Search Results for theorem

Biographies

1. Vandiver biography
   
   • For example he published two short papers in 1902 A Problem Connected with Mersenne's Numbers and Applications of a Theorem Regarding Circulants.
     
   • He also began reading papers on algebraic number theory and embarked on a study of the work of Kummer, in particular his contributions to solving Fermat's Last Theorem.
     
   • Over the next few years he published papers such as Theory of finite algebras (1912), Note on Fermat's last theorem (1914), and Symmetric functions formed by systems of elements of a finite algebra and their connection with Fermat's quotient and Bernoulli's numbers (1917).
     
   • for his several papers on Fermat's last theorem published in the Transactions of the American Mathematical Society and in the Annals of Mathematics during the preceding five years, with special reference to a paper entitled "On Fermat's last theorem".
     
   • In particular the paper entitled On Fermat's last theorem which was specially mentioned, was published in the Transactions of the American Mathematical Society in 1929.
     
   • Let us mention a few more of his early papers: The generalized Lagrange indeterminate congruence for a composite ideal modulus (1917); A property of cyclotomic integers and its relation to Fermat's last theorem (1919); A new type of criteria for the first case of Fermat's last theorem (1924); and A property of cyclotomic integers and its relation to Fermat's last theorem (1925).
     
   • He continued to work on extending Kummer's methods to show that the theorem was true for increasingly large exponents.
     
   • In 1952 he was able to implements his methods on early computers at the National Bureau of Standards Institute at Los Angeles and was able to prove the theorem true for all primes less than 2000.
     
   • It is his life-long work on Fermat's Last Theorem for which he is best known, but Vandiver also wrote papers on cyclotomic fields, Bernoulli numbers, the reciprocity laws, finite fields, techniques for factorisation, semigroups, semirings, and algebras.
     
   • Although the Cole Prize might be considered Vandiver's greatest distinction, we should also mention that he was vice-president of the American Mathematical Society in 1934-35 and was the Colloquium Lecturer at Ann Arbor in 1935 when he lectured on Fermat's Last Theorem.
     
   • History Topics: Fermat's last theorem .
     

2. Wiles biography
   
   • Wiles' interest in Fermat's Last Theorem began at a young age.
     
   • From that moment I tried to solve it myself, it was such a challenge, such a beautiful problem, this problem was Fermat's Last Theorem.
     
   • Wiles did not work on Fermat's Last Theorem for his doctorate.
     
   • about ten years ago, G Frey suggested and K Ribet proved (building on ideas of B Mazur and J-P Serre) that Fermat's Last Theorem follows from the Shimura-Taniyama conjecture that every elliptic curve defined over the rational numbers is modular.
     
   • is a counterexample to Fermat's Last Theorem, then the elliptic curve .
     
   • In fact Wiles abandoned all his other research when he heard what had been proved and, for seven years, he concentrated solely on attempting to prove the Shimura-Taniyama conjecture, knowing that a proof of Fermat's Last Theorem then followed.
     
   • Although less than the full Shimura-Taniyama conjecture, this result does imply that the elliptic curve given above is modular, thereby proving Fermat's Last Theorem.
     
   • In 1993 Wiles told two other mathematicians that he was close to a proof of Fermat's Last Theorem.
     
   • At the end of the final lecture he announced he had a proof of Fermat's Last Theorem.
     
   • His paper which proves Fermat's Last Theorem is Modular elliptic curves and Fermat's Last Theorem which appeared in the Annals of Mathematics in 1995.
     
   • for his proof of Fermat's last theorem.
     
   • History Topics: Fermat's last theorem .
     
   • AMS (An article about the proof of Fermat's last theorem) [registration required] .
     

3. Hirzebruch biography
   
   • One of his most famous results, now named the Hirzebruch-Riemann-Roch theorem, appeared in his 1954 paper Arithmetic genera and the theorem of Riemann-Roch for algebraic varieties.
     
   • Hirzebruch's impressive work on the Riemann-Roch theorem was fully set out in a book he published in 1956 entitled Neue topologische Methoden in der algebraischen Geometrie.
     
   • This monograph is an exposition of the main ideas and concepts centring around the fundamental Riemann-Roch theorem for algebraic manifolds of arbitrary dimension, a theorem whose proof has recently been found by the author and which is presented here in complete form for the first time.
     
   • In 1974 Hirzebruch, jointly with D Zagier, published The Atiyah-Singer theorem and elementary number theory.
     
   • the discovery of the signature theorem for differentiable manifolds and the formulation and proof of the Riemann-Roch theorem for algebraic varieties, .
     
   • the integrality theorem for characteristic classes of differentiable manifolds, .
     
   • the proportionality theorem for complex homogeneous manifolds and (with Armand Borel) the general theory of characteristic classes of homogeneous spaces of compact Lie groups, .
     
   • the 'topological' proof of the Dedekind reciprocity theorem through 4-manifold theory and other fascinating relations between differential topology and algebraic number theory .
     

4. Orlicz biography
   
   • Orlicz's name is associated not only with the Orlicz spaces but also with the Orlicz-Pettis theorem, Orlicz property, Orlicz theorem on unconditional convergence in Lp, Mazur-Orlicz bounded consistency theorem, Mazur-Orlicz theorem on inequalities, Mazur-Orlicz theorem on uniform boundedness in F-spaces, Orlicz category theorem, Orlicz interpolation theorem, Orlicz norm, Orlicz function, convexity in the sense of Orlicz, F-norm of Mazur-Orlicz, Drewnowski-Orlicz theorem on representation of orthogonal additive functionals and modulars, Orlicz theorem on Weyl multipliers, Matuszewska-Orlicz indices, Hardy-Orlicz spaces, Marcinkiewicz-Orlicz spaces, Musielak-Orlicz spaces, Orlicz-Sobolev spaces and Orlicz-Bochner spaces.
     
   • For example, the Orlicz-Pettis theorem says that in Banach spaces the classes of weakly subseries convergent and norm unconditionally convergent series coincide.
     
   • The Orlicz theorem on unconditional convergence in Lpis: .
     

5. De Rham biography
   
   • 38 (2) (1991), 114-115.',4)">4] Raoul Bott describes the context of de Rham's famous theorem:- .
     
   • In some sense the famous theorem that bears his name dominated his mathematical life, as indeed it dominates so much of the mathematical life of this whole century.
     
   • These are the natural extensions to manifolds of the distributions which had been introduced a few years earlier by Laurent Schwartz and of course it is only in this extended setting that both the de Rham theorem and the Hodge theory become especially complete.
     
   • The original theorem of de Rham was most probably believed to be true by Poincare and was certainly conjectured (and even used!) in 1928 by E Cartan.
     
   • The details of the de Rham theorem are given in [Notices Amer.
     
   • 38 (2) (1991), 114-115.',4)">4] but as far as this article is concerned it is sufficient to give the 'feel' for the type of theorem as nicely described there:- .
     
   • The theorem is then a sort of topological form of the particle-wave equivalence of quantum mechanics, and the quest for 'truly' understanding these and analogous dualities has been one of the great motivating forces in the mathematics of the last fifty years.
     
   • Of course de Rham produced much in the way of important mathematics in addition to the de Rham theorem.
     
   • He gave a reducibility theorem for Riemann spaces which is fundamental in the development of Riemannian geometry.
     

6. Menelaus biography
   
   • Book 3 deals with spherical trigonometry and includes Menelaus's theorem.
     
   • For plane triangles the theorem was known before Menelaus:- .
     
   • Menelaus produced a spherical triangle version of this theorem which is today also called Menelaus's Theorem, and it appears as the first proposition in Book III.
     
   • This was a direct proof of a theorem in Euclid's Elements and given Menelaus's dislike for reductio ad absurdum in his surviving works this seems a natural line for him to follow.
     
   • The new proof which Proclus attributes to Menelaus is of the theorem (in Heath's translation of Euclid):- .
     
   • Menelaus's theorem .
     
   • Menelaus's theorem .
     

7. Thales biography
   
   • Of course Thales could have used these geometrical methods for solving practical problems, having merely observed the properties and having no appreciation of what it means to prove a geometrical theorem.
     
   • On the other hand B L van der Waerden [cience Awakening (New York, 1954).',16)">16] claims that Thales put geometry on a logical footing and was well aware of the notion of proving a geometrical theorem.
     
   • The fifth theorem is believed to be due to Thales because of a passage from Diogenes Laertius book Lives of eminent philosophers written in the second century AD [Lives of eminent philosophers (New York, 1925).',6)">6]:- .
     
   • The theorem (iv) was attributed to Thales by Eudemus for less than completely convincing reasons.
     
   • [Eudemus] says that the method by which Thales showed how to find the distances of ships from the shore necessarily involves the use of this theorem.
     
   • Although theorem (iv) underlies this application, it would have been quite possible for Thales to devise such a method without appreciating anything of 'congruent triangles'.
     
   • As a final comment on these five theorems, there are conflicting stories regarding theorem (iv) as Diogenes Laertius himself is aware.
     
   • Others have attributed the story about the sacrifice of an ox to Pythagoras on discovering Pythagoras's theorem.
     

8. Roch biography
   
   • It appear in print in the following year and contains the theorem now known as the Riemann-Roch theorem.
     
   • As presented by Roch, the Riemann-Roch theorem related the topological genus of a Riemann surface to purely algebraic properties of the surface.
     
   • The Riemann-Roch theorem was so named by Max Noether and Alexander von Brill in a paper they jointly wrote 1874 when they refined the information obtained from the theorem.
     
   • It was extended to algebraic curves in 1929 and then in the 1950s an n-dimensional version, the Hirzebruch-Riemann-Roch theorem, was proved by Hirzebruch and a version for a morphism between two varieties, the Grothendieck-Riemann-Roch theorem, was proved by Grothendieck.
     
   • Roch's name will live on through the fundamental Riemann-Roch theorem, but it is a tragedy that the young man with so much mathematical promise died when he had only just commenced his career.
     

9. Atiyah biography
   
   • Subsequently (in collaboration with I M Singer) he established an important theorem dealing with the number of solutions of elliptic differential equations.
     
   • This 'index theorem' had antecedents in algebraic geometry and led to important new links between differential geometry, topology and analysis.
     
   • Combined with considerations of symmetry it led (jointly with Raoul Bott) to a new and refined 'fixed point theorem' with wide applicability.
     
   • The K-theory and the index theorem are studied in Atiyah's book K-theory (1967, reprinted 1989) and his joint work with G B Segal The Index of Elliptic Operators I-V in the Annals of Mathematics, volumes 88 and 93 (1968, 1971).
     
   • Atiyah also described his work on the index theorem in The index of elliptic operators given as an American Mathematical Society Colloquium Lecture in 1973.
     
   • The index theorem could be interpreted in terms of quantum theory and has proved a useful tool for theoretical physicists.
     
   • In 2004 he and Isadore Singer were awarded the Neils Abel prize of £480 000 for their work on the Atiyah-Singer Index Theorem.
     
   • Abel Prize Committee (The Atiyah-Singer Index theorem --pdf) .
     

10. Carleman biography
    
    • International success came, but his spectral theory was overshadowed by the abstract theory and he had also bad luck with his mean ergodic theorem.
      
    • One reason was that many of his results, for instance the extension of Holmgren's uniqueness theorem, the analysis of the Schrodinger operator, and the existence theorem for Boltzmann's equation, were two decades ahead of their time and therefore not immediately appreciated.
      
    • Names such as Carleman inequality, Carleman theorems (Denjoy-Carleman theorem on quasi-analytic classes of functions, Carleman theorem on conditions of well-definedness of moment problems, Carleman theorem on uniform approximation by entire functions, Carleman theorem on approximation of analytic functions by polynomials in the mean), Carleman singularity of orthogonal system, integral equation of Carleman type, Carleman operator, Carleman kernel, Carleman method of reducing an integral equation to a boundary value problem in the theory of analytic functions, Jensen-Carleman formula in complex analysis, Carleman continuum, Carleman linearization or Carleman embedding technique, Carleman polynomials, Carleman estimate in the unique continuation problem for solutions of partial differential equations and Carleman system in the kinetic theory of gas are well-known in mathematics (see [Encyclopaedia of Mathematics 2 (Kluwer 1988), 25-26.
      
    • Carleman is also one of the authors of a mean ergodic theorem (see [Ergodic theory in the 1930\'s : a study in international mathematical activity (manuscript, Jan 2000).',17)">17], where more is written about priority questions).
      

11. Selberg biography
    
    • Selberg is also well known for his elementary proof of the prime number theorem, with a generalisation to prime numbers in an arbitrary arithmetic progression.
      
    • The history of the prime number theorem is very interesting.
      
    • The theorem stating:- .
      
    • The necessary analytic tools were known by 1896 when Hadamard and de la Vallee Poussin independently proved the theorem using complex analysis.
      
    • Subsequent events are not entirely clear but Selberg published two papers An elementary proof of the prime number theorem and An elementary proof of Dirichlet's theorem about primes in an arithmetic progression in volume 50 of the Annals of Mathematics.
      
    • The following year he published An elementary proof of the prime number theorem for arithmetic progressions.
      
    • Prime Number Theorem .
      

12. Brianchon biography
    
    • This result is often called Brianchon's Theorem and it is the result for which he is best known.
      
    • In fact this theorem is simply the dual of Pascal's theorem which was proved in 1639:- .
      
    • In [Dictionary of Scientific Biography (New York 1970-1990).',1)">1] Greitzer points out that Pascal recognised that his theorem was projective in nature so it is surprising that it took 167 years before someone realised that its dual, which is Brianchon's Theorem, would also be true.
      
    • In this paper Recherches sur la determination d'une hyperbole equilatere, au moyen de quatres conditions donnee (1820) there appears a statement and proof of the nine point circle theorem.
      
    • Certainly they were not the first to discover this theorem, but they were the first to give a proper proof of the theorem and also they used, for the first time, the name "nine point circle".
      

13. De Giorgi biography
    
    • Influenced by methods which Caccioppoli had developed, De Giorgi went on to develop new techniques in geometric measure theory and he applied his results to the calculus of variations proving his regularity theorem for almost all minimal surfaces.
      
    • In the following year he proved what has become known as "De Giorgi's Theorem" concerning the Holder continuity of solutions of elliptic partial differential equations of second order.
      
    • Nash and I proved the same theorem, or, rather, two theorems very close to each other.
      
    • From the theorem of Nash one can deduce more or less immediately my theorem, following a quite different line of proof.
      
    • Thus, from my experiences, the discovery of a theorem can be made by different people, as if it were there waiting for someone to uncover it, and the statement of the theorem is always the same.
      

14. Vallee Poussin biography
    
    • His best known work, however, appeared four years later in 1896 when he proved the prime number theorem.
      
    • The prime number theorem had been conjectured in the 18th century, but in 1896 two mathematicians independently proved the result, namely Hadamard and Vallee Poussin.
      
    • Other than the prime number theorem, Vallee Poussin's only contributions to prime numbers were contained in two papers on the Riemann zeta function which he published in 1916.
      
    • Most of the additional material appeared in small type and covered topics such as set theory, in particular the Schroder-Bernstein theorem, the Lebesgue integral, functions of bounded variation, the Jordan curve theorem, polynomial approximation, Parseval's theorem on trigonometric series, results of Fejer, etc.
      
    • Prime Number Theorem .
      

15. Singer biography
    
    • His publications in these early years of his career include: (with Richard V Kadison) Some remarks on representations of connected groups (1952); Uniformly continuous representations of Lie groups (1952); ( with Warren Ambrose) A theorem on holonomy (1953); (with Richard Arens) Function values as boundary integrals (1954).
      
    • Singer is justifiably famous among mathematicians for his deep and spectacular work in geometry, analysis, and topology, culminating in the Atiyah-Singer Index theorem and its many ramifications in modern mathematics and quantum physics.
      
    • In the citation for the Steele Prize for Lifetime Achievement which Singer received in 2001, his work on the Atiyah-Singer Index theorem is highlighted [15]:- .
      
    • Singer's series of five papers with Michael F Atiyah on the Index Theorem for elliptic operators (which appeared in 1968 - 71) and his three papers with Atiyah and V K Patodi on the Index Theorem for manifolds with boundary (which appeared in 1975 - 76) are among the great classics of global analysis.
      
    • However, [the Index Theorem] represents only a small part of his contributions to geometry and analysis.
      
    • for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.
      

16. Birkhoff biography
    
    • His ergodic theorem transformed the Maxwell-Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure.
      
    • Birkhoff's discovery of what has come to be known as the "ergodic theorem" in 1931 - 32 is his most well-known contribution to dynamics.
      
    • It is, of course, not only the ergodic theorem that made Birkhoff the most famous mathematician in America in his day.
      
    • He had already achieved this distinction in most mathematicians eyes many years earlier when he proved Poincare's Last Geometric Theorem, a special case of the 3-body problem, in 1913.
      
    • Poincare had stated his theorem in Sur un theoreme de geometrie in 1912 but could only give a proof in certain special cases.
      
    • He also did important work on the four colour theorem.
      
    • History Topics: The four colour theorem .
      

17. Pythagoras biography
    
    • Of course today we particularly remember Pythagoras for his famous geometry theorem.
      
    • Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000 years earlier he may have been the first to prove it.
      
    • I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.
      
    • (ii) The theorem of Pythagoras - for a right angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.
      
    • Pythagoras's theorem .
      
    • History Topics: Pythagoras's theorem in Babylonian mathematics .
      

18. Bernstein Felix biography
    
    • It was at this time that he came up with the Schroder-Bernstein Theorem which we discuss below.
      
    • Two mathematicians who had heard Cantor heap praise on Bernstein for the Schroder-Bernstein Theorem, persuaded the student of fine arts at Pisa to become a mathematician.
      
    • Today Bernstein is best remembered by mathematicians for the Schroder-Bernstein Theorem.
      
    • This theorem states: .
      
    • We mentioned above that the result is known as the Schroder-Bernstein Theorem.
      
    • In fact the theorem was stated by Cantor in Beitrage zur Begrundung der transfiniten Mengenlehre but his justification of the result there is not a rigorous proof (as he himself was aware) and, as we explained above, it was while correcting the proofs of this work that Bernstein, still a high school student, constructed a correct proof.
      
    • His range of interests were remarkable and he worked on convex functions, isoperimetric problems, the Laplace transform, number theory (including Fermat's Last Theorem), differential equations and the mathematical theory of genetics.
      

19. Morishima biography
    
    • His passion was algebraic number theory and he had a particular love of Fermat's Last Theorem.
      
    • His first paper on Fermat's Last Theorem was published in the Proceedings of the Imperial Academy of Japan in 1928.
      
    • By 1935 he had published a total of 16 papers, the other 6 being: Uber den Fermatschen Quotienten (1931); On recent results about Fermat's last Theorem (Japanese) (1932); Uber die Einheiten und Idealklassen des Galoisschen Zahlkorpers und die Theorie der Kreiskorper der l-ten Einheitswurzein (1933); and Uber die Theorie der Kreiskorper der l-ten Einheitswurzein (1935).
      
    • He did publish the book Higher Algebra in 1940 (in Japanese) but this and one further paper on Fermat's Last Theorem (in 1952) was all in published in the 30 years between 1935 and 1965.
      
    • We mentioned above that Morishima published one paper on Fermat's Last Theorem in 1952.
      
    • This was On Fermat's last theorem (thirteenth paper) published in English in the Transactions of the American Mathematical Society.
      
    • In this paper he continued his investigations of the first case of Fermat's last theorem.
      

20. Grunsky biography
    
    • The final book he wrote was [The general Stokes\' theorem (Boston, MA, 1983).',1)">1] The general Stokes' theorem published in 1983.
      
    • Grunsky writes that the aim of the book is to give [The general Stokes\' theorem (Boston, MA, 1983).',1)">1]:- .
      
    • an intrinsic and easily comprehensible presentation of Stokes's theorem.
      
    • The treatise begins with an intuitive discussion of Stokes's theorem in the plane, which is then used as a model for generalising the result to higher dimensions.
      
    • Grunsky first proves Stokes's theorem for suitable k-dimensional region in Rk, and then for k-dimensional regions in Rn.
      
    • He then introduces the calculus of alternating multilinear forms and gives a proof of Stokes's theorem for manifolds.
      

21. Heine biography
    
    • He is best remembered for the Heine-Borel theorem:- .
      
    • The second part of this paper covers the history of the Heine-Borel theorem and is summarised in the following review:- .
      
    • The last half of the paper is devoted to a more systematic account of the gradual discovery and formulation of the so-called Heine-Borel theorem.
      
    • It begins with the implicit use of the theorem in various proofs of the theorem stating that a continuous function on a closed, bounded interval is uniformly continuous.
      
    • The first proof of this theorem was given by Dirichlet in his lectures of 1862 (published 1904) before Heine proved it in 1872.
      
    • Borel formulated his theorem for countable coverings in 1895 and Schonflies and Lebesgue generalized it to any type of covering in 1900 and 1898 (published 1904), respectively.
      

22. Ribenboim biography
    
    • Several texts followed in fairly quick succession: The Riemann-Roch theorem for algebraic curves (1965) and Linear representation of finite groups (1966).
      
    • The first of these is devoted to ramification theory in Galois extensions and the second to a proof of the theorem by Kronecker and Heinrich Weber on the abelian extensions of the field of rational numbers.
      
    • He began to write about this topic, one which clearly had an enormous fascination for him, in 1979 with his famous text 13 lectures on Fermat's last theorem.
      
    • Further number theory books are Catalan's Conjecture published in 1994, The new book of prime number records (1995) and Fermat's last theorem for amateurs (1999).
      
    • And on other topics included, the author has written complete books, such as on Catalan's problem, prime records and Fermat's Last Theorem.
      
    • His other works include 'The Little Book of Big Primes' (1991), '13 Lectures on Fermat's Last Theorem' (1995), 'The [New] Book of Prime Number Records' (1998), and 'Fermat's Last Theorem for Amateurs' (1999).
      

23. Legendre biography
    
    • Of course today we attribute the law of quadratic reciprocity to Gauss and the theorem concerning primes in an arithmetic progression to Dirichlet.
      
    • This is fair since Legendre's proof of quadratic reciprocity was unsatisfactory, while he offered no proof of the theorem on primes in an arithmetic progression.
      
    • This work resulted in his election to the Royal Society of London in 1787 and also to an important publication Memoire sur les operations trigonometriques dont les resultats dependent de la figure de la terre which contains Legendre's theorem on spherical triangles.
      
    • It is nevertheless certain that the theorem on the sum of the three angles of the triangle should be considered one of those fundamental truths that are impossible to contest and that are an enduring example of mathematical certitude.
      
    • Prime Number Theorem .
      
    • History Topics: Fermat's last theorem .
      
    • Kevin Brown (The Prime Number Theorem) .
      

24. Bishop biography
    
    • Examples are extensions of Mergelyan's approximation theorem and the theorem of Frigyes Riesz and Marcel Riesz concerning measures on the unit circle orthogonal to polynomials.
      
    • Here Bishop worked on uniform algebras (commutative Banach algebras with unit whose norms are the spectral norms) proving results such as antisymmetric decomposition of a uniform algebra, the Bishop-DeLeeuw theorem, and the proof of existence of Jensen measures.
      
    • He proved important results in this area such as the biholomorphic embedding theorem for a Stein manifold as a closed submanifold in Cn, and a new proof of Remmert's proper mapping theorem.
      
    • However, a good deal of Brouwer's intuitionism is rejected, notably his notions of free choice sequences, spreads and the bar theorem.
      

25. Kempe biography
    
    • But only in 1876 did Kempe prove a theorem on the possibility of reproducing any plane curve of degree n by means of an articulated mechanism.
      
    • In 1926 Gersgorin, basing his work on Kempe's considerations and using the complex variable method, proved a more general theorem on the possibility of constructing similar mechanisms for an arbitrary algebraic function.
      
    • Kempe published a false "proof" of the four colour theorem in 1879 which stood until Heawood found an error eleven years later.
      
    • Kempe was proposed for election to the Royal Society in the year he published his "proof" of the four colour theorem.
      
    • If Kempe's work on the four colour theorem is not mentioned in his obituary [Proc.
      
    • History Topics: The four colour theorem .
      

26. Gauss biography
    
    • At the academy Gauss independently discovered Bode's law, the binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the prime number theorem.
      
    • Gauss's dissertation was a discussion of the fundamental theorem of algebra.
      
    • showed that there can only be two poles in the globe and went on to prove an important theorem, which concerned the determination of the intensity of the horizontal component of the magnetic force along with the angle of inclination.
      
    • Prime Number Theorem .
      
    • History Topics: The fundamental theorem of algebra .
      

27. Hurwitz biography
    
    • We note that this first paper by Hurwitz, written jointly with Schubert, was on Chasles's theorem.
      
    • In 1893 the Swedish actuary and mathematics historian Gustaf Enestrom published a theorem on the complex roots of certain polynomials with real coefficients in a paper on pension insurance (in Swedish).
      
    • This result is now often called the Enestrom-Kakeya theorem, since S Kakeya published a similar result in 1912-1913.
      
    • But Kakeya's theorem contained a mistake, which was corrected by A Hurwitz in 1913.
      
    • Hurwitz informed E Landau about Kakeya's result (corrected); Landau needed the result in a proof of a theorem on infinite power series.
      
    • We mention a generalization of Enestrom's theorem and give an application to a similar result by Hurwitz.
      

28. Droz-Farny biography
    
    • In this he stated the following remarkable theorem without giving a proof: .
      
    • This is known as the Droz-Farny line theorem, but it is not known whether Droz-Farny had a proof of the theorem.
      
    • Looking at other work by Droz-Farny, one is led to conjecture that indeed he would have constructed a proof of the theorem.
      
    • The 1901 paper we mentioned above is, for example, one in which he gives a proof of a theorem stated by Steiner without proof.
      
    • The theorem is as follows: .
      

29. Schreier biography
    
    • His first paper in 1924 gave a simple algebraic proof of a theorem on knot groups, which generalised a theorem given by Dehn 10 years earlier.
      
    • He may have been directed towards the main theorem, which proves that certain torus knots were not isomorphic to their mirror images, by Reidemeister.
      
    • Schreier (1928) found an important refinement of the fundamental Jordan-Holder theorem, 39 years after the publication of Holder's paper.
      
    • It is rare that such a widely used and basic theorem can be deepened after such a long time.
      
    • Zassenhaus (1934) discovered a second improvement of the theorem.) .
      

30. Euler biography
    
    • I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer.
      
    • For I found that theorem about four years ago, at which time I also described its proof and application in greater detail to our Academy.
      
    • Further important results in number theory by Euler included his proof of Fermat's Last Theorem for the case of n = 3.
      
    • Although there were problems with his approach this eventually led to Kummer's major work on Fermats Last Theorem and to the introduction of the concept of a ring.
      
    • History Topics: Fermat's last theorem .
      
    • History Topics: The fundamental theorem of algebra .
      

31. Jordan biography
    
    • Jordan proved the Jordan-Holder theorem, namely that although groups can have different composition series, the set of composition factors is an invariant of the group.
      
    • The treatise contains the 'Jordan normal form' theorem for matrices, not over the complex numbers but over a finite field.
      
    • He studied primitive permutation groups and proved a finiteness theorem.
      
    • His finiteness theorem showed that for a given c there are only finitely many primitive groups with class c other than the symmetric and alternating groups.
      
    • Jordan is best remembered today among analysts and topologists for his proof that a simply closed curve divides a plane into exactly two regions, now called the Jordan curve theorem.
      
    • The second edition appeared in 1893 while the Jordan curve theorem appeared in the third edition of the text which appeared between 1909 and 1915.
      

32. Fermat biography
    
    • Fermat is best remembered for this work in number theory, in particular for Fermat's Last Theorem.
      
    • This theorem states that .
      
    • Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries.
      
    • Fermat posed further problems, namely that the sum of two cubes cannot be a cube (a special case of Fermat's Last Theorem which may indicate that by this time Fermat realised that his proof of the general result was incorrect), that there are exactly two integer solutions of x2 + 4 = y3 and that the equation x2 + 2 = y3 has only one integer solution.
      
    • A triangle theorem by Fermat .
      
