25. How many square ft. in a board 10ft. 8/ long and 1ft. 5′ broad? A. 15ft. 1' 4". 26. In a load of wood 8ft. 4' long, 2ft. 6′ high, and 3ft. 3' wide, how many solid feet? A. 67ft. 8' 6". NOTE.-Artificers compute their work by different measures Glazing and masons' flat work are computed by the square foot; painting, paving, plastering, &c. by the square yard; flooring, roofing, tiling, &c. by the square of 100 feet; brick work by the rod of 16 feet, whose square is 272; the contents of bales, cases, &c. by the ton of 40 cubic feet; and the tonnage of ships by the ton of 95ft 27. What will be the expense of plastering the walls of a room 8ft. 6' high, and each side 16ft. 3' long, at 62 cents per square yard? A. $38.368. 28. How many cubic feet in a block 4ft. 3′ wide, 4ft. 6' long, and 3ft. thick? A. 57ft. 4' 6". 29. How much will a marble slab cost, that is 7ft. 4' long and 1ft. 3' wide, at $1.25 per foot? A. $11.458. 30. How many cubic feet of wood in a load 6ft. 7 long, 3ft. 5′ high, and 3ft. 8' wide? A. 82ft. 5' 8" 4. 31. What will the paving of a court-yard, which is 70ft. long and 56ft. 4′ wide, come to, at $.20 per square yard? A. $87.63, nearly. 32. How many solid feet are there in a stick of timber 70ft. long, 15' thick, and 18' wide? A. 131ft. 3'. 33. A man built a house consisting of 3 stories; in the upper story there were 10 windows, each containing 12 panes of glass, each pane 14' long, 12' wide: the first and second stories contained 28 windows, each 15 panes, and each pane 16' long, 12′ wide: how many square feet of glass were there in the whole house? A. 700 sq. ft. INVOLUTION.* XCVII. 1. INVOLUTION is the process of finding powers. POWERS are the several products arising from multiplying any number by itself, and that product by the same number again, and so on. 2. Any number is called a first power of itself; but when it becomes repeatedly a factor in producing other powers, it is called their root, because they seem, as it were, to grow out of it. Q. By whom are Duodecimals used, and for what purpose? 19. How are the square and solid contents of any thing ascertained? 20. What is the Rule? 22,23. XCVII. Q. What is Involution? 1. What are meant by powers? 1. What, by first powers, second powers, &c.? 2. * INVOLUTION, from the Latin in, for in, and volvo, to roll, signifies the act of enrolling, enwrapping, or involving; the state of being mixed or complicated; the raising of powers, because a given number thereby becomes, by repeated multiplications, involved in other numbers. ROOT. The part of a plant in the ground. Figuratively, the bottom or lower part; the origin, cause, ancestry; a primitive word or theme. The first power, because it forms the basis of all the succeeding powers. 3. The first product, because the same factor is used twice, is called the second power, or square; the next product, because the same factor is used three times, is called the third power, or cube; and so on, as follows Thus 3= 3, 1st power, or root. 3×3= 9, 2d power, or square. 4. The INDEX or EXPONENT of a power denotes the number of times the root must be used as a factor to produce that power; consequently, index is only another name for the number of the power 5. Powers are frequently expressed by writing their indices in smaller figures on the right of their respective roots, as— The 8th power of 2 is 28=2×2×2×2×2×2×2×2=256. The 3d power of 4 is 43=4×4×4=64. RULE 6. Involve the given number or root, that is, multiply it by itself, and the product by the root again, and so on till the root has been used as a factor as many times as are indicated by the given power or its exponent. 7. What is the second power or square of 13? 8. What is the third power or cube of 18? A. 169. A. 5,832. 9. What is the fourth power or biquadrate of 11? A. 14,641. 10. What is the fifth power of 7 ?