‎Ueber den Dirichlet'schen biquadratischen Zahlenkörper. - [THE THEORY OF QUADRATIC NUMBER FIELDS]‎

‎Leipzig, B. G. Teubner, 1894. 8vo. Bound with the original wrappers in contemporary half calf. In ""Mathematische Annalen"", Volume 45., 1894. Entire volume offered. Library label to upper part of spine. Extremities with wear, internally very fine and clean. Pp. 309-340. [Entire volume: IV, 599 pp.].‎

‎First printing of Hilbert's influential paper on the theory of quadratic number fields. Hilbert's work enabled mathematicians to attack successfully the theory of quadratic forms with any number of variables and with any algebraic numerical coefficients. This lead in particular to the interesting problem: to solve a given quadratic equation with algebraic numerical coefficients in any number of variables by integral or fractional numbers belonging to the algebraic realm of rationality determined by the coefficients.The present volume contain several other papers by influential contemporary mathematicians. ‎

‎Ueber die Darstellung definiter Formen als Summen von Formenquadraten.‎

‎Leipzig, B.G. Teubner, 1888. 8vo. Original printed wrappers, no backstrip. In ""Mathematische Annalen. Begründet durch Rudolf Friedrich Alfred Clebsch. XXXII. [32] Band. 3. Heft."" Entire issue offered. [Hilbert:] Pp. 342-50. [Entire issue: Pp. 309-456].‎

‎First publication of Hilbert's fundamental and exceedingly important paper on real algebraic geometry. ""In 1888, David Hilbert published an influential paper [the present] which became fundamental for real algebraic geometry, and which remains an inspiring source for research even today."" (Pfister & Scheiderer). 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). ‎

‎Ueber die Darstellung definiter Formen als Summen von Formenquadraten.‎

‎Leipzig, B.G. Teubner, 1888. 8vo. Bound in recent full black cloth with gilt lettering to spine. In ""Mathematische Annalen"", Volume 32., 1888. 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. 342-350. [Entire volume: Pp. IV-600.]‎

‎First publication of Hilbert's fundamental and exceedingly important paper on real algebraic geometry. ""In 1888, David Hilbert published an influential paper [the present] which became fundamental for real algebraic geometry, and which remains an inspiring source for research even today."" (Pfister & Scheiderer). 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 volume contain several other papers by influential contemporary mathematicians such as Felix Klein, Hurwitz, Lie, Lilienthal and Peano.‎

‎Ueber die Endlichkeit des Invariantensystems für binäre grundformen (+) Ueber Büschel von binären Formen mit vorgeschriebener Functionaldeterminante. - [""WITHOUT DOUBT THIS IS THE MOST IMPORTANT WORK ON GENERAL ALGEBRA""]‎

‎Leipzig, B.G. Teubner, 1888. 8vo. Original printed wrappers, no backstrip and a small nick to front wrapper. In ""Mathematische Annalen. Begründet 1888 durch Rudolf Friedrich Alfred Clebsch. XXXIII.[33] Band. 2. Heft."" Entire issue offered. Internally very fine and clean. [Hilbert:] Pp. 223-6"" Pp.227-36 [Entire issue: Pp. 161-316].‎

‎First printing of Hilbert's exceedingly important and groundbreaking paper in which he proved his famous Basis Theorem that is, if every ideal in a ring R has a finite basis, so does every ideal in the polynomial ring R[x]. Hilbert had thus connected the theory of invariants to the fields of algebraic functions and algebraic varieties. When Felix Klein read the paper he wrote ""I do not doubt that this is the most important work on general algebra that the Mathematische Annalen has ever published.""Hilbert submitted a paper proving the finite basis theorem to Mathematische Annalen. However Gordan was the expert on invariant theory for the journal and he found Hilbert's revolutionary approach difficult to appreciate. He refereed the paper and sent his comments to Klein:""The problem lies not with the form ... but rather much deeper. Hilbert has scorned to present his thoughts following formal rules, he thinks it suffices that no one contradict his proof ... he is content to think that the importance and correctness of his propositions suffice. ... for a comprehensive work for the Annalen this is insufficient.""Gordan rejected the article. His - now famous - comment was: Das ist nicht Mathematik. Das ist Theologie. (i.e. This is not Mathematics. This is Theology).However, Hilbert had learnt through his friend Hurwitz about Gordan's letter to Klein and Hilbert wrote himself to Klein in forceful terms:""... I am not prepared to alter or delete anything, and regarding this paper, I say with all modesty, that this is my last word so long as no definite and irrefutable objection against my reasoning is raised.""At the time Klein received these two letters from Hilbert and Gordan, Hilbert was an assistant lecturer while Gordan was the recognised leading world expert on invariant theory and also a close friend of Klein's. However Klein recognised the importance of Hilbert's work and assured him that it would appear in the Annalen without any changes whatsoever, as indeed it did. Hilbert expanded on his methods in a later paper, again submitted to the Mathematische Annalen [1893] and Klein,after reading the manuscript, wrote to Hilbert saying:-I do not doubt that this is the most important work on general algebra that the Annalen has ever published.Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself said: ""I have convinced myself that even theology has its merits"".(Klein. Development of mathematics in the 19th century. P. 311).Sometimes Hilbert's first publication of the Basis Theorem is referred to as being published in the paper ""Zur Theorie der algebraischen Gebilde"" in Göottinger Nachrichten in 1888. This, however, was published in December 1888 and the present issue was published in March. ‎

