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Imperial College Press
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
THE LEGACY OF LEONHARD EULER
A Tricentennial Tribute
Published by
Imperial College Press
57 Shelton Street
Covent Garden
London WC2H 9HE
Distributed by
World Scientific Publishing Co. Pte. Ltd.
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All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Copyright © 2010 by Imperial College Press
LaiFun - The Legacy of Leonhard.pmd 9/4/2009, 3:04 PM
1

September 4, 2009 14:33 World Scientific Book - 9in x 6in LegacyLeonhard
Leonhard Euler (1707–1783)
ii

To my wife Sadhana, grandson Kirin, and
granddaughter Princess Maya,
with love and aﬀection.
v

Preface
Leonhard Euler (1707-1783) was a universal genius and one of the most
brilliant intellects of all time. He made numerous major contributions to
eighteenth century pure and applied mathematics, solid and fluid mechan-
ics, astronomy, physics, ballistics, celestial mechanics and optics. Among
the greatest mathematical and physical scientists of all time including New-
ton, Leibniz, Gauss, Riemann, Hilbert, Poincaré, and Einstein, Euler’s
monumental contributions are generally considered unique and fundamental
and have shaped much of the modern mathematical sciences. The Eulerian
universal view is the dominant influence in the fields of physics, astronomy,
continuum mechanics, natural philosophy, pure and applied mathematics.
He published almost 900 original research papers, memoirs, and 25 books
and treatises on mathematical and physical sciences. Even without the pub-
lication of his collected works, Leonhardi Euleri Opera Omnia, still in the
process of being edited by the Swiss Academy of Sciences, his voluminous
published works clearly demonstrate his amazing creativity, achievements
and contributions to a wide variety of subjects in mathematical, physical,
and engineering sciences. He also made contributions to other disciplines
including geography, chemistry, cartography, music, history and philosophy
of science.
The following quotations give some idea of the special veneration and
affection in which he was held by his contemporaries and successors. P.
S. Laplace wrote: “Read Euler, read Euler, he is the master of us all.”
It is a delight to quote Karl Friedrich Gauss: “... the study of Euler’s
works will remain the best school for different fields of mathematics and
nothing else can replace it.” On the other hand, the great twentieth century
mathematician André Weil said: “No mathematician ever attained such a
position of undisputed leadership in all branches of mathematics, pure and
applied, as Euler did for the best part of the eighteenth century.”
vii

viii The Legacy of Leonhard Euler — A Tricentennial Tribute
The tercentenary of Euler’s birth has recently been celebrated with glo-
rious success to pay a special tribute to this legendary mathematical and
physical scientist of the eighteenth century. There is absolutely no doubt
that Euler laid the solid foundations on which his contemporaries and suc-
cessors of the last three centuries were able to build new ideas, results, the-
orems and proofs. His extraordinary genius created a simple language and
style, unique symbols, and notations in which mathematical and physical
sciences have developed ever since. His name is also synoymously associ-
ated with a large number of results, terms, equations, theorems, and proofs
in mathematics and science.
Throughout his extensive research contributions and lucid writings, Eu-
ler was always influenced by his own thought as follows: “Since a general
solution must be judged impossible from want of analysis, we must be con-
tent with the knowledge of some special cases, and that all the more, since
the development of various cases seems to be the only way to bringing us at
last to a more perfect knowledge.” In addition, Euler’s quest of new knowl-
edge was simple and direct. His standards of mathematical rigor were far
more primitive than those of today, but as Richard Feynman (1918-1988),
an American genius, so cogently observed in the twentieth century: “...
However, the emphasis should be somewhat more on how to do the math-
ematics quickly and easily, and what formulas are true, rather than the
mathematician’s interest in methods of rigorous proof.” Euler has often
been criticized for his lack of mathematical clarity, elegance and rigor. In-
tuition played an important role in his discoveries. He was always interested
in creating a set of new ideas and results in the most diverse fields of math-
ematical and physical sciences. So, it is perhaps true that Euler’s work
met all requirements for rigor in his time. He was often satisfied when his
intuition gave him full confidence that the proof of results could be carried
through to complete mathematical rigor and then assigned the completion
of the proof to others.
In pure mathematics, his major research fields included differential and
integral calculus, infinite series and products, algebra, number theory, ge-
ometry of curves and surfaces, topology, graph theory, ordinary and par-
tial differential equations, calculus of variations, special functions, elliptic
functions, and integrals. In applied mathematics, he published papers on
the mechanics of particles and of solid bodies, elasticity and fluid mechan-
ics, optics, astronomy, lunar, and planetary motion. He also wrote many
textbooks on mechanics, mathematical analysis, algebra, analytic geome-
try, differential geometry, and the calculus of variations. In mathematical

Preface ix
physics, Euler discovered the fundamental partial differential equations for
the motion of inviscid incompressible and compressible fluid flows, and ap-
plied them to the blood flow in the human body. In the theory of heat,
he closely followed Daniel Bernoulli to describe heat as an oscillation of
molecules. He mathematically investigated the propagation of sound waves
and obtained many original results on refraction and dispersion of light.
Euler was one of the few scientists of the eighteenth century to favor the
wave theory as opposed to the particle theory of light. Euler also made
remarkable contributions to applied mathematics and engineering science.
For example, he studied the bending of beams and calculated the critical
load of columns. He described the perturbation effect of celestial bodies
on the orbits of planets. He obtained the paths of projectiles in a resisting
medium. He worked on the theory of tides and currents. His study on the
design of ships helped navigation. His three volumes on achromatic optical
instruments contributed to the design of microscopes and telescopes.
Euler maintained extensive contacts and correspondence with many of
the most eminent mathematical scientists of the time including Christian
Goldbach, A. C. Clairaut, Jean d’Alembert, Joseph Louis Lagrange, and
Pierre Simon Laplace. This led to the development of personal and pro-
fessional relationship between them. There was an amicable correspon-
dence between Euler and Goldbach, and Euler and Clairaut which dealt
with topical problems of number theory, mathematical analysis, differential
equations, fluid mechanics, and celestial mechanics. There were neither
any disagreements nor claims of one against the other. They discussed
all mathematical ideas and problems openly, often significantly prior to
their publication. Euler in Berlin and d’Alembert in Paris had an exten-
sive mathematical correspondence over many years. In 1757, they had a
strong disagreement, which eventually led to an estrangement, on whether
discontinuous or non-differentiable functions are admissible solutions of the
vibrating string problem. There was also a priority dispute between them
on the theory of the precession of the equinoxes and nutation of the axis of
the Earth. However, after d’Alembert visited Euler in Berlin in 1763, their
relation became more cordial. In 1759, the young Lagrange joined in the
discussion of solutions with a controversial article which was criticized by
both Euler and d’Alembert. However, Lagrange sided with most of Euler’s
views. In 1761, Lagrange, seeking to meet the criticisms of d’Alembert and
others, provided a different treatment of the vibrating string problem. The
debate continued for another twenty years with no resolution. The issues
in dispute were not resolved until Joseph Fourier picked up the subject in

x The Legacy of Leonhard Euler — A Tricentennial Tribute
the next century. Although Euler made an important and seminal con-
tribution to calculus of variations, Lagrange, at the age of 19, made the
first formulation of the equations of analytical dynamics according to the
principles of the calculus of variations, and his approach was superior to
Euler’s semi-geometric methods. Thus, the classical Euler-Lagrange vari-
ational problem of determining the extremum value of a functional led to
the celebrated Euler-Lagrange equation.
It has been calculated that his publications during his life averaged
about 800 pages a year. His complete works entitled Opera Omnia con-
sist of nearly 80 volumes, each approximately between 300 and 600 pages.
Euler was undoubtedly the most prolific mathematical and physical scien-
tists of all time. His whole working life was totally dedicated to the pur-
suit of fundamental discovery, dissemination of mathematical and scientific
knowledge, and popularization of their value to common people. His famous
three-volume Letters to a German Princess on Different Subjects in Natural
Philosophy was one of the most popular books on science ever written and
it was translated from German into eight different languages. The Letters
addressed a wide variety of subjects including optics, acoustics, mechanics,
astronomy, music, dioptrics, electricity and magnetism. This publication
was essentially a unique encyclopedia of physical and philosophical ideas
written in a popular style for the widest possible common audience. This
work formed the basis for the reform of the teaching of physics and science.
These are just a few examples of his prodigious contributions.
This volume is intended as a tricentennial memorial tribute to this uni-
versal mathematical scientist. My desire as well as interest in writing this
book commemorating Euler’s major contributions to mathematical and
physical sciences is founded on the deep respect and admiration for him
that I have gained from my own study and research of a small fragment of
his voluminous work. The origin of this book was essentially based on my
postgraduate course in the theory of elliptic functions and integrals with
applications in 1960s. Indeed, I was further stimulated by my own articles
and lectures for the last ten years on Euler and his major contributions.
These publications and presentations are intended for the great majority of
senior undergraduates and graduate students of mathematics, physics, and
engineering.
The intense and narrow specialization of contemporary mathematics is a
fairly recent phenomenon. The professional mathematical scientists spend
almost all of their time and energy on segments of mathematics or science
that seem to have little relationship to each other. They have hardly any

