Thus, take 15625, whose cube root is 25. The cube of the tens of the root is 203, or 8000. Now, 15625 8000 7625, which is equal to the sum of 3 × 202, 3 × 20 × 5, and 52, multiplied by 5

418. To find the cube root of a number.

(1) Find the cube root of 389017.

Explanation.-Separating the number into periods of three figures each, I find the root will be expressed by two figures.

I write 343, the greatest cube in the left-hand period, under the period, and 7, the cube root of 343, for the tens of the root.

Subtracting 343 from 389, and to the remainder annexing the next period, I have 46017. Now since 46017 (3t3tu + u2) × u, or 3702 plus 3 X 70 X units plus units 2, multiplied by the units, the units = 46017 ÷ (3 × 702 plus 3 X 70 × units units2).

The last two terms of this divisor, being small in comparison with the first, may be temporarily omitted, which gives 3 X 702, or 14700, for a trial divisor, and 3 for the units of the root. With this unit figure I now complete the divisor by adding the omitted terms, 3 X 70 units + units2 3 X 70 X 3 + 32, to the trial divisor. Multiplying this complete divisor, 15339, by 3, I obtain 46017, which is equal to the dividend.

Remark: Three times the square of the tens is the convenient trial divisor. This is in most instances a greater part of the complete divisor; for example, the least number of tens above one ten is 2, and the greatest figure in unit's place can not exceed 9; the cube of 29 is 24389, the first complete divisor is 1821, the first trial divisor being 1200, a greater part of it.

In extracting the cube root, only two periods of figures are considered in connection. Hence,

Any figure of the root after the second, is found in the same manner as the second figure of a root expressed by two figures.

This principle is fully illustrated in the solution of the following example.

RULE.-I. Separate the number into 3-figure periods, beginning at the decimal point.

II. Find the greatest cube in the left-hand period, and place its root on the right for the first root-figure. Subtract its cube from the left-hand period, and to the remainder annex the next period for a dividend.

III. Square the part of the root already found, multiply the result by 3, annex two ciphers, and place it on the left as a TRIAL DIVISOR: divide the dividend by the trial divisor, and take the quotient as the second figure of the root. To obtain the TRUE DIVISOR, add to the trial divisor 3 times the product of the last root-figure by the root previously found, annexing one cipher, and also add the square of the last root-figure.

IV. Multiply the true divisor by the last root-figure, subtract the product from the dividend, and to the remainder annex the next period for a new dividend.

V. Proceed with the second, and each succeeding dividend, in the same manner as with the first, until all the periods are used.

NOTE.-The notes under Art. 414 are applicable to cube root with the change of " square" to "cube".

Find the value of the following to three places of deci

24. A cubical block of granite contains 110592 cu. in.; what is its edge?

25. What is the height of a cubical box of the same capacity as a box 54 in. long, 49 in. wide, and 28 in. deep?

26. A cubical cistern having a capacity of 2197 cu. ft., is lined on the sides and bottom with copper, at 60 cts. per sq. foot; what is the cost of the lining?

27. The dimensions of a room are in the ratio of 3, 2, 1, and its capacity is 16464 cu. ft.; find the dimensions.

28. Find the edge of a cube whose volume is 3 times that of a cube whose edge is 2 ft.

29. I wish to construct a cubical bin that will contain 500 bushels of wheat; required one of its edges.

30. A cubical block of marble dropped into a rectangular reservoir 6 ft. long and 4 ft. wide, raises the water 1 in. ; what is the edge of the cube?

39

31. How many sq. in. of leather will it take to cover a cubical block of metal which weighs 94 lb. in water, and 100 lb. out of water?

128

MENSURATION.

419. Mensuration is the process of measuring lines, surfaces, solids, and angles.

In mensuration, the length of lines, the area of surfaces, the volume, capacity or solid contents of solids, and the size of angles are determined by means of the relations they sustain to certain given quantities. In general, the discovery of these relations is due to Geometry, Trigonometry, and Calculus. Hence, the rules of mensuration are the translations of formulas taken from the Higher Mathematics.

POLYGONS.

420. A polygon is a plane figure bounded by straight lines.

The perimeter of a polygon is the distance around it, or the sum of all its sides.

A polygon is regular when it has all its sides equal and all its angles equal.

The base of a polygon is the side on which it is supposed to stand; and the diagonal of a polygon is a straight line joining two angles not adjacent.

A polygon of three sides is a triangle; one of four sides, a quadrilateral; of five sides, a pentagon; of six sides, a hexagon; etc.

Triangles.

C

421. A triangle is scalene when it has no equal sides; isosceles, when it has two equal sides, and equilateral, when its three sides are equal.

The vertical angle of a triangle is the angle opposite the base, as the angle C; and the altitude is the perpendicular distance from the vertical angle to the. line of the base, as the line CD.

422. A right triangle is a triangle that

has one right angle.

The side AC, opposite the right angle, is the

06

B

hypothenuse; the side AB, the base; and BC, the perpendicular. 423. Any two sides of a right triangle being given, to find the third side.

THEOREM.-The square described on

the hypothenuse of a right triangle is equal to the sum of the squares described on the other two sides.

Hence, the square of either side about the right angle is equal to the square of the hypothenuse diminished by the square of the other side.

A

From this theorem, which is demonstrated by geometry, and illustrated by figure A, the following rules are derived: I. To find the hypothenuse:

RULE.-Extract the square root of the SUM of the square of the base and the square of the perpendicular.

II. To find the base or perpendicular:

RULE.-Extract the square root of the DIFFERENCE between the square of the hypothenuse and square of the given side.

NOTE. The square of the Difference may be found by the following simple rule: Multiply the sum of the given sides by their difference.