Precalculus Huygens Quiz Answers

Total Page:16

File Type:pdf, Size:1020Kb

Precalculus Huygens Quiz Answers Chapter Two Math History for Precalculus Christiaan Huygens 1. When did Huygens (HIE genz) live? 1629–1695 2. Where was he from? Holland (The Hague) 3. What was his marital status? lifelong bachelor 4. What was his most famous work and when was it published? Horologium Oscillatorium, 1673 This famous work discussed pendulum clocks. Pendulum motion is periodic and is described using trigonometric functions. The most important mathematical result in this work involves the study of a curve—the cycloid. He discovered a very important property of this curve. In later works he studied other important mathematical curves and their properties. Describe each curve below and explain what Huygens discovered about it. Curves Discovery 5. Cycloid Determined by a point on a It is the tautochrone (time needed for wheel as it rolls along the a ball to roll to lowest point is equal (level) ground regardless of starting point). 6. Catenary Determined by suspending a It is not a parabola. cable from two points 7. Cissoid Determined by secant It is rectifiable (found arc length segments of a circle formula). 8. Which of the three previous curves are functions? cycloid, catenary 9. Huygens wrote on the paraboloid, a surface formed by revolving a parabola about its axis. What did he discover? Its surface area 10. Huygens also wrote the first treatise on probability (following the correspondence of Pascal and Fermat). What concept did he contribute to this subject? expectation 11. Name some of his famous scientific theories. rings of Saturn (1654–1656, using his own improved telescope lenses); pendulum clocks (1658); wave theory of light (1678, 1690) 12. How did these discoveries result in a move to France? When? In 1665, Louis XIV offered him a pension to live in Paris based on his scientific reputation. 13. What were Huygens’s religious views? Protestant Christian 14. How does his departure from France shed light on his religious convictions? He fled to Holland because of Catholic intolerance for Protestants. When France revoked the Edict of Nantes, he severed even his communication on principle. Such actions show a commitment to Protestant convictions. .
Recommended publications
  • Pioneers in Optics: Christiaan Huygens
    Downloaded from Microscopy Pioneers https://www.cambridge.org/core Pioneers in Optics: Christiaan Huygens Eric Clark From the website Molecular Expressions created by the late Michael Davidson and now maintained by Eric Clark, National Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306 . IP address: [email protected] 170.106.33.22 Christiaan Huygens reliability and accuracy. The first watch using this principle (1629–1695) was finished in 1675, whereupon it was promptly presented , on Christiaan Huygens was a to his sponsor, King Louis XIV. 29 Sep 2021 at 16:11:10 brilliant Dutch mathematician, In 1681, Huygens returned to Holland where he began physicist, and astronomer who lived to construct optical lenses with extremely large focal lengths, during the seventeenth century, a which were eventually presented to the Royal Society of period sometimes referred to as the London, where they remain today. Continuing along this line Scientific Revolution. Huygens, a of work, Huygens perfected his skills in lens grinding and highly gifted theoretical and experi- subsequently invented the achromatic eyepiece that bears his , subject to the Cambridge Core terms of use, available at mental scientist, is best known name and is still in widespread use today. for his work on the theories of Huygens left Holland in 1689, and ventured to London centrifugal force, the wave theory of where he became acquainted with Sir Isaac Newton and began light, and the pendulum clock. to study Newton’s theories on classical physics. Although it At an early age, Huygens began seems Huygens was duly impressed with Newton’s work, he work in advanced mathematics was still very skeptical about any theory that did not explain by attempting to disprove several theories established by gravitation by mechanical means.
