Dear Math Sorcerer: We had a whole course in spherial trig in the US Army land survey school. We would go out at night and use our instruments to mesure angles and check our trig calculations. It was one to the most fun schools I ever had. we could predect the locations of stars. take care my friend
I wish math was taught like this more often. The practical applications are more enjoyable and easier to grasp when used in the real world. I remember old books about math teachers that would teach math techniques by measuring trees in in the yard using shadows, engineering techniques by building tunnels in mounds of straw and hay, etc
@@TheMathSorcerer I came to first know about the subject in high school when I used to watch Numb3rs an American T.V. series where F.B.I. takes help of a mathematician to solve various cases.
I already posted this on your previous video on the book that Ramanujan used to learn math, but spherical trigonometry is still taught at nautical academies to prospective navigators. Pilots learn the subject, too. Spherical trigonometry is seriously cool, you can't call yourself a navigator if you don't know this by heart 😄
Two stories where spherical trigonometry played a dramatic role: the spedition of Ernest Shakleton in Antartica and the Apollo 13 mission. In both cases people had to use this math subject to save their own lives, and had to perform calculations by hand.
@@aaryan6019I first Heard of Shackleton in a video documentary, but I have no link to It. en.m.wikipedia.org/wiki/Voyage_of_the_James_Caird there is no explicit mention in this article but this can give an idea; the crew had to use sun and stars to understand their position in this travel.
Back in the '60s, we had a brief intro to spherical trig in 10th grade. It was a combo course of plane & spherical trig and analytical geometry. No calculators except slide rules and really heavy with logarithms. Boy, did I learn a lot in that year.
I'm a Professional Land Surveyor and I had to do quite a bit of Spherical trig during my training. It can come up from time to time on very large projects!
@@trumanburbank6899 A well known book that includes astronomy and navigational math is named "The American Practical Navigator" originally by Nathaniel Bowditch. Most offshore mariners just call it Bowditch. My hard copy is 1000 8"1/2" X 11" pages, so it's a big book. It's a US govt. publication so it's not expensive in hardcover and it's also available as a free PDF.
@@trumanburbank6899 You're welcome. I've had editions from before satellites but when radar was a thing as was LORAN (1942). I've had an edition from when the GPS constellation was being built (1984). The most recent hard cover edition I have is 1995 and I just downloaded the 2019 PDF. I look forward to seeing the changes again. I thank you for asking your question about "...spherical trig textbooks which are disguised..." I think Bowditch is exactly that, and more.
I'm a math teacher and sometimes amateur astronomer. Several years ago I got really into spherical (positional) astronomy, essentially that last chapter in Brink's text. Two good books on the subject are W. M. Smart's "Text-Book on Spherical Astronomy" (Cambridge UP, 1931) and Robin M. Green's "Spherical Astronomy" (Cambridge UP, 1985).
In the Seventies, in India , Spherical Trig was there in my Civil Engineering mathematics. And used in Astronomy and Geodetic survey. We had thought it would be tougher than plane trig. But the formulas turned out to be quite similar to plane trig. 😀
Great video. I have a Plane Trigonometry book, my grandfather's, from the 40's, when he worked at Union Carbide. No cover, just pages of the book. Very delicate.
At a math camp, there were two courses taught by the same professor, one on Hyperbolic Geometry and the other on Spherical Trigonometry. They were meant to be taken consecutively, and were very interesting. This subject is important even to this day, so if you want to become any sort of astronomer or astrophysicist, you definitely need to learn this.
Hey professor! I’ve been watching your content for about two years now and it has been so rough the week before finals. I was in tears last night because i tried to cram for multivariable calculus triple integrals, div and curl, line integrals ect and laplace transformations for diff eq. These are my last math classes thats required. I do not want to stop studying math. You’ve inspired me to start a RU-vid channel and i want to make lecture series specifically on the first two years of math require for stem majors. I never want anyone to feel what i went through again. I’m just a engineering major tired of seeing his friends drop out of stem too. one day i want create my own math text books from pre calc to diff eq and linear algebra and make it free. Math and science are so freaking cool and i want to spread that love i have for the stem fields to everyone!
Another land surveyor here. I have a book in my library called "Spherical Astronomy" by Robin M. Green, Cambridge University Press, 1985. Similar subject as this video with a practical math application leaning towards astronomy. Back in the good old days before GPS we used to do star shots at night which involved setting a transit instrument on a control monument and measuring the angle to Polaris from a reference monument to derive North relative to our monument pair. These days I have a textbook on the Global Positioning System which delves as far into the geodetic mathematics as one cares to go. It's an interesting subject, I'll have to dig it out of the boxes in the garage as soon as I have time, LOL. By the way, as a fellow bibliophile, I love your videos.
