This work contains the first announcement of logarithms to the world, the first table of logarithms and the first use of the name logarithm, which was invented by Napier.
LOGARITHM (from Gr.
The calculation of a logarithm can be performed by successive divisions; evolution requires special methods.
2 In this treatise (which was written before Napier had invented the name logarithm) logarithms are called "artificial numbers."
Algebraic Forms; Binomial; Combinatorial Analysis; Determin Ants; Equation; Continued Fraction; Function; Theory of groups; Logarithm; Number; Probability; Series.
In n = a P, a is the root or base, p is the index or logarithm, and n is the power or antilogarithm.
For the purpose of thus simplifying the operations of arithmetic, the base is taken to be Io, 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.
The logarithms to base io of the first twelve numbers to 7 places of decimals are log 1 =0.0000000 log 5 log 2 = 0.3010300 log 6 log 3 =0.477 121 3 log 7 log 4 =0.6020600 log 8 The meaning of these results is that The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa.
The fact that when the base is io the mantissa of the logarithm is independent of the position of the decimal point in the number affords the chief reason for the choice of io as base.
The explanation of this property of the base io 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 io 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 a, io denotes o, &c. Logarithms thus increased are frequently referred to for the sake of distinction as tabular logarithms, so that the tabular logarithm =the true logarithm -IIo.
In tables of logarithms of numbers to base io the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure.
It follows very simply from the definition of a logarithm that logo b X logo a, = 1, logo m =log.
The exponential function, exp x, may be defined as the inverse of the logarithm: thus x =exp y if y= log x.
Napier's logarithms are not the logarithms now termed Napierian or hyperbolic, that is to say, logarithms to the base e where e= 2.7182818 ...; the relation between N (a sine) and L its logarithm, as defined in the Canonis Descriptio, being N=10 7 e L/Ip7, so that (ignoring the factors re, the effect of which is to render sines and logarithms integral to 7 figures), the base is C".
If 1 denotes the logarithm to base e (that is, the so-called "Napierian " or hyperbolic logarithm) and L denotes, as above, " Napier's " logarithm, the connexion between 1 and L is expressed by L = r o 7 loge 10 7 - 10 7 / or e t = I 07e-L/Ia7 Napier's work (which will henceforth in this article be referred to as the Descriptio) immediately on its appearance in 1614 attracted the attention of perhaps the two most eminent English mathematicians then living - Edward Wright and Henry Briggs.
The " liber posthumus " was the Constructio (1619), in the preface to which Robert Napier states that he has added an appendix relating to another and more excellent species of logarithms, referred to by the inventor himself in the Rabdologia, and in which the logarithm of unity is o.
Briggs pointed out in his lectures at Gresham College that it would be more convenient that o should stand for the logarithm of the whole sine as in the Descriptio, but that the logarithm of the tenth part of the whole sine should be Io,000,000,000.
But he considered that the change ought to be so made that o should be the logarithm of unity and io,000,000,000 that of the whole sine, which.
The only other mathematician besides Napier who grasped the idea on which the use of logarithm depends and applied it to the construction of a table is Justus Byrgius (Jobst Biirgi), whose work Arithmetische and geometrische Progress-Tabulen ...
The case of both tables, the connexion between N a number and L its " logarithm " is N = (e 1)L (Napier), L = (I.
The name logarithm is derived from the words X6 7 wv hp426s, the number of the ratios, and the way of regarding a logarithm which justifies the name may be explained as follows.
He then by means of a simple proportion deduced that log (I 00000 00000 00000 I)=o 00000 00000 00000 0 434 2 944 81 90325 1804, so that, a quantity 1.00000 00000 00000 x (where x consists of not more than seventeen figures) having been obtained by repeated extraction of the square root of a given number, the logarithm of I 00000 00000 00000 x could then be found by multiplying x by 00000 00000 00000 04342 To find the logarithm of 2, Briggs raised it to the tenth power, viz.
He obtained the logarithm of this quantity, viz.
0.00000 00000 00000 0 73 18 5593 6 90623 9336, which multiplied by 2 47 gave 0.01029 995 66 39811 95 265277444, the logarithm of 1.024, true to 17 or 18 places.
(-) 5 + &c., in which the series is always convergent, so that the formula affords a method of deducing the logarithm of one number from that of another.
The former of these equations gives a convergent series for logep, and the latter a very convergent series by means of which the logarithm of any number may be deduced from the logarithm of the preceding number.
By means of these tables and of a factor table we may very readily obtain the Briggian logarithm of a number to 61 or a less number of places or of its hyperbolic logarithm to 48 or a less number of places in the following manner.
Suppose the hyperbolic logarithm of the prime number 43,867 required.
The logarithm is then obtained by use of the formula d l d2 l d3 2 log e (x+d) = log e x-f- - x2+3 x3 - &c., in which of course the object is to render dlx as small as possible.
If the logarithm required is Briggian, the value of the series is to be multiplied by M.
An application to the hyperbolic logarithm of is given by Burckhardt in the introduction to his Table des diviseurs for the second million.
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 I - i r n, where n is one of the nine digits, and making use of subsidiary tables of logarithms of factors of this form.
All that is required therefore in order to obtain the logarithm of any number is a table of logarithms, to the required number of places, of n, 9n, 99 n, 999 n, &c., for n= I, 2, 3,
Taking as an example the calculation of the Briggian logarithm of the number 43,867, whose hyperbolic logarithm has been calculated above, we multiply it by 3, giving 131,601, and find by Gray's process that the factors of 1.31601 are (I) 1.316 (5) I.
Reference should also be made to Hoppe's Tafeln zur dreissigstelligen logarithmischen Rechnung (Leipzig, 1876), which give in a somewhat modified form a table of the hyperbolic logarithm of + Irn.
In the following list, which contains a few typical examples, the different formulae are arranged to give the logarithm of the saturation-pressure p in terms of the absolute temperature 0.
EJi= number whose common logarithm is 2.7288fa 7
A star is said to rise one unit in magnitude when the logarithm of its brightness diminishes by 0.4.
If we know n and N, then p is the logarithm of N to base n.
If N = a p, then p is called the logarithm of N to the base a, and is written logy N.
As the table of antilogarithms is formed by successive multiplications, so the logarithm of any given number is in theory found by successive divisions.
Thus, to find the logarithm of a number to base 2, the number being greater than i, we first divide repeatedly by 2 until we get a number between I and 2; then divide repeatedly by 10 12 until we get a number between I and 10 y2; then divide repeatedly by ioo v 2; and so on.
/ 2) 4, it may be written 23.5'4, and its logarithm to base 2 is 3.574.
The logarithm is also a function of frequent occurrence in analysis, being regarded as a known and recognized function like sin x or tan x; but in mathematical investigations the base generally employed is not 10, but a certain quantity usually denoted by the letter e, of value 2.71828 18284 ...
For example, suppose the logarithm of 543839 required to twelve places.