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make additions in the deficiencies. These additions will be 120 feet long, and 5 feet thick.

These additions are made in No. 6. The solid contents of one of these additions may be found thus: 120×5X5= 3000 feet. Three additions contain three times as many feet as one. 3000×3=9000 feet, solid contents of the three additions made in No. 6.

But the pile is not a cube. There must be an addition 5 feet wide, 5 feet thick, and 5 feet long. This addition is made in No. 7.

We have now made the following disposition of the brick : 1000000 solid feet.

In No. 1,

In No. 2,

600000 (added.)

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In pile No. 7 there are 1953125 feet of brick, and it is in a cubical form, and one of its sides is 125 feet long. Ans. 125 feet.

From the foregoing illustration of the principles of the cube root, the teacher will discover the propriety of referring frequently to XXI. SQUARE AND CUBIC MEASURE. The contents of the several additions should be exactly measured by the scholar, according to the rules of this section. Let this course be thoroughly attended to by the teacher, and by the aid of blocks the scholar will find little or no difficulty in fully comprehending all points relating to this rule.

No person can understand the nature of the operations performed in this rule, without an ocular illustration from the use of blocks. To aid the teacher and scholar in forming a set of blocks, and also in obtaining a more correct idea of the nature of the process required by the rule, reference has been made to each addition, according to the order in which it was made.

The mind of teacher and scholar. cannot be too deeply impressed with the necessity of devoting particular attention to this subject, if a correct idea of it be desired.

EXAMPLES FOR PRACTICE.

1. If a ball, 3 inches in diameter, weigh 4 pounds, what will be the weight of a ball that is 6 inches in diameter ? Ans. 32 lbs.

Spheres are to each other as the cubes of their diameters. 2. If the solid contents of a globe are 10648, what is the side of a cube of equal solidity? Ans. 22.

3. If a ball, weighing 4 pounds, be 3 inches in diameter, what will be the diameter of the same quality, weighing 32 pounds? Ans. 6 inches.

4. There is a cubical vessel whose side is 12 inches, and it is required to find the side of another vessel that is to contain three times as much? Ans. 17,306.

Cubes, and all similar solid bodies, are to each other as the cubes of their diameters.

5. There is a cellar dug, that contains a space of 1728 feet; what is the length of one side of the cellar?

Ans. 12 feet.

6. There is a cubical piece of timber, that contains 103823 solid inches; what is the length of one side? Ans. 47 inches.

INVOLUTION.

Involution is finding any required power. Thus, to find the square of any number, multiply that number by itself. Thus, 2×2 = 4, the second power of 2 4X4 16, the second power of 4; 8X8-64, the second power of 8.

To find the cube, or third power, of any number, multiply that number by itself twice. Thus,

2X2=4×2=8, third power of 2;
3x3=9x3=27, third power of 3;
4X4=16X4=64, third power of 4;
5X5=25X5=125, third power of 5.

To find the fourth power of any number, multiply that number by itself three times.

The following table of powers may be useful for occasional inspection.

[blocks in formation]

81| 256|

625

1296 24011 4096

6561

4th P. 1 16 5th P. 1 32 243 1024 3125 7776 16807 327681 59049 6th P. 1 64 729 4096| 15625| 46656 117649 262144 531441 7th P.1128/2187|16384| 78125| 279936 823543 2097152, 4782969 8th P. |1|256|6561|65536|390625|1679616|5764501|1677216|43046721|

EVOLUTION.

Evolution is finding the roots of powers. To distinguish roots, names may be employed to designate them, as in the case of powers. This sign denotes the square root. The square root may be called the second root, the cube root the third root, and so on.

4

The cube root may be denoted thus, s; the fourth root thus, ; fifth root thus, 5; sixth root thus, 6, and so on. Numbers, whose precise roots can be found, are called rational. Those, whose precise roots cannot be found, are called irrational, or surd numbers.

* Names of Powers. The first power is the root; the 2d, square; the 3d, cube; the 4th, biquadrate; the 5th, sursolid; the 6th, square cube; the 7th, second sursolid; the 8th, biquadrate squared, &c.

SUNDRY RULES IN EVOLUTION.

TO EXTRACT THE BIQUADRATE OR FOURTH ROOT.

and

RULE.-Extract the square root of the given number, then extract the square root of the root so found, and the last root so found will be the fourth, or biquadrate root.

What is the biquadrate root of 5308416?

Ans. 48.

TO EXTRACT THE SURSOLID OR FIFTH ROOT.

RULE.-Divide the given sum by five times the assumed root, whatever it may be, and to the quotient add one twentieth part of the fourth power of the same root.

From the square root of this sum subtract one fourth part of the square of the assumed root. To the square root of the remainder add one half of the assumed root, and the sum is the root required, or an approximation to it.

This rule gives the root to five places, and generally to eight or nine places at the first process.

What is the fifth root of 281950621875!

Ans. 195.

TO EXTRACT THE SIXTH ROOT.

RULE.-Extract the square root of the given number, and then extract the cube root of that root.

What is the sixth root of 191102976?

Ans. 24.

TO EXTRACT THE EIGHTH ROOT.

RULE.-Extract the square root of the given number continually, until you have three roots; the last of these is the root sought.

What is the eighth root of 1785793904896? Ans. 34.

TO EXTRACT THE NINTH ROOT.

RULE.-Extract the cube root of the given number, and you have the third power, whose cube root is the root sought. What is the ninth root of 5159780352 ? Ans. 12.

XXVII. Progression.

When unequal quantities occur, they may be considered in two different points of view: we may inquire, how much greater one quantity is than the other; or we may ask, how many times one quantity is greater than the other. The answer to the first question is found by subtraction, and the answer to the second by division. Although both results are ratios, yet mathematicians have distinguished them into the arbitrary divisions of ARITHMETICAL RATIO, and GEOMETRICAL RATIO. Hence, PROGRESSION is of two kindsARITHMETICAL and GEOMETRICAL.

ARITHMETICAL PROGRESSION.

When a series of numbers increases or decreases by the same quantity, it is called an arithmetical progression. When a series of numbers increases by the same quantity, it is called an ascending series. Thus, 1, 4, 7, 10, 13, 16, 19, 22, &c. are an ascending series of numbers. When a series of numbers decreases by the same number or quantity, it is called a descending series; thus, 22, 19, 16, 13, 10,7,4, 1. The number which expresses how much greater, or how much less, one term of the series is, than the next term in the same series, is called the difference. Numbers may be

terms in any series.

Thus, Indices,

employed to denote the number of These numbers are called indices.

1, 2, 3, 4, 5, 6, 7. Arith. Progression, 3, 7, 11, 15, 19, 23, 27, &c.

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