Denis Roegel

A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries

This document is a survey of the main European fundamental trigonometric tables printed in the 15th and 16th centuries. After a review of the work done before the 15th century in Greece, India and the Arab world, the starting points in Europe are examined. The seminal work of Regiomontanus is carefully studied and the lineage of all later works is established. c ,

Introduction

The purpose of this survey is to sort out the many fundamental European purely (i.e., non astronomical) trigonometric tables published in the 15th and 16th centuries, and specifically to clarify their relationships. 1 I am concerned here almost exclusively with tables of sines, tangents and secants, and not with more specialized trigonometric tables that might be used as auxiliary tables.

Although a study with a similar scope has been published by Glowatzki and Göttsche in 1990, 2 I feel that it is necessary to review the tables in the light of their ready access, and to see whether their understanding can be improved. I believe that my study brings new information and corrects some earlier mistakes. This new examination is also made in the context of the LOCOMAT project, 3 where a number of historical tables have been reconstructed (computationally and typographically) and analyzed, enabling a better assessment of their accuracy and lineage. However, it must be stressed that the absolute accuracy of the historical tables under consideration here is less important than their relationships and the process that led to their computation or organization.

In the following sections, I first give a short review of the history of purely trigonometric tables before the 15th century, and follow their development through Greece, India, the Arab world, and finally Western Europe. I am then considering the work of four great innovators, Johannes von Gmunden, Giovanni Bianchini, Georg von Peuerbach and Johannes Regiomontanus. The latter was the one who greatly expanded the world of trigonometric tables, and set the background for almost all future work until the end of the 16th century. I am therefore examining what are Regiomontanus's seminal tables, and then journey through a century of tables, from Regiomontanus's Tabulae directionum profectionumque of 1490 to Rheti-cus's Opus palatinum of 1596, which was itself the start of a new era, but the end of this survey. In this journey, I am in particular examining the genealogy of the tables. In other words, I am trying to find out who copied on whom, and I am also trying to shed a new light on the computations that were made, whenever possible.

Finally, this survey is also a companion document to a number of modern reconstructions, that is, reconstructions usually giving the exact values, but also trying to reproduce the original layout of the tables, so as to make their comparison straightforward. These reconstructions are those of Regiomontanus's table of tangents (1490), 4 Engel's table of sines (1490, but here reproduced from the 1504 edition), 5 Peuerbach's arctangent table (1516), 6 the tables of sines of Fine (1530), 7 Apian (1533), 8 Regiomontanus (1541), 9 Rheticus (1542) 10 and again Fine (1550), 11 and eventually the trigonometric tables of Rheticus (1551), 12 Reinhold (1554), 13 Maurolico (1558), 14 Viète (1579), 15 Fincke (1583), 16 Lansberge (1591), 17 Rheticus & Otho (1596) 18 and [START_REF] Pitiscus | Thesaurus mathematicus sive canon sinuum ad radium 1. 00000. 00000. 00000. et ad dena quaeque scrupula secunda quadrantis : una cum sinibus primi et postremi gradus, ad eundem radium, et ad singula scrupula secunda quadrantis : adiunctis ubique differentiis primis et secundis; atque, ubi res tulit[END_REF]. chords with a circle of radius R = 3438 and at intervals of 7.5

• , 34 and that this radius was then copied by Indian mathematicians, but this is still debated, by Toomer himself, 35 as well as by Klintberg in 2005 who believes that Hipparchus may have had instead a chord table with R = 3600. 36 On the other hand, Duke, also in 2005, and using the analysis of two eclipse trios, concurs with Toomer's original suggestion.

37

Later, Ptolemy (2nd century AD) gathered the earlier works and covered the computation and use of chords in the first book of the Almagest. His table gives the chords for every 30 ′ of the quadrant, using a circle of diameter 120 (figure 2).

38

The way Ptolemy computed his table of chords was to find first the sides of the inscribed regular triangle, quadrilateral, pentagon, hexagon and decagon in a circle divided in 60 parts, that is, of radius 60. 39 This gave 34 [Toomer (1974), p. 7] See [van Brummelen (2009), pp. 41-45] for a recent discussion on this topic. 35 [Ptolemaeus (1984), p. 215] 36 [Klintberg (2005)] 37 [Duke (2005)] 38 See [Ptolemaeus (1813[Ptolemaeus ( -1816), v. 1, pp. 38-45]), v. 1, pp. 38-45], [Ptolemaeus (1898[Ptolemaeus ( -1903), v. 1, pp. 48-63]), v. 1, pp. 48-63],

[ Ptolemaeus (1984), pp. 57-60].

39 Besides Toomer's edition of the Almagest [Ptolemaeus (1984), pp. 57 -60], see [Neugebauer (1975), pp. 21-24], [Pedersen (2011), ch. 3], [Bond (1921), pp. 301-303], [Clagett (1957), pp. 200-205], [Kneale (1965)], [START_REF] Glowatzki | Die Sehnentafel des Klaudios Ptolemaios[END_REF]], [START_REF] Thurston | Hugh Thurston. Early astronomy[END_REF], pp. 235-him the chords of 36 • , 60

• , 72

• , 90

• , 108

• , 120

• , and 144

• . Using the theorem known as Ptolemy's theorem (a relation between the four sides and two diagonals of a cyclic quadrilateral), Ptolemy was able to compute the chord of the difference of two arcs, when the chords of these arcs are known, and also the chord of their sum. He also was able to compute the chord of the half arc from that of the arc. Eventually, Ptolemy computed the chords of 0.75

• and of 1.5

• . Then Ptolemy used an interpolation to find the chord of 1

• :

Chd 1 • = 1 p 2 ′ 50
′′ This means that the chord of 1

• is a bit more than one part, given that the radius is equal to 60 parts. Of course, Chd 180

• = 2R = 120. The above value for Chd 1

• is correct, since we actually have Chd 1 • = 2 ⋅ 60 ⋅ sin 0.5 • = 1.047184 . . . ≈ 1 + 2/60 + 50/60 2 . This value will also be written 1; 2, 50, following a convention used by many authors.

40

After the computation of Chd 1

• , Ptolemy obtained Chd 0.5

• and eventually all the other values in his table of chords. Glowatzki and Göttsche recomputed Ptolemy's table using the procedure he described in the Almagest.

41

The beginning of Ptolemy's table of chords as given by Halma is shown in figure 2.

In the following excerpt of Ptolemy's table 236], [van Brummelen (2009), pp. 70-77], [START_REF] Buscherini | Stefano Buscherini and Antonio Clemente Domenico Panaino. The table of chords and Greek trigonometry[END_REF]], [Otero (2020)] and especially [van Brummelen (1993), pp. 46-73] for an extensive analysis of Ptolemy's chord table and its underlying mathematics. 40 Throughout this document, I will count decimal places beyond this radius, and not including it, so that the value of Chd 1

• given here will be considered given to two (sexagesimal) places and not three. 41 See [START_REF] Glowatzki | Die Sehnentafel des Klaudios Ptolemaios[END_REF]]. Glowatzki and Göttsche give the listings of the PL/I programs they used.

the values of Chd 0

• 30 ′ , Chd 1

• and Chd 1

• 30 ′ are given, together with differences. These differences are given in thirtieth of the actual differences, so that 31 ′ 25 ′′ becomes 31 ′ 25 ′′ 30 = 2×31 ′ 25 ′′

60

= 62 ′ 50 ′′ 60 = 62 ′′ 50 ′′′ = 1 ′ 2 ′′ 50 ′′′ . In Greek (right hand side), letters are used for numbers, in particular ο for 0, α for 1, β for 2, ι for 10, κ for 20, λ for 30, ν for 50, ιε for 15, κε for 25, λα for 31, λδ for 34, etc. Note however that Halma uses ς ′′ for 30 ′ , when actually Ptolemy used a symbol for the half degree.

42

As the chords are twice the sines of the half angles, Ptolemy's table would make it very easy to obtain the sines at intervals of 15 ′ .

Indian tables

The history of mathematics in India is complex and a lot of details are shady or lost. 43 As far as trigonometry is concerned, some elements of Greek chord tables were probably taken to India, but they were then converted to sines.

44 It seems that it was for practical reasons that Indian astronomers replaced the chords (jyā) by the sines, that is by half-chords (jyā-ardha, eventually shortened to jyā), with various values of the radius R of the base circle.

45

This move from chords to sines may seem to be a detail, but it had in fact far-reaching consequences, connecting trigonometric functions with right triangles and therefore to the Pythagorean theorem.

However, even though a transmission from Greece to India is compelling, there is no certainty about the origins of the calculations, and whether the values were borrowed from Greek sources or computed independently.

46

In any case, once the sine (jyā) had been defined for radius R, we had jyā(θ) = R sin θ = Sin θ.

Among the oldest sine tables, Neugebauer and Pingree mention the Paitāmahasiddhānta, possibly of the 1st century AD, which had a table based 42 [Ptolemaeus (1898-1903), v. 1, p. 48] 43 For summaries of the history of mathematics and astronomy in India and extensive discussions on trigonometry or tables, see [Srinivasiengar (1967)], [START_REF] Pingree | [END_REF]], [START_REF] Bag | Mathematics in ancient and medieval India[END_REF]], [START_REF] Katz | [END_REF]], [START_REF] Plofker | [END_REF]], [González-Velasco (2011), pp. 25-34], [van Brummelen (2009), pp. 94-134], [Puttaswamy (2012), pp. 108-116], [Divakaran (2018)], [START_REF] Montelle | [END_REF]] (especially page 57) and [START_REF] Ramasubramanian | [END_REF]].

44 [van Brummelen (2009), p. 99] 45 [van Brummelen (2009), p. 96] 46 [van Brummelen (2009), p. 99] Figure 2: The beginning of Ptolemy's table of chords as retranscribed by Halma in 1813[Ptolemaeus (1813-1816)]. The part on the right shows the numerical values as represented by Greek letters. The left part is the modern translation. One should take note that the layout of the table in the Greek manuscripts differs from that displayed here and half-degrees are marked with a special symbol [Ptolemaeus (1898[Ptolemaeus ( -1903), v. 1, p. 48]), v. 1, p. 48].

on R = 3438.

47

As mentioned above, Toomer has suggested that this radius 3438, which is 60 ⋅ 360 divided by an approximation of 2π, was actually borrowed from Hipparchus, 48 but this claim may now be questioned. In any case, the simultaneous choice in India of a radius of 3438 and a measure of the circumference of 360 ⋅ 60 ′ means that the radius was actually measured in the same units as the circumference, thus anticipating the concept of radians.

49

The S ūrya Siddhānta, a Sanskrit treatise on Indian astronomy, which in its original version goes back to the 4th century, may also have been one of the earliest texts giving a table of sine. The version now known of this work which had been heavily amended gives a table of sines with the same R = 3438 and for every multiple of 3

• 45 ′ . 50 This interval of 3

• 45 ′ may go back to an interval of 7

• 30 ′ for chords.

51

Around 499 AD, Āryabhat . a's Āryabhat . īya also used R = 3438 and had tables of sines (Sin x) and versines (or versed sines, utkramajyā, R -Cos x) for every x multiple of 3

• 45 ′ .

52

Then in the sixth century AD, Varāhamihira (c505-c587) gave a table of sines with R = 120 again for every multiple of 3

• 45 ′ . 53 Neugebauer and Pingree write that this table uses a terminology derived from that of the Paitāmahasiddhānta mentioned above. In any case, Varāhamihira's table may be much older than the 6th century as his work, the Pañca-siddhāntikā is a summary of five earlier siddhāntas. And since, as observed by Bag, 54 we have the chord of 7

• 30 ′ in a circle of radius R = 60 which is equal to the sine of 3

• 45 ′ in a circle of radius R = 120, and that therefore a table of chords can right away become a table of sines in a circle twice as large, it may be that Varāhamihira's table goes back to a table of chords with R = 60.

In his Brāhma-sphut . a-siddhānta, Brahmagupta (c598-668) 55 computed a table of sines but with the radius 3270 for every multiple of 3

• 45 ′ . 56 Gupta suggested that this peculiar value of R is rounded from 21600/6.6, where 6.6/2 is an approximation of 10.

57

Bag 58 gave a comparative view of the main early Indian tables of sines and examined how Varāhamihira and others may have computed their values.

Brahmagupta has also used the value R = 150 in his Khan . d . akhādyaka (665). 59 This value was then used again by Al-Khwārizmī.

Arabic tables

I merely sketch here the main milestones in the developement of trigonometric tables between the 8th and 13th centuries, before their wider transmission to Western Europe. [START_REF]being the value of R, and n being the number of additional sexagesimal places[END_REF] At the end of the 8th century, during the first years of the Abbasid Caliphate (750-1258), men of learning were gathered in Baghdad and they translated into Arabic the works of the Hindus and the Greeks. 61 In particular, excerpts of Brahmagupta's Brāhma-sphut . a-siddhānta were brought to the calif Al-Mansur (714-775) by a scholar named Ka ṅka 62 and a translation was made.

The Persian mathematician and astronomer Al-Khwārizmī (c780-850) wrote a revised edition of this translation, the Zīj al-Sindhind. 63 The word zīj is a generic name used for tabular astronomical works in Arabic and Persian, and it is derived from a Persian word meaning "cord" or "string", the tables with their columns and lines bearing some similarity with strings.

64

The Indian word jyā for the chord was translated to jib and was later probably incorrectly translated in the Latin sinus, based on the similar 56 [Plofker (2009), p. 81, 157] 57 [Gupta (1978)] 58 [START_REF] Bag | Sine table in ancient India[END_REF]] 59 See [Chatterjee (1970), p. 206], [Gupta (1978)], [Pingree (1996), p. 43], and[START_REF] Pingree | [END_REF]]. 60 For more extensive descriptions of Arabic mathematics and astronomy, see in particular the surveys of [von Braunmühl (1900[von Braunmühl ( , 1903), v. 1, pp. 42-86] ), v. 1, pp. 42-86] and [START_REF] Rashed | Encyclopedia of the history of Arabic science[END_REF]]. Heydari-Malayeri's short survey may also be of interest [Heydari-Malayeri (2007)]. On trigonometric tables in the Islamic world, see [Berggren (1986), p. 144] and[van Brummelen (2009), pp. 135-222].

61 [Bond (1921), p. 307] 62 [START_REF] Bag | Sine table in ancient India[END_REF], p. 84] 63 See [Kennedy (1956), pp. 148-154], [Pingree (1996), p. 41] and[van Dalen (1996)]. 64 [Heydari-Malayeri (2007)] unvocalized Arabic word jaib meaning "cavity".

65

For the Zīj al-Sindhind Al-Khwārizmī computed c820 a table of sines. In fact, according to McCarthy and Byrne, Al-Khwārizmī's treatise contained two sine tables. 66 The main table used the radius R = 60 and a step of 1 following the Hindu custom. 76 The shadows seem to have been viewed as apart from the cosines and they were gathered in the same category only by the Europeans in the 15th century.

77

Al-Khwārizmī's Zīj al-Sindhind was brought to Al-Andalus, the Muslimruled area of the Iberian Peninsula, sometime between 821 and 852, that is only a short time after its conception. The Umayyad dynasty, after their replacement by the Abbasid dynasty in 750, had reestablished itself there, first as an emirate, then as a caliphate.

During the 9th century, Ptolemy's Almagest was also translated in 65 See [Folkerts (2006), pp. 75-76], [Goldstein (2019), p. 132] and [Filliozat (1988) [START_REF] Hogendijk | [END_REF]] 73 [McCarthy and Byrne (2003), pp. 264-265] 74 [Suter (1914), tab. 58 and 58a] 75 [Suter (1914), tab. 60] 76 [START_REF] Bond | [END_REF], p. 307] 77 [Bond (1921), p. 308]. See also [START_REF] Moussa | [END_REF]] who considers the process by which the tangent and cotangent functions became more abstract, especially with Ab ū al-Wafā c .

Arabic, so that his table of chords was then also known in Arabic.

78

Around the year 860 the Iranian astronomer al-Marwazi (al-Hasib) (766-after 869) borrowed Ptolemy's table of chords and gave the sines for every 15 minutes. 79 He also constructed the first systematic [START_REF] Glowatzki | Die Sehnentafel des Klaudios Ptolemaios[END_REF], pp. 12-13] and [Folkerts (2006), p. 76].

79 [Debarnot (1996), p. 524] 80 [Joseph (2011), p. 497] 81 [Debarnot (1996), p. 512]. These "three places" probably include the radius. 82 [Debarnot (1996), p. 509] 83 See [Cullen (1982)], [Gupta (1987), p. 241], [START_REF] Qu | Qu Anjing. Revisiting an eighth-century Chinese table of tangents[END_REF]] and [Divakaran (2018), p. 209]. A recent summary of Indian and Islamic trigonometry in China is given in [van Brummelen (2021), pp. 185-191]. 84 See [Al-Battāni (1899-1907)] and [Kennedy (1956), pp. 154-156]. 85 [Al-Battāni (1899-1907), vol 2, pp. 55-56] 86 [Al-Battāni (1899-1907), vol 2, p. 60] 87 [Debarnot (1996), p. 524] 88 This is what Folkerts writes [Folkerts (2006), p. 76], but it may mean four places including the integer part, which would then mean three sexagesimal places with our conventions. [Debarnot (1996), p. 527] 90 [Bond (1921), p. 311] 91 [START_REF] Joseph | [END_REF], p. 497] 92 [Debarnot (1996), p. 524] 93 See [Berggren (1986), p. 150], [Berggren (2016), p. 181] and [King (1975), p. 43]. 94 See [Debarnot (1996), p. 525] and [Schoy (1923), pp. 382-383].

95 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. 9] 96 Note however that [Debarnot (1996), p. 524] misleadingly states that Ibn Y ūnus gave his sines only every 10 ′ . 97 [Schoy (1923), p. 394] approximation (the correct value is Sin 28

• 10 ′ = 28; 19, 20, 12, 0), but the values for Sin 28

• 1 ′ , Sin 28 Al-Zarqālī is often credited as the author of these tables, but this is not sure and he may also not be the author of their canons.

106

The original Arabic Toledan tables are no longer extant, but they are known through many Latin editions from the 12th century onward and they had an important influence on Western European astronomy. 107 The tables may have been organized (rather than translated) in Latin by Gerard of Cremona who died in 1187.

108

The Toledan tables contained a sine table with R = 150 (from 1

• to 180

• for every degree and with two sexagesimal places) 109 (see figures 3 and4 98 [Schoy (1923), p. 386] 99 [Schoy (1923), p. 396] 100 [START_REF] Moya | [END_REF]] 101 [START_REF] Lutz Richter-Bernburg | the Toledan Tables, and Andalusi science[END_REF]] 102 See [Pingree (1996), p. 46], [Chabás Bergón (2019), pp. 47-75], and [START_REF] Moya | [END_REF]]. 103 [Suter (1914)] 104 [Toomer (1968)] 105 [van Brummelen (2018), p. 547] 106 See [Busard (1971a), p. 74]. The canons of the tables were published by [START_REF] Curtze | [END_REF][Curtze (1900), p. 337].

107 [Zinner (1936), Toomer (1968), Pedersen (2002)] 108 [Zinner (1936), p. 747] 109 See [Zinner (1936), table 25, p. 749], [Toomer (1968), table 12, pp. 27-28], [Pedersen (2002), pp. 946-952], [Millás Vallicrosa (1950), pp. 62-63] and [START_REF] Kennedy | [END_REF] And finally, let's mention that at the beginning of the 15th century, the Persian mathematician Al-Kāshī (c1380-1429) was able to obtain 2,49,43,11,14,44,16,19,16 (correct value: Sin 1 2, 49, 43, 11, 14, 44, 16, 26, 18, . . .) by solving numerically the equation sin 3x = 3 sin x -4 sin 3 x for x = 1

Sin 1 • = 1;
• = 1;
• . 125 Hence, tables of sines and of chords were available to those willing to pick them up. Several relatively independent works appeared in the following centuries, of which a few can be mentioned. For instance, in 1220, Leonardo of Pisa (c1170-c1250), known as Fibonacci, published his Practica Geometriae where he gave a table of chords with a radius of 21 perticae and a circumference of 132 perticae (figure 9). 126 The pertica is a Roman length unit equal to 10 Roman feet or about 2.96 m. The ratio 132/21 corresponds to the approximation 22/7 for π. In Leonardo of Pisa's table, the arcs are measured with the circumference (first column), so that 90 degrees correspond to 33 perticae. In that case, the chord should have been 21 2 = 29.69 . . . but it is given as 29. For 180 degrees (66 perticae), the chord is 42, corresponding to twice the radius.

In that same century, Campanus of Novara (c1220-1296) also supposedly constructed a table of tangents for each degree.

127

In the 14th century, we should also note the work of Levi Ben Gershon (Gersonides) (1288-1344) who in 1342 independently constructed a table of sines for intervals of 15

′ with a radius R = 60 and two sexagesimal places.

128 And in the fist quarter of the 15th century, Jean Fusoris (c1365-1436) has independently recomputed tables of sines and chords, also at intervals of 15 ′ , with a radius R = 60 and with three to six sexagesimal places.

129

But the real starting point of new trigonometric computations in Europe were the investigations of Johannes von Gmunden and Giovanni Bianchini, Figure 9: Leonardo of Pisa's table of chords (1220) [Boncompagni (1862), p. 96].

which seem to have taken place independently at about the same time.

