Chapter 2

“Ego enim, cum meis auditoribus Londini, publice in Collegio Greshamensi horum doctrinam explicarem; animadverti multo futurum commodius, si Logarithmus sinus totius servaretur 0 (ut in Canone mirifico), Logarithmus autem partis decimae ejusdem sinus totius, nempe sinus 5 graduum, 44, m. 21, s., esset 10000000000. atque ea de re scripsi statim ad ipsum authorem, et quamprimum per anni tempus, et vacationem a publico docendi munere licuit, profectus sum Edinburgum; ubi humanissime ab eo acceptus haesi per integrum mensem. Cum autem inter nos de horum mutatione sermo haberetur; ille se idem dudum sensisse, et cupivisse dicebat: veruntamen istos, quos jam paraverat edendos curasse, donec alios, si per negotia et valetudinem liceret, magis commodos confecisset. Istam autem mutationem ita faciendam censebat, ut 0 esset Logarithmus unitatis, et 10000000000 sinus totius: quod ego longe commodissimum esse non potui non agnoscere. Coepi igitur, ejus hortatu, rejectis illis quos anteà paraveram, de horum calculo serio cogitare; et sequenti aestate iterum profectus Edinburgum, horum quos hic exhibeo praecipuos, illi ostendi, idem etiam tertia aestate libentissime facturus, si Deus illum nobis tamdiu superstitem esse voluisset.”

There is also a reference to the change of the logarithms on the title-page of the work.

These extracts contain all the original statements made by Napier, Robert Napier and Briggs which have reference to the origin of decimal logarithms. It will be seen that they are all in perfect agreement. Briggs pointed out in his lectures at Gresham College that it would be more convenient that 0 should stand for the logarithm of the whole sine as in theDescriptio, but that the logarithm of the tenth part of the whole sine should be 10,000,000,000. He wrote also to Napier at once; and as soon as he could he went to Edinburgh to visit him, where, as he was most hospitably received by him, he remained for a whole month. When they conversed about the change of system, Napier said that he had perceived and desired the same thing, but that he had published the tables which he had already prepared, so that they might be used until he could construct others more convenient. But he considered that the change ought to be so made that 0 should be the logarithm of unity and 10,000,000,000 that of the whole sine, which Briggs could not but admit was by far the most convenient of all. Rejecting therefore, those which he had prepared already, Briggs began, at Napier’s advice, to consider seriously the question of the calculation of new tables. In the following summer he went to Edinburgh and showed Napier the principal portion of the logarithms which he published in 1624. These probably included the logarithms of the first chiliad which he published in 1617.

It has been thought necessary to give in detail the facts relating to the conversion of the logarithms, as unfortunately Charles Hutton in his history of logarithms, which was prefixed to the early editions of hisMathematical Tables, and was also published as one of hisMathematical Tracts, has charged Napier with want of candour in not telling the world of Briggs’s share in the change of system, and he expresses the suspicion that “Napier was desirous that the world should ascribe to him alone the merit of this very useful improvement of the logarithms.” According to Hutton’s view, the words, “it is to be hopedthat his posthumous work” ... which occur in the preface to theChilias, were a modest hint that the share Briggs had had in changing the logarithms should be mentioned, and that, as no attention was paid to it, he himself gave the account which appears in theArithmeticaof 1624. There seems, however, no ground whatever for supposing that Briggs meant to express anything beyond his hope that the reason for the alteration would be explained in the posthumous work; and in his own account, written seven years after Napier’s death and five years after the appearance of the work itself, he shows no injured feeling whatever, but even goes out of his way to explain that he abandoned his own proposed alteration in favour of Napier’s, and, rejecting the tables he had already constructed, began to consider the calculation of new ones. The facts, as stated by Napier and Briggs, are in complete accordance, and the friendship existing between them was perfect and unbroken to the last. Briggs assisted Robert Napier in the editing of the “posthumous work,” theConstructio, and in the account he gives of the alteration of the logarithms in theArithmeticaof 1624 he seems to have been more anxious that justice should be done to Napier than to himself; while on the other hand Napier received Briggs most hospitably and refers to him as “amico mihi longè charissimo.”

Hutton’s suggestions are all the more to be regretted as they occur as a history which is the result of a good deal of investigation and which for years was referred to as an authority by many writers. His prejudice against Napier naturally produced retaliation, and Mark Napier in defending his ancestor has fallen into the opposite extreme of attempting to reduce Briggs to the level of a mere computer. In connexion with this controversy it should be noticed that the “Admonitio” on the last page of theDescriptio, containing the reference to the new logarithms, does not occur in all the copies. It is printed on the back of the last page of the table itself, and so cannot have been torn out from the copies that are without it. As there could have been no reason for omitting it after it had once appeared, we may assume that the copies which do not have it are those whichwere first issued. It is probable, therefore, that Briggs’s copy contained no reference to the change, and it is even possible that the “Admonitio” may have been added after Briggs had communicated with Napier. As special attention has not been drawn to the fact that some copies have the “Admonitio” and some have not, different writers have assumed that Briggs did or did not know of the promise contained in the “Admonitio” according as it was present or absent in the copies they had themselves referred to, and this has given rise to some confusion. It may also be remarked that the date frequently 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 until Mark Napier showed that the true date was 1617. When theDescriptiowas published Briggs was fifty-seven years of age, and the remaining seventeen years of his life were devoted with steady enthusiasm to extend the utility of Napier’s great invention.

