## Book Review: Elements Of Geometry

Elements Of Geometry, Plane And Spherical Geometry And Conic Sections, by Horatio Nelson Robinson

This is a book that I wish I had known about and been able to read when I was studying geometry as a high school student. The geometry class I took during my sophomore year of high school focused on theorems, and that is precisely what this book focuses on. The thought that I could have found relatively straightforward theorems to build up from a small set of fundamental principles to much more complex ones. It is not hard for me to understand how it is that Euclid’s works offered such an appeal to the mind of Abraham Lincoln as a young person, because of the fundamental and logical nature of building one’s understanding theorem from theorem recognizing that given a certain small set of assumptions that one can construct a great deal of understanding, using one proof as a premise for another one, building together one’s understanding of the world brick by brick, proof by proof. Such an approach to understanding has always been of great appeal to me, although admittedly I tend to combine deduction with other approaches to build together understanding. Still, depending on the strength of one’s assumptions, deduction can be a very powerful tool for understanding that is often not understood very well, especially at present.

This book is almost 300 pages long, and gets even longer if one includes the various log tables and other tables at the end of the book. The first part of the book is by far the longest part, dealing with plane geometry, which is divided into a variety of smaller “books” dealing with various matters. For example, book one deals with general principles as well as theorems dealing with angles and right lines, and this is followed by books that deal with proportions, circles and the measurement of angles, problems in geometrical constructions, measurements of polygons and circles and problems and exercises relating to algebra and geometric expression, the intersection of planes as well as solid geometry. This lengthy first book is followed by a book on plane trigonometry that covers elementary principles, the computation of sines and tangents, and various applications and explanations of oblique angles and related subjects. The third section of the book then covers spherical geometry, including plenty of applications for geography as well as astronomy. The fourth part of the book then covers conic sections with various principles and applications. After this there are many more pages of tables that remind the reader of the fate of math students before advanced calculation was at the fingertips of students.

There were at least a few aspects of this work that were a revelation. First of all, the approach of the book was very congenial to me, in that this book featured plenty of figures as well as very logical step-by-step proofs that were built by the author into very complex understandings of geometry. Also of interest is the way that this book discusses spherical geometry and its importance. At least from what I have read, it was once far more common for geometry classes to get into the field of spherical geometry and to explore its implications in understanding matters of geography. As we live on a mostly spherical world, it stands to reason that understanding spherical geometry would have practical benefits, but the failure to teach such subject matter has led to people not recognizing the importance of spherical geometry because it is something they have no familiarity with or understanding of whatsoever, and thus are unable to recognize the importance of it. This book, though, does deal with such practical applications in a way that is interesting and worthwhile, and that makes this worth the time it takes to read it even apart from this book’s general pleasure. 