BEZOUT, (ETIENNE). - BEZOUT'S THEOREMBiblio.com
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1779ref. 46989Paris Ph.-D. Pierres 1779. 4to. Nice recent vellum titlelabel with gilt lettering on spine. 4XXVIII471 pp. Wide-margined clean and fine. First edition of Bezout's main work - a fundamental contribution to algebraic geometry - in which he prooved the so called ´Bezout's theorem. The theorem was essentially stated by Isaac Newton in his proof of lemma 28 of volume 1 of his Principia where he claims that two curves have a number of intersection points given by the product of their degrees.Bézout's theorem is a statement in algebraic geometry concerning the number of common points or intersection points of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees. The work stimulated many investigations in the modern theory of elimination including Cauchy’s refinements of elimination procedure and Sylvester’s work on resultants and inertia forms. Bezout’s theorem is crucial to the study of the intersection of manifolds in algebraic geometry."It was not until 1779 that Bezout published his Théorie des équations algébriques his major work on elimination theory. Its best-known achievement is the statement and proof of Bezout’s theorem: "The degree of the final equation resulting from any number of complete equations in the same number of unknowns and of any degrees is equal to the product of the degrees of the equations." Bezout following Euler defined a complete polynomial as one that contains each possible combination of the unknowns whose degree is no more than the degree of the polynomial. Bezout also computed that the degree of the resultant equation is less than the product of the degrees for various systems of incomplete equations. Here we shall consider only the complete case.The proof makes one marvel at the ingenuity of Bezout who like Euler not only could manipulate formulas but also had the ability to choose those manipulations that would be fruitful. He was compelled to justify his nth-order results by a naive "induction" from the observed truth of the statements for 1 2 3 ···. Also numbered subscripts had not yet come into use and the notations available were clumsy."DSB. hardcover
1841ref. 49105Paris Bachelier 1841. 4to. Without wrappers. In "Comptes rendus hebdomadaires des séances de l’Académie des sciences" Vol. XII No 6. Pp. 267- 316. Entire issue offered. Cauchy's paper: pp. 283-298. Some scattered brownspots. First printing of an importent paper in information theory - the paper stating the earliest version of what will later be known as the "Nyquist Sampling Theorem" describing how many and what kind of samples are needed to construct a curve."The theorem will be formulated more completely in 1928 and become one of the cornerstones of information theory" Bryan Bunch 1841 M. unknown
1931ref. 48257Easton PA. Mack printing Compagny 1931. Royal8vo. Contemp. full cloth. Spine gilt and with gilt lettering. In: "Proceedings of the National Academy of Sciences of the United States of America" Vol. 17. VII710 pp. Entire volume offered. The papers: pp. 315-318 650-655 and 656-660. First editions of these importent papers in statistical mechanics. The so-called Koopman-von Neumann mechanics is a description of classical mechanics in terms of Hilbert space introduced by Bernard Koopman the paper offered and John von Neumann in 1931 and 1932. Ergodicity was introduced by Boltzmann but the modern theory started from the paper by Koopman and has been a cornerstone of statistical mechanics since. The ergodic method has found impressive applications in the fields of statistical mechanics number theory probability theory harmonic analysis and combinatorics.As Koopman and von Neumann demonstrated a Hilbert space of complex square integrable wavefunctions can be defined in which classical mechanics can be formulated as an operatorial theory similar to quantum mechanics.Birkhoff's proof in the third paper offered of "the ergodic theorem was deemed as importent as his proof of Poincare's geometric theorem" Landmarks Writing in Western Mathematics 1640-1940 p. 877. hardcover
1750ref. 42900Petropoli St. Petersbourg 1750. 4to. Uncut without wrappers. Extracted from "Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae" Tom. I. ad Annum 1747 et 1748. Pp. 3-19 a. 1 engraved plate. and pp. 20-48. First printing of both papers. The second is important as it contains Euler'is second proof of the Euler-Fermat theorem which Euler presents as a consequence of the theorem that abp = apbp mod p. This paper also includes results about possible divisors of a2n b2n and Euler uses this to show again that F5 is not prime. - Enestroem No. 133 a. 134. unknown
1858ref. 42295London Richard Taylor and William Francis 1858 and Taylor and Francis 1866. 4to. No wrappers as extracted from "Philosophical Transactions" Vol. 148 - Part I. Pp. 17-37 and Vol. 156 - Part I Pp. 25-35. Clean and fine. First appearance of this outstanding contribution to mathematics announcing his invention and developments of the ALGEBRA OF MATRICES what is now called the Cayley-Hamilton theorem for square matrices of any order. "The subject originated in a memoir of 1858 the paper offered and grew directly out of simple observations on the way in which the transformations linear of the theory of algebraic invariants are combined.a distinctive feature of these rules is that multiplication is not commutative.we get different results according to the order in which we do the multiplication. it seems about as far from anything of scientific or practical use as anything could possible be. Yet sixty seven years after Cayley's invented it HEISENBERG in 1925 recognized in the algebra of matrices exactly the tool which he neede for his revolutionary work in QUANTUM MECHANICS."Bell Men of Mathematics."It was in connection with the study of invariants under linear transformation that Cayley first introduced matrices to simplify the notation involved. Here he gave some basic notions. This was followed by his first major paper on the subject "A Memoir on the Theory of Matrices." the paper offered here. Kline Mathematical Thought.p. 806. unknown
