The Geometrical Work of Girard Desargues

244 Pages·1987·11.14 MB·English

by  J. V. Field

The Geometrical Work of Girard Desargues
The Geometrical Work
of Girard Desargues
J. V. Field
J. J. Gray
With 69 Illustrations
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
J. V. Field J. J. Gray
Science Museum The Open University
London SW7 2DD Walton Hall
England Milton Keynes MK7 6AA
England
AMS Classifications: 51-03, 01A40, OlA45, OlA75
Library of Congress Cataloging in Publication Data
Field, Judith
The geometrical work of Girard Desargues.
Bibliography: p.
Includes index.
1. Conics-Early works to 1800. 2. Perspective
Early works to 1800. 3. Desargues, Gerard, 1591-1661.
I. Gray, Jeremy. II. Title. III. Series.
QA485.G73 1987 516'.15 86-17748
© 1987 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1987
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Typeset by Mid-County Press, London.
9 8 7 6 543 2 1
ISBN -13:978-1-4613-8694-0 e-ISBN -13:978-1-4613-8692-6
DOl: 10.1007/978-1-4613-8692-6
Preface
Our main purpose in this book is to present an English translation of
Desargues' Rough Draft of an Essay on the results of taking plane sections of a
cone (1639), the pamphlet with which the modem study of projective geometry
began. Despite its acknowledged importance in the history of mathematics,
the work has never been translated before in its entirety, although short
extracts have appeared in several source books.
The problems of making Desargues' work accessible to modem
mathematicians and historians of mathematics have led us to provide a fairly
elaborate introduction, and to include translations of other relevant works.
The translation ofthe Rough Draft on Conics (as we shall call it) thus appears
in Chapter VI, the five preceding chapters forming an introduction and the
three following ones giving translations of other works by Desargues.
Chapter I briefly reviews parts of ancient geometrical works available to
Desargues which seem to be relevant to his own work, namely theorems in
Euclid's Elements, the first four books of Apollonius' Conics and some
remarks by Pappus in his Collection. These Hellenistic works belong to the
'high' mathematical tradition whose development has been the main theme of
all histories of mathematics. It is from these works that Desargues took the
theorems whose theory he was to reformulate in the Rough Draft on Conics.
However, although he was well read in the works of this 'high' mathematical
tradition, all Desargues' works other than the Rough Draft on Conics belong
to the 'low' mathematical tradition of practical geometry. For instance, he
wrote on such matters as drawing in perspective and setting up the gnomon of
a sundial. Moreover, he earned his living in professions much concerned with
practical mathematics, working as a military engineer and as an architect. It is
thus not surprising to find that the concerns of the 'low' mathematical
tradition seem to have played a significant part in the thinking behind the
Rough Draft on Conics. Since this 'low' mathematical tradition has received
relatively little attention from historians, our account of it, in Chapter II, is
somewhat more detailed than our summary of 'high' mathematics in Chapter I.
VI Preface
In Chapter III we discuss the reception accorded to the Rough Draft on
COllics by mathematicians of Desargues' own generation and of succeeding
generations up to that of Poncelet. Chapter IV provides a detailed outline of
the mathematical content of the Rough Draft on Conics, using modern
notation. Several passages in our translation of the work, in Chapter VI, have
been keyed to the summary in Chapter IV, so that the reader can follow
Desargues' reasoning without necessarily having to work through the treatise
line by line. The final introductory chapter, Chapter V, is the translators'
preface, which is mainly concerned with linguistic and textual problems. It
includes lists of the technical terms used in the Rough Draft on Conics
(Desargues' famous botanical vocabulary) and in his Perspective of 1636, the
terms being given in both French and English, This preface is followed by the
translation of the Rough Draft on Conics, in Chapter VI. The footnotes
labelled by a, b, c, etc, which are intended to provide a running commentary,
are our own. So too are the illustrations which accompany the text,
Desargues' own illustrations apparently being irretrievably lost. Chapter VII
contains a translation of Desa rgues' treatise on perspective of 1636. This work
has never been published in English before. (The corresponding French text is
given in Appendix 5.) Chapter VIII contains translations of the three
geometrical propositions by Desargues which were printed at the end of
Abraham Bosse's expanded account of Desargues' perspective method in
1648. The first of these propositions is Desargues' famous and beautiful
theorem on two triangles in perspective. Chapter IX gives a translation of
Desargues' newly rediscovered work on sundials, first published in 1640. The
diagrams (in the Notes to Chapter IX) are our own, since the single known
copy of the work contains only text.
