The Learning And Teaching Of Calculus Ideas Insights And Activities 1st Edition John Monaghan

The Learning And Teaching Of Calculus Ideas Insights And Activities 1st Edition John Monaghan
The Learning And Teaching Of Calculus Ideas Insights And Activities 1st Edition John Monaghan
The Learning And Teaching Of Calculus Ideas Insights And Activities 1st Edition John Monaghan

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6. “When we teach calculus, we need to know about di�erential and in�nitesimal ways
to understand calculus and about how calculus ideas evolved over time. We need to
know how calculus curricula look around the world and about the rich tradition
of calculus education research. We need to consider alternatives to the classic order
of a �rst calculus course. And, we must have answers to questions about the role
of calculus in solving real-world problems. Somewhat miraculously, this essential
volume in the IMPACT series does quite a bit of all of that.”
Elena Nardi, Professor of Mathematics Education (UEA, UK)
“John, Rob, Márcia, and Mike have accomplished a delicate balance among
broad historical, pedagogical, mathematical, cultural, and curricular perspectives
on matters related to the teaching and learning of calculus. Their book will be
important reading for teachers of calculus to understand the historical evolution of
the calculus curriculum they teach. Teachers will also become aware of the vibrant
discussions within mathematics education of how the calculus might be reshaped
to maintain intellectual integrity while simultaneously making central ideas more
accessible to students. Their book will also be important reading for graduate
students and researchers who are interested in inquiry into calculus learning and
teaching. The most impressive aspect of their book is that the authors cover so much
ground without taking a stance on any of the controversies they so carefully explain.”
Pat Thompson, Arizona State University
“The Learning and Teaching of Calculus presents a fascinating and compelling discussion
of calculus and the teaching of calculus. The authors have done an exceptional job
of writing in a way that is accessible and educative for a wide range of readers,
including teacher educators, those that teach calculus, and those that are interested
in conducting research on the learning and teaching of calculus. The book is full
of thought-provoking and engaging sample problems for students as well as clear
exposition of key theoretical ideas that is, paradoxically, both comprehensive and
concise. I highly recommend this book. You will undoubtably be inspired and awed
at the beauty of the ideas presented.”
Chris Rasmussen, San Diego State University

8. THE LEARNING AND TEACHING
OF CALCULUS
This book is for people who teach calculus – and especially for people who teach
student teachers, who will in turn teach calculus. The calculus considered is elementary
calculus of a single variable. The book interweaves ideas for teaching with calculus
content and provides a reader-friendly overview of research on learning and teaching
calculus along with questions on educational and mathematical discussion topics.
Written by a group of international authors with extensive experience in teaching
and research on learning/teaching calculus both at the school and university levels,
the book o�ers a variety of approaches to the teaching of calculus so that you can
decide the approach for you. Topics covered include
• A history of calculus and how calculus di�ers over countries today
• Making sense of limits and continuity, di�erentiation, integration and the fundamental
theorem of calculus (chapters on these areas form the bulk of the book)
• The ordering of calculus concepts (should limits come �rst?)
• Applications of calculus (including di�erential equations).
The �nal chapter looks beyond elementary calculus. Recurring themes across
chapters include whether to take a limit or a di�erential/in�nitesimal approach to
calculus and the use of digital technology in the learning and teaching of calculus.
This book is essential reading for mathematics teacher trainers everywhere.
John Monaghan is a professor at the University of Agder, Norway and an emeritus
professor at the University of Leeds, UK. He has taught in schools and universities, and
the learning and teaching of calculus has been a research interest throughout his career.
Robert Ely is a professor of mathematics education at the University of Idaho,
United States. He studies the reasoning of students with in�nitesimals, integrals,
variables, and argumentation, and he is particularly interested in the perspectives that
history can bring to such reasoning.
Márcia M.F. Pinto is Associate Professor at a public university in Brazil. She
has experience teaching mathematics to prospective teachers, mathematicians and
engineers and co-authoring textbooks for distance learning courses on calculus.
Michael O. J. Thomas is Professor Emeritus in the Mathematics Department at Auckland
University, New Zealand. His research explores advanced mathematical thinking at school
and university, including the role of representations, versatility and digital technology.

9. IMPACT (Interweaving Mathematics Pedagogy and
Content for Teaching)
The Learning and Teaching of Calculus
Ideas, Insights and Activities
John Monaghan, Robert Ely, Márcia M.F. Pinto and Michael O. J. Thomas
The Learning and Teaching of Number
Paths Less Travelled Through Well-Trodden Terrain
Rina Zazkis, John Mason and Igor’ Kontorovich
The Learning and Teaching of Mathematical Modelling
Mogens Niss and Werner Blum
The Learning and Teaching of Geometry in Secondary Schools
A Modeling Perspective
Pat Herbst, Taro Fujita, Stefan Halverscheid and Michael Weiss
The Learning and Teaching of Algebra
Ideas, Insights and Activities
Abraham Acravi, Paul Drijvers and Kaye Stacey

10. THE LEARNING
AND TEACHING
OF CALCULUS
Ideas, Insights and Activities
John Monaghan, Robert Ely,  
Márcia M.F. Pinto and Michael O.J. Thomas

11. Cover image: © Shutterstock
First published 2024
by Routledge
4 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
and by Routledge
605 Third Avenue, New York, NY 10158
Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2024 John Monaghan, Robert Ely, Márcia M.F. Pinto and Michael O. J. Thomas
The right of John Monaghan, Robert Ely, Márcia M.F. Pinto and Michael
O. J. Thomas to be identi�ed as authors of this work has been asserted in
accordance with sections 77 and 78 of the Copyright, Designs and Patents
Act 1988.
All rights reserved. No part of this book may be reprinted or reproduced or
utilised in any form or by any electronic, mechanical, or other means, now
known or hereafter invented, including photocopying and recording, or in
any information storage or retrieval system, without permission in writing
from the publishers.
Trademark notice: Product or corporate names may be trademarks or registered
trademarks, and are used only for identi�cation and explanation without
intent to infringe.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
ISBN: 978-1-032-06972-2 (hbk)
ISBN: 978-1-032-06973-9 (pbk)
ISBN: 978-1-003-20480-0 (ebk)
DOI: 10.4324/9781003204800
Typeset in Bembo
by Apex CoVantage, LLC

12. CONTENTS
Series foreword viii
Acknowledgement x
1 Introduction 1
2 Calculus across time and over countries 19
3 Making sense of limits and continuity 54
4 Making sense of di�erentiation 91
5 Integration and the fundamental theorem of calculus 133
6 Interlude: the ordering of chapters 3, 4 and 5 178
7 Calculus applications: di�erential equations and integration 183
8 Beyond elementary calculus 248
Index 286

13. SERIES FOREWORD
IMPACT, an acronym for lnterweaving Mathematics Pedagogy and Content for Teaching,
is a series of textbooks dedicated to mathematics education and suitable for teacher
education. The leading principle of the series is the integration of mathematics
content with topics from research on mathematics learning and teaching. Elements
from the history and the philosophy of mathematics, as well as curricular issues, are
integrated as appropriate.
In mathematics, there are many textbook series representing internationally ac­
cepted canonical curricula, but such a series has so far been lacking in mathematics
education. It is the intention of IMPACT to �ll this gap.
The books in the series will focus on fundamental conceptual understanding
of the central ideas and relationships, while often compromising on the breadth
of coverage. These central ideas and relationships will serve as organizers for the
structure of each book. Beyond being an integrated presentation of the central ideas
of mathematics and its learning and teaching, the volumes will serve as guides to
further resources.
Historically, the �eld of calculus has been one of the central areas of mathemat­
ics and is the basis for many other disciplines, especially in applied mathematics. Its
development was marked by a fruitful fundamental dispute between Newton and
Leibniz, which we can understand scienti�cally today. Again, it was Felix Klein who
successfully campaigned for Calculus to be established as part of the mathematical
school canon at the beginning of the 20th century. It has remained so to this day.
Moreover, both in terms of content and methods, Calculus is indispensable for
many university degrees. Hence Calculus constitutes a bridge from high school to
university mathematics, and this is why a Calculus book belongs into the IMPACT
series.

14. Series foreword ix
Series editors
Tommy Dreyfus (Israel), Ghislaine Gueudet (France), Nathalie M. Sinclair (Canada)
and G�nter Törner (Germany)
Series Advisory
Board Abraham Arcavi (Israel), Michèle Artigue (France), Jo Boaler (USA), Hugh
Burkhardt (Great Britain), Willi Dör�er (Austria), Francesca Ferrara (Italy), Koeno
Gravemeijer (The Netherlands), Angel Gutiérrez (Spain), Eva Jablonka (Germany).
Gabriele Kaiser (Germany), Carolyn Kieran (Canada), Kyeong-Hwa Lee (South
Korea), Frank K. Lester (USA), Fou-Lai Lin (Republic of China Taiwan), John
Monaghan (Great Britain/ Norway), Mogens Niss (Denmark), Alan H. Schoenfeld
(USA), Peter Sullivan (Australia), Michael O. Thomas (New Zealand) and Patrick
W. Thompson (USA).

15. ACKNOWLEDGEMENT
We thank the editors and two reviewers for their formative comments on drafts of
this book.

16. 1
INTRODUCTION
1.1 Introduction
Welcome to our book on calculus.
1
We are four mathematics education researchers,
but we have clocked-up many decades of practical teaching of calculus at school and
university level in Brazil, New Zealand, the United Kingdom and the United States.
This book is about teaching calculus, it is not a calculus textbook.
This book concerns introductory courses in calculus (elementary calculus) of a sin­
gle variable, which may be in the latter years of high school or the �rst year of uni­
versity, depending on country and/or institution. It is primarily a book for teacher
educators, i.e. people who teach teachers, secondly for (school or university) teach­
ers who wish to know more about ways to introduce their students to calculus and
thirdly for people, such as PhD students, who are contemplating doing mathematics
education research on an aspect of calculus. This book o�ers a variety of methods to
approach the teaching of calculus, provides a reader-friendly overview of research on
the learning and teaching of calculus and presents educational and mathematical mat­
ters for consideration (Edumatters and Mathematters) at various points in the chapters.
There are eight chapters:
1 Introduction
2 Calculus across time and over countries
3* Making sense of limits and continuity
4* Making sense of di�erentiation
5* Integration and the fundamental theorem of calculus
1 Or ‘analysis’ in some countries. The literature on calculus sometimes writes ‘the calculus’ or ‘Calculus’.
We have opted for ‘calculus’ to keep the writing style plain.
DOI: 10.4324/9781003204800-1

17. 2 Introduction
6 Interlude: the ordering of Chapters 3, 4 and 5
7* Calculus applications: di�erential equations and integration
8 Beyond elementary calculus.
The chapters marked with a star (*) address the substance of elementary calculus. We
now brie�y describe each chapter.
Chapter 1, which you are reading, introduces the book. This introduction is
followed by three sections: mathematical prerequisites for the study of calculus,
theoretical approaches mentioned in this book and a mathematical overview of dif­
ferential and in�nitesimal calculus. Mathematical prerequisites are important for
teaching any course, so we thought we’d put our considerations at the beginning of
the book. The section on theoretical approaches is there to help readers who do not
work in the �eld of mathematics education. You can be a brilliant calculus teacher
without knowing anything about the social-linguistic theory of commognition, but
if we mention, say, a commognitive study on teaching the derivative, we want a
quick way for you to �nd out what the theory of commognition is. The last sec­
tion provides a mathematical overview of di�erential and in�nitesimal calculus. This
overview is needed to fully understand parts of the following chapters as we often
present di�erential and/or in�nitesimal ways to understand calculus ideas.
Chapter 2 has two sections: a brief account of the history of calculus and a brief
account of calculus around the world. Both sections could be books in themselves,
so there is a need to focus on what is needed for the rest of the book. The history of
calculus section outlines major landmarks from Archimedes to Leibniz and Newton
to the arithmetisation of calculus in the 19th century to the invention of nonstand­
ard analysis in the 20th century. The calculus around the world section considers
school and beginning university calculus curricula from a number of countries. The
approach is synchronic not diachronic; it would be nice to present the development
of calculus curricula over time, but this is unrealistic given the length of the book.
Chapter 3 has four sections. The �rst looks at the place of limits and continuity in
elementary calculus curricula and one well know curriculum, Advanced Placement Cal­
culus, in particular. The second section provides an overview of education research on
limits and continuity. This is followed by a short history of limits and continuity. This
history is important in understanding that limits have not always been a part of calcu­
lus and that continuity was a central construct in the early days of calculus. The �nal
section looks at ways that limits and continuity can be introduced in your classroom.
Chapter 4 presents ways that (parts of) a �rst course in di�erential calculus can
be taught and learnt meaningfully. It does not tell you how to teach di�erentiation
but, rather, presents you with di�erent ways to do this. The chapter discusses prior
knowledge and curricula matters, ways that di�erentiation can be introduced, the
rules for di�erentiation, special functions, what derivatives tell us about functions
and their graphs, tasks and an overview of education research on di�erentiation.
Chapter 5 presents the main ideas of integral calculus and the fundamental theorem
of calculus (FTC). The focus is on signi�cant mathematical and conceptual elements
involved in making sense of these ideas. Di�erent approaches to teaching integration

