Part 3
by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines; and, as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only. It seems to be due to Wittich of Breslau, who was assistant for a short time to Tycho Brahe; and it was used by them in their calculations in 1582. Wittich in 1584 made known at Cassel the calculation of one case by this prosthaphaeresis; and Justus Byrgius proved it in such a manner that from his proof the extension to the solution of all triangles could be deduced.[8] Clavius generalized the method in his treatise _De astrolabio_ (1593), lib. i. lemma liii. The lemma is enunciated as follows:--
"Quaestiones omnes, quae per sinus, tangentes, atque secantes absolvi solent, per solam prosthaphaeresim, id est, per solam additionem, subtractionem, sine laboriosa numerorum multiplicatione divisioneque expedire."
Clavius then refers to a work of Raymarus Ursus Dithmarsus as containing an account of a particular case. The work is probably the _Fundamentum astronomicum_ (1588). Longomontanus, in his _Astronomia Danica_ (1622), gives an account of the method, stating that it is not to be found in the writings of the Arabs or Regiomontanus. As Longomontanus is mentioned in Anthony Wood's anecdote, and as Wittich as well as Longomontanus were assistants of Tycho, we may infer that Wittich's prosthaphaeresis is the method referred to by Wood.
It is evident that Wittich's prosthaphaeresis could not be a good method of practically effecting multiplications unless the quantities to be multiplied were sines, on account of the labour of the interpolations. It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and, analytically considered, it is not so different in principle from the logarithmic method. In fact, if we put xy = [phi](X + Y), X being a function of x only and Y a function of y only, we can show that we must have X = Ae^(qx), y = Be^(qy); and if we put xy = [phi](X + Y) - [phi](X - Y), the solutions are [phi](X + Y) = ¼(x + y)², and x = sin X, y = sin Y, [phi](X + Y) = -½cos(X + Y). The former solution gives a method known as that of quarter-squares; the latter gives the method of prosthaphaeresis.
An account has now been given of Napier's invention and its publication, the transition to decimal logarithms, the calculation of the tables by Briggs, Vlacq and Gunter, as well as of the claims of Byrgius and the method of prosthaphaeresis. To complete the early history of logarithms it is necessary to return to Napier's _Descriptio_ in order to describe its reception on the continent, and to mention the other logarithmic tables which were published while Briggs was occupied with his calculations.
John Kepler, who has been already quoted in connexion with Craig's visit to Tycho Brahe, received the invention of logarithms almost as enthusiastically as Briggs. His first mention of the subject occurs in a letter to Schikhart dated the 11th of March 1618, in which he writes-"Extitit Scotus Baro, cujus nomen mihi excidit, qui praeclari quid praestitit, necessitate omni multiplicationum et divisionum in meras additiones et subtractiones commutata, nec sinibus utitur; at tamen opus est ipsi tangentium canone: et varietas, crebritas, difficultasque additionum subtractionumque alicubi laborem multiplicandi et dividendi superat." This erroneous estimate was formed when he had seen the _Descriptio_ but had not read it; and his opinion was very different when he became acquainted with the nature of logarithms. The dedication of his _Ephemeris_ for 1620 consists of a letter to Napier dated the 28th of July 1619, and he there congratulates him warmly on his invention and on the benefit he has conferred upon astronomy generally and upon Kepler's own Rudolphine tables. He says that, although Napier's book had been published five years, he first saw it at Prague two years before; he was then unable to read it, but last year he had met with a little work by Benjamin Ursinus[9] containing the substance of the method, and he at once recognized the importance of what had been effected. He then explains how he verified the canon, and so found that there were no essential errors in it, although there were a few inaccuracies near the beginning of the quadrant, and he proceeds, "Haec te obiter scire volui, ut quibus tu methodis incesseris, quas non dubito et plurimas et ingeniosissimas tibi in promptu esse, eas publici juris fieri, mihi saltem (puto et caeteris) scires fore gratissimum; eoque percepto, tua promissa folio 57, in debitum cecidisse intelligeres." This letter was written two years after Napier's death (of which Kepler was unaware), and in the same year as that in which the _Constructio_ was published. In the same year (1620) Napier's _Descriptio_ (1614) and _Constructio_ (1619) were reprinted by Bartholomew Vincent at Lyons and issued together.[10]
Napier calculated no logarithms of numbers, and, as already stated, the logarithms invented by him were not to base e. The first logarithms to the base e were published by John Speidell in his _New Logarithmes_ (London, 1619), which contains hyperbolic log sines, tangents and secants for every minute of the quadrant to 5 places of decimals.
