A reconstruction of Adriaan Vlacq's tables in the Trigonometria

A reconstruction

of Adriaan Vlacqs tables in the

Trigonometria artificialis (1633)

Denis Roegel

26 November 2011

This document is part of the LOCOMAT project:http://locomat.loria.fr

[The Thesaurus Mathematicus], the Opus Palatinum,and Vlacqs Arithmetica Logarithmica, 1628

and Trigonometria Artificialis, 1633,may be said to be the four fundamental tables

of the mathematical sciences.

Glaisher, 1873 [24, p. 45]

1 Vlacqs Arithmetica logarithmica (16251628)Adriaan Vlacq (ca. 16001667) came to be involved in the publication of tables of loga-rithms through Ezechiel de Decker (ca. 1603ca. 1647), a Dutch surveyor and teacher ofmathematics.

De Decker had the idea of translating a number of works by Napier and others,but De Decker neither had a command of Latin, nor presumably the money to publishthe books. As Vlacq knew Latin and had money, they entered a contract to publish anumber of works. Two books were published in 1626, the Eerste deel van de nieuwetelkonst [10], containing several translations, and the Nieuwe telkonst [11], containing ashortened version of Briggs Arithmetica logarithmica, as well as a translation of Gunterstable of logarithms of sines and tangents. Both books had as their sole author De Deckerwho announced another more complete work, the Tweede deel , which for a long time hadseemed never to have been published.

The Tweede deel [12] was published in 1627, but only two or three copies of it seemto have survived, for reasons which are still not totally clear, and that we have tried toexpose in our study of Vlacqs 1628 table [41]. It was only in 1920 that the Tweede deelwas really rediscovered. It contained a table of the logarithms from 1 to 100000 to 10decimal places.

The same table was republished by Vlacq in 1628, with Briggs introduction to hisArithmetica logarithmica [6], and without any mention of De Deckers work or involve-ment [53]. Between 1628 and 1920, Vlacq was therefore considered as the author of awork of which he was only a participant, albeit one that did a great share of work.

The work on the Arithmetica logarithmica, of which he was actually the publisher(with Rammazeyn being the printer), apparently led Vlacq to learn enough mathematicsto compute tables by himself.

2 Vlacqs first trigonometric table (1628)A large part of De Deckers 1627 table was computed by Vlacq, as acknowledged by DeDecker in his preface [12]. In 1628, Vlacq added a table at the end of the Arithmeticalogarithmica, containing a canon of logarithms of trigonometric functions. This canongave the logarithms of the six trigonometric functions to 10 places for every minute ofthe quadrant. Like Gunters table, this table actually gave the values of 10 + log sin,10 + log cos, 10 + log tan, and 10 + log cot. It does not, however, use a radius of 1011 asclaimed by Vlacq and copied by others [37, p. 156].

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This table extended Gunters table which gave the logarithms of sines and tangentsto 7 places for every minute [27] and it prefigured Vlacqs 1633 Trigonometria artifi-cialis [54]. It was presumably computed by Vlacq who only needed to compute thelogarithms of sines from 0 to 45. Vlacq explained his method of construction in thepreface of the Arithmetica logarithmica. He actually based himself on Pitiscus Thesaurusmathematicus [38], computing logarithms of known sines using the radix method.1

If log sin has been computed for 45, then we can use log sin = log sin 2 +log sin 30 log sin(90 ) for < 45 and the logarithms of tangents and secants canbe obtained with

log tan = log sin log sin(90 )log sec = log sin(90 )

Such a method does however lead to an accumulation of errors. Delambre writesthat sometimes eight logarithms were used to compute a new one, which causes the lastdecimal to be sometimes incorrect [13, vol. 2, p. 421].

3 The Trigonometria artificialis (1633)Immediately after the completion of the Arithmetica logarithmica (1628), Vlacq musthave planned to extend this table and construct a more complete table. This was tobe the Trigonometria artificialis . This table, completed in 1630 [21, p. 298], was onlypublished in 1633. It gave the logarithms of the six trigonometric functions to 10 placesevery 10 seconds of the quadrant.

