And according to this system or scale, the common logarithmic tables now in use, are calculated*.

159. Now 0 being the logarithm of 1; 1 the logarithm of 10; 2 the logarithm of 100; &c. it follows that the logarithm of any number between 1 and 10 will be 0 with a fraction; between 10 and 100, 1 with a fraction; between 100 and 1000, 2 with a fraction, &c.

159. It is also evident from the nature of the progressions, that if any number of geometrical mean proportionals be interposed between any two terms of the geometrical series 1, 10, 100, 1000, &c. and the like number of arithmetical means between the corresponding indices 0, 1, 2, 3, &c. that the latter will be the indices or logarithms of the former.

Thus one geometrical mean proportional between 100, and 10000 is 1000 (151.)

And the arithmetical mean between the indices 2 and 4 is 3 (129), the logarithm of 1000.

In like manner the geometrical mean between 10 and 100 is. 1/1000 or 31-6227 &c.

* The invention of Logarithms is due to Lord Neper, Baron of Merchiston, in Scotland, who in 1614, published the first table of these numbers in a small treatise, entitled Mirifici Logarithmorum Canonis Descriptio. His logarithms, however, are of that form which has since been called hyperbolic logarithms. The present scale or system of logarithms we owe to Mr. Henry Briggs, at that time (1614) Professor of Geometry at Gresham College.

The modern Logarithmic tables, in most cs'eem at present foreneral use are, Gardener's, 4to. 1742. Taylor's, large 4to. 1792. Tables Portatives, par Callet, 8vo. (the stereotype edition). D. Hutton's Mathen atical Tables, 8vo. 1801; this also contains a very complete History of 10garithmns.

✔signifies the square root; thus V 15 x

or 36 is f

And the corresponding arithmetical mean between the indices 1 and 2 is 15, which is the logarithm or index of the term 31.6227 &c.

Therefore the business of computing the logarithm of a given number principally consists in finding a geometrical mean or term of the series equal to, or nearly equal to, the number proposed; then its corresponding arithmetical mean or index will be the logarithm sought.

Now, by repeated extractions of the square root, such an approximate mean proportional may be found, as in the following example:

160. Let it be required to find the logarithm of 2?

First. The number 2 lies between 1 and 10;

(151) and the geometrical mean between 1 and 10 is VI × 10 3.162278.

And the arithmetical mean between the indices 0 and 1 (the logarithms of 1 and 10) is 05:

therefore the index or logarithm of 3.162278 is 0.5.

Secondly. The number 2 now lies between 1 and 3.162278:

and the geometrical mean between those numbers is VIX 3.162278 = 1.778279.

And the arithmetical mean or half the sum of the indices 0 and 0.5 (the logarithms of 1 and 3 162278) is 0.25:

therefore the logarithm of 1-778279 is 0·25.

Thirdly.. The number 2 lies between 1778279 and 3·162278;
and the geometrical mean is ✔ 1·778279 × 3162278 = 2·371374

And the arithmetical mean between the indices 0.25 and 0.5 is 0.375: therefore the logarithm or index of 2:371374 is 0.375.

Fourthly. The terms next less and next greater than 2 are 1-778279 and 371374;

and the geom. mean is 1-778279 2·3713742·053525.

And half the sum of the corresponding indices or logarithmns 0.25 and 0.375 is 0.3125:

therefore the log. or index of 2-053525 is 0.3125.

LOGARITHMS

And in this manner by constantly making use of the resulting geometrical means next less and next greater than 2, after 22 extractions we get the term 1.999999, and the corresponding aritmetical mean or logarithm 0-3010299 for its index. Therefore as 1.999999 differs but 0·000 01 from 2, we may take 0-3010299 or 0-301030 (the nearest 6 decimals) for the logarithm of 2.

This is one of the methods by which logarithms were first computed. But more direct and expeditious rules have since been derived from algebraic formula, and the fluxion calculus.

161. Now from the logarithm of 2, the logarithms of 4, 8, 16, &c. the powers of 2, are obtained by multiplication.

Thus, 0-301030 X 2 = 0-301030 × 3 =

0.602060 the log. of 22 or 4.
0.903090 the log. of 23 or 8.

1204120 the log. of 24 or 16.
&c.

0-301030 X 4 =

&c.

162. And since 10 divided by 2 gives 5, if the logarithm of 2 be subtracted from the logarithm of 10, the remainder will be the logarithm of 5 (156).

Thus 1000000 log. of 10.

0.301030 log. of 2.

0698970 log. of 5.

163. And if the logarithm of 5 be multiplied by 2, 3, 4, &c. the products will be the logarithms of its powers; thus 0.698970 × 42.795880 the log of 54 or 625.

164. Hance in the common scale or system of logarithms, every number is supposed to be that power of 10 whose index is the logarithm of the number.

Thus by the foregoing operation 100-301030 is equal to 2, nearly.

100.903090 equal to 8,

165. The integral part of a logarithm is called its index or characteristic; thus in the logarithms 0.301030, 1.204120

2.795880, the indices are 0, 1, 2; the other figures being decimals. And as the indices are easily supplied by the computer himself, they are commonly omitted in the tables.

166. Since the logarithm of the divisor taken from that of the dividend gives the logarithm of the quotient (162), it follows that the index of the logarithm of a proper fraction will be negative.

Thus suppose the logarithm of 10, or the decimal 625 is required:

In this subtraction 1 is carried to the index 1, which together make 2, then I minus 2 gives 1 negative, marked with the negative sign (—) in the remainder.

167. But the logarithm of an improper fraction will have a positive index, because its value is greater than 1.

Thus to find the logarithm of 25 or 6:25.

168.

25, its log. 1:397910 (twice the log. of 5.)

4, its log. 0-602060 subtract.

0795880 log, of 625.

Because 625 x 10 = 6250; and 625 x 100 = 62500, if we add the logarithm of 10, and 100 to that of 625, we get 3-795880 the log. of 6250, and 4.795880 the log, of

62500.

169. Hence it appears, that the logarithm of a whole number and that of a mixed number, or a fraction, consisting of the same significant figure, differ in nothing but the index, which varies according to the place of the first figure.

Therefore the index or characteristic of any logarithm is always 1 less than the number of figures in the integral part of the natural number.

Explanation and use of the Table of Logarithms.

170. THE table contains the logarithms of the natural numbers from 1 to 10000, to 6 places of figures. The logarithms of the first 100 numbers are printed with the indices. Thus the logarithm of 8 is 0.903090: and the log. of 97 is 1-986772. The indices or characteristics of the other logarithms are to be annexed according to the value of the integral part of the number, as in art. 169.

171. To find the logarithm of a number consisting of 3 figures suppose 123.

Look in the left-hand column for the number 123; then 089905 in the next or 2d. column is the decimal part of its logarithm; and as the number 123 consists of 3 integers, the index will be 2 (169); therefore 2.089905 is the logarithm of 123.

172. To find the logarithm of a number consisting of 4 figures: suppose 2157.

The two first figures of the logarithm of 215 are 33; then under 7 at the top of the table, and in the horizo tal row an swering to 215 is 3850 which are the right hand figures of the