Denis Roegel

A reconstruction of Regiomontanus's table of tangents (1490)

Introduction

This document gives exact (modern) reconstructions of Johannes Regiomontanus's table of tangents published in 1490 [START_REF] Regiomontanus | Tabule directionum profectionumque famosissimi viri Magistri Joannis Germani de Regiomonte in nativitatibus multum utiles[END_REF] (figure 1).

I am providing two reconstructions, one (I) with truncated values, the second (II) with rounded values, but Regiomontanus's values were actually computed from his great sexagesimal table, by dropping two digits, rounding, then dividing, as explained in my survey.

The context of this table and details of the construction are found in my separate survey [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF].

Figure 1 :

 1 Figure 1: Excerpt of Regiomontanus's table of tangents [1] (1490) (source: The Budapest University of Technology and Economics, 85.211, www.manuscriptorium.com).


Denis Roegel

A reconstruction of Johannes Engel's table of sines in Regiomontanus's

Introduction

This document gives an exact (modern) reconstruction of Johannes Engel's sine table published in 1490 as part of Regiomontanus's Tabulae directionum profectionumque [START_REF] Regiomontanus | Tabule directionum profectionumque famosissimi viri Magistri Joannis Germani de Regiomonte in nativitatibus multum utiles[END_REF]. More precisely, the reconstructed table is that of the 1504 edition [START_REF] Regiomontanus | Tabule directionum profectionumque famosissimi viri Magistri Joannis Germani de Regiomonte in nativitatibus multum utiles: Una cum Tabella sinus recti[END_REF] (figure 2), which has less idiosyncrasies than that of 1490 (figure 1).

Engel's table was obtained from Regiomontanus's great sexagesimal table, merely by truncating its values. The details of the construction as well as the context of these tables are found in my separate survey [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF].

Figure 1: Excerpt of Engel's table [START_REF] Regiomontanus | Tabule directionum profectionumque famosissimi viri Magistri Joannis Germani de Regiomonte in nativitatibus multum utiles[END_REF] (1490) (source: The Budapest University of Technology and Economics, 85.211, (www.manuscriptorium.com).

Figure 2: Excerpt of Engel's table [START_REF] Regiomontanus | Tabule directionum profectionumque famosissimi viri Magistri Joannis Germani de Regiomonte in nativitatibus multum utiles: Una cum Tabella sinus recti[END_REF] (1504) (source: Bayerische Staatsbibliothek).

Incipit Tabella.


Denis Roegel

A reconstruction of Peuerbach's table for his Quadratum geometricum

Introduction

This document gives an exact (modern) reconstruction of Georg von Peuerbach's quadratum geometricum table published in 1516 [START_REF] Von | Quadratum geometricum[END_REF] (figures 1, 2, 3, 4, and 5). This table was later copied by Gemma Frisius (figures 6 and 7).

Peuerbach's table was printed in color, with columns alternating in black and red. This feature has not been reconstructed, but might be added in a future version. Of notice is the fact that not only the figures, but also the lines were in alternating colors.

The context of these tables, as well as the details of their construction are found in my separate survey [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF].

Figure 1: The title page of Peuerbach's Quadratum geometricum [START_REF] Von | Quadratum geometricum[END_REF] (source: e-rara).

Figure 2: The first page of Peuerbach's table [START_REF] Von | Quadratum geometricum[END_REF] (source: e-rara).

Figure 3: The second page of Peuerbach's table [START_REF] Von | Quadratum geometricum[END_REF] (source: e-rara).

6

Figure 4: The third page of Peuerbach's table [START_REF] Von | Quadratum geometricum[END_REF] (source: e-rara).

Figure 5: The fourth page of Peuerbach's table [START_REF] Von | Quadratum geometricum[END_REF] (source: e-rara).

Figure 6: Excerpt of Gemma Frisius's copy of Peuerbach's table [START_REF] Frisius | De radio astronomico et geometrico liber[END_REF] (source: Google Books).

Figure 7: Excerpt of Gemma Frisius's copy of Peuerbach's table [START_REF] Frisius | De radio astronomico et geometrico liber[END_REF] (source: Google Books).

      G. M. . G. M. . G. M. . G. M. . G. M. . G. M. .


Introduction

This document gives an exact (modern) reconstruction of Rheticus's sine table published in 1542 [START_REF] Copernicus | De lateribus et angulis triangulorum, etc[END_REF] as part of a "preview" of Copernicus's chapters on trigonometry (figures 1 and 2).

The context of this table, as well as the details of its construction are found in my separate survey [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF].

The table resembles that of Regiomontanus published in 1541, but the tabular differences are all given. However, the differences can hardly be reproduced exactly as in Rheticus's table. Rheticus does in particular not give all the changes of differences, and he does in particular ignore increases of the differences, which are mostly decreasing, but locally increasing. For instance, for 0 • 4 and 0 • 5 , Rheticus gives the correct sines 11636 and 14544, which differ by 2908, but he keeps the difference 2909, introducing 2909 only much later. This choice appears in numerous places. Rheticus had to choose when to introduce a given difference, given that its first mention is usually not its first tabular appearance, that it, it will not appear where it was expect to appear first.

In my reconstruction, I have tried to come close to Rheticus's choices, firstly by ignoring increasing sequences of differences, second by introducing differences when the rounded real difference between the sines, and not the actual tabular difference, change. A consequence of that choice is a relatively good positioning of the differences, but the differences may differ from the actual tabular differences, especially at the beginning of the table.

Rheticus may have smoothed the values of his table in order to ensure an even sequence of differences, at least in some parts of the table. In my reconstruction, I have always given the exact rounded values of the sines.

Finally, I have chosen to give the differences in full, whereas Rheticus avoids the repetition of certain figures, but does so inconsistently.

Figure 1: The first page of Rheticus's table [START_REF] Copernicus | De lateribus et angulis triangulorum, etc[END_REF] (source: Dresden).

Figure 2: The last page of Rheticus's table [START_REF] Copernicus | De lateribus et angulis triangulorum, etc[END_REF] (source: Dresden).

CANON SVBTENSARVM


. Rheticus died during a visit in Košice, now in Slovakia, in 1574.

3 Rheticus's first canon (1551)

.

Rheticus was the first to provide a uniform treatment of the six trigonometric functions. He was in particular the first to publish a table of secants. 3 Rheticus did not consider angles in circles, but considered triangles of which one of the side was constant, and he gave the lengths of the other sides as a function of the angle at the center. The name of "sine" was not used by Rheticus, nor those of tangent and secant, the latter two having not yet been introduced. 4 The sides were called "base," "perpendicular," and "hypothenuse." The tables span 14 pages, with 7 degrees on each double page. The last double page is not totally filled, with only the degrees 42 to 44. Each double page is split in three groups of two columns. The first group (hypothenuse = 10 7 ) gives the sines and cosines, with their differences. The second group (base = 10 7 ) gives the secants and the tangents. And the third group (perpendicular = 10 7 ) gives the cosecants and the cotangents.

The heading of the tables means "Canon of the doctrine of triangles in which in a plane triangle with right angle 10 7 parts are taken for 1) the (side) subtending the right angle, 2) the greatest of the two sides including the right angle, and 3) the smallest of the two sides including the right angle" [10, p. 13].

