References

The books and papers in this reference section include all items mentioned in name [date] format in the text. In addition, some are suitable as general introductions to the history of mathematics or as introductions to the history of particular topics. I would like to recommend the following:

Introductory Works

General

Boyer [1968], Burton [1985], Kline [1972].

Source Books (English translations of classic papers)

Struik [1969], Fauvel and Gray [1987].

Ancient Mathematics

Heath [1921], van der Waerden [1954, 1983].

Algebra

Edwards [1984], van der Waerden [1985].

334 References

Geometry

Gray [1980], Hildebrandt and Tromba [1985].

Algebraic Geometry

Boyer [1956], Brieskorn and Knorrer [1981], Dieudonne [1985].

Calculus

Baron [1969], Boyer [1949], Edwards [1979].

Number Theory

Weil [1984].

Mechanics

Dugas [1957, 1958].

Group Theory

Chandler and Magnus [1982], Wussing [1969].

Topology

Stillwell [1980].

References Cited

N.H.Abel [1826] Demonstration de l'impossibilite de la resolution algebrique des

equations generales qui passent Ie quatrieme degre.

[1827]

J. reine angew. Math. 1,65-84. Oeuvres Completes 1: 66-87. Recherches sur les fonctions elliptiques. J. reine angew. Math. 2,101-181; 3,160-190. Oeuvres Completes 1: 263-388.

References Cited

S. I. Adian [1957]

L. B. Alberti

335

The unsolvability of certain algorithmic problems in the theory of groups. (Russian) Tr. Moskov. Mater. Obsc. 6, 231-298.

[1436] Trattato della pittura. 1. Le R. D'Alembert [1746] Recherches sur Ie calcul integral.

Rist. Acad. Sci. Berlin 1,182-224. [1747] Recherches sur la courbe que forme une corde tendue mise en

vibration. Rist. Acad. Sci. Berlin 3, 214-219.

[1752] Essai d'une nouvelle theorie de la resistance des fluides.

AI-Khowarizmi [830] A. Amthor [1880]

R. Apery [1981]

J. R. Argand [1806]

R. Ayoub [1984]

Paris.

Al-jabr w'al muqdbala.

Das Problema bovinum des Archimedes. z. Math. Phys. (hist.-lit. Abt.)15, 153-171.

Interpolation de fractions continues et irrationalite de certains constantes. Mathematics, pp. 37-53, CTHS: Bull. Sec. Sci.; III, Bib. Nat. Paris.

Essai sur une maniere de representer les quantites imaginaires dans les constructions geometriques. Paris.

The lemniscate and Fagnano's contributions to elliptic integrals. Arch. Rist. Ex. Sci. 19, 131-149.

C. Bachet de Meziriac [1621] Arithmetica (Edition of Diophantus). A. Baillet [1691]

W. W.R. Ball [1890]

1. BaItrusaitis

Vie de Monsieur Descartes. Paris.

Newton's classification of cubic curves. Proc. London Math. Soc. 11, 104-143.

[1977] Anamorphic Art. Harry N. Abrams, New York.

S. Banach and A. Tarski [1924] Sur la decomposition des ensembles de points en parties respec­

tivement congruents.

1. Banville [1981]

M. E. Baron [1969]

I. G. Bashmakova [1981]

Fund. Math. 6, 244-277.

Kepler: A Novel. Seeker and Warburg, London.

The Origins of the Infinitesimal Calculus. Pergamon Press, Elmsford, N.Y.

Arithmetic of algebraic curves from Diophantus to Poincare. Rist. Math. 8, 393-416.

336

I. Beeckman [1615]

[1628]

E. Beltrami [1865]

[1868]

[1868']

D. Bernoulli [1738] [1743]

[1753]

James Bernoulli [1690]

[1692]

[1694]

[1697]

[1713]

Journal tenu par Isaac Beeckman de 1604 a 1634. Ed. C. de Waard, 4 vols. NijhotT, The Hague. Journal. Quoted in Oeuvres de Descartes 10: 344-346.

References

Risoluzione del problema: "Riportare i punti di una superficie sopra un piano in modo che Ie Iinee geodetiche vengano rap­presentate da Iinee rette." Ann. Mat. pura appl., ser. 1,7, 185-204. Opere Matematiche. 1: 262-280. Saggio di interpretazione della geometria non-euclidea. Giorn. Mat. 6, 284-312. Opere Matematiche. 1: 374-405. Partial English translation in Fauvel and Gray [1987]. Teoria fondamentale degli spazii di curvatura costante. Ann. Mat. pura appl., ser. 2,2,232-255. Opere Matematiche. 1: 406-429.

R ydrodynamica. Letter to Euler, 4 September 1743. In Enestrom [1906]. Refiexions et eclaircissemens sur les nouvelles vibrations des cordes exposees dans les memoires de I'academie de 1747 & 1748. Rist. Acad. Sci. Berlin 9,147-172.

Qua:stiones nonnullae de usuris cum solutione problema tis de sorte alearum propositi in Ephem. Gall. A. 1685. Acta Erud. 9, 219-233. Lineae cycloidales, evolutae, ant-evolute .... Spira mirabilis. Acta Erud. 11,207-213. Curvatura laminae elasticae. Acta Erud. 13,262-276. Solutio problematis fraternorum. Acta Erud. 16,211-217. Opera 1: 768-778. Ars Conjectandi. Opera 3: 107-286.

James and John Bernoulli [1689-1704] Uber unendliche Reihen.

John Bernoulli [1691]

[1691/2]

[1691/2']

[1696]

Ostwald's Klassiker, vol. 171. Engelmann, Leipzig, 1909.

Solutio problematis funicularii. Acta Erud. 10,274-276. Opera Omnia 1: 48-51. Lectiones de Calculo Differentialum. Ed. P. Schalheitlein. Basel, 1922. Lectiones mathematicae de methodo integralium aliisque. Opera Omnia 3: 385-558. Problema novum ad cujus solutionem Mathematici invitantur. Acta Erud. IS, 270. Opera Omnia 1: 161.

References Cited

[1697]

[1697']

[1699]

[1702]

[1712]

1. Bertrand [1860]

E. Bezout [1779]

G. BirkhotT (Ed.)

Curvatura radii in diaphanis non uniformibus. Acta Erud. 16,206-211. Opera Omnia 1: 187-193. Principia calculi exponentialum. Acta Erud. 16, 125-133. Opera Omnia 1: 179-187. Disputatio medico-physica de nutritione. Groningen. Opera Omnia 1: 275-306.

337

Solution d'un probleme concernant Ie calcul integral, avec quel­ques abreges par raport Ii ce calcul. Mem. Acad. Roy. Sci. Paris, pp. 289-297. Opera Omnia 1: 393-400. Angulorum arcuumque sectio indefinita. Acta Erud. 31, 274-277. Opera Omnia 1: 511-514.

