Making mathematical culture : university and print in the circle of Lefèvre d’Étaples

507 Pages·2018·7.101 MB·English

by  Lefèvre d’Étaples, Jacques, Oosterhoff, Richard J.

Acknowledgements
University Press Scholarship Online
Oxford Scholarship Online
Making Mathematical Culture: University and
Print in the Circle of Lefèvre d'Étaples
Richard Oosterhoff
Print publication date: 2018
Print ISBN-13: 9780198823520
Published to Oxford Scholarship Online: September 2018
DOI: 10.1093/oso/9780198823520.001.0001
(p.v) Acknowledgements
Richard Oosterhoff
Among many other things, Elora, dilectissima mea, has
convinced me that the convention of thanking those dearest
last is confusing. No one has given more to this book. So I
thank her first.
The practices of authorship in the modern university are quite
unlike those I uncover in this book. But we remain dependent
on others. Robert Goulding first shepherded the dissertation
that has been mostly rewritten as this book—he has seen this
work in more states of disarray than anyone else, signs of his
endurance and intellectual strength of character. Likewise,
Margaret Meserve, Lynn Joy, John van Engen, and Jill Kraye
are all exemplars of the well lived scholarly life, and early on
nudged me with questions that still haunt me. Isabelle Pantin
encouraged some very early thoughts with great kindness,
Emmanuel Faye and James Hirstein have graciously welcomed
me into the sodalitas Rhenani, and Jean-Marc Mandosio has
been an enduring source of encouragement, even offering
welcome suggestions on a late version. Benjamin Wardhaugh
commented in depth and at length, first on a chapter and then
on the entire thing, and gave wise and useful advice—most of
which I attempted to follow.
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Acknowledgements
Many friends have conspired to keep me closer to the
intellectual straight and narrow. Alex Marr continues to
inspire as a model of intellectual acuity, sagacity, and insight.
Raphaële Garrod and José Ramon Marcaida are congenial
spirits ministering to a soul in need of succour. Tim Chesters
and Richard Serjeantson offered their innate good sense with
their usual eloquence. All read parts of this work in one form
or another. Sundar Henny, Kate Isard, Arthur Keefer, and
Nathan Ristuccia also commented on sections at key moments,
with habitual good judgement and good cheer. Sachiko
Kusukawa, Sietske Fransen, and Katie Reinhart are the best of
cotravellers. Anthony Ossa-Richardson has been the
embodiment of scholarly friendship, reading each chapter with
care, saving me from countless solecisms and infelicities, and
commiserating to the end. I am profoundly grateful for such
friends, and sharply conscious of how little these words of
thanks are. The remaining errors are not theirs.
I owe special thanks for the gifts of institutions. Librarians at
the Bibliothèque Humaniste de Sélestat have been unfailingly
gracious every time I swept through with too many requests. I
found librarians of tremendous professionalism at the
Huntington Library, the Houghton Library, Duke Humphrey’s
Library (back in the day), Cambridge University Library, the
British Library, and indeed the Bibliotheque nationale de
France. I am grateful for fellowships from the Nanovic
Institute (Notre Dame) and the Notre Dame Institute for
Advanced Study, the Warburg Institute, the Houghton Library,
and the Huntington Library. For the last steps on this project,
the European Research Council provided wonderful support,
in the form of the project ‘Genius before Romanticism:
Ingenuity in Early Modern Art and Science’. Without Gaenor
Moore, who did the big kindness of (p.vi) obtaining many of
the final images, I might have forgotten many things, including
to say this: the research leading to these results has received
funding from the European Research Council under the
European Union’s Seventh Framework Programme (FP7/2007–
2013)/ERC grant agreement no. 617391. Not least, I must
thank the Maire of Sélestat, and Mr Laurent Naas of the
Bibliothèque Humaniste de Sélestat, for permission to use
images of annotations by Beatus Rhenanus. Finally, I am
grateful to Kavya Ramu, Howard Emmens, and Andrew
Hawkey for so carefully seeing these sometimes recalcitrant
pages through the press.
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List of Illustrations
4.2. Aristotle’s and Plato’s modes of philosophizing:
Lefèvre, Libri logicorum (1503), BHS K1047, endpage.
