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The Géométrie of 1637

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Descartes’s Mathematical Thought

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 237))

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Abstract

By the time that he composed his old Algebra and Regulae ad directionem ingenii, Descartes had established his basic program of algebraic analysis, which, even though immature, may be compared with that of François Viète. By that period his main interest, too, had already shifted from pure mathematics to natural philosophy (especially optics) and metaphysics. His studies carried out in the Netherlands during the years around 1630 first crystallized as Le Monde, ou Traité de la lumière, published posthumously in 1664, and the Dioptrique, one of the three essays added to the Discours de la méthode of 1637. The latter treatise exploited the theory of conic sections, which was excerpted partly in Beeckman’s Journal in 1628–1629 (see Chapter 4, § 1), whereas the former was a writing which can be evaluated to be “the first attempt to construct an entire physical universe on mechanical foundations.” 1

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References

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  17. AT, VI, p. 377; Olscamp, p. 182.

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  21. AT, VI, p. 370; Olscamp, p. 177. The historical significance of Descartes’s algebra of line segments is emphasized in M. S. Mahoney, “The Beginnings of Algebraic Thought in the Seventeenth Century,” Chapter 5 of Stephan Gaukroger, ed., Descartes: Philosophy,Mathematics and Physics (Brighton, Sussex/Totowa, New Jersey, 1980), pp. 141–155, esp. p. 145.

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  27. In this regard, Alexander Jones’s remark is significant. He believes that Pappus used the notion of compound ratios which might be interpreted as that of ‘products of ratios’ under the influence of his predecessors who admitted powers of more than three dimensions in arithmetical cases. He counts Diophantus as one of these predecessors. See Jones’s commentary on Pappus of Alexandria, Book 7 of the Collection, Edited with Translation and Commentary by Alexander Jones (New York/Berlin/Heidelberg/Tokyo, 1986), Part 1, p. 74 and Part 2, p. 404.

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  29. See M. S. Mahoney, “Mathematics,” Chapter 5 of David C. Lindberg, ed., Science in the Middle Ages (Chicago, 1978), p. 166.

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  32. AT, X, p. 463.

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  37. AT, VI, p. 18.

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  38. “Henry George Liddell and Robert Scott, A Greek-English Lexicon (Oxford, 1968), p. 112. See also our discussion on pp. 63–72.

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  39. Bos, Op. cit. (n. 15), p. 300.

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  42. Hultsch, I, pp. 54–47; Commandino, f. 4v; Ver Eecke, I, pp. 38–39. As to the translation we referred to A. G. Molland, “Shifting the Foundations: Descartes’s Transformation of Ancient Geometry,” Historia Mathematica, 3 (1976), pp. 27–28.

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  43. Jones, pp. 104–107; Hultsch, II, pp. 662–663; Commandino, f. 162v; Ver Eecke, II, pp. 495496.

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  47. Descartes, for example, states on his physics in Part Six of his Discours de la méthode: “[A]s soon as I had acquired some general notions in physics and had noticed, as I began to test them in various particular problems, where they could lead and how much they differ from the principles used up to now, I believed that I could not keep them secret without sinning gravely against the law which obliges us to do all in our power to secure the general welfare of mankind. For they opened my eyes to the possibility of gaining knowledge which would be very useful in life, and of discovering a practical philosophy which might replace the speculative philosophy taught in the schools. Through this philosophy we could know the power and action of fire, water, air, the stars, the heavens and all the other bodies in our environment, as distinctly as we know the various crafts of our artisans; and we could use this knowledge--as the artisans use theirs for all the purposes for which it is appropriate, and thus make ourselves, as it were, the lords and masters of nature. This is desirable not only for the invention of innumerable devices which would facilitate our enjoyment of the fruits of the earth and all the goods we find there, but also, and most importantly, for the maintenance of health, which is undoubtedly the chief good and the foundation of all the other goods in this life.” AT,VI, pp. 61–62. Descartes’s emphasis on the method of analysis itself may be interpreted as his preference for practice rather than mere speculation. In this regard, see the following literature. Gilson sees here an influence of Francis Bacon. Gilson, Texte et commentaire (Ch. 1, n. 1), pp. 443–446. Cf. Richard B. Carter, “Descartes’ Methodological Transformation of Homo Sapiens into Homo Faber,” Sudhoffs Archiv, 68 (1984), pp. 225229. See also Harcourt Brown, “The Utilitarian Motive in the Age of Descartes,” Annals of Science, 1 (1936), pp. 182–192. Incidentally, it was in this context that Karl Marx provided his single reference to Descartes in Das Kapital, Bd. 1 (1867): Marx-Engels Werke, Bd. 23 (Berlin, 1969), pp. 411–412, n. 111.

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  51. Molland, Op. cit. (n. 42), pp. 23–33.

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  52. AT, VI, pp. 396–397; Olscamp, p. 196. For the mathematical details of Descartes’s solution of the problem of Pappus, see Bos, Op. cit. (n. 15), pp. 298–303.

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  53. AT, VI, p. 19.

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  54. AT, VI, pp. 406–407; Olscamp, p. 203.

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  55. AT, I, p. 233; AM, I, pp. 212–213: “Datis quotcunque rectis lineis, puncta omnia ad illas iuxta tenorem quæstionis relata, contingent vnam ex lineis quæ describi possunt vnico motu continuo, & omni ex parte determinato ab aliquot simplicibus relationibus; nempe, à duobus vel tribus ad summum, si rectæ positione datæ non sint plures quam quatuor; à tribus vel quatuor relationibus ad summum, si rectæ positione datæ non sint plures quam otto; à quinque vel sex, si datæ rectæ non sint plures quam duodecim, atque ita in infinitum.”

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  56. AT, I, p. 233; AM, I, p. 213: “Et vice versâ nulla talis linea potest describi, quin possit inueniri positio aliquot rectarum, ad quas referantur infinita puncta, iuxta tenorem quwstionis, quw illam contingent. Quæ quidem rectæ non erunt plures quam quatuor, si curua descripta non pendeat à pluribus quam duobus simplicibus relationibus; nec plures quam octo, si curua non pendeat à pluribus quam quatuor relationibus; Si sic consequenter.”

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  57. AT, VI, p. 397; Olscamp, p. 196.

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  58. See D. T. Whiteside, ed., The Mathematical Papers of Isaac Newton, Vol. IV (Cambridge, 1971), Part II, 3, ¡ì 1: “Three Mistakes in Descartes’s Géométrie,” pp. 336–345, esp. pp. 340345. This memorandum is supposed to have been composed in the late 1670s. Incidentally, Bos has recently given a modern and more elegant demonstration. See Bos, Op. cit. (n. 15), “Appendix: On the Curves that Occur as Loci of Problems of Pappus,” pp. 332–338.

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  59. AT, I, p. 233; AM, I, p. 213: “Mc auteur simplices relationes illas appello, quarum singulæ non nisi singulas proportiones Geometricas inuoluunt.”

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  61. AT, I, p. 234, 1. 13, AM, I, p. 214.

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  62. AT, X, p. 456: “Advertendum est etiam, per numerum relationum intelligendas esse proportiones se continuo ordine subsequentes [—].”

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  63. AT, I, pp. 233–234; AM, I, pp. 213–214: “Atque hæc linearum quæsitarum definitio est, ni fallor, adæquata & sufficiens. Per hoc enim quod dicam illas vnico motu continuo describi, excludo Quadraticem & Spirales, aliasque eiusmodi, quæ non nisi per duos aut plures motus, ab inuicem non dependentes, describuntur. Et per hoc quod dicam ilium motum ab aliquot simplicibus relationibus debere determinari, alias innumeras excludo, quibus nulla nomina, quod sciam, sint imposita. Denique per numerum relationum singula genera definio; atque ita primum genus solas Conicas Sectiones comprehendit, secumdum ver¨° prter illas quas supra explicui, continet alias quam plurimas, quas longum esset recensere.”

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  64. As to the history and the text of this tract, see Descartes, The World, Translation and Introduction by M. S. Mahoney (n. 1).

