up to the given number, and the last product will be the number of changes required. 2. How many different ways may the first 5 letters of the alphabet be arranged? A. 120. 3. How many changes may be rung on 15 bells, and in what time may they be rung, allowing 3 seconds to every round? A. 1307674368000 changes; 3923023104090 seconds. 4. What time will it require for 10 boarders to seat themselves differently every day at dinner, allowing 365 days to the year? . 994138 years. 5. Of how many variations will the 26 letters of the alphabet admit? A. 403291161126605635584000000 POSITION Is a rule which teaches, by the use of supposed numbers, to find true ones. It is divided into two parts, called Single and Double. SINGLE POSITION. ¶ XCIII. This rule teaches to resolve those questions whose results are proportional to their suppositions. 1. A schoolmaster, being asked how many scholars he had, replied, "If I had as many more as I now have, one half as many more, one third, and one fourth as many more, I should have 296." How many had he? Let us suppose he had 24 We have now foul that we did not suppose the right number. If we had, the amount would have been 296. But 24 has been increased in the same manner to amount to 74, that some unknown number, the true number of scholars, must be, to amount to 296. Consequently, it is obvious, that 74 has the same ratio to 296 that 24 has to the true number. The question may, therefore, be solved by the following statement: As 74: 296 :: 24: 96, Ans. as many 12 as many 8 as many 6 74 This answer we prove to be right by increasing it by itself, one half 96 Itself, one third itself, and one fourth itself; 96 I. Suppose any number you choose, and proceed with it in the same manner you would with the answer, to see if it were right. : II. Then say, As this result the result in the question :: the supposed number: number sought. More Exercises for the Slate. 2. James lent William a sum of money on interest, and in 10 years it amount ed to $1600; what was the sum lent? A. $1000. 3. Three merchants gained, by trading, $1920, of which A took a certain sum, B took three times as much as A, and C four times as much as B; what share of the gain had each? A. A's share was $120; B's, $360, and C's, $1440. 4. A person, having about him a certain number of crowns, said, if a third, a fourth, and a sixth of them were added together, the sum would be 45; how many crowns had he? A. 60. 5. What is the age of a person, who says, that if of the years he has lived be multiplied by 7, and 3 of them be added to the product, the sum would be 292? A. 60 years. 6. What number is that, which, being multiplied by 7, and the product divided by 6, the quotient will be 14? A. 12. DOUBLE POSITION. TXCIV. This rule teaches to solve questions by means of two supposed numbers. In Single Position, the number sought is always multiplied or divided by some proposed number, or increased or diminished by itself, or some known part of itself, a certain number of times. Consequently, the result will be proportional to its supposition, and but one supposition will be necessary; but, in Double Position we employ two, for the results are not proportional to the suppositions. 1. A gentleman gave his three sons $10000, in the following manner : to the second $1000 more than to the first, and to the third as many as to the first and second. What was each son's part? Let us suppose the share of the first, 1000 Then the second 2000 Total, 6000 This, subtracted from 10000, leaves 4000 The shares of all the sons will, if our supposition be correct, amount to $10000; but, as they amount to $6000 only, we call the error 4000. Suppose, again, that the share of the first was 1500 Then the second=2500 Third 4000 8000 2000 We perceive the error in this case to be $2000. The first error, then, is $4000, and the second $2000. Now, the difference between these errors would seem to have the same relation to the difference of the suppositions, as either of the errors would have to the aifference between the supposition which produced it and the true number. We can easily make this statement, and ascertain whether it will produce such a result: As the difference of errors, 2000 500, difference of suppositions: either of the errors, (say the first) 4000 1000, the difference between its supposition and the true number. Adding this difference to 1000, the supposition, the amount is 2000 for the share of the first son; then $3000 that of the second, $5000 that of the third, Ans. For 2000+3000+-5000-10000, the whole estate. Had the supposition proved too great, instead of too small, it is manifest that we must have subtracted this difference. The differences between the results and the result in the question are called errors: these are said to be alike, when both are either too great or too small; unlike, when one is too great, and the other too small. From these illustrations we derive the following RULE. I. Suppose any two numbers, and proceed with each according to the manner described in the question, and see how much the result of each differs from that in the question. II. Then say, As the difference* of the errors: the difference of the suppositions either error difference between its supposition and the number sought. More Exercises for the Slate. 2. Three persons disputing about their ages, says B, "I am 10 years older than A;" says C, "I am as old as you both :" now what were their several ages, the sum of all of them being 100? Ans. A's, 20; B's, 30; C's, 50. 3. Two persons, A and B, have the same income; A saves of his yearly; but B, by spending $150 per annum more than A, at the end of 8 years finds himself $400 in debt; what is their income, and what does each spend per an num? First, suppose each had $200; secondly, $300; then the errors will be 400 and 200. A. Their income is $400; A spends $300, B $450. 