Encyclopædia Britannica, Ninth Edition/Logarithms

by James Whitbread Lee Glaisher

LOGARITHMS. The definition of a logarithm is 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

$\scriptstyle \log a\ \left(mn\right)=\log a\ m+\log a\ n$,	$\scriptstyle \log a\ {\frac {m}{n}}=\log a\ m-\log a\ n$,
$\scriptstyle \log a\ m^{r}=r\log a\ m_{1}$,	$\scriptstyle \log a\ {\sqrt[{r}]{m}}={\frac {1}{r}}\log a\ m$,

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/rth 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 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 log₁₀ m; but in analytical formulæ 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,	2 = 100.3010300,	3 = 100.4771213,	…
10 = 101.	11 = 101.0413927	12 = 101.0791812.

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 = 3.4084634, &e.

,

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

$\scriptstyle \log a\ b\times \log b\ a=1$,

$\scriptstyle \log b\ m=\log a\ m\times {\frac {1}{\log a\ b}}$.

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/log₁₀, which is called the modulus of the common system of logarithms. The numerical value of this modulus is 0.43429 44819 03251 82765 11289 &hellips;, and the value of its reciprocal, log, 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 + 1/1.2 + 1/1.2.3 + 1/1.2.3.4 + …

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

$\scriptstyle \sin x={\frac {1}{2i}}\left(e^{ix}-e^{-ix}\right),$

$\scriptstyle \cos x=\left(e^{ix}+e^{-ix}\right)$.

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 − 1/2x² + 1/3x³ − 1/4x⁴ + &c.;

the series, however, is convergent for real values of x only when x lies between + 1 and − 1. Other formulæ 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 formulæ:—

(i.)	Limit of $\scriptstyle {\frac {x^{h}-1}{h}}$, when h is indefinitely diminished, $\scriptstyle =\log x$;
(ii.)	$\scriptstyle \int {\frac {dx}{x}}=\log x+$ 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

$\int x^{n}dx={\frac {x^{n}+1}{n+1}}+$ const.

but if n = −1, this formula no longer gives a result. Putting, however, n = −1 + h, where h is indefinitely small, we have

$\int {\frac {dx}{x}}={\frac {x^{h}-1}{h}}+$ 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 a and b from their point of intersection is pro portional to log b/a

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 + ib) = log √(a² + b²) + i(α + 2nπ),

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 b = 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 shall 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 (1 + x) = x − 1/2x² + 1/3x³ − 1/4x⁴ + &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 lgx.

History.—The invention of logarithms has been accorded to [[Encyclopædia Britannica, Ninth Edition/John Napier|]], baron of Merchiston, 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 (i.e., logarithms to the base e), though closely connected with them, the relation between the two being

e¹ = 10⁷e^−L/107, or L = 10⁷log_e10⁷ − 10⁷l,

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

N = 10⁷e^−L/107,

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 prima 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 posthumus abunde nobis propediem satisfacturum.” The liber posthumus was the Canonis constructio already mentioned. This work 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 it, however, only a second edition of Briggs’s Arithmetica logarithmica, the title running Arithmetica logarithmica sive logarithmorum chiliades centum,… Editio secunda aucta per Adrianum Vlacq, Goudanum. 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, 1623).

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 wanting betwixt 20 and 90 calculated and printed, and I had done them all almost by my selfe, 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 book, 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 Logarithms (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 1′, 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 Arithimetique logarithmétique, containing seven-figure 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, inhoudende de Logarithms 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 Arithmetica of 1624, and Gunter’s log sines and tangents to 7 places for every minute. Vlacq rendered assistance in the publication of this work, and the privilege is made out to him.

The preceding paragraphs contain a brief account of the main facts relating to the invention of logarithms. In describing the contents of the works referred to the language and notation of the present day have been adopted, so that for example a table to radius 10,000,000 is described as a table to 7 places, and so on. Also, although logarithms have been spoken of as to the base e, &c., it is to be noticed that neither Napier nor Briggs. nor any of their successors till long afterwards, had any idea of connecting logarithms with exponents.

The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier’s Descriptio. The calculation of tables of the natural trigonometrical functions may be said to have formed the work of the last half of the 16th century, and the great canon of natural sines for every 10 seconds to 15 places which had been calculated by Rheticus was published by Pitiscus only in 1613, the year before that in which the Descriptio appeared. In the construction of the natural trigonometrical tables England had taken no part, and it is remarkable that the discovery of the principles and the formation of the tables that were to revolutionize or supersede all the methods of calculation then in use should have been so rapidly effected and developed in a country in which so little attention had been previously devoted to such questions.

The only possible rival to Napier in the invention of logarithms is Justus Byrgius, who about the same time constructed a rude kind of logarithmic or rather anti-logarithmic table; but there is every reason to believe that Napier s system was conceived and perfected before that of Byrgius; and in date of publication Napier has the advantage by six years. The title of the work of Byrgius is Arithmetische und geometrische Progress-Tabulen; in his table he has log 1 = 0 and log 10 = 230270022. The only contemporary reference to Byrgius is contained in the sentence of Kepler, “Apices logistici Justo Byrgio multis annis ante editionem Neperianam viam præjiverunt ad hos ipsissimos logarithmos,” which occurs in the “Præcepta” prefixed to the Tabulæ Rudolphinæ (1627); the apices are the signs °, ′, ″, used to denote the orders of sexagesimal fractions. The system of Byrgius is greatly inferior to that of Napier, and it is to the latter alone that the world is indebted for the knowledge of logarithms. The claims of Byrgius are discussed in Kästner’s Geschichte der Mathematik, vol. ii. p. 375, and vol. iii. p. 14; Montucla’s Histoire des Mathematiques, vol. ii, p. 10; Delambre’s Histoire de l’Astronomie moderne, vol. i. p. 560; De Morgan’s article on “Tables” in the English Cyclopædia; and Mr Mark Napier’s Memoirs of John Napier of Merchiston (1834).

