DOCSLIB.ORG

Indirect Proof

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Indirect Proof 95698_ch02_067-120.qxp 9/5/13 4:53 PM Page 76 76 CHAPTER 2 ■ PARALLEL LINES In Exercises 28 to 30, write a formal proof of each theorem. 34. Given: Triangle MNQ with M obtuse ∠MNQ 28. If two parallel lines are cut by a transversal, then the pairs Construct: NE Ќ MQ, with E of alternate exterior angles are congruent. on MQ. N Q 29. If two parallel lines are cut by a transversal, then the pairs 35. Given: Triangle MNQ with of exterior angles on the same side of the transversal are Exercises 34, 35 obtuse ∠MNQ supplementary. Construct: MR Ќ NQ 30. If a transversal is perpendicular to one of two parallel lines, (HINT: Extend NQ.) then it is also perpendicular to the other line. 36. Given: A line m and a point T not on m 31. Suppose that two lines are cut by a transversal in such a way that the pairs of corresponding angles are not congruent. T Can those two lines be parallel? 32. Given: Line ᐍ and point P P m not on ᐍ —! Suppose that you do the following: Construct: PQ Ќ ᐍ i) Construct a perpendicular line r from T to line m. ii) Construct a line s perpendicular to line r at point T. 33. Given: Triangle ABC with B three acute angles What is the relationship between lines s and m? Construct: BD Ќ AC, with D on AC. A C 2.2 Indirect Proof KEY CONCEPTS Conditional Contrapositive Indirect Proof Converse Law of Negative Inverse Inference Let P S Q represent the conditional statement “If P, then Q.” The following statements are related to this conditional statement (also called an implication). NOTE: Recall that ~P represents the negation of P. Conditional (or Implication) P S Q If P, then Q. Converse of Conditional Q S P If Q, then P. Inverse of Conditional ~P S ~Q If not P, then not Q. Contrapositive of Conditional ~Q S ~P If not Q, then not P. Consider the following conditional statement. If Tom lives in San Diego, then he lives in California. This true conditional statement has the following related statements: Converse: If Tom lives in California, then he lives in San Diego. (false) Inverse: If Tom does not live in San Diego, then he does not live in California. (false) Contrapositive: If Tom does not live in California, then he does not live in San Diego. (true) Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 95698_ch02_067-120.qxp 9/5/13 4:54 PM Page 77 2.2 ■ Indirect Proof 77 In general, the conditional statement and its contrapositive are either both true or both false! Similarly, the converse and the inverse are also either both true or both false. EXAMPLE 1 For the conditional statement that follows, give the converse, the inverse, and the con- trapositive. Then classify each as true or false. If two angles are vertical angles, then they are congruent angles. SOLUTION CONVERSE: If two angles are congruent angles, then they are vertical angles. (false) INVERSE: If two angles are not vertical angles, then they are not congruent angles. (false) CONTRAPOSITIVE: If two angles are not congruent angles, then they are not vertical angles. (true) “If P, then Q” and “If not Q, then not P” are equivalent. P Venn Diagrams can be used to explain why the conditional statement P S Q and its Q contrapositive ~Q S ~P are equivalent. The relationship “If P, then Q” is represented in Figure 2.11. Note that if any point is selected outside of Q (that is ~Q), then it cannot Figure 2.11 possibly lie in set P (thus, ϳ P). THE LAW OF NEGATIVE INFERENCE (CONTRAPOSITION) EXS. 1, 2 Consider the following circumstances, and accept each premise as true: 1. If Matt cleans his room, then he will go to the movie. (P S Q ) 2. Matt does not get to go to the movie. (~Q ) What can you conclude? You should have deduced that Matt did not clean his room; if he had, he would have gone to the movie. This “backdoor” reasoning is based on the fact that the truth of P S Q implies the truth of ~Q S ~P. LAW OF NEGATIVE INFERENCE (CONTRAPOSITION) 1. P S Q 2. ~ Q C. І ϳ P EXAMPLE 2 Use the Law of Negative Inference to draw a valid conclusion for this argument. 1. If the weather is nice Friday, we will go on a picnic. 2. We did not go on a picnic Friday. C. І ? SOLUTION The weather was not nice Friday. Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 95698_ch02_067-120.qxp 9/12/13 11:29 AM Page 78 78 CHAPTER 2 ■ PARALLEL LINES Like the Law of Detachment from Section 1.1, the Law of Negative Inference (Law of Contraposition) is a form of deduction. Whereas the Law of Detachment characterizes the method of “direct proof” found in preceding sections, the Law of Negative Inference char- acterizes the method of proof known as indirect proof. INDIRECT PROOF EXS. 3, 4 You will need to know when to use the indirect method of proof. Often the theorem to be proved has the form P S Q, in which Q is a negation and denies some claim. For instance, an indirect proof might be best if Q reads in one of these ways: c is not equal to d ᐍ is not perpendicular to m However, we will see in Example 5 of this section that the indirect method can be used to prove that line ᐍ is parallel to line m. Indirect proof is also used for proving existence and uniqueness theorems; see Example 6. Geometry in the Real World The method of indirect proof is illustrated in Example 3. All indirect proofs in this book are given in paragraph form (as are some of the direct proofs). In any paragraph proof, each statement must still be justified. Because of the need to order your statements properly, writing any type of proof may have a positive impact on the essays you write for your other classes! EXAMPLE 3 ! ! GIVEN: In Figure 2.12, BA is not perpendicular to BD When the bubble displayed on the level is not centered, the board PROVE:∠1 and ∠2 are not complementary A used in construction is neither PROOF: Suppose that ∠1 and ∠2 are vertical nor horizontal. complementary. Then m∠1 ϩ m∠2 ϭ 90Њ C because the sum of the measures of two complementary ∠s is 90. We also know that 1 m∠1 ϩ m∠2 ϭ m∠ABD by the Angle- 2 B Addition Postulate. In turn, m∠ABD ϭ 90Њ D by substitution.! ! Then ∠ABD is a right angle. In Figure 2.12 turn, BA Ќ BD. But this contradicts the given hypothesis; therefore, the supposition must be false, and it follows that ∠1 and ∠2 are not complementary. In Example 3 and in all indirect proofs, the first statement takes the form Suppose/Assume the exact opposite of the Prove Statement. By its very nature, such a statement cannot be supported even though every other statement in the proof can be justified; thus, when a contradiction is reached, the finger of blame points to the supposition. Having reached a contradiction, we may say that the claim involving ~Q has failed and is false; in effect, the double negative ~(~Q) is equivalent to Q. Thus, our only recourse is to conclude that Q is true. Following is an outline of this technique. Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 95698_ch02_067-120.qxp 9/5/13 4:54 PM Page 79 2.2 ■ Indirect Proof 79 STRATEGY FOR PROOF ■ Method of Indirect Proof EXS. 5–7 To prove the statement P S Q or to complete the proof problem of the form Given: P Prove: Q by the indirect method, use the following steps: 1. Suppose that ~Q is true. 2. Reason from the supposition until you reach a contradiction. 3. Note that the supposition claiming that ~Q is true must be false and that Q must therefore be true. Step 3 completes the proof. The contradiction found in an indirect proof often takes the form “Q and ~Q,” which cannot be true.
Load more

Recommended publications
  • Icons of Mathematics an EXPLORATION of TWENTY KEY IMAGES Claudi Alsina and Roger B
    AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 45 Icons of Mathematics AN EXPLORATION OF TWENTY KEY IMAGES Claudi Alsina and Roger B. Nelsen i i “MABK018-FM” — 2011/5/16 — 19:53 — page i — #1 i i 10.1090/dol/045 Icons of Mathematics An Exploration of Twenty Key Images i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page ii — #2 i i c 2011 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2011923441 Print ISBN 978-0-88385-352-8 Electronic ISBN 978-0-88385-986-5 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY-FIVE Icons of Mathematics An Exploration of Twenty Key Images Claudi Alsina Universitat Politecnica` de Catalunya Roger B. Nelsen Lewis & Clark College Published and Distributed by The Mathematical Association of America i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Frank Farris, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Thomas M. Halverson Patricia B. Humphrey Michael J. McAsey Michael J. Mossinghoff Jonathan Rogness Thomas Q. Sibley i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical As- sociation of America was established through a generous gift to the Association from Mary P.