Preface xi
time or opportunity to become familiar with the history of mathematics and
science. The emphasis on history may provide more broad perspective on
the whole subject and relate the subject matter of the courses not only to
each other, but also to the major developments of mathematical thoughts.
As Henri Poincaré eloquently wrote: “If you wish to foresee the future of
mathematics, our proper course is to study the history and present condi-
tion of science.” The writing of this volume was greatly influenced by the
above thought of Poincaré. This book may serve to some extent as a his-
torical introduction to mathematical sciences with the major emphasis on
selected Euler’s contributions. I hope that it will be helpful to professional
and prospective mathematical scientists.
While writing this book as an exposition and survey of history, three ma-
jor objectives have been kept in mind. The first is to focus each chapter on
a subject to which Euler made a significant research contribution. Included
are a short history of mathematical developments and discoveries before
Euler, and a brief sketch of the life, work, career, and major achievements
of Euler. The second is to present some historically significant, elegant, or
unexpected theorems, proofs and results with applications. The third is to
convey something of the fascination of mathematical sciences — of their
beauty, intellectual power, and wide variety. This book does not require a
graduate school mastery of any branch of mathematical sciences. It con-
tains a wide variety of material accessible to the widest possible audience
of mathematically literate readers.
It is my pleasure to express my grateful thanks to many friends, profes-
sional colleagues and students around the world who offered their sugges-
tions and help at various stages of the preparation of the book. I am par-
ticularly grateful to my graduate students, Arunabha Biswas and Arindam
Roy for helping me during the preparation of the book, especially for draw-
ing all the figures in the book. My special thanks to Ms. Veronica Chavarria
who cheerfully typed the manuscript with constant changes and revisions
and carefully checked all the names in the text. In spite of the best efforts
of everyone involved, some typographical errors will doubtlessly remain. I
wish to express thanks to Ms. Lai Fun Kwong and the Production Depart-
ment of Imperial College Press for their help and cooperation. Finally, I am
deeply indebted to my wife, Sadhana, for her understanding and tolerance
while the book was being written.
Lokenath Debnath
Edinburg, Texas

Leonhard Euler (1707-1783):
Chronology
April 15, 1707 Euler was born in Basel, Switzerland.
1720 At the age of 13, he graduated from the University
of Basel with Philosophy Major.
1724 At the age of 17, he received his Master’s degree with
a thesis comparing the philosophy of René Descartes
with that of Sir Isaac Newton.
1726-1727 Published first two research papers on the construc-
tion of isochronous curves in a resisting medium and
on reciprocal algebraic trajectories.
1726-1741 Maintained a regular contact with Johann Bernoulli
and his two sons Daniel and Nicholas.
1727-1741 Joined the Imperial Russian Academy of Sciences in
St. Petersburg and worked with Daniel Bernoulli and
Jacob Hermann. He was selected to become a Pro-
fessor of Mathematics at the age of 26, and to be in
charge of the Geography Department. This 14-year
stay in St. Petersburg was the first golden period of
his life.
1729 He first discovered the first fundamental function
in real and complex analysis, known as the gamma
function defined by the infinite integral
xiii

xiv The Legacy of Leonhard Euler — A Tricentennial Tribute
Γ(x) =
∞
0
e−t
tx−1
dt, Re x 0,
as a generalization of the factorial function.
He also introduced the Eulerian integral of the first
kind, known as the Euler beta function, in the form
B(x, y) =
1
0
tx−1
(1 − t)y−1
dt, x 0, y 0.
There is an elegant and beautiful relation between
these two functions given by
B(x, y) =
Γ(x)Γ(x)
Γ(x + y)
.
1730 Euler discovered his celebrated zeta function for real
s defined by an infinite series
ζ(s) =
∞

n=1
1
ns
, s 1.
The value of ζ(s) for s = 1 led him to discover the
divergent harmonic series
ζ(1) =
∞

n=1
1
n
=
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+· · ·+
1
n
+· · · ∞,
where each of its terms is the harmonic mean of the
two neighboring terms.
1732 Euler stated memorable the Euler-Maclaurin sum-
mation formula which was independently discovered
by Euler and Maclaurin. For a function f(x) with
continuous derivatives of all orders up to and includ-
ing (2m + 2) in 0 ≤ x ≤ n, then the sum
n

k=0
f(k) is
given by the Euler-Maclaurin summation formula

Leonhard Euler (1707-1783): Chronology xv
n

k=0
f(k) =
n
0
f(t)dt +
1
2
[f(0) + f(n)]
+
m

k=1
B2k
(2k)!

f(2k−1)
(n) − f(2k−1)
(0)

+ Rm,
where B2k are the Bernoulli numbers, and the re-
mainder term Rm is
Rm = 1
(2m+1)!
n
0
B2m+1(t)f(2m+1)
(t)dt.
1734 He married Catharina Gsell, daughter of a Swiss
artist then working in Russia and they had 13 chil-
dren and only 5 survived infancy.
1734-1737 Using the divergence of the harmonic series and the
identity
ζ(1) =
∞

n=1
1
n
=

p

1 −
1
p
−1
,
Euler proved that the number of primes is inﬁnite.
Euler proved another remarkable theorem, for s 1,
ζ(s) =
∞

n=1
1
ns
=

p

1 −
1
ps
−1
,
where p is a prime. This establishes an unexpected
link between the zeta function in analysis and the
distribution of prime numbers in number theory.
1735 He discovered four distinct solutions of the Basel
problem of ﬁnding the sum of the squares of the re-
ciprocals of the integers, that is,
ζ(2) =
1
12
+
1
22
+
1
32
+ · · · +
1
n2
+ · · · =
π2
6
.

xvi The Legacy of Leonhard Euler — A Tricentennial Tribute
1736 Euler first solved the famous problem of the Seven
Bridges of the city of Königsberg on the River Pregel
that to determine a route around the city so that
one can cross seven bridges once and only once. He
proved that such a route is impossible. But with an
extra bridge added, he proved that the solution is
possible. This marked the beginning of a new area
of mathematics known today as graph theory.
He also published his two large volumes, Mechanica
sive motus scientia analytice exposita (Mechanics or
the science of motion, expounded analytically). The
two-volume Mechanica dealt with a comprehensive
treatment of almost all aspects of mechanics includ-
ing the mechanics of rigid, flexible and elastic bodies
as well as fluid mechanics, celestial mechanics and
ballistics.
He first discovered his celebrated equations which
described the principles of conservation of mass,
momentum, and energy. He then formulated the
renowned Euler equations of motion for both incom-
pressible and compressible inviscid fluid flows.
1738-1740 He won the Grand Prix of the Paris Academy and be-
came an eminent mathematical scientist in the whole
of Europe. He became blind in the right eye in 1738.
1738-1741 Political conditions of Russia became very unstable
and the Russian Government was reluctant to sup-
port scientific research. He became concerned about
his future in St. Petersburg. Euler left St. Peters-
burg in 1741 for the Berlin Academy in Germany.
1739 He published his treatise on the theory of music en-
titled An attempt at a new theory of music, clearly
expounded on the most reliable principle of harmony.

