Who Discovered the Pythagorean Theorem? C 19
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The Theological Import of the Metaphor of the Book of Nature: Historical Hallmarks and Open Questions
Acta Pintériana 6. 2020. doi:10.29285/actapinteriana doi:10.29285/actapinteriana.2020.6.11 The theological import of the metaphor of the Book of Nature: historical hallmarks and open questions Giuseppe Tanzella-Nitti Facoltà di Teologia Scuola Internazionale Superiore per la Ricerca Interdisciplinare (SISRI) Pontificia Università della Santa Croce, Roma. [email protected] Tanzella-Nitti, G. (2020): The theological import of the metaphor of the Book of Nature: historical hallmarks and open questions. A „Természet könyve”-metafora teológiai hozadéka: történelmi fordulópontok és nyitott kérdések. Acta Pintériana, 6: 11-21. Abstract: A tanulmány célja összegezni a „Természet könyve”-metafora kutatásával kapcsolatos kérdések mai állását, különösképpen is a fundamentálteológus szempontjai szerint. A szerző a kérdés másodlagos irodalmának bemutatása nyomán hangsúlyozza, hogy a metafora iránti érdeklődés messze túlmutat a teológia területén. A liber naturae-hagyomány diakronikus szerkezetének elemzése után, a hagyomány aktualizálhatóságának problematikája köti le a szerző figyelmét. A szinkronikus és elemző tárgyaláson belül különösen érdekfeszítő a lex naturalis kérdéskör vizsgálata, amely természetes erkölcsi törvény problematikája szervesen illeszkedik az itt vizsgált tágabb hagyományba. A tanulmány a szerző korábbi írásainál terjedelmesebben kitér a „Természet könyve” – gondolat kortárs tanítóhivatali recepciójára (vö. AP 2016, pp. 55-75). A gondolatmenet a hagyomány aktualizálása szempontjából fontos hermeneutikai kérdéseket is felvázolja, miközben az egész témát elhelyezi a II. Vatikáni Zsinat által felelevenített klasszikus Logosz-teológia horizontján. Nyomtatott formában a jelen angol nyelvű szöveg magyar fordítása is elérhető (In: BAGYINSZKI Á. [ed.] [2019]: A „Természet könyve” mint a „Szentírás könyvének” analógiája. Konferenciakötet, Sapientia Szerzetesi Hittudományi Főiskola & L’Harmattan, Budapest, pp. 21-48). I. Introduction Recent times have witnessed a remarkable interest in the notion of the „Book of Nature” as a locus of divine presence and revelation. -
Geometry Course Outline
GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid -
Right Triangles and the Pythagorean Theorem Related?
Activity Assess 9-6 EXPLORE & REASON Right Triangles and Consider △ ABC with altitude CD‾ as shown. the Pythagorean B Theorem D PearsonRealize.com A 45 C 5√2 I CAN… prove the Pythagorean Theorem using A. What is the area of △ ABC? Of △ACD? Explain your answers. similarity and establish the relationships in special right B. Find the lengths of AD‾ and AB‾ . triangles. C. Look for Relationships Divide the length of the hypotenuse of △ ABC VOCABULARY by the length of one of its sides. Divide the length of the hypotenuse of △ACD by the length of one of its sides. Make a conjecture that explains • Pythagorean triple the results. ESSENTIAL QUESTION How are similarity in right triangles and the Pythagorean Theorem related? Remember that the Pythagorean Theorem and its converse describe how the side lengths of right triangles are related. THEOREM 9-8 Pythagorean Theorem If a triangle is a right triangle, If... △ABC is a right triangle. then the sum of the squares of the B lengths of the legs is equal to the square of the length of the hypotenuse. c a A C b 2 2 2 PROOF: SEE EXAMPLE 1. Then... a + b = c THEOREM 9-9 Converse of the Pythagorean Theorem 2 2 2 If the sum of the squares of the If... a + b = c lengths of two sides of a triangle is B equal to the square of the length of the third side, then the triangle is a right triangle. c a A C b PROOF: SEE EXERCISE 17. Then... △ABC is a right triangle. -
Pythagorean Theorem Word Problems Ws #1 Name ______