The only other mathematician besides Napier who grasped the idea on which the use of logarithm depends and applied it to the construction of a table is Justus Byrgius (Jobst Bürgi), whose workArithmetische und geometrische Progress-Tabulen... was published at Prague in 1620, six years after the publication of theDescriptioof Napier. This table distinctly involves the principle of logarithms and may be described as a modified table of antilogarithms. It consists of two series of numbers, the one being an arithmetical and the other a geometrical progression: thus

In the arithmetical column the numbers increase by 10, in the geometrical column each number is derived from its predecessor by multiplication by 1.0001. Thus the number 10x in the arithmetical column corresponds to 108(1.0001)xin the geometrical column; the intermediate numbers being obtained by interpolation. If we divide the numbers in the geometrical column by 108the correspondence is between 10x and (1.0001)x, and the table then becomes one of antilogarithms, the base being (1.0001)1/10, viz. for example (l.0001)1/10·990= 1.00994967. The table extends to 230270 in the arithmetical column, and it is shown that 230270.022 corresponds to 9.9999 9999 or 109 in the geometrical column; this last result showing that (1.0001)23027.022= 10. The first contemporary mention of Byrgius’s table occurs on page 11 of the “Praecepta” prefixed to Kepler’sTabulae Radolphinae(1627); his words are: “apices logistici J. Byrgio multis annis ante editionem Neperianam viam praeiverent ad hos ipsissimos logarithmos. Etsi homo cunctator et secretorum suorum custos foetum in partu destituit, non ad usus publicos educavit.” Another reference to Byrgius occurs in a work by Benjamin Bramer, the brother-in-law and pupil of Byrgius, who, writing in 1630, says that the latter constructed his table twenty years ago or more.4

As regards priority of publication, Napier has the advantage by six years, and even fully accepting Bramer’s statement, there are grounds for believing that Napier’s work dates from a still earlier period.

The power of 10, which occurs as a factor in the tables of both Napier and Byrgius, was rendered necessary by the fact that the decimal point was not yet in use. Omitting this factor in the case of both tables, the connexion between N a number and L its “logarithm” is

N = (e−1)L(Napier), L =(1.0001)1⁄10N (Byrgius),

viz. Napier gives logarithms to base e-1, Byrgius gives antilogarithms to base (1.0001)1/10.

There is indirect evidence that Napier was occupied with logarithms as early as 1594, for in a letter to P. Crügerus from Kepler, dated September 9, 1624 (Frisch’sKepler, vi. 47), there occurs the sentence: “Nihil autem supra Neperianam rationem esse puto: etsi quidem Scotus quidam literis ad Tychonem 1594 scriptis jam spem fecit Canonis illius Mirifici.” It is here distinctly stated that some Scotsman in the year 1594, in a letter to Tycho Brahe, gave him some hope of the logarithms; and as Kepler joined Tycho after his expulsion from the island of Huen, and had been so closely associated with him in his work, he would be likely to be correct in any assertion of this kind. In connexion with Kepler’s statement the following story, told by Anthony wood in theAthenae Oxonienses, is of some importance:—

“It must be now known, that one Dr Craig, a Scotchman ... coming out of Denmark into his own country, called upon Joh. Neper, Baron of Mercheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus, as ’tis said), to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it than that it was by proportional numbers. Which hint Neper taking, he desired him at his return to call upon him again. Craig, after some weeks had passed, did so, and Neper then showed him a rude draught of what he calledCanon mirabilis logarithmorum. which draught, with some alterations, he printing in 1614, it came forthwith into the hands of our author Briggs, and into those of Will. Oughtred, from whom the relation of this matter came.”

This story, though obviously untrue in some respects, gives valuable information by connecting Dr Craig with Napier and Longomontanus, who was Tycho Brahe’s assistant. Dr Craig was John Craig, the third son of Thomas Craig, who was one of the colleagues of Sir Archibald Napier, John Napier’s father, in the office of justice-depute. Between John Craig and John Napier a friendship sprang up which may have been due to their common taste for mathematics. There are extant three letters from Dr John Craig to Tycho Brahe, which show that he was on the most friendly terms with him. In the first letter, of which the date is not given, Craig says that Sir William Stuart has safely delivered to him, “about the beginning of last winter,” the book which he sent him. Now Mark Napier found in the library of the university of Edinburgh a mathematical work bearing a sentence in Latin which he translates, “To Doctor John Craig of Edinburgh, in Scotland, a most illustrious man, highly gifted with various and excellent learning, professor of medicine, and exceedingly skilled in the mathematics, Tycho Brahe hath sent this gift, and with his own hand written this at Uraniburg, 2d November 1588.” As Sir William Stuart was sent to Denmark to arrange the preliminaries of King James’s marriage, and returned to Edinburgh on the 15th of November 1588, it would seem probable that this was the volume referred to by Craig. It appears from Craig’s letter, to which we may therefore assign the date 1589, that, five years before, he had made an attempt to reach Uranienburg, but had been baffled by the storms and rocks of Norway, and that ever since then he had been longing to visit Tycho. Now John Craig was physician to the king, and in 1590 James VI. spent some days at Uranienburg, before returning to Scotland from his matrimonial expedition. It seems not unlikely therefore that Craig may have accompanied the king in his visit to Uranienburg.5In any case it is certain that Craig was a friend and correspondent of Tycho’s, and it is probable that he was the “Scotus quidam.”

