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QUEBEC : 1833. . m^ III » .i ii r' i»iiiwp ;»iir ,a«iiik> ^^Tt^/wT/i'io^aA -^n ^ \ ' ■M i .^r lUTmai^ m ,i,il^j^MM-^m 'If. - '. < i»; I » « SI (T^ttl ■rtjt'i Having encountered many difficulties during ten years as a Teacher in Quebec for want of a pro- per initiating book of arithmetic, I was induced to compile the following' work, which it is believed is suited to supply the existing demand of Ele- mentary schools in that department. My object has been to select only that which is useful and esssential to a good Accountant, Mea- surer and Book-Keeper. The arrangement is designed to meet the present state of our schools; and to facilitate the acquisition of common Arith- metic in the shortest time of which the study admits. ' / The public are indebted to the liberality of tlie Provincial Legislature for the sum oi fifty poundi^ which they voted toward the expenses of the pub- lication, that the price might be so reduced, that all the- children may be supplied with a copy. If the work should advance the cause of Education in our Elementary schools, the beneficence of the Parliament will be repaid, and the object of th« Compiler will be fiilly attained^ WiLLUU Morris. (luAu, IZlhJmtt 1833. CONTENTS. w t , t?> fj ' f rof * nn *;> r. i m H ruiinereUoB and Notation, . % a'(u>v. n T ^^^ ^ 3 7 Addition, . • r • * • rt 'r* i^ •!' v* Subtraction, . . ; ^ •^;»''^';}lol cm?1 r>h,|, ^ MuhipKcation, . . . '■ i'^'f . 'J > II Division, . • ... • • • • 17 Shop Accounts, • • • ,g/;tl^'>: • 22 Beduction, . • • *! ini^v^ifni hn* • 27 The Rule of Three Direct, • '}J-^'.mU* Un\* 40 The Rule of Three Inverse, •' 4^ r^^y • 46 The Double Rule of Three, . .^ , ^ . / 44 Simple Interest, • . »,*';"• • •' *® Compound Interest^ . *}4jii« t^ ^ ^^ Vulgar Fractions, . ;•,-•. ♦ .. ,• * • 4^ Decimal Fractions, ' . " "i^ 'i^^ ^^'^^^^l ^ 66 Duodecimals, or Cross MukiplicatioH^' ' ' t''i'>> . 61 Involution, . • . . * • lii r: 62 Evolution, • . . 'a/f ^lis i/uW ,tif »■ W Practical Geooietir, • ; * .>' :«1JJii 110 Book Keeping by Single Entm .; ' 'I.;!. '^ *^i Ha .iisaV^iSC ,-iiniwP t :\ ARITHMETIC. Question. What is Arithmcftic ? Answer. Arithmetic is the art of reckoning by num- bers ; and consists of five principal rules ; namely. Notation or Numeration, Addition, Subtraction, Multi- plication, and Division. Q. How are numbers expressed ) . A. All numbers are expressed by the ten following figures : 1 98 4569 89 10 O •m, or unit, two, Unree, four, fire, six, MTen, eight, nine, ten, cypher. anrBCB&Anaxr axtd xroTATzoir. Q. What is Notation? A. Notation is the method of writing down a number in figures. Q. What is Numeration ? A. Numeration is the art of reading a number ex- pressed in figures. Q. How must numbers be read ? A. From the lefl hand toward the right hand. , Q. How does the value of figuTies increase ? .^ A. In a ten-fold proportion from the right hand toward the left ; thus, the first figure on the right hand signifies soniany units; in the second place it represents so many tens ; and, in the third place, so many hundreds. Q. yf\ki is the use of the cypher ? A. The cipher serves to bring figures to their proper places, hy supplying vacant places. iThus, 7, seven ; 70« SQi^nty; 7Q0, seven hundred; 770, seven hundred and seventy ; 777) ^even bundre4 and seventy-aeyeD. 'K0t* i . ' M , 2 NUMERATION AND NOTATION. Q. Repeat your B H i B 9. 8 7, 6 A. Units, one ; tens, twenty-one ; hundreds, three hundred and twenty-one ; thousands, four thousand three hundred and twenty-one ; tens of thousands, fifty-four thousand three hundred and twenty-one; hundr«)ds of thousands, six hundred and fifly-four thousand, three hundred and twenty-one; millions, seven million, six hundred and filly-four thousand, three hundred and twenty- one ; tens of millions, eighty seven millions, six hundred and fifty-four thousand, three hundred and twenty-one ; hundreds of millions, nine hundred and eighty-seven mil- lions, six hundred and fifly-four thousand, three hundred and twenty-one. Examples. Write down in Figurea the following numbers : 1. Twenty-three f Ans. 88. 2. Two hundred and fifly-four. 3. One thousand eight hundred and thirty-two. 4. Twenty-fiv«) thousand, eight hundred and fifly-siz* 6. One hundred and twenty-threethousand, one hun- dred and twenty-three. 6. Eight hundred thousand, seven hundred and six. 7. Four millions, nine hundred and forty-one ^usand four hundred. 8. Twenty-seven millions, one hundred fifty-seven thousand, eight hundred thirty-two. 9. Seven hundred and twenty-two milfioni, tflB hmi- ^red Ihirty-oiie thousand, five hundred and leur. « ^ ,_ _.. . — , ... ._ ^ 1 • ADDITION. A 10. Six hundred and two millions, two hundred iad Ion thousand, five hundred. Write doi 11. 36. 12. 69 18. 172 14. 909 m in ffordi the following numbers : Ans. Thirty five. I 16. 2017 18. 2071909. | 16. 20760 19. 70064008. 17. 764068 20. 123466789. j v.> Q. What is Addition? A. Addition teaches to add two or more sums toge- ther, to make one whole or total sum. Q. Repeat your Addition Table. A. One and one are two, 1 and 2 are 3, 1 and 3 are 4, and so on. Addition Table. % lane 1 2 8 4 ^ 6 7 8 9 10 1= 2 3 4 6 6 7 2 „ = 3 4 6 6 7 8 9 10 11 3 „ 4 „ = 4 5 6 7 8 9 10 11 12 6 6 7 8 9 10 11 12 13 5 „ = 6 7 & d 10 11 12 13 14 6 „ = 7 6 9 10 11 12 13 14 15 7 „ = 8 9 10 11 12 13 14 15 16 8 „ s: 9 10 11 12 13 14 16 16 17 9 „ 10 11 12 13 14 15 16 17 18 A, •'■*(and . be- lling any and J*- Pence coLUMii.*-Abelow ; ap.d in this case cariy one to the under figl^ ^ the next name. £xamples« 10 90 la 4 1. Ans. £ 8. d. From 438 15 7^ Take 278 17 9) ^169 17 9| SUBTRACTION. Fartlltllp^.— IV t< 2 farthingf from 1 farthing I cannot, 2 farthings from 4 farthings and 2 remain — 2 and 1 are 3 farthings, set do?m }, and carry 1 to the pence. Pence* — 1 carried and 9 are 10, take 10 from 7, 1 cannot, 10 from 12 and 2 remain— 2 and 7 are 9 ; set down 9, and carry 1 to the shillings. Shillings.— 1 carried and 17 are 18 ; take 18 from 15 I cannot ; 18 from 20 and 2 remain — 2 and 15 are 17 ; set down 17 and carry 1 to the pounds. Pounds* — 1 carried to 8 are 9 ; take 9 from 8 I cannot, 9 from 10 and 1 remains — 1 and 8 are 9 ; set down 9 and carry 1 to the next column ; 1 carried and 7 are 8 ; take 8 from 3 I cannot ; 8 from 10 and 2 re- main — 2 and 3 are 5 ; set down 5, and carry 1 to the next column ; 1 carried and 2 are 3— take 3 from 4, and 1 remains ; set down 1. Remainder in words, one hundred and fifty-nine pounds* seventeen shillings and nine-pence three farthings. 2. From ^547 13 10 3. From iS7864 17 4| Take 326 10 9 Take 5412 11 l| 4. „ JS 21384 2 7i . „ 10120 1 2j 5. „ iB 721384 3 7^ „ 120123 4} 6. £53907 11 54 21302 id 10) 7. „ JB 38597 12 IJ „ 13270 10 8| 6. „ £32975 16 4) 9. „ £57384 13 7 12264 17 9| „ 27172 18 10| )» 10. „ £75432 3 8| 11. ,, , „ 14129 1 74 ;fc'37921 10 2| 12737 8 li 1 10 SUBTRACTION* 12. Froin;e37205 13 9\ Take 17921 17 9 13. From ;675082 4} Take 17392 16 6} 14. ' 15. •« >» ^£12764 19 7 139 11 10 £9999 0} . • ■ s 16. What is the difference between £73 0«. 6ld, and £19 13s. lOd. Ans. £53 6«. l^d. 17. A lends to B £100. How much is B in his debt, after A has received £73 12^. A^d, 1 Ans. £26 7«. l^d, 18. Subtract £17 2«. 6d. from iS500, and tell me the remainder ? Ans. £482 17«. 6d. 19. If £482 179. 6d. be taken from £500, what will be the remainder? Ans. £17 2«. 6d. 20. If I owe my friend £ 700, and I pay him i& 50 2«. 9|d. on account, what will remain due ? Ans. J&649 17». ^d. 21. What is the difference between £99 19«. ll|({. and £100? Ans. 4. 22. Take one farthing from £100. ^ Ans. £99 19«. il^d. 23. A merchant has in cash £474 89. 9d. ; goods, value £3443 15^. ; a house worth £713 II9. ; a ship £574; another £315; debts due to him £957 I89. Hid. Now he owes to A £115 7a. 8d. ; to B £327 18s. 4|(i. ; to C £74 139. 4d. I demand his net stock? ^ Ans. iS 5960 149. 3|^. 24. A borrowed from B, at sundry times, the following sums, viz. £781, £63 159., £52 IO9., £565 ; and has paid ab follows, at differert times in cash, £330 IO9., £54 13». 4d., £67 IO9. ; in goods £54 I89. 6d., £73 159. Sd. ; by a draft on John Steele, £63. What is A still due? Ans. £817 179. 6d. 25. Suppose that my rent for half a year is £10 129. ; and that I have laid out, for the land tax, 149. 6cl., and for several repairs £ 1 39. 3;|d., what have I to pay of my half year's rent.' Ans. £8 149. 2jd[. o| 1< ■' '<" i ^d. and 6«. 7|d. bis debt, 7a.7ld. I roe the 17*. 6d. ^at will ' 28. 6d, ^50 28. fa, 2id. . ll^rf. ina, j. . Hid. goods, a ship 7 I8s. £327 stock? ^ 3|rf. owing id has ) 10*., . , £73 ^ t is A *. 6d. 12*. ; id for »f my 2ji Q. What is Multiplication A. Multiplication teaches of two Numbers given, as ofte less; and compendiously perfori additions. Q. Repeat your eater in the of many Mnltiplieation Table. 14 16 as IS 34 34 34 27 33 36 IS 3128 33 44 48 35 55 60 35 43 42 40 48 54 73 16 18 34 37 33 36 40 45 48 54 49 56 6677 84 8 9 10 56. 63 64 79 73 81 9U so! 90100 88' 99 20 30 40 50 60 70 80 96108 110 120 11 13 93 34 33 44 55 66 77 88 99 110 121 133 36 48 13 60 73 84 96 i08 120 133 144 96 39 52 65 78 91 104 117 130 143 156 14 38 42 56 70 84 112 126 140 154 168 15 30 45 60 75 90 105 120 135 150 165 180 16 32 48 80 06 112 144 160 176 192 17 34 51 68 85 102 119 136 18 36 54 73 00 108 136 144 153163 170|180 1671198 204216 10 38 57 11 95 114 133 152 171 190 ■b 20 40 60 80 100 120 140 160 180 •200 200830 338340 Rule 1. — ^When the multiplier is not greater than 12, write it under the pence of the multiplicand, and in multi- plying, put down the overplus of farthings, pence, and shillingSff and carry as in addition. Examples. *' £ t. d. 1. MulUply 14 16 H MuHipKcand hf 7 Multiplier ;ei03 16 2j P»>duct fi ■■^»JM^*,^.^ -«^^t» ,.^j^ ik. :^f ■*«•■* ■*-,. 12 MULTIPLICATION. »? : - *» Farthings. — 7 times l are7— 7 farthings are Ijcf.; set down |, and carry 1 to the pence. Pence*— 7 times 7 are 49, and 1 carried are 50 — 60d. are 4f. 2d.; set down 2, and carry 4 to the shillings. ShlUingrs*—''^ times 16 are 112, and 4 are 116 — 116a. are £5 169.; set down 16 and carry 5 to the pounds. PonndS. — 7 times 4 are 28, and 5 carried are 33 ; set down 3, and carry 3. — 7 times 1 are 7, and 3 carried are 1.Q ; set down 10. Product in words, one hundred and three pounds, sixteen shillings, and two-pence three farthings. £ 8. d. £ 8, d, 2. Multiply 124321 2 4| 3. MoUipiy 23434 5 5^ by 2 by 2 4. n 234204 4 24 5. 3 T 135246 5 41 3 6. 432510 5 3} 7. 4 t* 274321 6 2| 4 8. „ 34523 12 6| 5 f* 9. I* 273534 13 3| 5 10. „ 417383 11 3^ 11. „ 543210 14 4; 6 ' .. 6 »» : • 12. „ 350214 15 4^ 13. 7 ft „ 215438 16 2i 7 MULTIPLICATION. 18 £ 9. d, £ i. d 14. Mdtipiy 521403 6 7} 15. MiUUpiy 488025 7 10) hj £ by 8 5 41 3 6 2| 4 3 3i 5 4 4J 6 6 I 16. „ 378210 10| 17. „ 321457 17 41 9 H 9 n 18. „ 527032 7 3j 19. „ 382721 14 3| 10 „ 10 »» rt 20. „ 387204 15 2J 21. '■9 22. „ 521432 13 4| 23. !♦ 12 „ 432579 10 4 11 „ 732173 4 10) It la To multiply by any number greater than 12, observe the following Rule* — Multiply the top line by 10, and that product again by the same number, until you have as many lines as there are figures in the multiplier ; then multiply the 1st line by the last figure, the second line by tne 2nd figure, and so on ; add these products together, and the sum will be the product of the number given. 4 m\ .-.■i : I i-ii 14 ■fh ,t ^ MULTIPLICATION. Examples. r^ « .*. i 24. What 18 the price of 345 yards of cloth, at £2 12«. 7}d. per yard? lit line £2 12 7^X5 10 2nd line drd line 26 6 3 x4 10 "B^M^lM--'^-* 4 r « k' H*: .' 263 2 6 x3 3 it, • 789 7 6 price of 300 yaMs. "105 5 price of 40 13 3 1} price of 5 »» ^,\ ' Ans. £907 15 7i price of 345 yards. i*< ■ li For 13, multiply by 10, and add 3 times the top line. 14, multiply by 10, and add 4 time&' the top line, &c. 24, multiply by 10 and by 2, and add 4 times the top line. 35, multiply by 10 and by 3, and 5 times the top line. 46, multiply by 10 and by 4, and 6 times the top line. 127, by 10 and by 10, twice the second line and 7 times the top line. 394, by 10 and by 10, 3 times the 3rd line, 9 times ■r:-^ *--- the 2nd, and 4 times the top line. K. B."- The pupil should be exercised for a few minutes in the plan abore, before he enters on the following questions :— What is the price of 25. 13 ib of sugar, at 1«. 3(2. per lb? Ans. 16*. 3d. 26. 14 moidores, at £1 7a, each? Ans. £18 18«. 27. 15 pistoles, at 17». ed. each? Ans. £13 2«. 6d. 28. 16 cwt of cheese, at £1 18«. Sd. per cwt. ? Ans. £30 18«. ed. MULTIPLICATION. 