    [Show full text]
  • On the Isochronism of Galilei's Horologium
    IFToMM Workshop on History of MMS – Palermo 2013 On the isochronism of Galilei's horologium Francesco Sorge, Marco Cammalleri, Giuseppe Genchi DICGIM, Università di Palermo, [email protected], [email protected], [email protected] Abstract − Measuring the passage of time has always fascinated the humankind throughout the centuries. It is amazing how the general architecture of clocks has remained almost unchanged in practice to date from the Middle Ages. However, the foremost mechanical developments in clock-making date from the seventeenth century, when the discovery of the isochronism laws of pendular motion by Galilei and Huygens permitted a higher degree of accuracy in the time measure. Keywords: Time Measure, Pendulum, Isochronism Brief Survey on the Art of Clock-Making The first elements of temporal and spatial cognition among the primitive societies were associated with the course of natural events. In practice, the starry heaven played the role of the first huge clock of mankind. According to the philosopher Macrobius (4 th century), even the Latin term hora derives, through the Greek word ‘ώρα , from an Egyptian hieroglyph to be pronounced Heru or Horu , which was Latinized into Horus and was the name of the Egyptian deity of the sun and the sky, who was the son of Osiris and was often represented as a hawk, prince of the sky. Later on, the measure of time began to assume a rudimentary technical connotation and to benefit from the use of more or less ingenious devices. Various kinds of clocks developed to relatively high levels of accuracy through the Egyptian, Assyrian, Greek and Roman civilizations.
    [Show full text]
  • Computer-Aided Design and Kinematic Simulation of Huygens's
    applied sciences Article Computer-Aided Design and Kinematic Simulation of Huygens’s Pendulum Clock Gloria Del Río-Cidoncha 1, José Ignacio Rojas-Sola 2,* and Francisco Javier González-Cabanes 3 1 Department of Engineering Graphics, University of Seville, 41092 Seville, Spain; [email protected] 2 Department of Engineering Graphics, Design, and Projects, University of Jaen, 23071 Jaen, Spain 3 University of Seville, 41092 Seville, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-953-212452 Received: 25 November 2019; Accepted: 9 January 2020; Published: 10 January 2020 Abstract: This article presents both the three-dimensional modelling of the isochronous pendulum clock and the simulation of its movement, as designed by the Dutch physicist, mathematician, and astronomer Christiaan Huygens, and published in 1673. This invention was chosen for this research not only due to the major technological advance that it represented as the first reliable meter of time, but also for its historical interest, since this timepiece embodied the theory of pendular movement enunciated by Huygens, which remains in force today. This 3D modelling is based on the information provided in the only plan of assembly found as an illustration in the book Horologium Oscillatorium, whereby each of its pieces has been sized and modelled, its final assembly has been carried out, and its operation has been correctly verified by means of CATIA V5 software. Likewise, the kinematic simulation of the pendulum has been carried out, following the approximation of the string by a simple chain of seven links as a composite pendulum. The results have demonstrated the exactitude of the clock.
    [Show full text]
  • ALBERT VAN HELDEN + HUYGENS’S RING CASSINI’S DIVISION O & O SATURN’S CHILDREN
    _ ALBERT VAN HELDEN + HUYGENS’S RING CASSINI’S DIVISION o & o SATURN’S CHILDREN )0g-_ DIBNER LIBRARY LECTURE , HUYGENS’S RING, CASSINI’S DIVISION & SATURN’S CHILDREN c !@ _+++++++++ l ++++++++++ _) _) _) _) _)HUYGENS’S RING, _)CASSINI’S DIVISION _) _)& _)SATURN’S CHILDREN _) _) _)DDDDD _) _) _)Albert van Helden _) _) _) , _) _) _)_ _) _) _) _) _) · _) _) _) ; {(((((((((QW(((((((((} , 20013–7012 Text Copyright ©2006 Albert van Helden. All rights reserved. A H is Professor Emeritus at Rice University and the Univer- HUYGENS’S RING, CASSINI’S DIVISION sity of Utrecht, Netherlands, where he resides and teaches on a regular basis. He received his B.S and M.S. from Stevens Institute of Technology, M.A. from the AND SATURN’S CHILDREN University of Michigan and Ph.D. from Imperial College, University of London. Van Helden is a renowned author who has published respected books and arti- cles about the history of science, including the translation of Galileo’s “Sidereus Nuncius” into English. He has numerous periodical contributions to his credit and has served on the editorial boards of Air and Space, 1990-present; Journal for the History of Astronomy, 1988-present; Isis, 1989–1994; and Tractrix, 1989–1995. During his tenure at Rice University (1970–2001), van Helden was instrumental in establishing the “Galileo Project,”a Web-based source of information on the life and work of Galileo Galilei and the science of his time. A native of the Netherlands, Professor van Helden returned to his homeland in 2001 to join the faculty of Utrecht University.
