Replacing Euclid’s Geometry with Algebra
The App++ encourages proofs of properties of the real numbers based on the axioms for the real numbers. These results, using the coordinate principles of Descartes are outlined below. They allow the replacement of Euclid’s geometric proofs with algebraic methods. Computational algorithms and thinking using the iPad are an integral component of this development.
The History of Analytical Geometry by Carl Boyer (Dover) presents the details of the transition of geometry from the geometric approach of Euclid to the algebraic approach of Descartes.
René Descartes (1596-1650) and Pierre de Fermat (1608-1665)
Descartes and Fermat independently developed the Foundations for analytic geometry. Their contributions are remarkable since Fermat was a lawyer and Descartes was a philosopher. Chapter V of Boyer’s book gives the highlights of their work.
Both used graphical methods to solve polynomial equations Descartes solved a cubic equation by finding the intersection of a parabola with a circle, Fermat solved a different cubic equation by finding the intersection of a parabola with a hyperbola.
Descartes stated his rule of signs for the minimal number of positive real roots of a polynomial equation of one variable, but without a proof.
Fermat gained further notoriety when Fermat and Blaise Pascal exchanged letters about the “fair game” problem in probability. They solved the problem and in doing so, established the basic foundations of probability.
Period of Commentaries – Clarifying Descartes Work
During the twenty year period after Descartes death, numerous mathematicians in Europe and England contributed further clarification and advances to analytical geometry.
This period is covered in Chapter VI of Boyer’s book. For example, Frans van Schooten wrote Geometric of Descartes. The first hundred pages concerns the work of Descartes, the next 900 pages contains additional locus problems, derivations of the equations of conics and a study of quadratic equations.
Some of the principal contributors to analytic geometry during this period were the Bernoulli’s, Pascal, Jan de Witt, Roberval, Debeaune, John Wallis and many others whose works are given in Boyer’s book.
René Descartes
Isacc Newton (1642 – 1727)
Although Newton did not endorse Descartes work, several of his papers used algebra freed from geometry. In 1664 Newton extended power series to fractional exponents and the binomial theorem to fractional powers. In 1707 he proposed to model algebra and algebraic operations on the real numbers by applying the usual operations of arithmetic. Newton is famous for his development of calculus and its application with his law of gravitation to deduce the three laws of Kepler’s motion of orbits of the planets about the sun.
Newton contributed to analytic geometry in the appendix of his work on optics in which he used coordinates to determine curves by an algebraic equation.
Leonhard Euler (1707 – 1783)
Euler studied the works of Descartes and Newton in his masters thesis. Euler was so prolific that he published 800 papers and 20 books on number theory, series, mathematical astronomy, calculus of variations, and differential equations (ordinary, partial, and numerical solutions).
His main contribution that is used in the App++ is the concept of a function and use of his notation, y = f(x), to represent a function. His books on arithmetic (1738), algebra (1770), precalculus (1748), and four books on calculus set the standard for excellence. His algebra book and 300 papers were written in the last 15 years of his life while blind.
Euler was not the first to discover that eiθ = cos(θ) + i sin(θ), but his many applications of the formula are now associated with his name.
Thomas Simpson (1710 – 1761)
Thomas Simpson in 1743 found that (b – a)/6 [f(a) + 4f(a + b)/2 + f(b)] would compute the integral from ∫ab f(x) of any cubic polynomial exactly. Simple cases of Simpson’s rule were obtained earlier by others. By subdividing an interval into smaller subintervals and applying Simpson’s rule on each subinterval, a composite rule is obtained with more accuracy. Simpson’s composite rule is straightforward and is applied on the iPad to obtain 15 decimal places of accuracy.
Benard Bolzano (1781-1848)
Bolzano was born and died in Prague, Bohemia (now Czech Republic). After a thesis on geometry, he won first place to fill the mathematics position at the University of Prague. At the same time, he also won first place as a Professor of Philosophy and religion.
The mathematics position was given to the substitute teaching the course and Bolzano was appointed to the Philosophy position in 1805.
In 1819 he was dismissed from the University for his liberal thinking (which he refused to recant). Bolzano not only opposed war in his lectures but advocated that the state spend no money on defense and instead feed the poor.
While teaching philosophy, he also sought the correct definitions and proofs in analysis.
In 1816, he published a proof of the Binomial Theorem. In 1817, he gave a rigorous definition for continuous functions and proved a version of the intermediate value theorem for continuous functions. In his proof of the intermediate value theorem, he proved a lemma which is equivalent to the least upper bound axiom.
