and tangents (K43,1888)
Denis Roegel
11 january 2021
This document is part of the LOCOMAT project:
http://locomat.loria.fr
and Jane (1834-1878). The result fills about 50 manuscript volumes, plus a number of transfer duplicates, which are kept at Edinburgh University Library and at the National Library of Scotland, Edinburgh. I have reconstructed a number of these tables and an overview of the tables and reconstructions can be found in a separate guide [49].
Sang’s purpose was in particular to provide fundamental tables, including for the decimal division of the quadrant. In 1890 [5, p. 189], he wrote that
In addition to the results being accurate to a degree far beyond what can ever be needed in practical matters, [the collection of computations] contains what no work of the kind has contained before, a complete and clear record of all the steps by which those results were reached. Thus we are enabled at once to verify, or if necessary, to correct the record, so making it a standard for all time.
For these reasons it is proposed that the entire collection be acquired by, and preserved in, some official library, so as to be accessible to all interested in such matters; so that future computers may be enabled to extend the work without the need of recomputing what has been already done; and also so that those extracts which are judged to be expedient may be published.
The present volume K43 gives the logarithmic sines and cosines.1
1 Direct computation of the logarithms of sines
Before Sang worked on the computation of volume K43, he gave a procedure for comput- ing the logarithms of sines in an article published in 1884 [78]. I am summarizing this article here.
Sang recalled that the first tables of logarithms of sines were computed from the sines, in particular by Napier [36, 37, 38, 40, 39, 47]. He stresses that if the computations can be done less laboriously without resorting to the sines, it should be adopted.
Sang refers to formulæ in Jean-François Callet’s 1795 edition of his Tables porta- tives [15, p. 48]. Callet2 starts with the series
cosa= 1−a2 2! + a4
4! − a6 6! +· · ·
1National Library of Scotland, Edinburgh, Acc.10780/53.
2Callet, 1744-1798, see Lalande’sBibliographie astronomique ; avec l’histoire de l’astronomie depuis 1781 jusqu’à 1802, 1803, p. 804-805.
iii
Similarly, one obtains sina=a
1− a2
22q2 1− a2
42q2 1− a2 62q2
· · ·
Taking the logarithms of these sums, and expanding ln(1−x) = −x− x22 −x33 − · · ·, we obtain series for ln secaand lnsinaa . Unfortunately, these series converge very slowly.3 Sang observes that although Callet’s result is correct, his reasoning is faulty and essentially takes for granted that the quotient of cosa by the product of (1− aq22)(1 −
a2
32q2)· · · tends to 1, which is not proven.
Then Sang obtains the series for ln seca and lnsinaa directly:
ln seca= 1
2a2+ 1
22 ·3a4+ 1
32·5a6 + 17
23·32·5·7a8+ 31 34 ·52·7a10
+ 691
2·35·52·7·11a12+ 2·43·127
35·52·72·11·13a14+ 257·3617
24·36·53·72·11·13a16 + 73·43867
38·53·72·11·13·17a18+ 31·41·283·617
2·38·54·72·11·13·17·19a20+· · · ln a
sina = 1
2·3a2+ 1
22·32·5a4+ 1
34·5·7a6+ 1
23·33·52·7a8+ 1
35·52·7·11a10
+ 691
2·37·53·72·11·13a12+ 2
36·52·72·11·13a14+ 3617
24·37·54 ·72·11·13·17a16
+ 43867
311·53·73·11·13·17·19a18+ 283·617
2·39·56·72·112·13·17·19a20+· · ·
As far as I know, these expressions have not been given before Sang, but Callet expressed the terms as sums of the reciprocals of even powers [15, p. 49]. For instance, for the first series, he basically wrote that
ln seca=
∞
X
n=1
∞
X
i=0
1 (2i+ 1)2n
!
× 22n nπ2na2n or rather he gave a series for Log cos mn · π2
where Log is a generic logarithm, and he used k for the modulus (in our casek = 1/ln 10).
Now, using these sums, Sang could write, multiplying the previous values by the modulus M = 1/ln 10:
3Yet, these are the series used by Andoyer for the table of logarithms of sines and tangents published in 1911 [1].
iv
+0.00011143152746952792×a14 +0.00003951630255383690×a16 +0.00001423587009856471×a18 +0.00000519262273891936×a20 +· · ·
log a
sina = 0.07238241365054197128×a2 +0.00241274712168473238×a4 +0.00015319029344030047×a6 +0.00001148927200802254×a8 +0.00000092842602085031×a10 +0.00000007833240298017×a12 +0.00000000680165583041×a14 +0.00000000060298012595×a16 +0.00000000005430574190×a18 +0.00000000000495207566×a20 +· · ·
Callet had given similar series up to the 24th power [15, p. 50]:
Log cos m
n ·π 2
=−km2
n2 ×1.233700550136. . .−. . . Log sinm
n ·π 2
= Logm−Logn+klnπ
2 −km2
n2 ×0.411233516. . .−. . . where 1.233700550136. . .= 12 · π222, and 0.411233516. . .= 16 · π222, etc.
In Sang’s numerical expressions,0.2171472409516. . .= 12·ln 101 ,0.0723824136505. . .=
1
6 · ln 101 , and therefore 1.233700550136...
0.2171472409516... = π222 ·ln 10.
