‎Mathématiques‎

‎[MENCKE, FRIEDRICH OTTO [EDT.]].‎

‎Nova Acta Eruditorum Anno MDCCXXXII. [1732].‎

‎Leipzig, Gleditsch & Martini, 1732. 4to. The entire volume offered in contemporary full vellum. Hand written title on spine. A yellow label pasted on to top of spine. A small stamp to title-page and free front end-paper. Library label to pasted down front free end-paper. As usual with brownspotting. (6), 594, (32) pp + 6 plates.‎

‎First issue of Nova Acta Eruditorum with Friedrich Otto Mencke as editor, the grandson of the founder of the journal, Otto Mencke. The offered volume contain many reviews by influential contemporary mathematicians, philosophers and historians.‎

Référence libraire : 46350

‎[BEL, KARL ANDREAS. [EDT.]].‎

‎Nova Acta Eruditorum. Anno MDCCLIX.‎

‎Leipzig, Gleditsch & Lanckisiii, 1759. 4to. The entire volume offered in contemporary full vellum. Hand written title on spine. A yellow label pasted on to top of spine. A small stamp to title-page and free front end-paper. Library label to pasted down front free end-paper. As usual with brownspotting. (4), 720, (30) pp + 1 plates.‎

‎The offered volume contain many reviews by influential contemporary mathematicians, philosophers and historians.‎

Référence libraire : 46352

‎"JACOBI, C.G.J.‎

‎Über die unmittelbare verification einer Fundamentalformel der Theorie der elliptischen Functionen (+) Über die partielle Differentialgleichung, welcher die Zähler und Nenner der elliptischen Functionen.‎

‎Berlin, G. Reimer, 1848. 4to. In the original wrappers, no backstrip. In ""Journal für die reine und angewandte Mathematik. Hrsg. von A.L. Crelle, 36. Band, Erstes [1] Heft"". Entire issue offered. A fine and clean copy. Pp. 75-89. [Entire volume: IV, (2), 96, (2) pp.].‎

‎First printing of these two papers influential papers by Jacobi published in the epoch-making year 1848.""In the revolutionary year of 1848 Jacobi became involved in a political discussion in the Constitutional Club. During an impromptu speech he made some imprudent remarks which brought him under fire from monarchists and republicans alike. Hardly two years before, in the dedication of volume I of his Opuscula mathematica to Friedrich Wilhelm IV, he had expressed his royalist attitude"" now he had become an object of suspicion to the government. A petition of Jacobi’s to become officially associated with the University of Berlin, and thus to obtain a secure status, was denied by the ministry of education. Moreover, in June 1849 the bonus on his salary was retracted. Jacobi, who had lost his inherited fortune in a bankruptcy years before, had to give up his Berlin home. He moved into an inn and his wife and children took up residence in the small town of Gotha, where life was considerably less expensive."" (DSB).‎

Référence libraire : 46355

‎[MENCKE, OTTO [EDT.]].‎

‎Actorum Eruditorum. Supplementa, tomus II [2].‎

‎Leipzig, Grosse & Fritschium, 1696. 4to. The entire volume offered in contemporary full vellum. Hand written title on spine. A yellow label pasted on to top of spine. A small stamp to title-page and free front end-paper. Library label to pasted down front free end-paper. As usual with various browning to leaves and plates. (2), 603 pp. + 10 engraved folded plates.‎

‎The offered volume contain many papers by influential contemporary mathematicians, philosophers and historians.‎

Référence libraire : 46396

‎"TSCHIRNHAUS, EHRENFRIED W. V. [FIRST PUBLICATION OF THE ""TSCHIRNHAUS TRANSFORMATION"".]‎

‎Methodus datae figurae, rectis lineis & curva Geometrica terminate, aut Quadraturam, aut impossibilitatem ejusdem Quadraturae determinandi. (+) Nova Methodus determinandi Maxima & Minima (+) Aufrendi omnes terminos intermedios ex data aequatione. - [FORCED LEIBNIZ TO PUBLISH THE CALCULUS]‎

‎Leipzig, Grosse & Gleditsch, 1683. 4to. Contemporary full vellum. Handwritten title on spine. Library label to pasted down front free end-paper and a small stamps on titlepage. In: ""Acta Eruditorum Anno MDCLXXXIII"". As usual with various browning to leaves and plates. Tschirnhaus' paper: pp. 122-124" Pp. 204-207" Pp. 433-437. [Entire volume: (8), 561, (7) pp + 13 plates].‎