The Trigonometria artificialis also contains the logarithms of numbers 120000 takenfrom Vlacqs Arithmetica logarithmica (1628).

3.1 The introduction to the tables

At about the same time as Vlacq, Briggs Trigonometria britannica [7] was completedand also printed at Gouda in 1633 by the same printer.

Given what Vlacq did with the Arithmetica logarithmica, it is not surprising to learnthat he also copied almost 50 pages from the Trigonometria britannica without givingany credit to Briggs [37, p. 159]. But, as shown below, Vlacq did not copy Briggs tables.

In any case, the identity of the introductions may explain why, according to Miller [37,p. 159], there are two slightly different title pages, one where the borrowing of Trigono-metria britannica is acknowledged, and another where it is not.

3.2 The sexagesimal division

BriggsTrigonometria britannica (1633) [7] uses a centesimal division of the degrees, andsince this is Briggs first and only trigonometric canon, there was no prior table by

1This was also observed by Delambre [13, vol. 2, p. 421], Glaisher [23, pp. 442443] and Wolf [58,vol. 1, pp. 169175].

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him with a similar division. It is likely that Vlacq took the traditional direction forthe Trigonometria artificialis (1633) [54], probably without knowing that Briggs woulddivide the degrees centesimally. We can theorize that Vlacq would have switched tothe centesimal division, had he known of Briggs projects soon enough, but Vlacq hadprobably learned of it too late for undertaking such a major change in the division of theangles.

Glaisher considered that if Vlacq had done the same in his Trigonometria artificialis(1633) [54], the switch to a centesimal division might have been easier [21, pp. 301302],because all the later tables eventually were based on Vlacqs tables. He was the lastperson who really could have made the switch to a centesimal division possible.

3.3 The construction of the tables

As far as we are aware, there have been very few investigations into the methods employedby Vlacq to construct his trigonometric tables. Only two articles shed a light in this area,namely Gauss remarks published in 1851 [18] and Glaishers further researches spurredby Gauss and published in 1873 [23].

Gauss had a closer look at the trigonometric part of Vegas Thesaurus logarithmorumcompletus [52]. These tables reproduce Vlacqs tables from 2 to 45, giving the loga-rithms of the trigonometric functions every 10 seconds. From 0 to 2, the functions wereinterpolated for every second by Lieutenant Dorfmund, under Vegas direction.

Gauss noticed that the tabular results in the sine column were almost without ex-ception equal to the sum of the corresponding tabular results in the cosine and tangentcolumns. [23, p. 440] However, if each value had been computed independently and cor-rectly rounded, this relation should only hold on average in three cases out of four.2 Forinstance, considering the equalities a+ b = c, rounding a, b and c will break the relationin certain cases: 0.13 + 0.24 = 0.37, but 0.1 + 0.2 6= 0.4.

Moreover, Gauss observed that Vlacqs logarithmic tangents are the most inaccurate,and that the logarithmic cosines are more accurate than the logarithmic sines. From this,Gauss concluded that the logarithmic tangents had been obtained from the logarithmicsines and cosines. Gauss thought that the logarithmic cosines were more accurate becausethe interpolation was simpler for them.

In fact, Vlacq explained in his preface how the tables were constructed [23, p. 441],but Gauss failed to notice it, perhaps because he did not have access to Vlacqs originalvolume and only to Vegas Thesaurus .

Vlacq first computed the logarithms of the cosines from 0 to 45. This was done usingthe sines given by Rheticus in the Opus palatinum (1596) [40] where they are tabulatedto 10 decimal places every 10 seconds, the same step as the one used by Vlacq. There are45 60 6 = 16200 logarithms and they could have been computed using Briggs radixmethod. If these logarithms were computed over a period of two years, they representedabout 20 logarithms computed per day, a number which could have been reduced if thecomputations were split between several computers.

2See Glaisher for a simple justification of this proportion [23, pp. 447448].