‎Ueber die gerade Linie als kürzeste Verbindung zweier Punkte. (Aus einem an Herrn F. Klein gerichteten Briefe). - [FIRST PUBLICATION OF THE HILBERT METRIC]‎

‎Leipzig, B.G. Teubner, 1895. 8vo. Original printed wrappers, no backstrip. In ""Mathematische Annalen. Begründet durch Alfred Clebsch und Carl Neumann. 46. Band. 1. Heft.""Entire issue offered. Internally very fine and clean. [Hilbert:] Pp. 91-96. [Entire issue: IV, 160 pp].‎

‎First printing of Hilbert's groundbreaking paper in which ""Hilbert's Metric"" (or Hilbert's projective metric) - and the metric in general - was introduced. The Hilbert metric an a closed convex cone that can be applied to various purposed in non-Euclidean geometryThe usefulness of Hilbert's metric were made clear in 1957 by Garrett Birkhoff who showed that the Perron-Frobenius theorem for non-negative matrices and Jentzch's theorem for integral operators with positive kernel could both be proved by an application of the Banach contraction mapping theorem in suitable metric spaces. (Serrin. Hilbert's Matric. P. 1). ‎

‎Ueber die reellen Züge algebraischer Curven.‎

‎Leipzig, B.G. Teubner, 1891. 8vo. Original printed wrappers, no backstrip. In ""Mathematische Annalen. Begründet durch Alfred Clebsch und Carl Neumann. 38. Band. 1. Heft."".(Entire issue offered). Titlepage to Bd. 38. IV,160 pp. a. 1 folded plate. Hilbert's paper: pp. 115-138.‎

‎First apperance of an importent paper on algebraic geometry.The issue contains also Felix Klein ""Ueber Normierung der linearen Differentialgleichungen zweiter Ordnung"", pp. 144-152.‎

‎Ueber die Theorie des relativquadratischen Zahlkörpers. - [A MAIN WORK ON ALGEBRAIC NUMBER-THEORY]‎

‎Leipzig, B.G. Teubner, 1898. Orig. printed wrappers, no backstrip. In: ""Mathematische Annalen begründet durch Alfred Clebsch und Carl Neumann."", 51. Bd., 1. Heft. The whole issue offered (=Heft 1). IV,160 pp. Hilbert's paper pp. 1-127.‎

‎First edition of Hilbert's famous report on algebraic numbers.""The work on algebraic number theory was climaxed in the nineteenth century by Hilbert's famous report on algebraic numbers. This report is primarely an account of what had been done during the century. However Hilbert reworked all of this earlier theory and gave a new, elegant and powerfull methods of securing these results. He had begun to create new ideas in algebraic number theory from about 1892 on and one of the new creations on Galoisian number fields was also incorporated in the report."" (Morris Kline in ""Mathematical Thoughts..."" pp. 825).‎

‎Ueber die Vollen Invariantensysteme.‎

‎Leipzig, B.G. Teubner, 1893. 8vo. Original printed wrappers, no backstrip. In ""Mathematische Annalen. Begründet 1893 durch Alfred Clebsch und Carl Neumann. 42. Band. 3. Heft."" Entire issue offered. [Hilbert:] Pp. 314-73. [Entire issue: Pp. 314-604].‎

