The following is a list of 47 papers written by John Kennedy. The topics vary from pure math and logic to computer graphics and probability, to parse trees and fundamental algorithms and techniques associated with numerical analysis and computing in general. Other miscellaneous topics include solving cubic and quartic equations, Huffman coding related to file compression, modular arithmetic, the basic geometry of camera lenses and magnifying glasses, the geometry of a sextant, a model of a piston, the mathematics of Vernier scales, rounding functions, regular polyhedra, trochoidal curves, dihedral groups and permutations, how to steer a pirate ship, and miscellaneous computational algorithms. For the most part these papers discuss applications of math or computer science. The papers listed below are in no particular order and vary in terms of a required math background, but most are short and simple. The major exception to this statement is the paper on error correcting codes.

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RPN Perspective
This paper provides a historical background for the Reverse Polish Notation and was originally published in the PPC Calculator Journal, Volume 9 Number 5, August 1982 pp. 26-29. This paper sets the historical record straight for when the Polish Notation was first invented. The time frame was in the year 1928, long before computers or calculators. This paper gives a logical background for understanding how a Reverse Polish Logic calculator works. It explains the relationship between logical and mathematical notations and explains the fundamentals of a parenthesis-free notation that can be used to represent mathematical expressions. It also discusses parallel meanings between logic and mathematics.
11 pages.
RPNBG.PDF  53,208 bytes   11/22/2009  4:40 pm.

A Brief Introduction To Bezier Curves
This paper describes what Bezier curves are and derives the basic cubic polynomial equations that are used to generate Bezier curves. It explains the amazing midpoint property of such curves and shows examples of different types of Bezier curves. It also shows how to make a smooth joining of two such curves and this discussion is a great aid in understanding why the mathematical definition of a smooth curve is one that requires a continuous first derivative. This paper also discusses the requirements for exact symmetry of Bezier curves and illustrates this with several more examples. The paper also shows how to construct points on a Bezier curve using a fixed ratio to make points that divide the lengths of several related vectors. The fixed ratio is a generalization of the midpoint property shown earlier. The algorithm described at the end of this paper is due to Casteljau and this algorithm is related to what are called the Bernstein polynomials. The paper concludes by showing relationships between text and Bezier curves.
32 pages.
BEZIER.PDF  142,739 bytes   12/23/2009  9:55 am.

The Basic Geometry Behind A Camera Lens And A Magnifying Glass
This paper describes the fundamental workings of a camera lens and a magnifying glass. The only math required is an understanding of the point-slope form for the equation of a line. This paper describes the physics behind a convex lens and derives what is called the Thin-Lens equation. It also describes the practical uses of the focal length. It explains the use of the magnification factor and its relationship to focal lengths.
11 pages.
CAMLENS.PDF  182,700 bytes   11/22/2009  4:43 pm.

A General Divisibility Test and the Use of Synthetic Substitution
This paper describes a little-known divisibility test for integer division that has wide applicability. In fact, the test can be applied to any integer with any integer divisor. The test makes use of synthetic substitution and takes advantage of viewing any integer as a special polynomial value. This paper also explains the relationship between this test and other popular divisibility tests such as the tests for even division by 2, 3, 5, 7, and 11.
10 pages.
DIVTEST.PDF  50,780 bytes   11/22/2009  4:43 pm.

A Gallery of Rounding and Truncating Functions
This paper discusses many ways in which numbers can be rounded or truncated. It contains graphs of the five fundamental functions that may be called Int(X), Frac(X), Round(X), Ceiling(X), and Floor(X). This paper carefully discusses and shows the relationships between all of these functions. It also shows how to define a robust modulo operator for real numbers and shows why we might want to consider Int(X) and Frac(X) as the two fundamental rounding-related functions. It shows how to use the Int(X) and Frac(X) functions to round X to any decimal position and how to extract specific digits from X. It also shows how to define several utility type functions related to Int(X) and Frac(X). In total, there are 15 rounding/truncating related functions and this paper explains how all of these are related to the two fundamental Int(X) and Frac(X) functions. This paper is a must read for anyone whose does almost any kind of either discrete or floating point arithmetic using any kind of a computer or any kind of mathematical software.
20 pages.
GALLERYROUND.PDF  120,021 bytes   11/22/2009  4:44 pm.

