‎Mathematics‎

‎"KOENIG (KÖNIG), SAMUEL. - GIVING RISE TO ONE OF THE UGLIEST SCIENTIFIC CONTROVERSIES.‎

‎De universali Principio aequilibrii & motus, in Vi viva reperto, deque nexu inter Vim Vivam & Sctionem, utriusque Minimo, Dissertatio. (+) ...continuata & finita. (2 Parts).‎

‎Leipzig, Gleditsch & Lanckis, 1751. 4to. No wrappers. In: ""Nova Acta Eruditorum, Anno MDCCLI"", March- Issue, Pars I-II. Entire issue in 2 parts offered. With titlepage to the volume 1751. Pp. 97-192. Koenig's paper: pp. 125-135 a. pp. 162-176. With 2 engraved plates. Titlepage with 2 stamps and a bit soiled. Leaves as usual a bit browned.‎

‎First printing of this important paper in which Koenig set forth his ""Law of least Action"". The law states that the kinetic energy of a system of mass points is equal to the sum of the kinetic energy of the motion of the system relative to the center of gravity and of the kinetic energy of the total mass of the system considered as a whole, which moves as the center of gravity of the system.""While still in Franeker, Koenig wrote the draft of his important essay on the principle of least action, which was directed against Maupertuis. The controversy touched off by this work, which was published in March 1751, resulted in perhaps the ugliest of all the famous scientific disputes. Its principal figures were Koenig, Maupertuis, Euler, Frederick II, and Voltaire" and, as is well known, it left an unseemly stain on Euler’s otherwise untarnished escutcheon. The quarrel occupied Koenig’s last years almost completely" moreover, he had been ill for several years before it started. Koenig emerged the moral victor from this affair, in which all the great scientists of Europe—except Maupertuis and Euler—were on his side. The later finding of Kabitz2 testifies to Koenig’s irreproachable character.""(DSB).‎

Bookseller reference : 46906

‎"HILBERT, DAVID. - METAMATHEMATICS.‎

‎Die logischen Grundlagen der Mathematik.‎

‎Berlin, Julius Springer, 1923. 8vo. Full cloth. Spine gone. In: ""Mathematische Annalen begründet durch Alfred Clebsch und Carl Neumann."", 88. Bd. (4),312 pp. (Entire volume offered). Hilbert's paper: pp. 151-165. Internally clean and fine.‎

‎First edition as a continuation of his paper from 1922 ""Neubegründung der Mathematik. Erste Mitteilung"".""This articlee, delivered as a lecture to the deutsche Naturforscher Gesellschaft in Leipzig, September 1922, is a sequel to (neubegründung...), and brings Hilbert's proof theory to maturity. Hilbert here introduces several technical refinements and clarification to his theory. Specifically: (i) he improves the formal system by adding a special sign for formal negation...(ii) he refines his account of the distinction between formal language and the metalanguage....(iii) he outlines a consistency proof for an elementary, quantifier-free formal system of number-theory. (iv) he begins to extend his proof theory to analysis and set theory....sketches a strategy for proving the consistency of a version of Zermel's axiom of choice for real numbers...(etc. etc). (William Ewald in from Kant to Hilbert, vol. II, pp.1134-35).‎

Bookseller reference : 47067

‎"TURING, A. M.‎

‎Rounding-off Errors in Matrix Processes. - [TURING'S LU FACTORIZATION]‎

‎Oxford, Clarendon Press, 1948. 8vo. Bound in contemporary full calf with gilt lettering to spine. In ""The Quarterly Journal of Mechanics and Applied Mathematics"", Vol. 1, 1948. Previous owner's name written to front free-endpaper. Ver fine and clean. Pp. 287-380. [Entire volume: (4), 474 pp.].‎