An account of the facts connected with the early history of logarithms is given by Hutton in his History of Logarithms, prefixed to all the early editions of his logarithmic tables, and also printed in vol. i. pp. 306–340 of his Tracts (1812); but unfortunately Hutton has interpreted all Briggs’s statements with regard to the invention of decimal logarithms in a manner clearly contrary to their true meaning, and unfair to Napier. This has naturally produced retaliation, and Mr Mark Napier has not only successfully refuted Hutton, but has fallen into the opposite extreme of attempting to reduce Briggs to the level of a mere computer. It seems strange that the relations of Napier and Briggs with regard to the invention of decimal logarithms should have formed matter for controversy. The statements of both agree in all particulars, and the warmest friendship subsisted between them. Napier at his death left his manuscripts to Briggs, and all the writings of the latter show the greatest reverence for him. The words that occur on the title page of the Logarithmicall arithmetike, of 1631 are “These numbers were first invented by the most excellent Iohn Neper, Baron of Merchiston; and the same were transformed, and the foundation and use of them illustrated with his approbation by Henry Briggs.” No doubt the invention of decimal logarithms occurred both to Napier and to Briggs independently; but the latter not only first announced the advantage of the change, but actually undertook and completed tables of the new logarithms. For more detailed information relating to Napier, Briggs, and Vlacq, and the invention of logarithms, the reader is referred to the life of Briggs in Ward’s Lives of the Professors of Gresham College, London, 1740; Thomas Smith’s Vitæ quorundam eruditissimorum et illustrium virorum (Vita Henrici Briggii), London, 1707; Mr Mark Napier’s Memoirs of John Napier already referred to, and the same author’s Naperi libri qui supersunt (1839); Hutton’s History; De Morgan’s article already referred to; Delambre’s Histoire de l’Astronomie Moderne; the report on mathematical tables in the Report of the British Association for 1873; and the Philosophical Magazine for October and December 1872 and May 1873. It may be remarked that the date usually assigned to Briggs’s first visit to Napier is 1616 and not 1615 as stated above, the reason being that Napier was generally supposed to have died in 1618; but it was shown by Mr Mark Napier that the true date is 1617.

For a description of existing logarithmic tables, and the purposes for which they were constructed, the reader is referred to the article Tables (Mathematical). In what follows only the most important events in the history of logarithms, subsequent to the facts connected with their invention and the original calculations, will be noticed.

Nathaniel Roe’s Tabulæ logarithmicæ (1633) was the first complete seven-figure table that was published. It contains seven-figure logarithms of numbers from 1 to 100,000, with characteristics unseparated from the mantissæ, and was formed from Vlacq’s table (1628) by leaving out the last three figures. All the figures of the number are given at the heads of the columns, except the last two, which run down the extreme columns, 1 to 50 on the left hand side, and 50 to 100 on the right hand side. The first four figures of the logarithms are printed at the tops of the columns. There is thus an advance half way towards the arrangement now universal in seven-figure tables. The final step was made by John Newton in his Trigononometria Britannica (1658), a work which is also noticeable as being the only extensive eight-figure table that has ever been published; it contains logarithms of sines, &c., as well as logarithms of numbers.

In 1705 appeared the original edition of Sherwin’s tables, the first of the series of ordinary seven-figure tables of logarithms of numbers and trigonometrical functions such as are in general use now. The work went through several editions during the last century, and was at length superseded in 1785 by Hutton’s tables, which have continued in successive editions to maintain their position up to the present time.

In 1717 Abraham Sharp published in his Geometry Improv’d the Briggian logarithms of numbers from 1 to 100, and of primes from 100 to 1100, to 61 places; these were copied into the later editions of Sherwin and other works.

In 1742 a seven-figure table was published in quarto form by Gardiner, which is celebrated on account of its accuracy and of the elegance of the printing. A French edition, which closely resembles the original, was published at Avignon in 1770.

In 1783 appeared at Paris the first edition of Callet’s tables, which correspond to those of Hutton in England. These tables, which form perhaps the most complete and practically useful collection of logarithms for the general computer that has been published, have passed through many editions, and are still in use.

In 1794 Vega published his Thesaurus logarithmorum completus, a folio volume containing a reprint of the logarithms of numbers from Vlacq’s Arithmetica logarithmica of 1628, and Trigonometria artificialis of 1633. The logarithms of numbers are arranged as in an ordinary seven-figure table. In addition to the logarithms reprinted from the Trigonometria, there are given logarithms for every second of the first two degrees, which were the result of an original calculation. Vega devoted great attention to the detection and correction of the errors in Vlacq’s work of 1628. He also published in 1797, in 2 vols. 8vo, a collection of logarithmic and trigonometrical tables which has passed through many editions, a very useful one volume stereotype edition having been published in 1840 by Hülsse. The tables in this work may be regarded as to some extent supplementary to those in Callet.