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The logarithms of the cosines were then used to compute the logarithms of the sinesfrom 0 to 45 using the formula given above

log sinx = log sin 2x+ log sin 30 log sin(90 x)= log cos(90 2x) log cosx log 2

The logarithms of the tangents were then obtained with

log tanx = log sinx log cosx

and the logarithms of the cotangents, secants and cosecants were easily derived fromthem.

Moreover, Glaisher observed that Briggs and Vlacqs calculations were independentand that Vlacq did not copy some values from Briggs Trigonometria britannica [7].3 Thisfollows from the fact that the values which are common to both tables are those corre-sponding to angles multiples of 3, and that Vlacqs Trigonometria artificialis coincideswith the trigonometric values given in Vlacqs Arithmetica logarithmica for all minutes,as checked by Glaisher [23, p. 444]. In other words, these values have not been copiedfrom Briggs, and the other values are not found in Briggs table.

3.4 The 20 first chiliades of the Arithmetica logarithmica

In addition to the trigonometric part, the Trigonometria artificialis contains a copy ofthe first 20 chiliades (120000) of the Arithmetica logarithmica, with an errata.4

4 ErrorsA number of errors were naturally given by Vega and the erratas to Vegas tables may beuseful for Vlacqs tables. A list of errors in Vlacqs 1633 table appears in Delambres His-toire de lastronomie moderne [13, vol.2, pp. 428432]. Delambre compared the commonvalues, every 3. Some errors in the Trigonometria artificialis were also given by Andoyerin 1911 [1]. More recently, an overview of the erratas of Vegas and Vlacqs tables wasgiven by Fletcher et al. [16, pp. 919925].

It is not our purpose to give here a list of all the errors in the original manuscript,but our reconstruction might drive another enthusiast to compile an updated list of allerrors in the Trigonometria artificialis . As an illustration to the real accuracy of Vlacqstables, we consider the first twelve values of log sin, for which the error can reach up tofour units of the last place:

3Delambre had also analyzed the accuracy of the Trigonometria artificialis [13, vol. 2, p. 423] andconcluded that Vlacq had made use of Rheticus table [13, vol. 2, p. 426], and not of Briggs table, fromwhich he could have interpolated.

4The copy of Vlacqs Arithmetica logarithmica from the Complutense University of Madrid is avail-able on google books (id: -XR-2oDEb9oC), but it only contains the Latin introduction, the trigonometricsupplement, as well as the fragment of the logarithms of numbers which was appended to Vlacqs Trigo-nometria artificialis. This may be the result of a binding error.

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Vlacq exact5,68557 48665 5,68557 486675,98660 48617 5,98660 486186,16269 61198 6,16269 612006,28763 48554 6,28763 485546,38454 48673 6,38454 486696,46372 61109 6,46372 611116,53067 28985 6,53067 289856,58866 48429 6,58866 484296,63981 73623 6,63981 736256,68557 48502 6,68557 484986,72696 75316 6,72696 753146,76475 60882 6,76475 60884

5 Vlacq-based tablesAlmost all the tables of logarithms published after Vlacqs Arithmetica logarithmica andTrigonometria artificialis until the 20th century were based on them, with various cor-rections, extensions, or simplifications.

As a most curious example, the 1628 Arithmetica logarithmica and Vlacqs 1633trigonometric table were even used as a basis for the Chinese tables of logarithms pub-lished around 1720 [48].

Probably the most popular such tables were those of Vega [2, 15], which went intomany editions. Vegas great Thesaurus logarithmorum completus [52] published in 1794gave the logarithms to 10 places, borrowing them from Vlacq from 1 to 100000, andperhaps from Briggs from 100000 to 101000. The logarithms of trigonometric functionswere taken from Vlacqs Trigonometria artificialis , only extending them for the firstdegree of the trigonometric functions.

