16th-17th Century Mathematics Notes PDF

16TH CENTURY MATHEMATICS The cultural, intellectual and artistic movement of the Renaissance, which saw a resurgence of learning based on classical sources, began in Italy around the 14th Century, and gradually spread across most of Europe over the next two centuries. Science and art we...

16TH CENTURY MATHEMATICS The cultural, intellectual and artistic movement of the Renaissance, which saw a resurgence of learning based on classical sources, began in Italy around the 14th Century, and gradually spread across most of Europe over the next two centuries. Science and art were still very much interconnected and intermingled at this time, as exemplified by the work of artist/scientists such as Leonardo da Vinci, and it is no surprise that, just as in art, revolutionary work in the fields of philosophy and science was soon taking place. It is a tribute to the respect in which mathematics was held in Renaissance Europe that the famed German artist Albrecht Dürer included an order-4 magic square in his engraving "Melencolia I". In fact, it is a so-called "supermagic square" with many more lines of addition symmetry than a regular 4 x 4 magic square (see image at right). The year of the work, 1514, The supermagic square shown in Albrecht Dürer's engraving "Melencolia I" is shown in the two bottom central squares. An important figure in the late 15th and early 16th Centuries is an Italian Franciscan friar called Luca Pacioli, who published a book on arithmetic, geometry and book-keeping at the end of the 15th Century which became quite popular for the mathematical puzzles it contained. It also introduced symbols for plus and minus for the first time in a printed book (although this is also sometimes attributed to Giel Vander Hoecke, Johannes Widmann and others), symbols that were to become standard notation. Pacioli also investigated the Golden Ratio of 1 : 1.618... (see the section on Fibonacci) in his 1509 book "The Divine Proportion", concluding that the number was a message from God and a source of secret knowledge about the inner beauty of things. During the 16th and early 17th Century, the equals, multiplication, division, radical (root), decimal and inequality symbols were gradually introduced and standardized. The use of decimal fractions and decimal arithmetic is usually attributed to the Flemish mathematician Simon Stevin the late 16th Century, although the decimal point notation was not popularized until early in the 17th Century. Stevin was ahead of his time in enjoining that all types of numbers, whether fractions, negatives, real numbers or surds (such as √2) should be treated equally as numbers in their own right. In the Renaissance Italy of the early 16th Century, Bologna University in particular was famed for its intense public mathematics competitions. It was in just such a competion that the unlikely figure of the young, self- taughtNiccolò Fontana Tartaglia revealed to the world the formula for solving first one type, and later all types, of cubic equations (equations with terms including x3), an achievement hitherto considered impossible and which had stumped the best mathematicians ofChina, India and the Islamic world. Building on Tartaglia‟s work, another young Italian, Lodovico Ferrari, soon devised a similar method to solve quartic equations (equations with terms including x4) and both solutions were published by Gerolamo Cardano. Despite a decade-long fight over the publication, Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what are now known as complex numbers, combinations of real and imaginary numbers (although it fell to another Bologna resident, Rafael Bombelli, to explain what imaginary numbers really were and how they could be used). Tartaglia went on to produce other important (although largely ignored) formulas and methods, and Cardano published perhaps the first systematic treatment of probability. With Hindu-Arabic numerals, standardized notation and the new language of algebra at their disposal, the stage was set for the European mathematical revolution of the 17th Century. ________________________________________________________________________________ - TARTAGLIA, CARDANO & FERRARI In the Renaissance Italy of the early 16th Century, Bologna University in particular was famed for its intense public mathematics competitions. It was in just such a competition, in 1535, that the unlikely figure of the young Venetian Tartaglia first revealed a mathematical finding hitherto considered impossible, and which had stumped the best mathematicians of China, India and the Islamic world. Niccolò Fontana became known as Tartaglia (meaning “the stammerer”) for a speech defect he suffered due to an injury he received in a battle against the invading French army. He was a poor engineer known for designing fortifications, a surveyor of topography (seeking the best means of defence or offence in battles) and a bookkeeper in the Republic of Venice. Niccolò Fontana Tartaglia But he was also a self-taught, but wildly ambitious, mathematician. He (1499-1557) distinguised himself by producing, among other things, the first Italian translations of works by Archimedes and Euclid from uncorrupted Greek texts (for two centuries, Euclid's "Elements" had been taught from two Latin translations taken from an Arabic source, parts of which contained errors making them all but unusable), as well as an acclaimed compilation of mathematics of his own. Tartaglia's greates legacy to mathematical history, though, occurred