The structure of the table was first described by De Morgan [24, 23]. A similar layout was used in Rheticus and Otho's Opus palatinum [55]. 5 Rheticus actually only computed two thirds of his table. The table of sines was taken from Regiomontanus [36, p. 152], but the tangents and secants were recomputed from the sines [36, p. 193]. Rheticus had therefore to compute about 1080 ratios (90 × 6 × 2). More precisely, the sines in Rheticus's table were copied from Regiomontanus's table for R = 10 7 [68], [36, p. 152]. Most of the values were not changed, but some of the

Trigonometric tables before Rheticus

As told by Gerhardt in his history of mathematics in Germany, Peuerbach (1423-1461) and his pupil Regiomontanus (1436Regiomontanus ( -1476) ) [START_REF] Folkerts | Regiomontanus als Mathematiker[END_REF][START_REF] Kaunzner | Über Regiomontanus als Mathematiker[END_REF][START_REF] Zinner | Regiomontanus: His life and work[END_REF] woke up the study of astronomy and built the necessary tables [33, p. 87]. Regiomontanus calculated tables of sines for every minute of arc for radiuses of 6000000 and 10000000 units. 1Regiomontanus had also computed a table of tangents [50], but with the smaller radius R = 10 5 and at intervals of 1 • . There were no printed tables of secants, but an ever growing number of tables copied from those of Regiomontanus or derived from his tables.

2 Rheticus (1514-1574) Georg Joachim was born in 1514 in Feldkirch (Austria) in what was formerly the Roman province of Raetia, from which the name Rheticus is derived. His father was a doctor. 2 After studying in Zurich from 1528 to 1532, Rheticus went to Wittenberg to complete his education. Wittenberg was then the center of the sciences. He obtained his doctorate in 1536, went to Nuremberg and Tübingen, then was called back to Wittenberg as professor of mathematics in 1536, along with Erasmus Reinhold. In 1539, he went to Frauenburg in order to assist Copernicus and pushed him to publish the details of his theory.

The first account of the Copernican system was not published by Copernicus, but in Rheticus's Narratio prima in 1540 [START_REF] Joachim | Ad clarissimum virum D. Ioannem Schonerum, de libris revolutionum eruditissimi viri & mathematici excellentissimi, reuerendi D. Doctoris Nicolai Copernici Torunnaei, canonici varmiensis[END_REF][START_REF] Rosen | Three Copernican treatises: The Commentariolus of Copernicus, the Letter against Werner, the Narratio prima of Rheticus[END_REF][START_REF] Beckmann | Rhetikus über Preußen und seine Gönner in Preußen[END_REF][START_REF] Beckmann | Zur Geschichte des kopernikanischen Systems[END_REF][START_REF] Beckmann | Zur Geschichte des kopernikanischen Systems[END_REF][START_REF] Beckmann | Zur Geschichte des kopernikanischen Systems[END_REF][START_REF] Beckmann | Zur Geschichte des kopernikanischen Systems[END_REF][START_REF] Tredwell | Michael Maestlin and the fate of the Narratio prima[END_REF]. In fact, Rheticus was more zealous than Copernicus, insisting on the physical truth of the motion of the earth. Rheticus's books were condemned by the catholic church in 1550 and put on the Index Expurgatorius, and this condemnation applied to his works on trigonometry. As observed by De Morgan, this severity was not so remarkable, if we remember that Rheticus held the chair of mathematics at Wittenberg, where Luther had taught theology [23, p. 518].

Rheticus resumed his teaching in Wittenberg in 1541 and in 1542, Copernicus's De lateribus et angulis triangulorum [START_REF] Copernicus | De lateribus et angulis triangulorum[END_REF][START_REF] Stamm | La géométrie de Nicolas Copernic[END_REF] was published by Rheticus in Wittenberg. This short treatise on trigonometry contains a table of sines for every minute of the quadrant and for a radius of 10 7 . This table was copied from Regiomontanus's decimal table [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF].

Copernicus's De Revolutionibus (1543), whose printing was partly overseen by Rheticus, contained the same chapter on trigonometry, but with a simpler table [16, book 1, chapter 12, ff. [START_REF] Cantor | Vorlesungen über Geschichte der Mathematik[END_REF][START_REF] Copernicus | De revolutionibus orbium coelestium[END_REF][START_REF] Copernicus | De lateribus et angulis triangulorum[END_REF][START_REF] Curtze | Zur Biographie des Rheticus[END_REF][START_REF] Richard | The first Copernican: Georg Joachim Rheticus and the rise of the Copernican Revolution[END_REF]. The sines were only given every 10 and for a radius of 100000. This table is also derived from Regiomontanus's works [36, pp. 178-179], [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF].

In 1542, Rheticus went to Nuremberg and Philip Melanchton [START_REF] Westman | The Melanchthon circle, Rheticus, and the Wittenberg interpretation of the Copernican theory[END_REF] then helped him to obtain a position at Leipzig University where he remained from 1542 to 1551. Rheticus typos were corrected, for instance the cosine of 38 • 40 whose value was still incorrect (as Sin 51 • 20 ) in Rheticus's 1542 table.

But Rheticus made new computations for the tangents and the secants using these sines. According to Glowatzki and Göttsche [36, p. 185], 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. 6It 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 [START_REF] Van Brummelen | Maurolico, Rheticus, and the birth of the secant function[END_REF][START_REF]Theodosii Sphaericorvm Elementorvm Libri[END_REF], although I argue in my survey [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF] that their suggestion is in fact not applicable to Maurolico's computations. However, in my survey I also suggest that Fincke used a similar procedure to compute his secants in 1583.

The 1551 canon has been reprinted in 1565 in Basel [36, p. 185]. 7Rheticus'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 does not yet seem to have understood that more accurate sines were needed. He has no problem giving cosecants and cotangents to 10 figures, when the sine is only given to 5 figures. This understanding of the need for more accurate sines only came later, and even the Opus palatinum [START_REF] Joachim | Opus palatinum de triangulis[END_REF] published in 1596 is still marred by this problem which will only fully be solved by Pitiscus in the early 17th century [48].

Rheticus's heirs and influence

After Rheticus's 1551 table, other authors such as Reinhold in 1554 [START_REF] Reinhold | Primus liber tabularum directionum[END_REF], [36, pp. 152-153], or Fincke in 1583 [START_REF] Fincke | Geometriae Rotundi Libri XIIII[END_REF], still continued to give simpler tables based on Regiomontanus's tables, but some authors were directly influenced by Rheticus. The first of them was François Viète.

Viète's Canon mathematicus (1579)

Inspired by Rheticus's table [START_REF] Hunrath | Des Rheticus Canon doctrinae triangulorum und Vieta's Canon mathematicus[END_REF], François Viète (1540-1603) constructed a new table, which he called the Canon mathematicus [START_REF] Viète | Canon mathematicus seu ad triangula cum appendicibus[END_REF]. The printing of the table was started in 1571 but it was only completed in 1579 [74, p. 205].

The Canon mathematicus contained a typographically sophisticated table of the six trigonometric functions for every minute of the quadrant and with a radius of 100000, with sometimes one or two additional figures (figure 1) [41] [26, pp. 455-456]. 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's first table. Viète took the sines from Regiomontanus, except for some parts which have been recomputed [36, pp. 154-155], [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF]. The tangents and secants were recomputed, not from Regiomontanus's values as suggested by Glowatzki and Göttsche [36, pp. 189, 196], but using Viète's own values, either in his table of sines, or from more accurate values, for instance for Sin 1 [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF].

Otho's Opus palatinum (1596)

After the publication of the 1551 canon, Rheticus continued to work on a more extensive project, where the six trigonometric functions would be given every 10 and for a radius of 10 10 . This work was only completed after Rheticus's death by Lucius Valentinus Otho and published in 1596 [START_REF] Joachim | Opus palatinum de triangulis[END_REF][START_REF] Roegel | A reconstruction of the tables of Rheticus's Opus Palatinum[END_REF]. It was further corrected by Pitiscus, in particular in his Thesaurus mathematicus published in 1613 [48].

Structure of the tables and recomputation

In addition to the tables described above, Rheticus's book contained a cover page, an introductory poem, and six pages of explanations which are not reproduced here. The original tables used two colors, and there are some overlaps between the red and black figures. Viète's tables also use two colors but their layout is much more successful. Some values were omitted and I have added them in my reconstruction. The sine, secant, and tangent values for 0 • , as well as the first difference of the tangents, were given as "....." or "........," and the value of the last cosecant of the table was merely left blank.

