Thus we see the question solved by the Single Rule of Three in plain and easy statements; for it is evident, if 100 dollars gain 6 dollars in one year, that 600 dollars must gain 36 dollars; and it is also obvious, that the gain must be proportional to the time; for as 12 months, are to the gain for that time, so 9 months, are to the gain for that time. Perhaps this rule will appear more simple to the young student, by carrying it still further back, and illustrating it by Multiplication and Division, as follows: The principles of this rule must appear plain to the student when he discovers that it is only an application of the Single Rule of Three, and from that, may be reduced back to Multiplication and Division ; because by reducing it back, all the difficulty vanishes. 2. If 8 persons expend 200 dollars in 9 months, how much will serve 18 persons 12 months? Ans. $600. NOTE. Of the three conditional terms, or terms of supposition, the student will discover, that the eight persons must be the cause of expense, and consequently put for the first term; and 9 months, the time, and therefore put in the second place; and 200 dollars is what was expended in that time, and consequently it is put in the third place; the other terms regularly fall under those of the same name. 3. If 10 men spend 9 dollars in 12 weeks, how much will 20 men spend in 24 weeks? Ans. $36. 4. If $100, in 12 months, gain 7 dollars interest, what will $600 gain, in 8 months?. Ans. $28. 5. If 20 bushels of oats be sufficient for 18 horses, 20 days, how many bushels will serve 60 horses, 36 days? Ans. 120 bushels. 6. If $100, in one year, gain 7 dollars interest, what sum will gain $38,50cts. in 15 months? Ans. $440. mo. $cts. 100:12: 7,00 15::38,50 NOTE. In this question, the proportion is inverse; therefore, the third and fourth terms are multiplied together for a divisor and the other terms for a dividend. 7. If $100 gain 7 dollars in one year, in what time will 440 dollars gain $38,50cts.? Ans. 15 months. 8. An usurer put out $650, to receive interest for the same; at the end of 6 months he received for principal and interest, $672,75cts. ; at what rate per cent did he receive interest? Ans. 7 per cent. 9. If 100 men, in 6 days of 10 hours each, can dig a trench 200 yards long,, 3 wide, and 2 deep; in how many days of 8 hours long, will 180 men dig a trench of 360 yards long, 4 wide, and 3 deep? Ans. 15 days. QUESTIONS ON THE DOUBLE KULE OF THREE. What does the Double Rule of Three teach? A. It teaches to resolve, by one statement, such questions as require two or more statements, when worked by the Single Rule of Three. What number of terms is generally given in the Double Rule of Three? A. Five.What rule do you observe, in stating questions, in this rule? A. Write that term which is the principal cause of gain, loss, or action, for the first term; time or distance for the second; and, gain, loss, or action, for the third: and then write the two remaining terms under their corresponding terms, that is, terms of the same name. When the blank falls under the third term, how do you proceed? A. Multiply the first and second terms together for a divisor, and the other three for a dividend, and the quotient will be the answer. But when the blank falls under the first or second terms, multiply the third and 4th terms for a divisor, and the other three for a dividend, and the quotient will be the answer. What term must be of the same kind with the answer? A. The term directly above the blank. EQUATION OF PAYMEMTS, Teaches how to reduce several stated times, at which money is payable, to one mean or equated time, for the payment of the whole. RULE.-Multiply each payment by its time, and add the several products together; then divide the sum of the products by the whole debt, and the quotient will be the equated time, or answer. PROOF. The interest of the whole sum to the equated time, at any given rate, will equal the interest of the several payments, for their respective times, at the same rate. NOTE. This rule is founded on the supposition that what is gained, by keeping some of the debts or payments after they are due, is lost by paying other debts or payments, before they are due. EXAMPLES. 1. A, owes B, $800, to be paid as follows, viz. $400 in 4 months, and $400 in 8 months; what is the equated time for the payment of the whole debt? $400X4=1600 Ans. 6 mo's. DEM. It is evident, if B wait on A, two months, after one half of his debt is due, that B should receive the other half, two months before it is due. And it is also evi dent, that the product of each payment, divided by the payment, gives the time it is due; then the sum of the products divided by the sum of the payments, must give the equated time. 2. B, owes C, $400, of which $200 are to be paid in two months, and $200 in four months; but they agree that the whole shall be paid at one time; at what time must it be paid? Ans. in 3 months. 