Leonhard Euler (1707-1783): Chronology xvii
1740 His other magnificent discovery was the universal
Euler constant γ defined by the limit
γ = lim
n→∞

1 +
1
2
+
1
3
+ · · · +
1
n
− ln n

= 0.577215665....
This constant was linked with the finite harmonic
series and the logarithm function.
Euler single-handedly created the theory of parti-
tions of numbers by a brilliant use of generating
functions and formal power series. He surprised the
mathematical community of the world with the re-
markable expansion
∞

n=1
(1 − xn
) =
∞

n=−∞
(−1)n
x
1
2 (3n2
−n)
.
This led him to discover the Euler Pentagonal Num-
ber Theorem in number theory.
1741 At the invitation of the King Frederick the Great
of Prussia, Euler joined the newly organized Berlin
Academy of Science (originally founded by G. W.
Leibniz in 1700).
1741-1766 Remained in Berlin Academy of Science for 25
years and completed his greatest work on Mechanics,
Physics, Pure and Applied Mathematics. His 25-year
stay in Berlin was regarded as the second golden pe-
riod of his life.
1743 Two of the greatest discoveries are Euler’s elegant
and beautiful formulas,
e±ix
= cos x ± i sin x,

xviii The Legacy of Leonhard Euler — A Tricentennial Tribute
and he then established two magniﬁcent formulas
eiπ
= −1 or eiπ
+ 1 = 0 and e2πi
− 1 = 0.
These simple formulas relate to six fundamental con-
stants e, i, π, 0, 1 and −1 in mathematics and sci-
ence.
1744 He was elected as Member of the Royal Society of
London and to the Paris Academy of Sciences, among
other many honors and awards.
He published his masterpiece treatise entitled Metho-
dus Inveniendi Lineas Curvas Maximi Minimive
proprietate gaudentes sive solutio problematis isoper-
imatrici Latissimo Sensu Acceptl (A method for dis-
covering curved lines that enjoy a maximum or min-
imum property, or the solution of the isoperimetric
problem taken in its widest sense) which contained
his memorable extensive research in the theory of
Calculus of Variations.
He published his major research monograph on Theo-
ria Motuum Planetarum et Cometarum (The Theory
of Motion of Comets and Planets) with solutions
of major problems of theoretical astronomy with
nature, structure, motion and action of comets and
planets.
1745 He translated Benjamin Robins’ 1742 treatise “New
Principles of Gunnery” in German with a large ex-
tensive commentary appended to it. His revised and
expanded version entitled Artillerie was published.
It was Euler who ﬁrst made a serious attempt to
study divergent series and integrals in a systematic
manner. From the mathematical and physical point
of view, Euler’s ingenious work was very useful and
served as the foundation of more modern theory of
divergent series and integrals with physical applica-
tions.

Leonhard Euler (1707-1783): Chronology xix
1746 Euler’s New Tables for Calculating the Position of
the Moon was published in Berlin.
1748 He published his two-volume masterpiece treatise
on mathematical analysis entitled Introductio in
Analysin Infinitorum (Introduction to the Analysis
of the Infinite).
1749 He completed the remarkable two-volume treatise
Scientia Navalis seu tractatus de construendis ac
dirigendis navibus (Naval Science) or Ship building
and Navigation in Berlin.
Euler proved a beautiful formula for ζ(2n), where
n(≥ 1) is a natural number, and he also discovered a
remarkable functional equation for the zeta function
in the form
ζ(1 − s) = 2(2π)−s
Γ(s)ζ(s) cos
πs
2
,
where Γ(s) is the gamma function discovered by Eu-
ler in 1729.
1751 He established a major milestone through his exten-
sive research and study of elliptic functions and el-
liptic integrals. He also proved many notable results
which dealt with the addition and multiplication the-
orems of elliptic integrals. His brilliant work stimu-
lated tremendous interest amongst many great math-
ematicians including Gauss, Lagrange, Jacobi, Abel,
Galois, Weierstrass and Riemann.
1753 Euler published his Memoir on Ballistics with the
first complete analysis of the equations of ballistic
motion in the atmosphere. He also published his
works on celestial mechanics and the fundamental
monograph on the theory of lunar motion.

xx The Legacy of Leonhard Euler — A Tricentennial Tribute
1755 He published his first comprehensive textbook on dif-
ferential calculus entitled Institutiones Calculi Dif-
ferentialis (Foundation of Differential Calculus).
1758 Euler discovered the celebrated formula
V − E + F = 2 (the Euler characteristic)
for a regular polyhedra and tried to prove it.
He completed his notable masterpiece Memoir on
the Calculus of Variations. In addition, Euler was
involved in major administrative duties of the
Academy including the Observatory, Botanic Gar-
dens, Calendars and Maps.
1759-1766 He served as President of the Berlin Academy under
the direct supervision of King Frederick the Great of
Prussia who did not respect and trust him. The King
Frederick offered the Presidency of the Academy to
d’Alembert who was Euler’s scientific rival. So, Euler
became very concerned about his future career in
Berlin.
1760 He first discovered the Euler phi-function φ(n) to
generalize the Fermat Little Theorem in the form
aφ(n)
≡ 1 (mod n), where (a, n) = 1. This function
has modern applications to a new area of mathemat-
ics known today as cryptography which deals with
secure system transmission of secret messages and
ciphers.
1760-1762 Euler wrote his famous Letters to a German
Princess, Anhalt-Dessau on different subjects in nat-
ural philosophy, astronomy, optics, music, acoustics,
electricity and magnetism that was one of the most
popular science books ever written in the history of
science.

Leonhard Euler (1707-1783): Chronology xxi
1765 Euler published his great third volume of his mechan-
ics book entitled Theoria motus corporum solidorum
seu rigidorum (Theory of Motion of Rigid Bodies).
It contained Euler’s differential equations of motion
of a rigid body under external forces. He introduced
the original idea of employing two coordinates — one
fixed, the other moving attached to the body, and
first derived differential equations for the angles be-
tween the respective coordinates axes, now called the
Euler angles. He worked out many major and inter-
esting examples including the intriguing motion of
the spinning of the top.
1766 At the age of 59, Euler received a cordial invitation
from the German Princess, Catherine the Great of
Russia, and moved back to St. Petersburg Academy.
1766-1783 His second St. Petersburg stay of 17 years can be
regarded as the third golden period of his life. This
period was very famous for his prolific and prodigious
scientific activities as he completed a large number of
epochal mathematical and scientific treatises and a
highly successful and popular work on mathematics,
science, and history and philosophy of science. It was
also a time that Euler suffered from several major
health problems and family disasters.
1768 Euler wrote his treatise on geometrical optics in three
volumes and his tract on the motion on the Moon.
1768-1770 His three-volume textbook on integral calculus enti-
tled Institutiones Calculi Integralis was published.
1769-1771 The three volumes of Euler’s Dioptrics were pub-
lished. This work dealt with his extensive research
in optical sciences and optical instruments including
telescopes and microscopes.

xxii The Legacy of Leonhard Euler — A Tricentennial Tribute
1770 Euler published his two-volume treatise on Voll-
ständige Anleitung zur Algebra (Elements of Alge-
bra).
1771 His house was badly burnt down in a ﬁre. He lost
his household, but most of his books and manuscripts
were saved. He became almost blind due to an un-
successful surgery to remove cataract in his left eye.
1773 He published his remarkable book Théorie compléte
de la construction et de la manoeuvre des vaisseaux
(The Complete Theory of Ship Building and Naviga-
tion) which contained the theory of the tides and the
sailing of ships.
1773-1776 His wife, Catherina, died in 1773 after 40 years of
their married life and he remarried to Catharina’s
half sister, Salmone Gsell, in 1776.
1776 Euler returned to mechanics with his seminal work
on deﬁnite formulation of the principles of linear and
angular momentum.
1776-1783 Euler completed almost half of his work during his
most productive second 18-year stay at St. Peters-
burg. He continued his research on optics, algebra,
geometry, celestial mechanics, naval science, lunar
and planetary motion. In addition, he did some ma-
jor research on probability theory and statistics, car-
tography, geography, chemistry, agriculture, pension
funds, history of mathematics and science, medical
and herbal remedies.
September 18,
1783
At the age of 76, Euler died in St. Petersburg as a
result of a stroke while playing with his grandson.