    • History Topics: Fermat's Last Theorem .
      

33. Carlson biography
    
    • Some of his most well-known contributions are a theorem connected to the Phragmen-Lindelof principle, a theorem about the zeros of the V-function and several theorems about power series with integer coefficients.
      
    • Such names as Carlson inequality, Carlson - Levin constants, Carlson theorem in complex analysis, Polya - Carlson theorem on rational functions and Carlson theorem on Dirichlet series are well-known in mathematics (see [Inequalities (Cambridge, 1934).
      
    • Carlson's theorem in complex analysis, says that if f (z) is an analytic function satisfying |f (z)| ≤ Cek|z|, where k < π for Re z ≥ 0, and if f (z) = 0 for z = 0, 1, 2, ..
      

34. Taylor biography
    
    • In fact the first mention by Taylor of a version of what is today called Taylor's Theorem appears in a letter which he wrote to Machin on 26 July 1712.
      
    • There are, in fact, two versions of Taylor's Theorem given in the 1715 paper which to a modern reader look equivalent but which, the author of [Arch.
      
    • James Gregory, Newton, Leibniz, Johann Bernoulli and de Moivre had all discovered variants of Taylor's Theorem.
      
    • The importance of Taylor's Theorem remained unrecognised until 1772 when Lagrange proclaimed it the basic principle of the differential calculus.
      
    • The main theorem in Taylor's theory of linear perspective is that the projection of a straight line not parallel to the plane of the picture passes through its intersection and its vanishing point.
      
    • A study of Brook Taylor's life and work reveals that his contribution to the development of mathematics was substantially greater than the attachment of his name to one theorem would suggest.
      

35. Ramanujam biography
    
    • To mention some of them, he has written proofs of the Castelnuovo theorem..
      
    • and the proof of Zariski's theorem ..
      
    • C P Ramanujam continuing my lectures at the Tata Institute lectured on and wrote up notes on Tate's theorem on homomorphisms between abelian varieties over finite fields.
      
    • We discussed many topics involving topology and algebraic geometry at that time, and especially Kodaira's Vanishing Theorem.
      
    • These include a characterization of C2, a version of the Kodaira vanishing theorem, a study of the automorphism group of a variety, a study of the purity of the discriminant locus, a proof that the invariance of the Milnor number implies the invariance of the topological type, and a geometric interpretation of multiplicity.
      
    • He felt the spirit of mathematics demanded of him not merely routine developments but the right theorem an any given topic.
      

36. Wu Wen-Tsun biography
    
    • The following year saw a wealth of papers from Wu: On the product of sphere bundles and the duality theorem modulo two; Sur l'existence d'un champ d'elements de contact ou d'une structure complexe sur une sphere; Sur les classes caracteristiques d'un espace fibre en spheres; Sur le second obstacle d'un champ d'elements de contact dans une structure fibree spherique; and Sur la structure presque complexe d'une variete differentiable reelle de dimension 4.
      
    • He wrote the important book Mechanical theorem proving in geometries (1984) in Chinese which was translated into English and published ten years later.
      
    • Mechanization of theorem proving in geometry and Hilbert's mechanization theorem.
      
    • The mechanization theorem of (ordinary) unordered geometry.
      
    • In 2000 Wu published Mathematics mechanization : Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equations-solving.
      

37. Yamabe biography
    
    • In On an arcwise connected subgroup of a Lie group published in 1950 in the Osaka Mathematical Journal Yamabe gave a direct proof of the theorem that an arcwise connected subgroup of a Lie group is a Lie subgroup.
      
    • Despite the large amount of progress that Yamabe was making to solve Hilbert's Fifth Problem, he was also producing many other papers such as: (with Zuiman Yujobo) On the continuous function defined on a sphere (1950); (with Morikuni Goto) On continuous isomorphisms of topological groups (1950); On an extension of the Helly's theorem (1950); and A condition for an abelian group to be a free abelian group with a finite basis (1951).
      
    • It was during his two years at Princeton that he published the two papers On the conjecture of Iwasawa and Gleason and A generalization of a theorem of Gleason.
      
    • A proof of a theorem on Jacobians (1957) published in the American Mathematical Monthly was a paper concerning the teaching of mathematics in which he gave a proof of the invertibility of mappings with non-zero Jacobian.
      
    • In the following year Yamabe published A unique continuation theorem for solutions of a parabolic differential equation written jointly with Seizo Ito.
      
    • The same theme was taken up in A unique continuation theorem of a diffusion equation which he published in 1959.
      

38. Nikodym biography
    
    • the Radon-Nikodym theorem and derivative, the Nikodym convergence theorem, the Nikodym-Grothendieck boundedness theorem), in functional analysis (the Radon-Nikodym property of a Banach space, the Frechet-Nikodym metric space, a Nikodym set), projections onto convex sets with applications to Dirichlet problem, generalized solutions of differential equations, descriptive set theory and the foundations of quantum mechanics.
      
    • The Radon-Nikodym theorem (Radon proved it in 1913 for Rn and Nikodym in 1930 for the general case) is now a fundamental theorem in analysis: .
      
    • His discoveries were very deep and will live for ever as long as Pythagoras's theorem, which has survived for centuries.
      

39. Liu Hui biography
    
    • This is achieved with an application of Pythagoras's theorem, which Liu Hui knew as the Gougu theorem.
      
    • But now we know AY and YX so we can compute AX using the Gougu theorem (Pythagoras) to be .
      
    • It shows how to use the Gougu theorem (Pythagoras theorem) to calculate heights of objects and distances to objects which cannot be measured directly.
      
    • D B Wagner (Pythagoras's theorem) .
      

40. Caccioppoli biography
    
    • In this work Caccioppoli began to investigate how to generalise Riesz's theorem on the representation of linear functionals by extending the initial definition set.
      
    • In the same year Caccioppoli considered the extension of the definition of linear functionals from the set of continuous functions to the set of Baire functions, anticipating a special case of the Hahn-Banach theorem.
      
    • Carrying on in this way Caccioppoli, in 1931, extended in some cases Brouwer's fixed point theorem, and applied his results to existence problems of both partial differential equations and ordinary differential equations.
      
    • To decide on both existence and uniqueness (and not only on existence, as Brouwer's theorem does) he provided the general concept of functional correspondence inversion, stating, in 1932, that a transformation between two Banach spaces is invertible only if it is locally invertible and if the compact sequences are the only ones to be transformed into convergent sequences.
      
    • In that period he successfully studied branches of functions defined on Cn, and in 1933 found the basic theorem on normal families of functions of complex variables, namely that if a family is normal to each complex variable, it is also normal to the whole set of variables.
      
    • Napoli, s.IV, v.4 (1934) ) the theorem on harmonicity of orthogonal functions to any Laplacian, best known as "Weyl's lemma".
      

41. Hippocrates biography
    
    • In his attempts to square the circle, Hippocrates was able to find the areas of lunes, certain crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii.
      
    • segment 1/segment 2 = AB2/AC2 = 1/2 since AB2+ BC2= AC2 by Pythagoras's theorem, and AB = BC so AC2= 2AB2.
      
    • In Hippocrates' study of lunes, as described by Eudemus, he uses the theorem that circles are to one another as the squares on their diameters.
      
    • This theorem is proved by Euclid in the Elements and it is proved there by the method of exhaustion due to Eudoxus.
      
    • However, Eudoxus was born within a few years of the death of Hippocrates, and so there follows the intriguing question of how Hippocrates proved this theorem.
      

42. Mazur Barry biography
    
    • Mazur's work in topology was outstanding and it led to the award of the Veblen Prize in 1966 for his work on the generalized Schoenflies theorem.
      
    • Ken Ribet's proof that the Taniyama-Shimura conjecture implies a proof of Fermat's Last Theorem, in "On modular representations of Gal(Q/Q) arising from modular forms" (1990).
      
    • Wiles's proof of the Taniyama-Shimura conjecture and of Fermat's Last Theorem, in "Modular elliptic curves and Fermat's last theorem" (1995), using results with R Taylor in "Ring-theoretic properties of certain Hecke algebras".
      
    • Mazur had much earlier received the Cole prize for work which would prove important in the solution of Fermat's last Theorem.
      

43. Germain biography
    
    • Among her work done during this period is work on Fermat's Last Theorem and a theorem which has become known as Germain's Theorem.
      
    • This was to remain the most important result related to Fermat's Last Theorem from 1738 until the contributions of Kummer in 1840.
      
    • History Topics: Fermat's last theorem .
      

44. Cauchy biography
    
    • Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations and Cauchy sequences.
      
    • Cauchy's theorem in Complex analysis .
      
    • History Topics: Fermat's last theorem .
      
    • History Topics: The fundamental theorem of algebra .
      

45. Chebotaryov biography
    
    • In this same year, 1922, Nikolai was to prove the theorem for which he is best known, the density theorem, which had been conjectured by Frobenius in a paper written in 1880 but only published in 1896.
      
    • The density theorem generalised Dirichlet's theorem on primes in an arithmetical progression giving a method used by Artin in 1927 in his reciprocity law, a result considered the main result of class field theory.
      
    • Chebotaryov received his doctorate from the Ukrainian Academy of Sciences with a doctoral thesis based on his 1922 result, the density theorem.
      

46. Fischer biography
    
    • Fischer is best known for one of the highpoints of the theory of Lebesgue integration, called the Riesz-Fischer Theorem.
      
    • The theorem is that the space of all square-integrable functions is complete, in the sense that Hilbert space is complete, and the two spaces are isomorphic by means of a mapping based on a complete orthonormal system.
      
    • Let us note again the major result, the Riesz-Fischer Theorem, for which he is best known as Weyl noted in the above quote.
      
    • His two papers of 1907 were Sur la convergence en moyenne and Applications d'un theorem sur la convergence en moyenne both published in Comptes rendus of the Academy of Sciences in Paris.
      
    • The theorem, now called the Riesz-Fischer theorem, is one of the great achievements of the Lebesgue theory of integration.
      

47. Maschke biography
    
    • He is best known today for Maschke's theorem, which he published in 1899, which states that if the order of the finite group G is not divisible by the characteristic of the field K, then the (finite-dimensional) K-representations of G are completely reducible.
      
    • Maschke proved a special case of his theorem in the paper Uber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen published in 1898.
      
    • In his proof Maschke used a theorem by Moore which he had announced to the Mathematics Club at the University of Chicago on 10 July 1896.
      
    • Moore's paper appeared in Mathematische Annalen two years later and the Loewy-Moore theorem provided Maschke with a critical step in the proof of his own theorem.
      

48. Al-Farisi biography
    
    • but he attempted no proof of this case of Fermat's Last Theorem.
      
    • In Tadhkira al-ahbab fi bayan al-tahabb (Memorandum for friends on the proof of amicability) al-Farisi gave a new proof of the following theorem by Thabit ibn Qurra on amicable numbers: .
      
    • In fact al-Farisi's approach is based on the unique factorisation of an integer into powers of prime numbers, and, according to Rashed, he states and attempts to prove this, the so-called fundamental theorem of arithmetic, in this work.
      
    • Whether al-Farisi proved or attempted to prove the fundamental theorem of arithmetic is also discussed in [Historia Math.
      
    • To check that Thabit's theorem gives amicable numbers with n = 4, al-Farisi has to show that p3, p4, and q4 are prime numbers.
      

49. Braikenridge biography
    
    • Braikenridge is well known for his geometrical theorems, in particular he discovered the following theorem now called the Braikenridge - Maclaurin theorem:- .
      
    • This theorem appeared in Exercitatio geometrica de descriptione linearum curvarum published in 1733 but there followed a dispute regarding priority with Maclaurin.
      
    • Braikenridge claimed to have discovered the theorem, and many other results, in 1726 when he was living in Edinburgh and that Maclaurin had learnt of them.
      
    • Some time before that, he showed me a theorem, which coincided with one in my book, though he seemed not to have observed that coincidence.
      

50. Church biography
    
    • Church is probably best remembered for 'Church's Theorem' and 'Church's Thesis' both of which first appeared in print in 1936.
      
    • Church's Theorem, showing the undecidability of first order logic, appeared in A note on the Entscheidungsproblem published in the first issue of the Journal of Symbolic Logic.
      
    • Church's Theorem extends the incompleteness proof given of Godel in 1931.
      
    • For the expert the chief interest in the tract is that it makes readily accessible careful detailed formulation and proofs of certain standard theorems, for example, the deduction theorem, the reduction to truth tables, the substitution rule for the functional calculus, Godel's completeness theorem, etc.
      

51. Gorenstein biography
    
    • Largely under the impetus of the odd order theorem, there was an awakening interest in finite group theory.
      
    • Tits (entering the field somewhat earlier) had deepened our understanding of the Chevalley groups and their Steinberg-Suzuki-Ree variations, Bender in Germany was to prove the fundamental strongly embedded subgroup classification theorem, and Harada was beginning his career in Japan.
      
    • Quickly assuming a leadership role in a single minded pursuit of the full classification theorem, he was to carry the entire "team" along with him over the following decade until the proof was completed.
      
    • The 255 page proof of the odd order theorem, filling an entire issue of the Pacific Journal, had set the tone, but it was by far not the longest paper.
      
    • No mathematical theorem could require the number of pages these fellows were taking! Surely they were missing some geometric interpretation of the simple groups that would lead to a substantially shorter classification proof.
      

52. Skolem biography
    
    • One example is the Skolem-Noether theorem.
      
    • Skolem extended work by Lowenheim (published in 1915) to give the Lowenheim-Skolem theorem, which he published in 1920.
      
    • Here he applied the Lowenheim-Skolem theorem to show what became known as Skolem's paradox: If the Zermelo's axiomatic system for set theory is consistent then it must be satisfiable within a countable domain.
      
    • We mentioned above that Skolem worked on algebra, and we also mentioned the Skolem-Noether theorem.
      
    • Skolem published this theorem in 1927 in a paper Zur Theorie der assoziativen Zahlensysteme.
      

53. Abel biography
    
    • Abel's theorem states that any such sum can be expressed as a fixed number p of these integrals, with integration arguments that are algebraic functions of the original arguments.
      
    • Abel's theorem is a vast generalisation of Euler's relation for elliptic integrals.
      
    • He called it only "A theorem": it had no introduction, contained no superfluous remarks, no applications.
      
    • It was a monument resplendent in its simple lines - the main theorem from his Paris memoir, formulated in few words.
      
    • In fact in a letter Abel had written to Crelle on 18 October 1828 he gave the theorem [Amer.
      

54. Moore Robert biography
    
    • To liven up a lecture he would run a race with his professor by seeing if he could discover the proof of an announced theorem before the lecturer had finished his presentation.
      
    • When the class returned for the next meeting he would call on some student to prove Theorem 1.
      
    • When a student stated that he could prove Theorem x, he was asked to go to the blackboard and present his proof.
      
    • Moore would then ask the next student to try or if he thought the difficulty encountered was sufficiently interesting, he would save that theorem until next time and go on to the next unproved theorem (starting again at the bottom of the class).
      

55. Bing biography
    
    • One has wild surfaces, the Schonflies theorem, triangulation, Dehn's lemma, the shrinking criterion, linking, the loop theorem, covering spaces, as well as the important side approximation theorem.
      
    • Many of these results are applications of the side approximation theorem.
      
    • For example, he never claimed to understand a theorem unless he personally knew a proof of it.
      

56. Polya biography
    
    • He wrote on the normal distribution and coined the term "central limit theorem" in 1920 which is now standard usage.
      
    • In 1921 he proved his famous theorem on random walks on an integer lattice.
      
    • His main contribution to combinatorics is his enumeration theorem, published in 1937.
      
    • a remarkable theorem in a remarkable paper, and a landmark in the history of combinatorial analysis.
      
    • The theorem solves the problem of how many configurations with certain properties exist.
      

57. Rouche biography
    
    • Although few today know who Rouche was, his name is very well known through Rouche's theorem which he published in the Journal of the Ecole Polytechnique 39 (1862).
      
    • Rouche later published a fuller version of this theorem in 1880 in the Journal de l'Ecole polytechnique.
      
    • When Frobenius discussed this result in his papers, for example in Zur Theorie der linearen Gleichungen published in Crelle's Journal in 1905, he gave credit for proving the theorem to both Rouche and Fontene.
      
    • However it is now often called the Rouche-Frobenius theorem, especially in the Spanish speaking world.
      
    • This is almost certainly because the Spanish/Argentinian mathematician Julio Rey Pastor referred to the theorem by this name.
      

58. Argand biography
    
    • However this is not so and, although he will always be remembered for the Argand diagram, his best work is on the fundamental theorem of algebra and for this he has received little credit.
      
    • He gave a beautiful proof (with small gaps) of the fundamental theorem of algebra in his work of 1806, and again when he published his results in Gergonne's Journal in 1813.
      
    • Certainly Argand was the first to state the theorem in the case where the coefficients were complex numbers.
      
    • ',6)">6], discusses the early proofs of the fundamental theorem and remarks that Argand gave an almost modern form of the proof which was forgotten after its second publication in 1813.
      
    • History Topics: The fundamental theorem of algebra .
      

59. Bendixson biography
    
    • As a young student Bendixson made his name by proving a theorem which he included in a letter which he wrote to Cantor, the letter being published in Volume 2 of Acta Mathematica.
      
    • This theorem states [Svenskt Biografiskt Lexikon 3 (Stockholm, 1922), 146-150.',1)">1]:- .
      
    • The proof of the theorem which Bendixson gave uses Cantor's notion of transfinite numbers.
      
    • Bendixson is probably best remembered for the Poincare- Bendixson theorem.
      
    • The Poincare-Bendixson theorem, which says an integral curve which does not end in a singular point has a limit cycle, was first proved by Poincare but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901.
      

60. Wedderburn biography
    
    • In the paper he published in that year he gave three proofs of this theorem which were all based on a clever use of the interplay between the additive group of a finite division algebra A, and the multiplicative group A* = A-{0}.
      
    • 33 (111) (1983), 274-299.',10)">10] Parshall discusses this theorem.
      
    • This theorem gave, as a corollary, the complete structure of all finite projective geometries.
      
    • Wedderburn and Veblen showed that in all these geometries Pascal's theorem is a consequence of Desargues' theorem.
      

61. Chowla biography
    
    • He wrote on additive number theory (lattice points, partitions, Waring's problem), analysis, Bernoulli numbers, class invariants, definite integrals, elliptic integrals, infinite series, the Weierstrass approximation theorem), analytic number theory (Dirichlet L-functions, primes, Riemann and Epstein zeta functions), binary quadratic forms and class numbers, combinatorial problems (block designs, difference sets, Latin squares), Diophantine equations and Diophantine approximation, elementary number theory (arithmetic functions, continued fractions, and Ramanujan's tau function), and exponential and character sums (Gauss sums, Kloosterman sums, trigonometric sums).
      
    • Among the theorems to which Chowla's name have been attached are the Bruck-Chowla-Ryser theorem on designs (1950); the Ankeny-Artin-Chowla theorem on the class number of real quadratic number fields (1952); the Chowla-Mordell theorem on Gauss sums (1962); and the Chowla-Selberg formula for the product of certain values of the Dedekind eta function.
      
    • Among a long list of other results we mention just a very few such as his generalisation of Wolstenholme's theorem; his work on classes of quintics not soluble by radicals; his closed form for the Bernoulli numbers; and his work on the length of the period of the continued fraction expansion of √N.
      

62. Frobenius biography
    
    • On Sylow's theorem.
      
    • This paper also gives a proof of the structure theorem for finitely generated abelian groups.
      
    • In 1884 he published his next paper on finite groups in which he proved Sylow's theorems for abstract groups (Sylow had proved his theorem as a result about permutation groups in his original paper).
      
    • In 1898 he introduced the notion of induced representations and the Frobenius Reciprocity Theorem.
      
    • History Topics: The fundamental theorem of algebra .
      

63. Helly biography
    
    • During this period, he undertook research on functional analysis and proved the Hahn-Banach theorem in 1912, fifteen years before Hahn published essentially the same proof and 20 years before Banach gave his new setting.
      
    • He is remembered for Helly's theorem, published in 1923, which states that if there are given n convex subsets of a d-dimensional Euclidean space with n ≥ d+1 and if each collection of d + 1 of the subsets has a point in common then there is a common point of the n subsets.
      
    • One is the fact that he gives the Hahn-Banach theorem for the space C[a, b], while he is providing a simpler proof of a theorem which Riesz had published the previous year.
      
    • He also gives the uniform boundedness principle for linear functionals, the Banach-Steinhaus theorem.
      

64. Autolycus biography
    
    • Theorem 2 of Euclid's Phaenomena consists of four propositions with proofs for only three of them while the missing one is replaced by the remark "that this is the case has been shown elsewhere"; indeed theorem and proof are found as Theorem 10 in Autolycus's 'Rotating Sphere'.
      
    • This means that a theorem in Autolycus's work has first a general statement, then a construction related to a particular figure with points in the figure denoted by letters, next comes the demonstration of the theorem, and finally a conclusion relating to the general statement is sometimes drawn.
      

65. Dirichlet biography
    
    • Dirichlet's first paper was to bring him instant fame since it concerned the famous Fermat's Last Theorem.
      
    • The theorem claimed that for n > 2 there are no non-zero integers x, y, z such that xn + yn = zn.
      
    • The cases n = 3 and n = 4 had been proved by Euler and Fermat, and Dirichlet attacked the theorem for n = 5.
      
    • We have already commented on his contributions to Fermat's Last Theorem made in 1825.
      
    • History Topics: Fermat's last theorem .
      

66. Goldie Alfred biography
    
    • This led Goldie to the results in universal algebra which he published in The Jordan-Holder theorem for general abstract algebras (1950) and The scope of the Jordan-Holder theorem in abstract algebra (1952).
      
    • The breakthrough which allowed him to complete his structure theorem for Noetherian prime rings came after a family holiday to Sedberg in the Yorkshire Dales National Park was abandoned early due to persistent rain [Amer.
      
    • Goldie published his results, now known as "Goldie's Theorem," in The structure of prime rings with maximum conditions (1958) and The structure of prime rings under ascending chain conditions (1958).
      
    • It is pertinent to note that a 1987 graduate text which aimed to describe the result and to survey some of its consequences ("Noncommutative Noetherian Rings" by J C McConnell and J C Robson) uses about 50 pages to establish Goldie's Theorem and over 500 pages to outline consequences.
      

67. Ruffini biography
    
    • Behold a very important theorem which I believe I am able to assert (if I do not err): to present the proof of it is the main reason for publishing this volume.
      
    • It is remarkable work and, except for one gap, proves the theorem as Ruffini claimed.
      
    • and recommend greatly the most important theorem which excludes the possibility of solving equations of degree greater than four.
      
    • Given the information in this article about the insolubility of the quintic, it is reasonable to ask why Abel has been credited with proving the theorem while Ruffini has not.
      

68. De Bruijn biography
    
    • In 1941 he published Ein Satz uber schlichte Funktionen followed by Common representative systems of two divisions of an aggregate into classes (1943) which generalised a theorem proved for finite sets by Denes Konig in 1916 and van der Waerden in 1927, then for infinite sets by Konig and Valko in 1925.
      
    • His work on combinatorics resulted in influential notions and results of which we mention the de Bruijn-sequences of 1946 and the de Bruijn-Erdos theorem of 1948.
      
    • The language is so designed that it is incorrect to state a theorem without first "constructing" a proof of the theorem.
      

69. Turan biography
    
    • The second was On a theorem of Hardy and Ramanujan which was published in the Journal of the London Mathematical Society.
      
    • His thesis On the number of prime divisors of integers, written in Hungarian, had been published in 1934 and contained his new proof of the theorem of Hardy and Ramanujan referred to above.
      
    • giving a systematic account of his methods for estimating "power sums", which he had developed (1941-53) into a versatile and powerful technique with numerous applications to Diophantine approximations, zero-free regions for the Riemann zeta function and the error term in the prime number theorem, and to problems in other parts of classical analysis.
      
    • As regards the latter, Turan found new approaches to such topics as quasi-analytic classes, Fabry's gap theorem and the theory of lacunary series, amongst others.
      

70. Arf biography
    
    • Later, I learned the Galois theorem and then I understood.
      
    • He completed his doctoral studies in 1938 obtaining, among other results, the theorem now known as the Hasse-Arf theorem.
      
    • His name is not only attached to Arf invariants but he is also remembered for the Hasse-Arf Theorem which plays an important role in class field theory and in Artin's theory of L-functions.
      

71. Bell John biography
    
    • During his time in Birmingham, Bell did work of great importance, producing his version of the celebrated CPT theorem of quantum field theory.
      
    • This theorem showed that under the combined action of three operators on a physical event: P, the parity operator, which performed a reflection; C, the charge conjugation operator, which replaced particles by anti-particles; and T, which performed a time reversal, the result would be another possible physical event.
      
    • Unfortunately Gerhard Luders and Wolfgang Pauli proved the same theorem a little ahead of Bell, and they received all the credit.
      
    • They were therefore pleased when John von Neumann proved a theorem claiming to show rigorously that it is impossible to add hidden variables to the structure of quantum theory.
      

72. Al-Khujandi biography
    
    • It remains for us to discuss the claim that al-Khujandi discovered the sine theorem.
      
    • Both Abu'l-Wafa and Abu Nasr Mansur claim to have discovered the sine theorem while, as far as we are aware, al-Khujandi makes no such claim.
      
    • Finally, although this really proves little, the theorem appears many times in the writings of Abu Nasr Mansur: both his writings on geometry as well as those on astronomy.
      
    • He stated Fermat's Last Theorem in the case n = 3 although, not surprisingly, his proof is wrong.
      

73. Beatty biography
    
    • We give a few examples: Derivation of the Complementary Theorem from the Riemann-Roch Theorem (1917), (with Muriel Wales) Theory of algebraic functions based on the use of cycles (1944), On the minimum value of the Riemann-Roch expression for order-bases in the large (1948), On the number of conditions to apply to a function R(Z, U) to build it on an assigned local order-basis t (1948), (with N D Lane) A symmetric proof of the Riemann-Roch theorem, and a new form of the unit theorem (1952), Upper and lower estimates for the area of a triangle (1954), and Difference methods in the theory of local order bases and their equivalent normalized function bases (1956).
      

74. Hua biography
    
    • By the time Hua returned to Jintan he was already engaged in mathematics and his first publication Some Researches on the Theorem of Sturm, appeared in the December 1929 issue of the Shanghai periodical Science.
      
    • This encouraged Hardy and Littlewood in 1920 to apply a similar method for general k, and they devised the so-called circle method to tackle the general Hilbert-Waring theorem and a host of other additive problems, Goldbach's problem among them.
      
    • During this time he studied Vinogradov's seminal method of estimating trigonometric sums and reformulated it, even in sharper form, in what is now known universally as Vinogradov's mean value theorem.
      
    • This famous result is central to improved versions of the Hilbert-Waring theorem, and has important applications to the study of the Riemann zeta function.
      

75. Osgood biography
    
    • In 1900 Osgood established, by methods due to H Poincare, the Riemann mapping theorem, namely that an arbitrary simply connected region of the plane with at least two boundary points, can be mapped uniformly and conformally onto the interior of a circle.
      
    • This theorem remains as Osgood's outstanding single result.
      
    • Some papers over the next few years included: Sufficient conditions in the calculus of variations (1900), On a fundamental property of a minimum in the calculus of variations and the proof of a theorem of Weierstrass's (1901), A Jordan curve of positive area (1903), On Cantor's theorem concerning the coefficients of a convergent trigonometric series, with generalizations (1909), On the gyroscope (1922), and On normal forms of differential equations (1925).
      

76. Rolle biography
    
    • Rolle's theorem, an important proposition of the calculus, also owes its origin to the method.
      
    • In fact Rolle is best remembered for 'Rolle's Theorem' which was published in Demonstration d'une Methode pour resoudre les Egalitez de tous les degrez in 1691.
      
    • The familiar Rolle's Theorem states: .
      
    • The name 'Rolle's Theorem' was given to this basic result by Giusto Bellavitis in 1846.
      

77. Pick biography
    
    • He is best remembered, however, for Pick's theorem which appeared in his eight page paper of 1899 Geometrisches zur Zahlenlehre published in Prague in Sitzungber.
      
