History of Mathematics - Entire List of Topics
(MATH 464 WI)

Subject to Change [Last Updated: Spring 2015]

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Katz

[1] Katz
The incommensurability of the side and diagonal of a square.

Katz
The three famous Greek construction problems and the restrictions under which they were to be solved.

[2] Dunham Chap. 1
Two Preliminaries (Triangle to Rectangle, to Square), followed by Great Theorem and Proof 1: Hippocrates of Chios Squares the Lune

Katz
Four logical forms of argument used in mathematical reasoning, with my "colloquial" names

Katz
The Greek distinction between Numbers (The Discrete) and Magnitude (The Continuous), and its implications

Katz
Statement & Explanation: Euclid's five postulates. Statement: Euclid's five common notions

Katz

Dunham Chap. 2

Katz

Katz
Statement & Use: Euclidean Algorithm, gcd of two numbers

[7] Katz
Statement & Proof: Euclid VI-12, "Construct x, so ax = bc" (Class notes)

[8] Dunham Chap. 3
Preliminaries (Defs of prime and composite), following by Great Theorem and Proof: Euclid's Proof of IX-20, infinitude of primes

Katz
Statement & Translation: Euclid IX-35, Partial sums of geometric sequences

Katz

[9] Katz
Statement & Proof: Euclid XII-2, Circles to diameters-squared (Method of Exhaustion)

Katz Addendum
Eratosthenes, Circumference of the Earth (Class notes)

Katz
Archimedes "On the Equilibrium of Planes".

[11] 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

Katz
Archimedes "The Method", Proposition 1, Area of a Parabolic Segment

[13] Katz
Apollonius' Prop. I-33, drawing a tangent to a parabola

[14] Katz
Archimedes' Trisection of an Angle with a marked straightedge (Class notes)

Katz
[15] Diophantus' Arithmetica I-28
[16] Diophantus' Arithmetica II-13

[17] Katz
Sun Zi's first example of the "Chinese Remainder Problem" & solution

-- EXAM 1 approximately here --

Katz
Succinctly detail the history of the Hindu-Arabic decimal place-value number system, as shown in class.
Etymology of the word "sine"

[18] Katz
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.

[19] Katz
Al-Khayyami's conic solution of the cubic x3 + cx = d, with sketches.

Katz
Al-Baghdadi's proof that between any two rational magnitudes (he uses 2 and 3) there exist infinitely many irrational magnitudes.

[20] Dunham Chap. 6
Great Theorem and Proof: Cardano's Solution of the Depressed Cubic. Be able to solve a cubic in the manner of Cardano.

[21] Katz
Bombelli’s proof that the sum of the two cube roots (involving the square root of –121) equal 4 (his "wild thought").

[22] Katz
Regiomontanus (Johannes Müller) Theorem II-1, The Law of Sines and proof

[23] Katz
Galileo's proof of his theorem on projectile motion (based on experiment and geometry).

[24] Katz
Fermat's straight line Theorem and proof.

[25] Katz
Pascal's proof of his solution to the de Mere problem on The Division of Stakes, Theorem, using induction, the arithmetical triangle, and his two "principles".

[26] Katz
Fermat's Method for finding a tangent line to a curve, illustrated for y = x1/2, using his "adequality" argument.

[27] Katz
Descartes' Circle Method for finding a normal line to a curve, as illustrated for the curve y = x2.

[28] Katz
Fermat's derivation of the area under the "higher parabola" y = pxk, from x = 0 to x = x0 > 0, using Roberval's inequality.

Katz
Roberval's proof that the area under half an arch of the cycloid is 3/2 times the area of the generating circle.

[29] Katz

-- EXAM 2 approximately here --

Katz
Newton's definitions of Fluxion and Fluent.

[30] Dunham Chap. 7
Great Theorem and Proof: Newton's Approximation of p.

[31] Katz
Newton's proof in "De Motu" of Theorem 1 (Kepler's 2nd Law)

[32] Katz
Leibniz' proof of the Transmutation Theorem (Handout), and
Leibniz' proof that p /4 = 1 - 1/3 + 1/5 -..., using his Transmutation Theorem.

[33] Dunham Chap. 8
Great Theorem and Proof: Johann Bernoulli's proof that the harmonic series diverges.

[34] Katz
Simpson's derivation of the rule d/dt(sin z) = cos(z) dz/dt using Roger Cotes' proof.

[35] Katz
MacLaurin's analytic derivation of part of the Fundamental Theorem of Calculus for xn.

[36] Dunham Chap. 9
Great Theorem and Proof: Euler's Proof that

[37] Katz
Euler's derivation of the quotient rule for derivatives.

Katz
George Berkeley's argument in The Analyst.

[38] Courant & Robbins
Proof that e is irrational.

[39] Katz
DeMoivre's Problem III and Solution from The Doctrine of Chances.

[40] 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.]

Katz
Saccheri's 1733 Proposition XVII.

[41] Katz
Euler's Solution to the Konigsberg Bridge Problem (Class notes)

[42] Dunham Chap. 11
Great Theorem and Proof: Cantor's proof that R is uncountable.

[43] 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)

-- Exam 3/Final Exam here --

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