Sang however observed that these series are inconvenient and can only be used with small arcs. He then concluded that using these series “would entail more labour than the simple plan of deducing the logarithm from the sine.” He therefore turned to the method expounded in section 3.
v
and observed that thelog sin of the lower half of the quadrant can be obtained from the upper half:
sina= sin 2a 2·sin(100c−a)
where 100c is the centesimal angle for the quadrant. Once log sin 100c to log sin 50c are obtained, the next values log sin 49c, log sin 48c, etc., can be obtained with
log sin 49c= log sin 98c−log 2−log sin 51c log sin 48c= log sin 96c−log 2−log sin 52c and so on.
Similarly, the values of log tan are easily derived, as well as the differences for log sin and log tan. The details are given in the next section.
According to Sang, this process gives more accurate values of log sin for small angles, than a direct process.
3 The procedure actually used
A summary of Sang’s procedure was given in the account drawn up in November 1890 [5, p. 188], but Sang gave more details in a seven page notice introducing his table of loga- rithmic sines and tangents.5 Sang wrote that
The following tables are founded on the Canon of Sines to fifteen places (1881)6 which had been shortened from scroll calculations to 18 places, and which are expected to be true throughout to within 0.503 in the fifteenth place. The logarithms of these sines from sin 100c to sin 50c were computed by help of the fifteen-place logarithms of numbers from 100000 to 370000 (1878),7 using the auxiliary table (1884).8
For this, the actual sines were divided by 3 or by 2 as such might be to bring the quotient within the range of the table, that quotient was then divided by
4National Library of Scotland, Edinburgh, Acc.10780/78.
5National Library of Scotland, Edinburgh, Acc.10780/53.
6These are the tables contained in volumes K41 and K42, see my reconstruction [60].
7See the reconstruction of the logarithms of numbers to 15 places [56].
8See my reconstruction of the auxiliary table [57].
vi
in the last place so that our log sines can hardly have an uncertainty of less than 4, or say even 5 units in the 15th place. But a change of 5 in the function causes changes of −5,+15, −15, +5 in the third differences so that if, of two consecutive logarithms, one were 5 in excess, the other, 5 in defect, an error of 30 would appear in the column of third differences; so that an irregularity of less than 30 does not necessarily imply an error in calculation. When occasion arose, the true third difference was computed from the formula10
∆3log sina= cosa
sin3a ×103.5271739.
The aside-paper having been thus filled up, the last differences were carefully inspected, and, whenever it was seen that a change of 1, 2 or even 3 units in the function would improve the sequence of the differences, that change was marked in red ink, as also were the consequent changes in the last column.
Thereafter the so corrected functions were copied on the present pages, but the differences were anew computed from the numbers as actually written; in this way a complete check was had on the accuracy of the work. This work was likewise a severe test of the truthfulness of the Canon of Sines and of Logarithms, in which no symptom of an error was found.
The second part of the Canon, that is from 50c to 0, was constructed es- sentially according to the process used by Nepair for his Wonder-Working
9For instance, if we compute log sin 70c, the table of sines gives the value 0.891006524188368, this value is divided by 3 to bring it below 0.370000:
log sin 70c= log 0.297002174729456 + log 3 But 0.297002174729456 = 0.297002174729456
297002 ×297002 = 0.000001000000588. . .× 297002 = 10−6 × 1.000000588×297002andlog 0.297002174729456 = log 1.000000588 + log 297002−6. Then, the auxiliary table gives log 1.000000588 = 0.000000255365080, and eventually log sin 70c = 0.477121254719662 + 0.000000255365080 + 5.472759373849331 − 6 = −0.050119116065927, and 10 + log sin 70c = 9.949880883934073, whereas the correct value is 9.949880884069011. The discrepancy is due to the fact that the quotient 0.000001000000588 was truncated. A more accurate value can be obtained by interpo- lation and we would have to add0.000000000134935to the above result, reaching −0.050119115930992 and hence a tabulated value of9.949880884069008, which is off by 3 units of the 15th place. This does of course ignore the errors in the logarithms of numbers mentioned by Sang below.
10This somewhat mysterious formula gives an approximation of the third difference in units of the 15th place. It can be established by help of the formulæ used in the computation of the Cadastre tables [48, p. 81]. Sang may have used the description of the tables by Lefort [31], or he may have established the results himself. In any case, we have ∆3log sina ≈ 2M(cota+ cot3a)× 0.01100 ×π23
. Observing that cota+ cot3a = sincos3aa and scaling this expression by 1015, we obtain Sang’s formula becauselogh
2M 0.01100 ×π23
×1015i
= 3.5271739. . ..
vii
tana= sina·seca;
Having written the logarithm of 12 on the lower edge of a card, and placed that card in position on the aside-paper, we write below, first the logarithm of the double arc, next the log-secant of the single arc (which is the arithmetical complement of the log-sine of the arc’s complement), and the sum of these three is the log-sine of the proposed arc. Again by adding to this the same log-secant, we obtain the log-tangent sought for. Thus, for the arc42c800, the computation is as under:12
9.6989 70004 336019 log12 85c600 9.9887 93819 381819 log sin 2a 57 20 0.1065 76259 114265 log seca 42 80 9.7943 40082 832103 log sina 9.9009 16341 946368 log tana
If this process alone were followed, each result would be independent of the others, and the only expedient for verification would be by repeating the work;
this again would need to be done by another computer in order to avoid the well known risk of falling into the same mistake even after a long lapse of time. Actually these calculations were made only for each tenth minute, in order to serve for verification; the computation from minute to minute being made by help of differences, in the following manner.
In computing downwards, say from 42c800, we observe13 that the difference of the log-sines of 42c800 and 42c79, is the sum of the differences for log-sines 85c600 and 85c580 and for those of 57c200 and 57c210. Now we readily get the former by taking the sum of the two contiguous differences already written in the preceding part of the table, while the latter is to hand. Hence the work is easily arranged as under:14
11See [36, 37, 38, 40, 39, 47].
