This video deserves a 'part B' where you work through an actual example. Good work on part A, best description I've found so far.

The line you draw from the observer to the GP of the star at 4:53 is *INCORRECT.* The line of sight between the observer and the star, and the line between the GP and the star are *PARALLEL* - it's not a triangle. The reason you see the star at 50 degrees elevation is because you are 40 degrees of a sphere away from the GP of the star. At 69 miles per degree, that makes the distance to the GP equal to 2760 miles.

Kudos to you sir! Most explanations of this process bury the pupil in jargon without first giving a simple conceptual explanation. I for one, thank you very much.

Thanks! This is the best overview I've seen so far. I watched a few other videos that went into great detail, but, they NEVER illustrated what those details were accomplishing. The drawing and simple explanation create context for the detailed math involved.

I was also in the Navy ( in the early ‘70’s) as a Quartermaster. Your presentation brought back some fond memories, for instance how satisfied we would get achieving a pinwheel fix when shooting stars. Not always easy when your vessel is rolling and pitching. I wonder if ships still carry Chronometers? In any case, nice job Sir.

In case anyone is wondering what "H0229" is, it's a Nautical Almanac. "A nautical almanac is a publication describing the positions of a selection of celestial bodies for the purpose of enabling navigators to use celestial navigation to determine the position of their ship while at sea. Wikipedia"

Good first-order explanation. Fifty three years ago I took two, three-hour classes on navigation over two semesters. One semester was devoted to celestial navigation. It was a part of my training as a midshipman at the US Naval Academy. Forty years ago, I served as Navigator on two nuclear submarines. To take celestial observations, we had to attach a specialized sextant to the bottom of our periscope and be at periscope depth for the duration of the observation at nearly constant course and speed. The fixes we obtained were usually very inaccurate when compared to modern electronic navigation fixes - errors of +/- 5 miles or more were common. Modern gps fixes with errors less than six feet are quicker and don’t require extended time at periscope depth. But, we’d take celestial fixes through the scope every year just to stay in practice. An operational difficulty for submarines is that celestial fixes are best taken near twilight (near sunrise or sunset) to be able to view both stars/moon and horizon at the same time. That meant that bad guys would know when a sub had to be at periscope depth to take these observations. Not good. So, we avoided celestial navigation in favor of electronic navigation fixes - GPS, NAVSAT, LORAN-C, OMEGA, bottom contour fixes. We also had two inertial navigation systems that maintained estimated position of the sub using gyroscopes and accelerometers. The more accurate explanation is that the LOP of a celestial observation represents the intersection of a cone extending from the star towards the earth that intersects with the surface of the ocean’s surface. In the open ocean that intersection describes a circle. And that circle is the LOP. Observation of the altitude of at least two other celestial objects, corrected to the time of the first observation, yields two more circular lines of position. With no measurement or instrument errors, these three LOPs intersect at a point on the ocean’s surface. The point is the fix position at the time of the first observation. Because the surface of the ocean is spherical, then linear geometry will not yield an accurate enough fix. Instead, spherical geometry is used to generate the tables in H. O. Pub 229. The navigator/quartermaster does not have to know spherical geometry, but they must know how to look up the star’s parameters for the time nearest the observation. And several corrections must be made in order to optimize the accuracy of the celestial fix.

Good 'general' explanation. As a surveyor and one who did astronomic observations for nuclear weapon delivery systems, we used this technique primarily to establish direction. We also used it to establish and or verify our location against predetermined markers on the ground (called trigstones). This method whereby you use two or more astronomic observations to locate your position in marine navigation, we also call 'resection'. Other points that have 'known' data sets could be church steeples for instance. I spent 12yrs in Europe doing exactly that, most of which was before 'GPS'. I 'trig traversed many kilometers, bringing directional and coordinate control to where I anticipated a 'firing point would be needed. Mostly this trig traverse occurred in winter where trig stones were covered in over 3ft of snow. Later as technology became more 'developed' they brought a navigation system (gyros and accelerometers) through my points previously established with trig traverse, and found my data to agree with a 1:50,000 Accuracy ratio. Yes we used T2 theodolites and 'math' as well as star finders, and emphemiris.

