9. Find the square root of 61504, and prove. 10. Find the square root of 43264, and prove.

11. Find the square root of 61928, to two places of decimals, and prove.

12. Find the square root of 363729'61, and prove. 13. Find the square root of 432'64, and prove.

14. Find the square root of 92165'4, and prove.

15. What is the value of 2, carried to three places of decimals? Prove.

16. Find the square root of 10 carried to two places of decimals, and prove.

EXTRACTION OF THE CUBE ROOT, OR DIVISION INTO THREE EQUAL FACTORS.

a. Pointing off Cubes into periods of three figures.

Exemplifications for the Black-board.

1. Write in columns, as follows, on the slate or black-board, the cubes of '01, 1, 1, 10, 100, &c., being the smallest significant figure, and the cubes of ‘09, ‘9, 9, 90, 900, &c., being the greatest significant figure. Write, also, a line of ciphers, and point them off into periods of three figures each, as under:

nor more than

Suggestive Questions, to be repeated till all can be answered without hesitation.--Of how many figures does the cube of any number of units consist? See 1 and 9 of table of cubes above. Ans. Not less than Which period will the cube of units occupy, then? How many ciphers are there in the cube of any number of tens? See the cube of 10 and of 90 above. Which period, then, will be occupied by the significant figures of the cube of tens? Which period will be occupied by those of the hundreds? Of the thousands? &c. Which period will be occupied by those of the tenths? Ans. The period to the right of that of the Which period

Of

by those of the hundredths? Where, then, should you look for roots of the units? For roots of the thousandths? the tenths? Of the tens ? Of the hundreds? &c. Why do we divide numbers whose roots are sought, into periods of three figures?

b. To find the Cube Root when it contains an integer of two

figures.

2. Find the cube root of 13824, and prove.

Suggestive Questions.-What is the greatest cube in 13? What is its root? its root? Is this root 2 units or 2 tens? Deducting, then, the cube of 20 [8000] from the given cube, what should be looked for next in order in the remainder [5824]? See the exemplification, in Involution, No. 1, cube of 24; or see the 10th principle, p. 166. Which of these numbers is now known? Ans. Three times the square of If a product [5824] and one of its factors [3.202] be found, how can the other factor be found? See p. 57, 5. What should be looked for next in the remainder [5824]? Is it known? What, lastly, will be found in that remainder? Is 13,824 an exact

cube, then?

3. Find the cube root of 373248, and prove by involution. 4. Find the cube root of 19683, and prove by involution. 5. Find the cube root of 262144, and prove by involution. 6. Find the cube root of 166375, and prove by involution.

c. To find the Cube Root, when it contains an integer of more than two figures, &c.

The directions given in treating of the extraction of the square root, when it consists of more than two figures, apply almost literally to that of the cube root, namely: Proceed with the two periods at the left, as if these were the whole, and then, bringing down another period, consider the two figures of the root that are thus found as the tens of the root, and find the units of the root as before. Should a remainder occur at the close, annex three ciphers as a period for tenths, unless there should be decimal places sufficient, and proceed to find

the root for the rank of tenths, in the same manner that the root for units was found, and so on, as far as may be considered necessary. The evolution of decimal fractions, in fact, does not differ from that of integers. The following example will make all this sufficiently plain.

7. Find the cube root of 14513286'7, and

prove.

8. Find the cube root of 84604519, and prove. 9. Find the cube root of 21024576, and prove.

10. Find the cube root of 28913245, to two places of decimals, and prove.

11. Find the cube root of 21036589, to two places of decimals, and prove.

12. Find the cube root of '000729, and prove.

13. Find the cube root of 2 to two decimal places, and prove.

14. Find the cube root of '02 to two decimal places, and prove.

15. Find the cube root of 20 to two decimal places, and prove.

16. Find the cube root of 3932586'4 to two decimal places, and prove.

Practical Exercises in Involution and Evolution.

DEFINITIONS.

I. A figure of three sides is called a triangle, and, if one of its corners, or angles, should be a square corner, like the angle at B in the annexed figure, it is called a right angle; and the figure is called a right angled triangle, and the two sides adjoining the right angle are said to be perpendicular to each other. The side A C, opposite the right angle, is called the hypothenuse. It is shown by Geometry, that the square of the hypothenuse is equal to the sum of the squares of the other two sides. It follows that the difference between the square of the hypothenuse and that of either of the other sides is equal to the square of the remaining side, since, if 9=4+5, then 9—5—4, and 9—4—5.

B

II. The round line which forms the boundary of a circle is called its circumference. Any straight line which passes through the centre, or middle point, of a circle, and is terminated in both ends by the circumference, is called its diameter. Now, we also learn by Geometry, that the areas, or contents, of circles, are not in proportion to their diameters, but to the squares of these diameters. Thus, a circle of 6 inches, or 6 feet, in diame

ter is 4 times as large as one of 3 inches, or 3 feet, in diameter, because the square of 6 [36] is 4 times as large as the square of 3 [9].

1. If a square field contain 2304 square rods, how many rods does it measure on each side; in other words, what is the square root of 2304? See Definitions 1 and 2, Involution, p. 162.

2. If each side of a square field be 48 rods long, how many square rods does it contain?

3. There are two square fields; the side of one being 20, and of the other 40 rods long. How many square rods in each, and how many times is the one larger than the other?