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multiplicand on the right of the figure multiplied by, writing down the products so that the figures on the right hand shall be in a vertical line, having carried to them from the products of the two preceding figures in the multiplicand, 1 for numbers from 5 to 15, 2 for numbers from 15 to 25, 3 from 25 to 35, &c.: and the sum of the products thus found will seldom differ from the true product, even in the last place of decimals used.

Ex. Required the product of 13.57483 and 84.8207, retaining only four places of decimals.

Contracted Multiplication.

1 3.5 7 4 8 3

7 0 2 8.4 8

1 0 8 5 9 8 6 4

542993

108599

2715

9 5

1 1 5 1.4 266

Compound Multiplication.

1 3.5 7 4 8 3

8 4.8207

9 50 2 3 8 1

2714966 0

1085986 4

542993 2

108 5 9 8 6 4

1 1 5 1.4 2 6 5 8 2 9 8 1

24. In Division, since the first figure of the quotient has always the same local value as the figure of the dividend which stands over the units' place of the first product, we have only to consider how many figures of the quotient will comprise the number of decimals proposed to be retained; then to take the same number of figures on the left of the divisor, and as many figures in the dividend as will contain them less than ten times: by these to find the first figure of the quotient, and for each of the following figures to divide the last remainder by the divisor wanting one figure more to the right than it did before, observing, as in the preceding Article, what numbers are to be carried in obtaining the partial contracted products.

Ex.

Divide 1254.46403 by 46.205175, so as to have four decimal places in the quotient.

Here, the quotient must contain six figures, and we

46.2051,75) 1 2 5 4.4 6 4, 0 3 (27.1498

92,410,4

330360

32 34 36

6924

4621

2303

1 8 4 8

455

416

39

37

and that the quotient 27.1498 is correct as far as it goes, may be seen by the common operation.

If the divisor do not contain as many figures as are required for the quotient, perform the ordinary division, till the numbers of figures in the divisor and of those remaining to be found in the quotient are equal, and then proceed according to this Rule.

VII. DIFFERENT SCALES OF NOTATION.

25. Numbers in the Common or Denary Scale of Notation may be expressed by means of the Arithmetical signs: thus,

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1849 1 × 1000 + 8 ×

100 + 4 × 10+9

=

= 1 x 103 +8 × 102 + 4 × 10 + 9:

but this method of representing numerical magnitudes is not confined to powers of 10: for

2316+ 4 + 2 + 1

= 1 × 24+0 × 23 + 1 × 2o + 1×2 +1:

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and hence arise the Binary, Ternary, &c., Scales of Notation, the digits employed, including the cipher 0, being equal in number to the Base of the System.

Hence, 23 in the common scale = 10111 in the binary : = 1202 in the ternary:

47

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again, if the base of the scale be 4, we shall have

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or 203 in the quaternary scale is equivalent to 35 in the common scale.

The following is the kind of operation necessary to transform a number from the common scale to any other: as, for instance, to find the numbers in the quinary and senary scales which shall be equivalent to 384:

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and the remainders taken in order form the digits of the numbers beginning at the right hand: thus,

384 in the common scale is equivalent to 3014 in the scale of 5, and to 1440 in the scale of 6.

In the fundamental operations, the processes are regulated by the base of the scale, instead of 10: thus, in the septenary and octenary scales, we have

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and in the nonary and duodenary scales, we shall have

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ANSWERS OMITTED IN PAGES 5, 8, 11.

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Page 8. NUMERATION. Article (19).

(1) Four thousand, three hundred and twenty.
(2) Eighty-seven thousand and fifty-four.

(3) Nine hundred and three thousand, seven hundred and fifty-six.

(4) Two million, seven hundred and fourteen thousand, three hundred and twenty-five.

(5) Eight million, forty-seven thousand, three hundred and twenty-eight.

(6) Twelve million eight hundred and seventy thousand and forty-five.

(7) Twenty million, eighty-four thousand, two hundred and sixteen.

(8) Seventy-nine million, thirty-thousand, two hundred and eighty-four.

(9) Three hundred and twenty-one million, four hundred and eight thousand, six hundred and fifty-three. (10) Four hundred and eight million, seventy-six thousand, and thirty-two.

(11) Three hundred and fourteen million, one hundred and fifty-nine thousand, two hundred and sixtyfive.

(12) Five hundred and seventy-one million, two hundred and sixty-eight thousand, four hundred and five.

Page 11. ADDITION. Article (24).

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(13) 14397.

(14) 1038206.

(15) 268491. (16) 74147863. (17) 330122351, read three hundred and thirty million, one hundred and twenty-two thousand, three hundred and fifty-one.

(1) 6.

Page 14. SUBTRACTION. Article (28).
(2) 34. (3) 154. (4) 6239.

(7) 127593.

(5) 14759.

(8) 4699. (9) 6713076. (11) 10942895. (12) 304924818.

Page 21. MULTIPLICATION. Article (37).

(2) 4425. (3) 11468.

(6) 327699.

(10) 60005393.

(1) 568.

(4) 392715.

(5) 246522.

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(19) 246913578, 493827156, 740740734, 987654312, 370370367, 617283945, 864197623, 1111111101. (20) 1287657, 1000055, 34381488, 8539410, 11216556, 46634205, 1013736849.

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(8) 795917072,3.

(7) 11979052,1.

(9) 315836,7. (10) 112233444,6.

(11) 823045,3. (12) 6267706,8. (13) 1272250,6.

(14) 87997,214.

(15) 19915,5559. (16) 3216,6886.

(17) 1453,2280.

(18) 43349,1080.

(19) 2384,6. 14888,6. 121446,37. 33478,35.

Page 31. GREATEST COMMON MEASURE. Article (52).

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