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'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 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 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).
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 TH-cam 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!
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.
The subject is sometimes taught within an undergraduate "applied mathematics for engineers" course. The subject appears in a 2-year college engineering technology curriculum.
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
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. 😀
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.
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.
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.
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.
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.
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.
I watched this a few days ago and was able to track down a copy for $12. I can’t wait for it to come. Thanks for the info and book. I wouldn’t have known it existed with out you.
@@TheMathSorcerer it is. Trig is the most beautiful math in my opinion. I know in my time the what, why and how’s of math being taught has changed a bit. It’s crazy to realize how much more is out there I am completely unfamiliar with. The evolution and older types really intrigue me. Side bar- I’m sure you are already familiar with Tom Lehrer, but if not, treat yourself and look up his song “new math”. He was a pretty cool guy and I really enjoy that song. Thanks again!
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.
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.
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.
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.
Very important for understanding spherical harmonics, which is totally essential to understand subatomic behavior and the quantum nature of chemical bonding, as well as Photonics.
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.
Spherical Trig is the basis of celestial navigation which was and to some extent still is studied in nautical colleges around the world by future ships deck officers. As a young merchant ship officer in the early 70’s predating satellite navigation we lived and died by celestial navigation. It is still a prerequisite to licensing.
Your post took me down memory lane. A retired Master Mariner myself, I remember studying spherical trigonometry as a cadet. It was essential for calculating sights and studying Principles of Navigation.
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.
My mom has two Trig books from college, basic and this one. I borrowed and played around with them after I finished college, but I only did a one-time read and work-through so I could get the gist of it . . . And, to be blunt, that was the very early 90s, by which point 3d computer modeling was a real thing. But I do like the idea of doing math for math's sake, just to keep brushed up on the rusty areas I don't use as much anymore, and to keep my brain in good working order. Thanks!
This book and your description remind me of marine and aviation navigation prior to GPS. Many years ago, I solved a system of spherical trig. equations for maritime great circle sailing. I used a handheld calculator and it still took hours. For the last six or more decades GPS provides solutions continuously.
Thank you. I took a math course my senior year of high school that included spherical geometry and trigonometry. There were no college or university courses in either area of mathematics at the various universities and colleges I attended. It is still used in surveying, navigation, plotting trajectories of rockets, artillery shells, satellites, etc.
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.
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.
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.
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.
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.
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
As @Rick Wilson said, this is taught to military land surveyors. This was used extensively to construct a highly accurate network of points (first order points) when surveying large areas. This network would then be filled in with a less accurate network of points (second order points) and so on until there was a sufficiently dense network.
was one of the books i promised to revisit when in school would have forgotten without your tube thanks of course i do not recall why i needed it so bad
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).
I learned spherical trigonometry while sailing around the Caribbean on a sailboat. It is the mathematics behind celestial navigation at sea. I took along a book my father used when studying navigation in the Army Air Corps during WWII, and I made a study of it over a period of a couple of months. We had three GPS units aboard as well as a sextant, navigation tables, and a nautical almanac. On passages, we would make celestial fixes and compare them to GPS fixes. As a result, if a lightning strike ever knocked out the electronics on a long passage, we knew we would be able to navigate. The study of spherical trig and Bowditch while cruising around the islands allowed me to understand the math and the approximations behind the navigation tables used in celestial navigation. It also gave me an appreciation for the power and convenience of GPS: four star sights and a half-hour of math could fix our position within a mile or two, and a glance at the GPS could fix our position within a boat length. (Although, especially in the Caribbean, most navigation charts were less accurate than the GPS.)
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.
Yes, I had read of the subject of Sphreical Trigonometry while reading the Horatio Hornblower novels by C.S. Forester. Spherical trig was something he needed to master to navigate in the 18th century. This was the first video I have seen that mentioned it and to be fair, I was not actually looking.
Spherical trig is used in GN&C (guidance, navigation, and control) for aerospace applications. Specifically the more "modern" version that makes use of quaternions.
Yes, I have taken it back in the University. I have a brief introduction into that when I was in college and had a deeper continuation of it in my Masters classes ... 2001 and 2009 respectively.