4. There is a fish whose head is 8 feet long, his tail is as long as his head and half his body, and his body is as long as his head and tail; what is the whole length of the fish? First, suppose his body 30; secondly, 28; the errors will then be 1 and 2. A. 32 feet 5. A labourer was hired 80 days upon this condition,-that for every day he was idle he should forfeit 50 cents, and for every day he wrought he should receive 75 cents; at the expiration of the time he received $25; now how many days did he work, and how many days was he idle? A. He worked 52 days, and was idle 28. MISCELLANEOUS EXAMPLES. 1. There is a room, one side of which is 20 feet long and 8 feet high; how many square feet are contained in that side? Hypothenise B This side is a regular parallelogram (T LXXIX.); and, to find the square contents, we have seen that we must multiply the length by the breadth; thus, 20 ft. 8 ft. = 160 sq. ft., Ans. But, had we been required to find the square contents of half of this parallelogram, as divided in the figure on the left, it is plain that, if we should multiply (20) the whole length by of (8) the width, or, in this case, the height, the product would be the square contents in this half, that is, in the figure B CD; thus, of 84; then, 4×20=80 sq. ft., which is precisely of 160, the square contents in the whole figure. The half BCD is called a triangle, because it has, as you see, 3 sides and 3 angles, and because the line B C falls perpendicularly on CD; the angle at C is called a right angle; the whole angle, then, BC Dinay properly be called a right-angled triangle. D Base. 20. C *The difference of the errors, when alike, will bo one subtracted from the other; when unlike, one added to the other. nuse. The line BC is called a perpendicular, CD the base, and D B the hypothe Nole. Both the base and perpendicular are sometimes called the legs of the triangle. Hence, to find the area of a right-angled triangle ;— Multiply the length of the base by the length of the perpendicular; the product will be the area required. 2. What is the area of a triangular piece of land, one side of which is 40 rods, and the distance from the corner opposite that side to that side 20 rods? Ans. 2040 400 rods. Note. To find the area of any irregular figure, divide it into triangles. A Hypothenuse. 50. Perpendicular. 30. In any right-angled triangle, it has been ascer tained, that the square of the hypothenuse is equal to the sum of the squares of the other two sides. Thus, in the adjacent figure, 402 1600, and 302900; then, √900+1600=50, the hypothenuse Base. 40. I. Hence, to find the hypothenuse, when the legs are given; Add the squares of the two legs together, and extract the square root of their sum. II. When the hypothenuse and one leg are given, to find the other leg ;— From the square of the hypothenuse subtract the square of the given leg, and the square root of the remainder will be the other. 3. A river 80 yards wide passes by a fort, the walls of which are 60 yards high; now, what is the distance from the top of the wall to the opposite bank of the river? In this example we are to find the hypothenuse. Ans. 100 yards. 4. There is a certain street, in the middle of which, if a ladder 40 feet long Le placed, it will reach a window 24 feet from the ground, on either side of said street; what is the width of the street? In this example, we are to find the length of the base of two triangles, and then the sum of these will be the distance required. Ans. 64 feet. 5. There is a certain elm, 20 feet in diameter, growing in the centre of a circular island; the distance from the top of the tree to the water, in a straight line, is 120 feet; and the distance from the foot 90 feet; what is the height of the tree? As the tree is 20 feet in diameter, the distance from its centre to the water is the length of the base, that is, 1090 100 feet. A. 66,332 ft.+. 6. Two ships sail from the same port; one goes due north 40 leagues, the other due east 30 leagues; how far are they apart? We are here to find the hypothenuse. A. 50 leagues. 8. What is the circumference of a wheel, the diameter of which is 8 feet? The circumference of a circle is greater, you are sensible, than the diameter, being a little more than 3 times, or, more accurately, 3,141592 times the diameter. A. 25,13 ft. 9. What is the diameter of a wheel, or circle, whose circumference is 12 feet? A. 4 ft., nearly. 10. If the distance through the earth be 8000 miles, how many miles around it? A. 25132,7 miles, nearly. 11. What is the area or contents of a circle, whose diameter is 6 feet, and its circumference 19 feet? NOTE. The area of a circle may be found by multiplying half the diameter by half the circumference, or by multiplying the square of half the diameter by 3,141592. A. 28 ft. 12 What is the area of a circle, whose diameter is 20 feet? 102 100 3,141592314,1592, Ans 13. What is the diameter of a circle, whose area is 314,1592? A. 20 ft. 14. What is the area, or square contents, of the earth, allowing it to be 8000 miles in diameter, and 25000 in circumference د NOTE. The area of a globe or ball is 4 times as much as the area of a circle of the same diameter; therefore. if we multiply the whole circumference into the whole diameter, the product will be the area. A. 200000000. 15. What are the solid contents of a globe or ball 12 inches in diameter ? The solid contents of a globe are found by multiplying its area by of its diameter. A. 904+ solid inches. 16. What are the solid contents of a round stick of timber, 10 inches in diameter, and 20 feet long? In this example, we may first find the area of one end, as before directed for a circle; then multiply by 20 feet, the length. A. 11 feet, nearly. Note. Solids of this form may be called cylinders. 17. What are the solid contents of a cylinder 4 feet in diameter, and 10 feet. fong? A. 125+feet. |