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Mathematical Table
1911 Encyclopedia Britannica
or In any table the results tabulated are termed the " tabular results " or " respondents, " and the corresponding numbers by which the table is entered are termed the " arguments." A table is said to be of single or double entry according as there are one or two arguments. For example, a table of logarithms is a table of single entry, the numbers being the arguments and the logarithms the tabular results; an ordinary multiplication table is a table of double entry, giving xy as tabular result for x and y as arguments. The intrinsic value of a table may be estimated by the actual amount of time saved by consulting it; for example, a table of square roots to ten decimals is more valuable than a table of squares, as the extraction of the root would occupy more time than the multiplication of the number by itself. The value of a table does not depend upon the difficulty of calculating it; for, once made, it is made for ever, and as far as the user is concerned the amount of labour devoted to its original construction is immaterial. In some tables the labour required in the construction is the same as if all the tabular results had been calculated separately; but in the majority of instances a table can be formed by expeditious methods which are inapplicable to the calculation of an individual result. This is the case with tables of a continuous quantity, which may frequently be constructed by differences. The most striking instance perhaps is afforded by a factor table or a table of primes; for, if it is required to determine whether a given number is prime or not, the only universally available method (in the absence of tables) is to divide it by every prime less than its square root or until one is found that divides it without remainder. But to form a table of prime numbers the process is theoretically simple and rapid, for we have only to range all the numbers in a line and strike out every second number beginning from 2, every third beginning from 3, and so on, those that remain being primes. Even when the tabular results are constructed separately, the method of differences or other methods connecting together different tabular results may afford valuable verifications. By having recourse to tables not only does the computer save time and labour, but he also obtains the certainty of accuracy.
The invention of logarithms in 1614, followed immediately by the calculation of logarithmic tables, revolutionized all the methods of calculation; and the original work performed by Henry Briggs and Adrian Vlacq in calculating logarithms in the early part of the 17th century has in effect formed a portion of every arithmetical operation that has since been carried out by means of logarithms. And not only has an incredible amount of labour been saved, 1 but a vast number of calculations and researches have been rendered practicable which otherwise would have been beyond human reach. The mathematical process that underlies the tabular method of obtaining a result may be indirect and complicated; for example, the logarithmic method would be quite unsuitable for the multiplication of two numbers if the logarithms had to be calculated specially for the purpose and were not already tabulated for use. The arrangement of a table on the page and all typographical details-such as the shape of the figures, their spacing, the thickness and placing of the rules, the colour and quality of the paper, &c.- are of the highest importance, as the computer has to spend hours with his eyes fixed upon the book; and the efforts of eye and brain required in finding the right numbers amidst a mass of figures on a page and in taking them out accurately, when the computer is tired as well as when he is fresh, are far more trying than the mechanical action of simple reading. Moreover, the trouble required by the computer to learn the use of a table need scarcely be considered; the important matter is the time and labour saved by it after he has learned its use.
In the following descriptions of tables an attempt is made to give an account 'of all those that a computer of the present day is likely to use in carrying out arithmetical calculations. Tables relating to ordinary arithmetical operations are first described, and afterwards an account is given of the most useful and least technical of the more strictly mathematical tables, such as factorials, gamma functions, integrals, Bessel's functions, &c. Nearly all modern tables are stereotyped, and in giving their titles the accompanying date is either that of the original stereotyping or of the tirage in question. In tables that have passed through many editions the date given is that of the edition described. A much fuller account of general tables published previously to 1872, by the present writer, is contained in the British Association Report for 1873, pp. 1-175.
Tables of Divisors (Factor Tables) and Tables of Primes.-The existing factor tables extend to 10,000,000. In 1811 L. Chernac published at Deventer his Cribrum arithmeticum, which gives all the prime divisors of every number not divisible by 2, 3, or 5 up to 1,020,000. In1814-1817J. C. Burckhardt published at Paris his Tables des diviseurs, giving the least divisor of every number not divisible by 2, 3, or 5 up to 3,036,000. The second million was issued in 1814, the third in 1816, and the first in 1817. The corresponding tables for the seventh, eighth, and ninth millions were calculated by Z. Dase and issued at Hamburg in 1862, 1863, and 1865. Dase died suddenly in 1861 during the progress of the work, and it was completed by H. Rosenberg. Dase's calculation was performed at the instigation of Gauss, and he began at 6,000,000 because the Berlin Academy was in possession of a manuscript presented by Crelle extending Burckhardt's tables from 3,000,000 to 6,000,000. This manuscript was found on examination to be so inaccurate that the publication was not desirable, and accordingly the three intervening millions were calculated and published by James Glaisher, the Factor Table for the Fourth 1 Referring to factor tables, J. H. Lambert wrote ( Supplementa tabularum, 1798, p. xv.): " Universalis finis talium tabularum est ut semel pro semper computetur quod saepius de novo computandum foret, et ut pro omni casu computetur quod in futurum pro quovis casu computatum desiderabitur." This applies to all tables.