Remarque Ii l'occasion de la Note precedente. Compo rend. 50, 781-782.

Theorie generale des equations algebriques. Paris.

[1973] A Source Book in Classical Analysis. Harvard University Press, Cambridge.

P. du Bois-Reymond .. [1875] Uber algebraische Werte, infinitiire Approxirnationen und

infinitiire Auflosung von Gleichungen.

V. G. Boltianskii [1978]

F. Bolyai [1832]

1. Bolyai [1832]

B. Bolzano [1817]

R. Bombelli [1572]

O. Bonnet [1848]

Math. Ann. 8, 363-414.

Hilbert's Third Problem. v. H. Winston, Silver Spring, Md.

Temamen. Juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi. Marosvasarhely.

Scientiam spatii absolute veram exhibens: A vertitate aut falsit­ate Axiomatis XI Euc1idei (a priori hand unquam decidanda) independentem. Appendix to F. Bolyai [1832]. English translation in Bonola [1912].

Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewiihren, wenigstens eine reelle Wurzel der Gleichung liege. Ostwald's Klassiker, vol. 153. Engelmann, Leipzig, 1905.

L'Algebra. Feltrinelli Editore, Milan, 19.66.

Memoire sur la theorie generale des surfaces. J. i.c. Poly tech. 19, 1-146.

338

R. Bonola [1912]

G. Boole [1847]

E. Borel [1898]

H.J. M. Bos [1981]

[1984]

A. Bosse

Noneuclidean Geometry. Dover, New York, 1955.

Mathematical Analysis of Logic. Basil Blackwell, London, 1948.

Lerons sur la theorie des fonctions. Gauthier-Villars, Paris.

References

On the representation of curves in Descartes' Geometrie. Arch. Hist. Ex. Sci. 24, 295-338. Arguments on motivation in the rise and decline of a mathematical theory: The "Construction of Equations," 1637-c. 1750. Arch. Hist. Ex. Sci. 30, 331-380.

[1648] Maniere universelle de Mr. Desargues. R. Bo~rgne and 1. P. Azra (Eds.) [1962] Ecrits et Memoires Mathematiques d'Evariste Galois: Edition

Critique Integrale de ses Manuscrits et Publications. Gauthier-Villars, Paris.

C. B. Boyer [1949]

[1956]

[1968]

H. R. Brahana [1921]

Brahmagupta

The History of the Calculus and Its Conceptual Development. Hafner Publishing Company, Inc.; Dover, New York, 1959. History of Analytic Geometry. Scripta Mathematica, New York. A History of Mathematics. Wiley, New York.

Systems of circuits on 2-dimensional manifolds. Ann. Math. 23,144-168.

[628] Brdhma-sphuta-siddMnta. Partial English translation in Colebrooke [1817].

E. Brieskorn and H. Knorrer [1981] Ebene algebraische Kurven.

H.Briggs [1624]

E. S. Bring [1786]

W. K. Buhler [1981]

D. M. Burton [1985]

F. Cajori [1913]

Birkhiiuser, Boston. English translation: Plane Algebraic Curves. Birkhiiuser, Basel, 1986.

Arithmetica logarithmica, sive logarithmorum chiliades triginta. London.

Meletemata quaedam mathematica circa transformationem aequ­ationum algebraicarum. Lund University. Promotionschrift.

Gauss. A Biographical Study. Springer-Verlag, New York.

The History 0'£ Mathematics. An Introduction. Allyn and Bacon, Newton, Mass.

History of the exponential and logarithmic concepts. Am. Math. Mon. 20, 5-14, 35-47, 75-84, 107-117, 148-151, 173-182,205-210.

References Cited

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[1874]

[1880]

[1883]

[1891]

G. Cardano [1545]

[1575]

A.-L. Cauchy [1813]

[1815]

[1825]

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[1846]

B. Cavalieri [1635]

A. Cayley [1854]

339

Uber die Ausdehnung eines Satzes aus der Theorie der trigono­metrischen Reihen. Math. Ann. 5,123-132. Gesammelte Abhandlungen, pp. 92-102. Uber eine Eigenschaft des InbegrifTes aller reellen algebraischen Zahlen. J. reine angew. Math. 77, 258-262. Gesammelte Abhandlungen, pp. 115-118. Uber unendlich lineare Punktmannigfaltigkeiten, 2. Math. Ann. 17(1880) 355-358. Gesammelte Abhandlungen, pp. 145-148. Grundlagen einer allgemeinen M annigfaltigkeitslehre. Leipzig. Gesammelte Abhandlungen, pp. 165-204. Uber eine elementare Frage der Mannigfaltigkeitslehre. Jahrb. Dtsch. Math. Ver.l, 75-78. Ges. Abh., pp. 278-280.

Ars Magna. English translation: The Great Art. M.I.T. Press, Cambridge, 1968. De Vita Propria Liber. English translation: The Book of My Life. Dutton, New York, 1930; Dover, New York, 1962.

Demonstration du theon:me general de Fermat sur les nombres polygones. Mem. Sci. Math. Phys. Inst. France, ser. 1,14,177-220. Oeuvres, ser. 2, 6: 320-353. Memoire sur Ie nombre des valeurs qu'une fonction peut acquerir, lorsqu'on y permute de to utes les manieres possibles les quantites qu'elle renferme. J. Bc. Poly tech. 18, 10: 1-28. Oeuvres, ser. 2, 1: 62-90. Memoire sur les integrales dejinies prises entre des limites imag­ina ires. Paris. Letter to Coriolis, 29 January 1837. Compo rend. 4, 214-218. Oeuvres, ser. I, 4: 38-42. Memoire sur les arrangements que I'on peut former avec des lettres donnees, et sur les permutations ou substitutions a I'aide desquelles on passe d'un arrangement a un autre. Ex. anal. phys. math. 3,151-252. Oeuvres, ser. 2, 13: 171-282. Sur les integrales qui s'etendent a to us les points d'une courbe fermee. Compo rend. 23, 251-255. Oeuvres, ser. 1, 10: 70-74.

Geometria indivisibilibus continuorum nova quadam ratione promota.

On the theory of groups, as depending on the symbolic equation 0" = 1.

340 References

Phil. Mag. 7,40-47. Collected Mathematical Papers 2: 123-130.

[1859] A sixth memoir upon quantics. Phil. Trans. Roy. Soc. 149,61-90. Collected Mathematical Papers 2: 561-592.

[1878] The theory of groups. Am. J. Math. 1,50-52. Collected Mathematical Papers 10: 401-403.

[1878'] The theory of groups: Graphical representation. Am. J. Math. 1, 174-176. Collected Mathematical Papers 10: 403-405.

B. Chandler and W. Magnus [1982] The History of Combinatorial Group Theory.

A. Church [1936]

[1938]

M. Clagett [1959]

[1968]

A.-C. Clairaut [1731]

[1740]

[1743]

A. Clebsch

Springer-Verlag, New York.

