PREHISTORIC MATHEMATICSOur prehistoric ancestors would have had a general sensibility about amounts, and would have instinctively known the difference between, say, one and two antelopes. But the intellectual leap from the concrete idea of two things to the invention of a symbol or word for the abstract idea of "two" took many ages to come about. The Ishango bone, a tally stick from central Africa, dates from about 20,000 years ago Even today, there are isolated hunter-gatherer tribes in Amazonia which only have words for "one", "two" and "many", and others which only have words for numbers up to five. In the absence of settled agriculture and trade, there is little need for a formal system of numbers. Early man kept track of regular occurrences such as the phases of the moon and the seasons. Some of the very earliest evidence of mankind thinking about numbers is from notched bones in Africa dating back to 35,000 to 20,000 years ago. But this is really mere counting and tallying rather than mathematics as such. Pre-dynastic Egyptians and Sumerians represented geometric designs on their artefacts as early as the 5th millennium BC, as did some megalithic societies in northern Europe in the 3rd millennium BC or before. But this is more art and decoration than the systematic treatment of figures, patterns, forms and quantities that has come to be considered as mathematics. Mathematics proper initially developed largely as a response to bureaucratic needs when civilizations settled and developed agriculture - for the measurement of plots of land, the taxation of individuals, etc - and this first occurred in the Sumerian and Babylonian civilizations of Mesopotamia (roughly, modern Iraq) and in ancientEgypt. SUMERIAN/BABYLONIAN MATHEMATICS Sumer (a region of Mesopotamia, modern-day Iraq) was the birthplace of writing, the wheel, agriculture, the arch, the plow, irrigation and many other innovations, and is often referred to as the Cradle of Civilization. The Sumerians developed the earliest known writing system - a pictographic writing system known as cuneiform script, using wedge-shaped characters inscribed on baked clay tablets - and this has Sumerian Clay Cones meant that we actually have more knowledge of ancient Sumerian and Babylonian mathematics than of early Egyptian mathematics. Indeed, we even have what appear to school exercises in arithmetic and geometric problems. As in Egypt, Sumerian mathematics initially developed largely as a response to bureaucratic needs when their civilization settled and developed agriculture (possibly as early as the 6th millennium BC) for the measurement of plots of land, the taxation of individuals, etc. In addition, the Sumerians and Babylonians needed to describe quite large numbers as they attempted to chart the course of the night sky and develop their sophisticated lunar calendar. They were perhaps the first people to assign symbols to groups of objects in an attempt to make the description of larger numbers easier. They moved from using separate tokens or symbols to represent sheaves of wheat, jars of oil, etc, to the more abstract use of a symbol for specific numbers of anything. Starting as early as the 4th millennium BC, they began using a small clay cone to represent one, a clay ball for ten, and a large cone for sixty. Over the course of the third millennium, these objects were replaced by cuneiform equivalents so that numbers could be written with the same stylus that was being used for the words in the text. A rudimentary model of the abacus was probably in use in Sumeria from as early as 2700 - 2300 BC. Sumerian and Babylonian mathematics was based on a sexegesimal, or base 60, numeric system, which could be counted physically using the twelve knuckles on one hand the five fingers on the other hand. Unlike those of theEgyptians, Greeks and Romans, Babylonian numbers used a true place-value system, where digits written in the left column represented larger values, much as in the modern decimal system, Babylonian Numerals although of course using base 60 not base 10. Thus, in the Babylonian system represented 3,600 plus 60 plus 1, or 3,661. Also, to represent the numbers 1 - 59 within each place value, two distinct symbols were used, a unit symbol ( ) and a ten symbol ( ) which were combined in a similar way to the familiar system of Roman numerals (e.g. 23 would be shown as ). Thus, represents 60 plus 23, or 83. However, the number 60 was represented by the same symbol as the number 1 and, because they lacked an equivalent of the decimal point, the actual place value of a symbol often had to be inferred from the context. It has been conjectured that Babylonian advances in mathematics were probably facilitated by the fact that 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 - in fact, 60 is the smallest integer divisible by all integers from 1 to 6), and the continued modern-day usage of of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, are all testaments to the ancient Babylonian system. It is for similar reasons that 12 (which has factors of 1, 2, 3, 4 and 6) has been such a popular multiple historically (e.g. 12 months, 12 inches, 12 pence, 2 x 12 hours, etc). The Babylonians also developed another revolutionary mathematical concept, something else that theEgyptians, Greeks and Romans did not have, a circle character for zero, although its symbol was really still more of a placeholder than a number in its own right. We have evidence of the development of a complex system of metrology in Sumer from about 3000 BC, and multiplication and reciprocal (division) tables, tables of squares, square roots and cube roots, geometrical exercises and division problems from around 2600 BC onwards. Later Babylonian tablets dating from about 1800 to 1600 BC cover topics as varied as fractions, algebra, methods for solving linear, quadratic and even some cubic equations, and the calculation of regular reciprocal pairs (pairs of number which multiply together to give 60). One Babylonian tablet e.g. Yet another gives an estimate for π of 3 1⁄8 (3. The Babylonian approach to solving them Babylonian Clay tablets from c. although the use of algebra and quadratic equations also appears.gives an approximation to √2 accurate to an astonishing five decimal places.1416). such as the ancient game of backgammon. as well as the volumes of simple shapes such as bricks and cylinders (although not pyramids). The idea of square numbers and quadratic equations (where the unknown quantity is multiplied by itself. and Babylonian mathematical tablets give us the first ever evidence of the solution of quadratic equations. suggests that the Babylonians may well have known the secret of rightangled triangles (that the square of the hypotenuse equals the sum of the square of the other two sides) many centuries before the Greek Pythagoras. The Babylonians used geometric shapes in their buildings and design and in dice for the leisure games which were so popular in their society. The tablet appears to list 15 perfect Pythagorean triangles with whole number sides. Their geometry extended to the calculation of the areas of rectangles. EGYPTIAN MATHEMATICS . Others list the squares of numbers up to 59. triangles and trapezoids. believed to date from around 1800 BC.125. and not deliberate manifestations of Pythagorean triples. The famous and controversial Plimpton 322 clay tablet. although some claim that they were merely academic exercises. x2) naturally arose in the context of the meaurement of land. 2100 BC showing a problem concerning usually revolved around a the area of an irregular shape kind of geometric game of slicing up and rearranging shapes. a reasonable approximation of the real value of 3. At least some of the examples we have appear to indicate problem-solving for its own sake rather than in order to resolve a concrete practical problem. the cubes of numbers up to 32 as well as tables of compound interest. a coil of rope for hundreds and a lotus plant for thousands. and a decimal numeric system was developed based on our ten fingers. is the Moscow Papyrus. there was no concept of place value. It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system at least as early as 2700 BC (and probably much early). though. which dates from the Egyptian Middle Kingdom around 2000 . a cubit Ancient Egyptian hieroglyphic numerals the measurement from elbow to fingertips) to measure land and buildings very early in Egyptian history.The early Egyptians settled along the fertile Nile valley as early as about 6000 BC. and they began to record the patterns of lunar phases and the seasons. Written numbers used a stroke for units. both for agricultural and religious reasons.1800 BC. so larger numbers were rather unwieldy (although a million required just one character. as well as other hieroglyphic symbols for higher powers of ten up to a million. a million minus one required fifty-four characters). The Pharaoh’s surveyors used measurements based on body parts (a palm was the width of the hand. . a heel-bone symbol for tens. However. The oldest mathematical text from ancient Egypt discovered so far. which dates from around 1300 BC. was achieved by a process of repeated doubling. The Berlin Papyrus. including unit fractions. It also contains evidence of other mathematical knowledge. dating from around 1650 BC. the first known example of a geometric series. over 3. and it gives us explicit demonstrations of how multiplication and division was carried out at that time. for example. Practical problems of trade and the market led to the development of a notation for fractions.e. The papyri which have come down to us demonstrate the use of unit fractions based on the symbol of the Eye of Horus. the combination of powers of two which add up to the number to be multiplied by was isolated. sixty-fourth). . This effectively made use of the concept of binary numbers. and many more years before the development of the computer was to fully explore its potential. Ancient Egyptian method of multiplication Multiplication. and how to solve first order linear equations as well as arithmetic and geometric series.000 years before Leibniz introduced it into the west. sixteenth. These corresponding blocks of counters could then be used as a kind of multiplication reference table: first. is a kind of instruction manual in arithmetic and geometry. thirty-second. of the number to be multiplied on one side and of one on the other. geometric and harmonic means. shows that ancient Egyptians could solve second-order algebraic (quadratic) equations. each half of the previous one (i. and then the corresponding blocks of counters on the other side yielded the answer. arithmetic. quarter. where each part of the eye represented a different fraction.The Rhind Papyrus. composite and prime numbers. half. eighth. so that the total was one-sixty-fourth short of a whole. They observed that the area of a circle of diameter 9 units. Thus. and Egyptian builders used ropes knotted at intervals of 3. For example. there is certainly evidence that they knew the formula for the volume of a pyramid .618 (which may have occurred for purely aesthetic. of the rule that a triangle with sides 3. if they needed to divide 3 loaves among 5 people.as well as of a truncated or clipped pyramid. was very close to the area of a square with sides of 8 units. Ancient Egyptian method of division The pyramids themselves are another indication of the sophistication of Egyptian mathematics. The Egyptians approximated the area of a circle by using shapes whose area they did know.Unit fractions could also be used for simple division sums. and not mathematical. This gives an effective approximation ofπ accurate to within less than one percent. so that the area of circles of other diameters could be obtained by multiplying the diameter by 8⁄9 and then squaring it. reasons). for example. Setting aside claims that the pyramids are first known structures to observe the golden ratio of 1 : 1.1⁄3 times the height times the length times the width . each person would receive one-third plus one-fifth plus one-fifteenth (which totals three-fifths. then they would divide the left over third from the second loaf into five pieces. as we would expect). GREEK MATHEMATICS . 4 and 5 units yields a perfect right angle. They were also aware. they would first divide two of the loaves into thirds and the third loaf into fifths. long before Pythagoras. the 3-4-5 right triangle is often called "Egyptian"). 4 and 5 units in order to ensure exact right angles for their stonework (in fact. By the Hellenistic period. 50. the Greeks had presided over one of the most dramatic and important revolutions in mathematical thought of all time. 10.As the Greek empire began to spread its sphere of influence into Asia Minor.000 repeated as many times needed to represent the desired number. But they soon started to make important contributions in their own right and. and in regular use possibly as early as the 7th Century BC.Mesopotamia and beyond. 100s. with symbols for 1. Addition was done by totalling separately the symbols (1s. . etc) in the numbers to be added. The ancient Greek numeral system. 10s. 500 and 1. 100. was fully developed by about 450 BC. the Greeks were smart enough to adopt and adapt useful elements from the societies they conquered. This was as true of their mathematics Ancient Greek Herodianic numerals as anything else. we can acknowledge contributions by individuals. for the first time. and multiplication was a laborious process based on successive doublings (division was based on the inverse of this process). 5. It was a base 10 system similar to the earlierEgyptian one (and even more similar to the later Roman system). and they adopted elements of mathematics from both the Babylonians and theEgyptians. known as Attic or Herodianic numerals. “the doubling (or duplicating) of the cube” and “the trisection of an angle”. These intransigent problems were profoundly influential on future geometry and led to many fruitful discoveries. where geometric elements corresponded with numbers. the ratios of the sides of similar triangles). and Greek mathematics was by no means limited to one man. Thales. one of the Seven Sages of Ancient Greece. also known as Thales' Theorem or the Intercept Theorem. as we will see. he is believed to have coined both the words "philosophy" ("love of wisdom") and "mathematics" ("that which is learned"). and all to be solved by purely geometric means using only a straight edge and a compass. whereby if a triangle is drawn within a circle with the long side as a diameter of the circle. Pythagoras’ Theorem (or the Pythagorean Theorem) is one of the best known of all mathematical theorems. about the ratios of the line segments that are created if two intersecting lines are intercepted by a pair of parallels (and.But most of Greek mathematics was based Thales' Intercept Theorem on geometry. To some extent. Pythagoras was perhaps the first to realize that a complete system of mathematics could be constructed. the legend of the 6th Century BC mathematician Pythagoras of Samos has become synonymous with the birth of Greek mathematics. He is also credited with another theorem. often referred to as the Three Classical Problems. however. Thales established what has become known as Thales' Theorem. The Three Classical Problems . But he remains a controversial figure. date back to the early days of Greek geometry: “the squaring (or quadrature) of the circle”. although what we know of his work (such as on similar and right triangles) now seems quite elementary. who lived on the Ionian coast of Asian Minor in the first half of the 6th Century BC. Indeed. is usually considered to have been the first to lay down guidelines for the abstract development of geometry. Three geometrical problems in particular. by extension. then the opposite angle will always be a right angle (as well as some other related properties derived from this). 