@The Math Sorcerer: I learned on my own Spherical Trigonometry because I always liked Celestial Mechanics. I obtained an online copy of Spher trig. With miltary and naval applications. All these texts are from the turn of the last century: 1880's to 1945. I even got one on Spherical Geometry and astronomy by several German and Hindu professors. These subjects are still taught in India and the East.
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.
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.
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!
Very, very interesting, thank you! Spherical trigonometry is the foundation of incredibly important things such as navigation. I wonder if fluency in this subject is however also critical or at least very helpful in other subjects as well such as physics. I say this because so many of the concepts we learn are in one dimension when in actuality they most frequently occur in three. If we turn to Quantum Mechanics, sometimes more complex problems are collapsed to one dimension by viewing the dynamics of the system through the lens of the norm of a radius, hence one dimension, as opposed to radial movement and scattering that in reality occur in three dimensions. Given the spherical or at least "spherical in a moment" nature of so much in nature, I'm thinking this is perhaps a "must-have" skill for many of us!
Spherical trigonometry can be used in the design and analysis of bevel gears. This was done by an engineer I worked with who learned spherical trig in the Navy, during WW-II
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. :)
"The VNR Concise Encyclopedia of Mathematics" is my goto for traditional math. Covers math pedagogy up thru middle college level for math majors, covering all elementary and practical topics as well as intros to senior level topics like functional analysis. Has its own chapter on spherical trig ... so yes I knew this topic existed. "Mathematics Form And Function" is my goto for senior level maths. "Structure And Interpretation of Computer Programs" is my goto for computer science. I also have the printed version of the CRC Concise Encyclopedia Of Mathematcs.
In the 1980s, I worked for a company that had a lot of retired military officers. We had a technical library in the building. Some of the titles were donated by employees who had acquired them along the way in their education or career path. As someone who loves to spend a lot of time doing math, I looked to see if there were any math books in QA (if they used Library of Congress system) or 510 (if they used Dewey Decimal) of interest to me. A book on Spherical Trigonometry caught my eye. I paged through it, but it was totally based on the dimensions of Planet Earth. I wanted a more theoretical treatment, with no flattened poles and north really at (0, 0, 1) in a one-unit radius sphere, not like our magnetic North Pole. I figured that one of our military veterans had studied from this book while in the service or preparing for it. It looks like the book you have has both the theoretical treatment I'd prefer and the more practical course of study for military troops and others who need it to do their jobs.
There is a selective high school program in the States since 1959, the "Summer Science Program". The astrophysics division deals with determining the orbit of asteroids. We were given an accelerated introduction to spherical trigonometry as one of the methods to complete our task. I remember getting strong headaches, as I usually do when I learn something new, but oh how fulfilling it was when I got the hang of it! Spherical trigonometry is a terrific subject for the visually-oriented math and physics lovers.
This is also known as, "3D Trigonometry". This involves extending the unit circle to 3 dimensions, so therefore, you will need 2 angles to input to get your trigonometric ratios. The first angle is usually called, "Theta", and the second is called, "Phi". It could be taught in multivariable calculus.
I remember it was covered in high school math in the mid 1970s. I still kinda remember how to convert polar coordinates to spherical coordinates. This type of math was covered and tested for on the final exams in Canada during that time period. I was born in 1960.
@@thelonegerman2314 I'm a old man, I can't remember all the funky names they called things back in Junior/Senior High School !! That was back in the 1970s!! Elementary School was in the late 1960s !!
Cylindrical geometry is also an interesting topic, useful for calculating forces and pressures inside of fluid couplings, torque converters, etc. (Using rectangular coordinates does not work.)
“The American Practice Navigator” aka “Bowditch” Still updated and in print today and free on line. First published in 1802 and written by Nathaniel Bowditch. This is where I learned about spherical trigonometry and celestial navigation. Very interesting history and subject in general:) Bowditch is was a fascinating and brilliant mathematician that was a major contributor to tons of stuff we take for granted today.