Johannes von Gmunden (c1384-1442)

Johannes von Gmunden (c1384-1442) founded the study of astronomy and trigonometry in Vienna in the early 1400s. 130 He had obtained his Master degree at the University in 1406. Johannes von Gmunden gave lectures on the construction of astronomical instruments and computed astronomical tables. 131 A few years before his death, he bequeathed his books to the University and thereby founded its first library. 132 Because very few of Johannes von Gmunden's works have been printed, he has been overshadowed by Georg Peuerbach and Regiomontanus.

133

In 1437, he wrote a treatise De sinibus, chordis et arcubus. 134 He described the computation of sines using the Arabic (in fact Indian) methods with the sines of multiples of 15 degrees, as well as the computation of chords using the methods given by Ptolemy in the Almagest. In particular, he described the computation of the sine of the half-angle α/2, as well as of the complementary angle 90

• -α, from the sine of α. The formulas are given without proof, like in the canons of the Toledan tables and in John of Lignères's canons. 135 This enabled von Gmunden to compute the sines for every multiple of 3

• 45 ′ for R = 150 and R = 60. Johannes von Gmunden's treatise is accompanied by several tables which, according to Glowatzki and Göttsche, were only computed in 1437 or later.

136

In the first part, a table of sines with R = 150 is given for each degree, and for minutes and seconds of the unit of the sinus totus. It is attributed 130 For summaries of Johannes von Gmunden's life and works, see [von Khauz (1755), pp. 27-32], [Aschbach (1865), pp. 455-467], [Klug (1943)], [Vogel (1973a)], [Grössing (1983), pp. 73-78], [START_REF] Gertrude | Johannes von Gmunden -der Astronom[END_REF]], [START_REF] Kaiser | [END_REF]], [START_REF] Shank | [END_REF]], [START_REF] Grössing | [END_REF]], [Folkerts (2006)], [van Brummelen (2009), pp. 248-249], and [Simek and Klein (2012)]. For a survey of his tables, see [Porres de Mateo (2003)] and [Chabás Bergón (2019), pp. 321-336]. See also [Durand (1952), pp. 54-56], and [Duhem (1959), pp. 349-367], especially for the scientific context in Vienna. Gessner mentions von Gmunden very briefly [Gessner and Simmler (1574), p. 375].

131 [Schmeidler (1977), p. 315] 132 [von Khauz (1755), p. 29] 133 [Sperl (1971a)] 134 This treatise was published in [Busard (1971a)]. See also [Kaiser (1988), pp. 91-96] and [Folkerts (2006), p. 71]. 135 See [Busard (1971a), p. 78] and [Folkerts (2006), p. 81].

136 [Glowatzki and Göttsche (1990), pp. 79-92] to Al-Zarqālī and must come from the Toledan tables. I assume it does not originate in John of Lignères's canons as these canons only give the sines every 15 degrees for that sinus totus.

137

Another table of sines, attributed to Ptolemy, with a sinus totus of 60, is also given for each degree. This table may be the Toledan table restricted to degrees, or it may be the table borrowed from Ptolemy's Almagest, but restricted to degrees. The two original tables are reproduced by Glowatzki and Göttsche. 138 In the second part, Johannes von Gmunden gives tables of chords and sines for every half-degree from 0

• to 180

• , with a radius (sinus totus) of 60.

The two original tables are also reproduced by Glowatzki and Göttsche. 151 [Folkerts (2006), p. 87] 152 [Busard (1971a), p. 76] 153 [START_REF] Pitiscus | Thesaurus mathematicus sive canon sinuum ad radium 1. 00000. 00000. 00000. et ad dena quaeque scrupula secunda quadrantis : una cum sinibus primi et postremi gradus, ad eundem radium, et ad singula scrupula secunda quadrantis : adiunctis ubique differentiis primis et secundis; atque, ubi res tulit[END_REF] 154 See [Barotti (1792), vol. 1, p. 119-132], [Birkenmajer (1911)], [START_REF] Vescovini | [END_REF]], [START_REF] Goldstein | [END_REF]] and [Chabás Bergón and Goldstein (2009), p. 13].

For a survey of Bianchini's tables, see [van Brummelen (2018)] and [Chabás Bergón (2019), pp. 337-364]. See also [Gruyer (1897), v. 2, pp. 428-430] for some background on his astronomical tables. Gessner mentions Bianchini very briefly [Gessner and Simmler (1574), p. 346].

155 [Rosi ńska (1994a), Rosi ńska (1997-1998)] 156 See [von Murr (1786), vol. 1, p. 74-205], [START_REF] Curtze | Urkunden zur Geschichte der Mathematik im Mittelalter und der Renaissance[END_REF]] and [Gerl (1989)]. 157 See [Rosi ńska (1981a)], [Rosi ńska (1981b)], [Rosi ńska (1987)], and [START_REF] Rosi Ńska | Mathematics for astronomy" at Universities in Copernicus' time: modern attitudes toward ancient problems[END_REF]].

A more complete summary of Bianchini's trigonometric tables was recently given by Chabás [START_REF] Bergón | José Chabás Bergón. An analysis of the Tabulae magistrales by Giovanni Bianchini[END_REF]].

158 [START_REF] Boffito | Le tavole astronomiche di Giovanni Bianchini (Da un codice della Coll. Olschki)[END_REF]] 159 [Birkenmajer (1911) Bianchini also computed decimal tables, that is tables not involving 60 and only based on powers of 10. These tables are found in the set of eight trigonometric tables named Tabulae magistrales.

167 Some of these tables give the values of trigonometric functions multiplied by certain astronomical factors (for instance the cosine of the obliquity of the ecliptic), but two of the tables are decimal tables (R = 10000) for the tangent and cosecant.

168 Among this set of tables, the Tabula magistralis quarta 169 gives the tangents at 10 ′ intervals and with R = 10 4 .

This table may have been the incentive for Regiomontanus to construct his own table of tangents in his Tabulae directionum profectionumque, 170 for every degree and with R = 10 5 (figure 15). He did not, however, use

Bianchini's values, but computed his tangents using his large sexagesimal 161 [Glowatzki and Göttsche (1990), pp. 95-114] 162 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF] [Toomer (1968), table 15, p. 32] 166 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. 105] 167 See [Rosi ńska (1984), pp. 476-477] and [Chabás Bergón (2019), p. 349]. 168 In 1981, [Rosi ńska (1981a)] wrote mistakenly that the tangents are given with R = 10 3 .

This error was repeated by Rosi ńska in 1987 [Rosi ńska (1987)] and 2002 [START_REF] Żyna | Przełom w trygonometrii połowy XV wieku: Kopernik jako spadkobierca i jako kontynuator tego przełomu[END_REF], p. 12], by Chabás and Goldstein [Chabás Bergón and Goldstein (2009), p. 20] and Brummelen in 2009[van Brummelen (2009) [Walsh (1996), pp. 289-291] 174 For summaries of Peuerbach's life and works, see in particular [Gassendi (1654), pp. 335-373], [von Khauz (1755), pp. 33-57], [Montucla (1758), v. 1, pp. 443-445], [Martin (1764), pp. 157-158], [Aschbach (1865), pp. 479-493], [Gallois (1890a), pp. 1-11], [Thorndike (1929), ch. 8], [Sperl (1971b)], [Vogel (1973b)], [START_REF] Rose | [END_REF]], [Hellman and Swerdlow (1978)], [Grössing (1983), pp. 79-116], [START_REF] Shank | [END_REF]], [START_REF] Samhaber | Die Zeitzither -Georg Peuerbach und das helle Mittelalter[END_REF]], [START_REF] Grössing | [END_REF]], [START_REF] Kaunzner | Über Schriften Georgs von Peuerbach mit einem mathematischen Hintergrund[END_REF]], [van Brummelen (2009), pp. 249-252], [Malpangotto (2020), pp. 19-34], and [Horst (2019)]. Several other references not cited here are given in the Geschichtsquellen des deutschen Mittelalters (https://www.geschichtsquellen.de/autor/749). One of the first biographical notices on Peuerbach was that of Tannstetter, when he published Peuerbach's table of eclipses [START_REF] Peuerbach | Tabulae eclypsium magistri Georgij Peurbachij, Tabula primi mobilis Joannis de Monteregio[END_REF]]. Gessner, on the other hand, only briefly mentions Peuerbach [Gessner and Simmler (1574), p. 231]. For the scientific context in Vienna, see [Durand (1952)] and [Duhem (1959), pp. 349-367]. 175 See [Hellman and Swerdlow (1978), p. 473], [START_REF] Grössing | Helmuth Grössing[END_REF], p. 80] and [Malpangotto (2020), p. 24].

176 See [von Khauz (1755), p. 38] and [Hellman and Swerdlow (1978), p. 473].

died in 1442. He can thus be considered as Gmunden's spiritual student. It seems unlikely that he knew him, but he certainly studied his works.

177

In 1454, after his return from Italy, Peuerbach completed a Theoricae Novae Planetarum which actually started as lectures on the theory of planetary motion. This work was published in 1473

178 by Regiomontanus (1436-1476),

Peuerbach's student and successor, who had certainly attended these lectures. 179 The Theoricae Novae Planetarum became a standard textbook of planetary theory for the next century. [START_REF] Shank | [END_REF], p. 124] 180 [Schmeidler (1977), p. 315] 181 [Hellman and Swerdlow (1978), p. 475] 182 [Malpangotto (2020), p. 33] 183 [START_REF] Glowatzki | Die Sehnentafel des Klaudios Ptolemaios[END_REF], p. 16] 184 Meskens writes that Bessarion had started the translation, but doesn't give any substantial proof of this statement [Meskens (2010), p. 136]. Meskens's statement may rest on [START_REF] Glowatzki | Die Sehnentafel des Klaudios Ptolemaios[END_REF], p. 17]. 185 See [Malpangotto (2020), p. 20] and [START_REF] Shank | [END_REF] 10). Peuerbach wrote that he used his now lost table with R = 600000 for the computation of the table. The possible values of the tangents range and [Zinner (1968), p. 36], [Zinner (1990), p. 23] about the radius of the table .  198 [Regiomontanus (1490)] 199 See [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF] 203 This incorrect statement is also found in [Martin (1764), p. 158] (and in [Lublink and Meijer (1763), pp. 183-198] which must have the same source), and it was more recently repeated by [Bendefy (1980), p. 245]. 204 [von Peuerbach (1516)]. See [Roegel (2021a)] for a modern reconstruction.

from 0 to 1200 and, for an entry x, Peuerbach's He established trigonometry as an independant field, separate from astronomy, in Western Europe, although the Persian al-T ūsī had already written a purely trigonometric treatise in the 13th century. Regiomontanus was the most famous Western mathematician of his time. After having studied in Leipzig, he came to Vienna around 1450 and became a friend and pupil of Georg von Peuerbach. In 1457, this is where he took his Master's degree and was appointed to the faculty, hence a colleague of Peuerbach.

211

Peuerbach was supposed to go to Italy with Cardinal Bessarion who had asked him to write an Epitome (summary) of Ptolemy's Almagest. But after Peuerbach's death in 1461, it was Regiomontanus who accompanied him to Italy. 212 Regiomontanus completed the Epitome, probably in 1462.

213

He also studied Greek and it was during the time of the completion of the Epitome that Regiomontanus studied the copy he had made of Trebizond's translation of the Almagest. [von Braunmühl (1900[von Braunmühl ( , 1903), v. 1, pp. 124-133] ), v. 1, pp. 124-133] 252 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. i] 253 [START_REF] Thorndike | Science and thought in the fifteenth century[END_REF] I also include in this section some tables which are not directly from Regiomontanus, for instance the tables of secants, but which are nevertheless based on Regiomontanus's other tables.

The following tables by Regiomontanus have been reconstructed in separate documents:

• the table of tangents, as published in 1490 (figure 15) 254 ;

• the table of sines with R = 6 ⋅ 10 6 , as published in 1541 (figure 23) 255 ;

• the table of sines with R = 10 7 , as published in 1541 (figure 24). 256 [Glowatzki and Göttsche (1990), pp. 72-125].

Fundamental tables

259 [Glowatzki and Göttsche (1990), pp. 10, 16, 22] which a more complete [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. 23] 262 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. 23] 263 [van Brummelen (2009), p. 263] gives Regiomontanus's implied value of Sin 1

• , but does not describe the actual interpolation process. See also [van Brummelen (2021), pp. 18-21], who hints at a procedure below 15

′ but without detailing it. Kästner gives also only a cursory description [Kästner (1796), pp. 540-560]. 264 See [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. 71] and [Mett (1996), p. 65]. 265 [von Peuerbach and Regiomontanus (1541)] 266 [START_REF] Regiomontanus | De triangulis planis et sphaericis libri quinque[END_REF]] 267 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], pp. 28-47]

Sine table with R = 60000

The Tabulae directionum profectionumque published in 1490 contains a 30 pages long sine table with R = 60000 giving the sines for every minute (figure 16),277 but this table was certainly computed by Johannes Engel for that edition (see § 6.1), and not by Regiomontanus. And the first table of secants with an interval of 1 ′ and R = 10 7 was published by Fincke in 1583. 295 Fincke was actually the one who named it secant. His secants were certainly computed from his tangents, which themselves go back to Regiomontanus, via Reinhold.

Table of tangents

296 293 [Glowatzki and Göttsche (1990), p. 193] Incidentally, there have also been surprising statements, such as the one of [START_REF] Davis | [END_REF], p. 21] who wrote that the first table of secants was that of Maurolico, and that Lansberge was wrong in ascribing this fact to Rheticus, when in fact Lansberge was right, and still is if one ignores manuscript tables.

294 [Viète (1579)] 295 [Fincke (1583)] 296 See [Reinhold (1554)] and [Glowatzki and Göttsche (1990), p. 193].

An analysis of Regiomontanus's great tables

One of my purposes has been to find out how Regiomontanus computed his two large tables of sines. We know rather well how he computed the sines at intervals of 45 ′ , but we know little beyond that, and no one seems to have investigated this matter so far, not even Glowatzki and Göttsche.

297

The first step in such an investigation is to clear the tables of the noise they contain, namely of the typos, both in the printed versions and in the manuscripts. Although I have not consulted manuscripts of Regiomontanus's table, I believe that it is possible to come very close to what Regiomontanus has actually computed.

Typos, accuracy and statistics

General principles

I have gone over each of the 2 × 5401 values of the sines (from 0

• to 90

• by steps of 1 ′ ) in the 1541 printing, trying to detect obvious typos. This work has been done independently of that of Glowatzki and Göttsche who had already reported a number of typos.

298 I have consequently made two tables where I corrected a number of typos, such as wrong digits in the left figures, swapped figures, or swapped lines. In the resulting tables, I carefully examined all the cases where Regiomontanus's tables were in error by more than 2 units of the last place. Every such case which appeared isolated was removed. The justification for correcting these seemingly small errors was that they would have been very easy to detect by computing differences between consecutive terms, and that almost always these anomalies were isolated, and could not have been Regiomontanus's real values, at least not his intended values. These decisions may be objectionable, but I have only corrected errors which are easy to detect by anybody working on tables. I did not correct any more fundamental issue. And these corrections are necessary in order to get a better understanding of the underlying computations.

Corrections to the tables

Apart from the very conspicuous typos already reported by Glowatzki and Göttsche (mostly not repeated here), I made the following smaller Note that my corrections do not always replace the 1541 printed values by the exact ones, but by the values I believe should have been printed. For instance, for 6

• 38 ′ , Glowatzki and Göttsche replaced 693009 by 693090, which does make sense as a typo. However, the value 693090 does not make much sense in its context (i.e., the surrounding values) and I believe that there was an error before that, and that Regiomontanus should have obtained 693088, which is the value I gave in my table. In this case, I believe that Regiomontanus accidently obtained the correct value 693090, and that the printer got it wrong by setting 693009.

The only values with a deviation of 3 units of the last place are those from 6

• 50 ′ to 6

• 53 ′ . I believe that the pivot 6

• 50 ′ was erroneously computed and should probably have been 713889 (with an error of 1). This has probably caused the sines of 6

• 51 ′ to 6

• 53 ′ to be also wrong by 3 units. I have however not fixed these deviations and these errors remain in the cleaned table, as they are not mere typos. But the truth is that these errors would not escape a close scrutiny by differencing.

The above table also contains corrections for a number of deviations of 2 units, when these were clearly isolated (1

• 29 ′ , 1 • 44 ′ , 6 • 38 ′ , 17 • 33 ′ , 50 • 59 ′ , 42 • 53 ′ , 45 • 42 ′ , 49 • 34 ′ , 51 • 35 ′ , 57 • 21 ′ , 70 • 2 ′ , 73 • 50 ′ , 77 • 17 ′ , 81 • 10 ′ , 81 • 29 ′ , 82
• 46 ′ ). I also corrected some suspicious transitions, where the error switched from 1 to -1 or from -1 to 1. These errors would have been very easy to detect by differencing and concern 5

• 1 ′ , 7 Among these corrections, the small ones for 37

• 42 ′ and 38

• 53 ′ have been made because their deviations appeared to be isolated. And like in the sexagesimal table, I have also corrected some suspicious transitions, where the error switched from 1 to -1 or from -1 to 1. These errors concern the values 4

• 19 ′ , 13

• 14 ′ , 60

• 17 ′ , 61

• 15 ′ , and 61

• 50 ′ . Some of these corrections may appear larger than these small transitions, but that may be because there may have been both printer typos and earlier errors, and that I first corrected the large errors, for instance for 13

• 14 ′ whose sine value in Regiomontanus's manuscript may have been 2289173, but which still can't have been the right one.

Some of these typos/errors were reported by Glowatzki and Göttsche, but not all of them, and, as I explained above, Glowatzki and Göttsche reported other errors which I have corrected, but not included in the above tables.

299 Moreover, my corrections do not always coincide with theirs, as I have tried to replace the incorrectly printed values by those that Regiomontanus has presumably computed, and not by the exact sines. I believe however that all the typos reported by Glowatzki and Göttsche have been taken care of in my versions.

Eventually, we end up with two tables which must be very close to Regiomontanus's calculations, and which have been cleared of probably almost all typos, both in the printed versions and in the manuscripts. What I mean by this is that Regiomontanus would have found all these errors by mere differencing and that the resulting cleaned tables provide a better start for the analysis of Regiomontanus's actual computations.

These tables are provided separately300 as text files for others to analyze, should they wish to.

The sexagesimal pivots

The cleaned tables now make it possible to have a closer look at computational errors and in particular at the accuracy of the pivots. It first appears that the sexagesimal table contains about 2223 errors of one unit or more, and none of more than 3 units. This does agree with the count given by Glowatzki and Göttsche who came up with 2232, but with slightly different corrections. I am of course writing "about," because in some cases I made adjustments which may or may not be correct. As far as the other pivots are concerned, two 15 ′ pivots are off by 2 and 78 are off by 1. 345 5

′ pivots are off by 1, and 11 by 2 or 3.

The decimal pivots

The decimal table contains about 1841 errors or one unit or more. Again, this is very close to Glowatzki and Göttsche's count which is 1833, but with slightly different corrections. There are also three 45 ′ pivots which are incorrect, but not the same ones as for the sexagesimal table. 22 15 ′ pivots are off by 1, and none by 2. 282 5

′ pivots are off by 1, and one is off by 2. It does therefore appear that the decimal table is somewhat more accurate than the sexagesimal table, but not by an order of magnitude.

Some general statistics

We can also observe that in the sexagesimal table the longest sequence without errors is of length 52 and starting at 56

• 4 ′ : once the typos are corrected, all the sines from 56

• 4 ′ to 56

• 55 ′ are correct. The longest sequence with a constant error of one unit of the last place (in the same direction) is of length 30 and starts at 3

• 52 ′ . The longest sequence with a constant error of two units is of length 6.

Similar results are obtained with the decimal table and the longest sequence without errors is of length 50, starting at 27

• 45 ′ . The average errors are -0.10 for the sexagesimal table and 0.12 for the decimal table, but it is difficult to analyze errors in more depth without taking into account the structure of the computations, namely the two trisections and the possibly final linear interpolation.

We can now try to answer a number of questions on the computation of the pivots:

• For instance, assuming two 45 ′ pivots are correct, how often are the 15 ′ pivots correct?

The answer to this question is surprising, because there is a clear difference between the sexagesimal and decimal tables. In the first case, 70 15 ′ pivots (for 114 ranges out of 120) are incorrect, but in the second case only 17 are incorrect (also for 114 ranges • 30 ′ and 32

• 15 ′ are correct.

• And assuming two 15 ′ pivots are correct, how often are the 5 ′ pivots correct?

We find that for the sexagesimal table ,119 5 ′ pivots are incorrect (for 231 ranges out of 360) when the 15 ′ pivots are correct, and that there are 30 incorrect twin pivots.

For the decimal table ,217 5 ′ pivots are incorrect (for 318 ranges out of 360), including 52 twin pivots. Under this perspective, the sexagesimal table appears more accurate than the decimal table.

An example of an incorrect 5

′ pivot in the sexagesimal table is that of 75

• 5 ′ .

• Finally, how often are the 5 ′ interpolations correct?

Again, we restrict ourselves to the cases where the two pivots are correct, as such a restriction is still representative. 301 What is the most common outcome? Is it 0, 0, 0, 0, 0, 0? In other words, if two adjacent 5 ′ pivots are correct, are the four intermediate values also usually correct?

