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. 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 log 10 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 Ό000000 log 5 = 0’6989700 log 9 = 0’9542425
log 2 = 0’3010300 log 6 = 07781513 log 10 = 1'0000000
log 3 = 0’4771213 log 7 = 0∙8450980 log 11 = 1 Ό413927
log 4 = 0’6020600 log 8 ≈ 0'9030900 log 12 = 1’0791812
The meaning of these results is that l = 10° 2 = 10°∙ 3oloaβ ° 3≈10° ' 4771213 · · ·
ιo=ιo 1 , ’ 11-IO 1 · 0413927 ’, 12=IO 1,0791818 ’
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 = Γ4084604, 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=T∙4084604, log ∙0025613 = 3∙40846β4, &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 l∙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 α b × log i a = 1, log⅛ m = log β m × —1— .
10g<ι o
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 —-—, which is called the modulus of log a b 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— which is called the modulus of 1 loge 10
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 i 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 ι i , 1 ι 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 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, arc cos x, &c., e x , and log x, which are universally treated in analysis as known functions. It is the inverse of the exponential function e, x , the theory of which may be regarded as including that of the circular functions, since sin x = -{d x -e~ ix ), cos x= ⅛(e i * +e~ ix ). 2λ
There is no series for log x proceeding either by ascending or descending powers of x, but there is an expansion for log (1+ae), viz. :—
log (l+x)=x-^ 2 +⅛≈ 3 -^z 4 + &c.; 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.
We have also the fundamental formulae—
(i.) Limit of X > when h is indefinitely diminished, = log x;
(h∙) log x+ const.
Either of these results might be regarded as the definition of log x; they may be readily connected with one another, for we have in general
∕∙ Q**∙+l JXl l dx = —— + const.; ∙ ’ n + 1
but if n= -1, this formula no longer gives a result. Putting, however, n = -1 + h, where h is indefinitely small, we have ∕ dx — 1
— = const. =—+ const. = log x + const, by (i.).
The result (ii.) establishes a relation, which is of historical interest, between the logarithmic function and the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy = const., we see at once 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 α and b from their point of intersection is proportional to log A .
(X>
The function log x is not a uniform function, that is to say, if x denotes a complex variable of the form a + ib, and if complex quantities are represented in the usual manner by points in a plane, then it does not follow that if x describes a closed curve log x also describes a closed curve; in fact we have log (a + iδ) = log √(α 2 + δ 2 ) + i( a +2nιr), where α is a determinate angle, and n denotes any integer. Thus, even when the argument is real, log x has an infinite number of values; for, putting 5 = 0 and taking a positive, in which case α=0, we obtain for log a the infinite system of values log a + 2nπi. It follows from this property of the function that we cannot have’for log x a series which snail be convergent for all values of x, as is the case with sin x, cos x, and e x ; as such a series could only represent a uniform function, and in fact the equation log (l+x)≈x-⅛x 2 +⅛x 3 -⅛x i + &c., is true only when the analytical modulus of x is less than unity.
The notation log x is generally employed in English works, but Continental writers usually denote the function by lx or lgar.
History,— The invention of logarithms has been accorded to John Napier, baron of Mcrchiston, in Scotland, with a unanimity which is rare with regard to important scientific discoveries. The first announcement was made in Napier’s Mirifici logarithmorum canonis descriptio (Edinburgh, 1614), which contains an account of the nature of logarithms, and a table giving natural sines and their logarithms for every minute of the quadrant to seven or eight figures. These logarithms are not what would now be called Napierian or hyperbolic logarithms (f.e., logarithms to the base e), though closely connected with them, the relation between the two being
_ L
e ’ =10 7 e 10’ j or L = 10 7 log e 10 7 -10 7 Z, where l denotes the logarithm to base e and L denotes Napier’s logarithm. The relation between N (a sine) and L its Napierian logarithm is therefore
_ L N = 10 7 e ÏÔ», and the logarithms decrease as the sines increase. Napier died in 1617, and his posthumous work Mirifici logarithmorum canonis constructio, explaining the mode of construction of the table, appeared in 1619, edited by his son.
Henry Briggs, then professor of geometry at Gresham College, London, and afterwards Savilian professor of geometry at Oxford, admired the Canon mirificus so much that he resolved to visit Napier. In a letter to Ussher he writes, “Naper, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder.” Briggs accordingly visited Napier in 1615, and stayed with him a whole month. 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, having explained them previously in his lectures at Gresham College, and written to Napier on the subject. Napier said that he had already thought of the change, and pointed out a slight 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 ρrimα was published, probably privately, in 1617, after Napier’s death, as in the short preface he states that why his logarithms are different from those introduced by Napier “sperandum, ejus librum posthumum abunde nobis propediem satisfacturum.” The liber posthumus was the Canonis constructio already mentioned. This wprk of Briggs’s, 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. 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 Ussher, 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 1 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 of the tables, and the applications of logarithms.
There was thus left a gap between 20,000 and 90,000, which was filled up by Adrian Vlacq, who published at Gouda, in Holland, in 1628, a table containing the logarithms of the numbers from unity to 100,000 to 10 places of decimals. Having calculated 70,000 logarithms and copied only 30,000, Vlacq would have been quite entitled to have called his a new work. He designates i⅛ however, only a second edition of Briggs’s Arithmetica logarithmica, the title running Arithmetica logarithmica sive logarith-mοrιι∕m chiliades centum, . . . Editio secunda aucta per Adrianum Vlacq, Goudaηum. This table of Vlacq’s was published, with an English explanation prefixed, at London in 1631 under the title Logarithmicall Arithmetike . . . London, printed by George Miller, 1631. There are also copies with a French title page and introduction (Gouda, 1628).