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  65. “Descartes, à (Deventer), à Mersenne, à Paris. (fin de fevrier 1634),” AT, I, pp. 285–286; AM, I, p. 253; CM, IV, pp. 50–51. Ovid, Tristia,III, iv, 25.

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  66. “Descartes, à Utrecht, à Mersenne, à Paris. (mars 1635),” AT, I, p. 322; AM, I, p. 284; CM, V, p. 125.

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  67. “Descartes, à Utrecht, à Constantin Huygens. ter novembre 1635,” AT, I, pp. 329–330; AM, I, pp. 290–291.

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  68. “Descartes, à (Leyde), à Mersenne, à Paris. (mars 1636), ” AT, I,pp. 339–340; AM, I, p. 301; CM, VI, p. 44.

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  69. bid.: “[E]n la Geometrie, ie tasche à donner vne façon generale pour soudre tous les Problemes qui ne l’ont encore iamais esté.”

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  70. “Descartes au P. Diriennes, 22 Février 1638,” AT, I, p. 458 (where the date of the letter is attributed to October, 1637); AM, II, p. 140: “C’est vn traitte que ie n’ay quasi compose que pendant qu’on imprimoit mes Météores [—].” The Father Deriennes taught mathematics at the college of La Flèche from 1635 to the year of his death 1661. A. Romano, La Contre-Reforme mathématique (Ch. 1, m. 13), pp. 599–600; Cf. Elie Denisoff, “Les Étapes de la rédaction du «Discours de la méthode»,” Revue philosophique de Louvain, 54 (1956), pp. 261–264; “Premier Essai” de Idem, Descartes, Premier théoricien de la physique mathématique (Louvain/Paris, 1970), pp. 15–18.

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  71. See the explanation of Van Schooten in the “Table des noms propres,” AM, I, 464. As wil be seen, Van Schooten edited the two Latin versions of the Géométrie. For his career, see Josef Ehrenfried Hofmann, Frans van Schooten der J¨¹ngere (Wiesbaden, 1962).

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  72. AT, VI, p. 368: “Aduertissement. Iusques icy i’ay tasché de me rendre intelligible a tout le monde; mais, pour ce traité, ie crains qu’il ne pourra estre leu que par ceux qui sçauent desia ce qui est dans les liures de Géométrie: car, d’autant qu’ils contienent plusieurs verités fort bien demonstrées, i’ay creu qu’il seroit superflus de les repeter, & n’ay pas laissé, pour cela, de m’en seruir.”

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  73. “Descartes, (près d’Alemaer), à Mersenne, à Paris. (fin décembre 1637),” AT, I,p. 478; AM, II, p. 65; CM, VI, p. 345.

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  74. É. Denisoff, Op. cit. (n. 70), p. 259, note 16. Denisoff refers to the item ‘géométrie’ in A. Lalande, Vocabulaire de la philosophie (Paris, 1932). See also Boyce Gibson, “La ((Géométrie» de Descartes au point de vue de sa méthode,” Revue de métaphysique et de morale, 4 (1896), pp. 386–398, esp. p. 390.

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  75. AT, VI, p. 369; Olscamp, p. 177.

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  76. According to Boyer, the terminology ‘analytic geometry’ was used probably for the first time by Michel Rolle in his “De l’evanovissement des quantitez inconnues dans la géométrie analytique,” Mémoires de l’Académie royale des sciences (Paris), Année 1709 (1709. 9. Aoust), pp. 419–450. See Carl B. Boyer, History of Analytic Geometry (New York, 1956), p. 155: “Incidentally, in Rolle’s discussion of this problem the term ‘analytic geometry’ appeared in print, perhaps for the first time, in a sense analogous to that of today.” On p. 141 of the same book Boyer also refers to Newton’s treatise titled “Artis Analyticae Specimina vel Geometria Analytica,” which was published in Opera quae exstant omnia, Vol. I, pp. 389–518, under the editorship of Samuel Horsley in London in 1779. The title is ascribed not to the author Newton but to a copyist William Jones. The tract which Horsley edited as Artis analyticae specimina vel Geometria analytica is basically identical to “De Methodus serierum et fluxionum,” written in Winter 1670–1671, in D. T. Whiteide, ed, The Mathematical Papers of Isaac Newton, Vol. III (1670–1673), (Cambridge, 1969). Jones copied Newton’s manuscript about 1710. See Whiteside’s Introduction, Op. cit., pp. 11–13. See also Whiteside’s “General Introduction,” in The Mathematical Papers of Isaac Newton, Vol. I (1664–1666), (Cambridge, 1967), pp. xxiii and xxviii, notes (26) and (34). I am skeptical about Newton’s use of ‘analytic geometry’ in other works as well.

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  77. H. J. M. Bos, “Arguments on Motivation in the Rise and Decline of a Mathematical Theory; the ‘Construction of Equations’, 1637¡ªca. 1750,” Archive for History of Exact Sciences, 30 (1984), p. 331.

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  78. In the section under the margin subtitle “Pourquoy les problesmes solides ne peuuent estre construits sans les sections coniques, ny ceux qui sont plus composés sans quelques autres lignes plus composées.” AT, VI, p. 475; Olscamp, p. 251.

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  79. AT, VI, p. 485; Olscamp, p. 259.

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  80. AT, VI, pp. 413–423; Olscamp, pp. 207--215.

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  81. Our simplified explanation owes much to the description by the editors of CM, VI, pp. 348–349.

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  82. AT, VI, pp. 424–425; Olscamp, pp. 215–216.

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  83. AT, VI, p. 428; Olscamp, p. 218.

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  84. Descartes gives no demonstration. For a detailed proof, see J. F. Scott, The Scientific Work of René Descartes (London, 1952), pp. 127–128. Scott has provided his proof

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  85. AT, VI, pp. 429–430; Olscamp, p. 219.

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  86. AT, VI, pp. 431–433; Olscamp, pp. 220–222. Cf. Scott, Op. cit. (n. 84), pp. 129–130. Figure 5.6 is taken from Scott, p. 129.

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  87. For the later development of the inverse method of tangents, see D. T. Whiteside, “Patterns of Mathematical Thought in the Later Seventeenth Century,” Archive for History of Exact Sciences,1 (1960), X. 3. “The concept of tangent,” pp. 348–365; Christoph J. Scriba, “The Inverse Method of Tangents: A Dialogue between Leibniz and Newton (1675–1677),” Archive for History of Exact Sciences, 2 (1963), pp. 113–137.

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  88. Kokiti Hara, “Comment Descartes a-t-il découvert ses ovales?”, Historia Scientiarum, No. 29 (1985), pp. 51–82. Importantly, Hara points out that the text and figures in the Adam-Tannery version are not a faithful reproduction of those of the Opuscule posthuma. For example, the titles of the “Exerpta mathematica,” X: “Ovales Opticae Qvatvor,” XI: “Earvm Descriptio et Tactio,” and XII: “Earvmdem Octo Vertices, Horvmqve Vsys” are not Descartes’s original but are taken from “Index Exerptorum” of the 1701 Opuscule version, namely by the editor of the editio princeps. (Cf. AT, X, “Avertissement” par Charles Adam, p. 280) The “Exerpta mathematica, X-XII” which we are discussing appear on pp. 9–17 in the 1701 edition and pp. 310–324 in the Adam-Tannery edition. Incidentally, Christiaan Huygens’s reconstruction of how Descartes found ovals can be seen in his Traité de la lumière (Leiden, 1690): OEuvres complètes de Christiaan Huygens, publiées par la Société Hollandaise des Sciences, t. XIX (La Haye, 1937; Amsterdam, 1967), pp. 529–530.

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  89. Paul Tannery, “Éclaircissements sur les ovales,” AT, X, p. 325.

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  90. We do not know for certain how Descartes acquired the knowledge of ovals in general. He does not seem to have mentioned these curves in his writings and letters during the 1620s, as far as I examined them. For mathematicians of the early seventeenth century, ovals of any kind might have been a natural extension of the conic sections, as can be seen in Kepler’s Astronomia nova (1609).

