bers, each 16 ft. by 20 ft. 8'; what did the work of flooring cost, at $'02 per square foot? Ans. $39'95. NOTE 2. - Masons' work is estimated by the perch of 16 feet in length, 14 feet in width, and I foot in hight. A perch contains 24'75 cubic feet. If any wall be 14 feet thick, its contents in perches may be found by dividing its superficial contents by 164; but if it be any other thickness than 14 feet, its cubic contents must be divided by 24'75, (=244,) to reduce it to perches. Joiners, painters, plasterers, brick-layers, and masons, make no allowance for windows, doors, &c. Brick-layers and masons make nc allowance for corners to the walls of houses, cellars, &c., but estimate their work by the girt, that is, the length of the wall on the outside. 8. The side walls of a cellar are each 32 ft. 6' long, the end walls 24 ft. 6', and the whole are 7 ft. high, and 13 ft. thick; how many perches of stone are required, allowing nothing for waste, and for how many must the mason be paid? Ans. {45r perches in the wall. The mason must be paid for 48 perches. 9. How many cord feet of wood in a load 7 feet long, 3 feet wide, and 3 feet 4 inches high, and what will it cost at $40 per cord foot? Ans. 4 cord feet, and it will cost $1'75. 10. How much wood in a load 10 ft. in length, 3 ft. 9' in width, and 4 ft. 8' in hight? and what will it cost at $1'92 per cord? Ans. 1 cord and 211⁄2 cord feet, and it wil cost $2'621. T 205. By some surveyors of wood, dimensions are taken in feet and decimals of a foot. For this purpose, make a rule or scale 4 feet long, and divide it into feet, and each foot into ten equal parts. Such a rule will be found very convenient for surveyors of wood and of lumber, for painters, joiners, &c.; for the dimensions taken by it being in feet and decimals of a foot, the casts will be no other than so many operations in decimal fractions. 1. How many square feet in a hearth stone, which, by a rule, as above described, measures 4'5 feet in length, and 2'6 feet in width? and what will be its cost, at 75 cents per square foot? Ans. 117 feet; and it will cost $8'775. 2. How many cords in a load of wood, 75 feet in length, 3'6 feet in width, and 4'8 in hight? Ans. 1 cord 1 cu. ft. 3. How many cord feet in a load of wood 10 feet long, 3'4 feet wide, and 3'5 high? Ans. 776. Questions. - 205. How do some surveyors of wood take dimen sions? Explain the rule used in measuring. How are dimensions take ov it estimated? INVOLUTION. First power. T 206. Three feet in length ( 111) are a yard, linear measure; 3 in length and 3 in width, 3x3=9 square feet, are a yard Second power. square measure; 3 in length, 3 in width, and 3 in hight, 3 x 3 x3=27 solid feet, are a yard, cubic measure, (T113.) When a number, as 3, is multiplied into itself, and the product by the original number, and so on, the several numbers produced are called powers, and the process of producing them is called Involution. The first number, represented by a line, is Third power. called the first power, or root; the second, represented by a square, is called the square, or 2d power; the third, represented by a cube, is called the cube, or 3d power. Fourth power. Fifth power. Sixth power. The 4th power of 3 is 3 times the 3d power, 3 blocks like that employed to represent the 3d power, and may be represented by a figure 3 times as large, that is, 3 feet wide, 3 feet high, and 9 feet long. The 5th power, by 3 times such a figure, or one 3 high, 9 wide, and 9 long. The 6th power, 3 times this, by a figure 9 long, 9 wide, and 9 high, or a cube. Thus it may be shown that the 9th, 12th, 15th, 18th, &c., powers, may be represented by cubes; the 7th, 10th, 13th, 16th, &c., by figures having greater length than width and hight; the 8th, 11th, 14th, 17th, &c., by figures having greater length and width than hight. To involve a number, take it as a factor as many times as is indicated by the required power. NOTE. 1 The number denoting the power is called the index, or exponant; thus, 51 den tes that 5 is raised or involved to the 4th power. EXAMPLES FOR PRACTICE. 1. What is the square, or 2d power, of 7? 2. What is the square of 30? Ans. 49. Ans. 900. Ans. 16000000. Ans. 64. Ans. 512000000. 3. What is the square of 4000? ད What is the cube, or 3d power, of 4? 5 What is the cube of 800? 6 What is the 4th power of 60 ? 7 What is the square of 1? of 2? Ans. 1, 8, 27, and 64. of ? Ans., 1, and 2. of ? Ans.,, and 313. the 5th power of ? the cube? Ans.and. Ans. 2'25, and 3'375. NOTE 2. A mixed number, like the above, may be reduced to an im proper fraction before involving; thus, 2; or it may be reduced to a decimal; thus, 2=2'25. Ans. $5612518. 15. What is the square of 47? Ans. 11: 1= 23. 16. What is the value of 74, that is, the 4th power of 7? The powers of the nine digits, from the first power to the fifth may be seen in the following 45? 65? Questions. 