The Great Mathematician, Leonard Euler (1707-1783)
Leonard Euler was born on April, 15 1707 in Basel, Switzerland. His father, Paul Euler had studied theology at the University of Basel and had attended Jacob Bernoulli's lectures. In fact Paul Euler and Johann Bernoulli had both lived in Jacob Bernoulli's house while undergraduates at Basel.
His family moved to Riehen when he was one year old and it
was in Riehen, not far from Basel, that Leonard was brought up. Paul
Euler had, some mathematical training and he was able to teach his son
elementary mathematics along with other subjects.
Leonhard entered the University in 1720, at the age
of 14, first to obtain a general education before going on to more
advanced studies. Johann Bernoulli soon discovered Euler's great
potential for mathematics in private tuition that Euler himself engineered. Euler's own account given in his unpublished autobiographical writings is as follows:-
... I soon found an opportunity to be introduced to a famous professor
Johann Bernoulli. ... True, he was very busy and so refused flatly to
give me private lessons; but he gave me much more valuable advice to
start reading more difficult mathematical books on my own and to study
them as diligently as I could; if I came across some obstacle or
difficulty, I was given permission to visit him freely every Sunday
afternoon and he kindly explained to me everything I could not
understand ...
In 1723 Euler completed his Master's degree in philosophy having
compared and contrasted the philosophical ideas of Descartes and
Newton. Euler completed his studies at the University of Basel in 1726.
He had studied many mathematical works during his time in Basel, and
Calinger has reconstructed many of the works that Euler read with the
advice of Johann Bernoulli. They include works by Varignon, Descartes,
Newton, Galileo, Van Schooten, Jacob Bernoulli, Hermann, Taylor and
Wallis. By 1726 Euler had already a paper in print, a short article on
isochronous curves in a resisting medium. In 1727 he published another
article on reciprocal trajectories and submitted an entry for the 1727
Grand Prize of the Paris Academy on the best arrangement of masts on a
ship. Daniel Bernoulli held the senior chair in mathematics at the
Academy but when he left St Petersburg to return to Basel in 1733 it
was Euler who was appointed to this senior chair of mathematics.
By 1740 Euler had a very high reputation, having won the Grand Prize of
the Paris Academy in 1738 and 1740. On both occasions he shared the
first prize with others. Euler's reputation was to bring an offer to go
to Berlin, but at first he preferred to remain in St Petersburg.
However political turmoil in Russia made the position of foreigners
particularly difficult and contributed to Euler changing his mind.
Accepting an improved offer Euler, at the invitation of Frederick the
Great, went to Berlin where an Academy of Science was planned to
replace the Society of Sciences. He left St Petersburg on 19 June 1741,
arriving in Berlin on 25 July.
In a letter to a friend Euler wrote:-
I can do just what I wish [in my research] ... The king calls me his professor, and I think I am the happiest man in the world.
In 1744 Euler was made director of mathematics department. We owe to
Euler the notation f(x) for a function (1734), e for the base of
natural logs (1727), i for the square root of -1 (1777), ∏ for pi,
for summation (1755), the notation for finite differences y and 2y
and many others.
After his death in 1783 the St Petersburg Academy continued to publish
Euler's unpublished work for nearly 50 more years. Euler's work in
mathematics is so vast that an article of this nature cannot but give a
very superficial account of it. He was the most prolific writer of
mathematics of all time. He made large bounds forward in the study of
modern analytic geometry and trigonometry where he was the first to
consider sin, cos etc. as functions rather than as chords as Ptolemy
had done.
Achievements and Contributions
Euler not only made advancements in mathematics, but also in astronomy,
mechanics, optics, and acoustics. He produced just more written work on
mathematics than anyone else. It was once said that he could write a
new paper in about thirty minutes and his desk was always covered with
his pending works. Over a period of about 25 years, Euler wrote over
380 articles on the following topics:
• calculus of variations
• calculation of planetary orbits
• artillery and ballistics (extending the book by Robins)
• shipbuilding and navigation
• motion of the moon
• differential calculus
Leonhard Euler was the first to prove that e is an irrational number.
He was also the first to consider sin, cos, and tan as abbreviations
for sine, cosine, and tangent. In 1732, Euler proved that Fermat's
conjecture, 2n + 1 is always prime if n is a power of 2, is not true
for all cases. He found it to be true if n=1, 2, 4, 8, and 16. The next
case, 232 + 1 = 4,294,967,297, does not
work because it is divisible by 641, which means that it is not prime.
