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Were the problems Fibonacci solved in his work “Flos” posed specifically for him?

Were the problems Fibonacci solved in his work “Flos” posed specifically for him?

According to Wikipedia, Fibonacci wrote "Flos", a work which contained solutions to problems posed by Johannes of Palermo. Did Johannes pose a challenge to all European mathematicians of the time, or were his problems directed at Fibonacci?

Does anybody know what are some specific examples of the problems posed?

Johannes of Palermo was a scholar in Frederick II's court. Frederick was aware of Fibonacci's work, and perhaps even an admirer. In 1225 when Frederick's court met in Pisa, Fibonacci was invited to demonstrate his works. I can't find a source for when exactly Johannes of Palermo posed his problems, but the two men certainly met in Pisa and Johannes posed his problems directly at Fibonacci:

A meeting was arranged between Fibonacci and Frederick at the Emperor's palazzo in Pisa, Frederick bringing with him an imposing retinue of people and animals. Frederick, who was about 30 years old, is described as "athletic-looking and of medium height, with reddish-blond hair and piercing blue eyes which are said to have made his courtiers tremble".

Mathematical questions for Fibonacci to solve were proposed by a scholar, Master John of Palermo. According to some writers, a mathematical tournament between Fibonacci and other mathematicians took place, but this does not seem to have been the case. Three of these problems are given later when I deal with Fibonacci's mathematical writings. At the time of his meeting with Frederick in the 1220's, Fibonacci was probably at the height of his prowess.

Source: 800 years young, A. F. Horadam, Department of Mathematics, University of New England

As for an example of the problem:

In Flos Fibonacci gives an accurate approximation to a root of 10x + 2x2 + x3 = 20, one of the problems that he was challenged to solve by Johannes of Palermo. This problem was not made up by Johannes of Palermo, rather he took it from Omar Khayyam's algebra book where it is solved by means of the intersection of a circle and a hyperbola. Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction. He then continues:-

And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.

Without explaining his methods, Fibonacci then gives the approximate solution in sexagesimal notation as (this is written to base 60, so it is 1 + 22/60 + 7/602 + 42/603 +… ). This converts to the decimal 1.3688081075 which is correct to nine decimal places, a remarkable achievement.

Source: Leonardo Pisano Fibonacci (short biography), School of Mathematics and Statistics, University of St Andrews, Scotland

There is another example from Flos in the source of my first quote, but unfortunately without Fibonacci's solution. Perhaps you would like to try and solve it on your own? ;)

Further reading:

  • The 800th birthday of the book that brought numbers to the west, Keith Devlin (Executive Director of the Center for the Study of Language and Information at Stanford University)
  • Fibonacci and Square Numbers, MathDL, The Mathematical Association of America


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’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’s incompleteness theorem, Turing desperately wanted to clarify and simplify Gödel’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.


Youth and studies Edit

Georg Cantor was born in 1845 in the western merchant colony of Saint Petersburg, Russia, and brought up in the city until he was eleven. Cantor, the oldest of six children, was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm's brother) was a well-known musician and soloist in a Russian imperial orchestra. [15] Cantor's father had been a member of the Saint Petersburg stock exchange when he became ill, the family moved to Germany in 1856, first to Wiesbaden, then to Frankfurt, seeking milder winters than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt his exceptional skills in mathematics, trigonometry in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the Technische Universität Darmstadt. [16] [17] In 1862, Cantor entered the Swiss Federal Polytechnic. After receiving a substantial inheritance upon his father's death in June 1863, [18] Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor was a good student, and he received his doctorate degree in 1867. [18] [19]

Teacher and researcher Edit

Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle University. [19] [20]

In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday.

Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. [19] [18] To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible. [21] Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague, [22] perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians. [23] Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work ever since he intentionally delayed the publication of Cantor's first major publication in 1874. [19] Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, and it usually involved Kronecker, [19] so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.

In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.

In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle. [24] Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta. [25] He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was ". about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party, "Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! . But of course I never want to know anything again about Acta Mathematica." [26]

Cantor suffered his first known bout of depression in May 1884. [18] [27] Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:

. I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures how much happier I would be to be scientifically active, if only I had the necessary mental freshness. [28]

This crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study of Elizabethan literature thinking there might be evidence that Francis Bacon wrote the plays attributed to William Shakespeare (see Shakespearean authorship question) this ultimately resulted in two pamphlets, published in 1896 and 1897. [29]

Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on December 29, 1891. [19] He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted.

In 1889, Cantor was instrumental in founding the German Mathematical Society [19] and chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time. Georg Cantor was also instrumental in the establishment of the first International Congress of Mathematicians, which was held in Zürich, Switzerland, in 1897. [19]

Later years and death Edit

After Cantor's 1884 hospitalization, there is no record that he was in any sanatorium again until 1899. [27] Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on December 16 (Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics. [30] Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. [31] Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God. [12] Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. [32] He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the Deutsche Mathematiker-Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but this did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.

Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. [33] The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life. [18]

Cantor's work between 1874 and 1884 is the origin of set theory. [34] Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics. [35]

In one of his earliest papers, [36] Cantor proved that the set of real numbers is "more numerous" than the set of natural numbers this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets). [37]

Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that the Cantor set, discovered by Henry John Stephen Smith in 1875, [38] is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. He also showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers.

Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter ℵ (aleph) with a natural number subscript for the ordinals he employed the Greek letter ω (omega). This notation is still in use today.

The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. [14] The US philosopher Charles Sanders Peirce praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Adolf Hurwitz and Jacques Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Cantor's religious ideas. This was later published, as were several of his expository works.

Number theory, trigonometric series and ordinals Edit

Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Peter Gustav Lejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the nth derived set Sn of a set S of zeros of a trigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S1 as its set of zeros, where S1 is the set of limit points of S. If Sk+1 is the set of limit points of Sk, then he could construct a trigonometric series whose zeros are Sk+1. Because the sets Sk were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets S, S1, S2, S3. formed a limit set, which we would now call Sω, and then he noticed that Sω would also have to have a set of limit points Sω+1, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ω, ω + 1, ω + 2, . [39]

Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics". [40] Cantor also published an erroneous "proof" of the inconsistency of infinitesimals. [41]

Set theory Edit

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, [34] "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"). [43] This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements). [44] Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable. His proof differs from diagonal argument that he gave in 1891. [45] Cantor's article also contains a new method of constructing transcendental numbers. Transcendental numbers were first constructed by Joseph Liouville in 1844. [46]

Cantor established these results using two constructions. His first construction shows how to write the real algebraic numbers [47] as a sequence a1, a2, a3, . In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence – that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more – namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers. [48] Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers. [49]

Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole. [50] Cantor also introduced the Cantor set during this period.

The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883, [51] was the most important of the six and was also published as a separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.

In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem. Cantor wrote on the Goldbach conjecture in 1894.

In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship these were his last significant papers on set theory. [52] The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis hence the name Cantor–Bernstein–Schröder theorem.

One-to-one correspondence Edit

Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Richard Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. About this discovery Cantor wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!") [53] The result that he found so astonishing has implications for geometry and the notion of dimension.

In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space R n has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.

This paper displeased Kronecker and Cantor wanted to withdraw it however, Dedekind persuaded him not to do so and Karl Weierstrass supported its publication. [54] Nevertheless, Cantor never again submitted anything to Crelle.

Continuum hypothesis Edit

Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety. [10]

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Kurt Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC"). [55]

Absolute infinite, well-ordering theorem, and paradoxes Edit

In 1883, Cantor divided the infinite into the transfinite and the absolute. [56]

The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it. [57] In 1883, Cantor also introduced the well-ordering principle "every set can be well-ordered" and stated that it is a "law of thought". [58]

Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an aleph. [59] First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem. [60] In 1932, Zermelo criticized the construction in Cantor's proof. [61]

Cantor avoided paradoxes by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form a set, the resulting contradiction only implies that the ordinals form an inconsistent multiplicity. On the other hand, Bertrand Russell treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is inconsistent. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: the Burali-Forti paradox (which was just mentioned), Cantor's paradox, and Russell's paradox. [62] Russell named paradoxes after Cesare Burali-Forti and Cantor even though neither of them believed that they had found paradoxes. [63]

In 1908, Zermelo published his axiom system for set theory. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the well-ordering theorem. [64] Zermelo had proved this theorem in 1904 using the axiom of choice, but his proof was criticized for a variety of reasons. [65] His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets. [66]

In 1923, John von Neumann developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. Von Neumann stated that a class is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets. This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class. [67] Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem. [68] In 1930, Zermelo defined models of set theory that satisfy von Neumann's axiom. [69]

The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's. [70] He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one. [71] To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications – he identified the Absolute Infinite with God, [72] and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world. [5] He was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science. [73] Joseph Dauben has traced the effect Cantor's Christian convictions had on the development of transfinite set theory. [74] [75]

Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. [76] Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind. [77] Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. [78] Mathematicians such as L. E. J. Brouwer and especially Henri Poincaré adopted an intuitionist stance against Cantor's work. Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set. [9]

Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. [6] In particular, neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity". [79] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake: [80] ". the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers." [81]

Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism – and was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs. [82]

It was important to Cantor that his philosophy provided an "organic explanation" of nature, and in his 1883 Grundlagen, he said that such an explanation could only come about by drawing on the resources of the philosophy of Spinoza and Leibniz. [83] In making these claims, Cantor may have been influenced by FA Trendelenburg, whose lecture courses he attended at Berlin, and in turn Cantor produced a Latin commentary on Book 1 of Spinoza's Ethica. FA Trendelenburg was also the examiner of Cantor's Habilitationsschrift. [84] [85]

In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim, [86] as well as theologians such as Cardinal Johann Baptist Franzelin, who once replied by equating the theory of transfinite numbers with pantheism. [7] Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him. [80]

Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his assertion that "the essence of mathematics is its freedom." [87] These ideas parallel those of Edmund Husserl, whom Cantor had met in Halle. [88]

Meanwhile, Cantor himself was fiercely opposed to infinitesimals, describing them as both an "abomination" and "the cholera bacillus of mathematics". [40]

Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering: ". I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers." [89]

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, René Descartes, George Berkeley, Gottfried Leibniz, and Bernard Bolzano on infinity. Instead, he always strongly rejected Kant's philosophy, both in the realms of the philosophy of mathematics and metaphysics. He shared B. Russell's motto "Kant or Cantor", and defined Kant "yonder sophistical Philistine who knew so little mathematics." [90]

Cantor's paternal grandparents were from Copenhagen and fled to Russia from the disruption of the Napoleonic Wars. There is very little direct information on his grandparents. [91] Cantor was sometimes called Jewish in his lifetime, [92] but has also variously been called Russian, German, and Danish as well.

Jakob Cantor, Cantor's grandfather, gave his children Christian saints' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they converted to Christianity. Cantor's father, Georg Waldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. Very little is known for sure about George Woldemar's origin or education. [93] His mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg and baptized Roman Catholic she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating:

Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr für Gleichberechtigung der Hebräer sein, im socialen Leben sind mir Christen lieber . [93]

("Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians. ") which could be read to imply that she was of Jewish ancestry. [94]

There were documented statements, during the 1930s, that called this Jewish ancestry into question:

More often [i.e., than the ancestry of the mother] the question has been discussed of whether Georg Cantor was of Jewish origin. About this it is reported in a notice of the Danish genealogical Institute in Copenhagen from the year 1937 concerning his father: "It is hereby testified that Georg Woldemar Cantor, born 1809 or 1814, is not present in the registers of the Jewish community, and that he completely without doubt was not a Jew . " [93]

It is also later said in the same document:

Also efforts for a long time by the librarian Josef Fischer, one of the best experts on Jewish genealogy in Denmark, charged with identifying Jewish professors, that Georg Cantor was of Jewish descent, finished without result. [Something seems to be wrong with this sentence, but the meaning seems clear enough.] In Cantor's published works and also in his Nachlass there are no statements by himself which relate to a Jewish origin of his ancestors. There is to be sure in the Nachlass a copy of a letter of his brother Ludwig from 18 November 1869 to their mother with some unpleasant antisemitic statements, in which it is said among other things: . [93]

(the rest of the quote is finished by the very first quote above). In Men of Mathematics, Eric Temple Bell described Cantor as being "of pure Jewish descent on both sides", although both parents were baptized. In a 1971 article entitled "Towards a Biography of Georg Cantor", the British historian of mathematics Ivor Grattan-Guinness mentions (Annals of Science 27, pp. 345–391, 1971) that he was unable to find evidence of Jewish ancestry. (He also states that Cantor's wife, Vally Guttmann, was Jewish).