Pythagorean Theorem word problems ws #1 Name __________________________ Solve each of the following. Please draw a picture and use the Pythagorean Theorem to solve. Be sure to label all answers and leave answers in exact simplified form. 1. The bottom of a ladder must be placed 3 feet from a wall. The ladder is 12 feet long. How far above the ground does the ladder touch the wall? 2. A soccer field is a rectangle 90 meters wide and 120 meters long. The coach asks players to run from one corner to the corner diagonally across the field. How far do the players run? 3. How far from the base of the house do you need to place a 15’ ladder so that it exactly reaches the top of a 12’ wall? 4. What is the length of the diagonal of a 10 cm by 15 cm rectangle? 5. The diagonal of a rectangle is 25 in. The width is 15 in. What is the area of the rectangle? 6. Two sides of a right triangle are 8” and 12”. A. Find the the area of the triangle if 8 and 12 are legs. B. Find the area of the triangle if 8 and 12 are a leg and hypotenuse. 7. The area of a square is 81 cm2. Find the perimeter of the square. 8. An isosceles triangle has congruent sides of 20 cm. The base is 10 cm. What is the area of the triangle? 9. A baseball diamond is a square that is 90’ on each side. -
5-7 the Pythagorean Theorem 5-7 the Pythagorean Theorem
55-7-7 TheThe Pythagorean Pythagorean Theorem Theorem Warm Up Lesson Presentation Lesson Quiz HoltHolt McDougal Geometry Geometry 5-7 The Pythagorean Theorem Warm Up Classify each triangle by its angle measures. 1. 2. acute right 3. Simplify 12 4. If a = 6, b = 7, and c = 12, find a2 + b2 2 and find c . Which value is greater? 2 85; 144; c Holt McDougal Geometry 5-7 The Pythagorean Theorem Objectives Use the Pythagorean Theorem and its converse to solve problems. Use Pythagorean inequalities to classify triangles. Holt McDougal Geometry 5-7 The Pythagorean Theorem Vocabulary Pythagorean triple Holt McDougal Geometry 5-7 The Pythagorean Theorem The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in Lesson 1-6, it states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. a2 + b2 = c2 Holt McDougal Geometry 5-7 The Pythagorean Theorem Example 1A: Using the Pythagorean Theorem Find the value of x. Give your answer in simplest radical form. a2 + b2 = c2 Pythagorean Theorem 22 + 62 = x2 Substitute 2 for a, 6 for b, and x for c. 40 = x2 Simplify. Find the positive square root. Simplify the radical. Holt McDougal Geometry 5-7 The Pythagorean Theorem Example 1B: Using the Pythagorean Theorem Find the value of x. Give your answer in simplest radical form. a2 + b2 = c2 Pythagorean Theorem (x – 2)2 + 42 = x2 Substitute x – 2 for a, 4 for b, and x for c. x2 – 4x + 4 + 16 = x2 Multiply. -
Natural Theology and the Qur'an
Natural Theology and the Qur’an Robert G. Morrison BOWDOIN COLLEGE Natural theology is reading the book of nature, not the book of revelation, for knowledge of God.1 Natural theology, as a category employed by practitioners, originated within the history of Christianity, as passages from the New Testament such as Romans 1:20 raised the possibility of knowledge about God without revelation.2 The best-known work of natural theology is William Paley’s (d. 1805 AD) Natural Theology: or, Evidences of the existence and attributes of the Deity, collected from the appearances of nature.3 Previous to this, Thomas Aquinas (d. 1275 AD) had made arguments that were in the spirit of natural theology and Robert Boyle (d. 1691 AD) was interested in the subject.4 The Qur’an, likewise, refers to the wonders of nature in the course of its arguments for God’s existence and power.5 The Qur’an’s discussion of humans’ fiṭra and ḥunafāʾ (s. ḥanīf),6 along with its evocation of the wonders of creation as evidence of God’s existence and power, meant that at least some of the Qur’an’s themes might be apprehended without revelation and that it is worth exploring the theme of natural theology in Islamic thought.7 But the general importance of revealed law in Islam and the specific doctrine of naskh are a reminder that the particulars of a revealed text matter. The fact that nature contains some things that are ḥarām8 suggests that the book of nature would not be on a par with al-sharʿ (‘revelation’). -
The Pythagorean Theorem and Area: Postulates Into Theorems Paul A