Million appearing at London in 1879, and those for the fifth and sixth millions in 1880 and 1883 respectively (all three millions stereotyped). The tenth million, though calculated by Dase and Rosenberg, has not been published. The nine quarto volumes ( Tables des diviseurs, Paris, 1814-1817; Factor Tables, London, 1879-1883; Factoren-Tafeln, Hamburg, 1862-1865) thus form one uniform table, giving the least divisor of every number not divisible by 2, 3, or 5, from unity to nine millions. The arrangement of the results on the page, which is due to Burckhardt, is admirable for its clearness and condensation, the least factors for 9000 numbers being given on each page. The tabular portion of each million occupies 112 pages. The first three millions were issued separately, and also bound in one volume, but the other six millions are all separate. Burckhardt began the publication of his tables with the second million instead of the first, as Chernac's factor table for the first million was already in existence. Burckhardt's first million does not supersede Chernac's, as the latter gives all the prime divisors of numbers not divisible by 2, 3, or 5 up to 1,020,000. It occupies 1020 pages, and Burckhardt found it very accurate; he detected only thirty-eight errors, of which nine were due to the author, the remaining twenty-nine having been caused by the slipping of type in the printing. The errata thus discovered are given in Burckhardt's first million. Other errata are contained in Allan Cunningham's paper referred to below.
Burckhardt gives but a very brief account of the method by which he constructed his table; and the introduction to Dase's millions merely consists of Gauss's letter suggesting their construction. The Introduction to the Fourth Million (pp. 52) contains a full account of the method of construction and a history of factor tables, with a bibliography of writings on the subject. The Introduction (pp. 103) to the Sixth Million contains an enumeration of primes and a great number of tables relating to the distribution of primes in the whole nine millions, portions of which had been published in the Cambridge Philosophical Proceedings and elsewhere. A complete list of errors in the nine millions was published by J. P. Gram ( Acta mathematics, 18 93, 1 7, p. 310). These errors, 141 in number, and which affect principally the second, third, eighth, and ninth millions, should be carefully corrected in all the tables. In 1909 the Carnegie Institution of Washington published a factor table by Prof. D. N. Lehmer which gives the least factor of all numbers not divisible by 2, 3, 5, or 7, up to ten millions. This table, which covers a range of 21,000 numbers on a single page, was reproduced by photography from a typewritten copy of the author's original manuscript. The introduction contains a list of errata in the nine millions previously published, completely confirming Gram's list.
