Table Of ContentSpringerBriefs in History of Science and Technology
Leo Corry
British Versions of
Book II of Euclid’s
Elements: Geometry,
Arithmetic, Algebra
(1550–1750)
SpringerBriefs in History of Science
and Technology
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Innovation/University of Ottawa, Ottawa, ON, Canada
Matteo Valleriani, Max-Planck-Institute for the History of Science, Berlin,
Germany
Tom Archibald, Simon Fraser University Burnaby, Burnaby, Canada
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Leo Corry
British Versions of Book II
of Euclid’s Elements:
Geometry, Arithmetic,
Algebra (1550–1750)
Leo Corry
Cohn Institute for History and Philosophy
of Science
Tel Aviv University
Tel Aviv, Israel
ISSN 2211-4564 ISSN 2211-4572 (electronic)
SpringerBriefs in History of Science and Technology
ISBN 978-3-031-11537-0 ISBN 978-3-031-11538-7 (eBook)
https://doi.org/10.1007/978-3-031-11538-7
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
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Acknowledgments
For useful editorial advice, I am indebted to Matteo Valleriani, to whom I thank for
continued support and interest. For help in locating material and useful comments
on various issues, I wish to express my thanks to Glen Van Brummelen, Inmaculada
Perez Martín, Nathan Sidoli and Benjamin Wardhaugh. Special thanks, I owe, to
Fabio Acerbi for sharing with me unpublished documents and forthcoming publica-
tions, as well as important insights on Barlaam’s treatises and their context. Michael
Barany read and commented on sections of a preliminary version, for which I am
especially thankful. I express my deep gratitude to Vincenzo De Risi and to Niccolò
Guicciardini for their respective, thoughtful, and detailed readings of my text, which
they helped improve with important comments and insightful suggestions.
v
Contents
1 Introduction: Euclidean Background .............................. 1
1.1 Book II .................................................... 3
1.2 The Early Printed Version of the Elements ...................... 5
1.3 The Rise of Viète’s Symbolic Algebra .......................... 8
References ...................................................... 11
2 The Main Figures: From Recorde to Wallis and Barrow ............ 13
2.1 Robert Recorde ............................................. 13
2.2 Billingsley and Rudd ......................................... 21
2.3 Oughtred and Harriot ........................................ 33
2.4 Wallis and Barrow ........................................... 38
References ...................................................... 45
3 Some Lesser-Known Figures ..................................... 49
3.1 Leeke and Serle ............................................. 49
3.2 Dechales Translated into English .............................. 52
3.3 Alingham .................................................. 54
3.4 Henry Hill ................................................. 57
3.5 Turn of the Eighteenth Century ................................ 59
References ...................................................... 62
4 Summary and Concluding Remarks .............................. 65
References ...................................................... 74
vii
Chapter 1
Introduction: Euclidean Background
Abstract The present book discusses the historically changing conceptions
concerning the relationship between geometry and arithmetic within the Euclidean
tradition that developed in the British context of the sixteenth and seventeenth century,
with a particular focus on Book II of the Elements. The then recently developed alge-
braic symbolism and methods, especially as promoted by François Viète and his
followers, took center stage as mediators between the two realms, thus offering ways
to work out that interaction that are not found in earlier editions of the Euclidean text
in other European contexts. The book discusses works written by prominent figures
in British mathematics, focusing on the way they handled results related with Book
II: Robert Recorde’s Pathway to Knowledge (1551), the first two English translations
of the Elements (by Henry Billingsley (1570) and Thomas Rudd (1651)), two remark-
able books published in 1631, Clavis Mathematicae by William Oughtred and Artis
Analyticae Praxis by Thomas Harriot, and the contributions of John Wallis and Isaac
Barrow. Also discussed are Euclidean versions written by somewhat lesser-known
and less influential, but no less interesting mathematicians, such as John Leeke and
George Serle, Reeve Williams and William Halifax, William Alingham and Henry
Hill. The introductory chapter comprises summary accounts of three background
issues that play a central role in the story: (a) Book II of Euclid’s Elements ; (b)
the early printed editions of the Elements; (c) the rise of the new symbolic algebra,
leading to the work of Viète.
· · ·
Keywords Renaissance mathematics British mathematics Euclidean tradition
· · ·
Euclid’s Elements Euclid’s Book II Early algebraic symbolism François Viète
The relationship between geometry and arithmetic throughout history has been a
recurring issue of interest for historians of mathematics working on various kinds of
mathematical cultures and contexts. The Euclidean traditions that developed from
Antiquity and up to the nineteenth century have traditionally offered a main focal
point that affords important insights on this topic. Book II of the Elements , more
particularly, because of the specific contents of its first ten propositions and the ways
in which they were approached through the ages, is of special interest in helping
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 1
L. Corry, British Versions of Book II of Euclid’s Elements: Geometry, Arithmetic,
Algebra (1550–1750), SpringerBriefs in History of Science and Technology,
https://doi.org/10.1007/978-3-031-11538-7_1
2 1 Introduction:EuclideanBackground
understand the intriguing historical questions involved here. In a previous article
(Corry 2013), I discussed this subject by examining versions of Book II in various
treatises, spanning the period from Late Antiquity to the Late Middle Ages. Later
on, I addressed related questions in the context of various medieval mathematical
cultures, Islamicate, Latin and Hebrew (Corry 2016, 2021).
