8. Daniel Gould and Albert Austin formed a partnership for 2 years, under the firm of Gould & Austin. Gould at first put in $3000, and at the end of 1 year put in $2000 more. Austin put in at first $2500, and at the end of 6 months put in $4000 more. Gould managed the business, for which he was to receive a salary of $800 per year. When the partnership espired, they closed up their affairs, and found that the property of the firm was worth $5000 more than the capital which had been contributed. How should it be divided ? 9. What would have been the answer to the last question had the property of the firm been worth $5000 less than the capital which had been contributed ? 133. Assessment of Taxes. (a.) Taxes are assessments laid on persons or property, usually for some public purpose. (b.) A tax on persons is called a Poll Tax, and a tax on property is called a PROPERTY Tax. (c.) The poll tax is only assessed on males between certain ages, as between twenty-one and seventy; and in some States it is pot assessed. (d.) In assessing taxes it is necessary 1st. To estimate the value of all the property to be taxed, and make a complete inventory of it. 2d. To find the number of polls, i. e. the number of persons liable to pay a poll tax. 3d. To determine what portion of the tax is to be raised upon the polls, and to divide it equally among them. 4th. To find how much must be paid on each dollar of the taxable property, in order to raise the remainder of the tax. (e.) This last may be done by dividing the amount to be raised by the estimated value of the property on which it is to be raised. It will then be easy to find the tax of any individual. 1. A tax of $2500 is to be raised in a certain town. The taxable property is valued at $850000, and there are 300 polls, each taxed $1.25. What will be the tax on each dollar, and what will be the tax of each of the following persons ? A, who pays a tax on $4480, and 2 polls. D, who pays a tax on $11500, and 1 poll. SOLUTION.—The tax on 300 polls at $1.25 each is $375, which, subtracted from $2500, leaves $2125 to be assessed on the property. Since $850000 is to be taxed $2125, one dollar will be taxed 350.oo of $2125, which is 27 mills. A's tax on 2 polls would be twice $1.25, or $2.50, and on $4480 property would be 4480 times 2} mills, which is $11.20. $2.50 + $11.20 gives $13.70 as the amount of A's tax. Find the tax of the others in the same way. 2. A tax of $6108.72 is to be raised by a certain town. The taxable property is valued at $1,824,600, and there are 240 polls, each taxed $1.121. How much is the tax on $1? How much is the tax on each of the following persons ? W, who pays a tax on $3620, and 1 poll. SECTION XIX. POWERS AND ROOTS. 134. Definitions. (a.) The product of a number taken any number of times as a factor is called a Power of the number. (See 68, j, k, and l.) (b.) A Root of a number is such a number as taken some number of times as a factor will produce the given number. (c.) If the rool must be taken twice as a factor to produce the number, it is the SQUARE Root, or the Second Root; if three times, it is the Cube Root, or the THIRD Root; if four times, it is the FOURTH Root, etc. ILLUSTRATION. — -3 is the square root of 9, the third root of 27, and the fourth root of 81; because 32 = 9, 38 = 27, 34 81. (d.) The character, V, called the RADICAL Sign, is used to indicate that the root of the number before which it is placed is to be extracted. (e.) The DEGREE of the root is indicated by a small figure called an Index, which is placed a little above and at the left of the radical sign. When no index is written, the square root is required. ILLUSTRATIONS. -V 49 or 49 means the square root of 49. (f.) We may also indicate that a root is to be extracted by using & FRACTIONAL EXPONENT. 811 ✓ 4; st ILLUSTRATIONS. 812 81; 48°, etc. (g.) The process of finding the powers of numbers is called INVOLUTION ; and the process of finding their roots is called Evolution, or the ExtRACTION OF Roots. 135. Relation of Square to its Root. (a.) TABLE OF SQUARES. 72 = 49 82 = 64 9' = 81 10000 100000000 100% = 10000 1000009 10000000000 1000o = 1000000 1000000 = 1000000000000 = 100 (b.) The preceding table shows, first, that there are below 100 bat 9 entire numbers which are perfect squares (c.) Second, that the entire part of the square root of any number below 100 will be less than 10, and therefore contain but one figure; of any number between 100 and 10000 will lie between 10 and 100, and therefore contain 2 figures; of any number bei tween 10000 and 1000000 will lie between 100 and 1000, and therefore contain 3 figures, etc.; or, in other words – (d.) If any number contains either 1 or 2 figures, the entire part of its root must contain 1 figure; if it contains 3 or 4 figures, the entire part of its root must contain 2 figures, etc. (e.) Hence, if we should begin at the right of any number, and separate it into periods of two figures each, the number of periods would be the same as the number of figures in its square root. The square of the highest denomination of the root would be found in the left-hand period; the square of the two highest de nominations would be found in the two left-hand periods, etc. (f.) In order to understand the method of extracting the square root, it is necessary to consider how the square of a number consisting of two parts is formed from those parts. To do this, let a represent any number whatever, and b represent any other number. Then will a + b represent the sum, and (a + b)', or (a + b) x (a + b), the square of the sum of any two numbers whatever. (g.) Performing the multiplication, we have a times a = aʼ; a times b = a × b, or, as it may be written, ab; 6 times a = a times 6 = a X b, or a b; b times b= 7. (h.) Writing the work as below, and adding the partial products, we have a to a to ab + b (a + b) x (a + b) = a + 2a times b + 13 a + 2 ab + b?. (i.) Hence (a + b)' = a + 2ab + b?, or, since a’ equals the square of the first number, and 2 a b equals twice the product of the first number by the second, and b' equals the square of the second, it follows that (j.) The square of the sum of any two numbers equals the square of the first number plus twice the product of the first number by the second, plus the square of the second. ILLUSTRATIONS. (7 + 3), or 10% = 72 + twice 7 times 3 + 32 49 + 42 + 9 100. (30 + 8), or 382 302 + twice 30 times 8 + 82 = 900 + 480 + 64 - 1444. (k.) The square a' + 2 ab + b may take another form ; for 2 a b + b = 2 a times b + b times b (2 a + b) x b, or, by omitting the sign of multiplication, which may be done without ambiguity, = (2 a + b) b. (1.) Hence (a + b)' = a + 2 a b + = a + (2 a + b) b, or since a” equals the square of the first number, and (2 a + b) b equals the product obtained by multiplying the sum of twice the first plus the second, by the second, it follows that – (m.) The square of the sum of any two numbers is equal to the square of the first number plus the product of two factors, one of which is the sum of twice the first number plus the second, and the other is the second number. (7 + 3)?, or 109 ILLUSTRATIONS. 136. Method of extracting the Square Root. 1. What is the square root of 877969 ? SOLUTION.—This number contains three periods of two figures each, which shows that there are three figures in its root. Moreover, the greatest square below 87 is the square of the highest denomination of the root, and the greatest square below 8779 is the square of the two highest denominations of the root. Hence, we may find the first two figures of the required root by finding the square root of 8779 as though it were units. The greatest square below 87, the left-hand period, is 81, the root of which, 9, is the first figure of the required root. Subtracting 81 from 87, and |