Contents
Preface vii
Leonhard Euler (1707-1783): Chronology xiii
1. Mathematics Before Leonhard Euler 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Pythagoras, the Pythagorean School and Euclid . . . . . . 2
1.3 The Major Impact of the European Renaissance on Math-
ematics and Science . . . . . . . . . . . . . . . . . . . . . 11
1.4 The Discovery of Calculus by Newton and Leibniz . . . . 21
2. Brief Biographical Sketch and Career of Leonhard Euler 31
2.1 Euler’s Early Life . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Euler’s Professional Career . . . . . . . . . . . . . . . . . 33
3. Euler’s Contributions to Number Theory and Algebra 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Euler’s Phi Function and Cryptography . . . . . . . . . . 57
3.3 Euler’s Other Work on Number Theory . . . . . . . . . . 60
3.4 Euler and Partitions of Numbers . . . . . . . . . . . . . . 68
3.5 Euler’s Contributions to Continued Fractions . . . . . . . 78
3.6 Euler’s Contributions to Classical Algebra . . . . . . . . . 83
4. Euler’s Contributions to Geometry and Spherical Trigonometry 101
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Euler’s Work in Plane Geometry . . . . . . . . . . . . . . 102
xxiii

xxiv The Legacy of Leonhard Euler — A Tricentennial Tribute
4.3 Incircle, Incenter and Heron’s Formula for an Area of a
Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.4 Centroid, Orthocenter and Circumcenter . . . . . . . . . . 115
4.5 The Euler Line and the Euler Nine-Point Circle . . . . . . 121
4.6 Euler’s Work on Analytic Geometry . . . . . . . . . . . . 126
4.7 Euler’s Work on Differential Geometry . . . . . . . . . . . 132
4.8 Spherical Trigonometry . . . . . . . . . . . . . . . . . . . 146
5. Euler’s Formula for Polyhedra, Topology and Graph Theory 153
5.1 Euler’s Formula for Polyhedra . . . . . . . . . . . . . . . . 153
5.2 Graphs and Networks . . . . . . . . . . . . . . . . . . . . 164
6. Euler’s Contributions to Calculus and Analysis 175
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.2 Euler’s Work on Calculus . . . . . . . . . . . . . . . . . . 177
6.3 Euler and Elliptic Integrals . . . . . . . . . . . . . . . . . 184
7. Euler’s Contributions to the Infinite Series and the Zeta
Function 197
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.2 Euler and the Infinite Series . . . . . . . . . . . . . . . . . 200
7.3 Euler’s Zeta Function . . . . . . . . . . . . . . . . . . . . 210
7.4 Euler and the Fourier Series . . . . . . . . . . . . . . . . . 224
7.5 Generalized Zeta Function . . . . . . . . . . . . . . . . . . 230
7.6 Applications of the Zeta Function to Mathematical Physics
and Algebraic Geometry . . . . . . . . . . . . . . . . . . . 231
8. Euler’s Beta and Gamma Functions and Infinite Products 235
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.2 Euler’s Beta and Gamma Functions . . . . . . . . . . . . 236
8.3 Applications of the Euler Gamma Functions . . . . . . . . 247
8.4 Euler’s Contributions to Infinite Products . . . . . . . . . 248
9. Euler and Differential Equations 255
9.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . 255
9.2 Euler’s Contributions to Ordinary Differential Equations . 261
9.3 Euler’s Work on Partial Differential Equations . . . . . . 275
9.4 Euler and the Calculus of Variations . . . . . . . . . . . . 288

Contents xxv
10. The Euler Equations of Motion in Fluid Mechanics 297
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 297
10.2 Eulerian Descriptions of Fluid Flows . . . . . . . . . . . . 298
11. Euler’s Contributions to Mechanics and Elasticity 309
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 309
11.2 Euler’s Work on Solid Mechanics . . . . . . . . . . . . . . 312
11.3 Euler’s Research on Elastic Curves . . . . . . . . . . . . . 320
11.4 Impact of Euler’s Work on Modern Aerodynamics . . . . 328
12. Euler’s Work on the Probability Theory 337
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 337
12.2 Euler’s Work on Probability . . . . . . . . . . . . . . . . . 341
12.3 Euler’s Beta and Gamma Density Distributions . . . . . . 345
13. Euler’s Contributions to Ballistics 349
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 349
13.2 Euler’s Research on Ballistics . . . . . . . . . . . . . . . . 352
14. Euler and his Work on Astronomy and Physics 359
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 359
14.2 Euler’s Contributions to Astronomy . . . . . . . . . . . . 363
14.3 Euler’s Work on Physics . . . . . . . . . . . . . . . . . . . 369
Bibliography 373
Index 383

Chapter 1
Mathematics Before Leonhard Euler
“Number rules the Universe.”
The Pythagoreans
“Geometry has two great treasures: one is the theorem of
Pythagoras; the other, the division of a line in extreme and
mean ratio. The first we may name as a measure of gold, the
second we may name as a precious jewel.”
Johann Kepler
“As long as algebra and geometry proceed along separate paths,
their advance was slow and their applications were limited. But
when these sciences joined company, they drew from each other
fresh vitality and hence forward marched on at a rapid pace
towards perfection.”
Joseph Louis Lagrange
1.1 Introduction
Historically, mathematics originated from the fundamentals of counting in
arithmetic. It is considered one of the greatest achievements of the human
endeavor. Originally, it was the study of numbers or symbols and their
relations. These symbols were created to stand for the natural numbers 1,
2, 3, · · · which form an infinite collection on which the basic arithmetic
operations of addition and multiplication could be performed. It was the
Ancient Hindus and Greeks who first discovered the natural numbers, but
they did not acknowledge negative numbers. The first systematic algebra
to use zero, negative numbers, and the decimal system was developed by
1

2 The Legacy of Leonhard Euler — A Tricentennial Tribute
Hindu mathematicians in India during the seventh century A.D. They used
positive and negative numbers to handle financial transactions involving
credit and debit. Subsequently, mathematics has successfully been used to
precisely formulate laws of nature.
Mathematics has more than 5000 years of history. By 3000 B.C., the
people of Babylonia, China, Egypt and India had developed early and prac-
tical number systems. They used the knowledge of number systems in busi-
ness, industry, government, science and indeed, in everyday life. Between
600 and 300 B.C., the Greeks took the next great step in advancing the
knowledge of arithmetic, algebra, geometry and astronomy. They appear
to have been the first to develop mathematical theory of arithmetic and
geometry. Subsequently, it was realized that all scientific problems depend
on mathematics for qualitative and quantitative descriptions and mathe-
matical formulas became very useful for experiments and observations.
1.2 Pythagoras, the Pythagorean School and Euclid
Most of our knowledge of mathematics of the classical age came from the
writings of many mathematicians and philosophers including Pythagoras
(580-500 B.C.), Euclid (330-275 B.C.), Archimedes (287-212 B.C.) and
Apollonius (260-200 B.C.). For the Greeks, mathematics was then largely
synonymous with geometry which dealt with the measurement of land. In-
deed, geometry was derived from two Greek words meaning measurement of
the earth. The Ancient Egyptians used geometry to measure the size of their
firm lands, and to find boundaries of these firm lands after yearly floods of
the Nile River washed away or covered old landmarks. Classically, geome-
try dealt with the size, shape, area, volume or position of any object. More
importantly, geometrical concepts and numerical ideas have been wrapped
up together for thousands of years and they cannot be separated at all.
In about 540 B.C., Pythagoras established a school of mathematics and
natural philosophy at Crotona in southern Italy. The influence of this great
master Pythagoras was simply remarkable as his students and followers
were very loyal to him and they formed themselves a society or brother-
hood. They were known as the Order of the Pythagoreans. Members of
the Pythagorean School were very obedient and loyal to their great master,
shared everything in common, held the same religious and philosophical
beliefs, made a commitment to the same pursuits and bound themselves
to an oath not to reveal their own secrets and teachings of the school.

Mathematics Before Leonhard Euler 3
Fig. 1.1 The star pentagon.
It is remarkable that the school discovered a beautiful star pentagon or
pentagram (see Figure 1.1), the most fitting badge of the Pythagorean
brotherhood. It was also a fitting symbol of mathematics and the Greek
emblem of health. In addition to their unique contributions to mathemat-
ics, particularly, to geometry and number theory; the Pythagoreans were
specially interested in the study of medicine and music. Figure 1.2 shows
an infinite sequence of nested pentagons.
They developed a large body of knowledge in geometry and properties
of numbers, and proved a large number of geometrical theorems including
one of the most famous theorems in geometry known as the Pythagorean
Theorem:
c2
= a2
+ b2
(1.2.1)
for any right-angled triangle of sides a and b adjoining the right angle and
c is the hypotenuse.
This theorem has probably received more diverse proofs than any other
theorem in all of mathematics. In the second edition of his book entitled
The Pyathgorean Proposition, E. S. Loomis (1968) has reported about 367
demonstrations (or proofs) of this famous theorem. Making reference to
Figure 1.3, a dissection type proof of this famous theorem can be given as

A
D
C
E
B
Fig. 1.2 Sequence of nested pentagons.
follows. The ﬁrst square of side (a + b) is dissected into four equal right
angled triangles of sides a and b and a square of side c so that (a + b)2
=
4(1
2 ab) + c2
= 2ab + c2
. The second ﬁgure is dissected into two squares
and four equal right angled triangles so that (a + b)2
= 4(1
2 ab) + a2
+ b2
.
Equating two equal expressions readily gives (1.2.1).
One of the Indian mathematicians, Bhaskara gave a second proof of the
Pythagorean theorem by drawing the altitude on the hypotenuse of the
a
a
a
c
c
c
c
c
c
a
b
b
b
b
a
a
b
b b
Fig. 1.3 Dissection of two equal squares.