The number of ranges to consider (where the two 5

′ pivots are correct) is similar in both tables: 576 ranges for the sexagesimal table and 604 ranges for the decimal table. Is the outcome the same? First, given that the table has been checked by differences, the only values which can appear between the two end 0s are 0 and ±1. There are therefore 3 4 = 81 different possible sequences, but the most common sequence is (0, 0, 0, 0, 0) with 277 cases in the sexagesimal table and 258 cases in the decimal table. Again, under this perspective, the sexagesimal table is in fact slightly more accurate than the decimal one.

A tentative analysis of Regiomontanus's construction

The procedure used by Regiomontanus to construct his large tables is a bit vague, but I believe that it can be clarified. As far as I know, no attempt has been made so far to explain this process. As mentioned above, Regiomontanus basically describes a subtabulation process, where from sine values at 45 ′ intervals he obtains values for every 15 ′ , then for every 5 ′ , and finally for every minute. Regiomontanus explicitely speaks of making the differences increase regularly, and it should be clear that the differences between values played a key role in this computation. It is also clear that what Regiomontanus has done was to interpolate values, more than merely to compute accurately thousands of sines.

Reading Regiomontanus's description, one can not avoid thinking of the works of Bürgi 302 and Briggs 303 and wonder if, perhaps, Regiomontanus had not anticipated them. I believe in fact that his computations were indeed forerunners of what Bürgi and Briggs did, a century or a century and a half later. Both Bürgi and Briggs analyzed how finite differences could be used not merely to find new values by adding differences, but also to subtabulate, and find intermediate values from larger differences. For instance, Briggs computed the logarithms of various integers as interpolations of logarithms given at larger intervals. Among the techniques he describes is the quinquisection, where he is able to divide an interval in five parts and obtain the intermediate logarithms.

The general setting

Here, I want briefly to test this hypothesis, which may be expanded later in the future. To be as general as possible, I will consider a sequence of sines v 0 , v 1 , v 2 , . . . , for angles a 0 , a 1 , a 2 , . . . , where a i+1 -a i is a constant interval, for instance 45 ′ . v i = Sin(a i ), with some radius R, which I will take here as 6 ⋅ 10

8 , but which could be different.

These values are used to define the finite differences ∆

1 0 = v 1 -v 0 , ∆ 1 1 = v 2 -v 1 , . . . , ∆ 2 0 = ∆ 1 1 -∆ 1 0 , ∆ 2 1 = ∆ 1 2 -∆ 1 1 , . . . , ∆ 3 0 = ∆ 2 1 -∆ 2 0 , etc.
What Regiomontanus sought to do was to find the sines v 1/3 , v 2/3 , etc., of the intermediate angles a 1/3 , a 2/3 , a 4/3 , etc. In other words, he was working on a trisection. For instance, if a 0 = 3

• , a 1 = 3

• 45 ′ , etc., then a 1/3 = 3

• 15 ′ and

v 1/3 = Sin 3 • 15 ′ .
The subtabulated differences are δ

1 0 = v 1/3 -v 0 , δ 2 0 = δ 1 1/3 -δ 1 0
, and so on.

302 [Roegel (2016a)] 303 [Roegel (2010a)]

I believe that during the first stage of his procedure, Regiomontanus tried to compute the smaller differences, that is the differences for intervals of 15 ′ , from the differences for intervals of 45 ′ . In other words, I believe that he tried to compute δ 1 0 and δ 2 0 , and these two values would then be sufficient to compute v 1/3 and v 2/3 .

I invite those who are unconvinced by this suggestion to consider for instance the trisection of the 45 ′ interval between Sin 20

• and Sin 20

• 45 ′ . The radius could be taken as R = 10

5 , and the sines to start with would then be 34202, 35429, 36650, 37865, etc. Merely manipulating these numbers without great thought leads to two approximations of the subtabulated first differences, namely 410 and 407. We would have three 410 differences and three 407 differences. This is of course not satisfactorily, it is not an even decrease, and looking at the second differences, we find 0,0,-3,0,0. This can be improved by starting with the first subtabulated difference 410 and spreading the -3 over five values, hence taking -0.6 instead of -3 for the second difference. We have here a very simple means to obtain the second differences. Adding up these differences, we end up with 36653 instead of 36650. It is not perfect, but it is not that bad. Since the second difference was correctly spred, we may want to improve the first difference 410, but it will actually be difficult to reach a better result with this radius. Such experiments are useful to convince oneself that it is practically unfeasible to get the differences to vary evenly merely by fiddling with the numbers, and at the same time they pave the way for the discovery of a relationship between certain values. And these are the key issues here. The first key is to notice that the second differences are practically proportional to the sines. This had actually been discovered long before Regiomontanus, for instance in India by Āryabhat . a in the 6th century 304 And Wagner and Hunziker recently suggested 305 that there was perhaps a transmission from India to Bürgi, although I am rather doubtful about such an assertion. In the above simplified example, one would readily find that the second differences are all equal to -6, at least around 20

• , and if this operation is done for other values, one can't be far from discovering that the second differences are proportional to the sines.

I will therefore assume that Regiomontanus first noticed that ∆

2 0 ≈ v 1 C 45
where C 45 is some constant, and that this is true on the entire sine table. I am also guessing that Regiomontanus knew that the constant depends on 304 See [Hayashi (1997)], [Bressoud (2002)], [START_REF] Raju | [END_REF], p. 132], [Lefort (2007)]

and [Gupta (2008)] for some references (among many others) describing Āryabhat . a's computation of sines and how the second differences are used.

305 [START_REF] Wagner | [END_REF] the size of the interval, hence my subscript. For intervals of 45 using an approximation of v 1/3 . Of course, v 1/3 is what we are looking for, but we can easily get an approximation of v 1/3 such as

v 1/3 ≈ v 0 + v 1 -v 0 3
and this is in fact sufficient to get a good approximation of δ The second key here is to see or guess that the second differences are involved in the approximations of the first differences. In any case, one may want to test whether

δ 1 0 ≈ ∆ 1 0 3 + αδ 2 0
for some value of α. Although the constancy of δ 2 makes this actually obvious, 308 it is also possible to observe experimentally that α = -1, and thus that

δ 1 0 ≈ ∆ 1 0 3 -δ 2 0
Again, in the simplified example given above, where the first differences are 1227, 1221, and 1215, and where the second differences are all about -6, 306 See [Roegel (2010b), § 2.4] for the computation of the exact values of ∆ sin x. The value 5836 is actually about 1/ sin 2 ∆x, that is 1/ sin 2 45 ′ . 307 This constant is given by 1/ sin 2 15 ′ ≈ 52525. 308 The three first differences are then xδ 2 , x, and x + δ 2 , and the average first difference is necessarily the median first difference.

it should not be difficult to notice that the first first difference is equal to the average first difference minus the second difference, 37865-34202 3 + 6 = 1227, or that the second (middle) first difference is also the mean first difference.

If these two observations are made, namely 1) the link between the second differences and the sines, and 2) the dependency of the subtabulated first differences on the subtabulated second differences, then it is possible to derive the values v 1/3 and v 2/3 .

An example

Let me show how to put this in practice on a small example. Let's for instance interpolate the sines between 39

• and 39

∆ 1 0 = 383663401 -377592235 = 6071166 δ 1 0 ≈ 6071166 3 + 7227 = 2030949 δ 1 1 ≈ δ 1 0 + δ 2 0 = 2023722 v 1/3 = Sin 39 • + 2030949 = 379623184 v 2/3 = v 1/3 + 2023722 = 381646906
and in fact Regiomontanus's table with R = 6 ⋅ 10 6 does have the values 3796232 and 3816469 which would have been obtained from the above computation. This procedure does unfortunately not work on all 45 ′ intervals, and Regiomontanus's pivots sometimes differ from those obtained with this procedure, although the difference never exceeds one unit of the last place. This does not prove that Regiomontanus did not use such a procedure, but it may be that some computations were lacking uniformity, and also that some errors were introduced in the computations. I also believe that the two guard digits, viz. those added when computing with R = 6 ⋅ 10 8 , were used throughout the interpolation, and not merely for the pivotal values.

The same procedure used to obtain the sines at 15 ′ intervals can be used to obtain the sines at 5 ′ intervals. The only difference is that δ 2 0 involves a new constant, which may have been guessed or computed by Regiomontanus, namely

δ 2 0 ≈ - v 1/3 52525 × 9
If for instance we want to compute Sin 39

• 5 ′ , we find

δ 2 0 ≈ - 378269218 52525 × 9 = -800 δ 1 0 ≈ 2030949 3 + 800 = 677783 δ 1 1 ≈ 676983 v 1/3 ≈ 378270018 v 2/3 ≈ 378947001
and these two values v 1/3 and v 2/3 , when rounded to R = 6 ⋅ 10 6 , are exactly the values given by Regiomontanus for the sines of 39

• 5 ′ and 39

• 10 ′ . But again, I must stress that although this procedure works on this example, it does (slightly) fail to give Regiomontanus's values on others.

Anyway, if Regiomontanus proceeded along these lines, he now has obtained the sines for all multiples of 5 ′ , using relatively simple techniques. In fact, the computations involved here (except those for the pivots) are more a matter of being clever than of being hard working.

What now remains is to divide the 5 ′ intervals in five parts. This is what Briggs called a quinquisection.

The same procedure could be applied here as for the trisection, but we would have δ

2 0 ≈ - v 1/5 52525 × 9 × 25 and 310 δ 1 0 ≈ ∆ 1 0 5 -2δ 2 0
When applying this procedure (which is left as an exercise) to the interval from 39

• to 39 then we end up with exactly Regiomontanus's values. This does not prove that Regiomontanus did such an interpolation in every case, but it does at least make it plausible that he proceeded that way in some cases.

Conclusion

Looking at Regiomontanus's tables, it is pretty clear that he had the means to compute the 45 ′ pivots correctly. The 15 ′ and 5 ′ pivots are relatively accurate, but less than the 45 ′ pivots. In the previous section, I have given a procedure which may be close to the one used by Regiomontanus to find his pivots.

For the 15 ′ pivots, we have seen earlier that Regiomontanus's sexagesimal table has 70 pivot errors. Now, if we use my algorithm using finite differences, we end up with 42 errors on the all the 15 ′ pivots. However, if we compare my pivot values with those of Regiomontanus, there appears to be about 85 differences. Regiomontanus's values do not perfectly agree with those of my algorithm for the first trisection, although the differences do not exceed one unit of the last place. This agreement can not be significantly improved even with a different constant C 15 . It is still possible that Regiomontanus made use of an algorithm close to the one I sketched, but perhaps he did not always use two guard digits, in addition of having made a few computation errors here and there.

I also believe that the last step was a linear interpolation, but that glitches came into play and that the computations were not done totally uniformly and rigorously.

To sum up, and in the absence of other convincing theories, I believe that it is plausible that Regiomontanus applied two trisections, computed the first subtabulated first and second differences in each range, derived the missing values, and interpolated linearly in the 5 ′ intervals, perhaps using only one guard digit, and eventually rounding all values to R = 6 ⋅ 10 6 . The same procedure could have been applied with the decimal table. I believe that Regiomontanus's tables contain the germs of several innovations, and that it was the quality of workmanship underlying these tables which is the true reason why they endured so long. They did contain errors and typos, but they provided a solid foundation for others to build upon, and only Bürgi, Briggs and a few others were able to develop similar skills to renew the computation of tables. Mention should also be made of Bürgi's sexagesimal sine table from c1587, which seems to be a totally independent and very accurate recomputation of sines, paralleling to some extent Rheticus's efforts that led to the Opus palatinum (1596) and the Thesaurus mathematicus (1613).

To sum up, the main new computations based on Regiomontanus's values are the following, which are detailed in the subsequent sections: In the following sections, I go into more detail for each of these tables copied from Regiomontanus's tables, or based on them. This list tries to be as complete as possible, but it is possible that some lesser known work containing a sine table or a more complete canon still escaped my attention.

Engel (1490)

Johannes Engel (or Johannes Angelus) (1453-1512) was an astronomer and astrologer from Aichach, near Augsburg. He published many almanachs and astronomical tables. [Fincke (1583)] 341 On Johannes Engel, see [Knobloch (1983)] and [START_REF] Dobrzycki | [END_REF]]. He is also mentioned by Gessner [Gessner and Simmler (1574), p. 336].

342 [Regiomontanus (1490)] 343 Not all editions seem to contain this sine table, and it is for instance absent from the copy at ULB Darmstadt (Inc II 357).

344 [Folkerts (1977), p. 234] 345 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF] 346 [Glowatzki and Göttsche (1990), pp. 65-71] 347 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF] Engel's table was used by Gaurico in 1524 (see below, § 6.2), in 1569 by Schreckenfuchs ( § 6.14) and reprinted in 1588 by Gallucci ( § 6.24).

Gaurico (1524)

Luca Gaurico (in Latin, Lucas Gauricus, in French Luc Gauric) (1475-1558) was an Italian astrologer, astronomer, and mathematician.

350

In 1524, as an appendix to Regiomontanus's Tabulae directionum profectionumque, 351 Gaurico published a table of sines with R = 100000 and at intervals of 10 ′ (figure 17). This In fact, it seems that Gaurico took the values in Engel's table for R = 60000, and merely multiplied them by 10/6, although this procedure will in a few cases give values that differ from those in Gaurico's table. 354 Gaurico's table was also certainly not the basis of Copernicus's table of sines (or semi-chords) published in 1543, although it uses the same radius and interval.

Copernicus (c1530?)

The earliest known decimal table of secants is a handwritten table by Nicolaus Copernicus (1473Copernicus ( -1543)), 355 included in his copy of Regiomontanus's

Tabulae directionum profectionumque published in 1490. 356 There Copernicus 350 For a summary of Gaurico's life and works, see [Gessner and Simmler (1574), p. 455] as well as [Moréri (1733), p. 243-244].

351 [START_REF] Regiomontanus | Tabule directionum[END_REF]] 352 [START_REF] Gaurico | Tabulae de primo mobili quas directionum vocitant[END_REF]] 353 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF] 355 For a summary of Copernicus's life and works, see [START_REF] Rosen | [END_REF]]. Note in passing that in 1574 Gessner only briefly mentions Copernicus [Gessner and Simmler (1574), p. 518]. 356 See [Curtze (1875), pp. 34-37], [Glowatzki and Göttsche (1990), pp. 190-192] • where Copernicus probably tried to obtain more accurate values. The discrepancy of these two values is not, in my opinion, a sufficient reason to look for a different source or a different computation for Copernicus's entire table of secants.

Fine (1530)

Oronce Fine (1494-1555) was a French mathematician and cartographer. After having learned his first lessons of mathematics from his father in Briançon, he matriculated at the University of Paris and from about 1531 until his death he occupied the chair of mathematics of the Collège Royal in Paris. 364 357 Stamm mistakenly wrote that the secants are given for every minute, but this is surely a typo [Stamm (1933)].

358 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF] [Glowatzki and Göttsche (1990), p. 192] 364 For summaries of Fine's life and works, see [START_REF] Gallois | [END_REF]], [START_REF] Poulle | Dictionary of Scientific Biography[END_REF]],

[ Marr (2009)], [Pantin (2013)] and [START_REF] Axworthy | Le mathématicien renaissant et son savoir : le statut des mathématiques selon Oronce Fine[END_REF], [START_REF] Axworthy | Oronce Fine and Sacrobosco: From the edition of the Tractatus de sphaera (1516) to the Cosmographia (1532)[END_REF]]. See also the accounts given by [Lindgren (2007)] (on land surveys) and [Fréchet (2009)], as well as the early notice by Gessner [Gessner and Simmler (1574), p. 534]. I have chosen to spell his name "Fine," in accordance with Poulle, but it is also sometimes spelled "Finé." • = 6975 instead of 6976. However, there are more deviations when Copernicus's table is compared to Gaurico's table published in 1524 (also with R = 10 5 ), with eight errors in the same interval. The most likely basic explanation would then be that Copernicus used Regiomontanus's table for R = 10 7 and made a few rounding errors.

407 [Stamm (1933), p. 2] 408 [Rosen (1975b)] 409 [Husson (2014)] 410 [START_REF] Roegel | [END_REF]] 411 For a first summary of Copernicus's life and works, see [START_REF] Rosen | [END_REF]]. A recent biography of Copernicus is that by [START_REF] Freely | Celestial revolutionary: Copernicus, the man and his universe[END_REF]]. For further study, one might turn to [Swerdlow and Neugebauer (1984)], to Owen Gingerich's works as well as to Copernicus's complete works. On the connections between Italy and Krakow before Copernicus, see [Walsh (1996)]. For Copernicus's trigonometric tables, see [START_REF] Żyna | Przełom w trygonometrii połowy XV wieku: Kopernik jako spadkobierca i jako kontynuator tego przełomu[END_REF]].

412 [Copernicus (1543)] 416 For instance, for 0

• 50 ′ , one obtains 8726/6 = 1454.333 . . . which is rounded to 1454. In case the result is a half integer, the rounding occurs to the integer below, except if the first rounding was by default, although there may be exceptions (such as 2

• 40 ′ ) taking account of how the first rounding was performed.

This procedure fails after 3

• 50 ′ and it seems that a different operation was involved. In fact, between 4

• and 5

• 20 ′ , there was apparently a truncation of the last two digits of Regiomontanus's table, the resulting value was multiplied by 10, divided by 6 and rounded. Between 5

• 40 ′ and 6

• , the initial procedure was again applied. These two procedures give a slightly better outcome than merely using Regiomontanus's table with R = 10

7 .

If 417 [START_REF] Apian | Instrumentum primi mobilis, a Petro Apiano nunc primum et inventum et in lucem editum, etc[END_REF]] 418 [Stamm (1933)] 419 [START_REF] Apian | Instrumentum primi mobilis, a Petro Apiano nunc primum et inventum et in lucem editum, etc[END_REF]] 420 [Folkerts (1977), p. 234] 421 [Swerdlow and Neugebauer (1984) • 20 ′ ) in Rheticus's 1542 table. Rheticus made new computations for the tangents and the secants using 423 For summaries on Gemma Frisius's life and works, see [START_REF] Cantor | [END_REF]], [START_REF] Hallyn | Fernand Hallyn. Trois notes sur Gemma Frisius[END_REF], Hallyn (1998), Hallyn (2004), Hallyn (2008)] and [START_REF] Kish | [END_REF]]. [Lindgren (2007)] gives some background on Gemma Frisius's work on land surveys. Note in passing that Gessner briefly mentions Gemma Frisius [Gessner and Simmler (1574), p. 221].

424 [Gemma Frisius (1545)] 425 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. 181] 426 [START_REF] Von Peuerbach | Georg von Peuerbach. Quadratum geometricum[END_REF]] 427 [Rosen (1975b)] 428 [START_REF] Joachim | Canon doctrinae triangulorum[END_REF]] See [De Morgan (1845a), De Morgan (1845b)]. 429 However, in the first treatise of trigonometry independent of astronomical applications, the Treatise on the Quadrilateral, the Persian al-T ūsī (1201-1274), already in the 13th century, had used all six trigonometric functions [Archibald (1949), p. 31]. 430 See [Rosi ńska (1994b)] and [Glowatzki and Göttsche (1990), p. 152] these sines. According to Glowatzki and Göttsche, 431 Rheticus merely computed the ratios for the tangents, but things are actually a bit more complicated.

First, it appears that the secants and cosecants were computed by dividing 1 (or rather 10 14 ) by the values of the cosines or sines, and truncating the results. This can readily be observed on the secants of 17

• 20 ′ , 29

• , 29

• 50 ′ , 43

• , etc., and practically every ratio whose decimal part is greater than 0.5. This is also true for the cosecants, an example being 35

• 30 ′ . But the tangents and cotangents are another story. I don't know exactly how Rheticus computed these values, but a close examination of Rheticus's values reveals that the tangents are more accurate than the cotangents and consequently one cannot have been computed from the other. They must have been computed differently. The tangents may have been computed by dividing the sines by the cosines, but this cannot have been the case for the cotangents.

432

It appears that the values of the cotangents are close to those obtained when computing

Cot x = Csc 2 x -R 2
but they are not totally identical. The agreement is however much better than that obtained by merely dividing the values of the cosines by the sines of Regiomontanus, and it may even be a little better if Csc 2 x is rounded to seven or eight significant digits. This hypothesis may need to be tested further, but it parallels a suggestion by van Brummelen and Byrne for the computation of secants by Maurolico, 433 although I argue below that their suggestion is in fact not applicable to Maurolico's computations. However, I also suggest below that Fincke used a similar procedure to compute his secants in 1583.