Humanistic Mathematics Network Journal Issue 25 Article 13 8-1-2001 The Pythagorean Theorem and Area: Postulates into Theorems Paul A. Kennedy Texas State University Kenneth Evans Texas State University Follow this and additional works at: http://scholarship.claremont.edu/hmnj Part of the Mathematics Commons, Science and Mathematics Education Commons, and the Secondary Education and Teaching Commons Recommended Citation Kennedy, Paul A. and Evans, Kenneth (2001) "The Pythagorean Theorem and Area: Postulates into Theorems," Humanistic Mathematics Network Journal: Iss. 25, Article 13. Available at: http://scholarship.claremont.edu/hmnj/vol1/iss25/13 This Article is brought to you for free and open access by the Journals at Claremont at Scholarship @ Claremont. It has been accepted for inclusion in Humanistic Mathematics Network Journal by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. The Pythagorean Theorem and Area: Postulates into Theorems Paul A. Kennedy Contributing Author: Department of Mathematics Kenneth Evans Southwest Texas State University Department of Mathematics, Retired San Marcos, TX 78666-4616 Southwest Texas State University [email protected] San Marcos, TX 78666-4616 Considerable time is spent in high school geometry equal to the area of the square on the hypotenuse. building an axiomatic system that allows students to understand and prove interesting theorems. In tradi- tional geometry classrooms, the theorems were treated in isolation with some of -
The Pythagorean Theorem
6.2 The Pythagorean Theorem How are the lengths of the sides of a right STATES triangle related? STANDARDS MA.8.G.2.4 MA.8.A.6.4 Pythagoras was a Greek mathematician and philosopher who discovered one of the most famous rules in mathematics. In mathematics, a rule is called a theorem. So, the rule that Pythagoras discovered is called the Pythagorean Theorem. Pythagoras (c. 570 B.C.–c. 490 B.C.) 1 ACTIVITY: Discovering the Pythagorean Theorem Work with a partner. a. On grid paper, draw any right triangle. Label the lengths of the two shorter sides (the legs) a and b. c2 c a a2 b. Label the length of the longest side (the hypotenuse) c. b b2 c . Draw squares along each of the three sides. Label the areas of the three squares a 2, b 2, and c 2. d. Cut out the three squares. Make eight copies of the right triangle and cut them out. a2 Arrange the fi gures to form 2 two identical larger squares. c b2 e. What does this tell you about the relationship among a 2, b 2, and c 2? 236 Chapter 6 Square Roots and the Pythagorean Theorem 2 ACTIVITY: Finding the Length of the Hypotenuse Work with a partner. Use the result of Activity 1 to fi nd the length of the hypotenuse of each right triangle. a. b. c 10 c 3 24 4 c. d. c 0.6 2 c 3 0.8 1 2 3 ACTIVITY: Finding the Length of a Leg Work with a partner. -
Faith After the Anthropocene
Faith after the the after Anthropocene Faith • Matthew Wickman and Sherman Jacob Faith after the Anthropocene Edited by Matthew Wickman and Jacob Sherman Printed Edition of the Special Issue Published in Religions www.mdpi.com/journal/religions Faith after the Anthropocene Faith after the Anthropocene Editors Matthew Wickman Jacob Sherman MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editors Matthew Wickman Jacob Sherman BYU Humanities Center California Institute of Integral Studies USA USA Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Religions (ISSN 2077-1444) (available at: https://www.mdpi.com/journal/religions/special issues/ Faith Anthropocene). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03943-012-3 (Hbk) ISBN 978-3-03943-013-0 (PDF) Cover image courtesy of Andrew Seaman. c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editors .............................................. vii Matthew Wickman and Jacob Sherman Introduction: Faith after the Anthropocene Reprinted from: Religions 2020, 11, 378, doi:10.3390/rel11080378 .................. -
The Euclidean Mousetrap
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PhilPapers Originally in Journal of Idealistic Studies 38(3): 209-220 (2008). Please quote from published version. THE EUCLIDEAN MOUSETRAP: SCHOPENHAUER’S CRITICISM OF THE SYNTHETIC METHOD IN GEOMETRY Jason M. Costanzo Abstract In his doctoral dissertation On the Principle of Sufficient Reason, Arthur Schopenhauer there outlines a critique of Euclidean geometry on the basis of the changing nature of mathematics, and hence of demonstration, as a result of Kantian idealism. According to Schopenhauer, Euclid treats geometry synthetically, proceeding from the simple to the complex, from the known to the unknown, “synthesizing” later proofs on the basis of earlier ones. Such a method, although proving the case logically, nevertheless fails to attain the raison