Several shorter items have been added as appendices. They are Descartes'
interesting, and apparently very carefully phrased, verdict on Desargues'
Rough Draft on Conics (Appendix 1), Beaugrand's defence of classical methods
(Appendix 2), Blaise Pascal's Essay on Conics (Appendix 3), a note on
Johannes Kepler's invention of points at infinity in 1604 (Appendix 4), and
finally, as already noted, the French text of Desargues' perspective treatise of
1636 (Appendix 5). This treatise has not been reprinted since it appeared in a
much emended version in Poudra's edition of Desargues' works in 1864.
Our bibliography is not intended to be exhaustive, but includes all primary
and secondary sources cited in our text.
It will probably be clear from the present authors' separate publication
records that in several chapters of this book the work of one or other author
predominates. We should, however, like to point out that neither of us,
working alone, could have produced what either of us would have regarded as
an adequate translation of the work with which we are chiefly concerned, the
Rough Draft on Conics. We hope that together we add up to a Desargues
scholar.
April 1986 J. V. F.
J. J. G.
Acknowledgements
We are grateful to Dr Jan Hogendijk for his detailed critical reading of an
earlier draft of our typescript from the point of view of a historian of
mathematics. We believe our work has benefited considerably from his
attention. Our study of linear perspective, in Chapter II, led us into the
territory of the art historian. Here we are grateful for the professional advice
and guidance kindly given us by Professor Martin J. Kemp, whose
forthcoming book, The Science of Art, sets perspective in a wider artistic
context than we have attempted to describe and gives an account that is not
only more detailed than our own mathematical sketch but also largely
complementary to it.
We are grateful to Dr S. S. Demidov and Professor A. P. Yushkevich for the
interest they showed in our work when a summary of it was presented (by
J. J. G.) at a seminar in the history of mathematics at Oberwolfach. This
interest led to the appearance of a Russian version of the lecture in I storiko
matematicheskie issledovaniya, XXIX (1985).
The text of Desargues' treatise on sundials of 1640 was believed lost until
Anthony J. Turner discovered a copy of it in 1983. We are grateful to Mr
Turner not only for giving us a photocopy of the work before his discovery
was published, but also for allowing us to include a translation of the treatise
in the present work (see Chapter IX).
Photographic credits are as follows: Bibliotheque nationale, Paris (App. 5);
British Library Board (Figs 8.1 to 8.4); Metropolitan Museum of New York
(Figs 7.1 and 7.2); Trustees of the Science Museum, London (Figs 2.6, 7.3 to
7.6 and A4.1); P. J. Booker, A History of Engineering Drawing, Northgate
Publishing, London, 1979 (Fig. 9.1), reproduced with permission.
Contents
Chapter I
The Greek Legacy 1
Chapter II
Applied Geometry 14
Chapter III
Mathematical Responses to Desargues' Rough Draft on Conics 31
Chapter IV
The Mathematical Content of the Rough Draft on Conics 47
Chapter V
Translators' Preface 60
Chapter VI
The Rough Draft on Conics (1639) 69
Chapter VII
The Perspective (1636) 144
Chapter VIII
The Three Geometrical Propositions of 1648 161
Chapter IX
The Sundial Treatise (1640) 170
Appendix 1
Letter from Descartes to Desargues (19 June 1639) 176
Appendix 2
Letter from Beaugrand to Desargues (25 July 1639) 178
x Contents
Appendix 3
Pascal's Essay on Conics (1640) 180
Appendix 4
Kepler's Invention of Points at Infinity 185
Appendix 5
The French Text of Desargues' Perspective (1636) 189
~otes 202
Bibliography 223
Index of the Technical Terms in Desargues' Rough Draft on Conics 233
Index 235
Chapter I
The Greek Legacy
When Desargues circulated fifty copies of his Brouillon project d'une
atteinte aux evenmens des rencontres du Cone avec un Plan (Rough Draft of an
Essay on the results of taking plane sections of a cone) in 1639, he was
contributing to a lively contemporary study of geometry. Descartes' novel
algebraic methods had been published two years before, and in 1639 Mydorge
published a more classical treatment of the conic sections. The classical
authors themselves were increasingly well studied. Desargues had available
Commandino's Latin edition of Euclid's Elements, published in 1572, as well
as his Latin edition of the first four books of Apollonius' Conics, published in
1566 with extensive commentaries by Eutocius, Pappus and Commandino
himself. The last four books of the Conics were unknown in Desargues' time.
