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ponding to any given Logarithm, therein : likewise for particular rules concerning the Indices, the reader will consult Table 1, with its explanation, at the end of this Treatise.

MULTIPLICATION, Two, or more numbers being given, to find their pro

duct by Logarithms.

RULE.

Having found the Logarithms of the given numbers in the Table, add them together, and their • sum is the Logarithm of the product; which Logarithm, being found in the Table, will give a natural number, that is, the product required.

Whatever is carried from the decimal part of the Logarithm is to be added to the affirmative indices; but subtracted from the negative. Likewise the indices must be added together, when they are all of the same kind, that is, when they are all affirmative, or all negative; but when they are of different kinds, the difference must be found, which will be of the same denomination with the greater.

Example 1. Required the product of 86.25 multiplied by 6.48

Log. of 86.25=1.935759
Log. of 6.48=0.811575

Product=558.9=2.747334

Example 2. Required the product of 46.75 and .3275

Log. of 46.75= 1.669782
Log. of .3275=--1.515211

Product = 15.31+ = 1.184993

Example 3. Required the product of 3.768, 2.053 and .007693.

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Example 4. Required the product of 27.63, 1.859, 7258 and 0.3591.

Log. of 27.63 1.441381
Log. of 1.859= 0.269279
Log. of .7258 =-1.860817
Log. of .03591=-2.555215

Product nearly=1.339 =

0.126692

DIVISION.

Two numbers being given, to find how many times

one is contained in the other, by Logarithms.

RULE.

From the Logarithm of the Dividend subtract the Logarithm of the Divisor, and the remainder will be the Logarithm, whose corresponding natural number will be the Quotient required.

In this operation, the Index of the Divisor must be changed from affirmative to negative, or from negative to affirmative; and then the difference of the affirmative and negative Indices must be taken for the index to the Logarithm of the Quotient. Likewise when one has been borrowed in the left hand place of the Decimal part of the Logarithm, add it to the Index of the Divisor, if affirmative; but subtract it, if negative ; and let the

Index, thence arising, be changed and worked with, as before.

Example 1. Divide 558.9 by 6.48.

Log. of 558.9 ==2.747334
Log. of 6.48 =0.811575

Quotient = 86.25 =1.935759

Example 2. Divide 15.31 by 46.75.

Log. of 15.31= 1.184975
Log. of 46.75= 1.669782

Quotient=.3275=-1.515193

Example 3. Divide .25951 by .007693.

Log. of .05951 =-2.774590
Log. of .007693=-3.886096

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Example 4. Divide .6651 by 22.5.

Log. of .6651=-1.822887
Log. of 22.5 1.352183

Quotient=.02956=–2.470704

PROPORTION,

Or the Rule of Three in Logarithms.

RULE.

Having stated the three given terms according to the rule in common Arithmetic, write them orderly under one another, with the signs of proportion; then add the Logarithms of the second and third terms together, and from their sum subtract

the Logarithm of the first term, and the remainder will be the Logarithm of the fourth term, or An

swer.

Or,-add together the Arithmetical Complement of the Logarithm of the first term, and the Logarithms of the second and third terms; the sum, rejecting 10 from the index, will be the Logarithm of the fourth term, or term required.

N.B. The Arithmetical Complement of a Logarithm is what it wants of 10,000000, or 20,000000; and the easiest way to find it is to begin at the left hand, and subtract every figure from 9, except the last, which should be taken from 10; but if the index exceed 9, it must be taken from 19.-It is frequently used in the rule of Proportion and Trigonometrical calculations, to change Subtrac: tions into Additions.

EXAMPLES.

1st. If a clock gain 14 seconds in 5 days 18 hours, how much will it gain in 17 days 15 hours? 5.75 days

: Log.=0.759668

17.625 days
14 Seconds

:: Log.=1.246129

: Log.=1.146128

2.392257

Answer=42". 91

1.632589

Or thus ; 5.75 days : Arith. Co. Log.=9.240332 17.625

Log.=1.246129 14 Seconds :

Log.=1.146128

Answer=42". 91

=1.632589

F

2d. Find a fourth proportional to 9.485, 1.969 and 347.2. 98.45 :

: Log.=1.993216

347.2 : & Log.=2.540580
1.969

: Log.=0.294246

2.834826

Answer=6.944 =0.841610 3d. What number will have the same proportion 10.8538 as .3275 has to .0131 .0131 :

Log.=--2.117271

.3275 ::
.8538

Log.=-1.515211
Log.=-1.931356

-1.446567

Answer=21.35= 1.329296 4th. Required a third proportional number to 9.642 and 4.821 9.642

Log.=0.984167

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To find any proposed power of a given number by

Logarithms. Rule. Multiply the Logarithm of the given number by the Index of the proposed power, and the

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