4th Century BC. etc. Democritus. so that in principle the swift Achilles can never catch up with the slow tortoise. The ancient Greek Aristotle was the first of many to try to disprove the paradoxes. was the first compilation of the elements of geometry. The most famous of his paradoxes is that of Achilles and the Tortoise.although their actual solutions (or. etc. We do know that he was among the first to observe that a cone (or . etc. Achilles gives the much slower tortoise a head start. Hippocrates of Chios (not to be confused with the great Greek physician Hippocrates of Kos) was one such Greek mathematician who applied himself to these problems during the 5th Century BC (his contribution to the “squaring the circle” problem is known as the Lune of Hippocrates). to infinity will never quite equal a whole. etc. His influential book “The Elements”. as it turned out. dating to around 440 BC. Paradoxes such as this one and Zeno's so-called Dichotomy Paradox are based on the infinite divisibility of space and time. although these works have not survived. but by the time Achilles reaches the tortoise's starting point. such as described in the well-known paradoxes attributed to the philosopher Zeno of Elea in the 5th Century BC. was also a pioneer of mathematics and geometry in the 5th . the tortoise has moved on again. most famous for his prescient ideas about all matter being composed of tiny atoms. and he produced works with titles like "On Numbers". from the false assumption that it is impossible to complete an infinite number of discrete dashes in a finite time. "On Mapping" and "On Irrationals". "On Geometrics". It was the Greeks who first grappled with the idea of infinity. however. which describes a theoretical race between Achilles and a tortoise. and his work was an important source for Euclid's later work. By the time Achilles Zeno's Paradox of Achilles and the Tortoise reaches that point. the tortoise has already moved ahead. "On Tangencies". although it is extremely difficult to definitively prove the fallacy. and rest on the idea that a half plus a quarter plus an eighth plus a sixteenth. the proofs of their impossibility) had to wait until the 19th Century. The paradox stems. particularly as he was a firm believer that infinity could only ever be potential and not real. Plato’s student Eudoxus of Cnidus is usually credited with the first implementation of the “method of exhaustion” (later developed by Archimedes). GREEK MATHEMATICS . Perhaps the most important single contribution of the Greeks.570-495 BC) .PYTHAGORAS Pythagoras of Samos (c. and which laid the foundations for the systematic approach to mathematics of Euclid and those who came after him. that is using repeated observations to establish rules of thumb. Older cultures. He also developed a general theory of proportion. though and Pythagoras. including Plato. whose work on logic was regarded as definitive for over two thousand years. an early method of integration by successive approximations which he used for the calculation of the volume of the pyramid and cone.pyramid) has one-third the volume of a cylinder (or prism) with the same base and height. and his protégé Aristotle. Plato the mathematician is best known for his description of the five Platonic solids. who established his famous Academy in Athens in 387 BC. which was applicable to incommensurable (irrational) magnitudes that cannot be expressed as a ratio of two whole numbers. and he is perhaps the first to have seriously considered the division of objects into an infinite number of cross-sections. However. it is certainly true that Pythagoras in particular greatly influenced those who came after him. but the value of his work as a teacher and popularizer of mathematics can not be understated.was the idea of proof. had relied on inductive reasoning. like the Egyptians and theBabylonians. as well as to commensurable (rational) magnitudes. thus extending Pythagoras’ incomplete ideas. It is this concept of proof that give mathematics its power and ensures that proven theories are as true today as they were two thousand years ago. and the deductive method of using logical steps to prove or disprove theorems from initial assumed axioms. Plato and Aristotle were all influential in this respect . . with at least 50 members killed in Croton alone. two represented opinion. The members were divided into the "mathematikoi" (or "learners"). etc. six. it is by no means clear whether many (or indeed any) of the theorems ascribed to him were in fact solved by Pythagoras personally or by his followers. and considered each number to have its own character and meaning. in 460 BC. justice. never marrying a woman who wears gold jewellery. and much of what we know about Pythagorean thought comes to us from the writings of Philolaus and other later Pythagorean scholars. He left no mathematical writings himself. and Pythagoras imposed his quasi-religious philosophies. four. and the "akousmatikoi" (or "listeners"). the number one was the generator of all numbers. strict vegetarianism. who extended and developed the more mathematical and scientific work that Pythagoras himself began. creation. etc) . never eating or even touching black fava beans. the seven planets or “wandering stars”. For example. There was always a certain amount of friction between the two groups and eventually the sect became caught up in some fierce local fighting and ultimately dispersed. harmony. secret rites and odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun. The over-riding dictum of Pythagoras's school was “All is number” or “God is number”. who focused on the more religious and ritualistic aspects of his teachings. he nevertheless remains a controversial figure. although his contribution was clearly important. Resentment built up against the secrecy and exclusiveness of the Pythagoreans and. three. and he is often called the first "true" mathematician. Odd numbers were thought of as female and even numbers as male.It is sometimes claimed that we owe pure mathematics to Pythagoras. never passing an ass lying in the street. seven. it was also profoundly mystical. communal living. marriage. But. and the Pythagoreans effectively practised a kind of numerology or number-worship. five. all their meeting places were burned and destroyed. The school he established at Croton in southern Italy around 530 BC was the nucleus of a rather bizarre Pythagorean sect. Although Pythagorean thought was largely dominated by mathematics. Indeed. where geometric elements corresponded with numbers. What Pythagoras and his followers did not realize is that this also works for any shape: thus.The holiest number of all was "tetractys" or ten. for any right-angled triangle. Before Pythagoras. Pythagoras discovered that a complete system of mathematics could be constructed. However. Pythagoras and his school . building from first principles using axioms and logic. three and four. Written as an equation: a2 + b2 = c2. two. for example. a triangular number composed of the sum of one. . and where integers and their ratios were all that was necessary to establish an entire system of logic and truth. the square of the length of the hypotenuse (the longest side. opposite the right angle) is equal to the sum of the square of the other two sides (or “legs”). the area of a pentagon on the hypotenuse is equal to the sum of the pentagons on the other two sides. as it does for a semi-circle or any other regular (or even irregular( shape.was largely responsible for introducing a more rigorous mathematics than what had gone before. geometry had been merely a collection of rules derived by empirical measurement.as well as a handful of other mathematicians of ancient The Pythagorean Tetractys Greece . He is mainly remembered for what has become known as Pythagoras’ Theorem (or the Pythagorean Theorem): that. It is a great tribute to the Pythagoreans' intellectual achievements that they deduced the special place of the number 10 from an abstract mathematical argument rather than from something as mundane as counting the fingers on two hands. 15. for example. thereby indicating the potential existence of a whole new world of numbers. though. because it is just a multiple of (3. dating from over a thousand years earlier. It was Pythagoras. was an . the irrational numbers (numbers that can not be expressed as simple fractions of integers). 4 and 5 units (32 + 42 = 52. it has become one of the best-known of all mathematical theorems. and as many as 400 different proofs now exist. 24. (7. who gave the theorem its definitive form. also has a right angle. Either way. (6. It soom became apparent. he found that it was not possible to express it as a fraction. and probably dates from well before Pythagoras' birth. (9. 10). 5). 10) is not what is known as a “primitive” Pythagorean triple. It should be noted. so that an isosceles triangle with sides 1. 12 13). some involving advanced differential equations. Poor Hippasus was apparently drowned by the secretive Pythagoreans for broadcasting this important discovery to the outside world. This discovery rather shattered the elegant mathematical world built up by Pythagoras and his followers. starting with (5. and it was touched on in some of the most ancient mathematical texts from Babylon andEgypt. (8. when Pythagoras’s student Hippasus tried to calculate the value of √2. etc. that non-integer solutions were also possible. but there are a potentially infinite number of other integer “Pythagorean triples”. 25). however that (6. However. One of the simplest proofs comes from ancient China. 1 and √2. 41). 4. although it is not clear whether Pythagoras himself definitively proved it or merely described it. though. as the Babylonians had discovered centuries earlier. some algebraic. Pythagoras’ Theorem and Pythagoras' (Pythagorean) Theorem the properties of rightangled triangles seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. as can be seen by drawing a grid of unit squares on each side as in the diagram at right).The simplest and most commonly quoted example of a Pythagorean triangle is one with sides of 3. 17). some geometrical. 8. 40. 8. etc. But the replacement of the idea of the divinity of the integers by the richer concept of the continuum. and the existence of a number that could not be expressed as the ratio of two of God's creations (which is how they thought of the integers) jeopardized the cult's entire belief system. for example to solve equations such as a(a .x) = x2 by geometrical means. the proper divisors of 220 are 1. a third of a length gives a different but harmonious note. 71. with their investigations of triangular. 42 = 16 = 1 + 3 + 5 + 7). on the other hand. each tuned in the ratio 3:2. but an octave higher. e. 2. They also discovered at least the first pair of amicable numbers. 4. the perfect fifth (3:2). and to use simple geometrical algebra. 11. 55 and 110. and it is based on a stack of perfect fifths. the Pythagoreans) also realized that the sum of the angles of a triangle is equal to two right angles (180°). 5. the perfect fourth (4:3) and the major third (5:4). such as that the square of a number n is equal to the sum of the first n odd numbers (e. etc.essential development in mathematics. They discovered several new properties of square numbers. and the proper divisors of 284 are 1. The Pythagoreans also established the foundations of number theory. It marked the real birth of Greek geometry.g. They were able to construct figures of a given area. 44. In this way. 10. 2. tend to give dissonant sounds. and probably also the generalization which states that the sum of the interior angles of a polygon with n sides is equal to (2n . The oldest way of tuning the 12-note chromatic scale is known as Pythagorean tuning. of which the sum is 284. For instance. playing half a length of a guitar string gives the same note as the open string.4) right angles. which deals with lines and planes and angles. Pythagoras is also credited with the discovery that the intervals between harmonious musical notes always have whole number ratios. 20. . 22.g. and that the sum of its exterior angles equals 4 right angles. Among his other achievements in geometry. all of which are continuous and not discrete. of which the sum is 220). square and also perfect numbers (numbers that are the sum of their divisors). Pythagoras Pythagoras is credited with the discovery of the ratios between described the first four harmonious musical tones overtones which create the common intervals which have become the primary building blocks of musical harmony: the octave (1:1). Pythagoras (or at least his followers. 4. Nonwhole number ratios. 220 and 284 (amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number. and 142. Plato played an important role in encouraging and inspiring Greek intellectuals to study mathematics as well as philosophy. Plato became known as the "maker of mathematicians". including Eudoxus. His Academy taught mathematics as a branch of philosophy. he was convinced that geometry was the key to unlocking the secrets of the universe. Plato was also one of ancient Greece’s most important patrons of mathematics. and his Academy boasted some of the most prominent mathematicians of the ancient world. Inspired byPythagoras. “doubling the cube” and “trisecting the angle”) and to some extent these problems have become identified with Plato. GREEK MATHEMATICS . Theaetetus and Archytas. clearly stated assumptions. including plane and solid geometry. Plato (c. and he insisted that geometric proofs be demonstrated with no aids other than a straight edge and a compass. and the first 10 years of the 15 year course at the Academy involved the study of science and mathematics.The mystical Pythagoras was so excited by this discovery that he became convinced that the whole universe was based on numbers.PLATO Although usually remembered today as a philosopher. and that the planets and stars moved according to mathematical equations. and thus produced a kind of symphony. which corresponded to musical notes.428-348 BC) He demanded of his students accurate definitions. and logical deductive proof. as Pythagoras had done. where he stressed mathematics as a way of understanding more about reality. Among the many mathematical problems Plato posed for his students’ investigation were the so-called Three Classical Problems (“squaring the circle”. The sign above the Academy entrance read: “Let no-one ignorant of geometry enter here”. although he was not the first to pose them. In particular. he founded his Academy in Athens in 387 BC. astronomy and harmonics. . the “Musical Universalis” or “Music of the Spheres”. But they nevertheless became popularly known as the Platonic Solids.Plato the mathematician is perhaps best known for his identification of 5 regular symmetrical 3-dimensional shapes. representing air). half a century later. cube and dodecahedron were probably familiar to Pythagoras. around 1600. and inspired mathematicians and geometers for many centuries to come. a contemporary of Plato. and which have become known as the Platonic Solids: the tetrahedron (constructed of 4 regular triangles. the cube (composed of 6 Platonic Solids squares. The tetrahedron. to prove that these were the only possible convex regular polyhedra. and representing earth). which Plato obscurely described as “the god used for arranging the constellations on the whole heaven”). For example. and the octahedron and icosahedron were probably discovered by Theaetetus. it fell to Euclid. the octahedron (composed of 8 triangles. Furthermore. and the dodecahedron (made up of 12 pentagons. which he maintained were the basis for the whole universe. and which for Plato represented fire). the icosahedron (composed of 20 triangles. and representing water). HELLENISTIC MATHEMATICS . the German astronomer Johannes Kepler devised an ingenious system of nested Platonic solids and spheres to approximate quite well the distances of the known planets from the Sun (although he was enough of a scientist to abandon his elegant model when it proved to be not accurate enough). Heron. Menelaus and Diophantus. Among the best known and most influential mathematicians who studied and taught at Alexandria wereEuclid. in the wake of the conquests of Alexander the Great. Alexandria in Egypt became a great centre of learning under the beneficent rule of the Ptolemies. He is perhaps best known as an engineer and inventor but. paid for their devotion to research. he is now considered of one of the greatest pure mathematicians of all time.By the 3rd Century BC. he devised the first system of latitude and longitude. The patrons of the Library were arguably the first professional scientists. his greatest legacy is the “Sieve of Eratosthenes” algorithm for identifying prime numbers. A mathematician. Eratosthenes of Alexandria was a near contemporary of Archimedes in the 3rd Century BC. and calculated the circumference of the earth to a remarkable degree of accuracy. Sicily. Euclid was the great chronicler of the mathematics of the time. in the light of recent discoveries. but also studied for a while in Alexandria. . mathematical breakthroughs were also beginning to be made on the edges of the Greek Hellenistic empire. As a mathematician. astronomer and geographer. In particular. The Sieve of Eratosthenes During the late 4th and early 3rd Century BC. Archimedes. He virtually invented classical (Euclidean) geometry as we know it. and its famous Library soon gained a reputation to rival that of the Athenian Academy. Archimedes spent most of his life in Syracuse. and one of the most influential teachers in history. Eratosthenes. and is considered an early innovator in the field of what would later become known as algebra. Diophantus of Alexandria was the first to recognize fractions as numbers. was the most prominent work on algebra in all Greek mathematics. but Alexandria remained an important intellectual centre for some centuries.It is not known exactly when the great Library of Alexandria burned down. and introduced the concept of spherical triangle (a figure formed of three great