The factor tables which have just been described greatly exceed both in extent and accuracy any others of the same kind, the largest of which only reaches 408,000. This is the limit of Anton Felkel's Tafel aller einfachen Factoren (Vienna, 1776), a remarkable and extremely rare book,' nearly all the copies having been destroyed. Georg Vega ( Tabulae, 1797) gave a table showing all the divisors of numbers not divisible by 2, 3, or 5 up to 502,000, followed by a list of primes from 102,000 to 400,313. In the earlier editions of this work there are several errors in the list, but these are no doubt corrected in J. A. Hiilsse's edition (1840). J. Salomon (Vienna, 1827) gives the least divisor of all numbers not divisible by 2, 3, or 5, up to 102,011, and B. Goldberg ( Primzahlen and Factoren-Tafeln, Leipzig, 1862) gives all factors of numbers not divisible by 2, 3, or 5 up to 251,650. H. G. Kohler ( Logarithmisch-trigonometrisches Handbuch, 1848 and subsequent editions) gives all factors of numbers not prime or divisible by 2, 3, 5, or II up to 21,525. Peter Barlow ( Tables, 1814) and F. Schaller ( Primzahlen-Tafel, Weimar, 1855) give all factors of all numbers up to 10,000. Barlow's work also contains a list of primes up to 100,103. Both the factor table and the list of primes are omitted in the stereotyped (1840) reprint. Full lists of errata in Chernac (181I), Barlow (1814), Hiilsse's Vega (1840), Kohler (1848), Schaller (1855), and Goldberg (1862) are contained in a paper by Allan Cunningham ( Mess. of Math., 1904, 34, P. 24; 1905, 35, p. 24). V. A. Le Besgue ( Tables diverses pour la decomposition des nombres, Paris, 1864) gives in a table of twenty pages, the least factor of numbers not divisible by 2, 3, or 5 up to 115,500. In Rees's Cyclopaedia (1819), article " Prime Numbers," there is a list of primes to 217,219 arranged in decades. The Fourth Million (1879) contains a list of primes up to 30,341. The fourth edition of the Logarithmic Tables (London, and Ithaca, N.Y., 1893) of G. W. Jones of Cornell University contains a table of all the factors of numbers not divisible by 2 or 5 up to 20,000. In the case of primes the ten-place logarithm is given. This table does not occur in the third edition (Ithaca, N.Y., 1891). On the first page of the Second Million Burckhardt gives the first nine multiples of the primes to 1423; and a smaller table of the same kind, extending only to 313, occurs in Lambert's Supplementa (1798). Several papers contain lists of high primes (i.e. beyond the range of the 2 For information about it, see a paper on " Factor Tables," in Camb. Phil. Proc. (1878), iii. 99-238, or the Introduction to the Fourth Million. factor tables). Among these may be mentioned two, by Allan Cunningham and H. J. Woodall jointly, in the Mess. of Math., 1902, 31, p. 165; 1905, 34, p. 72. See also the papers on factorizations of high numbers referred to under Tables relating to the Theory of Numbers. The Vienna Academy possesses the manuscript of an immense factor table extending to 100,000,000, constructed many years ago by J. P. Kulik (1793-1863) (see Ency. math. Wiss., 1900-1904, i. 952, and Lehmer's Factor Table, p. ix.).
Multiplication Tables.-A multiplication table is usually of double entry, the two arguments being the two factors; when so arranged it is frequently called a Pythagorean table. The largest and most useful work is A. L. Crelle's Rechentafeln (Bremiker's edition, 1857, stereotyped; many subsequent editions with German, French, and English title-pages), which gives in one volume all the products up to 1000 X 1000, so arranged that all the multiples of any one number appear on the same page. The original edition was published in 1820 and consisted of two thick octavo volumes. The second (stereotyped) edition is a convenient folio volume of 450 pages.' In 1908 an entirely new edition, edited by O. Seeliger, was published in which the multiples of 10, 20, ..., 990 (omitted in previous editions) are included. This adds 50 pages to the volume, but removes what has been a great drawback to the use of the tables. Other improvements are that the tables are divided off horizontally and vertically by lines and spaces, and that, for calculations in which the last two figures are rejected, a mark has been placed to show when the last figure retained should be increased. Two other tables of the same extent (moo X moo), but more condensed in arrangement, are H. C. Schmidt's Zahlenbuch (Aschersleben, 1896), and A. Henselin's Rechentafel (Berlin, 1897). An anonymous table, published at Oldenburg in 1860, gives products up to 500X509, and M. Cordier, Le Multiplicateur de trois cents carres (Paris, 1872), gives a multiplication table to 300X300 (intended for commercial use). In both these works the product is printed in full. The four following tables are for the multiplication of a number by a single digit. (I) A. L. Crelle, Erleichterungstafel fur