12I have kept Sang’s values, but the last figures are sometimes off by 1. Note that Sang writes the angle100c−42c80left oflog seca=−log sin(100c−42c80).
13This follows from
sinx
sin(x−∆x) = sin 2x
sin 2(x−∆x)×cos(x−∆x) cosx which is easily checked.
14The column on the right contains the first differences which are added to obtain new first differences.
For instance,31421 365585 + 54293 836215 = 85715 201800. Like above, some of the last figures are off by 1, but I did not correct them.
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85770 489825 31556 804511 4277 9.7940 82854 300122 54241 345712
85798 150223 4276 9.7939 97056 149899
an so on until we reach the logsine of 4270, which should agree with that found by the direct method. Yet this agreement does not give complete assurance of the accuracy of the intermediate work, because an error in one direction may happen to be balanced by another error in the opposite direction. Therefore the logsines only were copied on the actual table, and the differences were thence set down by actual subtraction.
The arrangement for the logtangents is even simpler, for we get their dif- ferences by augmenting each difference already found from the logsines by the corresponding difference from the logcosines; and, although these be not placed the one under the other, a very ordinary computer finds no difficulty in adding them as they stand. Thus, for the same arcs, the computation of the logtangents is15
Log tangents 4280 9.9009 16341 946368
1 40009 038015 4279 9.9007 76332 908353 1 40019 175363 4278 9.9006 36313 732990 1 40029 327999 4277 9.9004 96284 404991 1 40039 495935 4276 9.9003 56244 909056
and the same precautions were taken, while copying these upon the actual pages, as were used in the case of the logsines.
Since the logtangents for the other half of the quadrant are the arithmetical complements of these,16 it would be mere waste of labour to write them out in a preliminary table like the present.
15The values of thelog tanare also usually off by 1 in the last place.
16i.e.,log tan(100c−x) =−log tanx.
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δ1 = 101∆1− 2009 ∆2+ 200057 ∆3 δ2 = 1001 ∆2− 200018 ∆3 δ3 = 10001 ∆3. The successive values, found by the application of these differences being at 10 L
at 9 L−101∆1+2009 ∆2−200057 ∆3, at 8 L−102∆1+20016∆2−200096 ∆3, at 7 L−103∆1+20021∆2−2000119∆3, at 6 L−104∆1+20024∆2−2000128∆3, at 5 L−105∆1+20025∆2−2000125∆3, at 4 L−106∆1+20024∆2−2000112∆3, at 3 L−107∆1+20021∆2−200091 ∆3, at 2 L−108∆1+20016∆2−200064 ∆3, at 1 L−109∆1+2009 ∆2−200033 ∆3, and at 0 L− ∆1
from which we see that the greatest correction from second differences is 18∆2, while the maximum correction for third differences is less than the sixteenth part of ∆3.
The introduction to this volume abruptly ends here.
4 Content of Sang’s tables
Volume K43 was completed in 1888 and contains logarithmic sines and tangents. Sang’s table is a handwritten table, but it is preceded by a printed cover page (figure 1), perhaps meant to be published.
17See my (re)construction of the nine-place table where this spacing is used [43], as well as in the reconstruction of the 15-place table [56].
18The subtabulated values of the function are obtained using Newton’s forward difference formula:
zx=A+x∆1+x·x−1
2 ∆2+x·x−1 2 ·x−3
3 ∆3+· · ·
In this case, one should consider that the differences are −∆1, −∆2 and −∆3 and that x takes up the values 1/10 to 9/10. We then obtain the expressions given by Sang, which for some reason go backwards. The value of δ1 is the difference between z0.1 and z0 and should actually be of opposite sign for consistency. The values of δ2 and δ3 are easy to compute, but Sang had incorrectly written δ2= 1001 ∆2−200019 ∆3. See my reconstruction of Briggs’s 1624 table for details [45].
x
The third part gives the values of10 + log tanxfor centesimal angles from50cto0.01c and their differences.
Each part contains 41 values per page, with the last value of a page being identical to the first on the next page. Sang’s values of the logarithms of sines are correct to one or two units of the 15th place at the beginning of the table (starting at 100c), but the errors grow up to 15 units around 0c. The logarithms of tangents are similarly off up to two or three units of the 15th place at the beginning of the range (at 50c), and the errors grow up to 13 units or more at the end of range.
My reconstruction tries to follow faithfully the original tables, but there are some uncertainties, for instance regarding when the sizes of the prefixes of the logarithms of sines change. The prefixes start with 0.9999, but are later cut to three and two digits, perhaps not exactly when Sang did it. Sang may possibly not even have used a three-place prefix.
xi
les logarithmes des lignes trigonométriques. . .. Paris: Librairie A. Hermann et fils, 1911. [Reconstruction by D. Roegel in 2010 [44].]
[2] Marie Henri Andoyer. Fundamental trigonometrical and logarithmic tables. In Knott [30], pages 243–260.
[3] Anonymous. Note about Edward Sang’s project of computing a nine-figure table of logarithms. Nature, 10:471, 1874. [Issue of 8 October 1874. This note was reproduced in [73].]
[4] Anonymous. Correspondance. Comptes rendus hebdomadaires des séances de l’Académie des sciences, 80(22):1392–1393, janvier-juin 1875. [Minutes of the meeting of the 7 June 1875.]
