Part 1

Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally printed in subscript. Letter subscripts are preceded by an underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective paragraphs.

(4) Macrons and breves above letters and dots below letters were not inserted.

(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek letters.

(6) The following typographical errors have been corrected:

ARTICLE LOGARITHM: "... tangents and secants for every minute of the quadrant to 10 places; these were obtained by calculating the logarithms of the natural sines, &c. given in the Thesaurus mathematicus of Pitiscus (1613)." 'these' amended from 'there'.

ARTICLE LOGARITHM: "The final step was made by John Newton in his Trigonometria Britannica (1658) ..." 'Trigonometria' amended from 'Trigononometria'.

ARTICLE LOGIC: "S is not P." 'P' amended from 'M'.

ARTICLE LOGIC: "... Jevons proceeded to confuse analytic deduction from consequence to ground with hypothetical deduction from ground to consequence under the common term 'inverse deduction.'" 'consequence' amended from 'conseguence'.

ARTICLE LOGIC: "But in induction the given particulars are the evidence by which we discover the universal, e.g. particular magnets attracting iron are the origin of an inference that all do ..." 'attracting' amended from 'attracing'.

ARTICLE LOGIC: "... but which is to other regards deductive as syllogism, is set up in contrast to syllogism 906 and enumeration alike." 'contrast' amended from 'constrast'

ARTICLE LOGIC: "67a 39-63." '-63'-63' amended from '-b 3'.

ARTICLE LOIRE: "Owing to the extreme irregularity of the river in different seasons these canals form the only certain navigable way." 'extreme' amended from 'exteme'.

ARTICLE LOLLARDS: "He summed up their doctrines under eleven heads: they condemn the having and using of images in the churches ..." added 'of'.

ARTICLE LOMBARDY: "G. T. Rivoira in Origini dell' Architettura Lombarda (2 vols. Rome, 1901-1907) ..." 'Architettura' amended from 'Architetturo'.

ARTICLE LONDONDERRY: "The scenery of the Roe valley, with the picturesque towns of Limavady and Dungiven, is also attractive, and the roads from the latter place to Draperstown and to Maghera ..." 'attractive' amended from 'atrractive'.

ARTICLE LONGOMONTANUS: "... Problemata duo Geometrica (1638) ..." 'Geometrica' amended from 'Goemetrica'.

ARTICLE LOOM: "... and, as appears from Grew (Mus. Reg. Soc. p. 69), it was formerly given to the little grebe or dabchick (P. fluviatilis or minor)." 'as' amended from 'ns'.

ENCYCLOPAEDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION

ELEVENTH EDITION

VOLUME XVI, SLICE VIII

Logarithm to Lord Advocate

ARTICLES IN THIS SLICE:

LOGARITHM LONG, JOHN DAVIS LOGAU, FRIEDRICH LONG BRANCH LOGIA LONGCLOTH LOGIC LONG EATON LOGOCYCLIC CURVE, STROPHOID LONGEVITY LOGOGRAPHI LONGFELLOW, HENRY WADSWORTH LOGOS LONG FIVES LOGOTHETE LONGFORD (county of Ireland) LOGROÑO (province of Spain) LONGFORD (town of Ireland) LOGROÑO (Spanish town) LONGHI, PIETRO LOGROSCINO, NICOLA LONGINUS, CASSIUS LOGWOOD LONG ISLAND LOHARU LONG ISLAND CITY LÖHE, JOHANN KONRAD WILHELM LONGITUDE LOHENGRIN LONGLEY, CHARLES THOMAS LOIN LONGMANS LOIRE (river of France) LONGOMONTANUS, CHRISTIAN SEVERIN LOIRE (department of France) LONGSTREET, JAMES LOIRE-INFÉRIEURE LONGTON LOIRET LONGUEVILLE LOIR-ET-CHER LONGUEVILLE, ANNE GENEVIÈVE LOISY, ALFRED FIRMIN LONGUS LOJA LONGWY LOKEREN LÖNNROT, ELIAS LOKOJA LONSDALE, EARLS OF LOLLARDS LONSDALE, WILLIAM LOLLIUS, MARCUS LONS-LE-SAUNIER LOLOS LOO LOMBARD LEAGUE LOOE LOMBARDO LOOM (water-birds) LOMBARDS LOOM (weaving machine) LOMBARDY LOÓN LOMBOK LOOP LOMBROSO, CESARE LOOSESTRIFE LOMÉNIE, ÉTIENNE CHARLES DE LOOT LOMOND, LOCH LOPES, FERNÃO LOMONÓSOV, MIKHAIL VASILIEVICH LOPEZ, CARLOS ANTONIO LOMZA (government of Russian) LOPEZ DE GÓMARA, FRANCISCO LOMZA (town of Russia) LOP-NOR LONAULI LOQUAT LONDON (Canada) LORAIN LONDON (capital of England) LORALAI LONDON CLAY LORCA LONDONDERRY, EARLS OF LORCH (Prussian town) LONDONDERRY, STEWART (VANE) LORCH (town of kingdom of Württemberg) LONDONDERRY, ROBERT STEWART LORD, JOHN LONDONDERRY (county of Ireland) LORD LONDONDERRY (town of Ireland) LORD ADVOCATE LONG, GEORGE

