th century
play

th Century The 19 th The 19 Century Critical examination of - PDF document

Jul 24, 2023 •126 likes •254 views

The Saga of Mathematics A Brief History th Century The 19 th The 19 Century Critical examination of Euclidean Critical examination of Euclidean geometry. geometry. A Century of Surprises A Century of Surprises Especially the

  1. The Saga of Mathematics A Brief History th Century The 19 th The 19 Century � Critical examination of Euclidean Critical examination of Euclidean � geometry. geometry. A Century of Surprises A Century of Surprises � Especially the Especially the parallel postulate parallel postulate � which Euclid took for granted. which Euclid took for granted. Chapter 11 Chapter 11 � Euclidean versus Non Euclidean versus Non- -Euclidean Euclidean � geometry. geometry. Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 1 1 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 2 2 The Parallel Postulate The Parallel Postulate The Parallel Postulate The Parallel Postulate � If a straight line falling on two straight If a straight line falling on two straight � lines makes the interior angles on the L 1 lines makes the interior angles on the same side less than two right angles, same side less than two right angles, the two straight lines, if produced the two straight lines, if produced indefinitely, meet on that side on indefinitely, meet on that side on which are the angles less than the two which are the angles less than the two L 2 right angles. right angles. L Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 3 3 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 4 4 The Parallel Postulate Alternatives The Parallel Postulate Alternatives � Poseidonius Poseidonius (c. 135 (c. 135- -51 BC): Two 51 BC): Two � The most convincing evidence of The most convincing evidence of � � parallel lines are equidistant. parallel lines are equidistant. Euclid’s mathematical genius. Euclid’s mathematical genius. � Proclus Proclus (c. 500 AD): If a line (c. 500 AD): If a line � Euclid had no proof. Euclid had no proof. � � intersects one of two parallel lines, intersects one of two parallel lines, � In fact, no proof is possible, but he In fact, no proof is possible, but he � then it also intersects the other. then it also intersects the other. couldn’t go further without this couldn’t go further without this � Saccheri Saccheri (c. 1700): The sum of the (c. 1700): The sum of the � statement. statement. interior angles of a triangle is two right interior angles of a triangle is two right � Many have tried to prove it but failed. Many have tried to prove it but failed. angles. angles. � Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 5 5 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 6 6 Lewinter & Widulski 1

  2. The Saga of Mathematics A Brief History Alternatives Alternatives Alternatives Alternatives � Legendre Legendre (1752 (1752- -1833): A line 1833): A line � Farkas Farkas B Bó ólyai lyai (1775 (1775- -1856): There is 1856): There is � � through a point in the interior of an through a point in the interior of an a circle through every set of three a circle through every set of three angle other than a straight angle angle other than a straight angle non- non -collinear points. collinear points. B intersects at least one of the arms of intersects at least one of the arms of the angle. the angle. A P C Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 7 7 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 8 8 Playfair’s Axiom Playfair’s Axiom Non- Non -Euclidean Geometry Euclidean Geometry � Given a line and a point not on the Given a line and a point not on the � In the beginning of the last century, some In the beginning of the last century, some � � line, it is possible to draw exactly one line, it is possible to draw exactly one mathematicians began to think along more mathematicians began to think along more line through the given point parallel to line through the given point parallel to radical lines. radical lines. the line. th Postulate is not true! the line. � Suppose the 5 Suppose the 5 th Postulate is not true! � – – The Axiom is not The Axiom is not Playfair’s Playfair’s own invention. own invention. � “Through a given point in a plane, “Through a given point in a plane, two two lines, lines, � He proposed it about 200 years ago, but He proposed it about 200 years ago, but parallel to a given straight line, can be parallel to a given straight line, can be Proclus Proclus stated it some 1300 years earlier. stated it some 1300 years earlier. drawn.” drawn.” – Often substituted for the fifth postulate Often substituted for the fifth postulate – � This would change proposition in Euclid. This would change proposition in Euclid. � because it is easier to remember. because it is easier to remember. Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 9 9 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 10 10 Non- -Euclidean Geometry Euclidean Geometry Non- -Euclidean Geometry Euclidean Geometry Non Non � For example: The sum of the angles in a For example: The sum of the angles in a � After failing to prove the parallel postulate, After failing to prove the parallel postulate, � � triangle is less triangle is less than two right angles! than two right angles! mathematicians wondered if there was a mathematicians wondered if there was a � This of course led to Non This of course led to Non- -Euclidean Euclidean consistent “ “alternative alternative” ” geometry in which geometry in which consistent � geometry whose discovery was led by Gauss geometry whose discovery was led by Gauss the parallel postulate failed. the parallel postulate failed. (1777- -1855), 1855), Lobachevskii Lobachevskii (1792 (1792- -1856) and 1856) and (1777 � To their amazement, they found two! To their amazement, they found two! � Jonas B Bó ól lyai yai (1802 (1802- -1850). 1850). Jonas � The secret was to look at curved surfaces. The secret was to look at curved surfaces. � Later by Beltrami (1835 Later by Beltrami (1835- -1900), Hilbert 1900), Hilbert � � (1862- -1943) and Klein (1849 1943) and Klein (1849- -1945). 1945). – The plane is flat – The plane is flat – – it has no curvature or, as it has no curvature or, as (1862 mathematicians say, its curvature is 0. mathematicians say, its curvature is 0. � Fits Einstein’s Fits Einstein’s Theory of Relativity. Theory of Relativity. � Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 11 11 Lewinter & Widulski Lewinter & Widulski The Saga of Mathematics The Saga of Mathematics 12 12 Lewinter & Widulski 2

