Pythagoras and Zeno made early contributions to mathematics and philosophy. Pythagoras is credited with the first proof of the Pythagorean theorem, while Zeno conceived paradoxes to support Parmenides' view that motion is illusory. Archimedes made seminal advances in geometry, measurement of pi, and buoyancy. Euclid's Elements was a principal geometry text for over 2000 years, developing proofs from postulates including the parallel postulate. Later mathematicians like Descartes, Fermat, Pascal, Newton, Euler, Cantor further advanced fields like algebra, calculus, probability, and the theory of infinite sets.

1. Mathematicians
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, likely 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
"motion".
One of Zeno's most famous paradoxes posited a race between the popular Greek hero Achilles,
and a tortoise.
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 Achilles.
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.
Zeno's portrait is depicted in the style of an ancient Grecian amphora vase.
Overlaying Zeno is a graphical depiction of 'limits'.
Archimedes of Syracuse

2. 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.
Eukleides is pictured with what is perhaps his most famous postulate -- the fifth postulate, often
cited as the "parallel postulate". The parallel postulate states that one, and only one, line can be
drawn through a point parallel to a given line -- and it is from this postulate, and on this basis, that
what has come to be known as "Euclidean geometry" proceeds.
It was not until the 19th century that Euclid's fifth postulate -- the "parallel postulate" was
rigorously and successfully challenged.
The two parallel lines of Euclid meet and converge in the portrait of Johann Carl Friedrich Gauss
-- whose work led to the emergence of non-Euclidean geometry, where Euclid's fifth postulate
gave way to new mathematical universe, where 2 parallel lines could, in fact, meet.
The portrait of Gauss shares a common dominant color palette with the portrait of Euclid -- but

3. two different conceptions of 'geometry'.
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 constructions.
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' portrait is quadrisected by the axes of his great advance in analytical geometry: what
has come to be known as the Cartesian plane. It enabled an algebraic representation of
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 3-dimensional space could be determined by three
co-ordinates.
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 lawyer.
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

4. "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 Theorem.
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 color 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 powers.
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

5. 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.
Pascal's mechanical calculating device offered significant improvement over manual calculations,
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 analog 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 honor 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 color 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.

6. 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
theory.
Euler's portrait uses a similar color palette to those of Newton and Leibniz, whose work Euler built
on, expanded, and colored with valuable analytical insight.
Georg Cantor 1845 - 1918
Georg Cantor undertook the exploration of the "infinite", and developed modern theory on infinite
sets. which remains conceptually challenging.
Cantor's work provided an approach to problems that had beset mathematicians for centuries,
including Zeno's ancient paradoxes.
He gave the first clear and consistent definition of an infinite set.
Cantor's image is flanked by the "Aleph", the first letter of the Hebrew alphabet, which Cantor
used (accompanied by subscripts) in his descriptions of transfinite numbers -- quite simply
numbers which were not finite.
Cantor recognized and demonstrated that infinite sets can be of different sizes. He distinguished
between countable and uncountable sets, and was able to prove that the set of all rational
numbers Q is countable, while the set off all real numbers R is uncountable, and therefore,
though both were infinite, R was strictly larger.
Backing Cantor's image is a graphic generated from a "Cantor Set".
A "Cantor Set" is an infinite set constructed using only the numbers between 0 and 1.

7. A Cantor set is constructed by starting with a line of length 1, and removing the middle 1/3. Next,
the middle 1/3 of each of the pieces that are left are removed, and then the middle 1/3 of the
pieces that remain after that are removed.
The set that remains after continuing this process forever is called the Cantor set.
The Cantor set contains uncountably many points.
The graphic set which backs Cantor's image began with an algorithm to generate the Cantor set,
to which color was applied, and then universal operators related to color transition and
magnification, ultimately resulting in a unique image whose essence was the Cantor set.
The final problem which Cantor grappled with was the realization that there could be no set
containing everything, since, given any set, there is always a larger set -- its set of subsets.
Cantor came came to the conclusion that the Absolute was beyond man's reach, and identified
this concept with God.
In one of his last letters Cantor wrote:
I have never proceeded from any 'Genus supermum' of the actual infinite. Quite the contrary, I
have rigorously proved that there is absolutely no "Genus supremum' of the actual infinite. What
surpasses all that is finite and transfinite is no 'Genus'; it is the single, completely individual unity
in which everything is included, which includes the Absolute, incomprehensible to the human
understanding. This is the Actus Purissimus, which by many is called God."
- Georg Cantor