18. Introduction 3
and the FTC are presented along with notes on the pros and cons of these approaches.
Throughout the chapter, descriptions of research about student reasoning with integrals
and the FTC are presented to augment the discussion of concepts and ways of teaching.
Chapter 6 considers, and questions, the classic ordering of a �rst calculus course:
limits, di�erentiation and integration. It is a very short chapter, an interlude that
re�ects on the previous three chapters.
Chapter 7 discusses some applications of calculus to real-world problems such as the
spread of viral infections and simple harmonic motion. It begins with a consideration of
�rst and second order ordinary di�erential equations and covers the nature of their solu­
tions as functions, how these may be approximated and their graphical representation us­
ing slope �elds. There is also an overview of educational research on di�erential equations.
The second part of the chapter looks at applications of integration that involve �nding
volumes of revolution, surface area and lengths of curves. Finally, some methods of ap­
proximating the values of integrals are considered with applications such as ship stability.
Chapter 8 considers aspects of calculus and/or real analysis courses that come
after an elementary course in calculus. It does not attempt to describe the content
of these courses but raises matters that teachers of elementary calculus should, in our
opinion, note with regard to what calculus related things their students may do (or
not!) in their future studies.
1.2 Mathematical prerequisites for the study of calculus
It could be argued that, apart from giving them practical arithmetic skills, one of the
primary purposes of the secondary mathematics curriculum is to prepare students
for the study of calculus. In which case, describing the prerequisites for the study
of calculus ought to be easy. However, our goal here is not to give a list of school
mathematics topics, such as arithmetic techniques, that would have little value in this
context but rather to try to say what are some key constructs that will contribute to
student understanding of calculus.
In the preceding sentence, we have deliberately avoided use of the word ‘skills’ in
favour of constructs, whatever they are. We should explain why. One way to divide
mathematics content is into skills and processes, and another way to divide math­
ematics content is into objects, concepts and constructs. Of course, these two groups
are not totally disjoint but are related in a fundamental way, which will be explained
in Section 1.3 of this chapter. Su�ce it to say at this point that mathematical pro­
cesses can undergo a cognitive encapsulation into mathematical objects (Dubinsky &
McDonald, 2001; Tall et al., 2000). We would argue, with the support of consider­
able research evidence (see, for example, Kieran, 2007, and the review of Rakes et
al., 2010) that there has been, in many countries, a greater emphasis on the former
than on the latter. For example, students may be able to add or multiply decimals
without really understanding the concept of place value or be able to factorise bino­
mials without understanding what factors or quadratic functions are.
Hence, in the main we will place an emphasis here on what objects, concepts
and constructs students would be well advised to understand in order to do well in

19. 4 Introduction
calculus. Occasionally we may also mention a skill or process that would also help.
It will be di�cult to present them in the order that would usually be met in school
since this will vary from country to country, and even within countries, from school
to school. In addition, the approach employed and the level of formality or rigour
used will also vary considerably. So the key point is that they need to be understood
in the context of what that may mean in the local curriculum.
1.2.1 Important mathematical constructs
1.2.1.1 Number
Clearly the concept of number underpins much of mathematics, and while the set of real numbers as a complete ordered �eld and the corresponding real number line will be a step too far for the vast majority of secondary school students, some idea of the relationships between sets of numbers and how they might be represented will be of bene�t. Further, representing these using both interval and set notation is recommended. So, for example, there is value in having the sets
 ={12 3, .},, ,
 =.--3210123-,,,,. and relationships N ⊂ Z ⊂ Q ⊂R{ ,, . (without needing
a formal de�nition of ), along with interval notation, (--,1]={x : x E Rx: 1},,x
[– 1, 3), and so on. A good understanding of proportionality will be bene�cial along
1 1 1
with some basic familiarity with convergent sequences (e.g. that 1,, , , . . . gets
2 4 8
nearer and nearer to 0).
1.2.1.2 Variable
We have known for a number of years that many students’ understanding of letter or symbolic literal use in algebra does not extend to generalised number or variable (K�chemann, 1981). This may be because greater emphasis is placed in classrooms on the use of variables rather than on understanding what they are or that many school texts are silent on the de�nition of a variable. A key part of the di�culty here is to know what kind of de�nition to give to students. For example, both Schoenfeld and Arcavi (1988) and Wagner (1981) give a number of possible examples of symbol usage in mathematical statements.
We would, of course, expect variables to vary, but describing the manner in
which they do is not so straightforward. Skemp (1979)
says “In mathematics, an
unspeci�ed element of a given set is called a variable” (p. 228). So the statement ‘Let
x ∈ ’ is used in this way to say that x is an unspeci�ed real number. However, this
can appear to be a rather static idea that says nothing about exactly how x might vary.
Thompson and Harel (2021) stress the point that in order to understand both rate of change and accumulation, two fundamental ideas in calculus, students need to be conversant with the process of covarying two quantities. Addressing the thought of how variables vary, they are unconvinced by the idea that it means replacing one value with another. Instead they espouse the construct of ‘smooth variation’ and

20. Introduction 5
say that “If calculus students are to understand something akin to instantaneous rate
of change, they must envision that smooth variation happens in bits” (ibid, p. 512).
Thus, students would bene�t from experiencing quantities that do indeed vary. For
example, if a conical container is �lling with water such that the height of the water
in the container is given by x and the corresponding volume by V(x) then V varies
simultaneously with x, and it is this covariation that can provide valuable experience
(Thompson & Harel, 2021). Assisting students to see, for example, the smooth vari­
ation of the height quantity from x to x + h as the volume varies smoothly from V(x)
to V(x + h) and representing these on Cartesian axes as a continuous graph can help
build understanding in preparation for calculus. Other examples that could prove
useful are to think of how variables change over time. If students are familiar with
basic kinematics (formulas and graphs linking distance, velocity, acceleration and
average velocity), then this can also be useful for seeing this kind of variation, as well
as providing good applications for later di�erentiation and integration. For example,
if a velocity is changing with respect to time, then we may have
Vt()=30 +10t
and, if δt is a small change in t, and the corresponding small change in V is δV , then
V +8V = 30 + 10 (t +8t )
So
8V = 108t.
1.2.1.3 Function
Of course, we have already used function notation in the previous section, and we stress the point that what is provided is not an intended order for studying these concepts and neither should they be thought of as disjoint entities, but rather there is some overlap between them. While function is a fundamental concept of mathemat­ ics, there is a tendency, as with variable, for it to be used in many school classrooms without being de�ned. As a consequence many students may think of a function as an equation, a graph or an input-output process rather than as a correspondence, a mapping, a set of ordered pairs or a rule (Williams, 1998). This leads to erroneous ideas such as a function must have a formula, its graph needs to be smooth and con­ tinuous, or even that due to stressing the vertical line test, the graph, rather than any algebra, is the function (Thomas, 2003). In addition, often student experiences with functions, and their graphs, has been limited to construction of pointwise and global perspectives (Vandebrouck, 2011). Hence, they may evaluate the value of a function
at a speci�c point, �nding fa, say, deal with a function globally or on an inter­()
val by translating its graph or �nding its concavity, but they rarely consider a local

21. 6 Introduction
perspective that involves small intervals such as [xh-,xh+] (Thomas et al., 2017),
where the behaviour of the function on small intervals of decreasing size is a focus.
This becomes important for a study of, for example, when a function is continuous.
Working with functions is also important in calculus. We mention just a few key
aspects of this here. One is that a working understanding of simple polynomial func­ tions (linear, quadratic and cubic) and their Cartesian graphs seems essential, along with the concept of gradient (or slope) and its application to linear functions (e.g. an ability to identify positive and negative gradients and to ascribe approximate nu­ merical values to the gradients) and estimating gradients of quadratic functions using tangents. It is important to de�ne a polynomial (function) since research has shown that a number of students do not think of examples such as
05 2 x+1,, as polynomials
-1
+(since, they may reason, poly means greater than 1) but may also think that x x
is one.
Another important concept is the domain of a function, the values of the inde­
pendent variable for which the function is de�ned. This becomes important when thinking about di�erentiating functions given by a formula such as fx =ln gx
)() ((),
in order to consider when g(x) might be zero or negative, and hence f is not de�ned,
1
as well as thinking about the values of x for which functions such as y= can be
x
integrated. We note in passing that both notations, y=.and fx()=. will be of
value in di�erent areas of calculus
2
and so making students familiar with both is a
good idea (see Chapter 2 for historical information on why both are used). Know­ ing and understanding a wide range of di�erent functions is also valuable. Some other examples include trigonometric, rational, exponential and power functions. See Section 4.2.5 for a fuller discussion of some of these functions.
A second consideration is the concept of an inverse function and when it exists.
This is when the function, possibly on a restricted domain, is 1–1 (injective) and onto (surjective), also called a bijection. In this case if
y fx then x=fy=()
-1
(),
when it exists. This, of course, is why the standard process for �nding an inverse function, to make x the subject of the equation, works. Students may be taught
that the graphs of a function and its inverse are symmetric about the line y = x (see
Figure 1.1). If this is used then it is important, of course, to stress that this is only the case if the x- and y-axes have the same scales.
A third useful idea is that of composition of functions, and this is best accom­
plished with the fx , written fgcan() notation. A composite function of f and g 
be de�ned
3
as
fgx )()=f gx )( (()
2 It is worth noting here that we should use, say, f as the notation to represent a function, whereas f(x).
is the value of the function f at some point x.
3 gfis also a composite function.

22. Introduction 7
FIGURE 1.1 Illustrating the relationship between fa a()and f
-1
()
providing we also make sure that the domain of f contains the range of gso that
fgx()is de�ned. Then the domain of fgis the domain of g( )  (or a subset of
it) and the range of fgis a subset of (or equal to) the range of f. It is worth not­
ing that although real functions are strictly de�ned between two given sets (such as
f:\0 {}-) it is common practice in mathematics to give a rule for a function
and assume that the (natural) domain is the largest subset of the real numbers for
which the function is de�ned. Thus, instead of, say,
h:,4 o)-[
de�ned such that
hx= x-4()
we will often be given just
hx()= x-4
without the domain information, and we have to supply this ourselves. This makes the composition of functions trickier. For example, we might ask whether our

23. t
8 Introduction
x-1
2
.
students could say what the natural domains would be for the functions fgand
gf where 
(2x-1)
2
fx=()
x+1
and
2x-1
gx()=
In addition, it may also be useful if students have some experience working with
the binomial theorem, but it is possible to introduce this topic/concept during the
study of di�erentiation.
1.2.1.4 Geometry
An understanding of geometrical concepts can be very useful in calculus. One
important example is the measurement of angles using radians (see also Section 4.4).
If we de�ne a radian to be the angle subtended at the centre of a unit circle by an arc
2n
of unit length, then since the circumference of a unit circle is 2π.1 there are = 2n
1
radians in a circle. Thus 2π radians are equal to 360˚, and so, dividing by 2 and using
c
to represent radians,
t
c
= 180
.
and
°
180
c
=1
n
or
n
c
1
o
= .
180
Thus the length of an arc subtending an angle of θ
c
at the centre of a circle of radius
ris
e
n=e• 2 rr
2n
And the corresponding area of the sector is
e
2 1
2
•nr = re
2n 2
These ideas become essential when we start to di�erentiate trigonometric func­
tions such as fx=xand gx= x in order for the answers to be easier ()sin() ()cos()
functions, since, for example fx'º = cos xº ()when x() (), rather than cos x is
180

24. Introduction 9
in radians. This is because the limit arising in the formal definition of derivative (see
Chapter 4)
(h(
sin
| |
(2)
#
lim =
h->0(h(180
| |
(2)
when h is measured in degrees, but 1 when h is in radians.Mathematter
Questions to ponder – concepts versus procedures
(i) A cylinder has radius r and height h, while a sphere has radius r. If the
radius of each is increasing at the same rate, which is increasing faster,
the volume of the cylinder or the surface area of the sphere? Explain.
(ii) Why should a student think about before they begin a process to solve

23
64
1
x
x
+
+
=?
If they go ahead and use a procedure to ‘solve’ it, what would you say to
them about the answer x=-15.?
(iii) Which of these do your students think are equations?
yx=+21, 93 6=+,
23 25xx-() =- ,
ab ab-+ -35 8, 31 33xx+ ()=+
What reasons can you think of for why they may think that way?
(iv) Which of these represent functions? Which polynomials? Why or why
not?
a)
y
x
=
-
1
1
b) ht ()=#
c) 23 5
2
xy x+- = d)
pm m()=-
(
(
|
)
)
|
-
sin
1
4
#
(v) If 21 0x-= and 31 0x+= then 2131xx-= +, since both are equal to 0,
and hence x=-2. How would you discuss this reasoning with a student?
(vi) How would we get students to solve these equations?
a) (i) 25 9x-= (ii)
23 59x+()-=
b) (i) 25 72 1
22
xx xx-- =+ + (ii) 27 61
22
xx xx--=+ +
(vii) Does the function ff xx:,12 1
2
()-> ()=-()R where have a maximum
value?