If we consider only the logarithms of numbers, the main line of descent from the original calculation of Briggs and Vlacq is Roe, John Newton, Sherwin, Gardiner; there are then two branches, viz. , Hutton founded on Sherwin and Callet on Gardiner, and the editions of Vega form a separate offshoot from the original tables. Among the most useful and accessible of modern ordinary seven-figure tables of logarithms of numbers and trigonometrical functions may be mentioned those of Bremiker, Schrön, and Bruhns. For logarithms of numbers only perhaps Babbage’s table is the most convenient.

In 1871 Mr Sang published a seven-figure table of logarithms of numbers extending from 20,000 to 200,000; and the logarithms of the numbers between 100,000 and 200,000 were calculated de novo by Mr Sang as if logarithms had never been computed before. In tables extending from 10,000 to 100,000 the differences near the beginning of the table are large, and they are so numerous that the proportional parts must either be very crowded, or some of them have to be omitted; and to diminish this inconvenience many tables extend to 108,000. By beginning the table at 20,000 instead of at 10,000, the differences are halved in magnitude, while the number of them in a page is quartered. In this table multiples of the differences, instead of proportional parts, are given.

As regards the logarithms of trigonometrical functions, the next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to 10 places and then contracted to 7. On account of the great size of this table, and for other reasons, it never came into very general use, Bagay’s Nouvelles tables astronomiques (1829), which also contains log sines and tangents to every second, being preferred, but this work is now difficult to procure. The only other logarithmic canon to every second that has been published forms the second volume of Shortrede’s Logarithmic Tables (1849). It contains also proportional parts, and is the most complete and accessible table of logarithms for every second. Shortrede’s tables originally appeared in 1844 in one volume, during the author’s absence in India; but, not being satisfied with them in some respects, he made various alterations, and published a second edition in two volumes in 1849. There have been subsequent editions of the volume containing the trigonometrical canon. The work is an important one, and the pages are clear, although the number of figures on each is very great.

On the proposition of Carnot, Prieur, and Brunet, the French Government decided in 1784 that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant. Prony was charged with the direction of the work, and was expressly required “Non seulement & composer des tables qui ne laissassent rien à desirer quant & l’exactitude, mais à en faire le monument de calcul le plus vaste et le plus imposant qui eût jamais été exécuté on meme conçu.” Those engaged upon the work were divided into three sections: the first consisted of five or six mathematicians, including Legendre who were engaged in the purely analytical work, or the calculation of the fundamental numbers; the second section consisted of seven or eight calculators possessing some mathematical knowledge; and the third comprised seventy or eighty ordinary computers. The work, which was performed wholly in duplicate, and independently by two divisions of computers, occupied two years. As a consequence of the double calculation, there are two manuscripts in existence, one of which is deposited at the Observatory, and the other in the library of the Institute, at Paris. Each of the two manuscripts consists essentially of seventeen large folio volumes the contents being as follows:

Logarithms of numbers up to 200,000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

vols.

Natural sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

vols.„

Logarithms of the ratios of arcs to sines from 07.00000 to 07.05000, and log sines throughout the quadrant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

vols.„

Logarithms of the ratios of arcs to tangents from 07.00000 to 07.05000, and log tangents throughout the quadrant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

vols.„

The trigonometrical results are given for every hundred-thousandth of the quadrant (10″ centetimal or 3″.24 sexagesimal). The tables were all calculated to 14 places, with the intention that only 12 should be published, but the twelfth figure is not to be relied upon. The tables have never been published, and are generally known as the Tables du Cadastre, or, in England, as the great French manuscript tables.

A very full account of the Tables du Cadastre, with an explanation of the methods of calculation, formulæ employed, &c., has been published by M. Lefort in vol. iv. of the Annales de l’Observatoire de Paris. The printing of the table of natural sines was once begun, and M. Lefort states that he has seen six copies, all incomplete, although including the last page. Babbage compared his table with the Tables du Cadastre, and M. Lefort has given in his paper just referred to most important lists of errors in Vlacq’s and Briggs’s logarithms of numbers which were obtained by comparing the manuscript tables with those contained in the Arithmetica logarithmica of 1624 and of 1628. These are almost the only uses that have been made of the French tables, the calculation of which involved so great an expenditure of time and money.

It may be mentioned here that the late Mr John Thomson of Greenock made an independent calculation of the logarithms of numbers up to 120,000 to 12 places, and that the manuscript of the table was presented in 1874 to the Royal Astronomical Society by his sister. The table has been used to verify the errata which M. Lefort found in Vlacq and Briggs by means of the Tables du Cadastre. An account of Mr Thomson’s table, and of this and other comparisons between it and the printed tables, is to be found in the Monthly Notices of the Society, vol. xxxiv. pp. 447-75 (1874).

Although the Tables du Cadastre have never been published, other tables have appeared in which the quadrant is divided centesimally, the most important of these being Hobert and Ideler’s Nouvelles tables trigonometriques (1799), and Borda and Delambre’s Tables trigonometriques decimates (1800-1). The former work contains natural and log sines, cosines, tangents, and cotangents to 7 places, up to 3° (centesimal) at intervals of 10″ (centesimal), and thence to 50° at intervals of 1′. The latter gives log sines, cosines, tangents, and cosines for centesimal arguments, viz., from 0′ to 10′ at intervals of 10″, and from to 50° at intervals of 1′, to 11 places, and also, in another table, log sines, cosines, tangents, cotangents, secants, and cosecants from 0° to 3° at intervals of 10′, and thence to 50° at intervals of 1′ to 7 places. After the work was printed it was read by Delambre with the Tables du Cadastre, and a number of last-figure errors which are given in the preface were thus detected. Callet’s tables already referred to contain in a convenient form logarithms of trigonometrical functions for centesimal arguments.

