LOGARITHMS. The etymology of this word is Xiryiev cipleads, the amber of the ratios ; and the reason for the appellation will appear in the course of We article. We assume that the reader has the common knowledge of logarithms, and of the method of using them.
We have abandoned the intention of giving s view of the rise and progress of logarithms, for the following reasons. The subject is now one of such wide extent, when its theory and piactice are both incluffi d, that it would be like writing the history of a complete science to put together all that would be needed in an article pro fessing to show the past and present state of logarithmic algebra, as well as of logarithmic computation. If we were to confine oureelves to the latter only, the view of the subject would be too confined. And since the elements of the subject now usually given arc clothed in the most modern algebraical form, it would take considerable space to explain at length the process of the early writers in terms intelligible to those who are not conversant with their writings. We shall therefore devote the first part of this article to such explanations as will enable the student, fresh from modern books of algebra, to read the various histories which exist with facility : and we shall then point out how to deduce the principal fonnuhe connected with logarithms.
The early history of logarithms will be found at length in the preface to Dr. ilutton'a Tables ; in the History of Logarithms' com mined in the first volume of Dr. Huttotee Tracts, in. Delambre's Histoire des l'Aetronomie Moderne; vol. i. pp. 491-668. See also Testes, and, in the Moo: NAPIER, Humes, Greven, K mesa, MERCATOR, &c.
The idea of logarithms originally arose (in the mind of Napier) from the desire to make addition and subtraction supply the piece of multi plication and division. A table, in which are registered 1, a, a', &c., supplies this desideratum to a certain extent ; for since as multiplied by at gives a s+r, we find the product of the first by adding their exponents, and looking in the table for the (.r +y)th power. Thus for the act 1, 2, 4, 8, 16, kc., a table of logarithms is easily con.
strncted, a specimen of which is as follows :— Thus, to multiply 64 and 128, that is, to find the product of the sixth and seventh powers of 2, we must take the (6 + 7)th or 13th power, which, from the table, is 8192.
Such a table would be useless for general purposes, since it omits more numbers than it contains. But if we take a very little greater then unity. the powers will increase but slowly, and every whole number within given limits may be made either a power of a, or very near to a power of a. Suppose, for instance, that we wish for a table of logarithms which shall contain among its numbers either every whole number under a million or a fraction within h of every number under a million. Extract the square root of one million, the square root of that square root, and so on, until, say the rth root of one million has been extracted, and let this rth root he I + t. It is obvious that this extraction may be carried on until t Is mall as we please. Con sequently (1 e ie a million, and every lower power of 1 +1 is less than a million, so that ire standing for a million) no two consecutive powers (BM+ by so much as the ditffirence of 7R and se (1 + r), or by so much as mt. If then we proceed with the extraction until nit be lees than 1., we shall have t of the degree of smallness required : that is, since every whole number less than as lief between two powers of 1 + t, having exponents hiss than r, d fortiori every such whole number must be within h of some power of 1 This is in fact the first view which was taken of the teethed of constructing tables of logarithms ; and it must be remembered that Napier was not in possession of the modern way of expressing the powers of quantities. .On the methods of facilitating such enormous computations, and on the details which still remained for the first calculators after they had applied all the analysis which they had, we have not here to speak ; but we shall now show how the table may be formed by mere labour, and how the word logarithm arises.