In a letter written by Georg Cantor to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondence, Gauthier-Villars, Paris, 1934, p. 306), Cantor states that his paternal grandparents were members of the Sephardic Jewish community of Copenhagen. Specifically, Cantor states in describing his father: "Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde. " ("He was born in Copenhagen of Jewish (lit: 'Israelite') parents from the local Portuguese-Jewish community.") [95]

In addition, Cantor's maternal great uncle, [96] a Hungarian violinist Josef Böhm, has been described as Jewish, [97] which may imply that Cantor's mother was at least partly descended from the Hungarian Jewish community. [98]

In a letter to Bertrand Russell, Cantor described his ancestry and self-perception as follows:

Neither my father nor my mother were of German blood, the first being a Dane, borne in Kopenhagen, my mother of Austrian Hungar descension. You must know, Sir, that I am not a regular just Germain, for I am born 3 March 1845 at Saint Peterborough, Capital of Russia, but I went with my father and mother and brothers and sister, eleven years old in the year 1856, into Germany. [99]

Until the 1970s, the chief academic publications on Cantor were two short monographs by Arthur Moritz Schönflies (1927) – largely the correspondence with Mittag-Leffler – and Fraenkel (1930). Both were at second and third hand neither had much on his personal life. The gap was largely filled by Eric Temple Bell's Men of Mathematics (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the history of mathematics" and as "one of the worst". [100] Bell presents Cantor's relationship with his father as Oedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell – including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents. [101] A critique of Bell's book is contained in Joseph Dauben's biography. [102] Writes Dauben:

Cantor devoted some of his most vituperative correspondence, as well as a portion of the Beiträge, to attacking what he described at one point as the 'infinitesimal Cholera bacillus of mathematics', which had spread from Germany through the work of Thomae, du Bois Reymond and Stolz, to infect Italian mathematics . Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz and Veronese was to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible. [103]

  1. ^Grattan-Guinness 2000, p. 351.
  2. ^ The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources.
  3. ^Dauben 2004, p. 1.
  4. ^ Dauben, Joseph Warren (1979). Georg Cantor His Mathematics and Philosophy of the Infinite . princeton university press. pp. introduction. ISBN9780691024479 .
  5. ^ abDauben 2004, pp. 8, 11, 12–13.
  6. ^ abDauben 1977, p. 86 Dauben 1979, pp. 120, 143.
  7. ^ abDauben 1977, p. 102.
  8. ^Dauben 2004, p. 1 Dauben 1977, p. 89 15n.
  9. ^ abRodych 2007.
  10. ^ abDauben 1979, p. 280: ". the tradition made popular by Arthur Moritz Schönflies blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression.
  11. ^Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's mental illness as "cyclic manic-depression".
  12. ^ abDauben 1979, p. 248.
  13. ^Hilbert (1926, p. 170): "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können." (Literally: "Out of the Paradise that Cantor created for us, no one must be able to expel us.")
  14. ^ ab
  15. Reid, Constance (1996). Hilbert. New York: Springer-Verlag. p. 177. ISBN978-0-387-04999-1 .
  16. ^ru: The musical encyclopedia (Музыкальная энциклопедия).
  17. ^
  18. "Georg Cantor (1845-1918)". www-groups.dcs.st-and.ac.uk . Retrieved September 14, 2019 .
  19. ^
  20. Georg Cantor 1845-1918. Birkhauser. 1985. ISBN978-3764317706 .
  21. ^ abcde
  22. "Cantor biography". www-history.mcs.st-andrews.ac.uk . Retrieved October 6, 2017 .
  23. ^ abcdefgh
  24. Bruno, Leonard C. Baker, Lawrence W. (1999). Math and mathematicians: the history of math discoveries around the world. Detroit, Mich.: U X L. p. 54. ISBN978-0787638139 . OCLC41497065.
  25. ^
  26. O'Connor, John J Robertson, Edmund F (1998). "Georg Ferdinand Ludwig Philipp Cantor". MacTutor History of Mathematics.
  27. ^Dauben 1979, p. 163.
  28. ^Dauben 1979, p. 34.
  29. ^Dauben 1977, p. 89 15n.
  30. ^Dauben 1979, pp. 2–3 Grattan-Guinness 1971, pp. 354–355.
  31. ^Dauben 1979, p. 138.
  32. ^Dauben 1979, p. 139.
  33. ^ abDauben 1979, p. 282.
  34. ^Dauben 1979, p. 136 Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884.
  35. ^Dauben 1979, pp. 281–283.
  36. ^Dauben 1979, p. 283.
  37. ^ For a discussion of König's paper see Dauben 1979, pp. 248–250. For Cantor's reaction, see Dauben 1979, pp. 248, 283.
  38. ^Dauben 1979, pp. 283–284.
  39. ^Dauben 1979, p. 284.
  40. ^ ab
  41. Johnson, Phillip E. (1972). "The Genesis and Development of Set Theory". The Two-Year College Mathematics Journal. 3 (1): 55–62. doi:10.2307/3026799. JSTOR3026799.
  42. ^
  43. Suppes, Patrick (1972). Axiomatic Set Theory. Dover. p. 1. ISBN9780486616308 . With a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects. As a consequence, many fundamental questions about the nature of mathematics may be reduced to questions about set theory.
  44. ^Cantor 1874
  45. ^ A countable set is a set which is either finite or denumerable the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable".
  46. ^The Cantor Set Before Cantor Mathematical Association of America
  47. ^
  48. Cooke, Roger (1993). "Uniqueness of trigonometric series and descriptive set theory, 1870–1985". Archive for History of Exact Sciences. 45 (4): 281. doi:10.1007/BF01886630. S2CID122744778.
  49. ^ ab
  50. Katz, Karin Usadi Katz, Mikhail G. (2012). "A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography". Foundations of Science. 17 (1): 51–89. arXiv: 1104.0375 . doi:10.1007/s10699-011-9223-1. S2CID119250310.
  51. ^
  52. Ehrlich, P. (2006). "The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes" (PDF) . Arch. Hist. Exact Sci. 60 (1): 1–121. doi:10.1007/s00407-005-0102-4. S2CID123157068. Archived from the original (PDF) on February 15, 2013.
  53. ^ This follows closely the first part of Cantor's 1891 paper.
  54. ^Cantor 1874. English translation: Ewald 1996, pp. 840–843.
  55. ^ For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumerous – see
  56. Moore, A. W. (April 1995). "A brief history of infinity" (PDF) . Scientific American. 272 (4): 112–116 (114). Bibcode:1995SciAm.272d.112M. doi:10.1038/scientificamerican0495-112.
  57. ^ For this, and more information on the mathematical importance of Cantor's work on set theory, see e.g., Suppes 1972.
  58. ^ Liouville, Joseph (May 13, 1844). A propos de l'existence des nombres transcendants.
  59. ^ The real algebraic numbers are the real roots of polynomial equations with integercoefficients.
  60. ^ For more details on Cantor's article, see Georg Cantor's first set theory article and
  61. Gray, Robert (1994). "Georg Cantor and Transcendental Numbers" (PDF) . American Mathematical Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR2975129. . Gray (pp. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number.
  62. ^ Cantor's construction starts with the set of transcendentals T and removes a countable subset<tn> (for example, tn = e / n). Call this set T0. Then T = T0 ∪ <tn> = T0 ∪ <t2n-1> ∪ <t2n>. The set of reals R = T ∪ <an> = T0 ∪ <tn> ∪ <an> where an is the sequence of real algebraic numbers. So both T and R are the union of three pairwise disjoint sets: T0 and two countable sets. A one-to-one correspondence between T and R is given by the function: f(t) = t if tT0, f(t2n-1) = tn, and f(t2n) = an. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals (Cantor 1879, p. 4).
  63. ^Dauben 1977, p. 89.
  64. ^Cantor 1883.
  65. ^Cantor (1895), Cantor (1897). The English translation is Cantor 1955.
  66. ^
  67. Wallace, David Foster (2003). Everything and More: A Compact History of Infinity. New York: W. W. Norton and Company. p. 259. ISBN978-0-393-00338-3 .
  68. ^Dauben 1979, pp. 69, 324 63n. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.
  69. ^ Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself among the most prominent of these is W. Hugh Woodin. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.
  70. ^Cantor 1883, pp. 587–588 English translation: Ewald 1996, pp. 916–917.
  71. ^Hallett 1986, pp. 41–42.
  72. ^Moore 1982, p. 42.
  73. ^Moore 1982, p. 51. Proof of equivalence: If a set is well-ordered, then its cardinality is an aleph since the alephs are the cardinals of well-ordered sets. If a set's cardinality is an aleph, then it can be well-ordered since there is a one-to-one correspondence between it and the well-ordered set defining the aleph.
  74. ^Hallett 1986, pp. 166–169.
  75. ^ Cantor's proof, which is a proof by contradiction, starts by assuming there is a set S whose cardinality is not an aleph. A function from the ordinals to S is constructed by successively choosing different elements of S for each ordinal. If this construction runs out of elements, then the function well-orders the set S. This implies that the cardinality of S is an aleph, contradicting the assumption about S. Therefore, the function maps all the ordinals one-to-one into S. The function's image is an inconsistent submultiplicity contained in S, so the set S is an inconsistent multiplicity, which is a contradiction. Zermelo criticized Cantor's construction: "the intuition of time is applied here to a process that goes beyond all intuition, and a fictitious entity is posited of which it is assumed that it could make successive arbitrary choices." (Hallett 1986, pp. 169–170.)
  76. ^Moore 1988, pp. 52–53 Moore and Garciadiego 1981, pp. 330–331.
  77. ^Moore and Garciadiego 1981, pp. 331, 343 Purkert 1989, p. 56.
  78. ^Moore 1982, pp. 158–160. Moore argues that the latter was his primary motivation.
  79. ^ Moore devotes a chapter to this criticism: "Zermelo and His Critics (1904–1908)", Moore 1982, pp. 85–141.
  80. ^Moore 1982, pp. 158–160. Zermelo 1908, pp. 263–264 English translation: van Heijenoort 1967, p. 202.
  81. ^Hallett 1986, pp. 288, 290–291. Cantor had pointed out that inconsistent multiplicities face the same restriction: they cannot be members of any multiplicity. (Hallett 1986, p. 286.)
  82. ^Hallett 1986, pp. 291–292.
  83. ^Zermelo 1930 English translation: Ewald 1996, pp. 1208–1233.
  84. ^Dauben 1979, p. 295.
  85. ^Dauben 1979, p. 120.
  86. ^Hallett 1986, p. 13. Compare to the writings of Thomas Aquinas.
  87. ^
  88. Hedman, Bruce (1993). "Cantor's Concept of Infinity: Implications of Infinity for Contingence". Perspectives on Science and Christian Faith. 45 (1): 8–16 . Retrieved March 5, 2020 .
  89. ^
  90. Dauben, Joseph Warren (1979). Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton University Press. ISBN9780691024479 . JSTORj.ctv10crfh1.
  91. ^
  92. Dauben, Joseph Warren (1978). "Georg Cantor: The Personal Matrix of His Mathematics". Isis. 69 (4): 548. doi:10.1086/352113. JSTOR231091. PMID387662. S2CID26155985 . Retrieved March 5, 2020 . The religious dimension which Cantor attributed to his transfinite numbers should not be discounted as an aberration. Nor should it be forgotten or separated from his existence as a mathematician. The theological side of Cantor's set theory, though perhaps irrelevant for understanding its mathematical content, is nevertheless essential for the full understanding of his theory and why it developed in its early stages as it did.
  93. ^Dauben 1979, p. 225
  94. ^Dauben 1979, p. 266.
  95. ^
  96. Snapper, Ernst (1979). "The Three Crises in Mathematics: Logicism, Intuitionism and Formalism" (PDF) . Mathematics Magazine. 524 (4): 207–216. doi:10.1080/0025570X.1979.11976784. Archived from the original (PDF) on August 15, 2012 . Retrieved April 2, 2013 .
  97. ^
  98. Davenport, Anne A. (1997). "The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century". Isis. 88 (2): 263–295. doi:10.1086/383692. JSTOR236574. S2CID154486558.
  99. ^ abDauben 1977, p. 85.
  100. ^Cantor 1932, p. 404. Translation in Dauben 1977, p. 95.
  101. ^Dauben 1979, p. 296.
  102. ^
  103. Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind". American Catholic Philosophical Quarterly. 83 (4): 533–553. doi:10.5840/acpq200983444.
  104. ^
  105. Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind". American Catholic Philosophical Quarterly. 84 (3): 535.
  106. ^
  107. Ferreiros, Jose (2004). "The Motives Behind Cantor's Set Theory—Physical, Biological and Philosophical Questions" (PDF) . Science in Context. 17 (1–2): 49–83. doi:10.1017/S0269889704000055. PMID15359485. S2CID19040786.
  108. ^Dauben 1979, p. 144.
  109. ^Dauben 1977, pp. 91–93.
  110. ^ On Cantor, Husserl, and Gottlob Frege, see Hill and Rosado Haddock (2000).
  111. ^ "Dauben 1979, p. 96.
  112. ^ Russell, Bertrand The Autobiography of Bertrand Russell, George Allen and Unwin Ltd., 1971 (London), vol. 1, p. 217.
  113. ^E.g., Grattan-Guinness's only evidence on the grandfather's date of death is that he signed papers at his son's engagement.
  114. ^ For example, Jewish Encyclopedia, art. "Cantor, Georg" Jewish Year Book 1896–97, "List of Jewish Celebrities of the Nineteenth Century", p. 119 this list has a star against people with one Jewish parent, but Cantor is not starred.
  115. ^ abcdPurkert and Ilgauds 1985, p. 15.
  116. ^ For more information, see: Dauben 1979, p. 1 and notes Grattan-Guinness 1971, pp. 350–352 and notes Purkert and Ilgauds 1985 the letter is from Aczel 2000, pp. 93–94, from Louis' trip to Chicago in 1863. It is ambiguous in German, as in English, whether the recipient is included.
  117. ^ Tannery, Paul (1934) Memoires Scientifique 13 Correspondance, Gauthier-Villars, Paris, p. 306.
  118. ^Dauben 1979, p. 274.
  119. ^ Mendelsohn, Ezra (ed.) (1993) Modern Jews and their musical agendas, Oxford University Press, p. 9.
  120. ^Ismerjük oket?: zsidó származású nevezetes magyarok arcképcsarnoka, István Reményi Gyenes Ex Libris, (Budapest 1997), pages 132–133
  121. ^ Russell, Bertrand. Autobiography, vol. I, p. 229. In English in the original italics also as in the original.
  122. ^Grattan-Guinness 1971, p. 350.
  123. ^Grattan-Guinness 1971 (quotation from p. 350, note), Dauben 1979, p. 1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.)
  124. ^Dauben 1979
  125. ^ Dauben, J.: The development of the Cantorian set theory, pp.
  • Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR2708842. .
  • Dauben, Joseph W. (1979). Georg Cantor: his mathematics and philosophy of the infinite . Boston: Harvard University Press. ISBN978-0-691-02447-9 . .
  • Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF) . Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation.
  • Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN978-0-19-853271-2 . .
  • Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837. .
  • Grattan-Guinness, Ivor (2000). The Search for Mathematical Roots: 1870–1940. Princeton University Press. ISBN978-0-691-05858-0 . .
  • Hallett, Michael (1986). Cantorian Set Theory and Limitation of Size. New York: Oxford University Press. ISBN978-0-19-853283-5 . .
  • Moore, Gregory H. (1982). Zermelo's Axiom of Choice: Its Origins, Development & Influence. Springer. ISBN978-1-4613-9480-8 . .
  • Moore, Gregory H. (1988). "The Roots of Russell's Paradox". Russell: The Journal of Bertrand Russell Studies. 8: 46–56. doi: 10.15173/russell.v8i1.1732 . .
  • Moore, Gregory H. Garciadiego, Alejandro (1981). "Burali-Forti's Paradox: A Reappraisal of Its Origins". Historia Mathematica. 8 (3): 319–350. doi: 10.1016/0315-0860(81)90070-7 . .
  • Purkert, Walter (1989). "Cantor's Views on the Foundations of Mathematics". In Rowe, David E. McCleary, John (eds.). The History of Modern Mathematics, Volume 1. Academic Press. pp. 49–65. ISBN978-0-12-599662-4 . .
  • Purkert, Walter Ilgauds, Hans Joachim (1985). Georg Cantor: 1845–1918. Birkhäuser. ISBN978-0-8176-1770-7 . .
  • Suppes, Patrick (1972) [1960]. Axiomatic Set Theory. New York: Dover. ISBN978-0-486-61630-8 . . Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics.
  • Zermelo, Ernst (1908). "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/bf01449999. S2CID120085563. .
  • Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF) . Fundamenta Mathematicae. 16: 29–47. doi: 10.4064/fm-16-1-29-47 . .
  • van Heijenoort, Jean (1967). From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. ISBN978-0-674-32449-7 . .

