CIO IBC XIX. At the end, Napier's table is reprinted, but to two figures less. This work forms the earliest publication of logarithms on the continent.
' The title is Logarithmorum canonis descriptio, seu arithmeticarum supputationum mirabilis abbreviatio. Ejusque usus in utraque trigonometria ut etiam in omni logistica mathematica, amplissimi, facillimi & expeditissimi explicatio. Authore ac in ventore Ioanne Nepero, Barone Merchistonii, &c. Scoto. Lugduni . .
It will be seen that this title is different from that of Napier's work of 1614; many writers have, however, erroneously given it as the_ title of the latter.
numbers up to moo, and log sines and tangents from Gunter's Canon (162o). In the following year, 1626, Denis Henrion published at Paris a Traicte 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 z tot zo,000, which contained logarithms of numbers up to 1o,00o to to places, taken from Briggs's Arithmetica of 1624, and Gunter's log sines and tangents to 7 places for every minute.' 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 so 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 1\lorgan'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 afull 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 (162o) 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 sevenfigure
' In describing the contents of the works referred to, the language and notation of the present day have been adopted, so that for example a table to radius to,000,000 is described as a table to 7 places, and so on. Also, although logarithms have been spoken of as to the base e, &c., it is to be noticed that neither Napier nor Briggs, nor any of their successors till long afterwards, had any idea of connecting logarithms with exponents.table that was published. It contains sevenfigure logarithms of numbers from 1 to 1oo,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 columns1 to 50 on the lefthand side, and 50 to 100 on the righthand 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 sevenfigure tables. The final step was made by John Newton in his Trigononometria Britannica (1658), a work which is also noticeable as being the only extensive eightfigure 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 sevenfigure 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 Imjrov'd the Briggian logarithms of numbers from 1 to too, and of primes from too to 1too, to 61 places; these were copied into the later editions of Sherwin and other works.
In 1742 a sevenfigure 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 Francois 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 sevenfigure 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 o.ne volume stereotype edition having been published in 184o by Hulsse. 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 sevenfigure tables of logarithms of numbers and trigonometrical functions may be mentioned those of Bremiker, Schron and Bruhns. For logarithms of numbers only perhaps Babbage's table is the most convenient.'
In 1871 Edward Sang published a sevenfigure table of logarithms of numbers from 20,000 to 200,000, the logarithms between too,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.3 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 tenfigure 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 2895–z896 as Appendix 12, pp. 395722. 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 sevendecimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to to places and then contracted to 7. On account of the great size of this table, and for other reasons, it never
2 The smallest number of entries which are necessary in a table of logarithms in order that the intermediate logarithms may be calculable by proportional parts has been investigated by J. E. A. Steggall in the Proc. Edin. Math. Soc., 1892, 10, p. 35. This number is 1700 in the case of a sevenfigure table extending to too,000.
3 Accounts of Sang's calculations are given in the Trans. Roy. Soc. Edin., 1872, 26, p. 521, and in subsequent papers in the Proceedings of the same society.
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 titlepage 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. Prong was charged with the direction of the work, and was expressly required " non seulement a composer des tables qui ne laissassent Hen a desirer quanta l'exactitude, mais a en faire le monument de calcul le plus vaste at le plus imposant qui efit jamais ete executeou meme congu." 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 t >,
Logarithms of the ratios of arcs to sines from o4.00000
to 04.05000, and log sines throughout the quadrant 4
Logarithms of the ratios of arcs to tangents from
04.00000 to 04.05000, and log tangents throughout
the quadrant • 4
The trigonometrical results are given for every hundredthousandth of the quadrant (1o" 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 wm11ployed, &c., was published by Lefort in vol. iv. of the Annales de Cobservatoire 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 trigonometriques (1799), and Borda and Delambre's Tables trigonomeiriques dccimales (180018ol), both of which are sevenfigure tables. The latter work, which was much used, being difficult to procure, and greater accuracy being required, the French government in 1891 published an eightfigure 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 I igarithms arc 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 II places, corresponding to the logarithms from •00001 to •99999 at intervals of •0000l. 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 and 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 185o under the title Tafel der naturlichen Logarithmen der Zahlen.
Hyperbolic antilogarithms are simple exponentials, i.e. the hyperbolic antilogarithm of x is ez. Such tables can scarcely be said to come under the head of logarithmic tables. See TABLES, MATHEMATICAL: Exponential F1,rictions.