    [Show full text]
  • Mathematics, the Language of Watchmaking
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE 90LEARNINGprovidedL by Infoscience E- École polytechniqueA fédérale de LausanneRNINGLEARNINGLEARNING Mathematics, the language of watchmaking Morley’s theorem (1898) states that if you trisect the angles of any triangle and extend the trisecting lines until they meet, the small triangle formed in the centre will always be equilateral. Ilan Vardi 1 I am often asked to explain mathematics; is it just about numbers and equations? The best answer that I’ve found is that mathematics uses numbers and equations like a language. However what distinguishes it from other subjects of thought – philosophy, for example – is that in maths com- plete understanding is sought, mostly by discover- ing the order in things. That is why we cannot have real maths without formal proofs and why mathe- maticians study very simple forms to make pro- found discoveries. One good example is the triangle, the simplest geometric shape that has been studied since antiquity. Nevertheless the foliot world had to wait 2,000 years for Morley’s theorem, one of the few mathematical results that can be expressed in a diagram. Horology is of interest to a mathematician because pallet verge it enables a complete understanding of how a watch or clock works. His job is to impose a sequence, just as a conductor controls an orches- tra or a computer’s real-time clock controls data regulating processing. Comprehension of a watch can be weight compared to a violin where science can only con- firm the preferences of its maker.
    [Show full text]
  • The Birth of the Meter Randy K
    “Here is a story that ties together seemingly disparate threads in the development of trigonometry, calculus, and differential equations.” The Birth of the Meter Randy K. Schwartz Schoolcraft College, Livonia, MI t first thought, it might seem like the selection of units naval powers, since it allowed them to solve the age-old nau- for length and other physical quantities would be tical problem of finding one’s position at sea. Aarbitrary—so how could it involve any significant Nowadays, with GPS satellites and inertial guidance sys- mathematics? tems all around us, we tend to forget just how difficult it was But creating the world’s first system of measurements that for explorers like Columbus and Cartier to find their coordi- would not be arbitrary was one of the announced goals of the nates. Figuring out your latitude , or angular coordinate north leaders of the French Revolution after they seized power in or south of the Equator, was a fairly straightforward calcula- 1789. That is why they had to recruit some of the most eminent tion, based on using a quadrant or sextant to record the angle mathematicians of the time to work on the problem of defining of the Sun or other stars above the horizon. For example, at the meter, which would be the new system’s fundamental unit. night in the northern hemisphere, the angle of Polaris (the The contentious and painful birth of the meter turned out to be North Star) above the horizon, in degrees, is almost exactly an exciting chapter in the application of mathematics, a story the latitude of the observer, and an astronomical table could be that ties together seemingly disparate threads in the develop- consulted to improve accuracy.
    [Show full text]
  • The Cycloid and the Tautochrone Problem
    THE CYCLOID AND THE TAUTOCHRONE PROBLEM Laura Panizzi August 27, 2012 Abstract In this dissertation we will first introduce historically the invention of the Pendulum by Christiaan Huygens, in particular the cycloidal one. Then we will discuss mathematically the cycloid curve, related to the Tautochrone Problem and we’ll compare the circular and cycloidal pendulum. Finally we’ll introduce Abel’s Integral Equation as another way to attack and solve the Tautochrone Problem. 1HistoricalIntroduction Christiaan Huygens (14 April 1629 - 8 July 1695), was a prominent Dutch mathematician, astronomer, physicist and horologist. His work included early telescopic studies elucidating the nature of the rings of Saturn and the discovery of its moon Titan, the invention of the pendulum clock and other investigations in timekeeping, and studies of both optics and the cen- trifugal force. In 1657 patented his invention of the pendulum clock and in 1673 published his mathematical analysis of pendulums, Horologium Oscillatorium sive de motu pendulorum,hisgreatestworkonhorology.Ithadbeenobservedby Marin Mersenne and others that pendulums are not quite isochronous, that is, their period depends on their width of swing, wide swings taking longer than narrow swings. Huygens analysed this problem by finding the shape of the curve down which a mass will slide under the influence of gravity in the same amount of time, regardless of its starting point; the so-called 1 Tautochrone Problem.Bygeometricalmethodswhichwereanearlyuseof calculus, he showed that this curve is a Cycloid. ”On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after hav- ing departed from any point on the cycloid, are equal to each other...”1 2TheCycloid The cycloid is the locus of a point on the rim of a circle of radius R rolling without slipping along a straight line.
