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 nges, says B," I am 10 years oluler 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 nimself $100 in dept; what is their income, and what does each spend per annum? First, suppose each had $200 ; secondly, $309; then the errors will be-400 and 200. A. Their income is $400 ; A spends 300), B3 $150. 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? Firet, suppose his body 30; secondly, 23; the errors will then be l an.12. .1, 32 feet. 5. A laborer was hired 80 days upon this condition,--that for overy day he was idle he should forfeit 50 cents, and for every day he wronghil 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 idio? A. He worked 52 days, and was idle 28. MISCELLANEOUS EXAMPLES. A B : 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? This side is a regular parallelogram (TT LXXIX.); and, to find the squaro contents, we have seen that we must multiply the length by the breadth; thus, 20 ft. X $ ft. = 160 sq. ft., Ans. But, had we been required to find thó square contents of half of this parallelogram, is divided in the figure on the left, it is plain thai, if we should multiply (20) the whole length by of (8) the width, or, in this case, the height, the product would be trio square contents in this half, that is, in the figure BCD; thus, $ of 834; then, 4 x 20 = 80 sq. ft., wnich is D Base. 20. precisely of 160, the square contents in the whole figure. The half B C D is called a trian gle, because it has, as you see, 3 sides and 3 anglos, anil because the line B Ö falls perpendicularly on C]); the angie at C is called a rigki angle; the wholo angle, then, B C D, may properly bo called a right-angled triangle. The line B C is called a perpendicular, CD the base, and D B the hypothenuse. Note. --Both the base and perpendicular are sometimes called the legs of the triangle. · Hypothenuse.. che 3d & * The difference of the errors, when alike, will be ono subtracted from the other; when unlike, one added to the othor. 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 Tods, and the distance from the corner opposite that side to that side 20 roda ? Ans. X 40 = 400 rods, Note.-To find the area of any Irregular figure, divide it into triangles. A 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. When the Hypotheuse 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 froin 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 be 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 tlie 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, 10 + 90 = 100'feet. A. 66,332 it. +. 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 + feet. 9. What is the diameter of a wheel, or circle, whose circumference is 12 feet? A. 4 feet, nearly. 10. If the distance through the earth be 8000 miles, how inany miles arounu. 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. .. 28 feet. 12. What is 'he area of a circle whose diameter is 20 feet? = 109 x 33,141592 = 314,1592, 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 tho earth, ullowing it to be 8000 miles in diameter, and 25000 in circumference? 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 t of its diameter. A. 904-76 + solid inches. 16. What are the solid contents of a round stick of timber, 10 inchos 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 long? A. 125 + feet. When solids, being either round or square, taper regularly till they come to a point, they contain just j as much as if they were all the way as large as they are at the largest eni. When solids decrease regularly, as last described, they are called pyramids. When the base is square, they are called square pyramids; when triangular, triangular pyramids; and when round, circular pyramids, or cones. Hence, To find the Solid Contents of such Figures ;Multiply the area of the largest end by of the perpendicular height 18. What are the solid contents of a cone, the height of which is 30 feet, and its base 8 feet in diameter? A. 502,6+ ft. 19. There is a pyramid, whose base is 3 feet square, and its perpendicular height 9 feet; what are its solid contents ? A.32 xq = 27 ft. 20. What is the length of one side of a cubical block, which contains 9261 solid feet? A. 21 feet. 21. In a square lot of land, which contains 2648 acres, 3 toods, and 1 rod, what is the length of one side? A. 651 rods. 22. A grocer put 5 gallons of water into a cask containing 30 gallons of wine, worth 75 cents per gallon; what is a gallon of this mixture worth: A. 64% cts. 23. The first term of a geometrical series is 4, the last 56984, and the ratio 6; what is the sum of all the terms ? A. 68380. 24. “ The great bet, and when it will be paid.—The public mind has been considerably amused, for a few days past, with a singular bet, said to have been made between a friend of Mr. Adams and a friend of Gen. Jackson, on the eastern shore of Maryland. The bet was, that the Jackson man was to receive from the Adams man l cent for the first electoral vote that Jackson should receive over 130, 2 cents for the second, 4 for the third, and so on, doubling for every successive vote; and the Adams man was to have one hundred dollars if Jackson did not receive over 130 votes. According to the present appear. ances, Jackson will receive 173, 13 over 130, and the sum the Adams man will have to pay, in that event, will be $87960930222,07. “But the joke does not appear to be all on the Jackson man's side. The money is to be counted, and it will take a pretty long lifetime of any common man to count out the shiners.? Let's see :-allowing that a man can count sixty dollars a minute, and that he continues to count without ceasing, either to sleep, to take refreshment, or to keep the Sabbath, it will take him twentysecen hundred and eighty-nine years, nearly; but allow him to work eight hours a day, and rest on the Sabbath, he will be occupied 9789+ years ; so that the Adams man, when he is called upon for the cash, may tell his Jackson friend, Sit down, sir ; as soon as I can count the money, you shall have it; even the banks take time to count the money, you know.si Almost all porsons, in the ordinary avocations of life, unless they adopt some method of kcoping their accounts in a regular manner, will be subjected to continuul losses and inconveniences; to prevent which, the following plan or outline is composed, embracing the principles of Book-keeping in the most simple form. Before the pupil commences this study, it will not be necessary for him to have aftended to all the rules in the Arithmetic; but he shoukl make himself acquainted with the subject of Book-Keeping before he is suffered to leave school. A few examples only are given, barely sufficient to give the learner a view of the manner of keepiug books; it being intended that the pupil should be required to compose similar ones, and insert them in a book adapted to this purpose, Book-keeping is the method of recording business transactions. It is of two kinds—single and double entry; but we shall only notice the former, Single cntry is the simplest form of Book-keeping, and is-employed by retailers, mochanics, farmers, &c. It requires a Day-Book, Leger, and, where money is frequently received and paid out, a Cash-Book. DAY-BOOK This book should be a minute history of business transactions, in the order of time in which they occur ; it should be ruled with head lines, with one column on the left hand for post-márks and references, and two columns on the right for dollars and cents. The owner's name, the town or city, and the date of the first transaction, should stand at the head of the first page. It is the custom of many to continue inserting the name of the town on every page. This, however, is unnecessary. It is sufficient to write only the month, day, and year, at the head of each page after the first. This should be written in a larger hand than the entries. On commencing an account with any individual, his place of residence should be noted, provided it is not the same as that where the book is kept. If it be the same, this is unnecessary. As it often happens that different persons bear the same name, it is well, in such cases, to designate the individual with whom the account is opened, by stating his occupation, or particular place of residence. When the conditions of sale or purchase vary from the ordinary customs of the place, it should he stated. Every month, or oftener, the Day-Book should bo copied or posted into the Leger, as hereafter directed. The crosses, on the left hand column, show that the charge or credit, against which they stand, is posted, and the figures show the page of the Leger where the account is posted, Some use the figures only as post marks. Every article sold on credit, except when a note is taken, should be immediately charged, as it is always unsafe to trust to memory. Also, all labor per formoil, or any transaction whereby another is made indebted to us, should be immediately entered on the Day-Book. If farmers and mechanics would strictly observe this rule, they would not only save many quarrels, but much money. In this respect, at least, follow the example of Dr. Franklin, who nover omitted to make a charge as soon as it could be done. Never defer á charge till to-morrow, when it can be made to-day. |