206. What are powers? How is the first power rep resented? Why is the second power called the square? Why the third called the cube? How is the fourth power represented? - the fifth? the. sixth ? the tenth? - the fourteenth ? the twenty-first? the twentythird ? the twenty-fifth? What is involution? How is a number involved to any power? What is the index, and how written? How is a mixed num ber involved? EVOLUTION. T207. Evolution, or the extracting of roots, is the method of finding the root of any power or number. The root, as we have seen, is that number, which, by a continual multiplication into itself, produces the given power, and to find the square root of a number (one side of a square when the contents are given) is to find a number, which, being squared, will produce the given number; to find the cube root of a number (the length of one side of a cubic body when the solid contents are given) is to find a number, which, being cubed or involved to the 3d power, will produce the given number: thus, the square root of 144 is 12, because 122 = 144, and the cube root of 343 is 7, because 73, that is 7X7X7=343; and so of other numbers. NOTE. Although there is no number which will not produce a perfect power by involution, yet there are many numbers of whi h precise roots can never be obtained. But, by the help of decimals, we can approximate, or approach, towards the root to any assigned degree of exactness. Numbers, whose precise roots cannot be obtained, are called surd numbers, and those whose roots can be exactly obtained are called rational numbers. The square root is indicated by this character placed before the number; the other roots by the same character, with the index of the root placed over it. Thus, the square root of 16 is expressed 16; and the cube root of 27 is expressed 27; and the 5th root of 7776, 7767. When the power is expressed by several numbers, with the sign + or between them, a line, or vinculum, is drawn from the top of the sign over all the parts of it; thus, the square root of 21 —5 is No21—5. Extraction of the Square Root. T 208. 1. Supposing a man has 625 yards of carpeting, a yard wide, what is the length of one side of a square room, the floor of which the carpeting will cover? that is, what is one side of a square, which contains 625 square yards? SOLUTION.. We may find one side of a square containing 625 square vards, that is, the square root of 625, by a sort of trial; and, the Questions. 207. What is evolution? What is a root? square root, and how found? the cube root, and how found? Give exam ples. What do you say of perfect powers and perfect roots? Give the dis unction between surd and rational numbers. How is the square root indicated the c ibe rort? Describe the manner of using the vinculum. OPERATION. = 1st. We will endeavor to ascertain how many figures there wil be in the root. This we can easily do, by pointing off the number, from units, into periods of two figures each; for the square of any root always contairs just twice as many, or one figure less than twice as many figures, as are in the root. The square of 3 (3 × 3 = 9) contains 1 figure, the Square of 4 (4 × 4=16) contains 2 figures; the square of 9 (9 X 9 = 81) contains 2 figures; the square of 10 (10 X 10 = 100) contains 3 figures; the square of 32 (32 × 32= 1024) contains 4 figures; the square of 99 (99 .99 9801) contains 4 figures; the square of 100 (100 X 100: 10000) contains 5 figures, and so of any number. Pointing off the number, we find that the root will consist of two figures, a ten and a unit. d. 20 625 (2 4 225 FIG. I. A 22100 20 b = 2d. We will now seek for the first figure, that is, for the fens of the root, which we must extract from the left hand peri od, 6, (hundreds.) The greatest square in 6 (hundreds) we find to be 4, (hun dreds,) the root of which is 2, (tens, 20; therefore, we set 2 (tens) in the root. Since the root is one side of a square, let us form a square, (A, Fig. I.,) each side of which shall be regarded 2 tens, yards long. =20 The contents of this square are 20 X 20=400 yards, now disposed of, and which, consequently, are to be deducted from the whole number of yards, (625,) leaving 225 yards. This de duction is most readily performed by subtracting the square number, 4, (hundreds,) or the square of 2, (tens,) from the period 6, (hundreds,) and bringing down the next period to the remainder, making 225. 3d. The square A is now to be enlarged by the addition of the 225 remaining yards; and in order that the figure may retain its squarc form, the addition must be made on two sides. Now, if the 225 yards be divided by the length of the two sides, (20+20=40,) the quotient will be the breadth of this new addition of 225 yards to the sides c d and bc of the square A. But our root already found, 2 tens, is the length of one side of the figure A; we therefore take double this root, 4 tens, for a divisor. = The divisor, 4, (tens,) is in reality 40, and we are to seek how many times 40 is contained in 225, or, which is the same thing, we may seek how many times 4 (tens) is contained in 22, (tens,) rejecting the righ hand figure of the dividend, because we have rejected the cipher .n the -- Questions. 208. How may one side of a square, when the contents are given be found? Why, in the trial, must we point off the number into periods of two figures each? Illustrate. Why is the first root figure 2 tens? Of what number is 2 tens the root? What may now be formed? How large? How urst it be increased? What will be the divisor? the dividend?. the quotient? Why is the divisor too small? What is the entire divisor? How are the contents of the addition found? What does Fig. II. represent? What is the first method of proof? — the second method? |