Beatus contrasts the resolutive and compositive modes
of philosophizing, portrayed as variants of the
Porphyrian tree. The translation does not include the
diagrams of substances, which fill in layers of colour for
different composites: black for a substance with body,
or bodily nature; blue represents the addition of life, or
animate nature; red for the addition of soul, or sensible
nature; and white for rational nature. The descent from
universals is identified with Plato and the order of
nature (left, moving from substance to the individual
Plato and friends), and the ascent from particulars with
Aristotle and the order of cognition (right, moving from
individuals towards substance). 99
4.3. The table (formula) introducing Lefèvre’s Epitome
Boetii (1496), listing all the key terms of number theory
from broadest headings down to narrowest
subcategories. Note how each term is next to an
example, so that a numerus multiplex triplus is shown
as a ratio of 3:1. Lefèvre, Elementa arithmetica, etc.
(1496), h8r. By kind permission of the Syndics of
Cambridge University Library, Shelfmark: Inc.3.D.1.21.
110
4.4. The figura introductionis which outlines the basic
structure of sublunar physics in Aristotelian natural
philosophy, with major headings and minor
subdivisions. Lefèvre, Paraphrases philosophiae
naturalis (1492), b2r. By permission of the British
Library, Shelfmark IA.40121. 112
4.5. Typography of a textbook: list of theses, linked by
marginal numbers to discursive arguments in
paraphrases of Aristotle (see Figure 4.6). Lefèvre,
Paraphrases philosophiae naturalis (1492), a2r. By
permission of the British Library, Shelfmark IA.40121.
114
(p.x) 4.6. Typography of a textbook: a numbered
thesis. Lefèvre, Paraphrases philosophiae naturalis
(1492), b5r. By permission of the British Library,
Shelfmark IA.40121. 115
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List of Illustrations
4.7. Epitome of arithmetic. The formula of core
propositions collated from Boethius and Jordanus
shares a goal with the table from Ramus shown in
Figure 4.8: to epitomize a discipline. Lefèvre, Elementa
arithmetica, etc. (1496), i5r. By permission of the
British Library, Shelfmark C.106.a.8. 118
4.8. Epitome of dialectic: Peter Ramus, Dialecticae
institutiones (Paris: Jacques Bogard, 1543), 57r. By
permission of the British Library, Shelfmark C.106.a.8.
119
5.1. The ‘studious athlete’ in Bovelles, Liber de
intellectu, etc. (1511), 60v. Note the lines between
lectio, scriptio, and vision, which together with hearing
and speaking fill the scholar’s starry imagination. By
kind permission of the Syndics of Cambridge University
Library, Shelfmark: Acton.b.sel.32. 126
5.2. A typical page in Lefèvre’s Textus de sphera: table,
text, commentary (in smaller type), with diagrams and
section numbers both printed outside the forme.
Lefèvre, Textus de sphera (1495 [here 1516]), a5v. ©
Fitzwilliam Museum, University of Cambridge. 136
5.3. Frontispiece from Sacrobosco, Sphaera mundi
(Venice: Ottaviano Scoto, 1490). © Fitzwilliam
Museum, University of Cambridge. 138
5.4. Frontispiece from Lefèvre, Textus de sphera (Paris:
Estienne, 1516), a3v, reusing the woodblock from the
first edition of 1495. © Fitzwilliam Museum, University
of Cambridge. 139
5.5. The exaggerated realism of Sacrobosco, Sphaera
mundi (1490), a8r. © Fitzwilliam Museum, University of
Cambridge. 140
5.6. The introductoria additio in Lefèvre, Textus de
sphera (Paris: Estienne, 1500), BHS K 1046c, a2v–a3r.
Lefèvre’s primer explains the geometrical objects
needed to understand astronomy, as contemporary
editions of Sacrobosco sometimes also did (right). Here
Lefèvre goes further, explaining how to perform
sexagesimal arithmetic, which is the subject of Beatus
Rhenanus’ extensive notes on the facing page (left).
142–3
5.7. Illustration of a sphere on a lathe, with a semi-
circular cutting tool. Lefèvre, Textus de sphera (Paris:
Estienne, 1516), a4r. © Fitzwilliam Museum, University
of Cambridge. 145
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List of Illustrations
5.8. Lefèvre shows how to divide any interval equally,
‘without a certain or fixed ratio of number’. From his
Elementa arithmetica, musicalia, etc. (1496), G6v,
proposition III.35. By kind permission of the Syndics of
Cambridge University Library, Shelfmark: Inc.3.D.1.21.
