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François Viète’s revolution in algebra

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Abstract

Françios Viète (1540–1603) was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek geometric equalities. His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”.

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Notes

  1. “à barbaris defædata & conspurcata” (Viète 1591, fol. 2b.-7; 1646, p. xi.14). Translation from (Klein 1968, p. 318). Note: folio/page numbers are followed by line number, separated by a full stop. The first reference in this footnote is in the seventh line from the bottom of folio 2b, and the second is in the 14th line from the top of page xi.

  2. (Freguglia 2008a, p. 53) dates the composition to before 1593.

  3. The first sixteen folios were printed in Tours in 1593, and the remainder in Paris in 1600. Folio 16 ends in the middle of Zetetic IV.6 (Van Egmond 1985, p. 362).

  4. “...multiplicabimus 5 in 6, fient 30” (Gosselin 1577, fol. 49b.1); “duc 30 in 2, fient 60” (Gosselin 1577, fol. 70a.9).

  5. “...multiplicemus 1L in 20 P 1L, existent 20L P 1Q, ducamus 2L in 20 M 2L, fient 40L M 4Q” (Gosselin 1577, fol. 76a.13).

  6. For examples in Tartaglia, see (Tartaglia 1560, fol. 28a; Euclid 1543, fols. XCIIIb (labeled “XC”), CXXIIb). Even Viète uses both words in his Isagoge.

  7. “Ut 9 in 27 facit 243” (Diophantus 1575, p. 3.19).

  8. “Sic 8q per 4 faciunt 32q” (Ramus 1560, fol. 3a).

  9. “...solidum basim habens quadratum ex FD, & altitudinem DE” (Pappus 1588, fol. 282a.4).

  10. “...cubum AB, æqualem esse cubis AG & GB & triplo AG in quadratum GB, & triplo GB in quadratum AG” (Cardano 1545, fol 16b.7).

  11. “Ergo quod fit ex b,d in a,g æquale est his quæ fiunt ex a,d in b,g & ex a,b in d,g” (Ptolemy 1550, fol. Biiijb.22). George’s translation: “Erit ergo totium etiam quadrangulum quod est ex .A.G. In .B.D. æquale utrisque que sunt: ex .A.B. In .D.G. & ex .A.D. In .B.G.” (Ptolemy 1528, fol. 5a.-14).

  12. “...ergo totus ductus .a.g. in .b.d. est equalis ductui .a.b. in .g.d. & .a.d. in .b.g. simul.” (Ptolemy 1515, fol. 5b.-2).

  13. In a quantity like \(2+\sqrt{3}\) the ‘names’ or ‘terms’ are the 2 and the \(\sqrt{3}\).

  14. “Quando nomina quantitatum sunt ad invicem incommensurabilia: tunc congregatio haud aliter fieri potest, quàm aggregatis membris per adverbium Plus: nec etiam differentia aliter proferri, quàm per adverbium Minus: sicut ostendit Euclides in decimo, tam de binomijs, quàm de residuis.” (Maurolico 1575, p. 101.7). I italicize plus and minus because I am not translating them from Latin.

  15. “ut si iungendæ sint duæ quantitates r. 3. & r. 2. statim dicá, earum aggregatum esse r. 3. p\(^{\circ }\) r. 2. Si vero hæc ab illa subtrahenda fit, ilicet respondebo, residuum post subtractionem esse r. 3. \({\tilde{\mathrm{m}}}\). r. 2.” (Maurolico 1575, p. 94.10).

  16. “Si verò diversæ species addendæ sint inter se, vel subtrahendæ, quod accidit in hoc magisterio frequenter, tunc utimur duabis istis dictionibus, plus, & minus, quæ primoribus suis literis notantur, sic P, & M.” (Borrel 1559, p. 124.5).

  17. “excessu AC DE” (Pappus 1588, fol. 180a.-2).

  18. The Greek word was also sometimes translated more generally as quantitas. I am concerned specifically with the Latin magnitudo.

  19. “Arithmetica est scientia, quæ uim & naturam numerorum tradit”, “Geometria est scientia magnitudinum” (Dasypodius 1573, fols. 1a, 7a).