‎First printing of this important paper in which Turing for the very first time introduced the concept of LU factorization or LU decomposition. ""Turing's paper was one of the earliest attempts to examine the error analysis of the various methods of solving linear equations and inverting matrices. His analysis was basically sound. The main importance of the paper was that it was published at the dawn of the modern computing era, and it gave indications of which methods were 'safe' when solving such problems on a computer"". (Burgoyne, Collected Works of A M Turing).""In 1945, [Turing] declined an offer of a Fellowship at King's [College, Cambridge] in favour of joining the newly formed Mathematical Division at the National Physical Laboratory (NPL). His early work on computability, combined with his wartime experience in electronics, had fired him with an enthusiasm for working on the design of an electronic computer. ethe machine he designed, which was called the Automatic Computing Engine (ACE) in recognition of Babbage's pioneering work, was characteristically original…""While in the Mathematics Division of NPL, Turing became keenly interested in numerical analysis. His paper, ""Rounding-off Errors in Matrix Processes"", showed that the acute anxiety about the effect of rounding errors in Gaussian elimination was largely unjustified. This paper has been overshadowed to some extent by the von Neumann and Goldstine paper on matrix inversion, but it is a brilliant piece of work and would have repaid closer study at the time"". (""Turing, Alan M."" by James H. Wilkinson, p. 1803, in Encyclopedia of Computer Science, A. Ralston et al (eds.), 4th edition, Nature Publishing Group, 2000).In linear algebra, LU decomposition factorizes a matrix as the product of a lower triangular matrix and an upper triangular matrix. LU decomposition is a key step in several fundamental numerical algorithms in linear algebra such as solving a system of linear equations, inverting a matrix, or computing the determinant of a matrix. Not in Origins of Cyberspace nor The Erwin Tomash Library.‎

Bookseller reference : 47095

‎"HILBERT, DAVID.‎

‎Ueber einen allgemeinen Gesichtspunkt für invariantentheorietische Untersuchungen im binären Formengebiete.‎

‎Leipzig, B.G. Teubner, 1887. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 28., 1887. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of verso of title page. Very fine and clean. Pp. 381-446. [Entire volume: Pp. IV, 600.]‎

‎First printing of Hilbert's ""Habilitationsschrift"", a fundamental work on algebraic invariants. With this he meant to revolutionize the feld by several new methods thatplay no part in the 1888 proof but would reappear to some extent in Hilbert’s (1891-92" 1893) response to Gordan’s criticism. By that time Hilbert’s resultsplus further ones by Gordan would solve Gordan’s problem.‎

Bookseller reference : 47122

‎"KLEIN, FELIX.‎

‎Über hyperelliptische Sigmafunktionen.‎

‎Leipzig, B.G. Teubner, 1886. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 27., 1886. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp. 431-464 . [Entire volume: Pp. IV, 600.]‎

‎First printing of Klein's important work on hyperelliptic sigma functions.‎

Bookseller reference : 47125

‎"SCHEFFERS, GEORG.‎

‎Zurück-führung komplexer Zahlensysteme auf typische formen. - [MODERN ALGEBRA]‎

‎Leipzig, B. G. Teubner, 1891. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 39., 1891. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp. 293-390 . [Entire volume: IV, 600 pp. + 1 plate.]‎

‎First printing of Scheffers's important paper which contributed considerably to the development of modern algebra. Scheffers here distinguishes between a ""Nichtquaternion system"" (Nqss) and a Quaternion system (Qss). In today's language, Scheffers' Qss has the quaternion algebra as a subalgebra.Scheffers anticipates the concepts of direct product of algebras and direct sum of algebras with his section on reducibility, addition, and multiplication of systems. Thus Scheffers pioneered the structural approach to algebra. The volume contain several other papers by influential contemporary mathematicians.‎

Bookseller reference : 47127

‎"HARNACK, A.‎

‎Ueber die Vieltheiligkeit der ebenen algebraischen Curven. - [ANTICIPATION OF HILBERT'S SIXTEENTH PROBLEM]‎

‎Leipzig, B. G. Teubner, 1876. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 10., 1876. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp.189-198. [Entire volume: IV, 592 pp.].‎