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  91. A part of the title of Chapter II “Nullum Non Problema Solvere: Viète’s Analytic Program” of Michael S. Mahoney, The Mathematical Career of Pierre de Fermat (1601–1665) (Princeton, 1973; 21994). We have also learned much from his doctoral dissertation at Princeton University: “The Royal Road: The Development of Algebraic Analysis from 1550 to 1650, With Special Reference to the Work of Pierre de Fermat” (1967) and his paper “The Beginnings of Algebraic Thought etc.” (cited in n. 21 above).

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  92. We will discuss the Latin version of 1649 in Section 4.

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  93. The content and background of the debate is analyzed in Mahoney’s book on Fermat (n. 91), Chapter IV, Section IV, “Looking under the Bed: Descartes vs. Fermat, 163738,” pp. 170–193; A. I. Sabra, Theories of Light from Descartes to Newton (London, 1967; Cambridge, 1981), Chapter III, “Descartes’ Explanation on Reflection. Fermat’s Objections” & Chapter IV, “Descartes’ Explanation of Refraction. Fermat’s ‘Refutations’,” pp. 69–135.

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  94. For the situation in which Beaugrand made his criticism of the Géométrie public, see AM, I, p. 65, note 1; CM, VI, p. 344, the forward by the editors. As to the intricate relation between Beaugrand and Descartes which led to their mutual discord, see “Beaugrand, (Jean de),” (Table des noms propres), AM, I,pp. 429–430.

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  95. For Beaugrand’s career and publications, consult Henry Nathan, “Beaugrand, Jean,” DSB,I, pp. 541–542. His Géostatique was published in Paris in 1636 and its Latin version as Geostatica in Paris in 1637.

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  96. Op. cit. (n. 73), AT, I, pp. 478–479; AM, II, pp. 65–66; CM, VI, pp. 344–346: “Le iugement que l’autheur de la Geostatique fait de mes écrits me touche fort peu. Et ie ne suis pas bienaise s’estre obligé de parler auantageusement de moymesme; mais pour ce qu’il y a peu de gens qui puissent entendre ma Géométrie,et que vous desirez que ie vous mande quelle est l’opinion que i’en ay, je croy qu’il est à propos que ie vous die qu’elle est telle, que ie n’y souhaitte rien dauantage; et que i’ay seulement tasché par la Dioptrique et par les Météores de persuader que ma methode est meilleure que l’ordinaire, mais ie pretens l’auoir demonstré par ma Geometrie. Car dés le commencement i’y resous vne question, qui par le témoignage de Pappus n’a pû estre trouuée par aucun des Anciens; et l’on peut dire qu’elle ne l’a pû estre non plus par aucun des modernes, puis qu’aucun n’en a écrit, & que neantmoins les plus habiles ont tasché de trouuer les autres choses que Pappus dit au mesme endroit auoir esté cherchées par les anciens, comme l’Apollonius Redivivus, l’Apollonius Batavus, et autres, du nombre desquels it faut mettre aussi M. vostre Conseiller De maximis & minimis; mais aucun de ceux-là n’a rien sceu faire que les anciens ayent ignoré. Apres cela, ce que ie donne au second liure, touchant la nature & les proprietez des lignes courbes & la façon de les examiner, est, ce me semble, autant au delà de la geometrie ordinaire, que la rhetorique de Ciceron est au delà de l’a,b, c des enfans. Et ie croy si peu ce que promet vostre Geostaticien, qu’il ne me semble pas moins ridicule de dire qu’il donnera dans vne Preface des moyens pour trouuer les tangentes de toutes les lig[nes] courbes qui seront meilleurs que le mien, que le sont les Catitans de Comedies Italiennes.”

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  97. Marino Ghetaldi, Apollonius redivivus (Venice, 1607), “Ad Lectorem”: Opera amnia, Redactor Zarko Dadic (Zagreb, 1968), pp. 195–196. Pappus’s description on Apollonius’s Neuses can be seen in Alexander Jones, Book 7 of the Collection, Part 1, pp. 112–115 and 196229 (Lemmas to the Neuses); Hultsch, II, pp. 670–673 and 770–821; Commandino, ff. 163v-164r & 202v-219v; Ver Eecke, II, pp. 501–503 and 596--633. Cf. Heath, A History of Greek Mathematics, Vol. 2 (Oxford, 1921), pp. 189–192 and 412–416.

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  98. Ghetaldi, Apollonios redivivus seu restitutae Apollonii Pergaei de inclinationibus geometriae, liber secundus (Venice, 1613), “Ad Lectorem”: Opera omnia, p. 228.

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  99. With regard to Ghetaldi’s general attempt to restore Apollonius’s works, see Luigi Campedelli, “Ghetaldi (Ghettaldi), Marino,” DSB, V, pp. 381–383, esp. p. 382.

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  100. Dirk J. Struik, “Snel (Snellius or Snel van Royen), Willebrord,” DSB, XII, p. 500.

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  101. Wilebrordus Snellius, Hept a¨®ryov baroToµ1]ç etc. (Leiden, 1607), pp. 3 & 7.

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  102. For Pappus’s account of the Cutting off of a Ratio and the Cutting off of an Area, see Jones, Part 1, pp. 86–89 and 126–141 (Lemmas to the Cutting off of a Ratio and an Area); Hultsch, II, pp. 64–643 and 684–705; Commandino, ff. 158r-159r and 166r-173r; Ver Eecke, II, pp. 480–482 and 512–530. Cf. Heath, A History of Greek Mathematics, Vol. 2, pp. 175–180; Jones, Part 2, pp. 510–514.

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  103. Snellius, Op. cit. (n. 101), pp. 12–13 (p. 12 was numbered as p. 10 by mistake).

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  104. Jones, Part 1, pp. 88–91 & 142–195; Hultsch, II, pp. 642–645 & 704–771; Commandino, ff. 159r & 173r-202v; Ver Eecke, II, pp. 482–483 & 530–596.

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  105. See Jones, Part 2, p. 514. We modify only notation from the Greek to the Roman letters. Cf. Heath, Vol. 2, pp. 180–181.

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  106. Jones, Part 1, pp. 90–95 & 230–249; Hultsch, II, pp. 644–649 & 820–853; Commandino, ff. 159r-160r Si 219v-232r; Ver Eecke, II, pp. 483–485 & 633–658. Cf. Jones, Part 2, pp. 534539; Heath, Vol. 2, pp. 181–185

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  107. L. L. Busard, “Viète, François,” DSB, XIV, p. 22.

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  108. The editors of the various editions of the letter of Descartes have all agreed that Descartes mentions Fermat’s manuscript on Apollonius’s Plane loci. AT, I, p. 478, note d; AM, II, p. 65, note 4; CM, VI, p. 345, note 5.

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  109. As for Apollonius’s Plane loci, see Jones, Part 2, pp. 539–546.

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  110. See Mahoney, The Mathematical Career of Pierre de Fermat (n. 91), Chapter III, Section III, “The Origins of the Introduction: Apollonius’ Plane Loci and Conics,” pp. 92–117.

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  111. For the special word ‘adæquare’, see Mahoney, Fermat, pp. 163–164. The term ‘adæqualitas’ was first coined by Xylander as the Latin word for Diophantus’s ‘7rapur¨®Trlç’. Heath translated it into ‘near-as-possible equality’. See Heath, Diophantus of Alexandria (Cambridge, 1910; New York, 21964), pp. 95–98. Having examined various manuscript sources, the following paper has pointed out that Fermat thought ‘adæquare’ to be almost synonymous with ‘wquare’, meaning “to put equal.” I don’t believe this assertion to be definitive, but it must be certain that ‘adæquare’ had a very primitive sense close to “to put equal.” Herbert Breger, “The Mysteries of Adaequare: A Vindication of Fermat,” Archive for History of Exact Sciences, 46 (1993–1994), pp. 193–219.

