A reconstruction of the tables of Pitiscus' <i>Thesaurus Mathematicus</i> (1613)

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Denis Roegel

To cite this version:

Denis Roegel. A reconstruction of the tables of Pitiscus’ Thesaurus Mathematicus (1613). [Research

Report] 2010. �inria-00543933�

Pitiscus’ Thesaurus mathematicus

(1613)

Denis Roegel

may be said to be the four fundamental tables

of the mathematical sciences.

Glaisher, 1873 [28, p. 45]

1

Pitiscus’ initial work on trigonometry

Bartholomäus Pitiscus

1

_{was born in Silesia in 1561. Throughout his life, his main field}

was theology. He had however an interest for mathematics and he first published in 1595

an appendix on trigonometry to Scultetus’ treatise on spherical astronomy [65]. This was

the first time the word “trigonometry” appeared in print [32, 34]. An expanded version

of this appendix was published in 1600 [48], a second edition appeared in 1608 [49], and

a third one in 1612 [50], and we describe them below.

2

_{In 1613, Pitiscus published his}

Thesaurus mathematicus, and died the same year.

The initial appendix published in 1595 did not contain any trigonometric table, but

the expanded version published in 1600 contained a table of sines, tangents and secants

for every minute of the quadrant and to five decimal places [48, pp. 123–213]. The

table only had the headers for these three functions,

3

_{but it of course also gave the}

cosines, cotangents and cosecants without naming them.

4

_{The first table giving the}

six trigonometric functions was published by Georg Joachim Rheticus (1514–1574) in

1551 [52].

2

Corrections to the Opus palatinum (ca. 1603–1607)

2.1

Rheticus’ Opus palatinum (1596)

After having published his first trigonometric table in 1551, Rheticus started to work

on a much larger table, giving the six trigonometric functions for a radius of 10

10

and

at intervals of 10 seconds. Rheticus was the first to compute sine tables with intervals

smaller than one minute [24, p. 98]. Rheticus was not able to complete this table, but his

disciple Valentinus Otho continued the work and eventually published the table in 1596,

the Opus palatinum [53], more than twenty years after Rheticus’ death.

5

1

Outlines of Pitiscus’ life were given by Gerhardt [24, pp. 93–99], Zeller [73], and Archibald [4].

2

More detailed descriptions of the various editions and translations of Pitiscus’ works can be found

in Archibald’s notice [4] and Miura’s articles [44, 45]. See also Gravelaar’s article [29]. Miura’s second

article [45] compares in particular Pitiscus’ pre-logarithmic trigonometry with Norwood’s logarithmic

trigonometry published in 1631.

3

It was Thomas Fincke who in 1583 first used the names “tangent” and “secant” [20, 18]. See also [28,

p. 42].

4

The names “cosine” and “cotangent” were first used by Gunter in 1620 [30].

5

For more details on Rheticus’ first tables and work on the Opus palatinum, see our studies [59, 60]. For

biographical details on Rheticus, see in particular Burmeister [12], Danielson [16], and Schöbi-Fink [64].

it was found that the beginnings of the tables of cotangents and cosecants were corrupted

by a very important error. Adrianus Romanus (Adriaan van Roomen)

6

_{(1561–1615) was}

the first to notice the flaw [9]. In order to examine the accuracy of the table, Romanus

made use of the formula

sec a + tan a = tan

 a

2

+ 45

◦



_.

For instance, in the first cotangent, the last nine places were incorrect. Rheticus had

cot 10

′′

_{= 20626.4670574694}

csc 10

′′

_{= 20626.4670327177,}

the correct values being

cot 10

′′

_{= 20626.48060854917 . . .}

csc 10

′′

_{= 20626.48063278986 . . . .}

But the way in which the values were computed made this error gradually vanish.

The error eventually vanished in the 86th page.

Romanus did not only find the error, but he also went into a deeper error analysis,

establishing rules about the number of necessary extra decimals for a correct

computa-tion [9, p. 59]. He understood that Rheticus had not computed the sines with enough

digits to ensure an accuracy of 10 places in the last tangents and secants.

