E

Euclid: Reception in the Zamberti’s texts were published – at times in Renaissance combination. From the 1540s onward, revi- sions, selections, and vernacular Jens Høyrup began to appear, all based on the same two Section for and Science Studies, texts. In 1572, however, Commandino made a Roskilde University, Roskilde, Denmark new Latin from Zamberti’s text and a sounder manuscript, and in 1574 Clavius Abstract produced a didactically oriented redaction. Although the Latin Middle Ages received a These two set the scene for the next two number of versions of ’s Elements and centuries. several other Euclidean works, by the four- teenth century, only the Campanus redaction from c. 1259 was in circulation. In the four- Medieval Latin Background teenth and fifteenth century, this redaction was encountered by students of Arts or Medicine Until the beginnings of the twelfth century, the university faculties, even though we have scant Latin Middle Ages had access to Euclid only evidence that Euclid impressed their minds. In through Boethian translation of the Elements (or, the fifteenth century, other discovered quite likely, of an epitome of that work) – another him: Alberti took over the idea of elements, translation, also from c. 500 CE (Bohlin 2012), Regiomontanus used Euclid alongside Archi- seems not to have circulated. Part of the Boethian medes as an argument for the superiority of translation was conserved in coherent form – how mathematics over philosophy, and one Floren- much and for how long is disputed; Boethian tine abacus school tradition was able to give fragments were also integrated in gromatic writ- correct references to the Elements. ings (which served didactical purposes rather than A turn arrived with book printing. In 1482, surveying). the Campanus Elements were printed, and in In the twelfth century, a new translation 1498 and 1501, Giorgio Valla inserted pseudo- directly from the Greek was made (ed. Busard Euclidean and Euclidean material in two bulky 1987); it was used occasionally by Fibonacci volumes. A new though somewhat problematic (Folkerts 2006, IX) but left few traces beyond Latin translation from the Greek (including that. The translations from the Arabic made by also some minor works) was published by Gerard of Cremona (ed. Busard 1984) and Zamberti in 1505, and until 1540 a number of Hermann of Carinthia (ed. Busard 1967) also reprints or reeditions of Campanus’s and had limited circulation. The one prepared by

© Springer Nature Switzerland AG 2019 M. Sgarbi (ed.), Encyclopedia of Renaissance Philosophy, https://doi.org/10.1007/978-3-319-02848-4_918-1 2 Euclid: Reception in the Renaissance

Adelard of Bath, so-called Adelard I, on the other contact with the schools (“have seen its door”) and hand, became the basis for the didactically have looked into some vernacular booklets and adapted redaction known as Adelard II now wish to be held ; they are char- (ed. Busard and Folkerts 1992), probably due to acterized by citing authors they have never seen – Robert of Chester (or “of Ketton”), and for the “Priscian, , , Aristarchus, Euclid, epistemologically more sophisticated “Adelard , and others, most famous in the sciences” III” (ed. Busard 2001), probably the work of (Boccaccio 1564, 225r–v). This may be seen as a John of Tynemouth. Versions II and III were minimal list of those authors people of manners then used by Campanus of Novara for his redac- were supposed to know about. tion, written between 1255 and 1259 (ed. Busard In the fifteenth century, Battista Alberti 2005). This redaction was well adapted to use in went beyond name-dropping, manuscript posses- the scholastic university, where at least part of the sion, and possibly manuscript reading. His Elements was supposed to enter the Arts curricu- Elementi de pittura (ed. Grayson 1973,109–129) lum (in some places alternatively with Book I of from c. 1435 (as well as the parallel Latin Witelo’s Perspectiva). Campanus Elements came Elementa) not only borrow the Euclidean title. to dominate until well into the sixteenth century. “For the sake of brevity,” they also open with All of these