Newton's Notebook
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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. -
Introduction to Ultra Fractal
Introduction to Ultra Fractal Text and images © 2001 Kerry Mitchell Table of Contents Introduction ..................................................................................................................................... 2 Assumptions ........................................................................................................................ 2 Conventions ........................................................................................................................ 2 What Are Fractals? ......................................................................................................................... 3 Shapes with Infinite Detail .................................................................................................. 3 Defined By Iteration ........................................................................................................... 4 Approximating Infinity ....................................................................................................... 4 Using Ultra Fractal .......................................................................................................................... 6 Starting Ultra Fractal ........................................................................................................... 6 Formula ............................................................................................................................... 6 Size ..................................................................................................................................... -
Spinoza and the Sciences Boston Studies in the Philosophy of Science
SPINOZA AND THE SCIENCES BOSTON STUDIES IN THE PHILOSOPHY OF SCIENCE EDITED BY ROBERT S. COHEN AND MARX W. WARTOFSKY VOLUME 91 SPINOZA AND THE SCIENCES Edited by MARJORIE GRENE University of California at Davis and DEBRA NAILS University of the Witwatersrand D. REIDEL PUBLISHING COMPANY A MEMBER OF THE KLUWER ~~~.'~*"~ ACADEMIC PUBLISHERS GROUP i\"lI'4 DORDRECHT/BOSTON/LANCASTER/TOKYO Library of Congress Cataloging-in-Publication Data Main entry under title: Spinoza and the sciences. (Boston studies in the philosophy of science; v. 91) Bibliography: p. Includes index. 1. Spinoza, Benedictus de, 1632-1677. 2. Science- Philosophy-History. 3. Scientists-Netherlands- Biography. I. Grene, Marjorie Glicksman, 1910- II. Nails, Debra, 1950- Ill. Series. Q174.B67 vol. 91 OOI'.Ols 85-28183 101 43.S725J 100 I J ISBN-13: 978-94-010-8511-3 e-ISBN-13: 978-94-009-4514-2 DOl: 10.1007/978-94-009-4514-2 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, Holland. 2-0490-150 ts All Rights Reserved © 1986 by D. Reidel Publishing Company Softcover reprint of the hardcover 1st edition 1986 and copyright holders as specified on appropriate pages within No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner FROM SPINOZA'S LETTER TO OLDENBURG, RIJNSBURG, APRIL, 1662 (Photo by permission of Berend Kolk) TABLE OF CONTENTS ACKNOWLEDGEMENTS ix MARJORIE GRENE I Introduction xi 1. -
Rendering Hypercomplex Fractals Anthony Atella [email protected]
Rhode Island College Digital Commons @ RIC Honors Projects Overview Honors Projects 2018 Rendering Hypercomplex Fractals Anthony Atella [email protected] Follow this and additional works at: https://digitalcommons.ric.edu/honors_projects Part of the Computer Sciences Commons, and the Other Mathematics Commons Recommended Citation Atella, Anthony, "Rendering Hypercomplex Fractals" (2018). Honors Projects Overview. 136. https://digitalcommons.ric.edu/honors_projects/136 This Honors is brought to you for free and open access by the Honors Projects at Digital Commons @ RIC. It has been accepted for inclusion in Honors Projects Overview by an authorized administrator of Digital Commons @ RIC. For more information, please contact [email protected]. Rendering Hypercomplex Fractals by Anthony Atella An Honors Project Submitted in Partial Fulfillment of the Requirements for Honors in The Department of Mathematics and Computer Science The School of Arts and Sciences Rhode Island College 2018 Abstract Fractal mathematics and geometry are useful for applications in science, engineering, and art, but acquiring the tools to explore and graph fractals can be frustrating. Tools available online have limited fractals, rendering methods, and shaders. They often fail to abstract these concepts in a reusable way. This means that multiple programs and interfaces must be learned and used to fully explore the topic. Chaos is an abstract fractal geometry rendering program created to solve this problem. This application builds off previous work done by myself and others [1] to create an extensible, abstract solution to rendering fractals. This paper covers what fractals are, how they are rendered and colored, implementation, issues that were encountered, and finally planned future improvements. -
