MHF 3404  – HISTORY OF MATHEMATICS FLORIDA INT'L UNIV.
Review for Test #1 (April 9th)                                     FALL  2009

REMEMBER TO BRING AN 8’’x11” BLUE EXAM BOOKLET FOR THE TEST

KEY CONCEPTS AND MAIN DEFINITIONS:    Ch.1- Ch.7

The Egyptian method of multiplying and dividing numbers,  Egyptian fractions, the seked
     (reciprocal of the slope), Egyptian formulas for the area of the circle, area of a triangle,
     area of trapezoid, volume of a pyramid and volume of a frustrum.   The Babylonian
     method of finding square roots, Babylonian  formula for the area of a circle, Babylonian
     (Pythagorean) triples, Greek ratios, polygonal numbers, Pythagoras theorem, generating
     Pythagorean triples, The Eulidean algorithm for find greatest common divisors, The
     Arithmetic, Geometric, and Harmonic proportions;  The Bisection & Trisection of an
     arbitrary angle, Duplication of the Square & Cube, Squaring the Triangle & the Circle,
     Impossibility of solving the three classical problems using straight edge and compasses.
     The Quadratrix of Hippias and the trisection of an angle, Menaechmus Solution to the
     duplication of the cube, Dinostratus solution to squaring the circle, Axioms, Postulates,
     Hypothesis, and Theorems, Euclid's Deductive Geometry, Proof of the Pythagoras'  

     Theorem, Proof of the Euclid's formula for acute angled triangles, Curved surface area
     (CSA) of cones, truncated cones & sphere; and volume of sphere,  Areas of regions
     bounded by parabolas & the Archimedean spiral, Area of the triangle, The Julian and
     Gregorian Calendars, leap years, Heron's triangle formula, Solids formed by revolving
     plane regions, Surfaces formed by revolving plane curves, Centroids of plane regions,
     Centroids of plane curves, The two Theorems of Pappus.  

MAIN PROBLEM SOLVING TECHNIQUES:

1. Multiplying two numbers and diving one numb. by another using the Egyptian method
2.   Finding volumes of pyramids & cones, and frustrums (truncated pyramids & cones)
3.   Finding k-th approximation to the square root of a numb. using the Babylonian method

4.   Finding Pythagorean triples which include a given even or odd number.
5.   Finding the greatest common divisor of two integers by using Euclid's Algorithm.
6.   Finding the 3rd terms of the Arithmetic, Geometric, and Harmonic proportions
7.   Proving Pythagoras' Theorem and Euclid's Formula for acute-angled triangles

8.   Extracting square-roots digit by digit by the Theon of Alexandria Algorithm
9.   Justifying the formulas for the curved surface areas of the cone, frustrum, & sphere
10. Justifying the formulas for the volume of the solid frustrum and the solid sphere.
11. Determining which years are leap years according to the Gregorian Calendar of 1582.
12. Finding volumes and surface areas by using the two Theorems of Pappus.
13. Finding the centroids of a plane region R and of a plane curve C.

MAIN FORMULAS:

1.    AREA_Babylon(Circle) = (3D)²/12 = 3.r², AREA_Egypt(Circle) = (8D/9)² = (256/81).r²

2.    Vol.(pyramid or cone) = (h/3).(base B), Vol.(frustrum) = (h/3).[B₁ + (B₁.B₂)^1/2 + B₂]  
3.     A₀ = Goodish initial guess,    A₁ = (A₀ + n/A₀)/2,   . . . .,  A_k+1 = (A_k + n/A_k)/2  
4.     If a = u² - v², b = 2uv, c = u² + v²;  then  <a,b,c> & <b,a,c> are Pythagorean triples

5.     A_mean(a,b) = (a+b) / 2,      G_mean(a,b) = (a.b)^1/2,    H_mean(a,b) = (2.a.b) / (a+b)

6.     AC²  =  AB² + BC² (Pythagoras),   AC²  = AB² + BC² - 2.BC.BD  (Euclid)
7.     AREA_Archim (Circle) = r.(C/2),  C = 2.(Pi).r,    223/71 <  Pi  < 220/70 = 22/7   

8.     CSA(Cone) = (Pi).R.L, CSA(Frustrum) = (Pi).L.(R₁+R₂), CSA (Sphere) = 4(Pi).r²
9.     Volume(Sphere) = (4/3).Pi.r³,  Vol.(frustrum) =  (Pi)(h/3).[(R₁)² + (R₁.R₂)  +  (R₂)²]
10.   Area(Arch. Spiral) = (1/3).{a/(2.Pi).[Pi.(C.a)²],  Area (parabolic seg.) = (4/3) triangle           

11.   Heron's formula: Area (triangle) =  [s(s-a)(s-b)(s-c)]^1/2  where  s = (a+b+c)/2 

12.   Vol.(Solid of revol.) =  (area of region). (length of the path the centroid traces out)    
13.   Area(Surface of revol.) =  (length of curve).(length of the path the centroid traces out)
14.   x_c(R) = Int_{a to b}[x.(y₂-y₁)dx] / Area(R),     y_c(R) = Int_{a to b}[y.(x₂-x₁)dy] / Area(R),
15.   x_c(C) = Int[x.{1+(y')²}^1/2dx] / Length(C),  y_c(C) = Int[y.{1+(x')²}^1/2dy] / Length(C)