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1670)
Denis Roegel
To cite this version:
Denis Roegel. Mouton’s table of logarithms and its extensions (ca. 1670). [Research Report] 2011.
�hal-00654572�
and its extensions (ca. 1670)
Denis Roegel
20 December 2011
This document is part of the LOCOMAT project:
http://locomat.loria.fr
Gabriel Mouton (1618–1694) was a French abbot and scientist. In 1670, he published his book Observationes diametrorum solis et lunæ apparentium [14] in which he developed his ideas on interpolation.
It was perhaps at this time that Mouton constructed an extension of Vlacq’s Trigono- metria artificialis [19]. Vlacq’s table gave the logarithms of sines, cosines, tangents and cotangents for every ten seconds of the quadrant and to ten places.
Mouton’s table gave the same logarithms, but for every second of the first four degrees of the quadrant. Mouton’s original manuscript is located at the library of the Institut in Paris.
1There are apparently two other copies of this manuscript, but we have not seen them and we do not know where they are located.
2Mouton’s table follows Vlacq’s layout very faithfully. The table is totally handwritten and uses no printed forms. The copy at the Institut contains an additional page giving the logarithms for the first minute of the fifth degree. This page was not computed by Mouton.
2 The accuracy of Mouton’s table
An examination of Mouton’s table shows that his table is surprisingly accurate. We have compared several dozens of values, and we were unable to find any error. For instance, the first page of Mouton’s table (from 0
0to 1
0) gives 240 correct values, for the logarithms of sines, cosines, tangents and cotangents. The values are correctly rounded to ten places. We have not checked the differences, but they are presumably also correct.
The cotangents from 1
◦1
0to 1
◦2
0are also correct, and so are the sines from 3
◦56
0to 3
◦57
0, as well as the sines and tangents from 3
◦58
0to 3
◦59
0.
Contrasting with the first four degrees, the values from 4
◦0
0to 4
◦1
0display many errors, so that this alone shows that the computations were not done with the same rigor, and certainly not by the same person.
3 The source of Mouton’s table
The question then arises to Mouton’s source. At first, it is tempting to assume that Mouton interpolated between Vlacq’s values, especially since Vlacq’s layout was used.
But this cannot be the case, for the values given by Vlacq for every 10 seconds often differ from those given by Mouton. For instance, for 3
◦58
020
00, Vlacq gives log sin = . . . 480 (instead of . . . 482), for 3
◦58
050
00, Vlacq gives log sin = . . . 391 (instead of . . . 394), and so on. About every other value given by Vlacq is slightly wrong, but none of these errors show up in Mouton’s table.
1Ms. 921.
2In the second edition of hisAstronomie [3], published in 1771, Lalande wrote that Maraldi (probably Giovanni Domenico Maraldi (1709–1788)) had a copy of the table. The same year, he wrote that Jean- Jacques Dortous de Mairan (1678–1771) and Maraldi had copies [4]. But in the third edition, published in 1792 [5], he wrote that he and Cassini (probably Jean-Dominique Cassini (1748–1845)) have copies of Mouton’s table. Lalande may have inherited de Mairan’s copy. These copies may be among their papers in the archives of the Paris observatory.
3
not use a sexagesimal division and Briggs gives his values every 100th of a degree, that is, every 36 seconds. So, if Mouton used Briggs’ table, he must have interpolated his values every 36 seconds. This interval, however, is too large for an interpolation at the beginning of the logarithms of sines, for instance. At the end of the fourth degree, at least third differences would be needed to obtain ten correct decimal places, and the third differences would be far from sufficient at the beginning of the table.
Mouton could however have computed the logarithms of cosines that way, as they vary only very slowly, but he would also not have been able to obtain their values at 14 places from Briggs’ table, because he would have needed to use Briggs’ logarithms of tangents, which are only given to 10 places.
It therefore seems that Mouton did not even use Briggs’ values for the logarithms of sines, cosines, tangents and cotangents published in his Trigonometria britannica [1], but recomputed a number of values very accurately, perhaps every six seconds. This interval would have enabled him to compare his values with those of Briggs. Computing these values may have been done using Briggs’ methods, who first computed a number of sines, then their logarithms using the radix method.
Once Mouton had obtained accurate values of the logarithms of sines perhaps every six seconds, he may have subtabulated these intervals using Briggs’ method.