15 79, 18 cwt of tobacco, at £5 llf. 4d. percwtT Ana. £100 4f. 30. 20 cwt. of hops, at £4 7t. 2d, per cwt ? Ana. £87 8f. 4d., 31. 21 cwt. of hemp, at JSl 12«. per cwt.? Ana. ^33 12t. 82. 22 toua of hay, at £1 2«. per cwt? Ans. ;f24 4«. 33. 25 yda. of broad cloth, at 9«. 2d. per yard ? Ana. iSll 9a. 2d, 34. 28 yda. of auperfine do. at 19«. 4d, per yard? Ana. £27U,4d. 35. 32 yda. aerge, at 3«. 7d. per yard ? Ana. £5 14«. 8(2. 36. '48 acrea of land, at £2 3a, per acre? Ana. £103 4«. 37. 66 gallona of rum, at 8a. lOd. per gal. ? Ana. £29 3«. 38. 84 qra. of wheat, at ^1 12«. 8cl. per qr. ? Ana. £137 48. 39. 106 qra. of barley, at 149. 7^(2. per qr. ? Ana. £77 8a.0id. 40. 127 cwt. of hopa, at £3 Oa. 2d. per cwt. ? Ana. ^382 la. 2d. 41. 224 ft, of tea, at 79. 3|d. per lb? Ana. jf81 8a, 8d. 42. 336 fb of do. at 59. 2|(i. per ib? Ana. jf87 179. 43. 532 firkina of butter, at ^2 159. 6d. per firkin? Ana. ^1476 69. 44. 941 cwt. of augar, at £7 Oa. 4d. per cwt. ? Ana. ^6602 139. 8d, 45. 3918 yda. of brown cloth, at 129. 6d. per yard ? Ana. ^2448 159. 46. 6874 aeta of bucklea, at 159. 6d. per aet? Ana. ^^5327 79. 47. 9674 yda. of velvet, at 149. lOd. per yard ? Ana. £7174 179. Sd. 48. 10,000 yds. of ahalloon, at 1 Hd. per yard ? Ana. £479 39.4(1. 16 MULTIFUOATIOlf. Rnle 3* — To mulUpI/ a whole number by ft number eonsi^ og of two or more figures. Place the multiplier undo the multiplicand, then multiply by each figure se- parately, observing to put the first ngure of every product under its multiplier. Add these products together, and the sum will be the total product required. liXamples* 49. Multiply 472035 by 20034? Ans. 9456749190. Multiply 472035 Multiplicand by 20034 Multiplier % 1888140 1416105 944070 .04; Ans. 9456749190 Product. Ii. ir 50. Multiply 273580961 by 23? Ans. 6292362103. 51. Multiply 402097316 by 195? Ans. 78408976620. 52. MulUply 82164973 by 3027? Ans. 248713373271. V 53. Multiply 16358724 by 704006 ? Ans. 11516639848344. 54. How many letters are there in a page of a book which contains 45 lines, each line 59 letters ? Ans. 2655 letters. 55. How many grains of wheat will fill 987 bushels, when 1 bushel contains 675000 ? Ans. 666225000 grains. 56. How many strokes does the hammer of a clock strike in a year of 365 days, at 156 strokes in a day? V Ans. 56940 strokes. 57. How many feet will reach from Quebec to Mon- treal, if the distance be 180 miles, and 5280 feet in a »inile? Ans. 950400 feet. DIVISION. 17 Bivisioir. Q. What is Division ? A. Division is the method of finding how often one number is contained in another. The first number is called the Divisor, the second the Dividend, and the re- sult the Quotient. Q. Repeat your ' , DiTision Table. 2 into • 2 "3 1 i ; 6 8 1 n 6 9 12 8 12 16 20 24 ^ 32 36 40 44 t s 10 10 15 20 25 30 35 40 45 50 55 60 i 3 18 27 36 45 "54 63 ~72 ~81 "90 "99 108 i 20 lo 40 50 60 "to 80 90 100 no 120 1 ■a wm 32 I3 44 "55 8 S a ©» 24 36 "48 60 PS 26 39 65 i g 28 "42 56 "to 'S >n 30 45 60 75 8 "48 64 "so 1 a 34 51 I5 i a CD 36 54 "72 90 38 "57 1 40 3 .. 4 » 4 76 M 5 „ 5 10 12 14 16 18 20 If IB •21 '24 27 30 95 100 6 „ 6 66 77 88 99 110 121 132 72 84 96 108 120 132 144 78 91 104 117 130 143 156 84 "98 112 126 140 154 168 90 105 120 135 96 112 128 144 102 108 114 120 7 ., 7 119 136 126 144 133 140 160 8 „ 8 9 „ 9 153 162 171 180 10 „ 10 150 165 18'/ 160 176 192 170 187 204 180 190 20C 11 ,. 11 22 ■24 33 36 198 216 209 220 «2 „ 12 228 240 Rule !• — ^When tho divisor is not greater than 12, place the divisor on the left hand of the dividend, with a curve line between them ; then find how often the divisor is contained in the dividend, and place the numbers un- der the figures divided ; observing to reduce the remain- der in each name, if any, into the next inferior denomi- nation, adding the given number of that name, and so con- tinue to divide in the same manner to the lowest name placing the last remainder, if any, on the right. '*■' -T^.y: *2 "■J : 18 PIVISIOlf. ) ! i Bxamplet* 1. Difide £6207 3«. SJd by 6. £ «. d. 6) 6207 8 8^ Ana. £1241 8 8]— 2 over. Pounds* — 5 into 6, once and 1 over — set down I, and carry 10—10 and 2 are 12, 5 into 12, twice and 2 over— -set down 2, and carry 20 — 20 and are 20, 6 into 20, 4 times — set down 4 — 6 into 7, once and 2 over — set down 1, and carry 40 to the shillings. Shillings*— 40 carried and 3 are 43, 5 into 43, 8 times and 3 over — set down 8, and carry 36 to the pence. Pence*— 36 carried and 8 are 44, 5 into 44, 8 times and 4 over — set down 8, and carry 16 to the farthings. Farthinfl^S* — 16 carried and 1 are 17, 5 into 17, 3 times and 2 over— set dowa |, and 2 over on the right. £ ». d. £ ». d. 9* 2)2468 10 4 8. 2)26845 4 10 4. 3)36390 12 9 6. 4)48408 16 8 8. 5)56126 6 4i 10. tf^-iMSg 6 3 ^^^ M* 5. 3)43687 2 3 7. 4)57385 b 1 9. 5)62716 7 3i 11. 6)84037 7 4i 12. DIVISION. £ 9. d. £ i. d- 7)834576 2 8} It. 7)321407 4 6] 14. 8)123729 7 4 15. 8)78600« 7 8} 16. 9)387642 6 7i 17. 9)3072i ? t '} 18. 10).>27343 12 lOJ 19. 10)321785 5 8| 20. 11)387503 16 8J 21. 11)87927 9 5J 22. 12)32040 4} 23. 12)87980 lo;. Rule 2* — To divide by any number greater than 12. First ; draw a curve line on each side of the dividend, and put the divisor on the left hand side ; make a small table, by multiplying the divisor by the 9 digits, 1,2, 3, &c. respectively placing the products with their multi- pliers in horizontal rows under the divisor. 2d. If the first figure of the dividend be greater than the first figures of the divisor, count off as many figures from the led hand of the dividend as there are figures in the divisor; but if the first figures be less^ count 1 figure more from the dividend for the first member. Look in the tible for that product which is next less than the first member, and place it under the said member, and the figure which stands on the same line with the product, must be placed on the right hand of the dividend for the 1st quotitnt figure. Subtract the said product from the first member of the dividend, and bring down the next figure of the dividend to the remainder for a second mem- ber; [N'oceed with this member the same as before, and so contmioe till all the figures of the pounds are brou^t down. Multiply (Im remainder from the pounds, if any. 20 DIVISION. by 20, adding in the shillings of the dividend, and divide as before. Multiply the remainder from the shillings, if any, by 12, adding in the pence of the dividend, and divide again. Multiply the remainder from the pence, if any, by 4, adding in the farthings, and when divided once more the operation will be finished. Examples. 24. Divide ,^375683 17». 3irf. by 234. DIVISOR. DIVIDEND. (QUOTIENT. £ 8, d, £ 8. d. 1 . . 234) 375683 17 3J (1605 9 8|-32 rem. 234»»» 1416 1404 " wl^ 1 . . 234 2 . . 468 3 . . 702 4 . . 936 6 . '. 1170 6 . . 1404 7 . . 1638 8 . . 1872 1 9 . .2106 1283 1170 113 remainder from the pounds. • 20 ■ • 234) 2277 (9*. 2106 171 remainder from the shillings. 12 ' 234) 2055 {Sd, 1872 ;<*■ 183 remainder from the pence. 4 234) 734 (I 702 V - f, 32 remainder. Ans. JCieOS 0«. 8}d.=32 remain. DIVISION^ 21 25. Divide £14693 4f. e\d. by 13 ? Ans. £1130 4*. lli(/.-TV ' 26. Divide £17934 10#. TJd. by 14? Ans. £1281 9d.~t^* 27. Divide £37846 17*. lOfd. by 16 ? Ans. £2365 8*. 7}(f.-||. 2S. Divide 5 7384 19*. l^d, by 23? Ans. £2494 19*. ll}d.-,V 29. Divide £138457 14*. 2Jd. by 67 ? Ans. £2429 1*. 7}(f.--if 30. Divide £137586 13*. 5^^. by 124 ? Ans. £1109 11*. 4}-y',V * 31. Divide £321204 19*. lljd. by 674 ? Ans. £476 11*. 3irf.— 14|. 32. Divide £1875486 13*. 5Jrf. by 5374? Ans. £348 19*. 10(i.~|f^|. 33. Divide £49 14*. 6cl. equally between 39 men? Ans. £1 5*. 6cf. 34. If 27 cwt. of sugar cost £47 18*. 9(^., what cost 1 cwt. ? Ans. £1 15*. 3|(^.— 15 over. 35. If 72 yds. of cloth cost £85 5*. 6d., what cost lyard? Ans. £l3*.8Jd. 36. A prize of £7257 3*. 6d. is to be equally divided amongst 500 sailors. What is eacQ man's share ? Ans. £14 10*. 3|(|. 37. If a gentleman's income be £500 a year, what is he worm each day, counting 365 days in a year ? Ans, £1 7*. 4|d. 38. If a farm of 57 acres is let at £55 4*. 4|ci., what is the rent per acre? Ans. 19*. 4} '' 8 „ »» »» »♦ I M 3T 6 8 »» |io „ t« M »» 10 Rule 1 • — If the given price or rate be an aliquot part of a penny, shilling, or pound, divide the quantity by the aliquot part, which gives the answer in pence, shil- lings, or pounds respectively : if the answer be found in shillings, divide by 20, to bring it to pounds ; but if the answer be in pence, divide by 12, and then by 20. SHOP ACCOUNTS. 2S K. B.'It ii expected that the pupil will work each of the fol- lowing questions by multiplication and division, as in the folloiiin§ Examples. 1. What cost 100 ib of sugar, at 4d, per lb ? BT DIVISION. 8, 4• M IP. 1 .-r ■'•'■ l\l /•, -r t&* . - 1 >^ £ 8, d. Ans 7 6} # 13 2} II 1 10 3 fH 4 11 lOj n .^ 6 4 n ' m-' « 7 12 ^ « 9 16 11 31 18 r 60 2 1 II 69 15 4 II 86 10 II 103 1.5 0} II 123 18 7j II 213 17 II 308 18 ^ II 463 15 3 f* 607 11 8i II 943 16 Hi •9 1997 18 4 SHOP ACCOUNTS. 25 Rule 3* — When the quantity given consists of several denominationsi as Tons, Hundreds, Quarters, and Pounds ; multiply the price by the first or highest name, and take parts for the lower denominat'ions, as follows . Weight Table of Aliquot Part8. \ • X, .13 ■ ^ d9 • • a 2 w^ h h (*t pit o O O o OJib IS 1 s= Jh = tIt = ll^T Oi „ 1 « -* « 4 „ 7 „ »» »» u »» ft ft tt tV 1 j = TTlTt 1 8 „ tf ft ft = Vr = shf 14 „ »> ft i = i ~~ TiiT 16 „ 11 ft 1 = xiiT Iqr. M f) tt i = 1 2qrs. ft ft ft 2 = I Icwt » tt ft Tff a M »» t» It tV 4„ »» tf ft i 5 „ ♦» ft ft I 10 „ »♦ ft ff JL JV'iJ ' 1 .1. ^ 3 SHOP ACCOUNTS. ' '} Examples. 1. At i^3 17*. 6d. per cwt.,what is the value of 25 cwt. 2qrs. I41b of tobacco ? £ *. d. 3 17 6 6 JB19 7 6 5 t-^^:^^"^' — — Cwt. qr. ft). 9« 17 6 price of 25 2qrs.=J of lcwt.= 1 18 9 price of 2 141b =i of 2qrs. = 9 8^ price of 14 Ans. £99 5 11^ price of 25 2 14. 2. What cost 13cwt. Iqr. 7ib of molasses, at £l I2s. 4d. per cwt.? Ans. £21 10s. d^d. 3. What cost IScwt. 2qr. 81b of pearl ashes, at £ 2 5s. 6d. per cwt. ? Ans. £ 42 5s, 4. What cost 21cwt. 3qr. 14lb of starch, at £2 16s. 8d. per cwt.? Ans. iSei 19s. 7d. 5. What cost 32c wt. Iqr. 161b of soap, at £3 5». lid. per cwt.? ^ Aus. i2l06 15s. 2|d. 6. What cost 43cwt. 2qr. 21ib of madder, at £3 19». 4d, per cwt. ? £1 73 6s. 1 0^ d. 7. What cost 17cwt. Iqr. lift of cheese, at £3 14s. 8d. per cwt. ? Ans. £64 15s. 4d. 8. What cost 85c wt. Iqr. lOlb of butter, at £4 6s. 4d. per cwt.? Ans. i:368 7s. 7jd. 9. What cost 72cwt. Iqr. 181b of hops, at £4 5s. 8d. per cwt. ? Ans. iS 310 3s. 2d. 10. What cost 27cwt. 2qrs. 151b of raisins, at ie2 6s. 8d. per cwt.? Ans. £64 9s. 7d. 11. What cost 78cwt. 3qrs. 12ft» of currants, at ^22 17*. 9d. per cwt ? Ans. £227 14». REDUCTION. 27 12. What cost 56c wt. Iqr. 17Jb of sugar, at i^2 15f. 9d. per cwt. ? Ans. .£157 4*. 4^^. 13. What cost 97cwt. 151b of tobacco, at ^^3 17*. \0d. per cwt. ? Ans. £378 3d. 14. What cost 37cwt. 2qrs. 131t» of sugar, at i:4 14*. 6d. per cwi.? Ans. .£177 14«. 8jci. 15. What cost 15cwt. Iqr. lOlb of sugar, at c£3 14*. 6d. per cwt. ? Ans. £ 57 2s. 9d. 16. What cost 172cwt. 3qrs. 121b of madder, at .£4 15*. 4d. per cwt. ? Ans. £823 19«. O^d. 17. What cost 53cwt. I7tb of soap, at £3 1 1*. 6d. per cwt.? Ans. £190 4d. 18. What cost 45tons 17cwt. 2qrs. of iron, at ^7 18«. 4d. per ton? Ans. .£363 3*. 6id. nuBvcTzosr. Q. What is Reduction ? A. Reduction is the changing or reducing monies, weights, and measures, &c. out of one denomination into other numbers of another denomination, but equal to the same in value. Q. How are all great names brought into small ? A. Multiply by so many of the less as make one of the greater. Q. How are all small names brought into great? A. Divide by so many of the less as make one of the greater. * Money. 2 Farthings make 1 Halfpenny. 4 Farthings make 1 Pei\ny. ,.^. 12 Pence make 1 Shilling. ^ ^^ 20 Shillings make 1 Pound. Pounds multiplied by 20, are shillings ; Shillings multiplied by 12, are pence ; Pence multiplied by 4, are farthings ; ^'€ Fence multiplied by 2, are halfpence. I i •*« 28 REDUCTION. Farthings divided by 2, are halfpence ; Farthings divided by 4, are pence ; Pence divided by 12, are shillings; Shillings divided by 20, are pounds. ^Examples* -^ 1. In <£21 109. 6|d, how many shillings, pence, and farthings ? £ 8. d, 21 10 6^ Multiply by 20 and add in the 10a. Shillings=430 Multiply by 12 and add in the 6d. Pence=5166 Multiply by 4 and add in thejd. rartblnci = 20666 2. In 20666 farthings, how many pence, shillings, and pounds t 4) 20666 farthings 12) 5166} pence 20) 430 6| shillings. JS21 10 6J Proof of the first. £ s, d, A. 5166d[.430«.21106}. A. 430«. 