when he won the 1535 Bologna University mathematics competition by demonstrating a general algebraic formula for solving cubic equations (equations with terms including x3), something which had come to be seen by this time as an impossibility, requiring as it does an understanding of the square roots of negative numbers. In the competition, he beat Scipione del Ferro (or at least del Ferro's assistant, Fior), who had coincidentally produced his own partial solution to the cubic equation problem not long before. Although del Ferro's solution perhaps predated Tartaglia’s, it was much more limited, and Tartaglia is usually credited with the first general solution. In the highly competitive and cut-throat environment of 16th Century Italy, Tartaglia even encoded his solution in the form of a poem in an attempt to make it more difficult for other mathematicians to steal it. Tartaglia’s definitive method was, however, leaked to Gerolamo Cardano (or Cardan), a rather eccentric and confrontational mathematician, doctor and Renaissance man, and author throughout his lifetime of some 131 books. Cardano published it himself in his 1545 book "Ars Magna" (despite having promised Tartaglia that he would not), along with the work of his own brilliant student Lodovico Ferrari. Ferrari, on seeing Tartaglia's cubic solution, had realized that he could use a similar method to solve quartic equations (equations with terms including x4). In this work, Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what are now known as complex numbers, combinations of real and imaginary numbers of the type a + bi, where i is the imaginary unit √-1. It fell to another Bologna resident, Rafael Bombelli, to explain, at the end of the 1560's, exactly what imaginary numbers really were and how they could be used. Although both of the younger men were acknowledged in the foreword of Cardano's book, as well as in several places within its body, Tartgalia engaged Cardano in a decade-long fight over the publication. Cardano argued that, when he happened to see (some years after the 1535 competition) Scipione del Ferro's unpublished independent cubic equation solution, which was dated before Tartaglia's, he decided that his promise to Tartaglia could legitimately be broken, and he included Tartaglia's solution in his next publication, along with Ferrari's quartic solution. Ferrari eventually came to understand cubic and quartic equations much better than Tartaglia. When Ferrari challenged Tartaglia to another public debate, Tartaglia initially accepted, but then (perhaps wisely) decided Gerolamo Cardano (1501- not to show up, and Ferrari won by default. Tartaglia was thoroughly 1576) discredited and became effectively unemployable. Poor Tartaglia died penniless and unknown, despite having produced (in addition to his cubic equation solution) the first translation of Euclid’s “Elements” in a modern European language, formulated Tartaglia's Formula for the volume of a tetrahedron, devised a method to obtain binomial coefficients called Tartaglia's Triangle (an earlier version of Pascal's Triangle), and become the first to apply mathematics to the investigation of the paths of cannonballs (work which was later validated by Galileo's studies on falling bodies). Even today, the solution to cubic equations is usually known as Cardano’s Formula and not Tartgalia’s. Ferrari, on the other hand, obtained a prestigious teaching post while still in his teens after Cardano resigned from it and recommended him, and was eventually able to retired young and quite rich, despite having started out as Cardano’s servant. Cardano himself, an accomplished gambler and chess player, wrote a book called "Liber de ludo aleae" ("Book on Games of Chance") when he was just 25 years old, which contains perhaps the first systematic treatment of probability (as well as a section on effective cheating methods). The ancient Greeks, Romansand Indians had all been inveterate gamblers, but none of them had ever attempted to understand randomness as being governed by mathematical laws. The book described the - now obvious, but then revolutionary - insight that, if a random event has several equally likely outcomes, the chance of any individual outcome is equal to the proportion of that outcome to all possible outcomes. The book was far ahead of its time, though, and it remained unpublished until 1663, nearly a century after his death. It was the only serious work on probability until Pascal's work in the 17th Century. Cardano was also the first to describe hypocycloids, the pointed plane curves generated by the trace of a fixed point on a small circle that rolls within a larger circle, and the generating circles were later named Cardano (or Cardanic) circles. The colourful Cardano remained notoriously short of money thoughout his life, largely due to his gambling habits, and was accused of heresy in 1570 after publishing a horoscope of Jesus (apparently, his own son contributed to the prosecution, bribed by Tartaglia). 