LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK DEPARTMENT OF MATHEMATICS, ENGINEERING AND COMPUTER SCIENCE

MAT 132- HISTORY OF MATHEMATICS CATALOG DESCRIPTION 3 Periods, 3 Credits Prerequisite: MAT 096 OR WAIVER An examination of the theoretical developments of mathematics from antiquity to the end of the last century. Mathematical thought will be studied in relation to the social, economic, and technological forces of various crucial periods. Among the topics treated historically are systems of numeration, and geometry from Euclid through Riemann. The shifting geographic centers of mathematics will be traced in addition to the major mathematical movements from the introduction of rigor in analysis up to and including the arithmetization of analysis. COURSE OBJECTIVE: 1. To show how mathematics is one aspect of human activity, originating in specific historical situations and responding to practical and theoretical needs. 2. To trace the emergence of powerful formal mathematical methods from the empirical beginnings. 3. Through the study of several individual concepts and the role of particular mathematicians, to show mathematics evolves as an attempt to understand reality, both in its objectives and subjective aspects. Among these concepts are: a) Number systems (Sumerian, Babylonian, Egyptian, Roman, Arabic) b) Logic and the ‘laws of thought’ from Aristotle through Leibnitz and Boole to Frege and Russel c) Geometry as the science of space from Euclid through Kant until the discovery of non-Euclidean geometries in the 19th centuries d) Calculating devices, from the abacus to the modern computer 4. To explore the relationship of mathematics to science, art and philosophy up to comparatively modern times from a historical perspective.

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PERFORMANCE OBJECTIVE: 1. Explain both theoretically and concretely the number systems invented by the Babylonians, Egyptians, Romans and Arabic peoples. 2. Discuss the historical development of the concept of an axiom system, using Euclidean and non-Euclidean geometries as illustrative material. 3. Describe the development of rigor in analysis from Euler to Gauss and Cauchy 4. Describe the development of notation in mathematics through to the time of Euler. 5. Trace the history and methods used to solve the famous problems of antiquity and the discuss Zeno’s paradoxes and the solution to the Konigsburg bridge problem. 6. Attempt to explain the simultaneous occurrence of mathematical ideas in the minds of different mathematicians at the same time in history 7. Attempt to explain the near misses of mathematicians to discover the Calculus, more efficient notation and arithmetization before the credited founders. TEXT: Boyer, C. B., A History of Mathematics, 2nd ed. (New York: John Wiley & Sons, Inc.1991). EVALUATION: The educators in this course will be evaluated based on their responses to a take home midterm exam and the presentation of a written report to additional reading of their own selection.

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Lecture 1

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Chapters Ch. 1 Origins Ch. 2 Egypt Trigonometric Ratio P. 18 (optional)

Timeline -50,000 Evidence of Counting -25,000 Primitive Geometric Designs -4241 Hypothetical origin of the Egyptian Calendar

Ch. 3 Mesopotamia Cubic equations p. 33 (optional)

-2400 Positional notation in Mesopotamia -1850 Moscow Papyrus

Ch. 4 Ionia and the Pythagoreans Arithmetic and cosmology p. 53 (optional) Attic numeration p. 57 (optional) Ionian numeration p. 58 (optional) Ch.5 The Heroic Age Anaxagoras of Clazomenae p. 63 (optional) Hippias of Elis p. 68 (optional)

-585 Thales of Miletus; deductive geometry (?) -540 Pythagorean arithmetic and geometry (approx.) Rod numerals in China (approx.)

Ch. 6 the Age of Plato and Aristotle Socrates p. 83 (optional) Theodorus of Cyrene p.85 (optional) Menaechmus p. 93 (optional)

-428 Birth Archytas; death of Anaxagoras -427 Birth of Plato -420 Trisectrix of Hippias (approx.) Incommensurables (approx.)

Ch.7 Euclid of Alexandria Other works p. 102 (optional) Scope of book I p. 107 (optional) Books III and IV p. 113 (optional) Apocrypha p. 118 (optional)

-430 Death of Zeno; works of Democritis Elements of Hippocates of Chios (approx.)

Focus Problems Egyptian Calculation Unit Fractions Volume of a frustum Sexigesimal Fractions Quadratic Equations Division into mean and extreme ratio Figurate Numbers

Quadrature of Lunes Incommensurability Paradoxes of Zeno

-369 Death of Theatetus -360 Eudoxus on proportion and exhaustion (approx.) -350 Menaechmus on conic sections (approx.)

