#### Transcript Pythagoras - 2July

Pythagoras 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 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 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 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 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 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". Euclid 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. Euclid 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. Descarte 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 co-ordinates. Fermat 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 "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. Pascal 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 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 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. Newton 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. Newton 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. Liebniz 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 mathematics. 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. 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 work. 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. Euler 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. Lagrange 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. Lagrange 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 bodies. 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. Laplace 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 calculus. 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 hypothesis". Laplace 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. Gauss Johann Carl Friedrich Gauss 1777-1855 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. Gauss 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. Lovelace 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. Lovelace 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 subjects. 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 act. 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 This presentation has used images and text from http://www.mathematicianspictures.com/ Other historical links can be found at : http://www.phy.bg.ac.yu/web_projects/giants/pictures.html http://www.math.buffalo.edu/mad/madgreatest.html http://www-history.mcs.st-andrews.ac.uk/history/index.html http://www.geom.uiuc.edu/~banchoff/Flatland/