25. 10 Introduction
1.3 Theoretical approaches mentioned in this book
As part of the IMPACT series of books, this book aims to integrate mathemat­
ics content teaching with the broader research and theoretical base of mathematics
education. In particular we refer, at times, to approaches and ideas current in math­
ematics education research. We are aware that some readers will not be familiar with
these. This section is written to provide such readers with an overview of approaches
and ideas we mention. Our overview errs on the side of brevity; fuller descriptions
of approaches and ideas are available in the Encyclopedia of Mathematics Education
(EME, https://link.springer.com/referencework/10.1007/978-94-007-4978-8).
We start with constructivism.
4
This is a theory of knowledge development that
emerged, alongside other in�uences, from Jean Piaget’s developmental psychology.
Piaget considered knowledge as cumulative and cognitive development as mov­
ing through four stages: sensorimotor (0 to 2 years), preoperational (2 to 7 years),
concrete operational (7 to 11 years) and formal operational (12+ years). He posited,
amongst other things, three constructs in cognitive development:
• Schemas – cognitive structures representing a person’s knowledge about some
entity or situation, including its qualities and the relationships between these
5
• Assimilation – �tting new information into existing schemas, e.g. negative numbers
• Accommodation – modi�cation of existing schemas in the light of new informa­
tion assimilated.
Constructivist mathematics educators apply these constructs to all aspects of learn­
ers’ mathematical development. For example, Taback (1975) found that, at each
Piagetian stage, limit-related schemas held by the children were inherently contra­
dictory. In the late 20th century, two versions of constructivism emerged: radical
constructivism, cognition is the sole driver of knowledge development; and social
constructivism, the cultural-historical development of knowledge precedes individ­
ual knowledge development.
Social cultural (socio-cultural) approaches, including social constructivism and
activity theory,
6
view individual knowledge development within social structures,
historical development and the use of language and tools. Many social culturalists
subscribe to Vygotsky’s (1978, p. 57) statement, “Every function in the child’s cul­
tural development appears twice: �rst, on the social level, and later, on the individual
level; �rst between people . . . then inside the child”. The context of learning is
not a side issue for social culturalists as the who with, where, with what (tools) and why
of learning cannot be separated from the learning itself. For example, mathematics
done by a student using a computer algebra system (CAS) cannot, for a social cultur­
alist, be reduced to what the student alone can do or to what the CAS can do; the
4 EME: Constructivism in Mathematics Education
5 https://dictionary.apa.org/schema
6 EME: Activity Theory in Mathematics Education

26. Introduction 11
mathematics is done by the student-with-CAS. Bingolbali and Monaghan (2008) is
an example of a social cultural study on derivatives as it views students’ understand­
ings through students’ positional identities as engineers or as mathematicians.
Realistic Mathematics Education
7
(RME) is an approach to the design of teaching
mathematics that emerged in the Netherlands in the 1970s and is still developing. The
word ‘realistic’ in the title does relate to real-world situations, but it is also intended to
convey that the mathematical teaching sequences designed should connect with the
real-life experiences and imaginations of the students by o�ering them problem situ­
ations for the guided reinvention of mathematics. The word ‘mathematising’ was coined
by RME didacticians to emphasise the verb of doing mathematics rather than just
learning facts. RME distinguishes between horizontal and vertical mathematisations.
The former involves mathematising extra-mathematical phenomena from the real
world. Vertical mathematisation is inter-mathematical and involves building new (for
the student at a particular stage in their mathematical development) mathematical
connections between prior knowledge. RME has strong links with (and in�uenced
the development of) design research and stresses: starting from problems which are
meaningful to students, learning by doing and gradual mathematisation.
The Anthropological Theory of the Didactic
8
(ATD) was initiated by Yves Chevallard
in the 1980s and focuses on institutional aspects of mathematics education. A central
construct is the didactical transposition which traces the movement, over institutions,
of scholarly knowledge (produced by mathematicians, e.g. the limit notion) to curricula
knowledge (knowledge to be taught) to knowledge taught and knowledge learnt (by stu­
dents). ATD posits two knowledge blocks praxis and logos. Praxis consists of tasks and
techniques (e.g. integration by parts) to solve the tasks. Logos concerns the underlying
rationale for the praxis and has two levels: technology, which concerns the discourse
used in describing techniques; and theory, which provides the basis for the techno­
logical discourse. The theory is supposed to justify the technology by linking histori­
cally accumulated mathematical knowledge to knowledge taught but ATD analyses
often show that this linkage is often nebulous. For example, in an ATD study of limits
at high school, Barbé et al. (2005) reveal constraints that signi�cantly determine the
teacher’s practice and the mathematics taught. The development of ATD is ongoing
but the aforementioned description su�ces for references to ATD in this book.
Commognition
9
is a word made from the words ‘communication’ and ‘cogni­
tion’ and is a socio-cultural approach which views cognition and communication as
two sides of the same coin: thinking is communication with oneself, and learning
mathematics is participation in mathematical communication. Discourses are types of
communication which include words (e.g. functions); visual mediators (e.g. graphs);
narratives, stories about the objects of the discourse and routines, repetitive patterns in
the discourse. Routines in mathematics classes include explorations aimed at endorsing
narratives, deeds which involve practical action and rituals which are things done to
7 EME: Realistic Mathematics Education
8 EME: Anthropological Theory of the Didactic
9 EME: Commognition

27. 12 Introduction
create a common purpose in mathematics lessons (e.g. algebraic actions to establish a point of in�ection). Commognition views learning as ‘change in discourse’ which can occur at the object-level or the meta-level. At the object-level, change concerns the logical development of previously endorsed narratives of the discourse. At the meta-level, learning change does not follow logically from previously endorsed nar­ ratives but involves discursants (participants) making choices. Commognitive research on the teaching of learning of calculus thus pays close attention to what is said and done in classrooms with respect to the many constructs it introduces.
Embodied cognition
10
is a branch of cognitive psychology, but in mathematics edu­
cation it represents a view that learning mathematics is a mind-body activity, not just a mental act. This is not a new idea, but embodied cognition is a relatively new term (late 20th century). This is easy to appreciate in elementary mathematics – a child using her �ngers to count – but is it relevant to higher mathematics, calculus in particular? Carry out the following thought experiment: you are observing a student teacher teaching a �rst lesson on the derivative and introducing the gradient of the tangent to a function at a point – does the student teacher put her hand to graph at the point and incline the hand in the direction of the tangent at the point? Probably. There are now many di�erent views on the extent and importance of embodied cognition but reference to it in this book will keep to this basic idea.
Before introducing the next two theoretical approaches, we introduce the phrase/
construct, process-object encapsulation, which refers to making an object out of a pro­
cess. For example, the teaching and learning of functions usually begins with an input-output process, e.g.
x –2 –1 0 1 2
fx 1 0 1 4()
4
Students usually need to spend a considerable amount of time before this concep­
tion moves to viewing fx() as a single entity, an object. Process-object encapsula­
tion permeates much of school mathematics number and algebra curriculum. Gray
and Tall (1994) introduce the term ‘procept’ (pro cess-concept) for process-object
encapsulations where the student can move back and forth between process and object conceptions.
Actions, Processes, Objects, Schemas (APOS)
11
is a branch of constructivism which
embraces process-object encapsulation. Schemas are the end-point in learning a con­
cept, which starts with actions and moves to processes and then to objects and �nally schemas for the concept. Students are not always successful in reaching object or schema conceptions. APOS is used widely in studies of advanced mathematics. Asiala et al. (2001) is an example of an APOS study on students’ graphical under­ standings of the derivative.
David Tall (with others) developed the metaphor of the three worlds of mathematics
and has brought this into many accounts of learning calculus (Tall, 2013). The three
10 EME: Embodied Cognition 11 EME: Actions, Processes, Objects, Schemas (APOS) in Mathematics Education

28. Introduction 13
worlds are: an embodied world, a symbolic world (developed from the embod­
ied world) of actions into symbolic procedures (procepts) and a formal (axiomatic)
world where concepts are de�ned and their properties are deduced. He views these
three worlds as developing over time in the life of an individual and in the history
of mathematics.
1.3.1 Teacher knowledge
12
An import construct, pedagogical content knowledge (PCK), in teacher education
was introduced in Shulman (1986, p. 8). Good teachers, he argued, must not only
possess content knowledge and pedagogical knowledge; there must be interaction
(intersection in terms of Venn diagrams) between these two forms of knowledge –
pedagogical content knowledge. In the �eld of mathematics education this idea was
re�ned, by D. Ball and H. Bass (2002), to mathematical knowledge for teaching (MKT).
They sub-divide content (mathematical) knowledge into ‘common’, ‘specialised’ and
‘horizon’ content knowledge. Common content knowledge is required by everyone;
specialised content knowledge is unique to mathematics teaching; horizon content
knowledge is knowledge that would bene�t teaching but may not be taught because
it is beyond the level being taught. The topology of the real number line is an exam­
ple of horizon content knowledge in the teaching of calculus at the school level.
We end this section with a short account of conversions and treatments of math­
ematical representations, e.g. algebraic, graphic and numeric forms of a mathemati­
cal construct such as the derivative. The following does not present a theoretical
approach but introduces two constructs used in mathematics education research. A
seminal work on representations in mathematics is Duval (2006) where the di�erence
between conversions and treatments is considered. “Treatments are transformations
of representations that happen within the same register”
13
(ibid., p. 111), for example,
changing yx10 to yx 1. “Conversions are transformations of representa­--= =+
tion that consist of changing a register without changing the objects being denoted” (ibid., p. 112), for example, changing
yx 1 to a Cartesian graph. Duval (ibid.)=+
goes on to note “Conversion is more complex than treatment because any change of register �rst requires recognition of the same represented object between two rep­ resentations whose contents have very often nothing in common”. In the following chapters of this book, we often note that multiple representations are important for learning calculus, but Duval’s conversion/treatment distinction reminds us that alter­ native representations of the objects of calculus also add a level of di�culty.
1.4 A mathematical overview of differential and in�nitesimal
calculus
When it was developed in the 17th century, calculus was ‘the in�nitesimal calculus,’ –
a set of methods for working with in�nitesimal quantities. In the 19th century,
12 EME: Subject Matter Knowledge Within “Mathematical Knowledge for Teaching”
13 Duval (2006) de�nes ‘registers’ as semiotic systems that permit a transformation of representations.

29. 14 Introduction
in�nitesimals were jettisoned from the subject because they were seen to be not
rigorously de�ned, and calculus was reconceptualised in terms of limits. Calculus
classes followed suit, and today the vast majority of classes avoid in�nitesimals and
instead use limits when de�ning the principal ideas of calculus such as derivatives,
integrals and continuity. In the 1960s, Abraham Robinson developed the �eld of
nonstandard analysis (Robinson, 1966), which allows in�nitesimals to be formally
de�ned and for calculus to be conducted rigorously using them. He also proved that
essentially all the same things can be done with in�nitesimal calculus as with limits-
based calculus. Some calculus classes around the globe now use in�nitesimals, and
some researchers have argued for the bene�ts of such approaches (for a survey, see
Ely, 2021). One potential bene�t is that notations originally invented with in�ni­
tesimals in mind can more directly refer to quantities, rather than serving as token
dy
short hands for limit processes. Using in�nitesimals, dx has meaning on its own,
b dx
really is a quotient of small di�erences, and fxdx
f
() really is a sum of little bits.
a
We refer to in�nitesimal approaches periodically in this book, since one of our
goals is to stimulate the reader to conceptualise familiar elements of calculus in multiple ways and with multiple meanings. The purpose of this section is to provide some technical background for such references and to describe how in�nitesimals can be rigorously de�ned in such a way that calculus can be performed using them. Of course, an in�nitesimals-based calculus course need not formalise in�nitesimals at all, just as a standard calculus course need not formalise limits with the epsilon- delta de�nition. This section summarises the conceptual basis of such a formalisa­ tion, based on Keisler’s (1976) treatment, which should be consulted for further details, and which, at the time of this book’s publication, is available online for free on https://people.math.wisc.edu/~keisler/calc.html.
Robinson’s development of nonstandard analysis is one way of formalising Leib­
nizian in�nitesimals. It allows one to imagine a continuum where it is possible to ‘zoom in’ in�nitely to reveal di�erences between points that at a �nite scale appear identical. This continuum, the hyperreal numbers, is an extension of the real num­ ber line, and nonstandard analysis is the theory that works with this continuum. By saying that nonstandard analysis ‘formalises’ the idea of in�nitesimal quantities, we mean that it grounds these in the regular ZFC axiomatisation of modern math­ ematics. Robinson proved the transfer principle, that all (�rst-order logical) theo­ rems true in the hyperreal numbers (*
) are true in  and vice versa, which means
that standard analysis and nonstandard analysis are equivalent in scope, consistency and power. This actually allows current-day mathematicians to pursue results in *
 or , whichever they �nd handier, without sacri�cing rigour. For instance,
Terence Tao uses nonstandard analysis to avoid excessively complicated manage­ ment of epsilons, and he notes that “non-standard analysis is not a totally ‘alien’ piece of mathematics”, but is “basically only ‘one ultra�lter away’ from standard
analysis” (2007). Shortly we shall see what Tao means by “ultra�lter”.

30. Introduction 15
The image Robinson provides us for extending the reals to the hyperreals is to
start by picturing an in�nitesimal hyperreal number as a sequence of real numbers
that converges to 0. Sequences that converge faster to 0 are imagined to be smaller
in�nitesimals than those that converge slower. Thus we get an array of in�nitesimals
and comparing them amounts to comparing sequences of reals. The �rst technical­
ity is that we ought to view the two sequences {1, ½, ¼, 1/8, . . ., 1/2
n
. . .} and {3,
½, ¼, 1/8, . . ., 1/2
n
. . .} as the same hyperreal number, since they both converge
to 0 the same way. So Robinson begins by describing a hyperreal number not as a
sequence but as an equivalence class of sequences of real numbers. The idea is to
consider two sequences to be equivalent, (a
n
) ~ (b
n
), if a
n
= b
n
for “most” indices n.
Likewise, we want (a
n
) > (b
n
) if a
n
> b
n
for “most” n. But how do we know what
“most” means? In other words, how can we decide if an index set S = {n: a
n
= b
n
}
or Q = {n: a
n
> b
n
} is “large”?
One criterion for sets to be “large” is that it should allow the relation = (or >)
to be transitive. We want (a ) ~ (b ) and (b ) ~ (c ) to imply (a ) ~ (c ). Thus, if S =
n n n n n n
{n: a
n
= b
n
} is large, and T = {n: b
n
= c
n
} is large, then we need S∩T to be large too,
because it might be that a
n
= c
n
only for indices n∈S∩T. A second criterion for large
sets is that for any set P, exactly one of P or \P should be large. This is because we
want (a ) = (b ) when P = {n: a = b } is large, and we want (a ) ≠ (b ) when Q =
n n n n n n
{n: a
n
≠ b
n
} is large. Two more straightforward criteria for “largeness”:  is large and
∅ is not, and if A is large and A ⊆ B, then B is large. An ultra�lter is a collection of subsets of
 that satisfy these four criteria for largeness. In other words, an ultra�lter
is a collection of “large” sets of indices n, and on any of these index sets it is possible
to coherently compare two sequences a
n
= b
n
.
In order to de�ne the hyperreal numbers, we need to pick an ultra�lter that
meets an additional criterion to make sure the hyperreals do not end up being exactly the same as the reals: the ultra�lter must be nonprincipal. This means that every �nite set is small and every co�nite set is large. Proving that a nonprincipal ultra�lter even exists requires the axiom of choice, and ‘picking’ such an ultra�lter is a non-constructive endeavour. Nonetheless, this enables the hyperreals to be de­ �ned as follows: �x a nonprincipal ultra�lter. Let
R
N
be the set of all sequences of
real numbers and say (a ) ~ (b ) if {n: a = b } is large. The hyperreals are the set of
n n n n
equivalence classes *  = R
N
/∼.
The hyperreals are an extension of the reals, since we can identify any real number
r with the sequence {r, r, r , . . .}. It is worth checking that there indeed exist some
in&#6684777;nitesimal hyperreals that are not just 0. Consider ε = {1, 1/2, 1/3, 1/4, . . .}. This hyperreal number ε < 1/k for any integer k, because the set S of indices on
which {1, 1/2, 1/3, 1/4, . . .} < {1/k, 1/k, 1/k, . . .} is co&#6684777;nite, hence large. But ε > 0, since the set of indices for which {1, 1/2, 1/3, 1/4, . . .} > {0, 0, 0, . . .} is
,
which is large. Likewise, ∂ = {1, 1/2, 1/4, 1/8, . . .} is an even smaller in&#6684777;nitesimal, and it is not di&#438093348969;cult to show that there are uncountably many more. The reciprocals of all these in&#6684777;nitesimal hyperreals are all in&#6684777;nite numbers, which are also hyperreal.