We may infer therefore that as early as 1594 Napier had communicated to some one, probably John Craig, his hope of being able to effect a simplification in the processes of arithmetic. Everything tends to show that the invention of logarithmswas the result of many years of labour and thought,6undertaken with this special object, and it would seem that Napier had seen some prospect of success nearly twenty years before the publication of theDescriptio. It is very evident that no mere hint with regard to the use of proportional numbers could have been of any service to him, but it is possible that the news brought by Craig of the difficulties placed in the progress of astronomy by the labour of the calculations may have stimulated him to persevere in his efforts.

The “new invention in Denmark” to which Anthony Wood refers as having given the hint to Napier was probably the method of calculation called prosthaphaeresis (often written in Greek lettersπροσθαφαίρεσις), which had its origin in the solution of spherical triangles.7The method consists in the use of the formula

sin a sin b =1⁄2{cos (a − b) − cos (a + b)},

by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines; and, as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only. It seems to be due to Wittich of Breslau, who was assistant for a short time to Tycho Brahe; and it was used by them in their calculations in 1582. Wittich in 1584 made known at Cassel the calculation of one case by this prosthaphaeresis; and Justus Byrgius proved it in such a manner that from his proof the extension to the solution of all triangles could be deduced.8Clavius generalized the method in his treatiseDe astrolabio(1593), lib. i. lemma liii. The lemma is enunciated as follows:—

“Quaestiones omnes, quae per sinus, tangentes, atque secantes absolvi solent, per solam prosthaphaeresim, id est, per solam additionem, subtractionem, sine laboriosa numerorum multiplicatione divisioneque expedire.”

Clavius then refers to a work of Raymarus Ursus Dithmarsus as containing an account of a particular case. The work is probably theFundamentum astronomicum(1588). Longomontanus, in hisAstronomia Danica(1622), gives an account of the method, stating that it is not to be found in the writings of the Arabs or Regiomontanus. As Longomontanus is mentioned in Anthony Wood’s anecdote, and as Wittich as well as Longomontanus were assistants of Tycho, we may infer that Wittich’s prosthaphaeresis is the method referred to by Wood.

It is evident that Wittich’s prosthaphaeresis could not be a good method of practically effecting multiplications unless the quantities to be multiplied were sines, on account of the labour of the interpolations. It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and, analytically considered, it is not so different in principle from the logarithmic method. In fact, if we put xy = φ(X + Y), X being a function of x only and Y a function of y only, we can show that we must have X = Aeqx, y = Beqy; and if we put xy = φ(X + Y) − φ(X − Y), the solutions are φ(X + Y) =1⁄4(x + y)2, and x = sin X, y = sin Y, φ(X + Y) = −1⁄2cos(X + Y). The former solution gives a method known as that of quarter-squares; the latter gives the method of prosthaphaeresis.

An account has now been given of Napier’s invention and its publication, the transition to decimal logarithms, the calculation of the tables by Briggs, Vlacq and Gunter, as well as of the claims of Byrgius and the method of prosthaphaeresis. To complete the early history of logarithms it is necessary to return to Napier’sDescriptioin order to describe its reception on the continent, and to mention the other logarithmic tables which were published while Briggs was occupied with his calculations.

John Kepler, who has been already quoted in connexion with Craig’s visit to Tycho Brahe, received the invention of logarithms almost as enthusiastically as Briggs. His first mention of the subject occurs in a letter to Schikhart dated the 11th of March 1618, in which he writes-“Extitit Scotus Baro, cujus nomen mihi excidit, qui praeclari quid praestitit, necessitate omni multiplicationum et divisionum in meras additiones et subtractiones commutata, nec sinibus utitur; at tamen opus est ipsi tangentium canone: et varietas, crebritas, difficultasque additionum subtractionumque alicubi laborem multiplicandi et dividendi superat.” This erroneous estimate was formed when he had seen theDescriptiobut had not read it; and his opinion was very different when he became acquainted with the nature of logarithms. The dedication of hisEphemerisfor 1620 consists of a letter to Napier dated the 28th of July 1619, and he there congratulates him warmly on his invention and on the benefit he has conferred upon astronomy generally and upon Kepler’s own Rudolphine tables. He says that, although Napier’s book had been published five years, he first saw it at Prague two years before; he was then unable to read it, but last year he had met with a little work by Benjamin Ursinus9containing the substance of the method, and he at once recognized the importance of what had been effected. He then explains how he verified the canon, and so found that there were no essential errors in it, although there were a few inaccuracies near the beginning of the quadrant, and he proceeds, “Haec te obiter scire volui, ut quibus tu methodis incesseris, quas non dubito et plurimas et ingeniosissimas tibi in promptu esse, eas publici juris fieri, mihi saltem (puto et caeteris) scires fore gratissimum; eoque percepto, tua promissa folio 57, in debitum cecidisse intelligeres.” This letter was written two years after Napier’s death (of which Kepler was unaware), and in the same year as that in which theConstructiowas published. In the same year (1620) Napier’sDescriptio(1614) andConstructio(1619) were reprinted by Bartholomew Vincent at Lyons and issued together.10

Napier calculated no logarithms of numbers, and, as already stated, the logarithms invented by him were not to basee. The first logarithms to the base e were published by John Speidell in hisNew 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 theDescriptioof 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.

