The Golden Ratio by Mario Livio (science, math)

The number phi (Greek Φ, pronounced ‘fee’) has been known for millennia  Aside from it's direct use in mathematics, it is often called ‘the golden ratio’ that defines the most perfect esthetic proportions. That might sound pretty breathless (and it is), but it's also pretty accurate. In The Golden Ratio: The Story of Phi, the World's Most Astonishing Number Mario Livio addresses both the mathematical and esthetic features of phi, exploding historical myths and finding it in a number of less-than-obvious places. It's a fine little book and a very pleasant excursion after the disappointment of Kline's Mathematics and the Loss of Certainty.

This is a book about a number and about mathematics, but it is not a mathematics text. Livio writes for a  general reader, never getting into more mathematics or geometry than you got in high school (or at least, what you should have gotten in high school). His explanations are simple and clear, with good references for those who wish to go further into the topic. I could wish for a few more illustrations and appendices, but their lack didn't keep me from enjoying or finishing the book.

It's difficult to talk about the book without talking about the number, and that's to Livio's credit. He writes clearly enough that you focus on the topic rather than struggling to understand. Phi is the star, and it's place in history and mathematics is what you remember coming out of the book. Phi is found in  the logarithmic spiral and the chambered nautilus, in ferns and the Fibonacci sequence, in the corners of the dodecahedron and in the placement of branches on trees. Its use is as ancient as the early Greeks and as recent as Penrose tilings, with both beauty and math seen at every step of the way. In many ways Livio combines the two, with beauty in the math and mathematics in the beauty.

This was almost everything I want in a math or science book. It made me think, it educated me on both the science and the history, and it made me want to find more on this and similar topics. Recommended for both laymen and professionals.

Mathematics and the Loss of Certainty by Morris Kline (science)

When setting out the goal of reading and reviewing 52 books this year, I had to deal with the problem of bad books. I'm no longer one for killer reviews, reviews in which a book is eviscerated for the entertainment of those reading the reviews. Conversely, it's not fair to simply say "this book is bad" without saying why. So please, bear with me as I attempt to describe this most disappointing book without devolving into rant. Please do give me feedback, as I've recent encountered another book which is likely to get a bad review.

Morris Kline's Mathematics: The Loss of Certainty is a failure. It attempts to describe how the belief that mathematics and mathematical reasoning were complete and correct fell apart and why. Instead it is a disorganized history of mathematics, geometry, and proof across the ages. Kline never adequately describes the nature of geometric proof, nor the alternate geometries which shattered it. He dwells to no particular purpose on 1,500 years of irrelevancy during which algebra et. al. were developed and used without proof and without quest for proof. When he reached the 18th through 20th centuries, he jumps backwards and forwards discussing the changed feeling about proof and provability of mathematics, but leaves the causes of those changes for later chapters. In short, what should have been a fascinating history of mathematics and treatise about the nature of mathematical proof is a rambling mess. He focuses on nits, glides over critical but admittedly difficult items, and winds up leaving even this educated reader frustrated and uniformed.

In summary: not recommended.