  20. “Magnitudo est quæ crescit & augetur, atque secatur, dividique potest in infinitum usque. sunt autem tres species, linea, superficies, corpus” (Dasypodius 1573, fol. 22a.2).

  21. As Clavius does in (Euclid 1574, p. 14 of the Prolegomena).

  22. Some sixteenth-century mathematicians who apply the word magnitudo in these ways include Gregor Reisch, Johann Scheubel, Oronce Fine, Jacques Peletier, Pedro Monzón, François de Foix Candale, Petrus Ramus, Federico Commandino, Christoph Clavius, and Guillaume Gosselin (Reisch 1504, 2nd page of Book VI; Euclid 1550, p. 226.16; 1551, fol. 1a; 1557, p. 2; 1566, fol. 39a; 1572, fols. 1a, 6b, 57b, 124a; 1574, pp. 2, 14 of the Prolegomena; Monzón 1559, fols. 39a, 40a; Ramus 1569, pp. 2–3 of Geometriae; Gosselin 1577, fol. 1aff).

  23. Reisch and Commandino are two examples. See (Crapulli 1969, chapter I). Barocius’s 1560 translation of Proclus’s commentary on Book I of the Elements also restricts magnitudo to the three geometric dimensions (Proclus 1560, pp. 3, 21.4, 33.19, 34.4, 69.26), and in his investigation of the nature of angles he explains that they are not magnitudes (Proclus 1560, pp. 69ff).

  24. Following the uses of arithmos in Diophantus, Xylander and Viète use the word numerus with still another meaning as the name the first-degree unknown in logistice numerosa. There, too, the meaning is clear by the context. See Sect. 4.1.

  25. “...cubus ex IA plus solido sub AB & quadrato ex IA minùs solido duplo sub IA & quadrato ex AB æquatur cubo ex AB.” (Viète 1593d, fol. 19a.-1; 1646, p. 252.45).

  26. Translation adapted from (Klein 1968, p. 324).

  27. (Apollonius 1566, fol. 86b.13). Points b, d, l, and x are colinear. I wrote the letters from the diagram in italics to distinguish them from the surrounding text. Taliafero translates Apollonius’s Greek using the symbols “+” and “=”: “rect. BX, XD + rect. BL, LD + sq. XE + sq. LE = 2 sq. BE”, and Heiberg’s version is even more algebraic: “\(B\varXi \times \varXi \varDelta +B\varLambda \times \varLambda \varDelta +\varXi E^2+\varLambda E^2=2BE^2\)” (Apollonius 2000, p. 221; 1891, p. 381.18).

  28. (Viète 1593a, fol. 15a; 1595a, fol. 8; 1646, pp. 314ff, 371ff).

  29. (Viète 1595a, fol 8r; 1615a, p. 10; 1646, pp. 288, 315). Theorem I is equivalent to the related identities for \(\cos (a-b)\) and \(\sin (a-b)\).

  30. (Viète 1593a, fol. 15b; 1595a, fol. 9a; 1615a, p. 12; 1646, pp. 289, 316, 372). In all three books there are errors in the statements for the quintuple angle triangle, but the formulations in terms of logistice speciosa and in numbers are correct.

  31. (Viète 1595a, fol. 11b; 1646, p. 319). The vertical bar can be read as a kind of decimal place. The numbers are written as in (1595). In the 1646 printing they are written as “100,000,000,000,000” and “196,000,000,000,000”, and the numbers in Fig. 4 are shown as “184,160,000,000,000”, “164,953,600,000,000”, “139,159,056,000,000”, “107,778,549,760,000”, “72,096,901,529,600”, and “33,531,337,238,016”.

  32. Anderson modified Viète’s language by invoking the explicit multiplication and division of magnitudes using the preposition in and the division bar, both borrowed from logistice speciosa. He also multiplies magnitudes in his appendix of De Æquationum Emendatione.