I studied up on spherical trigonometry to write a sight reduction program for my programmable calculator, to use with my sextant. (just for fun. You could purchase a pre-programed navigators calculator at that time). I was crew on a charter yacht back then-- before satellite navigation and GPS. I used an out of print copy of the Bowditch's American Practical Navigator that I found at the New York Public Library. The coastal navigation volume was still in print but I couldn't find the volume covering Celestial Navigation at the time--and that was the one I needed to understand the spherical trig. It's a pretty cool study! You can do the math to find your latitude by measuring the height of the sun above the horizon, using plane geometry. To find your Longitude by taking sextant sights of the sun, moon and stars at other times of day--you need to use Spherical Geometry. (Actually, by using various tables, it can be done by the non-mathematical navigator without understanding the principles). But it is much more fun to actually understand how it all works!
Everyone's already talked about naval and civil uses for spherical trig, but I figured I'd shine light on another use: astronomy! In observational astronomy, we deal with many different spherical coordinate systems, and spherical trig is indispensable for coordinate conversions. It's also extremely useful for solving problems like the distance between two stars, "A star is directly overhead in Naples at [this time] and later directly overhead in Mexico City at [this time]. What is the greatest latitude at which it will be directly overhead?", And considering things like the motion of the stars, planets, and sun in our sky.
For my Mathematics undergraduate degree I did my fyp on Spherical Trig. It was called Measuring Heaven & Earth, the Mathematics of traversing the oceans.
If this book was first published in 1942, according to the copyright law, the copyright has expired and the book is now in the public domain. You should scan the book and make it available for everybody, so that the book is not lost. Also, you should send a scan of the book to Project Gutenberg for them to publish it on their website.
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’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 got interested in Spherical Trigonometry in the late 1980's as a result of reading material by R. Buckminster Fuller. Mr Fuller was trained by the U.S. Navy and worked with a boat during WW2. Mr. Fuller used Spherical geometry in his Geodesic Dome calculations and published some educational material on Spherical geometry in his 2 volume book 'Synergetics'. I located one of Mr. Brink's books at a local University library. Most University libraries have very few books on Spherical Trigonometry but there is some material in books on Geodesy and in New Foundations for Classical Mechanics by David Hestenes. I've seen that thin 1942 book at another University library that has another book by Mr. Brink that seems to be based on the 1942 book but printed especially for sailors. That library put most books from the 20th century in a high density archive off campus so it can't be stumbled across anymore.
I used to like maths until first year uni. Second semester maths had over 2000 students and ony 3 copies pf the text book in stock at the uni bookshop. Enquiries with the publisher found the book was out of print with no plans to do another print run. The only alternative was for the 2000 odd students to use the 4 copies available in the uni library. They did not change to a text that was readily available. The result was a very high failure rate, including me. That was the end of maths for me just as it was getting really interesting. I actually used spherical trigonometry quite a bit in my job. Fortunately I had already got that under my belt and I was very used to thinking in 3D.
It was not so easy to find references. I can relate to it also. Even if budget was not a problem , availability is already a huge one. No wonder, why some become math book collector. I wonder why the book are not printed anymore, making them so scarce, hence difficult to find. Could it be because of lack of sales on previous edition?
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.
i'm currently enrolled in a review program for civil engineering in our country, and surprisingly, this is one of the topics! really had fun with it because it's new for me. i didn't know this isn't common math anymore :(
I swear I stumbled across that exact book in my high school library (back in the 80s). Somebody must have donated it. I remember paging through it and looking at the diagrams. In the days before GPS spherical trigonometry was fundamental to navigation.
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?
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.
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 have an old Schaum's Outline Series math book by Frank Ayres that has a section covering spherical trigonometry. Of course Schaum's books are noted for having many worked problems, and accurate answers to all the exercise sets.
I don't know about schools these days, but when I was in high school trig in the late 90s we did a unit on spherical trigonometry. I assumed that college mathematics would go deeper into it, but I was a psych major so the only math classes I needed were statistics classes, which actually have been very useful in my life. I've been meaning to learn more math on my own, I never got into calculus, or even pre-calculus for that matter. But spherical trigonometry was helpful in combating the "arguments" of flat earthers.
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?
Hey! You just made me dig out my copy of "Plane and Spherical Trigonometry" McGraw-Hill 1934! I had to get out the box of math books I could not fit on my bookshelves... Thanks!
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.
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.
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.
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
Wow!!!
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
The Earth is approximately a sphere. This maths need to be taught - to anyone who lives on Earth.
@@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.
@@mrtienphysics666 a flat sphere though, right?
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!
oh that's awesome!!
