MATHEMATICAL WRINKLES FOR TEACHEES AND PRIVATE LEARNERS coN&idwirtar OF • ••* ' KNOTTY PROBLEMS; MATHEMATICAL RECREATIONS ANSWERS AND SOLUTIONS; RULES OF MENSURA- TION; SHORT METHODS; HELPS, TABLES, ETC. BY SAMUEL I. JONES PR0FSS80B OF MATHEMATICS IK THE OUNTBR BIBLICAL AND LITEBABT COLLEGE, OUNTEB, TEXAS PRICE $1.65, NET PUBLISHED BY SAMUEL L JONES GUNTER, TEXAS ^ fA I ti V\*\f () f: ^ i ? ' ^'^■"^f^-'^-^^^-^^ Copyright, 1912, By SAMUEL I. JONES. All rights reserved. NcriDoolr i^reaa J. 8. Cushing Co. — Berwick & Smith Co. Norwood, Masa., U.S.A. " Albert Smith, in one of his amusing novels, describes a woman who was convinced that she suffered from ' cob- wigs on the brain.* This may be a very rare complaint, but in a more metaphorical sense, many of us are very apt to suffer from mental cobwebs, and there is nothing equal to the solving of puzzles and problems for sweeping them away. They keep the brain alert, stimulate the imagination, and develop the reasoning faculties. And not only are they useful in this indirect way, but they often directly help us by teaching us some little tricks and 'wrinkles* that can be applied in the affairs of life at the most unexpected times, and in the most unexpected wavs." H. E. DUDENEY. Ill «00773 CONTENTS CHAPTBB PAOB I. Arithmetical Problems 1 II. Algebuaic Problems ........ 25 III. Geometrical Exercises 33 IV. Miscellaneods Problems 48 V. Mathematical Recreations 68 VL ExAMiNATiox Questions 113 Vn. Answers and Solutions 163 VIII. Short Methods 228 IX. Quotations on Mathematics 245 X. Mensuration 258 XI. Miscellaneous Helps 285 Xn. Tables 304 PREFACE The following pages contain many mathematical problems, puzzles, and amusements of past and present times. They have a long and interesting history and are part of the inher- itance of the school. This book is intended to be a helpful companion to teachers, and to impart to students a knowledge of the application of mathematical principles, which cannot be obtained from text- books. The present-day teacher has little time for the selection of suitable problems for supplementary work. This book is de- signed to meet the requirements of teachers who feel such extra assignments essential to thorough work. Whatever text is used, the necessity for a work of this kind is felt from the fact that fresh problems produce interest and stimulate inves- tigation. Originality is not claimed for all of the problems, but for many of them. They have been compiled from various sources. The author's aim has been to select problems not only instruc- tive, but also interesting and amusing. The rules of Mensuration and Short Methods have been included because of their usefulness. On account of the vari- ous helps placed in this book, it will serve as a handbook of mathematics to both teachers and pupils. The solutions to only part of the problems are given. In some cases solutions of considerable length are given, but at other times only the answers are given. Had the full solu- tions and proofs been given in every case, either half the prob- lems would have had to be omitted, or the size of the book greatly increased. vU viii PREFACE The author acknowledges his indebtedness to many friends for helpful suggestions. Specially is he under obligation to the late Dr. G-. B. M. Zerr, Philadelphia, Pa., for critically reading the manuscript. A few of his solutions published in the lead- ing Mathematical Journals have been used on account of their beauty and simplicity. He is indebted to Dr. H. Y. Benedict and Mr. J. W. Calhoun, of the University of Texas, for read- ing the manuscript and offering many valuable suggestions and criticisms. He is very thankful to Dr. George Bruce Hal- sted, head of the department of mathematics of the Colorado State Teachers' College at Greeley, for criticising the Defini- tions, Historical Notes, and Classifications. He is also specially indebted to Professor Dow Martin, of the Biblical and Lit- erary College of Gunter, Texas, for reading and correcting the proof-sheets. Any correction or suggestion relating to these problems and solutions will be most thankfully received. It is hoped that this small volume may produce higher and more noble results in awakening a real love and interest among the great body of teachers and students for the study of math- ematics, "the oldest and the noblest, the grandest and the most profound, of all sciences." SAMUEL I. JONES. GuNTEB, . Texas. MATHEMATICAL WRINKLES ARITHMETICAL PROBLEMS 1.* Between 3 and 4 o'clock I looked at my watch and noticed the minute hand between 5 and 6 ; within two hours I looked again and found that the hour and minute hands had exchanged places. What time was it when I looked the second time ? 2.* A tree 120 feet high was broken in a storm, so that the top struck the ground 40 feet from the foot of the tree. How long was tlie part of the tree that was broken over ? 3. How many acres does a square tract of land contain, which sells for $80 an acre, and is paid for by the number of silver dollars that will lie upon its boundary ? 4.* The area of a rectangular field is 30 acres, and its diag- onal is 100 rods. Find its length and breadth. 5.* Suppose two candles, one of which will burn in 4 hours and the other in 6 hours, are lighted at once. How soon will one be four times the length of the other ? 6.* While a log 2 feet in circumference and 10 feet long rolls 200 feet down a mountain side, a lizard on the top of the log goes from one end to the other, always remaining on top. How far did the lizard move ? 7. How many calves at $3.50, sheep at $1.50, and lambs at $ .60 per head, can be bought for $ 100, the total number bought being 100 ? • Problems denoted by (•) are algebraic or geometrical. They are placed here because arithmetical solutions are often demanded. 1 2 mathemat,ica;l wrinkles 8. A, mar. yfills to his- wife-i of. his estate, and the remain- ing I to ^iS'SOrr, if'SL4ch siie^ji'ld" be ^born ; but | of it to the wife and the other i to the daughter, if such should be born. After his death twins are born, a son and a daughter. How should the estate be divided so as to satisfy the will ? 9. What is the value of 4^^ , when n = 0? 10. A room is 30 feet long, 12 feet wide, and 12 feet high. On the middle line of one of the smaller side Avails and 1 foot from the ceiling is a spider. On the middle line of the oppo- site wall and 11 feet from the ceiling is a fly. The fly being paralyzed by fear remains still until the spider catches it by crawling the shortest route. How far did the spider crawl ? 11. I found $10; what was my gain per cent? 12.* A conical glass is 4 inches high and 6 inches across at the top. A marble is within the glass, and water is poured in till the marble is just immersed. If the amount of water poured in is ^ the contents of the glass, what is the diameter of the marble ? 13. A banker discounts a note at 9 % per annum, thereby getting 10 % per annum interest. How long does the note run ? 14. In extracting the square root of a perfect power the last complete dividend was found to be 1225. What was the power ? 15.* Mr. Smith has a lawn the dimensions of which are to each other as 3 to 2. If he should increase each dimension one foot, the lawn would cover 651 square feet of land. What are the dimensions of the lawn ? 16. A merchant marked his goods to gain 80 %, but on ac- count of using an incorrect yardstick, gained only 40 %. Find the length of the measure. ARtTHMETICAL PROBLEMS 3 17.* The area of a triangle is 24,276 square feet, and its sides are in proportion to the numbers 13, 14, and 15. Find the length of each side. 18. Between 2 and 3 o'clock, I mistook the minute hand for the hour hand, and consequently thought the time 55 min- utes earlier than it was. What was the correct time ? 19. A slate including the frame is 9 inches wide and 12 inches long. The area of the frame is \ of the whole area, or J of the area inside the frame. What is the width of the frame ? 20. If 6 acres of grass, together with what grows on the 6 acres during the time of grazing, keep 16 oxen 12 weeks, and 9 acres keep 26 oxen 9 weeks, how many oxen will 15 acres keep 10 weeks, the grass growing uniformly all the time ? 21. A boy on a sled at the top of a hill 200 feet long, slides down and runs half as far up another hill. He sways back and forth, each time going ^ as far as he came. How far will he have traveled by the time he comes to a halt ? 22. 3 + 3x3-3-3-3=? 23. 2h-2--2--2--2x2x 2x2-i-0x 2=? 24. 3^3-^3^3x3x3x0x 3= ? 25. A fly can crawl around the base of a cubical block in 4 minutes. How long will it take it to crawl from a lower corner to the opposite upper corner? 26. A squirrel goes spirally up a cylindrical post, making a circuit in each 4 feet. How many feet does it travel if the post is 16 feet high and 3 feet in circumference ? 27. If the cloth for a suit of clothes for a man weighing 216 pounds costs $ 16, what will be the cost of enough cloth of the same quality for a man of similar form weighing 512 pounds ? 28. A ball 12 feet in diameter when placed in a cubic room touches the floor, ceiling, and walls. What must be the diam- 4 MATHEMATICAL WEINKLES eter of 8 smaller balls, which, will touch this ball and the faces of the given cube ? 29. At what time between 3 and 4 o'clock is the minute hand the same distance from 8 as the hour hand is from 12 ? 30.* By cutting from a cubical block enough to make each dimension 2 inches shorter it is found that its solidity has been decreased 39,368 cubic inches. Find a side of the original cube. 31. A number increased by its cube is 592,788. Find the number. 32.* The difterence of two numbers is 40 ; the difference of their squares is 4800. What are the numbers ? 33. A man can row upstream in 3 hours and back again in 2 hours. Determine the distance, the rate of the current being 1 mile per hour. 34. A rented a farm from B, agreeing to give B i of all the produce. During the year A used 90 bushels of the corn raised, and at settlement first gave B 20 bushels to balance the 90 bushels and then divided the remainder as if neither had received any. How much did B lose ? x 35. A certain number increased by its square is equal to 13,340. Find the number. 36.* The cube root of a certain number is 10 times the fourth root. Find the number. 37. A number divided by one more than itself gives a quotient yL. What is the number ? 38. What do I pay for goods sold at a discount of 50, 25, and 100 % off, the list price being $ 50 ? 39. If an article had cost ^ less, the rate of loss would have been ^ less. Find the rate of loss. 40. A merchant having been asked for his lowest prices on shoes, replied, " I give a certain per cent off for cash, the same ARITHMETICAL PROBLEMS 5 per cent off the cash price tu ministers, and the same per cent off the price to ministers to widows." The price to widows is l^f of the marked price. What per cent does he give off for cash? 41. If James had $40 more money he could buy 20 acres of land, or with $ 80 less he could buy only 10 acres. How much money has he and what is the value of an acre ? 42. What is the least number of gallons of wine, expressed by a whole number, that will exactly fill, without waste, bottles containing either j, |, ^, or | gallons ? 43. I sold a house and gained a certain per cent on my in- vestment. Had it cost me 20 % less, I should have gained 30 % more. What per cent did I gain ? 44. Goods marked to be sold at 50 and 10 % discount were disposed of by an ignorant salesman at 60 % from the list price. What was the loss on cash sales amounting to S 15,000? 45. I paid $ 10 cash for a bill of goods. What was the list price, if I received a discount of 50, 25, 20, and 10 % off ? 46. My clock gains 10 minutes an hour. It is right at 4 P.M. What is the correct time when the clock shows mid- night of the same day ? 47. Two men working together can saw 5 cords of wood per day, or they can split 8 cords of wood when sawed. How many cords must they saw that they may be occupied the rest of the day in splitting it ? 48. A grocery merchant sells goods at 80 % profit and takes eggs in trade at market price. If 2 eggs in each dozen are bad, find his per cent gain. 49. A hollow sphere whose diameter is 6 inches weighs | as much as a solid sphere of the same material and diameter. How thick is the shell ? 50. If a bin will hold 20 bushels of wheat, how many bushels of apples will it hold ? 6 MATHEMATICAL WRmKLES 51. What per cent in advance of the cost must a merchant mark his goods so that after allowing 5 % of his sales for bad debts, and an average credit of 6 months, and 7 % of the cost of the goods for his expenses, he may make a clear gain of 12-1- ^ Qf lY^Q f^^.g^ cQs^ Qf ^\^Q goods, money being worth 6 % ? 52. A teacher in giving out the dividend 84,245,000 was mis- understood by his pupils, who reversed the order of the figures in millions period. The quotient obtained was 36,000 too small. What was the divisor ? 53. Three men bought a grindstone 20 inches in diameter. How much of the diameter must each grind off so as to share the stone equally, making an allowance of 4 inches waste for the aperture ? 54. James is 30 years old and John is 3 years old. In how many years will James be 5 times as old as John ? 55. A merchant sold a piano at a gain of 40 %. Had it cost him $400 more, he would have lost 40 %. What did it cost him? 56. A steamer goes 20 miles an hour dow^nstream, and 15 miles an hour upstream. If it is 5 hours longer in coming up than in going down, how far did it go ? 57. A and B together can do a piece of work in 24 days. If A can do only | as much as B, how long will it take each of them to do the work ? 58. The sum of two numbers is 80 ; the difference of their squares is 1600. What are the numbers ? 59. When a man sells goods at a price from which he re- ceived a discount of 33 J- %, what is his gain per cent ? 60. 6-6--6 + 6x2-2=? 61. 3--3--3^3--3--i--i--i--i=? 62. How much water will dilute 5 gallons of alcohol 90 % strong to 30 % ? ARITHMETICAL PROBLEMS ' 7 63. I bought a house and lot for $ 1000, to be paid for in 5 equal payments, interest at 10%, payable annually ; payments to be cash, 1, 2, 3, and 4 years from date of purchase. What was the amount of each payment ? 64. I buy United States 4% bonds at 106, and sell them in 10 years at 102. What is my rate of income ? 65. If a melon 20 inches in diameter is worth 20 cents, what is one 30 inches in diameter worth ? 66. The difference between the true discount and the bank discount of a note due in 90 days at 6 %, is $.90. What is the face of the note ? 67. A writing desk cost a merchant $ 20. At what price must it be marked so that the marked price may be reduced 40 % and still 50 % be gained ? 68. A man agreed to work 12 days for $ 18 and his board, but he was to pay $1 a day for his board for every day he was idle. He received $8 for his work. How many days did he work ? 69. A druggist, by selling 10 pounds of sulphur for a certain sum, gained 50%. If the cost of sulphur advances 20% in the wholesale mai-ket, what per cent will the druggist now gain by selling 7^ pounds for the same sum ? 70.* The head of a fish is 9 inches long. The tail is as long as the head and .V of the body, and the body is as long as the head and tail. What is the length of the fish ? 71. In a corner of a bin I pour some grain which extends up the wall 8 feet, and whose base is measured by a circular line 10 feet distant from the corner. How many bushels in the pile ? 72. A substance is weighed from both arms of a false bal- ance, and its apparent weights are 4 pounds and 16 pounds. Find its true weight. 8 MATHEMATICAL WEINKLES 73. When wheat is worth $.90 a bushel, a baker's loaf weighs 9 ounces. How many ounces should it weigh when wheat is worth $ .72 a bushel ? 74. The difference between the interest of $ 700 and $ 300 for the same time at 6 % is $ 84. Find the time. 75. What is the price of 10 % stocks that yield a profit equal to that of 5 % bonds bought at 80 ? 76. If I sell oranges at 8 cents a dozen, I lose 30 cents ; but if I sell them at 10 cents a dozen, I gain 12 cents. How many have I, and what did they cost me ? 77. If a man can swim across a circular lake in 20 minutes, how long will it take him to ride twice around it at twice his former rate ? 78. If f of the time past noon, plus 4 hours, equals f of the time to midnight plus 3 hours, what is the time ? 79. A horse steps more than 30 and less than 50 inches at each step. If he takes an exact number of steps in walking 259 inches and an exact number in walking 407 inches, what is the length of his step ? 80. I sold two horses for $ 200. I gained 10 % on the first and 20 % on the second. How much did each cost if the sec- ond cost $ 20 more than the first ? 81. A thief is 27 steps ahead of an officer, and takes 8 steps while the officer takes 5 ; but 2 of the officer's steps are equal to 5 of the thief's. In how many steps can the officer catch him ? 82. A tree is 60 feet high, which is f of f of the length of its shadow diminished by 20 feet. Eequired the length of its shadow. 83. What time is it if | of the time past noon is equal ta ■^ of the time to midnight ? 84. Between 2 and 3 o'clock the minute and hour hands of a clock are together. What time is it ? ARITHMETICAL PROBLEMS 9 85. Which weighs the more, a pound of feathers or a pound of gold ? 86. Four pedestrians whose, rates are as the numbers 2, 4, 6, and 8, start from the same point to walk in the same direc- tion ai'ound a circular tract 100 yards in circumference. How far has each gone when they are next together ? 87. If 2 miles of fence will inclose a square of 160 acres, how large a square will 3 miles of fence inclose ? 88. I bought a hojse for $ 90, sold it fon $ 100, and soon repurchased it for $ 80. How much did I make by trading ? 89. Considering the earth 8000, and the sun 800,000 miles in diameter, how many earths would it take to equal the sun ? 90. A merchant marks his goods to sell at an advance of 25%, and sells a book for $2.25, and allows the customer 10 % oif from the marked price. What did the book cost the merchant "/ 91. A merchant gives a discount of 10%, but uses a yard measure .72 of an inch too short. What rate of discount would allow him the same amount of gain if the measure were cor- rect? 92.* A merchant at one straight cut took off a segment of a cheese which weighed 2 pounds, and had \ of the circumfer- ence. W^hat was the weight of the whole cheese ? 93. What is the shortest distance that a fiy will have to go, crawling from one of the lower corners of the room to the op- posite upper corner — the room being 20 feet long, 15 feet wide, and 10 high ? 94. I buy goods at 50 % off and sell them at 40 and 10 % off. What is my per cent profit ? 95. A farmer goes to a store and says : " Give me as much money as I have and I will spend ten dollars with you." It is given him, and the farmer repeats the operation to a second, 10 MATHEMATICAL WRINKLES and a third store, and has no money left. What did he have in the beginning ? 96. A book and a pen cost $1.20; the book cost $1 more than the pen. What was the cost of each ? 97. A dealer asked 30% profit, but sold for 10 % less than he asked. What per cent did he gain ? 98. Suppose we leave the Pacific coast at sunrise, on Sep- tember 28, and cross the Pacific Ocean fast enough to have sun- rise all the way over to Manila, where it is sunrise September 29. How do you account for the lost day ? 99. A man was asked whether he had a score of sheep. He replied, " No, but if I had as many more, half as many more, and two sheep and a half, I should have a score." How many had he ? 100. What part of threepence is a third of twopence ? 101. Three boys met a servant maid carrying apples to market. The first took half of what she had, but returned to her 10 ; the second took J, but returned 2 ; and the third took away half those she had left, but returned 1. She then had 12 apples. How many had she at first ? 102. A person having about him a certain number of German coins, said, "If the third, fourth, and sixth of them were added together, they would make 54." How many did he have ? 103. If a log starts from the source of a river on Friday, and floats 80 miles down the stream during the day, but comes back 40 miles during the night with the return tide, on what day of the week will it reach the mouth of the river, which is 300 miles long ? 104. 1x2x3x4x5x6x7x8x9x0=? 105. One gentleman meeting another and inquiring the time past 12 o'clock, received for an answer, " One third of the time from now to midnight." What time in the afternoon was it ? ARITHMETICAL PROBLEMS 11 106. A said to B, " Give me $ 100, and then I shall have as much as you." B said to A, " Give me $ 100, and then I shall have twice as much as you." How many dollars had each ? 107. At the rate of 4 miles per hour, a raft floats past the lantliiig at 8 A.M.; the down-going steamer, at the rate of 16 miles per hour, passes the landing at 4 p.m. What time is it when the steamer overtakes the raft ? 108. A bought a horse for $ 80 and sold it to B at a certain rate per cent of gain. B sold it to C at the same rate per cent of gain. C paid $105.80 for the horse. What price did B pay, and what was the rate per cent of gain ? 109. The sum of two numbers is 582 and their difference is 218. What are the numbers ? 110. What are the contents and inside surface of a cubical box whose longest inside measurement is 2 feet ? 111. Three persons engaged in a trade with a joint capital of S 9000. A's capital was in trade 5 months, B's 2 months, and C's 1 month A's share of the gain was S 450, B's $ 270, and C's $ 180. What was the capital of each ? 112. A man was hired for a year for $ 100 and a suit of clothes, but at the end of 8 months he left and received his clothes and $ 60 in money. What was the value of the suit of clothes ? 1 13. A note for $ 100 was due on September 1, but on August 11, the maker proposed to pay as much in advance as would allow him 60 days after September 1, to pay the balance. How much did he pay August 11, money being worth 6 % ? 114. If I rent a house at $18 a month, payable monthly in advance, what amount of cash payable at the beginning of the year will pay the year's rent, interest at 5 % ? 115. If a house rents for $20 a month, payable at the close of each month, what amount is due if not paid till the end of year, interest at 6 % ? 12 MATHEMATICAL WEINKLES 116. A merchant sold a lease of $480 a year, payable quar- terly, having 8 years and 9 months to rim, for $ 2500. Did he gain or lose, and how much, interest at 8%, payable semi- annually ? 117. A box of oranges weighed 64 pounds by the grocer's scales, but being placed in the other scale of the balance, it weighed only 30 pounds. What was the true weight of the box of oranges ? 118. If a ball 5 inches in diameter weighs 8 pounds, what will be the weight of a similar ball 10 inches in diameter ? 119. A, B, and C dine on 8 loaves of bread. A furnishes 5 loaves, B 3 loaves, and C pays the others 8 cents for his share. How must A and B divide the money ? 120. A boy being asked how many fish he had, replied, " 11 fish are 7 more than -| of the number." How many had he ? 121. I have two lamps, one of 4-candle power, and one of 9-candle power. If the former is 30 feet distant, how far away must I place the latter to give me the same amount of light? 122. A merchant bought 90 boxes of lemons for $265, pay- ing $ 3.50 for first quality and $ 3 for second quality. How many boxes of each kind did he buy ? 123. A vessel after sailing due north and due east on alter- nate days, is found to be 16V2 miles northeast of the starting place. What distance has it sailed ? 124. Two teachers work together ; for 10 days' work of the first and 8 days' work of the second they receive $28, and for 5 days' work of the first and 11 days' work of the second they receive $21. What is each man's daily wages? 125. A hind wheel of a carriage 4 feet 6 inches high re- volved 720 times in going a certain distance. How many revolutions did the fore wheel make, which was 4 feet high ? ARITHMETICAL PROBLEMS 13 126. A farmer carried some eggs to market, for which he received $ 2.56, receiving as many cents a dozen as there were dozen. How many dozen were there ? 127. Three men, A, B, and C, ai-e to mow a circular meadow containing 9 acres. A is to receive $3, B S4, and C$5 for his work. What width must each man mow ? 128. If the diameter of a cannon ball is 100 times that of a bullet, how many bullets will it take to equal the cannon ball? 129. A man sells a cow and a horse for $ 120. He sells the horse for $100 more than the cow. What did he sell each for? 130. If a man 5J feet tall weighs 166.375 pounds, how much will a man 6 feet tall of similar proportions weigh ? 131. Having sold a house and lot at 4 % commission, I in- vest the net proceeds in merchandise after deducting my com- mission of 2% for buying. My whole commission is $50. For how much did I sell the house and lot ? 132. A teacher agreed to teach a 10-weeks school for $ 100 and his board. At the end of the term, on account of 3 weeks' absence caused by sickness, he received only $58. What was his board per week ? 133. In buying a bill of goods, I am offered my choice of 50, 25, and 5 % discount, or 5, 25, and 50 % discount. Which is better? 134. The product of two numbers exceeds their difference by their sum. Find one of the numbers. 135. Twice the sum of two numbers plus twice their differ- ence is 80. What is the greater number ? 136. One half the sum of two numbers exceeds one half their difference by 60. What is the smaller number? 137. What per cent is gained by sellir^g 13 ounces of coffee for a pound ? 14 MATHEMATICAL WKIKKLES 138. If I sell I of an acre of land for what an acre cost me, what per cent do I gain ? 139. I sold a horse for $ 200, losing 20 % ; I bought another and sold it at a gain of 25 % ; I neither gained nor lost on the two. What was the cost of each ? 140. At the time of marriage a wife's age was f of the age of her husband, and 24 years after marriage her age was \^ of the age of her husband. How old was each at the time of marriage ? 141. How much water is there in a mixture of 50 gallons of wine and water, worth $ 2 per gallon, if 50 gallons of the wine costs $250? 142. A Texas farmer keeps 2100 cows on his farm. For every 3 cows he plows 1 acre of ground and for every 7 cows he pastures 2 acres of land. How many acres are in his farm? 143. The divisor is 6 times the quotient. Find the quotient. 144. When gold was worth 25 % more than paper money, what was the value in gold of a dollar bill ? 145. I bought 15 yards of ribbon, and sold 10 of them for what I paid for all, and the remainder at cost. I gained $ .25 by the transaction. What did the ribbon cost me ? 146. If a ball of yarn 6 inches in diameter makes one pair of gloves, how many similar pairs will a ball 12 inches in diameter make ? 147. At what time between 4 and 5 o'clock do the hour and minute hands of a clock coincide ? 148. At what time between 2 and 3 o'clock do the hour and minute hands of a clock coincide ? 149. At what time between 2 and 3 o'clock are the hour and minute hands of the clock at right angles ? 150. At what time between 2 and 3 o'clock are the hands of a clock exactly opposite each other ? ARITHMETICAL PROBLEMS 15 151. From 200 hundredths take 15 tenths. 152. Find the sum of 2324 thousandths and 24,325 hun- dredths. 153. A lady at her marriage had her husband agree that if at his death they should have only a daughter, she should have J of his estate ; and if they should have only a son, she should have |. They had a son and a daughter. How much should each receive, if the estate was worth $ 23,375 ? 154. A crew can row 24 miles downstream in 3 hours, but requires 4 hours to row back. What is the rate of the current? 155. What minuend is 80 greater than the subtrahend, which is 20 greater than the remainder ? 156. The G. C. D. of two numbers is 60 and the L. C. M. is 720. Find the product of the numbers. 157. In extracting the cube root of a perfect power the oper- ator found the last complete dividend to be 132,867. Find the power. 158. A merchant marks his goods at an advance of 25 % on cost. After selling J of the goods, he finds that some of the goods on hand are damaged so as to be worthless ; he marks the salable goods at an advance of 10 % on the marked price and finds in the end that he has made 20 % on cost. What part of the goods was damaged ? 159. A king has a horse shod and agrees to pay 1 cent for driving the first nail, 2 cents for the second, 4 cents for the third, doubling each time. What will the shoeing with 32 nails cost? 160. I sold a book at a loss of 25 %. Had it cost me $1 more, my loss would have been 40%. Find its cost. 161. At noon the three hands — hour, minute, and second — of a clock are together. At what time will they first be to- gether again? 16 MATHEMATICAL WKINKi.ES 162. A train is traveling from one station to another. After traveling an hour it breaks down and is delayed for an hour. It then proceeds at f of its former speed, and arrives 3 hours late. Had it gone 50 miles farther before the breakdown, it would have arrived 1 hour and 20 minutes sooner. Find the rate of the train and the distance between the stations. 163. If a cocoanut 4 inches in diameter is worth 5 cents, what is the worth of one 6 inches in diameter ? 164. Prove that the product of the G.C.D. and L.C.M. of two numbers is equal to the product of the numbers. 165. Sum to infinity the series l + Y+i + B'+ **•• 166. Find the sum of 1 + i + i + 2V + • • • to infinity. 167. Find the sum of 4 -f- 0.4 -{- 0.04 + ... to infinity. 168. What is the distance passed through by a ball before it comes to rest, if it falls from a height of 100 feet and re- bounds half the distance at each fall ? 169. Two trains start at the same time, ouq from Jackson- ville to Savannah, the other from Savannah to Jacksonville. If they arrive at destinations 1 hour and 4 hours after passing, what are their relative rates of running ? 170. If sound travels at the rate of 1090 feet per second, how far distant is a thundercloud when the sound of the thun- der follows the flash of lightning after 10 seconds ? 171. The G.C.D. and the L.C.M. of two numbers between 100 and 200 are respectively 4 and 4620. Find the numbers. 172. What three equal successive discounts are equivalent to a single discount of 58.8 % ? 173. How much will the product of two numbers be in- creased by increasing each of the numbers by 1 ? 174. I can beat James 4 yards in a race of 100 yards, and James can beat John 10 yards in a race of 200 yards. How many yards can I beat John in a race of 500 yards ? a:^thmetical problems 17 175. Three ladies own a ball of yarn G inches in diameter. What portion of the diameter must each wind off in order to divide the yarn equally among them ? 176. Demonstrate the following : If the greater of two num- bers is divided by the less, and the less is divided by the remainder, and this process is continued till there is no re- mainder, the last divisor will be the greatest common divisor. 177. Find the volume of a rectangular piece of ice 8 feet long, 7 feet wide, and floating in water, with 2.4 inches of its thickness above water, the specific gravity of ice being .9. 178. Two trains, 400 and 200 feet long respectively, are moving with uniform velocities on parallel rails; when they move in opposite directions they pass each other in 5 seconds, but when they move in the same direction, the faster train passes the other in 15 seconds. Find the rate per hour at which each train moves. 179. A boy is running on a horizontal plane directly towards the foot of a tree 50 feet in height. When he is 100 feet from the foot of the tree, how much faster is he approaching it than the top ? 180. Express 77,610 in the duodecimal scale. 181.* In what scale is 6 times 7 expressed by 110? 182. Express Adam's age at his death in the binary scale. 183. Add 3152e, 4204e, 3241e, SlOSg. 184. Subtract 12,3125 from 23,024^. 185. Multiply 62,453; by 325;. 186. Divide 2,034,431, by 234,. 187. Extract the square root of 170». 188* Extract the cube root of 3I2O4. 189. How many trees can be set out upon a space 100 feet square, allowing no two to be nearer each other than 10 feet ? 18 MATHEMATICAL WKINKLES 190. How many stakes can be driven down upon a space 12 feet square, allowing no two to be nearer each other than 1 foot? 191. Multiply 789,627 by 834, beginning at the left to multiply. 192. Two fifths of a mixture of wine and water is wine ; but if 10 gallons of water be added to it, then only -^-^ of the mix- ture will be wine. How many gallons of each liquid is in the mixture ? 193. Simplify 10 4- — 1 + -1 1 + 1-i 194. 15,600 is the product of three consecutive numbers. What are they ? 195. Find a number which is as much greater than 1042 as it is less than 1236. 196. Multiply 729,038 by 105,357 using only 3 multipliers. 197. What is the smallest number to be subtracted from 10,697 to make the result a perfect cube ? 198. I wish to reach a certain place at a certain time ; if I walk at the rate of 4 miles an hour, I shall be 10 minutes late, but if I walk 5 miles an hour, I shall be 20 minutes too soon. How far have I to walk ? 199. A wineglass is half full of wine, and another twice the size is \ full. They are then filled up with water, and the contents mixed. What part of the mixture is wine, and what part water ? 200. A cork globe 2 feet in diameter, whose specific gravity is -Jg, is hollowed out and filled with lead whose specific gravity is 10. What must be the thickness of the shell of cork so that it will sink just even with the surface of the water? ARITHMETICAL PROBLEMS 19 201. What temperature will result from mixing 100 pounds of ice at 14° F. with 80 pounds of steam at 270° F. ? 202. It is 1800 miles from A to C, and the " Sunset Flyer " annihilates the distance in 50 hours. She averages 30 miles an hour from A to B, and 55 miles an hour from B to C. Locate B. 203. A square and its circumscribing circle revolve about the diagonal of the square as an axis. Compare the volumes and surfaces of the solids generated, the diagonal being 6 feet. 204. The aggregate area of two square fields is 8J acres. The side of the second is 10 rods longer than that of the first. Ascertain the length of the first. 205. How high above the earth's surface (radius 4000 miles) would a pound weight weigh but one ounce avoirdupois by a scale indicator, corrected for change of elasticity by tem- perature ? 206. On a west-bound freight train a man is running east- ward at the rate of 6 miles an hour, and likewise a man runs in the same direction 8 miles an hour on a train going east. If the trains pass while running 36 and 22 miles an hour, re- spectively, how many miles apart are the men at the end of one minute from the moment they pass each other ? 207. A drawer made of inch boards is 8 inches wide, 6 inches deep, and slides horizontally. How far must it be drawn out to put into it a book 4 inches wide and 9 inches long? 208. The dividend is 4352, the remainder 17, which is the G.C. D. of the quotient and divisor, whose difference you may find. 209. B paid S9 more than true discount by borrowing money at a bank for one year at 12 % . Find the face of the note. 20 MATHEMATICAL WRINKLES 210. How many feet of inch lumber in a wagon tongue 10 feet long, 4 inches square at one end and 2 inches by 3 inches at the other end ? 