In any case, Mouton was of course wise not to use Vlacq’s table, since even if Vlacq’s values had been correct, Mouton would never have been able to come up with such accurate tables. Every value in Vlacq’s table might have been wrong by as much as half a unit of the last place, and such errors would also take place on interpolated values. For the interval from 3
◦58
0to 3
◦59
0, five of Vlacq’s seven logarithms of sines, and five of Vlacq’s logarithms of tangents are incorrect, with the greatest errors being for log sin 3
◦58
050
00and log tan 3
◦58
050
00(3 units of the tenth place in each case).
3The last page of Mouton’s table is an addition, and almost every logarithm is given incorrectly. For instance, from 4
◦0
04
00to 4
◦0
013
00, the last digits of the logarithms of sines in the table are 429, 438, 426, 393, 339, 264, 168, 051, 913 and 754. These values are all incorrect, the correct digits being 430, 439, 427, 394, 340, 265, 169, 052, 914, 756.
Moreover, the value for 4
◦0
010
00in the table even differs from Vlacq’s value which is 167, and this also appears to be the case for other angles. We can therefore conclude that this additionnal page was also not based on Vlacq’s table, but was nevertheless computed much less accurately than Mouton’s main table. It might be of some interest to try to find the origin of the errors in this page, but we have not investigated the matter.
4 Further tables to one-second steps
After Mouton’s work, several other tables were given to one-second intervals. Perhaps the first such table was Gardiner’s, who in 1742 gave the sines to every second and to seven places for the first 72 minutes of the quadrant [7].
Gardiner’s original table was not based on Mouton’s, but the second edition edited by Pézenas and published in 1770 was [8]. In fact, Lalande wrote [5] that he had com- municated Mouton’s table to Pézenas [2, pp. 549–550]. It is however not clear if Lalande
3Incidentally, this suggests that Vlacq’s errors are not independent, but correlated, and this should certainly be investigated further.
4
used by Gardiner in 1742, but the table was extended to the first four degrees.
In 1792, Michael Taylor [17] published a table giving the logarithms for every second of the quadrant, also only to 7 places, and perhaps based on Pézenas’ table. In any case, Taylor did not mention Mouton’s table.
In 1794, Vega published his Thesaurus logarithmorum completus [18]
4which contained the logarithms of sines, cosines, tangents and cotangents for every second of the first two degrees and to ten places. The remaining part of the quadrant was borrowed and corrected from Vlacq’s table, but the first two degrees had been computed by Captain Friedrich Dorfmund.
5As a simple comparison shows, there are a number of discrepancies with Mouton’s table, and Vega’s is not as accurate as Mouton’s.
5 Reconstruction
The tables were recomputed using the GNU mpfr multiple-precision floating-point library developed at INRIA [6], and give the exact values. We have moreover extended Mouton’s table to the entire semi-quadrant.
4See Glaisher [10, pp. 138–139] for a short description of Vega’s table and an appraisal of its accuracy.
Gauss has also made some error analysis of Vega’s tables, but perhaps not clearly distinguishing between the work of Dorfmund and that of Vega [9].
5Vega writes that Dorfmund was Lieutenant des kayserl. königl. Artilleriekorps and that he was willing to compute the logarithms beyond the second degree [18, p.xxi].
5
6
7
8
The following list covers the most important references related to Mouton’s table. Not all items of this list are mentioned in the text, and the sources which have not been seen are marked so. We have added notes about the contents of the articles in certain cases.
[1] Henry Briggs and Henry Gellibrand. Trigonometria Britannica. Gouda: Pieter Rammazeyn, 1633.
[The tables were reconstructed by D. Roegel in 2010 [16].][2] Augustin de Backer and Aloïs de Backer. Bibliothèque des écrivains de la Compagnie de Jésus, etc. Liége: L. Grandmont-Donders, 1858.
[3] Joseph-Jérôme Lefrançois de Lalande. Astronomie, volume 1. Paris: Veuve Desaint, 1771.
[Second edition. A notice on Mouton appears on page 208.][4] Joseph-Jérôme Lefrançois de Lalande. Lettre sur des tables de sinus extrêmement rares. Journal des Sçavans, 55(2):291–303, October 1771.
[5] Joseph-Jérôme Lefrançois de Lalande. Astronomie, volume 1. Paris: Veuve Desaint, 1792.
[Third edition. A notice on Mouton appears on page 165.][6] 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.
[7] William Gardiner. Tables of logarithms, for all numbers from 1 to 102100, and for the sines and tangents to every ten seconds of each degree in the quadrant; as also, for the sines of the first 72 minutes to every single second: with other useful and necessary tables. London: G. Smith, 1742.