Rheticus's tables were recomputed using the GNU mpfr multiple-precision floatingpoint library developed at INRIA [START_REF] Fousse | MPFR: A multiple-precision binary floating-point library with correct rounding[END_REF], and give the exact values. The comparison of my table and Rheticus's will therefore immediately show where Rheticus's table contains errors. Differen: Hypotenusa Differen: Perpendicu: Differen:

Figure 1 :

 1 Figure 1: A page from Viète's Canon mathematicus (1579). This page gives (from left to right) the sines, cosines, tangents, secants, cotangents, and cosecants for angles from 4 • to 4 • 30 and from 85 • 30 to 86 • . (source: http://www.christies.com, book auctioned on June 17, 2008 in New York)

For a summary of fundamental trigonometric tables printed in the 15th and 16th centuries, see my survey[START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF]. The present report was first written in

2011, but it has been updated and corrected in light of this survey.2 For extensive biographies of Rheticus, see Burmeister[START_REF] Heinz | Eine bio-bibliographie[END_REF][START_REF] Heinz | Georg Joachim Rhetikus, zwischen Paracelsus und Kopernikus[END_REF], Kraai[START_REF] Kraai | Rheticus' heliocentric providence: a study concerning the astrology, astronomy of the sixteenth century -Die heliozentrische Providentia des Rheticus[END_REF], Danielson[START_REF] Richard | The first Copernican: Georg Joachim Rheticus and the rise of the Copernican Revolution[END_REF], and Schöbi-Fink et al 's collections[START_REF]Rheticus, Wegbereiter der Neuzeit (1514-1574[END_REF][START_REF]Wegbereiter der Neuzeit: Eine Würdigung[END_REF].

In 1819, Delambre, unaware of Rheticus's 1551 table, still wrote that Viète was the first to put together all these functions in 1579[26, p.

456].[START_REF] Clare | The Canon Doctrinae Triangvlorvm (1551) of Rheticus (1514-1576)[END_REF] It was Thomas Fincke who in 1583 first used the names "tangent" and "secant"[START_REF] Fincke | Geometriae Rotundi Libri XIIII[END_REF][START_REF] De | On the first introduction of the words Tangent and Secant[END_REF]. See also[35, p. 42].[START_REF] Beckmann | Zur Geschichte des kopernikanischen Systems[END_REF] De Morgan considers that the semi-quadrantal arrangement of the tables finds its roots in Rheticus's three cases of triangles. In the first case, the greatest side of the triangle is the cosine, and the smallest side is the sine. The divisions would not have been the same if Rheticus had given the sines for the whole quadrant. Rheticus's arrangement led to consider the duality of sine and cosine, tangent and cotangent, and secant and cosecant[23, pp. 523-524].

It will be interesting to see to what conclusions came[START_REF] Brooke | Determining the sine tables underlying early European tangent tables[END_REF] who seems to have conducted a similar investigation, but whose result is not yet published at the time I am writing this.

De Morgan wrote earlier that it had been reprinted in 1580[23, p. 521].

Note on the titles of the works: Original titles come with many idiosyncrasies and features (line splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore seemed pointless to capitalize works according to conventions which not only have no relation with the original work, but also do not restore the title entirely. In the following list of references, most title words (except in German) will therefore be left uncapitalized. The names of the authors have also been homogenized and initials expanded, as much as possible.The reader should keep in mind that this list is not meant as a facsimile of the original works. The original style information could no doubt have been added as a note, but I have not done it here.

Acknowledgements

It is my pleasure to acknowledge the help of Michela Malpangotto, who hinted to an earlier shortcoming in this study.


Tertia

  ,,, ,,  

 

 

  ,,   

 

 

  ,,   

 

 

 

,  

 

  ,, ,  

 

  ,, ,  

 

  ,, ,  

 

  

  

 

 

  ,,   

 

 

  ,, ,  

 

  ,, ,  

 

  

 

 

 

  

 

 

 

  ,, ,  

 

  ,, ,  

 

, ,, XXX 

Centesima Integri

 

 

  

 

 

 

  

 

 

 

  

 

 

  

 

          II

Keywords: lus rectus, part Hypotelus rectus, part

. I am first giving a brief overview of Viète's predecessors, then I describe the tables and how Viète constructed them, before giving a modern reconstruction of these tables, which can be compared to Viète's original tables.

Viète's predecessors

The history of trigonometry and the computation of tables goes back to Greece, and even before. I will here only highlight Viète's main predecessors, namely Regiomontanus and Rheticus, but a detailed survey is given separately for the fundamental trigonometric tables of the 15th and 16th centuries [41]. 1

Regiomontanus (1436-1476)

Regiomontanus is in fact the main source of Viète's Canon mathematicus. Regiomontanus, or rather Hans Müller, was probably born in 1436 in Königsberg, near Bamberg in Germany. 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.

He established trigonometry as an independant field, separate from astronomy, in Western Europe. After having studied in Leipzig, he came to Vienna around 1450 and became a friend and pupil of Georg von Peuerbach.

It seems that Regiomontanus started around 1460 to compute sines with a large radius in order to produce a table with R = 6 • 10 6 for his De triangulis (1462?). This table was certainly inspired by Peuerbach's table with R = 600000.

), or more groundbreaking in the case of Viète.

, in particular for a more thorough description of Regiomontanus's life and works.

Rheticus (1514-1574)

Rheticus was the first to really go beyond mere sine tables. Georg Joachim was born in 1514 in Feldkirch (Austria) in what was formerly the Roman province of Raetia, from which the name Rheticus is derived. His father was a doctor. 2 After studying in Zurich from 1528 to 1532, Rheticus went to Wittenberg to complete his education. Wittenberg was then the center of the sciences.

went to Nuremberg and Tübingen, then was called back to Wittenberg as professor of mathematics in 1536, along with Erasmus Reinhold. In 1539, he went to Frauenburg in order to assist Copernicus and pushed him to publish the details of his theory.

Rheticus resumed his teaching in Wittenberg in 1541 and in 1542, Copernicus's De lateribus et angulis triangulorum was published by Rheticus in Wittenberg [START_REF] Copernicus | De lateribus et angulis triangulorum[END_REF]. This short treatise on trigonometry contains a table of sines for every minute of the quadrant and for a radius of 10 7 . It was based on Regiomontanus's table with the same radius [START_REF] Von | Tractatus super propositiones Ptolemaei de sinubus & chordis[END_REF].

Copernicus's De Revolutionibus (1543), whose printing was partly overseen by Rheticus, contained the same chapter on trigonometry, but with a simpler table. The sines were only given every 10 and for a radius of 100000. The values were also derived from Regiomontanus's works [START_REF] Roegel | A survey of the main fundamental European trigonometric tables printed in the 15th and 16th centuries[END_REF].

In 1542, Rheticus went to Nuremberg and Philip Melanchton then helped him to obtain a position at Leipzig University where he remained from 1542 to 1551. Rheticus had to leave Leipzig after being accused of homosexual conduct and went to Prague in 1551. He gave up his academic career and completed his medicine studies. He settled in Cracow in 1554 where he practiced medicine. Cracow was located on the same meridian as Frauenburg and Rheticus had a 15-meter obelisk erected near the city for astronomical purposes which were never completed. Rheticus died during a visit in Košice, now in Slovakia, in 1574.

