Part 4
_Calculation of Logarithms._--The name logarithm is derived from the words [Greek: logon arithmos], the number of the ratios, and the way of regarding a logarithm which justifies the name may be explained as follows. Suppose that the ratio of 10, or any other particular number, to 1 is compounded of a very great number of equal ratios, as, for example, 1,000,000, then it can be shown that the ratio of 2 to 1 is very nearly equal to a ratio compounded of 301,030 of these small ratios, or _ratiunculae_, that the ratio of 3 to 1 is very nearly equal to a ratio compounded of 477,121 of them, and so on. The small ratio, or _ratiuncula_, is in fact that of the millionth root of 10 to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to 1 will be nearly the same as that of a^{301,030} to 1, and so on; or, in other words, if a denotes the millionth root of 10, then 2 will be nearly equal to a^{301,030}, 3 will be nearly equal to a^{477,121}, 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 fluxions. An account of the processes by which Napier constructed his table was given in the _Constructio Canonis_ 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.
10 1 3.162277... 0.5 1.778279... 0.25 1.333521... 0.125 1.154781... 0.0625
each quantity in the left-hand column being the square root of the one above it, and each quantity in the right-hand column being the half of the one above it. To construct this table Briggs, using about thirty places of decimals, extracted the square root of 10 fifty-four times, and thus found that the logarithm of 1.00000 00000 00000 12781 91493 20032 35 was 0.00000 00000 00000 05551 11512 31257 82702, and that for numbers of this form (i.e. for numbers beginning with 1 followed by fifteen ciphers, and then by seventeen or a less number of significant figures) the logarithms were proportional to these significant figures. He then by means of a simple proportion deduced that log (1.00000 00000 00000 1) = 0.00000 00000 00000 04342 94481 90325 1804, so that, a quantity 1.00000 00000 00000 x (where x consists of not more than seventeen figures) having been obtained by repeated extraction of the square root of a given number, the logarithm of 1.00000 00000 00000 x could then be found by multiplying x by .00000 00000 00000 04342....
To find the logarithm of 2, Briggs raised it to the tenth power, viz. 1024, and extracted the square root of 1.024 forty-seven times, the result being 1.00000 00000 00000 16851 60570 53949 77. Multiplying the significant figures by 4342 ... he obtained the logarithm of this quantity, viz. 0.00000 00000 00000 07318 55936 90623 9336, which multiplied by 2^47 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 10, he found (since 2 is the tenth root of 1024) log 2 = .30102 99956 63981 195. Briggs calculated in a similar manner log 6, and thence deduced log 3.
It will be observed that in the first process the value of the modulus is in fact calculated from the formula.
h 1 -------- = ---------, 10^h - 1 log(e) 10
the value of h being 1/2^54, and in the second process log10 2 is in effect calculated from the formula.
1 2^47 log(10) 2 = [2^(10/2^47) - 1] × --------- × ----. log(e) 10 10
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 = 10.000000 1.0000000 C = [root](AB) = 3.162277 0.5000000 D = [root](BC) = 5.623413 0.7500000 E = [root](CD) = 4.216964 0.6250000 F = [root](DE) = 4.869674 0.6875000 G = [root](DF) = 5.232991 0.7187500 H = [root](FG) = 5.048065 0.7031250 I = [root](FH) = 4.958069 0.6953125 K = [root](HI) = 5.002865 0.6992187 L = [root](IK) = 4.980416 0.6972656 M = [root](KL) = 4.991627 0.6982421 N = [root](KM) = 4.997242 0.6987304 O = [root](KN) = 5.000052 0.6989745 P = [root](NO) = 4.998647 0.6988525 Q = [root](OP) = 4.999350 0.6989135 R = [root](OQ) = 4.999701 0.6989440 S = [root](OR) = 4.999876 0.6989592 T = [root](OS) = 4.999963 0.6989668 V = [root](OT) = 5.000008 0.6989707 W = [root](TV) = 4.999984 0.6989687 X = [root](WV) = 4.999997 0.6989697 Y = [root](VX) = 5.000003 0.6989702 Z = [root](XY) = 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 (_Logarithmotechnia_, 1668), whose methods were arithmetical, and to Newton, Gregory, Halley and Cotes, who employed series. A full and valuable account of these methods is given in Hutton's "Construction of Logarithms," which occurs in the introduction to the early editions of 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 methods will be referred to which would now be used in the calculation of logarithms.