circle arcs. Heron’s Formula for finding the area of a triangle from its side lengths. a collection of problems giving numerical solutions of both determinate and indeterminate equations. who lived in the 1st . was the first to recognize geodesics on a curved surface as the natural analogues of straight lines on a flat plane.2nd Century AD. best known in mathematical circles for Heronian triangles (triangles with integer sides and integer area). which he named "trilaterals"). He was also the first mathematician to confront at least the idea of Menelaus of Alexandria introduced the concept of spherical triangle √-1 (although he had no idea how to treat it. In the 3rd Century AD. and his problems exercised the minds of many of the world's best mathematicians for much of the next two millennia. which deals with finding integer solutions to kinds of problems that lead to equations in several unknowns (Diophantine equations). Menelaus of Alexandria. In the 1st century BC. Diophantus’ “Arithmetica”. He applied himself to some quite complex algebraic problems. . including what is now known as Diophantine Analysis. something which had to wait for Tartaglia and Cardano in the16th Century). and Heron’s Method for iteratively computing a square root. His book “Sphaerica” dealt with the geometry of the sphere and its application in astronomical measurements and calculations. Heron (or Hero) was another great Alexandrian inventor. and the Christian regime that followed it even less so. who was also from Hellenistic Anatolia and who live in the 2nd Century BC. was perhaps the greatest of all ancient astronomers. It was Apollonius who gave the ellipse. Mention should also be made of Apollonius of Perga (a city in modernday southern Turkey) whose late 3rd Century BC work on geometry (and. only for its practical applications. the parabola. in particular. He went on to create the first table of chords (side lengths corresponding to different angles of a triangle). the first recorded female mathematician. By the time of the great Alexandrian astronomer Ptolemy in the 2nd Century AD. and showed how they could be derived from different sections through a cone. the Romans had tightened their grip on the old Greek empire. The final blow to the Hellenistic mathematical heritage at Alexandria might be seen in the figure of Hypatia. on conics and conic sections) was very influential on later European mathematicians. and the hyperbola the names by which we know them. The Romans had no use for pure mathematics.But Alexandria was not the only centre of learning in the Hellenistic Greek empire. Greek mastery of numerical procedures had progressed to the point where Ptolemy was able to include in his “Almagest” a table of trigonometric chords in a circle for steps of ¼° which (although expressed sexagesimally in the Babylonian style) is accurate to about five decimal places. however. He revived the use of arithmetic techniques first developed by the Chaldeans and Babylonians. Conic sections of Apollonius Hipparchus. however. He calculated (with remarkable accuracy for the time) the distance of the moon from the earth by measuring the different parts of the moon visible at different locations and calculating the distance using the properties of triangles. She was dragged to her death by a Christian mob in 415 AD. By the middle of the 1st Century BC and thereafter. and is usually credited with the beginnings of trigonometry. and a renowned teacher who had written some respected commentaries on Diophantus and Apollonius. ROMAN MATHEMATICS . and were the dominant number system for trade and administration in most of Europe for the best part of a millennium. and the Christian regime that followed it (after Christianity became the official religion of the Roman empire) even less so. only for its practical applications. based on earlier Babylonian and Greek abaci. L. Later. was replaced by IX (10 . It was decimal (base 10) system but not directly positional.g. VII = V + I + I = 7). but made calculation even more difficult. no mathematical innovations occurred under the Roman Empire and Republic. It was Roman arithmetic based on letters of the Roman alphabet . so that. a subtractive notation was also adopted. and did not include a zero. requiring conversion of the subtractive notation at the beginning of a sum and then its re-application at the end (see image at right). and there were no mathematicians of note.By the middle of the 1st Century BC. for arithmetic and mathematical purposes. Due to the difficulty of written arithmetic using Roman numeral notation. Despite all their advances Roman numerals in other respects. X. for example. and the mathematical revolution of theGreeks ground to halt. it was a clumsy and inefficient system. which simplified the writing of numbers a little. Roman numerals are well known today.combines to signify the sum of their values (e. V. calculations were usually performed with an abacus. the Roman had tightened their grip on the old Greek and Hellenistic e mpires. C. The Romans had no use for pure mathematics.I.1 = 9). where VIIII. D and M . MAYAN MATHEMATICS . CHINESE MATHEMATICS . and we have evidence of their working with sums up to the hundreds of millions. and with dates so large it took several lines just to represent them. 400. The Mayan and other Mesoamerican cultures used a vigesimal number system based on base 20 (and. 20.53059). compared to the modern value of 365. and the Maya constructed quite early a very sophisticated number system. compared to the modern value of 29.5308 days.242198). base 5). 8000. a dot. although in their calendar calculations they gave the third position a value of 360 instead of 400 (higher positions revert to multiples of 20). At its peak. 160000. one. they produced extremely accurate astronomical observations using no instruments other than sticks. due to the geographical disconnect. The importance of astronomy and calendar calculations in Mayan Mayan numerals society required mathematics. as well as the length of the lunar month (their estimate was 29.The Mayan civilisation had settled in the region of Central America from about 2000 BC. larger numbers were written in a kind of vertical place value format using powers of 20: 1. a bar. and were able to measure the length of the solar year to a far higher degree of accuracy than that used in Europe (their calculations produced 365. Mayan and Mesoamerican mathematics had absolutely no influence on Old World (European and Asian) numbering systems and mathematics. to some extent. The numerals consisted of only three symbols: zero.242 days. Despite not possessing the concept of a fraction. it was one of the most densely populated and culturally dynamic societies in the world. After the number 19. although the so-called Classic Period stretches from about 250 AD to 900 AD. and five. The pre-classic Maya and their neighbours had independently developed the concept of zero by at least as early as 36 BC. etc (see image above). represented as a shell shape. possibly more advanced than any other in the world at the time (although the dating of developments is quite difficult). However. Thus. probably originally developed from counting on fingers and toes. addition and subtraction was a relatively simple matter of adding up dots and bars. It was therefore a decimal place value system. Written numbers. magic squares .were regarded as having great spiritual and religious significance. hundreds. column and diagonal added up to the same total . or “suanpan”. we know of dates to about the 2nd Century BC). which were then places in columns representing units. adopted by the Chinese over a thousand years before it was adopted in the West . The Lo Shu Square. however. The simple but efficient ancient Chinese numbering system. and different numbers were believed to have cosmic significance. There was a pervasive fascination with numbers and mathematical patterns in ancient China. thousands. employed the slightly less efficient system of using a different symbol for tens. etc. and it had the effect of limiting the usefulness of the written number in Chinese.indeed it was the first such number system.squares of numbers where each row. probably much earlier than in China (the first Chinese abacus. tens. etc.Even as mathematical developments in the ancient Greek world were beginning to falter during the final centuries BC. an order three square where Lo Shu magic square. although some type of abacus was in use inMesopotamia. This was largely because there was no concept or symbol of zero. hundreds. which Ancient Chinese number system dates back to at least the 2nd millennium BC. very similar to the one we use today . Egypt and Greece. the burgeoning trade empire of China was leading Chinese mathematics to ever greater heights. with its traditional graphical representation . The use of the abacus is often thought of as a Chinese idea. In particular.and it made even quite complex calculations very quick and easy. thousands. used small bamboo rods arranged to represent the numbers 1 to 9. column and diagonal adds up to 15. circles and triangles of Yang Hui in the 13th Century (Yang Hui also produced a trianglular representation of binomial coefficients identical to the later Pascals’ Triangle. engineering and the payment of wages. with even greater magical and mathematical powers. But the main thrust of Chinese mathematics developed in response to the empire’s growing need for mathematically competent administrators. But soon. Among the greatest mathematicians of ancient China was Liu Hui. culminating in the elaborate magic squares. and was perhaps the first to use decimal fractions in the modern form).14159 (correct to .each row. taxation. probably by a variety of authors) became an important tool in the education of such a civil service. covering hundreds of problems in practical areas such as trade. A textbook called “Jiuzhang Suanshu” or “Nine Chapters on the Mathematical Art” (written over a period of time from about 200 BC onwards. It was particularly important as a guide to how to solve equations the deduction of an unknown number from other known information Early Chinese method of solving equations using a sophisticated matrix-based method which did not appear in the West untilCarl Friedrich Gauss re-discovered it at the beginning of the 19th Century (and which is now known as Gaussian elimination). he also formulated an algorithm which calculated the value of π as 3. bigger magic squares were being constructed. who produced a detailed commentary on the “Nine Chapters” in 263 AD. was one of the first mathematicians known to leave roots unevaluated. giving more exact results instead of approximations. is perhaps the earliest of these. dating back to around 650 BC (the legend of Emperor Yu’s discovery of the the square on the back of a turtle is set as taking place in about 2800 BC). By an approximation using a regular polygon with 192 sides. INDIAN MATHEMATICS . as well as developing a very early forms of both integral and differential calculus. Perhaps the most brilliant Chinese mathematician of this time was Qin Jiushao. A technique for solving such problems. who explored solutions to quadratic and even cubic equations using a method of repeated approximations very similar to that later devised in the West by Sir Isaac Newton in the 17th Century. such as 3. initially posed by Sun Tzu in the 3rd Century AD and considered one of the jewels of mathematics. the Golden Age of Chinese mathematics. and even today it has practical uses. By the 13th Century. though. The Chinese went on to solve far more complex equations using far larger numbers than those outlined in the “Nine Chapters”. there were over 30 prestigious mathematics schools scattered across China. including what has become known as the Chinese Remainder Theorem. 5 and 7. such as in Internet cryptography.five decimal places). They also started to pursue more abstract mathematical problems (although usually couched in rather artificial practical terms). extraordinarily complex mathematics for its time. a rather violent and corrupt imperial administrator and warrior. The Chinese Remainder Theorem was being used to measure planetary movements by Chinese astronomers in the 6th Century AD. Qin even extended his technique to solve (albeit approximately) equations involving numbers up to the power of ten. This uses the remainders after dividing an unknown number by a succession of smaller numbers. in order to calculate the smallest value of the unknown number. fractions. sometimes considered one of the greatest intellectual innovations of all time. and give a remarkably accurate figure for the square root of 2. correct to 5 decimal places. and were certainly using it before about the 3rd Century AD. uncountable and infinite. As early as the 3rd or 2nd Century BC. Like the Chinese. The Sutras also contain geometric solutions of linear and quadratic equations in a single unknown. They refined and perfected the system. which comes remarkably close to the actual size of a carbon atom (about 70 trillionths of a metre). creating the ancestors of the nine numerals that (thanks to its dissemination by medieval Arabic mathematicans) we use across the world today. cubes and roots. in order to demonstrate the size of an atom. long before Pythagoras. some very advanced mathematical discoveries were made at a very early time in India.4142156. Ancient Buddhist literature also demonstrates a prescient awareness of indeterminate and infinite numbers. as well as describing six more numbering systems over and above these. squares. in two directions. subtraction. A 4th Century AD Sanskrit text reports Buddha enumerating numbers up to 1053. obtained by adding 1 + 1⁄3 + 1⁄(3 x 4) + 1⁄(3 x 4 x 34). in area. a text known as the “Sulba Sutras” (or "Sulva Sutras") listed several simple Pythagorean triples. multiplication. and provide evidence of the use of arithmetic operations such as addition. as well as a statement of the simplified Pythagorean theorem for the sides of a square and for a rectangle (indeed. leading to a number equivalent to 10 421. It also describes a series of iterations in decreasing size. Mantras from the early Vedic period (before 1000 BC) invoke powers of ten The evolution of Hindu-Arabic numerals from a hundred all the way up to a trillion. which yields a value of 1. As early as the 8th Century BC. infinite everywhere and perpetually infinite.Despite developing quite independently of Chinese (and probably also ofBabylonian mathematics) . particularly the written representation of the numerals. Jain mathematicians recognized five different types of infinities: infinite in one direction. Given that there are an estimated 1080 atoms in the whole universe. . the Indians early discovered the benefits of a decimal place value number system. with numbers deemed to be of three types: countable. this is as close to infinity as any in the ancient world came. it seems quite likely that Pythagoras learned his basic geometry from the "Sulba Sutras"). Brahmagupta established the basic mathematical rules for dealing with zero: 1 + 0 = 1. it seems that Indian contributions to mathematics have not been given due acknowledgement until very recently in modern history. one of which could be negative.even though it may well have been in practical use for centuries before that. He even attempted to write down these rather abstract concepts. The use of zero as a number which could be used in calculations and mathematical investigations. and pointed out that quadratic equations could in theory have two possible solutions. as it had been treated until that time) is usually credited to the 7th Century Indian mathematicians Brahmagupta . 