of a number of five figures is therefore taken out at once, and two more figures may be interpolated for as in logarithms. R. Picarte, La Division réduite à une addition (Paris, 1861), gives to ten significant figures the reciprocals of the numbers from 10,000 to 100,000, and also the first nine multiples of these reciprocals. J. C. Houzeau gives the reciprocals of numbers up to 100 to 20 places and their first nine multiples to 12 places in the Bulletin of the Brussels Academy, 1875, 40, p. 107. E. Gélin (Recueil de tables numériques, Huy, 1894) gives reciprocals of numbers to 1000 to 10 places.

Tables for the Expression of Vulgar Fractions as Decimals.— Tables of this kind have been given by Wucherer, Goodwyn and Gauss. W. F. Wucherer, Beyträge zum allgemeinern Gebrauch der Decimalbrüche (Carlsruhe, 1796), gives the decimal fractions (to 5 places) for all vulgar fractions whose numerator and denominator are each less than 50 and prime to one another, arranged according to denominators. The most extensive and claborate tables that have been published are contained in Henry Goodwyn's First Centenary of Tables of all Decimal Quotients (London, 1816), A Tabular Series of Decimal Quotients (1823), and A Table of the Circles arising from the Division of a Unit or any other Whole Number by all the Integers from 1 to 1024 (1823). The Tabular Series (1823), which occupies 153 pages, gives to 8 places the decimal corresponding to every vulgar fraction less than whose numerator and denomi nator do not surpass 1000. The arguments are not arranged according to their numerators or denominators, but according to their magnitude, so that the tabular results exhibit a steady increase from 001 (bo) to 09989909 (=). The author intended the table to include all fractions whose numerator and denominator were each less than 1000, but no more was ever published. The Table of Circles (1823) gives all the periods of the circulating decimals that can arise from the division of any integer by another integer less than 1024. Thus for 13 we find 876923 and 153846, which are the only periods in which fraction whose denominator is 13 can circulate. The table occupies 107 pages, some of the periods being of course very long (e.g., for 1021 the period contains 1020 figures). The First Centenary (1816) gives the complete periods of the reciprocals of the numbers from 1 to 100. Goodwyn's tables are very scarce, but as they are nearly unique of their kind they deserve special notice. A second edition of the First Centenary was issued in 1818 with the addition of some of the Tabular Series, the numerator not exceeding 50 and the denominator not exceeding 100. A posthumous table of C. F. Gauss's, entitled "Tafel zur Verwandlung gemeiner Brüche mit Nennern aus dem ersten Tausend in Decimalbrüche," occurs in vol. ii. pp. 412-434 of his Gesammelte Werke (Göttingen, 1863), and resembles Goodwyn's Table of Circles. On this subject see a paper "On Circulating Decimals, with special reference to Henry Goodwyn's Table of Circles and Tabular Series of Decimal Quotients," in Camb. Phil. Proc., 1878, 3, p. 185, where is also given a table of the numbers of digits in the periods of fractions corresponding to denominators prime to 10 from 1 to 1024 obtained by counting from Goodwyn's table. See also under Circulating Decimals (below).

Sexagesimal and Sexcentenary Tables.-Originally all calculations were sexagesimal; and the relics of the system still exist in the division of the degree into 60 minutes and the minute into 60 seconds. To facilitate interpolation, therefore, in trigonometrical and other tables the following large sexagesimal tables were constructed. John Bernoulli, A Sexcentenary Table (London, 1779), gives at once the fourth term of any proportion of which the first term is 600" and each of the other two is less than 600"; the table is of double entry, and may be described as giving the value of xy/600 correct to tenths of a second, x and y each containing a number of seconds less than 600. Michael Taylor, A Sexagesimal Table (London, 1780), exhibits at sight the fourth term of any proportion where the first term is 60 minutes, the second any number of minutes less than 60, and the third any number of minutes and seconds under 60 minutes; there is also another table in which the third term is any absolute number under 1000. Not much use seems to have been made of these tables, both of which were published by the Commissioners of Longitude. Small tables for the conversion of sexagesimals into centesimals and vice versa are given in a few collections, such as Hülsse's edition of Vega. H. Schubert's Fünfstellige Tafeln und Gegentafeln (Leipzig, 1897) contains a sexagesimal table giving xy/60 for x=1 to 59 and y = 1 to 150.

Trigonometrical Tables (Natural).-Peter Apian published in 1533 a table of sines with the radius divided decimally. The first complete canon giving all the six ratios of the sides of a right angled triangle is due to Rheticus (1551), who also introduced the semiquadrantal arrangement. Rheticus's canon was calculated for every ten minutes to 7 places, and Victa extended it to every minute (1579). In 1554 Reinhold published a table of tangents to every minute. The first complete canon published in England was by Thomas Blundeville (1594), although a table of sines had appeared four years earlier. Regiomontanus called his table of tangents (or rather cotangents) tabula foecunda on account of its great use; and till the introduction of the word " tangent "by Thomas Finck (Geometriae rotundi libri XIV., Basel, 1583) a table of tangents was called a tabula foecunda or