17TH CENTURY MATHEMATICS In the wake of the Renaissance, the 17th Century saw an unprecedented explosion of mathematical and scientific ideas across Europe, a period sometimes called the Age of Reason. Hard on the heels of the “Copernican Revolution” of Nicolaus Copernicus in the 16th Century, scientists like Galileo Galilei, Tycho Brahe and Johannes Kepler were making equally revolutionary discoveries in the exploration of the Solar system, leading to Kepler‟s formulation of mathematical laws of planetary motion. The invention of the logarithm in the early 17th Century by John Napier (and later improved by Napier and Henry Briggs) contributed to the advance of science, astronomy and mathematics by making some difficult calculations relatively easy. It was one of the most significant mathematical developments of the age, and 17th Century physicists like Kepler and Newton could never have performed the complex calculatons needed for their innovations without it. The Logarithms were invented by John Napier, early in the 17th French astronomer and mathematician Pierre Century Simon Laplace remarked, almost two centuries later, that Napier, by halving the labours of astronomers, had doubled their lifetimes. The logarithm of a number is the exponent when that number is expressed as a power of 10 (or any other base). It is effectively the inverse of exponentiation. For example, the base 10 logarithm of 100 (usually written log10 100 or lg 100 or just log 100) is 2, because 102 = 100. The value of logarithms arises from the fact that multiplication of two or more numbers is equivalent to adding their logarithms, a much simpler operation. In the same way, division involves the subtraction of logarithms, squaring is as simple as multiplying the logarithm by two (or by three for cubing, etc), square roots requires dividing the logarithm by 2 (or by 3 for cube roots, etc). Although base 10 is the most popular base, another common base for logarithms is the number e which has a value of 2.7182818... and which has special properties which make it very useful for logarithmic calculations. These are known as natural logarithms, and are written loge or ln. Briggs produced extensive lookup tables of common (base 10) logarithms, and by 1622 William Oughted had produced a logarithmic slide rule, an instrument which became indispensible in technological innovation for the next 300 years. Napier also improved Simon Stevin's decimal notation and popularized the use of the decimal point, and made lattice multiplication (originally developed by the Persian mathematician Al-Khwarizmi and introduced into Europe by Fibonacci) more convenient with the introduction of “Napier's Bones”, a multiplication tool using a set of numbered rods. Although not principally a mathematician, the role of the Frenchman Marin Mersenne as a sort of clearing house and go-between for mathematical thought in France during this period was crucial. Mersenne is largely remembered in mathematics today in the term Mersenne primes - prime numbers that are one less than a power of 2, e.g. 3 (22-1), 7 (23-1), 31 (25-1), 127 (27-1), 8191 (213-1), etc. In modern times, the largest known prime number has almost always been a Mersenne prime, but in actual fact, Mersenne‟s real connection with the numbers was only to compile a none-too-accurate list of the smaller ones (when Edouard Lucas devised a method of checking them in the 19th Century, he pointed out that Mersenne had incorrectly included 267-1 and left out 261-1, 289-1 and 2107-1 from his list). The Frenchman René Descartes is sometimes considered the first of the modern school of mathematics. His development of analytic geometry and Cartesian coordinates in the mid-17th Century soon allowed the orbits of the planets to be plotted on a graph, as well as laying the foundations for the later development of calculus (and much later multi-dimensional geometry). Although not principally a mathematician, the role of the Frenchman Marin Mersenne as a sort of clearing house and go-between for mathematical thought in France during this period was crucial. Mersenne is largely remembered in mathematics today in the term Mersenne primes - prime numbers that are one less than a power of 2, e.g. 3 (22-1), 7 (23-1), 31 (25-1), 127 (27-1), 8191 (213-1), etc. In modern times, the largest known prime number has almost always been a Mersenne prime, but in actual fact, Mersenne‟s real connection with the numbers was only to compile a none-too-accurate list of the smaller ones (when Edouard Lucas devised a method of checking them in the 19th Century, he pointed out that Mersenne had incorrectly included 267-1 and left out 261-1, 289-1 and 2107-1 from his list). The Frenchman René Descartes is sometimes considered the first of the modern school of mathematics. His development of analytic geometry and Cartesian coordinates in the mid-17th Century soon allowed the orbits of the planets to be plotted on a graph, as well as laying the foundations for the later development of calculus (and much later multi-dimensional geometry). Descartes is also credited with the first use of superscripts for powers or exponents. Two other great French mathematicians were close contemporaries of Descartes: Pierre de Fermat andBlaise Pascal. Fermat formulated several theorems which greatly extended our knowlege of number theory, as well as contributing some early work on infinitesimal calculus. Pascal is most famous for Pascal‟s Triangle of binomial coefficients, although similar figures had actually been produced by Chinese and Persian mathematicians long before him. It was an ongoing exchange of letters between Fermat and Pascal that led to the development of the concept of expected values and the field of probability theory. The first published work on probability theory, however, and the first to outline the concept of mathematical expectation, was by the Dutchman Christiaan Huygens in 1657, although it was largely based on the ideas in the letters of the two Frenchmen. The French mathematician and engineer Girard Desargues is