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The ellipse : a historical and mathematical journey
- Author / Creator
- Available as
- Online
- Physical
Details
Checking for availability...
- Creator
- Arthur Mazer
- Format
- Books
- Language
- English
- Publication
-
- Hoboken, N.J. : Wiley, [2010]
- ©2010
- Physical Details
-
- xi, 303 pages : illustrations ; 24 cm
- ISBNs
- 0470587180, 1283203359, 047059103X, 9786613203359, 9780470591031, 9780470591024, 0470591021, 9780470587188
- OCLC
- ocn436387163, ocn746324241
- Includes bibliographical references and index.
- Chapter 1. Introduction -- Chapter 2. The Trial: Starting Out -- 2.1. A Sticky Matter -- 2.2. Numbers -- 2.2.1. Integers, Rational Numbers, and Irrational Numbers -- 2.2.2. The Size of the Irrational Numbers -- 2.2.3. Suitability of Rationals and the Decimal System -- 2.2.4. Rational and Irrational Outcomes -- Chapter 3. The Space: Geometry -- 3.1. Euclidean Space, Dimension and Rescaling -- 3.1.1. Euclidean Space and Objects -- 3.1.2. Euclidean Space in Higher Dimensions -- 3.1.3. Unit Measurements and Measures of Objects -- 3.1.4. Rescaling, Measurement, and Dimension -- 3.1.5. Koch's Snowflake, a Fractal Object -- 3.2. Measurements of Various Objects -- 3.2.1. Pythagorean Theorem, Length of the Hypotenuse -- 3.2.2. Cavalieri's Theorem in Two Dimensions -- 3.2.3. Cavalieri's Theorem, Archimedes Weighs In -- 3.2.4. Simple Applications of Cavalieri's Theorem -- 3.2.5. The Circle -- 3.2.6. Surface Area of the Cone -- 3.2.7. Cavalieri's Theorem a Stronger Version in Three Dimensions -- 3.2.8. Generalized Pyramids -- 3.2.9. The Sphere as a Generalized Pyramid -- 3.2.10. The Surface Area and Volume of the Sphere -- 3.2.11. Equal-Area Maps, Another Excursion -- Chapter 4. The Language: Algebra -- 4.1. Cartesian Coordinates and Translation of the Axes -- 4.1.1. Intersections of Geometric Objects as Solutions to Equations -- 4.1.2. Translation of Axis and Object -- 4.2. Polynomials -- 4.2.1. Lines -- 4.2.2. Parabolas and the Quadratic Equation -- 4.3. Circles -- 4.3.1. Equations for a Circle -- 4.3.2. Archimedes and the Value of π -- 4.3.3. Tangent Lines to a Circle -- 4.4. The Four-Dimensional Sphere -- 4.4.1. Pythagorean Theorem in Higher Dimensions -- 4.4.2. Measurements in Higher Dimensions and n-Dimensional Cubes -- 4.4.3. Cavalieri's Theorem -- 4.4.4. Pyramids -- 4.4.5. The n-Dimensional Sphere as an n-Dimensional Pyramid -- 4.4.6. The Three-Dimensional Volume of the Four-Dimensional Sphere's Surface -- 4.5. Finite Series and Induction -- 4.5.1. A Simple Sum -- 4.5.2. Induction -- 4.5.3. Using Induction as a Solution Method -- 4.6. Linear Algebra in Two Dimensions -- 4.6.1. Vectors -- 4.6.2. The Span of Vectors -- 4.6.3. Linear Transformations of the Plane Onto Itself -- 4.6.4. The Inverse of a Linear Transformation -- 4.6.5. The Determinant -- 4.7. The Ellipse -- 4.7.1. The Ellipse as a Linear Transformation of a Circle -- 4.7.2. The Equation of an Ellipse -- 4.7.3. An Excursion into the Foci of an Ellipse -- Chapter 5. The Universal Tool: Trigonometry -- 5.1. Trigonometric Functions -- 5.1.1. Basic Definitions -- 5.1.2. Triangles -- 5.1.3. Examples -- 5.2. Graphs of the Sine, Cosine, and Tangent Functions -- 5.3. Rotations -- 5.4. Identities -- 5.4.1. Pythagorean Identity -- 5.4.2. Negative of an Angle -- 5.4.3. Tan(θ) in Terms of Sin(θ) and Cos(θ) -- 5.4.4. Sines and Cosines of Sums of Angles -- 5.4.5. Difference Formulas -- 5.4.6. Double-Angle Formulas -- 5.4.7. Half-Angle Formulas -- 5.5. Lucky 72 -- 5.6. Ptolemy and Aristarchus -- 5.6.1. Construction of Ptolemy's Table -- 5.6.2. Remake of Aristarchus -- 5.7. Drawing a Pentagon -- 5.8. Polar Coordinates -- 5.9. The Determinant -- Chapter 6. The Slayer: Calculus -- 6.1. Studies of Motion and the Fundamental Theorem of Calculus -- 6.1.1. Constant Velocity and Two Problems of Motion -- 6.1.2. Differential Calculus, Generalizing the First Problem -- 6.1.3. Integral Calculus, Generalizing the Second Problem -- 6.1.4. Relations Between Differentiation and Integration and the Fundamental Theorem of Calculus -- 6.1.5. Integration, Leibniz' Notation, and the Fundamental Theorem of Calculus -- 6.2. More Motion: Going in Circles -- 6.3. More Differential Calculus -- 6.3.1. Differentiation Rules -- 6.3.2. Notation and the Derivative at a Specified Point -- 6.3.3. Higher Order Differentiation and Examples -- 6.3.4. Differentiation and the Enquirer -- 6.4. More Integral Calculus -- 6.4.1. The Antiderivative and the Fundamental Theorem of Calculus -- 6.4.2. Methods of Integration -- 6.5. Potpourri -- 6.5.1. Cavalieri's Theorem and the Fundamental Theorem of Calculus -- 6.5.2. Volume of the Sphere and Other Objects with Known Cross-Sectional Areas -- Chapter 7. Eight Minutes that Changed History -- 7.1. Newton's Laws of Motion -- 7.2. Galilean Checkpoint -- 7.3. Constants of Motion and Energy -- 7.3.1. Energy of a Tossed Object -- 7.3.2. Energy of a System Moving in a Single Dimension -- 7.4. Kepler and Newton: Aristarchus Redeemed -- 7.4.1. Polar Coordinates -- 7.4.2. Angular Momentum -- 7.4.3. The Ellipse