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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. -
Greeks Doing Algebra
Greeks Doing Algebra There is a long-standing consensus in the history of mathematics that geometry came from ancient Greece, algebra came from medieval Persia, and the two sciences did not meet until seventeenth-century France (e.g. Bell 1945). Scholars agree that the Greek mathematicians had no methods comparable to algebra before Diophantus (3rd c. CE) or, many hold, even after him (e.g. Szabó 1969, Unguru and Rowe 1981, Grattan-Guinness 1996, Vitrac 2005. For a survey of arguments see Blåsjö 2016). The problems that we would solve with algebra, the Greeks, especially the authors of the canonical and most often studied works (such as Euclid and Apollonius of Perga), approached with spatial geometry or not at all. This paper argues, however, that the methods which uniquely characterize algebra, such as information compression, quantitative abstraction, and the use of unknowns, do in fact feature in Greek mathematical works prior to Diophantus. We simply have to look beyond the looming figures of Hellenistic geometry. In this paper, we shall examine three instructive cases of algebraic problem-solving methods in Greek mathematical works before Diophantus: The Sand-reckoner of Archimedes, the Metrica of Hero of Alexandria, and the Almagest of Ptolemy. In the Sand-reckoner, Archimedes develops a system for expressing extremely large numbers, in which the base unit is a myriad myriad. His process is indefinitely repeatable, and theoretically scalable to express a number of any size. Simple though it sounds to us, this bit of information compression, by which a cumbersome quantity is set to one in order to simplify notation and computation, is a common feature of modern mathematics but was almost alien to the Greeks. -
Abstract 1. Introduction 2. Robert Stirling
Stirling Stuff Dr John S. Reid, Department of Physics, Meston Building, University of Aberdeen, Aberdeen AB12 3UE, Scotland Abstract Robert Stirling’s patent for what was essentially a new type of engine to create work from heat was submitted in 1816. Its reception was underwhelming and although the idea was sporadically developed, it was eclipsed by the steam engine and, later, the internal combustion engine. Today, though, the environmentally favourable credentials of the Stirling engine principles are driving a resurgence of interest, with modern designs using modern materials. These themes are woven through a historically based narrative that introduces Robert Stirling and his background, a description of his patent and the principles behind his engine, and discusses the now popular model Stirling engines readily available. These topical models, or alternatives made ‘in house’, form a good platform for investigating some of the thermodynamics governing the performance of engines in general. ---------------------------------------------------------------------------------------------------------------- 1. Introduction 2016 marks the bicentenary of the submission of Robert Stirling’s patent that described heat exchangers and the technology of the Stirling engine. James Watt was still alive in 1816 and his steam engine was gaining a foothold in mines, in mills, in a few goods railways and even in pioneering ‘steamers’. Who needed another new engine from another Scot? The Stirling engine is a markedly different machine from either the earlier steam engine or the later internal combustion engine. For reasons to be explained, after a comparatively obscure two centuries the Stirling engine is attracting new interest, for it has environmentally friendly credentials for an engine. This tribute introduces the man, his patent, the engine and how it is realised in example models readily available on the internet. -
Water, Air and Fire at Work in Hero's Machines
Water, air and fire at work in Hero’s machines Amelia Carolina Sparavigna Dipartimento di Fisica, Politecnico di Torino Corso Duca degli Abruzzi 24, Torino, Italy Known as the Michanikos, Hero of Alexandria is considered the inventor of the world's first steam engine and of many other sophisticated devices. Here we discuss three of them as described in his book “Pneumatica”. These machines, working with water, air and fire, are clear examples of the deep knowledge of fluid dynamics reached by the Hellenistic scientists. Hero of Alexandria, known as the Mechanicos, lived during the first century in the Roman Egypt [1]. He was probably a Greek mathematician and engineer who resided in the city of Alexandria. We know his work from some of writings and designs that have been arrived nowadays in their Greek original or in Arabic translations. From