LOGARITHM (from Gr. [Greek: logos], word, ratio, and [Greek: arithmos], number), in mathematics, a word invented by John Napier to denote a particular class of function discovered by him, and which may be defined as follows: if a, x, m are any three quantities satisfying the equation a^x = m, then a is called the base, and x is said to be the logarithm of m to the base a. This relation between x, a, m, may be expressed also by the equation x = log(a) m.

_Properties._--The principal properties of logarithms are given by the equations

log(a) (mn) = log(a)m + log(a)n, log(a)(m/n) = log(a)m - log(a)n, log(a)m^(r) = r log(a)m, log(a)[root r]m = (1/r)log(a)m,

which may be readily deduced from the definition of a logarithm. It follows from these equations that the logarithm of the product of any number of quantities is equal to the sum of the logarithms of the quantities, that the logarithm of the quotient of two quantities is equal to the logarithm of the numerator diminished by the logarithm of the denominator, that the logarithm of the rth power of a quantity is equal to r times the logarithm of the quantity, and that the logarithm of the rth root of a quantity is equal to (1/r)th of the logarithm of the quantity.

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 to be 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. 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....

Thus in arithmetical calculations if the base is not expressed it is understood to be 10, so that log m denotes log10 m; but in analytical formulae it is understood to be e.

The logarithms to base 10 of the first twelve numbers to 7 places of decimals are

log 1 = 0.0000000 log 5 = 0.6989700 log 9 = 0.9542425 log 2 = 0.3010300 log 6 = 0.7781513 log 10 = 1.0000000 log 3 = 0.4771213 log 7 = 0.8450980 log 11 = 1.0413927 log 4 = 0.6020600 log 8 = 0.9030900 log 12 = 1.0791812

The meaning of these results is that

1 = 10^0, 2 = 10^(0.3010300), 3 = 10^(0.4771213), ... 10 = 10^1, 11 = 10^(1.0413927), 12 = 10^(1.0791812).

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; thus, for example,

log 2.5613 = 0.4084604, log 25.613 = 1.4084604, log 2561300 = 6.4084604, &c.

In the case of fractional numbers (i.e. numbers in which the integral part is 0) the mantissa is still kept positive, so that, for example, _ _ log .25613 = 1.4084604, log .0025613 = 3.4084604, &c.

the minus sign being usually written over the characteristic, and not before it, to indicate that the characteristic only, and not the whole expression, is negative; thus _ 1.4084604 stands for -1 + .4084604.

The fact that when the base is 10 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 10 as base. 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. Logarithms thus increased are frequently referred to for the sake of distinction as _tabular logarithms_, so that the tabular logarithm = the true logarithm + 10.

In tables of logarithms of numbers to base 10 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

log(a) b × log(b) a = 1, log(b) m = log(a) m × (1/log(a) b).

The second of these relations is an important one, as it shows that from a table of logarithms to base a, the corresponding table of logarithms to base b may be deduced by multiplying all the logarithms in the former by the constant multiplier 1/log(a)b, which is called the _modulus_ of the system whose base is b with respect to the system whose base is a.

The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is 10; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier 1/log(e)10, which is called the modulus of the common system of logarithms. The numerical value of this modulus is 0.43429 44819 03251 82765 11289 ..., and the value of its reciprocal, log^(e) 10 (by multiplication by which Briggian logarithms may be converted into Napierian logarithms) is 2.30258 50929 94045 68401 79914 ....

The quantity denoted by e is the series,

1 1 1 1 1 + --- + --- + ----- + ------- + ... 1 1·2 1·2·3 1·2·3·4

the numerical value of which is,

2.71828 18284 59045 23536 02874 ....