Primary literature in English Edit

Primary literature in German Edit

  • Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF) . Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID199545885.
  • Cantor, Georg (1878). "Ein Beitrag zur Mannigfaltigkeitslehre". Journal für die Reine und Angewandte Mathematik. 1878 (84): 242–258. doi:10.1515/crll.1878.84.242. .
  • Georg Cantor (1879). "Ueber unendliche, lineare Punktmannichfaltigkeiten (1)". Mathematische Annalen. 15 (1): 1–7. doi:10.1007/bf01444101. S2CID179177510.
  • Georg Cantor (1880). "Ueber unendliche, lineare Punktmannichfaltigkeiten (2)". Mathematische Annalen. 17 (3): 355–358. doi:10.1007/bf01446232. S2CID179177438.
  • Georg Cantor (1882). "Ueber unendliche, lineare Punktmannichfaltigkeiten (3)". Mathematische Annalen. 20 (1): 113–121. doi:10.1007/bf01443330. S2CID177809016.
  • Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (4)". Mathematische Annalen. 21 (1): 51–58. doi:10.1007/bf01442612. S2CID179177480.
  • Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen. 21 (4): 545–591. doi:10.1007/bf01446819. S2CID121930608. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
  • Georg Cantor (1891). "Ueber eine elementare Frage der Mannigfaltigkeitslehre" (PDF) . Jahresbericht der Deutschen Mathematiker-Vereinigung. 1: 75–78.
  • Cantor, Georg (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512. doi:10.1007/bf02124929. S2CID177801164. Archived from the original on April 23, 2014.
  • Cantor, Georg (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. doi:10.1007/bf01444205. S2CID121665994.
  • Cantor, Georg (1932). Ernst Zermelo (ed.). "Gesammelte Abhandlungen mathematischen und philosophischen inhalts". Berlin: Springer. Archived from the original on February 3, 2014. . Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.

Secondary literature Edit

  • Aczel, Amir D. (2000). The Mystery of the Aleph: Mathematics, the Kabbala, and the Search for Infinity. New York: Four Walls Eight Windows Publishing. . 0-7607-7778-0. A popular treatment of infinity, in which Cantor is frequently mentioned.
  • Dauben, Joseph W. (June 1983). "Georg Cantor and the Origins of Transfinite Set Theory". Scientific American. 248 (6): 122–131. Bibcode:1983SciAm.248f.122D. doi:10.1038/scientificamerican0683-122.
  • Ferreirós, José (2007). Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought. Basel, Switzerland: Birkhäuser. . 3-7643-8349-6 Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory.
  • Halmos, Paul (1998) [1960]. Naive Set Theory. New York & Berlin: Springer. . 3-540-90092-6
  • Hilbert, David (1926). "Über das Unendliche". Mathematische Annalen. 95: 161–190. doi:10.1007/BF01206605. S2CID121888793.
  • Hill, C. O. Rosado Haddock, G. E. (2000). Husserl or Frege? Meaning, Objectivity, and Mathematics. Chicago: Open Court. . 0-8126-9538-0 Three chapters and 18 index entries on Cantor.
  • Meschkowski, Herbert (1983). Georg Cantor, Leben, Werk und Wirkung (Georg Cantor, Life, Work and Influence, in German). Vieweg, Braunschweig.
  • Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind"[1], American Catholic Philosophical Quarterly, 83 (4): 532–553, https://doi.org/10.5840/acpq200983444. With acknowledgement of Dauben's pioneering historical work, this article further discusses Cantor's relation to the philosophy of Spinoza and Leibniz in depth, and his engagement in the Pantheismusstreit. Brief mention is made of Cantor's learning from F.A.Trendelenburg.
  • Penrose, Roger (2004). The Road to Reality. Alfred A. Knopf. . 0-679-77631-1 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist.
  • Rucker, Rudy (2005) [1982]. Infinity and the Mind. Princeton University Press. . 0-553-25531-2 Deals with similar topics to Aczel, but in more depth.
  • Rodych, Victor (2007). "Wittgenstein's Philosophy of Mathematics". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. .
  • Leonida Lazzari, L'infinito di Cantor. Editrice Pitagora, Bologna, 2008.
    at Internet Archive
  • O'Connor, John J. Robertson, Edmund F., "Georg Cantor", MacTutor History of Mathematics archive, University of St Andrews
  • O'Connor, John J. Robertson, Edmund F., "A history of set theory", MacTutor History of Mathematics archive, University of St Andrews Mainly devoted to Cantor's accomplishment.
  • Stanford Encyclopedia of Philosophy: Set theory by Thomas Jech. The Early Development of Set Theory by José Ferreirós.
  • "Cantor infinities", analysis of Cantor's 1874 article, BibNum(for English version, click 'à télécharger'). There is an error in this analysis. It states Cantor's Theorem 1 correctly: Algebraic numbers can be counted. However, it states his Theorem 2 incorrectly: Real numbers cannot be counted. It then says: "Cantor notes that, taken together, Theorems 1 and 2 allow for the redemonstration of the existence of non-algebraic real numbers …" This existence demonstration is non-constructive. Theorem 2 stated correctly is: Given a sequence of real numbers, one can determine a real number that is not in the sequence. Taken together, Theorem 1 and this Theorem 2 produce a non-algebraic number. Cantor also used Theorem 2 to prove that the real numbers cannot be counted. See Cantor's first set theory article or Georg Cantor and Transcendental Numbers.

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Were the problems Fibonacci solved in his work &ldquoFlos&rdquo posed specifically for him? - History

In the previous chapter, we mentioned the emperor Frederick II, whose court was located in Sicily. His encouragement of arts and sciences gave a voice to one of the most remarkable mathematicians of the Middle Ages, Leonardo of Pisa, with whom we begin our discussion of late Medieval mathematics.

29.1 Leonardo of Pisa (Fibonacci)

As soon as translations from Arabic into Latin became generally available in the twelfth and thirteenth centuries, Western Europeans began to learn about algebra. The first work translated (by Robert of Chester in 1145) was al-Khwarizmi's Algebra. Several talented mathematicians appeared early on who were able to make original contributions to the development of algebra. In some cases the books that they wrote were not destined to be published for many centuries, but at least one of them formed part of an Italian tradition of algebra that continued for several centuries. That tradition begins with Leonardo, who wrote several mathematical works, the best known of which is the Liber abaci. 1

29.1.1 The Liber abaci

Many of the problems in the Liber abaci (Book of Computation) reflect the routine computations that must be performed when converting currencies. These are applications of the Rule of Three that we have found in Brahmagupta and Bhaskara. Many of the other problems are purely fanciful. Leonardo's indebtedness to Arabic sources was detailed by Levey (1966), who listed 29 problems in the Liber abaci that are identical to problems in the Algebra of Abu Kamil. In particular, the problem of separating the number 10 into two parts satisfying an extra condition occurs many times. For example, one problem is to find x such that .

29.1.2 The Fibonacci Sequence

The most famous (not the most profound) of Leonardo's achievements is a problem from his Liber abaci, whose second edition appeared in 1202: How many pairs of rabbits can be bred from one pair in one year, given that each pair produces a new pair each month, beginning two months after its birth?

By enumeration of cases, the author concludes that there will be 377 pairs, and “in this way you can do it for the case of infinite numbers of months.” The reasoning is simple. Each month, those pairs that were alive two months earlier produce duplicates of themselves. Hence the total number of rabbits after n + 2 months is the number alive after n + 1 months plus the number alive after n months. That is, each term in the sequence is the sum of the two preceding numbers.

Assuming the original pair was a mature pair, ready to reproduce, the sequence generated in this way&mdashstarting at the beginning of the year, when 0 months have elapsed&mdashis (1, 2, 3, 5, 8,. . .), and its 13th term is 377. This sequence has been known as the Fibonacci sequence since the printing of the Liber abaci in the nineteenth century. The Fibonacci sequence has been an inexhaustible source of identities. Many curious representations of its terms have been obtained, and there is a mathematical journal, the Fibonacci Quarterly, named in its honor and devoted to its lore.

A Practical Application

In 1837 and 1839 the crystallographer Auguste Bravais (1811–1863) and his brother Louis (1801–1843) published articles on the growth of plants. 2 In these articles they studied the spiral patterns in which new branches grow out of the limbs of certain trees and classified plants into several categories according to this pattern. For one of these categories they gave the amount of rotation around the limb between successive branches as 137° 30&prime 28”. Now, one could hardly measure the limb of a tree so precisely. To measure within 10° would require extraordinary precision. To refine such crude measurements by averaging to the claimed precision of 1”, that is, 1/3600 of a degree, would require thousands of individual measurements. In fact, the measurements were carried out in a more indirect way, by counting the total number of branches after each full turn of the spiral. Many observations convinced the brothers Bravais that normally there were three branches in a little less than two turns, five in a little more three turns, eight in a little less than five turns, and thirteen in a little more than eight turns. For that reason they took the actual amount of revolution between successive branches to be the number we call of a complete (360°) revolution, since

Observe that 360° ÷ &Phi ≈ 222 . 4922359° ≈ 222° 29&prime 32 ” = 360° &minus (137° 30&prime 28 ”). An illustration of this kind of growth is shown in Fig. 29.1. The picture shows three views of a branch of a flowering crab apple tree with the twigs cut off and the points from which they grew marked by pushpins. When these pins are joined by string, the string follows a helical path of nearly constant slope along the branch. By simply counting, one can get an idea of the average number of twigs per turn. For example, the fourth intersection is between pins 6 and 7, indicating that the average number of pins per turn up to that point is between and . One can see that the pins that fall nearest to the intersection of this helical path with the meridian line marked along the length of the branch are pins numbered 3, 5, 8, and 13, which are Fibonacci numbers, and that the intersections they are near come at the end of 2, 3, 5, and 8 revolutions, respectively, also Fibonacci numbers. Thus the average number of twigs per turn is approximately or or or . The brothers Bravais knew that the ratios of successive Fibonacci numbers are the terms in the continued-fraction expansion of the Golden Ratio , and hence they chose this elegant way of formulating what they had observed. By looking at the side of the intersection where the corresponding pins are in Fig. 29.1, you can see that the first and third of these approximations are underestimates and the second and fourth are overestimates. You can also see that the approximation gets better as the number of turns increases.

Figure 29.1 Three views of a branch of a flowering crab apple tree.

This pattern is not universal among plants, although the brothers Bravais were able to find several classes of plants that exhibit a pattern of this type, with different values for the first two terms of the sequence.

29.1.3 The Liber quadratorum

In his Liber quadratorum [Book of Squares (Sigler, 1987)] Leonardo speculated on the difference between square and nonsquare numbers. In the prologue, addressed to the Emperor Frederick II, Leonardo says that he had been inspired to write the book because a certain John of Palermo, whom he had met at Frederick's court, had challenged him to find a square number such that if 5 is added to it or subtracted from it, the result is again a square. 3 This question inspired him to reflect on the difference between square and nonsquare numbers. He then notes his pleasure on learning that Frederick had actually read one of his previous books and uses that fact as justification for writing on the challenge problem.

The Liber quadratorum is written in the spirit of Diophantus and shows a keen appreciation of the conditions under which a rational number is a square. Indeed, the ninth of its 24 propositions is a problem of Diophantus: Given a nonsquare number that is the sum of two squares, find a second pair of squares having this number as their sum. This problem is Problem 9 of Book 2 of Diophantus, as discussed in Section 4 of Chapter 9. Leonardo's solution of this problem, like that of Diophantus, involves a great deal of arbitrariness, since the problem does not have a unique solution. The resemblance in some points is so strong that one is inclined to think that Leonardo saw a copy of Diophantus, or, more likely, an Arabic work commenting and extending the work of Diophantus. This question is discussed by the translator of the Liber quadratorum (Sigler, 1987, pp. xi–xii), who notes that strong resemblances have been pointed out between the Liber quadratorum and a book by Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji (953–1029) called the Fakhri, 4 parts of which were copied from the Arithmetica, but that there are also parts of the Liber quadratorum that are original.

