log (ξ + iη) = log √(ξ2 + η2) + i(α + 2nπ),
where α is the numerically least angle whose cosine and sine are ξ/√(ξ2 + η2) and η/√(ξ2 + η2), and n denotes any integer. Thus even when the argument is real log x has an infinite number of values; for putting η = 0 and taking ξ positive, in which case α = 0, we obtain for log ξ the infinite system of values log ξ + 2nπi. 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. Also
exp (ξ − iη) = eξ (cos η + i sin η).
An alternative method of developing the theory of the exponential function is to start from the definition
exp x = 1 + x + x2/2! + x3/3! + ...,
the series on the right-hand being convergent for all values of x and therefore defining an analytical function of x which is uniform and regular all over the plane.
Invention and Early History of Logarithms.—The invention of logarithms has been accorded to John Napier, baron of Merchiston in Scotland, with a unanimity which is rare with regard to important scientific discoveries: in fact, with the exception of the tables of Justus Byrgius, which will be referred to further on, there seems to have been no other mathematician of the time whose mind had conceived the principle on which logarithms depend, and no partial anticipations of the discovery are met with in previous writers.
The first announcement of the invention was made in Napier’s Mirifici Logarithmorum Canonis Descriptio ... (Edinburgh, 1614). The work is a small quarto containing fifty-seven pages of explanatory matter and a table of ninety pages (see [Napier, John]). The nature of logarithms is explained by reference to the motion of points in a straight line, and the principle upon which they are based is that of the correspondence of a geometrical and an arithmetical series of numbers. The table gives the logarithms of sines for every minute of seven figures; it is arranged semi-quadrantally, so that the differentiae, which are the differences of the two logarithms in the same line, are the logarithms of the tangents. 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 = 107 e−L/(l07), so that (ignoring the factors 107, the effect of which is to render sines and logarithms integral to 7 figures), the base is e−1. Napier’s logarithms decrease as the sines increase. If l 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 l and L is expressed by