Yes, I studied spherical trig as an officer of the Royal Australian Navy in the 1970s learning astronomical navigation. Years later, one of my Sailors asked me to teach it to him. I re-studied, and did that to his and my satisfaction.
Thank you so much! My father loved math and went to the U.S. Merchant Marine Academy where I think he majored in Celestial Navigation. He talked about Spherical Trig from time to time. He might have even used this textbook since he went to college from the late forties to the early fifties.
The subject is sometimes taught within an undergraduate "applied mathematics for engineers" course. The subject appears in a 2-year college engineering technology curriculum.
A great topic. I was first exposed it reading the Bowditch (American practical navigator) in the Navy. You can find both volumes for free online from NGA. Bored on watch as quartermaster and not knowing any calculus at the time, I was trying to estimate the total waterspace assigned to our unit, which although rectangular, the calculation of it was not so simple when you take into account the deformation of the shape as it is superimposed on a sphere. Turns out there are great analytical solutions with Napiers' rules/spherical excess to these types of problems that completely avoid the use of messy double integrals and polar coordinates.
Spherical trig is used in GN&C (guidance, navigation, and control) for aerospace applications. Specifically the more "modern" version that makes use of quaternions.
I have Kells, L, Kern, W, Bland, J. (1942, 1st ed., 5th imp), "Spherical Trigonometry with Naval and Military Applications with Tables". I bought it at a library book sale when I was in High School, and it's great. Just for fun, the contents: 1. Logarithms 2. Review of Plane Trigonometry 3. The Right Spherical Triangle 4. Elementary Applications (e.g. course and distance, Mercator charts) 5. The Oblique Spherical Triangle 6. Applications (find the time of sunrise, time of day, misc exercises Tables of Computed Altitude and Azimuth Lines of Position Circles of Equal Altitudes Aerial Navigation App A - The Mil App B -The Range Finder App C - Stereographic projections, etc App D - Vectors, Relative Movement Index ANSWERS !!! Five Place Logarithmic and Trigonometric Tables (116 pages of tables) Table 1 - Common Logarithms Table 2 - Logarithms of Trigonometric Functions Table 3 - Trigonometric Functions
Most excellent! Thanks for taking the time to make this video. I'd like to find that book. I came across spherical trigonometry while working on a personal project. I used it to determine cardinal East as it rises on the horizon given my location on Earth.
Indeed I have heard of spherical trigonometry. I found a book titled "Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry", by Glen Van Brummelen.The author is described as "coordinator of mathematics and the physical sciences at Quest University Canada and president of the Canadian Society for History and Philosophy of Mathematics." It was copyrighted 2013 and was published by Princeton University Press. I have not worked my way through the book yet. I will do so in the future.
I had occasion to delve into spherical geometry when I had to write programs to trawl geographical coordinates to find points with various bounded shapes, points outside boundaries, distances between points and smallest enclosing circles. And yeah, I found this small but well defined little world of maths I’d never encountered before.
I found a copy of Dr. Bruhns Logarithm Tables. His writing in the preface is much more lyrical. Also found a book of logarithmic sines, cosines, tangents and cotangents. Fun stuff, ingenious stuff they have in there.
One of my favorite books is Burington's Handbook of Mathematical Tables and Formulas, published post-WW2 for the training of new engineers and scientists in and out of the military. It wasn't exactly this book that convinced Dad to change majors but he found it odd that a PoliSci major planning to be a town manager needed to take Drafting and learn surveying. I have this book, my Mom's slide-rule, and my F-i-L's CRC HB of Chem & Phys, inter alia. I taught myself trig to work out bills of materials for Bucky Fuller Geodesic Domes that were never built, so i was starting with Spherical and working backwards! The history of geodesy and related arts in which the curvature of the earth is non-trivial is something of an ongoing interest. When next you're in Boston, we've got a Spherical Trig history site for you.
If you're interested a different - and classical - approach to spherical trigonometry, "Heavenly Mathematics" by Brummelen develops the subject ruler and compass style while discussing the history of Ancient Greek and Arabic astronomy. The book mostly follows Ptolemy's Almagest. I also have the first edition of Schaum's Outline of Trigonometry (1954) which is actually subtitled "Plane and Spherical" - the chapters dedicated to spherical trigonometry were simply cut out at a later edition (I have the 5th edition too).