  33. “26  Ecquis verò, cum magnitudines omnes sint lineæ, superficies, vel corpora, tantus proportionum suprà triplicatam, aut demum quadruplicatam rationem potest esse usus in rebus humanis, nisi fortè in sectionibus angulorum, ut ex lateribus figurarum anguli, vel ex angulis latera consequamur? | 27  Ergo à nemine hactenus adgnitum mysterium angularium sectionum, sive ad Arithmetica, sive Geometrica aperit, & edocet | Data ratione angulorum dare rationem laterum. | Facere ut numerum ad numerum, ita angulum ad angulum.” (Viète 1591, fol. 9a; 1646, p. 12). Translation adapted from (Viète 1983, p. 32).

  34. Anderson clarifies this in Problem I in Ad Angularium Sectionum Analyticen. Given two angles that have to one another the ratio of an integer to an integer, to find the sides either by construction or by numerical calculation. This follows from the main theorem (Viète 1615a, p. 40; 1646, p. 300).

  35. Anderson clarifies this in Problem II in Ad Angularium Sectionum Analyticen (Viète 1615a, p. 40; 1646, p. 300). The ratio is of an integer to an integer, and it asserts effectively that one can divide a given angle into n equal parts. This is equivalent to constructing the perpendicular of the smaller angle triangle [given in Theorem V of Ad Problema (Viète 1595a, fol. 10b; 1646, p. 318)], which amounts to constructing the solution to an nth degree polynomial equation. Viète poses the problems of dividing an angle into 3, 5, and 7 parts in his Theorematia I, II, and III in Ad Problema, which he solves by numerically calculating the perpendiculars (Viète 1595a, fol. 12aff; 1646, pp. 320ff). Anderson repeats these problems in Ad Angularium Sectionum Analyticen, giving instead geometric constructions (Viète 1615a, pp. 42ff; 1646, pp. 301ff).

  36. (Pappus 2010, p. 125).

  37. (Euclid 1572, fol. 229b.24), from Commandino’s commentary to Proposition XIII.1.

  38. (Regiomontanus 1584, fols. 141aff). For the table of tangents, see fol. 31a.

  39. “Si fuerint quatuor lineæ rectæ continuè proportionales. Dato adgregato mediarum & adgregato extremarum, dantur singulæ mediæ vel extremæ.” (Viète 1595b, fol. 9a; 1646, p. 270).

  40. (Viète 1593a, fol. 11a; 1646, p. 364). In the 1593 printing line AB is misprinted twice as ZB, and the 100,000,000 is mistakenly written as 200,000,000. Only the first ZB is corrected in the 1646 edition.

  41. “Quadratum numeri cuiusdam ductum in latus & in 10,000 facit 57,732,824. In notis 10,000N + 1C æquatur 57,732,824. Quæritur quis fit numerus ille. | Numerus 57,732,824 est Cubus adiunctus Solido sub lateris Quadrato & datâ longitudine 10,000. [\(\ldots \)] ita tamen ut cum solidum dividatur per longitudinem, quod inde oritur non intelligatur radix ipsa, sed radicis Quadratum. Illud enim est legi homogeneorum attendisse.” (Viète 1600a, fol. 11b.-9; 1646, p. 182.11). As Witmer notes, both editions mistakenly write the number as 57,732,824. The translation of the last part, “Thus since the solid [\(\ldots \)]” is taken from (Viète 1983, p. 331).

  42. “rhetics or exegetics [\(\ldots \)] eáque potissimum ad artis ordinationem pertinere, cùm reliquæ duæ exemplorum sint potius quam præceptorum, ut logicis iure concedendum est, suum éxercet officium, tam circa numeros, si de magnitudine numero explicandâ quæstio est, quàm circa longitudines, superficies, corporáve, si magnitudinem re ipsa exhiberi oporteat.” (Viète 1591, fol. 8a.27; 1646, p. 10.29). The “rhetics or exegetics” is written in Greek. Translation modified slightly from (Klein 1968, p. 346).

  43. (Christianidis and Oaks 2013, pp. 136–137; Oaks 2010, p. 28; 2012, p. 43; Marre 1880, p. 738; Borrel 1559, pp. 123ff). These are just samples. I have not tried to be exhaustive.

  44. “...logisticem sub specie noviter inducendam” (Viète 1591, fol. 4a.21; 1646, p. 1.25).

  45. (Viète 1615b, p. 72.2; 1646, p. 131.4). The “Z” in (1615b) is mistakenly shown in lower case. I have restored it.