One advance in the Liber quadratorum is the use of general letters in an argument. Although in some proofs Leonardo argues much as Diophantus does, using specific numbers, he becomes more abstract in others. For example, Proposition 5 requires finding two numbers, the sum of whose squares is a square that is also the sum of the squares of two given numbers. He says to proceed as follows. Let the two given numbers be. a . and. b . and the sum of their squares. g . . Now take any other two numbers. de . and. ez . [not proportional to the given numbers] the sum of whose squares is a square. These two numbers are arranged as the legs of a right triangle. If the square on the hypotenuse of this triangle is. g ., the problem is solved. If the square on the hypotenuse is larger than. g ., mark off the square root of. g . on the hypotenuse. The projections (as we would call them) of this portion of the hypotenuse on each of the legs are known, since their ratios to the square root of. g . are known. Moreover, that ratio is rational, since they are the same as the ratios of. a . and. b . to the hypotenuse of the original triangle. These two projections therefore provide the new pair of numbers. Being proportional to. a . and. b ., which are not proportional to the two numbers given originally, they must be different from those numbers.

This argument is more convincing, because it is more abstract, than proofs by example, but the geometric picture plays an important role in making the proof comprehensible.

29.1.4 The Flos

Leonardo's approach to algebra begins to look modern in other ways as well. In one of his works, called the Flos super solutionibus quarumdam questionum ad numerum et ad geometriam vel ad utrumque pertinentum [The Full Development 5 of the Solutions of Certain Questions Pertaining to Number or Geometry or Both (Boncompagni 1854, p. 4)] he reports the challenge from John of Palermo mentioned above, which was to find a number satisfying x 3 + 2x 2 + 10x = 20 using the methods given by Euclid in Book 10 of the Elements, that is, to construct a line of this length using straightedge and compass. In working on this question, Leonardo made two important contributions to algebra, one numerical and one theoretical. The numerical contribution was to give the unique positive root in sexagesimal notation correct to six places. The theoretical contribution was to show by using divisibility properties of numbers that there cannot be a rational solution or a solution obtained using only rational numbers and square roots of rational numbers.

29.2 Hindu–Arabic numerals

The Liber abaci advocated the use of the Hindu–Arabic numerals that we are familiar with. Partly because of the influence of that book, the advantages of this system came to be appreciated, and within two centuries these numerals were winning general acceptance. In 1478, an arithmetic was published in Treviso, Italy, explaining the use of Hindu–Arabic numerals and containing computations in the form shown in Fig 26.1 of Chapter 26. In the sixteenth century, scholars such as Robert Recorde (1510–1558) in Britain and Adam Ries (1492–1559) in Germany, advocated the use of the Hindu–Arabic system and established it as a universal standard.

The system was explained by the Flemish mathematician and engineer Simon Stevin (1548–1620) in his 1585 book De Thiende (Decimals). Stevin took only a few pages to explain, in essentially modern terms, how to add, subtract, multiply, and divide decimal numbers. He then showed the application of this method of computing in finding land areas and the volumes of wine vats. He wrote concisely, as he said, “because here we are writing for teachers, not students.” His notation appears slightly odd, however, since he put a circled 0 where we now have the decimal point, and thereafter he indicated the rank of each digit by a similarly encircled number. For example, he would write 13.4832 as . Here is his explanation of the problem of expressing 0.07 ÷ 0.00004:

When the divisor is larger [has more digits] than the dividend, we adjoin to the dividend as many zeros as desired or necessary. For example, if is to be divided by , I place some 0s next to the 7, namely 7000. This number is then divided as above, as follows:

Hence the quotient is 1750 0 (Gericke and Vogel, 1965, p. 19).

From a 1535 illustration to theMargarita philosophica (Philosophical Pearl) published byGregor Reisch (1467–1525) in 1503. Copyright © FotoMarburg/Art Resource.

Except for the location of the digits and the cross-out marks, this notation is essentially what is now used by school children in the United States. In other countries&mdashRussia, for example&mdashthe divisor would be written just to the right of the dividend and the quotient just below the divisor.

Stevin also knew what to do if the division does not come out even. He pointed out that when is divided by , the result is an infinite succession of 3s and that the exact answer will never be reached. He commented, “In such a case, one may go as far as the particular case requires and neglect the excess. It is certainly true that , or , and so on, are exactly equal to the required result, but our goal is to work only with whole numbers in this decimal computation, since we have in mind what occurs in human business, where [small parts of small measures] are ignored.” Here we have a clear case in which the existence of infinite decimal expansions is admitted, without any hint of the possibility of irrational numbers. Stevin was an engineer, not a theoretical mathematician. His examples were confined to what is of practical value in business and engineering, and he made no attempt to show how to calculate with an actually infinite decimal expansion.

Stevin did, however, suggest a reform in trigonometry that was ignored until the advent of hand-held calculators, remarking that, “if we can trust our experience (with all due respect to Antiquity and thinking in terms of general usefulness), it is clear that the series of divisions by 10, not by 60, is the most efficient, at least among those that are by nature possible.” On those grounds, Stevin suggested that degrees be divided into decimal fractions rather than minutes and seconds. Modern hand-held calculators now display angles in exactly this way, despite the scornful remark of a twentieth-century mathematician that this mixture of sexagesimal and decimal notation proves that“it required four millennia to produce a system of angle measurement that is completely absurd.”

29.3 Jordanus Nemorarius

The translator and editor of Jordanus' book De numeris datis (On Given Numbers, Hughes, 1981, p. 11) says, “It is reasonable to assume. . .that Jordanus was influenced by al-Khwarizmi's work.” This conclusion was reached on the basis of Jordanus' classification of quadratic equations and his order of expounding the three types, among other resemblances between the two works.

De numeris datis is the algebraic equivalent of Euclid's Data. Where Euclid says that a line is given (determined) if its ratio to a given line is given, Jordanus Nemorarius says that a number is given if its ratio to a given number is given. The well-known elementary fact that two numbers can be found if their sum and difference are known is generalized to the theorem that any set of numbers can be found if the differences of the successive numbers and the sum of all the numbers is known. This book contains a large variety of data sets that determine numbers. For example, if the sum of the squares of two numbers is known, and the square of the difference of the numbers is known, the numbers can be found. The four books of De numeris datis contain about 100 such results. These results admit a purely algebraic interpretation. For example, in Book 4 Jordanus Nemorarius writes:

If a square with the addition of its root multiplied by a given number makes a given number, then the square itself will be given. [p. 100] 6

Where earlier mathematicians would have proved this proposition with examples, Jordanus Nemorarius uses letters representing abstract numbers. The assertion is that there is only one (positive) number x such that x 2 + &alphax = &beta, and that x can be found if &alpha and &beta are given.

29.4 Nicole d'Oresme

A work entitled Tractatus de latitudinibus formarum (Treatise on the Latitude of Forms) was published in Paris in 1482 and ascribed to Oresme, but probably written by one of his students. It contains descriptions of the graphical representation of “intensities.” This concept finds various expressions in physics, corresponding intuitively to the idea of density. In Oresme's language, an “intensity” is any constant of proportionality. Velocity, for example, is the “intensity” of motion.

We think of analytic geometry as the application of algebra to geometry. Its origins in Europe, however, antedate the high period of European algebra by a century or more. The first adjustment in the way mathematicians think about physical dimensions, an essential step on the way to analytic geometry, occurred in the fourteenth century. The crucial idea found in the representation of distance as the “area under the velocity curve” was that since the area of a rectangle is computed by multiplying length and width and the distance traveled at constant speed is computed by multiplying velocity and time, it follows that if one line is taken proportional to time and a line perpendicular to it is proportional to a (constant) velocity, the area of the resulting rectangle is proportional to the distance traveled.

Oresme considered three forms of qualities, which he labeled uniform, uniformly difform, and difformly difform. We would call these classifications constant, linear, and nonlinear. Examples are shown in Fig. 29.2, which can be found in another of Oresme's works. Oresme (or his students) realized that the “difformly difform” constituted a large class of qualities and mentioned specifically that a semicircle could be the representation of such a quality.

Figure 29.2 Nicole Oresme's classification of motions.

The advantage of representing a distance by an area rather than a line appeared in the case when the velocity changed during a motion. In the simplest nontrivial case the velocity was uniformly difform. This is the case of constant acceleration. In that case, the distance traversed is what it would have been had the body moved the whole time with the velocity it had at the midpoint of the time of travel. This is the case now called uniformly accelerated motion. According to Clagett (1968, p. 617), this rule was first stated by William Heytesbury (ca. 1313–ca. 1372) of Merton College, Oxford around 1335 and was well known during the Middle Ages. Boyer (1949, p. 83) says that the rule was stated around this time by another fourteenth-century Oxford scholar named Richard Suiseth, 7 known as Calculator for his book Liber calculatorum. Suiseth shares with Oresme the credit for having proved that the harmonic series () diverges.

The rule just stated is called the Merton rule. In his book De configurationibus qualitatum et motuum, Oresme applied these principles to the analysis of such motion and gave a simple geometric proof of the Merton Rule. He illustrated the three kinds of motion by drawing a figure similar to Fig. 29.2. He went on to say that if a difformly difform quality was composed of uniform or uniformly difform parts, as in the example in Fig. 29.2, its quantity could be measured by (adding) its parts. He then pushed this principle to the limit, saying that if the quality was difform but not made up of uniformly difform parts, say being represented by a curve, then “it is necessary to have recourse to the mutual measurement of curved figures” (Clagett, 1968, p. 410). This statement must mean that the distance traveled is the “area under the velocity curve” in all three cases. Oresme unfortunately did not give any examples of the more general case, but he could hardly have done so, since the measurement of figures bounded by curves was still very primitive in his day.

29.5 Trigonometry: Regiomontanus and Pitiscus

In the late Middle Ages, the treatises translated into Latin from Arabic and Greek were made the foundation for ever more elaborate mathematical theories.

29.5.1 Regiomontanus

Analytic geometry as we know it today would be unthinkable without plane trigonometry. Latin translations of Arabic texts of trigonometry, such as the text of Nasir al-Din al-Tusi, began to circulate in Europe in the late Middle Ages. These works provided the foundation for such books as De triangulis omnimodis (On General Triangles) by Regiomontanus, published in 1533, more than half a century after the author's death. This book contained trigonometry almost in the form still taught. Book 2, for example, contains as its first theorem the law of sines for plane triangles, which asserts that the sides of triangles are proportional to the sines of the angles opposite them. The main difference between this trigonometry and ours is that a sine remains a line rather than a ratio. It is referred to an arc rather than to an angle. It was once believed that Regiomontanus discovered the law of sines for spherical triangles (Proposition 16 of Book 4) as well, 8 but we now know that this theorem was known at least 500 years earlier to Muslim mathematicians whose work Regiomontanus must have read.

29.5.2 Pitiscus

A more advanced book on trigonometry, which reworked the reasoning of Heron on the area of a triangle given its sides, was Trigonometriæ sive de dimensione triangulorum libri quinque (Five Books of Trigonometry, or, On the Size of Triangles), published in 1595 and written by the Calvinist theologian Bartholomeus Pitiscus (1561–1613). This was the book that established the name trigonometry for this subject even though the basic functions are called circular functions (Fig. 29.3). Pitiscus showed how to determine the parts into which a side of a triangle is divided by the altitude, given the lengths of the three sides, or, conversely, to determine one side of a triangle knowing the other two sides and the length of the portion of the third side cut off by the altitude. To guarantee that the angles adjacent to the side were acute, he stated the theorem only for the altitude from the vertex of the largest angle.

Figure 29.3 The three basic trigonometric functions: The secant OB, which cuts the circle the tangent AB, which touches the circle the sine CD, which is half of a chord.

Pitiscus' way of deriving his fundamental relation was as follows. If the shortest side of the triangle ABC is AC and the longest is BC, let the altitude to BC be AG, as in Fig. 29.4. Draw the circle through C with center at A, so that B lies outside the circle, and let the intersections of the circle with AB and BC be E and F, respectively. Then extend BA to meet the circle at D, and connect CD. Then &angBFE is the supplement of &angCFE. But &angEDC is also supplementary to &angCFE, since the two are inscribed in arcs that partition the circle. Thus, &angBFE = &ang CDB, and so the triangles BCD and BEF are similar. It follows that , and since , , , and , we find

Observe that . When this substitution is made, we obtain what is now known as the law of cosines:

Figure 29.4 Pitiscus' derivation of the proportions in which an altitude divides a side of a triangle.

Pitiscus also gave an algebraic solution of the trisection problem discovered by an earlier mathematician named Jobst Bürgi (1552–1632). The solution had been based on the fact that the chord of triple an angle is three times the chord of the angle minus the cube of the chord of the angle. This relation makes no sense in terms of geometric dimension it is a purely numerical relation. It is interesting that it is stated in terms of chords, since Pitiscus surely knew about sines.

29.6 A Mathematical Skill: ProsthaphÆresis

Pitiscus needed trigonometry in order to do astronomy, especially to solve spherical triangles. Since the computations in such problems often become rather lengthy, Pitiscus discovered (probably in the writings of other mathematicians) a way to shorten the labor. While the difficulty of addition and subtraction grows at an even, linear rate with the number of digits being added, multiplying two n-digit numbers requires on the order of 2n 2 separate binary operations on integers. Thus the labor becomes excessive and error-prone for integers with any appreciable number of digits. As astronomy becomes more precise, of course, the number of digits to which quantities can be measured increases. Thus a need arose some centuries ago for a shorter, less error-prone way of doing approximate computations.

The ultimate result of the search for such a method was the subject of logarithms. That invention, however, required a new and different point of view in algebra. Before it came along, mathematicians had found a way to make a table of sines serve the purpose that was later fulfilled by logarithms. Actually, the process could be greatly simplified by using only a table of cosines, but we shall follow Pitiscus, who used only a table of sines and thus was forced to compute the complement of an angle where we would simply look up the cosine. The principle is the same: converting a product to one or two additions and subtractions&mdashhence the name prosthaphæresis, from prosthæresis(taking forward, that is, addition) and aphæresis (taking away, that is, subtraction).