31. 16 Introduction
An operation on real numbers has its analogue in *  by just performing the
same operation coordinate by coordinate. It can be shown that *  is a &#6684777;eld under
standard arithmetic operations *+ and *×. This means some of the heuristics Leib­ niz used with in&#6684777;nitesimal and in&#6684777;nite numbers can be proven as theorems in the hyperreals, e.g. “an in&#6684777;nitesimal number times an in&#6684777;nitesimal number is in&#6684777;nitesi­ mal” and “the reciprocal of an in&#6684777;nitesimal number is in&#6684777;nite”.
In general any statement about real numbers has a ‘starred’ statement for hy­
perreals, which means that it is true on a ‘large’ set of coordinates. The transfer principle establishes that this relationship goes both ways: a (&#6684777;rst-order) statement is true for all real numbers if and only if its starred version is true for all hyperreal numbers. An example is the following statement, which is true in
: “For any
positive x, there is a natural number n (∈) such that x > 1/n ”. This sounds like
it is not true in *, since there exist in&#6684777;nitesimals there. But the way to ‘star’ this
statement is “for any positive (hyper)real x , there is a (hyper)natural number n
(∈*) such that x > 1/n ”. This is true because *  contains in&#6684777;nite numbers.
Another very important example is the Archimedean axiom for the real numbers: for any positive a and b , there exists n (∈
) s.t. na > b . The spirit of this state­
ment is that a magnitude can always be iterated some number of times to exceed another given magnitude. This statement is not true in the hyperreals, which are a nonarchimedean &#6684777;eld. Nonetheless, the statement becomes true when you allow n to be an in&#6684777;nite hypernatural (∈ *
).
Using the least upper bound property of the reals, it is not hard to show that each
&#6684777;nite hyperreal number p is in&#6684777;nitely close to exactly one real number r. This real r
is called the shadow of p (“sh(p)”), or standard part of p (“st(p)”). Likewise, each real number r has a cloud or monad
14
of hyperreal numbers that are in&#6684777;nitely close to it.
Using ≈ to mean in&#6684777;nitely close, p ≈ st(p). The monad of hyperreals around the real
number p formalises the idea that there are numbers that become visible only when
you zoom in in&#6684777;nitely on p.
When you prove a theorem in the hyperreals, and then ‘unstar’ it to get the corre­
sponding statement in the reals, this unstarring often entails ‘rounding’ &#6684777;nite hyper- real numbers to their shadow or standard part. For example, consider the function y = x
2
in *. Choose an in&#6684777;nitesimal non-zero increment of x, calling it “dx”. We
can &#6684777;nd the corresponding increment of y, represented as dy, as follows:
dy =xdx
2
x
2
( + )-()
2 2 2
dy = x + 2xdx + dx -x
dy = xdxdx
2
2 +
14 Robinson picked this term as a tribute to Leibniz. For Leibniz, a monad was a fundamental meta­
physical particle, not a mathematical entity used in calculus.

32. Introduction 17
We can go further if we wish, and divide both sides by the in&#6684777;nitesimal dx, since
division works normally in the &#6684777;eld *:
dy
= 2xd+ x
dx
(dy J
st 2xdxThus we can de&#6684777;ne the derivative function fx' ()=st


dx

=( +)=2x.
This is an example of how transferring between * and  is done not by “pre­
tending in&#6684777;nitesimals are 0” (a complaint against in&#6684777;nitesimal techniques levelled
by some philosophers in the early days of calculus) but rather by taking standard
parts when we wish to create a statement in
. Typically transferring to  by taking
standard parts does the same work as taking a limit in standard analysis.
De&#6684777;ning the de&#6684777;nite integral provides another example of how this unstarring
n
replaces the use of a limit. First, we note that if a sum L
r
k
is de&#6684777;ned for any natural
n, by the transfer principle we can de&#6684777;ne
n
=
n
k 》
k=0
*r
k
for any hypernatural n (includ­
ing in&#6684777;nite n), and this will have all the same properties as an ordinary sum. Now
b
suppose we wish to de&#6684777;ne fxdx
J
() . First, we can &#6684777;nd an in&#6684777;nitesimal dx and an
a
in&#6684777;nite hypernatural n so that b = a + n·dx. Then we partition the interval [a, b] into n increments of size dx. On each increment, &#6684777;nd an x, and calculate f(x)dx. The
b
integral fxdx
(
() is the standard part of the in&#6684777;nite sum of these f(x)dx.
a
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34. 2
CALCULUS ACROSS TIME
AND OVER COUNTRIES
This chapter has two sections which position calculus over time and at a point in
time (2022). It provides a base for events and concepts that we refer to in the follow­
ing chapters. Both sections could be books in themselves, but this is a short book,
so we apologise for the brevity. The &#6684777;rst section visits the history of calculus and
outlines major landmarks from Archimedes to Leibniz and Newton to the arith­
metisation of calculus in the 19th century. This section does not include the devel­
opment of nonstandard analysis in the 1960s as this has been covered in Section 1.4.
The second section considers school and beginning university calculus curricula in
a number of countries.
2.1 Where did calculus come from?
Our purpose in this part is not to go through the historical development of calculus;
there are books, e.g. Edwards (1979), that already do this. Instead, our goal in this
section is to tell a short story about the problems and questions that people worked
on for many centuries that informed the development of calculus. We partition this
story into three sub-sections: the area problem, tangents and optimisation and the
calculus of Newton and Leibniz.
2.1.1 The area problem
For millennia, mathematicians around the globe tackled the general problem of
how to &#6684777;nd the area of regions with curved edges. Before seeing examples, it is
worth asking what exactly it means to &#6684777;nd the area of a curvy shape. For example,
for ancient Greek mathematicians it was a problem of construction: given a curved
region, construct a rectilinear region that takes up the same amount of space. The
DOI: 10.4324/9781003204800-2

35. 20 Calculus across time and over countries
term quadrature for area-&#6684777;nding re&#6684780;ects this idea – you ‘quadrate’ a region if you
make it a quadrilateral without changing its area. But in general, how would you
know if you have succeeded in accomplishing this?
Mathematter
What is the area of this blob?
How would you go about finding it?
2.1.1.1 Area of a circle
How do you &#6684777;nd the area of a circle? Today that is a simple question that many
school students will be able to answer immediately. But that was not always the
case. Mathematicians in Babylon, Greece, Egypt, China, India and other parts of
the ancient world tried in various ways. In so doing, all of them developed ways to
estimate the value of the circumference of a circle of diameter 1, a value that we
now call π.
In some of these ancient cases, this estimation was probably done through careful
measurement (see Figure 2.1). In other cases, the estimate of π was done by trap­
ping a circle inside and outside of polygons with many sides and then using other
geometric facts to determine the areas of these polygons. For instance, Archimedes
(c.287–c.212 BC) began with hexagons inside and outside a circle (see Figure 2.2),
reasoning that the true area of the circle must lie between the areas of the two hexa­
gons. He then doubled the number of sides to get 12-gons, using known formulas
for areas of isosceles triangles. He kept doubling the number of sides, eventually
FIGURE 2.1 Some estimates of π in the ancient world

36. Calculus across time and over countries 21
FIGURE 2.2 Trapping a circle between polygons with more and more sides
stopped with 96-gons. With some other reasoning, he could show that the circum­
1 10
ference of a circle of diameter 1 must be between 3 and 3 , which is actually
7 71
within 0.0002 of the value of π as we know it today. This illustrates the true area of
a circle as the limit of these upper and lower bounds as the number of sides in the
polygons increased to in&#6684777;nity.
This technique for estimating the circle’s area was used throughout the ancient
world. Chinese mathematician Liu Hui also used 96-gons to estimate π in c.263 AD.
Two centuries later in India, Aryabhatta used a 384-gon and in China, Zu Chongzhi
used a 12288-gon (!) to estimate π ≈ 355/113, correct to seven digits.
2.1.1.2 Area of a parabola
Archimedes also tackled the problem of &#6684777;nding the area of a piece of a parabola.
In his Method, he determined the exact area of a piece of a parabola. He &#6684777;rst sur­
rounded the parabola piece ABV with a triangle ABC (as in Figure 2.3) and then
imagined chopping it into in&#6684777;nitely many indivisible slices (e.g., XX‴ ). Then he
imagined a lever PB, whose midpoint and fulcrum D was also the midpoint of AC.
He used some geometric facts about parabolas along with the law of lever moments
(which he had earlier in his life discovered) to deduce the following: each indivisible
slice XX′ of the parabola, when moved to the end of the lever at point P, balances
with its corresponding slice XX‴ of the triangle, left where it is. Thus the entire
parabola sector ABV, hanging at P, balances with the entire triangle ABC left where
it is. Since the triangle’s centre of mass is at its centroid, which is 1/3 of the way
from the fulcrum D to the lever end B, this means the triangle must be three times
bigger than the parabola sector.
In Quadrature of the Parabola, Archimedes used a di&#6684774;erent technique to prove this
answer was correct; he made a double exhaustion argument that relied on approxi­
mating the parabolic region’s area as closely as wanted by packing enough triangles
inside the region (see Figure 2.4). This is quite similar to the method he used for
the circle’s area.
Archimedes’ Method was lost for at least a millennium until the Archimedes Pal­
impsest was found in the 20th century. Without knowing about it, Bonaventura

37. 22 Calculus across time and over countries
FIGURE 2.3 Archimedes’ Method used to determine the area of a parabola sector, from
Boyer (1949, pp. 49–50)
FIGURE 2.4 Archimedes’ proof involves &#6684777;lling a parabolic region with triangles
Cavalieri (1598–1647) invented in the 1630s a similar technique for slicing a region
into indivisible pieces and putting these into one-to-one correspondence with slices
of another known &#6684777;gure to determine the region’s area (or volume). Cavalieri’s col­
league Torricelli, in his work de Dimensione Parabolae, presents 11 di&#6684774;erent ways to
&#6684777;nd the area of the parabola using Cavalieri’s method of indivisibles. In one of these,
Proposition 20, he even hangs pieces on an imagined lever!
At almost the same time, Fermat and Roberval were writing letters back and forth
in 1636 generalising Archimedes’ quadrature question to “parabolas” of the form
yx
k
. Instead of Archimedes’ triangles, Fermat and Roberval put N rectangles of =

38. Calculus across time and over countries 23
uniform width inside the region (see Figure 2.5). Since the curve is increasing, the
right side of each subinterval gives a height for an over-estimating rectangle, and the
left side gives a height for an underestimating rectangle. This means that the area A
is bounded between the left-hand sum and the right-hand sum:
k k k k
(0 J1 (1 J1 (N;1J1 (1 J1

+

+ +

< A <

N N N N N N N N
k k
(2 J1 (N J1
+

+ +

N N N N
This simpli&#6684777;es to:
1
[
k k k J
1 k k k
1 2 .+N 1 ) A [1 + +.+N

J(Equation 2.1)++(- << 2
k+1 k +1 N N
FIGURE 2.5 Trapping a parabolic region between under- and over-estimating rectangles

39. 24 Calculus across time and over countries
With such a general underestimate and overestimate, how would one arrive at an
exact answer? In Chapter 5 we will see the ‘front-door’ approach to answering that
question, which Fermat and Roberval used by developing general formulas for
these sums.
Edumatter
Calculus students are told that the exact area A of the curved region is achieved
by letting the number of rectangles N grow without bound in Figure 2.5. What
kinds of mental images do you think students have of the ultimate result of that
growing process? What mental image do you have?
2.1.2 Tangents and optimisation
2.1.2.1 Tangent lines
Given a curve with a point on it, how can we construct a tangent line to the curve
through that point? This question sounds completely unrelated to the quadrature
question, but calculus reveals them to be, surprisingly, two sides of the same coin.
It is easy enough to construct the tangent line to a circle because it is perpen­
dicular to the radius. For other curves it is less obvious. Apollonius &#6684777;gured out how
to construct tangent lines to parabolas, hyperbolas and ellipses in Book I of Conics,
in the late 200s BC. For instance, to &#6684777;nd the tangent line to a parabola at a point
C: make a perpendicular CD to the diameter and extend the diameter the same
distance so that AE = DE (Figure 2.6). AC is tangent to the curve at C.
FIGURE 2.6 Tangent line to a parabola, from Apollonius’ Conics