In 1624 Benjamin Ursinus published at Cologne a canon of logarithms exactly similar to Napier's in the _Descriptio_ of 1614, only much enlarged. The interval of the arguments is 10´´, and the results are given to 8 places; in Napier's canon the interval is 1', and the number of places is 7. The logarithms are strictly Napierian, and the arrangement is identical with that in the canon of 1614. This is the largest Napierian canon that has ever been published.
In the same year (1624) Kepler published at Marburg a table of Napierian logarithms of sines with certain additional columns to facilitate special calculations.
The first publication of Briggian logarithms on the continent is due to Wingate, who published at Paris in 1625 his _Arithmétique logarithmétique_, containing seven-figure logarithms of numbers up to 1000, and log sines and tangents from Gunter's _Canon_ (1620). In the following year, 1626, Denis Henrion published at Paris a _Traicté des Logarithmes_, containing Briggs's logarithms of numbers up to 20,001 to 10 places, and Gunter's log sines and tangents to 7 places for every minute. In the same year de Decker also published at Gouda a work entitled _Nieuwe Telkonst, inhoudende de Logarithmi voor de Ghetallen beginnende van 1 tot 10,000_, which contained logarithms of numbers up to 10,000 to 10 places, taken from Briggs's _Arithmetica_ of 1624, and Gunter's log sines and tangents to 7 places for every minute.[11] Vlacq rendered assistance in the publication of this work, and the privilege is made out to him.
The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the _Principia_ of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier's _Descriptio_. The calculation of tables of the natural trigonometrical functions may be said to have formed the work of the last half of the 16th century, and the great canon of natural sines for every 10 seconds to 15 places which had been calculated by Rheticus was published by Pitiscus only in 1613, the year before that in which the _Descriptio_ appeared. In the construction of the natural trigonometrical tables Great Britain had taken no part, and it is remarkable that the discovery of the principles and the formation of the tables that were to revolutionize or supersede all the methods of calculation then in use should have been so rapidly effected and developed in a country in which so little attention had been previously devoted to such questions.
For more detailed information relating to Napier, Briggs and Vlacq, and the invention of logarithms, the reader is referred to the life of Briggs in Ward's _Lives of the Professors of Gresham College_ (London, 1740); Thomas Smith's _Vitae quorundam eruditissimorum et illustrium virorum_ (Vita Henrici Briggii) (London, 1707); Mark Napier's _Memoirs of John Napier_ already referred to, and the same author's _Naperi libri qui supersunt_ (1839); Hutton's _History_; de Morgan's article already referred to; Delambre's _Histoire de l'Astronomie moderne_; the report on mathematical tables in the _Report of the British Association_ for 1873; and the _Philosophical Magazine_ for October and December 1872 and May 1873. It may be remarked that the date usually assigned to Briggs's first visit to Napier is 1616 and not 1615 as stated above, the reason being that Napier was generally supposed to have died in 1618; but it was shown by Mark Napier that the true date is 1617.
