282 PERMUTATION AND COMBINATION. Sec. 106. 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 D E F ? The permutations of three are 6X5×4=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 per mitted 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 fije, 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, THEORETIC ARITHMETIC IS THE SCIENCE WHICH TREATS OF NUMBERS, and PRACTICAL ARITHMETIC IS THE ART OF COMPUTING BY NUMBERS. Theoretic arithmetic, then, calls into exercise the reasoning pow. ers, 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 figures 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 OF 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, 3 are full, 3 half full, and 3 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 1934 16 25 437 746 224716411|35| 4 |22|48|14|40 6 824 50 16 32 34 10 26 42 18 2036 228 44 46 1238 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. 12 b. c. at the same time? 13 As 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. 12 d. 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 being completed? Ans. 32 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. |