Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry by Glen Van Brummelen (PDF)

Description

An unparalleled illustrated history of spherical trigonometry from antiquity to todayHeavenly Mathematics traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. Heavenly Mathematics is illustrated throughout with stunning historical images and informative drawings and diagrams. This unique compendium also features easy-to-use appendixes as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book was written by an author who knew the subject of spherical trigonometry and presented it well. But, it’s not a book for the weak of heart. Some of the symbols date to the Greek days and later, but way before modern technology. I studied celestial and terrestrial navigation and the spherical trigonometry needed for that. If not always understandable, it was easy enough for someone dedicated to the arts of navigation using a sextant and sight reduction tables. This book goes beyond what is needed and answers the “Why that equation?” questions; so it’s quite in depth. It’s a book that takes some study and is intended to be a textbook, including end-of-chapter exercises. It’s not a book that allows one to skim through.

⭐I have just re-read this book as part of a study of non-Euclidean geometry, and I’m more impressed on second reading.This is both a book of history and of math. Both sides of the coin are covered nicely. The historical perspective is absorbing, and math is thoroughly done with proofs given or put in the exercises for every theorem. Which brings up the exercises! These are not at all like almost every other math book I’ve read. These are a guided tour through a part of the subject that draws the reader deeper and deeper into the web of spherical trigonometry until he is caught. Each exercise builds on the previous until the proof or example is completed. Very little of the “plug and grind” type of exercise: each is a true learning experience. They are worth reading even if you do not choose to actually do them. This handholding may not appeal to those with advanced math skills, but I found the approach very refreshing.The math is developed from the standpoint of a person living at the time. You start with the problems that motivated them, and start by constructing your own table of trig functions. Then, the theorems are developed using only what was known at the time. This approach is really interesting: the math and the history hold hands and walk forward together. This gave me a greater appreciation for the tenacity and brilliance of my mathematical predecessors.However, if you are math challenged, you are not forgotten. The sections that dive into math are marked with arrows, and the reader who only wants to read on the history is advised they can skip these parts, along with the exercises. If you do, there are parts near the end that will be more difficult to appreciate.Al in all, I found this book to be a very different approach from that taken by many books. I found both the history and the math illuminating, and wish the author would take a shine to hyperbolic trig. I enjoyed it a great deal and recommend it. It’s a great book!

⭐A great book on a great topic! Recommend this book for all math lovers, cartographers, navigators, and astronomers!

⭐This is one of those books I wish I could get autographed by the author. Why oh why don’t they teach this in high school anymore?

⭐From a 3rd year math major:I would definitely recomend this book to anyone interested in learning about spherical geometry, spherical trig, or anyone interested in mathematics generally. It starts with really good background information and progresses fluidly through spherical geometry, then into spherical trig, and practical applications. It is written in a way that if you just want the overview of the material you can get it, but if you really want to dig in there are numerous proofs and an abundance of in depth material as well. It is also nice that the author distinguishes where the detailed explanations are so you can skip over them if you so choose. There are also exercises at the end of each section that challence your knowedge of what you just read and require you to use some creative mathematical skills as well. This could easily be used as a text book on this material. If you take the shorter route, this book can be read through quite quickly and easily, but if you choose to actually study the information and related proofs you can spend quite a bit of time on this material. I read it through completely once, taking time to understand the proofs and examples, and grasped the bulk of the information. I plan to read through it again shortly and try to make sense of the things I couldn’t get through the first time. The better your beginning knowledge in Euclidean geometry, algebra, and trigonometry, the more you will gain from this book.

⭐This book can’t quite decide whether it wants to be a history of the neglected field of spherical trigonometry, or a textbook on it (complete with derivations and problem sets). Fortunately, it succeeds reasonably well at both. If you’re more interested in the history, you can just skim lightly over the mathematical details and it still reads well. If you want to learn spherical trig, the history may be a bit of a digression but it’s an interesting digression. The later chapters on applications are particularly noteworthy, since few math books nowadays condescend to discuss such mundane matters as how the mathematical methods might get used. An interesting and rewarding book.