jeden, der zu rechnen hat (Berlin, 1836), a work extending to woo pages, gives the product of a number of seven figures by a single digit, by means of a double operation of entry. Each page is divided into two tables: for example, to multiply 93 $2 477 by 7 we turn to page 825, and enter the right-hand table at line 77, column 7, where we find 77339; we then enter the left-hand table on the same page at line 93, column 7, and find 656, so that the product required is 6567-7339. (2) C. A. Bretschneider, Produktentafel (Hamburg and Gotha, 1841), is somewhat similar to Crelle's table, but smaller, the number of figures in the multiplicand being five instead of seven. (3) In S. L. Laundy, A Table of Products (London, 1865), the product of any five-figure number by a single digit is given by a double arrangement. The extent of the table is the same as that of Bretschneider's, as also is the principle, but the arrangement is different, Laundy's table occupying only to pages and Bretschneider's 99 pages. (4) G. Diakow's Multiplikations-Tabelle (St Petersburg, 1897) is of the same extent as Bretschneider's table but occupies woo pages. Among tables extending to Ioo X moo (i.e. giving the products of two figures by three) may be mentioned C. A. Muller's Multiplications-Tabellen (Karlsruhe, 1891). The tables of L. Zimmermann ( Rechentafeln, Liebenwerda, 1896) and J. Riem ( Rechentabellen fur Multiplication, Basel, 1897) extend to 100X10,000. In a folio volume of 500 pages J. Peters ( Rechentafeln fur Multiplikation and Division mit einbis vierstelligen Zahlen, Berlin, 1909) gives products of four figures by two. The entry is by the last three figures of the multiplicand, and there are 2000 products on each page. Among earlier tables, the interest of which is mainly historical, mention may be made of C. Hutton's Table of Products and Powers of Numbers (London, 1781), which containsa table up to too X 1000, and J. P. Gruson's Grosses Einmaleins von Eins bis Hunderttausend (Berlin, 1799)-a table of products up to 9 X to,000. The author's intention was to extend it to 100,000, but only the first part was published. In this book there is no condensation or double arrangement; the pages are very large, each containing 125 lines.
Quarter-Squares.-Multiplication may be performed by means of a table of single entry in the manner indicated by the formula ab =±(a+b)2-±(a-b)'. ' Only one other multiplication table of the same extent as Crelle's had appeared previously, viz. Herwart von Hohenburg's Tabulae arithmeticae lrporeackupkQews universales (Munich, 1610), a huge folio volume of more than a thousand pages. It appears from a correspondence between Kepler and von Hohenburg, which took place at the end of 1608, that the latter used his table when in manuscript for the performance of multiplications in general, and that the occurrence of the word prosthaphaeresis on the title is due to Kepler, who pointed out that by means of the table spherical triangles could be solved more easily than by Wittich's prosthaphaeresis. The invention of logarithms four years later afforded another means of performing multiplications, and von Hohenburg's work never became generally known. On the method of prosthaphaeresis, see Napier, John, and on von Hohenburg's table, see a paper " On multiplication by a Table of Single Entry," Phil. Mag., 1878, ser. v., 6, p. 331.
Thus with a table of quarter-squares we can multiply together any two numbers by subtracting the quarter-square of their difference from the quarter-square of their sum. The largest table of quarter-squares is J. Blater's Table of Quarter-Squares of all whole numbers from z to 200,000 (London, 1888), 2 which gives quarter-squares of every number up to 200,000 and thus yields directly the product of any two five-figure numbers. This fine table is well printed and arranged. Previous to its publication the largest table was S. L. Laundy's Table of Quarter-Squares of all numbers up to I oo,000 (London, 1856), which is of only half the extent, and therefore is only directly available when the sum of the two numbers to be multiplied does not exceed too,000.
Smaller works are J. J. Centnerschwer, Neuerfundene Multiplicationsand Quadrat-Tafeln (Berlin, 1825), which extends to 20,000, and J. M. Merpaut, Tables arithmonomiques (Vannes, 1832), which extends to 40,000. In Merpaut's work the quarter-square is termed the " arithmone." L. J. Ludolf, who published in 1690 a table of squares to 100,000 (see next paragraph), explains in his introduction how his table may be used to effect multiplications by means of the above formula; but the earliest book on quartersquares is A. Voisin, Tables des multiplications, ou logarithmes des nombres entiers depuis I jusqu'a 20,000 (Paris, 1817). By a logarithm Voisin means a quarter-square, i.e. he calls a a root and 4a' its logarithm. On the subject of quarter-squares, &c., see Phil. Mag. [v.] 6, p. 331.