40. Calculus across time and over countries 25
The advent of coordinate geometry in the 1630s led to the study of a range of
curves generated by simple algebraic equations (see Figure 2.7). Mathematicians
asked about the areas bounded by these curves and about how to construct lines
tangent to them, among other questions. One clever method for &#6684777;nding tangent
lines was Fermat’s method of adequality.
We illustrate Fermat’s method on his example of the parabola y = x . Sketch
a tangent line to the curve at B, which intersects the diameter at E (seen in Fig­ ure 2.8). Put point A on this line near B and drop perpendiculars from A and B to the diameter. Fermat used similar triangles to notice that EI/EC = AI/BC ≈ FI/BC (where F is the intersection of AI with the parabola). If C and I are a small distance
xe+ te+
e apart, this means that ~ . Cross-multiplying, squaring both sides, and
x t
combining terms, we get:
tx +~ +e te() x
tx +e ~+ex
2
() t
2
()
2 2
te ~2etxex +. (Equation 2.2)
FIGURE 2.7 A few examples from the range of curves studied in the mid-1600s
FIGURE 2.8 Fermat’s method of adequality for &#6684777;nding a tangent line (from Katz, 2009)

41. 26 Calculus across time and over countries
In order for the ≈ to become =, we would need the small distance e to actually
be zero. But so far, if we replace e with 0, our whole equation will collapse into
0 = 0. However, from Equation 2.2, we can divide both sides by e (since e ≠ 0) to
get t
2
~ txex. Now we can imagine what happens when e becomes zero, so the 2 +
points F and B become ‘adequated’, made equal or indistinguishable from each
other. This gives us the result that t = 2x, which is exactly what Apollonius found.
The strange thing about Fermat’s technique is that it requires us to &#6684777;rst imagine
two points on a curve that are some distance e apart and then later imagine that they have become the same point or at least two points that are somehow indistinguisha­ ble from each other. How does this work? This is the strange idea that ends up being at the heart of calculus, which is further considered in the sub-section In&#6684777;nitesimals and limits for Newton and Leibniz that follows.
2.1.2.2 Optimisation
Another type of problem of long-standing interest to mathematicians is the problem of optimisation: what is the largest or smallest value obtained by a variable quantity under some given constraints? These problems can often be solved using Fermat’s idea of adequality also. The method is based on the idea that, if some quantity A has a
maximum value when x = 3, then A will be almost at its maximum when x is near 3.
To show how adequality works in such a case, Fermat considered a classic prob­
lem, which we adapt a tiny bit: bend a given chunk of wire into a rectangle that has the greatest possible area.
What is the maximum area
you can make this box if
you keep it a rectangle and
don’t change its perimeter?
Suppose the entire wire has &#6684777;xed length B, and imagine the rectangle has a vari­
able width w. Then the rectangle’s height will have to be B/2 – w. This makes its
area A = w(B/2 – w). Fermat noticed that, for the value of w that maximises this
area A, making a small change in w does not really have any e&#6684774;ect on A. In other
words, for some small e,
wB /2 -w w + eB /2 - + .( )~( )( (we ))
Simplifying as we did in the tangent line example, and dividing both sides by e , we get
4wB~-e.

42. Calculus across time and over countries 27
Now if we adequate the two widths w and w + e, assuming they are the same, we get
the width w = B/4. This is the width that makes the largest possible rectangle, and
it happens when the rectangle is a perfect square.
The fact that this same technique works for both the tangent problem and the
optimisation problem gives a hint that these are somehow two sides of the same
coin. A few decades later, this calculus coin was minted.
2.1.3 The calculus of Newton and Leibniz
We have glanced at some techniques developed by mathematicians like Fermat,
Descartes and Roberval in the mid-1600s for solving the area problem and tangent
line/optimisation problems. Those techniques look quite a bit like integration and
di&#6684774;erentiation respectively. So why do historians usually credit the development of
calculus to Newton and Leibniz, rather than these other earlier mathematicians?
One immediate answer to this question is that these earlier mathematicians did
not realise the fundamental theorem of calculus (FTC), which provides the deep
connection between the solutions to these two very di&#6684774;erent-looking problems.
But this is not quite the whole story either because both Isaac Barrow and James
Gregory were aware of the FTC and provided demonstrations of it prior to Newton
and Leibniz. Both adapted the method of adequality to make use of the general
connection between the area and tangent problems. Nonetheless, they did not fully
understand the signi&#6684777;cance of this connection nor did they develop generalised
methods to exploit that connection (Katz, 2009). Their treatments of the FTC
were geometrical and lacked the generalisability of Newton’s and Leibniz’s analyti­
cal approaches. Newton and Leibniz both developed highly generalised methods
for employing the FTC, although they conceptualised, represented and developed
these methods quite di&#6684774;erently. Although there arose during their lifetimes a heated
controversy over who stole the method from the other, the consensus now is that
Newton and Leibniz independently proved and conceptualised the FTC. Plenty
has been written about the life and work of these two mathematicians, and of their
controversy, so we do not delve into all that here. Our focus is to summarise the dif­
ferent ways they each thought about the fundamental ideas of calculus.
2.1.3.1 Isaac Newton
Newton &#6684777;rst developed his idea of the FTC while stuck at home in Woolsthorpe
during the plague years of 1665–1666. The imagery Newton appealed to was
dynamic, grounded in the idea of smooth motion. Consider the area y under a
curve up to a moving vertical bar of height q, where both y and q are determined by
the varying position x of the vertical bar (as in Figure 2.9). Newton noticed that as
the sliding bar moves, the rate at which the area y grows is the height q. In modern
terminology, dy/dx = q. This means that if we know a formula for the area y (in
terms of x), we can get a formula for the curve’s height q by di&#6684774;erentiating. It also

43. -
28 Calculus across time and over countries
FIGURE 2.9 Newton’s illustration for the FTC
means that if we instead know a formula for the height q, we can get a formula for
the area y by antidi&#6684774;erentiating (Bressoud, 2011).
The reason Newton’s idea was so powerful was that at the same time he de­
veloped very general ways to di&#6684774;erentiate algebraically loads of new functions. For instance, Fermat’s approach already allowed for di&#6684774;erentiation of polynomials, but Newton’s generalisation of the binomial theorem ultimately allowed one to expand expressions of the form
(xh)
q
for any rational q . This allowed him to +
make a general di&#6684774;erentiation power rule and di&#6684774;erentiate functions with radicals. He also developed general product and quotient rules, so he could tackle rational functions. He illustrated the power of these general techniques in his October 1666 tract on &#6684780;uxions; he used an example where the area under a curve was
ax
given by the complicated radical quotient y = , and found its derivative
2 2
a x
(‘&#6684780;uxion’) to get the height q (Newton, 1967). Newton also &#6684777;gured out ways to
write many functions using power series, which he could then di&#6684774;erentiate and integrate term by term.
2.1.3.2 Gottfried Wilhelm Leibniz
G.W. Leibniz’s idea of the FTC has its roots in the connection between sums and
di&#6684774;erences of &#6684777;nite quantities. He noticed that “the sums of the di&#6684774;erences between successive terms, no matter how great their number, will be equal to the di&#6684774;erence between the terms at the beginning and the end of the series” (Leibniz, translated in Child, 1920, pp. 30–31). For instance, consider a sequence such as z
n
= 1, 5, 7,
8, 8.5, where the di&#6684774;erences between the terms make another sequence y
n
= 4, 2,
5
1, 0.5. The sums of these di&#6684774;erences y
n
42 10.5 is just z
5
–z
1
= 8.5–1.L
=+++
1
Leibniz imagined that this would also be true for in&#6684777;nitesimal quantities as well, and he developed notations like dz to denote in&#6684777;nitesimal di&#6684774;erences (di&#6684774;erentials) between values of some varying quantity y as well as ∫ (big S for ‘summa’) to denote sums of in&#6684777;nitesimal bits. Consider a curve de&#6684777;ned over an interval chopped into

44. Calculus across time and over countries 29
FIGURE 2.10 Leibniz’s sums of di&#6684774;erences of a curve’s height
subintervals of in&#6684777;nitesimal width dx, where the curve’s height at each point in the division is z. If you create the sequence of di&#6684774;erences {dz} between these heights, then the sum ∫dz of these di&#6684774;erences will be the di&#6684774;erence between the last and &#6684777;rst heights, z – z (see Figure 2.10).
last &#6684777;rst
Leibniz was able to use this idea because, like Newton, he developed general
analytic techniques. His techniques related to calculating with differentials. In particular, he developed general rules for starting with an equation relating two or more variable quantities (e.g.
yx
3
) and deriving from this an equation =
2
that relates their differentials (e.g. dy = 3xdx ). We shall discuss more of how
he derived these differential equations in Section 2.3.3 about infinitesimals and limits. For now, the important thing is that this correspondence between vari­ able equations and their differential equations enabled him to envision and use the FTC.
Although Leibniz did not use the same example as Newton, it is easy to illustrate
how his idea of the FTC applies to it. Imagine an area y = A(x) accumulated up
to some variable x under a curve of height q . We can imagine it formed as a sum
of little areas of height q and width dx, so any value of y is ∫q·dx. The di&#6684774;erence dy
between any two successive y values is then given by dy = d ∫ q·dx, meaning that
dy = q·dx for that position in question, as seen in Figure 2.11. This means if we have an equation for the height q as a function of x , then we can reconstruct an equation
for the area y up to any point x . We must antidi&#6684774;erentiate the equation dy = q·dx,

45. 30 Calculus across time and over countries
FIGURE 2.11 Di&#6684774;erences of successive increments of area
which is something Leibniz could often do because he had developed systematic
ways of di&#6684774;erentiating equations like y = A(x) to get their di&#6684774;erential equations
dy = q·dx.
The imagery Leibniz used was much more static than Newton’s dynamic im­
agery. Leibniz treated curves as already-existing, rather than being traced out in
time. As Fauvel et al. (1990) summarise,
Newton’s aims were seemingly more speci&#6684777;c and concrete than those of Leibniz,
who was more interested in broad philosophical principles. Newton’s patterns
of thought were more physical. His variable quantities varied with time – they
were &#6684780;owing quantities (&#6684780;uents) with a velocity, or rate of change (a &#6684780;uxion) – as
opposed to Leibniz’ more formal and static in&#6684777;nitesimals.
(p. 67)
In the next section we explore how this di&#6684774;erence between the two mathematicians’
approaches manifests in the way they treated limits and in&#6684777;nitesimals.
2.1.3.3 In&#6684777;nitesimals and limits for Newton and Leibniz
Both main types of problems from which calculus emerged – the area problem and
the tangent/optimisation problem – require grappling in some way with the in&#6684777;nite
and the in&#6684777;nitesimal in order to get exact, not just approximate, answers. Not surpris­
ingly, Newton and Leibniz dealt with this issue in fundamentally di&#6684774;erent ways,
grounded in the fundamentally di&#6684774;erent imagery they used when reasoning about
the big ideas of calculus.
Leibniz’s ideas about the in&#6684777;nitesimal are areas of lively scholarly debate. Some
historians believe Leibniz predominately saw in&#6684777;nitesimals as &#6684777;ctional in&#6684777;nitely
small entities (e.g., Bos, 1974; Katz & Sherry, 2013). Others believe Leibniz,
particularly in his later work, primarily treated in&#6684777;nitesimals as syncategorematic

46. Calculus across time and over countries 31
or formal shorthands for variable &#6684777;nite quantities that can be taken as small as desired (e.g. Arthur, 2013; Ishiguro, 1990). Either way, Leibniz’s notation and language are consistent with an image of in&#6684777;nitesimal di&#6684774;erentials as static enti­ ties that become apparent by zooming. For instance, by zooming in in&#6684777;nitely on some variable z , it becomes possible to distinguish in&#6684777;nitesimal di&#6684774;erences dz
in the values of z . One would need to zoom in in&#6684777;nitely again in order to see
second order in&#6684777;nitesimal di&#6684774;erences d (dz) (aka d
2
z) between the values of the
di&#6684774;erentials dz. Similarly, squaring a di&#6684774;erential dx gives a di&#6684774;erential (dx)
2
that
would require second order zooming to see. This process can be repeated inde&#6684777;­ nitely, and Leibniz discussed such a hierarchy of in&#6684777;nitely many orders of in&#6684777;ni­ tesimal numbers.
1
Using this idea, Leibniz developed heuristics for neglecting
higher-order in&#6684777;nitesimals when deriving di&#6684774;erential equations and derivatives. We illustrate how Leibniz’s approach to di&#6684774;erentiation works using the example of y = x
3
we mentioned earlier.
Given some in&#6684777;nitesimal increment dx of x, the goal is to &#6684777;nd the magnitude of
the corresponding increment dy of y:
dy =xdx )-()(+
3
x
3
3 2 2 3 3
dy = x + 3x dx + 3xdx + dx -x
2 2 3
dy = 3xdx + 3xdx + dx
Since the di&#6684774;erentials dx
2
and dx
3
are higher-order in&#6684777;nitesimal, they are negligible
at the in&#6684777;nitesimal scale at which dy and dx are apparent. Thus they can be ignored,
giving the di&#6684774;erential equation
2
dy = 3xdx . This derivation establishes the correspondence between the equation yx
3
, which=
2
relates full-scale amounts of x and y, and its di&#6684774;erential equation dy = 3xdx, which
relates in&#6684777;nitesimal di&#6684774;erences in x and y. This form of the di&#6684774;erential calculus,
which established such correspondences, became a powerful tool for the continental mathematicians like the Bernoulli brothers, who used and championed Leibniz’s
2
approach. For these mathematicians, di&#6684774;erential equations like dy = 3xdx were
dy
2
more commonly used than derivatives like = 3x because they treated an inte­
dx
2 2
gral such as ∫ 3xdx as an in&#6684777;nite sum of di&#6684774;erentials 3xdx.
Although Leibniz’s in&#6684777;nitesimal calculus produced powerful results on the con­
tinent, they were the object of some debate and contention. For instance, one
1 Leibniz also realised that for each such order there should also be a corresponding order of in&#6684777;nitely-
large numbers too, in order to allow for division by (non-zero) in&#6684777;nitesimals.