[5] Anonymous. Dr Edward Sang’s logarithmic, trigonometrical, and astronomical tables. Proceedings of the Royal Society of Edinburgh, 28:183–196, 1908. [Possibly by Cargill Gilston Knott, reprinted in [28]. Reprints [81].]
[6] Raymond Clare Archibald. Tables of trigonometric functions in non-sexagesimal arguments. Mathematical Tables and other Aids to Computation, 1(2):33–44, April 1943.
[7] Raymond Clare Archibald. Arithmetic, logarithmic, trigonometric, and
astronomical tables, computed, 1848, 1869–89, by Edward Sang, and his daughters Jane Nicol Sang, Flora Chalmers Sang, and presented in 1907 to the Royal Society of Edinburgh, in custody for the British Nation. Mathematical Tables and other Aids to Computation, 1(9):368–370, 1945.
[8] Henry Briggs. Arithmetica logarithmica. London: William Jones, 1624. [The tables were reconstructed by D. Roegel in 2010. [45]]
[9] Johann Karl Burckhardt. Table des diviseurs pour tous les nombres du deuxième million, ou plus exactement, depuis 1020000 à 2028000, avec les nombres premiers qui s’y trouvent. Paris: Veuve Courcier, 1814. [also published in [11] together with [12]
and [10]]
19Note on the titles of the works: Original titles come with many idiosyncrasies and features (line splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore seemed pointless to capitalize works according to conventions which not only have no relation with the original work, but also do not restore the title entirely. In the following list of references, most title words (except in German) will therefore be left uncapitalized. The names of the authors have also been homogenized and initials expanded, as much as possible.
The reader should keep in mind that this list is not meant as a facsimile of the original works. The original style information could no doubt have been added as a note, but I have not done it here.
xv
trouvent. Paris: Veuve Courcier, 1817. [each part was also published separately as [12], [9], and [10]]
[12] Johann Karl Burckhardt. Table des diviseurs pour tous les nombres du premier million, ou plus exactement, depuis 1 à 1020000, avec les nombres premiers qui s’y trouvent. Paris: Veuve Courcier, 1817. [also published in [11] together with [9] and [10]]
[13] Florian Cajori. A history of mathematics. New York: Macmillan and co., 1894.
[14] Jean-François Callet. Tables portatives de logarithmes, publiées à Londres par Gardiner, augmentées, et perfectionnées dans leur disposition par M. Callet, etc.
Paris: Firmin Ambroise Didot, 1783. [This edition does not yet contain tables for the decimal division.]
[15] Jean-François Callet. Tables portatives de logarithmes, contenant les logarithmes des nombres, depuis 1 jusqu’à 108000 ; etc. Paris: Firmin Didot, 1795. [There have been numerous later printings of these tables.]
[16] Alexander Duncan Davidson Craik. Edward Sang (1805–1890): calculator extraordinary. Newsletter of the British Society for the History of Mathematics, 45:32–43, Spring 2002.
[17] Alexander Duncan Davidson Craik. The logarithmic tables of Edward Sang and his daughters. Historia Mathematica, 30(1):47–84, February 2003.
[18] Alexander Duncan Davidson Craik. Sang, Knott and Spence on logarithmic and other tables, 2016. [article written for a joint meeting of the James Clerk Maxwell Society and the British Society for the History of Mathematics in celebration of the 400th anniversary of the publication of John Napier’s Mirifici Logarithmorum Canonis Descriptio, 4th April 2014, Clerk Maxwell House, Edinburgh,https://www.collectanea.eu/napier400memorial]
[19] Alan Fletcher, Jeffery Charles Percy Miller, Louis Rosenhead, and Leslie John Comrie. An index of mathematical tables. Oxford: Blackwell scientific publications Ltd., 1962. [2nd edition (1st in 1946), 2 volumes]
[20] Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, and Paul Zimmermann. MPFR: A multiple-precision binary floating-point library with correct rounding. ACM Transactions on Mathematical Software, 33(2), 2007.
[21] James Whitbread Lee Glaisher. On errors in Vlacq’s (often called Brigg’s or Neper’s) tables of ten-figure logarithms of numbers. Monthly Notices of the Royal Astronomical Society, 32(7):255–262, May 1872.
xvi
meeting of the British Association for the advancement of science,” London: John Murray, 1874.
A review by R. Radau was published in theBulletin des sciences mathématiques et astronomiques, volume 11, 1876, pp. 7–27]
[25] G. Govi. Rapport sur l’utilité des tables de logarithmes à plus de sept décimales ; à propos d’un projet publié par M. Sang. Atti della Reale Accademia di Scienze di Torino, 8:157–170, 1873. [Reprinted in [71].]
[26] Albert Hatzfeld. La division décimale du cercle. Revue scientifique, 48:655–659, 1891.
[27] James Henderson. Bibliotheca tabularum mathematicarum, being a descriptive catalogue of mathematical tables. Part I: Logarithmic tables (A. Logarithms of numbers), volume XIII of Tracts for computers. London: Cambridge University Press, 1926.
[28] Ellice Martin Horsburgh, editor. Modern instruments and methods of calculation: a handbook of the Napier tercentenary exhibition. London: G. Bell and sons, 1914.
[29] Cargill Gilston Knott. Edward Sang and his logarithmic calculations. [30], pages 261–268.
[30] Cargill Gilston Knott, editor. Napier Tercentenary Memorial Volume. London:
Longmans, Green and company, 1915.
[31] Pierre Alexandre Francisque Lefort. Description des grandes tables logarithmiques et trigonométriques, calculées au bureau du cadastre, sous la direction de Prony, et exposition des méthodes et procédés mis en usage pour leur construction. Annales de l’Observatoire impérial de Paris, 4 (supplément):123–150, 1858.