In any case, Rheticus's work remains based on Regiomontanus's tables, and although he was the first to construct a table giving all six triangle ratios, he did not compute the cosecants and cotangents sufficiently accurately for small angles, and seems to have not yet understood that more accurate sines were needed. He had no problems giving cosecants and cotangents to 10 figures, when the sines were only given to 5 figures. This understanding of the need for more accurate sines only came later, and even the Opus 431 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF] The sines were copied from Regiomontanus's sines, 441 probably from the 1541 printing, but the tangents were recomputed at intervals of 1 ′ , using these sines. 442 Moreover, in the range from 89

• to 89

• 59 ′ 50 ′′ , Reinhold gave the tangents at intervals of 10 ′′ . The tangents seem to have been computed in a non systematic way. For the angles which are found in Regiomontanus's table, Reinhold has apparently mostly taken the ratio of the sines given by Regiomontanus, but sometimes the result was truncated (for instance for Tan 1

• where Reinhold gives 174550 instead of 174551), and sometimes the sines were rounded to the tens (for instance for Tan 10

• where 173648/984808 was computed 434 [START_REF] Rheticus | [END_REF]] 435 [START_REF] Pitiscus | Thesaurus mathematicus sive canon sinuum ad radium 1. 00000. 00000. 00000. et ad dena quaeque scrupula secunda quadrantis : una cum sinibus primi et postremi gradus, ad eundem radium, et ad singula scrupula secunda quadrantis : adiunctis ubique differentiis primis et secundis; atque, ubi res tulit[END_REF]] 436 [Roegel (2010c)] 437 For a summary of Reinhold's life and works, see [Gingerich (1975)]. Note in passing that Gessner briefly mentions Reinhold [Gessner and Simmler (1574), p. 184].

438 [Gingerich (1973)] 439 [Reinhold (1554)] 440 See [van Brummelen (2021), pp. 5-7].

441 [Glowatzki and Göttsche (1990), pp. 152-153] 442 [Glowatzki and Göttsche (1990), p. 185] instead of 1736482/9848078). In the case of Tan 80 444 The forthcoming study [Pritchard (2021)] may contain some interesting clues on this matter.

445 [Fincke (1583)] 446 [START_REF] Clavius | Theodosii tripolitae sphaericorum libri III[END_REF]] 447 [Glowatzki and Göttsche (1990), p. 185] The "Fraction" column gives the ratios actually used by Reinhold for the tangents, and the column on the right gives the values of these fractions. These can be compared with those in Reinhold's table.

It turns out that the values given by these fractions are almost exactly those of Reinhold, with the occasional rounding errors or typos. For instance, Reinhold table has Tan 89

• 0 ′ 10 ′′ = 574493507 which must be a typo for 574493501. In most cases, the fractions seem to have been truncated (and for instance for 89

• 30 ′ 10 ′′ this resulted in the incorrect value), but in some cases they were rounded (for instance for 89 • 10 ′′ , something obviously went wrong, because the value of Sin 89

• 10 ′′ used is smaller than that for Sin 89

• , although the resulting value is still acceptable. Of course, given the limited number of significant digits for the sines, especially at the end of the range, most of the figures in the tangents end up being meaningless. It doesn't make much sense to give Tan 89

• 59 ′ 50 ′′ to 12 places, when the value of Sin 10 ′′ used only has three places. . . We can also see that at the end of the range Reinhold moved to cosine values of 10

7 , but that he did not try to do a finer interpolation. In any case, such an analysis can be made for all 60 × 5 = 300 values which are not multiples of 1 ′ , but this is left as an exercise. There may be other errors such as the one mentioned for 89

• 10 ′′ and it might be interesting to do some detailed statistics about these errors.

It is precisely for these reasons that Viète's tangents and secants published in 1579 are much more accurate than those of Reinhold, because Viète used sufficiently accurate sines for the number of figures he was trying to compute for the tangents and secants.

Reinhold's table of tangents was the first table of tangents at intervals of 1

′ . Secants at this interval would only be published 25 years later by later became a Benedictine. Maurolico edited the works of classical authors including Archimedes, Apollonius, Autolycus, Theodosius and Serenus. He also composed his own unique treatises on mathematics and mathematical science.

In 1558, Maurolico published his commentary on the spherics of Theodosius. 458 It contains short tables of sines (figure 39), tangents (figure 40)

and secants (figure 41), all using the radius R = 10 5 and only giving values every degree. Maurolico also gave the tangents and secants for 89 See [van Brummelen and Byrne (2021)] as well as [van Brummelen (2021), p. 22]. 473 See the second page of the preface of Magini's Primum mobile [START_REF] Magini | [END_REF]], of which an excerpt is translated in [van Brummelen and Byrne (2021)], but mistakenly attributed to Magini's De planis triangulis.

474 [van Brummelen and Byrne (2021), p. 206] 475 [Glowatzki and Göttsche (1990), p. 193] In fact, the procedure suggested by van Brummelen and Byrne (which echoes Birkenmajer's suggestion for Copernicus's table of secants) does not always work, and fails to give Maurolico's values for 86

• , 87

• , or 89

• 30 ′ . The solution is actually much simpler, and Maurolico must have proceeded like for the tangents, using an additional digit for a number of sines. Doing so from 85

• to the last angle gives the values in Maurolico's table, the only exception being 86

• . But in that case, computing the secant from the tangent also does not yield the value in Maurolico's table. It is possible that Maurolico only used one additional digit for the secants from 87

• , or that he made a mistake, or perhaps that the table contains a typo there.

Incidentally, Fincke seems to have used the procedure suggested by van Brummelen and Byrne in order to compute his secants in 1583 (and in fact van Brummelen and Byrne claim so, 476 but with no references).

I have also given a modern reconstruction of Maurolico's tables in a separate document. 477

Eisenmenger (1562)

Samuel Eisenmenger (1534Eisenmenger ( -1585)), known as Siderocrates, was a German physician, theologian and astronomer. He was professor of astronomy at the University of Tübingen in 1557-1568.

In

Witekind (1576)

Hermann Witekind (or Wilken) (1522-1603), a student of Melanchthon, was a German humanist and mathematician. In 1585, under the pseudonym of Augustine Lercheimer he published a book against the persecution of witches.

484

In 1576, he published his work Conformatio horologiorum sciotericorum etc.

485 in which he included a table of sines for a radius of 100000 and for every minute of the quadrant (figure 46). This table was presumably copied from one of Apian's tables (1533, 1534 or 1541), 486 as the values all seem to agree. 487 The layout, however, is different. Each page has six columns for degrees and 30 rows for 30 minutes. Six degrees therefore span two pages.

482 [Malpangotto (2020), pp. 221-232] 483 [Schreckenfuchs (1569)] 484 [START_REF] Binz | [END_REF]] 485 [START_REF] Witekind | Conformatio horologiorum sciotericorum in superficiebus planis utcunque sitis, iacentibus, erectis, reclinatis, inclinatis, & quocunque spectantibus, compendiaria & facilis, cum quadrantis horologici & geometrici conformatione & usibus, ac tabulis sinuum[END_REF]] 486 [START_REF] Apian | obtained from Regiomontanus (10 7 ) by mere truncation, without rounding[END_REF]], [START_REF] Apian | Instrumentum primi mobilis, a Petro Apiano nunc primum et inventum et in lucem editum, etc[END_REF]] and [START_REF] Apian | Instrumentum sinuum, seu primi mobilis, nuper a Petro Apiano inventum[END_REF]].

487 [Glowatzki and Göttsche (1990), p. 169] Around 1587, Bürgi devised a new way (his so-called "Kunstweg") to compute sines iteratively, without any geometrical construction 543 and he constructed at least two tables, one giving the sines at intervals of 2 ′′ and another giving the sines at intervals of 1 ′ . However, I believe that Bürgi did not use his new algorithm to construct these tables, and instead built up the tables by finite differences. Although Bürgi's work represents a new computation of sines, it is therefore possible that he reinvented some techniques already used by Regiomontanus, and even before in India, as mentioned earlier (see § 5).

The 60) gives the sines at intervals of 1 ′ , with a radius R = 60, and to four sexagesimal places, except for the last two degrees where they are given to five and six sexagesimal places. These four sexagesimal places correspond to a radius of 10 9 with a sine usually correct to 9 decimal places. For instance, Sin 75 The later years of Magini's life were devoted to cartography and geography. He worked in particular on an atlas of Italy.

Clavius (1593)

In 1593, Clavius published his work Astrolabium 564 which contained a sine table with R = 10 7 and at intervals of 1 ′ (figure 68). This table was copied from Clavius's earlier tables published in 1586, 565 but with some corrections.

For instance, as mentioned previously, the value of Sin 89

• 30 ′ was given incorrectly in Clavius's 1586 table, and was corrected here, perhaps after the discovery of the typo by Magini. Ceulen departed somewhat from the previous tables, in that he did not separate sines, tangents and secants in different tables, but put them together, for a range of two degrees, on each page.

Fale (1593)

Rheticus/Otho (1596)

After the publication of his Canon doctrinae triangulorum in 1551 which was based on Regiomontanus's tables, 582 With the exception of Bürgi's work, this was the first new computation of trigonometric values in the 16th century, since most of the trigonometric tables printed in the 16th century actually use values or computations inherited from Regiomontanus's tables 586 (see figure 13).

However, even a cursory examination of the Opus palatinum reveals that it contains two overlapping tables. On one hand, there is a table giving all six functions with a radius R = 10

10 and an interval of 10 ′′ . This table spans 540 pages. On the other hand, there is a table giving only the cosecants and cotangents, with a radius R = 10 7 and the same interval of 10 ′′ . This second table spans 180 pages. One might expect the second table to be an abridgement of the first, but this is not the case, as is apparent when comparing the first values of the cosecants and cotangents. These two tables obviously correspond to two different computations. This has actually been noticed before, and Glaisher wrote that "there seems no reason why it should have been printed at all, as the great ten-decimal canon completely supersedes it." 587 582 [START_REF] Joachim | Canon doctrinae triangulorum[END_REF]] 583 [Rosen (1975b)] 584 [Zeller (1944), p. 62] 585 See [START_REF] Rheticus | [END_REF]] and [START_REF] Roegel | [END_REF]]. See also [Glaisher (1873), p. 43] and [van Brummelen (2009), pp. 273-282] for descriptions of the Opus palatinum.

586 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. i] 587 [START_REF] Glaisher | [END_REF] ), but adjustments that did not always produce more accurate results (as for Csc 4 ′ ). It is possible that some of these "adjustments" were in fact typos. Rheticus did the same for the cotangents, taking the sines from Regiomontanus. It seems that the adjustments made for the sines in the case of cosecants were also used for the cotangents, but this should be checked throughout the table.

For the 10 ′′ intervals, Rheticus merely interpolated the sines. For instance, Csc 10 ′′ is obtained using Sin 10 ′′ = 485, Csc 20 ′′ uses Sin 20 ′′ = 970, and so on. There may be the usual typos, such as for Cot 10

′′ which is given as 206085546390, but should be 206185546392, and was merely obtained by dividing 9999999 by 485.

Sometime after that first computation, Rheticus must have realized that the cosecants and cotangents could not be computed accurately with such a scheme, because Regiomontanus's sines were not accurate enough for small angles. He must therefore have decided to construct a larger table, and he computed this time the sines and cosines with a radius of 10 15 and an interval of 10 ′′ . This was probably done around 1570. This work was used to produce the table for R = 10 10 published in 1596. However, the cosecants and cotangents were not computed using these accurate values of the sines, but those from the Opus palatinum itself. For instance, for Csc 1 ′ , Rheticus (or Otho) used the sine value 2908882, instead of Regiomontanus's 2909, but not the more accurate 290888204563 in the R = 10 15 table.

When the Opus palatinum was published, Otho must have decided to include Rheticus's earlier computation of cosecants and cotangents, but the reason for publishing it remains unclear, as Otho must have realized that these first calculations were inadequate. On the other hand, it was much more difficult for him than for us to realize it, and he perhaps decided to include these tables in case they contained some valuable results. Of course, computing the cosecants and cotangents with the sines given in the Opus palatinum is still not enough for small angles, as the sines are still not sufficiently accurate. This led Bartholomaeus Pitiscus (1561-1613) to correct the Opus palatinum and to publish Rheticus's sine table with R = 10 15 (figure 73) as well as other tables that he computed himself in his Thesaurus mathematicus.

589 Incidentally, Pitiscus was the one who first coined the word "trigonometry."

Conclusion

This marks the end of our journey through 15th and 16th century fundamental trigonometric tables. But this end is also a beginning. Rheticus's Opus palatinum and its amendments by Pitiscus were the start of a new era and these tables would themselves last until the 20th century. And the first years of the 17th century were the place of a bifurcation. On one hand trigonometric tables would continue their path, with little changes beyond Rheticus's masterpiece, 590 and on the other hand they made their foray into the world of logarithms, as if logarithms naturally absorbed the trigonometric functions.

591

Logarithms first appeared in public in 1614, and they started in association with sines. Indeed, when Napier published 592 the first table of logarithms in 1614, it was a table of logarithms of sines, and these sines were either those of Fincke 593 or those of Lansberge. 594 Napier's work was therefore based again on that of Regiomontanus, and not yet on Rheticus's work. 589 See [START_REF] Pitiscus | Thesaurus mathematicus sive canon sinuum ad radium 1. 00000. 00000. 00000. et ad dena quaeque scrupula secunda quadrantis : una cum sinibus primi et postremi gradus, ad eundem radium, et ad singula scrupula secunda quadrantis : adiunctis ubique differentiis primis et secundis; atque, ubi res tulit[END_REF]] and [Roegel (2010f)]. 590 De Morgan wrote that "There have been no trigonometrical tables of note published since the invention of logarithms, except those which contain logarithms" [De Morgan (1842), p. 497].

591 See [van Brummelen (2021), pp. 62-109] for a recent survey of the development of logarithms as a continuation of trigonometry. 592 See [Napier (1614)] and [Roegel (2010g)].

593 [Fincke (1583)]

19 2 Figure 1 :

 21 Figure1: Chords and sines. AB is the chord of α and BC is its sine, for a radius R. We have Chd R α = 2R sin(α/2) and BC = Sin R α.

Figure 3 :

 3 Figure 3: The beginning of the table of sines to R = 150 in a Latin edition of the tables of Toledo (BNF Ms. Latin 16211, f • 26v).

Figure 4 :

 4 Figure 4: The beginning of the table of sines to R = 150 in a Latin edition of the tables of Toledo (BNF Ms. Latin 16655, f • 24v).

Figure 5 :

 5 Figure 5: The beginning of the table of sines to R = 60 in a Latin edition of the tables of Toledo (BNF Ms. Latin 16211, f • 28r).

Figure 6 :

 6 Figure 6: The beginning of the table of sines to R = 60 in a Latin edition of the tables of Toledo (BNF Ms. Latin 16655, f • 27v).

Figure 7 :

 7 Figure 7: The table of shadows in a Latin edition of the tables of Toledo (BNF Ms. Latin 16211, f • 24v).

Figure 8 :

 8 Figure 8: The table of shadows in a Latin edition of the tables of Toledo (BNF Ms. Latin 16655, f • 24v).

  160

Figure 11 :

 11 Figure 11: Regiomontanus's probable birthplace in Königsberg, Bavaria. (photographs by the author)

Figure 12 :

 12 Figure 12: The list of pivots for Regiomontanus's large sexagesimal table [von Peuerbach and Regiomontanus (1541)] (source: Dresden).

  563

Figure 15 :

 15 Figure 15: Excerpt of Regiomontanus's table of tangents [Regiomontanus (1490)] (source: The Budapest University of Technology and Economics, 85.211, www.manuscriptorium.com).

Figure 16 :

 16 Figure 16: Excerpt of Engel's table of sines with R = 60000 [Regiomontanus (1490)] (source: The Budapest University of Technology and Economics, 85.211, www.manuscriptorium.com).

Figure 17 :

 17 Figure 17: Excerpt of Gaurico's table of sines, with R = 100000 [Regiomontanus (1524)] (source: Google books).

Figure 18 :

 18 Figure 18: Excerpt of Fine's table of sines [Fine (1530)] (source: Google books).

Figure 19 :

 19 Figure 19: An excerpt of Apian's table of sines [Apian (1533)] (source: Bayerische Staatsbibliothek).

Figure 22 :

 22 Figure 22: The first page of a manuscript of Regiomontanus's first great table of sines (R = 6 ⋅ 10 6 ) (source: Kislak Center for Special Collections, Rare Books and Manuscripts, University of Pennsylvania, LJS 172, ca1476).

Figure 23 :

 23 Figure 23: The first page of the first printing of Regiomontanus's first great table of sines (R = 6 ⋅ 10 6 ) [von Peuerbach and Regiomontanus (1541)](source: Dresden).

Figure 24 :

 24 Figure 24: The first page of the first printing of Regiomontanus's second great table of sines (R = 10 7 ) [von Peuerbach and Regiomontanus (1541)](source: Dresden).

Figure 25 :

 25 Figure 25: The first page of Rheticus's table of sines in Copernicus's De lateribus [Copernicus (1542)] (source: Dresden).

Figure 26 :

 26 Figure 26: The first page of Copernicus's table of sines in the De revolutionibus [Copernicus (1543)] (source: e-rara).

Figure 29 :

 29 Figure 29: Excerpt of Frisius's table of arctangents [Gemma Frisius (1545)] (cont'd).

Figure 30 :

 30 Figure 30: Excerpt of Fine's table of sines [Fine (1550)] (source: Google books).

Figure 31 :

 31 Figure 31: Excerpt of Rheticus's table of the six trigonometric functions [Rheticus (1551)] (source: Dresden).

Figure 32 :

 32 Figure 32: An excerpt from Reinhold's table of tangents [Reinhold (1554)] (source: e-rara).

Figure 33 :

 33 Figure 33: The end of Reinhold's table of tangents, with values every 10 seconds [Reinhold (1554)] (source: e-rara).

Figure 34 :Figure 35 :

 3435 Figure 34: An excerpt from Reinhold's table of sines [Reinhold (1554)] (source: e-rara).

Figure 36 :

 36 Figure 36: An excerpt of Gaurico's sine table with the heading tabula fecunda [Gaurico (1557)].

Figure 38 :

 38 Figure38: An excerpt of Gaurico's sine table[START_REF] Gaurico | Tabulae de primo mobili quas directionum vocitant[END_REF]].

Figure 39 :

 39 Figure 39: Maurolico's table of sines [Maurolico (1558)] (source: Österreichische Nationalbibliothek).

Figure 41 :

 41 Figure 41: Maurolico's table of secants [Maurolico (1558)] (source: Österreichische Nationalbibliothek).

Figure 43 :

 43 Figure 43: Excerpt of Schreckenfuchs's table of tangents [Schreckenfuchs (1569)].

Figure 44 :

 44 Figure 44: Excerpt of Schreckenfuchs's first table of sines [Schreckenfuchs (1569)].

Figure 45 :

 45 Figure 45: Excerpt of Schreckenfuchs's second table of sines [Schreckenfuchs (1569)].

Figure 46 :

 46 Figure 46: Excerpt of Witekind's table of sines [Witekind (1576)].

Figure 47 :

 47 Figure 47: Excerpt of Peucer's table of sines [Peucer (1579)].

Figure 48 :

 48 Figure 48: Excerpt of Viète's Canon mathematicus [Viète (1579)] (source: e-rara).

Figure 49 :

 49 Figure 49: Excerpt of Bressieu's table of sines [Bressieu (1581)] (source: Google Books).

Figure 50 :

 50 Figure 50: Excerpt of Bressieu's table of tangents (odd columns) and secants (even columns) [Bressieu (1581)] (source: Google Books). Note that the faded parts are artefacts of the way Google Books stores images.

Figure 51 :

 51 Figure 51: An excerpt from Giuntini's table of sines [Giuntini (1581)] (source: Google books).

Figure 52 :

 52 Figure 52: An excerpt from Padovani's table of sines [Giuntini (1581)] (source: Google books).

Figure 53 :

 53 Figure 53: An excerpt from Fincke's table of tangents [Fincke (1583)] (source: e-rara).

Figure 54 :

 54 Figure 54: An excerpt from Fincke's table of sines [Fincke (1583)] (source: e-rara).

Figure 55 :Figure 56 :

 5556 Figure 55: An excerpt from Fincke's table of secants [Fincke (1583)] (source: e-rara).

Figure 57 :

 57 Figure 57: An excerpt from Clavius's table of tangents [Clavius (1586)].

Figure 58 :

 58 Figure 58: An excerpt from Clavius's table of secants [Clavius (1586)].

Figure 59 :

 59 Figure 59: Another excerpt from Clavius's table of secants [Clavius (1586)].

Figure 60 :

 60 Figure 60: An excerpt of Bürgi'stable of sines (1587). Bürgi's Fundamentum Astronomiae manuscript is kept at the Biblioteka Uniwersytecka Wrocław, under call number IV Qu 38a. This excerpt of Bürgi's sine table was provided by Dieter Launert and is included in LOCOMAT (http://locomat.loria.fr) with permission.

Figure 61 :

 61 Figure 61: An excerpt from Gallucci's table of sines [Gallucci (1588)] (source: Google books).

Figure 62 :

 62 Figure 62: An excerpt from Lansberge's table of sines [van Lansberge (1591)] (source: e-rara).

Figure 63 :

 63 Figure 63: An excerpt from Lansberge's table of tangents [van Lansberge (1591)] (source: e-rara).

Figure 64 :

 64 Figure 64: An excerpt from Lansberge's table of secants [van Lansberge (1591)] (source: e-rara).

Figure 66 :

 66 Figure 66: An excerpt from Magini's table of tangents [Magini (1592)].

Figure 67 :

 67 Figure 67: An excerpt from Magini's table of secants [Magini (1592)].

Figure 70 :

 70 Figure 70: The first page from Ceulen's table of trigonometric functions [Ceulen (1596)].

Figure 71 :

 71 Figure 71: The last page from Ceulen's table of trigonometric functions [Ceulen (1596)].

Figure 72 :

 72 Figure 72: Excerpt of Rheticus's Opus palatinum [Rheticus and Otho (1596)] (source: e-rara).

Figure 73 :

 73 Figure 73: Excerpt of Pitiscus's Thesaurus mathematicus [Pitiscus (1613)] (source: École des Ponts ParisTech, Paris, photograph by the author).