47. 32 Calculus across time and over countries
outspoken skeptic of in&#6684777;nitesimals was Michel Rolle, who began his paper Du Nou­
veau Systême de l’In&#6684777;ni by saying:
Geometry had always been considered as an exact science, and indeed as the
source of the exactness which is widespread among the other parts of mathemat­
ics. Among its principles one could only &#6684777;nd true axioms and all the theorems
and problems proposed were either soundly demonstrated or capable of a sound
demonstration. And if any false or uncertain propositions were slipped into it they
would immediately be banned from this science. But it seems that this feature of
exactness does not reign anymore in geometry since the new system of in&#6684777;nitely
small quantities has been mixed to it. I do not see that this system has produced
anything for the truth and it would seem to me that it often conceals mistakes.
(Rolle, 1703, p. 312)
Rolle proposed that one of the biggest incoherencies of in&#6684777;nitesimals was the ambi­
guity of when they are treated as non-zero quantities and when they are ignored as
zeroes. If at any moment in a derivation one assumes that x + dx = x, then dx was 0
all along. So how could any of these di&#6684774;erential equations mean anything?
Lacking any formalised development of the continuum as a domain that can in­
clude non-zero quantities that were smaller than any &#6684777;nite number, the champions
of Leibniz’s calculus resorted to several other defences against Rolle’s objections.
These included
(i) In&#6684777;nitesimals are &#6684777;ctive but useful entities (but then how do we know they
won’t give rise to contradictions?).
(ii) Every manipulation with in&#6684777;nitesimals is just a syncategorematic shorthand for
a lengthy but rigorous Greek-style contradiction argument that uses only &#6684777;nite
quantities (but could there be any guarantee of this?).
(iii) In&#6684777;nitesimals are not &#6684777;xed but changing quantities that get smaller and smaller;
a di&#6684774;erential equation captures the way in which they evanesce or vanish (Man­
cosu, 1996).
This last idea re&#6684780;ects much more Newton’s imagery than Leibniz’s imagery. Two
decades after he &#6684777;rst developed his in&#6684777;nitesimal calculus, Newton sought to put it
on a more secure footing in Principia Mathematica (1687) with the idea of “&#6684777;rst and
ultimate ratios”. (We will discuss his idea here using Leibniz’s notation, although
Newton did not use it.) Newton’s idea was essentially to de&#6684777;ne in&#6684777;nitesimal quanti­
dy dy
2
ties in ratio with one another, e.g. . To say = 3x meant to imagine dx and dy
dx dx
as quantities diminishing in tandem, and where the ratio between them is captured
as they together approach 0 and evanesce. Newton speci&#6684777;es, “I wish it always to be
understood that I have in mind not indivisibles but evanescent divisibles, and not
sums and ratios of de&#6684777;nite parts but the limits of such sums and ratios”.

48. Calculus across time and over countries 33
He goes on to defend this idea:
It may be objected that there is no such thing as an ultimate proportion of van­
ishing quantities, inasmuch as before vanishing the proportion is not ultimate, and
after vanishing it does not exist at all. But by the same argument it could equally
be contended that there is no ultimate velocity of a body reaching a certain place
at which the motion ceases; for before the body arrives at this place, the velocity
is not the ultimate velocity, and when it arrives there, there is no velocity at all.
But the answer is easy: to understand the ultimate velocity as that with which
a body is moving, neither before it arrives at its ultimate place and the motion
ceases, nor after it has arrived there, but at the very instant when it arrives, that
is, the very velocity with which the body arrives at its ultimate place and with
which the motion ceases. And similarly the ultimate ratio of vanishing quanti­
ties is to be understood not as the ratio of quantities before they vanish or after
they have vanished, but the ratio with which they vanish. . . . There exists a limit
which their velocity can attain at the end of the motion, but cannot exceed. This
is the ultimate velocity.
(Newton, 1999, pp. 441–443, emphasis ours)
Newton does not de&#6684777;ne the term “limit” in the Principia, nor does he provide vari­
ables or indices by which to operate with limits. Nonetheless, the concept of limit
grew from his idea and became formalised to capture Newton’s imagery. For exam­
dy fx(+h)-f ()
ple, by de&#6684777;ning =lim
x
, we can imagine the limit of the ratio
dx h-0 h
between increments of y and of x as both increments shrink in tandem, captures as
they ‘vanish’. The idea is that there should be some e&#6684774;ective unambiguous way to deduce the behaviour of these shrinking quantities at the limit by their behaviour at the &#6684777;nite stages.
The ambiguity of Newton’s idea was famously mocked by George Berkeley
in 1734, who described such in&#6684777;nitesimals as “ghosts of departed quantities” (Ely, 2021). To address criticisms like Berkeley’s, Maclaurin sought to develop Newton’s idea of limits in his A Treatise of Fluxions (1742) by trying to remove in&#6684777;nitesimals
entirely: “We shall not consider any part of space or time as indivisible, or in&#6684777;nitely little; but we shall consider a point as a term
2
or limit of a line, and a moment as a
term or limit of time” (MacLaurin, 1742, p. 3). Mathematicians during the follow­ ing century, such as Euler and Cauchy, continued to use in&#6684777;nitesimals in their work, and it was not until the mid-1800s that mathematicians such as Weierstrass formally developed the limit idea in a manner that avoided in&#6684777;nitesimals (Katz, 2009). We leave further discussion of limits and in&#6684777;nitesimals to Chapter 3.
2 MacLaurin’s usage of “term” re&#6684780;ects an older meaning from the medieval Latin terminus: boundary
or limit.

49. 34 Calculus across time and over countries
2.1.3.4 The birth of functions
One of the key mathematical concepts emerging from Newton’s and Leibniz’s
development of calculus was the early idea of function. Leibniz consistently assumed
that variables can always be written in terms of other related variables in the same
situation, which provided a robust basis for fearlessly substituting variables as much
as needed in a context. Leibniz and his colleagues explicitly highlighted the impor­
tance of this idea and coined the term “function” – a variable was a function of
other variables if it could be algebraically expressed in terms of them (Bos, 1974).
We point out two of the key ways in which this early idea of function di&#6684774;ers from
our current de&#6684777;nitions of the term.
First, in Leibniz’s day there was developing a sense that if, say, y was a function
of x, then each value of y was determined by the value of x on which it depended.
Consider a comment of Johann Bernoulli in a letter to Leibniz in 1698: “To denote
a function of some indeterminate quantity x, I like to use the corresponding capital
letters X or Greek ξ, so that we can see at the same time on which indeterminate
quantity the function depends” (Leibniz, 1849–1863, Abth. 1, Band III, pp. 531–2).
This idea of dependence, however, did not require a function to have only one out­
put value for any input value in its domain (Kleiner, 1989). One just needed to be
unambiguous about what values were being talked about in a given situation.
Second, a function needed to be given by an appropriate analytic expression or
formula. Which expressions were seen as appropriate broadened during the early
years of calculus. For instance, Euler’s 1748 de&#6684777;nition is pretty loose: “A function
of a variable quantity is an analytic expression composed in any way whatsoever of
the variable quantity and numbers or constant quantities” (Euler, 1748/1988, p. 3).
Thus an in&#6684777;nite power series could be a function, but something piecewise-de&#6684777;ned
could not be.
The f(x) notation Euler introduced highlighted the dependence of a function
value on its domain value, as well as the importance of their being just one such
value. One bene&#6684777;t of function notation over just variable notation is that it allows
such values to be referred to without necessarily being evaluated or calculated. For
instance, f(3) means the function’s value when the underlying domain variable is 3,
whether or not we can determine what that value is. With variables alone it is harder
to succinctly say “the value of y when x is 3” (although the notation y|
3 can help).
It is because of these kinds of bene&#6684777;ts that the main objects with which calculus is
conducted are functions rather than curves.
Mathematter
Imagine something that is now considered a function, but Leibniz would not
have. Also imagine something that is now not considered a function, but Leib­
niz would have.

50. Calculus across time and over countries 35
2.2 Calculus around the world
This section considers school and beginning university calculus curricula in a sample of countries around the world. A very short history of calculus teaching is presented but the section mainly looks at these things at a moment in time, 2022. We start with four questions which relate to curricula issues covered.
Edumatter
(i) Should calculus be a part of high school mathematics curricula?
• If the answer is ‘yes’, then should calculus be a part of the curriculum
for all students or for a subset of students?
(ii) What issues/problems related to calculus are students likely to encounter
in the school to university transition?
(iii) Should the teaching of calculus be a speci&#6684777;c focus in a teacher education
programme?
• If the answer is ‘yes’, then what aspects of learning/teaching calculus
should be addressed?
(iv) How much freedom do you, as a teacher, have in selecting what to teach
in a calculus course and how to teach it?
• What factors (curricula, institutions, examinations, people, etc.) limit
your freedom?
This section has four foci:
• High school
• The transition from high school to university
• The preparation of teachers to teach calculus
• Who decides what is taught.
We hope that our coverage of these foci provides you with food for thought in
considering the aforementioned questions. It must be said that the evidence-base
3
is
not extensive. We &#6684777;rst note that we, the authors, have a pretty good idea of calcu­
lus curricula in Brazil, England, New Zealand, the United States and the Interna­
tional Baccalaureate
4
(IB) and have discussed calculus curricula with colleagues from
around the world. We also have access to a nascent database on calculus around the
3 We are writing in 2022, and there may, of course, be more information by the time you read this.
4 The IB is interesting from a calculus-around-the-world perspective because it has schools in 150
countries.

51. 36 Calculus across time and over countries
world. This is not in the public domain at the time of writing, but it may be by the
time you read this. As we write, it has reports on calculus (at school, at the beginning
of university and in teacher education) in the following countries: Canada (Que­
bec), Denmark, England, France, Germany, Greece/Cyprus, Israel, Japan, Mexico,
New Zealand, Norway, Peru and Sweden.
Moving to the public domain, there are three useful sources:
• Törner et al. (2014), which includes brief overviews of calculus curricula and
teaching in Belgium, Cyprus, England, France, Germany, Greece and Italy.
• Bressoud et al. (2016), which includes overviews of calculus instruction in France,
Germany, Uruguay and the United States.
• A special issue of the journal ZDM – Mathematics Education (Volume 53, Issue 3,
2021) on “Calculus in High School and College Around the World”. It includes
reports on calculus in Germany, Israel, Singapore, South Korea, Tunisia and the
United States.
All of these sources provide references to speci&#6684777;c published works that deal with
educational matters related to calculus in speci&#6684777;c countries. We do not list these
references here.
2.2.1 A very short history of calculus teaching
Zuccheri and Zudini (2014) provide a succinct history of calculus teaching in Europe
but by ‘teaching’ they mean the ‘to whom’, ‘what’ (curriculum content) and ‘where’
(school or university), rather than the matter how teachers communicated the ideas
of calculus. We summarise this work up to the beginning of the 19th century.
Calculus was taught at the university level and in some schools soon after its crea­
tion by Newton and Leibniz. The &#6684777;rst textbook on the calculus was written in 1696
by the Marquis de l’H&#7275266;pital. Shortly after this, other textbooks appeared (at the
school level in some countries). A French textbook of the mid-18th century started:
with a de&#6684777;nition of “di&#6684774;erential” as an in&#6684777;nitely small step, both di&#6684774;erentia­
tion and integration were discussed (most of the attention, however, was paid
to calculations). Integration was de&#6684777;ned formally as the inverse operation of
di&#6684774;erentiation.
(Zuccheri & Zudini, 2014, p. 495)
Still in this era, calculus was mainly conducted with equations rather than functions.
Di&#6684774;erentiation was the derivation of a di&#6684774;erential equation, an equation describing
the relative sizes of di&#6684774;erentials of multiple variables (see Section 1.4). By the end of
the 18th century, particularly in France, the focus had changed to the derivation of a
derivative function. In a popular French textbook for universities and some schools
from 1802, the derivative of a function was called the di&#6684774;erential coe&#438093348969;cient and was

52. Calculus across time and over countries 37
de&#6684777;ned using an intuitive limit notion (as the limit of the ratio of concurrent incre­
ments of the function and its variable).
Moving to the end of the 19th century,
The strong need to reform the teaching of mathematics was realized in several
countries. . . . It seemed important to introduce the notions of variable and func­
tion and to establish the inclusion of calculus as the main item on the reform
agenda along with the modernization of the teaching of mathematics.
(Zuccheri & Zudini, 2014, p. 497)
Before 1902, in France, the derivative was reserved for higher education courses, but
the new programmes of 1902 saw the introduction of calculus in secondary schools.
Poincaré believed the derivative should be introduced “in the way of Lagrange” and
that Leibniz’s di&#6684774;erential notation was dangerous because it could instil the errone­
ous idea of a ratio to pupils. In Germany, Felix Klein was an important &#6684777;gure in
school reform. Klein supported functional thinking and saw function as a key concept
in reform curricula. Klein advocated using Cartesian graphs to represent functional
dependencies and to introduce the ideas of calculus. Klein also considered it useful
to introduce the &#6684777;rst notions of in&#6684777;nitesimal calculus via applications to the natural
sciences and to problems in the &#6684777;eld of insurance, thereby establishing more con­
nections with real-world practice. In Klein (1908/1932, p. 240) he wrote:
We want the concepts, expressed via the symbols y = f(x), dy/dx, ∫ ydx , to
become familiar to the students with these characters, i.e. not as a new abstract discipline, but in an organic construction, in the context of the whole teaching, starting from the simplest examples and going up, step by step.
In the United Kingdom, at a meeting of the British Association, the mechanical engi­ neer John Perry, an ardent mathematics education reformer, proposed a curriculum in which teachers could teach the elements of calculus using squared paper and sim­ ple algebra and laboratory experiments, resources he employed in an 1897 textbook for engineering students. A 1914 international survey (that included Austria, Bel­ gium, Denmark, France, Germany Holland, Hungary, Italy, Norway and the United Kingdom) summarised the content of calculus taught in schools:
• Nearly everywhere, only functions of one variable were dealt with.
• The di&#6684774;erentiation of polynomial and rational functions was considered every­ where, while in some places it was supplemented by the di&#6684774;erentiation of expo­ nential and trigonometric functions with their inverse functions.
• Generally, Lagrange’s notation was preferred to that of Leibniz.
• In most countries, the notion of integral was also introduced, with a clear prefer­ ence for beginning with the inde&#6684777;nite integral and then dealing with the de&#6684777;nite integral.