[32] Pierre Alexandre Francisque Lefort. Observations on Mr Sang’s remarks relative to the great logarithmic table compiled at the Bureau du Cadastre under the
direction of M. Prony. Proceedings of the Royal Society of Edinburgh, Session 1874–1875, 8:563–581, 1875. [See [74] for Sang’s answer.]
[33] Percy Alexander MacMahon. Sang’s seven-place logarithms. Nature, 97(2442):499, 1916. [Review of the 1915 reprint of Sang’s tables.]
[34] Charles E. Manierre. The decimal system for time and arc for use in navigation.
Popular Astronomy, 28:99–103, 1920. [Only on practical aspects of a switch to a decimal system, not historical ones.]
xvii
[37] John Napier. A description of the admirable table of logarithmes. London, 1616.
[English translation of [36] by Edward Wright, reprinted in 1969 by Da Capo Press, New York. A second edition appeared in 1618.]
[38] John Napier. Mirifici logarithmorum canonis constructio. Edinburgh: Andrew Hart, 1619. [Reprinted in [39] and translated in [40]. A modern English translation by Ian Bruce is available on the web.]
[39] John Napier. “Logarithmorum canonis descriptio” and “Mirifici logarithmorum canonis constructio”. Lyon: Barthélemi Vincent, 1620. [Reprint of Napier’s descriptio andconstructio. At least theconstructio was reprinted by A. Hermann in 1895.]
[40] John Napier. The construction of the wonderful canon of logarithms. Edinburgh:
William Blackwood and sons, 1889. [Translation of [38] by William Rae Macdonald.]
[41] National Library of Scotland. Inventory Acc.10780: Papers and manuscripts of Edward Sang, 2003. [6 pages,
http://www.nls.uk/catalogues/online/cnmi/inventories/acc10780.pdf]
[42] David Bruce Peebles. Edward Sang. Proceedings of the Royal Society of Edinburgh, 21:xvii–xxxii, 1897.
[43] Denis Roegel. A construction of Edward Sang’s projected table of nine-place logarithms to one million (1872). Technical report, LORIA, Nancy, 2010. [This construction is based on the specimen pages [71].]
[44] Denis Roegel. A reconstruction of Henri Andoyer’s table of logarithms (1911).
Technical report, LORIA, Nancy, 2010. [This is a reconstruction of [1].]
[45] Denis Roegel. A reconstruction of the tables of Briggs’Arithmetica logarithmica (1624). Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of [8].]
[46] Denis Roegel. A reconstruction of the tables of Thompson’s Logarithmetica Britannica (1952). Technical report, LORIA, Nancy, 2010. [This is a unpublished reconstruction of the tables in [86], not available for copyright reasons.]
[47] Denis Roegel. Napier’s ideal construction of the logarithms. Technical report, LORIA, Nancy, 2010.
[48] Denis Roegel. The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation. Technical report, LORIA, Nancy, 2010.
xviii
[52] Denis Roegel. Edward Sang’s steps for the construction of the logarithms of the primes (K1-K3). Technical report, LORIA, Nancy, 2020.
[53] Denis Roegel. A reconstruction of Edward Sang’s table of logarithms of the first 10000 primes (K4). Technical report, LORIA, Nancy, 2020.
[54] Denis Roegel. A reconstruction of Edward Sang’s table of logarithms of the first myriad of integers (K5). Technical report, LORIA, Nancy, 2020.
[55] Denis Roegel. A reconstruction of Edward Sang’s table of logarithms of the second myriad of integers (K6). Technical report, LORIA, Nancy, 2020.
[56] Denis Roegel. Introduction to Edward Sang’s table of logarithms to 15 places.
Technical report, LORIA, Nancy, 2020. [This document is supplemented by 90 volumes of tables, as well as a volume gathering the entire table.]
[57] Denis Roegel. A reconstruction of Edward Sang’s auxiliary table for logarithms of almost unitary values (K39,1884). Technical report, LORIA, Nancy, 2020.
[58] Denis Roegel. A reconstruction of Edward Sang’s canon of sines (K40/1,1876).
Technical report, LORIA, Nancy, 2020.
[59] Denis Roegel. A reconstruction of Edward Sang’s canon of sines (K40/2,1877).
Technical report, LORIA, Nancy, 2020.
[60] Denis Roegel. A reconstruction of Edward Sang’s canon of sines (K41-K42,1881).
Technical report, LORIA, Nancy, 2020.
[61] Denis Roegel. A reconstruction of Edward Sang’s table of sines in degrees (K44,1879). Technical report, LORIA, Nancy, 2020.
[62] Denis Roegel. A reconstruction of Edward Sang’s table of circular segments (K45,1879). Technical report, LORIA, Nancy, 2020.
[63] Denis Roegel. A reconstruction of Edward Sang’s table of mean anomalies: volume A (K46,1880). Technical report, LORIA, Nancy, 2020.
[64] Denis Roegel. A reconstruction of Edward Sang’s table of mean anomalies: volume B (K47,1880). Technical report, LORIA, Nancy, 2020.
[65] Ralph Allen Sampson. Logarithmic, trigonometrical, and astronomical tables:
forty-seven quarto volumes in manuscript (1848 to 1890). By Edward Sang, LL.D.
In Knott [30], pages 236–237.
xix
200 000. London: Charles and Edwin Layton, 1871.
[69] Edward Sang. Account of the new table of logarithms to 200000. Transactions of the Royal Society of Edinburgh, 26:521–528, 1872.