‎First appearance of Tschirnhaus's three exceedingly important papers which were to to initiate one of the most famous mathematical discoveries. In the papers he used infinitisimal methods which were very close to Leibniz's method and where he tried to lay down criteria for rational quadratures in the case of conic, cubic and quadratic curves, papers that led Leibniz to publish his first paper on the differential calculus, the ""Nova Methoda"" in the Acta for 1684 in order to secure his priority over Tschirnhaus concerning the calculus. Leibniz discovered, when he read Tschirnhaus' papers, that Tschirnhaus had here published results showing similarity with Leibniz's invention of the calculus as he had confided to Tschirnhaus earlier, during their Parisian stay, and this without references to Leibniz.The present volume of Acta also contain the first edition of Tschirnhaus' ""Tschirnhaus Tranformation"". Tschirnhaus work intensively on finding a general method for solving equations of higher of higher degree. ""His transformations constituted the most promising contribution to the solution of equations during the seventeenth century" but his elimination of the second and third coefficients by means of such transformation was far from adequate for the solution of the quintic.(Boyer. A History of Mathematics, 1968, 472 p.).Tschirnhaus (1651-1708) , a Saxon nobleman, had as wide interest as acquaintances: He studied in Leyden, served in the Dutch army, visited England and Paris several times. He set up a glassworks in Italy and is said to have introduced Porcelain to Europe. He wrote about philosophy and mathematics and was a close friend of Leibniz.‎

Référence libraire : 46399

‎"HERMITE, C.‎

‎Note sur la réduction des fonctions homogénes á coefficients entiers et á deux indéterminées.‎

‎Berlin, G. Reimer, 1848. 4to. In ""Journal für die reine und angewandte Mathematik, 1848. Band, 4 Heft"". In the original printed wrappers, without backstrip. Fine and clean. [Hermite:] Pp. 357-364. [Entire volume: Pp. 275-364 + 3 folded plate.].‎

‎First printing.Charles Hermite (1822-1901), French mathematician, made significant contributions to pure mathematics, and especially to number theory and algebra. In 1858 he solved the equation of the fifth degree by elliptic functions, and in 1873 he proved that e (the base of natural logarithms) is transcendental. The legacy of his work can be shown in the large number of mathematical terms which bear the adjective 'Hermitian'.‎

Référence libraire : 46404

‎"JACOBI, C. G. J.‎

‎Elementarer Beweis einer merkwürdigen analytischen Formel, nebst einigen aus ihr folgenden Zahlensätzen.‎

‎Berlin, G. Reimer, 1840. 4to. In ""Journal für die reine und angewandte Mathematik, 1848."". Without backstrip. Fine and clean. Pp.: 13-32 [Entire volume: 13-100 pp.].‎

‎First printing of Jacobi's proof an analytical formula. ""Most of Jacobi's fundamental research articles in the theory of elliptic functions, mathematical analysis, number theory, geometry, and me-chanics were published in Crelle's Journal fue die reine and angewandte Mathematik. With an average of three articles per volume, Jacobi was one of its most active contributors and quickly helped to establish its international fame."" (DSB)‎

Référence libraire : 46406

‎"RICHELOT, F.‎

‎De integralibus quibusdam definitis, quorum summa ad quadraturam divisionemque circuli revocatur.‎

‎Berlin, G. Reimer, 1840. 4to. In ""Journal für die reine und angewandte Mathematik, 21. band, 4. heft."". Entire issue offered in the original printed wrappers, without backstrip. Fine and clean. Pp. 293-327 [Entire volume: Pp. 293-380, (2) + 1 folded plate.].‎

‎First printing.Richelot was a student of Jacobi. Richelot authored numerous publications among them - with his 1832 dissertation - the first known guide to the Euclidean construction of the regular 257-gon with compass and straightedge.‎

Référence libraire : 46408

‎"BABBAGE, CHARLES.‎

‎On some new Methods of investigating the Sums of several Classes of infinite Series. Read April 1, 1819.‎

‎(London, W. Bulmer & Co.), 1819. 4to. No wrappers as issued in ""Philosophical Transactions of the Royal Society of London for the Year MDCCCXIX."", pp. 249-282.‎

