Sandifer is a renowned expert on Euler. In 2001 he was co-founder and later secretary of the Euler Society. He wrote several books and contributed and promoted the Euler centenary year in 2007. Since November 2003, he also had a monthly column called How Euler Did It hosted by the Euler Archive website in collaboration with the Mathematical Association of America (MAA). These columns are stand-alone items elucidating a particular aspect of Euler's work. They are available online at the Euler Archive. A first collection of these columns appeared as a book How Euler Did it (MAA, 2007). Unfortunately in 2009, Sandifer had to recover from a severe stroke, but some colleagues filled up some of the gap so that the column kept appearing until early 2010. This book contains 35 of these columns from March 2007 till February 2010. September 2009 does not exist for the reason just explained. October, November, and December 2009 were filled up by guest author Rob Bradley.
In the previous collection, the items were ordered chronologically, but in this volume, they are grouped by topic: Geometry, Number Theory, Combinatorics, Analysis (the largest part), Applied Mathematics, and some miscellaneous part called Euleriana. The columns are explaining indeed how Euler did prove some of his results. Euler got his problems for example from marginal notes by Fermat who, as we know, announced theorems by scribbling some notes in the margin of a book, or some problems were formulated by Euler himself and there are many other sources. Some historical background is given, but the main contribution is just explaining how Euler indeed constructed his proof. Sometimes original drawings are reproduced. These columns are certainly welcomed by readers not familiar with the original language of the papers or the correspondence which was often Latin, German, or French. Some translations of the original texts by Euler can be found at the Euler Archive but many are not translated yet.
There are too many different topics to be discussed in detail in this review. They include prime numbers, trigonometry, probability theory, mortality tables and actuarial science, the zeta and gamma functions, formulas to approximate π, partial fractions, complex analysis, optics, fluid dynamics, gravity and many more. The Euleriana part deals with Euler as a teacher, but there are also contributions about Euler and the hollow earth, errors that Euler made, and about Euler and pirates. I leave it to your imagination what these are all about. You will have to read the column to know.
Let me take just one example from the applied mathematics chapter to illustrate how the items are treated by Sandifer in his blog. In 1756 Euler while working in Berlin publishes a paper on modeling of saws. The column starts by explaining that Euler worked in 1728 as a physician for the Russian Navy in St. Petersburg where he learned the importance on lumber and the operation of sawmills. When 30 years later the Prussian King Frederik II was about to embark in a war, Euler, remembering the importance of lumber, wrote his paper. The saw model is about a vertical blade that cuts when moving down. Each tooth should cut the same amount of wood, which means the teeth side of the saw should be slanted, the next tooth cutting another peel where the previous one had just removed its part. This models the shape of the saw. Then the motion is formalized, the middle part of the saw blade that really cuts the wood, the speed and the energy needed, and finally the manpower needed to lift the saw (it was supposed to move down by gravity), and the amount that was cut per worker and per hour. It beautifully illustrates the genius of Euler in an easily understandable mathematical language as it is brought to us by Sandifer.
As you can see from my previous enumeration, there are enough topics to make the book of interest to many. Simple mathematics suffice to illustrate the brilliant mind of Euler. They are also mathematical gems as a column: well written and documented, mainly addressing mathematicians or teachers, but understandable with a minimum of training. If fits perfectly well in the MAA Spectrum series that targets the general mathematically-interested reader. Given the dynamics and volatility of websites, it is a good initiative that the MAA has made these columns available as a book.