Abstract
In the previous chapters, I have shown that the question about the solvability of the problem of the indefinite and definite quadratures of the circle, the ellipse, and the hyperbola was a source of studies and debates among geometers from the second half of the Seventeenth Century. More particularly, I have chosen to study one episode within precise temporal bounds (1667–1676). The example I chose to analyze revolves around the claim made by James Gregory, in his Vera circuli et hyperbolae quadratura, regarding the impossibility of solving the quadrature of the circle in an exact way, and the criticism to which this claim was later exposed.
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Notes
- 1.
- 2.
Archimedes was ill at ease with computations involving infinite series, thus he used the more rigorous method of exhaustion. Several centuries later, Viète studied the same theorem, and hinted at the possibility of using the geometric progression as a shorter method of proof (Viète 1593, p. 29). Like Van Heuraet’s proof, the non-finite portion of Viète’s solution only occurs during the heuristic step of analysis, and the fact that the infinite sum reaches a finite value allows the synthetic portion of the solution to be carried out by a “classical” constructivist approach.
- 3.
GPU, p. 3.
- 4.
“And if the geometer, after the proper application of this method according to the properties of the figure, cannot find any solution, he must resort to convergent series, whose limit is the unknown figure or another one, in a given ratio with this one.” See Malet (1996, p. 226), for a further discussion.
- 5.
Cf., among several manuscripts, VII6, p. 504: “nam cum linearum curvarum, aut spatiorum ipsis conclusorum magnitudo quaeritur (…) neque aequationes neque curvae Cartesianae nos expedire possunt; opusque est novi plane generis aequationibus, constructionibus curvisque novis; denique et calculo novo, nondum a quoquam tradito, cujus si nihil aliud saltem specimina quaedam, mira satis, jam nunc dare possem.”
- 6.
See also Debuiche (2013).
- 7.
Leibniz (2011, pp. 8–9): “Archimedes quidem Polygona Circulo inscribens & circumscribens, quoniam major est inscriptis, & minor circumscriptis, modum ostendit, exhibendi limites intra quos circul[u]s cadat, sive exhibendi appropinquationes: esse scilicet rationem circumferentiae ad diametrum, majorem quam 3 ad 1, seu quam 21 ad 7, & minorem quam 22 ad 7. Hanc Methodum alii sunt prosecuti, Ptolemaeus, Vieta, Metius, sed maxime Ludolphus Coloniensis, qui ostendit esse circumferentiam ad diametrum, ut 3.14159265358979323846&c. ad 1, 00000000000000000000. Verum hujusmodi Appropinquationes, etsi in Geometria practica utiles, nihil tamen exhibent, quod menti satisfaciat, avidae veritatis, nisi progressio talium numerorum in infinitum continuandorum reperiatur.”
- 8.
Leibniz (2011, p. 10): “Et licet uno numero summa ejus seriei exprimi non possit, et series in infinitum producatur, quoniam tamen una lege progressionis constat, tota satis mente percipitur. Nam siquidem circulus non est quadrato commensurabilis, non potest uno numero exprimi, sed in rationalibus necessario per seriem exhiberi debet; quemadmodum & Diagonalis quadrati, & sectio extrema & media ratione facta, quam aliqui divinam vocant, aliaeque multae quantitates, quae sunt irrationales.”
- 9.
Cf. AII2, p. 252.
- 10.
The fact that this procedure could not lead to true knowledge was stressed by Stevin: “Encore qu’il nous fust possible, de soubstraire par action, plusieurs cent mille fois la moindre grandeur de la majeure, & le continuer plusieurs milliers d’années, toutefois (estant les deux nombres proposez incommensurables) l’on travailleiroit eternellement, demeurant toujours ignorant, de ce qui encore à la fin en pourroit encore avenir. Cette manière donc de cognition n’est pas légitime, ainsi position de l’impossible, afin d’ainsi aucunement declairer, ce qui consiste veritablement en la Nature …” (Stevin 1958, Vol. 2, p. 723).
- 11.
Wallis (2014, p. 426).
- 12.
A similar view might have influenced Newton, as shown in Blåsjö (2017, pp. 56–57).
- 13.
See also Jesseph (1999, pp. 37 and 38).
- 14.
Wallis (1685, p. 316).
- 15.
The terms in bold represent the terms obtained by successive interpolations.
- 16.
Wallis (1685, p. 316).
- 17.
Cf. Wallis to Leibniz, 6/01/1698 or 1699 (LSG, IV, p. 57).
- 18.
Cf. Lützen (2014).
- 19.
LSG, IV, p. 38: “Quodsi supponatur hoc numerus n, numerus fractus, surdus, vel utcumque arretos, comminiscendae sunt novae extractionum methodi casibus hujusmodi congruae. Quippe (quod ego saepe moneo) in omnibus operationibus Resolutoriis (quales sunt Subtractio, Divisio, Extractio radicum, Aequationum solutio, Interpolatio etc.) semper pervenitur ad id quod stricto sensu fieri non potest, sed quod utcumque designetur quasi-factum (ut sunt − 1, \(\frac {3}{2}\), \(\sqrt {2}\) etc). Adeoque continue procedetur ad alios aliosque gradus…seu Inexplicabilitatis, in infinitum, ut nunquam desitura sit materia ultra ultraque procedendi, volentibus id aggredi.”
