Euler: The Mathematics of Musical Sadness

2014 ◽  
Author(s):  
Peter Pesic
Keyword(s):  
Number Theory ◽  
Lower Degree ◽  
Web Browsers ◽  
Great Mathematician ◽  
Mathematical Basis ◽  
Bridge Problem ◽  
Leonhard Euler ◽  
Basic Concepts ◽  
New Criterion ◽  
Numerical Complexity

Throughout his life, the great mathematician Leonhard Euler spent most of his free time on music, to which he devoted his first book. This chapter discusses how he reformulated the ordering of musical intervals on a new mathematical basis. For this purpose, Euler devised a “degree of agreeableness” that numerically indexed musical intervals and chords, replacing ancient canons of numerical simplicity with a new criterion based on pleasure. Euler applied this criterion (and Aristotle’s teachings about the pleasure of tragedy) to argue that minor intervals and chords evoke sadness through their greater numerical complexity, hence lower degree of agreeableness than the major. This work involved extensive attention to ratios and numerical factorization immediately preceding his subsequent interest in continued fractions and number theory. Having devised a new kind of index, Euler was prepared to put forward indices that would address novel problems like the Königsberg bridge problem and the construction of polyhedra, basic concepts of what we now call topology. Throughout the book where various sound examples are referenced, please see http://mitpress.mit.edu/musicandmodernscience (please note that the sound examples should be viewed in Chrome or Safari Web browsers).

Euler's contribution to number theory

2007 ◽  
Vol 91 (522) ◽  
pp. 453-461 ◽  
Author(s):  
Peter Shiu
Keyword(s):  
Eighteenth Century ◽  
Number Theory ◽  
Scientific Output ◽  
Scholarly Work ◽  
Long Period ◽  
Peter The Great ◽  
Complete Work ◽  
Leonhard Euler ◽  
The Subject ◽  
Historic Background

Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

scholarly journals On known and less known relations of Leonhard Euler with Poland

2016 ◽  
Vol 15 ◽  
pp. 75-110
Author(s):  
Roman Sznajder ◽  
Keyword(s):  
Graph Theory ◽  
18Th Century ◽  
Mathematical Community ◽  
Geometric Interpretation ◽  
European Countries ◽  
Complex Numbers ◽  
And Topology ◽  
Bridge Problem ◽  
Leonhard Euler ◽  
The City

In this work we focus on research contacts of Leonhard Euler with Polish scientists of his era, mainly with those from the city of Gdańsk (then Gedanum, Danzig). L. Euler was the most prolific mathematician of all times, the most outstanding mathematician of the 18th century, and one of the best ever. The complete edition of his manuscripts is still in process (Kleinert 2015; Kleinert, Mattmüller 2007). Euler’s contacts with French, German, Russian, and Swiss scientists have been widely known, while relations with Poland, then one of the largest European countries, are still in oblivion. Euler visited Poland only once, in June of 1766, on his way back from Berlin to St. Petersburg. He was hosted for ten days in Warsaw by Stanisław II August Poniatowski, the last king of Poland. Many Polish scientists were introduced to Euler, not only from mathematical circles, but also astronomers and geographers. The correspondence of Euler with Gdańsk scientists and officials, including Carl L. Ehler, Heinrich Kühn and Nathanael M. von Wolf, originated already in the mid-1730s. We highlight the relations of L. Euler with H. Kühn, a professor of mathematics at the Danzig Academic Gymnasium and arguably the best Polish mathematician of his era. It was H. Kühn from whom Euler learned about the Königsberg Bridge Problem; hence one can argue that the beginning of the graph theory and topology of the plane originated in Gdańsk. In addition, H. Kühn was the first mathematician who proposed a geometric interpretation of complex numbers, the theme very much appreciated by Euler. Findings included in this paper are either unknown or little known to a general mathematical community.

scholarly journals Considerations for Goldbach's Strong Conjecture

2021 ◽  
Author(s):  
Carleilton Severino Silva
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Weak Version ◽  
Strong Version ◽  
Leonhard Euler

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

Traversing Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Chinese Postman Problem ◽  
Round Trip ◽  
Eulerian Graphs ◽  
Bridge Problem ◽  
Chinese Postman ◽  
Leonhard Euler ◽  
The City ◽  
East Prussia

This chapter considers Eulerian graphs, a class of graphs named for the Swiss mathematician Leonhard Euler. It begins with a discussion of the the Königsberg Bridge Problem and its connection to Euler, who presented the first solution of the problem in a 1735 paper. Euler showed that it was impossible to stroll through the city of Königsberg, the capital of German East Prussia, and cross each bridge exactly once. He also mentioned in his paper a problem whose solution uses the geometry of position to which Gottfried Leibniz had referred. The chapter concludes with another problem, the Chinese Postman Problem, which deals with minimizing the length of a round-trip that a letter carrier might take.

