It is certainly not easy to define the content and nature of mathematics
briefly. Formal explanations, which are possible nowadays due to the general
notion of structure and other logical notions, neglect not only the historical
development, but also the instinct and experience of a mathematician, who
knows what is “substantial” and “interesting” and what is not. However,
within the given understanding of mathematics, it is even more complicated
to explain what geometry is and what, in turn, is part of its history. The
dominant views on the subject of geometry, as well as its position and meaning within mathematics, have not only changed repeatedly over the course
of time. As mathematics became increasingly sophisticated, mathematicians
also took opposing positions while trying to find answers to these questions.
We will look at all these aspects in this book.
Even though geometry was mainly considered as one application of a primarily arithmeticaly oriented mathematics amongst many others in the earliest
cultures (such as in ancient Egypt, Mesopotamia, India, China, etc.), it became the core and main interest of mathematics in Ancient Greece. It was
there and then that vague notions and procedures justified only by trial and
error were transformed into a theory with definitions, axioms, theorems and
proofs. The heritage stemming from this period was so powerful for over
two thousand years that mathematicians were usually called geometricians.
Furthermore, the axiomatic-deductive method of cognition assurance, which
was based on the Greeks’ methods of dealing with geometric matters, was
referred to as “mos geometricus”, and the implementation in other sciences,
including other realms of mathematics, “more geometrico”, in other words
‘in geometrical fashion’, became a rarely achieved scientific and theoretical
program. This agenda influenced, for example, Newton in the 17th century, as
he re-founded mechanics, Galois at the beginning of the 19th century, when
criticising the contemporary situation of algebra, and Hilbert, while encouraging the scientific community to axiomatise further branches of physics in
his famous speech in 1900.
As a result of the European Renaissance, geometry was flooded by an extraordinary wave of inspiration and applications in the fields of astronomy, geodesy,
cartography, mechanics, optics, architecture, visual arts and, hence, leading
to a wealth of new challenges. The efforts made to solve these new challenges
essentially led to the development of the four pillars of the “modern” mathematics in the 17th century. These pillars are: the concept of function, coordinate-systems, differential calculus and integral calculus. Geometry gave
birth to these pillars, and then was superseded and lost its leading position to
them in a very subtle manner. Formulae and calculus took over increasingly
in the 18th century and pushed visualisation and logical argumentation aside.
The 19th century led to an enormous growth in the size and meaning of geometry. Projective, descriptive and n-dimensional geometry, vector calculus,
12 Introduction
non-Euclidean geometry, intrinsic differential geometry, topology, and also
numerous “buds” in other areas that would only come to blossom in the 20th
century, such as geometrical probability and measure theory, graph theory
and general polyhedral theory, began developing at first without any recognisable relationship to one another. This “explosion” of geometrical disciplines,
which led to the century being named the “geometrical century” according
to mathematicians, was accompanied by the disintegration of the then dominant understanding of geometry as a science of “true physical space”. We will
look at how the different approaches for dealing intellectually with the new
situation in geometry crucially coined the whole view of mathematics that
was dominant until the invention of the computer and its rising popularity.
However, we must also examine how geometry lost its central position within
mathematics over the course of the first half of the 20th century. This has
been a development that still negatively influences the organisation of mathematics in secondary and further education nowadays, despite the fact that
geometry has achieved a higher than ever level in regards to its theoretical
width and depth as well as its practical significance.
At the end of the 20th century, geometry was, on the one hand, a huge pool
of facts on the “ordinary two and three dimensional Euclidean space” and
an even bigger pool of unanswered questions on those. On the other hand,
geometry was not really thought of as being part of mathematics in the ordinary sense nowadays, but rather considered a way of thinking, which is more
or less useful and necessarily found in almost every realm of mathematics,
depending on the scientist’s personal approach. Thus, there is a geometrical
theory of numbers, a geometrical theory of functions, algebraic geometry and
geometrical stochastics. There are geometrical methods within variational
calculus, discrete and combinatorial geometry, as well as computer geometry. The latter is not to be confused with computational geometry, which
basically refers to a “theory of complexity of geometrical algorithms”.