Two tables of logarithms of numbers which have been recently published may be noticed, as they involve points of novelty. The first of these is Pineto’s Tables de logarithmcs (St Petersburg, 1871). The tables are intended to give in a small space (56 pages) all the results that can be obtained from a complete ten-figure table by means of the following principle:—only the logarithms of the numbers from 1,000,000 to 1,011,000 are given directly, all other numbers being brought within the range of this table by multiplication by a factor, the logarithm of which factor is to be subtracted from the logarithm in the table. A list of the most convenient factors and their logarithms is given in a separate table. The principle of multiplying by a factor which is subsequently cancelled by subtracting its logarithm is one that is frequently employed in the calculation of logarithms, but the peculiarity of the present work is that it forms part of the process of using the table. The other tables, which occupy only ten pages, were published in a tract entitled Tables de logarithmes à 12 décimales jusqu’ à 434 milliards by MM. Namur and Mansion at Brussels in 1877. The fact that the differences of the logarithms of numbers near to 434294 (these being the first figures of the modulus of the Briggian logarithms) commenced with the figures 100…, so that the interpolations in this part of the table are very easily and accurately performed, is ingeniously made use of. A table is given of logarithms of numbers near to 434294, and other numbers are brought within the range of the table by multiplication by one or two factors which are indicated.

In the ordinary tables of logarithms the natural numbers are all integers, while the logarithms tabulated are incommensurable. In an antilogarithmic table, the logarithms are exact quantities such as .00001, .00002, &c., and the numbers are incommensurable. The earliest and largest table of this kind that has been constructed is Dodson’s Antilogarithmic canon (1742), which gives the numbers to 11 places, corresponding to the logarithms from .00001 to .99999 at intervals of .00001. The only other extensive tables of the same kind that have been published occur in Shortrede’s Logarithmic tables already referred to, and in Filipowski’s Table of antilogarithms (1849). Both are similar to Dodson’s tables, from which they were derived, but they only give numbers to 7 places.

The most elaborate table of hyperbolic logarithms that exists is due to Wolfram, a Dutch lieutenant of artillery. His table gives the logarithms of all numbers up to 2200, and of primes (and also of a great many composite numbers) from 2200 to 10,009, to 48 decimal places. The table appeared in Schulze’s Neue und erweiterte Sammlung logarithmischer Tafeln (1778), and was. reprinted in Vega’s Thesaurus (1794), already referred to. Six logarithms omitted in Schulze’s work, and which Wolfram had been prevented from computing by a serious illness, were published subsequently, and the table as given by Vega is complete. The largest hyperbolic table as regards range was published by Zacharias Dase at Vienna in 1850 under the title Tafel der natürlichen Logarithmen der Zahlen. It gives hyperbolic logarithms of numbers from 1000.0 to 10500.0 at intervals of 1 to 7 places, with differences and proportional parts, arranged like an ordinary seven-figure table of Briggian logarithms. The table appeared in the thirty-fourth part (new series, vol. xiv.) of the Annals of the Vienna Observatory (1851); but separate copies were issued.

Hyperbolic antilogarithms are simple exponentials, i.e., the hyperbolic antilogarithm of x is e^x. A seven-figure table of e^x and its Briggian logarithm from x = .01 to x = 10 at intervals of .01 is given in Hülsse’s edition of Vega’s Sammlung, and in other collections of tables; but by far the most complete table that has been published occurs in Gudermann’s Theorie der potential- oder cyklisch-hyperbolischen Functionen, Berlin, 1833. This work consists of reprinted papers from Crelle’s Journal, and one of the tables contains the Briggian logarithms of the hyperbolic sine, cosine, and tangent of x from x = 2 to x = 5 at intervals of .001 to 9 places, and from x = 5 to x = 12 at intervals of 0.01 to 10 places. Since the hyperbolic sine and cosine of x are respectively 1/2(e^x − e^−x) and 1/2(e^x + e^−x), the values of e^x and e^−x may be deduced from the results given in the table by simple addition and subtraction.

Logistic numbers is the old name for what would now be called ratios or fractions. Thus a table of log a/x, where x is the argument and a a constant, is called a table of logistic or proportional logarithms; and since log a/x = log a - log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign. The first table of this kind appeared in Kepler’s Chilias logarithmorum (1624) already referred to. The object of a table of log a/x is to facilitate the working out of proportions in which the third term is a constant quantity a. In most collections of tables of logarithms, and especially those intended for use in connexion with navigation, there occurs a small table of logistic logarithms in which a = 3600″ (−1 or 1{{sup|h), the table giving log 3600 − log x, and x being expressed in minutes and seconds. It is also common to find tables in which a= 10800″(= 3° or 3^h), and x is expressed in degrees (or hours), minutes, and seconds. Such tables are generally given to 4 or 5 places. The usual practice in books seems to be to call logarithms logistic when is 3600″, and proportional when a has any other value.