⭐Excellent book! This is a fountain of mathematical history, and easy to understand. I have quoted it many times in giving explanations of concepts. It’s a book that is easy to get into, but hard to put down. I do a lot of volunteer work at a museum, trying to explain the science behind popular historical and archaeological finds, to remove the mysticism behind how the “ancients” accomplished important achievements. It’s important to know how the application of basic mathematical and scientific concepts can realize seemingly complex goals. This book shows clear scientific/mathematical ideas and how they influenced history. Plus, it’s a great way to “remember” sphericaI trigonometry, and how it is applied. I hope the author continues with more books in this vein.

⭐The author suffers the same condition as many poor mathematics writers: leave the proof to the reader. The pages are a rollercoaster where he’s either going through a formula with a fine-tooth comb, or making wild assumptions between steps, or rather, giant leaps, leaving the reader confused. I could’ve finished this book sooner, but I spent so many hours going back and forth between diagrams and theorems wondering from which sleeve he pulled his latest result necessary for the proof, that the pages started to crumple.The way he explains the history is actually interesting and easy to follow, which makes me wonder if he didn’t miss his calling as a historian rather than a mathematician.Edit: my bad, I just did some Googling and it turns out he IS a historian. Meaning he has no reason to be proving theorems.

⭐(Written by Brenda W’s husband who likes some of the more obscure topics of mathematics)Not an easy book to review because it is neither a formal, text book primer on the subject or a pure history. It is in fact an interesting compromise between the two ,intermixed with some other minor but relevant topics thrown in for good measure. Heavenly Mathematics could in fact mean anything from, I just want to devote my life to the study of mathematics forsaking all other, to some sort of ‘ breaking the secret code of the Bible’. In fact it does run true to form and gives an interesting general account of spherical trigonometry and enough of how to apply it and how it was applied in the ages before the computer took over everything and people didn’t have to think anymore! Not a book for the absolute beginner since an O Level (I am showing my age) or better still, a second attempt with an E grade, A Level Mathematics is assumed. Without this background the book really would be more akin to Hellish Mathematics, which it most certainly is not.The easiest way to describe its contents is to write them down for all to see and decide whether this is the book for them.1.Heavenly Mathematics, (a sort of very elementary celestial mechanics). 2. Exploring the Sphere. 3. The Ancient Approach. 4. The Medieval Approach. 5. The Modern Approach: Right Angled Triangles. 6. The Modern Approach: Oblique Triangles. 7. Areas, Angles, and Polyhedra. 8. Stereographic Projection. 9. Navigating by the Stars.Especially for anyone whose principle interest might be Astronomy, Astrology or for those who are very adventurous or just tired of living who might dream of navigating a small boat across the Atlantic would benefit by reading this book and absorbing its contents. The book is American Canadian but has a good British English flavour and feel throughout. I have to say that I rather enjoyed it but its style and content would not be for everybody. It depends on your philosophy, how you view this present world and how much you enjoy the obscure!.

⭐The learning curve starts off very steep! But the combination of the history of the development of the art is fascinating. I guess it will not suddenly equip you for a career in the subject, or things associated with it, but it is loaded with examples, and tests for you at the end of each chapter, to see how much of it you have taken in! I cannot say it will help me much with what I was interested in, the “Heavenly” side of the mathematics, but I am glad to have bought it for the interest alone.

⭐Exellent read. Very good to get a grasp of planetary astronomy.

⭐L’idea di “raccontare” la trigonometria sferica è originale. Inquadra il contesto storico in cui è stata usata. Gli esempi di calcolo sono utili ma poi finisce li. L’autore poteva benissimo invogliare il lettore interessato fornendo più esercizi e la relativa risposta. Il rimando alla bibliografia potrebbe sanare le curiosità del lettore interessato ma visto il prezzo si poteva fare di più.

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