In the same year (1624) Kepler 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 hisArithmétique logarithmétique, containing seven-figure logarithms ofnumbers up to 1000, and log sines and tangents from Gunter’sCanon(1620). In the following year, 1626, Denis Henrion published at Paris aTraicté 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 entitledNieuwe Telkonst, inhoudende de Logarithmi voor de Ghetallen beginnende van 1 tot 10,000, which contained logarithms of numbers up to 10,000 to 10 places, taken from Briggs’sArithmeticaof 1624, and Gunter’s log sines and tangents to 7 places for every minute.11Vlacq rendered assistance in the publication of this work, and the privilege is made out to him.

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 thePrincipiaof 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’sDescriptio. 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 theDescriptioappeared. In the construction of the natural trigonometrical tables Great Britain 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.

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’sLives of the Professors of Gresham College(London, 1740); Thomas Smith’sVitae quorundam eruditissimorum et illustrium virorum(Vita Henrici Briggii) (London, 1707); Mark Napier’sMemoirs of John Napieralready referred to, and the same author’sNaperi libri qui supersunt(1839); Hutton’sHistory; de Morgan’s article already referred to; Delambre’sHistoire de l’Astronomie moderne; the report on mathematical tables in theReport of the British Associationfor 1873; and thePhilosophical Magazinefor 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 Mark Napier that the true date is 1617.

In the years 1791-1807 Francis Maseres published at London, in six volumes quarto “Scriptores Logarithmici, or a collection of several curious tracts on the nature and construction of logarithms, mentioned in Dr Hutton’s historical introduction to his new edition of Sherwin’s mathematical tables ...,” which contains reprints of Napier’sDescriptioof 1614, Kepler’s writings on logarithms (1624-1625), &c. In 1889 a translation of Napier’sConstructioof 1619 was published by Walter Rae Macdonald. Some valuable notes are added by the translator, in one of which he shows the accuracy of the method employed by Napier in his calculations, and explains the origin of a small error which occurs in Napier’s table. Appended to the Catalogue is a full and careful bibliography of all Napier’s writings, with mention of the public libraries, British and foreign, which possess copies of each. A facsimile reproduction of Bartholomew Vincent’s Lyons edition (1620) of theConstructiowas issued in 1895 by A. Hermann at Paris (this imprint occurs on page 62 after the word “Finis”).

It now remains to notice briefly a few of the more important events in the history of logarithmic tables subsequent to the original calculations.

