18th Century Mathematics: Bernoulli Brothers & Euler PDF

**18TH CENTURY MATHEMATICS - BERNOULLI BROTHERS** --------------------------------------------------------------------------------------- http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Jacob (1654-1705) and Johann Bernoulli (1667-1748)\ *![http://www.storyofmathematics.c...

**18TH CENTURY MATHEMATICS - BERNOULLI BROTHERS** --------------------------------------------------------------------------------------- http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Jacob (1654-1705) and Johann Bernoulli (1667-1748)\ * --------------------------------------------------------------------------------------- Unusually in the history of mathematics, a single family, the Bernoulli's, produced half a dozen outstanding mathematicians over a couple of generations at the end of the 17th and start of the 18th Century. The Bernoulli family was a prosperous family of traders and scholars from the free city of Basel in Switzerland, which at that time was the great commercial hub of central Europe.The brothers, Jacob and Johann Bernoulli, however, flouted their father\'s wishes for them to take over the family spice business or to enter respectable professions like medicine or the ministry, and began studying mathematics together. After Johann graduated from Basel University, the two developed a rather jealous and competitive relationship. Johann in particular was jealous of the elder Jacob\'s position as professor at Basel University, and the two often attempted to outdo each other. After Jacob\'s early death from **tuberculosis**, Johann took over his brother\'s position, one of his young students being the great Swiss mathematician [[Leonhard Euler]](http://www.storyofmathematics.com/18th_euler.html). However, Johann merely shifted his jealousy toward his own talented son, Daniel (at one point, Johann published a book based on Daniel\'s work, even changing the date to make it look as though his book had been published before his son\'s). Johann received a taste of his own medicine, though, when his student **Guillaume de l\'Hôpital** published a book in his own name consisting almost entirely of Johann\'s lectures, including his now famous rule about 0 ÷ 0 (a problem which had dogged mathematicians since [[Brahmagupta]](http://www.storyofmathematics.com/indian_brahmagupta.html)\'s initial work on the rules for dealing with zero back in the 7th Century). This showed that 0 ÷ 0 does not equal zero, does not equal 1, does not equal infinity, and is not even undefined, but is \"indeterminate\" (meaning it could equal any number). The rule is still usually known as l\'Hôpital\'s Rule, and not Bernoulli\'s Rule. Despite their competitive and combative personal relationship, though, the brothers both had a clear aptitude for mathematics at a high level, and constantly challenged and inspired each other. They established an early correspondence with [[Gottfried Leibniz]](http://www.storyofmathematics.com/17th_leibniz.html), and were among the first mathematicians to not only study and understand infinitesimal calculus but to apply it to various problems. They became instrumental in disseminating the newly-discovered knowledge of calculus, and helping to make it the cornerstone of mathematics it has become today. --------------------------------------------------------------------------------------------------- http://www.storyofmathematics.com/images/transparent\_blank.gif*\ The Bernoulli's first derived the brachistrochrone curve, using his calculus of variation method\ * --------------------------------------------------------------------------------------------------- But they were more than just disciples of [[Leibniz]](http://www.storyofmathematics.com/17th_leibniz.html), and they also made their own important contributions. One well known and topical problem of the day to which they applied themselves was that of designing a sloping ramp which would allow a ball to roll from the top to the bottom in the fastest possible time. The brachistochrone problem was posed by [[Johann Bernoulli]](https://www-history.mcs.st-and.ac.uk/Mathematicians/Bernoulli_Johann.html) in *Acta Eruditorum* [[Ⓣ]](javascript:trans('%20Learned%20articles',73)) in June 1696. He introduced the problem as follows:- Johann Bernoulli demonstrated through calculus that neither a straight ramp or a curved ramp with a very steep initial slope were optimal, but actually a less steep curved ramp known as a brachistochrone curve (a kind of upside-down cycloid, similar to the path followed by a point on a moving bicycle wheel) is the curve of fastest descent. This application was an example of the "calculus of variations", a generalization of infinitesimal calculus that the Bernoulli brothers developed together, and has since proved useful in fields as diverse as engineering, financial investment, architecture and construction, and even space travel. Johann also derived the equation for a catenary curve, such as that formed by a chain hanging between two posts, a problem presented to him by his brother Jacob. --------------------------------------------------------------------------------------- http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Bernoulli Numbers\ * --------------------------------------------------------------------------------------- Jacob Bernoulli's book "The Art of Conjecture", published posthumously in 1713, consolidated existing knowledge on probability theory and expected values, as well as adding personal contributions, such as his theory of permutations and combinations, Bernoulli trials and Bernoulli distribution, and some important elements of number theory, such as the Bernoulli Numbers sequence. He also published papers on transcendental curves, and became the first person to develop the technique for solving separable differential equations (the set of non-linear, but solvable, differential equations are now named after him). He invented polar coordinates (a method of describing the location of points in space using angles and distances) and was the first to use the word "integral" to refer to the area under a curve. Jacob Bernoulli also discovered the appropximate value of the irrational number *e* while exploring the compound interest on loans. When compounded at 100% interest annually, \$1.00 becomes \$2.00 after one year; when compounded semi-annually it ppoduces \$2.25; compounded quarterly \$2.44; monthly \$2.61; weekly \$2.69; daily \$2.71; etc. If it were to be compounded continuously, the \$1.00 would tend towards a value of \$2.7182818\... after a year, a value which became known as *e*. Alegbraically, it is the value of the infinite series (1 + ^1^⁄~1~)^1^.(1 + ^1^⁄~2~)^2^.(1 + ^1^⁄~3~)^3^.(1 + ^1^⁄~4~)^4^\... Johann's sons Nicolaus, Daniel and Johann II, and even his grandchildren Jacob II and Johann III, were all accomplished mathematicians and teachers. Daniel Bernoulli, in particular, is well known for his work on fluid mechanics (especially Bernoulli's Principle on the inverse relationship between the speed and pressure of a fluid or gas), as much as for his work on probability and statistics. Bernoulli Numbers **18TH CENTURY MATHEMATICS - EULER** ---------------------------------------------------------------------------------------- *\ Leonhard Euler (1707-1783)\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- Leonhard Euler was one of the giants of 18th Century mathematics. Like the [[Bernoulli's]](http://www.storyofmathematics.com/18th_bernoulli.html), he was born in Basel, Switzerland, and he studied for a while under [[Johann Bernoulli]](http://www.storyofmathematics.com/18th_bernoulli.html) at Basel University. But, partly due to the overwhelming dominance of the [[Bernoulli]](http://www.storyofmathematics.com/18th_bernoulli.html) family in Swiss mathematics, and the difficulty of finding a good position and recognition in his hometown, he spent most of his academic life in Russia and Germany, especially in the burgeoning St. Petersburg of Peter the Great and Catherine the Great. Despite a long life and thirteen children, Euler had more than his fair share of tragedies and deaths, and even his blindness later in life did not slow his prodigious output - his collected works comprise nearly 900 books and, in the year 1775, he is said to have produced on average one mathematical paper every week - **as he compensated for it with his mental calculation skills and photographic memory** (for example, he could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last). Today, **Euler is considered one of the greatest mathematicians of all time**. His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights and measures and even the theory of music. ---------------------------------------------------------------------------------------- *\ Mathematical notation created or popularized by Euler\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- Much of the notation used by mathematicians today - including *e*, *i*,*f*(*x*), ∑, and the use of *a*, *b* and *c* as constants and *x*, *y* and *z* as unknowns - was either created, popularized or standardized by Euler. His efforts to standardize these and other symbols (including π and the trigonometric functions) helped to internationalize mathematics and to encourage collaboration on problems. He even managed to combine several of these together in an amazing feat of mathematical alchemy to produce one of the most beautiful of all mathematical equations, *e^i^*^π^ = -1, sometimes known as Euler's Identity. This equation combines arithmetic, calculus, trigonometry and complex analysis into what has been called \"the most remarkable formula in mathematics\", \"uncanny and sublime\" and \"filled with cosmic beauty\", among other descriptions. Another such discovery, often known simply as Euler's Formula, is *e^ix^* = cos*x* + *i*sin*x*. In fact, in a recent poll of mathematicians, three of the top five most beautiful formulae of all time were Euler's. He seemed to have an instinctive ability to demonstrate the deep relationships between trigonometry, exponentials and complex numbers. The discovery that initially sealed Euler's reputation was announced in 1735 and concerned the calculation of infinite sums. It was called the Basel problem after the [[Bernoulli's]](http://www.storyofmathematics.com/18th_bernoulli.html) had tried and failed to solve it, and asked what was the precise sum of the of the reciprocals of the squares of all the natural numbers to infinity i.e. ^1^⁄~1~^2^+ ^1^⁄~2~^2^ + ^1^⁄~3~^2^ + ^1^⁄~4~^2^ \... (a zeta function using a zeta constant of 2). Euler's friend [[Daniel Bernoulli]](http://www.storyofmathematics.com/18th_bernoulli.html) had estimated the sum to be about 1^3^⁄~5~, but Euler's superior method yielded the exact but rather unexpected result of π^2^⁄~6~. He also showed that the infinite series was equivalent to an infinite product of prime numbers, an identity which would later inspire [[Riemann]](http://www.storyofmathematics.com/19th_riemann.html)'s investigation of complex zeta functions. ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ The Seven Bridges of Königsberg Problem\ * --------------------------------------------------------------------------------------- Also in 1735, Euler solved an intransigent mathematical and logical problem, known as the Seven Bridges of Königsberg Problem, which had perplexed scholars for many years, and in doing so laid the foundations of graph theory and presaged the important mathematical idea of topology. The city of Königsberg in Prussia (modern-day Kaliningrad in Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem was to find a route through the city that would cross each bridge once and only once. In fact, Euler proved that the problem has no solution, but in doing so he made the important conceptual leap of pointing out that the choice of route within each landmass is irrelevant and the only important feature is the sequence of bridges crossed. This allowed him to reformulate the problem in abstract terms, replacing each land mass with an abstract node and each bridge with an abstract connection. This resulted in a mathematical structure called a "graph", a pictorial representation made up of points (vertices) connected by non-intersecting curves (arcs), which may be distorted in any way without changing the graph itself. In this way, Euler was able to deduce that, because the four land masses in the original problem are touched by an odd number of bridges, the existence of a walk traversing each bridge once only inevitably leads to a contradiction. If Königsberg had had one fewer bridges, on the other hand, with an even number of bridges leading to each piece of land, then a solution would have been possible. ---------------------------------------------------------------------------------------- The Euler Characteristic *\ The Euler Characteristic\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- The list of theorems and methods pioneered by Euler is immense, and largely outside the scope of an entry-level study such as this, but mention could be made of just some of them:  the demonstration of geometrical properties such as Euler's Line and Euler's Circle;  the definition of the Euler Characteristic *χ* (chi) for the surfaces of polyhedra, whereby the number of vertices minus the number of edges plus the number of faces always equals 2 (see table at right);  a new method for solving quartic equations;  the Prime Number Theorem, which describes the asymptotic distribution of the prime numbers;  proofs (and in some cases disproofs) of some of Fermat's theorems and conjectures;  the discovery of over 60 amicable numbers (pairs of numbers for which the sum of the divisors of one number equals the other number), although some were actually incorrect;  a method of calculating integrals with complex limits (foreshadowing the development of modern complex analysis);  the calculus of variations, including its best-known result, the Euler-Lagrange equation; a proof of the infinitude of primes, using the divergence of the harmonic series;  the integration of [[Leibniz]](http://www.storyofmathematics.com/17th_leibniz.html)\'s differential calculus with [[Newton]](http://www.storyofmathematics.com/17th_newton.html)\'s Method of Fluxions into a form of calculus we would recognize today, as well as the development of tools to make it easier to apply calculus to real physical problems;  etc, etc. In 1766, Euler accepted an invitation from Catherine the Great to return to the St. Petersburg Academy, and spent the rest of his life in Russia. However, his second stay in the country was marred by tragedy, including a fire in 1771 which cost him his home (and almost his life), and the loss in 1773 of his dear wife of 40 years, Katharina. He later married Katharina\'s half-sister, Salome Abigail, and this marriage would last until his death from a brain hemorrhage in 1783.  **18TH CENTURY MATHEMATICS** ---------------------------------------------------------------------------------------- Calculus of variations *\ Calculus of variations\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- Most of the late 17th Century and a good part of the early 18th were taken up by the work of disciples of [[Newton]](http://www.storyofmathematics.com/17th_newton.html)and [[Leibniz]](http://www.storyofmathematics.com/17th_leibniz.html), who applied their ideas on calculus to solving a variety of problems in physics, astronomy and engineering. The period was dominated, though, by one family, the [[Bernoulli]](http://www.storyofmathematics.com/18th_bernoulli.html)'s of Basel in Switzerland, which boasted two or three generations of exceptional mathematicians, particularly the brothers, Jacob and Johann. **They were largely responsible for further developing [[Leibniz]](http://www.storyofmathematics.com/17th_leibniz.html)'s infinitesimal calculus - paricularly through the generalization and extension of calculus known as the \"calculus of variations\" - as well as [[Pascal]](http://www.storyofmathematics.com/17th_pascal.html) and[[Fermat]](http://www.storyofmathematics.com/17th_fermat.html)'s probability and number theory**. Basel was also the home town of the greatest of the 18th Century mathematicians, [[Leonhard Euler]](http://www.storyofmathematics.com/18th_euler.html), although, partly due to the difficulties in getting on in a city dominated by the[[Bernoulli]](http://www.storyofmathematics.com/18th_bernoulli.html) family, [[Euler]](http://www.storyofmathematics.com/18th_euler.html) spent most of his time abroad, in Germany and St. Petersburg, Russia. He excelled in all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, and was able to find unexpected links between the different fields. He proved numerous theorems, pioneered new methods, standardized mathematical notation and wrote many influential textbooks throughout his long academic life. In a letter to [[Euler]](http://www.storyofmathematics.com/18th_euler.html) in 1742, the German mathematician Christian Goldbach proposed the Goldbach Conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes (e.g. 4 = 2 + 2; 8 = 3 + 5; 14 = 3 + 11 = 7 + 7; etc) or, in another equivalent version, every integer greater than 5 can be expressed as the sum of three primes. Yet another version is the so-called "weak" Goldbach Conjecture, that all odd numbers greater than 7 are the sum of three odd primes. They remain among the oldest unsolved problems in number theory (and in all of mathematics), although the weak form of the conjecture appears to be closer to resolution than the strong one. Goldbach also proved other theorems in number theory such as the Goldbach-Euler Theorem on perfect powers. Despite [[Euler]](http://www.storyofmathematics.com/18th_euler.html)'s and the [[Bernoullis]](http://www.storyofmathematics.com/18th_bernoulli.html)' dominance of 18th Century mathematics, many of the other important mathematicians were from France. In the early part of the century, Abraham de Moivre is perhaps best known for de Moivre\'s formula, (cos*x* + *i*sin*x*)*^n^* = cos(*nx*) + *i*sin(*nx*), which links complex numbers and trigonometry. But he also generalized [[Newton]](http://www.storyofmathematics.com/17th_newton.html)'s famous binomial theorem into the multinomial theorem, pioneered the development of analytic geometry, 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. France became even more prominent towards the end of the century, and a handful of late 18th Century French mathematicians in particular deserve mention at this point, beginning with "the three L's". Joseph Louis Lagrange collaborated with [[Euler]](http://www.storyofmathematics.com/18th_euler.html) in an important joint work on the calculus of variation, but he also contributed to differential equations and number theory, and he is usually credited with originating the theory of groups, which would become so important in [[19th]](http://www.storyofmathematics.com/19th.html) and [[20th Century]](http://www.storyofmathematics.com/20th.html) mathematics. His name is given an early theorem in group theory, 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. ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Lagrange's Mean value Theorem\ * --------------------------------------------------------------------------------------- Lagrange is also credited with the four-square theorem, that any natural number can be represented as the sum of four squares (e.g. 3 = 1^2^ + 1^2^ + 1^2^ + 0^2^; 31 = 5^2^ + 2^2^ + 1^2^ + 1^2^; 310 = 17^2^ + 4^2^ + 2^2^ + 1^2^; etc), as well as another theorem, confusingly also known as Lagrange's Theorem or Lagrange's Mean Value Theorem, which states that, given a section of a smooth continuous (differentiable) curve, there is at least one point on that section at which the derivative (or slope) of the curve is equal (or parallel) to the average (or mean) derivative of the section. Lagrange's 1788 treatise on analytical mechanics offered the most comprehensive treatment of classical mechanics since [[Newton]](http://www.storyofmathematics.com/17th_newton.html), and formed a basis for the development of mathematical physics in the [[19th Century]](http://www.storyofmathematics.com/19th.html). Pierre-Simon Laplace, sometimes referred to as "the French [[Newton]](http://www.storyofmathematics.com/17th_newton.html)", was an important mathematician and astronomer, whose monumental work "Celestial Mechanics" translated the geometric study of classical mechanics to one based on calculus, opening up a much broader range of problems. Although his early work was mainly on differential equations and finite differences, he was already starting to think about the mathematical and philosophical concepts of probability and statistics in the 1770s, and he developed his own version of the so-called Bayesian interpretation of probability independently of Thomas Bayes. Laplace is well known for his belief in complete scientific determinism, and he maintained that there should be a set of scientific laws that would allow us - at least in principle - to predict everything about the universe and how it works. ---------------------------------------------------------------------------------------- The first six Legendre polynomials (solutions to Legendre's differential equation) *\ The first six Legendre polynomials (solutions to Legendre's differential equation)\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- Adrien-Marie Legendre also made important contributions to statistics, number theory, abstract algebra and mathematical analysis in the late 18th and early 19th Centuries, athough much of his work (such as the least squares method for curve-fitting and linear regression, the quadratic reciprocity law, the prime number theorem and his work on elliptic functions) was only brought to perfection - or at least to general notice - by others, particularly[[Gauss]](http://www.storyofmathematics.com/19th_gauss.html). His "Elements of Geometry", a re-working of [[Euclid]](http://www.storyofmathematics.com/hellenistic_euclid.html)'s book, became the leading geometry textbook for almost 100 years, and his extremely accurate measurement of the terrestrial meridian inspired the creation, and almost universal adoption, of the metric system of measures and weights. Yet another Frenchman, Gaspard Monge was the inventor of descriptive geometry, a clever method of representing three-dimensional objects by projections on the two-dimensional plane using a specific set of procedures, a technique which would later become important in the fields of engineering, architecture and design. His orthographic projection became the graphical method used in almost all modern mechanical drawing. After many centuries of increasingly accurate approximations, Johann Lambert, a Swiss mathematician and prominent astronomer, finally provided a rigorous proof in 1761 that π is irrational, i.e. 