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

**Katz 1.5**

- How the Egyptians may have found their approximation to the area of a circle, (8/9)d
^{2}, where d is its diameter. (pp.20-21) - 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. (p.21) - 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.) (pp.20-21)

**Katz 1.8**

The Chinese geometric solution to problem #6 in the Jiu zhang (the reed in the pond). (p.33)

**Katz 1.9**

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)

**Katz 2.1.1**

The incommensurability of the side and diagonal of a square. (p.51)

**Katz 2.1.2**

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

**Katz 2.3.1**

Four logical forms of argument used in mathematical reasoning, with my "colloquial" names (p.55)

**Katz 2.3.2**

The Greek distinction between Numbers (The Discrete) and Magnitude (The Continuous), and its implications (pp.56-57)

**Katz 2.4.1**

Statement & Explanation: Euclid's five postulates (p.61). Statement: Euclid's five common notions (p.62)

**Katz 2.4.2**

- Statement & Proof: Euclid I-6 (p.65)
- Statement & Proof: Euclid I-29, Euclid's first use of Postulate 5 (p.66)
- Statement & Proof: Euclid I-32, Angle sum is two right angles (Ex.11, p.96)

**Dunham Chap. 2**

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

**Katz 2.4.3**

- Statement: Euclid II-4, Example of Geometric Algebra (p.68)
- Proof: Euclid II-11 (p.70)

**Katz 2.4.5**

Statement & Use: Euclidean Algorithm, gcd of two numbers (pp.78-79)

**Katz 2.4.6**

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

**Katz 2.4.7**

Statement & Translation: Euclid IX-35, Partial sums of geometric sequences (pp.87-88)

**Katz 2.4.8**

- Statement: Euclid X-1 (p.89)
- Statement: Euclid X-9 (p.89)

**Katz 2.4.9**

Statement & Proof: Euclid XII-2, Circles to diameters-squared (Method of Exhaustion) (pp.91-92)

**Katz Addendum**

Eratosthenes, Circumference of the Earth (Ex.41, p.99 & Class notes)

**Katz 3.1.1**

Archimedes "On the Equilibrium of Planes".

- Statements: Postulate 1 [Principle of Insufficient Reason], and Postulates 2-3 (p.105)
- Statement & Proof: Proposition 1 (p.105)
- Statement & Proof: Proposition 6, The Law of the Lever (p.106) (Commensurable case only)

**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 3.2**

Quadratrix of Hippias, description (p.110); Use in trisecting an angle. (Class notes)

**Katz 3.3.1**

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

- Ideas of 1st Proof: By the Law of the Lever (pp.111-112)
- Ideas of 2nd Proof: By the Sum of Series (pp.113-114)

**Katz 3.5.1**

Apollonius' Prop. I-33, drawing a tangent to a parabola (pp.123-124)

**Katz 3.5.3**

Archimedes' Trisection of an Angle with a marked straightedge (Class notes)

**Katz 4.2.1**

Ptolemy's Theorem and proof (p.148)

**Katz 5.2.1**

Diophantus' Arithmetica II-13 (Class notes & near p.177)

**Katz 6.3.1**

Sun Zi's first example of the "Chinese Remainder Problem" & solution (pp.197-198)

-- EXAM 1 approximately here --

**Katz 6.6.1**

Etymology of the word "sine" (p.213, sidebar)

**Katz 6.9**

Succinctly detail the history of the Hindu-Arabic decimal place-value number system, as shown in class. (pp.230-232)

**Katz 7.2.4 **

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)

**Katz 7.2.5**

Al-Khayyami's conic solution of the cubic x^{3} + cx = d, with sketches. (pp.260-261)

**Katz 7.4.3**

Al-Baghdadi's proof that between any two rational magnitudes (he uses 2 and 3) there exist infinitely many irrational magnitudes. (pp.272-273)

**Katz 8.4.2**

Statement of the Mean Speed Rule (Theorem) by Heytesbury (p.319), and Oresme's geometric proof of it (pp.320-321)

**Katz 8.4.2**

Oresme's statement and geometric proof in terms of time, velocity, and total distance of the fact that the infinite series (1/2)×
1 + (1/2^{2})×
2 + (1/2^{3})×
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.

**Katz 9.3.2**

Bombelli’s proof that the sum of the two cube roots (involving the square root of –121) equal 4 (his "wild thought"). (p.367)

**Katz 10.3.1**

Regiomontanus (Johannes Müller) Theorem II-1, The Law of Sines and proof (p.400)

**Katz 10.5.1**

Galileo's proof of the Mean Speed Rule (Theorem). (p.421)

**Katz 10.5.2**

Galileo's proof of his theorem on projectile motion (based on experiment and geometry). (p.424)

**Katz 11.1.1**

Fermat's straight line Theorem and proof. (p.435)

**Katz 11.3.2**

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).

**Katz 12.1.1**

Fermat's Method for finding a tangent line to a curve, illustrated for y = x^(1/2), using his "adequality" argument. (p.471)

**Katz 12.1.2**

Descartes' Circle Method for finding a normal line to a curve, as illustrated for the curve y = x^{2} (p.473).

**Katz 12.2.1**

Statement of Cavalieri's Principle (in the plane) (p.477).

Fermat's derivation of the area under the "higher parabola" y = px

**Katz 12.2.5**

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.

**Katz 12.2.6**

- 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. (pp.491-492)

**Katz 12.4.1**

Van Heuraet's construction to show that arc __length__ can be interpreted as the __area__ under a certain curve (pp.496-497).

**Katz 12.5.2**

Newton's derivation of the arcsin(x) series. (pp.508-509)

-- EXAM 2 approximately here --

**Katz 12.5.3 **

Newton's definitions of Fluxion and Fluent. (p.510)

**Katz 12.5.6**

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.

**Katz 12.6.2**

Leibniz' proof of the Transmutation Theorem (pp.525-526)

**Katz 12.6.2**

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.

**Katz 13.1.1**

Johann Bernoulli's solution of the brachistochrone problem. (pp.547-548)

**Katz 13.2.1**

Simpson's derivation of the rule d/dt(sin z) = cos(z) dz/dt using Roger Cotes' proof. (pp.561-562)

**Katz 13.2.2**

MacLaurin's analytic derivation of part of the Fundamental Theorem of Calculus for x^{n}. (p.564)

**Katz 13.2.5**

Euler's derivation of the quotient rule for derivatives. (p.571)

**Katz 13.5.1**

George Berkeley's argument in The Analyst. (pp.582-583)

**Katz 14.1.1**

Bernoulli's Law of Large Numbers (Class notes only)

**Katz 14.1.2**

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.]

**Katz 14.3.7**

Euler's Solution to the Konigsberg Bridge Problem (Class notes only)

**Dunham Chap. 11**

Great Theorem and Proof: Cantor's proof that **R** is __un__countable. 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)

-- Exam 3/Final Exam here --

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