LOGARITHMS. The definition of a logarithm is as follows i - if a, x, m are any three quantities satisfying the equation az= 771, 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=1084 ft.

Properties. - The principal properties of logarithms are given by the equations log (inn)-log. m+logo TL, log,2-n-' m -log, logc, -s• , log. Van= --r-log,.en, 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 nth root of a quantity is equal to –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 en, 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 ea denotes logo en ; but in analytical formulre 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=03010300 log 6=0.7781513 log 10-1.0000000 log 3-0-4771213 log 7-013450980 log 11=1'0413927 log 4-0-6020600 log 8-0.9030900 log 12=1'0791812 The meaning of these results is that , 11-161.041392, 12_10.0791812.

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, 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 -.40846€14, 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 T.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, S 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„ 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 lo-l_, which is called the modulus of log„ 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 log110 which is called the modulus of , the common system of logarithms. The numerical value of this modulus is 0.13129 41819 03251 82765 11289 ... , and the value of its reciprocal, loge 10 (by multiplication by which Briggian logarithms may be converted into Napierian logarithms) is 2.30258 50929 91045 6S401 79911. . .

The quantity denoted by e is the series, 1+ + + the numerical value of which The mathematical function log x, or log, 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, are cos x, &c., e' and log x, which are universally treated in analysis as known functions. It is the inverse of the exponential function the theory of which may be regarded as including that of the circular functions, since sin x= . c-lx), cue x=Yeix+e-1-9.

2 There is no series for lo,"° x proceeding either by ascending or descending powers of x, but there is an expansion for log (1+x), viz.: - log (1 +x)-x - ix2+ Ax2- &c.; the series, however, is convergent for real values of x only when x lies between +1 and -1. Other formulre which are deducible from this equation are given in the portion of this article relating to the calenlation of logarithms.

We have also the fundamental formulre - 4 Limit of '''1-1 1 , when h is indefinitely diminished, =log x ; dx f 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 fec/x=n +1 + const. ; but if at= - 1, this formula no longer gives a result. rutting, however, n= -1 +h, where is is indefinitely small, we have d.r - + 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 a and b from their point of intersection is proportional to log b .

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 1/(a7+ b2)+ i(a 2n7r), where a 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 8-0 and taking a positive, in which case a= 0, we obtain for log a the infinite system of values log a+ 2niri. 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 ; as such a series could only represent a uniform function, and in fact the equation log (1 + x)=x- .x2+ Ax3-4x4+ &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 John Napier, baron of Merchiston, in Scotland, with a unanimity which is rare with regard to important scientific 1611), 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 _ L.

em =107e 10' , or L= 107 log, 107-1074 where 1 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 -10'e 10 , and the logarithms decrease as the sines increase. Napier died in 1617, and his posthumous work Miriflei logarithmorum eanonis construed°, 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 Canonfieus 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 lie had accomplished, and, Ile says, he would gladly have paid him a third visit in 1617 had Napier's life been spared.

Briggs's Loriarithnzorion datias 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 posthumum abunde nobis propediem satisfacturum."

Thee liber postImmus was the C(010711.3 COTISITUCtiO 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 1621 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 site logarithmorunz chiliades centum, . . Editio secuncla aucta per Aclrictnum Vlacy, Goudanunz. This table of Vlacq's was published, with an English explanation prefixed, at London in 1631 under the title Logarithmieall 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 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 Arlacque, an Hollander, who bathe done all the whole hundred chiliades and printed them in Latin, Dutche, and Frenche, 1000 bookes in these 3 languages, and bathe sould them almost all. But he bathe cutt off 4 of my figures throughout ; and bathe 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 Song's table (1871), part of which was the result of an originafcalculation.

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, triangulorunz, 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, STc., given in the Thesaurus Mathematieus 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 Trigononzetria 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 Arithnzetica 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 - A rithnzetica logarithmica (Briggs), 1624 ; Arithmetica logarithmica (Vlacq), 1628 ; Trigonometria Briton-niece (Briggs), 1633 ; Trigon.onzetria ctrtificialis (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 Logaritlaznes (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 Yew 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 S 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 Arithmetique logarithmaique, 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 Traide des Logarithnzes, 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 Logarithmi voor de Chetallen beginnende van The preceding paragraphs contain abrief 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 Itheticus 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 antilogarithmic 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 and 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 prveiverunt ad hos ipsissimos logarithmos," which occurs in the " Prfecepta " prefixed to the Tabulte Ruclolphinx (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 Kiistner's Geschichte der Nathe»utak, vol. ii. p. 375, and vol. iii. p. 14 ; Montucla's Histoire des Mathematiques, vol. ii, p. 10; Delambre's IJistoire de l' Astronomic moderne, vol. i. p. 560 ; De Morgan's article on " Tables " in the English Cyclopedia ; and Mr Mark Napier's _Memoirs of Joker 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 corn puter. 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 Logarithnzicall arith• metike 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 Vitm quormulant eruditissimoruin et ill ustrium V2.1'071011 (Vita Henrici Briggii), London, 1707 ; Mr Mark Napier's Memoirs of John, Napier already referred to, and the same author's Naperi libri qni supersunt (1839); Hutton's History ; De Morgan's article already referred to ; Delambre's Ilistoire de l'Astrono»zie 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 Tabulx logarithuac (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 mantissce, 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 Ilutton'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 Lamed 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 ,4eithnmetica logarithneiea 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 Trigoitoinetrict, 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 Hiilsse. 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, Schron, and Brulins. 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 arc 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 Trigonometric 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 Trigonometric 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 astronomhgoaes (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, lie 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, Pricur, and Brunet, the French Government decided in 1784 that new tables of sines, tangents, &e., and their logarithms, should be calculated in relation to the centesimal division of the quadrant. Prosy was charged with the direction of the work, and was expressly required "Non seulement is composer des tables qui ne laissassent rien is deusirer (plant is l'exactitude, mail is en faire le monument de caleul le plus vaste et le plus hnposant qui eat jamais etc execute ou mime coueu." Those engaged upon the work were divided intolthree 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 arc 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.