1 . at least some of whom were probably aware of the earlier Indian work.0 = 1. would revolutionize mathematics. using the initials of the names of colours to represent unknowns in his equations. Certainly. . and many of its mathematical discoveries predated similar discoveries in the West by several centuries. Brahmagupta also established rules for dealing with negative numbers. the 12th Century mathematician Bhaskara II). one of the earliest intimations of what we now know as algebra. a blank or empty space within a number. The earliest recorded usage of a circle character for the number zero is usually attributed to a 9th Century engraving in a temple in Gwalior in central India.The Indians were also responsible for another hugely important development in mathematics. Bhaskara I . The so-called Golden Age of Indian mathematics can be said to extend from the 5th to 12th Centuries. which has led to some claims of plagiarism by later European mathematicians.or possibly another Indian. But the brilliant conceptual leap to The earliest use of a circle character for the number zero was in India include zero as a number in its own right (rather than merely as a placeholder. and 1 x 0 = 0 (the breakthrough which would make sense of the apparently non-sensical operation 1 ÷ 0 would also fall to an Indian. tangents and secants. perhaps even more astonishing. Although the Greeks had been able to calculate the sine function of some angles. in 3. only 70 miles off its true value. a method of linking geometry and numbers first developed by the Greeks. versine and inverse sine. For instance. cosines. navigate the seas and even chart the Indian astronomers used trigonometry tables to estimate the relative heavens. to an accuracy of 4 decimal places. They realized that.835 miles. As early as the 6th Century AD. But. then the Sun. the great Indian mathematician and astronomer Aryabhata produced categorical definitions of sine. and that any calculation can only ever be an approximation.75° intervals from 0° to 90°. cosine. cosine and tangent functions (which relate the angles of a triangle to the relative lengths of its sides) to survey the land around them. Their sine tables gave a ratio for the sides of such a triangle as 400:1. and specified complete sine and versine tables. indicating that the Sun is 400 times further away from the Earth than the Moon. . and were able to accurately measure the angle as 1⁄7°. when the Moon is half full and directly opposite the Sun. contains the roots of modern trigonometry. Aryabhata also demonstrated solutions to simultaneous quadratic equations. something not proved in Europe until 1761. Moon and Earth form a right angled triangle. He used this to estimate the circumference of the Earth. inverse sines. the Indian astronomers wanted to be able to calculate the sine function of any given angle. and produced an approximation for the value of π equivalent to 3. distance of the Earth to the Sun and Moon Indian astronomers used trigonometry to calculated the relative distances between the Earth and the Moon and the Earth and the Sun. They used ideas like the sine. including the first real use of sines. by unknown authors and dating from around 400 AD. A text called the “Surya Siddhanta”. he seems to have been aware that π is an irrational number. correct to four decimal places. arriving at a figure of 24.1416.Golden Age Indian mathematicians made fundamental advances in the theory of trigonometry. therefore. who lived in the 12th Century. cubic and quartic equations (including negative and irrational solutions) to solutions of Diophantine equations of the second order to preliminary concepts of infinitesimal calculus and mathematical analysis to spherical trigonometry and other aspects of trigonometry. dividing 1 by smaller and smaller factions yields a larger and larger number of pieces. He developed infinite series approximations for a range of trigonometric functions. 1 ÷ 1⁄3 = 3. etc. So. so 1 ÷ 1⁄2 = 2. He is credited with explaining the previously misunderstood operation of division by zero. He noticed that dividing one into two pieces yields a half. sine. and he made important contributions in terms of the systemization of (then) current knowledge and improved methods for known solutions.Bhaskara II. Some of his contributions to geometry and algebra and his early forms of differentiation and integration for simple functions may have been transmitted to Europe via Jesuit missionaries. sometimes called the greatest mathematician-astronomer of medieval India. Bhaskara II also made important contributions to many different areas of mathematics from solutions of quadratic. dividing one into pieces of zero size would yield infinitely many pieces. including π. Similarly. ISLAMIC MATHEMATICS . and it is possible that the later European development of calculus was influenced by his work to some extent. Illustration of infinity as the reciprocal of zero However. Some of his findings predate similar discoveries in Europe by several centuries. The Kerala School of Astronomy and Mathematics was founded in the late 14th Century by Madhava of Sangamagrama. indicating that 1 ÷ 0 = ∞ (the symbol for infinity). Ultimately. was one of the most accomplished of all India’s great mathematicians. One consequence of the Islamic prohibition on depicting the human form was the extensive use of Some examples of the complex symmetries used in Islamic temple complex geometric decoration patterns to decorate their buildings.The Islamic Empire established across Persia. The outstanding Persian mathematician Muhammad Al-Khwarizmi was an early Director of the House of Wisdom in the 9th Century. and allowed a much more general way of analyzing problems other than just the specific problems previously considered by the Indians and Chinese. Iberia and parts of India from the 8th Century onwards made significant contributions towards mathematics.9 and 0). Central Asia. In fact. and one of the greatest of early Muslim mathematicians. Al-Khwarizmi's other important contribution was algebra. over time. Perhaps Al-Khwarizmi’s most important contribution to mathematics was his strong advocacy of the Hindu numerical system (1 . later. the Middle East. The Qu’ran itself encouraged the accumulation of knowledge. Muslim artists discovered all the different forms of symmetry that can be depicted on a 2-dimensional surface. and later by Europe as well. and he introduced the fundamental algebraic methods of “reduction” and “balancing” and provided an exhaustive account of solving polynomial equations up to the second degree. Western) mathematics. . North Africa. The House of Wisdom was set up in Baghdad around 810. which he recognized as having the power and efficiency needed to revolutionize Islamic (and. and work started almost immediately on translating the major Greek and Indian mathematical and astronomy works into Arabic. he helped create the powerful abstract mathematical language still used across the world today. and which was soon adopted by the entire Islamic world. and a Golden Age of Islamic science and mathematics flourished throughout the medieval period from the 9th to 15th Centuries. They were able to draw on and fuse together the mathematical developments of both Greece and India. In this way. raising mathematics to the form of an art. AlKaraji used mathematical induction to prove the binomial theorem. and then proving that. Omar Khayyam (perhaps better known as a poet and the writer of the “Rubaiyat”. Among other things. he was held back from further advances by his inability to separate the algebra from the geometry. such as (x +y)2. but an important mathematician and astronomer in his own right) generalized Indian methods for extracting square and cube roots to include fourth. and a purely algebraic method for the solution of cubic equations had to wait another 500 years and the Italian mathematicians del Ferro and Tartaglia.The 10th Century Persian mathematician Muhammad Al-Karaji worked to extend algebra still further. and introduced the theory of algebraic calculus. Al-Karaji was the first to use the method of proof by mathematical induction to prove his results. including Al-Karaji. fifth and higher roots in the early 12th Century. A binomial is a simple type of Binomial Theorem algebraic expression which has just two terms which are operated on only by addition. Although he did in fact succeed in solving cubic equations. He carried out a systematic analysis of cubic problems. and although he is usually credited with identifying the foundations of algebraic geometry. Some hundred years after Al-Karaji. The co-efficients needed when a binomial is expanded form a symmetrical triangle. by proving that the first statement in an infinite sequence of statements is true. subtraction. usually referred to as Pascal’s Triangle after the 17th Century French mathematician Blaise Pascal. . multiplication and positive whole-number exponents. Persia. although many other mathematicians had studied it centuries before him in India. then so is the next one. China andItaly. if any one statement in the sequence is true. revealing there were actually several different sorts of cubic equations. freeing it from its geometrical heritage. and devised what is now known as "Alhazen's problem" (he was the first mathematician to derive the formula for the sum of the fourth powers. 4. distinct from astronomy. the 11th Century Persian Ibn al-Haytham (also known as Alhazen). 44. 20. of which the sum is 220). a⁄(sin A) = b⁄(sin B) =c⁄( sin C). Other medieval Muslim mathematicians worthy of note include: the 9th Century Arab Thabit ibn Qurra. 55 and 110. 10. 4. including listing the six distinct cases of a right triangle in spherical trigonometry. who. 2. in addition to his groundbreaking work on optics and physics. e. who developed a general formula by which amicable numbers could be derived. established the beginnings of the link between algebra and geometry. scientist and mathematician Nasir AlDin Al-Tusi was perhaps the first to treat trigonometry as a separate mathematical discipline.g. 5. as well as on tangents of a circle. re-discovered much later by both Fermat and Descartes(amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number. the 10th Century Arab mathematician Abul Hasan al-Uqlidisi. and the proper divisors of 284 are 1.The 13th Century Persian astronomer. 7. the proper divisors of 220 are 1. and particularly the use of decimals instead of fractions (e. using a method that is readily generalizable). although the sine law for spherical triangles had been discovered earlier by the 10th Century Persians Abul Wafa Buzjani and Abu Nasr Mansur. who wrote the earliest surviving text showing the positional use of Arabic numerals. 71. of which the sum is 284.375 insead of 73⁄8). and 142.g. 22. One of his major mathematical contributions was the formulation of the famous law of sines for Al-Tusi was a pioneer in the field of spherical trigonometry plane triangles. 11. 2. who continued Archimedes' investigations of areas and volumes. Building on earlier work byGreek mathematicians such as Menelaus of Alexandria and Indian work on the sine function. and . he gave the first extensive exposition of spherical trigonometry. the 10th Century Arab geometer Ibrahim ibn Sinan. and further developments moved to Europe. Scholastic scholars only valued studies in the humanities. based on the Roman/Greek model Centuries. whose works included topics such as computing square roots and the theory of continued fractions. Europe. such as "How many angels can stand on the point of a needle?" From the 4th to 12th Medieval abacus. Robert . who applied the theory of conic sections to solve optical problems. factorization and combinatorial methods. geometry. mathematics and almost all intellectual endeavour stagnated. in which science. astronomy and music was limited mainly to Boethius’ translations of some of the works of ancient Greek masters such as Nicomachus and Euclid. though. and spent much of their energies quarrelling over subtle subjects in metaphysics and theology. the 13th Century Moroccan Ibn al-Banna al-Marrakushi. and Eastern knowledge gradually began to spread to the West. All trade and calculation was made using the clumsy and inefficient Roman numeral system. Islamic mathematics stagnated. such as philosophy and literature. the 13th Century Persian Kamal al-Din al-Farisi. as well as pursuing work in number theory such as on amicable numbers. later re-discovered by Fermat) and the the first use of algebraic notation since Brahmagupta.296 and 18.416. Europe had fallen into the Dark Ages. as well as the discovery of the first new pair of amicable numbers since ancient times (17. and particularly Italy. With the stifling influence of the Turkish Ottoman Empire from the 14th or 15th Century onwards. was beginning to trade with the East. By the 12th Century. European knowledge and study of arithmetic. Indian and Isl amicmathematicians had been in the ascendancy. MEDIEVAL MATHEMATICS During the centuries in which theChinese. and with an abacus based on Greekand Roman models. not tending to a limit. Also.of Chester translated Al-Khwarizmi's important book on algebra into Latin in the 12th Century. . and the complete text of Euclid's “Elements” was translated in various versions by Adelard of Bath. leading from his research into musicology.. being the first to prove that the harmonic series 1⁄1 + 1⁄2 + 1⁄3 +1⁄4 + 1⁄5. he was the first to use fractional exponents. The advent of the printing press in the mid-15th Century also had a huge impact. and also worked on infinite series. and opened the way for great advances in European mathematics. his main contribution to mathematics being in the area of . The great expansion of trade and commerce in general created a growing practical need for mathematics. other than infinity). Europe’s first great medieval mathematician was the Italian Leonardo of Pisa. Oresme was one of the first to use graphical analysis The German scholar Regiomontatus was perhaps the most capable mathematician of the 15th Century. and arithmetic entered much more into the lives of common people and was no longer limited to the academic realm. is a divergent infinite series (i. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their commercial needs and mathematics gradually began to acquire a more important position in education. He used a system of rectangular coordinates centuries before his countryman René Descartespopularized the idea.e. An important (but largely unknown and underrated) mathematician and scholar of the 14th Century was the Frenchman Nicole Oresme. better known by his nicknameFibonacci. perhaps his most important contribution to European mathematics was his role in spreading the use of the Hindu-Arabic numeral system throughout Europe early in the 13th Century. which soon made the Roman numeral system obsolete. as well as perhaps the first time-speeddistance graph. Although best known for the so-called Fibonacci Sequence of numbers. Herman of Carinthia and Gerard of Cremona. and about the elliptical orbits of the planets and relative motion. and it was largely through his efforts that trigonometry came to be considered an independent branch of mathematics. mathematician and astronomer. which foreshadowed the later discoveries of Copernicus and Kepler. His book "De Triangulis". Mention should also be made of Nicholas of Cusa (or Nicolaus Cusanus). whose prescient ideas on the infinite and the infinitesimal directly influenced later mathematicians like Gottfried Leibniz and Georg Cantor. in which he described much of the basic trigonometric knowledge which is now taught in high school and college. 16TH CENTURY MATHEMATICS The supermagic square shown in Albrecht Dürer's engraving "Melencolia . a 15th Century German philosopher. He helped separate trigonometry from astronomy. He also held some distinctly non-standard intuitive ideas about the universe and the Earth's position in it.trigonometry. was the first great book on trigonometry to appear in print. 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). and it is no surprise that. An important figure in the late 15th and early 16th Centuries is an Italian Franciscan friar called Luca Pacioli. (see the section on Fibonacci) in his 1509 book "The Divine Proportion". 1514. In fact.. . concluding that the number was a message from God and a source of secret knowledge about the inner beauty of things. The year of the work. revolutionary work in the fields of philosophy and science was soon taking place. which saw a resurgence of learning based on classical sources. Pacioli also investigated the Golden Ratio of 1 : 1. began in Italy around the 14th Century. symbols that were to become standard notation. as exemplified by the work of artist/scientists such as Leonardo da Vinci.The cultural. intellectual I" and artistic movement of the Renaissance. is shown in the two bottom central squares. Johannes Widmann and others). geometry and bookkeeping at the end of the 15th Century which became quite popular for the mathematical puzzles it contained.618.. and gradually spread across most of Europe over the next two centuries. 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. who published a book on arithmetic. 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". just as in art. Science and art were still very much interconnected and intermingled at this time. India and the Islamic world. Building on Tartaglia’s work. with dates of first use particular was famed for its intense public mathematics competitions. an achievement hitherto considered impossible and which had stumped the best mathematicians ofChina. and later all types. Tartaglia. and Cardanopublished perhaps the first systematic treatment of probability. Rafael Bombelli.Tartaglia went on to produce other important (although largely ignored) formulas and methods. Cardano and Ferrari between them demonstrated the first uses of what are now known as complex numbers.During the 16th and early 17th Century. another young Italian. Bologna University in Basic mathematical notation. self-taughtNiccolò Fontana Tartaglia revealed to the world the formula for solving first one type. soon devised a similar method to solve quartic equations (equations with terms including x4) and both solutions were published by Gerolamo Cardano. The use of decimal fractions and decimal arithmetic is usually attributed to the Flemish mathematician Simon Stevin the late 16th Century. multiplication. negatives. radical (root). Despite a decade-long fight over the publication. . combinations of real and imaginary numbers (although it fell to another Bologna resident. Lodovico Ferrari. the equals. It was in just such a competion that the unlikely figure of the young. although the decimal point notation was not popularized until early in the 17th Century. whether fractions. to explain what imaginary numbers really were and how they could be used). real numbers or surds (such as √2) should be treated equally as numbers in their own right. Stevin was ahead of his time in enjoining that all types of numbers. In the Renaissance Italy of the early 16th Century. division. decimal and inequality symbols were gradually introduced and standardized. of cubic equations (equations with terms including x3). 