Rheticus published his first trigonometrical canon, the Canon doctrinae triangulorum, in 1551 in Leipzig [START_REF] Joachim | Canon doctrinae triangulorum[END_REF]. It gave the six trigonometrical functions at intervals of 10 , semiquadrantically arranged. Each function was given to 7 places, or more exactly as integers for a radius of 10 7 . This is a very rare table, and it was practically unknown when De Morgan happened to find a copy of it in the 1840s [START_REF] De | On the almost total disappearance of the earliest trigonometrical canon[END_REF][START_REF] De | On the almost total disappearance of the earliest trigonometrical canon[END_REF].

Rheticus was the first to provide a uniform treatment of the six trigonometric functions. He was in particular the first to publish a table of secants. 3 Rheticus only computed two thirds of his table. The table of sines was taken from Regiomontanus [23, p. 152], but the tangents and secants were recomputed from the sines [23, p. 193]. Rheticus had therefore to compute about 1080 ratios (90 × 6 × 2).

Rheticus did not consider angles in circles, but triangles of which one of the side was constant, and he gave the lengths of the other sides as a function of the angle at the center. The name of "sine" was not used by Rheticus, nor those of tangent and secant, the latter two having not yet been introduced. 4 The sides were called "base," "perpendicular," and "hypothenuse." The tables span 14 pages, with 7 degrees on each double page. The last double page is not totally filled, with only the degrees 42 to 44. Each double page is split in three groups of two columns. Each group gives the values of two functions and their differences. The first group (hypothenuse = 10 7 ) gives the sines and cosines, with their differences. The second group (base = 10 7 ) gives the secants and the tangents. And the third group (perpendicular = 10 7 ) gives the cosecants and the cotangents. The structure of the table was first described by De Morgan [START_REF] De | On the almost total disappearance of the earliest trigonometrical canon[END_REF][START_REF] De | On the almost total disappearance of the earliest trigonometrical canon[END_REF].

The 1551 canon has been reprinted in 1565 in Basel [23, p. 185]. 5 2 Viète's Canon mathematicus (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" [53, p. 73]. 6 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).

Also inspired by Rheticus's 1551 Canon doctrinae triangulorum [START_REF] Joachim | Canon doctrinae triangulorum[END_REF][START_REF] Hunrath | Des Rheticus Canon doctrinae triangulorum und Vieta's Canon mathematicus[END_REF], Viète constructed a new table, which he called the Canon mathematicus [START_REF] Viète | Canon mathematicus seu ad triangula cum appendicibus[END_REF]. This work contained a typographically sophisticated table of the six trigonometric functions for every minute of the quadrant and with a radius of 100000, with sometimes one or more additional figures [START_REF] Hunrath | Des Rheticus Canon doctrinae triangulorum und Vieta's Canon mathematicus[END_REF], [16, pp. 455-456]. 7 (see figure 1). The printing of the table was started in 1571 but it was only completed in 1579 [43, p. 205].

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's 1551 table (which however only had an interval of 10 ).

The first double column gives the sines and cosines. They correspond to the perpendicular and the base of a triangle whose hypotenuse is 100000. The second double column gives the tangent and the secant. They correspond to the perpendicular and the hypotenuse of a triangle whose base is 100000. The third double column gives the cotangent and the cosecant. They correspond to the base and the hypotenuse of a triangle whose perpendicular is 100000.

The values of the cosines and secants up to 2 are given with four additional digits. From 3 to 24 , only three additional digits are given, from 25 to 4 • 5 only two, and finally only one. This is explained in a marginal note on the second page of the table.

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 • [23, pp. 154-155], but in fact the cosines from 0 • to 30 have been computed from Regiomontanus's (or Reinhold's) sines, probably with cos x ≈ 1-sin 2 x 2 , which, with a radius R other than 1, becomes Cos x ≈ R -Sin 2 x 2R . For instance, for 6 , R = 10 For the tangents and secants, Glowatzki and Göttsche wrote that they were recomputed from Regiomontanus's values [23, pp. 189, 196]. 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.

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 • , Viète used more accurate values of the sines than those printed in his table, with 1 to 5 more figures than Regiomontanus. For instance, for Cot 1 , Viète used Sin 1 = 29.0888204 (here with R = 10 5 ), that is 5 more figures than Regiomontanus. This enabled him to obtain (reducing the cosine and sine to R = 10 and this is precisely the value given in the Canon mathematicus (the exact value being 343774667).

The same applies for Csc 1 . For Sin 2 , Viète took 58.1776385, whereas Regiomontanus only has 5818. And so on.

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 [START_REF] Viète | Ad angulares sectiones theoremata καθολικωτερα[END_REF], [53, pp. 79-80], 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 • 36 . Bisecting 6 • 40 we find Sin 3 • 20 . Using the two values Sin 3 • 36 and Sin 3 • 20 , we obtain the sine of the difference of the angles, namely sin 16 . And bisecting 16 four times, we obtain Sin 1 . But as a matter of fact Viète seems to have proceeded slightly differently. He actually found two approximations of Sin 1 , one greater and one smaller than the sought value. An interpolation between these two values then gave a better approximation of Sin 1 [46, pp. 62-67], [45, pp. 22-23].

Viète made a special case of the first value, which is actually not 0 degrees. Viète wanted to avoid infinite values, as alluded in the marginal note on the first page. 8 In fact, Viète computed with an angle whose sine and tangent was close to 10 -5 (one unit with an hypotenuse of 100000) and whose cotangent and cosecant was therefore close to 10 5 (10 10 units with a perpendicular of 100000). An angle satisfying these conditions is arctan (10 -5 ) (or arcsin (10 -5 ) which is almost identical), We have arctan (10 -5 ) ≈ 0 2 3 3 4 , but Viète's table incorrectly gives it as 0 2 3 2 4 , although this is probably a typographical error. If Viète had really meant 2 4 , he would certainly have written 1 2 instead. This angle is not difficult to obtain, as it is given to a good approximation in degrees by 1 10 5 × 180 π , even if we only take π ≈ 3.14.

As observed by Tannery, if Viète's tables are rare it is because of the success of Rheticus's Opus palatinum (1596), of Pitiscus's Thesaurus mathematicus (1613), and because of the introduction of logarithms (1614). They all made Viète's tables obsolete.

A persistent legend is 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 have explained why this book is of great rarity [START_REF] Eneström | Vorlesungen über Geschichte der Mathematik (review)[END_REF] [7, pp. 583-584]. But according to Ritter's biography of Viète [33, p. 54], this legend rests on the editor of Viète's 1646 Opera omitting the Canon mathematicus, on the grounds that the computations would have to be redone [43, p. 208] [START_REF] Tannery | Vorlesungen über Geschichte der Mathematik, von Moritz Cantor (review)[END_REF]. Moreover, as I have shown, Viète's table is actually very accurate.

The Canon mathematicus was also published with a London imprint in 1589 (Opera mathematica, London: Bouvier) and there is an edition dated 1609, but Bosmans showed that it is not a reprint. It is the 1579 edition rebound [4, pp. 111-114] [3, pp. 297-298]. Cantor and von Braunmühl had mistakenly thought that it was a new edition [START_REF] Von | Vorlesungen über Geschichte der Trigonometrie[END_REF] [7, p. 583], probably after Eneström led them to think so [START_REF] Eneström | Vorlesungen über Geschichte der Mathematik (review)[END_REF].

Reconstruction and analysis of errors

I have reconstructed the first two tables of the Canon mathematicus. The second table (Canonion triangulorum) is described separately [START_REF] Roegel | A reconstruction of Viète's Canonion triangulorum[END_REF].

The main trigonometrical table was recomputed using the GNU mpfr multiple-precision floating-point library developed at INRIA [START_REF] Fousse | MPFR: A multiple-precision binary floating-point library with correct rounding[END_REF], and gives the exact values. The comparison of my table and Viète's therefore immediately shows where Viète's table contains errors. I have not checked Viète's entire table, but only the values of the trigonometric lines on the first and last pages out of 90. The differences have not been checked. On the first page, there are two typographical errors and 14 last digit errors, out of 186 values. On the last page, there is one typographical error, and there are 13 last digit errors. Based on these samples, Viète's table may have an average of one or two typographical errors per page, and about a dozen last-digit errors, which were never greater than two units on the first and last pages. These observations counter the allegations according to which Viète's table was full of errors, and could therefore not be printed.

