A Source Book in Mathematics

Contents PAGE AUTHOR'S PREFACE. . . . . . . . . . . . . . . . . . . . . . . . v I. THE FIELD OF NUMBER THE FIRST PRINTED ARITHMETIC. TREVISO, 1478 . . . . . . . . . 1 Selection translated from the Italian by David Eugene Smith ROBERT RECORDE on "THE DECLARATION OF THE PROFIT OF ARITH- METICKE" . . . . . . . . . . . . . . . . . . . . . . . . . 13 Selected from The Ground of Artes, by David Eugene Smith STEVIN ON DECIMAL FRACTIONS . . . . . . . . . . . . . . . . . 20 Translated from the French by Vera Sanford DEDEKIND ON IRRATIONAL NUMBERS . . . . . . . . . . . . . . . . 35 Translated from the German by Wooster Woodruff Beman. Selec- tion made and edited by Vera Sanford JOHN WALLIS ON IMAGINARY NUMBERS. . . . . . . . . . . . . . . 46 Selected and edited by David Eugene Smith WESSEL ON COMPLEX NUMBERS . . . . . . . . . . . . . . . . . . 55 Translated from the Danish by Martin A. Nordgaard PASCAL ON THE ARITHMETIC TRIANGLE. . . . . . . . . . . . . . . 67 Translated from the French by Anna Savitsky BOMBELLI AND CATALDI ON CONTINUED FRACTIONS . . . . . . . . . 80 Translated from the Italian by Vera Sanford BERNOULLI ON "BERNOULLI NUMBERS" . . . . . . . . . . . . . . 85 Translated from the Latin by Jekuthiel Ginsburg EULER ON EVERY INTEGER AS A SUM OF FOUR SQUARES. . . . . . . 91 Translated from the Latin by E. T. Bell EULER ON THE USE OF e TO REPRESENT 2.718· · · . . . . . . . . 95 Selection translated from the Latin by Florian Cajori HERMITE ON THE TRANSCENDENCE OF e . . . . . . . . . . . . . . 99 Translated from the French by Laura Guggenbühl GAUSS ON THE CONGRUENCE OF NUMBERS . . . . . . . . . . . . . . 107 Translated from the Latin by Ralph G. Archibald GAUSS ON THE THIRD PROOF OF THE LAW OF QUADRATIC RECIPROCITY 112 Translated from the Latin by D. H. Lehmer KUMMER ON IDEAL NUMBERS. . . . . . . . . . . . . . . . . . . . 119 Translated from the German by Thomas Freeman Cope CHEBYSHEV (TCHEBYCHEFF) ON THE TOTALITY OF PRIMES . . . . . 127 Translated from the French by J. D. Tamarkin NAPIER ON THE TABLE OF LOGARITHMS. . . . . . . . . . . . . . . 149 Selected and edited by W. D. Cairns ix DELAMAIN ON THE SLIDE RULE . . . . . . . . . . . . . . . . . . . 156 Edited by Florian Cajori OUGHTRED ON THE SLIDE RULE . . . . . . . . . . . . . . . . . . . 160 Edited by Florian Cajori PASCAL ON HIS CALCULATING MACHINE . . . . . . . . . . . . . . . . 165 Translated from the French by L. Leland Locke LEIBNIZ ON HIS CALCULATING MACHINE. . . . . . . . . . . . . . . . 173 Translated from the Latin by Mark Kormes NAPIER ON THE NAPIER RODS. . . . . . . . . . . . . . . . . . . . 182 Translated from the Latin by Jekuthiel Ginsburg GALILEO GALILEI ON THE PROPORTIONAL OR SECTOR COMPASSES . . . 186 Translated from the Italian by David Eugene Smith D'OCAGNE ON NOMOGRAPHY. . . . . . . . . . . . . . . . . . . . . . 192 Translated from the French by Nevin C. Fisk II. THE FIELD OF ALGEBRA CARDAN ON IMAGINARY ROOTS. . . . . . . . . . . . . . . . . . . . 201 Translated from the Latin by Vera Sanford CARDAN ON THE CUBIC EQUATION . . . . . . . . . . . . . . . . . . 203 Translated from the Latin by R. B. McClenon FERRARI-CARDAN ON THE BIQUADRATIC EQUATION. . . . . . . . . . . 207 Translated from the Latin by R. B. McClenon, with additional notes by Jekuthiel Ginsburg FERMAT ON THE EQUATION x^n + y^n = z^n. . . . . . . . . . . . . . 213 Translated from the French by Vera Sanford FERMAT ON THE SO-CALLED PELL EQUATION . . . . . . . . . . . . . 214 Translated from the Latin by Edward E. Whitford JOHN WALLIS ON GENERAL EXPONENTS. . . . . . . . . . . . . . . . . 