53. 38 Calculus across time and over countries
Törner et al. (2014) regard these initiatives as the &#6684777;rst period in the curricular his­
tory of school calculus. This period lasted until the 1960s when modern mathemat­
ics gained broad in&#6684780;uence over the curriculum. The second period was dominated
by the modern mathematics approach and lasted until the beginning of the 1980s
where the focus moved from computational techniques to the fundamental con­
cepts of functions, limits and continuity and on their structural dimension. Teaching
emphasised ε−δ de&#6684777;nitions and proofs of theorems and focused on de&#6684777;nitions and
foundations rather than on applications. The third period, in reaction to the failed
second period and promoted by mathematics education research (e.g. students’ di&#438093348969;­
culties with the limit concept) and educational ideas (student-orientation, problem-
solving, applications, modelling and the role of digital technologies), returned to
intuitive/informal approaches to calculus promoted in the &#6684777;rst period.
2.2.2 High school
We begin by considering to whom calculus is taught. We take the term ‘high school’
(or upper secondary school) to mean a school that provides general education for
students aged 14–18 years of age; actual ages will vary slightly from country to coun­
try. We write ‘general education’ to distinguish this from ‘vocational education’;
many countries divide their schooling on general/vocational lines at 14 or 16 years
of age. For example, in the European Union, 52% of high school pupils in 2018 fol­
lowed a general programme with the remainder following vocational programmes.
Calculus is rarely taught in vocational programmes (exceptions being those that
prepare students for applied STEM subjects at university).
With the exception of state schools in Peru, calculus is taught in high schools in
every country named in the previous section and it is generally taught in the last two
years of schooling. But it is not taught to all students in this category; to whom it is
taught depends on the programmes o&#6684774;ered. We brie&#6684780;y describe curricula in Singa­
pore and the United States to show variation in these programmes.
Singapore
5
At the end of Grade 10 (16 years of age) students take Ordinary-level exami­
nations, O-levels. O-level mathematics does not include calculus but there is
a second O-level (Additional Mathematics – AM) that does include calculus.
Singapore and Singaporeans take education seriously, and about 50% of all
Singaporean students take AM. In Years 11–12 higher level mathematics has
two strands: H1 and H2 Mathematics (H1M and H2M) as, respectively, prepa­
ration for non-STEM and STEM undergraduate courses. H2M requires AM as
5 Based on Toh (2021).

54. Calculus across time and over countries 39
a prerequisite. It is a more thorough treatment of mathematics compared to
H1M, which is more basic and does not require AM as a prerequisite. The aims
of H1M and H2M are to acquire mathematical concepts and skills to
• support their tertiary studies in business and the social sciences
• prepare for their tertiary studies in mathematics, sciences, engineering and
other related disciplines.
H2M students are exposed to a wide repertoire of techniques, including the dif­
ferentiation and integration of trigonometric functions and advanced di&#6684774;eren­
tiation techniques such as implicit di&#6684774;erentiation. H1M students are exempted
from these advanced techniques, which are mainly relevant to STEM related
courses.
So, high school students in Singapore e&#6684774;ectively make their decision to study, or
not, a STEM subject at university, at the beginning of Grade 9 (14 years of age), and
this is a factor in whether they study AM/calculus or not. At the higher level there
are two courses, but they are distinct; H1M does not lead on to H2M.
United States
The main high school calculus course is called Advanced Placement (AP). It has
two levels called AB and BC. Anyone can enrol in an AP calculus course provid­
ing that they have done a course called Precalculus. The curriculum content
of Calculus AB is a subset of that of Calculus BC, and students who take the
Calculus BC examination receive a Calculus AB sub-score. Students can just
take Calculus AB, go straight into Calculus BC or they can take Calculus AB and
then go on to Calculus BC. Limits, derivatives, integrals and di&#6684774;erential equa­
tions are covered in both courses, but Calculus BC has additional topics, e.g.
further integration and Euler’s method for approximate solution of di&#6684774;erential
equations. Calculus BC also has two additional topics: derivatives and integrals
with parametric equations, polar equations, as well as vector functions; and
in&#6684777;nite sequences and series.
In contrast to Singapore: (i) a decision to study calculus at high school in order
to study a STEM subject at university does not need to be made at 14 years of
age; (ii) whereas H1M does not lead in to H2M, Calculus AB can lead to Cal­
culus BC.

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content Scribd suggests to you:

59. The Project Gutenberg eBook of Die
Hohkönigsburg: Eine Fehdegeschichte aus
dem Wasgau

60. This ebook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg License included with this
ebook or online at www.gutenberg.org. If you are not located in the
United States, you will have to check the laws of the country where
you are located before using this eBook.
Title: Die Hohkönigsburg: Eine Fehdegeschichte aus dem Wasgau
Author: Julius Wolff
Release date: April 1, 2019 [eBook #59185]
Language: German
Credits: Produced by The Online Distributed Proofreading Team at
http://www.pgdp.net
*** START OF THE PROJECT GUTENBERG EBOOK DIE
HOHKÖNIGSBURG: EINE FEHDEGESCHICHTE A US DEM WASGAU
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61. Anmerkungen zur Transkription
Das Original ist in Fraktur gesetzt. Im Original gesperrter Text ist so markiert. Im Original
in Antiqua gesetzter Text ist so ausgezeichnet.
Weitere Anmerkungen zu Transkription befinden sich am Ende des Buches.

62. In der
Grote'schen Sammlung
von
Werken zeitgenössischer Schriftsteller
erschienen von Julius Wolff:
Till Eulenspiegel redivivus. Ein Schelmenlied. Mit
Illustrationen. 24. Tausend. br. 4 M., geb. 4 M. 80 Pf.
Der Rattenfänger von Hameln. Eine Aventiure. Mit
Illustrationen von P. Grot Johann. 70. Tausend. br. 4 M., geb. 4
M. 80 Pf.
Schauspiele. (Kambyses. – Die Junggesellensteuer.) Zweite
Auflage. br. 4 M., geb. 4 M. 80 Pf.
Der wilde Jäger. Eine Waidmannsmär. 89. Tausend. br. 4 M.,
geb. 4 M. 80 Pf.
Tannhäuser. Ein Minnesang. Mit Porträtradirung nach einer
Handzeichnung von Ludwig Knaus. Zwei Bände. 39. Tausend. br.
8 M., geb. 9 M. 60 Pf.
Singuf. Rattenfängerlieder. 16. Tausend. br. 4 M., geb. 4 M. 80 Pf.
Der Sülfmeister. Eine alte Stadtgeschichte. Zwei Bände. 39.
Tausend. br. 8 M., geb. 9 M. 60 Pf.
Der Raubgraf. Eine Geschichte aus dem Harzgau. 47.
Tausend. br. 6 M. 60 Pf., geb. 7 M.
Lurlei. Eine Romanze. 55. Tausend. br. 5 M. 60 Pf., geb. 6 M.
Das Recht der Hagestolze. Eine Heirathsgeschichte aus
dem Neckarthal. 34. Tausend. br. 6 M. 60 Pf., geb. 7 M.

63. Die Pappenheimer. Ein Reiterlied. 23. Tausend. br. 5 M. 60
Pf., geb. 6 M.
Renata. Eine Dichtung. 28. Tausend. br. 5 M. 60 Pf., geb. 6 M.
Der fliegende Holländer. Eine Seemannssage. 29.
Tausend. br. 4 M. 60 Pf., geb. 5 M.
Das schwarze Weib. Roman aus dem Bauernkriege. 21.
Tausend. br. 6 M. 60 Pf., geb. 7 M.
Aus dem Felde. Nebst einem Anhang: Im neuen Reich.
Dritte vermehrte Auflage. geb. 2 M. 50 Pf.
Assalide. Dichtung aus der Zeit der provençalischen
Troubadours. 15. Tausend. br. 5 M. 60 Pf., geb. 6 M.
Der Landsknecht von Cochem. Ein Sang von der
Mosel. 17. Tausend. br. 5 M. 60 Pf., geb. 6 M.
Der fahrende Schüler. Eine Dichtung. 14. Tausend. br. 5 M.
60 Pf., geb. 6 M.
Die Hohkönigsburg. Eine Fehdegeschichte aus dem
Wasgau. br. 5 M. 60 Pf., geb. 6 M.

64. Grote'sche Sammlung
von
Werken zeitgenössischer Schriftsteller.
Siebenundsiebzigster Band.
Julius Wolff, Die Hohkönigsburg.

65. Die Hohkönigsburg.
Eine Fehdegeschichte aus dem Wasgau
von
Julius Wolff.
Berlin,
G. Grote'sche Verlagsbuchhandlung.
1902.

66. Alle Rechte, insbesondere das der Uebersetzung in andere Sprachen,
vorbehalten.
Druck von Fischer & Wittig in Leipzig.

67. Die Hohkönigsburg.
Charlottenburg, 1902.

69. I
I.
m Augustsonnenschein des Tages St. Bartholomäi 1483 wehte auf
dem Bergfried der Hohkönigsburg, des größten Schlosses im
ganzen Elsaß, eine Fahne in den Thierstein'schen Farben, gelb
und roth, denn die Grafen dieses Namens führten in ihrem
Wappenschilde sieben rothe Rauten in goldenem Felde.
Die Burg lag auf einem von Osten nach Westen gestreckten
Bergrücken, der aber nach der Ebene zu mit seiner Schmalseite als
ein alle anderen sichtbaren Höhen übersteigender, spitzer Kegel
erschien und, Mauern und Thürme gleich einer zackigen Krone
tragend, den Blick aus der Ferne schon auf sich zog und
unwiderstehlich fesselte.
Die Umwallung der sehr ausgedehnten Werke bestand aus zwei,
durch einen breiten Zwischenraum getrennten Ringmauern, deren
äußere mit einer Anzahl vorspringender Rundthürme bewehrt war,
und drei, in gemessenen Abständen aufwärts folgende Thore hatte
zu durchschreiten, wer zum Hochschlosse hinan wollte. An jedem
dieser Thore stand heut ein Doppelposten von geharnischten
Knechten, die mit ihren Hellebarden in kerzengrader Haltung den
nahenden Gästen des Burgherren salutirten. Hinter dem zweiten
Thore gelangte man auf einen geräumigen Hof, wo sich die
Stallungen, Sattel- und Geschirrkammern und die Schmiede
befanden. Dort mußten die Berittenen vom Pferde steigen, denn von
hieraus hatten sie den in mehreren Absätzen über Treppenstufen
führenden Weg zum dritten und höchsten Thore zu Fuß zu machen.
Es hieß das Löwenthor, weil über seinem Bogen zu beiden Seiten
eines stark beschädigten, nicht mehr erkennbaren Wappens –
vermuthlich das der Hohenstaufen – zwei in Stein gehauene Löwen
ruhten. Hier stand außer den zwei Reisigen noch ein Herold mit dem
Stab, in Federbarett und gesticktem Wappenrock, um die

70. Ankommenden im Namen seines Herren zu empfangen und sie bis
zum Eingange des Saalbaues zu geleiten.
Man erwartete heut viel Besuch, denn es galt, das nach seiner
Erstürmung völlig ausgebrannte, jetzt aber mächtig und prächtig
wieder aufgerichtete Schloß durch ein glänzendes Fest einzuweihen,
zu dem Einladungen an die im weiteren Umkreis wohnende
Ritterschaft ergangen waren.
Wechselvolle Schicksale hatten die Hohkönigsburg seit ihrer
Entstehung heimgesucht.
Ursprünglich geschaffen war sie im zwölften Jahrhundert von den
Hohenstaufen. Nach ihnen hatten die Herzöge von Lothringen die
Lehenshoheit und belehnten nach einander die Landgrafen von
Werd, die Grafen von Öttingen und die Bischöfe von Straßburg mit
der vielumworbenen Feste, die zeitweilig auch an die Rappoltstein,
von Rathsamhausen und von Hohenstein als Afterlehen überging.
Dann kam sie an das Habsburgische Kaiserhaus, in dessen Besitz sie
lange verblieb. Um die Mitte des fünfzehnten Jahrhunderts aber
hatte sich eine Schaar wüster Placker und Pracher, unter denen auch
einige von Adel waren, dort widerrechtlich eingenistet und trieb als
Wegelagerer und Buschklepper ihr freches Räuberhandwerk in einer
für die ganze Umgegend so unerträglichen Weise, daß sich endlich
der Bischof und der Rath von Straßburg, die Grafen von Rappoltstein
und die Bürgerschaft von Schlettstadt zum Kampfe gegen die
streitbaren Schnapphähne und ihre zahlreichen Spießgesellen
verbündeten, die Burg belagerten und einnahmen, das Gesindel, das
leider durch die Flucht entkam, verjagten und das zum Raubnest
gewordene Schloß zerstörten.
Über ein halbes Menschenalter lang starrten die gewaltigen
Trümmer öde und obdachlos auf dem hohen Bergrücken gen
Himmel, bis 1479 Kaiser Friedrich III. die Grafen Oswald und
Wilhelm von Thierstein mit der Burg belehnte und denen, die sie
gebrochen hatten, dem Bischof und der Stadt Straßburg, gebot, sie
zu Schutz und Trutz fest und wohnlich wieder herzustellen. Der
Obermeister der im ganzen deutschen Reiche berühmten und