A B
C
D
a
c
n
h
m
b
Fig. 1.4 A right angled triangle with ∠ACB = 90◦.
right angled triangle ABC with ∠ACB = 90◦
. It follows from similar right
angled triangles as shown in Figure 1.4 that
a
c
=
m
a
and
b
c
=
n
b
.
Bhaskara gave another proof using dissection in which the square on
the hypotenuse is divided into four equal triangles, (see Figure 1.5) each is
congruent to the given right angled triangle of sides a, b, and c and a square
with side b − a. Clearly, a simple algebra supplies the proof as follows:
c2
= 4

1
2
ab

+ (b − a)2
= a2
+ b2
or
a2
= cm and b2
= cn
so that
a2
+ b2
= c(m + n) = c2
.
c
c
c
c
c
c
a
a
a
a
a a
b-a
b-a
b
b
b-a
Fig. 1.5 Dissection of a square into four triangles and a square.

A famous British mathematician, John Wallis (1616-1703) rediscovered
this ancient proof in the seventeenth century. For several other proofs of the
Pythagorean theorem, the reader is referred to Loomis’ book (1940) and
to a book entitled Great Moments in Mathematics Before 1650 by Howard
Eves (1911-2004) published in 1983.
When a = b = 1, c =
√
2, an irrational number which cannot be
expressed as the ratio of two integers. In other words, the rational numbers
are not adequate for measuring the hypothenuse of a right-angled triangle
whose base and height are unity. The discoveries of Pythagoras theorem and
the irrational numbers were the greatest achievements of the Pythagoreans
in the history of mathematics. Indeed, the Pythagorean discovery of an
irrational number led to the solution of equations such as
x2
= 2. (1.2.2)
Although x =
√
2 is irrational, but it can be expressed in terms of approx-
imate rational numbers 1.4, 1.41, 1.414, · · · with finite number of decimal
places.
The Pythagoreans also proved many geometrical theorems including the
equality of the base angles of an isosceles triangle, and the sum of three
angles of a triangle is equal to two right angle. They also proved the famous
algebraic identities
(a ± b)2
= a2
± 2ab + b2
(1.2.3ab)
using purely geometrical arguments.
More remarkably, they made three great discoveries: the first, one was
the introduction of proof in mathematics, that mathematical proof must
proceed from given assumptions, the second one was that the natural num-
bers were insufficient for the construction of mathematics, and the third
one was that the set of natural, rational and irrational numbers form the
complete set of real numbers with the geometrical interpretation. Geometri-
cally, to each real number corresponds to one and only one point on the real
line. In addition, there were three famous unsolved problems that exerted
so great influence on the development of Greek mathematics. The original
idea was to solve them by ruler and compasses constructions. However, the
impossibility of solutions by a ruler and a compass kept these problems at
the center of the mathematical stage for many centuries.
The first problem was known as the Delian problem which dealt with
the doubling of a cube, that is, to construct a cube whose volume is twice
that of a given cube. Mathematically, the problem is to find a solution of
x3
= 2. (1.2.4)

In about 400 B.C., Archytas brilliantly solved the problem by finding the
point of intersection of three surfaces in three-dimensional space: a cylinder,
a cone, and a torus generated by rotating a circle about one of its tangents.
This was indeed a most remarkable achievement of Archytas as there was
little known then about three-dimensional (or solid) geometry.
The second problem was the trisection of a given angle θ by a ruler and
a compass. Mathematically, it reduces to a solution for θ which satisfies
the equation
4x3
− 3x − cos θ = 0. (1.2.5)
For θ = 60◦
so that cos θ = 1
2 . The polynomial on the left of (1.2.5) is
irreducible over the field Q of rational numbers. It can be shown that θ
cannot belong to a field extension E of Q of degree 2m
. Consequently, the
trisection of an angle θ = 60◦
is not possible with a ruler and a compass.
For the construction of regular polygons with a ruler and a compass, the
set of complex solution of the well-known cyclotomic equation
xn
− 1 = 0 (1.2.6)
contains the number one and divides the unit circle into n equal parts. The
solution is possible with the aid of the following theorem due to Gauss:
Theorem of Gauss: A regular n-gon can be constructed with ruler and
compass if and only if
n = 2m
p1p2 · · · pr, (1.2.7)
where m is a natural number and p
rs are pair distinct Fermat’s primes of
the form
Fk = 22k
+ 1, k = 0, 1, 2, · · · . (1.2.8)
It is probably known that for k = 0, 1, 2, 3, 4, the above number is prime.
Consequently, a regular n-gon can be constructed for n in the list of primes
2, 3, 5, 17, 257, 65, 537. For n ≤ 20, the construction of all regular n-gons
with n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17,20 is possible using only a ruler
and a compass.
Finally, the third problem was to square the circle, that is, to construct
a square with ruler and compass whose area is equal to that of a given unit
circle. The length x for the sides of a square is a solution of the equation
x2
= π. (1.2.9)
In 1882, Ferdinand Lindemann (1852-1939), David Hilbert’s (1862-1943)
teacher, proved the transcendence of the number π over the field Q of

rational numbers. Consequently, number x or π cannot be an element of
any algebraic field extension of Q. So the problem has no solution. Clearly,
the first two problems are algebraic so that they require the solution of
a cubic equation. The third problem is totally different as it involves the
transcendental number π.
Indeed, in mathematics, the Pythagoreans made great progress, partic-
ularly in the theory of numbers, and in geometry of line, plane and solid
figures, and also, lengths, areas, and volumes associated with them. It is
the most appropriate to recall the delightful quotation of Johann Kepler
(1571-1630): “Geometry has two great treasures: one is the theorem of
Pythagoras; the other, the division of a line in extreme and mean ratio.
The first we may name as a measure of gold, the second we may name as
a precious jewel.”
In Greek mathematics, there was another remarkable number, the so
called the golden number (or golden ratio) that is defined in geometry by
dividing a straight line segment in such a way that the ratio of the total
length l to the larger segment x is equal to the ratio of the larger to the
smaller segment. In other words, the golden ratio, g = (l/x) is determined
by the equation
l
x
=
x
l − x
(1.2.10)
or, equivalently,
g2
− g − 1 = 0. (1.2.11)
The positive solution of quadratic equation (1.2.11)
g =
l
x
=
1
2
√
5 + 1 = 1.618. (1.2.12)
The inverse ratio of g is
1
g
=
x
l
=
1
2
√
5 − 1 = 0.618 (1.2.13)
so that 1
g = g − 1.
In geometry, the Pythagoreans developed the theory of space filling fig-
ures, whatever the motivation for their work, the Pythagoreans evidently
considered the geometrical figures to be very important for space filling. For
example, one of the diagrams (see Figure 1.6) shows six equal equilateral
triangles filling space around their central point. But five such equilateral
triangles can similarly be fitted together to generate a bell-tent-shaped fig-
ure around a central vertex so that their bases form a regular pentagon.

Fig. 1.6 Pythagorean or Six Equilateral Triangles filling space around the center.
Such a figure becomes a solid figure with the vertex of a regular icosa-
hedron. This process can be repeated by surrounding each vertex of the
original equilateral triangles with five triangles. Exactly twenty equilateral
triangles are required to generate the beautiful solid figure of the icosa-
hedron of twelve vertices and twenty faces. It is remarkable that in solid
geometry there are exactly five such regular figures, known as regular poly-
hedra (or Platonic solids), and that in the plane there is a very limited
number of regular space-filling geometric figures. The first three sim-
plest regular polyhedra including tetrahedron, cube and octahedron were
found by Egyptian mathematicians. Pythagorean discovered the remain-
ing two — the icosahedron, and the dodecahedron with twenty vertices and
twelve faces.
It is important to point out that a study of the properties of the regular
pentagon led to the discovery of the golden ratio, the ratio in this case being
that of the diagonal of the pentagon to its side. In Figure 1.7, the diagonal
AC of the pentagon divides the diagonal BE into two unequal segments BP
and PE such that the ratio of the smaller segment to the larger is equal to
the ratio of the larger segment to the whole diagonal. In fact, any diagonal
of the regular pentagon divides any other interesting diagonal in this way.
Such division was known to the Greek mathematicians as “division of a
line in mean and extreme ratio”. We have already stated that this ratio
is the golden ratio g = (BE/PE), where BE = and PE = x so that
the algebraic formulation is ( − x)/x = (x/). This leads to the quadratic
equation (1.2.11) in the golden ratio g.

