UNIVERSITY OF CALIFORNIA AT LOS ANGELES GIFT OF I-.IISS BETTY JA1ISS WENTWORTH-SMITH MATHEMATICAL SERIES JUNIOR HIGH SCHOOL MATHEMATICS BOOK I BY GEORGE WENTWORTH DAVID EUGENE SMITH AND JOSEPH CLIFTON BROWN GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON ATLANTA DALLAS COLUMBUS SAN FRANCISCO COPYRIGHT, 1917, BY GEORGE WENTWORTH DAVID EUGENE SMITH, AXD JOSEPH CLIFTON BROWN ENTERED AT STATIONERS' HALL ALL RIGHTS RESERVED 520.1 tgfre gtfttnacum CINN AND COMPANY PRO- PRIETORS BOSTON U.S.A. V- PEEFACE A proper curriculum for junior high schools and six-year T high schools demands, in the opinion of many teachers, a course in> in mathematics which introduces concrete, intuitional geometry c^ and the simple uses of algebra in the lower classes. This book 9= is intended to meet such a demand for the lowest class. ^. Wiphi ifMH ,.., 3ffc -UnWW. effpM>:. >^ll ..-, .. ..... ! ....... .... HOME ACCOUNTS 23 Exercise 15. Expense Accounts In ike foUoidng family expense account for 4 mo-, the interne being $125 a montk, foul: 1. The amount saved each month. 2. The total of each hem for 4 mo., inclnding savings. 3. The per cent which each total is of the grand total EXPENSE Accoryr FEB. H . i Ar*. TOTAL Food 24.10 23.30 24.80 22.15 Household Rent 18.00 laoo 18.00 18.00 Fuel (average) . . Gas (cooking) . . Electricity . . . Help 4.20 2.20 1.60 1.80 4.20 4,20 2.35 L8fl 2.25 .65 4.20 2.15 1.30 1.30 2.30 4.20 2.25 1.45 1.00 9^60 Furnishings ... Personal Clothing .... Carfare . . . . 7.00 .80 2,00 1.25 .90 24.00 1.20 Insurance Health and accident 1.10 L10 1.10 1.10 Life (average) . . Benevolences 4.30 4.30 4.30 4.30 Church .... 3.70 4.20 2.60 iQ Charity .... Education, Recreation 1.30 1.90 -.-: I 2.30 .80 4.70 3.20 3.60 Incidentals .... 7.30 1.30 2.60 7.38 Savings Total .... These per cents AanW be carried to the nearest tenth. 24 ARITHMETIC OF THE HOME Problem Data. The following price list may be used in solving the problems on page 25 and similar problems. The data may also be secured by the students through inquiry at home or at some grocery. This list may be made the basis of prac- tical problems in simple domestic bookkeeping. The object is, of course, to make arithmetic as real as possible, and when this purpose has been served, the student should proceed to other topics. Allspice, 10 $ per can ; $1 per dozen cans. Asparagus, 35 $ per can ; $4 per dozen cans. Bacon, American, 280 per pound. Sliced, in jars, 30$ per pound; $3.25 per dozen jars. Breakfast cereal, 14 $ per package ; $1.60 per dozen packages. Cinnamon, 10 $ per can ; $1 per dozen cans. Cloves, 30 $ per pound ; 50 $ per 2 pound box. Cocoa, half-pound cans, 25 $; $2.75 per dozen cans. Coffee, Maracaibo, 20 $ per pound ; 5 Ib. for 85 $. Java and Mocha, 35 $ per pound ; 5 Ib. for $1.60. Old Government Java, green, 27$ per pound ; 5 Ib. for $1.30. Crackers, Salines, 25 $ per tin ; $2.75 per dozen tins. Ginger snaps, 8 $ per carton ; 90 $ per dozen cartons. Domino sugar, 5 Ib. for 60 $. Flour, buckwheat, 6 $ per pound ; a bag of 24^ Ib., $1.30. Self-raising, 3 Ib. for 19 $ ; 6 Ib. for 35 $. Wheat, 5$ per pound ; $6 per barrel of 196 Ib. ; 90$ per sack of 24|lb. Granulated sugar, 8 $ per pound. Herring, 15 $ per can ; $1.75 per dozen cans. Honey, 8-ounce bottles, 30 $; $3.25 per dozen bottles. Loaf sugar, 11 $ per pound. Macaroni, 12 $ per package ; 25 packages for $2.75. Maple sirup, pints, 25$; gallon cans, $1.45 ; $16.50 per dozen cans. Olive oil, 40 $ per pint. Olives, 32 $ per bottle ; $3,.75 per dozen bottles. Soups, half -pint cans, 10 $; pint cans, 16 0; quart cans, 28$; $3.25 per dozen quart cans. Sugar sirup, half-gallon cans, 50$; 5-gallon cans, $4. Tea, Black India, 50$ per pound. English Breakfast, 48$ per pound. HOUSEHOLD ECONOMICS 25 Exercise 16. Household Economics 1. If a family wishes a dozen cans of cocoa, what per cent is saved in buying at the dozen rate ? In such cases reckon the per cent on the higher price. 2. If a family wishes 5 gal. of sugar sirup, how much is saved in buying a 5-gallon can instead of 10 half -gallon cans ? What per cent is saved ? 3. How much does a hotel manager save in buying 120 gal. of maple sirup by the dozen gallon cans instead of by the single can ? What per cent does he save ? 4. In which is the per cent of saving greater, in buying honey by the dozen bottles instead of by the bottle, or in buying maple sirup by the dozen cans instead of by the can? 5. If a woman wishes 24^ Ib. of buckwheat flour, how much does she save in buying it by the bag? 6. What per cent is saved in buying self-raising flour by the 6-pound package instead of by the 3-pound package ? 7. By inquiry at home, make out a grocery list for a week, from page 24. Make two pages of a home account, the left-hand page showing the amount received, and the right-hand page showing the amounts spent for groceries. 8. Make out a bill for six items of groceries, making the proper extensions and footing. Receipt the bill. Unless the students recall this from their preceding work in arith- metic, the teacher should take it up at the blackboard. 9. If a man uses 2320 cu. ft. of gas in April, how much is his gas bill for that month at 80$ per 1000 cu. ft. ? 10. At the beginning of the month a gas meter registers 14,260, and at the end of the month 17,140. How much is the gas bill for the month, at fl per 1000 cu. ft.? 26 ARITHMETIC OF THE HOME " Exercise 17. Heating the House 1. A man put a hot-water heater in his house at a cost of $540, and found that he used 12 T. of coal last season, the coal costing $7.60 per ton. How much did he spend for the heater and fuel ? 2. If the house in Ex. 1 was heated for 204 da., what was the average cost of the fuel per day ? 3. If the house in Exs. 1 and 2 had 9 rooms, what was the average cost of the fuel per room per day ? 4. A man has a steam-heating plant in his house. Last winter it consumed 22-T. of coal costing $7.25 per ton. How much did the coal cost ? 5. If 15% of the coal in Ex. 4 was lost in ashes, how many pounds of coal were lost in ashes? 6. If the house in Ex. 4 has 14 rooms and is heated for 190 da. in a year, what was the average cost of the fuel per day and the average cost per room per day ? 7. If 85% of the weight of coal is used in producing heat in a furnace, how many tons of coal are transformed into heat by a furnace that burns 1 7 T. in a season ? 8. A man used 14 T. of coal in his furnace in a season, but on buying a new furnace he used 8-% less coal. At $6.75 a ton, how much did he save on the coal? 9. A heating plant costing $525 averages 12 T. of coal per year at $7. 25 'a ton and furnishes the same amount of heat as a plant costing $375 and averaging 14 T. of coal per year at the same price. Counting as part of the cost an annual depreciation of 10% of the original cost price, and not considering interest, which plant costs the more money in 4 yr., and how much more ? HOUSEHOLD ECONOMICS 27 Exercise 18. The Family Budget 1. Last year Mr. Stone received an income of $3000. He set aside certain per cents of his income as follows: rent, 15%; heat, 3%; light, 1J% ; food, 28%; wages, 5-|% ; incidentals, 7% ; other personal expenses, 15% ; books, music, church, and pleasure, 8%. How much money did Mr. Stone allow for each of these purposes? 2. Mr. Stone in Ex. 1 really paid for rent, $320; for heat, $52.75; for light, $28.50; for food, $608.75; for wages, $135; for incidentals, $175.50; for other personal expenses, $302.80 ; and for books, music, church, and pleasure, $167.75. How much did each item of expendi- ture differ from the estimate and how much did Mr. Stone save during the year? 3. In Ex. 2 what per" cent of the amount spent for rent and food was spent for rent and what per cent for food ? 4. Mr. Sinclair has an income of $175 a month. He pays during the year for rent, $480; for heat and light, $85.75; for food, $675.80; for clothing, $168.40; for in- surance, $54.75; and for other expenses, $250. What per cent of his income does he save ? 5. In Ex. 4 what per cent of his income does Mr. Sinclair pay for rent ? for food ? for heat and light ? 6. If a man's income is $225 a month and he spends $600 a year for rent, what per cent of his income does he spend for rent and what per cent is left for other purposes ? 7. If a family with an income of $2200 a year spends 16% of its income for rent and 26% for food, what amount does it spend for each of these items ? Students should be encouraged to prepare family budgets at home, with the help of their parents. 28 ARITHMETIC OF THE HOME Exercise 19. Household Economics 1. A grocer sells coffee in half-pound packages at 19 $ a package and in 4-pound cans at $1.40 a can. If a woman wishes 4 lb., what per cent does she save in purchasing by the can ? 2. If you can buy Dutch cocoa in ^-pound boxes at 24$ a box or in 4-pound cans at $2.65 a can, and you wish 4 lb., what per cent do you save in purchasing by the can ? 3. If you can buy maple sirup at 480 a quart or in gallon cans at $1.50 a can, and you wish 1 gal. of sirup, what per cent do you save in purchasing by the can ? 4. If a woman can buy corn at 15 $ a can or $1.50 per dozen cans, what per cent does she save on 4 doz. cans in buying by the dozen ? 5. If a woman can buy soup at 20$ a can or $2.10 per dozen cans, what per cent does she save on 2 doz. cans in buying by the dozen ? 6. A woman can buy a bushel of potatoes for 80$ or a peck for 25$. If she needs a bushel of potatoes, what per cent does she save if she buys by the bushel ? 7. A woman can buy -|- doz. cans of tomatoes for 75 $ or 1 can for 15$. If she needs ^ doz. cans, what per cent does she save if she buys by the half dozen? 8. If flour costs $7.40 a barrel (196 lb.) or 5$ a pound, what per cent does a family save in purchasing flour by the barrel if it requires 196 lb. ? 9. If a family's ice bill averages $1.75 a month, and ice costs 35$ per 100 lb., how many pounds does the family use ? If by having a better ice box there is a saving of 10% in the amount of ice used, how many pounds are used ? How much is now the average ice bill per month ? HOUSEHOLD ECONOMICS 29 10. If you can buy some chairs for $24 cash or $3 down and $3 a month for 8 mo., what per cent do you save if you pay cash ? In Exs. 10-12 interest is not to be considered at this time. It should be mentioned incidentally as a subject to be studied later. 11. If you can buy a sewing machine for $40 cash or by paying $4 a month for a year, what per cent do you save if you pay cash ? 12. If a reduction of 10% is allowed on all electric- light bills paid before the tenth of each month, what amount would be saved in 4 mo. if advantage is taken of this rule in the account on page 23 ? 13. After the holidays the price of toys in a certain store was reduced 40%. How much would you save by waiting until after the holidays to buy a mechanical toy that was marked $3.50 before the reduction? In the following problems use the current market price as found by inquiry at home or at the store : 14. Find the per cent which you can save in purchasing each of the following in 5-pound packages instead of by the pound: sugar, starch, prunes, raisins. The teacher may omit Exs. 14-16 if desired. A few such problems, in which the students supply the data, serve, however, to make the subject more real. 15. Find the per cent saved in purchasing each of the following by the dozen cans : tomatoes, corn, peaches. As a matter of economy it should be noticed that it is not always good policy to purchase in large amounts because the material may deteriorate or be wasted. 16. Find the per cent saved in purchasing potatoes by the bushel instead of by the peck. 30 ARITHMETIC OF THE HOME Exercise 20. Miscellaneous Problems 1. Mr. Anderson earns $28 a week. He spends 20% of his income for rent, 26% for food for the family, 6% for fuel and lights, 18% for clothing for the family, 10% for church and charity, and 2% for incidentals. How much is left each year for other expenses and for savings ? Although 1 yr. = 52^- wk., or 52f wk. in leap years, 52 wk. is always to be taken as a year in problems of this type. 2. The girls in a class in millinery need 20 yd. of a certain quality of ribbon. They can buy this ribbon at 220 a yard, or 5yd. for $1. What per cent will be saved if they take the latter plan ? 3. The goods for a certain dress cost $7.80 and the buttons and trimmings cost $2.20. The cost of making the dress is 60% of the cost of the materials. If a dress of like quality and style can be bought for $15, what per cent is saved by buying the dress ready made ? 4. Make out a blank like the one shown below, but extended to include your entire school program, and com- pute the per cent of time devoted to each subject: TIME OF RECITATION SUBJECT MINUTES OF RECITATION PER CENT OF TOTAL MINUTES OF STUDY PERCENT OF TOTAL 9-935 940-1015 Arithmetic English 35 35 5. Some girls made 30 pieces of candy from the follow- ing recipe: 3 cups granulated sugar, 150; 1^- cups milk, 30; ^ cake chocolate, 2^-0; an inch cube of butter, 20. The fuel cost them 20, and they sold the candy at the rate of 3 pieces for 50. What per cent was gained on the cost? EEVIEW 31 Exercise 21. Review Drill Add, and also subtract, the following : 1. 2. 3. 4. $750.68 $680.01 $630.27 5 ft. 4 in. 298.98 297.56 429.68 2 ft. 6 in. Multiply the following : 5. 6. 7. 8. $298.63 $342.80 $674.39 2ft. 7 in. 27 92 129 8_ Divide as indicated, to two decimal places : 9. $426.34 H- 7. 10. 3469.1 -=-16. 11. 427jn-0.6. 12. $275 is what per cent more than $200 ? 13. $200 is what per cent less than $275? what per cent less than $300 ? 14. If 17 cars cost $13,600, how much will 9 cars cost? 15. How much is 72% of 350 lb.? of $3500? of 3? 16. How much is 175% of $2500? 12% of $625? % of $2000? |% of $1200? 11% of $2400? Write the answers to the following : 17. 39,987 + 46,296. 22. CDXL = (?> 18. 73,203-59,827. 23. 321 ft. - (?) yd. 19. 34J x $42,346. 24. 18 gal. - (?) qt. 20. 429. 75 -s- 25. 25. 4|bu. = (?)pk. 21. J of 25 ft. 4 in. 26. 16 mL= (?) yd. In all such drill work the teacher should keep a record of the time required by the students to solve the problems. Each student should strive to improve his record when reviewing the page later in the year. 32 ARITHMETIC OF THE HOME Exercise 22. Problems without Numbers 1. If you have an account with several items of income and several items of expenses, how do you proceed to balance the account ? 2. How would you proceed to make out a household account for a week ? 3. How do you find what per cent of the week's income is spent for household expenses ? 4. If you know the income of a household and know what per cent of the income is allowed for food, how do you find the amount allowed for food ? 5. If you know what fraction of his income a man spends for rent, how do you find what per cent he spends for rent? 6. If you wish to know before you receive the gas bill the amount of gas consumed at your home next month, how will you proceed to read the meter ? 7. If you know the cost of tomatoes per dozen cans and the cost per can, how will you find the per cent of saving of a person who purchases a dozen cans at the dozen rate instead of by the can ? 8. If you know the recipe for making cake for a certain number of persons, how will you change the recipe if you are making enough for a certain other number of persons? 9. If a man wishes a set of dining-room furniture and finds that, by waiting a week, he can buy it at a mark- down sale at a certain rate per cent off. the regular price, how will you find the amount he will save by waiting? 10. If you know how much a man paid for rent last year and how much more he pays this year, how will you find the per cent of increase ? ARITHMETIC OF THE STORE 33 II. ARITHMETIC OF THE STORE Nature of the Work. Fred Dodge applied for a position in a store. The manager asked him if he could add a column of figures quickly and correctly, and if he could compute quickly in his head. Fred thought he could, but when the manager tested him it was found that Fred was lacking in two things: he had not been taught to check his work, and he did not know the common short cuts in figuring that are used in all stores. Fred found that the arithmetic work which he needed most was addition, making change, and multiplication. We shall briefly review these subjects. In this review special attention will be given only to such topics as are not generally treated in the elementary arithmetic which precedes this course. Oral Addition. In adding two numbers like 48 and 26 mentally, it is better to begin at the left. Simply think of 68 (which is 48 + 20) and 6, the sum being 74. This is the way the clerk in the store adds 48$ and 26$. Exercise 23. Addition All work .oral Add the following, beginning at the left and stating only the results : 1. 2. 3. 4. 5. 6. 29 68 75 38 45 68 15 23 21 24 25 27 7. 8. 9. 10. 11. 12. 75 76 88 95 75 80 25 26 75 25 30 75 34 ARITHMETIC OF THE STORE Exercise 24. Addition See how long it takes to copy and add these numbers, check- ing the additions and writing the total time : 1. 2. 3. 4. 4287.75 4349.08 4476.82 4495.53 425.90 346.58 345.46 228.69 381.92 238.46 38.69 642.95 360.58 190.84 248.60 72.38 5. 6. 7. 8. 4458.65 4394.38 4298.05 4482.60 97.86 26.95 342.60 234.65 230.95 700.00 20.07 381.90 48.38 83.56 78.68 32.83 621.04 75.08 . 380.95 300.00 9. 10. 11. 12. 42080.60 43064.45 44148.20 43275.25 3679.70 817.66 876.42 842.35 909.36 4239.58 3192.68 3065.06 517.38 86.38 2124.45 2095.05 2310.25 3098.07 629.00 812.40 3096.65 2901.94 5082.00 3028.60 13. 14. 15. 16. 46240.45 44063.45 43083.95 42438.65 239.76 398.43 498.76 480.00 3865.42 2200.75 298.80 3557.76 396.37 4346.68 3763.84 463.48 900.48 328.93 . 2989.95 3086.75 1637.00 4043.68 4263.49 2945.50 ADDITION AND SUBTRACTION 35 Oral Subtraction. In subtracting mentally it is better to begin at the left except in making change. In the case of 52-28 think simply of 32 (which is 52-20) and take 8 from it, leaving 24. This subtraction may be treated by the process of making change, next described. Students should be familiar with both processes. Making Change. If you owe 640 to a merchant and give him $1, he says, " 64 and 1 are 65, and 10 are 75, and 25 are $ 1," or, briefly, " 64, 65, 75, $1," at the same time laying down 10, 100, and 250. The merchant will first lay down the coin or coins that will bring the amount up to a multiple of 5 ; then the largest coin or coins that will bring it up to a multiple of 25 ; and so on. Exercise 25. Subtraction All ivork oral Subtract the following numbers : 1. 2. 3. 4. 5. 6. 7. 47 47 47 56 73 83 95 30 33 39 28 34 36 48 Imagine yourself selling goods at a store and receiving in each case the first amount given, the goods costing the second amount. State the amount of change due in each case, and state what coins and bills you would give in change : 8. $1; 840. 11. $3; $2.20. 14. $5; $2.65. 9. $2; $1.25. 12. $4; $3.56. 15. $10; $7.75. 10. $2; $1.78. 13. $5; $2.28. 16. $5; $2.35. The teacher should ask the students to find how a cash drawer is arranged, and should describe the cash register. A little work in making change with real or toy money may profitably be given. 36 ARITHMETIC OF THE STORE Exercise 26. Subtraction See how long it takes to copy these numbers, to subtract, and to check by adding each difference to its subtrahend; ivrite the total time with the last result : 1. 2. 3. 4. 74,856 24,965 44,430 34,008 36,278 18,986 36,898 30,975 5. 6. 7. 8. 75,500 38,990 78,006 60,900 34,965 29,009 38,869 36,969 9. 10. 11. 12. $275.68 $220.85 $308.06 $600.04 46.99 165.90 88.79 189.86 13. 14. 15. 16. $278.00 $470.41 $309.20 $202.70 149.96 82.64 67.64 32.96 17. 18. 19. 20. $402.64 $300.00 $408.72 $472.92 89.85 183.75 45.86 88.96 21. 22. 23. 24. $309.92 $482.60 $300.00 $425.30 43.48 193.84 285.68 226.98 25. 26. 27. 28. $329.80 $408.73 $496.05 $506.00 49.96 229.84 309.78 329.80 MULTIPLICATION 37 Oral Multiplication. When Fred went to work in the store he found that he often needed to multiply rapidly. For example, if he sold 7 yd. of cloth at 45$ a yard, he needed to find the total selling price at once, without using a pencil. lie found that it was usually easier to begin at the left to multiply. In the case of 7 x 45$ he simply thought of 7 X 40$, or $2.80, and 35$, making $3.15 in all. Exercise 27. Multiplication Examples 1 to 12, oral Multiply the following, beginning at the left :. 1. 2. 3. 4. 5. 6. 45 38 32 56 56 65 _6 _4 _7 _8 _9 _7 7. 8. 9. 10. 11. 12. 72 77 56 45 55 35 _ -1 _ _Z _ Multiply the following : 13. 43 x 473. 21. 355 x 926. 29. 35 x 6464. 14. 38 x 308. 22. 280 x 628. 30. 42 x 8480. 15. 29 x 247. 23. 84 x 6088. 31. 68 x 9078. 16. 66 x 385. 24. 29 x 4756. 32. 39 x 4030. 17. 425 x 736. 25. 63 x 2798. 33. 203 x 3405. 18. 520 x 826. 26. 42 x 4802. 34. 330 x 4143. 19. 332 x 509. 27. 34 x 3006. 35. 223 x 6062. 20. 477 x 805. 28. 23 x 3989. 36. 447 x 3095. Teachers who care to give the check of casting out nines may do so at this time. Algebra is required, however, for its explanation. 238174. 38 ARITHMETIC OF THE STORE Short Cuts in Multiplication. You have already learned in arithmetic that there are certain short cuts in multipli- cation. These short cuts can be used extensively in the store. The most important ones are as follows : To multiply by 10, move the decimal point one place to the right ; annex a zero if necessary. To multiply by 100 or 1000, move the decimal point to the right two or three places respectively ; annex zeros if necessary. To multiply by 3, multiply by 10 and divide by 2. To multiply by 25, multiply by 100 and divide by 4. To multiply by 125, multiply by 1000 and divide by 8. To multiply by 33^, multiply by 100 and divide by 3. To multiply by 9, multiply by 10 and subtract the multi- plicand. To multiply by 11, multiply by 10 and add the multiplicand. Exercise 28. Short Cuts Find the results mentally whenever you can Multiply, in turn, by 10, by 100, by 5, by 25, and by 125 : 1. 6456 9248 25,192 23,848 22,200 2. 8168 9.376 19,920.8 25.088 58.752 3. 5776 24.432 56,246.4 23.048 46.832 Multiply, in turn, by 33 j, by 9, and by 11 : 4. 46,977 67,053 15,240 17,604 13,806 5. 441.54 466.74 1639.2 457.05 96.816 6. 483.66 190.56 1804.5 20.166 69.306 Multiply, in turn, by 5, by 25, and by 50 : 7. 15,384 56,812 87,824 756,52 73.728 8. 86,988 47,752 93,104 527.24 43.332 SHOKT CUTS IN MULTIPLICATION 39 Multiply the following : 9. 10 x 10.35. 20. 331 x 45. 31: 1000 x $7.62. 10. 10 x $225. 21. 33 x 288. 32. 50 x $4220. 11. 10 x $7.75. 22. 33J x 585. 33. 125 x $3200. 12. 50 x $652. 23. 100 x $45. 34. 331 x $345. 13. 50 x $345. 24. 100 x $33. 35. 16| x 186. 14. 25 x $544. 25. 1000 x $65. 36. 16| x $696. 15. 25 x $280. 26. 25 x $85.35. 37. 5 x 40,364. 16. 25 x $428. 27. 12| x $4400. 38. 5 x $15,680. 17. 675 x $35. 28. 12| x $4088. 39. 125 x $408. 18. 25 x $5.20. 29. 125 x $560. 40. 125 x $4.08. 19. 10 x $4.80. 30. 125 x $5600. 41. 16| x 7200. 42. How much will 25 books cost at 80$ each? 43. How much will 25 yd. of cloth cost at 16$ a yard ? 44. How much will 50 cans of corn cost at 14$ each? 45. How much will 4 books cost at 75$ each? 46. How much will 25 yd. of cloth cost at 24$ a yard ? 47. How much will 12^ yd. of cloth cost at 48$ a yard ? 48. How much will 80 doz. pencils cost at 56$ a dozen ? 49. How much will 75 books cost at 60$ each? 50. How much will 75 coats cost at $5 each ? 51. How much will a man earn in 48 wk. at $25 a week ? 52. How will 3^- doz. cans of tomatoes cost at 12$ a can ? 