considered one of the founders of the field of projective geometry, later developed further by Jean Victor Poncelet and Gaspard Monge. Projective geometry considers what happens to shapes when they are projected on to a non-parallel plane. For example, a circle may be projected into an ellipse or a hyperbola, and so these curves may all be regarded as equivalent in projective geometry. In particular, Desargues developed the pivotal concept of the “point at infinity” where parallels actually meet. His perspective theorem states that, when two triangles are in perspective, their corresponding sides meet at points on the same collinear line. Desargues’ perspective theorem By “standing on the shoulders of giants”, the Englishman Sir Isaac Newton was able to pin down the laws of physics in an unprecedented way, and he effectively laid the groundwork for all of classical mechanics, almost single-handedly. But his contribution to mathematics should never be underestimated, and nowadays he is often considered, along with Archimedes and Gauss, as one of the greatest mathematicians of all time. Newton and, independently, the German philosopher and mathematician Gottfried Leibniz, completely revolutionized mathematics (not to mention physics, engineering, economics and science in general) by the development of infinitesimal calculus, with its two main operations, differentiation and integration. Newtonprobably developed his work before Leibniz, but Leibniz published his first, leading to an extended and rancorous dispute. Whatever the truth behind the various claims, though, it is Leibniz‟s calculus notation that is the one still in use today, and calculus of some sort is used extensively in everything from engineering to economics to medicine to astronomy.  Both Newton and Leibniz also contributed greatly in other areas of mathematics, including Newton‟s contributions to a generalized binomial theorem, the theory of finite differences and the use of infinite power series, and Leibniz‟s development of a mechanical forerunner to the computer and the use of matrices to solve linear equations. However, credit should also be given to some earlier 17th Century mathematicians whose work partially anticipated, and to some extent paved the way for, the development of infinitesimal calculus. As early as the 1630s, the Italian mathematician Bonaventura Cavalieri developed a geometrical approach to calculus known as Cavalieri's principle, or the “method of indivisibles”. The Englishman John Wallis, who systematized and extended the methods of analysis of Descartes and Cavalieri, also made significant contributions towards the development of calculus, as well as originating the idea of the number line, introducing the symbol ∞ for infinity and the term “continued fraction”, and extending the standard notation for powers to include negative integers and rational numbers. Newton's teacher Isaac Barrow is usually credited with the discovery (or at least the first rigorous statrement of) the fundamental theorem of calculus, which essentially showed that integration and differentiation are inverse operations, and he also made complete translations of Euclid into Latin and English. ______________________________________________________________________________________ FERMAT Another Frenchman of the 17th Century, Pierre de Fermat, effectively invented modern number theory virtually single-handedly, despite being a small-town amateur mathematician. Stimulated and inspired by the “Arithmetica” of the Hellenistic mathematician Diophantus, he went on to discover several new patterns in numbers which had defeated mathematicians for centuries, and throughout his life he devised a wide range of conjectures and theorems. He is also given credit for early developments that led to modern calculus, and for early progress in probability theory. Although he showed an early interest in mathematics, he went on study law at Orléans and received the title of councillor at the High Court of Judicature in Toulouse in 1631, which he held for the rest of his life. He was fluent in Latin, Greek, Italian and Spanish, and was praised for his written verse in several languages, and eagerly sought for advice on the emendation of Greek texts. Pierre de Fermat (1601- 1665) Fermat's mathematical work was communicated mainly in letters to friends, often with little or no proof of his theorems. Although he himself claimed to have proved all his arithmetic theorems, few records of his proofs have survived, and many mathematicians have doubted some of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat One example of his many theorems is the Two Square Theorem, which shows that any prime number which, when divided by 4, leaves a remainder of 1 (i.e. can be written in the form 4n + 1), can always be re-written as the sum of two square numbers (see image at right for examples). His so-called Little Theorem is often used in the testing of large prime numbers, and is the basis of the codes which protect our credit cards in Internet transactions today. In simple (sic) terms, it says that if we have two numbers a and p, where p is a prime number and not a factor of a, then amultiplied by itself p-1 times and then divided by p, will always leave a remainder of 1. In mathematical terms, this is written: ap-1 = 1(mod p). For example, if a = 7 and p = 3, then 72 ÷ 3 should leave a remainder of 1, and 49 ÷ 3 does in fact leave a remainder of 1.  