COMPOUND PROPORTION. COMPOUND PROPORTION is the rule by means of which such questions as would require two or more statings in simple proportion (Rule of Three) can be resolved in one. As the rule, however, is but little used, and not easily acquired, it is deemed preferable to omit it here, and to show the operation by two or more statings. EXAMPLE.-How many men can dig a trench 135 feet long in 8 days, when 16 men can dig 54 feet in 6 days? Feet. Men. Days Men. Feet. Men. Days. Men. EXAMPLE. If a man travel 130 miles in 3 days of twelve hours each, in how many days of 10 hours each would he require to travel 360 miles? EXAMPLE.-If 12 men in 15 days of 12 hours build a wall 30 feet long, 6 wide, and 3 deep, in how many days of 8 hours will 60 men build a wall 300 feet long, 8 wide, and 6 deep? Ans. 120 days. INVOLUTION. INVOLUTION is the multiplying any number into itself a certain number of times... The products obtained are called POWERS. The number is called the RooT, or first power. When a number is multiplied by itself once, the product is the square of that number; twice, the cube; three times, the biquadrate, &c. Thus, of the number 5. 5 is the Root, or 1st power. 66 5X5= 25 5X5X5=125 66 66 5X5X5X5=625 Square, or 2d power, and is expressed 52. Cube, or 3d power, and is expressed 53. The little figure denoting the power is called the INDEX or Exponent. EVOLUTION is finding the Roor of any number. The sign✓ placed before any number, indicates the square root of that numbe is required or shown. The same character expresses any other root by placing the index above it. Thus, 255, and 4+2=√36. Roots which only approximate are called Surd Roots. 4. TO EXTRACT THE SQUARE ROOT. RULE.-Point off the given number from units' place into periods of two figures each. Find the greatest square in the left-hand period, and place its root in the quotient; subtract the square number from the left-hand period, and to the remainder bring down the next period for a dividend. Double the root already found for a divisor; find how many times the divisor is Contained in the dividend, exclusive of the right-hand figure, place the result in the quotient, and at the right hand of the divisor. Multiply the divisor by the last quotient figure, and subtract the product from the dividend; bring down the next period, and proceed as before. NOTE.-Mixed decimals must be pointed off both ways from units. SQUARE ROOTS OF VULGAR FRACTIONS. RULE. Reduce the fractions to their lowest terms, and that fraction to a decimal, and proceed as in whole numbers and decimals. NOTE. When the terms of the fractions are squares, take the root of each and set one above the other; as, & is the square root of 3. EXAMPLE.-What is the square root of 12? Ans. 0.86602540. To find the 4th root of a number, extract the square root twice, and for the 8th root thrice, &c., &c. TO EXTRACT THE CUBE ROOT. RULE. From the table of Roots (page 99) take the nearest cube to the given number, and call it the assumed cube. Then say, as the given number added to twice the assumed cube is to the assumed cube added to twice the given number, so is the root of the assumed cube to the required root, nearly. And, by using in like manner the root thus found as an assumed cube, and proceeding as above, another root will be found still nearer; and in the same manner as far as may be deemed necessary. EXAMPLE.-What is the cube root of 10517.9 ? Nearest cube, page 99, 10648, root 22. Then say, as the sum of n+1XA and n-1XP is to the sum of n+1XP and B-1XA, so is the assumed root to the required root R. EXAMPLE.-What is the cube root of 1500? The nearest cube, page 99, is 1331, root 11. then, P1500, n=3, A= 1331, r=11; n+1XA5324, n+1XP = 6000 8324 : 8662: 11: 11.446 Ans. ARITHMETICAL PROGRESSION. ARITHMETICAL PROGRESSION is a series of numbers increasing or decreasing by a constant number or difference; as, 1, 3, 5, 7, 9, 15, 12, 9, 6, 3. The numbers which form the series are called Terms; the first and last are called the Extremes, and the others the Means. When any three of the following parts are given, the remaining two can be found, viz.: The First term, the Last term, the Number of terms, the COMMON DIFFERENCE, and the SUM of all the terms. When the First Term, the Common Difference, and the Number of Terms are given, to find the Last Term. RULE.-Multiply the number of terms less one, by the common difference, and to the product add the first term. EXAMPLE. A man travelled for 12 days, going 3 miles the first day, 8 the second, and so on; how far did he travel the last day? 12-1x5+358 Ans. When the Number of Terms and the Extremes are given, to find the Common Difference. RULE.-Divide the difference of the extremes, by one less than the number of terms. EXAMPLE. The extremes are 3 and 15, and the number of terms 7; what is the common difference? 15-3-(7-1)= When the Extremes and Number of Terms are given, to find the Sum of all the Terms. RULE.