Huffman Coding
This paper introduces the concept of Huffman coding that is concerned with file compression. This paper also discusses Morse Code, an early example of file compression that pre-dates the computer by 100 years. The discussion includes the algorithm that can be used to create an efficient coding scheme for any kind of data.
10 pages.
HUFFMAN.PDF  41,839 bytes   11/22/2009  4:45 pm.

The Pure Cubic Polynomial and Intersections With Linear Functions
This paper contains five fundamental theorems that describe the three distinct ways in which a linear function may intersect the graph of Y=X^3. The results are simple, but somewhat surprising because we have never seen them summarized as they are here, in any PreCalculus text. This paper includes a ruler and compass construction method for drawing a tangent line the curve Y=X^3 at any point on that curve. After reading this paper you will be better prepared to read the paper on Solving Cubic Equations.
8 pages.
CUBICANDLINEARRELATIONSHIP.PDF  50,652 bytes   12/09/2009  5:54 pm.

Solving Cubic Equations
This paper demonstrates the techniques used by del Ferro, Cardano, Bombelli and Viete to solve cubic equations. This paper contains examples showing exact radical solutions and shows other solutions that are difficult to put in an exact form. The paper finishes by showing a technique that uses both trigonometric and hyperbolic functions to solve cubics. See also the paper that discusses the relationship between linear functions and the pure cubic curve Y=X^3.
10 pages.
SOLVINGCUBICS.PDF  65,012 bytes   01/24/2010  3:06 pm.

Solving Quartic Equations
This paper demonstrates using completing the square techniques to solve 4th degree polynomial equations. Such solutions also involve using the techniques to solve cubic equations. See also the paper that discusses how the pure quartic Y=X^4 can intersect with a general quadratic polynomial.
6 pages.
SOLVINGQUARTICEQUATIONS.PDF  39,886 bytes   11/20/2009  3:03 pm.

The Pure Quartic and Intersections With Quadratic Polynomials
This paper contains six fundamental theorems that describe and summarize the various ways in which a quadratic polynomial can intersect with the pure quartic Y=X^4. These results should be in any PreCalculus text, but we have never seen them. This paper is related to another paper on solving quartic equations.
15 pages.
QUARTICQUADRATICINTERSECTIONS.PDF  93,030 bytes   01/25/2010  10:50 am.

Prime Factoring The Factorial of an Integer
This paper shows how to predict the multiplicity of any prime factor of n!. It shows practical applications in terms of how to compute large combinations and large binomial coefficients using multiple precision arithmetic. It also contains an intellectually interesting formula for the prime factorization of n!
7 pages.
NFACT.PDF  42,404 bytes   11/22/2009  4:47 pm.

An Efficient Algorithm For Computing Large Integer Powers of Any Base
This paper discusses an efficient technique for computing b^n where n is a large integer, and b can be any type of base including a real number, a polynomial, a matrix, or any other base type for which multiplication is defined. The algorithm depends on the bit pattern found in the binary representation of the integer n. The paper also describes the relationship between this algorithm and one used to multiply any two integers that involves only shifting bits and adding and subtracting. This algorithm is also known as the Fast Exponential Method and has applications in modular arithmetic and cryptography.
9 pages.
NPWR.PDF  43,864 bytes   11/22/2009  4:48 pm.

Parse Trees
This paper discusses parse trees and shows how to construct on paper the parse tree for any mathematical or logical expression. Parse trees are recursive data structures and this paper also discusses recursive algorithms that can be used to traverse parse trees. Two examples are given of functions that can be used to compute any math expression or any symbolic logic expression. Another part of this paper discusses the relationship between a parse tree and a stack for the purpose of performing calculations like a Reverse Polish logic calculator.
10 pages.
PARSETREES.PDF  62,098 bytes   11/22/2009  4:49 pm.

How Does A Sextant Work?
This paper describes the basic geometry behind a sextant which is a simple navigational device with mirrors that is used to determine the angle between an object in the sky and the horizon.
5 pages.
SEXTANT.PDF  252,465 bytes   11/22/2009  4:50 pm.