‎First printing of Harnack's Curve Theorem which formed the background for Hilbert's sixteenth problem. In real algebraic geometry, Harnack's curve theorem, named after Axel Harnack, describes the possible numbers ofconnected components that an algebraic curve can have, in terms of the degree of the curve.Hilbert's sixteenth problem was posed as the ""Problem of the topology of algebraic curves and surfaces"", he presented his problem as follows: ""The upper bound of closed and separate branches of an algebraic curve of degree n was decided by Harnack (the present paper)"" from this arises the further question as of the relative positions of the branches in the plane.As of the curves of degree 6, I have - admittedly in a rather elaborate way - convinced myself that the 11 branches, that they can have according to Harnack, never all can be separate, rather there must exist one branch, which have another branch running in its interior and nine branches running in its exterior, or opposite. It seems to me that a thorough investigation of the relative positions of the upper bound for separate branches is of great interest, and similarly the corresponding investigation of the number, shape and position of the sheets of an algebraic surface in space - it is not yet even known, how many sheets a surface of degree 4 in three-dimensional space can maximally have."" (Rohn, Flächen vierter Ordnung, Preissschriften der Fürstlich Jablonowskischen Gesellschaft, Leipzig 1886).Hilbert had investigated the M-curves of degree 6, and found that the 11 components always were grouped in a certain way. His challenge to the mathematical community now was to completely investigate the possible configurations of the components of the M-curves.Furthermore he requested a generalization of Harnack's Theorem to algebraic surfaces and a similar investigation of surfaces with the maximum number of components.The problem is still unsolved today.‎

Bookseller reference : 47129

‎"KILLING, WILHELM.‎

‎Die Zusammensetzung der stetigen endlichen Transformationsgruppen.‎

‎Leipzig, B. G. Teubner, 1889. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 34., 1889. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp.57-122 [Entire volume: IV, 600 pp.].‎

‎First publication of Killing's important third paper (of a total of four) in which he laid the foundation of a structure theory for Lie algebras.""In particular he classified all the simple Lie algebras. His method was to associate with each simple Lie algebra a geometric structure known as a root system. He used linear transformation, to study and classify root systems, and then derived the structure of the corresponding Lie algebra from that of the root system.""(Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences)Unfortunately for Killing a myth arose that his work was riddled with error, which later has been proved untrue. ""As a result, many key concepts that are actually due to Killing bear names of later mathematicians, including ""Cartan subalgebra"", ""Cartan matrix"" and ""Weyl group"". As mathematician A. J. Coleman says, ""He exhibited the characteristic equation of the Weyl group when Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born.""The theory of Lie groups, after the Norwegian mathematician Sophus Lie, is a structure having both algebraic and topological properties, the two being related.‎

Bookseller reference : 47131

‎"BACHARACH, I.‎

‎Ueber den Cayley'schen Schnittpunktsatz. - [THE CAYLEY-BACHARACH THEOREM]‎

‎Leipzig, B. G. Teubner, 1886. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 26., 1886. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. With a marginal tear affecting pp. 275-282 and 2 cm of the text, otherwise very fine and clean. Pp. 275-299. [Entire volume: IV, 606 pp.].‎

‎First printing of Bacharach's paper which gave the final solution to what later was to be known as the Cayley-Bacharach Theorem: That when a plane curve of degree r is drawn through the mn points common to two curves of degrees m and n (both less than r), these do not count for mn conditions in the determination of the curve but for mn reduced by (m + n - r - 1) (m + n - r - 2).""Cayley wrote copiously on analytical geometry, touching on almost every topic then under discussion. Although, as explained elsewhere, he never wrote a textbook on the subject, substantial parts of Salmon’s Higher Plane Curves are due to him"" and without his work many texts of the period, such as those by Clebsch and Frost, would have been considerably reduced in size. One of Cayley’s earliest papers contains evidence of his great talent for the analytical geometry of curves and surfaces, in the form of what was often known as Cayley’s intersection theorem (C. M. P., I, no. 5 [1843], 25-27). There Cayley gave an almost complete proof (to be supplemented by Bacharach, in Mathematische Annalen, 26 [1886], 275-299)"" (DSB).‎

Bookseller reference : 47134

‎"BOIS-REYMOND, PAUL DAVID GUSTAV DU.‎

‎Über asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichunge. - [ANTICIPATING CANTOR'S ""DIAGONAL ARGUMENT"".]‎

‎Leipzig, B. G. Teubner, 1875. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 8., 1875. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Title page missing a small piece of paper to the right margin, not affecting text. Very fine and clean. Pp. 363-414. [Entire volume: IV, 576 pp.].‎

