E Euclid: Reception in the Zamberti’s texts were published – at times in Renaissance combination. From the 1540s onward, revi- sions, selections, and vernacular translations Jens Høyrup began to appear, all based on the same two Section for Philosophy and Science Studies, texts. In 1572, however, Commandino made a Roskilde University, Roskilde, Denmark new Latin translation 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 Euclid’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 circles 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 philosophers; 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, Aristotle, Cicero, Aristarchus, Euclid, epistemologically more sophisticated “Adelard Ptolemy, 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, Leon 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 Hypsicles and a Book XV written by rest they consist of specified elementary tasks Isidore of Miletus; 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 Data 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 mathematicians but adapting what they say Catoptrics (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 Archimedes, 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 astronomer 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 Megara men- Regiomontanus) – most famously of all probably tioned in Plato’s Phaedo (and by Diogenes 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 Proclus’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.