8.2 Notes - Summary The History of Mathematics: an Introduction PDF

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Megan Troutman History of Math 10/26/17 Chapter 8: The Mechanical World: Descartes and Newton 8.2 Descartes: The Discours de la Méthode  The Writings of Descartes o René Descartes (1596-1650)  Analytic geometry (gives algebraic equations to curves)  Foundations for the growth of math in modern times (turning point between medieval and modern times)  1st: led to development of calc by Newton and Leibniz  Said the only way he could do good math was by staying in bed as long as he liked (very delicate as a child, wasn’t supposed to live long)  Friends with Marin Mersenne from school  Enlisted in army, had 3 dreams while he shut himself in an oven to be warm  Wrote Le Monde (encyclopedia of physics) but didn’t publish it until after his death bc it supported heliocentric theory (and he would’ve been shunned like Galileo) and he was a devoted Roman Catholic  Invited by Queen of Sweden to start a school  led to his death (she made him get up at 5 am in the cold) o The Discours: autobiographical resume of Descarte’s progress, highly complicated philosophy and science, in French  “Systematic doubt” = pursuit of certainty (of math)  Math starts with one simple thing, then proves another (so math should be a model for other studies)  Math reasoning, not results, are what interested Descartes  Wanted to find what the simplest true things were, so he started by rejecting everything  Proved that he himself exists: “I think; therefore, I am.”  Nothing can exist without reasoning  Inventing Cartesian Geometry o Diotropics: nature and properties of light (refraction, human eye, telescopes) o Meterology: snow crystals, rain drops, thunder and lightning, rainbows o Geometry: combined algebra and geometry to make analytic geometry  Uses x, y, z for unknowns and beginning letters of alphabet for constants, exponents  Same letter for both positive and negative numbers  Thought of exponents as lines (x3 was in a proportion, not a volume)  Generalizations of the 3 and 4 line locus problems of Pappus’s Conics  Germinal idea of the coordinate system (but never did perfectly perpendicular ones like ours is); only 1st quadrant  “Coordinate” is Leibniz’s work  Distinguished between geometrical and mechanical curves (“algebraic” and “transcendental”)  Most curves were geometric, quadratrix and spiral were mechanical

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Tangent is perpendicular through the point to the normal (on a curve)  Used circles The Algebraic Aspect of La Géométrie o Nature of equations and principles of their solutions o Recommends all terms be taken to same side so equation equals 0  f(x) is divisible by (x-a) if a is a root, equation of n degree has n roots o Use of negative and imaginary roots: Descartes’s rule of signs  By looking at an equation, you can assign an upper bound to the number of its positive roots  The number of positive roots is equal either to the number of variations in the signs of its coefficients or to this number decreased by a positive integer  x3 + x2 – x +2 = 0 either has 2 positive roots or none  John Wallis said it doesn’t work for imaginary roots  Negative roots of f(x) are also positive roots of f(-x)  law for negative roots o Simple solution for the quartic o Not clear to follow/read, says he purposely omitted some things so people discovered for themselves o A later edition (commentary by van Schooten) had a more convenient way to find the double roots of Descartes’s tangent method (Johann Hudde’s Rule) Descartes’s Principia Philosophiae o Replace medieval (Aristotle and Ptolemy) with scientific system o Principal phenomena of nature could be explained with math and mechanics  Solar system as giant clock o Used vortices to account for the origin and current state of the universe  Solar system was filled with primordial matter (plenum or ether)  Asserted earth was at rest so Church didn’t get mad at him (stationary within an ether that moved) o Compared solar system to whirlpools of water so people understood o Theory of vortices wasn’t supported by experimental evidence (contradicted Kepler’s laws of elliptical orbits) o First serious effort to separate natural knowledge from theology  Although Newton’s laws of gravity shattered his ideas later o Was allowed by Church bc he put it forth hypothetically  prohibited later o Descartes: dominant thinker of 1600’s (math, philosophy, optics, meteorology, science) Perspective Geometry: Desargues and Poncelet o Girard Desargues (1593-1662): geometric study of perspective  Wanted to draw a 3D world on a flat canvas  Projective geometry: study of properties of figures that stay the same under projection (intersection of a cone and a plane)  Extended Euclidean plane by adding certain “infinitely distant” points  Line at infinity  eliminated special cases caused by parallelism  Desargues’s Theorem: if the lines joining corresponding vertices of two triangles pass through a point, then the intersection points of corresponding sides lie on the line

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Lots of people didn’t accept it, hard to read (twigs and knots for lines and points), no symbolism  publisher destroyed all copies, eventually an original was found  Did come up with word “involution” o Jean Victor Poncelet (1788-1867): rescued projective geometry  Gave geometry first full presentation as separate math (written while in prison)  Controversial principle of geometric continuity (transformations)  Law of duality in projective planes (only need to prove 1 of dual statements bc then other is true)  “Father of modern geometry”...