Fermat identified a subset of numbers, now known as Fermat numbers, which are of the form of one less than 2 to the power of a power of 2, or, written mathematically, 2 2n + 1. The first five such numbers are: 21 + 3 = 3; 22 + 1 = 5; 24 + 1 = 17; 28 + 1 = 257; and 216 + 1 = 65,537. Interestingly, these are all prime numbers (and are known as Fermat primes), but all the higher Fermat numbers which have been painstakingly identified over the years are NOT prime numbers, which just goes to to show the value of inductive proof in mathematics. Fermat's pièce de résistance, though, was his famous Last Theorem, a conjecture left unproven at his death, and which puzzled mathematicians for over 350 years. The theorem, originally described in a scribbled note in the margin of his copy of Diophantus' “Arithmetica”, states that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n greater than two (i.e. squared). This seemingly simple conjecture has proved to be one of the world‟s hardest mathematical problems to prove. There are clearly many solutions - indeed, an infinite number - when n = 2 (namely, all the Pythagorean triples), but no Fermat’s Last Theorem solution could be found for cubes or higher powers. Tantalizingly, Fermat himself claimed to have a proof, but wrote that “this margin is too small to contain it”. As far as we know from the papers which have come down to us, however, Fermat only managed to partially prove the theorem for the special case of n = 4, as did several other mathematicians who applied themselves to it (and indeed as had earlier mathematicians dating back to Fibonacci, albeit not with the same intent). Over the centuries, several mathematical and scientific academies offered substantial prizes for a proof of the theorem, and to some extent it single-handedly stimulated the development of algebraic number theory in the 19th and 20th Centuries. It was finally proved for ALL numbers only in 1995 (a proof usually attributed to British mathematician Andrew Wiles, although in reality it was a joint effort of several steps involving many mathematicians over several years). The final proof made use of complex modern mathematics, such as the modularity theorem for semi-stable elliptic curves, Galois representations and Ribet‟s epsilon theorem, all of which were unavailable in Fermat‟s time, so it seems clear that Fermat's claim to have solved his last theorem was almost certainly an exaggeration (or at least a misunderstanding). In addition to his work in number theory, Fermat anticipated the development of calculus to some extent, and his work in this field was invaluable later to Newton and Leibniz. While investigating a technique for finding the centres of gravity of various plane and solid figures, he developed a method for determining maxima, minima and tangents to various curves that was essentially equivalent to differentiation. Also, using an ingenious trick, he was able to reduce the integral of general power functions to the sums of geometric series. Fermat‟s correspondence with his friend Pascal also helped mathematicians grasp a very important concept in basic probability which, although perhaps intuitive to us now, was revolutionary in 1654, namely the idea of equally probable outcomes and expected values. PASCAL The Frenchman Blaise Pascal was a prominent 17th Century scientist, philosopher and mathematician. Like so many great mathematicians, he was a child prodigy and pursued many different avenues of intellectual endeavour throughout his life. Much of his early work was in the area of natural and applied sciences, and he has a physical law named after him (that “pressure exerted anywhere in a confined liquid is transmitted equally and undiminished in all directions throughout the liquid”), as well as the international unit for the meaurement of pressure. In philosophy, Pascals‟ Wager is his pragmatic approach to believing in God on the grounds that is it is a better “bet” than not to.  But Pascal was also a mathematician of the first order. At the age of sixteen, he wrote a significant treatise on the subject of projective geometry, known as Pascal's Theorem, which states that, if a hexagon is inscribed in a circle, then the three intersection points of opposite sides lie on a single line, called the Pascal line. As a young man, he built a functional calculating machine, able to perform additions and subtractions, to help his father with his tax calculations. He is best known, however, for Pascal‟s Triangle, a convenient tabular presentation of binomial co-efficients, where each number is the sum of the two numbers directly above it. A binomial is a simple type of algebraic expression which has just two terms operated on only by addition, subtraction, multiplication and positive whole-number exponents, such as (x +y)2. The co-efficients produced when a binomial is expanded form a symmetrical triangle (see image at right). Blaise Pascal (1623-1662) Pascal was far from the first to study this triangle. The Persian mathematician Al-Karaji had produced something very similar as early as the 10th Century, and the Triangle is called Yang Hui's Triangle in China after the 13th Century Chinese mathematician, and Tartaglia‟s Triangle in Italy after the eponymous 16th Century Italian. But Pascal did contribute an elegant proof by defining the numbers by recursion, and he also discovered many useful and interesting patterns among the rows, columns and diagonals of the array of numbers. For instance, looking at the diagonals alone, after the outside "skin" of 1's, the next diagonal (1, 2, 3, 4, 5,...) is the natural numbers in