5166(1. 20666qrs. 3. In £8, how many shillings ? Ans. 160«. 4. In 160 shillings, how many pounds? Ans. BQ. 5. In i^ 12, how noany farthings? Ans. 11520far. 6. In 11520 farthings, how many pounds ? Ans. iS12. 7. \n£\l ba, S^d, how many farthings ? Ans. 16573. 8. In 16573 farthings, how many pounds ? Ans. £11 bs^ Z\d, 9. In £36 7». 9d., how many pence ? Ans. 8733 pence. 10. In 8733({., how many pounds ? Ans. jf 36 7«. Ocf. 11. £^lb 175. lOjd., how many farthings ? ^ Ans. 360859. 12. In 360S59 iarthings, how many pounds ? Ans. iS376 17«. 10j<;. ¥« \ ^w Troy weight. 29 Id. In 100 crowns of 5«. eachthow many farthingst Am. 24000 far. 14. In 36 guineas of 21«. each, how many pence ? Ana. 9072d. 15. In 9072d., how many guineas of 2l9. each? Ans. 36 guineas. •■■-%^M ■ .m'nl VaOT WBXOBT. 24 Grains make 1 Pennyweight. 20 Pennyweights make 1 Ounce. 12 Ounces make 1 Pound. Pounds troy multiplied by 12, are ounces. Ounces „ multiplied by 20, are pennyweights. Pennyweights multiplied by 24, are grains. j Grains divided by 24, are pennyweights. Pennyweights divided by 20, are ounces. Ounces divided by 12, are pounds troy. ^*v Examples. 1. In lib troy, how many grains ? lib. 12 12 ounces. 20 240 pennyweights. 24 960 480 Ans. 5760 grains. 2. In 5760 grains how many pounds troy 1 24) 6760 (grains. 48 I 240 96 96 20) 240 pennyweights. 12) 12 ounces. Ans. 1 pound troy. 3* 5. Uow many grains are there in STfts. t Ans. 213120 grs. 6. In 213120 grains, how many lbs. 7 Ans. 37tts. 7. lo 484% lloz. ITdwts. 23 grs., bow many grains? Ans. 2793561 grs. 8. Reduce 2793551 grs. into pounds ? Ans. 4841b. lloz. ITdwts. 23 grs. AVOniDUVOZS WBZOBV. 16 Drams make 1 16 Ounces make 1 14 Pounds make 1 2 Stone or 28ib make 1 4 Qrs. or 1121b make 1 20 Hundred wt. make 1 Ounce. Pound. Stone. Quarter. Hundred wt. Ton. Tons multiplied by 20, are hundreds. Hundreds multiplied by 4, are quarters. Quarters multiplied by 28, are [jounds. Pound multiplied by 16, are ounces. Ounces multiplied by 16, are drams. Drams divided by 16, are ounces. Ounces divided by 16, are pounds. Pounds divided by 28, are quarters. Quarters divided by 4, are hundreds. Hundreds divided by 20» are tons» I I AVOIRDUPOIS WEIGHT. 81 Examplet* 1. In 1 tODf how manj dnunft 1 ton. SO 20 hundreds. 4 80 quarters. 38 ' «40 160 2240 pounds. 16 13440 2240 35840 le 215040 35840 Ans. JS573440 drams. a. Reduce 573440 drams into tons. 16) 673440 drams. 48 35840 93 80 134 128 , 64 64 16) 35840 ounces. 28) 2240 pounds; 20) Ans. 80 quarters. 20 hundreds. 1 ton. 3. In 15 tons how many pounds ? Ans. 33600]bs. 4. Reduce 3d6001bs. into tons. Ans. 15 tons. 5. In 27cwt. 2qrs. 12Ib» how many &s. 1 Ans. 3092ibs. 6. Reduce 3092ibs. into cwts. Ans. 37cwt. 2qrs. 12ib. 7. In 3 qrs. 14!b» how many ounces ? Ans. 1568 ounces. 1 m 82 I't CLOTH MEASURe. 8. Reduce 1568 ounces into quarters. Afiii. Sqrs. 14tbs. 9. In 35 tons, 17cwt. Iqr. 23ib. 7oz. ISdrs., how many drams ? Ans. 20571005drs. 10. Reduce 20571005drs. into tons. Ans. 25ts. 17cwt. Iqr. 23Ib. 7oz. iSdrs. o&OTB miAsraB. V li. 2| Inches make 1 Nail. 4 Nails make 1 Quarter of a yard. 3 Quarters make 1 Flemish ell. 4 Quarters make 1 Yard. 5 Quarters make 1 English ell. J 6 Quarters make 1 French ell. Yards multiplied by 4, are quarters. t., Quarters multiplied by 4, are nails. Nails divided by 4, are quarters. Quarters divided by 4, are yards. ,..•... .1., Exainplesi 1. In 1 yard, how many nails ? 1 yard. 4 4 quarters. 4 .^'^•''m •- Ans. 16 nalL 2. Reduce 16 nails into yards ? Ans. 4) 16 nails. 4) 4 quarters. 1 yard. \X i^ .*.f$. *-: < , »i •5;V; ,"^^1 .\-^V S (!) LONG MEASURE. 3. In 87 yds.f how many imilt ? Ans. 592 naili. 4. How many yds. are in 592 aails ? Ans. 37 yds. 5. Reduce 15 yds. 3qrs. 1 n. to nails. ? Ans. 253 nails. 0. How many yds. aro there in 253 nails ? Ans. 15yds. Sqrs. Inl. 7. In 73 ells Flemish, how many qrs. ? Ans. 219qrs. 8. In 73 ells English, how many qrs. ? Ans. 36iSqrs. 9. In 73 ells French, how many qrs. ? Ans. 438qr8. 10. Reduce 352 nails into ells English ? Ans. 17 ells, Sqrs. m XiONa MSASvaa. 12 Lines make I Inch. 12 Inches make 1 Foot. 3 Feet make 1 Yard. 5| Yards make I Pole or Rod 40 Poles make 1 Furlong. 8 Furlong 9 make 1 Mile* 3 Miles make 1 League. 69^ Miles make 1 Degree. .'■i' I m Leagues multiplied by 3, are miles— miles multiplied by 8, are furlongs — furlongs multiplied by 40, are poles — poles multiplied by 5J, are yards — yards multiplied by 3, are feet — feet multiplied by 12, are inches. Inches divided by 12, are feet — feet divided by 3, are yards — ^half-yards divided by 11, are poles — poles divided by 40, are furlongs — furlongs divided by 8, are miles- miles divided by 3, are leagues. :;|f '1# > i •f I :<%/" .d^k' .'tl I 34 LONG MEASURE. Examples. 1. In 1 mile, how many inches ? 1 . 8 8 furlongs. 40 J of=320 poles. 1600 160 1760 yards. 3 5280 feet. 12 Ans. 63360 inches. 2. In 63360 inches, how many miles? 12) 63360 inches. 3) 6280 feet. 5J 2 1760 yards. 2 11) 3520 40) 320 poles. ■ 8) 8 furlongs. Ans. 1 mile. 3. In 273 miles, how many inches ? Ans. 17297280 inches. 4. Reduce 17297280 inches, how many miles ? Ans. 273 miles. 5. Reduce 5 m. 6 fur. 3 yds. into inches ? Ans. 364428 inches. 6. In 364428 inches, how many miles ? Ans. 5m. 6f. 3yds. 7. Reduce 2m. If. 8pls. 3yrds. 2inch. into inches ? Ans. 136334 inches. 8. In 136334 inc\ i, how many miles ? Ans. 2m. If. 8 p. 3yds. 2 inch. LAND MEASURE. i LAND BIBASUllB. 35 144 Square inches make 1 Square Foot. 9 Square feet make 1 Square Yard. 30^ Square yards make 1 Sq. Pole or Perch. 40 Poles make 1 Rood. 4 Roods, or 10 chains make 1 Acre. Acres multiplied by 4, are roods — roods multiplied by 40, are perches. Perches divided by 40, are roods — roods divided by 4, are acres. ^ V examples* 1. In 1 acre, how many perches ? 1 acre. 4 ;ofe 4 roods. 40 Ans. 160 perches. 2. In 160 perches, how many acres? 40) 160 perches. 4) 4 roods. Ans. 1 acre. 8. In 15 acres, how many poles or perches ? Ans. 2400 poles. 4. How many acres are there in 2400 poles ? . Ana* 15 acres. 5. Reduce 27&. Ir. 32p. into pole^. Ans. 4392 poles 6. Reduce 4392 poles into acres. Ans. 27a. Ir. 32p LZanD XMBASVMI. 2 Glasses make 1 Gill. . 4 Gills make 1 Pint. 2 Pints make 1 Quart 4 Quarts make 1 Gallon. % Ki 'H I ^ I '.li |, >if '4 'i n it At ( 36 LIQUID MEASURE. 42 Gallons make 1 Tierce. 63 Gallons make 1 Hogshead. 126 Gallons make 1 Pipe. ;^. , 252 Gallons make 1 Tun. Tuns multiplied by 4, are hogsheads — tuns multiplied by 2, are pipes or butts — pipes multiplied by 2, are hogs- heads — ^hogsheads multiplied by 63, are gallons — gallons multiplied by 4, are quarts— quarts multiplied by 2, are pints — pints multiplied by 4, are gills. Gills divided by 4, are pints — pints divided by 2, are quarts— quarts divided by 4, are gallons — gallons divided by 63, are hogsheads — hogsheads divided by 2, are pipes — hogsheads divided by 4, are tuns. XSxamples* r 1. In 1 tun, how many glasses ? 1 tun. 4 4 hogsheads. 262 gallons. 4 1008 quarts. 9 2016 pints. 4 8064 giUs. 2 Ans. 16128 glaise^. I 2. In 16128 glasses, how many tuns ? .Xi':. 2) 16128 glasses. 4) 8064 gills. 2) 2016 pinti. 4) 1008 quarts. 68) 4) Ans. hi:'-' 252 gallons. 252 (4 4 1 tUD. it DRY MEASURE. 87 3. In 19 hogsheads, how many pints ? Ans. 9576 pints. 4. How many hogsheads are there in 9576 pints ? Ans. 19hhd. 5. Reduce 13 1. 1 p. Ihhd. 17 gal. 5pts. into pints. Ans. 27S61 pints. €. Reduce 27861 pints into tuns. Ans^ 13t. Ip. Ihhd. 17gal. 5pt8. ;5vaOft^ ,1*?*^ Us-- av* ' 2 Pints make 1 Quart. 4 Quarts make 1 Gallon. 2 Gallons make 1 Peck. 4 Pecks make 1 Bushel. 8 Bushels make 1 Quarter. 5 Quarters make 1 Load. 2 Loads make 1 Last. •1 .3 Lasts multiplied by 80, are bushels-^bushels multiplied by 4, are pecks — pecks divided by 4, are bushels — bush- els divided by 80, are lasts. Examples. 1. tni 1 fast,' how many pecks? 1 last. 2. In 320 pecker, how many lasts? 'XIR -^ ' - - - 80 4) 320 pecks. Ans. 80 bushels. 320 pecks. 80) 80 bushels. Ans. 1 last. 3. In 128 bushels, how many pecks ? Ans. 512 pecks. .1 '-"4 h 'h i I'd ^1 I' I n *ik Tl TIIIE. 4. In 612 pecki, how many bushels ? Ans. 128 bushels. 5. In 20 lasts,) 3 bush. 3 pecks, how many pecks ? Ans. 6415 pecks. 6. Reduce 6415 pecks into lasts? Ans. 201. 3 b. 3pks. Villi a. 1 Minute. 1 Hour. 1 Day. 1 Week. 1 Lunar month. 60 Seconds . . . make 60 Minutes . • make 24 Hours . . . make 7 Days . . make 4 Weeks . • make 18 Lunar months, \ 12 Calendar months, or > make 1 Common year. 365 Days, . • . j 365d. 5h. 48m. 48s. . make 365d. 6h. . , • • make 366 Days . • . make 1 Solar year. 1 Julian year. 1 Leap year. Tears multiplied by 365^, are days — days multiplied by 24, are hours — ^hours multiplied by 60, are minutes — minutes multiplied by 60, are seconds — seconds divided by 60, are minutes — minutes divided by 60, are hours — hours divided by 24, are days — days divided by 365J, are years. i f >'^J.\,h ' (ii. Y I .^?lA. .x.f...^ ■^mmm^-l ,• ' ;|i'?:ftt "ifii lu S. TIME. Exampleg. 1. In 1 year, how many 2. In 31557600 secondfl, seconds ? V •v. 1 year. 365J 365} days. 24 1460 730 6 is the} of 24. 8766 hours. 60 625960 minutes. As. 31657600 seconds. how many years ? 60) 31557600 sec. 60) 525960 min. 24) 8766 hn. 365}) 365} days. 4 4 1461) Ans. 1461 1461 (I 1 year. 3. In 13 years, how many days ? Ans. 4748} days. 4. Reduce 4748} days into years. Ans. 13 years. 5. In 28 years, how many hours? Ans. 245448 hours. 6. Reduce 245448 hours into years. Ans. 28 years. 7. In lOyrs. 26days. 12h., how many minutes? ' ^^ Ans. 5297760 min. 8. In5297760minutes, how many years? Ans. lOyrs. 26ds. 12fas. 9. How many days since the birth of Christ, this year being 1832? Ans. 669138 days. 10. How many years are there in 669138 days? Ans. 1832 years. NoTi.— In quettiont which are perfonatd br Multiplication and DiTision, the operatioa mar often be abridf ed» by nmpljr adding or lubtractinf a part of the given number^uiin, To reduce sterling into cunrtncy, multiply by 60, imI diride by 64r-»or add ^* ^? |t»i* j^uwrjt'n tnn bk ^m- u i3 f> ft •8? i *f:' u I 40 THE RULE OP THREE DIRECT. To reduce currency into Bterling* multipl/ by 54, and divide by 60 — or deduct •j'y. To reduce guineas into pounds, add ■^^, To reduce pounds into guineas, deduct ^\. To reduce ells English into yards, add ^, '-^-'i^*. To reduce yards into ells English, deduct j, &c. &e« !Examples* 1. In ^9 ster. how many pounds currency X I) i£9 sterling. f,% 'i h Ans. ^10 currency. 2. In ;f 10 currency, how many pounds sterling? y^^) £10 currency. 1 Ans. £9 sterling. 3. In i£l8 ster. how many pounds currency? Ans. .£20 curr. 4. Reduce <£20 curr. into sterl. Ans. £ 18 sterl. 5. How much currency must be paid for an English bill of ^100 ? Ans. iClll 29. ^^dL 6. How much sterl. is equal to £11 1 2^. 2|d. curr. ? ^^ Ans. £100 sterl. *7. How much must be paid in Quebec, to receive in London <£18Q? Ans. £200. 8. How much sterl. must be paid in London to receive in Quebec £200 ? Ans. jf 180. 9. In 20 guineas how many pounds? Ans. £21. 10. Reduce £21 into guineas. Ans. 20gs. 11. In 20 ells English, how many yards ? Ans. 25yard8. 1.3. ReduCQ 25 yards into ells English. Ans. ^OellSf ^^•i«t ,," VBB ILir&B or TBBJBII 9X1L1ICT. H-S!i«f '^ Q. What is taught by the Rule of Threat '***^""' A. The Rule of Three teaches, by three numbers given to find a fba^th, which shall have the same proportion to the third, as the second has to the first. ^^ n\ m*\mf n THK RULB or TRRIE DIRECT. 41 Q. WbMi if the proportiott Mud to be ^reet ? A* Direct proportion require! iho fourth term to he grMUwr than the itcimd, when the third ib greater than the firat ; or the fourth to be leu than the secondi when the third is lesa than the firat Role*— First state the question ; that is, place the numbers in such order, that the first and third be of one kind, and the second the same as the number required : then bring the first and third numbers into one name, and the second into the lowest term mentioned. Multiply the second and third numbers together, and divide the pro- duct by the first; the quotient will be the answer to the question in the same denomination you left the second number in* .