217 Translated from the Latin by Eva M. Sanford WALLIS AND NEWTON ON THE BINOMIAL THEOREM FOR FRACTIONAL AND NEGATIVE EXPONENTS. . . . . . . . . . . . . . . . . . . . . . 219 Selection from Wallis's Algebra, by David Eugene Smith NEWTON ON THE BINOMIAL THEOREM FOR FRACTIONAL AND NEGATIVE EXPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Translated from the Latin by Eva M. Sanford LEIBNIZ AND THE BERNOULLIS ON THE POLYNOMIAL THEOREM . . . . 229 Translated from the Latin by Jekuthiel Ginsburg HORNER ON NUMERICAL HIGHER EQUATIONS . . . . . . . . . . . . . . 232 Selected and edited by Margaret McGuire ROLLE ON THE LOCATION OF ROOTS . . . . . . . . . . . . . . . . . 253 Translated from the French by Florian Cajori ABEL ON THE QUINTIC EQUATION . . . . . . . . . . . . . . . . . . 261 Translated from the French by W. H. Langdon, with notes by Oystein Ore LEIBNIZ ON DETERMINANTS . . . . . . . . . . . . . . . . . . . . . 267 Translated from the Latin by Thomas Freeman Cope BERNOULLI, VERSES ON INFINITE SERIES. . . . . . . . . . . . . . 271 Translated from the Latin by Helen M. Walker CONTENTS xi PAGE BERNOULLI ON THE THEORY OF COMBINATIONS . . . . . . . . . . . 272 Translated from the Latin by Mary M. Taylor GALOIS ON GROUPS AND EQUATIONS . . . . . . . . . . . . . . . . 278 Translated from the French by Louis Weisner ABEL'S THEOREM ON THE CONTINUITY OF FUNCTIONS DEFINED BY POWER SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Translated from the German by Albert A. Bennett GAUSS ON THE FUNDAMENTAL THEOREM OF ALGEBRA . . . . . . . . 292 Translated from the Latin by C. Raymond Adams III. THE FIELD OF GEOMETRY DESARGUES ON PERSPECTIVE TRIANGLES . . . . . . . . . . . . . 307 Translated from the French by Lao G. Simons DESARGUES ON THE 4-RAYED PENCIL. . . . . . . . . . . . . . . 311 Translated from the Latin by Vera Sanford PONCELET ON PROJECTIVE GEOMETRY. . . . . . . . . . . . . . . 315 Translated from the French by Vera Sanford PEACELLIER'S CELL . . . . . . . . . . . . . . . . . . . . . 324 Translated from the French by Jekuthiel Ginsburg PASCAL, "ESSAY POUR LES CONIQUES" . . . . . . . . . . . . . . 326 Translated from the French by Frances Marguerite Clarke BRIANCHON'S THEOREM. . . . . . . . . . . . . . . . . . . . . 331 Translated from the French by Nathan Altshiller-Court BRIANCHON AND PONCELET ON THE NINE-POINT CIRCLE THEOREM . . 337 Translated from the French by Morris Miller Slotnick FEUERBACH ON THE THEOREM WHICH BEARS HIS NAME. . . . . . . . 339 Translated from the German by Roger A. Johnson THE FIRST USE OF π FOR THE CIRCLE RATIO. . . . . . . . . . . 346 Selection made by David Eugene Smith from the original work GAUSS ON THE DIVISION OF A CIRCLE INTO n EQUAL PARTS. . . . 348 Translated from the Latin by J. S. Turner SACCHERI ON NON-EUCLIDEAN GEOMETRY. . . . . . . . . . . . . . 351 Translated from the Latin by Henry P. Manning LOBACHEVSKY ON NON-EUCLIDEAN GEOMETRY. . . . . . . . . . . . 360 Translated from the French by Henry P. Manning BOLYAI ON NON-EUCLIDEAN GEOMETRY. . . . . . . . . . . . . . . 371 Translated from the Latin by Henry P. Manning FERMAT ON ANALYTIC GEOMETRY . . . . . . . . . . . . . . . . . 389 Translated from the French by Joseph Seidlin DESCARTES ON ANALYTIC GEOMETRY. . . . . . . . . . . . . . . . 397 Translated from the French by David Eugene Smith and Marcia L. Latham PONCELET'S THEOREM. . . . . . . . . . . . . . . . . . . . . . 403 Translated from the German by Arnold Emch RIEMANN ON SURFACES AND ANALYSIS SITUS . . . . . . . . . . . 404 Translated from the German by James Singer RIEMANN ON THE HYPOTHESES WHICH LIE AT THE FOUNDATIONS OF GEOMETRY. . . . . . . . . . . . . . . . . . . . . . . . . 