errors. Multiply the first supposed number by the last error, and the last supposed number by the first error; and if the errors be alike (that is, both too great or both too small), divide the difference of the products by the difference of the errors; but if unlike, divide the sum of the products by the sum of the errors, and the quotient will be the answer. NOTE. This rule is founded on the supposition that the first error is to the second, as the difference between the true and first supposed is to the difference between the true and second supposed number; when that is not the case, the exact answer to the question cannot be found by this rule. 7. There is a fish, whose head is 10 inches long, his tail is as long as his head, and half the length of his body, and his body is as long as his head and tail both; what is the length of the fish? Suppose the fish to be 40 inches long, then 40 Again sup. 60 40 10 The above operation is called Double Position. The above question, and most others belonging to this rule, may be solved by fractions, thus: The body of the whole length; the tail of }+10=} +10, and the head 10: then +1+10+10 the length; but 44, and 4-11-10+10-20 in. and 20x480 in. Ans. 126 PERMUTATION OF QUANTITIES. 292, 293, 294. and B loses $87, then A's | than A, at the end of 4 years money is double that of B; what did each lay out? Ans. $300. 5. A and B have both the game income; A saves one fifth of his yearly, but B, by spending $50 per annum more finds himself $100 in debt; Bermutation of Quantities. 292. Permutation of Quantities is a rule, which enables us to deter mine how many different ways the order or position of any given number of things may be varied. 293. 1. How many changes may be made of the letters in the word and? The letter a can alone have only one position, a, denoted by 1, a and n can have two positions, an and na, denoted by 1X2=2. The three letters, a, n, and d, can, any two of them, leaving out the third, have two changes, Ix2, consequently when the third is taken in, there will be 1×2×3=6 changes, which may be thus expressed: and, adn, nda, nad, dan and dna, and the same may be shown of any number of things. Hence, 294. To find the number of permutations that can be made of a given number of different things. RULE. Multiply all the terms of the natural series of numbers from 1 up to the given number, continually together, and the last product will be the answer required 2. How many days can 7 persons be placed in a different position at dinner? 5040. | 3. How many changes may be rung on 6 bells? Ans. 720. 4. How many changes can be made in the position of the 8 notes of music? Ans. 40320. 5. How many changes may be rung on 12 bells, and how long would they be in ringing, supposing 10 changes to be rung in one minute, and the year to consist of 365 days, 5 hours and 49 minutes? Ans. 479001600 changes, and 91 years, 26d. 22h. 41m. time. Periodical Decimals. 295. The reduction of vulgar fractions to decimals (129) presents two cases, one in which the operation is terminated, as -0.375, and the other in which it does not terminate, as 0.272727, &c. In fractions of this last kind, whose decimal value cannot be exactly found, it will be observed that the same figures return periodically in the same order. Hence they have been denominated periodical decimals. 296. Since in the reduction of a vulgar fraction to a decimal, there can be no remainder in the successive divisions, except in one of the series of the numbers, 1, 2, 3, &c. up to the divisor, when the number of divisions exceeds that of this series, some one of the former remainders must recur, and consequently the partial dividends must return in the same order. The fraction 0.333+. Here the same figure is repeated continually; it is therefore called a single repetend. When two or more figures are repeated, as 0.2727+ (295), or 324324, it is called a compound repetend. A single repetend is denoted by a dot over the repeating figure, as 0.3, and a compound repetend by a dot over the first and last of the repeating figures, as 0324324. 297. The fractions which have 1 for a numerator, and any number of 9's for the denominator, can have no significant figure in their periods except 1. Thus -0.1111+. -0.01010+.5-0.001001001. This fact enables us easily to ascertain the vulgar fraction from which a periodical decimal is derived. As the 0.1111+ is the developement of †, 0.22+=3, 0.3=3, &c. Again, as 0.010101, or 0.01, is the developement of, 0.02, and so on, and in like manner of, &c. Hence, 298. To reduce a periodical, or circulating decimal, to a vulgar fraction. RULE. Write down one period for a numerator, and as many nines for a denominator as the number of figures in a period of the decimal. 1. What is an Arithmetical Progression? When is the series ascending? When descending? What is meant by the extremes? The means? When the first and last terms are given, how do you find the common difference? How the number of terins? How the sum of the series ? 2. What is a Geometrical Progression? What is an ascending series? What a descending What is the ratio? When the first term and the ratio are given, how do you find any other term? When the first and last term and the ratio are given, how do you find the sum of the series! 3. What is annuity? When is it in arrears? What does an anuity at compound interest form? How do you find the amount of an annuity at compound interest? 4. What is the common division of a foot? What are these called? What kind of series do these frac tions form? What is the ratio? What is the rule for the multiplication of duodecimals? How are all denominations less than a foot to be regarded? 5. What is Position? What does it suppose when single? When double? What kind of questions may be solved by the former? by the latter? 6. What is meant by the permutation of quantities? How do you find the number of permutations? Explain the reason. 7. What is meant by a periodical decimal? By a single repetend By a compound repetend? How is a repetend denoted? How is a poriodical decimal changed to an equivalent vulgar fraction? PART III. PRACTICAL EXERCISES SECTION I. Exchange of Currencies. 299. In £13, how many dollars, cents and mills? Now, as the pound has different values in different places, the amount in Federal Money will vary according to those values. In England, $1=4s. 6d.=4.5s.=£45=£0.225, and there £13-13 0.225-$57.777. In Canada, $15s= £0.25, and there £13-13-0.25-$52. In New England, $1-6s. ££0.30, and there, £13-13-0.3-$43.333. In New York, $1-8s.--£0.4, and there, £13-13÷0.4 32.50. In Pennsylvania, $1-7s. 6d. 7.58. £.75 £0.375, and there, £13—13÷0.375—$34.666. And in Georgia, $1= 4.6+ 4s. 8d. 4.6s. ££0.2333+, and there, £13-13÷0.2333 =$55.722. 8d.-4.6+s.£20.0 200 300. In £16 7s. 8d. 2qr., how many dollars, cents and mills? Before dividing the pounds, as above, 7s. 8d. 2qr., must be reduced to a decimal of a pound, and annexed to £16. This may be done by Art. 143, or by inspection, thus, shillings being 20ths of a pound, every 2s. will be 1 tenth of a pound therefore write half the even number of shillings for the tenths= £0.3. One shilling being 1 20th £0.05; hence, for the odd shilling we write £0.05. Farthings are 960ths of a pound, and if 960ths be increased by their 24th part, they are 1000ths. Hence 8d. 2qr.(-34qr.+1)=£0.035; and 16+0.3+0.05+0.035 =£16.385, which, divided as in the preceding example, give for English currency, $72.822, Can. $65.54, N. Y. $40.962, &c. Hence, 301. To change pounds, shillings, pence and farthings to Federal Money, and the reverse. RULE. Reduce the shillings, &c. to the decimal of a pound; then, if it is English currency, divide by 0.225; if Canada, by |