Lecture 9: Newton Method 9.1 Motivation 9.2 History
10-725: Convex Optimization Fall 2013 Lecture 9: Newton Method Lecturer: Barnabas Poczos/Ryan Tibshirani Scribes: Wen-Sheng Chu, Shu-Hao Yu Note: LaTeX template courtesy of UC Berkeley EECS dept. 9.1 Motivation Newton method is originally developed for finding a root of a function. It is also known as Newton- Raphson method. The problem can be formulated as, given a function f : R ! R, finding the point x? such that f(x?) = 0. Figure 9.1 illustrates another motivation of Newton method. Given a function f, we want to approximate it at point x with a quadratic function fb(x). We move our the next step determined by the optimum of fb. Figure 9.1: Motivation for quadratic approximation of a function. 9.2 History Babylonian people first applied the Newton method to find the square root of a positive number S 2 R+. The problem can be posed as solving the equation f(x) = x2 − S = 0. They realized the solution can be achieved by applying the iterative update rule: 2 1 S f(xk) xk − S xn+1 = (xk + ) = xk − 0 = xk − : (9.1) 2 xk f (xk) 2xk This update rule converges to the square root of S, which turns out to be a special case of Newton method. This could be the first application of Newton method. The starting point affects the convergence of the Babylonian's method for finding the square root. Figure 9.2 shows an example of solving the square root for S = 100. x- and y-axes represent the number of iterations and the variable, respectively. -
Newton.Indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag
omslag Newton.indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag. 1 e Dutch Republic proved ‘A new light on several to be extremely receptive to major gures involved in the groundbreaking ideas of Newton Isaac Newton (–). the reception of Newton’s Dutch scholars such as Willem work.’ and the Netherlands Jacob ’s Gravesande and Petrus Prof. Bert Theunissen, Newton the Netherlands and van Musschenbroek played a Utrecht University crucial role in the adaption and How Isaac Newton was Fashioned dissemination of Newton’s work, ‘is book provides an in the Dutch Republic not only in the Netherlands important contribution to but also in the rest of Europe. EDITED BY ERIC JORINK In the course of the eighteenth the study of the European AND AD MAAS century, Newton’s ideas (in Enlightenment with new dierent guises and interpre- insights in the circulation tations) became a veritable hype in Dutch society. In Newton of knowledge.’ and the Netherlands Newton’s Prof. Frans van Lunteren, sudden success is analyzed in Leiden University great depth and put into a new perspective. Ad Maas is curator at the Museum Boerhaave, Leiden, the Netherlands. Eric Jorink is researcher at the Huygens Institute for Netherlands History (Royal Dutch Academy of Arts and Sciences). / www.lup.nl LUP Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 1 Newton and the Netherlands Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 2 Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. -
The Newton-Leibniz Controversy Over the Invention of the Calculus
The Newton-Leibniz controversy over the invention of the calculus S.Subramanya Sastry 1 Introduction Perhaps one the most infamous controversies in the history of science is the one between Newton and Leibniz over the invention of the infinitesimal calculus. During the 17th century, debates between philosophers over priority issues were dime-a-dozen. Inspite of the fact that priority disputes between scientists were ¡ common, many contemporaries of Newton and Leibniz found the quarrel between these two shocking. Probably, what set this particular case apart from the rest was the stature of the men involved, the significance of the work that was in contention, the length of time through which the controversy extended, and the sheer intensity of the dispute. Newton and Leibniz were at war in the later parts of their lives over a number of issues. Though the dispute was sparked off by the issue of priority over the invention of the calculus, the matter was made worse by the fact that they did not see eye to eye on the matter of the natural philosophy of the world. Newton’s action-at-a-distance theory of gravitation was viewed as a reversion to the times of occultism by Leibniz and many other mechanical philosophers of this era. This intermingling of philosophical issues with the priority issues over the invention of the calculus worsened the nature of the dispute. One of the reasons why the dispute assumed such alarming proportions and why both Newton and Leibniz were anxious to be considered the inventors of the calculus was because of the prevailing 17th century conventions about priority and attitude towards plagiarism. -