    • Pick's theorem is on reticular geometry.
      
    • Pick's theorem states that the area of a reticular polygon is L + B/2 - 1 where L is the number of reticular points inside the polygon and B is the number of reticular points on the edges of the polygon.
      
    • From that time on Pick's theorem has attracted much attention and admiration for its simplicity and elegance.
      

78. Riesz biography
    
    • Riesz was to publish many papers in this journal, the first in 1922 being on Egorov's theorem on linear functionals.
      
    • His theorem, now called the Riesz-Fischer theorem, which he proved in 1907, is fundamental in the Fourier analysis of Hilbert space.
      
    • Riesz made many contributions to other areas including ergodic theory where he gave an elementary proof of the mean ergodic theorem in 1938.
      

79. Godel biography
    
    • The theorem did not destroy the fundamental idea of formalism, but it did demonstrate that any system would have to be more comprehensive than that envisaged by Hilbert.
      
    • Karlis Podnieks (A hypertext introduction to Godel's theorem) .
      
    • Torkel Franzen (The incompletenes theorem) .
      
    • Sheffield University (Godel's Theorem) .
      

80. Moufang biography
    
    • Reversing a development going from Euclid to Descartes in which geometry is replaced by algebra as a fundamental discipline of mathematics, Hilbert had shown that a subset of his axioms for plane geometry (essentially the incidence axioms) together with the incidence theorem of Desargues permits the introduction of coordinates on a straight line which are elements of a skew field.
      
    • If Desargues' theorem is replaced by that of Pappus, the coordinates become elements of a field.
      
    • Moufang (1933) showed that another incidence theorem, called the theorem of the complete quadrilateral (or of the invariance of the fourth harmonic point), allows one to introduce coordinates which are elements of an alternating division algebra.
      

81. Eisenstein biography
    
    • As a boy of six I could understand the proof of a mathematical theorem more readily than that meat had to be cut with one's knife, not one's fork.
      
    • He was working on a variety of topics at this time including quadratic forms and cubic forms, the reciprocity theorem for cubic residues, quadratic partition of prime numbers and reciprocity laws.
      
    • At this time Eisenstein was working on a variety of topics including quadratic and cubic forms and the reciprocity theorem for cubic residues.
      
    • The work of both Kummer and Eisenstein, and the rivalry which existed between the two in their work published in 1850 on the higher reciprocity laws, is discussed in [Number theory related to Fermat\'s last theorem (Boston, Mass., 1982), 31-43.',7)">7].
      

82. Grosswald biography
    
    • Topics covered include: the fundamental theorem of arithmetic, congruences, the quadratic reciprocity theorem, the standard arithmetical functions, the prime number theorem, Fermat's last theorem, and the theory of partitions.
      

83. Bezout biography
    
    • He eschewed the frightening terms "axiom", "theorem", "scholium", and tried to avoid arguments that were too close and detailed.
      
    • This work includes a result known as Bezout's theorem:- .
      
    • Despite this Bezout, who was prepared to enter long and difficult algebraic manipulations, proved his theorem with just a little hand waving over an inductive argument.
      
    • Other Web sitesSheffield University (Bezout's Theorem) .
      

84. Castelnuovo biography
    
    • They pursued the idea that a Riemann-Roch theorem for surfaces would be a powerful tool.
      
    • Castelnuovo is also remembered for the Kronecker-Castelnuovo theorem which states:- .
      
    • Kronecker had first stated a version of this theorem in a lecture which he gave to the Accademia dei Lincei in 1886.
      
    • Kronecker never published the theorem and it was Castelnuovo's version which appeared in print.
      

85. Lob biography
    
    • It is the second of these papers which is the most famous and contains what is now known as Lob's theorem.
      
    • The problem really goes back to Godel's incompleteness theorem of 1931.
      
    • Lob's theorem is in some sense gives rise to a paradox called Lob's paradox.
      
    • Examples of papers he published in the 1970s are A model theoretic characterization of effective operations (1970), Hierarchies of number-theoretic functions (1970), A reduction theorem for predicate logic (1972) and Embedding first order predicate logic in fragments of intuitionistic logic (1976).
      

86. Haret biography
    
    • The present paper, dedicated to the 140th anniversary of the birthday of the Romanian mathematician, mechanicist, and astronomer Spiru Haret (1851 - 1912), is an attempt to state as a theorem his famous result concerning the well-known problem on the invariability of the major axes of planetary orbits, related to the stability of the solar-planetary system.
      
    • "Spiru Haret's theorem" is to be naturally added to the logical succession of theorems with respect to this problem known as "Laplace-Lagrange theorem" and "Poisson's theorem".
      

87. Dougall biography
    
    • In 1952 he published The double six of lines and a theorem in Euclidean plane geometry in Proc.
      
    • Representing the lines of projective 3-space by the points of a quadric fourfold Q in projective 5-space, the author shows that Schlafli's "double six" theorem is equivalent to the following: .
      
    • By regarding Q as a 4-sphere in complex Euclidean 5-space, and making some projections, he relates this to a simple theorem of plane geometry: .
      
    • He deduces this from the classical invariant relation, thus providing a new proof for the double six theorem itself.
      

88. Schauder biography
    
    • Schauder's main achievement consists in transferring some topological notions and theorems to Banach spaces (the fixed point theorem, invariance of domain, the concept of index).
      
    • In particular, Schauder's formulation of a fixed point theorem originated a new, extremely fruitful method in the theory of differential equations, known as Schauder's method ..
      
    • Schauder's fixed point theorem and his skillful use of function space techniques to analyse elliptic and hyperbolic partial differential equations are contributions of lasting quality.
      
    • Existence proofs for complicated nonlinear problems using his fixed point theorem have become standard.
      

89. Kummer biography
    
    • In 1843 Kummer, realising that attempts to prove Fermat's Last Theorem broke down because the unique factorisation of integers did not extend to other rings of complex numbers, attempted to restore the uniqueness of factorisation by introducing 'ideal' numbers.
      
    • Not only has his work been most fundamental in work relating to Fermat's Last Theorem, since all later work was based on it for many years, but the concept of an ideal allowed ring theory, and much of abstract algebra, to develop.
      
    • In fact the prize of 3000 francs was offered for a solution to Fermat's Last Theorem but when no solution was forthcoming, even after extending the date, the Prize was given to Kummer even though he had not submitted an entry for the Prize.
      
    • History Topics: Fermat's last theorem .
      

90. Smithies biography
    
    • Smithies looks in detail at the development of the concept of an adjoint operator in the years before the Hahn-Banach theorem.
      
    • Smithies last paper was A forgotten paper on the fundamental theorem of algebra published in the Notes and Records of the Royal Society of London in 2000.
      
    • In 1798, there appeared in the Philosophical Transactions of the Royal Society a paper by James Wood, purporting to prove the fundamental theorem of algebra, to the effect that every non-constant polynomial with real coefficients has at least one real or complex zero.
      
    • After putting Wood's work in context, I conclude by showing how his idea can be used to prove the complex form of the fundamental theorem of algebra, stating that every non-constant polynomial with complex coefficients has at least one zero in the complex field.
      

91. Kellogg biography
    
    • This paper includes what today is called 'Kellogg's theorem' on harmonic and Green's functions.
      
    • This contains the Birkhoff-Kellogg Theorem which generalises the Brouwer fixed point theorem.
      
    • After his death Converses of Gauss' theorem on the arithmetic mean was published in the Transactions of the American Mathematical Society.
      

92. Lagrange biography
    
    • The paper draws an analogy between the binomial theorem and the successive derivatives of the product of functions.
      
    • In 1771 he proved Wilson's theorem (first stated without proof by Waring) that n is prime if and only if (n -1)! + 1 is divisible by n.
      
    • History Topics: The fundamental theorem of algebra .
      
    • Sheffield University (Lagrange's Four-square Theorem) .
      

93. Laguerre biography
    
    • Deep relations between elliptic functions and Cartesian ovals were also established in 1867, with the geometrical proofs of the addition theorem of elliptic functions given by Darboux and Laguerre.
      
    • When Darboux proved the orthogonality of systems of homofocal ovals, he also showed that ovals provide a geometrical interpretation of the addition theorem and that they constitute the algebraic form of the integral solution.
      
    • Laguerre, on the other hand, proved the addition theorem with the help of anallagmatic curves using Poncelet's theorem on inscribed and circumscribed polygons to two conics.
      

94. Banach biography
    
    • There is the Hahn-Banach theorem on the extension of continuous linear functionals, the Banach-Steinhaus theorem on bounded families of mappings, the Banach-Alaoglu theorem, the Banach fixed point theorem and the Banach-Tarski paradoxical decomposition of a ball.
      

95. Herbrand biography
    
    • He made contributions to mathematical logic where Herbrand's theorem on the theory of quantifiers appears in his doctoral thesis.
      
    • Herbrand's theorem establishes a link between quantification theory and sentential logic which is important in that it gives a method to test a formula in quantification theory by successively testing formulas for sentential validity.
      
    • Since testing for sentential validity is a mechanical process, Herbrand's theorem is today of major importance in software developed for theorem proving by computer.
      

96. Mittag-Leffler biography
    
    • His best known work concerned the analytic representation of a one-valued function, this work culminated in the Mittag-Leffler theorem.
      
    • This study began as an attempt to generalise results in Weierstrass's lectures where he had described his theorem on the existence of an entire function with prescribed zeros each with a specified multiplicity.
      
    • He eventually assembled his findings on generalising Weierstrass's theorem to meromorphic functions into a paper which he published (in French) in 1884 in Acta Mathematica .
      
    • Mittag-Leffler became the sole proprietor of a theorem that later became widely known and with this he took his place in the circle of internationally known mathematicians.
      

97. Lehmer Emma biography
    
    • Another joint project which Lehmer undertook with her husband was assisting Vandiver with his work on Fermat's Last Theorem, and together the Lehmers computed many Bernoulli numbers which Vandiver required in his work.
      
    • As an example of other contributions Lehmer made to Fermat's Last Theorem she published On a resultant connected with Fermat's Last Theorem in 1935, then, jointly with her husband, On the first case of Fermat's Last Theorem in 1941.
      

98. Faltings biography
    
    • Faltings has been closely linked with the work leading to the final proof of Fermat's Last Theorem by Andrew Wiles.
      
    • However, Faltings was the natural person that Wiles turned to when he wanted an opinion on the correctness of his repair of his proof of Fermat's Last Theorem in 1994.
      
    • History Topics: Fermat's last theorem .
      
    • AMS (An article about the proof of Fermat's last theorem) [registration required] .
      

99. Landau biography
    
    • On 9 June 1900 he wrote a letter from Paris, where he was studying, to Hilbert giving an outline of his ideas for proving the prime ideal theorem for algebraic number fields.
      
    • One he claimed was trivial and another could be trivially deduced from a theorem of Mittag-Leffler.
      
    • He gave a proof of the prime number theorem in 1903 which was considerably simpler that the ones given in 1896 by Vallee Poussin and Hadamard.
      
    • Prime Number Theorem .
      

100. Ceva Giovanni biography
     
     • The theorem states that lines from the vertices of a triangle to the opposite sides are concurrent precisely when the product of the ratio the sides are divided is 1.
       
     • Ceva also rediscovered and published Menelaus's theorem.
       
     • Ceva's theorem .
       
     • Menelaus's theorem .
       

101. Al-Haytham biography
     
     • In number theory al-Haytham solved problems involving congruences using what is now called Wilson's theorem: .
       
     • Using Wilson's theorem, this is divisible by 7 and it clearly leaves a remainder of 1 when divided by 2, 3, 4, 5, and 6.
       
     • Ibn al-Haytham's second method gives all the solutions to systems of congruences of the type stated (which of course is a special case of the Chinese Remainder Theorem).
       
     • Wilson's theorem .
       

102. Hilbert biography
     
     • Hilbert's first work was on invariant theory and, in 1888, he proved his famous Basis Theorem.
       
     • Twenty years earlier Gordan had proved the finite basis theorem for binary forms using a highly computational approach.
       
     • He discovered a completely new approach which proved the finite basis theorem for any number of variables but in an entirely abstract way.
       
     • Hilbert submitted a paper proving the finite basis theorem to Mathematische Annalen.
       

103. Hahn biography
     
     • However to many mathematicians he is best remembered for the Hahn-Banach theorem which we mention again below.
       
     • These include a report on integral equation he wrote in 1911, his modification of Hellinger's theory of invariants of quadratic forms, in which he dispensed with the use of the Hellinger integral, and his work on duality in Banach spaces, culminating with his proof of the Hahn-Banach theorem in 1927.
       
     • He wrote papers on the theory of curves including one which gave a rigorous proof of the Jordan's theorem for simple closed polygons which he based on Veblen's geometrical axioms.
       
     • Other papers in this area characterise topological spaces that are continuous images of a line segment and related to this topic is what is now known as the Hahn-Mazurkiewicz theorem.
       

104. Lefschetz biography
     
     • He published his fixed point theorem for compact orientable manifolds in 1923, giving a fuller account of his famous fixed point theorem in Intersections and transformations of complexes and manifolds published in the Transactions of the American Mathematical Society in 1926.
       
     • In 1927 he fulfilled his promise extending his fixed point theorems to manifolds with boundary, and by this stage Brouwer's fixed point theorem became a special case.
       
     • Even if there is little truth in a joke which circulated about Lefschetz, namely that he never wrote a correct proof or stated an incorrect theorem, there is an underlying truth in it reflecting on his style of mathematics.
       

105. Erdos biography
     
     • Another result on prime numbers associated with Erdos is the Prime Number Theorem, namely:- .
       
     • The theorem was conjectured in the 18th century, Chebyshev himself came close to a proof, but it was not proved until 1896, when Hadamard and de la Vallee Poussin independently proved it using complex analysis.
       
     • Erdos did receive the Cole Prize of the American Mathematical Society in 1951 for his many papers on the theory of numbers, and in particular for the paper On a new method in elementary number theory which leads to an elementary proof of the prime number theorem published in the Proceedings of the National Academy of Sciences in 1949.
       
     • Prime Number Theorem .
       

106. Larmor biography
     
     • Today, however, Larmor is widely remembered by scientists for just two formulae and one theorem which, although correctly attributed to him, have been seen by historians of science as tangential to his main research interests.
       
     • Indeed, none of the recent scholarly studies of Larmor's scientific work even mention the now famous formulae and theorem.
       
     • These are the 'Larmor precession', the 'Larmor frequency', 'Larmor's theorem' and 'Larmor's formula'.
       
     • Larmor's theorem is a related result concerning how a certain transformation can negate the magnetic field for a charged particle subject to electric and magnetic fields.
       

107. Kerekjarto biography
     
     • The first was "Poincare's last geometric theorem", later proved by G D Birkhoff, and the second was "Brouwer's translation theorem".
       
     • In fact, while the first theorem played an important role in Poincare's research in dynamics, the second was applied by Brouwer in the theory of continuous groups.
       
     • The introduction states that the author essentially follows Hilbert; it lists as major deviations only that angle does not appear as primitive concept, but is defined, and that the existence of motions (based on congruence of segments) is postulated instead of the first congruence theorem for triangles.
       

108. Dandelin biography
     
     • Dandelin has an important theorem on the intersection of a cone and its inscribed sphere with a plane, discovered in 1822, named after him.
       
     • This theorem shows that if a cone is intersected by a plane in a conic, then the foci of the conic are the points where this plane is touched by the spheres inscribed in the cone.
       
     • In 1826 he generalised his theorem to a hyperboloid of revolution, rather than a cone, relating Pascal's hexagon, Brianchon's hexagon and the hexagon formed by the generators of the hyperboloid.
       
     • Other Web sitesFree University of Brussels Belgium (A colour picture illustrating Dandelin's theorem) .
       

109. Petersen biography
     
     • In 1891 Julius Petersen published a paper that contained his now famous theorem: any bridgeless cubic graph has a 1-factor.
       
     • These days Petersen's theorem is always proven indirectly using major results such as Hall's theorem from 1935 and Tutte's theorem on 1-factors from 1947.
       

110. Parseval biography
     
     • It was the second of these, dated 5 April 1799, which contains the result known today as Parseval's theorem.
       
     • Today this theorem is seen in the context of Fourier series, and often also in more abstract settings which are quite far removed from Parseval's original ideas.
       
     • The original theorem was concerned with summing infinite series.
       
     • Of course we have modernised the notation, for example subscript notation was not used in Parseval's time, and we have also corrected his theorem for he omitted the first 2 on the left hand side.
       

111. Montel biography
     
     • The idea of compactness had emerged as a fundamental concept in analysis during the nineteenth century; provided a set is bounded in Rn, it is possible to define for and sequence of points, a subsequence which converges to a point of Rn (the Bolzano-Weierstrass theorem).
       
     • But at the beginning of the twentieth century Ascoli's theorem had very few applications, and it was Montel who made it popular by showing how useful it could be for analytic functions of a complex variable.
       
     • Montel introduced a set of functions called a normal family and used these ideas to simplify classical results in function theory such as the mapping theorem of Riemann and Hadamard's characterisation of entire functions of finite order.
       

112. Lame biography
     
     • A professor at the Institute where Lame taught had written a book which contained a proof of Taylor's theorem.
       
     • which, with a = b is xn + yn = an so he was led to Fermat's last theorem.
       
     • History Topics: Fermat's last theorem .
       

113. Waring biography
     
     • His result that the product of the degrees of the original equations is the degree of the single reduced equation is known as the Generalised Theorem of Bezout.
       
     • Waring also stated, without giving a proof, what is now known as 'Waring's theorem':- .
       
     • It is reasonable to assume that Waring had this type of result in mind when he stated 'Waring's theorem'.
       

114. Kober biography
     
     • Kober was a highly productive mathematician working on special functions, functional analysis (in this area Kober's Theorem which appeared A theorem on Banach spaces (1939) is named after him), approximation theory and the theory of functions of a real variable.
       
     • In explaining a theorem he divided the proof into a large number of tiny steps and then he let the class take these steps by patient and suggestive questioning.
       

115. Gordan biography
     
     • For the next twenty years Gordan tried to prove the finite basis theorem conjecture for n-ary forms.
       
     • In 1888 Hilbert proved the finite basis theorem, only giving an existence proof, not one which allowed the basis to be constructed.
       
     • The style of Gordan's mathematics, which lead to his difficulties with Hilbert's basis theorem, is described in [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       

116. Kaluznin biography
     
     • It was here that he obtained his first research result - a generalisation of what is today a well-known theorem of Kurosh on the classification of abelian groups.
       
     • A particularly important result is the well-known theorem of Krasner and Kaluznin concerning the embeddings of a group with a subnormal series into the wreath product of the factors of the series.
       
     • This theorem is widely used in the theory of group varieties, combinatorial group theory, and permutation group theory.
       

117. Wolf Frantisek biography
     
     • The first was An extension of the Phragmen-Lindelof theorem while the second was On summable trigonometrical series: an extension of uniqueness theorems.
       
     • Chapter IV deals with trigonometric integrals summable over sets of positive measure and extends Kuttner's theorem and some similar results of Marcinkiewicz and Zygmund.
       
     • Finally in the last chapter there are some very general inversion theorems which the author asserts will yield every known inversion theorem for trigonometric integrals.
       

118. Bolza biography
     
     • Papers which appeared in the Transactions of the American Mathematical Society over the next few years were: New proof of a theorem of Osgood's in the calculus of variations (1901); Proof of the sufficiency of Jacobi's condition for a permanent sign of the second variation in the so-called isoperimetric problems (1902); Weierstrass' theorem and Kneser's theorem on transversals for the most general case of an extremum of a simple definite integral (1906); and Existence proof for a field of extremals tangent to a given curve (1907).
       

119. Goursat biography
     
     • The Cauchy-Goursat theorem states the integral of a function round a simple closed contour is zero if the function is analytic inside the contour.
       
     • Cauchy had established the theorem with the added condition that the derivative of the function was continuous.
       
     • Goursat removed this extra condition in Demonstration du theorem de Cauchy (1884).
       
     • In [History of topology (North-Holland, Amsterdam, 1999), 111-122.',4)">4] Katz notes that it was Goursat who first noted the generalized Stokes theorem.
       

120. Clifford Alfred biography
     
     • as an assistant professor, a paper famous in the semigroup community about union of groups semigroups, that Clifford learned about Rees' Theorem determining the structure of completely 0-simple semigroups, generalizing the Wedderburn theory of rings.
       
     • The impact was profound, first because [Clifford's first paper] was a special case (in fact of the Suschkiewitsch paper), second, its application to paper 'Semigroups admitting relative inverses' in hand, where Clifford proved S is a union of groups if and only if S is a semilattice of completely simple semigroups (to which Rees structure theorem applies), and finally because of its intrinsic beauty and importance.
       
     • In the mid 70's Clifford became very excited by the work of Nambooripad on the structure of regular semigroups in terms of their idempotent ordering and "sandwich matrices" and wrote several expository papers on Nambooripad structure theorem for regular semigroups.
       

121. Bayes biography
     
     • What may the reader expect to find in this Essay? As regards probability, he will expect, of course, some or other version of what has become known as 'Bayes's theorem': and such expectation will indeed be met.
       
     • Bayes Group (Bayes' theorem) .
       
     • Stanford Encyclopedia of Philosophy (Bayes' theorem) .
       

122. Jones Vaughan biography
     
     • Jones worked on the Index Theorem for von Neumann algebras, continuing work begun by Connes and others.
       
     • The central connecting link in all this mathematics was a tower of nested algebras which Jones had discovered some years earlier in the course of proving a theorem which is known as the "Index Theorem".
       

123. Prufer biography
     
     • This paper introduces the concepts of height and purity, and investigates how far the basis theorem for finite abelian groups generalises to countable p-groups.
       
     • It also contains the 'Theorem of Prufer': .
       
     • Prufer gives a decomposition theorem for 'linearly compact groups' and shows that those of finite rank are the direct product of rank 1 groups.
       

124. Nash biography
     
     • His famous theorem, that any compact real manifold is diffeomorphic to a component of a real-algebraic variety, was thought of by Nash as a possible result to fall back on if his work on game theory was not considered suitable for a doctoral thesis.
       
     • If you're so good, why don't you solve the embedding theorem for manifolds.
       
     • This paper contains his famous deep implicit function theorem.
       

125. Krull biography
     
     • In 1925 he proved the Krull-Schmidt theorem for decomposing abelian groups of operators.
       
     • In 1928 he defined the Krull dimension of a commutative Noetherian ring and brought ring theory into in new setting in which he was able to show that the principal ideal theorem held.
       
     • The principal ideal theorem [European Mathematical Society Newsletter 32 (1999), 13.',2)">2]:- .
       

126. Poncelet biography
     
     • He illustrated this technique by first noting the theorem from Euclidean geometry which states that the product of segments of intersecting chords in a circle is constant.
       
     • Poncelet then used his principle to show that if the point of intersection is considered to be outside the circle, one obtains the theorem that the product of the secants and their external segments are constant.
       
     • No proof is required, Poncelet says, for one simply uses the Euclidean theorem and invokes his principle of continuity.
       

127. Saunderson biography
     
     • For example consider his theorem:- .
       
     • An application of Pythagoras's Theorem reduces this to saying that (a2 + b2) + 2ab and (a2 + b2) - 2ab are perfect squares.
       
     • In Book 9 Saunderson presents the binomial theorem and the theory of logarithms.
       

128. Alexander biography
     
     • Soon after arriving in Princeton, Alexander generalised the Jordan curve theorem and continued his work, now exclusively on topology, with an important paper on the Jordan-Brouwer separation theorem.
       
     • This latter paper contains the Alexander Duality Theorem and Alexander's lemma on the n-sphere.
       

129. Bott biography
     
     • Solving this problem led to a joint Bott-Duffin paper containing what is now known as the Bott-Duffin theorem.
       
     • Included is the famous Bott periodicity theorem (1956) and the Morse-Bott functions, an important generalization of Morse functions which Bott introduced in the course of this work.
       
     • His style is typically the antithesis of the Definition-Theorem-Proof approach so favoured among mathematical speakers.
       

130. Hadamard biography
     
     • Perhaps his most important result proved during this time was the prime number theorem which he proved in 1896.
       
     • This theorem was conjectured in the 18th century, but it was not proved until 1896, when Hadamard and (independently) Charles de la Vallee Poussin, used complex analysis.
       
     • Prime Number Theorem .
       

131. Paramesvara biography
     
     • One of Paramesvara's most remarkable mathematical discoveries, no doubt influenced by Madhava, was a version of the mean value theorem.
       
     • He states the theorem in his commentary Lilavati Bhasya on Bhaskara II's Lilavati.
       
     • There are other examples of versions of the mean value theorem in Paramesvara's work which we now consider.
       

132. Kingman biography
     
     • The beginnings of the subject was a 1965 paper by J M Hammersley and D J A Welsh in which they made conjectured the subadditive ergodic theorem.
       
     • Kingman succeeded in proving the theorem: he published The ergodic theory of subadditive stochastic processes in the Journal of the Royal Statistical Society in 1968 and An ergodic theorem in the Bulletin of the London Mathematical Society in the following year.
       

133. Fields biography
     
     • The main purpose of the book was to present the Riemann-Roch Theorem, the Weierstrass Gap Theorem and the related Hurwitz Theorem, and theorems of Brill and Max Noether.
       

134. Cartwright biography
     
     • Her theorem, now known as Cartwright's Theorem, gave an estimate for the maximum modulus of an analytic function which takes the same value no more than p times in the unit disc.
       
     • To prove the theorem she used a new approach, applying a technique introduced by Ahlfors for conformal mappings.
       

135. Tait biography
     
     • One of the problems he considered after that was the colouring of graphs since he claimed to have a correct proof of the four colour theorem.
       
     • In this work he gave what Thomson considered the first proof of the Waterston-Maxwell equipartition theorem.
       
     • History Topics: the four colour theorem .
       

136. Hunt biography
     
     • Theorem 12, for example, contains as a special case the law of the iterated logarithm for the Wiener process.
       
     • For example A theorem of Elie Cartan (1956) in which Hunt states:- .
       
     • Andre Weil and Hopf and Samelson have given a topological proof of the following theorem of Elie Cartan.
       

137. Goldbach biography
     
     • 7 (1954), 625-629.',14)">14]) they discuss Fermat numbers, Mersenne numbers, perfect numbers, the representation of natural numbers as a sum of four squares, Waring's problem (which Euler solved before Waring), polynomials representing numerous primes, Fermat's Last Theorem, and the representation of any odd numbers in the form 2n2 + p where p is prime.
       
     • History Topics: Fermat's last theorem .
       
     • History Topics: The fundamental theorem of algebra .
       

138. Malcev biography
     
     • Among these results we mention the local theorem for the class of groups representable by matrices of a given order, and also the theorem on residual finiteness of finitely generated linear groups.
       
     • This last theorem implies, in particular, the proposition that free groups are residually finite.
       

139. Chudakov biography
     
     • In 1947 Chudakov published On Goldbach-Vinogradov's theorem in the Annals of Mathematics.
       
     • In this paper he proves Vinogradov's theorem that every large odd integer is representable as a sum of three odd primes.
       
     • It studies the zeros of L-functions, primes in arithmetic progressions and, as its high point, proves the three-prime theorem of Goldbach and Vinogradov.
       

140. Herstein biography
     
     • The first of these papers generalised the theorem of Wedderburn that shows that every finite division ring is commutative.
       
     • His first paper on this topic was A generalization of a theorem of Jacobson (1951) in which he proved the following theorem: .
       

141. Linnik biography
     
     • One of these, raised by Markov, the creator of the theory of chains, was: to find the conditions for the application of the integral limit theorem to the case of a singular chain.
       
     • The second problem concerned the conditions under which the local limit theorem for lattice type variables forming a chain holds.
       
     • Typical in this respect is the theorem on the asymptotic distribution and the ergodic behaviour of the set of lattice points on the sphere x2 + y2 + z2 = m with increasing m.
       

142. Mordell biography
     
     • During this time he discovered the result for which he is best known, namely the finite basis theorem, which proved a conjecture of Poincare.
       
