From Plato to Pythagoras: The Scholae mathematicae

Goulding, Robert

doi:10.1007/978-90-481-3542-4_3

From Plato to Pythagoras: The Scholae mathematicae

Robert Goulding²

Chapter
First Online: 18 November 2009

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Part of the book series: Archimedes ((ARIM,volume 25))

Abstract

Ramus’s great contribution to the history of mathematics, the Prooemium mathematicum, was written in a time of confessional strife, both in France and in Ramus’s own life. Some time in 1561 or 1562, Ramus converted to the Reformed religion – to no one’s surprise, since many had long suspected that he had secretly embraced Protestantism. In 1562, when Calvinists were expelled from Paris, Ramus – by then one of the most famous scholars in the world – was given royal safe-passage to Fontainebleau.¹ He worked in the library there for several months, reworking the lectures on mathematics that he had developed over several years. These mature thoughts on mathematics and history would eventually be published in two versions: the Prooemium mathematicum of 1567, and the Scholae mathematicae of 1569, which contained the three books of the Prooemium scarcely altered, plus another 28 books of criticism of Euclid, extending his brief remarks of the 1555 Arithmetic in exhausting detail.² I will argue in this chapter that Ramus’s elaborate reworking of his history of mathematics bears witness to the religious and civil strife that overtook France in those years, as well as to the academic conflicts in which Ramus found himself embroiled in the University of Paris and the Collège Royale.