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Method of exhaustion Duplication of the cube Squaring of the circle Pythagorean Theorem Golden Section Infinity of primes

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Ch.8 Archimedes of Syracuse Measurement of the circle p. 125 (optional) Area of parabolic segment p.128 (optional) Volume of a parabolic segment p. 129 (optional) Segment of a Sphere p. 130 (optional) Ch. 9 Apollonius of Perga Cycles and Epicycles p. 143 (optional) Conjugate diameters p. 149 (optional) Tangents and Harmonic Division p. 150 (optional) Ch. 10 Greek Trigonometry and Mensuration Aristarchus of Samos p. 159 (optional) The 360degree circle p. 166 (optional) Construction of tables p. 167 (optional) Ch. 11 Revival and Decline of Greek Mathematics Nicomachus of Gerasa p. 178 (optional) Proclus of Alexandria p. 190 (optional) Boethius p.191 (optional) Ch. 12 China and India Magic Squares p. 197 (optional) Rod numerals p. 198 (optional) Hindu Multiplication p. 215 (optional) Ch. 13 Arabic Hegemony ‘Abd al-Hamid ibn-Turk p. 234 (optional) Thabit ibn-Qurra p.235 (optional)

-300 Euclid’s Elements (approx.) -230 Sieve of Eratosthenes (approx.) -225 Conics of Appolonius (approx.) -212 Death of Archimedes

Angle Trisection

-180 Cissoid of Diocles (approx.) Conchoid of Nicomedes (approx.) -140 Trigonometry of Hipparchus

The double napped cone

+75 Works of Heron of Alexandria (approx.) 100 Nicomachus: Arithmetica (approx.) Menelaus: Spherics (approx.)

Ptolemy’s Formulas

The method

Volume of a Sphere

The three and four line locus

150 Ptolemy: Almagest Trisections (approx.) 250 Diophantus: Arithmetica (approx.?) Theory of Means 320 Pappus: Mathematical Collections (approx.)

415 Death of Hyapatia 470 Tsu ch’ung-chi’s value of pi (approx.) 485 Death of Proclus 529 Closing of schools at Athens 662 Bishop Sebokht mentioned Hindu numerals 775 Hindu works translated into Arabic 830 al-kwarizmi: algebra

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Algebra and Horner’s Method

Quadratic equations Geometric Foundations Heron’s Problem

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Al-Biruni and Alhazen p.239 (optional) Ch. 14 Europe in the Middle Ages Campanus of Novara p. 259 (optional) Medieval Kinematics p. 261 (optional)

Ch. 15 The Renaissance Nicolas Chuquet’s Triparty p. 276 (optional) Georg Joachim Rheticus p. 292 (optional) Ch. 16 Prelude to Modern Mathematics Reconstruction of Apollonius’ On Tangencies p. 322 (optional)

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Ch. 17 The Time of Fermat and Descartes

Ch. 18 A Transitional Period Georg Mohr p. 370 (optional) Pietro Mengali p. 370 (optional) Jan De Witt p.372 (optional) Rene Francoise de Sluse p.373 (optional) Ch. 19 Newton and Leibniz

(approx.) 1114 Birth of Bhaskara 1123 Death of Omar Khayyam 1142 Adelard of Bath translated Euclid 1202 Fibonacci: Liber abaci 1270 Wm. of Moerbeke translated Achimedes (approx.) 1360 Oresme’s Latitude of Forms (approx.) 1476 Death of Regiomontanus 1489 Use of + and – by Widman 1492 Use of decimal point by Pellos 1545 Cardan: Ars Magna 1564 Birth of Galileo 1579 Viete: Canon Mathematicus 1614 Napier’s Logarithms 1635 Cavalieri: Geometria indivisibilibus 1629 Fermat’s Method of Maxima and Minima 1637 Descartes: Discours de la Method 1639 Deargues: Brouillon Project 1642 Birth of Newton; Death of Galileo 1655 Wallis: Arithmetica Infinitorum 1667 Gregory: Geometriae Pars 1670 Barrow Lectiones Geometriae 1684 Leibniz’s first paper on the calculus

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Graphs of Linear Equations Infinite Series

Solution of the Cubic Briggsian Logarithms

Theorem of Cavalieri

Fermat’s Differentiations Fermat’s Integrations Pascal’s Triangle Gregory’s Series Continued Fractions Barrow’s Tangents