2.3

Pitiscus’ recomputations

When the errors were found, Otho was ill and no longer capable of correcting them. It

was Pitiscus who undertook the recomputation of the 86 incorrect pages, probably taking

into account Romanus’ analysis. This apparently only happened after the death of Lucius

Valentinus Otho in 1603, as Pitiscus first tried to obtain a manuscript copy of Rheticus’

sine table to 15 decimal places, thinking that Rheticus’ unpublished tables might help

for the correction.

Either by his own analysis or as a consequence of Romanus’ analysis, Pitiscus

under-stood that Rheticus’ computations were not sufficient and he decided to compute new

tables.

6

Romanus was one of the mathematicians who had embarked on the calculation of trigonometric

tables. In 1593, he mentioned in particular Rheticus’ letter from 1568 (reproduced in [26, p. 228]), and

he was aware of the planned completion of the Opus palatinum [9, pp. 55–56]. Romanus’ own plans were

to compute a sine table for the radius 10

16

, but he did not have calculators to second him. Romanus

was probably one of the first persons to buy the Opus palatinum, as he eagerly awaited its publication.

As he was corresponding with Clavius, he also sent him a copy, and they exchanged about the accuracy

of the tables. The correspondence with Clavius also reveals that Christoph Grienberger [28, p. 45] was

preparing a table of sines, but this table was never published, perhaps as a consequence of Romanus’

insights.

title page, was reissued in 1607.

Pitiscus’ corrections were studied in detail by Prony [54, 55]

8

_{and then by Mollweide,}

who was unaware of Prony’s work [46].

3

The new editions of Pitiscus’ trigonometry (1608–

1612)

In the second and third editions of his trigonometry [49, 50], Pitiscus continued to expand

his tables.

9

_{These improvements were mainly a consequence of Pitiscus’ work on the Opus}

palatinum, and Pitiscus explains how his computations were done. Pitiscus first computed

the sines of 30

◦

_{, 15}

◦

_{, 5}

◦

_{, 1}

◦

_{, 30}

′

_{, 10}

′

_{, 5}

′

_{, 1}

′

_{, 30}

′′

_{, 10}

′′

_{, 5}

′′

_{, 1}

′′

_{, all for the radius 10}

25

[49,

p. 62], [50, p. 71]. The computations were later detailed by Bernoulli [8, pp. 28–30].

After that, Pitiscus used the sines to compute the tangents and secants. As observed by

Gerhardt, the accuracy of the computations can be checked with the formulæ given by

Pitiscus, or by using the first, second and third differences. In these tables, Pitiscus does

not give the differences, but the proportional parts for 1

′

_{or 1}

′′

_{[8, p. 33].}

Finally, Pitiscus’ book contained a table of sines, tangents and secants for every second

of the first and last minutes of the quadrant, for every two seconds from 0

◦

₁

′

to 0

◦

₁₀

′

and

from 89

◦

₅₀

′

_{to 89}

◦

₅₉

′

_{, for every ten seconds from 0}

◦

₁₀

′

_{to 1}

◦

_{and from 89}

◦

_{to 89}

◦

₅₀

′

_{, and}

for every minute from 1

◦

_{to 89}

◦

_{with 5 to 12 decimal places. The values were given with}

their differences. The tables in the 1608 and 1612 editions are mostly identical, differing

only by the headers, and in some rare cases by the number of decimal places, which is

sometimes one less in the 1612 version.

4

Pitiscus’ Thesaurus mathematicus (1613)

Finally, Pitiscus began to work on a new project incorporating his own work with that of

Rheticus. The Thesaurus mathematicus [51] was eventually published in 1613 and

con-tained the new tables on which the corrected Opus palatinum was resting.

10

_{It contained}

7

The corrected edition is very rare, but can easily be identified without looking at the actual values,

first because the new pages are of a lower paper quality, and then as there is a small layout error at the

bottom of page 7 of the corrected copies, where the words basis and hypothenusa have been interchanged.

Copies of the 1607 edition seem to be located at Göttingen and Jena. In 1949, Archibald had also located

three copies and there are probably a few more [5, pp. 558–559]. Prony bought one copy which is now

in the library of the Ponts et chaussées, Fol. 415, see [54, 55].