     • This theorem, concerning the finite generation of the group of rational points on an elliptic curve, is beautifully surveyed in [Math.
       
     • In Mordell's paper in which his finite basis theorem appeared he conjectured that there are only finitely many rational points on any curve of genus greater than one.
       

143. Feldman biography
     
     • In addition to his work on the measure of transcendence of numbers, Feldman also produced many results strengthening Liouville's theorem on the rational approximation of algebraic numbers.
       
     • In An effective power sharpening of a theorem of Liouville in 1971 he proved the following theorem:- .
       

144. Douglas biography
     
     • Another five papers by Douglas appeared in 1940: Theorems in the inverse problem of the calculus of variations; Geometry of polygons in the complex plane; On linear polygon transformations; A converse theorem concerning the diametral locus of an algebraic curve and A new special form of the linear element of a surface.
       
     • The second and third of these papers generalise the following elementary geometrical theorem: If on each side of any triangle as base an isosceles triangle with 120° as vertex-angle is constructed (always outward or always inward), then the vertices of these isosceles triangles form an equilateral triangle.
       
     • He also presented a series of papers On the basis theorem for finite abelian groups.
       

145. Gergonne biography
     
     • in three articles in the Annales [1824-27], Gergonne [gave] the general principle that every theorem in the plane, connecting points and lines, corresponds to another theorem in which points and lines are interchanged, provided no metrical relations are involved.
       
     • Gergonne's theorem .
       

146. Hipparchus biography
     
     • He then goes on to show that the table can be computed from some basic formulae which would be known to Hipparchus, one of which is the supplementary angle theorem, essentially Pythagoras's theorem, and the half-angle theorem.
       

147. Mertens biography
     
     • Many people are aware of Mertens contributions since his elementary proof of the Dirichlet theorem appears in most modern textbooks.
       
     • He proved these results using Chebyshev's theorem, a weak version of the prime number theorem.
       

148. Stone biography
     
     • This article included the celebrated Stone-von Neumann uniqueness theorem.
       
     • One particularly important result proved by Stone during this period was a substantial generalisation of Weierstrass's theorem on uniform approximation of continuous functions by polynomials.
       
     • This result is now known as the Stone-Weierstrass theorem.
       

149. Allan Graham biography
     
     • The main object of the exposition is the construction of the holomorphic functional calculus in several variables and the application of this calculus to the Silov idempotent theorem, the local maximum modulus theorem and the Arens-Royden theorem.
       

150. Lax Peter biography
     
     • First of all, it was the experience of being part of a scientific team - not just of mathematicians, but people with different outlooks - with the aim being not a theorem, but a product.
       
     • Another important cornerstone of modern numerical analysis is the 'Lax Equivalence Theorem'.
       
     • Inspired by Richtmyer, Lax established with this theorem the conditions under which a numerical implementation gives a valid approximation to the solution of a differential equation.
       

151. Pontryagin biography
     
     • By 1927, although he was still only 19 years old, Pontryagin had begun to produce important results on the Alexander duality theorem.
       
     • lies not merely in its effect on the further development of topology; of equal significance is the fact that his theorem enabled him to construct a general theory of characters for commutative topological groups.
       
     • A fundamental theorem concerning characteristic classes of a manifold deals with special classes called the Pontryagin characteristic class of the manifold.
       

152. Gregory biography
     
     • In February 1671 he discovered Taylor's theorem (not published by Taylor until 1715), and the theorem is contained in a letter sent to Collins on 15 February 1671.
       
     • However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor expansions more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that π and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals.
       

153. Griffiths Brian biography
     
     • He published three papers based on the ideas in his thesis Local topological invariants (1953), A mapping theorem in "local" topology (1953), and A contribution to the theory of manifolds (1954).
       
     • We do not forget, however, that 'the most important existence theorem in mathematics is the existence of people'.
       
     • We give the titles of a few of his mathematical education article which give an overview of his interests in that topic: Pure mathematicians as teachers of applied mathematicians (1968); Mathematics Education today (1975); Successes and failures of mathematical curricula in the past two decades (1980); Simplification and complexity in mathematics education (1983); The implicit function theorem: technique versus understanding (1984); A critical analysis of university examinations in mathematics (1984); Cubic equations, or where did the examination question come from? (1994); The British Experience of Teaching Geometry since 1900 (1998); and The Divine Proportion, matrices and Fibonacci numbers (2008).
       

154. Wilder biography
     
     • The best known example of such a positional invariant is embodied in the Jordan curve theorem: A simple closed curve in the 2-sphere has precisely two complementary domains and is the boundary of each of them.
       
     • A converse to the Jordan curve theorem, proved by Schonflies, states that a subset of the 2-sphere is a simple closed curve if it has two complementary domains, is the boundary of each of them, and is accessible from each of these domains.
       
     • He continued to undertake research with this aim and in 1930, in A converse of the Jordan-Brouwer separation theorem in three dimensions, Wilder showed that a subset of Euclidean 3-space whose complementary domains satisfied certain homology conditions was a 2-sphere.
       

155. Urbanik biography
     
     • The main theorem asserts that the cake can be divided so that the value of each person's share is, in his own estimation, exactly p times the value of the whole cake, where p is strictly greater than 1/n, and, moreover, a best possible p can be attained.
       
     • Urbanik published a number of highly significant papers on this topic such as Representation theorem for Marczewski's algebras (1959), A representation theorem for Marczewski's algebra (1960), (with E Marczewski) Abstract algebras in which all elements are independent (1960), and Linear independence in abstract algebras (1966).
       

156. Dyson biography
     
     • there are so many proofs of the theorem that every equation has a root that it seems almost criminal to produce another.
       
     • He returned to Trinity College in 1946 as a fellow having written a dissertation from which he published three papers; A theorem on the densities of sets of integers (1945), A theorem in algebraic topology (1948), and On the product of four non-homogeneous linear forms (1948).
       

157. Haselgrove biography
     
     • The first was A Note on Fermat's Last Theorem and the Mersenne Numbers in the January/February issue of 1949 and the second was Telepathy Experiment in the October issue of 1950.
       
     • This was A connection between the zeros and the mean values of z(s) (1949) followed by Some theorems in the analytic theory of numbers (1951), On Ingham's Tauberian theorem for partitions (1952), and (with H N V Temperley) Asymptotic formulae in the theory of partitions (1954).
       
     • Among the results proved by Haselgrove in the 1951 paper is an extension of Linnik's method for proving the Goldbach-Vinogradov three prime theorem, namely that any sufficiently large odd integer is the sum of at most three primes.
       

158. Jensen biography
     
     • Jensen contributed to the Riemann Hypothesis, proving a theorem which he sent to Mittag-Leffler who published it in 1899.
       
     • The theorem is important, but does not lead to a solution of the Riemann Hypothesis as Jensen had hoped.
       
     • History Topics: Fermat's last theorem .
       

159. Sturm biography
     
     • Sturm's theorem had the good fortune of immediately becoming a classic and of finding a place in teaching that it will hold forever.
       
     • Strangely although the theorem quickly became a classic it was soon relegated to history and, contrary to what Hermite believed, vanished from textbooks.
       
     • As the title indicated, two events in the history of the algebraic theorem of Sturm are examined in [Rev.
       

160. Mazur biography
     
     • The theorem we proved - that a transformation preserving distances is linear - is now part of the standard treatment of the geometry of function spaces.
       
     • The mean ergodic theorem in Banach spaces was announced by Mazur in 1932 but a proof does not appear in print until 1938 when Yosida and by Kakutani published the result.
       
     • For example, the weak-basis theorem, due to Mazur, is given by Banach in his book but no proof appears.
       

161. Schwarz biography
     
     • In 1870 he produced work related to the Riemann mapping theorem.
       
     • Although Riemann had given a proof of the theorem that any simply connected region of the plane can be mapped conformally onto a disc, his proof involved using the Dirichlet problem.
       
     • Then, by approximating an arbitrary simply connected region by polygons he was able to give a rigorous proof of the Riemann mapping theorem.
       

162. Hensel biography
     
     • Not only is the term p-adic integer due to Hensel but also in Zahlentheorie he uses the description "Fermat's Little Theorem" for the first time:- .
       
     • There is a fundamental theorem holding in every finite group, usually called Fermat's Little Theorem because Fermat was the first to have proved a very special part of it.
       

163. Artin biography
     
     • By this stage he had proved, using very clever arguments with Galois theory and Cauchy's theorem on subgroups of prime order, that O had to be an extension of K of degree 2 and that the subfield K had to have the property that -1 could not be expressed as a sum of squares.
       
     • In my opinion, the main importance of Artin's Reciprocity Law is that it opens a new viewpoint on those classical laws, formulating it as an isomorphism theorem.
       
     • Similarly, Artin's Reciprocity Law opens the way to new applications and progress.The most striking application was given by Furtwangler's proof of the principal ideal theorem of class field theory, given one year after the publication of Artin's Reciprocity Law.
       

164. Quillen biography
     
     • The work on cohomology led to Quillen giving a structure theorem for mod p cohomology rings of finite groups, this structure theorem solving a number of open questions in the area.
       
     • He has a somewhat retiring life-style, appearing rarely in public, and then almost invariably with some extraordinary new theorem or idea in hand.
       

165. Desargues biography
     
     • Desargues' famous 'perspective theorem' - that when two triangles are in perspective the meets of corresponding sides are collinear - was first published in 1648, in a work on perspective by Abraham Bosse.
       
     • Desargues' theorem .
       
     • University of Minnesota (Desargues theorem and its relationship to one of Monge's geometry theorems) .
       

166. Bers biography
     
     • He spoke at the University so I went there and sure enough, he proved this theorem.
       
     • In 1958 Bers address the International Congress of Mathematicians in Edinburgh, Scotland, where he lectured on Spaces of Riemann surfaces and announced a new proof of the measurable Riemann mapping theorem.
       
     • In his talk Bers summarised recent work on the classical problem of moduli for compact Riemann surfaces and sketched a proof of the Teichmuller theorem characterizing extremal quasiconformal mappings.
       

167. Chebyshev biography
     
     • Chebyshev also came close to proving the Prime Number Theorem, proving that if .
       
     • Twenty years later Chebyshev published On two theorems concerning probability which gives the basis for applying the theory of probability to statistical data, generalising the central limit theorem of de Moivre and Laplace.
       
     • Prime Number Theorem .
       

168. Shoda biography
     
     • The second chapter is on the theory of free systems, including the fundamental theorem and the theorem of change of generators (of Tietze).
       
     • Structural theory of abstract ring-systems is developed, under chain conditions, including (generalized) Peirce decompositions and Wedderburn's theorem; for the latter the notion of matrices is also generalized.
       

169. Sharkovsky biography
     
     • He is perhaps best known for an important theorem on continuous functions which he proved in 1964.
       
     • Although the result did not attract a great deal of interest at the time of its publication, during the 1970s other surprising results were proved which turned out to be special cases of Sharkovsky's theorem.
       

170. Liouville biography
     
     • He proved a major theorem concerning the measure preserving property of Hamiltonian dynamics.
       
     • History Topics: Fermat's last theorem .
       

171. Kutta biography
     
     • The former contains the Runge-Kutta method for solving ordinary differential equations while the latter contains the Zhukovsky- Kutta (Joukowski -Kutta) theorem giving the lift on an aerofoil.
       
     • Let us mention, in particular, his two important publications on this topic by the Koniglich Bayerischen Akademie der Wissenschaften of Munich (Royal Bavarian Academy of Sciences) in 1910 and 1911, one of which was entitled Uber ebene Zirkulationsstromungen nebst flugtechnischen Anwendungen and contains his famous mapping theorem from function theory.
       

172. Book biography
     
     • it is certainly difficult to forget Ron's smile, Ron's gentleness from which he never departed whoever he was talking to and which hid an iron will to understand and to grow, theorem after theorem, nice scientific crops.
       

173. Thurston biography
     
     • Although this is a natural analogue of the situation for 2-manifolds, where such a result is given by Riemann's uniformisation theorem, it is much less plausible - even counter-intuitive - in the 3-dimensional situation.
       
     • Kleinian groups, which are discrete isometry groups of hyperbolic 3-space, were first studied by Poincare and a fundamental finiteness theorem was proved by Ahlfors.
       

174. Ferrers biography
     
     • I learn from Mr Ferrers that this theorem was brought under his cognizance through a Cambridge examination paper set by Mr Adams of Neptune notability.
       
     • The above proof of the theorem of reciprocity is due to Dr Ferrers, the present head of Gonville and Caius College, Cambridge.
       

175. Bochner biography
     
     • Among much else the book contains Bochner's most famous theorem, characterising the Fourier-Stieltjes transforms of positive measures as positive-definite functions ..
       
     • Bochner found that the Riemann Localisation Theorem was not valid for Fourier series of several variables (1935 - 1936), which led him indirectly to consider functions of several complex variables (1937).
       

176. Noether Emmy biography
     
     • Hilbert's basis theorem of 1888 had given an existence result for finiteness of invariants in n variables.
       
     • Emmy Noether's first piece of work when she arrived in Gottingen in 1915 is a result in theoretical physics sometimes referred to as Noether's Theorem, which proves a relationship between symmetries in physics and conservation principles.
       

177. Koebe biography
     
     • Koebe's proof of the uniformisation theorem has been described as:- .
       
     • These are not, however, a collection of great works on a par with his proof of the uniformisation theorem.
       

178. Tarski biography
     
     • The paper considers Godel's incompleteness theorem as well as Tarski's undefinability theorem and look at their consequences for the axiomatic method in mathematics.
       

179. Naimark biography
     
     • In 1943 he proved the Gelfand-Naimark theorem on self-adjoint algebras of operators in Hilbert space.
       
     • In the same year he generalised von Neumann's spectral theorem to locally compact abelian groups.
       

180. Bienayme biography
     
     • He also worked on independent binomial trials and his most important contribution was his statement of the criticality theorem for simple branching processes which he gave in 1845.
       
     • He stated, and gave a proof which leaves something to be desired, a sophisticated limit theorem which was studied again by von Mises in 1919.
       

181. Thomason biography
     
     • He produced a theorem published in 1988 said to be:- .
       
     • Much work has been done in attempts to extend Quillen's localization theorem to more general contexts, but none has even begun to approach the complete generality achieved in this paper.
       

182. Qin Jiushao biography
     
     • Chapter 1 is on indeterminate analysis; it contains remarkable work on the Chinese remainder theorem which occurs right at the beginning of the text.
       
     • As we have mentioned, the most remarkable method in the text is the method for solving simultaneous integer congruences, the Chinese Remainder Theorem.
       

183. De Morgan biography
     
     • When I send you one, you take it from me, generalise it at a glance, bestow it thus generalised upon society at large, and make me the second discoverer of a known theorem.
       
     • History Topics: The four colour theorem .
       

184. Suss biography
     
     • In 1947 Suss proved the following theorem in Kennzeichnende Eigenschaften der Kugel als Folgerung eines Brouwerschen Fixpunktsatzes.
       
     • In 1954 Suss's paper Eine characteristische Eigenschaft der Ellipse proved the following theorem: Let M be a set of ovals equivalent to each other under the group of affine transformations, and let M have the property that any two of its members meeting in more than four points coincide.
       

185. Vinogradov biography
     
     • He introduced and developed two fundamental methods, which could be briefly described as 'the bilinear form technique' and 'the mean value theorem'.
       
     • The papers included: On the distribution of power residues and nonresidues (1918); On the distribution of fractional parts of values of a function of one variable (1926); On Waring's theorem (1928); and Representation of an odd number as a sum of three primes (1937).
       

186. Coates biography
     
     • Wiles had studied for his doctorate under Coates at Cambridge from 1974 and this proved an important link in the various strands which led to Wiles' proof of Fermat's Last Theorem.
       
     • During the 1980s Coates's work was concerned with elliptic curves, Iwasawa theory and p-adic L-functions, all work closely related to the direction that would eventually yield the proof of Fermat's Last Theorem.
       

187. Heawood biography
     
     • Heawood spent 60 years of his life working on the four colour theorem.
       
     • History Topics: The four colour theorem .
       

188. Wilson John biography
     
     • He is best known among mathematicians for Wilson's theorem which states that:- .
       
     • Almost certainly Wilson's theorem was a guess made by him, based on the evidence of a number of special cases, which neither he nor Waring knew how to prove.
       

189. Waterston biography
     
     • This was a special case of what later became known as the "equipartition theorem." There is no evidence that any physical scientist read the book; perhaps it was overlooked because of its misleading title, 'Thoughts on the Mental Functions'.
       
     • The published abstract of that paper clearly states that in gas mixtures, the average kinetic energy of each kind of molecule is the same; thus he established his priority for the first statement of the equipartition theorem.
       

190. Kurosh biography
     
     • The second of these papers appeared in Mathematische Annalen and contains a proof of the celebrated Kurosh subgroup theorem, which describes subgroups of a free product of groups.
       
     • The book includes many of Kurosh's own results on groups, in particular the Kurosh Subgroup Theorem mentioned above.
       

191. Bohr Harald biography
     
     • In 1914 they proved the Bohr-Landau theorem on the distribution of zeros of the zeta function.
       
     • The fundamental theorem for almost periodic functions is a generalisation of the Parseval identity for Fourier series.
       

192. Simson biography
     
     • However the Simson line does not appear in his work but Poncelet in Proprietes Projectives says that the theorem was attributed to Simson by Servois in the Gergonne's Journal.
       
     • It appears that the theorem is due to William Wallace.
       

193. Milne-Thomson biography
     
     • Apart from rearrangements and new methods of presentation this edition differs from its predecessor in three important particulars: the introduction of the circle theorem [Milne-Thomson (1940)], whereby the disturbance of a given two-dimensional flow by the introduction of a circular cylinder can be written down without calculation; the corresponding theorem for the sphere [P Weiss (1944)]; the addition of a chapter on the flow of compressible fluids.
       

194. Lang biography
     
     • His early publications were in the area of his thesis, for example in 1952 he published three papers: On quasi algebraic closure; Hilbert's Nullstellensatz in infinite-dimensional space; and (with John Tate) On Chevalley's proof of Luroth's theorem.
       
     • Your famous theorem in Diophantine equations earned you the distinguished Cole Prize of the American Mathematical Society.
       

195. Holder biography
     
     • Holder proved the uniqueness of the factor groups in a composition series, the theorem now called the Jordan-Holder theorem.
       

196. Jones Burton biography
     
     • His first publication A theorem concerning locally peripherally separable spaces was written at this time and appeared in the Bulletin of the American Mathematical Society in 1935.
       
     • It is a good plan to encourage students to change a theorem until they can prove it; weaken the conclusion or strengthen the hypothesis or both.
       

197. Tauber biography
     
     • This all came out of his work on Abel's limit theorem which dated back to 1826.
       
     • The conditions which Tauber gave to allow him to prove the converse of Abel's limit theorem on power series are now known as 'Tauberian conditions' and appeared in Ein Satz aus der Theorie der unendlichen Reihen (1897).
       

198. Gronwall biography
     
     • Such names as Gronwall's inequality or Gronwall's lemma, Gronwall's summability method and Gronwall's theorem are known in the mathematical literature.
       
     • In 1913 Gronwall proved (Gronwall's theorem) that .
       

199. Girard Albert biography
     
     • In algebra he had some early thoughts on the fundamental theorem of algebra and translated the works of Stevin in 1625.
       
     • History Topics: The fundamental theorem of algebra .
       

200. Wall biography
     
     • The book builds up to a proof of the Alexander duality theorem in the plane; a result which generalises the Jordan curve theorem.
       

201. Al-Biruni biography
     
     • These include: theoretical and practical arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles.
       
     • Nauk (3)(89) (1983), 16-19.',10)">10] contains a letter that al-Biruni wrote to al-Sijzi (translated into English in [Beyruni\'ye Armagan (Ankara, 1974), 169-207.',63)">63]) which contains proofs of both the plane and spherical versions of the sine theorem.
       

202. Machin biography
     
     • Taylor wrote to Machin on 26 July 1712 stating what we now call Taylor's theorem.
       
     • This was the spark which led Taylor to one of two versions of his theorem which he published three years later.
       

203. Morley biography
     
     • He is perhaps best known, however, for a theorem which is now known as Morley's Theorem:- .
       

204. Vranceanu biography
     
     • In Rome Vranceanu studied under Levi-Civita, obtaining his doctorate on 5 November 1924 for a dissertation Sopra una teorema di Weierstrass e le sue applicazioni alla stabilita which gave a new proof of a theorem on the decomposition of analytical functions of more variables and also studied applications of the theorem to mechanics.
       

205. Bertini biography
     
     • The first theorem is a statement about singular points of members of a pencil of hypersurfaces in an algebraic variety.
       
     • The second theorem is about the irreducibility of a general member of a linear system of hypersurfaces.
       

206. Zeeman biography
     
     • There is an elementary discussion of the cusp and the pitchfork and a statement of the classification theorem for elementary catastrophes.
       
     • History Topics: Pythagoras's theorem in Babylonian mathematics .
       

207. Fermi biography
     
     • For him, even at this time, to know a theorem or law meant chiefly to know how to use it.
       
     • Fermi submitted his doctoral thesis Un teorema di calcolo delle probabilita ed alcune sue applicazioni (A theorem on probability and some of its applications) to the Scuola Normale Superiore and was examined on 7 July 1922.
       

208. Siegel biography
     
     • These include his improvement of Thue's theorem, described above, given in his 1920 dissertation, and its application to certain polynomial Diophantine equations in two unknowns, proving an affine curve of genus at least 1 over a number field has only a finite number of integral points in 1929.
       
     • the n-body problem and the theorem of Bruns on algebraic integrals.
       

209. Guccia biography
     
     • Guccia himself had four articles appear in the first volume of this publication, the first on Cremona transformations and a generalisation of a theorem due to Hirst, while the second was on a generalisation of a theorem due to Max Noether.
       

210. Jacobi biography
     
     • One of the prettiest results in the global theory of curves is a theorem of Jacobi (1842): The spherical image of the normal directions along a closed differentiable curve in space divides the unit sphere into regions of equal area.
       
     • The statement of this theorem is an afterthought to a paper in which Jacobi responds to the published correction by Thomas Clausen (1842) of an earlier paper by Jacobi (1836).
       

211. Archimedes biography
     
     • His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes' principle.
       
     • His most famous theorem which gives the weight of a body immersed in a liquid, called Archimedes' principle, is contained in this work.
       

212. Casey biography
     
     • While resident in Kilkenny, Casey was introduced to advanced mathematics by a former student of Trinity College, Dublin, who proposed that Casey should study a theorem of Poncelet, concerning polygons inscribed in a circle of a coaxial system.
       
     • Casey's short proof of Poncelet's theorem was communicated to Richard Townsend, Fellow of Trinity College, in 1858, and was published in the Quarterly Journal of Mathematics 5 (1862), 43-53.
       

213. Denjoy biography
     
     • In 1934 he wrote that his greatest achievements had been the integration of derivatives, the computation of the coefficients of a converging trigonometric series, a theorem on quasi-analytic functions, and differential equations on a torus.
       
     • Similarly Denjoy's theorem on quasi-analytic functions has been the foundation of studies by Mandelbrojt and has proved important in the development of large areas of current research.
       

214. Lobachevsky biography
     
     • Since Euclid's axiomatic formulation of geometry mathematicians had been trying to prove his fifth postulate as a theorem deduced from the other four axioms.
       
     • Lobachevsky did not try to prove this postulate as a theorem.
       

215. Higman biography
     
     • It is this paper which introduced many important ideas but the most significant result was a reduction theorem for the restricted Burnside problem which essentially reduced the problem to looking only at groups of prime power exponent.
       
     • As a corollary to this theorem Higman proved the existence of a universal finitely presented group containing every finitely presented group as a subgroup.
       

216. Rado biography
     
     • However unlike Rado, who had only just begun his university studies, Helly was already a research mathematician who had made remarkable progress in his work on functional analysis, proving the Hahn-Banach theorem in 1912.
       
     • He used conformal mappings of polyhedra, applying a limit theorem to certain approximations to obtain the minimal surface required.
       

217. Bernstein Sergi biography
     
     • In 1911 he introduced what are now called the Bernstein polynomials to give a constructive proof of Weierstrass's theorem (1885), namely that a continuous function on a finite subinterval of the real line can be uniformly approximated as closely as we wish by a polynomial.
       
     • He generalised Lyapunov's conditions for the central limit theorem, studied generalisations of the law of large numbers, worked on Markov processes and stochastic processes.
       

218. Theodorus biography
     
     • using Pythagoras's theorem.
       
     • ., √17 without obtaining a general theorem long before he got to 17.
       

219. Tamarkin biography
     
     • For example they published: On the summability of Fourier series (two papers), On a theorem of Hahn-Steinhaus, On a theorem of Paley and Wiener, On the theory of linear integral equations.
       

220. Al-Umawi biography
     
     • In it al-Umawi gives rules for calculating: lengths of chords and lengths of arcs of circles (using Pythagoras's theorem); areas of circles, areas of segments of circles, areas of triangles and quadrilaterals; volumes of spheres, volumes of cones and volumes of prisms.
       
     • This theorem is attributed to Pascal three hundred years after al-Umawi, and indeed al-Umawi only gives the special cases mention here.
       

221. Bombieri biography
     
     • First among Bombieri's achievements is his remarkable theorem on the distribution of primes in arithmetical progressions, which is obtained by an application of the methods of the large sieve.
       
     • Bombieri applied his improved large sieve method to prove what is now called "Bombieri's mean value theorem", which concerns the distribution of primes in arithmetic progressions.
       
     • One gets the sense that every lemma, every theorem, every remark has been carefully considered, and every proof has been thought through in every detail.
       

222. Borel biography
     
     • And it contained the explicit statement and proof of the famous covering theorem which, quite inappropriately, acquired the name of the Heine-Borel theorem ..
       

223. Brouwer biography
     
     • His first fixed point theorem, which showed that an orientation preserving continuous one-one mapping of the sphere to itself always fixes at least one point, came out of his researches on Hilbert's fifth problem.
       
     • He also introduced the idea of the degree of a mapping, generalised the Jordan curve theorem to n-dimensional space, and defined topological spaces in 1913.
       

224. Kleene biography
     
     • chapter one of this book provides the first systematic exposition of the foundations of intuitionist analysis set out as an axiomatic system treating Brouwer's fan theorem, the bar theorem, and the continuity principle (called Brouwer's principle).
       

225. Morawetz biography
     
     • with both elliptic and hyperbolic regions, to prove a striking new theorem for boundary value problems for partial differential equations.
       
     • This theorem was motivated by applications and leads to a startling practical prediction.
       

226. Urysohn biography
     
     • The letter discusses Urysohn's metrization theorem and his construction of a universal separable metric space.
       
     • Urysohn's main contributions, in addition to the theory of dimension discussed above, are the introduction and investigation of a class of normal surfaces, metrization theorems, and an important existence theorem concerning mapping an arbitrary normed space into a Hilbert space with countable basis.
       

227. Turing biography
     
     • Turing was elected a fellow of King's College, Cambridge, in 1935 for a dissertation On the Gaussian error function which proved fundamental results on probability theory, namely the central limit theorem.
       
     • Although the central limit theorem had recently been discovered, Turing was not aware of this and discovered it independently.
       

228. Durell biography
     
     • Further topics encompass the special hyperbolic functions; projection and finite series; complex numbers; de Moivre's theorem and its applications; one- and many-valued functions of a complex variable; and roots of equations.
       
     • He contributed The teaching of loci in the elementary geometry course to school certificate stage (1936), On differentials (1936), Differentials (1937), A theorem in solid geometry (1941), The transition from school to university mathematics (1948), and The nature of main-school geometry (1949).
       

229. Weierstrass biography
     
     • Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals.
       
     • History Topics: The fundamental theorem of algebra .
       

230. Lesokhin biography
     
     • Other exercises give a general theorem such as the theorem that a characteristic subgroup is normal or that a finite p-group has a centre.
       

231. Van Vleck biography
     
     • For example he published On the determination of a series of Sturm's functions by the calculation of a single determinant (1899), On linear criteria for the determination of the radius of convergence of a power series (1900), On the convergence of continued fractions with complex elements (1901), A determination of the number of real and imaginary roots of the hypergeometric series (1902), On an extension of the 1894 memoir of Stieltjes (1903), and On the extension of a theorem of Poincare for difference-equations (1912).
       
