C. Edward Sandifer has been studying Euler for decades and is one of the world's leading experts on his work. This book is the second collection of Sandifer's 'How Euler Did It' columns. Each is a jewel of historical and mathematical exposition. The sum total of years of work and study of the most prolific mathematician in history, this volume will leave the reader marveling at Euler's clever inventiveness, which Sandifer explicates and puts in context.

Review

C. Edward Sandifer's How Euler Did Even More is the second collection of his monthy columns from MAA Online, "How Euler Did It." The first collection, also titled How Euler Did It, appeared in 2007 as part of the five-volume set published by the MAA in recognition of the tercentenary of Euler's birth. It contained Sandifer's columns from November 2003 through February 2007. This second collection contains his columns from March 2007 through February 2010, with the addition of two guest columns by Rob Bradley and one by Dominic Klyve. (Bradley assisted Sandifer with the details of publication of this collection.)

There are several ways to read this book. First, one may choose simply to open it at random to read Sandifer's discussion of how Euler attacked and thought about certain problems. Sandifer places Euler's work into context of the mathematics of his time, then describes what Euler did and how he did it and why it mattered, keeping in mind the advice of John Fauvel that Sandifer references in How Euler Did It : "Content, Context and Significance." An alternative would be to read the columns for particular topics that Euler considered; the columns are organized into sections on geometry, number theory, combinatorics, analysis, applied mathematics, and Euleriana. This last section includes two columns reflecting on Euler as teacher, two on light-hearted topics (Euler and the hollow earth and Euler and pirates), and on discussing Euler's fallibility.

A third way to read this book would seem to summarize a great deal of Sandifer's writing on Euler. That is, one could use individual sections as invitations and guides to read Euler's texts in their original languages or in translations. The background that Sandifer provides in each column, along with the sense of "here's what Euler's doing" will make reading Euler much more accessible. (As an aside, I am particularly appreciative of the way Sandifer consults what Euler actually wrote, rather than relying on secondary sources, in his discussions of what Euler did.)

If you already have How Euler Did It , I can't imagine that you'd not also enjoy How Euler Did Even More. If you haven't yet dipped into these books, I'd encourage you to do so. --Joel Haack, MAA Reviews

This volume collects 35 "How Euler Did It" columns from MAA Online, of which 32 were authored by C. Edward Sandifer (with two "guest columns" by Rob Bradley and one by Dominic Klyve). The assembled essays are a sequel to the author's 2007 book [How Euler Did It, MAA Spectrum, Math. Assoc. America, Washington, DC, 2007;MR2321397 (2008h:01015)], which assembled 39 of his MAA Online columns on Euler. The most salient feature of these collected essays is the breadth of Euler's mathematical vision and the significance of his results.

The essays on Euler's contributions to pure mathematics are grouped into four sections (Geometry, Number Theory, Combinatorics, and Analysis). Given Euler's status as "analysis incarnate", it is no surprise that more than half of these columns are devoted to geometry situate Euler in the context of the rise of analytic techniques and the decline of more traditional "synthetic" methods in the style of Euclid. The three columns on Euler's study of combinatorics deal (as one would expect) with issues of probability and statistics. The 13 columns dealing with analysis range quite widely over topics in the field. The fifth part of this collection contains columns dealing with various topics in applied mathematics, notably fluid mechanics, gravitation theory, optics, pneumatics, and a remarkable Eulerian treatment of the mechanics of sawblades. The sixth and final section bears the title "Euleriana" and collects five columns concerned with generally non-mathematical aspects of Euler and his work.

On the whole, this collection is notable for the clarity of its exhibition, the wide range of subjects, and the sophistication of its mathematical treatment. One comes away with a renewed appreciation for the genius of Euler, as well as an improved understanding of what mathematical practice in the eighteenth century really looked like. Anyone with an interest in Euler or the development of mathematics in the eighteenth century will find a wealth of important material here. --Douglas M. Jesseph, Mathematical Reviews Clippings

The author was one of the driving forces behind the formation of the Euler Society founded at the beginning of the 21st century and he published the first of the "Euler Volumes" -- the MAA Euler Tercentenary book series. In 2003, he began to write a series of monthly columns for MAA Online entitled How Euler Did It. From this column several books derived. The present book collects the final 35 columns of How Euler Did It from which 32 were written by the author.

The articles within the book are not sorted chronologically but grouped together thematically. The first group appears under the heading "Geometry," then follow "Number Theory," "Combinatorics," "Analysis, " "Applied Mathematics." The last (sixth) part contains some "Euleriana," among other things an interesting two-part essay on Euler as a teacher. Like the other books in the series this one is also very readable and gives once more insights into the works of Leonhard Euler. --Zentrallblatt

Book Description

A collection of lively expository columns on the work of Euler, written by a leading scholar.

About the Author

C. Edward Sandifer is Professor of Mathematics at Western Connecticut State University in Danbury, Connecticut. He is Secretary of The Euler Society (www.EulerSociety.org). His first book, The Early Mathematics of Leonhard Euler, was published by the MAA in December 2006, as part of the celebrations of Euler's tercentennial in 2007. The MAA published a collection of forty 'How Euler Did It' columns in June 2007.