Recommend

The Foundations of 19 th Century Literature are, not surprisingly, in the 18 th century English

The Foundations of 19 th Century Literature are, not surprisingly, in the 18 th century English

The Foundations of 19 th Century Literature are, not surprisingly, in the 18 th century English Literature through the ages 1. 1400s Chaucer - verse 2. 1600s Shakespeare stage plays 3. 1720 1770 18 th century classics - books 4.

518 views • 39 slides
17 th century French Tapastry Celestial map from 17 th century by Frederik de Wit ( Dutch

17 th century French Tapastry Celestial map from 17 th century by Frederik de Wit ( Dutch

17 th century French Tapastry Celestial map from 17 th century by Frederik de Wit ( Dutch cartographer) Saturn seems to have impressed the seal of melancholy on me from the beginning. Marsilio Ficino (1433-1499) the Dweller on the Threshold,

378 views • 36 slides
st Century 21 st 21 ry Questions For r & From st Century 21 st 21 ry Cli lients Seattle

st Century 21 st 21 ry Questions For r & From st Century 21 st 21 ry Cli lients Seattle

st Century 21 st 21 ry Questions For r & From st Century 21 st 21 ry Cli lients Seattle AIA CAE October 19, 2016 Jeanne L. Narum Principal Learning Spaces Collaboratory Overview Questions From Clients, For Clients

546 views • 43 slides
21 st Century Skills 21 Century Skills Instructional Services Department Board Update, April

21 st Century Skills 21 Century Skills Instructional Services Department Board Update, April

21 st Century Skills 21 Century Skills Instructional Services Department Board Update, April 2010 Agenda 21 st Century Skills National/State/Local Initiatives Partnership for 21 st Century Skills Framework Framework

178 views • 13 slides
The 21 st Century Conservation Vision To meet the challenges of the 21st Century, we must:

The 21 st Century Conservation Vision To meet the challenges of the 21st Century, we must:

The 21 st Century Conservation Vision To meet the challenges of the 21st Century, we must: Make strategic, cost-effective conservation investments; Increase efficiency, transparency, accountability; and Design/manage for landscapes

710 views • 30 slides
21 st Century Disaster Readiness Kevin Yeskey, M.D. March 15, 2018 21 st Century: An

21 st Century Disaster Readiness Kevin Yeskey, M.D. March 15, 2018 21 st Century: An

21 st Century Disaster Readiness Kevin Yeskey, M.D. March 15, 2018 21 st Century: An Increasingly Complex & Dangerous World Saving Lives. Protecting Americans. 2 New Madrid Earthquake No notice event Consequences eclipse