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  112. OEuvres de Fermat, éd. Charles Henry et Paul Tannery, t. I (Paris, 1891), pp. 134–136; Varia opera mathematica, ed. Samuel de Fermat (Toulouse, 1679), pp. 63–64. An English translation is in D. J. Struik, ed., A Source Book in Mathematics, 1200–1800 (Cambridge, Mass., 1969), pp. 222–224. Cf. Mahoney, Fermat (n. 91), pp. 166–167.

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  113. Beaugrand’s technique can be seen in one of his facta, which is reproduced in CM, VIII, pp. 90–92.

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  114. A more interesting story about the debate between Descartes and Fermat which developed in 1638 can be seen in Mahoney, Fermat, pp. 170–192. Mahoney states his observation on the personalities of the two French mathematicians: “Descartes learned nothing from it [the debate], even though he was in error; Fermat, even though he was correct, gained new mathematical insights that led him to revise his methods and sharpen his tools” (p. 171).

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  115. AT, I, pp. 479–480; AM, II, p. 66; CM,VI, p. 346: “Et tant s’en faut que les choses que i’ay écrites puissent estre aisément tirées de Viète, qu’au contraire, ce qui est cause que mon traitté est difficile à entendre, c’est que i’ay tasché à n’y rien mettre que ce que i’ay crû n’auoir point esté sceu ny par luy, ny pas aucun autre. Comme on peut voir, si on confere ce que i’ay écrit du nombre des racines qui sont en chaque équation dans la page 372, qui est l’endroit o¨´ ie commence à donner les regles de mon Algebre, auec ce que Viète en a écrit tout à la fin de son liure de Emendatione eequationum; car on verra que ie le determine generalement en toutes équations, au lieu que luy n’en ayant donné que quelques exemples particuliers, dont il fait toutesfois si grand estat qu’il a voulu conclure son liure par là, il a monstré qu’il ne le pouuoit determiner en general. Et ainsi i’ay commencé o¨´ il auoit acheué; ce que i’ay fait toutesfois sans y penser, car i’ay plus feuilleté Viète depuis que i’ay receu vostre derniere, que ie n’auois jamais fait auparavant, l’ayant trouué icy par hazard entre les mains d’vn de mes amis; et entre nous ie ne trouue pas qu’il en ait tant sceu que ie pensois, nonobstant qu’il fust fort habile.”

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  116. CM, VI, p. 346, note 2.

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  117. Viète, Opera mathematica (1646), p. 158. Cf. Witmer’s translation, p. 309. Both AT, I, p. 479, note a & AM, II, p. 66, note 2 mistakenly refer to “Chap. XVI” instead of “Chap. XIV”.

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  118. For example, see Frédéric Ritter, “François Viète, Inventeur de l’algèbre moderne 15401603: Essai sur sa vie et son oeuvre,” Revue occidentale philosophique, sociale et politique, seconde série, 10 (1895), p. 401; H. L. L. Busard, “Viète, François,” DSB, XIV, pp. 23–24.

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  119. Opera mathematica, p. 158; Witmer, p. 310.

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  120. The first edition, p. 372; AT, VI, p. 445; Olscamp, pp. 229–230.

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  121. The first edition, p. 373; AT, VI, p. 446; Olscamp, p. 230. See Smith and Latham, Op. cit. (n. 84), p. 160, note [196].

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  122. The first edition, p. 380; AT, VI, p. 453; Olscamp, p. 236. See our discussion on p. 196.

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  123. M. S. Mahoney, “The Beginnings of Algebraic Thought in the Seventeenth Century” (n. 21), p. 146.

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  124. According to Joseph E. Hofmann, this fact, which is equivalent to the fundamental theorem of algebra, was known, for example, by P. Roth (1608), Albert Girard (1629), and Thomas Harriot (1631). Hofmann, Geschichte der Mathematik, Erster Teil (Berlin, 21963), p. 166. Cf. Mahoney, Fermat, p. 150, note 15. On the development since Girard, see Christian Gilan, “Sur l’histoire du théorème fondamental de l’algèbre: Théorie des équaions et calcul intégral,” Archive for History of Exact Sciences, 42 (1991), pp. 91–136.

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  125. See Paul Tannery, “La Correspondance de Descartes dans les Inédits du Fonds Libri,” esp., Chapter III, “Les pamphlets mathématiques contre Descartes,” Mémoires scientifiques, publiés par J.-L. Heiberg et H.-G. Zeuthen, t. IV: Sciences modernes, éd. par Gino Loria (Toulouse/Paris, 1926), pp. 203–228.

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  126. “(Jean Beaugrand), à (Paris), à Mersenne, à Paris. (début de mars 1638),” CM, VII, pp. 87–103, esp. p. 88. For Beaugrand’s charge that Descartes plagiarized Harriot, see the editors’ note to CM, VII, pp. 201–202.

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  127. “Descartes, à (Santpoort), à Mersenne, à Paris. 31 mars 1638,” AT, II, p. 82; AM, II, p. 213; CM, VII, p. 120: “[I]e commence en cela par ou Viète auoit fini.”

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  128. “Descartes, à (Santpoort), à Mersenne, à Paris. (20 février 1639),” AT, II, p. 524; AM, III, p. 195; CM, VIII, p. 325: “le n’ay aucune connoissance de ce Geometrie dont vous m’écrituez, et ie m’étonne de ce qu’il dit, que nous auons estudié ensemble Viète à Paris; car c’est vn livre dont ie ne me souuiens pas auoir seulement lamais vû la couuerture, pendant que i’ay esté en France.”

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  129. CM, VIII, p. 328: the editor’s note for 1. 17.

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  130. r7–77 VI, pp. 471 & 473; Olscamp, pp. 249–250 for Cardano, AT, VI, p. 472; Olscamp, p. 249 for Ferro.

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  131. A. Rupert Hall, Philosophers at War: The Quarrel between Newton and Leibniz (Cambridge, 1980), p. 6.

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  132. Mahoney, Fermat, p. 49.

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  133. See H. Nathan, DSB, I (n. 95), p. 541. An almost exhaustive catalog of Viète’s works is now available: Warren Van Egmond, “A Catalog of François Viète’s Printed and Manuscript Works,” in Menso Folkerts und Uta Lindgren, hrsg., Mathernata: Festschrift f¨¹r Helmut Gericke, Boethius, Bd. XII (Stuttgart, 1985), pp. 359–396. For Vaulézard’s translations, see pp. 383–384. Recently a reprinted edition was published: Vaulézard, La nouvelle algèbre de M. Viète (Tours, 1986). For Vaulézard’s dedication to Beaugrand, see p. 69.

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  134. For example, a scholium added to Chapter IIII: “On the Rules of Specious Logistic” of the Isagoge in the 1646 edition was originally Beaugrand’s. The scholium contains a concordance which compares the propositions of Diophantus’s Arithmetic and those of Viète’s Zetetica. Francisci Vietce in artem analyticem isagoge, ed. I. de Beaugrand (Paris, 1631), pp. 69–71 correspnds to the Opera mathematica (1646), p. 4. Cf. J. Winfree Smith’s translation in Jacob Klein’s Greek Mathematical Thought and the Origin of Algebra (Ch. 4, n. 79), pp. 329–330. As to Beaugrand’s 1631 edition, I refer to a copy in the Bibliothèque Nationale of Paris (Rés. V. 2075 (1)).

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  135. Vietce Ad logisticem speciosam notae priores (Paris, 1631): Opera mathematica, pp. 1341. See also Witmer’s English translation, “Preliminary Notes on Symbolic Logistic,” pp. 3382; F. Ritter’s French translation, “Notes premières pour la logistice spécieuse,” Bulletino de bibliografia e di storia delle scienze matematiche e fisiche, 1 (1868), pp. 245–276.

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  136. Mahoney, Fermat, p. 26.

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  137. Beaugrand wrote to Mersenne about geometry and astronomy on June 28 and July 31, 1630 (CM, II, pp. 504–507 & 509–525). On February 20, 1932 he also addressed to Mersenne a letter mentioning the problem of Pappus and his edition of Viète (CM, III, pp. 254–262). Mersenne asked Rivet to transmit exemplars of Viète’s book to Golius and Descartes in his letter on March 14, 1632 (CM, III, pp. 268–272).