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Notes

1.
Waddington (1855, pp. 136, 149–150).
2.
Ramus (1567) and Ramus (1569). In what follows on the Prooemium mathematicum, all references will be to the more widely-accessible Scholae mathematicae.
3.
Ramus (1599, p. 410).
4.
Ramus (1569, p. 1): “Haec magni philosophi magna prorsus sententia est, artes sunt aeternarum et immutabilium rerum, at ipsarum apud homines notitia nequaquam est aeterna.”
5.
Ibid.; see Pliny, Natural History XVIII.211.
6.
Pace Popper (2006, p. 96), there does not seem to be any influence of Annius of Viterbo in Ramus’s Prooemium mathematicum. While Ramus was undoubtedly aware of Annius’s fictions (which played an important role in his work on the ancient Gauls) all of Ramus’s references to Berosus in the Prooemium are to the genuine Berosus cited by Josephus, not the Annian pseudo-Berosus.
7.
Ramus (1569, pp. 2–4).
8.
Ibid., p. 3. Ramus discovered this fact in Pliny, Natural History VII.123.
9.
Ibid., p. 4.
10.
Ramus (1569, p. 4): “ut a rege Pharaone sacerdotibus ager in stipendium mathematicae professionis esset assignatus.”
11.
Genesis 47:20–22. Ramus incorrectly cited the passage as Genesis 27.
12.
In Jewish Antiquities II.189, he simply says that Pharaoh took possession of all the land of Egypt “save only the priests, for these kept their domains.”
13.
Ramus (1569, p. 4).
14.
Ramus (1569, p. 109): “Ergo artes mathematicae divinitus vel oblatae vel inventae, quae Dei potentiam in mundi creatione, sapientiam in administratione, pietatem ex infinita bonorum omnium erga genus humanum largitate demonstrarent.”
15.
Ramus (1576, p. 2).
16.
Ramus (1569, pp. 54–55). Much of this passage is taken from one of Ramus’s mathematical actiones against Jacques Charpentier, discussed below.
17.
Ibid., pp. 82–83.
18.
See the previous chapter, at p. 31.
19.
On Ramus’s “socratism,” see Walton (1970).
20.
Ramus (1569, p. 77): “cum … in Academiam Parisiensem induxissem, Socratis miseriam et calamitatem mirabilem mihi conciliavi, judiciis omnibus, pro impudente et ignaro calumniatore, pro impio etiam damnatus; religionis enim fundamentum nonnulli tum in scholasticis sophismatis collocarant; scribere quicquam aut loqui publice privatimque prohibitus. Id fuit manus et linguam velut amputare; denique Socratis praeter cicutam nihil nobis admodum abfuit.”
21.
Ibid., pp. 5–6. It is a little unusual that Ramus said nothing here of Pythagoras’s own travels in Egypt, as so many other historians did – even Ramus himself, who mentioned Pythagoras’s journey to Egypt in his second actio against Charpentier, written the previous year. See Ramus (1599, p. 419). Rather, he gave the impression that Pythagoras was a linear successor of Thales.
22.
Ibid., p. 12: “Sed unus mathematicorum omnium, tanquam Homerus, habetur Plato, qui non solum a Theodoro Cyreneo in Graecia, a Pythagoreis in Italia, a sacerdotibus in Aegypto mathematica tum inventa didicit, sed per sese multa exprompsit.” Plato’s supposed admiration of Pythagoras’s “elements” will be explained below.
23.
Ibid., p. 15.
24.
Ramus (1569, p. 13).
25.
Ibid.
26.
Philoponus (1542, p. 36) (commentary to Post. An. I. 7): “Quod autem duo cubi si[n]t unus cubus, hoc dicit, quomodo possibile est cubum duplicare et rursus manere cubi figuram, videtur autem pro hoc vulgata innui historia. Daliis [sic] enim peste laborantibus, respondit Apollo fore ut liberarentur a peste, si aram duplicarent cubicam habentem formam, hi autem edificarunt addentes priori arae, alterum cubum aequalem. At duorum cuborum compositio cubi formam permutavit, fuit enim pro cubo trabs. Peste autem non cessante, Respondit dominus, non fecisse eos quod praeceptum fuerat. … Venerunt autem ad Platonem quaerentes viam quomodo cubum duplicarent, hic autem ad ipsos ait, videtur vobis improperare dominus, veluti negligentibus geometriae. Duplicatio enim cubi invenietur inquit, si duarum rectarum duae mediae proportionaliter inveniantur, et hoc problema discipulis proposuit, qui de hoc scripserunt ut potuit unusquisque quorum nihil servatur usque modo, sed neque geometra hoc significavit …”