    [Show full text]
  • Simple Andi Complete K-Points in Continuous and in Modular Projective Spaces
    SIMPLE ANDI COMPLETE K-POINTS IN CONTINUOUS AND IN MODULAR PROJECTIVE SPACES by Beulah Armstrong Submitted to the Department of Mathematica and the faculty of the Graduate School of the University of Kansas in partial fulfillment.of I the requirements for the degree of Master of Arts. · Approved . 'UI/;~ _ Depar.tment of~ B I B L I ·o G R A P H Y. Allman, G.J. - Greek Geometry from Thales to Euclid - Longmans, Green & Co., London - 1889. Beman, W.W. and Smith, D.E. - Kls.in's Famous. Problems in ·Elementary Geometry - Ginn & Co., Boston - 1897. , , n Cantor, Moritz - Vorlesungen Uber Geschichte der Math- ematik - B.G. !l:eubner - Leipzig - 1894. Daviesf Iviathematical Dictionary and Cyclopedia of Math- ematical ~cieilce - A.s. Barnes & Co., New. York - 1855. Eisenlohr, Augu.st - Mathematisches Hendbuch der Alten Aegypter, Erst er Band - j. c. Hinrich, Le~pzig - 1877/ . _ Gow, James - A Short History of Greek Mathematics, University Press, Cambridge - 1884. Heath, T.L. - The i;chirteen Books of Euclid 1 s Elements, University Press; Cambridge - 1908. Lachlan, R. :.. Modern Pure Geometry - MacMillan & Co., New York - 1893. Tropfke, Johannes, Gesohichte der E1ementar Mathematik, Veit und Comp, Leipzig, 1903. Veblen and Young - Projective Geometry - Ginn & Co., New York - 1910. Young, J.W.A. - Monographs on Modern Mathematics, ·Longmans, Green & Co., New York - 1911. 1. INTRODUCTION. It is the pil:rpose of this paper, (1) ¥0 trace the development of the concepts of the simple· k-point ·and the complete k-point as they appear in mathematical literature and to £ormulate their definitions in a modular or an ordinary projective n-space (Se.ctioris 1 and 2); and (2) To determine the number of complete k-points in a modular projective plane (Section 3).
    [Show full text]
  • Polygrammal Symmetries in Bio-Macromolecules: Heptagonal Poly D(As4t) . Poly D(As4t) and Heptameric -Hemolysin
    Polygrammal Symmetries in Bio-macromolecules: Heptagonal Poly d(As4T) . poly d(As4T) and Heptameric -Hemolysin A. Janner * Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen The Netherlands Polygrammal symmetries, which are scale-rotation transformations of star polygons, are considered in the heptagonal case. It is shown how one can get a faithful integral 6-dimensional representation leading to a crystallographic approach for structural prop- erties of single molecules with a 7-fold point symmetry. Two bio-macromolecules have been selected in order to demonstrate how these general concepts can be applied: the left-handed Z-DNA form of the nucleic acid poly d(As4T) . poly d(As4T) which has the line group symmetry 7622, and the heptameric transmembrane pore protein ¡ -hemolysin. Their molecular forms are consistent with the crystallographic restrictions imposed on the combination of scaling transformations with 7-fold rotations. This all is presented in a 2-dimensional description which is natural because of the axial symmetry. 1. Introduction ture [4,6]. In a previous paper [1] a unifying general crystallography has The interest for the 7-fold case arose from the very nice axial been de®ned, characterized by the possibility of point groups of view of ¡ -hemolysin appearing in the cover picture of the book in®nite order. Crystallography then becomes applicable even to on Macromolecular Structures 1997 [7] which shows a nearly single molecules, allowing to give a crystallographic interpre- perfect 7-fold scale-rotation symmetry of the heptamer. In or- tation of structural relations which extend those commonly de- der to distinguish between law and accident, nucleic acids have noted as 'non-crystallographic' in protein crystallography.