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  138. Beaugrand called Viète’s symbolic algebra “analyse specieuse” in his letter to Mersenne on February 20, 1632 (CM, III, 256).

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  139. n Beaugrand’s edition (n. 134), for example, pp. aij“, 73–74 & 99.

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  140. Viète, In artem analyticem isagoge (Tours, 1591), f. 9r; Opera mathematica, p. 12.

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  141. “Descartes, à (Amsterdam), à Mersenne, à Paris. (3 mai 1632),” AT, I, p. 245; AM, I,p. 222; CM, III, pp. 296–297: “Ie vous remercie du liure d’Analyse que m’auez enuoyé; mais entre nous, ie ne vois pas qu’il soit de grande vtilité, ny que personne puisse apprendre en le lisant la façon, ie ne dis pas de nullum non problema soluere, mais de soudre aucun probleme, tant puisse-t-il estre facile. Ce n’est pas que ie ne veuille bien croire que les auteurs en sont fort sçauans, mais ie n’ay pas assez bon esprit pour juger de ce qui est dans ce livre, non plus que de ce que vous me mandez du probleme de Pappus: car il faut bien aller au delà des sections coniques & des lieux solides, pour le resoudre en tout nombre de lignes données, ainsi que le doit resoudre vn homme qui se vante de nullum non problema soluere, et que ie pense l’auoir resolu.”

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  142. “Descartes, à Amsterdam, à Mersenne, à Paris. (5 avril 1632),” AT I, p. 244; AM, I p. 221; CM III, p. 291.

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  143. For a Latin text and its French translation, see Muhammad ibn Musa Al-Khwarizmi, Le Calcul Indien (Algorismus): Versions latines du XIF siècle, par André Allard, Préface de Roshdi Rashed (Paris, 1992). For a historical background, see Ahmad S. Saidan, “Numeration and Arithmetic,” Enclyclopedia of the History of Arabic Science, edited by Roshdi Rashed in collaboraion with Régis Morelon, Vol. 2 (London/New York, 1996), pp. 331–348.

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  144. Prolégomèmes d’Ebn-Khaldoun, texte arabe, par M. Quartermère, Tome premier, Troisième partie (Paris, 1858), p. 96; Ibn Khaldun, The Mugaddimah: An Introduction to History, translated from the Arabic by Franz Rosenthal, Vol. 3 (London, 1958), p. 122.

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  145. Prolégomèmes, p. 97; Rosenthal’s translation, p. 124.

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  146. Khwarizmi, Al-Kitâb al-mukhtasar fti’l-hisâb al-jabr wa’l-m,ugabalah ed. by ‘Ali Mustafa. Masharrafa and Muhammad Mursi Ahmad (Cairo, 1939); The Algebra of Mohammed ben Musa edited and translated by Frederic Rosen (London, 1830). A critical Arabic text is a desideratum, and a version by Prof. Roshdi Bashed will be published in near future.

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  147. Roshdi Rashed et Hélène Bellosta, Ibrähim ibn Sinan: Logique et Géométrie au Xe Siècle (Leiden/Boston/Köln, 2000).

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  148. Hélène Bellosta, “Ibrahim ibn Sinan: On Analysis and Synthesis,” Arabic Sciences and Philosophy Vol. 1 (1991), pp. 211–232, at p. 229.

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  149. R. Rashed, “La philosophie des mathématiques d’Ibn al-Haytham, I: L’analyse et la synthèse,” Mélanges de l’Insitut Dominicain d’Études Orientales du Caire (MIDEO) 20 (1991), pp. 31–231, & “La philosophie des mathématiques d’Ibn al-Haytham, II: Les Connus,” MIDEO 21 (1993), pp. 87–273. Both the texts have been reproduced with elegant commentaries by Rashed in R. Rashed, Les mathématiques infinitésimales du IX’ au XI’ siècle Vol. IV: Ibn al-Haytham Méthodes géométiques, Transformations ponctuelles et Philosophie des mathématiques (London, 2002). This volume is one of the most beautiful achievements which may be called an ‘objet d’art’ in recent scholarship of the history of Arabic mathematics. The dual concept of analysis and synthesis can be also seen in R. Rashed, Les mathématiques infinitésimales du IX’ au XP siècle Vol. III: Ibn al-Haytham, Théorie des coniques, Constructions géométriques et Géométrie pratique (London, 2000).

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  150. R. Rashed, “L’analyse et la syntèse selon Ibn al-Haytham,” Mathématiques et philosophie de l’Antiquité à l’âge classique Hommage à Jules Vuillemin, sous la direction de R. Rashed (Paris, 1991), pp. 131–162, at p. 147.

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  151. R. Rashed et B. Vahabzadeh, Al-Khayyàm mathématicien (Paris, 1999), pp. 12–29. Cf. Idem “Fermat and Algebraic Geometry,” Historia Scientiarum Vol. 11, No. 1 (2001), pp. 1–23, at p. 23.

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  152. Al-Khayyam Mathématicien pp. 293–296, & 298; pp. 372–373; Ibid. pp. 205–209, & 210–211; p. 250.

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  153. A1-Bahir en Algèbre d’As-Samaw’al édition, notes et introduction par Salah Ahmad et Roshdi Rashed (Damas, 1972), Arabic text, p. 73. The translation is taken from George A. Saliba, “The Meaning of al-jabr wa’l-muqabalah,” Centaurus 17 (1973), p. 195. For alSamaw’al’s career and works, see Adel Anbouba, “Al-Samaw’al, Ibn Yahya Al-Maghribi,” DSB XII, pp. 91–95.

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  154. Op. cit. (As-Samaw’al), p. 74. The translation has been aided by Prof. Suzuki Takanori of Tokai University, but is essentially mine. See the following historical remark by S. Ahmad and R. Roshed in their French introduction (p. 74): “L’élément commun à l’algèbre et à la géométrie, ne subsiste que dans une démarche qui va en négligeant la synthèse pour ne garder que l’analyse. Algèbre et géométrie sont analogues, selon l’auteur, puisquue dans l’une et l’autre, le mathématicien part des prémisses connus, des conditions nécessaires, consécutives à ces prémisses et ainsi de suite jusqu’à la détermination des inconnuses. Or cet élément commun semble subordonner implicitement la géométrie à l’algèbre, dans la mesure o¨´ il ne retient que la voie analytique des géométres et o¨´ il néglige, dans la description la synthèse. As-Sawam’al saura plus tard tirer de cette concption la conséquence qui s’impose et identifiera précidément, et pour la première fois dans l’histoire de la philosophie mathématique, algèbre et analyse, géométrie et synthèse.”

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  155. Adel Anbouba, Op. cit. (n. 153), p. 93: “The book [Al-bdhir] was apparently altogether unknown in the west.” For the problem of transmission of Arabic mathematics to Latin West in general, see André Allard, “The Influence of Arabic Mathematics in the Medieval West,” Enclyclopedia of the History of Arabic Science Vol. 2 (n. 143), pp. 539–580.

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  156. Kurt Vogel, “Fibonacci, Leonardo, or Leonardo of Pisa,” DSB, IV, pp. 604–613 at p. 605.

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  157. For the development of algebra from a practical art to a theoretical discipline, see, for example, Paul Benoit, “Calcul, algèbre et marchandise,” Éléments d’histoire des sciences sous la direction de Michel Serres (Paris, 1989), pp. 197–221,& Giovanna C. Cifoletti, “La Question de l’Algèbre: Mathématique et rhétorique des hommes de droit dans la France du 16’ siècle,” Annales: Histoire, sciences sociales, 50’ Année, n¡ã 6 (1995), pp. 1385–1416. This trend can be witnessed in the mathematical disciplines in general. See Mario Biagioli, “The Social Status of Italian Mathématicians, 1450–1600,” History of Science 27 (1989) pp. 41–95.