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, although the obsession with obtaining more and more accurate approximations continues to this day. (Over a hundred years later, in 1882, Ferdinand von Lindemann would prove that π is also transcendental, i.e. it cannot be the root of any polynomial equation with rational coefficients). 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. **19TH CENTURY MATHEMATICS - GALOIS** ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Évariste Galois (1811-1832)\ * --------------------------------------------------------------------------------------- Évariste Galois was radical republican and something of a romantic figure in French mathematical history. He died in a duel at the young age of 20, but the work he published shortly before his death made his name in mathematical circles, and would go on to allow proofs by later mathematicians of problems which had been impossible for many centuries. It also laid the groundwork for many later developments in mathematics, particularly the beginnings of the important fields of abstract algebra and group theory. Despite his lacklustre performance at school (he twice failed entrance exams to the École Polytechnique), the young Galois devoured the work of Legendre and Lagrange in his spare time. At the tender age of 17, he began making fundamental discoveries in the theory of polynomial equations (equations constructed from variables and constants, using only the operations of addition, subtraction, multiplication and non-negative whole-number exponents, such as *x*^2^ - 4*x* + 7 = 0). He effectively proved that there can be no general formula for solving quintic equations (polynomials including a term of *x*^5^), just as the young Norwegian Niels Henrik Abel had a few years earlier, although by a different method. But he was also able to prove the more general, and more powerful, idea that there is no general algebraic method for solving polynomial equations of any degree greater than four. ---------------------------------------------------------------------------------------- An example of Galois' rather undisciplined notes *\ An example of Galois' rather undisciplined notes\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- Galois achieved this general proof by looking at whether or not the "permutation group" of its roots (now known as its Galois group) had a certain structure. He was the first to use the term "group" in its modern mathematical sense of a group of permutations (foreshadowing the modern field of group theory), and his fertile approach, now known as Galois theory, was adapted by later mathematicians to many other fields of mathematics besides the theory of equations. Galois' breakthrough in turn led to definitive proofs (or rather disproofs) later in the century of the so-called "Three Classical Problems" problems which had been first formulated by[[Plato]](http://www.storyofmathematics.com/greek_plato.html) and others back in ancient[[Greece]](http://www.storyofmathematics.com/greek.html): the doubling of the cube and the trisection of an angle (both were proved impossible in 1837), and the squaring of the circle (also proved impossible, in 1882). Galois was a hot-headed political firebrand (he was arrested several times for political acts), and his political affiliations and activities as a staunch republican during the rule of Louis-Philippe continually distracted him from his mathematical work. He was killed in a duel in 1832, under rather shady circumstances, but he had spent the whole of the previous night outlining his mathematical ideas in a detailed letter to his friend Auguste Chevalier, as though convinced of his impending death. Ironically, his young contemporary Abel also had a promising career cut short. He died in poverty of tubercolosis at the age of just 26, although his legacy lives on in the term "abelian" (usually written with a small \"a\"), which has since become commonplace in discussing concepts such as the abelian group, abelian category and abelian variety. **19TH CENTURY MATHEMATICS - BOLYAI AND LOBACHEVSKY** ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ János Bolyai (1802-1860) and Nikolai Lobachevsky (1792-1856)\ * --------------------------------------------------------------------------------------- János Bolyai was a Hungarian mathematician who spent most of his life in a little-known backwater of the Hapsburg Empire, in the wilds of the Transylvanian mountains of modern-day Romania, far from the mainstream mathematical communities of Germany, France and England. No original portrait of Bolyai survives, and the picture that appears in many encyclopedias and on a Hungarian postage stamp is known to be unauthentic. His father and teacher, Farkas Bolyai, was himself an accomplished mathematician and had been a student of the great German mathematician [[Gauss]](http://www.storyofmathematics.com/19th_gauss.html) for a time, but the cantankerous [[Gauss]](http://www.storyofmathematics.com/19th_gauss.html) refused to take on the young prodigy János as a student. So, he was forced to join the army in order to earn a living and support his family, although he persevered with his mathematics in his spare time. He was also a talented linguist, speaking nine foreign languages, including Chinese and Tibetan. ---------------------------------------------------------------------------------------- Euclid\'s parallel postulate *\ Euclid\'s parallel postulate\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- In particular, Bolyai became obsessed with [[Euclid]](http://www.storyofmathematics.com/hellenistic_euclid.html)\'s fifth postulate (often referred to as the parallel postulate), a fundamental principle of geometry for over two millennia, which essentially states that only one line can be drawn through a given point so that the line is parallel to a given line that does not contain the point, along with its corollary that the interior angles of a triangle sum to 180° or two right angles. In fact, he became obsessed to such an extent that his father warned him that it may take up all his time and deprive him of his \"health, peace of mind and happiness in life\", a tragic irony given the unfolding of subsequent events. Bolyai, however, persisted in his quest, and eventually came to the radical conclusion that it was in fact possible to have consistent geometries that were independent of the parallel postulate. In the early 1820s, Bolyai explored what he called "imaginary geometry" (now known as hyperbolic geometry), the geometry of curved spaces on a saddle-shaped plane, where the angles of a triangle did NOT add up to 180° and apparently parallel lines were NOT actually parallel. In curved space, the shortest distance between two points *a* and *b* is actually a curve, or geodesic, and not a straight line. Thus, the angles of a triangle in hyperbolic space sum to less than 180°, and two parallel lines in hyperbolic space actually diverge from each other. In a letter to his father, Bolyai marvelled, "Out of nothing I have created a strange new universe". Although it is easy to visualize a flat surface and a surface with positive curvature (e.g. a sphere, such as a the Earth), it is impossible to visualize a hyperbolic surface with negative curvature, other than just over a small localized area, where it would look like a saddle or a Pringle. So the very concept of a hyperbolic surface appeared to go against all sense of reality. It certainly represented a radical departure from Euclidean geometry, and the first step along the road which would lead to Einstein's Theory of Relativity among other applications (although it still fell well short of the multi-dimensional geometry which was to be later realized by[[Riemann]](http://www.storyofmathematics.com/19th_riemann.html)). Between 1820 and 1823, Bolyai prepared, but did not immediately publish, a treatise on a complete system of non-Euclidean geometry. His work was, however, only published in 1832, and then only a short exposition in the appendix of a textbook by his father. On reading this, [[Gauss]](http://www.storyofmathematics.com/19th_gauss.html) clearly recognized the genius of the younger Bolyai's ideas, but he refused to encourage the young man, and even tried to claim his ideas as his own. Further disheartened by the news that the Russian mathematician Lobachevski had published something quite similar two years before his own paper, Bolyai became a recluse and gradually went insane. He died in obscurity in 1860. Although he only ever published the 24 pages of the appendix, Bolyai left more than 20,000 pages of mathematical manuscripts when he died (including the development of a rigorous geometric concept of complex numbers as ordered pairs of real numbers). ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Hyperbolic Bolyai-Lobachevskian geometry\ * --------------------------------------------------------------------------------------- Completely independent from Bolyai, in the distant provincial Russian city of Kazan, Nikolai Ivanovich Lobachevsky had also been working, along very similar lines as Bolyai, to develop a geometry in which [[Euclid]](http://www.storyofmathematics.com/hellenistic_euclid.html)'s fifth postulate did not apply. His work on hyperbolic geometry was first reported in 1826 and published in 1830, although it did not have general circulation until some time later. This early non-Euclidean geometry is now often referred to as Lobachevskian geometry or Bolyai-Lobachevskian geometry, thus sharing the credit.[[Gauss]](http://www.storyofmathematics.com/19th_gauss.html)' claims to have originated, but not published, the ideas are difficult to judge in retrospect. Other much earlier claims are credited to the 11th Century Persian mathematician Omar Khayyam, and to the early 18th Century Italian priest Giovanni Saccheri, but their work was much more speculative and inconclusive in nature. Lobachevsky also died in poverty and obscurity, nearly blind and unable to walk. Among his other mathematical achievements, largely unknown during his lifetime, was the development of a method for approximating the roots of algebraic equations (a method now known as the Dandelin-Gräffe method, named after two other mathematicians who discovered it independently), and the definition of a function as a correspondence between two sets of real numbers (usually credited to Dirichlet, who gave the same definition independently soon after Lobachevsky).  **19TH CENTURY MATHEMATICS - GAUSS** ---------------------------------------------------------------------------------------- Carl Friedrich Gauss *\ Carl Friedrich Gauss (1777-1855)\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- Carl Friedrich Gauss is sometimes referred to as the \"Prince of Mathematicians\" and the \"greatest mathematician since antiquity\". He has had a remarkable influence in many fields of mathematics and science and is ranked as one of history\'s most influential mathematicians. Gauss was a child prodigy. There are many anecdotes concerning his precocity as a child, and he made his first ground-breaking mathematical discoveries while still a teenager. At just three years old, he corrected an error in his father payroll calculations, and he was looking after his father's accounts on a regular basis by the age of 5. At the age of 7, he is reported to have amazed his teachers by summing the integers from 1 to 100 almost instantly (having quickly spotted that the sum was actually 50 pairs of numbers, with each pair summing to 101, total 5,050). By the age of 12, he was already attending gymnasium and criticizing Euclid's geometry. Although his family was poor and working class, Gauss\' intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at 15, and then to the prestigious University of Göttingen (which he attended from 1795 to 1798). It was as a teenager attending university that Gauss discovered (or independently rediscovered) several important theorems. ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Graphs of the density of prime numbers\ * --------------------------------------------------------------------------------------- At 15, Gauss was the first to find any kind of a pattern in the occurrence of prime numbers, a problem which had exercised the minds of the best mathematicians since ancient times. Although the occurrence of prime numbers appeared to be almost competely random, Gauss approached the problem from a different angle by graphing the incidence of primes as the numbers increased. He noticed a rough pattern or trend: as the numbers increased by 10, the probability of prime numbers occurring reduced by a factor of about 2 (e.g. there is a 1 in 4 chance of getting a prime in the number from 1 to 100, a 1 in 6 chance of a prime in the numbers from 1 to 1,000, a 1 in 8 chance from 1 to 10,000, 1 in 10 from 1 to 100,000, etc). However, he was quite aware that his method merely yielded an approximation and, as he could not definitively prove his findings, and kept them secret until much later in life. ---------------------------------------------------------------------------------------- 17-sided heptadecagon constructed by Gauss *\ 17-sided heptadecagon constructed by Gauss\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- In Gauss's annus mirabilis of 1796, at just 19 years of age, he constructed a hitherto unknown regular seventeen-sided figure using only a ruler and compass, a major advance in this field since the time of [[Greek]](http://www.storyofmathematics.com/greek.html) mathematics, formulated his prime number theorem on the distribution of prime numbers among the integers, and proved that every positive integer is representable as a sum of at most three triangular numbers. Although he made contributions in almost all fields of mathematics, number theory was always Gauss' favourite area, and he asserted that "mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics". An example of how Gauss revolutionized number theory can be seen in his work with complex numbers (combinations of real and imaginary numbers). ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Representation of complex numbers\ * --------------------------------------------------------------------------------------- Gauss gave the first clear exposition of complex numbers and of the investigation of functions of complex variables in the early 19th Century. Although imaginary numbers involving *i*(the imaginary unit, equal to the square root of -1) had been used since as early as the [[16th Century]](http://www.storyofmathematics.com/16th.html) to solve equations that could not be solved in any other way, and despite [[Euler]](http://www.storyofmathematics.com/18th_euler.html)'s ground-breaking work on imaginary and complex numbers in the [[18th Century]](http://www.storyofmathematics.com/18th.html), there was still no clear picture of how imaginary numbers connected with real numbers until the early 19th Century. Gauss was not the first to intepret complex numbers graphically (Jean-Robert Argand produced his Argand diagrams in 1806, and the Dane Caspar Wessel had described similar ideas even before the turn of the century), but Gauss was certainly responsible for popularizing the practice and laos formally introduced the standard notation a + b*i* for complex numbers. As a result, the theory of complex numbers received a notable expansion, and its full potential began to be unleashed. At the age of just 22, he proved what is now known as the Fundamental Theorem of Algebra (although it was not really about algebra). The theorem states that every non-constant single-variable polynomial over the complex numbers has at least one root (although his initial proof was not rigorous, he improved on it later in life). What it also showed was that the field of complex numbers is algebraically \"closed\" (unlike real numbers, where the solution to a polynomial with real co-efficients can yield a solution in the complex number field). Then, in 1801, at 24 years of age, he published his book "Disquisitiones Arithmeticae", which is regarded today as one of the most influential mathematics books ever written, and which laid the foundations for modern number theory. Among many other things, the book contained a clear presentation of Gauss' method of modular arithmetic, and the first proof of the law of quadratic reciprocity (first conjectured by [[Euler]](http://www.storyofmathematics.com/18th_euler.html) and Legendre). ---------------------------------------------------------------------------------------- Line of best fit by Gauss' least squares method *\ Line of best fit by Gauss' least squares method\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- For much of his life, Gauss also retained a strong interest in theoretical astrononomy, and he held the post of Director of the astronomical observatory in Göttingen for many years. When the planetoid Ceres was in the process of being identified in the late 17th Century, Gauss made a prediction of its position which varied greatly from the predictions of most other astronomers of the time. But, when Ceres was finally discovered in 1801, it was almost exacly where Gauss had predicted. Although he did not explain his methods at the time, this was one of the first applications of the least squares approximation method, usually attributed to Gauss, although also claimed by the Frenchman Legendre. Gauss claimed to have done the logarithmic calculations in his head. As Gauss' fame spread, though, and he became known throughout Europe as the go-to man for complex mathematical questions, his character deteriorated and he became increasingly arrogant, bitter, dismissive and unpleasant, rather than just shy. There are many stories of the way in which Gauss had dismissed the ideas of young mathematicians or, in some cases, claimed them as his own. ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Gaussian, or normal, probability curve\ * --------------------------------------------------------------------------------------- In the area of probability and statistics, Gauss introduced what is now known as Gaussian distribution, the Gaussian function and the Gaussian error curve. He showed how probability could be represented by a bell-shaped or "normal" curve, which peaks around the mean or expected value and quickly falls off towards plus/minus infinity, which is basic to descriptions of statistically distributed data. He also made ths first systematic study of modular arithmetic - using integer division and the modulus - which now has applications in number theory, abstract algebra, computer science, cryptography, and even in visual and musical art. While engaged on a rather banal surveying job for the Royal House of Hanover in the years after 1818, Gauss was also looking into the shape of the Earth, and starting to speculate on revolutionary ideas like shape of space itself. This led him to question one of the central tenets of the whole of mathematics, Euclidean geometry, which was clearly premised on a flat, and not a curved, universe. He later claimed to have considered a non-Euclidean geometry (in which [[Euclid]](http://www.storyofmathematics.com/hellenistic_euclid.html)\'s parallel axiom, for example, does not apply), which was internally consistent and free of contradiction, as early as 1800. Unwilling to court controversy, however, Gauss decided not to pursue or publish any of his avant-garde ideas in this area, leaving the field open to [[Bolyai and Lobachevsky]](http://www.storyofmathematics.com/19th_bolyai.html), although he is still considered by some to be a pioneer of non-Euclidean geometry. ---------------------------------------------------------------------------------------- Gaussian curvature *\ Gaussian curvature\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- The Hanover survey work also fuelled Gauss\' interest in differential geometry (a field of mathematics dealing with curves and surfaces) and what has come to be known as Gaussian curvature (an intrinsic measure of curvature, dependent only on how distances are measured on the surface, not on the way it is embedded in space). All in all, despite the rather pedestrian nature of his employment, the responsibilities of caring for his sick mother and the constant arguments with his wife Minna (who desperately wanted to move to Berlin), this was a very fruitful period of his academic life, and he published over 70 papers between 1820 and 1830. Gauss' achievements were not limited to pure mathematics, however. During his surveying years, he invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances to mark positions in a land survey. In later years, he collaborated with Wilhelm Weber on measurements of the Earth\'s magnetic field, and invented the first electric telegraph. In recognition of his contributions to the theory of electromagnetism, the international unit of magnetic induction is known as the gauss. **19TH CENTURY MATHEMATICS - POINCARÉ** ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Henri Poincaré (1854-1912)\ * --------------------------------------------------------------------------------------- Paris was a great centre for world mathematics towards the end of the 19th Century, and Henri Poincaré was one of its leading lights in almost all fields - geometry, algebra, analysis - for which he is sometimes called the "Last Universalist". Even as a youth at the Lycée in Nancy, he showed himself to be a polymath, and he