[70] Edward Sang. On mechanical aids to calculation. Journal of the Institute of Actuaries and Assurance Magazine, 16:253–265, 1872. [The article was published in the July 1871 issue, but the volume is dated 1872.]
[71] Edward Sang. Specimen pages of a table of the logarithms of all numbers up to one million...: shortened to nine figures from original calculations to fifteen places of decimals, 1872. [These specimen pages were reprinted in 1874 in a booklet which contained also a reprint of Govi’s report [25], a reprint of Sang’s article on Vlacq’s errors [72], and several other letters by eminent scientists supporting the publication of Sang’s table. The specimen pages were used to construct [43].]
[72] Edward Sang. On last-place errors in Vlacq’s table of logarithms. Proceedings of the Royal Society of Edinburgh, 8:371–376, 1875. [First printed in the 1874 edition of [71].]
[73] Edward Sang. Remarks on the great logarithmic and trigonometrical tables
computed by the Bureau du Cadastre under the direction of M. Prony. Proceedings of the Royal Society of Edinburgh, Session 1874–1875, 8:421–436, 1875. [This article reproduces [3].]
[74] Edward Sang. Reply to M. Lefort’s Observations (with a Postscript by M. Lefort).
Proceedings of the Royal Society of Edinburgh, Session 1874–1875, 8:581–587, 1875.
[This is a reply to [32].]
[75] Edward Sang. On the construction of the canon of sines, for the decimal division of the quadrant. Proceedings of the Royal Society of Edinburgh, 9:343–349, 1878.
[76] Edward Sang. On the precautions to be taken in recording and using the records of original computations. Proceedings of the Royal Society of Edinburgh, 9:349–352, 1878.
[77] Edward Sang. Description of new astronomical tables for the computation of anomalies. Proceedings of the Royal Society of Edinburgh, 10(107):726–727, 1880.
[78] Edward Sang. On the construction of the canon of logarithmic sines. Proceedings of the Royal Society of Edinburgh, 12:601–619, 1884.
xx
[81] Edward Sang. List of trigonometrical and astronomical calculations, in manuscript, 1890. [Dated July 1890. National Library of Scotland: Acc10780/10. Reprinted in [5].]
[82] Edward Sang. On last-place errors in Vlacq. Nature, 42(1094):593, 1890.
[83] Robert Shortrede. Logarithmic tables, to seven places of decimals, containing logarithms to numbers from 1 to 120,000, numbers to logarithms from .0 to 1.00000, logarithmic sines and tangents to every second of the circle, with arguments in space and time, and new astronomical and geodesical tables.
Edinburgh: Adam and Charles Black, 1844.
[84] Robert Shortrede. Logarithmic tables, containing logarithms to numbers from 1 to 120,000, numbers to logarithms from ·0 to 1·00000, to seven places of decimals; etc.
Edinburgh: Adam and Charles Black, 1849.
[85] James Francis Tennant. Note on logarithmic tables. Monthly Notices of the Royal Astronomical Society, 33:563–565, 1873.
[86] Alexander John Thompson. Logarithmetica Britannica, being a standard table of logarithms to twenty decimal places of the numbers 10,000 to 100,000. Cambridge:
University press, 1952. [2 volumes, unpublished reconstruction by D. Roegel in 2010 [46].]
[87] Shane F. Whelan. Edward Sang: actuary of the Millennium. Newsletter of the Society of Actuaries in Ireland, November 1999. [A slightly edited version was published in The Actuary, April 2000, page 27.]
xxi
TABLE 1: log sin from 100
cto 50
c00 00 000000 5 357893 10 715788 −1 99 99 99994 642107 16 073681 10 715787 3 98 99978 568426 26 789468 10 715790 1 97 99951 778958 37 505258 10 715791 2 96 99914 273700 48 221049 10 715793 4 99 95 99866 052651 58 936842 10 715797 3 94 99807 115809 69 652639 10 715800 3 93 99737 463170 80 368439 10 715803 6 92 99657 094731 91 084242 10 715809 3 91 99566 010489 101 800051 10 715812 8 99 90 99464 210438 112 515863 10 715820 5 89 99351 694575 123 231683 10 715825 6 88 99228 462892 133 947508 10 715831 8 87 99094 515384 144 663339 10 715839 8 86 98949 852045 155 379178 10 715847 6 99 85 98794 472867 166 095025 10 715853 12 84 98628 377842 176 810878 10 715865 6 83 98451 566964 187 526743 10 715871 12 82 98264 040221 198 242614 10 715883 10 81 98065 797607 208 958497 10 715893 10 99 80 97856 839110 219 674390 10 715903 12 79 97637 164720 230 390293 10 715915 12 78 97406 774427 241 106208 10 715927 12 77 97165 668219 251 822135 10 715939 13 76 96913 846084 262 538074 10 715952 13 99 75 96651 308010 273 254026 10 715965 15 74 96378 053984 283 969991 10 715980 15 73 96094 083993 294 685971 10 715995 13 72 95799 398022 305 401966 10 716008 18 71 95493 996056 316 117974 10 716026 14 99 70 95177 878082 326 834000 10 716040 18 69 94851 044082 337 550040 10 716058 17 68 94513 494042 348 266098 10 716075 17 67 94165 227944 358 982173 10 716092 20 66 93806 245771 369 698265 10 716112 16 99 65 93436 547506 380 414377 10 716128 22 64 93056 133129 391 130505 10 716150 18 63 92665 002624 401 846655 10 716168 21 62 92263 155969 412 562823 10 716189 21 61 91850 593146 423 279012 10 716210 22 99 60 91427 314134 433 995222 10 716232 21