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  158. Peter of Spain (Petrus Hispanus Portugalensis), Tractatus called afterwards Summule Logicales, First Critical Edition from the Manuscripts with an Introduction by L. M. De Rijk (Assen, 1972), p. 1: “ars artium et scientia scientiarum” (a variant).

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  159. For the detail of Ramus’s program for the reform of logic, consult, among others, Walter J. Ong, Ramus, Method, and the Decay of Dialogue: From the Art of Discourse to the Art of Reason (Cambridge, Mass., 1958); Wilbur Samuel Howell, Logic and Rhetoric in England, 1500–1700 (Princeton, 1956). I have made an English translation of Book II, Chapter 1, “On the Definition of Logic,” of Ramus’s Scholae dialecticae (formerly called Aristotelicae animadversiones, which was published in Paris in 1543) in Scholae in liberales artes (Basel, 1569) with a historical introduction. See Chikara Sasaki, “Ramus’ Struggle against Scholastic Logic: Historical Introduction and Translation,” The Journal of the Department of Liberal Arts, The College of Arts and Sciences, The University of Tokyo, No. 15 (1982), pp. 119–136.

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  160. F. A. Yates, The Art of Memory (Ch. 3, n. 35), pp. 234–235

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  161. Petrus Ramus, Scholae mathematicae (Basel, 1569): Lazarus Schöner’s edition (Frankfurt am Main, 1627), p. 107: “[¡­] non solum a doctis hominibus [¡­] sed â pueris, a mechanicis, ab architectis, â pictoribus facilius ediscentur, facilius excercebuntur.”

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  162. See my paper “The Acceptance of the Theory of Proportion in the Sixteenth and Seventeenth Centuries ¡ª Barrow’s Reaction to the Analytic Mathematics,” Historia Scientiarum No. 29 (1985), pp. 93–126, esp. VI. 2: “Ramus’s Criticism of the Euclidean Tradition,” pp. 117–123.

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  163. [Euclidis elementa mathematical (Paris, 1545). Cf. W. J. Ong, Ramus and Talon Inventory (Cambridge, Mass., 1958), p. 68, no. 34*; Charles Thomas-Stanford, Early Editions of Euclid’s Elements (London, 1926), p. 35, no. 27.

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  164. See Peter Scharratt, “La Ramée’s Early Mathematical Teaching,” Bibliothèque d’Humanisme et Renaissance 28 (1966), pp. 605–614.

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  165. Ramus, Geometria (1569): Lazarus Schöner’s edition (Frankfurt am Main, 1599), p. 86. See Mahoney’s explanation of this part in his Fermat p. 32, note 16; “The Royal Road,” p. 186.

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  166. Ibid.: analyseos usus sperest, ut postea in cubo, cum alius in totis elementis nullus sit.“ Interestingly, in the English translation by William Bedwell, this part reads: ”For it is the same way from Cambridge to London, that it is from London to Cambridge.“ Via Regia ad Geometriam: The Way to Geometry, Being Necessary and Useful, from Astronomers, Geographers, Landmeaters, Sea-men, Engineres, Architecks, Carpenters, Paynters, Carvers, €éc. ed. John Clerke (London, 1636), p. 160. Note that the title is an exiting one: The Royal Road to Geometry A description as the circumstances in which the Via Regia ad Geometriam was published can be seen in Howell, Op. cit. (n. 159), p. 246.

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  167. This is pointed out persuasively in M. S. Mahoney’s “The Royal Road,” Chapter III: “The Formula Found: Algebraic Analysis in the Sixteenth and Seventeenth Centuries,” pp. 145–225. As to Ramus’s notion of analysis, especially non-mathematical, see Ong, Op. cit. (n. 159), Chapter XI, 13: “Analysis and Genesis (Synthesis),” pp. 263–267.

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  168. [Ramus], Algebra (Paris, 1560), f. 2r: “Algebra est pars Arithmeticæ, quæ é figuratis continué proportionalibus numerationem quandam propriam instituit. Varies siquidem figurato-rum species in eam progressionem succedunt, quarum numeratio quædam ab antecedentium numerorum numeratione diversa est.” In his conmperhensive study of Ramean mathematics, J. J. Verdonk has pointed out that there are many parallel structures between Ramus’s Algebra and Johannes Scheubel’s Algebrae compendiosa facilisque descriptio published in Paris in 1551. See Verdonk, Petrus Ramus en de Wiskunde (Assen, 1966), 411, “De Algebra van Ramus,” pp. 209–224. For the life and works of Scheubel, see Mary S. Day, Scheubel as an Algebraist (New York, 1926). This book examines a manuscript of Scheubel’s algebraic work preserved at the Columbia University Library.

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  169. Cf. W. J. Ong, Ramus and Talon Inventory (n. 163), pp. 335–336: “There is nothing about the relation of analysis and synthesis to algebra or to arithmetic. [¡­] The fact that this first, tentative edition is anonymous suggests how Ramus’ desire for fame as a mathematician was tempered by foreboding concerning his initial qualifications.”

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  170. Ong, Ibid. p. 333.

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  171. Petri Rami Algebra a Lazaro Schonero emendata (Francofvrti, 1599), p. 190. “A quare resoltione Algebra Græcis dicta fuit analytica, quibus absoluta Arithmetica dicebatur synthetica.” Cf. Mahoney, “The Royal Road,” p. 187, which mistakenly ascribes this statement to Ramus, not Schöner.

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  172. Jacokob Klein, “Die griechische Logistik und die Entstehung der Algebra,” Quellen und Studien zur Geschichte der Mathemtik, Astronomie und Physik Abteilung B: Studien, Bd. 3–1 (1934), pp. 18–105 & Bd. 3–2 (1936), pp. 122–235. Its Eglish translation by Eva Brann is Greek Mathematical Thought and the Origin of Algebra (n. 134). Note that J. Klein was under the influence of Edmund Husserl’s phenomenological philosophy, which can be seen in Jacob Klein, “Phenomenology and the History of Science,” Philosophical Essays in Memory of Edmund Husserl ed. by Marvin Farber (Cambridge, Mass, 1940), pp. 143--163.

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  173. Mahoney, “The Royal Road” (n. 91), which was written under the supervision of Prof. Thomas S. Kuhn at Princeton University.

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  174. J. J. Verdonk, Op. cit. (n. 168), p. 370.

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  175. Hardin Craig has named Ramus “the greatest master of the short cut the world has ever seen” in his The Enchanted Glass: The Eizabethan Mind in Literature (Oxford, 1936), p. 143. Ramus himself claimed that his method to reorganize the liberal arts is an “adrèsse et abrègement de chemin.” See Dialectique (1555), Édition critique par Michel Dassonville (Gèneve, 1964), p. 145. Father Ong characterized Ramus’s work in general as something similar to ‘the calculus without pain’ or ‘topology without pain’ in his lecture “Milton’s Logic and the Pacification of Academia,” delivered at Princeton University on May 4, 1979, at which I was in the audience.

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  176. E. Hofmann, Op. cit. (Ch. 2, n. 105), p. XII*.

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  177. Viète, Universalium inspectionurn ad canonum mathematicum liber singularis (Paris, 1579), pl. 71.