canon foecundus. Besides tangent," Finck also introduced the
word secant," the table of secants having previously been called
tabula benefica by Maurolycus (1558) and tabula foecundissima by
Vieta.
By far the greatest computer of pure trigonometrical tables is
George Joachim Rheticus, whose work has never been superseded.
His celebrated ten-decimal canon, the Opus palatinum, was pub-
lished by Valentine Otho at Neustadt in 1596, and in 1613 his
fifteen-decimal table of sines by Pitiscus at Frankfort under the
title Thesaurus mathematicus. The Opus palatinum contains a
complete ten-decimal trigonometrical canon for every ten seconds
of the quadrant, semiquadrantally arranged, with differences for
all the tabular results throughout. Sines, cosines, and secants are
given on the left-hand pages in columns headed respectively "Per-
pendiculum," Basis,' Hypotenusa," and on the right-hand
pages appear tangents, cosecants, and cotangents in columns
headed respectively Perpendiculum," "Hypotenusa," "Basis."
At his death Rheticus left the canon nearly complete, and the
trigonometry was finished and the whole edited by Valentine
Otho; it was named in honour of the elector palatine Frederick IV.,
who bore the expense of publication. The Thesaurus of 1613 gives
natural sines for every ten seconds throughout the quadrant, to
15 places, semiquadrantally arranged, with first, second, and third
differences. Natural sines are also given for every second from
0° to 1° and from 89° to 90°, to 15 places, with first and second
differences. The rescue of the manuscript of this work by Pitiscus
forms a striking episode in the history of mathematical tables.
The alterations and emendations in the earlier part of the cor-
rected edition of the Opus palatinum were made by Pitiscus, who
had his suspicions that Rheticus had himself calculated a ten-
second table of sincs to 15 decimal places; but it could not be
found. Eventually the lost canon was discovered amongst the
papers of Rheticus which had passed from Otho to James Christ-
mann on the death of the former. Amongst these Pitiscus found
(1) the ten-second table of sines to 15 places, with first, second,
and third differences (printed in the Thesaurus); (2) sines for
every second of the first and last degrees of the quadrant, also
to 15 places, with first and second differences; (3) the commence-
ment of a canon of tangents and secants, to the same number of
decimal places, for every ten seconds, with first and second differ-
ences; (4) a complete minute canon of sines, tangents, and secants,
also to 15 decimal places. This list, taken in connexion with
the Opus palatinum, gives an idea of the enormous labours under-
taken by Rheticus; his tables not only remain to this day the
ultimate authorities but formed the data from which Vlacq calcu
lated his logarithmic canon. Pitiscus says that for twelve years
Rheticus constantly had computers at work.

A history of trigonometrical tables by Charles Hutton was prefixed to all the early editions of his Tables of Logarithms, and forms Tract xix. of his Mathematical Tracts, vol. i. p. 278, 1812. A good deal of bibliographical information about the Opus palatinum and earlier trigonometrical tables is given in A. De Morgan's article "Tables in the English Cyclopaedia. The invention of logarithms the year after the publication of Rheticus's volume by Pitiscus changed all the methods of calculation; and it is worthy of note that John Napier's original table of 1614 was a logarithmic canon of sines and not a table of the logarithms of numbers. The logarithmic canon at once superseded the natural canon; and since Pitiscus's time no really extensive table of pure trigonometrical functions has appeared. In recent years the employment of calculating machines has revived the use of tables of natural trigonometrical functions, it being found convenient for some purposes to employ such a machine in connexion with a natural canon instead of using a logarithmic canon. A. Junge's Tafel der wirklichen Länge der Sinus und Cosinus (Leipzig, 1864) was published with this object. It gives natural sines and cosines for every ten seconds of the quadrant to 6 places. F. M. Clouth, Tables pour le calcul des coordonnées goniométriques (Mainz, n.d.), gives natural sines and cosines (to 6 places) and their first nine multiples (to 4 places) for every centesimal minute of the quadrant. Tables of natural functions occur in many collections, the natural and logarithmic values being sometimes given on opposite pages, sometimes side by side on the same page.

The following works contain tables of trigonometrical functions other than sines, cosines, and tangents. J. Pasquich, Tabuiae logarithmico-trigonometricae (Leipzig, 1817), contains a table of sinx, cosx, tanx, cot'x from x=1° to 45° at intervals of 1' to 5 places. J. Andrew, Astronomical and Nautical Tables (London, 1805), contains a table of "squares of natural semichords," ie. of sinx from x=0° to 120° at intervals of 10" to 7 places. This table was greatly extended by Major-General Hannyngton in his Haversines, Natural and Logarithmic, used in computing Lunar Distances for the Nautical Almanac (London, 1876). The name "haversine," frequently used in works upon navigation, is an abbreviation of "half versed sine"; viz., the haversine of x is equal to (1-cos x), that is, to sinx. The table gives logarithmic haversines for every 15" from o° to 180°, and natural haversines for every 10" from 0° to 180°, to 7 places, except near the beginning, where the logarithms are given to only 5 or 6 places. It occupies