  46. “Magnitudines quæ ex genere ad genus suâ vi proportionaliter ad scendunt vel descendunt vocentur Scalares.” (Viète 1591, fol. 4b.37; 1646, p. 3.1). Translation adapted from (Viète 1983, p. 16).

  47. For examples (Bos 2001, p. 149; Freguglia 2008b, p. 154). Freguglia’s explanation of an occurrence of “B quad.” on p. 158 is incorrect. On the other hand Louis Charbonneau and Michel Serfati each explain the two series correctly. (Charbonneau 2006, p. 68; Serfati 2005, pp. 214ff).

  48. (Viète 1615b, p. 27.12; 1646, p. 99.4). I will give the forms of the equations from the 1646 edition. The only difference is the arrangement of the terms on the page.

  49. “Quum autem ipsum E quadratum Radix statuetur plana, hæc erit æquationis enunciato. \(\left. \begin{array}{l}~~~\hbox {E plani-quadratum.}\\ +~\hbox {B quadratum in E planum}\end{array}\right\} ~{\ae }\hbox {quabitur B quadrato in Z planum.}\)” (This is how the equation appears in (1615).) (Viète 1615b, p. 27.-3; 1646, p. 99.11). Translation adapted from (Viète 1983, p. 192).

  50. “designatibur commodè vocabulo

    figure c

    vel

    figure d

    , veluti A in B.” (Viète 1591, fol. 5b.17; 1646, p. 5.-12). The passage continues: “[\(\ldots \)] by which it will be signified that the one has been multiplied in the other; or as others say (ut alii), that it is produced sub A & B”. Translation adapted from (Klein 1968, p. 333). Viète always multiplies species with the preposition in. He may have included the geometric term sub to strengthen the connection with geometry.

  51. “De præceptis Logistices speciosæ. Caput. IIII. | Logistice numerosa est quæ per numeros, Speciosa quæ per species seu rerum formas exhibetur, ut pote per Alphabetica elementa. | Logistices speciosæ canonica præcepta sunt quatuor, ut numerosæ. | PRÆCEPTUM I. | Magnitudinem magnitudini addere. | Sunto duæ magnitudines A & B. Oportet alteram alteri addere.” (Viète 1591, fol. 5a; 1646, p. 4). Translation adapted from (Viète 1983, pp. 17–18).

  52. “...adgregatæ erunt A plus B, siquidem sint simplices longitudines latitudinés-ue”. Translation adapted from (Viète 1983, p. 18).

  53. “Et hîc se prȩbet Geometram Analysta opus verum efficiundo post alîus similis vero resolutionem:illic Logistam, potestates quascumque numero exhibitas, sive puras, sive adfectas resolvendo.” (Viète 1591, fol. 8a.-10; 1646, p. 10.-13). The translation of the second sentence is taken from (Viète 1983, p. 29).

  54. “Neque verò in Geometricâ phrasi hic erit magna dissimilitudo: Enim verò dicet Geometra, B planum esse aggregatum quadratorum à tribus proportionalibus lineis rectis, D vero solidum quod fit ab aggregato extremarum in mediæ quadratum” (Viète 1615b, p. 11.-6; 1646, p. 91.5). Translation modified slightly from (Viète 1983, p. 173).

  55. From Proposition III, Chapter XVIII in De Recognitione Aequationum (Viète 1615b, p. 54; 1646, p. 111). The equation in species is shown as it appears in the 1646 Opera Mathematica. In the original printing it is

    $$\begin{aligned} \left. \begin{array}{l}~~~\hbox {B solidum in A}\\ -\hbox {A quadrato-quadrato.}\end{array}\right\} ~{\ae }\hbox {quetur Z plano-plano}. \end{aligned}$$

  56. “Forma autem Zetesim ineundi ex arte propriâ est, non iam in numeris suam logicam exercente, quæ fuit oscitantia veterum Analystarum, sed per logisticem sub specie noviter inducendam, feliciorem multò & potiorem numerosâ ad comparandum inter se magnitudines” (Viète 1591, fol. 4a.19; 1646, p. 1.23). Translation modified from (Viète 1983, p. 13).