d’être of the entity. In order to obtain this, a separate method is required, which Schopenhauer refers to as “analysis”, thus echoing a method already in practice among the early Greek geometers, with however some significant differences. In this essay, I here discuss Schopenhauer’s criticism of synthesis in Euclid’s Elements, and the nature and relevance of his own method of analysis. The influence of philosophy upon the development of mathematics is readily seen in the practice among mathematicians of offering a demonstration or “proof” of the many theorems and problems which they encounter. This practice finds its origin among the early Greek geometricians and arithmeticians, during a time in which philosophy and mathematics intermingled at an unprecedented level, and a period in which rationalism enjoyed preeminence. -
Philosophy and the Mirror of Nature
Philosophy and the Mirror of Nature RICHARD RORTY Princeton University Press Princeton, New Jersey Copyright © 1979 by Princeton University Press Published by Princeton University Press, Princeton, New Jersey All Rights Reserved Library of Congress Cataloging-in-Publication Data Rorty, Richard. Philosophy and the mirror of nature. Includes index. 1. Philosophy. 2. Philosophy, Modern. 3. Mind and body. 4. Representation (Philosophy) 5. Analysis (Philosophy) 6. Civilization-Philosophy. I. Title. B53·R68 190 79- 84013 ISBN 0-691-07236-1 ISBN 0-691-02016-7 pbk. Publication of this book has been aided by a grant from The National Endowment for the Humanities This book has been composed in Linotype Baskerville Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America Second printing, with corrections, 1980 First Princeton Paperback printing, 1980 20 19 18 17 16 15 14 13 12 I I 10 TO M. V. R. When we think about the future of the world, we always have in mind its being at the place where it would be if it continued to move as we see it moving now. We do not realize that it moves not in a straight line, but in a curve, and that its direction constantly changes. Philosophy has made no progress? If somebody scratches where it itches, does that count as progress? If not, does that mean it wasn't an authentic scratch? Not an authentic itch? Couldn't this response to the stimulus go on for quite a long time until a remedy for itching is found? Wenn wir an die Zukunft der Welt denken, so meinen wir immer den Ort, wo sie sein wird, wenn sie so weiter Hiuft, wie wir sie jetzt laufen sehen, und denken nieht, da�s sie nieht gerade lauft, sondern in einer Kurve, und ihre Riehtung sieh konstant andert. -
Sophie's World
Sophie’s World Jostien Gaarder Reviews: More praise for the international bestseller that has become “Europe’s oddball literary sensation of the decade” (New York Newsday) “A page-turner.” —Entertainment Weekly “First, think of a beginner’s guide to philosophy, written by a schoolteacher ... Next, imagine a fantasy novel— something like a modern-day version of Through the Looking Glass. Meld these disparate genres, and what do you get? Well, what you get is an improbable international bestseller ... a runaway hit... [a] tour deforce.” —Time “Compelling.” —Los Angeles Times “Its depth of learning, its intelligence and its totally original conception give it enormous magnetic appeal ... To be fully human, and to feel our continuity with 3,000 years of philosophical inquiry, we need to put ourselves in Sophie’s world.” —Boston Sunday Globe “Involving and often humorous.” —USA Today “In the adroit hands of Jostein Gaarder, the whole sweep of three millennia of Western philosophy is rendered as lively as a gossip column ... Literary sorcery of the first rank.” —Fort Worth Star-Telegram “A comprehensive history of Western philosophy as recounted to a 14-year-old Norwegian schoolgirl... The book will serve as a first-rate introduction to anyone who never took an introductory philosophy course, and as a pleasant refresher for those who have and have forgotten most of it... [Sophie’s mother] is a marvelous comic foil.” —Newsweek “Terrifically entertaining and imaginative ... I’ll read Sophie’s World again.” — Daily Mail “What is admirable in the novel is the utter unpretentious-ness of the philosophical lessons, the plain and workmanlike prose which manages to deliver Western philosophy in accounts that are crystal clear.