‎First edition, the journal printing, of one of the importent works in pure mathematics, which occupied Babbage from 1815-20. The item offered is the authors 5th work.""Babbage brings his great powers of inventiveness to this subject (i.e. infinite series) but like many of his contemporaries, appeared insensitive to problems arising out of convergence, and accepted unquestionably several absurd results."" (Dubbey 1979, pp. 135-36). - Hook and Norman No 24.‎

Référence libraire : 46465

‎"HURWITZ, A.‎

‎Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. - [HURWITZ DETERMINANT]‎

‎Leipzig, B. G. Teubner, 1895. 8vo. In the original printed wrappers, no backstrip. In ""Mathematische Annalen. Begründet durch Alfred Clebsch and Carl Neumann. 46. Band. 2. Heft."" Entire issue offered. Internally very fine and clean. Pp. 273-285. [Entire volume: Pp. 161-320].‎

‎First printing of Hurwitz's important paper in which he presented the ""Hurwitz determinant"" for the first time. They are used to give a criterion for all roots of a polynomial to have negative real part.‎

Référence libraire : 46504

‎[CLEBSCH, ALFRED (+) CARL NEUMANN. EDT.].‎

‎Mathematische Annalen. Begründet durch Alfred Clebsch and Carl Neumann. 51. Band. 2. Heft. 1898.‎

‎Leipzig, B. G. Teubner, 1898. 8vo. In the original printed wrappers, no backstrip. Entire issue offered. Internally very fine and clean. Pp. 161-480‎

‎First printing.‎

Référence libraire : 46506

‎"LIPSCHITZ, R.‎

‎Beiträge zur Theorie der Variation der einfachen Integrale.‎

‎Berlin, G. Reimer, 1865. 4to. In the original printed wrappers, without backstrip. In ""Journal für die reine und angewandte Mathematik, 65 Band, 1. Heft, 1865"". Minor browning, otherwise fine. Pp. 24-41. [Entire volume: IV, 96, (2) pp..].‎

‎First printing of Lipschitz's paper on integrals. ""At the age of fifteen he began the study of mathematics at the University of Königsberg, where Franz Neumann was teaching. He then went to Dirichlet in Berlin, and he always considered himself Dirichlet’s student. After interrupting his studies for a year because of illness, he received his doctorate from the University of Berlin on 9 August 1853. After a period of taining and teaching at the Gymnasiums in Königsberg and Elbing, Lipschitz became a Privatdozent in mathematics at the University of Berlin in 1857. In the same year he married Ida Pascha, the daughter of a neighboring landowner. In 1862 he became an associate professor at Breslau, and in 1864 a full professor at Bonn, where he was examiner for the dissertation of nineteen-year-old Felix Klein in 1868. He rejected an offer to succeed Clebsch at Göttingen in 1873. Lipschitz was a corresponding member of the academies of Paris, Berlin, Göttingen, and Rome.Lipschitz distinguished himself through the unusual breadth of his research. He carried out many important and fruitful investigations in number theory, in the theory of Bessel functions and of Fourier series, in ordinary and partial differential equations, and in analytical mechanics and potential theory."" (DSB)‎

Référence libraire : 46521

‎"CAMUS, CHARLES ÈTIENNE LOUIS. - THE GEAR OF TIMEKEEPERS.‎

‎Sur la Figure des Dents des Roues, et des Aisles des Pignons, pour rendre les Horloges plus parfaites.‎

‎(Paris, L'Imprimerie Royale, 1735). 4to. Without wrappers. Extracted from ""Mémoires de l'Academie des Sciences. Année 1733"". Pp. 117-140 and 4 folded engraved plates. Fine and clean.‎

‎By this paper Camus was the first to work out the mathematical theory of gearteeth into a systematical and general theory of the mechanism.‎

Référence libraire : 46584

‎"OZANAM, (JACQUES).‎

‎L'Usage du Compas de Proportion, expliqué et demontré d'une maniere courte & facile, & augmenté d'un Traité de la division des Champs. Derniere édition, corrigée & augmentée.‎

‎Paris, Charles-Antoine Jombert, 1736. 8vo. Contemp. full vellum. 150,(2) pp. and 1 folded engraved plate, many textdiagrams. Leaves somewhat browned.‎