Common or Briggian Logarithms of Numbers.—Nathaniel Roe’sTabulae logarithmicae(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 mantissae, 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 head 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 top 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 hisTrigonometriaBritannica(1658), a work which is also noticeable as being the only extensive eight-figure table that until recently had 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 18th century, and was at length superseded in 1785 by Hutton’s tables, which continued in successive editions to maintain their position for a century.In 1717 Abraham Sharp published in hisGeometry Improv’dthe 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 François 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, passed through many editions.In 1794 Vega published hisThesaurus logarithmorum completus, a folio volume containing a reprint of the logarithms of numbers from Vlacq’sArithmetica logarithmicaof 1628, andTrigonometria artificialisof 1633. The logarithms of numbers are arranged as in an ordinary seven-figure table. In addition to the logarithms reprinted from theTrigonometria, 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. Vega’sThesaurushas been reproduced photographically by the Italian government. Vega 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.12In 1871 Edward Sang published a seven-figure table of logarithms of numbers from 20,000 to 200,000, the logarithms between 100,000 and 200,000 being the result of a new calculation. 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.13John Thomson of Greenock (1782-1855) made an independent calculation of logarithms of numbers up to 120,000 to 12 places of decimals, and his table has been used to verify the errata already found in Vlacq and Briggs by Lefort (seeMonthly Not. R.A.S.vol. 34, p. 447). A table of ten-figure logarithms of numbers up to 100,009 was calculated by W. W. Duffield and published in theReport of the U.S. Coast and Geodetic Survey for 1895-1896as Appendix 12, pp. 395-722. The results were compared with Vega’sThesaurus(1794) before publication.Common or Briggian 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 theTrigonometriato 10 places and then contracted to 7. On account of the great size of this table, and for other reasons, it nevercame into very general use, Bagay’sNouvelles tables astronomiques(1829), which also contains log sines and tangents to every second, being preferred; this latter work, which for many years was difficult to procure, has been reprinted with the original title-page and date unchanged. The only other logarithmic canon to every second that has been published forms the second volume of Shortrede’sLogarithmic Tables(1849). In 1784 the French government decided 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 à composer des tables qui ne laissassent rien à désirer quant à l’exactitude, mais à en faire le monument de calcul le plus vaste et le plus imposant qui eût jamais été exécuté ou même 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, one 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 200,0008vols.Natural sines1”Logarithms of the ratios of arcs to sines from 0q.00000to 0q.05000, and log sines throughout the quadrant4”Logarithms of the ratios of arcs to tangents from0q.00000 to 0q.05000, and log tangents throughoutthe quadrant4”The trigonometrical results are given for every hundred-thousandth of the quadrant (10″ centesimal 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 theTables du Cadastre, or, in England, as the great French manuscript tables.A very full account of these tables, with an explanation of the methods of calculation, formulae employed, &c., was published by Lefort in vol. iv. of theAnnales de l’observatoire de Paris. The printing of the table of natural sines was once begun, and Lefort states that he has seen six copies, all incomplete, although including the last page. Babbage compared his table with theTables du Cadastre, and 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 theArithmetica logarithmicaof 1624 and of 1628.As theTables du Cadastreremained unpublished, other tables appeared in which the quadrant was divided centesimally, the most important of these being Hobert and Ideler’sNouvelles tables trigonométriques(1799), and Borda and Delambre’sTables trigonométriques décimales(1800-1801), both of which are seven-figure tables. The latter work, which was much used, being difficult to procure, and greater accuracy being required, the French government in 1891 published an eight-figure centesimal table, for every ten seconds, derived from theTables du Cadastre.Decimal or Briggian Antilogarithms.—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’sAntilogarithmic canon(1742), which gives the numbers to 11 places, corresponding to the logarithms from .00001 to .99999 at intervals of .00001. Antilogarithmic tables are few in number, the only other extensive tables of the same kind that have been published occurring in Shortrede’sLogarithmic tablesalready referred to, and in Filipowski’sTable of antilogarithms(1849). Both are similar to Dodson’s tables, from which they were derived, but they only give numbers to 7 places.Hyperbolic or Napierian logarithms(i.e.to basee).—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’sNeue und erweiterte Sammlung logarithmischer Tafeln(1778), and was reprinted in Vega’sThesaurus(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 titleTafel der natürlichen Logarithmen der Zahlen.Hyperbolic antilogarithmsare simple exponentials,i.e.the hyperbolic antilogarithm ofxisex. Such tables can scarcely be said to come under the head of logarithmic tables. SeeTables, Mathematical:Exponential Functions.Logistic or Proportional Logarithms.—The old name for what are now called ratios or fractions arelogistic numbers, so that 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 work of 1624 which has been 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 1h), 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 3h), 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 a is 3600″, and proportional when a has any other value.Addition and Subtraction, or Gaussian Logarithms.—Gaussian logarithmsare 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 entitledSupplément logarithmique, printed at Bordeaux in the year XI. (1802/3); he calculated a table to 14 places, but only a specimen of it which appeared in theSupplémentwas printed. The first table that was actually published is due to Gauss, and was printed in Zach’sMonatliche Correspondenz, xxvi. 498 (1812). Corresponding to the argument log x it gives the values of log (1 + x−1) and log (1 + x).Dual Logarithms.—This term was used by Oliver Byrne in a series of works published between 1860 and 1870. Dual numbers and logarithms depend upon the expression of a number as a product of 1.1, 1.01, 1.001 ... or of .9, .99, .999....In the precedingrésuméonly those publications have been mentioned which are of historic importance or interest.14For fuller details with respect to some of these works, for an account of tables published in the latter part of the 19th century, and for those which would now be used in actual calculation, reference should be made to the articleTables, Mathematical.Calculation of Logarithms.