A SQUARE NUMBER is the product of two equal factors; or, the product of a number multiplied by itself. The SQUARE ROOT is the number, which, being multiplied by itself, produces the square number. A CUBE is the product of three equal factors; or, the product of a number twice multiplied by itself. The CUBE ROOT is the number, which, being twice multiplied by itself, produces the cube. Property 1. The sum, or the difference of any two even numbers, is an even number. Prop. 2. The sum, or difference, of two odd numbers is even; but the sum of three odd numbers is odd. Prop. 3. The sum of an even number of odd numbers is even; but the sum of an odd number of odd numbers is odd. Prop. 4. The sum, or the difference of an even number and an odd number, is odd. Prop. 5. The product of an even, and an odd number, or of two even numbers, is even. Prop. 6. An odd number cannot be divided by an even number, without a remainder. Prop. 7. A square number, or a cube number, arising from an even root, is even. Prop. 8. The product of any two odd numbers is an odd number.. Prop. 9. The product of any number of odd nunbers is odd: hence the square, and the cube of an odd number are odd. Prop. 10. If an odd number measure an even number, it will also measure the half of it. Prop: 11. If a square number he either multiplied or divided by a square, the product or quotient is a square. . Prop. 12. If a square number be either multiplied or divided by a number that is not a square, the product or quotient is not a square. Prop. 13. The difference between an integral cube and its rooi, is always divisible by 6. Prop. 14. The product arising from two different prime numbers cannot be a square. Prop 15. The product of no two different numbers, prime to each other, can make a square, unless each of ihose numbers be a square. Prop. 16. Every prime number above 2, is either I greater or 1 less than some multiple of 4. Prop. 17. Every prime number above 3, is either 1 greater or 1 less than some multiple of 6. Prop. 18. The number of prime numbers is unlimited. The first ten are, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23. The learner may find the succeeding ten. VIII. PROBLEMS. A PROBLEM is a proposition or a question requiring something to be done; either to investigate some truth or property, or to perform some operation. The following Problems and Rules are founded in the correspondence of the four principal operations of arithmetic; viz. Addition, Subtraction, Multiplication, and Division. PROBLEM I. The sum of two numbers, and one of the numbers being given, to find the other. RULE. Subtract the given number from the given sum; the remainder will be the number required. 1. Suppose 37486 to be the sum of two numbers, one of which is 8602; what is the other? 2. 33000 news-papers are sold in London, daily: of these, 17500 are morning papers, the rest, evening: how many of the latter? PROBLEM II. The difference between two numbers, and the greater number being given, to find the smaller. RULE. Subtract the difference from the greater number; the remainder will be the number required. 3. If 1406 be the difference between two numbers, and the greater number be 4879, what is the smaller ? 4. The area of North and South America is 18000000 square miles: that of North America is 11000000: what is that of South America? PROBLEM III. The difference between two numbers, and the smaller number being given, to find the greater. RULE. Add the smaller number and the difference together; the sum will be the number required. 5. Suppose 86974 to be the difference beiween two numbers, and the smaller number to be 7064; what is the greater number? 6. The British House of Lords consists of 427 members; the number in the House of Commons is 131 greater. How many are there in the House of Commons? PROBLEM IV. The sum and difference of two numbers being given, to find the numbers. RULE. Subtract the difference from the sun, and divide the remainder by 2; the quotient will be the smaller number. Then add the given difference to the smaller nuniber, and this sum will be the greater number. 7. What are the two numbers whose sum is 1094, and whose difference is 154? 8. The United States Congress, consisting of a Senate and House of Representatives, has 255 members. The House has 192 members more than the Senate. How many in each branch? PROBLEM V. The product of two factors, and one of the factors being given, to find the other. RULE Divide the product by the given factor, and the quotient will be the required factor. 9. 1246038349 is the product of some two numbers, one of wliich is 269191: what is the other? 10. Suppose a session of Congress which continues 180 days, to cost 504000 dollars; what is the expense per day, to the United States? PROBLEM VI. The dividend and quotient being given to find the divisor. RULE. Divide the dividend by the given quotient, and the quotient thence arising will be the number sought. 11. Suppose 101442075 to be a dividend, and 4025 the quotient; what is the divisor? 12. 17155 pounds of beef having been equally divided among a number of soldiers, each one found that his share was 47 pounds. What was the number of soldiers ? PROBLEM VII. The divisor and quotient being given, to find the dividend. RULE. Multiply the divisor and quotient together; the product will be the required dividend. 13. If 800027 be a divisor, and 97563 the quotient, what number is the dividend ? 14. A quantity of beef was divided equally among 2742 soldiers, and each soldier received for his share 152 pounds. What quantity was divided ? PROBLEM VIII. The product of three factors, and two of those factors being given, to find the third factor. RULE. Find the product of the two given. factors, and by this number divide the given product; the quotient will be the factor required. 15. Suppose the product of three factors to be 1344, one of these factors being 12, and another 8; what is the third factor? 16. How many days will 9720 pounds of hay last 12 horses; allowing each horse to eat 45 pounds a day? PROBLEM IX. Two numbers being given, to find their greatest common measure; that is, the greatest number which will divide them both without a remainder. RULE. Divide the greater number by the 'smaller, and this divisor by the remainder, and thus continue dividing the last divisor by the last remainder, till nothing remains. The divisor last used will be the number required. When the greatest common measure of more than two numbers is required, first, find the greatest common mearure of any two of the numbers, then find the greatest common measure of the number found and another of the given numbers, and thus proceed, till all the given numbers are brought in. 17. What is the greatest common measure of 918, 1998, and 522 ? 918)1998(2 54)522(9 1836 486 162)9185 36) 54 (1 810 36 108) 162 1 18) 36(2 36 Ans. 18. The truth of the rule in this problem will be discovered by retracing the first of the above operations, as follows. Since 54 (the last divisor] measures 108, it also measures 103 +54, or 162. Again, since 54 measures 108 ana 162, it also measures 5 X 162 +108, or 913. In the same manner it will be found to measure 2 X 918 +162, or 1998. Therefore, 54 measures both 918 and 1998. It is also the greatest common measure; for, suppose there be a greater then, since the greater measures 918 and 1998, it also measures the remainder, 162; and since it measures 162 and 918, it also measures the remainder 108; in the same manner it will be found to measure the remuinder, 54; that is, the greater measures the less, which is absurd. 13. What is the greatest common measure of the numpers, 323 and 425 ? 19. What is the greatest common measure of 2310 and 4626 ? 20. What is the greatest common measure of 1092, 1428, 1197 and 805 ? 21. Suppose a hall to be 154 feet long, and 55 wide; what is the length of the longest pole, that will exactly measure both the length and width of the hall ? 22. A owns 720 rods of land, B owns 336 rods, and C 1736 rods. They agree to divide their land into equal house lots, fixing on the greatest nurnber of rods for a lot, |