‎Ozanam's work on the proportional compas - first printed in 1688 - became very popular and successfull and it was lastly printed in 1794.""By almost any criterion Ozanam cannot be regarded as a first-rate mathematician, even of his own time. But he had a flair for writing and during his career wrote a number of books, some of which were very popular, passing through many editions.""(DSB).‎

Référence libraire : 46755

‎KARSTEN, WENCESL. JOHANN GUSTAV.‎

‎Auszug aus den Anfangsgründen und dem Lehrbegriffe der Mathematischen Wissenschaften. Die zweyte Auflage. 2 Bde.‎

‎Greifswald, Anton Ferdinand Röse, 1785. Small8vo. 2 contemp. hcalf. Gilt spines and titlelabels with gilt lettering. Spines slightly rubbed. XVIII,(3),456(6),445,(19) pp. and 25 folded engraved plates. 2 plates with a small hole, taped. Some scattered brownspots.‎

‎Deals with Arithmetics, Geometry, Trigonometry, Algebra, Statics, Hydraulics, Perspective, Machine theory, Artillery, Optics, Dioptrics, Catroptics, Astronomy, Gnomonics, Civil- and military Architecture. - Karsten was a pupil of Leonhard Euler.‎

Référence libraire : 46847

‎HAMILTON, WILLIAM ROWAN.‎

‎On Quaternions" or on a new System of Imaginaries in Algebra. - [MAKING THE GENERAL THEORY OF RELATIVITY POSSIBLE]‎

‎London, Richard and John Taylor, 1844. Contemp. hcalf. Gilt lettering to spine ""Philosophical Magazine"" - Vol. XXV. In: ""The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Conducted by David Brewster et al."". Vol. XXV. A stamp to titlepage and a few other pages. Entire volume offered (July-December 1844). VIII,552 pp., textillustr. Hamilton's paper: pp. 10-13, 141-145 and 241-246.‎

‎First printing of this landmark paper in which Hamilton published his creation of a new algebra of quaternions (a noncommunicative algebra), a turning point in the development of mathematics and a discovery which made possible the creation of the general theory of relativity. His algebra was later to form the basis of quantum mechanics and for the proper understanding of the atom.""Gauss had treated imaginary numbers in combination with real ones as representing points on a plane and showed the methods by which such complex numbers could be manipulated. Hamilton tried to extend this to threee dimensions and found himself unable to work out a self-consistent method of multiplication, until it occurred to him that the cummutative law of multiplication need not necessarily hold. It is taken for granted that A times B is equal to B times A... and this is an example of what seems to be an eternal and inescapable truth. Hamilton, however, showed that he could built up a logical algebra for his quaternions only when B times A was not made to equal - A times B. This seems against common sense but, like Lobachevski, Hamilton showed that the truth is relative and depends on the axioms you choose to accept.""(Asimov).The creation of quaternions is one of the famous moments in the history of mathematics. ""The quaternions came to Hamilton in one of those flashes of understanding that occasionally occur after long deliberation on a problem. He was walking into Dublin on 16 October 1843 along the Royal Canal to preside at a meeting of the Royal Irish Academy, when the discovery came to him. As he described it, ""An electric circuit seemed to close.""18 He immediately scratched the formula for quaternion multiplication on the stone of a bridge over the canal. His reaction must have been in part a desire to commemorate a discovery of capital importance, but it was also a reflection of his working habits. Hamilton was an inveterate scribbler. His manuscripts are full of jottings made on walks and in carriages. He carried books, pencils, and paper everywhere he went. According to his son he would scribble on his fingernails and even on his hard-boiled egg at breakfast if there was no paper handy.""(DSB).Hamilton later developed his invention in his book from 1853 ""Lectures on Quaternions"" - see PMM: 334 and Grattan-Guiness ""Landmark Writings in Western Mathematics 1640-1940"", pp. 460 ff.In this volume other importent papers by Gassiot, Sylvester, Joule, Draper.‎

Référence libraire : 46875

‎"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).‎

Référence libraire : 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).‎

Référence libraire : 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.‎

Référence libraire : 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.‎

Référence libraire : 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.‎

Référence libraire : 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.‎

Référence libraire : 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.‎

Référence libraire : 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.‎

Référence libraire : 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).‎

Référence libraire : 47134