Leibniz on the Parallel Postulate and the Foundations of Geometry: The Unpublished Manuscripts

199 Pages·2016·1.541 MB·English

by  De Risi, Vincenzo, Leibniz, Gottfried Wilhelm

Science Networks
Historical Studies
51
Vincenzo De Risi
Leibniz on the
Parallel Postulate
and the Foundations
of Geometry
The Unpublished Manuscripts
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Vincenzo De Risi
Leibniz on the
Parallel Postulate
and the Foundations
of Geometry
The Unpublished Manuscripts
Vincenzo De Risi
MPI for the History of Science
Berlin, Germany
ISSN 1421-6329 ISSN 2296-6080 (electronic)
Science Networks. Historical Studies
ISBN 978-3-319-19862-0 ISBN 978-3-319-19863-7 (eBook)
DOI 10.1007/978-3-319-19863-7
Library of Congress Control Number: 2015960187
Mathematics Subject Classification 2010: 01-02, 01A45, 01A50, 02-03, 51-03
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Foreword
In this book, I provide an edition of Leibniz’ writings on the theory of parallels,
which include several attempts to prove the Parallel Postulate. A few of these papers
were published by Leibniz himself in his lifetime, while others were only printed in
modern editions of his mathematical works. The most important essays, however,
were still unpublished, and I have transcribed them from the Leibnizian manuscripts
in the Niedersächsische Landesbibliothek in Hannover. Given the enormous amount
of Leibnizian papers preserved in the Leibniz-Archiv, I cannot claim to have found
all the relevant material, and to have a complete picture of Leibniz’ endeavors in this
direction we have to wait for the publication of the related volumes by the Academy
Edition of Leibniz’ Werke; since however the scholars at the Leibniz-Archiv have
just begun the first surveys of Leibniz’ geometrical writings after 1676, it is likely
that a full edition will require many years. In any case, I am confident that the pres-
ent collection of papers on the theory of parallels is comprehensive enough to give
a quite good idea of Leibniz’ work in this field.
Most of the texts presented here are sections and paragraphs of longer essays on
the foundations of geometry, while a few others are self-contained notes and remarks
on the Parallel Postulate that Leibniz penned from time to time. Given the highly
fragmentary character of these drafts and private notes, their meaning and signifi-
cance may easily be missed and reading them requires a careful study of Leibniz’
intellectual development and environment. To this end, I introduce them with an
essay commenting the most relevant passages and outcomes, while dealing with
the history of the attempts to prove the Parallel Postulate at the time of Leibniz, the
main epistemological tenets of Leibniz’ philosophy of geometry, and the historical
reception of Leibniz’ ideas on the subject. On the one hand, my introductory essay
is strictly related to my previous book Geometry and Monadology, published in this
Birkhäuser series in 2007; and while the latter book dealt with Leibniz’ philosophy
of space and metaphysical foundations of geometry, the present essay complements
those researches expounding Leibniz’ geometrical epistemology (albeit from a very
specific perspective). On the other hand, this volume may also be read in connection
with my commented editions of Saccheri and Lambert on the theory of parallel lines
(both published by Birkhäuser), and the three books together offer a comprehensive
account of the prehistory of non-Euclidean geometry in the eighteenth century.
I would like to thank the Leibniz-Archiv and the Niedersächsische Landesbib-
liothek for allowing me to read, transcribe and publish Leibniz’ manuscripts on the
v
vi Foreword
theory of parallels. My deepest gratitude goes to Siegmund Probst, whose help in
finding and deciphering Leibniz’ papers was invaluable for the present edition.
I began to work on Leibniz’ theory of parallels in 2009, while I was Alexander
von Humboldt Fellow at the Technische Universität Berlin. I would like to thank
the Alexander von Humboldt Stiftung for financial support, and my generous host
in Berlin, Eberhard Knobloch, who also carefully read and commented on the first
draft of this book. His suggestions and advice saved me from several mistakes and
considerably enhanced the final version. I am also very grateful to Richard Arthur,
Gideon Freudenthal, Mattia Mantovani, and Victor Pambuccian, whose illuminating
remarks on further drafts of the volume were crucial to my understanding of several
passages.
My studies were presented and discussed in a few seminars from 2010 onwards,
in Paris, Hannover, Pisa, Urbino, Leipzig, Ghent, and Princeton, and I am grate-
ful to all the participants who helped me in understanding Leibniz’ mathematics
and epistemology; in particular, I mention here Herbert Breger, Daniel Garber, Tal
Glezer, Pierluigi Graziani, Jürgen Jost, Massimo Mugnai, Enrico Pasini, Francesco
Piro, and David Rabouin, whose comments and remarks substantially improved the
present study.
Finally, I would like to thank Fred Sengmueller and James Garahan for a linguistic
revision of the manuscript, David Merry for having helped me with the translation
of Leibniz’ texts, and Chiara Fabbrizi for the general editing.