211. How many inch balls can be put in a box which meas- ures inside 10 inches square and 5 inches deep ? 212. If the posts of a wire fence around a rectangular field twice as long as wide were set 16 feet apart instead of 12 feet, it would save 66 posts. How many acres in the field ? 213. If gold is 19.3 times as heavy as water and copper 8.89 as heavy, how many times as heavy is a coin composed of 11 parts of gold and 1 part of copper ? 214. A ball falls 15 feet and bounces back 5 feet. How far will it bound before it comes to rest ? 215. A borrows $500 from a building and loan association and agrees to pay $9.50 per month for 72 months, the first payment to be made at the end of the first month. What rate of interest does he pay ? The association claims to charge only 8 % (the legal rate in Alabama). How can the per cent be figured out ? 216. A rope 50 feet long is fastened to two stakes, driven 40 feet apart. A calf is fastened to a ring which moves freely on this rope. Over what area can the calf graze ? 217. A metal dog made of gold and silver weighs 8.75 ounces. Its specific gravity is 14.625, that of gold 19.25, and that of silver 10.5. Find the number of ounces of gold in it. 218. By drilling an inch hole through a cubical block of wood parallel to the faces of the block, -J^ of the wood was cut away. What were the dimensions of the block ? 219. Find two numbers whose G. CD. is 24, and L. C. M. 288. 220. Find the greatest number that will divide 364, 414, and 539, and leave the same remainder in each case. ARITHMETICAL PROBLEMS 21 221. Had an article cost me 8% less, the number of per cent gain would have been 10 % more. What was the gain ? 222. At what time between 3 and 4 o'clock will the minute hand be as far from 12 on the left side of the dial plate as the hour hand is from 12 on the right side ? 223. A ball whose specific gravity is 3| measures a foot in diameter. Find the diameter of another ball of the same weight but with a specific gravity of 2J^. 224. A owes $ 2500 due in two years. He pays $ 500 cash and gives a note payable in 8 months, for the balance. Find the face of the note, money being worth 6 %. 225. A man bought a horse for $201, giving his note due in 30 days. He at once sold the horse, taking a note for $224.40, due in 4 months. What was his rate of gain at the time of the sale, interest 6 % ? 226. The minute hand and the hour hand coincide every 65 minutes. Does the clock gain or lose, and how much ? 227. A ball weighing 970 ounces, weighs in water 892 ounces, and in alcohol 910 ounces. What is the specific gravity of alcohol ? 228. A steamer moves through 8° of longitude daily in ply- ing to and fro across the Atlantic. How long is it from one noon to the next ? 229. A, B and C raise 165 acres of grain. A owns 100 acres of the land and B 65 acres. C pays the others $110 rent. How must A and B divide this money if the grain is shared equally ? 230. A silver cup is a hemisphere filled with wine worth $1.20 a quart. The value of the cup is 2 dimes for every square inch of internal surface, and the cup is worth just as much as the wine. What is the value of the cup ? 231. A ball 12 inches in diameter is rolled around a circular room 12 feet in diameter in such a way that it always touches 22 MATHEMATICAL WRINKLES both wall and floor. How many revolutions does the ball make in rolling once around the room ? 232. A man desires to purchase eggs at 5 cents, 1 cent, and ^ cent, respectively, in such numbers that he will obtain 100 eggs for a dollar. How many solutions in rational inte- gers ? 233. How many board feet in a piece of lumber, 2 inches square at one end and at the other end 1 inch by 12 inches, if the ends are parallel ? 234. How many board feet in the above piece of lumber if it is 24 feet long ? 235. Is anything expressed by .^ ? If so, what ? 236. A man bequeathed to his son all the land he could in- close in the form of a right-angled triangle with 2 miles of fence, the base of the triangle to be 128 rods. How many acres did he get? 237. The distance around a rectangular field is 140 rods, and the diagonal is 50 rods. Find its length, breadth, and area. 238. The specific gravity of ice being .918 and of sea water 1.03, find the volume of an iceberg floating with 700 cubic yards above water. 239. A room is 30 feet long, 12 feet wide, and 12 feet high. At one end of the room, 3 feet from the floor, and midway from the sides, is a spider. At the other end, 9 feet from the floor, and midway from the sides, is a fly. Determine the shortest path by way of the floor, ends, sides, and ceiling, the spider can take to capture the fly. 240. A and B are engaged in buying hogs, each paying out of his individual funds for hogs purchased by him, and each retaining as his individual funds the money received from sales made by him. They now wish to form a partnership to cover ARITHMETICAL PROBLEMS 23 all past transactions and to share equally in the settlement for sales and purchases, and also to be equally interested in hogs which they have on hand unsold. The following data given : A has paid for hogs $1183.35, and received from sales of hogs $434.35. ' B has paid for hogs $241.55, and received from sales of hogs $619.00. Invoice of hogs on hand at this time $511.35. How much does A owe B, or B owe A, so that they will have shared equally in payments and receipts, and be equally inter- ested in the hogs on hand ? 241. The hour, minute, and second hands of a clock turn on the same center. At what time after 12 o'clock is the hour hand midway between the other two ? The second hand mid- way between the other two? The minute hand midway be- tween the other two ? 242. My agent sold pork at a commission of 7%. The pro- ceeds being increased by $6.20, I ordered him to buy cattle at a commission of 3J%. Cattle now declined in price 33 J %, and I found my total loss, including commissions, to be exactly $1002.20. Find the value of the pork. 243. A owes $900, due December 10, but he makes two equi- table payments, one September 8 and the other January 10. Find each payment. 244. A man, dying, left an estate of $23,480 to his three sons, aged 15, 13, and 11 years, to be so divided that each share placed at interest shall amount to the same sum as the sons, respectively, become 21 years of age. What was each son's share, money being worth 5 % ? 245. A man spent $ 100 in buying two kinds of silk at $ 4.50 and $4.00 a yard; by selling it at $4.25 per yard he gained 2 ojo • How many yards of each did he buy ? 24 MATHEMATICAL WKIKKLES 246. A lady being asked the time of day replied, "It is between 4 and 5 o'clock, and the hour and minute hands are together." What was the time ? 247. Three men A, B, and C can do a piece of work in 60 days. After working together 10 days, A withdraws and B and C work together at the same rate for 20 days, then B with- draws and C completes the work in 96 days, working i longer each day. Working at his former rate, C alone could do the work in 222 days. Find how long it would take A and B each separately to do the work. 248. In a class there are twice as many girls as boys. Each girl makes a bow to every other girl, to every boy, and to the teacher. Each boy makes a bow to every other boy, to every girl, and to the teacher. In all there are 900 bows made. How many boys are in the class ? 249. A boy weighing 96 pounds is seated on one end of a see- saw 16 feet long, and a boy weighing 120 pounds is seated on the other end. Find the distance of each boy from the point of support, the lengths of the two arms of the plank being inversely proportional to the weights at their ends. 250. Two men are on opposite sides of the center of the earth. Find the shortest distance that each will be required to go in order to exchange places, provided they travel different routes and so travel as to enjoy each other's company for 500 miles of the distance. (Radius of earth = 4000 miles.) 251. A conical wine glass 2 inches in diameter and 3 inches deep is ^ full of water. What is the depth of the water ? 252. A hollow sphere 8 inches in diameter is filled with water. How many hollow cones, each 8 inches in altitude, and 8 inches in diameter at the base, can be filled with the water in the sphere ? ALGEBRAIC PROBLEMS 1. I am now twice as old as you were when I was your age. When you are as old as I now am, the sum of our ages will be 100. What are our ages ? 2. A starts from Gunter to Denton, and at the same time B starts from Denton to Gunter ; A reaches Denton 32 hours, and B reaches Gunter 60 hours, after they meet on the way. In how many hours do they make the journey? 3. At what time between 10 and 11 o'clock is the second hand of a clock one minute space nearer to the hour hand than it is to the minute hand? 4. In walking along a street on which electric cars are running at equal intervals from both ends, I observe that I am overtaken by a car every 12 minutes, and that I meet one every 4 minutes. What are the relative rates of myself and tjie cars, and at what intervals of time do the cars start ? 6. What are eggs per dozen when 2 less in a shilling's worth raise the price one penny per dozen? 6. Two men agree to build a walk 100 yards in length for S200. They divide the work so that one man should receive 60 cents more per yard than the other. How many yards does each man build, if he receives $100? 7. Two boats start from opposite sides of a river at the same instant, and throughout the journeys to be described maintain their respective speed. They pass one another at a point just 720 yards from the left shore. Continuing on their respective journeys, they reach opposite banks, where each boat remains 10 minutes and then proceeds on its return trip. 26 MATHEMATICAL WRINKLES This time the boats meet at a point 400 yards from the right shore. What is the width of the river ? 8. How many acres does a square tract of land contain, which sells for $ 160 an acre, and is paid for by the number of silver dollars that will lie upon its boundary ? 9. Two girls, 4 feet apart, walk side by side around a circular park. How far does each walk if the sum of their distances is 1 mile ? 10. How many acres are there in a field, the number of rails used in fencing the field equaling the number of acres — each rail being 11 feet long and the fence 4 rails high ? 11. Three men are going to make a journey of 40 miles. The first can walk at the rate of 1 mile per hour, the second walks at the rate of 2 miles per hour, and the third goes in a buggy at the rate of 8 miles per hour. The third takes the first with him and carries him to such a point as will allow the third time to drive back to meet the second, and carry him the remaining part of the 40 miles, so as all may arrive at the same time. How long will it require to make the journey ? 12. Two trains, 400 and 200 feet long respectively, are mov- ing with uniform velocities on parallel rails ; when they move in opposite directions, they pass each other in 5 seconds, but when they move in the same direction, the faster train passes the other in 15 seconds. Find the rate per hour at which each train moves. 13. How many minutes is it until 6 o'clock, if 50 minutes ago it was 4 times as many minutes past 3 o'clock? 14. A man bought a gun for a certain price. Now, if he sells it for $ 9, he will lose as much per cent as the gun cost. Required the cost of the gun. 15. In a nest were a certain number of eggs; if I had brought 1 egg that I didn't bring, I should have brought | of ALGEBRAIC PROBLEMS 27 thera, and if I had left 2 eggs that I did bring, I should have brought half of them. How many eggs were in the nest? 16. A man sold a lot for $ 144. The number of dollars the lot cost was the same as the number of per cent profit. What did the lot cost ? 17. What is the side of a cube which contains as many cubic inches as there are square inches in its surface ? 18. What is the length of one edge of that cube which con- tains as many solid units as there are linear units in the diag- onal through the opposite corners ? 19. The sum, the product, and the difference of the squares of two numbers are all equal. Find the numbers. 20. Upon inquiring the time of day, a gentleman was in- formed that the hour and minute hands were together between 4 and 5. What was the time of day ? 21. An officer wishing to arrange his men in a solid square, found by his first arrangement that he had 39 men over. He then increased the number of men on a side by 1, and found 50 men were needed to complete the square. How many men did he have ? 22. A young lady being asked what she paid for her eggs, replied, "Three dozen cost as many cents as I can buy eggs for 36 cents." What was the price per dozen ? 23. A cube is formed out of a lot of cubical blocks, 1 foot each, and it is found by using 448 more another cube is formed, the edge of which is 8 feet. What was the length of an edge of the original cube ? 24. Find two numbers whose product is equal to the differ- ence of their squares, and the sum of their squares equal to the difference of their cubes. 25. A young lady being asked her age, answered, " If you add the square root of my age to | of my age, the sum will be 10." Required her age. 28 MATHEMATICAL WRINKLES 26. There is a fish whose head is 9 inches long ; the tail is as long as the head and | the body ; and the body is as long as the head and the tail together. What is the length of the fish? 27. I bought 2 horses for S 80 ; I sold them for $ 80 apiece, the gain on the one being 20 % more than on the other. What was the cost of each ? 28. A man has a square lot upon which he wishes to build a house facing the street, with a driveway around the other three sides. He wants the house to cover the same amount of land as the driveway. How wide shall he make the driveway, the lot being 100 feet each way ? 29. An officer can form his men into a hollow square 4 deep, and also into a hollow square 6 deep ; the front in the latter formation contains 12 men fewer than in the former formation. Find the number of men. 30. How must a line 12 inches long be divided into two parts so that the rectangle of the whole line and one part shall equal the square on the other side ? 31. Two miners, B and C, have the same monthly wages. B is employed 7 months in the year, and his annual expenses are $350; C is employed 5 months in the year, and his annual expenses are $250. In 5 years B saves the same amount that C saves in 7 years. What were the monthly wages of each? 32. Simplify: '^ , : • 33. Find the value of x in the equation : 2(l + ^') = (l + a;)^ 34. Solve the equation : iB* + 4 m^x —m* — 0. ALGEBRAIC PROBLEMS 29 Solve the following equations : 36. r^ + 3/=ll, 39. Vx+V2/ = 5, 36. x'-y^f-^x, 4Q x-hy = 13, ^ + y = Kx-fy ^_^^_^g^ «^- ^ + 3/ = 10, ,, a- + a:y + 2r' = 39, yVa;=12. ar^4-a:z + z2 = i9, 38. x^4-r = 13, y2^y2-f-z* = 49. y + X2/ = 9. 42. 5t/(a;« + l)-3ar'(2/«+l)=0, 15y«(ar^4-l)-x(/ + l)=0. 43. A farmer being asked how many acres he had, replied, " My land is square. I have plowed just 2 rods wide around, and have plowed just \ my land." How many acres has he ? 44. From a 10-gallon keg of wine a man filled a jug. He then filled the keg with water, and repeated the operation a second time, when he found the keg contained equal amounts of water and wine. Find the capacity of the jug. 45. If a certain number is divided by 32, the remainder is 25 ; if divided by 25, the remainder is 19 ; and if divided by 19, the remainder is 11. What is the number? 46. If Dr. A loses 3 patients out of 7 ; Dr. B, 4 out of 13 ; and Dr. C, 5 out of 19 ; what chance has a sick man for his life, who is dosed by the three doctors for the same disease ? 47. Said Robin to Richard, " If ever I come To the age that you are, brother mine, Our ages united would amount to the sum Of years making ninety-nine." Said Richard to Robin, " That's certain, and if it be fair For us to look forward so far, I then shall be double the age that you were, When I was the age that you are." 30 MATHEMATICAL WRINKLES 48. A tells the truth 2 times out of 3, B 6 times out of 7, and C 4 times out of 5. What is the probability of the truth of an assertion that A and B affirm and C denies ? 49. A plank 16 feet long with a weight of 196 pounds, on one end balances across a fulcrum placed 1 foot from the 196-pound weight. What is the weight of the plank ? 50. A man desires to purchase eggs at 5 cents, 1 cent, and ■i- cent, respectively, in such numbers that he will obtain 100 eggs for a dollar. How many solutions in rational integers ? 51. Ann's brother started to school. On the first day the teacher asked him his age. He replied, " When I was born, Ann was ^ the age of mother and is now ^ as old as father, and I am J of mother's age. In 4 years I shall be i as old as father." How old is Ann's brother ? 52. Solve for x : Om+6 Qm-1 Q(n+1)* ^6 ^ _J_ !_» i.y ^ ,21^-^-n . 32-1-^-1). 53. My wife was born ,, lL(f)UL 2^^' 2«-ijL(2 . 3 . 3K e^ijl What was her age August 10, 1904 ? Note. — Problems 54-67, inclusive, are from Bowser's "College Al- gebra." 54. Express with positive exponents ■V(a + &)'x(a + 6)-t. 55. Extract the square root of 6-f.2V2 + 2V3-|-2V6. 66. Extract the square root of 54.V10- V6- Vi5. 57. Solve a;~^ + a;~^ = 6. xy =2*. 61. x^ + y^ = 14a^/, x-{-y = a. 62. m'<^)'-^- xy — x — y — bAi. ALGEBRAIC PROBLEMS 31 58. Solve a;^-|-x^ = 1056. 69. Solve -Jt— {a^-h^)x= -^ -. 60. Solve the following : 6(a^ + y* + 2^) = 13(0; ^- 2/ -h 2) = *fL, 63. a? + y{xy-l) = 0, f-x(xy + l) = 0. 65. (x^-hl)y = (f-¥l)^y {f^l)x=9(x^ + l)f, 66. A offers to run three times round a course while B runs twice round, but A gets only 150 yards of his third round fin- ished when B wins. A then offers to run four times round to B three times, and now quickens his pace so that he runs 4 yards in the time he formerly ran 3 yards. B also quickens his so that he runs 9 yards in the time he formerly ran 8 yards, but in the second round falls off to his original pace in the first race, and in the third round goes only 9 yards for 10 he went in the first race, and accordingly this time A wins by 180 yards. Determine the length of the course. 67. On the ground are placed n stones ; the distance between the first and second is 1 yard, between the second and third 3 yards, between the third and fourth 5 yards, and so on. How far will a person have to travel who shall bring them one by one to a basket placed at the first stone ? 68. Sionius and his wife Lionius sip from the same bowl filled with milk. Lionius sips during f of the time which Sionius would take to empty the bowl ; then Lionius stops and 32 MATHEMATICAL WRINKLES hands it to Sionius to finish. If both had sipped together, the bowl would have been emptied 6 minutes sooner, and Lionius would have received | of the milk which Sionius sipped after receiving the bowl from Lionius. In what time would Sionius and Lionius sipping together empty the bowl ? 69. Once, in classic days, Silenus lay asleep, a goatskin filled with wine near him. Dionysius passing by, profited by seizing the skin, and drinking for | of that time in which Silenus alone could have emptied said skin. At this point Si- lenus awoke, and seeing what was happening, snatched away the precious skin, and finished it. Now, had both started together, and drunk simultaneously, they would have consumed the wine skin in 2 hours less time. And, in this case, Dionysius' share would have been J as much as Silenus did secure, by waking and snatching the skin. In what time would either one of them alone finish the goatskin ? 70. Three regiments move north as follows : B is 20 miles east of A ; C is 20 miles south of B, and each marches 20 miles between the hours of 5 a.m. and 3 p.m. A horseman with a message from C starts at 5 a.m. and rides north till he overtakes B, then sets a straight course for the point at which he calculates to overtake A, then sets a straight course for the next point at which he will again overtake B, then rides south to the point where he first overtook B, reaching that point at the same time as C, namely, 3 p.m. What uniform rate of travel enabled the messenger to do this ? 71. Three men and a boy agree to gather the apples in an orchard for $ 50. The boy can shake the apples in the same time that the men can pick them, but any one of the men can shake them 25 % faster than the other two men and boy can pick them. Find the amount due each. GEOMETRICAL EXERCISES 1. Construct a trapezoid having given the sum of the parallel sides, the sum of the diagonals, and the angle formed by the diagonals. 2. If three equal circles are tangent to each other, each to each, and inclose a space between the three arcs equal to 200 square feet, find the diameter of each circle. 3. An iron rod of a certain length stands against the side of a house ; if it is pulled out 4 feet at the bottom, the top moves down the side of the house a distance equal to ^ the rod. Find the length of the rod. 4. A circle whose area is 1809.561 square feet is described upon the perpendicular of a right triangle as a diameter. From the point where the circumference cuts the hypotenuse a tangent to the circle is drawn, which cuts the base. If the shortest distance from the point of intersection of the tangent with the base to the perpendicular is 18 feet, what is the length of the hypotenuse? 5. The number of cubic inches contained by two equal opposite spherical segments, together with the number of cubic inches contained by the cylinder included between these segments, is 600. If this be J of the number of cubic inches contained by the whole sphere, find the height of the cylinder. 6. The sum of the sides of a right-angled triangle is 200 feet. What is its area, the hypotenuse being 4 times the per- pendicular let fall upon it from the right angle ? 33 34 MATHEMATICAL WRINKLES 7. In a right-angled triangle the hypotenuse is 100 feet, and a line bisecting the right angle and terminating in the hypotenuse is 14.142 feet. Eind the length of each of the other two sides. 8. Two posts, one of which is 24, and the other 16 feet high are 100 feet apart. What is the length of a rope just long enough to touch the ground between them, the ends of the rope being fastened to the top of each post? 9. A ladder 30 feet long leans against a perpendicular wall at an angle of 30°. How far will its middle point move, pro- vided the top moves down the wall until it reaches the ground ? 10. A man owns a piece of land in the form of a right- angled triangle. The sum of the sides about the right angle is 70 feet and their difference equals the length of a line parallel to the shorter side, dividing the triangle into two equal parts. Determine the length of the shorter side. 11. Required the greatest right triangle which can be con- structed upon a given line as hypotenuse. 12. A man has a lot the shape of which is an equilateral triangle, with an area of 60 square rods. How long a rope will be required to graze his horse over ^ the lot, provided he ties the rope to a corner post? 13. An iron ball 3 inches in diameter weighs 8 pounds. Eind the weight of an iron shell 3 inches thick, whose external diameter is 30 inches. 14. Eind the altitude of the maximum cylinder that can be inscribed in a cone whose altitude is 9 feet and whose base is 6 feet. 15. Construct a plane triangle having given the base, the vertical angle, and the bisector of the vertical angle. 16. How much of the earth's surface would a man see if he WQre rg^ised, to the height of the diameter above it ? GEOMETRICAL EXERCISES 35 17. To what height must a man be raised above the earth in order that he may see \ of its surface ? 18. What part of the surface of a sphere 20 feet in diameter is illuminated by a lamp 100 feet from the surface of the sphere ? 19. If the earth is assumed to be a sphere of 4000 miles radius, how far at sea can a lighthouse 110 feet high be seen ? 20. Determine the sides of an equilateral triangle, having given the lengths of the three perpendiculars drawn from any point within to the sides. 21. Find the number of cubic inches of water that a bowl will hold, whose shape is that of a spherical segment, 10 inches in height, the diameter of the top being 40 inches. 22. Find the side of the lai-gest cube that can be cut from a globe 24 inches in diameter. 23. Which is the greater — 3 solid inches, or 3 inches solid ? 24. Three men living 60 miles from one another wish to dig a well that will be the same distance from each of their homes. Where must they dig the well ? 25. Bisect a given quadrilateral by a straight line drawn through a vertex. 26. One arm of a right triangle is 30 feet and the perpen- dicular from the vertex of the right triangle to the hypotenuse is 24 feet. Find the area of the triangle. 27. Three chords, lengths 6, 8, and 10, just go around in a semicircle. Find the radius of the circle. 28. A cone, a half globe, and a cylinder, of the same base and altitude, are as 1 : 2 : 3. 29. Two sides of a triangle are 3 feet and 8 feet, respec- tively, and inclose an angle of 60°. Find the third side. 36 MATHEMATICAL WRINKLES 30. A rectangular garden is 40 feet by 60 feet. It is sur- rounded by a road of uniform width, the area of which is equal to the area of the field. Find the width of the road. 31. The sum of the two crescents made by describing semi- circles outward on the two sides of a right triangle and a semi- circle toward them on the hypotenuse, is equivalent to the right triangle. 32. Prove that the circle through the middle points of the sides of a triangle passes through the feet of the perpendicu- lars from the opposite vertices, and through the middle points of the segments of the perpendiculars included between their point of intersection and the vertices. 33. What is the volume of the frustum of a sphere, the radius of whose upper base is 3 feet and lower base 4 feet, and altitude 1 foot ? 34. If a circle rolls on the inside of a fixed circle of double the radius, find the length of the path that any fixed point in the circumference of the moving circle will trace out. 35. Find the diameter of a circle inscribed in a triangle whose sides are 6, 8, and 10 feet, respectively. 36. Find the diameter of a circle circumscribed about a triangle whose sides are 6, 8, and 10 feet, respectively. 37. What is the area of an equilateral triangle whose sides are 100 inches ? 38. What is the area of a tetragon (square) whose sides are 100 inches ? 39. What is the area of a regular pentagon whose sides are 100 inches ? 40. What is the area of a regular hexagon whose sides are 10 feet? GEOMETRICAL EXERCISES 37 41. What is the area of a regular heptagon whose sides are 10 feet ? 42. What is the area of a regular octagon whose sides are 10 feet ? 43. What is the area of a regular nonagon whose sides are 10 feet ? 44. What is the area of a regular decagon whose sides are 10 feet ? 45. What is the area of a regular undecagon whose sides are 10 feet? 46. What is the area of a regular dodecagon whose sides are 10 feet? 47. Find the side of an inscribed square of a triangle whose base is 10 feet and altitude 4 feet. 48. Find the diameter of a circle of which the height of an arc is 6 inches and the chord of half the arc is 10 inches. 49. Find the height of an arc, when the chord of the arc is 10 inches and the radius of the circle is 8 inches. 50. Find the chord of half an arc, when the chord of the arc is 20 feet and the height of the arc is 2 feet. 51. Find the chord of half an arc, when the chord of the arc is 10 inches and the radius of the circle is 8 inches. 52. Find the side of a circumscribed polygon, when the side of a similar inscribed polygon is 10 feet and the radius of the circle is 30 feet. 53. A log 10 feet long, 2 feet in diameter at one end and 3 feet at the other, is rolled along till the larger end describes a circle. Find the diameter of the circle. 54. At the extremities of the diameter of a circular park stand two electric light posts, one 12 feet high and the other 18 feet high. What points on the circumference of the park 38 MATHEMATICAL WEINKLES are equidistant from the tops of the posts, the diameter of the park being 100 feet ? 55. What is the circumference of the largest circular ring that can be put in a cubical box whose edge is 4 feet ? 56. What is the side of the largest square that can be in- scribed in a semicircle whose diameter is 2Vo feet? 57. What is the volume of the largest cube that can be inscribed in a hemisphere whose diameter is 3 feet ? 58. In a triangle whose base is 30 inches and altitude 18 inches a square is inscribed. Find its area. 59. Two equal circles of 10-inch radii are described so that the center of each is on the circumference of the other. Find the area of the curvilinear figure intercepted between the two circumferences. 60. Two equal circles of 8-inch radii intersect so that their common chord is equal to their radius. Find the area of the curvilinear figure intercepted between the two cir- cumferences. 61. Find the area of a zone whose altitude is 4 feet on a sphere whose radius is 10 feet. 62. Find the volume of a segment of a sphere whose altitude is 1 foot and the radius of the base 2 feet. 63. Mr. Brown has a plank of uniform thickness 10 feet long, 12 inches wide at one end and 5 inches at the other. How far from the large end must it be cut straight across so that the two parts shall be equal ? 64. Having given the lesser segment of a straight line divided in extreme and mean ratio, to construct the whole line. 65. Find the volume of a spherical shell whose two surfaces are 64 tt and 36 tt. 66. To construct a triangle having given the three medians. GEOMETRICAL EXERCISES 39 67. Two sides of a quadrilateral lot run east 216 feet and north 63 feet. If the other two sides measure 135 and 180 feet, respectively, what is its area in square yards ? 68. If the perimeter of a right triangle is 240 rods and the radius of the inscribed circle 20 rods, what are the sides ? 69. On a hillside which slopes 11 feet in 61 feet of its length, stands an upright pole. If this pole should break at a certain point and fall up hill, the top would strike the ground 61 feet from the base of the pole ; but if it should fall down hill, its top would strike the ground 4S^ feet from the base of the pole. Find the length of the pole. 70. A house and barn are 25 rods apart. The house is 12 rods and the barn 5 rods from a brook running in a straight line. What is the shortest distance one must walk from the house to get a pail of water from the brook and carry it to the barn ? 71. Construct geometrically the square root of any number, n. 72. Construct a triangle having given the base, the median upon the base, and the difference between the base angles. 73. A man owning a rectangular field 300 feet by 600 feet, wishes to lay out driveways of equal width having the diago- nals of the field as center lines, and such that the area of the driveways shall be J of the area of the field. Determine the width of the driveways. 74. Two ladders 14 feet apart at their base touch each other at the top. Each is inclined the same, and a round 10 feet up on either side is as far from the top as it is from the base of the other ladder. Get the length of the ladders. uouse^ """"'•^ Bam 1 Brook iO 40 MATHEMATICAL WRINKLES 75. A tree 123 feet high breaks off a certain distance up, and the moment the top strikes a stump 15 feet high the broken part points to a spot 108 feet from the base of the tree. Find the length of the part broken off. 76. Divide a triangle into three equivalent parts by lines drawn from a point P within the triangle. 77. From a point P without a circumference, to draw a secant which is bisected by the circumference. 78. To construct a triangle having given the three feet of the altitudes. 79. If from any point in the circumference of a circle per- pendiculars be dropped upon the sides of an inscribed triangle (produced, if necessary), the feet of the perpendiculars are in a straight line. 80. Inside a square 10-acre lot a cow was tethered to the fence at a point 1 rod from the corner by a rope just long enough to allow her to graze over an acre of ground. How long was the rope ? 81. From any point P in the bisector of the angle A in a triangle ABCy perpendiculars PA\ PB\ PC are drawn to the three sides. Prove PA' and JB'C" intersect in the median from A. 82. If the bisectors of two angles of a triangle are equal, the triangle is isosceles. 83. In a right triangle the bisector of the right angle also bisects the angle between the perpendicular and the median from the vertex of the right angle to the hypotenuse. 84. Find the locus of a point the sum or the difference of whose distances from two fixed straight lines is given. 85. The bisector of an angle of a triangle is less than half the sum of the sides containing the angle. GEOMETRICAL EXERCISES 41 86. The difference between the acute angles of a right triangle is equal to the angle between the median and the perpendicu- lar drawn from the vertex of the right angle to the hypotenuse. 87. A hollow rubber ball is 2 inches in diameter and the rubber is -^jr inch thick. How much rubber would be used in the manufacture of 1000 such balls ? 88. Having given two concentric circles, draw a chord of the larger circle, which shall be divided into three equal parts by the circumference of the smaller circle. 89. The distances from a point to the three nearest corners of a square are 1 inch, 2 inches, and 2J inches. Construct the square. 90. Draw a chord of given length through a given point, within or without a given circle. 91. Find the greatest segment of a line 10 inches long, when it is divided in extreme and mean ratio. 92. In a quadrilateral ABCD, AB = 10, BC = 17, CD = 13, DA = 20, and AC = 21. Find the diagonal BD. 93. To divide a trapezoid into two similar trapezoids by a line parallel to the base. 94. From a given point in a circumference, to draw a chord that is bisected by a given chord. 95. In a given line AB, to find a point C such that AC: BC = 1 : V2: 96. From a given rectangle to cut off a similar rectangle by a line parallel to one of its sides. 97. Find the locus of a point in space the ratio of whose distances from two given points is constant. 98. Find the locus of a point whose distance from a fixed straight line is in a given ratio to its distance from a fixed plane perpendicular to that line. 42 MATHEMATICAL WEINKLES 99. Any point in the bisector of a spherical angle is equally distant from the sides of the angle. 100. If any number of lines in space meet in a point, the feet of the perpendiculars drawn to these lines from another point lie on the surface of a sphere. 101. If the angles at the vertex of a triangular pyramid are right angles, and the lateral edges are equal, prove that the sum of the perpendiculars on the lateral faces from any point in the base is constant. 102. A plane bisecting two opposite edges of a regular tetraedron divides the tetraedron into two equal polyedrons. 