Gaussian logarithms are intended to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers themselves being unknown; and on this account they are frequently called addition and subtraction logarithms. The object of the table is in fact to give log (a±b) by only one entry when log a and log b are given. The utility of such logarithms was first pointed out by Leonelli in a book entitled Supplement logarithmique, printed at Bordeaux in the year XI. (1802-3); this work being very scarce, a reprint of it was published by M. J. Houel in 1876. Leonelli calculated a table to 14 places, but only a specimen of it which appeared in the Supplement was printed. The first table that was actually published is due to Gauss, and was printed in Zach’s Monatliche Correspondenz, vol. xxvi. p. 498 (1812). Corresponding to the argument $\mathrm {A}$ it gives, to 5 places, $\mathrm {B}$ and $\mathrm {C}$, where

$\mathrm {A} =\log x$,

$\mathrm {B} =\log \left(1+{\frac {1}{x}}\right)$,

$\mathrm {C} =\log \left(1+x\right)$.

so that $\mathrm {C} =\mathrm {A} +\mathrm {B}$.

We have identically

$\log \left(a+b\right)=\log a+\log \left(1+{\frac {b}{a}}\right)=\log a+\mathrm {B}$ (for argument loga/b);

and, in using the table, the rule is to take log a to be the larger of the two logarithms, and to enter the table with log a − log b as argument; we then have log (a + b) = log a + $\mathrm {B}$, or, if we please, = log b + $\mathrm {C}$. The formula for the difference is log (a − b) = log b + $\mathrm {B}$ (argument sought in column $\mathrm {C}$) if log a − log b is greater than .30103 and = log b − $\mathrm {A}$ (argument sought in column $\mathrm {B}$ ) if log a − log b is less than .30103.

The principal tables of Gaussian logarithms are (1) Mathiessen, Tafel zur bequemern Berechnung (Altona, 1818), giving B and C for argument A to 7 places,—this table is not a convenient one; (2) Peter Gray, Tables and Formulæ (London, 1849), and Addendum (1870), giving full tables of C and log (1 − x) for argument A to 6 places; (3) Zech, Tafeln dur Additions und Subtractions—logarithmen (Leipsic, 1849), giving 7-place values of B for argument A, and 7-place values of C for argument B. These tables appeared originally in Hülsse’s edition of Vega’s Sammlung (1849); (4) Wittstein, Logarithmes de Gauss (Hanover, 1866), giving values of B for argument A to 7 places. This is a large table, and the arrangement is similar to that of an ordinary seven-figure table of logarithms.

In 1829 Widenbach published at Copenhagen a small table of modified Gaussian logarithms giving log x + 1/x - 1(= D) corresponding to A as argument; A and D are thus reciprocal, the relation between them being in fact 10^{A +D} = 10^A + 10^D + 1, so that either A or D may be regarded as the argument.

Gaussian logarithms are chiefly useful in the calculations connected with the solution of triangles in such a formulæ as cot 1/2C = a + b/a - btan (A - B), and in the calculation of life contingencies.

Calculation of Logarithms.—The name logarithm is derived from the words λόϒων άριΘμός, the number of the ratios, and the way of regarding a logarithm which justifies the name may be explained as follows. Suppose that the ratio of 10, or any other particular number, to 1 is compounded of a very great number of equal ratios, as for example 1,000,000, then it can be shown that the ratio of 2 to 1 is very nearly equal to a ratio compounded of 301,030 of these small ratios, or ratiunculæ, that the ratio of 3 to 1 is very nearly equal to a ratio compounded of 477,121 of them, and so on. The small ratio, or ratiuncula, is in fact that of the millionth root of 10 to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to 1 will be nearly the same as that of a^301,030 to 1 and so on; or, in other words, if a denotes the millionth root of 10, then 2 will be nearly equal to a^301,030, 3 will be nearly equal to a^477,121, and so on.

Napier’s original work, the Descriptio canonis of 1614, contained, not logarithms of numbers, but logarithms of sines, and the relations between the sines and the logarithms were explained by the motions of points in lines, in a manner not unlike that afterwards employed by Newton in the method of fluxions. An account of the processes by which Napier constructed his table is given in the Constructio canonis of 1619. These methods apply, however, specially to Napier’s own kind of logarithms, and are different from those actually used by Briggs in the construction of the tables in the Arithmetica logarithmica, although some of the latter are the same in principle as the processes described in an appendix to the Constructio. It may be observed that in the Constructio logarithms are called artificials, and this seems to have been the name first employed by Napier, but which he subsequently replaced by logarithms. It is to be presumed that he would have made the change of name also in the Constructio, had he lived to publish it himself.

The processes used by Briggs are explained by him in the preface to the Arithmetica logarithmica (1624). His method of finding the logarithms of the small primes, which consists in taking a great number of continued geometric means between unity and the given primes, may be described as follows. He first formed the table of numbers and their logarithms:

