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10 has over any other, was first seen and applied by Briggs, who was Professor at Oxford about the year 1670; the logarithms are, therefore, sometimes called the 'Briggian Logarithms."

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The student will perceive that the base 10 has this advantage, in consequence of our system of notation being decimal. If our system were duodecimal, our logarithms would then have to be calculated to the base 12, to be possessed of a like advantage, and so on for any other system.

N.B. The decimal part of a logarithm is called the mantissa,-the whole number is called the characteristic.

18. To show that every number has a Calculable Logarithm.

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may not be (for aught we have yet shown) convergent, unless a 2. Now, if a be any number, a root of it can always be found, say the nth, which shall be less than 2. Now,

1

1

1

1

loge an = (an — — (an
− 1) — («2 − 1) + (« − 1)%, — &c.,

2

- -

3

1

1

-1 is less than 1, is convergent, Now, log.. ar = loge a, ..

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which, since a

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convergent, if n is properly chosen, for every value of a, the value of loge a might be found from it, for all values of a. To calculate a table from such a formula, would be most laborious; but as, the calculation of the common tables presupposes that we know loge 10, if we treat the subject in its logical order, it would be necessary to calculate loge 10 from this series, before going further. We shall find, if we do, so, that...........

Hence

loge 10

1

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log. 10

434294481.

In future pages we shall denote this number by u. It is called the modulus of the tabular logarithm. Its actual calculation can be scen on p. 273, Vol. II., of Peacock's Algebra.

The logarithmic series to the base 10 we have already seen to be....

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19. To derive from the Logarithmic series, others from which the numerical values of

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From this we can calculate successively the values of log. 2, log. 3, log. 4.... as

far as we please.

Thus, remembering that log. 1

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1

1 1

1 1

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+

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3

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In the ordinary tables the logarithms are given calculated to 7 places of decimals. Hence, in making the calculation from the above series we must take in every term 7.00000001; or, since each term is greater than the one that comes after it, we must reduce each term to decimals until we find one 00000001, which we can omit together with all that come after it; all that go before it being reduced to decimals and added together, give the required logarithm. Thus, to find log. 11:-here p 10 .. 2p+1=21.

=

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.. log. 1110413927. To seven places of decimals, which is the logarithm you will find registered in the table. From the formula

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=log. (x=1)(x+1)

2 log x-log (x − 1) — log (x + 1).

=

— 2μ

.. log (x + 1) = 2 log x-log (x-1)-2μ {2x2-1+&c.}.

20. To explain the use of this last Series.

The use of this formula, which converges very rapidly, gives us any logarithm in terms of the two that precede it. It will be observed in this formula that

and all the terms following it can be omitted, provided

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1

1

3° (2x2 - 1)3

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.. if x 7

17 334.

169. 13.

Hence, if we employ this formula to calculate logarithms, we have, for all numbers greater than 13

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log (x + 1) = 2 log x — log (x — 1) — 2x2.

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2x

Hence, the formula finally reduces itself to

log (x + 1) = 2 log x — log (x — 1) — ——- ...

Or log (x+1)—log x=log x— log (x — 1) - — —

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(IX.)

u

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Hence, if we only calculate to 7 places of decimals, n must be at least many hundreds

μ

for to differ sensibly from

μ

(.; + n)3 •

Now, the formula would give us

μ

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log. (x+n+1)-log. (x)=log. (an)—log. (x n − 1) — which, unless is several hundreds, we have seen is practically the same as log. (x + n) — log. (xn−1)=log. (xn−1) — log. (x + n − 2) —

So that

log. (+1)—log. x = log. x=log. (x — 1)

X=

log. (x+2)-log,a= log. (+1)-log.

log. (x+3)-log. (x+2)=log. (x+2)-log. (x+1)—

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log. (x + 1),—19g. (x+n.— 1) = log. (x + x — 1) — log. (a + n − 2) — — n (x

.. adding together,

log. (x+n)—log.x=log. (xn−1) — log. (x — 1) — n

μ

.. log. (x + n) —log. (x + n − 1) = log. x.—log. (x—1); — n.

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Now, if n is sufficiently small for to be 0000001, it is clear that we may omit

μ

n so long as this is the case, and hence the differences between the successive

22

logarithms will continue the same within that limit.

For instance, we can show by formula (VIII.) that log. (10000) = 4.00000000. log. (10001)=4·00004342945.

.. log. (10001) log 1000000004343.

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Hence n1000000000001.

until n = 20. So that the logarithms of 10001, 10002, 10003.... 10020. can be found the one from the one before it by merely adding .0000434294.

We shall then have to calculate log. (10020) and log. (10021) from the original formula (VIII.) and find how far we can use the difference between these for deducing log. (10022) by log. (10023), &c. After some 20, or 30.logarithms are thus found by simple addition, a fresh calculation will become necessary: by proceeding in this manner, without any exorbitant labour, a table which gives the logarithm for every number from 10000 up to 99999 can be constructed; which is practically the same as from 1 up to 100000.

It is to be observed that with such a table, by means of a very simple subsidiary calculation, we can obtain the logarithm of any number from 1 to 10000000,

Thus, the tables give us log. 73894, log, 73895. i.e., log, 7389400, log. 7389500, for these only differ from those in the characteristic.

The subsidary calculation referred to enables us to fill up any one of the logarithms of 7389401, &c., up to 7389499.

21. To explain the construction and use of the Table of Proportional Parts.

Suppose N to be a number such as that above referred to, 71000000; and suppose its log. to be given in the table; then log. (N+100) is also given in the tables. From these data we want to find log. (N+8) where 8 lies between 0 and 100.

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which can be omitted, since we only take in the first seven places of logarithms.

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Now, log; (N +100) — log. N = 4 (suppose) is given by the tables, and we see that—

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We have already seen that the difference between two consecutive logarithms is the same for several logarithms together; accordingly, a small subsidiary table, giving A X 2 AX 3

Δ

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A X 9
10 ".

is, calculated for each different value of A, and

is printed, as ▲ occurs, in the margin of the table. For instance, ▲ corresponding to to log. 28568, is 0000152; or, as it is written, 152, it being understood that the last figure, 2, falls under the seventh decimal of the logarithm. In this case the subsidiary table is the accompanying. It is called a Table of Proportional'

152

115

▲ a Parts. Since: a and b are digits, this table gives at once and 10

Ab
10

2 30

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therefore

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Hence, by means of this table, we can determine log,

5 76.

10 10

6 91

7106

(N. + 8), from log, N. by addition only.

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Thus, log. 2856800 — 6·4558798..

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