319 views • 12 slides
Task 1: If 20th century-type droughts (i.e., 1950s) under 21st century temperatures are the new

Task 1: If 20th century-type droughts (i.e., 1950s) under 21st century temperatures are the new

Task 1: If 20th century-type droughts (i.e., 1950s) under 21st century temperatures are the new normal for drought, how might such droughts impact upper Colorado River flow (UCRB)? Approach : Assess 20th century-type droughts in the

904 views • 34 slides
The Evolution of Regulation 20 th Century Lessons and 21 st Century Opportunities John W. Mayo

The Evolution of Regulation 20 th Century Lessons and 21 st Century Opportunities John W. Mayo

The Evolution of Regulation 20 th Century Lessons and 21 st Century Opportunities John W. Mayo Twelfth ACCC Regulatory Conference Regulating for the Future July 28, 2011 John Stuart Mill, On Liberty , 1860 There is, in fact, no recognized

501 views • 21 slides
OKLAHOMA 21 st CENTURY COMMUNITY LEARNING CENTERS The Purpose 21 st Century Community Learning

OKLAHOMA 21 st CENTURY COMMUNITY LEARNING CENTERS The Purpose 21 st Century Community Learning

OKLAHOMA 21 st CENTURY COMMUNITY LEARNING CENTERS The Purpose 21 st Century Community Learning Center programs provide important supports to the traditional school day, including: 1. Academic enrichment activities that help students meet state

371 views • 9 slides
Transforming Learning for the 21 st Century From Philosophy to Physical Reality Barbara Stone AM

Transforming Learning for the 21 st Century From Philosophy to Physical Reality Barbara Stone AM

Transforming Learning for the 21 st Century From Philosophy to Physical Reality Barbara Stone AM : CEFPI Conference, Sydney, June 17, 2011 barbarastoneam@gmail.com Changing Paradigms for the New Millennium 20th Century 21st Century

341 views • 17 slides
TPPA: A 21 ST Century Trade Agreement? Dr Erik Monasterio A Deal For The 21 st Century? The

TPPA: A 21 ST Century Trade Agreement? Dr Erik Monasterio A Deal For The 21 st Century? The

TPPA: A 21 ST Century Trade Agreement? Dr Erik Monasterio A Deal For The 21 st Century? The free-trade oxymoron NEGOTIATORS GUARANTEES No additional cost to medications. No effect on PHARMAC Fundamentals of PHARMAC untouched

298 views • 17 slides
LANGUAGE & TOOLS WHY GOVERNMENT CANNOT MANAGE THE 21ST CENTURY 21st Century Problems

LANGUAGE & TOOLS WHY GOVERNMENT CANNOT MANAGE THE 21ST CENTURY 21st Century Problems

LANGUAGE & TOOLS WHY GOVERNMENT CANNOT MANAGE THE 21ST CENTURY 21st Century Problems Climate, for example, is hard The Big Threats Nukes, Biotech, Nanotech, Asteroids, Weird Physics, Environment Mostly government funded

300 views • 28 slides
Confucian Ethics and the 21st Century Global Business Reg Little A Lost Half Century

Confucian Ethics and the 21st Century Global Business Reg Little A Lost Half Century

Confucian Ethics and the 21st Century Global Business Reg Little A Lost Half Century Superiority of Confucian business culture was evident 50 years ago in Japan Australias Beijing Embassy foreshadowed Chinas rise in 1976 Asia

414 views • 7 slides
A New Vision for 21st Century 21 st Century Education [Insert Presenter Name] [Insert Presenter

A New Vision for 21st Century 21 st Century Education [Insert Presenter Name] [Insert Presenter

Preparing Students for the A New Vision for 21st Century 21 st Century Education [Insert Presenter Name] [Insert Presenter Title & Company] [Insert Event Name] KTEC Enrollment Meeting [Insert Date] January 09 PLEASE NOTE: This is only a

606 views • 39 slides
The Creffield Road Conservation Area: From Prehistoric Times to the Twentieth Century Dr

The Creffield Road Conservation Area: From Prehistoric Times to the Twentieth Century Dr