order. The next diagonal within that (1, 3, 6, 10, 15,...) is the triangular numbers in order. The next (1, 4, 10, 20, 35,...) is the pyramidal triangular numbers, etc, etc. It is also possible to find prime numbers, Fibonacci numbers, Catalan numbers, and many other series, and even to find fractal patterns within it. Pascal also made the conceptual leap to use the Triangle to help solve problems in probability theory. In fact, it was through his collaboration and correspondence with his French contemporary Pierre de Fermat and the Dutchman Christiaan Huygens on the subject that the mathematical theory of probability was born. Before Pascal, there was no actual theory of probability - notwithstanding Gerolamo Cardano‟s early exposition in the 16th Century - merely an understanding (of sorts) of how to compute “chances” in dice and card games by counting equally probable outcomes. Some apparently quite elementary problems in probability had eluded some of the best mathematicians, or given rise to incorrect solutions. It fell to Pascal (with Fermat's help) to bring together the separate threads of prior knowledge (includingCardano's early work) and to introduce entirely new mathematical techniques for the solution of problems that had hitherto resisted solution. Two such intransigent problems which Pascal and Fermat applied themselves to were the Gambler‟s Ruin (determining the chances of winning for each of two men playing a particular dice game with very specific rules) and the Problem of Points (determining how a game's winnings should be divided between two equally skilled players if the game was ended prematurely). His work on the Problem of Points in particular, although unpublished at the time, was highly influential in the unfolding new field. be illustrated by a simple game of “winner take all” involving the tossing of a coin. The first of the two players (say,Fermat and Pascal) to achieve ten points or wins is to receive a pot of 100 francs. But, if the game is interrupted at the point where Fermat, say, is winning 8 points to 7, how is the 100 franc pot to divided? Fermat claimed that, as he needed only two more points to win the game, and Pascal needed three, the game would have been over after four more tosses of the coin (because, if Pascal did not get the necessary 3 points for your victory over the four tosses, then Fermat must have gained the necessary 2 points for his victory, and vice versa. Fermat then exhaustively listed the possible outcomes of the four tosses, and concluded that he would win in 11 out of the 16 possible outcomes, so he suggested that the 100 francs be split 11⁄16 (0.6875) to him and 5⁄16 (0.3125) to Pascal. Pascal then looked for a way of generalizing the problem that would avoid the tedious listing of possibilities, and realized that he could use rows from his triangle of coefficients to generate the numbers, no matter how many tosses of the coin remained. As Fermat needed 2 more points to win the game and Pascal needed 3, he went to the fifth (2 + 3) row of the triangle, i.e. 1, 4, 6, 4, 1. The first 3 terms added together (1 + 4 + 6 = 11) represented the outcomes where Fermat would win, and the last two terms (4 + 1 = 5) the outcomes where Pascal would win, out of the total number of outcomes represented by the sum of the whole row (1 + 4 + 6 +4 +1 = Fermat and Pascal’s solution to the Problem of Points 16). Pascal and Fermat had grasped through their correspondence a very important concept that, though perhaps intuitive to us today, was all but revolutionary in 1654. This was the idea of equally probable outcomes, that the probability of something occurring could be computed by enumerating the number of equally likely ways it could occur, and dividing this by the total number of possible outcomes of the given situation. This allowed the use of fractions and ratios in the calculation of the likelhood of events, and the operation of multiplication and addition on these fractional probabilities. For example, the probability of throwing a 6 on a die twice is 1⁄6x 1⁄6 = 1⁄36 ("and" works like multiplication); the probability of throwing either a 3 or a 6 is 1⁄6 + 1⁄6 = 1⁄3 ("or" works like addition). Later in life, Pascal and his sister Jacqueline strongly identified with the extreme Catholic religious movement of Jansenism. Following the death of his father and a "mystical experience" in late 1654, he had his "second conversion" and abandoned his scientific work completely, devoting himself to philosophy and theology. His two most famous works, the "Lettres provinciales" and the "Pensées", date from this period, the latter left incomplete at his death in 1662. They remain Pascal‟s best known legacy, and he is usually remembered today as one of the most important authors of the French Classical Period and one of the greatest masters of French prose, much more than for his contributions to mathematics. NEWTON In the heady atmosphere of 17th Century England, with the expansion of the British empire in full swing, grand old universities like Oxford and Cambridge were producing many great scientists and mathematicians. But the greatest of them all was undoubtedly Sir Isaac Newton. Physicist, mathematician, astronomer, natural philosopher, alchemist and theologian, Newton is considered by many to be one of the most influential men in human history. His 1687 publication, the "Philosophiae Naturalis Principia Mathematica" (usually called simply the "Principia"), is considered to be among the most influential books in the history of science, and it dominated the scientific view of the physical universe for the next three centuries. Although largely synonymous in the minds of the general public today with gravity and the story of the apple tree, Newton remains a giant in the minds of mathematicians everywhere (on a par with the all-time greats like Archimedes and Gauss), and he greatly influenced the subsequent path of mathematical development.  