A
P Q
D
C
E
B
Fig. 1.7 A Regular Pentagon ABCDE.
Some of the angles associated with Figure 1.7 follow from the con-
struction of the triangle ACD with angles ∠ACD = ∠ADC = 72◦
and
∠CAD = 36◦
. It is then a simple matter to construct the complete pen-
tagon so that ∠ABC = 108◦
, ∠BAC = 36◦
= ∠BCA, and hence, all angles
of the pentagon are known.
It was Euclid of Alexandria (365-300 B.C.) made the first systematic
development of Euclidean geometry in his famous treatise, The Elements in
13 volumes. These volumes represented a standard reference of geometry
and number theory and a great model for the first axiomatic method in
mathematics. He first proved that the number of primes is infinite which
is one of the fundamental results in mathematics. However, the first com-
pletely rigorous axiomatic method of mathematics from a modern point of
view was given by David Hilbert in his Principle of Geometry that was
published in 1899. The Greek mathematicians also advanced other areas
of mathematics and astronomy. Archimedes (287-212 B.C.) of Syracuse
also made many other major contributions to mathematics and mathe-
matical physics. He determined the center of mass of bodies and simple
surfaces and derived the formula for the workings of levers and equilibrium
of floating bodies. His major work for finding areas and volumes marked
the birth of calculus. Archimedes was probably the last great mathemat-

ical scientists of ancient times. Another Greek mathematician, Claudius
Ptolemy (85-169 A.D.) of Alexandria became famous for his major contri-
butions to plane and spherical trigonometry and astronomy. Diophantus
of Alexandria worked on theory of equations and earned the title of Father
of Algebra. During the Middle Ages, the greatest discoveries in India were
natural numbers including zero and the decimal number system. Accord-
ing to P. S. Laplace (1749-1827): “It is India that gave us the ingenious
method of expressing all numbers by ten symbols, each symbol receiving
a value of position, as well as an absolute value. We shall appreciate the
grandeur of the achievement when we remember that it escaped the genius
of Archimedes and Apollonius.”
1.3 The Major Impact of the European Renaissance on
Mathematics and Science
During the middle ages, the Italian mathematician Leonardo of Pisa (Fi-
bonacci (1170-1250)) published his major book Liber Abaci in 1202, an in-
ﬂuential book which introduced the Hindu-Arabic number system to West-
ern Europe. The European Renaissance, from the 1400 to the 1600s pro-
duced many great advances in physics, astronomy, pure and applied math-
ematics. Michael Stifel (1487-1567), Nicolò Tartaglia (1506-1557), Giro-
lamo Cardano (1501-1576), and Francois Viète (1540-1603) made major
contributions to algebra, trigonometry and quadratic and cubic equations.
Viète introduced the use of letters to stand for unknown quantities. Nico-
laus Copernicus (1473-1543), the great astronomer who boldly rejected the
fourteen-hundred year old Ptolemy’s mathematical theory of astronomy
with the Earth at the center of the universe and discovered the revolu-
tionary modern heliocentric picture of the universe with the Sun at the
center and made contributions to mathematics through his great work in
astronomy with the publishing of De revolutionibus orbium coelestium in
1543.
Thoroughly convinced by the beauty and harmony of the Copernicus
heliocentric system that the planets revolve in orbits about the sun at
the center of the Universe, a great German mathematical scientist and as-
tronomer, Johann Kepler used his brilliant imagination and amazing per-
severance to modernize the Copernicus model in mathematical astronomy.
As a research assistant to the famous Danish-Swedish astronomer, Tycho
Brahe (1546-1601), Kepler had the rare opportunity to utilize Brahe’s pre-

cise and extensive observational data. Based on these observational data,
Kepler first discovered his three famous laws of planetary motion, the first
two founded in 1609 and the third one ten years later in 1619. Kepler’s
laws of planetary motion are considered as major landmarks in the history
of mathematical science and astronomy, for in the effort to justify them,
Newton was led to discover modern celestial mechanics during 1660-1666.
In his celebrated work of 1619, Harmony of the World, Kepler expressed
his great satisfaction with the following statement in the preface:
“I am writing a book for my contemporaries or — it does not matter
— for posterity. It may be that my book will wait for a hundred years for
a reader. Has not God united for 6000 years for an observer?”
The 1600s brought many major discoveries in mathematics and astron-
omy. Two British mathematicians, John Napier (1550-1617) and Henry
Briggs (1556-1631), first Savilian Professor of Geometry at the University
of Oxford, invented logarithms to the base of 10. Logarithms to the base of
10 are usually known as Briggian logarithms, through the advantage of us-
ing this base appears to have occurred independently to Napier and Briggs.
Napier published his book Mirifici logarithmorum canonis descriptions, in
which logarithms are introduced in great detail. On the other hand, two En-
glishmen, Thomas Harriot (1560-1621), and William Oughtred (1557-1660)
developed new methods for classical algebra. Galileo Galilei (1564-1642) an
Italian astronomer and physicist and Johann Kepler, a German mathemati-
cian and astronomer tremendously expanded knowledge of mathematics
and physics through their studies of astronomy, physics and mathematics.
Galileo discovered the famous law of falling bodies which marked the be-
ginning of modern experimental physics. He suggested that all bodies are
attracted to the Earth by the constant gravitational acceleration regardless
of their weights. His famous experiment dealt with dropping two unequal
weights from the top of the Leaning Tower of Pisa. This became controver-
sial because it contradicted Aristotle’s (384-322 B.C.) old views that heavy
bodies fell faster than lighter ones. It is also important to mention Galileo’s
work on the curve cycloid in 1630 and his suggestion that arches of bridges
should be built in the shape of cycloid. The quadrature (or finding an
area) of the cycloid has been calculated in 1630 by Evangelista Torricelli
(1608-1647), a student of Galileo. About the same time, Pascal proved
many new theorems about properties of cycloid and calculated the area of
the segment of cycloid. This was followed by another remarkable discovery
of a great Dutch mathematical scientist, Christian Huygens (1629-1695)
in 1658 that was concerned with the solution of the problem of the tan-

tochronous motion. Indeed, the cycloid is a true tantochrone that is, if a
particle is allowed to slide from rest down a cycloid, it takes exactly the
same time to reach the bottom, no matter where it starts from. Huygens
also made another discovery that a pendulum bob swinging along a cycloid
curve takes exactly the same time to make a complete oscillation whether
it swings through a small or large arc. He made many new and sensational
discoveries in physics and astronomy including his strong support for the
Copernicus heliocentric model of the universe. In 1609, he build a telescope
that has opened new worlds in astronomy, and has become an indispensable
instrument for centuries for astronomy.
Galileo discovered the laws of pendulum and was credited for his most
remarkable discovery of four bright satellites of the planet Jupiter in 1610.
In the same year, he observed some peculiar form of the planet Saturn.
His historic achievements in astronomy dealt with the discovery of many
more and more powerful telescopes that were sold in Europe. This instru-
ment has made it possible to study, observe and photograph many heavenly
bodies which were formerly unknown. His name and fame as the greatest
experimental scientist of his time attracted many scholars from all parts
of Europe. Christian Huygens also built a powerful telescope which made
possible his new discovery of satellites and the rings of Saturn. He was the
first one who used a pendulum to regulate a clock and then applied the
basic principles of pendulum in building astronomical clocks. In addition,
he investigated the wave theory of light and discovered the polarization of
light.
Galileo is universally considered as the founder of methodology of mod-
ern science. His radical departure from the Greeks, the medieval and con-
temporary scientists led him to establish the fact that matter as well as
motion were only the first step to a new approach to nature. In 1632,
he published his beautifully written masterpiece, A Dialogue on the Two
Principal Systems of the World in which he gave a critical evaluation of
the comparative merits of the old and new theories of motions of the ce-
lestial bodies. He spent considerable amount of time in writing on force
and motion. In particular, his firm helicentric views of the universe was in
severe disagreement with religion doctrines of the Inquisition. In 1638, he
published his other greatest classic, Dialogues on the Two New Sciences in
which he founded the modern science of mechanics. It contained his life’s
work on motion, acceleration, and gravity and provided a sound basis for
the three laws of motion formulated by Sir Isaac Newton (1642-1727) in
1687.