53. At $7.50 each, how much will 11 tables cost? 54. At $8.25 each, how much will 9 desks cost? 55. At $9.60 each, how much will 25 chairs cost? 56. At $42.50 each, how much will 11 typewriters cost? 40 ARITHMETIC OF THE STORE Product of an Integer and a Fraction. In the store we frequently have to find the product of an integer and a fraction. For example, we may need to find the cost of :f yd. of velvet at $2 a yard. As we have learned, To find the product of a fraction and an integer, multiply the numerator of the fraction by the integer and write the product over the denominator. Before actually multiplying, indicate the multiplication and cancel common factors if possible. Reduce the result to an integer, a mixed number, or a common fraction in lowest terms. For example, to multiply -g-|- by 18. Since we have -g-|> if we have 18 times as much we shall have 22 or T* r While we use ^ as an illustration, we seldom find in a store any need for a common fraction with a denominator larger than 8. Exercise 29. Multiplication All work oral Multiply the following, using cancellation when possible : 1. -| of 6. 6. 48 x 11. jL of 48. 16. 4 of 960. 2. f of 8. 7. 50 x 9 T^' 12. 24 x 7 8- 17. fof 272. 3. 48 xf. 8. 40 x 3^. 13. 128 x f . 18. 864 x -J. 4. 80' x|. 9. *of 35. 14. 132) 71 8, 7|, 3|, 620; No. 5: 8, 8, 8, 8, 8, 4, 63J0. THE PAY ROLL 71 6. Fill each space marked with an asterisk in the fol- lowing pay roll, allowing double pay for all overtime: PAY ROLL For the week ending Jan . 10 , 1920 No. NAME No. OF HOURS PER DAY TOTAL TIME WAGES PER HOUR TOTAL WAGES M. T. W. T. F. S. 1 R. S. Jones V/ \J V/ ^ t V/ 54 40 * * 2 3 M. L. King J. M. Mead V x/ \J V I I/ * 1 46 50 * * * * Totals * * * * * * * * * Before assigning Ex. 6 the teacher should explain that from one and a half to two times the regular hourly wage is usually paid for overtime, and that the check (v/) in the above pay roll means full time for the day. In this pay roll the full time is 8 hr. except on Saturday, when it is 4 hr. The symbol %/ means 8 hr. + 2 hr. overtime. A dash ( ) indicates absence. Part time, like 6-| hr., is indicated as above on Friday for King. Since the allowance for overtime is double that for regular work, Jones's time is 8 + 8 + 8 + 8 + 8 + 4 regular time and 2 x (2 + 1 + 1|) overtime, or 54 hr. in all. Explain the significance of the parentheses. Make out pay rolls (inserting names') when the men's numbers, the hours per day, and the wages per hour are as follows, 8 hr. constituting a day's work except on Saturday, ivhen it is 4 hr., and double pay being given for overtime : 7. No. 1: 8, 9, 8, 9, 8, 5, 67^0; No. 2: 8J, 9, 9, 8, 8, 4, 650; No. 3: 8, 8, 8, 10, 8, 6, 62; No. 4: 8, 9, 9, 9, 8, 4, 60^; No. 5: 8J, 8, 9, 8, 8, 5, 600. 8. No. 1: 8, 10, 8/10, 8, 6, 600; No. 2: 9J, 8, 8, 9, 8, 4, 62? ; No. 3: 10, 10, 10, 10, 8, 5, 620; No. 4: 8, 8, 8, 8, 10, 9, 62^0; No. 5: 8, 8, 8, 9, 8J, 6J, 650. 72 ARITHMETIC OF INDUSTRY Exercise 54. The Iron Industry 1. What is the weight of a steel girder that is 18' 10' r long and weighs 46^ Ib. to the running foot ? 2. What is the cost of 16' 4" of iron rod, 4^ Ib. to the foot, at !$ a pound? 3. The wooden pattern from which an iron casting is made weighs 6^% as much as the iron. The pattern weighs 67^-lb. How much does the casting weigh? 4. If steel rails weighing 180 Ib. to the yard are used between New York and Chicago, a distance of 980 mi., how many tons of rails will be required for a double- track road between these cities? 5. An iron tire expands ^-^Q% on being heated for shrinking on a wheel. A certain wooden wheel needs a tire 16' 8" in circumference. How much longer will 'the tire be when thus heated? 6. If 3.5% of metal is lost in casting, how much metal must be melted to make a casting to weigh 77.2 Ib.? Since 100% - 3.5% = 96.5%, 77.2 is 96.5% of the weight. ?. In a certain blast furnace the casting machine turns out 40 pigs of iron per minute, averaging in weight 110 Ib. each. If this machine runs for 312 da., 16 hr. a day, how many long tons (2240 Ib.) of pig iron will it turn out ? 8. Some years ago the average daily wages paid to em- ployees in a certain mill was $1.90, and the men worked 11 hr. a day, 6 da. in the week. At present the average daily wage is $2.60 and the men work 8 hr. a day, 5 da. in the week and 5 hr. on Saturday. Considering the wages per hour, what has been the per cent of increase :' Considering the hours per dollar of wages, what has been the per cent of decrease ? MISCELLANEOUS PROBLEMS 73 Exercise 55. Miscellaneous Problems 1. Sea Island cotton is usually shipped in bags of 150 lb., while Alabama cotton is shipped in bales of 500 lb. How many bags of Sea Island cotton at 280 a pound will equal in value 42 bales of Alabama cotton at 11$ a pound? 2. The average number of wage earners engaged in the manufacture of cotton goods during a recent year was 379,366. The value of the materials was $431,602,540 and the value of the finished products was $676,569,335. What per cent of value was added by manufacture ? 3. The 'United States produced 10,102,102 bales of cotton in the year 1900 and 11,068,173 bales in the year 1915. What was the per cent of increase ? 4. The total value of the cotton raised in the United States in a recent year was $627,861,000, and the number of bales was 11,191,820. Find the average value of a bale. 5. During a recent year the United States produced 11,000,000 bales of cotton and used only 7,000,000 bales. The amount used in this country was what per cent of the total amount produced ? 6. During a recent year 86,840 sq. mi. of cotton terri- tory was invaded by the boll weevil. The total area .infected at the end of the year was 409,014 sq. mi. What was the per cent of increase for the year? 7. In the days when cotton cloth was woven by hand an experienced weaver could turn out 45 yd. of cloth per week. At present a workman operating six power looms in a cotton mill will produce 1500 yd. per week. How long would it have taken the worker to do this with the hand loom? What is the per cent of increase in output per man with the power looms ? 74 ARITHMETIC OF INDUSTRY 8. Before the invention of the cotton gin a laborer could separate in a day only 1 Ib. of lint from the seed. At the present some gins turn out 10 bales of 500 Ib. each per day. Such a machine does the work of how many men ? 9. In making a silk lamp shade the following materials were used : 1J yd. silk @ $1.10, 2J yd. silk fringe @ $1.84, | yd. silk.net @ $2.20, 1 frame costing 60<. The labor and overhead charges amounted to $3.25. The shade was marked $14.50 but was sold at a discount of 10%. Find the gain per cent over the total cost. Overhead charges, also called overhead or lurden, means the general expense of doing business. 10. By repairing an automobile engine a mechanic in- creased its horse power 7^% and reduced the amount of gasoline necessary to run it 3%. Before the repairs were made the engine developed 40 H.P. and used 2 gal. of gasoline on a 20-mile trip. How much gasoline per horse power did it use on a 50-mile trip after it was repaired ? The letters H.P. are commonly used for horse power. 11. It is desired to construct an engine that will generate 102.5 H.P. net, that is, actually available for use. It is found that 18 % of the horse power generated is lost. This being the case, what horse power must be generated ? 12. How many fleeces of wool averaging 6^ Ib. each must be used to make a bale of wool weighing 250 Ib., and how many pounds will be left over? 13. If a wool sorter can sort 80 Ib. of wool in a day, how many days will it take him to sort a shipment of 24 bales of 250 Ib. each ? 14. After scouring (cleaning) a shipment of 12,000 Ib. of wool it weighed only 5240 Ib. What per cent of the original weight was lost by scouring? EEVIEW 75- Exercise 56. Review Drill 1. Add 147.832, 29.68, 575, 0.387. 2. From 1000 subtract the sum of 148.9 and 9.368. 3. Multiply 78.4 by 9.86. 4. Divide 0.8 by 0.13 to three decimal places. 5. How much is a profit of 14^-% on a sale of cotton- goods which cost $1275.50? 6. Find the commission at ^% on goods sold for $15,000. 7. Goods listed at $1450 are sold at a discount of 6%, 10%. Find the selling price. 8. How much is the profit, at 12^% on cost plus over- head charges, on the sale of goods which cost $1645.75, the overhead charges being $268.50 ? 9. How much is the loss on a house which cost $4500, including all charges, and which was sold at a loss of 6 % ? 10. Some goods which cost $750, including all charges,, were sold for $675. What was the per cent of loss ? Write the results of the folloiving : 11. 84in. = (?)ft. 16. 2sq. ft. = (?)sq. in. 12. 84oz. = (?)lb. 17. 2cu. ft. = (?)cu. in. 13. 84ft. =(?)yd. 18. 2 sq. yd. = (?) sq. ft. 14. 84pt. = (?)qt. 19. 72in. = (?)ft. 15. 84qt. = (?)pt. 20. 72in. = (?)yd. 21. Make out an imaginary personal account of six items on each side, and balance the account. 22. Write a bill for silver purchased; a cash check for merchandise ; a receipted bill for furniture bought ; an. invoice of a wholesale dealer. 76 ARITHMETIC OF INDUSTRY Exercise 57. Problems without Numbers 1. If you know the wages per hour and the number of hours worked each day by each man, without overtime, how do you find the total wages due all the men in a shop in a week? 2. If you know the regular wages due a man per hour, the wages for overtime, and the number of hours he works each day in a week, some of these being overtime, how do you find his total wages for a week ? 3. If you know the weight of a steel girder per running foot and the length of the girder, how do you find the total weight? 4. If you know the weight of a girder and its length, how do you find its weight per running foot ? 5. If you know the weight of each of several bales of cotton and the price paid per pound, how do you find the gain or loss to a firm that buys this cotton on a basis of 500 Ib. to the bale ? 6. If you know the shipping rate per hundredweight from where you live to Liverpool, and know the weight of a shipment, how do you find the total charge for freight ? 7. If a farmer has wheat to ship to Chicago and knows the freight rate per bushel or carload and the number of bushels or carloads, how does he find the freight charges ? 8. If a farmer knows the charges for shipping a certain number of bushels of corn to Chicago, how does he find the freight rate per bushel? 9. If you know the average weight of a fleece of wool and the number of fleeces a sheep grower has, how do you find the number of bales of wool weighing 250 Ib. ach and the amount, if any, that will be left over ? ARITHMETIC OF THE BANK 77 V. ARITHMETIC OF THE BANK Saving. A boy who puts 1$ each day into a toy bank will have enough in six months to buy a catcher's mitt or three baseball bats. A girl who saves 100 a day will have enough in a month or two to buy a good pair of shoes. A man who saves $1 a day will have enough in a few years to buy a building lot in some place where he may care to live. We often get the things that we need for comfortable living or the things that give us legitimate pleasure, by saving a little at a time. The following suggestion to teachers will be found helpful : Begin with a brief discussion of the need and value of saving money. There are many of us who never learn how to save money wisely. Many of us prefer to gratify our immediate desires rather than to provide for the future.. Why is this a bad plan? On the other hand, there are others of us who, in order to save for the future, deny ourselves the things which it would be real economy to buy. There is always the temptation to live extravagantly. Extravagance includes not only living beyond our means but also spending money foolishly. Every boy and every girl should begin early in life to form the habit of saving, no matter if it be but a few dollars a year. i Exercise 58. Saving 1 A man wishes to buy an automobile that costs $780. If he saves |2 every week day, in how many weeks will he save enough to buy the car ? 2. A boy wishes to buy a camera that costs |6.60. If he saves 10$ every week day, in how many wieeks will he save enough to buy the camera ? 3. A girl wishes to buy a purse that costs 60$. If she saves 5$ every week day, in how many weeks will she save enough to buy the purse ? 78 ARITHMETIC OF THE BANK 4. A man who has been smoking six cigars a day, which he buys at the rate of three for a quarter, decides to give up smoking and save the money. How much will this saving amount to in 5 yr. ? In all such problems the year is to be considered as 365 da. (or 313 da., excluding Sundays), although in 5 yr. there will probably be one leap year and there may be two leap years, and although a year need not have exactly 52 Sundays. 5. A boy earns some money by selling papers. He finds that he can easily save 150 a day, excluding Sundays. If he does this for 5 yr., how much will he save in all ? 6. A woman in a city has a telephone for which she is charged 5$ for each call. She finds that she can economize by reducing the number of her telephone calls on an average four a day, including Sundays. If she does this for 3 yr., how much will she save ? Make out accounts, inserting dates, items of receipts and payments, and the balances, given the following: 7. On hand, $4.20. Receipts: 200, 400, 250, $1, 300, 650. Payments: 250, 300, 100, 750. 8. On hand, $4.30. Receipts: 100, 150, '600, 550. Payments: 300, 420, 750, 50, 300, 600, 200, 50. 9. On hand, $4.63. Receipts: 100, 120, 320, 120, $1.10. Payments: 250, 350, 50, 240, $1.50, 250, 500. 10. On hand, $5.10. Receipts: $1.25, 350, 750, 420, 680, 700, $1.22. Payments: $1.50, 700, 50, 100. 11. On hand, $1.30. Receipts: $1, 400, 50, 150, 100, 250, 120, 160, 100. Payments: 100, 200, 50, 50, 220, 30. 12. On hand, $3.20. Receipts: 700, 600, 100, 30, 150, 100, 250, 50, 50, 300, 520, 300. Payments: $1,50, 500, 900, 700, 180, 120, 300, 450, 250. SAVINGS BANKS 79 Bank Account Essential. One thing that is essential at some time to everyone who hopes to succeed is a bank account. A reliable person may "open an account," as it is called, as soon as he begins to save even small amounts. People who are saving money usually keep it in a bank until they have enough for investing permanently. Certain kinds of banks, such as savings banks and trust companies, not only guarantee to take care of all money left with them by depositors but also pay a certain per cent of interest. National banks also generally allow interest on what are called inactive accounts ; that is, deposits that remain undis- turbed for some time. Many schools have found it interesting and profitable to organize school banks, electing the officers and carrying on a regular banking business, either with small amounts of real money placed on deposit by students and transferred by the teacher to some bank or trust company, or with imitation money. Such exercises should not, how- ever, interfere with the work in computing. Savings Bank. To deposit money a person goes to a ank, says that he wishes to open an account, and leaves his money with the officer in charge. The officer gives him a book in which is written the amount deposited, and the depositor writes his name in a book or on a card, for identification. When he wishes to draw out money, he takes his book to the bank, signs a receipt or a check for the amount he desires, and receives the money, the amount being entered in his book. Students should be told of the advantages of opening even small accounts at a savings bank. A boy who deposits $1 a week for 10 yr. in a bank paying 2% every 6 mo. and adding it to the account, will have $631.54 in 10 yr., and a man who puts in $10 a week will have about ten times as much, or $6317.16. The class should be told about trust companies, which take charge of funds, manage estates, and pay interest on deposits. 80 ARITHMETIC OF THE BANK Exercise 59. Saving 1. How much will 25$ saved each working day, 310 such days to the year, amount to in 10 yr.? 2. If a boy, beginning at the age of 14 yr., saves 25$ a day for 310 da. a year and deposits it in a bank, how much has he when he is 21 yr. old, not counting interest ? Interest and withdrawals are not to be considered in such cases. 3. If a man saves $3.25 a week out of his wages and continues to do this 52 wk. in a year for 12 yr., how much money will he save ? 4. If a father gives his daughter on each birthday until and including the day she is 25 yr. old as many dollars as she is years old, depositing it for her in a savings bank, how much has she when she is 25 yr. old ? 5. A merchant saves $750 the first year he is in business. The second year he saves one third more than in the first year. The third year his savings are only 85% as much as the second year. The fourth year they increase 30 % over the third year. How much does he save in the four years? 6. A man works on a salary of $18 a week for 52 wk. in a year. His expenses are $3.75 a week for house rent, 60% as much for clothing, 300% as much for food as for clothing, and 20 % as much for other necessary expenses as for food. How much of his salary can he deposit each year in the savings bank ? 7. A clerk had a salary of $12 a week two years ago and a commission of 2 % on goods he sold. That year he worked 50 wk. and sold $4800 worth of goods. Last year his salary was increased 25%, his rate of commission remaining the same. He worked 48 wk. and sold goods to the amount of $5000. How much was his income increased ? INTEREST 81 Interest. If Mr. James has a house and lot worth |5000, and rents it to Mr. Jacobs at $42 a month, his income from the rent is 12 x $42, or $504 a year. This is a little more than 10% of the value of the property, but out of it Mr. James has to pay for various expenses, such as insurance, repairs, and taxes. If Mr. James has $5000 and lends it to Mr. Jacobs at the rate of 6% a year, his income from this transaction is 6% of $5000, or $300 a year. Money paid for the use of money is called interest. In the above illustration about the lending of money $300 is the interest for 1 yr., 6% is the rate of interest, and $5000 is the principal. Schools do not require, as formerly, the learning of many defini- tions. What is necessary is that the student should use intelligently such terms as interest, rate, and principal. Men often have to borrow money to carry on their busi- ness. For example, a merchant may wish to buy a lot of holiday goods, feeling sure that he can sell them at a profit. In this case it is good business for him to borrow the money, say in November, taking advantage of all cash discounts allowed, and then to repay the money in January after the goods are sold. If, for example, he needs $1000 for 2 mo. and can borrow it from a bank at the rate of 6% a year, he will have to pay ^ of 6% of $1000, or 1% of $1000, or $10, a sum which he can e'asily afford to pay for the use of the money. To find the interest on any sum of money for part of a year first find the interest for 1 yr. and then find it for the given part of a year. Formal rules for such work need not be memorized. An example ^r two may profitably be worked on the blackboard before studying the next page. 82 ARITHMETIC OF THE BANK Exercise 60. Interest Examples 1 to 27, oral Find the interest on the following amounts for 1 yr. at the given rates : 1. $1000, 5%. 6. $150, 4%. ll. $400, 6%. 2. $1000, 6%. 7. $250, 4%. 12. $400, 3. $1000,4^%. 8. $500, 5%. 13. $1000, 4. $2000, 5%. 9. $600, 6%. 14. $1000, 5. $3000, 6%. 10. $800, 3%. 15. $1000, 4f %. .Frnrf iAe interest on the folloiving amounts for 6 mo. at the given rates : 16. $100, 6%. 18. $1000, 5%. 20. $3000, 4%. 17. $300, 6%. 19. $2000, 5%. 21. e interest .on the following amounts for 1 yr. 6 mo. at the given rates : 22. $1000, 4%. 24. $2000, 6%. 26. $5000, 4%. 23. $1000, 5%. 25. $5000, 5%. 27. $2500, 4%. 28. A man having $17,250 invested in business has found that his net profits average 16% a year on the investment. He is offered $25,000 for the business, and he could invest the money at 4^-%. If he sells out and retires, what is his annual loss in income ? 29. In April a coal dealer borrowed $66,420 at 5%. With this he purchased his summer's supply of coal at $5.40 a ton, his overhead charges being 30 < a ton. He sold the coal at $6.68 a ton, the buyers paying for the unloading and delivery, and he paid his debt in October after keeping the money 6 mo. How much did he gain ? INTEREST FOE MONTHS AND DAYS 88 Interest for Months and Days. Suppose that a man bor- rows from a bank $400 on Sept. 10, 1919, at 6%. What will the interest amount to Aug. 7, 1920 ? yr. mo. da. 1920 8 7 = second date 1919 9 10 = first date 10 27 = difference in time Taking, as is usual, 30 da. to the month, the difference in time is 327 da. We therefore have 109 Hence the interest due August 7, 1920, is $21.80. Banks usually lend money for a definite number of days or else require payment to be made on demand. In either case they com- pute the interest for the days that the borrower has the money and not for months and days. To enable them to compute the interest easily they have interest tables. Private individuals, however, occa- sionally have to compute interest for months and days, and in that case they may proceed as in the above problem. It is a waste of time for the student to find the interest on very small or very large sums of money, for very short or very long periods, or at more than legal rates. A few such examples may be given, however, for practice in computation. In general, interest is now reckoned on such a sum as $750 rather than $749.75, and for periods not exceeding 90 da. rather than one involving years, months, and days. Teachers should advise the students that if the interest is for more than 1 yr. they should first find it for the given number of years, and then, by the above method, for parts of a year. Such cases are, however, rapidly becoming obsolete. Banking facilities make it rare to find interest periods for years, months, ad days. 84 ARITHMETIC OF THE BANK Interest for 30, 60, and 90 Days. In borrowing money at a bank the time for which the money is borrowed is usually 30 da., 60 da., or 90 da., except when repayment is to be made on demand. Since 6 % is the most common rate, it is convenient to be able to work mentally the common types of interest examples. How much interest must you pay if you borrow $500 from a bank at 6% for 60 da.? for 30 da.? for 90 da.? Since 60 da. = -$$ yr. = ^ yr., the interest on $500 for 60 da. is of 6% of $500, or 1% of $5( For 30 da. the interest is \ of 1% of $500, or $2.50. For 90 da. the interest is f of 1% of $500, or $7.50. From this work state a simple rule for finding the interest at 6% for 60 da.? for 30 da.? for 90 da.? Exercise 61. Interest Examples 1 to 15, oral Find the interest at 6% on the following amounts: 1. |400, for 60 da. 6. $840, for 60 da. 2. $650, for 30 da. 7. $350, for 90 da. 3. $725, for 60 da. 8. $450, for 30 da. 4. $875, for 60 da. 9. $950, for 30 da. 5. $900, for 90 da. 10. $860, for 30 da. 11. Find the interest on $600 for 60 da. at 5%. The interest at 6% is $6, and so at 5% it is $ of $6. 12. Find the interest on $3000 for 30 da. at 5%. 13. Find the interest on $240 for 90 da. at 5%. 14. Find the interest on $600 for 60 da. at 4%. 15. Find the interest on $1200 for 30 da. at 3%. INTEREST 85 16. Find the interest on |400 for 2 yr. 10 mo. 27 da. at 6%. The interest for 2 yr. is 2 x 6% of $400, or $48, and for 10 mo. 27 da. is $21.80, as found on page 83. Hence the total interest is $69.80. As already stated, such examples are becoming more rare. A few are given on this page, chiefly as exercises in computation. Find the interest on the following : 17. $1250 for 2 mo. 17 da. at 5%. 18. |1500 for 7 mo. 23 da. at 6% ; at 5J%. 19. $2400 for 8 mo. 11 da. at 5% ; at 6% ; at 5J$. 20. $575 for 2 yr. 9 mo. 15 da. at 5% ; at 5J%. 21. $850 for 3 yr. 10 mo. 6 da. at 5J% ; at 6%. 22. $925 for 4 yr. 10 mo. 6 da. at 6% ; at 5%. 