I was just wondering if there are spherical trig textbooks which are disguised as such by the use of "Surveying" in the book title.
@@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.
@@beachbum77979 Just took a look at that book (Bowditch). Wow. What an amazing book. Thank you.
@@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 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 😄
very cool:)
In your perspective could a person pick up Spherical Trig immediately after Plane Trig?
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.
@@marcopaolovaleriovezzoli5776 Ooo Do you have any more details you could share? Perhaps a link to a website?
@@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 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).
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 TH-cam 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!
All the best brother
make your dreams come true, i believe in you
Fantastic idea!!! Do it!!! Thanks you so much for your passion and drive, the world will benefit from it.
People who know Spherical Trig
1. Professional Surveyors, Navigators, and Astronomers
2. People really into Kerbal Space Program
We use spherical trigonometry in orbital mechanics. I used it when designing a satellite constellation for my senior project!
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.
The subject is sometimes taught within an undergraduate "applied mathematics for engineers" course. The subject appears in a 2-year college engineering technology curriculum.
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra??
@@thelonegerman2314 No, it is the Riemannian geometry of surfaces with positive curvature.
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
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. 😀
in surveying we study photogrammetry in advanced now
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra??
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.
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.
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 really like this book. I love the way it's written - everything is so clear and easy to understand. Thanks for sharing this priceless beauty!
☺️☺️
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra Using The Triangle Equality ??
@@thelonegerman2314 No, absolutely not. Just like elementary geometry is not the same as analytic geometry.
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.
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.
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.
I watched this a few days ago and was able to track down a copy for $12. I can’t wait for it to come. Thanks for the info and book. I wouldn’t have known it existed with out you.
Wow that is awesome
@@TheMathSorcerer it is. Trig is the most beautiful math in my opinion.
I know in my time the what, why and how’s of math being taught has changed a bit. It’s crazy to realize how much more is out there I am completely unfamiliar with. The evolution and older types really intrigue me.
Side bar- I’m sure you are already familiar with Tom Lehrer, but if not, treat yourself and look up his song “new math”. He was a pretty cool guy and I really enjoy that song.
Thanks again!
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.
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.
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra Using The Triangle Equality ?? Or Multivariate Calculus?
I was introduced to spherical trig in an introductory astronomy course for astronomy majors. The text we used was Spherical Astronomy by W M Smart.
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.
The figures that you see in the book are done on a drafting table with straight edge, dividers and compass. I used to teach these methods.
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.
YOu might want to contact the Internet Archive to see if they'd like to add the book (digitally) so others could enjoy it.
Very important for understanding spherical harmonics, which is totally essential to understand subatomic behavior and the quantum nature of chemical bonding, as well as Photonics.
Excellent - just pulled out my 1964 copy of Principles of Marine Navigation (D. A. Moore). Back to rehab math! 👍
Nice 👍
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.
Do you Mean NP Completeness of Computational Sets??
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra Using The Triangle Equality ??
@@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.
Spherical Trig is the basis of celestial navigation which was and to some extent still is studied in nautical colleges around the world by future ships deck officers. As a young merchant ship officer in the early 70’s predating satellite navigation we lived and died by celestial navigation. It is still a prerequisite to licensing.
Your post took me down memory lane. A retired Master Mariner myself, I remember studying spherical trigonometry as a cadet. It was essential for calculating sights and studying Principles of Navigation.
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.
My mom has two Trig books from college, basic and this one. I borrowed and played around with them after I finished college, but I only did a one-time read and work-through so I could get the gist of it . . . And, to be blunt, that was the very early 90s, by which point 3d computer modeling was a real thing. But I do like the idea of doing math for math's sake, just to keep brushed up on the rusty areas I don't use as much anymore, and to keep my brain in good working order. Thanks!
its a part of Space and Tech 3rd term lesson.Replaced by W. M. SMART's Celestial Mechanics
This book and your description remind me of marine and aviation navigation prior to GPS. Many years ago, I solved a system of spherical trig. equations for maritime great circle sailing. I used a handheld calculator and it still took hours. For the last six or more decades GPS provides solutions continuously.
Thank you.
I took a math course my senior year of high school that included spherical geometry and trigonometry.