As just pointed out, the amount of labor involved in multiplying two numbers increases in direct proportion to the product of the numbers of digits in the two factors, while the labor of adding increases in proportion to the number of digits in the smaller number. Thus, multiplying two 15-digit numbers requires over 200 one-digit multiplications, and another 200 or so one-digit additions, while adding the two numbers requires only 15 such operations (not including carrying). It was the large number of digits in the table entries that caused the problem in the first place, but the key to the solution turned out to be in the structural properties of the sine function.

There are hints of this process in several sixteenth-century works, but we shall quote just one example. In his Trigonometria, first published in Heidelberg in 1595, Pitiscus posed the following problem: To solve the proportion in which the first term is the radius, while the second and third terms are sines, avoiding multiplication and division. The problem here is to find the fourth proportional x, satisfying r : a = b : x, where r is the radius of the circle and aand b are two sines (half-chords) in the circle. We can see immediately that x = ab/r, but as Pitiscus says, the idea is to avoid the multiplication and division, since in the trigonometric tables the time a and b might easily have seven or eight digits each.

The key to prosthaphæresis is the well-known formula

This formula is applied as follows: If you have to multiply two large numbers, find two angles having the numbers as their sines. Replace one of the two angles by its complement. Next, add the angles and take the sine of their sum to obtain the first term then subtract the angles and take the sine of their difference to obtain a second term. Finally, divide the sum of these last two sines by 2 to obtain the product. To take a very simple example, suppose that we wish to multiply 155 by 36. A table of trigonometric functions shows that sin (8° 55 ') = 0.15500 and sin (90° &minus 68° 54 ') = 0.36000. Hence, since we moved the decimal points a total of five places to the left in the two factors, we obtain

In general, some significant figures will be lost in this kind of multiplication. Obviously, no labor is saved in this simple example, but for large numbers this procedure really does make things easier. In fact, multiplying even two seven-digit numbers would tax the patience of most modern people, since it would require about 100 separate multiplications and additions. A further advantage is that prosthaphæresis is less error-prone than multiplication. Its advantages were known to the Danish astronomer Tycho Brahe (1546–1601), 9 who used it in the astronomical computations connected with the precise observations he made at his observatory during the latter part of the sixteenth century.

This process could be simplified by using the addition and subtraction formula for cosines rather than sines. That formula is

29.7 Algebra: Pacioli and Chuquet

The fourteenth century, in which Nicole d'Oresme made such remarkable advances in geometry and nearly created analytic geometry, was also a time of rapid advance in algebra, epitomized by Antonio de’ Mazzinghi (ca. 1353–1383). His Trattato d'algebra (Treatise on Algebra) contains some complicated systems of linear and quadratic equations in as many as three unknown (Franci, 1988). He was one of the earliest algebraists to move the subject toward the numerical and away from the geometric interpretation of problems.

29.7.1 Luca Pacioli

In the fifteenth century, Luca Pacioli wrote Summa de arithmetica, geometrica, proportioni et proportionalita (Encyclopedia of Arithmetic, Geometry, Proportion, and Proportionality), which was closer to the elementary work of al-Khwarizmi and more geometrical in its approach to algebra than was the work of Mazzinghi. Actually, (Parshall, 1988) the work was largely a compilation of the works of Leonardo of Pisa, but it did bring the art of abbreviation closer to true symbolic notation. For example, what we now write as was written by Pacioli as

Here co means cosa (thing), the unknown. It is a translation of the Arabic word used by al-Khwarizmi. The abbreviation ce means censo (power), and Rv is probably a printed version of Rx, from the Latin radix, meaning root. 10 Pacioli's work was both an indication of how widespread knowledge of algebra had become by this time and an important element in propagating that knowledge even more widely. The sixteenth-century Italian algebraists who moved to the forefront of the subject and advanced it far beyond where it had been up to that time had all read Pacioli's treatise thoroughly.

29.7.2 Chuquet

According to Flegg (1988), on whose work the following exposition is based, there were several new things in the Triparty. One is a superscript notation similar to the modern notation for the powers of the unknown in an equation. The unknown itself is called the premier or “first,” that is, power 1 of the unknown. In this work, algebra is called the rigle des premiers “rule of firsts.” Chuquet listed the first 20 powers of 2 and pointed out that when two such numbers are multiplied, their indices are added. Thus, he had a clear idea of the laws of integer exponents. A second innovation in the Triparty is the free use of negative numbers as coefficients, solutions, and exponents. Still another innovation is the use of some symbolic abbreviations. For example, the square root is denoted R 2 (R for the Latin radix, or perhaps the French racine). The equation we would write as 3x 2 + 12 = 9x was written . Chuquet called this equation impossible, since its solution would involve taking the square root of &minus63.

His instructions are given in words. For example (Struik, 1986, p. 62), consider the equation

Chuquet says to subtract from both sides, so that the equation becomes

Next he says to square, getting

Subtracting (that is, 4x 2 ) from both sides and adding to both sides then yields

Chuquet's approach to algebra and its application can be gathered from one of the illustrative problems in the second part (Problem 35). This problem tells of a merchant who buys 15 pieces of cloth, spending a total of 160 ecus. Some of the pieces cost 11 ecus each, and the others 13 ecus. How many were bought at each price?

If x is the number bought at 11 ecus apiece, this problem leads to the equation 11x + 13(15 &minus x) = 160. Since the solution is , this means the merchant bought pieces at 13 ecus. How does one set about buying a negative number of pieces of cloth? Chuquet said that these pieces were bought on credit!

Problems and Questions

Mathematical Problems

29.1 Carry out Leonardo's description of the way to find two numbers the sum of whose squares is a square that is the sum of two other given squares in the particular case when the given numbers are. a . = 5 and. b . = 12 (the sum of whose squares is 169 = 13 2 ). Take. de . = 8 and. ez . = 15. Draw the right triangle described by Leonardo, and also carry out the numerical computation that produces the new pair for which the sum of the squares is again 169.

29.2 Use Pitiscus' law of cosines to find the third side of a triangle having sides of length 6 cm and 8 cm and such that the altitude to the side of length 8 cm divides it into lengths of 5 cm and 3 cm. (There are two possible triangles, depending on the orientation.)

29.3 Use prosthaphæresis to find the product 829.038 × 66.9131. (First write this product as 10 5 × 0.829038 × 0.669131. Find the angles that have the last two numbers as cosines, and use the addition and subtraction formula for cosines given above.)

Historical Questions

29.4 What parts of the algebraic work of Leonardo of Pisa were compilations of work in earlier sources, and what parts were advances on that earlier work?

29.5 In what ways did the geometric work of Nicole of Oresme prefigure modern analytic geometry?

29.6 How did Regiomontanus and Pitiscus change the way mathematicians thought about trigonometry? How did their trigonometry continue to differ from what we use today?

Questions for Reflection

29.7 Was there scientific value in making use of the real (irrational, infinitely precise) number &Phi, as the Bravais brothers did, even though no actual plant grows exactly according to the rule they stated? Why wouldn't a rational approximation have done just as well?

29.8 How did the notion of geometric dimension (length, area, volume) limit the use of numerical methods in geometry? How did Oresme's latitude of forms help to overcome this limitation?

29.9 Does it make sense to interpret the purchase of a negative number of items as an amount bought on credit? Would it be better to interpret such a “purchase” as returned merchandise?

1. Devlin (2011), who has looked at the old manuscripts of this work, says that it is properly spelled Liber abbaci. The spelling we are using merely preserves a long-standing traditional usage.

2. See the article by I. Adler, D. Barabe, and R. V. Jean, “A history of the study of phyllotaxis,” Annals of Botany, 80 (1997), pp. 231–244, especially p. 234. The articles by Auguste and Louis Bravais are “Essai sur la disposition générale des feuilles curvisériées,” Annales des sciences naturelles, 7 (1837), pp. 42–110, and “Essai sur la disposition générale des feuilles rectisériées,” Congrès scientifique de France, 6 (1839), pp. 278–330.

3. Leonardo gave a general discussion of problems of this type, asking when m 2 + kn 2 and m2 + 2kn 2 can both be squares.

4. Apparently, this word means something like glorious and the full title might be translated as The Glory of Algebra.

5. The word flos means bloom and can used in the figurative sense of “the bloom of youth.” That appears to be its meaning here.

6. This translation is my own and is intended to be literal Hughes gives a smoother, more idiomatic translation on p. 168.

7. Also known as Richard Swyneshed and as Swineshead with a great variety of first names. There is uncertainty whether the works ascribed to this name are all due to the same person.

8. This law says that the sines of the sides of spherical triangles are proportional to the sines of their opposite angles. (Both sides and angles in a spherical triangle are measured in great-circle degrees.)

9. The formula for the product of two sines had been discovered in 1510 by Johann Werner (1468–1522). This formula and the similar formula for cosines were first published in 1588 in a small book entitled Fundamentum astronomicum written by Nicolai Reymers Baer (dates uncertain), known as Ursus, which is the Latin translation of Baer. Brahe, however, had already noticed their application in spherical trigonometry and had been using them during the 1580s. He even claimed credit for developing the technique himself. The origin of the technique of prosthaphæresis is complicated and uncertain. A discussion of it was given by Thoren (1988).

10. The symbol Rx should not be confused with the same symbol in pharmacy, which comes from the Latin recipe, meaning take.


The Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1986. [8] [10] [11] In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm + 1 . [9]

Variations of two earlier meters [is the variation]. For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]. In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations]. [a]

Hemachandra (c. 1150) is credited with knowledge of the sequence as well, [7] writing that "the sum of the last and the one before the last is the number . of the next mātrā-vṛtta." [15] [16]

Outside India, the Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci [6] [17] where it is used to calculate the growth of rabbit populations. [18] [19] Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: how many pairs will there be in one year?

  • At the end of the first month, they mate, but there is still only 1 pair.
  • At the end of the second month they produce a new pair, so there are 2 pairs in the field.
  • At the end of the third month, the original pair produce a second pair, but the second pair only mate without breeding, so there are 3 pairs in all.
  • At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.

At the end of the n th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month n – 2 ) plus the number of pairs alive last month (month n – 1 ). The number in the n th month is the n th Fibonacci number. [20]

The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas. [21]

The first 21 Fibonacci numbers Fn are: [2]

F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765

The sequence can also be extended to negative index n using the re-arranged recurrence relation

which yields the sequence of "negafibonacci" numbers [22] satisfying

Thus the bidirectional sequence is

F−8 F−7 F−6 F−5 F−4 F−3 F−2 F−1 F0 F1 F2 F3 F4 F5 F6 F7 F8
−21 13 −8 5 −3 2 −1 1 0 1 1 2 3 5 8 13 21

Closed-form expression Edit

Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed form expression. It has become known as Binet's formula, named after French mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre and Daniel Bernoulli: [23]

It follows that for any values a and b , the sequence defined by

If a and b are chosen so that U0 = 0 and U1 = 1 then the resulting sequence Un must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations:

Taking the starting values U0 and U1 to be arbitrary constants, a more general solution is:

Computation by rounding Edit

for all n ≥ 0 , the number Fn is the closest integer to φ n 5 >>>> . Therefore, it can be found by rounding, using the nearest integer function:

In fact, the rounding error is very small, being less than 0.1 for n ≥ 4 , and less than 0.01 for n ≥ 8 .

Fibonacci numbers can also be computed by truncation, in terms of the floor function:

As the floor function is monotonic, the latter formula can be inverted for finding the index n(F) of the largest Fibonacci number that is not greater than a real number F > 1 :

Limit of consecutive quotients Edit

Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio φ : [26] [27]

This convergence holds regardless of the starting values, excluding 0 and 0, or any pair in the conjugate golden ratio, − 1 / φ . [ clarification needed ] This can be verified using Binet's formula. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, . The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.

Decomposition of powers Edit

Since the golden ratio satisfies the equation

This expression is also true for n < 1 if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule F n = F n − 1 + F n − 2 . =F_+F_.>

A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is

Equivalently, the same computation may performed by diagonalization of A through use of its eigendecomposition:

The matrix A has a determinant of −1, and thus it is a 2×2 unimodular matrix.

This property can be understood in terms of the continued fraction representation for the golden ratio:

The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for φ , and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:

Taking the determinant of both sides of this equation yields Cassini's identity,

Moreover, since A n A m = A n+m for any square matrix A , the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing n into n + 1 ),

In particular, with m = n ,

These last two identities provide a way to compute Fibonacci numbers recursively in O(log(n)) arithmetic operations and in time O(M(n) log(n)) , where M(n) is the time for the multiplication of two numbers of n digits. This matches the time for computing the n th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with memoization). [28]

The question may arise whether a positive integer x is a Fibonacci number. This is true if and only if at least one of 5 x 2 + 4 +4> or 5 x 2 − 4 -4> is a perfect square. [29] This is because Binet's formula above can be rearranged to give

which allows one to find the position in the sequence of a given Fibonacci number.

This formula must return an integer for all n, so the radical expression must be an integer (otherwise the logarithm does not even return a rational number).

Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that Fn can be interpreted as the number of sequences of 1s and 2s that sum to n − 1. This can be taken as the definition of Fn, with the convention that F0 = 0, meaning no sum adds up to −1, and that F1 = 1, meaning the empty sum "adds up" to 0. Here, the order of the summand matters. For example, 1 + 2 and 2 + 1 are considered two different sums.

For example, the recurrence relation

or in words, the nth Fibonacci number is the sum of the previous two Fibonacci numbers, may be shown by dividing the Fn sums of 1s and 2s that add to n − 1 into two non-overlapping groups. One group contains those sums whose first term is 1 and the other those sums whose first term is 2. In the first group the remaining terms add to n − 2, so it has Fn-1 sums, and in the second group the remaining terms add to n − 3, so there are Fn−2 sums. So there are a total of Fn−1 + Fn−2 sums altogether, showing this is equal to Fn.