23. A dealer bought 24 sets of furniture on Nov. 1, at $50 a set, promising to pay for them later, with interest at 6%. He paid the bill on the following Jan. 16. What was the amount of principal and interest ? 24. A man borrowed $750 on Mar. 10, at 6%, and $1600 on Apr. 10, at 5%. He paid the entire debt on July 10 of the same year. How much did he pay in all ? 25. A man borrowed $750 on May 1, at 5%, and $1800 on July 5, at 4^%. He paid both debts with interest on^ ( Dec. 16 of the same year. How much did he pay in all ? 26. What is the total amount of principal and interest on $950 borrowed Mar. 10, at 6%, and $1600 borrowed May 15, at 5%, the payment in both cases being made on Oct. 20 of the same year? 27. A man borrowed $750 on May 9, at 5%, and $625 on June 15, at 6%, each loan to run for 60 da. When was each due, and how much was the total interest? 86 ARITHMETIC OF THE BANK $2000.= first principal .02 ).= int. first 6 mo. Interest at Savings Banks. Savings banks usually pay interest every six months or every three months. This interest is added to the principal, and the total amount then draws interest. Compound Interest. When interest as it becomes due is added to the principal and the total amount then draws interest, the investor is said to receive compound interest on his money. Compound interest is not commonly used, but if one collects in- terest when due and at once reinvests it, he practically has the ad- vantage of compound interest. The method of finding compound interest is substantially the same as that used in simple interest. For example, how much is the amount of $2000 in 2 yr., deposi- ted in a savings bank that pays 4% annu- ally, the interest being compounded semiannu- ally? How much is the compound interest ? 2000. $2040.= amt. after 6 mo. .02 $40.80 = int. second 6 mo. 2040. $2080.80 = amt. after 1 yr. .02 $41.62 = int. third 6 mo. 2080.80 $2122.42 = amt. after 11 yr. .02. $42.45 = int. fourth 6 mo. 2122.42 $2164.87 = amt. after 2 yr. 2000. $164.87 = int. for 2 yr. Simple interest for the same time is $160, or $4.87 less than the compound interest. Here the compound interest has been found exactly, but savings banks pay interest only on the dollars and not on the cents. 8T Savings Bank Account. The following is a specimen account at a savings bank which pays interest at the rate of 4 f a year, . the interest being payable semiannually, on January 1 and July 1, on the smallest balance on deposit at any time during the previous interest period: DATE DEPOSITS INTEREST PAYMENTS BALANCE 1922 July 1 600 50 600 50 July 20 75 675 50 Sept. 6 120 555 50 Dec. 7 60 615 50 Dec. 20 65 550 50 1918 Jan. 1 11 561 50 May 9 200 761 50 July 1 11 22 772 72 The smallest balance during the first interest period is 1550.50. Interest is computed on the dollars only, the cents being neglected. At 4% per year the interest for 6 mo. on $550 is 2% of $550, or $11. In the second period the smallest balance is $561.50, and therefore the interest is 2% of $561, or $11.22. Some banks allow interest from the first of each month; others from the first of each quarter ; others, as above, from the first of each half year. The interest is computed on the smallest balance on hand between this day and the next interest day, and is usually added every half year, although it is sometimes added every quarter. Students should ascertain the local custom as to savings banks. 88 AEITHMETIC OF THE BANK Exercise 62. Compound Interest Find the amount of principal and interest at simple interest, and also at interest compounded in a savings bank annually : 1. 13000, 2yr., 5%. 6. $2750, 4 yr., 2. $3000, 4yr., 6%. 7. $825.50, 5yr., 3. $2000, 4yr., 4%. 8. $2000, 6 yr., 4%. 4. $3250, 4yr., 3%. 9. $625.50, 4 yr., 5. $3750, 4yr., 3%. 10. $875.50, 3 yr., the amount of principal and interest, the interest beiny compounded in a savings bank st-miannually : 11. $400, 3yr., 4%. 16. $600, 2 yr., 4%. 12. $600, 2yr., 4%.jte,ffU1*17. $2000, 2 yr., 4J%. 13. $850, 2yr., 6%. (p$2000, 3 yr., 4%. 14. $900, 3yr., 3%. , 19. $3000, 2 yr., 3%. 15. $900, 3yr., 4%.*Sb l 'tfeo. $3000, 4 yr., 4%. 21. If you deposited $140 in a savings bank on July 17, 1919, and $35 on Feb. 9, 1920, and if you have made no withdrawals, to how much interest are you entitled July 1, 1920? In this bank on July 1 and Jan. 1 interest on each deposit at 4% per year is credited from the day of deposit if on the first day of a month, and otherwise from the first day of the following month. 22. If a man deposits $1500 in a savings bank on Jan. 1, $215 on Feb. 1, $140 on May 7, $270 on Sept. 11, and $243 on Dec. 3, and makes no withdrawals, how much will he have to his credit on the following Jan. 1 ? In this bank on July 1 and Jan. 1 interest on each deposit at 4% per year is credited from the day of deposit if on the first day of a month, and otherwise from the first day of the following month. POSTAL SAVINGS BANKS 89 Postal Savings Bank. The United States government conducts a savings bank in connection with the post office. Although all savings banks are carefully regulated and in- spected by the state governments, there are many persons who are willing to take the smaller rate of income which the postal savings bank pays, because of the fact that our government guarantees the payment of their money. Any person of the age of 10 yr. or over may deposit money in amounts of not less than $1, but no fractions of a dollar are accepted for deposit. No one can deposit more than $1000 in any one calendar month or have a balance at any time of more than $1000, exclusive of accumulated interest. Deposits may be made at the larger post offices, and a depositor receives a postal savings certifi- cate for the amount of each deposit. Interest is paid by the government at the rate of 2% for each full year that the money remains on deposit, beginning on the first day of the month next following the one in which the deposit is made. Interest is not paid for any fraction of a year. A person may exchange his deposits in sums of $20 or multiples of $20 for bonds bearing interest at Exercise 63. Postal Savings Bank All work oral Find the, interest for 1 yr. on the following deposits : 1. $30. 2. $40. 3. $75. 4. $300. 5. $500. Find the interest for 2 yr. on the following deposits: 6. $50. 7. $60. 8. $100. 9. $200. 10. $500. Find the interest for 1 yr. on a %\]o bond of: 11. $80. 12. $240. 13. $360. 14. $480. 15. $500. 90 ARITHMETIC OF THE BANK Bank of Deposit. When a man has money enough ahead to pay liis bills by checks, he will find it convenient to have an account with a bank such as merchants commonly use, sometimes called a bank of deposit. Such banks do not pay interest on small accounts, the deposit being a matter of convenience and safety. If a man wishes to open an account he sometimes has to give references, for banks do not wish to do business with unreliable persons. A man's credit in business is always a valuable asset. In some sections of the country banks receive deposits under two classes of accounts, savings accounts and check- ing accounts. In the former case they act as savings banks ; in the latter, as banks of deposit. For the purposes of the school it is not necessary to consider this difference further. Students should, however, investi- gate the local cus- tom in the matter. Deposit Slip. A man, when he de- posits money or checks in a bank, fills out a deposit slip similar to the one here shown. Sometimes the depositor enters the name of the bank on which each check is drawn ; sometimes the receiving teller at the bank does this by writing the bank's number ; and sometimes it is not entered at all. These are technicalities that do not concern the school. DEPOSITED FOR CREDIT OF IN THE SECOND NATIONAL BANK OF THE CITY OF NEW YORK IQ1 RTTT.S DOLLARS CENTS roiN CT-fFCTC ON R'K BANKS OF DEPOSIT 91 Exercise 64. Deposit Slips Write or fill out deposit slips for the following deposits, inserting the name of the depositor and of the bank : 1. Bills, $375; silver, $60; check on Garfield Bank, $87.50; check on Miners Bank, $627.75. 2. Bills, $423; gold, $175; silver, $235.75; check on Corn Exchange Bank, $736.90. 3. Bills, $135; check on Second National Bank of New York, $425 ; check on Chase National Bank, $75.40. 4. Bills, $1726; gold, $100; silver, $200; check on Merchants Bank, $245.50; check on Union Bank, $275.40. 5. Bills, $1275 ; checks on Harriman National Bank, $146.50, $200 ; checks on Jefferson Bank, $325, $86.50. 6. Gold, $100 ; checks on First National Bank, $175, $240, $32.80 ; checks on Sherman Bank, $37.42, $61.85. 7. Bills, $2475 ; silver, $275.50 ; check on Case Bank, $43.50 ; check on Miners National Bank, $250. 8. Bills, $345; silver, $350.75; gold, $480; check on Merchants National Bank, $455 ; check on Farmers Trust Co., $262.50 ; check on City Bank, $1000. 9. A man deposited $475.75 in cash to-day, a check for 50% of a debt of $675 due him, and a check in payment for 45yd. of velvet at $2.25 a yard less 33|-% discount. Make out a deposit slip. 10. A merchant received cash for 8 doz. forks @ $14.75, 5-|- doz. teaspoons @ $13, a watch costing $40.50, and 4 clocks @ $7.75. He also received a check on the Lincoln Trust Co. for 3 doz. dessert spoons @ $17.75 and 4-| doz. nutcrackers @ $9. He deposited all this in a bank. Make out a deposit slip. JMl 92 ARITHMETIC OF THE BANK Check. A check book containing checks and stubs, substan- tially as follows, although often varying in certain details, is given the depositor when he opens an account. No. 8 using a short method. 4. Divide it by $1.08. 5. Find 12J% of it; 37% of it; 62|% of it This Material for Daily Drill is so arranged as to give daily practice in the fundamental operations. By first going through all the exercises with the number denoted by (a), and then with the one denoted by (b), and so on, more than a hundred different exercises will result, or more than one exercise for each school day of the half year, giving enough for a selection. 105 106 MATERIAL FOE DAILY DKILL EXERCISE 3 Taking (a) $8.96, (b) $13.44, (c) $17.92, (d) $22.40, (e) $26.88, or (f) $31.36, as the teacher directs : 1. Add it to |19 + 1287.30 + $2.75 + 148.60 -f 142.86. 2. Subtract it from $4.63 + $10.14 + $27.82 + $9.86. 3. Multiply it by 750, using a short method. 4. Divide it by $1.12. 5. Find 25% of it; 2J% of it; 250% of it. EXERCISE 4 Taking (a) $9.28, (b) $13.92, (c) $18.56, (d) $23.20, (e) $27.84, or (f) $32.48, as the teacher directs : 1. Add it to $37.62 + $0.27 + $150 + $3.98 + $48.60. 2. Subtract it from $25.37 + $17.26 + $14.96 + $5.04. 3. Multiply it by 125, using a short method. 4. Divide it by $1.16. 5. Find | of it; 75% of it; f of it; 37^% of it; 3.75% of it. EXERCISE 5 Taking (a) $9.92, (b) $14.88, (c) $19.84, (d) $24.80, (e) $29.76, or (f) $34.72, as the teacher directs : 1. Add it to $3.09 + $17+ $0.75 + $27.68 + $9.32. 2. Subtract it from $2.80 + $15.06 + $19.87+ $10.13. 3. Multiply it by 37-|, using a short method. 4. Divide it by $1.24. 5. Divide it by f ; by 2|. MATERIAL FOR DAILY DRILL 10T EXERCISE 6 Taking (a) $10.56, (b) $15.84, (c) $21.12, (d) $26.40, (e) $31.68, or (f) $36.96, as the teacher directs : 1. Add it to $0.29 + $28.70 + $15 + $3.28 + $4.96. 2. Subtract it from $140 + $72.36 + $27.64. 3. Multiply it by 33-^, using a short method. 4. Divide it by 8 ; by 33 ; by $2.64. 5. Divide it by EXERCISE 7 Taking (a) $1038, (b) $1632, (c) $21.76, (d) $27.20, (e) $32.64, or (f ) $38.08, as the teacher directs : 1. Add it to $7.33 + $26 + $0.48 + $7.88 + $2.94. 2. Subtract it from $75 + $37.42 + $12.58. 3. Multiply it by 6.25. 4. Divide it by 8 ; by 17; by 34; by $2.72; by $1.36. 5. Divide it by 6 j ; by ll. EXERCISE 8 Taking (a) $11.20, (b) $16.80, (c) $22.40, (d) $33.60, (e) $28, or (f) $39.20, as the teacher directs : 1. Add it to 9 times itself, using a short method. 2. Subtract it from 11 times itself. 3. Multiply it by 37^. 4. Divide it by 8 ; by 7 ; by 5 ; by 35 ; by $1.40. . 5. Divide it by 8f; by 4|; by 108 MATERIAL FOR DAILY DRILL EXERCISE 9 Taking (a) $11.52, (b) $17.28, (c) $23.04, (d) $28.80, (e) $34.56, or (f ) $40.32, as the teacher directs : 1. Add it to $1.20 + $0.92 + $17 + $3.75 + $28.67. 2. Subtract it from $32.75 + $19.82 + $10'.18. 3. Multiply it by 62.5. 4. Divide it by 2 ; by 4 ; by 8 ; by 16 ; by 32 ; by 36. 5. Multiply it by -|. Divide it by 2|^. EXERCISE 10 Taking (a) $11.84, (b) $77.76, (c) $23.68, (d) $29.60, (e) $35.52, or (f) $41.44, as the teacher directs : 1. Add it to $12 + $16.75 + $0.82 + $2.98 -f $48.20. 2. Subtract it from $75 + $37.80 + $42.60 + $17.90. 3. Multiply it by 37.85. 4. Divide it by 2; by 4 ; by 8 ; by $0.37. 5. Multiply it by 0.12J; by J; by EXERCISE 11 Taking (a) 1216, (b) 182.4, (c) 24.32, (d) 0.4256, (e) 3.648, or (f) 0.304, as the teacher directs : 1. Add it to 9 + 15.75 + 21 + 5J. 2. Subtract it from 1300. 3. Multiply it by 0.365. 4. Divide it by 3.04 ; by 0.7 ; by 40 ; by 400 ; by 4000. 5. Divide it by 12 J; by 6J; by 3. MATERIAL FOR DAILY DRILL 109 EXERCISE 12 Taking (a) 124.8, (b) 18.72, (c) 24.96, (&) 0.312, (e) 3.744, or (f) 0.4368, as the teacher directs: 1. Add it 'to 3.848 + 148.276+175 + 48.76 + 9.009. 2. Subtract it from 4.6273+74.896 + 56.215. 3. Multiply it by 12.5, using a short method. 4. Divide it by 2 ; by 4 ; by 8 ; by 13 ; by 3.12 ; by 6. 5. Find 12-|% of it, using a- short method. EXERCISE 13 Taking (a) 13.44, (b) 20.16, (c) 2.688, (d) 0.4704, (e) 4.032, or (f) 0336, as the teacher directs: 1. Add it to 72.8796+182.08 + 7.087+72.6 + 0.983. 2. Subtract it from 2.786 + 46.93 + 53.17. 3. Multiply it by 342.87. 4. Divide it by 3 ; by 7 ; by 8 ; by 0.042 ; by 3.36. 5. Of what number is it 75% ? f ? 7J%? f% ? EXERCISE 14 Zfc&ingr (a) 1.408, (b) 07..Z0, (c) .&?, (d) 0.352, (e) 4.004, or (f) 0.4928, as the teacher directs: 1. Add it to 482.76894 + 9 + 0.987 + 0.7236 + 483. 2. Subtract it from 0.7+276.93 + 14.963 + 5.037. 3. Multiply it by f of J. 4. Divide it by 0.8; by 4.4; by 0.352; by 2.2; by f. 5. Of what number is it 80% ? 120% ? 1J ? f ? 110 MATERIAL FOR DAILY DRILL EXERCISE 15 Taking (a) 15.36, (b) 23.04, (c) 3.072, (d) 0.384, (e) 4.608, or (f) 0.5376, as the teacher directs: 1. Add it to 0.2702 + 298.742 + 0.7298 + 7017+ 2983. 2. Subtract it from 36.7+921.006+78.239 + 21.761. 3. Multiply it by 122J. 4. Divide it by 9|. 5. Of what number is it 125%? 1J? ? 12j%? EXERCISE 16 Taking (a) 15.68, (b) 2.352, (c) 313.6, (d) 0.392, (e) 470.4, or (f) 5488, as the teacher directs: 1. Add it to 0.1271+ 2789.762 + 2936 + 7064 + 0.8729. 2. Subtract it from 48.789 + 968.32 + 3429 + 6571. 3. Multiply it by itself. 4. Divide it by 0.0784. 5. What per cent is it of 5 times itself ? of half itself ? EXERCISE 17 Taking (a) 2.88, (b) 26.4, (c) 124.8, (d) 17.04, (e) 34.56, or (f) 69.12, as the teacher directs: 1. Add it to 125% of itself. 2. Subtract it from 200% of itself. 3. Multiply it by 0.5% of itself. 4. Divide it by 0.15 ; by 0.075 ; by 1.5. 5. Of what number is it 12%? 1.2%? 120%? PART II. GEOMETRY I. GEOMETRY OF FORM First Steps in Geometry. Thousands of years ago, when people began to study about forms, they were interested in pictures showing the shapes of objects ; these they used in decorating their walls, and later in showing the plans of their houses and their temples and in representing animals and human beings. As land became valuable they showed an interest in measuring objects, fields, and building ma- terial. When they wished to locate places on the earth's surface and when they began to study the stars, it was necessary that they should consider position. From very early times, therefore, the ideas of form, size, and position have interested humanity. There are three things which we naturally ask about an object : What is its shape ? How large is it ? Where is it ? It is these three questions that form the bases of the kind of geometry which we are now about to study. There are also other questions which we might ask about the object, such as these : How much is it worth ? What is its color ? Of what is it made ? None of these questions, however, has to do with geometry. The teacher will recognize that demonstrative geometry is not touched upon directly by the three questions above set forth. Another question might be asked relating to all three, namely, How do you know that your statement is true? It is this question which leads to the proof of propositions. For the present we are concerned almost exclusively with intuitional and observational geometry as related to the questions of shape, size, and position. Ill 112 GEOMETRY OF FORM Geometric Figures. You are already familiar with such common forms as the square, triangle, circle, arc, and cube. Such forms are generally known as geometric figures. Angle. Two straight lines drawn from a point form an angle. The two straight lines are called the sides of the angle, and the point where they meet is called the vertex. The three most important angles are the right angle, the acute angle, which is less than a right angle, and the obtuse angle, which is greater than a right angle. RIGHT ANGLE ACUTE ANGLE OBTUSE ANGLE If necessary, the teacher should explain what we mean when we say that an angle is greater than or less than another angle. This is easily done by slowly opening a pair of compasses. Acute angles and obtuse angles are called oblique angles. Triangle. A figure bounded by three straight lines is called a triangle. EQUILATERAL ISOSCELES RIGHT ACUTE OBTUSE The five most important kinds of triangles are the equi- lateral triangle, having all three sides equal; the isosceles triangle, having two sides equal ; the right triangle, having one right angle ; the acute triangle, having three acute angles; and the obtuse triangle, having one obtuse angle. The side opposite the right angle in a right triangle is called the hypotenuse of the right triangle. The sum of the sides of a triangle is called the perimeter. ANGLES AND TRIANGLES 113 Exercise 1. Angles and Triangles Examples 1 to 6, oral 1. Point to three right angles in the room. 2. Point, if possible, to two straight lines on the wall or on a desk which form an acute angle. 3. Point, if possible, to two straight lines hi the school- room which form an obtuse angle. 4. Which is the greater, an acute angle or an obtuse angle ? 5. How many right angles all lying flat on the top of a table will completely fill the space around a point on the table? 6. If one side of an equilateral triangle is 6 in., what .is the perimeter of the triangle ? 7. Draw a right angle as accurately as you can by the aid of a ruler. 8. Draw an acute angle and an obtuse angle, writing the name under each. 9. In this figure name by capital letters the tri- angles which seem to you to be right triangles. 10. In the same figure name by a small letter each of the acute angles. 11. In the same figure name by a capital letter each of the obtuse triangles. 12. What kinds of angles are represented hi the figure by the letters o, p, r, s ? Write the name after each letter. 13. If two straight lines intersect, what can you say as to any equal angles ? 114 GEOMETRY OF FORM Quadrilateral. A figure bounded by four straight lines is called a quadrilateral. The rectangle, square, parallelogram, and trapezoid, the four most important kinds of quadrilaterals, are shown below. RECTANGLE SQUARE PARALLELOGRAM TRAPEZOID A quadrilateral which has all its angles right angles is called a rectangle. A rectangle which has its sides all equal is called a square. A quadrilateral which has its opposite sides parallel is called a parallelogram. A quadrilateral which has one pair of opposite sides parallel is called a trapezoid. It is not necessary at this time to give a formal definition of parallel lines. The students are familiar with the term. We shall hereafter use the word line to mean straight line unless we wish to use the word straight with line for purposes of emphasis. Polygon. A figure bounded by straight lines is called a polygon. The quadrilaterals shown above are all special kinds of polygons, and a triangle is also a polygon. Polygons may have three, four, five, six, or any other number of sides greater than two. The side on which a polygon appears to rest is called the base of the polygon. The sum of all the sides of a polygon is called the perimeter of the polygon. The points in which each pair of adjacent sides intersect are called the vertices of the polygon. In the case of a triangle the vertex of the angle opposite the base is usually called the vertex of the triangle. 115 Congruent Figures. If two figures have exactly the same shape and size, they are called conc/ruent figures. Drawing Instruments. The instruments commonly used in drawing the figures in geometry are the compasses, the ruler, the protractor, and the right triangle. The compasses are used for drawing circles as here shown and also for laying distances on paper. A protractor of the general type here .shown is convenient for use by students, and with its aid angles ot any number of de- grees can be drawn. 0^2 \\ i\ \\ For work out of doors a surveyor measures angles and finds levels by means of a transit such as is here shown. Each student should have a ruler, a pair of compasses, and a protractor, since the constructions studied in this book can be made only by their use. If necessary such familiar terms as circle, radius, diameter, arc, and circumference should be explained informally. They are more for- mally stated later. On pages 117 and 119 and later in the work some interesting illus- trations of ancient instruments are given. Students often make similar instruments for use in geometry. 116 GEOMETRY OF FORM Constructing Triangles. We often have to construct triangles of various shapes and sizes. We shall first con- sider the following case : Construct a triangle having its sides respectively equal to three given lines. Let Z, m, n be the given lines. It is required to construct a triangle with I, m, n as sides. Draw a line with the ruler and on it mark off with the compasses a line AB equal to I. in It is more nearly accurate to do this with the compasses than with a ruler. With A as center and m as radius draw a circle; with B as center and n as radius draw another circle cutting the first at C. Draw AC and BC. Then because AB=l, AC=m, and BC=n it follows that ABC is the required triangle. Show why it is not necessary to draw the whole circle in either case. Teachers should informally explain to the students the methods commonly used in lettering a line, an angle, and a triangle. Exercise 2. Triangles 1. Construct a triangle with sides 2 in., 3 in., 4 in. Construct triangles with sides as follows : 2. 3 in., 4 in., 5 in. 5. -| in., ^ in., 1 in. 3. lin., 2 in., 2^ in. 6. 2Jin., 2J in., 2 in. 4. 1J in., 21 in., 3 in. 7. 3 in., 31 in., 3J in. In schools in which the metric system is taught it is desirable to use the system in this work. The necessary metric measures often will be found on protractors such as the one shown on page 115. ANCIENT INSTRUMENTS 117 uadrants used for measuring angles hundreds of years ago. German, Italian, and Hindu specimens, 118 GEOMETRY OF FOKM Isosceles Triangle. In the case studied on page 116 we see that the three sides need not all be equal. If two sides are equal we have to construct an isosceles triangle. Construct a triangle having two sides each equal to a given line and the base equal to another given line. The base of an isosceles triangle is always taken as the side which is not equal to one of the other sides. Let AB be the given base and let I be the given line. Then with center A and radius I draw a circle, and with center B and radius I draw another circle, or pref- erably only an arc in each case. Let the two arcs or the two circles intersect at the point C. Then ABC is the triangle required. l> Equilateral Triangle. From the preceding case we see that if the base is equal to each of the other sides, we shall have an equilateral triangle. Exercise 3. Isosceles and Equilateral Triangles 1. In making a pattern for the tiles used in the floor shown below it is necessary to draw an equilateral triangle of side 1 in. Draw such a triangle. Construct isosceles triangles with bases 1 in. and equal sides as follows : 2. fin. 3. -Jin. 4. 