Pythagoras - 2July

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Transcript Pythagoras - 2July

Pythagoras of Samos
c. 569 - 500 B. C. E.
Pythagoras of Samos was the leader of a
Greek religious movement whose central
tenet was that all relations could be
reduced to number relations ("all things
numbers"), a generalization that stemmed
from their observations in music,
mathematics, and astronomy.
The movement was responsible for
advancements in mathematics,
astronomy, and music theory. Because
the movement practiced secrecy, and
because no records survived, precisely
which contributions were made by
Pythagoras himself, and which were
made by his followers, cannot be
determined with certainty.
Pythagoras is pictured with a visual
representation of the proof of the theorem
which has come to bear his name. The
use of triangles with sides bearing a ratio
of 3:4:5 to construct a right angle was
known to antiquity. And the Pythagorean
theorem was known and used by the
Babylonians. Pythagoras is credited with
the first recorded proof of the theorem
that bears his name.
Euclid, possibly independently of the work of
the Pythagoreans, developed and
recorded, in his Elements, his own proof
of the same theorem.
Zeno of Elea
c. 495 - 430 B.C.E.
Zeno of Elea conceived a number of "paradoxes".
Zeno conceived these not as mathematical
amusement, but as an attempt to support the
doctrine of his teacher, the ancient Greek
philosopher Parmenides, that all evidence of the
senses is illusory, particularly the illusion of
One of Zeno's most famous paradoxes posited a race
between the popular Greek hero Achilles, and a
Zeno set out to logically show that, with the tortoise
given a head start, Achilles, speedy as he might be,
could, in fact, never overtake the plodding reptile.
Zeno reasoned that when Achilles reached the starting
point of the tortoise, the tortoise would have
advanced incrementally further. Achilles would
continually reach a point the tortoise had already
reached, while the tortoise would at the same time
have reached a slightly further point. Thus, Zeno
reasoned, the tortoise could never be overtaken by
Zeno's paradox provided an early entree into
the science and mathematics of limits.
Zeno's paradox is resolved with the insight
that a sum of infinitely many terms can
nevertheless yield a finite result, an
insight of calculus. It was not until
Cantor's development of the theory of
infinite sets in the mid-nineteenth century
that, after more than two millennia,
Zeno's Paradoxes could be fully resolved.
Archimedes of Syracuse
287 - 212 B.C.E.
Archimedes of Syracuse is generally
regarded as the greatest mathematician
and scientist of antiquity, and widely
considered, along with Newton and
Gauss, as one of the greatest
mathematicians of all time.
Archimedes' inventions were diverse -compound pulley systems, war machines
used in the defense of Syracuse, and
even an early planetarium.
His major writings on mathematics included
contributions on plane equilibriums, the
sphere, the cylinder, spirals, conoids and
spheroids, the parabola, "Archimedes
Principle" of buoyancy, and remarkable
work on the measurement of a circle.
Archimedes is pictured with the methods he
used to find an approximation to the area of a
circle and the value of pi. Archimedes was
the first to give a scientific method for
calculating pi. to arbitrary accuracy. The
method used by Archimedes -- the
measurement of inscribed and circumscribed
polygons approaching a 'limit" (described as
'exhaustion') -- was one of the earliest
approaches to "integration". It preceded by
more than a millennia Newton, Leibniz, and
modern calculus.
Archimedes was killed in the aftermath of the
Battle of Syracuse -- a siege won by the
Romans using war machines many of which
had been invented by Archimedes himself.
Archimedes was killed by a Roman soldier
who likely had no idea who Archimedes
was. At the time of his death Archimedes
was reputedly sketching a geometry problem
in the sand, his last words to the Roman
soldier being "don't disturb my circles".
Eukleides (Euclid)
c. 330 - 275 B.C.E
Eukleides (Euclid of Alexandria), although
little is known about his life, is likely the
most famous teacher of mathematics of
all time. His treatise on mathematics, The
Elements, endured for two millennia as a
principal text on geometry.
The Elements commences with definitions
and five postulates. The first three
postulates deal with geometrical
construction, implicitly assuming points,
lines, circles, and thence the other
geometrical objects.
Postulate four asserts that all right angles are
equal -- a concept that assumes a
commonality to space, with geometrical
constructs existing independent of the
specific space or location they occupy.
