9. A man would mix 4 bushels of wheat at $1'50 per bushel, rye at $1'16, corn at $75, and barley at $'50, so as to sell the mixture at 84 per bushel; how much of each must he use ? 10. A goldsmith would mix gold 17 carats fine, with some 19, 21, and 24 carats fine, so that the compound may be 22 carats fine; what proportions of each must he use ? Ans. 2 of the 3 first sorts, to 9 of the last. 11. If he use 1 oz. of the first kind, how much must he use of the others ? What would be the quantity of the compound ? Ans. to last, 74 ounces. 12. If he would have the whole compound consist of 15 oz., how much must he use of each kind ? if of 30 oz., how much of each kind ? of 37} oz., how much? Ans. to the last, 5 oz. of the 3 first, and 224 oz. of the last. Hence, when the quantity of the compound is given, we may say, -As the sum of the PROPORTIONAL quantities, found by the ABOVE RULE, is to the quantity REQUIRED, so is each PROPORTIONAL quantity, found by the rule, to the REQUIRED quantity of EACH. 13. A man would mix 100 pounds of sugar, some at 8 cents, some at 10 cts. and some at 14 cts. per pound, so that the compound may be worth 12 cents per pound; how much of each kind must he use ? We find the proportions to be 2, 2, and 6. Then, 2+2 +6= 10, and 2 : 20 lbs. at 8 cts. 10 : 100 :: 2 : 20 lbs. at 10 cts. Ans. 6 : 60 lbs. at 14 cts. 14. How many gallons of water, of no value, must be mixed with brandy åt $1“20 per gallon, so as to fill a vessel of 75 gallons, which may be worth 92 cents per gallon ? Ans. 174 gallons of water to 575 gallons of brandy. 15. A grocer has currants at 4 d., 6 d., 9 d., and 11 d. per lb.; and he would make a mixture of 240 lbs., so that the mixture may be sold at 8 d. per lb.; how many pounds of each sort may he take? Ans. 72, 24, 48, and 96 lbs., or 48, 48, 72, 72, &c. Note. This question may have five different answers. QUESTIONS. 1. What is alligation? 2. medial? 3. the rule for operating? 4. What is alligation alternate ? 5. When the price of the mixture, and the price of the several simples, are given, how do you find the proportional quantities of each simple ? 6. When the quan tity of one simple is given, how do you find the others ? 7. When the quantity of the whole compound is given, how do you find the quantity of each simple ? DUODECIMALS. 1 103. Duodecimals are fractions of a foot. The word is derived from the Latin word duodecim, which signifies twelve. A foot, instead of being divided decimally into ten equal parts, is divided duodecimally into twelve equal parts, called inches, or primes, marked thus, ('). Again, each of these parts is conceived to be divided into twelve other equal parts, called seconds, ("). In like manner, each second is conceived to be divided into twelve equal parts, called thirds, ("'); each third into twelve equal parts, called fourths, ("'); and so on to any extent. In this way of dividing a foot, it is obvious that I' inch, or prime, is of a foot. 1 second is 1 of 1 of a foot, 114 of of of a foot. T728 1" fourth is of of a foot. TE Its of 20736 I fifth is of 1 of T2 of a foot, &c Ta of is ofte 248832 Duodecimals are added and subtracted in the same manner as compound numbers, 12 of a less denomination making 1 of a greater, as in the following TABLE. 1" second, 12 inches, or primes, 1 foot. Note. The marks,', ", "", "', &c., which distinguish the different parts, are called the indices of the parts or denominations. MULTIPLICATION OF DUODECIMALS. Duodecimals are chiefly used in measuring surfaces and solids. 1. How many square feet in a board 16 feet 7 inches long, and I foot 3 inches wide ? Note. Length x breadth = superficial contents, (T 25.) OPERATION. 7 inches, or primes, = 72 of a foot, and 3 in. ft. ches = 12 of a foot; consequently, the product Length, 16 7 of 7' X 3'= of a foot, that is, 21" =l' and Breadth, 1 3 9"; wherefore, we set down the 9', and reserve 4 l' 9" the l' to be carried forward to its proper place. 16 71 To multiply 16 feet by 3', is to take pa of 16 Ans. 20 8 g 13, that is, 48'; and the l' which we re served makes 49, = 4 feet l'; we therefore set down the l', and carry forward the 4 feet to its proper place. Then, multiplying the multiplicand by the 1 foot in the multiplier, and adding the two products together, we obtain the Answer, 20 feet, 8', and 9". The only difficulty that can arise in the multiplication of duodecimals, is, in finding of what denomination is the product of any two denomination This may be ascertained as above, and in all cases it will be found to hold true, that the product of any two denominations will always be of the denomination denoted by the sum of their INDICES. Thus, in the above example, the sum of the indices of " X 3' is "; consequently, the product is 21";. and thus primes multiplied by primes will produce seconds; primes multiplied by seconds produce thirds ; fourths multiplied by fifths produce ninths, &c. It is generally most convenient, in practice, to multiply the multiplicand first by the feet of the multiplier, then by the inches, &c., thus : ft. 16 ft. X 1 ft. = 16 ft., and 7' X 1 ft. = 7'. Then, 16 1 3 ft. X 3 = 48' = 4 ft., and 7' X 3' = 21" = 1'9". The two products, added together, give for the Answer, 20 ft. 16 7 8' 9', as before. 