An unsolvable problem of elementary number theory. Am. J. Math. 58, 345-363. Review. J. Symb. Logic 3, 46.

The Science of Mechanics in the Middle Ages. University of Wisconsin Press, Madison. Nicole Oresme and the Medieval Geometry of Qualities and Motions. University of Wisconsin Press, Madison.

Sur les courbes que ron forme en coupant un surface courbe quelconque par un plan donnee de position. Mem. Acad. Sci. Paris, pp. 490-493. Sur l'integration ou la construction des equations differentielles du premier ordre. Mem. Acad. Sci. Paris, p. 294. Theorie de la figure de la Terre tiree des principes de l'hydro­dynamique. Durand, Paris.

[1864] Uber einen Satz von Steiner und einige Punkte der Theorie der Curven dritter Ordnung. J. reine angew. Math. 63, 94-121.

M. R. Cohen and I. E. Drabkin [1958] Source Book in Greek Science.

P. Cohen [1963]

H. T. Colebrooke [1817]

R. Connelly [1978]

J. L. Coolidge [1945]

Harvard University Press, Cambridge.

The independence of the continuum hypothesis I, II. Proc. Nat. Acad. Sci. 50,1143-1148; 51,105-110.

Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhtiscara. John Murray, London. Martin Sandig, Wiesbaden, 1973.

A counterexa!Dple to the rigidity conjecture for polyhedra. lnst. Hautes Etudes Sci., Publ. Math. 47, 333-338.

A History of the Conic Sections and Quadric Surfaces. Clarendon Press, Oxford.

References Cited

N. Copernicus [1543] R. Cotes [1714]

[1722]

D.A.Cox

De revolutionibus orbium coelestium.

Logometria. Phil. Trans. 29, 5-45. Harmonia Mensurarum. Cambridge.

[1984] The arithmetic-geometric mean of Gauss. Enseign. Math. 30, 275-330.

H. S. M. Coxeter and W. o. 1. Moser [1980] Generators and Relations for Discrete Groups, 4th ed.

G. Cramer [1750]

1. N. Crossley [1987]

D. van Dalen [1972]

Springer-Verlag, New York.

Introduction Ii l'analyse des lignes courbes algebriques. Geneva.

The Emergence of Number, 2nd ed. World Scientific, Singapore.

Set theory from Cantor to Cohen.

341

In Sets and Integration (D. van Dalen and A. F. Monna, eds.). Wolters-NoordhotT, Groningen, pp. 1-74.

1. H. Davenport [1981]

F. David [1962]

M. Davis [1965]

[1973]

R. Dedekind [1872]

[1876]

On the Integration of Algebraic Functions. Lecture Notes in Computer Science. Springer-Verlag, New York.

Games, Gods and Gambling. Charles Griffin, London.

The Undecidable. Raven Press, New York. Hilbert's tenth problem is unsolvable. Am. Math. Mon. 80, 233-269.

Stetigkeit und die Irrationalzahlen. English translation in Essays on the Theory of Numbers. Open Court, Chicago, 1901. Bernhard Riemann's Lebenslauf. In Riemann, Werke 2nd ed., pp. 539-558.

P. Dedron and 1. Itard [1973] Mathematics and Mathematicians, vol. 1.

M.Dehn [1900]

[1910]

[1912]

[1987]

Open University Press, Milton Keynes.

Dber raumgleiche Polyeder. Gattinger Nachrichten, pp. 345-354. Dber die Topologie des dreidimensionalen Raumes. Math. Ann. 69,137-168. English translation in Dehn [1987], pp. 92-126. Dber unendliche diskontinuierliche Gruppen. Math. Ann. 71,116-144. English translation in Dehn [1987], pp. 133-178. Papers on Group Theory and Topology. Springer-Verlag, New York.

342 References

M. Dehn and P. Heegaard [1907] Analysis situs.

G. Desargues [1639]

R. Descartes [1637]

[1638]

L. E. Dickson [1903]

[1919, 1920, 1923]

J. Dieudonne [1985]

P. G. L. Dirichlet [1829]

[1837]

P. Dombrowski [1979]

S. K. Donaldson [1983]

S. Dostrovsky [1975]

R. Dugas [1957]

[1958]

A. Diirer [1525]

Enzyklopadie Mathematische Wissenschaft, vol. IIIAB3, pp. 153-220. Leipzig.

Brouillon projet d'une atteinte aux evenemens des recontres du cone avec un plan. In Taton [1951], pp. 99-180. English translation in Field and Gray [1987].

La Geometrie. English translation: The Geometry. Dover, New York, 1954. Letter to Mersenne, 18 January 1638. Oeuvres 1: 490.

Introduction to the Theory of Algebraic Equations. Wiley, New York. History of the Theory of Numbers, 3 vols. Carnegie Institute of Washington. Chelsea, New York, 1951.

History of Algebraic Geometry. Wadsworth, Belmont, Calif.

Sur la convergence des series trigonometriques qui servent a representer une fonction arbitraire entre des Iimites donnees. J. reine angew. Math., 4,157-169. Werke 1: 117-132. Beweis des Satzes, dass jede unbegrenzte arithmetische Progres­sion, deren erstes Glied und DitTerenz ganze Zahlen ohne gemein­schaftlichen Factor sind, unendliche viele Primzahlen enthiilt. Abh. Akad. Wiss. Berlin, pp. 45-81. Werke 1: 315-342.

150 Years after Gauss' "Disquisitiones generales circa superficies curveas." Asterisque 62.

An application of gauge theory to four dimensional topology. J. DijJ. Geom. 18,279-315.

Early vibration theory: Physics and music in the seventeenth century. Arch. Hist. Ex. Sci. 14,169-218.

A History of Mechanics. Routledge and Kegan Paul, London. Mechanics in the Seventeenth Century. Editions du GritTon, Neuchatel.

Underweysung der Messung, mit dem Zirckel und Richtscheyt. Verlag Waiter Uhl, Unterscheidheim, 1972.

References Cited

W.Dyck [1882]

[1883]

C. H. Edwards [1979]

H. M. Edwards [1974]

[1977]

[1984]

G. Eisenstein [1847]

G. Enestrom [1906]

s. B. Engelsman [1984]

L. Euler [1728]

[1728']

[1734/5]

[1736]

[1743]

[1746]

[1748]

[1748']

[1750]

[1752]

Gruppentheoretische Studien. Math. Ann. 20, 1-44. Gruppentheoretische Studien II. Math. Ann. 22, 70-108.

The Historical Development of the Calculus. Springer-Verlag, New York.

Riemann's Zeta Function. Academic Press, New Y rirk. Fermat's Last Theorem. Springer-Verlag, New York. Galois Theory. Springer-Verlag, New York.

Beitriige zur Theorie der elliptischen Functionen. J. reine angew. Math. 35,137-274.