- 20.
One example that Wallis had in mind was probably that of Cardano and Bombelli: cf. Gavagna (2014, pp. 178–179).
- 21.
Wallis (2004, p. 162).
- 22.
Wallis (1685, pp. 264–267).
- 23.
AVI4, 124, p. 518.
- 24.
AVI4, 124, p. 520: “Interdum etiam operatio actu ipso facienda vel pro tempore vel omnino est impossibilis, saltem in nostris characteribus etsi construendo exhiberi possit aut a natura jam exhibita sit. Ita impossibile est subtrahi cum nihil adest, et tamen hoc in natura repraesentatur, cum quis plus debet, quam habet in bonis. Item numerum integrum primitivum impossibile est actu dividi per alium; unde fit fractio, quae repraesentat divisionem esse faciendam, re quae isto numero designatur divisa in partes ad eam divisionem exhibendam aptiores. Eodem modo oriuntur quantitates incommensurabiles, seu radices surdae ubi extractio non habet locum.” See Leibniz (2018, pp. 98ff.) for a French translation.
- 25.
AVI, 4, 124, p. 522: “Reales vero licet incommensurabiles quantitates possunt in natura exhiberi: eaeque sunt vel Algebraicae vel Transcendentes. Algebraicae cum inveniuntur extractione radicis certi gradus. At Transcendentes cum gradus aequationis aut est incertus, aut non est enuntiabilis. Et ad Transcendentes pertinent Logarithmi. Et varii sunt modi Quantitatem Transcendentem exprimendi, tum ad instar Logarithmorum, tum certis operationibus quae supponuntur in infinitum continuatae, eae finities actu ipso praestitae quantitati quaesitae quantumlibet accedere debent, unde oriuntur appropinquationes.”
- 26.
AVI, 4, 271, p. 1448: “Nam \(\sqrt {-1}\) aliquam notionem involvit, licet ea non possit exhiberi, et si quis eam circulo exhibere volet, inveniet circulum illum a recta quae ad hoc requiritur non attingi. Multum tamen interest inter quaestiones quae insolubiles sunt ob radices imaginarias, et quae [in]solubiles sunt ob absurditatem. Ut si quis quaerat numerum, qui in se ductus faciat 9: et ad 5 additus faciat etiam 9; talis numerus implicat contradictionem, debet enim simul esse 3 et 4, seu 3 et 4 debent esse aequales pars toti. Verum si quis numerum quaerat talem cujus quadratum additum ad 9, faciat idem quod numeri triplum. Is quidem nunquam ostendet totum esse majus sua parte, tali numero admisso, sed tamen et illud ostendet, talem numerum non posse designari.” (English translation in Leibniz 1984, p. 21).
- 27.
Cf. also AVI4, 124, p. 521; cf. AVI4, 271, p. 1448.
- 28.
For a further discussion, see Cortese (2016).
- 29.
AVI4, 124, p. 520: “Et quaedam extractiones tales sunt, ut radices illae surdae nec in natura rerum extent, tunc dicuntur imaginariae …utile tamen est eam perinde considerari ad realem, quia radices imaginariae computandae cum realibus; ut integer numerus radicum alicujus aequationis habeatur; aliaque multa de ea utilia possint determinari.” A similar argument can be found in Girard’s Invention nouvelle en l’Algebre: “Donc il se faut resouvenir d’observer tousjours cela: on pourroit dire à quoy sert ces solutions qui sont impossibles, je respond pour trois choses, pour la certitude de la reigle generale, & qu’il ny a point d’autre solutions, & pour son utilité”(Girard 1629, p. 47; also quoted in Gavagna 2014, p. 186).
- 30.
Let us recall, for instance, Gauss’s polemical remarks against the common habit of calling certain numbers or roots “impossible” when they are not (logically) impossible at all. According to Gauss, this is bad terminology because complex numbers are not logically impossible, unlike, say, “rectilinear equilateral right-angle triangles”: “If someone would say a rectilinear equilateral right triangle is impossible, there will be nobody to disagree. But, if he intended to consider such an impossible triangle as a new species of triangle and to apply to it other qualities of triangles, would anyone refrain from laughing? That would be playing with words, or, rather, misusing them” (in Detlefsen 2008, pp. 112–113).
- 31.
A discussion of impossibility proofs in the Nineteenth Century can be found in Lützen (2009).
- 32.
Especially for the impossibility of solving the quintic equation and for the case of geometric constructions, a long discussion can be found in Vuillemin (1963, Chapters 2–4).
- 33.
Crippa (2014), especially the Introduction.
- 34.
Heeffer (2008, p. 149), with smaller modifications.
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Crippa, D. (2019). Conclusion. In: The Impossibility of Squaring the Circle in the 17th Century. Frontiers in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01638-8_4
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