Over two miraculous years, during the time of the Great Plague of 1665-6, the young Newton developed a new theory of light, discovered and quantified gravitation, and pioneered a revolutionary new approach to mathematics: infinitesimal calculus. His theory of calculus built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental mathematicians as René Descartes, Pierre de Fermat, Bonaventura Cavalieri, Johann van Waveren Hudde and Gilles Personne de Roberval. Unlike the static geometry of the Greeks, calculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us, such as the orbits of planets, the motion of fluids, etc. The initial problem Newton was confronting was that, although it was easy enough to represent and calculate the average slope of a curve (for example, the increasing speed of an object on a time-distance graph), the slope of a curve was constantly varying, and there was no method to give the exact slope at any one individual point on the curve i.e. effectively the slope of a tangent line to the curve at that point. Intuitively, the slope at a particular point can be approximated by taking the average slope (“rise over run”) of ever smaller segments of the curve. As the segment of the curve being considered approaches zero in size (i.e. an infinitesimal change in x), then the calculation of the slope approaches closer and closer to the exact slope at a point (see image at right). Without going into too much complicated detail, Newton (and his contemporary Gottfried Leibnizindependently) calculated a derivative function f „(x) which gives the slope at any point of a function f(x). This process of calculating the slope or derivative of a curve or function is called differential calculus or differentiation (or, in Newton‟s terminology, the “method of fluxions” - he called the instantaneous rate of change at a particular point on a curve the "fluxion", and the changing values of x and y the "fluents"). For instance, the derivative of a straight line of the type f(x) = 4x is just 4; the derivative of a squared function f(x) = x2 is 2x; the derivative of cubic functionf(x) = x3 is 3x2, etc. Generalizing, the derivative of any power function f(x) = xr is rxr-1. Other derivative functions can be stated, according to certain rules, for exponential and logarithmic functions, trigonometric functions such as sin(x), cos(x), etc, so that a derivative function can be stated for any curve without discontinuities. For example, the derivative of the curve f(x) = x4 - 5p3 + sin(x2) would be f ‟(x) = 4x3 - 15x2 + 2xcos(x2). Having established the derivative function for a particular curve, it is then an easy matter to calcuate the slope at any particular point on that curve, just by inserting a value for x. In the case of a time-distance graph, for example, this slope represents the speed of the object at a particular point. The “opposite” of differentiation is integration or integral calculus (or, in Newton‟s terminology, the “method of fluents”), and together differentiation and integration are the two main operations of calculus. Newton‟s Fundamental Theorem of Calculus states that differentiation and integration are inverse operations, so that, if a function is first integrated and then differentiated (or vice versa), the original function is retrieved. The integral of a curve can be thought of as the formula for calculating the area bounded by the curve and the xaxis between two defined boundaries. For example, on a graph of velocity against time, the area “under the curve” would represent the distance travelled. Essentially, integration is based on a limiting procedure which approximates the area of a curvilinear region by breaking it into infinitesimally thin vertical slabs or columns. In the same way as for differentiation, an integral function can be stated in general terms: the integral of any power f(x) = xr isxr+1⁄r+1, and there are other integral functions for exponential and logarithmic functions, trigonometric functions, etc, so that the area under any continuous curve can be obtained between any two limits. Newton chose not to publish his revolutionary mathematics straight away, worried about being ridiculed for his unconventional ideas, and contented himself with circulating his thoughts among friends. After all, he had many other interests such as philosophy, alchemy and his work at the Royal Mint. However, in 1684, the German Leibniz published his own independent version of the theory, whereas Newton published nothing on the subject until 1693. Although the Royal Society, after due deliberation, gave credit for the first discovery to Newton (and credit for the first publication to Leibniz), something of a scandal arose when it was made public that the Royal Society‟s subsequent accusation of plagiarism against Leibniz was actually authored by none other Newton himself, causing an ongoing controversy which marred the careers of both men. Despite being by far his best known contribution to mathematics, calculus was by no means Newton‟s only contribution. He is credited with the generalized binomial theorem, which describes the algebraic