DUODECIMALS. 462. A Duodecimal is a denominate number in which twelve units of any denomination make a unit of the next higher denomination. A duodecimal may be regarded as a fraction whose denominator is a power of 12; or a number whose scale is 12. The term is derived from the Latin duodecim, twelve. 463. Duodecimals are used by artificers in measuring surfaces and solids. The foot is divided into primes, marked'; the primes into seconds ("); the seconds into thirds (""), etc., as is shown in the following TABLE. The accents used to mark the different denominations, are called Indices. 464. The prime denotes the twelfth of a foot; the second, the twelfth of the twelfth of a foot, etc. When a duodecimal denotes the area of a surface, the foot is a square foot; the prime, the twelfth of a square foot; the second, the twelfth of a twelfth of a square foot, etc. When a duodecimal denotes the contents of a solid, the foot is a cubic foot; the prime, the twelfth of a cubic foot, etc. 465. ADDITION AND SUBTRACTION. PROBLEMS. 1. Add 12 ft. 8' 11", 16 ft. 10' 9", and 24 ft. 6". 2. Add 12 ft. 9′ 11′′ 4"", 23 ft. 7′′ 10′′′′, and 10′ 6′′ 9"". 3. From 21 ft. 7' 10" take 15 ft. 9' 4". 4. From the sum of 30 ft. 8" 4"" and 14 ft. 7' 10"", take their difference. 466. MULTIPLICATION OF DUODECIMALS. 5. Multiply 13 ft. 7' 8" long and 6 ft. 5′ wide? PROCESS. 13 ft. 7/ 8" 6 ft. 5' 5 ft. 8' 2/ 4/ 81 ft. 10' 0" Multiply first by 5′ and then by 6 ft., and add the partial products. Since 1X21⁄2 12×12=144, 144×12=1728, etc., feet X primes (or twelfths) must produce primes; primes by primes, seconds; seconds by primes, thirds; and, generally, the denomination of the product of any two denominations is denoted by the sum of their indices. 87 ft. 6' 2' 4''', Ans. 6. What are the superficial contents of a board 9 ft. 7' 4" long and 10′ 6′′ wide? 7. What are the solid contents of a block of marble 7 ft. 6' long, 2 ft. 8' wide, and 1 ft. 4' thick? NOTE. The answers to the 5th and 6th problems are in square feet and duodecimal parts of a square foot, and the answer to the 7th problem is in cubic feet and duodecimal parts of a cubic foot (Art. 464). 467. DIVISION OF DUODECIMALS. 8. Divide 87 ft. 6′ 2′′ 4"" by 13 ft. 7′ 8′′. 9. Divide 62 ft. 11" 3"" by 8 ft. 6′ 9′′. 10. Multiply 10 ft. 5' 8" by 3 ft. 10', and divide the product by 5 ft. 2' 10". PERMUTATIONS. 468. Permutations are the changes of order, which a number of objects may undergo, and each object enter once and but once in each result. 469. The diagram at the right shows the number of permutations of 1, 2, and 3 letters. The letter a permits no change of order. The letter b may be placed before and after the letter a, giving two (1×2) permutations of two letters-ba, ab. The letter e may be placed before, between, and after the two letters ab; and the same for ba, giving six (1 × 2 × 3) permutations of three letters. A fourth letter, as d, may evidently occupy four different positions in each of the six combinations of these letters, giving twenty-four (1×2×3×4) permutations of four letters. In like manner it may be shown that the number of permutations of any number of objects is equal to the continued product of all the integers from 1 to the given number of objects inclusive. PROBLEMS. 1. In how many different orders may 6 boys sit on a bench? 2. In how many different orders may all the letters in the word permutation be written? 3. How many permutations may be made of the nine digits? 4. How many different combinations of eight notes each may be made of the octave? ANNUITIES. 470. An Annuity is a sum of money, payable annually, for a given number of years, for life, or forever. The term is also applied to sums of money payable at any regular intervals of time. 471. A Certain Annuity is an annuity that is payable for a given number of years. A Contingent Annuity is an annuity payable for an uncertain period, as during the life of a person. A Perpetual Annuity is one that continues forever. 472. An Immediate Annuity is an annuity whose payment begins at once. A Deferred Annuity is an annuity whose payment begins at a future time. 473. The Forborne or Final Value of an annuity is the sum of the compound amounts of all its payments, from the time each is due to the end of the annuity. The Present Value of an annuity is the present worth of the forborne or final value. NOTE. The principal applications of the subject of annuities are in leases, life estates, rents, dowers, life insurance, etc.; and the problems arising are readily solved by means of tables which give the present and final values of $1 at the usual rates of interest. A full discussion of the principles involved in the construction of these tables, can not well be presented in a school arithmetic. RULES OF MENSURATION. 474. SURFACES AND LINES. 1. To find the area of a rectangle, Multiply the length by the width. 2. To find either side of a rectangle, Divide the area by the other side. 3. To find the area of a triangle, Multiply the base by one half of the altitude. 4. To find the area of any quadrilateral having two sides parallel, Multiply one half of the sum of the two parallel sides by the perpendicular distance between them. 5. To find the circumference of a circle, 1. Multiply the diameter by 3.1416. Or, 6. To find the area of a circle, 1. Multiply the square of the diameter by .7854. Or, 7. To find the diameter of a circle, whose area is given, Divide the area by .7854, and extract the square root of the quotient. 8. To find the side of the largest square that can be inscribed in a circle, Multiply the radius by the square root of 2. 9. To find the side of the largest equilateral triangle that can be inscribed in a circle, Multiply the radius by the square root of 3. 10. To find the area of an ellipse, the two diameters being given, Multiply the product of the two diameters by .7854. 11. To find the surface of a sphere, 1. Multiply the circumference by the diameter. Or, 12. To find the entire surface of a right prism or right cylinder, Multiply the perimeter or circumference of the base by the height, and, to the product, add the surface of the two bases. 13. To find the convex surface of a pyramid or cone, Multiply the perimeter or circumference of the base by one half the slant height. 14. To find the hypotenuse of a right-angled triangle, |