4 1 9" 20 8' 9" 2. How many solid feet in a block 15 ft. 8 long, 1 st. 5' wide, and 1 ft. 44 thick? OPERATION. ft. Length, 15 8 The length multiplied by the breadth, and Breadth, 15 that product by the thickness, gives the solid contents, (T 36.) 15 8 6 6' 4" 22 2' 4" Thickness, 1 4' 22 24" 74' 9" 4" Ans. 29 7'1" 4" From these examples, we derive the following Rule :-Write down the denominations as compound numbers, and in multiplying remember that the product of any two denominations will always be of that denomination denoted by the sum of their indices. EXAMPLES FOR PRACTICE. 3. How many square feet in a stock of 15 boards, 12 ft. 8' in length, and 13 wide ? Ans. 205 ft. 10%. 4. What is the product of 371 ft. 2'6" multiplied by 181 ft. 1'9" ? Ans. 67242 ft. 10 1" 4' 6m. Note. Painting, plastering, paving, and some other kinds of work, are done by the square yard. If the contents in square feet be divided by 9, the quotient, it is evident, will be square yards. 5. A man painted the walls of a room 8 ft. 2' in height, and 72 ft. 4' in compass ; (that is, the measure of all its sides ;) how many square yards did he paint ? Ans. 65 yds. 5 ft. 8'S". 6. There is a room plastered, the compass of which is 47 st. 3', and the height 7 ft. 6'; what are the contents ? Ans. 39 yds. 3 ft. 4'6". 7. How many cord feet of wood in a load 8 feet long, 4 feet wide, and 3 seet 6 inches high? Note. It will be recollected that 16 solid feet make a cord foot. Ans. 7 cord feet. 8. In a pile of wood 176 feet in length, 3ft. 9 wide, and 4 ft. 3 high, how many cords? Ans. 21 cords, and 75 cord feet over 9. How many feet of cord wood in a load 7 feet long, 3 feet wide, and 3 feet 4 inches high; and what will it come to at $40 per cord foot ? Ans. 4. cord feet, and it will come to $1'75. 10. How much wood in a load 10 ft. in length, 3 ft. 9 in width, and 4 ft. 8' in height; and what will it cost at $1'92 per cord ? Ans. 1 cord and 212 cord feet, and it will come to $24624. 1 104. Remark. 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. On one end of the rule, for 1 foot, let each of these parts be divided into 10 other equal parts. The former division will be 10ths, and the latter 100ths of a foot. Such å 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 deci. mals of a foot, the casts will be no other than so many operations in decimal fractions. 11. How many square feet in a hearth stone, which, by a rule as above described, measures 4-5 feet in length, and 26 feet in width; and what will be its cost, at 75 cents per square foot? Ans. 117 feet; and it will cost $3*775. 12. How many cords in a load of wood 7'5 feet in length, 36 feet in width, and 4'8 feet in heighi ? Ans. 1 cord, 1 ft. 13. How many cord feet in a load of wood 10 feet long, 34 wide, and 3-5 feet high? Ans. 77. 16 QUESTIONS. 1. What are duodecimals? 2. From what is the word derived ? 3. Into how many parts is a foot usually divided, and what are the parts called ? 4. What are the other denominations? 5. What is understood by the indices of the denominations? 6. In what are duode. cimals chiefly feed ? 7. How are the contents of a surface bounded by straight lines found 7 8. How are the contents of a solid found ? 9. How is it known of what denomination is the product of any two denominations? 10. How may a scale or rule be formed for taking di. mensious in feet and decimal parts of a foot ? - of ? INVOLUTION. T 105. Involution, or the raising of powers, is the multiplying any given number into itself continually a certain number of times. The products thus produced are called the powers of the given number. The number itself is called the first power, or root. If the first power be multiplied by itself, the product is called the second power, or square; if the square be múltiplied by the first power, the product is called the third power, or cube, &c.; thus, 5 is the root, or 1st power, of 5. 