343

Der Briefwechsel zwischen Leonhard Euler und Daniel Bernoulli. Bibl. Math., ser. 3,7,126-156.

Families of Curves and the Origins of Partial Differentiation. North-Holland, Amsterdam.

Letter to John Bernoulli, 10 December 1728. Bibl. Math., ser. 3,4,352-354. De linea brevissima in superficie quacunque duo quaelibet puncta iungente. Comm. Acad. Sci. Petrop. 3, 110-124. Opera Omnia, ser. I, 25: 1-12. De summis serierum reciprocarum. Comm. Acad. Sci. Petrop. 7, 123-134. Opera Omnia, ser. I, 14: 73-86. Theorematum quorundam ad numeros primos spectantium de­monstratio. Comm. Acad. Sci. Petrop. 8,141-146. Opera Omnia, ser. 1,2: 33-37. Additamentum I de curvis elasticus. Lausanne and Geneva. Opera Omnia, ser. 1,24: 231-297. English translation: Isis 20 (1933), 72-160. Letter to Goldbach, 14 June 1746. Briefwechsel Opera Omnia, ser. quarta A, I, 152. Introductio in Analysin Infinitorum I. Opera Omnia, ser. I, 8. Introductio in Analysin Infinitorum II. Opera Omnia, ser. I, 9. Letter to Goldbach, 9 June 1750. Corresp. Math. Phys. I, (1843), pp. 521-524. Elementa doctrinae solidoTum. Novi Comm. Acad. Sci. Petrop. 4,109-140. Opera Omnia, seT. I, 26: 71-93.

344

[1758] Theoremata arithmetica nova methodo demonstrata. Novi Comm. Acad. Sci. Petrop. 8, 74-104. Opera Omnia, ser. 1,2: 531-555.

[1760] Recherches sur la courbure des surfaces. Mem. Acad. Sci. Berlin 16,119-143. Opera Omnia, ser. 1,28: 1-22.

[1768] Institutiones Calculi Integralis. Opera Omnia, ser. I,ll.

[1770] Vollstandige Einleitung zur Algebra. English translation: Elements of Algebra. Longman, Orme and Co., London, 1840. Springer-Verlag, New York, 1984.

References

[1777] De repraesentatione superficiei sphaericae super plano. Acta Acad. Sci. Imper. Petrop. 1, 107-132.

[1849] De numeris amicabilibus. Comm. Arith. 2, 627-636. Opera Omnia, ser. 1,5: 353-365.

G. C. de'T. di Fagnano [1718] Metodo per misurare la lemniscata.

Giorn. lett. d'Italia 29. Opere Matematiche. 2: 293-313.

G. Faltings [1983] Endlichkeitssatze fiir abelsche Varietaten iiber Zahlkorpern.

Inv. Math. 73, 349-366. J. Fauvel and 1. Gray [1987] The History of Mathematics. A Reader.

P.1. Federico [1982]

S. Feferman [1986]

P. Fermat [1629]

[1640]

Macmillan, New York.

Descartes on Polyhedra. Springer-Verlag, New York.

Preface to Kurt Godel Collected Works, vol. l. Oxford University Press, Oxford.

Ad locos pIanos et solidos isagoge. Oeuvres 1: 92-103. English translation in D. E. Smith, A Source Book in Mathematics 2: 389-396. Dover, New York, 1959. Letter to Frenic1e, 18 October 1640. Oeuvres 2: 209.

[1670] Observations sur Diophante. Oeuvres 3: 241-276.

Fibonacci (Leonardo of Pisa) [1202] Liber Abaci. [1225] Flos Leonardi Bigolli Pisani super solutionibus quarundam quaes­

tionum ad numerum et ad geometriam pertinentium. J. Field and J. Gray [1987]

1. Fourier [1822]

The Geometrical Work of Girard Desargues. Springer-Verlag, New York.

La theorie analytique de la chaleur. Didot, Paris. English translation: The Analytical Theory of Heat.

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[1982]

G. Frege [1879]

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[1638]

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[1831']

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[c. 1819]

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345

Cambridge University Press, Cambridge, 1878. (Dover, New York, 1955.)

Book II of Euclid's Elements and a pre-Eudoxan theory of ratio. Arch. Hist. Ex. Sci. 22, 5-36. Book II of Euclid's Elements and a pre-Eudoxan theory of ratio. Part 2: Sides and diameters. Arch. Hist. Ex. Sci. 26,193-209.

Begriffschrift. English translation in van Heijenoort [1967], pp. 5-82

The transcendence of 7t has been known for about a century-but who was the man who discovered it? Results Math. 7,164-183.

Letter to Paolo Sarpi, 16 October 1604. Works of Galileo 10: 115. Dialogues Concerning Two New Sciences. Dover, New York, 1952.

Memoire sur les conditions de resolubilite des equations par radicaux. Bcrits Mem. Math., pp. 43-71. Analyse d'un memoire sur la resolution algebrique des equations. Bcrits Mem. Math., pp. 163-165.

Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. Helmstedt dissertation. Werke 3: 1-30. Disquisitiones Arithmeticae. English translation, Yale University Press, New Haven, Conn., 1966. Letter to Bessel, 18 December 1811. Briefwechsel mit F. W. Bessel. Georg Olms Verlag, Hildesheim, 1975, pp. 155-160. Demonstratio nova altera theorematis omnem functionem al­gebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. Comm. Recentiores (Gottingae) 3, 107-142. Werke 3: 31-56. Determinatio attractionis quam in punctum quod vis positionis datae exerceret planeta si eius massa per totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dis­pertita. Comm. Soc. Reg. Sci. Gpttingensis Rec. vol. 4. Werke 3: 331-355. Die Kugel. Werke 8: 351-356. Allgemeine Auflosung der Aufgabe; Die Theile einer gegebenen Flache so abzubilden, dass die Abbildung dem Abgebildeten in

346

[1825]

[1827]

[1828]

[1831]

[1832]

[1832']

[1846]

[1846']

A. O. Gelfond [1960]

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[1938]

[1946]

H. H. Goldstine [1977]

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Review of "The mathematical career of Pierre de Fermat, by M. S. Mahoney." Bull. Am. Math. Soc. 79,1138-1149. Elliptic Functions According to Eisenstein and Kronecker. Springer-Verlag, New York. Number Theory. An Approach Through History. Birkhauser, Basel.

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Siyuan yujian (Jade mirror offour unknowns).

