Logarithm

“ The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa. When the base is 10, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa ”

“ The error, however, although continually increasing, yet retains always the same ratio to the logarithm, except for a very small disturbing element to be afterwards referred to, so that the true logarithm will always be very nearly equal to the logarithm found by Napier’s method of computation less a six hundred billionth part. ”

“ Logarithms were originally invented for the sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and division may be replaced by those of addition and subtraction, and the operations of raising to powers and extraction of roots by those of multiplication and division. For the purpose of thus simplifying the operations of arithmetic, the base is taken equal to 10, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated. ”

“ For in the same manner as numbers are considered as powers of 10, so their indices, whether integral or decimal, being, in fact, numbers, may again be regarded as powers of 10; and their secondary indices, or logarithmic logarithms, may be laid down as logometric intervals, to which other logometric intervals may be added or subtracted mechanically, so as to present to view any multiples or aliquot part of a logarithm, and consequently any powers whatever of the number to which that logarithm is index ”

“ Other formulae which are deducible from this equation are given in the portion of this article relating to the calculation of logarithms.
The function log x as x increases from 0 towards ∞ steadily increases from −∞ towards +∞. It has the important property that it tends to infinity with x, but more slowly than any power of x, i.e. that x−m log x tends to zero as x tends to ∞ for every positive value of m however small. ”

“ Difficulties are indeed involved in the application of the calculus of probabilities to the exact sciences. Why are the decimals of a table of logarithms, why are those of the number π distributed in accordance with the laws of chance? Elsewhere I have already studied the question in so far as it concerns [Pg 412] logarithms, and there it is easy. It is clear that a slight difference of argument will give a slight difference of logarithm, but a great difference in the sixth decimal of the logarithm. Always we find again the same criterion. ”

“ The only other logarithmic canon to every second that has been published forms the second volume of Shortrede’s Logarithmic Tables (1849) . In 1784 the French government decided that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant. ”

“ In his twentieth year (1826) he engaged in researches respecting exponential functions, which conducted him to important results. They were printed in the ‘Philosophical Transactions’ for 1829 under the title ‘An Attempt to Rectify the Inaccuracy of some Logarithmic Formulæ.’ Of these results one of the principal is the discovery of the existence of two arbitrary and independent integers in the complete expression of an imaginary logarithm. ”

“ It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation
log (1 + x) = x − 1/2x2 + 1/3x3 − 1/4x4 + ... is true only when the analytical modulus of x is less than unity. The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x. ”

“ The best general method of calculating logarithms consists, in its simplest form, in resolving the number whose logarithm is required into factors of the form 1 - 14n, where n is one of the nine digits, and making use of subsidiary tables of logarithms of factors of this form. ”

“ A list of the most convenient factors and their logarithms is given in a separate table. The principle of multiplying by a factor which is subsequently cancelled by subtracting its logarithm is one that is frequently employed in the calculation of logarithms, but the peculiarity of the present work is that it forms part of the process of using the table. ”

“ The explanation of this property of the base 10 is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact. It should be mentioned that in most tables of trigonometrical functions, the number 10 is added to all the logarithms in the table in order to avoid the use of negative characteristics, so that the characteristic 9 denotes in reality 1, 8 denotes 2, 10 denotes 0, &c. ”

“ From the above example we see that the error in the logarithm found by Napier’s method amounts only to unity in the 15th place, so that his method of computation clearly gives accurate results far in excess of his requirements, But it is easy to show that Napier’s method may be adapted to meet any requirements of accuracy. In sec. 60, Napier, in suggesting the construction of a table of logarithms to a greater number of places, proposes to take 100000000 as radius. ”

“ For as in statics, from weights of 1, of 2, of 4, of 8, and of other like numbers of pounds in the same proportion, every number of pounds weight, which to us now are Logarithms, may be formed by addition; so, from the proportionals V, T, S, R, &c., which correspond to them, and from others also to be formed in duplicate ratio, the proportionals corresponding to every proposed Logarithm may be formed by corresponding multiplication of them among themselves, as experience will show. ”

“ Vlacq rendered assistance in the publication of this work, and the privilege is made out to him.
The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier’s Descriptio. ”

“ The calculation of a logarithm can be performed by successive divisions; evolution requires special methods.
The above definitions of logarithms, &c., relate to cases in which n and p are whole numbers, and are generalized later.
44. Law of Indices.—If we multiply np by nq, we multiply the product of p n’s by the product of q n’s, and the result is therefore np+q. Similarly, if we divide np by nq, where q is less than p, the result is np−q. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa. ”

“ Just at this time, the invention of logarithms, by a Scotch laird, John Napier of Merchiston, near Edinburgh, was becoming known in the then comparatively restricted scientific world. Logarithms are prepared tables of numbers, by which complex problems in trigonometry, and the tedious extraction of roots, can be performed by the simpler rules of arithmetic. To the well-educated, they save much time and labour; in the art of navigation, they enable the mariner who may be unskilled in mathematics, to work out the most intricate calculations. ”

“ It is important to notice that in the Constructio logarithms are called artificial numbers; and Robert Napier states that the work was composed several years (aliquot annos) before Napier had invented the name logarithm. The Constructio therefore may have been written a good many years previous to the publication of the Descriptio in 1614. ”

“ It has now, therefore, been sufficiently shown that there are Logarithms, what they are, and of what use they are: for by their help without the trouble of multiplication, division, or extraction of roots we have both demonstrated clearly and shown by examples in both kinds of Trigonometry that the arithmetical solution of every Geometrical question may be very readily obtained. ”

“ The earlier methods proposed were, like those of Briggs, purely arithmetical, and for a long time logarithms were regarded from the point of view indicated by their name, that is to say, as depending on the theory of compounded ratios. The introduction of infinite series into mathematics effected a great change in the modes of calculation and the treatment of the subject. Besides Napier and Briggs, special reference should be made to Kepler (Chilias, 1624) and Mercator (Logarithmotechnia, 1668) , whose methods were arithmetical, and to Newton, Gregory, Halley and Cotes, who employed series. ”

“ Since the scale of the instrument presents to view the proportion of logarithms to each other, while the slider represents the ratios of their respective numbers, it becomes a means of illustrating many points relative to the general theory of logarithms, whether to exhibit a series of logarithms formed according to any particular modulus, or by an inverted position of the slider to find the moduli of all different systems, and accordingly in that, for instance, in which the modulus is equal to the basis. ”