—of 9 ? A. 16,807; 59,049. 11. What is the sixth power of 5 ?—of 4? A. 15,625; 4,096. 12. What number is meant by 321-by 53?-by 205 ?-by 74?— by 65? A. 9; 125; 3,200,000; 2,401; 7,776. 13. What is the numerical difference between 26 and 83 ?-between 95 and 48 ?-between 1010 and 205? A. 448; 6,487; 9,996,800,000. 14. What is the 2d power of?—of .75 ?—of }? What is the 3d power of ?—of 4.22 ? A. .5625; 2; 33; 16 2 9 15. What is the square of 5? 16. What is the square of 16? 9 8 75.151448. A. 301. A. 2721. 17. What is the difference between the cube of ube of and the biquad rate of of? 4 3 81 A. 499 8000 18. What is the numerical value of ?-of 51? A. 2; 1663. 5 19. What number is equal to 32× 43?-to 35 × 24? A. 576; 3,888. 20. What is the difference between 45 and 43× 42? A. 0. Q. What is meant by Index or Exponent? 4. Give an example. What is the rule? 6. What are the second, third and fourth powers sometimes called? 3. What is the square of 20?-cube of 3?-biquadrate of 3?-fifth power of 2?— seventh power of 2? * BIQUADRATE, from two Latin words, bis, twice, and quadra, a square, is so called because that number which is used twice as a factor in producing a square is used twice more in producing the biquadrate or fourth power 21. In the last example the exponents of 43 and 42 added together make 5, the exponent of 45; therefore 22. The powers of the same root are multiplied by adding their exponents. 23. 2153×2151×215° are equal to what? A. 21513. 24. Involve 24; that is, raise it to the power denoted by its exponent. A. 16. Involve 26. A. 64. Involve 22. A. 4. 25. What is the quotient of 26 (=64) divided by 22 (=4)? A. 16=24. 26. Hence powers of the same root may be divided by subtracting the exponent of the divisor from the exponent of the dividend. 27. Divide 31521 by 31516, and 8239 by 8218. 28. What is the square of the 9 digits? A. 315*; 823. A. 15,241,578,750,190,521. 29. What is the sum of the squares of all the composite numbers between 1 and 20? A. 1,442. 30. What is the sum of the cubes of all the prime numbers between 1 and 20. A. 15,803. 31. Suppose there is a pile of wood, whose dimensions, that is, its length, breadth, and depth, are each 17 feet; how many cords does the pile contain? A. 38c. 49ft. 32. Suppose a piece of land lies in the form of a square, and each side measures 135 rods; how many acres does it contain? A. 113A. 145rd. 33. Suppose a pile of wood, whose dimensions are each 18 feet, be sold for $8 per cord; what will the pile bring? A. $398.672, nearly. 34. If the amount of $1 at compound interest for 1 year is $1.06, what is the amount for 4 years ?-for 5 years ?-for 7 years ?-for 10 years? A. $1.262477+; $1.338225+; $1.50363+; $1.790848+. 35. What is the difference in value, at $183 per acre, between a quantity of land containing 250 square miles and one which is 250 miles square? [See vii. 44.] A. $747,000,000. feet wide, and 131 .19 of each part, at A. $29.2169+. 36. If a solid block of granite, 27 feet long, 13 feet thick, be halved, what will be the value of the rate of 183 cents for 3 solid feet? Q. What is the numerical difference between 5 whose index shall be 3, and 12 whose index shall be 2? What is the product of 15 whose index shall be 8, if multiplied by 23 whose index shall be 5? What is the rule for it? 22. What is the quotient of 46 divided by 42? What is the rule? 26. How many square rods in a piat of ground 12 rods square? What is the difference in square yards, between 11 square yards and 11 yards square? [See vii. 43, 44.] XCVIII. EVOLUTION.* 1. EVOLUTION is the finding of the root from having the power given, and is therefore the converse of Involution, which is the finding of the power from having the root given. 