103. The volume of a truncated triangular prism is equal to the product of the lower base by the perpendicular on the lower base from the intersection of the medians of the upper base. 104. The point of intersection of the perpendiculars erected at the middle of each side of a triangle, the point of intersec- tion of the three medians, and the point of intersection of the three perpendiculars from the vertices to the opposite sides are in a straight line ; and the distance of the first point from the second is half the distance of the second from the third. 105. Three circles are tangent externally at the points A, B, and C, and the chords AB and AC are produced to cut the circle BC at D and E. Prove that DE is a diameter. 106. A cylindrical bucket without a top is 6 inches in cir- cumference and 4 inches high. On the inside of the vessel 1 inch from the top is a drop of honey, and on the opposite side of the vessel 1 inch from the bottom, on the outside, is a fly. How far will the fly have to go to reach the honey ? 107. P is any point on the circumcircle of an equilateral triangle ABC; AP, BP meet BC, CA respectively in X, Y. Prove BX - AY is constant. GEOMETRICAL EXERCISES 43 108. Find the locus of all points from which two unequal circles subtend equal angles. 109. Show that any two perpendicular lines terminated by the opposite sides of a square are equal to one another, and by this property show how to escribe a square to a given quadri- lateral. 110. If the incircle passes through the centroid of the tri- angle, find the relation between the sides a, 6, and c. 111. If through a point O within a triangle ^BC parallels EFy GHy IK to the sides be drawn, the sum of the rectangles of their segments is equal to the rectangle contained by the segments of any chord of the circumscribing circle passing through 0. 112. If two chords intersect at right angles within a circle, the sum of the squares on their segments equals the square on the diameter. 113. If from a point A^ without a circle, two secants, ACD and AGKy are drawn, the chords C/iTand DG intersect on the chord of contact of the tangents from the point A to the circle. 114. If from a given point without a given circle any num- ber of secants are drawn, the chords joining the points of intersection of the secants with the circle all cross on the same straight line. 115. To draw a tangent from a given external point to a given circle by means of a ruler only. 116. Of all polygons constructed with the same given sides, the cyclic polygon is the maximum. 117. The square on the side of a regular inscribed pentagon is equal to the square on the side of a regular inscribed hexa- gon, plus the square on the side of a regular inscribed decagon. 118. The area of an inscribed regular dodecagon is three times the square of the radius of the circle. 44 MATHEMATICAL WRINKLES - 119. The square of the side of an inscribed equilateral tri- angle is equal to the sum of the squares of the sides of an inscribed square and inscribed regular hexagon. 120. Construct a circumference equal to three times a given circumference. 121. Construct a circle equivalent to three times a given circle. 122. If ABCD be a cyclic quadrilateral, and if we describe any circle passing through the points A and B, another through B and C, a third through G and D, and a fourth through D and A ; these circles intersect successively in four other points, E, F, G, H, forming another cyclic quadrilateral. 123. Construct a triangle, given the altitude, the median, and the angle bisector, all from the same vertex. 124. Prove that the circumcircle of a triangle bisects each of the six segments determined by the incenter and the three excenters of the triangle. 125. If A, B, G are three collinear points, and if K is any other point, prove that the circumcenters of the triangles KBG, KCA, and KAB are concyclic with K. 126. If the diameter of a circle be divided into any number of segments, and circumferences be de- scribed upon these segments as diameters, the sum of these circumferences is equal to the circumference of the original circle. 127. I own a square garden as shown in the above diagram. Within the garden stands a tree 30 feet, 40 feet, and 50 feet respectively from three successive corners. How much land have 1 ? GEOMETRICAL EXERCISES 45 The Famous Nine-Point Circle. 128. (a) If a circle be described about the pedal triangle of any triangle, it will pass through the middle points of the lints drawn from the orthocenter to the vertices of the triangle, and through the middle points of the sides of the triangle, in all, through nine points. (6) The center of the nine-point circle is the middle point of the line joining the orthocenter and the center of the circum- circle of the triangle. (c) The radius of the nine-point circle is half the radius of the circumcircle of the triangle. (d) The centroid of the triangle also lies on the line join- ing the orthocenter and the center of yl the circumcircle of \ / ; the triangle, and \ / I divides it in the \ X j ratio of 2:1. \ ^,^^^,^ j (e) The sides of \ -- - ^ the pedal triangle intersect the sides of the given tri- /^ angle in the radi- i a'< cal axis of the cir- \ cumscribing and nine-point circles. (/) The nine- point circle is tan- gent to the in- \ / scribed and es- "'v / cribed circles of \ ^-- ^-^ the triangle. Let ABD be any triangle, A\ B\ D\ the projections of the vertices on the opposite sides; //, «/, K, the mid-points of OAy / \ •Y --' >> A II ^-, O ^ / \ ; K 1 46 MATHEMATICAL WRINKLES OB, OD, respectively, being the orthocenter. Let L, M, N" be the mid-points of the sides. Join F, E, and D. The A A'B'D' is called the pedal triangle. The nine points A', N, K, D', H, B', J, L, M are concyclic ; and the circle through them is the nine- point circle of the triangle. For the proofs of these theorems, see "Finkel's Mathematical Solution Book " and the monograph, " Some Noteworthy Prop- erties of the Triangle and Its Circles," by Dr. W. H. Bruce, president of the North Texas State Normal School, Denton. 129. If from any point in either side of a right triangle, a line is drawn perpendicular to the hypotenuse, the product of the segments of the hypotenuse is equal to the product of the segments of the side plus the square of the perpendicular. 130. A, B, and C are fixed points. Describe a square with one vertex at A, so that the sides opposite to A pass through B and G. 131. If ABCD is a cyclic quadrilateral, prove that the cen- ters of the circles inscribed in triangles ABC, BCD, CDA, DAB are the vertices of a rectangle. 132. A round hole one foot in diameter is cut through a sphere 20 inches in diameter. Find the volume of the part remaining, the axes of the hole passing through the center of the sphere. 133. Given the incenter, circumcenter, and one excenter of a triangle, construct it. 134. Divide the triangle whose sides are 7, 15, 20 into two equivalent parts by a radius of the circumcircle. 135. Construct a triangle, given its altitude and the radii of the inscribed and circumscribed circles. 136. In the semicircle ABCD express the diameter AD in terms of the chords AB, BC, and CD. GEOMETRICAL EXERCISES 47 137. On one side of an equilateral triangle describe out- wardly a semicircle. Trisect the arc and join the points of division with the vertex of the triangle. Find the ratio of the segments of the diameter. 138. If a, 6, c are the sides of a triangle, and 5 (a^ -\-b^-\-c^ = G {ab + bc -\- ac), show that the incircle passes through the centroid of the triangle. 139. If through the vertices of any inscribed polygon tan- gents are drawn forming a circumscribed polygon, the con- tinued product of the perpendiculars from any point in the circle on the sides of the inscribed polygon is equal to the con- tinued product of the perpendiculars from the same point on the sides of the circumscribed polygon. 140. A lot 100 feet long and 60 feet wide has a walk ex- tending from one corner halfway around it, and occupying one third of the area. Required the width of the walk. A geometrical construction is desired. 141. Construct a triangle, having given the vertical angle, the sum of the tlfiree sides, and the perpendicular. 142. Prove that the dihedral angle of a regular octahedron is the supplement of the dihedral angle of a regular tetrahedron. 143. Given the three diagonals of an inscriptible quadrilat- eral, to construct the quadrilateral. 144. Pis a point on the minor arc AB of the circumcircle of the regular hexagon ABCDEF; prove that PE + PD = PA 4- P5 + PC 4- PF. 145. In a right triangle the hypotenuse is 17 and the diam- eter of the inscribed circle 6. Another equal circle is described touching the base produced and the hypotenuse ; how far apart are the centers of the two circles ? 146. Two equal circular discs are to be cut out of a rectan- gular piece of paper, 9 inches long and 8 inches wide. What is the greatest possible diameter of the discs ? MISCELLANEOUS PROBLEMS 1. A seed is planted. Suppose at the end of 2 years it produces a seed, and one each year thereafter ; each of these when 2 years old produces a seed yearly. All the seeds produced do likewise. How many seeds will be produced in 20 years ? 2. If a 4-inch auger hole be bored diagonally through a 12- inch cube, what will be the volume displaced, the axis of the auger hole coinciding with the diagonal of the cube ? 3. I have a circular orchard 110 yards in diameter. How many trees can be set in it so that no two shall be within 16 feet of each other, and no tree within 5 feet of the fence ? 4. What is the convex surface and voluifte of a cylindric ungula whose least length is 5 feet, greatest length 13 feet, the radius of the base being 1^- feet ? 5. AYhat is the length of the arc whose chord is 16 feet and height 6 feet ? 6. Find the area of a sector, having given the chord of the arc equal to 16 feet, and the height of the arc equal to 6 feet. 7. What is the area of a segment whose base is 6 feet and height 2 feet ? 8. Find the volume of an iron rod 2 inches in diameter and 10 feet from end to end containing a loop whose inner diameter is 4 inches. 9. What is the area of a circular zone, one side of which is 30 inches and the other 40 inches, and the distance between them 10 inches ? 48 MISCELLANEOUS PROBLEMS 49 10. The shell of a hollow iron ball is 4 inches thick, and contains \ of the number of cubic inches in the whole ball. Find the diameter of the ball. 11. A rope 60 feet long wraps around two trees 6 feet and 10 feet in diameter, respectively, and crosses between them. Find the distance between their centers. 12. On the tire of a wheel 4 feet in diameter is a black spot. How far does the spot move while the wheel makes 4 revolutions ? 13. A fly lights on the spoke of a carriage wheel 4 feet in diameter, 1 foot up from the ground. How far will the fly have traveled when the wheel has made 2 revolutions on a level plane? 14. An eagle and a sparrow are in the air ; the eagle is 100 feet above the sparrow. If the sparrow flies straight forward in a horizontal line, and the eagle flies twice as fast directly towards the sparrow, how far will each fly before the sparrow is caught ? 15. A cow is tethered to the corner of a barn 25 feet square, by a rope 100 feet long. How many square feet can she graze ? 16. A solid cube weighs 300 pounds. If a power is applied at an angle of 45° at an upper edge of the cube, how many foot pounds will be required to overturn the cube ? 17. A tree 110 feet high, standing by the side of a stream 100 feet wide, is broken by a storm ; the fallen part is unde- tached from the stump, and its top rests 10 feet above the water and points directly to the opposite shore. How high is the stump ? 18. At the edge of a circular lake 1 acre in area stands a tree. What length of rope, tied to this tree, will allow a horse to graze upon \ of an acre ? 50 MATHEMATICAL WRINKLES 19. A horse is tied to a stake in the circumference of a 6-acre field. How long must the rope be to allow him to graze over just 1 acre inside the field ? 20. What is the longest piece of carpet 3 feet wide, cut square at the ends, that can be put in a room 16 feet by 20 feet ? 21. The fore wheel and the hind wheel of a carriage are 12 feet and 15 feet in circumference, respectively; a rivet in the tire of each is observed to be up when the carriage starts. How far will each rivet have moved when they are next up together ? 22. A log 40 inches in diameter is to be sawed by four men. What part of the diameter must each man saw to do ^ of the work ? 23. What is the length of a chord cutting off the fourth part of a circle whose radius is 10 feet ? 24. Find the length of a chord cutting off the third part of a circle whose diameter is 40 feet. 25. A tree 80 feet high was broken in a storm so that the top struck the ground 40 feet from the foot of the tree. If the tree remained in contact, what was the length of the path through which the top of the tree passed in falling to the ground ? 26. By boring through the center of a wooden ball, with an auger 4 inches in diameter, i of the solid contents of the ball is displaced. Eind the diameter of the ball. 27. Find the diameter of an auger that will displace i of the solid contents of a ball 5 feet in diameter, by boring through its center. 28. Three horses are tethered each to a rope 42 feet in length to the corners of an equilateral triangle whose side is 80 feet. Over how many square feet can each graze, provided they are at no time upon the same ground ? MISCELLANEOUS PROBLEMS 51 29. How many acres of water can a man see, standing on a ship, with his eyes just 14 feet above the water, when there is no land in sight ? 30. In a farmer's pasture is located a triangular house, the length of each side being 10 yards. The farmer wishing to graze his horse finds that stakes are not plentiful and decides to tie the rope to one corner of the house. If the rope is long enough to allow the horse to graze 30 yards from the corner of the house, over how much ground can the horse graze ? 31. Three men wish to carry each J of an 8-foot log of uni- form size and density. Where must the hand stick be placed so tliat the one at the end of the log and the others at the ends of the stick shall each carry equal weights ? 32. If three equal circles are tangent to each other, each to each, and inclose a space between the three arcs equal to 100 square inches, find their radius. 33. If three equal circles are tangent to each other, each to each, with a radius of 10 inches, find the area of the space inclosed between the three arcs. 34. If 4 acres pasture 40 sheep 4 weeks, and 8 acres pasture 66 sheep 10 weeks, how many sheep will 20 acres pasture 50 weeks, the grass growing uniformly all the time ? 35. A rabbit 60 yards due east of a hound is running due south 20 feet per second ; the hound gives chase at the rate of 25 feet per second. How far will each run before the rabbit is caught ? 36. How many fruit trees can be set out upon a space 100 feet square, allowing no two to be nearer each other than 10 feet ? 37. How many stakes can be driven down upon a space 12 feet square, allowing no two to be nearer each other than 1 foot? 52 MATHEMATICAL WRINKLES 38. The sum of the sides of a triangle is 100. The angle at A is double that at B, and the angle at B is double that at C. Find the sides. 39. A conical glass 4 inches in diameter and 6 inches in altitude, is filled with water. How much water will run out if it be turned through an angle of 45° ? 40. At what latitude is the circumference of a parallel half that of the equator, regarding the earth a perfect sphere ? 41. The difference between the circumscribed and inscribed squares of a circle is 72. What is the area of the circle ? 42. A drawer made of inch boards is 8 inches wide, 6 inches deep, and slides horizontally. How far must it be drawn out to put into it a book 4 inches thick, 6 inches wide, and 9 inches long ? 43. With what velocity must a pail of water be whirled over the head to prevent the water from falling out, the radius of the circle of revolution being 4 feet ? 44. Two hunters killed a deer, and wishing to ascertain its weight they placed a rail across a fence so that it balanced with one on each end. They then exchanged places, and the lighter man taking the deer in his lap, the rail again balanced. Find the weight of the deer, the hunters' weights being 160 and 200 pounds. 45. At each corner of a square pasture whose sides are 100 feet a cow is tied with a rope 100 feet long. What is the area of the part common to the four cows ? 46. Find the volume generated by the revolution of a circle 10 feet in diameter about a tangent. 47. Find the volume generated by revolving a semicircle 20 inches in diameter about a tangent parallel to its diam- eter. MISCELLANEOUS PROBLEMS 63 48. A circle of 10 inches radius, with an inscribed regular hexagon, revolves about an axis of rotation 20 inches distant from its center and parallel to a side of the hexagon. Find the difference in area of the generated surfaces. 49. Find the difference in the volumes of the two generated solids. 50. An equilateral triangle rotates about an axis without it, parallel to, and at a distance 10 inches from one of its sides. Find the surface thus generated, a side of the triangle being 4 inches. 61. A rectangle whose sides are 6 inches and 18 inches is revolved about an axis through one of its vertices, and parallel to a diagonal. Find the surface thus generated. 52. Find the surface of a square ring described by a square foot revolving round an axis parallel to one of its sides and 4 feet distant. 53. Find the volume generated by an ellipse whose axes are 40 inches and 60 inches, revolving about an axis in its own plane whose distance from the center of the ellipse is 100 inches. 54. AVhat power acting horizontally at the center of a wheel 4^ feet in diameter and weighing 270 pounds, will draw it over a cylindrical log 6 inches in diameter, lying on a horizontal plane ? 55. Find the volume generated by the revolution of a circle 2 feet in diameter about a tangent. 56. Find the surface generated by the revolution of a circle 2 feet in diameter about a tangent. 57. Find the surface and volume of a cylindric ring, the diameter of the inner circumference being 12 inches and the diameter of the cross section 16 inches. 54 MATHEMATICAL WRINKLES 58. Eind the surface and volume of the segment of the same cylindric ring, if a plane is passed perpendicular to its axis, and at a distance of -4 inches from the center. 59. A galvanized cistern is 8 feet in diameter at the top, 10 feet at the bottom, and 10 feet deep. A plane passes from the top on one side to the bottom on the other side. What is the volume of the part contained between this plane and the base? 60. A wineglass in the form of a frustum of a cone is 4 inches in diameter at the top, 2 inches at the bottom, and 5 inches deep. If, when full of water, it is tipped just so that the raised edge at the bottom is visible, what is the volume of the water remaining in the glass ? 61. To what depth will a sphere of cork, 2 feet in diameter, sink in water, the specific gravity of cork being .25 ? 62. The diameter of two equal circular cylinders, intersecting at right angles, is 3 feet. What is the surface common to both? 63. In digging a well 4 feet in diameter, I come to a log 4 feet in diameter lying directly across the entire well. What was the contents of the part of the log removed ? 64. What is the volume of a solid formed by two cylindric rings 2 inches in diameter, whose axes intersect at right angles and whose inner diameters are 10 inches ? 65. Find the area of a circular lune or crescent ABCD; the chord ^0=10 feet; the height EB = S feet ; and the height ED =2 feet. 66. Find the circumference of an ellipse, the transverse and conjugate diameters being 80 inches and 80 inches. 67. The axes of an ellipse are 60 inches and 20 inches. What is the difference in area between the ellipse and a circle having a diameter equal to the conjugate axis ? MISCELLANEOUS PROBLEMS 55 68. What is the area of a parabola whose base, or double ordinate, is 30 inches and whose altitude, or height, is 20 inches ? 69. What is the area of a cycloid generated by a circle whose radius is 6 feet ? 70. Two men, A and B, started from the same point at the same time ; A traveled southeast for 10 hours, and at the rate of 10 miles per hour, and B traveled due south for the same time, going 6 miles per hour; they turned and traveled directly towards each other at the same rates respectively, till they met. How far did each man travel ? 71. In front of a house stand two pine trees of unequal height; from the bottom of the second to the top of the first a rope 80 feet in length is stretched, and from the bottom of the first to the top of the second a rope 100 feet in length is stretched. If these ropes cross 10 feet above the ground, find the distance between the trees. 72. To trisect any angle. 73. A grocer has a platform balance the ratio of whose arms is 9 to 10. If he sells 20 pounds of merchandise to one man, weighing it on the right-hand pan, and 20 pounds to another man, weighing it on the left-hand pan, what per cent does he gain or lose by the two transactions ? 74. A and B carry a fish weighing 54 pounds hung between them from the middle of a 10-foot oar. One end of the oar rests on A's shoulder, but the other end is pushed 1 foot be- yond B's shoulder. What part of the weight does each carry ? 75. A half-ounce bullet is fired with a velocity of 1400 feet per second from a gun weighing 7 pounds. Find the velocity in feet per second with which the gun begins to recoil, and the mean force in pounds' weight that must be exerted to bring it to rest in 4 inches. 56 MATHEMATICAL WRINKLES 76. A bullet fired with a velocity of 1000 feet per second penetrates a block of wood to a depth of 12 inches. If it were fired through a plank of the same wood, 2 inches thick, what would be its velocity on emergence, assuming the resistance of the wood to the bullet to be constant ? 77.* A horse is tied to one corner of a rectangular barn 30 by 40 feet. What is the surface over which the horse can range if the rope with which he is tied is 80 feet long ? 78.* How many acres are there in a circular tract of land, containing as many acres as there are boards in the fence inclosing it, the fence being 5 boards high, the boards 8 feet long, and bending to the arc of a circle ? 79.* A thread passes spirally around a cylinder 10 feet high and 1 foot in diameter. How far will a mouse travel in unwind- ing the thread if the distance between the coils is 1 foot ? 80. A string is wound spirally 100 times around a cone 100 feet in diameter at the base. Through what distance will a duck swim in unwinding the string, keeping it taut at all times, the cone standing on its base at right angles to the sur- face of the water ? 81.* After making a circular excavation 10 feet deep and 6 feet in diameter, it was found necessary to move the center 3 feet to one side, the new excavation being made in the form of a right cone having its base 6 feet in diameter and its apex in the surface of the ground. Required the total amount of earth removed. 82.* A 20-foot pole stands plump against a perpendicular wall. A cat starts to climb the pole, but for each foot it ascends, the pole slides one foot from the wall ; so that when the top of the pole is reached, the pole is on the ground at right angles to the wall. Required the distance through which the cat moved. * These problems are from *' Finkel's Solution Book." MISCELLANEOUS PROBLEMS 57 83. A tree 96 feet high was broken by the wind in such a manner that the top struck the ground 36 feet from the foot of the tree. If the parts remained connected at the place of breaking, forming with the ground a right triangle, how high was the stump ? 84. The distance around a rectangular field is 140 rods, and the diagonal is 50 rods. Find its length, breadth, and area. 85. The area of a rectangular field is 30 acres, and its diag- onal is 100 rods. FiQd its length and breadth. 86. Two trees of equal height stand upon the same level plane, 60 feet apart and perpendicular to the plane. One of them is broken off close to the ground by the wind, and in fall- ing it lodges against the other tree, its top striking 20 feet below the top of the other. Find the height of the trees. 87. A square field contains 10 acres. From a point in one side, 10 rods from the corner, a line is drawn to the opposite side cutting off 6J acres. How long is the line ? 88. Find the edge of the largest hollow cube, having the shell three inches in thickness, that can be made from a board 42J feet long, 2 feet wide, and 3 inches thick. 89. A circular farm has two roads crossing it at right angles 40 rods from the center, the roads being 60 and 70 rods re- spectively, within the limits of the farm. Find the area of the farm. 90. The longest straight line that can be stretched in a cir- cular track is 200 feet in length. Find the area of the track. 91. From the two acute angles of a right triangle lines are drawn to the middle points of the opposite sides ; their respec- tive lengths are V73 and V52 feet. Find the sides of the triangle. 92. A wheel of uniform thickness, 4 feet in diameter, stands in the mud 1 foot deep. What fraction of the wheel is out of the mud ? MATHEMATICAL RECREATIONS 1. Mary is 24 years old. She is twice as old as Ann was when Mary was as old as Ann is now. How old is Ann ? 2. There is a great big turkey that weighs 10 pounds and a half of its weight besides. What is its weight? 3. With 6 matches form 4 equilateral triangles, the side of each being equal to the length of a match. 4. One tumbler is half full of wine, another is half full of water. From the first tumbler a teaspoonful of wine is taken out and poured into the tumbler containing the water. A teaspoonful of the mixture in the second tumbler is then trans- ferred to the first tumbler. As the result of this double trans- action is the quantity of wine removed from the first tumbler greater or less than the quantity of water removed from the second tumbler ? 5. (i) Take any number; (ii) reverse the digits; (iii) find the difference between the number formed in (ii) and the given number; (iv) multiply this difference by any number you please ; (v) cross out any digit except a naught ; (vi) give me the sum of the remaining digits, and I will give you the figure struck out. 6. (i) Take any number; (ii) add the digits; (iii) sub- tract the sum of the digits from the given number ; (iv) cross out any digit except a naught; (v) give me the sum of the remaining digits, and I will give you the figure struck out. 58 MATHEMATICAL RECREATIONS 59 7. Given a plank 12 inches square, required to cover a hole in a floor 9 inches by 16 inches, cutting the plank into only two pieces. 8. Place four 9's in such a manner that they will exactly equal 100. 9. The square is 8 inches by 8 inches. By forming the latter figure out of the four parts of the square it is found to be _i II/__-__ r w iO b small squares. Each of the figures are made up of A, B, and O. In the rectangle A -{- B -{- C = 65. In the square A -\- B -\- C = 64:. .-. 65 = 64. .-. 1 = 0. "" "1 ■" *" 7 y \^ y 8 y A ^ ,^ 1 l<< ^ y n ^ ± -L L. 13 221. To prove that 1 Let Then and Note. — If a = 1, 1 = 200. a = 6 = 10. . •.! = «» -1-61 .-. 1 = 10^ + 102. .-. 1 = 200. if a = 2, 1 = 8 ; if a = 3, 1 = 18 ; etc. MATHEMATICAL RECREATIONS 95 * 222. To prove that 1 = 2000. Let a = 6 = 10. Then a^ - h^ = 0, and a« - 6« = 0. (Things equal to the same things are equal to each other.) .-. l = ci3-f-6^ (Dividing by a' — 6^) .-. 1 = W + 10». .-. 1 = 2000. Note. — If a = 1, 1 = 2; if a = 2, 1 = 16; if a = 3, 1 = 54; etc. Also many other problems may be made similar to problems Nos. 221 and 222. 223. Another Geometrical Fallacy To prove that it is possible to let fall two perpendiculars to a line from an external point. Take two intersecting circles with centers and 0'. Let one point of intersection be P, and draw the diameters PJfandPxV. Draw MN cutting the circumferences at A and B. Then draw PA and PB. Since Z PBM is inscribed in a semicircle, it is a right angle. Also since /.PAN is inscribed in a semicircle, it is a right angle. .'.PA and PB are both ± to MN. 224. Given three or more integers, as 30, 24, and 16; re- quired to find their greatest integral divisor that will leave the same remainder. • The exposing of fallacies has been left to the student. They should be studied in every High School and College.. 96 MATHEMATICAL WRINKLES 225. To Prove that You are as Old as Methuselah Proof : Let X = Methuselah's age. Let y = your age. Let s = the sum. Then x-\-y = s. ... (x-\-y)(x-y)=s(x-y). . • . x^ — y^ = sx— sy. .'. x^— sx — y^ — sy. s- s^ 4 4 ■■■('-S"-(-iJ- s s .-. x = y. 226. How many shoes would it take for the people of a town if one third of them had but one foot and one half the remainder went barefoot ? 227. The Spider and the Four Gnats On a suspended piece of glass 10 inches long, 4 inches wide, and 4 inches high is a spider and four gnats. The spider is on one end ^ inch from the bottom and midway between the sides. The gnats are on the other end. Three of them are \ inch, I inch, and 1 inch, respectively, from the top and mid- way between the sides. The fourth is 1|^ inches from the top and on an edge. Determine the shortest path possible, by way of the six faces of the piece of glass, for the spider to catch the four gnats and return to the place from which he started. 228. What difference would there be in the weight of a per- fectly air-tight bird cage, depending on whether the bird were sitting on the perch or flying about ? MATHEMATICAL RECREATIONS 97 A- V 229. To prove that part of a line equals the whole line. Take a triangle ABC^ and draw CP ± to AB. From C draw CX, making Z Ar\ = /-B. Then A ABC and ACX are ^. similar. .-. A ABC'. A ACX=BC': CX\ Furthermore, A ABC'. A ACX^^AB.AX. .'.BC^.CT^AB.AX, WJ':AB = ~CT'.AX. W^AC^+A^-^AB'AP, ~CX'' = AC''-^AT-2AX'AP, 2AB'AP_~AC'-^AX*-2AX'AP or But and AC*-^Aff or or AC' AB AB -\-AB-2AP AX AX + AX-2AP. iB-'^^'-AX ^^' AC'-AB'AX ^ AC^-ABAX AB AX ..\AB = AX. — From Wentworth and Smith's " Geometry/ 230. To prove that part of an angle equals the whole angle. Take a square ABCD, and draw MM'P, the ± bisector of CD. Then MM'P is also the ± bisector of AB. From B draw any line BX equal to AB. -^X Draw DX and bisect it by the ± NP. Since '/ DX intersects CD, Js to these lines cannot be / parallel, and must meet as at P. ^ Draw PA, PD, PC, PX, and PB. Since MP is the ± bisector of CD, PD = PC p 98 MATHEMATICAL WRINKLES Similarly, PA = PB, and PD = PX. ..PX=PD=Pa But BX=BC by construction, and PB is common to A PBX and P5(7. .'.A PBX is congruent to A PBC, and Z X5P = Z CBP. .'. the whole Z XBP equals the part, Z (7J5P. — From Wentworth and Smith's '^ Geometry." 231. The Four-color Map Problem Not more than four colors are necessary in order to color a map of a country, divided into districts, in such a way that no two contiguous districts shall be of the same color. Probably the following argument, though not a formal dem- onstration, will satisfy the reader that the result is true. Let A, B, C be three contiguous districts, and let X be any other district contiguous with all of them. Then X must lie either wholly outside the external boundary of the area ABO or wholly inside the internal boundary ; that is, it must occupy a position either like X or like X'. In either case every re- maining occupied area in the figure is inclosed by the boun- daries of not more than three districts; hence there is no possible way of drawing another area Y which shall be contiguous with A, B, C, and X. In other words, it is possible to draw on a plane four areas which are con- tiguous, but it is not possible to draw five such areas. If A, B, C are not contiguous, each with the other, or if X is not contiguous with A B, and C, it is not necessary to color them all differently, and thus the most unfavora- ble case IS that already treated. Moreover, any of the above areas may diminish to a point and finally disappear without affecting the argument. That we may require at least four colors is obvious from. MATHEMATICAL RECREATIONS % the above diagram, since in that case the areas Aj B, C, and X would have to be colored differently. A proof of the proposition involves difficulties of a high order, which as yet have baffled all attempts to surmount them. — From Ball's " Mathematical Recreations." 232. RoMEO AND Juliet On a checker board are located two snails. They are Romeo and Juliet. Juliet is on her balcony waiting the arrival of her lover, but Romeo has been dining and forgets, for the life of him, the number of her house. The squares represent sixty-four houses, and the amorous swain visits every house once and only once before reaching his beloved. Now make him do this with the fewest possible turnings. The snail can move up, down, and across the board and through the diagonals. Mark his track. — From " Canterbury Puzzles." 233. Find the exact dimensions of two cubes the sum of whose volumes will be exactly 17 cubic inches. Of course the cubes may be of different sizes. 234. I have two balls whose circumferences are respectively 1 foot and 2 feet. Find the circumferences of two other balla different in size whose combined volumes will exactly equal the combined volumes of the given balls. 235. Can the number 11,111,111,111,111,111 be divided by any other integer except itself and unity ? ■^ Vik- 100 MATHEMATICAL WKIKKLES 236. My friend owns a house containing 16 rooms as indicated in the diagram. While visiting him one day, he said to me, " Can you enter at the door A and pass out at the door B and enter every one of the 16 rooms once and only once ? " Show how I might have done this. 237. Given a plank contain- ing 169 square inches as shown below. Show how a hole 13 inches square may be covered by cutting the plank into three pieces. 238. Given a piece of cloth in the shape of an equilateral tri- angle. Required to cut it into four pieces that may be put together and form a perfect square. 239. A Shokt Method of Multiplication ^a^ampZe. —Multiply 41,096 by 83. The answer is found to be 3,410,968 by inspection. It will be observed that the answer is found by placing the last figure of the multiplier before the number and the first after it. Also if we prefix to 41,096 the number 41,095,890, repeated any num- ber of times, the result may always be multiplied by 83 in this peculiar manner. 8 multiplied by 86 = 688. Also to multiply 1,639,344,262,295,081,967,213,114,754,098,- 360,655,737,704,918,032,787 by 71, all you have to do is to place another 1 at the beginning and another 7 at the end. MATHEMATICAL ^T^CiSliXTIONS 101 ♦ 240. The SquaI^k fAh%j^*)y' ' " ' , To prove that the diagonal of any square field equals the sum of any two sides. 100 rd Fia. 1. FiQ. 2. Fig. 3. Given the square field ABCD with a side equal to 100 rods. The distance from Aio C along two sides is 200 rods. Now in Fig. 1 the distance from Ato C along t;he diagonal path is 200 rods. In Fig. 2 the steps are -smaller, yet the di- agonal path is 200 rods long. In Fig. 3 the steps are very small, yet the distance must be 200 rods and would yet be if we needed a microscope to detect the steps. In this way we may go on straightening out the zigzag path until we ulti- mately reach a perfect straight line, and it therefore follows that the diagonal of a square equals the sum of any two sides. Can you expose the fallacy ? 