Number.	Logarithm.
10.162277 …‍	1
3.162277 …	0.5
1.778279 …	0.25
1.333521 …	0.125
1.154781 …	0.0625

each quantity in the left hand column being the square root of the one above it, and each quantity in the right hand column being the half of the one above it. To construct this table Briggs, using about thirty places of decimals, extracted the square root of 10 fifty-four times, and thus found that the logarithm of 1.00000 00000 00000 12781 91493 20032 35 was 0.00000 00000 00000 05551 11512 31257 82702, and that for numbers of this form (i.e., for numbers beginning with 1 followed by fifteen ciphers, and then by seventeen or a less number of significant figures) the logarithms were proportional to these significant figures. He then by means of a simple proportion deduced that log (1.00000 00000 00000 1) = 0.00000 00000 00000 04342 94481 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 1.00000 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., 1024, and extracted the square root of 1.024 forty-seven times, the result being 1.00000 00000 00000 16851 60570 53949 77. Multiplying the significant figures by 4342… he obtained the logarithm of this quantity, viz., O.OOOOO 00000 00000 07318 55936 90623 9336, which multiplied by 2⁴⁷ gave 0.01029 99566 39811 95265 277444, the logarithm of 1.024, true to 17 or 18 places. Adding the characteristic 3, and dividing by 10, he found (since 2 is the tenth root of 1024) log 2 = .30102 99956 63981 195. Briggs calculated in a similar manner log 6, and thence deduced log 3.

It will be observed that in the first process the value of the modulus is in fact calculated from the formula

h/10^h − 1 = 1/log_e10,

the value of h being 1/2⁵⁴, and in the second process log₁₀ 2 is in effect calculated from the formula

$\log _{10}2=\left(2^{\frac {10}{2^{4}7}}\right)\times {\frac {1}{\log _{e}10}}\times {\frac {2^{4}7}{10}}$.

Briggs also gave methods of forming the mean proportionals or square roots by differences; and the general method of constructing logarithmic tables by means of differences is due to him.

The following calculation of log 5 is given as an example of the application of a method of mean proportionals. The process consists in taking the geometric mean of numbers above and below 5, the object being to at length arrive at 5.000000. To every geometric mean in the column of numbers there corresponds the arithmetical mean in the column of logarithms. The numbers are denoted by A, B, C, &c., in order to indicate their mode of formation.

	Numbers.		Logarithms.
A =	1	.000000	0.000000
B =	10	.000000	1.000000
C = √(AB) =	3	.162277	0.500000
D = √(BC) =	5	.623413	0.750000
E = √(CD) =	4	.216964	0.625000
F = √(DE) =	4	.869674	0.687500
G = √(DF) =	5	.232991	0.718750
H = √(FG) =	5	.048065	0.703125
I = √(FH) =	4	.958069	0.6953125
K = √(HI) =	5	.002865	0.6992187
L = √(HK) =	4	.980416	0.6972656
M = √(KL) =	4	.991627	0.6982421
N = √(KM) =	4	.997242	0.6987304
O = √(KN) =	5	.000052	0.6989745
P = √(NO) =	4	.998647	0.6988525
Q = √(OP) =	4	.999350	0.6989135
R = √(OQ) =	4	.999701	0.6989440
S = √(OR) =	4	.999876	0.6989592
T = √(OS) =	4	.999963	0.6989668
V = √(OT) =	5	.000008	0.6989707
W = √(TV) =	4	.999984	0.6989687
X = √(WV) =	4	.999997	0.6989697
Y = √(VX) =	5	.000003	0.6989702
Z = √(XY) =	5	.000000	0.6989700

Great attention was devoted to the methods of calculating logarithms during the 17th and 18th centuries. The earlier methods proposed were, like those of Briggs, purely arithmetical, and for a long time logarithms were regarded from the point of view indicated by their name, that is to say, as depending on the theory of compounded ratios. The introduction of infinite series into mathematics effected a great change in the modes of calculation and the treatment of the subject. Besides Napier and Briggs, special reference should be made to Kepler (Chilias, 1624) and Mercator (Logarithmotechnia, 1668), whose methods were arithmetical, and to Newton, Gregory, Halley, and Cotes, who employed series. A full and valuable account of these methods is given in Hutton’s “Construction of Logarithms,” which occurs in the introduction to the early editions of his Mathematical Tables, and also forms tract 21 of his Mathematical Tracts (vol. i., 1812). Many of the early works on logarithms were reprinted in the Scriptores logarithmici of Baron Maseres (6 vols. 4to, 1791-1807).

In the following account only those formulæ and methods will be referred to which would now be used in the calcula tion of logarithms.

Since

$\log _{e}\left(1+x\right)=x-{\frac {1}{2}}x^{2}+{\frac {1}{3}}^{3}-{\frac {1}{4}}x^{4}+\And c.$,

we have, by changing the sign of x,

$\log _{e}\left(1-x\right)=-x-{\frac {1}{2}}x^{2}-{\frac {1}{3}}x^{3}-{\frac {1}{4}}x^{4}-\And c.$;

whence

$\log _{e}{\frac {1+x}{1-x}}=2\left(x+{\frac {1}{3}}x^{3}-{\frac {1}{5}}x^{5}+\And c.\right)$,

and, therefore, replacing x by p - q/p + q,

$\log _{e}{\frac {p}{q}}=\left\{{\frac {p-q}{p+q}}+{\frac {1}{3}}\left({\frac {p-q}{p+q}}\right)^{3}+{\frac {1}{5}}\left({\frac {p-q}{p+q}}\right)^{5}+\And c.\right\}$,

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.

As particular cases we have, by putting q = 1,

$\log _{e}p=2\left\{{\frac {p-1}{p+1}}+{\frac {1}{3}}\left({\frac {p-1}{p+1}}\right)^{3}+{\frac {1}{5}}\left({\frac {p-1}{p+1}}\right)^{5}+\And c.\right\}$,

and by putting q = p + 1,

$\log _{e}\left(p+1\right)-\log _{e}p=2\left\{{\frac {1}{2p+1}}+{\frac {1}{3}}{\frac {1}{\left(2p+1\right)^{3}}}+{\frac {1}{5}}{\frac {1}{\left(2p+1\right)^{5}}}+\And c.\right\}$;

the former of these equations gives a convergent series for log_ep, 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.