8

Some drafts of Prony’s analysis are kept at the library of the Ponts et chaussées, Ms. 1745. See also

Archibald [4, pp. 394–396].

9

Among the changes, we can note that Pitiscus published some results due to Bürgi. See Leopold for

the exact references to the 1608 and 1612 editions [43, p. 33], [24, pp. 94–95], [15, pp. 619, 646–647]. One

should however be cautious concerning the claims of some authors that Pitiscus used a decimal point in

his tables. An examination of his tables shows that this is not true. See our article on Napier [61].

10

with first and second differences, again by Rheticus to fifteen decimal places (60

pages of tables);

3. values for the basic sines from which the others were calculated to 22 decimal places

by Pitiscus; these sines were given to 25 places and already included in Pitiscus’

Trigonometriæ mentioned above (8 pages of tables); this section also contains tables

for each basic sine, for instance for the angle 15

◦

:

56414 33335 51156 24855 31186

0 51763 80902 05041 52469 77977

1 14020 67661 22214 48822 73383

1 03527 61804 10083 04939 55954

1 62445 02006 73461 64888 25580

1 55291 42706 15124 57409 33931

. . .

The second line above is the chord of 30

◦

_{(0.51763 . . .), and the fourth and sixth}

lines are multiples of this chord.

What however is not clear are the odd lines. The first line is probably related to

the second one, the third one to the fourth one, and so on, but the meaning of these

lines was not given by Pitiscus, and the problem eluded both Jean Bernoulli [8,

p. 30] and Kaestner [40, pp. 619–620] who admitted that they could not find what

Pitiscus meant. Kaestner assumed that it was some kind of proof by nine. Perhaps

someone can unravel this 400 year-old mystery?

4. The fourth part was a table of sines to 22 decimal places by Pitiscus for 10

′′

, 30

′′

,

50

′′

_{, 1}

′

₁₀

′′

_{, 1}

′

₃₀

′′

_{, 1}

′

₅₀

′′

_{, etc., until 35}

′

_{, with the first five differences, together with}

the sines of the complements with the first four differences (4 pages of tables); these

sines were needed for the corrections to the Opus palatinum [8, p. 27].

There are three sets of paginations, one for the first table, one for the second table,

and one for the last two parts together. As a consequence, some copies may be bound

differently and give the tables in a different order.

11

Pitiscus seems to have computed several other tables, such as tables of tangents and

secants at intervals of 10

′′

_{for the first two degrees, but these tables were not published [17,}

p. 300].

Partial lists of errors in the corrected Opus palatinum and the Thesaurus

mathe-maticus have been established, and Archibald gives several references which should be

consulted [4, pp. 395–396].

below.

11

The above order is the one at the École Nationale des Ponts et Chaussées, but the copy at the

University of Goettingen, for instance, is different.

Pitiscus’ Thesaurus mathematicus was published the year before Napier’s logarithms.

Logarithms were going to change the way computations were done, and natural tables

of sines were less needed than accurate tables of logarithms. This may explain why

Rheticus and Pitiscus’ tables remained unsuperseded until Henri Andoyer’s 17-place

12

tables published in 1915–1918 [2].

5

Further editions and erratas

Pitiscus’ Thesaurus mathematicus was used by Vlacq to compute the trigonometric table

appended to his Arithmetica logarithmica in 1628 [68], [27, pp. 442–443].

Later, in 1897, Wilhelm Jordan (1842–1899) published sine and cosine tables excerpted

from the Opus palatinum, with the title Opus Palatinum : Sinus- u. Cosinus-Tafeln von

10

′′

_{zu 10}

′′

_{[38]. This was not, however, a reprint of the Opus palatinum.}

13

_{For errors in}

the Opus palatinum and later derived editions, see Fletcher et al. [21].

6

Structure of the tables and recomputation

The three main tables were recomputed using the GNU mpfr multiple-precision

floating-point library developed at INRIA [22], and give the exact values. The comparison of our

table and Pitiscus’ will therefore immediately show where Pitiscus’ table contains errors.