Logistic or Proportion si Logarithms.—The old name for what arenow 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 as= 3600"( =1 ° or 1h), the table giving log 3600 — log x, and x being expressed in minutes and seconds. It is also common to find tables in which a= Io800"(=30 or 3^),and xis xpressed 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 360o", 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 tb) by only one entry when log a and log b are given. The utility of such logarithms was first pointed out by Leonelli ina book entitled Supplement 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 Supplement 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+x1) and log (1+x).
Dual Logarithms.—This term was used by Oliver Byrne in a series of works published between 186o and 1870. Dual numbers and logarithms depend upon the expression of a number as a product of 1.1, Pot, Pool . . . or of .9, .99, '999
In the preceding resume only those publications have been mentioned which are of historic importance or interest.' 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.
Calculation of Logarithms.—The name logarithm is derived from the words X ryssv bptO iµ s, the number of the ratios, and the way of regarding a logarithm which justifies the name may be explained as follows. Suppose that the ratio of 10, or any other particular number, to 1 is compounded of a very great number of equal ratios, as, for example, 1,000,000, then it can be shown that the ratio of 2 to I is very nearly equal to a ratio compounded of 301,030 of these small ratios, or ratiunculae, that the ratio of 3 to I is very nearly equal to a ratio compounded of 477,121 of them, and so on. The small ratio, or ratiuncula, is in fact that of the millionth root of Io to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to I will be nearly the same as that of a301'°3° to 1, and so on; or, in other words, if a denotes the millionth root of to, then 2 will be nearly
equal to a'''lt o 3 will be nearly equal to 07'121, and so on.
Napier's original work, the Descriptio Canonis of 1614, contained, not logarithms of numbers, but logarithms of sines, and the relations between the sines and the logarithms were explained by the motions of points in lines, in a manner not unlike that afterwards employed by Newton in the method of fiuxions. An account of the processes by which Napier constructed his table was given in the Constructio Canons of 1619. These methods apply, however, specially to Napier's own kind of logarithms, and are different from those actually used by Briggs in the construction of the tables in the Arithmetica Logarithmica, although some of the latter are the same in principle as the processes described in an appendix to the Constructio.
The processes used by Briggs are explained by him in the preface to the Arithmetica Logarithmica (1624). His method of finding the logarithms of the small primes, which consists in taking a great number of continued geometric means between unity and the given primes, may be described as follows. He first formed the table of numbers and their logarithms:
Numbers. Logarithms.
Io I
3.162277. . . 0.5
1.778279 . . o•25
P33352I . . . 0.125
1'154781 . . . 0.0625
each quantity in the lefthand column being the square root of the one
above it, and each quantity in the righthand column being the half
In vol. xv. (1875) of the Verhandelingen of the Amsterdam Academy of Sciences, Bierens de Haan has given a list of 553 tables of logarithms. A previous paper of the same kind, containing notices of some of the tables, was published by him in the Verslagen en Mededeelingen of the same academy (Aid. Natuurkunde) deel. iv. (1862), p. 15.
of the one above it. To construct this table Briggs, using about thirty places of decimals, extracted the square root of to fiftyfour times, and thus found that the logarithm of I•00000 00000 00000 12781 91493 20032 35 was 0.00000 00000 00000 05551 11512 31257 82702, and that for numbers of this form (i.e. for numbers beginning with i followed by fifteen ciphers, and then by seventeen or a less number of significant figures) the logarithms were proportional to these significant figures. He then by means of a simple proportion
deduced that log (1•00000 00000 00000 1)=o•00000 00000 00000 04342 94481 90325 1804, so that, a quantity 1.00000 00000 00000 X (where x consists of not more than seventeen figures) having been obtained by repeated extraction of the square root of a given number, the logarithm of 1.00000 00000 00000 x could then be found by multiplying x by .00000 00000 00000 04342
To find the logarithm of 2, Briggs raised it to the tenth power, viz. 1024, and extracted the square root of I•024 fortyseven times, the result being 1 00000 00000 00000 16851 60570 53949 77. Multiplying the significant figures by 4342. ..he obtained the logarithm of this quantity, viz. 0.00000 00000 00000 07318 55936 90623 9336, which multiplied by 247 gave 0.01029 99566 39811 95265 277444, the logarithm of 1.024, true to 17 or 18 places. Adding the characteristic 3, and dividing by to, he found (since 2 is the tenth root of 1024) log 2 = .30102 99956 63981 195. Briggs calculated in a similar manner log 6, and thence deduced log 3.
