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Articles 1 - 17 of 17
Full-Text Articles in Physical Sciences and Mathematics
Euler's Navigation Variational Problem, Sylvio R. Bistafa
Euler's Navigation Variational Problem, Sylvio R. Bistafa
Euleriana
In a 1747 publication, De motu cymbarum remis propulsarum in fluviis (“On the motion of boats propelled by oars in rivers”), Leonhard Euler (1707-1783) works out various instances of a boat moving at constant speed across a stream flowing in straight streamlines at assigned speeds, in which one of these gives rise to a variational problem consisting of finding the quickest crossing path between two points on opposite side of the river banks, which is generally known as the navigation variational problem. This problem together with the well-known catenary and brachistochrone problems, are considered classical examples in the calculus …
Basel Problem: Historical Perspective And Further Proofs From Stochastic Processes, Uwe Hassler, Mehdi Hosseinkouchack
Basel Problem: Historical Perspective And Further Proofs From Stochastic Processes, Uwe Hassler, Mehdi Hosseinkouchack
Euleriana
In this note, we offer a historical perspective on solutions of the Basel problem. In particular, we have a closer look at some of the less famous results by Euler E41 and provide a review of a selection of the assemblage of earlier proofs. Moreover, we show how to generate further proofs using Karhunen-Lo\`{e}ve expansions of stochastic processes.
Answer To A Question Concerning Euler's Paper "Variae Considerationes Circa Series Hypergeometricas", Alexander Aycock
Answer To A Question Concerning Euler's Paper "Variae Considerationes Circa Series Hypergeometricas", Alexander Aycock
Euleriana
We solve a problem concerning Euler's paper "Variae considerationes circa series hypergeometricas" (\cite{E661}), as suggested by G. Faber in the preface to Volume 16,2 of the first series of Euler's Opera Omnia. Our solution employs methods introduced by Euler at other places.
Euler Archive Spotlight, Cynthia Huffman Ph.D.
Euler Archive Spotlight, Cynthia Huffman Ph.D.
Euleriana
This issue we spotlight the translations of Jordan Bell, one of the most prolific translators for the Euler Archive.
Learning Mathematics From The Master: A Collection Of Euler-Based Primary Source Projects For Today’S Students, Part Ii, Janet H. Barnett, Dominic Klyve, Dave Ruch
Learning Mathematics From The Master: A Collection Of Euler-Based Primary Source Projects For Today’S Students, Part Ii, Janet H. Barnett, Dominic Klyve, Dave Ruch
Euleriana
This article and its prequel together highlight a set of nine classroom-ready projects that draw on the remarkable writing of Leonhard Euler (1707--1783) as a means to help students develop an understanding of standard topics from today’s undergraduate mathematics curriculum. Part of a larger collection of primary source projects intended for use in a wide range of undergraduate mathematics courses, these projects are freely available to students and their instructors. We provide a general description of the pedagogical design underlying these projects, more detailed descriptions of the individual projects themselves, and instructions for obtaining downloadable copies for classroom use.
On The Motion Of Boats Propelled By Oars In Rivers, Sylvio R. Bistafa
On The Motion Of Boats Propelled By Oars In Rivers, Sylvio R. Bistafa
Euleriana
Euler considers the following problem: A boat with a perfect rudder moves at constant speed across a stream flowing in straight streamlines at assigned speeds. Assuming that the downstream velocity of the boat equals that of the river, how should the rudder be set so that the boat traverses a given path? He works out various instances, one of which gives rise to a variational problem, in detail. (From Clifford Truesdell's An idiot's fugitive essays on science: methods, criticisms, training, circumstances.)
A Method For Calculating The Equation Of Noon (An English Translation Of Methodus Computandi Aequationem Meridiei), Patrick T. Headley
A Method For Calculating The Equation Of Noon (An English Translation Of Methodus Computandi Aequationem Meridiei), Patrick T. Headley
Euleriana
In this paper Euler presents a method for determining solar noon, the time at which the Sun crosses the meridian. The method requires the times of two observations of the Sun, one in the morning and one in the afternoon, at equal altitudes above the horizon. Solar noon is approximately the midpoint between two such observations, but, since the declination of the Sun will have changed during the day, a correction term, called the equation of noon, is required. Euler explains that this term is too large to ignore and discusses the table of values constructed by de la Hire; …
Review Of The History Of Mathematics: A Source-Based Approach (Vol. 2), Part I, Erik R. Tou
Review Of The History Of Mathematics: A Source-Based Approach (Vol. 2), Part I, Erik R. Tou
Euleriana
Review of The History of Mathematics: A Source-Based Approach (Vol. 2), Part I, by June Barrow-Green, Jeremy Gray, and Robin Wilson. MAA Press, 2022, 330 + xiv pages.