The present book is devoted to expanding the discussion initiated in my previous
publications, by focusing now on the case of the British context of the sixteenth
and seventeenth century. Like in other contexts, also here we find a remarkable
variety of approaches to the ways in which the role of geometry and arithmetic
in mathematics was understood and the interaction between the two realms was
implemented. But the British context also involves some peculiar traits which make
it worthy of special attention. Prominent amongst them is that—unlike in many
earlier contexts—algebraic symbolism and methods that had been developed in the
previous decades, especially by François Viète (1540–1603) and his followers, took
center stage here as mediators between the two realms, thus offering new ways to
work out that interaction. Moreover, the ideas that developed in this regard—and
that I am about to discuss—had considerable impact on subsequent developments
not only in the mathematical realms that I will analyze, but also in the foundational
debates around the emerging field of British calculus.
This chapter provides a general background introduction, dealing with the devel-
opment of the Euclidean corpus following the early printed editions of the Elements.
Chapter 2 discusses several works written by prominent figures in British mathe-
matics, focusing on the way they handle results related with Book II: (2.1) Robert
Recorde’s Pathway to Knowledge (1551), not itself a version of the Elements, but
closely associated with the Euclidean tradition broadly understood; (2.2) the first
two English translations of the Elements, one by the highly influential one of Henry
Billingsley (1570), and another one by Thomas Rudd (1651); (2.3) two remark-
able books published in 1631, Clavis Mathematicae by William Oughtred and Artis
Analyticae Praxis by Thomas Harriot, both marking the beginning of an increased
interest in symbolic algebraic methods in the British context; and (2.4) the contri-
butions of John Wallis and Isaac Barrow, who espoused opposing and complemen-
tary views about the relative primacy of arithmetic and geometry in mathematical
discourse. Chapter 3 discusses Euclidean versions written by somewhat lesser-known
and less influential, but no less interesting mathematicians, such as John Leeke and
George Serle (who published a new translation in 1661), Reeve Williams and William
Halifax, William Alingham and Henry Hill. The latter two in particular, whose
versions were published in 1695 and 1726 respectively, help us realize the extent
to which algebraic methods had been fully absorbed at the turn of the century in the
British mathematical context.
The present, introductory chapter comprises summary accounts of three back-
ground issues that play a central role in the story: (a) Book II of Euclid’s Elements;
(b) the early printed editions of the Elements; (c) the rise of the new symbolic algebra,
leading to the work of Viète.
1.1 BookII 3
1.1 Book II
Book II of the Elements is a brief collection of only fourteen propositions. The first
ten provide elementary tools that are used for proving the last four propositions in
the book, which are more mathematical significant. Thus, for instance, II.1 is a kind
of distributivity law of area-formation over concatenation, which is proved on purely
geometric grounds. Euclid’s formulation states that “If there are two straight lines,
and one of them is cut into any number of segments whatever, then the rectangle
contained by the two straight lines equals the sum of the rectangles contained by the
uncut straight line and each of the segments.” The original accompanying diagram
is as in Fig. 1.1.
This proposition, as just indicated, is proved by purely geometric arguments,
based particularly on I.34. It is then used in the proof of II.11, which teaches how to
obtain the mean and extreme section of any given segment. Other results of Book II
help proving propositions in later books of the Elements, as well as in Apollonius’
Conica. In the proofs of each of the first ten propositions of Book II, Euclid relied
directly on results taken from Book I, and not from any other result in Book II itself.
Mathematicians of later periods followed the alternative possibility of proving first
II.1, based on I.34, and then proving the next nine based on II.1.
Two propositions that attract particular attention in the historiography are II.5 and
II.6. They will also appear prominently in my account here. Euclid formulated them
as follows (with accompanying diagrams as shown in Figs. 1.2 and 1.3 respectively):
II.5: If a straight line be cut into equal and unequal segments, the rectangle contained by
the unequal segments of the whole together with the square on the straight line between the
points of section is equal to the square on the half.
The segment AB is cut here into equal segments at C and into unequal segments
at D. Proposition II.5 states that the rectangle on AD, DB together with the square
on CD is equal to the square on CB. The diagram is built by taking AK = DB, BF =
CB.
Fig. 1.1 The canonical
version of Euclid’s II.1