It will be observed that in the first process the value of the modulus is in fact calculated from the formula.
h I
Ioh — i_ loge lo'
the value of h being 1/254, and in the second process logic 2 is in effect calculated from the formula.
10 I 267 logto2= (2'''47 —I) Xloga IOX Io'
Briggs also gave methods of forming the mean proportionals or square roots by differences; and the general method of constructing logarithmic tables by means of differences is due to him.
The following calculation of log 5 is given as an example of the application of a method of mean proportionals. The process consists in taking the geometric mean of numbers above and below 5, the object being to at length arrive at 5.000000. To every geometric mean in the column of numbers there corresponds the arithmetical mean in the column of logarithms. The numbers are denoted by A, B, C, &c., in order to indicate their mode of formation.
Numbers. Logarithms.
A= 1.000000 0.0000000
B = 0.000000 1.0000000
C= I/ (AB) = 3.162277 0.5000000
D = J (BC) = 5.623413 0.7500000
E= J (CD) = 4.216964 0.6250000
F= J (DE) = 4.869674 0.6875000
G = (D F) = 5.232991 0.7187500
H= (FG) = 5.048065 0.7031250
I = J (FH) = 4.958069 0'6953125
K = (HI) = 5.002865 0.6992187
L= (IK) = 4.980416 0.6972656
 4'991627 0.6982421
• 4'997242 0'6987304
 5.000052 0.6989745
 4.998647 0.6988525
= 4'999350 6'6989135
 4'999701 0.6989440
 4'999876 0.6989592
= 4'999963 0.6989668
 5.000008 0.6989707
 4'999984 o•6989687
4'999997 0'6989697
5.000003 0.6989702
5.000000 0.6989700
Great attention was devoted to the methods of calculating logarithms during the 17th and 18th centuries. The earlier methods proposed were, like those of Briggs, purely arithmetical, and for a long time logarithms were regarded from the point of view indicated by their name, that is to say, as depending on the theory of compounded ratios. The introduction of infinite series into mathematics effected a great change in the modes of calculation and the treatment of the subject. Besides Napier and Briggs, special reference should be made to Kepler (Chilias, 1624) and Mercator (Logarilhnsotechnia, 1668), whose methods were arithmetical, and to Newton, Gregory, Halley and Cotes, who employed series. A full and valuable account of these methods is given in Hutton's " Construction of Logarithms," which occurs in the introduction to the early editions of his Mathematical Tables, and also forms tract 21 of his Mathematical Tracts (vol. i., 1812). Many of the early works on logarithms were reprinted in the Scriptores logarithmici of Baron Maseres already referred to.
In the following account only those formulae and methodswill be referred to which would now be used in the calculation o) logarithms.
Since
log.(l+x) =x—9x2+3x3—9x4+&c.,
we have, by changing the sign of x,
lop(' —x) = —x — ~x2 — 3x3 — 4x6 —&c. ;
log, ±2(x+3x3+6x5+&c.), and, therefore, replacing x by p+q,
loge 4=2 p+q+3 (p~q7) +) ( q) 5+8c. ,
in which the series is always convergent, so that the formula affords a method of deducing the logarithm of one number from that of another.
As particular cases we have, by putting q= 1,
loge p= 2 p+1+ (p+l) 3+6 (p+1) 5+&c. and by putting q=p+I,
log,(p+1)—log.p =2 2p+1 2p h1)1v(2p 1)5+&c. ;
the former of these equations gives a convergent series for log.p,and the latter a very convergent series by means of which the logarithm of any number may be deduced from the logarithm of the preceding number.
From the formula for log.(p/q) we may deduce the following very convergent series for loge 2, log, 3 and log. 5, viz.:
log.2=2(7P +5Q +3R), log,3=2(11P+8Q +5R), log. 5 =2 (16P +12Q +7R) ,
where,
P—31+ '~31)af'c(31)5 &c. Q=49+11'(491 )3+6'(49)5+&c. R 161+ 3 (161)3+ 6 (161)5+&c.
The following still more convenient formulae for the calculation of log,2, log.3, &c. were given by J. Couch Adams in the Proc. Roy. Soc., 1878, 27, p. 91. If
10 1 \II 25 4
a=log9 =—log (1 —IT)) , o 1og24=—log 1—100)'
81 1 50 2 c=1og0=log(1+R.)), d=log49=—log (1 100),
126 8 e= log125 = log (1 +1000)
then
log 2=7a—2b+3c, log 3=11a—3b+5c, log 5=16a—4b+7c, and
log 7=1(39a—1ob+17c—d) or=19a—4b+8c+e, and we have the equation of condition,
a—2b+c=d+2e.