9.9999
99 60 91427 314134 433 995222 10 716232 21 59 90993 318912 444 711454 10 716253 22 58 90548 607458 455 427707 10 716275 24 57 90093 179751 466 143982 10 716299 24 56 89627 035769 476 860281 10 716323 23 99 55 89150 175488 487 576604 10 716346 25 54 88662 598884 498 292950 10 716371 25 53 88164 305934 509 009321 10 716396 25 52 87655 296613 519 725717 10 716421 28 51 87135 570896 530 442138 10 716449 25 99 50 86605 128758 541 158587 10 716474 28 49 86063 970171 551 875061 10 716502 28 48 85512 095110 562 591563 10 716530 28 47 84949 503547 573 308093 10 716558 28 46 84376 195454 584 024651 10 716586 30 99 45 83792 170803 594 741237 10 716616 30 44 83197 429566 605 457853 10 716646 31 43 82591 971713 616 174499 10 716677 30 42 81975 797214 626 891176 10 716707 32 41 81348 906038 637 607883 10 716739 31 99 40 80711 298155 648 324622 10 716770 34 39 80062 973533 659 041392 10 716804 32 38 79403 932141 669 758196 10 716836 34 37 78734 173945 680 475032 10 716870 35 36 78053 698913 691 191902 10 716905 32 99 35 77362 507011 701 908807 10 716937 38 34 76660 598204 712 625744 10 716975 34 33 75947 972460 723 342719 10 717009 37 32 75224 629741 734 059728 10 717046 37 31 74490 570013 744 776774 10 717083 36 99 30 73745 793239 755 493857 10 717119 39 29 72990 299382 766 210976 10 717158 38 28 72224 088406 776 928134 10 717196 39 27 71447 160272 787 645330 10 717235 39 26 70659 514942 798 362565 10 717274 41 99 25 69861 152377 809 079839 10 717315 39 24 69052 072538 819 797154 10 717354 41 23 68232 275384 830 514508 10 717395 43 22 67401 760876 841 231903 10 717438 41 21 66560 528973 851 949341 10 717479 43 99 20 65708 579632 862 666820 10 717522 42
9.9999
99 20 65708 579632 862 666820 10 717522 42 19 64845 912812 873 384342 10 717564 46 18 63972 528470 884 101906 10 717610 42 17 63088 426564 894 819516 10 717652 45 16 62193 607048 905 537168 10 717697 46 99 15 61288 069880 916 254865 10 717743 45 14 60371 815015 926 972608 10 717788 47 13 59444 842407 937 690396 10 717835 47 12 58507 152011 948 408231 10 717882 46 11 57558 743780 959 126113 10 717928 49 99 10 56599 617667 969 844041 10 717977 48 09 55629 773626 980 562018 10 718025 49 08 54649 211608 991 280043 10 718074 50 07 53657 931565 1001 998117 10 718124 48 06 52655 933448 1012 716241 10 718172 53 99 05 51643 217207 1023 434413 10 718225 50 04 50619 782794 1034 152638 10 718275 52 03 49585 630156 1044 870913 10 718327 51 02 48540 759243 1055 589240 10 718378 53 01 47485 170003 1066 307618 10 718431 54 99 00 46418 862385 1077 026049 10 718485 53 98 99 45341 836336 1087 744534 10 718538 55 98 44254 091802 1098 463072 10 718593 54 97 43155 628730 1109 181665 10 718647 55 96 42046 447065 1119 900312 10 718702 56 98 95 40926 546753 1130 619014 10 718758 57 94 39795 927739 1141 337772 10 718815 56 93 38654 589967 1152 056587 10 718871 58 92 37502 533380 1162 775458 10 718929 58 91 36339 757922 1173 494387 10 718987 59 98 90 35166 263535 1184 213374 10 719046 57 89 33982 050161 1194 932420 10 719103 61 88 32787 117741 1205 651523 10 719164 60 87 31581 466218 1216 370687 10 719224 60 86 30365 095531 1227 089911 10 719284 62 98 85 29138 005620 1237 809195 10 719346 60 84 27900 196425 1248 528541 10 719406 64 83 26651 667884 1259 247947 10 719470 62 82 25392 419937 1269 967417 10 719532 63 81 24122 452520 1280 686949 10 719595 64 98 80 22841 765571 1291 406544 10 719659 64
9.9999