Wallis’s Principle of Intercalculation

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Ch. 20 The Bournoulli Era Rodger Cotes p. 426 (optional) James Stirling p. 427 (optional)

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Ch. 21 The Age of Euler Differential Equations p. 451 (Optional) The Claurauts p. 452 (optional) Ch. 22 Mathematicians of the French Revolution Publication before 1789 p. 469 (optional) Lagrange and Determinants p. 470 (optional) Lacroix on Analytic Geometry p. 478 (optional) Ch. 23 The time of Gauss and Cauchy

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Ch. 24 Geometry Projective Geometry: Poncelet and Chasles p. 535 (optional) Synthetic non-metric geometry: Von Staudt p. 539 (optional) Analytic Geometry p. 551 (optional)

1696 Brachistochrone (The Bernoullis) L’Hospital’s Rule 1733 Saccheri: Euclid Vindicated 1734 Berkeley: The Analyst 1742 Maclaurin Treatise of Fluxions 1743 D’ Alembert: Traite de dynamique. 1748 Euler: Introductio; Agnesi : Instituzioni 1750 Cramer’s rule; Fagnono’s ellipse 1777 Buffon’s needle problem 1779 Bezout on Elimination

Infinite Series

Taylor’s Series

Fermat’s Lesser Theorem

1788 Lagrange: Geometry of cords Mecanique Analytique 1794 Legendre: Elemens de Geometrie Geometry of 1796 Laplace: Systeme du Transversals monde 1797 Carnot: Metaphysique du Calcul

1801 Disquisitiones Arithmeticae 1817 Bolzano: Rein analytischer Beweis 1829 Lobachevskian Geometry 1832 Bolyai: Absolute Science of space; death of Galois 1854 Riemann’s Habilitationsschrift; Boole: Laws of Thought

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Fundamental Theorem of Algebra Prime number theorem Definition of Limit Postulates of NonEuclidean Geometry

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Ch. 25 Analysis Mathematical Physics in the English Speaking Countries p. 556 (optional) Modern Mathematics and the Mathematics of Computers

1827 Dedekind: Stetigkeit und irrational Zahlen

Teacher Lectures on topics in the history of mathematics

Antiquity through until the present day

1874 Cantor’s Mengenlehre 1903 John Von Neuman

Denumerable and Non-denumerable sets Density of Subsets of R Matrices and Numerical Representation in the Computer Teacher Exposition

There will be two major evaluations for you grade in this course: 1. A take home exam based on essay questions from the Boyer text. The take home exam must be typed and will count for 50% of your grade. 2. A one half-hour presentation on a topic or figure in the history of mathematics. The presentation must be based on a typewritten paper approximately 5 pages in length. The paper and presentation together will count for 50% of your grade. Suggested Bibliography for Projects 1. 2. 3. 4.

Aiton, E. J., Leibniz: A Biography (Bristol and Boston: A. Hilger, 1984). Bonola, R., Non-Euclidean Geometry (New York: Dover, 1955) Cardan, J. The Book of My Life, Trans. by J. Stoner (New York: Dover, 1962) Coolidge, J. L., History of the Conic Sections and Quadratic Surfaces (New York: Dover, 1968). 5. Dantzig, Tobias, The Bequest of the Greeks (New York: Scribner, 1955). 6. Dauben, J. L., Georg Cantor. His Mathematics and philosophy of the Infinite (New Jersey: Princeton University, 1979). 7. Dunham, W., Journey Through Genius. The Great Theorems of Mathematics (New York: Penguin, 1990). 8. Dunnington, G. W., Carl Friedrich Gauss. Titan of Science (New York: Exposition Press, 1955). 9. Gleick, J. Isaac Newton (New York: Pantheon, 2003) 10. Hardy, G. H., A Mathematicians Apology (Cambridge: Cambridge University Press, 2001). 11. Knorr, W. R., The Evolution of Euclidean Elements (Dordrech and Boston: Reidel, 1975). 12. Lagrange, J. L. Lectures on Elementary Mathematics, trans. by T.J. McCormack (Chicago: Open Court, 1901) 13. Smith, D. E. and L. C. Karpinski, The Hindu-Arabic Numerals (Boston: Ginn, 1911). 14. Singh, S., Fermat’s Enigma (New York: Anchor Books, 1997). 15. Van der Waerden, B.L., Science Awakening (New York: John Wiley, 1963). 16. Zaslavsky, C., Africa Counts. Number and Pattern in African Culture (Boston: Prindle, Weber and Schmidt, 1973).

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