Index

A Abel, 58, 66, 159-164, 166, 215, 218,

272, 278, 279, 290 theorem 162, 163, 166

Abelian function, 166 integral, 166

Adams, 171 Addition theorems, 149, 155-158, 222,

230 Adian, 309 Alberti, 78, 80-82, 84 Algebra, 48, 49, 73 Algebraic

curves, 25, 67, 69, 92, 169, 171, 198-200,204,231,237,240,295

functions, 120, 125, 126, l31, 154,291 geometry, 49, 73, 217 topology, 311

Algorithm, 5, 35, 48, 309, 323 euclidean, 30, 32, 34, 276

Al-Haytham (Alhazen), 77, 102, 255 Al-Khwiirizmi, 48, 49, 52, 54 Al-Kuji,21 Analytic geometry, 9, 11,49,54,65,69,

73, 76, 86, 150 Anamorphosis, 81, 82 Angeli, l31 Angle division, 55, 56, 192-195 Angular defect, 250, 257, 258

Angular excess, 248, 249, 258, 259 Anthyphairesis, 32, 34 Apollonius, 21, 24, 53, 65, 66, 169 Arc length, 67, 153, 155, 174, 176,

237-240 Archimedes, 34,44,46,47,61, 102, 103,

118, 123, 167, 171 cattle problem, 34 Method, 38,47, 103, 167 spiral, 103, 171

Argand, 196, 197 Aristotle, 9, 37, 38,46,112,168 Arithmetization of geometry, 73, 74 Asymptotic lines, 256, 257, 263, 264 Axiom of choice, 318-320, 330 Axioms, 13, 324, 326-328

of infinity, 327, 328

B Babylonians, 3, 51, 52 Bachet, 28, 36, 140, 141, 144 Banach,319

-Tarski theorem, 319 Barrow, 110, 114 Bartels, 252, 259, 273 Beeckman, 71, 76, 176 Beltrami, 218, 246, 258-266, 271, 303 Berkeley, 38 Bernays, 326

364

Bernoulli, Daniel, 132, 178, 181, 184-187,202

Hydrodynamica, 179, 185-187 Bernoulli, James, 117, 124, 132, 154,

172-174,181-184,240-242,246, 318

Bernoulli, John, 117, 124, 132, 172-173, 181,183-184,186,187,193,195, 220, 221, 242, 246

Bertrand, 300 Bessel, 160, 224 Betti, 218 Bezout's theorem, 71, 72, 93-96, 200,

205 Binomial

coefficients, 135-139 series, 109, 114 theorem, 107, 122

Birkhoff, 312 Bolyai, F., 42, 259,271, 272 Bolyai, J., 66, 259, 260, 262, 266, 271,

272,274 Bolzano, 196, 197 Bombelli,36, 190, 191,221 Bonnet, 246, 247 Bonola, 259 Booie,324 Borel, 317, 318

sets, 317, 318 Bosse, 83, 96, 97 Boyle, 181, 185 Brachistochrone, 173 Brahana, 296 Brahmagupta, 34, 48, 51, 52 Branching, 125, 210, 211 Briggs, 123 Bnng, 58 Brouncker, 31, 34, 107 Brunelleschi, 78

C Calculus, 54, 100-110, 167, 170, 183,

237 of variations, 174, 183

Cantor, 179,314-316,320,321 Cardano, 54, 57, 59-62, 189, 190

formula, 55-57, 189-191 Cardinality, 315

Cartesian parabola, 71 Cassini, 24, 179

ovals, 24

Index

Catenary, 171-173, 183, 240, 242 Cauchy, 29, 163, 181,223,225,234-,236,

279, 289, 290 integral theorem, 224-226, 236 -Riemann equations, 222

Cavalieri, 102, 103, 131 Cayley, 262, 279, 280, 285

theorem, 280 Celestial mechanics, 169-171, 234, 311 Chain rule, 107 Chinese remainder theorem, 135 Choice function, 318, 319 Chord and tangent construction, 35, 49,

147,230,279 Christina of Sweden, 76 Church,323 Cicero, 46, 47 Circularfunctions, 56,108,109,120,131,

152, 154, 221, 258 Cissoid, 22, 23, 67 Clairaut, 179, 180, 202 Clebsch, 147 Closed form, 110, 116, 154 Cohen, 316, 320, 330 Combinatorics, 101, 116, 126, 136 Complex functions, 73, 181,220-232,

269 Complex numbers, 56, 57, 59, 72, 90,

131, 153, 159,222 in algebra, 188-220 geometric interpretation, 191, 192,

196-198 Composition

of forces, 169 of functions, 275

Computable, 321-323 functions, 321-323 real numbers, 321 set of defining relations, 323

Condorcet, 203, 234 Conformal mapping, 223, 224, 254 Congruence, 284, 319 Conic sections, 20, 24, 65, 66, 69, 73, 83,

86, 88-90,95, 98 Connelly, 236 Construction of equations, 71, 72

Index

Continued fractions, 34, 106, 107, 289 Continuity, 169, 179, 196, 197, 198, 2oo,

292, 317 Continuum hypothesis, 316, 318, 319, 330 Convergence of series, 131, 228 Copernicus, 169 Costruzione legittima, 80 Cotes, 194, 195, 221 Countable additivity, 318, 319 Countable sets, 314-320 Counting board, 50 Coverings, 210, 212, 302-308

universal, 302-308 Cramer, 50, 72, 279

rule, 50 Crelle, 163 Cross-ratio, 83-85 Cryptography, 62, 64, 1I2, 138 Cube, 15, 16, 282

duplication of, 18, 20, 23, 53, 54 Cubic curves, 67, 69, 90, 1I4, 144-149,

152, 153, 213, 230, 231 Newton's classification, 69-71, 90,

1I4, 1I5, 147, 230 projective equivalence, 231

Cubic equations, 54-56, 58, 189-191 Curvature, loo, 1I4, 217, 241-246, 253,

297-301, 306 centre of, 241 constant, 244-246, 258, 260, 301 Gaussian, 244-248, 250, 254 geodesic, 246-248 negative, 245-247, 258, 261, 301 principal, 244 radius of, 241 of surfaces, 243-246

Cusp, 23, 67, 70, 71, 90 Cycloid,99, 171, 173, 174,240,242

D D'Alembert, 69, 177, 180, 196-203,222,

223,233 lemma, 197, 199

Decimals, 48, 107 Dedekind, 9, 38, 40, 217, 218, 254

cut, 40, 197,314 Degen, 162 Degree of a curve, 22, 67

Dehn, 43, 295, 308 De la Hire, 97 De Moivre, 57, 109, 113, 126, 127,

192-195,220 formula, 57, 193, 221

365

Desargues, 81-83, 86, 88, 92, 95-97, 250 theorem, 83

Descartes, 14,22,49,58,66-68,71, 73-7,86,103,104,107, 1I2, 171, 181,271, 293, 297-3oo

factor theorem, 59, 124, 195 folium, 67, 68, 104, 145-147 La Geometrie. 66,71, 77, 104, 171

