find the square root of the given area, and that root will be the side of the area required.

544. A sphere is a solid bounded by a continued convex surface, every part of which is equally distant from the point within called the centre.

The diameter of a sphere is a straight line passing through the centre, and terminated by the surface; as A B.

B

545. A CONE is a solid having a circle for its base, and tapering uniformly to a point, called the vertex.

The altitude of a cone is its perpendicular height, or a line drawn from the vertex perpendicular to the plane of the base, as B C. The diameter of its base is a straight line drawn through the centre of the plane of the base from one side of the circle to the other; as A D.

546. Spheres are to each other as the cubes of their diameters, or of their circumferences.

Similar cones are to each other as the cubes of their altitudes, or of the diameters of their bases.

All similar solids are to each other as the cubes of their homologous or corresponding sides, or of their diameters.

A

D

NOTE. - Cones and other solids are said to be similar when their corresponding parts are in direct proportion to each other.

547. To find the contents of any solid which is similar to a given solid.

State the question as in Proportion, and cube the given sides, diameters, ultitudes, or circumferences, and the fourth term of the proportion is the answer required.

548, To find the side, diameter, altitude, or circumference of any solid, which is similar to a given solid.

State the question as in Proportion, and cube the given sides, diameters, altitudes, or circumferences, and the cube root of the fourth term of the proportion is the answer required.

549. To find the side of a cube that shall be equal in solidity to any given solid.

Find the cube root of the contents of the given solid, and that root will be the side of the cube required.

550. To find a mean proportional (Art. 333) between any two numbers.

Find the square root of the product of the two numbers, and that root will be the mean proportional required.

551. To find two mean proportionals between two given numbers.

Find the cube root of the quotient of the greater of the two numbers divided by the less. The product of the less number by that root will be the least mean proportional, and ihe quotient of the greater number by the same root will be the other mean proportional.

552. To find any two numbers, whose sum and product are given.

From the square of half the sum of the two numbers subtract their product, and the square root of the remainder will equal half the difference of the two numbers, which added to half their sum will give the larger, and subtracted from half their sum will give the smaller, of the numbers required.

553. To find any two numbers, when their sum and the difference of their squares are given.

The difference of their squares diviileil by the sum of the numbers will give their difference; and half of their difference aildeıl to half of their sum will give the larger, and half of their difference subtracted from half of their sum will give the smaller, of the numbers required.

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