Bolzano was also the first to produce an example of a function which was everywhere continuous on an interval but was nowhere differentiable.
In our development of the real numbers, we assume the least upper bound property as an axiom. That is, if a set of real numbers has an upper bound then it has a least upper bound. This axiom fills in all the “holes” in the real numbers that are not rational numbers with irrational numbers. This results in a 1-1 correspondence between all real numbers and all geometric points on the x-axis.
Julius Plücker (1801 – 1868)
Julius Plücker was a German mathematician and experimental physicist. Boyer states that, “No single person has contributed more to analytic geometry, both as to volume and power, than did Plücker.” In addition to over 600 pages of articles in the standard journals, Plücker published a half-dozen book length articles devoted to coordinate methods. Each volume produced a “new geometry”. For example, his first volume was on Cartesian coordinates, while the second volume contributed to homogenous coordinates.
From the start, Plücker was convinced that he could obtain all the results of synthetic geometry (Euclid, Hilbert) by means of coordinates. He fulfilled the goals of Descartes in spades.
Felix Klein (1849 – 1925)
Felix was Plücker’s student and his thesis was on line geometry. In 1872 Klein published his famous “Erlanger Programm” in which geometry should be viewed as the study of the invariants of a group of transformations acting on a set.
For Euclidean geometry the group of transformations is the rigid motions, that is translations, rotations, and reflections, and the invariant is the distance between points. Klein noted that each geometry could be characterized by a group of transformations. By characterizing the groups of transformations the classification showed that projective geometry is the most basic and the other geometries (affine, hyperbolic, Euclidean, etc.) are contained beneath it.
Bernhard Riemann (1826 – 1866)
Riemann was a German mathematician whose chief contributions were to complex analysis. Our interest is his work is the numerical evaluation of the Riemann integral which he formulated in 1854 to find the area under the graph of a non-negative continuous function f(x) on the interval [a, b]; Area = ∫ab f(x) dx.
If one divides [a, b] into sub-intervals and approximates the sub-area with the largest rectangle under the graph, then the sum of the sub-intervals is called a lower sum. The least upper bound of the lower sums is the area under the graph. An easy numerical approximation on the iPad computes the lower sums.
H.S. Wall (1902 – 1972)
H.S. Wall was a student of Van Vleck at Madison, Wisconsin. His thesis was on continued fractions, and he published Analytic Theory of Continued Fractions in 1948 and his book Creative Mathematics in 1968. Wall moved to the University of Texas in 1946 and his method of teaching is used in Creative Mathematics.
Using Riemann’s definition of area under a curve and following Wall’s book on Creative Mathematics, we define the natural logarithm ln(x) as the area under the graph from one to positive x of the reciprocal function 1/t; Area = ∫1x 1/t dt.
ln(x) is a 1-1 and onto function from the positive numbers to the real numbers, and hence has an inverse function exp(x) (alternately written as ex). Using ln(x) and ex, we then define for positive a, ax for all real numbers x, and obtain all the standard properties of exponents.
In Summary
Our approach will not only prepare students with the mathematical and coding skills for the increasing digital trends in society, but will alter the present STEM and coding courses in high school. For example, there should no longer be a ninth grade geometry class. It should be replaced with a linear algebra class and complex variables which will contain algebraic proofs of standard geometry results.
High School geometry teachers should take a geometry course based on “Geometry: A Metric Approach with Models” by Millman and Parker. A robotics class and lab must be offered on a regular basis to prepare students the opportunity to work in an area of expanding digital opportunities.
Teachers and their students will benefit from an introduction to the history of mathematics. A great history source is found in Part II – The Origins of Modern Mathematics, The Princeton Companion to Mathematics. The editor is Timothy Gowers. Published by The Princeton University Press.
Early mathematical contributions of the civilizations of Egypt, Mesopotainia, Greece, Babylonian, China and India are unveiled.
These early developments were extended by midieval Islamic mathmeticians, especially in the development of Algebra.
The Islamic contributions were adopted in Europe and exploded in all directions.
Finally, the modern view of number systems, non-Euclidian geometry, modern algebra, analysis and algorithms is discussed.
PHOTO CREDITS:
– André Hatala [e.a.] (1997) De eeuw van Rembrandt, Bruxelles: Crédit communal de Belgique, ISBN 2-908388-32-4.
– File:Portrait of Sir Isaac Newton, 1689.jpg from https://exhibitions.lib.cam.ac.uk/linesofthought/artifacts
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