Squares, Cubes, & g c., and Square Roots and Cube Roots.-The most convenient table for general use is P. Barlow's Tables (Useful Knowledge Society, I r ondon, from the stereotyped plates of 1840), which gives squares, cubes, square roots, cube roots, and reciprocals to t 0,000. These tables also occur in the original edition of 1814. The largest table of squares and cubes is J. P. Kulik, Tafeln der Quadratand Kubik-Zahlen (Leipzig, 1848), which gives both as far as 100,000. Blater's table of quarter-squares already mentioned gives squares of numbers up to 100,000 by dividing the number by 2; and up to 200,000 by multiplying the tabular result by 4. Two early tables give squares as far as 100,000, viz. Maginus, Tabula tetragonica (Venice, 1592), and Ludolf, Tetragonometria tabularia (Amsterdam, 1690); G. A. Jahn, Tafel der Quadratund Kubikwurzeln (Leipzig, 1839), gives squares to 27,000, cubes to 24,000, and square and cube roots to 25,500, at first to fourteen decimals and above 1010 to five. E. Gelin ( Recueil de tables numeriques, Huy, 1894) gives square roots (to 15 places) and cube roots (to to places) of numbers up to boo. C. Hutton, Tables of Products and Powers of Numbers (London, 1781), gives squares up to 25,400, cubes to 10,000, and the first ten powers of the first hundred numbers. P. Barlow, Mathematical Tables (original edition, 1814), gives the first ten powers of the first hundred numbers. The first nine or ten powers are given in Vega, Tabulae (1797), and in Hulsse's edition of the same (1840), in Kohler, Handbuch (1848), and in other collections. C. F. Faa de Bruno, Calcul des erreurs (Paris, 1869), and J. H. T. Muller, Vierstellige Logarithmen (1844), give squares for use in connexion with the method of least squares. Four-place tables of squares are frequently given in fiveand four-figure collections of tables. Small tables often occur in books intended for engineers and practical men. S. M. Drach ( Messenger of Math., 1878, 7, p. 87) has given to 33 places the cube roots (and the cube roots of the squares) of primes up to 127. Small tables of powers of 2, 3, 5, 7 occur in various collections. In Vega's Tabulae (1797, and the subsequent editions, including Hulsse's) the powers of 2, 3, 5 as far as the 45th, 36th, and 27th respectively are given; they also occur in Kohler's Handbuch (1848). The first 25 powers of 2, 3, 5, 7 are given in Salomon, Logarithmische Tafeln (1827). W. Shanks, Rectification of the Circle (1853), gives every 12th power of 2 up to 2 721. A very valuable paper (" Power-tables, Errata ") published by Allan Cunningham in the Messenger of Math., 1906, 35, p. 13, contains the results of a careful examination of 27 tables containing powers higher than the cube, with lists of errata found in each. Before using any power table this list should be consulted, not only in order to correct the errata, but for the sake of references and general information in regard to such tables. In an appendix (p. 23) Cunningham gives errata in the tables of squares and cubes of Barlow (1814), Jahn (1839), and Kulik (1848).
Triangular Numbers.-E. de Joncourt, De natura et praeclaro usu simplicissimae speciei numerorum trigonalium (The Hague, 1762), contains a table of triangular numbers up to 20,000: viz. 2n(n+I) is given for all numbers from n=1 to 20,000. The table occupies 224 pages.
Reciprocals.-P. Barlow's Tables (1814 and 1840) give reciprocals up to 10,000 to 9 or 10 places; and a table of ten times this extent is given by W. H. Oakes, Table of the Reciprocals of Numbers from I to 100,000 (London, 1865). This table gives seven figures of the reciprocal, and is arranged like a table of seven-figure logarithms, differences being added at the side of the page. The reciprocal 2 The actual place of publication (with a German title, &c.) is Vienna. The copies with an English title, &c., were issued by Triibner; and those with a French title, &c., by Gauthier-Villars. All bear the date 1888.
