A reconstruction of Viète's <i>Canonion triangvlorvm</i> (1579)

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(1579)

Denis Roegel

To cite this version:

Denis Roegel. A reconstruction of Viète’s Canonion triangvlorvm (1579). [Research Report] 2011.

�hal-00654459�

of Viète’s

Canonion triangvlorvm

(1579)

Denis Roegel

20 December 2011

François Viète was probably the greatest French mathematician of the 16th century. He

was a lawyer, but worked in parallel on mathematics. He developed a new algebra with

the use of letters as parameters in equations.

2

The

Canonion triangulorum (1579)

In his Canonion triangulorum, which is part of the Canon mathematicus (1579), Viète

gave a number of right-angled triangles with rational sides. More precisely, Viète gave

the Pythagorean triples



4n

n

2

_{+ 4}

,

n

2

_{− 4}

n

2

_{+ 4}

, 1



(1)



4n

n

2

_{− 4}

, 1,

n

2

_{+ 4}

n

2

_{− 4}



(2)



1,

n

2

_{− 4}

4n

,

n

2

+ 4

4n



(3)

for 1422 values of n and scaled by a factor 100000. These triples are of the form (b, p, h),

where b

2

_{+ p}

2

_{= h}

2

_{, b being the basis, p the perpendicular, and h the hypotenuse of a}

right triangle. The above formulæ were given by Viète at the end of the Canonion. They

were of course not expressed in modern terms, but using more verbose expressions.

In the first sequence of triples (series I ), the hypotenuse is made constant and equal

to 100000. In the second sequence (series II ), the perpendicular is constant, and in the

third sequence (series III ), it is the basis. These sequences are the same as those used

in Viète’s trigonometric table [14].

The layout of the table alternates between odd and even pages, with symmetrical

sequences of triple series. The value of n is given in the right column of odd pages, and in

the left column of even pages. The first value of the first page corresponds to n = 999424,

and the three corresponding triples are

 19988480000

49942416589

, 99999 +

49942376589

49942416589

, 100000



,

(4)

 99942400000

249712082943

, 100000, 100000 +

200000

249712082943



,

(5)



100000, 24985599999 +

28107

31232

, 24985600000 +

3125

31232



(6)

The last value of the table corresponds to n = 1, but this value leads to negative

values in the triples, and Viète therefore replaced all values by

f.

For n = 2, only the first and third triples are meaningful, and are (100000, 0, 100000)

and (100000, 0, 100000). Viète replaced the two zeros by

f, and put

f

for the second

triple.

1, 2, 3, . . . , 200,

202, 204, 206, . . . , 400,

404, 408, 412, . . . , 800,

808, 816, 824, . . . , 1600,

. . .

100 · 2

m−1

_{+ 2}

m−1

_{, 100 · 2}

m−1

_{+ 2 · 2}

m−1

_{, . . . , 100 · 2}

m

. . .

413696, 417792, 421888, . . . , 819200,

827392, 835584, 843776, . . . , 999424

Each sequence, except the first and the last, contains 100 terms. n takes a total of

200 + 100 × 12 + 22 = 1422 values.

According to Ritter [9, p. 47], Viète may have computed this table in order to provide

better approximations of the equation of time, or of the difference between the true

and mean motion of a planet. Viète himself acknowleged the little utility of his table.

Delambre wrote that the triangles are expressed with fractions which are so large that

their practical utility is not clear [3, p. 456].

Viète’s table seems to be rather accurate, according to our samples. The only error

found among the several dozen values examined is that for the index 29 (last page). Viète

gave the pair 13727 +

137

₁₆₉

, 99054 +

114

169

instead of 13727 +

137

169

, 99053 +

43

169

.

Viète’s work made its way to Thomas Harriot through Nathaniel Torporley [12, 11].

3

Reconstruction

The values of the table were computed using the GNU mpfr multiple-precision

floating-point library developed at INRIA [5]. In this case, the library was only used with integers

and fractions were simplified using an implementation of Euclid’s GCD algorithm.

We have tried to remain as close as possible to the original layout, but we currently

didn’t make use of colors. The original tables may have some parts in color. As soon as

we get a chance to consult the original table, we will adapt the coloring if necessary.

The following list covers the most important references

1

related to Viète’s tables. Not all

items of this list are mentioned in the text, and the sources which have not been seen are

marked so. We have added notes about the contents of the articles in certain cases.

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

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

and co., 1842.