■*»*rp »y,X ;. £xamples« J. 1. If one ib of sugar cost 4|d., what will 54ib cost t .H ^i .1^.1/. . If 1 -41 • 64 18 l4 -i^k • .;«5«» I b2Xr>*'|* ^3 v^f V farthmgs . K' Or 7,: - ^ * IS 432 54 • "*< • 4) 972 farthings. "J 12) 243 pence. .4' }'- ' ^'' ■ . i 2,0) 2,0 3 . Ans. £10 3 IT <* ' •4 S mi ■i' > u 1 • m ^ 2. If 4 yards of doth cost 3«., what will 24ydfl. cost? Ana. 18*. , 8. If 24 yds. of cloth cost 18»., what will 4yd8. cost? « Ana. 3#. 4» 't.4 42 THE RULE OF THREE DIRECT. I i! 4. If I buy 4 yds. of oloth for 3«., how many yardi will IBt, buvl Ana. 24yd8. 5. If 24 ydi. cost 18*., how many will I get for da, 1 Ans. 4yds. 6. If 1 yard cost 159. 6c2., what will 32yds. cost ? Ans. iS24 16«. ( 1 7. If 82yds. cost jf 24 16«., what is the value of 1 yrd. ? Ans. 15«. 6d. 8. What will aScwt. Sqrs. 141b of tobacco come to» at 15^(f.perib? Ans. £187 Z8,dd. 9. Bought 27|yd8. of muslin, at 6a. 9}d. per yrd., what is the amount of the whole ? Ans. £9 5«. Of^tL . J. 10. Bought 17cwt. Iqr. 141b of iron, at Sid per lb. what was t^ price of the whole ? Ans. x26 7«. 0^, 11. If coffee is sold for b\d, 'per ounce, what will be the price of 2cwt. 1 Ans. £82 2a, 8d, 12. How many yards of cloth may be bought for ;£21 lit. Hd„ when SJyds. cost £2 14«. 3d. ? Ans. 27yds. 3qrs. l-^jvl. 13. If lowt. of Cheshire cheese cost £1 149. Bd., what roust I give for S^ft,'i Ans. 1«. Id. 14. If a gentleman's income be £ 500 a year, and he spend 199. id. per day, what is his annual saving ? Ans. £ 147 39. 4d. 15. If 504 Fleir^jsh ells, 2qrs. cost iS283 179. 6d., what is the cost of 14yds. ? Ans. £ 10 109. 16. If 1 English ell, 2qrs. cost 49. 7(2., what will 39}yds. cost at the same rate ? Ans. £5 ds. 5^d, . 4. 17. If 27yds of Holland cost iS5 129. 6d., how many English ells can I buy for iS 100 ? Ans. 384 ells. 18. A draper bought 420yds. of broad cloth, at the rate of 149. 10|d. per ell English, what was the whole amount ? Ans. jS250 59. 19. What must be paid fbr 7 casks of prunes, each weighing 2cwt. Iqr. I4]b» at £2 199. 8d. per cwt. ? Ans. Jg49 1l9. Uj^d. 20. At £\9 199. ll|d. the ton, what will 19 tons» 19c wt. 3qrs. 27\fb come to»at that rate ? ,f. . Ans. ^99 19». 6|f f f^. It; u (^ THE RULE or THEEE INVRftSE. 43 ^ „_I ,ai. T ^. _ 4 (■ a. < k.^ tf .^M-WlK ^-«^^^i ' '*iu^ t I I * VBB BiVAS or TB&mi xanramsa. .♦^ Jl^ Q. TV bat 18 Inverse proportion ? A. Inverse proportion requires the fourth term to be Ui» than the second^ when the third is greater than the first ; or the fourth to be greater than the second, when the third is less than the first. Rule* — State the question, and reduce the terms as in the rule of three dirwt ; then multiply the first and second terms together, and divide their product by the third; the quotient witl be the answer* aa in the last rule. Examples* 1. If 8 men can do a piece of work in 12 days, in bow many days can 16 men do the same I m. d. m. If 8 : 12 : : 16 ;>;r»^ r^flt j — ■ i a'. m*'> < ' 16) 96 (6 day». Ans. 'mo W. 96 •J ■\'f i i 2. If 54 men can build a hou9G in 90 days, how many men can do the same in 50 days ? Ans. 97} men. 3. How many sovereigns, of 208, each, are equivalent to 240 pieces of 129. each ? Ans. 144. ' 4. How many yards of stuff three quarters wide, are equal in measure to 30yds. of f^ quarters wide ? Ans. 50yds.^ 5. If I lend a friend i&200 for 12 months, how long ought he to lend me £150 ? Ans. 16 months. 6. If for 24«. I have 12001b carried 36 miles, what weight can I have carried 24 miles for the same money ? Ans. 18001b. 7. If I have a right to keep 45 sheep on a common 20 days, how long may I keep 50 upon it ? Ans. 18 days.. -4 DOUBLE EVLE OF TBEEE. r 8. If 1000 foldMra have proviiions for 8 months, how maax mutt be eent Awa/» uai the prevkioM may last Smooths? Ans. 400. 9. A courier makes a journey^in 34 davs» by traveling 12 hours a day : how roan^ days will he be in gofaig the •ame joumey* traveling 16 hours a day ? Ans. 18 da vs. 10. How much will line a cloke* which is made of 4* yards of plush* 7 quarters wide, the stuff for the lining being but 3 quarters wide ? Ans. 9| yards.. TMM BOVBU mu&a or Q. What is the Double Rule of Three ? A. The Double Rule of Three has five terms given» three of supposition and two of demand, to find a aixth, in the same proportion with the terms of demand, as that of the terms of supposition. It is performed by two stat- ings of the single rule of three. Rule* — Put the terms of demand one under another in the third place ; the terms of supposition in the same order in the firtt place, except that which is of the same kind as the term required, which must be in the second place. Examine the statings separately, using the middle term in each, to know if the proportion is direct or tn- ver«e. When the stating is direct, mark ihe first term with an asterisk : when inverse, mark the third term*, then multiply the marked terms together for a divisor, and multiply oU the other terras for a dividend; divide, and the quoti«;:t will be the answes.. If J/ \ s'^?s,i& 'I Sf*'' DOUBLE RULE OP THREE. 45 Examples. 1. If 14 horses eat 56 bushels of oats in 16 days, how many bushels will serve 20 horses 24 days ? * Examine the sttUngs, thee: 1ft. If 14 honei eat 56 bushels, 20 horses, being more, will eat more ; the stating is, there* fore, Direct. " 2nd. If 16 days consume 56 bushels, 24 days, being more, will consume more; the slating is, therefore, Direct 14 16 84 14 66 20 1120 24 4480 2240 26880(1206. Ans. 224 J^m^ 448 448 2. If 8 men in 14 days can mow 112 acres of grass* how many men can mow 2000 acres in 10 days ? Ans. 200 men. 3. If iSlOO in 12 months gain £6 interest, how much will £75 gain in 9 months ? Ans. £3 7«. 6d, ,4. If £100 in 12 months gain £6 interest, what prin- cipal will gain £ 3 7«. 6d, in 9 months ? Ans. i)75. $. If JS 100 gain £ 6 interest in 12 months, in what time will £ 75 gain £ 3 7«. 6d» interest ? Ans. 9 months. 6. If a carrier charges £ 2 28. for the carriage of 3cwt., 150 miles ; how much ought he to charge for the carriage of7cwt. 3qrs. 141b., 50 miles? Ans. £1 16«. 9d. 7. If 40 acres of grass be mown by 8 men in 7 days, how many acres can be mown uy 24 men in 28 days ? Abb. 480. i 111 '* ' r *■ A •J 'i) K\ M 46 SIMPLE INTBRE8T. 8. If £2 will pay 8 men for 6 days* work, how much will pay 32 men for 24 dayi' work ? Ana. £9S St. 9. If I pay i6 14 10«. for the carriage of 60cwt. 20 miles, what weight can I have carried 30 miles for £6 8«. 9d, ? Ans. 15 cwt. 10. If 144 threepenny loaves serve 18 men for 6 days, how many fourpenny loaves will serve 21 men for 9 days ? Ans. 189. ■j^. ,j SZMV&a ZSVTB&BST. Q. What is Simple Interest ? A. Simple Interest is the premium allowed for the loan of money for a given time. The money lent, is called . . The Principal. The premium for the loan of £ 100 J j^ ^ ^^ for 1 year, is called . . j '^ The Principal and Interest together, \ ja ^,„umni is called ... ] * Rule* — To find the interest of any sum of money, for any given time ; multiply the principal by the rate per cent, and the product divided by 100 will give the interest for i year ; multiply the interest for 1 year by the number of years given, to which add the interest of tbe given months and days, if any, which may be found by tdcing the aliquot parts of a year's interest, and their sum will be the interest required. vif ^h m,:- 8IMPLB INTESMT. Examplef. 1. What if the interest of £384 2«. 6d. for 6 yn. 7 m. 15 dayi, at the rate of 5 per cent, per annum t £ t, d, 384 2 6 Principal. 6 1,00) iB 19,20 12 20 6 f. 4,12 d. 1,50 4 ' gr. 2,00 » iB19 4 6 ;.ui^ «• r,- • ■ - 96 74 interest for 5 jrean. 6 mo. I year ss 9 1 j3 ^ Interest for 6 montha. lmo.Iof6m.^ 1 12 Interest for 1 „ 15dys.| oflm.ss 16 Interest for 15 days. Ana. Jf 108 8} Intst. for yiB.5 7 15 o *;y. m 2. What ii tiie interest of ie375 for 1 year, at 6 per cent per annum 7 Ans. i618 1 St. 3. What is the interest of £945 lOs. for 1 year, at £4 per cent, per Annum ? Ans. £ 37 16». 4}f * * 3 f Numerators. . r Ans. yds. 69J 10) 15 (li yanb. ' ;;' 10 ' • i. Add 14), 16}, 20} yards together. Ans. 5I|yds. , 3. Add 18{, 12|, 19j, 20} yds. together. Atas. 71yds. 4. Add 142^, 6^, 20|t, 81|{ yds. togetfier. " Ans. 250{yds. AM>ITI0N OF FRACTIONS* 51 5. Add together 27^, 18^, ^\, 187 yds. together. Ans. 252f|yd8. 6* Add 17^, 87/r* 146|^ 50}|, and d|f Ans. 305}fyds. 1 /it* Rule 2* — ^When the fractions have not a common denominator, arrange the denominators in a line, and divide any two or more of them by any common divisor, placing the quotients and the undivided numbers below ; proceed with them in the same manner, and repeat the process till there remain not any two numbers common- surable; then multiply the undivided numbers by the quotients and divisors, and the last product will be the least common denominator — then divide this common denominator by each of the denominators of the given fraction, and multiply the quotient by the numerator, net- ting down the ^^luct opposite each fraction — add Uiese products toge.r > ind proceed as in the last examples. .-j-j -- Examples. 7. Add together 12^ yds. 14|, 15j, 18}, and 17tV 60 C. den. jrds. 12i (To (U>d tha laut Com. Donon.) 2) 2 . 3 . 6 . 4 . 10 ••^kfS 3) 1 . 3 . 3 . 2 . 5 2 5 2 141 . 18| . 17iV V 30 , , 45 6 — Au. 77^1 yds. 60) 11 l(myds. ' i^i hiv 10 3 30 2 60 ■h a-rii 3) H-\h C.p. 60 ^mi ji^. , i ft .d ■:t : ^ i II I t 1 i »ii< \^Q 52 SUBTRACTION OF 7RACTT0NS. 8. Add 40}, 27|, 34|, 48^, and 39) yaiils. Ans. 185^ yd^. 9. Add 150^ lb. 139|, 162|, and ITOjlh. together. Ans. 623,Vlb. 10. Add 16^, 1^ ^, 13j, 20j, 25^^, 30|, and ll^Ib. Ans. 136;{lb. 11. Add 124|, lOlf, 79}, and 17 together. Ans. ' 12. Add 132}5, 507J, 384|. and 18,^. Ans. ■J h sirBTiLjt.oTzoar or nuLoszoxni. Rule* — Prepare the fractions the same as for Addi- tion, when necessary ; then subtract the one numerator from the other, and set the remainder over the common denominator, for the difference of the fraction sought. Note. When the numerator of the fractional part in the niMr». htnd is greater than the other numerator, lubtract it from the com* mon denominator, and add the remainder to the other numerci tqr— < set down the fraction, and ^rry 1. —.,...-.. • ► <*,? « ■ »■ ■' * examples* 1. From 18} yards of cloth take I4g yards, yds. ""f K , From 18} . . . 4 Take 14| . . .6.^.4. f r Ans. 3| yds. } 2. What is the difference between 20} and 16) ? Ans. 4|. 3. What is the difference between 37f and 29| t Ans. 8). 4. From 26^ take 18). Ans. 8^. 6. If 59/t be taken from 102), what will remain ? Ans. 42). MULTIFUOATiON OF ffEAOflONS. 93 6. Lent £123^ and racttved £ a7)f wliat ia yet due ? r iCAns. 96^. ' 7. Borrowed jS87^, and paid 84) ; what do I yet owet Ana. jf2|f. uvibTzvuoATzoir or mAonom. " R^le* — Multiply all the numeratora together for a nume) vtor, and all the denominatora together for a deno- minator, which will give the product required. Note.— If there be a mixed number, multiply the whole number by the denominator of the fraction, adding in the numerator, and place the sum over the denominator. Examples. V f 1. Required the product of 3| and a* ..I 10) 28 (2f Ans. 20 ^i' ^2. 3. 4. 5. off. 6. 7. 8. Ana. J. Ans. /y. Ans. j"^. 9. Required tbo product of J and |, Required the product of i^ and |. Required the product of ^, |, and |f. ^^. Required the continued product of ft 3^, 5, and | Ana. 42. What is the \ of £20? Ans. £6 J3«. 4 105) 2366 (^22 lOf. Sd. Ans. 210 266 210 56 20 105) 1120 105 70 12 106) 840 840 (10*. (8d. ill 56 DECIMAL rBACTIONS. ^ i. IfiofayaideoitiBftWhatwiUiVdrtjrinleMtT Ana, 15ff. 3. Iff yrd. coat iS}, what will H T^- 908$ T JUs, lit. S. 6. If 48 men can build a wall in 24| days, how many men can do the same in 192 days ? Ans. 6^ men. 7. If } yrd. of Holland cost J6|, what will 12f ells cost at the same rate ? Ans. £7 6|a. . |. 8. If 3} yards of cloth« that is 1| yard wide, be suffi- cient to make a cloak, how much that is | of a yard wide, will make another of the same size t Ans. 4^ yds. 9< If 12| yards of cloth cost 15«. 9d., what will 48 J yards cost at the same rate ? Ans. ^3 9} Ans, 15 yds. Q. What is a Decimal Fraction ? A. A decimal fraction is that which has for its denomi- nator 1, with as many ciphers annexed as the numerator has places ; and it is usually expressed by setting down the numerator only, with a point before it, on the lefl blind. Thus, fy is .4« and f^^ is .24, and j^^^ is .074, &c. &c. ;VM5,)- vfed ADDITIOK OP DECIMALS. 57 ABBRZON 09 BBOZBKA&I. Role* — Set down the numbers in such order that the separating points may stand exactly under each other, then add up the columns, and place the point in the sum directly under the other points. liXamples. 1 Add together 14.25+121.372+107.3804-26. 14 . 25 121 . 