Hands on History a Resource for Teaching Mathematics © 2007 by the Mathematical Association of America (Incorporated)
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Mathematics Is a Gentleman's Art: Analysis and Synthesis in American College Geometry Teaching, 1790-1840 Amy K
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2000 Mathematics is a gentleman's art: Analysis and synthesis in American college geometry teaching, 1790-1840 Amy K. Ackerberg-Hastings Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Higher Education and Teaching Commons, History of Science, Technology, and Medicine Commons, and the Science and Mathematics Education Commons Recommended Citation Ackerberg-Hastings, Amy K., "Mathematics is a gentleman's art: Analysis and synthesis in American college geometry teaching, 1790-1840 " (2000). Retrospective Theses and Dissertations. 12669. https://lib.dr.iastate.edu/rtd/12669 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margwis, and improper alignment can adversely affect reproduction. in the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. -
The Origin and Evolution of Calipers
History of Precision Measuring Instruments The Origin and Evolution of Calipers MAP1481_E12029_2_MeasuringHistory_Calipers.indd 1 18/5/21 1:34 PM Contents 1. About the Nogisu ...............................................................................................................................1 2. About the Caliper ...............................................................................................................................2 3. Origin of the Name Nogisu and Vernier Graduations ..........................................................................4 4. Oldest Sliding Caliper: Nogisu .............................................................................................................8 5. Scabbard Type Sliding Caliper till the Middle of 19th Century .............................................................9 6. World's Oldest Vernier Caliper (Nogisu) in Existence .........................................................................11 7. Theory That the Vernier Caliper (Nogisu) Was Born in the U.S. ..........................................................12 8. Calipers before Around 1945 (the End of WWII) in the West ...........................................................13 8. 1. Slide-Catching Scale without Vernier Graduations: Simplified Calipers ......................................14 8. 2. Sliding Calipers with Diagonal Scale ..........................................................................................18 8. 3. Sliding Caliper with a Vernier Scale: Vernier Caliper ..................................................................19 -
Surveying and Drawing Instruments
SURVEYING AND DRAWING INSTRUMENTS MAY \?\ 10 1917 , -;>. 1, :rks, \ C. F. CASELLA & Co., Ltd II to 15, Rochester Row, London, S.W. Telegrams: "ESCUTCHEON. LONDON." Telephone : Westminster 5599. 1911. List No. 330. RECENT AWARDS Franco-British Exhibition, London, 1908 GRAND PRIZE AND DIPLOMA OF HONOUR. Japan-British Exhibition, London, 1910 DIPLOMA. Engineering Exhibition, Allahabad, 1910 GOLD MEDAL. SURVEYING AND DRAWING INSTRUMENTS - . V &*>%$> ^ .f C. F. CASELLA & Co., Ltd MAKERS OF SURVEYING, METEOROLOGICAL & OTHER SCIENTIFIC INSTRUMENTS TO The Admiralty, Ordnance, Office of Works and other Home Departments, and to the Indian, Canadian and all Foreign Governments. II to 15, Rochester Row, Victoria Street, London, S.W. 1911 Established 1810. LIST No. 330. This List cancels previous issues and is subject to alteration with out notice. The prices are for delivery in London, packing extra. New customers are requested to send remittance with order or to furnish the usual references. C. F. CAS ELL A & CO., LTD. Y-THEODOLITES (1) 3-inch Y-Theodolite, divided on silver, with verniers to i minute with rack achromatic reading ; adjustment, telescope, erect and inverting eye-pieces, tangent screw and clamp adjustments, compass, cross levels, three screws and locking plate or parallel plates, etc., etc., in mahogany case, with tripod stand, complete 19 10 Weight of instrument, case and stand, about 14 Ibs. (6-4 kilos). (2) 4-inch Do., with all improvements, as above, to i minute... 22 (3) 5-inch Do., ... 24 (4) 6-inch Do., 20 seconds 27 (6 inch, to 10 seconds, 403. extra.) Larger sizes and special patterns made to order. -
Maps and Diagrams. Their Compilation and Construction