17TH CENTURY MATHEMATICS In the wake of the Renaissance. had doubled their lifetimes. The value of logarithms arises from the fact that multiplication of two or more numbers is equivalent to adding their logarithms. a period sometimes called the Age of Reason. a much simpler operation. by halving the labours of astronomers. Tycho Brahe and Johannes Kepler were making equally revolutionary discoveries in the exploration of the Solar system. The French astronomer and mathematician Pierre Simon Laplace remarked. that Napier. squaring is as simple as multiplying the logarithm by two (or by three for .With Hindu-Arabic numerals. Hard on the heels of the “Copernican Revolution” of Nicolaus Copernicus in the 16th Century. scientists like Galileo Galilei. leading to Kepler’s formulation of mathematical laws of planetary motion. because 102 = 100. early in the 17th Century 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. Logarithms were invented by John Napier. The logarithm of a number is the exponent when that number is expressed as a power of 10 (or any other base). In the same way. It was one of the most significant mathematical developments of the age. standardized notation and the new language of algebra at their disposal. almost two centuries later. division involves the subtraction of logarithms. the base 10 logarithm of 100 (usually written log10 100 or lg 100 or just log 100) is 2. the 17th Century saw an unprecedented explosion of mathematical and scientific ideas across Europe. For example. It is effectively the inverse of exponentiation. the stage was set for the European mathematical revolution of the 17th Century. and 17th Century physicists like Kepler and Newton could never have performed the complex calculatons needed for their innovations without it. astronomy and mathematics by making some difficult calculations relatively easy. the largest known prime number has almost always been a Mersenne prime. etc). the role of the Frenchman Marin Mersenne as a sort of clearing house and gobetween for mathematical thought in France during this period was crucial. Although base 10 is the most popular base. 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”. etc). Mersenne is largely remembered in mathematics today in the term Mersenne primes prime numbers that are one less than a power of 2. but in actual fact. 127 (27-1). square roots requires dividing the logarithm by 2 (or by 3 for cube roots. another common base for logarithms is the number e which has a value of 2. 31 (25-1). Although not principally a mathematician. Briggs produced extensive lookup tables of common (base 10) logarithms. and are written loge or ln. Graph of the number of digits in the known Mersenne primes Mersenne’s real connection with the numbers was only to compile a none-tooaccurate list of the smaller ones (when Edouard Lucas devised a method of checking them in the 19th Century.cubing. he pointed out that Mersenne had incorrectly included 2 671 and left out 261-1.. These are known as natural logarithms. and by 1622 William Oughted had produced a logarithmic slide rule. 7 (23-1). Napier also improved Simon Stevin's decimal notation and popularized the use of the decimal point. In modern times. 289-1 and 2107-1 from his list).7182818. a multiplication tool using a set of numbered rods.. as well as laying the foundations for the later development of . etc. and which has special properties which make it very useful for logarithmic calculations. 3 (22-1). The Frenchman René Descartes is sometimes considered the first of the modern school of mathematics. 8191 (213-1). 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. an instrument which became indispensible in technological innovation for the next 300 years.g. e. and nowadays he is often considered. their corresponding sides meet at points on the same collinear line. and he effectively laid the groundwork for all of classical mechanics. almost single-handedly. . Fermat formulated several theorems which greatly extended our knowlege of number theory. as one of the greatest mathematicians of all time. 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. Descartes is also credited with the first use of superscripts for powers or exponents. Pascal is most famous for Pascal’s Triangle of binomial coefficients. when two triangles are in perspective. By “standing on the shoulders of giants”.calculus (and much later multi-dimensional geometry). as well as contributing some early work on infinitesimal calculus. Projective geometry considers what happens to shapes when they are projected on to a non-parallel plane. His perspective theorem states that. In particular. The French mathematician and engineer Girard Desargues is considered one of the founders of the field of projective geometry. the Englishman Sir Isaac Newton was able to pin down the laws of physics in an unprecedented way. however. Two other great French mathematicians were close contemporaries of Descartes: Pierre de Fermat and Blaise Pascal. was by the Dutchman Christiaan Huygens in 1657. although it was largely based on the ideas in the letters of the two Frenchmen. Desargues’ perspective theorem Desargues developed the pivotal concept of the “point at infinity” where parallels actually meet. For example. although similar figures had actually been produced by Chinese and Persian mathematicians long before him. and so these curves may all be regarded as equivalent in projective geometry. a circle may be projected into an ellipse or a hyperbola. along with Archimedes and Gauss. later developed further by Jean Victor Poncelet and Gaspard Monge. But his contribution to mathematics should never be underestimated. and the first to outline the concept of mathematical expectation. Newton's teacher Isaac Barrow is usually credited with the discovery (or at least the first rigorous statrement of) the fundamental theorem of calculus.Newton and. and Leibniz’s development of a mechanical forerunner to the computer and the use of matrices to solve linear equations. which essentially showed that integration and differentiation are inverse operations. the development of infinitesimal calculus. and to some extent paved the way for. the theory of finite differences and the use of infinite power series. engineering. As early as the 1630s. with its two main operations. introducing the symbol ∞ for infinity and the term “continued fraction”. as well as originating the idea of the number line. Whatever the truth behind the various claims. differentiation and integration. though. and extending the standard notation for powers to include negative integers and rational numbers. and he also made complete translations of Euclid into Latin and English. 18TH CENTURY MATHEMATICS . including Newton’s contributions to a generalized binomial theorem. the Italian mathematician Bonaventura Cavalieri developed a geometrical approach to calculus known as Cavalieri's principle. Both Newton and Leibniz also contributed greatly in other areas of mathematics. credit should also be given to some earlier 17th Century mathematicians whose work partially anticipated. economics and science in general) by the development of infinitesimal calculus. Newtonprobably developed his work before Leibniz. However. it is Leibniz’s calculus notation that is the one still in use today. who systematized and extended the methods of analysis of Descartes and Cavalieri. completely revolutionized mathematics (not to mention physics. independently. also made significant contributions towards the development of calculus. or the “method of indivisibles”. leading to an extended and rancorous dispute. and calculus of some sort is used extensively in everything from engineering to economics to medicine to astronomy. the German philosopher and mathematician Gottfried Leibniz. but Leibniz published his first. The Englishman John Wallis. Jacob and Johann. although the weak form of the conjecture appears to be closer to resolution than the strong one. . although. astronomy and engineering. which states that every even integer greater than 2 can be expressed as the sum of two primes (e. partly due to the difficulties in getting on in a city dominated by theBernoulli family. who applied their ideas on calculus to solving a variety of problems in physics. Euler spent most of his time abroad. pioneered new methods. 8 = 3 + 5.as well asPascal and Fermat’s probability and number theory. Leonhard Euler. the German mathematician Christian Goldbach proposed the Goldbach Conjecture. from geometry to calculus to trigonometry to algebra to number theory. 14 = 3 + 11 = 7 + 7. They were largely responsible for further developingLeibniz’s Calculus of variations infinitesimal calculus paricularly through the generalization and extension of calculus known as the "calculus of variations" . etc) or. and was able to find unexpected links between the different fields. in Germany and St.g. standardized mathematical notation and wrote many influential textbooks throughout his long academic life. the Bernoulli’s of Basel in Switzerland. He proved numerous theorems. In a letter to Euler in 1742. 4 = 2 + 2. in another equivalent version. that all odd numbers greater than 7 are the sum of three odd primes. Goldbach also proved other theorems in number theory such as the Goldbach-Euler Theorem on perfect powers. They remain among the oldest unsolved problems in number theory (and in all of mathematics). which boasted two or three generations of exceptional mathematicians. by one family. though. every integer greater than 5 can be expressed as the sum of three primes. The period was dominated. particularly the brothers. Russia.Most of the late 17th Century and a good part of the early 18th were taken up by the work of disciples of Newtonand Leibniz. Basel was also the home town of the greatest of the 18th Century mathematicians. Yet another version is the so-called “weak” Goldbach Conjecture. He excelled in all aspects of mathematics. Petersburg. 31 = 52 + 22 + 12 + 12. and he is usually credited with originating the theory of groups. France became even more prominent towards the end of the century.g. and a handful of late 18th Century French mathematicians in particular deserve mention at this point. as well as another theorem. Lagrange’s 1788 treatise on analytical mechanics offered the most comprehensive treatment of classical mechanics since Newton. confusingly also known as Lagrange’s Theorem or Lagrange’s Mean Value Theorem. Abraham de Moivre is perhaps best known for de Moivre's formula.Despite Euler’s and the Bernoullis’ dominance of 18th Century mathematics. given a section of a smooth continuous (differentiable) curve. But he also generalized Newton’s famous binomial theorem into the multinomial theorem. which links complex numbers and trigonometry. Pierre-Simon Laplace. 3 = 12 + 12 + 12 + 02. that any natural number can be represented as the sum of four squares (e. and formed a basis for the development of mathematical physics in the 19th Century. sometimes referred to as “the French Newton”. but he also contributed to differential equations and number theory. etc). In the early part of the century. which would become so important in 19th and 20th Century mathematics. beginning with “the three L’s”. His name is given an early theorem in group theory. which states that. many of the other important mathematicians were from France. (cosx + isinx)n = cos(nx) + isin(nx). Lagrange is also credited with the four-square theorem. whose monumental work “Celestial Mechanics” translated the geometric study of classical mechanics to one based on . there is at least one point on that section at which the Lagrange’s Mean value Theorem derivative (or slope) of the curve is equal (or parallel) to the average (or mean) derivative of the section. and his work on the normal distribution (he gave the first statement of the formula for the normal distribution curve) and probability theory were of great importance. Joseph Louis Lagrange collaborated with Euler in an important joint work on the calculus of variation. which states that the number of elements of every sub-group of a finite group divides evenly into the number of elements of the original finite group. 310 = 172 + 42 + 22 + 12. was an important mathematician and astronomer. pioneered the development of analytic geometry. athough much of his work (such as the least squares method for curve-fitting and linear regression. Laplace is well known for his belief in complete scientific determinism. Gaspard Monge was the inventor of descriptive geometry. became the leading geometry textbook for almost 100 years. Ferdinand von Lindemann would prove that π is also .at least in principle .calculus. (Over a hundred years later.or at least to general notice . a Swiss mathematician and prominent astronomer. abstract algebra and mathematical analysis in the late 18th and early 19th Centuries. After many centuries of increasingly accurate approximations. Yet another Frenchman. His orthographic projection became the graphical method used in almost all modern mechanical drawing. opening up a much broader range of problems. in 1882. architecture and design. a re-working of Euclid’s book.to predict everything about the universe and how it works.by others. he was already starting to think about the mathematical and philosophical concepts of probability and statistics in the 1770s. and his extremely accurate measurement of the terrestrial meridian inspired the creation. a technique which would later become important in the fields of engineering. The first six Legendre polynomials (solutions to Legendre’s differential His “Elements of equation) Geometry”. and almost universal adoption. Although his early work was mainly on differential equations and finite differences. the quadratic reciprocity law. finally provided a rigorous proof in 1761 that π is irrational. Johann Lambert. the prime number theorem and his work on elliptic functions) was only brought to perfection .e. number theory. although the obsession with obtaining more and more accurate approximations continues to this day. a clever method of representing three-dimensional objects by projections on the twodimensional plane using a specific set of procedures. Adrien-Marie Legendre also made important contributions to statistics. and he developed his own version of the so-called Bayesian interpretation of probability independently of Thomas Bayes. i. particularlyGauss. of the metric system of measures and weights. it can not be expressed as a simple fraction using integers only or as a terminating or repeating decimal. This definitively proved that it would never be possible to calculate it exactly. and he maintained that there should be a set of scientific laws that would allow us . Fourier and Galois. as exemplified by “the three L’s”. Approximation of a periodic function by the Fourier Series Joseph Fourier's study. 