Berkeley's Case Against Realism About Dynamics
Lisa Downing [Published in Berkeley’s Metaphysics, ed. Muehlmann, Penn State Press 1995, 197-214. Turbayne Essay Prize winner, 1992.] Berkeley's case against realism about dynamics While De Motu, Berkeley's treatise on the philosophical foundations of mechanics, has frequently been cited for the surprisingly modern ring of certain of its passages, it has not often been taken as seriously as Berkeley hoped it would be. Even A.A. Luce, in his editor's introduction to De Motu, describes it as a modest work, of limited scope. Luce writes: The De Motu is written in good, correct Latin, but in construction and balance the workmanship falls below Berkeley's usual standards. The title is ambitious for so brief a tract, and may lead the reader to expect a more sustained argument than he will find. A more modest title, say Motion without Matter, would fitly describe its scope and content. Regarded as a treatise on motion in general, it is a slight and disappointing work; but viewed from a narrower angle, it is of absorbing interest and high importance. It is the application of immaterialism to contemporary problems of motion, and should be read as such. ...apart from the Principles the De Motu would be nonsense.1 1The Works of George Berkeley, Bishop of Cloyne, ed. A.A. Luce and T.E. Jessop (London: Thomas Nelson and Sons, 1948-57), 4: 3-4. In this paper, all references to Berkeley are to the Luce-Jessop edition. Quotations from De Motu are taken from Luce's translation. I use the following abbreviations for Berkeley’s works: PC Philosophical Commentaries PHK-I Introduction to The Principles of Human Knowledge PHK The Principles of Human Knowledge DM De Motu A Alciphron TVV The Theory of Vision Vindicated and Explained S Siris 1 There are good general reasons to think, however, that Berkeley's aims in writing the book were as ambitious as the title he chose. -
Newton and Kant on Absolute Space: from Theology to Transcendental Philosophy
Newton and Kant on Absolute Space: From Theology to Transcendental Philosophy Michael Friedman Abstract I argue that Einstein’s creation of both special and general relativity instantiates Reichenbach’s conception of the relativized a priori. I do this by show- ing how the original Kantian conception actually contributes to the development of Einstein’s theories through the intervening philosophical and scientific work of Helmholtz, Mach, and Poincaré. In my previous work on Newton and Kant I have primarily emphasized methodo- logical issues: why Kant takes the Newtonian Laws of Motion (as well as certain related propositions of what he calls “pure natural science”) as synthetic a priori constitutive principles rather than mere empirical laws, and how this point is inti- mately connected, in turn, with Kant’s conception of absolute space as a regulative idea of reason – as the limit point of an empirical constructive procedure rather than a self-subsistent “container” existing prior to and independently of all perceptible matter. I have also argued that these methodological differences explain the circum- stance that Kant, unlike Newton, asserts that gravitational attraction must be con- ceived as an “action at a distance through empty space,” and even formulates a (rare) criticism of Newton for attempting to leave the question of the “true cause” of gravitational attraction entirely open. In this paper I emphasize the importance of metaphysical and theological issues – about God, his creation of the material world in space, and the consequences different views of such creation have for the metaphysical foundations of physics. I argue, in particular, that Kant’s differences with Newton over these issues constitute an essential part of his radical transforma- tion of the very meaning of metaphysics as practiced by his predecessors. -
Determinism Is False