S. Pineto's Tables de logarithmes vulgaires à dix décimales, construites d'après un nouveau mode (St Petersburg, 1871), though a tract of only 80 pages, may be usefully employed when Vlacq and Vega are unprocurable. Pineto's work consists of three tables: the first, or auxiliary table, contains a series of factors by which the numbers whose logarithms are required are to be multiplied to bring them within the range of table 2; it also gives the logarithms of the reciprocals of these factors to 12 places Table 1 merely gives logarithms to 1000 to 10 places. Table 2 gives logarithms from 1,000,000 to 1,011,000, with proportional parts to hundredths. The mode of using these tables is as follows. If the logarithm cannot be taken out directly from table 2, a factor M is found from the auxiliary table by which the number must be multiplied to bring it within the range of table 2 Then the logarithm can be taken out, and, to neutralize the effect of the multiplication, so far as the result is concerned, log 1/M must be added; this quantity is therefore given in an adjoining column to M in the auxiliary table. A similar procedure gives the number answering to any logarithm, another factor (approximately the reciprocal of M) being given, so that in both cases multiplication is used. The laborious part of the work is the multiplication by M; but this is somewhat compensated for by the ease with which, by means of the proportional parts, the logarithm is taken out. The factors are 300 in number, and are chosen so as to minimize the labour, only 25 of the 300 consisting of three figures all different and not involving o or 1. The principle of multiplying by a factor which is subsequently cancelled by subtracting its logby A. Namur and P. Mansion at Brussels in 1877 under the title Tables de logarithmes à 12 décimales jusqu'à 434 milliards. Here a table is given of logarithms of numbers near to 434,294, and other numbers are brought within the range of the table by multiplication by one or two factors. The logarithms of the numbers near to 434,294 are selected for tabulation because their differences commence with the figures 100. and the presence of the zeros in the difference renders the interpolation easy.

327 folio pages, and was suggested by Andrew's work, a copy | superintendent of the survey. The table was compared with Vega's of which by chance fell into Hannyngton's hands Hannyng- Thesaurus before publication. ton recomputed the whole of it by a partly mechanical method, a combination of two arithmometers being employed. A table of haversines is useful for the solution of spherical triangles when two sides and the included angle are given, and in other problems in spherical trigonometry. Andrew's original table seems to have attracted very little notice. Hannyngton's was printed, on the recommendation of the superintendent of the Nautical Almanac office, at the public cost. Before the calculation of Hannyngton's table R. Farley's Natural Versed Sines (London, 1856) was used in the Nautical Almanac office in computing lunar distances. This fine table contains natural versed sines from o° to 125° at intervals of 10 to 7 places, with proportional parts, and log versed sines from o° to 135° at intervals of 15" to 7 places. The arguments are also given in time. The manuscript was used in the office for twenty-five years before it was printed. Traverse tables, which occur in most collections of navigation tables, contain multiples of sines and cosines. Common or Briggian Logarithms of Numbers and Trigonometrical Ratios. For an account of the invention and history of logarithms, see LOGARITHM. The following are the fundamental works which contain the results of the original calculations of logarithms of numbers and trigonometrical ratios:-Briggs, Arithmetica logarithmica (London, 1624), logarithms of numbers from I to 20,000 and from 90,000 to 100,000 to 14 places, with interscript differences; Vlacq, Arithmetica logarithmica (Gouda, 1628, also an English edition, London, 1631, the tables being the same), ten-figure logarithms of numbers from 1 to 100,000, with differences, also log sines, tangents, and secants for every minute of the quad-arithm is used also in a tract, containing only ten pages, published rant to 10 places, with interscript differences; Vlacq, Trigonometria artificialis (Gouda, 1633), log sines and tangents to every ten seconds of the quadrant to 10 places, with differences, and ten-figure logarithms of numbers up to 20,000, with differences; Briggs, Trigonometria Britannica (London, 1633), natural sines to 15 places, tangents and secants to 10 places, log sines to 14 places, and tangents to 10 places, at intervals of a hundredth of a degree from 0 to 45°, with interscript differences for all the functions. In 1794 Vega reprinted at Leipzig Vlacq's two works in a single folio volume, Thesaurus logarithmorum completus. The arrangement of the table of logarithms of numbers is more compendious than in Vlacq, being similar to that of an ordinary seven-figure table, but it is not so convenient, as mistakes in taking out the differences are more liable to occur. The trigonometrical canon gives log sines, cosines, tangents, and cotangents, from o° to 2° at intervals of one second, to 10 places, without differences, and for the rest of the quadrant at intervals of ten seconds. The trigonometrical canon is not wholly reprinted from the Trigonometria artificialis, as the logarithms for every second of the first two degrees, which do not occur in Vlacq, were calculated for the work by Lieutenant Dorfmund. Vega devoted great attention to the detection of errors in Vlacq's logarithms of numbers, and has given several important errata lists. F. Lefort (Annales de l'Observatoire de Paris, vol. iv.) has given a full errata list in Vlacq's and Vega's logarithms of numbers, obtained by comparison with the great French manuscript Tables du cadastre (see LOGARITHM; comp. also Monthly Notices R.A.S., 32, pp. 255, 288; 33, p. 330; 34. P. 447). Vega seems not to have bestowed on the trigonometrical canon anything like the care that he devoted to the logarithms of numbers, as Gauss' estimates the total number of last-figure errors at from 31,983 to 47.746, most of them only amounting to a unit, but some to as much as 3 or 4.