-Multiply the number of terms by half the sum of the extremes. EXAMPLE.-How many times does the hammer of a clock strike in 12 hours? 12X(13÷2): =78 Ans. When the Common Difference and the Extremes are given, to find the Number of Terms. RULE.-Divide the difference of the extremes by the common difference, and add one to the quotient. EXAMPLE.-A man travelled 3 miles the first day, 5 the second, 7 the third, and so on, till he went 57 miles in one day. How many days had he travelled at the close of the last day? 57-3+2+1=28 Ans. To find two Arithmetical Means between two given Extremes. RULE.-Subtract the less extreme from the greater, and divide the difference by three, the quotient will be the common difference, which, being added to the less extreme, or taken from the greater, will give the means. EXAMPLE. Find two arithmetical means between 4 and 16. 16-4÷3 4 com. dif. 4+4 8 one mean. 16-412 second mean. To find any Number of Arithmetical Means between two Extremes. RULE-Subtract the less extreme from the greater, and divide the difference by one more than the number of means required to be found, and then proceed as in the foregoing rule. GEOMETRICAL PROGRESSION. GEOMETRICAL PROGRESSION is any series of numbers continually increasing by a constant multiplier, or decreasing by a constant divisor. As, 1, 2, 4, 8, 16, and 15, 71, 34. The constant multiplier or divisor is the RATIO. When any three of the following parts are given, the remaining two can be found, viz.: The FIRST term, the LAST term, the NUMBER of terms, the RATIO, and the SUM OF ALL THE TERMS. When the Ratio, Number of Terms, and the First Term are given, to find the Last Term. RULE. Write a few of the leading terms of the series, and place their indices over them, beginning with a cipher. Add together the most convenient indices, to make an index less by one than the number of the term sought. Multiply together the terms of the series or powers belonging to those indices, and the product, multiplied by the first term, will be the answer. NOTE. When the first term is equal to the ratio, the indices must begin with a unit. EXAMPLE. The first term is 1, the ratio 2, and the number of terms 23; what is the last term? Indices. 0 1 2 3 4 5 6 7 Terms. 1, 2, 4, 8, 16, 32, 64, 128. 1+2+3+4+5+7=22. 128X32X16X8X4X2=4194304X14194304 Ans. EXAMPLE. If one cent had been put out at interest in 1630, what would it have amounted to in 1834 if it had doubled itself every 12 years? 1834--1630204+12= 17.17+1=18. 0 1 2 3 4 7 1, 2, 4, 8, 16, 128, 1+2+3+4+7= 17. When the First Term, the Ratio, and the Number of Terms are given, to find the Sum of the Series. RULE.-Raise the ratio to a power whose index is equal to the number of terms, from which subtract 1; then divide the remainder by the ratio less 1, and multiply the quotient by the first term. EXAMPLE. If a man were to buy 12 horses, giving 2 cents for the first horse, 6 cents for the second, and so on, what would they cost him? 312=531441-1 = 531440÷÷(3-1)=2=265720×2=$5.314.40 Ans. By another Method, the greater Extreme being known. A TABLE OF GEOMETRICAL PROGRESSION, Whereby any questions of Geometrical Progression proceeding from 1, an of double ratio, may be solved by inspection, if the number of terms ex ceed not 50. PERMUTATION is a rule for finding how many different ways, any given number of things may be varied in their position. RULE.-Multiply all the terms continually together, and the last product will be the answer. EXAMPLE.-How many variations will the nine digits admit of? 1X2×3×4X5×6×7×8×9=362880 Ans. COMBINATION. COMBINATION is a rule for finding how often a less number of things, can be chosen from a greater. RULE.-Multiply together the natural series, 1, 2, 3, &c., up to the number to be taken at a time. Take a series of as many terms, decreasing by 1, from the num ber out of which the choice is to be made, and find their continued product Di vide this last product by the former, and the quotient is the answer. EXAMPLE.-How many combinations may be made of 7 letters out of 12? 1X 2X 3X4X5X6X7 5040. 12X11X10X9X8X7X6: 39916805040792 Ans. EXAMPLE.-How many combinations can be made of 5 letters out of 10? POSITION is of two kinds, SINGLE and DOUBLE, and is determined by the number of SUPPOSITIONS. SINGLE POSITION. RULE. Take any number, and proceed with it as though it were the correct one; then say, as the result is to the given sum, so is the supposed number to the number required. EXAMPLE.-A commander of a vessel, after sending away in boats, †, and 4 of his crew, had left 300; what number had he in command? Suppose he had 600. of 600 is 200 EXAMPLE.-A person being asked his age, replied, if of my age be multiplied by 2, and that product added to half the years I have lived, the sum will be 75. How old was he? Ans. 37 years. DOUBLE POSITION. RULE. Take any two numbers, and proceed with each according to the condi |