[2] Henri Bosmans. Le traité des sinus de Michiel Coignet. Annales de la Société

Scientifique de Bruxelles, 25 (seconde partie, mémoires):91–170, 1901–1902.

[3] Jean-Baptiste Joseph Delambre. Histoire de l’astronomie du moyen âge. Paris:

Veuve Courcier, 1819.

[pp. 455–483 on Viète]

[4] Gustaf Eneström. M. Cantor. Vorlesungen über Geschichte der Mathematik

(review). Bibliotheca mathematica, 6 (new series):91–92, 1892.

[5] Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, and Paul

Zimmermann. MPFR: A multiple-precision binary floating-point library with

correct rounding. ACM Transactions on Mathematical Software, 33(2), 2007.

[6] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.

London: Taylor and Francis, 1873.

[Also published as part of the “Report of the forty-third

meeting of the British Association for the advancement of science,” London: John Murray, 1874.

A review by R. Radau was published in the Bulletin des sciences mathématiques et

astronomiques, volume 11, 1876, pp. 7–27]

[7] Karl Hunrath. Des Rheticus Canon doctrinæ triangulorum und Vieta’s Canon

mathematicus. Zeitschrift für Mathematik und Physik, 44 (supplement):211–240,

1899.

[= Abandhandlungen zur Geschichte der Mathematik, 9th volume]

[8] Charles Hutton. Mathematical tables. London: G. G. J. Robinson, J. Robinson and

R. Baldwin, 1785.

[The same text on Viète was reproduced in later editions.]

[9] Frédéric Ritter. François Viète, inventeur de l’algèbre moderne, 1540–1603, notice

sur sa vie et son œuvre. Paris: Dépôt de la Revue occidentale, 1895.

[Ritter wrote

much more material on Viète and his manuscripts are at the library of the Institut in Paris,

Ms. 2004–2012.]

1

_{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 we have not done it here.

[11] Jacqueline Stedall. Notes made by Thomas Harriot on the treatises of François

Viète. Archive for history of exact sciences, 62(2):179–200, 2008.

[12] Rosalind Cecilia Hildegard Tanner. Nathaniel Torporley’s ‘Congestor analyticus’

and Thomas Harriot’s ‘De triangulis laterum rationalium’. Annals of Science,

34:393–428, 1977.

[13] Paul Tannery. Ritter (Frédéric). François Viète, notice sur sa vie et son œuvre

(review). Bulletin des sciences mathématiques, 20:204–211, 1896.

[14] François Viète. Canon mathematicus seu ad triangula cum appendicibus. Paris:

Jean Mettayer, 1579.

[The main table was reconstructed in [10].]

[15] François Viète. Canonion triangulorum laterum rationalium. Paris: Jean Mettayer,

1579.

[part of the Canon mathematicus [14]]

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

Pitiscus. PhD thesis, University of Michigan, 1944.

[published in 1946]

CANONION TRIANGVLORVM LATERVM RATIONALIVM

_

SERIES

SERIES

SERIES

I.

II.

III.

&c.



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



_{,,,},,,

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,}

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

 ,

,



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



_{,,,},,,

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,}

,



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

 ,

,



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,

CANONION TRIANGVLORVM



SERIES

SERIES

SERIES

III.

II.

I.

,

,,,

_{,}

,,,

_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



_{,,,},,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

,,,

_{,}

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



_{,,,},,,

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

 ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

,,,

_{,}

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



_{,,,},,,

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

,,,

 ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

_

,,,

_

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



_{,,,},,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

,,,

_{,},

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

,,,

_{,}

,,,

,_{,}

,

_{,,,},



_{,,,},,,

,

,,,_{,,,}



_{,,,},,,

,

,,,

, ,

,,,

, ,

,

, ,,,



,,, ,,,

,

,,, ,,,



,,, ,,,

LATERVM RATIONALIVM

_

SERIES

SERIES

SERIES

I.

II.

III.



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



_{,,,},,,

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,}

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

 ,

,



,,,_{,,,}

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



_{,,,},,,

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



_{,,,},,,

,

,,,_{,,,}



_{,,,},,,

,

_{,,,},

,,,

,_{,}

,,,

_{,}

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

 

,,,

 

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,}

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,}

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

, ,

,



,,,_{,,,}

,

,,,_{,,,}



,,_{,,}

,

_{,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,,_{,,,}

,

,,,_{,,,}



,,,_{,,,}

,

_{,,,},

,,,

,_{,}

,,,

_{,},

,



,,, ,,,

,

,,, ,,,



,,, ,,,

,

, ,,,

,,,

, ,

,,,

 ,

,