"

A copy of Vlacq's Arithmetica logarithmica (1628 or 1631), with the errors in numbers, logarithms, and differences corrected, is still the best table for a calculator who has to perform work requiring ten-figure logarithms of numbers, but the book is not easy to procure, and Vega's Thesaurus has the advantage of having log sines, &c., in the same volume. The latter work also has been made more accessible by a photographic reproduction by the Italian govern ment (Riproduzione fotozincografica dell' Istituto Geografico Militare, Florence, 1896). In 1897 Max Edler von Leber published tables for facilitating interpolations in Vega's Thesaurus (Tabularum ad faciliorem et breviorem in Georgii Vegae "Thesauri logarithmorum magnis canonibus interpolationis computationem utilium Trias, Vienna, 1897). The object of these tables is to take account of second differences. Prefixed to the tables is a long list of errors in the Thesaurus, occupying twelve pages. From an examination of the tabular results in the trigonometrical canon corresponding to 1060 angles von Leber estimates that out of the 90,720 tabular results 40,396 are in error by 1, 2793 by #2, and 191 by 3. Thus his estimated value of the total number of last-figure errors is 43,326, which is in accordance with Gauss's estimate. A table of ten-figure logarithms of numbers up to 100,009, the result of a new calculation, was published in the Report of the U.S. Coast and Geodetic Survey for 1895-6 (appendix 12, pp. 395-722) by W. W. Duffield, See his " Einige Bemerkungen zu Vega's Thesaurus logarithmorum, in Astronomische Nachrichten for 1851 (reprinted in his Werke, vol. iii. pp. 257-64); also Monthly Notices R.A.S., 33, p. 440.

|

The tables of S. Gundelfinger and A. Nell (Tafeln zur Berechnung neunstelliger Logarithmen, Darmstadt, 1891) afford an easy means of obtaining nine-figure logarithms, though of course they are far less convenient than a nine-figure table itself. The method in effect consists in the use of Gaussian logarithms, viz., if N=n+p, log N=log n+log (1+p/n)=log n+B where B is log (1+p/n) to argument A-log p-log n. The tables give log n from n = 1000 to n = 10,000, and values of B for argument A.2

Until 1891, when the eight-decimal tables, referred to further on, were published by the French government, the computer whe could not obtain sufficiently accurate results from seven-figure logarithms was obliged to have recourse to ten-figure tables, for, with only one exception, there existed no tables giving eight or nine figures. This exception is John Newton's Trigonometria Britannica (London, 1658), which gives logarithms of numbers to 100,000 to 8 places, and also log sines and tangents for every centesimal minute (ie. the nine-thousandth part of a right angle), and also log sines and tangents for the first three degrees of the quadrant to 5 places, the interval being the onethousandth part of a degree. This table is also remarkable for giving the logarithms of the differences instead of the actual differences. The arrangement of the page now universal in seven-figure tables-with the fifth figures running horizontally along the top line of the page-is due to John Newton.

As a rule seven-figure logarithms of numbers are not published separately, most tables of logarithms containing both the logarithms of numbers and a trigonometrical canon. Babbage's and Sang's logarithms are exceptional and give logarithms of numbers only. C. Babbage, Table of the Logarithms of the Natural Numbers from 1 to 108,000 (London, stereotyped in 1827; there are many tirages of later dates), is the best for ordinary use. Great pains were taken to get the maximum of clearness. The change of figure in the middle of the block of numbers is marked by a change of type in the fourth figure, which (with the sole exception of the asterisk) is probably the best method that has been used. Copies of the book were printed on paper of different colours-yellow, brown, green, &c. as it was considered that black on a white ground was a fatiguing combination for the eye. The tables were also issued with title-pages and introductions in other languages. In 1871 E. Sang published A New Table of Seven-place Logarithms of all Numbers from 20,000 to 200,000 (London). In an ordinary table extending from 10,000 to 100,000 the differences near the beginning are so numerous that the proportional parts are either very crowded or some of them omitted; by making the table extend from 20,000 to 200,000 instead of from 10,000 to 100,000 the differences are halved in magnitude, while there are only oneA fourth as many in a page. There is also greater accuracy.