The Creffield Road Conservation Area: From Prehistoric Times to the Twentieth Century Dr Jonathan Oates Oakley Avenue, 1970s John Allen Brown, 1833-1903 St. Marys church, 18 th century Fordhook, 18 th century Fielding Road, 1970s Henry

715 views • 32 slides
Center for 21 st Century Health Professionals and Researchers = Center for 21 st Century Health

Center for 21 st Century Health Professionals and Researchers = Center for 21 st Century Health

Center for 21 st Century Health Professionals and Researchers = Center for 21 st Century Health Professions & Center for Gene Regulation in Health and Disease (GRHD) http://www.csuohio.edu/sciences/grhd.html Signature theme of Health

253 views • 4 slides
Presentation August 2019 Century Processing Plant in operation Cautionary Statement New Century

Presentation August 2019 Century Processing Plant in operation Cautionary Statement New Century

Diggers & Dealers Mining Forum Presentation August 2019 Century Processing Plant in operation Cautionary Statement New Century Resources believes that the production target, forecast financial information derived from that target and other

917 views • 35 slides
5G IS SO AST CENTURY CONNECTIVITY 5G IS SO LAST CENTURY Presented By: Zunaid Bulbulia 24

5G IS SO AST CENTURY CONNECTIVITY 5G IS SO LAST CENTURY Presented By: Zunaid Bulbulia 24

5G IS SO LAST CENTURY. THE CASE FOR SERVICES 5G IS SO AST CENTURY CONNECTIVITY 5G IS SO LAST CENTURY Presented By: Zunaid Bulbulia 24 October 2017 Presented by Zunaid Bulbulia Presented on 24 October 2017 GAME PLAN By the end of

732 views • 45 slides
Pretest: 21 st Century Acronymese FDA NDA FDAAA CDER IND FR CPA

Pretest: 21 st Century Acronymese FDA NDA FDAAA CDER IND FR CPA

Pretest: 21 st Century Acronymese FDA NDA FDAAA CDER IND FR CPA CDISC ADaM PDUFA BLA CPI PRO HL7 ARSI ICH CDASH IMI CAMD E9 CRO 21 st Century Review at the

1.26k views • 82 slides
Twentieth artistic language (techniques of Century representation) versus the image and its

Twentieth artistic language (techniques of Century representation) versus the image and its

20th century part 1 F11.shw the 20th century dialectic: Twentieth artistic language (techniques of Century representation) versus the image and its source Ruptures and [or] Continuities disintegration of the real versus the hallucination

461 views • 7 slides
From Frege to Gosling: 19th Century Logic and 21st Century Programming Languages Philip

From Frege to Gosling: 19th Century Logic and 21st Century Programming Languages Philip

From Frege to Gosling: 19th Century Logic and 21st Century Programming Languages Philip Wadler Avaya Labs Gottlob Frege, Begriffsschrift , 1879 Part I The Curry-Howard Isomorphism Modus ponens Frege, 1879 Gentzen, 1934 B A

735 views • 60 slides
Global Leadership Initiative The Global Century Strategic Issues: 1. Partnering for Student

Global Leadership Initiative The Global Century Strategic Issues: 1. Partnering for Student

Global Leadership Initiative The Global Century Strategic Issues: 1. Partnering for Student Success 2. Education for the Global Century 3. Dynamic Learning Environment The Global Century The challenges that our students will confront at

702 views • 15 slides
21st Century Computer Architecture A CCC community white paper

21st Century Computer Architecture A CCC community white paper

21 st Century Computer Architecture Mark D. Hill, Univ. of Wisconsin-Madison 6/2016 @ Architecture 2030 @ ISCA 1. Whitepaper Content (tell a story) 2. Process, Impact & Computing Community Consortium (CCC) 21st Century Computer

558 views • 27 slides
Risk Communication in Risk Communication in st Century the 21 st Century the 21 Ragnar L

Risk Communication in Risk Communication in st Century the 21 st Century the 21 Ragnar L

Risk Communication in Risk Communication in st Century the 21 st Century the 21 Ragnar L fstedt fstedt Ragnar L Professor and Director Kings Centre for Risk Management Kings College, London 1 Classified - Internal use I n this

464 views • 28 slides

More recommend