MATH 400- HISTORY OF MATHEMATICS- SUMMER 2006- COURSE LOG

6/5       discussion of syllabus and course outline/ main periods in Ancient Greek history/
Py/thagoras and his school/ Pythagoras' theorem (Euclid I.47) and converse (Euclid I.48)

reading assignment: Struik Ch. III (Greece), handout below
Notes on Ancient Greek Mathematics

6/6        Pythagorean triples (also Plato's method); an example of geometric algebra (Euclid II.11)/
Zeno's paradoxes/ resolution by Eudoxus of Cnidos: proportions for incommensurable
magnitudes/ The three problems of Antiquity/ Hipparchus: area of a lune (exercise),
reduction of doubling the cube to two mean proportions (exercise) / Solution by Menaechmus
as the intersection of parabolas/ Euclid's Elements: the five postulates.

A book review: Dirk Struik's `Concise History of Mathematics'
(appeared in Notices of the American Mathematical Society)

6/7       Euclid's Elements/ Book II: a^2-b^2 (exercise) /Books I-IV: propositions leading to construction of a regular pentagon/
facts about constructibility; Gauss and Fermat `primes'/the number theory books, VII-X: existence of infinitely many primes/
perfect numbers and Mersenne primes; Euler's converse/ the solid geometry books, XI-XIII: proof that there are only five regular solids.

reading assignment (for Friday): Struik ch. VI (the seventeenth century), and the handout.

6/8        Archimedes: On the sphere and the cylinder (two exercises- tomb problem (1pt), spherical segment (1pt)/ Area of a disk by the mehtod
of exhaustion/ area of a parabolic segment (exercise: error term in geometric series-1pt)
Method of exhaustions in Archimedes: two examples
(these are the examples done in class; without the diagrams for now)
Other work: Archimedean polyhedra, On Floating Bodies, The Sand Reckoner (handout)
ARCHIMEDES
Link to an interesting Drexel U web site- be sure to check out the Archimedean solids and `On Floating Bodies'!

6/9        Seventeenth Century: Analysis before Newton and Leibniz: outline of European history/ Descartes and analytic geometry/
Cavalieri and integration of `higher parabolas'/ Fermat: maxima and minima
European history in a nutshell, 1648-1727
Calculus before Newton and Leibniz
(selections by Descartes, Fermat, Cavalieri, from Struik's `source book'.)

BIOSKETCH ASSIGNMENT

6/12         A paradox in Cavalieri's infinitesimal approach/ Newton chronology/Newton's general binomial theorem/
quadrature of rational powers/inversion of power series and the series for sin x.
Isaac Newton (1642-1727)-chronology
Newton's Principia-selection of results

6/13        Leibniz chronology/ the `transmutation theorem' and integration by parts/Leibniz's series for pi/4
The structure of Newton's Principia
Leibniz chronology( 1646-1716)

6/14         TEST 1  (with some answers)
Newton's Principia:  comments on selected results from Books 1 and 2
Some results from the Principia in modern notation  (with 4 exercises.)

6/15         Johann and Jakob Bernouilli:  sum of series related to the geometric series/the Brachystochrone problem
(Jakob's first solution- handout from Struik's `source book'). Graded test 1 returned.

6/16         Logarithms: Napier, Briggs, the Newton/Mercator expansion of hyperbolic area/ Euler: introduction of the constant e;

(sources: [Dunham-Euler, Havil-Gamma])/ logarithmic divergence of the harmonic series and the Euler-Mascheroni constant
(sources: same).  Euler's summation method via infinite products: solution of the Basel problem [Dunham-Gallery] (exercise: fourth powers)

Reading assignment (for Monday):  Struik ch. VII (eighteenth century);  Derbyshire on Euler (p. 55-62, p 99-107)
Mathematics in the 18th century- selective chronology

6/19         The gamma function (sources: [Dunham-Gallery, Havil-Gamma])/ The zeta function, primes and Euler's product formula  Sources: [Dunham-Euler],
[Derbyshire- p.99-107]

6/20         Euler's 1744 paper on the calculus of variations/ the least action principle in mechanics/ Maupertuis-  measurement of the
shape of the earth and the least action principle priority dispute/ Lagrange's 1755 letter to Euler and 1760 paper- the Euler-Lagrange equations
in  Lagrange's formulation

6/21         d'Alembert , Euler and D.Bernouilli on the 1-D wave equation/ development of the function concept (handout)/ The equations of fluid mechanics
in the Euler and Lagrange formulations
Reading assignment:  `first interlude' in [Dunham-Gallery]

6/22          Elliptic integrals and elliptic functions- from Fagnano and Euler  to Legendre
Elliptic functions-historical introduction (translation)

6/23          Gauss and  the heptadecagon/ Laplace and the central limit theorem
First biosketch due
Timeline- selected mathematicians, 1800-1900
C.F. Gauss (1777-1855)- chronology

6/26           TEST 2- material from 6/14 to 6/23 (as test 1- 15 min closed book, 30 min open book)
Fourier series and transforms- Fourier, Poisson, Dirichlet
See [Struik VIII.6, VIII.12]
TEST 2, part 1          TEST 2, part 2 (with answers)

6/27           Cauchy- mean value theorem, approach to integration, fundamental theorem of calculus, Cauchy-Riemann equations

6/28           Riemann's habilitation thesis: structure of the paper/  the Riemann integral, integrability condition;
example of an integrable function with dense discontinuity set/  decay of Fourier coefficients of integrable functions.
[Dunham ch. 7]  G.F.B. Riemann (1826-1866)- chronology
Riemann's papers   (in German; if you find them in translation, let me know.)

6/29           Liouvillle- Liouville's inequality, existence of transcendental numbers/ transcendence of e (Hermite)

[Dunham ch. 8] transcendence of e  (this proof was presented in class, except for part (5); the handout has 4 exercises-due 7/7).

6/30           Weierstrass- uniform continuity and uniform convergence/ a continuous, nowhere-differentiable function [Dunham ch.9]

7/3            The Prime Number Theorem, from Gauss and Legendre to Hadamard and de la Vallee Poussin
[Derbyshire, part 1]  Prime Number Theorem- key sections in text
Riemann's 1859 paper on the Prime Number Theorem (in English):
On The Number of Prime Numbers Less Than a Given Quantity

7/4            INDEPENDENCE DAY (no class)
Reading assignment: `second interlude' in [Dunham-Gallery]

7/5            Sets of discontinuity, discontinuity of derivatives and of integrable functions: 5 examples and 3 questions.
Volterra's theorem on sets of discontinuity; Darboux's results on the intermediate value theorem and the FTC;
Baire's results:  nowhere dense sets and meager sets; characterization of sets of discontinuity for functions continuous on a
dense set;  continuity of the derivative on a dense set. [Dunham- 2nd interlude, chapters on Volterra and Baire]

7/6             Cantor: uncountability of intervals, transcendental numbers are dense. Remarks on meager sets and Baire's results.
Cantor's and Dedekind's constructions of the real numbers from the rationals. Sets of measure zero. Lebesgue's
characterization of the discontinuity set of a Riemann-integrable function. The Cantor middle-thirds set: construction
and properties (no proofs.)  [Dunham- chapters on Cantor and Lebesgue]   Second biosketch collected.

7/7            TEST 3 (45 min)- material from 6/26 to 7/6
TEST 3, part 1(closed book)
TEST 3, part 2 (open book)