2A seven-figure table of the same kind is contained in S. Gundelfinger's Sechsstellige Gaussische und siebenstellige gemeine Logarithmen (Leipzig, 1902).

in the leading figures, when it occurs in a column, is not marked at all; and the table must be used with very great caution. In fact it is advisable to go through the whole of it, and fill in with ink the first o after the change, as well as make some mark that will catch the eye at the head of every column containing á change. The table always be correct. Partly on account of the absence of a mark to was calculated by interpolation from the Trigonometria artificialis denote the change of figure in the column and partly on account of to 10 places and then reduced to 7, so that the last figure should the size of the table and a somewhat inconvenient arrangement, the work seems never to have come into general use. Computers have always preferred V. Bagay's Nouvelles Tables astronomiques et hydrographiques (Paris, 1829), which also contains a complete very clearly marked by a large black nucleus, surrounded by a logarithmic canon to every second. The change in the column is circle, printed instead of o. Bagay's work having become rare and costly, was reprinted with the errors corrected. The reprint, howto be no means of distinguishing it from the original work except by turning to one of the errata in the original edition and examining ever, bears the original title-page and date 1829, and there appears whether the correction has been made.

further peculiarity of this table is that multiples of the differences, instead of proportional parts, are give at the side of the page. Typographically the table is exceptional, as there are no rules, the numbers being separated from the logarithms by reversed commas -a doubtful advantage. This work was to a great extent the result of an original calculation; see Trans. Roy. Soc. Edin., 1871, 26. Sang proposed to publish a nine-figure table from 1 to 1,000,000, but the requisite support was not obtained. Various papers of Sang's relating to his logarithmic calculations will be found in the Proc. Roy. Soc. Edin. subsequent to 1872. Reference should here be made to Abraham Sharp's table of logarithms of numbers from 1 to 100 and of primes from 100 to 1100 to 61 places, also of numbers from 999,990 to 1,000,010 to 63 places. These first appeared in Geometry Improv'd. have been republished in Sherwin's, Callet's, and the earlier editions by A. S. Philomath (London, 1717). They of Hutton's tables. H. M. Parkhurst, Astronomical Tables (New York, 1871), gives logarithms of numbers from 1 to 109 to 102 places. In many seven-figure tables of logarithms of numbers the values of S and T are given at the top of the page, with V, the variation of each, for the purpose of deducing log sines and tangents. S and T denote log (sin x/x) and log (tan x/x) respectively, the argument being the number of seconds denoted by certain numbers (sometimes only the first, sometimes every tenth) in the number column on each page. Thus, in Callet's tables, on the page on which the first number is 67200, S-log (sin 6720°/6720) and T-log (tan 6720"/6720), while the V's are the variations of each for 10. To find, for example, log sin 1°52'12"-7, or log sin 6732"-7, we have S-4-6854980 and log 6732-7-3,8281893, whence, by addition, we obtain 8-5136873; but for 10" is 2-29, whence the variation for 12-7 is -3, and the log sine required is 8-5136870. Tables of S and T are frequently called, after their inventor, "Delambre's tables." Some seven-figure tables extend to 100,000, and others to 108,000, the last 8000 logarithms, to 8 places, being given to ensure greater accuracy, as near the beginning of the numbers the differences are large and the interpolations more laborious and less exact than in the rest of the table. The eight-figure logarithms, however, at the end of a seven-figure table are liable to occasion error; for the computer who is accustomed to three leading figures, common to the block of figures, may fail to notice that in this part of the table there are four, and so a figure (the fourth) is sometimes omitted in taking out the logarithm. In the ordinary method of arranging a seven-figure table the change in the fourth figure, when it occurs in the course of the line, is a source of frequent error unless it is very clearly indicated. In the earlier tables the change was not marked at all, and the computer had to decide for himself, each time he took out a logarithm, whether the third figure had to be increased. In some tables the line is broken where the change occurs; but the dislocation of the figures and the corresponding irregularity in the lines are very awkward. Babbage printed the fourth figure in small type after a change; and Bremiker placed a bar over it. The best method seems to be that of prefixing an asterisk to the fourth figure of each logarithm after the change, as is done in Schrön's and many other modern tables. This is beautifully clear and the asterisk at once catches the eye. Shortrede and Sang replace o after a change clear in the case of the o's, but leaves unmarked the cases in which by a nokta (resembling a diamond in a pack of cards). This is very the fourth figure is 1 or 2. A method which finds favour in some recent tables is to underline all the figures after the increase, or to place a line over them.

contained in R. Shortrede's Logarithmic Tables (Edinburgh). This
work was originally issued in 1844 in one volume, but being dis-
The only other canon to every second that has been published is
satisfied with it Shortrede issued a new edition in 1849 in two
volumes. The first volume contains logarithms of numbers, anti-
logarithms, &c., and the second the trigonometrical canon to every
second. The volumes are sold separately, and may be regarded as
independent works; they are not even described on their title-
pages as vol. i. and vol. ii. The trigonometrical canon is very com-
plete in every respect, the arguments being given in time as well as
in arc, full proportional parts being added, &c. The change of
the change occurs. The page is crowded and the print not very
clear, so that Bagay is to be preferred for regular use.
figure in the column is denoted by a nokta, printed instead of o where

divided centesimally were J. P. Hobert and L. Ideler, Nouvelles tables
trigonométriques (Berlin, 1799), and C. Borda and J. B. J. Delambre,
Previous to 1891 the only important tables in which the quadrant is
Tables trigonométriques décimales (Paris, 1801). The former give,
cotangents, to 7 places, the arguments proceeding to 3° at intervals
of 10 and thence to 50° at intervals of 1 (centesimal), and also
among other tables, natural and log sines, cosines, tangents, and
of a right angle to 10 places. The latter gives long sines, cosines,
tangents, cotangents, secants, and cosecants from o° to 3° at intervals
natural sines and tangents for the first hundred ten-thousandths
50° at intervals of 1 (centesimal) to 7 places. There is also a table
of log sines, cosines, tangents, and cotangents from o' to 10' at
of 10" (with full proportional parts for every second), and thence to
intervals of to and from 0 to 50° at intervals of 10' (centesimal) to
II places. Hobert and Ideler give a natural as well as a logarithmic
line being broken when a change of figure takes place in it.
canon, but Borda and Delambre give only the latter. Borda and
Delambre give seven-figure logarithms of numbers to 10,000, the

