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MULTIPLICATION BY LOGARITHMS.
TAKE out the logarithms of the factors from the table, then add them together, and their sum will be the logarithm of the product required. Then, by means of the table, take out the natural number, answering to the sum, for the product sought.
Observing to add what is to be carried from the decimal part of the logarithm to the affirmative index or indices, or else subtract it from the negative.
Also, adding the indices together when they are of the same kind, both affirmative or both negative ; but subtracting the less from the greater, when the one is affirmative and the other negative, and prefixing the sign of the greater to the remainder.
DIVISION BY LOGARITHMS:
From the logarithm of the dividend subtract the logarithin of the divisor, and the number answering lo the remainder will be the quotient required.
Observing to change the sign of the index of the divisor, from affimative to negative. or from negative to affirmative; then take the sum of the indices if they be of the same name, or their difference when of different signs, with ihe sign of the greater, for the index to the logarithm of the quotient.
And also, when I is borrowed, in the left-hand place of the decimal part of the logarithm, add it to the index of the divisor when ihat index is affirmative, but subtract it. when negative; then let the sign of the index arising from hence be changed, and worked with as before.
1. To divide 24163 by ·4567. (2. To divide 37 149 by 523.76 Numbers. Logs.
Numbers. Logs Dividend 24163 4.383151 Dividend 37 149 . 1.569947 Divisor 4567 3.659631 Divisor 523 76 - 2 719132
0723520 Quot. •0709275
3. Divide :06314 by 007241 4. To divide 7438 by 12 9476 Numbers. Logs.
Numbers. Logs. Divid. 06314 2 800305 Divid. •7438 1.871456 Divisor .007241 3-859799 Divisor 12.9476 1.112189
0.940506 Quot. •057447
Here I carried from the Here the 1 taken from the decimals to the 3, makes it -1, makes it become - 2, to become -2, which taken from set down. the other - 2, leaves o remaining
Note. As to the Rule-of-Three, or Rule of Proportion, it is performed by adding the logarithms of the 2d and 3d terms, and subtracting that of the first term from their sum.
INVOLUTION BY LOGARITHMS,
Take out the logarithm of the given number from the table. Multiply the log thus found, by the index of the power proposed. Find the number answering to the product, and it will be the power required.
Note. In multiplying a logarithm with a negative index, by an affirmative number, the product will be negative. But what is to be carried from the decimal part of the logarithm, will always be affirmative. And therefore their difference will be the index of the product, and is always to be made of the same kind with the greater,
EXAMPLES 1. To square the number 2. To find the cube of 2.5791.
3.07146. Numb. Log.
Log. Root 2.5791 0 411468 Root 3.07 146 0.487345 The index, - - 2
The index 3
9750 11700 5850
Here 4 times the negative index being -8,and 3 to carry, the difference - 5 is the index Power 5.14932* of the product.
* This answer 5:11932 though found strictly according to the general rule, is not correct in the last two figures 32 ; nor can the answers to such questions relating to very high powers be generally found true to 6 places of figures by the table of logarithms in this work: if any power above the hundred thousandth were required, not one figure of the answer found by the table of logarithms here given could be do pended on.
The logarithm of 1.0045 is 00194994108 true to eleven places, which multiplied by 365 gives -7117285 true to 7 places, and the corresponding number true to 7 places is 5:149067. VOL. I.
EVOLUTION BY LOGARITHMS. TAKE the leg. of the given number out of the table.
Divide the log. thus found by the index of the root. Then the number answering to the quotient, will be the root.
Note. When the index of the logarithm, to be divided, is negative, and does not exactly contain the divisor. wii hout some remainder, increase the index by such a number as will make it exactly divisible by the index, carrying the units borrowed, as so many tens, to the left-hand place of the decimal, and then divide as in whole numbers
Ex. 1. To find the square root Ex. 2. To find the 3d root of of 365
12345. Numb. Log.
Log. Power 365 2)2 562293 Power 12345 3) 4 09 1491 Root 19.10496 1.281146 Root 23.1116 1.3638303
Ex. 3. To find the 10th root | Ex. 4. To find the 365th root of 2.
of 1 045. Numb. Log
Numb. Log. Power 2 -- 10 ) 0:301030 Power 1 045 365 ) 0·019116 Root 1•071773 0:030103 Root 1000121 O 000052
Ex. 5. To find ✓ 093. Ex, 6. To find the •00048. Numb. Log.
Log. Power .093 2) -- 2.968483 Power .00048 3)— 4.681241 Root .304959 - 1:4842414 Root •0782973 — 2.893747 Here the divisor 2 is con
Here the divisor 3 not being extained exactly once in the need by 2, to make up 6, in which the
actlycontained in-4, it is augmentgative index — 2, and there- divisor is contained just 2 times 1 fore the index of the quotient then the 2, thus borrowed, being is -- 1.
carried to the decimal figure 6, makes 26, which divided by 3, gives 8, &c.
DEFINITIONS AND NOTATION.
LGEBRA is the science of computing by symbols, It is sometiines also called Analysis; and is a general kind of arithmetic, or universal way of computation.
2. In this science, quantities of all kinds are represented by the letters of the alphabet. And the operation to be per, formed with them, as addition or subtraction, &c. are denoted by certain simple characters, instead of being expressed by words at length.
3. In algebraical questions, some quantities are known or given, viz those whose values are known : and others unknown, or are to be found out, viz. those whose values are not known. The former of these are represented by the leading letters of the alphabet, a, b, c, d, &c; and the latter, or unknown quantities, by the final letters, z, y, x, u,&c.
4. The characters used to denote the operations, are chiefly the following:
+ signifies addition, and is named plus.
signifies the square root; the cube root; the 4th root, &c.; and the nth root.
::: : signifies proportion.
Thus a + b denotes that the number represented by o is to be added to that represented by a
Q-6 denotes, that the number represented by 6 is to be subtracted from that represented by a.
ain 6 denotes the difference of a and b, when it is not known Which is the greater,