19TH CENTURY MATHEMATICS The 19th Century saw an unprecedented increase in the breadth and complexity of mathematical concepts. Even though the Dane Caspar Wessel had produced a very similar paper at the end of the 18th Century. Periodic functions that can be expressed as the sum of an infinite series of sines and cosines are known today as Fourier Series. Lambert was also the first to introduce hyperbolic functions into trigonometry and made some prescient conjectures regarding non-Euclidean space and the properties of hyperbolic triangles. Laplace and Legendre (see the section on 18th Century Mathematics).defining it in terms of a correspondence between elements of the domain and the range .e. Fourier (following Leibniz. and even though it .transcendental. but the two countries treated mathematics quite differently. although the definition that is found in texts today . Lagrange and others) also contributed towards defining exactly what is meant by a function. Both France and Germany were caught up in the age of revolution which swept Europe in the late 18th Century. Lagrange. where i is √-1) could be represented on geometric diagrams and manipulated using trigonometry and vectors. of infinite sums in which the terms are trigonometric functions were another important advance in mathematical analysis. at the beginning of the 19th Century. After the French Revolution. it cannot be the root of any polynomial equation with rational coefficients). Euler.is usually attributed to the 19th Century German mathematician Peter Dirichlet. and they are still powerful tools in pure and applied mathematics. i. Jean-Robert Argand published his paper on how complex numbers (of the form a + bi. In 1806. Napoleon emphasized the practical usefulness of mathematics and his reforms and military ambitions gave French mathematics a big boost. Riemann. took a rather different approach. breaking away completely from all the limitations of 2 and 3 dimensional geometry. This left the field open for János Bolyai and Nikolai Lobachevsky (respectively. The Frenchman Évariste Galois proved in the late 1820s that there is no general algebraic method for solving polynomial equations of any degree greater than four. number theory. however. detached from the demands of the state and military. group theory. Germany. Gauss also claimed to have investigated a kind of nonEuclidean geometry using curved space but. whether flat or curved. just a few years earlier. algebra and probability.was Gauss who popularized the practice. The German Bernhard Riemann worked on a different kind of non-Euclidean geometry called elliptic geometry. on the other hand. including geometry. modules. soon took this even further. a Hungarian and a Russian) who both independently explored the potential of hyperbolic geometry and curved spaces. Later in life. still unproven after 150 years. rings. including areas like algebraic geometry. under the influence of the great educationalist Wilhelm von Humboldt. shown the impossibility of solving quintic equations. hyperbolic and elliptic geometry decided not to pursue or publish any of these avantgarde ideas. going further than the Norwegian Niels Henrik Abel who had. vector spaces and noncommutative algebra. sometimes called the “Prince of Mathematics”. and touched on many different parts of the mathematical world. received his education at the prestigious University of Göttingen. they are still known today as Argand Diagrams. along with Archimedes and Newt on. unwilling to court controversy. remains one of the world’s great unsolved mathematical mysteries and the testing ground for new generations of mathematicians. supporting pure mathematics for its own sake. He is widely regarded as one of the three greatest mathematicians of all times. His exploration of the zeta function in multi-dimensional complex numbers revealed an unexpected link with the distribution of prime numbers. he Euclidean. Some ofGauss’ ideas were a hundred years ahead of their time. and his famous Riemann Hypothesis. and breaching an impasse which had existed for centuries. It was in this environment that the young German prodigy Carl Friedrich Gauss. . Galois' work also laid the groundwork for further developments such as the beginnings of the field of abstract algebra. and began to think in higher dimensions. as well as on a generalized theory of all the different types of geometry. calculus. fields. OR and NOT. Quaternions.e. and which could be applied to the solution of logical problems and mathematical functions. in which. a machine was built almost 200 years later to his specifications and worked perfectly. 0 and 1. Another 19th Century Englishman. etc). provided the first example of a noncommutative algebra (i. In the mid-19th Century. This recognition of the possible existence of nonarithmetical algebras was an important stepping stone toward future developments in abstract algebra. in which the only operators were AND. where a quantity representing a 3dimensional rotation can be described by just an angle and a vector). He also described a kind of binary system which used just two objects. Although never actually built in his lifetime. He also designed a much more sophisticated machine he called the "analytic engine". Boolean algebra was the starting point of modern mathematical logic and ultimately led to the development of computer science. and its later generalization by Hermann Grassmann. whose 1843 theory of quaternions (a 4dimensional number system. and showed that several different consistent algebras may be derived Hamilton’s quaternion . complete with punched cards. "on" and "off" (or "true" and "false". and the extension of the scope of algebra beyond the ordinary systems of numbers. the British mathematician George Boole devised an algebra (now called Boolean algebra or Boolean logic). is usually credited with the invention of symbolic algebra.British mathematics also saw something of a resurgence in the early and mid-19th century. and was the true forerunner of the modern electronic computer. one in which a x b does not always equal bx a). His large "difference engine" of 1823 was able to calculate logarithms and trigonometric functions. Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th Century. famously. 1 + 1 = 1. George Peacock. it was Charles Babbage in 19th Century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. The concept of number and algebra was further extended by the Irish mathematician William Hamilton. printer and computational abilities commensurate with modern computers. and early consideration of sets (collections of objects defined by a common property. and especially his theory of periodic functions and elliptic functions and their relation to the elliptic theta function. But it also saw a re-visiting of some older methods and an emphasis on mathematical rigour. the theory of higher singularities. Throughout the 19th Century. . In 1845. and was a pioneer of modern group theory. But Cayley was one of the most prolific mathematicians in history. leading to the development of mathematical analysis. Cauchy also proved Cauchy's theorem. infinite series and analytic functions. a continuous curve possessing no tangent at any of its points). In the first decades of the century. matrix algebra. etc). Weierstrass completely reformulated calculus in an even more rigorous fashion. integration. the Bohemian priest Bernhard Bolzano was one of the earliest mathematicians to begin instilling rigour into mathematical analysis. such as "all the numbers greater than 7" or "all right triangles". The Englishman Arthur Cayley extended Hamilton's quaternions and developed the octonions. determinants and matrices. a fundamental theorem of group theory.by choosing different sets of axioms. as well as the theory of invariants. and higher dimensional geometry (anticipating the later ideas of Klein). the Frenchman Augustin-Louis Cauchy. When the German mathematician Karl Weierstrass discovered the theoretical existence of a continuous function having no derivative (in other words. particularly. from which all the basic concepts of analysis could be derived. mathematics in general became ever more complex and abstract. he saw the need for a rigorous “arithmetization” of calculus. Carl Jacobi also made important contributions to analysis. Along with Riemann and. a branch of pure mathematics largely concerned with the notion of limits (whether it be the limit of a sequence or the limit of a function) and with the theories of differentiation. which he discovered while examining permutation groups. as well as giving the first purely analytic proof of both the fundamental theorem of algebra and the intermediate value theorem. only in four or more dimensions. It can be best visualized as a cylinder looped back through itself to join with its other end from the "inside". who had identified over 60 such numbers in the 18th Century. and which has . and applied it to the geometric theory of differential equations by means of continuous groups of transformations known as Lie groups. Klein’s 1872 Erlangen Program. but it has come to hold Möbius' name). the Möbius function and the Möbius inversion formula. some of them huge). and his work was very important in the later development of group theory and function theory. Johann Benedict Listing. He also introduced homogeneous coordinates and discussed geometric and projective transformations. an unknown 16-year old Italian. In an unusual occurrence in 1866. include the Klein bottle. Non-orientable surfaces with no identifiable "inner" and "outer" sides Felix Klein also pursued more developments in non-Euclidean geometry. Niccolò Paganini. which had been completely overlooked by some of the greatest mathematicians in history (including Euler. The Norwegian mathematician Marius Sophus Lie also applied algebra to the study of geometry. discovered the second smallest pair of amicable numbers (1.184 and 1210). including the Möbius configuration. Many other concepts are also named after him. Georg Cantor established the first foundations of set theory. He largely created the theory of continuous symmetry.August Ferdinand Möbius is best known for his 1858 discovery of the Möbius strip. devised the same object just a couple of months before Möbius. Möbius transformations. which classified geometries by their underlying symmetry groups (or their groups of transformations). the Möbius transform of number theory. a non-orientable twodimensional surface which has only one side when embedded in threedimensional Euclidean space (actually a German. which enabled the rigorous treatment of the notion of infinity. In the later 19th Century. was a hugely influential synthesis of much of the mathematics of the day. a one-sided closed surface which cannot be embedded in threedimensional Euclidean space. In 1881. gaps or discontinuities. Building on Riemann’s deep ideas on the distribution of prime numbers. every location on the number line continuum contains either a rational or an irrational number. now called a Dedekind cut which is now a standard definition of the real numbers. Dedekind also came up with the notion. In the face of fierce resistance from most of his contemporaries and his own battle against mental illness. the upper class being strictly greater than all the members of the other lower class.since become the common language of nearly all mathematics. He showed that any irrational Venn diagram number divides the rational numbers into two classes or sets. some of which were larger than others. one by Jacques Hadamard and one by Charles de la Vallée Poussin. the Englishman John Venn introduced his “Venn diagrams” which become useful and ubiquitous tools in set theory. Cantor’s work on set theory was extended by another German. who defined concepts such as similar sets and infinite sets. Cantor explored new mathematical worlds where there were many different infinities. the year 1896 saw two independent proofs of the asymptotic law of the distribution of prime numbers (known as the Prime Number Theorem). with no empty locations. Richard Dedekind. . Thus. which showed that the number of primes occurring up to any number x is asymptotic to (or tends towards) x⁄log x. In his attempt to show that mathematics grows out of logic. its implications led to the early intimations of what would later become known as chaos theory. universals and existentials. He is sometimes referred to as the “Last Univeralist” as he was perhaps the last mathematician able to shine in almost all of the various aspects of what had become by now a huge. it was Minkowski who realized that the Einstein’s 1905 special theory of relativity could be best understood in a fourdimensional space. a deceptively simple problem which had stubbornly resisted resolution since the time ofNewton. The 20th Century would belong to the specialists. in 1907. Later. In between his important work in theoretical physics. set the stage for the radical advances of Giuseppe Peano. involving complex concepts such as convex sets. often referred to as Minkowski space-time. lattice points and vector space. a great friend ofDavid Hilbert and teacher of the young Albert Einstein. leaving behind a knotty problem known as the Poincaré conjecture which remined unsolved until 2002. Although his solution actually proved to be erroneous. in so doing. which championed a faith in human intuition over rigour and formalism. he also greatly extended the theory of mathematical topology. Bertrand Russelland David Hilbert in the early 20th Century. over two hundred years earlier. he devised techniques that took him far beyond the logical traditions of Aristotle (and even of George Boole). as well as the notions of quantifiers. and perhaps the last of the great mathematicians to adhere to an older conception of mathematics. Poincaré was also an engineer and a polymath. including a rigorous treatment of the ideas of functions and variables.Hermann Minkowski. He extended Boole's "propositional logic" into a new "predicate logic" and. developed a branch of number theory called the "geometry of numbers" late in the 19th Century as a geometrical method in multi-dimensional space for solving number theory problems. . He was the first to explicitly introduce the notion of variables in logical statements. Minkowski space-time Gottlob Frege’s 1879 “Begriffsschrift” (roughly translated as “Concept-Script”) broke new ground in the field of logic. Henri Poincaré came to prominence in the latter part of the 19th Century with at least a partial solution to the “three body problem”. encyclopedic and incredibly complex subject. using just a compass and straight edge as Euclid would have done.s each year and jobs in both teaching and industry. singularity theory. In 1904. Hardy and his young Indian protégé Srinivasa Ramanujan.H. Johann Gustav Hermes completed his construction of a regular polygon with 65. model theory. in which the notion of axioms as “selfevident truths” was largely discarded in favour of an emphasis on such logical concepts as consistency and completeness. category theory. sheaf theory. and the development of hundreds of specialized areas and fields of study. chaos theory. complexity theory and many more. were just two of the great mathematicians of the early 20th Century who applied themselves in earnest to solving problems of the previous century. which came to fruition in the hands of Giuseppe Peano. Brouwer. but Hardy is credited with reforming British mathematics.J. Fields of Mathematics topology.David Hilbert and.537 sides (216 + 1). It also saw mathematics become a major profession. and Ramanujan proved himself to be one of the most brilliant (if somewhat undisciplined and unstable) minds of the century.E. game theory. which had sunk to something of a low ebb at that time. involving thousands of new Ph. catastrophe theory. Although they came close.20TH CENTURY MATHEMATICS The 20th Century continued the trend of the 19th towards increasing generalization and abstraction in mathematics. such as the Riemann hypothesis. graph theory. a feat that took him over ten years. L. such as group theory. building on the earlier advances of Gottlob Frege. they too were defeated by that most intractable of problems. The early 20th Century also saw the beginnings of the rise of the field of mathematical logic. functional analysis.D. . knot theory. The eccentric British mathematicianG. Others followed techniques dating back millennia but taken to a 20th Century level of complexity. However. He was unfailingly optimistic about the future of mathematics. . Bertrand Russell and A. and turned mathematics on its head with his famous incompleteness theorem. spent his pre-war years trying to clarify and simplify Gödel’s rather abstract proof. and 2 (the Riemann hypothesis and the Kronecker-Weber theorem on abelian extensions) are still open. But. where axioms are not taken to be self-evident truths. in which he set out his generations of 23 problems mathematicians to come. Hilbert's approach signalled the shift to the modern axiomatic method. Hilbert was himself a brilliant mathematician. as well as overseeing the development of what amounted to a whole new style of abstract mathematical thinking.particularly. including the idea that there was no way of telling beforehand which problems were provable and which unprovable. famously declaring in a 1930 radio interview “We must know. the Austrian Kurt Gödel was soon to put some very severe constraints on what could and could not be solved. His methods led to some conclusions that were perhaps even more devastating than Gödel’s. We will know!”. 10 have now been solved. These “Hilbert problems” effectively set the agenda for 20th Century mathematics. as a spin-off. with the remaining 4 being too loosely formulated to be stated as solved or not. Whitehead. The century began with a historic convention at the Sorbonne in Paris in the summer of 1900 which is largely remembered for a lecture by the young German mathematician David Hilbert in which he set out what he saw as the 23 greatest unsolved mathematical problems of the day. perhaps best known for his war-time work in breaking the German enigma code. and was a well-loved leader of the mathematical community during the first part of the century. Of these original 23 problems. and laid down the gauntlet for Part of the transcript of Hilbert’s 1900 Paris lecture. Alan Turing. responsible for several theorems and some entirely new mathematical concepts. 