In his tables, Pitiscus had omitted some values and we supplied the correct ones: in

the first table, the differences were not given for 0

◦

_{, the second difference of the sine was}

not given for 0

◦

₁₀

′′

and the third differences were not given for 0

◦

₁₀

′′

; in the second table,

the second differences for 0

◦

₀

′

₁

′′

_{and 89}

◦

₅₉

′

₅₉

′′

_{were not given; and in the third table,}

the first difference of the sines was not given for 34

′

₅₀

′′

_{, the second differences of the}

sines were not given for 0

′

₁₀

′′

and 34

′

₅₀

′′

, the third differences of the sines were not given

for 0

′

₁₀

′′

_{, 34}

′

₃₀

′′

_{, and 34}

′

₅₀

′′

_{, the fourth differences of the sines were not given for 0}

′

₁₀

′′

_,

0

′

₃₀

′′

_{, 34}

′

₃₀

′′

_{, and 34}

′

₅₀

′′

_{, the fifth differences of the sines were not given for 0}

′

₁₀

′′

_{, 0}

′

₃₀

′′

_,

34

′

₁₀

′′

, 34

′

₃₀

′′

, and 34

′

₅₀

′′

; the first differences of the cosines was not given for 0

′

₁₀

′′

and

34

′

₅₀

′′

_{, the second differences of the cosines were not given for 0}

′

₁₀

′′

_{, 34}

′

₃₀

′′

_{, and 34}

′

₅₀

′′

_,

the third differences of the cosines were not given for 0

′

₁₀

′′

_{, 0}

′

₃₀

′′

_{, 34}

′

₃₀

′′

_{, and 34}

′

₅₀

′′

_{, the}

fourth differences of the cosines were not given for 0

′

₁₀

′′

, 0

′

₃₀

′′

, 34

′

₁₀

′′

, 34

′

₃₀

′′

, and 34

′

₅₀

′′

.

12

van Brummelen mistakenly writes that Andoyer’s tables are 20-place tables, but only a small part

of the tables are to 20 places. The general tables are to 17 places.

13

At least five more editions were published in 1913, 1923, 1925, 1929, and 1936. The sines and cosines

were only given to seven places, with differences, with 10 minutes per page, from 0

◦

_{to 45}

◦

_{, like in}

the Opus palatinum. The angles are also given in the centesimal division. In the five pages preface,

Jordan announced his plans for the inclusion of tangents, cotangents, secants and cosecants in a second

edition, but Jordan’s death has probably canceled these plans. These tables would presumably also have

been derived from the Opus palatinum. Jordan cites Regiomontanus, Rheticus, Otho and Pitiscus, but

Regiomontanus’ work was actually not used. Jordan seems to have translated in German the introduction

of the Opus palatinum, and perhaps planned to publish it in the future, what apparently never happened.

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] Melchior Adam. Vitæ germanorum theologorum, qui superiori secvlo ecclesiam

christi voce scriptisqve propagarunt et propugnarunt, congestæ et ad annum usque

MDCXVIII deductæ. Heidelberg: Jonas Rosa, 1620.

[pp. 833–841 on Pitiscus, reprinted

in 1653]

[2] Marie Henri Andoyer. Nouvelles tables trigonométriques fondamentales contenant

les valeurs naturelles des lignes trigonométriques. . . . Paris: Librairie A. Hermann

et fils, 1915–1918.

[3 volumes, reconstruction by D. Roegel in 2010 [58].]

[3] Anonymous. Article “Table”. In The Penny Cyclopædia of the Society for the

Diffusion of Useful Knowledge, volume 23, pages 496–501. London: Charles Knight

and co., 1842.

[4] Raymond Clare Archibald. Bartholomäus Pitiscus (1561–1613). Mathematical

Tables and other Aids to Computation, 3(25):390–397, 1949.

[5] Raymond Clare Archibald. Rheticus, with special reference to his Opus palatinum.

Mathematical Tables and other Aids to Computation, 3(28):552–561, 1949.

[A short

biography of Rheticus, but with some inaccuracies.]

[6] Raymond Clare Archibald. The Canon Doctrinae Triangvlorvm (1551) of Rheticus

(1514–1576). Mathematical Tables and other Aids to Computation, 7(42):131, 1953.