372 107 . 380 .26 Ans. 243 . 262 2. \Vhat is the sum of 276. +39.213+72014.9+417. and 5032? Ans. 7779.113. 3. Add together 7530.+ 16.201+3 -0142+957.13. Ans. 8506.3452. 4. Add together 1.5+85.07+121.321+23.17. Ans. 181.081. HI i SUBTmAOTZOBT or BBOZaiEALS. ' Rule* — Place the numbers under each other as in addition ; then subtract and point off the decimals as in the last rule. Examples. 1. What is the difference between 91.73 and 3.138. 91 . 73 . 2 . 138 . 'titl Ans. 89 . 592 the diflbrenee. M wA ■"(, m 58 DIVISION OF DECIMALS. 2. Find the difference between 1.9185 and 2. 73. Am. 0.8115. 8. Subtract 4.90142 from 214.81. Ana. 209.90858. 4. Find the difference between 2714 and *916. Ans. 2713.084. wvxinvuojLVzoir or dboziba&b. '•■ Rule* — Place the factors under each other, and mul- , tiply them together. Then point off in the product just as many places of decimals as there are decimals in both factors. But if there be not so many figures in the pro- duct, supply the defect by prefixing cyphers. Examples. . 1. Multiply 24.5 by 1.6. 24 1 5 6 147 . 245 L -^ *l^-^.^^' An9f 39 . 20 the Product. ■ i. KS *.- ««? 2. Multiply 79 . 347 by 23 . 15. Ans. 1836 . 88305. 8. Multiply . 63478 by . 8204. Ans. . 520773512, 4. Multiply. 385746 by. 004(54. Ans. .00178986144. T Hulc* — Divide as in the rule of division, and point off in the quotient as many places for decimals, as the decimal pltces in the dividend exceed those in the divisor. REDUCTION OF DECIMALS. 59 Sxamples* 1. Divide 34.80 by 1.6. 1.5) 34.80 (23.2 Ans. 30 48 45 "lo J** 8. Divide 123.70536 by 54.25. 3. Divide 12 by .7854. 4. Divide 4195.6i3 by 100. Ans. 2.2803. Ans. 15.278. Ans. 41.9568. RUDVoTXoir or dbozhaui. TO REDUCE A FRACTION TO A DECIMAL. Rtlle* — Divide the numerator by the denominator annexing cyphers to the numerator as far as necessaryi so shall the quotient be the decimal required. fiXamples. 1 . Reduce | to a decimal. 4) 100 .25 Ans. .4. * '. > ' ,. XM A ,m 2. Reduce 1 to a decimal. ' 3. Reduce } to a decimal. 4 4. Reduce | to a decimal. .^ 6. Reduce A to a decimal. Ans. •6.*' Ans. !75'. Ans. •«5. Ans.^lS. 0, / M ^iC. 00 REDUCTION or DECIMALS. 6. Reduce jfj to a decimal. Ana. .031360. 7. Reduce 9«. to the decimal of a pound. Ana. '46. 8. Reduce 9d. to the decimal of a shilling. Ana. "76. 9. Reduce J to the decimal of a penny. Ana. *26. TO RBDCCE A DECIMAL TO ITS PROPER VALUE. Rule. — Multiply the given decimal by the number of parts of the next mferior denominationt cutting off the decimals from the product ; then multiply the remainder by the next inferior denomination ; thus proceedingt till you have taken in the least known parts of an integer. Examples. 1. What is the value of .8323 of a ;f 7 8323 20 9. 16.6460 12 ■r ft i ->•■ v.. m. ..f . /, .r( , «^- '''•'''*20 Ana. 16». 7}d. .*■ •ti'irj/i-i ! }008 •» 1 a. What is the value of *775 of a £ ? Ans. 15«. 6. znvo&wxoifa Q. What is Involution ? A. Involution is the multiplying any number by itself, and that product by the former multiplier ; and the pro- ducts which aiise are called powers. Table of the Squares and Cubes of the nine digits. Roots 1 1 2 3 4 5 6 • 7 8 9 Squares. . . 1 4 9 16 25 36 49 64 81 Cubes 1 8 27 64 • 125 216 343 512 729 • Examples. 1. What is the fifth power of 8 ? *i|X' 4: \ ■P ■n, M'^ " \- 8 Root or 1st power. 8 64 Square or 2nd power. 8 ■"^ 612 Cube or 3rd power. 8 4090 Biquadrate or 4th power. 8 32768 Sursolid or 5th power. 2. lYhatis the square of .085? 3. What is the cube of 25.4 ? 4. What is the biquadrate of 1 .2 ? Ans. .007225. j^. 16387.064. Ans. 2.0736. Q THE SQUARE EOOT. Bvosunoir. Q. What is Evolution? # > A. Evolution is the method of finding the first powers or iroots of any given numbers : and it is commonly called extraction of the square* cube, biquadrate, sursolid, roots, &c. i- f TBS SQVA&B ROOT. Q. What is the extraction of the Square Root ? A. To extract the square root, is to find out such a number as being multiplied by itself, the product will be equal to the given number. Rule* — Divide the given number into periods of two figures each, beginning at the right hand ; find the great- est square in the first period on the lefl hand, and set its root on the right hand of the given number, after the man- ner of a quotient figure in division ; then subtract the square thus found from the said period, and to the remain- der annex the two figures of the next following period, for a dividend — double the root above mentioned for a divisor, and find how often it is contained in the said dividend, exclusive of the right hand figure, and set that quotient figure both in the quotient and divisor. Multiply the whole augmented divisor by this last quotient figure, and sub- tract the product from the said dividend, bringing down to it the next period of the given numbers for a new dividend, and proceed as before. NoTB.— Whtv, thcfigaret of the whole number are ezhanited. If there be a remainder, peiiodi of cipben may be uied at pleaaure to continue the extraction { but tba figures produced in the quotient will be deeimala* iT^.^W ' V, M THE CUBE ROOT. Examples. 1. What is the square root of 119025? 119025 (345 the root. Ans. *'■' ,..,,. 9 64) 290 256 685) ■ 1 * k. 3425 3425 2. 3. 4. 5. 6. 7. 8. 9. 10. What What What What What What What What What is the square root of 106929 ? is the square root of 22071204 ? is the square root of 17.3056? is the square root of 000729 ? is the square root of 3 ? Ans. is the square root of 5 ? Ans. is the square root of 6 ? is the square root of 10 ? is the square root of 12 ? Ans. Ans. Ans. Ans. Ans. 327. Ans. 4698. Ans. 4.16. Ans. .027. 1.732050. 2.236068. 2.449489. 3.162277. 3.464101. TBB CVBB ROOT. Q. What is the extraction of the Cube Root ? A. To extract the cube root» is to find out a number, ivhich being multiplied into itself, and then into that pro- duct, producetb the given number. Rule* — Divide the given number into periods of three figures each, beginning at units' place; find the greatest cube in the first period, and subtract it there- from ; put the root in the quotient, and bring down the iiffures in the next period to the remainder, for a JRe- solvend ; multiply the square of the root found by 300, for a divisor, and annex to the root the number of times which that is contained in the Reaolvend ; add 30 times THE CUBE ROOT. the preceding figure, or figures, multiplied by the last, and the square of the last, to the divisor ; and multiply the sum by the last for a Subtrahend : subtract it from the Re9olvendi and repeat the process as far as necessary. '1^ )ds of id the there- vn the a JRe- 300, times times Examples. 1. What is the cube root of 99252847? .15^ 99252847 4'=64 (463 4^X800=4800) 35252 Resolvend. j-fy*.^ 720=4X30X6 36=6'* 4800 5556 6 33336 Subtrahend. 46''x300=634800) 1916847 Resolvend. , ' 4140=46x30x3 9=3« 634800 ^' -■*» 638949 3 1916847 Subtrahend. /^ f "S- U!i r i .1' ... 2. What is the cube root of 389017 ? Ans. 73. 3. What is the cube root of 673373097125 ? Ans. 8765. 4. What is the cube root of 12-977875 ? Ans. 2.35. 6* lip, I 'J >•■ PRACTICAL GEOMETRY* .fr-'tf' t r h PBJionoA& OBOBsaTrnv. Practical Geometry is a mechanical method of de» scribing mathematical figures by means of a ruler and compasses, or other instruments proper for the purpose. 1. A point is that which has no parts, or dimensions; as A. 2. A line is length without breadth ; and its bounds or extremes are points. 3. A right, or straight Une, is that which lies evenly between its extreme points; A. as A B. 4. A superficies is that which , has length and breadth only ;. and its bounds or extremes are tines ; as A BCD. .A .B I 5. A plane, or plane superficies, is that which is every where perfectly flat and even. 6. A body or solid, is that which has length, breadth, and thickness, and its bounds or extremes are superficies ; as ABCD. 7. A plane recHlinetU angle is the inclination or opening of two right lines» which meet in a point with- out cutting each other; as ABC. 8. One right line is said to be perpendicular to another, when the angles on each side of it are equal. Thus A B is perp. to C D. , % # PRACTICAL OEOHETRT« 67 9. A right angle is that which is formed by two right lines, that are perpendicular to each other; as ABC. B 10. An acute angle is that which is less thaii e right angle ; as A B C. 11. An obtuse angle is that which is greater than a right angle ; as ABC. 12. A circle is a plane figure, formed by the revolution of a right line about one of its extremities, which remain ^1 fixed ; a&j ^ IS, The centre of a circle is the point 0, about which it is described ; and th<3 circumference is the line or boundary A B C A, by which it is contained. "' 14. The radiu3 of a circle is a right line drawn from the centre to the cir- cumference ; as A. 'n 15. The diameter of a circle is a right line passing through the centre, and terminated on each side by the circumference ; as A B. ;i^ ^"^1 PRACTICAL OEOMBTRY. ,-y 16. An are of a circle is any part its periphery, or circumference ; as A D. ( 17. A chord is a right line which joins the / extremities of an arc ; as A B. \ * 18. All rthne figures, bounded by three right Hues, are called iricn^^!efi; nnd receive different denominations according to th^ nature ot their sides and angles* 'J •■i«' 19. An equilateral irianf^le is that which has all its sides •equal ; as A B C. 20. An isosceles triangle is that which haB only two of its sides equal , as A B C. 27, A rhombus is a quadrilateral, whose sides are all equal, but its angles are not right angles; as AB C D. 70 PRACTICAL GEOMETRY. 28. A parallelogram is a . quadrilateralt whose opposite sides are parallel; as A £ C D. 1 I I I 29. A rectangle is a parallelo- gram, whose angles are all right angles ; as A B C D. As 'Jfrr' 30. A rhomboid is a paral- A^ lelogram, whose angles are / not right angles ; as A B / C D. B^ / 31. All other four-sided figures, besides these, are called trapeziums. 32. Parallel right lines are such as are every whore at an equal dis- -^ tance from each other ; thus A B is q parallel to CD. 3 .D 33. An angle is usually de* noted by three letters, the one which stands at the angular point being always to be read in the middle; as A B C, a- CBD, DB£, &c. PRACTICAL GEOIIETEY. 71 ■ ■:rr < Problem 1* To divide a given line A B into two equal parta. »' , 7K v»J >f! -B 1. From the points A and B, as centres, with any dis- tance greater than half of A B, describe arcs cutting each other in n and m. 2. Through these points draw the line ncmi and the point c, where it cuts A B, will be the middle of the linot as required. * j^ Problem 2. To divide a given angle ABC into two equal parts* ■%}■ ^-s-#( :*Jt'.- 1. From the point B, with any radius, describe the arc A C ; and from the points A, C, with the same, or any other radius, describe arcs cutting each other in n. 2. Then through the point n, draw the line B n^ and it will bisect the angle A B C, as was required. irii^: 72 PRACTICAL GEOMETRY, Problem 3* From a given point C, in a given right line A B, to erect a perpendicular. Case !• — When the point is in or near the middle of the line. M ■>^- '.*■ ' V ^.- ^ 'ii'v';''j .i At a given point £, to make an angle equal to a given angle ABC. ti>liif! b ^H'i '.'j^' '1'» '>ff' ^^ 1. From the point B, with any radius, describe the arc n m, cutting the legs B A, B C m the points m n. »si I '«jl i J i' 74 PRACTICAL 0£OMET|tY« 2. Draw the line E D ; and from the point E, with tbf same radius as before, describe the arc r «. 3. Take thedifConce m r, on the fonnmr arc, and appljr it to the arc r «, from r to ». 4. Then through the points E, S, draw the line E Ft and the angle D £ F will b^ equal to the angle ABC, f Problem 6. - -^ as was required. To draw a line parallel to a given line A B. 1. From any tWd poinfe r, !», in the line A B with any radius describe the arcib am. 2. Then draw the line C D, to touch these arcs, with- out cutting them, and it will be parallel to A B, as was required. T^roblem 7. r \' ^ dt^« To divide a given line A B into any proposed number of equal parts. I #^ — ^^. / / ' "-^- ' ^ / 1. From one end A of the line draw A''in, making any angle with A B ; and from the other end B, draw B n, making an equal angle A iB n. . ■ f^fe- O. eJi|Wl »»M ^i-^ .iy^ r«^ « Ui. PRACTICAL OEOMlRTRY. 