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6.5960563,* the subtraction of 10 from 13:4684762—that is, of 3-4039437+6.5960563 from 6-8724199+6.5960563—leaves the same remainder as the subtraction of 3.4039437 from 6-8724199. So that the logarithm of the fourth term of the last proportion can be obtained as follows, 10 being subtracted mentally from 13—the integral part of the sum of the three addends: Log 19327:56

4.2861770 Log 385.694=

2-5862429 Log 2534:8 =34039437 ; arith. comp. =6.5960563 Log (19327-56x385.694 - 2534-8) = 34684762 EXAMPLE IV._Find the 17th power of :85764.

The logarithm of 85764 being Log •85764= 1'9333050 1°9333050, the logarithm of 8576417

17 is 1'9333050 X 17=2.8661850; so that the required power is '073483

Log -85764"=2.8661850 (nearly).

2.8661850=log '073483;

073483 ="85764". In multiplying 19333050 by 17, we obtain ('9333050 X 17=) 15.8661850, which is additive, from the mantissa ; and (IX 17=) 17, which is subtractive, from the characteristic. We therefore write the product (17+15.8661850) under the form 2.8661850; 17+15 being equivalent to 2. Because, diminishing a number by 2 is obviously the same as performing the twofold operation of (a) adding 15 to the number, and (b) subtracting 17 from the result.

EXAMPLE V.-Extract the 7th root of 316.4758.

The logarithm of 316-4758 be. Log 316'4758=2.5003405; ing 2-5003405, the logarithmn of 2.5003405 +7==0'3571915; V 316-4758 is of 2'5003405- 0-3571915=log 2-2761; i.e., O‘3571915; so that the re- 2.27615V 316-4758 quired root is 2-2761.

EXAMPLE VI.-Extract the 5th root of '000000813972.

The logarithm of .000000813972 being 7.9106095, the logarithm of N 000000813972 is: of 7.9106095—i.e., 2°7821219; so that the required root is '060551 :

7

* This is most easily obtained when we subtract the last figure (7) of 3:4039437 from 10, and each of the others from 9—the “carrying" of I being dispensed with.

Log 'oooooo813972=7'9106095

3+3
5)10+39106095

27821219
2-7821219=log '060551 ;

•060551=000000813972. The manner in which 7.9106095 is divided by 5 requires a word of explanation. One-fifth of 7'9106095 being 1.5821219, one-fifth of 7:9106095 would, at first sight, appear to be 1°5821219; but if we multiply 1.5821219 by 5, we shall obtain (5+2'9106095=) 39106095, instead of 7.9106095. The divi. sion of 7 by 5 would give ī for quotient, and leave 2-equivalent to 20 tenths—for remainder. If 9 tenths were added to this remainder, the result would be (not 29, but) II tenths, the division of which by 5 would give '2 for quotient, and leave I for remainder. So that, if the division were continued, the mantissa would be marked minus, as well as the characteristic; whereas A MANTISSA IS NEVEK MARKED MINUS* (unless when written, as a subtrahend, after another logarithm). In every

such case as the one under consideration, we begin by so altering the form of the logarithm as to have, for characteristic, a multiple of the divisor; and we naturally select the lowest multiple which answers our purpose. The first multiple of 5, after 7, being 10, which exceeds 7 by 3, we convert 7.9106095 into 10+3'9106095 by adding 3 to the characteristic, and +3 to the mantissa; 3+3 being=o.f We then find, with the greatest facility, that } of 79106095 is 2.7821219.

Again : in dividing 4.7689317 by 3, we first add 2 to the characteristic, and +2 to the mantissa ; the lowest multiple of 3, after 4, being 6, 4.7689317 which exceeds 4 by 2; and 2+2 being 2+2

We then find that } of 4.7689317 is 2-9229772.

3) 6+2.7689317

2.9229772 * This explains why the sign minus is always placed above (instead of before) the characteristic of the logarithm of a decimal. If placed before the characteristic, the sign would be understood to indicate that the whole of the logarithm was subtractive.

† It is obvious that if :9106095 were added to a number, and a subtracted from the sum, the result would be exactly the same as if 3.9106095 were added to the number, and 10 subtracted from the sum.

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by 2+

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3} of the fraction being multiplied by 5) or (the terms 33

16 of this last fraction being multiplied by 16)

37 The exact value of

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being the original fraction can

152 be written under the form

Dividing its terms by 2+1 3+

529 7, we find that the fraction lies between į and }, and that, consequently, it is more accurate to substitute & for than to reject zy altogether; so that, as a fourth and still closer approximation, we may set down

or (the terms of the

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of this last fraction being multiplied by 67)

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2+

The exact value of being the original fraction can

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4+7 admits of no alteration, its numerator being 1, the work of

decomposition ”—as it is sometimes called—terminates here;

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485

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and so on,

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and when } is taken into account, we necessarily obtain, instead of another approximation, the original fraction ( 18.5. itself :7

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i
5+

5+2 152
zo
3+

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3+ 4+1

152

4+1 152.

485

152 II22 2+

I 3+

I 5+

4+1 Of such approximations as the foregoing,-each closer than the preceding one,—the first is too large; the second, too small; the third, too large; the fourth, too small;

alternately. Thus, in writing } for 485, which is really equivalent to we take a fraction whose denominator is too

2153 small; so that the first approximation is too large. Again, } being a larger fraction than 2+} is a larger number than

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;
and is a smaller fraction than
2+

2+ 485

s; so that the second approximation is too small. Moreover, } being a larger fraction than 3+} is a larger number than 3+1 3+

is a smaller fraction than 59' 1

5.75 2+ is a smaller number than 2+

I 3+}

3+

2+

3+3 is a larger fraction than

485 or than

; so that the 2+

520 third approximation is too large. Lastly, I being a larger

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fraction than 5 5+is a larger number than 5+

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is a smaller fraction than

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is a larger number than 2+!
3+
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or than

2+
3+
5+}

3+! 485

4-
; so that the fourth approximation is too small.
291. An expression like i

2+1
3+
5+

4+ is called a CONTINUED FRACTION, or a CHAIN FRACTION.

292. To convert an ordinary fraction* into a Continued fraction : Divide—first, the terms of the given fraction by the numerator; next, the terms of the fractional part of the resulting denominator by its numerator; then, the terms of the fractional part of the new denominator by the numerator of this part; and so on, the process being continued until a fractional unit is obtained.

In converting 18. into a continued fraction, we divided first, the terms of the given fraction by 485; next, the terms of the fraction 153 by 152; then, the terms of the fraction in

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a

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* That is, an ordinary fraction which is proper, in its simplest form, and not a fractional unit.

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