except the Hermann translation con- 5 sets of definitions, 22 in total in each of the tain 15 books, that is, they include a Book XIV 2 versions (not fully identical, however); for the written by and a Book XV written by rest they consist of specified elementary tasks Isidore of ; of the Hermann translation, similar to Euclidean problems – firstly “to describe only 12 books are extant. a straight line from one point to another one.” As Euclid’s and Optics were also translated stated in the opening of De pictura (ed. Grayson in the twelfth century (from the Greek as well as 1973,6–107), Alberti does so borrowing from the Arabic) and so was the pseudo-Euclidean but adapting what they say (Murdoch 1971, 444). The (probably about the merely intelligible to a topic interested pseudo-)Euclidean De ponderoso et levi only in that which can be seen. This is probably (ed. trans. Moody and Clagett 1952,21–31) was the first Renaissance example of use of the “geo- translated from the Arabic. However, none of metric method”–inspired by Euclid and not by these works had an influence coming close to , whose axiomatic-deductive works that of the Elements. Alberti does not know. In 1564, Regiomontanus – first trained and active in the University of Vienna and then under the influence of Bessarion drawn into Ital- Early Humanist Interest and Knowledge ian Humanism – held a series of lectures on the al-Farghānī in Padua. The inaugural The fourteenth-century humanists who had lecture (ed. Schmeidler 1972,43–53), an oration frequented one of the integrated Arts and Medi- “explaining the mathematical sciences and their cine faculties of Italian universities or an Arts or a utility,” mixes the two currents of thought (Byrne Medicine faculty elsewhere were likely to have 2006) but also shows traces of that pride of Italian gained some familiarity with the first books of the mathematical practitioners which made them Elements, but sources give little more than hints claim priority of mathematics over philosophy. (cf. Siraisi 1973,74–77). A copy of the Campanus Regiomontanus uses Euclid and Archimedes for version in the library of S. Spirito in Florence that purpose – while philosophy is split into war- listed in a catalogue from 1451 may have been ring schools, “Euclid’s theorems have the same part of the legacy from Boccaccio (Ullman 1964, certitude today as a thousand years ago, and 285). In any case, in his Della genealogia de gli Archimedes’s inventions will call forth no less Dei,wefind an attack on those who have had brief Euclid: Reception in the Renaissance 3 admiration in a thousand centuries than pleasure Euclid in Print in us when reading them.” In the beginning of the oration, Euclid has already been declared “the The fifteenth century is also the time when some father of all geometers.” pure-bred humanists started collecting mathemat- We also find in this oration an early instance of ical manuscripts for their libraries (Alberti, though the mistaken identification of Euclid the geometer a humanist, was more than that and so was with the philosophers Euclid of men- Regiomontanus) – most famously of all probably tioned in ’s Phaedo (and by Bessarion. However, Euclid was not their first Laërtius). This Renaissance mistake may go choice. Accordingly, the first printed Elements back to Theodoros Melichita in the early four- (Campanus 1482) were made by Erhard Ratdolt teenth century (Heath 1926, 3) (there is no reason in Venice in Campanus’ version. Ratdolt was no to believe Theodoros to have been inspired by a humanist but an outstanding and innovative Ger- similar but oblique reference in Valerius Maximus man printer. His dedicatory letter to Duke nor to assume that the fifteenth-century Italian Mocenigo of Venice shows him to be more closely writers took over from Valerius something that linked to the university tradition than to the medieval authors, eager readers of his, had not humanist current. thought of ). But Latin humanist readers, finding This edition can be seen to have fulfilled a the name Euclid in Plato, can also have reinvented need. Ratdolt produced a reprint in the same the mistake independently. year, another one was made in Ulm in 1486, and Around the same time, two encyclopedic trea- a third was in Basel in 1491 (Cantor 1892, 266f ). tises coming from the Florentine abacus school The first Euclidean texts produced by a human- environment testify of interest in the Elements. ist are found in an anthology collected by Giorgio One is the anonymous Florence, Biblioteca Valla (1498,cvir–dvr) – namely, Elements XIV Nazionale Centrale, Palat. 