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The author is so wonderfully entranced and entrancing with this "lost art" that he kindof misses the fact that some of the "old" arts like quaternions (revolving around a world centered on the square root of negative 1 instead of 1!), triangular polygons on surfaces, Napier's pentagon, etc. are being reborn today in the art and craft of modeling, game programming, simulation, digital art and computer graphics (eg. Maya, ZBrush).
Sure, there are now algorithms and calculus functions that "eclipse" the ancient navigational methods, not to mention GPS, but no self respecting GIS teacher can ignore Spherical Trig even today! Wolfram in particular has spent a lot of time tinkering with Java and other applets in spherical trig, and many GIS (Geographical Information Systems) teachers I know will LOVE this text.
This book is really about the beauty of mathematics, and in a Platonic sense, the translation of angular dimensions and fractals into the "real spherical" world not only of planets and stars, but more recently, molecules and RNA folding. If you're a math amateur, you'll love the beauty, and the trig is doable with a little review. If you're a pro, you might just find relationships that newer methods have obscured, but give you many "aha" moments about limits and even hyper modern applications like inverse kinematics, joints and robotics that often have to translate angular into circular momentum and are full of what we'd call trig functions and ODE's today. You'll love this whether you're doing robotics, or working on prosthetic limbs, as well as the more obvious celestial and navigational applications.
"Al-Jabr" (Muḥammad ibn Mûsâ al-Khwârizmî) is often given as the major example of Islamic contributions to math, with less reference to the Spanish Moor salvation of many Greek discoveries Justin tried to destroy, and nearly none to spherical trig. The author reprises these contributions with notes on how the lunar calendar posed problems whose Islamic solutions contributed to far more than the calendar. I'm not Islamic, but it is interesting to see the system presented in a science frame here in the West once in a while.
There are also some very cool and accurate "corrections" to the history of astronomy. As many of you probably know, Kepler was credited with a lot of work actually done by Tycho Brahe, who's area of expertise was, among many others -- you guessed it-- Spherical trig. Although the theory of ST is deep, vast and ancient, applying it, without Maple, was far from easy, and the top math minds of the ancients were baffled by the "tiny" details (read calculus) when one tried to apply theory to the reality of polygon to sphere. Python programmers will smile at this when considering the brute force needed to x/y/z a asteroid pocked sphere in Maya, vs. using code.
All in all, this book is highly recommended. The other reviewers give the glowing historical value, but I wanted to add another facet-- the fact that this art is far from irrelevant to today's most exciting topics in math in addition to the historical beauty and importance. A related field that might interest you as well (search it on Wiki) is Orthographic or Orthogonal Projection. This "old" cartography art also is being reborn in 2D to 3D and vice versa projection in as far ranging fields as biomedical visualization, galaxy modeling and gaming. The Etymology of some of these terms is fascinating, for example, Ortho-doxy translates as "the straight path to glory!" Certainly in the spirit of this volume, with wonder and beauty as important to the author as the math.
I'm a technical consultant, digital artist and mathematician at ShaderJoes dot com and have nothing to do with this book's author, publisher, or Amazon. My review is solely for Amazon shoppers, and we always buy the books we review here.