Logarithms of the ratios of arcs to sines from 07'00000 to 07'05000, 07'05000, and log tangents throughout the quadrant 4 ,, The trigonometrical results are given for every hundred-thousandth of the quadrant (10" centesimal or 3".24esimal). The tables were all calculated to 14 places, with the fntention 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, formulae 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 lie 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 Arithmetiea loyarithmiea 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 Greeimck made an independent calculation of the logarithms of numbers Imp 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 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 ldeler's Nouvelles tables trigonometriques (1799), and Borda and Delambre's Tables trigonomarigmes cleeimales (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 0° to 50" at intervals of 10', to 11 places, and also, in another table, log sines, cosines, tangents, cotangents, secants, and cosecants from 00 to 3° at intervals of 10", and thence to 50° atintervals of 1' to 7 places. .After the work was printed it was read by Delambre with the Tables dm 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 logarithmes (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 : - ouly 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 nmost 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 a 12 deei2nales jusqu' a 434 milliards by NM. 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 arc exact quantities such as '00001, '00002, &e., and the numbers are incommensurable. The earliest and largest table of this kind that has been constructed is Dodson's Antilogarithmic MUM (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 Wolfi am, 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 Acute mind 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 front 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 Base at Vienna in 1850 under the title Tafel der watarlichen 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 Obserratory (1851); but separate copies were issued.

Hyperbolic antilogarithms are simple exponentials, i.e., the hyperbolic antilogarithm of x is e. A seven-figure table of e and its Briggian logarithm from x='01 to x=10 at intervals of '01 is given in Hidsse's edition of Vega's Samsulnng, and in other collections of tables ; but by far the most complete table that has been published occurs in Gudermann's Theorie der potenzial- oder eykl isch-h yperbolischc F anet loam, 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 :!-,(e –e-°) and + -x), the values of e' and e-' may be deduced from the results given in the table by simple addition and subtraction.

Logistic ?luaus is the old name for what would now be called (1624) already referred to. The object of a table of log Z.- 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 15), the table giving log 3600- log a, and x being expressed in minutes and seconds. It is also common to find tables in which a -10800"( = 3° or 35), and a 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 a is 3600", and proportional when a has any other value.

Gaussian. /(Jua•ielyas 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 en try when log a and log b are given. The utility of such logarithms was first pointed out by Leonelli in a book entitled Supplement logarithmigue, 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 Su/T/6,4ent was printed. The first table that was actually published is due to Gauss, and was printed. in Zach's 11onalliehe Curiesponden; vol. xxvi. p. 498 (1812). Corresponding to the argument A it gives, to 5 places, 11 and C, where - log .,, li=log (1 + 17) , C = log (1 + a).

so that C= A + B.

We have identicallylog (a +b)= log + log 71-b )= log a + B (for argument log c:); and, in nsing 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+ B, or, if we please, -log 8+ C. The formula for the difference is log (a - b)- log b+ A (argument sought in column C) if log a - log b is greater than '30103 and = log b - A (argument sought in column B) if log a - log b is less than '30103.

The principal tables of Gaussian logarithms are (1) Mathiessen, Tafel ear boinemeol. Th:reehn any (Altona, 1818), giving 11 and C fur argument A to 7 places, - this table is not a convenient one ; (2) Peter Grav, rthles amt. Formula; (London, 1849), and Addendum (1370), gi■-im, full tables of C and log (1 -a) for argument A to 6 places; (3) Ze(.1i, 7'';,•/a (h•...lila/lions. and Subtraetions-logarithmen (Leipsie, 1849), giving 7-plaee values of 11 for argument A, and 7-place values of C for argument B. These tables appeared originally in Ilitisse's edition of Vega's Stentinlung (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 (-D) corresponding to A as argument ; A and D are thus reciprocal, the relation between them being in fact 10^+D =10^+ 101)+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 formulae as cot IC a + b tan (A- B), and in the calculation of life contingencies.

Calculation of Logarithms. - The name logarithm is derived from the words A.Orov apc.Op.Os, 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 t-o 1 is very nearly equal to a ratio compounded of 301,030 of these small ratios, or ratiuncuke, 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 471121, 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 logaritlimica., 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 loyarithmica (1624). Ilis 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.

3-162277 . . 0.5 1.778279 . . . 0'25