A Piston Model
This paper shows a model of a piston head and derives the basic equation that governs the motion of a piston head as it moves in a periodic way, back and forth, inside a cylinder. This paper explains why the periodic motion of a piston head is NOT a pure sine wave, even though at first glance the graph may look very similar to a sine wave. We also calculate and analyze the velocity of the piston head and determine where it moves the fastest and where it moves the slowest. We also show how the angular velocity of a turning crankshaft relates to the linear speed of the piston head as it traverses back and forth. A piston moving back and forth inside a cylinder provides a means of converting linear motion to circular motion and vice versa. Such a model is fundamental to understanding just one aspect of both fluid pumps and internal combustion gasoline engines.
5 pages.
PISTONMODEL.PDF  60,304 bytes   11/22/2009  4:51 pm.

A Remarkable Half-Angle Relationship
This paper shows a simple figure related to the unit circle and a special line that intersects that circle. The slope of that line is related to points on the unit circle and three application examples are given. One of those applications involves a special integration technique by Karl Weierstrass (1815-1897), commonly taught in 2nd semester calculus. Another application is related to finding rational points on the unit circle and suggests a way of thinking about a class of Diophantine equations. A third application is a ruler and compass construction using the unit circle and a couple of special lines to construct the reciprocal of a number.
8 pages.
HALFANGLESPECIAL.PDF  69,119 bytes   11/22/2009  4:56 pm.

The Simplex Algorithm Divorced From Inequalities
This paper was written to explain the basis for the Simplex Algorithm and to show how the algorithm arises naturally in the context of using matrices to solve linear systems. A comparison is made with the steps used to find the row reduced echelon form of a matrix. The algorithm is carefully explained and an example is given of cycling. This paper also discusses nonstandard problems and the use of artificial variables.
18 pages.
SIMPLEXP.PDF  96,001 bytes   11/22/2009  4:57 pm.

Subtraction Using Complements
This short paper describes a simple algorithm for performing subtraction of integers. The algorithm is unusual because it describes subtraction as a special process involving only complements and addition. Examples are given in base 10 and the general algorithm is explained using any base b where b is 2 or larger. When b=2 the algorithm is used to explain how a computer actually performs subtraction by performing a special kind of addition. This paper also briefly discusses problems that must be avoided when computers perform computations with numbers.
4 pages.
SUBTRACTIONUC.PDF  35,850 bytes   11/22/2009  4:58 pm.

Signed Angles and the Perpendicular Vector Concept Applied to Find the Area of a Polygon
This paper introduces the concept of the signed angle between two vectors and combines this with the idea of a particular vector perpendicular to a given vector to obtain a most simple and efficient formula that computes the area of any polygon in two-dimensional space. A special sum of dot products using the polygon's vertex point coordinates is all that is required to compute the area enclosed by the polygon.
12 pages.
VECTORPERP.PDF  70,683 bytes   11/22/2009  4:59 pm.

How Does A Vernier Scale Work?
This paper describes the basic math behind a Vernier scale. Vernier scales are often found on scientific instruments and make it possible to read values accurately to within 1/100 of a unit.
7 pages.
VERNIERSCALE.PDF  480,840 bytes   11/22/2009  5:00 pm.

Fractions And Decimals
This paper was published in the PPC Calculator Journal, Volume 5 Number 6, July, 1978 pages 17-19. This paper explains the fundamental relationships between fractions and decimals and explains the relationship between finite decimals and periodic decimals. The results can be used to predict both the non-periodic and the periodic lengths of any repeating decimal and it explains when, why and how certain fractions result in finite decimals. It also contains the surprising result that any integer which does not have 2 or 5 as a prime factor has some multiple which consists of all 9's. This paper also mentions applications related to multiple precision arithmetic and cryptography.
8 pages.
FRACDEC.PDF  52,065 bytes   11/22/2009  5:01 pm.

Some Polynomial Theorems
This paper contains a collection of 31 theorems, lemmas, and corollaries that help explain some fundamental properties of polynomials. The statements of all these theorems can be understood by students at the precalculus level, even though a few of these theorems do not appear in any precalculus text. However, to understand the proofs requires a much more substantial and more mature mathematical background, including proof by mathematical induction and some simple calculus. Of significance are the Division Algorithm and theorems about the sum and product of the roots, two theorems about the bounds of roots, a theorem about conjugates of irrational roots, a theorem about integer roots, a theorem about the equality of two polynomials, theorems related to the Euclidean Algorithm for finding the G.C.D. of two polynomials, and theorems about the Partial Fraction Decomposition of a rational function and Descartes' Rule of Signs. It is rare to find proofs of either of these last two major theorems in any precalculus text.
30 pages.
POLYTHEOREMS.PDF  122,515 bytes   11/22/2009  5:03 pm.

