Simson on Porisms: An Annotated Translation of Robert Simson’s Posthumous Treatise on Porisms and Other Items on this Subject

257 Pages·2000·8.49 MB·English

by  Ian Tweddle (auth.)

Sources
in the History of Mathematics and
Physical Sciences
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Ian Tweddle
Simson on Porisms
An Annotated Translation of
Robert Simson's Posthumous Treatise on
Porisms and Other Items on this Subject
With 227 Figures
~
T
Springer
lan Tweddle
Department of Mathematies
University of Strathclyde
Livingstone Tower
26 Riehmond Street
Glasgow G1 lXH
UK
Sources Editor:
Gerald J. Toomer
2800 South Oeean Boulevard, 21F
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USA
British Library Cataloguing in Publication Data
Tweddle, lan
Simson on porisms an annotated translation of Robert
Simson's posthumous treatise on porisms and other items on
this subject. - (Sources in the history ofmathematics and
physical sciences)
1. Porisms - History
1. Title II. Simson, Robert
516.2'2
ISBN 978-1-84996-862-1 ISBN 978-1-4471-3673-6 (eBook)
DOI 10.1007/978-1-4471-3673-6
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© Springer-Verlag London 2000
Originally published by Springer-Verlag London Berlin Heidelberg in 2000
The use ofregistered names, trademarks, etc. in this publication does not imply, even in the absence of a
specific statement, that such names are exempt from the relevant laws and regulations and therefore free
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The publisher makes no representation, express or implied, with regard to the accuracy of the information
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that may be made.
Robert Simson (1687-1768)
Portrait by Peter De Nune, 1746
(Courtesy of the Collins Gallery, University of Strathclyde)
Preface
Robert Simson is generally recognised as the first person to have made sig
nificant progress in resolving the mysteries of Euclid's lost work on Porisms.
He published a paper on this topic in the Philosophical Transactions in 1723;
but, although he continued to investigate Porisms throughout the rest of his
life, he published nothing more on this topic during his lifetime, and it was
left to his executor to assemble his various manuscripts into the posthumous
Tractatus de Porismatibus. This treatise, the paper and various extracts from
Simson's notebooks and correspondence form the subject matter of Simson
on Porisms. My aim has been to produce an accurate translation supple
mented by historical and mathematical notes; I hope thereby to add some
thing to the reputation of Simson, who, I believe, has been rather neglected
by mathematical historians, possibly because of his obsession with ancient
geometry at a time when great strides were being made in contemporary
mathematics by mathematicians such as Euler, the Bernoullis and Simson's
fellow-countrymen Colin MacLaurin and James Stirling. Simson's claim to
fame rests upon his scholarship, which is no less worthy of recognition.
Several people have helped me directly or indirectly with this project. I
would like to record my thanks to the following in particular:
(i) Peter Ottaway, a former undergraduate student, whose need for an hon
ours project in the history of mathematics led to our working on Simson's
paper, which in turn inspired me to study the treatise;
(ii) Professor R.A. Rankin of the University of Glasgow for mutual exchange
of ideas and information on Simson;
(iii) the Special Collections staff of the Andersonian Library, University of
Strathclyde, for help with material and references;
(iv) the staff of the Department of Special Collections at Glasgow University
Library for providing me with the bulk of my material;
(v) Professor G.J. Toomer, the series editor, for his helpful comments and
encouragement and, in particular, for saving me from a few gaffs in
translation and mathematical interpretation;
(vi) my wife, Grace, and my son, Edward, for their interest and support.
I am grateful to the Department of Special Collections, Glasgow Univer
sity Library, and to the Library of the Royal Society for permission to quote
from manuscripts in their possession. The portrait of Robert Simson appears
V III Preface
by kind permission of the Curator of the Collins Gallery, University of Strath
clyde. I am likewise indebted to the Librarian of Glasgow University Library
concerning the reproduction of four pages from Simson's notebooks.
Finally, it is a pleasure to acknowledge my indebtedness to all the members
of the editorial and production staff at Springer with whom I have dealt.
IT
Glasgow
April 2000
Contents
Introduction
Background 1
Some Remarks on the Work .................................... . 5
Mathematical Requirements, Terminology, Notation ............ . 6
Simson's Title ..................................................... . 9
Part I. Introductory Material
Summary .......................................................... 11
Preface ............................................................ 13
Definitions ......................................................... 17
Propositions 1-6 ................................................... 18
Pappus's Account of the Porisms ................................... 33
Notes on Part I .................................................... 38
Part II. Pappus's Two General Propositions and Euclid's First
PorisIll
Summary .......................................................... 45
Propositions 7-25 .................................................. 47
Notes on Part II ................................................... 91
Part III. LeIllIllas and Restorations
Summary .......................................................... 97
Propositions 26-79 ................................................. 99
Notes on Part III .................................................. 188
x Contents
Part IV. Various Porisms: Fermat, Simson and Stewart
Summary .......................................................... 199
Propositions 80-93 ................................................. 201
Notes on Part IV 242
Appendices
AI. A Translation of Simson's 1723 Paper along with some
Comments .................................................... 251
A2. "That this goes to a given point" 263
A3. Correspondence between Pappus's Lemmas and
Simson's Propositions ......................................... 265
A4. Corrections to Simson's Text ................................... 266
References ......................................................... 269
Index .............................................................. 273
Introduction
Background
Robert Simson (1687-1768) was Professor of Mathematics at the University
of Glasgow from 1711 until his retirement in 1761. He had originally studied
for the ministry at Glasgow but mathematics, which he first came to as a re
laxation from theological arguments, gradually became his dominant interest.
During 1710-11 he spent about a year in London pursuing his mathematical
studies and becoming acquainted with several eminent mathematicians, in
particular Edmund Halley, who was largely responsible for Simson's dedicat
ing himself to the study of Greek geometry.
Early on Simson became intrigued by Euclid's Porisms. What was known
about these was outlined by Pappus of Alexandria in Book 7 of his Math
ematical Collection and became known to Simson through the edition of
Commandino.1 According to Pappus the three books of Porisms have thirty
eight Lemmas and there are one hundred and seventy-one Theorems. How
ever, although he gave statements of theorems from all three books, most of
these are in very contracted form and have become mutilated, with the result
that only the first Porism recorded from the first Book and his statements
of two general propositions, one incorporating ten loci from the first Book,
the other a proposed generalisation of the former, have come down to us in
tact. It is not even clear from Pappus what kind of result should be regarded
as a Porism. He gives one definition which he attributes to geometers who
came after Euclid and who, he asserts, did not understand the true nature of
Porisms. He then gives a definition which he attributes to the ancientsj but
it is so general as to be almost all-embracing. In 1706 Halley published his
Greek and Latin translations, made from Arabic manuscripts, of Apollonius's
The Cutting off of a Ratio [13], a work also discussed by Pappus in Book 7,
and included with it a translation of Pappus's description of the Porismsj
Halley concluded that Pappus's account would be of little value to anyone
trying to understand the nature of the Porisms.
After much frustrated effort punctuated by attempts to close his mind to
the Porisms, Simson finally made a break-through in the early 1720s. The
1 Federicus Commandinus Urbinas [5]. Halley presented Simson with a copy of this
work which contained Halley's own notes.
I. Tweddle, Simson on Porisms
© Springer-Verlag London 2000