~r HJ.Mo Mouse andHR Wilkinson MAPS AND DIAGRAMS 8 his third edition does not form a ramatic departure from the treatment of artographic methods which has made it a standard text for 1 years, but it has developed those aspects of the subject (computer-graphics, quantification gen- erally) which are likely to progress in the uture. While earlier editions were primarily concerned with university cartography ‘ourses and with the production of the- matic maps to illustrate theses, articles and books, this new edition takes into account the increasing number of professional cartographer-geographers employed in Government departments, planning de- partments and in the offices of architects pnd civil engineers. The authors seek to ive students some idea of the novel and xciting developments in tools, materials, echniques and methods. The growth, mounting to an explosion, in data of all inds emphasises the increasing need for discerning use of statistical techniques, nevitably, the dependence on the com- uter for ordering and sifting data must row, as must the degree of sophistication n the techniques employed. New maps nd diagrams have been supplied where ecessary. HIRD EDITION PRICE NET £3-50 :70s IN U K 0 N LY MAPS AND DIAGRAMS THEIR COMPILATION AND CONSTRUCTION MAPS AND DIAGRAMS THEIR COMPILATION AND CONSTRUCTION F. J. MONKHOUSE Formerly Professor of Geography in the University of Southampton and H. R. WILKINSON Professor of Geography in the University of Hull METHUEN & CO LTD II NEW FETTER LANE LONDON EC4 ; © ig6g and igyi F.J. Monkhouse and H. R. Wilkinson First published goth October igj2 Reprinted 4 times Second edition, revised and enlarged, ig6g Reprinted 3 times Third edition, revised and enlarged, igyi SBN 416 07440 5 Second edition first published as a University Paperback, ig6g Reprinted 5 times Third edition, igyi SBN 416 07450 2 Printed in Great Britain by Richard Clay ( The Chaucer Press), Ltd Bungay, Suffolk This title is available in both hard and paperback editions. -
1 Functions, Derivatives, and Notation 2 Remarks on Leibniz Notation
ORDINARY DIFFERENTIAL EQUATIONS MATH 241 SPRING 2019 LECTURE NOTES M. P. COHEN CARLETON COLLEGE 1 Functions, Derivatives, and Notation A function, informally, is an input-output correspondence between values in a domain or input space, and values in a codomain or output space. When studying a particular function in this course, our domain will always be either a subset of the set of all real numbers, always denoted R, or a subset of n-dimensional Euclidean space, always denoted Rn. Formally, Rn is the set of all ordered n-tuples (x1; x2; :::; xn) where each of x1; x2; :::; xn is a real number. Our codomain will always be R, i.e. our functions under consideration will be real-valued. We will typically functions in two ways. If we wish to highlight the variable or input parameter, we will write a function using first a letter which represents the function, followed by one or more letters in parentheses which represent input variables. For example, we write f(x) or y(t) or F (x; y). In the first two examples above, we implicitly assume that the variables x and t vary over all possible inputs in the domain R, while in the third example we assume (x; y)p varies over all possible inputs in the domain R2. For some specific examples of functions, e.g. f(x) = x − 5, we assume that the domain includesp only real numbers x for which it makes sense to plug x into the formula, i.e., the domain of f(x) = x − 5 is just the interval [5; 1). -
Plenary Lectures
Plenary lectures A. D. Alexandrov and the Birth of the Theory of Tight Surfaces Thomas Banchoff Brown University, Providence, Rhode Island 02912, USA e-mail: Thomas [email protected] Alexander Danilovich Alexandrov was born in 1912, and just over 25 years later in 1938 (the year of my birth), he proved a rigidity theorem for real analytic surfaces of type T with minimal total absolute curvature. He showed in particular that that two real analytic tori of Type T in three-space that are iso- metric must be congruent. In 1963, twenty-five years after that original paper, Louis Nirenberg wrote the first generalization of that result, for five times differentiable surfaces satisfying some technical hypothe- ses necessary for applying techniques of ordinary and partial differential equations. Prof. Shiing-Shen Chern, my thesis advisor at the University of California, Berkeley, gave me this paper to present to his graduate seminar and challenged me to find a proof without these technical hypotheses. I read Konvexe Polyeder by A. D. Alexandrov and the works of Pogoreloff on rigidity for convex surfaces using polyhedral methods and I decided to try to find a rigidity theorem for Type T surfaces using similar techniques. I found a condition equivalent to minimal total absolute curvature for smooth surfaces, that also applied to polyhedral surfaces, namely the Two-Piece Property or TPP. An object in three- space has the TPP if every plane separates the object into at most two connected pieces. I conjectured a rigidity result similar to those of Alexandroff and Nirenberg, namely that two polyhedral surfaces in three-space with the TPP that are isometric would have to be congruent. -