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 reduite a une addition (Paris, 1861), gives to ten significant figures the reciprocals of the numbers from io,000 to ioo,000, and also the first nine multiples of these reciprocals. J. C. Houzeau gives the reciprocals of numbers up to loo to 20 places and their first nine multiples to 12 places in the Bulletin of the Brussels Academy, 1875, 40, p. 107. E. Gain ( Recueil de tables numeriques, Huy, 1894) gives reciprocals of numbers to loon to to 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, Beytrage zum allgemeinern Gebrauch der Decimalbruche (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 elaborate 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 I to 1024 (1823). The Tabular Series (1823), which occupies 153 pages, gives to 8 places the decimal corresponding to every vulgar fraction less than M whose numerator and denominator 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
ool (= i sa o) to
0 99 8 99 0 9 (=). The author intended the table to include all fractions whose numerator and denominator were each less than woo, 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
676923 and
153846, which are the only periods in which a 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 loo. 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 loo. A posthumous table of C. F. Gauss's, entitled " Tafel zur Verwandlung gemeiner Briiche mit Nennern aus dem ersten Tausend in Decimalbruche," occurs in vol. ii. pp. 412-434 of his Gesammelte Werke (Gottingen, 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 Jo from I 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 moo. 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 Hillsse's edition of Vega. H. Schubert's Fiinfstellige Tafeln and 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 rightangled 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 Vieta 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 XI V., 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 published 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 " Perpendiculum," " 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 o° 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 corrected edition of the Opus palatinum were made by Pitiscus, who had his suspicions that Rheticus had himself calculated a tensecond table of sines 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 Christmann on the death of the former. Amongst these Pitiscus found (I) 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 commencement of a canon of tangents and secants, to the same number of decimal places, for every ten seconds, with first and second differences; (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 undertaken by Rheticus; his tables not only remain to this day the ultimate authorities but formed the data from which Vlacq calculated 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 Lange der Sinus and 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 coordonnees goniometriques (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, Tabulae logarithmico-trigonometricae (Leipzig, 1817), contains a table of sin 2 x, cos 2 x, tan 2 x, cot 2 x from x =1 ° to 45° at intervals of I' to 5 places. J. Andrew, Astronomical and Nautical Tables (London, 1805), contains a table of " squares of natural semichords," i.e. of sin 2 2x from x=o° to 120° at intervals of To" 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 2(1 - cos x ), that is, to sin 2 2x. The table gives logarithmic haversines for every 15" from 0° to 180°, and natural haversines for every 10" from o° to 180°, to 7 places, except near the beginnng, where the logarithms are given to only 5 or 6 places. It occupies 327 folio pages, and was suggested by Andrew's work, a copy of which by chance fell into Hannyngton's hands. Hannyngton 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 0° 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 Ioo,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 I to Ioo,000, with differences, also log sines, tangents, and secants for every minute of the quadrant 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 o° 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 io 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., 3 2, pp. 2 55, 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 government ( 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 t I, 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, 1 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.
xxvi. II a superintendent of the survey. The table was compared with ega's Thesaurus before publication.
S. Pineto's Tables de logarithmes vulgaires a dix decimales, construites d'apres 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 I merely gives logarithms to woo to to places. Table 2 gives logarithms from i,000,000 to I,oir,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 r/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 0 or 1. The principle of multiplying by a factor which is subsequently cancelled by subtracting its logarithm is used also in a tract, containing only ten pages, published by A. Namur and P. Mansion at Brussels in 1877 under the title Tables de logarithmes a 12 decimales jusqu'a 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 loo. .. and the presence of the zeros in the difference renders the interpolation easy.
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 (I +p/n) =log n+B where B is log (I +p/n) to argument A=log p - log n. The tables give log n from n= moo to n= Io,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 who 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 (i.e. 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 Io,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 Io,000 to Ioo,000 the differences are halved in magnitude, while there are only onefourth as many in a page. There is also greater accuracy. A 2 A seven-figure table of the same kind is contained in S. Gundelfinger's Sechsstellige Gaussische and siebenstellige gemeine Logarithmen (Leipzig, 1902).
further peculiarity of this table is that multiples of the differences, instead of proportional parts, are given 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 I 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 toj872. Reference should here be made to Abraham Sharp's table of logarithms of numbers from I to loo and of primes from loo to Imo to 61 places, also of numbers from 999,990 to 1,000,010 to 63 places. These first appeared in Geometry Improv'd. .. by A. S. Philomath (London, 1717). They have been republished in Sherwin's, Callet's, and the earlier editions of Hutton's tables. H. M. Parkhurst, Astronomical Tables (New York, 1871), gives logarithms of numbers from I 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 I °52'12"
7, or log sin 6732". 7, we have S = 4.68 549 80 and log 6732.7 =3,8281893, whence, by addition, we obtain 8.5136873; but V 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 Schron'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 by a nokta (resembling a diamond in a pack of cards). This is very clear in the case of the o's, but leaves unmarked the cases in which the fourth figure is I 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.