Pictured over Euclid's right shoulder is a
small drawing which is taken from
Euclid's proof of the right angled triangle
which has come to be known as the
theorem of Pythagoras. While very little
is known about the lives of either
Pythagoras or Eukleides, it is both
plausible and likely that Euclid and
Pythagoras independently discovered
and "proved" this basic theorem. Euclid's
proof of this theorem relies on most of his
46 theorems which preceded this proof.
Central to Euclid's portrait is a circle with its
radius drawn. Euclid's geometry was one
of construction, and the circle and radius
were central elements to Euclid's
René Descartes
1596 - 1650
René Descartes viewed the world with a cold
analytical logic. He viewed all physical bodies,
including the human body, as machines operated
by mechanical principles. His philosophy
proceeded from the austere logic of "cogito ergo
sum" -- I think therefore I am.
In mathematics Descartes chief contribution was in
analytical geometry.
Descartes saw that a point in a plane could be
completely determined if its distances
(conventionally 'x' and 'y') were given from two
fixed lines drawn at right angles in the plane, with
the now-familiar convention of interpreting positive
and negative values.
Conventionally, such co-ordinates are referred to as
"Cartesian co-ordinates".
Descartes asserted that, similarly, a point in 3dimensional space could be determined by three
Pierre de Fermat
1601 - 1665
Pierre de Fermat is perhaps the most famous number theorist in
history. What is less widely known is that for Fermat
mathematics was only an avocation: by trade, Fermat was a
He work on maxima and minima, tangents, and stationary points,
earn him minor credit as a father of calculus.
Independently of Descartes, he discovered the fundamental
principle of analytic geometry.
And through his correspondence with Pascal, he was a co-founder
of probability theory.
But he is probably most well-known for his famous "Enigma".
Fermat's portrait is inscribed with this famous "Enigma", which is
also known as Fermat's Last Theorem. It states that xn + yn =
zn has no whole number solution when n > 2.
Fermat, having posed his theorem, then wrote
"I have discovered a truly remarkable proof which this margin is too
small to contain."
The proof Fermat referred to was not to be found, and thus began a
quest, that spanned the centuries, to prove Fermat's Last
Fermat's image is also overlaid by Fermat's spiral. Fermat's spiral
(also known as a parabolic spiral), is a type of Archimedean
spiral, and is named after Fermat who spent considerable time
investigating it.
Blaise Pascal
1623 - 1662
Blaise Pascal, according to contemporary observers, suffered
migraines in his youth, deplorable health as an adult, and
lived much of his brief life of 39 years in pain.
Nevertheless, he managed to make considerable contributions in
his fields of interest, mathematics and physics, aided by
keen curiosity and penetrating analytical ability.
Probability theory was Pascal's principal and perhaps most
enduring contribution to mathematics, the foundations of
probability theory established in a long exchange of letters
between Pascal and fellow French mathematician
Fermat. While games of chance long preceded both of
them, in the wake of probability theory the vagaries of such
games could be viewed through the lens of a measurable
percentage of certainty, which we have come to refer to as
the "odds".
Pascal is pictured overlaid by a Pascal's triangle in which the
numbers have been translated to relative colour densities.
Pascal created his famous triangle as a ready reckoner for
calculating the "odds" governing combinations.
Each number in a Pascal triangle is calculated by adding together
the two adjacent numbers in the wider adjacent row. The sum
bf the numbers in any row gives the total arrangement of
combinations possible within that group. The numbers at the
end of each row give the the "odds" of the least likely
combinations, with each succeeding pair of triangles giving
the chances of combinations which are increasingly likely.
Though apparently simple and relatively simple to generate,
Pascal's triangle holds within itself a complex depth of
numerical patterns, applicable to the physical world and
beyond, and the theory of probabilities has found increasingly
wide application in modern mathematics and sciences,
extending well beyond seemingly simple games of chance.
Pascal also did seminal work in the field of binomial coefficients
which in some senses paved the way for Newton's discovery
of the general binomial theorem for fractional and negative
Pascal is also considered the father of the "digital" calculator. In
1642, at the age of 19, Pascal had invented the first digital
calculator, the "Pascaline".
Mechanical calculators based on a logarithmic principle had
already been constructed years previously by the
mathematician Shickard, who had built machines to calculate
astronomical dates, Hebrew grammar, and to assist Kepler
with astronomical calculations.