Since
log(e)(1 + x) = x - ½x² + (1/3)x³ - ¼x^4 + &c.,
we have, by changing the sign of x,
log(e)(1 - x) = -x - ½x² - (1/3)x³ - ¼x^4 - &c.;
whence
1 + x log(e) ----- = 2(x + (1/3)x³ + (1/5)x^5 + &c.), 1 - x
p - q and, therefore, replacing x by -----, p + q _ _ p | p - q /p - q\³ /p - q\^5 | log(e) --- = 2 | ----- + (1/3)( ----- ) + (1/5)( ----- ) + &c. |, q |_ p + q \p + q/ \p + q/ _|
in which the series is always convergent, so that the formula affords a method of deducing the logarithm of one number from that of another.
As particular cases we have, by putting q = 1, _ _ | p - 1 /p - 1\³ /p - 1\^5 | log(e) p = 2 | ----- + (1/3)( ----- ) + (1/5)( ----- ) + &c. |, |_ p + 1 \p + 1/ \p + 1/ _|
and by putting q = p + 1, _ _ | 1 1 1 | log(e)(p + 1) - log(e)(p) = 2 | ------ + (1/3)--------- + (1/5)----------- + &c. |; |_ 2p + 1 (2p + 1)³ (2p + 1)^5 _|
the former of these equations gives a convergent series for log(e)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(e)(p/q) we may deduce the following very convergent series for log(e)2, log(e)3 and log(e)5, viz.:--
log(e)2 = 2( 7P + 5Q + 3R), log(e)3 = 2(11P + 8Q + 5R), log(e)5 = 2(16P + 12Q + 7R),
where
1 1 1 P = -- + (1/3) · ------ + (1/5) · ------ + &c. 31 (31)^3 (31)^5
1 1 1 Q = -- + (1/3) · ------ + (1/5) · ------ + &c. 49 (49)^3 (49)^5
1 1 1 R = --- + (1/3) · ------- + (1/5) · ------- + &c. 161 (161)^3 (161)^5
The following still more convenient formulae for the calculation of log(e)2, log(e)3, &c. were given by J. Couch Adams in the _Proc. Roy. Soc._, 1878, 27, p. 91. If
10 / 1 \ 25 / 4 \ a = log -- = -log ( 1 - -- ), b = log -- = -log ( 1 - --- ), 9 \ 10 / 24 \ 100 /
81 / 1 \ 50 / 2 \ c = log -- = log ( 1 + -- ), d = log -- = -log ( 1 - --- ), 80 \ 80 / 49 \ 100 /
126 / 8 \ e = log --- = log ( 1 + ---- ), 125 \ 1000 /
then
log 2 = 7a - 2b + 3c, log 3 = 11a - 3b + 5c, log 5 = 16a - 4b + 7c,
and
log 7 = ½(39a - 10b + 17c - d) or = 19a - 4b + 8c + e,
and we have the equation of condition,
a - 2b + c = d + 2e.
By means of these formulae Adams calculated the values of log(e)2, log(e)3, log(e)5, and log(e)7 to 276 places of decimals, and deduced the value of log(e)10 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 16360 77384 90514 28443 48666 76864 65860 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,
log(10) (1 + x) = M (x - ½x² + (1/3)x³ - ¼x^4 + &c.),
and so on.