—The name logarithm is derived from the wordsλόγων ἀριθμός, 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, orratiunculae, 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, orratiuncula, 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 a301,030to 1, and so on; or, in other words, if a denotes the millionth root of 10, then 2 will be nearly equal to a301,030, 3 will be nearly equal to a477,121, and so on.Napier’s original work, theDescriptio Canonisof 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 was given in theConstructio Canonisof 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 theArithmetica Logarithmica, although some of the latter are the same in principle as the processes described in an appendix to theConstructio.The processes used by Briggs are explained by him in the preface to theArithmetica 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:—Numbers.Logarithms,1013.162277...0.51.778279...0.251.333521...0.1251.154781...0.0625each 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 halfof 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. 0.00000 00000 00000 07318 55936 90623 9336, which multiplied by 247gave 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=1,10h− 1loge10the value of h being 1/254, and in the second process log102 is in effect calculated from the formula.log102 =(210/2247− 1)×1×247.loge1010Briggs 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 =1.0000000.0000000B =10.0000001.0000000C = √(AB)=3.1622770.5000000D = √(BC)=5.6234130.7500000E = √(CD)=4.2169640.6250000F = √(DE)=4.8696740.6875000G = √(DF)=5.2329910.7187500H = √(FG)=5.0480650.7031250I = √(FH)=4.9580690.6953125K = √(HI)=5.0028650.6992187L = √(IK)=4.9804160.6972656M = √(KL)=4.9916270.6982421N = √(KM)=4.9972420.6987304O = √(KN)=5.0000520.6989745P = √(NO)=4.9986470.6988525Q = √(OP)=4.9993500.6989135R = √(OQ)=4.9997010.6989440S = √(OR)=4.9998760.6989592T = √(OS)=4.9999630.6989668V = √(OT)=5.0000080.6989707W = √(TV)=4.9999840.6989687X = √(WV)=4.9999970.6989697Y = √(VX)=5.0000030.6989702Z = √(XY)=5.0000000.6989700Great 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 hisMathematical Tables, and also forms tract 21 of hisMathematical Tracts(vol. i., 1812). Many of the early works on logarithms were reprinted in theScriptores logarithmiciof Baron Maseres already referred to.In the following account only those formulae and methods will be referred to which would now be used in the calculation of logarithms.Sinceloge(1 + x) = x −1⁄2x2+1⁄3x3−1⁄4x4+ &c.,we have, by changing the sign of x,loge(1 − x) = −x −1⁄2x2−1⁄3x3−1⁄4x4− &c.;whenceloge1 + x= 2 (x +1⁄3x3+1⁄5x5+ &c.),1 − xand, therefore, replacing x by (p − q)/(p + q),logep= 2{p − q+1⁄3(p − q)3+1⁄5(p − q)5+ &c.},qp + qp + qp + qin 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,logep = 2{p − 1+1⁄3(p − 1)3+1⁄5(p − 1)5+ &c.},p + 1p + 1p + 1and by putting q = p + 1,loge(p + 1) − logep = 2{1+1⁄31+1⁄51+ &c.};2p + 1(2p + 1)3(2p + 1)5the former of these equations gives a convergent series for logep, 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 loge(p/q) we may deduce the following very convergent series for loge2, loge3 and loge5, viz.:—loge2 = 2 (7P + 5Q + 3R),loge3 = 2 (11P + 8Q + 5R),loge5 = 2 (16P + 12Q + 7R),whereP =1+1⁄3·1+1⁄5·1+ &c.31(31)3(31)5Q =1+1⁄3·1+1⁄5·1+ &c.49(49)3(49)5R =1+1⁄3·1+1⁄5·1+ &c.161(161)3(161)5The following still more convenient formulae for the calculation of loge2, loge3, &c. were given by J. Couch Adams in theProc. Roy. Soc., 1878, 27, p. 91. Ifa = log10= −log(1 −1), b = log25= −log(1 −4),91024100c = log81= log(1 +1), d = log50= −log(1 −2),808049100e = log126= log(1 +8),1251000thenlog 2 = 7a − 2b + 3c, log 3 = 11a − 3b + 5c, log 5 = 16a − 4b + 7c,andlog 7 =1⁄2(39a − 10b + 17c − d) or = 19a − 4b + 8c + e,and we have the equation of condition,a − 2b + c = d + 2e.By means of these formulae Adams calculated the values of loge2, loge3, loge5, and loge7 to 276 places of decimals, and deduced the value of loge10 and its reciprocal M, the modulus of the Briggian system of logarithms. The value of the modulus found by Adams isMo = 0.4342944819032518276511289189166050822943970058036665661144537831658646492088707747292249493384317483187061067447663037336416792871589639065692210646628122658521270865686703295933708696588266883311636077384905142844348666768646586085135561482123487653435434357317253835622186825which is true certainly to 272, and probably to 273, places (Proc. Roy. Soc., 1886, 42, p. 22, where also the values of the other logarithms are given).If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus,log10(1 + x) = M (x −1⁄2x2+1⁄3x3−1⁄4x4+ &c.),and so on.As has been stated, Abraham Sharp’s table contains 61-decimalBriggian 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’sTable des diviseursfor a number near to this which shall have no prime factor greater than 10,009, it appears that2,193,349 = 23 × 47 × 2029;thus43,867 =1⁄50(23 × 47 × 2029 + 1),and thereforeloge43,867 = loge23 + loge47 + loge2029 − loge50+1−1⁄21+1⁄31− &c.2,193,349(2,193,349)2(193,349)3The first term of the series in the second line is0.00000 04559 23795 07319 6286;dividing this by 2 × 2,193,349 we obtain0.00000 00000 00103 93325 3457,and the third term is0.00000 00000 00000 00003 1590,so that the series =0.00000 04559 23691 13997 4419;whence, taking out the logarithms from Wolfram’s table,loge43,867 = 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 formulaloge(x + d) = logex +d−1⁄2d2+1⁄3d3− &c.,xx2x3in 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 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 π is given by Burckhardt in the introduction to hisTable des diviseursfor 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 − .1rn, 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 105and by 5 the number becomes 1.087678, and resolving this number into factors of the form 1 − .1rn we find that543839 = 105× 5(1 − .128) (1 − .146) (1 − .156) (1 − .163) (1 − .173)× (1 − .185) (1 − .197) (1 − .1109) (1 − .1113) (1 − .1122),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 − .1rn is generally known as Weddle’s method, having been published by him inThe Mathematicianfor 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 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 devised a method of applying a table of Hearn’s form (i.e.of factors of the form 1 + .1rn) to the construction of logarithms, and calculated a table of logarithms of factors of the form 1 + (.001)rn to 24 places. This was published in 1876 under the titleTables 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 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)7340Taking the logarithms from Gray’s tables we obtain the required logarithm by addition as follows:—522878745280337562704972 = colog 3119255889277936685553913 = log (1)3040050733157610239 = log (2)259708022525453597 = log (3)338749695752424 = log (4)868588964 = log (5)261445278 = log (6)178929 = log (7)148 = log (8)4.642137934655780757288464 = log1043,867In Shortrede’sTablesthere 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 hisTables de Logarithmes à 27 Décimales(Paris, 1867) Fédor Thoman gives tables of logarithms of factors of the form 1 ± .1rn. In theMessenger of Mathematics, vol. iii. pp. 66-92, 1873, 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 ± .1rn 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 theArithmetica logarithmicaof 1624 a table of the logarithms of 1 + .1rn up to r = 9 to 15 places of decimals. It was first formally proposed as an independent method, with great improvements, by Robert Flower inThe Radix,a new way of making Logarithms, which was published in 1771; and Leonelli, in hisSupplement logarithmique(1802-1803), already noticed, referred to Flower and reproduced some of his tables. A complete bibliography of this method has been given by A. J. Ellis in a paper “on the potential radix as a means of calculating logarithms,” printed in theProceedings of the Royal Society, vol. xxxi., 1881, pp. 401-407, and vol. xxxii., 1881, pp. 377-379. Reference should also be made to Hoppe’sTafeln zur dreissigstelligen logarithmischen Rechnung(Leipzig, 1876), which give in a somewhat modified form a table of the hyperbolic logarithm of 1 + .1rn.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 theTables du Cadastreis given by Lefort in vol. iv. of theAnnales de l’Observatoire de Paris.

