New and Noteworthy Titles on our Bookshelf

Archimedes, Fulcrum of Science

The University of Chicago Press, 2024, 240 pp.

By Nicholas Nicastro

Cover is courtesy of The University of Chicago Press.

We know Archimedes to be a highly regarded inventor and mathematician from Syracuse in Ancient Greece. What little of his written work that survived and was translated over the years is nothing compared to the surviving stories (and perhaps myths) of his life and work. Well-documented references to his life make apparent that Archimedes served as an inspiration to Galileo, Newton, Tesla, and many others. Archimedes has been the subject of many pieces of art throughout history, confirming that his impact on Western culture has persisted since his death in 212 BCE.

This book is a history of the life and work of Archimedes. Some of the book focuses on political history, as Archimedes had friends in high places. However, much of the book focuses on his mathematics. While not overly technical, it does include figures and drawings of his mathematical work, along with references to the work of other ancient mathematicians. Archimedes is known most for his work in geometry, including planes, parabolas, cylinders, circles, spheres, and spirals. His estimation of $\pi$ would not be improved upon for centuries. His last written work was called The Sand-Reckoner, and in it, he estimates how much sand it would take to fill the universe. Many understand this work to be a foreshadowing of infinity and infinitesimals, which continue to be utilized in calculus centuries later. Truly, the impact and influence of Archimedes cannot be understated. This book was an enjoyable way to discover more about the life and work of one of the most foundational mathematicians.

Geometric Gems: An Appreciation for Geometric Curiosities. Volume 1: The Wonders of Triangles

World Scientific, 2024, 396 pp.

By Alfred S. Posamentier and Robert Geretschläger

Cover is courtesy of World Scientific.

I fear I may have underestimated triangles, those easy-to-draw polygons with three sides. There is so much more to know beyond similarity and the law of sines! The first volume of the series is titled The Wonders of Triangles and wow, are they wonderful! This volume goes beyond the nine-point circle and the concurrence of the centroid, orthocenter, and circumcenter (three central points of a triangle) by detailing 175 “curiosities” that involve altitudes, bisectors, medians, and more. This book is likely one of the largest collections of its kind. An example of a curiosity is #91, which states that “a triangle with one-third the area of a given triangle results by trisecting each of the sides of the original triangle.” The curiosity explains how to form a triangle of one-third the area of the original by using certain points on the trisected sides. Neat!

Each curiosity is clearly written, illustrated, and proven. I cannot emphasize how much I enjoyed the figures. The visual components of this volume add to the overall understanding of each curiosity, and I found it vital there was a figure for each curiosity and each proof. All 175 curiosities appear first in the book, followed by a section containing the proofs. At the end, the authors include a Toolbox that covers the geometric necessities for the proofs. Most of these tools are covered in a typical secondary education, which makes this book accessible to advanced high schoolers and their teachers as well as undergraduate students or the general public. Trying to prove some of the curiosities on your own could be a fun challenge or math club activity. I’m curious to see what future volumes hold, as the authors promise subsequent volumes on quadrilaterals and circles. If you have an appreciation for geometric curiosities, you are likely to take delight in this book.