Surveying Is the Art of Determining the Relative Position
CE6304-SURVEYING-I UNIT-I INTRODUCTION AND CHAIN SURVEYING DEFINITION: Surveying is the art of determining the relative positions of points on, above or beneath the surface of the earth by means of direct or indirect measurements of distance, direction & elevation. Plane Survey: Surveying which the mean surface of earth regarded as plain surface and not curve it really is known as plain surveying. A following Assumption are made: (i) A level line is considered a strait line thus the plump line at a point is parallel plump line at any after point. (ii) The angles between two such lines that intersect is a plain angle and not a sphere angle. (iii) The meridian through any two points parallel. (iv) When we deal with only a small portion earths surface the above assumptions can justify. (v) The error induced for a length of an 18.5 kms it‘s only 0.0152 ms grater than sub dented chord 1.52 cm. Geodetic survey : Survey is which the shape (curvature) of the earth surface is taken in the account a higher degree of precision is exercised in linear and angular measurement is tanned as Geodetic Survey. A line connecting two points is regarded as an arc. Such surveys extend over large areas PRINCIPLES OF SURVEYING Location of a point by measurement from 2 points of reference Working from whole to part. Location of a point by measurement from 2 points of reference There should be 2 points of reference say P & Q P,Q are the ground reference points and permanent points. -
Instruments and Methods of Physical Measurement
INSTRUMENTS AND METHODS OF PHYSICAL MEASUREMENT By J. W. MOORE, . i» Professor of Mechanics and Experimental Philosophy, LAFAYETTE COLLEGE. Easton, Pa.: The Eschenbach Printing House. 1892. Copyright by J. W. Moore, 1892 PREFACE. 'y'HE following pages have been arranged for use in the physical laboratory. The aim has been to be as concise as is con- sistent with clearness. sources of information have been used, and in many cases, perhaps, the ipsissima verba of well- known authors. j. w. M. Lafayette College, August 23,' 18g2. TABLE OF CONTENTS. Measurement of A. Length— I. The Diagonal Scale 7 11. The Vernier, Straight 8 * 111. Mayer’s Vernier Microscope . 9 IV. The Vernier Calipers 9 V. The Beam Compass 10 VI. The Kathetometer 10 VII. The Reading Telescope 12 VIII. Stage Micrometer with Camera Lucida 12 IX. Jackson’s Eye-piece Micrometer 13 X. Quincke’s Kathetometer Microscope 13 XI. The Screw . 14 a. The Micrometer Calipers 14 b. The Spherometer 15 c. The Micrometer Microscope or Reading Microscope . 16 d. The Dividing Engine 17 To Divide a Line into Equal Parts by 1. The Beam Compass . 17 2. The Dividing Engine 17 To Divide a Line “Originally” into Equal Parts by 1. Spring Dividers or Beam Compass 18 2. Spring Dividers and Straight Edge 18 3. Another Method 18 B. Angees— 1. Arc Verniers 19 2. The Spirit Level 20 3. The Reading Microscope with Micrometer Attachment .... 22 4. The Filar Micrometer 5. The Optical Method— 1. The Optical Lever 23 2. Poggendorff’s Method 25 3. The Objective or English Method 28 6. -
The Pope's Rhinoceros and Quantum Mechanics
Bowling Green State University ScholarWorks@BGSU Honors Projects Honors College Spring 4-30-2018 The Pope's Rhinoceros and Quantum Mechanics Michael Gulas [email protected] Follow this and additional works at: https://scholarworks.bgsu.edu/honorsprojects Part of the Mathematics Commons, Ordinary Differential Equations and Applied Dynamics Commons, Other Applied Mathematics Commons, Partial Differential Equations Commons, and the Physics Commons Repository Citation Gulas, Michael, "The Pope's Rhinoceros and Quantum Mechanics" (2018). Honors Projects. 343. https://scholarworks.bgsu.edu/honorsprojects/343 This work is brought to you for free and open access by the Honors College at ScholarWorks@BGSU. It has been accepted for inclusion in Honors Projects by an authorized administrator of ScholarWorks@BGSU. The Pope’s Rhinoceros and Quantum Mechanics Michael Gulas Honors Project Submitted to the Honors College at Bowling Green State University in partial fulfillment of the requirements for graduation with University Honors May 2018 Dr. Steven Seubert Dept. of Mathematics and Statistics, Advisor Dr. Marco Nardone Dept. of Physics and Astronomy, Advisor Contents 1 Abstract 5 2 Classical Mechanics7 2.1 Brachistochrone....................................... 7 2.2 Laplace Transforms..................................... 10 2.2.1 Convolution..................................... 14 2.2.2 Laplace Transform Table.............................. 15 2.2.3 Inverse Laplace Transforms ............................ 15 2.3 Solving Differential Equations Using Laplace -
The Tautochrone/Brachistochrone Problems: How to Make the Period of a Pendulum Independent of Its Amplitude