There were no college or university courses in either area of mathematics at the various universities and colleges I attended. It is still used in surveying, navigation, plotting trajectories of rockets, artillery shells, satellites, etc.
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.
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.
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra??
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra Using The Triangle Equality ??
@@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.
I accidentally went from 85kg to 65kg when i started pushing math really hard. So this resonates with me.
Nice
What’s a mathematicians favourite exercise?
‘Curl’ing a Vector Function
So you didn't eat when you were hungry?
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.
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.
a similar thing happened in my physics class as well
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.
Its not forgotten I learned it in college about 8 years ago.
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
This video brings to mind the lost calculator -- the slide rule. How about a video on that subject?
As @Rick Wilson said, this is taught to military land surveyors. This was used extensively to construct a highly accurate network of points (first order points) when surveying large areas. This network would then be filled in with a less accurate network of points (second order points) and so on until there was a sufficiently dense network.
was one of the books i promised to revisit when in school would have forgotten without your tube thanks of course i do not recall why i needed it so bad
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).
I learned spherical trigonometry while sailing around the Caribbean on a sailboat. It is the mathematics behind celestial navigation at sea. I took along a book my father used when studying navigation in the Army Air Corps during WWII, and I made a study of it over a period of a couple of months.
We had three GPS units aboard as well as a sextant, navigation tables, and a nautical almanac. On passages, we would make celestial fixes and compare them to GPS fixes. As a result, if a lightning strike ever knocked out the electronics on a long passage, we knew we would be able to navigate.
The study of spherical trig and Bowditch while cruising around the islands allowed me to understand the math and the approximations behind the navigation tables used in celestial navigation. It also gave me an appreciation for the power and convenience of GPS: four star sights and a half-hour of math could fix our position within a mile or two, and a glance at the GPS could fix our position within a boat length. (Although, especially in the Caribbean, most navigation charts were less accurate than the GPS.)
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.
In the 70's I frequented Holmes Bookstore in Oakland, California. So many old books! I really miss that place.
Yes, I had read of the subject of Sphreical Trigonometry while reading the Horatio Hornblower novels by C.S. Forester. Spherical trig was something he needed to master to navigate in the 18th century. This was the first video I have seen that mentioned it and to be fair, I was not actually looking.
Spherical trig is used in GN&C (guidance, navigation, and control) for aerospace applications. Specifically the more "modern" version that makes use of quaternions.
Yes, I have taken it back in the University. I have a brief introduction into that when I was in college and had a deeper continuation of it in my Masters classes ... 2001 and 2009 respectively.
@The Math Sorcerer: I learned on my own Spherical Trigonometry because I always liked Celestial Mechanics. I obtained an online copy of Spher trig. With miltary and naval applications. All these texts are from the turn of the last century: 1880's to 1945. I even got one on Spherical Geometry and astronomy by several German and Hindu professors. These subjects are still taught in India and the East.
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.
Please tell us what NGA is an acronym for.
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.
Project Gutenberg has online a book on spherical trigonometry by Todhunter.
Also, have you ever looked into Bowditch's American Practical Navigator?
I haven’t I should check it out !
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!
haha
Very, very interesting, thank you! Spherical trigonometry is the foundation of incredibly important things such as navigation. I wonder if fluency in this subject is however also critical or at least very helpful in other subjects as well such as physics. I say this because so many of the concepts we learn are in one dimension when in actuality they most frequently occur in three. If we turn to Quantum Mechanics, sometimes more complex problems are collapsed to one dimension by viewing the dynamics of the system through the lens of the norm of a radius, hence one dimension, as opposed to radial movement and scattering that in reality occur in three dimensions. Given the spherical or at least "spherical in a moment" nature of so much in nature, I'm thinking this is perhaps a "must-have" skill for many of us!
Spherical trigonometry can be used in the design and analysis of bevel gears. This was done by an engineer I worked with who learned spherical trig in the Navy, during WW-II
We appreciate how much information we receive from videos like this. May God bless you no matter what.
There are several books on Spherical Trig offered on Amazon, some of which are recent; so the subject is hardly forgotten.
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra Using The Triangle Equality ??