     • By following Van Vleck's own steps in deriving consequences of his zero-one law, a result ("the extended Van Vleck theorem") is given which is directly comparable to Borel's law of normal numbers.
       

232. Peterson biography
     
     • The dissertation contains a derivation of two equations equivalent to those of Mainardi and Codazzi, and in it Peterson outlined a proof of the fundamental theorem of surface theory.
       
     • The thesis also contains what has become known as Bonnet's theorem, published by Pierre Bonnet fifteen years later in 1867.
       

233. Codazzi biography
     
     • Bonnet used Codazzi's formulas to prove an existence theorem in the theory of surfaces.
       
     • 6 (2) (1979), 137-163.',3)">3] shows that, in 1853, Karl M Peterson, then a student of Minding at the University of Dorpat (now named Tartu), submitted a dissertation containing a derivation of two equations equivalent to those of Mainardi and Codazzi and outlining a proof of the fundamental theorem of surface theory.
       

234. Plucker biography
     
     • He stated the first theorem on STSs, which says that not every m has an STS(m), but only those which are of the form m = 6n+3.
       
     • So, since he missed m = 6n+1, the first theorem on STSs was wrong, or at least incomplete.
       

235. Baker biography
     
     • He wrote the important Abel's Theorem and the Allied Theory of Theta Functions in 1897 and another major contribution Multiply Periodic Functions in 1907.
       
     • Its contents are as follows: Euclid's theory of parallel lines; Propositions of incidence; The symbolic representation and Pappus' theorem; Theorems proved from the propositions of incidence; The fundamental hypothesis; The symbols of the real points of a line; Involution and harmonic ranges; Related ranges and pencils; Conics; Assignment of two absolute points, properties of circles; The parabola; The rectangular hyperbola; Theorems on conics; Length and distance; Equation of conic and line.
       

236. Steiner biography
     
     • The 'Steiner theorem' states that the two pencils by which a conic is projected from two of its points are projectively related.
       
     • Another famous result is the 'Poncelet-Steiner theorem' which shows that only one given circle and a straight edge are required for Euclidean constructions.
       

237. Plancherel biography
     
     • In his work he achieved fundamental results, one of them is the famous Plancherel theorem in harmonic analysis and which is now known in many generalizations (Plancherel measures).
       
     • In algebra Plancherel obtained results on quadratic forms and their applications, to the solvability of systems of equations with infinitely many variables and to the theory of commutative Hilbert algebras (theorem of Plancherel-Godement).
       

238. Ladyzhenskaya biography
     
     • He started teaching his daughters mathematics in the summer of 1930 beginning with giving explanations of the basic notions of geometry, then he formulated a theorem and in turn made his daughters prove it.
       
     • Find the least restrictive conditions on the behaviour of parabolic equations under which the uniqueness theorem holds for the Cauchy problem.
       

239. Sokhotsky biography
     
     • First of all, there is the famous theorem on the behaviour of an analytic function in a neighbourhood of an essential singularity.
       
     • This theorem was published by Sokhotskii (in his magister's thesis) and by Casorati in 1868, whereas Weierstrass published it eight years later - in 1876.
       

240. Campbell biography
     
     • However, he is most frequently remembered for the Campbell-Baker- Hausdorff theorem which gives a formula for multiplication of exponentials in Lie algebras.
       
     • His next contribution to the subject was a "Proof of the Third Fundamental Theorem in Lie's Theory".
       

241. Wexler-Kreindler biography
     
     • The object of this note is to include in the language of the theory of abelian categories some results known in the theory of modules over principal rings, in particular the theorem concerning the submodules of free modules over principal rings, as well as the theorem on the invariants factors of a submodule of finite type of a free module over a principal ring (non-commutative).
       

242. Jung biography
     
     • Part II is concerned with proof of the Riemann-Roch theorem.
       
     • The last and seventh part is concerned with more special problems: a new expression for the Zeuthen-Segre invariant, a detailed study of fields of the form C(x,y,(W(x, y))1/2, and the study of surfaces with a pencil of rational curves, leading to Enriques' theorem.
       

243. Kuperberg biography
     
     • In 1972, the year Krystyna register as a research student at Rice University in Houston, Texas, she published An isomorphism theorem of the Hurewicz-type in Borsuk's theory of shape in Fundamenta Mathematicae.
       
     • In 1975 she published A note on the Hurewicz isomorphism theorem in Borsuk's theory of shape which improved on the results of her paper from 1972 published in Fundamenta Mathematicae which we mentioned above.
       

244. Rogers James biography
     
     • The Rogers inequality was proved in 1888 in his paper An extension of a certain theorem in inequalities, Messenger of Math.
       
     • Then in 1934 in the well known Inequalities book of Hardy-Littlewood-Polya [Inequalities (Cambridge, 1934).',1)">1] on page 25 it was stated in an footnote that "Holder states the theorem in a less symmetrical form given a little earlier by Rogers".
       

245. Klein biography
     
     • Klein initiated a correspondence with Poincare, and soon a friendly rivalry ensued as both sought to formulate and prove a grand uniformization theorem that would serve as a capstone to this theory.
       
     • Working under great stress, Klein succeeded in formulating such a theorem and in sketching a strategy for proving it.
       

246. Von Neumann biography
     
     • In game theory von Neumann proved the minimax theorem.
       
     • An idea of Koopman on the possibilities of treating problems of classical mechanics by means of operators on a function space stimulated him to give the first mathematically rigorous proof of an ergodic theorem.
       

247. Bolyai biography
     
     • All theorems we state without explicitly specifying the system Σ or S in which the theorem is valid are meant to be absolute, that is, valid independently of whether Σ or S is true.
       
     • In his comment to Theorem 35 he remarks that the proofs of Lobachevsky concerning spherical trigonometry bear the impress of genius and his work should be esteemed as a masterly achievement.
       

248. White biography
     
     • In [A semicentennial history of the American Mathematical Society 1888-1938 (New York, 1980), 158-161.',1)">1] Archibald describes in detail a theorem proved by White in 1915:- .
       
     • This theorem is more strictly fundamental than von Staudt's ..
       

249. Mercer biography
     
     • Mercer's theorem about the uniform convergence of eigenfunction expansions for kernels of operators appears in his 1909 paper Functions of positive and negative types and their connection with the theory of integral equations published in the Philosophical Transactions.
       
     • There have been many papers written since then generalising Mercer's theorem to various other settings.
       

250. Schramm biography
     
     • The paper Existence and uniqueness of packings with specified combinatorics (1991), containing work done for his doctoral thesis, gave a highly significant generalization of the Andreev-Thurston circle packing theorem.
       
     • (The spheres must have disjoint interiors, but they don't have to be the same size.) It's a standard theorem in classical geometry, also related to important work in hyperbolic geometry and complex analysis, that you can realize any planar simple graph by kissing circles in R2, i.e., the circles are the vertices and the kissing pairs are the edges.
       

251. Halsted biography
     
     • His work on the foundations of geometry led him to publish Demonstration of Descartes's theorem and Euler's theorem in the Annals of Mathematics in 1885, the year after he arrived at Austin, and then, in the same journal, Klein's Evanston lectures in 1893.
       

252. Calugareanu biography
     
     • His investigations on Picard's fundamental theorem, on theorems of Borel and Nevanlinna in connection with the study of exceptional values of meromorphic functions of finite genus, made him already in the third and fourth decades of this century one of the first important Romanian mathematicians as well as a mathematician of European stature and distinguished member of the Romanian school of complex analysis founded by David Emmanuel and Dimitrie Pompeiu.
       
     • This elementary textbook covers all the standard topics usually covered in such an elementary text, but also has a chapter on elliptic functions with applications to cubic curves in the plane, and a final section on modular functions which contains the theorem of Picard.
       

253. Geminus biography
     
     • Geminus considers the concepts of 'hypothesis', 'theorem', 'postulate', 'axiom', 'line', 'surface', 'figure', 'angle' etc.
       
     • But Geminus proves an interesting classification theorem, namely that the helix, the circle and the straight line are the only curves with the property that any part of the curve will coincide with any other part of the same length.
       

254. Bevan-Baker biography
     
     • It is pointed out that to extend the theorem of Kirchhoff to the space outside a closed surface it is necessary to prescribe the asymptotic behaviour of the wave-functions under consideration.
       
     • The peculiarities of wave-propagation in two dimensions are next indicated and Weber's analogue of Helmholtz's theorem is given.
       

255. Frechet biography
     
     • In 1907 he discovered an integral representation theorem for functionals on the space of quadratic Lebesgue integrable functions.
       
     • The topics discussed are: the Jordan curve theorem, the map colouring problem, the Euler characteristic and the classification of surfaces.
       

256. Konig Julius biography
     
     • Konig's proof contains an error in that he applied a theorem due to Felix Bernstein in a case where it does not hold.
       
     • It was a little while before Zermelo found the error in the proof and then in 1905 Felix Bernstein published a short note correcting his theorem.
       

257. Taniyama biography
     
     • This conjecture proved to be a major factor in the proof of 'Fermat's Last Theorem' by Andrew Wiles.
       
     • In this paper, the author not only gives a new proof, free from artificial restrictions, of his earlier theorem on the zeta-function of Abelian varieties with sufficiently many complex multiplications, but develops, in relation with that problem, a number of new ideas of far-reaching importance.
       
     • History Topics: Fermat's last theorem .
       

258. Szego biography
     
     • From these beginnings he moved to prove a number of limit theorems, now known as the Szego limit theorem, the strong Szego limit theorem and Szego's orthogonal polynomials and on the unit circle.
       

259. Montgomery biography
     
     • These include: Periodic one-parameter groups in three-space (1936); Translation Groups of Three-Space (1937); Compact Abelian transformation groups (1938); Non-Abelian Compact Connected Transformation Groups of Three-Space (1939); A theorem on the rotation group of the two-sphere (1940); Topological group foundations of rigid space geometry (1940); Topological transformation groups.
       
     • I (1940); and A theorem on Lie groups (1942).
       

260. Spencer Tony biography
     
     • By means of the Cayley-Hamilton theorem it is shown that any product of symmetric 3 × 3 matrices can be expressed as a linear combination of matrix products of limited total degree, and of certain explicitly determined types, with coefficients expressible in terms of the traces of matrix products.
       
     • The basis for the orthogonal invariants of a larger number of matrices is then deduced by using Peano's theorem.
       

261. Christoffel biography
     
     • This is now called Christoffel's theorem.
       
     • The Christoffel reduction theorem, so named by Klein, solves the local equivalence problem for two quadratic differential forms.
       

262. Burkill biography
     
     • The book covers: sets and functions, metric spaces, continuous functions on metric spaces, real and complex limits and series, uniform convergence, Riemann-Stieltjes integration, multivariable differential and integral calculus, Fourier series, Cauchy's theorem, Laurent expansions, residue calculus, infinite products, the factor theorem of Weierstrass, asymptotic expansions, and applications to special functions in particular the gamma function.
       

263. Dickson biography
     
     • He proved many interesting results in number theory, using results of Vinogradov to deduce the ideal Waring theorem in his investigations of additive number theory.
       
     • One of these contains generalizations of the classical theorem on representing a natural number as the sum of three squares.
       

264. Al-Tusi Nasir biography
     
     • In his model Nasir, for the first time in the history of astronomy, employed a theorem invented by himself which, 250 years later, occurred again in Copernicus, "De Revolutionibus", III 4.
       
     • The theorem referred to in this quotation concerns the famous "Tusi-couple" which resolves linear motion into the sum of two circular motions.
       

265. Hopf Eberhard biography
     
     • In particular he studied the theory of measure and invariant integrals in ergodic theory and his paper On time average theorem in dynamics which appeared in the Proceedings of the National Academy of Sciences is considered by many as the first readable paper in modern ergodic theory.
       
     • This paper is concerned with some extensions of Jentzsch's theorem on the existence of a positive eigenfunction for a positive integral operator.
       

266. Mackey biography
     
     • His interest in physics continued, however, and he published A theorem of Stone and von Neumann (1949) in which he generalised a theorem about quantum mechanics proved by Stone and von Neumann in 1930.
       

267. Rado Richard biography
     
     • These important combinatorial results were in the area of Hall's theorem, Ramsey's theorem, the Rado selection principle, matroids and theory of transversals, and partitions.
       

268. Shewhart biography
     
     • Among the topics considered are measurements presented as original data, characteristics of original data, summarizing original data (both by symmetric functions and by Chebyshev's theorem), measurement presented as meaningful predictions, and measurement presented as knowledge.
       
     • Every mathematical theorem involving this mathematically undefined concept can then be given the following predictive form: If you do so and so, then such and such will happen.
       

269. Hindenburg biography
     
     • His ideas centred around the so-called polynomial theorem which was a generalisation of the binomial theorem.
       

270. Begle biography
     
     • He continued to produce papers of significance such as A note on local connectivity (1948), Topological groups and generalized manifold (1948), A note on S-spaces (1949), A fixed point theorem (1950), and The Vietoris mapping theorem for bicompact spaces (1950).
       

271. Specker biography
     
     • Geneve, Geneva, 1982), 11-24.',5)">5] where his 32 publications up to 1979 are divided into 10 categories: topology, recursive analysis, combinatorial set theory, type theory, axiomatic set theory, Ramsey's theorem, arithmetic, logic of quantum mechanics, algorithms, and miscellaneous.
       
     • Examples of his later papers are The fundamental theorem of algebra in recursive analysis (1969), Die Entwicklung der axiomatischen Mengenlehre (1978), (with H Kull) Direct construction of mutually orthogonal Latin squares (1987), Application of logic and combinatorics to enumeration problems (1988).
       

272. Markov biography
     
     • Especially remarkable is his research relating to the theorem of Jacob Bernoulli known as the Law of Large Numbers, to two fundamental theorems of probability theory due to Chebyshev, and to the method of least squares.
       
     • He proved the central limit theorem under fairly general assumptions.
       

273. Smullyan biography
     
     • Beginning with fun-filled monkey tricks and classic brain-teasers with devilish new twists, Professor Smullyan spins a logical labyrinth of even more complex and challenging problems as he delves into some of the deepest paradoxes of logic and set theory, including Godel's revolutionary theorem of undecidability.
       
     • This book deals primarily with the proofs of, and the interconnections between, various formulations of the completeness theorem for first-order logic.
       

274. Ledermann biography
     
     • The little book (152 pages) discusses the group axioms, isomorphisms, cyclic groups, coset decompositions, Lagrange's theorem, permutation groups, normal subgroups, quotient groups, homomorphisms, the first and second isomorphism theorems, and the Jordan-Holder theorem.
       

275. Patodi biography
     
     • An analytic approach, via the heat equation yields easily a formula for the index of an elliptic operator on a compact manifold: but, the formula involves an integrand containing too many derivatives of the symbol, while from the Atiyah-Singer index theorem one would expect only two derivatives to figure.
       
     • The second paper which came from his thesis was An analytic proof of the Riemann- Roch- Hirzebruch theorem for Kaehler manifolds which extended the methods of his first paper to a much more complicated situation.
       

276. Iwasawa biography
     
     • A proof of the Riemann-Roch theorem is given, and the theory of Riemann surfaces and their topology is studied.
       
     • today it is no exaggeration to say that Iwasawa's ideas have played a pivotal role in many of the finest achievements of modern arithmetical algebraic geometry on such questions as the conjecture of B Birch and H Swinnerton-Dyer on elliptic curve; the conjecture of B Birch, J Tate, and S Lichtenbaum on the orders of the K-groups of the rings of integers of number fields; and the work of A Wiles on the modularity of elliptic curves and Fermat's Last Theorem.
       

277. Zygmund biography
     
     • One of the novel features is the early introduction of Runge's theorem and its application to facilitate the proof of Cauchy's theorem and other results.
       

278. Moser Leo biography
     
     • These include On the sum of digits of powers (1947), Some equations involving Euler's totient (1949), Linked rods and continued fractions (1949), On the danger of induction (1949) and A theorem on the distribution of primes (1949).
       
     • He also indicates how the proof can be modified to give a proof of Bertrand's theorem.
       

279. Joachimsthal biography
     
     • He has a theorem named after him which concerns the intersection of surfaces.
       
     • He is also remembered for another theorem on the four normals to an ellipse from a point inside it.
       

280. Varignon biography
     
     • The book contains what is today known as Varignon's parallelogram theorem: The figure formed when the mid-points of the sides of a quadrilateral are joined in order is a parallelogram.
       
     • He gives a completely rigorous proof of this theorem, being the first to do so.
       

281. Kolchin biography
     
     • Well before he had completed his research he was writing papers such as On certain ideals of differential polynomials (1939), written jointly with Ritt, and On the basis theorem for infinite systems of differential polynomials (1939).
       
     • Despite serving his country during the war, Kolchin was still able to publish papers such as On the basis theorem for differential systems (1942) and begin his fundamental work on the Galois theory of differential fields in the three part paper Extensions of differential fields.
       

282. Pic biography
     
     • Among the papers he published in Romanian, we mention the following (where we give an English translation) On the structure of quasi-Hamiltonian groups (1949), On a new generalisation of nilpotency of a group (1954), On a theorem of B H Neumann (1960), and On a theorem of L Fejer (1962).
       

283. Dye biography
     
     • Dye's first paper was The Radon-Nikodym theorem for finite rings of operators which was published in the Transactions of the American Mathematical Society in 1952.
       
     • One of the main results of the paper is a non-commutative analogue of the Radon-Nikodym theorem which [Pacific J.
       

284. Skopin biography
     
     • In particular, he found and proved independently a result that is now known as Kawada's theorem [m (Russian), Izv.
       
     • Nauk SSSR (N.S.) 95 (1954), 29-32.',2)">2] was essentially used by I R Shafarevich in the proof of his famous theorem.
       

285. Kuttner biography
     
     • at the age of only 26, Kuttner proved a basic theorem in the general theory of trigonometric series, a result delightful for both the deceptive simplicity of its statement and the elegance of its proof.
       
     • Zygmund greatly admired this theorem of Kuttner, which now occupies an honoured place in Zygmund's monumental work on trigonometric series.
       

286. Roth biography
     
     • The typical Schedule A question was a triple-decker: first the candidate would be asked to prove a theorem; then would come a problem based more or less on the theorem; and thirdly, another problem even less based than the first.
       

287. Carleson biography
     
     • In 1967 Hormander introduced some ideas to simplify Carleson's proof and Carleson lectured on The corona theorem to the Fifteenth Scandinavian Congress in Oslo in 1968.
       
     • He explained in [Mathematics unlimited - 2001 and beyond (Springer, Berlin, 2001), 455-461.',4)">4] how he was led to prove the theorem:- .
       

288. Thabit biography
     
     • After giving nine lemmas Thabit states and proves his theorem: for n > 1, let pn = 3.2n -1 and qn = 9.22n-1 -1.
       
     • Thabit generalised Pythagoras's theorem to an arbitrary triangle (as did Pappus).
       

289. Dinghas biography
     
     • Examples are the formula of Plana-Abel-Cauchy, the theorem of Julia-Wolff-Caratheodory, and the theory of Nevanlinna and of Hallstrom.
       
     • Some topics treated which are not always found in the older short elementary texts are the homotopy concept for closed curves, cluster sets of meromorphic functions, removable compact sets of singularities, the monodromy theorem, and the Mittag-Leffler partial fraction expansion of a meromorphic function.
       

290. Riesz Marcel biography
     
     • In it he gave the correct generalisation of Cantor's uniqueness theorem for convergent trigonometric series to trigonometric series summable by the Cesaro method.
       
     • Another highlight from this period is his beautiful proof of Fatou's theorem which give conditions under which the power series of an analytic function converges to a point on its circle of convergence.
       

291. Penrose biography
     
     • I learnt about Turing machines and Godel's theorem ..
       
     • In 1965, using topological methods, Penrose proved an important theorem which, under conditions which he called the existence of a trapped surface, proved that a singularity must occur in a gravitational collapse.
       

292. Serre biography
     
     • Serre's theorem led to rapid progress not only in homotopy theory but in algebraic topology and homological algebra in general.
       
     • Among these texts, which show the topics Serre has worked on, are Homologie singuliere des espaces fibres (1951), Faisceaux algebriques coherents (1955), Groupes d'algebriques et corps de classes (1959), Corps locaux (1962), Cohomologie galoisienne (1964), Abelian l-adic representations (1968), Cours d'arithmetique (1970), Representations lineaires des groupes finis (1971), Arbres, amalgames, SL2 (1977), Lectures on the Mordell-Weil theorem (1989) and Topics in Galois theory (1992).
       

293. Dilworth biography
     
     • The main topics in lattice theory to which Dilworth contributed are: Chain partitions in ordered sets, in particular his chain decomposition theorem for partially ordered sets; Uniquely complemented lattices; Lattices with unique irreducible decompositions; Modular and distributive lattices, in particular his covering theorem for modular lattices; Geometric and semimodular lattices; and Multiplicative lattices, where he studied, among other topics, abstract ideal theory, and the representation and embedding theorems for Noether lattices and r-lattices.
       

294. Schottky biography
     
     • Schottky's Theorem (1904) is related to Picard's Theorem.
       

295. Diophantus biography
     
     • Certainly Fermat was inspired by this work which has become famous in recent years due to its connection with Fermat's Last Theorem.
       
     • History Topics: Fermat's last theorem .
       

296. Parry biography
     
     • In 1963 he published An ergodic theorem of information theory without invariant measure generalising the individual version of McMillan's ergodic theorem of information theory without the hypothesis of an invariant probability function.
       

297. Knapowski biography
     
     • Among Knapowski's other number theory papers we mention: On prime numbers in arithmetical progression (1958), On the Mobius function (1958), Contributions to the theory of the distribution of prime numbers in arithmetical progressions (1961, 1962), On Linnik's theorem concerning exceptional L-zeros (1961), and Further developments in the comparative prime number theory (8 papers).
       
     • He also wrote on other mathematical topics such as On some criteria for indecomposability of polynomials (1955) and A theorem from finite group theory (1956).
       

298. Al-Karaji biography
     
     • One of the results on which al-Karaji uses this form of induction comes from his work on the binomial theorem, the binomial coefficients and the Pascal triangle.
       
     • This is a beautiful description of the binomial theorem using the Pascal triangle.
       

299. Adleman biography
     
     • Martin Gardner had written an article on Godel's theorem in Scientific American which overwhelmed Adleman with its deep philosophical implications: .
       
     • Adleman decided to join graduate school and come away with an understanding of Godel's theorem at a level beyond the superficial.
       

300. Gentzen biography
     
     • Of course Godel published his incompleteness theorem just at the time Gentzen was beginning his work.
       
     • By Godel's unprovability theorem, such a proof as Gentzen gave had to make use of tools stronger than those of S; extending ordinary mathematical induction, Gentzen employed transfinite induction up to Cantor's first epsilon number, and he also showed that this was the minimum required for such proof.
       

301. Kelly biography
     
     • Kelly showed that this theorem went over to entire subsets of Minkowski n-space.
       
     • Examples of papers by Kelly on graph theory are A congruence theorem for trees published in the Pacific Journal in 1957, On some mappings related to graphs published in the same journal in 1964, and The minimal regular graph containing a given graph co-authored with Paul Erdos and published in 1967.
       

302. Mahler biography
     
     • He proved important results about polar convex bodies, compound convex bodies and the very useful Mahler Compactness Theorem.
       
     • For example Lectures on diophantine approximations : g-adic numbers and Roth's theorem (1961) was prepared from notes by R P Bambah of lectures given by Mahler at the University of Notre Dame in autumn 1957 and was described as an "extremely valuable contribution".
       

303. Thom biography
     
     • His work on topology, in particular on characteristic classes, cobordism theory and the Thom transversality theorem led to his being awarded a Fields medal in 1958.
       
     • When it comes down to it, this extension resulted from B Malgrange's extension of the preparation theorem.
       

304. Hamilton biography
     
     • History Topics: The four colour theorem .
       
     • History Topics: The fundamental theorem of algebra .
       

305. Milne William biography
     
     • These included: The geometrical meaning of the triad of points (1910); A property of the complete quadrangle (1911); The teaching of limits and convergence to scholarship candidates (1911); The teaching of limits and convergence to scholarship candidates (1912); The teaching of limits and convergence to scholarship candidates (1913); Another proof and generalisation of the theorem given in note 339 (1913); The teaching of modern analysis in secondary schools (1915); The graphical treatment of power series (1918); The uses and functions of a school mathematical library (1918); Mathematics and the pivotal industries (1919); The training of the mathematical teacher (1920); and Noether's canonical curves (1920).
       
     • Further papers read by Milne to the Society include: The Focal Circles of Circular Cubics on 10 February 1911; The system of cubic curves circumscribing two triangles and a polar to them (communicated by Neil McArthur to the meeting of 10 November 1911); An easy geometrical representation of the Sextic Covariant of a Binary Quartic (communicated by Neil McArthur to the meeting of 10 November 1911); Investigations on Circular Cubics and Bi-circular Quartics on 10 May 1912; Nonagons nonuply in perspective (communicated by N McArthur on 9 May 1913); Easy Proof of von Staudt's Theorem (communicated by P Comrie on 15 January 1915); The apolar locus of two tetrads of points (communicated by P Ramsay on 12 January 1917); and The co-apolars of a cubic curve (communicated by Archibald Milne on 9 February 1917).
       

306. Pappus biography
     
     • It was discussed by Descartes and Newton and what is now known as Guldin's theorem is was proved by Pappus in Book VII of the Mathematical Collection.
       
     • as Archimedes showed, and as is proved by us in the commentary on the first book of the ["Almagest"] by a theorem of our own.
       

307. Cramer Harald biography
     
     • We should give two specific results which we have not mentioned previously which will be remembered as major contributions, namely his work on the central limit theorem and his beautiful theorem that if the sum of two independent random variables is normal then all are normal.
       

308. Seidenberg biography
     
     • He continued to publish papers such as On the Lasker-Noether decomposition theorem (1984) which asks:- .
       
     • When does the Lasker-Noether decomposition theorem, which says that an ideal in a commutative Noetherian ring is the intersection of a finite number of primary ideals, hold in a constructive sense? .
       

309. Fejer biography
     
     • In 1900 Fejer published a fundamental summation theorem for Fourier series.
       
     • He seemed to relive the birth of the theorem; we were present at the creation.
       

310. Farey biography
     
     • Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before ..
       
     • The article [The Mathematical Intelligencer 17 (2) (1995), 64-67.',3)">3] contains other interesting information on Farey's sequence, its relation to Pick's area theorem, and the inaccurate historical comments made about the sequence over many years.
       

311. Koszul biography
     
     • The main topics on which Koszul undertook research included: homology and cohomology of Lie algebras; relative cohomology; reductive subalgebras and the transgression theorem; the formalism of spectral sequences; "Koszul complexes"; proper and differentiable actions of Lie groups; slices; hermitian forms on complex homogeneous domains; bounded domains; locally flat manifolds; convex homogeneous domains; simplicial spaces; themes related to Gelfand-Fuks theory and supergeometry.
       
     • The superb lecture notes were published in 1957 and covered: Cech cohomology with coefficients in a sheaf; resolutions; a theorem concerning the cohomology with coefficients in a sheaf for a paracompact space; isomorphism of ordinary Cech cohomology with de Rham-cohomology, Alexander-Spanier- cohomology, and singular cohomology.
       

312. Huhn biography
     
     • First he found a new proof for my theorem that the ideal lattice of a distributive lattice is the congruence lattice of a lattice.
       
     • for this proof he gave a very interesting representation theorem for finite distributive lattices.
       

313. Plemelj biography
     
     • He was the first to discover the sharp formulation of Koebe's distortion theorem.
       
     • Another contribution that we should mention was Plemelj's simple proof of the n = 5 case of Fermat's Last Theorem which he published in 1912.
       

314. Gershgorin biography
     
     • His 1931 paper Uber die Abgrenzung der Eigenwerte einer Matrix ('About the limits of Eigenvalues in a Matrix', his only one not in Russian) gave powerful estimates for matrix eigenvalues, known as his Circle Theorem.
       