his own writings, it is possible to gather that he knew the works of Archimedes and of Philo the Byzantian, who was a contemporary of Ctesibius [2]. It is almost certain that Heron taught at the Museum, a college for combined philosophy and literary studies and a religious place of cult of Muses, that included the famous Library. For this reason, Hero claimed himself a pupil of Ctesibius, who was probably the first head of the Museum of Alexandria. Most of Hero’s writings appear as lecture notes for courses in mathematics, mechanics, physics and pneumatics [2]. In optics, Hero formulated the Principle of the Shortest Path of Light, principle telling that if a ray of light propagates from a point to another one within the same medium, the followed path is the shortest possible. -
15 Famous Greek Mathematicians and Their Contributions 1. Euclid
15 Famous Greek Mathematicians and Their Contributions 1. Euclid He was also known as Euclid of Alexandria and referred as the father of geometry deduced the Euclidean geometry. The name has it all, which in Greek means “renowned, glorious”. He worked his entire life in the field of mathematics and made revolutionary contributions to geometry. 2. Pythagoras The famous ‘Pythagoras theorem’, yes the same one we have struggled through in our childhood during our challenging math classes. This genius achieved in his contributions in mathematics and become the father of the theorem of Pythagoras. Born is Samos, Greece and fled off to Egypt and maybe India. This great mathematician is most prominently known for, what else but, for his Pythagoras theorem. 3. Archimedes Archimedes is yet another great talent from the land of the Greek. He thrived for gaining knowledge in mathematical education and made various contributions. He is best known for antiquity and the invention of compound pulleys and screw pump. 4. Thales of Miletus He was the first individual to whom a mathematical discovery was attributed. He’s best known for his work in calculating the heights of pyramids and the distance of the ships from the shore using geometry. 5. Aristotle Aristotle had a diverse knowledge over various areas including mathematics, geology, physics, metaphysics, biology, medicine and psychology. He was a pupil of Plato therefore it’s not a surprise that he had a vast knowledge and made contributions towards Platonism. Tutored Alexander the Great and established a library which aided in the production of hundreds of books. -
James Hutton, the Scottish Enlightenment and the North West
by Vivien As old as the hills Martin ECENTLY we had a display of fossils Rin the library. A young woman with two small children examined a fossilised James Hutton, the Scottish dinosaur tooth with great interest. She then turned to me and asked if cavemen would have kept dinosaurs as pets. And to my surprise the question was serious. Enlightenment and the It brought home to me just how difficult the concept of time can be. Especially the further back you go. All those billions of years that have gone into creating North West Highlands Geopark the planet we know today, including the millions it’s taken for our particular bit of it, Scotland, to reach its present form. Such a vast span of time can be hard, if not impossible, for our minds to grasp. This ‘deep time’, as it’s called, is measured in eons, eras, periods and epochs. Geologists believe that many of these eras were brought to an end by specific cataclysmic events. Like, for example, the one 64 million years ago, when a gigantic meteor strike is thought to have set off a chain reaction so destructive that it led to mass extinctions on Earth, the dinosaurs included. Extinctions that occurred long, long before the arrival of humans. So no, if you were a caveman you most certainly wouldn’t have had a dinosaur as a pet! Fred Flintstone has a lot to answer for! “Go to the mountains to read the immeasurable course of time.” James Hutton, 1788 Fred Flintstone has a lot to Fred & Dino Credit Hanna-Barbera Gruinard Bay answer for! Prof Lorna The North West Highlands Dawson of the Geopark welcomes you! James Hutton Institute So how do we know how old the earth is? After all, humankind is one of the more recent additions to the planet and people weren’t around to witness what happened. -
EM Waves, Ray Optics, Optical Instruments Mar