573; the other is and XV – and thus actually pseudo-Euclidean; Benedetto da Firenze’s Trattato di praticha Valla presents the former as Euclid’s 14th book d’arismetrica from 1463, the autograph of which and the latter as Hypsicles’s interpretation of the is Siena, L.IV.21. Both are described with copious same book. Scattered properly Euclidean frag- extracts in Arrighi (2004). That they mention ments taken from a Greek manuscript were to be Euclid’s name is not very informative – many found in his posthumous encyclopedia De abacus books excelled in dropping names in the expetendis et fugiendis rebus opus (Valla 1501). way described by Boccaccio, often in fully mis- In this work he draws much on ’s com- guided ways. But these two treatises are different: mentary to Elements I, which gives him access to they refer repeatedly correctly to Euclidean books Eudemus’s catalogue of geometers and to the (II, V, VII, IX, X), at times with quotations dating of Euclid to the time of the first Ptolemy; pointing (as could be expected) to Campanus. nonetheless, the identification of the geometer and The citations are so similar that the two writers the from Megara survived not only must be assumed to have drawn on the same Valla’s insight but also the publication of Pro- intermediate source – which shows that their clus’s commentary in 1533 (infra). branch of abacus culture (descending from Paolo The first full Renaissance text of the Elements dell’Abbaco and Antonio de’ Mazzinghi) had a translated from the Greek was produced by tradition for such interest in Euclid. How far this Bartolomeo Zamberti in 1505 (reprinted in 1510 tradition goes back we cannot know, but it seems and 1517 by the same Venetian printer), with likely that it was a product of the fifteenth century. copious attacks on Campanus. The (unidentified) From the fifteenth century, there are also two manuscript he used was in the Theonine tradition, manuscripts (Siena, L.IV.16 and 17) containing and Zamberti supposed that the demonstrations Italian translations of the Campanus version were due to Theon; Campanus instead had relied (Kristeller 1965, VI, 158; Folkerts 2006,XI through Adelard on Arabic, pre-Theonine manu- 223), which might come from the same tradition. scripts, which is one of several reasons for the 4 Euclid: Reception in the Renaissance divergences which provoked Zamberti’s anger. Zamberti- texts together. The first edition of this The volume also contained Books XIV–XV, pre- kind was made by Jacques Lefèvre d’Étaples in sented as “Hypsicles report of a volume supposed 1516, containing only the Elements (15 books). to be by Euclid,” respectively, Book XIV of Lefèvre d’Étaples supposes only the definitions, Euclid’s Elements “in the report of Hypsicles” postulates, and common notions to be due to (Zamberti 1505, X iiiir, viiv); moreover, Euclid’s Euclid, while he ascribes the proofs to Campanus, Phenomena; the pseudo-Euclidean Catoptrics; respectively, Zamberti. Another combined edition Euclid’s Optics; and his Data, with Marinus’s was made by Herwagen in Basel in 1537 (Euclidis introduction. A long dedication also serves as Megarensis 1537), with an introduction where introduction, presenting the history of mathemat- Philip Melanchthon speaks at length about the ics in its relation to philosophy from Homer to utility and the moral implications of mathematics. and Proclus; where relevant, it draws on It included the other Euclidean works translated Proclus’s commentary (and thus on Eudemus). by Zamberti (the Data still with Marinus’s intro- Identifying Euclid with Euclid of Megara, duction) and the pseudo-Euclidean De levi et Zamberti states