    [Show full text]
  • Council for Innovative Research Peer Review Research Publishing System Journal: Journal of Advances in Mathematics
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by KHALSA PUBLICATIONS ISSN 2347-1921 Some Geometrical Calculations for the Spherical Indicatrices of Involutes of a Timelike Curve in Minkowski 3-Space Mustafa Bilici, Mustafa Çalışkan Ondokuz Mayıs University, Educational Faculty, Department of Mathematics 55200 Kurupelit, Samsun, Turkey [email protected] Gazi University, Faculty of Sciences, Department of Mathematics, 06500 Teknik Okullar,Ankara, Turkey [email protected] ABSTRACT In this paper, we introduce the spherical indicatrices of involutes of a given timelike curve. Then we give some important 3 2 2 relationships between arc lengths and geodesic curvatures of the base curve and its involutes on E1 , S1 , H0 . Additionally, some important results concerning these curves are given. Keywords: Minkowski space; involute evolute curve couple; spherical indicatrices; geodesic curvature. Mathematics Subject Classification 2010: 51B20, 53B30, 53C50 Council for Innovative Research Peer Review Research Publishing System Journal: Journal of Advances in Mathematics Vol 5, No. 2 [email protected] www.cirworld.com, member.cirworld.com 668 | P a g e D e c 2 3 , 2 0 1 3 ISSN 2347-1921 INTRODUCTION The notion of the involute of a curve was introduced by Christiaan Huygens in his work titled Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae (1673). In the theory of curves in Euclidean space, one of the important and interesting problems is the characterizations of a regular curve. In particular, the involute of a given curve is a well known concept in the classical differential geometry (see [5]).
    [Show full text]
  • Simple Pendulum
    The (Not So) Simple Pendulum ©2007-2008 Ron Doerfler Dead Reckonings: Lost Art in the Mathematical Sciences http://www.myreckonings.com/wordpress December 29, 2008 Pendulums are the defining feature of pendulum clocks, of course, but today they don’t elicit much thought. Most modern “pendulum” clocks simply drive the pendulum to provide a historical look, but a great deal of ingenuity originally went into their design in order to produce highly accurate clocks. This essay explores horologic design efforts that were so important at one time—not gearwork, winding mechanisms, crutches or escapements (which may appear as later essays), but the surprising inventiveness found in the “simple” pendulum itself. It is commonly known that Galileo (1564-1642) discovered that a swinging weight exhibits isochronism, purportedly by noticing that chandeliers in the Pisa cathedral had identical periods despite the amplitudes of their swings. The advantage here is that the driving force for the pendulum, which is difficult to regulate, could vary without affecting its period. Galileo was a medical student in Pisa at the time and began using it to check patients’ pulse rates. Galileo later established that the period of a pendulum varies as the square root of its length and is independent of the material of the pendulum bob (the mass at the end). One thing that surprised me when I encountered it is that the escapement preceded the pendulum—the verge escapement was used with hanging weights and possibly water clocks from at least the 14th century and probably much earlier. The pendulum provided a means of regulating such an escapement, and in fact Galileo invented the pin-wheel escapement to use in a pendulum clock he designed but never built.