241. Given a rectangular block of wood 8 inches by 4 inches by 3J inches. Required to cut it into similar blocks 2\ inches by IJ inches by 1\ inches with the least possible waste. How many blocks can be had ? A Time Problem 242. A man Who carries a watch in which the hour, minute, and second hands turn upon the same center was asked the time of day. He replied, " The three hands are at equal dis- tances from one another and the hour hand is exactly 20- minute spaces ahead of the minute hand." Can you tell the time? • See footnote, page 95. 102 MATBEMATICAL WRINKLES -' ' Tnti: 'Tj^ze Planter 243. Are you a practical tree planter? If so, you are requested, (a) to show how sixteen trees may be planted in twelve straight rows, with four trees in every row, (b) to show how sixteen trees may be planted in fifteen straight rows, with four trees in every row. 244. Five persons can be seated in six different ways around a table in such a manner that any one person is seated only once between the same two persons. Show the manner of seating. 245. Seven persons may be seated in fifteen different ways around a table in such a manner that any one person is seated only once between the same two persons. Show the ways in which they might be seated. 246. On his morning stroll, Mr. Busybody encountered a laborer digging a hole. ^' How deep is that hole ? " he asked. " Guess," replied the workingman, who stood in the hole. " My height is exactly five feet and ten inches." " How much deeper are you going ? " "I am going twice as deep," rejoined the laborer, "and then my head will be twice as far below ground as it now is above ground." Mr. Busybody wants to know how deep that hole will be when finished. 247. One night three men. A, B, and C, stole a bag of apples and hid them in a barn over night, intending to meet in the morning to divide them equally. Some time before morning A went to the barn, divided the apples into three equal shares and had one apple too many, which he threw away. A took one share and put the others back into the bag. Soon after B came and did exactly as A had done. Then came C, who re- peated what A and B had done before him. In the morning the three met, saying nothing of what they had done during MATHEMATICAL KECREATIONS 103 the night. The remaining apples were divided into three equal shares, with still one apple too many. How many apples were there in the bag at the beginning ? 248. The following diagram represents a section of a rail- way track with a siding. Eight cars are standing on the main line in the order 1, 2, 3, 4, 5, 6, 7, 8, and an engine is standing on the side track. The siding will hold five cars, or four cars and the engine. The main line will hold only the eight cars and the engine. Also when all the cars and the engine are on the • main line, only the one occupying the place of 8 can be moved on the siding. With 8 at the extremity, as shown, there is just room to pass 7 on the siding. The cars can be moved without the aid of the engine. You are required to reverse the order of the cars on the main line so that they will be numbered 8, 7, 6, 5, 4, 3, 2, 1 ; and to do this by means which will involve as few transfer- ences of the engine, or a car to or from the siding as are possible. 249. The Mysterious Addition To express the sum of five numbers, having given only the first. Have a person write a number, say 55,369. Subtract two from the number, and place it before the remainder, giving 255,367, which is the sum of the numbers to be added. Each 104 MATHEMATICAL WRINKLES number is to contain the same number of figures as kk oqq the first. _ 3g|4g^ After the first number is expressed have the per- g-i ^o* son write the second, say 38,465. Then write the 03 461 third yourself, using such figures in the number, rrn koq that if added to the figures in the number above — j^- — will make nine. Have the person write the fourth ^i^^,obi number. Then write the fifth yourself in the same way as the third. These numbers added will give the required sum, 250. At the close of four and a half months' hard work, the ladies of a certain Dorcas Society were so delighted with the completion of a beautiful silk patchwork quilt for the dear curate that everybody kissed everybody else, except, of course, the bashful young man himself, who kissed only his sisters, whom he had called for, to escort home. There were just a gross of osculations altogether. How much longer would the ladies have taken over their needlework task if the sisters of the curate referred to had played lawn tennis instead of at- tending the meetings? Of* course we must assume that the ladies attended regularly, and I am sure that they all worked equally well. A mutual kiss counts two osculations. — From " Canterbury Puzzles." 251. The Arithmetical Triangle This name has been given to a contrivance said to have originated or to have been perfected by the famous Pascal. 1 2 1 3 3 1 4 6 4 1 5 10 10 5 1 6 15 20 15 6 1 7 21 35 35 21 7 1 8 28 56 70 56 28 8 etc. etc. MATHEMATICAL RECREATIONS 105 This peculiar series of numbers is thus formed : Write do^vn the numbers 1, 2, 3, etc., as far as you please, in a vertical row. On the right hand of 2 place 1, add them together, and place 3 under the 1 ; then 3 added to 3 = 6, which place under the 3 ; 4 and 6 are 10, which place under the 6, and so on, as far as you wish. This is the second vertical row, and the third is formed from the second in a similar way. This triangle has the property of informing us, without the trouble of calculation, how many combinations can be made, taking any number at a time, out of a larger number. Suppose the question were that just given ; how many selec- tions can be made of 3 at a time, out of 8 ? On the horizontal row commencing with 8, look for the third number ; this is 56, which is the answer. 252. Twelve nests are in a circle. In each nest is only one egg. Required to begin at any nest, always going in the same direction, and pick up an egg, pass it over two other eggs, and place it in the next nest. This process is to be continued until six eggs have been removed and then six of the nests should contain two eggs each, and the other six should be empty. Show how this can be done by making the fewest possible revolutions around the nests. 253. A man in a city skyscraper, in a time of fire, made his escape by descending on a rope. He was 300 feet above the ground and had a rope only 150 feet long and 1^ inches in diameter. Show how he made his escape without jumping from the window or dropping from the end of the rope. 254. A German farmer while visiting town bought a cask of wine containing 100 pints of pure wine. After reaching home he hid the cask in his barn thinking no one would find it. While away from home his neighbor found the cask and drew out 30 pints. Each time he drew out a pint he replaced it with a pint of pure water before drawing the next pint. How much wine was stolen ? 106 MATHEMATICAL WRINKLES 255. While out fishing on a lake in a small boat I found myself without oars. I was two miles from shore. I had nothing to use to row the boat. Besides this there was no current to help me, for the water was perfectly smooth. I had nothing in the boat but a heavy trot-line one inch in diameter and six large fish. I could not swim and had no way of securing assistance. Was it possible for me to reach the shore under such circumstances ? If so, how ? 256. C's age at A's birth was 5i times B's age and now is equal to the sum of xV's age and B's age. If A were 3 years younger or B 4 years older, A's age would be | of B's age. Find the ages of A, B, and C. (Solve by arithmetic.) 257. What is the smallest sum of money in pounds, shil- lings, pence, and farthings that can be expressed by using each of the nine digits, 1, 2, 3, 4, 5, 6, 7, 8, and 9, once and once only ? 258. A Eeversible Magic Square The digits 0, 1, 2, 6, and 8, when turned upside down, can be read, 0, 1, 7, 9, and 8. It will be observed that this square when turned upside down is still magic. 259. To prove that part of an angle equals the whole angle. Take a right triangle ABO and construct upon the hypote- nuse BC an equilateral triangle BCD, as shown. On CD lay off OP equal to CA. Through X, the mid-point of AB, draw PX to meet CB pro- duced at Q. Draw QA. Draw the ± bisectors of QA and QP, as YO and ZO. These 29 IZ 61 Z2 Zl 62 19 2Z 12 21 ZZ 69 6Z Z9 22 II MATHEMATICAL RECREATIONS 107 must meet at some point O because they are ± to two inter- secting lines. Draw OQ, OA, OP, and OC. Since O is on the ± bisector of QA, .'. OQ = OA, Similarly OQ=OP, and .'. OA = OP. But CA = CP, by construction, and CO = CO. .-. A AOC is congruent to A POC, and Z ^CO = Z PCO. 260. Another Triangle Fallacy To prove that the sum of two sides of a triangle is equal to the third side. Let ABC be a triangle. -^ ^ ^ -,Z> Complete the parallelo- gram and divide the diag- ^^ ^iv\"'""/ / onal AC into n equal parts. / /____N=the ra- dius of the largest marble which could be covered in the glass. Area of A ABC = area of 3 A of which OD is the altitude. Area of A ABC = 12. Hence one half the radius of the largest marble = 12 h- (5+5 -f 6) = |. The diameter of the largest ball which could be covered in the glass = 3. = 12 7r. . •. V of cone ABC = - iri^h = ^ 3 3 V of largest marble = - ird? = — . 6 2 3.2 TT ^ = amount of water it takes to cover the largest ANSWERS AND SOLUTIONS 167 marble. Now, the water which would cover the largest marble is to the water covering the required marble as the largest marble is to the required marble. .'. a; = 2.433 inches, Ans. 13. The discount is yj^ of the face of the note. The interest is 10 % of the proceeds. Hence, 10 % of the proceeds = 9 % of the face. jIj^ of the proceeds = ^ of yf^ = -j-A^ of the face. {%% of the proceeds = 100 x -j-J^^ = ^^%% of the face, jj jj _ ^o^<^ — ^Y^^ = ^^ of the face, which is the discount for the required time. .-. the time is ^ -i- yj^ = IJ years = 400 days. 14. Since it was a perfect power, the right-hand period must have been 25, and the last figure of the root must have been 5. Hence the last trial divisor was 1225 -*- 5 = 245. Then by the rule for extracting the square root, we know 5 to have been annexed to 24, which must have been double the root already found. That portion of the root, then, must have been ^ of 24 = 12. The entire root was 125. Therefore the power was 125^=15,625, Ans. 15. The length of the lawn is f of its width, and if J of it be taken off by a line parallel to the end, a square will be left, the side of which is the width of the lawn. The area of the lawn = f the area of the square. If the dimensions of the lawn be increased 1 ft., its area will be equivalent to the area of f of the square -ff of a strip 1 foot wide-|-|^ of a strip 1 foot wide + a square with an area of 1 square foot = 651 square feet. Area of f of the square -+- f of a strip 1 foot wide = 650 square feet. Taking J of this quantity, we have the square + 1 of a strip 1 foot wide = J of 650 square feet = 62,400 square inches. But | of a strip 1 foot wide = a strip of the 168 MATHEMATICAL WRINKLES same length | of a foot wide = two strips the same length |- of a foot wide, or 10 inches wide. Now, if we place these two strips on adjacent sides of the square and also a square containing 100 square inches at the corner, we will have a new square the area of which = 62,400 square inches + 100 square inches = 62,500 square inches. A side of this square = 250 inches. Therefore the width of the lawn = 250 inches — 10 inches = 240 inches, or 20 feet, and the length = 30 ft., Ans. 16. Let 100 % = cost. - 180 % = marked price. 140 % = selling price. .-. i|-2. = 1|- = length in yards. 17. The area of a triangle whose sides are 13, 14, and 15 feet may be found by the rule : " Add the three sides together and take half the sum; from the half sum subtract each side separately ; multiply the half sum and the remainders together and extract the square root of the product.'' 13 feet -f 14 feet + 15 feet = 42 feet = sum of sides, i of 42 feet = half sum of sides. 21 feet - 13 feet = 8 feet. 21 feet - 14 feet = 7 feet. 21 feet - 15 feet = 6 feet. 7056 = product of the half sum and the three remainders. •\/7056 = 84 square feet, area of triangle with sides 13, 14, and 15 feet. Area of given triangle is 24,276 square feet. The problem now becomes merely a comparison of areas, the larger triangle having sides in the same proportion as the smaller. Similar surfaces are to each other as the squares of their like dimensions, therefore, 84 : 24,276 : : 13^ : the square of the corresponding side. Or, V24,276 x 169 -- 84 = 221, length of the corresponding side. ANSWERS AND SOLUTIONS 169 Similarly with 14 and 15 we find the other corresponding sides = 238 and 255. Aiis. 221, 238, and 255. 18. Since my mistake was 55 minutes, the hands must have been 5 minute spaces apart. At 2 o'clock they were 10 spaces apart, hence the minute hand had gained 5 spaces. It gained 55 spaces in 1 hour, hence to gain 5 spaces requires -jij- of an hour, or 5^\ minutes. Therefore, it was 5^ minutes past 2 o'clock, Ans. 19. The area of the whole slate = 108 square inches. The area of the frame = ^^ of 108 square inches = 27 square inches. Now, suppose 4 slates so placed as to form a square 9 + 12, or 21 inches, on a side. The whole area of this square = 441 square inches. 441 square inches — 4 x 27 square inches = 108 square inches, the area of the frames of the four slates = 338 square inches, the area of a square formed by the four slates without frames. V338 square inches = 18.242 inches, a side of the square. Then since 21 inches includes 4 widths of the frame, 21 inches — 18.242 inches = 4 times the width of the frame. Therefore the frame is .6895 inch wide. 20. 6 acres + 72* grdwths keep 16 oxen 12 weeks, or 1 ox for 192 weeks. 3 acres + 36 growths keep 16 oxen 6 weeks, or 1 ox for 96 weeks. Adding the above, we have, 9 acres -h 108 growths keep 16 oxen 18 weeks, or 1 ox for 288 weeks. 9 acres + 81 growths keeps 26 oxen 9 weeks, or 1 ox 234 weeks. Subtracting, 27 growths keep 1 ox 288 weeks — 234 weeks, or 54 weeks. .*. 1 growth keeps 1 ox 2 weeks. .•. IpO growths keep 1 ox 300 weeks. • A growth is the weekly growth on one acre. 170 MATHEMATICAL WBINKLES Also, 72 growths keep 1 ox 144 weeks. Then, 6 acres keep 1 ox 192 weeks — 144 weeks, or 48 weeks. 6 acres keep 1 ox 48 weeks, 1 acre keeps 1 ox 8 weeks. 15 acres keep 1 ox 120 weeks. .-. 15 acres + 150 growths keep 1 ox 120 weeks + 300 weeks, or 420 weeks. Hence the number of oxen required is 420 ^ 10 = 42, Ans. 21. 400. 22. Precedence is given to the signs X and ^ over the signs + and — ; hence the operations of multiplication and division should always be performed before addition and subtraction. Ans. = 8. 23. 00 . 24. 0. 25. 2.236 minutes. 26. 20 feet. 27. 28.44+. 28. The distance from the extreme point of the given ball to the corner is to the distance of the nearest point of the given ball from the corner, as the diamfeter of the given ball is to the diameter of the required ball. 12 feet = an edge of the cube. Then V3 x 144 = 20.7846, the distance from a lower to the opposite upper corner of the room. 20.7846 — 12 = twice the distance from the given ball to the corner. 4.3923 = the distance of the nearest point of the given ball from the corner. Then, the distance from the extreme point of the ball to the corner = 20.7846 - 4.3923 = 16.3923. .-. 16.3923 feet : 4.3923 feet : : 12 feet : (3.215 feet), Ayis. 29. Let I = number of minutes past 3 o'clock. 40 — |- = distance the minute hand is from 8. y\ = number of minute spaces the hour hand is from 3. 15 4- y\ = distance the hour hand is from 12. But since the minute hand is the same distance from 8 that the hour hand is from 12, then 40— 24_-IK_|_ 2 .-. I, or 11 = 23yL minute past 3 o'clock, Ans. ANSWERS AND SOLUTIONS 171 30. Since an edge of the given cube differs from an edge of the original cube by 2 inches, the difference in the solidity of the cubes will be the solidity of 7 blocks 2 inches thick — a corner cube, 3 narrow blocks, and 3 square blocks. The con- tents of these 7 solids = 39,368 cubic inches. By taking away the 8 cubic inches, the number of cubic inches in the corner cube, there remains 39,360 cubic inches, the solidity of the 3 narrow blocks and 3 square blocks. Then 1 square block and 1 narrow block contain 13,120 cubic inches. Now, since these blocks are 2 inches in thickness, the sum of the areas of 1 face in each of the 2 = 6560 square inches. That is, the area of a square and a rectangle 2 inches in width = 6560 square inches. This rectangle is equivalent to 2 rect- angles of equal length and 1 inch wide. Now, if we place these rectangles on adjacent sides of the square and also add a square 1 square inch in area to complete the square, we will have a square = 6561 square inches. A side of this square = V6561 = 81 inches = an edge of the original cube after the reduction, increased by 1 inch. .-. an edge of the original cube = 82 inches, Aiis. 31. The required number is the remainder left after sub- tracting the largest cube. In extracting the cube root of 592,788 we find 84 to be a side of the largest cube, and 84 to be the remainder. .-. 84 is the required number, Ans, 32. 80; 40. 33. In 1 hour A can row upstream J of the distance. In 1 hour A can row downstream | of the distance. ^ — i = J, or twice the distance the stream flows in 1 hour. Hence, the stream flows -^ of the distance in 1 hour. ^ of the distance = 1 mile. .*. the distance = 12 miles, Ans. 34. B's share = i(90 4- 20) = 22 ; 22 - 20 = 2, the loss. 35. The required number is the remainder left after sub- tracting the largest square. In extracting the square root of 172 MATHEMATICAL WRINKLES 13,340 we find 115 to be the whole mimber of the root and 115 the remainder. .-. 115 is the required number, Ans. 36. v^number = 10 Vnumber. Raising to the 12th power, (number)* = 10^^ (number)^ Dividing by (number)^, we have the number = 10^^ ^ 1,000,000,000,000. 37. -^. 38. 0. 39. 66|%. 40. This problem may be solved by Geometrical Progres- sion. I = ar""'^. I = Iff, the last term, a = 1, the first term, w = 4, the number of terms. .•.|A| = r«,andr = |. .-.1-1 = 1, or 121%. 41. $200; $12. 42. 30. 43. 20%. 44. $750. 45. $37,037. 46. 10: 40 o'clock. 47. 3iij cords. 48. Let 100 % = cost of goods. 180 % = marked price of goods, iof 180% =30%, loss. . 180 % - 30 % = 150 % = selling price. 150 % - 100 % = 50 %, gain, Ans. 49. ^i : -^ : : 6 : (5.738), the diameter of the inside sphere. 6 inches — 5.738 inches = .262 inch, twice the thickness of the shell. .*. .131 inch = the thickness of the shell. 50. The number of bushels of apples = | of 20 bushels = 16 bushels. 51. Let 100 % = present worth of sales. 103 % of present worth of sales = 95 % of sales. 1 % of present worth of sales = ^% % of sales. 100 % of present worth of sales = ^2^o% % of sales. ... 9232^4_ ^^ of sales = 1191% of cost of goods. 1 % of sales = 1.29if J % of cost of goods. 100 % of sales = 129if ^ % of cost of goods. /. 29|f J % = the per cent advance of the cost. ^^•90 [ All 2 190 a|i ANSWERS AND SOLUTIONS 173 52. J84,245,000 - 48,245,000 = 36,000,000. 36,000,000 H- 36,000 = 1000, the divisor, Ans. 53. 1.754 inches; 2.246 inches; 4 inches. 54. 3| years. 56. 300 miles. 56. $300. 57. 60 days ; 40 days. 58. 1600 -^ 80 = 20, the difference of the two numbers. The sum -h the difference = twice the greater number. Hence, 80 -f 20 = 100 = twice the greater number. .-. 50 = the greater number, and 30 = the smaller number. 59. 50%. 60. 15. 61. 3. 10 5 10 gallons of water, Ans. 63. Solve by means of Progression : Let P =■ principle ; r= rate of interest ; n= number of years ; A = amount of each payment. Then .^ r.P(l-f r)V (l + r)*-l Since one amount is paid at the beginning of the year, the principal less that amount will be the money to reckon as the new principal for the term of 4 years. $1000-^=(P-^). (1+rr-l (1+tV)*-1 ^ tV($10 0-^)(1.4641) . .4641 Clearing of fractions, $.4641 A=r^{$U64.1 - 1.4641 A). 6.1051^= $146.41. .♦.^=$239.81, Ans. 64. 3^%. 65. $.67^. 174 MATHEMATICAL WRINKLES 66. The true discount on $lis $!-($ 1-^1.015) = $.014/^0^. The bank discount on $ 1 is $ .015. Then $ .015 - $ .014^\\\ = mOj\\\, the difference. $ .90 ^ .000y2^ = $ 4060, the face of the note, Ans. 67. $50. 73. 11^ ounces. 68. 8 days. 74. 3|- years. 69. 66|%. 75. $160. 70. 6 feet. 76. 25 dozen ; 92 cents. 71. 168.298+ bushels. 77. 62.832 minutes. 72. 8 pounds. 78. 5-^j hours. 79. 37 inches. 80. $76.52, first; $ 96.52, second. 81. 30 steps. 82 104 feet. 83. 27^ minutes after 5 o'clock. 84. lOif minutes past 2 o'clock. 85. A pound of feathers. 86. 600; 1200; 1800; 2400 yards. 87. 360 acres. 88. $20. 89. 1,000,000. 90. $2. 91. Let 100 %= the marked price. He receives 100 % - 10 % = 90 % . Since he uses a yard measure .72 of an inch too short, he gives only 35 2V inches for 1 yard. He sells 35 Jg^ inches for 90 fc of the marked price. Therefore he would sell 36 inches for 91^ % of the marked price. .-. 100 % — 91fi % = 8489 %, the required discount. 92. 69.36286+ pounds. 95. $8.75. 93. 32feet+. 96. Book, $1.10; pen, $.10. 94. 8%. 97. 17%. ANSWERS AND SOLUTIONS 175 98. Each new day begins at the 180th meridian, which was crossed in the Pacific Ocean before reaching Manila. 99. 7 sheep. 101. 40. 103. Friday. 100. f. 102. 72. 104. 0. 105. 3 P.M. 107. G:40 p.m. 106. A, S500; B, $700. 108. B paid $92; 15% gain. 109. The greater -f the less = 582. The greater — the less = 218. .-. 2 times the less = 364, and the less = 182. The gi-eater = 400. 110. 2760.4288+ cubic inches; 1152 square inches. 111. A's, $90; B's, $135; G% $180. 112. $20. 113. August 11 was 21 days before the note was due. The use of any sum of money for 21 days, or -j^^^ of a month, at 6 % is equal to -j-J^ of it. Then, since he promised to pay such a sum that the use of it for 21 days was to equal the use of the sum unpaid for 2 months, y^ of the sum unpaid = y^yVrr of the sum paid. Hence the sum unpaid = -,^j^ of the sum paid .-. f^ of the sum paid + ^%\ of the sum paid = $ 100. .-. the sum paid = $ 74.07. 114. $212.12. 118. 64 pounds. 115. $246.60. 119. 7 cents to A ; 1 cent to B 116. $50 gain. 120. 10. 117. 43.817 pounds. 121. 45 feet. 122. 30 of first quality ; 16 of second quality. 123. 32 miles. 125. 810 revolutions. 124. $2; SI. 126. 16 dozen. 176 MATHEMATICAL WRINKLES 127. A, 2.87 rods; B, 4.72 rods; C, 13.82 rods. 128. 1,000,000. 134. 2. 129. Horse, $110; cow, $10. 135. 20. 130. 216 pounds. 136. 60. 131. $850. 137. 20%. 132. $4. 138. 20%. 133. They are the same. 139. First, $250; second, $200. 140. Husband's age, 24 years ; wife's age, 20 years. 141. 20 gallons of wine ; 30 gallons of water. 142. 1300. 143. 4. 144. $.80. 145. $.75. 146. 8. 147. 21-j9j- minutes past 4 o'clock. 148. lOif minutes past 2 o'clock. 149. 27^ minutes past 2 o'clock ; 3 o'clock. 150. 43^^ minutes past 2 o'clock. 151. .50. 152. 245.574. 153. Wife, $8500; son, $12,750; daughter, $2125. 154. 1 mile. 158. 9||%, or 9.69+. 155. 180. 159. $42,949,672.95. 156. 43,200. 160. $4. 157. 1,860,867. 161. Midnight. 162. 1 hour and 20 minutes is lost in going 50 miles. .-. 80 minutes is lost in going 50 miles. .-. 1 minute is lost in going | mile. .-. 120 minutes is lost in going 75 miles. .-. 2 hours is lost in going 75 miles. But 2 hours is the entire time lost. .*. the distance traveled after the breakdown is 75 miles. ANSWERS AND SOLUTIONS 177 Again, the train at its original speed goes as far in 3 hours as it went in 5 hours at its speed after the breakdown, .-. in 3 hours at the original speed it goes 75 miles. .-. ill 1 hour at the original speed it goes 25 miles. .-. the length of the line is 75 miles +25 miles = 100 miles. 163. llj cents. 166. 1^. 168. 300 feet. 165. 2. 167. 4f 169. 2:1. 170. 2 miles 340 feet. 171. 132 and 140. 172. 20%. 173. By their sum. 174. James's speed = ^ of my speed. John's speed = -J^J of James's speed. .-. James's speed = 1^ of (^ of my speed). .-. James's speed = ^4|, or ^ of my speed. .-. James's speed and my speed are in the ratio of 456 to 500. .-. in running 500 yards I beat James 500 yards — 456 yards = 44 yards, Ans. 175. First, .759 inch; second, 1.08+ inches ;, third, 4.16 + inches. 176. First Method. 1. Any remainder which exactly di- vides the previous divisor is a common divisor of the two given quantities. 2. The greatest common divisor will divide each remainder, and cannot be greater than any remainder. 3. Therefore, any remainder which exactly divides the pre- vious divisor is the greatest common divisor. Second Method. 1. Each remainder is a number of times the greatest common divisor. For a number of times the greatest common divisor, subtracted from another number of times the greatest common divisor, leaves a number of times the greatest common divisor. 2. A remainder cannot exactly divide the previous divisor unless such remainder is once the greatest common divisor. 178 MATHEMATICAL WRINKLES 3. Hence, the remainder which exactly divides the previous divisor, is once the greatest common divisor. 177. 112 cubic feet. 178. In 5 seconds both trains travel 600 feet, .-.in 1 hour both trains travel Sl^ miles. In 15 seconds the faster train gains 600 feet. .-. in 1 hour the faster train gains 27^ miles. Now, we have the sum of their rates = 81^^ miles and the difference of their rates = 27^ miles. .-. rate of faster + rate of slower = 81^^ miles, and rate of faster — rate of slower = 27^ miles. .-. 2 times rate of faster = 109 jij- miles. .-. rate of faster == 54^^ miles. Also, 2 times rate of slower = 54^^ miles. .-. rate of slower == 27^^ miles. 179. 1.118 times. 185. 3042315V 180. SSte6. 186. 3424^. 181. Senary. 187. 139. 182. 1,110,100,010 years. 188. 124. 183. 221446. 189. 128. 184. 10212^. 190. 180. 191. 658,548,918. 192. 28 gallons wine ; 42 gallons water. 193. lOf 194. Since the numbers are consecutive, each r the cube root of 15,600 ; in other words, the numbers must lie between 20 and 30. Now, 15,600 is divisible by 25, since it ends in two ciphers, hence 25 may be one of the numbers. By trial, we find that 624 would be the product of the other two, which themselves must end in 4 and 6 to give a product ending in 4. Ans. 24; 25; 26. ANSWERS AND SOLUTIONS 179 195. Such a number must lie halfway between 1042 and 1236. .-. 1236 — 1042 = 194, which divided by 2, gives 97. .-.1042 + 97 = 1139, Ans. 196. 76,809,256,566. 197. 49. The remainder left over after subtracting the largest cube is the number. 198. At 4 miles per hour = 1 mile in 15 minutes, and 5 miles per hour = 1 mile in 12 minutes. .-. in going 1 mile there is a difference of 3 minutes, but the actual difference is 10 minutes + 5 minutes = 15 minutes. .-. 15 minutes -r- 3 minutes = 5. Ans. 5 miles. 199. i of small glass = i of total, and since the large glass is J of both, J of the large glass = J of total, and J -f J = ^ = wine. .-. |-J = water. — From « Arithmetical Wrinkles." 200. When the ball just floats, its specific gravity is 1. Then by Allegation, we have \10| 9 I 35 I' ^^ ^ ^ of t ^ (12)» = Agf fi^ TT, and ^1[I|||^TTJ7) =5.52 inches, radius of ball, and 12 — 5.62, or 6.48, inches is the thickness of the shell. — From " The School Visitor." 201. 14° F. = - 10° C. and 270° F. = 132|° C. The specific heat of ice is .505, that of steam is .48, latent heat of fusion is 80, and that of evaporation is 537 ; then, 100(10 X .505 + 80 -f 100) = 18,505 heat units, required to melt the ice and raise its temperature to 100° C. There are 80 x 32| x .48 = 1237^ heat units given off in reducing the steam at 132|° C. to steam at 100° C. There are (18,505-1237^)^537 = 32.16 pounds of steam 180 MATHEMATICAL WRINKLES at 100° C. to be condensed to water at 100° C. The result would be 132.16 pounds of water at 100° C, and 80-32.16 = 47.84 pounds of steam at 100° C. — From " The School Visitor." 202. If the average for the entire distance were 30 miles an hour, 50 X 30 or 1500 miles would be run, but this lacks 300 miles which must be made up running 55 miles per hour, or 25 miles an hour faster, taking 300 -^ 25, or 12 hours. Hence, the distance from 5 to C is 12 x 55, or 660 miles, and (50 — 12) times 30, gives 1140 miles from A to B, — From " The School Visitor." 203. Volume of sphere = 2 times volume of double cone. Surface of sphere = V2 times surface of double cone. 204. 20 rods. 205. For bodies above the earth's surface, the weight varies inversely as the square of the distances from the center. Hence, to weigh yL as much as at the surface, the body must be Vl6 = 4 times as far from the center, or 16,000 miles, and the required height above the surface is 16,000 - 4000 = 12,000 miles. 206. 1 mile. 207. 7.2 inches. 208. 34. 209. The difference between the bank and the true discount is always the interest on the true discount. Hence S9 is 12% of the true discount, which is $ 75. The bank discount is $ 9 more, or S84, which is 12 % of the face of the note, and then $84 divided by .12 gives $ 700, the face. 210. By the prismoidal formula, the volume F is |- of (upper base + lower base + 4 times middle section) x length. Therefore F= i (4 x 4 -f 2 x 3 + 4 X 3 x 3J) x 120 -- 144 = 8f feet, Ans. 211. Place the box on its end and put in 11 rows of 5 and 4 balls, alternately, making a total of 50 balls in the first layer. ANSWERS AND SOLUTIONS 181 Place the second layer in the hollows of the first, and it has 6 rows of 4 each and 5 rows of 5 each, making 49 balls in the second layer. In this manner 12 layers may be placed, making a total of (50 + 49) x 6 = 594 balls. — From " The Ohio Teacher." 212. If the field were 48 feet wide, it would take one post less at each end and two less at each side, or 6 less ; but to make 66 less, the field must be 11 x 48 = 528 feet, or 32 rods wide, and 64 rods long ; area, 12.8 acres. 213. 17.584, specific gravity. 214. 7^ feet, the distance the ball bounds. 30 feet equals the whole distance the ball moves. 215. Let r = rate per month, 12 r = rate per annum, p = sum borrowed, n = number of payments, q = cash payment. Then, from algebra, we get ^ = 0^^> 9 = 9J, p=$500, n = 72. ••• to -Pr){^ - ry = q, and (19 - IQOO r)(l + r)" = 19. .-. r = .00911, and 12 r = .10932 = 10.932 %. 216. 1178.1 square feet. 218. 6.864+ inches. 217. 5JJJ ounces. 219. 72 and 96. 220. Since the numbers have a common factor plus the same remainder, if the numbers are subtracted from one another, the results will contain the common factor without the remainder, thus : 364 414 539 364 414 50 125 The largest number that will divide all of these numbers is 25, Ans. 221. I. Let S = selling price and (7= cost. 182 MATHEMATICAL WRINKLES S — C Then, S — C= gain and — = rate of gain. O Also, S — -f^-^ C = supposed gain, and J§ 9 2 (J 1_0_0_ g Q — gJ^J, — = ^^ „ = supposed rate of gain. .-. S- 0=1^0, or 15%, ^ns. 11. A jSJiort Solution. 8 : 10 = 92 : 115. 115 - 100 = 15 % gain. 222. Let |- = distance the hour hand moves past 3 o'clock. ^ = distance the minute hand moves in the same time. Then -2^ + 1 = -^i = distance they both move. But the distance they both move = 45 minutes. .-. -^ = 45 minutes. 1 = ^ of 45 minutes = l^f minutes. -2/- = 24 X l|f minutes = 41/^ minutes. .-. It is 41 j7^ minutes past 3 o'clock. 223. The weight of the first ball is 3| times an equal bulk of water, and that of the second is 2^^ times the equal bulk of water; hence, 3|- times the volume of the first equals 2^ times the volume of the second ball. But the volumes vary as the cubes of the diameters ; hence, the required diameter is, d = -^"(3f -i- 210) =, 1^ feet, Ans. 224. The amount of $500 for 2 years at 6% is $560; $ 2500 — $ 560, or $ 1940, is the amount of the note, the pres- ent worth of which, for 24 - 8, or 16 months, is $ 1796.30. 225. The present worth of $201 for 30 days at 6% is $200; the present worth of $224.40 for 4 months at 6% is $220. Hence, the present rate of gain is (220 - 200) -- $200 = 10%, Ans, ANSWERS AND SOLUTIONS 183 226. If the 65 minutes be counted on the face of the same clock, then the problem would be impossible, for the hands must coincide every 65^-^ minutes as shown by its face, and it matters not if it runs fast or slow ; but if it is measured by true time it gains j\ of a minute in (yo minutes, or yY^ ^^ ^ minute per hour, Ans. 227. The loss of weight of an immersed body equals the weight of the fluid displaced. Hence 970 — 892 = 78 ounces, weight of water displaced, and 970 — 910 = 60 ounces weight of alcohol displaced. But as water is taken as the standard of comparison, the specific gravity of alcohol is 60 -r- 78 = ^§ = .769+, Ans. 228. The rate of the ship is J* per hour, while that of the sun is 15°. When they both move west, the sun gains 14|° ; but when the ship moves east the sun gains 15J°. Therefore since the sun must make a gain of 360° in each case, the time from noon to noon is 360° -5- 14| = 24^^ hours, west and 360° -5- 15 i° = 23^ hours east. 229. ^ of 165 acres = 65 acres, the amount of land each man should furnish. 100 acres — 55 acres = 45 acres, the number of acres A fur- nishes C. 65 acres — 55 acres = 10 acres, the number of acres B fur- nishes C. Hence, ^ ot $ 110 = $90, the amount A should receive, and ^ of $ 110 = $ 20, the amount B should receive. 230. Let r be the internal radius of the cup ; and the volume of a quart of wine,' 57f inches. Then 240 ttt^ -t- (3 X 57J) = value of wine in cents. Also 40 ttt* = value of cup in cents. . A(\^ 2407rr» ••^^'^ = 3-^^- .-. r = 28 J in. .-. 40 irr* = $ 1047.74, Ans. 184 MATHEMATICAL WRINKLES 231. 11 times. 232. Eleven integral solutions, as follows : Average price = 1 SI 1 2 3 4 5 6 7 8 9 10 11 91 82 73 64 55 46 37 28 19 10 1 8 16 24 32 40 48 56 64 72 80 88 233. The volume is found by the prism oidal formula. 1 Z (2 X 2 4- 1 X 1 2 + 4 X f X V ) - 144 = ^ Z feet, or if I be the length in feet, the board measure is || of the length in feet. 234. l|i board feet. , 235. Since .0^ == .05, .^ must be .5. 236. 96 acres. 7|- acres. 237. 40 rods ; 30 rods ; 238. A liter of ice weighs 918 grams and a liter of sea water weighs 1030 grams. Then 918 divided by 1.03 equals 891.262 cubic centimeters displaced by one liter of ice, and 1000 — 891.262 is 108.738 cubic centimeters above water. Now 108.738 divided by 1000 gives .108738 of the whole above water, and 700 divided by .108738 equals 6437.5 cubic yards, the volume of the iceberg. Side-wall^^-^'^ 1 Ei^d s^ ^-^ 1 End Floor 239. The distance SFis the hypotenuse of a right triangle = V(15)2 + (39)2 ^ 41.78+ ft. 240. $ 563.23 due A. 241. (a) The least time required is 59f| seconds past 12. (6) The least time required is 30y\^2T seconds past 12. (c) The least time required is 1^%V minutes past 12. ANSWERS AND SOLUTIONS 185 242. S 2500.00. 243. S225 = first payment; $675 = second payment. 244. First, $8400; second, S7800; third, $7280. 245. 8 yards of first kind; 16 yards of second kind. 246. 21^ minutes past 4 o'clock. 247. A, 261^ days; B, 120 days. 248. 10. 251. 2 inches. 249. 7ifeet; 8| f eet. 252. 2 cones. 250. 