14

by 2, the product will be 8, which is the logarithm of 256, or the square of 16.

Again if the logarithm of any number be divided by the index of its root, the quotient will be equal to the logarithm of that root: thus, the index or logarithm of 256 is 8; now, 8 divided by 2 gives 4, which is the logarithm of 16, or the square root of 256, according to the first series. The logarithms most convenient for practice are such as are adapted to a geometrical series increasing in a tenfold ratio, as in the last of the foregoing series; being those which are generally found in most mathematical works, and which are usually called common logarithms, in order to distinguish them from other species of logarithms.

In this system of logarithms, the index or logarithm of I is o; that of IO is I; that of 100 is 2; that of 1000 is 3; that of 10000 is 4; etc., etc.; whence it is manifest that the logarithms of the intermediate numbers between 1 and 10 must be o, and some fractional parts; that of a number between 10 and 100 must be 1, and some fractional parts; and so on for any other number; those fractional parts may be computed by the following

Between

Rule.--To the geometrical series 1, 10, 100, 1000, 10000 etc., apply the arithmetical series 0, 1, 2, 3, 4, etc., as logarithms. Find a geometrical mean between 1 and 10, or between 10 and 100, or an other two adjacent terms of the series between which the proposed nun. r lies. the mean thus found and the nearest extreme, find another geometrical mean in the same manner, and so on till you arrive at the number whose logarithm is sought. Find as many arithmetical means, according to the order in which the geometrical ones were found, and they will be the logarithms of the said geometrical means, the last of which will be the logarithm of the proposed number.

EXAMPLE. To compute the Log. of 2 to Eight Places of Decimals— Here the proposed number lies between 1 and 10.

FIRST. The log. of I is o, and the log. of 10 is 1; therefore o + I ÷ 2 = 5 is the arithmetical mean, and √1 × 10 = 3.1622777 is the geometrical mean hence the log. of 3.1622777 is ·5.

SECOND. The log. of I is o, and the log. of 3.1622777 is 5; therefore 0+ ·5 ÷ 2 = 25 is the arithmetical mean, and √1 × 3.1622777 1.7782794 the geometrical mean: hence the log. of 1-7782794 is 25.

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THIRD. The log. of 1.7782794 is 25, and the log. of 3.1622777 is 5; therefore 25+ 5 ÷ 2 = .375 is the arithmetical mean, and V1-7782794 X 3.1622777 =2.3713741 the geometrical mean hence the log. of 2.3713741 is 375.

FOURTH. The log. of 1.7782794 is 25, and the log. of 2.3713741 is 375; therefore 25+375 ÷ 2 = 3125 is the arithmetical mean, and √1.7782794 × 2·3713741 = 2·0535252 the geometrical mean: hence the log. of 2.0535252 is 3125.

FIFTH.-The log. of 1.7782794 is 25, and the log. of 2·0535252 is ·3125 ; therefore 253125 ÷ 2 = 28125 is the arithmetical mean, and

√1-7782794 × 2·0535252

log. of 1-9109530 is 28125.

=

1-9109530 the geometrical mean: hence the

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SIXTH.-The log. of 1-9109530 is 28125, and the log. of 2.0535252 is 3125; therefore 28125 +31252.296875 is the arithmetical mean, and 1-9109530 X 2.0535252 = I-9809568 the geometrical mean: hence the log. of 1.9809568 is 296875.

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SEVENTH. The log. of 1-9809568 is 296875, and the log. of 2.0535252 is 3125; therefore 296875 + ·3125 ÷ 2 3046875 is the arithmetical mean, and √1-9809568 × 2·0535252 = 2.0169146 the geometrical mean: hence the log. of 2.0169146 is ⚫3046875.

EIGHTH.-The log. of 2.0169146 is ·3046875, and the log. of 1.9809568 is 296875; therefore 3046875 +296875230078125 is the arithmetical mean, and √2·0169146 × 1·9809568 1.9988548 the geometrical

=

mean hence the log. of 1-9988548 is 30078125.

Proceeding in this manner, it will be found, after 25 extractions, that the 1g. of 1.9999999 is 30103000; and since 19999999 may be considered as being essentially equal to 2 in all the practical purposes to which it can be applied, therefore the log. of 2 is 30103000.

If the log. of 3 be determined, in the same manner, it will be found that the twenty-fifth thmetical mean will be 47712125, and the geometrical mean 2.9999999; and since this may be considered as being in every respect equal to 3, therefore the log. of 3 is 47712125.

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Now, from the logs. of 2 and 3, thus found, and the log. of 10, which is given I, a great many other logarithms may be readily raised; because the sum of the logs. of any two numbers gives the log. of their product; and the difference of their logs. the log. of the quotient; the log. of any number, being multiplied by 2, will give the log. of the square of that number; or, multiplied by 3, will give the log. of its cube; as in the following examples—

Since the odd numbers 7, 11, 13, 17, 19, 23, 29, etc., cannot be exactly deduced from the multiplication or division of any two numbers, the logs. of those must be computed in accordance with the rule by which the logs. of 2 and 3 were obtained; after which the labour attending the construction of a table of logarithms will be greatly diminished, because the principal part of the numbers may then be very readily found by addition, subtraction, and composition.