  Latin translation (1126) of Maslama al-Majriti (c950-c1007)'s late 10th century Cordova edition of the original table. It was reproduced by Suter in 1914.

			• ,	67
	and was likely based on Ptolemy's table of chords, 68 but Al-Khwārizmī
	also used another simpler table with R = 150,	69 that is Brahmagupta's
	70 This simpler table only contained the radius from the Khan . d . akhādyaka. sines at intervals of 15 • and was especially known from a commentary
	by Al-Biruni. 71 In particular, McCarthy and Byrne convincingly discard
	Hogendijk's suggestion	72 of a possible candidate for a full R = 150 sine
	table that could be attributed to Al-Khwārizmī.

73

Al-Khwārizmī's main table survives in Adelard of Bath (c1080-1152)'s

74

The table with R = 150 is not found in Adelard of Bath's translation, but its radius R = 150 made it to the tables of Toledo.

Al-Khwārizmī also had a table of shadows with a gnomon of 12, 75

  From then on, the tangent could have a place comparable to that of the sine. But in the West, tangents were only rediscovered in the 15th century byBianchini and Regiomontanus. It is interesting to note that a table equivalent to a table of tangents has appeared elsewhere before al-Hasib's table, namely in China. Indian mathematics had actually been exported to China and such a table was constructed there in the 8th century in the form of a shadow-list, but this table was a false start for Chinese trigonometry.

	table of
	tangents/cotangents 80 (from 0 • 30 ′ to 89 • at intervals of 30 ′ and to three
	places 81 ), although the tangent function had been tabulated before with-
	out being identified as such.

82 83 In the S . ābi Zīj, 84 the Syrian astronomer Al-Battāni (Albategnius) (c858-929) put forward the advantages of sines. His table gave the sines for R = 60 and for every half-degree and to two sexagesimal places. 85 Al-Battāni also computed a table of cotangents (table of shadows) for every degree. 86 But the first original arabic constructions of sine tables were the works of Ab ū al-Wafā c and Ibn Y ūnus. 87 The Persian mathematician and astronomer Ab ū al-Wafā c (940-998) gave a better method for the computation of trigonometric tables and his sine table for R = 60 has a step of 15 ′ and the values were computed on four sexagesimal places. 88 It should be observed, however, as already mentioned, that Ptolemy's table itself already gave the means to construct a 78 See

  He constructed a table of tangents and cotangents with the radius 1, but it was still subdivided sexagesimally. He also recomputed a table of sines. According to Debarnot, Ibn Y ūnus's table of sines is more directly based on the Almagest Ibn Y ūnus gave the sines for every minute, for R = 60 and to four sexagesimal places. An excerpt of that table is shown byBerggren and King.

	As an indication of Ab ū al-Wafā c	's work, let us mention that he found 89
	Sin 30 ′ = 0; 31, 24, 55, 54, 55.
	The correct value is
	Sin 30 ′ = 0; 31, 24, 55, 54, 0, 12, . . . ,
	that is
	Sin 30 ′ = 0 + 31/60 + 24/60 2	+ 55/60 3	+ 54/60 4	+ 0/60 5	+ 12/60 6	+ ⋯
	Ab ū al-Wafā
					92 than that of Ab	ū
	al-Wafā

table of sines at intervals of 15 ′ , since the sine of 15 ′ is half the chord of 30 ′ . c was also the first to take the radius as unity. 90 He seems also to have been the one who introduced the secant and cosecant. 91 Ibn Y ūnus (950-1009), astronomer of Cairo, wrote the Hakemite tables. c . 93 Ibn Y ūnus obtained Sin 1 • = 1; 2, 49, 43, 4, 94 while the correct value is Sin 1 • = 1; 2, 49, 43, 11, 14, 44, . . . As observed by Glowatzki and Göttsche, the values of Ibn Y ūnus's table were obtained by interpolation. 95 Ibn Y ūnus very likely computed the sines at intervals of 10 ′ and filled the intermediate values by interpolation. 96 For instance, he gives 97 Sin 28 • 10 ′ = 28; 19, 20, 11, 0 which is a rather good 89

  In 1031, the Córdoban caliphate came to an end, the state was divided in a number of smaller kingdoms, and it is during the period 1031 to 1085 that Andalusian science flourished.

		• 2 ′ , etc., Sin 28 • 9 ′ , are all less accurate, with the
	least accurate being that for Sin 28 • 5 ′ .
	A table with such a small interval would not be available in Western
	Europe before the work of Regiomontanus in 1462 (see § 3.4). Incidentally,
	Regiomontanus's table was also obtained by interpolation, albeit certainly
	using a more elaborate scheme.
	At about the same time as Ibn Y ūnus, the Iranian scholar Al-Biruni (973-c1050) had obtained the very accurate value Sin 1 • = 1; 2, 49, 43, 11, 14. 98
	His table	99 gives the sines at intervals of 15

′ and uses R = 1. 100 In particular, around 1070 or 1080 a group of astronomers in Toledo, including Al-Zarqālī (c1028-1087) and perhaps also S . āc id al-Andalusī (1029-1070), 101 put together the "Toledan tables." 102 The tables of Toledo were closely based on those of Al-Khwārizmī 103 and Al-Battāni 104 which had been available in Al-Andalus since the 10th century.

105

  originates neither in Al-Khwārizmī's treatise (because Al-Khwārizmī's table only gives the sines at intervals of one degree), nor in Al-Battāni's treatise (because of distinctive discrepancies). , but the original tables are not. These Alfonsine tables arrived in Paris in the early 14th century and they spread in a modified form in Latin,120 becoming the Parisian Alfonsine tables. These tables were only superseded in the 16th century by the Prutenic tables based on Copernicus's theory.More accurate trigonometric tables were constructed in the Arabic world after those of Al-Khwārizmī and Al-Battāni. Chabás and Goldstein mention for instance a 14th century manuscript giving a table of sines for 2700 arguments, at 1′ intervals, 121 so presumably up to 45• and giving sines and cosines.And during the next century in Samarkand (now in Uzbekistan), Ulugh Beg (1394-1449) also computed a table of sines for intervals of one minute.

		half degree
	of the quadrant and with two sexagesimal places) 110 (see figures 5 and 6 for
	Latin editions). As mentioned above, the radius 150 of this table 111 (but not the values)
	possibly goes back to Al-Khwārizmī's table, and consequently to Brah-
	magupta's Khan . d . akhādyaka. However, as observed by van Dalen,	112 the
	idiosyncrasies of the table indicate that it was likely derived from a ta-
	ble with R = 60 (probably by Al-Battāni	113 ) by multiplying the values by
	2.5 and McCarthy and Byrne 114 believe that Al-Zarqālī was the one who
	made this transformation, perhaps in the hope of restoring a table which
	he thought to be that of Al-Khwārizmī.		122

, p. 128]. for Latin editions) and another sine table with R = 60 (for every 115 The second sine table in the Toledan tables, with R = 60, 116 It may possibly be based on Ptolemy's table of chords. The table of shadows of the Toledan tables (see figures 7 and 8) is the same as that in Al-Khwārizmī and Al-Battāni's tables. manuscript

3 The starting point in Western Europe: from von Gmunden to Regiomontanus

	In the 13th and 14th centuries, many writings appeared based on the canons
	of the Toledan tables, in particular the canons of John of Lignères (1322).
	These canons borrowed the details of the computation of the sines from the
	canons of the Toledan tables, but they also gave a sine table for R = 60 and
	for every half degree, as well as a table of shadows. 124
	Moreover, in 1175 Gerard of Cremona (c1114-87) translated Ptolemy's
	Almagest from the Arabic in Latin.

2 Giovanni Bianchini (c1410-c1469)

	Johannes von Gmunden's treatise was heavily used and Peuerbach
	borrowed much from it. Eventually, Peuerbach's own treatise made its way
	into Regiomontanus's works and was printed in 1541. 151
	Although Johannes von Gmunden's treatise did not contain any signifi-
	cant novelty, it brought the impetus for a new computation of sine tables,	152
	which would find its completion in Pitiscus' Thesaurus mathematicus in
	1613.	153
	139 3.Giovanni Bianchini was a merchant and businessman, probably born in Incidentally, Klug writes incorrectly 140 that Johannes von Gmunden was Bologna or Florence around 1410. He later went to Ferrara, but also visited
	the first to compute a table of sines at intervals of 30 other cities. He became interested in astronomical calculations at an early ′ . The first of the tables in the second part is the table given by Ptolemy. 141 age. 154 His scientific works were written between 1440 and 1460 and he is
	in particular the author of one of the few treatises of algebra written in the The second table may be the result of Johannes von Gmunden's computa-tion, as it goes slightly beyond the table found in the Toledan tables. As a fifteenth century in Latin. 155 He corresponded with Regiomontanus during
	matter of fact, the sines are given to three sexagesimal places, 142 but the last the latter's stay in Italy. 156
	place is always 0 (actually not shown at all) or 30. It may be derived from Rosi ńska was the first to describe Bianchini's purely trigonometric ta-
	another table. Glowatzki and Göttsche bles, which consist in two decimal and two sexagesimal tables. 143 drew the attention of a number of incorrect 157 Earlier writers such as Boffito 158 and Birkenmajer 159 mentioned some of Bianchini's statements on Johannes von Gmunden's tables, in particular by von Braun-mühl. 144 The latter, and later Bond 145 and Zeller trigonometric tables, but did not describe them in detail. 146 for instance incorrectly stated that Gmunden had a table with radius 600000. Bianchini's table of sines for R = 60 ⋅ 10 3 appears in his Tabulae primi 147 Cantor and Ene-ström also made the mistake. 148 Some typos of Busard's transcription 149 are mobilis and was reproduced and transcribed by Glowatzki and Göttsche.
	also corrected by Glowatzki and Göttsche. 150

.3 Georg von Peuerbach (1423-1461)

	table of sines. Another of Bianchini's tables, the Tabula magistralis quinta	171 gives the
	cosecants at 10 ′ intervals and with R = 10 4 .
	Rosi ńska assumed that Bianchini's extant decimal tables were derived from a decimal table of sines for R = 10 4 , and this would make sense. Unfortunately, such a table is no longer extant. 172 This table for R = 10 4
	may itself have been computed from Bianchini's sine table found in his
	Tabulae primi mobilis.
	Bianchini's work did not stay confined in Italy but circulated until
	Krakow, as described by Walsh.
	Georg Aunpekh, known as Georg von Peuerbach (1423-1461), was an
	Austrian mathematician and astronomer.	174 He was born in Peuerbach,
	Austria. In 1446 he registered at the University of Vienna and between
	1448 and 1451, he travelled to Italy. There he met Nicholas of Cusa (1401-
	1464) who had been papal legate in Germany since 1446 and cardinal since
	1448. In Ferrara, Peuerbach may also have met Giovanni Bianchini. 175 The
	latter wanted to obtain positions for Peuerbach in Bologna or Padua, but
	Peuerbach did not accept them. 176 He then returned to Vienna.
	Peuerbach was first influenced by Johannes von Gmunden who had
	171 [Chabás Bergón (2019), p. 351]
	172 As mentioned above, Rosi ńska actually wrote that the decimal table of tangents used
	R = 10 3 and therefore also posited a sine table with that radius.
	173

173 3

  180 It contains solid sphere representations of Ptolemaic planetary models, and this work was of great importance until the solid sphere hypothesis was disproved by Tycho Brahe at the end of the 16th century.

181

Peuerbach was acquainted with Cardinal Bessarion (1403-1472) who was then papal legate in Germany. In 1460, Bessarion spent more than one year in Vienna 182 in order to gain imperial support for the war against the Turks and during this stay he became friends with Peuerbach. Bessarion, a Greek, wanted to produce a new translation of the Almagest, because he considered Trebizond's work to be flawed. George of Trebizond (c1395-c1472) was of Greek origin and has translated many works from Antiquity in Latin. In particular in 1451 he composed a Commentary on the Almagest, which has never been printed.

183

Bessarion had himself considered translating the Almagest from the Greek, but his duties didn't let him the time to.

184 Bessarion asked Peuerbach (who did not know Greek) to write an Epitome (summary) of Ptolemy's Almagest.

185

He also wanted him to accompany him to Italy for further investigations on the Almagest. Peuerbach certainly wanted to take Regiomontanus with him to Italy, but Peuerbach died in 177

[Vogel (1973a)

, p. 120]

178

See

[START_REF] Malpangotto | Theoricae novae planetarum Georgii Peurbachii dans l'histoire de l'astronomie[END_REF]

, pp. 116-119 & 678-679]. Many sources give the date of publication as 1472, but I follow Malpangotto here. Note that Khauß wrote that the Theoricae were published in 1460 [von Khauz (1755), p. 46]. 179

  Peuerbach and Regiomontanus were in fact the most important members of the first Viennese mathematical school of the 15th century. 189 Peuerbach's work on the Epitome led him to work on reforming Ptolemy's astronomy. Gassendi later wrote that Peuerbach resurrected an almost dying astronomy and that without him, we would have neither Copernicus nor Brahe. table with R = 60000 is again found in the 1490 edition of Regiomontanus's Tabulae directionum profectionumque, 198 but it happens to be a table derived from Regiomontanus's large sexagesimal table, and not Peuerbach's table. Moreover, the 1490 table gives the sines at intervals of 1 ′ . Around 1450, Peuerbach took R = 600000 and a step of 10 ′ and went beyond what Johannes von Gmunden and Bianchini had done. But this table with R = 600000 is no longer extant. 199 We know of its existence because Peuerbach mentioned it in the Propositio prima of his Quadratum geometricum (or Canones gnomonis 200 ) written in 1455 and published in 1516.

, p. 183]. 32 1461 before the journey began.

186 By that time, Peuerbach had written six chapters of his Epitome, and not based on the Greek text.

187

Regiomontanus, who learned Greek, added the seven missing chapters to Peuerbach's work after Peuerbach's death. This Epitome was only printed in 1496 and was very influential, in particular on Copernicus. 188 Together with Johannes von Gmunden, 190 Or, as others have put it, Peuerbach and his pupil Regiomontanus 191 woke up the study of astronomy and built the necessary tables. 192 And Hellman and Swerdlow wrote that the "Epitome is the true discovery of ancient mathematical astronomy in the Renaissance because it gave astronomers an understanding of Ptolemy that they had not previously been able to achieve." 193 But as Thorndike notes, "it very likely never occurred to Peurbach that his name would go down to posterity as the reviver of the mathematics of classical antiquity or as the reformer of the mathematics of his own time." Figure 10: The first page of Peuerbach's table of arctangents (1516) [von Peuerbach (1516)] (e-rara). A 201 And another work of Peuerbach confirms that the step of the table was 10 ′ . Gassendi 202 also mentions a table of sines by Peuerbach with R = 6000000 and a step of 10 ′ , and that this table had been extended to a step of 1 ′ by Regiomontanus, but this is probably a typo, no such table with R = 6000000 being known of Peuerbach. 203 Peuerbach was the one who provided the impetus for the replacement of Ptolemy's chords with the sines from Arabic mathematics, and Regiomontanus computed tables of sines for every minute of arc for radiuses of 6000000 and 10000000 units. Among Peuerbach's other works is also his Quadratum geometricum 204 already mentioned, written in 1455 and published in 1516. This work describes the geometrical square, an instrument for measuring heights. A similar instrument was also described by Oronce Fine in his De re et praxi geometrica published in 1556. Peuerbach's treatise contains what is basically a table of arctangents (figure

  . The value 1200 used in this table may have been influenced by the radius 12000 in Bianchini's table of cotangents.

	205
	This table of arctangents was reprinted by Gemma Frisius 206 in 1545
	and a similar table was given by Magini in 1592.

table actually gives the value arctan(x/1200) in degrees. For instance, for x = 1200, Peuerbach's table gives 45 • . For x = 500, Peuerbach's table gives arctan(5/12) = 22 • 37 ′ 12 ′′ 207 3.4 Johannes Regiomontanus (1436-1476) Regiomontanus, or rather Hans Müller, was probably born in 1436 in Königsberg, near Bamberg in Germany (figure 11). 208 He had latinized his name as Johannes de Monte Regio and it was only half a century after his death in 1476 that he became known as Regiomontanus. 209

"

  Regiomontanus envisaged an exchange of problems and answers to be based on friendly emulation, but the older Italian was speedily scared away by the precocity of the enthusiastic German." and is useful for solving problems in spherical trigonometry. The table was computed using Regiomontanus's sine table with R = 6 ⋅ 10 6 .218 Glowatzki and Göttsche gave a survey of similar tables or variants published until the 19th century. to Rome at the invitation of Pope Sixtus IV in order to work on a reform of the Julian calendar, and this is where he died in 1476, probably from the plague. During all these years, Regiomontanus worked on a critique of Trebizond's translation of the Almagest, his Theonis Alexandrini Defensio in sex voluminibus contra Georgium Trapezuntium, a work which was probably only completed in the 1470s and still remains only in manuscript form.227It seems that Regiomontanus started around 1460 to compute sines with a large radius in order to produce a table with R = 6⋅10 by Bianchini, but greatly developped by Regiomontanus, made it much simpler to use the tables. With the new decimal radius, there is therefore no longer any need to mix the bases 10 and 60, as was the case in the older tables.Regiomontanus's Tabulae directionum profectionumque from 1467 and published in 1490 also contained a table of tangents (figure15) which was probably inspired by Bianchini's table of tangents. De triangulis omnimodis, was completed about 1464 but only printed in 1533, without any table.Delambre was critical of Regiomontanus and wrote that except for his observations and trigonometrical work, Regiomontanus had hardly the time to do more than show his good intentions. Regiomontanus are very modern and could still be used now, only the decimal point would have to be shifted.Thorndike thought that Peuerbach and Regiomontanus's importance had perhaps been overestimated, among other things because Regiomontanus was more than a mathematician. He was a mathematical publisher, and he came at just the right time.

	250 Delambre stresses that
	Regiomontanus was less advanced as a calculator than Ibn Y ūnus and Ab al-Wafā c . However, this opinion may need to be revised in the light of my ū
	analysis of the construction of his tables.
	Braunmühl 251 considered that Regiomontanus's work on triangles was
	′ intervals (figure 16). But contrary to what Bond, influential, even if it didn't contain anything original. 237 De-lambre, 238 or more recently Folkerts, 239 Zinner, 240 North, 241 Brummelen, 242 And as observed by Glowatzki and Göttsche, 252 the tables computed
	Husson, 243 and Chabás and Goldstein wrote, giomontanus nor borrowed from Bianchini. It was appended to Regiomon-244 this table is neither by Re-tanus's book, probably by Johannes Engel (or Johannes Angelus) (1453-1512), by 253
	250 [Delambre (1819), p. 365]
	251

216

It was during this time that Regiomontanus constructed his Tabula primi mobilis which was only published in 1514.

217

This table gives the values of arcsin(sin x sin y) for 0 ≤ x, y ≤ 90 • 219 Regiomontanus returned from Italy around 1465, 220 he went to Pozsony (Pressburg, Bratislava) in 1467, at the invitation of Matthias Corvinus (1443-1490), King of Hungary, 221 of whom he became an astronomical adviser. Some time later, he was called to Buda. 222 It was during this time in Hungary that Regiomontanus worked with the Polish astronomer Marcin Bylica (c1433-1493) 223 whom Regiomontanus met in Rome. Together they computed some tables, in particular Regiomontanus's Tabulae directionum profectionumque. returned 6 for his De triangulis (1462?) (figure 22). This table was certainly inspired by Peuerbach's table with R = 600000, although Hallam claimed 228 that Regiomontanus was ignorant of that table. Glowatzki and Göttsche 229 give Regiomontanus's description of the computations, the Compositio tabularum sinuum rectorum, as well as a German translation. Regiomontanus's description is contained in the 1541 edition of Peuerbach's treatise on sines. 230 In section 4 below I analyze how Regiomontanus may have computed his table. Around 1468, Regiomontanus composed another table with a radius of 10000000. Both the sexagesimal and the decimal tables were given at intervals of 1 ′ . These tables were first printed in 1541 (figures 23 and 24). 231 They were however not the first tables with such intervals, and they came after those of Ibn Y ūnus and Ulugh Beg (see § 2.3). Regiomontanus's table of sines with R = 10 7 was accessible in Cracow at the end of the 15th century 232 and was undoubtly one of the sources of Copernicus's trigonometric tables. The move from a sexagesimal division to a decimal division, initiated 233 Cardano consid-ered that Regiomontanus's entire Tabulae directionum profectionumque was largely drawn from Bianchini. 234 Folkerts, 235 however, considered that Regiomontanus's table of tangents was influenced by Al-Battāni. In fact, Regiomontanus's table of tangents was certainly computed using his large sexagesimal table as I shall show later. A modern reconstruction of this table of tangents is given separately. 236 The Tabulae directionum profectionumque also contains a table of sines with R = 60000 and at 1 245 and was derived from Regiomontanus's table with R = 6000000. Moreover, as observed by Glowatzki and Göttsche, the appended table was never used by Regiomontanus. 246 A modern reconstruction of Engel's table is given separately. 247 In fact, most of Regiomontanus's writings were only published after his death. His main work on trigonometry, 248 It is the first systematic such treatise published in Europe and it was probably used by Copernicus. However, as observed by Stamm, 249 it is unlikely that Copernicus had access to Regiomontanus's treatise in manuscript form and he probably only saw the 1533 edition in the 1530s.