13,066.4 miles. ALGEBRAIC PROBLEMS 1. Let X = your age. y = difference between our ages. Then a; -h y = my age. and (x-^y)-\-(x-\-2y) = 100. Solving, X = 33^ and x-}-y = 44 J. 2. A, 72 hours ; B, 90 hours. 3. Let the time be x minutes past 10 o'clock. We assume that at the beginning of every minute the second hand points at 12 on the dial. The distance of the second hand from the minute hand at the required time is 60 a; — a; = 59 a; ; and that of the second hand from the hour hand is 60 - 60 a; - (10 -^ tV x) = 50 - 60 X - -jJj a;. .-. 59 x = 51 - 60 X 4- iV ^• Solving, x = y^:j^ minutes = 25| Jff seconds. — From " American Mathematical Monthly." 186 MATHEMATICAL WRINKLES 4. Let s = distance between cars going in the same direction. Let i = interval of time between cars going in the same direction. Let X = rate of car. Let y = rate of man. Then, x — y = rate of approach when both travel in the same direction. x-\-y = rate of approach when they travel in opposite directions. By conditions of problem, 12(x-y)=^s = 4.{x + y). .'.x = 2y. Also, ^ = ^ = ii^±l)=6. X X Therefore the interval between cars is 6 minutes, and my rate is half the speed of the cars. — F.rom " School Science and Mathematics." 5. Let X = the number of eggs for a shilling. Then - = the cost of one Qgg in shillings. X 12 and — = the cost of one dozen in shillings. X 12 But if X — 2 — the number of eggs for a shilling, then - X — 2 would be the cost of one dozen in shillings. 12 12 1 .*. = -— (1 penny being y^ of a shilling). X — 2 X 12 Solving, a; = 18 or — 16. Then if 18 eggs cost a shilling, 1 dozen will cost if of a shilling, or 8 pence, Ans. 6. Let X = amount per yard received by one. Then ic -j- 1 = amount per yard received by the other. ANSWERS AND SOLUTIONS 187 Solving, «= 1.7808. One builds 56.15 yards at $1.7808; other builds 43.85 yards at $2.2808, Ans. 7. 1760 yards, or 1 mile. 8. Let X = number of acres. 160 X = number of dollars for which the land sold. Then since 1^ inches = diameter of a dollar, 1^(160 x) = 240 X = perimeter of square in inches. -y or 60 a: = length of one side of the square in inches. 4 -— ^, or 5 a; = length of one side of the square in feet. (5 xy = the number of square feet in the square. = number of acres. 25 X* • • 43560 "• Solving, x = 1742.4, ^715. 9. 2652.5+ feet ; 2627.4-^ feet. 10. Let X = one side of the square in feet. Then 43560 = the number of acres. 16aj = = the number of feet of boards in the fence 16 X 11 = the number of boards in the fence. a? 16x **' 43560 11 Solving, X: = 63,360. Then X^ 43560 = 92,160 acres, or 144 sections. 11. 10^ hours. 188 MATHEMATICAL WRINKLES 12. Let X = rate of faster train per hour in miles. y = rate of slower train per hour in miles. In 5 seconds both trains travel 600 feet. .-. in one hour they travel 81^ miles, or x + y = Sl^\. (1) In seconds the fast train gains 600 feet. .'. in one hour the fast train gains 27 ^^ miles, or x-y=2T^. (2) Solving, X = 54y6_, and y = 27 ^^y, Ayis. 13. Let X = number of minutes until 6 o'clock. Then 6 hours — x= time past noon. 3 hours -f- 4 a; = time past noon 50 minutes ago. .♦. 360 - .T = 180 + 4 a; + 50. Solving, X = 26, Ans, 14. Let X = cost of the gun in dollars. -— - = per cent of loss. lUU Then (^y.=ioss. x' ^ 100 Solving, a; = 90, or 10. ••• $ 90, or $ 10 = the cost of the gun. 15. Let X = number of eggs he brought. Then a; + 1 = 1 of them. and f (a; + 1) = number of eggs in the nest. Also a; — 2 = i of them. and 2 (a; — 2) = number of eggs in the nest. ...2(x- 2) = 1(0^ + 1). Solving, aj = ll. Then 2 (a; -2) = 18, Ans. 16. Let X = cost of lot in dollars. ANSWERS AND SOLUTIONS 189 X 100 = per cent of gain. ) 100 = gain. ^ + 100 = = 144. Then Solving, X = 80, or - 180. $ 80 = Arts. 17. 6 inches. 18. VS. 19. | ± jVS and J ± hV5. 20. 21 minutes 49^ seconds past 4 o'clock. 21. Let X = number of men in a side of the first square. Then xr = number of men in the first square, and ic^ + 39 = number of men. Also X -h 1 = number of men in a side of the second square Then (x -f 1)^ = number of men in the second square, and (a; + 1)* — 60 = number of men. ... (x 4- 1)'- 50 = ar^-f 39. Solving, x= 4A. Then a:* 4- 39 = 1975, Ans. 22. 12 cents. 23. 4 feet. 24. iV5 and i(VE ± 5), 25. 16. 26. 6 feet. 27. Let X = cost of first horse. 80 — a; = cost of second horse. Then 80 — a; = gain on first horse. and 80 — (80 — x) = gain on second horse. 80-a; x X = rate of gain on first horse. 80 -a; 80 -X 1 = rate of gain on second horse. 80-a; X 5 Solving a; = 41.905, and 80 -x = 38.005. 190 MATHEMATICAL WRINKLES 28. The lot is 100 feet x 100 feet = 10,000 square feet. The house and the driveway each covers 5000 square feet. Let X = the width of the driveway. On each side of the lot it extends from front to rear 100 feet ; total, 200 x square feet. The house is 100 feet — 2 x feet ; the driveway behind the house is 100 feet — 2x feet long by x feet wide ; the total number of square feet is 100 x — 2 x^. The total number of square feet in the driveway (at sides of lot and rear of house) is 200 x + lOOx-2 x\ .'. - 2 a;2 4- 300 a? = 5000. Solving, x = 19.1. 29. 672. 30. 7.416 inches from either end. 31. Their monthly wages may be any number of dollars. If they receive more than $ 50 a month they will each lay up the same sum. If they receive less than $50, they will be- come equally indebted. 32. 1 33. 2(l-}-x')=(l-j-xy. 2-\-2x^ = x^-j-4:x' + 6x'-{-4x + l. Transposing, aj4_4^_6^_ 4.x -{-1 = 0. Adding 12 x^ to both sides, x*-4.a^ + 6x^-4.x + l = 12x^ But x^-4:a^-\-6x'-4.x-\-l = (x-iy. Then (x-iy-12x' = 0, or (x^ -2x^iy-12x'=0. Factoring, (aj2_2a; + l-f 2cc-\/3)(ic2-2a^ + l-2a;V3) = 0. ... a^_ 2a; + l + 2a;\/3 = 0, and x'-2x-^l- 2 a;V3 =_ 0. Solving, » = 1 - V3 ± V3 - 2 V3, or 1 + V3 ± V3 + 2 V3. ANSWERS AND SOLUTIONS 191 34. X* -^ 4: m^x — m* = 0. Factoring, (a^ 4- mxy/2 - mV2 + m*)(aj* - mxy/2 + m'y/2 + m*) = 0. .-. x^ ^_ wa;V2 - w2V2 + m* = 0. o I • w . mV2V2-l Solving, x = =: ± — V2 V2 35. a^ + 3/=ll. (1) y2-fx= 7. (2) (3) y - 2 = 9 - ar^, from (1). (4) / _ 4 = 3 - a:, from (2). (5) 2/ - 2 = -^ ^, from (4) by dividing by y + 2. ' 2/ +^ y -^ ^ 3 a; •. 9 - ar^ = 2/ + 2 y + 2 a^ ^, by transposing. 2/ + 2 2/ + 2 Then ^"7+2"^V27T4) ='^"^rf2"^(,27T4j' completing the square. 3 = x , extracting the square root. 22/ + 4 2^ + 4' Then canceling, a; = 3, and substituting, y = 2. Note.— From Horner's method we find a; =- 2.803, 3.681,-3.778. Hence y = 3.131, - 1.849, - 3.283. 36. x = 2; 2/=l. 39. a: = 4; 2/ =9. 37. a; = 4; 2/ = 6. 40. a; = 4; y = 9. 38. a; = 2 ; 2/ = 3. 41. x = ±2, ±-^;2/ = ±5, ±-^;2 = ±3, ±-l3. V7 Vi V7 42. 52/(a^ + l)-3ar3(2/» + l) = 0. (1) 15/(ar' + l)-x(2/« + l)=0. (2) 192 MATHEMATICAL WRINKLES (3) lB(^^yl±l,iror.(2). (5) 5(^-^^\ = s(y-^ ^\ from (4). (6) 15('aj + ^') = 2/' + -3,from(3). a;y SV 2/^ (8) 3(x + ±] = ^(/-fA), from(6). ^ (6). adding (5) and (6) Then \/^{x -\- -) = 2/ + -, extracting the cube root. \ ^J y But since y-{-- = 7?-\- — y ^ Dividing by a; + - ? we have a; a;" + 2 + ^ = •^+ 3, by adding + 3. ANSWERS AND SOLUTIONS 193 * + - T= V \/5 + 3, by extracting the square root. Also ^ - - = ^y/'5 - 1, by subtracting 1. ••• ^ = i[V^5 + 3 4- Vs/5-1]. But y + l = (^^-^l) in. ^^Q, in. 208. Cut as indicated in first figure, and rearrange the pieces as shown in the second figure. 209 9 ..8 > 2 " JO 210. Count and mark every ninth one, marking it "Turk" until 15 are marked. Mark the remaining ones, " Christians." 211. 16|. 213. 7 and 5. 214. 10| hours. 217. 60. 215. 72. 218. 28. 216. PRECAUTION. 219. Divide tlie cross as indicated in the first figure and rearrange as shown in the latter. \ \ \ rz 7 Ti I -\ 1 ^"^^'-^ '' ANSWERS AND SOLUTIONS 223 224. Subtract the smallest number from each of the others. The G. CD. of the differences is the required divisor. An- swer, 2. 226. The number of shoes equals the number of persons. 227. 27.3083+ inches. 228. There would be no difference in the weight when the bird perched or flew. The air which supports the bird rests on the bottom of the cage. If the same cage had no top, the same would hold. If it had no bottom, there would still be no difference in weight. In this case the flight of the bird would tend to produce a vacuum just under the top, and the air above the cage would press downward with a force equal to the weight of the bird. If both top and bottom were removed, there would be a difference equal to the weight of the bird. — From " School Science and Mathematics." 232. rri w$ ^■" ^m ■^ ^^^ ^P losm^ t y ? /, A / aq- 7 / / / # / A V / / / / : v. i. , / / / ^ -^ V / z / 1 1 233. 2||U|. inches and J^ff inches. (2itHf)' + (ttftt)' = 17. 234. Similar solids are to each other as the cubes of cor- responding lengths. Therefore the volumes of the balls are to 224 MATHEMATICAL WKINKLES each other as 1^ is to 2^ or as 1 to 8. By adding 1 to 8 we get 9. 9 is the sum of these two perfect cubes. We must now find two other numbers whose cubes added together make 9. These numbers must be fractionah They are lifffrg-fl-lf-J fppt and 676702467503 fppj- iet!b ana 3 4 8 6TT6^8T6"6"0 ^^^^• 235. Yes. By 2,071,723 and 5,363,222,357. T — I — r ^- + + + +[+ J I L 236. I entered the room C because I put my foot and part of my body in it, and I did not enter the other room twice, because after once going in I never left it until I made my exit at B. This is the only possible solution. 237. AI^SWERS AND SOLUTIONS 225 238. Bisect AB at D and BC at E ; produce AE to F making EF equal to EB-, bisect AF at O and describe the arc AHF ; produce £5 to H, and J&fT is tlie length of the side of the required square; from E with distance EIIj describe the arc JIJ and make JK equal to BE ; now from the points D and K drop perpendiculars on EJ at L and 3f. If you have done this accu- rately, you will now have the required directions for the cuts. 241. 24. Keduce. the length of the block by half an inch. The small block constitutes the waste. Cut the other piece into three pieces each IJ inches thick. Each of these may then be cut into eight blocks. 242. There are eleven times in twelve hours when the hour hand is exactly twenty minute spaces ahead of the minute hand. If we start at four o'clock and keep on adding 1 hour 5 minutes 27^ seconds, we shall get all these eleven times, the last being 2 hours, 54 minutes, 32^ seconds past twelve. Another addition brings us back to four o'clock, but at this time the second hand is nearly twenty-two minute spaces be- hind the minute hand, and if we examine all our eleven times, we shall find that only in one case is the second hand the required distance. This time is 54 minutes, 32-j^ seconds past 2. ^ 243. A (6) 226 MATHEMATICAL WEINKLES 244. 1 — 2 — 3 — 4 — 5; 1 — 2 — 4 — 5—3; 1 — 3 — 2 — 5 — 4; 1—3 — 4— 2 — 5; 1—4 — 2 — 3 — 5; 1 — 4-3 — 5 — 2. 245. Let A, B, C, T>, E, F, and G represent the seven men. The way of arranging them is as follows : — ABC D E F G A C D B G E F A D B C F G E A G B F E C D AFC E G D B A E D G F B C ACE B G F D A D G C F E B A B F D E G C A E F D C G B AGE B D F C A F G C B E D A E B F C D G A G C E D B F A F D G B C E 246. 3ift. 247. The bag contained either 79, 160, 241, 322, or 403, etc. 248. Twenty-six transfers are necessary. Move the cars so as to reach the following positions : — £-567 8 1234 E 56 123 87 56 ^312 87 E = 10 transfers = 2 transfers = 5 transfers 9 transfers. 8765432 1 250. If there were twelve ladies in all, there would be 132 kisses among the ladies alone, leaving twelve more to be ex- ANSWERS AND SOLUTIONS 227 changed with the curate — six to be given by him and six to be received. Therefore of the twelve ladies, six would be his sisters. Consequently, if twelve could do the work in four and a half months, six ladies would do it in nine months. 252. Only three revolutions are necessary. Number the nests from 1 to 12 in the direction the person travels. Transfer the egg in nest No. 1 to nest No. 2, in No. 5 to nest No. 8, in No. 9 to No. 12, in No. 3 to No. 6, in No. 7 to No. 10, in No. 11 to No. 2, and complete the last revolution to nest No. 1. This can also be done by transferring the egg in nest No. 4, to No. 7, in No. 8 to No. 11, in No. 12 to No. 3, in No. 2 to No. 5, in No. 6 to No. 9, in No. 10 to No. 1. 253. He divided the rope in half. He simply untwisted the strands and divided it into two ropes, each being of the original length of the rope. He then tied these two ropes together and had a rope almost twice as long as the original rope. 254. 26.0299626611 71957726998490768328505774732373764 7323555652999. 255. I reached the shore with little difficulty. I fastened one end of the trot line to the stern of the boat, and then while standing in the bow, gave the line a series of violent jerks thus propelling the boat forward. 256. C*s age at A's birth 4- A's present age = A's present age -f B's ; then C's age at A*s birth = B's present age. By the second condition, A's age — 3 = | (B's 4- 4), from which A's age = f B's age + 6 years. The difference between A's and B's present ages = B's age at birth of A. Therefore \ of B's present age — 6 years = B's age at A's birth, and 5^ (J B's age — 6 years) = -y- of B's present age, from which -y- B's age — 33 years = B's age, or 88 years. C's age at A's birth was also 88 ; B's, 88 -i- 5J, or 16 years. A's present age is 88 — 16 = 72 years ; B's 88, and C's 88 -f 72 = 160 years. 257. £2,567 18 s. 9|d. SHORT METHODS Business men everywhere complain that the schools teach neither accuracy nor rapidity in calculations. They claim that the pupils must learn facts and principles and have much practice in the application of principles; that because a boy can apply a principle to-day is no guarantee that he will have the same knowledge and ability tomorrow ; that eternal vigi- lance is not only the price of liberty, but also the price of proficiency. " The mechanic who is not skillful in the use of his tools will never rise above poor mediocrity; the pupils' arithmetical tools are figures, and unless he can handle these with facility and accuracy, he must ever remain a plodder, a waster of time, and a blunderer upon whose results none can depend." We are living in a fast age, an age of steam and electricity, when results are attained by lightning methods. ADDITION There are no short cuts in addition ; every figure in every column must be added to ascertain the amount. Nevertheless the time required to perform an operation in addition can be substantially shortened in the following ways : 1. By making plain, legible figures. 2. By placing units of a certain order immediately beneath units of a like order. 3. By omitting the " ands " and " ares.'* 4. By making combinations of 10. 5. By double column adding. SHORT METHODS 229 1. Civil Service Method When long columns are to be added, the following method will be found practical. 485 576 324 449 625 264 33 29 24 To. insure accuracy, add each column from top downwards as well as from bot- tom upwards. 2723 OPBBATION 24 21 83 62 63 49 Two-Column Adding Explanation. To add 2 columns at a time, begin with the number at the bottom and add the units of the number next above, and then add the tens, naming the totals only. Continue in this way until all the numbers are added. Thus, the given example would read 49, 62, 102, 104, 164, 167, 247, 248, 268, 274, 304, 808, 328. 3. OPERATION 142 881 212 468 1203 To add Tliree or More Columns Explanation. Three columns or more may be added at one time by extending the two-column method to include all the columns desired. Thus, 468, 470, 480, 680, 681, 761, 1061, 1063, 1103, 1203. 4. A Jap Method of Adding Illustrative Example. — Find the sum of 382, 498, 364, 899, 842, and 789. 230 MATHEMATICAL WRINKLES OPERATION Explanation. — Write 382 as read. To add 498, say 4 + 3 = 7 ; 9 + 8 = 17, write .7, the dot (.) shows that 1 ten is to be carried ; 8 + 2 = .0. To add 364, say 3 + 7. = 11, the dot following the 7 increases its value 1 and is read 8. Continuing this method, we obtain 2.6.6.4 as the result, which would be read 3774. 382 7.7.0 1 1.4 4 2 0.3.3 2985 2.6.6.4 Ans. Note. — To be an adder of any consequence, one ought to be able to add at least one hundred figures per minute. SUBTRACTION There are three common methods of subtraction. In the following example, we may say, (1) 6 from 15, 9 ; 2 from 3, 1 ; 4 from 13, 9 ; 1345 (2) 6 from 15, 9 ; 3 from 4, 1 ; 4 from 13, 9 ; 426 (3) 6 and 9, 15 ; 2 and 1 and 1, 4 ; 4 and 9, 13. 919 Each of these methods is easily understood. The first is the simplest of explanation, and hence it is generally taught to children. The second is slightly more rapid than the first. But the third, familiar to all as the common method of "mak- ing change," is so much more rapid than either of the others that it is recommended to all computers. This method is called the " Addition Method." MULTIPLICATION The squares of all numbers up to 30 should be memorized. They become the basis of further knowledge of numbers. Thus : 13 X 13 = 169 14 X 14 =3 196 15 X 15 = 225 16 X 16 = 256 17x17 = 289 18 X 18 = 324 19 X 19 = 361 20 X 20 = 400 21 X 21 = 441 22 X 22 = 484 23 X 23 = 529 24 X 24 = 576 25 X 25 = 625 26 X 26 = 676 27 X 27 = 729 28 X 28 = 784 29 X 29 = 841 30 X 30 = 900 SHORT METHODS 231 1. When the Multiplicand and Multiplier are Alter- nating Numbers Alternating numbers are those having in their regular order a number between them; as 7 and 9; 19 and 21 ; 32 and 34. Rule. — Write the square of the intermediate number less one. Example.— 15 x 17 = 16^ - 1 = 256 - 1 = 255. 17 X 19 = 182 - 1 = 324 - 1 = 323. 39 X 41 = 40»--l = 1600 - 1 = 1599. Note. — The product of two numbers having three intermediate num- bers between them is equal to the square of the central number less 4. Thus 9 X 13 = 112 - 4 = 117. 2. When the multiplier is a composite number. Multiply 328 by 42. OPERATION 328 7 Explanation. — The factors of 42 are 7 and 6. We 2296 multiply 328 by 7, and this result by 6 and obtain 13,776. 6 13776 Ans. 3. When the right-hand figure of the multiplier is 1. Multiply 23,425 by 41. OPERATION Explanation. — Multiply the units' figure of the mul- 23426 by 41 tiplicand by the tens' figure of the multiplier and se^ the 93700 figure of the product obtained one place to the left of 060425 Ans. units' figure of the multiplicand. Continue in this man- ner until all the figures of the multiplicand have been multiplied by the figures of the multiplier, and add the product, or products, thus found to the multiplicand and the result will be the product desired. 4. When the multiplier is a unit of any order. Rule. — Annex as many ciphers to the multiplicand as there are ciphers in the multiplier. Thus 42 X 10 = 420 ; 21 X 100 = 2100, etc. 232 MATHEMATICAL WKINKLES 5. When the multiplier is 11. Rule. — Beginning with units, add each term of the multipli- cand to the one preceding, carrying as in the regular rule. Multiply 1328 by 11. OPERATION Explanation.— +8 = 8 and we write 8 for the units' 1328 figure of the product ; 8 + 2 = 10, we write for tens' 11 place ; 2 + 3 = 5 and 1 carried = 6 ; we write 6 ; 3 + 1=4, 14608 Ans. we write 4 ; 1+0=1, we write 1 and the product is 14,608. 6. When the multiplier is 9, 99, or any number of 9's. Rule. — Annex to the multiplicand as many ciphers as the mul- tiplier contains O's, and subtract the multiplicand from the result. Thus 43561 x 999 = 43,561,000 - 43,561 = 43,517,439, Ans. 7. " To multiply any two figures by 11. Rule. — Add the figures and place the result between them. Thus 42 X 11 = 462, 29 X 11 = 319, etc. 8. To multiply by any number which ends with 9. Multiply 327 by 39. OPERATION Explanation. — The next number higher than 39 is 327 40. Multiplying the multiplicand by 40 produces a re- 40 suit of 13,080. The real multiplier is one less than 40, 13080 therefore by subtracting once the multiplicand from the 327 result we get the desired product. 12753 Ans. 9. To multiply by 15, 150, and 1500. Multiply 324 by 15. Explanation. — Annex a cipher to the multiplicand, OPKHATION" take one half of that number and add to it and you have the desired product. 1^=^ To multiply by 150, annex two ciphers, and to multiply 4860 Ans. ^^ ^^^^ axvne^ three ciphers. 10. To multiply two numbers ending in 5. Rule. — To multiply two small numbers each ending in 5, such SHORT ^METHODS 233 as S5 and 75, take the product of the left-hand figures (the S and 7), increased by Imlf their sunij and prefix the result to 25. Thus 35 5 X 5 = 25. 16 3 X 7 4- 1(3 + ") = 26. 2625, Ans. 11. To square any number of two digits. Rule. — Square the figure in units^ place to obtain the figure ill units^ place of the answer and carry as in multiplication. TJien take twice tJie product of the figures in units' and tens* ])lace, plus the amount carried. To the jjart of the square thus far obtained prefix the square of the figure in tens' place plus the amount carried. Thus (84)2 = 7056. 4^ = 16. Put down 6 and carry 1. 2 (8 X 4) -h 1 = 65. Put down 5 and carry 6. 82 + 6 = 70. Prefix 70 to 56. This also applies to numbers of more than two digits, though not so readily performed mentally, 12. To square a number ending in 5. Rule. — To square a number ending in 5, such as 85, take the product of 8 by the next higher figure (9) and annex 25 to the result. Thus 85- = 7225. 13. To square any number consisting of 9's. Rule. — Write as many 9*s less one as there are in the given number, an 8, as unany ciphers as 9*s, and a I. Thus 9992 = 998001. 14. To multiply by complements. Complements are useful not only in addition and subtraction, but also in multiplication. When the complements are small and the numbers of which they are complements are large, there is a great advantage in this method. 234 MATHEMATICAL WRINKLES Multiply 98 by 95. OPERATION Explanation. — The product of the comple- 98 complement 2 merits gives the two right-hand figures, 10, and 95 complement _5_ subtracting either complement from the other fac- 9310 10 tor gives the other two figures, 93. Multiply 198 by 192. Explanation. — When the numbers to be OPERATION ,.. 1. 1 , , , 1 , multiplied are between one hundred and two ,^- , ^ - hundred, the remainder found by subtracting 192 complement 8 .,, ' , . .u .u v. Z. ..,-3-, , — either complement from the other number must be doubled. Note. — If the numbers to be multiplied are between two hundred and three hundred, the remainder must be multiplied by three ; between three hundred and four hundred by four ; between four hundred and five hun- dred by five ; and so on. 15. To multiply by excesses. Rule. — From the sum of the numbers subtract 100 or 1000, as required, and annex the product of the excesses. Note. — An excess is the amount greater than 100, 1000, etc. Example. — 112 x 103 = 11536. 112 + 03 = 115. To 115 annex 12 x 3, or 36 = 11536. Example. — 1009 x 1007 = 1016063. 1009 + 007 = 1016. To 1016 annex 063 = 1016063. DIVISION When the divisor is an aliquot part of some higher unit. 1. To divide by 2\, multiply the dividend by 4 and point off one place. 2. To divide by 5, multiply the dividend by 2 and point off one place. 3. To divide by 10, point off one place. SHORT METHODS 235 4. To divide by 12^, multiply the dividend by 8 and point off two places. 5. To divide by 16|, multiply the dividend by 6 and point off two places. 6. To divide by 20, multiply the dividend by 5 and point off two places. 7. To divide by 25, multiply the dividend by 4 and point off two places. 8. To divide by 33J, multiply the dividend by 3 and point off two places. 9. To divide by 50, multiply the dividend by 2 and point off two places. 10. To divide by 66|, multiply the dividend by 3, point off two places, and divide by 2. 11. To divide by 100, point off two places. 12. To divide by 125, multiply the dividend by 8 and point off three places. 13. To divide by 200, multiply the dividend by 5 and point off three places. 14. To divide by 250, multiply the dividend by 4 and point off three places. 15. To divide by 500, multiply the dividend by 2 and point off three places. 16. To divide by 1000, point off three places. FRACTIONS 1. To add two fractions which have 1 for their numerator. Rule. — Write the sum of the given denominators over the prod- uct of the given denominators. Thus i + i = jV 236 MATHEMATICAL WKINKLES 2. To subtract two fractions which have 1 for their numerator. Rule. — Write the difference of the given denominators over the product of the given denominators. ±nus ^ -g- _ 2^-j. _ ^^. 3. To multiply two mixed numbers when the whole numbers are the same and the sum of the fractions is 1. Rule. — Multiply the lohole number by the next highest whole number J and to the product thus obtained add the product of the fractions. Thus 94 X 91 = 9O2V 4. To multiply two mixed numbers when the difference of the whole numbers is 1, and the sum of the fractions is 1. Rule. — Multiply the larger number increased by 1, by the smaller number; then square the fraction belonging to the larger mimber and subtract its square from 1. Add the whole number and the fraction and you have the desired product. Thus 54 X 44 = 24Jt. 5. To multiply two mixed numbers ending in J. Rule. — To the product of the whole numbers, add half their sum plus \. (If the sum be an odd number, call it one less, to make it even, and annex |.) Thus 81 X 64 = ^b\, ^x^ = 35f , etc. 6. To square any number ending in one half. Rule. — Midtiply the number by itself increased by unity, and annex \. 7. To square any number ending in one fourth. Rule. — Multiply the number by itself increased by ^, and annex 8. To square any number ending in three fourths. Rule. — Multiply the number by itself increased by 1-|-, and annex ^^. SHORT METHODS 237 9. To square any number ending in one third. Rule. — Multiply the number by itself increased by J, and annex J. 10. To square any number ending in two thirds. Rule. — Multiply the number i>y itself increased by 1\, and annex ^. 11. To multiply two numbers ending with the same fraction. Rule. — To the product of the whole numbers^ add that fraction of their sum, and the square of the fraction. Thus lof X 6f = 90 -f 6 H- A = 96:^. 12. To square any mixed number. Rule. — Multiply the whole number by itself increased by twice the fraction, and add the square of the frojctimx. INTEREST 1. The Thirty-six Per Cent Method. Rule. — Multiply the principal by the time in days, move the decimal point three plox^es to the left, and divide: If at 1 % by 36. If at 7 % by 5.143. If at 2 % by 18. If at 8 % by 4.5. If at 3 % by 12. If at 9 % by 4. If at 4 % by 9. If at 10 % by 3.6. If at 5 % by 7.2. If at 11 % by 3.273. If at 6% by 6. If at 12% by 3. 2. The Bankers' Sixty-day Method. Rule. — (a) Moving the decimal point in the principal three places to the left gives the interest ai 6 fo for 6 days. Moving the decimal point in the principal two places to the left gives the interest at 6% for 60 days. Moving the decimal point in the principal one place to the left gives the interest at 6% for 600 days. 238 MATHEMATICAL WRINKLES Writing the principal for the interest gives the interest at 6 (Jo for 6000 days. (h) The interest for any other time or rate can easily be found by using convenient multiples or aliquot parts. Thus Interest on $36 for 6 days at 6 % = $ .036. Interest on $ 36 for 60 days at 6 % = S .36. Interest on $36 for 600 days at 6 % = $ 3.60. Interest on $ 36 for 6000 days at 6 % = $ 36.00. Example. — Find the interest on $ 300 for 4 yr. 6 mo. 18 da. at 6%. OPERATION $72.00 = interest for the number of years. $ 9.00 = interest for the number of months, $ .90 = interest for the number of days. $81.90 = the required interest. Explanation. — 6 % of $300 = $ 18, the interest for one year. 4 x $ 18 = $72, the interest for 4 years. $3 = the interest for 2 months. 3 x $3 = $9, the interest for 6 months. 3 x $ .30 = $ .90, the interest for 18 days. 3. The Six Per Cent Method. Interest on $ 1 for 1 year = $ .06. Interest on $1 for 1 month = $ .OOJ. Interest on $ 1 for 1 day = $ .OOOi. Rule. — Multiply 6 cents by the 7iuniber of years, \ a cent by the number of months, ^ of a mill by the number of days, and multi- ply their sum by the principal. Example. — Find the interest on $400 at 6 % for 6 yr. 4 mo. 12 da. OPERATION $ .36 = interest on.$ 1 for number of years. .02 = interest on $ 1 for number of months. .002 = interest on $ 1 for number of days. $.382 = interest on $1 for the given time. 400 $152.80 = the required interest. SHORT METHODS 239 4. The Cancellation Method. (1) When the time is in years. Formula : J ■ _ Principal X Rate X Time (2) When the time is in months. Formula : T ^ . Principal x Rate x Time ^"*^'^^' = 100102 (3) When the time is in days. Formula : T t t — ^^^"c^P^^ X ^^^^ X Time n eres - ^^^ ^ ^^^ Exact Interest = Principal x Rate x Time, 100 X 365 OPERATION Example. — Find the interest on SOOO at 12% for 1 year, 3 months, 12 days. ^L2iiL>L462^ ^02.40, interest. 5 5. The New Cancellation Method. Rule. — Wnte the principal, timey and rate at the right of a vertical line; at the lejl of this line write a year in the same de- nomination in which the time is expressed. Cancel and reduce. Tlie result will be the interest for the given time and rate. OPERATION Example. — Find the interest on $ 1080 for 3 yr. 4 mo. 12 da. at 6 %. Example. — Find the interest on $ 540 for 2 yr. 4 mo. 12 da. at 10 %. $ 90 im ;? 40.4 .06 $218.16 = interest. OPERATION 6 213 m .10 $127.80 = interest. 240 MATHEMATICAL WRINKLES 6. The Cancellation-Thirty-six Per Cent Method. Formula : Interest = -QQl Qf Principal x Number of Days x Rate 36 This method is a combination of the Cancellation Method and Thirty-six Per Cent Method and should be very popular on account of its simplicity. OPERATION Example. — Find the interest '^* on S5112 at 4 % for 100 days. 3^ =$56.80, interest. 9 7. The Twelve Per Cent Method. To find the interest for 1 month on any principal at 12 %, simply remove the decimal point two places to the left in the principal ; in other words, divide the principal by 100. This gives the interest for 1 month at 12 %. Rule. — Poi7}t off two places in the principal, and multiply by the time expressed in months and decimals, or fractions of a month. Example.— What is the operation interest on $185 at 12% $1.85 = interest at 12% for 1 month. for 3 months, 15 days ? ^^^^'"^^ '"^ T""!^"' -^ $6.47| = interest for 3^ months, Ans. APPROXIMATE RESULTS In scientific investigations exact results are rarely possible, since the numbers used are obtained by observation or by experiments and are only approximate. There is a degree of accuracy beyond which it is impossible to go. The student should always bear in mind that it is a waste of time to carry out results to a greater degree of accu- racy than the data on which they are founded. Results beyond two or three decimal places are seldom desired in business. SHORT METHODS 241 I. Multiplication. Rule. — I. Write the terms of the multiplier in a reverse order, placing the units' term under that tei-vi of the multiplicand which is of the lowest order m the required product. II. Multiply each term of the multiplicand by the multiplier y rejecting those terms that are on the right of the term used as a multiplier, increasing each partial product by as many units as would have been caiTied to it from the product of the rejected part of the multiplicand, and one more when the second term toioards the light in the product of the rejected terms is 5 or more than 5 ; and place the right-hand terms of these partial products in the same column. III. Add the partial products, and point off in the sum the required number of decimal places. OPERATION 4.78567 95141.3 Example.— Multiply 4.78567 14.3570 = 4.7856 x 3 + .0002. by 3.14159, correct to four 'I'^ = '''' ^ ! + -^i* /. , ,' .1914 = 4.78 X .04 + .0002. decimal places. 48 = 4.7 x .001 + .0001. 24 = 4 X .0005 + .0004. 4 = 0+ .00009 + .0004. 15.0346 2. Division. Rule. — I. Compare the divisor with the dividend to ascertain the number of terms in the quotient. II. For the first contracted divisor, take as many terms of the divisor, beginning with the first significant term on the left, as there are terms in the quotient; and for each successive divisor, reject the right-hand term of the previous divisor, until all the terms of the divisor have been rejected. III. In multiplying by the several terms of the quotient, carry from the rejected terms of tJie divisor as in contracted multiplica- tion. 242 MATHEMATICAL WEINKLES Example. — Divide 35.765342 by 8.76347, correct to four deci- mal places. OPERATION 8.76347)35.765342(4.0811 35 053 9 = 4 X 87634 + 7114 7010:^8 X 876 + 2 104 88 = 1 X 87 + 1 16 9=1x8+1 7 3. Square Root. Rule. — Find, as visual, more than one-half the terms of the root, and then divide the last remainder by the last divisor, using the contracted method. Example. — Extract the square root of 10. 61 OPERATION 10(3.16227766+ 9 100 61 626 3900 3756 6322 14400 12644 63242 175600 126484 632447 4911600 4427129 6324547 48447100 44271820 63245546 417527100 379473276 632455526 3805382400 3794733156 CONTRACTED METHOD 10(3.16227766+ 9 61 1 100 61 626 3900 3756 6322 14400 12644 63242 175600 126484 49116 44269 4847 4427 "420 379 "41 38 4. Cube Eoot. Rule. — Extract the cube root, as usual, until one more than half the terms required in the root have been found; then with SHORT METHODS 243 the trial divisor and last remainder proceed, as in contracted division of decimals, to find the other terms of the root, dropping two figures instead of one from the divisor at each step, and one from each remainder. Example. — Extract tlie cube root of 2 to four decimal places. OPERATIOX 2.000000 1 1.2599 1 300 60 4 1000 304 728 43200 1800 25 272000 45025 225125 Next trial divisor, 40i5T^ | 4687^ remainder. 4219 = 9 X 408 + 9 X 75 46^ _42 = 4x 9 + 6 4 6. Extraction of Any Root. Rule. — Obtain one less than half of the figures required in the root as the nde directs; then, instead of annexing ciphers and bringing down a period to the last numbers in the columns, leave the remainder in the right-hand column for a dividend; cutoff the right-hand fuf a re from the last number of the j^revious column, two right-hand figures fro7n the last number in the column before that, and so on, always cutting off one more figure for every col- umn to the left. With the number in the right-hand column and the one in the previous column, determine the next figure of the root, and use it as directed in the rule, recollecting that the figures cut off are not used except in carrying the tens they produce. TJiis process is continued until the required number of figures 244 MATHEMATICAL WRINKLES is obtained, observing that when all the figures in the last number of any column are cut off, that column will be no longer used. Remark. — Add to the 1st column mentally ; multiply and add to the next column in one operation : multiply and subtract from the right-hand column in like manner. Example. — Extract the cube root of 44.6 to six decimals. 9 2 700 3 17 5 367 500 37 17 16 37 594^ 37 659 37/^3 OPERATION 4 4 . 6 (3 17 600 1725000 238 136 12182 865 111 546323 3 6 90 95 100 1050 1054 1058 Remark. — The trial divisors may be known by ending in two ciphers ; the complete divisors stand just beneath them. After getting 3 figures of the root, contract the operation by last rule. — From Ray's " Higher Arithmetic." MARKING GOODS To find the selling price of a single article at a certain per cent profit when the price per dozen and rate per cent gain are given. Thus, to make 5 per cent, multiply the cost per dozen by .08f . multiply by .11| multiply by .12-j3- multiply by .121 multiply by .12|i multiply by .13^ multiply by .13} multiply by .13f multiply by .14.