While studying other unproved conjectures of Fermat, Euler introduced
the phi function, Φ (n), where the numbers of integers k with 1 ≤ k ≤ n
and k is coprime to n. He also found in 1749, that if a and b are
coprime, then a2 + b2 has no divisor of the form 4n + 1. Euler found
the solution to a problem that many top mathematicians could not figure
out. This was called the Basel problem. The problem was to find a
closed form for the sum of the infinite series ζ (2) = (1/n). Euler
showed in 1735 that ζ (2) = ∏2/6 but he did not stop at just this, he
also found that ζ (4) = ∏4/90, ζ (6) = ∏6/945, ζ (8) = ∏8/9450, ζ(10) =
∏10/93555 and ζ (12) = 691∏12/638512875. Two years later, he proved
that there was a connection between the zeta function and the series of
prime numbers, thus discovering this famous relation:
ζ (s) = ∑(1/ns) = ∏ (1 - p-s)-1
Here the sum is divided by all natural numbers n, while the product is divided by all prime numbers.
Euler had found the rational coefficients C in ζ (2n) = C ∏2n in 1739,
in terms of the Bernoulli numbers. Euler introduced his famous constant
γ, in 1735, which he showed to be the limit of 1/1 + 1/2 + 1/3 + ... +
1/n - ln n as n tends to infinity. He calculated the constant γ to 16
decimal places. In a letter to Goldbach in 1744, he was the first to
express an algebraic function using Fourier series when he gave the
result: ∏/2 - x/2 = sin x + (sin 2x)/2 + (sin 3x)/3 + ... This,
however, was not published until 1755. Leonhard Euler also found a
proof of Fermat's Last Theorem for n = 3. He introduced a proof
involving numbers of the form A + B.
People claim that Euler began mathematical analysis. In 1748, he wrote
Introductio in analysin infinitorum where he says that mathematical
analysis is the study of functions. Dealing with this work, Euler gave
the formula: eix = cos x + i sin x. He eventually published his
complete theory of logarithms of complex numbers in 1751. In 1729, he
first introduced the beta and gamma functions.
Algebra had been for a considerable period of time, a very limited
Science. This method was used to consider the idea of dimension as the
distillation of abstraction which the human mind can attain only by the
rigorous application with which one separates this notion by occupying
the imagination which otherwise might benefit from some assistance or
some rest to one's intelligence. Finally, the over usage of notations
that this Science employs rendering it in certain ways too foreign to
our nature, too far from our pedestrian concepts, so that the human
spirit might easily enjoy itself and acquire some ease in its practice.
Even the direction of algebraic methods rebuffed those who meditated on
such things and if the point were complicated, it forced them to forget
it entirely or to think only of the formulas. The road which we follow
is sure, however the goal where we wish to go and the point from which
we left disappears in the eyes of the Geometer. It certainly took a
great deal of courage to lose sight of the earthly trappings and so be
exposed to an entirely new science. As we cast our looks, towards the
works of the great mathematicians of the last century, these very same
ones to which algebra owes its greatest discoveries, we will see how
little they knew and how best to employ these very same methods that
they perfected. At the same time one will not be able to deny the very
revolutionary aspect of Euler's transformation of algebraic analysis
into a shinning, universal method applicable in all its aspects and
easy to use.
After having provided the steps to the roots of algebraic equations,
and their general solvability, numerous new theories and some ingenious
and insightful views, Mr. Euler's research was directed to the
calculation of transcendental quantities. Leibniz and the two Bernoulli
each share the glory for having introduced exponential and logarithmic
functions into algebraic analysis. Cotes had already provided the way
in which to represent the roots of certain algebraic equations by sine
and cosine.
These discoveries led Euler to an important discovery by observing the
unique characteristics of exponential and logarithmic quantities born
within the circle and following methods by which the solutions make the
problems disappear, the terms of the imaginaries which would then be
present and which would have complicated the calculation, even though
the are known to collapse, reduced the formulas to simpler and more
convenient expressions. He was able to provide an entirely new
understanding to the part of analysis which concerns itself with the
questions of Astronomy and Physics. This process has been adopted by
all mathematicians and has become a useful and basic tool and has
produced in this section of mathematics about the same revolutionary
effect as the discovery of logarithms had into ordinary calculations.