Had it loosely mentioned in high school. Back in late 80s / early 90s used it in cartography and astronomy course in college. It's beautiful and very useful in this context and even when I used GIS like Esri geoprocessing tools, decades later.
@@thelonegerman2314 Nope, much less than this, just concepts and basic applications, it was in basic years and there was a lot of ground to cover. But maybe for inferring ridge lines (I remember we used polynomial interpolation, so it would be easy to differentiate), I really do not remember.
Those diagrams were made by a draftsman with ink and paper on a drawing table with many unique drawing tools. I did drawings just like that in 1968 in a drawing course for college. And the professional guys at the research center at college made drawings even more perfect. Also, there are some Spherical Trigonometry books scanned for free download from the 1880's and the drawings are equally good. You young whipper snappers would be amazed what we could do back in the days. :)
I have a similar 1940 book that my father learned spherical trig during officer training in the Navy. Myself, I needed it to understand celestial mechanics. Before computers, we had to do calculations by hand to know where to point the telescope.
For my Mathematics undergraduate degree I did my fyp on Spherical Trig. It was called Measuring Heaven & Earth, the Mathematics of traversing the oceans.
I’ve heard of spherical trigonometry because I’m interested in physics history. I first heard about it in reference to Caroline Herschel being a very gifted at spherical trigonometry. Which was a vital mathematics for astronomers, as the book discusses. Caroline did nearly all the calculations and recording while William, her brother, did the observing. This was the sinking duo who first knowingly discovered Uranus.
I also used spherical trig in land surveying classes. The earth is a sphere (or at least a spheroid) and any measurements on its surface, except for very small areas, will need to be treated accordingly.
I did a term paper in my advanced Math 10 class in 1984 on Spherical Trig. My father, a land surveyor, had daily experience with the subject and several textbooks. I didn’t see the subject again until a third-year university geometry course.
I'm actually taking a course on the history of astronomy and gravitational motion. The celestial sphere was conceived in ancient Greece, when the Earth was thought to be the unmoving center of the universe (but obviously still a round sphere, not flat). One could conceive of all the stars in the sky as lying on an invisible sphere that domes over the earth and rotates once every 24 hrs. The Sun, the Moon and the planets were in a special class (ancient Greek "planetos", wandering stars) that sat on different rings within the sphere. This model was used up to Copernicus. The coolest math used by ancient astronomers involving the celestial sphere has to be Ptolemy, who contrived of all kinds of tools to measure star angles and model orbits before people came around to the heliocentric system.
While it is true that this isn't taught much anymore, anyone who has to deal with maps will know the special problems that they present in spherical coordinates. I vividly recall a chemistry lecture where the professor derived spherical coordinates from Cartesian coordinates. It took the entire lecture, and I was in awe when he finished. I have those notes squirreled away somewhere.
Out of curiosity, why did your chem professor do that? Does trig or spherical trig play a role understanding molecular bonds and chemical reactions? Im not trying to be snarky, Ive never taken a chemistry course, so I honestly dont know.
I seem to recall seeing a Chinese university-entrance exam (this would have been about a decade ago now) which featured a lot of pretty intense-looking spherical trig questions. So maybe it’s still a big focus of secondary-level maths there, or was until quite recently? Maybe someone who has experience of the system out there could let us know?
What a surprise to hear this is no longer taught in schools. When I was 14 or 15, in the late 1970s, we learned Spherical Trigonometry using an extremely old textbook by an English academic called Isaac Todhunter. I found the whole subject to be quite eye-opening at the time. What a shame it is no longer taught to teenagers.
As a high school student who loves astronomy, this topic is so freaking important to us! All the time we have to calculate the position of the Sun or any star im general, we use it!
I grew up with spherical trig; used it to find my way across the ocean before handheld GPS. I learned from my father how to navigate with a sextant and the old government printing office tables for sight reduction. I use it now to make sundials, astrolabes, and where to point my telescope.
I am a chess player; used to prepare for tournaments & take the game very serious. I can attest to the fact that playing chess can reach levels of intensity that are insane & burns massive amounts of calories and can be particularly hard on the body, especially the nervous system. Physical training has been tenet of professional chess at least since Botvinnik’s Russian school for this very reason. Competitively, physical conditioning can give one an edge if long games test endurance. Now, I don’t find this so much with math. Here is my theory: on a visceral level chess is a fight against an opponent. On a deep level this translates into greater allocation of energy resources to the brain & nervous system. We get greater stress, cortisol, & autonomic activity paired with rushes of dopamine. It’s a fight.