     • Other Web sitesMathWorld (Gerschgorin Circle Theorem) .
       

315. Eckmann biography
     
     • Peter Hilton, who had been a personal friend of Eckmann's for many years spoke in detail of Eckmann's research in topology: continuous solutions of systems of linear equations, a group-theoretical proof of the Hurwitz-Radon theorem, complexes with operators, spaces with means, simple homotopy type.
       

316. Bunyakovsky biography
     
     • One would have to note, however, that the terminology of mathematics is not universal and in some countries his theorem is correctly named, or named after Cauchy, Bunyakovskii and Schwarz.
       

317. Behnke biography
     
     • I was invited because I had published a note in the Comptes Rendus de l'Academie des Sciences about circled domains, where I had proved quite easily a theorem which had been proved earlier by Behnke, but under certain conditions, in a particular case.
       

318. Shnirelman biography
     
     • For the proof of this theorem the authors used a method, which they broadly generalised, that had been devised by G D Birkhoff, who in 1919 showed the existence of one closed geodesic.
       

319. Leibniz biography
     
     • History Topics: The fundamental theorem of algebra .
       

320. Castigliano biography
     
     • In his dissertation there appears a theorem which is now named after Castigliano.
       

321. James Ralph biography
     
     • James was awarded a doctorate from Chicago in 1932 for his thesis Analytical Investigations in Waring's Theorem on number theory.
       

322. Weil biography
     
     • History Topics: Fermat's last theorem .
       

323. Lyapunov biography
     
     • In particular in two papers published in 1900 and 1901, he proved the central limit theorem using a technique based on characteristic functions.
       

324. Huntington biography
     
     • This is now known as Huntington's theorem.
       

325. Picken biography
     
     • He read papers to the Society such as A Proof of the Addition Theorem in Trigonometry to the meeting on Friday 9 December 1904, On a Direct Method of Obtaining the Foci and Directrices from the General Equation of the Second Degree to the meeting on 9 June 1905, On Simson Line and Related Theory: and An Exercise in Geometric Generality (communicated by A W Young) on 8 May 1914.
       

326. Clifford biography
     
     • A student, having problems with Ivory's theorem on the attraction of an ellipsoid, describes Clifford's response to his questions:- .
       

327. Schlafli biography
     
     • He then developed the fundamental theorem on class and degree of an algebraic manifold, theorems that attracted the interest of the Italian school of geometers.
       

328. Sylow biography
     
     • After proving Cauchy's theorem that a group of order divisible by a prime p has a subgroup of order p, Sylow asks whether it can be generalised to powers of p.
       

329. Lipschitz biography
     
     • Peano gave an existence theorem for this differential equation, giving conditions which guarantee at least one solution.
       

330. Egorov biography
     
     • Egorov also worked on integral equations and a theorem in the theory of functions of a real variable is named after him.
       

331. Rosanes biography
     
     • In 1870 he provided a demonstration that each plane Cremona transformation can be factored as a product of quadratic transformations, a theorem that Max Noether also proved independently at about the same time.
       

332. Chrystal biography
     
     • History Topics: The fundamental theorem of algebra .
       

333. Farkas biography
     
     • In fact he is remembered for Farkas theorem which is used in linear programming and also for his work on linear inequalities.
       

334. Bombelli biography
     
     • History Topics: The fundamental theorem of algebra .
       

335. Fibonacci biography
     
     • There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series.
       

336. Salmon biography
     

337. Lyndon biography
     
     • In the following year the paper New proof for a theorem of Eilenberg and Mac Lane appeared, then, in 1950, the paper The representation of relational algebras which resulted from his early interest in that topic.
       

338. Amitsur biography
     
     • starts with the famous Amitsur-Levitzki theorem.
       

339. Friedrichs biography
     
     • Open problems and unresolved difficulties are carefully noted, and the reader is never left in doubt as to whether he is presented with a mathematical theorem or with a conjecture based on physical experience.
       

340. Simon biography
     
     • After a preliminary section, the book covers the main partial regularity results, due to Schoen and Uhlenbeck, then a fairly comprehensive discussion of minimizing tangent maps including a somewhat simplified proof of the author's celebrated theorem on the uniqueness of tangent maps, and finally an account of recent results on the rectifiability of the singular set.
       

341. Ostrowski biography
     
     • His work on aglebraic equations involved a study of the fundamental theorem of algebra, Galois theory, and estimating the roots of algebraic equations.
       

342. Li Rui biography
     
     • This was a work by Qin Jiushao which presents a general algorithm for the solving simultaneous congruences, that is the Chinese remainder theorem.
       

343. Koch biography
     
     • Von Koch also wrote papers on number theory, in particular he wrote several papers on the prime number theorem such as Sur la distribution des nombres premiers in 1901 and Contribution a la theorie des nombres premiers in 1910.
       

344. Bachet biography
     
     • He is most famous for his Latin translation of Diophantus's Greek text Arithmetica (1621) in which Fermat wrote his famous 'Last Theorem' marginal note.
       

345. Chernikov biography
     
     • By 1938 he already had two papers published on generalisations of results from finite group theory to infinite group theory, in particular generalising Frobenius's theorem to infinite groups.
       

346. Chung biography
     
     • Then she flipped the book open to a key theorem and said, gently, "I think I can do a little better with the proof." My eyes were bulging.
       

347. Gluskin biography
     
     • I must confess it was vexing, for twice during these years I was standing with one foot in the next world, and I would rather not leave behind a published theorem which was not true.
       

348. Pascal biography
     
     • Although Pascal was not the first to study the Pascal triangle, his work on the topic in Treatise on the Arithmetical Triangle was the most important on this topic and, through the work of Wallis, Pascal's work on the binomial coefficients was to lead Newton to his discovery of the general binomial theorem for fractional and negative powers.
       

349. Al-Samawal biography
     
     • Also in this book is al-Samawal's description of the binomial theorem where the coefficients are given by the Pascal triangle.
       

350. Pade biography
     
     • This led him to work on the connection bewteen Sylvester's formulas on the polynomials arising in the application of Sturm's theorem and the theory of continued fractions.
       

351. Budan de Boislaurent biography
     
     • Sturm's theorem had the good fortune of immediately becoming a classic and of finding a place in teaching that it will hold forever.
       

352. Allardice biography
     
     • For example at the meeting held on Friday 14 March 1884 he read a paper on the geometry of the spherical surface; at the meeting on Friday 8 January 1886 he discussed a problem of symmetry in an algebraical function; on 11 February 1887 he communicated a note on a theorem in algebra; on 11 January 1889 he contributed a note on a formula in quaternions; on 13 December 1889 he discussed some theorems in the theory of numbers; on 13 November 1891 his paper Barycentric Calculus of Mobius was read by John Alison; on 14 December 1901 his paper Four Circles Touching a Common Circle was communicated to the meeting by Mr George Duthie; and on 13 January 1911 his paper On the envelope of the directrices of a system of similar conics through three points was communicated by E D Williamson.
       

353. Foster biography
     
     • In 1944, together with his colleague Benjamin Bernstein, he published Symmetric approach to commutative rings, with duality theorem : Boolean duality as special case in the Duke Mathematical Journal.
       

354. Janiszewski biography
     
     • He gave a topological characterisation of the plane which simplified considerably the Jordan curve theorem.
       

355. Sun Zi biography
     
     • This, of course, is important for it is a problem which is solved using the Chinese remainder theorem.
       

356. Minding biography
     
     • Minding also studied the bending of surfaces proving what is today called Minding's theorem in 1839.
       

357. Calderon biography
     
     • for his ground-breaking work on singular integral operators leading to their application to important problems in partial differential equations, including his proof of uniqueness in the Cauchy problem, the Atiyah-Singer index theorem, and the propagation of singularities in nonlinear equations..
       

358. Bugaev biography
     
     • The theorem relating convergence almost everywhere and uniform convergence by D F Egorov, one of Bugaev's pupils, in 1911 is seen as marking the beginning of the Moscow school of the theory of functions of a real variable.
       

359. Bernoulli Nicolaus(II) biography
     
     • History Topics: The fundamental theorem of algebra .
       

360. Mostowski biography
     
     • The theory of models, especially models for set theory and for arithmetic; model products and their theories; families of models as topological spaces; models with indiscernible elements and the Ehrenfeucht-Mostowski theorem.
       

361. Nelson biography
     
     • She gave a talk Recent results on continuous ordered algebras in which, among other things, she described all free continuous algebras, as well as free continuous semilattices, and gave the Birkhoff theorem for these algebras.
       

362. Warner biography
     
     • She published her first research paper A note on Borsuk's antipodal point theorem in the Oxford Quarterly Journal of Mathematics in 1956.
       

363. Mendelsohn biography
     
     • He wrote papers on a wide variety of combinatorial problems, for example: Symbolic solution of card matching problems (1946), Applications of combinatorial formulae to generalizations of Wilson's theorem (1949), Representations of positive real numbers by infinite sequences of integers (1952), A problem in combinatorial analysis (1953), The asymptotic series for a certain class of permutation problems (1956), and Some elementary properties of ill conditioned matrices and linear equations (1956).
       

364. Ore biography
     
     • He then worked on non-commutative ring theory and proved his celebrated embedding theorem for a non-commutative integral domain into a division ring.
       

365. Sommerville biography
     
     • the classification of all types on non-euclidean geometry (including those usually excluded as bizarre), the extension, involving the measurement of generalised angles in higher space, of Euler's Theorem on polyhedra, space filling figures, the classification of polytopes (i.e.
       

366. Peirce Charles biography
     
     • History Topics: the four colour theorem .
       

367. Cayley biography
     
     • History Topics: The four colour theorem .
       

368. Macdonald William biography
     
     • Examples of papers Macdonald read to the Society are: An account of Pascal's "Essais pour les Coniques" (Friday 14 March 1884); A proof of a geometrical theorem (Friday 11 February 1887); and A Suggestion for Improvement of Mathematical Tables (Friday 8 March 1895).
       

369. Cavalieri biography
     
     • D Wagner (Cavalieri's theorem) .
       

370. Oresme biography
     
     • Oresme was the first to prove Merton's theorem, namely that the distance travelled in a fixed time by a body moving under uniform acceleration is the same as if the body moved at a uniform speed equal to its speed at the midpoint of the time period.
       

371. Feit biography
     
     • Although he published around 100 other papers, his name will always be most closely associated with this one result, described by Zelmanov as "easily the best single theorem in group theory." However, his other contributions on finite group theory, character theory, and modular representation theory, are also impressive.
       

372. Cramer biography
     
     • He states a theorem by Maclaurin which says that an equation of degree n intersects an equation of degree m in nm points.
       

373. Watson Henry biography
     
     • This paper was written after a correspondence between the authors in 1873 and the paper contains a version of the 'Criticality Theorem' which is the foundation of the modern theory of branching processes.
       

374. Maclaurin biography
     
     • Grabiner gives five areas of influence of Maclaurin's treatise: his treatment of the fundamental theorem of the calculus; his work on maxima and minima; the attraction of ellipsoids; elliptic integrals; and the Euler-Maclaurin summation formula.
       

375. Steinfeld biography
     
     • Two new sets of conditions are obtained for unique prime factorisation in a partially ordered semigroup (not necessarily commutative), generalising the fundamental theorem or commutative ideal theory.
       

376. Cotes biography
     
     • Cotes discovered an important theorem on the nth roots of unity, gave the continued fraction expansion of e, invented radian measure of angles, anticipated the method of least squares, published graphs of tangents and secants, and discovered a method of integrating rational fractions with binomial denominators.
       

377. Mineur biography
     
     • He was awarded his doctorate in 1924 for a thesis on functional equations in which he established an addition theorem for Fuchsian functions.
       

378. Cantelli biography
     
     • He continued to develop these ideas in Sulla probabilita comme limite di frequenza (On probability seen as a limit of frequencies) and Su due applicazioni di un teorema di G Boole alla statistica matematica (On two applications of a Boole's theorem to mathematical statistics) both published in the following year.
       

379. Macaulay biography
     
     • The papers look at algebraic curves, the Riemann-Roch theorem and algebraic polynomials.
       

380. Libri biography
     
     • To neglect the path by which human nature ought to have passed to arrive at such and such a discovery - for example, not to stop at a mathematical theorem, until at the hands of a Lagrange or a Gauss it has received a definitive form - would be to act as a naturalist who attempted only to study insects under the shape of beautiful butterflies, without giving the slightest attention to the caterpillars, to those less perfect larvae which at a later period are to be transformed into those self-same lepidoptera ..
       

381. Szasz biography
     
     • In fact Szasz worked on problems associated with both Riesz brothers, and he gave a very simple proof a theorem by Marcel Riesz on rational functions with given bounds on the unit circle.
       

382. Saurin biography
     
     • In reply [Reponse a ecrit de M Rolle de l'Academie Royale des Sciences insere dans le Journal du 13 Avril 1702, sous le titre de Regles et Remarques pour le Probleme general des Tangentes par M Saurin (1702), Remarques sur les courbes des deux premiers exemples proposes par M Rolle dans le Journal du jeudi 13 Avril 1702 (1703), and Remarques sur un cas singulier du probleme general des tangentes (1716)], Saurin explicated the nature and treatment of such indeterminate expressions on the basis of de L'Hopital's theorem ..
       

383. Zhukovsky biography
     
     • Today it is known as the Kutta-Joukowski theorem, since Kutta pointed out that the equation also appears in his 1902 dissertation.
       

384. Riemann biography
     
     • Beings living on the surface may discover the curvature of their world and compute it at any point as a consequence of observed deviations from Pythagoras' theorem.
       

385. Rennie biography
     
     • In On dominated convergence he proves a converse of Lebesgue's theorem of dominated convergence and gives an application to Fourier series.
       

386. Al-Khazin biography
     
     • Al-Khujandi claimed to have proved that x3 + y3 = z3 is impossible for whole numbers x, y, z which of course is the n = 3 case of Fermat's Last Theorem.
       

387. Petryshyn biography
     
     • Petryshyn's main work has been in iterative and projective methods, fixed point theorems, nonlinear Friedrichs extension, approximation-proper mapping theorem, and topological degree and index theories for multi-valued condensing maps.
       

388. Hammersley biography
     
     • Hammersley published a variety of papers in 1951 including A theorem on multiple integrals, On a certain type of integral associated with circular cylinders, The sums of products of the natural numbers, and The total length of the edges of the polyhedron.
       

389. Jabir ibn Aflah biography
     
     • He also gave his name to a theorem in spherical trigonometry, and his criticisms of Ptolemy's Almagest are well known.
       

390. Casorati biography
     
     • Casorati is best remembered for the Casorati- Weierstrass theorem which says that an analytic function comes arbitrarily close to any given value in any neighbourhood of an essential singularity.
       

391. Kuczma biography
     
     • Many of the theorems are of general interest; the occasional theorem requiring lengthy and tedious proof should not discourage the general reader.
       

392. Descartes biography
     
     • History Topics: The fundamental theorem of algebra .
       

393. Kodaira biography
     
     • One of the themes running through much of his work is the Riemann-Roch theorem and this plays an important role in much of his research.
       

394. Torricelli biography
     
     • In De motu gravium which was published as part of Torricelli's 1644 Opera geometrica, Torricelli also proved that the flow of liquid through an opening is proportional to the square root of the height of the liquid, a result now known as Torricelli's theorem.
       

395. Witt biography
     
     • Having seen a remarkably simple proof by Witt of Wedderburn's theorem that every finite skew field is commutative, Herglotz encouraged him to submit it for publication and it became Witt's first paper appearing in 1931.
       
     • However, it was Emmy Noether who suggested a topic related to the Riemann-Roch theorem and this was indeed the topic on which his dissertation Riemann-Rochscher Satz und Z-Funktion im Hyperkomplexen was written.
       
     • This, together with results of Poincare from 1899 and Birkhoff in 1937 (independently of Witt), led to the famous Poincare-Birkhoff-Witt theorem (see [Expo.
       
     • 22 (2) (2004), 145-184.',3)">3] for the history of the theorem).
       
     • The Poincare-Birkhoff-Witt theorem gives an explicit description of the universal associative enveloping algebra of any Lie algebra over any field, and thereby establishes a remarkable relation between associative and nonassociative algebras.
       
     • He lectured in Spain and, as a consequence of these lectures, published two expository papers in Spanish: On Zorn's theorem (1950), and Intuitionistic mathematics (1951).
       
     • Kurosh had lectured about a theorem of Witt's and, when told that, Witt smiled and said "I proved that theorem when I was in the USSR".
       

396. Walsh Joseph biography
     
     • He continued to publish a steady stream of papers with On the location of the roots of the derivative of a polynomial appearing in 1920 and then two papers A generalization of the Fourier cosine series and A theorem on cross-ratios in the geometry of inversion in 1921.
       

397. Stevin biography
     
     • It is famous for containing the theorem of the triangle of forces which gave impetus to statics.
       

398. Ulugh Beg biography
     
     • This excellent book records the main achievements which include the following: methods for giving accurate approximate solutions of cubic equations; work with the binomial theorem; Ulugh Beg's accurate tables of sines and tangents correct to eight decimal places; formulae of spherical trigonometry; and of particular importance, Ulugh Beg's Catalogue of the stars, the first comprehensive stellar catalogue since that of Ptolemy.
       

399. Boys biography
     
     • History Topics: The four colour theorem .
       

400. Pless biography
     
     • For example she published On Witt's theorem for nonalternating symmetric bilinear forms over a field of characteristic 2 in 1964 and On the invariants of a vector subspace of a vector space over a field of characteristic two in the following year, both publications being in the Proceedings of the American Mathematical Society.
       

401. Herschel biography
     
     • Also in 1813 he was elected as a fellow of the Royal Society of London, having published a mathematics paper On a remarkable application of Cotes's theorem in the Transactions of the Royal Society.
       

402. Al-Khwarizmi biography
     
     • History Topics: The fundamental theorem of algebra .
       

403. Lopatynsky biography
     
     • His research interests then moved towards differential equations with his first paper on this topic Solution of the equation y ' = f (x, y) published in 1939, proving a general existence theorem.
       

404. Noether Max biography
     
     • This theorem gives [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       

405. Cowling biography
     
     • The next three chapters are concerned with Boltzmann's H-theorem, the Maxwellian velocity distribution, the free path, persistence of velocities and the elementary theory of the transport phenomena.
       

406. Remez biography
     
     • For example in 1940 his publications included On some estimates of best approximation and, in particular, on a fundamental theorem of de la Vallee-Poussin (Russian) (1940), Principe des moindres puissances, 2k-iemes et principe des moindres carres dans les problemes d'approximation (1940), Sur certaines classes de fonctionnelles lineaires dans les espaces Cp et sur les termes complementaires des formules d'analyse approximative (1940), and Sur les termes complementaires de certaines formules d'analyse approximative (1940).
       

407. Plessner biography
     
     • During his time in Marburg he published a paper containing what is today called Plessner's theorem, concerning the boundary behaviour of functions meromorphic in the unit disk.
       

408. Poinsot biography
     
     • History Topics: The fundamental theorem of algebra .
       

409. Blackwell biography
     
     • He had found a game theory proof of the Kuratowski Reduction theorem and connecting the areas of game theory and topology [Mathematical People (Boston, 1985), 18-32.',2)">2]:- .
       

410. Nash-Williams biography
     
     • Of course, such necessary and sufficient conditions must be of a psychologically satisfactory kind, and we should not, for example, want a theorem which merely said, perhaps in a slightly disguised form, that a graph has a Hamiltonian circuit if and only if it has a Hamiltonian circuit.
       

411. Laplace biography
     
     • History Topics: The fundamental theorem of algebra .
       

412. Solitar biography
     
     • In 1957 they published Note on a theorem of Schreier then in the following year the two papers On free products and Subgroup theorems in the theory of groups given by defining relations.
       

413. Bolzano biography
     
     • The paper gives a proof of the intermediate value theorem with Bolzano's new approach and in the work he defined what is now called a Cauchy sequence.
       

414. Dantzig biography
     
     • is a fine example of mathematical style: it consists of a concise string of definitions and theorems organised in such a way that in this context each theorem is obvious and none needs a proof.
       

415. Kovalevskaya biography
     
     • Bruns in which she gave a new, simpler proof of Bruns' theorem on a property of the potential function of a homogeneous body.
       

416. Fasenmyer biography
     
     • 1 which he suggests is of equal difficulty to proving Fermat's Last Theorem:- .
       

417. Ehrenfest-Afanassjewa biography
     
     • It consists of a critical discussion of the foundations of (classical) statistical mechanics, in particular, the use of the concept 'probability', Boltzmann's H-theorem, the objections of Loschmidt and Zermelo and the various attempts to overcome them, and the difference between the approaches of Boltzmann and Gibbs.
       

418. Novikov Sergi biography
     
     • In 1965 Novikov proved his famous theorem on the invariance of Pontryagin classes and stated the conjecture, now known as the Novikov conjecture, concerning the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group.
       

419. Renyi biography
     
     • Other papers published early in his career include: On a Tauberian theorem of O Szasz (1946); Integral formulae in the theory of convex curves (1947); On the minimal number of terms of the square of a polynomial (1947); On some new applications of the method of Academician I M Vinogradov (1947); (with Yu V Linnik) On certain hypotheses in the theory of Dirichlet characters (Russian) (1947).
       

420. Beltrami biography
     
     • He also used them in giving a generalisation of Green's theorem.
       

421. Possel biography
     
     • We mention Sur l'indetermination de la puissance d'un torseur reparti in which he gave proofs of some formulas of use in the mechanics of continuous media, where the differential elements are subjected to couples as well as forces per unit volume; Les principes mathematiques de la mecanique classique which was based in ideas due to Brelot; Sur la definition d'un torseur reparti et sur l'evaluation de sa puissance which examines when external forces on part of a body are equivalent to couples alone; Initiation a la topologie resulting from work carried out in Portugal; Sur les systemes derivants et l'extension du theoreme de Lebesgue relatif a la derivation d'une fonction a variation bornee extending the classical theorem for linear Lebesgue measure; and La notion physique d'energie vis-a-vis des definitions du travail et de la force which considers the formulation of classical mechanics given by Brelot.
       

422. Stirling biography
     
     • and he also gives a theorem to treat convergence of an infinite product.
       

423. Schmidt Harry biography
     
     • The equation of continuity, the impulse theorem and the Navier-Stokes equations.
       

424. James Ralph biography
     

425. Rajagopal biography
     
     • He also studied functions of a complex variable giving an analogue of a theorem of Edmund Landau on partial sums of Fourier series.
       

426. Lasker biography
     
     • He proved the primary decomposition theorem for an ideal of a polynomial ring in terms of primary ideals in a paper Zur Theorie der Moduln und Ideale published in volume 60 of Mathematische Annalen in 1905.
       

427. Posidonius biography
     
     • 'theorem' and 'problem', and subjects belonging to elementary geometry.
       

428. Schmidt biography
     
     • He found a new proof of the Jordan curve theorem which quickly became a classic.
       

429. Cherry biography
     
     • His first papers On the form of the solution of the equations of dynamics, On Poincare's theorem of 'the non-existence of uniform integrals of dynamical equations', and Note on the employment of angular variables in celestial mechanics were all published in 1924 and Some examples of trajectories defined by differential equations of a generalised dynamical type in the following year.
       

430. Green biography
     
     • The formula connecting surface and volume integrals, now known as Green's theorem, was introduced in the work, as was "Green's function" the concept now extensively used in the solution of partial differential equations.
       

431. Rutherford biography
     
     • These papers were The Cayley-Hamilton theorem for semi-rings, The eigenvalue problem for Boolean matrices, Orthogonal Boolean matrices and On certain numerical coefficients associated with partitions.
       

432. Moriarty biography
     
     • At the age of twenty-one he wrote a treatise upon the binomial theorem, which has had a European vogue.
       

433. Littlewood biography
     
     • Cartwright and Littlewood's analysis of the van der Pol equation and its generalizations led them to explore some interesting topological methods, including the development of a fixed-point theorem for continua invariant under a homeomorphism of the plane.
       

434. Regiomontanus biography
     
     • And where a theorem may present some problem, he may always look down to the numerical examples for help.
       

435. Dedekind biography
     
     • This gave powerful results such as a purely algebraic proof of the Riemann-Roch theorem.
       

436. Pringsheim biography
     
     • He gave a very simple proof of Cauchy's integral theorem.
       

437. Whyburn biography
     
     • These concern Moore's arcwise connectivity theorem which he had first proved for connected open sets in 1916 but had still not published by 1930.
       

438. Kuiper biography
     
     • Following the publication of his thesis, Kuiper published papers such as On differentiable line systems of one dual variable (1948), On conformally-flat spaces in the large (1949), A closure theorem (1949), On compact conformally Euclidean spaces of dimension > 2 (1950), On linear families of involutions (1950), Compact spaces with a local structure determined by the group of similarity transformations in En (1950), Einstein spaces and connections (1950), and Distribution modulo 1 of some continuous functions (1950).
       

439. Mason biography
     
     • He published seven papers in the Transactions of the American Mathematical Society between 1904 and 1910: Green's theorem and Green's functions for certain systems of differential equations (1904), The doubly periodic solutions of Poisson's equation in two independent variables (1905), A problem of the calculus of variations in which the integrand is discontinuous (1906), On the boundary value problems of linear ordinary differential equations of second order (1906), The expansion of a function in terms of normal functions (1907); The properties of curves in space which minimize a definite integral (1908) and Fields of extremals in space (1910).
       

440. Monge biography
     
     • Minnesota (One of Monge's geometry theorems and its relationship to Desargues theorem) .
       

441. Smith biography
     
     • Smith also extended Gauss's theorem on real quadratic forms to complex quadratic forms.
       

442. Peschl biography
     
     • A generalisation of a theorem by E Study).
       

443. Glenie biography
     
     • The was an attempt to base Newton's fluxional calculus on the binomial theorem rather than on the concept of motion.
       

444. McDuff biography
     
     • I became much less passive: I applied to the Institute for Advanced Study and got in, and even had a mathematical idea again, which grew into a joint paper with Segal on the group-completion theorem.
       
     • Among her early contributions to the field is a paper on the flux homomorphism and a celebrated theorem on the classification of rational and ruled symplectic 4-manifolds.
       
     • Her work includes fundamental theorems on the symplectic blowup construction, a theorem on the symplectic packing problem (joint with Leonid Polterovich), and a series of seminal joint papers with Francois Lalonde on the symplectic energy and the stability of Hamiltonian flows.
       
     • Among the many applications of this work is an important extension of Gromov's non-squeezing theorem.
       
     • This work also introduced a technique called symplectic inflation, an application of which is a theorem of McDuff on the existence of two symplectic structures in the same cohomology class that can be connected by a path of symplectic forms, but not by a path representing the same cohomology class.
       

445. Sierpinski biography
     
     • It happened when he came across a theorem which stated that points in the plane could be specified with a single coordinate.
       

446. Laszlo biography
     
     • There is some discussion of the Farkas theorem (used in linear programming) and, more generally, of Farkas' research on linear inequalities.
       

447. Mirsky biography
     
     • Combinatorics, where he also wrote an important book Transversal Theory and he developed ideas coming from Hall's theorem:- .
       

448. Sturm Rudolf biography
     
     • I [EFR] certainly remember examination questions which one of my lecturers would set asking "without using such and such a theorem, prove that ..
       

449. Mohr biography
     
     • Mohr proves in the book that a line segment can be divided in golden section with compass alone, and the historical and pedagogical importance of this theorem is discussed by Zuhlke in [Math.-Phys.
       

450. Murnaghan biography
     
     • produced by the Rice Institute, namely Hubert Evelyn Bray whose thesis A Green's Theorem in Terms of Lebesgue Integrals was submitted in 1918, the year Murnaghan left.
       

451. Guldin biography
     
     • Volume 2 contains Guldin's Theorem: .
       

452. Ingham biography
     
     • He generalised work on the prime number theorem of Hadamard and Vallee Poussin.
       

453. Gelfond biography
     
     • This result is now known as Gelfond's theorem and solved Problem 7 of the list of Hilbert problems.
       