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  178. Ramus quoted and commented on the well-known interpolations to Propositions 1–5 of Euclid XIII (see Chapter 2, ¡ì 3, p. 72) on analysis and synthesis. In Ramus’s text, analysis is defined as “assumptio quaesti tanquam concessi per consequentia ad verum concessu.” Scholae mathematicae (n. 161), p. 300. It is totally identical with Viète’s (see Ch. 4, n. 75: “Assumptio” in Ramus is “adsumptio” in Viète). Incidentally, one of the texts of Euclid that Ramus used for writing his Scholae mathematicae was doubtlessly the editio princeps of the Greek text of Theon of Alexandria, published by Simon Grynaeus the elder in Basel in 1533 (see Ch. 2, n. 46). See p. 239 of that edition. Cf. Commandino’s definition of analysis (resolutio): “Resolutio est sumptio quaesti tanquam concessi per ea, quae consequuntur in aliquod uerum concessum.” Commandino, Op. cit. (Ch. 2, n. 47), f. 229r. Note that Commandino’s Latin translation appeared in 1572 after the publication of the Scholae mathematicae in 1569. 179For example, in Chapter VI, “On the Examination of Theorems by Poristics,” of the Isagoge Viète uses the Ramist three laws of method “icaTà 7rc’vToç” “MAT’ avTÒ (sic!),” and “ka9’ öiov 7rpOrou”. Isagoge (1591), f. 8r; Opera mathematica (1646), p. 10. Ramus explained these three laws (in his own spelling) “Kara 7rcivreN”, “kar’avro” and “kaT’ 8.ov 7rpûOrov” in the Scholae mathematicae pp. 76–77 and 109 (see also his Scholae dialecticae (n. 159), p. 32.). These conceptions were originally used in Aristotle’s Posterior Analytics (Book A, Ch. IV, 73a21–74a3). Thus T. R. Witmer, for example, mistakenly ascribes them directly to Aristotle in his translation of Viète’s works (p. 28, note 47). But, as has been at times pointed out, Ramus distorted Aristotle’s original meanings and applied them to his reorganization of the liberal arts. For Ramus’s misunderstanding of the Aristotelian conceptions, see Howell, Op. cit. (n. 159), pp. 149–153; Ong, Op. cit. (n. 152), pp. 258–262. Viète clearly follows Ramus’s use and misspelling (“kar’a&TÓ” instead of “icaO’avr¨®”) For a discussion about Ramus’s influence on Viète’s thought, see Mahoney, “The Royal Road,” esp. pp. 178–180 & 196–197.

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  179. Viète, Isagoge (1591), f. 4r; Opera mathematica p. 1: “Doctrina bene inveniendi in mathematicis.” In his Scholae mathematicae (p. 108), Ramus defined mathematics as follows: “What, then, is mathematics? I would say that it is the doctrine of quantity: As, in fact, arithmetic is defined as the doctrine of computing well; geometry, the doctrine of measuring well, so the art of mathematics might be defined by the general term and from its end be named, so to speak, the art of ‘quantifying’.” (“Quid igitur est mathematica? Est (inquam) doctrina quantitatis: ut vero definitur arithmetica bene nume-randi, geometria bene metiendi doctrina, sic generali verbo, modo ex suo fine definiretur mathematica ars, tamquam diceretur, quantitandi.” Such a manner of defining the arts by Ramus reminds us of Cicero’s Topica one of Ramus’s favorite books. Cicero called ‘roiruc)’ as ‘ars inveniendi’. Topica 6.

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  180. “The royal road” was, of course, used in Euclid’s reply to the king Ptolemy the First. “It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry than through the Elements and Euclid replied that there was no royal road to geometry.” See Proclus, A Commentary on the First Book of Euclid’s Elements Translated, with Introduction and Notes, by Glenn R. Morrow (Princeton, 1970; With a new foreword by Ian Mueller, 1992), pp. 56–57 (Friedlein, p. 68). I learned from Mahoney’s Ph. D. thesis “The Royal Road” that this expression is applicable to algebraic analysis of early modern period. The construction of such a road, however, was consciously aimed at by contemporary mathematicians such as Galileo (see p. 124) and the English Ramist William Bedwell (see n. 166).

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  181. George Sarton, Six Wings: Men of Science in the Renaissance (Bloomington/London, 1957), p. 44. See also Van Egmond, Op. cit. (n. 133)

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  182. H. L. L. Busard, “Viète, François” (n. 118), pp. 23–24. It is known that Jacques Alleaume was residing in Holland and taught military sciences at the Military Academy there during the early seventeenth century. See Gustave Cohen, Écrivans français en Hollande dans la première moitié du XVII’ siècle (n. 3), pp. 341, 373 & 381; C. Laplatte, “Aleaume (Jacques) (1562–1627),” Dictionnaire de Biographie Française t. 1 (Paris, 1933), cols. 1371–1372. For Anderson, see A. M. C[lerke], “Anderson, Alexander (1582–1619?),” Dictionary of National Biography Vol. 1 (Oxford, 1921), pp. 371–372.

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  183. Ghetaldi, De Resolutione et compositione mathematica (Rome, 1630): Opera omnia (n. 97), pp. 353–707. This book is examined in Mahoney, “The Royal Road”, pp. 206–217; Ernest Stipanié, “L’OEuvre principale de Getaldié ((De resolutione et compositione mathematica»,”Actes (Ch. 2, n. 107), pp. 91–104.

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  184. See Van Egmond, Op. cit. (n. 133), p. 383.

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  185. We have already mentioned this translation on p. 246.

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  186. For Harriot as a mathematician, see J. A. Lohne, “Thomas Harriot als Mathematiker,” Centaurus, 11 (1965), pp. 19–45, esp. pp. 39–42;“Dokumente zu Revalidierung von Thomas Harriot als Algebraiker,” Archive for History of Exact Sciencesre 3 (1966), pp. 185–205; Jacqueline A. Stedall, “Rob’d of Glories: The Posthumous Misfortunes of Thomas Harriot and His Algebra,” Archive for History of Exact Sciencesre 54 (2000), pp. 455–497. In the latter two papers, the authors have pointed out that Harriot’s Praxis was published with many additions by the editor.

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  187. An examination of Oughtred’s Clavis mathematicx can be seen in Mahoney, “The Royal Road,” pp. 203–206. By examining the first edition of Oughtred’s Clavis H. Bosmans speculated about the possibility of its influence upon Descartes’s Géométrie in his “La première édition de la Clavis mathematica (sic!) d’Oughtred. Son influence sur la Géométrie de Descartes,” Annales de la société scientifique de Bruxelles 35’ année, 1910–1911 (Louvain, 1911), 1“ partie, p. 121 and 2’ partie, pp. 24–78. As Bosmans has admitted, there is no clear evidence to claim that Oughtred influenced Descartes. For Oughtred, see Florian Cajori, William Oughtred: A Great Seventeenth Century Teacher of Mathematics (Chicago/London 1916).

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  188. Adrianus Romanus, Idea mathematicce pars prima (Antwerpen, 1593), f. **iij“. For Van Roomen’s bibliography, see H. Bosmans, ”Romain (Adrien),“ Biographie nationale publiée par l’Académie royale des sciences, des lettres et de beaux-arts de Belgique, t. XIX (Bruxelles, 1907), cols. 848–889; P. Bockstaele, Nationaal Biografisch Woordenboek Vol. 2 (Brussels, 1966), cols. 751–765.

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  189. Viète, Opera mathematica (Leiden, 1646), pp. 305–324.

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  190. This work is examined in H. Bosmans, “Analyse de trois ouvrages célèbres d’Adrien Romain,” Annales de la société scientifique de Bruxelles 22’ Année, 1904–1905 (Louvain, 1905), pp. 68–79, esp. pp. 68–74.

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  191. G. Sarton, Six Wings (n. 182), p. 46. Sarton took this story from Jacques Auguste de Thou’s Historia sui temporis (1604–1608). See notes 63 on pp. 251–252 and 76 on p. 253. See also P. Bockstaele, “The Correspondence of Adriaan van Roomen,” LIAS, III (1976)/Separatum, p. 72, note 10; H. L. L. Busard, “Roomen, Adriaan van,” DSB XI, p. 532.

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  192. Viète, Opera mathematica p. 307. pp. 315–316 & 320–321

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  193. Adrianus Romanus, In Archimedis circuli dimensionem expositio et analysis. Apologia pro Archimede ad clarissimum virum Josephum Scaligerum. Exercitationes cycliae, contra Josephum Scaligerum, Orontium Finaeum et Raymarum Ursum (W¨¹rzburg, 1597). The actual place of publication was Geneva instead of W¨¹rzburg. See Bockstaele, Op. cit. (n. 192), p. 44, note 3. J. J. Scaliger’s Cyclometria elernenta duo (Louvain, 1594) is examined briefly in Michel Chasles, Aperçu historique sur l’origine et dévelopment des méthode en géométrie, Seconde éd. (Paris, 1875), p. 443. The following article locates Van Roomen’s work in the history of the quadrature of the circle in the sixteenth century: D. Bierens de Haan, “Notice sur quelques quadratures du cercle dans les Pays-Bas,” Bulletino di bibliografia e di storia delle scienze matematiche e fisiche, 7 (1874), pp. 99–140, esp. pp. 127–138.