[7] Raymond Clare Archibald and Leslie John Comrie. (Erratas in Rheticus’ Opus

palatinum and Pitiscus’ Thesaurus mathematicus). Mathematical Tables and other

Aids to Computation, 6(39):163–166, 1952.

[8] Jean Bernoulli. Analyse de l’Opus Palatinum de Rheticus & du Thesaurus

mathematicus de Pitiscus : ouvrages très rares, qui se trouvent dans la

bibliothèque de l’Académie. Nouveaux mémoires de l’Académie royale des sciences

et belles-lettres, Année 1786:10–33, 1788.

[9] Paul Petrus Bockstaele. Adrianus Romanus and the trigonometric tables of Georg

Joachim Rheticus. In Serge˘ı Sergeevich Demidov et al, editor, Amphora: Festschrift

für Hans Wussing zu seinem 65. Geburtstag, pages 55–66. Basel: Birkhäuser, 1992.

14

_{Note 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.

Bayerischen Akademie der Wissenschaften, editor, Allgemeine Deutsche Biographie,

volume 14, pages 93–94. Leipzig: Duncker & Humblot, 1881.

[See also the second

biography of Rheticus in the same dictionary [31].]

[12] Karl Heinz Burmeister. Georg Joachim Rheticus, 1514-1574 : Eine

bio-bibliographie. Wiesbaden: G. Pressler, 1967–1968.

[3 volumes]

[13] Hubertus Lambertus Ludovicus Busard. Pitiscus, Bartholomeo. In

Charles Coulston Gillispie, editor, Dictionary of Scientific Biography, volume 11,

pages 3–4. New York: Charles Scribner’s Sons, 1975.

[14] Florian Cajori. Did Pitiscus use the decimal point? Archeion, 4(4):313–318, 1923.

[15] Moritz Cantor. Vorlesungen über Geschichte der Mathematik. Leipzig:

B. G. Teubner, 1900.

[volume 2, pp. 600–604 on Rheticus and 603–604 and 646–647 on

Pitiscus]

[16] Dennis Richard Danielson. The first Copernican: Georg Joachim Rheticus and the

rise of the Copernican Revolution. New York: Walker & Company, 2006.

[17] Joseph-Jérôme Lefrançois de Lalande. Lettre sur des tables de sinus extrêmement

rares. Journal des Sçavans, 55(2):291–303, October 1771.

[18] Augustus De Morgan. On the first introduction of the words Tangent and Secant.

Philosophical Magazine, Series 3, 28(188):382–387, May 1846.

[19] Jean-Baptiste Joseph Delambre. Histoire de l’astronomie moderne. Paris: Veuve

Courcier, 1821.

[two volumes, see in particular volume 2, pp. 1–35 on Rheticus and Pitiscus]

[20] Thomas Fincke. Geometria rotundi. Basel: Henric Petri, 1583.

[not seen]

[21] 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. For Rheticus and Pitiscus, see pp. 793, 865,

866, 884.]

[22] 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.

[23] Wilhelm Gaß. Bartholomäus Pitiscus. In Historische Kommission bei der

Bayerischen Akademie der Wissenschaften, editor, Allgemeine Deutsche Biographie,

volume 26, pages 204–205. Leipzig: Duncker & Humblot, 1888.

[24] Carl Immanuel Gerhardt. Geschichte der Mathematik in Deutschland, volume 17 of

Geschichte der Wissenschaften in Deutschland. Neuere Zeit. München:

Conrado Gesnero, deinde in Epitomen redacta et novorum librorum accessione

locupletata, jam vero postremo recognita et in duplum post priores editiones aucta,

per Josiam Simlerum. Zurich: Christoph Froschauer, 1574.

[a second edition was

published in 1583] [not seen]

[27] James Whitbread Lee Glaisher. On logarithmic tables. Monthly notices of the

Royal Astronomical Society, 33(7):440–458, 1873.

[28] 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.]

[see pp. 44–45 for Pitiscus]

[29] Nicolaas Lambertus Willem Antonie Gravelaar. Pitiscus’ trigonometria. Nieuw

Archief voor Wiskunde, 3 (series 2):253–278, 1898.

[30] Edmund Gunter. Canon triangulorum. London: William Jones, 1620.