By means of these formulae Adams calculated the values of log. 2, log.3, log.5, and log.7 to 276 places of decimals, and deduced the value of log.to and its reciprocal M, the modulus of the Briggian system of logarithms. The value of the modulus found by Adams is
Mo =0.43429 44819 03251 82765 11289
18916 60508 22943 97005 80366
65661 14453 78316 58646 49208
87077 47292 24949 33843 17483
18706 10674 47663 03733 64167
92871 58963 90656 92210 64662
81226 58521 27086 56867 03295
93370 86965 88266 88331 1636o
77384 90514 28443 48666 76864
6586o 85135 56148 21234 87653
43543 43573 17253 83562 21868
25
which is true certainly to 272, and probably to 273, places (Proc. Roy. Soc., 1886, 42, p. 22, where also the values of the other logarithms are given).
If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus,
logio (I +x) = M (x —axe + 3x3 — 4x4 +&c.) ,
and so on.
As has been stated, Abraham Sharp's table contains 61decimal
111= J (Kt) N= J (KM) 0 = J (K N) P=J(NO) Q= J (OP) R = s/ (OQ) S= (OR) T= J (OS) V = J (OT) W= J (TV) X= J (WV) Y= J (VX) Z=J(XV)
whence
Briggian logarithms of primes up to 1 too, so that the logarithms of all composite numbers whose greatest prime factor does not exceed this number may be found by simple addition; and Wolfram's table gives 48decimal hyperbolic logarithms of primes up to I0,009. By means of these tables and of a factor table we may very readily obtain the Briggian logarithm of a number to 61 or a less number of places or of its hyperbolic logarithm to 48 or a less number of places in the following manner. Suppose the hyperbolic logarithm of the prime number 43,867 required. Multiplying by 5o, we have 50X43,867=2,193350, and on looking in Burckhardt's Table des diviseurs for a number near to this which shall have no prime factor greater than 10,009, it appears that
2,193,349 = 23 X 47 X2029 ;
thus
43,867=,5(23X47X2029+1),
and therefore
loge 43,867=log. 23+loge 47+log; 2029—log, 50
1 1 1
+2,193,349—1
+ ' (2,193,349)3—&c.
(2,193,349)2
The first term of the series in the second line is
0.00000 04559 23795 07319 6286;
dividing this by 2X2,193,349 we obtain 93325 3457,
0.00000 00000 00103
and the third term is 00003 1590,
0.00000 00000 00000
so that the series = 13997 4419;
0.00000 04559 23691
whence, taking out the logarithms from Wolfram's table,
log. 43,867 =10.68891 76079 60568 10191 3661.
The principle of the method is to multiply the given prime (supposed to consist of 4, 5 or 6 figures) by such a factor that the product may be a number within the range of the factor tables, and such that, when it is increased by 1 or 2, the prime factors may all be within the range of the logarithmic tables. The logarithm is then obtained by use of the formula
d d2 d3
loge (x+d)=logx} x 4 +}z6—&c.,
in which of course the object is to render d/x as small as possible. If the logarithm required is Briggian, the value of the series is to be multiplied by M.
If the number is incommensurable or consists of more than seven figures, we can take the first seven figures of it (or multiply and divide the result by any factor, and take the first seven figures of the result) and proceed as before. An application to the hyperbolic logarithm of g is given by Burckhardt In the introduction to his Table des diviseurs for the second million.
The best general method of calculating logarithms consists, in its simplest form, in resolving the number whose logarithm is required into factors of the form i — •i'n, where n is one of the nine digits; and making use of subsidiary tables of logarithms of factors of this form. For example, suppose the logarithm of 543839 required to twelve places. Dividing by 1o5 and by 5 the number becomes 1.087678, and resolving this number into factors of the form — • 1'n we find that
543839=1O5X5(I•I28)(1—•1'6)(1166)(1.163)(I•173)
X 1 .185)(1 •197)(1 .1109)(1 .1113)(1 .1122), where 1—•128 denotes i—•o8, I—•146 denotes 1—•0006, &c., and so on. All that is required therefore in order to obtain the logarithm of any number is a table of logarithms, to the required number of places, of •n, •91f, •99n, .999n, &e., for n =I, 2, 3, . . . 9.
