390 



THE POPULAE EDUCATOR. 



DP found, and in all respects treated as are the vanishing lines of 

 retiring sides. 



PROBLEM XXXVI. (Fig. 60). Two square slabs of different 

 dimensions, the smaller of ivhich is lying v/pon the otlier ; the 

 plans of their centres coincide; the nearest angle of the lower 

 (me touches the pp. The side of the larger slab is 4'5 feet ; the 

 smaller, 3 feet. Thiclmess of each, 1 foot. Angle of sight, distance, 

 and height of tlie eye, as in the last problem. 



A portion of the subject represented by the plan A must be 

 constructed, for the purpose of obtaining the length of that part 

 of the diagonal line between a and b. As the angles of the 

 object are right angles, therefore the angle formed by the 

 vanishing lines from E to the HL will be a right angle. Bisect 

 it by the line E o ; E o will then be the vanishing line of the 

 diagonal of the slabs, and o the VP. Find its distance point by 

 drawing from o the arc E DO. After the lower slab, m c d e, 

 is drawn according to previous instructions, produce the per- 

 pendicular m c through v ; make m c and c v equal to the 

 thickness of the slabs ; in other words, mark their heights on the 

 line of contact from m. Draw the diagonals m o, c o, and 

 wo; also the diagonal d e. Our object now is to determine the 

 nearest angle of the upper slab. Upon the diagonal of the base, 

 m o, we must cut off the distance of a b, in the plan A. Make 

 m n equal to the lino a b, and from n draw a line to DO, 

 cutting the retiring diagonal m o in h; m h will then bo the 

 perspective distance of a b. From h draw the perpendicular 

 hsr; this perpendicular, cutting the diagonal from c, gives the 

 nearest angle of the upper slab in s ; c v being the measured 

 thickness of the upper slab, therefore s r is the perspective 

 thickness. The diagonal d e, cutting the retiring base of the 

 upper slab from s each way, gives the perpendicular edges at I 

 and fe. The remaining retiring lines must be directed to their 

 respective vanishing points. 



LESSONS IN ARITHMETIC. XLVL 



EXERCISE 64. MISCELLANEOUS EXAMPLES. 

 1. Eeduce to a proper fraction 



^? v fc_nrf ^ HLI 



x - - 



2. How many bricks 9 inches long, 4^ broad, and 4 thick, 

 will be required for a wall 30 feet long, 20 feefe high, and 4 feet 

 thick, allowing 6 per cent, of the space for mortar ? 



3. When hay was 5 per ton, a well-to-do farmer hid him- 

 self in a load, and his weight, of course, was added in that of 

 the hay. Before the hay was shipped, the trick was detected ; 

 and after another weighing, 7s. 6d. was deducted from the price. 

 Find the farmer's weight. 



4. A room is 20 feet long, 16 broad, and 12 high. If pure 

 gold be worth .4 5s. per oz. Troy, and a cubic foot of gold 

 weigh 19260 oz. avoirdupois, what is the value of the gold 

 which will exactly fill the room ? 



5. A wine-merchant buys 12 dozen of port at 84s. per dozen, 

 and 60 dozen more at 48s. per dozen ; he mixes them, and sells 

 the mixture at 72s. per dozen : what profit per cent, does he 

 realise on his original outlay? 



6. How many cubes of which the sides are 2| inches can be 

 cut out of a cube of which the side is 22 inches ? 



7. A young lady desires to paper her room with postage- 

 stamps, but being herself unable to calculate the number which 

 will be required, she supplies the following data : Her room is 

 14 feet 9 inches long, 9 feet 3 inches broad, and 10 feet 6 

 inches high ; it contains two windows, each 5| feet by 4 feet, 

 and three doors, each 6 feet by 3 feet ; a postage-stamp is -J-jj of 

 an inch long, and f of an inch broad. Make the calculation for 

 her. 



8. The daily issue of the Times is 60,000 copies. Three days 

 of the week it consists of 3 sheets, and for the remaining three 

 of 4 sheets. If a sheet be 3 feet long and 2 feet broad, find the 

 number of acres which the weekly issue of the Times would 

 cover. 



9. Express n O f 7s. 6d. + -625 of 10s. -545 of 9s. 2d. as a 

 decimal fraction of .10. 



10. Among how many men must .73 be divided in order that 

 the share of each may be ,4 17s. 4d. ? And among how many 

 must .79 17s. 6d. be divided in order that half of them may 

 have 10s. 7d, each, and the other half 7s. 2d. each ? 