As has been stated, Abraham Sharp's table contains 61-decimal Briggian logarithms of primes up to 1100, so that the logarithms of all composite numbers whose greatest prime factor does not exceed this number may be found by simple addition; and Wolfram's table gives 48-decimal hyperbolic logarithms of primes up to 10,009. By means of these tables and of a factor table we may very readily obtain the Briggian logarithm of a number to 61 or a less number of places or of its hyperbolic logarithm to 48 or a less number of places in the following manner. Suppose the hyperbolic logarithm of the prime number 43,867 required. Multiplying by 50, we have 50 × 43,867 = 2,193,350, and on looking in Burckhardt'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 × 47 × 2029;
thus
43,867 = (1/50)(23 × 47 × 2029 + 1),
and therefore
log(e) 43,867 = log(e) 23 + log(e) 47 + log(e) 2029 - log(e) 50
1 1 1 + --------- - ½ ------------ + (1/3) ---------- - &c. 2,193,349 (2,193,349)² (193,349)³
The first term of the series in the second line is
0.00000 04559 23795 07319 6286;
dividing this by 2 ×2,193,349 we obtain
0.00000 00000 00103 93325 3457,
and the third term is
0.00000 00000 00000 00003 1590,
so that the series =
0.00000 04559 23691 13997 4419;
whence, taking out the logarithms from Wolfram's table,
log(e) 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 log(e)(x + d) = log(e)x + --- - ½ -- + (1/3) -- - &c., x x² x³
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 [pi] 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 1 - .1^(r)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 10^5 and by 5 the number becomes 1.087678, and resolving this number into factors of the form 1 - .1^(r)n we find that
543839 = 10^5 × 5(1-.1²8)(1-.1^(4)6)(1-.1^(5)6)(1-.1^(6)3)(1-.1^(7)3) × (1-.1^(8)5)(1-.1^(9)7)(1-.1^(10)9)(1-.1^(11)3)(1-.1^(12)2),
where 1-1²8 denotes 1-.08, 1-.1^(4)6 denotes 1-.0006, &c., and so on. All that is required therefore in order to obtain the logarithm of any number is a table of logarithms, to the required number of places, of .n, .9n, .99n, .999n, &c., for n = 1, 2, 3, ... 9.
The resolution of a number into factors of the above form is easily performed. Taking, for example, the number 1.087678, the object is to destroy the significant figure 8 in the second place of decimals; this is effected by multiplying the number by 1-.08, that is, by subtracting from the number eight times itself advanced two places, and we thus obtain 1.00066376. To destroy the first 6 multiply by 1 - .0006 giving 1.000063361744, and multiplying successively by 1 - .00006 and 1 - .000003, we obtain 1.000000357932, and it is clear that these last six significant figures represent without any further work the remaining factors required. In the corresponding antilogarithmic process the number is expressed as a product of factors of the form 1 + .1^(n)x.
This method of calculating logarithms by the resolution of numbers into factors of the form 1 - .1^(r)n 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 1 + (.1)^(r)n is known by the name of Hearn, who published it in the same journal for 1847. In 1846 Peter Gray constructed a new table to 12 places, in which the factors were of the form 1-(.01)^(r)n, so that n had the values 1, 2, ... 99; and subsequently he constructed a similar table for factors of the form 1 + (.01)^(r)n. He also devised a method of applying a table of Hearn's form (i.e. of factors of the form 1 +.1^(r)n) to the construction of logarithms, and calculated a table of logarithms of factors of the form 1 + (.001)^(r)n to 24 places. This was published in 1876 under the title _Tables for the formation of logarithms and antilogarithms to twenty-four or any less number of places_, and contains the most complete and useful application of the method, with many improvements in points of detail. Taking as an example the calculation of the Briggian logarithm of the number 43,867, whose hyperbolic logarithm has been calculated above, we multiply it by 3, giving 131,601, and find by Gray's process that the factors of 1.31601 are
(1) 1.316 (5) 1.(001)^(4)002 (2) 1.000007 (6) 1.(001)^(5)602 (3) 1.(001)²598 (7) 1.(001)^(6)412 (4) 1.(001)³780 (8) 1.(001)^(7)340
Taking the logarithms from Gray's tables we obtain the required logarithm by addition as follows:--
522 878 745 280 337 562 704 972 = colog 3 119 255 889 277 936 685 553 913 = log (1) 3 040 050 733 157 610 239 = log (2) 259 708 022 525 453 597 = log (3) 338 749 695 752 424 = log (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 = log(10)43,867
In Shortrede's _Tables_ there are tables of logarithms and factors of the form 1 ± (.01)^(r)n to 16 places and of the form 1 ± (.1)^(r)n to 25 places; and in his _Tables de Logarithmes à 27 Décimales_ (Paris, 1867) Fédor Thoman gives tables of logarithms of factors of the form 1 ± .1^(r)n. In the _Messenger of Mathematics_, vol. iii. pp. 66-92, 1873, Henry Wace gave a simple and clear account of both the logarithmic and antilogarithmic processes, with tables of both Briggian and hyperbolic logarithms of factors of the form 1 ± .1^(r)n 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 1 + .1^(r)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_ (1802-1803), already noticed, referred to Flower and reproduced some of his tables. A complete bibliography of this method has been given by A. J. Ellis in a paper "on the potential radix as a means of calculating logarithms," printed in the _Proceedings of the Royal Society_, vol. xxxi., 1881, pp. 401-407, and vol. xxxii., 1881, pp. 377-379. 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 1 + .1^(r)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.)