him to have listened to ponderoso (sic; supra), presented as a fragment and to be a contemporary of Plato and at the same and indeed lacking the fifth and final proposition time takes Proclus as his witness that Euclid lived of the medieval text (and formulated in very dif- at the time of the first Ptolemy. After the introduc- ferent words). New editions of this collection tion comes Euclid’s vita, consisting of excerpts were published by Herwagen in 1546 and 1558 from Suidas, Diogenes Laërtius, , Aulus with only modest changes. Gellius (all speaking of Euclid of Megara), Heron Already in 1533, Simon Grynaeus had brought of , Proclus, and Marinus (speaking of out the , also at the printing house the ). In the end, Zamberti attempts of Herwagen – unfortunately from two low- to find from available chronicles the epoch of quality manuscripts (Heath 1926, 101). It also Ptolemy I, coming to 291 BC; he does nothing to contains Proclus’s commentary to Elements determine that of Socrates and appears not to be I. Books XIV and XV are presented as “according aware of any contradiction. In any case, this vita to others, by Hypsicles.” There is no trace of demonstrates to the full Zamberti’s humanist cre- Euclid of Megara, not even a polemical refutation. dentials. Unfortunately, as Maurolico was to The number of editions and reprints may seem observe in a letter from 1556 (Napoli 1876, 27), to suggest that there was strong interest at least in while translating faithfully, Zamberti lacked the the Elements. A letter from Maurolico to Pietro necessary mathematical insight and did not dis- Bembo (ed. Spezi 1862, 80) gives the opposite cover the mistakes of his manuscript. impression –“Galen flourishes everywhere, the Zamberti’s attacks against Campanus were academies resound with Justinian, the marble is answered by Luca Pacioli, who produced a new shattered in dialectical disputes. Why is the excel- edition of the Campanus Elements (Pacioli 1509), lent Euclid silent? Why are Archimedes and also in Venice. It promised to correct the errors Theodosius silent? [...] Of Euclid, hardly six that had crept in because of the negligence of books are read.” Compared to what we have copyists. Actually, Pacioli’s text and his diagrams encountered in fourteenth- and even fifteenth- are very close to those of Ratdolt’s edition, but he century Humanism, however, “hardly six books” added a number of commentaries (Folkerts 2006, constitute a qualitative jump – and six books XI, 227f ) – 138 in total, of which 42 over 10 lines remained a standard school book for long. long, according to Folkerts. Already a decade or The first (bilingual) six-book edition was so before publishing this edition, Pacioli had published in the same year by Oronce Finé (Finé made a vernacular translation of Campanus, 1536). It presents itself as Finé’s “demonstrations which however has been lost. of the first six books of the Elements of Euclid of A phase followed where publishers might Megara, augmented and emended, together with play safe and print the Campanus- and the the same Euclid’s Greek text, and Zamberti’s Euclid: Reception in the Renaissance 5

Latin interpretation,” all examined by Finé him- the needs of artisans (that is, largely without pro- self. What this means is that the part of the text ofs) in 1562 (Heath 1926, 106–108; Murdoch which Zamberti had ascribed to Euclid – that is, 1971, 449); and in 1570, Henry Billingsley made definitions, postulates, common notions, and an impressive English translation, prefaced by enunciations – are given as by Grynaeus and John Dee, and including “Scholies, Annotations, Zamberti. The demonstrations are rephrased ped- and Inventions, of the Best Mathematiciens, both agogically by Finé (and marked “Orontius”). in time past, and in this our age” (Heath 1926,109). A particular six-book version of “Euclides Megarensis” was published by Johann Scheubel (1550). It also borrows the formulation of the Euclid for the Future: Commandino and matters which were regarded as properly Euclid- Clavius ean from Grynaeus and Zamberti. The proofs, however, are not only Scheubel’s own – the dia- All