75 2. In each of the linet A m, B n, beginning at A and B, set off as many equal parU, of any convenient length, M A B is to be aivided mto. 3. Then join the points A, 5 ; 1 , 4 ; 2, 3, &€., and A B will be divided as was required. xt Problem 8. To find the centre of a given circle, or of one ab^tdj described. y v , f»» a ./ a • Jt'ifJ .ill Ifi/^ I. ims . .OI- Mi'twii*rr-A 1. Draw any dhord A B« and bisect it with the perpen- dicular C D. 2. Bisect C D, in like manner, with the chord E F, ind their intersection, o, will be the centre required ; obsei*v- ing that the bisection of the chords, and the raising of the perpendicular, may be performed as in problems Ist Aod drd. . ^ / A St- — 'A '^il 1?J u "311 ifcl ^C 75 fRAOTICAL OSOMETRT. ■ Problem 9rvriafc5«]f^f"'" To describe the dircumference 9f a circle through an/ three ^ven points A, B, C, provided the/ are not in a right line. ;^J^ vhftytff; oaa '\h 10 4<33 mil bfui .;T 1. From the middle point B, draw the lines or chords B A, B C ; and bisect them perpendicularly, with lines meeting each other in O. 2. Then from the point of intersection, O, with the distance O A, O B, or O C, describe the circle ABC, and it will be that required. Problem lO, '1 Upon a given right Une A B, to make an equilateral triangle, ' Wr * ■ '^ '*sf» vtds 10 KHiiaLM -Mil biifi ,8i ./ . \ i j'4u*i)ouai(f vHlrii>ifi *itu 1. From the poinU A and B, with a radiua equal to A B, describe aros cutting each other in C. 2. Draw thb lines A CJB C, and the agui« A C B will be the triangl6 .equired. PRACTICAL cmbMtttiit. 77 Problem 11. To make a triangle^ the three sides of which shall be respectively equal to thffis ^ven lines, A, B, C. vIX ; .-. /lv;j;ix;,_;^J) "j/; 1. Draw a line D E equal to one of the given lines C. 2. From the point D, with a radius equal to A, describe an arc ; and from the point E, with a radius equal to B, describe another arc, cutting the former in F. 3. Then draw the lines D F, E' F, and D F E, will be the triangle required. ^-k Problem 12. Upon a given line A li to describe a square. &di il^ bm f-Am^am '\ ,-,', .J^ . A. iMn 'A\ vMnl .1: \S. i '\r'> h From the point B, draw B C perpendicular, and equal to AB. 2. From the points A and C, with the radius A B, or C B, describe two arcs cutting each other in B, and draw the lines A D, C D, and the figure A B C D will be the square required. 7* ■I. %\ '•''M il 78 PRACTICAL OFOMETRY. Problem 13. lo A given triangle, A B C, to inscribe a circle. -.•* t^* r**" f:'«'*-y»* i u;--pj -f^:,-7>2-^- . -/ Bisect any two of the angles, as A and B, with the lines A and B o, 2. Then from the point of intersection o, let fall the perpendicular o n upon either of the sides, and it will be the radius of the circle required. ,jf ,, Arf,t(T . I In a given circle, to inscribe any regular polygon. n»?! ni)'/l^^ 'i a. 1. Draw the diameter A B, which divide into as many equal parts as the figure has sides. 2. From the points A, B^ as centres, and with the radius A B, describe arcs crossing each other in C. 3. From the point C, through the second division of the diameter, draw the line C D ; then, if the points A, D, be joined, the line A D will be the side of the polygon required. NoTi;.-— It ii to be obienred that in the construction here giTea, A D ii the side of a pentagon, or a figure of Jive sides. PRACTICAL GEOMETRY. 79 W Problem 15. On a giTen line A B, to make a regular pentagon. i 1. Produce A B towards n, and at the point B make the perpendicular B m equal to A B. 2. Bisect A B in r, and from r as a centre, with the radius r m describe the arc m, n, cut ing the produced line A B in n. 3. From the points A and B, with the radius A n, de- scribe arcs cutting each other in D, and from the points A D and B D, with the radius A B, describe arcs cutting each other in C and E. 4. Then if the line B C, C D, D E, and E A be drawn* ABODE will be the pentagon required. (itlU -^V Problem 16. On a given line A B to make a regular hexagon. 1. From the points A, B, as centres, with the radius A B, describe arcs cutting each other in ; and from the point 0, with the diiL^tance A or OB, describe the circle A B C D E F. 2. Then if the line A 6 be applied six times round tlie circumference, it will form the hexagon requhred. m Il 1 V 80 P&ACTICAL GEOMETRY. Problem 17. Ib a given ciicle to iascribe a regidar heptagon. >^ .'4-/f« il inioij: ^3 •^^^^:::i^ • 1, From any point A in the circumference, with the radius A of the circle; describe the arc B O C, cutting the circumference in B and C. ' 2. Draw the chord B C, cutting A in D, and B D, or C D, carried seven times round the circle from A, will form the heptagon required. '^^'''^^^^'^^•-r:, Problem 18. On a given line A B, to form a regular octagon. .HO^li< ^^iflf 1<\V2, 1. On the extremities of the given line A B ere^^i the indefinite perpendiculars A F and B E. 2. Produce A B both ways to m and. n, and bisect the angles m A F and n B £ with the lines A H and B G. tj» 'IX aiicj"*..') PRACTICAL QEOMETRT* 91 3. Make A H and B C each eoual to A B, and draw H 6, C D parallel to A F or B £, and make each of them equal to A B. 4. From G, D, as centres, i?:th a radiua eqa&i to A B, describe arcs crossing A F, B £« in F and E ; then if 6 F, F E and £ D be drawn, ABCDEFGUwiU be the octagon required. ... A. Problem 19. \ ^ ■ - k M To make an angle of any proposed number of degrees. So flJ^flMwS)?) '-'"• 1. Draw any line A B ; and having taken the first 60 degrees from Uie scale of chords, describe, with radius, the arc n m. 2. Take in like manner the chord of the proposed number of degrees from the same scale, and apply it from n to m. 3. Then, if the lino A C be drawn from the poiat A through m, the angle CAB will be that required. \f \ r. "«C.- ■"^^ . fi V K-t, 1-^ 82 PRACTICAL GEOMETRY. fT f Problem 20* HV a.Vi X n ' To find a risht line that shall be Qearly equal io.^aj given arc A D B of a circle. 4, n /, ^1 nty^ifyo ftiO f» J .';'U'>''"i > vydiu iljufi 06 ;;ii.^(.} oT 1. Divide the chord A B into four equal parts, nnd set ^ne of the parts A C, on the arc from B to D. 2. Draw C D, and the double of this line will be equal to the arc A D B nearly. Note. — If a right line be made equal to 3^ times the diameter of a circle, it will be equal to the circumference nearly. Upon a given line A B> to describe az^ oval, or a i%ur6 resembling an ellipse. ■ 7;5cvt>pij h» irmituiti -T .«» ot i\ moi\ i i:u M, -uU 1.;. ■. /'\ /'^ \-U'U ,if'MYV ,8 'Mlt -itt i*^m!'5fiJ 1. Divide A B into three equal parts A C, C D, D B ; and from the points C, D, with the radii C A, D B, d^ scribe the circles A G D E and C H E F. 2. Through the intersections m, n, and centres €, D, draw the lines m H, E n ; and from the points n, m, with .Ka. m m .H.^ PRACTICAL GEOlfETRY. 83 loh the radii n E, tn H, describe the arcs £ F» H 6, and A O U B F £ will be the oval required* . i^_. ,^ Or ihuii -^VtB't-^ The transverse and conjugate diameters being given. t*, IV U01S y , bisecting each other perpendicularly in the centre 0« 2. With the radius A O, and centre C, describe area cutting A B in F/; and these two points will be the foci of[ the ellipse. ft^-— < — — — j -" " " ' " " " ^'^' n /3. Take any number of points n,«, &c. in the tr«>.ns- ^Verse diameter A B, and with the radiii An, ftB, and centres F /, describe arcs intersecting each other in .;, ,4^ Through the points S, S, &c. draw the curve A^ C S 1! Dt and it will be the circumference required ; or, having found the foci, F/, as before, take a thread, of the length of the transverse diameter, and put it round two pins fixed at the points F/,* then stretch the thread F S/ to its greatest extent, and it will reach to the point S in the curve; and by moving a pencil round within tha .thread, keeping it always stretched, it will trace out the curve required. m ?*i->d.:ft vj:.: i£Rt nt iRVB %nun w* ' I •''^ i I ml ^:i: 84 IIKNSORATIOlf or 8UPSRnCIE8. i U xursmuinoif or ■rmmriozBS. Q. VHiat is Mensuration of Superfkies ? A. Mensuration of superficies teaches how to find the area or superficial content of any figure, without anj regard to its thickness. ^^ . Rule* — ^To multiply the side by itself gives the area. .1 .* Examples* ' 1. WhatistheareaofthesquareABCDiWhoMfide 18 8 feet? .«! ^ > No».— Iftachofthe . tides of the square A B 3 feet C D be divided into 8 3 P*i^t*t '^^^ ^* opposite " ■ , points be connected, it A«. o ' ■«* ...y. ''4v^' t . ' 9.5 6875 12375 10) 130.625 acres 13.0625 ^ 4 roods 0.2500 40 poles 10.0000 n Ans. 13 ac. Oro. 10 po, 2. Required the area of a rectangular board, whose length is 12 feet, and breadth 2 feet. Ans. 24 feet. 3. Required the area of a rectangular board, whose length is 12| feet, and the breadth 10 inches. Ans. 10/^ sq. feet. 4. Required the superficial content of a rectangular field, whose length is 12.25 6hains, and bieadth 8.5 chains. Ans. 10 ac. 1 ro. 26 po. 5. How many square yards of painting in a partition, whose length is 20 feet, and height 8 feet ? Ans. 17 yds. 7 feet. 6. What is the area of a rectangle, whose length is 14 feet, 6 inches, and breadth 4 feet, 9 inches ? Ans. 68 .ft. 126 sq. in. h Wi ^^% ^1 ■11 n 86 MENSURATION OP SUPERFICIES* or iL ABOMBVf. Rllle* — Multiply the side by the pcrpcndic>a]ar breadth) and the product will be the area. J. £xample8« * !^ * sfn; 1. Required the area of the rhombus A B C D, whose side A B is 12 ft. 6 in. and its perpendicular breadth D £ 9ft. 3in. ft. in. i 12 6=A B ' 9 3=D£ 112 6 3 1 6 •W Ans. Feet 115 7 6 p. 2. What is the area of a rhombus, whose side is 14 feet, and perpendicular breadth 5 feet ? Ans. 70 feet. 3. Required the area of a rhombus, the length of whose side is 12 ft. 9 in. and its height 10 ft. 6 in. ;: Ans. 133 ft. 10 in. 6 p. 4. Required the content of a rhombus, the side being 4 ft. 10 in. and the perpendicular 18 inches. Ans. 7 ft. 3 in. 5. Find the content of a piece of land in the form of a rhombus, its length being 6.20 chains, the perpendicular 5.45 ch. Ans. 3 ac. 1 ro. 20 po. .i^-^:. -M ''■K' *^%fe,^. ¥-Al le MENSURATION OF SUPERFICIES. 87 ;if or BHOMBOZO. R\ille.— Multiply the length by the perpendicular heigh', gives the area. Examples. 1. What is the area of the rhoml • length A B is 10.52 chains, and its p< D £ 7.63 chains ? t ^, whose jlt height 10.52=A B 7.63 3156 6312 7364 10);80.2676 acres 8.02676 4 roods 0.10704 40 Ans. 8a. Or. 4 p. poles 4.28160 2. Required the area of a rhomboid, whoso length is 10.51 chains, and breadth 4.28 chains. Ans 4 ac. 1 ro. 39 po. 3. What is the area of a rhomboid, whose length is 7 ft. 9 in. and height 3 ft. 6 in. ? Ans. 27 ft. 18 sq.- in. 4. How many square yards in a rhomboid, whose length is 37 ft. and height 5 ft. 3 in. ? Ans. 21/j sq. yds. ^. >. o \>.^. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I ^1^ 1^ us 140 : 1— III— llll^ < 6" ». Hiotographic Sdeaces Corporalion \ ,V \\ 23 VVIST JMAIN STRUT WIBSTEt,N.Y. 145S0 (716) t72-4S03 ■45 ^ 88 MENSURATION OF SUPERFICIES. or A Rule 1* — Multiply the boUie by the perpendicular height, and take half the product for the area. ^ Examples* 1. Required the area of the triangle ABC, whose base A B is 10 ft. 9 in. and its perpendicular height D C 7 (I ^ ft. in. S ' 10 9— 10.76=AB 7 3= 7.25=C D - ni. Vj: ?:^^t p»*iv)'£f/ 1250 4375 625 Ans. 1075.00 2. Required the area ofa hexagon, one of whose equal sides is 14.6 feet» and the perpendicular from the centre 12.64 feet? Ans. 553.632 sqr. feet. 3. Required the area of a heptagon, one of whose equal sides is 19.38» and the perpendicular from the- centre 20. Ans. 1356.6. 4. Required the areu of an octagon^^ one of whose equal sides is 9.941, and the perpendicular from the centre 12. Ans. 477.168. 5. Required the erea of a regular octagon, each of whose equal sides is 16. Ans. 1236.0773. 6. Required the area of a regular decagon, each of its equal sides being 20}. Ans. 3233.491125. OF za&BavziAR BzaaT-ZiZNSD rzouRxis. Rule* — Divide the figure into triangles and trape- iaums» and fmd the areas of each of them separately, then MENSURATION OF SUPERFICIES. 93 add these areas together, and their sum will give the area of the whole figure. « . .1.. Examples. 1. Required the area of the irregular right-lined figure A B G D £ F, the dimensions of which are as follows : F B=20.75, F C=27.48, E C-18.5, B n= 14.25, £ jn=:9.35, D r=12.8, and A «=8.6. -^ <» •^.s.li ■.1 'YU ...^^' / . -i •r A B F. »f D E C. 20.75X8.6 =178.450 -7-2=89.225 .r« 18.5 X 12.8 =236.80 ^^2=118.40 am 14.25-^9.85=23.60X27.48=648.5280-1-2=324.264 .re. of F B C E. _- ^— ^ Abs. 531.889 trM «ri B C D £ F. 2. Required the area of an irregular hexagon, like that in the last example, supposing the dimensions of the dif- ferent lines to be the halves of those before given. OF OIROZJB. "Rule 1* — Multiply the diameter by 3. 