FOOTNOTES:
[1] Dr Thomas Smith thus describes the ardour with which Briggs studied the _Descriptio_: "Hunc in deliciis habuit, in sinu, in manibus, in pectore gestavit, oculisque avidissimis, et mente attentissima, iterum iterumque perlegit,..." _Vitae quorundam eruditissimorum et illustrium virorum_ (London, 1707).
[2] William Lilly's account of the meeting of Napier and Briggs at Merchiston is quoted in the article NAPIER.
[3] It was certainly published after Napier's death, as Briggs mentions his "librum posthumum." This _liber posthumus_ was the _Constructio_ referred to later in this article.
[4] Frisch's _Kepleri opera omnia_, ii. 834. Frisch thinks Bramer possibly relied on Kepler's statement quoted in the text ("Quibus forte confisus Kepleri verbis Benj. Bramer...."). See also vol. vii. p. 298.
The claims of Byrgius are discussed in Kästner's _Geschichte der Mathematik_, ii. 375, and iii. 14; Montucla's _Histoire des mathématiques_, ii. 10; Delambre's _Histoire de l'astronomie moderne_, i. 560; de Morgan's article on "Tables" in the _English Cyclopaedia_; Mark Napier's _Memoirs of John Napier of Merchiston_ (1834), p. 392, and Cantor's _Geschichte der Mathematik_, ii. (1892), 662. See also Gieswald, _Justus Byrg als Mathematiker und dessen Einleitung in seine Logarithmen_ (Danzig, 1856).
[5] See Mark Napier's _Memoirs of John Napier of Merchiston_ (1834), p. 362.
[6] In the _Rabdologia_ (1617) he speaks of the canon of logarithms as "a me longo tempore elaboratum."
[7] A careful examination of the history of the method is given by Scheibel in his _Einleitung zur mathematischen Bücherkenntniss_, Stück vii. (Breslau, 1775), pp. 13-20; and there is also an account in Kästner's _Geschichte der Mathematik_, i. 566-569 (1796); in Montucla's _Histoire des mathématiques_, i. 583-585 and 617-619; and in Klügel's _Wörterbuch_ (1808), article "Prosthaphaeresis."
[8] Besides his connexion with logarithms and improvements in the method of prosthaphaeresis, Byrgius has a share in the invention of decimal fractions. See Cantor, _Geschichte_, ii. 567. Cantor attributes to him (in the use of his prosthaphaeresis) the first introduction of a subsidiary angle into trigonometry (vol. ii. 590).
[9] The title of this work is--_Benjaminis Ursini_ ... _cursus mathematici practici volumen primum continens illustr. & generosi Dn. Dn. Johannis Neperi Baronis Merchistonij &c. Scoti trigonometriam logarithmicam usibus discentium accommodatam_ ... _Coloniae_ ... _CI[~C] I[~C]C 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.
[10] 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 inventore 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.
[11] 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 10,000,000 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.
[12] 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 seven-figure table extending to 100,000.
[13] 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.
[14] 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 (Afd. Natuurkunde) deel. iv. (1862), p. 15.