‎First printing of Paul du Bois-Reymond's important paper in which he anticipate Cantor's famous ""diagonal argument"". Although Cantor proved that the real numbers are uncountable one year earlier he did not find the much clearer diagonal argument until some years later.""In 1875, he described dense sets under the title 'pantachisch,' from the Greek for 'everywhere.' He later claimed, against Georg Cantor, priority in their discovery. In his textbook, Hobson awarded the laurels to du BoisReymond. Although Cantor presented his first diagonal proof to the public in On an elementary question of set theory [Cantor 1891], Paul du Bois-Reymond had been there well before him, having published a plainly diagonal argument in an 1875 article on approximation by in?nitesimals. [P. du Bois-Reymond 1875] Arguably his greatest invention was the Infinitärcalcül or infinitary calculus, an original, nonCantorean account of infinite and infinitesimal sizes as first-classentities that represent not the extents of collections but the rates of growth of real-valued functions."" (McCarty, David Hilbert and Paul du Bois-Reymond: Limits and Ideals).‎

Bookseller reference : 47136

‎"HURWITZ, ADOLF.‎

‎Über algebraische Gebilde mit Eindeutigen Transformationen in sich. - [HURWITZ SURFACE]‎

‎Leipzig, B. G. Teubner, 1893. 8vo. Bound with the original wrappers in contemporary half calf. In ""Mathematische Annalen"", Volume 41., 1893. Entire volume offered. Library label to upper part of spine. Extremities with wear, internally very fine and clean. Pp. 403-442. [Entire volume: IV, 604 pp.].‎

‎First printing of Hurwitz's paper on Riemann surface theory and hyperbolic geometry, today known as a Hurwitz surface" essentially a compact Riemann surface with precisely 84(g - 1) automorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms.The present volume contain several other papers by influential contemporary mathematicians.‎

Bookseller reference : 47139

‎"HELMHOLTZ, HERMANN von. - A SEMINAL PAPER ON HYDRODYNAMICS.‎

‎Ueber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen,‎

‎Berlin, Georg Reimer, 1858. 4to. Extracted from ""Journal für die reine und angewandte Mathematik Crelle/Borchardt"", 55. Band, 1. Heft.: 3. Pp. 25-55 and with titlepage to volume 55. A bit of browning to outer corners of titlepage.‎

‎Firsi printing of Helmholtz' seminal paper on the hydrodynamics of vortex motion, which is the first detailled analysis of fluid motion not constrained to being irrotational - being his most importent contribution to mathematical physics. The paper was the direct inspiration for J.J. Thomson's ""vortex atom"" (See Pais ""Inward Bound"", pp. 176 ff).""In 1857, in a work of genius that proved him to be a mathematician of first rank 'On the Integrals of the Hydrodynamic Equations which express Vortex-motion' (the paper offered), he gave the solution of some extremely difficult hydrodynamical problems. He rejected the earlier hypotheses, and followed up the analogies between the motion of fluids and the electromagnetic action of electrical currents, which were of such much importence for his subsequent work on the Theory of Electricity and Magnetism.""Leo Koenigsberger, p. 167 ff.)""In 1858 Helmholtz published his seminal memoir ""Ueber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen,"" important for both its physical results and its mathematical methods. His motivations for taking up this new research interest remain unclear. One motive seems, however, to have been his interest in frictional phenomena, carried over from his interest in energetics"" another was his growing awareness of the power of Green’s theorem.""(DSB).Parkinson ""Breakthroughs"", 1858 P.‎

Bookseller reference : 47146

‎"PEARSON, KARL. - INTRODUCING THE ""CHI-SQUARED GOODNESS-OF-FIT TEST""‎

‎On the Criterion that a given System of Deviations from the Probable in the Case of a Correlated System of Variables is such that it can be reasonably supposed to have arisen from Random Sampling.‎

‎London, Taylor and Francis, 1900. Contemp. hcalf, spine gone and covers loose. In: ""The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science"", Vol. 50, Fifth Series. VI,(2),624 pp. a. 5 plates. (Entire volume offered). Pearson's paper: pp. 157-175. A stippled stamp on titlepage. Internally clean and fine.‎