Gen. Phys. II Exam 3 - Chs. 24,25,26 - EM Waves, Ray Optics, Optical Instruments Mar. 26, 2018 Rec. Time Name For full credit, make your work clear. Show formulas used, essential steps, and results with correct units and significant figures. Points shown in parenthesis. For TF and MC, choose the best answer. OpenStax Ch. 24 - Electromagnetic Waves 1. (3) Which type of electromagnetic (EM) waves has the highest frequency in vacuum? a. x-rays. b. infrared. c. red light. d. blue light. e. ultraviolet. f. AM radio. g. all tie. 2. (3) An EM wave is traveling vertically upward with its magnetic field vector oscillating north-south. Its electric field vector is oscillating a. north-south. b. east-west. c. vertically up and down. 3. (3) The first physicist to confirm the generation and detection of EM waves by using LC oscillator circuits was a. Alexander Bell. b. James Watt. c. Andr´e-Marie Amp`ere. d. Heinrich Hertz. e. Carl Friedrich Gauss. 4. (3) TF In vacuum, electromagnetic waves of higher frequencies travel faster than lower frequencies. 5. (3) TF EM waves in vacuum can be considered to be transverse waves. 6. (3) TF Earth's ozone layer is important in blocking dangerous infrared light from the sun. 7. (3) Which physical effect did James Clerk Maxwell add into the equations of electromagnetism that carry his name, based on theoretical reasoning? a. changing magnetic fields produce electric fields. b. changing electric fields produce magnetic fields. c. moving electric charges produce magnetic fields. d. moving electric charges experience magnetic forces. -
Bridges Conference Paper
Bridges 2018 Conference Proceedings Landmarks in Algebra Quilt Elaine Krajenke Ellison Sarasota, Florida, USA; [email protected]; www.mathematicalquilts.com Abstract The Landmarks in Algebra quilt was an effort to include as many cultures that contributed significantly to the development of algebra as we know it today. Each section of the quilt illustrates a culture or a mathematician that made important advances in the development of algebra. Space did not allow more than four cultures, even though other nationalities made contributions. Babylonian mathematicians, Greek mathematicians, Indian mathematicians, and Persian mathematicians are highlighted for their important work. Figure 1: The Landmarks in Algebra quilt, 82 inches by 40 inches. First Panel: Plimpton 322 The first panel of the quilt illustrates the Plimpton 322 tablet. The Plimpton 322 tablet inspired me to design a quilt based on algebra. Recent historical research by Eleanor Robson, an Oriental Scholar at the University of Oxford, has shed new light on the ancient work of algebra [8]. The 4,000 year old Babylonian cuneiform tablet of Pythagorean Triples was purchased by George Arthur Plimpton in 1923 from Edgar J. Banks. Banks said the tablet came from a location near the ancient city of Larsa in Iraq [4]. Our first knowledge of mankind’s use of mathematics comes from the Egyptians and the Babylonians [1]. The Babylonian “texts” come to us in the form of clay tablets, usually about the size of a hand. The tablets were inscribed in cuneiform, a wedge-shaped writing owing its appearance to the stylus that was used to make it. Two types of mathematical tablets are generally found, table-texts and problem-texts [1]. -
Memoir 216 3300 Penrose Place, P.O
Revising the Revisions: James Hutton’s Reputation among Geologists in the Late Eighteenth and Nineteenth Centuries by A. M. Celâl Şengör İTÜ Avrasya Yerbilimleri Enstitüsü ve Maden Fakültesi Jeoloji Bölümü Ayazağa 34469 İstanbul Turkey Memoir 216 3300 Penrose Place, P.O. Box 9140 Boulder, Colorado 80301-9140, USA 2020 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/5198018/mwr216-00.pdf by guest on 01 October 2021 Copyright © 2020, The Geological Society of America (GSA), Inc. All rights reserved. Copyright is not claimed on content prepared wholly by U.S. government employees within the scope of their employment. Individual scientists are hereby granted permission, without fees or further requests to GSA, to use a single figure, a single table, and/or a brief paragraph of text in other subsequent works and to make unlimited photocopies of items in this volume for noncommercial use in classrooms to further education and science. Permission is also granted to authors to post the abstracts only of their articles on their own or their organization’s website providing that the posting cites the GSA publication in which the material appears and the citation includes the address line: “Geological Society of America, P.O. Box 9140, Boulder, CO 80301-9140 USA (https://www.geosociety.org),” and also providing that the abstract as posted is identical to that which appears in the GSA publication. In addition, an author has the right to use his or her article or a portion of the article in a thesis or dissertation without requesting permission from GSA, provided that the bibliographic citation and the GSA copyright credit line are given on the appropriate pages. -
Enlightened Scotland: How the Age of Reason Made an Impact on the Country's Thinkers