That was most simple explanation for celestial navigation I have ever heard! Thank you!

Armed only with a sextant, a book, some math, AND A CLOCK!

Hello, I know absolutely nothing about navigation. Isaw this film"Adrift" a true story about a Couple (man and women) who were given the Job of sailing this well equipped yacht across the Pacific. Sadly they ran into a fierce storm, the boat got completely wrecked, the man (an English guy) was thrown overboard and sadly died and the American lady came to to find all navigation equipment completely destroyed except a sextant.She is seen looking through this sextant to try and find her position and it was the vastness of the sea that she was in that made me realise the importance of knowing something about knowing where you are and which direction to want to go. Your simple but clear description of finding our roughly where you are at sea was great. Thank you. I'll have to get that book with the funny title..

Thank you for your service. I was a Marine NCO and later in my career an Army Officer. I did operate a pair of LPC's (leather personnel carriers). I am new to sailing and will use D.R. plots with running fixes to support electronic navigation. Great information. Regards Jarhead 66

Finally someone xplained this in a way that the average joe can understand.Hats off to you sir.

5:00 Isn't this pretty misleading? I mean, the base of that triangle isn't a straight line, but follows the curvature of the earth. And the horizons at the observation point and the point underneath the star aren't parallel. I think it would make more sense to draw the earth as a circle and then you can see that the observed angle of the star above the horizon will be 90 degrees minus the angle at the center of the earth between the two points, which is how you get the distance. Another misleading feature of the diagram in the video is that the star is in reality essentially infinitely far away (compared to distances on earth), so the light from the star hitting the two points on earth is essentially parallel, which is important for the geometry to work.

I have tried to learn celestial navigation for years reading books ...failed..videos...failed and then you came along, Your video is OUTSTANDING and easy to follow and I finally get it.........THANK YOU.....

Thank you for that wonderful comment. Clearly you know what you are talking about and bring back a lot of "rusty" memories for me when I was a Navy Navigator. The goal of my video was to give students a general sense of the mathematics that go into celestial navigation. Of course I find it amazing that we can use the starts to locate our location on earth so I try to motivate students by using real life examples of math to become more interested in the subject.

@1:45 skip to

I watched your video to educate myself about sextant use and celestial navigation because it relates to the controversy about whether the oceans are curved - or flat. Based on your video, I'm inclined to infer that the LOP (line of position) and the 900 mile leg of your triangle - are flat. In addition, based on your video illustrations I infer that the ocean must be flat in order for the sextant/navigation method to be effective.

I like idea of part B - an example. I want to see how you look stuff up in the book. I want to know how to derive the “900mi” LOP circle. Having said all that - BRAVO to you for making this easy to understand. I love navigation. Always have. Always will. My dad once took me for a ride in a heavy snowstorm in NH. He pulls off a dirt road and parks. He shows me on a topo map roughly where on that road we were. He then showed me a roughly parallel road a few miles distant from the one we were on. He produced a compass. Oriented it to the map. Showed me how to find a heading. He said “you think you can walk that heading and come out on that other road?” I mean SURE - it was a long way away but if I could come close to following that heading it was gonna be as easy as shooting fish in a barrel. Out I went into the woods all alone with that compass. Heart beating. Sweating too. But I followed that heading. I came out on that road probably 3 hours later. It was getting dark in the sky and I just waited. I heard a car coming. It was dad in the old Chrysler Newport. He pulls up and rolls the window down and says “I think you know how to keep from getting lost now right?” YUP. Every kid should be given that experience. I loved him for that. What a great thing to build confidence. Thanks Dad!!

