100 books like Euclid's Elements

David Hilbert was the most important mathematician at the dawn of the 20th century. In 1900, he gave the mathematical community its homework for the next 100 years setting out the list of open problems that had to be solved by 2000. While to the rest of the mathematicians, he may have appeared as their professor, he was also the class clown. As colorful and funny as he was brilliant, you cannot but come away loving this great mathematical genius.

Ulam was a Polish mathematical prodigy, publishing significant mathematics by the time he was 20. He was part of the rich Polish math community centered around Stefan Banach. Unlike most, he was heading to the United States in 1939 (with his younger brother) when Germany invaded Poland. All the rest of his family were murdered by the Nazis. He on the other hand ended up in Los Alamos, providing critical help on the Manhattan Project. Later in life, he wrote this book, his autobiography. Based on his history, one could well think that it would be a book full of tragic grief. Instead, it is a pean to the joys of doing mathematics and of living a life full of mathematics, without downplaying the horrors of the mid-twentieth century. 

Most mathematicians believe that the Riemann Hypothesis is the most important open question in mathematics, including me. But it is almost impossible to explain why this is such a central concern. This book is one of the attempts to explain to the non-mathematician why the Riemann Hypothesis is so important. As a partial spoiler alert, it has to do with the nature of prime numbers, which in part explains the title. It is not a book to read in one sitting, but it with a little work is great for seeing, at least in part, the big picture.

Gamma is a number, though little understood. Even its most basic properties are still unknown. We don’t even know if it is a rational number (a ratio of integers). This wonderful book explains why anyone would care. While it does require some mathematical background, anyone who has had calculus and is willing to read the book with a notepad and pen next to them in order to check and explore the formulas on their own will come away with a true appreciation of gamma and its impact. 

Baruch Spinoza was the philosophical flower of the Dutch Golden Age. Bertrand Russell called him the "noblest and most lovable of the great philosophers", and I am certainly not going to disagree. Like many of the innovators of the Golden Age, his ideas still seem fresh. Expelled from his Jewish community in Amsterdam for his ‘heresies’, we now find his conception of God as nature highly congenial. We probably share his dislike of ritual and perhaps aspire to his renunciation of materialism. His advice neither to fear nor to hope when it concerns things we can do nothing about is as good now as it was when it appeared in his most famous work, Ethics, in 1677.

Spinoza’s philosophy is hard to approach in the original – his arguments are rigorously constructed in the style of ancient Greek mathematics proofs. But Steven Nadler, as well as producing a towering biography… show more.

I find this book fascinating and pioneering in more than one way. Not only does it uncover the historical development of the important science of optics in terms of theories of vision, the predominant form of the subject in premodern times.

But the book also covers key concepts and models in visual theory starting from ancient and medieval periods, including the extramission models of ancient Greek mathematicians and physicians, the intromission models of medieval Islamic and European natural philosophers, and the critical combination of those models and disciplines in generating dependent breakthroughs.

This is among a few monographs devoted to the subject that successfully draw on our common heritage in documenting and explaining the course of the field’s transmission and transformation. 

Robin Evans was a versatile architectural historian and theorist who died too young. This highly original and unusual book, published after his death, is about the relationship of geometry to architecture, and how methods of drawing, including perspective and orthographic projection, can influence what is conceived and built. I admire the way in which Evans, unlike many architectural historians, is able to combine deep scholarship with a working practical understanding of how buildings are made, and how they are used in practice. There has been no other recent writer on architecture with so subtle a mind.

Peter Sis is an acclaimed international writer, artist, and filmmaker. He’s created several award-winning children’s books. In The Three Golden Keys, Sis recaptures his lost childhood in Prague and takes the reader on a dark, mysterious adventure through Prague to find the three keys that will unlock the three rusty padlocks of his childhood home and of his own childhood memories.

I like his art more than his storytelling which is often so personal as to be cryptic. So why am I recommending this book? Because his illustrations of Prague are so beautifully detailed, surreal, and mysterious. It’s a work of art that’s personal, poignant, and poetic. It’s a gorgeous book but perhaps enjoyed by adults more than children.

I stumbled on this in a used bookstore. What a find! The old-school, kid-friendly illustrations lead swiftly from simple beginnings (“What happens when you stretch a painting?”) to the depths of undergraduate topology. I haven’t used this in the classroom yet, but honestly, I could imagine busting it out with anyone from first-graders to first-year PhD candidates.

This may seem an odd choice, but as a maths popularizer I need to know all that I can about why some people find the main elements of the subject so difficult. I found Doug French's book exceptionally helpful in this respect, even though it is aimed principally at high school teachers. This is partly because he focuses throughout on the most important mathematical ideas and difficulties. Moreover, the scope is wider than the title suggests, for he also ventures imaginatively into both geometry and calculus.