Babbage printed a subscript point under the last figure of each logarithm that had been increased. Schrön used a bar subscript, which, being more obtrusive, seems less satisfactory. In some tables the increase of the last figure is only marked when the figure is increased to a 5, and then a Roman five (v) is used in place of the Arabic figure. Hereditary errors in logarithmic tables are considered in two papers "On the Progress to Accuracy of Logarithmic Tables" and On Logarithmic Tables," in Monthly Notices R.A.S., 33, pp. 330, 440. See also vol. 34, p. 447; and a paper by Gernerth, Ztsch. f. d. Österr. Gymm., Heft vi. p. 407. Passing now to the logarithmic trigonometrical canon, the first great advance after the publication of the Trigonometria artificialis in 1633 was made in Michael Taylor's Tables of Logarithms (London, 1792), which give log sines and tangents to every second of the quadrant to 7 places. This table contains about 450 pages with an average number of 7750 figures to the page, so that there are altogether nearly three millions and a half of figures. The change Legendre (Traité des fonctions elliptiques, vol. ii., 1826) gives a table of natural sines to 15 places, and of log sines to 14 places, for every 15 of the quadrant, and also a table of logarithms of uneven numbers from 1163 to 1501, and of primes from 1501 to 10,000 to 19 places. The latter, which was extracted from the Tables du cadastre, is a continuation of a table in W. Gardiner's Tables of Logarithms (London, 1742; reprinted at Avignon, 1770), which gives logarithms of all numbers to 1000, and of uneven numbers from 1000 to 1143. Legendre's tables also appeared in

his Exercices de calcul intégral, vol. iii. (1816).

procure, and seven-figure tables being no longer sufficient for the
The tables of Borda and Delambre having become difficult to
arithms of numbers to 120,000) log sines and tangents for every ten
accuracy required in astronomy and geodesy, the French govern-
from the Tables du cadastre of Prony (see LOGARITHM). The title
ment in 1891 issued an eight-figure table containing (besides log-
l'armée: Tables des logarithmes à huit décimales
seconds (centesimal) of the quadrant, the latter being extracted
ordre du ministre de la guerre (Paris, Imprimerie Nationale, 1891).
of this fine and handsomely printed work is Service géographique de
These tables are now in common use where eight figures are required.
centesimally, and but for the appearance in the same year of Vlacq's
publiées par
Trigonometria artificialis, in which the degree is divided sexagesi-
In Brigg's Trigonometria Britannica of 1633 the degree is divided
mally, this reform might have been effected. It is clear that the
most suitable time for making such a change was when the natural
canon was replaced by the logarithmic canon, and Briggs took
divided it centesimally instead of sexagesimally, thus ensuring the
advantages of decimal division (a saving of work in interpolations,
advantage of this opportunity. He left the degree unaltered, but
multiplications, &c.), with the minimum of change. The French
mathematicians at the end of the 18th century divided the right
angle centesimally, completely changing the whole system, with no
appreciable advantages over Briggs's system. In fact the centesimal
degree is as arbitrary a unit as the nonagesimal and it is only the
venienee. Briggs's example was followed by Roe, Oughtred, and
other 17th-century writers; but the centes mal division of the
non-centesimal subdivision of the degree that gives rise to incon-
degree seemed to have entirely passed out of use, till it was revived
by C. Bremiker in his Logarithmisch-trigonometrische Tafeln mit fünf
Decimalstellen (Berlin, 1872, 10th ed. revised by A. Kallius, 1906).
This little book of 158 pages gives a five-figure canon to every
hundredth of a degree with proportional parts, besides logarithms
of numbers, addition and subtraction logarithms, &c.

The eight-figure table of 1891 has now made the use of a cen-
tesimal table compulsory, if this number of figures is required.

logarithms, strictly so called, have entirely passed out of use and are of purely historic interest; it is therefore sufficient to refer to the article LOGARITHM, where a full account is given. Apart from the inventor's own publications, the only strictly Napierian tables of importance are contained in Ursinus's Trigonometria (Cologne, 1624-1625) and Schulze's Sammlung (Berlin, 1778), the former being the largest that has been constructed. Logarithms to the base e where e denotes 2.71828..., were first published by J. Speidell, New Logarithmes (1619).