Euler CycleAccording to NIST (National institute of standards and Technology), Euler Cycle is defined as
A path through a graph which starts and ends at the same vertex and includes every edge exactly once.
Where path is defined as a list of vertices of a graph where each vertex has an edge from it to the next vertex.
Where graph is defined as a set of items connected by edges. Each item
is called a vertex or node. Formally, a graph is a set of vertices and
a binary relation between vertices, adjacency.
Formal Definition: A graph G can be defined as a pair (V, E), where V
is a set of vertices, and E is a set of edges between the vertices E =
{(u, v) | u, v ∈ V}. If the graph is undirected, the adjacency relation
defined by the edges is symmetric, or E = {{u, v} | u, v ε V} (sets of
vertices rather than ordered pairs). If the graph does not allow
self-loops, adjacency is irreflexive.
Where vertex is defined as an item in a graph. Sometimes referred to as a node.
Where a node is defined as (1) A unit of reference in a data structure.
Also called a vertex in graphs and trees. (2) A collection of
information which must be kept at a single memory location)
Where edge is defined as a connection between two vertices of a graph.
In a weighted graph, each edge has a number, called a "weight." In a
directed graph, an edge goes from one vertex, the source, to another,
the target, and hence makes connection in only one direction.
The Königsberg bridge problem
If the seven bridges of the city of Königsberg (left figure; Kraitchik
1942), formerly in Germany but now known as Kaliningrad and part of
Russia, over the river Preger can all be traversed in a single trip
without doubling back, with the additional requirement that the trip
ends in the same place it began. This is equivalent to asking if the
multigraph on four nodes and seven edges has an Eulerian circuit. This
problem was answered in the negative by Euler (1736), and represented
the beginning of graph theory.
J. Kåhre observes that bridges bb and dd no longer exist and that
aa and cc are now a single bridge passing above A with a stairway in
the middle leading down to A. Even so, there is still no Eulerian cycle
on the nodes A,B,C, and D using the modern Königsberg bridges, although
there is an Eulerian trail. An example of Eulerian trail is illustrated
in the right figure above where, as a last step, the stairs from A to
aacc can be climbed to cover not only all bridges but all steps as
well.
To mathematicians, though, Königsburg is best known for a puzzle
associated with its seven bridges, its citizens pondered for a long
time whether it was possible to walk about the city in such a way that
you cross all seven bridges over the river Pregel exactly once.This
might seem like a fun but inconsequential problem. However it was the
inspiration for a 1736 paper by the great Swiss mathematician Leonhard
Euler (1707-1783) which arguably began the field of topology. In this
paper, Euler proved that there was no such path around the city. In
fact Euler gives a simple criterion which determines whether or not
there is a solution to any similar problem with any number of bridges
connecting any number of landmasses.
Euler first simplified the problem, replacing each landmass by a vertex
(yellow dot) and each bridge by an arc (black path). Such a
configuration of vertices and connecting arcs is nowadays known as a
graph. The original problem is equivalent to the problem of travelling
around the graph in such a way that you cross each arc exactly once.
Defining the degree of a vertex to be the number of arcs that lead to
it, Euler proved:
Theorem There exists a (at least one) path on a graph which travels
along each arc exactly once if and only if the graph has at most two
vertices of odd degree.
Such a path on a graph, if it exists, is now called an Euler path in
his honor. The Königsburg example has three vertices of degree three,
and one of degree five, so it has no Euler path and the original puzzle
has no solution.
The key to the proof of (one direction of) Euler's result is the
realisation that if a vertex is not the initial or final vertex of a
path and each bridge is only used once, then the number of arcs that
are traversed leading to/from that vertex must be even (each time you
go through a vertex, you use one arc to get there and another to
leave). This also shows that if there are two vertices of odd degree,
one must be the initial vertex of the Euler path and the other the
final vertex. Incidentally, notice that the number of vertices of odd
degree in any graph must be even since the sum of all the degrees is
double the number of vertices.This problem could have been solved by
adding or removing one bridge.