Pascal's device, capable of adding two decimal numbers, was
based on a design described in Greek antiquity by Hero of
Alexandria. It employed the principle of a one tooth gear
engaging a ten-tooth gear once every time it revolved. Thus, it
took ten revolutions of the first gear in order to make next gear
rotate once. The train of gears produced mechanically an
answer equivalent to that obtained using manual arithmetic.
Unfortunately, Pascal's invention served primarily as an early
lesson in the vagaries of business, and the problems of new
technology. Pascal himself was the only one who could repair
the device, and the cost of the machine cost exceeded the
cost of the people it replaced. The people themselves
objected to the very idea of the machine, fearing loss of their
skilled jobs.
Pascal worked on the "Pascaline" digital calculator for three years - from 1642 to 1645 -- and produced approximately 50
machines, before giving up.
The world would have to wait another 300 years for the electronic
computer. The principle used in Pascal's calculator was
eventually used in analogue water meters and odometers.
Sir Isaac Newton
1642 [1643 New Style Calendar] - 1727
Sir Isaac Newton stated that "If I have seen further it is by
standing upon the shoulders of giants." Newton's
extraordinary abilities enabled him to perfect the
processes of those who had come before him, and to
advance every branch of mathematical science then
studied, as well as to create some new subjects.
Newton himself became one of those giants to whom
he had paid homage.
Newton's image is set against the cover of a tome easily
recognizable to those familiar with the history of
mathematics -- his Principia Mathematica, The
Mathematical Principles of Natural Philosophy, first
published in 1687. Its first two parts, prefaced by
Newton's "Axioms, or Laws of Motion", dealt with the
"Motion of Bodies". The third part dealt with "The
System of the World" and included Newton's writings
on the Rules of Reasoning in Philosophy, Phenomena
or Appearances, Propositions I-XVI, and The Motion
of the Moon's Nodes.
Inscribed over Newton's image is Newton's binomial
theorem, which dealt with expanding expressions of
the form (a+b) n. This was Newton's first epochal
mathematical discovery, one of his "great theorems".
It was not a theorem in the same sense as the
theorems of Euclid or Archimedes, insofar as Newton
did not provide a complete "proof", but rather
furnished, through brilliant insight, the precise and
correct formula which could be used stunningly to
great effect.
Newton is widely regarded as the inventor of modern
calculus. In fact, that honour is correctly shared with
Leibniz, who developed his own version of calculus
independent of Newton, and in the same time frame,
resulting in a rancorous dispute.
Leibniz's calculus had a far superior and more elegant
notation compared to Newton's calculus, and it is
Leibniz's notation which is still in use today.
Newton's portrait shares a colour palette with Leibniz, the
other acknowledged "inventor" of calculus, Lagrange,
a pioneer of the "calculus of variations", and Laplace
and Euler, two of those who built on what had been
so ably begun.
Gottfried Wilhelm Leibniz
1646 - 1716
Gottfried Wilhelm Leibniz was a philosopher, mathematician,
physicist, jurist, and contemporary of Newton. He is
considered one of the great thinkers of the 17th century. He
believed in a universe which followed a "pre-established
harmony" between mind and matter, and attempted to
reconcile the existence of a material world with the existence
of a supreme being.
The twentieth century philosopher and mathematician Bertrand
Russell considered Leibniz's greatest claim to fame to be his
invention of the infinitesimal calculus -- a remarkable
achievement considering that Leibniz was self-taught in
Leibniz is portrayed overlaid with integral notation from his
calculus which he developed coincident with but
independently of Newton's development of calculus.
Although the historical record suggest that Newton developed his
version of calculus first, Leibniz was the first to
publish. Unfortunately, what emerged was not fruitful
collaboration, but a rancorous dispute that raged for decades
and pitted English continental mathematicians supporting
Newton as the true inventor of the calculus, against
continental mathematicians supporting Leibniz.
Today, Leibniz and Newton are generally recognized as 'co-inventors'
of the calculus.
But Leibniz' notation for calculus was far superior to that of Newton,
and it is the notation developed by Leibniz, including the integral
sign and derivative notation, that is still in use today.