Number Theory: A Very Short Introduction

2020 ◽  
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Short Introduction ◽  
New Developments ◽  
Leonhard Euler

Number Theory: A Very Short Introduction explains the branch of mathematics primarily concerned with the counting numbers, 1, 2, 3, …. Long considered one of the most ‘beautiful’ areas of mathematics, number theory dates back over two millennia to the Ancient Greeks, but has seen some startling new developments in recent years. Trailblazers in the field include mathematicians Euclid of Alexandria, Pierre de Fermat, Leonhard Euler, and Carl Friedrich Gauss. Number theory has intrigued and attracted amateurs and professionals alike for thousands of years, appearing in both recreational contexts (puzzles) and practical concerns (cryptography). Some problems in number theory are easy, whereas others are notorious with no known solutions to date.

Graph Theory: A Look Back—The Road Ahead

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Social Networks ◽  
Graph Theory ◽  
The Internet ◽  
The Road ◽  
Eulerian Graphs ◽  
Bridge Problem ◽  
Look Back ◽  
Leonhard Euler ◽  
The Subject ◽  
The 1960S

This book concludes with an epilogue, which traces the evolution of graph theory, from the conceptualization of the Königsberg Bridge Problem and its generalization by Leonhard Euler, whose solution led to the subject of Eulerian graphs, to the various efforts to solve the Four Color Problem. It considers elements of graph theory found in games and puzzles of the past, and the famous mathematicians involved including Sir William Rowan Hamilton and William Tutte. It also discusses the remarkable increase since the 1960s in the number of mathematicians worldwide devoted to graph theory, along with research journals, books, and monographs that have graph theory as a subject. Finally, it looks at the growth in applications of graph theory dealing with communication and social networks and the Internet in the digital age and the age of technology.

Young’s Musical Optics

2014 ◽  
Author(s):  
Peter Pesic
Keyword(s):  
Natural Philosophy ◽  
Wave Theory ◽  
Scientific Development ◽  
Scientific Instrument ◽  
Web Browsers ◽  
Acoustic Analogy ◽  
Wave Nature ◽  
Light Waves ◽  
Leonhard Euler ◽  
Optical Phenomena

Building on the work of Leonhard Euler, Thomas Young advanced the wave theory of sound and light. This chapter describes how Young found his way to music against the strictures of his Quaker milieu. His new-found passions for music and dance informed his studies of sound and languages. His early work on the accommodation of the eye remained a touchstone for his later scientific development. At many points, his understanding of sound influenced and shaped his approach to light, including the decisive experiments that established its wave nature. His early investigations into the sounds of pipes led him to make an acoustic analogy that could explain optical phenomena such as Newton’s rings. He introduced a new system of temperament and used the piano as a scientific instrument. His comprehensive Lectures on Natural Philosophy included many plates that juxtaposed acoustic and optical phenomena. When Young turned to the decipherment of Egyptian hieroglyphics, he relied on sound and phonology. His final suggestions about the transverse nature of light waves again turned on the comparison with sound. Throughout the book where various sound examples are referenced, please see http://mitpress.mit.edu/musicandmodernscience (please note that the sound examples should be viewed in Chrome or Safari Web browsers).

scholarly journals Considerations for Goldbach's Strong Conjecture

2021 ◽  
Author(s):  
Carleilton Severino Silva
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Weak Version ◽  
Strong Version ◽  
Leonhard Euler

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

3. Prime-time mathematics

2020 ◽  
pp. 37-57
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Prime Time ◽  
Leonhard Euler ◽  
Mersenne Primes ◽  
Fermat Primes

‘Prime-time mathematics’ explores prime numbers, which lie at the heart of number theory. Some primes cluster together and some are widely spread, while primes go on forever. The Sieve of Eratosthenes (3rd century BC) is an ancient method for identifying primes by iteratively marking the multiples of each prime as not prime. Every integer greater than 1 is either a prime number or can be written as a product of primes. Mersenne primes, named after French friar Marin de Mersenne, are prime numbers that are one less than a power of 2. Pierre de Fermat and Leonhard Euler were also prime number enthusiasts. The five Fermat primes are used in a problem from geometry.

Sign in / Sign up

Export Citation Format

Share Document