Descriptive geometry, 82 Diagonal argument, 320--325 Dichotomy of leno, 37, 38, 1I8 Diderot, 203 Differentiability, 169, 178, 223, 224, 226 Differential equations, 170, 174,270, 31I

partial, 176, 177 Differential geometry, 73, 1I4, 164, 217,

218, 237-250, 259, 260 Differentiation, loo, 103, 107, 1I0, 169,

237 partial, 183

Diocles, 22 Diophantine equations, 5, 27, 31, 48, 144,

147 Diophantus, 4,27,28,35,36,49, 135,

140, 141, 144, 147, 189 Dirichlet, 140, 165, 179,217,218,317,

320 function, 317, 320 principle, 217, 224

Distance (See also Metric) definition of, 9, 11 euclidean, 9, ll, 266 hyperbolic, 261, 262, 264-266, 306

Divergence of series, 119 Division of stakes, 137, 138 Dodecahedron, 15, 17, 282 Donaldson, 293 Double periodicity, 159, 160,217,

226-228, 230 Du Bois-Reymond, 321 Duplication of the cube, 18-20,23,53,

54 Durer, 78 Dyck, 285, 287

366

E Einstein's equations, 330 Eisenstein, 217, 227, 228, 254 Elastic curve (elastica), 154, 171, 174, 175 Elementary functions, 154 Elimination, 49-50, 72 Ellipse, 20, 21, 88, 89, 114,154, 170, 174 Elliptic curves, 154, 228-231 Elliptic functions, 66, 110, 147-149,

152-161, 164-166,214,217, 226-230,279, 302, 311

Elliptic integrals, 153-155, 158, 159, 174 Empty set, 315 Epicycles, 24, 25, 97, 169-171 Equiangular spiral, 237-241 Equidistant curve, 263, 264, 266 Euclid, 3, 25, 26, 42, 44, 53, 82, 250, 257

Elements, 3, 13, 14, 25, 26, 29, 30, 39, 40, 46, 52, 53, 61, 73, 113, 116, 255-257,262,316

Euclidean algorithm, 30, 32, 33, 276 Euclidean geometry, 73, 246, 255, 260,

266, 272, 273, 301, 306 Eudoxus, 25, 39,40, 44, 47 Euler, 27, 29, 31, 69, 72, 107, 124, 128,

129, 132-134, 139, 140, 144, 147, 149, 157, 158, 161, 162, 164, 166, 170, 174, 184, 186, 193, 194,202, 215,221-223,227,228,233,236, 244, 246, 252, 276, 293, 295, 312

Algebra, 134, 191 characteristic, 293-297, 300-304, 306,

307 pentagonal number theorem, 29, 134 polyhedron formula, 293, 294, 300, 301

Exhaustion, (See Method of exhaustion) Exponential function, 108, 109, 120, 154,

184,221,222 complex, 193,221,222

Extreme value theorem, 197, 198

F Fagnano, 147, 149, 157, 161, 174 Faltings, 141 Fermat, 5, 27, 28, 35, 36, 49, 66, 71, 76,

103,104,140,141,143,144,147, 149-151,236,276

last theorem, 19, 140, 141, 144 little theorem, 138-140, 276, 277

Ferrari,54,57, 59,60, 62 Ferro, Scipione del, 54, 59 Fibonacci, 53, 126

sequence, 126-128 Fields, 279 Fior, Antonio Maria, 54

Index

Folium of Descartes, 67, 104, 145-147 Fourier, 179, 316, 317

series, 176, 179,217,314,316,317 Frege, 324 Fuchs, 310, 311 Fuchsian functions, 311 Function, 110, 123, 125, 178, 179, 222,

234 many-valued, 125

Fundamental group, 307-309, 311 Fundamental theorem

of algebra, 110, 195-200,205,206,252 of arithmetic, 31 of calculus, 110, 131

Fiirtwangler, 329

G Galileo, 76, 168, 169, 172 Galois, 19, 58, 278, 279, 288-291

theory, 16, 58, 279, 291 Gauss, 19,27,58, 106, 159-164, 174,

196-200,206,208,216-218,221, 224,225,244,246-250,252-254, 258-260,268,270,271-274,291, 297-300,306

-Bonnet theorem, 247-250, 258, 297-302, 306

curvature, 244-250, 254, 298-302, 306 sphere, 208

Generating functions, 126, 137 Genus, 141, 144-147; 162,208,215,217,

297 Geodesic, 246, 247, 260, 306

curvature, 246-248 Geometric series, 44, 46, 108, 118, 120,

127, 129, 183 GOdel, 316, 321, 323-331

incompleteness theorem, 324-328, 330, 331

Index

second theorem, 326-328 Golden

ratio, 128 rectangle, 16

Goursat, 226 Gravity, 114, 241 Green, 225, 226 Gregory, 107, 114, 116, 121-123,

130-132, 161 -Newton interpolation formula, 121-123

Groups, 232, 275-288 definition, 275 finite, 280

H

fundamental, 307-309, 311 isomorphism problem, 309 of permutations, 277-279 polyhedral,280-284 presentation by generators and relations,

283, 285-288 of rigid motions, 271, 284 of transformations, 232, 276, 284 word problem, 323

Hadamard, 194 Hahn, 329 Halley, 114, 170 Halting problem, 322, 323 Hamilton, 283 Harnack, 317, 318 Harrington, 327 Harriot, 123, 223, 237-239, 249-252,

257, 294, 300 Hausdorff, 319 Heath, 141, 143 Heegaard, 295 Hermite, 19, 58, 279, 289, 291 Higman, 323 Hilbert, 9, 11, 14, 43, 73, 74, 83, 224,

244, 245, 326 problems, 43, 316

Hipparchus, 65 Hobbes, 13, 25, 73, 103, 104, 106 Holbein, 81 Holder, 191 Holmboe, 162 Homeomorphisms, 292, 295

Homogeneous coordinates, 91-95 Homotopic paths, 307-308 Hooke, 114, 115, 172, 181 Horocyc1e, 262-264, 303 Horosphere, 260, 266 Hudde, 104, 107 Hurewicz, 311

367

Huygens, 67,97, 100, 116, 172, 173,240, 242,245

Hydrodynamics, 179-181,222 Hydrostatics, 46, 167 Hyperbola, 20, 65, 71, 88, 89, 107,204 Hyperbolic

functions, 168, 182, 183, 258 geometry, 259-266, 269-271, 285,

301,311 conformal models, 263-266, 269 projective model, 260-263

plane, 258, 260, 269, 306 Hypergeometric, 106

differential equation, 270 Hypotenuse, I

I Icosahedron, 15, 16, 17,267,282 Indivisibles, 103, 106 Inertia, 168 Infinite descent, 141, 143, 144 Infinite series, 100, 107-109, 118-129,

178 Infinitesimals, 101, 104 Infinity, 37, 39

axioms of, 327 line at, 83, 89, 92 points at, 72, 82, 83, 88, 91-93,

206-208 Inflexion, 67, 90 Integration, 100, 101, 108, 110, 116, 154,

169, 193, 195,220 complex, 221, 224-226

Intermediate value theorem, 196, 197 Interpolation, 106, 107, 116, 121-123 Inverse square law, 114, 170 Inversion

of integrals, 120, 121, 153, 154, 158, 159, 162, 164

of series, 108, 109, 121

368

Involute, 242 Irrational numbers, 2, 8, 9, 14, 25, 32 Isometric surfaces, 244, 246 Isoperimetric problem, 183

