WORLDS OUT OF NOTHING: HISTORY OF MATHEMATICS

By BARBARA SANCHEZ

CONTENTS 1 MATHEMATICS IN THE FRENCH REVOLUTION THE FRENCH REVOLUTION SOME MATHEMATICIANS MONGE DESCRIPTIVE GEOMETRY 2 PONCELET (AND POLE AND POLAR) PONCELET REMINISCES PONCELET’S MATHEMATICS FROM PONCELET’S TRAIT´E COMMENTARY POLE, POLAR, AND DUALITY 3 THEOREMS IN PROJECTIVE GEOMETRY THE THEOREMS OF PAPPUS, DESARGUES AND PASCAL SOME PROPERTIES OF SOME TRANSFORMATIONS CROSS-RATIO PORISMATA 4 PONCELET’S TRAIT´E PONCELET’S SINGULAR CLAIMS MEETING CAUCHY RESPONDS OTHER RESPONSES PONCELET’S MORE CONVENTIONAL METHODS 5 DUALITY AND THE DUALITY CONTROVERSY POLE AND POLAR GERGONNE VERSUS PONCELET CURVES OF HIGHER DEGREE GERGONNE 6 PONCELET AND CHASLES WHAT WAS DONE – DI?ERING OPINIONS INSTITUTIONS AND CAREERS CHASLES WHAT WAS DONE? CHASLES, STEINER AND CROSS-RATIO EXTRACTS FROM CHASLES’ APER¸CU HISTORIQUE ON DESCRIPTIVE GEOMETRY ON MONGE AND HIS SCHOOL ON MONGE’S WORK A QUICK INTRODUCTION TO MODERN PROJECTIVE GEOMETRY THE REAL PROJECTIVE PLANE PROJECTIVE SPACES 7 LAMBERT AND LEGENDRE SACCHERI LAMBERT LEGENDRE 8 GAUSS GAUSS SCHWEIKART AND TAURINUS WHAT GAUSS KNEW GAUSSIAN CURVATURE 9 J´ANOS BOLYAI J´ANOS AND WOLFGANG BOLYAI J´ANOS BOLYAI’S GEOMETRY J´ANOS BOLYAI’S SECTION 32 10 LOBACHEVSKII 10 1 LOBACHEVSKII AND KASAN 10 2 LOBACHEVSKII’S NEW GEOMETRY 10 2 1 ASTRONOMICAL EVIDENCE 10 3 LOBACHEVSKII’S BOOKLET OF 1840 10 3 1 OPENING REMARKS 10 3 2 CONCLUDING REMARKS 11 TO 1855 11 1 MINDING’S SURFACE 11 2 THE BOLYAIS READ LOBACHEVSKII 11 3 FINAL YEARS OF J´ANOS BOLYAI 11 4 FINAL YEARS OF LOBACHEVSKII 11 5 GAUSS’S DEATH, GAUSS’S NACHLASS 12 WRITING 12 1 ASSESSMENT AND ADVICE 12 1 1 READING AND WRITING THE HISTORY OF MATHEMATICS 12 1 2 PRACTICE QUESTIONS 12 2 REFERENCES AND FOOTNOTES 12 2 1 APPENDICES 12 2 2 NAMES 12 2 3 YOUR ESSAYS 12 3 ASSESSMENT QUESTION 12 3 1 ADVICE 13 M¨OBIUS 13 1 M¨OBIUS’S BARYCENTRIC CALCULUS 13 1 1 BARYCENTRIC COORDINATES 13 1 2 PROJECTIVE TRANSFORMATIONS 13 1 3 DUALITY 13 1 4 CENTRAL PROJECTION FROM ONE PLANE TO ANOTHER 13 2 A NOTE ON DUALITY 13 3 M¨OBIUS’S COORDINATES 14 THE DUALITY PARADOX 14 1 HIGHER PLANE CURVES 14 1 1 CUBIC CURVES 14 2 PL¨UCKER’S RESOLUTION OF THE DUALITY PARADOX 14 3 CON?RMATION BY OTHERS 14 4 PL¨UCKER 14 5 HESSE 15 THE PL¨UCKER FORMULAE 15 1 SINGULAR POINTS 15 1 1 THE NON-SINGULAR CUBIC CURVE IN THE PLANE 15 1 2 THE NON-SINGULAR QUARTIC CURVE IN THE PLANE 15 1 3 28 REAL BITANGENTS 16 HIGHER PLANE CURVES 16 1 NON-SINGULAR POINTS AND TANGENTS 16 2 DOUBLE POINTS 16 3 HOMOGENEOUS COORDINATES 16 4 FIRST AND SUBSEQUENT