A commentary on the first book of Euclid’s Elements. Translated, with introduction and notes, by Glen R. Morrow.

406 Pages·1970·16.947 MB·English

by  Proclus

PROCLUS
A COMMENTARY ON THE
FIRST BOOK OF
EUCLID’S ELEMENTS
PROCLUS
A COMMENTARY ON THE
FIRST BOOK OF
EUCLID’S ELEMENTS
Translated
with Introduction and Notes by
GLENN R. MORROW
Adam Seybert Professor Emeritus
of Moral and Intellectual Philosophy
University of Pennsylvania
PRINCETON UNIVERSITY PRESS 1970
PRINCETON, NEW JERSEY
Copyright © 1970 by Princeton University Press
All Rights Reserved
L.C. Card: 73-90955
ISBN 0-691-07160-8
This book has been composed in Linotype Times Roman
Printed in the United States of America by
Princeton University Press
Princeton, New Jersey
TO MY WIFE
τν Ψνχν Λου
151063

PREFACE
Hellenic intelligence and precision, together with a Platonic devo¬
tion to mathematics as the liberating science, he is uniquely
qualified to transmit to us something of this Hellenic and Platonic
enthusiasm.
To this I must add my personal conviction that the student of
Plato’s later thought, if he hopes ever to comprehend the part that
mathematics played in the development of Plato’s thinking, and
particularly the puzzling statements of Aristotle regarding the latest
form of the theory of Ideas, must enter with sympathy and under¬
standing into this climate of thought created by the mathematicians
of Plato’s and the following centuries. As a devout Platonist,
Proclus is naturally a resource of the first order for enabling us to
do this.
The purpose of this book, then, is to make available to English
readers a treatise of unique value for the history of mathematics
and of philosophy. Like the mathematics with which it deals, it has
universally been treated with respect, and it has often been cited
as evidence for this or that fact in the history of mathematics; but
it has seldom been looked at as a whole, chiefly because of the
obstacles presented by a Greek text of such length. These obstacles,
I venture to say, are appreciated even by those who can read
Proclus’ language with facility; and for those who cannot they are
insuperable barriers. I hope this translation will help to overcome
these hindrances to acquaintance with a treatise of outstanding
quality, almost the only one from antiquity that deals with what
we today would call the philosophy of mathematics.
My notes do not presume to be a full commentary on the text.
I have not essayed to deal with all the mathematical questions pre¬
sented, still less with the issues involved in Proclus’ excursions into
Neoplatonic ontology and cosmology. What I have tried to do in
them is to provide the explanations essential to understanding my
translation (and the text of Friedlein on which it is based), to
identify the passages in previous writers to which Proclus refers, to
characterize briefly the personages that figure in his historical
comments, and to furnish some preliminary aids to the further
exploration of the philosophical and mathematical contents so
richly deployed before us. I have labored to make my translation
as clear a presentation of Proclus’ thought as my powers and
— viii —
PREFACE
insight permitted; yet those who make use of it will certainly
find that there is still much to explore.
In addition to my debt to previous translators and scholars,
which is evident in my footnotes, I acknowledge with grateful
appreciation my indebtedness to my colleague at the University of
Pennsylvania, Charles H. Kahn, for constant advice and encourage¬
ment; to Robert S. Brumbaugh, of Yale University, who read my
manuscript for the Princeton University Press and made many
helpful suggestions, in particular calling attention to the significance
of Iamblichus’ De Communi Mathematica Scientia\ and to Ian
Mueller, of the University of Chicago, who read the manuscript
in its entirety, saved me from many errors, and made numerous
pertinent comments on the text, some of which I have presented
in my notes, with his initials “I.M.” to identify them.
To Mr. Sanford G. Thatcher, Social Science Editor of the
Princeton University Press, I am greatly indebted for the assiduous
attention he has given to this manuscript, and for the advice and
assistance he has so generously rendered in editing it and in over¬
seeing its progress through the Press. To him and to his associates
I hereby express my warm and sincere thanks. Mrs. Georgia
Minyard, of the University of Pennsylvania, has also given me much
competent help in editing and proofreading, for which I am most
grateful.
Glenn R. Morrow
Swarthmore, Pennsylvania
May 30, 1969
■— IX —
Table of Contents
Preface vii
Abbreviations xii
Introduction xv
Proclus: His Life and Writings xv
Euclid and the Elements xxi
Proclus as a Commentator on Euclid xxiv
Proclus’ Philosophy of Mathematics xxxii
Translator’s Note xliv
The Commentary
Prologue: Part One 3
Prologue: Part Two 39
Definitions 7 0
Postulates and Axioms 140
Propositions: Part One 156
Propositions: Part Two 276
Supplementary Note 344
Index 347
— xi —
Abbreviations
The following abbreviations have been used in the footnotes for
authors and works frequently cited.
Barocius = Procli Diadochi Lycii in Primum Euclidis Elementorum
Commentariorum Libri IV a Francisco Barocio Patritio Veneto
Editi, Padua, 1560.
CAG = Commentaria in Aristotelem Graeca, 23 vols., Berlin,
1882-1909.
Diels6 = Hermann Diels and Walther Kranz, Fragmente der Vor-
sokratiker, 6th edn., Berlin, 1951-1952.
Dodds = E. R. Dodds, Proclus: The Elements of Theology, 2nd
edn., Oxford, 1963.
Friedlein = Procli Diadochi in Primum Euclidis Elementorum
Librum Commentarii ex Recognitione Godofredi Friedlein,
Leipzig, 1873.
Gow = James Gow, History of Greek Mathematics, Cambridge,
1884; reprinted, New York, 1923.
Grynaeus = Commentariorum Procli editio prima quae Simonis
Grynaei opera addita est Euclidis dementis graece editis, Basel,
1533.
Heath = Thomas Heath, Greek Mathematics, 2 vols., Oxford,
1921.
Heath, Euclid = Thomas Heath, The Thirteen Books of Euclid’s
Elements, Translated from the text of Heiberg with Introduction
and Commentary, 2nd edn., 3 vols., Cambridge, 1926.
Heiberg = Euclidis Elementa edidit et Latine interpretatus est /. L.
Heiberg, 5 vols., Leipzig, 1883-1888. Supp. Vol. vi (1896), ed.
H. Menge.
Kroll = Wilhelm Kroll, De Oraculis Chaldaicis, Breslau, 1894.
RE = Pauly-Wissowa-Kroll, Realencyclopddie der Classischen
Altertumswissenschaft, Stuttgart, 1894-1952.
Rosan = Lawrence J. Rosan, The Philosophy of Proclus, New
York, 1949.
Schonberger = Leander Schonberger, Proklus Diadochus: Kom-
mentar zum Ersten Buch von Euklids Elementen, a translation
— xii —
ABBREVIATIONS
into German, with Introduction, Commentary, and Notes by
Max Steck, Halle, 1945.
Tannery = Paul Tannery, Mimoires Scientifiques, 17 vols., Paris,
1912-1950.
Taylor = Thomas Taylor, The Philosophical and Mathematical
Commentaries of Proclus on the First Book of Euclid’s Elements,
2 vols., London, 1788, 1789.
Van der Waerden = B. L. Van der Waerden, Science Awakening,
New York, 1961.
Ver Eecke = Paul ver Eecke, Proclus de Lycie: Les Commentaires
sur le Premier Livre des Elements d’Euclide, a translation into
French, with an Introduction and Notes, Bruges, 1949.
Von Arnim = Ioannes von Arnim, ed., Stoicorum Veterum Frag-
menta, 4 vols., Leipzig, 1903-1924.
Xlll —