duct of two factors, one of which is the sum obtained by adding three times the square of the first figure of the root to the product of three times the first figure of the root plus the second multiplied by the second, and the other is the second figure of the root. If we knew the first of these factors, which we will call the TRUE DIVISOR, we could find the other by dividing 63726 by it. But as we only know the value of the highest denomination, we have to make three times its square 3 times 6° = 108, but as 6 is tens with reference to the next denomination of the root, the 108 (being 3 times the square of 6 tens) must be hundreds, and in dividing by it we must drop the two right-band figures of the dividend. Dividing the 637 by 108 gives 5 for a quotient, which is either equal to or larger than the next figure of the root.* To determine it, we complete the true divisor by adding to the trial divisor 3 times the first figure of the root plus the second, multiplied by the second (see 138, i). Three times 6 = 18, which we regard as tens. 18 tens + 5 = 185, and this multiplied by 5 is 925, which, added to the trial divisor, 108 hundreds, gives 11725 for the true divisor. Multiplying the true divisor by 5 gives 58625, which, being less than the dividend, shows that 5 is really the second figure of the root. Hence, 65 represents the two highest denominations of the root, and, as the root contains three figures, the 65 must be tens. Subtracting 58625 from 63726, and annexing the next period to the remain der, gives 5101264 for a new dividend, which contains the remaining part of the cube of the tens and units, and must therefore contain the product of two factors, such as described in 138, i. We therefore proceed to form a trial divisor. We may do this the most easily by adding together the square of the last root figure found, the last true divisor, and the number which was added to the last trial divisor, which gives 25 + 11725 + 925 = 12675 = 3 times 65%, for the next trial divisor. This trial divisor, since the 65 is tens, must be hundreds. Dropping the two right-hand figures of the dividend, as before, and diri. ding the remaining part, 51012, by the trial divisor, gives 4 for a quotient, which is either equal to or larger than the units figure of the root. To complete the true divisor, we multiply 65, the part of the root before found by 3, annex the last root figure, 4, to the product, and multiply the result by 4. This gives 7816, which added to the trial divisor, 12675 hundreds, gives 1275316 for the true divisor. Multiplying this true divisor by 4 gives 5101264 for a product, which, being equal to the dividend, shows that 4 is really the last figure of the root, or that the required root is 654. * For the trial divisor is always smaller than the true divisor. The work may be written thus: 279726264 ( 654 = Cube Root. 216 108 ) 63726 = First dividend. 925 25 12675 ) 5101264 = Second dividend. 7816 1275316 5101264 = Second true divisor multiplied by 4. (a.) These principles are of universal application, and furnish the following rules — 1st. Divide the given number into periods of three figures each, beginning with the units. The left-hand period may contain one, two, or three figures. 2d. Find the greatest cube in the left-hand period, and place its root as the first figure of the required root. 3d. Subtract this cube from the left-hand period, and to the remainder bring down the next period, calling the result a dividend. 4th. Find three times the square of the part of the root already found, and make it a TRIAL DIVISOR. 5th. See how many times the trial divisor is contained in the dividend, excepting the two right-band figures, and write the quotient as the next figure of the root. 6th. To three times the part of the root previously found, annex the last root figure, multiply the result by the last figure, and, placing the product under the trial divisor, but two places further to the right, add it to the trial divisor. This will give the true divisor. 7th. Multiply the true divisor by the last root figure, placing the product under the dividend. 8th. Subtract the product from the dividend, and to the remainder annex the next period for a new dividend. 9th. Add together the square of the last quotient figure, the last true divisor, and the number standing over the last true divisor. The sum will equal three times the square of the root already found, and will be the second trial divisor. 10th. Now proceed as with the first trial divisor. 10. 10460353203 ? 3. 912673? 7. 340068392 ? 11. 125300240064? 4. 59319? 8. 6434856 ? 12. 106429035419 ? 5. 17546 ? 9. 145531576? 13. 1371330631 ? 140. Cube Root of Fractions. (a.) To extract the cube root of a fraction, we extract the cube root of its numerator for a new numerator, and the cube root of its denominator for a new denominator. (b.) If its numerator and denominator are not both perfect cubes, we can only get the approximate value of its cube root. (c.) If the denominator is not a perfect cube, both terms should be multiplied by the square of the denominator, or by such other number as will make it so. (d.) Find the exact or approximate cube roots of the following fractions 3. 21 343 125 125 5. 18 ST. (e.) In order that a decimal fraction may have a perfect cube for its denominator, its numerator must contain three decimal places, or some multiple of three. ILLUSTRATIONS.—The denominators of .008, 2.017, and .571632 are perfect cubes, but the denominators of .08, .2017, and 57.1632 are not. (f.) Hence, to extract the cube root of a decimal fraction, annex zeroes if necessary to make its number of decimal places some multiple of 3. Then extract its root as in whole numbers, observing that there will be one decimal place in the root for every three in the given fraction. The root may be found to any number of decimal places by annexing three zeroes to the fraction for every additional figure desired in its root. What is the cube root to 5 places of 7. .6? 10. 2? 13. .00007 ? 14. 3.21 ? 15. 543.768? SECTION XX. MENSURATION. 141. Plane Figures. (a.) A Triangle is a figure having three sides and three angles. (b.) A Right-ANGLED TRIANGLE is a triangle having a right angle. (c.) The ANGLE, RECTANGLE, and SQUARE are defined on pages 25 and 26. (d.) Lines are PARALLEL when they lie in the same direction, as, for instance, the lines forming the sign of equality. (e.) A PARALLELOGRAM is a four-sided figure, having its opposite sides parallel. (f.) A TRAPEZOID is a four-sided figure, having two of its sides parallel. (g.) A Polygon is a figure bounded on all sides by straight lines. the us ming sa (h.) Similar figures are those which have the same shape, i. e. which have the angles of the one equal to the corresponding angles of the other, and the sides about the equal angles proportional. (i.) In similar figures, then, the first side of the one is to the first side of the other as the second side of the one is to the second side of the other. (j.) The Base of a figure is the side on which it is supposed to stand. Any side of a figure may be regarded as its base. The side opposite the base of a parallelogram is often called its upper base. (k.) The ALTITUDE of a triangle is the perpendicular distance from the side assumed as its base to the vertex of the opposite angle. ILLUSTRATION. — In figure 2, when A B is the base, A C is the altitude; when B C is the base, A D is its altitude; and when A C is the base, A B is the altitude. (1.) The ALTITUDE of a rectangle, a parallelogram, or a trapezoid, is the perpendicular distance between its parallel bases. ILLUSTRATIONS. — A B is the altitude of figure 3, ZQ is the altitude of figure 4, and U V of figure 5. (m.) A Diagonal of a polygon is a line connecting the vertices of two angles not adjacent. its origine straight |