Without being too precise, mechanics is simply the study motion of ma-
terial bodies (or particles) that can be described by mathematical models.
In mechanics, a body (particle) is supposed to be subject to certain forces,
which affect its motion according to certain laws. Expressed in the language
of mathematics, these laws usually take the form of differential equations,
that is, they connect the position, velocity, momentum, acceleration of the
body at a particular instant of time. They do not primarily describe the
whole motion, but merely the laws governing it. It is the motion as a whole
which has to be derived from the law. In other words, this is a problem
of solving differential equations with time as the independent variable, and
there are one or more dependent variables which determine the position of
the body.
Galileo’s two greatest classics are not only two profound books of all
time, but they are clear, direct, truly powerful and fascinating in the history
of science and philosophy. In general, his scientific philosophy and scientific
method were in agreement with those of Descartes, Huygens, Newton and
others. His new methodology of science led him to believe in the total refor-
mulation that not only imparted expected and unprecedented power to sci-
ence, but bound it indissolubly to mathematics. It was Galileo who remark-
ably discovered the more radical, more effective and more practical meth-
ods for modern science. He demonstrated the profound effectiveness of his
approach to science through his own work. It is a delight to quote a philoso-
pher, Thomas Hobbes (1588-1678) who said of Galileo: “He has been the
first to open to us the door to the whole physics.” Galileo himself was con-
vinced that nature is simple, orderly, and mathematically designed which
can be documented by his own famous 1610 quotation: “Philosophy [na-
ture] is written in that great book which ever lies before our eyes — I mean
the universe — but we cannot understand it if we do not first ... labyrinth.”
Both Galileo and Newton strongly emphasized that mathematical prin-
ciples are quantitative principles which played a vital role in providing the
correct physical explanation of natural phenomena. They also believed that
experiments are needed to establish basic laws of science. In the preface to
his Principia, Newton expressed his firm views on the intimate relationship
between the mathematical principles (or laws) and the natural phenomena
as follows:
“Since the ancients (as we are told by Pappus) esteemed the science
of mechanics of greatest importance in the investigation of natural things,
and the moderns, rejecting substantial forms and occult qualities, have en-
deavored to subject the phenomena of nature to the laws of mathematics, I

have in this treatise cultivated mathematics as far as it relates to philosophy
[science] ... and therefore I offer this work as the mathematical principles
of philosophy, for the whole burden in philosophy seems to consist in this
— from the phenomena of motions to investigate the forces of nature, and
then from these forces to demonstrate the other phenomena....”
Finally, we close this section by adding the most tragic event of Galileo’s
life. In 1633, after a long and painful trial by a tribunal of the Inquisition
because of his heliocentric view of the universe contrary to church teachings,
he was sentenced to house imprisonment for the rest of his life. He remained
a prisoner in Florence until his death in 1642.
During the Renaissance period, two great French mathematicians, René
Descartes (1596-1650) and Pierre de Fermat (1601-1665) created a new
branch of mathematics which is now known as analytic geometry. By the
1630s, both men discovered the basic idea of using algebraic equations to
represent curves and surfaces and investigated their fundamental proper-
ties. Descartes’ major objective was to unify the hitherto largely two sepa-
rate disciplines of algebra and geometry, in particular to use the algebraic
method to solve geometrical construction problems. His great mathematical
work dealing with applications of algebra to geometry was La Géométrie.
On the other hand, based on the work of Apollonius on conic sections,
Fermat discovered the fundamental principle of geometry, which he ex-
pressed thus: “Whenever in a final equation two unknown quantities are
found, we have a locus, the extremity of one of these describing a line,
straight or curved.” This profound statement was written at least one year
before the publication of Descartes’ La Géométrie. Fermat formulated his
major ideas further in his short book entitled Ad locus planos et solidos is-
agoge (Introduction to Loci Consisting of Straight Lines and Curves of the
Second Degree) which was published in 1679 – almost fourteen years after
his death: That is why Descartes is widely recognized as the sole creator
of coordinate geometry. However, it clearly follows from Fermat’s above
quotation that his approach is undoubtedly more simple, direct and more
systematic than that of Descartes. In the eighteenth century, the view that
the algebraic approach to geometry was more than a mathematical tool.
Algebra itself is a fundamental method of introducing and investigating
curves and surfaces in general. All these simply mean that the analytic
geometry paved the way for a complete unification of algebra and geometry
from a modern point of view.
Based on the great work of the classical masters, Apollonius and Dio-
phantus on geometry, in general and conic sections, in particular, Girard

Desargues (1591-1661) created a totally new branch of geometry in 1639
which is now know as the Projective Geometry. It deals with the study
of the descriptive properties of geometrical figures. In other words, it is
basically concerned with those geometrical properties which are unchanged
by the operation of section and projection. The basic metrical properties in
geometry which include distance, areas, angles, congruence, and similarity
are not considered in projective geometry. For example, the Pythagorean
Theorem for a right angled triangle (c2
= a2
+ b2
), the area of a triangle of
base a and height h = 1
2 ah and the sum of the three angles of a triangle
ABC (A + B + C = π) are famous metrical theorems. Projective geometry
has grown into a vast and beautiful branch of geometry through the major
works of great French mathematicians including Desargues, Blaise Pascal
(1623-1662), Gaspard Monge (1746-1818) and Jean Victor Poncelet (1788-
1867). Like several other branches of geometry, it has become a new source
of mathematical knowledge for the study of descriptive geometry.
The observed symmetry between points and lines in a projective plane
leads to the so-called principle of duality which is one of the most elegant
properties of the projective geometry. This basic principle states that, in
a projective plane, every theorem (or proposition) remains true when the
words point and line are consistently interchanged. Thus, given a theorem
and its proof, we can immediately formulate the dual theorem whose proof
can be written down mechanically by the use of the duality principle in
every step of the proof of the original theorem.
Desargues not only introduced many ideas, notably the point and the
line at infinity and gave elegant proofs of many new theorems. Above all,
he first discovered the concepts of section, projection and cross-ratio of four
points which were used to give a new method of proof. He then developed a
unified approach to several types of conic sections through projections and
sections. It may be appropriate to give some examples of basic theorems
in projective geometry. One such example is the Desargues’ famous two-
triangle theorem which is illustrated in Figure 1.8 with a vortex O and
the triangle A

B

C

is obtained from the triangle ABC by projection and
section from the vertex O. Desargues’ theorem then states that the three
pairs of the corresponding sides AB and A

B

, BC and B

C

, and AC
and A

C

(or their extensions) of two triangles perspective from a point
meet in three colinear points L, M, N as shown in Figure 1.8. Conversely,
if the three pairs of corresponding sides of the two triangles meet in three
points that lie on one straight line, then the line joining corresponding
vertices meet in one point. In other words, making reference to Figure 1.8,

A
A'
B'
B
O
L
N
C
C'
M
Fig. 1.8 The Desargues two triangles.
the converse theorem asserts that since AA

, BB

, and CC

intersect at
a point O, the sides AB and A

B

intersect at a point L, AC and A

C

meet in a point M and BC and B

C

meet at N; then L, M, and N lie
on a straight line. It is important to note that both theorems are true
whether the triangles ABC and A

B

C

lie on the same or diﬀerent planes.
Desargues gave an elegant proof of his theorem and its converse for both
two- and three-dimensional cases.
The second major contributor to projective geometry was Pascal. In
1640, at the age of sixteen, Pascal gave a pleasant surprise to the world
by publishing a short book entitled Essai pour les coniques in which he
described his celebrated theorem of the hexagrammum mysticum (Mystic
Hexagram). It is universely known as the Pascal Theorem which is illus-
trated in Figure 1.9, and it states that the three pairs of opposite sides of a
hexagon inscribed in a conic meet in three collinear points. In other words,
making reference to Figure 1.9, if BA and DE intersect at L, CD and AF
intersect at M, and BC and FE intersect at N, then L, M, N lie a straight
line. Conversely, if a hexagon is such that the points of intersection of its
three pairs of opposite sides lie on a straight line, then the vertices of the