_The logarithmic Function._--The mathematical function log x or log(e) x is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or sin^{-1} x,&c., log x and e^(x) which are universally treated in analysis as known functions. The notation log x is generally employed in English and American works, but on the continent of Europe writers usually denote the function by lx or lg x. The logarithmic function is most naturally introduced into analysis by the equation _ / x dt | log x = ---, (x > 0). _/ 1 t

This equation defines log x for positive values of x; if x <= 0 the formula ceases to have any meaning. Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions. A connexion with the circular functions, however, appears later when the definition of log x is extended to complex values of x.

A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy = const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log b/a.

The following fundamental properties of log x are readily deducible from the definition

(i.) log xy = log x + log y.

(ii.) Limit of (x^(h)-1)/h = log x, when h is indefinitely diminished.

Either of these properties might be taken as itself the definition of log x.

There is no series for log x proceeding either by ascending or descending powers of x, but there is an expansion for log (1 + x), viz.

log (1 + x) = x - 1/2 x^2 + 1/3 x^3 - 1/4 x^4 + ...;

the series, however, is convergent for real values of x only when x lies between +1 and -1. 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 [oo] steadily increases from -[oo] towards +[oo]. 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 [oo] 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 [oo] as y increases from -[oo] towards +[oo]. As y tends towards [oo], exp y tends towards [oo] more rapidly than any power of y.

The exponential function possesses the properties

(i.) exp (x + y) = exp x × exp y.

d (ii.) --- exp x = exp x. dx

(iii.) exp x = 1 +x + x²/2! + x³/3! + ...

From (i.) and (ii.) it may be deduced that

exp x = (1 + 1 + 1/2! + 1/3! + ... )^x

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 e^x and the result

e^x = 1 + x + x²/2! + x³/3! ...

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 = [xi] + i[eta] _ / x dt log x = | --- _/ 1 t

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

log ([xi] + i[eta]) = log [root]([xi]² + [eta]²) + i([alpha] + 2n[pi]),

where [alpha] is the numerically least angle whose cosine and sine are [xi]/[root]([xi]² + [eta]²) and [eta]/[root]([xi]² + [eta]²), and n denotes any integer. Thus even when the argument is real log x has an infinite number of values; for putting [eta] = 0 and taking [xi] positive, in which case [alpha] = 0, we obtain for log [xi] the infinite system of values log [xi] + 2n[pi]i. 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 - ½x^2 + {1/3}x^3 - ¼x^4 + ...

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 ([xi] - i[eta]) = e^{[xi]}(cos [eta] + i sin [eta]).

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 = 10^7e^{-L/(l0^7)}, so that (ignoring the factors 10^7, the effect of which is to render sines and logarithms integral to 7 figures), the base is e^{-l}. 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

L = 10^7 log(e) 10^7 - 10^7 l or e^(l) = 10^7 e^(-L/10^7)

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 calculations he had made, and suggested to Napier the advantages that would result from the choice of 10 as a base, an improvement which he had explained in his lectures at Gresham College, and on which he had written to Napier. Napier said that he had already thought of the change, and pointed out a further improvement, viz., that the characteristics of numbers greater than unity should be positive and not negative, as suggested by Briggs. In 1616 Briggs again visited Napier and showed him the work he had accomplished, and, he says, he would gladly have paid him a third visit in 1617 had Napier's life been spared.

Briggs's _Logarithmorum chilias prima_, which contains the first published table of decimal or common logarithms, is only a small octavo tract of sixteen pages, and gives the logarithms of numbers from unity to 1000 to 14 places of decimals. It was published, probably privately, in 1617, after Napier's death,[3] and there is no author's name, place or date. The date of publication is, however, fixed as 1617 by a letter from Sir Henry Bourchier to Usher, dated December 6, 1617, containing the passage--"Our kind friend, Mr Briggs, hath lately published a supplement to the most excellent tables of logarithms, which I presume he has sent to you." Briggs's tract of 1617 is extremely rare, and has generally been ignored or incorrectly described. Hutton erroneously states that it contains the logarithms to 8 places, and his account has been followed by most writers. There is a copy in the British Museum.

Briggs continued to labour assiduously at the calculation of logarithms, and in 1624 published his _Arithmetica logarithmica_, a folio work containing the logarithms of the numbers from l to 20,000, and from 90,000 to 100,000 (and in some copies to 101,000) to 14 places of decimals. The table occupies 300 pages, and there is an introduction of 88 pages relating to the mode of calculation, and the applications of logarithms.