Geometry

Bibliography

BIBLIOGRAPHY

Euclidean geometry:

The Thirteen Books of Euclid's Elements, 3 vol. (1908; 2nd ed. rev., 1926), is the standard English translation, with extensive commentary by T.L. HEATH; DAVID HILBERT, Grundlagen der Geometrie (1899 and later editions; Eng. trans., The Foundations of Geometry, 1902), gives a logical account of Euclid's own methods and many new results. The later editions include revised reprints of very important papers of the author. The axiomatic approach now common in all branches of mathematics is due to the influence of this book and to the work of Giuseppe Peano and his school. HENRY G. FORDER, Foundations of Euclidean Geometry (1927), expounds in full detail the work of Hilbert, his followers, and of the American and Italian schools. H.S. MacDONALD COXETER and S.L. GREITZER, Geometry Revisited (1967), is a pleasant account of theorems in Euclidean geometry. For Euclidean geometry in its setting in general geometry, see HENRY G. FORDER, Geometry (1950); H.S. MacDONALD COXETER, Introduction to Geometry (1961); DANIEL PEDOE, A Course of Geometry for Colleges and Universities (1970); and HENRY P. MANNING, Geometry of Four Dimensions (1914), an excellent work. See also HAROLD ABELSON, Turtle Geometry (1981), computer programs for creating geometrical graphics. For transformation geometry, I.M. YAGLOM, Geometric Transformations (1962; orig. pub. in Russian, 1955), gives a good elementary introduction. Much more detailed is FRIEDRICH BACHMANN, Aufbau der Geometrie aus dem Spiegelungsbegriff (1959), based on papers by Hjelmslev. H.P. HUDSON, Ruler and Compasses (1916), is a thorough treatment of constructions. For the theory of congruence, see J.F. RIGBY, "Axioms for Absolute Geometry," Can. J. Math., 20:158-181 (1968). T. BONNESEN and W. FENCHEL, Theorie der konvexen Körper (1934), is excellent for the classical theory. For more recent developments the following books are representative: H.G. EGGLESTON, Convexity (1958); FREDERICK A. VALENTINE, Convex Sets (1964); and I.M. YAGLOM and V.G. BOLTYANSKY, Convex Figures (1961; orig. pub. in Russian, 1951). For an interesting collection of modern research papers, see the Proceedings of the Seventh Symposium in Pure Mathematics (1963), which includes "Helly's Theorem and Its Relatives," by L. DANZER, B. GRUNBAUM, and V. KLEE; and "Convex Curves of Constant Minkowski Breadth," by P.C. HAMMER. There exists a variety of books dealing with specialized topics in convexity. HERBERT BUSEMANN, Convex Surfaces (1958); and R.T. ROCKAFELLAR, Convex Analysis (1970), are typical. Additional references on convexity are: EDWIN F. BECKENBACH, "Convex Functions," Bull. Am. Math. Soc., 54:439-460 (1948); RUSSELL V. BENSON, Euclidean Geometry and Convexity (1966); L. FEJES TOTH, Lagerungen in der Ebene auf der Kugel und im Raum (1953); B. GRUNBAUM, Convex Polytopes (1967); HUGO HADWIGER, Altes und Neues über konvexe Körper (1955); HUGO HADWIGER and HANS DEBRUNNER, Kombinatorische Geometrie in der Ebene (1959; Eng. trans., Combinatorial Geometry in the Plane, 1964); HERMANN MINKOWSKI, Gesammelte Abhandlung (1911); EDWIN E. MOISE, Elementary Geometry from an Advanced Standpoint (1963); T. MOTZKIN, Beiträge zur Theorie der linearen Ungleichungen (1936); and CLAUDE A. ROGERS, Packing and Covering (1964). ( H.G.F./F.A.V.)

Analytic geometry:

D.M.Y. SOMMERVILLE, Analytical Geometry of Three Dimensions (1934); ALAN ROBSON, An Introduction to Analytical Geometry, 2 vol. (1940-47); H.S.M. COXETER, The Real Projective Plane, 2nd ed. (1955), and Introduction to Geometry, 2nd ed. (1969); NIKOLAI V. EFIMOV, An Elementary Course in Analytical Geometry, 2 vol. (1966; orig. pub. in Russian, 1962). HENRY F. BAKER, Principles of Geometry, 2nd ed., vol. 2 (1929); LUTHER P. EISENHART, Coordinate Geometry (1939); and ROSS R. MIDDLEMISS, Analytic Geometry, 3rd ed. (1968), are three general references on conic sections. For the history of the conic sections, see JULIAN L. COOLIDGE, A History of Conic Sections and Quadric Surfaces (1945); and THOMAS L. HEATH, A History of Greek Mathematics, 2 vol. (1921). For special curves, see JULIAN L. COOLIDGE, A Treatise on Algebraic Plane Curves (1931, reprinted 1959); HAROLD HILTON, Plane Algebraic Curves, 2nd ed. (1932); and ROBERT C. YATES, A Handbook on Curves and Their Properties, rev. ed. (1959). A considerable amount of information on special curves may be found in a variety of books and textbooks on analytic geometry, calculus, differential equations, the calculus of variations, and also mechanics. ( R.C.A./N.A.Ct./R.G.S.)

Algebraic geometry:

BARTEL L. VAN DER WAERDEN, Einführung in die algebraische Geometrie (1939), contains a fuller introductory account of the geometry of varieties in projective space. The birational geometry of algebraic varieties over the field of complex numbers is dealt with in many works by Italian geometers. OSCAR ZARISKI, Algebraic Surfaces (1935), contains a comprehensive account of this, with a full bibliography. The notion of an abstract variety is due to ANDRE WEIL, Foundations of Algebraic Geometry, rev. ed. (1962); SERGE LANG, Introduction to Algebraic Geometry (1958), contains an elementary account. For the theory of sheaves, see ROGER GODEMONT, Topologie algébrique et théorie des faisceaux (1958). For information about schema, see ALEXANDRE GROTHENDIECK, Eléments de géométrie algébrique, 4 pt. (1960-66). Useful introductions include SHIGERU IITAKA, Algebraic Geometry (1982); and ROBIN HARTSHORNE, Algebraic Geometry (1977). (Ed.)

Non-Euclidean geometry:

ROBERTO BONOLA, La geometría noneuclidea (1906; Eng. trans., Non-Euclidean Geometry: A Critical and Historical Study of Its Developments, 1955), includes translations of the original papers by Bolyai and Lobachevsky; JULIAN L. COOLIDGE, A Treatise on the Circle and the Sphere (1916), an early account of inversive geometry; H.S. MacDONALD COXETER, Non-Euclidean Geometry, 5th ed. (1965), the projective approach, with an extensive bibliography; Introduction to Geometry, 2nd ed. (1969), see ch. 15-16 for absolute geometry and hyperbolic; H.S. MacDONALD COXETER and S.L. GREITZER, Geometry Revisited, ch. 5-6 (1967), on the development of inversive geometry; PATRICK DU VAL, Homographies, Quaternions, and Rotations (1964), spherical space explored with the aid of quaternions; L. FEJES TOTH, Regular Figures (1964), includes good drawings of Euclidean and non-Euclidean tessellations; HANS SCHWERDTFEGER, The Geometry of Complex Numbers (1962), the analytic approach to inversive geometry; DUNCAN M.Y. SOMMERVILLE, The Elements of Non-Euclidean Geometry (1914, reprinted 1958), the simplest exposition.

(H.S.MacD.C.)