Topology
Topology is a branch of geometry that deals with the way shapes can
be distorted and while the shape and size of the shape change, the
basic physical structure remains the same. Certain properties are
invariant even under continuous transformations. In other words,
Topology is the study of properties of shapes that remain the same when
the shapes are stretched or compressed. For example, a triangle is
topologically equivalent to a square which is topologically equivalent
to a circle because although the shape and size of these shapes change,
the order of the points is the same. Also, a bowl is topologically
equivalent to a plate since if it is flattened without being broken,
the points would still be in the same order.
Euler also made great advances in Topology. One of the great advances
Euler made was the discovery of the "Euler Characteristic". The Euler
Characteristic is, in math, is a number which is a characterization of
the various classes of geometric figures. This is based only on the
topological relationship between the number of vertices, edges, and
faces of a geometric figure. This number is given by; characteristic
equals vertices minus (edges plus faces) which is represented as:
C = v-(e + f)
Topological ideas are present in almost all areas of today's
mathematics. The subject of topology itself consists of several
different branches, such as point set topology, algebraic topology and
differential topology, which have relatively little in common. We shall
trace the rise of topological concepts in a number of different
situations.
Perhaps the first work which deserves to be considered as the
beginnings of topology is due to Euler. In 1736 Euler published a paper
on the solution of the Königsberg bridge problem entitled Solutio
problematis ad geometriam situs pertinentis which translates into
English as The solution of a problem relating to the geometry of
position. The title itself indicates that Euler was aware that he was
dealing with a different type of geometry where distance was not
relevant
The paper not only shows that the problem of crossing the seven bridges
in a single journey is impossible, but generalises the problem to show
that, in today's notation,
A graph has a path traversing each edge exactly once if exactly two vertices have odd degree.
The next step in freeing mathematics from being a subject about
measurement was also due to Euler. In 1750 he wrote a letter to
Christian Goldbach which, as well as commenting on a dispute Goldbach
was having with a bookseller, gives Euler's famous formula for a
polyhedron
v - e + f = 2
where v is the number of vertices of the polyhedron, e is the number of
edges and f is the number of faces. It is interesting to realise that
this, really rather simple, formula seems to have been missed by
Archimedes and Descartes although both wrote extensively on polyhedra.
Again the reason must be that to everyone before Euler, it had been
impossible to think of geometrical properties without measurement being
involved.
Euler published details of his formula in 1752 in two papers, the first
admits that Euler cannot prove the result but the second gives a proof
based dissecting solids into tetrahedral slices. Euler overlooks some
problems with his remarkably clever proof. In particular he assumed
that the solids were convex, that is a straight line joining any two
points always lies entirely within the solid.
The route started by Euler with his polyhedral formula was followed by
a little known mathematician Antoine-Jean Lhuilier (1750 -1840) who
worked for most of his life on problems relating to Euler's formula. In
1813 Lhuilier published an important work. He noticed that Euler's
formula was wrong for solids with holes in them. If a solid has g holes
the Lhuilier showed that
v - e + f = 2 - 2g.
This was the first known result on a topological invariant.
Conclusion
Euler became almost entirely blind after an illness. His home
was destroyed by fire in 1771, and he was able to save only himself and
his mathematical manuscripts. He had a cataract operation after
the fire and his sight was restored for a few days. He, however, did
not take care of himself, so he became completely blind. He was still
able to continue his work while blind because of his remarkable memory,
and he produced almost half of all of his works while blind. On
September 18, 1783 in Saint Petersburg, Russia, Leonhard Euler said, "I
die." before he died of a stroke. Leonhard Euler made large bounds in
analytic geometry, trigonometry, calculus, and the number theory. It
has been known that after certain periods of great efforts, the
mathematical sciences appeared to have exhausted human capabilities and
to have reached their limits. When all of a sudden new ways to
calculate arrived at the very moment that it seemed that they have
reached the limit of their progress; a new method was introduced into
the Sciences and provided them with new impetus. They are quickly
enriched by the solutions to a great number of problems that the
Mathematicians dared not deal with because of the difficulty and the
physical impossibility to conduct their calculations to a satisfactory
conclusion. Does one think that justice should be reserved to the one
who knew to introduce these methods and make them useful or that a
portion of the glory should go to all those who use them with success
will at least have the recognition of priority so that they might
quibble without being ungrateful. Leonard Euler was a Great Calculator and his discoveries led to modern inventions and exploration.
|
