I've been relearning trigonometry lately by myself for navigation and astronomy; not for work, just curiosity I guess. One book I've really enjoyed is Heavenly Mathematics by Van Bremmelen. It's a spherical trig textbook, but it's written by a math historian who describes how trigonometry was gradually developed over human history and he discusses its early proofs, methods and applications. I have to confess that the historical approach has really helped me develop a more complete mental picture and appreciation of the math itself. Understanding the "how" and "why" of its development, and seeing the early practical need and implementation for some of this stuff has made the topic a lot more engaging.
Is spherical trig still a thing? Calculation is so cheap now. If you want to find the spherical distance between cities from first principles, it's easiest to convert to 3D rectangular coordinates and find the angle ABC from the vector dot product, where A and C are the cities and B is the center of the earth. Same for other types of navigational calculations like headings. But, I've never looked into it really.
Yep, it's mostly used in for navigation, especially as a backup in case GPS fails (in the case of a lightning strike on a sailboat, for instance, probably all your electronics are toast). The military still uses it with slide rules/log tables, in case of jamming, etc. It kind of "disappeared" sometime after WWII, and there aren't many new books on it, but you'll see it come up in computer graphics and animation quite a bit, and I'm pretty sure it's used extensively in surveying still.
This is a great book if you already know good amount of Math. It helps you fit things into a bigger picture. Really appreciate the fact that something like this exists.
This book is really fascinating because it contains a surprising amount of Soviet ideology. The authors repeatedly state that mathematics is posterior to the material world, not prior to it. That is, mathematics is just the observation of regularity in the world, particularly those discovered by people working to create things. Contrast this with the still heavily idealistic world of western mathematics, where mathematicians are more likely to sympathize with the notion that numbers are real things somewhere out there whose structures the real world supervenes upon in some way.
Interesting stuff!
Even though I favor the Soviet view of mathematics personally (I do not think numbers "exist" out there independent of the material world), I think this approach hampers the didactic goals of the text and probably hurt Soviet mathematics as well. The examples in the text are all highly concrete (literally things like rubber mats when discussing curvature). This very down to earth style makes the abstract notions of curvature in other contexts (for example, general relativity) more difficult to grasp, in my opinion.
On the other hand, some people prefer strong, material, examples of mathematical ideas. This book definitely provides that. The section on affine maps in terms of fixing the plane of a surveillance airplane photograph is beautifully concrete.
I'm looking for mathematics books that take the time to explain with words and sentences what is actually going on when they introduce a new theorem, something that focuses on meaning.
This is a great series. I was awed by it in high school. Some parts were too advanced for me (and probably still are), but I got a lot out of other parts.
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[ 2.8 ms ] story [ 42.1 ms ] threadhttps://archive.org/details/MathematicsItsContentsMethodsAnd...
It compounds, as you learn more history, you're able to connect more and more dots, and learning new information becomes easier on average.
Interesting stuff!
Even though I favor the Soviet view of mathematics personally (I do not think numbers "exist" out there independent of the material world), I think this approach hampers the didactic goals of the text and probably hurt Soviet mathematics as well. The examples in the text are all highly concrete (literally things like rubber mats when discussing curvature). This very down to earth style makes the abstract notions of curvature in other contexts (for example, general relativity) more difficult to grasp, in my opinion.
On the other hand, some people prefer strong, material, examples of mathematical ideas. This book definitely provides that. The section on affine maps in terms of fixing the plane of a surveillance airplane photograph is beautifully concrete.
On Teaching Mathematics by V.I.Arnold - https://archive-dsweb.siam.org/The-Magazine/All-Issues/vi-ar...
Mathematics and Physics: mother and daughter or sisters? by V.I.Arnold - https://ufn.ru/en/articles/1999/12/c/
Anyone knows if such books exist?