3. A, owes B, $760, to be paid as follows; $200 in 6 months, 240, in 7 months, and 320, in 10 months; what is the equated time for the payment of the whole debt? Ans. 8 months. 4. A, holds B's note for $1200, which is to be paid in the following manner; in 6 months, in 8 months, and the remainder in 10 months; what is the equated time for the payment of the whole? Ans. 7 months. QUESTIONS ON EQUATION OF PAYMENTS What does Equation of Payments teach? A. It teaches to find a mean time for the payment of a debt, which is made payable by instalments. How do you proceed in the work? A. First, multiply each payment by its time, then divide the sum of the products, by the whole debt, and the quotient will be the answer. On what supposition is this rule founded? A. On the supposition that what is gained by putting off some payments after they are due, is lost by paying others, before they are due. INVOLUTION, Teaches how to find the powers of numbers. A power is a product or number produced by multiplying any given number, called the root, a certain number of times, continually by itself. Thus 2 is the root or 1st power of 2; 4 is the 2d power, or square of 2, produced thus, 2X2-4; 8 is the 3d power, or cube of 2, produced thus, 2x2x2=8; 16 is the 4th power, or biquadrate of 2, produced thus, 2×2×2×2= 16, and so on. The power is often denoted by a figure placed at the right hand of the number, and a little above it, which figure is called the index or exponent of that power. This index or exponent is always one more than the number of multiplications, to produce the power; or it is equal to the number of times the given number is taken as a factor, in producing the power; thus, 3 is used twice to produce the square, or 2d power, 3×3=32, or 9; and the cube, or 3d power, 3×3×3=33, or 27; and so on. Thus the student will perceive, in finding the square of 3, there is only one multiplication, or two factors; in finding the cube there are two multiplications, or three factors, and so on. Involution is performed by the following RULE.-Multiply the given number, or first power, continually by itself, till the number of multiplications be one less than the index, or exponent of the power to be found, and the last product will be the power required. The powers of the nine digits, from the 1st power to the 5th, may be found in the following Table; 16 EXAMPLES. 1. What is the 5th power of 4? 81 Cubes, 343 512 729 Biquadrates, or 4th Powers. |1|16|81| 256 625|1296| 2401| 4096| 6561 or 5th Powers. |1|32|243|1024|3125|7776|16807|32768|59049| 64 125 216 4 The 1st power or root. 7. What is the cube of 60? 8. What is the square of 1? Ans. 216,000. Ans. 1. Ans. 1. 9. What is the cube of 1? NOTE.-A decimal fraction is involved or raised to any power, the same as a whole number, and the same rules are observed in pointing off as in Multiplication of Decimals. A vulgar Fraction is raised to any power by multiplying the numerator of the fraction by itself, and the denominator by itself, till the number of multiplications be one less than the index, or exponent of the power to be found; then the power of the numerator, placed over the power of the denominator, gives the power of the fraction sought. If it be required to raise a mixed number to a certain power, first reduce it to an improper fraction, and then proceed as with a simple fraction. But those who desire it, may first reduce the given fraction to a decimal, and then raise the decimal to the power required. 10. What is the square or 2d power of ,5? 11. What is the cube, or 3d power of,5? 12. What is the square of of 21? 13. What is the cube, or 3d power of 4? square, or 2d power 16. How much is 93, that is, the 3d Ans. ,25. Ans.,125. Ans.. Ans. 64 343 Ans. 25 Ans. 289-18. 16 power of 9? Ans. 729. Ans. 7776. 17. How much is 65 ? 18. How much is 104 ? Ans. 10,000. NOTE. It will be seen from the preceding examples, that raising a simple fraction, whether vulgar or decimal, to a higher power, diminishes it in the same proportion as a whole number becomes increased. EVOLUTION, Is the extracting or finding the roots of any given powers; or it is exactly the reverse of Involution. The root of any number or power is such a number, as being multiplied into itself a certain number of times, wil produce that power. Thus, 2 is the square root, or 2d root of 4, because 22-2×2-4; and 3 is the cube root, or 3d root of 27, because 333×3×3—27. The power of any given number or root may be found exactly by multiplying the number continually into itself. But there are numbers, of which a proposed root can never be exactly found. Yet, by means of decimals, we may approximate or approach towards the root, to any degree of exactness. Those numbers whose roots only approximate towards the true roots, are called surd numbers; but those whose roots can be exactly found, are called rational numbers. The Roots are sometimes denoted by writing the character, be |