Similarly, it may be shown that the sum of the first Fibonacci numbers up to the nth is equal to the (n + 2)-nd Fibonacci number minus 1. [30] In symbols:

This is done by dividing the sums adding to n + 1 in a different way, this time by the location of the first 2. Specifically, the first group consists of those sums that start with 2, the second group those that start 1 + 2, the third 1 + 1 + 2, and so on, until the last group, which consists of the single sum where only 1's are used. The number of sums in the first group is F(n), F(n − 1) in the second group, and so on, with 1 sum in the last group. So the total number of sums is F(n) + F(n − 1) + . + F(1) + 1 and therefore this quantity is equal to F(n + 2).

A similar argument, grouping the sums by the position of the first 1 rather than the first 2, gives two more identities:

A different trick may be used to prove

Symbolic method Edit

Numerous other identities can be derived using various methods. Some of the most noteworthy are: [33]

Cassini's and Catalan's identities Edit

Cassini's identity states that

D'Ocagne's identity Edit

Putting k = 2 in this formula, one gets again the formulas of the end of above section Matrix form.

The generating function of the Fibonacci sequence is the power series

This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum:

for s(x) results in the above closed form.

Setting x = 1/k , the closed form of the series becomes

In particular, if k is an integer greater than 1, then this series converges. Further setting k = 10 m yields

Some math puzzle-books present as curious the particular value that comes from m = 1 , which is s ( 1 / 10 ) 10 = 1 89 = .011235 … <10>>=<89>>=.011235ldots > [35] Similarly, m = 2 gives

Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, we can write the sum of every odd-indexed reciprocal Fibonacci number as

and the sum of squared reciprocal Fibonacci numbers as

If we add 1 to each Fibonacci number in the first sum, there is also the closed form

and there is a nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio,

is known, but the number has been proved irrational by Richard André-Jeannin. [36]

The Millin series gives the identity [37]

Divisibility properties Edit

Every third number of the sequence is even and more generally, every kth number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property [38] [39]

Any three consecutive Fibonacci numbers are pairwise coprime, which means that, for every n,

gcd(Fn, Fn+1) = gcd(Fn, Fn+2) = gcd(Fn+1, Fn+2) = 1.

Every prime number p divides a Fibonacci number that can be determined by the value of p modulo 5. If p is congruent to 1 or 4 (mod 5), then p divides Fp − 1, and if p is congruent to 2 or 3 (mod 5), then, p divides Fp + 1. The remaining case is that p = 5, and in this case p divides Fp.

These cases can be combined into a single, non-piecewise formula, using the Legendre symbol: [40]

Primality testing Edit

The above formula can be used as a primality test in the sense that if

Fibonacci primes Edit

A Fibonacci prime is a Fibonacci number that is prime. The first few are:

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, . OEIS: A005478 .

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many. [42]

Fkn is divisible by Fn, so, apart from F4 = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number. [43]

The only nontrivial square Fibonacci number is 144. [44] Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers. [45] In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers. [46]

1, 3, 21, 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming. [47]

No Fibonacci number can be a perfect number. [48] More generally, no Fibonacci number other than 1 can be multiply perfect, [49] and no ratio of two Fibonacci numbers can be perfect. [50]

Prime divisors Edit

With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem). [51] As a result, 8 and 144 (F6 and F12) are the only Fibonacci numbers that are the product of other Fibonacci numbers OEIS: A235383 .

The divisibility of Fibonacci numbers by a prime p is related to the Legendre symbol ( p 5 ) <5>> ight)> which is evaluated as follows:

If p is a prime number then

It is not known whether there exists a prime p such that

Such primes (if there are any) would be called Wall–Sun–Sun primes.

Also, if p ≠ 5 is an odd prime number then: [54]

Example 1. p = 7, in this case p ≡ 3 (mod 4) and we have:

Example 2. p = 11, in this case p ≡ 3 (mod 4) and we have:

Example 3. p = 13, in this case p ≡ 1 (mod 4) and we have:

Example 4. p = 29, in this case p ≡ 1 (mod 4) and we have:

For odd n, all odd prime divisors of Fn are congruent to 1 modulo 4, implying that all odd divisors of Fn (as the products of odd prime divisors) are congruent to 1 modulo 4. [55]

All known factors of Fibonacci numbers F(i) for all i < 50000 are collected at the relevant repositories. [56] [57]

Periodicity modulo n Edit

If the members of the Fibonacci sequence are taken mod n, the resulting sequence is periodic with period at most 6n. [58] The lengths of the periods for various n form the so-called Pisano periods OEIS: A001175 . Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular n, the Pisano period may be found as an instance of cycle detection.

More generally, in the base b representation, the number of digits in Fn is asymptotic to n log b ⁡ φ . varphi .>

The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation, and specifically by a linear difference equation. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients.

Some specific examples that are close, in some sense, from Fibonacci sequence include:

  • Generalizing the index to negative integers to produce the negafibonacci numbers.
  • Generalizing the index to real numbers using a modification of Binet's formula. [33]
  • Starting with other integers. Lucas numbers have L1 = 1, L2 = 3, and Ln = Ln−1 + Ln−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.
  • Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have Pn = 2Pn − 1 + Pn − 2. If the coefficient of the preceding value is assigned a variable value x, the result is the sequence of Fibonacci polynomials.
  • Not adding the immediately preceding numbers. The Padovan sequence and Perrin numbers have P(n) = P(n − 2) + P(n − 3).
  • Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as n-Step Fibonacci numbers. [59]

The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient): [60]

Mathematics Edit

These numbers also give the solution to certain enumerative problems, [61] the most common of which is that of counting the number of ways of writing a given number n as an ordered sum of 1s and 2s (called compositions) there are Fn+1 ways to do this. For example, there are F5+1 = F6 = 8 ways one can climb a staircase of 5 steps, taking one or two steps at a time:

5 = 1+1+1+1+1 = 2+1+1+1 = 1+2+1+1 = 1+1+2+1 = 2+2+1
= 1+1+1+2 = 2+1+2 = 1+2+2

The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied recursively until a single step, of which there is only one way to climb.

The Fibonacci numbers can be found in different ways among the set of binary strings, or equivalently, among the subsets of a given set.

  • The number of binary strings of length n without consecutive 1 s is the Fibonacci number Fn+2 . For example, out of the 16 binary strings of length 4, there are F6 = 8 without consecutive 1 s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Equivalently, Fn+2 is the number of subsets S of <1, . n> without consecutive integers, that is, those S for which <i, i + 1> ⊈ S for every i .
  • The number of binary strings of length n without an odd number of consecutive 1 s is the Fibonacci number Fn+1 . For example, out of the 16 binary strings of length 4, there are F5 = 5 without an odd number of consecutive 1 s – they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets S of <1, . n> without an odd number of consecutive integers is Fn+1 .
  • The number of binary strings of length n without an even number of consecutive 0 s or 1 s is 2Fn . For example, out of the 16 binary strings of length 4, there are 2F4 = 6 without an even number of consecutive 0 s or 1 s – they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets. was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his solvingHilbert's tenth problem. [62]
  • The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
  • Moreover, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding.
  • Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple, obtained from the formula

Computer science Edit

  • The Fibonacci numbers are important in computational run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers. [65]
  • Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers – by dividing the list so that the two parts have lengths in the approximate proportion φ. A tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming.
  • A Fibonacci tree is a binary tree whose child trees (recursively) differ in height by exactly 1. So it is an AVL tree, and one with the fewest nodes for a given height — the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees. [66]
  • Fibonacci numbers are used by some pseudorandom number generators.
  • Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.
  • A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers. [67]
  • The Fibonacci number series is used for optional lossy compression in the IFF8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as μ-law. [68][69]
  • They are also used in planning poker, which is a step in estimating in software development projects that use the Scrum methodology.

Nature Edit

Fibonacci sequences appear in biological settings, [70] such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, [71] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone, [72] and the family tree of honeybees. [73] [74] Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers. [75] Field daisies most often have petals in counts of Fibonacci numbers. [76] In 1754, Charles Bonnet discovered that the spiral phyllotaxis of plants were frequently expressed in Fibonacci number series. [77]

Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars. [78]

A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel [de] in 1979. [79] This has the form

where n is the index number of the floret and c is a constant scaling factor the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1) , the nearest neighbors of floret number n are those at n ± F(j) for some index j , which depends on r , the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers, [80] typically counted by the outermost range of radii. [81]

Fibonacci numbers also appear in the pedigrees of idealized honeybees, according to the following rules:

  • If an egg is laid by an unmated female, it hatches a male or drone bee.
  • If, however, an egg was fertilized by a male, it hatches a female.

Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, Fn , is the number of female ancestors, which is Fn−1 , plus the number of male ancestors, which is Fn−2 . [82] This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

The pathways of tubulins on intracellular microtubules arrange in patterns of 3, 5, 8 and 13. [84]


In Aufzeichnungen, die Newton lieber nicht der Veröffentlichung preisgegeben hätte, beschreibt er den Prozess für die Lösung von simultanen Gleichungen, den spätere Autoren speziell für lineare Gleichungen anwandten. Diese Methode — welche Euler nicht empfahl, welche Legendre “ordinaire” nannte, und welche Gauß “gewöhnlich” nannte — wird nun nach Gauß benannt: Gaußsches Eliminationsverfahren. Die Verbindung des Gaußschen Namens mit Elimination wurde dadurch hervorgebracht, dass professionelle Rechner eine Notation übernahmen, die Gauß speziell für seine eigenen Berechnungen der kleinsten Quadrate ersonnen hatte, welche zuließ, das Elimination als eine Sequenz von arithmetischen Rechenoperationen betrachtet wurde, die wiederholt für Handrechnungen optimisiert wurden und schließlich durch Matrizen beschrieben wurden.


Pythagorean origins Edit

The Pythagorean equation, x 2 + y 2 = z 2 , has an infinite number of positive integer solutions for x, y, and z these solutions are known as Pythagorean triples (with the simplest example 3,4,5). Around 1637, Fermat wrote in the margin of a book that the more general equation a n + b n = c n had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, and no proof by him has ever been found. His claim was discovered some 30 years later, after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries. [4]

The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics.

Subsequent developments and solution Edit

The special case n = 4 , proved by Fermat himself, is sufficient to establish that if the theorem is false for some exponent n that is not a prime number, it must also be false for some smaller n, so only prime values of n need further investigation. [note 1] Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach that was relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible (meaning that mathematicians generally considered a proof impossible, exceedingly difficult, or unachievable with current knowledge). [5]

Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura conjecture (eventually as the modularity theorem), it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was (like Fermat's theorem) widely considered completely inaccessible to proof. [6]

In 1984, Gerhard Frey noticed an apparent link between these two previously unrelated and unsolved problems. An outline suggesting this could be proved was given by Frey. The full proof that the two problems were closely linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture" (see: Ribet's Theorem and Frey curve). [2] These papers by Frey, Serre and Ribet showed that if the Taniyama–Shimura conjecture could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would also follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could also be used to contradict the Taniyama–Shimura conjecture. So if the modularity theorem were found to be true, then by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well.

Although both problems were daunting and widely considered to be "completely inaccessible" to proof at the time, [2] this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers. Unlike Fermat's Last Theorem, the Taniyama–Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics. [7] However, general opinion was that this simply showed the impracticality of proving the Taniyama–Shimura conjecture. [8] Mathematician John Coates' quoted reaction was a common one: [8]

"I myself was very sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn’t see it proved in my lifetime."

On hearing that Ribet had proven Frey's link to be correct, English mathematician Andrew Wiles, who had a childhood fascination with Fermat's Last Theorem and had a background of working with elliptic curves and related fields, decided to try to prove the Taniyama–Shimura conjecture as a way to prove Fermat's Last Theorem. In 1993, after six years of working secretly on the problem, Wiles succeeded in proving enough of the conjecture to prove Fermat's Last Theorem. Wiles's paper was massive in size and scope. A flaw was discovered in one part of his original paper during peer review and required a further year and collaboration with a past student, Richard Taylor, to resolve. As a result, the final proof in 1995 was accompanied by a smaller joint paper showing that the fixed steps were valid. Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the Taniyama–Shimura–Weil conjecture, now proven and known as the modularity theorem, were subsequently proved by other mathematicians, who built on Wiles's work between 1996 and 2001. [9] [10] [11] For his proof, Wiles was honoured and received numerous awards, including the 2016 Abel Prize. [12] [13] [14]

Equivalent statements of the theorem Edit

There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem.

In order to state them, we use mathematical notation: let N be the set of natural numbers 1, 2, 3, . let Z be the set of integers 0, ±1, ±2, . and let Q be the set of rational numbers a/b , where a and b are in Z with b ≠ 0 . In what follows we will call a solution to x n + y n = z n where one or more of x , y , or z is zero a trivial solution. A solution where all three are non-zero will be called a non-trivial solution.

For comparison's sake we start with the original formulation.

  • Original statement. With n , x , y , z ∈ N (meaning that n, x, y, z are all positive whole numbers) and n > 2 , the equation xn + yn = zn has no solutions.

Most popular treatments of the subject state it this way. It is also commonly stated over Z : [15]

  • Equivalent statement 1:xn + yn = zn , where integer n ≥ 3, has no non-trivial solutions x , y , z ∈ Z .

The equivalence is clear if n is even. If n is odd and all three of x, y, z are negative, then we can replace x, y, z with −x, −y, −z to obtain a solution in N . If two of them are negative, it must be x and z or y and z . If x, z are negative and y is positive, then we can rearrange to get (−z) n + y n = (−x) n resulting in a solution in N the other case is dealt with analogously. Now if just one is negative, it must be x or y . If x is negative, and y and z are positive, then it can be rearranged to get (−x) n + z n = y n again resulting in a solution in N if y is negative, the result follows symmetrically. Thus in all cases a nontrivial solution in Z would also mean a solution exists in N , the original formulation of the problem.