"

through several more. J. Galbraith and S. Haughton, Manual of | Mathematical Tables (London, 1860), give five-figure logarithms to 10,000 and log sines and tangents for every minute, also a small table of Gaussian logarithms. J. Houel, Tables de logarithmes à cing décimales (Paris, 1871; new edition 1907), is a very convenient collection of five-figure tables; besides logarithms of numbers and circular functions, there are Gaussian logarithms, least divisors of numbers to 10,841, antilogarithms, &c. The work (118 pp.) is printed on thin paper. A. Gernerth, Fünfstellige gemeine Logarithmen (Vienna, 1866), gives logarithms to 10,800 and a ten-second canon. The most copious table of hyperbolic logarithms is Z. Dase, Tafel There are sixty lines on the page, so that the double page contains | der natürlichen Logarithmen (Vienna, 1850), which extends from log sines, cosines, tangents, and cotangents extending over a minute. to 1000 at intervals of unity and from 1000 to 10,500 at intervals C. Bremiker, Logarithmisch-trigonometrische Tafeln mit fünf Decimal- | of 1 to 7 places, with differences and proportional parts, arranged stellen (10th edition by A. Kallius, Berlin, 1906), which has been as in an ordinary seven-figure table. By adding log 10 to the results already referred to, gives logarithms to 10,009 and a logarithmic the range is from 10,000 to 105,000 at intervals of unity. The table canon to every hundredth of a degree (sexagesimal), in a handy formed part of the Annals of the Vienna Observatory for 1851, but volume; the lines are divided into groups of three, an arrangement separate copies were printed. The most elaborate table of hyperabout the convenience of which there is a difference of opinion. bolic logarithms is due to Wolfram, who calculated to 48 places the H. Gravelius, Fünfstellige logarithmisch-trigonometrische Tafeln für logarithms of all numbers up to 2200, and of all primes (also of a die Decimalteilung des Quadranten (Berlin, 1886), is a well-printed great many composite numbers) between this limit and 10,009. five-figure table giving logarithms to 10,009, a logarithmic canon to Wolfram's results first appeared in Schulze's Sammlung (1778). every centesimal minute (i.e. ten-thousandth part of a right angle), Six logarithms which Wolfram had been prevented from computing by and an extensive table (40 pp.) for the conversion of centesimally a serious illness were supplied in the Berliner Jahrbuch, 1783, p. 191. expressed arcs into sexagesimally expressed arcs and vice versa. The complete table was reproduced in Vega's Thesaurus (1794). Among the other tables is a four-place table of squares from 0 to 10 where several errors were corrected. Tables of hyperbolic logarithms at intervals of oor with proportional parts. E. Becker, Logarith- are contained in the following collections:-Callet, all numbers to misch-trigonometrisches Handbuch auf fünf Decimalen (2nd stereo. 100 and primes to 1097 to 48 places; Borda and Delambre (1801). ed., Leipzig, 1897), gives logarithms to 10,009 and a logarithmic canon all numbers to 1200 to 11 places; Salomon (1827), all numbers to for every tenth of a minute to 6° and thence to 45° for every minute. 1000 and primes to 10,333 to 10 places; Vega, Tabulae (including There are also Gaussian logarithms. V. E. Gamborg, Logaritmetabel Hülsse's edition, 1840), and Köhler (1848), all numbers to 1000 and (Copenhagen, 1897), is a well-printed collection of tables, which primes to 10,000 to 8 places; Barlow (1814), all numbers to 10,000; contains a five-figure logarithmic canon to every minute, five-figure Hutton, Mathematical Tables, and Willich (1853), all numbers to logarithms of numbers to 10,000, and five-figure antilogarithms, 1200 to 7 places; Dupuis (1868), all numbers to 1000 to 7 places. viz., five-figure numbers answering to four-figure mantissae from Hutton also gives hyperbolic logarithms from 1 to 10 at intervals 0000 to 9999 at intervals of 0001. H. Schubert, Fünfstellige of 01 to 7 places. Rees's Cyclopaedia (1819), art, Hyperbolic Tafeln und Gegentafeln (Leipzig, 1896), is peculiar in giving, Logarithms," contains a table of hyperbolic logarithms of all numbers besides logarithms of numbers and a logarithmic and natural to 10,000 to 8 places. canon, the three converse tables of numbers answering to logarithms, and angles answering to logarithmic and natural trigonometrical functions. The five-figure tables of F. G. Gauss (Berlin, 1870) have passed through very many editions, and mention should also be made of those of T. Wittstein (Hanover, 1859) and F. W: Rex (Stuttgart, 1884). S. W. Holman, Computation Rules and Logarithms (New York, 1896), contains a well-printed and convenient set of tables including five-figure logarithms of numbers to 10,000 and a five-figure logarithmic canon to every minute, the actual characteristics (with the negative sign above the number) being printed, as in the tables of Dupuis, 1868, referred to above. There is also a four-place trigonometrical canon and four-place_antilogarithms, reciprocals, square and cube roots, &C. G. W. Jones, Logarithmic Tables (4th London, and Ithaca, N.Y., 1893), contains a five-place natural trigonometrical canon and a six-place logarithmic canon to every minute, six-place Gaussian and hyperbolic logarithms, besides a variety of four-place tables, including squares, cubes, quarter-squares, reciprocals, &c. The factor table has been already noticed. It is to be observed that the fourth edition is quite a distinct work from the third, which contained much fewer tables. J. B. Dale, Five-figure Tables of Mathematical Functions (London, 1903), is a book of 92 pages containing a number of small five-figure tables of functions which are not elsewhere to be found in one volume. Among the functions tabulated are elliptic functions of the first and second kind, the gamma function, Legendre's coefficients, Bessel's functions, sine, cosine, and exponential integrals, &c. J. Houel's Recueil de formules et de tables numériques (Paris, 1868) contains 19 tables, occupying 62 pages, most of them giving results to 4 places; they relate to very varied subjects-antilogarithms, Gaussian logarithms, logarithms of 1+x/1-x elliptic integrals, squares for use in the method of least squares, &c. C. Bremiker, Tafel vierstelliger Logarithmen (Berlin, 1874), gives fourfigure logarithms, of numbers to 2009, log sines, cosines, tangents, and cotangents to 8° for every hundredth of a degree, and thence to 45° for every tenth of a degree, to 4 places. There are also Gaussian logarithms, squares from 0·000 to 13,500, antilogarithms, &c. The book contains 60 pages. It is not worth while to give a list of fourfigure tables or other tables of small extent, which are very numerous, but meation may be made of J. M. Peirce, Mathematical Tables chiefly to Four Figures (Boston, U.S., 1879), 42 pp., containing also hyperbolic functions; W. Hall, Four-figure Tables and Constants (Cambridge, 1905), 60 pp., chiefly for nautical computation; A. du P. Denning, Five-figure Mathematical Tables for School and Laboratory Purposes (12 pp. of tables, large octavo); A. R. Hinks, Cambridge Four-figure Mathematical Tables (12 pp.). C. M. Willich, Popular Tables (London, 1853), is a useful book for an amateur; it gives Briggian and hyperbolic logarithms to 1200 to 7 places, squares, &c., to 343, &c.