%&5&3.*/*4. Barry Loewer %FUFSNJOJTNJTBDPOUJOHFOUNFUBQIZTJDBMDMBJNBCPVUUIFGVOEBNFOUBMOBUVSBMMBXT UIBUIPMEJOUIFVOJWFSTF*UTBZT The natural laws and the way things are at time t determine the way things will be at later times. 5IF NBUIFNBUJDJBO 1JFSSF4JNPO -BQMBDF FYQSFTTFE IJT CFMJFG UIBU EFUFS- minism is true this way: 8F PVHIU UP SFHBSE UIF QSFTFOU TUBUF PG UIF VOJWFSTF BT UIF FGGFDU PG JUT antecedent state and as the cause of the state that is to follow. An intel- MJHFODF LOPXJOH BMM UIF GPSDFT BDUJOH JO OBUVSF BU B HJWFO JOTUBOU BT XFMM as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were suf!ciently QPXFSGVMUPTVCKFDUBMMEBUBUPBOBMZTJTUPJUOPUIJOHXPVMECFVODFSUBJO UIF future as well as the past would be present to its eyes. The perfection that the human mind has been able to give to astronomy affords but a feeble outline of such intelligence. 5IF QIZTJDT PG -BQMBDFT EBZ UIF àSTU EFDBEFT PG UIF OJOFUFFOUI DFOUVSZ XBT /FXUPOJBO DMBTTJDBM NFDIBOJDT*TBBD/FXUPOGPSNVMBUFEQSJODJQMFTUIBUIFUIPVHIU FYQSFTTUIFMBXTEFTDSJCJOHIPXGPSDFTEFUFSNJOFUIFNPUJPOTPGCPEJFT F ma) and IPXUIFQPTJUJPOTPGCPEJFTBOEPUIFSGBDUPSTEFUFSNJOFHSBWJUBUJPOBMBOEPUIFSLJOET PGGPSDFT6TJOHUIFTFQSJODJQMFT /FXUPOBOEQIZTJDJTUTGPMMPXJOHIJNXFSFBCMFUP QSFEJDUBOEFYQMBJOUIFNPUJPOTPGDFMFTUJBMBOEUFSSFTUSJBMCPEJFT'PSFYBNQMF UIFTF laws account for the orbits of the planets, the trajectories of cannon balls, and the QFSJPET PG QFOEVMVNT-JLF/FXUPO -BQMBDFEJE OPULOPX BMM UIFGPSDFTUIFSF BSF but he envisioned that, once those forces (and the corresponding force laws) were LOPXO /FXUPOJBO QIZTJDT XPVME CF B complete physical theory. That is, its laws would account for the motions of all material particles. And since he thought that FWFSZUIJOH UIBU FYJTUT JO TQBDF JT DPNQPTFE PG WBSJPVT LJOET PG WFSZ TNBMM NBUFSJBM #"33:-0&8&3 QBSUJDMFT PSBUPNT IFUIPVHIUUIBU/FXUPOJBONFDIBOJDT PODFBMMUIFGPSDFTXFSF LOPXO XPVMECFXIBUUPEBZXFXPVMEDBMMthe theory of everything. -
A Short History of Stochastic Integration and Mathematical Finance
A Festschrift for Herman Rubin Institute of Mathematical Statistics Lecture Notes – Monograph Series Vol. 45 (2004) 75–91 c Institute of Mathematical Statistics, 2004 A short history of stochastic integration and mathematical finance: The early years, 1880–1970 Robert Jarrow1 and Philip Protter∗1 Cornell University Abstract: We present a history of the development of the theory of Stochastic Integration, starting from its roots with Brownian motion, up to the introduc- tion of semimartingales and the independence of the theory from an underlying Markov process framework. We show how the development has influenced and in turn been influenced by the development of Mathematical Finance Theory. The calendar period is from 1880 to 1970. The history of stochastic integration and the modelling of risky asset prices both begin with Brownian motion, so let us begin there too. The earliest attempts to model Brownian motion mathematically can be traced to three sources, each of which knew nothing about the others: the first was that of T. N. Thiele of Copen- hagen, who effectively created a model of Brownian motion while studying time series in 1880 [81].2; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, [1, 2, 11]); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter, in 1905 [21](See [64] for a discussion of Einstein’s model and his motivations.) Of these three models, those of Thiele and Bachelier had little impact for a long time, while that of Einstein was immediately influential. -
Notes on Stochastic Processes
Notes on stochastic processes Paul Keeler March 20, 2018 This work is licensed under a “CC BY-SA 3.0” license. Abstract A stochastic process is a type of mathematical object studied in mathemat- ics, particularly in probability theory, which can be used to represent some type of random evolution or change of a system. There are many types of stochastic processes with applications in various fields outside of mathematics, including the physical sciences, social sciences, finance and economics as well as engineer- ing and technology. This survey aims to give an accessible but detailed account of various stochastic processes by covering their history, various mathematical definitions, and key properties as well detailing various terminology and appli- cations of the process. An emphasis is placed on non-mathematical descriptions of key concepts, with recommendations for further reading. 1 Introduction In probability and related fields, a stochastic or random process, which is also called a random function, is a mathematical object usually defined as a collection of random variables. Historically, the random variables were indexed by some set of increasing numbers, usually viewed as time, giving the interpretation of a stochastic process representing numerical values of some random system evolv- ing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule [120, page 7][51, page 46 and 47][66, page 1]. Stochastic processes are widely used as math- ematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including physical sciences such as biology [67, 34], chemistry [156], ecology [16][104], neuroscience [102], and physics [63] as well as technology and engineering fields such as image and signal processing [53], computer science [15], information theory [43, page 71], and telecommunications [97][11][12].