Briggs had himself been engaged in filling up the gap, and in a letter to Pell, written after the publication of Vlacq’s work, and dated October 25, 1628, he says :—
“My desire was to have those chiliades that are wantinge betwixt 20 and 90 calculated and printed, and I had done them all almost by my seife, and by some frendes whom my rules had sufficiently informed, and by agreement the busines was conveniently parted amongst us; but I am eased of that charge and care by one Adrian Vlacque, an Hollander, who hathe done all the whole hundred chiliades and printed them in Latin, Dutche, and Frenche, 1000 bookes in these 3 languages, and hathe sould them almost all. But he hathe cutt off 4 of my figures throughout; and hathe left out my dedication, and to the reader, and two chapters the 12 and 13, in the rest he hath not varied from me at all.”
The original calculation of the logarithms of numbers from unity to 101,000 was thus performed by Briggs and Vlacq between 1615 and 1628. Vlacq’s table is that from which all the hundreds of tables of logarithms that have subsequently appeared have been derived. It contains of course many errors, which have gradually been discovered and corrected in the course of the two hundred and fifty years that have elapsed, but no fresh calculation has been published. The only exception is Mr Sang’s table (1871), part of which was the result of an original calculation.
The first calculation or publication of Briggian or common logarithms of trigonometrical functions was made in 1620 by Gunter, who was Briggs’s colleague as professor of astronomy in Gresham College. The title of Gunter’s oo , which is very scarce, is Canon triangulorum, and it contains logarithmic sines and tangents for every minute of the quadrant to 7 places of decimals.
The next publication was due to Vlacq, who appended to his logarithms of numbers in the Arithmetica logarithmica of 1628 a table giving log sines, 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).
During the last years of his life Briggs devoted himself to the calculation of logarithmic sines, &c., and at the time of his death in 1631 he had all but completed a logarithmic canon to every hundredth of a degree. This work was published by Vlacq at his own expense at Gouda in 1633, under the title Trigonometria Britannica. It contains log sines (to 14 places) and tangents (to 10 places), besides natural sines, tangents, and secants, at intervals of a hundredth of a degree. In the same year Vlacq published at Gouda his Trigonometria artificialis, giving log sines and tangents to every 10 seconds of the quadrant to 10 places. This work also contains the logarithms of the numbers from unity to 20,000 taken from the Arithmetica logarithmica of 1628. Briggs appreciated clearly the advantages of a centesimal division of the quadrant, and by dividing the degree into hundredth parts instead of into minutes, made a step towards a reformation in this respect, and but for the appearance of Vlacq’s work the decimal division of the degree might have become recognized, as is now the case with the corresponding division of the second. The calculation of the logarithms not only of numbers but also of the trigonometrical functions is therefore due to Briggs and Vlacq; and the results contained in their four fundamental works— Arithmetica logarithmica (Briggs), 1624; Arithmetica logarithmica (Vlacq), 1628; Trigonometria Britannica (Briggs), 1633; Trigonometria artificialis (Vlacq), 1633—have never been superseded by any subsequent calculations.
A translation of Napier’s Descriptio was made by Edward Wright, whose name is well known in connexion with the history of navigation, and after his death published by his son at London in 1616 under the title A Description of the admirable Table of Logarithmes (12mo); the edition was revised by Napier himself. Both the Descriptio (1614) and the Constructio (1619) were reprinted at Lyons in 1620 by Bartholomew Vincent, who thus was the first to publish logarithms’on the Continent.
Napier calculated no logarithms of numbers, and, as already stated, the logarithms invented by him were not to base e. The first logarithms to the base e were published by John Speidell in his New Logarithmes (London, 1619), which contains hyperbolic log sines, tangents, and secants for every minute of the quadrant to 5 places of decimals.
In 1624 Benjamin Ursinus published at Cologne a canon of logarithms exactly similar to Napier’s in the Descriptio of 1614, only much enlarged. The interval of the arguments is 10", and the‘results are given to 8 places; in Napier’s canon the interval is Γ, and the number of places is 7. The logarithms are strictly Napierian, and the arrangement is identical with that in the canon of 1614. This is the largest Napierian canon that has ever been published.
Kepler took the greatest interest in the invention of logarithms, and in 1624 he published at Marburg a table of Napierian logarithms of sines, with certain additional columns to facilitate special calculations.
The first publication of Briggian logarithms on the Continent is due to Wingate, who published at Paris in 1625 his Arithmétique logarithmétique, containing sevenfigure logarithms of numbers up to 1000, and log sines and tangents from Gunter’s Canon (1620). In the following year, 1626, Denis Henrion published at Paris a Traicté des Logarithmes, containing Briggs’s logarithms of numbers up to 20,001 to 10 places, and Gunter’s log sines and tangents to 7 places for every minute. In the same year De Decker also published at Gouda a work entitled Nieuwe Telkonst, inhoudeηde de Logarithmi voor de Ghetallen beginnende van 1 tot 10,000, which contained logarithms of numbers up to 10,000 to 10 places, taken from Briggs’s...