5 X 5 = 25 is the 2d power, or square, of 5, '5 X 5 X 5 = 125 is the 3d power, or cube, of 5, 5 X 5 X 5 X 5= 625 is the 4th power, or biquadrate, of 5, The number denoting the power is called the index or exponent; thus 54 denotes that 5 is raised or involved to the 4th power. 1. What is the square, or 2d power, of 7 ? Ans. 49. 2. What is the square of 30 ? Ans. 900. 3. What is the square of 4000 ? Ans. 16000000, 4. What is the cube, or 3d power, of 4 ? Ans. 64. 5. What is the cube of 800 ? Ans. 512000000. 6. What is the 4th power of 60 ? Ans. 12960000. 7. What is the square of 1? - of 2 ? of 3 ? of 1? Ans. 1, 4, 9, and 16. 8. What is the cube of 1 ? of 2 ? of 4? Ans. 1, 8, 27, and 64. 9. What is the square of } ? of ? Ans: $, 25, and the 10. What is the cube of } ? of? of ?? Ans. 245, and 313. 11. What is the square of 1? the 5th power of } ? Ans. #, and 32. 12. What is the square of 1'5? the cube? Ans. 2-25, and 3375 13. What is the 6th power of 162? Ans, 2'985984. 14. Involve 24 to the 4th power. Note. A mixed number, like the above, may be reduced to an improper fraction before involving; thus, 24 =; or it may be reduced to a decimal, thus, 21 = 2'25. Ans. 6_561 25:16 6. 256 15. What is the square of 43 ? 1=2341 64 16. What is the the value of 74, that is, the 4th power of 7 ? Ans. 2401. 17. How much is 93 ? 10+? Ans. 729, 7776, 10000. 18. How much is 27 } 36 ? 65 ? 108 ? Ans. to last, 100000000. The powers of the 9 digits, from the first power to the 5th, may be seen in the following TABLE. Roots or Ist Powers 1 | 21 31 41 51 6 71 81 91 Squares or 2d Powers | 1 | 41 91 161 251 36 49 64 81! (Cubes or 3d Powers 1 | 8| 271 641 125) 216 3431 729 | Biquadrates or 4th Powers | 1 | 16/ 81! 256 625|1296 2401| 4096 6561 Sursolids or 5th Powers | 1 |32|243|1024|312517776|16807|32768|59019 Ans. 1521 512 EVOLUTION. 1106. 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. The square root is a number which, being squared, will produce the given number; and the cube, or third root, is a number which, being cubed or involved to the third 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 79, that is, 7 X7 X7,=343; and so of other numbers. Although there is no number which will not produce a perfect power by involution, yet there are many numbers of which precise roots can never be obtained. But, by the help of decimals, we can aproximate, or approach, towards the root lo any assigned degree of exactness. Numbers, whoso pre. cise 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 w 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 v16; and the cube root of 27 is expressed 3/27 ; and the 5th root of 7776, 577776. 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 21 – 5, &c. EXTRACTION OF THE SQUARE ROOT. T 107. To extract the square root of any number, is to find a number which, being multiplied into itself, shall produce the given number. 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 co-. ver? that is, what is one side of a square which contains 625 square yards ? We have seen, (9 35,) that the contents of a square surface is found by multiplying the length of one side into itself, that is, by raising it to the second power; and hence, having the contents (625) given, we must extract its square root to find one side of the room. This we must do by a sort of trial; and, Ist. We will endeavor to ascertain how many figures there will be in the root. This we can easily di, by pointing off the number, from units, into periods of two figures each; for the square of any root always contains just iwice as many, or 1 figure less than twice as many figures as are in the root; of which truth the pupil may easily satisfy himself by trial. Pointing off the number, we find that the root will consist OPERATION. of two figures, a ten and a unit. 2d. We will now seek for the first figure, 4 that is, for the tens of the root, and it is plain that we must extract it from the left 225 hand period, 6, (hundreds.) The greatest square in 6 (hundreds) we find, by trial, FIG. I. to be 4, (hundreds,) the root of which is 2, (tens, = 29 ;) therefore, we set 2 (Jens) in the root. The root, it will be recolleci. ed, is one side of a square. Let us, then, А form a square, (A, Fig. I.) each side of which shall be supposed 2 tens, = 20 yards, expressed by the root now obtained. 20 20 The contents of this square are 20 x 20 20 = 400 yards, now disposed of, and which, 400 consequently, are to be deducied from the whole number of yarus, (625,) leaving 225 yards. This deduction is most readily performed by subtracting the square num. ber 4, (hundreds.) or the square of 2, (the figure in the root already found,) from the 625 ( 2 20 |