6 Structure of the tables and recomputationThe original Trigonometria artificialis contains an introduction of 52 pages, followed bya first section of tables for the logarithms of the trigonometric functions. This sectionis followed by a second one which is an excerpt of Vlacqs Arithmetica logarithmica, butonly from 1 to 20000. The last page of that table, for the interval 1995120000, has beenreset and is followed by an errata. This section is actually a leftover from the Arithmeticalogarithmica, as proven by the signatures of the pages, which have also been reproducedhere. The signatures of the first part were not reproduced.

These two tables are reproduced here. The tables were recomputed using the GNUmpfr multiple-precision floating-point library developed at INRIA [17], and give the exactvalues. The comparison of our table and Vlacqs will therefore immediately show whereVlacqs table contains errors, and this is of course one of the purposes of this reconstruc-tion. Apart from the change in accuracy, we have tried to be as faithful as possible to

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the original tables. Among the slight changes, we mention the uniformization of head-ers. In the original table, there are sometimes variations from one page to the other andthese idiosyncrasies were not included here. A few values have also been changed for apurpose of uniformity. The first logarithms of sines and tangents were 0 in Vlacqs table,and we have written Infinita instead, like for the logarithm of the first cotangent. Thecorresponding differences which had been left empty by Vlacq were also adapted. In thesecond table, only the first difference was missing and was supplied.

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ReferencesThe following list covers the most important references5 related to Vlacqs tables. Notall items of this list are mentioned in the text, and the sources which have not been seenare marked so. We have added notes about the contents of the articles in certain cases.

[1] Marie Henri Andoyer. Nouvelles tables trigonomtriques fondamentales contenantles logarithmes des lignes trigonomtriques. . . . Paris: Librairie A. Hermann et fils,1911. [Reconstruction by D. Roegel in 2010 [43].]

[2] Raymond Clare Archibald. (Vegas tables). Mathematical Tables and other Aids toComputation, 2(16):161165, 1946.

[3] David Bierens de Haan. Notice sur des tables logarithmiques hollandaises.Bulletino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche,6:203238, May 1873.

[4] David Bierens de Haan. Adriaan Vlack en zijne logarithmentafels. Verslagen enmededeelingen der Koninklyke Akademie van wetenschappen, Afd. Natuurkunde, 8(2nd ser.):163199, 1874.

[5] K. Boecklen. Die Interpolation in Vegas Thesaurus Logarithmorum Completus.Astronomische Nachrichten, 216(5180):379384, 1922.

[6] Henry Briggs. Arithmetica logarithmica. London: William Jones, 1624. [The tableswere reconstructed by D. Roegel in 2010. [45]]

[7] Henry Briggs and Henry Gellibrand. Trigonometria Britannica. Gouda: PieterRammazeyn, 1633. [The tables were reconstructed by D. Roegel in 2010 [44].]

[8] Moritz Cantor. Adriaen Vlacq. In Historische Kommission bei der BayerischenAkademie der Wissenschaften, editor, Allgemeine Deutsche Biographie, volume 40,page 86. Leipzig: Duncker & Humblot, 1896.

[9] Moritz Cantor. Vorlesungen ber Geschichte der Mathematik. Leipzig:B. G. Teubner, 1900. [volume 2, pp. 743747 on Vlacq]

[10] Ezechiel de Decker. Eerste deel van de nieuwe telkonst, inhoudende verscheydemanieren van rekenen, waerdoor seer licht konnen volbracht worden de geometrischeende arithmetische questien. Gouda: Pieter Rammazeyn, 1626. [not seen]

5Note on the titles of the works: Original titles come with many idiosyncrasies and features (linesplitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has thereforeseemed pointless to capitalize works according to conventions which not only have no relation with theoriginal work, but also do not restore the title entirely. In the following list of references, most titlewords (except in German) will therefore be left uncapitalized. The names of the authors have also beenhomogenized 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.