Permutations, Regular Polygons, Symmetries, Rotations, Reflections, Dihedral groups, Orthogonal Matrices, Isometries, Cycles, Transpositions, Parity, And All That Jazz
This paper develops some of the fundamental properties of permutations by considering the properties of linear transformations that map regular polygons onto themselves. This paper shows relationships among permutations, regular polygons, symmetries, rotations, reflections, orthogonal matrices, isometries, and dihedral groups. It also discusses permutations as products of cycles and transpositions and introduces the idea of a Cauchy number for a permutation. This in turn leads to the parity properties of permutations. This paper assumes a minimal background in linear algebra. Some experience with groups or group theory would also be helpful.
24 pages.
PERMSANDROTS.PDF  421,210 bytes   12/11/2009  12:19 pm.

Equivalence Relations And Partitions
This paper develops the fundamental properties of equivalence relations. It also establishes the connections between equivalence relations and partitions of a set. This paper shows how every equivalence relation determines a unique partition, and vice versa, it shows how every partition determines a unique equivalence relation. In fact, this paper begins by defining an ordered pair in terms of sets. This paper is for mature students who would like to see applications within the theory of sets.
4 pages.
EQUIVALENCERELATIONS.PDF  41,326 bytes   01/09/2010  1:59 pm.

Abstract Algebraic Theories Applied To Error Correcting Codes
This paper is an extensive set of notes that develops basic properties of the two abstract algebra constructs known as groups and fields. In particular, it develops some of the basic properties of finite fields known as Galois fields. After establishing a mathematical foundation, these notes discuss examples of error detecting and error correcting codes. Included are linear codes, Hamming codes, cyclic codes, BCH (Bose-Chadhuri-Hocquenghem) codes, and Reed-Solomon codes. This is a very long paper and requires a strong math background and some mathematical maturity. It is recommended for students who have completed at least one course in Linear Algebra and who are interested in advanced applications of mathematical theories.
189 pages.
ERRORCODESTHEORY.PDF  837,416 bytes   02/07/2010  11:35 am.

Algorithm To Convert A Decimal To A Fraction
This paper describes an algorithm that can be used to convert an arbitrary decimal to a fraction. This technique depends on continued fractions and is of major importance in numerical analysis. The algorithm has been implemented on Hewlett-Packard scientific calculators and is also in Mathematica and MatLab.
4 pages.
DEC2FRAC.PDF  26,131 bytes   11/22/2009  5:07 pm.

Some Very Special Trigonometric Function Values
This paper develops the exact values of both sine and cosine functions for angles that are multiples of 9 degrees. The values are given in an exact radical form. Then this paper shows how these values relate to the Golden Ratio and to a pentagon and also how they relate to 3D polyhedra such as the dodecahedron and the icosahedron.
10 pages.
SPECIALTRIGVALUES.PDF  67,585 bytes   11/22/2009  5:08 pm.

Establishing Regular Polyhedra Information
This paper develops formulas for the surface area and volume of regular polyhedra as well as formulas for the radii of both inscribed and circumscribed spheres. Even more important it develops the exact values of the dihedral angles for all five of the regular Platonic solids. The five regular polyhedra are the tetrahedron, the hexhedron or cube, the octahedron, the dodecahedron, and the icosahedron.
28 pages.
ESTABLISHINGPOLYHEDRAINFO.PDF  187,969 bytes   11/22/2009  5:09 pm.

Earth Navigation and the Estimation of Dip
This paper describes some basic earth navigational facts and uses some mathematics to derive an estimation for what is called Dip. Dip is a phenomena that is associated with the refraction of light. It is essential to know about Dip in order to make effective use of a sextant.
6 pages.
NAVIGATION01.PDF  70,488 bytes   11/22/2009  5:10 pm.

How To Steer A Pirate Ship
This paper is all about setting up and solving a differential equation related to a classical problem of pursuit. Although the problem is stated in terms of a pirate ship, the same result could be applied to the case of a dog chasing a cat or a lion chasing a zebra. This paper assumes you have completed two semesters of calculus.
13 pages.
PIRATESHIP.PDF  78,857 bytes   11/22/2009  5:11 pm.