LV           VI , ,  f , ,  f ,, ,, LIIII           VII , ,  , ,  ,, ,, LIII           VIII , ,  , ,  ,, ,, LII           IX , ,  , ,  ,, ,, LI           X , ,  , ,  ,, ,, L           XI , ,  , ,  ,, ,, XLIX           XII , ,  , ,  ,, ,, XLVIII           XIII , ,  , ,  ,, ,, XLVII           XIIII , ,  , ,  ,, ,, XLVI           XV , ,  , ,  ,, ,, XLV           XVI , ,  , ,  ,, ,, XLIIII           XVII , ,  , ,  ,, ,, XLIII           XVIII , ,  , ,  ,, ,, XLII           XIX , ,  , ,  ,, ,, XLI           XX , ,  , ,  ,, ,, XL           XXI , ,  , ,  ,, ,, XXXIX           XXII , ,  , ,  ,, ,, XXXVIII           XXIII , ,  , ,  ,, ,, XXXVII           XXIIII , ,  , ,  ,, ,, XXXVI           XXV , ,  , ,  ,, ,, XXXV           XXVI , ,  , ,  ,, ,, XXXIIII           XXVII , ,  , ,  ,, ,, XXXIII           XXVIII , ,  , ,  ,, ,, XXXII           XXIX , ,  , ,  ,, ,, XXXI           XXX , ,  , ,  ,, ,, XXX PRIMA SECVNDA SERIE TERTIA scrvp.
LXXXV.

          XXXII , ,  , ,  ,, ,, XXVIII           XXXIII , ,  , ,  ,, ,, XXVII           XXXIIII , ,  , ,  ,, ,, XXVI           XXXV , ,  , ,  ,, ,, XXV           XXXVI , ,  , ,  ,, ,, XXIIII           XXXVII , ,  , ,  ,, ,, XXIII           XXXVIII , ,  , ,  ,, ,, XXII           XXXIX , ,  , ,  ,, ,, XXI           XL , ,  , ,  ,, ,, XX           XLI , ,  , ,  ,, ,, XIX           XLII , ,  , ,  ,, ,, XVIII           XLIII , ,  , ,  ,, ,, XVII           XLIIII , ,  , ,  ,, ,, XVI           XLV , ,  , ,  ,, ,, XV           XLVI , ,  , ,  ,, ,, XIIII           XLVII , ,  , ,  ,, ,, XIII           XLVIII , ,  , ,  ,, ,, XII           XLIX , ,  , ,  ,, ,, XI           L , ,  , ,  ,, ,, X           LI , ,  , ,  ,, ,, IX           LII , ,  , ,  ,, ,, VIII           LIII , ,  , ,  ,, ,, VII           LIIII , ,  , ,  ,, ,, VI           LV , ,  , ,  ,, ,, V           LVI , ,  , ,  ,, ,, IIII           LVII , ,  , ,  ,, ,, III           LVIII , ,  , ,  ,, ,, II           LIX , ,  , ,  ,, ,, I           LX , ,  , ,  ,, ,, f PRIMA SECVNDA SERIE TERTIA scrvp.
LXXXV.

          II , ,  , ,  ,, ,, LVIII           III , ,  , ,  ,, ,, LVII           IIII , ,  , ,  ,, ,, LVI           V , ,  , ,  ,, ,, LV           VI , ,  , ,  ,, ,, LIIII           VII , ,  , ,  ,, ,, LIII           VIII , ,  , ,  ,, ,, LII           IX , ,  , ,  ,, ,, LI           X , ,  , ,  ,, ,, L           XI , ,  , ,  ,, ,, XLIX           XII , ,  , ,  ,, ,, XLVIII           XIII , ,  , ,  ,, ,, XLVII           XIIII , ,  , ,  ,, ,, XLVI           XV , ,  , ,  ,, ,, XLV           XVI , ,  , ,  ,, ,, XLIIII           XVII , ,  , ,  ,, ,, XLIII           XVIII , ,  , ,  ,, ,, XLII           XIX , ,  , ,  ,, ,, XLI           XX , ,  , ,  ,, ,, XL           XXI , ,  , ,  ,, ,, XXXIX           XXII , ,  , ,  ,, ,, XXXVIII           XXIII , ,  , ,  ,, ,, XXXVII           XXIIII , ,  , ,  ,, ,, XXXVI           XXV , ,  , ,  ,, ,, XXXV           XXVI , ,  , ,  ,, ,, XXXIIII           XXVII , ,  , ,  ,, ,, XXXIII           XXVIII , ,  , ,  ,, ,, XXXII           XXIX , ,  , ,  ,, ,, XXXI           XXX , ,  , ,  ,, ,, XXX PRIMA SECVNDA SERIE TERTIA scrvp.
LXXXIV.

          XXXII , ,  , ,  ,, ,, XXVIII           XXXIII , ,  , ,  ,, ,, XXVII           XXXIIII , ,  , ,  ,, ,, XXVI           XXXV , ,  , ,  ,, ,, XXV           XXXVI , ,  , ,  ,, ,, XXIIII           XXXVII , ,  , ,  ,, ,, XXIII           XXXVIII , ,  , ,  ,, ,, XXII           XXXIX , ,  , ,  ,, ,, XXI           XL , ,  , ,  ,, ,, XX           XLI , ,  , ,  ,, ,, XIX           XLII , ,  , ,  ,, ,, XVIII           XLIII , ,  , ,  , ,, XVII           XLIIII ,

9 )

 9

Figure 1 :

 1 Figure 1: A page from Viète's Canon mathematicus (1579). This page gives (from left to right) the sines, cosines, tangents, secants, cotangents, and cosecants for angles from 4 • to 4 • 30 and from 85 • 30 to 86 • . (source: http://www.christies.com, book auctioned on June 17, 2008 in New York)

Figure 2 :

 2 Figure 2: An excerpt of the same page from the Gallica digital library. This excerpt is representative of the bad digital quality of most of the books scanned from microfilms at Gallica, at least in 2011 when I first wrote this.

  Viète gives the value 99999848. Viète did not use the values of the sines in his table for this purpose, and using 175 in the previous example would not give a sufficiently accurate cosine.

			8 and Regiomontanus's
	sine value 17453:
	Cos 6 ≈ 10 8 -	174530 2 2 • 10 8 = 10 8 -	17453 2 2 • 10 6 = 99999847.69 . . .

and

The first version of this document was made available in

2011, and it has been reworked to blend with this new survey.2 For extensive biographies of Rheticus, see Burmeister[START_REF] Heinz | Eine bio-bibliographie[END_REF], and Danielson[START_REF] Richard | The first Copernican: Georg Joachim Rheticus and the rise of the Copernican Revolution[END_REF].

In 1819, Delambre, unaware of Rheticus's 1551 table, still wrote that Viète was the first to put together all these functions in 1579[16, p.

456].[START_REF] Bosmans | Le traité des sinus de Michiel Coignet[END_REF] It was Thomas Fincke who in 1583 first used the names "tangent" and "secant"[START_REF] Fincke | Geometriae Rotundi Libri XIIII[END_REF][START_REF] De | On the first introduction of the words Tangent and Secant[END_REF].

For summaries of Viète's life and works, see[START_REF] De | The Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge[END_REF],[START_REF] Ritter | inventeur de l'algèbre moderne[END_REF],[START_REF] Hubertus Lambertus | Dictionary of Scientific Biography[END_REF], and[45, pp. 9-11].

