Elements of Algebra

74 ELEMENTS SECT. I. 253. As the first part, or characteristic of a logarithm, is subject to no difficulty, it is seldom expressed in the Tables; the second part only is written, or the seven figures of the decimal fraction. There is a set of English Tables in which we find the logarithms of all numbers from $1$ to $100000$, and even those of greater numbers; for small additional Tables shew what is to be added to the logarithms, in proportion to the figures, which the proposed numbers have more than those in the Tables. We easily find, for example, the logarithm of $379456$, by means of that of $37945$ and the small Tables of which we speak *. 254. From what has been said, it will easily be perceived, how we are to obtain from the Tables the number corresponding to any logarithm which may occur. Thus, in multiplying the numbers $343$ and $2401$; since we must add * The English Tables spoken of in the text are those which were published by Sherwin in the beginning of the last century, and have been several times reprinted; they are likewise to be found in the tables of Gardener, which are commonly made use of by astronomers, and which have been reprinted at Avignon. With respect to these Tables it is proper to remark, that as they do not carry logarithms farther than seven places, independently of the characteristic, we cannot use them with perfect exactness except on numbers that do not exceed six digits; but when we employ the great Tables of Vlacq, which carry the logarithms as far as ten decimal places, we may, by taking the proportional parts, work, without error, upon numbers that have as many as nine digits. The reason of what we have said, and the method of employing these Tables in operations upon still greater numbers, is well explained in Saunderson's "Elements of Algebra," Book IX. Part II. It is farther to be observed, that these Tables only give the logarithms answering to given numbers, so that when we wish to get the numbers answering to given logarithms, it is seldom that we find in the Tables the precise logarithms that are given, and we are for the most part under the necessity of seeking for these numbers in an indirect way, by the method of interpolation. In order to supply this defect, another set of Tables was published at London in $1742$, under the title of "The Antilogarithmic Canon, &c. by James Dodson." He has arranged the decimals of logarithms from $0,0001$ to $1,0000$, and opposite to them, in order, the corresponding numbers carried as far as eleven places. He has likewise given the proportional parts necessary for determining the numbers, which answer to the intermediate logarithms that are not to be found in the Table. F. T.