Variety And Variation, Christopher Goff, Erik R. Tou
Variety And Variation, Christopher Goff, Erik R. Tou
Euleriana
Welcome to Volume 2 of Issue 2 of Euleriana. While we occasionally discover themes that emerge from the articles and translations presented in a given issue, the wide range of Euler’s work more often results in a variety of topics for each issue. This is no less true for Issue 2.
Euler Archive Spotlight, Cynthia Huffman Ph.D.
Euler Archive Spotlight, Cynthia Huffman Ph.D.
Euleriana
A spotlight on the Euler Archive focusing on Euler and the Basel Problem.
Leonhard Euler’S Correspondence Schedule, Dominic Klyve
Leonhard Euler’S Correspondence Schedule, Dominic Klyve
Euleriana
In addition to his large number of published articles and books, Leonhard Euler engaged in a prolific correspondence with scientists, mathematicians, and administrators throughout his career. By compiling the dates of all of his known letters, as described in the Opera Omnia, we can get some understanding of Euler’s weekly schedule. We report here for the first time Euler’s preference, particularly during his Berlin Period, of writing letters on Tuesdays and Saturdays.
Learning Mathematics From The Master: A Collection Of Euler-Based Primary Source Projects For Today’S Students, Part I, Janet Heine Barnett, Dominic Klyve, Kenneth M. Monks, Adam E. Parker
Learning Mathematics From The Master: A Collection Of Euler-Based Primary Source Projects For Today’S Students, Part I, Janet Heine Barnett, Dominic Klyve, Kenneth M. Monks, Adam E. Parker
Euleriana
This article and its sequel will together highlight a set of nine classroom ready projects that draw on the remarkable writing of Leonhard Euler (1707–1783) as a means to help students develop an understanding of standard topics from today’s undergraduate mathematics curriculum. Part of a larger collection of primary source projects intended for use in a wide range of undergraduate mathematics courses, these projects are freely available to students and their instructors. We provide a general description of the pedagogical design underlying these projects, more detailed descriptions of the individual projects themselves, and instructions for obtaining downloadable copies for classroom …
The Solution Of A Problem Of Searching For Three Numbers, Of Which The Sum, Product, And The Sum Of Their Products Taken Two At A Time, Are Square Numbers, Mark R. Snavely, Philip Woodruff
The Solution Of A Problem Of Searching For Three Numbers, Of Which The Sum, Product, And The Sum Of Their Products Taken Two At A Time, Are Square Numbers, Mark R. Snavely, Philip Woodruff
Euleriana
This paper first appeared in Novi Commentarii academiae scientiarum Petropolitanae, Volume 8, pp. 64-73 and is reprinted in Opera Omnia: Series 1, Volume 2, pp.519-530. Its Eneström number is E270. Euler improves his results significantly in "On Three Square Numbers, of Which the Sum and the Sum of Products Two Apiece will be a Square" (E523).
On A New Class Of Oscillations, Sylvio R. Bistafa
On A New Class Of Oscillations, Sylvio R. Bistafa
Euleriana
In this publication, Euler derived for the first time, the differential equation of the (undamped) simple harmonic oscillator under harmonic excitation, namely, the motion of an object subjected to two acting forces, one proportional to the distance travelled, the other one varying sinusoidally with time.
On A New Class Of Oscillations, Sylvio R. Bistafa
On A New Class Of Oscillations, Sylvio R. Bistafa
Euleriana
This publication was motivated by Krafft’s accidental observation of a suspended clock setting itself in constant motion as a pendulum. His analysis of the phenomenon led him to conclude that the vibration of the clock was solely due to imbalances in the clock's balance wheel. Next, he conceives a 'little machine' as he called it, in which a straight bar loaded by small weights at its extremities is free to oscillate about the center of gravity of a regular suspended clock. He then investigates different oscillating conditions, by calculating moments with different weights and lengths of the bar arms, to …
Review Of Change And Variations, Robert E. Bradley
Review Of Change And Variations, Robert E. Bradley
Euleriana
Review of Change and Variations: A History of Differential Equations to 1900, by Jeremy Gray, Springer Undergraduate Mathematics Series, 2021, 419 + xxii pages.
Conversations On Change, Christopher Goff, Erik Tou
Conversations On Change, Christopher Goff, Erik Tou
Euleriana
Welcome to Volume 2 of Euleriana! One of the pleasures of studying the his- tory of mathematics is reading how historical figures conversed with each other on topics of their era. In this volume, those conversations focus on differential equations—a subject which occupied Euler’s attention for much of his 56-year career. Throughout, we will see how he corresponded with many scholars of the day as he developed and refined his mathematical ideas.