^ multiply by .15 multiply by .16-| 6 % multiply by .OSf 40% 8 % multiply by .09 45% 10 % multiply by .09^ 50% 121 cf^ multiply by .09f ^^% 15 % multiply by M^^ 60% 20 % multiply by .10 " 65% 25 % multiply by .lO^^ 66|^ 30 % multiply by .lOf 75% 33^% multiply by .11^ 80% 35 % multiply by .Hi 100% QUOTATIONS ON MATHEMATICS " Mathematics, the queen of the sciences." — Gauss. " Mathematics, the science of the ideal, becomes the means of investigating, understanding, and making known the world of the real." — White. " Mathematics is the glory of the human mind." — Leibnitz. " The two eyes of exact science are mathematics and logic." — De Morgan. " Mathematics is the science which draws necessary conclu- sions from given premises." — Pierce. " The advance and the perfecting of mathematics are closely joined to the prosperity of the nation." — Napoleon. " Geometry is the perfection of logic, and excels in training the mind to logical habits of thinking. In this respect it is superior to the study of logic itself, for it is logic embodied in the science of tangible form." — Brooks. "God geoiuetrizes continually," was Plato's reply when questioned as to the occupation of the Deity. " There is no royal road to geometry." — Euclid. " Let no one who is unacquainted with geometry enter here," was the inscription over the entrance into the academy of Plato the philosopher. " All scientific education which does not commence with mathematics is, of necessity, defective at its foundation." — COMTE. " A natural science is a science only in so far as it is mathe- matical." — Kant. 246 246 MATHEMATICAL WRINKLES "Mathematics is the language of definiteness, the necessary vocabulary of those who know." — White. "The laws of nature are but the mathematical thoughts of God." — Kepler. " Mathematics is the most marvelous instrument created by the genius of man for the discovery of truth." — Laisant. " Euclid has done more to develop the logical faculty of the world than any book ever written. It has been the inspiring influence of scientific thought for ages, and is one of the cornerstones of modern civilization." — Brooks. "Mathematics is thinking God's thought after Him. When anything is understood, it is found to be susceptible of mathematical statement. The vocabulary of mathematics is the ultimate vocabulary of the material universe." — White. " Geometry is regarded as the most perfect model of a de- ductive science, and is the type and model of all science." — Brooks' "Mental Science." " I have always treated and considered puzzles from an edu- cational standpoint, for the reason that they constitute a species of mental gymnastics which sharpen the wits, clear fog and cobwebs from the brain, and school the mind to concentrate properly. Comparatively but few people know how to think properly. As a school for mechanical ingenuity, for stirring up the gray matter in the brain, puzzle practice stands unique and alone." — Sam Loyd. " Geometry not only gives mental power, but it is a test of mental power. The boy who cannot readily master his geometry will never attain to much in the domain of thought. He may have a fine poetic sense that will make a writer or an orator; but he can never reach any eminence in scientific thought or philosophic opinion. AH the great geniuses in the realm of science, as far as known, had fine mathematical QUOTATIONS ON MATHEMATICS 247 abilities. So valuable is geometry as a discipline that many lawyers and preachers review their geometry every year in order to keep the mind drilled to logical habits of thinking." — Brooks* " Mental Science." "Mathematics is the very embodiment of truth. No true devotee of mathematics can be dishonest, untruthful, unjust. Because, working ever with that which is true, how can one develop in himself that which is exactly opposite ? It would be as though one who was always doing acts of kindness should develop a mean and groveling disposition. Mathematics, there- fore, has ethical value as well as educational value. Its prac- tical value is seen about us everyday. To do away with every one of the many conveniences of this present civilization in which some mathematical principle is applied, would be to turn the finger of time back over the dial of the ages to the time when man dwelt in caves and crouched over the bodies of wild beasts." — B. F. Fixkel. " As the drill will not penetrate the granite unless kept to the work hour after hour, so the mind will not penetrate the secrets of mathematics unless held long and vigorously to the work. As the sun's rays burn only when concentrated, so the mind achieves mastery in mathematics, and indeed in every branch of knowledge, only when its possessor hurls all his forces upon it. Mathematics, like all the other sciences, opens its door to those only who knock long and hard. No more damaging evidence can be adduced to prove the weakness of character than for one to have aversion to mathematics ; for whether one wishes so or not, it is nevertheless true, that to have aversion for mathematics means to have aversion to ac- curate, painstaking, and persistent hard study, and to have aversion to hard study is to fail to secure a liberal education, and thus fail to compete in that fierce and vigorous struggle for the highest and the truest and the best in life which only the strong can hope to secure." — B. F. Finkel. 248 MATHEMATICAL WRINKLES "Mathematics develops step by step, but its progress is steady and certain amid the continual fluctuations and mis- takes of the human mind. Clearness is its attribute, it combines disconnected facts and discovers the secret bond that unites them. When air and light and the vibratory phenomena of electricity and magnetism seem to elude us, when bodies are removed from us into the infinitude of space, when man wishes to behold the drama of the heavens that has been- enacted cen- turies ago, when he wants to investigate the effects of gravity and heat in the deep, impenetrable interior of our earth, then he calls to his aid the help of mathematical analysis. Mathe- matics renders palpable the most intangible things, it binds the most fleeting phenomena, it calls down the bodies from the in- finitude of the heavens and opens up to us the interior of the earth. It seems a power of the human mind conferred upon us for the purpose of recompensing us for the imperfection of our senses and the shortness of our lives. Nay, what is still more wonderful, in the study of the most diverse phenomena it pursues one and the same method, it explains them all in the same language, as if it were to bear witness to the unity and simplicity of the plan of the universe.'^ — Fourier. " The practical applications of mathematics have in all ages redounded to the highest happiness of the human race. It rears magnificent temples and edifices, it bridges our streams and rivers, it sends the railroad car with the speed of the wind across the continent; it builds beautiful ships that sail on every sea ; it has constructed telegraph and telephone* lines and made a messenger of something known to mathematics alone that bears messages of love and peace around the globe ; and by these marvelous achievements, it has bound all the nations of the earth in one common brotherhood of man." B. F. FiNKEL. " Mathematics is the indispensible instrument of all physical research." — Berthelot. QUOTATIONS ON MATHEMATICS 249 " It is in mathematics we ought to learn the general method always followed by the human mind in its positive researches." — COMTE. " All my physics is nothing else than geometry." — Descartes. "If the Greeks had not cultivated conic sections, Kepler could not have superseded Ptolemy." — Whewell. " There is nothing so prolific in utilities as abstractions." — Faraday. " I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history." — Glaisher. "The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development." — Cajori. " If we compare a mathematical problem with a huge rock, into the interior of which we desire to penetrate, then the work of the Greek mathematicians appears to us like that of a vigorous stonecutter who, with chisel and hammer, begins with indefatigable perseverance, from without, to crumble the rock slowly into fragments; the modern mathematician ap- pears like an excellent miner who first bores through the rock some few passages, from which he then bursts it into pieces with one powerful blast, and brings to light the treasures within." — Hankel. "The world of ideas which mathematics discloses or illu- minates, the contemplation of divine beauty and order which it induces, the harmonious connection of its parts, the infinite hierarchy and absolute evidence of truths with which mathe- matical science is concerned, these, and such like, are the surest grounds of its title to human regard." — Sylvester. " I often find the conviction forced upon me that the increase of mathematical knowledge is a necessary condition for the advancement of science, and if so, a no less necessary condition 250 MATHEMATICAL WRINKLES for the improvement of mankind, I could not augur well for the enduring intellectual strength of any nation of men, whose education was not based on a solid foundation of mathematical learning and whose scientific conception, or in other words, whose notions of the world and of things in it, were not braced and girt together with a strong framework of mathematical reasoning." — H. J. Stephen Smith. " If the eternal and inviolable correctness of its truths lends to mathematical research, and therefore also to mathematical knowledge, a conservative character on the other hand, by the continuous outgrowth of new truths and methods from the old, progressiveness is also one of its characteristics. In mar- velous profusion old knowledge is augmented by new, which has the old as its necessary condition, and, therefore, could not have arisen had not the old preceded it. The indestructibility of the edifice of mathematics renders it possible that the work can be carried to ever loftier and loftier heights without fear that the highest stories shall be less solid and safe than the foundations, which are the axioms, or the lower stories, which are the elementary propositions. But it is necessary for this that all the stones should be properly fitted together ^ and it would be idle labor to attempt to lay a stone that belonged above in a place below." — Schubert. "As the sun eclipses the stars by his brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more if he solves them." — Brahmagupta. " Mathematical reasoning may be employed in the inductive sciences; indeed some of their greatest achievements have been obtained through mathematics. By it Newton demonstrated the truth of the theory of gravitation; by it Leverrier dis- covered a new planet in the heavens ; by it the exact time of an eclipse of the sun or moon is predicted centuries before it comes to pass. Mathematics is the instrument by which the QUOTATIONS ON MATHEMATICS 251 engineer tunnels our mountains, bridges our rivers, constructs our aqueducts, erects our factories and makes them musical with the busy hum of spindles. Take away the results of the reasoning of mathematics, and there would go with it nearly all the material achievements which give convenience and glory to modern civilization." — Brooks' " Mental Science and Culture." " The science of geometry came from the Greek mind almost as perfect as Minerva from the head of Jove. Beginning with definite ideas and self-evident truths, it traces its way, by the processes of deduction, to the profoundest theorem. For clear- ness of thought, closeness of reasoning, and exactness of truths, it is a model of excellence and beauty. It stands as a type of all that is best in the classical culture of the thoughtful mind of Greece. Geometry is the perfection of logic ; Euclid is as classic as Homer." — Brooks' " Philosophy of Arithmetic." "Only a limited number of people are capable of appreciat- ing the beauties of this oldest of all sciences." — Locke. " The value of mathematical instruction as a preparation for those more difficult investigations consists in the applicability, not of its doctrines, but of its methods. Mathematics will ever remain the past-perfect type of the deductive method in general ; and the applications of mathematics to the simpler branches of physics furnish the only school in which philoso- phers can effectually learn the most difficult and important portion of their art, the employment of the laws of simpler phenomena for explaining and predicting those of the more complex. These grounds are quite sufficient for deeming mathe- matical training an indispensable basis of real scientific educa- tion, and regarding, with Plato, one who is dyew/u-cTpryTos, as wanting in one of the most essential qualifications for the suc- cessful cultivation of the higher branches of philosophy." — From J. S. Mill's " Systems of Logic." 252 MATHEMATICAL WRINKLES " Hold nothing as certain save what can be demonstrated." — Newton. " To measure is to know." — Kepler. " It may seem strange that geometry is unable to define the terms which it uses most frequently, since it defines neither movement, nor number, nor space — the three things with which it is chiefly concerned. But we shall not be surprised if we stop to consider that this admirable science concerns only the most simple things, and the very quality that renders these things worthy of study renders them incapable of being de- fined. Thus the very lack of definition is rather an evidence of perfection than a defect, since it comes not from the ob- scurity of the terms, but from the fact that they are so very well known." — Pascal. " The method of making no mistake is sought by every one. The logicians profess to show the way, but the geometers alone ever reach it, and aside from their science there is no genuine demonstration." — Pascal. " We may look upon geometry as a practical logic, for the truths which it studies, being the most simple and most clearly understood of all truths, are on this account the most suscepti- ble of ready application in reasoning." — D'Alembekt. " Without mathematics no one can fathom the depths of philosophy. Without philosophy no one can fathom the depths of mathematics. Without the two no one can fathom the depths of anything." — Bordas Demoulin. "The taste for exactness, the impossibility of contenting one's self with vague notions or of leaning upon mere hypoth- eses, the necessity for perceiving clearly the connection between certain propositions and the object in view, — these are the most precious fruits of the study of mathematics." — Lacroix. " God is a circle of which the center is everywhere and the circumference nowhere." — Rabelais. QUOTATIONS ON MATHEMATICS 253 "The sailor whom an exact observation of longitude saves from shipwreck owes his life to a theory developed two thou- sand years ago by men who had in mind merely the specula- tions of abstract geometry." — Condorcet. " The statement that a given individual has received a sound geometrical training implies that he has segregated from the whole of his sense impressions a certain set of these impres- sions, that he has then eliminated from their consideration all irrelevant impressions (in other words, acquired a subjective command of these impressions), that he has developed on the basis of these impressions an ordered and continuous system of logical deduction, and finally that he is capable of express- ing the nature of these impressions and his deductions there- from in terms simple and free from ambiguity. Now the slightest consideration will convince any one not already conver- sant with the idea, that the same sequence of mental processes underlies the whole career of any individual in any walk of life if only he is not concerned entirely with manual labor ; consequently a full training in the performance of such se- quences must be regarded as forming an essential part of any education worthy of the name. Moreover, the full apprecia- tion of such processes has a higher value than is contained in the mental training involved, great though this be, for it in- duces an appreciation of intellectual unity and beauty which plays for the mind that part which the appreciation of schemes of shape and color plays for the artistic faculties ; or, again, that part which the appreciation of a body of religious doctrine plays for the ethical aspirations. Now geometry is not the sole possible basis for inculcating this appreciation. Logic is an alternative for adults, provided that the individual is pos- sessed of sufficient wide, though rough, experience on which to base his reasoning. Geometry is, however, highly desirable in that the objective bases are so simple and precise that they can be grasped at an early age, that the amount of training for 254 MATHEMATICAL WBINKLES the imagination is very large, that the deductive processes are not beyond the scope of ordinary boys, and finally that it affords a better basis for exercise in the art of simple and exact expression than any other possible subject of a school course/' — Carson. " Geometry is a mountain. Vigor is needed for its ascent. The views all along the paths are magnificent. The effort of climbing is stimulating. A guide who points out the beauties, the grandeur, and the special places of interest, commands the admiration of his group of pilgrims." — David Eugene Smith. " If mathematical heights are hard to climb, the fundamental principles lie at every threshold, and this fact allows them to be comprehended by that common sense which Descartes declared was ' apportioned equally among all men.' " — Collet. " The wonderful progress made in every phase of life during the last hundred years has been possible only through the increasing use of symbols. To-day, only the common laborer works entirely with the actual things. Those who occupy more remunerative positions in the business world work very largely with symbols, and in the professional world the possession of and ability to use a set of symbols is a prerequisite of even moderate success. The work of a man's hands remains after the worker has gone, but the products of mental labor are lost unless they are preserved to the world through some symbolic medium. It may be said without fear of successful contradic- tion that the language of mathematics is the most widely used of any symbolism. The man who has command of it possesses a clear, concise, and universal language. Fallacies in reason- ing and discrepancies in conclusions are easily detected when ideas are expressed in this language. The most abstruse prob- lem is immediately clarified when translated into mathematics. To quote from M. Berthelot, ' Mathematics excites to a high degree the conceptions of signs and symbols — necessary in- QUOTATIONS ON MATHEMATICS 255 struments to extend the power and reach of the human mind by summarizing. Mathematics is the indispensable instrument of all physical research.' But not only physical but all scien- tific research must avail itself of this same instrument. In- deed, so completely is nature mathematical that to him who would know nature there is no recourse but to be conversant with the language of mathematics." — Carpenter. No less an astronomer than J. Herschel has said of as- tronomy: "Admission to its sanctuary and to the privileges and feelings of a votary is only to be gained by one means — sound and sufficient knowledge of mathematics, the great in- strument of all exact inquiry, without which no man can ever make such advances in this or any other of the higher depart- ments of science as can entitle him to form an independent opinion on any subject of discussion within their range." "It is only through mathematics that we can thoroughly understand what true science is. Here alone can we find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain. Any sci- entific education setting forth from any other point is faulty in its basis." — Comte. " The enemies of geometry, those who know it only imper- fectly, look upon the theoretical problems, which constitute the most difficult part of the subject, as mental games which consume time and energy that might better be employed in other ways. Such a belief is false, and it would block the progress of science if it were credible. But aside from the fact that the speculative problems, which at first sight seem barren, can often be applied to useful purposes, they always stand as among the best means to develop and to express all the forces of the human intelligence." — Abbe Bossut. " We study music because music gives us pleasure, not neces- sarily our own music, but good music, whether ours, or, as is 256 MATHEMATICAL WEINKLES more probable, that of others. We study literature because we derive pleasure from books ; the better the book, the more subtle and lasting the pleasure. We study art because we re- ceive pleasure from the great works of the masters, and prob- ably we appreciate them the more because we have dabbled a little in pigments or in clay. We do not expect to be com- posers, or poets, or sculptors, but we wish to appreciate music and letters and the fine arts, and to derive pleasure from them and to be uplifted by them. At any rate these are the nobler reasons for their study. " So it is with geometry. We study it because we derive pleasure from contact with a great and an ancient body of learning that has occupied the attention of master minds dur- ing the thousands of years in which it has been perfected, and we are uplifted by it. To deny that our pupils derive this pleasure from the study is to confess ourselves poor teachers, for most pupils do have positive enjoyment in the pursuit of geometry, in spite of the tradition that leads them to proclaim a general dislike for all study. This enjoyment is partly that of the game, — the playing of a game that can always be won, but that cannot be won too easily. It is partly that of the aes- thetic, the pleasure of symmetry of form, the delight of fitting things together. But probably it lies chiefly in the mental up- lift that geometry brings, the contact with absolute truth, and the approach that one makes to the Infinite. We are not quite sure of any one thing in biology ; our knowledge of ge- ology is relatively very slight, and the economic laws of society are uncertain to every one except some individual who at- tempts to set them forth ; but before the world was fashioned the square on the hypotenuse was equal to the sum of the squares on the other two sides of a right triangle, and it will be so after this world is dead ; and the inhabitant of Mars, if he exists, probably knows its truth as we know it. The uplift of this contact with absolute truth, with truth eternal, gives pleasure to humanity to a greater or less degree, depending QUOTATIONS ON MATHEMATICS 257 upon the mental equipment of the particular individual ; but it probably gives an appreciable amount of pleasure to every student of geometiy who has a teacher worthy of the name." — From "The Teaching of Geometry," by David Eugene Smith. Mathematics has not only commercial value, but also educa- tional, rhetorical, and ethical value. No other science offers such a rich opportunity for original investigation and dis- covery. While it should be studied because of its practical worth, which can be seen about us every day, the primary object in its study should be to obtain mental power, to sharpen and strengthen the powers of thought, to give pen- etrating power to the mind which enables it to pierce a subject to its core and discover its elements ; to develop the power to express one's thoughts in a forcible and logical manner; to develop the memory and the imagination; to cultivate a taste for neatness and a love for the good, the beautiful, and the true ; and to become more like the gi*eatest of mathematicians, the Mathematician of the Universe. MENSURATION Mensuration is that branch of mathematics which treats of the measurement of geometrical magnitudes. Annulus, or Circular Ring An annul us is the figure included between two concentric circumferences. (1) To find the area of an annulus. Rule. — Multiply the sum of the two radii by their difference^ and the product by it. Formula. — A = (i\ + r^ (r^ — rg) tr. (2) To find the area of a sector of an annulus. Rule. — Multiply the sum of the bounding arcs by half the difference of their radii. Belts Length of belts. (a) For a crossed belt, L = 2Vc'-(r,-r,y + {r,-r,)f^-2sm-'':i±LA (b) For an uncrossed belt, L = 2^\c^-(r,-r,yi + 7r(r, + r,) + 2(r,-r,)sm-'''^^, where r^ is the greater radius and rj the less, and c the distance between the parallel axes. 258 MENSURATION 250 Bins, Cisterns, Etc. (1) To find the exact capacity of a bin in bushels. Rule. — Divide the contents in cubic feet by .83, or {1728-i- 2150.42); the quotient will represent the number of bushels of grain, etc. Four fifths of this number of bushels is the number of bushels of coal, apples, potatoes, etc., that the bin will hold. (2) To find the approximate capacity of a bin in bushels. Rule. — Any number of cubic feet diminished by ^ will represent an equivalent number of bushels. (3) To find the contents of a cistern, vessel, or space in gallons. Rule. — Divide the contents in cubic inches by 231 for liquid gallons, or by 268.8 for dry gallons. Brick and Stone Work Stonework is commonly estimated by the perch ; brickwork by the thousand bricks. (1) In estimating the work of laying stone, take the entire outside length in feet, thus measuring the corners twice, times the height in feet, times the thickness in feet, and divide by 24|, to obtain the number of perches. No allowance is to be made for openings in the walls unless specified in a written contract. (2) In estimating the material in stonework, deduct for all openings and divide the exact number of cubic feet of wall by 24|, to obtain the number of perches of material. To obtain the number of perches of stone, deduct ^ for mor- tar and filling. (3) In estimating the work of laying common bricks (com- mon bricks are 8 inches x 4 inches x 2 inches and 22 are as- sumed to build 1 cubic foot), take the entire outside length in feet, thus measuring the corners twice, times the height in feet, times the thickness in feet, and multiply by .022, to obtain the 260 MATHEMATICAL WRIKKLES number of thousand bricks. No allowance is to be made for openings in the walls unless specified in a written contract. (4) In estimating the material in brickwork, deduct for all openings and multiply the exact number of cubic feet of wall by 22, to obtain the number of brick required. Carpeting Carpets are usually either 1 yard or J yard in width. The amount of carpet that must be bought for a room de- pends upon the length and number of strips, and the waste in matching the patterns. (1) To obtain the number of strips. A fraction of a strip cannot be bought. Thus, if the num- ber of strips is found to be 6^, make it 7. (a) When laid lengthwise. — ^Divide the width of the room in yards by the width of the carpet in yards. (b) When laid crosswise. — Divide the length of the room in yards by the width of the carpet in yards. (2) To obtain the number of yards of carpet needed to car- pet a room. Rule. — Multiply the length of a strip in yards {-\- the fraction of a yard allowed for waste, when considered) by the number of strips. Casks and Barrels To find the contents in gallons. Rule. — Add to the head diameter (inside) two thirds of the difference between the head and b^ing diameters ; but if the staves are only slightly curved, add six tenths of this difference ; this gives the mean diameter ; express it in inches, square it, multiply it by the length in inches and this product by .008 J/. : the product will be the contents in liquid gallons. MENSURATION 261 Circle A circle is a portion of a plane bounded by a curved line every point of which is equally distant from a point within called the center. (1) Formulae. — Area = irr^, or \ ird^. Area = cPx .7854, or circumference^ X .07958. Circumference = diameter x 3.1416. Circumference = radius x 6.2832. Diameter = circumference x .31831. Diameter = circumference -^ tt. Radius = circumference x .159155. Radius = .56419 x Varea. Side of inscribed square = dx .707107. Side of inscribed square = circumference x .22508. Area of inscribed square = | cT'. Side of an equal square = circumference X .282. Area of an equal square = f cP. Side of inscribed equilateral triangle = d x .86. (2) Given the area inclosed by three equal circles, to find the diameter of a circle that will just inclose the three equal circles. Rule. — Divide the given area by .03473265, extract the square root of the quotient, and multiply by 2, and the result will be tlie diameter required. Formula. — Diameter =2\£ area .03473265 (3) To find the diameter of the three largest equal circles that can be inscribed in a circle of a given diameter. Rule. — Multiply the given diameter by .J^G^l, or divide by 2.1557, and the result will be the required diameter. D Formula. — d = .4641 x D, or 2.1557 262 MATHEMATICAL WRINKLES (4) Given the radius a, b, c, of the three circles tangent to each other, to find the radius of a circle tangent to the three circles. Formula. — r or r' = — ^^ ^ , the 2 V[a6c (a + 6 + c)] T (ab -f ac -f- be) minus sign giving the radius of a tangent circle circumscribing the three given circles, and the plus sign giving the radius of a tangent circle inclosed by the three given circles. — Fi'om " The School Visitor." (5) Given the chord of an arc and the radius of the circle, to find the chord of half the arc. Formula. — k= v 2 r^ — r V4 r^ — c^, where r = radius and c = the given chord. (6) Given the chord of an arc and the radius of the circle, to find the height of the arc. Rule. — From the radius, subtract the square root of the differ- ence of the squares of the radius and half the chord. Formula. — h = r — Vr^ — \ c^, where r = radius and c = the given chord. (7) Given the height of an arc and a chord of half the arc, to find the diameter of the circle. Rule. — Divide the square of the chord of half the arc by the height of the chord. Formula. — d = c^-r-hj where c = chord of half the arc and h = height. (8) Given a chord and height of the arc, to find the chord of half the arc. Rule. — Extract the square root of the sum of the squares of the height of the arc and half the chord. Formula. — A; = V^N-T^> where h = height and c = the given chord, MENSURATION 263 (9) Given the radius of a circle and a side of an inscribed polygon, to find the side of a similar circumscribed polygon. 2 sr Formula. — s' = ■ . where s' = the side required and 8 = the side of the inscribed polygon. Cone A cone is a solid bounded by a conical surface and a plane. (1) To find the lateral ar6a of a right circular cone. Rule. — Multiply the circumference of its base by half the slant height. Formula. — Lateral area = irrh, where r = the radius of the base and h = the slant height. (2) To find the volume of any cone. Rule. — Multiply the base by one third the altitude. Formula. — V=i aB. Formula, when base is a circle. — F= ^ ar^Tr, where a = alti- tude, B = base, and r = radius of the base. Crescent A crescent is a portion of a plane included between the cor- responding arcs of two intersecting circles, and is the difference between two segments having a common chord, and on the same side of it. Cube or Hexahedron Diagonal = V3 x edge', or Varea -h 2. Diagonal = edge x 1.7320508. Surface = 6 x edge*, or 2 x diagonal*. Volume = edge'. Cycloid A cycloid is the curve generated by a point in the circum- ference of a circle which rolls on a straight line. 264 MATHEMATICAL WRINKLES (1) To find the length of a cycloid. Rule. — Multiply the diameter of the generating circle by 4- (2) To find the area of a cycloid. Rule. — Multiply the area of the generating circle by 3. (3) To find the surface generated by the revolution of a cycloid about its base. Rule. — Multiply the area of the generating circle by -^^. (4) To find the volume of the solid formed by revolving the cycloid about its base. Rule. — Multiply the cube of the radius of the generating circle bydir", (5) To find the surface generated by revolving the cycloid about its axis. Rule. — Midtiply eight times the area of the generating circle by tr minus |. (6) To find the volume of the solid formed by revolving the cycloid about its axis. Rule. — Multiply | of the volume of a sphere whose radius is that of the generating circle by ^ ir^ — |. (7) To find the surface formed by revolving the cycloid about a tangent at the vertex. Rule. — Multiply the area of the generating circle by ^^. (8) To find the volume formed by revolving a cycloid about a tangent at the vertex. Rule. — Multiply the cube of the radius of the generating circle byl-r^. Cylinder A cylinder is a solid bounded by a cylindric surface and two parallel planes. (1) To find the lateral area of a right circular cylinder. Rule. — Multiply its length by the circumference of its base. MENSURATION 266 (2) To find the volume of any cylinder. Rule. — Multiply the altitude of the cylinder by the area of its base. Formula. — V= a x B. Formula when base is a circle. — V= airi^. (3) To find the surface common to two equal circular cylin- ders whose axes intersect at right angles. Rule. — Multiply the square of the radius of the intersecting cylinders by 16. (4) To find the volume common to two equal circular cylin- ders whose axes intersect at right angles. Rule. — Multiply the cube of the radius of the intersecting cylin- ders by iy\. (5) To find the length of the maximum cylinder inscribed in a cube, the axis of the cylinder coinciding with the diagonal of the cube. Formula. — Length = ^aV3, where a is the edge of the cube. (6) To find the volume of the maximum cylinder inscribed in a cube, the axis of the cylinder coinciding with the diagonal of the cube. Formula. — F=^?ra^V3, where a is the edge of the cube. Density of a Body The density of any substance is the number of times the weight of the substance contains the weight of an equal bulk of water. To find the density of a body. Rule. — Divide the weight in grams by the bulk in cubic cen- timeters. 