Leibniz considered symbols to be critical for human understanding of
all things. So much so that he attempted to develop an entire
'alphabet of human thought', in which all fundamental concepts
would be represented by symbols which could be combined to
represent more complex thoughts. Leibniz never finished this
Leibniz, who had strong conceptual differences with Newton in other
areas, notably with Newton's concept of absolute space, also
develop bitter conceptual differences with Descartes over what
was then referred to as the "fundamental quantity of motion", a
precursor of the Law of Conservation of Energy.
Much of Leibniz' work went unpublished during his lifetime. He died
embittered, in ill health, and without achieving the considerable
wealth, fame, and honour accorded to Newton.
Leibniz' diverse writings -- philosophical, mathematical, historical, and
political -- were resurrected and published in the late 19th and
20th centuries.
But calculus -- with Leibniz notation still in use today -- remains his
towering legacy.
Leonhard Euler
1707 - 1783
Leonhard Euler's intellect was towering and his work in
mathematics panoramic. In the words of the eminent
mathematical historian, W.W. Rouse Ball, Euler "created a
good deal of analysis, and revised almost all the branches of
pure mathematics which were then known filling up the
details, adding proofs, and arranging the whole in a
consistent form."
Euler's image is incised with a very elegant and symbolically rich
formula, a consequence of Euler's famous equation. It
incorporates the chief symbols in mathematical history up to
that time -- the principal whole numbers 0 and 1, the chief
mathematical relations + and =, pi the discovery of
Hippocrates, i the sign for the "impossible" square root of
minus one, and the logarithm base e.
The intricate shadow cast on Euler's image is in fact a view of
the city of Königsberg as it was in Euler's day, showing the
seven bridges over the River Pregel. Euler enjoyed solving
puzzling problems for recreational amusement, and tackled
the problem of whether all seven bridges of Konigsberg could
be crossed without re-crossing any one of them. In solving
the problem, which he did by mathematically representing
and formalizing it -- Euler gave birth to modern graph
Joesph-Louis Lagrange
1736 - 1813
Joseph-Louis Lagrange not only provided
brilliant analyses which were eventually to
facilitate, among other things, modern-day
satellites, but reveled in and put on display
the sheer beauty of mathematics. One of
Lagrange's works, Mécanique Analytique,
has been described as a "scientific poem".
Lagrange's image is inscribed with the "EulerLagrange equation", a seminal differential
equation in the 'calculus of variations', which
concerns itself with paths, curves, and
surfaces for which a given function has a
stationary value.
Lagrange's image is backed by a color plot of
fields surrounding points in space, overlaid by
a triangle identifying and connecting 3 critical
"Lagrangian points", named after Lagrange
who first showed their existence.
These 3 Lagrangian points define a position in
space where the pulls of two rotating
gravitational bodies (such as the EarthMoon, or Earth-sun) combine to form a
point at which a third body of
comparatively negligible mass would
remain stationary relative to the two
Lagrangian points have proven invaluable in
positioning satellites for synchronous
orbit, and more recently, other
Lagrangian points first thought unstable,
have become the basis for 'chaotic
control'. This is a relatively new technique
being explored for space flight, similar to
gravity assist, which may enable practical
interplanetary missions -- flown with
much smaller amounts of fuel.
Pierre-Simon Laplace
1749 - 1827
Pierre-Simon Laplace was a mathematician who
firmly believed the world was entirely
deterministic. Like a man with a hammer to
whom everything was a nail, to Laplace the
universe was nothing but a giant problem in
Laplace's Mécanique Céleste (Celestial Mechanics),
essentially translated the geometrical study of
mechanics by Newton to one based on
calculus. Napoleon asked Laplace why there
was not a single mention of God in Laplace's
entire five volume explaining how the heavens
operated. (Newton, a man of science who
believed in an omnipresent God, had even
posited God's periodic intervention to keep the
universe on track.) Laplace replied to Napoleon
that he had "no need for that particular
Laplace also used calculus, among other things, to
explore probability theory. Laplace considered
probability theory to be simply "common sense
reduced to calculus", which he systematized in
his "Essai Philosophique sur les Probabilités"
(Philosophical Essay on Probability, 1814).
Laplace's contention that the universe and all it
contained were deterministic machines was
thoroughly over-turned by the discoveries of
twentieth century physics.