1-J in. 5. 2 in. rTTTTTTl YYYYYTY rYYYYYY 1 ! 6. Construct equilateral triangles with sides as follows : 7. 4- in. 8. -Jin. 9. 1A in. 10. II in. 11. 24 in. 4 O O ~ ANCIENT INSTRUMENTS 119 curious illustration from an Italian v>or^ of the seventeenth century showing the use of the ancient quadrant. The distance was required for the purpose of properly fixing the guns. The computations may be made in "various ways. 120 GEOMETRY OF FORM 12. Cut three isosceles triangles of different shapes from paper and fold each through the middle so that one of the equal sides lies exactly on the other. What inference can you make as to the equality or inequality of the angles which are opposite the equal sides? Write the statement as follows : In an isosceles triangle the angles opposite the equal sides are equal. 13. Draw three equilateral triangles of different sizes. With a protractor measure each angle in each of the triangles. What inference can you make as to the number of degrees in each angle ? Write the statement, beginning as follows: The number of degrees in each angle of, etc. 14. From Ex. 13 what inference can you make as to the number of degrees in the sum of the three angles of an equilateral triangle ? This is the same as the number of degrees in how many right angles ? 15. Draw three triangles of various shapes and investi- gate for each the conclusion drawn in Ex. 14. This is most easily done by cutting them from paper and then cutting off the three angles in each case and fitting them A B x Y together. Write the statement, beginning as follows: In any triangle the sum of the three angles is equal to, etc. 16. From the truth discovered in Ex. 15, find the third angle of a triangle in which two angles are 75 and 45. 17. In a certain right triangle one acute angle is 30. How many degrees are there in the other acute angle ? In this work the student is led to discover by experiment various important propositions to be proved later in his work in geometry. Teachers may occasionally find it advantageous to develop simple proofs in connection with this intuitional treatment. PEBPENDICULARS Perpendicular. A line which makes a right angle with another line is said to be perpendicular to that line. One of the best practical methods of constructing a line perpendicular to a given line and passing through a given point is shown in this illustration. Place a right tri- angle AB C so that BC lies along the given line. Lay a straight- edge or ruler along AC, as in the left-hand figure. Since you wish the per- pendicular line, or perpendicular, to pass through the point P, slide the triangle along MN until AB passes through the point P, as shown in the right-hand figure. Then draw a line along AB, and it will be perpendicular to the line XY and will pass through, the point P. Exercise 4. Perpendiculars 1. Draw a line XY and mark a point P about -|in. below it. Through P construct a line perpendicular to XY, by the above method. 2. Through a point P on the line XY construct a line perpendicular to XY, by the above method. 3. Construct a right triangle in which the two shorter sides shall be 1|- in. and 2 in. 4. Construct a square having its side 2 in. 5. Draw a picture showing how two carpenter's squares can be tested by standing them on any flat surface with two edges coinciding and two other edges extending in opposite directions. 122 GEOMETRY OF FORM Other Methods of Constructing Perpendiculars. There are other convenient methods of constructing perpendiculars. From a given point on a given straight line construct a perpendicular to the line. Let AB be the given line and P be the given point. With P as center and with any con- venient radius draw arcs intersecting A I \ -B A. P" AB at X and Y. u * With X as center and XY as radius draw a circle, and with Y as center and the same radius draw another circle, and call one intersection of the circles C. p With a ruler draw a line from P to (7. From a given point outside a given straight line construct a perpendicular to 3" the line. Let AB be the given line and P be the given point. How are the points X and Y fixed ? Then how is the point C fixed ? Draw the perpendicular PC. Exercise 5. Perpendiculars 1. In making a pattern for a tiled floor- like the one here shown it becomes necessary to draw a square 1 in. on a side. Construct such a square, using the first of the above methods. 2. Construct a rectangle as in Ex. 1, using the second of the above methods. 3. Given two points on a given line, construct perpen- diculars to the line from each of them. ANCIENT INSTRUMENTS 123 arly leveling instruments^ wit A a picture from a published in 1624 showing their use. 124 GEOMETRY OF FORM Bisecting a Line. To divide a line into two equal parts is to bisect it. In constructing the common figures we often have to bisect a line. We can bisect a line ^x> roughly by measuring it with a ruler, but for accurate work we have a much better method. Bisect a given line. M Let AB be the given line. What is now required ? With A and B as centers and with radius greater than \AB draw "^" arcs. The most convenient radius is usually AB itself. Call the points of intersection X and Y. Draw the straight line XY, and call the point where it cuts the given line M. Then XY bisects AB at M. This is much more nearly accurate than it is to measure the line with a ruler and then take half the length. Bisecting an Angle. To draw a line from the vertex of an angle dividing it into two equal angles is to bisect it. Bisect a given angle. Let AOB be the given angle. What is now required ? With as center and with any con- venient radius draw an arc cutting OA at X and OB at Y. With X and Y respectively as centers and with a radius greater than half the distance from X to Y draw arcs and call their point of intersection P. Draw OP. Then OP is the required bisector. This is much more nearly accurate than it is to measure the angle with a protractor and then take half the number of degrees. SIMPLE CONSTRUCTIONS 125 Exercise 6. Simple Constructions 1. Draw a line 4.5 in. long. Bisect this line with ruler and compasses. Check the construction by folding the paper at the point of bisection, making a fine pinhole through one end of the line to see if it strikes the other end. 2. Construct a triangle having two of its sides 3 in., the third side being less than 6 in. 3. Construct a triangle having its sides respectively 2 in., 2.5 in., and 3 in. 4. Draw a line 4 in. long, and at a point 1 in. from either end construct a perpendicular to the line. 5. Is it possible to construct a triangle having its sides respectively 3 in., 2 in., and 1 in. ? If not, what is there in the general nature of these lengths which makes such a triangle impossible ? 6. With a protractor draw an angle of 35. Bisect this angle and check the work with the protractor. To draw the angle of 35 draw a line, mark a point O upon it, lay the hypotenuse of a triangular protractor on it, sliding it down slightly so that the center of the circle rests on O. Lay a ruler on the protractor from along the line of 35 and mark a point on the paper. Remove the protractor and draw a line from to the point. Construct the triangles ivhose sides are as follows and bisect all three angles of each triangle: 7. 4 in., 3 in., 41 in. 10. 3|- in., 4^ in., 5^ in. 8. 5 in., 7 in., 8 in. 11. 7-| in., 4-| in., 5 in. 9. 6 in., 3 in., 5 in. 12. 3-| in., 3^ in., 3-| in. 13. In Exs. 7-12 what do you observe as to the way in which the three bisectors meet ? Write a statement of your conclusion, beginning .as follows: The bisectors of the three angles of a triangle, etc. 126 GEOMETRY OF FORM Constructing an Angle equal to a Given Angle. In copy- ing figures we often have to construct an angle equal to a given angle. This leads to the following construction : From a given point on a given line construct a line which shall make with the given line an angle equal to a given angle. 15 Q Let P be the given point on the given line PQ and let angle AOB be the given angle. What is now required ? With as center and with any radius draw an arc cut- ting OA at C and OB at D. \Vith P as center and with OC as radius draw an arc cutting PQ at M. With M as center and with the straight line joining C and D as radius draw an arc cutting the arc just drawn at JV, and draw PN. Then the angle MPN is the required angle. Exercise 7. Simple Constructions Construct triangles with sides as follows and bisect all three of the sides of each triangle : 1. 5 in., 6 in., Tin. 4. 3 in., 3^ in., 4J in. 2. 4 in., 4 in., 7 in. 5. 2J in., 4 in., 4 in. 3. 3^ in., 4 in., 7-| in. 6. 3 in., 3^ in., 4 in. Interesting figures may be formed by connecting the points of bisection and shading in various ways the parts thus formed. SIMPLE CONSTRUCTIONS 127 7. Construct a triangle ABC with AB = 1 in., AC\\ in., angle A = 30, and then construct another triangle XYZ with XF=1 in., XZ=lJin., angle X= 30. Are the triangles ABC and XYZ congruent? 8. From Ex. 7 write a complete statement of the truth inferred, beginning as follows: Two triangles are congruent if two sides and the included angle of one are respectively equal to, etc. 9. Construct a triangle ABC in which angle A 30, angle B = 60, AB = 1^- in., and then construct another triangle XYZ in which angle X= 30, angle r=60, -YF=l^in. Are these triangles congruent? What is the reason ? Write a complete statement of the truth inferred, as in Ex. 8. 10. Construct a triangle ABC in which AB = 1 in., 2? (7=1^ in., CL4=l|-in., and then construct another triangle XYZ in which XY= 1 in., YZ=l^ in., ZX= 1 in. Are these triangles congruent? Write a complete statement of the truth inferred, as in Ex. 8. 11. Construct a triangle with angles 30, 60, and 90, and another triangle with sides twice as long but with angles the same. Are these triangles congruent? Are triangles in general congruent if the angles of one are respectively equal to the angles of the other ? 12. As in Ex. 11, construct two triangles with angles 45, 45, and 90, one with sides three times as long as the other. 13. Try to construct a triangle with angles 45, 60, and 90. If you have any difficulty in making the con- struction, write a statement of the cause. 14. Try to construct a triangle with angles 45, 45, and 100. If you have any difficulty in making the construction, write a statement of the cause. 128 GEOMETRY OF FOKM Parallel Lines. One of the most common constructions in making architectural and mechanical drawings is to draw one line parallel to another line. For ^ practical purposes one of the best plans is to place a wooden or celluloid tri- angle ABC with one side BC on the given line, lay a ruler along another side AB, and then slide the triangle along the ruler to the position A'B'C' (read .4-prime, J5-prime, (7-prime). Then B'C' is parallel to BC. A triangular protractor like the one shown on page 115 of this book may be used for the above purpose. Draftsmen in offices of architects or in machine shops often use a T-square as here shown. As the part MN slides along the edge CD of a draw- ing board, the part OP moves parallel to its original position. Drawing EF and sliding the T-square along, we can easily draw lines parallel to EF. A second T-square may slide along BD if the board is rectangular, and thus lines can be drawn perpendicular to the line EF or to any lines parallel to it. When the lines are very long, this is the best method. Draftsmen also use a parallel ruler like the one here shown. They also use a cylindric ruler, rolling it along the paper as a guide for r-r i parallel lines. In gen- VJ /? eral, however, the plan J/ II of sliding a triangle along a ruler is one of the simplest and at the same time is accurate. It should be used in the exercises which follow. D ANCIENT INSTRUMENTS 129 Early uses of geometry in studying the stars, ofe, an astrolabe used in measuring the angles of stars abo'Ve the orizon. "Below, an ancient Hindu bronze sphere of the AeaTens, -with stars inlaid in silver. 130 GEOMETRY OF FORM L M N O Dividing a Line. We often need to divide a line into a given number of equal parts; that is, to solve this problem: Divide a given line into any given number of equal parts. Let AB be the given line, and let it be required to divide AB into five equal parts. Draw any line from A, as AX. Mark off on AX with the com- passes any five equal lengths AP, PQ, QR, RS, and ST. Draw TB, and then, by sliding a triangle along a ruler, draw SO, RN, QM, and PL parallel to TB. Then AB is divided into five equal parts, AL, LM, MN, NO, and OB. The material for another very simple method may be easily prepared by the student. Let him rule a large sheet of paper with several parallel lines at equal intervals, and number these lines as shown on the edge. If it is desired to divide the line AB into five equal parts, place the paper on which AB is drawn over the ruled paper so that the line passes through A and the line 5 through B. Lay the ruler along each ruled line in turn and mark each point of division. In this way the four required points of division may be accurately found. SIMPLE CONSTRUCTIONS 131 Exercise 8. Simple Constructions 1. Draw a line 5 in. long and divide it into nine equal parts by using ruler, triangle, and compasses. Construct triangles whose sides are as follows, and con- struct a perpendicular to each side at its midpoint: 2. 4-| in., 4^ in., 5 in. 4. 5-^- in., 5^ in., 6^ in. 3. 3J in., 31 in., 51 in. 5. 4 in., 5J in., 6J in. The teacher should ask for the inference as to the meeting of the three perpendicular bisectors of the sides of a triangle. 6. With a protractor draw an angle of 45. With ruler and compasses bisect this angle. Check the construction by folding the paper ; by using the protractor. 7. Draw any triangle, bisect the sides, and join the points of bisection, thus forming another triangle. With ruler and triangle test to see whether the sides of the small triangle are parallel to those of the large triangle. 8. Repeat Ex. 7 for a triangle of different shape. What general law do you infer from these two cases ? v 9. Draw a line 4^ in. long and divide it into seven equal parts. '' 10. Construct a square 3 in. on a side. If the figure is correctly drawn, the two diagonals will be equal. Check by measuring the diagonals with the compasses. If such words as diagonal are not familiar they should be explained by the teacher when they are met. It is desirable to avoid formal definitions at this time, provided the students use the terms properly. 11. Construct two parallel lines and draw a slanting line cutting these lines so that eight oblique angles are formed. Name the various pairs of angles in the figure that appear to be equal. 132 GEOMETRY OF FORM Geometric Patterns. By the aid of the constructions described on pages 116-130 it is possible to construct a large number of useful and interesting patterns, designs for decorations, and plans for buildings or gardens. To secure the best results in this work the pencil should be sharpened to a fine point and should contain rather hard lead, and the lines should be drawn very fine. Exercise 9. Geometric Patterns 1. By the use of compasses and ruler construct the following figures : The lines made of short dashes show how to fix the points needed in drawing a figure, and they should be erased after the figure is completed unless the teacher directs that they be retained to show how the construction was made. 2. By the use of compasses and ruler construct the following figures : It is apparent from the figures in Exs. 1 and 2 that the radius of the circle may be used in drawing arcs which shall divide the circle into six equal parts by simply stepping round it. GEOMETRIC PATTERNS 133 3. By the use of compasses and ruler construct the following figures, shading such parts as will make a pleasing design in each case : 4. By the use of compasses and ruler construct the following figures, shading such parts as will make a pleasing design in each case : 5. By the use of compasses and ruler construct the following figures : In such figures artistic patterns may be made by coloring portions of the drawings. In this way designs are made for stained-glass windows, for oilcloths, for colored tiles, and for other decorations. 134 GEOMETKY OF FORM 6. By the use of compasses and ruler construct the following figures, leaving the dotted construction lines: As stated on page 133, artistic patterns may be made by coloring various parts of these drawings. Interesting effects are also pro- duced in black and white, as in the designs in Ex. 9 on page 135. 7. Draw a line 1^ in. long and divide it into eighths of an inch, using the ruler. Then with the compasses construct this figure. It is easily shown, when we come to the measurement of the circle, that these two curve lines divide the space inclosed by the circle into parts that are exactly equal in area. By continuing each semicircle to make a complete circle another inter- esting figure is formed. Other similar designs are easily invented, and stu- dents should be encouraged to make such original designs. 8. In planning a Gothic window this drawing is needed. The arc BC is drawn with A as center and AB as radius. The small arches are drawn with A, D, and B as centers and AD as radius. The center P is found by using A and B as centers and AE as radius. How may the points Z>, E, and F be found ? Draw the figure. A GEOMETRIC PATTERNS 135 9. Copy each of the following designs, enlarging each to twice the size shown on this page : This example and the following examples on this page may be omitted by the class at the discretion of the teacher if there is not enough time for such work in geometric drawing. 10. This figure shows a piece of inlaid work in an Italian church. Construct a design of this general nature, changing it to suit your taste. Construct the figures as accu- rately as you can. 11. Construct a design for par- quetry flooring, using only com- binations of squares. 12. Repeat Ex. 11, using com- binations of squares and equilat- eral triangles. 13. Repeat Ex. 11, using combinations of squares, rec- tangles, and equilateral triangles. 14. Construct a design for a geometric pattern for lino- leum, using only combinations of circles and squares. 15. Repeat Ex. 14, using only combinations of circles and equilateral triangles. 16. Repeat Ex. 14, using only combinations of circles, squares, and equilateral triangles. 136 GEOMETRY OF FOKM Drawing to Scale. The ability to understand drawings, maps, and other graphic representations depends in part upon knowing how to draw to scale. Thus, if your schoolroom is 30 ft. long and 20 ft. wide, and you make a floor plan 3 in. long and 2 in. wide, you draw the plan to scale, 1 in. representing 10 ft. We indi- cate this by writing: "Scale, 1 in. = 10 ft." We may also write this: "Scale, 1 in. = 120 in.," or "Scale ^5," We often write 1' for 1 ft. and 1" for 1 in., so that the scale may also be indicated as 1" = 10'. The following shows a line AS drawn to different scales : ^J. j The line AB drawn to the scale The line AB drawn to the scale The line AB drawn to the scale The figures shown below illustrate the drawing of a rectangle to scale. In this case the lower rectangle is a drawing of the upper one to the scale ' -^-, or 1 to 2, or 1" to 2". Notice that the area of the lower rectangle is only that of the upper one. When we draw to the scale -| we mean that the length of every line is ^ the length of the corresponding line in the original. Whatever the shape of the figure, the area will then be ^ the area of the original figure. Maps are figures drawn to scale. The scale is usually stated on the map, as you will see in any geography. The scale used on a map is often expressed by means of a line' divided to represent miles, and sometimes by such a statement as that 1 in. = 100 mi. 1ST Exercise 10. Drawing to Scale 1. Measure the cover of this book. Draw the outline to the scale ^. This means that the four edges are to be drawn to form a rec- tangle like the front cover, with no decorations. 2. Measure the top of your desk. Draw a plan to the scale ^. 3. If a line 1 in. long in a drawing represents a dis- tance of 8 ft., what distance is represented by a line 3f in. long ? by a line 4|- in. long ? by a line 1.5 in. long ? 4. If the scale is 1 in. to 1 ft, what distance on a drawing will represent 6 ft. 3 in. in the object drawn ? 5. A drawing of a rectangular floor 20 ft. by 28 ft. is 5 in. by 7 in. What scale was used ? 6. A farmer plotted his farm as here shown, using the scale of 1 in. to 40 rd. Find the dimensions of each plot. WHKAT OATS CORN PASTURE WOOD LOT ! O [DWELLING ! j AND i S BARNS 7. A plan of a rectangular school garden is drawn to the scale of 1 in. to 2 ft. 6 in. The plan is 18 in. long and 12^ in. wide. What are the dimensions of the garden? 8. The infield of a baseball diamond is 90ft. square. Draw a plan to the scale of 1 in. to 20 ft. 138 GEOMETRY OF FORM 9. The field of play of a football field is 300 ft. long and 160 ft. wide. Lines parallel to the ends of the field are drawn at intervals of 5 yd., and the goals, 18 ft. 6 in. wide, are placed at the middle of the ends of the field. Draw a plan to the 'scale of 1 in. to 60 ft. and indicate the position of the goals and of the 5-yard lines. 10. A double tennis court is 78 ft. long and 36 ft. wide. Lines are drawn parallel to the longer sides and 4 ft. 6 in. from them, and the service lines are parallel to the ends and 18 ft. from them. The net is halfway between the ends. Draw a plan to any convenient scale. 11. The drawing here shown is the floor plan of a certain type of barn. Determine the scale to which the i i Cfl i i TTLE I 1 1 STRL i LS I 1 1 i FHSSRGE box STHLL -*-- j > x c |8 -v, DRIUEWHY SLIDING ^ FEED rHSSHGE MO RSE STHL LS ROOH 1 1 i i | * 1 1 | \ plan is drawn, find the width of the driveway in the barn, the width bf each horse stall, the width of each cattle stall, and the dimensions of the box stall and the feed room. DRAWING TO SCALE 139. 12. A class in domestic science drew a plan for a model kitchen in an apartment house, using the scale -^-. If the plan is 3 in. long and 2 in. wide, what are the actual dimensions of the kitchen ? 13. A drawing was made of a lamp screen 20- in. high.. The drawing being 2^- in. high, what scale was used? 14. The drawing below is the plan for a concrete bungalow. Find the scale used in drawing the plan. 15. In Ex. 14 find the dimensions of the living roon\ dining room, and smaller bedroom including wardrobe. 140 GEOMETRY OF FORM Accurate Proportions. Suppose that you measure a rec- tangular room and. find it to be 20 ft. long and 16 ft. wide, and suppose that you measure a drawing of the room and find it to be" 10 in. long and 6 in. wide. You would con- clude that the drawing is not a good one, because the width should be, as in the room, |- of the length. An accurate drawing or picture must maintain the pro- portions of the object. That is, if the width of the object is ^ of the length, the width of the object shown in the drawing must be ^ of the length in the drawing ; if the width of the object is of the length in one case, it must be -g- of the length in the other case ; and so on for other proportions. It is better at this time to explain informally the meaning of proportion, as is done above. A more formal explanation of the subject of proportion is given later in the book when it is needed. Exercise 11. Accurate Proportions 1. A house is 36 ft. high and the garage is 20 ft. high. If the house is represented in a drawing as 18 in. high, how high should the drawing of the garage be ? In all such cases the objects are supposed to be at approximately the same distance from the eye, so that the element of perspective does not enter. 