The cosines and secants are given more accurately than the other values throughout the table. They are given to four more places from 0 • to 0 • 2 , to three more places from 0 • 3 to 0 • 24 , to two more places from 0 • 25 to 4 • 5 , and to one more place from 4 • 6 to 45 • .

I are indebted to Marisa Addomine for helping me translate the two marginal notes.

Note on the titles of the works: Original titles come with many idiosyncrasies and features (line splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore seemed pointless to capitalize works according to conventions which not only have no relation with the original work, but also do not restore the title entirely. In the following list of references, most title words (except in German) will therefore be left uncapitalized. The names of the authors have also been homogenized and initials expanded, as much as possible.The reader should keep in mind that this list is not meant as a facsimile of the original works. The original style information could no doubt have been added as a note, but I have not done it here.


Dr Yi-Xiang Wang
email: yixiang_wang@cuhk.edu.hk

Fragility fracture prevalence among elderly Chinese is less than half of that of Caucasians

2

Fragility (osteoporotic) fracture prevalence difference among various ethnic groups has been well noted. In this letter, I summarize the evidence that fragility fracture prevalence among elderly Chinese is no more than half of that of Caucasians. I have systematically searched and read relevant literature. However, the current letter is not a systematic review.

There are reports with conflicting results, it is not reasonable to simply synthetise them together. It can be noted that some reports are limited by sampling bias, study artifacts, and small sample size. I primarily review original reports where Chinese and Caucasians were compared in the same study. This will allow a more direct comparison of risk factors and be less affected by methodological differences. This letter reflects my personal interpretation of the published data, which is expected also to reflects the mainstream conclusion of most reports. For the reports from the USA, an important limitation is that Chinese is often grouped under the category of 'Asians'. As there are some high-quality comparative studies conducted between Caucasians and Japanese, I also use some of these data to support my arguments for the risks with the Chinese population.

Hip fragility fracture

Hip fracture is the most important fragility fracture. Among all osteoporotic fractures, hip fracture incurs the greatest morbidity, mortality, and costs. Hip fracture typically requires hospitalization, making hip fracture incidence ascertainment more reliable than for other types of fractures.

With a database of all hospitalizations for the State of California, in 1987 Silverman and Madison (1) examined the incidence of hip fracture in Caucasians (non-Hispanic White) and Asian Americans for the years 1983 and 1984. They reported the Asian American populations in California had rate ratios of hip fracture relative to Caucasians of 0.61 for women and 0.54 for men.

In 1993 Ho et al [START_REF] Ho | Hip fracture rates in Hong Kong and the United States, 1988 through 1989[END_REF] examined the hospital discharge data for hip fracture in Hong Kong and in the USA for 1988 through 1989. Hong Kong data were obtained from six government hospitals and three government-assisted hospitals. The USA data were obtained from the National Hospital Discharge Survey. Incidence rates of hip fracture for the 2-year interval were calculated for men and women. The age-adjusted fracture rates per 100,000 were significantly lower in Hong Kong than in the USA for men (100 vs.187, ratio 0.53 ) and 3 3 women (247 vs 535, ratio: 0.46). Ho et al [START_REF] Ho | Hip fracture rates in Hong Kong and the United States, 1988 through 1989[END_REF] concluded that hip fracture rates in the USA were typically 1.5 to 2.5 times those in Hong Kong across the age and sex groups.

In 1996 Lauderdale et al (3) estimated US national age-and sex-specific nontraumatic hip fracture incidence rates for elderly Chinese Americans, Japanese Americans, and Korean Americans (all ≥ 65 years). Based on a 50% sample of 1992 Medicare enrollees, cohorts of persons with distinctive Chinese (n=24,366), Japanese (n=28,762), and Korean (n=5,470) names were followed passively for two years for hospitalization with a diagnostic code indicating hip fracture. Cohorts of Caucasians (10 % sample) were followed for comparison.

They reported that age-adjusted hip fracture incidence was lower for all three East Asian-American groups than for Caucasians, with the fracture ratio relative to Caucasians being 0.30 for Chinese women and 0.42 for Chinese men. Lauderdale et al (3) further explored whether risk varied by year of immigration, and observed a trend toward higher risk of hip fracture among the more recent immigrants in all three ethnic groups. This suggests that the westernization of the lifestyles, such as higher intake of milk and Calcium etc, may protect against hip fragility fracture.

In 2003 Zingmond et al (4) evaluated the change in standardized hip fracture incidence from 1983 through 2000 in California Hispanics relative to other racial groups. With Caucasians' rate as the reference, the hip fracture rate was approximately 0.5 for Asian women, and 0.4 for Asian men.

In 2004, Fang et al [START_REF] Fang | Variations in hip fracture hospitalization rates among different race/ethnicity groups in New York City[END_REF] analysed the hip fracture hospitalization rates for Asian and Caucasian patients aged 50 and older in New York City from 1988 to 2002. They reported annual ageadjusted hip fracture hospitalization rates per 100,000 among Asian women and Caucasian women were 174 and 459 respectively (Asian vs Caucasian ratio: 0.379); and the corresponding rates for men were 104 and 230 respectively (ratio: 0.452).

In 2011, with a comparison made with their own data (2,302 women and 1,810 men) and published Japanese and Swedish data, Bow et al reported that Japanese and Hong Kong Chinese had very similar age-specific hip fragility fracture prevalence; while the hip fracture rates for Hong Kong men and women aged 65 to 69 years old were 49% and 33% of those of Caucasian men and women in the same age group.

In 2012 Wright et al [START_REF] Wright | Recent trends in hip fracture rates by race/ethnicity among older US adults[END_REF] The standardized rate of hip fracture among women of Hawaii Japanese and Okinawa were both 0.48 (95% CI: 0.38-O.60 and 0.41-0.57) of the value for Caucasian women. The standardized rates of hip fracture among men of Hawaii Japanese and Okinawa Japanese were 0.33 (95%:0.21-0.50) and 0.32 (95%: 0.23-0.45) of the value for Caucasian men. It was noted that, although diet and other cultural attributes of the Hawaii Japanese were more westernized than the Okinawan population, there were no detectable differences in hip fracture rates between Hawaii ethnic Japanese and Okinawa Japanese.

General fragility fractures

By analysing the data from the National Osteoporosis Risk Assessment (NORA) which is an observational study of postmenopausal women in the USA, in 2005 Elizabeth Barrett-Connor et al [START_REF] Barrett-Connor | Osteoporosis and fracture risk in women of different ethnic groups[END_REF] reported the baseline characteristics and 1-year fracture incidents in the mean age was 67.0 (±9.0) years and 66.4 (±9.5) years for Chinese (n=74)and Caucasian men (n=2044). Any low trauma fracture history was recorded in 7.7% of the Chinese women, while in 22.2% of the Caucasian women; and recorded in 6.8% of the Chinese men and 17.2% of the Caucasian men. For the 5-year incident fracture, 6.4% of Chinese women reported any incident fractures compared with 7.8% in Caucasian women. 3.0% of Chinese men reported any incident fractures compared with 4.1% any incident fractures in Caucasian men. In the meantime, Morin et al [START_REF] Morin | Differences in fracture prevalence and in bone mineral density between Chinese and White Canadians: the Canadian Multicentre Osteoporosis Study (CaMos)[END_REF] noted that the comparisons between the incident fractures between Chinese and Caucasians were inconclusive due to the small numbers of events.