    [Show full text]
  • The Logarithmic Spiral by Donald G
    THE PENTAGON Volume XVI SPRING, 1957 Number 2 CONTENTS Page National Officers 66 The Logarithmic Spiral By Donald G. Johnson 67 Consequences of A* + B8 = Ca By Alfred Moessner 76 Paradox Lost—Paradox Regained By Peggy Steinbeck •— 78 Watch That Check—It Might Bounce! By Dana R. Sudborough 84 Construction of the Circle of Inversion By Shirley T. Loeven 85 Mathematical Notes By Merl Kardatzke 86 Creating Interest for Gifted Students By Ati R. Amir-Moez „ 88 Order Among Complex Numbers By Merl Kardatzke 91 Problem Corner , 94 The Mathematical Scrapbook 97 The Book Shelf 101 Installation of New Chapter 109 Kappa Mu EpsOon News . ._ 110 Program Topics 112 National Officers C. C. Richtmeyer President Central Michigan College, Mt. Pleasant, Michigan R. G. Smith Vice-President Kansas State Teachers College, Pittsburg, Kansas Laura Z. Greene Secretary Washburn Municipal University, Topeka, Kansas M. Leslie Madison Treasurer Colorado A&M College, Port Collins, Colorado Frank Hawthorne .... - Historian The State Education Department, Albany, N.Y. Charles B. Tucker -- - Past President Teachers College, Emporia, Kansas Kappa Mu Epsilon, national honorary mathematics society, was founded in 1931. The object of the fraternity is fourfold: to further the interests of mathematics in those schools which place their primary emphasis on the undergraduate program; to help the undergraduate realize the important role that mathematics has played in the develop ment of western civilization; to develop an appreciation of the power and beauty possessed by mathematics, due, mainly, to its demands for logical and rigorous modes of thought; and to provide a society for the recognition of outstanding achievements in the study of mathe matics at the undergraduate leveL The official journal, THE PENTA GON, is designed to assist in achieving these objectives as well as to aid in establishing fraternal ties between the chapters.
    [Show full text]
  • Tessellated and Stellated Invisibility
    Tessellated and stellated invisibility Andr´eDiatta,1 Andr´eNicolet,2 S´ebastien Guenneau,1 and Fr´ed´eric Zolla2 September 20, 2021 1Department of Mathematical Sciences, Peach Street, Liverpool L69 3BX, UK Email addresses: [email protected]; [email protected] 2Institut Fresnel, UMR CNRS 6133, University of Aix-Marseille III, case 162, F13397 Marseille Cedex 20, France Email addresses: [email protected]; [email protected]; Abstract We derive the expression for the anisotropic heterogeneous matrices of per- mittivity and permeability associated with two-dimensional polygonal and star shaped cloaks. We numerically show using finite elements that the forward scat- tering worsens when we increase the number of sides in the latter cloaks, whereas it improves for the former ones. This antagonistic behavior is discussed using a rigorous asymptotic approach. We use a symmetry group theoretical approach to derive the cloaks design. Ocis: (000.3860) Mathematical methods in physics; (260.2110) Electromagnetic the- ory; (160.3918) Metamaterials; (160.1190) Anisotropic optical materials 1 Introduction Back in 2006, a team led by Pendry and Smith theorized and subsequently experimen- tally validated that a finite size object surrounded by a spherical coating consisting of a metamaterial might become invisible for electromagnetic waves [1, 2]. Many authors have since then dedicated a fast growing amount of work to the invisibility cloak- arXiv:0912.0160v1 [physics.optics] 1 Dec 2009 ing problem. However, there are alternative approaches, including inverse problems’s methods [3] or plasmonic properties of negatively refracting index materials [4]. It is also possible to cloak arbitrarily shaped regions, and for cylindrical domains of arbi- trary cross-section, one can generalize [1] to linear radial geometric transformations [5] ′ r = R1(θ)+ r(R2(θ) R1(θ))/R2(θ) , 0 r R2(θ) , ′ − ≤ ≤ θ = θ , 0 < θ 2π , (1) ′ ≤ x = x , x IR , 3 3 3 ∈ ′ ′ ′ where r , θ and x3 are “radially contracted cylindrical coordinates” and (x1, x2, x3) is the Cartesian basis.
