Société de Calcul Mathématique, SA

We introduce the new book by Bernard Beauzamy:


Archimedes' Modern Works


A book you will be proud to have and to show to your friends !

ISBN : 978-2-9521458-7-9, ISSN : 1767-1175. Size 15,3 x 24 cm. Hardcover, 220 pages.

This book is devoted to Archimedes' modern works: direct consequences of his ideas, leading to progresses in presently unexplored, or poorly explored, domains of science.

This, at first sight, may look quite strange, because Archimedes died more than 2 200 years ago, and most people think that everything he did, no matter how clever he was, has been long ago incorporated by modern science, and that he belongs to the past. Certainly, he is considered as one of the greatest geniuses of all times, but people think that his heritage must be simply of historical value.

But this is not true at all, as this book clearly demonstrates. Two main ideas, namely the "Archimedes Maps" and "the Method", lead to completely new approaches of present problems, with far more powerful tools than the ones which were produced otherwise.

The present book is organized as follows:

We start with a quick and simple historical presentation, just in order to remind the reader about Archimedes' times. This book is not intended to address any preoccupation about history of mathematics. In the First Part, we present Archimedes Maps, and their applications to modern problems, such as monitoring, surveillance, resource allocation, and many others, with concrete examples to present concerns.

Archimedes' approaches provide interesting and relevant research subjects, the kind of research subject which will be appreciated by all Governments and funded by all Agencies. It relates deep old mathematics to new socially important preoccupations. The most hostile project director, hearing such topics, will fall onto his knees, start crying, and draw his checkbook.

The impatient reader, who wants to return to his modern computations, will ask: "Is that all?". Of course, a vast majority of scientists would be more than happy to see that their work is still useful 2 200 years after their death, and that it is used for socially important questions, carries unsolved problems and generates new research topics. A vast majority of scientists… But remember that here we deal with one of the greatest geniuses of all times. So it is not all, you have seen nothing yet. The benediction of the Governments and of the Agencies is not enough; Satan is in charge and Archimedes drives the show.

A second line of thought, developed in the second part of the present book, deals with the "weighing methods". The idea is to compare an unknown information to an artificially produced one (thus well known). Archimedes himself considered the "Method" as his masterpiece, and asked that the result (a sphere and a cylinder) should be engraved on his tomb. The Method was lost for more than 2 000 years, and, strangely enough, the mathematical contents were not rediscovered by anyone else during that time. This is strange indeed, because all classical concepts in mathematics, such as equations, polynomials, differential or integral calculus, were discovered independently many times, by many people, at many places. The Method is certainly not a classical concept; it does not fit with anything we know. It was discovered only once, and then lost.

The Method is a special case of general Greek Comparison Methods. But it is very special. Nobody knows how such an idea came to his mind, because there are no other examples. But the Method, as we see, is extremely powerful, and brings direct solutions to modern problems, such as electricity production. We present it in modern terms (which has never been done before) and investigate its modern applications to probabilities, systems of polynomial equations, optics, non destructive testing, and so on.

Then we give, for the reader's convenience, the few texts which present Archimedes life (mostly about the siege of Syracuse, 212 BC, when he died), and some comments about his abilities, compared to the abilities of modern mathematicians.

About the author : Bernard Beauzamy is a French mathematician, born 1949. He has been University Professor, 1979-1995. He established SCM SA in 1995, a company which does mathematical models. He has been chairman and CEO of this company since then. Among his previous books, "Introduction to Banach Spaces and their Geometry" and "Introduction to Operator Theory and Invariant Subspaces" have become classics.

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Table of contents

I. Introduction 9
1. Comparison methods 9
2. Weighing Methods 11
3. Organization of the book 11
Historical Facts 13

First Part : Archimedes Maps 17
Chapter I: Sphere and great circle 19
I. Introduction 19
II. A direct extension of the construction 43
III. Comments 44
Chapter II: The Maps 49
I. Introduction 49
II. The Archimedes-Lambert projection 49
III. Archimedes Transformations and Maps 54
IV. Using A-maps 55
A. Optimal location of resources 55
B. Optimal location of monitoring points 56
1. These zones should be in one piece 57
2. No zone should be too long or too large. 57
V. Simple construction of A-maps in 2d 57
A. General territories 57
1. Homothety 57
2. Rotation 57
B. The case of a square (or cube) 59
VI. Constructing Archimedes Transforms 60
A. General use 60
B. Simple structures in two dimensions 61
C. Analytic structures in higher dimensions 65
D. Constructing an Archimedes Transform 68
E. An explicit 3 dimensional example 70
F. A general attempt for A Transform 71
1. Archimedes Floating Bodies 72
2. Building Archimedes Transforms 74
G. Uniformly distributed points in a set 75

Second Part: Archimedes "Method" 79
Introduction 81
Chapter I: The Method 85
I. Statement of the results 85
II. Archimedes ideas and their evolution 89
Chapter II: Understanding the Method 91
I. Introduction 91
II. A direct application 91
III. A first statement in modern terms 93
IV. Enlarging the scope 94
A. Changing the scale on the axis 95
B. Modification of one of the solids 95
V. A direct application to the cone 96
VI. Cone and cylinder 99
VII. The Method: Extended statement 102
VIII. Four Dimensional Extension 103
Chapter III: Modern Times and Beyond 107
I. More computing power ! 107
II. Weighing methods 107
Chapter IV: Weights and Robustness 111
I. Introduction 111
II. The robustness, extended statement 112
III. The weight of a solid 113
A. Determination of the weight 113
B. Finding the center of gravity 115
IV. Finding the composition of a solid 116
A. Hiero's crown 116
B. What if the solid is not homogeneous ? 119
C. Using rotations 120
D. The moment of inertia 122
1. Replacing by an homogeneous solid 122
2. An invariant 123
Chapter V: Weights and Probabilities 127
I. Introduction 127
II. Basic results 128
III. An estimate for the volume of a solid 132
Chapter VI: Weights and Polynomials 139
I. Introduction 139
II. A pedagogical example 140
A. Description of the example 140
B. The theory behind the example 142
C. Practical implementation 142
III. General theory 144
A. Polynomials in one variable 144
B. Case of a system of polynomial equations 145
IV. Robustness of the method 146
Chapter VII: Optics 147
I. Introduction 147
II. General approach 148
III. The "Easy to Reach" Hyperbola 151
A. The construction 151
B. Approximate computations 154
C. Other similar hyperbolas: 155
D. Another approach 157
IV. Generating artificial information 159
V. Weights of solids 165
A. Basic construction 166
B. Various uncertainties 169
1. Uncertainty upon the refraction index 169
2. Uncertainty upon the the target point 170
3. Imprecise equality of moments 170
Chapter VIII: Nondestructive testing 175
I. Introduction 175
II. Eddy Currents 175
III. Archimedes comes in again 178

Third Part: What is known about A ? 181
Introduction 183
Plutarch 187
Polybius 192
Titus Livius 196
Marcus Vitruvius Pollio 198
Marcus Tullius Cicero 200
Marcus Tullius Cicero 201
The Tomb of Archimedes, modern times 205
The modern siege of Syracuse 208
Archimedes abilities 211
1. Visualisation in 2d and 3d 211
2. Conceptualisation 211
3. Intuition 213
4. Technical abilities 213
5. Conceptual abilities 214
6. Modern views about Archimedes 214

References 217

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