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  194. Romanus, Apologia pro Archimede “CAPVT SEXTVM. GEOMETRIAE, & Arithmetic communem esse scientiam, quæ quantitatem generaliter vti mensurabilem considerat,” p. 23: “[¡­] Nimirum scientiam esse quandam Mathematicam communem Arithmetic & Geometriæ, ad quam spectarent affectiones omnibus quantitatibus: c¨´m auteur proportio sit omnibus quantatibus communis, non abstractis tantum vt numeris et magnitudinibus, sed concretis etiam, vti temporibus, sonis, vocibus, lotis, motibus, potentiis (Nam hnc omnia & plura alia proportionem dicuntur habere, si eorum habitudo consideretur secundum quantitatem) [¡­].”

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  195. Ibid.: “Eutochius (sic!) vocat absolutè Mathematicam.”

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  196. Apollonii Pergaei Conicorvm Libri Qvattvor ed. F. Commandino (Bononiae, 1566), f. 14r: “Proportio ex proportionibus componi dicitur, quando proportionum quantitates inter se multiplicata aliquam producunt.” Cf. Heiberg, Apollonii Pergaei quae Graece exstant cum commentarijs antiquis Vol. 2 (Leipzig, 1893; Stuttgart, 1974), pp. 218–219.

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  197. Commandino, f. 14v: “Taira ryàp rei pa r)µara (5okoOvre eiµev àöea(pâ, hoc est, hæ enim mathematics disciplina germana esse uidentur.” Cf. Heiberg, pp. 220–221.

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  198. See a historically just interpretation. Ian Mueller, Op. cit. (n. 16), p. 136. As to Book V, Mueller asserts, “thus it seems quite certain that, for Euclid, magnitudes do not include numbers.”

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  199. Romanus, Apologia p. 43: “EXEMPLVM. DATA. Proposita sit fractio B, cuius denominator numeratore maior non est. QVAESITVM. Oportet eandem reducere ad numerum integrum vel mixtum. PRAXIS. Diuidatur A per B. quotiens fit C â. Dico numerum C1- 3 2 esse numerum mixtum fracto â quivalentem. Sic si propositi fracti ddenominator B diuidens numeratorem A faciat quotientem numerum integrum C, is erit æqualis fractioni.“

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  200. Ibid.

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  201. M. Chasles, “Histoire de l’algèbre Note sur la nature des opérations algébriques (dont la connaissance a été attribuée, à tort, à Fibonacci).¡ªDes driots de Viète méconnus),” Comptes rendus des séances de l’Académie des sciences t. XII (Séance du Mercredi 5 mai 1841), p. 755: “Il semble donc que c’est Romanus qui a le plus approché de la conception de Viète, dans ce sens qu’il en a eu l’idée; mais il n’a pas su appliquer cette idée heureuse. Il faut donc reconnaître entre lui et Viète, dans l’histoire de l’algèbre, une distance immense, comme celle qui sépare Pythagore de Copernic et de Galilée dans la découverte et les preuves du mouvement de la Terre, ou Kepler de Newton dans l’histoire du principe de la gravitation universelle. Néanmoins la tentative de Romanus lui fait honneur, et rehausse le mérite et la gloire de Viète.”

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  202. See Philippe Gilbert, “Notice sur le mathématicien louvaniste Adrianus Romanus, Professeur à l’ancienne Université de Louvain. (XVIe siècle),” Revue catholique t. 17 (Louvain/Brussels, 1859), pp. 277–286, 394–409 & 522–527, esp. p. 409: “Nous arrivons ainsi à cette conclusion, déjà justifiée d’ailleurs par le contenu du livre, que les idées de Romanus sur l’arithmétique universelle lui appartiennnent en propre, puisqu’il n’a pu connaitre Viète et profiter de ses idées que vers 1597 ou 1598, tandis que la partie de son ouvrage que nous avons examinée plus haut doit évidement se rapporter, quant à la date de la composition, vers l’année 1595 ou peut-étre m¨ºme à une époque antérieure.”

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  203. A. Romanus, Mathernaticce analyseos triurnphus (Louvain, 1609), p. 3: “Et sanè conatus ij veterum tum recentionum omnes irriti, nos qvoque ab ea investigatione terrere potuissent, nisi pleraque Problemata, qva Geometria viribus solvi neqveunt, Analytica (Algebra vulgo dicta) beneficio expediri posse animadvertissem, ideoque num & Polygona eidem doctrine subessent investigare cepi.” I refer to a copy of this very rare work at the Staat-und Stadtbibliothek Augusburg. See also A. Ruland, “Adrien Romain, premier professeur à la faculté de médicine de Wurzbourg,” Bibliophile belge (Bulletin trimestriel publie par la Société des bibliophiles de Belgique), II (1867), pp. 56–100, 161–187 & 256–269, esp. pp. 86–87.

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  204. H. Bosmans, Op. cit.(n. 191), pp. 77–79, where the author mathematically examines Van Roomen’s present work.

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  205. Romanus, Mathematicx analyseos triumphus p. 4.

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  206. /bid. p. 6.

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  208. H. Bosmans, “Le fragment du Commentaire d’Adrien Romain sur l’Algèbre de Mahumed ben Musa el-Chowârezmî,” Annales de la société scientifique de Bruxelles XXXe Année, 1905–1906 (Louvain, 1906), 2e partie, pp. 267–287, esp. p. 180.

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  209. lbid., p. 271, note (***): “Nos itaque maluimus Algebram sive Analyticam scientiam revocare ad Mathesin primam, quae quantitatem universalem considerat.” In the same manuscript, A. Romanus also refers to Viète’s Zetetica of 1593. See Bosmans, p. 273.

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  210. See W. J. Ong, Ramus and Talon Inventory (n. 163), Nos. 615 & 616: Geometriae libri septern et viginti, ad clarissimum Adrianum Romanum mathematicorum ocellum (Hanover, 1604 & 1612).

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  249. Ibid. Vol. VIII (1697–1722) (Cambridge, 1981), pp. 450–453.

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  250. See my paper, Op. cit. (n. 162), esp. pp. 123–125, where I discuss the tension between the two and its historical background.

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  254. See various papers contained in Donald Gillies, ed., Revolutions in Mathematics (Oxford, 1992) and Elena Ausejo and Mariano Hormig¨®n, eds., Paradigms and Mathematics (Madrid, 1996). The late Prof. Kuhn wrote to me “There must have been revolutions in mathematics” in his comment on my paper submitted to his course “Introduction to Philosophy of Science” of the academic year of 1976–1977. Previously, he had been skeptical of the possibility of applying his viewpoint of science to mathematics. It wouldn’t be necessary to be pointed out that discussions should be made critically by defining the meaning of the word ‘revolution’ precisely. On the meaning of ‘revolution’, see Ilan Rachum, “Revolution”: The Entrance of a New Word into Western Political Discourse (Lanham/New York/Oxford, 1999).

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  256. The concept of ‘paradigm’ is taken, of course, from Thomas S. Kuhn’s Structure of Scientific Revolutions (Chicago, 1962; 21970; 31996). Significantly, in his last years Kuhn rehabilitated and used the term ‘paradigm’ as ‘hermeneutic bases’. See Thomas S. Kuhn, The Road Since Structure: Philosophical Essays, 1970–1993, with an Autobiographical Interview edited by James Conant and John Haugeland (Chicago/London, 2000), p. 221.

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    Chikara Sasaki

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Sasaki, C. (2003). The Géométrie of 1637. In: Descartes’s Mathematical Thought. Boston Studies in the Philosophy of Science, vol 237. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1225-5_6

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