2. A landscape gardener is drawing to scale a plan for a rectangular flower garden 18 ft. long and 14 ft. wide. In the drawing the length is represented by 6^ in. By what should the width be represented ? 3. Draw a right triangle whose sides are 3 in., 4 in., and 5 in. respectively, and draw another right triangle of the same shape but with the hypotenuse l^in. long. SIMILARITY OF SHAPE 141 Similarity of Shape. As we have already seen, it is frequently necessary to draw a figure of the same shape as another one, but not of the same size. For example, an architect or a map drawer may reduce the original by using a small scale, but if we are making a drawing of a small object seen through a micro- scope we use a large scale. But whether the drawing reduces or enlarges the original, the shape remains the same. Figures which have the same shape are said to be similar. For example, here are two drawings of a hand mirror. In outline each drawing is similar to the mirror itself, and each is also similar to the other. Figures which are similar to the same figure are similar to each other. Two maps of a state are not only similar in outline to the state itself, but each is similar in outline to the other. Exercise 12. Similarity of Shape 1. Construct three equilateral triangles whose sides are respectively 2 in., 3|- in., and 5 in. Are they similar ? 2. Construct three rectangles, the first being 1^- in. by 2|- in. ; the second, 3 in. by 5 in. ; and the third, 2 in. by 2-^- in. If they are not all of the same shape, discuss the exception. 3. Construct a right triangle of the same shape as this triangle but twice as high, and another of the same shape but three times as high. 142 GEOMETRY OF H)KM Angles in Similar Figures. Here are two similar right triangles, ABC and A'B'C', and in each triangle a perpen- dicular (jt?, p' respectively) is drawn from the vertex of the right angle to the hypotenuse. B' Are the figures still similar ? Are the sides proportional ? What can be said as to the corresponding angles? This brings us to another property of two similar figures, namely, that the angles of one are equal respectively to the angles of the other. That is, in similar figures, corresponding lines are in proportion and corresponding angles are equal. A close approximation to similar figures may be seen in the case of moving pictures. The large picture shown on the screen is sub- stantially similar to the small picture on the reel, although there is some distortion, particularly around the edges. Exercise 13. Similar Figures All work oral State which of the following pairs of figures are necessarily similar and state briefly the reasons in each case: 1. Two squares. 4. Two rectangles. 2. Two triangles. 5. Two isosceles triangles. 3. Two circles. 6. Two equilateral triangles. 7. State whether two parallelograms, each side of one being 3 in. and each side of the other being 4 in., must always be similar, and give the reason for your answer. SIMILAR FIGURES 143 Similar Figures in Photographs. If you have ever used a plate camera you have seen that there is a piece of ground glass in the back and that an object in front of the camera ap- pears inverted on this ground glass. The reason is clear, for the ray of light from the point A of the flower passes through the lens of the camera and strikes the plate at A. That is, On a photographic plate the figure is similar in outline to the original, but is inverted. There is, of course, a slight distortion on account of the refrac- tion of the rays of light in passing through the lens. If the camera is 8 in. long and the object is 16 in. away from the lens 0, an object 5 in. high will appear as 2* in. high on the plate. That is, since the length of the ' "] 5in B _J in> ~-~ 16 in. camera, B'O, is half the 2 * in -JI--~- ~~~~o = ~' distance of the object from 0, or half of OB, we see that A'B', the height of the object on the plate, is half of AB, that is, half the real height. Similarly, if the length of the camera is 10 in., arid the height of an object 18 ft. away is 5 ft., we can easily find the height of the object on the plate as follows : Reducing all the measurements to inches, we have 18 ft. = 18 X 12 in., and 5 ft. = 5 x 12 in. 10 Then x 5 x 12 in. = 2| in. 18 x 12 The teacher is advised to solve this on the blackboard. 144 GEOMETRY OF FORM Exercise 14. Similar Figures in Photographs 1. A man 5 ft. 8 in. tall stands 16 ft. from a camera which is 8 in. long. What will be the height of his photograph? Explain by drawing to scale. 2. The photograph of a man who is 5 ft. 8 in. tall is 6 in. high, and the camera is 10 in. long. How far did the man stand from the- camera ? 3. If a boy's face is 8 ft. from a camera which is 10 in. long, the height of the photograph of his face is what pro- portion to the height of his face? If he places his hand 2 ft. nearer the camera, the length of the photograph of his hand is what proportion to the length of his hand ? One of the first things a beginner has to learn in using a camera is that objects appear distorted unless they are at about the same distance from the camera, especially if they are relatively near to it. 4. A tree photographed by a 4-inch camera at a distance of 10 ft. appears on the photograph as 6 in. high. How high is the tree? We see by this problem that heights and distances can often be found by photography ; and, in fact, much difficult engineering work is now done with the aid of photographs. 5. A camera is held directly in front of the middle of a door and at a distance of 8 ft. from it. The door is 4 ft. by 7 ft. 6 in. and the length of the camera is 8 in. Find the dimensions of the door in the photograph. 6. A 10-inch camera is placed at a certain distance from a tree which is 50 ft. high, and a boy 5 ft. tall stands between the tree and the camera. The height of the boy in the photograph is 1-J- in., and the height of the tree 8 in. Find the distance of both the boy and the tree from the camera. THE PANTOGRAPH 145 The Pantograph. Probably you have seen an instru- ment which is extensively used by architects, draftsmen, designers, and map makers in drawing plane figures simi- lar to other plane figures. It usually consists of four bars parallel in pairs, and is known as a pantograph. In explaining the pantograph it becomes necessary to speak of the ratio of two lines. By the ratio of 2 ft. to 5 ft. is meant the quotient 2 ft. -f- 5 ft., or |-, and by the ratio of ^ in. to 3^ in. is meant \ in. -s- 3^ in., or -^. Likewise, by the ratio of a line AB to a line CD is meant the quotient found by dividing the length of the line AS by the length of CD. This ratio is written AB/CD, or AS: CD. If AB is half CD, then AB:CD = \. This is read "the ratio of AB to CD is equal to one half." In the figure the bars are adjustable at B and E. The end A is fixed, that is, it remains in the same place while the pantograph is c being used. A trac- ing point is placed at T and a pencil at P, and BP and PE are so adjusted as to form a parallelogram PECB such that any required ratio AB-.AC is equal to CE-.CT. Then as the tracer T traces a given figure, the pencil P draws a similar figure. If the given figure is to be enlarged instead of reduced, the pencil and the tracing point are interchanged. This discussion of the pantograph has little value unless the in- strument is actually used by the students. A fairly good one can be made of heavy cardboard or of strips of wood, and school-supply houses will furnish ,a school with the instrument at a low cost. A simple pantograph can be made by fastening a rubber elastic at one end, sticking a pencil point through the other end, and placing a pin for a tracer anywhere along the band. 146 GEOMETKY OF FORM Exercise 15. The Pantograph 1. Draw a 'plan of your schoolroom to scale and then enlarge it to twice the size with the aid of a pantograph. This exercise should be omitted in case the school is not supplied with a pantograph. 2. Find the map of your state in a geography and re- duce it to half the size by using a pantograph. 3. By using a pantograph reduce the size of this plan of a cottage to two thirds its present size. This can be done by laying this page flat on the drawing board while someone holds the book. It is better, however, to copy the plan on paper and use the pantograph with the drawing. It is desired that the student should use the pantograph a few times in connection" with Various kinds of work in which it is really used in practical life. 4. By using a pantograph enlarge this sketch for a child's coat to three times the given size. In addition to this, other similar drawings should be made and then enlarged. Of late the pantograph has come into extensive use by dressmakers for the purpose of enlarging designs of this kind. 5. Draw a sketch of any object in the room and reduce the sketch to one third its size by using, a pantograph. 6. Draw a sketch of a tree near the school and enlarge the sketch to five times its size by using a pantograph. FIRST FLOOR FMN SYMMETRY 14T Symmetry. If we place a drop of ink on a piece of paper and at once fold the paper so as to spread the ink, we shall often find curious and interest- ing forms frequently resembling flowers, leaves, or butterflies. These forms are even more interesting if we use a drop of black ink and a drop of red ink. The interest in such figures comes from the fact that they are symmetric, that is, that one side is exactly like the other. In this case we say that the figure is symmetric with respect to an axis, this axis being the crease in the paper or, more generally, the line which divides the figure into two parts that will fit each other if folded over. In architecture we often find symmetry with respect to an axis. For example, in this picture of the interior of a great cathedral we see that much of the beauty and gran- deur is due to symmetry. This case is evidently one of symmetry with respect to a plane instead of with respect to a line. We may also have symmetry with respect to a center, that is, a figure may turn halfway round a point and appear exactly as at first. This is seen in a circle, or, among solids, in a sphere. It is also seen in the Gothic window shown on page 148. Symmetry of all kinds plays a very important .part in art, not merely in architecture, painting, and sculpture, but in all kinds of decoration. 148 GEOMETKY OF FOKM Exercise 16. Symmetry 1. Has this Gothic window an axis of symmetry? If so, draw the circle and indicate the axis of symmetry. If it has more than one axis of symmetry, draw each axis of symmetry. 2. If the figure has a center of symmetry, indicate this center in your rough sketch by the letter 0. 3. Draw an equilateral triangle and draw all its axes of symmetry. 4. Draw a square and draw all its axes of symmetry. 5. Draw a plane figure with no axis of symmetry ; one having only one axis of symmetry ; one having two axes of symmetry; one having any number of axes of symmetry. 6. Draw the following designs in outline and indicate by letters all the axes of symmetry in each design: 7. Write a list of three windows in churches in your locality which have axes of symmetry. If you know of any window which has a center of symmetry, mention it. The class should be asked to mention other illustrations of axes of symmetry, as in doors and in linoleum patterns. There should also be questions concerning planes of symmetry, as in a cube, a sphere, a chair, animals, and vases. Objects in the schoolroom offer a good field for inquiry. CUKVES 149 Plane Figures formed by Curves. We have already men- tioned a number of figures formed by curve lines without attempting to define them. We shall now mention these again and shall discuss more fully a few of those which occur most frequently in drawing, pattern making, architecture, measuring, and the like. This figure represents a circle with center 0, radius OA, and diameter BC. The circle is sometimes thought of as the space inclosed and sometimes as the curve line inclosing the space. The length of this curve is called the circumference, and sometimes the curve itself is called by this name. It is not expected that the above statement will be considered as a formal definition to be learned. All that is needed at this time is that the terms shall be used properly. Teachers should recognize that circle and circumference both have two meanings, as stated above. Another interesting figure, but one which is used not nearly so often as the circle, is the ellipse. If we place two thumb tacks at A and B, say 3 in. apart, and fasten to them the ends of a string which is more than 3 in. long, draw the string taut with a pencil point P, and then draw the pencil round while keep- ing the string taut, we shall trace the ellipse. It is evident that an ellipse has two axes of symmetry and one center of symmetry. The orbits of the planets about the sun are ellipses. When facilities for drawing permit, the student should draw ellip- ses of various sizes and shapes and should satisfy himself that two ellipses are not in general similar. 150 GEOMETRY OF FOKM Solids bounded by Curved Surfaces. We have often mentioned the sphere, and shall now speak of it and of other solids bounded in whole or in part by curved surfaces. A sphere is a solid bounded by a surface whose every point is equidistant from a point within, called the center. We also speak of the radius and diameter of a sphere, just as we speak of the radius and diameter of a circle. A cylinder is a solid bounded by two equal circles and a curved surface as shown in this figure. The two circles which form the ends are called the bases of the cylinder and the radius and diameter of either base are called respectively the radius and diameter of the cylinder. The line joining the centers of the two bases is called the axis of the cylinder, and its length is called the height or altitude of the cylinder. A cone is a solid like the one here shown. It has a circular base and an axis of symmetry from the center of the base to the vertex of the cone. The per- pendicular distance from the vertex to the base is called the height or altitude of the cone. The words radius and diameter are used as with the cylinder. In this book we shall consider only cylinders and cones in which the axes are perpendicular to the bases. SOLIDS BOUNDED BY CURVES 151 Exercise 17. Solids bounded by Curves 1. If you cut off a portion of a sphere, say a wooden ball, by sawing directly through it, but not necessarily through the center, what is the shape of the flat section ? Illustrate by a drawing. 2. In Ex. 1 is. the section always a plane of symmetry? If not, is it ever a plane of symmetry, and if so, when ? 3. A cylinder is symmetric with respect to what line or lines ? with respect to what plane or planes ? 4. Could a cylindric piece of wood, say a broom handle, be so cut that the section would be a circle ? If so, how should it be cut ? Could it be so cut that the section would seem to be an ellipse ? Illustrate each answer. The section last mentioned is really an ellipse, and this is proved in higher mathematics. 5. Could a cylindric piece of wood be so cut that the section would be a rectangle ? a trapezoid ? Illustrate. 6. Three cylinders of the same height, 4 in., have as diameters 3 in., 4 in., and 5 in. respectively. Can a section in any one of them be a square ? Illustrate the answer. 7. Is a cone symmetric with respect to any line ? to any plane ? Illustrate each answer. 8. How could a cone be cut so as to have the section a circle ? a triangle ? apparently an ellipse ? Illustrate. The section last mentioned is really an ellipse, and this is proved in higher mathematics. This is the reason why an ellipse is called a conic section. Other conic sections are studied in higher mathe- matics, and they are important in the study of astronomy, mechanics, and other sciences. 9. How is the largest triangle obtained by cutting a cone ? Illustrate the answer. 152 GEOMETRY OF FORM Exercise 18. Review 1. If the rays of light from any object ABC pass through a small aperture of an opaque screen and fall upon another screen parallel to the object, an inverted image A'B'C' c will be formed as here shown. If B the object is 5 ft. long and 9 ft. A from 0, how far from must the second screen be placed so that the image shall be 6 in. long ? How far, so that the image shall be 8 in. long ? 2. With the aid of ruler and compasses, construct figures similar to each of the following figures, but twice as large, and indicate the axis, axes, or center of symmetry of each. 3. Draw the following figure about half as large again and make it the basis for a pattern for lino- leum, using other lines as necessary. 4. Draw a square and cut it into four tri- angles by means of two diagonals. Describe the triangles with respect to their being equi- lateral, isosceles, or right. 5. Draw this figure about half ,as large r \ again and make it the basis for a pattern J^ J-^ for a church window, using other lines as ( j may be necessary for the purpose. ^_^^_x OUTDOOR PROBLEMS 153 Exercise 19. Optional Outdoor Work 1. Collect, if possible, several leaves of each of the following kinds of tree : oak, elm, maple, pine, and poplar. Are the leaves of each kind of tree approximately similar to the other leaves of the same tree ? Has each of the leaves an axis of symmetry ? 2. Do you know of any building lots or fields that are triangular? If so, make rough outline drawings of them. 3. Do any of the public buildings of your community have cylindric columns ? If so, which buildings ? 4. Do you know of any church spires that are conic in shape ? If so, which spires ? 5. What is the shape most frequently used in decorat- ing the interiors of churches in your vicinity ? 6. Can you find an illustration of a Gothic window in any of the churches in your vicinity ? If so, where ? 7. If there is a standpipe in your vicinity, what is its shape ? What is the shape of most of the smokestacks of the factories in your community ? 8. Notice the designs in the carpeting, wall paper, and linoleum exhibited by various stores. What general pattern or design is most frequently used ? 9. If convenient, inspect a house that is being built and compare the floor plan with the plans of the contractor or architect. What scale was used in drawing the plan? 10. Name illustrations of each of the following forms that you have seen in your community: circle, rectangle, cylinder, cone, sphere, trapezoid, and triangle. As stated above, this work is purely optional. It is suggestive of a valuable line of local questions. 154 GEOMETRY OF FORM Exercise 20. Problems without Figures 1. How do you construct a triangle, having given the lengths of the three sides ? 2. How do you construct an isosceles triangle, having given the base and one of the equal sides ? 3. How do you construct an equilateral triangle, having given one of the sides ? 4. State two methods of drawing from a given point a perpendicular to a given line. 5. If you have a line drawn on paper, what is the best way you know to bisect it ? 6. How do you bisect an angle ? 7. How do you construct an angle exactly equal to a given angle ? 8. How do you draw a line parallel to a given line ? 9. If you have a line drawn on paper and wish to divide it into five equal parts, how do you proceed ? 10. How do you construct a six-sided figure in a circle, the sides all being equal ? 11. How do you draw to a given scale the rectangular outline of the printed part of this page ? 12. How do you draw a plan of the top of your desk to the scale of a certain number of inches to a foot ? 13. When you know the scale which was used in draw- ing a map, how do you find the actual distance between two cities which are shown on the map ? 14. How can you enlarge a drawing by the aid of a pantograph ? 15. How do you determine whether a figure is symmetric with respect to an axis ? GEOMETRY OF SIZE 155 II. GEOMETRY OF SIZE Size. On page 111 we found that geometry is concerned with three questions about any object : What is its shape ? How large is it ? Where is it ? Thus far we have con- sidered the shape of objects; we shall now consider size. There are several ideas to be considered when we think and speak of the size of objects, such as length, area, and volume, all of which we may include in the single expres- sion geometric measurement. That is, we shall not think of size as including the measurement of weight, of value, of hardness, and the like, but only as including the length (width, height, depth, and distance in general), area (surface), and volume (capacity) of figures. Length. It seems very easy to measure accurately the length of anything, but it is not so easy as it seems. Linen tape lines stretch, steel tape lines contract in cold weather, ordinary wooden rulers shrink a little when they get very dry, and chains wear at the links and thus become longer with age. But these matters are of less moment than the carelessness of those who make the measurements. If the members of your class, each by himself, should measure the length of the walk in front of your school, to the nearest sixteenth of an inch, and not compare results until they had finished, it is likely that each would have a different result. In fact, all measurement is simply a close approximation. One of the best ways of securing a close approximation to the true result is to make the measurement in two different ways. Never fail to check a measurement. Just as we should always check an addition by adding in the opposite direction, so we should always check a measurement of length by measuring, if possible, in the opposite direction. 156 GEOMETRY OF SIZE Outdoor Work. In connection with the study of the size and position of common forms we shall first suggest a certain amount of work to be done out of doors. 1. Measure the length of the school grounds. To do this, drive two stakes at the appropriate corners, putting a cross on top of each stake so as to get two points between which to measure. Measure from A to B by ^^ holding the tape taut and ^^^^ '~\__ n level, drawing perpendic- ulars when necessary by means of a plumb line as shown in the figure. Check the work by measuring from B back to A in the same way. 2. Run a straight line along the sidewalk in front of the school yard. Of course for a short distance this is easily done by stretching a string or a measuring tape, but for longer distances another plan is necessary. x~ ~p Q R Y If we wish to run a line from X to F, say 300 ft., we drive stakes at these points and mark a cross on the top of each so as to have exact points from which to work. Now have one student stand at X and another at Y", each with a plumb line marking the exact points. Then have a third student hold a plumb line at some point P, the student at X motioning him to move his plumb line to the right or to the left until it is exactly in line with X and Y. A stake is then driven at P, and the student at X moves on to the point P. The point Q is then located in the same way. In this manner we stake out or " range " the line from X to Y, checking the work by ranging back from Y to X. OUTDOOR WORK 157 3. Measure the height of a tree by making on the ground a right triangle congruent to a right triangle which has the tree as one side. To do this, sight along an upright piece of cardboard so as to get the angle from the ground to the top of the tree. Mark tne angle on the cardboard and then turn the cardboard down flat so as to have an equal angle on the ground. A right triangle can now easily be laid out on the ground so as to be congruent to the one of which the tree is one side. By measuring a certain side of this right triangle, the height of the tree can be found. 