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     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
 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 3-
       dimensional 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 "Euler-
       Lagrange 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 Earth-
        Moon, 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 'non-
     Euclidean 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", so-
     named 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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