The resolution of a number into factors of the above form is easily performed. Taking, for example, the number i •087678, the object is to destroy the significant figure 8 in the second place of decimals; this is effected by multiplying the number by 1—.o8, that is, by subtracting from the number eight times itself advanced two places, and we thus obtain 1.00066376. To destroy the first 6 multiply by I—•0006 giving 1.000063361744, and multiplying successively by 1—•00006 and 1—.000003, we obtain 1.000000357932, and it is clear that these last six significant figures represent without any further work the remaining factors required. In the corresponding antilogarithmic process the number is expressed as a product of factors of the form I+•1^x.
This method of calculating logarithms by the resolution of numbers into factors of the form I—•Irn is generally known as Weddle's method, having been published by him in The Mathematician for November 1845, and the corresponding method for antilogarithms by means of factors of the form i+(•1)rn is known by the name of Hearn, who published it in the same journal for 1847. In 1846 Peter Gray constructed a new table to 12 places, in which the factors were of the form i —(•o1)rn, so that n had the values 1, 2, . . . 99; and subsequently he constructed a similar table for factors of the form +(•oi )'n. He also devised a method of applying a table of Hearn'sform (i.e. of factors of the form I+•Irn) to the construction of logarithms, and calculated a table of logarithms of factors of the form I +(•ooI)'n to 24 places. This was published in 1876 under the title Tables for the formation of logarithms and antilogarithms to twentyfour or any less number of places, and contains the most complete and useful application of the method, with many improvements in points of detail. Taking as an example the calculation of the Briggian logarithm of the number 43,867, whose hyperbolic logarithm has been calculated above, we multiply it by 3, giving 131,601, and find by Gray's process that the factors of 1.31601 are
(I) I.316 (5) •(cot)5002
(2) 1.000007 (6) I•(00I)5602
(3) 1061)2598 (7) 1•(001)6412
(4) 1•(001)3780 (8) I•(ooi)734o Taking the logarithms from Gray's tables we obtain the required logarithm by addition as follows:
522 878 745 280 337 562 704 972 =colo 3
119 255 889 277 936 685 553 913 =log (i)
3 040 050 733 157 610 239 =log 2)
259 708 022 525 453 597 =log (3)
338 749 695 752 424 =10g (4)
868 588 964 =log (5)
261 445 278 =log (6)
178 929 =log (7)
148 =log (8)
4.642 137 934 655 780 757 288 464=10gio43,867
In Shortrede's Tables there are tables of logarithms and factors of the form i r(•oi)'n to 16 places and of the form r =(•i)rn to 2 places; and in his Tables de Logarithmes a 27 De'cimales (Paris, 1867}5 Fedor Thoman gives tables of logarithms of factors of the form =. rn. In the Messenger of Mathematics, vol. iii. pp. 6692, 1873, Henry Wace gave a simple and clear account of both the logarithmic and antilogarithmic processes, with tables of both Briggian and hyperbolic logarithms of factors of the form I t • Irn to 20 places.
Although the method is usually known by the names of Weddle and Hearn, it is really, in its essential features, due to Briggs, who gave in the Arithmetica logarithmica of 1624 a table of the logarithms of I +•I'n up to r=9 to 15 places of decimals. It was first formally proposed as an independent method, with great improvements, by Robert Flower in The Radix, a new way of making Logarithms, which was published in 1771; and Leonelli, in his Supplement logarithmique (18021803), already noticed, referred to Flower and reproduced some of his tables. A complete bibliography of this method has been given by A. J. Ellis in a paper "on the potential radix as a means of calculating logarithms,' printed in the Proceedings of the Royal Society, vol. xxxi., 188r, pp. 401407, and vol. xxxii., 1881, pp. 377379. Reference should also be made to Hoppe's Tafeln zur dreissigstelligen logarithmischen Rechnung (Leipzig, 1876), which give in a somewhat modified form a table of the hyperbolic logarithm of i +.1'n.
The preceding methods are only appropriate for the calculation of isolated logarithms. If a complete table had to be reconstructed, or calculated to more places, it would undoubtedly be most convenient to employ the method of differences. A full account of this method as applied to the calculation of the Tables du Cadastre is given by Lefort in vol. iv. of the Annales de l'Observatoire de Paris.
(J. W. L. G.)
End of Article: CIO IBC XIX 

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