411 Translated from the German by Henry S. White xii CONTENTS PAGE MONGE ON THE PURPOSE OF DESCRIPTIVE GEOMETRY . . . . . . . . 426 Translated from the French by Arnold Emch REGIOMONTANUS ON THE LAW OF SINES FOR SPHERICAL TRIANGLES . . 427 Translated from the Latin by Eva M. Sanford REGIOMONTANUS ON THE RELATION OF THE PARTS OF A TRIANGLE. . 432 Translated from the Latin by Vera Sanford PITISCUS ON THE LAWS OF SINES AND COSINES . . . . . . . . . . 434 Translated from the Latin by Jekuthiel Ginsburg PITISCUS ON BÜRGI'S METHOD OF TRISECTING AN ARC. . . . . . . 436 Translated from the Latin by Jekuthiel Ginsburg DE MOIVRE'S FORMULA. . . . . . . . . . . . . . . . . . . . . 440 Translated from the Latin and from the French by Raymond Clare Archibald CLAVIUS ON PROSTHAPHAERESIS AS APPLIED TO TRIGONOMETRY. . . . 455 Translated from the Latin by Jekuthiel Ginsburg CLAVIUS ON PROSTHAPHAERESIS . . . . . . . . . . . . . . . . . 459 Translated from the Latin by Jekuthiel Ginsburg GAUSS ON CONFORMAL REPRESENTATION . . . . . . . . . . . . . . 463 Translated from the German by Herbert P. Evans STEINER ON QUADRATIC TRANSFORMATION BETWEEN TWO SPACES . . . 476 Translated from the German by Arnold Emch CREMONA ON GEOMETRIC TRANSFORMATIONS OF PLANE FIGURES . . . 477 Translated from the Italian by E. Amelotti LIE'S MEMOIR ON A CLASS OF GEOMETRIC TRANSFORMATIONS . . . . 485 Translated from the Norwegian by Martin A. Nordgaard MÖBIUS, CAYLEY, CAUCHY, SYLVESTER, AND CLIFFORD ON FOUR OR MORE DIMENSIONS. . . . . . . . . . . . . . . . . . 524 Note by Henry P. Manning CAYLEY ON HIGHER SPACE . . . . . . . . . . . . . . . . . . . 527 Selected by Henry P. Manning CAUCHY ON HIGHER SPACE . . . . . . . . . . . . . . . . . . . 530 Translated from the French by Henry P. Manning SYLVESTER ON HIGHER SPACE. . . . . . . . . . . . . . . . . . 532 Selected by Henry P. Manning CLIFFORD ON HIGHER SPACE . . . . . . . . . . . . . . . . . . 540 Selected by Henry P. Manning IV. THE FIELD OF PROBABILITY FERMAT AND PASCAL ON PROBABILITY . . . . . . . . . . . . . . . 546 Translated from the French by Vera Sanford DE MOIVRE ON THE LAW OF NORMAL PROBABILITY . . . . . . . . . 566 Selected and edited by Helen M. Walker LEGENDRE ON LEAST SQUARES. . . . . . . . . . . . . . . . . . . 576 Translated from the French by Henry A. Ruger and Helen M. Walker CHEBYSHEV (TCHEBYCHEFF) ON MEAN VALUES . . . . . . . . . . 580 Translated from the French by Helen M. Walker CONTENTS xiii PAGE LAPLACE ON THE PROBABILITY OF ERRORS IN THE MEAN RESULTS OF A GREAT NUMBER OF OBSERVATIONS, ETC . . . . . . . . . . . . . 588 Translated from the French by Julian L. C. A. Gÿs V. FIELD OF THE CALCULUS, FUNCTIONS, QUATERNIONS CAVALIERI ON AN APPROACH TO THE CALCULUS . . . . . . . . . . . 605 Translated from the Latin by Evelyn Walker FERMAT ON MAXIMA AND MINIMA. . . . . . . . . . . . . . . . . . 610 Translated from the French by Vera Sanford NEWTON ON FLUXIONS. . . . . . . . . . . . . . . . . . . . . . 613 Translated from the Latin by Evelyn Walker LEIBNIZ ON THE CALCULUS. . . . . . . . . . . . . . . . . . . . 619 Translated from the Latin by Evelyn Walker BERKELEY'S "ANALYST". . . . . . . . . . . . . . . . . . . . . 627 Selected and edited by Florian Cajori CAUCHY ON DERIVATIVES AND DIFFERENTIALS . . . . . . . . . . . . 635 Translated from the French by Evelyn Walker EULER ON DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. . . . . . 638 Translated from the Latin by Florian Cajori BERNOULLI ON THE BRACHISTOCHRONE PROBLEM . . . . . . . . . . . 644