  • Equivalent statement 2:xn + yn = zn , where integer n ≥ 3, has no non-trivial solutions x , y , z ∈ Q .

This is because the exponent of x, y, and z are equal (to n ), so if there is a solution in Q , then it can be multiplied through by an appropriate common denominator to get a solution in Z , and hence in N .

  • Equivalent statement 3:xn + yn = 1 , where integer n ≥ 3, has no non-trivial solutions x , y ∈ Q .

A non-trivial solution a , b , c ∈ Z to x n + y n = z n yields the non-trivial solution a/c , b/cQ for v n + w n = 1 . Conversely, a solution a/b , c/dQ to v n + w n = 1 yields the non-trivial solution ad, cb, bd for x n + y n = z n .

This last formulation is particularly fruitful, because it reduces the problem from a problem about surfaces in three dimensions to a problem about curves in two dimensions. Furthermore, it allows working over the field Q , rather than over the ring Z fields exhibit more structure than rings, which allows for deeper analysis of their elements.

  • Equivalent statement 4 – connection to elliptic curves: If a , b , c is a non-trivial solution to xp + yp = zp , p odd prime, then y2 = x(xap )(x + bp ) (Frey curve) will be an elliptic curve. [16]

Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form. However, the proof by Andrew Wiles proves that any equation of the form y 2 = x(xa n )(x + b n ) does have a modular form. Any non-trivial solution to x p + y p = z p (with p an odd prime) would therefore create a contradiction, which in turn proves that no non-trivial solutions exist. [17]

In other words, any solution that could contradict Fermat's Last Theorem could also be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true, then it would follow that no contradiction to Fermat's Last Theorem could exist either. As described above, the discovery of this equivalent statement was crucial to the eventual solution of Fermat's Last Theorem, as it provided a means by which it could be "attacked" for all numbers at once.

Pythagoras and Diophantus Edit

Pythagorean triples Edit

In ancient times it was known that a triangle whose sides were in the ratio 3:4:5 would have a right angle as one of its angles. This was used in construction and later in early geometry. It was also known to be one example of a general rule that any triangle where the length of two sides, each squared and then added together (3 2 + 4 2 = 9 + 16 = 25) , equals the square of the length of the third side (5 2 = 25) , would also be a right angle triangle. This is now known as the Pythagorean theorem, and a triple of numbers that meets this condition is called a Pythagorean triple – both are named after the ancient Greek Pythagoras. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples, [18] and methods for generating such triples have been studied in many cultures, beginning with the Babylonians [19] and later ancient Greek, Chinese, and Indian mathematicians. [1] Mathematically, the definition of a Pythagorean triple is a set of three integers (a, b, c) that satisfy the equation [20] a 2 + b 2 = c 2 . +b^<2>=c^<2>.>

Diophantine equations Edit

Fermat's equation, x n + y n = z n with positive integer solutions, is an example of a Diophantine equation, [21] named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:

Diophantus's major work is the Arithmetica, of which only a portion has survived. [22] Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica, [23] that was translated into Latin and published in 1621 by Claude Bachet. [24]

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x 2 + y 2 = z 2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). [25] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC). [26] Many Diophantine equations have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no cross terms mixing two letters, without sharing its particular properties. For example, it is known that there are infinitely many positive integers x, y, and z such that x n + y n = z m where n and m are relatively prime natural numbers. [note 2]

Fermat's conjecture Edit

Problem II.8 of the Arithmetica asks how a given square number is split into two other squares in other words, for a given rational number k, find rational numbers u and v such that k 2 = u 2 + v 2 . Diophantus shows how to solve this sum-of-squares problem for k = 4 (the solutions being u = 16/5 and v = 12/5). [27]

Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica next to Diophantus's sum-of-squares problem: [28]

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain. [29] [30]

After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father's comments. [31] Although not actually a theorem at the time (meaning a mathematical statement for which proof exists), the margin note became known over time as Fermat’s Last Theorem, [32] as it was the last of Fermat's asserted theorems to remain unproved. [33]

It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely. Only one related proof by him has survived, namely for the case n = 4, as described in the section Proofs for specific exponents. While Fermat posed the cases of n = 4 and of n = 3 as challenges to his mathematical correspondents, such as Marin Mersenne, Blaise Pascal, and John Wallis, [34] he never posed the general case. [35] Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van der Poorten [36] suggests that while the absence of a proof is insignificant, the lack of challenges means Fermat realised he did not have a proof he quotes Weil [37] as saying Fermat must have briefly deluded himself with an irretrievable idea.

The techniques Fermat might have used in such a "marvelous proof" are unknown.

Taylor and Wiles's proof relies on 20th-century techniques. [38] Fermat's proof would have had to be elementary by comparison, given the mathematical knowledge of his time.

While Harvey Friedman's grand conjecture implies that any provable theorem (including Fermat's last theorem) can be proved using only 'elementary function arithmetic', such a proof need be 'elementary' only in a technical sense and could involve millions of steps, and thus be far too long to have been Fermat's proof.

Proofs for specific exponents Edit

Exponent = 4 Edit

Only one relevant proof by Fermat has survived, in which he uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer. [39] [40] His proof is equivalent to demonstrating that the equation

has no primitive solutions in integers (no pairwise coprime solutions). In turn, this proves Fermat's Last Theorem for the case n = 4, since the equation a 4 + b 4 = c 4 can be written as c 4 − b 4 = (a 2 ) 2 .

Alternative proofs of the case n = 4 were developed later [41] by Frénicle de Bessy (1676), [42] Leonhard Euler (1738), [43] Kausler (1802), [44] Peter Barlow (1811), [45] Adrien-Marie Legendre (1830), [46] Schopis (1825), [47] Olry Terquem (1846), [48] Joseph Bertrand (1851), [49] Victor Lebesgue (1853, 1859, 1862), [50] Théophile Pépin (1883), [51] Tafelmacher (1893), [52] David Hilbert (1897), [53] Bendz (1901), [54] Gambioli (1901), [55] Leopold Kronecker (1901), [56] Bang (1905), [57] Sommer (1907), [58] Bottari (1908), [59] Karel Rychlík (1910), [60] Nutzhorn (1912), [61] Robert Carmichael (1913), [62] Hancock (1931), [63] Gheorghe Vrănceanu (1966), [64] Grant and Perella (1999), [65] Barbara (2007), [66] and Dolan (2011). [67]

Other exponents Edit

After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be established for all odd prime exponents. [68] In other words, it was necessary to prove only that the equation a n + b n = c n has no positive integer solutions (a, b, c) when n is an odd prime number. This follows because a solution (a, b, c) for a given n is equivalent to a solution for all the factors of n. For illustration, let n be factored into d and e, n = de. The general equation

a n + b n = c n

implies that (a d , b d , c d ) is a solution for the exponent e

(a d ) e + (b d ) e = (c d ) e .

Thus, to prove that Fermat's equation has no solutions for n > 2, it would suffice to prove that it has no solutions for at least one prime factor of every n. Each integer n > 2 is divisible by 4 or by an odd prime number (or both). Therefore, Fermat's Last Theorem could be proved for all n if it could be proved for n = 4 and for all odd primes p.

In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect. [69] In 1770, Leonhard Euler gave a proof of p = 3, [70] but his proof by infinite descent [71] contained a major gap. [72] However, since Euler himself had proved the lemma necessary to complete the proof in other work, he is generally credited with the first proof. [73] Independent proofs were published [74] by Kausler (1802), [44] Legendre (1823, 1830), [46] [75] Calzolari (1855), [76] Gabriel Lamé (1865), [77] Peter Guthrie Tait (1872), [78] Günther (1878), [79] [ full citation needed ] Gambioli (1901), [55] Krey (1909), [80] [ full citation needed ] Rychlík (1910), [60] Stockhaus (1910), [81] Carmichael (1915), [82] Johannes van der Corput (1915), [83] Axel Thue (1917), [84] [ full citation needed ] and Duarte (1944). [85]

The case p = 5 was proved [86] independently by Legendre and Peter Gustav Lejeune Dirichlet around 1825. [87] Alternative proofs were developed [88] by Carl Friedrich Gauss (1875, posthumous), [89] Lebesgue (1843), [90] Lamé (1847), [91] Gambioli (1901), [55] [92] Werebrusow (1905), [93] [ full citation needed ] Rychlík (1910), [94] [ dubious – discuss ] [ full citation needed ] van der Corput (1915), [83] and Guy Terjanian (1987). [95]

The case p = 7 was proved [96] by Lamé in 1839. [97] His rather complicated proof was simplified in 1840 by Lebesgue, [98] and still simpler proofs [99] were published by Angelo Genocchi in 1864, 1874 and 1876. [100] Alternative proofs were developed by Théophile Pépin (1876) [101] and Edmond Maillet (1897). [102]

Fermat's Last Theorem was also proved for the exponents n = 6, 10, and 14. Proofs for n = 6 were published by Kausler, [44] Thue, [103] Tafelmacher, [104] Lind, [105] Kapferer, [106] Swift, [107] and Breusch. [108] Similarly, Dirichlet [109] and Terjanian [110] each proved the case n = 14, while Kapferer [106] and Breusch [108] each proved the case n = 10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for n = 3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for n = 14 was published in 1832, before Lamé's 1839 proof for n = 7. [111]

All proofs for specific exponents used Fermat's technique of infinite descent, [ citation needed ] either in its original form, or in the form of descent on elliptic curves or abelian varieties. The details and auxiliary arguments, however, were often ad hoc and tied to the individual exponent under consideration. [112] Since they became ever more complicated as p increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proved by building upon the proofs for individual exponents. [112] Although some general results on Fermat's Last Theorem were published in the early 19th century by Niels Henrik Abel and Peter Barlow, [113] [114] the first significant work on the general theorem was done by Sophie Germain. [115]

Early modern breakthroughs Edit

Sophie Germain Edit

Ernst Kummer and the theory of ideals Edit

In 1847, Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation x p + y p = z p in complex numbers, specifically the cyclotomic field based on the roots of the number 1. His proof failed, however, because it assumed incorrectly that such complex numbers can be factored uniquely into primes, similar to integers. This gap was pointed out immediately by Joseph Liouville, who later read a paper that demonstrated this failure of unique factorisation, written by Ernst Kummer.

Kummer set himself the task of determining whether the cyclotomic field could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by developing the ideal numbers.

(Note: It is often stated that Kummer was led to his "ideal complex numbers" by his interest in Fermat's Last Theorem there is even a story often told that Kummer, like Lamé, believed he had proven Fermat's Last Theorem until Lejeune Dirichlet told him his argument relied on unique factorization but the story was first told by Kurt Hensel in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources. Harold Edwards says the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken". [120] See the history of ideal numbers.)

Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all regular prime numbers. However, he could not prove the theorem for the exceptional primes (irregular primes) that conjecturally occur approximately 39% of the time the only irregular primes below 270 are 37, 59, 67, 101, 103, 131, 149, 157, 233, 257 and 263.

Mordell conjecture Edit

In the 1920s, Louis Mordell posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions, if the exponent n is greater than two. [121] This conjecture was proved in 1983 by Gerd Faltings, [122] and is now known as Faltings's theorem.

Computational studies Edit

In the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregular primes. In 1954, Harry Vandiver used a SWAC computer to prove Fermat's Last Theorem for all primes up to 2521. [123] By 1978, Samuel Wagstaff had extended this to all primes less than 125,000. [124] By 1993, Fermat's Last Theorem had been proved for all primes less than four million. [125]

However, despite these efforts and their results, no proof existed of Fermat's Last Theorem. Proofs of individual exponents by their nature could never prove the general case: even if all exponents were verified up to an extremely large number X, a higher exponent beyond X might still exist for which the claim was not true. (This had been the case with some other past conjectures, and it could not be ruled out in this conjecture.) [126]

Connection with elliptic curves Edit

The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding" [127] : 211 Taniyama–Shimura–Weil conjecture, proposed around 1955—which many mathematicians believed would be near to impossible to prove, [127] : 223 and was linked in the 1980s by Gerhard Frey, Jean-Pierre Serre and Ken Ribet to Fermat's equation. By accomplishing a partial proof of this conjecture in 1994, Andrew Wiles ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now known as the modularity theorem.

Taniyama–Shimura–Weil conjecture Edit

Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.

The link was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist André Weil found evidence supporting it, though not proving it as a result the conjecture was often known as the Taniyama–Shimura–Weil conjecture. [127] : 211–215

Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof. [127] : 203–205, 223, 226 For example, Wiles's doctoral supervisor John Coates states that it seemed "impossible to actually prove", [127] : 226 and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]." [127] : 223

Ribet's theorem for Frey curves Edit

In 1984, Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution (a, b, c) for exponent p > 2, then it could be shown that the semi-stable elliptic curve (now known as a Frey-Hellegouarch [note 3] )

y 2 = x (xa p )(x + b p )

would have such unusual properties that it was unlikely to be modular. [128] This would conflict with the modularity theorem, which asserted that all elliptic curves are modular. As such, Frey observed that a proof of the Taniyama–Shimura–Weil conjecture might also simultaneously prove Fermat's Last Theorem. [129] By contraposition, a disproof or refutation of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture.

In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of 4 numbers (a, b, c, n) capable of disproving Fermat's Last Theorem, could also be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore, if the latter were true, the former could not be disproven, and would also have to be true.

Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to prove the modularity theorem – or at least to prove it for the types of elliptical curves that included Frey's equation (known as semistable elliptic curves). This was widely believed inaccessible to proof by contemporary mathematicians. [127] : 203–205, 223, 226 Second, it was necessary to show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could not be modular. Frey showed that this was plausible but did not go as far as giving a full proof. The missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was identified by Jean-Pierre Serre who also gave an almost-complete proof and the link suggested by Frey was finally proved in 1986 by Ken Ribet. [130]

Following Frey, Serre and Ribet's work, this was where matters stood:

  • Fermat's Last Theorem needed to be proven for all exponents n that were prime numbers.
  • The modularity theorem – if proved for semi-stable elliptic curves – would mean that all semistable elliptic curves must be modular.
  • Ribet's theorem showed that any solution to Fermat's equation for a prime number could be used to create a semistable elliptic curve that could not be modular
  • The only way that both of these statements could be true, was if no solutions existed to Fermat's equation (because then no such curve could be created), which was what Fermat's Last Theorem said. As Ribet's Theorem was already proved, this meant that a proof of the Modularity Theorem would automatically prove Fermat's Last theorem was true as well.