11. Find the length of the side of a square which is equal in 

 area to the rectangle, the sides of which are 513 yards 1 foot 

 11 inches and 1628 yards 11 inches. 



How much would it cost to cover the area with turf at 4Jd. 

 per square yard ? 



12. If 100 Ibs. of tea be bought at 4s. 4d. and sold at 5s., 

 and 100 Ibs. of sugar bought at 6d. and sold at 7d., what profit 

 per cent, will be realised on the whole outlay ? 



13. The removal of a quantity of brick earth, 32 square yards 

 in area and of a uniform depth of 2 yards, costs ^62 2s. 8d. ; 

 what is tho cost of the removal of a cubic yard ? 



14. A person's average annual expenditure, from the year 

 1830 to the year 1850 inclusive, is .391 9s. 2d. He finds that 

 in 1830 he spent ,391 16s., and in 1851, .445 8s. 9d. What 

 was his average annual expenditure from 1831 to 1851 in- 

 clusive ? 



15. In Austria 120 gulden (paper currency) are worth 100 

 silver gulden. What amount of paper money should be obtained 

 for 10 sterling if the value of 1 be 9 gulden 30 kreutzers in 

 silver (60 kreutzers = 1 gulden) P 



16. What sum at .4 per cent, compound interest will amount 

 in two years to .405 12s. ? 



17. A room is 60 feet long by 29 feet wide; how many people 

 can be seated in it on chairs 1^ feet wide, and placed 2 feet 

 apart from back to back, allowing a clear passage 3 feet wido 

 down the middle of the room and a space of 15 feet deep at one 

 end? 



18. Divide 296-293 by 41-967 so as to have six decimal places 

 in the quotient. 



19. Find the length of the longest chain in terms of which 88 

 yards 2 feet 5 inches and 1 19 yards 2 feet 1 inch can both be 

 expressed as integers. 



20. By selling at 4 per cent, profit, a tradesman gained 

 .47 14s. ; what was the prime cost of his goods ? 



21. The interest on a certain sum of money for two years is 

 .71 16a. 7^d., and the discount on the same sum for the same 

 time is 63 17s., simple interest being reckoned. Find tho 

 rate per cent, and tho sum. 



22. A barters some tea with B for flour which is worth 

 2s. 3Jd. a stone, but uses a false pound weight of 15f oz. 

 What value should B sot upon his flour that tho exchange may 

 be fair ? 



23. A garrison consisting of 2000 men, finds during a siege 

 that it has provisions for six weeks ; at the end of a fortnight 

 400 men are killed in a sally. How long will the provisions last ? 



24. Find the expense of painting a room 23 feet 6 inches 

 long, 12 feet 7 inches high, and 17 feet 6 inches broad, at 

 3s. 6?d. a square yard. 



25. The gold procured from Australia in nine months in 1851 

 amounted to 313644 ounces. In 1861, the New Zealand gold- 

 fields yielded 314438 ounces in the same time. What is the 

 excess in weight and value (at .3 17s. lO^d. per ounce) of the 

 average monthly return from New Zealand over that from 

 Australia ? 



26. Find tho difference between the amount of .247 10s. for 

 two years, and the present worth of the same sum due after two 

 years at 5 per cent. 



27. How many bricks, of which the length, breadth, and 

 thickness are 9, 6, 3 inches respectively, will be required to build 

 a wall whereof the length, height, and thickness are 72, 8, and 

 lifeet? 



28. Express as the fraction of .10 the difference between 

 -8jj and .8 X ; and find the value of f of a ton of sugar 

 when T | of a ton is worth 6 5s. 



29. Find two numbers whose greatest common measure is 

 179, and least common multiple 56385. 



30. In which way had one better buy sugar, at 3 guineas per 

 cwt., or at 2 16s. 4d. per quintal of 100 Ibs. ? and how much 

 is one buying when the gain by the more advantageous way is a 

 guinea? 



31. On what sum is the daily interest at 4 per cent, one 

 penny ? 



32. If 16 darics make 17 guineas, 19 guineas make 24 pistoles, 

 31 pistoles make 38 sequins, how many sequins are there in 1851 

 darics ? 



33. Which way had one better buy coffee, at 6 guineas a cwt., 

 or at 5 12s. 4d. per quintal of 100 Ibs. ? And how much is one 

 buying, when the loss on the less advantageous way is 1 it 