The Tautochrone/Brachistochrone Problems: How to make the Period of a Pendulum independent of its Amplitude Tatsu Takeuchi∗ Department of Physics, Virginia Tech, Blacksburg VA 24061, USA (Dated: October 12, 2019) Demo presentation at the 2019 Fall Meeting of the Chesapeake Section of the American Associa- tion of Physics Teachers (CSAAPT). I. THE TAUTOCHRONE A. The Period of a Simple Pendulum In introductory physics, we teach our students that a simple pendulum is a harmonic oscillator, and that its angular frequency ! and period T are given by s rg 2π ` ! = ;T = = 2π ; (1) ` ! g where ` is the length of the pendulum. This, of course, is not quite true. The period actually depends on the amplitude of the pendulum's swing. 1. The Small-Angle Approximation Recall that the equation of motion for a simple pendulum is d2θ g = − sin θ : (2) dt2 ` (Note that the equation of motion of a mass sliding frictionlessly along a semi-circular track of radius ` is the same. See FIG. 1.) FIG. 1. The motion of the bob of a simple pendulum (left) is the same as that of a mass sliding frictionlessly along a semi-circular track (right). The tension in the string (left) is simply replaced by the normal force from the track (right). ∗ [email protected] CSAAPT 2019 Fall Meeting Demo { Tatsu Takeuchi, Virginia Tech Department of Physics 2 We need to make the small-angle approximation sin θ ≈ θ ; (3) to render the equation into harmonic oscillator form: d2θ rg ≈ −!2θ ; ! = ; (4) dt2 ` so that it can be solved to yield θ(t) ≈ A sin(!t) ; (5) where we have assumed that pendulum bob is at θ = 0 at time t = 0. -
The Cycloid Scott Morrison
The cycloid Scott Morrison “The time has come”, the old man said, “to talk of many things: Of tangents, cusps and evolutes, of curves and rolling rings, and why the cycloid’s tautochrone, and pendulums on strings.” October 1997 1 Everyone is well aware of the fact that pendulums are used to keep time in old clocks, and most would be aware that this is because even as the pendu- lum loses energy, and winds down, it still keeps time fairly well. It should be clear from the outset that a pendulum is basically an object moving back and forth tracing out a circle; hence, we can ignore the string or shaft, or whatever, that supports the bob, and only consider the circular motion of the bob, driven by gravity. It’s important to notice now that the angle the tangent to the circle makes with the horizontal is the same as the angle the line from the bob to the centre makes with the vertical. The force on the bob at any moment is propor- tional to the sine of the angle at which the bob is currently moving. The net force is also directed perpendicular to the string, that is, in the instantaneous direction of motion. Because this force only changes the angle of the bob, and not the radius of the movement (a pendulum bob is always the same distance from its fixed point), we can write: θθ&& ∝sin Now, if θ is always small, which means the pendulum isn’t moving much, then sinθθ≈. This is very useful, as it lets us claim: θθ&& ∝ which tells us we have simple harmonic motion going on. -
A Tale of the Cycloid in Four Acts
A Tale of the Cycloid In Four Acts Carlo Margio Figure 1: A point on a wheel tracing a cycloid, from a work by Pascal in 16589. Introduction In the words of Mersenne, a cycloid is “the curve traced in space by a point on a carriage wheel as it revolves, moving forward on the street surface.” 1 This deceptively simple curve has a large number of remarkable and unique properties from an integral ratio of its length to the radius of the generating circle, and an integral ratio of its enclosed area to the area of the generating circle, as can be proven using geometry or basic calculus, to the advanced and unique tautochrone and brachistochrone properties, that are best shown using the calculus of variations. Thrown in to this assortment, a cycloid is the only curve that is its own involute. Study of the cycloid can reinforce the curriculum concepts of curve parameterisation, length of a curve, and the area under a parametric curve. Being mechanically generated, the cycloid also lends itself to practical demonstrations that help visualise these abstract concepts. The history of the curve is as enthralling as the mathematics, and involves many of the great European mathematicians of the seventeenth century (See Appendix I “Mathematicians and Timeline”). Introducing the cycloid through the persons involved in its discovery, and the struggles they underwent to get credit for their insights, not only gives sequence and order to the cycloid’s properties and shows which properties required advances in mathematics, but it also gives a human face to the mathematicians involved and makes them seem less remote, despite their, at times, seemingly superhuman discoveries.