‎In this fundamental paper in statistics, Pearson introduced his chi-squared test, the statistical procedure whose results are evaluated by reference to the chi-squared distribution, or the formula yielding a measure of how well a set of observations fits a theoretical hypothesis, the test of goodness of fit. A founding seminal paper in statistical testing theory.""Pearson’s many contributions to statistical theory and practice, many contributions to statistical theory and practice, this X2 text for goodness of fit is certainly one of his greatest"" and in its original and extended forms it has remained one of the most useful of all statistical tests."" (DSB).Parkinson ""Breakthroughs"" 1900 M.‎

Bookseller reference : 47148

‎"KLEIN, FELIX.‎

‎Ueber die Transformation elfter Ordnung der elliptischen Functionen. (i.e. ""On the eleventh order transformation of elliptic functions""). - [BELYI'S THEOREM]‎

‎Leipzig, B. G. Teubner, 1879. 8vo. In the original wrappers without backstrip. In ""Mathematische Annalen"", Volume 15, heft 3 + 4, 1879. Entire issue offered. Very fine and clean. Pp. 533-554. [Entire issue: 305-576 + 1 folded plate].‎

‎First printing of what later was to be known af Belyi's theorem or Belyi functions named after G. V. Belyi in 1979. Belyi functions and dessins d'enfants dates to the work of Felix Klein" he used them in this study an 11-fold cover of the complex projective line with monodromy group PSL.‎

Bookseller reference : 47155

‎"HILBERT, DAVID.‎

‎Über eine Darstellungsweise der invarianten Gebilde in binären Formengebiete (+) Ueber die Singularitäten der Discriminantenfläche (+) Ueber binäre Formenbüschel mit besonderen Combinanteneigenschaften.‎

‎Leipzig, B. G. Teubner, 1887. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 30, 1887. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp. 15-29" 437-441" Pp. 561-570. [Entire volume: IV, 596 pp.].‎

‎First printing of these early three papers by Hilbert. David Hilbert, one of the most influential mathematicians of the 19th and early 20th centuries, is probably best known for the ""Hilbert Problems"" - a list of twenty-three problems in mathematics all unsolved at the time, and several of them were very exceedingly influential for 20th century mathematics.He is regarded as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.""Hermann Weyl described his teacher Hilbert's style: ""It is as if you were on a swift walk through a sunny open landscape" you look freely around, demarcation lines and connecting roads are pointed out to you, before you must brace yourself to climb the hill" then the path goes straight up."" (Princeton Companion to Mathematics). The present volume contain several other papers by influential contemporary mathematicians.‎

Bookseller reference : 47158

‎"CANTOR, GEORG.‎

‎Über unendliche, lineare Punktmannichfaltigkeiten. - [THE THEORY OF SETS AND INFINITE SETS - THE SECOND PAPER]‎

‎Leipzig, B.G. Teubner, 1880. 8vo. Bound in a nice contemporary half calf with five raised gilt bands. Red leather title label with gilt lettering to spine. All edged gilt. In ""Mathematische Annalen"", Band 17, 1880. Entire volume offered. Corners with wear, otherwise a very fine and clean copy. Pp. 355-358. [Entire volume: IV, 576 pp.].‎

‎First printing of Cantor's important second paper of the landmark series consisting of a total of six papers which together constitute the foundation Theory of Sets (Mengenlehre) and Transfinite Set Theory. Cantor here introduces his new Set Theory with which he created an entirely new field of mathematical research and is widely regarded as being one of the most important mathematical conquests in the 19th century. ""Cantor's second paper of 1880 was brief. It continued the bricklaying work of the article of 1879, and it too sought to reformulate old ideas in the context of linear point sets. It also introduced for the first time an embryonic form of Cantor's boldest and most original discovery: the transfinite numbers. As a preliminary to their description, however, Cantor introduced several definitions. He also pointed out that first species sets could be completely characterized by their derived sets."" (Dauben, P. 80)Hilbert spread Cantor's ideas in Germany and praised Cantor's transfinite arithmetic as ""the most astonishing product of mathematical thought, one of the most beautiful realizations of human activity in the domain of the purely intelligible"". He is famously quoted for saying ""No one shall expel us from the paradise which Cantor created for us"". Bertrand Russel described Cantor's work as ""probably the greatest of which the age can boast"".""The major achievement of the ""Grundlagen"" was its presentation of the transfinite ordinal numbers as a direct extension of the real numbers. Cantor admitted that his new ideas might seem strange, even controversial, but he had reached a point in his study of the continuum where the new numbers were indispensable for further progress. Cantor had finally come to the realization that his 'infinite symbols' were not just indices for derived sets of the second species, but could be regarded as actual transfinite numbers that were just as real mathematically as the finite natural numbers."" (Grattan-Guinness, Landmark Writings in Western Mathematics, Pp. 604-5).Dauben: (Cantor)1880d.‎