4 Regiomontanus's seminal tables

  , p. 150] We can now pause and summarize the situation of Regiomontanus's tables at the end of the 15th century. There are four different trigonometric tables usually associated with Regiomontanus: a large table of sines with radius 6000000, another one with radius 10 7 , a table of tangents with R = 10 5 and a smaller table of sines with R = 60000, but of which Regiomontanus is actually not the author. Most of the tables published during the 16th century are ultimately based on the table for R = 10 7 .

  table for R = 6 ⋅ 10 6 could be computed.

			260 This
	auxiliary table is only partially extant.
	Once he had his pivots, Regiomontanus computed the sines at intervals
	of 15 ′ , dividing the sines at intervals of 45 ′ obtained earlier in three parts in
	such a way that the sines vary smoothly.	261 Then Regiomontanus trisected
	each interval, again by ensuring that the differences vary smoothly. This
	gave him the sines at intervals of 5 ′ . 262 The same procedure was again
	applied to obtain the sines at intervals of 1 ′ .	263
	For the table with R = 10 7 , Regiomontanus possibly also first computed
	a number of pivot values with R = 10 9 , but these pivots have not been kept.

4.2 Sine table with R = 6000000

Regiomontanus's first large complete sine table was for a radius of 6000000 and was probably computed around 1462 in Rome.

264

It gives the sines for every minute. Figure

22

shows an excerpt of a manuscript of that table. This table is based on the computations made with R = 6 ⋅ 10 8 as described in the previous sections. After Regiomontanus's death, Regiomontanus's table was long kept in manuscript form. It was only published in 1541 with Peuerbach's Tractatus super propositiones Ptolemae etc., 265 and together with the table for R = 10 7 (figures 23 and 24). These two tables were then again published in 1561 in Regiomontanus's De triangulis. 266 Glowatzki and Göttsche gave a facsimile of the 1541 sexagesimal table and listed its errors. 267 Regiomontanus's sine table appears rather accurate, although it is probably slightly less accurate than the table for R = 10 7 . Sampling only the values for whole degrees, there are 25 last-place errors and one typo (for 260 However, as observed by Glowatzki and Göttsche, an error in the computation of sin 45 ′ caused other (small) errors, in particular in the interpolation leading to sin 1 • [Glowatzki and Göttsche (1990), pp. 26-27]. 261

  40• , 3856796 which should be 3856726). Of the last-place errors, all are of one unit, and one (80• ) is of two units. I have given separately a modern reconstruction of this table with the exact values which can be used for comparison with Regiomontanus's table.268 And in § 5, I am giving a more detailed analysis of Regiomontanus's errors and computation procedure. In Regiomontanus's table, the column of differences does not give the actual difference ∆, but the difference per second, in other words ∆/60. These differences are given to one decimal place which is separated by a space.269 For instance, the first difference is ∆ = 1745 and it is given as 29 1, because 1745/60 = 29.08 . . .. But this value can also be read 291, in which case it is the sixth of the actual difference. These differences follow a layout similar to those in Bianchini's table with R = 60 ⋅ 10 Like in the previous table, sines are given for every minute. This table was also published in 1541 and 1561 together with the sexagesimal table (figures 23 and 24). Glowatzki and Göttsche gave a facsimile of the entire 1541 edition of the table, and listed its typos. 274 The differences are expressed like in Regiomontanus's sexagesimal table and the first difference is for instance ∆ = 2909 and it is given as 48 5, corresponding to ∆/60 = 48.48 . . . I have given separately a modern reconstruction of this table. to note that Regiomontanus's table is slightly more accurate than the previous one with R = 6 ⋅ 10 6 . 276 Sampling only the sines for whole degrees, we can for instance see that there are only seven incorrect values, one of which (for 25 • ) being an obvious typo (4226583 which should be 4226183), and the other six values being only off by one unit of the last place. This suggests of course that the decimal table was not merely obtained from the sexagesimal table, but must have been obtained either from the pivots of the sexagesimal table, or from newly computed pivots, as described above.

3 , so that it is possible that Regiomontanus borrowed this layout. 270 4.3 Sine table with R = 10000000 Regiomontanus's second large sine table was for a radius of 10 7 and was completed in 1468. 271 It came shortly after the smaller decimal table of tangents which was computed in 1467. This large decimal table is probably not the first decimal table of sines, although Folkerts claimed so. 272 It has been assumed that Bianchini had a decimal table of sines, probably with a radius R = 10 4 (see § 3.2), but this table is no longer extant. Regiomontanus's table is also not based on his large sexagesimal table. 273 Regiomontanus may have computed a number of pivot values, perhaps with R = 10 9 , or he may have reused the sexagesimal pivots by multiplying them by 10/6. In any case, these pivots have not been kept. Then, Regiomontanus must have proceeded by interpolation as in the sexagesimal table. 275 It is interesting

  tangents for every degree, and for a radius of 100000. The tangents were computed from the 1462 table of sines (with R = 6000000), by first dropping two digits and rounding the values, and then by mere division. . In these four cases, Regiomontanus very likely got the computations wrong, or these are typos. Incidentally, the same procedure fails miserably when using the decimal table of sines, and it is almost impossible to obtain the values of the table of tangents with this starting point. Regiomontanus's table was not the first table of tangents, as tangents had already been used in eastern Islam, as mentioned above (see § 2.3). Regiomontanus's table was reproduced in subsequent editions of his Tabulae directionum profectionumque, by Gemma Frisius in 1545 (but as cotan-And finally, mention should be made of Bendefy who mistakenly wrote in 1980 that Regiomontanus had constructed a table of tangents for a radius R = 10

	actually gives exactly Regiomontanus's values, except for the angles 43 • , 4.6 Secant tables
	73 • , 85 • and 89 Regiomontanus did not compute tables of secants, but the first tables of
	secants are based on his tables of sines. This is the case of Copernicus's
	table of secants, which might have been computed around 1530. Bianchini
	had computed a table of cosecants ( § 3.2). But I am not aware of earlier such tables, although, as mentioned before ( § 2.3), Ab ū al-Wafā c introduced
	the notion of secant in Baghdad in the 10th century.
	The tables of secants published by Rheticus in 1551 [Rheticus (1551)]
	and by Maurolico in 1558 [Maurolico (1558)] also ultimately derive from
	gents) 282 (also with Peuerbach's 1516 quadratum table Regiomontanus. 293 283 ), by Gaurico in 1557, It was Viète 294 who in 1579 was the first to compute a table of secants 284 by Maurolico in 1558 (at least partially, and he called it umbra ver-sa), 285 by Schreckenfuchs in 1569, 286 and in subsequent editions of these with an interval of 1
	works.
	292
	281 This procedure
	of Hungarian mathematics (A magyarországi matematika története, 1970), but I was not able
	to check this source. Bendefy also cites Zinner's article on Regiomontanus in Hungary,
	published in Hungarian, and which does not seem to contain such a statement (Ernő Zin-
	ner, Regiomontanus Magyarországon, Matematikai és Természettudományi Értesítő, volume
	55, 1936?, pp. 280-288).

Regiomontanus's Tabulae directionum profectionumque, 278 from 1467 and printed in 1490, also contained a short table of tangents, which he called tabula fecunda (figure 15). 279 The name "tangent" was as a matter of fact only introduced in 1583 by Fincke. 280 Regiomontanus's table is only one page long and gives the • 287 Gaurico's table (1557) 288 only goes up to 50 • , and is attributed to Campanus. But neither Glowatzki and Göttsche, 289 nor von Braunmühl, 290 nor Zinner 291 were able to understand this attribution.

Curiously, Gaurico also gives a sine table with the heading tabula fecunda and also only up to 50

• .

7

and for every minute, and that it was only Reinhold who published it in 1554. ′ albeit with a variable radius between R = 10 5 and R = 10 9 .

  , there is an obvious typo, and the original value may have been correct. The value for 45 ′ may also be a typo. In any case, none of these small errors have any serious impact on the values in the sexagesimal table.Consequently, the 45 pivot is correctly given in the table for R = 6.10 8 , but there is a different value in the final table. The neighboring values would make things worse if I gave the correct value to Sin 6• 45 ′ , so that I suspect that Regiomontanus made an error when copying his own (correct) value of Sin 6 These three sines are off by one in the sexagesimal table.

The same remark applies to the decimal table. The pivots at 45 ′ intervals (for R = 6 ⋅ 10 8 ) for the sexagesimal table appear very accurate. There are only 17 values which are not correct, and among them all are off by one unit of the last place, except those for 45 ′ , 27 • , 57 • 75 ′ , and 59 • 25 ′ . In the case of 57 • 75 ′ ′ pivots in the sexagesimal table (for R = 6 ⋅ 10 6 ) are mostly correct. In fact, they should even all be correct. But there are three exceptions. The 6 • 45 ′ • 45 ′ . The same observations apply to Sin 8 • 15 ′ and Sin 44 • 15 ′ .

  ). The 15 ′ pivots of the decimal table appear clearly more accurate than in the sexagesimal table. interval) are wrong, and the cases where only one of them is wrong. Surprisingly, there are 24 cases of wrong twin pivots in the sexagesimal table, and none in the decimal table.

	Morevover, in the decimal table, 13 out of 17 wrong (non twin) 15 ′
	pivots concern the second 15 ′ pivots. Things are more even in the
	sexagesimal table, where 13 out of 22 wrong (non twin) 15 ′ pivots
	concern the second 15 ′ pivots.
	An example of incorrect 15 ′ pivot in the decimal table is that of 32 • ,
	where 31
	An example of incorrect 15 ′ pivot in the sexagesimal table is that of
	45 • 15 ′ , where the sines of 45 • and 45 • 45 ′ are correct.
	One should actually distinguish the cases where the two twin/double
	15 ′ pivots (in a 45

′

  Regiomontanus may or may not have noticed that C 15 /C 45 ≈ 9. But he must certainly have noticed that the third differences ∆ 3 vary only very slowly and that their variations can be neglected on small ranges. At this stage, Regiomontanus could have had a means to compute δ

			′ , we have
	C 45 ≈ -5836. The exact expression behind this value matters little here,	306
	but what is important is that by playing with differences, any serious
	table computer would eventually find out that there is some constant ratio
	involved, and perhaps think of using it backwards. By computing a few
	exact values of sines at 15 ′ intervals, Regiomontanus may have found
	that another constant is involved: 307 δ 0 ≈ 2	v 1/3 C 15	and that C 15 ≈ -52525.
				2
				0

  Note that in Rheticus's 1551 table, the sines were copied fromRegiomontanus (or Rheticus 1542); inReinhold's table (1554), the sines were copied from Regiomontanus (or Rheticus 1551); inSchreckenfuchs's table (1569), one table was copied from Engel (in an edition of the Tabulae directionum profectionumque) and another was copied from the 1490 table of tangents (or another edition of the Tabulae directionum profectionumque); in Bressieu (1581), the sines were copied from Fine (1530 or 1550).bother checking the values using the differences, and Clavius in 1586 was apparently the first to get rid of the typographical errors of earlier tables. Note however that Glowatzki and Göttsche do not always give the direct predecessor of a table, and do seldom consider the layouts of the tables as an indication for their source. Fine's tables are the only fully sexagesimal tables based on Regiomontanus's tables, apart from those of Schreckenfuchs published in 1569 and of Bressieu published in 1581.

										J. v. Gmunden	sin (1437)
										(c1384-1442)	30 60; 60 3	Bianchini	cot	Bianchini	sin
										(1410-1469) 10 12000	(1410-1469) 10 10 4
										Peuerbach	sin
										Peuerbach	sin (1450?)	(1423-1461) 10 60000
					(1423-1461) 10 600000 Among the tables with radius 10 5 are the tables of Bassantin (1557), Peuerbach atan (1455) (1423-1461) 1/1200 ; 60 3 326
				Regiomontanus sin (1462) (1436-1476) 1 6 • 10 6 327 Peucer (1579) Witekind (1576), 328 , Giuntini (1581) Regiomontanus sin (1468) (1436-1476) 1 10 7 and Fale (1593) 331 which were taken directly or indirectly from Apian Regiomontanus tan (1467) (1436-1476) 1 • 10 5 330 329 , Padovani (1582)
				Peuerbach (1423-1461) 1/1200 ; 60 3 atan (1516) 332 which itself goes back to Regiomontanus's table of 1468, merely Regiomontanus tan (1490,1504,. . . ) (1436-1476) 1 • 10 5 Engel? sin (1490) (1533), (1455-1512) 1 60000 by dropping two digits and no rounding. 333
					Copernicus (m) sec (1530?) (1473-1543) 1 • 10 4 Engel's table with R = 60000 (figure 16) is derived from Regiomon-Fine sin (1530) Gaurico sin (1524) (1475-1558) 10 10 5
				(1494-1555) 1 60; 60 2 tanus's large sexagesimal table and was used by Schreckenfuchs in 1569
				(see § 6.14).		Regiomontanus sin (1541,1561) (1436-1476) 1 6 • 10 6	Regiomontanus sin (1541,1561) (1436-1476) 1 10 7
				Copernicus sin (1543) (1473-1543) 10 10 5 Immediately following the table of sines for radius 60000 published Rheticus Fine sin (1542) (1494-1555) 1 60; 60 2 sin (1542) (1514-1574) 1 10 7 Fine Gemma Frisius cot (1545) (1508-1555) 1 • 10 5 in 1524, 334 there is an additional table of sines with radius 100000 and for sin (1550) (1494-1555) 1 60; 60 2 Rheticus sec/csc,tan/cot (1551) (1514-1574) 10 10 7 Rheticus sin/cos (1551) (1514-1574) 10 10 7 every 10	Gemma Frisius (1508-1555)	atan (1545) 1/1200 ; 60 3
										Reinhold	sin (1554)	Reinhold	tan (1554)
										(1511-1553) 1	10 7	(1511-1553) 1	10 7
								Maurolico	sin,sec,tan (1558)
	Rheticus (m) sin/cos,sec/csc,tan/cot (1560?)				(1494-1575) 1 •	10 5
	(1514-1574)	10	10 7
		Schreckenfuchs	sin (1569)	Schreckenfuchs tan (1569)	Schreckenfuchs	sin (1569)	Eisenmenger sin (1562)
		(1511-1579)	15 60000	(1511-1579)	1 •	10 5	(1511-1579)	15 60; 60 3	(1534-1585) 1	10 7
										Witekind	sin (1576)
										(1522-1603) 1	10 5
		Rheticus (m) sin/cos (1570?)						Peucer	sin (1579)
		(1514-1574)	10	10 15						Viète	sin/cos,tan/cot,sec/csc (1579)	(1525-1602) 1	10 5
										(1540-1603) 1	10 5 -10 9	Bressieu	tan,sec (1581)	Bressieu	sin (1581)
										Giuntini	sin (1581)	Padovani sin (1582)	(c1546-1617) 1	60; 60 2	(c1546-1617) 1 60; 60 2
										(1523-1590) 1	10 5	(c1512-?) 1	10 5
								Fincke	sin (1583)	Fincke	sec (1583)	Fincke	tan (1583)
							(1561-1656) 1	10 7	(1561-1656) 1	10 7	(1561-1656) 1	10 7
										Bürgi (m)	sin (1587)
					Gallucci	sin (1588)			Clavius	sin,sec,tan (1586)	(1552-1632) 1 60; 60 4
					(1538-1621) 1 60000			(1538-1612) 1	10 7
										Lansberge	sin,sec,tan (1591)
										(1561-1632) 1	10 7
										Magini	sin/cos,sec/csc,tan/cot (1592)
										(1555-1617) 1	10 7
										Clavius	sin (1593)	Fale sin (1593)
										(1538-1612) 1	10 7	1	10 5
										Blundeville	sin,sec,tan (1594)
										(c1522-c1606) 1	10 7
	Rheticus/Otho csc/cot (1596)	Rheticus/Otho sin/cos,sec/csc,tan/cot (1596)	Ceulen	sin,sec,tan (1596)
		(1514-1574)	10	10 7	(1514-1574)	10		10 10		(1540-1610) 1	10 7
				Figure 13: The interrelationships between the main 15th and 16th century
				fundamental trigonometric tables. Corner squares (⬜) indicate no longer
				extant tables, unfilled corner circles (◯) indicate new computations, and
				filled corner circles (•) indicate computations based on earlier tables.

Tables marked "(m)" in the lower part are manuscript tables. See figure

14

for details on the links.

Figure

14

: The interrelationships between the main 15th and 16th century fundamental trigonometric tables (cont'd, see figure

13

). The number of places of sexagesimal tables is shown as 60; 60 n , the first 60 being the value of R, and n being the number of additional sexagesimal places. ′ (figure

17

). As mentioned by

Delambre, 335

this table was added by Gaurico (see § 6.2). Fine's tables from 1530 and 1550 are not mentioned by Glowatzki and Göttsche.

336

Fine's table published in 1530 and reprinted in 1550 gives the sines for a radius R = 60, at intervals of 1 ′ and to two sexagesimal places.

  Regiomontanus's table contains a column of differences, whose values can be interpreted as the sixths of the differences of the sines. For instance, (5197024 -5196152)/6 = 145.333 . . . and that column starts with 145. Engel's table contains exactly the same value 145, although it is basically meaningless in that reduced table. By analyzing the correspondence of Regiomontanus with Bianchini and others, Glowatzki and Göttsche have also shown that Regiomontanus actually did not himself use the table R = 60000 printed in 1490, and whenever he used this radius, he drew the values by dropping two digits from his large table and rounding the value. Such a table may have been held at the Seitenstetten Abbey until 1924, but it was then sold and had not been located by the authors. If this table surfaces again, one should check whether its values are truncated or rounded. Engel's table (figure 16) is found again in the 1504 edition, 348 where the title of the work explicitely mentions this sine table. It is also found in later editions of the Tabulae directionum profectionumque, where it is often attributed to Regiomontanus. My modern reconstruction 349 is based on the 1504 edition which has less idiosyncrasies than the 1490 version.

	Angle	R	E
	60 • 0 ′	5196152 51961
	60 • 1 ′	5197024 51970
	60 • 2 ′	5197896 51978
	60 • 3 ′	5198768 51987
	60 • 4 ′	5199639 51996
	60 • 5 ′	5200510 52005
	60 • 6 ′	5201380 52013
	60 • 7 ′	5202350 52022
	60 • 8 ′	5203119 52031
	In this sample, we can also see that Engel introduced an additional error
	for 60 • 7 ′ .
	Moreover,
	, pp. 48-49]

346

This is an additional proof for the fact that the table for R = 60000 actually did not exist before it was prepared for printing in 1490. Instead, Glowatzki and Göttsche 347 assume that Regiomontanus had made a table with R = 60000 for himself, but different from the printed one.

  table was reprinted in 1557, 352 together with Regiomontanus's table of tangents but the latter only up to 50 • . It is tempting to consider that Gaurico took his sines from a manuscript of Regiomontanus's table for R = 10 7 and truncated or rounded the values (this was also suggested by Glowatzki and Göttsche 353 ), but this is actually not the case. Gaurico's values differ both from the truncated values and from the rounded ones of the 10 7 table.