Laplace is portrayed with what is possibly the most
celebrated differential equation ever devised -Laplace's partial differential equation, commonly
referred to as Laplace's Equation, shown here
in the form of a Laplacian operator.
Laplace's partial differential has been successfully
used for tasks as diverse as describing the
stability of the solar system, the field around an
electrical charge, and the distribution of heat in a
pot of food in the oven.
Johann Carl Friedrich Gauss
Gauss, a stickler for perfection, lived by the motto
"pauca sed matura" (few but ripe). He published
only a small portion of his work. It is from a scant
19 page diary, published only after Gauss's death,
that many of the results he established during his
lifetime were posthumously gleaned.
Gauss is portrayed with one of his most important
results -- the refutation of Euclid's fifth postulate,
the 'parallel postulate', which posited that parallel
lines would never meet.
Gauss discovered that valid self-consistent geometries
could be created in which the parallel postulate
did not hold. These geometries came to be known
as 'non-Euclidean geometries".
Gauss chose not to publish his results in alternative
geometries, and credit for the discovery of 'nonEuclidean geometry' was accorded to others
(Bolyai, and Lobachevski) who arrived at similar
results independently.
Overlying Gauss's portrait the Gaussian
distribution curve is incised. This
probability distribution curve is commonly
referred to as the "normal distribution" by
statisticians, and, because of its curved
flaring shape, as the "bell curve" by social
scientists. The Gaussian distribution has
found wide application in numerous
experimental situations, where it
describes the deviations of repeated
measurements from the mean. It has the
characteristics that positive and negative
deviations are equally likely, and small
deviations are much more likely than
large deviations.
Gauss is also known for Gaussian primes,
Gaussian integers, Gaussian integration,
and Gaussian elimination -- to name only
a few of the achievements directly named
after an individual who was, perhaps, the
most gifted mathematician of all time.
Ada Byron, Lady Lovelace
1815 - 1852
Ada Byron, Lady Lovelace aspired to be "an Analyst (&
Metaphysician)", a title she presciently invented for herself at
a time when the notion of "professional scientist" had not
even taken full form. She not only met her expectations, but
is generally regarded as the first person to anticipate the
general purpose computer, and in many senses the world's
first "computer programmer".
A complex intellect, Ada was the daughter of the romantic poet
Lord Byron -- who separated from her mother only weeks
after Ada's birth, and never met his daughter Ada -- and
Annabella (Lady Byron), who was herself educated as both a
mathematician and a poet.
By the age of 8 Ada was adept at building detailed model boats.
By the age of 13 she had produced the design for a flying
machine. At the same time she was becoming an
accomplished musician, learning to play piano, violin, and
harp, and had a passion for gymnastics, dancing, and riding.
Ada set her sights on meeting Mary Somerville, a mathematician who had translated the works
of Laplace into English. And it was through her acquaintance with Mary Sommerville that,
in 1834, Ada met Charles Babbage, Lucasian professor of mathematics at Cambridge -- a
post once held by Sir Isaac Newton.
Babbage was the inventor of a calculating machine known as the "Difference Engine", sonamed because it operated based on the method of finite differences.
Ada was struck by the "universality" of Babbage's ideas -- something few others saw at the
time. What was to become a life-long friendship blossomed, with correspondence that
started with the topics of mathematics and logic, and burgeoned to include all manner of
In 1834 Babbage had already begun planning for a new type of calculating machine -- the
"Analytical Engine", conjecturing a calculating machine that could not only foresee, but
When Babbage reported on his plans for this new "Analytical Engine" at a conference in
Turin in 1841, one of the attendees, Luigi Menabrea, was so impressed that he wrote an
account of Babbage's at lectures. Ada, then 27, married to the Earl of Lovelace, and the
mother of three children under the age of eight, translated Menabrea's article from French
into English. Babbage suggested she add her own explanatory notes.
What emerged was "The Sketch of the Analytical Engine", published as an article in 1843,
with Ada's notes being twice as long as the original material. It became the definitive work
on the subject of what was to eventually become "computing".
In 1852, Ada Byron, Lady Lovelace, died from cervical cancer. She was 36 years old.
At her own request, Ada Byron was buried at the family estate, beside her father whom she
never met.
In 1980, the United States Department of Defense completed a new computer language.
This advanced new computer language was named "Ada".
The End
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