In the years 1791-1807 Francis Maseres published at London, in six volumes quarto "Scriptores Logarithmici, or a collection of several curious tracts on the nature and construction of logarithms, mentioned in Dr Hutton's historical introduction to his new edition of Sherwin's mathematical tables ...," which contains reprints of Napier's _Descriptio_ of 1614, Kepler's writings on logarithms (1624-1625), &c. In 1889 a translation of Napier's _Constructio_ of 1619 was published by Walter Rae Macdonald. Some valuable notes are added by the translator, in one of which he shows the accuracy of the method employed by Napier in his calculations, and explains the origin of a small error which occurs in Napier's table. Appended to the Catalogue is a full and careful bibliography of all Napier's writings, with mention of the public libraries, British and foreign, which possess copies of each. A facsimile reproduction of Bartholomew Vincent's Lyons edition (1620) of the _Constructio_ was issued in 1895 by A. Hermann at Paris (this imprint occurs on page 62 after the word "Finis").
It now remains to notice briefly a few of the more important events in the history of logarithmic tables subsequent to the original calculations.
_Common or Briggian Logarithms of Numbers._--Nathaniel Roe's _Tabulae logarithmicae_ (1633) was the first complete seven-figure table that was published. It contains seven-figure logarithms of numbers from 1 to 100,000, with characteristics unseparated from the mantissae, and was formed from Vlacq's table (1628) by leaving out the last three figures. All the figures of the number are given at the head of the columns, except the last two, which run down the extreme columns--1 to 50 on the left-hand side, and 50 to 100 on the right-hand side. The first four figures of the logarithms are printed at the top of the columns. There is thus an advance half way towards the arrangement now universal in seven-figure tables. The final step was made by John Newton in his _Trigonometria Britannica_ (1658), a work which is also noticeable as being the only extensive eight-figure table that until recently had been published; it contains logarithms of sines, &c., as well as logarithms of numbers.
In 1705 appeared the original edition of Sherwin's tables, the first of the series of ordinary seven-figure tables of logarithms of numbers and trigonometrical functions such as are in general use now. The work went through several editions during the 18th century, and was at length superseded in 1785 by Hutton's tables, which continued in successive editions to maintain their position for a century.
In 1717 Abraham Sharp published in his _Geometry Improv'd_ the Briggian logarithms of numbers from 1 to 100, and of primes from 100 to 1100, to 61 places; these were copied into the later editions of Sherwin and other works.
In 1742 a seven-figure table was published in quarto form by Gardiner, which is celebrated on account of its accuracy and of the elegance of the printing. A French edition, which closely resembles the original, was published at Avignon in 1770.
In 1783 appeared at Paris the first edition of François Callet's tables, which correspond to those of Hutton in England. These tables, which form perhaps the most complete and practically useful collection of logarithms for the general computer that has been published, passed through many editions.
In 1794 Vega published his _Thesaurus logarithmorum completus_, a folio volume containing a reprint of the logarithms of numbers from Vlacq's _Arithmetica logarithmica_ of 1628, and _Trigonometria artificialis_ of 1633. The logarithms of numbers are arranged as in an ordinary seven-figure table. In addition to the logarithms reprinted from the _Trigonometria_, there are given logarithms for every second of the first two degrees, which were the result of an original calculation. Vega devoted great attention to the detection and correction of the errors in Vlacq's work of 1628. Vega's _Thesaurus_ has been reproduced photographically by the Italian government. Vega also published in 1797, in 2 vols. 8vo, a collection of logarithmic and trigonometrical tables which has passed through many editions, a very useful one volume stereotype edition having been published in 1840 by Hülsse. The tables in this work may be regarded as to some extent supplementary to those in Callet.
