NOTE. Of course, the discount should be made as directed in § LXXXVIII, page 230, article COMPOUND INTEREST. Or, the present worth may be calculated as though the annuity were to commence immediately, and to continue to the end of the time of the given annuity: if, from this sum, the present worth for the time of reversion be subtracted, the remainder will be the present worth required. 16. If an annuity of $100 be 14 years in reversion, to continue 20 years afterwards, what is its present worth, discounting at 5 per A. $629.426. cent. ? 17. What is the present worth, at 6 per cent. of an annuity of $120, to continue forever? A. $2,000. NOTE. The answer is evidently a sum whose annual interest is $120. 18. Which is preferable, an annuity of $100 for 15 years, to commence immediately, or the reversion of the same annuity, forever, after the 15 years have expired? also, what is the difference? A. The term of 15 yrs. is better than the reversion forever after, by $75.928+ NOTE. If the time extend beyond the limits of the table, calculate as far as the table will allow, and consider the rest as an annuity in reversion. 19. Find the present worth of an annuity of $400, to continue 34 yrs. at 6 per cent. A. $6,477.16. 20. Find the present worth of a $70 annuity, to continue 59 yrs. at 5 per cent. A. $1,321.3021. NOTE. To give a complete developement of the subject of annuities, is not the province of arithmetic. Contingent annuities, or those whose continuance depends on uncertainties, as the duration of the life, or lives, of one, or of several persons, involve the doctrines of CHANCES, and are, in many cases, complex and tedious in calculation. PERMUTATION AND COMBINATION. § CV. The two letters, A B, may be written a в, or в A. Any two things, therefore, have two orders of succession, or relative positions, in which they may be placed, in a single line. The word permutation, means change, and in mathematics, CHANGES IN THE ORDER IN WHICH THINGS SUCCEED EACH OTHER, ARE CALLED PERMUTATIONS. 1. What number of permutations can be made on the letters A B C ? If c be left out, A and B, as seen above, admit of 2 permutations. So, if в be left out, a and c admit of 2 permutations. And if a be left out, в and c admit of 2 permutations. But before each of these permutations, the letter left out may be placed; and as there were 2 permutations, 3 times, there are 6 in the whole. The pupil may make them for himself. From the above it will be seen that, of 2 things, there may be 1X2 =2 permutations; of 3, 1X2Xx3=6 permutations, and by the same mode of reasoning, it may be shown that, Sec. 105. PERMUTATION AND COMBINATION. 281 THE PERMUTATIONS, WHICH CAN BE MADE OF ANY NUMBER OF THINGS, ARE EQUAL TO THE CONTINUED PRODUCT OF THE NATURAL SERIES OF NUMBERS, FROM 1, UP TO THE NUMBER OF THINGS GIVEN. 2. Four gentlemen agreed to remain together, could arrange themselves differently at dinner. did they remain ? as long as they How many days A. 24 days. 3. 10 gentlemen made the same agreement, but they all died before it could be fulfilled. The last survivor lived 53 yrs. 98 days, after the agreement. How much did the bargain then want of being fulfilled, allowing 365 days to the year? A. 9,888 yrs. 237 d. 4. How many years will it take to ring all the possible changes on 12 bells, supposing that 10 can be rung in a minute, and that the year contains 365 d. 5 h. 49 m.? A. 91 yrs. 26 d. 22 h. 41 m. 5. How many permutations may be made of the figures, 1, 2, 3, 4, 5, taken two at a time? Let 1 be placed by itself. To this, each other figure may be joined, making 4 permutations. Then 2 may be taken in the same way; and so with every other figure, there being 4 permutations each time. Then, as there are 5 figures, there will be 5×4-20 permu. tations of two figures. 6. How many permutations can be made on the figures above, taken three at a time? Here, if we set apart each arrangement of 2 figures, found as above, we may join to every one, each of the 3 remaining figures, which will make 3 times as many permutations. Now the permutations by twos, we have seen, are 5X4, and 3 times this number =5×4×3=60 permutations of three figures. By extending this mode of reasoning, we obtain the following. THE PERMUTATIONS WHICH CAN BE MADE OF ANY NUMBER OF THINGS, TAKEN A GIVEN NUMBER AT A TIME, ARE EQUAL TO THE CONTINUED PRODUCT OF A DECREASING NATURAL SERIES, WHOSE GREATEST TERM IS THE WHOLE NUMBER OF THINGS, AND WHOSE NUMBER OF TERMS, THE NUMBER TO BE TAKEN AT A TIME. 7. How many numbers can be expressed by the nine digits, taken four at a time? A. 3,024. 