454. Bohl biography
     
     • Among Bohl's achievements was, rather remarkably, to prove Brouwer's fixed-point theorem for a continuous mapping of a sphere into itself, see [Uspekhi matematicheskikh nauk (NS) 10 (3) (65) (1955), 188-192.',6)">6].
       

455. Fock biography
     
     • Some we have mentioned but now let us list a few: Fock space; Fock vacuum; the Fock method of quantisation; the Fock proper time method; the Hartree-Fock method; Fock symmetry; the Klein-Fock-Gordon equation; the Fock-Krylov theorem; and Dirac-Fock-Podolsky formalism.
       

456. Hermite biography
     
     • Hermite may have still been an undergraduate but it is likely that his ideas from around 1843 helped Liouville to his important 1844 results which include the result now known as Liouville's theorem.
       

457. Thompson Robert biography
     
     • He discussed: quantitative prediction; high and low roads; the numerical range; similarity invariants of principal submatrices; commutators; the triangle inequality; the facial structure of the unit ball; the Gershgorin circle theorem; matrices, graphs, inertia, number theory; power embeddings and dilatations; the Schubert calculus; the spectrum of a sum of Hermitian matrices; the Hadamard-Schur product; the exponential function; the exponential function and commutativity; integral quadratic forms; the matrix-valued numerical range; inequalities with subtracted terms; and further uses of the computer.
       

458. Hay biography
     
     • believed in teaching the logic of the subject rather than just following the theorem-proof format of the text (which of course was straight out of Euclid).
       

459. Robinson Julia biography
     
     • This result has been described as the most important theorem in elementary game theory.
       

460. Veblen biography
     
     • History Topics: The four colour theorem .
       

461. Curry biography
     
     • He had examined simplified methods of deriving the paradoxes (such as those of Richard and Russell) in systems of logic which are inconsistent, and had also developed a method of introducing into combinatory logic undefined notions of generality, such as quantification or formal implication, in such a way that a consistency theorem like that of Church and Rosser could still be derived.
       

462. Chernoff biography
     
     • A list of other topics treated follows: D-optimality and the Kiefer-Wolfowitz equivalence theorem; hypothesis-testing in a treatment which is largely, although not whole-heartedly, decision-theoretical; the large-sample evaluation of risk in terms of the Chernoff bounds (a term not used in the text) and the various information numbers; optimisation of sample size in the case of low-cost experimentation; the sequential probability ratio test, no-overshoot approximations, optimality; the Chernoff "procedure A" for sequential design, and its asymptotic optimality; adjacent hypotheses, and the Schwarz boundaries; testing for the sign of a normal mean, with a general consideration of dynamic programming ideas, and of helpful asymptotics; some discussion of one- and two-armed bandits.
       

463. Ramsey biography
     
     • rests on a theorem he didn't need, proved in the course of trying to do something we now know can't be done! .
       

464. Cohn biography
     
     • He generalised a theorem due to Magnus, and worked on the structure of tensor spaces.
       

465. Nassau biography
     
     • Finally we list a few of Nassau's papers: Questions and Discussions: Discussions: Evaluation of the Determinant |1/(r + s - 1)! | (1924); Some extensions of the generalized Kronecker symbol (1926); Questions and Discussions: Discussions: Concerning a Theorem in Determinants (1927); and (with O E Brown) A Navigation Computer (1947).
       

466. Frohlich biography
     
     • His research progressed exceptionally well and he published five papers in 1954: On fields of class two; On the absolute class group of Abelian fields; A note on the class field tower; The generalization of a theorem of L Redei's; and A remark on the class number of Abelian fields.
       

467. Euclid biography
     
     • someone who had begun to learn geometry with Euclid, when he had learnt the first theorem, asked Euclid "What shall I get by learning these things?" Euclid called his slave and said "Give him threepence since he must make gain out of what he learns".
       

468. Hsiung biography
     
     • With the fundamental material in place, the author discusses the theory of smooth curves, including the Frenet formulas, the isoperimetric inequality and the four-vertex theorem.
       

469. Drach biography
     
     • The proof of trancendency which he gave, and also the proof of the general theorem of Lindemann, is essentially that of Weierstrass.
       

470. Sundman biography
     
     • In the 1940 paper Sundman gave a new and elegant demonstration of the theorem of Poisson on the invariability of the major axes of the planetary orbits which had been demonstrated in several ways.
       

471. Landsberg biography
     
     • Here he studied the Riemann-Roch theorem.
       

472. Nielsen Jakob biography
     
     • Among them is his fundamental paper, in the Matematisk Tidsskrift in 1921, on free groups, in which the Nielsen-Schreier theorem (or rather Nielsen's part of it) is proved for the first time: now there is no further excuse for misquoting this paper, as has happened repeatedly in the past.
       

473. Mathews biography
     
     • The book also discusses prime numbers and Riemann's memoir on primes but, since it was written two or three years before the prime number theorem was proved, this part of the work became dated rather quickly.
       

474. Poisson biography
     
     • In his final year of study he wrote a paper on the theory of equations and Bezout's theorem, and this was of such quality that he was allowed to graduate in 1800 without taking the final examination.
       

475. Roth Klaus biography
     
     • Roth's theorem settles a question which is both of a fundamental nature and of extreme difficulty.
       

476. Theodosius biography
     
     • In many cases the extant diagrams show an axial symmetry which is not wrong but which is not required by the theorem or proof in question.
       

477. Wielandt biography
     
     • Among his contributions was a shorter, elegant proof of the Perron-Frobenius Theorem.
       

478. Steenrod biography
     
     • Perhaps the most important result is the covering homotopy theorem, which has many consequences.
       

479. Samuel biography
     
     • covers the basic facts of algebraic number theory starting with the divisibility theory in principal ideal domains and ending with the unit theorem, finiteness of the class number and the more elementary theorems of the Hilbert ramification theory.
       

480. Koksma biography
     
     • The approximation theorem of Kronecker is discussed at length.
       

481. Kruskal Martin biography
     
     • He submitted his thesis The Bridge Theorem for Minimal Surfaces and was awarded his doctorate in 1952.
       

482. Fine Henry biography
     
     • He gave his retiring address as president on An unpublished theorem of Kronecker respecting numerical equations.
       

483. Aepinus biography
     
     • 15 (1) (1988), 9-31.',4)">4] relate that in 1763 Aepinus published in Latin in the Commentaries of the St Petersburg Academy a proof of the binomial theorem for real values of the exponent.
       

484. Wallace Alexander biography
     
     • Each day, in the early afternoon, he swept into the department making the rounds of all his students and colleagues with his usual greeting, "Ah, Mr Koch"; or, "Ah, Professor Mostert; a theorem a day brings promotion and pay!.
       

485. Al-Mahani biography
     
     • Al-Mahani was one of the modern authors who conceived the idea of solving the auxiliary theorem used by Archimedes in the fourth proposition of the second book of his treatise on the sphere and the cylinder algebraically.
       

486. Tinseau biography
     
     • Tinseau wrote on the theory of surfaces, working out the equation of a tangent plane at a point on a surface, and he generalised Pythagoras's theorem proving that the square of a plane area is equal to the sum of the squares of the projections of the area onto mutually perpendicular planes.
       

487. Tapia biography
     
     • In this paper we show that the Kantorovich theorem gives useful results when stated in terms of a Newton-like method called the weak Newton method.
       

488. Cassels biography
     
     • He next worked on Vinogradov's theorem on uniform distribution and, in 1957, he published his first book Introduction to Diophantine approximation (1957) which was reprinted in 1972.
       

489. Deligne biography
     
     • initially on the generalisation of Zariski's main theorem.
       

490. Von Dyck biography
     
     • He made significant contributions to the Gauss-Bonnet theorem.
       

491. Scheffers biography
     
     • For example Scheffers' most important work was a paper in 1903 on Abel's theorem which still showed Lie's influence.
       

492. Collingwood biography
     
     • It was at this time that he completed his work on generalising Nevanlinna's second fundamental theorem, which became his first paper in 1924.
       

493. Zeckendorf biography
     
     • This is called Zeckendorf's theorem, and the subsequence of Fibonacci numbers which add up to a given integer is called its Zeckendorf representation.
       

494. Pars biography
     
     • He based his treatment on the theorem of Lagrange that he called the fundamental equation, which he proceeded to translate into six different forms, each exploited in appropriate contexts.
       

495. Grassmann biography
     
     • He proves the Steinitz Exchange Theorem, named for the man who published it in 1913 ..
       

496. Bauer biography
     
     • The final part of the text proceeds at a much faster pace and covers topics such as the central limit theorem, conditional expectation, martingales, and some topics in stochastic processes.
       

497. Baker Alan biography
     
     • succeeded in obtaining a vast generalisation of the Gelfond-Schneider Theorem ..
       

498. Newton biography
     
     • ' he only gave a proof of the converse theorem that if the orbit is an ellipse the force is inverse square.
       

499. Walsh biography
     
     • He takes up some known method or formula of analysis, makes in it a slight and quite unimportant change (for every theorem admits of some variety in the mode of its expression) and views the result to which he is led as an original discovery.
       

500. Green Sandy biography
     
     • He was only there for two years but during this time he published perhaps his best-known work A transfer theorem for modular representations (1964) published in the first volume of the Journal of Algebra.
       

501. Post biography
     
     • As for any claims I might make perhaps the best I can say is that I would have proved Godel's Theorem in 1921 - had I been Godel.
       

502. Delsarte biography
     
     • A proof of this result was first sketched in his paper Note sur une propriete nouvelle des fonctions harmoniques (1958) and is explained in more detail in Lectures on Topics in Mean Periodic functions and the Two Radius Theorem published in Bombay in 1961.
       

503. Kneser Hellmuth biography
     
     • History Topics: The fundamental theorem of algebra .
       

504. Robinson Raphael biography
     
     • In 1939 he published On numerical bounds in Schottky's theorem in the Bulletin of the American Mathematical Society, and in the following year published On the mean values of an analytic function in the same journal.
       

505. Thompson John biography
     
     • Work in this area was started by Hilbert with his proof of the irreducibility theorem, and the authors of [Notices Amer.
       

506. Macdonald biography
     
     • the relations between convergent series and asymptotic expansions, the zeros and the addition theorem of the Bessel functions, various Bessel integrals, spherical harmonics and Fourier series.
       

507. Peano biography
     
     • Peano's continuous space-filling curves cannot be 1-1 of course, otherwise Netto's theorem would be contradicted.
       

508. Cardan biography
     
     • History Topics: The fundamental theorem of algebra .
       

509. Hankel biography
     
     • History Topics: The fundamental theorem of algebra .
       

510. Roy biography
     
     • A sample of the papers Roy published during this period follows: The use and distribution of the Studentized D2-statistic when the variances and covariances are based on k samples (1940), On hierarchical sampling, hierarchical variances and their connexion with other aspects of statistical theory (1940), The distribution of the root-mean-square of the second type of the multiple correlation co-efficient (1940), Analysis of variance for multivariate normal populations: the sampling distribution of the requisite p-statistics on the null and non-null hypotheses (1942), Bernoulli's theorem and Tshebycheff's analogue (1945), On a certain class of multiple integrals (1945), Notes on testing of composite hypotheses (1947), and On the construction of an unbiassed and most powerful critical region out of any given statistic (1948).
       

511. Galileo biography
     
     • He then described an experiment using a pendulum to verify his property of inclined planes and used these ideas to give a theorem on acceleration of bodies in free fall:- .
       

512. Ince biography
     
     • is similar to that previously used by [Ince] to prove the corresponding theorem for Mathieu functions.
       

513. Lakatos biography
     
     • to reduce it to other conjectures) followed by criticism via attempts to produce counter-examples both to the conjectured theorem and to the various steps in the proof.
       

514. Redei biography
     
     • The paper was Existence theorem for the primitive root of the congruence xφ(pa) - 1 equiv 0 (mod pα) (Hungarian).
       

515. Catalan biography
     
     • I beg you, sir, to please announce in your journal the following theorem that I believe true although I have not yet succeeded in completely proving it; perhaps others will be more successful.
       

516. Bliss biography
     
     • They were An existence theorem for a differential equation of the second order, with an application to the calculus of variations and Sufficient condition for a minimum with respect to one-sided variations.
       

517. Newman biography
     
     • The first was an early inroad on Hilbert's Fifth Problem, in which he proved that abelian continuous groups do not have arbitrarily small subgroups, the second was a simplified proof of a difficult fixed point theorem of Cartwright and Littlewood arising in the study of differential equations.
       

518. Crofton biography
     
     • In [The life and times of the central limit theorem (New York, 1974).',1)">1] Crofton's work on the hypothesis of elementary errors is discussed.
       

519. Ikeda biography
     
     • In May 1968 Arf and Langlands visited Ege University to deliver talks on The Cartan-Dieudonne Theorem and Automorphic Forms.
       

520. Bonnet biography
     
     • A formula for the line integral of the geodesic curvature along a closed curve is known as the Gauss-Bonnet theorem.
       

521. Courant biography
     
     • In fact this method first appeared in an existence proof of a version of the Riemann mapping theorem in the Hurwitz-Courant book of 1922.
       

522. Whitney biography
     
     • Following this he published a number of papers on graph theory such as A theorem on graphs (1931), Non-separable and planar graphs (1932), Congruent graphs and the connectivity of graphs (1932), The coloring of graphs (1932), A numerical equivalent of the four color map problem (1937).
       

523. Buffon biography
     
     • However, he was more interested in mathematics than he was in the law and at the age of 20 Buffon (he was now calling himself Georges-Louis Leclerc De Buffon) discovered the binomial theorem.
       

524. Mansur biography
     
     • The work is in three books: the first book studies properties of spherical triangles, the second book investigates properties of systems of parallel circles on a sphere as they intersect great circles, while the third book gives a proof of Menelaus's theorem.
       

525. Guinand biography
     
     • Guinand worked on summation formulae and prime numbers, the Riemann zeta function, general Fourier type transformations, geometry and some papers on a variety of topics such as computing, air navigation, calculus of variations, the binomial theorem, determinants and special functions.
       

526. Mochizuki biography
     
     • We should make special mention of Mochizuki's paper Unipotent matrix groups over division rings (1978) where he presented a non-commutative version of "Kolchin's Theorem" which solved a famous problem of Kaplansky.
       

527. Gruenberg biography
     
     • Before the award of his doctorate Gruenberg had published a number of papers such as Some theorems on commutative matrices (1951), A note on a theorem of Burnside (1952), Two theorems on Engel groups (1953), and Commutators in associative rings (1953).
       

528. Conon biography
     
     • cited as the propounder of a theorem about the spiral in a plane which Archimedes proved: this would, however, seem to be a mistake, as Archimedes says at the beginning of his treatise that he sent certain theorems, without proofs, to Conon, who would certainly have proved them had he lived.
       

529. Bremermann biography
     
     • In 1954 in Uber die Aquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum von n komplexen Veranderlichen Bremermann proved the general theorem obtaining Oka's result for n complex variables where n ≥ 2.
       

530. Reiner biography
     
     • He was awarded a Master's Degree for a thesis on binary quadratic forms in 1945 and his doctorate in 1949 for a thesis on a generalisation of Meyer's theorem.
       

531. Clausius biography
     
     • the merit of first establishing [Sadi Carnot's theorem] upon correct principles is entirely due to Clausius.
       

532. Hardy biography
     
     • He reasoned that God would not allow the boat to sink on the return journey and give him the same fame that Fermat had achieved with his "last theorem".
       

533. Chevalley biography
     
     • Chevalley's theorem was important in applications made in 1954 to quasi-algebraically closed fields and applications made the following year to algebraic groups.
       

534. Chow biography
     
     • Chow's theorem that a compact analytic variety in a projective space is algebraic was published in 1949.
       

535. Kolmogorov biography
     
     • This was published jointly with Khinchin and contains the 'three series' theorem as well as results on inequalities of partial sums of random variables which would become the basis for martingale inequalities and the stochastic calculus.
       

536. Zassenhaus biography
     
     • During this time he proved Zassenhaus's lemma, a beautiful result on subgroups which can be used to give a simple proof of the Jordan-Holder theorem.
       

537. Fagnano Giovanni biography
     
     • One theorem on the triangle which he discovered is worth quoting.
       

538. Heron biography
     
     • History Topics: Pythagoras's theorem in Babylonian mathematics .
       

539. Freudenthal biography
     
     • He wrote important papers on a spectral theorem for Riesz spaces in 1936 and on the suspension theorems in 1937.
       

540. Savary biography
     
     • Savary also developed a theorem (named after him) on the curvature of a roulette, the curve traced out by a point on a fixed curve which rolls on a second curve.
       

541. Al-Maghribi biography
     
     • He wrote Book on the theorem of Menelaus and Treatise on the calculation of sines.
       

542. Gateaux biography
     
     • In a last paragraph, Gateaux mentions the possible applications of such an integration of functionals, such as the residue theorem ..
       

543. Kronecker biography
     
     • It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature.
       

544. Grothendieck biography
     
     • He gave an algebraic proof of the Riemann-Roch theorem.
       

545. Hirsch biography
     
     • However he kept up his mathematics attending a study group where he studied Emmy Noether's work and read Schreier's paper on the Jordan-Holder theorem.
       

546. Dieudonne biography
     
     • Let us not pass judgement on whether the text is too sophisticated to fulfil its intended purpose but we do note that in introducing the real numbers in the first chapter Dieudonne assumes they are an ordered field in which the intermediate value theorem is valid for polynomials of degree 3.
       

547. Moser Jurgen biography
     
     • This work introduced techniques which could be applied to almost any dynamical system of Hamiltonian type and the "Moser twist stability theorem".
       

548. Tutte biography
     
     • Moreover, the more customary topics are leavened with some 'pleasant surprises', such as the author's attractive theory of decomposition of graphs into 3-connected '3-blocks', an interesting and remarkable approach to electrical networks, and - perhaps particularly - the classification theorem for closed surfaces.
       

549. Zaanen biography
     
     • During the war years he published papers such as On some orthogonal systems of functions (1939), A theorem on a certain orthogonal series and its conjugate series (1940), On some orthogonal systems of functions (1940), Uber die Existenz der Eigenfunktionen eines symmetrisierbaren Kernes (1942), Uber vollstetige symmetrische und symmetrisierbare Operatoren (1943), Transformations in Hilbert space which depend upon one parameter (1944), and On the absolute convergence of Fourier series (1945).
       

550. Harriot biography
     
     • History Topics: The fundamental theorem of algebra .
       

551. Iyanaga biography
     
     • Iyanaga managed to solve a question of Artin on generalising the principal ideal theorem and this was published in 1939.
       

552. Straus biography
     
     • Together they obtained some of the first results which went beyond Turan's theorem on graphs containing no complete subgraphs.
       

553. Feuerbach biography
     
     • Feuerbach's theorem .
       

554. Hironaka biography
     
     • His work generalised that of Zariski who had proved the theorem concerning the resolution of singularities on an algebraic variety for dimension not exceeding 3.
       

555. Van Kampen biography
     
     • Van Kampen solved the problem, showing that Zariski's relations were sufficient, and the result is now known as the Zariski-van Kampen theorem.
       

556. Carnot biography
     
     • He showed that several of the theorems of Euclid's Elements can be established from a single theorem.
       

557. Tichy biography
     
     • He was awarded his doctorate in 1959 for his thesis An Exposition of Godel's Incompleteness Theorem in the Simple Theory of Types (Czech).
       

558. Steinitz biography
     
     • He proved that every field has an algebraically closed extension field, perhaps his most important single theorem.
       

559. Bernoulli Jacob biography
     
     • History Topics: Fermat's Last Theorem .
       

560. Sargent biography
     
     • provided a simple and direct proof for a theorem which is fundamental in the development of the Cesaro-Perron scale of integration.
       

561. Milnor biography
     
     • It discusses Milnor's theorem, which shows that the total curvature of a knot is at least 4π.
       

562. Fenyo biography
     
     • The authors prove Titchmarsh's theorem, which is central to the theory, and include a number of worked examples.
       

563. Franklin biography
     
     • History Topics: The four colour theorem .
       

564. James Ralph biography
     
     • James was awarded a doctorate from Chicago in 1932 for his thesis Analytical Investigations in Waring's Theorem on number theory.
       

565. Tikhonov biography
     
     • In fact Tikhonov's work led from topology to functional analysis with his famous fixed point theorem for continuous maps from convex compact subsets of locally convex topological spaces in 1935.
       

566. Vitali biography
     
     • His significant mathematical discoveries include a theorem on set-covering, the notion of an absolutely continuous functions and a criteria for the closure of a system of orthogonal functions.
       

567. Gromoll biography
     
     • This paper contains a result known as the "soul theorem" which [New York Times (19 June 2008).',1)">1]:- .
       

568. Viete biography
     
     • History Topics: The fundamental theorem of algebra .
       

569. Thue biography
     
     • Thue's Theorem states that:- .
       

570. Spitzer biography
     
     • A theorem by Kruskal is used to prove that each line of force in such a system generates a toroidal surface; ideally the wall is such a surface.
       

571. Dini biography
     
     • Dini looked at infinite series and generalised results such as a theorem of Kummer and one of Riemann, the ideas for which had first emerged in work of Dirichlet.
       

572. Al-Nasawi biography
     
     • One discusses the theorem of Menelaus while the other is [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
       

573. Zorn biography
     
     • The characterization theorem which we are going to derive will serve to show that the functions which fall under the first definition but not under the second are, from a certain point of view, to be considered as freaks, counter examples rather than examples.
       

574. Levi biography
     
     • One year later, he wrote On Sines, Chords and Arcs which examined trigonometry, in particular proving the sine theorem for plane triangles and giving 5 figure sine tables.
       
     • Gino Loria suggested that the sine theorem be named after Levi but he was not the first to present the theorem, which was known to Jabir ibn Aflah in the 12th century, but he may have rediscovered it.
       

575. Wallis biography
     
     • His interpolation used Kepler's concept of continuity, and with it he discovered methods to evaluate integrals which were later used by Newton in his work on the binomial theorem.
       

576. Krasnosel'skii biography
     
     • Following this he achieved a remarkable publication record with papers (all written in Russian) such as On the deficiency numbers of closed operators (1947), (with M G Krein) On the centre of a general dynamical system (1947), (with M G Krein) Fundamental theorems on the extension of Hermitian operators and certain of their applications to the theory of orthogonal polynomials and the problem of moments (1947), On the extension of Hermitian operators with a nondense domain of definition (1948), On self-adjoint extensions of Hermitian operators (1949), (with M G Krein) On a proof of the theorem on category of a projective space (1949), and On a fixed point principle for completely continuous operators in functional spaces (1950).
       

577. Bauer Mihaly biography
     
     • Kronecker called this a 'boundary value problem' (Randwertproblem) because of a (vague) analogy with Cauchy's theorem computing the values of an analytic function on a disc from its values taken at the boundary.
       

578. Connes biography
     
     • (1) general classification and a structure theorem for factors of type III, obtained in his thesis .
       

579. Al-Baghdadi biography
     
     • Thabit ibn Qurra's theorem is as follows: for n > 1, let pn = 3.2n -1 and qn = 9.22n-1 -1.
       

580. Al-Sijzi biography
     
     • The letter contains proofs of both the plane and spherical versions of the sine theorem, which al-Biruni says were due to his teacher Abu Nasr Mansur ibn Ali ibn Iraq.
       

581. Pfaff biography
     
     • He developed Taylor's Theorem using the form with remainder as given by Lagrange.
       

582. Lowenheim biography
     
     • Lowenheim analysed and improved upon the customary methods of solving equations in the calculus of classes or domains (that is, set theory in its Peirce-Schroder [Charles Peirce and Ernst Schroder] setting) and proved what is now known as Lowenheim's general development theorem for functions of functions.
       

583. Heisenberg biography
     
     • In fact by this time he had become interested in number theory and he read Kronecker's work and tried to find a proof of Fermat's Last Theorem.
       

584. Mandelbrot biography
     
     • Many of those problems have been definitively solved, or shown to be insoluble, culminating as we all know most recently in the mid-nineties with the discovery of the proof of Fermat's Last Theorem.
       

585. D'Alembert biography
     
     • History Topics: The fundamental theorem of algebra .
       

586. Aleksandrov Aleksandr biography
     
     • In 1933 he published A theorem on convex polyhedra and An elementary proof of the existence of a centre of symmetry in a three-dimensional convex polyhedron.
       

History Topics

1. Fermat's last theorem
   
   • Fermat's last theorem .
     
   • In this way the famous 'Last theorem' came to be published.
     Go directly to this paragraph
     
   • Fermat's Last Theorem states that .
     
   • Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat.
     Go directly to this paragraph
     
   • From this it is easy to deduce the n = 4 case of Fermat's theorem.
     Go directly to this paragraph
     
   • It is worth noting that at this stage it remained to prove Fermat's Last Theorem for odd primes n only.
     Go directly to this paragraph
     
   • Euler wrote to Goldbach on 4 August 1753 claiming he had a proof of Fermat's Theorem when n = 3.
     Go directly to this paragraph
     
   • Hence Fermat's Last Theorem splits into two cases.
     Go directly to this paragraph
     
   • Sophie Germain proved Case 1 of Fermat's Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197.
     Go directly to this paragraph
     
   • In 1832 Dirichlet published a proof of Fermat's Last Theorem for n = 14.
     Go directly to this paragraph
     
   • Lame's proof is exceedingly hard and makes it look as though progress with Fermat's Last Theorem to larger n would be almost impossible without some radically new thinking.
     Go directly to this paragraph
     
   • The year 1847 is of major significance in the study of Fermat's Last Theorem.
     Go directly to this paragraph
     
   • On 1 March of that year Lame announced to the Paris Academie that he had proved Fermat's Last Theorem.
     Go directly to this paragraph
     
   • Cauchy supported Lame but, in rather typical fashion, pointed out that he had reported to the October 1847 meeting of the Academie an idea which he believed might prove Fermat's Last Theorem.
     Go directly to this paragraph
     
   • Kummer had used his new theory to find conditions under which a prime is regular and had proved Fermat's Last Theorem for regular primes.
     Go directly to this paragraph
     
   • By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular (and so Fermat's Last Theorem is true for that prime) if p does not divide the numerators of any of the Bernoulli numbers B2 , B4 , ..
     Go directly to this paragraph
     
   • More powerful techniques were used to prove Fermat's Last Theorem for these numbers.
     Go directly to this paragraph
     
   • Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved.
     Go directly to this paragraph
     
   • It has the dubious distinction of being the theorem with the largest number of published false proofs.
     Go directly to this paragraph
     
   • Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993.
     Go directly to this paragraph
     
   • The final chapter in the story began in 1955, although at this stage the work was not thought of as connected with Fermat's Last Theorem.
     Go directly to this paragraph
     
   • In 1986 the connection was made between the Shimura-Taniyama- Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrucken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space.
     Go directly to this paragraph
     
   • Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter -example to the Shimura-Taniyama-Weil Conjecture.
     Go directly to this paragraph
     
   • The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA.
     Go directly to this paragraph
     
   • In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat's Last Theorem as a corollary to his main results.
     Go directly to this paragraph
     
   • Having written the theorem on the blackboard he said I will stop here and sat down.
     Go directly to this paragraph
     
   • In fact Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.
     Go directly to this paragraph
     
   • To some extent, proving Fermat's Theorem is like climbing Everest.
     
   • However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem.
     Go directly to this paragraph
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Fermat's_last_theorem.html .
     

2. Fermat's last theorem references
   
   • References for: Fermat's last theorem .
     
   • A D Aczel, Fermat's last theorem : Unlocking the secret of an ancient mathematical problem (New York, 1996).
     
   • D A Cox, Introduction to Fermat's last theorem, Amer.
     
   • H M Edwards, Fermat's last theorem : A genetic introduction to algebraic number theory (New York, 1996).
     
   • H M Edwards, The background of Kummer's proof of Fermat's last theorem for regular primes, Arch.
     
   • D R Heath-Brown, The first case of Fermat's last theorem, Math.
     
   • R de Castro Korgi, The proof of Fermat's last theorem has been announced in Cambridge, England (Spanish), Lect.
     