of these editions and translations, from grams are not lettered; instead the lines often carry Zamberti onward, can be said to define and even numbers corresponding to a supposed length; constitute the sixteenth-century Euclid. In the there are also regularly numerical computations – early 1570s, however, two very different trans- and the whole is preceded by a 76-page-long lations appeared that were to define the Euclid of introduction to algebra (including the theory of the next two centuries. One was Commandino irrationals). Rather than a humanist attempt to (1572). Federico Commandino based this Latin restore , this edition is thus an translation on Grynaeus as well as another, better early, still groping attempt at synthesis of Greek Greek manuscript (still in the Theonine tradition), theory with the higher level of Rechenmeister and also included many previously unknown mathematics. Greek scholia; moreover, being an outstanding A number of other editions based (sometimes mathematician, he understood the text much bet- faithfully, sometimes with innovations) on ter than any sixteenth-century predecessor. There Grynaeus, Campanus, and Zamberti are listed in were thus very good reasons that Commandino’s Heath (1926, 101f ) and Murdoch (1971, 449). translation became the direct or indirect basis for Taken together they confirm that there was much many new translations and editions (Murdoch more interest in Euclid in the sixteenth than in 1971,44)– also editions combining with Grynaeus previous centuries – Maurolico’s complaints Greek text. It was superseded only when François notwithstanding. Peyrard discovered and published an apparently Further confirmation comes from the vernacu- pre-Theonine text (Peyrard 1804). lar translations of the Elements that appeared; at Commandino was also a better historian than the same time, these make clear the diversity of his predecessors. Whereas Valla and Grynaeus groups that partook in this interest. Earliest was appear not to have believed in the identification (Tartaglia 1543), “according to the two transla- with Euclid of Megara but did not discuss why, tions” (Campanus and Zamberti) but presenting a and while Zamberti did not notice that the evi- single text. Different from Finé and the combined dence he draws on excludes it, Commandino editions of Lefèvre d’Étaples and Herwagen, wants (fourth page of the unfoliated preface) to Tartaglia presents enunciations and demonstra- “free those many from their error who firmly tions as belonging on an equal footing to the believe our Euclid to be the same as both the Elements and adds his own discussions under the philosopher from Megara and the geometer” and heading “The translator.” sets out briefly the reasons – both chronological A French translation followed (Forcadel 1564) and from what Diogenes Laërtios tells about the (six books only), in which the demonstrations are work of Euclid of Megara. (justly) ascribed to Forcadel. Scheubel made a The other was Clavius (1574). In an initial German translation of Books VII–IX in 1558; address to the reader, Christophorus Clavius Xylander a full German translation adapted to explains why all preceding editions are deficient – 6 Euclid: Reception in the Renaissance

“excepting that of Federico Commandino, a Clavius, C., (ed., Trans.). 1574. Euclidis Elementorum libri highly skillful geometer, by whose work and dil- XV. 2 vols. Roma: Vincenzio Accolto. Commandino, F., (ed., Trans.). 1572. Euclidis igence Euclid has been rendered in Latin in its Elementorum libri XV, unà cum Scholiis antiquis. pristine splendor.” However, Clavius’s aim is Pesaro: Camillo Franceschino. (as he explains) to produce a book that can serve Euclidis Megarensis mathematici clarissimi Elementorum those who progress in the sweet study of mathe- Geometricorum libri XV [...]. 