The Early Mathematics of Leonhard Euler

by Sandifer, C. Edward

• ISBN13: 9780883855591
• ISBN10: 0883855593
• Format: Hardcover
• Copyright: 2006-12-14
• Publisher: Mathematical Assn of Amer

Summary

Describes Euler's early mathematical works - the 50 mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin. These works contain some of Euler's greatest mathematics: the Konigsburg bridge problem, his solution to the Basel problem, his first proof of the Euler-Fermat theorem. Also presented are important results that we seldom realize are due to Euler: that mixed partial derivatives are equal, our f(x) notation, and the integrating factor in differential equations. The book is a portrait of the world's most exciting mathematics between 1725 and 1741, rich in technical detail, woven with connections within Euler's work and with the work of other mathematicians in other times and places, laced with historical context.

Author Biography

Ed Sandifer is Professor of Mathematics of Western Connecticut State University. He earned his PhD at the University of Massachusetts under John Fogarty, studying ring theory. He became interested in Euler while attending the Institute for the History of Mathematics and Its Uses in Teaching, IHMT, several summers in Washington DC, under the tutelage of Fred Rickey, Victor Katz and Ron Calinger. Because of a series of advising mistakes, as an undergraduate he studied more foreign languages than he had to, so now he can read the works of Euler in their original Latin, French and German. Occasionally he reads Spanish colonial mathematics in its original as well. Now he is secretary of the Euler Society, and he writes a monthly on-line column, How Euler Did It, for the MAA. He and hi wife Theresa, live in a small town in western Connecticut, and he has run the Boston Marathon every year since 1973.

Table of Contents

Preface

1725+1727

Construction of isochronal curves in any kind of resistant

Method of finding reciprocal algebraic trajectories

1728

Solution to problems of reciprocal trajectories

A new method of reducing innumerable differential equations of the second degree to equations of the first degree: Integrating factor

1729+1731

On transcendental progressions, or those for which the general term cannot be given algebraically

On the shortest curve on a surface that joins any two given points

On the summation of innumerably many progressions

1732

General methods for summing progressions

Observations on theorems that Fermat and others have looked at about prime numbers

An account of the solution of isoperimetric problems in the broadest sense

1733

Construction of differential equations which do not admit separation of variables

Example of the solution of a differential equation without separation of variables

On the solution of problems of Diophantus about integer numbers

Inferences on the forms of roots of equations and of their orders

Solution of the differential equation axn dx = dy + y2dx

1734

On curves of fastest descent in a resistant medium

Observations on harmonic progressions

On an infinity of curves of a given kind, or a method of finding equations for an infinity of curves of a given kind

Additions to the dissertation on infinitely many curves of a given kind

Investigation of two curves, the abscissas of which are corresponding arcs and the sum of which is algebraic

1735

On sums of series of reciprocals

A universal method for finding sums which approximate convergent series

Finding the sum of a series from a given general term

On the solution of equations from the motion of pulling and other equations pertaining to the method of inverse tangents

Solution of a problem requiring the rectification of an ellipse

Solution of a problem relating to the geometry of position

1736

Proof of some theorems about looking at prime numbers

Further universal methods for summing series

A new and easy way of finding curves enjoying properties of maximum or minimum

1737

On the solution of equations

An essay on continued fractions

Various observations about infinite series

Solution to a geometric problem about lunes formed by circles

1738

On rectifiable algebraic curves and algebraic reciprocal trajectories

On various ways of closely approximating numbers for the quadrature of the circle

On differential equations which sometimes can be integrated

Proofs of some theorems of arithmetic

Solution of some problems that were posed by the celebrated

1739

On products arising from infinitely many factors

Observations on continued fractions

Consideration of some progressions appropriate for finding the quadrature of the circle

An easy method for computing sines and tangents of angles both natural and artificial

Investigation of curves which produce evolutes that are similar to themselves

Considerations about certain series

1740

Solution of problems in arithmetic of finding a number, which, when divided by given numbers leaves given remainders

On the extraction of roots of irrational quantities: gymnastics with radical signs

1741

Proof of the sum of this series 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/ 36 + etc

Several analytic observations on combinations

On the utility of higher mathematics

Topically related articles

Index

About the author

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