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[11] Ezechiel de Decker. Nieuwe telkonst inhoudende de logarithmi voor de ghetallenbeginnende van 1 tot 10000, ghemaeckt van Henrico Briggio Professor van deGeometrie tot Ocxfort. Gouda: Pieter Rammazeyn, 1626. [not seen]

[12] Ezechiel de Decker. Tweede deel van de nieuwe tel-konst ofte wonderliicke konstighetafel, inhoudende de logarithmi, voor de getallen van 1 tot 100000 toe. Gouda:Pieter Rammazeyn, 1627. [500 partial facsimile copies were printed in 1964 by Nieuwkoop &B. de Graaf with an introduction by Alphonse Johannes Emile Marie Smeur.]

[13] Jean-Baptiste Joseph Delambre. Histoire de lastronomie moderne. Paris: VeuveCourcier, 1821. [2 volumes, see vol. 1, p. 545, vol. 2, pp. 420432]

[14] Johannes Faulhaber. Canon Triangulorum Logarithmicus, Das ist KnstlicheLogarithmische Tafeln der Sinuum, Tangentium, und Secantium. Augsburg:Andreas Aperger, 1631.

[15] Gerlinde Faustmann. Jurij Vega the most internationally distributed logarithmtables. [8 pages, no date, no later than 2004, http://www.rechenschieber.org/vega.pdf]

[16] Alan Fletcher, Jeffery Charles Percy Miller, Louis Rosenhead, and Leslie JohnComrie. An index of mathematical tables. Oxford: Blackwell scientific publicationsLtd., 1962. [2nd edition (1st in 1946), 2 volumes]

[17] Laurent Fousse, Guillaume Hanrot, Vincent Lefvre, Patrick Plissier, and PaulZimmermann. MPFR: A multiple-precision binary floating-point library withcorrect rounding. ACM Transactions on Mathematical Software, 33(2), 2007.

[18] Carl Friedrich Gauss. Einige Bemerkungen zu Vegas Thesaurus Logarithmorum.Astronomische Nachrichten, 32(756):181188, 1851.

[19] David Gibb. A course in interpolation and numerical integration for themathematical laboratory, volume 2 of Edinburgh Mathematical Tracts. London:G. Bell & sons, Ltd., 1915.

[20] James Whitbread Lee Glaisher. Addition to a paper on errors in Vlacqs ten-figurelogarithms, published in the last number of the Monthly Notice. Monthly Noticesof the Royal Astronomical Society, 32(8):288290, June 1872.

[21] James Whitbread Lee Glaisher. Notice respecting some new facts in the earlyhistory of logarithmic tables. The London, Edinburgh and Dublin PhilosophicalMagazine and Journal of Science, Series 4, 44:291303, 1872.

[22] James Whitbread Lee Glaisher. On errors in Vlacqs (often called Briggs orNepers) tables of ten-figure logarithms of numbers. Monthly Notices of the RoyalAstronomical Society, 32(7):255262, May 1872.

[23] James Whitbread Lee Glaisher. On logarithmic tables. Monthly notices of theRoyal Astronomical Society, 33(7):440458, 1873.

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[24] 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-thirdmeeting of the British Association for the advancement of science, London: John Murray, 1874.]

[25] Herman Heine Goldstine. A history of numerical analysis from the 16th through the19th century. New York: Springer, 1977.

[26] Hugo Grotius. De imperio summarum potestatum circa sacra. Leiden: Brill, 2001.[Introduction, English translation and commentary by Harm-Jan Van Dam, 2 volumes.]

[27] Edmund Gunter. Canon triangulorum, sive tabul sinuum et tangentiumartificialium ad radium 10000,0000.& ad scrupula prima quadrantis. London:William Jones, 1620. [The tables were reconstructed by D. Roegel in 2010. [42]]

[28] Albert Hatzfeld. La division dcimale du cercle. Revue scientifique, 48:655659,1891.

[29] James Henderson. Bibliotheca tabularum mathematicarum, being a descriptivecatalogue of mathematical tables. Part I: Logarithmic tables (A. Logarithms ofnumbers), volume XIII of Tracts for computers. London: Cambridge UniversityPress, 1926.