POLARS 16 4 1 THE ?RST POLARS OF A CIRCLE 16 4 2 IN?ECTION POINTS 16 5 HESSIANS 16 6 ADDENDUM 16 7 REFERENCES 17 COMPLEX CURVES 17 1 COMPLEX BY NECESSITY 17 1 1 COMPLEX NUMBERS IN GEOMETRY 17 1 2 THE INTRODUCTION OF COMPLEX CURVES 17 2 ELLIPTIC FUNCTIONS 18 RIEMANN 18 1 RIEMANN 18 2 RIEMANN’S PUBLICATIONS 18 3 RIEMANN ON GEOMETRY 18 3 1 SURFACES 18 4 RIEMANNIAN GEOMETRY 18 5 FROM RIEMANN’S HABILITATIONSVORTRAG 19 DI?ERENTIAL GEOMETRY OF SURFACES 19 1 BASIC TECHNIQUES 19 1 1 GEODETIC PROJECTION 19 2 INTRODUCING BELTRAMI’S SAGGIO 19 2 1 BELTRAMI’S TEORIA OF 1868 19 3 THE SAGGIO 19 3 1 FROM BELTRAMI’S SAGGIO OF 1868 19 4 LEGENDRE’S ERROR 19 5 REFERENCES 20 NON-EUCLIDEAN GEOMETRY ACCEPTED 20 1 BELTRAMI’S VERSION 20 2 GAUSS’S POSTHUMOUS CONTRIBUTION 20 2 1 KANT? 20 3 FELIX KLEIN 20 3 1 KLEIN AT ERLANGEN 20 3 2 AND BEYOND 20 4 KLEIN’S CAYLEY METRIC 20 5 KLEIN’S UNI?CATION OF GEOMETRY 20 6 THE ERLANGEN PROGRAM IN THE 1890S 20 7 WEIERSTRASS AND KILLING 21 WRITING 21 1 ASSESSMENT QUESTIONS 21 2 ADVICE 21 2 1 CREMONA 21 2 2 SALMON 21 2 3 LOBACHEVSKII’S ACCOUNT IN 1840 22 FUNDAMENTAL GEOMETRY 22 1 THE RISE OF PROJECTIVE GEOMETRY 22 2 CREMONA 22 2 1 CREMONA’S PROJECTIVE GEOMETRY 22 3 SALMON 22 4 ANXIETY – PASCH 22 5 HELMHOLTZ 22 5 1 FREE MOBILITY 23 HILBERT 23 1 HILBERT 23 2 HILBERT AND GEOMETRY 23 2 1 THE GRUNDLAGEN DER GEOMETRIE 23 2 2 DESARGUES’ THEOREM 23 3 IMPACT 23 4 REFERENCES 24 ITALIAN FOUNDATIONS 24 1 PEANO AND SEGRE 24 2 ENRIQUES 24 3 PIERI 24 4 CONCLUSIONS 24 5 VERONESE 25 THE DISC MODEL 25 1 POINCAR´E 25 1 1 A PRIZE COMPETITION 25 1 2 POINCAR´E’S DISCOVERY OF NON-EUCLIDEAN GEOMETRY 25 1 3 THE POINCAR´E AND BELTRAMI DISCS 25 2 POINCAR´E AND KLEIN 25 3 CIRCUMCIRCLES 25 4 INVERSION AND THE POINCAR´E DISC 25 4 1 INVERSION 25 5 REFERENCES 26 THE GEOMETRY OF SPACE 26 1 HOW TO DECIDE? 26 2 POINCAR´E’S CONVENTIONALISM 26 2 1 ENRIQUES DISPUTES 26 3 “SPACE AND GEOMETRY” 26 4 POINCAR´E’S ARGUMENTS 27 SUMMARY: GEOMETRY TO 1900 27 1 REFERENCES 28 THE FORMAL SIDE 28 1 NAGEL’S THESIS 28 2 FROM HILBERT’S GRUNDLAGEN DER GEOMETRIE 29 THE PHYSICAL SIDE 29 1 GEOMETRY AND PHYSICS 29 2 EINSTEIN 29 2 1 THE SPECIAL THEORY OF RELATIVITY 29 2 2 THE PARADOXES OF SPECIAL RELATIVITY 29 3 MINKOWSKI 29 4 EINSTEIN, GRAVITY AND THE ROTATING DISC 29 5 EINSTEIN’S RELATIVITY 30 IS GEOMETRY TRUE? 30 1 TRUTH 30 1 1 MATHEMATICAL TRUTHS 30 2 PROOF 30 2 1 FREGE VERSUS HILBERT 30 3 RELATIVE CONSISTENCY 30 4 POINCAR´E ON “NON-EUCLIDEAN GEOMETRIES” 31 WRITING 31 1 ASSESSMENT QUESTIONS 31 2 ADVICE ON WRITING SUCH ESSAYS 31 3 HOW THE ESSAYS WILL BE GRADED