In the 1950s Spherical Geometry was a popular optional 'O' level subject. It aimed to give a mathematical background, on such items as Napier's Rules, to those preparing for entry to the Mechantile Marine at the deck cadet/apprentice entry point. Changes to technology and the contraction of the Merchant Navy led to a reduced popularity for this subject. Up until the early 2000s I was still running occasional courses on the subject for those starting to work on navigation systems and orbital dynamics at Higher Graduate level. I tended to have very bright, well qualified students who grasped the concepts very easily. Questions on Great Circles were frequently used on Selection Boards for personnel applying for jobs in these spheres of work.
I have a book from 1942 called Plane and Spherical Trigonometry by Paul R. Rider. It has a nice faded blue cover. There is a book that has been on my wish list for about 10 years called Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. It always looked interesting its just a little pricey for me and I already have so many books. It looks like it covers a lot of the history of it which I always enjoy. If I can get it at a lower price I'll probably grab it. I noticed in a previous video you had The Probability Tutor by Carol Ash on your lower book shelf. I was wondering how you liked that book?
Some of the very best books are quite small . Geodesic Geometry and mathematics I believe I recall reading draws upon both Ballistics math (?) and Spherical Trigonometry . ( A very long time ago calculated a nine frequency ( icosahedral ) Geodesic structure with proof because I needed to understand Synergetics better ) . Am not mathematically inclined and this was steeply remedial for me . I used a programable calculator to crucially carry the decimal point ( school had been a disaster ) . So now one realizes it has changed the way one thinks about things forever and supplied a measure of confidence .
Heavenly Mathematics is a modern popular book that covers it. It mainly fell out of favor when navigation didn't require it. With GPS, it's really not used now.
Garrett birhoff’s “ a survey of modern algebra” was published in the 1930’s . That book if I saw properly was published in 1942. My impression is that during that time period a number of things that were once considered separate subjects are now taught as a special case of vector calculus, vector spaces, or general algebra.
Back in the early '60s, when I was learning trigonometry, I bought off a supermarket rack a book in the Barnes and Noble College Outline Series entitled Plane and Spherical Trigonometry by Kaj L. Nielsen and John H. Vanlonkhuyzen. The presence of the spherical trigonometry section made perfect sense to me since we live on the surface of a sphere, real life trigonometry problems for anyone navigating a ship or plane over some substantial portion of the earth's surface require spherical trigonometry. I never really got into it, but simply used the first half of the book for reference. I can't find the book now. I probably unhauled it sometime in the last 20 years since I didn't really need it. Most of the trigonometry reference I used it for 60 years ago I can pull out of my head; the rest I can look up on Wikipedia in seconds. The likelihood of my needing to pick up spherical trigonometry for navigation at this point in my life is minimal, though I'm sure I could do it if I were motivated. However, using an app for that would probably be quicker and more reliable. I would also note that some of the narrative I usually heard about the parallel postulate and how surprising the existence of non-Euclidean geometries was frequently rang a little hollow when I recalled how spherical trigonometry was in common use by many non-mathematicians for everyday applied problems in navigation.
We studied some spherical geometry in highschool astrophysics class. I think this isn't tought in a maths degree because most of the knowledge is probably contained in complex analysis/ differential geometry. We also don't really study normal highschool geometry in uni anymore, I guess for similar reasons.
Mass Maritime grad and trig tutor. Awesome refresher! I think we learned from Spherical Trigonometry with Naval and Military Applications. I wish I kept the book. Nautical miles, kids.
The first words spoken by my lecturer in Principles of Navigation at Leith Nautical College to the class in 1976 were: "For the purposes of navigation, we can consider the Earth to be at the centre of a sphere of infinite proportions." I wasn't keen on Maths at the time, but these words got me hooked and I loved this subject.
My step father told me stories of his Navy days when he was a quarter master on an aircraft carrier. He said he barely made it through high school and basic algebra. He said that he was befriended by a highly educated young officer. The officer helped him learn math up through Spherical Trigonometry. This helped him in his job to track and plot position the fleet and other ships of interest. The stories gave me hope I could attain similar levels of understanding of math. I was in middle school at the time. I went on to learn what is now called STEM. I really enjoy your videos and especially this one as it takes me back to a seminal and sentimental time in my life. I am 57 now. Thanks
Dad had spherical geometry in conjunction to learning celestial navigation. Before GPS celestial navigation was based on spherical geometry. In practice, tables and construction were used to solve for position.