158
5.9. Beatus Rhenanus records the performance of
calculations of the distances between the various
planetary spheres: on the bottom of the page, he
records various ‘modern’ opinions on the number of
these heavenly spheres—the omission of what is
evidently ‘Regiomontanus’ in the first line suggests he
missed the name during dictation and intended to fill it
in later. Detail from Beatus Rhenanus’ copy of Lefèvre,
Textus de sphera (1500), BHS K 1046c, a1v. 166
(p.xi) 5.10. Beatus draws a table for converting units
of measurement. Detail from Beatus Rhenanus’ copy of
Lefèvre, Textus de sphera (1500), BHS K 1046c, a2r.
167
5.11. Beatus recalls the terminology of Greek music
theory: BHS K 1046, end papers. 168
5.12. Beatus and the ‘Lullian pyramid’: BHS K 1046,
end papers. 169
5.13. Bovelles links cognitive modes with objects in the
world. Bovelles, De intellectu, etc. (1511), 28v. A page
later, Bovelles offers a cone of the senses that matches
the cone of reality, which contracts into God. Bovelles,
De intellectu, etc. (1511), 29v. By kind permission of the
Syndics of Cambridge University Library, Shelfmark:
Acton.b.sel.32. 170
5.14. Beatus ruminating on a series of triangular
numbers. BHS K 1046a, 20r. 172
5.15. Beatus brings together series to test a rule of
inference. BHS K 1046, end page, detail. (This is the
bottom of the page shown above in Figure 5.12.) 173
5.16. Beatus classifies forms of ‘proportionality’
between lines, surfaces, and bodies. Lefèvre, Clichtove,
Bovelles, Epitome Boetii, etc. (1503), BHS K 1046a, 81v.
177
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List of Illustrations
5.17. The point of analogy in Beatus’ notes is to justify
inferences from one domain to another. BHS K 1046a,
6v, detail of a schema between figurate numbers and
geometrical forms. Inset: BHS K 1046, end page, detail
of the schema for an analogy between geometrical
forms and material objects. 178
6.1. How to visualize past, present, and future along a
continuum. Lefèvre, Totius philosophiae naturalis
paraphrases (1492), M3r. By permission of the British
Library, Shelfmark IA.40121. 187
6.2. The behaviour of mathematical objects in physics.
Sequence of diagrams illustrating the behaviour of
points in dividing lines, lines in dividing surfaces, and
surfaces in dividing bodies. Detail from Lefèvre and
Clichtove, Totius philosophae naturalis paraphrases
(1502), 134v. FC5 L5216 502t. Houghton Library,
Harvard University. 193
6.3. Seeing density. In the margin, a line a–k helps
visualize a ten foot object, then compressed into line l,
to illustrate the concept of density. Detail from Lefèvre
and Clichtove, Totius philosophae naturalis paraphrases
(1502), 136v. FC5 L5216 502t. Houghton Library,
Harvard University. 194
6.4. The mathematically absurd and the physically
impossible. The argument ad absurdum visualizes why
qualities can only be mixed within an absolute
maximum. Note the references to gradus and the
Pythagorean denarium in the margin. Lefèvre and
Clichtove, Totius philosophiae naturalis paraphrases
(1502), 147r. FC5 L5216 502t. Houghton Library,
Harvard University. 196
7.1. God as a circle, enfolding the creation, in which
creatures revolve from centre to circumference.
Lefèvre, Quincuplex Psalterium (1509), 185r. By kind
permission of the Syndics of Cambridge University
Library, Shelfmark: Adams.4.50.6. 216
(p.xii)
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Conventions and Abbreviations
University Press Scholarship Online
Oxford Scholarship Online
Making Mathematical Culture: University and
Print in the Circle of Lefèvre d'Étaples
Richard Oosterhoff
Print publication date: 2018
Print ISBN-13: 9780198823520
Published to Oxford Scholarship Online: September 2018
DOI: 10.1093/oso/9780198823520.001.0001
(p.xiii) Conventions and
Abbreviations
Richard Oosterhoff
Conventions
All translations, unless indicated differently, are my own. Titles
of Latin books are left in the original, unless they seemed
more commonly familiar in English translation.