   • F Nemenzo, Fermat's last theorem : a mathematical journey, Matimyas Mat.
     
   • A van der Poorten, Notes on Fermat's last theorem (New York, 1996).
     
   • A van der Poorten, Remarks on Fermat's last theorem, Austral.
     
   • P Ribenboim, 13 lectures on Fermat's last theorem (New York, 1979).
     
   • P Ribenboim, Kummer's ideas on Fermat's last theorem, Enseign.
     
   • P Ribenboim, Fermat's last theorem, before June 23, 1993, in Number theory (Providence, RI, 1995), 279-294.
     
   • P Ribenboim, The history of Fermat's last theorem (Portuguese), Bol.
     
   • P Ribenboim, Recent results on Fermat's last theorem, Canad.
     
   • R Schoof, Fermat's last theorem, in Jahrbuch uberblicke Mathematik (Braunschweig, 1995), 193-211.
     
   • S L Singh, Fermat's last Theorem (London 1997) .
     
   • S Wagon, Fermat's last theorem, Math.
     
   • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Fermat's_last_theorem.html] .
     

3. Fermat's last theorem references
   
   • References for: Fermat's last theorem .
     
   • A D Aczel, Fermat's last theorem : Unlocking the secret of an ancient mathematical problem (New York, 1996).
     
   • D A Cox, Introduction to Fermat's last theorem, Amer.
     
   • H M Edwards, Fermat's last theorem : A genetic introduction to algebraic number theory (New York, 1996).
     
   • H M Edwards, The background of Kummer's proof of Fermat's last theorem for regular primes, Arch.
     
   • D R Heath-Brown, The first case of Fermat's last theorem, Math.
     
   • R de Castro Korgi, The proof of Fermat's last theorem has been announced in Cambridge, England (Spanish), Lect.
     
   • F Nemenzo, Fermat's last theorem : a mathematical journey, Matimyas Mat.
     
   • A van der Poorten, Notes on Fermat's last theorem (New York, 1996).
     
   • A van der Poorten, Remarks on Fermat's last theorem, Austral.
     
   • P Ribenboim, 13 lectures on Fermat's last theorem (New York, 1979).
     
   • P Ribenboim, Kummer's ideas on Fermat's last theorem, Enseign.
     
   • P Ribenboim, Fermat's last theorem, before June 23, 1993, in Number theory (Providence, RI, 1995), 279-294.
     
   • P Ribenboim, The history of Fermat's last theorem (Portuguese), Bol.
     
   • P Ribenboim, Recent results on Fermat's last theorem, Canad.
     
   • R Schoof, Fermat's last theorem, in Jahrbuch uberblicke Mathematik (Braunschweig, 1995), 193-211.
     
   • S L Singh, Fermat's last Theorem (London 1997) .
     
   • S Wagon, Fermat's last theorem, Math.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Fermat's_last_theorem.html .
     

4. The four colour theorem
   
   • The four colour theorem .
     
   • At Cayley's suggestion Kempe submitted the Theorem to the American Journal of Mathematics where it was published in 1879.
     Go directly to this paragraph
     
   • Tait addressed the Royal Society of Edinburgh on the subject and published two papers on the (what we should now call) Four Colour Theorem.
     Go directly to this paragraph
     
   • The Four Colour Theorem returned to being the Four Colour Conjecture in 1890.
     Go directly to this paragraph
     
   • Percy John Heawood, a lecturer at Durham England, published a paper called Map colouring theorem.
     Go directly to this paragraph
     
   • The year 1976 saw a complete solution to the Four Colour Conjecture when it was to become the Four Colour Theorem for the second, and last, time.
     Go directly to this paragraph
     
   • The Four Colour Theorem was the first major theorem to be proved using a computer, having a proof that could not be verified directly by other mathematicians.
     
   • Despite some worries about this initially, independent verification soon convinced everyone that the Four Colour Theorem had finally been proved.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/The_four_colour_theorem.html .
     

5. Fund theorem of algebra references
   
   • References for: The fundamental theorem of algebra .
     
   • J Pla i Carrera, The fundamental theorem of algebra before Carl Friedrich Gauss, Publ.
     
   • A Fryant and V L N Sarma, Gauss' first proof of the fundamental theorem of algebra, Math.
     
   • R C F Kooistra, Gauss and the fundamental theorem of algebra (Dutch), Nieuw Tijdschr.
     
   • S S Petrova, From the history of the analytic proofs of the fundamental theorem of algebra (Russian), History and methodology of the natural sciences XIV : Mathematics, mechanics (Moscow, 1973), 167-172.
     
   • S S Petrova, The first proof of the fundamental theorem of algebra (Bulgarian), Fiz.-Mat.
     
   • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Fund_theorem_of_algebra.html] .
     

6. Fund theorem of algebra references
   
   • References for: The fundamental theorem of algebra .
     
   • J Pla i Carrera, The fundamental theorem of algebra before Carl Friedrich Gauss, Publ.
     
   • A Fryant and V L N Sarma, Gauss' first proof of the fundamental theorem of algebra, Math.
     
   • R C F Kooistra, Gauss and the fundamental theorem of algebra (Dutch), Nieuw Tijdschr.
     
   • S S Petrova, From the history of the analytic proofs of the fundamental theorem of algebra (Russian), History and methodology of the natural sciences XIV : Mathematics, mechanics (Moscow, 1973), 167-172.
     
   • S S Petrova, The first proof of the fundamental theorem of algebra (Bulgarian), Fiz.-Mat.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Fund_theorem_of_algebra.html .
     

7. Fund theorem of algebra
   
   • The fundamental theorem of algebra .
     
   • The Fundamental Theorem of Algebra (FTA) states .
     
   • Then since an arbitrary polynomial can be converted to a monic polynomial by multiplying by axk for some k the theorem would follow by iterating the decomposition.
     
   • Now in the later paper Reflexions sur la nouvelle theorie d'analyse Argand simplifies d'Alembert's idea using a general theorem on the existence of a minimum of a continuous function.
     Go directly to this paragraph
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Fund_theorem_of_algebra.html .
     

8. The four colour theorem references
   
   • References for: The four colour theorem .
     
   • D A Holton and S Purcell, The four colour theorem-a short history, Austral.
     
   • P C Kainen, Is the four color theorem true?, Geombinatorics 3 (2) (1993), 41-56.
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/The_four_colour_theorem.html .
     

9. The four colour theorem references
   
   • References for: The four colour theorem .
     
   • D A Holton and S Purcell, The four colour theorem-a short history, Austral.
     
   • P C Kainen, Is the four color theorem true?, Geombinatorics 3 (2) (1993), 41-56.
     
   • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/The_four_colour_theorem.html] .
     

10. Babylonian Pythagoras
    
    • Pythagoras's theorem in Babylonian mathematics .
      
    • Pythagoras's theorem in Babylonian mathematics .
      
    • In this article we examine four Babylonian tablets which all have some connection with Pythagoras's theorem.
      
    • Certainly the Babylonians were familiar with Pythagoras's theorem.
      
    • This shows a nice understanding of Pythagoras's theorem.
      
    • Is it the earliest known mathematical classification theorem? Although I cannot believe that Zeeman has it quite right, I do feel that his explanation must be on the right track.
      
    • we prove that in this tablet there is no evidence whatsoever that the Babylonians knew the Pythagorean theorem and the Pythagorean triads.
      
    • I feel that the arguments are weak, particularly since there are numerous tablets which show that the Babylonians of this period had a good understanding of Pythagoras's theorem.
      
    • Now the triangle ABD is a right angled triangle so, using Pythagoras's theorem AD2 = AB2 - BD2, so AD = 40.
      
    • Using Pythagoras's theorem again on the triangle OBD we have .
      

11. Word problems
    
    • This is not a theorem, it is simply saying that the vague notion of "computable function" can be given the rigorous definition of l-definability.
      
    • Higman proved the Higman embedding theorem in 1961: .
      
    • Higman embedding theorem: A finitely generated group H can be embedded in a finitely presented group if and only if H is recursively presented.
      
    • We note that Higman also used his embedding theorem to prove that there exists a finitely presented group U such that given any finitely presented group G then G is isomorphic to subgroup of U.
      
    • Adian proved the following theorem in 1957: .
      
    • Theorem: .
      
    • One also obtains as a corollary to Adian's theorem that the isomorphism problem is insoluble, since one cannot determine whether a group is isomorphic to the trivial group.
      
    • In 1959 Gilbert Baumslag, Bill Boone and Bernhard Neumann proved the following theorem: .
      
    • Theorem: .
      
    • Theorem: .
      

12. Ring Theory
    
    • However, the motivation for generalising arithmetic came mostly from attempts-to prove Fermat's Last Theorem.
      
    • This theorem, proved as recently as 1995, states: .
      
    • In 1847 Lame announced that he had a solution of Fermat's Last Theorem and sketched out a proof.
      
    • In fact, numbers of the form a + b +c2 where a, b, c are integers and is a complex cube root of 1, also have unique factorisation, and this can be used to prove the n = 3 case of Fermat's last Theorem.
      
    • He then saw the relevance of his theory to Fermat's Last Theorem.
      
    • The popular story that Kummer invented "ideal complex numbers" in an attempt to correct an error in this proof of Fermat's Last Theorem is almost certainly false; see Edwards [Fermat\'s Last Theorem, (Berlin 1977).
      
    • In 1847, just after Lame's announcement, Kummer used his "ideal complex numbers" to prove Fermat's Last Theorem for all n < 100 except n = 37, 59, 67 and 74.
      
    • Hilbert, motivated by studying invariant theory, studied ideals in polynomial rings proving his famous "Basis Theorem" in 1893.
      
    • Special cases of this theorem had been studied by Gordan from 1868 and on seeing Hilbert's proof Gordan is said to have exclaimed "This is not mathematics, it's theology".
      

13. Matrices and determinants
    
    • In the 1812 paper the multiplication theorem for determinants is proved for the first time although, at the same meeting of the Institut de France, Binet also read a paper which contained a proof of the multiplication theorem but it was less satisfactory than that given by Cauchy.
      Go directly to this paragraph
      
    • I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree.
      
    • That a matrix satisfies its own characteristic equation is called the Cayley-Hamilton theorem so its reasonable to ask what it has to do with Hamilton.
      
    • In fact he also proved a special case of the theorem, the 4 cross 4 case, in the course of his investigations into quaternions.
      
    • Despite the fact that Cayley only proved the Cayley-Hamilton theorem for 2 cross 2 and 3 cross 3 matrices, Frobenius generously attributed the result to Cayley despite the fact that Frobenius had been the first to prove the general theorem.
      

14. Bolzano publications.html
    
    • Contains reprints of the following papers by Bolzano: Considerations on some points in elementary geometry (1804), Contributions to a better founded exposition of mathematics (1810), The binomial theorem (1816), Pure analytical proof of the intermediate value theorem (1817), and The three problems of curve length, surface area and volume (1817).
      
    • In these entries Bolzano comments on mathematical texts he has read in preparation for writing his work on the binomial theorem, Der binomische Lehrsatz (1816).
      
    • In these entries Bolzano comments on mathematical texts he has read in preparation for writing his work on the binomial theorem, Die drey Probleme der Rectification, der Complanation und der Cubirung (1817).
      
    • He also jots down his ongoing ideas on trigonometric series, the binomial theorem, Taylor's theorem, the mean-value theorem, and convergence of infinite series.
      

15. Indian Sulbasutras
    
    • The first result which was clearly known to the authors is Pythagoras's theorem.
      
    • The Baudhayana Sulbasutra gives only a special case of the theorem explicitly:- .
      
    • While thinking of explicit statements of Pythagoras's theorem, we should note that as it is used frequently there are many examples of Pythagorean triples in the Sulbasutras.
      
    • It is a construction, based on Pythagoras's theorem, for making a square equal in area to two given unequal squares.
      
    • This follows from Pythagoras's theorem since SX2 = PX2 + PS2.
      
    • The Baudhayana Sulbasutra offers no proof of this result (or any other for that matter) but we can see that it is true by using Pythagoras's theorem.
      

16. Babylonian Pythagoras references
    
    • References for: Pythagoras's theorem in Babylonian mathematics .
      

17. Prime numbers
    
    • Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.
      Go directly to this paragraph
      
    • He proved what has come to be known as Fermat's Little Theorem (to distinguish it from his so-called Last Theorem).
      
    • Fermat's Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers.
      
    • He extended Fermat's Little Theorem and introduced the Euler φ-function.
      
    • The statement that the density of primes is 1/log(n) is known as the Prime Number Theorem.
      Go directly to this paragraph
      

18. Babylonian Pythagoras references
    
    • References for: Pythagoras's theorem in Babylonian mathematics .
      

19. Arabic mathematics
    
    • He discovered a beautiful theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other.
      Go directly to this paragraph
      
    • Al-Baghdadi (born 980) looked at a slight variant of Thabit ibn Qurra's theorem, while al-Haytham (born 965) seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2k-1(2k - 1) where 2k - 1 is prime.
      Go directly to this paragraph
      
    • Al-Haytham, is also the first person that we know to state Wilson's theorem, namely that if p is prime then 1+(p-1)! is divisible by p.
      Go directly to this paragraph
      
    • It is called Wilson's theorem because of a comment made by Waring in 1770 that John Wilson had noticed the result.
      Go directly to this paragraph
      
    • Al-Farisi (born 1260) gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorisation and combinatorial methods.
      Go directly to this paragraph
      
    • The discovery of the binomial theorem for integer exponents by al-Karaji (born 953) was a major factor in the development of numerical analysis based on the decimal system.
      Go directly to this paragraph
      

20. General relativity
    
    • Hilbert applied the variational principle to gravitation and attributed one of the main theorem's concerning identities that arise to Emmy Noether who was in Gottingen in 1915.
      Go directly to this paragraph
      
    • No proof of the theorem is given.
      Go directly to this paragraph
      
    • In fact Emmy Noether's theorem was published with a proof in 1918 in a paper which she wrote under her own name.
      Go directly to this paragraph
      
    • This theorem has become a vital tool in theoretical physics.
      Go directly to this paragraph
      
    • A special case of Emmy Noether's theorem was written down by Weyl in 1917 when he derived from it identities which, it was later realised, had been independently discovered by Ricci in 1889 and by Bianchi (a pupil of Klein) in 1902.
      Go directly to this paragraph
      

21. Chinese overview
    
    • The Zhoubi suanjing contains a statement of the Gougu rule (the Chinese version of Pythagoras's theorem) and applies it to surveying, astronomy, and other topics.
      
    • Although it is widely accepted that the work also contains a proof of Pythagoras's theorem, Cullen in [Astronomy and Mathematics in Ancient China (Cambridge, 1996).',3)" onmouseover="window.status='Click to see reference';return true">3] disputes this, claiming that the belief is based on a flawed translation given by Needham in [Science and Civilisation in China 3 (Cambridge, 1959).',13)" onmouseover="window.status='Click to see reference';return true">13].
      
    • In it Liu uses Pythagoras's theorem to calculate heights of objects and distances to objects which cannot be measured directly.
      
    • However, it does contains a problem solved using the Chinese remainder theorem, being the earliest known occurrence of this type of problem.
      
    • The treatise contains remarkable work on the Chinese remainder theorem, gives an equation whose coefficients are variables and, among other results, Heron's formula for the area of a triangle.
      

22. Nine chapters references
    
    • J W Dauben, The "Pythagorean theorem" and Chinese mathematics : Liu Hui's commentary on the gou-gu theorem in Chapter Nine of the J'iu zhang suan shu', in Amphora (Basel, 1992), 133-155.
      
    • Z J Liang, From the elimination method in the Jiu zhang suanshu (Arithmetic in nine sections) to automated theorem proving (Chinese), J.
      
    • D B Wagner, A proof of the Pythagorean theorem by Liu Hui (third century AD), Historia Math.
      

23. Cubic surfaces
    
    • At the end of his 1865 treatise The Geometry of Three Dimensions Salmon described how the two had collaborated over finding the Cayley-Salmon theorem.
      
    • Steiner already knew of Cayley-Salmon theorem about 27 straight lines when he started his own work on cubic surfaces.
      
    • It was Steiner who communicated to Schlafli the Cayley-Salmon theorem on 27 lines on a cubic surface.
      
    • He also established connections between the Cayley-Salmon theorem on 27 lines on a cubic surface and Pascal's Mystic Hexagram:- .
      

24. Burnside problem
    
    • Theorem (Burnside, 1905 [Proc.London Math.
      
    • Theorem (Schur, 1911 [Sitzungsber.
      
    • The 1956 Hall-Higman paper contains a remarkable reduction theorem for the Restricted Burnside Problem: .
      
    • Theorem (Hall-Higman, 1956 [p-length of p-soluble groups and reduction theorems for Burnside\'s Problem, Proc.
      

25. Perfect numbers
    
    • In this letter he gave a generalisation of results in the earlier letter stating the result now known as Fermat's Little Theorem which shows that for any prime p and an integer a not divisible by p, ap-1- 1 is divisible by p.
      
    • Certainly Fermat found his Little Theorem as a consequence of his investigations into perfect numbers.
      
    • Using special cases of his Little Theorem, Fermat was able to disprove two of Cataldi's claims in his June 1640 letter to Mersenne.
      
    • Note that (i) is trivial while (ii) and (iii) are special cases of Fermat's Little Theorem.
      

26. Ten classics
    
    • Perhaps the most important mathematics which is included in the Zhoubi suanjing is related to the Gougu rule, which is the Chinese version of the Pythagoras Theorem.
      
    • The Haidao suanjing shows how to use the Gougu theorem (Pythagoras theorem) to calculate heights of objects and distances to objects which cannot be measured directly.
      
    • This, of course, is important for it is a problem which is solved using the Chinese remainder theorem.
      

27. Set theory
    
    • Cantor now remarks that this proves a theorem due to Liouville, namely that there are infinitely many transcendental (i.e.
      Go directly to this paragraph
      
    • This theorem was also proved by Felix Bernstein and independently by E Schroder.
      Go directly to this paragraph
      
    • This theorem had been conjectured by Cantor.
      Go directly to this paragraph
      
    • Emile Borel pointed out that the Axiom of Choice is in fact equivalent to Zermelo's Theorem.
      Go directly to this paragraph
      

28. Mathematical classics
    
    • Perhaps the most important mathematics which is included in the Zhoubi suanjing is related to the Gougu rule, which is the Chinese version of the Pythagoras Theorem.
      
    • The Haidao suanjing shows how to use the Gougu theorem (Pythagoras theorem) to calculate heights of objects and distances to objects which cannot be measured directly.
      
    • This, of course, is important for it is a problem which is solved using the Chinese remainder theorem.
      

29. Nine chapters references
    
    • J W Dauben, The "Pythagorean theorem" and Chinese mathematics : Liu Hui's commentary on the gou-gu theorem in Chapter Nine of the J'iu zhang suan shu', in Amphora (Basel, 1992), 133-155.
      
    • Z J Liang, From the elimination method in the Jiu zhang suanshu (Arithmetic in nine sections) to automated theorem proving (Chinese), J.
      
    • D B Wagner, A proof of the Pythagorean theorem by Liu Hui (third century AD), Historia Math.
      

30. Prime numbers references
    
    • J Echeverria, Observations, problems and conjectures in number theory-the history of the prime number theorem, in The space of mathematics (Berlin, 1992), 230-252.
      
    • L J Goldstein, A history of the prime number theorem, Amer.
      
    • W Schwarz, Some remarks on the history of the prime number theorem from 1896 to 1960, in Development of mathematics 1900-1950 (Basel, 1994), 565-616.
      

31. Bourbaki 2
    
    • This instalment of Bourbaki's super-textbook gives a notable account of Rolle's theorem and Taylor's theorem with remainder; of the indefinite integral, as anti-derivative, for a function having only discontinuities of the first kind, such a function being a uniform limit of a function which is constant by intervals, an "interval" being an open interval or a point; of Cauchy limits for such integrals; of integration and differentiation with respect to a parameter under the integral sign; and of the elementary logarithmic, exponential and trigonometric functions.
      
    • There were attempts at homotopy theory, at spectral theory of operators, at the index theorem, at symplectic geometry.
      

32. Prime numbers references
    
    • J Echeverria, Observations, problems and conjectures in number theory-the history of the prime number theorem, in The space of mathematics (Berlin, 1992), 230-252.
      
    • L J Goldstein, A history of the prime number theorem, Amer.
      
    • W Schwarz, Some remarks on the history of the prime number theorem from 1896 to 1960, in Development of mathematics 1900-1950 (Basel, 1994), 565-616.
      

33. Calculus history
    
    • In fact, although Barrow never explicitly stated the fundamental theorem of the calculus, he was working towards the result and Newton was to continue with this direction and state the Fundamental Theorem of the Calculus explicitly.
      
    • He also calculated areas by antidifferentiation and this work contains the first clear statement of the Fundamental Theorem of the Calculus.
      

34. Bolzano's manuscripts references
    
    • A Canada and S Villegas, Bolzano's theorem in several variables? (Spanish), Gac.
      
    • S B Russ, A translation of Bolzano's paper on the intermediate value theorem, Historia Math.
      

35. Bolzano's manuscripts references
    
    • A Canada and S Villegas, Bolzano's theorem in several variables? (Spanish), Gac.
      
    • S B Russ, A translation of Bolzano's paper on the intermediate value theorem, Historia Math.
      

36. Bourbaki 1
    
    • The names for the theorems were taken from French generals, and the final and most ridiculous theorem he presented he had named "Bourbaki's theorem", taking the name from General Bourbaki.
      

37. Orbits
    
    • Liouville made a number of very important mathematical discoveries while working on the theory of perturbations including the discovery of Liouville's theorem "when a bounded domain in phase space evolves according to Hamilton's equations its volume is conserved".
      Go directly to this paragraph
      
    • In 1890 Poincare proved his famous recurrence theorem, namely that in any small region of phase space trajectories exist which pass through the region infinitely often.
      Go directly to this paragraph
      

38. U of St Andrews History
    
    • In February 1671 he discovered Taylor's theorem (not published by Taylor until 1715), and the theorem is contained in a letter sent to Collins on 15 February 1671.
      Go directly to this paragraph
      

39. Ledermann interview
    
    • Then he came to this theorem, that the sum of the angles of the triangle is exactly 180 degrees, and this should be true for any triangle at any time in the future.
      
    • He would be walking along dreaming that one day he would prove Herr Fermat's theorem, or this or that other conjecture.
      

40. Real numbers 1
    
    • Usually when Euclid wants to illustrate a theorem about magnitudes he gives a diagram representing the magnitude by a line segment.
      
    • Although this theorem is valid, nevertheless we cannot recognise by such experience the incommensurability of two given magnitudes.
      

41. Fair book
    
    • He then finds the third side of the triangle using Pythagoras's theorem, and computes the area using Heron's formula.
      

42. Knots and physics
    
    • In fact it is not, but for alternating knots, it is an invariant and this fact is a consequence of Tait's second conjecture (a theorem since 1993).
      

43. Group theory
    
    • He defines isomorphism of permutation groups and proves the Jordan-Holder theorem for permutation groups.
      Go directly to this paragraph
      

44. Kepler's Laws
    
    • Because of its importance the proof has been reproduced several times [Mathematical Gazette 82, no.493 (1998), p.42.',2)" onmouseover="window.status='Click to see reference';return true">2]; though modernized in style, and reordered - to increase its impact - clearly it has not been altered in substance, since it relies on nothing more 'advanced' than Pythagoras' Theorem, and properties of similar triangles.
      

45. Nine chapters
    
    • The first 13 problems are solved using an application of Pythagoras's theorem, which the Chinese knew as the Gougu rule.
      

46. Indian mathematics
    
    • The Hindu astronomers did not possess a general method for solving problems in spherical astronomy, unlike the Greeks who systematically followed the method of Ptolemy, based on the well-known theorem of Menelaus.
      

47. Special relativity
    
    • Einstein called it a theorem that if two synchronous clocks C1 and C2 start at a point A and C2 leaves A moving along a closed curve to return to A then C2 will run slow compared with C1.
      

48. Golden ratio
    
    • The definition appears in Book VI but there is a construction given in Book II, Theorem 11, concerning areas which is solved by dividing a line in the golden ratio.
      

49. Topology history
    
    • Weierstrass in 1877 in a course of unpublished lectures gave a rigorous proof of the Bolzano-Weierstrass theorem which states .
      Go directly to this paragraph
      

50. Babylonian mathematics
    
    • In our article on Pythagoras's theorem in Babylonian mathematics we examine some of their geometrical ideas and also some basic ideas in number theory.
      

51. Mathematics and Art
    
    • The main theorem in Taylor's theory of linear perspective is that the projection of a straight line not parallel to the plane of the picture passes through its intersection and its vanishing point.
      

52. Mathematical games
    
    • This is the book which Fermat was reading when he inscribed the margin with his famous Last Theorem.
      Go directly to this paragraph
      

53. Elliptic functions references
    
    • M Rosen, Abel's theorem on the lemniscate, Amer.
      

54. Ten classics references
    
    • L Wang, The date of the Sunzi suanjing and the Chinese remainder theorem, in Proc.
      

55. Perfect numbers references
    
    • C M Taisbak, Perfect numbers : A mathematical pun? An analysis of the last theorem in the ninth book of Euclid's Elements, Centaurus 20 (4) (1976), 269-275.
      

56. Ring Theory references
    
    • Edwards, H.M., Fermat's Last Theorem, (Berlin 1977).
      

57. Chinese overview references
    
    • K S Shen, Historical development of the Chinese remainder theorem, Arch.
      

58. Orbits references
    
    • S R Valluri, C Wilson and W Harper, Newton's apsidal precession theorem and eccentric orbits, J.
      

59. Planetary motion
    
    • Applying Pythagoras' theorem to ΔAPH, we derive: .
      

60. Greek sources II
    
    • The whole work is written in a language used for conics which predated Apollonius's contributions except for one theorem, Proposition 8, where Apollonius's terminology is used.
      

61. Fractal Geometry
    
    • In fact, it is "asymptotically similar to Julia sets near any point on its boundary," as proved in a theorem by the Chinese mathematician Tan Lei.
      

62. Maxwell's House
    
    • Other than the topics of Maxwell's described above by Tait, there are also manuscripts by Tait on Vanishing Fractions which is l'Hopital's rule, a manuscript on Maclaurin's Theorem and On the imaginary roots of negative quantities by the Rt Rev Terrot.
      Go directly to this paragraph
      

63. Elliptic functions references
    
    • M Rosen, Abel's theorem on the lemniscate, Amer.
      

64. Quantum mechanics history
    
    • In 1859 Gustav Kirchhoff proved a theorem about blackbody radiation.
      Go directly to this paragraph
      

65. Infinity
    
    • This is merely a modern phrasing of what Euclid actually stated as his theorem which, according to Heath's translation, reads:- .
      

66. Real numbers 3
    
    • Godel proved some striking theorem in 1930.
      

67. The number e
    
    • He used the binomial theorem to show that the limit had to lie between 2 and 3 so we could consider this to be the first approximation found to e.
      

68. Ten classics references
    
    • L Wang, The date of the Sunzi suanjing and the Chinese remainder theorem, in Proc.
      

69. Mathematical classics references
    
    • L Wang, The date of the Sunzi suanjing and the Chinese remainder theorem, in Proc.
      

70. Perfect numbers references
    
    • C M Taisbak, Perfect numbers : A mathematical pun? An analysis of the last theorem in the ninth book of Euclid's Elements, Centaurus 20 (4) (1976), 269-275.
      

71. Ring Theory references
    
    • Edwards, H.M., Fermat's Last Theorem, (Berlin 1977).
      

72. Chinese overview references
    
    • K S Shen, Historical development of the Chinese remainder theorem, Arch.
      

73. Orbits references
    
    • S R Valluri, C Wilson and W Harper, Newton's apsidal precession theorem and eccentric orbits, J.
      

74. Trigonometric functions
    
    • De Moivre published his famous theorem .
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75. History overview
    
    • Realising that there are abstract properties 2 and 3 which apply to every sets with 2 and 3 elements and that 2 + 3 = 5 is a general theorem which applies whether they are sets of apples, books or trees moves from counting into the realm of mathematics.