[8] William Gardiner. Tables de logarithmes, contenant les logarithmes des nombres, depuis 1 jusqu’à 102100, & les logarithmes des sinus & des tangentes, de 10 en 10 secondes, pour chaque degré du quart de cercle, avec différentes autres tables, publiées ci-devant en Angleterre. Nouvelle édition, augmentée des logarithmes des sinus & tangentes, pour chaque seconde des quatre premiers degrés. Avignon:
J. Aubert, 1770.
[edited by Esprit Pézenas][9] Carl Friedrich Gauss. Einige Bemerkungen zu Vega’s Thesaurus Logarithmorum.
Astronomische Nachrichten, 32(756):181–188, 1851.
[10] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.
London: Taylor and Francis, 1873.
[Also published as part of the “Report of the forty-third 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]
6Note 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 we have not done it here.
9
applications, la démonstration générale et complète de la méthode de quinti-section de Briggs et de celle de Mouton, quand les indices sont équidifférents, et du
procédé exposé par Newton, dans ses Principes, quand les indices sont quelconques.
In Additions à la Connaissance des temps ou des mouvements célestes, à l’usage des astronomes et des navigateurs, pour l’an 1847, pages 181–222. Paris: Bachelier, 1844.
[A summary is given in theComptes rendus hebdomadaires des séances de l’Académie des sciences, 19(2), 8 July 1844, pp. 81–85, and the entire article is translated in theJournal of the Institute of Actuaries and Assurance Magazine, volume 14, 1869, pp. 1–36.][12] Louis Gabriel Michaud, editor. Biographie universelle, ancienne et moderne, volume 30. Paris: L. G. Michaud, 1821.
[Mouton’s notice is on page 346][13] Louis Gabriel Michaud, editor. Biographie universelle, ancienne et moderne, nouvelle édition, volume 29. Paris: Ch. Delagrave et Cie, 1843.
[Mouton’s notice is on page 485][14] Gabriel Mouton. Observationes Diametrorum Solis Et Lunæ Apparentium, Meridianarúmque aliquot altitudinum Solis & paucarum fixarum. Cum tabulâ declinationum Solis constructa ad singula graduum Eclipticæ scrupula prima. Pro cujus, & aliarum tabularum constructione seu perfectione, quædam numerorum proprietates non inutiliter deteguntur. Huic Adjecta Est Brevis Dissertatio De dierum naturalium inæqualitate; & de temporis æquatione. Una Cum Nova Mensurarum Geometricarum Idea: nováque methodo eas communicandi, &
conservandi in posterùm absque alteratione. Lugduni: ex typographiâ Matthæi Liberal, 1670.
[pages 368–396 entitled “De nonnullis numerorum proprietatibus” concern the interpolation by differences][15] Denis Roegel. A reconstruction of Adriaan Vlacq’s tables in the Trigonometria artificialis (1633). Technical report, LORIA, Nancy, 2010.
[This is a recalculation of the tables of [19].][16] Denis Roegel. A reconstruction of the tables of Briggs and Gellibrand’s
Trigonometria Britannica (1633). Technical report, LORIA, Nancy, 2010.
[This is a recalculation of the tables of [1].][17] Michael Taylor. Tables of logarithms of all numbers, from 1 to 101000; and of the sines and tangents to every second of the quadrant. London: Christopher Buckton, 1792.
[18] Georg Vega. Thesaurus logarithmorum completus. Leipzig: Weidmann, 1794.
[19] Adriaan Vlacq. Trigonometria artificialis. Gouda: Pieter Rammazeyn, 1633.
[The tables were reconstructed by D. Roegel in 2010. [15]][20] Franz Xaver von Zach. Lettre XIII. Correspondance astronomique, géographique, hydrographique et statistique, 1:215–222, 1818.
10
TRIGONOMETRIÆ ARTIFICIALIS,
SIVE
MAGNI CANONIS
TRIANGULORUM
LOGARITHMICI
SUPPLEMENTUM,
Exhibens Logarithmos Sinuum & Tangentium Singulis Scrupulis Secundis debitos, tàm in quatuor primis Quadrantis gradibus, quàm in
eorum complementis.
M S Logarithmici. Differen. Logarithmici. Diffe. Logarithmi. Communes Logarithmi.
Infinita ,, Infinita Infinita
Infinita Infinita
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