[31] Siegmund Günther. Rheticus, G. J. In Historische Kommission bei der Bayerischen

Akademie der Wissenschaften, editor, Allgemeine Deutsche Biographie, volume 28,

pages 388–390. Leipzig: Duncker & Humblot, 1889.

[See also the second biography of

Rheticus in the same dictionary [11].]

[32] Martin Hellmann. Pitiscus und seine kleine >Trigonometrie<. Mannheimer

Geschichtsblätter, 4 (Neue Folge):107–129, 1997.

[33] Martin Hellmann. Bartholomäus Pitiscus (1561–1613) : Geometrie als Zeitvertreib

— Rechnen als Aufgabe. In Rainer Gebhardt, editor, Rechenbücher und

mathematische Texte der frühen Neuzeit : Tagungsband zum Wissenschaftlichen

Kolloquium „Rechenbücher und Mathematische Texte der Frühen Neuzeit” anläßlich

des 440. Todestages des Rechenmeisters Adam Ries vom 16.—18. April 1999 in der

Berg- und Adam-Ries-Stadt Annaberg-Buchholz, volume 11 of Schriften des

Adam-Ries-Bundes e.V. Annaberg-Buchholz, pages 196–202. Annaberg-Buchholz:

Adam-Ries-Bund e.V., 1999.

[34] Martin Hellmann. Bartholomäus Pitiscus (1561–1613) und seine kleine

Trigonometrie. In Michael Toepell, editor, Mathematik im Wandel : Anregungen zu

einem fächerübergreifenden Mathematikunterricht, volume 2, pages 118–126.

Hildesheim: Franzbecker, 2001.

[35] Samuel Herrick, Jr. Natural-value trigonometric tables. Publications of the

Astronomical Society of the Pacific, 50(296):234–237, 1938.

[36] Erna Hilfstein. Was Valentinus Otho a mathematics professor at the University of

Heidelberg? Organon, 22/23:221–225, 1986/1987.

[38] Wilhelm Jordan. Opus Palatinum : Sinus- u. Cosinus-Tafeln von 10

′′

_{zu 10}

′′

_.

Hannover, Leipzig: Hahn’sche Buchhandlung, 1897.

[39] Louis Charles Karpinski. The decimal point. Science (new series),

45(1174):663–665, 1917.

[issue dated 29 June 1917]

[40] Abraham Gotthelf Kästner. Geschichte der Mathematik seit der Wiederherstellung

der Wissenschaften bis an das Ende des achtzehnten Jahrhunderts, volume 1.

Göttingen: Johann Georg Rosenbusch, 1796.

[pp. 590–611 are devoted to the Opus

palatinum, and pp. 564–565 and 612–626 to Pitiscus’ work]

[41] Cargill Gilston Knott, editor. Napier Tercentenary Memorial Volume. London:

Longmans, Green and company, 1915.

[42] Dionysius Lardner. Babbage’s calculating engine. The Edinburgh review,

59(120):263–327, July 1834.

[The Opus palatinum is described on page 280.]

[43] John H. Leopold. Astronomen, Sterne, Geräte : Landgraf Wilhelm IV. und seine

sich selbst bewegenden Globen. Luzern: Joseph Fremersdorf, 1986.

[44] Nobuo Miura. The applications of trigonometry in Pitiscus: a preliminary essay.

Historia scientiarum: international journal of the History of Science Society of

Japan, 30:63–78, 1986.

[45] Nobuo Miura. The applications of logarithms to trigonometry in Richard Norwood.

Historia scientiarum: international journal of the History of Science Society of

Japan, 37:17–30, 1989.

[46] Karl Brandan Mollweide. Nachricht von den durch Bartholom. Pitiscus in dem

Canon des Rhäticus gemachten Verbesserungen. Allgemeine Literatur-Zeitung,

1(61):484–488, March 1810.

[47] Jean-Étienne Montucla. Histoire des mathématiques. Paris: Charles Antoine

Jombert, 1758.

[two volumes, pp. 470–471 of the first volume describe Rheticus’ work]

[48] Bartholomaeus Pitiscus. Trigonometriæ sive de dimensione triangulorum libri

quinque. Augsburg: Michael Manger, 1600.