Thanks for the great feedback. I agree with your comments. When I made the video I just kind of created it as I was think about it. Nevertheless the topic of navigation and teaching the mathematics around the subject is huge and offers several approaches. Thanks for watching!

It's easy to measure "altitude" of a celestial body, correct according to height above horizon, upper or lower limb, etc. Subtract from 90 degrees. This is your distance from the body in nautical miles. Best to shoot body's that are due north or south for latitude, and due east or west for longitude. Position of bodies are listed in the "nautical almanac" in longitude and latitude according to date and time. Each "fix" results in a triange on the map, or a distance in nautical miles when subtracted from 90 degrees, to the body. Problems develop because maps are not true d/t curvature of the earth. The map triangle can have three 90 degree angles which is 270 degrees (or more!), so conventional trig is worthless except for figuring the distance to the body with your calculated position on a distance line from the GP (geographical position, aka long. and lat.). Also, you must assume longitude to get latitude, and vice versa. So, at best, especially with a time that is off (one second = 1 nautical mile), problems develop regarding actual position vs. calculated position. Also, arcs drawn on a map are not true circles (grid lines on the map do not make a true square). There are "cheat sheets" that can be used for celestial navigation, which seem more accurate than the actual hand computed fixes. Corrections are often made using a "trafrail log" which measures distance covered (not corrected for current!). In short, oh brave explorer, be sure to have a modern system such as GPS to guarantee accuracy of your navigational skill and provide information when the weather is bad to keep you out of trouble!. There's a reason celestial navigation has taken a back burner. It's fun to play with to see the difficulties, but that's about it. Thanks for the site!

flat earthers sure talk big when they dont even have a map yet

Hello and Thank you for your service as well. Before I was a Navy Officer I was in the Marine Corps (enlisted 87 -91) - Semper Fi. As you stated navigation uses several skills and methods. Knowing how to DR is key as it's too difficult to update your position with celestial nav only.

Hi Ken- thanks for watching the video I'm glad you enjoyed it. I'm a math teacher however I love engineering and other science topics as well....

The two LOP circles would intersect at two points, two fixes. I guess I third star would tell you which fix was the right one.

Flat Earth brought me here. Thank you for your video

I think for what you're trying to show here it would have made more sense to take the simplest case so the principles are apparent, and then show how the more difficult cases relate to the simple main idea. As the earth rotates, there is one line of longitude that is closest to the sun, halfway between the dark edges where night begins. This is local noon and in theory it is a razor-sharp line, zipping along about 1000mph near the equator. Everything on the 'morning' side of it is seeing the sun rising, everything on the 'afternoon' side of it sees the sun going down. Since the earth is turning at a steady rate, if you know the clock-time difference between your noon locally, and a reference longitude - Greenwich, by convention - then you know your longitude. Measuring the highest point of the sun's arc in your sky gives you your local noon. The other even more simple case is your latitude, from Polaris. Since Polaris is above the north pole, it is straight overhead at the pole (90 degrees) - and just on the horizon at the equator (0 degrees). All the time. And everything in between follows perfectly. At 37N latitude, Polaris is at a 37 degree angle. Once you understand these basic ideas, their more complicated manifestations relate back to something pretty intuitive. The sun moves up and down in the sky according to season, so you need to add or subtract declination if you use it for latitude. Sounds a bit weird, but when you see that you're just trying to measure an angle, the fact that the angle shifts because the earth tilts relative to the sun, isn't so hard to come to terms with. Starting right in with random stars and charts and almanacs and trig is too much for a beginner. In my humble opinion.

Hi! Listen, I am from Ukraine, and my native language is Russian. I just wanted to thank you and tell that I couldn’t find more simple and obvious explanation for this topic on my native language! You are awesome, thanks again mate! :)

5:05 “I’m not going to get into the math because that’s not the purpose of the video.”. Huh? Why title your video like that then?