[30] Charles Hutton. Mathematical tables: containing common, hyperbolic, and logisticlogarithms, also sines, tangents, secants, and versed-sines, etc. London: G. G. J.,J. Robinson, and R. Baldwin, 1785.

[31] G. Huxley. Briggs, Henry. In Charles Coulston Gillispie, editor, Dictionary ofScientific Biography, volume 2, pages 461463. New York: Charles Scribners Sons.

[32] Graham Jagger. The making of logarithm tables. In Martin Campbell-Kelly, MaryCroarken, Raymond Flood, and Eleanor Robson, editors, The history ofmathematical tables: from Sumer to spreadsheets, pages 4877. Oxford: OxfordUniversity Press, 2003.

[33] Graham Jagger. The will of Henry Briggs. BSHM Bulletin: Journal of the BritishSociety for the History of Mathematics, 21(2):127131, July 2006.

[34] Johannes Kepler, John Napier, and Henry Briggs. Les milles logarithmes ; etc.Bordeaux: Jean Peyroux, 1993. [French translation of Keplers tables and Nepers descriptioby Jean Peyroux.]

[35] Frdric Maurice. Mmoire sur les interpolations, contenant surtout, avec uneexposition fort simple de leur thorie, dans ce quelle a de plus utile pour lesapplications, la dmonstration gnrale et complte de la mthode de quinti-sectionde Briggs et de celle de Mouton, quand les indices sont quidiffrents, et duprocd expos par Newton, dans ses Principes, quand les indices sont quelconques.In Additions la Connaissance des temps ou des mouvements clestes, lusagedes astronomes et des navigateurs, pour lan 1847, pages 181222. Paris: Bachelier,

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1844. [A summary is given in the Comptes rendus hebdomadaires des sances de lAcadmie dessciences, 19(2), 8 July 1844, pp. 8185, and the entire article is translated in the Journal of theInstitute of Actuaries and Assurance Magazine, volume 14, 1869, pp. 136.]

[36] Erik Meijering. A chronology of interpolation: from ancient astronomy to modernsignal and image processing. Proceedings of the IEEE, 90(3):319342, March 2002.

[37] Leo Miller. Milton and Vlacq. Papers of the Bibliographical Society of America,73:145208, 1979.

[38] Bartholomaeus Pitiscus. Thesaurus mathematicus sive canon sinuum ad radium1. 00000. 00000. 00000. et ad dena quque scrupula secunda quadrantis : una cumsinibus primi et postremi gradus, ad eundem radium, et ad singula scrupulasecunda quadrantis : adiunctis ubique differentiis primis et secundis; atque, ubi restulit, etiam tertijs. Frankfurt: Nicolaus Hoffmann, 1613. [The tables were reconstructedby D. Roegel in 2010. [46]]

[39] Jean-Charles Rodolphe Radau. tudes sur les formules dinterpolation. BulletinAstronomique, Srie I, 8:273294, 1891.

[40] Georg Joachim Rheticus and Valentinus Otho. Opus palatinum de triangulis.Neustadt: Matthaeus Harnisch, 1596. [This table was recomputed in 2010 byD. Roegel [47].]

[41] Denis Roegel. A reconstruction of De Decker-Vlacqs tables in the Arithmeticalogarithmica (1628). Technical report, LORIA, Nancy, 2010. [This is a recalculation ofthe tables of [53].]

[42] Denis Roegel. A reconstruction of Gunters Canon triangulorum (1620). Technicalreport, LORIA, Nancy, 2010. [This is a recalculation of the tables of [27].]

[43] Denis Roegel. A reconstruction of Henri Andoyers table of logarithms (1911).Technical report, LORIA, Nancy, 2010. [This is a reconstruction of [1].]

[44] Denis Roegel. A reconstruction of the tables of Briggs and GellibrandsTrigonometria Britannica (1633). Technical report, LORIA, Nancy, 2010. [This is arecalculation of the tables of [7].]

[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[6].]

[46] Denis Roegel. A reconstruction of the tables of Pitiscus Thesaurus Mathematicus(1613). Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of[38].]