27.
Euclid, Elements XI.33 and corollary.
28.
Ramus (1569, p. 13): “proindeque e vestigio volare Platonis literae ad omnes familiares in Italiam, in Aegyptum, in Graeciam universam, omnesque praestantes Geometras excitare ad hoc problema demonstrandum.”
29.
Ramus portrayed Plato in a similar way in his second actio against Charpentier, written the previous year. Telling the story of Delos in order to underline Plato’s preeminence in Greek mathematics, Ramus concluded by saying, “And so Plato’s letters on this subject flew not only to Italy and the whole of Egypt, but through the whole Greek world, in order to inflame all people to the study of mathematics.” (Ramus (1599, p. 419): “Itaque Platonis epistolae ex hoc argumento volare non in Italiam solum et Aegyptum, sed in universam Graeciam, ad omnes mortales Mathematicis studiis inflammandum.”)
30.
See, for instance, his letter to John Dee, in Ramus (1599, pp. 174–175); on p. 14 of the Scholae mathematicae, he related the information he obtained from their correspondence (that there were no university professors of mathematics in England) and urged Queen Elizabeth to appoint Dee to a royally-funded chair. On p. 66 he recalled his correspondence with Joachim Rheticus over the possibility of an “astronomy without hypotheses” – a project that was so close to his heart that he offered his own chair in the Collège to any astronomer who succeeded in it (ibid., pp. 49–50). The letter to Rheticus may be found in Ramus (1599, pp. 213–218). On Ramus’s search for a non-hypothetical astronomy, see Jardine (2001); and Grafton (1997, pp. 261–262), which sets out the historical basis for Ramus’s critique of Ptolemy.
31.
Ramus (1569, p. 13): “P. Ramus Veromanduus sum, non Atheniensis Plato.” He would make a parallel statement later in the Scholae mathematicae (at p. 110), injecting himself into the narrative: “I am Peter Ramus, regius professor at Paris, anxious about the future of the teaching of mathematics.” The context here was again Ramus’s efforts to build an international coalition of mathematicians.
32.
Ibid., p. 15: “Plato mundum universum mathematicae studio per deliacum illud duplicandi cubi problema incendit atque inflammavit.”
33.
Ramus (1569, p. 18): “ista pene muliebris zelotypia.”
34.
Ibid.: “Vilescet philosophia, si mathesis mechanicis opificum manibus exponatur.”
35.
Ibid.: “Sic pontifices Romani, fastus quondam suos; sic theologi plerique nostri theologiam populo ignotam esse voluerunt.” Ramus repeated this charge later (at p. 54), recalling the jealousy of Plato and the “common arrogance of priests, theologians and philosophers” (“ambitionem pontificum, theologorum, philosophorum communem”).
36.
Ibid., p. 19: “Maxima igitur Platonis in mathematicis gloria foedissima ejusmodi maculam sibi aspersit.”
37.
Ibid., pp. 27–28: “Vetus illa jam inde a Platone mathematicis perversa et praepostera opinio fuit, mathematicae usum non esse vulgo communicandum…”
38.
Ibid., pp. 28–29.
39.
Ramus’s odium for demonstration and his quest for an authentic, undemonstrated mathematics will be considered in fifth and sixth chapters.
40.
Ramus (1569, p. 30): “caeca ambitio.”
41.
Ramus (1569, p. 110): “Tum impietates e christiana religione et immanes sectas Christiani sublatas esse laetabuntur. Romam tum denique vere triumphantem omnes confitebuntur … hunc aureum pontificatum omnes mortales complectentur, fovebunt, osculabuntur.”
42.
Ibid., p. 35: “Quamobrem iste mihi imprimis placet author, qui Platonis geometriam cum Archimedis mechanica, qui artem cum usu artis tam solerter atque industrie conjunxerit.”
43.
Gesner (1545), s.v. “Hero Alexandrinus.”
44.
Ramus (1569, p. 35).
45.
On the nature of Proclus’s list, see p. 1 above.
46.
See, for instance Ramus (1569, pp. 77, 100) for two examples of a list of elementators. The term “elementator” translates Proclus’s stoikheiôtês.
47.