The radius of your “Circle of Equal Altitude” for star A would be 2400 NM, not 900 NM. ZD = 90* - Ho x 60NM/*

As a former ships captain (merchant), and a navigator for many years, before the advent of GPS, although we did have DECCA and in the western hemisphere LORAN, these did not reach out to sea, and were only useful near land. Thus, celestial navigation at sea. Basically, three items are needed: a quality sextant, a good accurate timepiece (Chronometer), and sight tables, I prefer to use the Air Almanac because it was much simpler than HO229 and sights can be worked out much faster. A stopwatch is also helpful. Although I have been retired for 14 years, my sextant and stopwatch reside on a shelf in my study.

Hi Bruce. Thanks for watching the video. Yes indeed I think you should study the process more and it's really a joy to be able to navigate by the stars. I wish you all the best and thanks once more for watching my video... John

Congratulations TabletClass Math, you have now become a flat earth poster boy! I understand that it was never your intention and you tried to prevent it by writing: "SPECIAL NOTE: The earth is round/oval, so my use of right triangles is a simple math model to explain the basic concepts of navigating by the stars. However, the actual math needed to perform celestial navigation involves trigonometry on a sphere- which is very advance math- as such you should watch the video knowing that right angle trigonometry is not actually used in navigation because the earth is a sphere.". This however is of no relevance in the flat earth cult, so congratulations to your new found fame!

minute and a half in - still waiting for some information. any more waffle?

A very simple conceptual explanation. As a sailor, I love it. I also like that it works if I don't have a phone or a GPS or any other technology. Except I need a sextant, and a timepiece, and knowledge of stars, and a book (H0229), and a pen. The lesson I'm taking from this is to keep my Garmin charged and drop it in the water. Just giving you grief. Your explanation is great.

Very well done!! Being a total novice with this skill I have been looking for some decent & basic information re positioning/sextant use for a while.....I found your KZfaq clip and now understand the principles involved. You ARE a legend. Thank you from a layman in Australia

I think this is completly wrong.... how can you calculate this '900' miles having ONLY three angles? Please note the distance to any star is -comparing to the size of the Earth- practically infinite. So I think the author of this video is showing wrong triangle.

hi you use flat earth model. Interesting.

Hi Brian- I agree with your comments; finding your position using a sextant really is an amazing feeling as it connects you with so much history of how people navigated in the past

hello demmy- maybe sometime in the future I can put together a video on various coordinate systems. Thanks for the suggestion and feedback- keep learning!

Proof the earth is flat

need to be more to the point too long intro

if indeed you saw a star at 50 degrees above the horizon you would be 2400 nm from its ground point, not 900 miles

On your question about the use of a sextant at night. It's easy to use a sextant however it takes alot of skill to take accurate "shots" of the stars/moon at night especially when your ship is rolling and pitching ; I was never that good as it takes YEARS to develop the skills of master navigators.

Uh...that's all wrong...there are NO triangles in celestial navigation...

Hello, QM rate is excellent! You will be on the bridge (pilot house) of the ship and see all the action; also you wont feel trapped inside the ship. As to the math I would get really good at working with decimals. Also you could learn how about angle measurement in terms of degrees-minutes-seconds. However the post important thing is to learn from those senior QM; listen to what they say. It will take a few deployments but you will master all the critical skills of nav at sea-good luck!

Celestial navigation was developed over 1000s of years. At first navigators used more basic methods before developing the sextant, book or clock. However these methods could not produce a fix so navigators never knew their exact position. It was not until the 1600s where accurate clocks where made to handle the rough movements for ships at sea that navigators could determine longitude - then the art of celestial navigation became more like what the video described.