98 80 22841 765571 1291 406544 10 719659 64 79 21550 359027 1302 126203 10 719723 65 78 20248 232824 1312 845926 10 719788 66 77 18935 386898 1323 565714 10 719854 64 76 17611 821184 1334 285568 10 719918 69 98 75 16277 535616 1345 005486 10 719987 64 74 14932 530130 1355 725473 10 720051 70 73 13576 804657 1366 445524 10 720121 67 72 12210 359133 1377 165645 10 720188 68 71 10833 193488 1387 885833 10 720256 70 98 70 09445 307655 1398 606089 10 720326 69 69 08046 701566 1409 326415 10 720395 70 68 06637 375151 1420 046810 10 720465 71 67 05217 328341 1430 767275 10 720536 71 66 03786 561066 1441 487811 10 720607 72 98 65 02345 073255 1452 208418 10 720679 71 64 00892 864837 1462 929097 10 720750 75 63 99429 935740 1473 649847 10 720825 71 62 97956 285893 1484 370672 10 720896 76 61 96471 915221 1495 091568 10 720972 72 98 60 94976 823653 1505 812540 10 721044 76 59 93471 011113 1516 533584 10 721120 76 58 91954 477529 1527 254704 10 721196 75 57 90427 222825 1537 975900 10 721271 77 56 88889 246925 1548 697171 10 721348 77 98 55 87340 549754 1559 418519 10 721425 77 54 85781 131235 1570 139944 10 721502 79 53 84210 991291 1580 861446 10 721581 78 52 82630 129845 1591 583027 10 721659 78 51 81038 546818 1602 304686 10 721737 82 98 50 79436 242132 1613 026423 10 721819 78 49 77823 215709 1623 748242 10 721897 82 48 76199 467467 1634 470139 10 721979 81 47 74564 997328 1645 192118 10 722060 82 46 72919 805210 1655 914178 10 722142 81 98 45 71263 891032 1666 636320 10 722223 84 44 69597 254712 1677 358543 10 722307 83 43 67919 896169 1688 080850 10 722390 85 42 66231 815319 1698 803240 10 722475 82 41 64533 012079 1709 525715 10 722557 87 98 40 62823 486364 1720 248272 10 722644 85
9.9998
98 40 62823 486364 1720 248272 10 722644 85 39 61103 238092 1730 970916 10 722729 86 38 59372 267176 1741 693645 10 722815 86 37 57630 573531 1752 416460 10 722901 87 36 55878 157071 1763 139361 10 722988 88 98 35 54115 017710 1773 862349 10 723076 89 34 52341 155361 1784 585425 10 723165 87 33 50556 569936 1795 308590 10 723252 91 32 48761 261346 1806 031842 10 723343 88 31 46955 229504 1816 755185 10 723431 91 98 30 45138 474319 1827 478616 10 723522 90 29 43310 995703 1838 202138 10 723612 93 28 41472 793565 1848 925750 10 723705 91 27 39623 867815 1859 649455 10 723796 92 26 37764 218360 1870 373251 10 723888 93 98 25 35893 845109 1881 097139 10 723981 93 24 34012 747970 1891 821120 10 724074 96 23 32120 926850 1902 545194 10 724170 92 22 30218 381656 1913 269364 10 724262 96 21 28305 112292 1923 993626 10 724358 96 98 20 26381 118666 1934 717984 10 724454 95 19 24446 400682 1945 442438 10 724549 98 18 22500 958244 1956 166987 10 724647 96 17 20544 791257 1966 891634 10 724743 98 16 18577 899623 1977 616377 10 724841 99 98 15 16600 283246 1988 341218 10 724940 98 14 14611 942028 1999 066158 10 725038 99 13 12612 875870 2009 791196 10 725137 100 12 10603 084674 2020 516333 10 725237 100 11 08582 568341 2031 241570 10 725337 102 98 10 06551 326771 2041 966907 10 725439 100 09 04509 359864 2052 692346 10 725539 104 08 02456 667518 2063 417885 10 725643 100 07 00393 249633 2074 143528 10 725743 104 06 98319 106105 2084 869271 10 725847 103 98 05 96234 236834 2095 595118 10 725950 106 04 94138 641716 2106 321068 10 726056 102 03 92032 320648 2117 047124 10 726158 107 02 89915 273524 2127 773282 10 726265 104 01 87787 500242 2138 499547 10 726369 108 98 00 85649 000695 2149 225916 10 726477 105
9.9997
98 00 85649 000695 2149 225916 10 726477 105 97 99 83499 774779 2159 952393 10 726582 109 98 81339 822386 2170 678975 10 726691 106 97 79169 143411 2181 405666 10 726797 109 96 76987 737745 2192 132463 10 726906 109 97 95 74795 605282 2202 859369 10 727015 109 94 72592 745913 2213 586384 10 727124 110 93 70379 159529 2224 313508 10 727234 111 92 68154 846021 2235 040742 10 727345 110 91 65919 805279 2245 768087 10 727455 112 97 90 63674 037192 2256 495542 10 727567 112 89 61417 541650 2267 223109 10 727679 112 88 59150 318541 2277 950788 10 727791 114 87 56872 367753 2288 678579 10 727905 113 86 54583 689174 2299 406484 10 728018 115 97 85 52284 282690 2310 134502 10 728133 113 84 49974 148188 2320 862635 10 728246 116 83 47653 285553 2331 590881 10 728362 117 82 45321 694672 2342 319243 10 728479 114 81 42979 375429 2353 047722 10 728593 118 97 80 40626 327707 2363 776315 10 728711 117 79 38262 551392 2374 505026 10 728828 119 78 35888 046366 2385 233854 10 728947 117 77 33502 812512 2395 962801 10 729064 119 76 31106 849711 2406 691865 10 729183 120 97 75 28700 157846 2417 421048 10 729303 120 74 26282 736798 2428 150351 10 729423 120 73 23854 586447 2438 879774 10 729543 121 72 21415 706673 2449 609317 10 729664 122 71 18966 097356 2460 338981 10 729786 123 97 70 16505 758375 2471 068767 10 729909 121 69 14034 689608 2481 798676 10 730030 123 68 11552 890932 2492 528706 10 730153 125 67 09060 362226 2503 258859 10 730278 124 66 06557 103367 2513 989137 10 730402 124 97 65 04043 114230 2524 719539 10 730526 127 64 01518 394691 2535 450065 10 730653 123 63 98982 944626 2546 180718 10 730776 129 62 96436 763908 2556 911494 10 730905 125 61 93879 852414 2567 642399 10 731030 129 97 60 91312 210015 2578 373429 10 731159 127
9.9996