[expanded version of Pitiscus’ appendix in [65]]

[49] Bartholomaeus Pitiscus. Trigonometriæ sive de dimensione triangulorum libri

quinque. Augsburg: Johannes Prætorius, 1608.

[second edition of [48]]

[50] Bartholomaeus Pitiscus. Trigonometriæ sive de dimensione triangulorum libri

quinque. Frankfurt: Nicolaus Hoffmann, 1612.

[third edition of [48]]

secunda quadrantis : adiunctis ubique differentiis primis et secundis; atque, ubi res

tulit, etiam tertijs. Frankfurt: Nicolaus Hoffmann, 1613.

[52] Georg Joachim Rheticus. Canon doctrinæ triangulorum. Leipzig: Wolfgang

Gunter, 1551.

[This table was recomputed in 2010 by D. Roegel [59].]

[53] Georg Joachim Rheticus and Valentinus Otho. Opus palatinum de triangulis.

Neustadt: Matthaeus Harnisch, 1596.

[This table was recomputed in 2010 by

D. Roegel [60].]

[54] Gaspard-Clair-François-Marie Riche de Prony. Notice sur les grandes tables

logarithmiques et trigonométriques, calculées au Bureau du cadastre, sous la

direction du citoyen Prony, membre de l’Institut national, et directeur de l’École

des Ponts et Chaussées et du Cadastre, avec le rapport sur ces tables et sur

l’introduction qui y est jointe, fait à la classe des sciences physiques et

mathématiques de l’Institut national. Par les citoyens Lagrange, Laplace et

Delambre. Paris: Baudouin, 1801.

[This document actually contains three parts: 1) the

notice itself, dated 1st germinal 9 (pp. 1–8), 2) a supplement on the corrections to Rheticus’ Opus

palatinum, dated 21 germinal 9 (pp. 8–12), actually a summary of [55], 3) a report on the tables

by Lagrange, Laplace and Delambre, dated 11 germinal 9 (pp. 13–26).]

[55] Gaspard-Clair-François-Marie Riche de Prony. Eclaircissemens sur un point de

l’histoire des tables trigonométriques. Mémoires de la classe des sciences

mathématiques et physiques de l’Institut de France, pages 67–93, 1803–1804.

[A

summary of this article appeared in The Monthly magazine, volume 12, 1801, page 231, and in

L’esprit des journaux, septembre 1801, pp. 166–167. See also [54].]

[56] 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 [69].]

[57] Denis Roegel. A reconstruction of De Decker-Vlacq’s tables in the Arithmetica

logarithmica (1628). Technical report, LORIA, Nancy, 2010.

[This is a recalculation of

the tables of [68].]

[58] Denis Roegel. A reconstruction of Henri Andoyer’s trigonometric tables

(1915–1918). Technical report, LORIA, Nancy, 2010.

[This is a reconstruction of [2].]

[59] Denis Roegel. A reconstruction of the tables of Rheticus’s Canon doctrinæ

triangulorum (1551). Technical report, LORIA, Nancy, 2010.

[This is a recalculation of

the tables of [52].]

[60] Denis Roegel. A reconstruction of the tables of Rheticus’s Opus Palatinum (1596).

Technical report, LORIA, Nancy, 2010.

[This is a recalculation of the tables of [53].]

[61] Denis Roegel. Napier’s ideal construction of the logarithms. Technical report,

[63] Ralph Allen Sampson. The great tables preceding the discovery of logarithms. In

Knott [41], pages 213–218.

[64] Philipp Schöbi-Fink. Rheticus — der erste Kopernikaner. In Gerhard Wanner and

Philipp Schöbi-Fink, editors, Rheticus — Wegbereiter der Neuzeit. Eine

Würdigung, pages 7–44. Feldkirch: Rheticus-Gesellschaft, 2010.

[65] Abraham Scultetus. Sphæricorum libri tres methodice conscripti & utilibus scholiis

expositi. Heidelberg, 1595.

[pp. 157–213 are Pitiscus’ Trigonometria: sive de solutione

triangulorum, which was expanded in [48]]

[66] Johannes Tropfke. Geschichte der Elementar-Mathematik in systematischer

Darstellung. Leipzig: Veit & Comp., 1902–1903.