7 are partially solved. whose monumental joint work the “Principia Mathematica” was so influential in mathematical and philosophical logicism.that there could be solutions to mathematical problems which were true but which could never be proved.N. which proved the unthinkable . André Weil was another refugee from the war in Europe. With the gradual and wilful destruction of the mathematics community of Germany and Austria by the anti-Jewish Nazi regime in the 1930 and 1940s. geometry and topology. a general architecture that most electronic computers follow even today. interested in the hidden structures beneath all mathematics. fled the Nazis to this safe haven. Many of the brightest European mathematicians. particularly to the Institute for Advanced Study in Princeton. John von Neumann. His theorems. under the secret nom-deplume of Nicolas Bourbaki.his work also led to the development of computers and the first considerations of such concepts as artificial intelligence. and . wrote many influential books on the mathematics of the 20th Century. including Hermann Weyl. In addition to his physical work in quantum theory and his role in the Manhattan Project and the development of nuclear physics and the hydrogen bomb. Kurt Gödel and Albert Einstein. which allowed connections to be made between number theory. are considered among the greatest achievements of modern mathematics. the focus of world mathematics moved to America. he is particularly remembered as a pioneer of game theory. John von Neumann is considered one of the foremost mathematicians in modern history. after narrowly avoiding death on a couple of occasions. another mathematical child prodigy who went on to make major contributions to a vast range of fields. a charismatic and beloved figure in 20th Century French mathematics. Perhaps the greatest heir to Weil’s legacy was Alexander Grothendieck. and particularly for his design model for a storedprogram digital computer that uses a processing unit and a separate storage Von Neumann’s computer architecture design structure to hold both instructions and data. which attempted to reproduce the collegiate atmosphere of the old European universities in rural New Jersey. algebra. Grothendieck was a structuralist. He was also responsible for setting up a group of French mathematicians who. the most famous example of a fractal closely related. Julia and Mandelbrot fractals are The Mandelbrot set. and his “theory of topoi” is highly relevant to mathematical logic. classical analysis.in the 1950s he created a powerful new language which enabled mathematical structures to be seen in a new way. Benoît Mandelbrot. thus allowing new solutions in number theory. (where z is a number in the complex plane of the form x + iy). his achievements in algebraic geometry have fundamentally transformed the mathematical landscape. geometry. graph theory. Grothendieck all but abandoned mathematics for radical politics. and provided an algebraic definition of the fundamental group of a curve. Although. and which are . in which smaller parts exhibit approximate reduced-size copies of the whole. and he is considered by some to be one of the dominant figures of the whole of 20th Century mathematics. he gave an algebraic proof of the Riemann-Roch theorem. The Mandelbrot set involves repeated iterations of complex quadratic polynomial equations of the form zn+1 =zn2 + c. even in fundamental physics. perhaps no less than those of Cantor. some of which are still active after his death. As a humorous tribute. of the Mandelbrot sets of yet another French mathematician. But it only really gained much attention in the 1970s and 1980s with the beautiful computer plottings of Julia sets and. approximation theory. and probability theory. set theory. particularly. The immensely prolific and famously eccentric Hungarian mathematician worked with hundreds of different collaborators on problems in combinatorics. He was also known for offering small prizes for solutions to various unresolved problems (such as the Erdös conjecture on arithmetic progressions). In addition. number theory. an "Erdös number" is given to mathematicians according to their collaborative proximity to him. and who became known as the father of fractal geometry. The iterations produce a form of feedback based on recursion. The field of complex dynamics (which is defined by the iteration of functions on complex number spaces) was developed by two Frenchmen. Paul Erdös was another inspired but distinctly non-establishment figure of 20th Century mathematics. Pierre Fatou and Gaston Julia. after the 1960s. His “theory of schemes” allowed certain of Weil's number theory conjectures to be solved. early in the 20th Century. and it was Mandelbrot who coined the term fractal. Gödel and Hilbert. e. His work rocked the mathematical world in the 1960s. Another of Hilbert’s problems was finally resolved in 1970. The possible behaviours that a chaotic system may have can also be mapped graphically. all modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis. a fractal structure corresponding to the behaviour of the Lorenz oscillator (a 3-dimensional dynamical system that exhibits chaotic flow). and particulary from its ability to carry out a huge number of repeated iterations of simple mathematical formulas which would be impractical to do by hand. Paul Cohen is an example of a second generation Jewish immigrant who followed the American dream to fame and success. the more detail can be seen.infinitely complex (so that. Since this result. was chaos theory. that there is no general method for determining when polynomial equations have a solution in whole numbers. are fractal in nature (the more you zoom in. when he proved that Cantor's continuum hypothesis about the possible sizes of infinite sets (one of Hilbert’s original 23 problems) could be both true AND not true. Lorenz's discovery came in 1961. . and that there were effectively two completely separate but valid mathematical worlds. In additon to complex dynamics. i. one in which the continuum hypothesis was true and one where it was not. when a computer model he had been running was actually saved using three-digit numbers rather than the six digits he had been working with. another field that benefitted greatly from the advent of the electronic computer. Matiyasevich built on decades of work by the American mathematician Julia Robinson. however much one zooms in and magifies a part. known as "strange attractors". and this tiny rounding error produced dramatically different results.and he demonstrated this with his Lorenz attractor.a phenomenon he described by the term “butterfly effect” . when the young Russian Yuri Matiyasevich finally proved that Hilbert’s tenth problem was impossible. in a great show of internationalism at the height of the Cold War. Chaos theory tells us that some systems seem to exhibit random behaviour even though they are not random at all. and conversely some systems may have roughly predictable behaviour but are fundamentally unpredictable in any detail. He discovered that small changes in initial conditions can produce large changes in the long-term outcome . whose interest in chaos came about accidentally through his work on weather prediction. although the overall pattern remains the same). In arriving at his proof. and it was discovered that these mappings. it exhibits just as much complexity). An early pioneer in modern chaos theory was Edward Lorenz. In 1936. It was an achievement Wiles had set his sights on early in life and pursued doggedly for many years. including Goro Shimura. and states that. Petersburg. Jean-Pierre Serre and Ken Ribet. turned . Also in the 1970s. Margherita Piazzola Beloch had shown how a length of paper could be folded to give the cube root of its length. who lives a frugal life with his mother in a suburb of St. and his unconventional folding techniques have demonstrated many unexpected geometrical results. Yutaka Taniyama. The eventual proof that only four colours suffice turned out to be significantly harder. The Japanese origami expert Kazuo Haga has at least three mathematical theorems to his name. One proof Example of a four-colour map was given by Alfred Kempe in 1879. The most recent of the great conjectures to be proved was the Poincaré Conjecture. it was a joint effort of several steps involving many mathematicans over several years. The proof itself is over 100 pages long. Appel and Haken’s solution required some 1. the final proof of the Taniyama-Shimura Conjecture for semi-stable elliptic curves. in some cases more powerful than Euclidean geometry. However.1976 saw a proof of the four colour theorem by Kenneth Appel and Wolfgang Haken. which was solved in 2002 (over 100 years after Poincaré first posed it) by the eccentric and reclusive Russian mathematician Grigori Perelman. An origami proof of the equally intractible "trisecting the angle" problem followed in 1986. In reality.500 configurations. The four colour conjecture was first proposed in 1852 by Francis Guthrie (a student of Augustus De Morgan). specifically. The British mathematician Andrew Wiles finally proved Fermat’s Last Theorem for ALL numbers in 1995. some 350 years after Fermat’s initial posing. though. in any given separation of a plane into contiguous regions (called a “map”) the regions can be coloured using at most four colours so that no two adjacent regions have the same colour. but it was shown to be incorrect by Percy Heawood in 1890 in proving the five colour theorem. Gerhard Frey. with Wiles providing the links and the final synthesis and.200 hours of computer time to examine around 1. but it was not until 1980 that an origami method was used to solve the "doubling the cube" problem which had defeated ancient Greek geometers. Perelman. origami became recognized as a serious mathematical method. the first major theorem to be proved using a computer. down the $1 million prize, claiming that "if the proof is correct then no other recognition is needed". The conjecture, now a theorem, states that, if a loop in connected, finite boundaryless 3-dimensional space can be continuously tightened to a point, in the same way as a loop drawn on a 2-dimensional sphere can, then the space is a three-dimensional sphere. Perelman provided an elegant but extremely complex solution involving the ways in which 3-dimensional shapes can be “wrapped up” in even higher dimensions. Perelman has also made landmark contributions to Riemannian geometry and geometric topology. John Nash, the American economist and mathematician whose battle against paranoid schizophrenia has recently been popularized by the Hollywood movie “A Beautiful Mind”, did some important work in game theory, differential geometry and partial differential equations which have provided insight into the forces that govern chance and events inside complex systems in daily life, such as in market economics, computing, artificial intelligence, accounting and military theory. The Englishman John Horton Conway established the rules for the so-called "Game of Life" in 1970, an early example of a "cellular automaton" in which patterns of cells evolve and grow in a gridm which became extremely popular among computer scientists. He has made important contributions to many branches of pure mathematics, such as game theory, group theory, number theory and geometry, and has also come up with some wonderful-sounding concepts like surreal numbers, the grand antiprism and monstrous moonshine, as well as mathematical games such as Sprouts, Philosopher's Football and the Soma Cube. Other mathematics-based recreational puzzles became even more popular among the general public, including Rubik's Cube (1974) and Sudoku (1980), both of which developed into full-blown crazes on a scale only previously seen with the 19th Century fads of Tangrams (1817) and the Fifteen puzzle (1879). In their turn, they generated attention from serious mathematicians interested in exploring the theoretical limits and underpinnings of the games. Computers continue to aid in the identification of phenomena such as Mersenne primes numbers (a prime number that is one less than a power of two - see the section on 17th Century Mathematics). In 1952, an early computer known as SWAC identified 2257-1 as the 13th Mersenne prime number, the first new one to be found in 75 years, before going on to identify several more even larger. With the advent of the Internet in the 1990s, the Great Internet Mersenne Prime Search (GIMPS), a collaborative project of volunteers who use freely available computer software to search for Mersenne primes, has led to another leap in the discovery rate. Currently, the 13 largest Mersenne primes were all discovered in this way, and the largest (the 45th Mersenne prime number and also the largest known prime number of any kind) was discovered in 2009 and contains nearly 13 million digits. The search also continues for ever more Approximations for π accurate computer approximations for the irrational number π, with the current record standing at over 5 trillion decimal places. The P versus NP problem, introduced in 1971 by the American-Canadian Stephen Cook, is a major unsolved problem in computer science and the burgeoning field of complexity theory, and is another of the Clay Mathematics Institute's million dollar Millennium Prize problems. At its simplest, it asks whether every problem whose solution can be efficiently checked by a computer can also be efficiently solved by a computer (or put another way, whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure). The solution to this simple enough sounding problem, usually known as Cook's Theorem or the Cook-Levin Theorem, has eluded mathematicians and computer scientists for 40 years. A possible solution by Vinay Deolalikar in 2010, claiming to prove that P is not equal to NP (and thus such insolulable-but-easilychecked problems do exist), has attracted much attention but has not as yet been fully accepted by the computer science community. LIST OF IMPORTANT MATHEMATICIANS This is a chronological list of some of the most important mathematicians in history and their major achievments, as well as some very early achievements in mathematics for which individual contributions can not be acknowledged. Where the mathematicians have individual pages in this website, these pages are linked; otherwise more information can usually be obtained from the general page relating to the particular period in history, or from the list of sources used. A more detailed and comprehensive mathematical chronology can be found athttp://wwwgroups.dcs.st-and.ac.uk/~history/Chronology/full.html. Date 35000 BC 3100 BC 2700 BC 2600 BC 20001800 BC 18001600 BC 1650 BC 1200 BC 1200900 BC 800400 BC 650 BC 624546 BC 570495 BC 500 Thales Name Nationality African Major Achievements First notched tally bones Sumerian Earliest documented counting and measuring system Egyptian Earliest fully-developed base 10 number system in use Sumerian Multiplication tables, geometrical exercises and division problems Earliest papyri showing numeration system and basic arithmetic Egyptian Babylonian Clay tablets dealing with fractions, algebra and equations Egyptian Rhind Papyrus (instruction manual in arithmetic, geometry, unit fractions, etc) First decimal numeration system with place value concept Chinese Indian Early Vedic mantras invoke powers of ten from a hundred all the way up to a trillion Indian “Sulba Sutra” lists several Pythagorean triples and simplified Pythagorean theorem for the sides of a square and a rectangle, quite accurate approximation to √2 Lo Shu order three (3 x 3) “magic square” in which each row, column and diagonal sums to 15 Early developments in geometry, including work on similar and right triangles Chinese Greek Pythagoras Greek Expansion of geometry, rigorous approach building from first principles, square and triangular numbers, Pythagoras’ theorem Discovered potential existence of irrational numbers while Hippasus Greek volume of a cone Plato Greek Platonic solids. especially on cones and conic sections (ellipse. proofs and theorems including Euclid’s Theorem on infinitude of primes Formulas for areas of regular shapes. parabola. influential teacher and popularizer of mathematics. hyperbola) Chinese “Nine Chapters on the Mathematical Art”. use of axioms and postulates. Lune of Hippocrates Democritus Greek Developments in geometry and fractions. comparison of infinities 287212 BC 276195 BC 262190 BC 200 BC Archimedes Greek Eratosthenes Greek “Sieve of Eratosthenes” method for identifying prime numbers Apollonius of Perga Greek Work on geometry. insistence on rigorous proof and logical methods Method for rigorously proving statements about areas and volumes by successive approximations Eudoxus of Cnidus Greek Aristotle Greek