  , p. 178] 354 This procedure anticipates what Copernicus has probably done in some places in the sine table included in his 1543 opus, although on the basis of Regiomontanus's full sexagesimal table.

  the values of the secant for each degree357 and for R = 10000, overlayed toRegiomontanus's table of tangents. A reproduction of Copernicus's table was given by Glowatzki and Göttsche. Sec x = Tan x +R 2 . Rosi ńska also observed that Copernicus's table of secants is not copied from Bianchini's table of cosecants. 361 Rosi ńska concluded therefore that Copernicus used neither Bianchini's nor Regiomontanus's tables for his table of secants. Glowatzki and Göttsche observed that since Copernicus's values are very accurate, he must have used a manuscript of Regiomontanus's tables with R = 6 ⋅ 10 6 or R = 10 7 , later published in 1541. to my experiments, using either of Regiomontanus's tables, computing the exact secants, by mere division, as Curtze suggested, and rounding, will give almost always Copernicus's values, except for 88

	358
	Curtze considered that Copernicus had computed the secants from
	the cosines 359 but Birkenmajer 360 thought that the secants were computed
	using the formula

and [Folkerts et al. (2019)].

gave 362 But still, 363 Now, according • and 89

  table of sines. Fine's table gives the sines for a radius R = 60, at intervals of 1 ′ and to two sexagesimal places (figure 18). This table is not based on Engel's 1490 table 366 as one might think. Instead it must be based directly on a manuscript of Regiomontanus's large sexagesimal table, without truncating. In 1542, Fine published his De sinibus libri II which was the first treatise solely on trigonometry to be printed in France. 367 This work is an appendix of Fine's De mundi sphaera. 368 Ross is very critical of this work and considers that it was unoriginal and out of date, 369 because it does not contain any contributions to trigonometric mathematics, and because it lags behind the developments soon introduced by Rheticus for the unification of sines, shadows, etc., in a same framework. Fine also appears to be unaware of Regiomontanus's De triangulis omnimodis 370 published in 1533 and which laid the foundations of modern trigonometry. But on the other hand, Fine's purpose with this book was pedagogical and he succeeded in contributing to the revival of mathematics in Paris. The 1542 De sinibus libri II reprinted the table of sines published in 1530, with only minor variations. The layout is the same, although the table was obviously reset, as can be observed on the last lines of each page. The second edition of De sinibus libri II was published in 1550, is the only sexagesimal table based on Regiomontanus's tables, apart from those of Schreckenfuchs 374 published in 1569 and of This table appears to have been obtained by merely dropping the last two digits of Regiomontanus's table for R = 10 7 , without any rounding. 384 Regiomontanus's table having only been printed in 1541, Apian must have had access to one of the manuscripts of the 1468 table. Moreover, as observed by Glowatzki and Göttsche, the manuscript table used by Apian is in fact the same as the one used in the 1541 printed edition of Regiomontanus's table. visit, he also made it possible for Erasmus Reinhold to become closely acquainted with Copernicus's theory, leading to the publication of the Prutenic tables in 1551. But in 1542, before the publication of Copernicus's De revolutionibus, Rheticus published its trigonometrical chapters under the title De lateribus et angulis triangulorum. 396 This work contains a table of sines at intervals of 1 ′ and for a radius of 10 7 (figure 25). The sines were in fact not called sines, but half-chords. And the table is actually by Rheticus and not by Copernicus. 397 More precisely, Rheticus took the sines from Regiomontanus's table 398 published in 1541 (or from a common manuscript source). This is in particular confirmed by printing errors found in both editions. 399 Some of the typos that remain in Rheticus's table are in fact so conspicuous that they should have been corrected by Rheticus. Rheticus however did not carry over Regiomontanus's differences, but introduced the actual differences. In her article on Copernicus's tables, Rosi ńska 400 hypothesizes that Copernicus had first planned to append to his work a table of sines with R = 10 6 but that this table was eventually replaced by Rheticus's one with seems to be the first table of sines where a value can easily and explicitely be read in two different ways. This novelty was As we have seen earlier, Copernicus (1473-1543) 411 had computed a small table of secants, perhaps around 1530, and in 1542 the trigonometric chapters of De revolutionibus orbium coelestium were published separately by Rheticus, together with a sine table. But in Copernicus's famous De revolutionibus orbium coelestium published in 1543 shortly before his death,

	observed by Stamm,	407 Rosen 408 and more recently by Husson. 409 For each
	value, there is both a reading using the first line and the first column
	(giving the sines), and a second reading using the last line and the last
	column (giving the cosines). Similar features are found again in the tables
	of Rheticus (1551), Reinhold (1554), Viète (1579), Clavius (1583), Magini
	(1592) and Rheticus/Otho (1596). Of course, earlier tables, including those
	385 of Regiomontanus, can also be read that way, but not explicitely, and it is
	And, as remarked by Kish, necessary to perform a (simple) computation to find the cosine of an angle, 386 Apian's sine table is the first printed table giving sines every minute and divided decimally, Regiomontanus's table having only been printed in 1541. for instance. I am giving separately a modern reconstruction of Rheticus's table. 410
	I am giving separately a modern reconstruction of Apian's 1533 table. 387
	Apian's tables seem to have been copied by directly or indirectly by Bas-6.7 Copernicus (1543)
	santin in 1557, 388 by Witekind in 1576,	389 by Peucer in 1579, 390 , by Giuntini
	in 1581,	391 , by Padovani in 1582, 392 and indirectly by Fale 393 in 1593.
	R = 10 7 . In the past von Braunmühl, 401 Cantor,	402 Busard,	403 Rosen 404 and Fol-
	kerts

371

but the sine table now uses a different layout (figure

30

). In the two editions of this work (1542 and 1550), Fine's introduction gives the sines up to 90

• at intervals of 3 • 45 ′ . In a second table, he gives the sines up to 7 • 30 ′ at intervals of 15 ′ . These two tables were not given in the 1530 De geometria 372 and were presumably not used for the computation of the table published in 1530. None of these tables are mentioned by Glowatzki and Göttsche. 373 Fine's table his 405 were of the opinion that Rheticus was the real author (computer) of the sine table. And Zinner thought that the author of the table was Copernicus himself and that he may have been inspired to construct a table with R = 10 7 by a glimpse of Regiomontanus's table. 406 Rheticus's table 412 Copernicus included another table of sines, with an interval of 10 ′ and a radius R = 10 5 (figure 26). Like in the excerpt published in 1542, the sines were actually not called sines, but half-chords. Copernicus's table shows a few deviations from the values obtained from Regiomontanus's table with R = 10 7 when the values are rounded to R = 10 5 . For instance there are three errors in the first 36 values (from 0 • to 6 • ) and Copernicus gives Sin 0 • 40 ′ = 1163 instead of 1164, Sin 1 • 30 ′ = 2617 instead of 2618, and Sin 4

  Glowatzki and Göttsche were of the same opinion and concluded that Copernicus must have used a manuscript version of Regiomontanus's table of sines with R = 10 7 , 413 as he certainly did for his table of secants. 414 Copernicus probably did not take the sines directly from the table published in 1542, although it is almost identical to that of Regiomontanus, because Copernicus's manuscript must have been ready long before its publication. In any case, if Copernicus took his values from Regiomontanus, he also made some corrections to that table, as Regiomontanus's typos for the sines of 36 think that it is possible to get a somewhat better understanding of the elaboration of Copernicus's table. I have said above that among the first 36 values of Copernicus's table, there are three obvious errors when comparing them to the rounded values from Regiomontanus's table with R = 10 7 . For instance, for 0 • 40 ′ , Regiomontanus's table gives 116353, and Copernicus has 1163, which looks like a truncation, but for almost every other angle Copernicus's sine is the rounded and not truncated value obtained from Regiomontanus's table. Now, if we start with Regiomontanus's sexagesimal table, that is, the table with R = 6 ⋅ 10 6 , the decimal values can be obtained by dividing

	• 10 ′ and 51 • 20 ′ , reprinted in 1542, have been corrected in the
	De revolutionibus orbium coelestium.
	Regiomontanus's values by 6. Considering only the first 36 values in
	Copernicus's table (from 0 • to 6 • ), it appears that until 3 • 50 ′ , one obtains
	Copernicus's values by dropping one digit of Regiomontanus's table and
	rounding, then dividing by 6, then rounding.

415

However, I

  we look at the last page of Copernicus's table, also containing 36 values, a comparison with Regiomontanus's table with R = 10 course only sampled the first and last 36 values of Copernicus's table, and this should be further investigated. It suggests however that different computations may have been involved in the making of Copernicus's sine table, and probably that some parts of Copernicus's table are based on Regiomontanus's sexagesimal table, whereas others are based on the table with R = 10 7 , in addition of involving different rounding schemes from the same source. It is also possible that some values were based on other tables. But among the 16 values that are incorrectly rounded on the last page of Copernicus's table when starting with Regiomontanus's great sexagesimal table, only 10 of Copernicus's values are identical with those published by Apian in 1534. 417 It is therefore not possible to conclude that Copernicus used Apian's table. Perhaps for some comparisons, but not for all values. Given this somewhat confused situation, it is understandable that Copernicus's table led to other opinions or conclusions. For instance,

		7 reveals five
	rounding errors (84 • 30 ′ , 85 • , 86 • 40 ′ , 87 • 40 ′ and 89 • 50 ′ ). But if we start with
	the great sexagesimal table as above, there are in fact even more errors, 16
	altogether. For instance, for 86 • one obtains 99757, and not Copernicus's
	99756.
	I have of Stamm 418 wrote that Copernicus probably compared his values to those
	published by Apian in 1534, 419 but I have just shown that this is not conclu-
	sive. Folkerts	420 thought that Copernicus had computed the table himself,
	since he could not have been able to use Regiomontanus's table for R = 10 7
	which was only published in 1541. Looking for Copernicus's source, Swerd-
	low and Neugebauer 421 excluded most sources, including Regiomontanus's
	tables printed in 1541, but they did not conclude further. And according
	to Rosi ńska, 422 Copernicus did not use Regiomontanus's table for his table
	of sines, although she did not provide another theory for the origin or
	calculation of the table.

  Theoricae Novae Planetarum. When Rheticus came back from his visit to Copernicus, Reinhold studied Copernicus's theory closely and after the publication of Copernicus's De revolutionibus orbium coelestium, Reinhold made detailed annotations of this work. 438 Between 1544 and 1551, Reinhold worked on recasting Copernicus's theory in handier tables and in 1551 he finally published his Tabulae prutenicae coelestium motuum (Prutenic tables).In his Primus liber tabularum directionum published in 1554 after his death,439 Reinhold gave a table of sines (figure34) and a table of tangents (figures 32 and 33), both with radius R = 10 7 and at intervals of 1

	palatinum 434 published in 1596 is still marred by this problem which will
	only fully be solved by Pitiscus in the early 17th century. 435
	I am giving separately a modern reconstruction of Rheticus's table. 436
	6.10 Reinhold (1554)
	Erasmus Reinhold (1511-1553) was a German astronomer and mathemati-
	cian. He was born in Saalfeld, Germany. In 1536 he became professor of
	mathematics at the university of Wittenberg.	437 In 1542, Reinhold published
	a commentary on Peuerbach's
	, p. 185]
	432 It will be interesting to see to what conclusions came [Pritchard (2021)] who seems to
	have conducted a similar investigation, but whose result is not yet published at the time I
	am writing this.
	433 [van Brummelen and Byrne (2021)]

′ . 440

  • , there must have been a computation error, as Reinhold has actually computed 9848085/1736482 instead of 9848078/1736482. This error is not present in Rheticus's table. In the last part of the table, Reinhold added the tangents at intervals of 10 ′′ . The values themselves are not as accurate as one might wish, but this matters little here. What does interest us is to find out how Reinhold computed these values. These computations seem so far not to have been analyzed, not even by Glowatzki and Göttsche. 447 At first, this part of the table suggests a new computation of the sines and cosines of 10 ′′ , 20 ′′ , etc., up to 59 ′ 50 ′′ , but this was most certainly not the case. It is in fact very easy to see what Reinhold has done, because the ratios behind each tangent value can be reconstructed. I am giving here only some samples:

			443
	My samples may or may not be representative of the entire table, and it
	would be useful to conduct a thorough analysis of Reinhold's table of tan-
	gents. 444 It seems in particular that Reinhold did not copy Rheticus's values
	given in 1551 at intervals of 10 ′ . Reinhold's error on Tan 80 • , incidentally, is
	found again in Fincke's table, 445 as well as in Clavius's table.
	Angle	Fraction	Value
	89 • 0 ′ 10 ′′	9998370 174038	57.44935014 . . .
	89 • 0 ′ 30 ′′	9998502 173070	57.77143352 . . .
	89 • 0 ′ 50 ′′	9998519 172101	58.09680943 . . .
	89 • 30 ′ 10 ′′	9999625 86780	115.22960359 . . .
	89 • 30 ′ 20 ′′	9999628 86296	115.87591543 . . .
	89 • 59 ′ 10 ′′	9999999 2424	4125.41212871 . . .
	89 • 59 ′ 20 ′′	9999999 1939	5157.29706034 . . .
	89 • 59 ′ 30 ′′	10000000 1455	6872.85223367 . . .
	89 • 59 ′ 40 ′′	10000000 970	10309.27835051 . . .
	89 • 59 ′ 50 ′′	10000000 485	20618.55670103 . . .
	443 [Rheticus (1551)]

446

  and these values are obtained by linear interpolation of Regiomontanus's sines. There may again be some slight inaccuracies, and the value of Sin 59′ 50 ′′ would for instance have been better at 174039 than 174038. The numerators used by Reinhold were also obtained by interpolation from Regiomontanus's values. For instance, for 89• 30 ′′ , 9998502 is just halfway between Regiomontanus's sine values 9998477 and 9998527. But for Tan 89

			• 59 ′ 30 ′′ ).
	We can see in this sample that Reinhold uses
	Sin 10 ′′ = 485	Sin 20 ′′ = 970	Sin 30 ′′ = 1455
	Sin 40 ′′ = 1939	Sin 50 ′′ = 2424	. . .

  the difference between Maurolico's values and those computed from Regiomontanus's sines becomes much larger. The last four values are more accurate than the values that could have been obtained from Regiomontanus's table with R = 10 7 . Van Brum-and all that Maurolico needed was to take 2908.9 instead of Regiomontanus's 2909. There was no need to resort to bisections, trisections, etc., and to recompute the sines of small angles. There was also no need to interpolate as Regiomontanus did to construct his tables. And proceeding the above way does not require a very accurate value of π, as Ptolemy's value 3.1416 . . . is sufficient. Maurolico's third table is his table of secants, which he called tabella benefica. 469 It was actually Fincke who first named that function secant. There Maurolico claims that he worked on this matter in 1550, and it would then very likely be a work independent of that of Rheticus. Magini had also assumed that Maurolico's work was an independent one, not influenced by Rheticus. 473 One might therefore assume that it was directly computed from Regiomontanus, probably from the 1541 edition. Maurolico's table of secants is in fact very accurate, and more accurate than what would have been obtained by a mere use of Regiomontanus's tables. 474 In order to explain this accuracy, Brummelen and Byrne claim that Maurolico computed the secants from the tangents, and not directly from Regiomontanus's sines, as claimed by Glowatzki and Göttsche. Tan θ = 631375 and 631375 2 + 100000 2 = 639245.172 . . . and Maurolico gives Sec θ = 639245.

	• 15 470 ′ , Fincke thought that Maurolico had copied his secants from Rheticus. 89 • 30 ′ , 89 • 45 ′ , 89 • 55 ′ and 89 • 59 471 ′ . But Brummelen recently gave an edition of Maurolico's short manual The values of the sines differ from those of the earlier tables with R = 10 5 , namely those of Gaurico (1524) 459 and Apian (1533). 460 Instead, of his tabella benefica.
	Maurolico seems to have taken his values from Regiomontanus's table by
	dropping two digits and rounding.	461 And, contrary to what von Braun-
	mühl wrote, 462 Maurolico was unaware of Rheticus's Canon doctrinae trian-gulorum 463 published in 1551. 464
	As far as the tangents are concerned, Glowatzki and Göttsche wrote
	that Maurolico took his values from Regiomontanus's Tabulae directionum
	profectionumque 465 and recomputed those above 45 • using Regiomontanus's
	466 sines. But according to Brummelen, 467 the tangents were copied from Re-giomontanus's 1490 table up to about 60 • . Above 60 475 • , the values of the tangents seem to have been recomputed from Regiomontanus's sines and According to van Brummelen and Byrne, Maurolico used the relation
	Brummelen implies (his table 3) that they have been recomputed that way until 89 • 15 sec 2 θ = tan 2 θ + 1
	which transcribes into	Sec 2 θ = Tan	2 θ + R
	This appears to work for 81 • , but merely computing 1/1564345 would have
	given the correct result too.
	469 [Delambre (1819), p. 440]
	470 [Fincke (1583)]
	471 [van Brummelen and Byrne (2021), p. 200]
	472

′ inclusive. Beyond 89 • 15 ′ 472 2

when the radius is R. For instance, for θ = 81

• ,

  Commentaria in Sphaeram Ioannis de Sacrobusto, 483 he reprinted Regiomontanus's table of tangents from one of the editions of the Tabulae directionum profectionumque (figure 43). But Schreckenfuchs also gave two tables of sines. His first table of sines covers two pages and uses the radius R = 60000 and an interval of 15 ′ (figure 44). This table is likely based on the table of sines found in Regiomontanus's Tabulae directionum profectionumque (1490, 1504, 1550, 1552 or 1559), namely Johannes Engel's table (figure16), as the values are truncated and not rounded from the large table with R = 6 ⋅ 10 6 .Schreckenfuchs's second table spans six pages, also with 15 ′ intervals, and uses a radius R = 60 and three sexagesimal places (figure45). However, the last place is always given as 0 or 30. This second table cannot have been obtained from the first one. For instance, for 15 ′ , Schreckenfuchs gives the sine as 261 (60000 × sin 15 . It is therefore to assume that now Schreckenfuchs used the sines in Regiomontanus 1541 (or 1561). For 15 ′ , Regiomontanus gave 26180, and this then would lead to 0

	magest of Ptolemy and in 1556, he published a commentary on Peuerbach's
	Theoricae Novae Planetarum.
			p 15 ′ 42 ′′ 28.8 ′′′ that
	Schreckenfuchs could have rounded to 0 p 15 ′ 42 ′′ 30
	1562, Eisenmenger published his Libellus geographicus 478 in which he
	gave a table of sines with R = 10 7 and at intervals of 1 ′ (figure 42). This
	table was certainly also copied from Regiomontanus,	479 and probably from
	the 1541 printing.	480 The layout and headings of Eisenmenger's table are
	practically identical to those of Regiomontanus's published table, except
	that Eisenmenger put only half a degree in each column.

482

In 1569, in his ′ = 261.7 . . .), but 261 would give a sine of 0 p 15 ′ 39 ′′ 36 ′′′ , not 0 p 15 ′ 42 ′′ 30 ′′′ ′′′ .

  2 ′′ table does not seem to have survived, but the 1 ′ table resurfaced a few years ago. At that time, I made modern reconstructions of both tables.

544

Bürgi's surviving sine table

(figure

  • is given as 57; 57, 19, 58, 43 which corresponds to the decimal value 0.965925827, the correct value being 0.96592582628906 . . .. Rheticus must have had such accurate values already in the 1570s, before Bürgi, but with the exception of Rheticus, Bürgi's table was probably the most accurate sine table constructed at the end of the 16th century. Bürgi's table can be compared to those of Fine, Schreckenfuchs and Bressieu which are also sexagesimal tables, but which are less accurate and based on Regiomontanus's sines. Magini gives another table with R = 10 7 and at 1 ′ intervals, with sines, versines, tangents and secants, but the values are not those of the 1592 table. Instead, Magini took the values from Rheticus and Othos' Opus palatinum (1596).

	In contrast, Rheticus and Otho's Opus palatinum (1596) 545 gives the value
	9659258263 for Sin 75 • , and this is correct to 10 places. Rheticus also gives
	the sines every 10 ′′ .
	And in 1613 Pitiscus	546 gave 96592,58262,89067, instead of the correct
	96592,58262,89068.

  Thomas Fale (born c1560?) was an English mathematician. Very little is known of him. = 8803, when Witekind gave the correct 8802 (compare figures 46 and 69). As observed by De Morgan and Goodwin, Fale's table may be the earliest sine table printed in England. new names introduced by Fincke in 1583.

566

In 1593, Fale published his Horologiographia.

567

This work, which is the only one known of him, appears to be the first book in English on sundials.

568

It contains in particular a table of sines (figure

69

) which was presumably copied from Witekind's Conformatio horologiorum sciotericorum etc.

569 published in 1576 and with which it shares the values and the layout.

570

There are however some slight differences, and Fale gives for instance Sin 5

• 3 ′

  Rheticus (1514-1574) 583 continued to work on a more extensive project, where the six trigonometric functions would be given every 10 ′′ and for a larger radius. As observed by Zeller, 584 "Rheticus built his trigonometry on the foundation established by Regiomontanus." Rheticus embarked on totally new computations, but his work was only completed after his death by Lucius Valentinus Otho (c1545-1603) and published in 1596 in the Opus palatinum 585 (figure 72). Otho had met Rheticus in 1573 and Rheticus had asked him to complete his work.

  , p. 43] I have therefore assumed that the shorter table is in fact an older table, perhaps computed by Rheticus around 1560. 588 I believe that after his Canon doctrinae triangulorum (1551), which already used R = 10 7 and an interval of 10 ′ , Rheticus decided first to compute the functions with an interval 60 times smaller, that is 10 ′′ , but with the same radius. This is what I have shown in figure 13. It is in fact easy to see what were the computations in this first attempt at a 10 ′′ table (and it would consequently be rather straightforward to complete this table with those for the sines, cosines, tangents and secants, which presumably existed). We can observe that the cosecants at 1 ′ intervals were merely obtained by the fractions 10 14 /2909, 10 14 /5818, 10 14 /8727, 10 14 /11635, 10 14 /14544, etc. In other words, Rheticus merely used the sines found in Regiomontanus's table, apparently sometimes with slight adjustments (as for Csc 4 ′ or Csc 10 ′

There is a vast literature on numerical tables, and I am directing the reader to a number of general surveys, such as[START_REF] Hutton | Mathematical tables: containing common, hyperbolic, and logistic logarithms, also sines, tangents, secants, and versed-sines, etc[END_REF]],[START_REF] Morgan | Augustus De Morgan. Article "Table[END_REF]],[De Morgan (1851)],[START_REF] Glaisher | [END_REF]],[START_REF] Davis | [END_REF], pp. 1-40],[Campbell-Kelly et al. (2003)], etc. This document also mentions many people, and I am not always directing to specific biographical information for each of them. Valuable informations can in particular be found in the notices of[Hockey (2014)], in particular on Al-Battāni, Ab ū al-Wafā

In the 12th century, the Christians, assisted by Jewish scholars, translated many Arabic works. In particular, Gerard of Cremona (c1114-1187) translated in Latin the canons of the tables of Toledo and, as mentioned above, Adelard of Bath made Al-Khwārizmī's astronomical tables accessible to the Latins.Around 1272, the Alfonsine tables were constructed in Toledo under the guidance of King Alfonso X of Castile (1221-1284).