A
B
C
D
F
E
M
L
N
Fig. 1.9 The Pascal hexagon in a conic.
hexagon lie on a conic. However, Pascal did not give an explicit proof of
his theorem and its converse. He simply stated that his theorem is true for
a circle and it must also be true for a conic by the method of projection.
Desargues regarded Pascal’s 1640 essay was so brilliant that he could
not believe it was written by such a young man. He encouraged Pascal to
do more research on projective geometry in order to develop the method of
projection and section further. At the advice of Desargues, Pascal began
working on conics and used projective methods, that is, projection and sec-
tion. He admired Desargues’ work and acknowledged his debt to Desargues
by saying : “I should like to say that I owe the little that I have found on
this subject to his writings.”
In addition to his contribution to projective geometry, Pascal made ma-
jor contributions to the mathematical theory of probability. In 1654, a
French man, the Chevalier de Méré, suggested some problems associated
with games of chance. During that time, Pascal had some correspondence
with Pierre de Fermat dealing with these problems of games of chance and
gambling in general. Thus, the ﬁrst research collaboration of Pascal and
Fermat on problems of games and chance led to mark the birth of the math-
ematical theory of probability which is now widely used in mathematical
statistics. Based on some results of Fermat and Pascal, Christian Huygens
wrote the ﬁrst treatise on probability in 1657. About the same time, Pascal
wrote a treatise entitled Traité du triangle arithmétique which included the

coefficients of the binomial expansion
(a + b)n
=
n

r=0
(n
r ) an−r
br
, (1.3.1)
so that, when a = b = 1,
n

r=0
(n
r ) = 2n
, (1.3.2)
the number of combinations of n objects taken r at a time is,
n
Cr = (n
r ) =
n!
(n − r)! r!
, (1.3.3)
the Pascal recurrence relation
n
Cr +n
Cr−1 = n+1
Cr, (1.3.4)
and the coefficients of the binomial formulas organized into famous Pascal’s
triangle. Pascal made an extensive study of the properties of his triangle, in
the course of which he discovered the principle of mathematical induction.
This principle, which states the validity of the mathematical argument by
recurrence, is now considered as one of the fundamental axioms of modern
mathematics. Many proofs in mathematics are based on the famous prin-
ciple of induction. Pascal became a renowned scientist in Europe for his
fundamental works in geometry, hydrostatistics and probability theory. He
also invented a new calculating machine which is still preserved in a French
museum.
After a century of slow progress, the revival of the projective geome-
try received considerable attention by Gaspard Monge (1746-1818) and his
school at the École Polytechnique. Monge’s extensive work in descriptive
geometry, ordinary and partial differential equations won the remarkable
admiration from mathematical scientists of the world. His greatest student
was Poncelet who published his famous Treatise on the Projective Proper-
ties of Figures in 1822 which he subsequently expanded and revised this
treatise and later published in two volumes entitled Applications d’analyse
et de géométrie (1862-1864). All these published works were his major con-
tributions to projective geometry and to the creation of modern projective
geometry. He was the first mathematician to recognize fully that projective
geometry was a new branch of geometry with methods and results of its
own. He formulated the general problem of seeking all properties of geo-
metrical figures which were common to all sections of any projection of a
figure, that is, remained unchanged by projection and section. His work

was essentially based on three major ideas: homologous figure, principle
of continuity, and pole and polar with respect to a conic. Two figures are
called homologous if one can be derived from the other by one projection
and a section. In his 1822 Traité, Poncelet phrased the principle of conti-
nuity as: “If one figure is derived from another by a continuous change and
the latter is as general as the former, then any property of the first figure
can be asserted at once for the second figure.” He advanced the principle
as an absolute truth and used it in his Traité to prove many theorems, and
then generalized the principle to make assertions about imaginary figures.
The concept of pole and polar with respect to a conic was the third major
idea in Poncelet’s work. He gave a general formulation of the transforma-
tion form pole to polar and conversely. His major objective in studying
polar reciprocation with respect to a conic was to establish the principle
of duality in projective geometry. By virtue of this principle, lines can be
as fundamental as points in the development of plane projective geometry.
Like others, Poncelet recognized that theorems dealing with figures lying
in one plane when interchanged the word point by line and vice versa not
only made sense but proved to be true in general. It is fair to say that all
major contributors to projective geometry made the significant efforts to
elevate the subject to hew heights of rigor, clarity, elegance and generality.
The Renaissance mathematical scientists including Copernicus, Brahe,
Kepler, Galileo, Pascal, Huygens, Descartes, Newton and Leibniz spoke
repeatedly of the cohesiveness and harmony that God imparted to the Uni-
verse through His mathematical laws and design. These men discovered
mathematical knowledge that would reveal the grandeur and glory of God’s
creation. Once Galileo said: “Nor does God less admirably discover Himself
to us in Nature’s action than in the Scriptures’ sacred dictions.” Towards
the end of the Renaissance period, many European scientists became very
active in the formation of scientific societies or research institutes in order
to stimulate more scientific research and to increase communication among
mathematical scientists. Although the Italian academies and professional
societies were founded in the early seventeenth century with Galileo and
his students as members, but, unfortunately, they were disbanded after a
while. For example, in France, several mathematical scientists including
Desargues, Descartes, Fermat and Pascal met informally under the leader-
ship of Marin Mersenne (1588-1648) to organize the Academie Royale des
Sciences in 1630s. In England, John Wallis began in 1645 to hold meetings
in Greshan College, London in order to establish a similar organization in
England. This informal group was chartered by Charles II in 1662 and es-

tablished the Royal Society of London for the promotion and dissemination
of scientific knowledge. Its first president was a famous mathematician,
Lord William Brouncker (1620-1684). The Philosophical Transactions of
the Royal Society began its publication in 1665 and it was one of the first
research journals to include mathematical and scientific papers. The French
Academy of Sciences was founded by Colbert in 1666. The famous Lucasian
Chair of Mathematics was established at the University of Cambridge in
1663 by Henry Lucas (1610-1667) who was a former student of Cambridge.
The first professorship in mathematics was established at the University of
Oxford in 1619. John Wallis became of professor of mathematics at Oxford
in 1649 and held the Chair of mathematics until 1702. On the other hand,
Gottfried Wihelm Leibniz (1646-1716) in Germany provided a major lead-
ership role for some years to establish the Berlin Academy of Sciences in
1700 with Leibniz as its first President. In Russia, Peter the Great founded
the Academy of Sciences at St. Petersburg in 1724. These academies and
their scientific journals opened new outlets for mathematical and scientific
communication first in Europe and then in other nations of the world. They
not only promoted new scientific research, but also supported scientists for
the cultivation of mathematics and sciences and for making mathematics
and science more useful for the society. These professional organizations
played the major role in advanced study and research, and in dissemination
of scientific and mathematical knowledge throughout the world.
1.4 The Discovery of Calculus by Newton and Leibniz
Historically, Sir Isaac Newton and Gottfried Wihelm Leibniz independently
discovered the calculus in the seventeenth century. In recognition of this
remarkable discovery. John Von Neumann’s (1903-1957) thought seems to
worth quoting. “... the calculus was the first achievement of modern math-
ematics and it is difficult to overestimate its importance. I think it defines
more equivocally than anything else the inception of modern mathematics,
and the system of mathematical analysis, which is its logical development,
still constitute the greatest technical advance in exact thinking.”
Both Newton and Leibniz recognized that calculus can be regarded as
the branch of mathematical study which treats change and the rate of
change. They also observed the close connection between algebra and ge-
ometry, epitomized by the fact that every equation has a graph and every
graph an equation.

By 1664, the young Newton became familiar with all mathematical ideas
and results of his predecessors and was fully ready to discover his own. In
his new analytical methods, Newton remarkably combined the ideas, results
and methods of three largely separate branches of mathematics: coordinate
geometry, calculus and infinite series (or more precisely, the representation
of functions by infinite series). During 1664-1666, Newton developed all
the basic ideas and methods in his first version dealing with the fluxional
calculus. In this work, he treated variables as moving quantities changing
with time and introduced the concept of velocity and acceleration at any
instant of time. He then considered exposition of his ideas and results
in the book Methodus Fluxionum et Serieum Infinitarum (The Method of
Fluxions and Infinite Series) which was not published until 1736, after
his death. In his book, Newton treated variables as flowing quantities
generated with time by the continuous motion of points, lines and planes,
rather than as static infinitesimal quantities as in his first version of the
calculus. He defined a variable quantity x or y as the fluent, and its rate
of change with respect to time as the fluxion which was denoted by ẋ and
ẏ (the Newtonian dot notation) which is now known as the derivative or
the velocity. Subsequently, he stated more clearly the fundamental problem
of calculus by introducing any two variables rather than the time as the
independent variable. For example, y = xn
so that, in modern notation,
(dy/dx) = n xn−1
. One of the outstanding problems of the seventeenth
century was that of finding the tangent to a curve at an arbitrary point.
It was solved by Newton’s teacher at Cambridge University, Isaac Barrow
(1630-1677). Newton developed the idea of the rate of change from the
analytic point of view. He also demonstrated his ideas by examples of
finding tangents to well-known plane curves including cycloid and spirial.
He gave another example of a plane curve with its algebraic equation in the
form
x3
− ax2
+ axy − y3
= 0, (1.4.1)
to derive the fluxional equation
3x2
ẋ − 2a x ẋ + a(ẋy + xẏ) − 3y2
ẏ = 0. (1.4.2)
This gives the slope (or gradient) of the tangent to the curve at any
point (x, y) so that
dy
dx
=
ẏ
ẋ
=
(3x2
− 2ax + ay)
(3y2 − ax)
. (1.4.3)
In his book, Newton not only developed a general method for finding
the instantaneous rate of change of one variable with respect to another,

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