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 18th century, and was at length superseded in 1785 by Hutton’s tables, which continued in successive editions to maintain their position for a century.

In 1717 Abraham Sharp published in hisGeometry Improv’dthe 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 François 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, passed through many editions.

In 1794 Vega published hisThesaurus logarithmorum completus, a folio volume containing a reprint of the logarithms of numbers from Vlacq’sArithmetica logarithmicaof 1628, andTrigonometria artificialisof 1633. The logarithms of numbers are arranged as in an ordinary seven-figure table. In addition to the logarithms reprinted from theTrigonometria, 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. Vega’sThesaurushas been reproduced photographically by the Italian government. Vega 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.12

In 1871 Edward Sang published a seven-figure table of logarithms of numbers from 20,000 to 200,000, the logarithms between 100,000 and 200,000 being the result of a new calculation. 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.13John Thomson of Greenock (1782-1855) made an independent calculation of logarithms of numbers up to 120,000 to 12 places of decimals, and his table has been used to verify the errata already found in Vlacq and Briggs by Lefort (seeMonthly Not. R.A.S.vol. 34, p. 447). A table of ten-figure logarithms of numbers up to 100,009 was calculated by W. W. Duffield and published in theReport of the U.S. Coast and Geodetic Survey for 1895-1896as Appendix 12, pp. 395-722. The results were compared with Vega’sThesaurus(1794) before publication.

Common or Briggian 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 theTrigonometriato 10 places and then contracted to 7. On account of the great size of this table, and for other reasons, it nevercame into very general use, Bagay’sNouvelles tables astronomiques(1829), which also contains log sines and tangents to every second, being preferred; this latter work, which for many years was difficult to procure, has been reprinted with the original title-page and date unchanged. The only other logarithmic canon to every second that has been published forms the second volume of Shortrede’sLogarithmic Tables(1849). In 1784 the French government decided 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 à composer des tables qui ne laissassent rien à désirer quant à l’exactitude, mais à en faire le monument de calcul le plus vaste et le plus imposant qui eût jamais été exécuté ou même 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, one 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:—

The trigonometrical results are given for every hundred-thousandth of the quadrant (10″ centesimal 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 theTables du Cadastre, or, in England, as the great French manuscript tables.

A very full account of these tables, with an explanation of the methods of calculation, formulae employed, &c., was published by Lefort in vol. iv. of theAnnales de l’observatoire de Paris. The printing of the table of natural sines was once begun, and Lefort states that he has seen six copies, all incomplete, although including the last page. Babbage compared his table with theTables du Cadastre, and 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 theArithmetica logarithmicaof 1624 and of 1628.

As theTables du Cadastreremained unpublished, other tables appeared in which the quadrant was divided centesimally, the most important of these being Hobert and Ideler’sNouvelles tables trigonométriques(1799), and Borda and Delambre’sTables trigonométriques décimales(1800-1801), both of which are seven-figure tables. The latter work, which was much used, being difficult to procure, and greater accuracy being required, the French government in 1891 published an eight-figure centesimal table, for every ten seconds, derived from theTables du Cadastre.

Decimal or Briggian Antilogarithms.—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’sAntilogarithmic canon(1742), which gives the numbers to 11 places, corresponding to the logarithms from .00001 to .99999 at intervals of .00001. Antilogarithmic tables are few in number, the only other extensive tables of the same kind that have been published occurring in Shortrede’sLogarithmic tablesalready referred to, and in Filipowski’sTable of antilogarithms(1849). Both are similar to Dodson’s tables, from which they were derived, but they only give numbers to 7 places.

Hyperbolic or Napierian logarithms(i.e.to basee).—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’sNeue und erweiterte Sammlung logarithmischer Tafeln(1778), and was reprinted in Vega’sThesaurus(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 titleTafel der natürlichen Logarithmen der Zahlen.

Hyperbolic antilogarithmsare simple exponentials,i.e.the hyperbolic antilogarithm ofxisex. Such tables can scarcely be said to come under the head of logarithmic tables. SeeTables, Mathematical:Exponential Functions.

Logistic or Proportional Logarithms.—The old name for what are now called ratios or fractions arelogistic numbers, so that 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 work of 1624 which has been 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 1h), 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 3h), 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 a is 3600″, and proportional when a has any other value.

Addition and Subtraction, or Gaussian Logarithms.—Gaussian logarithmsare 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 entitledSupplément logarithmique, printed at Bordeaux in the year XI. (1802/3); he calculated a table to 14 places, but only a specimen of it which appeared in theSupplémentwas printed. The first table that was actually published is due to Gauss, and was printed in Zach’sMonatliche Correspondenz, xxvi. 498 (1812). Corresponding to the argument log x it gives the values of log (1 + x−1) and log (1 + x).

Dual Logarithms.—This term was used by Oliver Byrne in a series of works published between 1860 and 1870. Dual numbers and logarithms depend upon the expression of a number as a product of 1.1, 1.01, 1.001 ... or of .9, .99, .999....