Wiles's general proof Edit

Ribet's proof of the epsilon conjecture in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success, Andrew Wiles, an English mathematician with a childhood fascination with Fermat's Last Theorem, and who had worked on elliptic curves, decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves. [131]

Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife. [127] : 229–230 His initial study suggested proof by induction, [127] : 230–232, 249–252 and he based his initial work and first significant breakthrough on Galois theory [127] : 251–253, 259 before switching to an attempt to extend horizontal Iwasawa theory for the inductive argument around 1990–91 when it seemed that there was no existing approach adequate to the problem. [127] : 258–259 However, by mid-1991, Iwasawa theory also seemed to not be reaching the central issues in the problem. [127] : 259–260 [132] In response, he approached colleagues to seek out any hints of cutting-edge research and new techniques, and discovered an Euler system recently developed by Victor Kolyvagin and Matthias Flach that seemed "tailor made" for the inductive part of his proof. [127] : 260–261 Wiles studied and extended this approach, which worked. Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January 1993 he asked his Princeton colleague, Nick Katz, to help him check his reasoning for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly. [127] : 261–265 [133]

By mid-May 1993, Wiles felt able to tell his wife he thought he had solved the proof of Fermat's Last Theorem, [127] : 265 and by June he felt sufficiently confident to present his results in three lectures delivered on 21–23 June 1993 at the Isaac Newton Institute for Mathematical Sciences. [134] Specifically, Wiles presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves together with Ribet's proof of the epsilon conjecture, this implied Fermat's Last Theorem. However, it became apparent during peer review that a critical point in the proof was incorrect. It contained an error in a bound on the order of a particular group. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz (in his role as reviewer), [135] who alerted Wiles on 23 August 1993. [136]

The error would not have rendered his work worthless – each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected. [127] : 289, 296–297 However without this part proved, there was no actual proof of Fermat's Last Theorem. Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor, without success. [137] [138] [139] By the end of 1993, rumours had spread that under scrutiny, Wiles's proof had failed, but how seriously was not known. Mathematicians were beginning to pressure Wiles to disclose his work whether it was complete or not, so that the wider community could explore and use whatever he had managed to accomplish. But instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve. [140]

Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and fix the error. He adds that he was having a final look to try and understand the fundamental reasons for why his approach could not be made to work, when he had a sudden insight – that the specific reason why the Kolyvagin–Flach approach would not work directly also meant that his original attempts using Iwasawa theory could be made to work, if he strengthened it using his experience gained from the Kolyvagin–Flach approach. Fixing one approach with tools from the other approach would resolve the issue for all the cases that were not already proven by his refereed paper. [137] [141] He described later that Iwasawa theory and the Kolyvagin–Flach approach were each inadequate on their own, but together they could be made powerful enough to overcome this final hurdle. [137]

"I was sitting at my desk examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn’t work. Suddenly I had this incredible revelation. I realised that, the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much." — Andrew Wiles, as quoted by Simon Singh [142]

On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" [143] [144] and "Ring theoretic properties of certain Hecke algebras", [145] the second of which was co-authored with Taylor and proved that certain conditions were met that were needed to justify the corrected step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.

Subsequent developments Edit

The full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996), [9] Conrad et al. (1999), [10] and Breuil et al. (2001) [11] who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem.

Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. For example: no cube can be written as a sum of two coprime n-th powers, n ≥ 3. (The case n = 3 was already known by Euler.)

Fermat's Last Theorem considers solutions to the Fermat equation: a n + b n = c n with positive integers a , b , and c and an integer n greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent n to be a negative integer or rational, or to consider three different exponents.

Generalized Fermat equation Edit

The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions a, b, c, m, n, k satisfying [146]


A short review of the book The Modulor can be found in Ostwald (2001), the account on the development of the Modulor is given in Matteoni (1986).

For one of the earliest discussions see Pevsner (1957).

Sequence (2.1)–(2.2) can be found in Le Corbusier (2000: I, 51) Sequence (2.3)–(2.4) in Le Corbusier (2000: I, 67) Sequence (2.5)–(2.6) in Le Modulor étude 1945, Document 32285, FLC Sequence (2.7)–(2.8) in Document 21007, FLC.

Substitute (a_n=) into (a_=a_+ a_) to obtain (<>>=<>>+<>>) . Divide both sides by (< q^>) to get the quadratic equation (q^2=q+1) , which has two irrational roots (q=<(1mp sqrt<5>)>/<2>) . Thus, if a geometric progression has the Fibonacci recursion property, the common ratio is necessarily (q=<(1mp sqrt<5>)>/<2>) . Following the argument in the other way, it is clear that this condition is also sufficient.

See this observation also in Evans (1995: 275).

Evans (1995: 395, remark 7) mentions that doubling of series was Le Corbusier’s idea, while introduction of Fibonacci numbers could be Jerzy Soltan’s contribution.

See this observation also in Tell (2019: 32, 34).

See Fischler (1979) on Le Corbusier’s relations with golden ratio.

Evans (1995: 279) states that these instructions themselves contain a mathematical contradiction, but this is not the case. As we will see below, there exists a solution of this problem. The errors were made in the proposed solutions.

See this observation also in Linton (2004: 56). My approach in this section has a lot of common points with careful geometric analysis of Linton and I agree with most of his statements excluding few observations that I mention further in the text.

See Linton (2004: 62) who provides another proof and makes a remark on the proof provided here.

We use the following statement: suppose that (angle B) is the right angle of a right triangle ABC inscribed in a circle. Then AC is a diameter of this circle.

I conjecture that this mismatch between the problem and the solutions is the result of Le Corbusier’s denial of the true extent of an independent research of foundations of the norms by his assistant. See also Loach (1998: 207).

Linton (2004: 59–63) investigates three diagrams, attributing the last one to Taton. However, I doubt that the mathematician created a diagram of his own he may simply provide an explanation of the diagram of Maillard and Le Corbusier.

See also Fischler (1979: 100). The phrasing in the citation by André Wogenscky could also be an implicit evidence: ‘The research resumed at a brisk pace after the Second World War, and it was at this time that, with the help of collaborators and as a result of slow, tentative process, the worksite grid was abandoned and the Modulor was invented’ (Wogenscky 1987: 124). According to Soltan (1987: 2), Gerald Hanning left the atelier around this time in 1945, and that could be one of the reasons that geometry was abandoned for the new direction towards anthropomorphic scales.

The first proposal by Hanning in his letter 25 August 1943 (FLC B317) seems to contain a similar error within his own diagram: an inscribed angle is erroneously marked as being right angle. This indicates that Hanning’s discovery of the flaws of both diagrams was not immediate.

For the details on the golden ratio myth I refer the reader to these texts written by art historians and mathematicians: Gamwell (2015), Gardner (1994), Frascari and Volpi Ghirardini (2015), Herz-Fischler (2005), Frings (2002), and Markowsky (1992).

Van Ceulen

Ludolf van Ceulen (1540-1610) was born in Hildesheim, Germany, to a large family who were not particularly wealthy. Consequently, Van Ceulen received only an elementary education and could not read Latin or Greek, the languages in which the majority of mathematical texts were published (O’Connor and Robertson, 2009). As a young man with a keen interest in mathematics he relied on friends to translate important texts for him.

Ludolf van Ceulen, Ludolphi à Ceulen de circulo et adscriptis liber…. Omnia é vernaculo latina fecit, et annotationibus illustravit Willebrordus Snellius (Leiden, 1619), title page portrait.

There were many approximations of π before Van Ceulen’s (including those attributed to Ptolemy, Al-Khwārizmī and Fibonacci) but it was Archimedes who was a major influence on Van Ceulen’s work and findings. Archimedes published the first theoretical calculation of π around 250 BC which he found using a regular polygon of 96 sides (Vajta 2000). Van Ceulen was fixated on Archimedes’ work and when eventually a friend translated it for him, Van Ceulen was inspired to spend the rest of his life looking for a better approximation of π using Archimedes’ method (O’Connor and Robertson, 2009). By the time of his death Van Ceulen had determined π to 35 places of decimals and as Vajta notes ‘The Germans were so much impressed by Van Ceulen’s achievement that they began to call [π] the Ludolph’s number.’ (Vajta)

Due to his lack of Latin, he was unable to publish any new findings himself, and so he focused on reviewing and criticising the work of others which resulted in a series of mathematical disputes. William Goudaan, a teacher from Haarlem, posed a geometric problem which Van Ceulen solved, yet Goudaan did not accept his solution and published his own findings. Van Ceulen realised these findings were incorrect and so in 1584 published his own side of the dispute (O’Connor, and Robertson, 2009).

When Simon van der Eycke (1584-1603) published an incorrect proof of the quadrature of the circle in 1584, Van Ceulen released two publications highlighting Van Der Eycke’s error. Subsequently, while Van Ceulen was teaching in Leiden, a leading professor published work in which he claimed that π was equal to . Van Ceulen knew this was incorrect and when he approached the professor, Van Ceulen was challenged to put his objections in writing. However he could not participate in this dispute, his inability to write in Latin prohibited him from engaging under the usual terms (O’Connor, J and Robertson, E.F, 2009).

Van Ceulen’s most famous student was Willebrord Snell (1580-1626) who translated two of Van Ceulen’s works into Latin after his death (biography.com, 2008). This made them more accessible to the mathematicians around the world.

Ludolf van Ceulen, Ludolphi à Ceulen de circulo et adscriptis liber…. Omnia é vernaculo latina fecit, et annotationibus illustravit Willebrordus Snellius (Leiden, 1619), p. 6.

Shown above is a diagram of a circle with an inscribed equilateral triangle and regular pentagon along with some constructions generating regular polygons with more than 5 sides. Notice the handwritten square roots along some of the sides denoting the lengths required for the construction.

In the table below, Van Ceulen lists some lengths of interest again using square roots. Specifically, he starts with an equilateral triangle which has three sides and each time calculates the length of a side of a regular polygon with double the number of sides as the previous one.

Ludolf van Ceulen, Ludolphi à Ceulen de circulo et adscriptis liber…. Omnia é vernaculo latina fecit, et annotationibus illustravit Willebrordus Snellius (Leiden, 1619), p. 22.

The next table shows a similar calculation in which Van Ceulen has used a circle with very large diameter (200000000000000) and given the length of a side of each inscribed regular polygon. For example, an inscribed hexagon (six sides) has side length exactly half the diameter of the circle.

Ludolf van Ceulen, Ludolphi à Ceulen de circulo et adscriptis liber…. Omnia é vernaculo latina fecit, et annotationibus illustravit Willebrordus Snellius (Leiden, 1619), p. 48.

Van Ceulen died on 31 December 1610 and was buried in St Peter’s Church in Leiden. His approximations for π were engraved on his original tombstone which went missing (Vajta 2000). Today, a modern version stands in St Peter’s Church and carved on its tombstone is his lower bound of 3.14159265558979323846264338327950288 and his upper bound of 3.14159265558979323846264338327950289 this honours his contribution to improving the accuracy of geometry and trigonometry (O’Connor and Robertson, 2009) and also features on the cover of Worth’s copy of Ludolphi à Ceulen De circulo et adscriptis liber.

Vajta, M., Fourier Transform and Ludolph van Ceulen, University of Twente (Netherlands).

O’Connor, J. J. and E. F. Robertson, Ludolf van Ceulen, MacTutor History of Mathematics (University of St Andrews, 2009).


Oaks is responsible for the content of this article, and Alkhateeb for the translations from the Arabic. The authors express their thanks to Barnabas Hughes and an anonymous referee for their thoughtful comments on an earlier version of this article.

General notes. Notation for references to al-Khwārizmī: R 3/210, M&A 161 means Rosen's edition [al-Khwārizmī, The Algebra of Mohammed ben Musa, Oriental Translation Fund, London, 1831], English translation page 3, Arabic text page 2, line 10, and Musharrafa's and Aḥmad's Arabic edition [al-Khwārizmī, Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa'l-muqābala, edited by ‘Alī Muṣṭafā Musharrafa and Muḥammad Mursī Aḥmad, Cairo, 1939] page 16, line 1. References to Abū Kāmil: A 937, L 2576, H 15813 means Arabic text [Abū Kāmil, Kitāb fī al-jabr wa'l-muqābala, Institute for the History of Arabic-Islamic Science, Frankfurt am Main, 1986] page 93, line 7, Latin text [Sesiano, La version latine médiévale de lɺlgèbre d➫ū Kāmil, Rodopi, Amsterdam, 1993, pp. 315–452] line 2576, and Hebrew edition [Levey, The Algebra of Abū Kāmil: Kitāb fī al-jābr wa’l-muqābala in a Commentary by Mordecai Finzi. University of Wisconsin, Madison, 1966] page 158, line 13 of the English translation. References to Ibn Badr: IB 52/3619 refers to [Sánchez Pérez, Compendio de Álgebra de Abenbéder, Impr. Ibérica, Madrid, 1916], Spanish translation page 52, Arabic text page 36, line 19. A semicolon separates the page number from the line number in other references as well. The line number indicates the beginning of the referred passage, which may run on to several lines. In texts in which the lines are already numbered, we defer to them. Translations. Because Rosen and Levey misinterpreted the meanings of many words, we felt it necessary to produce new translations directly from the Arabic. For al-Khwārizmī we use mainly Musharrafa's and Aḥmad's edition, but with an eye also on Rosen's edition and the Latin translations. We also translate Ibn Badr from the Arabic.