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hands of some to whom it may be useful to have one, I subjoin the following general RULE -

Draw a line or hook at the right, and another at the left of the dividend, and write the divisor outside that on the left.

By trial find how many times the divisor is contained in as many of the nearest figures of the dividend thereto as may be necessary, which will not exceed one figure more than the number of figures in the divisor, place the figure denoting this number of times at the right hand, outside the other line or hook, which will give the highest figure of the quotient, multiply the divisor thereby, and place the product under that part of the dividend which thus contained the divisor, substract it therefrom, and to the remainder bring down the next figure of the dividend; again find by trial how often the divisor is contained in this remainder, with the figure thus annexed, and place the figure denoting the number of times in the quotient beside the former, and multiply and subtract as before; but if the first or any subsequent remainder with the figure of the dividend annexed thereto, be less than the divisor, place a cipher in the quotient and bring down another figure or cipher of the dividend, which annex to the former, and if it should still be less than the divisor place another cipher in the quotient, always taking care to put a figure or a cipher in the quotient, for every figure or cipher of the dividend thus brought down, and so on until the remainder is equal to, or exceeds the divisor, or until all the figures or ciphers in the dividend are exhausted, taking care as at first, that there shall not be more than one figure above the number of figures in the divisor, and thus proceed until all the figures in the dividend are brought down, when if there is a remainder, it is to be added to the quotient, as a fraction as before noticed.*

* The above rule is best illustrated by examples. 1. Let 85394 be divided by 7. 7) 85394 10000 x 7 = 70000 10000 = first quotient.

15394
2000 x 7 = 14000 2000. = second quotient
1394
100 x 7 = 700 100 = third quotient.
694 -
90 × 7 = 630 90 = fourth quotient.
64
9 x 7 = 63 9 = fifth quotient.

--, 1.
T

Remainder 1 12199; sum of all the quotients and theanswer.

After division is performed, by multiplying the quotient by the divisor, and adding the remainder, if any to the product, the former dividend will be obtained if the work is right; or by subtracting the remainder from the dividend, and dividing this remainder by the quotient found, the former divisor will be obtained, which furnishes two methods of proof.

Čramples. Let 47384 be divided by 4 4)47384 (11846 4 4.

~7 47384 first proof.
4 -
33
32
18 11846)47384 (4
16 47384
24 I. : : second proof.
24

2 Let 5760075 be divided by 375
- 375 ) 57.60075 ( 10000

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And the same reasoning will apply to any other numbers.

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For a further demonstration, and other observations on Division, see Walker's Philosophy of Arithmetic, pages 19 to 29.

Let 85394 be divided by 7.
7) 85394 ( sing
7

15" 85394 first proof. 14 . . . . 13 From 85394 7 Take. ... 1 69 12199) 85393 (7 63 85393 64 second proof. 63 + - --- 1. 384634 -i- 2 2. 34554 -i- 3 3. 475023 -i- 3 4. 275484 .. 4 5. 86964 ... 4 6. 8734716 .. 6 7. 87475 . , 5 8. 61343821 .. 7 9. 684.796 .. 8 10. 5046848 .. 8 11. 801621 ... 9. 12. 372059] ... 9 Let 6477007 be divided by 47 47) 6477007 (137808}}r. 47 47. 177 964657 141 551235 367 6477007 first proof. 329 . - T380 - From 6477007. 376. . . Take. . 31 T07 . 137808) 6476976 (47. 376 551232 31 964656 -- 964656 - . . . . . . . second proof. . . . = 13. 658.394 - 14" 14. . 57358495 - 15 15. 3748964 . . 16. 16. 67384908 ... 18 17. 8497.8384 ... 17 18. 5738429 . . 19. 19. 6738494 ... 24. 20. 8937463 ... 29. 21. 5300732 ... 31 - 22. 43580764 . . 43 23. 10000000 .. 81 24, 53.980785 . . 49

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25. 39.48084 -- 47 26, 8940898 + 57 27. 43116473 .. 53 28. 64086937 .. 77 29. 143143143 .. 65 30, 29.83064 . . 85 31. 24396084 .. 87 32. 31789482 . . 95 33. 43116473 ... 93 34. 58106878 .. 67 35. 43410769 . . 97 36. 42108760 .. 78 37. 5830076 ... 98 38. 57385964 .. 69 39, 5847089 ... 99 40. 84968087 . . 99 Let 5760075 be divided by 375 - 375).5760075 ( 15360 or 375 375 2010 - 76805 1875 107527 T350 46080 T 1125 576.0075 proof. 2257 2250 75 41. 3184738 -- 142 42. 506594082 -i- 185 43. 10000000 ... I 11 44. 123456789 ... l II 45. 8730748 . . 486 46. 987654321 ... 999 47. 43987396 . . 473 48. 510084398 .. 537 49. 10000000 . . 729 50. 41241.2412 . . .333 51. 68.184784 .. 675 52. 9410087996.. 976 53. 111111111111 .. 864 54. 939473842 . . 894 55. 29843008 .. 873 56. 718410884 .. 756

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57. 231426845, -i- 1125, 58, 5987308697 +-3508 59.4371062904, .. 2314 60. 6710075798 ... 5763 61. 7391082763. ... 7854, 62.468.1007042 ... 6577 63. 1000000000 ... llll 64. 17814637409 ... 7359 65.8532784001 .. 4272 66. 394864.1842 ... 1732 67.3161642984 .. 6429 68, 59.8439.4805 ... 9845 69. 757869396 .. 7076 70. 1000000000000 ... 11111 71. 8493740754: .. 8465 72.5555555555 ... 123456 73.543.108343. . . 4310474. 5555555555 ... 654321

75. 6859384075 .. 56943 76. 101101101101101.. 111111 77. 61087107384 ... 6710878. 101101101101101... 99999 79. 571064249386... 75321 80. 1234567894 ... 22222. 81. 732062028073.. 6931882. 727727727727 ... 31725. * 83 431076394.10676 -- 9344584 84. 681241043.10.1874 .. 47386.107 85.543961543961543961 ... 194187946

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When the divisor does not exceed 12, the result may be obtained in one line, by mentally performing the successive 'multiplications and subtractions as we proceed, attending only to that part of the dividend, which ascertains the successive digits of the quotient, and only writing those digits.

€rampleń,

2) 47536 5)479575 3)374013

23768 95915 124671

47536 479575 374013 86. 102396 -- 2 87. 5625903 -- 3 88. 30389628 . . 4. 89, 39504020 .. 4 90. 27499875 .. 5 91. 23750135 .. 5 92. 567750252 .. 6 93. 74936430 .. 6 94. 1927772. .. 7 95. 594.7867128 .. 7 96. 381483784.. 8 97. 806767533 .. 8 98. 611681454 ... 9. 99. 6795080752 ... 9 100. 2063644 ... 11. 101. 93.46425 ... ll 102. 39486756 ... 12 103, 757312128 ... 12

104, 493486424. 12 105, 830490840 ... 12

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