proved to be one of the top students in every topic he studied. He continued to excel after he entered the École Polytechnique to study mathematics in 1873, and, for his doctoral thesis, he devised a new way of studying the properties of differential equations. Beginning in 1881, he taught at the Sorbonne in Paris, where he would spend the rest of his illustrious career. He was elected to the French Academy of Sciences at the young age of 32, became its president in 1906, and was elected to the Académie française in 1909. Poincaré deliberately cultivated a work habit that has been compared to a bee flying from flower to flower. He observed a strict work regime of 2 hours of work in the morning and two hours in the early evening, with the intervening time left for his subconscious to carry on working on the problem in the hope of a flash of inspiration. He was a great believer in intuition, and claimed that \"it is by logic that we prove, but by intuition that we discover\". It was one such flash of inspiration that earned Poincaré a generous prize from the King of Sweden in 1887 for his partial solution to the "three-body problem", a problem that had defeated mathematicians of the stature of [[Euler]](http://www.storyofmathematics.com/18th_euler.html), Lagrange and Laplace. [[Newton]](http://www.storyofmathematics.com/17th_newton.html) had long ago proved that the paths of two planets orbiting around each other would remain stable, but even the addition of just one more orbiting body to this already simplified solar system resulted in the involvement of as many as 18 different variables (such as position, velocity in each direction, etc), making it mathematically too complex to predict or disprove a stable orbit. Poincaré's solution to the "three-body problem", using a series of approximations of the orbits, although admittedly only a partial solution, was sophisticated enough to win him the prize. -------------------------------------------------------------------------------------------------- Computer representation of the paths generated by Poincaré's analysis of the three body problem *\ Computer representation of the paths generated by Poincaré's analysis of the three body problem\ *http://www.storyofmathematics.com/images/transparent\_blank.gif -------------------------------------------------------------------------------------------------- But he soon realized that he had actually made a mistake, and that his simplifications did not indicate a stable orbit after all. In fact, he realized that even a very small change in his initial conditions would lead to vastly different orbits. This serendipitous discovery, born from a mistake, led indirectly to what we now know as chaos theory, a burgeoning field of mathematics most familiar to the general public from the common example of the flap of a butterfly's wings leading to a tornado on the other side of the world. It was the first indication that three is the minimum threshold for chaotic behaviour. Paradoxically, owning up to his mistake only served to enhance Poincaré's reputation, if anything, and he continued to produce a wide range of work throughout his life, as well as several popular books extolling the importance of mathematics. Poincaré also developed the science of topology, which [[Leonhard Euler]](http://www.storyofmathematics.com/18th_euler.html) had heralded with his solution to the famous Seven Bridges of Königsberg problem. Topology is a kind of geometry which involves one-to-one correspondence of space. It is sometimes referred to as "bendy geometry" or "rubber sheet geometry" because, in topology, two shapes are the same if one can be bent or morphed into the other without cutting it. For example, a banana and a football are topologically equivalent, as are a donut (with its hole in the middle) and a teacup (with its handle); but a football and a donut, are topologically different because there is no way to morph one into the other. In the same way, a traditional pretzel, with its two holes is topological different from all of these examples. --------------------------------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ A 2-dimensional representation of the 3-dimensional problem in the Poincaré conjecture\ * -------------------------------------------------------------------------------------------------------------- In the late 19th Century, Poincaré described all the possible 2-dimensional topological surfaces but, faced with the challenge of describing the shape of our 3-dimensional universe, he came up with the famous Poincaré conjecture, which became one of the most important open questions in mathematics for almost a century. The conjecture looks at a space that, locally, looks like ordinary 3-dimensional space but is connected, finite in size and lacks any boundary (technically known as a closed 3-manifold or 3-sphere). It asserts that, if a loop in that 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 just a three-dimensional sphere. The problem remained unsolved until 2002, when an extremely complex solution was provided by the eccentric and reclusive Russian mathematician Grigori Perelman, involving the ways in which 3-dimensional shapes can be "wrapped up" in higher dimensions. Poincaré's work in theoretical physics was also of great significance, and his symmetrical presentation of the Lorentz transformations in 1905 was an important and necessary step in the formulation of Einstein's theory of special relativity (some even hold that Poincaré and Lorentz were the true discoverers of relativity). He also made important contribution in a whole host of other areas of physics including fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory and cosmology. **19TH CENTURY MATHEMATICS - BOOLE** ---------------------------------------------------------------------------------------- George Boole *\ George Boole (1815-1864)\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- The British mathematician and philosopher George Boole, along with his near contemporary and countryman Augustus de Morgan, was one of the few since [[Leibniz]](http://www.storyofmathematics.com/17th_leibniz.html) to give any serious thought to logic and its mathematical implications. Unlike [[Leibniz]](http://www.storyofmathematics.com/17th_leibniz.html), though, Boole came to see logic as principally a discipline of mathematics, rather than of philosophy. His extraordinary mathematical talents did not manifest themselves in early life. He received his early lessons in mathematics from his father, a tradesman with an amateur interest in in mathematics and logic, but his favourite subject at school was classics. He was a quiet, serious and modest young man from a humble working class background, and largely self-taught in his mathematics (he would borrow mathematical journals from his local Mechanics Institute). It was only at university and afterwards that his mathematical skills began to be fully realized, although, even then, he was all but unknown in his own time, other than for a few insightful but rather abstruse papers on differential equations and the calculus of finite differences. By the age of 34, though, he was well respected enough in his field to be appointed as the first professor of mathematics of Queen\'s College (now University College) in Cork, Ireland. But it was his contributions to the algebra of logic which were later to be viewed as immensely important and influential. Boole began to see the possibilities for applying his algebra to the solution of logical problems, and he pointed out a deep analogy between the symbols of algebra and those that can be made to represent logical forms and syllogisms. In fact, his ambitions stretched to a desire to devise and develop a system of algebraic logic that would systematically define and model the function of the human brain. His novel views of logical method were due to his profound confidence in symbolic reasoning, and he speculated on what he called a "calculus of reason" during the 1840s and 1850s. ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Boolean logic\ * --------------------------------------------------------------------------------------- Determined to find a way to encode logical arguments into a language that could be manipulated and solved mathematically, he came up with a type of linguistic algebra, now known as Boolean algebra. The three most basic operations of this algebra were AND, OR and NOT, which Boole saw as the only operations necessary to perform comparisons of sets of things, as well as basic mathematical functions. Boole's use of symbols and connectives allowed for the simplification of logical expressions, including such important algebraic identities as: (*X* or *Y*) = (*Y* or *X*); not(not*X*) = *X*; not(*X* and *Y*) = (not *X*) or (not *Y*); etc. He also developed a novel approach based on a binary system, processing only two objects ("yes-no", "true-false", "on-off", "zero-one"). Therefore, if "true" is represented by 1 and "false" is represented by 0, and two propositions are both true, then it is possible under Boolean algebra for 1 + 1 to equal 1 ( the "+" is an alternative representation of the OR operator) Despite the standing he had won in the academic community by that time, Boole's revolutionary ideas were largely criticized or just ignored, until the American logician Charles Sanders Peirce (among others) explained and elaborated on them some years after Boole's death in 1864. Almost seventy years later, Claude Shannon made a major breakthrough in realizing that Boole\'s work could form the basis of mechanisms and processes in the real world, and particularly that electromechanical relay circuits could be used to solve Boolean algebra problems. The use of electrical switches to process logic is the basic concept that underlies all modern electronic digital computers, and so Boole is regarded in hindsight as a founder of the field of computer science, and his work led to the development of applications he could never have imagined. **19TH CENTURY MATHEMATICS - RIEMANN** ---------------------------------------------------------------------------------------- Bernhard Riemann *\ Bernhard Riemann (1826-1866)\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- Bernhard Riemann was another mathematical giant hailing from northern Germany. Poor, shy, sickly and devoutly religious, the young Riemann constantly amazed his teachers and exhibited exceptional mathematical skills (such as fantastic mental calculation abilities) from an early age, but suffered from timidity and a fear of speaking in public. He was, however, given free rein of the school library by an astute teacher, where he devoured mathematical texts by Legendre and others, and gradually groomed himself into an excellent mathematician. He also continued to study the Bible intensively, and at one point even tried to prove mathematically the correctness of the Book of Genesis. Although he started studying philology and theology in order to become a priest and help with his family\'s finances, Riemann\'s father eventually managed to gather enough money to send him to study mathematics at the renowned University of Göttingen in 1846, where he first met, and attended the lectures of, [[Carl Friedrich Gauss]](http://www.storyofmathematics.com/19th_gauss.html). Indeed, he was one of the very few who benefited from the support and patronage of [[Gauss]](http://www.storyofmathematics.com/19th_gauss.html), and he gradually worked his way up the University\'s hierarchy to become a professor and, eventually, head of the mathematics department at Göttingen. ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Elliptic geometry\ * --------------------------------------------------------------------------------------- Riemann developed a type of non-Euclidean geometry, different to the hyperbolic geometry of [[Bolyai and Lobachevsky]](http://www.storyofmathematics.com/19th_bolyai.html), which has come to be known as elliptic geometry. As with hyperbolic geometry, there is no such thing as parallel lines, and the angles of a triangle do not sum to 180° (in this case, however, they sum to more than 180º). He went on to develop Riemannian geometry, which unified and vastly generalized the three types of geometry, as well as the concept of a manifold or mathematical space, which generalized the ideas of curves and surfaces. A turning point in his career occurred in 1852 when, at the age of 26, have gave a lecture on the foundations of geometry and outlined his vision of a mathematics of many different kinds of space, only one of which was the flat, Euclidean space which we appear to inhabit. He also introduced one-dimensional complex manifolds known as Riemann surfaces. Although it was not widely understood at the time, Riemann's mathematics changed how we look at the world, and opened the way to higher dimensional geometry, a potential which had existed, unrealized, since the time of [[Descartes]](http://www.storyofmathematics.com/17th_descartes.html). ---------------------------------------------------------------------------------------- 2-D representation of Riemann's zeta function *\ 2-D representation of Riemann's zeta function\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- With his "Riemann metric", Riemann completely broke away from all the limitations of 2 and 3 dimensional geometry, even the geometry of curved spaces of [[Bolyai and Lobachevsky]](http://www.storyofmathematics.com/19th_bolyai.html), and began to think in higher dimensions, extending the differential geometry of surfaces into *n*dimensions. His conception of multi-dimensional space (known as Riemannian space or Riemannian manifold or simply "hyperspace") enabled the later development of general relativity, and is at the heart of much of today's mathematics, in geometry, number theory and other branches of mathematics. He introduced a collection of numbers (known as a tensor) at every point in space, which would describe how much it was bent or curved. For instance, in four spatial dimensions, a collection of ten numbers is needed at each point to describe the properties of the mathematical space or manifold, no matter how distorted it may be. Riemann's big breakthrough occurred while working on a function in the complex plane called the Riemann zeta function (an extension of the simpler zeta function first explored by [[Euler]](http://www.storyofmathematics.com/18th_euler.html) in the previous century). He realized that he could use it to build a kind of 3-dimensional landscape, and furthermore that the contours of that imaginary landscape might be able to unlock the Holy Grail of mathematics, the age-old secret of prime numbers. ----------------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ 3-D representation of Riemann's zeta function and Riemann's Hypothesis\ * ---------------------------------------------------------------------------------------------- Riemann noticed that, at key places, the surface of his 3-dimensional graph dipped down to height zero (known simply as "the zeroes") and was able to show that at least the first ten zeroes inexplicably appeared to line up in a straight line through the 3-dimensional landscape of the zeta-function, known as the critical line, where the real part of the value is equal to ½. With a huge imaginative leap, Riemann realized that these zeroes had a completely unexpected connection with the way the prime numbers are distributed. It began to seem that they could be used to correct [[Gauss]](http://www.storyofmathematics.com/19th_gauss.html)' inspired guesswork regarding the number of primes as numbers as one counts higher and higher. The famous Riemann Hypothesis, which remains unproven, suggests that ALL the zeroes would be on the same straight line. Although he never provided a definitive proof of this hypothesis, Riemann's work did at least show that the 15-year-old [[Gauss]](http://www.storyofmathematics.com/19th_gauss.html)' initial approximations of the incidence of prime numbers were perhaps more accurate than even he could have known, and that the primes were in fact distributed over the universe of numbers in a regular, balanced and beautiful way. The discovery of the Riemann zeta function and the relationship of its zeroes to the prime numbers brought Riemann instant fame when it was published in 1859. He too, though, died young at just 39 years of age, in 1866, and many of his loose papers were accidentally destroyed after his death, so we will never know just how close he was to proving his own hypothesis. Over 150 years later, the Riemann Hypothesis is still considered one of the fundamental questions of number theory, and indeed of all mathematics, and a prize of \$1 million has been offered for the final solution. **19TH CENTURY MATHEMATICS - CANTOR** ---------------------------------------------------------------------------------------- Georg Cantor *\ Georg Cantor (1845-1918)\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- The German Georg Cantor was an outstanding violinist, but an even more outstanding mathematician. He was born in Saint Petersburg, Russia, where he lived until he was eleven. Thereafter, the family moved to Germany, and Cantor received his remaining education at Darmstradt, Zürich, Berlin and (almost inevitably) Göttingen before marrying and settling at the University of Halle, where he was to spend the rest of his career. He was made full professor at Halle at the age of just 34, a notable accomplishment, but his ambitions to move to a more prestigious university, such as Berlin, were largely thwarted by Leopold Kronecker, a well-established figure within the mathematical community and Cantor\'s former professor, who fundamentally disagreed with the thrust of Cantor\'s work. Cantor's first ten papers were on number theory, after which he turned his attention to calculus (or analysis as it had become known by this time), solving a difficult open problem on the uniqueness of the representation of a function by trigonometric series. His main legacy, though, is as perhaps the first mathematician to really understand the meaning of infinity and to give it mathematical precision. Back in the 17th Century, Galileo had tried to confront the idea of infinity and the apparent contradictions thrown up by comparisons of different infinities, but in the end shied away from the problem. He had shown that a one-to-one correspondence could be drawn between all the natural numbers and the squares of all the natural numbers to infinity, suggesting that there were just as many square numbers as integers, even though it was intuitively obvious there were many integers that were were not squares, a concept which came to be known as Galileo's Paradox. He had also pointed out that two concentric circles must both be comprised of an infinite number of points, even though the larger circle would appear to contain more points. However, Galileo had essentially dodged the issue and reluctantly concluded that concepts like less, equals and greater could only be applied to finite sets of numbers, and not to infinite sets. Cantor, however, was not content with this compromise. -------------------------------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Cantor's procedure of bijection or one-to-one correspondence to compare infinite sets\ * ------------------------------------------------------------------------------------------------------------- Cantor\'s starting point was to say that, if it was possible to add 1 and 1, or 25 and 25, etc, then it ought to be possible to add infinity and infinity. He realized that it was actually possible to add and subtract infinities, and that beyond what was normally thought of as infinity existed another, larger infinity, and then other infinities beyond that. In fact, he showed that there may be infinitely many sets of infinite numbers - an infinity of infinities - some bigger than others, a concept which clearly has philosophical, as well as just mathematical, significance. The sheer audacity of Cantor's theory set off a quiet revolution in the mathematical community, and changed forever the way mathematics is approached. His first intimations of all this came in the early 1870s when he considered an infinite series of natural numbers (1, 2, 3, 4, 5, \...), and then an infinite series of multiples of ten (10, 20 , 30, 40, 50, \...). He realized that, even though the multiples of ten were clearly a subset of the natural numbers, the two series could be paired up on a one-to-one basis (1 with 10, 2 with 20, 3 with 30, etc) - a process known as bijection - to show that they were the same "sizes" of infinite sets, in that they had the same number of elements. This clearly also applies to other subsets of the natural numbers, such as the even numbers 2, 4, 6, 8, 10, etc, or the squares 1, 4, 9, 16, 25, etc, and even to the set of negative numbers and integers. In fact, Cantor realized that he could, in the same way, even pair up all the fractions (or rational numbers) with all the whole numbers, thus showing that rational numbers were also the same sort of infinity as the natural numbers, despite the intuitive feeling that there must be more fractions than whole numbers. ---------------------------------------------------------------------------------------- Cantor's diagonal argument for the existence of uncountable sets *\ Cantor's diagonal argument for the existence of uncountable sets\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- However, when Cantor considered an infinite series of decimal numbers, which includes irrational numbers like π,e and √2, this method broke down. He used several clever arguments (one being the \"diagonal argument\" explained in the box on the right) to show how it was always possible to construct a new decimal number that was missing from the original list, and so proved that the infinity of decimal numbers (or, technically, real numbers) was in fact bigger than the infinity of natural numbers. He also showed that they were "non-denumerable" or \"uncountable\" (i.e. contained more elements than could ever be counted), as opposed to the set of rational numbers which he had shown were technically (even if not practically) "denumerable" or \"countable\". In fact, it can be argued that there are an infinite number of irrational numbers in between each and every rational number. The patternless decimals of irrational numbers fill the \"spaces\" between the patterns of the rational numbers. Cantor coined the new word "transfinite" in an attempt to distinguish these various levels of infinite numbers from an absolute infinity, which the religious Cantor effectively equated with God (he saw no contradiction between his mathematics and the traditional concept of God). Although the cardinality (or size) of a finite set is just a natural number indicating the number of elements in the set, he also needed a new notation to describe the sizes of infinite sets, and he used the Hebrew letter aleph (). He defined Aleph~0~ (aleph-null or aleph-nought) as the cardinality of the countably infinite set of natural numbers; ~1~ (aleph-one) as the next larger cardinality, that of the uncountable set of ordinal numbers; etc. Because of the unique properties of infinite sets, he showed that Aleph~0~ + ~0~ = Aleph~0~, and also that ~0~ x Aleph~0~ = ~0~ All of this represented a revolutionary step, and opened up new possibilities in mathematics. However, it also opened up the possibility of other infinities, for instance an infinity - or even many infinities - between the infinity of the whole numbers and the larger infinity of the decimal numbers. This idea is known as the continuum hypothesis, and Cantor believed (but could not actually prove) that there was NO such intermediate infinite set. The continuum hypothesis was one of the 23 important open problems identified by[[David Hilbert]](http://www.storyofmathematics.com/20th_hilbert.html) in his famous 1900 Paris lecture, and it remained unproved - and indeed appeared to be unprovable - for almost a century, until the work of [[Robinson and Matiyasevich]](http://www.storyofmathematics.com/20th_robinson.html) in the 1950s and 1960s. ---------------------------------------------------------------------------------------- Modern set theory notation *\ Modern set theory notation\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- Just as importantly, though, this work of Cantor\'s between 1874 and 1884 marks the real origin of set theory, which has since become a fundamental part of modern mathematics, and its basic concepts are used throughout all the various branches of mathematics. Although the concept of a set had been used implicitly since the beginnings of mathematics, dating back to the ideas of Aristotle, this was limited to everyday finite sets. In contradistinction, the "infinite" was kept quite separate, and was largely considered a topic for philosophical, rather than mathematical, discussion. Cantor, however, showed that, just as there were different finite sets, there could be infinite sets of different sizes, some of which are countable and some of which are uncountable. Throughout the 1880s and 1890s, he refined his set theory, defining well-ordered sets and power sets and introducing the concepts of ordinality and cardinality and the arithmetic of infinite sets. What is now known as Cantor\'s theorem states generally that, for any set *A*, the power set of *A* (i.e. the set of all subsets of *A*) has a strictly greater cardinality than *A* itself. More specificially, the power set of a countably infinite set is uncountably infinite. Despite the central position of set theory in modern mathematics, it was often deeply mistrusted and misunderstood by other mathematicians of the day. One quote, usually attributed to [[Henri Poincaré]](http://www.storyofmathematics.com/19th_poincare.html), claimed that \"later generations will regard Mengenlehre (set theory) as a disease from which one has recovered\". Others, however, were quick to see the value and potential of the method, and [[David Hilbert]](http://www.storyofmathematics.com/20th_hilbert.html) declared in 1926 that \"no one shall expel us from the Paradise that Cantor has created\". Cantor had few other mathematicians with whom he could discuss his ground-breaking work, and most were distinctly unnerved by his contemplation of the infinite. During the 1880s, he encountered resistance, sometimes fierce resistance, from mathematical contemporaries such as his old professor Leopold Kronecker and [[Henri Poincaré]](http://www.storyofmathematics.com/19th_poincare.html), as well as from philosophers like Ludwig Wittgenstein and even from some Christian theologians, who saw Cantor\'s work as a challenge to their view of the nature of God. Cantor himself, a deeply religious man, noted some annoying paradoxes thrown up by his own work, but some went further and saw it as the wilful destruction of the comprehensible and logical base on which the whole of mathematics was based. As he aged, Cantor suffered from more and more recurrences of mental illness, which some have directly linked to his constant contemplation of such complex, abstract and paradoxical concepts. In the last decades of his life, he did no mathematical work at all, but wrote extensively on his two obsessions: that Shakespeare's plays were actually written by the English philosopher Sir Francis Bacon, and that Christ was the natural son of Joseph of Arimathea. He spent long periods in the Halle sanatorium recovering from attacks of manic depression and paranoia, and it was there, alone in his room, that he finally died in 1918, his great project still unfinished. **20TH CENTURY MATHEMATICS - HILBERT** ---------------------------------------------------------------------------------------- *\ David Hilbert (1862-1943)\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- David Hilbert was a great leader and spokesperson for the discipline of mathematics in the early 20th Century. But he was an extremely important and respected mathematician in his own right. Like so many great German mathematicians before him, Hilbert was another product of the University of Göttingen, at that time the mathematical centre of the world, and he spent most of his working life there. His formative years, though, were spent at the University of Königsberg, where he developed an intense and fruitful scientific exchange with fellow mathematicians Hermann Minkowski and Adolf Hurwitz. Sociable, democratic and well-loved both as a student and as a teacher, and often seen as bucking the trend of the formal and elitist system of German mathematics, Hilbert's mathematical genius nevertheless spoke for itself. He has many mathematical terms named after him, including Hilbert space (an infinite dimensional Euclidean space), Hilbert curves, the Hilbert classification and the Hilbert inequality, as well as several theorems, and he gradually established himself as the most famous mathematician of his time. His pithy enumeration of the 23 most important open mathematical questions at the 1900 Paris conference of the International Congress of Mathematicians at the Sorbonne set the stage for almost the whole of 20th Century mathematics. The details of some of these individual problems are highly technical; some are very precise, while some are quite vague and subject to interpretation; several problems have now already been solved, or at least partially solved, while some may be forever unresolvable as stated; some relate to rather abstruse backwaters of mathematical thought, while some deal with more mainstream and well-known issues such as the Riemann hypothesis, the continuum hypothesis, group theory, theories of quadratic forms, real algebraic curves, etc. ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Hilbert's algorithm for space-filling curves\ * --------------------------------------------------------------------------------------- As a young man, Hilbert began by pulling together all of the may strands of number theory and abstract algebra, before changing field completely to pursue studies in integral equations, where he revolutionized the then current practices. In the early 1890s, he developed continuous fractal space-filling curves in multiple dimensions, building on earlier work by Guiseppe Peano. As early as 1899, he proposed a whole new formal set of geometrical axioms, known as Hilbert\'s axioms, to substitute the traditional axioms of[[Euclid]](http://www.storyofmathematics.com/hellenistic_euclid.html). But perhaps his greatest legacy is his work on equations, often referred to as his finiteness theorem. He showed that although there were an infinite number of possible equations, it was nevertheless possible to split them up into a finite number of types of equations which could then be used, almost like a set of building blocks, to produce all the other equations. Interestingly, though, Hilbert could not actually construct this finite set of equations, just prove that it must exist (sometimes referred to as an existence proof, rather than constructive proof). At the time, some critics passed this off as mere theology or smoke-and-mirrors, but it effectively marked the beginnings of a whole new style of abstract mathematics. -------------------------------------------------------------------------------------------- Among other things, Hilbert space can be used to study the harmonics of vibrating strings *\ Among other things, Hilbert space can be used to study the harmonics of vibrating strings\ *http://www.storyofmathematics.com/images/transparent\_blank.gif -------------------------------------------------------------------------------------------- This use of an existence proof rather than constructive proof was also implicit in his development, during the first decade of the 20th Century, of the mathematical concept of what came to be known as Hilbert space. Hilbert space is a generalization of the notion of Euclidean space which extends the methods of vector algebra and calculus to spaces with any finite (or even infinite) number of dimensions. Hilbert space provided the basis for important contributions to the mathematics of physics over the following decades, and may still offer one of the best mathematical formulations of quantum mechanics. Hilbert was unfailingly optimistic about the future of mathematics, never doubting that his 23 problems would soon be solved. In fact, he went so far as to claim that there are absolutely no unsolvable problems - a famous quote of his (dating from 1930, and also engraved on his tombstone) proclaimed, "We must know! We will know!" - and he was convinced that the whole of mathematics could, and ultimately would, be put on unshakable logical foundations. Another of his rallying cries was "in mathematics there is no *ignorabimus*", a reference to the traditional position on the limits of scientific knowledge. Unlike [[Russell]](http://www.storyofmathematics.com/20th_russell.html), Hilbert's formalism was premised on the idea that the ultimate base of mathematics lies, not in logic itself, but in a simpler system of pre-logical symbols which can be collected together in strings or axioms and manipulated according to a set of "rules of inference". His ambitious program to find a complete