In the precedingrésuméonly those publications have been mentioned which are of historic importance or interest.14For fuller details with respect to some of these works, for an account of tables published in the latter part of the 19th century, and for those which would now be used in actual calculation, reference should be made to the articleTables, Mathematical.

Calculation of Logarithms.—The name logarithm is derived from the wordsλόγων ἀριθμός, 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, orratiunculae, 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, orratiuncula, 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 a301,030to 1, and so on; or, in other words, if a denotes the millionth root of 10, then 2 will be nearly equal to a301,030, 3 will be nearly equal to a477,121, and so on.

Napier’s original work, theDescriptio Canonisof 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 was given in theConstructio Canonisof 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 theArithmetica Logarithmica, although some of the latter are the same in principle as the processes described in an appendix to theConstructio.

The processes used by Briggs are explained by him in the preface to theArithmetica 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:—

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 halfof 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. 0.00000 00000 00000 07318 55936 90623 9336, which multiplied by 247gave 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.

the value of h being 1/254, and in the second process log102 is in effect calculated from the formula.

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.

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 hisMathematical Tables, and also forms tract 21 of hisMathematical Tracts(vol. i., 1812). Many of the early works on logarithms were reprinted in theScriptores logarithmiciof Baron Maseres already referred to.

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

Since

loge(1 + x) = x −1⁄2x2+1⁄3x3−1⁄4x4+ &c.,

we have, by changing the sign of x,

loge(1 − x) = −x −1⁄2x2−1⁄3x3−1⁄4x4− &c.;

whence

and, therefore, replacing x by (p − q)/(p + q),

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,

and by putting q = p + 1,

the former of these equations gives a convergent series for logep, 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 loge(p/q) we may deduce the following very convergent series for loge2, loge3 and loge5, viz.:—

loge2 = 2 (7P + 5Q + 3R),loge3 = 2 (11P + 8Q + 5R),loge5 = 2 (16P + 12Q + 7R),

loge2 = 2 (7P + 5Q + 3R),

loge3 = 2 (11P + 8Q + 5R),

loge5 = 2 (16P + 12Q + 7R),

where

The following still more convenient formulae for the calculation of loge2, loge3, &c. were given by J. Couch Adams in theProc. Roy. Soc., 1878, 27, p. 91. If

then

log 2 = 7a − 2b + 3c, log 3 = 11a − 3b + 5c, log 5 = 16a − 4b + 7c,

and

log 7 =1⁄2(39a − 10b + 17c − d) or = 19a − 4b + 8c + e,

and we have the equation of condition,

a − 2b + c = d + 2e.

By means of these formulae Adams calculated the values of loge2, loge3, loge5, and loge7 to 276 places of decimals, and deduced the value of loge10 and its reciprocal M, the modulus of the Briggian system of logarithms. The value of the modulus found by Adams is

which is true certainly to 272, and probably to 273, places (Proc. Roy. Soc., 1886, 42, p. 22, where also the values of the other logarithms are given).

If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus,

log10(1 + x) = M (x −1⁄2x2+1⁄3x3−1⁄4x4+ &c.),

and so on.

As has been stated, Abraham Sharp’s table contains 61-decimalBriggian 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’sTable des diviseursfor 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 =1⁄50(23 × 47 × 2029 + 1),

and therefore

loge43,867 = loge23 + loge47 + loge2029 − loge50

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,

loge43,867 = 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

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 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 π is given by Burckhardt in the introduction to hisTable des diviseursfor 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 − .1rn, 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 105and by 5 the number becomes 1.087678, and resolving this number into factors of the form 1 − .1rn we find that

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 − .1rn is generally known as Weddle’s method, having been published by him inThe Mathematicianfor 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 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 devised a method of applying a table of Hearn’s form (i.e.of factors of the form 1 + .1rn) to the construction of logarithms, and calculated a table of logarithms of factors of the form 1 + (.001)rn to 24 places. This was published in 1876 under the titleTables 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 Gray’s process that the factors of 1.31601 are

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

In Shortrede’sTablesthere 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 hisTables de Logarithmes à 27 Décimales(Paris, 1867) Fédor Thoman gives tables of logarithms of factors of the form 1 ± .1rn. In theMessenger of Mathematics, vol. iii. pp. 66-92, 1873, 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 ± .1rn 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 theArithmetica logarithmicaof 1624 a table of the logarithms of 1 + .1rn up to r = 9 to 15 places of decimals. It was first formally proposed as an independent method, with great improvements, by Robert Flower inThe Radix,a new way of making Logarithms, which was published in 1771; and Leonelli, in hisSupplement logarithmique(1802-1803), already noticed, referred to Flower and reproduced some of his tables. A complete bibliography of this method has been given by A. J. Ellis in a paper “on the potential radix as a means of calculating logarithms,” printed in theProceedings of the Royal Society, vol. xxxi., 1881, pp. 401-407, and vol. xxxii., 1881, pp. 377-379. Reference should also be made to Hoppe’sTafeln zur dreissigstelligen logarithmischen Rechnung(Leipzig, 1876), which give in a somewhat modified form a table of the hyperbolic logarithm of 1 + .1rn.

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 theTables du Cadastreis given by Lefort in vol. iv. of theAnnales de l’Observatoire de Paris.

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