The dichotomy of geometry suggested here has established itself very well
in the meantime. The three dimensional Euclidean space remains the appropriate model for all “ordinary” problems, even though it is only a very
rough approximation of reality according to the findings of physics. Within
the Euclidean plane we create “pictures” of everything we want to “look at”
and understand. Their meanings are associated with the dominance of seeing
amongst the human senses. Inside the n-dimensional Euclidean space, mathematics embeds functions, relations and, almost all other examined objects
by using coordinates, for example. Furthermore, geometry predominates in
all those areas where a number of possibly very abstract objects are viewed
as a “space” by using in broader sense terms taken from geometry, such as
topology, metrics, dimension and linearity, with the intention of inspiring our
imagination and to use analogies. To what extent one may want to practise
this is – as already pointed out – a matter of style. It is an intellectual technique, without which modern mathematics in the form described here could
not have developed.Introduction 3
To what degree the latter can really be considered geometry and to what extent the applied branches of geometry belong to mathematics or are already
part of engineering is debatable. In the following, we will also defend the concept that there is an “unconscious” unprofessional mathematics that coexists
with professional, deductive mathematics. The former manifests itself in the
intuitive use of notions, shapes, methods, knowledge and know-how, which
is difficult to put into words, but exists as a material product of engineering,
handcrafts and the arts. Hence, this book will also serve as a reflection on
the historical development of geometry, which will include many, often unusual aspects. We intend to contribute to the clarification of the position and
meaning of geometry within mathematics and to raise interest in it.
The critical reader, that we would like to have, may pose the question how a
history of geometry fits in a series called ‘From Pebbles to Computers’. What
computers have to do with geometry is investigated in detail in Chapter 8.5.
With regards to ‘pebbles’ (accounting tokens) we refer to the Pythagoreans,
who got some simple pre-numbertheoretical results from patterns of geometrically ordered stones. Thus they could realize why ab is forever equal to ba
and why the distance between two square numbers n2 and (n + 1)2 is always
2n + 1.
This book features problems added chapter by chapter, most of which are
not historical problems strictly speaking, but problems that result from the
history presented here. For instance, questions without answers when they
first occurred; questions that just simply did not come to mind, but were
possible; old problems that nowadays are much easier to solve given modern
methods; and suggestions that result from old problems. Most of the problems
are reduced to special cases, contain hints or are asked in a manner that will
require only a highschool or slightly more advanced mathematical background
to be solved. However, a few questions are more difficult and “open-ended”.
Here, the reader is invited to probe and explore.
We have avoided the use of first names and the inclusion of the dates births
and deaths within the main text apart from a few, well-reasoned exceptions.
As far as we could determine those data, they are available in the index of
names at the end of the book.
The pictures of the people at the beginning of each chapter are of different
styles. We cannot rely on authentic portraits from antiquity or the nonEuropean Middle Ages. (One reason being that people in Islamic countries
were often not portrayed due to religious reasons.) However, we must acknowledge that later eras felt the necessity to make pictures of their most
important personalities. In this book, a “picture” can be an imagined portrait or a symbolic graphic representation. In this respect, stamps can also
serve as a cultural document of the history of science. Multiple books have
been devoted to this exact subject [Gjone 1996, Schaaf 1978, Schreiber, P.
1987, Wußing/Remane 1989]. For example, a picture of Euclid (not shown
here) was taken from a manuscript of Roman field surveyors (agrimensores).
Here, two things are striking. First, these agrimensores thought of Euclid, the4 Introduction
master of the logical-axiomatic approach, as their forefather, and, second, the
picture has an almost oriental ambience. Considering the mix of peoples and
cultures in Alexandria at 300 BC, this may appear more realistic than some
neo-classically influenced pseudo-antique art.
From the European Middle Ages onwards, portraits began to appear intentionally more similar to the individual persons, as artists started relying on
themselves as models. For example, the portrait of Piero della Francesca is an
alleged self-portrait. It comes from his Fresco “Resurrection” (around 1465)
located in his hometown of Borgo Sansepolcro.
The picture of Ren´e Descartes presented here was painted by Frans Hals
shortly before the philosopher departed for Sweden. It is not only one of the
very few cases in which a genuinely famous painter portrayed a genuinely famous mathematician (a second example is the portrait of Felix Klein painted
by Max Liebermann), but multiple copies of this painting were subsequently
made in the 17th century reflecting varying facial expressions, which since
then partially even flipped horizontal have haunted encyclopaedias and the
science-historical literature as images of Descartes.
Peter Schreiber