For the sake of readability, I have regularized Latin
transcriptions, notably capitalization, i/j, u/v, punctuation, and
I have silently expanded abbreviations. I have counted
signatures in arabic numerals, so that signature e, folio IV,
verso is e4v.
Dates have been normalized to modern standards. Before the
gradual adoption of the Gregorian calendar (in France in
1582, in England not fully until 1752) the legal year normally
began on 25 March. Therefore a date given as February 1494 I
record as February 1495.
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Conventions and Abbreviations
Jacques Lefèvre d’Étaples (ed.), Euclidis
Megarensis mathematici clarissimi Elementorum
geometricorum libri xv, Campani Galli transalpini in
eosdem commentariorum libri xv, Theonis
Alexandrini (p.xiv) Bartholamaeo Zamberto
Veneto interprete, in tredecim priores,
commentariorum libri xiii, Hypsiclis Alexandrini in
duos posteriores, eodem Bartholomaeo Zamberto
Veneto interprete, commentariorum libri ii (Paris:
Henri Estienne, 1517)
Liber de intellectu etc.
Charles de Bovelles, Que hoc volumine continentur:
Liber de intellectu; Liber de sensu; Liber de nichilo;
Ars oppositorum; Liber de generatione; Liber de
sapiente; Liber de duodecim numeris; Epistole
complures. Insuper mathematicum opus
quadripartitum: De numeris perfectis; De
mathematicis rosis; De geometricis corporibus; De
geometricis supplementis (Paris: Henri Estienne,
1511)
Libri logicorum
Jacques Lefèvre d’Étaples, Libri logicorum ad
archteypos recogniti cum novis ad litteram
commentariis ad felices primum Parhisiorum et
communiter aliorum studiorum successus in lucem
prodeant ferantque litteris opem (Paris: Wolfgang
Hopyl and Henri Estienne, 1503)
PE
Eugene F. Rice Jr. (ed.), The Prefatory Epistles of
Jacques Lefèvre d’Étaples and Related Texts (New
York: Columbia University Press, 1972)
Politica etc.
Jacques Lefèvre d’Étaples, Contenta. Politicorum
libro octo. Commentarii. Economicorum duo.
Commentarii. Hecatonomiarum septem.
Economiarum publ. unus. Explanationis Leonardi
[Bruni] in Oeconomica duo (Paris: Henri Estienne,
1506)
Renaudet
Augustin Renaudet, Préréforme et humanisme à
Paris pendant les premières guerres d’Italie, 1494–
1517 (1916; 2nd edn, Paris: Édouard Champion,
1953)
Textus de sphera
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Conventions and Abbreviations
Jacques Lefèvre d’Étaples, Textus de sphera
Johannis de Sacrobosco, cum additione (quantum
necessarium est) adiecta: novo commentario nuper
edito ad utilitatem studentium philosophice
parisiensis academie: illustratus (Paris: Wolfgang
Hopyl, 1495)
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Introduction
University Press Scholarship Online
Oxford Scholarship Online
Making Mathematical Culture: University and
Print in the Circle of Lefèvre d'Étaples
Richard Oosterhoff
Print publication date: 2018
Print ISBN-13: 9780198823520
Published to Oxford Scholarship Online: September 2018
DOI: 10.1093/oso/9780198823520.001.0001
Introduction
Richard Oosterhoff
DOI:10.1093/oso/9780198823520.003.0001
Abstract and Keywords
Jacques Lefèvre d’Étaples is now best known as an Aristotelian
humanist and a founder of the French Reformation. In his day,
however, Lefèvre was at the centre of a circle of scholars
invested in university reform, then widely known for their
interest in mathematics. Among his closest collaborators, first
as students and then as university masters, were Josse
Clichtove and Charles de Bovelles. After outlining the
development of a new mathematical culture, this chapter
orients the reader to the circle’s collective biography around
the notion of friendship, and positions the book’s argument in
relation to the historiographies of mathematics, print, and the
university.
Keywords:   Jacques Lefèvre d’Étaples, Josse Clichtove, Charles de Bovelles,
friendship, universities, mathematics, print
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