Development and standardization of logic (although not then considered part of mathematics) and deductive reasoning Euclid Greek Definitive statement of classical (Euclidean) geometry. statement of the Three Classical Problems. many formulas. including guide to how to solve equations using sophisticated matrix-based methods Develop first detailed trigonometry tables 190120 BC 36 BC Hipparchus Greek Mayan Pre-classic Mayans developed the concept of zero by at least .BC 490430 BC 470410 BC 460370 BC 428348 BC 410355 BC 384322 BC 300 BC Zeno of Elea Greek trying to calculate the value of √2 Describes a series of paradoxes concerning infinity and infinitesimals Hippocrates of Chios Greek First systematic compilation of geometrical knowledge. “method of exhaustion” for approximating areas and value of π. leaving roots unevaluated. Heron’s Method for iteratively computing a square root Develop even more detailed trigonometry tables 90-168 Ptolemy AD 200 AD 200 AD 200284 AD 220280 AD Diophantus Sun Tzu Chinese First definitive statement of Chinese Remainder Theorem Indian Refined and perfected decimal place value number system Greek Diophantine Analysis of complex algebraic problems. negative roots of quadratic equations. accurate approximation for π (and recognition that π is an irrational number) Basic mathematical rules for dealing with zero (+. remarkably accurate approximation of the sine function Advocacy of the Hindu numerals 1 . solutions to simultaneous quadratic equations. complete and accurate sine and versine tables. solution of quadratic equations with two unknowns First to write numbers in Hindu-Arabic decimal system with a circle for zero. calculated value of π correct to five decimal places.and x). cosines.Persian Karaji First use of proof by mathematical induction.this time 10-70 AD Heron (or Hero) Greek of Alexandria Greek/Egyptian Heron’s Formula for finding the area of a triangle from its side lengths. including first real use of sines. to find rational solutions to equations with several unknowns Liu Hui Chinese Solved linear equations using a matrices (similar to Gaussian elimination). including to prove the binomial theorem .9 and 0 in Islamic world. tangents to a circle 400 AD Indian 476550 AD Aryabhata Indian 598668 AD 600680 AD 780850 AD Brahmagupta Indian Bhaskara I Indian Muhammad Al. tangents and secants Definitions of trigonometric functions. solution of polynomial equations up to second degree Continued Archimedes' investigations of areas and volumes. early forms of integral and differential calculus “Surya Siddhanta” contains roots of modern trigonometry. including algebraic methods of “reduction” and “balancing”. negative numbers.Persian Khwarizmi 908946 AD 9531029 Ibrahim ibn Sinan Arabic Muhammad Al. foundations of modern algebra. inverse sines. . “Alhazen's problem”. such as for a time-speeddistance graph. factorization and combinatorial methods Use of infinite series of fractions to give an exact formula for π. noted existence of different sorts of cubic equations Established that dividing by zero yields infinity. sine formula and other trigonometric functions. cubic and quartic equations (including negative and irrational solutions) and to second order Diophantine equations. Tartaglia’s Triangle (earlier version of Pascal’s Triangle) Published solution of cubic and quartic equations (by Tartaglia Omar Khayyam Persian 11141185 Bhaskara II Indian 11701250 Leonardo of Pisa (Fibonacci) Italian 12011274 12021261 12381298 Nasir al-Din al. fifth and higher roots. found solutions to quadratic. formulated law of sines for plane triangles Solutions to quadratic. first to use fractional exponents.AD 9661059 AD 10481131 Ibn al-Haytham Persian/Arabic (Alhazen) Derived a formula for the sum of fourth powers using a readily generalizable method. advocacy of the use of the Hindu-Arabic numeral system in Europe. important step towards development of calculus System of rectangular coordinates. also introduced standard symbols for plus and minus Formula for solving all types of cubic equations. explored amicable numbers. also worked on infinite series Influential book on arithmetic. Fibonacci's identity (product of two sums of two squares is itself a sum of two squares) Developed field of spherical trigonometry.Persian Tusi Qin Jiushao Chinese Yang Hui Chinese 12671319 Kamal al-Din al-Farisi Persian 13501425 Madhava Indian 13231382 Nicole Oresme French 14461517 14991557 Luca Pacioli Italian Niccolò Fontana Tartaglia Italian 1501- Gerolamo Italian . geometry and book-keeping. Yang Hui’s Triangle (earlier version of Pascal’s Triangle of binomial co-efficients) Applied theory of conic sections to solve optical problems. involving first real use of complex numbers (combinations of real and imaginary numbers). cubic and higher power equations using a method of repeated approximations Culmination of Chinese “magic” squares. introduced some preliminary concepts of calculus Fibonacci Sequence of numbers. circles and triangles. established beginnings of link between algebra and geometry Generalized Indian methods for extracting square and cube roots to include fourth. transcendental curves Further developed infinitesimal calculus. Bernoulli Numbers sequence. introduced symbol ∞ for infinity. Pascal’s Triangle of binomial coefficients Development of infinitesimal calculus (differentiation and integration). greatly extending knowlege of number theory. Mersenne primes (prime numbers that are one less than a power of 2) Early development of projective geometry and “point at infinity”. acknowledged existence of imaginary numbers (based on √-1) Italian Devised formula for solution of quartic equations 15221565 15501617 15881648 Lodovico Ferrari John Napier British Invention of natural logarithms. Napier’s Bones tool for lattice multiplication Clearing house for mathematical thought during 17th Century. solved linear equations using a matrix Helped to consolidate infinitesimal calculus. developed a technique for solving separable differential equations. Two-Square Thereom and Last Theorem).1576 Cardano and Ferrari). functions for curve of fastest descent Marin Mersenne French 15911661 15961650 Girard Desargues René Descartes French French 15981647 16011665 Bonaventura Cavalieri Pierre de Fermat Italian French 16161703 John Wallis British 16231662 16431727 Blaise Pascal French Isaac Newton British 16461716 Gottfried Leibniz German 16541705 Jacob Bernoulli Swiss 16671748 Johann Bernoulli Swiss . originated idea of number line. also contributed to probability theory Contributed towards development of calculus. added a theory of permutations and combinations to probability theory. generalized binomial theorem. also practical calculating machine using binary system (forerunner of the computer). perspective theorem Development of Cartesian coordinates and analytic geometry (synthesis of geometry and algebra). popularized the use of the decimal point. infinite power series Independently developed infinitesimal calculus (his calculus notation is still used). developed standard notation for powers Pioneer (with Fermat) of probability theory. laid ground work for almost all of classical mechanics. including the “calculus of variation”. also credited with the first use of superscripts for powers or exponents “Method of indivisibles” paved way for the later development of infinitesimal calculus Discovered many new numbers patterns and theorems (including Little Theorem. least squares method for curve-fitting and linear regression. orthographic projection 16901764 17071783 Christian Goldbach German Leonhard Euler Swiss 17281777 Johann Lambert Swiss 17361813 Joseph Louis Lagrange Italian/French 17461818 17491827 Gaspard Monge Pierre-Simon Laplace French French Celestial mechanics translated geometric study of classical mechanics to one based on calculus. introduced hyperbolic functions into trigonometry. prime number theorem. Gaussian curvature Early pioneer of mathematical analysis. Gaussian distribution. Lagrange’s theorem of finite groups. proved numerous theorems. construction of heptadecagon. made conjectures on nonEuclidean space and hyperbolic triangles Comprehensive treatment of classical and celestial mechanics. Möbius inversion formula 17521833 Adrien-Marie Legendre French 17681830 17771825 Joseph Fourier French Carl Friedrich Gauss German 17891857 Augustin-Louis French Cauchy 17901868 August Ferdinand Möbius German . Cauchy's theorem (a fundamental theorem of group theory) Möbius strip (a two-dimensional surface with only one side). standardized mathematical notation and wrote many influential textbooks Rigorous proof that π is irrational. four-square theorem. Gaussian error curve.(brachistochrone) and catenary curve 16671754 Abraham de Moivre French De Moivre's formula. first statement of the formula for the normal distribution curve. least squares approximation method. exposition of complex numbers. probability theory Goldbach Conjecture. reformulated and proved theorems of calculus in a rigorous manner. Möbius configuration. Möbius transform (number theory). mathematical analysis. elliptic functions Studied periodic functions and infinite sums in which the terms are trigonometric functions (Fourier series) Pattern in occurrence of prime numbers. mean value theorem Inventor of descriptive geometry. Gaussian function. Goldbach-Euler Theorem on perfect powers Made important contributions in almost all fields and found unexpected links between different fields. quadratic reciprocity law. belief in scientific determinism Abstract algebra. Bayesian interpretation of probability. Fundamental Theorem of Algebra. non-Euclidean geometry. Möbius function. calculus of variations. pioneered new methods. Möbius transformations. development of analytic geometry. etc Devised Boolean algebra (using operators AND. Galois theory. higher dimensional geometry. matrix algebra. Riemannian geometry (differential geometry in multiple dimensions). led to the development of computer science Discovered a continuous function with no derivative. theory of invariants. abelian groups. complex manifold theory. Riemann Hypothesis Defined some important concepts of set theory such as similar sets and infinite sets. ring theory. extended Hamilton's quaternions to create octonions Non-Euclidean elliptic geometry. group theory. group theory. laid groundwork for abstract algebra. forerunner of programmable computer. proposed Dedekind cut (now a standard definition of the real numbers) Introduced Venn diagrams into set theory (now a ubiquitous British 17921856 18021829 18021860 18041851 18051865 18111832 Nikolai Lobachevsky Niels Henrik Abel János Bolyai Russian Norwegian Hungarian Carl Jacobi German William Hamilton Irish Évariste Galois French 18151864 George Boole British 18151897 Karl Weierstrass German 18211895 Arthur Cayley British 18261866 Bernhard Riemann German 18311916 Richard Dedekind German 1834- John Venn British . Developed theory of hyperbolic geometry and curved spaces independendly of Bolyai Proved impossibility of solving quintic equations. abelian categories. starting point of modern mathematical logic. determinants and matrices Theory of quaternions (first example of a non-commutative algebra) Proved that there is no general algebraic method for solving polynomial equations of degree greater than four. abelian variety Explored hyperbolic geometry and curved spaces independently of Lobachevsky Important contributions to analysis. zeta function. pioneer in development of mathematical analysis Pioneer of modern group theory. OR and NOT). theory of periodic and elliptic functions. advancements in calculus of variations. theory of higher singularities. reformulated calculus in a more rigorous fashion.17911858 17911871 George Peacock Charles Babbage British Inventor of symbolic algebra (early attempt to place algebra on a strictly logical basis) Designed a "difference engine" that could automatically perform computations based on instructions stored on cards or tape. Riemann surfaces. continuous symmetry. developer of mathematical logic and set theory notation. work on group theory and function theory Partial solution to “three body problem”. contributed to modern method of mathematical induction Co-wrote “Principia Mathematica” (attempt to ground mathematics on logic) 23 “Hilbert problems”. first rigorous treatment of the ideas of functions and variables in logic. investigated iterative and recursive processes Proved several theorems marking breakthroughs in topology (including fixed point theorem and topological invariance of dimension) 18481925 Gottlob Frege German 18491925 Felix Klein German 18541912 Henri Poincaré French 18581932 Giuseppe Peano Italian 18611947 18621943 Alfred North Whitehead David Hilbert British German 18641909 Hermann Minkowski German 18721970 18771947 Bertrand Russell G. foundations of modern chaos theory. Hardy British British 18781929 18811966 Pierre Fatou French L. finiteness theorem. major contributor to study of the foundations of mathematics Klein bottle (a one-sided closed surface in four-dimensional space).H. theory of types Progress toward solving Riemann hypothesis (proved infinitely many zeroes on the critical line). logic and statistics) Applied algebra to geometric theory of differential equations. encouraged new tradition of pure mathematics in Britain. “Entscheidungsproblem“ (decision problem). Cantor's theorem (which implies the existence of an “infinity of infinities”) One of the founders of modern logic. rigorous treatment of the notion of infinity and transfinite numbers. Hilbert space. Poincaré conjecture Peano axioms for natural numbers. Brouwer Dutch .E.1923 18421899 18451918 Marius Sophus Norwegian Lie Georg Cantor German tool in probability. Erlangen Program to classify geometries by their underlying symmetry groups. formalism Geometry of numbers (geometrical method in multidimensional space for solving number theory problems). Lie groups of transformations Creator of set theory. Minkowski space-time Russell’s paradox.J. co-wrote “Principia Mathematica” (attempt to ground mathematics on logic). developed modern axiomatic approach to mathematics. extended theory of mathematical topology. taxicab numbers Pioneer in field of complex analytic dynamics. contributions to algebraic topology.e. classical analysis. provided insight into complex systems in daily life such as economics.000 theorems. computer plottings of Mandelbrot and Julia sets Mathematical structuralist. identities and equations. and mock theta functions Developed complex dynamics. theory of schemes. partition function and its asymptotics. revolutionary advances in algebraic geometry. Julia set formula 18931978 19031957 19061978 Gaston Julia French John von Neumann Kurt Gödel Hungarian/ American Austria Pioneer of game theory. differential geometry and partial differential equations. number theory. etc Work in game theory. number theory. set theory and probability theory Pioneer in modern chaos theory. computing and military Proved that continuum hypothesis could be both true and not true (i. logic and set theory Theorems allowed connections between algebraic geometry and number theory. Turing test of artificial intelligence Set and solved many problems in combinatorics. Weil conjectures (partial proof of Riemann hypothesis for local zeta functions). founding member of influential Bourbaki group Breaking of the German enigma code. category theory. geometry and (especially) recreational mathematics. fractals. Lorenz oscillator.18871920 Srinivasa Ramanujan Indian Proved over 3. work in quantum and nuclear physics Incompleteness theorems (there can be solutions to mathematical problems which are true but which can never be proved). Lorenz attractor. design model for modern computer architecture. independent from Zermelo-Fraenkel set theory) Important contributions to game theory. coined term “butterfly effect” Work on decision problems and Hilbert's tenth problem. Robinson hypothesis Mandelbrot set fractal. notably with the invention of the cellular automaton called the "Game of Life" 19061998 André Weil French 19121954 19131996 Alan Turing British Paul Erdös Hungarian 19172008 19191985 19242010 1928- Edward Lorenz American Julia Robinson American Benoît Mandelbrot Alexander Grothendieck French French 1928- John Nash American 19342007 1937- Paul Cohen American John Horton Conway British . number theory. graph theory. group theory. Turing machine (logical forerunner of computer). including on highly composite numbers. approximation theory. Gödel numbering. contributions to Riemannian geometry and geometric topology 1953- Andrew Wiles British 1966- Grigori Perelman Russian .1947- Yuri Matiyasevich Russian Final proof that Hilbert’s tenth problem is impossible (there is no general method for determining whether Diophantine equations have a solution) Finally proved Fermat’s Last Theorem for all numbers (by proving the Taniyama-Shimura conjecture for semistable elliptic curves) Finally proved Poincaré Conjecture (by proving Thurston's geometrization conjecture).