Part 261

For instance, the series 1 . 3 . 9 . 27 . 81 . 243, &c. is in geometrical progression, since each term contains that which precedes it the same number of times, which is 3.

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.

For instance, by adding the terms 2 and 3 which answers to 100 and 1000, I have 5, which answers to 100000; whence I conclude that the product of 100 by 1000 is 100000, which in fact it is.

It is always easy to ascertain the logarithm of unity followed by any given number of ciphers; for such logarithm will invariably be expressed by as many units as there may be ciphers in the given number. In order to extend this practice to the formation of intermediate logarithms, it may be conceived, that, although any given number, for instance 3, may not apparently form any part of the geometrical progression 1 . 10 . 100, yet if we were to insert a great number of geometrical means, suppose 1,000,000, between the two first terms, we should either find the number 3 itself, as one of such means, or a number of very near approximation to it. The intermediate terms between 10 . 100 and between 100 . 1000 might be found in like manner, as well as a corresponding number of intermediate terms, in arithmetical proportion, between 0 and 1, and between 1 and 2, 2 and 3, &c. The whole of the geometrical terms being then arranged upon the same line, and the whole of the arithmetical terms upon another line, under the former, it is obvious that the lower series would contain units, or decimal fractions, corresponding with the numbers in the upper series, or, in other words, the logarithmic relation of the two series would be complete and exactly similar to that of the fundamental progressions.

It is thus, that, in the tables most in use, the number of decimal places in the logarithmic quantities is 7, than which, however, many more are used by men of science with a view to the attainment of a corresponding degree of precision. Nevertheless, in certain tables which were made a few years ago for the use of accompting houses, the number of decimal places is reduced to 5, and the rather, as a greater degree of precision is not necessary in those calculations of business which do not require more than approximate results.

It should be remarked, in respect to the tables of logarithms, that the first figure to the left of each logarithm is called the characteristic; since it is that figure which denotes the class of the geometrical progression which comprises the number to which the logarithm relates. For instance, if the characteristic of a number be 2, I know that it relates to the second class, or the hundreds, the logarithm of 100 being 2; and, as that of 1000 is 3, every number from 100 to 999 inclusively, cannot have any other logarithm than 2 and a decimal fraction.

Thus, the characteristic of a logarithm is a number corresponding to the natural numbers, namely, 1 to 10, 2 to 100, 3 to 1000, 4 to 10000, &c. &c. The characteristic of the logarithm of any number under 10 is 0.

It happens by this progressive correspondence, that a number being 10 times, 100 times, or 1000 times greater than another number, has the same logarithm as the lesser number, as far as relates to the decimal fractions of each. The characteristic alone is susceptible of variation, as will be seen by the logarithms of the following numbers:

Numbers Logarithms 3 0,47712 30 1,47712 300 2,47712 3000 3,47712

the characteristics of which are separated by a comma, being 0, 1, 2, 3.

It is this property by which the extraction of logarithms is facilitated, since, if we know the logarithm of the number 30, and are desirous of finding that of 300, of 3000, or of 3, it is requisite merely to add to the characteristic of 30, or to deduct from it, as many units as there may be more or less ciphers in the number whose logarithm is sought.

LOGEMENT, Fr. means generally any place occupied by military men, for the time being, whether they be quartered upon the inhabitants of a town, or be distributed in barracks. When applied to soldiers that have taken the field, it is comprehended under the several heads of huts, tents, &c.

LOGIS, Fr. Quarters.

Marquer les LOGIS, Fr. To mark the officer’s rooms according to their respective ranks.

LOGEMENT d’une attaque, Fr. See Lodgment in FORTIFICATION.

LONG BOAT, the largest boat belonging to a ship: it serves to bring goods, provisions, &c. to or from the ship, to land men, to weigh the anchor, &c.

Le long de la Côte, Fr. Along the coast.

Tout du long de l’année, Fr. All the year round.

Long à la guerre, Fr. An expression used in the French service.