In 2021, by analysing the above-mentioned Manitoba BMD Program registry data, Leslie et al [START_REF] Bow | Ethnic difference of clinical vertebral fracture risk[END_REF] reported that, at baseline prior fracture history was recorded in 20.1% of the Caucasian women (total included n= 68,907, mean age: 64.7±10.9 years) and 11.7% of the Asian women (total: 1910, mostly Chinese and Filipino, mean age: 62.8±9.9 years). During the follow-up of 8.8±5.1 years for Caucasians and 6.5±5.0 years for Asians, the incident rate for major osteoporotic fracture (MOF, a composite of hip, humerus, forearm, and clinical vertebral fractures) per 1000 person-years was 11.4 (95%CI:11.2-11.7) for Caucasians and 5.0 (95%CI: 3.8-6.2) for Asians. They noted that adjusted for baseline risk using the Canadian 6 FRAX tool with BMD, Asian women compared with Caucasian women were at much lower risk for MOF (hazard ratio 0.46, 95% CI: 0.35-0.59) and hip fracture (0.16, 95% CI 0.08-0.34).

Vertebral fragility fracture

Many earlier studies reported that, compared with those of Caucasians, East Asians' prevalence of radiographic OVF (osteoporotic vertebral fracture) was similar or even higher [START_REF] Bow | Ethnic difference of clinical vertebral fracture risk[END_REF][START_REF] Kung | Epidemiology and diagnostic approaches to vertebral fractures in Asia[END_REF]. However, these results are likely due to study artifacts. Recent evidence suggests that, compared with Caucasians, the relative prevalence of OVF follows the same pattern as other clinical fractures.

In 2012 Kwok et al [START_REF] Kwok | Prevalence and risk factors of radiographic vertebral fractures in elderly Chinese men and women: results of Mr. OS (Hong Kong) and Ms. OS (Hong Kong) studies[END_REF] estimated that OVF prevalence in Chinese women is similar to those of Japanese, Korean, and Latin Americans, but lower than those of age-matched European and American Caucasian women. It is interesting to note that it has been demonstrated in many USA studies show Asian Americans have a hip fragility prevalence similar to that of Hispanics (4, 5,7). However, the results of Kwok et al [START_REF] Kwok | Prevalence and risk factors of radiographic vertebral fractures in elderly Chinese men and women: results of Mr. OS (Hong Kong) and Ms. OS (Hong Kong) studies[END_REF] were derived with various criteria for defining radiographic OVF.

With spine radiographs from two epidemiological studies conducted in Hong Kong and in

Rome, a recent study compared OVF prevalence in age-matched elderly Chinese women and Italian women (both mean age: 74.1 years, 15). The results show Hong Kong Chinese subjects had endplate and/or cortex OVF (ECF) in 26% cases involving 3.54% of the vertebrae, while Italian subjects had ECF in 47% cases involving 8.21% of the vertebrae; 9.5% Chinese women and 26% Italian women had OVF with ≥40% height loss [START_REF] Morin | Differences in fracture prevalence and in bone mineral density between Chinese and White Canadians: the Canadian Multicentre Osteoporosis Study (CaMos)[END_REF]. OVF in Italian subjects tended to be more severe, more likely to be multiple. A trend suggested an earlier onset of OVF among Italians.

In MrOS (Hong Kong) and MsOS(Hong Kong) studies, from baseline on, clinical spine fractures (mostly fragility fractures) were followed-up for 10 years for 1954 male participants, and for 9 years for 1953 female participants. Clinical spine fracture ≥ 1 time (i.e., at least one fracture incident) were recorded 133 cases and 273 cases per 100,000 person-years in men and women, respectively [START_REF] Su | The predictive power of vertebral endplate/cortical fracture (ECF) for further osteoporotic fracture risk among community dwelling elderly Chinese. The 11th meeting of Chinese Society of Osteoporosis and Bone Mineral Research[END_REF]. In MrOS(USA) study which has the same enrolment strategy for participants as MrOS (Hong Kong) study, with 5995 cases followedup for 4.7 years, Freitas et al [START_REF] Kwok | Prevalence and risk factors of radiographic vertebral fractures in elderly Chinese men and women: results of Mr. OS (Hong Kong) and Ms. OS (Hong Kong) studies[END_REF] recorded clinical spine fracture incidence of 216 cases/100,000 person-years. Note that, at the end of these follow-ups, MrOS(Hong Kong) subjects were older than the USA subjects. Moreover, for the population-based study in Rochester, Minnesota, for the 5-year study period of 1985-1989, Cooper et al [START_REF] Cooper | Incidence of clinically diagnosed vertebral fractures: a population-based study in Rochester, Minnesota, 1985-1989[END_REF] estimated clinical vertebral fracture incidences with moderate trauma of 238 for men and 975 for women per 100,000 person-years for the age group of 75-84 years, which is much higher than the MrOS (Hong Kong) and MsOS (Hong Kong) rates. However, the study of Cooper et al [START_REF] Cooper | Incidence of clinically diagnosed vertebral fractures: a population-based study in Rochester, Minnesota, 1985-1989[END_REF] is limited by its small sample size for the elderly subjects.

To support the discussion above that Chinese have a lower OVF prevalence than that of Caucasians, for the MrOS (Hong Kong) year-4 follow-up study, with 1500 elderly men followed-up for four years, participants with baseline OVF had little increased risk for further OVF [START_REF] Wang | Elderly males with or without existing osteoporotic vertebral fracture have much lower future vertebral fracture risk than elderly females: the MrOS (Hong Kong) year-4 follow-up spine radiograph study[END_REF][START_REF] Wang | Existing severe osteoporotic vertebral fractures in elderly Chinese males were only weakly associated with higher further vertebral fracture risk at year-4 follow-up[END_REF]. This result differs from Caucasian studies on men where participants with baseline OVF had an increased risk for further OVF [START_REF] Karlsson | Characteristics of Prevalent Vertebral Fractures Predict New Fractures in Elderly Men[END_REF][START_REF] Lentle | Vertebral Fractures: Which Radiological Criteria Are Better Associated With the Clinical Course of Osteoporosis?[END_REF]. Moreover, all reports suggest that, compared with Caucasians, Asians and Chinese in particular have a lower incident rate of back pain. Mailis-Gagnon et al (22) collected data on new patients over a three-year period at the Comprehensive Pain Program (CPP) in downtown Toronto. They noted the east Asian group (primarily Chinese) was the most underrepresented (1.6% of the CPP population), despite that this group accounted for 9% of the population in Toronto and 6.01% of the Greater Toronto Area population. Waterman et al [START_REF] Waterman | Low back pain in the United States: incidence and risk factors for presentation in the emergency setting[END_REF] analysed the National Electronic Injury Surveillance System (USA) for all cases of low back pain presenting to emergency departments between 2004 and 2008. They found that the per 1,000 personyears low back pain incident rates were 1.23 among whites, while only 0.20 among Asians.

Bone mass decline rate

There is evidence that elderly East Asians lose bone mass more slowly than Caucasians. In 1998 Dennison et al [START_REF] Dennison | Bone loss in Great Britain and Japan: a comparative longitudinal study[END_REF] published a study on bone loss rate in a British cohort and a Japanese cohort. Annual bone loss rates by BMD measure were obtained for 172 Hertfordshire (UK) men (mean age: 66 years) and 143 Hertfordshire women (mean age: 65.6 years). The follow-up study was performed 4 years after the baseline measure. Eighty-six Japanese men (mean age: 68.7 years) and 90 Japanese women (mean age: 69.3 years) completed a similar study in Taiji, Japan. The follow-up study was performed 3 years after the baseline measure. For British subjects, femoral neck annual bone loss was 0.36±1.98% in men and 1.05±1.54% in women; trochanter annual bone loss was 0.20±1.23% in men and 1.43±1.33% in women. For Japanese subjects, femoral neck annual bone change was an increase of 0.33±2.22% in men and an increase of 0.14±2.11% in women; trochanter annual bone change was an increase of 0.59±1.57% in men, and a loss of 0.20 ±1.71% in women.