  57. “Fiat tertii hypotenusa similis ei quod fit ex hypotenusa primi in hypotenusam secundi, nempe Z in X.” (Viète 1646, p. 34.17).

  58. See footnote 8 in (Viète 1983, p. 13), and (Klein 1968, pp. 172ff).

  59. (Euclid 1572, fol. 57a.-4). Others who use the same phrase are Campanus, Zamberti, Scheubel, Peletier, and Clavius.

  60. (Viète 1591, fol. 4b.-10; 1646, p. 3.1). The word appears in other places in his works with the same meaning.

  61. “Prima & perpetua lex æqualitatum seu proportionum, quæ, quoniam de homogeneis concepta est, dicitur lex homogeneorum, hæc est [\(\ldots \)]” (Viète 1591, fol. 4b.26; 1646, p. 2.-13).

  62. “Quibus non attendisse causa fuit multæ caliginis & cæcutiei veterum Analystarum.” (Viète 1591, fol. 4b.-11; 1646, p. 2.-2). Translated in (Viète 1983, p. 16).

  63. “Cuius inventi lætitiâ adfectus, ô Diva Melusinis, tibi oves centum pro unâ Pythagoræâ immolavi.” (Viète 1595a, fol. 8b; 1646, p. 315). English translation from (Klein 1968, p. 253 n. 197). Pythagoras got only one sheep because his triangle has only has one angle at the vertex. Klein calls this a “playful remark”.

  64. “Datâ summâ laterum & summâ Cuborum distinguere latera”.

  65. The two exceptions among Viète’s zetetics are the last two of Book V. There the questions are already posed in terms of species.

  66. The “\(+\)” is mistakenly shown as “−” in 1593.

  67. The equation is shown as “ æquatur E quadrato” in the 1646 Opera Mathematica. In Witmer’s notation it is “\((4D^S-B^3)/3B\) is equal to \(E^2\)”.

  68. Zetetic I.1.

  69. “Invenire numero tria Latera, sub quibus binis quod fit Planum adscito dato plano, eveniat Quadratum.” (Viète 1593c, fol. 22a; 1646, p. 78).

  70. (Stifel 1544, fol. 103a). Translation taken mainly from (Nunn 1914, p. 412).

  71. (Viète 1615a, p. 47; 1646, pp. 303–304). Antonio Garibaldi also mentions the connection between the theorem on angular sections and Viète’s interest in astronomy (Garibaldi 1992, p. 168).

  72. “Les relations entre les parties d’une même figure sont ou des relations de position ou des relations de grandeurs” (Marie 1884, p. 3).

  73. Marco Panza makes the same observation about the quantitative nature of Viète’s zetetics (Panza 2006, p. 279; 2007, p. 115).

  74. “Theon, Apollonius, Papus, & alii\(\ldots \)” (Viète 1600b, fol. 8b.7; 1646, p. 8).

  75. “Zeteticem autem subtilissimè omnium exercuit Diophantus in iis libris qui de re Arithmeticâ conscripti sunt. Eam verò tanquam per numeros, non etiam per species, quibus tamen usus est, institutam exhibuit, quò sua esset magis admirationi subtilitas & solertia, quando quæ Logistæ numeroso subtiliora adparent, & abstrusiora, ea utique specioso familiaria sunt & statim obvia.” (Viète 1591, fol. 8a.10; 1646, p. 10.6). Translation modified from (Viète 1983, p. 27).

  76. “il cuore del ‘programma viètano’ ” (Freguglia 1989, p. 52).

  77. Examples can be found in: Pappus, Book II of The Collection; Jordanus, throughout his De Numeris Datis; Part 2 of Chapter 14 of Fibonacci’s Liber Abaci; many proofs in al-Fārisī’s Foundations of Rules on Elements of Benefits; Tartaglia’s Part II Book VII; and Cardano’s Arte Magna. Some specific references: (Fibonacci 1857, pp. 358–360, 376–377; 2002, pp. 497–499, 517; al-Fārisī 1994, pp. 501–510; Tartaglia 1556, fol. 109b; Cardano 1545, fols 54b–55b).