Most particularly in the inclusion of Pythagoras, which will be discussed in detail below. Compare also the list on p. 35, which includes Geminus as well as Pythagoras.
48.
Ramus (1569, p. 39): “ut Euclidi praeter inane nomen nihil admodum relinquatur.” In both fifth and sixth chapters, I shall consider why it was that Ramus questioned the significance of Euclid, and how he reimagined the authorship of the Elements.
49.
See p. 20 above.
50.
On the professors of mathematics at the Collège Royal, see Pantin (2004), especially the table of holders of the chairs on p. 200. On Fine and reform, see Margolin (1976).
51.
Charpentier (1566, sig. D4v).
52.
Ramus, La Remonstrance faite au conseil privé en la chambre du Roy, au Louvre le 18 janvier 1567, Paris (1567, pp. 14–15); cited by Skalnik (2002, p. 83).
53.
In his 1551 speech Pro philosopica disciplina, Ramus (1599, pp. 255–323). On Charpentier’s association with Galland and his animus against Ramus, see Waddington (1855, p. 41; and pp. 73–75) on the circumstances of the speech itself. Charpentier’s obsessive hatred for Ramus was attested even by a sympathetic biographer like Masson (1638, pp. 272–274), who judges that Charpentier’s greatest vice was his implacable anger, which he himself admitted could not be assuaged by any philosophical remedies.
54.
Ong (1958a, p. 220).
55.
In Charpentier (1564, fols 3v and 11r–v).
56.
For instance, in one of his orations he wrote that if Ramus demanded wide mathematical knowledge from a professor, then he should not be a teacher himself “since not only is he not well established in this subject through long and assiduous practice, but (as his teachers will attest) he can hardly even parrot faithfully what has been dictated to him at home.” (Charpentier (1566, sig. G2r): “Quoniam hic non modo in ea non est longo usu et assiduo confirmatus, sed vix adhuc potest, quod eius magistri testantur, domi dictata, suis fideliter recitare.”)
57.
See particularly his Actio secunda against Charpentier, where he “admits, or rather proclaims” that he has received help from others (Ramus 1599, p. 431); and his preface to the 1569 Arithmetica, acknowledging the assistance of Pena, Forcadel and Risner (ibid., pp. 135–136).
58.
This is according to the testimony of Charpentier, in Charpentier (1566, sig. E4v). Charpentier is not, of course, an objective source on Ramus’s teaching. But he can be trusted here since he is admitting, almost despite himself, that Ramus has adopted an ambitious mathematical curriculum. (In a final, catty remark, he laments that Ramus had made such an effort for so few students).
59.
The chronology of the case is complicated by the fact that Ramus made several errors in dating his orations according to the Roman style. The order of the speeches as I give them here is based on internal cross-references in the speeches.
60.
Ramus (1599, pp. 420–422). See n. 16 above.
61.
The text of the arrêt is in Waddington (1855, pp. 176–178). See also Pantin (2004, pp. 193, 202).
62.
See particularly the second oration: Charpentier (1566, sig. D4v). He noted that even Ramus, Professor of “Philosophy and Eloquence” had taken the chair of a Professor of Hebrew.
63.
Waddington (1855, pp. 178–179); Girot (1998, pp. 70–71).
64.
Ramus, La Remonstrance, extracts from which are edited in Waddington (1855, pp. 411–417).
65.
Waddington (1855, pp. 178–181). The only complete and accurate chronology of the case and the subsequent pamphlet war is found in Girot (1998).
66.
Waddington (1855, p. 181).
67.
Ong (1958a, p. 27).
68.
Ong (1958b, p. 357).
69.
Skalnik (2002, pp. 81–87).
70.
Ibid., p. 83.
71.
Ibid., p. 87.
72.
See, for instance, Charpentier (1566, sigs B2r, C2r–v, H3r–v).
73.
Girot (1998, pp. 73–74). Note, for instance, in Charpentier’s first oration before Parlement (Charpentier, 1566, sig. B2r) that he equated Ramus’s insistence on holding an examination with a desire to usurp regal powers for himself. In the third oration (ibid., sig. D4r) he compared Ramus’s tenure as dean to the madman who, just the other day, had gone running through the streets proclaiming himself king of France. Through this comparison he associated Ramus, as usual, with unrestrained passions and delusions of grandeur, but also with treasonous ambitions.