In 2011 Sheu et al [START_REF] Sheu | Osteoporotic Fracture in Men (MrOS) Research Group. Age-related decline in bone density among ethnically diverse older men[END_REF] reported a USA-based study with an average follow-up of 4.6 years in 3,869 Caucasian men and 145 Asian men aged ≥65 years (mean ages: 73±5 and 72±5 years, respectively). The annual rate of decline in BMD at the femoral neck was 0.32% and 0.09%/year respectively for Caucasians and American Asians. They conclude that Asian men experience a slower rate of decline in BMD compared with Caucasians.

With the CaMos data and comparing Caucasian Canadian and Chinese Canadian, in 2020

Morin et al [START_REF] Morin | Differences in fracture prevalence and in bone mineral density between Chinese and White Canadians: the Canadian Multicentre Osteoporosis Study (CaMos)[END_REF] reported that during a 5-year period Chinese lost less bone mass than Caucasian men and women. For the participants aged >50 years, 5-year BMD change (g/cm 2 ) was -0.002 and -0.000 for Chinese women femoral neck and total hip, while was -0.011 and -0.013 for Caucasian women femoral neck and total hip. 5-year BMD change (g/cm 2 ) was -0.005 and -0.003 for Chinese men femoral neck and total hip, while -0.011 and -0.013 for Caucasian men femoral neck and total hip. The current letter is more focused on the reports from North America. It can be argued that the most reliable comparative studies are those when the two ethnic groups are compared by the same investigators and using the same criteria or methods. While fragility fracture prevalence is certainly influenced by a combination of genetic and environmental/lifestyle factors, if Chinese in North America and Caucasians in North America are compared, then environmental and lifestyle for the Chinese living in China would not be fully counted.

However, I argue that it is the biological risk factors that dominate. For example, Hong Kong Chinese and Chinese in USA both have a hip fracture prevalence of no more than half of that of Caucasians. Ross et al [START_REF] Ross | A comparison of hip fracture incidence among native Japanese, Japanese Americans, and American Caucasians[END_REF] described that, although the lifestyle of the Hawaii Japanese is more westernized than Okinawa Japanese, the hip fracture rates of Hawaii Japanese and Okinawa Japanese are almost the same.

The three East Ascian ethnic groups, i.e., Chinese, Japanese, and Korean, are under the same category of 'Asians' in most of the USA reports. A detailed comparison of fragility fracture prevalence among the three East Ascian ethnic groups is beyond the scope of this letter. However, I take the assumption that the fragility fracture prevalence of the three 9 East Ascian ethnic groups is broadly similar. Bow et al reported that Japanese and Hong Kong Chinese have very similar age-specific hip fragility fracture prevalence (6). Shin et al [START_REF] Shin | Osteoporotic Fractures in Men (MrOS) Research Group. Race/ethnic differences in associations between bone mineral density and fracture history in older men[END_REF] reported the prevalence of self-reported non-traumatic fracture for men aged 65 years and older was Hong Kong Chinese 5.6 %, and Korean 5.1 %. Kwok et al [START_REF] Kwok | Prevalence and risk factors of radiographic vertebral fractures in elderly Chinese men and women: results of Mr. OS (Hong Kong) and Ms. OS (Hong Kong) studies[END_REF] estimated OVF in Chinese women is very similar to those of Japanese, Korean. Lau et al [START_REF] Lau | The incidence of hip fracture in four Asian countries: the Asian Osteoporosis Study (AOS)[END_REF] analysed the Hong Kong and Singapore hospitals' hip fracture discharge data for the year 1997. For the patients who were 50 years of age and older, they noted that the age-adjusted incident rates for men and women (per 100,000) were: Hong Kong 180 and 459; Singapore, 164 and 442; thus being quite similar.

For the reports cited in this letter, fragility fracture prevalence among elderly Chinese is mostly no more than 50% of that of Caucasians, and much lower rates (such as 30% of that of Caucasians) have also been reported. A much lower bone mass decline rate among elderly Chinese may at least partially contribute to their lower fragility fracture prevalence.

Based on the evidence presented in this letter (which are themselves rather heterogeneous), it may be a reasonable summary that fragility fracture prevalence among elderly Chinese is probably 40% of that Caucasians, which is the same for both women and for men. It has been reported that hip fragility fracture prevalence in Taiwan is substantially higher than those in other East Asian Territories [START_REF] Cheung | An updated hip fracture projection in Asia: The Asian Federation of Osteoporosis Societies study[END_REF]. Since the population in Taiwan is not ethnically distinct, the reported higher hip fragility fracture prevalence in Taiwan could be either due to 1) study artefacts, or 2) some kinds of unique lifestyle factors. The actual cause for this discrepancy should be further clarified. The summary of this letter will also be relevant for estimating the osteoporosis burden for Chinese, and setting the BMD threshold for defining osteoporosis status among Chinese [START_REF] Wang | Fine turning the cutpoint T-score as an epidemiological index with high specificity for osteoporosis: methodological considerations for the Chinese population[END_REF].

  investigated hip fracture incidence trends in ethnic subgroups of older Medicare beneficiaries (≥65 years old ), and analysed annual hip fracture incidence 4 rates from 2000 through 2009. They reported that, for both elderly men and women, Asians had approximately half of hip fracture incidence of those of Caucasians. By analysing the Canadian population-based Manitoba BMD (bone mineral density) Program registry data, in 2021 Leslie et al (8) reported a study included 68,907 Caucasian women (mean age: 64.7±10.9 years) and 1910 Asian women (mostly Chinese and Filipino, mean age: 62.8±9.9 years). During a follow-up period of 8.8±5.1 years for Caucasians and 6.5±5.0 years for Asians, incident hip fracture rate per 1000 person-years was 0.6 (95% CI[confidence interval]: 0.2-1.0%) for Asians and 3.6 (95% CI: 3.5-3.8) for Caucasians (Asian vs Caucasian ratio: 0.17).For Japanese data, by analysing hospital records, in 1991 Ross et al[START_REF] Ross | A comparison of hip fracture incidence among native Japanese, Japanese Americans, and American Caucasians[END_REF] described hip fracture incidence rates from 1979 to 1981 among men and women (ages 50-84 years) of Japanese ancestry living in Hawaii, and compared with hip fracture incidence rates from 1984 to 1985 among Japanese living on Okinawa, Japan. Both rates were further compared with those from 1978 to 1982 among American Caucasians of Rochester, Minnesota, USA.

  Caucasians and Asians. At baseline, among 179,470 Caucasians, 23.3% had a maternal history of facture and history of any fracture since 45 years old was recorded in 11.2% of the cases; among 1,912 Ascians, 11.6% had a maternal history of facture and history of fracture since 45 years old was recorded in 7.2% of the cases. During the follow-up, any fracture was recorded in 1.5% of 149,524 Caucasians and in 0.7% of 1,258 Asians. They noted that after 5 adjusting for BMD, weight, and other covariates, with Caucasian women as the referent group (relative risk: 1.0), Asian American women had a relative risk of 0.32 (95% CI, 0.15-0.66) for fracture. With cross-sectional analysis of the datasets of the Osteoporotic Fractures in Men Study (MrOS[USA], n=5342 Caucasians), MrOS Hong Kong (n=1968), Namwon Study and Dong-gu study in Korea (n=3891), in 2014 Shin et al (11) reported the prevalence of self-reported non-traumatic fracture for men aged 65 years and older was US white 17.1 %, US Asian 10.5 %, Hong Kong Chinese 5.6 %, and Korean 5.1 %. Analysing the Canadian Multicentre Osteoporosis Study (CaMos) data, in 2020 Morin et al (12) compared the prevalence of low-trauma fractures between Chinese and Caucasian participants at the time of recruitment into the cohort, and 5-year incident fractures and BMD (bone mineral density) changes between both groups. At baseline, the mean age was 63.2 (±8.7) years and 67.0 (±7.1) years for Chinese (n=104) and Caucasian women (n=5361);