    [Show full text]
  • Chapter 1 Plane Figurate Numbers
    December 6, 2011 9:6 9in x 6in Figurate Numbers b1260-ch01 Chapter 1 Plane figurate numbers In this introductory chapter, the basic figurate numbers, polygonal numbers, are presented. They generalize triangular and square numbers to any regular m-gon. Besides classical polygonal numbers, we consider in the plane cen- tered polygonal numbers, in which layers of m-gons are drown centered about a point. Finally, we list many other two-dimensional figurate numbers: pronic numbers, trapezoidal numbers, polygram numbers, truncated plane figurate numbers, etc. 1.1 Definitions and formulas 1.1.1. Following to ancient mathematicians, we are going now to consider the sets of points forming some geometrical figures on the plane. Starting from a point, add to it two points, so that to obtain an equilateral triangle. Six-points equilateral triangle can be obtained from three-points triangle by adding to it three points; adding to it four points gives ten-points triangle, etc. So, by adding to a point two, three, four etc. points, then organizing the points in the form of an equilateral triangle and counting the number of points in each such triangle, one can obtain the numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, . (Sloane’s A000217, [Sloa11]), which are called triangular numbers. 1 December 6, 2011 9:6 9in x 6in Figurate Numbers b1260-ch01 2 Figurate Numbers Similarly, by adding to a point three, five, seven etc. points and organizing them in the form of a square, one can obtain the numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, .
    [Show full text]
  • Secondary Schools Curriculum Guide, Mathematics, Grades 10-12, Levels 87-112
    DOCUMENT RESUME ED 077 740 SE 016 300 AUTHOR Rogers, Arnold R., Ed.; And Others . TITLE Secondary Schools Curriculum Guide, Mathematics, Grades 10-12, Levels 87-112. INSTITUTION Cranston School Dept., R.I. SPONS AGENCY Office of Education (DREW), Washington, D.C. Projects to Advance Creativity in Education. PUB DATE 72 NOTE 145p.; Draft Copy EDRS PRICE MF-$0.65 HC-$6.58 DESCRIPTORS *Behavioral Objectives; Calculu, Computers; *Curriculum; Curriculum Guides; !ometry; Mathematics Education; Number Concepts; Objectives; Probability Theory; *Secondary School Mathematics IDENTIFIERS ESEA Title III; Objectives Bank ABSTRACT Behavioral objectives for geometry, algebra, computer mathematics, trigonometry, analytic geometry, calculus, and probability are specified for grades 10 through 12. General objectives are stated for major areas under each topic and are followed by a list of specific objectives for that area. This work was prepared under an ESEA Title III contract. (DT) FILMED FROM BEST AVAILABLE COPY U S DEPARTMENT OF HE AL TN EDUCATiON & WELF ARE NAT.ONAL INSTITUTE OF EDUCATION :'F D :, %. .. E,., . v .F c . i Secondary Schools F C- A _. .. t . CURRICULUM GUIDE Cranston School Department Cranston, Rhode Island 1972 MATHEMATICS Grades 10-12 Levels 87-112 Secondary School CURRICULUM GUIDE DRAFT COPY Prepared By a curriculum writing team of secondary teachers Project PACESETTER Title III, E. S. E. A., 1965 Cranston School Department 845 Park Avenue Cranston, R.I. 02910 1972 ACKNOWLEDGEMENTS Dr. Joseph J. Picano Superintendent of Schools Mr. Robert S. Fresher Mr. Josenh A. murrav Assistant Superintendent Assistant Sunerintendent Dr. Guy N. DiBiasio Mr. Carlo A. samba Director of Curriculum Director of (rant Programs Mr.