Hyperbolic or Napierian or Natural Logarithms.-The logarithms invented by Napier and explained by him in the Descriptio (1614) were not the same as those now called natural or hyperbolic (viz., to base e), and very frequently also Napierian, logarithms. Napierian

Logarithms to base e are generally termed Napierian by English writers, and natural by foreign writers. There seems no objection to the former name, though the logarithms actually invented by Napier depended on the base e, but it should be mentioned in text-books that so-called Napierian logarithms are not identical with those originally devised and calculated by Napier. Tables to convert Briggian into Hyperbolic Logarithms, and vice versa.-Such tables merely consist of the first hundred (sometimes only the first ten) multiples of the modulus 43429 44819... and its reciprocal 2.30258 50929... to 5, 6, 8, 10, or more places. They are generally to be found in collections of logarithmic tables, but rarely exceed a page in extent, and are very easy to construct. Schrön and Bruhns both give the first hundred multiples of the modulus and its reciprocal to 10 places, and Bremiker (in his edition of Vega and in his six-figure tables) and Dupuis to 7 places. C. F. Degen, Tabularum Enneas (Copenhagen, 1824), gives the first hundred multiples of the modulus to 30 places.

Antilogarithms.-In the ordinary tables of logarithms the natural numbers are integers, while the logarithms are incommensurable. In an antilogarithmic canon the logarithms are exact quantities, such as 00001, 00002, &c., and the corresponding numbers are incommensurable. The largest and earliest work of this kind is J. Dodson's Antilogarithmic Canon (London, 1742), which gives numbers to 11 places corresponding to logarithms from 0 to 1 at intervals of 00001, arranged like a seven-figure logarithmic table, with interscript differences and proportional parts at the bottom of the page. This work was the only large antilogarithmic canon for more than a century, till in 1844 Shortrede published the first edition of his tables; in 1849 he published the second edition, and in the same year Filipowski's tables appeared. Both these works contain seven-figure antilogarithms: Short rede gives numbers to logarithms from 0 to 1 at intervals of 00001, with differences and multiples at the top of the page, and H. E. Filipowski, A Table of Antilogarithms (London, 1849), contains a table of the same extent, the proportional parts being given to hundredths.

Small tables of antilogarithms to 20 places occur in several collections of tables, as Gardiner (1742), Callet, and Hutton, Fourand five-place tables are not uncommon in recent works, as e.g. in Houel (1871), Gamborg (1897), Schubert (1896), Holman (1896). Addition and Subtraction, or Gaussian Logarithms.-The object of such tables is to give log (ab) by only one entry when log e and log b are given. Let

A=log x, Blog (1+1), C=log (1+x). Leaving out the specimen table in Z. Leonelli's Théorie des logarithmes additionnels et déductifs (Bordeaux, 1803), in which the first suggestion was made, the principal tables are the following: Gauss, in Zach's Monatliche Correspondenz (1812), gives B and C for argument A from o to 2 at intervals of 001, thence to 3.40

'Leonelli's original work of 1803, which is extremely scarce, was reprinted by J. Houël at Paris in 1875