‎First printing of Hilbert's fundamental landmark paper in which he ""INTRODUCED STUNNING NEW IDEAS WHICH HAVE DEEPLY INFLUENCED THE DEVELOPMENT OF MODERN ALGEBRA AND ALGEBRAIC GEOMETRY."" (Buchberger. Gröbner bases and applications. P. 63). The ideas presented in the present paper was introduced in his 1890-paper, but here he ""called attention to the fact that his earlier results failed to give any idea of how a finite basis for a system of invariants could actually be construted. [...] To show how these drawbacks could be overcome, Hilbert thus adopted an even more general standpoint [...]. He described the guiding idea of this culminating paper of 1893 as invariants could actually be constructed"". (Hendricks. Proof theory: history and philosophical significance. P. 59) Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem. Twenty years earlier, Paul Gordan had demonstrated the theorem of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the exceedingly complicated calculations involved. Hilbert realized that it was necessary to take a completely different path. Hilbert sent his results to the Mathematische Annalen. Gordan, the expert on the theory of invariants for the Mathematische Annalen, did not appreciate the revolutionary nature of Hilbert's theorem and rejected the article. His - now famous - comment was: Das ist nicht Mathematik. Das ist Theologie. (i.e. This is not Mathematics. This is Theology).Klein, on the other hand, recognized the importance of the work immediately, and guaranteed that it would be published without the slightest alterations. Encouraged by Klein and by the comments of Gordan, Hilbert extended his method in a second article, providing estimations on the maximum degree of the minimum set of generators, and he sent it once more to the Annalen. After having read the manuscript, Klein wrote to him, saying: ""WITHOUT DOUBT THIS IS THE MOST IMPORTANT WORK ON GENERAL ALGEBRA that the Annalen has ever published.""Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself said: ""I have convinced myself that even theology has its merits"".(Klein. Development of mathematics in the 19th century. P. 311).‎

‎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.‎

‎Zur Variationsrechnung. - [HILBERT'S 23RD PROBLEM]‎

‎Leipzig, B. G. Teubner, 1906. 8vo. In the original printed wrappers, without backstrip. In ""Mathematische Annalen, 62. Band., 3. Heft., 1906."". A fine and clean copy. Pp. 351-370. [Entire issue: Pp. 329-448.].‎

‎First printing of Hibert's important paper in which he addressed a number of topics in the Calculus of Variations and thereby extended the ideas given in his account of the 23rd problem. In contrast with Hilbert's other 22 problems, his 23rd is not so much a specific ""problem"" as an encouragement towards further development of the calculus of variations. His work in the present paper led to a modern definition of the field.‎

‎Über den Begriff der Klasse von Differentialgleichungen.‎

‎Leipzig, B.G. Teubner, 1912. 8vo. Bound in half cloth with the original wrappers. Marbled boards. In ""Mathematische Annalen. Herausgegeben von A. Clebsch und C. Neumann. 73. Band, Heft 1-4, 1912"". Black leather title label to spine with gilt lettering. Library label pasted on to top of spine and library stamp to title page. Light writing in pencil to front wrapper. [Hilbert:] Pp. 95-108. [Entire issue: IV, (2), 599, (1)].‎

‎First printing of Hilbert's paper on on the concept of the class of differential equations.The issue contain many other papers by contemporary mathematicians.‎

‎Über die Transcendenz der Zahlen e und pi. - [ANTICIPATION OF HILBERT'S SEVENTH PROBLEM]‎

‎Leipzig, B.G. Teubner, 1893. 8vo. Original printed wrappers, no backstrip and a small nick to front wrapper. In ""Mathematische Annalen. Begründet 1868 durch Rudolf Friedrich Alfred Clebsch. 43. Band. 2. und 3. (Doppel-)Heft.""Entire issue offered. Internally very fine and clean. [Hilbert:] Pp. 216-19. [Entire issue: Pp. 145-456].‎

‎First publication of Hilbert's important contribution to transcendental number theory which anticipates Hilbert's seventh problem, the seventh of twenty-three problems proposed by Hilbert in 1900 which became of seminal importance to 20th century mathematics. A transcendental number is a number which is not algebraic-that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are pi and e. Euler was the first person to define transcendental numbers - The name ""transcendentals"" comes from Leibniz in his 1682 paper where he proved sin x is not an algebraic function of x.‎

‎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. ‎

‎Über ternäre definite Formen.‎

‎(Berlin, Stockholm, Paris, 1893). 4to. Without wrappers as extracted from ""Acta Mathematica. Hrsg. von G. Mittag-Leffler."", Vol. 17, pp. 169-197.‎

‎First edition. This importent paper constitutes Hilbert's own version of part of the solution of his ""Mathematische Probleme"" listed at the International mathematical Congress in Paris 1900 (as problem 17) on the ""Expression of definite forms by squares"". ""....But since as I have shown, not every definite form can be compounded by addition from squares of forms, the question arises - which I have answered affirmatively for ternary forms (the paper offered here)- whether every definite form may be expressed as a quotient of sums of squares of forms....""‎

‎Über ternäre definite Formen.‎

‎Stockholm, Beijer, 1893. 4to. As extracted from ""Acta Mathematica, 17. Band]. No backstrip. Fine and clean. Pp. 169-197.‎

‎First edition. This important paper constitutes Hilbert's own version of part of the solution of his ""Mathematische Probleme"" listed at the International mathematical Congress in Paris 1900 (as problem 17) on the ""Expression of definite forms by squares"". ""....But since as I have shown, not every definite form can be compounded by addition from squares of forms, the question arises - which I have answered affirmatively for ternary forms (the paper offered here)- whether every definite form may be expressed as a quotient of sums of squares of forms....""‎