3-Dimensional Graphing Transformations
This paper discusses the fundamental mathematics and algorithms behind making 3-dimensional perspective drawings. It describes how to transform world coordinates to eye coordinates and how to perform 3D clipping with respect to a viewing pyramid and how to map eye coordinates onto the viewing plane and from there to the final display device plane.
18 pages.
3DSETUP.PDF  215,765 bytes   08/21/2009  10:32 am.

Bresenham's Integer Only Line Drawing Algorithm
This is a fundamental paper on how to efficiently draw a line on any display device that is composed of discrete pixels. The fundamental ideas in this paper can also be applied to drawing circles and ellipses. This contains what is perhaps the first and most fundamental techniques of all computer-based graphics. See also the related papers for drawing circles and ellipses.
8 pages.
BRESENL.PDF  45,305 bytes   11/22/2009  5:12 pm.

A Fast Bresenham Type Algorithm For Drawing Circles
This paper describes how to quickly and efficiently draw a circle on any display device that is composed of discrete pixels. It contains an algorithm that takes advantage of the circle's 8-point symmetry and minimizes the errors associated with finding the nearest pixel for each next point to be plotted. The algorithm uses exact integer arithmetic only. See also the related papers for drawing ellipses and lines.
7 pages.
BCIRCLE.PDF  43,325 bytes   11/22/2009  5:13 pm.

A Fast Bresenham Type Algorithm For Drawing Ellipses
This paper describes how to quickly and efficiently draw an ellipse on any display device that is composed of discrete pixels. It contains an algorithm that takes advantage of the ellipse's 4-point symmetry and minimizes the errors associated with finding the nearest pixel for each next point to be plotted. The algorithm uses exact integer arithmetic only. See also the related papers for drawing circles and lines.
7 pages.
BELIPSE.PDF  47,801 bytes   11/22/2009  5:14 pm.

Continued Fractions
This paper is an introduction to continued fractions. It introduces the notation and terminology related to continued fractions and presents two Fundamental Theorems about continued fractions. It shows how to solve a quadratic equation using continued fractions. It also illustrates how to convert an arbitrary decimal to a fraction. This algorithm is of fundamental importance. This paper also introduces a matrix notation for both simple and general continued fractions and shows examples of constants and function expressions in continued fraction forms. These forms are interesting alternatives when compared to infinite series for the same constants and functions.
20 pages.
CONFRAC.PDF  81,625 bytes   11/22/2009  5:15 pm.

The Theory of Identity
This is a technical paper written primarily in the first order predicate calculus. It explains in a very subtle way the difference in meaning of “equals” in mathematics and logic. Using only one axiom this paper proves the Reflexive Law for equality (i.e., x=x) and it proves the Law of Symmetry (i.e., if x=y then y=x), and the Transitive Law for Equality (i.e. if x=y and y=z then x=z. Yes these need to be proved! It also discusses the difference between equality in logic and equality in mathematics.
3 pages.
IDENTITY.PDF  34,427 bytes   11/22/2009  5:18 pm.

Implementing Trigonometric and Hyperbolic Functions and Their Inverses For Both Real and Complex Arguments
This paper discusses how to implement all the trigonometric and hyperbolic functions with either real or complex arguments. The paper basically uses only Tan to generate all the inverse functions. This paper carefully discusses the complex exponential and logarithmic functions and shows how to compute complex nth roots and how to define a general complex power function.
11 pages.
INVTH.PDF  66,012 bytes   11/22/2009  5:19 pm.

Modular Arithmetic For RSA Cryptography
This paper contains the theory of modular arithmetic that is required to understand the RSA cryptographic algorithm. It includes theorems by Fermat and Euler and Wilson and builds the fundamental properties of the Euler Phi-function. It also provides a high-level description of the RSA cryptographic scheme and explains why and how the RSA decoding algorithm works.
10 pages.
MODULARRSA.PDF  68,430 bytes   11/22/2009  5:19 pm.

The Number e and Compound Interest
This paper discusses how the number e arises naturally in the computation of compound interest. Using a limit formula from calculus, this paper shows how e arises by considering instantaneous compounding. It also shows how to compute the Future Value of a series of payments that results in an integral formula. It also discusses the case where the compounding and payment periods are different.
6 pages.
NUMBERE.PDF  47,000 bytes   11/22/2009  5:20 pm.