74.
Girot (1998, pp. 79–81). Skalnik also acknowledges that Charpentier was correct on this and other points of institutional history and practice. See Skalnik (2002, n. 57 on pp. 85–86).
75.
Girot (1998, p. 74).
76.
First mathematical preface, Ramus (1599, pp. 120–121): “Hinc tot, tamque excellentia ingenia excitari, Thaletis, Pythagorae, Hippocratis, Platonis, Eudoxi, Ptolemaei, Euclidis, Archimedis, aliorumque innumerabilium coeperunt.” Second mathematical preface, ibid., p. 121: “haec tandem Graecorum et Italorum, Thaletis, Pythagorae, Anaxagorae, Hippocratis, Platonis, Archytae, Aristotelis, Euclidis, Philolai, Archimedis, reliquorum omnium (de quibus Proclus scripsit) celebrata gymnasia fuerunt.”
77.
In his early neglect of Pythagoras, Ramus was following the lead, it seems, of Regiomontanus who, in his 1464 oration on the history of mathematics, passed over Pythagoras in a single sentence (as noted in first chapter above).
78.
Ramus (1599, pp. 419–420).
79.
See Heninger (1974); Riedweg (2005, especially pp. 129–132); and Joost-Gaugier (2006, especially pp. 66–76). The figure of Pythagoras was quite malleable; not long before Ramus, Johannes Reuchlin had claimed he had brought Pythagoras back to life in his presentation of Kabbalistic wisdom, insisting that “Kabbalah and Pythagoreanism are of the same stuff” (Jones, 1983, p. 19).
80.
Charpentier (1566, sig. B4r).
81.
Charpentier (1566, sig. G3v): “Certo sciebam hunc in Cathedra Mathematica saepe obmutuisse, quod in via de manibus excidissent ea quae a magistris paulo ante acceperat; millies etiam inter docendum coactum fateri, Mathematicam descriptionem parum feliciter succedere, quod in hac non esset satis exercitatus; nec minus frequenter posteriore lectione ea omnino invertisse, quae superiore magna animi confidentia videbantur esse constituta.” In the same vein, in the aftermath of the case one of Charpentier’s anonymous supporters recorded how Ramus lost the thread of a geometrical proof in front of his class, and, entirely out of resources, stood agape and “dumber than a fish” in front of his bemused students. Anonymous (1567, p. 9): “… dum videlicet susceptae propositionis demonstrationem nulla ratione potuisti exponere, sed pisce mutior factus, illico de cathedra descendisti.”
82.
Ibid., sigs G3r–v.
83.
Ramus, Scholae mathematicae, p. 7: “… quod mathematicam philosophiam in speciem liberalis et ingenuae doctrinae primus redegerit, ludumque aperuerit, in quo juventus tam honestas, tamque nobiles exercitationes haberet.” See Proclus (1992, pp. 52–53).
84.
Ibid., p. 7: “Non quosvis ait Gellius libro primo capite nono in disciplinam admittebat, sed ephuseognômonei ex oris et vultus ingenio … ne amousoi, atheôrêtoi, ageômetrêtoi otio et ludo disciplinae tam liberalis abuterentur.” In Gellius, the anecdote about physiognomy occurs at the beginning of Noctes atticae I. 9; at the end of this chapter on the Pythagoreans, Gellius records a saying of his friend Taurus, that modern philosophers were amousoi, atheôrêtoi and ageômetrêtoi in comparison with the followers of Pythagoras.
85.
Scholae mathematicae, p. 12.
86.
See Vitae, VIII. 6.
87.
Gellius, Noctes atticae I. 9: “Hi dicebantur in eo tempore mathêmatikoi, ab his scilicet artibus quas iam discere atque meditari inceptaverant.”
88.
Scholae mathematicae, p. 8: “Cuius utinam paideutikon illud liberalis et ingenuae institutionis fundamentum, paulo diligentius ab hominibus attenderetur, propria humanitatis elementa tandiu a scholis nostris nequaquam abessent.”
89.
In his Ramus (1559, fols 44v–45r), he claimed that the ancient Gauls taught the liberal arts in their native language; if they had written down their teachings, it would be possible for the French to learn the arts in the vernacular, without the years now needed for the study of Latin and Greek grammar.