Having shown the construction of the Tables of Logarithms, we shall now proceed to the manner of using them.

PROPERTIES OF LOGARITHMS

It being understood that when we write or say 4, or 56, or 749, or 8476, and so forth to any extent, we mean that we are giving expression to integers (or whole numbers) which may consist of any number of figures (or digits) ranging from one to millions or more.

On the other hand, if we say or write five-tenths and express it by a figure, thus, 5; or again, if we say five-hundredths and write it as -05; and yet further, if we say thirty-seven hundredths and write it as 37, we are giving expression to a decimal fraction.

And again, if we write 37.04 or 349 67, we have numbers that are partly integers and partly decimals; the figures to the left of the dot (or decimal point) being the integers, and those to the right of the dot being decimals. All these are natural numbers, or such as appertain to common arithmetic.

The table in Norie's "Epitome" to which the following remarks apply is the Table of Logs., and to it we direct special attention. In the first part of it you will see a column marked "No." and by its side another column marked "Log." The meaning of "No." is "number," and "Log is the abbreviation for logarithm. You will also see that the numbers run from 1 to 100, and that there is a logarithm corresponding to each number, and standing by its side

It is important to note this, for you will see that each logarithm consists of two parts, viz., a figure to the left of the decimal point, and six figures to the right of the decimal point. The figure to the left is called the index, and the figures to the right are the mantissa, or, in a plain word, decimals.

If you now look in Table cf Logs to the numbers beyond 100, you will see that the logarithmic columns by the side of the numbers have no index, but only the decimal part, and for this reason,-that though no logarithm is complete without an index and a decimal part, the index is readily supplied according to a fixed rule, and one very easily to be remembered.

RULE FOR THE INDEX.--The index of a logarithm is always less by one than the number of figures (or digits) in the integer or whole number. On this basis you will see that the number 8 has index o; number 49 has index I; and number 100 has index 2. In a similar way number 300 or 465 would have index 2; number 4876 would have index 3; and number

75687 would have index 4; also number 3400757 would have index 6. Now, using 3400756 as a special illustration of the rule, we see that it consists of seven figures or digits; one less than seven is six, hence 6 is the logarithmic index of the number. Similarly 49 consists of two figures; one less than two is one; hence 49 has I for its logarithmic index.

If the number is partly an integer and partly a decimal, as 84.674, then the number of figures in the integer (or to the left hand of the dot or decimal point) only is counted to give the index; and as 84 consists of two figures, we have I for the index: similarly 846.74 gives index 2; and 8.4674 gives index o.

If the number is entirely decimal, the index of the logarithm is properly negative; thus the index of the logarithm of 9, or of 94, or of 949, is - I (minus one); the index of 09, or 094, or of 0949, is 2; also of 009, or 0094, the index is 3. But to avoid confusion in the addition and subtraction of indices of different characters it is customary to use the arithmetical complement (that is, the number subtracted from 10) of the negative indices, and consider these complements as positive; hence for I we say 9; for 2 we say 8; for 3 we say 7, and so on; since 9 is the arithmetical complement of 10 lessened by 1; and 8 is the arithmetical complement of 10 lessened by 2, etc.

When you read further on you will see that the decimal part of the logarithm of 8764 is 942702, and if we complete the logarithm by prefixing its index, we get

From these numbers with their accompanying logarithms we also learn another fact, viz., that the same significant figures (as 8764), whether they are whole numbers, or partly whole and partly decimal, or entirely decimal, give the same logarithm in so far as its decimal part is concerned : the only change is in the index of the logarithm.

It, therefore, follows that the index of a logarithm must be known before any definite value can be assigned to the corresponding natural number. For example, the logarithm of 8764 is 942702, and if the index of the logarithm be 3 the natural number will be 8764, and if the index be I the natural number will be 87.64; but if the index be 6 the natural number will be 8764000. It will thus be seen that the number of figures in the whole number is one more than the number indicated by the index.

Example. Required the logarithm of 25047. First, look in the lefthand column for 250; opposite to this, in the column with 4 at the top, is the decimal part of the logarithm, which is 398634; in the right-hand column is the difference 173; this multiplied by 7, the fifth figure, gives 1211; from which cut off the last figure, and the remainder 121 added to .398634 will give, with the proper index prefixed, 4·398755, the logarithm required.

In the same manner the logarithm of 598765 will be found to be 5777256; obtained as follows

Here, in getting the correction for 65, two figures are cut off to the right because we have multiplied the difference by two figures; and having added 47 (the figures to the left) to the logarithm of 5987 we get the decimal part of the required logarithm; and finish by prefixing the proper index, which in this case is 5 because 598765 is a whole number, and consists of six figures. But the logarithm of 598-765 would be 2.777256.

N.B. When correcting logarithms for the number of figures in excess of four, always cut off as many figures on the right-hand side as you multiply by.

The index 5 being supplied on the basis that there are six figures in the whole number, and the proper decimal part of the logarithm being found through the column of differences, as before. Similarly, the logarithm of 374.909 would be 2.573925.