Two editions of Pappus' Collection had also been published by Commandino
(posthumously) in 1588 and 1602. In this chapter we sketch what in these
ancient works forms the background to Desargues' remarkable text. The
mathematical details of his reformulation of those ideas is described in more
detail in Chapter IV. Ironically, the modern (Greek-less) English reader is in
some ways scarcely more able than Desargues was to approach the originals.
There is, of course, Heath's three-volume edition of Euclid's Elements (see
Bibliography for details). But only the first three books of Apollonius' Conics
exist in English, translated by R. C. Taliaferro in 1939; the first seven books
exist in the French translation of Ver Eecke, 1923. Heath provided an
extensive detailed commentary on the Conics in 1896. Finally, Book VII of
Pappus' Collection is only now translated into English, by A. Jones (see the
Bibliography); happily Ver Eecke put all of it that survives into French in 1933.
The Classical Geometry of Conic Sections
The ancient geometers presented not only a body of results, but a way of
deriving them and hence of formulating the basic concepts of geometry. Since
2 The Geometrical Work of Girard Desargues
some of these methods and ideas were accepted by Desargues while others
were deliberately rejected, it will be necessary to look at them both in a little
detail. We start by asking: What was geometry supposed to be about? The
answer is that it was about the concept of magnitude in a rather general and
elusive sense. Informally, it is clear that magnitudes like line segments, plane
figures and angles are the staple diet of geometry; but formally and
philosophically it is not easy to state what these concepts mean. What Euclid's
Elements embodies is one way of systematizing the study of them so that one
can reason deductively about them. While the approach he so successfully
presented in that book certainly does not resolve the philosophical problems,
it is more relevant for us to pursue the mathematical subtleties it contains. The
reader is referred to Mueller's book (1981) for a thorough discussion of many
aspects of the Elements; we shall be highly selective.
Euclid distinguished between the cOflcepts 'line segment' and 'length' in a
way one tends to blur today. To Euclid, one line segment is equal to another if
they can be made to coincide exactly, and one is shorter than another if it can
be made to coincide exactly with a piece of the second segment. Line
segments may be added (by juxtaposition) and subtracted (by inserting one in
the other). Indeed, any strict modern definition of length for line segments
would recognize that length is a function defined on the set of segments and
satisfying some obvious intuitive rules (invariant under motion, additive, etc.).
The point to grasp is that when, in the Elements, one segment is said to be
equal to another it means that they can be made to coincide with one another
exactly (as his fourth Common Notion says). It follows that they have the same
length; one does not first measure the lengths and deduce that the segments
are equal. To a Greek geometer the logical absurdity of such a manoeuvre
would have been obvious.
The same is true of area. The area of a figure is a primitive concept in the
Elements, not reducible to that of a product of lengths. Euclid did not show
two figures were equal in area by computing their areas, but by a dissection
and motion argument, as the famous proof of Pythagoras' theorem literally
shows. As is well known, the theorem says that if ABC is a triangle right
angled at A, then the sum of the squares on AB and AC is equal to the square
on BC. In Elements (I,47)-by which we mean Book I, Proposition 47-this is
proved as shown in Fig. 1.1.
The square on AC (ACKH) is equal in area to twice the area of triangle
BCK, since they are on the same base, CK, and between the same parallels.
Triangles BCK and ECA have the same area because they are congruent, and
that area is half the area of the rectangle CELP. So the areas of the square
ACKH and the rectangle CELP are equal. The square ABFG can likewise be
cut up and fitted on to BLPD, so the theorem is proved
In the second book of the Elements Euclid showed how, given any
rectilineal figure, a square can be found equal in area to the given figure. This
result establishes that all rectilineal figures are comparable in size or, as we
might say, that they are ordered magnitudes, and it establishes a 'simplest' one