The series 243 . 81 . 27 . 9 . 3 . 1 is also in geometrical progression, each of the terms being contained by the preceding the same number of times.
Of the formation of logarithms.—Logarithms are numbers in arithmetical progression, corresponding, term by term, with a similar series of numbers in geometrical progression. If, for instance, we have a geometrical series and an arithmetical series as follows,
| 1 | . | 3 | . | 9 | . | 27 | . | 81 | . | 243 |
| 1 | . | 3 | . | 5 | . | 7 | . | 9 | . | 11 |
we shall call each term of the lower series the logarithm of the corresponding term in the upper series.
Any given quantity may therefore have an infinite number of different logarithms, since the same geometrical progression may be made to correspond with an infinite diversity of series in arithmetical progression.
In the formation, however, of tables of logarithms, it has been found convenient to adopt a ten-fold progression, as the geometrical progression, and the series of natural numbers as the arithmetical progression. It will be remarked, that, in respect to the latter, the ratio, or common measure of increase, is always unity, while the former has the advantage of being adapted to the mode of notation which is in universal use. The following, therefore, are the progressions chosen:
| 1 | . | 10 | . | 100 | . | 1000 | . | 10000 | . | 100000 | . | 1000000 |
| 0 | . | 1 | . | 2 | . | 3 | . | 4 | . | 5 | . | 6 |
It follows from the nature and correspondence of these progressions, that, as often as the ratio of the former may have been used as a factor in the formation of any one of the terms of that progression, so often will the ratio of the second progression have been added to form the corresponding term of this identical second progression. For instance, in the term 10000, the ratio 10 is 4 times a factor, and in the term 4 the ratio is added 4 times.
If any two terms of the geometrical progression be intermultiplied, and if the corresponding terms of the arithmetical progression be added, the product and the sum will be two terms which will correspond with each other in the same progressions.
Upon this principle it is, that, by the simple addition of any two or more terms of the arithmetical progression, we can ascertain the product of the corresponding terms of the geometrical progression.