## Description:

C. Edward Sandifer has been studying Euler for decades and is one of the world's leading experts on his work. This book is the second collection of Sandifer's 'How Euler Did It' columns. Each is a jewel of historical and mathematical exposition. The sum total of years of work and study of the most prolific mathematician in history, this volume will leave the reader marveling at Euler's clever inventiveness, which Sandifer explicates and puts in context.

## Review

C. Edward Sandifer's

How Euler Did Even Moreis the second collection of his monthy columns from MAA Online, "How Euler Did It." The first collection, also titledHow Euler Did It,appeared in 2007 as part of the five-volume set published by the MAA in recognition of the tercentenary of Euler's birth. It contained Sandifer's columns from November 2003 through February 2007. This second collection contains his columns from March 2007 through February 2010, with the addition of two guest columns by Rob Bradley and one by Dominic Klyve. (Bradley assisted Sandifer with the details of publication of this collection.)There are several ways to read this book. First, one may choose simply to open it at random to read Sandifer's discussion of how Euler attacked and thought about certain problems. Sandifer places Euler's work into context of the mathematics of his time, then describes what Euler did and how he did it and why it mattered, keeping in mind the advice of John Fauvel that Sandifer references in

How Euler Did It: "Content, Context and Significance." An alternative would be to read the columns for particular topics that Euler considered; the columns are organized into sections on geometry, number theory, combinatorics, analysis, applied mathematics, and Euleriana. This last section includes two columns reflecting on Euler as teacher, two on light-hearted topics (Euler and the hollow earth and Euler and pirates), and on discussing Euler's fallibility.A third way to read this book would seem to summarize a great deal of Sandifer's writing on Euler. That is, one could use individual sections as invitations and guides to read Euler's texts in their original languages or in translations. The background that Sandifer provides in each column, along with the sense of "here's what Euler's doing" will make reading Euler much more accessible. (As an aside, I am particularly appreciative of the way Sandifer consults what Euler actually wrote, rather than relying on secondary sources, in his discussions of what Euler did.)

If you already have

How Euler Did It, I can't imagine that you'd not also enjoyHow Euler Did Even More. If you haven't yet dipped into these books, I'd encourage you to do so. --Joel Haack, MAA ReviewsThis volume collects 35 "How Euler Did It" columns from MAA Online, of which 32 were authored by C. Edward Sandifer (with two "guest columns" by Rob Bradley and one by Dominic Klyve). The assembled essays are a sequel to the author's 2007 book [How Euler Did It, MAA Spectrum, Math. Assoc. America, Washington, DC, 2007;MR2321397 (2008h:01015)], which assembled 39 of his MAA Online columns on Euler. The most salient feature of these collected essays is the breadth of Euler's mathematical vision and the significance of his results.

The essays on Euler's contributions to pure mathematics are grouped into four sections (Geometry, Number Theory, Combinatorics, and Analysis). Given Euler's status as "analysis incarnate", it is no surprise that more than half of these columns are devoted to geometry situate Euler in the context of the rise of analytic techniques and the decline of more traditional "synthetic" methods in the style of Euclid. The three columns on Euler's study of combinatorics deal (as one would expect) with issues of probability and statistics. The 13 columns dealing with analysis range quite widely over topics in the field. The fifth part of this collection contains columns dealing with various topics in applied mathematics, notably fluid mechanics, gravitation theory, optics, pneumatics, and a remarkable Eulerian treatment of the mechanics of sawblades. The sixth and final section bears the title "Euleriana" and collects five columns concerned with generally non-mathematical aspects of Euler and his work.

On the whole, this collection is notable for the clarity of its exhibition, the wide range of subjects, and the sophistication of its mathematical treatment. One comes away with a renewed appreciation for the genius of Euler, as well as an improved understanding of what mathematical practice in the eighteenth century really looked like. Anyone with an interest in Euler or the development of mathematics in the eighteenth century will find a wealth of important material here. --Douglas M. Jesseph, Mathematical Reviews Clippings

The author was one of the driving forces behind the formation of the Euler Society founded at the beginning of the 21st century and he published the first of the "Euler Volumes" -- the MAA Euler Tercentenary book series. In 2003, he began to write a series of monthly columns for MAA Online entitled

How Euler Did It.From this column several books derived. The present book collects the final 35 columns ofHow Euler Did Itfrom which 32 were written by the author.The articles within the book are not sorted chronologically but grouped together thematically. The first group appears under the heading "Geometry," then follow "Number Theory," "Combinatorics," "Analysis, " "Applied Mathematics." The last (sixth) part contains some "Euleriana," among other things an interesting two-part essay on Euler as a teacher. Like the other books in the series this one is also very readable and gives once more insights into the works of Leonhard Euler. --Zentrallblatt

## Book Description

A collection of lively expository columns on the work of Euler, written by a leading scholar.

## About the Author

C. Edward Sandifer is Professor of Mathematics at Western Connecticut State University in Danbury, Connecticut. He is Secretary of The Euler Society (www.EulerSociety.org). His first book, The Early Mathematics of Leonhard Euler, was published by the MAA in December 2006, as part of the celebrations of Euler's tercentennial in 2007. The MAA published a collection of forty 'How Euler Did It' columns in June 2007.