[67] Glen van Brummelen. The mathematics of the heavens and the Earth: the early

history of trigonometry. Princeton: Princeton University Press, 2009.

[68] Adriaan Vlacq. Arithmetica logarithmica. Gouda: Pieter Rammazeyn, 1628.

[The

introduction was reprinted in 1976 by Olms and the tables were reconstructed by D. Roegel in

2010. [57]]

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

[The

tables were reconstructed by D. Roegel in 2010. [56]]

[70] Anton von Braunmühl. Vorlesungen über Geschichte der Trigonometrie. Leipzig:

B. G. Teubner, 1900, 1903.

[2 volumes]

[71] Rudolf Wolf. Geschichte der Astronomie, volume 16 of Geschichte der

Wissenschaften in Deutschland. München: R. Oldenbourg, 1877.

[see pp. 343–346 on

Rheticus and Pitiscus]

[72] Rudolf Wolf. Handbuch der Astronomie : ihrer Geschichte und Litteratur. Zürich:

Friedrich Schulthess, 1890.

[2 volumes]

[73] Mary Claudia Zeller. The development of trigonometry from Regiomontanus to

Pitiscus. PhD thesis, University of Michigan, 1944.

[published in 1946]

CANON SINUUM

AD DECADES

SCRUPULO-RUM SECUNDOSCRUPULO-RUM,

Et ad partes Radij

1 00000 00000 00000.

Unà cum differentiis primis, secundis, tertiis.

PRIMVS ET POSTREMVS NVMERVS

VNIVS CVIVS QVE PAGINÆ GRADVS DENOTAT:

secundus scrupula prima reliquorum maiores, scrupula prima ;



Sinus

Diff. I.

II.

III.

Sinus complementi

Diff. I.

II.

III.



    .. −  −



 ..    ..      ..    ..   −   ..    ..      ..    ..   −   ..    ..    



..    ..   



 ..    ..   −   ..    ..      ..    ..   −   ..    ..      ..    ..    



..    ..   



 ..    ..   −   ..    ..      ..    ..      ..    ..      ..    ..   − 



..    ..   



 ..    ..   −   ..    ..      ..    ..      ..    ..   −   ..    ..    



..    ..   



 ..    ..   −   ..    ..      ..    ..      ..    ..      ..    ..   − 



..    ..   



 ..    ..      ..    ..   −   ..    ..      ..    ..      ..    ..   − 



..    ..   



 ..    ..      ..    ..      ..    ..      ..    ..      ..    ..   − 



..    ..   



 ..    ..      ..    ..      ..    ..      ..    ..   −   ..    ..    



..    ..   −



 ..    ..      ..    ..      ..    ..   −   ..    ..      ..    ..    



..    ..   −



 ..    ..      ..    ..      ..    ..   −   ..    ..      ..    ..   − 

Sinus complementi

Diff. I.

II.

III.

Sinus

Diff. I.

II.

III.





Sinus

Diff. I.

II.

III.

Sinus complementi

Diff. I.

II.

III.



..    ..   



 ..    ..      ..    ..   −   ..    ..      ..    ..      ..    ..    



..    ..   



 ..    ..   −   ..    ..      ..    ..      ..    ..      ..    ..    



..    ..   



 ..    ..   −   ..    ..      ..    ..      ..    ..      ..    ..    



..    ..   



 ..    ..      ..    ..      ..    ..   −   ..    ..      ..    ..    



..    ..   



 ..    ..   −   ..    ..      ..    ..      ..    ..      ..    ..    



..    ..   



 ..    ..   −   ..    ..      ..    ..      ..    ..      ..    ..    



..    ..   −



 ..    ..      ..    ..      ..    ..      ..    ..   −   ..    ..    



..    ..   



 ..    ..      ..    ..      ..    ..      ..    ..      ..    ..    



..    ..   



 ..    ..      ..    ..      ..    ..      ..    ..      ..    ..    



..    ..   



 ..    ..      ..    ..      ..    ..      ..    ..      ..    ..    

Sinus complementi

Diff. I.

II.

III.

Sinus

Diff. I.

II.

III.