See [Bond (1920), p. 319] and[START_REF] Curtze | [END_REF]].

[Glowatzki and Göttsche (1990), p. 81]

[Glowatzki and Göttsche (1990), pp. 85 and 89]

[Klug (1943), p.57]

[Ptolemaeus (1984), pp. 57-60]

In figure13, the number of places of sexagesimal tables is shown as 60; 60 n , the first

This table is reproduced byGlowatzki and Göttsche [Glowatzki and Göttsche (1990), pp. 116-123]. They draw the attention to incorrect statements by[Cantor (1900), p. 182]

In Italy, Regiomontanus also became associated with Giovanni Bianchini. Part of their correspondence still survives.

 215 Durand writes that211 See[Rosen (1975a), p. 348] and[Schmeidler (1977), p. 316].212 See[Rose (1975), pp. 90-117],[Schmeidler (1977), p. 316],[Grössing (1980)],

In 1471 Regiomontanus moved to Nuremberg. There he set up a printing press for the purpose of publishing the most important classical scientific works,

See [Shank (2007)] for some excerpts.

[Hallam (1837), p. 259]

[Glowatzki and Göttsche (1990), pp. 11-24]

[von Peuerbach and Regiomontanus (1541)]

[von Peuerbach and Regiomontanus (1541),[START_REF] Roegel | A reconstruction of Regiomontanus's great tables of sines[END_REF]

See [Rosi ńska (1984), pp. 503-504] and[Rosi ńska (1987), pp. 421-422].

[van Brummelen (2018)]

[START_REF] Roegel | A reconstruction of Regiomontanus's great tables of sines[END_REF]

[Glowatzki and Göttsche (1990), p. 27]

[Glowatzki and Göttsche (1990), p. 94]

See [Folkerts (1977), p. 234] and[Mett (1996), p. 96].

[Folkerts et al. (2016), p. 136]

[Glowatzki and Göttsche (1990), p. 126]

[Glowatzki and Göttsche (1990), pp. 127-147]

[START_REF] Roegel | A reconstruction of Regiomontanus's great tables of sines[END_REF]

[Glowatzki and Göttsche (1990), p. 147]

[Regiomontanus (1490)]

[Regiomontanus (1490)]

[Chabás Bergón (2019), p. 383]

[Fincke (1583)]

[Glowatzki and Göttsche (1990), p. 183]

[START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF]

[Glowatzki and Göttsche (1990), p. 46-47 and 145-147]

The errors which have not been reported can easily be found either by a careful comparison of my cleaned tables with Regiomontanus's tables, or by checking the tables given by Glowatzki and Göttsche.

See the files roegel2021regio6.txt and roegel2021regio10.txt.

We could also consider the computation of 5 ′ pivots from incorrect 15′ pivots, for instance by shifting these pivots, but I don't think we would reach significantly different results.

The two values I am using are in fact given in[von Peuerbach and Regiomontanus (1541)].

This is in fact also pretty obvious, because like in the case of the trisection, we have a sequence of values of which the median is necessarily equal to the average, and the first first difference is obtained by subtracting twice the second difference from the median value.

[Glowatzki and Göttsche (1990), pp. 178-179]

[Glowatzki and Göttsche (1990), p. 192]

[Glowatzki and Göttsche (1990), p. 150]

This procedure is reminiscent from that probably used by Gaurico in 1524, although Gaurico started with Engel's table.

[van Brummelen and Byrne (2021), p. 205]

This concurs with De Morgan who considered that "it is clearly nothing but a previous attempt made before the larger plan was resolved on."([De Morgan (1851), p. 599] and[Glaisher (1873), p. 43])

recomputed in 2021 by D.[START_REF] Roegel | A reconstruction of Fine's Tabula proportionalis[END_REF]].]

Figure 20: An excerpt of Apian's table of sines [Apian (1534)].

Figure 21: An excerpt of Apian's table of sines [Apian (1541)].

After Regiomontanus

Most of the trigonometric tables printed in the 16th century actually use values or computations inherited from Regiomontanus's tables 311 (see fig-

ures 13 and 14). Rheticus (1514Rheticus ( -1574) ) was the only one to compute really new values which were eventually published in 1596 by Otho 312 and in 1613 by Pitiscus. 313 Bürgi also computed sines anew, but his table was not published and was not used by others. Among all these tables, Glowatzki and Göttsche distinguished those which retain the radius R = 10

7 and those for which R = 10 5 . 314 But we should also consider separately the few sexagesimal tables based on Regiomontanus's tables, namely those of Engel, Fine, Schreckenfuchs and Bressieu.

The tables with radius 10 7 include those of Rheticus (1542 and1551),

Bressieu published in 1581. 375 Bressieu has actually copied Fine's table, from either of the three editions I have mentioned.

Certainly in order to help for the work with sexagesimal numbers, Fine had also published a sexagesimal multiplication table, his tabula proportionalis.

376

I am giving separately a modern reconstruction of Fine's 1530 and 1550 tables. 377

Apian (1533)

Peter Apian (1495Apian ( -1552)), also known as Petrus Apianus, was actually born Peter Bennewitz, or Peter Bienewitz, in Leisnig, Germany. 378 He was active in astronomy and geography and was a popularizer of astronomical and geographical instrumentation. 379 Apian studied at the University of Leipzig from 1516 to 1519 and then for two years in Vienna. His first major work was his Cosmographia (1524), which was later revised by Gemma Frisius (1508-1555), Apian's student. Apian's second major work was his Astronomicum Caesareum (1540) which displayed an elaborate typography and the use of sophisticated volvelles.

From 1526 until his death he occupied the chair of mathematics and astronomy at the University of Ingoldstadt.

Apians's mathematical work is linked to Regiomontanus's writings. 380 He published his work on sines in 1533.

In his Introductio geographica published in 1533, 381 Apian provides a table of sines with R = 10 5 and for every minute of the quadrant (figure 19). The same table was reprinted in 1534 in Apian's Instrvmentvm primi mobilis 382 (figure 20) and in 1541 in his Instrumentum sinuum 383 (figure 21).

375 [START_REF] Bressieu | Maurice Bressieu. Metrices astronomicaee libri quatuor[END_REF]] 376 [START_REF] Roegel | A reconstruction of Fine's Tabula proportionalis[END_REF], [START_REF] Roegel | A reconstruction of Fine's Tabula proportionalis[END_REF]] 377 See [Roegel (2021g)] and [Roegel (2021h)]. 378 For surveys of Apian's life and works, see in particular [Günther (1882)], [Gallois (1890a), pp. 102-116], [START_REF] North | [END_REF]], [Kish (1970)] and [Röttel (1995)]. See also the early notice by Gessner [Gessner and Simmler (1574), p. 552]. 379 See in particular [Lindgren (2007)] for some background on land surveys. 380 See [START_REF] Kaunzner | Zur Mathematik Peter Apians[END_REF]], [Folkerts (1995)] and [Lindgren (2007) melen therefore suggested some kind of independent computation.

468

I believe however that the threshhold for that "independent" computation occurs earlier than 89

• 15 ′ . In fact, it is easy to see what Maurolico has done for the last values of the table. He basically recomputed the required sines with one more digit and used them for the tangents. For instance,

• for 89 ′ to be 87265.3. This procedure was applied at least as early as 85

• and we can see for instance that the ratio 9961947/871557.4 gives exactly Maurolico's value, but that 9961947/871557 (Regiomontanus's values) does not.

Maurolico's accurate computation of the last tangents then boils down to a one digit more accurate value of sin 1

′ than that provided by Regiomontanus. I don't know how Maurolico obtained that value, but there is a very simple way, which is to observe that the sine of a small angle measured on the circumference (in radians) is almost equal to the angle itself. Therefore, with R = 10

Viète (1579)

François Viète (1540-1603) was a French mathematician whose work on the new algebra was an important step towards modern algebra. According to Zeller, Viète "was the foremost mathematician of France in the sixteenth century." 492 Viète received a bachelor's degree in law in 1560 and held a number of official positions. In 1573, the King Charles IX made him counselor to the parlement of Brittany. He came back to Paris in 1580. Among his many works is his In artem analyticem isagoge, the earliest work on symbolic algebra (1591).

488 For a summary of Peucer's life and works, see [Kolb (1976)].

489 [START_REF] Peucer | De dimensione terrae et geometrice numerandis locorum particularium intervallis ex Doctrina triangulorum sphaericorum & canone subtensarum[END_REF]]. Earlier editions from 1550 [START_REF] Chassagnette | La géométrie appliquée à la sphère terrestre. Le De Dimensione terrae (1550) de Caspar Peucer[END_REF]] and 1554 do not include the table of sines. 490 [START_REF] Apian | obtained from Regiomontanus (10 7 ) by mere truncation, without rounding[END_REF]], [START_REF] Apian | Instrumentum primi mobilis, a Petro Apiano nunc primum et inventum et in lucem editum, etc[END_REF]] and [START_REF] Apian | Instrumentum sinuum, seu primi mobilis, nuper a Petro Apiano inventum[END_REF]].

491 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. 169] 492 [Zeller (1944), p. 73]. For summaries of Viète's life and works, see [START_REF] Morgan | Augustus De Morgan. Article "Vieta, Francis[END_REF]], [Ritter (1895)], [START_REF] Busard | [END_REF] 48).

The printing of the table was started in 1571 but it was only completed in 1579.

496

This was the first published canon giving the trigonometric functions every minute, but on the other hand it gave them to less places than Rheticus' 1551 table (which however only had an interval of 10 ′ ). The sines from 0

• to 45

• and the cosines from 0

• 30 ′ to 45

• were taken from Regiomontanus (or Reinhold) and rounded or not, depending on the range of the table. Glowatzki and Göttsche had observed that Viète had recomputed the sines from 89

• 61 ′ to 90 For the tangents and secants, Glowatzki and Göttsche wrote that they were recomputed from Regiomontanus's values. 498 But we can actually tell a bit more. First, we can see that Viète computed the secants from 0

• to 45

• by inverting his cosines (and not those of Regiomontanus). The tangents between 0

• to 45

• were computed by using Regiomontanus's full values. 493 See [START_REF] Joachim | Canon doctrinae triangulorum[END_REF]] and [Hunrath (1899)].

494 [Viète (1579)] 495 See [Hunrath (1899)] and [Delambre (1819) [Glowatzki and Göttsche (1990), pp. 154-155] 498 [Glowatzki and Göttsche (1990), pp. 189, 196] 89 Then, the cotangents from about 5

• to 45

• were also computed from the ratios cos / sin using Viète's values (and not by inverting the tangents). The cosecants from about 5

• to 45

• were computed by inverting Viète's values of the sines.

But for the cotangents and cosecants between 0

• and about 5 These values are much more accurate than the tangents and secants given by Reinhold in 1554 and Fincke in 1583 for large angles, and obviously Viète had a much better understanding of the requirements for exact computations.

In his treatise on angular sections, 499 Viète describes a way to compute the sine of 1 ′ and other values he needed. This sine can be obtained as follows. First, like Ptolemy before, one can compute the sines of 18

• and 60

• . Trisecting 60

• twice, we obtain Sin 20

• and then Sin 6

• 40 ′ . Using quinquisection with 18

• , we obtain Sin 3 A persistent legend is also that the Canon mathematicus contained many errors, and that Viète consequently withdrew or re-purchased all the copies he could find and had them destroyed. This would then explain why this book is of great rarity. 503 In his Metrices astronomicae published in 1581, 513 Bressieu first gives a sexagesimal table of sines, with an unusual layout (figure 49). The sines and sines of the complementary angle (cosines) are given in two adjacent columns and the table therefore only runs up to 45

• . But Bressieu's layout is in fact very unusual, in that it doesn't use a footer line. In figure 49, the first column (headed 18) gives from top to bottom the sines from 18

• to 19

• . The second column (headed 71) gives the sines from 71

• to 72

• , but from bottom to top. Consequently, the second column actually also gives the cosines from 18

• to 19

• , from top to bottom. The values are given in degrees (or parts) with a radius of 60. For instance, the sine of 45

• is given as 42; 25, 35 as Sin 45 In figure 13, the accuracy of Bressieu's tables is indicated as 60; 60 2 , by which I mean a radius of 60 and two sexagesimal places. However, for 89

• and above, the values of the tangents and secants are given to only one sexagesimal place.

Bressieu is mentioned by Zeller 517 but not by Glowatzki and Göttsche.

Giuntini (1581)

Francesco Giuntini (1523-1590) was an Italian theologian and one of the most famous astrologer of the second half of the 16th century.

518

In his Speculum astrologiae 519 published in 1581, Giuntini included a table of sines with R = 10 5 and an interval of 1 ′ (figure 51). This table was most certainly copied from one of Apian's tables (1533, 1534 or 1541), 520 or perhaps from one of its derivatives.

The 1573 edition of the Speculum astrologiae does not contain this sine table.

Fincke (1583)

Thomas Fincke (1561Fincke ( -1656) ) This book does in particular contain tables of sines (figure 54), tangents (figure 53) and secants (figure 55) with R = 10 7 and intervals of 1 ′ . 527 And it was precisely Fincke who coined the names "tangent" and "secant" which had not been used before. Incidentally, Viète did apparently not approve of these names.

528

Fincke's sines do slightly differ from those of Reinhold, hence from those of Regiomontanus. It seems that Fincke made a number of small last figure adjustments to either Reinhold's or Regiomontanus's tables.

529

Given that the tangents were certainly taken from Reinhold (1554), 530 I assume that this was also the case for the sines.

As far as the tangents are concerned, we can see for instance that the last values agree with those of Reinhold, except for 89

• 53 ′ , 89

• 56 ′ , and 89

• 57 ′ . In the first case, Fincke's tangent is less accurate than Reinhold's, but in the two other cases the tangents are slightly more accurate.

Finally, Fincke's secants are the result of new computations. The val-524 For summaries of Fincke's life and works, see [Thorndike (1958), p. 140], [START_REF] Verdonk | [END_REF]], [Moesgaard (1972), p. 119-120] and [van Brummelen (2021), pp. 13-16]. Some authors have wrongly attributed some works of Kaspar Finck (1578-1631), who was a German theologian, to Thomas Fincke. In particular, the Methodica tractatio doctrinae sphaericae published in 1626, and cited by Moesgaard, is not by Thomas Fincke.

525 [Fincke (1583)] 526 See [START_REF] Schönbeck | Thomas Fincke und die Geometria rotundi[END_REF]] and [Zeller (1944), pp. 88-90]. 527 See also [Glaisher (1873), p. 42].

528 [Zeller (1944), p. 88] 529 [Glowatzki and Göttsche (1990), pp. 157-158] 530 [Reinhold (1554)] 94 ues differ from those of Rheticus's Canon doctrinae triangulorum. 531 Fincke, however, did not use Regiomontanus's sines, nor his own version to compute the secants. Instead, it seems that he computed the secants using his tangents. Fincke most certainly used the formula

to compute the secants, and when computing Tan 2 x, he must have kept only seven or eight significant figures and rounded, although the procedure may not have been systematic. This is reminiscent of the computation of cotangents by Rheticus in 1551, and also echoes a recent suggestion by van Brummelen and Byrne for the computation of secants by Maurolico. 532 In fact, it is only after I concluded the above that I noticed that van Brummelen and Byrne claimed that Fincke used this formula to compute the secant.

533

Fincke's tangents and secants, as well as Reinhold's tangents, are less accurate than those published by Viète in 1579. For instance, Fincke and Reinhold gave Tan 89

where only the first four figures are correct. Instead, Viète gives a value whose error is about 10000 times smaller. This is so because Viète took more accurate values for the sines and understood that this was necessary in order to obtain tangents with such an accuracy. I am giving separately a modern reconstruction of Fincke's tables.

534

After the publication of his Geometriae Rotundi, Fincke began to study medicine in Basel, Padua, Siena and Pisa. He became MD in 1587. He then returned to Denmark where he held the chair of mathematics at the University of Copenhagen from 1591 until 1602, but afterwards was more active as a physician and his mathematical activity never reached again the level of his 1583 book.

Euclid) in 1574 and was a supporter of the Ptolemaic system, and at the same time a friend of Galileo. He also helped develop algebra in Italy and introduced Stifel's symbols "+" and "-."

He was also a member of the Vatican commission that accepted the proposed calendar invented by Aloysius Lilius, that is known as Gregorian calendar.

In his last years he was probably the most respected astronomer in Europe and his textbooks were used for astronomical education for over fifty years in and even out of Europe.

In 1586, Clavius published an edition of Theodosius's sphaerics, 536 in which he included tables of sines (figure 56), tangents (figure 57) and secants (figures 58 and 59) with R = 10 7 and at intervals of 1 ′ .

537

Clavius's sines and tangents were taken from Reinhold (1554), 538 as they do not show the alterations made by Fincke. 539 But the secants instead were taken from Fincke's work (1583). 540 And in fact Clavius used the new names "tangent" and "secant" coined by Fincke. Clavius, however, corrected all the typos in the earlier editions (but not the last digit deviations).

541

On the other hand, Clavius's table has at least one typo, namely for sin 89

• 30 ′ which he gives as 9999616 instead of the correct 9999619. This error was corrected by Magini in 1592, and by Clavius himself in 1593.

Clavius's 1586 table, without the corrections of the typos, was copied by Blundeville in 1594.

Bürgi (1587)

This survey of 15th and 16th century trigonometrical tables based on Regiomontanus's work would not be complete without mentioning Jost Bürgi . Bürgi is well known as a (very) skillful mechanician, clockmaker and instrument maker, and also as an inventor of a table of progressions which could be used for the same purpose as logarithms.

542 536 [START_REF] Clavius | Theodosii tripolitae sphaericorum libri III[END_REF]] 537 [Zeller (1944), pp. 91-94] 538 [Reinhold (1554)] 539 [Fincke (1583)] 540 [Fincke (1583)] 541 [Glowatzki and Göttsche (1990), p. 158] 542 On Bürgi's table of progressions, see [START_REF] Roegel | [END_REF]]. The most recent overview of Bürgi's work, which contains many other references, is [Staudacher (2018)]. For reasons explained in [Roegel (2017)], I do not view Bürgi as a coinventor of logarithms.

Gallucci (1588)

Giovanni Paolo Gallucci (1538-1621) was an Italian astronomer and translator.

547 Among his notable translations, Gallucci published in 1591 his Della simmetria dei corpi humani, a translation of Dürer's "Four books on human proportion" (Vier bücher von menschlicher Proportion, 1528). He was also a private teacher to the Venetian nobility and a founding member of the second Venetian Academy.

Gallucci's most famous works are probably his Theatrum mundi, et temporis 548 published in 1588, and his Speculum Uranicum published in 1593, both featuring some volvelles. In the Theatrum mundi Gallucci also included a table of sines with a radius R = 60000 and an interval of 1 ′ (figure 61). This table was most certainly copied from Engel's table ( § 6.1), in one of the editions of the Tabulae directionum profectionumque where it appears, not necessarily the 1490 edition. Gallucci uses exactly the same layout, with six half-degrees per page, but he has dropped the differences. The values seem to agree, with the exception of a few transcription errors. separately a modern reconstruction of Lansberge's tables.

554

In 1632, Lansberge published his best known work, his Tabulae motuum coelestium perpetuae, for the prediction of planetary positions. Lansberge was a follower of Copernicus and his work is based on an epicyclic theory, but he did not accept Kepler's theories.

Lansberge died that same year in Middelburg in the Netherlands.

Magini (1592)

Giovanni Antonio Magini (1555-1617) 555 For a summary of Magini's life and works, see [START_REF] Campedelli | Dictionary of Scientific Biography[END_REF]].

556 [Magini (1592)] 557 [START_REF] Magini | [END_REF]] 558 [Zeller (1944), pp. 97-100] 559 [START_REF] Clavius | Theodosii tripolitae sphaericorum libri III[END_REF]] 560 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF]

Ceulen (1596)

Ludolph van Ceulen (1540-1610) was a German-Dutch mathematician born in Hildesheim. At some point he settled in Holland. In the 1580s and 1590s he was a fencing master as well as a mathematics teacher. He died in 1610 in Leiden.

578

In 1596 he published his main work, Vanden circkel etc.

579 where he gave among other things a 20-place approximation of π.

Ceulen's book also contains tables of sines, tangents and secants for R = 10 7 and at intervals of 1 ′ (figure 71). Ceulen's tables are certainly based on those of Lansberge 580 who is mentioned by Ceulen. 581 Ceulen uses the Three years later, the decimal logarithms were introduced by Briggs, 595 and expanded in 1624 and 1628. 596 They were however unrelated to trigonometric tables.

Edmund Gunter was the first to compute and publish tables of decimal logarithms of sines and tangents in 1620. 597 595 [Briggs (1617)] 596 [Briggs (1624), [START_REF] Vlacq | Arithmetica logarithmica[END_REF]] 597 [Gunter (1620)] 598 [START_REF] Briggs | Henry Briggs and Henry Gellibrand[END_REF]] 599 [START_REF] Glowatzki | Die Tafeln des Regiomontanus : ein Jahrhundertwerk, volume 2 of Algorismus[END_REF], p. ii] 600 [START_REF] Vlacq | Trigonometria artificialis[END_REF]]