If we consider only the logarithms of numbers, the main line of descent from the original calculation of Briggs and Vlacq is Roe, John Newton, Sherwin, Gardiner; there are then two branches, viz. Hutton founded on Sherwin and Callet on Gardiner, and the editions of Vega form a separate offshoot from the original tables. Among the most useful and accessible of modern ordinary seven-figure tables of logarithms of numbers and trigonometrical functions may be mentioned those of Bremiker, Schrön and Bruhns. For logarithms of numbers only perhaps Babbage's table is the most convenient.[12]
In 1871 Edward Sang published a seven-figure table of logarithms of numbers from 20,000 to 200,000, the logarithms between 100,000 and 200,000 being the result of a new calculation. By beginning the table at 20,000 instead of at 10,000 the differences are halved in magnitude, while the number of them in a page is quartered. In this table multiples of the differences, instead of proportional parts, are given.[13] John Thomson of Greenock (1782-1855) made an independent calculation of logarithms of numbers up to 120,000 to 12 places of decimals, and his table has been used to verify the errata already found in Vlacq and Briggs by Lefort (see _Monthly Not. R.A.S._ vol. 34, p. 447). A table of ten-figure logarithms of numbers up to 100,009 was calculated by W. W. Duffield and published in the _Report of the U.S. Coast and Geodetic Survey for 1895-1896_ as Appendix 12, pp. 395-722. The results were compared with Vega's _Thesaurus_ (1794) before publication.
_Common or Briggian Logarithms of Trigonometrical Functions._--The next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the _Trigonometria_ to 10 places and then contracted to 7. On account of the great size of this table, and for other reasons, it never came into very general use, Bagay's _Nouvelles tables astronomiques_ (1829), which also contains log sines and tangents to every second, being preferred; this latter work, which for many years was difficult to procure, has been reprinted with the original title-page and date unchanged. The only other logarithmic canon to every second that has been published forms the second volume of Shortrede's _Logarithmic Tables_ (1849). In 1784 the French government decided that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant. Prony was charged with the direction of the work, and was expressly required "non seulement à composer des tables qui ne laissassent rien à désirer quant à l'exactitude, mais à en faire le monument de calcul le plus vaste et le plus imposant qui eût jamais été exécuté ou même conçu." Those engaged upon the work were divided into three sections: the first consisted of five or six mathematicians, including Legendre, who were engaged in the purely analytical work, or the calculation of the fundamental numbers; the second section consisted of seven or eight calculators possessing some mathematical knowledge; and the third comprised seventy or eighty ordinary computers. The work, which was performed wholly in duplicate, and independently by two divisions of computers, occupied two years. As a consequence of the double calculation, there are two manuscripts, one deposited at the Observatory, and the other in the library of the Institute, at Paris. Each of the two manuscripts consists essentially of seventeen large folio volumes, the contents being as follows:--
Logarithms of numbers up to 200,000 8 vols.
Natural sines 1 "
Logarithms of the ratios of arcs to sines from 0^q.00000 to 0^q.05000, and log sines throughout the quadrant 4 "
Logarithms of the ratios of arcs to tangents from 0^q.00000 to 0^q.05000, and log tangents throughout the quadrant 4 "
The trigonometrical results are given for every hundred-thousandth of the quadrant (10´´ centesimal or 3´´.24 sexagesimal). The tables were all calculated to 14 places, with the intention that only 12 should be published, but the twelfth figure is not to be relied upon. The tables have never been published, and are generally known as the _Tables du Cadastre_, or, in England, as the great French manuscript tables.
A very full account of these tables, with an explanation of the methods of calculation, formulae employed, &c., was published by Lefort in vol. iv. of the _Annales de l'observatoire de Paris_. The printing of the table of natural sines was once begun, and Lefort states that he has seen six copies, all incomplete, although including the last page. Babbage compared his table with the _Tables du Cadastre_, and Lefort has given in his paper just referred to most important lists of errors in Vlacq's and Briggs's logarithms of numbers which were obtained by comparing the manuscript tables with those contained in the _Arithmetica logarithmica_ of 1624 and of 1628.
As the _Tables du Cadastre_ remained unpublished, other tables appeared in which the quadrant was divided centesimally, the most important of these being Hobert and Ideler's _Nouvelles tables trigonométriques_ (1799), and Borda and Delambre's _Tables trigonométriques décimales_ (1800-1801), both of which are seven-figure tables. The latter work, which was much used, being difficult to procure, and greater accuracy being required, the French government in 1891 published an eight-figure centesimal table, for every ten seconds, derived from the _Tables du Cadastre_.