8. How many words of five letters each, may be made from an alphabet of 26 letters, supposing that a number of consonants may make a word? A. 7,893,600. From the letters, A, B and C, we can make three assemblages of two letters, of which no one shall contain exactly the same letters as another. These are A B, A C and B c. A combination means a collection of things, and in mathematics, COLLECTIONS OF WHICH NO TWO ARE EXACTLY ALIKE, CONSISTING EACH OF A GIVEN NUMBER OF THINGS, ARE CALLED COMBINATIONS. 9. How many combinations of two letters can be made from A B C D? The permutations of two, we have seen to be 4X3=12. But on each combination of two, we have likewise seen, there can be made 1X2=2 permutations of two. Hence, the permutations are twice as many as the combinations. 12÷2-6 Ans. 10. How many combinations of three can be made of A B C DEF? The permutations of three are 6X5X4=120, which, as before, are 1×2×3=6 times the number of combinations. 120÷6=20 Ans. Hence, to find the number of combinations, which can be made of a given number of things taken from a given set, FIND THE NUMBER OF PERMUTATIONS WHICH CAN BE MADE FROM THE PROPOSED SET, TAKING THE GIVEN NUMBER OF THINGS AT A TIME, and DIVIDE IT BY THE NUMBER OF PERMUTATIONS WHICH CAN BE MADE ON ANOTHER SET, consisting ONLY OF AS MANY THINGS AS ARE TO BE TAKEN AT A TIME. 11. How many combinations of two letters can be made from 24? A. 276. 12. A successful general was asked by his king, what reward he should give him for his services. The general's modesty only permitted him to ask a cent for every file of 12 men which he could make with 100 men. The king graciously granted the request, requiring only that the general should actually parade his several files in front of the palace, before payment was made. What was the amount of the reward, and how long was the general obliged to wait for it, allowing 2 minutes to draw up a file, 6 hours a day to the parade, and 365 days to the year, Sundays being excepted? A. Reward $10,504,210,511,067.-Time 186,528,344 centuries, 94 yrs. 196 d. 2 h. 20 m. On the principles above explained, are formed combination lotteries. 13. How many tickets in a lottery formed by ternary combination, from 60 numbers? A. 34,220. 14. In the same lottery, are 9 drawn numbers; how many prize tickets, having each 3 drawn numbers on them? A. 84. § CVI. We have now illustrated all the fundamental principles of ARITHMETIC, and investigated those subjects which fall within the scope of a practical treatise. It will be seen, that one part of the preceding exercises consists of problems to be performed by numerical operations, and of rules to serve as guides in performing them. These, taken by themselves, constitute what is called PRACTICAL ARithmetic. Another part investigates the principles of numbers, or demonstrates their properties and relations; and this is denominated THEORETIC ARITHMETIC. Hence, the subject may properly be said to embrace both an art and a science, of which, and THEORETIC ARITHMETIC IS THE SCIENCE WHICH TREATS OF NUMBERS, PRACTICAL ARITHMETIC IS THE ART OF COMPUTING BY NUMBERS. Theoretic arithmetic, then, calls into exercise the reasoning powers, and is well adapted to mental discipline; practical arithmetic exercises the judgment in an inferior degree, but is of great utility in the transactions of business. OBSERVATIONS ON VARIOUS TOPICS. The learner has, no doubt, perceived that many of our present advantages for calculation, result from our system of notation. The ancient Greeks and Romans made their calculations by means of small pebbles, as is probable from the frequent use of the word meaning pebble, by the classic writers, as connected with calculation. The Romans afterwards employed, in their ordinary calculations, a small board or table, on which beads were strung on wires. This was called the ABACUS. The Chinese use something very similar at the present day, cal. led the SWAN-PAN. The properties of numbers are of two kinds, the essential, and the accidental; the essential, existing in numbers from their very nature, and the accidental, depending on the mode of representing them. Thus, it is an essential property of 9, that it is a square number, and of 7, that it is prime. It is an accidental property of 9, that it will divide a number, when it will divide the sum of the fig ures composing it. It is from the accidental properties, that we derive the greatest advantage in calculation. Among the Greeks, the disciples of PYTHAGORAS turned their attention to the essential properties of numbers. They divided them into many different classes, as perfect or imperfect, redundant or deficient, &c. Perfect numbers, are those which are equal to the sum of their aliquot parts: 6, 28 and 