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

**Katz**

- How the Egyptians may have found their approximation to the area of a circle, (8/9)d
^{2}, where d is its diameter. - How the Babylonians and Chinese may have found their approximation to the area of a circle, (3/4)d
^{2}, where d is its diameter. - How the Babylonians and Chinese may have found the correct formula for the area of a circle, (C/2)(d/2), where C is circumference and d diameter. (Two different methods were shown.)

**[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**

**[3]**Statement & Proof: Euclid I-5- Statement & Proof: Euclid I-29, Euclid's first use of Postulate 5
- Statement & Proof: Euclid I-32, Angle sum is two right angles

**Dunham Chap. 2**

**[4]**One Preliminary (Euclid I-41), followed by Great Theorem and Proof: Euclid's Proof of I-47, the Pythagorean Theorem**[5]**Proof of Euclid I-48, the Pythagorean Theorem Converse

**Katz**

- Statement: Euclid II-4, Example of Geometric Algebra
**[6]**Proof: Euclid II-11

**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**

- Statement: Euclid X-1
- Statement: Euclid X-9

**[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".

- Statements: Postulate 1 [Principle of Insufficient Reason], and Postulates 2-3
- Statement & Proof: Proposition 1
**[10]**Statement & Proof: Proposition 6, The Law of the Lever (Commensurable case only)

**[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

**[12]**Ideas of 1st Proof: By the Law of the Lever- Ideas of 2nd Proof: By the Sum of Series

**[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 x^{3} + 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 = x^{1/2}, using his "adequality" argument.

**[27] **Katz

Descartes' Circle Method for finding a normal line to a curve, as illustrated for the curve y = x^{2}.

**[28] ****Katz**

Fermat's derivation of the area under the "higher parabola" y = px^{k}, from x = 0 to x = x_{0 }> 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

- Gregory of St. Vincent's Theorem and proof about a relationship between certain points under the hyperbola xy = 1 and certain areas under that curve.
- The derivation of de Sarasa's observation about how this relates to the logarithmic product rule.

-- 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 x^{n}.

**[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 __un__countable.

**[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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