266 MATHEMATICAL WBINKLES DODECAEDRON A dodecaedron is a polyedron of twelve faces. (1) To find the area of a regular dodecaedron. Rule. — Multiply the square of an edge by 20.6^578. (2) To find the volume of a regular dodecaedron. Rule. — Multiply the cube of an edge by 7.66312. Ellipse An ellipse is a plane curve of such a form that if from any point in it two straight lines be drawn to two given fixed points, the sum of these straight lines will always be the same. (1) To find the circumference of an ellipse, the transverse and conjugate diameters being known. Rule. — Multiply the square root of half the sum of the squares of the two diameters by 3.141592. (2) To find the area of an ellipse, the transverse and con- jugate diameters being given. Rule. — Multiply the product of the diameters by .785398. Frustum of a Cone or Pyramid A frustum of a cone or pyramid is the portion included be- tween the base and a parallel section. (1) To find the lateral surface. Rule. — Midtiply the sum of the perimeters, or circumferences, by one half the slant height. (2) To find the entire surface. Rule. — Add to the lateral surface the areas of both ends, or bases. (3) To find the volume of a frustum of a cone or pyramid. Rule. — To the sum of the areas of both bases add the square root of the product, and midtiply this sum by 07ie third of the altitude. MENSURATION 267 Grain and Hay (1) To find the quantity of grain in a bin. Rule. — Multiply the contents in cubic feet by .83 j and the result will be the contents in bushels. (2) To find the quantity of corn in a wagon bed or in a ^in. Rule. — (i) For shelled corny mxdtiply the contents in cubic feet by .83, and the result will be the contents in bushels. Rule. — (S) For com on the cob, deduct one half for cob. Rule. — (3) For com ,not ^^ shucked " deduct two thirds for cob and shuck. (3) To find the quantity of hay in a stack or rick. Rule. — Divide the contents in cubic feet by 550 for clover or by 450 for timothy; the quotient will be the number of tons. (4) In well-settled stacks 15 cubic yards make one ton. (5) When hay is baled, 10 cubic yards make one ton. Hexaedron (See Cube.) Hyperbola A hyperbola is a section formed by passing a plane through a cone in a direction to make an angle at the base greater than that made by the slant height. To find the area of a hyperbola, the transverse and conjugate axes and abscissa being given. Rule. — (1) To the product of the transverse diameter and absci-fsa add ^ of the square of the abscissa, and multiply the square root of the sum by 21. (2) Add 4 times the square root of the product of the trans- verse diameter and abscissa to the product last found, and divide the sum by 75. (3) Divide 4 times the product of the conjugate diameter and abscissa by the transverse diameter, and this last quotient multi- plied by the former will give the area required^ nearly. 268 MATHEMATICAL WRINKLES ICOSAEDRON An icosaedron is a polyedron of twenty faces. (1) To find the area of a regular icosaedron. Rule. — Multiply the square of an edge by 8.66025. (2) To find the volume of a regular icosaedron. Rule. — Multiply the cube of an edge by 2.18169. Irregular Polyedron To find the volume of any irregular polyedron. Rule. — Cut the polyedron into prismatoids by passing parallel planes through all its summits. Irregular Solids To find the volume of any irregular solid. Rule. — Immerse the solid in a vessel of water and determine the quantity of water displaced. Logs (1) To find the side of the squared timber that can be sawed from a log. Rule. — Multiply the diameter of the smaller end by .707. (2) To find the number of board feet in the squared timber that can be sawed from a log. Rule. — Multiply together one half the length in feet, the diameter of the smaller end in feet, and the diameter of the smaller end in inches. Problem. — Find the side, and the number of board feet, in the squared timber that can be sawed from a log whose length is 16 feet, and diameter of the smallest end 15 inches. Solution. — By (1) the side is 15 inches x .707, or 10.606 inches. By (2) the number of the board feet is \^- x ^f x 15 = 150, Ans. MENSURATION 269 Lumber When boards are 1 inch thick or less, they are estimated by the square foot of surface, the thickness not being considered. Thus a board 10 feet long, 1 foot wide, and 1 inch (or less) thick contains 10 square feet. Hence, to find the number of board feet in a plank. Rule. — Multiply the length in feet by the width in feet by the thickness in inches. Note. — The average width of a board that tapers uniformly is one half the sum of the end widths. LUNE A lune is that portion of a sphere comprised between two great semicircles. To find the area of a lune. Rule. — Multiply its angle in radians by twice the square of the radius. OCTAEDRON An octaedron is a polyedron of eight faces. (1) To find the area of a regular octaedron. Rule. — Multiply the square of an edge by S.j^G^I' (2) To find the volume of a regular octaedron. Rule. — Multiply the cube of an edge by .4714- Painting and Plastering Painting and plastering are usually estimated by the square yard. The processes of calculating the cost of painting and plastering vary so much in different localities that it is impos- sible to lay down any rule. Usually some allowance is made for doors, windows, etc., but there is no fixed rule as to how much should be deducted. Sometimes one half the area of the openings is deducted. 270 MATHEMATICAL WRINKLES Papering Wall paper is sold only by the roll, and any part of a roll is considered a whole roll. The amount of wall paper required to paper a room depends upon the area of the walls and ceiling and the waste in matching. (1) American paper is commonly 18 inches wide, and has 8 yards in a single roll, and 16 yards in a double roll. Foreign papers vary in width and length to the roll. (2) Wall paper is usually put up in double rolls, but the prices quoted are for single rolls. (3) Borders and friezes are sold by the yard and vary in width. (4) The area of a single roll is 36 square feet, and allowing for all waste in matching, etc., will cover 30 square feet of wall. (5) There is no fixed rule as to how much should be deducted for doors and windows. Some dealers deduct the exact area of the openings, while others deduct an approximate area, allowing 20 square feet for each. (6) The number of single rolls required for the ceiling and for the walls must be estimated separately. (7) To obtain the number of single rolls required for the ceiling. Rule. — Divide its area in square feet by 30. (8) To obtain the number of single rolls required for the walls. Rule. — From the area of the walls in square feet deduct the area of the openings, and divide by 30. Parabola A parabola is the locus of a point whose distance from a fixed point is always equal to its distance from a fixed straight line. MENSURATION 271 (1) To find the length of any arc of a parabola cut off by a double ordinate. Rule. — When the abscissa is less than half the ordinate : To t/ie square of the ordinate add J of the square of the abscissa, and twice the square root of the sum will be the length of the arc. (2) To find the area of the parabola, the base and height being given. Rule. — Multiply the bcLse by the heighty and ^ of the product will be the area. (3) To find the area of a parabolic frustum, having given the double ordinates of its ends and the distance between them. Rule. — Divide the difference of the cubes of the two ends by the difference of their squares and multiply tlie quotient by J of the altitude. Parallelogram A parallelogram is a quadrilateral whose opposite sides are parallel. To find the area of any parallelogram. Rule. — Multiply the base by the altitude. Parallelopiped A parallelopiped is a prism whose bases are parallelograms. To find the volume of any parallelopiped. Rule. — Multiply its altitude by the area of its base. Prism A prism is a polyedron whose ends are equal and parallel polygons, and its sides parallelograms. (1) To find the lateral area of a prism. Rule. — Multiply a lateral edc/e by the perimeter of a right sec- tion. 272 . MATHEMATICAL WEINKLES (2) To find the volume of any prism. Rule. — Multiply the area of the base by its altitude. Prismatoid A prismatoid is a polyedron whose bases are any two poly- gons in parallel planes, and whose lateral forces are triangles determined by so joining the vertices of these bases that each lateral edge with the preceding forms a triangle with one side of either base. (1) To find the volume of any prismatoid. Rule. — Add the areas of the two bases and four times the mid c7^oss section; multiply this sum by one sixth the altitude. Old Prismoidal Formula. — (2) To find the volume of a prismatoid, or of any solid whose section gives a quadratic. Rule. — Multiply one fourth its altitude by the sum of one ba^e and three times a section distant from that base two thirds the altitude. New Prismoidal Formula. — V=-(B + 3T). — From Halsted's "Metrical Geometry." Pyramid A pyramid is a polyedron of which all the faces except one meet in a point. (1) To find the lateral area of a regular pyramid. Rule. — Multi2)ly the perimeter of the base by half the slant height. (2) To find the volume of any pyramid. Rule. — Multiply the area of the base by one third of the altitude. MENSURATION 273 Pyramid, Spherical A spherical pyramid is the portion of a sphere bounded by a spherical polygon and the planes of its sides. Rule. — Multiply the area of the hose by one third of the radius of the sphere. Note. — The area of a spherical polygon Is equivalent to a lune whose angle is half the spherical excess .of the polygon. Quadrilateral A quadrilateral is a polygon of four sides. To find the area of any quadrilateral. Rule. — Multiply half the diagonal by the sum of the perpen- diculars upon it from the opposite angle. Rhombus A rhombus is a parallelogram whose sides are all equal and whose angles are oblique. To find the area of a rhombus. Rule. — Take half the product of Us diagonals. Rings If a plane curve lying wholly on the same side of a line in its own plane revolves about that line, the solid thus generated is called a ring. (1) Theorem of Pappus. (a) If a plane curve lying wholly on the same side of a line in its own plane revolves about that line, the area of the solid thus generated is equal to the product of the length' of the re- volving line and the path described by its center of mass. (6) If a plane figure lying wholly on the same side of a line in its own plane revolves about that line, the volume of the solid thus generated is equal to the product of the revolving area and the length of the path described by its center of mass. 274 MATHEMATICAL WRINKLES (2) To find the surface of an elliptic ring. Formula. — Surface = 2 tt^ c V|((2a)^ + (26)2). (3) To find the volume of an elliptic ring. Formula. — Volume = 2 -n-^abc, where 2 a and 2 b are the axes of the ellipse and c the distance of the center of the ellipse from the axis of rotation. (4) To find the surface of a cylindric ring. Formula. — Surface = 4 ii^ra. (5) To find the volume of a cylindric ring. Formula. — Volume = 27rVa, where a = distance of the center of the generating curve from the axis of rotation, and r = the radius of the circle. Roofing and Flooring A square 10 feet on a side, or 100 square feet, is the unit of measure in roofing, tiling, and flooring. The average shingle is taken to be 16 inches long and 4 inches wide. Shingles are usually laid about 4 inches to the weather. When laid 4^ inches to the weather, the exposed surface of a shingle is 18 square inches. Allowing for waste, about 1000 shingles are estimated as needed for each square, but if the shingles are good, 850 to 900 are sufficient. There are 250 shingles in a bundle. Sector A sector is that portion of a circle bounded by two radii and the intercepted arc. To find the area of a sector. Rule. — (a) Multiply the length of the arc by half the radius. (b) If the arc is given in degrees, take such a part of the whole area of the circle as the number of degrees in the arc is of 360°. MENSURATION 275 Sector, A Spherical A spherical sector is the volume generated by any sector of a semicircle which is revolved about its diameter. To find the volume of a spherical sector. Rule. — Multiply the area of its zone by one third the radius. Formula. — V= | irar^y where r = radius of the sphere and a = altitude of the spherical segment. Segment of Circle A segment of a circle is the portion of a circle included be- tween an arc and its chord. (1) To find the area of a segment less than a semicircle. Rule. — From the sector having the same arc a.9 the segment subtract the area of the triangle formed by the chord and the two radii from its extremities. (2) An approximate rule for finding the area of a segment. Rule. — Take two thirds the product of its chord and height. (3) To find the area of a segment of a circle, having given the chord of the arc and the height of the segment, i.e. the versed sine of half the arc. Rule. — Divide the cube of the height by twice the base and in- crease the quotient by two thirds of the product of the height and base. (4) To find the volume of the solid generated by a circular segment revolving about a diameter exterior to it. Rule. — Multii)ly one sixth the area of the circle tvhose radius is the chord of the segment by the projection of that chord upon the axis. Formula. — F= ; TT^LB'-^ X ^I'B', where AB is the chord of the segment and A'B' is its projection upon the axis. 276 MATHEMATICAL WRINKLES Segment, A Spherical A spherical segment is a portion of a sphere contained be- tween two parallel planes. To find the volume of any spherical segment. Rule. — To the product of one half the sum of its bases by its altitude add the volume of a sphere having that altitude for its diameter. Shell, A Cylindric A cylindric shell is the difference between two circular cylinders of the same length. To find the volume of a cylindric shell. Rule. — Multiply the sum of the inner and outer radii by their difference, and this product by ir times the altitude of the shell. Shell, A Spherical A spherical shell is the difference between two spheres which have the same center. To find the volume of a spherical shell. Formula. — V= f ir (r^^ — r^), where rj and r denote the radii. Similar Solids Similar solids are solids which have the same form, and dif- fer from each other only in volume. Rule. — Any two similar solids are to each other as the cubes of any two like dimensions. Similar Surfaces Similar surfaces are surfaces which have the same shape, and differ from each other only in size. Rule. — Any two similar surfaces are to each other as the squares of any two like dimensions. MENSURATION 277 Sphere A sphere is a closed surface all points of which are equally distant from a fixed point within called the center. (1) Formulae. — Area = 4 ttt^, or vd^. Area = 7^x12.5664. Area = d2x 3.1416. Area = circumference' x .3183. Volume = I nr^f or J 7rd\ Volume = J d X area. Volume = circumference'' x .0169. Volume = r« x 4.1888, or d» x .5236. (2) Side of an inscribed cube ( r X 1.1547, = -J or ( d X .5774. (3) To find the edge of the largest cube that can be cut from a hemisphere. Formula. — Edge =d x .408248. (4) To find the volume of a frustum of a sphere, or the por- tion included between two parallel planes. Rule. — To three times the sum of the squared radii of the two ends add the square of the altitude ; multiply this sum by .5235987 times the altitude. (5) To find the edge of the largest cube that can be inscribed in a hemisphere of given diameter. Rule. — Multiply the radius by ^ of the square root of 6, Spheroid A spheroid is a solid formed by revolving an ellipse about one of its axes as an axis of revolution. Spheroid, Oblate An oblate spheroid is the spheroid formed by revolving an ellipse about its conjugate diameter as an axis of revolution. ^ 278 MATHEMATICAL WRINKLES To find the volume of an oblate spheroid. Rule. — Multiply the square of the semitransverse diameter by the semiconjugate diameter and this product by ^ tt. Spheroid, Prolate A prolate spheroid is the spheroid formed by revolving an ellipse about its transverse diameter as an axis of revolution. To find the volume of a prolate spheroid. Rule. — Multiply the square of the semiconjugate diameter by the semitransverse diameter and this product by ^ ir. Spindle, A Circular A circular spindle is the solid formed by revolving the seg- ment of a circle about its chord. (1) To find the volume of a circular spindle. Rule. — Multiply the area of the generating segment by the path of its center of gravity. (2) To find the volume formed by revolving a semicircle about a tangent parallel to its diameter. Rule. — Multiply one fourth of the volume of a sphere ivhose radius is that of the generating semicircle by (10 — S ir). Spindle, A Parabolic A parabolic spindle is a solid formed by revolving a parabola about a double ordinate perpendicular to the axis. To find the volume of a parabolic spindle. Rule. — Multiply the volume of its circumscribed cylinder by ^. Square A square is a rectangle whose sides are all equal, (1) To find the area of a square. Rule. — /Square an edge. MENSURATION 279 (2) Given the diagonal, to find the area. Rule. — Take one half the square of the diagonal. (3) Given the diagonal, to find a side. Rule. — Extract the square root of one half the square of the diayonal. (4) To find the side of the largest square inscribed in a semicircle of given diameter. Rule. — Multiply the radius of the given circle by | of the square root of 5. Tetraedron A tetraedron is a polyedron of four faces. (1) To find the surface of a tetraedron. Rule. — Mtdtiply the square of an edge by V^, or 1.73205. (2) To find the volume of a tetraedron. Rule. — Multiply the cube of an edge by ■^'^2^ or .11785. Trapezium and Irregular Polygons To find the area of a trapezium or any irregular polygon. Rule. — Divide the figure into triangles^ find the area of the triangles, and take their sum. Trapezoid A trapezoid is a quadrilateral two of whose sides are par- allel. (1) To find the area of a trapezoid. Rule. — Multiply the altitude by one half the sum of the parallel sides. (2) Width = area -t- (^ of the sum of the parallel sides). (3) Sura of the parallel sides = (area -^ width) x 2. (4) To find the length of a line parallel to the bases of a trapezoid that shall divide it into equal areas. 280 MATHEMATICAL WRINKLES Rule. — Square the bases and extract the square root of half their sum. Triangle A triangle is a portion of a plane bounded by three straight lines. (1) To find the area of a triangle. Rule. — Multiply the base by half the altitude. (2) To find the area of a triangle, having given the three sides. Rule. — From half the sum of the three sides subtract each side separately; multiply half the sum and the three remainders to- gether : the square root of the product will be the area. (3) To find the radius of the inscribed circle. Rule. — Divide the area of the triangle by half the sum of its sides. (4) To find the radius of the circumscribing circle. Rule. — Divide the product of the three sides by four times the area of the triangle. (5) To find the radius of an escribed circle. Rule. — Divide the area of the triangle by the difference between half the sum of its sides and the tangent side. (6) To cut off a triangle containing a given area by a line running parallel to one of its sides, having given the area and base. Rule. — The area of the given triangle is to the area of the tri- angle to be cut off, as the square of the given base is to the square of the required base. Extract the square root of the result. (Equilateral) Triangle (1) Area = one half the side squared and multiplied by V3, or 1.732050+. MENSURATION 281 (2) Altitude = one half the side multiplied by V3, or 1.732050^. (3) Center of the inscribed and circumscribed circle is a point in the altitude one third of its length from the base. (4) Radius of the circumscribed circle = two thirds of the altitude. (5) Radius of the inscribed circle = one third of the altitude. (6) Side = 2 Varea^W3. Side = radius of the circumscribed circle multiplied by vs. (7) All equilateral triangles are similar. (8) Each angle = 60°. (Right) Triangle (1) B&ae^y/ih'-f). (2) Perpendicular = V(/i* - 6*). (3) Hypotenuse =V6N-p. (4) Diameter of inscribed circle = (b+p) — h. (6) Side opposite an acute angle of 30° = one half of the hypotenuse. (6) Similar, if an acute angle of one = an acute angle of another. (7) Altitude of an isosceles triangle forms two right triangles. (8) To find a point in a right-angled triangle equidistant from its vertices. Rule. — Divide the hypotenuse by 2; the point will lie in the hypotenuse. (9) To find the perpendicular height of a right triangle when the base and the sum of the perpendicular and hypotenuse are known. 282 MATHEMATICAL WRINKLES Rule. — From the square of the sum of the perpendicular and hypotenuse take the square of the base, and divide the difference by twice the sum of the perpendicular and hypotenuse. (Spherical) Triangle A spherical triangle is a spherical polygon of three sides. To find the area of a spherical triangle. Rule. — Find the area of a lune whose angle is half the spheri- cal excess of the triangle. Note. — The spherical excess of a triangle is the excess of the sum of its angles over 180°. Ungula, a Conical A conical ungula is a portion of a cone cut off by a plane oblique to the base and contained between this plane and the base. To find the volume of a conical ungula, when the cutting plane passes through the opposite extremes of the ends of the frustum. Rule. — Multiply the difference of the square roots of the cubes of the radii of the bases by the square root of the cube of the radius of the lower base and this product by ^tt times the altitude. Divide this last product by the difference of the radii of the two bases, and the quotient will be the volume of the ungida. Ungula, A Cylindric A cylindric ungula is any portion of a cylinder cut off by a plane. (1) To find the convex surface of a cylindric ungula, when the cutting plane is parallel to the axis of the cylinder. Rule. — Multiply the arc of the base by the altitude. (2) To find the volume of a cylindric ungula whose cutting plane is parallel to the axis. MENSURATION 283 Rule. — Multiply the area of the base by the altitude. (3) To find the convex surface of a cylindric ungula, when the plane passes obliquely through the opposite sides of the cylinder. Rule. — Multiply the circumference of the base by half the sum of the greatest and least lengths of the ungula. (4) To find the volume of a cylindric ungula, when the plane passes obliquely through the opposite sides of the cylinder. Rule. — Multiply the area of the base by half the least and greatest lengths of the ungula. Ungula, A Spherical A spherical ungula is a portion of a sphere bounded by a lune and two great semicircles. To find the volume of a spherical ungula. Rule. — Multiply the area of the lune by one third the radius; OTy multiply the volume of the sphere by the quotient of the angle of the lune divided by 360°. Wedge A wedge is a prismatoid whose lower base is a rectangle, and upper base a sect parallel to a basal edge. To find the volume of any wedge. Rule. — To twice the length of the ba^e add the opposite edge; mtdtiply the sum by the width of the base, and this product by one sixth the altitude of the wedge. Wood Measure The unit of wood measure is the cord. The cord is a pile of wood 8 feet by 4 feet by 4 feet. A pile of wood 1 foot by 4 feet by 4 feet is called a cord foot. 284 MATHEMATICAL WRINKLES A cord of stove wood is 8 feet long by 4 feet high. The length of stove wood is usually 16 in. Zone A zone is the curved surface of a sphere included between two parallel planes or cut off by one plane. (1) To find the area of a zone. Rule. — Multiply the altitude of the spherical segment by tivice IT times the radius of the sphere. (2) To find the area of a zone of one base. Rule. — Tlie area of a zone of one base is equivalent to the area of a circle whose radius is the chord of the generating arc. (Circular) Zone A circular zone is the portion of a plane inclosed by two parallel chords and their intercepted arcs. (1) If both chords are on the same side of the center. Rule. — Find the difference between the areas of the tioo seg- ments. (2) If the chords are on opposite sides of the center. Rule. — Subtract the sum of the areas of the two segments from the area of the circle. MISCELLANEOUS HELPS 1. Pi (tt) = 3.1416, or 3|. Its value to seven hundred and seven places is 3.14159265358979323846264338327950288419716939937510582 09749445923078164062862089986280348253421170679821480 86513282306647093844609550582231725359408128481117450 28410270193852110555964462294895493038196442881097566 59334461284756482337867831652712019091456485669234603 48610454326648213393607260249141273724587006606315588 17488152002096282925409171536436789259036001133053054 88204665213841469519415116094330572703657595919530921 86117381932611793105118548074462379834749567351885762 72489122793818301194912983367336244193664308602139501 60924480772309430285530966202755693979869502224749962 06074970304123668861995110089202383770213141694119029 88582544681639799904659700081700296312377381342084130 791451183980570985. 2. The contents of a spheroid equals the square of the re- volving axis X the fixed axis x .5236. 3. To find the distance a spot on the tire of a revolving wheel moves, multiply the distance traveled by 4 and divide by TT. 4. Sound travels 1087 feet per second at 0** C. or 1126 feet per second at 20° C. 6. Electricity travels about 186,000 miles per second. 6. To find the approximate number of bushels of corn in a crib, take the dimensions in feet, and multiply their product 286 286 MATHEMATICAL WEINKLES by .8, if the corn is shelled ; by A, if shucked ; by .3, if in the shuck. 7. Eoofing, flooring, and slating are often estimated by the square, which contains 100 square feet. 8. The long ton of 2240 pounds and the long hundredweight of 112 pounds are used in the United States custom houses and in weighing coal and iron in the mines. 9. The term carat is sometimes used to express the fineness of gold, each carat meaning a twenty -fourth part. 10. It takes 1000 shingles to cover 100 square feet laid 4 inches to the weather. It takes 900 shingles to cover 100 square feet laid 4|- inches to the weather. 11. The area of an ellipse is a mean proportional between the circumscribed and inscribed circles. 12. Gunter's chain is 66 feet long, consisting of 100 links. 13. The first 24 periods of numeration are — units, thousands, millions, billions, trillions, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, undecillions, duode- cillions, tredecillions, quartodecillions, quintodecillions, sexde- cillions, septodecillions, octodecillions, nonodecillions, vigin- tillions, primo-vigintillions, and secundo vigintillions. 14. Mathematicians have given the signs X and -;- precedence over the signs + and — ; hence the operations of multiplication and division should always be performed before addition and subtraction. 15. The true weight of an article weighed on false scales is a mean proportional between the two apparent weights. 16. To find any term of an arithmetical progression. Rule. — Any ter^n of an arithmetical series is equal to the first term, increased or diminished by the common difference multiplied by a number one less than the number of terms. MISCELLANEOUS HELPS- 287 17. To find the sum of an arithmetical series. Rule. — Multiply half the sum of the extremes by the number of terms. 18. To find any term of a geometrical series. Rule. — Multiply the first term by the ratio raised to a power one less than the number of terms. 19. To find the sum of a geometrical series. Rule. — Multiply the greater extreme by the ratio, subtract the less extreme from the product, and divide the remainder by the ratio less 1. 20. To sum a geometrical series to infinity. Rule. — When the ratio is a proper fraction, divide the first term by 1 less the ratio. 21. To find the harmonic mean between two numbers. Rule. — Divide twice their product by their sum. 22. To find the mean proportional between two numbers. Rule. — Take the square root of their product. 23. A body immersed in a liquid is buoyed up by a force equal to the weight of the liquid displaced. That is, it loses a portion of its weight just equal to the weight of the water dis- placed. 24. If we have the sum and difference of two numbers given, add the sum and difference and take half of it for the greater, subtract and take half of it for the smaller. 25. To find the day of the week for any date. Rule. — To the given year of the century add its \, neglecting remainder ; to this add the day of the month, the raJtio of the cen- tury, and the ratio of the month; then divide by 7, and the re- mainder will be the number of the day of tJie week, counting Sunday 1st, Monday 2d, and so on. Monthly Ratios January- = 3 or 2. August = 5. February = 6 or 5. September = 1. March = 6. October = 3. April = 2. November = 6. May = 4. December = 1. June = 0. In leap years July = 2. Jan. = 2. reb.:= 288 MATHEMATICAL WRINKLES Centennial Ratio 200, 900, 1800, 2200 = 0. 300, 1000 . . . . = 6. 400, 1100, 1900, 2300 = 5. 500, 1200, 1600, 2000 = 4. 600, 1300 . . . . = 3. 700, 1400, 1700, 2100 = 2. 100, 800, 1500 . . =1. Examples. — March 4, 1877, was on [77 -}- 19 + 4 +0 + 6]^ 7, remainder 1 = Sunday. Jan. 31, 1845 was on [45 + 11 + 31 + + 3] ^ 7, remainder 6 = Friday. Oct. 12, 1492, was on [92 + 23 + 12 + 2 + 3] -7- 7, remainder 6 = Friday. Leap years are known by being divisible by 4, except those centuries that can- not be divided by 400 ; hence 1900 was not a leap year. 26. To find the day's length at any latitude (for example, 71° N. Lat.). Let t be the time before 6 o'clock for sunrise ; then the length of the day is (2 1 plus 12) hours. If d be the sun's declination and I the latitude, then sin \ t equals cot (90° — T) tan d. For longest day d equals 23° 27', and I equals 71°. Therefore, sin \t equals cot 19° tan (23° 27'). \t must be expressed in degrees. log cot 19° = 10.463028 log tan (23° 27')= 9.637265 log i^ = 10.100293 As the logarithm of the sine of an angle cannot be greater than 10, this shows that the person's latitude is within the limits of the Arctic circle, and on the longest day there the sun does not rise and set. — From " The School Visitor." 27. To find the G. C. D. of fractions. Rule. — Find the G. C. D. of the iiumerators of the fractions, and divide it by the L. C. M. of their denominators. 28. To find the L. C. M. of fractions. Rule. — Divide the L. C M. of the numerators by the O. C D. of the denominators. MISCELLANEOUS HELPS 289 29. To find the height of a stump of a broken tree. Rule. — From the square of the height of the tree subtract the square of the distance the top rests from the base of the tree, and divide the remainder by twice the height of the tree. 30. To find how many board feet in a round log. Rule. — Subtract 4 from the diameter of the log in inches, and the square of this remainder equals the number of board feet in a log 16 feet long. 31. To find the velocity of a nailhead in the rim of a mov- ing wheel. Rule. — Divide tivice the height of the nailhead above the plane tipon which the wheel rolls, by the radius, and multiply this product by the velocity of the center; then extract the square root. Note. — Its velocity at the bottom is zero ; at the top, twice that of the center ; and when its height is half the radius, its velocity equals that of the center. 32. To find the distance to the horizon. Rule. — Take one and one half times the height the observer is above the s^irface of the ground in feet. TJie square root of this number is the number of miles an object on the surface can be seen, 33. Extraction of any root. Horner^s Method, invented by Mr. Horner, of England, is the best general method of extracting roots. Any root whose index contains only the factors 2 or 3 can be extracted by means of the square and cube root. Rule. — I. Divide the number into periods of as many figures each as there are units in the index of the root, and at the left of the given number arrange the same number of columns, ivriting 1 at the head of the left-hand column and ciphers at the head of the ^ others. II. Find the required root of the first period, for the first figure of the root, multiply the number in the 1st col. by this first term of the root and add it to the 2d col., multiply this sum by the root and add it to the 3d col., and thus continue, writing the last prod- 290 MATHEMATICAL WRINKLES uct U7ider the first period; subtract and bring down the next period for a dividend. III. Repeat this process, stopping one column sooner at the right each time until the sum falls in the 2d col. Then divide the dividend by the number in the last column, which is the trial divisor; the result is the second figure of the root. IV. Use the second figure of the root precisely as the first, remembering to place the products one place to the right in the 2d col., two in the 3d col., etc.; continue this operation until the root is completed or carried as far as desired. Notes. — 1. Only a part of the dividend is used for finding a root figure, according to the principle of place value. The partial dividend thus used always terminates w^ith the first figure of the period annexed. 2. If any dividend does not contain the trial divisor, place a cipher in the root, and bring down the next period ; annex one cipher to the last term of the 2d column, two ciphers at the last term of the 3d, three to the 4th, and then proceed according to the rule. Example. — Extract the fourth root of 5636405776. OPERATION 2 4 8 12 12 (1) 8 24 32 t. d. 21063 56-3640.5776(274 16 2 4 2 i03640 6 (1) 24 609 3009 658 (2) 53063 T. D. 25669 78732 t. d. 1766944 371441 2 (1) 8 7 321995776 87 3667 707 80498944 x. d. 321995776 7 94 7 (2) 4374 4336 101 7 (2) 108 441736 4 1084 — From Brooks' "Higher Arithmetic." MISCELLANEOUS HELPS 291 SCIENTIFIC TRUTHS 1. The intensity of light varies inversely as the square of the distance from the source of illumination. 2. The intensity of sound varies inversely as the square of the distance from the source of the sound. 3. Gravitation varies inversely as the square of the distance between the centers of gravity. 