4. Run a line through a point P parallel to a given line AB for the purpose of laying out one of the two sides of a tennis court. P O Stretch a tape line from P to any point M on AB, bisect the line PM at 0, and from any point N on -^ N ,, ^ AB draw NO. Pro- long NO to Q, making OQ equal to NO, and draw PQ. Sup- pose that ON is 20 ft. Then sight from N through 0, and place a stake at Q just 20 ft. from 0. Then P and Q determine a line parallel to AB. The proof of this fact, like the proofs of many other facts inferred from certain of the exercises, is part of demonstrative geometry, which the student will meet later in his course in the high school^ Outdoor work will be given at intervals and always by itself, so that it can easily be omitted. The circumstances vary so much in dif- ferent parts of the country as to climate, location of the school, and other conditions, that a textbook can merely suggest work of this kind which may or may not be done, as the teacher directs. A good tape line, three plumb lines (lines with a piece of lead at one end), and a pole abovit 10 ft. long will serve for an equipment for beginners. 158 GEOMETRY OF SIZE Exercise 21. Practical Measurements of Length 1. Measure the length of this page to the nearest thirty- second of an inch, checking the work. If a ruler is used which is, as usual, divided only to eighths of an inch, the student will have to use his judgment as to the nearest thirty-second of an inch. The protractor illustrated on page 115 has an edge on which lengths are given to sixteenths of an inch, and such a scale may be used if laid along the edge of a ruler. 2. Measure the length of this page to the nearest twen- tieth of an inch, checking the work. The protractor illustrated on page 115 has an edge divided into tenths of an inch. There is advantage in the student becoming familiar with the units of the metric system, even before he studies the subject on page 205, since these units have come into use in our foreign trade and in all our school laboratories. It is desirable to know that 10 millimeters (mm.) = 1 centimeter (cm.) = 0.4 in., nearly; 10 cm. = 1 decimeter (dm.); 10 dm. = 1 meter (m.) = 39.37 in. 3. Measure the length of the longest line of print on this page, to the nearest sixteenth of an inch, checking the work. If the student has a pair of dividers (compasses with sharp points), this may be used to transfer the length to a ruler. This method is usually more nearly accurate than to lay the ruler on the page. 4. Measure the length of your schoolroom to the nearest eighth of an inch, checking the work. If the class works in groups of two, and each group checks its result with care, there may still be some difference. In that case an average may be taken. This is commonly done in surveying. 5. In the upper part of the opposite picture a man is sighting across the stream in line with the front part of his cap. He then turns and sights along the ground, as shown by the other man standing near him. How does he find the width of the stream by this method ? LENGTH 159 (Curious illustrations from old geometries if- the XVI century. The first one shows howto measure the distance across a stream. , west 8 ft. to E, south 19 ft. to F, and east to A. Find the cost of cementing the floor at 150 per square foot. 166 GEOMETRY OF SIZE 16. Construct a rectangle and then construct a similar rectangle the area of which is three times that of the first. 17. The screens A, B, and C are 1 ft., 2 ft., and 3 ft. respectively from an electric light L. If screen A should be removed, the quantity of light which fell on it would fall on B. If screens A and B should be removed, the same quantity of light would fall on screen C. How would the intensity of light compare for a given area on each of the screens ? 18. How many paving blocks each 4 in. by 4 in. by 10 in., placed on their sides, will be required to pave a street 1800 ft. long and 34 ft. 8 in. wide ? 19. Printers usually cut business cards from sheets 22 in. by 28 in. How many cards 2 in. by 4 in. can be cut from one of these sheets ? Draw a plan. 20. At 6$ a square foot find the cost of enough wire screen for the 8 windows of a gymnasium, each being 28 in. by 64 in. inside the frame. Allowance should be made for the wire to overlap the frames 1 in. on every side. 21. Find the area of a double tennis court. A standard double tennis court is 36 ft. by 78 ft. 22. What is the meaning of the statement A=bh? 23. How many acres are there in a football field ? A standard football field is 100 yd. by 53 yd. 1 ft. An acre is 160 sq. rd., and a rod is 5^ yd. The teacher should give plenty of practical work in finding the areas of floors and the like. 24. A farm team plowing a field walks at the rate of 2 mi. per hour and is actually plowing -| of the time. What area will be plowed from 7 A.M. to noon if a plow is used which turns a furrow 14 in. wide ? AREA OF A PARALLELOGRAM 167 Area of a Parallelogram. It is often convenient or nec- essary to find the area of a parallelogram. If from any parallelogram, like ABCD in the first figure, \ve cut off the shaded triangle T by a line perpendicular D to DC, and place the triangle at the other end of the paral- lelogram, as shown in the figure at the right, the resulting figure is a rectangle. That is, the area of a parallelogram, is equal to the area of a rectangle of the same base and the same height. But the formula for the area of a rectangle is A = bh. Therefore the area of a parallelogram is equal to the product of the base and height. This may be expressed by the formula A = bh. The teacher should make sure that the students understand the meaning of this formula. The purpose is to introduce algebraic forms as needed. The students should see that the value of A depends upon the values of b and h. In the language of more advanced mathematics, A is called a function of b and h. The students should see that all formulas are expressions of functions. 1 1 1 1 1 1 1 1 1 1 1 [ 1 J 1 J 1 Rectangular pieces of cardboard, as in the figures shown just above, may be arranged to lead the student to infer that when the base and height of a rectangle are equal respectively to the base and height of a parallelogram, the areas are equal. 168 GEOMETRY OF SIZE Exercise 24. Area of a Parallelogram 1. Draw parallelograms of the shapes and sizes of the following and show, by cutting off triangles and placing them as explained on page 167, that each parallelogram can be transformed into a rectangle of the same area. 2. On squared paper draw four parallelograms and a rectangle, all having equal bases and equal heights, but all of different shapes. By cutting off a triangle from each parallelogram and moving it to the other side, trans- form each into a rectangle of the same area. Count the squares and compare the areas of the resulting rectangles. Draw the following parallelograms to scale and find the area of each: 3. Base 30 in.; other side, 20 in.; height, 15 in.; scale ^-. 4. Base 12 in. ; other side, 8 in. ; height, 6 in. ; scale 4-. 5. On squared paper draw a rectangle and a parallelo- gram with equal bases and equal heights. Compute the area of each by counting the included squares, and thus compare the areas. 6. A floor is paved with six-sided tiles, as here shown. The tiles have been divided by dotted lines in the picture to suggest a method of measuring them. What measurements would you take to find the area of each tile ? What other divisions of the tiles can you suggest for convenience in finding the area of each? AEEA OF A TRIANGLE 169 Exercise 25. Area of a Triangle 1. How is the area of a parallelogram found? 2. In the parallelogram here shown how do the areas of the triangles ABC and CD A compare ? A triangle is what part of a parallelogram of the same base and height ? / ^7 / 9*** / The parallelogram should be cut out of paper / ,,---'' I and then divided into two congruent triangles 1^- J by cutting along one of the diagonals. 3. If the parallelogram in Ex. 2 is 6 in. wide and 3 in. high, what is its area? What is the area of each of the triangles formed by drawing the diagonal AC? 4. If a parallelogram is 5 ft. wide and 2 ft. high, what is its area ? What is the area of each of the triangles ? 5. If a parallelogram is 8 yd. wide and 3 yd. high, what is its area ? What is the area of each of the triangles ? 6. Find the area of a rectangle with base 7 ft. and height 10ft.; of a triangle with base 7ft. and height 10 ft. Notice that a rectangle is one kind of a parallelogram. 7. Considering the above examples, state a rule for finding the area of a triangle. 8. Draw to scale a triangle with sides 6 in., 7 in., and 8 in. respectively. Draw lines to show that the area of the triangle is half the area of a rectangle with the same base and height. 9. What is the area of a triangular garden with base 32ft. and height 16ft.? Find the areas of triangles with bases and heights as follows : 10. 4in.,3.6in. 12. 8yd.,9yd. 14. 9ft., 4 ft. 4 in. 11. 9 in., 7.4 in. 13. 7.5 in., 8.4 in. 15. 6ft 3 IP... 8ft. 170 GEOMETRY OF SIZE Area of a Triangle. From the illustrations given and the questions asked on page 169 it is easily seen that The area of a triangle is equal to half the product of the base and height. This may be expressed by the formula A = i bh. 4 For example, what is the area of a triangle of base 14 in. and height 9 in. ? ^ of 14 x 9 sq. in. = 63 sq. in. Exercise 26. Area of a Triangle Examples 1 to 9, oral State the areas of triangles with these bases and heights : 1. 12 in., 9 in. 4. 28 in., 3 in. 7. 32 in., 8 in. 2. 14 in., 11 in. 5. 8 in., 4.5 in. 8. 3.5 in., 4 in. 3. 9 in., 10 in. 6. 3.5 in., 6 in. 9. 8 in., 9.5 in. 10. How many square yards of bunting are there in a triangular school pennant of base 56 in. and height 2 yd. ? Find the areas of triangles with these bases and heights : 11. 17ft., 46ft. 14. 36ft, 17.6ft. 12. 19.5 ft., 18.3 ft. 15. 18.3 ft., 14.4 ft. 13. 22.7ft, 16.4ft. 16. 29.7yd., 24.8yd. 17. The span AB of a roof is 40 ft., the rise MC is 15 ft., the slope CB is 25 ft., and the length BE is 60 ft. Find the area of each gable end and the area of the roof. AREA OF A TRIANGLE 171 i> 18. On squared paper draw a right triangle with the two sides respectively 1.5 in. and 2.5 in. Estimate the area by counting the squares, compute the area accurately, and then find what per cent the first result is of the second. When we speak of the two sides of a right triangle we always mean the two perpendicular sides. 19. A field 65 rd. by 140 rd. is cut by a diagonal into two equal right triangles. A railway runs along this diagonal and takes 3 A. off each triangular field. How much is the rest of the field worth at $140 an acre ? 20. In this figure ABCD represents an 8-inch square, E, F, G, and H being the mid-points of the sides. In the square AEOH, AP = QE = ER = SO = OT = . = \AE. Find the area of each of the small triangles and also of the octagon PQRSTUVW. The dots ( ) mean " and so on " and, in this case, take the place of " UH = H V = WA." An octagon is a figure of eight sides. 21. The triangle ABC is made by driving pins at A and B, running a rubber band around them, and stretching this band to the point C. Now imagine C to move along CE parallel to AB, stopping first at D and then at E. Have ABC, ABD, and ABE different areas'? State your reasons fully. Since any field may be cut into triangles by drawing certain diagonals, the students are now prepared to find the area of any piece of land that admits of easy measurement. JH1 U TO V w 7 sp~ \ \ R J IP QE I D 172 GEOMETRY OF SIZE Area of a Trapezoid. If a trapezoid T has its double cut from paper and turned over and fitted to it, like Z>, the two together make a parallelo- , <- -, gram. How does the area / T \ D / of the whole parallelogram *- ^ ' compare with the area of the trapezoid T? How does the base of the parallelogram compare with the sum of the upper and lower bases of the trapezoid? How do you find the area of the parallelogram ? Then how do you find the area of the trapezoid ? D r If from the trapezoid A BCD, here shown, the shaded portion is cut off and is fitted into the space marked by the dotted lines, what kind of figure is formed ? How is the area of the resulting figure found ? If the shaded portions of this trapezoid are fitted into the spaces marked by the dotted lines, what kind of figure is formed ? How is the area of the resulting figure found? From these illustrations we infer the following: To find the area of a trapezoid, multiply the sum of the parallel sides by one half the height. This may be expressed by the formula where A stands for the area, h for the height, B for the lower base, and b for the upper base. The parentheses show that B and b are to be added before the sum is multiplied by ^ h. ' For example, if h = 4, B=7, and 5 = 5, we have ^=-|-x4x(7 + 5) = 2 x 12 = 24. AREA OF A TRAPEZOID 173 Exercise 27. Area of a Trapezoid Examples 1 to 6, oral Find the area of each of the trapezoids whose height is first given below, followed by the parallel sides : 1. 6 in.; 9 in., 11 in. 7. 18 in.;' 9.5 in., 27.3 in. 2. 8 in.; 14 in., 6 in. 8. 24 in.; 11 in., 9} in. 3. 12 in.; 4 Jin., 3^ in. 9. 17 in.; 18 in., 26 in. 4. 11 in.; 8 in., 12 in. 10. 14ft; 6 ft. 4 in., 9ft. 5. 9 in.; 4^ in., 5Jin. 11. 42yd.; 19|yd., 37|yd. 6. 13 in.; 11 in., 7 in. 12. 127ft; 96f ft, 108J ft 13. Find the number of acres in a field in the form of a trapezoid, the parallel sides being 33^ rd. and 17^ rd. and the distance between these parallel sides being 14 rd. 14. If the area of a trapezoid is 396 sq. in. and the parallel sides are 19 in. and 17 in., what is the height? 15. In this figure show that we may find the area of the trapezoid by adding the areas of two triangles. This should be taken up at the blackboard. b The teacher should show that in this case we /\ \ have hB + %hb = -| h (B + b), just as / \ x \ \ h A little algebra may thus be introduced as necessity requires and the way made easier for more elaborate algebra later. 16. Suppose that the upper and lower bases of a trape- zoid are equal, does the formula for the trapezoid still hold true ? The trapezoid becomes what kind of a poly- gon ? The formula becomes the formula for what figure ? Practical outdoor work in measuring fields and in computing areas may now be given, or it may be postponed until after page 174 has been studied. 174 GEOMETRY OF SIZE K Area of any Polygon. A polygon like ABCDEF may be divided into triangles, parallelograms, and trapezoids as here shown, and the areas of these parts may be found separately and then added. As an exercise the teacher may assign to the class the finding of the area of the field here represented, the figure being drawn to the scale 1 in. = 200 rd. Area found from Drawing. Suppose that the area of a field ABC has to be found, and that there is a large swamp as indicated in the figure. In such a case it is not easy to find the height of the triangle ; that is, the dis- tance CD. The lengths of the three c sides may, however, be measured, and then the area may be found by draw- ing the outline to scale and measuring the height of this triangle. Only the drawing to scale is here shown. If the scale is 1 in. = 100 rd., A 5 ~B we see that CD is 90 rd., because CD is 0.9 in. If AB is 100 rd. the area of the triangle is ^ X 90 x 100 sq. rd., or 4500 sq. rd., which is equal to 28-J- A. Therefore, to find the area of a field from a drawing, Draw the plan to scale; divide the plan into triangles; from the base and height of each triangle on the plan com- pute the base and height of each triangle in the field; from these results find the areas of the several triangles and thus find the area of the field. It must be understood that surveyors have better methods, but this method is sufficient for our immediate purposes. The immediate object in view is not to make practical surveyors tut to show the general power of mathematics. AREAS Exercise 28. Areas 1. This plan represents a space 150 ft. long and 75 ft. wide, with two triangular flower beds, in a city park. Around the inside of the space is a sidewalk 6^ ft. wide. Meas- ure the figure, determine the scale used in drawing the plan, and find the area of each of the flower beds in the park. 2. This map is drawn to the scale 1 in. = 520 mi. Care- fully measure the map and determine approximately the length of each side of each state, and then find the approximate area of each state. The results obtained will be, of course, merely approximate, since the map is so small. The method is, however, the one which is employed in practical work with larger maps. 3. Each side of a brick building with a slightly sloping roof is in the form of a trapezoid, as here shown. The building is 57 ft. wide, 57 ft. high on the front, and 52 ft. high on the rear. On this side there are 4 windows each 4 ft. wide and 9 ft. high and 4 windows of the same width but 6^ ft. high. If it takes 14 bricks per square foot of outside surface to lay the wall, how many bricks will be needed to lay this' wall, deducting for the 8 windows ? 4. The sides of a triangular city lot are respectively 72 ft., 60 ft., and 48 ft. Draw a plan of the lot to the scale of 1 in. to 12 ft., measure the altitude of the scale drawing, and find the altitude and area of the lot. WYOMING UTAH COLORADO 176 GEOMETRY OF SIZE 12' 20' 11' 20' 25' T u III IV V 16 20' 32' 20' ' VT .! VI i 4! VIII coj /Vtoi OOl J 5. This sketch shows the plan of some small suburban garden plots which are offered for sale at 20 $ a square foot. Find the price of each lot. 6. In a certain city Washing- ton Street runs east and west and intersects Third Avenue at right angles. Using the scale 1 in. = 100 ft., draw a plan of the property on the southeast corner from the following description : Beginning at the corner, run south 160 ft., then east 75 ft., then north 15 ft., then east 50 ft., then by a slanting line to a point on Washington Street 100 ft. from the corner, and then to the corner. Find the area of the plot and the value at $2.20 per square foot. 7. In order to measure the distance AB across a swamp some boys measure a line CD, drawn as shown in the figure, and find it to be 280 ft. & long. They find that DA = 40 ft. and CB = 90 ft., DA and CB being perpendicular to CD. Draw the plan to some convenient scale and determine the distance AB. Find also the area of the trapezoid A BCD. 8. A swimming tank is 60 ft. long and 35 ft. wide. Draw a plan to the scale 1 in. = 10 ft., determine the length of the diagonal by measurement, and then compute the num- ber of yards that a student will swim in swimming along the diagonal of the tank eight times. 9. A lot has a frontage of 65 ft. and a depth of 150 ft., and a path runs diagonally across it. Draw the plan to scale and find, by measurement, the distance saved by using the path instead of walking round the two sides at a distance of 2 ft. outside the edges of the lot. D 280ft. AEEAS 177 10. Suppose that you have 360 ft. of wire screen to inclose a plot in which to keep chickens. If you wish to inclose the largest possible area in the form of a parallelo- gram, triangle, or trapezoid, which form would you use ? Show by a drawing on squared paper that the form which you choose incloses a larger area than the others. Remem- ber that there are several kinds of triangles, several kinds of parallelograms, and several kinds of trapezoids. 11. Draw three different triangles, each with base 2 in. and height 1 in. Find the area of each. What do you infer as to the equality of the areas of triangles having equal bases and equal heights ? Write the statement in full. 12. Upon the same base of 2 in. draw three different parallelograms, each having a height 1^- in. Find the area of each parallelogram. What do you infer as to the equal- ity of the areas of parallelograms on the same base and with equal heights? Write the statement in full. 13. For computing the area covered by 1000 ft. of a river, some boys at wish to find the width OB of the river, as here shown. They know that the distance AB is 300 ft. and that the angle at B is 90. Show how, by sighting along YB and XA and by making certain measurements, the boys can find the distance OB without crossing the river. 14. Find the area of an equilateral triangle 3.1 in. on a side. This may be done by drawing the triangle on squared paper and counting the squares, or, more accurately, by first approximating the height by measurement on the squared paper. 15. Find the area of an isosceles triangle with sides 2 in., 2 in., and 11 in. 178 GEOMETRY OF SIZE Exercise 29. Optional Outdoor Work 1. Determine the area of your school grounds by care- fully making the necessary measurements and dividmg the grounds into triangles, if necessary. 2. In Ex. 1 determine the area by drawing the plan to scale. 3. Drive two stakes in the ground at A and j#, 12 ft. apart. Fasten one end of a 15-foot line at A and one end of a 9-foot line at B. Draw the loose ends taut and drive a stake where they meet, at C. What kind of an angle is formed at 4. Draw the figure of Ex. 3 to the scale which is four times the one here used v and determine from your figure some other measurements which might be used to lay out the same kind of angle. Try this on the school grounds. 5. What is the largest scale on which a plan of your school grounds could be drawn on a piece of paper 12 in. by 14 in., if you allow for a margin of at least 1 in.? 6. Draw to scale a plan of the lot on which your home stands and indicate the ground plan of the house. 7. Draw to scale a floor plan of some public building in your vicinity. Compute the area covered. 8. Lay off on your school grounds an isosceles triangle, an equilateral triangle, and a right triangle, each with a perimeter of 30 ft. Compute and compare the areas. 9. Lay off on your school grounds several rectangles, each of which has a perimeter of 30 ft. Compute and compare the areas. ANCIENT PROOFS 179 f/mo ri'ht lines cut the one the other, the hed angles jklk equdl iheone to the othe page from the first English edition of the great geometry written by Euclid of ^Alexandria, about 300 2?. . ^ow r/4 ancient Gr.eeks prated their statements. 180 GEOMETRY OF SIZE 10. If the street is to be paved in front of your school- house, what measurements are necessary to determine the area to be covered ? Make the measurements for the block in which your schoolhouse stands, draw the plan to scale, and compute the area. 11. If a sidewalk is to be laid in front of your school- house, what measurements are necessary and what prices must be known in order that you may find the cost of the walk ? Make the measurements, find by inquiry the prices, and compute the cost of the walk. Such examples are merely typical of the work which many schools will wish to have done. It is impossible, however, to anticipate the practical cases which may arise in any given locality. They may relate to some building in process of erection, to the laying of a water main in the street, to the reservoir of the city water supply, or to the cost of stone steps for a schoolhouse. The important thing is that the problem should be real and interesting to the class. 12. Compute the number of square feet of the surface of some building which needs to be painted, find the average cost per square yard for painting it one coat, and then compute the cost of painting the building. 