Bookseller reference : 47160

‎"SEGRE, CORRADO.‎

‎Etude des différentes surfaces du 4e ordre à conique double ou cuspidale (générale ou décomposée) considérées comme des projections de l'intersection de deux variétés quadratiques de l'espace à quatre dimensions. - [SEGRE SURFACES]‎

‎Leipzig, B.G. Teubner, 1884. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 24, 1884. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. Pp. 313-444. [Entire volume: IV, 592 pp.].‎

‎First printing of Segre's paper containing what would later be known as Segre Surfaces: An intersection of two quadrics in 4-dimensional projective space with In algebraic geometry. The term ""Segre surface"" is also occasionally used for various other surfaces, such as a quadric in 3-dimensional projective space, or the hypersurface‎

Bookseller reference : 47180

‎"PEANO, GIUSEPPE.‎

‎Demonstration de l’intégrabilité des équations différentielles ordinaires. - [PEANO EXISTENCE THEOREM]‎

‎Leipzig, B.G. Teubner, 1890. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 37, 1890. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Fine and clean. Pp. 182-228. [Entire volume: IV, 604 pp.].‎

‎First printing of Peano's correct and complete Existence Theorem. Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations. The theorem is also known as Peano existence theorem, Peano theorem or Cauchy-Peano theorem.‎

Bookseller reference : 47182

‎"CANTOR, GEORG (+) WILHELM KILLING.‎

‎Bemerkung mit Bezug auf den Aufsatz: Zur Weierstrass'-Cantor'schen Theorie der Irrationalzahlen (+) Die Zusammensetzung der stetigen endlichen Transformationsgruppen.‎

‎Leipzig, B. G. Teubner, 1889. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 33., 1889. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Very fine and clean. P. 476"" Pp. 1-48. [Entire volume: IV, 604 pp.].‎

‎First printing of CANTOR'S important comment to Illigens paper from the same year: ""Zur Weierstrass'-Cantor'schen Theorie der Irrationalzahlen"". He states that: ""The squareroot of 3 is thus only a symbol for number which has yet to to be found, but is not its definition. The definitions is, however, satisfactorily given by my method as, say (1.7, 1.73, 1.732, ...). [From the present paper]. First publication of KILLING'S important second paper (of a total of four) in which he laid the foundation of a structure theory for Lie algebras.""In particular he classified all the simple Lie algebras. His method was to associate with each simple Lie algebra a geometric structure known as a root system. He used linear transformation, to study and classify root systems, and then derived the structure of the corresponding Lie algebra from that of the root system.""(Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences)Unfortunately for Killing a myth arose that his work was riddled with error, which later has been proved untrue. ""As a result, many key concepts that are actually due to Killing bear names of later mathematicians, including ""Cartan subalgebra"", ""Cartan matrix"" and ""Weyl group"". As mathematician A. J. Coleman says, ""He exhibited the characteristic equation of the Weyl group when Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born.""The theory of Lie groups, after the Norwegian mathematician Sophus Lie, is a structure having both algebraic and topological properties, the two being related.‎

Bookseller reference : 47188

‎"BRILL, ALEXANDER VON (+) MAX NOETHER.‎

‎Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie. - [THE BRILL-NOETHER THEORY]‎

‎Leipzig, B. G. Teubner, 1874. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 7., 1874. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Title-page and index-page with nicks and repairs. Otherwise fine and clean. Pp. 269-316. [Entire volume: IV, 638 pp.].‎

‎First printing of this important paper in which the Brill-Noether theory was presented for the first time. In the theory of algebraic curves, Brill-Noether theory, is the study of special divisors, certain divisors on a curve C that determine more compatible functions than would be predicted. In classical language, special divisors move on the curve in a ""larger than expected"" linear system of divisors.‎