From the formula for log_ep/q we may deduce the following very convergent series for log_e2, log_e3, and log_e5, viz.:—

log_e2 = 2(7P +5Q + 3R),
log_e3 = 2(11P + 8Q + 5R),
log_e5 = 2(16P + 12Q + 7R),

where

P = 1/31 + 1/3 · 1/(31)³ + 1/5 · 1/(31)⁵ + &c.
Q = 1/49 + 1/3 · 1/(49)³ + 1/5 · 1/(49)⁵ + &c.
R = 1/161 + 1/3 · 1/(161)³ + 1/5 · 1/(161)⁵ + &c.

The following still more convenient formulæ for the calculation of log_e 2, log_e 3, &c. are given by Professor J. C. Adams in the Proceedings of the Royal Society, vol. xxvii. (1878), p. 91. If

$a=\log {\frac {10}{9}}=\log \left(1-{\frac {1}{10}}\right)$,	$b=\log {\frac {25}{24}}=\log \left(1-{\frac {4}{100}}\right)$,
$c=\log {\frac {81}{80}}=\log \left(1+{\frac {1}{80}}\right)$,	$d=\log {\frac {50}{49}}=-\log \left(1-{\frac {2}{100}}\right)$,
$e=\log {\frac {126}{125}}=\log \left(1+{\frac {8}{1000}}\right)$,

then

$\log 2=7a-2b+3c$,

$\log 3=11a-3b+5c$,

$\log 5=16a-4b+7c$,

and

$\log 7={\frac {1}{2}}\left(39a-10b+17c-d\right)$ or $=19a-4b+8c+d$,

and we have the equation of condition,

$a-2b+c=d+2c$,

By means of these formula Professor Adams has calculated the values of log_e 2, log_e 3, log_e 5, and log_e 7 to 260 places of decimals, and he has deduced the value of log, 10 and its reciprocal $\mathrm {M}$, the modulus of the Briggian system of logarithms. The value of the modulus found by Professor Adams is

$\mathrm {M} =$.	43429	44819	03251	82765	11289
	18916	60508	22943	97005	80366
	65661	14453	78316	58646	49208
	87077	47292	24949	33843	17483
	18706	10674	47663	03733	64167
	92871	58963	90656	92210	64662
‍
	81226	58521	27086	56867	03295
	93370	86965	88266	88331	16360
	77384	90514	28443	48665	76864
	65860	85135	56148	21234	87653
	43543	43573	17247	48649	05993
	55353	05

The values of the other logarithms are given in the paper referred to.

If the logarithms are Briggian all the series in the preceding formulæ must be multiplied by $\mathrm {M}$, the modulus; thus, for example,

$\log _{10}\left(1+x\right)=\mathrm {M} \left(x-{\frac {1}{2}}x^{2}+{\frac {1}{3}}x^{3}-{\frac {1}{4}}x^{4}+\And c.\right)$,

and so on.

As has been stated, Abraham Sharp’s table contains 61-decimal Briggian logarithms of primes up to 1100, so that the logarithms of all composite numbers whose greatest prime factor does not exceed this number may be found by simple addition; and Wolfram’s table gives 48-decimal hyperbolic logarithms of primes up to 10,009. 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. Multiplying by 50, we have 50 × 43,867 = 2,193,350, and on looking in Burckhardt’s Table des diviscurs for a number near to this which shall have no prime factor greater than 10,009, it appears that

2,193,349 = 23 × 47 × 2029;

thus

43,867 = 1/36(23 × 47 × 2029 + 1),

and therefore

$\log _{e}43,867=\log _{e}23+\log _{e}47+\log _{e}2029-\log _{e}50$

$+{\frac {1}{2,193,349}}-{\frac {1}{2}}{\frac {1}{\left(2,193,349\right)^{2}}}+{\frac {1}{3}}{\frac {1}{\left(2,193,349\right)^{3}}}-\And c.$

The first term of the series in the second line is

0.00000 04559 23795 07319 6286;

dividing this by 2 × 2,193,349 we obtain

0.00000 00000 00103 93325 3457,

and the third term is

0.00000 00000 00000 00003 1590,

so that the series =

0.00000 04559 23691 13997 4419;

whence, taking out the logarithms from Wolfram’s table,

log_e 43,867 = 10.68891 76079 60568 10191 3661.

The principle of the method is to multiply the given prime (supposed to consist of 4, 5, or 6 figures) by such a factor that the product may be a number within the range of the factor tables, and such that, when it is increased by 1 or 2, the prime factors may all be within the range of the logarithmic tables. The logarithm is then obtained by use of the formula

$\log _{e}\left(x+d\right)=\log _{e}x+{\frac {d}{x}}-{\frac {1}{2}}{\frac {d^{2}}{x^{2}}}+{\frac {1}{3}}{\frac {d^{3}}{x^{3}}}=\And c.$,

in which of course the object is to render d/x as small as possible.

If the logarithm required is Briggian, the value of the series is to be multiplied by $\mathrm {M}$.

