OF LOGARITHMS.

216. Logarithms are numbers in arithmetical progression, which answer, term for term, to a like series of numbers in geometrical progression. If we take, for example, the following geometrical and arithmetical progressions

2:4:8: 16: 32: 64; 128: 256, etc.

<3 5 7 9.11.13. 15. 17, etc.

Each term of the lower series is called the logarithm of the term which is in the same place in the upper series.

217. The same number may then have an infinity of different logarithms, since to the same geometrical progression we can make correspond an infinity of different arithmetical progressions. As we shall here consider logarithms only with regard to the use which we can make of them in numerical calculations, we will not stay to consider the different geometrical and arithmetical progressions, which we might compare to each other; we shall directly proceed to those which have been considered in the formation of the tables of logarithms.

218. The decuple progression is chosen for the geometrical progression, and the natural series of numbers for the arithmetical progression; that is to say, the two following progressions

1:10:100: 1000 10000 100000 1000000

219. Thus, it is easy to perceive that the logarithm of a unit followed by any number of ciphers has always as many units as there are ciphers on the right of the unit.

We shall not here teach the method which was followed to find the intermediate logarithms of the decuple progression; it depends upon principles that we cannot here explain; but we shall show their formation in a way, which, truly, would not be the most expeditious for calculating these logarithms, but which suffices as well for the understanding of this formation as to give a reason for the uses to which we apply these artificial numbers.

220. According to the definition that we have given of logarithms, we see that to have the logarithm of any number,

of 3, for example, this number must make a part of the fundamental geometrical progression. Now, although we do not see that 3 can make a part of the geometrical progression 1: 10: 100, etc.; we see, however, that if, between 1 and 10, we should insert an exceeding great number of geometrical means, (214) as we should then ascend from 1 to 10 by degrees, so much the more compact as the number of these means should be greater, one of two things would happen; either that some one of these means would be found precisely the number 3, or that there should be found two successive numbers, between which the number 3 would be comprised, and the difference of each of which from 3 would be less in proportion, as the number of the means inserted should be greater.

This established, if we should insert, in like manner, between 0 and 1 as many arithmetical means as we have inserted geometrical means between 1 and 10, each term of the geometrical progression having for logarithm the corresponding term of the arithmetical progression, we should take in this, for the logarithm of 3, the number which should here be found in the same place as 3 is found in the geometrical progression; or, if 3 were not exactly one of the terms of this, we should take, in the arithmetical progression, the term answering to that in the geometrical progression, which approaches the nearest to the number 3.

It is thus that we should proceed, in effect, if we had not more expeditious means. At any rate, this is the amount of the calculation of logarithms.

221. We must then represent that having inserted 1000000 geometrical means between 1 and 10, a like number between 10 and 100, a like number between 100 and 1000, etc. we have also inserted the same number of arithmetical means between 0 and 1, the same number between 1 and 2, the same number between 2 and 3 etc.; that having ranged all the first in the same line, and all the second underneath, we have sought in the first the number nearest to 2, and have taken in the lower series the corresponding number; that we have likewise sought in the first the number nearest to 3, and that

we have taken in the lower series the corresponding number; that we have done the same successively, for the numbers 4, 5, 6, etc.; that, finally, having transported into the same column, as we see in the table, the numbers 1, 2, 3, 4, 5, etc. we have written in the adjacent column the terms of the arithmetical progression corresponding to these, or at least, those which approach the nearest to them; then we shall have the idea of the formation of logarithms, and of their disposition in the ordinary tables.

The table at the end of the book contains only the decimal part of the logarithm of each number as far as 10000, except the part from 1 to 100.

222. Let us remark, with regard to this Table, that the first figure which should stand on the left of each logarithm, that is to say, the figure on the left of the comma, is called the index, because it indicates in what decade the number is comprised to which this logarithm belongs; for example, if the number be between 1000 and 10000, because the logarithm of 1000 is 3, and that of 10000 being 4, any number between 1000 and 10000 can only have for its logarithm 3 and a fraction; the logarithm has then 3 for its index, and the other figures express this fraction reduced to decimals.

From what we have here said, it is plain that the index of a logarithm is always one less than the number of figures contained in the whole number.

Hence, the index of 116 is 2, being one less than the number of figures contained in 116; also, the index of 126,3 is 2, for although this number contains four figures, yet there are only three belonging to the whole number, the figure 3 being a decimal.

To find the logarithm of a number consisting of three figures, I seek this number in the left hand column of the table, and opposite to it, in the column marked Q at the top or bottom, I find the decimal part of its logarithm, to which I prefix the index as signified above: thus the logarithm of 276 is 2,440909.

To find the logarithm of a number consisting of four figures, seek the first three figures in the left hand column; opposite to these, in the column marked with the fourth figure

at the top or bottom, is the decimal part of the logarithm required, to which prefix the index as before: thus the log. of 1263 is 3,101403; that of 126,3 is 2,101403; and that of 12,63 is 1,101403.

PROPERTIES OF LOGARITHMS.

223. As we here only treat of logarithms such as they are in the ordinary tables, the properties we are about to explain relate only to the geometrical progressions which have unity for the first term, and the arithmetical progressions which have a cipher for the first term.

Let us then compare, term for term, any geometrical progression, having a unit for the first term, with any arithmetical progression having a cipher for the first term; for example, the two following progressions:

1:39:27: 81: 243: 729: 2187: 6561, etc.

0.4 8:12. 16. 20. 24. 28

32, etc.

It follows from the nature and perfect correspondence of these two progressions, that as many times as the ratio of the first is factor in any one of the terms of that progression, so many times the ratio of the second is contained in the corresponding term of the second; for example, in the term 2187, the ratio 3 is 7 times factor, and in the term 28, the ratio 4 is contained seven times.

In effect, according to what has been said (206 and 212,) the ratio is factor in any term whatever of the first as many times as there are terms preceding it; and in the second, any term whatever is, composed of as many times the ratio as there are terms preceding it. Now there is the same number of terms on the one hand and on the other.

Let us thence conclude, that any term of the geometrical progression will always have for its correspondent in the arithmetical progression, a term that will contain the ratio of this as many times as the ratio of the first is factor in the term first mentioned.

224. Therefore, if we multiply the one by the other two terms of the geometrical progression, and if we add at the same time the two corresponding terms of the arithmetical progression,