Probability Notes
This is a series of notes that lay the foundation for studying probability. This paper consists of 44 items that are mostly definitions and theorems. Of significance are the definitions for effectively impossible events and effectively certain events. The paper gives two examples of events, one of which has probability 0 and the other has probability 1. The first event is not impossible and the second event is not certain to happen.
11 pages.
PROB.PDF  70,624 bytes   11/22/2009  5:21 pm.

Pythagorean Triples
This paper contains 12 items related to Pythagorean Triples. Of significance is a formula and technique for generating primitive Pythagorean triples.
4 pages.
PYTHAGT.PDF  30,386 bytes   11/22/2009  5:22 pm.

Sequences and Infinite Series of Functions
This paper introduces the limit concept for a sequence of functions and pre-supposes a knowledge of infinite series. Examples are given which show that a sequence of continuous functions can have a discontinuous limit and that the limit of the derivatives or integrals do not necessarily converge to the derivative or integral of the limit. The concept of uniform convergence is presented and then six theorems are given that establish the correct limit properties that one would expect. The paper concludes by giving a proof of the Weierstrass M-test for an infinite series of functions.
8 pages.
SEQUENCESFOFXP.PDF  59,075 bytes   11/22/2009  5:23 pm.

Series Notes
This is a collection of 97 theorems and definitions related to infinite series.  This set of notes includes most of the standard results but also contains proofs of several results that are left incomplete in many calculus textbooks.  These notes are fundamental for studying real analysis in more detail.  Included are proofs that the positive and the negative terms of a conditionally convergent series must both be divergent, that all re-arrangements of an absolutely convergent series converge to the same sum, that some re-arrangement of a conditionally convergent series can be made to converge to any real number, that a function defined by a power series is continuous and that it is differentiable and integrable term by term and that such a function has an infinite number of derivatives.  An example is also given of an infinitely differentiable function that cannot be represented by a power series.  Cauchy products and quotients of series are developed and both the Nested Interval Property and the Cauchy Convergence Criterion are given.  This paper also has a short and elegant proof that the sum of the reciprocals of the squares of consecutive positive integers converges to pi-squared divided by 6.
42 pages.
SERIESP.PDF  210,714 bytes   11/22/2009  5:24 pm.

Statistical Summations
This paper develops a set of alternative formulas which can be used to accumulate standard statistical sums. The formulas are recursive in nature. The paper first summarizes all the standard sums that are used in performing linear regression. Then it introduces the alternative formulas and thereby a means of accumulating sums that is more accurate than using the standard formulas. Next it derives the equivalences of the standard regression formulas and demonstrates how to compute the slope and y-intercept of the line of best fit.
11 pages.
STATSUM.PDF  56,348 bytes   11/22/2009  5:24 pm.

Synthetic Substitution Versus Long Division
This is a short paper whose only purpose is to explain the connection between long division and synthetic substitution. In fact, this may be considered to form the basis for the technique popularly known and misunderstood as Synthetic Division.
3 pages.
SYNTHD.PDF  26,603 bytes   11/22/2009  5:25 pm.

Trochoidal Curves
This paper develops the parametric formulas that can be used to generate 2D curves that are in the family of what are called trochoids. Such curves include hypotrochoids, epitrochoids, peritrochoids and trochoidal roses. Special cases of these curves include what are called deltoids, astroids, cardioids, nephroids, and limacons. These curves can be used to make some stunning artistic figures by varying some of the graph parameters for the curves. A free computer program called Koch is available that can be used to make computer art related to these curves.
13 pages.
TROCHOIDS.PDF  196,108 bytes   11/22/2009  5:26 pm.

An Angle Problem and an Analytic Geometry Solution
This paper presents a simple geometry problem in which a point P lies in the first quadrant but on the unit circle in the xy-plane. The point's x-coordinate is (1+SQRT(5))/4. The problem is to prove that the angle between the positive x-axis and the line connecting the origin point to P must measure 36 degrees exactly. The solution to this problem involves constructing a regular pentagon inscribed in the unit circle. The points of that regular pentagon are constructed one at a time, using analytic geometry. This paper demonstrates the power of analytic geometry because it is very difficult to solve this problem using only plane geometry.
7 pages.
ANANGLEPROBLEM.PDF  47,901 bytes   11/22/2009  5:26 pm.

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