BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY CONTINUATION OF THE BULLETIN OF THE NEW YORK MATHEMATICAL SOCIETY. A HISTORICAL AND CRITICAL REVIEW OF MATHEMATICAL SCIENCE EDITED BY F.N. COLE, ALEXANDER ZIWET, F. MORLEY, E.O. LOVETT, C.L. BOUTON, D.E. SMITH. VOL. VIII. OCTOBER 1901 TO JULY 1902. PUBLISHED FOR THE SOCIETY BY THE MACMILLAN COMPANY, LANCASTER, PA., AND NEW YORK, 1902. MATHEMATICAL PROBLEMS LECTURE DELIVERED BEFORE THE INTERNATIONAL CONGRESS OF MATHEMATICIANS AT PARIS IN 1900. BY PROFESSOR DAVID HILBERT. [Pg 1] CONTENTS PROBLEM PAGE 1. CANTOR'S PROBLEM OF THE CARDINAL NUMBER OF THE CONTINUUM. 10 2. THE COMPATIBILITY OF THE ARITHMETICAL AXIOMS. 11 3. THE EQUALITY OF THE VOLUMES OF TWO TETRAHEDRA OF EQUAL BASES AND EQUAL ALTITUDES. 13 4. PROBLEM OF THE STRAIGHT LINE AS THE SHORTEST DISTANCE BETWEEN TWO POINTS. 13 5. LIE'S CONCEPT OF A CONTINUOUS GROUP OF TRANSFORMATIONS WITHOUT THE ASSUMPTION OF THE DIFFERENTIABILITY OF THE FUNCTIONS DEFINING THE GROUP. 15 6. MATHEMATICAL TREATMENT OF THE AXIOMS OF PHYSICS. 18 7. IRRATIONALITY AND TRANSCENDENCE OF CERTAIN NUMBERS. 19 8. PROBLEMS OF PRIME NUMBERS. 20 9. PROOF OF THE MOST GENERAL LAW OF RECIPROCITY IN ANY NUMBER FIELD. 21 10. DETERMINATION OF THE SOLVABILITY OF A DIOPHANTINE EQUATION. 22 11. QUADRATIC FORMS WITH ANY ALGEBRAIC NUMERICAL COEFFICIENTS. 22 12. EXTENSION OF KRONECKER'S THEOREM ON ABELIAN FIELDS TO ANY ALGEBRAIC REALM OF RATIONALITY. 22 13. IMPOSSIBILITY OF THE SOLUTION OF THE GENERAL EQUATION OF THE 7TH DEGREE BY MEANS OF FUNCTIONS OF ONLY TWO ARGUMENTS. 25 14. PROOF OF THE FINITENESS OF CERTAIN COMPLETE SYSTEMS OF FUNCTIONS. 26 15. RIGOROUS FOUNDATION OF SCHUBERT'S ENUMERATIVE CALCULUS. 28 16. PROBLEM OF THE TOPOLOGY OF ALGEBRAIC CURVES AND SURFACES. 28 17. EXPRESSION OF DEFINITE FORMS BY SQUARES. 29 18. BUILDING UP OF SPACE FROM CONGRUENT POLYHEDRA. 30 19. ARE THE SOLUTIONS OF REGULAR PROBLEMS IN THE CALCULUS OF VARIATIONS ALWAYS NECESSARILY ANALYTIC? 32 20. THE GENERAL PROBLEM OF BOUNDARY VALUES. 34 21. PROOF OF THE EXISTENCE OF LINEAR DIFFERENTIAL EQUATIONS HAVING A PRESCRIBED MONODROMIC GROUP. 34 22. UNIFORMIZATIOM OF ANALYTIC RELATION'S BY MEANS OF AUTOMORPHIC FUNCTIONS. 35 23. FURTHER DEVELOPMENT OF THE METHODS OF THE CALCULUS OF VARIATIONS. 36 MATHEMATICAL PROBLEMS[1] Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?