EXAMPLES. 1. What are the two mean proportionals between 4 and 256? 256÷4=64; then√/64=4, and 4×4=16, the lesser, and 16×4=64, the greater. Proof, 4: 16:: 64: 256. 2. What are the two mean proportionals between 5 and 625 ? Ans. 25 and 125. 3. What are the two mean proportionals between 7 and 2401 ? Ans. 49 and 343. EXTRACTION OF ROOTS IN GENERAL. RULE. Art. 235.-I. Point the given number into periods of as many figures as the index of the root directs. Thus, for the square root, two figures; cube root, three; fourth root, four, etc. II. Find, by trial, the greatest root in the left-hand period, and subtract its power from that period, and to the remainder bring down the first figure of the next period, for a dividend. III. Involve the root, already found, to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor, by which find the second figure of the root. IV. Involve the whole root now found to the given power; subtract it from the given number, as before, and bring down the first figure of the next period to the remainder, for a new dividend, and proceed as before, till the work is finished. OBS.-The roots of most of the powers may be found by repeated extractions of the square and cube root-Thus: For the 4th root, take the square root of the square root. For the 6th take the square root of the cube root. 66 QUESTIONS.-1. Rule for finding a mean proportional between two numbers? 2. What is a cube? 3. What is a cube root? 4. What is it to extract the cube root? 5. What is the rule? 6. Why do you distinguish the given number into periods of three figures each? 7. Why do you multiply the square of the quotient by 300? 8. Why the quotient by 30? 9. Why the triple square by the last quotient figure? 10. Why the triple quotient by the square of the last quotient figure? 11. Explain the process of illustrating this rule by blocks. 12. What proportion have solids to one another? 13. Rule for finding two mean proportionals between two numbers? 14. Rule for extracting roots in general? For the 8th root, take the square root of the 4th root. take the cube root of the cube root. For the 9th 66 take the cube root of the 4th root. EXAMPLES. 1. What is the square root of 7569 ? 8 X 8= Operation. 7569(87 64 square, or 2d power, of the quotient. 21 x 21 x 21 X21 × 21=4084101-5th power of the quotient. 3. What is the fourth root of 140283207936? Ans. 612. 4. What is the seventh root of 4586471424? Ans. 24. 5. What is the ninth root of 1352605460594688 ? Ans. 48. ARITHMETICAL PROGRESSION. Art. 236. ARITHMETICAL PROGRESSION is when a series of numbers increases by a common excess, or decreases by a common difference. When numbers increase by a common excess, they form the ascending series, as 2, 4, 6, 8, 10, 12, etc. QUESTION.-1. What is Arithmetical Progression? When numbers decrease by a common difference, they form the descending series, as 12, 10, 8, 6, 4, 2, etc. The numbers forming the series are called the terms; the first and last terms are called the extremes, and the other terms the means. When any even number of terms differs by Arithmetical Progression, the sum of the two extremes will equal the sum of any two means equally distant from the extremes; as 2, 4, 6, 8, 10, 12. The two extremes, 2+12=6+8, the two means. When the number of terms is odd, the double of the mean will equal the sum of the two extremes, or the sum of any two numbers equally distant from the extremes; as 1, 2, 3, 4, 5. The double of the mean 3x2=5+1=6. In Arithmetical Progression, five things are to be considered, viz. the first and last terms, the number, common difference, and sum of all the terms; any three of which being given, the other two may be found. 1. If I buy 4 books, giving 2 cents for the first, 4 for the second, and so on, with a common difference of 2, what do I pay for the last book? It is evident, that if we add 2 cents, the common difference, to the price of the first book, we shall have the price of the second, and so on to the last; thus, 2+2=4, 4+2=6, 6+2 8 cents, the answer. It will be seen that 2, the common difference, is added to every term but the last. If, then, we multiply the number of terms, less 1, by the common difference, we have the difference between the cost of the first book and the last; thus, 3×2=6, and 6+2=8, as before. Therefore, Art. 237.-When the first term, the number of terms, and common difference are given, to find the last term: RULE. Multiply the number of terms, less 1, by the common difference, and to the product add the first term, and the sum will be the last term. 2. If the first term of a series be 5, the number of terms 35, and the common difference 3, what is the last term? Ans. 35-1x3=102+5=107. QUESTIONS.--2. When is the series ascending? 3. When descending? 4. What is meant by the terms? 5. What is meant by the extremes? 6. By the means? 3. If I buy 80 yards of cloth, giving 6 cents for the first, 10 for the second, and so on, with a common difference of 4, what do I pay for the last yard? Ans. 322 cents. 4. Suppose a man purchase 40 sheep, paying 3 pence for the first, 10 for the second, and so on, with a common difference of 7, what does he pay for the last sheep? Ans. 276 pence. 5. If 96 acres of land be sold at the rate of 10 cents for the first acre, 19 for the second, and so on, with a common difference of 9, for how much is the last acre sold? Ans, 865 cents. 6. If I buy 4 books, the prices of which are, in Arithmetical Progression, giving 2 cents for the first, and 8 for the last, what is the common difference in the prices of the books? This question is the reverse of question 1st. 8-2=6, 632, the common difference. It is plain, that the difference between the price of the first and last book, is the whole addition made to the price of the first book; and as the addition is made equally to the three books, it is equally plain that the whole addition, divided by the number of additions, will be the addition made to the price of each book. Therefore— Art. 238.-When the extremes and number of terms are given, to find the common difference, we have this RULE. Divide the difference of the extremes by the number of terms less 1, and the quotient will be the common difference. 7. If the first term of a series be 3, the last term 276, and the number of terms 40, what is the common difference? Ans. 7. 8. A man on a journey travels the first day 2 miles, and increases his travel daily by an equal excess for 15 days, so that the last day he travels 72 miles. What was the daily increase? Ans. 5 miles. 9. Bought books, paying 2 cents for the first, and 8 cents for the last, with a common difference of 2. What number of books did I buy? As the difference of the extremes, divided by the number of the terms less 1, will give the common difference, it is evident that the difference of the extremes divided by the common difference, will give the number of the terms less 1. Then, 8-2-6, the difference of the extremes, and 6÷2=3, which is one less than the number of terms; then 3+1=4, the number of books purchased. Therefore Art. 239.-When the first and last terms, and the common difference are given, to find the number of terms— RULE. Divide the difference of the extremes by the common difference, and the quotient will be 1 less than the number of terms. 10. If the first term of a series be 2, and the last term 72, the common difference 5, what is the number of terms? Ans. 15. 11. A man bought sheep, paying at the rate of 3 pence for the first, and 276 for the last, with a common difference of 7. What number did he buy? Ans. 40. 12. A man has a number of sons, the common difference of whose ages is 4 years; the youngest is 8, the eldest 40 years old. How many sons has he? Ans. 9. 13. If I buy 4 books, paying 2 cents for the first, and 8 cents for the last, how many cents do I pay in all ? If the price of the first book is 2 cents, and the price of the last is 8 cents, it is evident that the average price of the books is half way between 2 cents, the price of the first, and 8 cents, the price of the last book: 2+8÷2=5. Then 5, the average price, multiplied by the number of books, will give the whole cost: 5X4=20 cents. The same may also be shown by writing the double series, thus: 2 + 4 + 6 + 8 8+ 6 + 4 + 2 10 10 10 10 It will be seen by this formula, that the sum of any two corresponding terms in the double series is equal to the sum of the two extremes in the simple series; if, therefore, we multiply the sum of the extremes by the number of terms, we shall obtain a sum twice too large. Therefore Art. 240.-When the first and last terms, and the number of terms are given, to find the sum of the series |