13. Suppose that a water main is to be laid in the street in front of the schoolhouse. Ascertain by inquiry the usual width of a trench for such a purpose, and draw a plan of the street to scale, showing the location of the trench and giving it the proper width to scale on the drawing. 14. In the upper picture on the opposite page can you see how the height of the tower could be measured by simply tipping the quadrant over flat and making certain measure- ments on the ground? Try this plan in measuring the height of some tree or building. In this case also it may be noticed that the angle is exactly 45, and so there is another and better way of finding the height. . ANCIENT INSTRUMENTS 181 Illustrations from old books on geometry, showing hov> the height of a tower or the distance to an island can be found by the aid of a simple instrument t>hich can easily be made. 182 GEOMETRY OJ SIZE Ratio. We often hear of the ratio of one number to another, as when some one speaks of the ratio of the width of a tennis court to its length, or the ratio of daylight to darkness in the winter, or the ratio of a man's expenses to his income. By the ratio of 3 to 4 we mean 3-5-4, or !> while the ratio of 1 in. to 1 ft. is -j^r and the ratio of \ to | is 1 -J- f , or f . The relation of one number to another of the same kind, as expressed by the division of the first number by the second, is called the ratio of the first to the second. A few examples of ratio should be given on the blackboard. Tims the ratio of $3 to $6 is > or, in its simplest form, \ ; the ratio of 1 yd. to 1 ft. is the same as the ratio of 3 ft. to 1 ft., or 3 ; the ratio of 5 to 2 is -|, or 2^ ;. and the ratio of any number to itself is 1. The ratio of 2 to 3 may be written in the fraction forms, ^ or 2/3, or it may be written with a colon between the numbers ; that is, as 2 : 3. The teacher should explain to the class that the ratio of 12 ft. to 12 ft. 12 4 ft., for example, may be written ' , > 12 : 4, or simply 3. The "X it. "I word "ratio " is used for each of these forms. The expression 12 : 4 is read "the ratio of 12 to 4," or "as 12 is to 4," 12 and 4 being called the terms of the ratio. Since any number divided by a number of the same kind, as inches by inches or dollars by dollars, has an abstract quotient, we see that A ratio is always abstract, and its terms may therefore be written as abstract numbers. That is, instead of labeling our numbers, as in 2 f t. : 4 ft., we may omit all labels and write simply 2 : 4, or |, or |. Teachers should use the familiar fraction form first. Indeed, the special symbol (:) is slowly going out of use because it is not neces- sary. We often see 2 : 3 written as 2/3 instead of ^. Ratios are little more than fractions and may -be treated accordingly. EATIO 183 Exercise 30. Ratio All work oral 1. Expressed' in simplest form, what is the ratio of 6 to 12? of 12 to 6? When a ratio is asked for, the result should always be stated in the simplest form unless the contrary is expressly stated. 2. What is the ratio of $4 to $12? of 4ft. to 12ft.? 3. What is the ratio of 4-J to 9 ? of 15 to 7| ? 4. In the figure below, what is the ratio of E to Z>? What is the ratio of E to C ? When we speak of the ratio of E to D we mean the ratio of their number values ; that is, of 1 to 2, the ratio being ^. When we speak of the ratio of E to 2 B we mean the ratio of 1 to 2 x 4. This is 4- o 5. In the figure below, what is the ratio of E to What is the ratio of E to A? of E to & + C? Referring to the figure, state the following ratios : 6. E to %B. 7. D to 2 A. 8. 2 E to D. 9. 2 D to B. 10. <7 to 3 E. 11. A to E, 12. C to 5. 13. D to C. 14. C to ^. 15. A to 2D. D E 16. What is the ratio of any number to twice itself ? 17. What is the ratio of a foot to a yard ? of an inch to a foot ? of 8 oz. to 1 Ib. ? of 1 pt. to 1 qt.? of 2 qt. to 1 gal.? In every such case the measures must be expressed in the same units before the ratio is found. Thus the ratio of 1 yd. to 7 ft. is the ratio of 3 ft. to 7 ft., or of 1 yd. to 2^ yd., either of which is ^. 184 GEOMETRY OF SIZE Proportion. An expression of equality between two ratios is called a, proportion. For example, $5 : $8 = 10 ft. : 16 ft. is a proportion. This proportion is read " $5 is to $8 as 10 ft. is to 16 ft." or " the ratio of |5 to |8 is equal to the ratio of 10 ft. to 16 ft." It may, of course, be written simply 5 : 8 = 10 : 16, or J- ^|. The first and last terms of a proportion are called the extremes; the second and third terms are called the means. These expressions are unnecessary, however, in the treatment of the subject in the junior high school. We often have three terms of a proportion given and wish to find the fourth. For example, we may have the proportion n: 14 =27: 63, where n represents some number whose value we wish to find. We may write the proportion in the more familiar fraction form, thus: 07 . n i If, now, ^ of n is equal to |-|, we see that n must be equal to 14 x --|, or 6. The teacher should show on the blackboard that we need merely multiply the two equal ratios by 14, canceling as much as possible, and we have n = 6. If we have 4 : w = 12 : 6, we may simply take the ratios the other way, and have n: 4 = 6 :12, and then solve as above. The old method of solving business problems by ratio and pro- portion is no longer used to any considerable extent. The subject of ratio has a value of its own, however, and proportion is peculiarly useful in geometry. It is interesting to notice that in any proportion of abstract numbers the product of the first and fourth terms is equal to the product of the second and third terms. PKOPORTION 185 Exercise 31. Proportion Find the value of n in each of the following proportions: 1. n:18 = 7:9. 3. 7:w=9:72. 2. 7i:42=13:14. 4. 15:13 = ra:65. 5. A certain room is 24 ft. by 32 ft. and the width is represented on a drawing by a line 9 in. long. How long a line should represent the length ? 6. When a tree 38 ft. high casts a shadow 14 ft. long, how long is the shadow cast by a tree 64 ft. high ? In all such cases the trees are supposed to be in the same locality and perpendicular to a level piece of ground. 7. If a picture 42 in. by 96 in. is reduced photographi- cally so that the length is 7-| in., what is the width ? 8. By means of a pantograph a student enlarges the floor plans for a house in the ratio of 8 : 3. If the dining room in the original plans measures 2-| in. by 3 in., what are the dimensions in the enlarged drawing? 9. The sides of a triangle are 9 in., 7 in., and 6 in. Construct a triangle the corresponding sides of which are to the sides of the given triangle as 3:4. 10. A map is drawn to the scale of 1 in. to 0.8 mi. How many acres of land are represented by a portion of the map 1 in. square ? 1 mi. = 320 rd., and 1 A. = 160 sq. rd. 11. The floor of a schoolroom is 24ft. by 30ft. The total window area is to the floor area as 1:5, and the 6 windows have equal areas, each window being 3-^ ft. wide. Determine the height of each window to the nearest quarter of an inch. 186 GEOMETRY OF SIZE Proportional Numbers. Numbers which form a propor- tion are called proportional numbers. Similar Figures. As stated on page 141, figures which have the same shape are called similar figures and are said to be similar. For example, these two tri- angles are similar. Likewise triangles ABC and XYZ on page 187 are similar. x A B Proportional Lines. The lengths of corresponding lines in similar figures are proportional numbers ; that is, corre- sponding lines in similar figures are proportional. For example, in the above triangles XY : YZ = AB : BC. In two circles the circumferences and radii are proportional, the circum- ference of the first being to the circumference of the second as the radius of the first is to the radius of the second. Exercise 32. Similar Figures Examples 1 to 4, oral 1. In the above triangles, if XY is twice as long as AB, how does ZX compare in length with CA ? 2. In the figure below state two proportions that exist among AB, AD, AC, and AE. 3. In this figure, if AB is -| of AD, C, what is the ratio of AC to AE? 4. In the same figure, if DE repre- sents the height of a man 6 ft. tall, BC the height of a 'boy, DA the length of the shadow cast by the man, and BA the length of the shadow cast by the boy, show how to find the height of the boy by measuring the lengths of the shadows. PROPOKTIONAL LIKES 187 5. If a tree BC casts a shadow 35 ft. long at the same time that a post YZ which is 12 ft. high casts a shadow 15 ft. long, how high is the tree ? Suppose YZ to be the post, XY to be its shadow, and A B to be the shadow of the tree. Since the triangles ABC and XYZ are similar, we may find h, the height of the tree, from the proportion BC _ YZ AB~XY' _e or by writing the values, 35 ~15' 7 9 whence h = & That is, the tree is 28 ft. high. 6. If a tree casts a shadow 58 ft. long at the same time that a post 8 ft. high casts a shadow 14 ft. 6 in. long, how high is the tree ? Draw the figure to scale. 7. If a telephone pole casts a shadow 27 ft. long at the same time that a boy 5 ft. tall casts a shadow 4 ft. 6 in. long, how high is the pole? Draw the figure to scale. 8. A boy threw a ball directly upward and watched its shadow on the sidewalk. When the ball began to descend, the shadow of the ball was at a fence post- 32 ft. away. The boy was 4 ft. 6 in. tall and his shadow was 2 ft. 3 in. long. How high did the boy throw the ball above the level of the ground? Draw the figure to scale. 9. A water tower casts a shadow 87 ft. 8 in. long at the same time that a baseball bat placed vertically upright casts a shadow twice its own length on a level sidewalk. Find the height of the water tower. Draw the figure to scale. 188 GEOMETRY OF SIZE 10. This man is holding a right triangle ABC in which AB = BC. What is the height of the tree in the picture if the base of. the triangle is 5 ft. 3 in. from the ground and if AD is 32 ft. ? This is a common way employed by woodsmen for measuring the heights of trees. The man backs away from the tree until, holding the triangle ABC so that AB is level, he can just see the top of the tree along the side A C. In all problems involving heights and distances the student should estimate the result in advance. This will serve as a check on the accuracy of the work. 11. In Ex. 10 suppose that a triangle is used which has AB equal to twice BC, that AD is 62 ft., and that the point B is 5 ft. 7 in. above the ground ; find the height of the tree. 12. A woodsman wishes to determine the distance from the ground to the lowest branch of a tree. He finds that if he places a stick vertically in the ground at a distance of 32 ft. from the tree, lies on his back with his feet against the stick, and sights over the top of the stick, the line of sight will strike the tree at the lowest limb, as shown in the figure. The woodsman's eye is 5 in. above the ground, the distance EF, as shown in the figure, is 5 ft 6 in., and the top of the stick is 4 ft. 9 in. above the ground. Determine the distance BC from the ground to the lowest branch. PROPORTIONAL LINES 189 13. A boy whose eye is 15 ft. from the bottom of a wall sights across the top and bottom of a stick 8 in. long and just sees the top and bottom of the wall, the stick being held parallel to the wall as shown. If the bottom of the stick is 18 in. from the eye, what is the height of the wall? 14. A woodsman steps off a distance of 30 ft. from a tree, faces the tree, and holds his ax handle at arm's length in front of him parallel to the tree. His hand is 2 7 'in. from his eye, and 2 ft. 4 in. of the ax handle just covers the distance from the ground to the lowest limb of the tree, How high is the lowest limb of the tree ? This method suffices for a fair approximation to the height. 15. Wishing to find the length AB of a pond, some boys choose a point C in line with A and B, and at B and C draw lines perpendicular to BC, and draw AD. By measuring they then find B C to be 84 ft., DE to be 112 ft., and EA to be 154 ft. What is the length of the pond ? 16. In Ex. 15 what other measurements may be used to find the distance AB? 17. If ^ in. on a map represents a distance of 375 mi., how many miles will 2- in. represent ? 18. If a tree casts a shadow 40 ft. long when a post 5 ft. high casts a shadow 6^ ft. long, how high is the tree ? 19. If 1-| in. on a map represents a distance of 325 mi., how many inches represent a distance of 340 mi. ? 190 GEOMETRY OF SIZE 20. In one of the upper illustrations on the opposite page suppose the length of the shadow of the post to be 1 ft. 6 in. shorter than the height of the post, and suppose the shadow of the tower to be 69 ft. 4 in. and the height of the post to be 5 ft. 2 in. Find the height of the tower. 21. In one of the upper illustrations on the opposite page there is also shown a very old method of finding the height by means of a mirror placed level on the ground. Can you see two similar triangles in the picture? If so, describe the method by which you could find the height of a tree in this way. 22. Some members of a class made a right triangle with one side 9 in. and the other side 12 in. One of them held the triangle so that the longer side was vertical and then backed away from a tree until he could just see the top by sighting along the hypotenuse. The class then measured and found that the eye of the observer was 45 ft. in a hori- zontal line from the tree and 4 ft. 10 in. from the ground. Draw the figure to scale and find the height of the tree. 23. Draw a plan of the top of your desk to scale, rep- resenting the length by 3 in. What will be the width of the drawing, and how can it be found? 24. The extreme length of a new leaf is 2 in. and the extreme width is 1 in. After the leaf has grown 1 in. longer, maintaining the same shape, what is its width ? 25. A girl is making an enlarged drawing from a photo- graph of a friend. In the photograph the distance between the eyes is |^ in. and the length of the nose is T ^- in. If the distance between the eyes in the drawing is 50% more than it is in the original, what is the length of the nose in the drawing ? PROPORTIONAL LINES 191 Saptrr 1eJe$f con Mm . A This is the way we locate a place on a map of the world. We say it is so many degrees north or south of the equator and so many degrees east or west of the meridian of Greenwich. Thus we say that a place is 40 N. and 70 W., meaning that it is 40 north of the equator and 70 west of the prime meridian, the one which passes through Greenwich. The principle applies whether we use a Mercator's projection or a globe, except that the curvature of the lines is seen in the latter case. The lines we use need not meet at right angles. For example, if we know that two water mains join at a point 60 ft. south of the road AB, as here shown, and 90 ft. from the road CD and on the side towards J5, we can locate the point P by A'. simply drawing two lines. How are these lines drawn ? Tf we know that P is 60 ft. south of AB and 90 ft. from CD, but do not know on which side, there would be two possible points. Under what circumstances would there be three possible points, or four possible points? 228 GEOMETRY OF POSITION Exercise 56. Position fixed by Two Lines 1. Suppose that we do not know the side of the road CD on which the two water mains mentioned on page 227 join, but we do know that it is on the south side of the road AB. Draw a figure showing all the possible positions of the water mains. 2. Suppose that we know that the water mains join to the west of (7Z>, but do not know whether the point is to the north or to the south of AB. Draw a figure showing all the possible positions. 3. Suppose that we are uncertain as to the side of each road on which the water mains join. Draw a figure show- ing all the possible positions. 4. Each side of a square park ABCD is 150 yd. long. A monument is to be erected in the park at a point 130 ft. from AB and 170 ft. from BC. Draw the figure to scale and determine the distance of the monument from each corner of the park. 5. There are three survivors of a shipwreck. The first says that the ship lies between 2 mi. and 2-| mi. from a straight coast line which runs from a lighthouse L to the west; the second says that the ship lies between l^mL and 2 mi. from a straight coast line which runs from L to the north ; and the third says that it lies 2^ mi. from L. Can they all be right? If so, draw the figure to scale and indicate where to dredge for the wreck. 6. In a rectangular field ABCD, 90 rd. by 130 rd., there is a water trough which is 290 ft from the side AB and 320 ft. from BC. Draw the figure to scale and thus find the distance of the water trough from the point D. POSITION FIXED BY TWO LINES 229 Points Equidistant from Two Lines. Suppose that two roads AB and CD intersect at and that it is desired to place a street lamp at a point equidistant from the two streets. How many possible positions are there for the lamp ? If we think of ourselves as walk- ing in such a direction as to be always equidistant from OB and OD, we see that we shall be walking along ON, Similarly, to be equidistant from OD and OA we must walk along OP. In general, any point on any of the dotted lines in the figure is equi- distant from AB and CD. Therefore we may locate the lamp anywhere on either line, these lines evidently bisect- ing the angles formed by the roads. Exercise 57. Points Equidistant from Two Lines 1. By using this figure draw a line containing points equidistant from two lines. , D The dotted lines are each ^in. from AB * ^ ^ and CD respectively. They intersect at P, and OP is drawn and prolonged to M. Z. Draw a line containing points equidistant from two lines by using a figure somewhat like the one used in Ex. 1, page 228. 3. In a park P, which lies between two streets M and JV, as shown in the figure, an electric light is to be placed so as to be equidis- tant from M and N and 80 ft. from the corner C. Copy the figure and show how to find the position of the light. In this case a circle intersects a straight line. 230 GEOMETRY OF POSITION 4. The manager of an amusement park decides to place six lights at intervals of 80 ft., so that each light .shall be equidistant from two intersecting drive- ways M and N, as shown in the figure. The first light is to be placed 15 ft. from the inter- section of the driveways. Copy the figure to scale and indicate the position of each of the six lights. 5. A fountain is to be erected in the space S between the two streets M and N. The contract provides that the fountain shall be 50 ft. from the nearest side of each of these streets. Copy the figure and indicate the position of the fountain. 6. A monument is to be so placed in a triangular city park that it shall be equally distant from the three sides. Draw a plan on any convenient scale and show on the plan all the lines necessary to find the point 0. Is equidistant from the sides AB and BCt Is it equidistant from AB and CA? Is it necessary to draw a line from (7? 7. A flagpole is to be so placed that it shall be equally distant from the three sides of a triangular park whose sides are 380 ft., 270 ft., and 300 ft. respectively. Draw a plan on a convenient scale and find the distance of the flagpole from each side of the park. 8. A water main has a gate located at a point 7 ft. from a certain lamp-post which stands on the edge of a straight sidewalk. The gate is placed 3 ft. from the edge of the walk, towards the street. Draw a plan showing every possible position of the gate. 9. Consider Ex. 8 when the gate is located at a point 3 ft. from the lamp-post. USES OF ANGLES 231 Use of Angles. There is still another method by which a man may locate valuables which he has buried. Suppose as before that there are two trees, A and B, and that he has buried his valuables at X. If he knows the exact direction A* B and distance from A to X, he can easily find the place where the valuables are buried. Both the distance and direction can be recorded on paper. For example, the man might write : " 30 north of line joining the trees, 150 ft. from the west tree," and this would recall to his mind that the angle of 30 is to be measured at A, namely, the angle BAX, and that X will be found 150 ft. from A. The man can lay off the angle on a piece of paper by the aid of a protractor like the one described on page 115, and can then sight along the arms of this angle. The use of the protractor, as given on page 115, may be reviewed at this time if necessary. Exercise 58. Uses of Angles 1. Estimate the number of degrees in each of the following angles, then check your estimate by use of a protractor. 2. Without using a protractor draw several angles of as near 45 as you can, and in different positions. Check the ac- curacy of your drawings by measuring the angles with a protractor. Similarly, draw angles of 10, 80, 75, 15, 50, and 30. Check the accuracy of your drawing in each case. You will find that practice will enable you to estimate lengths and the size of angles with a fair degree of accuracy. 232 GEOMETRY OF POSITION 3. Indicate on paper a direction 30 west of south of the school ; 20 east of north ; 20 east of south. By 30 west of south is meant a direction making an angle of 30 with a line running directly south, and to the west of that line. 4. It is found that a submarine cable is broken 5 mi. from a certain lighthouse and 30 west of south. Draw a plan, using the scale of 1 in. to 1 mi., and show where the repair ship must grapple to bring up the ends for splicing. 5. A boy starts to walk on a straight road which runs 20 east of north. After he has gone 3 mi. he turns on an- other straight road and walks 2 mi. due west. Is he then east or west of his starting point, and how many degrees ? 6. A flagstaff CD stands on the top of a mound the height BC of which is known to be 30 ft. From the point A the angle BAD is ob- served to be 50 and the angle CAD to be 20. Draw a diagram to scale and find ap- proximately the height of the flagstaff CD. <_ B 7. Wishing to find the length of a pond, some boys staked out a right triangle as shown in the figure. Using a pro- tractor they found that the angle A was 40, and they measured AB, finding it to be 500 ft. Draw the figure to the scale of 1 in. to 200 ft. and thus find the length of the pond. 8. Some men buried a chest 150 yd. from a tree A and 250 yd. from another tree B, which was 200 yd. due north of A. On returning some years later they found that the tree B had disappeared. Draw a plan to scale and show how they can find the buried chest, provided that they have a compass and remember on which side of AB they hid the chest. USES OF ANGLES 233 9. A seaport is on a straight coast which runs due north and south. A steamer sails from it in a direction 30 west of north at the rate of 15 mi. an hour. When will she be 25 mi. from the coast, and how far will she be from the seaport at that time ? 10. Some boys wished to determine the height of a cliff. They found the distance AC to be 90 ft. and the angle BAC to be 45. They then drew the figure to scale and determined the height of the cliff. What is the height of the cliff ? 11. While a ship is steaming due east at the rate of 20 knots an hour, the lookout observes a light. At 9 P.M. the light is due north, at 9.15 P.M. it is 10 west of north, and at 9.30 P.M. it is 20 west of north. Determine by a drawing whether the light observed is stationary. A knot is generally taken as 6080 ft., this being approximately the sea mile ; the statute mile used on land is 5280 ft. 12. There is a seaport A on a straight coast running east and west. A rock B lies 3000 yd. from A and 30 north of east from the coast line. A ship steams from A in a direction 20 north of east from the coast line. What is the nearest approach of the ship to the rock? How far is the ship from the coast at that time ? 