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. :)
"The VNR Concise Encyclopedia of Mathematics" is my goto for traditional math. Covers math pedagogy up thru middle college level for math majors, covering all elementary and practical topics as well as intros to senior level topics like functional analysis. Has its own chapter on spherical trig ... so yes I knew this topic existed. "Mathematics Form And Function" is my goto for senior level maths. "Structure And Interpretation of Computer Programs" is my goto for computer science. I also have the printed version of the CRC Concise Encyclopedia Of Mathematcs.
In the 1980s, I worked for a company that had a lot of retired military officers. We had a technical library in the building. Some of the titles were donated by employees who had acquired them along the way in their education or career path. As someone who loves to spend a lot of time doing math, I looked to see if there were any math books in QA (if they used Library of Congress system) or 510 (if they used Dewey Decimal) of interest to me.
A book on Spherical Trigonometry caught my eye. I paged through it, but it was totally based on the dimensions of Planet Earth. I wanted a more theoretical treatment, with no flattened poles and north really at (0, 0, 1) in a one-unit radius sphere, not like our magnetic North Pole. I figured that one of our military veterans had studied from this book while in the service or preparing for it.
It looks like the book you have has both the theoretical treatment I'd prefer and the more practical course of study for military troops and others who need it to do their jobs.
There is a selective high school program in the States since 1959, the "Summer Science Program". The astrophysics division deals with determining the orbit of asteroids. We were given an accelerated introduction to spherical trigonometry as one of the methods to complete our task. I remember getting strong headaches, as I usually do when I learn something new, but oh how fulfilling it was when I got the hang of it! Spherical trigonometry is a terrific subject for the visually-oriented math and physics lovers.
This is also known as, "3D Trigonometry". This involves extending the unit circle to 3 dimensions, so therefore, you will need 2 angles to input to get your trigonometric ratios. The first angle is usually called, "Theta", and the second is called, "Phi". It could be taught in multivariable calculus.
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra Using The Triangle Equality ?? Delta
I remember it was covered in high school math in the mid 1970s.
I still kinda remember how to convert polar coordinates to spherical coordinates. This type of math was covered and tested for on the final exams in Canada during that time period. I was born in 1960.
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra Using The Triangle Equality ??
@@thelonegerman2314 I'm a old man, I can't remember all the funky names they called things back in Junior/Senior High School !! That was back in the 1970s!! Elementary School was in the late 1960s !!
I have learned it on my own! It's a nice subject but I'd like to apply in navigation but my engineering degree is far away from that. Nice video Sir!!
Cylindrical geometry is also an interesting topic, useful for calculating forces and pressures inside of fluid couplings, torque converters, etc. (Using rectangular coordinates does not work.)
“The American Practice Navigator” aka “Bowditch”
Still updated and in print today and free on line. First published in 1802 and written by Nathaniel Bowditch.
This is where I learned about spherical trigonometry and celestial navigation. Very interesting history and subject in general:) Bowditch is was a fascinating and brilliant mathematician that was a major contributor to tons of stuff we take for granted today.
I studied up on spherical trigonometry to write a sight reduction program for my programmable calculator, to use with my sextant. (just for fun. You could purchase a pre-programed navigators calculator at that time).
I was crew on a charter yacht back then-- before satellite navigation and GPS. I used an out of print copy of the Bowditch's American Practical Navigator that I found at the New York Public Library. The coastal navigation volume was still in print but I couldn't find the volume covering Celestial Navigation at the time--and that was the one I needed to understand the spherical trig.
It's a pretty cool study! You can do the math to find your latitude by measuring the height of the sun above the horizon, using plane geometry.
To find your Longitude by taking sextant sights of the sun, moon and stars at other times of day--you need to use Spherical Geometry. (Actually, by using various tables, it can be done by the non-mathematical navigator without understanding the principles). But it is much more fun to actually understand how it all works!
It's folded pretty well into astronomy courses. Always seemed like an intuitive extension of plane trig.
Everyone's already talked about naval and civil uses for spherical trig, but I figured I'd shine light on another use: astronomy! In observational astronomy, we deal with many different spherical coordinate systems, and spherical trig is indispensable for coordinate conversions. It's also extremely useful for solving problems like the distance between two stars, "A star is directly overhead in Naples at [this time] and later directly overhead in Mexico City at [this time]. What is the greatest latitude at which it will be directly overhead?", And considering things like the motion of the stars, planets, and sun in our sky.