Babbage printed a subscript point under the last figure of each logarithm that had been increased. Schron 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, 44 o. See also vol. 34, P
447; and a paper by Gernerth, Ztsch. f. d. osterr. 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 1 Legendre ( Traite 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 uneveh 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 moo, and of uneven numbers from woo to 1143. Legendre's tables also appeared in his Exercices de calcul integral, vol. iii. (1816).
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 a change. The table was calculated by interpolation from the Tr i gonometria artificialis to 10 places and then reduced to 7, so that the last figure should always be correct. Partly on account of the absence of a mark to denote the change of figure in the column and partly on account of 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 logarithmic canon to every second. The change in the column is very clearly marked by a large black nucleus, surrounded by a circle, printed instead of 0. Bagay's work having become rare and costly was reprinted with the errors corrected. The reprint, however, bears the original title-page and date 1829, and there appears to be no means of distinguishing it from the original work except by turning to one of the errata in the original edition and examining whether the correction has been made.
The only other canon to every second that has been published is contained in R. Shortrede's Logarithmic Tables (Edinburgh). This work was originally issued in 1844 in one volume, but being dissatisfied with it Shortrede issued a new edition in 1849 in two volumes. The first volume contains logarithms of numbers, antilogarithms, &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 titlepages as vol. i. and vol. ii. The trigonometrical canon is very complete in every respect, the arguments being given in time as well as in arc, full proportional parts being added, &c. The change of figure in the column is denoted by a nokta, printed instead of o where the change occurs. The page is crowded and the print not very clear, so that Bagay is to be preferred for regular use.
Previous to 1891 the only important tables in which the quadrant is divided centesimally were J. P. Hobert and L. Ideler, Nouvelles tables trigonometriques (Berlin, 1799), and C. Borda and J. B. J. Delambre, Tables trigonometriques decimales (Paris, 1801). The former give, among other tables, natural and log sines, cosines, tangents, and cotangents, to 7 places, the arguments proceeding to 3° at intervals of 10" and thence to 50° at intervals of I' (centesimal), and also natural sines and tangents for the first hundred ten-thousandths of a right angle to Do places. The latter gives long sines, cosines, tangents, cotangents, secants, and cosecants from o° to 3°at intervals of 10" (with full proportional parts for every second), and thence to 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 intervals of Io" and from 0° to 50° at intervals of Io' (centesimal) to II places. Hobert and Ideler give a natural as well as a logarithmic canon; but Borda and Delambre give only the latter. Borda and Delambre give seven-figure logarithms of numbers to 10,000, the line being broken when a change of figure takes place in it.
The tables of Borda and Delambre having become difficult to procure, and seven-figure tables being no longer sufficient for the accuracy required in astronomy and geodesy, the French government in 1891 issued an eight-figure table containing (besides logarithms of numbers to 120,000) log sines and tangents for every ten seconds (centesimal) of the quadrant, the latter being extracted from the Tables du cadastre of Prony (see Logarithm). The title of this fine and handsomely printed work is Service geographique de l'armee: Tables des logarithmes a huit decimates. publiees par ordre du ministre de la guerre (Paris, Imprimerie Nationale, 1891). These tables are now in common use where eight figures are required.
In Brigg's Trigonometria Britannica of 1633 the degree is divided centesimally, and but for the appearance in the same year of Vlacq's Trigonometria artificialis, in which the degree is divided sexagesimally, 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 advantage of this opportunity. He left the degree unaltered, but divided it centesimally instead of sexagesimally, thus ensuring the advantages of decimal division (a saving of work in interpolations, 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 non-centesimal subdivision of the degree that gives rise to inconvenience. Briggs's example was followed by Roe, Oughtred, and other 17th-century writers; but the centesimal division of the degree seemed to have entirely passed out of use, till it was revived by C. Bremiker in his Logarithmisch-trigonometrische Tafeln mit flint Decimalstellen (Berlin, 1872, Loth 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 centesimal table compulsory, if this number of figures is required.
The Astronomische Gesellschaft are, however, publishing an eightfigure table on the sexagesimal system, under the charge of Dr. J. Bauschinger, the director of the k. Recheninstitut at Berlin. The arrangement is to be in groups of three as in Bremiker's tables.
