History of Mathematics - Entire List of Topics
The Chinese geometric solution to problem #6 in the Jiu zhang (the reed in the pond). (p.33)
The Babylonian geometric solution to the generic problem "Given x+y=b and xy=c, b and c known constants, find x and y". (pp.36-37)
The incommensurability of the side and diagonal of a square. (p.51)
The three famous Greek construction problems and the restrictions under which they were to be solved. (pp.51-52)
Dunham Chap. 1
Two Preliminaries (Triangle to Rectangle, to Square), followed by Great Theorem and Proof 1: Hippocrates of Chios Squares the Lune
Four logical forms of argument used in mathematical reasoning, with my "colloquial" names (p.55)
The Greek distinction between Numbers (The Discrete) and Magnitude (The Continuous), and its implications (pp.56-57)
Statement & Explanation: Euclid's five postulates (p.61). Statement: Euclid's five common notions (p.62)
Dunham Chap. 2
Statement & Use: Euclidean Algorithm, gcd of two numbers (pp.78-79)
Statement & Proof: Euclid VI-12, "Construct x, so ax = bc" (Class notes & p.83)
Dunham Chap. 3
Preliminaries (Defs of prime and composite), following by Great Theorem and Proof: Euclid's Proof of IX-20 infinitude of primes
Statement & Translation: Euclid IX-35, Partial sums of geometric sequences (pp.87-88)
Statement & Proof: Euclid XII-2, Circles to diameters-squared (Method of Exhaustion) (pp.91-92)
Eratosthenes, Circumference of the Earth (Ex.41, p.99 & Class notes)
Archimedes "On the Equilibrium of Planes".
Dunham Chap. 4
Four Preliminaries (See Class notes), followed by Great Theorem and Proof: Archimedes "On the Measurement of The Circle", Proposition 1, The Area of a Circle
Quadratrix of Hippias, description (p.110); Use in trisecting an angle. (Class notes)
Archimedes "The Method", Proposition 1, Area of a Parabolic Segment
Apollonius' Prop. I-33, drawing a tangent to a parabola (pp.123-124)
Archimedes' Trisection of an Angle with a marked straightedge (Class notes)
Ptolemy's Theorem and proof (p.148)
Diophantus' Arithmetica II-13 (Class notes & near p.177)
Sun Zi's first example of the "Chinese Remainder Problem" & solution (pp.197-198)
-- EXAM 1 approximately here --
Etymology of the word "sine" (p.213, sidebar)
Succinctly detail the history of the Hindu-Arabic decimal place-value number system, as shown in class. (pp.230-232)
Al-Karaji's theorem (for n=10) that the sum of the cubes equals the square of the sum, & his "inductive style" proof with picture.(p.255)
Al-Khayyami's conic solution of the cubic x3 + cx = d, with sketches. (pp.260-261)
Al-Baghdadi's proof that between any two rational magnitudes (he uses 2 and 3) there exist infinitely many irrational magnitudes. (pp.272-273)
Statement of the Mean Speed Rule (Theorem) by Heytesbury (p.319), and Oresme's geometric proof of it (pp.320-321)
Oresme's statement and geometric proof in terms of time, velocity, and total distance of the fact that the infinite series (1/2)× 1 + (1/22)× 2 + (1/23)× 3 +... = 2.
Dunham Chap. 6
Great Theorem and Proof: Cardano's Solution of the Depressed Cubic. [Corresponds to Katz 9.3.1, pp.362-363.] Be able to solve a cubic in the manner of Cardano.
Bombelli’s proof that the sum of the two cube roots (involving the square root of –121) equal 4 (his "wild thought"). (p.367)
Regiomontanus (Johannes Müller) Theorem II-1, The Law of Sines and proof (p.400)
Galileo's proof of the Mean Speed Rule (Theorem). (p.421)
Galileo's proof of his theorem on projectile motion (based on experiment and geometry). (p.424)
Fermat's straight line Theorem and proof. (p.435)
Pascal's proof of his solution to the de Mere problem on The Division of Stakes, Theorem (pp.455- 456), using induction, the arithmetical triangle, and his two "principles" (from p.451, bottom).
Fermat's Method for finding a tangent line to a curve, illustrated for y = x^(1/2), using his "adequality" argument. (p.471)
Descartes' Circle Method for finding a normal line to a curve, as illustrated for the curve y = x2 (p.473).
Statement of Cavalieri's Principle (in the plane) (p.477).
Roberval's proof that the area under half an arch of the cycloid is 3/2 times the area of the generating circle (pp 489-490), as presented in class.
Van Heuraet's construction to show that arc length can be interpreted as the area under a certain curve (pp.496-497).
Newton's derivation of the arcsin(x) series. (pp.508-509)
-- EXAM 2 approximately here --
Newton's definitions of Fluxion and Fluent. (p.510)
Newton's proof in "De Motu" of Theorem 1 (Kepler's 2nd Law) (pp.516-517)
Dunham Chap. 7
Great Theorem and Proof: Newton's Approximation of p.
Leibniz' proof of the Transmutation Theorem (pp.525-526)
Leibniz' proof that p /4 = 1 - 1/3 + 1/5 -..., using his Transmutation Theorem. (p.526)
Dunham Chap. 8
Great Theorem and Proof: Johann Bernoulli's proof that the harmonic series diverges.
Johann Bernoulli's solution of the brachistochrone problem. (pp.547-548)
Simpson's derivation of the rule d/dt(sin z) = cos(z) dz/dt using Roger Cotes' proof. (pp.561-562)
MacLaurin's analytic derivation of part of the Fundamental Theorem of Calculus for xn. (p.564)
Euler's derivation of the quotient rule for derivatives. (p.571)
George Berkeley's argument in The Analyst. (pp.582-583)
Bernoulli's Law of Large Numbers (Class notes only)
DeMoivre's Problem III and Solution from The Doctrine of Chances. (p.601)
Dunham Chap. 9
Great Theorem and Proof: Euler's Proof that
Courant & Robbins
Proof that e is irrational.
Dunham Chap. 10
Great Theorem and Proof: Euler refutes Fermat's conjecture (4 theorems). [Preliminary Theorems A, B, C can be collapsed into one general result.]
Euler's Solution to the Konigsberg Bridge Problem (Class notes only)
Dunham Chap. 11
Great Theorem and Proof: Cantor's proof that R is uncountable. Def of denumerable (countable); Proof that Q is denumerable (countable) (pp.251-258)
Dunham Chap. 12
Great Theorem and Proof: Cantor's proof that the cardinality of any set is strictly less than the cardinality of its set of subsets (its power set)
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