    [Show full text]
  • Curvature in the Calculus Curriculum Jerry Lodder
    Curvature in the Calculus Curriculum Jerry Lodder 1. INTRODUCTION. My first duty as a new graduate student at a major research university was the repeated correction of the same problem on 150 calculus exams. The course material was the calculus of curves and surfaces in three-space, and the problem was a routine calculation of curvature, requiring the memorization of a par- ticular formula and the mechanics of differentiation. Being conscientious in my duties, I devised a grading scheme whereby a mistake in algebra would cost a few points and a mistakenly computed derivative a few more points, with the largest point deduction reserved for an incorrect formula for curvature. Not only had the material been reduced to rote calculation, but that method of instruction, at least for the calculus sequence, was being promulgated to the next generation of instructors. Possibly the difficulty lies in the subject itself—how can a critical understanding of curvature be tested on a one-hour exam, especially when curvature is merely one of several topics to be tested? Should the students be asked to explain the definition of curvature as the magnitude of the rate of change of the unit tangent with respect to arclength? In subsequent courses I have spent more time motivating that particular definition than actually computing curvature, but find the definition rather far removed from an intuitive idea of curvature. Should the students be asked to sketch the osculat- ing circle to a given curve at a given point, and then notice that as the point of contact changes, the radius of the circle is inversely proportional to the measure of curva- ture? This seems more intuitive, and remains quite a viable introduction to curvature, although the concept of an osculating circle requires some explanation.
    [Show full text]
  • DISCOVERING HUYGENS C.D. Andriesse Institute for the History
    DISCOVERING HUYGENS C.D. Andriesse Institute for the History and Foundations of Science, Universiteit Utrecht The delicacies of our conference dinner, ladies and gentlemen, may have brought you in the right mood to hear about the dinner that was served to Christiaan Huygens at the occasion of his sister’s marriage – served not only to this frail little man, of course, but to forty healthy stomachs of his family as well. What would you think of a swine’s head and a hundred partridges, capons, turkeys, pheasants and hares around, all stuffed with lamb’s meat and lardoon? When the aromas of the meat become too much for you, sniff then at one of those strongly perfumed candles. And when there is not enough Burgundy to rinse your throat, quell the pepper with the sugar and marzipan in that astonishing quantity of pastries. Since it’s a wedding party, you bang your plate with the cutlery when two youngsters, coyly hiding behind a napkin, alternately kiss and sip from each other’s glass of wine. Is it Christiaan’s turn to dive behind a napkin? He is fond of young women, and being just thirty-one … But wait a moment. Despite his merrymaking – Christiaan uses this word (réjouissances) when he writes a learned friend the other day – it is already clear to him that he is not a man to raise a family. And to others for that matter. They all sense it, the young women he pays attention to, sends letters and little presents, asks to sit for him and let him draw their portrait: beautiful Marianne Petit, or Haasje Hooft, and not to forget the affective Suzette Caron who will inherit half his capital – to mention only these.
    [Show full text]
  • From Ciphers to Confidentiality: Secrecy, Openness and Priority In
    BJHS, Page 1 of 21. © British Society for the History of Science 2012 doi:10.1017/S0007087412000088 From ciphers to confidentiality: secrecy, openness and priority in science MARIO BIAGIOLI* Abstract. I make three related claims. First, certain seemingly secretive behaviours displayed by scientists and inventors are expression neither of socio-professional values nor of strategies for the maximization of the economic value of their knowledge. They are, instead, protective responses to unavoidable risks inherent in the process of publication and priority claiming. Scientists and inventors fear being scooped by direct competitors, but have also worried about people who publish their claims or determine their priority: journal editors or referees who may appropriate the claims in the manuscript they review or patent clerks who may claim or leak the inventions contained in the applications that cross their desks. Second, these protective responses point to the existence of an unavoidable moment of instability in any procedure aimed at establishing priority. Making things public is an inherently risky business and it is impossible, I argue, to ensure that priority may not be lost in the very process that is supposed to establish it. Third, I offer a brief archaeology of regimes and techniques of priority registration, showing the distinctly different definitions of priority developed by each system. The temporality of secrecy and openness – the different ways in which time frames them as concepts – illuminates their mutual relationship as well as their fundamental link to priority.1 It also explains certain seemingly secretive behaviours displayed by scientists and inventors, showing that they are, in fact, protective responses to risks inherent in the process of publication and priority claiming.
    [Show full text]