13. A man knows a tree across a stream to be 60 ft. high ; he finds the angle at his eye as shown in the figure to be 30, and he knows that his eye is 5 ft. from the ground. Make a drawing to scale and de- termine the width of the stream. 14. Two towers of equal height are " 300 ft. apart. From the foot of each tower the top of the other tower makes an angle of 30 with a horizontal line. Determine the height of the towers. 234 GEOMETRY OF POSITION Exercise 59. Miscellaneous Problems 1. A ship steams from a seaport A in a direction 50* east of north. A dangerous rock lies northeast from A and 5000 yd. from the coast running east and west through A. Find how near the ship approaches to the rock. 2. Two sides of a rectangular field ABCD are 80 rd. and 95 rd. A tree in the field stands 120 ft. from AB and 160 ft. from BC. Draw the figure to scale and show the position of the tree. 3. A man computes that he should buy 16 T. of soft coal for his winter supply. His cellar is so arranged that he can make a coal bin 18 ft. long, and the height of the cellar is 8^ ft. How wide a bin should be constructed if the top of the coal is to be a foot from the ceiling when the bin contains 16 T. of coal ? Allow 35 cu. ft. of coal to the ton. 4. A boy standing due south of a flagpole finds its angle of elevation, that is, the angle to the top, to be 20. After he has walked 280 ft. to the northwest on level ground, he sees the flagpole to the northeast. Draw the figure to scale and determine the heig t of the flagpole. 5. How long a shadow will the flagpole mentioned in Ex. 4 cast when the sun's angle of elevation is 40 ? 6. A tree 90 ft. high casts a shadow 140 ft. long. Draw the figure to scale and find the sun's angle of elevation. 7. A military commander standing 1000 ft. -from a fort finds its angle of elevation to be 30. Draw the figure to scale and determine the height of the fort. 8. In order to check his work in Ex. 7 the commander took the angle of elevation from a point 1200 ft. from the fort. What did he find this angle to be ? OUTDOOR WORK 235 Exercise 60. Optional Outdoor Work 1. Determine the height of a tree on or near your school grounds by finding the angle of elevation of the top from two different positions and measuring the distance of each position from the tree. 2. Two boys wishing to determine which of two smoke- stacks is the taller computed the height of each stack by three different methods and then took the average of the results as the correct height. What three methods might they have used ? Use three methods to determine the height of some high object near your school. 3. A graduating class decides to present a drinking- fountain to the school. The fountain is to be placed 30 ft. from a straight street which runs in front of the school grounds and 15 ft. from the front door of the school build- ing. How could the members of the class determine the desired position ? Locate, if possible, such a position on your school grounds or on some lot in the vicinity. 4. Suggest two methods of determining the distance between two points when the distance cannot be measured directly. Use both methods to determine the distance be- tween two easily accessible points on your school ground. Check the accuracy of each method by actually measuring the required distance. 5. A hawk's nest is observed in a high tree and some distance below the top of the tree. Suggest two methods by which the height of the nest may be determined other than by direct measurement. Check the accuracy of the two methods by applying them to a similar situation where the required distance can be actually measured as a check upon the accuracy of the methods. JMl 236 GEOMETRY OF POSITION Exercise 61. Problems without Figures 1. A workman has a circular disk of metal and wishes to find its exact center. How should he proceed? 2. A man wishes to set out a tree so that it shall be equally distant from three trees which are not in the same straight line. How should he proceed to find the position ? 3. A contractor wishes to tap a water main at a point equidistant from two hydrants. How should he proceed to find the required point? 4. A city engineer is asked to place an electric-light pole at a point equidistant from two intersecting streets and at a given distance from the corner. How does he do it ? 5. A man wishes to build on a corner lot a house at a given distance from one street and at another given distance from the other street. How does he lay out the plan ? 6. A drinking-fountain is to be placed in a park at a given distance from a hydrant which is at the side of the street in front of the park, and at a given shorter distance back from the street. Show that there are two possible points, and show how to find them. 7. If you know that two water mains join somewhere under a certain road, but you do not know where, what measurements could be given you with respect to one or more trees along the side of the road that would tell you where to dig to find the point? 8. A straight electric-light wire runs under the floor of a room. The distances of one point of the wire from the northeast corner and from the north wall are known, and also the distances of another point from the southwest corner and from the south wall. Show how to mark on the floor the course of the wire. SUPPLEMENTARY WORK 237 IV. SUPPLEMENTARY WORK Squares and Square Roots. If a square has a side 4 units long, it has an area of 16 square units. Therefore 16 is called the square of 4, and 4 is called the square root of 16. Square Roots of Areas. Therefore, consider- ing only the abstract numbers which represent the sides and area, The side of a square is equal to the square root of the area. Symbols. The square of 4 is written 4 2 , and the square root of 16 is written Vl6. Perfect Squares. Such a number as 16 is called a per- fect square, but 10 is not a perfect square. We may say, however, that VlO is equal to 3.16 -f , because 3.16 2 is very nearly equal to 10. Square Roots of Perfect Squares. Square roots of perfect squares may often be found by simply fac- toring the numbers. For example, V441 = V3 X 3 x 7 X 7 = V3 x 7x 3 x.7 = V21 x21=21. That is, we separate 441 into its factors, and then separate these factors into two equal groups, 3x7 and 3x7. Hence we see that 3x7, or 21, is the square root of 441. We prove this by seeing that 21 x 21 = 441. To find the square root of a perfect square, separate it into two equal factors. The work in square root may be omitted if there is not time for it. 238 SUPPLEMENTARY WORK Square of the Sum of Two Numbers. Since 47= 40 + 7, the square of 47 may be obtained as follows : 40 + 7 40 + 7 40 2 40 2 + 2 x (40 x 7) + 7 2 = 1600 + 2x280 +49 = 1600 + 560 + 49 = 2209. + u This relationship is conveniently seen in the above figure, in which the side of the square is 40 + 7. Every number consisting of two or more figures may be regarded as composed of tens and units. Therefore The square of a number contains the square of the tens, plus twice the product of the tens and units, plus the square of the units. This important principle in square root should be clearly under- stood, both from the multiplication and from the illustration. Separating into Periods. The first step in extracting the square root of a number is to separate the figures of the number into groups of two figures each, called periods. Show the class that 1 = I 2 , 100 = 10 2 , 10,000 = 100 2 , and so it is evident that the square root of any number between 1 and 100 lies between 1 and 10, and the square root of any number between 100 and 10,000 lies between 10 and 100. In other words, the square root of any integral number expressed by one figure or two figures is a number of one figure ; the square root of any integral number ex- pressed by three or four figures is a number of two figures ; and so on. Therefore, if an integral number is separated into periods of two figures each, from the right to the left, the number of figures in the square root is equal to the number of the periods of figures. Th* last period at the left may have one figure or two figures. SQUAEE EOOT 239 Extracting the Square Root. The process of extract- ing the square root of a number will now be considered, although in practice such roots are usually found by tables. For example, required the square root of 2209. Show the class that if we separate the figures of the number into periods of two figures each, beginning at the right, we see that there will be two integral places in the square root of the number. The first period, 22, contains the square of the tens' number of the root. Since the greatest square in 22 is 16, then 4, the square root of 16, is the tens' figure of the root. Subtracting the square of the tens, the re- 2209(47 16 80 87 609 609 mainder contains twice the tens x the units, plus the square of the units. If we divide by twice the tens (that is, by 80, which is 2 x 4 tens), we shall find approximately the units. Dividing 609 by 80 (or 60 by 8), we have 7 as the units' figure. Since twice the tens x the units, plus the square of the units, is equal to (twice the tens + the units) x the units, that is, since 2 x 40 x 7 + 7 2 = (2 x 40 + 7) x 7, we add 7 to 80 and multiply the sum by 7. The product is 609, thus completing the square of 47. Checking the T/ork, 47 2 = 2209. Exercise 62. Square Root Find the square roots of the following numbers : 1. 3249. 2. 3721. 3. 3969. 4. 5041. Find the sides of squares, given the following areas : 5. 6724 sq. ft. 7. 9025 sq. ft. 9. 7921 sq. yd. 6. 7569 sq. ft. 8. 9409 sq. ft. 10. 6889 sq. ft. Find the square roots of the following fractions l>y taking the square root of each term of each fraction : 240 SUPPLEMENTARY WORK Square Root with Decimals. Find the value of V151.29. Show the class that the greatest square of the tens in 151.29 is 100, and that the square root of 100 is 10. Then 51.29 contains 2 x 10 x the units' number of the root, plus the square of the units' number. Ask why this is the case. Dividing 51 by 2 x 10, or 20, we find that the next figure of the root is 2. We have now found 12, the square being 100 + 44 = 144. Then 7.29 contains 2 x 12 x the tenths' number of the root, plus the square of the tenths' number, because we have subtracted 144, which is the square of 12. Dividing by 24, we find that the tenths' figure of the root is 3. Hence the square root of 151.29 is 12.3. If the number is not a perfect square, we may annex pairs of zeros at the right of the decimal point and find the root to as many decimal places as we choose. 151. 1 29(12.3 20 22 51 44 240 243 7 7 29 29 Summary of Square Root. We now see that the following are the steps to be taken in extracting square root : Separate the number into periods of two figures each, be- ginning at the decimal point. Find the greatest square in the left-hand period and write its root for the first figure of the required root. Square this root, subtract the result from the left-hand period, and to the remainder annex the next period for a dividend. Divide the new dividend thus obtained by twice the part of the root already found. Annex to this divisor the figure thus found and multiply by the number of this figure. Subtract this result, bring down the next period, and pro- ceed as before until all the periods have been thus annexed. The result is the square root required. SQUARE ROOT 241 9. 63,001. 10. 21,224,449. 11. 49,112,064. 12. 96,275,344. Exercise 63. Square Root Find the square roots of the following : 1. 12,321. 5. 19.4481. 2. 54,756. 6. 0.2809. 3. 110.25. 7. 1176.49. 4. 8046.09. 8. 82.2649. In Exs. 13-17 give the square roots to two decimal places only. 13. 2. 14. 5. 15. 7. 16. 8. 17. 11. 18. Find, to the nearest hundredth of an inch, the side of a square whose area is 3 sq. in. Square on the Hypotenuse. As we learned on page 112, in a right triangle the side opposite the right angle is called the hypotenuse. If a floor is made up of triangu- lar tiles like this, it is easy to mark out a right triangle. In the figure it is seen that the square on the hypotenuse contains eight small tri- angles, while the square on each side contains four such triangles. Hence we see that The square on the hypotenuse is equal to the sum of the squares on the other two sides. This remarkable fact is proved in geometry for all right triangles. Given that AB = 12 and AC= 9, find BC. Since JBC 2 = AB 2 + ~AC 2 , G therefore lJC 2 = 12 2 + 9 2 , or ~BC Z = 144 + 81 = 225, and BC = V225 = 15. A. 12 242 SUPPLEMENTARY WORK Exercise 64. Square Root 1. How long is the diagonal of a floor 48 ft. by 75 ft. ? On this page state results to two decimal places only. 2. Find the length of the diagonal of a square that contains 9 sq. ft. 3. The two sides of a right triangle are 40 in. and 60 in. Find the length of the hypotenuse. 4. What is the length of a wire drawn taut from the top of a 75-foot building to a spot 40 ft. from the foot ? 5. A telegraph pole is set perpendicular to the ground, and a taut wire, fastened to it 20 ft. above the ground, leads to a stake 15 ft. 6 in. from the foot of the pole, so as to hold it in place. How long is the wire ? 6. A derrick for hoisting coal has its arm 27 ft. 6 in. long. It swings over an opening 22 ft. from the base of the arm. How far is the top of the derrick above / 52 the opening? Reversing the procedure on page 241, the square on either side is equal to the difference between what two squares ? 7. The foot of a 45-foot ladder is 27 ft. from the wall of a building against which the top rests. How high does the ladder reach on the wall ? _B 8. To find the length of this pond a class laid off the right triangle ACE as shown. It was found that .4(7=428 ft., BC = 321 ft., and AD = 75 ft. Find DB. / ? 9. How far from the wall of a house must the foot of a 36-foot ladder be placed so that the top may touch a window sill 32 ft. from the ground ? , . PYRAMIDS 243 Prism. A solid in which the bases are equal polygons and the other faces are rectangles is called a prism, Volume of a Prism. It is evident that we may find the volume of a prism in the same way that we found the volume of a cylinder (page $00). That is, The volume of a prism is equal to the product of the base and height. Pyramid. A solid of this shape in which the base is any polygon and the other faces are triangles meeting at a point is called a pyramid. The point at which the triangular faces meet is called the vertex of the pyramid, and the distance from the vertex to the base is called the altitude of the pyramid. The faces not including the base are called lateral faces. Volume of a Pyramid. If we fill a hollow prism with water and then pour the water into a hollow pyramid of the same base and the same height, as here shown, it will be found that the pyramid has been filled exactly three times with the water that filled the prism. Therefore, Ttie volume of a pyramid is equal to one third the product of the base and height. Lateral Surface of a Pyramid. The height of a lateral face of a pyramid is called the slant height of the pyramid. Since the area of each lateral face of a pyramid is equal to half the product of the base and altitude, The area of the lateral surface of a pyramid is equal to the perimeter of the base multiplied by half the slant height. 244 SUPPLEMENTARY WORK Lateral Surface of a Cone. If we should slit the surface of a cone and flatten it out, we would have part of a circle. The terms "lateral surface " and "slant height" will be understood from the study of the pyramid. From our study of the circle we infer that The lateral surface of a cone is equal to the circumference of the base multiplied by half the slant height. Volume of a Cone. In the way that we found the volume of a pyramid we may find the volume of a cone. Then The volume of a cone is equal to one third the product of the base and height. Find the volume of a cone of height 5 in. and radius 2 in. Area of base is -^ x 4 sq. in. Volume is x 5 x '^? x 4 cu. in. = 20.95 cu. in. This method of calculation gives 20.95 5 5 T cu. in. as the volume, but 20.95 cu. in. is a much more practical form for the answer. Teachers will observe that only the simplest forms of prisms, pyramids, and cones have been considered in this book. Exercise 65. Lateral Surfaces and Volumes Find the volumes of prisms and also the volumes of pyra- mids with the following bases and heights : 1. 36sq.in., 7in. 2. 48 sq.in.,5|dn. 3. 5. 7 sq. in., 4.8 in. Find the lateral surfaces of pyramids with the following perimeters of bases and slant heights : 4. 30 in., 18 in. 5. 3 ft. 3 in., 8 in. 6. 5 ft. 9 in., 10 in. Find the volumes of cones with the following radii of bases and heights : 7. 14 in., 6 in. 8. 5.6 in., 15 in. 9. 49 in., 15 in. CONES AND SPHERES 245 Surface of a Sphere. If we. wind half of the surface of a sphere with a cord as here shown, and then wind with exactly the same length of the cord the surface of a cylinder whose radius is equal to the radius of the sphere and whose height is equal to the diameter, we find that the cord covers half the curved surface of the cylinder. Therefore the surface of a sphere is equal to the curved surface of a cylinder of the same radius and height. We can now easily show that surface of sphere = ^- X 2 x radius x 2 x radius. Hence the surface of a sphere is equal to 4 times -j- times the square of the radius. Volume of a Sphere. It is shown in geometry that The volume of a sphere is equal to ~ times %j- times the cube of the radius. The cube of r means r x r x r and is written r 8 . 1. Considering the earth as a sphere of 4000 mi. radius, find the surface. 4 x 3j x 4000 2 = 4 x 3,? x 16,000,000 = 201,142,857}. Therefore the surface is about 201,143,000 sq. mi. 2. If a ball is 4 ft. in diameter, find the volume. The radius is of 4 ft., or 2 ft. The volume is f x ^ x 2 x 2 x 2 cu. ft., or 33.52 cu. ft. 246 SUPPLEMENTARY WORK Exercise 66. Surfaces and Volumes 1. If a ball has a radius of 1^ in., find the surface. 2. Find the surface of a tennis ball of diameter 2$ in. 3. If a cubic foot of granite weighs 165 lb., find the weight of a sphere of granite 4 ft. in diameter. 4. A bowl is in the form of a hemisphere 4.9 in. in diameter. How many cubic inches does it contain ? 5. A ball 4' 6" in diameter for the top of a tower is to be gilded. How many square inches are to be gilded ? 6. A pyramid has a lateral surface of 400 sq. in. The slant height is 16 in. Find the perimeter of the base. 7. A conic spire has a slant height of 34 ft. and the circumference of the base is 30 ft. Find the lateral surface. 8. What is the entire surface of a cone whose slant height is 6 ft. and the diameter of whose base is 6 ft. ? 9. What is the weight of a sphere of marble 3 ft. in circumference, marble being 2.7 times as heavy as water and 1 cu. ft. of water weighing 1000 oz. ? 10. Taking the earth as an exact sphere with radius 4000 mi., find the volume to the nearest 1000 cu. mi. 11. If 1 cu. ft. of a certain quality of marble weighs 173 lb., what is the weight of a cylindric marble column that is 12ft. high and 18 in. in diameter? 12. How many cubic yards of earth must be removed in digging a canal 8 mi. 900 ft. long, 180 ft. wide, and 18ft. deep? 13. A marble 1^ in. in diameter is dropped into a cylindric jar 5 in. high and 4 in. in diameter, half full of water. How much does the marble cause the water to rise? TABLES FOR REFERENCE LENGTH 12 inches (in.) =1 foot (ft.) 3 feet = 1 yard (yd.) 5| yards, or 16 feet = 1 rod (rd.) 320 rods, or 5280 feet = 1 mile (mi.) SQUARE MEASURE 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd.) 30| square yards = 1 square rod (sq. rd.) 160 square rods = 1 acre (A.) 640 acres = 1 square mile (sq. mi.) CUBIC MEASURE 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) 128 cubic feet = 1 cord (cd.) WEIGHT 16 ounces (oz.) = 1 pound (Ib.) 2000 pounds = 1 ton (T.) LIQUID MEASURE 4gills(gi.)=l pint(pt.) 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) 31| gallons =1 barrel (bbl.) 2 barrels =1 hogshead (hhd.) 247 248 TABLES FOE REFERENCE DRY MEASURE 2 pints (pt.) = 1 quart (qt.) 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.) TIME 60 seconds (sec.) 1 minute (min.) 60 minutes = 1 hour (hr.) 24 hours = 1 day (da.) 7 days = 1 week (wk.) 12 months (mo.) =1 year (yr.) 365 days = 1 common year 366 days = 1 leap year VALUE 10 mills = 1 cent ( or ct.) 10 cents = 1 dime (d.) 10 dimes = 1 dollar ($) ANGLES AND ARCS 60 seconds (60") =1 minute (I') 60 minutes = 1 degree (1) COUNTING 12 units = 1 dozen (doz.) 12 dozen, or 144 units =l-gross (gr.) 12 gross, or 1728 units =1 great gross PAPER 24 sheets = 1 quire 500 sheets = 1 ream Formerly 480 sheets of paper were called a ream. The word "quire " is now used only for folded note paper, other paper being usually sold by the pound. INDEX PAGE Account .... 2, 4, 23, 57, 87 Accurate proportions . . . 140 Acute angle 112 triangle 112 Addition 33 Aliquot parts 42 Altitude 150, 200, 243 Amount of a note 94 Angle 112 Arc 115 Area 162,164,167,169,172,174,196 Balance 2 Bank 79, 86, 90, 97 Base 114, 150, 200 Bill . 48 Bisection 124 Cash check 43 Center 149, 150 Check 2, 43, 92, 155 Circle .... 115, 149, 194, 196 Circumference . . 115, 149, 194 Commercial paper 99 Compound interest .... 86 Cone 150, 244 Congruent figures 115 Constructing triangles . . . 116 Creditor 49 Cube 198 Cylinder 150, 200 Debtor 49 Deposit slip 90 PAGE Diameter . . 115, 149, 150, 194 Discount 44, 46, 97 Distances 222 Dividing a line 130 Drawing instruments . . . 115 to scale 136, 204 Ellipse 149 Equilateral triangle . . 112, 118 Face of a note 94 Formulas 164,167,170,172,194, 196, 200, 201 Fractions 67 Geometric figures 112 measurement 155 patterns 132 Height 150,200 Hypotenuse 112, 241 Indorsement 92, 95 Instruments, drawing . . . 115 Interest 81 Invoice 50 Isosceles triangle . . . 112, 118 Lateral surface .... 243, 244 Lathing 203 Length 165 Line 114 List price 44 Locating points 219 249 250 INDEX FACE Maker of a note 96 Map drawing 217 Marked price 44 Material for daily drill . . . 105 Metric measures 205 Miscellaneous problems . . 30, 73, 102, 212, 234 Multiplication 37 Net price 44 Note 94 Oblique angle 112 Obtuse angle 112 triangle 112 Outdoor work 153, 156, 178, 192 213, 235 Overhead charges 74 Pantograph 145 Parallel lines 128 Parallelogram .... 114, 167 Payee 92, 95 Per cent 6, 7 Percentage problems .... 17 Perimeter 112, 114 Perpendicular .... 121, 222 Photograph 143 Plane figures 149 Plastering 203 Polygon 114, 174 Position . . . 215, 219, 227, 229 Postal savings bank .... 89 Price list 24, 47 Principal 81, 94 Problems without figures . . 154 214, 236 without numbers 32, 54, 66, 76, 104 Proceeds 97 PACK Promissory note 94 Proportion 184 Proportional 186 Protractor 115 Pyramid 243 Quadrilateral 114 Radius . . . 115, 149, 150, 194 Rate of interest 81, 95 Ratio 145, 182 Receipted bill 49 Rectangle 114, 164 Rectangular solid 198 Review drill 31, 53, 65, 75, 103, 152 Right angle 112 triangle 112 Savings bank 79, 86 Several discounts 46 Short cuts in multiplication . 38 Similar figures .... 141, 186 Six per cent method .... 100 Size 155 Sphere 150, 245 Square 114, 196, 237 roots 237 Squared paper 162 Subtraction 35 Symmetry 147 Tables for reference .... 247 Trapezoid 114, 172 Triangle 112, 169 Units of area 164 Uses of angles 231 Vertex . . . 112, 114, 150, 243 Volume . 198, 200, 243, 244, 245 JAM 3 1939 Y 3 19 49- OCT 3 i 'a 30 1 4 1958 UNIVERSITY OF CALIFORNIA AT LOS ANGELES THE UNIVERSITY LIBRARY This book is DUE on the last date stamped below APR 22 1 6 1974 -fitC'0 mi JUL 7615 ;C 1 1 1940 MAR 3 o 1' UNIVKKS1TY of CAUl'OKNlA AT LOS ANGELES TJRRARY I A 000 934 527 3 PLEA* DO NOT REMOVE THIS BOOK CARD s o University Research Library 00