_Decimal or Briggian Antilogarithms._--In the ordinary tables of logarithms the natural numbers are all integers, while the logarithms tabulated are incommensurable. In an antilogarithmic table, the logarithms are exact quantities such as .00001, .00002, &c., and the numbers are incommensurable. The earliest and largest table of this kind that has been constructed is Dodson's _Antilogarithmic canon_ (1742), which gives the numbers to 11 places, corresponding to the logarithms from .00001 to .99999 at intervals of .00001. Antilogarithmic tables are few in number, the only other extensive tables of the same kind that have been published occurring in Shortrede's _Logarithmic tables_ already referred to, and in Filipowski's _Table of antilogarithms_ (1849). Both are similar to Dodson's tables, from which they were derived, but they only give numbers to 7 places.
_Hyperbolic or Napierian logarithms_ (i.e. to base e).--The most elaborate table of hyperbolic logarithms that exists is due to Wolfram, a Dutch lieutenant of artillery. His table gives the logarithms of all numbers up to 2200, and of primes (and also of a great many composite numbers) from 2200 to 10,009, to 48 decimal places. The table appeared in Schulze's _Neue und erweiterte Sammlung logarithmischer Tafeln_ (1778), and was reprinted in Vega's _Thesaurus_ (1794), already referred to. Six logarithms omitted in Schulze's work, and which Wolfram had been prevented from computing by a serious illness, were published subsequently, and the table as given by Vega is complete. The largest hyperbolic table as regards range was published by Zacharias Dase at Vienna in 1850 under the title _Tafel der natürlichen Logarithmen der Zahlen_.
_Hyperbolic antilogarithms_ are simple exponentials, i.e. the hyperbolic antilogarithm of x is e^x. Such tables can scarcely be said to come under the head of logarithmic tables. See TABLES, MATHEMATICAL: _Exponential Functions_.
_Logistic or Proportional Logarithms._--The old name for what are now called ratios or fractions are _logistic numbers_, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since log (a/x) = log a - log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign. The first table of this kind appeared in Kepler's work of 1624 which has been already referred to. The object of a table of log (a/x) is to facilitate the working out of proportions in which the third term is a constant quantity a. In most collections of tables of logarithms, and especially those intended for use in connexion with navigation, there occurs a small table of logistic logarithms in which a = 3600´´ (= 1° or 1^h), the table giving log 3600 - log x, and x being expressed in minutes and seconds. It is also common to find tables in which a = 10800´´ (= 3° or 3^h), and x is expressed in degrees (or hours), minutes and seconds. Such tables are generally given to 4 or 5 places. The usual practice in books seems to be to call logarithms logistic when a is 3600´´, and proportional when a has any other value.
_Addition and Subtraction, or Gaussian Logarithms._--_Gaussian logarithms_ are intended to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers themselves being unknown; and on this account they are frequently called addition and subtraction logarithms. The object of the table is in fact to give log (a ± b) by only one entry when log a and log b are given. The utility of such logarithms was first pointed out by Leonelli in a book entitled _Supplément logarithmique_, printed at Bordeaux in the year XI. (1802/3); he calculated a table to 14 places, but only a specimen of it which appeared in the _Supplément_ was printed. The first table that was actually published is due to Gauss, and was printed in Zach's _Monatliche Correspondenz_, xxvi. 498 (1812). Corresponding to the argument log x it gives the values of log (1 + x^-1) and log (1 + x).
_Dual Logarithms._--This term was used by Oliver Byrne in a series of works published between 1860 and 1870. Dual numbers and logarithms depend upon the expression of a number as a product of 1.1, 1.01, 1.001 ... or of .9, .99, .999....
In the preceding _résumé_ only those publications have been mentioned which are of historic importance or interest.[14] For fuller details with respect to some of these works, for an account of tables published in the latter part of the 19th century, and for those which would now be used in actual calculation, reference should be made to the article TABLES, MATHEMATICAL.