496, are examples; only 10 are known. Besides this, they entertained the most absurd notions, with respect to the qualities of numbers. They considered even numbers, as feminine, and terrestrial in their nature; while odd numbers were esteemed masculine, and celestial. The sum of the first four even, and of the first four odd numbers, viz. 36, was supposed to combine all virtue, celestial and terrestrial; and being, at the same time, the square of 6, the first perfect number was thought to possess wonderful properties. To swear by the TETRACTYS or QUATERNION, as this number was called, "was to contract the most solemn of all obligations." Before the introduction of Arabic figures into England, arithmetical operations, (which, of course, were performed by means of Roman characters,) were difficult, particularly those of division. The science had, however, considerably advanc ed, and treatises were written upon it, of one of which, ALCUIN, a disciple of the "venerable BEDE," who, himself, wrote on the subject, was the author. In this work, were first proposed the well known puzzles, of conveying three jealous husbands, with their wives, across a river, in a boat which can carry but two at a time, so that no woman shall be in company with any of the men, unless her own husband be present; of dividing equally, among three persons, 21 casks, of which, 7 are full, 7 half full, and 7 are empty; and many others of the same nature. About the middle of the 15th century, a man named EMANUEL MOSCOPULIUS invented MAGIC SQUARES, of which the following are examples. 31 2228 13 40 143 10 19341625 437 746 |22|47|16|41|10|35|| 4| |48|14|40 61 824501632 34 10 26 42 18 2036 228 44 461238 430 If the columns of these squares be added, either perpendicularly, horizontally, or diagonally, the sum will be the same, throughout each square. The numbers arranged in them, form, in each square, a complete arithmetical progression. Any arithmetical progression, in which the number of terms is a square number, will admit of a similar arrangement. Numbers in geometrical progression, may be arranged so that the continued products of the columns shall be equal. Numbers bearing to each other the relation called harmonical proportion, may be likewise arranged so that these continued products shall be in harmonical proportion. The pupil will find the study of the structure of these squares amusing, but of little practical importance. He will probably discover the law by which the terms of the progressions are arranged in the squares above; but there are a great variety of methods in which the arrangement may be made. It is more difficult when the number of terms is even, than when it is odd. MISCELLANEOUS EXAMPLES. 1. If a staff 3 ft. 8 in. 15 b. c. casts a shadow 2 ft. 8 in. 2 b. c., what is the height of a spire, that cast a shadow 163 ft. 7 in. 1 b. c. at the same time? Ans 222 ft. 9 in. 21131 b. c. 2. A merchant having mixed sugars, of which the first kind was worth as much pr. lb. as the second, in quanties to make the mixture worth the first, found that the second kind had been injured, so as to be worth only of the value he had supposed it worth. What part of the value of the second per lb. was that of the mixture, and what was its price; the first kind having been worth 11d.? Ans. -12d. 3. 25 men are engaged on a large building which would have af. forded them employment for 64 days, but after they had been working 15 d. 13 others joined them. How long was the building in be. ing completed? Ans. 329 d. 38 4. A cistern is supplied by a pipe, which alone will fill it in 3 hours, and by another, which, alone, will fill it in 2 hours. How long will both running together, be, in filling it? Ans. 1 h. 12 m. 5. A cistern receives in an hour water enough to fill of it, and discharges, in the same time, enough to fill of it. How long is the cistern in being filled? Ans. 3 days. 6. A cistern receives water from a pipe, which, alone will fill it in 11 hours; but after the water has been running 5 hours, another pipe is opened, and both together fill the rest of the cistern in 2 hours. How long would it take the second pipe alone to fill the cistern? Ans. 5 h. 30 m. 7. A pipe pours water enough into a cistern to fill of it in an hour; another is opened 2 hours after the first, which would have hastened the filling of the cistern 1 hour, but 2 hours having elapsed, a third commences discharging, by which the filling of the cistern is accomplished in the same time it would have been, if the first had run alone. How long would it take the second pipe alone to fill the cistern, or the third alone to empty it? Ans. Second, 40 h. Third, 26 h. 40 m. |