1537. Basel: Johannes ’ Herwagen. matics; therefore he does not propose Euclid s Finé, O., ed. 1536. In sex priores libros geometricorum naked words, which are often more concise than Euclides Megarensis demonstrationes. Paris: Simon illuminating. He provides them with extra expla- Colinée. nations, sometimes his own, sometimes borrowed Forcadel, P., ed. 1564. Les six premiers livres des Elements ’ ’ d Euclide, commentez. Paris: Hierosme de Marnef & from Proclus (Clavius uses Francesco Barozzi s Guillaume Cavellat. translation from 1560), Campanus, or others. Grayson, C., ed. 1973. Leon Battista Alberti, Opere Rommevaux (2005,31–58) analyzes the addi- volgari. III. Trattati d’arte, Ludi rerum tions and occasional changes of the Euclidean mathematicarum, grammatica della lingua toscana, Opuscoli amatori, Lettere. Bari: Laterza. text. Grynaeus, S., ed. 1533. Εuklιdoustoιweιov bιbl. ιe. Εk Clavius published new editions of these Ele- tov yeovoBsuvousιov. Basel: Herwagen. ments in 1589, 1591, 1603, 1607, and 1612. They Lefèvre-d’Étaples, J. 1516. Euclidis Megarensis were used for the teaching of Jesuit recruits and Geometricorum elementorum libri XV. Paris: Henricus Stephanus. hence became the book behind the teaching of Moody, E.A., and M. Clagett, eds. 1952. The medieval much of the social elite of Catholic Europe for science of weights (Scientia de ponderibus). Treatises long. They were also the Euclid which Matteo Ascribed to Euclid, Archimedes, Thabit ibn Qurra, Ricci and Xu Guangqi translated into Chinese in Jordanus de Nemore, and Blasius of Parma. Madison, Wisconsin: University of Wisconsin Press. 1607 (the usual six books), in time for Clavius to Napoli, F., ed. 1876{a}. Scritti inediti di Francesco see the Chinese printed book. Maurolico. Bullettino di Bibliografia e di Storia delle Scienze matematiche e fisiche 9: 23–121. Pacioli, L., ed. 1509. Euclidis Megarensis philosophi acutissimi mathematicorumque omnium sine References controversia principis opera a Campano interprete fidissimo tralata. Venezia: Paganinus de Paganino. ’ Primary Literature Peyrard, F., (ed., Trans.). 1804. Les Élémens d Euclide, traduits littéralement. Paris: F. Louis. Boccaccio, G. 1564. Della genealogia degli dei. Venezia: Scheubel, J., (ed., Trans.). 1550. Euclidis Megarensis, Francesco Lorenzini. Philosophi et mathematici excellentissimi, Sex libri Busard, H.L.L., ed. 1967. The translation of the Elements priores, De Geometricis principiis, Graeci et latini, of Euclid from the Arabic into Latin by Hermann of una cum demonstrationibus propositionum.... – 59 Carinthia (?). Janus 54: 1 142; Janus (1972), Algebrae porro regulae, propter numerorum exempla, – 125 187; Amsterdam: Mathematisch Centrum, 1977. passim propositionibus adiecta, his libris præmissae Busard, H.L.L., ed. 1984. The Latin Translation of the sunt, eademque demonstratae. Basel: Herwagen. ’ Arabic Version of Euclid s Elements Commonly Schmeidler, F., ed. 1972. Joannis Regiomontani Opera Ascribed to Gerard of Cremona.Leiden:Brill. collectanea. Faksimiledrucke von neun Schriften Busard, H.L.L., ed. 1987. The mediaeval Latin translation Regiomontans und einer von ihm gedruckten Schrift ’ of Euclid s Elements made directly from the Greek. seines Lehrers Purbach. Osnabrück: Otto Zeller. Wiesbaden: Franz Steiner. Spezi, G., ed. 1862. Lettere inedite del Card. Pietro Bembo ’ Busard, H.L.L., ed. 2001. Johannes de Tinemue s redac- e di altri scrittori del secolo XVI. Roma: Tipografia ’ tion of Euclid s elements, the so-called Adelard III delle Scienze matematiche e fisiche. version. 2 vols. Stuttgart: Franz Steiner. Tartaglia, N., (ed., Trans.). 1543. Euclide Megarense Busard, H.L.L., ed. 2005. Campanus of Novara and philosopho: diligentemente rassettato, et alla integrità ’ Euclid s Elements. Vol. 2. Stuttgart: Franz Steiner. ridotto. Secondo le due tradottioni. Venezia: Guilielmo Busard, H.L.L., and M. Folkerts, eds. 1992. Robert of de Monserra & Pietro di Facolo. ’ ’ Chester s (?) redaction of Euclid s Elements, the Valla, G., (ed., Trans.). 1498. Hoc in volumine hec so-called Adelard II version. Vol. 2. Boston: Birkhäuser. continentur: Nicephori logica/Georgii Valle libellus Campanus. (ed., Trans.). 1482. Preclarissimus liber de argumentis/Euclidis quartus decimus elementorum/ elementorum Euclidis perspicacissimi. Venezia: Ratdolt. Euclid: Reception in the Renaissance 7

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