90.
See n. 53 above for this oration (the text of which is in Ramus 1599, pp. 255–323).
91.
Ramus (1599, p. 170): “Nec in isto rhetorico studio grammaticas regulas permiscemus…;” p. 171: “… et Dialecticae inventionis dispositionisque praecepta, quae Rhetores in rhetoricis artibus parum distincte confuderant, in dialectica arte proprie et perspicue tradimus.”
92.
Ibid., p. 177.
93.
Ibid.. “… [volumus] Physicam veram, mathematicis rationibus fundatam doceri et exerceri.”
94.
Two years later, Ramus would use a similar line against the Aristotelian Jakob Schegk, whom he enjoined to keep a modest, “Pythagorean silence” until he had mastered sufficient mathematics to express a worthwhile opinion on philosophy. See Ramus (1599, pp. 205–206).
95.
Ong (1958a, chapters 5 and 10).
96.
See Ong (1974), for Ramus’s last emendations to his mathematics, made shortly before his death.
97.
Proclus (1992, p. 54): “Hippocrates wrote a book on elements, the first of whom we have any record who did so.”
98.
Scholae mathematicae p. 10: “Primus mathematicae in schola magister Pythagoras fuit, sed ut de primis initiis credi par est, minus distinctus, ut stoikheiôtês ideo non appelletur: sed tamen quidquid sit, Hippocrates Pythagorae magnitudine minime deterritus mathematicum magisterium auxit et exornavit elementis ordine, viaque pleniore et uberiore deductis.”
99.
Knorr connects Hippocrates’s systematization of geometry with the problem of squaring plane figures. In his view, Hippocrates was concerned to catalog the techniques already known for squaring rectilinear figures, in order to narrow down the approaches to squaring curvilinear figures, especially lunules (of which Hippocrates squared three of the five quadrable types) and the circle itself. Knorr (1986, pp. 40–41).
100.
Ramus wrote of Theudius, the third elementator in Proclus’s catalog, that he “did not consider it odious or invidious to correct the Elements of Pythagoras and Hippocrates, or of Leon.” (Ramus 1569, p. 19: “… Theudius … nec odiosum sibi, nec invidiosum putavit Pythagorae, Hippocratis, Leontisque stoikheiôsin corrigere et emendare.”)
101.
Charpentier (1566, sig. G2r).
102.
Ramus stressed that mathematical progress was continuous, by noting that Pythagoras himself was unaware of the more general, superior theorem that became Elements VI.31; if Pythagoras’s theorem was worth the sacrifice of a bull, then, Ramus thought, the anonymous VI.31 deserved at least a thousand (Ramus 1569, p. 7).
103.
Ramus (1569, p. 17): “Leo igitur tertius mathematicae philosophiae non solum magister et doctor, sed scriptor Pythagora et Hippocrate usus laude perfectior et accuratior fuit.”
104.
See n. 100 above.
105.
Ramus (1569, p. 19): “Pythagoras, ut hunc etiam tanquam stoikheiôtên numerem … Hippocrates istam laudem aemulatus, elementa demonstrationibus exornata descripsit et publicavit.”
106.
Ramus (1569, p. 77).
107.
See Goulding (2005).
108.
Scholae mathematicae, p. 96.
109.
See Ong (1974).
110.
Ramus (1569, p. 7): “Amores nempe mathematici sunt illi acerbi primum difficilesque, tandem voluptatis plenissimi.”
111.
Text at n. 49 of second chapter above.
112.
Scholae mathematicae, p. 32.
113.
Scholae mathematicae, p. 112: “a quibus mathematicas artes pueris faciles, opificum vulgo familiares, cognitione denique et usu non tantum mirabiles, sed etiam populares factas esse videam.”
114.
[scholmath] p. 13: “Ergo Pythagoras Academiae Parisiensi mathematicas optabit: Ergo Plato in Academia Parisiensi mathematicas artes desiderabit; et uterque Parisiensem Academiam, tum Pythagoream et Platonicam esse judicabit, cum mathematicis primas in philosophia detulerit.”
115.
See passage quoted at n. 61 of first chapter.
116.
Scholae mathematicae p. 46.
117.
Ibid., p. 49.
118.
Ibid., p. 71.
119.
That is, the Cardinal of Lorraine, who was Ramus’s patron until Ramus converted to the reformed religion. Throughout the case, even as Charpentier mocked him for having lost his patron, Ramus affected that he and the Cardinal remained close.