This book is dedicated to my mother Laura, whose unfailing care made everything
possible.
Berlin, January 2015
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
2 The Theory of Parallel Lines in the Age of Leibniz . . . . . . . . . .7
2.1 From Antiquity to the Renaissance.. . . . . . . . . . . . . . . . .7
2.2 Jesuit and French Attempts in the Seventeenth Century. . . . . . 11
2.3 Italian and British Attempts in the Seventeenth Century. . . . . . 14
3 Leibniz’ Epistemology of Geometry and the Parallel Postulate. . . 21
3.1 Proving Axioms. . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Leibniz’ Theory of Definition. . . . . . . . . . . . . . . . . . . 31
3.3 Geometry as the Science of Space. . . . . . . . . . . . . . . . . 40
3.4 Philosophy and the Parallel Postulate. . . . . . . . . . . . . . . 49
4 Leibniz’ Attempts to Prove the Parallel Postulate . . . . . . . . . . 57
4.1 Preliminary Definitions.. . . . . . . . . . . . . . . . . . . . . . 57
4.2 Leibniz’ Youth and the Parisian Years (Texts 1–4). . . . . . . . . 65
4.3 The First Studies on the Parallel Postulate: 1677–1689
(Texts 5–16). . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Leibniz’ Studies in the 1690s (Texts 17–29 and 34). . . . . . . . 80
4.5 Leibniz’ Late Studies (Texts 30–39). . . . . . . . . . . . . . . . 91
5 Reception and Legacy . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1 Toward a Geometry of Space in the Eighteenth Century. . . . . 103
5.2 Wolff, Kästner and the Early Reactions. . . . . . . . . . . . . 105
5.3 The Concept of Lage, Direction, and the Actual Infinite. . . . . 109
5.4 The Principle of Reason. . . . . . . . . . . . . . . . . . . . . 115
6 Leibniz’ Texts on Parallel Lines. . . . . . . . . . . . . . . . . . . 125
6.1 From Characteristica geometrica. Reading Notes to Fabri’s
“Synopsis geometrica” (1673) . . . . . . . . . . . . . . . . . 125
6.2 From De secandis parallelis (1676). . . . . . . . . . . . . . . 125
6.3 From Linea infinita & extensio interminata
(January–April 1676) . . . . . . . . . . . . . . . . . . . . . . 127
6.4 From Dissertationis de arithmetica circuli quadratura
propositiones septem and Quadraturae circuli arithmetica
pars prima (April–June 1676). . . . . . . . . . . . . . . . . . 129
6.5 From Leibniz’ Marginal Notes to Barrow’s Edition of Euclid
(1676–1677). . . . . . . . . . . . . . . . . . . . . . . . . . . 129
vii
viii Contents
6.6 From Characteristica geometrica (January 1677) . . . . . . . 129
6.7 From Characteristica geometrica, Scheda 1 (1679) . . . . . . 129
6.8 From Linea est via puncti … (1679) . . . . . . . . . . . . . . 131
6.9 Axioma 13 Euclidis … (Around 1679) . . . . . . . . . . . . . 131
6.10 From De calculo algebraico et constructiones lineares
optime conciliandis (January 1680). . . . . . . . . . . . . . . 133
6.11 From Elementa nova matheseos universalis (Summer 1683) . . 135
6.12 From Definitiones (1685) . . . . . . . . . . . . . . . . . . . . 137
6.13 From Si linea recta moveatur … (1685). . . . . . . . . . . . . 139
6.14 From De curvis similibus et similiter positis et parallelis (1685). 139
6.15 From Logica de notionibus. Annotata circa schedas
Jungianas (1685) . . . . . . . . . . . . . . . . . . . . . . . . 141
6.16 From Phoranomus (1689). . . . . . . . . . . . . . . . . . . . 143
6.17 From Demonstrationes Euclideas (Around 1690). . . . . . . . 143
6.18 Parallelae (1690) . . . . . . . . . . . . . . . . . . . . . . . . 143
6.19 From Generalia de natura linearum
(Acta eruditorum, September 1692). . . . . . . . . . . . . . . 145
6.20 Rectae parallelae (around 1692) . . . . . . . . . . . . . . . . 145
6.21 From Lineam specie datam describere (Around 1693) . . . . . 147
6.22 From Leibniz’ Notes to Arnauld’s Nouveaux elemens
de geometrie (Around 1693) . . . . . . . . . . . . . . . . . . 147
6.23 From Elementa geometriae generalia (Around 1694) . . . . . 153
6.24 From De novo usu centri gravitatis
(Acta eruditorum, November 1695). . . . . . . . . . . . . . . 153
6.25 From Expediendi Laboris causa … (Around 1695). . . . . . . 153
6.26 From Duae rectae parallelae … (1695). . . . . . . . . . . . . 155
6.27 Leibniz to Johann Bernoulli (January 29th/February 8th, 1697) . . 157
6.28 From Specimen analyseos anagogicae (1698) . . . . . . . . . 159
6.29 From Justification du Calcul des infinitesimales par celuy
de l’Algebre ordinaire (1702). . . . . . . . . . . . . . . . . . 159
6.30 From Attentius examinans … (Around 1702) . . . . . . . . . . 159
6.31 From Calculum situs (Around 1702) . . . . . . . . . . . . . . 161
6.32 From Table de définitions (1702–1704). . . . . . . . . . . . . 165
6.33 From Nouveaux Essais sur l’entendement humain ,
II, xxxi, § 3; and III, iii, § 18 (1703–1705). . . . . . . . . . . 165
6.34 From De lineae super linea incessu (Acta eruditorum,
January 1706) , and Letter to Johann Bernoulli (July 15th, 1706) . 167
6.35 From Demonstratio omnimoda … (1712). . . . . . . . . . . . 167
6.36 From In Euclidis πρῶτα (1712) . . . . . . . . . . . . . . . . . 169
6.37 From Spatium absolutum … (August–September 1714) . . . . 179
6.38 From Rectam definio … (1715) . . . . . . . . . . . . . . . . . 179
6.39 From Calculi situs fundamenta … (1715). . . . . . . . . . . . 179
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
I Leibniz on the Parallel Postulate
and the Foundations of Geometry