Bookseller reference : 47190

‎"ABEL, NIELS HENRIK.‎

‎Mathematische Bruchstücke aus Herrn N. H. Abels Briefen.‎

‎(Berlin, G. Reimer, 1830). 4to. No wrappers. Extracted from""Journal für die reine und angewandte Mathematik. Hrsg. von A.L. Crelle"", Bd. 5, Fine and clean. Pp. 336-343. [Offered pages: Pp. 319-418.].‎

‎First printing.‎

Bookseller reference : 47199

‎"HILBERT, DAVID.‎

‎Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem). Dem Andenken an Hermann Minkowski Gewidmet. - [Hilbert-Kamke problem]‎

‎Leipzig, B. G. Teubner, 1909. 8vo. Bound in contemporary half calf with gilt lettering to spine. In ""Mathematische Annalen"", 67 band. 1909. Bookplates to pasted down front free end-paper and library stamp to verso of title page. Top half of spine is detached. Bookblock, however, still firmly attached. Fine and clean. Pp. 281-300. [Entire volume: IV, 575 pp.].‎

‎First printing of a groundbreaking work in Number Theory. Edward Waring (1734-98) stated, in his ""Meditationes Algebraicae"" (1770), the theorem known now as ""Waring's Theorem"", that every integer is either a cube or the sum of at most nine cubes"" also every integer is either a fourth power of the sum of at most 19 fourth powers. He conjectured also that every positive integer can be expressed as the sum of at most r kth powers, the r depending on k. These theoremes were not proven by him, but by David Hilbert in the paper offered.Hilbert proves that for every integer n, there exists an integer m such that every integer is the sum of m nth powers. This expands upon the hypotheis of Edward Waring that each positive integer is a sum of 9 cubes (n=3, m=9) and of 19 fourth powers (n= 4, m=19).This issue also contains F. Hausdorff's ""Zur Hilbertschen Lösung des Waringschen Problems"", pp. 301-305.(Se Kline p. 609).‎

Bookseller reference : 47248

‎"STEINER, J.‎

‎Sur le maximum et le minimum des figures dans le plan, sur la sphére et dans l'espace en général. Second mémoire.‎

‎(Berlin, G. Reimer, 1842) 4to. As extracted from ""Journal für die reine und angewandte Mathematik, 34 Band, Heft, 1842"". Without wrappers and backstrip. Fine and clean. Pp.189-250.‎

‎First printing of Swiss mathematician Steiner's ""outstandingly clear and brilliant exposition"" on the sphere in general (Klein, Development of Mathematics in the 19th Century P 118).""Students and contemporaries wrote of the brilliance of Steiner's geometric research and of the fiery temperament he displayed in leading others into the new territory he had discovered. Having set himself the task of reforming geometry, Steiner sought to discover simple principles from which many seemingly unrelated theorems in the subject could be deduced in a natural way."" (DSB)‎

Bookseller reference : 47252

‎"JACOBI, C.G.J.‎

‎De motu puncti singularis. - [SOLUTION TO THE INVERSE-SQUARE PROBLEM]‎

‎Berlin, G. Reimer, 1842. 4to. No wrappers. In ""Journal für die reine und angewandte Mathematik. Hrsg. von A.L. Crelle, 24. Band, erstes Heft"". No backstrip. A fine and clean copy. Pp. 5-27. [Entire issue: (6), 99 pp.].‎

‎First printing of this important memoire by Jacobi in which Jacobi presented his solution to the inverse-square problem. ""In a memoir written in Latin, Jacobi published in 1842 a simple, concise, and elegant solution of the inverse-square problem. This note argues that Jacobi's solution has not attracted the attention that it deserves and that it should prove valuable to teachers and students in introductory mechanics courses. A translation of the excerpt in which Jacobi describes his solution is given, and the method is then recast in the now more familiar language of vectors."" (Gauthier)‎

Bookseller reference : 47254

‎"JACOBI, C. G. J.‎

‎De usu legitimo formulae summatoriae Maclaurinianae.‎

‎Berlin, G. Reimer, 1834. 4to. As extracted from ""Journal für die reine und angewandte Mathematik, 12 Band, 1834"". Fine and clean. Pp. 263-72.‎

‎First printing of Jacobi's short but highly important paper in which he prooved Faulhaber's formula.‎

Bookseller reference : 47256