J Jacobi, 29, 58, 66, 147, 149, 153,

157-160,164-166,217,228,229, 291

Fundamenta Nova, 159, 166 Jade Mirror, 50 Jill Xian, 136 Jordan, 279, 291, 295, 317

K Kac, 54 Kaestner, 243, 252, 253 Kelvin, 217 Kepler, 16, 17,21,24,77,82,83,92,

114, 170 Klein, 19,58,91,92,160,206,232,261,

279,283-285,295,296,301,303, 304, 306, 310, 311

Erlanger Programm, 284, 292 Koebe,232 Kolmogorov, 318 Kronecker, 128 Kummer, 140

L Lacroix. 73 Lagrange, 24, 27, 28, 34, 73, 160, 162,

164,170,203,223,232-235,252, 277, 278, 289

theorem, 278 Lambert, 257, 258, 359 Lame, 140 Laplace, 164, 170, 171,203,234,235 Large cardinals, 318, 319, 327 Lattice, 229 Laurent, 226 Lavoisier, 234 Lebesgue, 217, 318

integral, 317, 318 measure, 318, 319

Legendre, 44, 140, 158, 215, 289, 294, 301

Index

Leibniz, 57, 66, 98, 107, 109, 110, 115-117,123,139,149,154,170, 172, 174, 182-184, 186, 192,220, 221,293,324

Lemniscate (of Bernoulli), 24, 163, 174 arc length, 155-157 division of, 161 integral, 155, 157, 159, 174

Leonardo da Vinci, 81 Leverrier, 171 Levi ben Gershon, 135, 136, 139, 277 L'Hopital, 173, 183 Lindemann, 19, 120 Linear equations, 49, 50 Linear fractional transformations, 232,

271,283,316 Liouville, 154, 265, 266, 291 Listing, 295 Liu Hui, 49 Lobachevsky, 66, 259, 260, 262, 266,

272-274 Logarithm, 108, 114, 116, 120, 121, 131,

184, 220-222, 225 . complex 193, 220-222, 225

Logarithmic spiral (See Equiangular spiral)

M Maclaurin, 72 Markov, 309 Mathematical induction, 136, 137, 143,

144 Mathematical logic, 116, 324 Matiasevich, 5, 32 Matrices, 50 Mean

arithmetic-geometric, 131, 160, 252 geometric harmonic, 131

Measure, 316-320, 327 zero, 317-319

Mechanical curves, 67, 171-176,237 Menaechmus, 20, 65, 71, 205 Mengoli, 123, 124 Mercator, G., 223

projection, 223 Mercator, N., 107, 108, 120 Mersenne, 29, 66, 76, 81, 96, 151, 169,

176 primes, 29

Index

Merton acceleration theorem, 168 Method of exhaustion, 9, 25,40-47, 100 Metric (See also Distance), 217 Minding, 246, 258 Mobius, 84, 86, 215, 284, 295, 296

band, 93, 295 Modular functions, 58, 160 Monge, 73 Mordell, 35, 141 Multinomial theorem, 139 Multiplicity, 204-206

N Neil, 67, 70, 240 Neumann, 211, 224 Newton, 5, 12,21,24,56,66,69-73,77,

90, 101, 104, 107-109, 110, 112-116, 121-123, 125, 131, 147, 162,168-171,173,174,179,184, 197,202,210,212,226,240-242, 252, 289, 31I

classification of cubic curves, 69-71, 90, 1I4, 1I5, 147, 228

first law, 168 Principia, 1I4, 1I5, 170, 173, 181 second law, 168, 177

Niceron, 81, 82 Nielsen, 308 Nine Chapters, 49 Non-euclidean geometry, 14, 188, 218,

255-274,31I Normal subgroups, 278, 279 Novikov, 323

o Octahedron, 15, 16, 282 Ordinal numbers, 314-316 Oresme, 65, 66, 119, 168 Orientability, 295 Ostrogradsky, 225 Ostrowski, 200

p

Parabola, 20, 65, 71,88, 102, 168, 172, 205

area of segment, 44-46

369

Parallels, 73, 80, 82, 83, 93, 255-257, 262, 272, 273, 316, 318

Parametrization of circle, 6, 153 of cubic curves, 145, 149, 152, 153,

229-231 of ellipse, 154 of folium, 68, 145

Paris, 327 Pascal, 76, 86,95,97-99, 136, 138, 139,

151,250 theorem, 95, 98 triangle, 98, 116, 135, 136

Peano arithmetic, 326-328 Pell's equation, 31-4 Pendulum, 154, 173, 242 Pentagon, 19 Perfect numbers, 29, 138 Perseus, 23, 214 Perspective, 78-80, 82 Pi, 18, 19, 106, 107, 120, 131,237 Plato, 3 Plimpton-322, 3 Pliicker, 91, 95 Plutarch, 46, 47 Poincare, 35, 147, 171,228,232,

269-271,285,286,294,295,301, 304, 306, 308-312

conjecture, 311, 312 last theorem, 312

Polya, 134, 298 Polygonal numbers, 28, 29 Polyhedra,

classification of, 295 convex, 235, 293, 297, 301 generalized, 295 regular, 15-17,25,53,267,280-284 volumes, 260

Poncelet, 83 Post, 321, 325, 326, 328 Potential, 179, 180, 224 Power series, 120, 121,223,226,234

fractional, 125, 126 with negative powers, 226

Prime numbers, 29, 30, 31, 128, 129, 179, 218,219

infinitely many, 29, 129, 179 Mersenne, 29

Probability, 62, 127, 137,203,318

370

Projection, 82, 85 Projective

completion, 92, 93, 207 geometry, 72, 73, 82, 86 line, 92, 93, 206-208 plane, 92, 206, 296, 303, 304 transformations, 82, 83, 230, 262, 276,

284 Pseudosphere, 240, 245, 247, 258,

261-263, 265, 303 Ptolemy, 24, 77, 169, 170,223 Puiseux, 126, 197, 210, 212, 226

expansions, 126 Pulverizer, 34 Pythagoras, 3, II, 12, 179

converse theorem, I theorem, 1,2,7-10,13,73

Pythagoreans, 8, 9, II, 28, 32, 33, 38, 176 Pythagorean triples, 3,4, 31, 140, 142

Q Quadratic equations, 51-53, 55, 189 Quadrature, 107, 108 Quantum mechanics, 167 Quintic equations, 58, 162, 164,278,289,