120.
Ibid., p. 74.
121.
Ibid., p. 75.
122.
Horace, Epistulae I.17: Omnis Aristippum decuit color et status et res.
123.
Charpentier (1567, fol. 18v).
124.
Ibid., fol. 22r.
125.
Ramus (1569, pp. 46–47).
126.
See p. 14 above.
127.
Charpentier (1567, fol. 58r): “Sicque Pythagoreorum institutio et paedia, erat posita in mathematicis, quoniam, ut dixi, Philosophiae mysteria, quae volebant a suis tantum intelligi, illi per numeros et figuras explicabant.”
128.
Ibid., fols 58r–59v.
129.
See Pesic (1997).
130.
Charpentier (1567, 60r–v): “An ad id confugies quod audio nuper in explicatione geometriae tibi factum esse familiare? Arithmeticae scilicet et geometriae subiectum, quod quantum dicitur, non esse affectionem eius substantiae ad quam naturale corpus refertur, sed eius principium atque fundamentum. Equidem etsi permulti fide dignissimi, mihi testati sunt, hoc a te saepe in tuis praelectionibus esse praedicatum facileque suspicer hac nova opinione quorsum in his quae ad religionem pertinent velis evadere, haec tamen mihi tam absurda est tamque monstrosa, ut non audeam tibi eam hoc loco ascribere.” The religious implication was the denial of substance and accident, which would undermine the doctrine of transsubstantiation of the elements of the Eucharist.
131.
Traces of Ramus’s late lectures on Pythagoras may be found in his Scholae metaphysicae and Scholae physicae, where his analysis and critique of the third and sixth books of the Physics and the tenth book of the Metaphysics (the latter Aristotelian text one of the classic attacks on Pythagoreanism) 1ed him to a kind of Pythagorean atomism.
132.
Proclus (1992, p. 57).
133.
See fifth chapter.
134.
Ramus (1569, p. 1): “Tertius, Ptolemaei regis problema adversus Euclidem disputabit, de magis perspicua, magisque compendiaria via matheseos instituendae.”
135.
Ramus (1569, p. 24): “stoikheiôseôs tamen Euclideae rationem et viam videtur rex ille non probasse, neque Euclides ipse satis liberaliter regi fecisse. Rex enim Euclidem aliquando interrogasse fertur, num qua ad Geometriam via magis compendiaria esset, quam stoikheiôseôs ab eo compositae; cui Euclides, Semita (inquit) ô rex, ad geometriam regia nulla est; quo responso videtur significasse viam elementorum a se compositorum esse latam, apertam, simplicem, directam et tanquam militarem, ideoque regiam esse. Semitam autem breviorem esse lubricam et ancipitem, neque ideo regiam. Sed istud problema tertio libro plenius edisseretur, Regisne hac in re judicium, an Euclidis logikôteron fuerit.
136.
Ibid., p. 28.
137.
Ibid., p. 79.
138.
Ibid., p. 104: “Quapropter totam regii problematis querimoniam concludamus. Ptolemaeus queritur Euclidis stoikheiôsin obscuram et difficilem esse; Euclides contra confirmat esse perspicuam et facilem, ut si quis illustriorem aut expeditiorem requirat, lubricam semitam quaerat, non viam regiam. Stoikheisis Euclidis defenditur a Proclo, caussa regis a nobis suscepta et hactenus secundum leges consensu partium laudatas ac probatas acta est.” Proclus’s defense is, of course, the Commentary, which Ramus claimed supported in principle the three laws of Ramist method – laws that he argued Euclid had violated (hence the reference to “laws agreed upon by the consent of all parties.”)

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Program of Liberal Studies, University of Notre Dame, 313 Decio Faculty Hall, Notre Dame, IN, 46556, USA
Prof. Robert Goulding

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Goulding, R. (2010). From Plato to Pythagoras: The Scholae mathematicae . In: Defending Hypatia. Archimedes, vol 25. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3542-4_3

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DOI: https://doi.org/10.1007/978-90-481-3542-4_3
Published: 18 November 2009
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3541-7
Online ISBN: 978-90-481-3542-4
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