2. Thus, the second or square root of 36 is 6, because the second power or square of 6 is 36; the third or cube root of 27 is 3, because the third power or cube of 3 is 27; the fourth root of 16 is 2, because the fourth power of 2 is 16, &c. 3. A Rooт, then, of any number is that factor which, multiplied into itself a certain number of times, will produce the given number. The process of finding it is called its EXTRACTION. 4. The number or name of the root corresponds with the number or name of its power. 5. That is, if 4 be the second power or square of 2, then 2 is the second or square root of 4; and if 27 be the cube of 3, then 3 is the cube root of 27. 6. Find by trial the square root of 64?-of 144?-of 3,600 ?—of 25 ?—of 42.25 ?—of 9% ?—of } of {} ?—of 61 ?—of 3? 16 3 ; }; (61=25) 1⁄2, or 21; 12. 7. Find by trial the cube root of 1?—of 8?—of 27?—of 64 ?—of 2 10 A. 1; 2; 3; 4; .5; 3; % (=}); 11⁄2. 8. Find by trial the biquadrate or fourth root of 16 ?—of 10,000?— of 11⁄2?—of 1 ?—of .0016?—of 5 of 1000? 16 81 1 10 2 A. 2; 10; ; 1; 2; 16. 9. In Involution, the required power of any number may be exactly XCVIII. Q. What is Evolution? 1. Give an example. Give an example. What is the root of any number? 3. Whence their names? 4. Give an example. Give the answers to the examples (on being read aloud by the teacher) in No. 6--in No. 7-in No. 8. Are all numbers susceptible of exact powers and roots? 9, 10. What classification is made in reference to such numbers? 11, 12. Give an example. 13. ascertained, because it is done by multiplication, which produces an exact product. 10. On the contrary, in Evolution there are many numbers whose roots cannot be accurately expressed, as the square root of 2, there being no factor that, multiplied into itself, will produce it. 11. Numbers whose roots can be exactly ascertained are called PERFECT POWERS, and their roots RATIONAL numbers. 12. But other numbers are called IMPERFECT POWERS, and their oots IRRATIONAL NUMBERS, or surds. 13. Thus 16 is a perfect square, because its root is a rational number; but 16 is an imperfect cube, because there is no factor the third power of which is that number. Its root, then, is a surd. 14. By the means of decimals, however, we can come nearer and nearer to the desired root; that is, approximate towards it to any assignable degree of exactness; as the square root of 2, which is nearly 1.41421356+. 15. Roots are often indicated after the manner of powers in Involution, the numerators of which show the powers of the given numbers, and the denominators the required roots; thus 1 4 3 1 273 1 means the square root of 41 or 4; then 42=2. means the cube root of 27, which is 3; then 273=3. 161 means the fourth root of 16, which is 2; then 16a=2. 4 31 means the square root of the fourth power of 3; then 32–9. 6 6 23 means the cube root of the sixth power of 2; then 23=4. 16. The square root is also indicated by the radical sign ✓, and other roots by placing before the same sign their respective indices. 17. Thus, √9, 3√8, *√16, denote the square, cube, and fourth roots, respectively. 18. Since √25=5, therefore √25× √25=25. 3 4. 4 3 4 4 20. Since ✓16=2, therefore *✓ 16×1✓ 16×*✓ 16×*✓ 16—16. 21. But *✓ 16× * ✓ 16 × *✓ 16=8, since *√/16=2, and 2×2×2=8. 22. When numbers have a line, called a vinculum, drawn over them, or are enclosed in a parenthesis, they are to be taken together. 23. Thus, 3√30–3 or 3√(30-3) means that 3 is first to be taken from 30, leaving 27, of which the cube root is to be extracted. 24. Find by trial the difference between the square of 81 and the A. 6,552. square root of 81. 1 4 6 1 Q. How may surd roots be expressed with a tolerable degree of exactness? 14. How are roots indicated? 15. What is meant by 42? [Read 4 with the index 1.] What by 273?—by 164?—by 32?—by 23? What other indications of roots are there? 16. Give an example. See 17. Jointly? 22 When are numbers to be taken |