expansion of powers of a binomial (an algebraic expression with two terms, such as a2 -b2); he made substantial contributions to the theory of finite differences (mathematical expressions of the formf(x + b) - f(x + a)); he was one of the Newton's Method for approximating the roots of a curve by successive interations after an initial guess first to use fractional exponents and coordinate geometry to derive solutions to Diophantine equations (algebraic equations with integer-only variables); he developed the so-called “Newton's method” for finding successively better approximations to the zeroes or roots of a function; he was the first to use infinite power series with any confidence; etc. In 1687, Newton published his “Principia” or “The Mathematical Principles of Natural Philosophy”, generally recognized as the greatest scientific book ever written. In it, he presented his theories of motion, gravity and mechanics, explained the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis and the motion of the Moon. Later in life, he wrote a number of religious tracts dealing with the literal interpretation of the Bible, devoted a great deal of time to alchemy, acted as Member of Parliament for some years, and became perhaps the best-known Master of the Royal Mint in 1699, a position he held until his death in 1727. In 1703, he was made President of the Royal Society and, in 1705, became the first scientist ever to be knighted. Mercury poisoning from his alchemical pursuits perhaps explained Newton's eccentricity in later life, and possibly also his eventual death. LEIBNIZ The German polymath Gottfried Wilhelm Leibniz occupies a grand place in the history of philosophy. He was, along with René Descartes and Baruch Spinoza, one of the three great 17th Century rationalists, and his work anticipated modern logic and analytic philosophy. Like many great thinkers before and after him, Leibniz was a child prodigy and a contributor in many different fields of endeavour. But, between his work on philosophy and logic and his day job as a politician and representative of the royal house of Hanover, Leibniz still found time to work on mathematics. He was perhaps the first to explicitly employ the mathematical notion of a function to denote geometric concepts derived from a curve, and he developed a system of infinitesimal calculus, independently of his contemporary Sir Isaac Newton. He also revived the ancient method of solving equations using matrices, invented a practical calculating machine and pioneered the use of the binary system.  Like Newton, Leibniz was a member of the Royal Society in London, and was almost certainly aware of Newton‟s work on calculus. During the 1670s (slightly later than Newton‟s early work), Leibniz developed a very similar theory of calculus, apparently completely independently. Within the short period of about two months he had developed a complete theory of differential calculus and integral calculus (see the section on Newtonfor a brief description and explanation of the development of calculus). Unlike Newton, however, he was more than happy to publish his work, and so Europe first heard about calculus from Leibniz in 1684, and not from Newton(who published nothing on the subject until 1693). When the Royal Society was asked to adjudicate between the rival claims of the two men over the development of the theory of calculus, they gave credit for the first discovery to Newton, and credit for the first publication to Leibniz. However, the Royal Society, by then under the rather biassed presidency of Newton himself, later also accused Leibniz of plagiarism, a slur from which Leibniz never really recovered. Ironically, it was Leibniz‟s mathematics that eventually triumphed, and Leibniz’s and Newton’s notation for Calculus his notation and his way of writing calculus, not Newton‟s more clumsy notation, is the one still used in mathematics today. In addition to calculus, Leibniz re-discovered a method of arranging linear equations into an array, now called a matrix, which could then be manipulated to find a solution. A similar method had been pioneered by Chinese mathematicians almost two millennia earlier, but had long fallen into disuse. Leibniz paved the way for later work on matrices and linear algebra by Carl Friedrich Gauss. He also introduced notions of self-similarity and the principle of continuity which foreshadowed an area of mathematics which would come to be called topology. During the 1670s, Leibniz worked on the invention of a practical calculating machine, which used the binary system and was capable of multiplying, dividing and even extracting roots, a great improvement on Pascal‟s rudimentary adding machine and a true forerunner of the computer. He is usually credited with the early development of the binary number system (base 2 counting, using only the digits 0 and 1), although he himself was aware of similar ideas dating back to the I Ching of Ancient China. Because of the ability of binary to be represented by the two phases "on" and "off", it would later become the foundation of virtually all modern computer systems, and Leibniz's documentation was essential in the development process. Leibniz is also often considered the most important logician between Aristotle in Ancient Greece and George Boole and Augustus De Morgan in the19th Century. Even though he actually published nothing on formal logic in his lifetime, he enunciated in his working drafts the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion and the empty set.