291

R Ramsey, 327, 328 Rational

functions, 121, 128, 145, 152,217,230 numbers, 8, 9, 40, 314, 316 points on a cubic, 35, 144-149 points on a curve, 5, 141, 144 points on the circle, 4-6, 35 right-angled triangle, 4, 6, 141-144

Real numbers, 39, 40, 197,314 completeness of, 314 uncountabilityof, 314, 318

Recurrence relations, 32-34, 127, 128 Recursively enumerable sets, 325, 326 Relativity, 167, 330 Resultant, 72 Rhumb line, 239 Riemann, 129, 160, 181,210,215-219,

224,227,228,254,266,270,294, 295,311

hypothesis, 129, 218, 219

integral, 217, 317, 318, 320 mapping theorem, 224

Index

surfaces, 210, 212-215, 217, 226, 295 zeta function, 129, 218

Right-angled triangle side lengths, 1 with rational sides, 4, 141-144

Rigid motion, 246, 262, 269, 285, 319 Roberval, 66, 103, 169 Roses of Grandi, 69 Ruffini, 58, 278 Ruler and compass constructions, 18, 53,

157, 161 Russell, 26, 324

S Saccheri, 256, 257, 264 Salmon, 231 Schwarz, 224, 270, 304 Scott; 293 Secular variation, 170 Seifert and Threlfall, 308 Set theory, 39, 176, 179,314-316,318,

319, 324, 327 axioms, 327

Shelah,319 Sluse, 104, 114 Snell, 77 Solovay, 319 Solution by radicals, 58, 125, 277-279 Spherical geometry, 257, 258 Spira, 23 Squares, 28, 29, 142-144 Squaring the circle, 18, 120 Statics, 47, 168 Steiner, 217 Stereographic projection, 207, 223 Stevin, 169 Stirling, 69 Surfaces

curvature of, 243, 244 non-orientable, 295 of constant curvature, 244-246 topological classification, 295-297

Symptoms 22

T Tarski,319 Tartaglia, 54, 59-62

Index

Taurinus, 258, 259 Tautochrone, 173 Taylor, 122, 177, 178, 184

series, 122 theorem, 122

Tessellations, 17, 267-270, 285-287, 295, 304-308

Tetrahedron, 15, 16, 17,272,280 Thales, 11, 13 Theaetetus, 25 Theon of Alexandria, 36 Theory of equations, 58, 188, 277-279,

283, 289-291 Theory of proportions, 9,39,41,47 Theta functions, 29, 160 Three-body problem, 170, 312 Thurston, 293 Tietze, 309 Topology, 73, 116,206-208,212-215,

218,232,285,287,292-309,311 algebraic, 311 general, 292 geometric, 292

Torricelli, 103, 119, 239 Torus, 23, 152, 213, 226, 229, 302-306,

311 geometry, 306

Tractrix, 240, 242, 245 Trajectory of projectile, 61, 168, 169 Transcendental

curves, 67, 171,237 functions, 120, 131, 160, 173, 291 numbers, 19, 120, 131, 237

Triangular numbers, 28, 30 Trisection, 18, 56 Turing, 321-323

machines, 321-323, 325, 326

U Ularn,319 Uncountability, 314-316, 318, 320 Uniformization, 231, 232 Unsolvability, 309, 323

of algorithmic problems, 323 of halting problem, 322 of homeomorphism problem, 309, 323 involving Diophantine equations, 5

V

of isomorphism problem, 309 of word problem, 323

Vandermonde, 277, 278 Van Heuraet, 67, 240 Van Roomen, 64 Vibrating string, 8, 169, 176-179 Vienna circle, 329, 330

371

Viete, 36, 55-58, 63, 64, 106, 107, 113, 130, 192, 193

Vitali, 319 Vitruvius, 46 Von Neumann, 315

W Wachter, 260, 266, 271 Wallis, 73,105-107,109-113, 121, 123,

161,191, 192,256 Arithmetica infinitorum 105-107

Wantzel, 19, 53 Wave equation, 177 Weierstrass, 38, 197, 198, 228, 230

P-function 158, 228, 230 Weil, 143 Well-ordered sets, 319 Wessel, 197 Whitehead, 324 Word problem, 323 Wren, 114, 240

X Xylander,36

Y Yang Hui, 136

Z Zeno, 37, 38 Zermelo, 319 Zeta function, 129, 218 Zeuthen, 141, 143 Zhu Shijie, 50, 136

Undergraduate Texts in Mathematics

(continued from page ii)

Lang: Calculus of Several Variables. Third edition.

Lang: Introduction to Linear Algebra. Second edition.

Lang: Linear Algebra. Third edition.

Lang: Undergraduate Algebra. Second edition.

Lang: Undergraduate Analysis. LaxlBursteinlLax: Calculus with

Applications and Computing. Volume 1.

LeCuyer: College Mathematics with APL.

LidllPilz: Applied Abstract Algebra. Macki-Strauss: Introduction to Optimal

Control Theory. Malitz: Introduction to Mathematical

Logic. MarsdenIWeinstein: Calculus I, II, ill.

Second edition. Martin: The Foundations of Geometry

and the Non-Euclidean Plane. Martin: Transformation Geometry: An

Introduction to Symmetry. MillmanlParker: Geometry: A Metric

Approach with Models. Second edition.

Moschovakis: Notes on Set Theory. Owen: A First Course in the

Mathematical Foundations of Thermodynamics.

Palka: An Introduction to Complex Function Theory.

Pedrick: A First Course in Analysis. PeressinilSullivanlUhl: The Mathematics

of Nonlinear Programming. PrenowitzlJantosciak: Join Geometries. Priestley: Calculus: An Historical

Approach. Protter/Morrey: A First Course in Real

Analysis. Second edition.

Protter/Morrey: Intermediate Calculus. Second edition.

Roman: An Introduction to Coding and Information Theory.

Ross: Elementary Analysis: The Theory of Calculus.

Samuel: Projective Geometry. Readings in Mathematics.

Scharlau/Opolka: From Fermat to Minkowski.

Sethuraman: Rings, Fields, and Vector Spaces: An Approach to Geometric Constructability .

Sigler: Algebra. Silvermanffate: Rational Points on

Elliptic Curves. Simmonds: A Brief on Tensor Analysis.

Second edition. Singertrhorpe: Lecture Notes on

Elementary Topology and Geometry. Smith: Linear Algebra. Second edition. Smith: Primer of Modem Analysis.

Second edition. StantonlWhite: Constructive

Combinatorics. Stillwell: Elements of Algebra:

Geometry, Numbers, Equations. Stillwell: Mathematics and Its History. Strayer: Linear Programming and Its

Applications. Thorpe: Elementary Topics in

Differential Geometry. Toth: Glimpses of Algebra and

Geometry. Troutman: Variational Calculus and

Optimal Control. Second edition. Valenza: Linear Algebra: An

Introduction to Abstract Mathematics. Whyburn/Duda: Dynamic Topology. Wilson: Much Ado About Calculus.