‎Über ternäre definite Formen.‎

‎Stockholm, Beijer, 1893. 4to. Bound in contemporary half cloth with gilt lettering to spine. In ""Acta Mathematica"", Vol, 17, 1883. Entire volume offered. Stamps to title page, otherwise a fine and clean copy. Pp. 169-197. [Entire volume: (6), 416 pp].‎

‎First edition. This important paper constitutes Hilbert's own version of part of the solution of his ""Mathematische Probleme"" listed at the International mathematical Congress in Paris 1900 (as problem 17) on the ""Expression of definite forms by squares"". ""....But since as I have shown, not every definite form can be compounded by addition from squares of forms, the question arises - which I have answered affirmatively for ternary forms (the paper offered here)- whether every definite form may be expressed as a quotient of sums of squares of forms....""‎

‎Ueber einen allgemeinen Gesichtspunkt für invariantentheorietische Untersuchungen im binären Formengebiete.‎

‎Leipzig, B.G. Teubner, 1887. 8vo. Original printed wrappers, no backstrip. In ""Mathematische Annalen. Begründet durch Alfred Clebsch und Carl Neumann. 28. Band. 3. Heft."". (Entire issue offered). Pp. 309-456. Hilbert's paper: pp. 381-446.‎

‎This is Hilbert's ""Habilitationsschrift"", a fundamental work on algebraic invariants.‎

‎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).‎

‎Ueber die stetige Abbildung einer Linie auf ein Flächenstück.‎

‎Leipzig, B.G. Teubner, 1891. 8vo. Original printed wrappers, no backstrip. In ""Mathematische Annalen. Begründet durch Alfred Clebsch und Carl Neumann. 38. Band. 3. Heft."". (Entire issue offered). Pp. 315-460 a. 5 lithographed colourplates. Plates with a dampstain. The plates does not belong to Hilbert's paper). Hilbert's paper: pp. 459-460.‎

‎First appearace of Hilbert's importent papaer in which he introduces his space-filling curve or the Hilbert-Curve. He constructed an example of a curve that passed through every point of a square by a series of successive approximations. At each stage every square is divided into four equal smaller squares and the curve replaced.Although it was Peano [1890] that produced the first space-filling curves, it was Hilbert (in the paper offered) who first popularized their existence and gave an insight into their generation. Space-filling curves are commonly used to reduce a multidimensional problem to a one-dimensional problem" the curve is essentially a linear transversal of the discrete multidimensional space. ‎

‎Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl nter Potenzen (Waringische Problem).‎

‎Leipzig, B.G. Teubner, 1909. Orig. printed wrappers. No backstrip. In. ""Mathematische Annalen. Hrsg. von Felix Klein, Walther v. Dyck, David Hilbert, Otto Rosenthal"", 67. Bd., 3. Heft. Pp. 281-432 (=3. Heft). Hilbert's paper: pp. 281-300.‎

‎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).‎

‎Über die Theorie der relativ-Abel'schen Zahlkörper [Hilbert] (+) Sur les fonctions abéliennes [Poincaré].‎

‎Berlin, Stockholm, Paris, Beijer, 1902. 4to. Bound in contemporary half cloth with gilt lettering to spine. In ""Acta Mathematica"", Vol. 26, 1802. Entire volume offered. Stamps to title page, otherwise a fine and clean copy. pp. 99-132" Pp. 48-98. [Entire volume: (4), 400 pp.].‎

‎First printing of these two important papers on the theory of quadratic number fields.‎

‎Grundlagen der Mathematik. Erster Band.‎

‎Berlin, Springer, 1934. Lex8vo. Bound in boards with orig. printed wrappers withbound. XII,471 pp.‎

‎First edition. Volume one only.‎

‎Anschauliche Geometrie. Mit 330 Abbildungen.‎

‎Berlin, Julius Springer, 1932. Orig. full cloth. Small stamp to foot of titlepage. VIII,310 pp., textillustr. From the library of the Danish logician and philosopher Jørgen Jørgensen with his name on front free endpaper. Internally fine and clean.‎

‎First edition.‎

‎Grundzüge der theoretischen Logik. Dritte, verb. Aufl.‎

‎Berlin, Springer, 1949. Orig. full cloth. A few brownspots to covers. A small stamp on foot of titlepage. VIII,156 pp.‎

‎Grundzüge der theoretischen Logik. Vierte Auflage.‎

‎Berlin, Göttingen..., Springer-Verlag, 1959. Orig. full cloth. VIII,188 pp. A few underlinings and notes.‎