  78. The letter C is not avoided because it is used in logistice numerosa. Sometimes the letter N designates a species, as in Proposition XLVII of Ad Logisticem Speciosam, Notæ Priores and in zetetics IV.4 and IV.5.

  79. (Viète 1593d, fol. 17b.-2; 1646, p. 249.-4).

  80. From De Æquationum Emendatione (Viète 1615b, p. 122.17; 1646, p. 155.16).

  81. “...quantitates Geometricas” (Borrel 1559, p. 120).

  82. Translated in (Oaks 2011, p. 61).

  83. Translated in (Oaks 2011, p. 61).

  84. “Il e st peu de mathématiciens à qui cette science doive plus qu’à cet homme illustre” (Montucla 1758, p. 488).

  85. “M. Viete d’avoir établi l’usage des lettres pour désigner non seulement les quantités inconnues, mais même celles qui sont connues.” (Montucla 1758, p. 488).

  86. “Un autre avantage plus estimable encore, est la facilité qu’elle procure de pénétrer dans la nature & la composition des équations” (Montucla 1758, p. 489).

  87. “La question, telle que se la posa Viète, était d’introduire les grandeurs elles-mêmes, sous leur formes concrète, dans les équations algébriques” (Marie 1884, p. 6).

  88. “Es sind Grössen, nicht Zahlen”.

  89. (Klein 1968, p. 7), translation adjusted in (Hopkins 2011, p. 96).

  90. (Klein 1968, p. 171). Emphases in quotations from Klein are all his.

  91. (Klein 1968, p. 172), translation adjusted in (Hopkins 2011, p. 282).

  92. (Klein 1968, p. 175), translation adjusted in (Hopkins 2011, p. 255).

  93. (Klein 1968, p. 176), translation adjusted in (Hopkins 2011, pp. 288, 289).

  94. (Klein 1968, p. 174), translation adjusted in (Hopkins 2011, p. 286).

  95. (Klein 1968, p. 175), translation adjusted in (Hopkins 2011, pp. 255–256); (Klein 1936, p. 183).

  96. Chapter 8 in (Bos 2001).

  97. “...in ultima analisi ricondursi alla teoria delle proporzioni, cioè, tenendo presente il libro V degli Elementi euclidei, alla forma più astratta della geometria classica” (Freguglia 1989, p. 53).

  98. “le lettere (che non esprimono solo incognite) sono suscettibili di varie interpretazioni.” (Freguglia 1989, p. 51).

  99. (Panza 2006, p. 280). He expresses these ideas also in (Panza 2005, pp. 19ff).

  100. “Ma le lettere non si limitano a sostituire e a rinviare ai numeri; esse denotano anche delle grandezze geometriche: linee, piani, solidi” (Giusti 1990, p. 425).

  101. “\(\ldots \)le lettere vengono ad assumere una posizione intermedia tra numeri e grandezze, spezzando cosí quel legame diretto che aveva determinato la separazione tra l’algebra e la geometria, ed instaurando invece un rapporto complesso, che passa attraverso il tramite della rappresentazione letterale” (Giusti 1990, p. 425).

  102. Passages where the species are identified with numbers can be found in (Serfati 2005, pp. 158.10, 159.4, 159.12), and the whole discussion of Viète’s innovation is set against the representation of givens in geometry.

  103. (Pappus 1982, p. 7). He follows up the proposition with a numerical example.

  104. Translated in (Jordanus de Nemore 1981, p. 129). Again, a numerical example follows. This is the same problem as Viète’s zetetic II.5.

  105. “n’était pas astronome, mais il était le plus grand géomètre de son temps” (Delambre 1819, p. 455).

  106. “il créa l’Algèbre nouvelle, en représentant tous les éléments d’une question, connus ou inconnus, par des lettres de l’alphabet, les opérations à effectuer sur elles par des signes et enfin le résultat par une formule, dans laquelle il suffisait, si la même question était posée avec des données différentes, de les substituer pour obtenir immédiatement le nouveau résultat demandé” (Ritter 1895, p. 21).

  107. Translated in (Descartes 1954, p. 2).

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Oaks, J.A. François Viète’s revolution in algebra. Arch. Hist. Exact Sci. 72, 245–302 (2018). https://doi.org/10.1007/s00407-018-0208-0

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