and consistent set of axioms for all of mathematics (which became known as Hilbert's Program), received a severe set-back, however, with the incompleteness theorems of [[Kurt Gödel]](http://www.storyofmathematics.com/20th_godel.html) in the early 1930s. Nevertheless, Hilbert\'s work had started logic on a course of clarification, and the need to understand [[Gödel]](http://www.storyofmathematics.com/20th_godel.html)\'s work then led to the development of recursion theory and mathematical logic as an autonomous discipline in the 1930s, and later provided the basis for theoretical computer science. For a time, Hilbert bravely spoke out against the Nazi repression of his Jewish mathematician friends in Germany and Austria in the mid 1930s. But, after mass evictions, several suicides, many deaths in concentration camps, and even direct assassinations, he too eventually lapsed into silence, and could only watch as one of the greatest mathematical centres of all time was systematically destroyed. By the time of his death in 1943, little remained of the great mathematics community at Göttingen, and Hilbert was buried in relative obscurity, his funeral attended by fewer than dozen people and hardly reported in the press. **20TH CENTURY MATHEMATICS - GÖDEL** ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Kurt Gödel (1906-1978)\ * --------------------------------------------------------------------------------------- Kurt Gödel grew up a rather strange, sickly child in Vienna. From an early age his parents took to referring to him as "Herr Varum", Mr Why, for his insatiable curiosity. At the University of Vienna, Gödel first studied number theory, but soon turned his attention to mathematical logic, which was to consume him for most of the rest of his life. As a young man, he was, like [[Hilbert]](http://www.storyofmathematics.com/20th_hilbert.html), optimistic and convinced that mathematics could be made whole again, and would recover from the uncertainties introduced by the work of [[Cantor]](http://www.storyofmathematics.com/19th_cantor.html) and [[Riemann]](http://www.storyofmathematics.com/19th_riemann.html). Between the wars, Gödel joined in the cafe discussions of a group of intense intellectuals and philosophers known as the Vienna Circle, which included logical positivists such as Moritz Schlick, Hans Hahn and Rudolf Carnap, who rejected metaphysics as meaningless and sought to codify all knowledge in a single standard language of science. Although Gödel did not necessarily share the positivistic philosophical outlook of the Vienna Circle, it was in this enviroment that Gödel pursued his dream of solving the second, and perhaps most overarching, of[[Hilbert]](http://www.storyofmathematics.com/20th_hilbert.html)'s 23 problems, which sought to find a logical foundation for all of mathematics. The ideas he came up with would revolutionize mathematics, as he effectively proved, mathematically and philosophically, that [[Hilbert]](http://www.storyofmathematics.com/20th_hilbert.html)'s (and his own) optimism was unfounded and that such a foundation was just not possible. His first achievement, which actually served to advance [[Hilbert]](http://www.storyofmathematics.com/20th_hilbert.html)\'s Program, was his completeness theorem, which showed that all valid statements in Freges\'s \"first order logic\" can be proved from a set of simple axioms. However, he then turned his attention to \"second order logic\", i.e a logic powerful enough to support arithmetic and more complex mathematical theories (essentially, one able to accept sets as values of variables). Gödel's incompleteness theorem (technically \"incompleteness theorems\", plural, as there were actually two separate theorems, although they are usually spoken of together) of 1931 showed that, within any logical system for mathematics (or at least in any system that is powerful and complex enough to be able to describe the arithmetic of the natural numbers, and therefore to be interesting to most mathematicians), there will be some statements about numbers which are true but which can NEVER be proved. This was enough to prompt John von Neumann to comment that \"it\'s all over\". ---------------------------------------------------------------------------------------- Gödel's Incompleteness Theorem *\ Gödel's Incompleteness Theorem\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- His approach began with the plain language assertion such as "this statement cannot be proved", a version of the ancient "liar paradox", and a statement which itself must be either true or false. If the statement is false, then that means that the statement can be proved, suggesting that it is actually true, thus generating a contradiction. For this to have implications in mathematics, though, Gödel needed to convert the statement into a \"formal language\" (i.e. a pure statement of arithmetic). He did this using a clever code based on prime numbers, where strings of primes play the roles of natural numbers, operators, grammatical rules and all the other requirements of a formal language. The resulting mathematical statement therefore appears, like its natural language equivalent, to be true but unprovable, and must therefore remain undecided. The incompleteness theorem - surely a mathematician's worst nightmare - led to something of a crisis in the mathematical community, raising the spectre of a problem which may turn out to be true but is still unprovable, something which had not been even considered in the whole two millennia plus history of mathematics. Gödel effectively put paid, at a stroke, to the ambitions of mathematicians like [[Bertrand Russell]](http://www.storyofmathematics.com/20th_russell.html) and [[David Hilbert]](http://www.storyofmathematics.com/20th_hilbert.html) who sought to find a complete and consistent set of axioms for all of mathematics. His work PROVED that any system of logic or numbers that mathematicians ever come up with will always rest on at least a few unprovable assumptions. His conclusions also imply that not all mathematical questions are even computable, and that it is impossible, even in principle, to create a machine or computer that will be able to do all that a human mind can do. ------------------------------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Representation of the Gödel Metric, an exact solution to Einstein\'s field equations\ * ------------------------------------------------------------------------------------------------------------ Unfortunately, the theorems also led to a personal crisis for Gödel. In the mid 1930s, he suffered a series of mental breakdowns and spent some significant time in a sanatorium. Nevertheless, he threw himself into the same problem that had destroyed the mental well-being of [[Georg Cantor]](http://www.storyofmathematics.com/19th_cantor.html)during the previous century, the continuum hypothesis. In fact, he made an important step in the resolution of that notoriously difficult problem (by proving that the the axiom of choice is independence from finite type theory), without which [[Paul Cohen]](http://www.storyofmathematics.com/20th_cohen.html) would probably never have been able to come to his definitive solution. Like [[Cantor]](http://www.storyofmathematics.com/19th_cantor.html) and others after him, though, Gödel too suffered a gradual deterioration in his mental and physical health. He was only kept afloat at all by the love of his life, Adele Numbursky. Together, they witnessed the virtual destruction of the German and Austrian mathematics community by the Nazi regime. Eventually, along with many other eminent European mathematicians and scholars, Gödel fled the Nazis to the safety of Princeton in the USA, where he became a close friend of fellow exile Albert Einstein, contributing some demonstrations of paradoxical solutions to Einstein\'s field equations in general relativity (including his celebrated Gödel metric of 1949). But, even in the USA, he was not able to escape his demons, and was dogged by depression and paranoia, suffering several more nervous breakdowns. Eventually, he would only eat food that had been tested by his wife Adele, and, when Adele herself was hospitalized in 1977, Gödel simply refused to eat and starved himself to death. Gödel's legacy is ambivalent. Although he is recognized as one of the great logicians of all time, many were just not prepared to accept the almost nihilistic consequences of his conclusions, and his explosion of the traditional formalist view of mathematics. Worse news was still to come, though, as the mathematical community (including, as we will see, [[Alan Turing]](http://www.storyofmathematics.com/20th_turing.html)) struggled to come to grips with Gödel's findings. **20TH CENTURY MATHEMATICS - TURING** ---------------------------------------------------------------------------------------- Alan Turing *\ Alan Turing (1912-1954)\ *http://www.storyofmathematics.com/images/transparent\_blank.gif ---------------------------------------------------------------------------------------- The British mathematician Alan Turing is perhaps most famous for his war-time work at the British code-breaking centre at Bletchley Park where his work led to the breaking of the German enigma code (according to some, shortening the Second World War at a stroke, and potentially saving thousands of lives). But he was also responsible for making[[Gödel]](http://www.storyofmathematics.com/20th_godel.html)'s already devastating incompleteness theorem even more bleak and discouraging, and it is mainly on this - and the development of computer science that his work gave rise to - that Turing's mathematical legacy rests. Despite attending an expensive private school which strongly emphasized the classics rather than the sciences, Turing showed early signs of the genius which was to become more prominent later, solving advanced problems as a teenager without having even studied elementary calculus, and immersing himself in the complex mathematics of Albert Einstein\'s work. He became a confirmed atheist after the death of his close friend and fellow Cambridge student Christopher Morcom, and throughout his life he was an accomplished and committed long-distance runner. In the years following the publication of [[Gödel]](http://www.storyofmathematics.com/20th_godel.html)'s incompleteness theorem, Turing desperately wanted to clarify and simplify [[Gödel]](http://www.storyofmathematics.com/20th_godel.html)'s rather abstract and abstruse theorem, and to make it more concrete. But his solution - which was published in 1936 and which, he later claimed, had come to him in a vision - effectively involved the invention of something that has come to shape the entire modern world, the computer. ---------------------------------------------------------------------------------------  http://www.storyofmathematics.com/images/transparent\_blank.gif*\ Representation of a Turing Machine\ * --------------------------------------------------------------------------------------- During the 1930s, Turing recast incompleteness in terms of comput