Collections of Tables
For a computer who requires in one volume logarithms of numbers and a ten-second logarithmic canon, perhaps the two best books are L. Schron, Seven-Figure Logarithms (London, 1865, stereotyped, an English edition of the German work published at Brunswick), and C. Bruhns, A New Manual of Logarithms to Seven Places of Decimals (Leipzig, 1870). Both these works (of which there have been numerous editions) give logarithms of numbers and a complete ten-second canon to 7 places; Bruhns also gives log sines, cosines, tangents, and cotangents to every second up to 6° with proportional parts. Schron contains an interpolation table, of 75 pages, giving the first 100 multiples of all numbers from 40 to 420. The logarithms of numbers extend to 108,000 in Schron and to 100,000 in Bruhns. Almost equally convenient is Bremiker's edition of Vega's Logarithmic Tables (Berlin, stereotyped; the English edition was translated from the fortieth edition of Bremiker's by W. L. F. Fischer). This book gives a canon to ever y ten seconds, and for the first five degrees to every second, with logarithms of numbers to 100,000. Schron, Bruhns, and Bremiker all give the proportional parts for all the differences in the logarithms of numbers. In Babbage's, Callet's, and many other tables only every other table of proportional parts is given near the beginning for want of space. Schron, Bruhns, and most modern tables published in Germany have title-pages and introductions in different languages. J. Dupuis, Tables de logarithmes a sept decimates (stereotyped, third tirage, 1868, Paris), is also very convenient, containing a ten-second canon, besides logarithms of numbers to 100,000, hyperbolic logarithms of numbers to moo, to 7 places, &c. In this work negative characteristics are printed throughout in the tables of circular functions, the minus sign being placed above the figure; for the mathematical calculator these are preferable to the ordinary characteristics that are increased by to. The edges of the pages containing the circular functions are red, the rest being grey. Dupuis also edited Callet's logarithms in 1862, with which this work must not be confounded. J. Salomon, Logarithmische Tafeln (Vienna, 1827), contains a ten-second canon (the intervals being one second for the first two degrees), logarithms of numbers to 108,000, squares, cubes, square roots, and cube roots to woo, a factor table to 102,011, ten-place Briggian and hyperbolic logarithms of numbers to woo and of primes to 10,333, and many other useful tables. The work, which is scarce, is a well-printed small quarto volume.
Of collections of general tables among the most useful and accessible are Hutton, Callet, Vega, and Kohler. C. Hutton's well-known Mathematical Tables (London) was first issued in 1785, but considerable additions were made in the fifth edition (1811). The tables contain seven-figure logarithms to 108,000, and to 1200 to 20 places, some antilogarithms to 20 places, hyperbolic logarithms from i to to at intervals of
oi and to 1200 at intervals of unity to 7 places, logistic logarithms, log sines and tangents to every second of the first two degrees, and natural and log sines, tangents, secants, and versed sines for every minute of the quadrant to 7 places. The natural functions occupy the left-hand pages and the logarithmic the right-hand. The first six editions, published in Hutton's lifetime (d. 1823), contain Abraham Sharp's 61-figure logarithms of numbers. Olinthus Gregory, who brought out the 1830 and succeeding editions, omitted these tables and Hutton's introduction, which contains a history of logarithms, the methods of constructing them, &c. F. Callet's Tables portatives de logarithmes (stereotyped, Paris) seems to have been first issued in 1783, and has since passed through a great many editions. In that of 1853 the contents are seven-figure logarithms to 108,000, Briggian and hyperbolic logarithms to 48 places of numbers to 100 and of primes to 1097, log sines and tangents for minutes (centesimal) throughout the quadrant to 7 places, natural and log sines to 15 places for every ten minutes (centesimal) of the quadrant, log sines and tangents for every second of the first five degrees (sexagesimal) and for every ten seconds of the quadrant (sexagesimal) to 7 places, besides logistic logarithms, the first hundred multiples of the modulus to 24 places and the first ten to 70 places, and other tables. This is one of the most complete and practically useful collections of logarithms that have been published, and it is peculiar in giving a centesimally divided canon. The size of the page in the editions published in the 19th century is larger than that of the earlier editions, the type having been reset. G. Vega's Tabulae logarithmo-trigonometricae was first published in 1797 in two volumes. The first contains seven-figure logarithms to 101,000, log sines, &c., for every tenth of a second to I', for every second to I° 30', for ev
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Chisholm, Hugh, General Editor. Entry for 'Mathematical Table'. 1911 Encyclopedia Britanica. https://www.studylight.org/encyclopedias/eng/bri/m/mathematical-table.html. 1910.