Proving earth is flat earth. Very nice 👍🏻

John.. Please delete this video. You are embarrassing the rest of us who know how this is done. Your use of 'thing' and the totally wrong distance in your example is just poor. If you are going to reference an example, use the proper terms and correct information. 50 deg observed altitude equates to 2400 nm from the stars geographic position. Which is tabulated in a book by-the-way. And, You do not need a major in mathematical study to determine this fact. It's simple multiplication. Oh.. I think it's HO pub 229? really? maybe it's something else. Maybe a Nautical Almanac? Maybe Sight Reduction Table? Sorry, I couldn't watch past the 900 mile thing... 17yrs ocean navigation, U.S. Navy Instructor (Celestial Navigation) Good luck

Where does your radial distance from the GP come from? You take a sextant reading of 50 degrees and you know that you are 900 miles from the geographical position. How do you know this? david

Sod sitting in this fella's class all day.....zzz

Best explanation I've heard wasn't even looking for this but I just got hooked.

Awesome class. If current teachers would use this method of teaching more students would grasp the skills of mathematics and value it's usefulness. Great job!

hi there, nice introduction to celestial navigation but i think it could be improved by using a model of a round earth where all rays coming from a star are parallel due to the infinite distance he's away. Then you could simply show that you get different angles on different positions due to the bend of the earth's surface and as a little supplémentaire you then could show that the difference in degrees you obtain can directly be converted in nautical miles and everybody would be satisfied ;)

Oh wow MoBoards- maybe I will do a video. MoBoards are a great way to show how vectors in math are used in navigation. I know what you mean being fast- if you do MoBoards as part of your job you end up being good. I actually used to work my MoBoard problems right on the radar screen using a marker pencil- could do all CPA problems that way; only used actual MoBoard for formations, Screen Kilo, Form 1, etc. Thanks for watching!

navigating by the stars has evolved over 1000s of years; some of the ancient methods used basic tools just to hold study the latitude of the north start; this keep navigators on a one Lat. until they reached the coast; at which time they switched to piloting along the coast; the use of a sextant was introduced much later on as science and math evolved as well..

Thanks for your comments. During the day you can use the sun to get LAN (local apparent noon) which gives you a LOP (line of position). Celestial navigation is part of the overall navigation effort on board ships. Navigators also need to DR (dead reckon) which is updating your position by using your course, speed and last fix. Of course the more you estimate a your position using DR the less accurate your position; hence using the sun during the day can help update your fix;

If you measure a star at a height of 50 °, it means that you are on a circle whose center is vertical to the star and whose radius is exactly 2400 nautical miles, not 900 .... Because Zenith distance = 90°-50° = 40° and 40° = 2400' (=2400 NM)

Superb video, the ability to break down a sophisticated method of navigation for laymen to understand shows high intelligence. Thank you!

Hi A nice explanation. One comment. You describe that with two stars you get a fix. Technically, this is not true. Two stars will give two possible location points because the circles of equal altitude intersect at two points not one. It takes three stars to uniquely determine a position. In practise, you have been keeping track of approximately where you are (dead reckoned position) so for practical purposes you can pick out the correct point with two stars.

These arcs of reference is how the Astrolabes and chronometers came about. The polynesians were very accurate and they used memory, songs, stories and 3d stick charts to remember star nav. In fact all the accounts I've read of early western mariners meeting the south seas people were blown away by their level and ability of natural navigation, their culture and deep connection with the stars far surpassing our mariners and their scientific gadgets and gizmos and charts and books. Sadly most of that knowledge is gone.................... A simple quick arc reference for latitude would be 90* minus 50* gives you 40*. 40* can then be used to calculate distance etc away from point A and the celestial sphere rotation plus the location of star A will give you an accurate reference for direction. This can be done super quick with outstretched hands and is very accurate. My location here in the UK is 55.5* which is 5 and a half hands up from the horizon or down from the pole star. It's accurate too if you know the suns arc for the four seasons too. But it's better to learn it as the Polynesian and the desert Bedouins did by memory. Never rely on kit.....................

If the earth is flat, explain hemispheres. Why can't New Zealand see the big dipper? Why is it summer in the northern hem but winter in the southern?