If the number is incommensurable or consists of more than seven figures, we can take the first seven figures of it (or multiply and divide the result by any factor, and take the first seven figures of the result) and proceed as before. An application to the hyperbolic logarithm of $\pi$ 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 1 - 1⁴n, where n is one of the nine digits, and making use of subsidiary tables of logarithms of factors of this form. For example, suppose the logarithm of 543839 required to twelve places. Dividing by 10⁵ and by 5 the number becomes 1.087678, and resolving this number into factors of the form 1 − .1⁴n we find that

$543839=10^{5}$	$\times 5\left(1-.1^{3}8\right)\left(1-.1^{4}6\right)\left(1-.1^{5}6\right)\left(1-.1^{6}3\right)\left(1-.1^{7}3\right)$
	$\times \left(1-.1^{8}5\right)\left(1-.1^{9}7\right)\left(1-.1^{10}9\right)\left(1-.1^{11}3\right)\left(1-.1^{12}2\right)$

where 1 - .1²8 denotes 1 - .08, 1 - .1⁴6 denotes 1 - .0006, &c., and so on. 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, .99n, .999n, &c., for n = 1, 2, 3,… 9.

The resolution of a number into factors of the above form is easily performed. Taking, for example, the number 1.087678, the object is to destroy the significant figure 8 in the second place of decimals; this is effected by multiplying the number by 1 − .08, that is, by subtracting from the number eight times itself advanced two places, and we thus obtain 1.00066376. To destroy the first 6 multiply by 1 − .0006 giving 1.000063361744, and multiplying successively by 1 − .00006 and 1 - .000003, we obtain 1.000000357932, and it is clear that these last six significant figures represent without any further work the remaining factors required. In the corresponding antilogarithmic process the number is expressed as a product of factors of the form 1 + .1ⁿx.

This method of calculating logarithms by the resolution of numbers into factors of the form 1 − .1^rn is generally known as Weddle’s method, having been published by him in The Mathematician for November 1845, and the corresponding method for antilogarithms by means of factors of the form 1 + (.1)^rn is known by the name of Hearn, who published it in the same journal for 1847. In 1846 Mr Peter Gray constructed a new table to 12 places, in which the factors were of the form 1 − (.01)^rn, so that n had the values 1, 2, … 99; and subsequently he constructed a similar table for factors of the form 1 + (.01)^rn. He also discovered a method of applying a table of Hearn’s form (i.e., of factors of the form 1 + (.1^rn) to the construction of logarithms, and calculated a table of logarithm’s of factors of the form 1 + (.001)^rn to 24 places. This was published in 1876 under the title Tables for the formation of logarithms and antilogarithms to twenty-four or any less number of places, and contains the most complete and useful application of the method, with many improvements in points of detail. 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 Mr Gray’s process that the factors of 1.31601 are

(1) 1.316	(5) 1.(001)4002
(2) 1.000007	(6) 1.(001)5602
(3) 1.(001)2598	(7) 1(001)6412
(4) 1.(001)3780	8) 1-(001)7340.

Taking the logarithms from Mr Gray’s tables we obtain the required logarithm by addition as follows:

522	878	745	280	337	562	704	972	= colog3
119	255	889	277	936	685	553	913	= log (1)
	3	040	050	733	157	610	239	= log (2)
		259	708	022	525	453	597	= log (3)
			338	749	695	752	424	= log (4)
					868	588	964	= log (5)
					261	445	278	= log (6)
						178	929	= log (7)
							148	= log (8)
‍
4.642	137	934	655	780	757	288	464	= log10 43,867

In Shortrede’s Tables there are tables of logarithms and factors of the form 1 ± (.01)^rn to 16 places and of the form 1 ± (.1)^rn to 25 places; and in his Tables de Logarithmes à 27 Décimales (Paris, 1867) Fédor Thoman gives tables of logarithms of factors of the form 1 ± .1^r n. In the Messenger of Mathematics, vol. iii. pp. 66-92, 1873, Mr Henry Wace gave a simple and clear account of both the logarithmic and antilogarithmic processes, with tables of both Briggian and hyperbolic logarithms of factors of the form 1 ±.1^rn to 20 places.

Although the method is usually known by the names of Weddle and Hearn, it is really, in its essential features, due to Briggs, who gave in the Arithmctica logarithmica of 1624 a table of the logarithms of 1 + .1^rn up to r = 9 to 15 places of decimals. It was first formally proposed as an independent method, with great improvements, by Robert Flower in The Radix, a new way of making Logarithms, which was published in 1771; and Leonelli, in his Supplément logarithmique (1802-3), already noticed, referred to Flower and reproduced some of his tables. A complete bibliography of this method has been given by Mr A. J. Ellis in a paper “on the potential radix as a means of calculating logarithms” printed in the Proceedings of the Royal Society, vol. xxxi., 1881, pp. 401-407, and vol. xxxii., 1881, pp. 377-379. Reference should also be made to Hoppe’s Tafeln zur dreissigstelligen logarithmischen Rechnung (Leipsic, 1876), which give in a somewhat modified form a table of the hyperbolic logarithm of 1 + .1^rn.

The preceding methods are only appropriate for the calculation of isolated logarithms. If a complete table had to be reconstructed, or calculated to more places, it would undoubtedly be most convenient to employ the method of differences. A full account of this method as applied to the calculation of the Tables du Cadastre is given by M. Lefort in vol. iv. of the Annales de l’Observatoire de Paris. (J. W. L. G.)