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.
The exponential function, exp x, may be defined as the inverse of the logarithm: thus x = exp y if y = log x. It is positive for all values of y and increases steadily from 0 toward ∞ as y increases from -∞ towards +∞. As y tends towards ∞, exp y tends towards ∞ more rapidly than any power of y.
The exponential function possesses the properties
| (i.) | exp (x + y) = exp x × exp y. |
| (ii.) | (d/dx) exp x = exp x. |
| (iii.) | exp x = 1 + x + x2/2! + x3/3! + ... |
From (i.) and (ii.) it may be deduced that
where the right-hand side denotes the positive xth power of the number 1 + 1 + 1/2! + 1/3! + ... usually denoted by e. It is customary, therefore, to denote the exponential function by ex and the result
is known as the exponential theorem.
The definitions of the logarithmic and exponential functions may be extended to complex values of x. Thus if x = ξ + iη
where the path of integration in the plane of the complex variable t is any curve which does not pass through the origin; but now log x is not a uniform function, that is to say, if x describes a closed curve it does not follow that log x also describes a closed curve: in fact we have
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
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
An alternative method of developing the theory of the exponential function is to start from the definition
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/(107), 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
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 former translated the work into English; the latter was concerned with Napier in the change of the logarithms from those originally invented to decimal or common logarithms, and it is to him that the original calculation of the logarithmic tables now in use is mainly due. Both Napier and Wright died soon after the publication of the Descriptio, the date of Wright’s death being 1615 and that of Napier 1617, but Briggs lived until 1631. Edward Wright, who was a fellow of Caius College, Cambridge, occupies a conspicuous place in the history of navigation. In 1599 he published Certaine errors in Navigation detected and corrected, and he was the author of other works; to him also is chiefly due the invention of the method known as Mercator’s sailing. He at once saw the value of logarithms as an aid to navigation, and lost no time in preparing a translation, which he submitted to Napier himself. The preface to Wright’s edition consists of a translation of the preface to the Descriptio, together with the addition of the following sentences written by Napier himself: “But now some of our countreymen in this Island well affected to these studies, and the more publique good, procured a most learned Mathematician to translate the same into our vulgar English tongue, who after he had finished it, sent the Coppy of it to me, to bee seene and considered on by myselfe. I having most willingly and gladly done the same, finde it to bee most exact and precisely conformable to my minde and the originall. Therefore it may please you who are inclined to these studies, to receive it from me and the Translator, with as much good will as we recommend it unto you.” There is a short “preface to the reader” by Briggs, and a description of a triangular diagram invented by Wright for finding the proportional parts. The table is printed to one figure less than in the Descriptio. Edward Wright died, as has been mentioned, in 1615, and his son, Samuel Wright, in the preface states that his father “gave much commendation of this work (and often in my hearing) as of very great use to mariners”; and with respect to the translation he says that “shortly after he had it returned out of Scotland, it pleased God to call him away afore he could publish it.” The translation was published in 1616. It was also reissued with a new title-page in 1618.
Henry Briggs, then professor of geometry at Gresham College, London, and afterwards Savilian professor of geometry at Oxford, welcomed the Descriptio with enthusiasm. In a letter to Archbishop Usher, dated Gresham House, March 10, 1615, he wrote, “Napper, lord of Markinston, hath set my head and hands a work with his new and admirable logarithms. I hope to see him this summer, if it please God, for I never saw book which pleased me better, or made me more wonder.[1] I purpose to discourse with him concerning eclipses, for what is there which we may not hope for at his hands,” and he also states “that he was wholly taken up and employed about the noble invention of logarithms lately discovered.” Briggs accordingly visited Napier in 1615, and stayed with him a whole month.[2] He brought with him some
- ↑ Dr Thomas Smith thus describes the ardour with which Briggs studied the Descriptio: “Hunc in deliciis habuit, in sinu, in manibus, in pectore gestavit, oculisque avidissimis, et mente attentissima, iterum iterumque perlegit,...” Vitae quorundam eruditissimorum et illustrium virorum (London, 1707).
- ↑ William Lilly’s account of the meeting of Napier and Briggs at Merchiston is quoted in the article Napier.