1416 — gives . the circumference. 2. Divide the circumference by 3.1416 — gives the diameter. 3. Square the diameter, and multiply it by .7854 — gives the area. 4. Square the circumference, and multiply it by .07958 — gives the area. 5. Divide the area by .7854, and extract the square root— "gives the diameter. 6. Divide the area by .07958, and extract the square root— gives the circumference. m ii 94 MENSURATION OF SUPERFICIES. Examples* ... ^ :> 1. What is the circumference and area of a circle* whose diameter is 3 feet ? 3.1416 3 f«et dlamcUr. Ant. 9.4248 fMt einumf. .7854 Au. 7.0686tq«ft«arM. 2. If the diameter be 26, what is the circumference ? Ans. 81.6816. 3. If the circumference be 75» what is the diameter ? * Ans. 23.873. 4 What is the circumference, when the diameter is 7 ? - Ans. 21.9912. 5. What is the diameter of a circle, whose circumfer- ence is 50? Ans. 15.9156.. 6. What is the area of a circle, whose diameter is 5} feet? Ans. 23.758350 feet. ' 7. How many square yards are there in a circle, whose circumference is 10| yds.? Ans. 9.19646375. 8. How many square yards are there in a circle, whose radius is 15^ feet ? Ans. 81.1798. 9. How many square feet are there in a circle, whose circumference is 20y*7 yards ? Ans. 289.36. ^ 10. It is required to find the radius of a circle, whose area is an acre. Ans. 39^ yrds. Rule 7. — To find the length ofany arc of a circle : Multiply the chord of half the arc by 8, from that product subtract the chord of the whole arc, and one-third of the remainder will be the lengths of the arc nearly ; or multi- ply the decimal .01745 by the degrees in the given arc, and that product by the radius of the circle, for the length of the arc. MENSURATION OF SUPERFICIES. dd Examples. ' 1. The chord A B of the whole are A B C is 48.74, and the chord A C of half the arc 30.25, what is the length of the arc ? ■-'{ 242.00 48.74 3) 193.26 jV-> AnS. 64.42 length of the tK A B C. 2. Required the length of the arc A B, which con- tains 60 degrees, and the radius B O of the circle being 7 feet. ,i ■ ^''^, 1.04700 7 Ans. 7.32900 fMt tba m a b. -•..•»♦■/> '' 3. The chord of the whole arc is 30, and the cL.nl of half the arc is 17, what is the length of the arc ? Ans. 35^. 4. The chord of the whole arc is 50|, and the chord of half the arc is 30f, what is the length of the arc ? Ans. 64.6. 5. What is the length of an arc of 30 degrees, the radius of its circle being 9 feet ? Ans. 4.7115. 6. What is the length of an arc of 12^ degrees, the radius being 10 feet? Ans. 2.1231. Rule 8* — To find the area of a sector of a circle: Multiply the radius of ^e circle, by half the length of the A /■/ H 96 MENSURATION OF SUPERFICIES. arc of the sector, and the product will be the area : or, as 260 degrees is to the number of degrees in the arc of the sector, so is the area of the circle to the area of the sector. Examples. 1. What is the area of the sector O B C A 0, the radius A being 10 feet, and the arc A CB, 18 feet? 1st Example. 10=AO 9=iAGB > SOfMtuMorOBG AO. 2. Required the area of the sector, the arc A G B being 30 degrees, and the diameter 3 feet. "'r' 2d Example. .7854X 9=7.0686=»1!;^:.*« Ana. no : 80 1 : 70686 : .SBtOS WM .f the MCtort hi 8. What is the area of a sector, whose arc contains 18 degrees, Uie diameter being 3 feet ? Ans. .35343. 4. What is the area of a sector, whose radius is 10, and arc 20 ? Ans. 100. 5. Required the area of a sector, whose radius is 25, and its axz containing 147° 29'. Ans. 804.3986. Rule 9. — To find the area of a segment of a circle : Find the area of the sector, having the same arc as the segment ; then find the area of the triangle formed by the chord of the segment and the two radii of the sector, and the sum or difference of these areas, according as the seg- ment is greater or less than a semicircle, will be the area of the segment required. MENSURATION OF SUPCRIiCIES. 97 Examples. 1. The chord A B is 24, and the height or versed sine C D of half the arc A C B is 5 ; what is the area of the segment A B C A ? ^■* , First find the requintes thus : y/ A D»+D C« :s:A C 13 the chord of half the arc. A C>^5=C £ 33.8 the diameter of the circle C £ 4-2=C O 16.9 the radius of the circle. AC X8— AB-r3a:A C B 26.666 the length of the arc. CO-CD=*DO 11.9 the perp. of the A A OB. D 0=11.9 ^, , A B= 12 i L* 142. Sana of ^ CAB. 13.333halfthearc ACB 16.9 the radius. / 119997 / 79998 13333 From 225.3277 «if,;'» B C A 0. Take 142.8 ««ofu.. ^, A B. Hem. 82.6277 •:;^'^ A B C A. Mgnmil r«(.t"3»wi 2. Required the area of a segment, its chord being 12, and the radius of the circle 10. Ans. 16.3504. 3. What is the area of a segment, whose versed sine m 18, and the diameter of the circle 50 ? Ans. 636.375. 4. Required the area of a segment, whose chord is 16, and the diameter 20. Ans. 44.728. 9 A 06 MENSURATION OF 6UPERPICIC8. 6. What is tho area of a leffment of a circle, wboM arc is 60", and the diameter of the circle 10 feet ? Ana. 2.2647. 6. Required the orea of the segment of a circle, whose ▼ersed sine is 5, and the diameter of the circle 20 feet. Ans. 61.4184. r or AN BXAZPSIf OS OVAXi. Rule* — Multiply the transverse diameter by the con- gregate, and this product again by .7854, and the result will be the area. t Examples* 1. Required the area of an ellipsis, whose tranverse diameter A B is 24, and the congregate C D, 18. A B 24 SB tosMvtrM diMMltr. G D 18^^ coajogtu tltamtttr. 192 24 482 .7864 1.728 2160 8456 3024 Ans. 389.2928 ss >morik«tiiipri«. S. Required the area of an ellipse, whose transrerse and conjugate diameters are 70 and 50. Ans. 2748.9. 3. Find the erea of an oyalt whose two axes are 24 and 18. Ans. 889.2928. •ft m.N8UBATIOIf or aOLIIMU Qi Whtt if Meaauration of Mlidi ? A. Mensuration of loUdi teaches how to find the whole eapecity or content of any solid, considered under the triple dunensions of length, breadth, and thicknesa. Deflnitioiu* Q. What is a cube ? A. A cube is a solid contained by six equal square sides, or faces, as A D. Q. What is a parallelo- / piped? "^i^ A. A parallelopiped is a solid contained by six rectan- gular plane faces, every oppo- site two of which are equal and parallel ; aa A D. n Q. What is a prism? > A. A prism is a solid, whose ends are two equal, parallel, and similar plane figures, and its sides parallelograms; as A B C D £ A. 100 MENSURATION OF SOLIDS. Q. What 18 a oylinder ? A. A cylinder is a solid, whose surface is circular, and its ends two circular planes as AC. i,» Q. What is a pyramid ? A. A pyramid is a solid, whose sides are all triangles, meeting in a point as the vertex, and the base any plane figure ; as A B C D £ A. Q. What is a sphere ? A. A sphere or globe is a solid described by the revolution of a semicircle about its diameter, which remains fixed, as A B C D. OF A OUBS, PBZSSS OH OlTXiXNDBR. ; ? Rule !• — To find the solid content of a cube, prism, or cylinder : find the area of the base, or end, and multiply it by the height or length of the figure, gives the solid content. Rule 2* — To find the superficial content of a cube, prism, or cylinder : Multiply the perimeter or circum- ference of the base or end of the figure, by the height or length of the cube, prism, or cylinder, to which product add the areas of the two ends, if required, gives the whole surface of the figure. MENSURATION OF SOLIDS. 101 £xamples« 1. What is the solid and superficial content of a paral* lelopiped A B C G H £, whose length A B is 8 feet, its breadth A £ 4^ leet, and its depth A D 6} feet. H a 4e kreadlh ••^ A E. 6.75 'i'^ 225 315 270 9(«ic« Ihc brtadih. 1 q e twice tiM 22.5 iMriawtof. 8 30.375 area of the end. 8 length A B. B 180.0 "^/•» 60.75 Ans. 240.75 !i".S:'' Am. 243.000 feet solid content. •conual. 3. What id the superficial content of a cube, the length of each side being 20 feet ? Ans. 2400 feet. 3. What is the superficial content of a triangular prism, whose length is 20 feeif and each side of its end or base 18 inches? ' Ans. 91.948 feet. 4. Find the convex surface of a round prism, or cylin- der, whose length is 20 feet, and the diameter of its base 2 feet. Ans. 125.664. 5. What is the solid content of a centre, whose side is 24 inches? Ans. 13824. 6. How many cubic feet are in a block of marble, its length being 3 ft. 2 in., breadth 2 ft. 8 in., and thickness 2ft. 6 in.? Ans. 21^ feet. 7. Required the solidity of a triangular prism, whose length is 10 feet, and the Uiree sides of its triangular end or base at 3, 4, 5 feet. Ans. 60. 8. Required the content of a round pillar or cylindw, whose length is 20 feet, and circumference 5 ft. 6 in. Ans. 48.1459. OF PTAAmD OR ooira. Rule 1 • — To find the solid content of a pyramid or cone : Multiply the area of the base by one-third of the perpendicular height, gives the solid content. 9* ri'i 102 MENSURATION OF SOLIDS. Rule 2* — To find the superficial content of a pyra* mid or cone : Multiply the perimeter of the base by half the slant height, to which add the area of the ba&e, if re- quisite, gives the superficial content. • '^ Examples. 1. What is the solid and superficial content of the cone A B C, the diameter of whose base A B is 6 feet, and the slant height A C or B C is 10 feet ? M'^* C V.i.''. '^.'i:. ;"!',i=^. v5< ■;;. .1 in-i '■' _, f\;j}- AC«=10xlO=100 ^ABs: ax 3= 9 '■.» ' ri .i''^'. ^'i V* 91 ^i pevp. o€s= 9.539a WA. .7854 AB*=:6X6= 36 47124 23562 8,1416 6 18.8496 circumfer. 5 i slant ht. 94.2480 arMofthabedy. Si8.2744 arMofthabut. Ans^ 19^^5224 feetNp>eMta«t 28.2744 area of the base, multiplied by 3.1797 | of the perp. C. ^ Ans. 89.9041 feet solid coat. «r- MENSURATION OF SOLIDS. 103 2. What is : erficial content of a triangular pyramid, the slant heigbi .:«eing 20 feet, and each side of the base 3 feet? Ans. 90 feet. 8. Required the surface of a cone, the slant height being 50 feet, and the diameter of the base 8| feet. ^ Ans. 667.69. 4. Requi.ed the solidity of a square pyramid, each side of its base being 30, and its perpendicular height 25. Ans. 7500. 5. Find the solid content of a triangular pyramid, whose perpendicular height is 30, and each side of the base 3. Ans. 38.97117. 6. What is the content of a pentagonal pyramid, its height being 12 feet, and each side of its base U feet ? Ans. 27.5276. 7. Required the content of a cone, its height being 10| feet, and the circumference of its base 9 feet. Ans. 22.56093. 1 or THB FBirSTUM OF A OR ooira. Rule 1* — To the areas of the two ends of the frus- tum add the square root of their product, and this sum being multiplied hj a tl^rd of the height, will give the iolidity. IB Rule 2* — Multiply the sum of the perimeterr or circumference of the two ends, by the slant height of the frustum, and \M the product will give the superficial eoDtent. r^ -- •-, — J ■• ,»^:tftf« . ii .'Vi « .:tUii '<■• -'^A^OF^^' 104 MENSURATION ' OF SOLIDS. Examples. 1. Required the solid and superficial content of the frustum of the cone £ A B D, the diameter of whose greater end A B is 5 feet, that of the less end £ D 3 feet, and the perpendicular height, S S, 9 feet. Note 1. — When the ■ItDt keigklU notgiven, it may be found by ex- tracting the aquare root of theium of the iquarea of the perpendicular height and diySbreBce of the radii ; .7854 5x5= 25 39270 15708 areaof AB==:19.6350 Tktu: 3x3= .7854 9 A8=ai-ES5=U=«l 9»=9X9=81 areaof£D= 7.0686 19.6350 7.0686 V» 82 9.0553 aleiit bt. A 8=5 X 3.1416=!15.7080 ED=3X 3.1416= ^'*^^ 25.1328 w'"«rfe«'*. 9.0553 alant bt. ^« 138.79196100 11 'YOI Muarerootef l.#01 protluct. 19.6350 arMofA B. 7.0686 "i^ of ED. Aim. 226.58504 aup con. 38.4846 \ of perp. 3 Ans. 115.4538 MUdeontMit. ■■* ;t ■0. ,5 H Note 2.— To the superficial content add the areai of both endi, if the whole surface be required. 2. What is the solidity of the frustum of a cone, the diameter of the greater end heing 4 feet, and of the less end 2 feet, and the altitude 9 feet? Ans. 65.9736. MENSURATION OF SOLIDS. 