[47] Denis Roegel. A reconstruction of the tables of Rheticuss Opus Palatinum (1596).Technical report, LORIA, Nancy, 2010. [This is a recalculation of the tables of [40].]

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[48] Denis Roegel. Introduction to Chinese and Japanese tables of logarithms, with areview of secondary sources. Technical report, LORIA, Nancy, 2010. [Thisintroduction is supplemented by three volumes of reconstructed tables patterned after Vlacqstables.]

[49] Dirk Jan Struik. Vlacq, Adriaan. In Charles Coulston Gillispie, editor, Dictionaryof Scientific Biography, volume 14, pages 5152. New York: Charles ScribnersSons.

[50] Glen van Brummelen. The mathematics of the heavens and the Earth: the earlyhistory of trigonometry. Princeton: Princeton University Press, 2009.

[51] Otto E. van Poelje. Adriaen Vlacq and Ezechiel de Decker: Dutch contributors tothe early tables of Briggsian logarithms. Journal of the Oughtred Society,14(1):3040, 2005.

[52] Georg Vega. Thesaurus logarithmorum completus. Leipzig: Weidmann, 1794.

[53] Adriaan Vlacq. Arithmetica logarithmica. Gouda: Pieter Rammazeyn, 1628. [Theintroduction was reprinted in 1976 by Olms and the tables were reconstructed by D. Roegel in2010 [41].]

[54] Adriaan Vlacq. Trigonometria artificialis. Gouda: Pieter Rammazeyn, 1633.

[55] Anton von Braunmhl. Vorlesungen ber Geschichte der Trigonometrie. Leipzig:B. G. Teubner, 1900, 1903. [2 volumes]

[56] Derek Thomas Whiteside, editor. The Mathematical Papers of Isaac Newton:Volume I, 16641666. Cambridge: Cambridge University Press, 1967.

[57] J. Hill Williams. Briggss method of interpolation; being a translation of the 13thchapter and part of the 12th of the preface to the Arithmetica Logarithmica.Journal of the Institute of Actuaries and Assurance Magazine, 14:7388, 1869.

[58] Rudolf Wolf. Handbuch der Astronomie : ihrer Geschichte und Litteratur. Zrich:Friedrich Schulthess, 1890. [2 volumes]

[59] Z. Adrien Vlacq. In Louis Gabriel Michaud, editor, Biographie universelle,ancienne et moderne, nouvelle dition, volume 44, page 1. Paris: Ch. Delagrave etCie, (no year, ca. 1865).

[60] Mary Claudia Zeller. The development of trigonometry from Regiomontanus toPitiscus. PhD thesis, University of Michigan, 1944. [published in 1946]

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Vlacqs 1633 table (reconstruction, D. Roegel, 2010)

MAGNVS CANONTRIANGVLORVM,

Continens LOGARITHMOS Sinuum& Tangentium ad dena Scrupu-

la Secunda.


0M S

SINVVMLogarithmi. Differ.

Sin. Comple.Logarithmi. Diff.

TANGEN.Logarithmi.

Differ.commu-nes.

Tang. compl.Logarithmi.

Infinita ,, Infinita Infinita Infinita Infinita

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,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

Sin. Comple. SINVVM Tang. compl. TANGEN. S M

89


0M S



TANGEN.Logarithmi.

Differ.commu-nes.


,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,


89


1M S



TANGEN.Logarithmi.

Differ.commu-nes.


,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,


88


2M S



TANGEN.Logarithmi.

Differ.commu-nes.


,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,


87


3M S



TANGEN.Logarithmi.

Differ.commu-nes.


,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,


86


4M S



TANGEN.Logarithmi.

Differ.commu-nes.


,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,


85


5M S



TANGEN.Logarithmi.

Differ.commu-nes.


,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,


84


6M S



TANGEN.Logarithmi.

Differ.commu-nes.


,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , , ,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, , ,

,, ,, ,, ,, ,

Documents

A reconstruction of Adriaan Vlacq's tables in the Trigonometria