For my Mathematics undergraduate degree I did my fyp on Spherical Trig. It was called Measuring Heaven & Earth, the Mathematics of traversing the oceans.
If this book was first published in 1942, according to the copyright law, the copyright has expired and the book is now in the public domain. You should scan the book and make it available for everybody, so that the book is not lost. Also, you should send a scan of the book to Project Gutenberg for them to publish it on their website.
Ochoa is not a common name. Are you related to Frank Ochoa? Frank is a well known stock trader.
@@geraldwellborn5047 I am not related. It is a common surname in Spain and Spanish-speaking countries.
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 still remember how to use a drafting kit. It is a lost art.
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 got interested in Spherical Trigonometry in the late 1980's as a result of reading material by R. Buckminster Fuller. Mr Fuller was trained by the U.S. Navy and worked with a boat during WW2. Mr. Fuller used Spherical geometry in his Geodesic Dome calculations and published some educational material on Spherical geometry in his 2 volume book 'Synergetics'. I located one of Mr. Brink's books at a local University library. Most University libraries have very few books on Spherical Trigonometry but there is some material in books on Geodesy and in New Foundations for Classical Mechanics by David Hestenes. I've seen that thin 1942 book at another University library that has another book by Mr. Brink that seems to be based on the 1942 book but printed especially for sailors. That library put most books from the 20th century in a high density archive off campus so it can't be stumbled across anymore.
I used to like maths until first year uni. Second semester maths had over 2000 students and ony 3 copies pf the text book in stock at the uni bookshop. Enquiries with the publisher found the book was out of print with no plans to do another print run. The only alternative was for the 2000 odd students to use the 4 copies available in the uni library. They did not change to a text that was readily available. The result was a very high failure rate, including me. That was the end of maths for me just as it was getting really interesting. I actually used spherical trigonometry quite a bit in my job. Fortunately I had already got that under my belt and I was very used to thinking in 3D.
It was not so easy to find references. I can relate to it also. Even if budget was not a problem , availability is already a huge one. No wonder, why some become math book collector. I wonder why the book are not printed anymore, making them so scarce, hence difficult to find. Could it be because of lack of sales on previous edition?
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.
Yes I've heard about it and learned about it... use case: celestial navigation and astronomy.
i'm currently enrolled in a review program for civil engineering in our country, and surprisingly, this is one of the topics! really had fun with it because it's new for me. i didn't know this isn't common math anymore :(
Wow that's awesome!
what is your country?
@@HuyPham-li8gk 🇵🇭
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra Using The Triangle Equality ??
I swear I stumbled across that exact book in my high school library (back in the 80s). Somebody must have donated it. I remember paging through it and looking at the diagrams. In the days before GPS spherical trigonometry was fundamental to navigation.
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?
There is one on Abe books for $13.71 + $2.99 for shipping.
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.
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.
By Spherical Geometry Do you mean Jacobian Matrix and Vector Algebra Using The Triangle Equality ??
I have an old Schaum's Outline Series math book by Frank Ayres that has a section covering spherical trigonometry. Of course Schaum's books are noted for having many worked problems, and accurate answers to all the exercise sets.
The celestial sphere part is so freaking cool
I know! I love old books!
@@TheMathSorcerer sir can you provide us a with a source to get old books
I don't know about schools these days, but when I was in high school trig in the late 90s we did a unit on spherical trigonometry. I assumed that college mathematics would go deeper into it, but I was a psych major so the only math classes I needed were statistics classes, which actually have been very useful in my life. I've been meaning to learn more math on my own, I never got into calculus, or even pre-calculus for that matter. But spherical trigonometry was helpful in combating the "arguments" of flat earthers.
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?
9:40 love the dieresis in 'coordinate'.
Hey! You just made me dig out my copy of "Plane and Spherical Trigonometry" McGraw-Hill 1934! I had to get out the box of math books I could not fit on my bookshelves... Thanks!
Barron's Trigonometry has a chapter devoted to spherical trigonometry.
Oh wow that's cool.
My dad studied spherical trig in public high school, he graduated 1929.
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.
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.
It is also used in setting up ball bearing grinding machines.
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.
What you do now?