    [Show full text]
  • A Greek Sepher Sephiroth
    Supplement to a Greek Sepher Sephiroth LIBER MCCCLXV-777 "Liber MCCLXIV The Greek Qabalah." A complete dictionary of all sacred and important words and phrases given in the Books of the Gnosis and other important writings both in the Greek and the Coptic. Unpublished. A.'. A.'. Publication in Class C By Frater Apollonius ▫ 4° = 7 AN CV 2009 e.v. Sacred Geometry Extracted from a course on ‘Geometry in Art & Architecture Dartmouth College Polygons The plane figures, the polygons, triangles, squares, hexagons, and so forth, were related to the numbers (three and the triangle, for example), were thought of in a similar way, and in fact, carried even more symbolism than the numbers themselves, because they were visual. Polygons are directly related to the Pythagoreans; the equilateral triangle (Sacred tetractys), hexagon, triangular numbers, and pentagram. A polygon is a plane figure bounded by straight lines, called the sides of the polygon. From the Greek poly = many and gon = angle The sides intersect at points called the vertices. The angle between two sides is called an interior angle or vertex angle. A regular polygon is one in which all the sides and interior angles are equal. A polygram can be drawn by connecting the vertices of a polgon. Pentagon & Pentagram, hexagon & hexagram, octagon & octograms Equilateral Triangle There are, of course, an infinite number of regular polygons, but we'll just discuss those with sides from three to eight. In this unit we'll cover just those with 3, 5, and 6 sides. We'll start with the simplest of all regular polygons, the equilateral triangle.
    [Show full text]
  • Package 'Teachingdemos'
    Package ‘TeachingDemos’ April 7, 2020 Title Demonstrations for Teaching and Learning Version 2.12 Author Greg Snow Description Demonstration functions that can be used in a classroom to demonstrate statistical con- cepts, or on your own to better understand the concepts or the programming. Maintainer Greg Snow <[email protected]> License Artistic-2.0 Date 2020-04-01 Suggests tkrplot, lattice, MASS, rgl, tcltk, tcltk2, png, ggplot2, logspline, maptools, R2wd, manipulate LazyData true KeepSource true Repository CRAN Repository/R-Forge/Project teachingdemos Repository/R-Forge/Revision 76 Repository/R-Forge/DateTimeStamp 2020-04-04 15:55:45 Date/Publication 2020-04-07 16:00:10 UTC NeedsCompilation no Depends R (>= 2.10) R topics documented: TeachingDemos-package . .3 bct..............................................5 cal..............................................6 char2seed . .8 chisq.detail . .9 ci.examp . 10 clipplot . 12 clt.examp . 13 1 2 R topics documented: cnvrt.coords . 14 coin.faces . 17 col2grey . 18 cor.rect.plot . 19 dice ............................................. 20 digits . 22 dots ............................................. 23 dynIdentify . 24 evap............................................. 26 faces............................................. 27 faces2 . 29 fagan.plot . 30 gp.open . 32 hpd ............................................. 34 HWidentify . 35 lattice.demo . 36 ldsgrowth . 37 loess.demo . 38 mle.demo . 40 ms.polygram . 41 my.symbols . 43 outliers . 46 pairs2 . 48 panel.my.symbols . 50 petals . 52 plot2script . 53 power.examp . 55 put.points.demo . 56 Pvalue.norm.sim . 57 rgl.coin . 59 rgl.Map . 61 roc.demo . 62 rotate.cloud . 63 run.cor.examp . 65 run.hist.demo . 66 SensSpec.demo . 67 shadowtext . 68 sigma.test . 69 simfun . 70 slider . 74 sliderv . 78 SnowsCorrectlySizedButOtherwiseUselessTestOfAnything . 79 SnowsPenultimateNormalityTest . 81 spread.labs . 82 squishplot . 84 steps .