4. The heating effect of a small radiant mass upon a dis- tant object varies inversely as the square of the distance. MATHEMATICAL DEFINITIONS Algebra is that branch of mathematics in which mathemat- ical investigations and computations are made by means of letters and other symbols. Analytical Geometry is that branch of geometry in which the properties and relations of geometrical magnitudes are investi- gated by the aid of algebraic analysis. Analytical Trigonometry is that branch of trigonometry which treats of the properties and relations of the trigonometrical functions. Applied, or Mixed, Mathematics is the application of pure mathematics to the mechanic arts. Arithmetic is the science that treats of numbers, the methods of computing by them, and their applications to business and science. Astronomy is that branch of applied mathematics in which mathematical principles are used to explain astronomical facts. Calculus is that branch of algebraic analysis which com- mands, by one general method, the most difficult problems of geometry and physics. Calculus of Variations is that branch of calculus in which the 292 MATHEMATICAL WRINKLES laws of dependence which bind the variable quantities together are themselves subject to change. Conic Sections is that branch of Platonic geometry which treats of the curved lines formed by the intersection of the surface of a right cone and a plane. Descriptive Geometry is that branch of geometry which treats of the graphic solutions of all problems involving three dimen- sions by means of projections upon auxiliary planes. Differential Calculus is that branch of calculus which investi- gates mathematical questions by using the ratio of certain indefinitely small quantities called differentials. Geometry is the science which treats of the properties and relations of space. Gunnery is that branch of applied mathematics which treats of the theory of projectiles. Integral Calculus is that branch of calculus which determines the relations of magnitudes from the known differentials of these magnitudes. It is the reverse method of the differential calculus. Mathematics is that science which treats of the measurement of and exact relations existing between quantities and of the methods by which it draws necessary conclusions from given premises. Mechanics is that branch of applied mathematics which treats of the action of forces on material bodies. Mensuration is that branch of applied mathematics which treats of the measurement of geometrical magnitudes. Metrical Geometry is that branch of geometry which treats of the length of lines and the magnitudes of angles, areas, and solids. Navigation is that branch of applied mathematics which treats of the art of conducting ships or vessels from one place to an- other. MISCELLANEOUS HELPS 293 Optics is that branch of applied mathematics which treats of the laws of light. Plane Geometry is that branch of pure geometry which treats of figures that lie in the same plane. Plane Trigonometry is that branch of trigonometry which treats of the solution of plane triangles. Platonic Geometry is that branch of metrical geometry in which the argument, or proof, is carried forward by a direct in- spection of the figures themselves, or pictured before the eye in drawings, or held in the imagination. Pure Geometry is that branch of Platonic geometry in which the argument, or proof, uses compasses and ruler only. Pure Mathematics treats of the properties and relations of quantity without relation to material bodies. Quaternions is that branch of algebra which treats of the relations of magnitude and position of lines or bodies in space by means of the quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadri- nomial form. Solid Geometry, or Geometry of Space, is that branch of pure geometry which treats of figures which do not lie wholly within the same plane. Spherical Trigonometry is that branch of trigonometry which treats of the solution of spherical triangles. Surveying is that branch of applied mathematics which teaches the art of determining and representing areas, lengths and direc- tions of bounding lines, and the relative position of points upon the earth's surface. Trigonometry is that branch of Platonic geometry which treats of the relations of the angles and sides of triangles. 294 MATHEMATICAL WRINKLES HISTORICAL NOTES The oldest known mathematical work, a papyrus manuscript deciphered in 1877, and preserved in the British Museum, was written by Ash-mesu (the moon-born), commonly called Ahmes, an Egyptian, sometime before 1700 b.c. This work was entitled " Directions for obtaining the Knowledge of All Dark Things." This work contains problems in arithmetic and geometry and contains the first suggestions of algebraic notation and the solution of equations. This work was founded on another work believed to date back as far as 3400 b.c. Pythagoras, who died about 580 e.g., raised mathematics to the rank of a science. He was one of the most remarkable men of antiquity. The study of geometry was introduced into Greece about 600 B.C. by Thales. Thales founded a school of mathematics and philosophy at Miletus, known as the Ionic School, Euclid's " Elements," tlie greatest textbook on geometry, was published about 300 b.c. Euclid taught mathematics in the great university at Alexandria, Egypt. The name Mathematics is said to have first been used by the Pythagoreans. About 440 B.C. Hippocrates of Chios wrote the first Greek textbook on geometry. To the great philosophic school of Plato, which flourished at Athens (429-348 b.c), is due the first systematic attempt to create exact definitions, axioms, and postulates, and to distin- guish between elementary and higher geometry. Diophantus, who died about 330 a.d., was the first writer on algebra worthy of recognition. His "Arithmetical is the earliest treatise on algebra now extant. He was the first to state that " a negative number multiplied by a negative number gives a positive number." MISCELLANEOUS HELPS 295 Al Hovarezmi, who died about 830, published the first book known to contain the word " algebra " in the title. The first edition of Euclid was printed in Latin in 1482, and the first one in English appeared in 1570. Robert Recorde published the first arithmetic printed in the English language in 1540. The first arithmetic published in America was written by Isaac Greenwood and issued in 1729. Chauncey Lee published in 1797 an arithmetic called " The American Accomptant." This work contains the dollar mark, though in much ruder form than the character now in use. Descartes, the French philosopher, invented the method of computing graphs from equations about 1637. On June 8, 1637, he published the first analytical geometry. The differential calculus was invented by Newton and Leibniz about 1670. In 1686 Leibniz published in a paper, " The Acta Erudi- torum," the rudiments of the integral calculus. Hipparchus, who lived sometime between 200 and 100 b.c, was the greatest astronomer of antiquity and originated the science of trigonometry. The symbols of the Hindu or Arabic notation, except the zero, originated in India before the beginning of the Christian era. The zero appeared about 500 a.d. Nearly 4000 years ago Ahmes solved problems involving the area of the circle and found results that gave ir = 3.1604. The Babylonians and Jews used tt = 3. The Romans used 3 and sometimes 4, or for more accurate work S^. About 500 a.d. the_Hindus used 3.1416. The Arabs about 830 a.d. used Vj VlO, 3.1416. In 1596 Van Ceulen computed v to over 30 deci- mal places. In 1873 Shanks computed v to 707 decimal places. Logarithms were invented by John Napier, of Scotland, about 1614 A.D. His logarithms were not of ordinary numbers, 296 MATHEMATICAL WRINKLES but of the ratios of the legs of a right-angled triangle to the hypotenuse. Later Briggs constructed tables of logarithmic numbers to the base 10. The first publication of Briggian logarithms of trigonometric functions was made in 1620 by Gunter. Gunter was a colleague of Briggs. He invented the words cosine and cotangent, and found the logarithmic sines and tangents for every minute to seven places. HISTORICAL NOTES ON ARITHMETIC " The Science of Arithmetic is one of the purest products of human thought. Based upon an idea among the earliest which spring up in the, human mind, and so intimately associated with its commonest experience, it became interwoven with man's simplest thought and speech, and was gradually un- folded with the development of the race. The exactness of its ideas, and the simplicity and beauty of its relations, attracted the attention of reflective minds, and made it a familiar topic of thought ; and, receiving contributions from age to age, it continued to develop until it at last attained to the dignity of a science, eminent for the refinement of its principles and the certitude of its deductions. " The science was aided in its growth by the rarest minds of antiquity, and enriched by the thought of the profoundest thinkers. Over it Pythagoras mused with the deepest enthu- siasm; to it Plato gave the aid of his refined speculations; and in unfolding some of its mystic truths, Aristotle employed his peerless genius. In its processes and principles shines the thought of ancient and modern mind — the subtle mind of the Hindu, the classic mind of the Greek, the practical spirit of the Italian and English. It comes down to us adorned with the offerings of a thousand intellects, and sparkling with the MISCELLANEOUS HELPS 297 gems of thought received from the profoundest minds of nearly every age." — From Brooks' "Philosophy of Arithmetic." The first step in the historical development of arithmetic was in counting things. How far back this operation dates is not known. Counting among primitive people was of a very elementary nature, as it is now among people of a low grade of civilization. A knowledge of arithmetic is coeval with the race. Every people, no matter how uncivilized, has some crude knowledge of numbers and employs them in its transactions with one another. Some of them have no real numeral words, while others have very few. The Chiquitos of Bolivia have no real numerals. The Campas of Peru have only three, but can count to ten. The Bushmen of South Africa have but two numerals. The natives of Lower California can- not count above five. Very few of the Esquimos can count above five. The more intelligent can count to twenty or more. The Egyptians stand at the beginning of the first period in the historical development of arithmetic. Menes, their first king, changed the course of the Nile, made a great reservoir, and built the temple of Phthah at Memphis. They built the pyramids at a very early period. Surely a people who were en- gaged in enterprises of such magnitude must have known some- thing of mathematics — at least of practical arithmetic. To them all Greek writers are unanimous in ascribing, without envy, the priority of invention in the mathematical sciences. Aristotle says that mathematics had its birth in Egypt, be- cause there the priestly class had the leisure needful for the study of it. In Herodotus we find this (lie 109): "They said also that this king (Sesostris) divided the land among all Egyptians so as to give each one a quadrangle of equal size and to draw from each his revenues, by imposing a tax to be levied yearly. But every one from whose part the river tore away anything, had to go to him and notify what had happened ; he then sent the overseers, who had to measure out by how much 298 MATHEMATICAL WRINKLES the land had become smaller, in order that the owner might pay on what was left, in proportion to the entire tax imposed. In this way, it appears to me, geometry originated." One of the oldest known works on mathematics, a manuscript copied on papyrus, a kind of paper used about the Mediter- ranean in early times, is still preserved and is now in the Brit- ish Museum. It was deciphered in 1877 and found to be a mathematical manual containing problems in arithmetic and geometry. It was written by Ahraes sometime before 1700 b.c, and was founded on an older work believed to date back as far as 3400 B.C. This work is entitled " Directions for obtaining the Knowledge of All Dark Things." In the arithmetical part it teaches operations with whole numbers and fractions. Some problems in this papyrus seem to imply a rudimentary knowl- edge of proportion. The area of an isosceles triangle, of which the sides measure 10 ruths and the base 4 ruths, is erroneously given as 20 square ruths, or half the product of the base by one side. The area of a circle is found by deducting from the diameter -i of its length and squaring the remainder, tt is taken = Q^^-f = 3.1604. According to Herodotus the ancient Egyptian computation consisted in operating with pebbles on a reckoning board whose lines were at right angles to the user. There is reason to believe the Babylonians used a similar device. The earli- est Greeks, like the Egyptians and Eastern nations, counted on the fingers or with pebbles. The E-omans employed three methods, reckoning upon the fingers, upon the abacus (a me- chanical contrivance with columns for counters), and by tables prepared for the purpose. The method of finger reckoning seems to have prevailed among savage tribes from the be- ginning of time, and every observer knows how exceedingly common its use is among children learning to count. They perhaps adopt this method instinctively. MISCELLANEOUS HELPS 299 The Egyptiajis used the decimal scale. The Greeks and Egyptians made exclusive use of unit fractions, or fractions having one for the numerator. They kept the numerator con- stant and dealt with variable denominators. The Babylonians kept the denominators constant and equal to 60. Also the Romans kept them constant, but equal to 12. The Greeks also had much to do with the advancement of mathematics. They discriminated between the science of numbers and the art of calculation. They were among the first writers on arithmetic. About twenty-five centuries ago Pythagoras classified numbers into perfect and imperfect, even and odd, solid, square, cubical, etc. " He regarded num- bers as of divine origin — the fountain of existence — the model and archetype of things — the essence of the universe." He regarded even numbers as feminine, and allied to the earth ; odd numbers were supposed to be endued with mascu- line virtues, and partook of the celestial nature. He consid- ered "number as the ruler of forms and ideas, and the cause of gods and daemons " ; and again that " to the most ancient and all-powerful creating Deity, number was the canon, the efficient reason, the intellect also, and the most undeviating of the composition and generation of all things." Philolaus declared "that number was the governing and self-begotten bond of the eternal permanency of mundane natures." Another ancient said that number was the judicial instrument of the Maker of the universe, and the first para- digm of mundane fabrication. Plato ascribed the invention of numbers to God himself. In the " Phaedrus " he said, " The name of the Deity himself was Theuth. He was the first to invent numbers, and arithmetic, and geometry, and astronomy." In the " Timaeus," he said, "Hence, God ventured to form a certain movable image of eternity; and thus while he was disposing the parts of the 300 MATHEMATICAL WEINKLES universe, he, out of that eternity which rests in unity, formed an eternal image on the principle of numbers, and to this we give the appellation of time." Euclid, who lived about 300 b.c, was one of the early Greek writers upon arithmetic. In his " Elements " he treats of the theory of numbers, including prime and composite numbers, greatest common divisor, least common multiple, continued proportion, geometrical progressions, etc. Archimedes, who was born about 287 e.g., was one of the most noted Greek mathematicians. He discovered the ratio of the cylinder to the inscribed sphere, and in commemoration of this the figure of a cylinder was engraved upon his tomb. He also wrote two papers on arithmetic. In the first he explained a convenient system of representing large numbers. In the second he showed that this method enabled a person to write any number however large, and as proof gave his celebrated illustration that the number of grains of sand required to fill the universe is less than 10^. In 1202 Leonardo of Pisa published his great work " Liber Abaci." This work contained about all the knowledge the Arabs possessed in arithmetic and algebra and furnished the most lasting material for the extension of Hindu methods. In 1540 Kobert Kecorde published the first arithmetic printed in the English language. He invented the present method of extracting the square root. In 1729 Isaac Greenwood published the first arithmetic pub- lished in America. In 1788 Nicolas Pike's arithmetic was published at New- buryport, Mass. It was a very popular book and was highly recommended by George Washington. / In 1797 Chauncey Lee published " The American Accomp- tant." MISCELLANEOUS HELPS 301 In 1799 Daboll published at New London, Conn., " The School- master's Assistant," which was indorsed by Noah Webster. In this book the comma is used in place of the decimal point. In 1821 Warren Colburn's " First Lessons in Intellectual Arithmetic " appeared. This book met with remarkable suc- cess. About two million copies were sold in twenty -five years. It revolutionized the teaching of arithmetic, and its influence is felt to this day. MATHEMATICAL SIGNS The symbols -h and — were used by Widmann in his arith- metic published at Leipzig in 1489, = by Kobert Recorde in his "Whetstone of Witte" published in 1557, x by William Oughtred in 1631, the dot (•) as a symbol of multiplication by Harriot in 1631, the absence of a sign between two letters to indicate multiplication by Stifel in 1544, : as a symbol of divi- sion by Leibniz, / as a symbol of division was used very early by the Hindus and Arabs and is supposed to be the oldest of all the mathematical signs, -f- as a symbol of division by Rahn, a Swiss, in an algebra published at Zurich in 1659, > and < by Harriot in 1631, : : by Oughtred in 1631, V was first used in this form by Rudoltf in 1525, oo and fractional exponents by Wallis and Newton in 1658, dx and J by Leibniz on October 29, 1675. The symbols :^, >, <, indicating "not equal," etc., are recent. Parentheses were first used as symbols of aggregation by Girard in 1629. The decimal point came into use in the seventeenth century ; it seems to have appeared first in a work published by Pitiscus in 1612. Positive integral exponents in the present form were first used by Chuquet in 1484. The Greek letter v was first used to represent the ratio of the circumference to the diameter by William Jones in his "Synopsis Palmariorum Matheseos," in 1706, and came into general use through the influence of Euler. 302 MATHEMATICAL WRINKLES BB e g o o be bh AhoqO ^-^ a -43 Ph Q H W .2 W 2 5 'S a 1l c a> rt S t*, tc S? -g S « « ee C2 " ■r^ 2^ CO rjt lO «0 t- C be ^ OQ V. -2 Tl (M CO ^ s 1 1: MISCELLANEOUS HELPS 303 '4 Bg« 3 V =-9 •3 5 = S c 1 I I. I fi e II E IT fc I- — a e = 4*^ e 2 s 03 a vj I I .5 5 a . I H H oc^tnoh- E ® ft. 1 1 «l ^ . Ph O O (S H II TABLES O >0 iOO»CO»OQiCO»COiOO»00»OOiOO»riOiOO»r 1 C4 00 T-iT-iT-ir-i(Mc^j(MC-J CM o rH S S§§Sg88|g| Siligiii §liii o rH o» oo ^ §S ij iS 8 ?3 S3 8 8 1 3§I3§ISI 8^gS^ rH r-( (N (M iM Oi 00 CO rH ^^^^§^SS28S Igggl^ss ||g|8 00 t* T*< ?^^^^^5§Sg^^ g g 1 g s 1 1 g rh S CO S t^ rH rH rH T-H r-< t- CD c-i S^g§8^^^S8?2 E2^88§8^g ^ ^^ ^ ^ g co I i . ^ 1 . 1 ( lO o J3 g ^^ ^ 1-? ^ ^ g i§ S §gl2SS8g^8 3S J28^ rH rH r-1 rH rH lO 1 cc A3Sg^§5^§8^^^ Sg8^SS£5g ^28^88 «# "1 o ^?3J5^5^^^8?S^ ^^^^^^^S ?S 8 8 ^ ^ 1 « «! ■* ^'^^E2:^SSgS^c5 C^(M8cO^COCOTtl ^ ^ '^ ^ 8 1 « rH C4 CO^WatOl'OOCSOrHC* eo^>ocot*ooo>0 rH « CO Tt< <0 C4 (M 47 2. «).'>'-' 2.288 1.456 .930 4.948 7.645 4.900 1.917 .947 SfBSTANCES Lead, cast . . Lead, white . Lead, ore . . Lignum vitse . Lime .... Lime, stone . Mahogany . . Manganese Maple . . . Marble . . . Men (living) . Mercury, pure Mica .... Milk .... Nickel . . . Niter. . . . Oil, castor . . Oil, linseed . Opal .... Opium . . . Pearl. . . . Pewter ... Platinum (native) Platinum, wire Poplar . . . Porcelain . • Quartz . . . Rosin . . . Salt .... Sand .... Silver, cast Silver, coin Slate .... Steel .... Stone . . . Sulphur, fused Tallow . . . Tar .... Tin . . . . Turpentine, spirits of i Vinegar . . 1 Walnut . . . ; Water, distilled ! Water, sea ! Wax .... Zinc, cast . . Si'ECinc GBA^ ll.;i5() 7.2;« 7.250 1.333 .804 2.386 1.063 3.700 .750 2.716 .891 14.000 2.750 1.032 8.279 1.900 .970 .940 2.114 l..'i37 2.510 7.471 17.000 21.041 .383 2.385 2.500 1.100 , 2.130 1.5O0 to 1 .800 10.474 10.534 2.110 7.816 2.000 to 2.700 um .941 1.015 7.291 .870 1.013 .671 1.000 1.028 .897 7.190 310 MATHEMATICAL WEINKLES Approximate Values of Foreign Coins in United States Monet Value in Country Standard Monetary Unit Terms of U. S. Gold Dollars Argentine Republic . Gold & Silver Peso .965 Austria- Hungary . . Gold Crown .203 Belgium Gold & Silver Franc .193 Bolivia Silver Boliviano .441 Brazil Gold Milreis .546 British Possessions in N. A. [except New- foundland^ . . . Gold Dollar 1.00 Central Am. States Guatemala"] Honduras Silver Peso .441 Nicaragua ( Salvador J Chili Gold Peso .365 [ Shanghai .661 China Silver TaeU Haikwan .736 [ Canton .722 Colombia .... Gold Dollar 1.00 Costa Rica .... Gold Colon .465 Cuba Gold Peso .91 Denmark .... Gold Crown .268 Ecuador Gold Sucre .487 Egypt Gold Pound [100 Piastres] 4.943 Finland Gold Mark .193 France Gold & Silver Franc .193 German Empire . . Gold Mark .238 Great Britain . . . Gold Pound Sterling 4.866^ Greece Gold & Silver Drachma .193 Haiti Gold & Silver Gourde .965 India Gold Pound Sterling 4.866^ Italy Gold & Silver Lira .193 Japan Gold Yen, Gold .498 Liberia Gold Dollar 1.00 Mexico Gold Peso .498 Netherlands . . . Gold & Silver Florin .402 Newfoundland . . . Gold Dollar 1.014 Norway Gold Crown .268 Peru Gold Sol .487 Portugal Gold Milreis 1.08 Russia Gold Rouble, Gold .515 Spain Gold & Silver Peseta .193 Sweden Gold Crown .268 Switzerland . . . Gold & Silver Franc .193 Tripoli Silver Mahbub [20 Piastres] .413 Turkey Gold Piastre .044 Venezuela .... Gold & Silver Bolivar .193 TABLES 311 WEIGHTS AND MEASURES Avoirdupois Weight 16 ounces (oz.) = 1 pound (lb.) 100 pounds = 1 hundredweight (cwt.) 20 hundredweight, or 2000 pounds = 1 ton (T.) 1 ton = 20 cwt. = 2000 lb. = 32,000 oz. 1 pound Avoirdupois weight = 7000 grains. 1 ounce Avoirdupois weight = 437 J gr. Troy Weight 24 grains (gr.) = 1 pennyweight (pwt.) 20 pennyweights = 1 ounce (oz.) 12 ounces = 1 pound (lb.) 1 lb. = 12 oz. = 240 pwt. = 5760 gr. 1 ounce Troy weight = 480 gr. Apothecaries* Weight 20 grains (gr. xx) = 1 scruple (sc., or 3) 3 scruples (3iij) = 1 dram (dr., or 3) 8 drains (3viij) = 1 ounce (oz., or 3) 12 ounces (3xij) =1 pound (lb., or ft.) 1 ft. = 12 3 = 96 3 = 288 3 = 5760 gr. Medicines are bought and sold in quantities by Avoirdupois weight. Apothecaries' Fluid Measure 60 minims, or. drops (HI, or gtt.) = 1 fluidrachm (f 3) 8 fluidrachms = 1 fluidounce (f 3 ) 16 fluidounces = 1 pint (O.) 8 pints = 1 gallon (Cong.) 1 Cong. = 80. = 128 f 3 = 1024 f 3 = 61,440 m. O. is an abbreviation of octans, the Latin for one eighth ; Cong, for con- giarium, the Latin for gallon. 312 MATHEMATICAL WRINKLES Linear Measure 12 inches (in.) = 1 foot (ft.) 3 feet = 1 yard (yd.) 5^ yards, or 16| feet = 1 rod (rd.) 320 rods = 1 mile (mi.) 1 mi. = 320 rd. = 1760 yd. = 5280 ft. = 63,300 in. Mariners' Linear Measure 9 inches (in.) =1 span (sp.) 8 spans, or 6 feet = 1 fathom (fath.) 120 fathoms =1 cable's length (c. 1.) 7^ cable lengths = 1 nautical mile (or knot) (mi.) 3 miles = 1 league Geographical and Astronomical Linear Measure 1 geographic mile = 1.15 statute miles 3 geographic miles = 1 league 60 geographic miles, or 1 _ f of latitude on a meridian, 69.16 statute miles J ~ [or of longitude on the equator 360 degrees = the circumference of the earth Surveyor's Linear Measure 7.92 inches = 1 link (1.) 25 links = 1 rod (rd. ) \ 4 rods = 1 chain (ch.) 80 chains = 1 mile (mi.) 1 mile = 80 ch. = 320 rd. = 8000 1. = 63,360 in. Jewish Linear Measure cubit = 1.824 ft. I mile (4000 cubits) = 7296 ft. Sabbath day's journey = 3648 ft. | day's journey = 33.164 mi. Square Measure 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd.) SO^ square yards = 1 square rod or perch (sq. rd.; P.) 160 square rods = 1 acre (A.) 640 acres = 1 square mile (sq. mi. ) TABLES 313 •q. mL A. sq. rd. sq. yd. sq. ft. sq. in. 1 = 640 = 102,400 = 3,097,600 = 27,878,400 = 4,014,489,600 1 = 160 = 4840 = 43,660 = 6,272,640 1 = 30J = 272J = 89,204 1 = 9 = 1290 Surveyor's Square Measure 626 square links (sq. 1.) = 1 square rod (sq. rd.) 16 square rods = 1 square chain (sq. ch.) 10 square chains = 1 acre (A.) 640 acres = 1 square mile (sq. mi.) 36 square miles = 1 township (Tp.) Tp. sq. mi. A. sq. ch. sq. rd. sq. 1. 1 = 36 = 23,040 = 230,400 = 3,686,400 = 2,304,000,000 1 = 640 = 6400 = 102,400 = 6,400,000 1 = 10 = 160 = 100,000 Cubic Measure 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) 1 cu. yd. = 27 cu. ft. = 46,666 cu. in. Wood Measure 16 cubic feet = 1 cord foot (cd. ft.) 8 cord feet, or ] , , ... 128 cubic feet 1 = 1'=""^ ^'^'^ 24J cubic feet = I PercMPch.) of etone * [ or of masonry Drt Measure 2 pints (pt.) = 1 quart (qt.) 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.) 1 bu. = 4 pk. = 32 qt. = 64 pt. Liquid Measure 4 gills = 1 pint (pt.) 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) 31i gallons = 1 barrel (bbl.) 1 bbl. = 31 J gal. = 126 qt. = 262 pt. = 1008 gi. 3l4 MATHEMATICAL WRINKLES Circular Measure 60 seconds = 1 minute (') 60 minutes = 1 degree (°) 360 degrees = 1 circle Commercial Weight 16 drams = 1 ounce (oz.) 16 ounces = 1 pound (lb.) 2000 pounds = 1 ton (T.) Paper 24 sheets = 1 quire 20 quires = 1 ream 2 reams = 1 bundle 6 bundles = 1 bale English Money 4 farthings (far.) = 1 penny (d.) 12 pence = 1 shilling (s.) 20 shillings = 1 pound (£) 1 £ = 20s. = 240d. = 960 far. 1 £ = $4.8665 in U. S. money Measure of Time 60 seconds (sec. ) = 1 minute (min. ) 60 minutes = 1 hour (hr.) 24 hours = 1 day (da. ) 7 days = 1 week (wk.) 365 days = 1 year (yr.) 366 days = 1 leap year 1 da. = 24 hr. = 1440 min. = 86,400 sec. THE METRIC SYSTEM (^The Acme of Simplicity} The following prefixes are used in the Metric System : (Greek) (Latin) deka, meaning 10 deci, meaning .1 hekto, meaning 100 centi, meaning .01 kilo, meaning 1000 milli, meaning .001 myria, meaning 10,000 TABLES 315 Linear Measure 10 millimeters (mm.) = 1 centimeter (cm.) 10 centimeters = 1 decimeter (dm.) 10 decimeters = 1 meter (m. ) 10 meters = 1 dekameter (Dm.) 10 dekametere = 1 hektometer (Hm.) 10 hektometers = 1 kilometer (Km.) 10 kilometers = 1 myriameter (Mm.) SgcAUK Measure 100 square millimeters (sq. mm.) = 1 square centimeter (sq. cm.) 100 square centimeters = 1 square decimeter (sq. dm.) 100 square decimeters = 1 square meter (sq. m.) 100 square meters = 1 square dekameter (sq. Dm.) 100 square dekameters = 1 square hektometer (sq. Hm.) 100 square hektometers = 1 square kilometer (sq. Km.) The area of a farm is expressed in hektares. The area of a country is expressed in square kilometers. Cubic Measure 1000 cubic millimeters (cu. mm.) = 1 cubic centimeter (cu. cm.) 1000. cubic centimeters = 1 cubic decimeter (cu. dm.) 1000 cubic decimeters = 1 cubic meter (cu. m.) Table of Capacity 10 milliliters (ml.) = 1 centiliter (cl.) 10 centiliters = 1 deciliter (dl.) 10 deciliters = 1 liter (1.) 10 liters = 1 dekaliter (Dl.) 10 dekaliters = 1 hektoliter (HI.) .10 hektoliters = 1 kiloliter (Kl.) 10 kiloliters = 1 myrialiter (Ml.) The hektoliter is used in measuring grain, vegetables, etc. The liter is used in measuring licjuids and small fruits. Table of Weight 10 milligrams (mg.) = 1 centigram (eg.) 10 centigrams = 1 decigram (dg.) 10 decigrams = 1 gram (g.) 316 MATHEMATICAL WRINKLES 10 grams = 1 dekagram (Dg. ) 10 dekagrams = 1 hektogram (Hg.) 10 hektograms = 1 kilogram (Kg.) 1000 kilograms = 1 metric ton (T.) A myriagram = 10,000 grams A quintal (Q.) = 100,000 grams TABLE OF EQUIVALENTS Long Measure 1 inch = 2.54 centimeters 1 centimeter = .3937 of an inch 1 foot = .3048 of a meter 1 decimeter = .328 of a foot 1 yard = .9144 of a meter 1 meter = 1.0936 yards 1 rod = 6.029 meters 1 dekameter = 1.9884 rods 1 mile = 1.6093 kilometers 1 kilometer = .62137 of a mile Square Measure 1 square inch = 6.462 square centimeters 1 square foot = .0929 of a square meter 1 square yard = .8361 of a square meter 1 square rod = 26.293 square meters 1 acre = 40.47 ares 1 square mile = 259 hectares 1 square centimeter = .155 of a square inch 1 square decimeter = .1076 of a square foot 1 square meter = 1.196 square yards 1 are = 3.954 square rods 1 hektare = 2.471 acres 1 square kilometer = .3861 of a square mile Cubic Measure 1 cubic inch = 16.387 cubic centimeters 1 cubic foot = 28.317 cubic decimeters 1 cubic yard = .7645 of a cubic meter 1 cord = 3.624 steres 1 cubic centimeter := .061 of a cubic inch 1 cubic decimeter = .0353 of a cubic foot 1 cubic meter = 1.308 cubic yards 1 stere = .2759 of a cord TABLES 317 Measures of Capacity 1 liquid quart = .9463 of a liter 1 dry quart = 1.101 liters 1 liquid gallon = .3785 of a dekaliter 1 peck = .881 of a dekaliter 1 bushel = .3524 of a hektoliter 1 liter = 1.0567 liquid quarts 1 liter = .908 of a dry quart 1 dekaliter = 2.6417 liquid gallons 1 dekaliter =1.135 pecks 1 hektoliter = 2.8375 bushels Measures of Weight 1 grain Troy = .0648 of a gram 1 ounce Avoirdupois = 28.35 grams 1 ounce Troy = 31.104 grams 1 pound Avoirdupois = .4536 of a kilogram 1 pound Troy = .3732 of a kilogram 1 ton (short) = .9072 of a tonneau 1 gram = .03527 of an ounce Avoirdupois 1 gram = .03215 of an ounce Troy 1 gram = 15.432 grains Troy 1 kilogram = 2.2046 pounds Avoirdupois 1 kilogram = 2.679 pounds Troy 1 tonneau = 1.1023 tons (short) CONVENIENT MULTIPLES FOR CONVERSION To Convert Grains to Grams, multiply by .066 Ounces to Grams, multiply by 28.35 Pounds to Grams, multiply by 453.6 Pounds to Kilograms, multiply by .46 Hundredweights to Kilograms, multiply by 50.8 Tons to Kilograms, multiply by 1016. Grams to Grains, multiply by 15.4 Grams to Ounces, multiply by .36 Kilograms to Ounces, multiply by 35.3 Kilograms to Pounds, multiply by 2.2 Kilograms to Hundredweights, multiply by .02 318 MATHEMATICAL WEINKLES Kilograms to Tons, multiply by- .001 Inches to Millimeters, multiply by 25.4 Inches to Centimeters, multiply by 2.54 Feet to Meters, multiply by .3048 Yards to Meters, multiply by .9144 Yards to Kilometers, multiply by .0009 Miles to Kilometers, multiply by 1.6 Millimeters to Inches, multiply by .04 Centimeters to Inches, multiply by .4 Meters to Feet, multiply by 3.3 Meters to Yards, multiply by 1.1 Kilometers to Yards, multiply by 1093.6 Kilometers to Miles, multiply by .62 MISCELLANEOUS Acre = 5645.376 square varas. Acre (square) is 209f feet each way. Ampere (unit of current) is that current of electricity that decom- poses .00009324 gram of water per second. Are = a square dekameter. Barleycorn = ^ inch. Barrel (flour) weighs 196 pounds. Barrel (wine) holds 31 gallons. Bushel (imperial) = 2216.192 cubic inches. Bushel (U. S.) = 2150.42 cubic inches. Cable length = 120 fathoms. Calorie = 42,000,000 ergs = .428 kilogrammeter. Carat (assayer's weight) =: 10 pennyweight. Carat (of diamond) = 3^ grains. , Centare = 1 square meter. Century = 100 years. Chaldron = 36 bushels. Coulomb (unit of quantity) is a current of 1 ampere during 1 second of time. Crown = 5 shillings. Cubic foot of water weighs 62| pounds. Cubit = 18 inches. Cycle (metonic) = 19 years. Cycle (of indiction) = 15 years. Cycle (solar) =28 years. TABLES 319 Degree (1°) = ^ of a right angle = ■—- radian. 180 Dozen = 12. Dozen (baker's) = 13. Eagle = a 10. Farthing = S .00503. Fathom = 6 feet. Firkin (wine measure) = 9 gallons. Florin (Austrian) = 84.53. Fortnight = 2 weeks. Furlong = J mile. Gallon (drj') = 268 cubic inches. Gallon (liquid) = 231 cubic inches. Gram = weight of 1 cubic centimeter of distilled water at its maximum density. Great gross = 12 gross. Gross = 12 dozen. Gross ton, long ton = 2240 pounds. Guilder (Holland) = S.402. Guinea = 21 shillings. Half section = 320 acres. Hand = 4 inches. Heat of fusion of ice at 0° C. is 80 calories per gram. Heat of vaporization of water at 100° C. is 536 calories per gram. Hectare = 1 square hectometer. Hogshead = 2 barrels. Kilo = a kilogram. Knot = 6086 feet, or 1.15 miles. Labor = 177.136 acres. League or Sitio (Spanish) = 4428.4 acres. Leap year. The centennial years divisible by 400 and all other years divisible by 4 are leap years. Light travels 300,000,000 meters, or 180,000 miles, per second. Liter = 1 cubic decimeter. Long hundredweight =112 pounds. Mill = 8 .001. Minim = a drop of pure water. Mite = f .0187. Nautical mile = 1 knot. Ohm (unit of resistance) is the electrical resistance of a column of mercury 106 centimeters long and of 1 square millimeter section. 320 MATHEMATICAL WEINKLES Pace (common) = 3 feet. Pace (military) = 2^ feet. Pack = 240 pounds. Parcian (Spanish) = 5314.08 acres. Penny =$.02025. Period (Dionysian, or Paschal) = 532 years. Quarter (English) = 8 bushels; U. S. = 8^ bushels. Quarter section = 160 acres. Quintal = 100,000 grams. Radian = — = 57.2957796°. IT Eod = 5| yards, or 16^ feet. Score = 20. Section = 640 acres. Sextant = 60°. Shilling = 1 .243. Sign = 30 degrees. Span = 9 inches. Specific heat of ice is about 0.506. Square = 100 square feet. . Stere = .2759 cord, or 1 cubic meter. Stone = 14 pounds. Strike (dry measure) = 2 bushels. Ton (long) = 2240 pounds. Ton (register) = 100 cubic feet. Ton (shipping) = 40 cubic feet. Ton (short) = 2000 pounds. Tonneau = 1.1023 tons. Township = 36 square miles. Vara (California) = 33 inches. Vara (Texas) = .9259+ yard, or 33^ inches. Volt (unit of electromotive force) is 1 ampere of current passing through a substance having 1 ohm of resistance. Watt (unit of power) is the power of 1 ampere current passing through a resistance of 1 ohm. Year (common) = 365 days. Year (leap, or bissextile) = 366 days. Year (lunar) = 354 days. Year (sidereal) = 365 days, 6 hours, 9 minutes, 9 seconds. Year (solar) = 365 days, 5 hours, 48 minutes, 46.05 seconds. INDEX Age Table Algebraic Problems Answers and Solutions to Algebraic Problems . . . . Arithmetical Problems . . . Greometrical Exercises . . . Mathematical Recreations . . Miscellaneous Problems . . . Approximate Results Arithmetical Problems . . . . Arithmetical Series Belts Bins, Cisterns, etc Brick and Stone Work . . . . Carpeting Casks and Barrels Compound Interest Tables . . . Cube Roots of Integers . . . . Density of a Body Examination Questions . . . Extraction of Any Root . . . . Foarth Dimension Fractions Classified Geometrical Exercises . . . . Geometrical Magnitudes Classi- fied Grain and Hay G. C. D. of Fractions . . . . . Harmonic Mean Historical Notes Historical Notes on Arithmetic . Homer's Methotl Interest L. C. M. of Fractions . . . . I^ogs Lumber Marking Goods Mathematical Branches Defined Mathematical Recreations . . , Mathematical Signs Mathematics Classified . . . , Mean Proportional 75 25 185 163 im 205 201 240 1 286 258 259 259 260 260 305 307 265 113 289 108 302 33 303 267 288 287 294 296 289 237 288 »)8 244 291 58 301 302 287 Mensuration 258 Metric Tables 314 Miscellaneous Helps 285 Miscellaneous Problems ... 48 Multiplication Table 304 Nine Point Circle 45 Numbers Classified 302 Painting and Plastering . . . 269 Papering 270 Periods of Numeration .... 286 Pi (it) to 707 places 285 Quotations on Mathematics . . 245 Right Triangles whose Sides are Integral 306 Roofing and Flooring .... 274 Scalene Triangles whose Areas are Integral 306 Scientific Truths 291 Short Methods 228 Addition 228 Approximate Results .... 240 Division 234 Fractions 235 Multiplication 230 Interest 237 Subtraction 230 Similar Solids 276 Similar Surfaces 276 Specific Gravities 309 Squares of Integers 306 Square Roots of Integers . . . 307 Table of Equivalents 316 Table of Prime Numbers ... 308 Tables 304 To find the Day of the Week . . 287 To find the Day's Length at Any Longitude 288 To find the Height of a Stump . 289 To Sura to Infinity 287 Values of Foreign Coins . 310 Weights and Measures .... 311 Wood Measure 283 321 v^ msmE^mi^S^sMS^ 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. Thi. book is due on the last date stamped below, ot Th.sbook.sa^^^^^^^^^i^j,„ d. I ^ Reoewedbook^are^ibiect^^ Th RE -1 Uat^^^^e^-rwiH g(^to=0— ' — jq{jy"2r8'^i^ ^TJI 44 -REcnrt©- jjim-UJ99a ^RTO'DOD H(W17*66-2PM -APrat-^96^ LD 2lA-50m-4,'59 (A1724sl0)476B General Library . Uaiversity of California Berkeley LD 21-100m-6,'56 (B9311sl0)476 General Library University of California Berkeley H w \:K