My dad taught me planar trig when I was 8-9. I barely remember it, and Google stuff when I need it, but I remember him saying how spherical trig is the really difficult one...
When I was about 20 I deduced that there must be 12 universes touching ours, and then I learnt later that newton deduced the same thing based on his work in stacking spheres
We covered a bit of spherical trig when dealing with multivariable calculus. I don't remember a lot, but I do remember that it wasn't as bad as I thought and lots of spherical things can be greatly simplified with trig.
Seeing the thumbnail, I was expecting this to introduce something analog to regular trigonometric functions but based on steradians instead of normal (flat) angle measures. During introduction I switched to thinking it will be about non-euclidean trigonometry.
Just as an example of another unusual and interesting book I saw in 2009 at NYU: "Geometry with an Introduction to Cosmic Topology" by Michael P. Hitchman. 🙂
It's pretty funny that you mention how math makes you hungry, because I was doing some review for my upcoming analysis exam and I decided to take my math snack break as I watch this video!
Put that book back on the shelf and instead get a book on celestial navigation. That's where this stuff is taught. We still use spherical trig, but it's all part of navigating the oceans. To find a school that taught this you would need a school that trains ships' officers. Yes, we use GPS for most of this stuff, but you still need to know how to do the sightings and perform the calculations for those times when the computer is down, the batteries go dead, there's a magnetic surge due to solar storms, etc.
When learning celestial navigation (finding our location on the ocean by sighting stars), the basics start out with this. But rather than have to perform all the calculations by hand, often we use 'sight reduction tables', which are pre-calculated solutions for distance and bearing between our location and the geographic position of the sighted body. Of course now days, software and calculators can be used, but still something for using the Nautical Almanac and tables. :)
I'm right now in the 11th grade and we started with this topic 3 weeks ago in the advanced math class, our teacher said it is the first time he does this topic with his students, because he didn't learn it at school but wants that we learn it.
Current World Chess Champion Magnus Carlsen Plays football to be in Shape besides liking the Sport. 😊 Plus there are Chess GMs who are Chubby and even Thicc.
I have encountered allusions to spherical trigonometry and quite a lot of work from the field or adjacent to it, but never heard it called this until today. Glad to be able to put a name to the work I've been researching for fun
Probably the diagrams would have been hand-drawn with ruling pens. I was taught how to do it in my graphic design schooling in 1999, might have been one of the last classes in that program to still learn how to do that since computers were just about to take over and the skill of using a ruling pen was largely useless by then. Makes very clear and clean lines with precise width that doesn't vary. Combine with stencils, rulers and compass and you can draw any complex design you like. Using a ruling pen was incredibly difficult and required a very steady hand and rigid attention to detail and process.
I really love your videos, they really relax me. I am math book collector as well. Regarding the eternal debate about books having answers to the excises, I have noticed that the old Romanian math books (those are quite similar in structure and content with Russians ones) do provide answers in 95% of cases to all exercises.For me it really makes sense as it allows you to check your result(s) with the ones of the author. I think all books should follow this rule..after all one is innocent until proven guilty :)))
I remember reading a book called "Gödel's Proof" by Nagel and Newman. I bought it in Heffer's in Cambridge when I was seventeen It used the three geometries Euclidean, hyperbolic and spherical as analogies for axiomatic systems. Spherical geometry was barely mentioned but that was my only contact with it. 43 years ago.
@@thelonegerman2314 No in the book which I still have on two pages in this thin popularisation of Gödel's Proof in the chapter on "The Problem of Consistency" There is reference to Riemannian geometry being reducible to the geometry of a Euclidean sphere. I in my seventeen year old mind see that as spherical geometry. I recommend reading the book it is short and clear explanation of Gödel's Proofs.
@@thelonegerman2314 No I mean the reduction of non-Euclidean geometry to Euclidean via spherical Euclidean geometry. The book isn't even 100 pages long and aimed at the general public. It referenced spherical Euclidean geometry and that was my sole encounter with it.
Yes, I've not only heard of spherical trigonometry, I took a class on it. Well, not specifically that topic as such, but it was a course on using the sextant for navigation and it's basically all celestial sphere trig.