105 3. What in the solidity of the fruatoin of a sqare pyni- at of dide of the end inches the less end 15 inches^ and the height 60 inches ? Ans. 9.479 cu. ft. 4. What is the solidity of the frustum of an hexagonrvl pyramid, the side of whose greater end is 3 feet, that of the less end 2 feet, and the length 12 feet 7 Ans. 197.463776 cu. ft. 5. What is the superficial content of the frustum of a square pyramid, whose slant height is 10 feet, one side of the greater end being 3 ft. 4 in. and of the less end 2 ft. 2 in.? Ans. 110 ft. 6. To find the convex surface of the frustum of a cone, the slant height of the frustum being 12^ feet, and the circumference of the two ends 6 and 8.4 feet. Ans. 90 feet. 4 OF JL 8PBSBB OR OZiOBS. Rule 1 • — To find the solid content of a sphere : Cube the diameter, and multiply it by .5236, and the pro- duct will be the solidity. '' Rule 2. — To find the superficial content of a sphere : Square the diameter, and multiply it by 3.1416, or multiply the diameter by the circumference — either of these methods will give the superficial content. ^Examples* 1. What is the solid and superficial content of the sphere A D E B, whose diameter A £ is 17 inches ? « A^ 17'=289 17^=4913 .^^^^^^ 3.1416 .5236 29478-*^! 14739 9826 24565 Ans. 2572.4468 solid inches. , 1734 B 289 1156 289 867 Ans. 907.9224 ;:„»;: if i' I KM MENSURATION OF 80LID8. 2. What if the loUdity of a tphertf whose diameter is l^feet? Ans. 1.2411. 3. What is the solidity of the earth, supposing it to b« perfectly spherical, its diameter being 7957} miles ? Ans. 263858149120 cu.m. • 4. What is the convex superficies of a sphere, whose diameter is 1| feet, and the circumference 4.1888 feet ? r Ans. 5.58506 sq. feet. 5. If the diameter, or axis of the earth, be 7957 j miles ; what is the whole surface, supposing it to be a perfect sphere? Ans. 198944286.35235 sq. miles. is > or TBB SfiOMSKT OF A svBsma. Rule 1 • — To three times the square of the radius of its base add the square of its height ; and this sum mul- tiplied by the height, and the product again by .5236, will give the solidity. Rule !2«— -Multiply the circumference of the whole sphere by the height of the segment, and the product will be the sup trficial content. '■': i.yi.. i i ^. ^ . V-J . '*■ I': ' '}■/'■■' ■ ^j»-.. -* ~.n4.^^^ aV." A*^-^ TETRAEDRON. " " i*'t^..k-'\. .' HEXAEDRON d- J rS- J . ■ i 4rif OCTAEDROM. na.fc :, n-^ -* 1 .t 1 .';::, it- >.|i "fo DODECAEDROrV .-..*. '^^ ICOSABDROIT. 'H MENSURATION OF SOLIDS. 100 Table or THS SURrACEB AND SOLIDITIES O^ tAK RCQULAR BODIES, WHEN TUE LINEAR EDGE IS 1. No. of SIDES. 4 6 8 12 20 NAMES. 1 SURFACES. SOLIDITIES. Tetraedron . . Hdxasdroa Octaedron Dodecaedron . Icosaedron . . . 1.73205 6.0000 3.46410 20.64578 8.66025 0.11785 1.0000 0.47140 7.66312 2.18169 Rule 1 • — To find the superficies : Multiply the square of the linear edge by the tabular area, opposite its name, and the product will be the superficial content. K- Rule 2« — To find the solidity : Multiply the cube of the linear edge, b)' the tabular solidity, opposite its name, and the product will be the solid content. .•lb ;i^ lit liXampleso 1. What is the superficial and solid content of a Tetraedron, whose linear edge is 4 ? Tabular area 1.73205 4«= 16 Tab* solidity 0.11786 1039230 173205 Ana. 17.71280,-Pi,. m 64 47140 70710 7.54240 -"1 •onUat. / m 110 artificers' work. 2. Required the solidity of a tetraedron, whose linear edge is 6. Ana. 25.452. 3. What is the superficial content of an octaedron, whose linear side is 4 ? Ans. 65.4256. 4. The linear side of a dodecaedron being 3, what is the solidity? Ans. 206.90424. 5. What is the solid content of ao icosaedron, whose sMeisS? - Ans. 58.90563 I Artificers estimate, or compute the value of their works by different measures, viz. : Olazingt Maaoni? flat work, and sotne parts of joiners' work, are computed at so much per square foot, Painter8\ Pl€uterers\ Paver8\ ana some descriptions of joiners' work, are estimated by the square yard, Roofst Floorst PartitionSf &c., by the square of 100 feet. Bricklayers' work, by the square rod, containing 272 J feet. ! NoTB 1.— The roof of a house it said to be of a true pUchf when the rafters are 1 of the breadth of the building. In this case, there- fore, the breadth must be accounted equal to the breadth and half breadth of the building. NoTK 2.— Bricklayers compute their work at the rate of a brick and a half thick ; therefore, if the thickness of a wall be more or lest, it must be reduced to the standard thickness by roultiplyiug the area of the wall by the number of half bricks in the thickness, and dividing the product by 3. i:t . .^ lExamples. \ 1. A certain house'jias 3 tiers of windows, 3 in a tier, the height of the first tier being 7 feet 10 inches, the second 6 feet 8 inches, and the third 5 feet 4 inchet ; artificers' work. in and the breadth of each window is 3 feet 11 inches What will the glazing cost at 14 d. per s .- % ^.i ,!♦'' '■•'■ ■' -■" • •h ■I-' '■■.- ■r.)^<- •i , 'y^ S^ >*i ;^' ^ '^7/^' 1* - .■ iff f . A"' ^. * , . >■ '' ' '' '■ I'li-i' ! U '1 its per :he8 5ing per ong iach I BOOK IkKKPING, BY SINGI. r KMHY. Question. What is Book knrping ? Jlnswer. Book keeping is the tirt of recordm g pocuniary or commci'cial traiisactiuns in a regular and H^r.euiutic manner. Q What are the names of the books u»i. , to all sums received ; and credited, on the right hand siil •, by all sums paid. The excess of the Dr. side above the Cr. shows the balance or amount of cash in chest. Q. What is the Ledger? A, In the Ledger are collected the dispersed accounts of each person from the Day-Book and Cash-Book, and entered in a concise manner in one folio ; the sums in which he is Dr. being arranged on the left hand, and those in which he is Cr. on the right hand page of the folio ; the balance of each is ascertained by taking the difference between the Dr. and Cr. sides. Q. What is the Bill-Book I Ji. In the Bill-Book are copied the particulars of ail Bills *sj ilxchange, whether Receivable or Payable. 10* m ji^ DAY BOOK Inventory. Jmn'y i. j have in ready money, Biilff receivable, No. t, on S. Johnston, due 291 h iiist. Tea, 3 chests, wt. 2cwt. 3qrs. lOIb. at 68. 2d. per lb. Raw Sugar, 2hlids. wt. 27ct. 3qr. 181b. at 3£. 14s. 8d. per cwt. James Taylor owes me on bond, dated Au- gust I4j 1828, with interest, at £5 per Ct. per ann. tf I owe as folloi¥s; John Herdson, a balance of accounts, Bernard Mason, for purchase of my house, by auction, to be paid Isl Feb. next, £800 Duty on do. at £6 per ct. 40 Bills payable, viz. No. 1, Wm. Homes' bill to H. Williams or order, accepted by me, due I9th inst Allen, Wild, St Co. Leeds, Cr. By 3 pieces superfine blue cloth, each 36 yds at 25s. 6d. per yard, „ 2 piecr t narrow brown, 84 yds. at 43. 9d. „ Wrappers, £. 1500 24 9S 104 70 1796 8. » 3 1 4 n 8 d. » 8 >» 8 Bernard Masun, lo 2 s». raw sugar at 9i per lb. To 3i lbs. green lea, at 8s. 6d. per lb. „ 3| yds. blue cloth, at 2ds. Dr, 37 840 45 922 >i 10 15 If 9 10 »» 10 137 19 157 14 19 5 18 »» 6 1 1 7 5 6 7114 2 "9i 8. d. » *f 3 8 1 »» 4 >t >« » 9 10 If »» 15 10 t U » 19 »» 6 6 18 6 2 7 7i o 9i DiAY ROOK. 115 J«n> ». 10 »» 14 » 19 u » 23 23 Samiul Fletcher, Cr. ^y Keni hops, let. Ick- 5lbs. at £5 7t.0d. „ Worcester, do. 12 at & 11 6 Six innnths' credit, or £ 6 per Ct. diicoont for present payment. Simmonds & Co. Liverpool, Qy yellow soap. 2cwt. at 76c. „ 12doz. candies, at Ss. 6d. 4 doz. mould do. at lis. 3d. Cc. i» William Tomlingon, To narrow cloth, 7 yds. at 5s. 6d. „ callicoi 15 yds. at Os. 8^ per yd. Dr Hazard and Jones, To 1^ St. yellow soap, at 9d. per lb. ^ St. mottled do. at 9^d. 9 lb. candies, at 9d. Dr >» »» 3 lb. moulds, at Is. 0||d. Jameit Sanderson, By goods, as per invoice, Cr. Hazard and Jones, To 17.^ lbs. loaf sugar, at Is. Id. „ 12 ll)s. raw, do. at lOd. „ li lbs. Congou tea, at 78. 6d. „ i lb. Hyson, at 12s. Dr. £. j •■ 6 IS 8 15 d. 6 3 5 9 7 12 6 2 14 19 1119 10 9 15 0! 5 Oi 6 Oi 3 1 II n 31 .fohn Hcrdson, 'lo Hops, 10 lbs. ,, i| ream cap paper. Br. at Is. Id. p. at 7d. per quire, O'lO 6 7i 9 9 6 Hi 10 10 ic| 8 * I k 116 DAY BOOK. Feb. 1 10 10 12 William Tomlinson, To 2 8k yellow soap, at 9d. per lb. „ 6 lbs. mould candles, at ]«. Id. „ Id lb(. lump sugar, at la. O^d. Dr Tames Taylor, Dr To half a year's interest on £70, at 6 per cent, per ann. William Tomlinson, Dr. To 1 piece sup. blul cloth, ^6 yds. at 27s. 6d. For bill at 1 month. James Sanderson, Cr, By cheese, 25Ct. 3qrs. 171bs. at £3, 2s. 6d per cwt. ~ Allen, Wild, & Co. Leeds, Dr. To my acceptance of their Bill at 2 mon. > drawn 3d Jan. B. P. Book, No. 2, 5 Oats' Purveyance, in partnership with J. Henderson, Dr. To cash for oats purchased by me. Do. do. by Henderson, Do. for warehouse room, &c. Cr. By cash received for oats sold, Do. do. by J. H. Dr. To Profit, i (£42, 13s. S^d.) being my share, ? „ J. H. i (£42, 13 6i) being his share, 5 X i. 1 6 16 2 4 15 49 10 80 80 26 449 18 II I 13 477 4 507 9 65 d. 6 8 21 562i 1 1 3 8 11 2 8 10 85 6 11 DAY BOOK. 117 1832. Feb. 12 John Henderson, To cash advanced him on oatt' concern, „ Oats sold and reeceived for by him, By oats purchased by him, „ his sh^re or profit, £. i«. Dr. t 433 17 56 ^'i Cr. 488 19 449 42 !3 491 13 d. 3 H 11 * f .^.'iP^^Fy I i CASH BOOK. ^ V ( ' 1832. Ca8H, Dr. F £. 8 d. 1332. Contra. Cr.lF . £. 8 .!d. Jaa.l To stock, 1500- 1 Jun. 9. By S. Fletcher, paid 1 " Bills RcceivaUle, him bill No. 1. 24 3; 9 No. 1, on S. John- 'IIisBCc't.l5l.5s.9d. son, 24 : 1 9 less ISs.Sd. " 9 " S. Fletcher, cheque Discount,! 41. 10s. 6d _ — ■ — on Smith &, Co. 9K 5 3 Diif. sou Dr. side, 91.138.3d. ^,^ ^ . ^ .' " 19 "Bills payable No. 1. / W. Holmes's, 45 lOI — - , >: ,■ ' " 31 " T. Henderson, bal. ■ i'. of accU. (ab. 4d.) 37 5 6 Bulauco, 142C17 9 Feb.l 1533 1 7~ Feb.l. 1533 17 To balance, per op- By Bernard Mason, posite. 14261 7 9 (total, e:m. Se. 2id. " * " W. Tomlinson, abt. 5s. 2}d.) 832 («b8t.3i,) 41 3 " 8 " W. Tomlinson, pd. " J. Taylor, i year's to J. Sims, by his or- int. on 701. 11 5 der, 10 " W. Tomlincon, bill on Jones Sc Co. Lon. " 16 James Sanderson, 881. Is. 8|d. abtmt. ! No. 2, duo 10 May, 60 Is. 8}d. GS t I Should have been at 1 Irao.deljit him with Balance, 565 1 2 1U i discount 10a. — - — — His account 491. 10, diff. lOi. see Or. ' side. — - - — " 12 To Hazard & Jones's Assignee, composi- tion on 31. 15b. 6d. at I 1 i 128. 6d in the £• 1 Loss, 11. 88.3}. 2 7 2k '■ 1495 12 Tu - ' * 1495 12 lU Index to the Ledger* A. Allen, Wild, & Co. B. Bills payable. F. Fletcher, Samuel. H. Heudervon, John. Hazard & jonefl. M. Maaon, Bernard. O. Oats' Purveyance. S. Slock, Simmonds & Co. Sanderson, Jamet, T. Taylor, James, Tomlinson, Wm. 1832. Jan. 1 LEDGER. '^' 6 9 lU 1633. 1 Stock, J8n.l Dr. Feb.l2 1832. Jan. 1 1832. Jan.31 To mindriei, amount of my debta, " Balance acc't« James Tavlor, Dr. To money on bond, " Haifa yr'a interest, £■ 922 460 JOBN HCNDERSONiDr. To cash, 371. 5s. 6d. abt. 4d. Feb.l2 1838. Jan. 8 " Sundries, " Do. on oats' concern, ^ance, 70 1 71 13 18 •1 488 d. 10 0* 1833. Jan.l Contra, 1833. Feb. 4 Cr. By sundries, amoant of property. Cr. 491 Bernard Masoit, Dr. To sundries, '• Cash, 833, abat. 5s. Sid 1831. Bills Payable, Dr Jan.l9Tocas]i, Feb.l8 10 8 8 Jli 8J Contra, By cash for interest, " Balance, £• s. 1796 d. 9 cb 1 1332. Jan.l Feb.l2 Balunce, flfo. 1833. ALLBit,Wii.o,&Co. Dr. Feb.lOlTo bills payable, " ISj" Balance, ffo 7 832 840 45 80 14 5 00 10 00 9i 3i Contra, Cr. By balance of acc'ts. By sundries on oats* concern. 70 "ti 37 401 "491 n 15 10 13 8* 13 84 1832. Jan. 1 Contra, Cr. By purchase of house, and duty, due Feb- ruary 1st. 80 77 157 00 18 18 1832. Jan. 1 Feb.l0 ..-I— 1832. Jan. 5 Contra, Cr. ByHolmea'tb01,No.l Allen, Wild, & Co's. bill. No. 3, bb 840 I40 45 10 80 Contra, By cloth, Cr. 157 18 157 18 i 120 LEDGER. 1833. 'Samuel Fletcuir, Dr Jan. 9 To bill «nd diacount ou bopi, c. b. Feb.l2 1832. Jan.U Feb. 1 1833. SiMMONOS & Co. Dr, To balance, ffo. Wm. ToMLiNfoir, Dr To good*, Do. To cloth, Coib, 101. diac. 10s. (Iazard Sc Jones, Dr. Jan. 19 To soap and candles, " 28 " Sugar and tea, 14 1830. Feb.lO 1830. Feb.l3 1832. Feb.l2 Ja's Sanderson, Dr To cash, 881. abat. Ic. 8). Oats' Purveyanre iu Co. with J. Herdson, Dr. To 8undrio8, ProHi, it0 8elf,421.13i.5|fd. JT.H. 42l.13i.5id CfENERAL Dr. To cash in hand, James Taylor owes me s. 19 19 19 49 10 60 88 477 8.' 1833. Contra, Jan. 9 Cr. 56-2 8} 11 10 5fir) 12 70 635 12 Jan. 10 By hops, Choquo on Smith and Co. Contra, By foods. Cr. 1832. Feb.l 1832. Fcbl2 Contra. By casli. Abatement, BybHI, Cr. c.b. c.b lit »u 1832. Jan. 23 Feb.lO 1832 Feb.12 Contra, Cr. By cash .for composi- tion. Remainder lost. CbNTRA, By goods. " Choose, Cr. £ 15 9 34 14 Contra. By sundries, Cr 1832. Balance, Cr. Feb.12 By T. Herdson, I owe, Bills payHlilc, Allen, Wild, & Co. Simmonds &, Co. Stock acc't. debited. 60 60 s. d. 5 13 19 19 13 13 9 3 3i 3J 7 80 88 5C2 15 2J 3* 11 6 2* 8i 10 562 11 10 1 17 80 " 77 14 460 635 II 6 04 m 15 9 24 14 5 13 13 60 3i 60 2* 3* 15 7 80 3 16 6 2i 88 84 5C2 11 10 562 11 10 1 80 77 H 46U 17 »> 18 19 18 4i »» 6 Oi 635 12 lU •*#