1873ref. 38036Paris: Gauthier-Villars 1873. 4to. 282x225mm. Entire volume 1628 pp. offered here in original blank wrappers unopened. An exceptionally fine copy. 4 parts First edition of Hermite's epoch-making memoir in which he proved the transcendence of e and thus initiated a new era in number theory. A decade later Lindemann used the method of Hermite's work to establish the transcendence of pi. unknown
1879ref. 41917London Edward Stanford 1879. Without wrappers in "Proceedings of the Royal Geographical Society and monthly Record of Geography" April issue with titlepage to vol. 1 1879. Pp.2 225-288 a. 2 folded maps. Cayley's paper: pp. 259-261 Fitrst appearance of Cayley's famous paper on the Four-Colour-Problem"The four-colour map problem to prove that on any map only four colours are needed to separate countries is celebrated in mathematics. It resisted the attempts of able mathematicians for over a century and when it was successfully proved in 1976 the ‘computer proof’ was controversial: it did not allow scrutiny in the conventional way. At the height of his influence in 1878 Arthur Cayley had drawn attention to the problem at a meeting of the London Mathematical Society and it was duly ‘announced’ in print. the paper offered. He made a short contribution himself and he encouraged the young A. B. Kempe to publish a paper on the subject. Though ultimately unsuccessful the work of Cayley and Kempe in the late 1870s brought valuable insights. Francis Galton is revealed as the ‘go-between’ in suggesting Cayley publish his observations in Proceedings of the Royal Geographical Society." Tony Crilly.The Four-Colour-Theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major mathematical theorem to be proved using a computer. unknown
1900ref. 51494Paris Gauthier-Villars 1900. 4to. No wrappers. In: "Comptes Rendus Hebdomadaires des Séances de L'Academie des Sciences" Tome 131 No 24. Pp. 975- 1017. Entire issue offered. Fejér's here spelled Téjer ! paper: pp. 984-987. Clean and fine. First printing of this importent paper in which Féjer states the "Summation Theorem" that bears his name."Fejér’s main works deal with harmonic analysis. His classic theorem on C 1 summability of trigonometric Fourier series 1900 not only gave a new direction to the theory of orthogonal expansions but also through significant applications became a starting point for the modern general theory of divergent series and singular integrals. Through a Tauberian theorem of G. H. Hardy’s the convergence theory of Fourier series was considerably affected by Fejér’s theorem as well; it is closely connected with Weierstrass’ approximation theorems and with the more advanced theory of power series and harmonics potential theory and makes possible a number of analogues for related series such as Laplace series."DSB. unknown
1770ref. 41596Berlin Haude et Spener 1770. 4to. No wrappers as issued in "Mémoires de l'Academie Royale des Sciences et Belles-Lettres" Tome V pp. 203-221 1 plate and pp. 222-288 1 engraved plate. Both papers first edition. The first paper is Euler's discussion of "Cramers Paradox" and it contains his inventions of 2 kinds of curves "Cusps of first kind" or keratoid cusp and "Cups of second kind" or ramphoid cusp. - Enestroem E 169.The second paper contains Euler's famous proof of "The fundamental Theorem of Algebra". - Enestroem E 170. unknown
1832ref. 41657Berlin G. Reimer 1832. 4to. Without wrappers. Extracted from "Journal für die reine und angewandte Mathematik. Hrsg. von A.L. Crelle" 9. Bd. pp. 99-104. First printing. unknown
1907ref. 48911Paris Gauthier-Villars 1907. 4to. No wrappers. In: "Comptes Rendus Hebdomadaires des Séances de L'Academie des Sciences" Tome 144 No 11 No. 19 and No. 21. Pp. 593- 664 pp. 1009- 1080. pp. 1137- 1192.3 entire issues offered.Reesz' paper: pp. 615-619. Fischer's paper: pp. 1022-1024 a. 1148-51. Nos 19 a. 21 with some small tears to outher margins. paper fragile. Sewing loose. First apperance of two fundamental papers - Riesz setting forth the theorem and Fischer proving it - the mathematics of which later made it clear that there is an equivalence between matrix mechanics Heisenberg and wave mechanics Schrödinger in quantum physics.The Riesz-Fischer theorem of 1907 concerning the equivalence of the Hilbert space of sequences of convergent sums of squares with the space of functions of summable squares formed the mathematical basis for demonstrating the equivalence of matrix mechanics and wave mechanics. unknown
1780ref. 44931Paris Moutard Panckoucke 1780. 4to. Extract from "Mémoires fe Mathematique et de Physique Présentés à l'Academie des Sciences par divers Savans" Tome IX. Pp. 593-624 and 2 folded engraved plates. Clean and fine. First appearance of an importent papwer in the history of analytic geometry."In this article Tinseau gave an interesting generalization of the Pythagorean theorem for space of three dimensions: the square of the area of a plane surface is equal to the sum of the squares of the projection of this surface upon three mutually perpendicular coordinate planes.To Tinseau it appears that the use of the word "conoid" in the modern sense is due."Boyer "History of Analytic Geometry p. 207."Two of the three memoirs that constitute Tinseau’s oeuvre deal with topics in the theory of surfaces and curves of double curvature: planes tangent to a surface contact curves of circumscribed cones or cylinders various surfaces attached to a space curve the determination of the osculatory plane at a point of a space curve problems of quadrature and cubature involving ruled surfaces the study of the properties of certain special ruled surfaces particularly conoids and various results in the analytic geometry of space. In these two papers the equation of the tangent plane at a point of a surface was first worked out in detail the equation had been known since Parent methods of descriptive geometry were used in determining the perpendicular common to two straight lines in space and the Pythagorean theorem was generalized to space the square of a plane area is equal to the sum of the squares of the projections of this area on mutually perpendicular planes. DSB. Although Tinseau published very little his papers are of great interest as additions to Monge’s earliest works. Indeed Tinseau appears to have been Monge’s first disciple. unknown
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ref. 1334697620.Gpaperback. Good. Access codes and supplements are not guaranteed with used items. May be an ex-library book. paperback