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Enlighten Riach, A. (2017) Enlightened Scotland: How the Age of Reason made an impact on the country's thinkers. National, 2017, 21 Apr. This is the author’s final accepted version. There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it. http://eprints.gla.ac.uk/161602/ Deposited on: 01 May 2018 Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk The Enlightenment Alan Riach From the moment in 1660 when Thomas Urquhart dies laughing to the publication of Walter Scott’s first novel Waverley in 1814, Scottish literature becomes increasingly aware of itself. That is, Scottish writers – some of them – become increasingly self-conscious of their own literary ancestors. This is brought about centrally in the work of Allan Ramsay (1684-1758), in his anthologies of earlier poets, and in the edition of Gavin Douglas’s translation of Virgil’s Aeneid (1710) published by Thomas Ruddiman (1674-1757), which was probably what Burns was quoting from in the epigraph to “Tam o’ Shanter”: “Of Brownyis and Bogillis full is this Buke.” The evolutionary turn towards greater self-awareness, the process of questioning the formation of national identity, the position of that identity in an ongoing struggle for economic prosperity and international colonial and imperial power, is central in the history of Scotland from around 1660 to 1707. It underlies and suffuses the writing of philosophers, politicians, dramatists, novelists, poets, and travel writers. -
7. Documentation
7. Documentation Papahänaumokuäkea Marine National Monument 7. Documentation 7.a Photographs, Image Inventory and Other Audiovisual Materials (Photo: James Watt) Table 7.1: Image inventory and authorization Id. No. Format Caption Date Photographer / Copyright owner Contact details: Non- Director of (if different than copyright owner exclusive the video photographer/ cession of director of video) rights Birds 1 JPEG French Frigate Shoals 2005 James Watt Sue Watt Sue@Seapics 1 - Red Footed Boobie 808-329-4253 Sunset Birds 2 JPEG Kure - Laysan 2005 NOAA Andy.Collins@ Y Albatross NOAA.gov Birds 3 JPEG Laysan - Great 2005 James Watt Sue Watt Sue@Seapics 1 frigatebird 808-329-4253 Birds 4 JPEG Laysan - Laysan Duck 2005 James Watt Sue Watt Sue@Seapics 1 808-329-4253 Birds 5 TIF Midway Atoll - White 2007 Sandra Hall USFWS Barbara_ Y Tern Chick [email protected] Cetaceans 1 JPEG Humpback Whale 2007 Doug Perrine HIHWNMS Naomi.Mcintosh@ 2 224 Mother and Calf NOAA.gov Cetaceans 2 JPEG Leaping Dolphin 2005 Andy Collins NOAA Andy.Collins@ Y NOAA.gov Cetaceans 3 JPEG Midway - Spinner 2005 James Watt Sue Watt Sue@Seapics 1 Dolphin bottom view 808-329-4253 Coral & JPEG French Frigate Shoals 2007 JE Maragos USFWS Barbara_ Y Invertebrates - Acropora Coral [email protected] 1 Coral & JPEG French Frigate Shoals 2005 James Watt Sue Watt Sue@Seapics 1 Invertebrates - Table coral 808-329-4253 2 Coral & JPEG Hertwigia Sponge 2007 NOWRAMP NOAA Andy.Collins@ Y Invertebrates NOAA.gov 3 Coral & JPEG Kure - Triton Trumpet 2005 James Watt Sue Watt Sue@Seapics 1 Invertebrates 808-329-4253 4 Coral & JPEG Kure-Banded Spiny 2005 James Watt Sue Watt Sue@Seapics 1 Invertebrates Lobster 808-329-4253 5 7. -
Electronics 101
Electronics 101 NANT – New York John Sweeny OFF Saturday – September 28th, 2013 Is this your understanding of a Dialysis Machine? 2 Electricity is intimidating because… Human aren’t equipped to detect it …until it’s to late ! You have five senses: ◦ See (10,000 volts) ◦ Hear (5,000 volts) ◦ Taste (???) ◦ Smell (ozone from movement in air) ◦ Feel ( a/c > 20 volts, static > 3,000 volts) Skin is your best protection if it’s dry. Electricity tends to flow on the outside of the body. The heart is mid-torso. 3 Basic Electron Characteristics 4 Electrical Terminology Coulomb (Q) – mks system’s unit of charge. 1 coulom b of charge passing a point in a wire in 1 second equals one ampere. 1 coulomb = 6.24 x 1018 electrons ◦ Charles Augustin Coulomb (1736 – 1806) Volt (()V) – measure of ppgyotential energy in an electric field. 1 volt = one joule of energy per one coulomb of charge. (1 watt = 1 joule/second) ◦ Alessandro Volta (1745 – 1827) ◦ Created the first chemical battery 5 Electrical Terminology Ampere (I) – The unit of current = a “flow rate” of one coulomb per second in a wire ◦ Andre Marie Ampere (1775 – 1836) ◦ Established the relationshippy between electricity and magnetism. Ohm (R or Ω) – the unit of resistance = 1 ohm is the ratio of one volt/one ampere. The reciproca l of resistance is conductance. ◦ Georgg( Simon Ohm (1789 – 1854) ◦ Discovered the direct proportionality of current in a conductor to the voltage applied (V = IR, R = V/I, I = V/R) 6 Electrical Terminology Watt – Unit of power in the mks system.