I used to be able to do Celestial Navigation with a Sextant, a clock, and a current Nautical Almanac - the book of tables for the positions of the closest stars, our solar system's other planets, and our sun during the day. The math wasn't hard. The more difficult math was in the construction of the tables for each celestial object for the coming year. I'm pretty rusty now, but it wasn't hard. Using a Sextant requires some practice to get good readings, and is especially difficult on rough seas, in windy conditions, and/or in small boats. Get regular practice, take readings for 3 stars, and you will soon get your position amazingly accurately.

best explanation so far for this topic found on this platform

How did you define the 900 mi distance?

Very nice illustration! I see your example is on the TIGHAR article abouit searching for Amelia Earhart In one of their articles they have suggested that FN was confused and was reading the wrong dated charts and instead of using Jul 3 data he may have been using Jul 2 data or vice versa. In fact one person goes into a very long dissertation, which is almost convincing, until you realize he fails to mention one very important detail, which would have been quite obvious with the use of the naked eye alone. (the MOON was available that night) The argument has also been supposedly confirmed by some navigator and that these chart errors could have put them 59.14 nm farther east or west of where they actually were. That may be OK for star error, however the MOON behaves quite differently as it cycles is every 28 days. The stars only vary a few minutes from day to day.. As example lets just say near the IDL where they would have been the coordinates are Latitude S0* 06.1’ Longitude E 179* 56.1’ If FN had been reading the wrong day's charts the MOON would have shown them to be over a thousand miles off course The Moon From July 2 and July 3rd 1937 shows a difference in Hc as 14* 40’ and Dec 14* 14’. This difference is at total of 28 degrees 54 minutes and error of 1734 nm.

I'm studying the "Sextant Crop CIrcle" and this information exchange was most helpful for the linguistic studies being performed.

I know what you mean- MoBoard problems were great especially when doing things like winds for a helo. A radar screen can be used as a MoBoard so you can plot contacts on your screen and get a CPA; then determine what course correction is needed to open up a safe distance; this stuff is really important for large vessels as you need lots of time and distance to manuver

hello- I have not anymore videos however I may look into creating more- I love navigation and find it very interesting to see how math is used to find your location

First time I watch a video about navigation that I actually understand. Thanks

Hi Eric, Yes indeed you are correct; thanks for watching the video and it's great getting feedback from people the understand the skill involved to celestial navigate

I'm the same way- more data the better. Thanks for the great input on the videos, I think I might brush up on MoBoards and do a intro video; all the best! John

Nice work! I've been studying cel. nav. for a couple of years now and as you say, it's a great way to learn about trigonometry, history and other subjects. Everything in the video is essentially correct. I heard you say you weren't sure about a couple of things, so here's some very minor corrections: The book you were referring to is actually the Nautical Almanac. The circles of possible positions are called "circles of equal altitude", but when plotting them on a chart, only a small...(cont.)

You reference Ho229 as the book containing the celestial data. That is incorrect. The book you are attempting to reference is the Nautical Almanac. Ho229 is the book (actually a series of books) for the Sight Reduction tables, so you can do the math without a calculator.

In addition to a sextant, the book and some math you also need CHARTS and an accurate CLOCK to plot where you are.

Take care. I'm an old Jarhead that went through ITR with an M1 and used an M14 before the Matty Matel special was forced on us. Best regards and thanks Jarhead

indeed; 2 LOPs are not enough; 3 LOPs are the min to have a basic idea of your location; also to your point navigators need to know how to DR using estimated speed and direction- all skills navigators need to really find an accurate fix

Very good basic explanation. However, you would use a nautical almanac to find the present position of the celestial body for the date and time, new is issued each year. Then go to the HO 229 to get for lat and long variation or declination. Also, the Bowditch tables will quickly simplify the math for finding the angles. The mariner needs to understand that this is a very simplified explanation due to the exactness of time and high of eye, etc.