    [Show full text]
  • Crowley's Greek Qabalah From
    A Greek Sepher Sephiroth LIBER MCCCLXV-777 "Liber MCCLXIV The Greek Qabalah." A complete dictionary of all sacred and important words and phrases given in the Books of the Gnosis and other important writings both in the Greek and the Coptic. Unpublished. A.'. A.'. Publication in Class C By Frater Apollonius ▫ 4° = 7 AN CV 2009 e.v. Preface The Bible, by various authors unknown. The Hebrew and Greek Originals are of Qabalistic value. It contains also many magical apologues, and recounts many tales of folk-lore and magical rites. This small and simple quote by Crowley in his ―Curriculum of A.'. A.'.‖ says more than may be readily apparent to most Thelemites. But then again, most Thelemites don‘t really seem to even know about Crowley‘s deep knowledge of the Bible. And without reading both John‘s Apocalypse and Liber CDXVIII very carefully, one can‘t easily see how Thelema is truly a further development of the New Testament revelation; indeed as Motta says, a correction to the distortions that have come through time. Crowley indeed takes this one step further in his description of that lack of understanding in light of the message of John‘s Apocalypse. He viewed the controversial document, which itself, barely made it into the Christian canon, as authentic prophecy; the problem being that its interpretation by people at the start of the Piscean Age was too difficult for a document that actually addressed the start of the Aquarian Age, which he heralded as the Aeon of Horus. He writes in the Book of Thoth: The seers in the early days of the Aeon of Osiris foresaw the Manifestation of this coming Aeon in which we now live, and they regarded it with intense horror and fear, not understanding the precession of the Aeons, and regarding every change as catastrophe.
    [Show full text]
  • Gems of Geometry
    Gems of Geometry John Barnes Gems of Geometry John Barnes Caversham, England [email protected] ISBN 978-3-642-05091-6 e-ISBN 978-3-642-05092-3 DOI 10.1007/978-3-642-05092-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009942783 Mathematics Subject Classification (2000): 00-xx, 37-xx, 51-xx, 54-xx, 83-xx © Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To Bobby, Janet, Helen, Christopher and Jonathan Preface This book is based on lectures given several times at Reading University in England at their School of Continuing Education, from about 2002. One might wonder why I gave these lectures.
    [Show full text]
  • ED102023.Pdf
    DOCUMENT RESUME ED 102 023 88 SE 018 746 TITLE Secondary Schools Curriculum Guide, Mathematics, Gravies 10-12. Revised. INSTITUTION Cranston School Dept., R.I. SPONS AGENCY Bureau of Elementary and Secondary Education (DREW /OED, Washington, D.C. PUB DATE 74 NOTE 198p.; This is a revision to ED 077 740; Best Copy Available EDRS PRICE MF-$0.76 HC-$9.51 PLUS POSTAGE DESCRIPTORS Algebra; *Behavioral Objectives; Calculus; Computers; Curriculum; *Curriculum Guides; Geometry; Mathematics Education; Number Concepts; *Objectives; Probability Theory; Secondary Education; * Secondary School Mathematics; Trigonometry IDENTIFIERS Elementary Secondary Education Act Title III; ESEA Title III; General Mathematics; *Objectives Bank ABSTRACT Behavioral objectives for grades 10 through 12 are specified for plane geometry, algebra, general mathematics, computer mathematics, slide rule mathematics, basic college mathematics, trigonometry, analytic geometry, calculus and probability. Most sections present material in terms of portions of a school year. At least one major objective is stated for each Section of the guide, encompassing the entire uork for the unit. Several more specific objectives are also included in each section, intended to indicate level of learning, content, and means of evaluation. Each objective is followed by a list of suggested activities. (SDI U SDEPARTMENT OF KALTN, EDUCATION &WELFARE NATIONAL INSTITUTE OF EDUCATION THIS 00CuMINT HAS BEV, REPRO OUCfO E xAc rt. v As REM, f0 cRom tH PL:RSON OW ORGANI/ATIOu ORIGIN AtINo if POINTS OF Ow OPINIONS STATE° 00 N91 NICESSARII v NEPAL Oc ictAk. NAtIONAL INSTITUTE Ot UOUCAtION POSITION OR POLICY ] Secondary Schools ,;CURRICULUM GUIDE Cranston School Department Cranston, Rhode Island 1973-74 MATH Grades 1042 SECONDARY SCHOOLS CURRICULUM GUIDES Prepared by: THE FIFTH QUARTER PLAN FUNDED BY: TITLE III, E.S.E.A.
    [Show full text]