71. entscheidenden Bauhütte des Münsters empfahl zu dem Zwecke
einen tüchtigen, erfahrenen Mann, und der Erwählte, Meister
Ebhardt, baute und besserte mit Straßburgischen Werkleuten und
Straßburgischem Gelde Jahre lang, ehe die Grafen von Thierstein mit
ihren Familien, einem auserlesenen Gesinde und einer ansehnlichen
Besatzung in die herrlich wieder erstandene Hochburg einziehen
konnten. Und heute, kaum zwei Wochen nach deren Übersiedelung
von ihrem Herrenhofe zu Straßburg, waren die Thore des alten
Hohenstaufenschlosses laubgeschmückt und gastlich geöffnet, um
die Menge der Geladenen einzulassen.
Nur ein Thierstein'sches Familienglied fehlte bei dem heutigen
Feste, Graf Oswalds einziger, noch unmündiger Sohn Heinrich, der
als Edelknabe auf der Burg eines alten Adelsgeschlechtes in der
Schweiz war, um dort, wie das so Brauch war, unter fremder Zucht
und Obhut ritterliches Wesen und höfischen Dienst zu lernen.
Die beiden Reisigen, die am Löwenthor die Ehrenwache hatten
und reicher gekleidet und gewappnet waren als die Knechte an den
unteren Thoren, waren Dienstleute aus der nächsten Umgebung des
Schloßherren, der eine, Marx, der Falkonier, der andere, Herni, der
Armbrustspanner des Grafen Oswald, der als der ältere der zwei
Brüder Thierstein der eigentliche machthabende Lehensträger war.
Der Dritte hier an dem Thore, der in Heroldstracht, Ottfried Isinger,
nahm als Stallmeister eine Vertrauensstellung auf der Burg ein und
kannte viele der Herren, die nach und nach mit ihren Gemahlinnen,
Söhnen und Töchtern oder auch allein die Treppen heraufkamen. Er
nannte seinen Gesellen die Namen von Fleckenstein, Müllenheim,
Andlau, Geroldseck, Dürkheim, Kageneck, Zorn von Bulach, und der
eine und der andere der Herren hatte ein freundliches Wort für ihn,
aber die meisten schritten ohne Gruß durch das Thor und würdigten
den sich tief Verbeugenden keines Blickes.
Als nun wieder einmal eine Gesellschaft von Herren und Damen so
achtlos eingetreten war, meinte Herni, der Armbrustspanner: »Es will
mich bedünken, als kämen unsere vornehmen Gäste nicht alle mit

72. fröhlichen Gesichtern. Manche schauen fast mürrisch und
unzufrieden darein.«
»Hab ich auch schon gemerkt,« stimmte der Falkonier ihm zu.
»Und wißt ihr, was ich glaube? – sie gönnen uns die schöne, große
Burg nicht; manch Einer von ihnen hauste gern selber hier oben als
hochmögender Herr und Landvogt im Wasigen.«
»Damit könntest Du Recht haben, Marx!« lachte Isinger. »Dieser
und Jener mag auf das Lehen gehofft haben, denn keine von allen
ihren Burgen ist so groß und stark wie diese außer Girbaden
vielleicht, das den Müllenheim gehört. Aber unser Herr hat beim
Kaiser einen Stein im Brett, denn er hat dem Haus Österreich gute
Dienste geleistet, und Bischof Albrecht von Straßburg, Pfalzgraf bei
Rhein und Herzog in Bayern, hat als sein Fürsprecher beim
Habsburger eine gewichtige Stimme.«
»Wer waren denn die Letzten, die so hochnäsig vorübergingen?«
fragte Herni. »Der Eine, der Gedrungene, Breitschultrige, sah Dich
ganz übermeßlich an, Ottfried!«
»Ja, der kennt mich, und ich kenne ihn auch,« erwiederte der
Stallmeister mit besonderem Nachdruck. »Es war Herr Burkhard von
Rathsamhausen mit seiner Sippe, die auf den beiden Ottrotter
Schlössern sitzen.«
»Aha!« machte Herni, »darum der böse Blick. Die haben auch
einmal hier oben gesessen, vom Kaiser Wenzel mit der Burg belehnt.
Es geht die Sage, ihrer sieben Rathsamhausen hätten sich einst, als
sie hier die Herren waren, durch Handfeste unter einander gelobt
und verpflichtet, daß kein Einziger etwas von seinem Besitz
veräußern sollte ohne Willfahren aller Übrigen.«
»So? woher weißt Du denn das?«
»Hat mir unser Graf einmal auf einem Pirschgang erzählt.«
»Ja, dann wird es sie wohl wurmen, daß sie nicht wieder die
Belehnten sind,« meinte Isinger, »denn die Rathsamhausen sind das
stolzeste Geschlecht im ganzen Wasgau.«

73. »Stolz! Graf Oswald ist auch stolz, und das wahrhaftig nicht
wenig,« sagte Marx.
»Hat auch Ursach dazu als Schloßherr von Hohkönigsburg, aber so
trotzig und starrköpfig wie Herr Burkhard ist er doch nicht. Das ist
ein abenteuriger Mann und hat ein gar grimmig Gemüth; ich könnte
euch mehr als ein verwegenes Stücklein von ihm erzählen.«
»O, unser Graf läßt nicht mit sich spaßen,« bemerkte Herni. »Wer
ihm steifnackig entgegentritt, den weiß er zu ducken, wenn's nöthig
ist.«
»Gewiß! aber in diesen letzten Tagen, wo ich viel mit ihm zu
berathen hatte, wollte er mir garnicht gefallen. Er war unruhig,
aufgeregt und schien sich auf das Bankett nicht recht zu freuen, als
sorgte er um den Verlauf und das gute Gelingen.«
»Dann konnte er es ja unterlassen,« sagte Marx.
»Das ging nicht; er ist es sich und seiner Stellung schuldig, sich
bei seinem Einzuge hier als Herr und Gebieter der mächtigsten Burg
im Lande den anderen Edelleuten zu zeigen und ihnen seinen hohen
Rang von vornherein klar zu machen. Begreifst Du das?«
»Hm! deßhalb! ja natürlich!«
Sie mußten das Gespräch abbrechen, denn jetzt nahte Seine
Hochwürden der Abt von St. Pilt mit mehreren seiner Chorherren
und einigen Chorknaben, die zur Weihe der Schloßkapelle geladen
waren und von den Wachthabenden in schweigender Ehrfurcht
gegrüßt wurden.
Es war Nachmittag. Die Sonne stand noch ziemlich hoch über dem
Walde, der mit seinen alten, mächtigen Tannen, seinen Eichen und
Buchen die Berge und Thäler unabsehbar bedeckte und aus dem
sich, hell beleuchtet, die benachbarten Burgen erhoben. Den
schroffen Gipfel zur Rechten hielt Hohrappoltstein wie eine Wacht
besetzt, zur Linken funkelte die Frankenburg und weiterhin am
steilen Bergeshang die Scherweiler Schlösser Ortenberg und
Ramstein. Tief unten aber, gradaus ergoß sich weit und breit mit

74. Städten und Dörfern und Rebengeländen das Ried, die fruchtbare
Ebene zum Rheine hin, dessen Spiegel man bei Breisach blitzen und
blinken sah. Jenseits des Stromes lagerte deutlich das
Kaiserstuhlgebirge, und im Hintergrunde schimmerten langgezogen
und wolkenhoch die Umrisse des Schwarzwaldes. Aber die äußerste
Ferne war dunstig, und die Alpen, die bei ganz klarem Wetter ihre
schneeigen Häupter über den Horizont emporrecken, waren nicht
sichtbar.
So bot der Ausblick von hier oben ein herrliches Bild, und einer der
Herren, die sich sammt ihren Damen soeben im Stallhof aus den
Sätteln geschwungen hatten, schien es vom untersten
Treppenabsatz über die Ringmauern hinweg so aufmerksam zu
betrachten, als suchte er darin einen bestimmten Punkt. Es war der
Graf Maximin, genannt Schmasman, von Rappoltstein, in dem
Geschlecht der zweite seines Namens, der mit seiner Gemahlin
Herzelande und seiner Tochter Isabella heraufgeritten war. Sie
wohnten auf der St. Ulrichsburg über dem Städtchen Rappoltsweiler,
und in ihrer Begleitung waren sein Bruder Kaspar und dessen noch
junge Gemahlin Imagina, die von ihrem ganz nahe dabei
befindlichen Felsenhorst Burg Giersberg den gleichen Weg mit ihnen
hatten, während der dritte Bruder, der im Alter zwischen jenen
beiden stand, Graf Wilhelm und seine Gemahlin von dem höher
liegenden Hohrappoltstein noch fehlten oder vielleicht schon vor
ihnen eingetroffen waren.
Gräfin Herzelande trat zu dem Umschauhaltenden heran und
fragte: »Wonach spähst Du, Schmasman?«
»Mich verdrießt es,« erwiederte der Graf, »daß von Egenolf noch
immer nichts zu sehen ist; er hätte heute pünktlich sein sollen.«
»Unser lieber Sohn wird schon nachkommen,« suchte die Gattin
den Grollenden zu beruhigen. »Ich habe ihm sein Festgewand bereit
legen lassen, daß er nur hineinzuschlüpfen braucht, wenn er vom
Gejaide heimkehrt.«

75. »Schon den dritten Tag ist er von früh bis spät auf der Pirsch.
Welches seltenen Wildes Fährte mag er so eifrig verfolgen, daß er
Alles darüber vergißt?«
»Ei, laß ihn doch pirschen, Schwager!« sprach mit anmuthiger
Gebärde Gräfin Imagina und streichelte dem Familienoberhaupte die
bärtige Wange. »Das edle Waidwerk ist nun einmal Egenolfs größte
Freude.«
»Die Freude gönn' ich ihm,« sagte der Graf, »aber heute mußte er
Rücksicht nehmen. Die Thiersteiner werden denken, er früge nichts
danach, bei dem Antrittsfest ihr Gast zu sein. Graf Oswald ist
ohnehin mißtrauisch und wittert bald hier, bald dort einen Gegner
und Neider.«
»Es fehlt ihm auch wohl an solchen nicht,« fiel Graf Kaspar ein.
»Mag sein,« antwortete der ältere Bruder. »Er hat keinen leichten
Stand und wird noch um Gunst werben müssen, ehe es ihm gelingt,
sich unter uns Alteingesessenen hier heimisch und beliebt zu
machen, falls ihm überhaupt etwas daran gelegen ist.«
»Die Thiersteiner sind selber ein altes Rittergeschlecht,« sprach
Gräfin Herzelande.
»Aber Eingewanderte, Schweizer, aus dem Aargau und ehemals
Lehensträger der Baseler Bischöfe. Der hohen Clerisei verdanken sie
zumeist den kaiserlichen Lehensbrief.«
»Schmasman, Du hast mit keinem Auge nach der Hohkönigsburg
geschielt?« neckte ihn die allzeit muntere Imagina.
»Ich?! nein, Du fürwitziges Weiblein!« lachte der Graf hell auf,
»aber ich glaube, ich hätte sie haben können, wenn ich ernsthaft
danach getrachtet hätte.«
»Und Du hättest keinen Neider gehabt,« fügte Herzelande mit
einem innigen Blick auf ihren stattlichen, ritterlichen Gemahl hinzu.
»Wer weiß? aber laßt uns hier nicht länger stehen bleiben,«
mahnte Schmasman, »ich höre neue Gäste anreiten.«

76. Sie stiegen langsam die Stufen hinan, doch nach einer kleinen
Weile sagte Schmasman zu der neben ihm gehenden Herzelande:
»Soll mich nur wundern, ob die Ottrotter heute kommen werden.«
»Du zweifelst daran?« fragte sie, wie erschrocken wieder stehen
bleibend.
»Sicher bin ich nicht. Burkhard war wenig geneigt dazu, und ich
habe ihm stark zureden müssen. Er fühlt sich durch die Art der
Einladung verletzt, weil es Graf Oswald nicht der Mühe werth
gehalten, ihm seinen Besuch zu machen, sondern nur seinen
jüngeren Bruder Wilhelm geschickt hat, der einen etwas kühlen
Empfang auf Schloß Rathsamhausen gefunden haben mag, wie ich
aus Burkhards Reden schließen muß.«
»Ist das sein einziger Grund, heut auf der Hohkönigsburg nicht
erscheinen zu wollen? Da könnten wir uns ja gleichfalls beklagen,
denn wenn auch Graf Oswald bei uns auf der Ulrichsburg war, seine
Frau und Tochter haben sich mir und Isabella nicht präsentirt, so
nahe wir ihnen auch wohnen. Wir kennen die Damen noch gar
nicht.«
»Das schadet ja nichts, Mutter« sprach hinter ihren Eltern Isabella.
»Ich freue mich auf das Fest und werde mich mit der jungen Gräfin
schon zu stellen wissen.«
»Sie haben auch in der kurzen Zeit, die sie hier sind, mehr zu thun
gehabt als nach allen umliegenden Burgen zu reiten,« entschuldigte
Herzelande selbst die ihr bisher noch Ferngebliebenen. »Wer wird
denn unter diesen Umständen so empfindlich sein!«
»So denk' ich auch,« sagte Schmasman, »aber Du kennst doch
unsern Freund Burkhard. Wenn der in übler Laune ist, ärgert ihn die
Fliege an der Wand, daß ihm die Zornader schwillt. Ich bin sehr
neugierig, ob er hier sein wird, und wenn nicht, so wird zwischen
ihm und Thierstein wenig Liebe wachsen.«
Inzwischen waren sie, bald auf einem Treppenabsatz anhaltend,
bald gemächlich weiterschreitend, an das Löwenthor gekommen.

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