At 3.32 in the video, he mentions a book,-- Ho 229,-- that navigators use to find the ground position of a star. Actually the book used to find the ground position of a star is not Ho 229 but either The Nautical Almanac or the Air Almanac. Ho 229 (sight reduction tables that precompute the complicated math) are other tables that are used to find the altitude of that same star from an assumed position. The star's attitude from this assumed position is then compared with the altitude as found on the sextant. The adoption of an assumed position is a contrivance used in order to be able to plot a position line on a chart. Otherwise the intercepting arcs (lines of position) cover such vast areas that they are not amenable to be plotted other than on a globe. For instance for a sextant altitude of 50 degrees the line of position (arc of the circle) would have a radius of 2400 nautical miles.

Shoot, not going to go into the math, eh? The math is why I'm here.

Where, on the surface of the Earth, do I place that first 900 mile circle?

Brilliant little vid..! You explained it so well. Greetings, and many thanks, from Australia.

This is so concise -- a perfect refresher!

to actually plot a fix on a map/chart it rather involved; my goal was to give a general understanding of the process but in actual practice it takes training to develop ad transfer fix data onto a chart

Great starting point, thanks ! So to cap it off celestial map to know one or two stars sextant the HO-229 Accurate time (know what time zone) What would it happen if I want to take the measurement during the day? Again Thanks,

I really enjoyed your video. I liked the way you drew it out on an actual picture of a horizon. I recently went on a cruise, and I took my sextant with me. I shot star fixes that were within .75 of a nm of the GPS fix. I commented this to the Captain of the ship, and he called it a fluke; that Sextants were inaccurate; and that he hasn't touched one in 20 years. I think he just stinks at Celestial Navigation and discounts it out of hand. Its sad he's grown lazy because of GPS.

Congratulations!!! Your video is a lot easy to understand and well explained. Please keep with the classes, I am learning a lot. Thank you

The reference to the circle drawn as an LOP isn't correct. It's a "circle of equal altitude". Using Pub. No. 229 or Pub. No. 249 (which has the more common and useful navigational stars) alone will not provide you with anything useful unless you have an Ap. Latitude and LHA. based, of course, on GMT/UT1 time. And without The Nautical Almanac you can't know much of anything position wise except a rough approximation of Latitude (so long as you know the sun's declination for the day and time of the sight). You only can obtain an LOP once the sight reduction is accomplished and Hc is compared to Ho and plotted on a UPS. The video, while nice, overly simplifies celestial navigation.

Some important steps missing: Then calc the Zenith Distance based on Sextant Altitude (Hs), 90-Hs: 90-50= 40 deg. For the Northern Hemisphere, in your example, the Zenith Distance is 40 degrees, which converts to 2,400 NM (40 deg x 60). [1 degree of arc on a Great Circle is 60 NM]. This means that if you suddenly went 2,400 NM toward the Star you’d be at the ZENITH (Star directly above or Hs = 90 deg). From the Almanac at the Fix Time: 1) Obtain the Star’s DECLINATION - equivalent to LATITUDE AND obtain the Star’s Greenwich Hour Angle (GHA) from the Almanac - equivalent to LONGITUDE measured Westward from the Greenwich Meridian (Zero Deg Long) from 0- 360 (No East or West). To plot the curved LOP you’d have to draw an arc 2,400 NM Radius from Star’s ZENITH position (Declination and GHA). As you discuss another separate observation with another Star can determine your FIX with a second (and third) curved position line. Although tricky, due to Convergence (of Longitude), and depending upon the type of projection of the plotting chart you could take a True Bearing TO the Star at Fix time, then back-plot a Great Circle line FROM the star, compensating for longitudinal difference from the Star and your approximate Longitude? That way you’d have a Curved LOP and an Azimuthal (straight) LOP for a 2-position line FIX My background: I was an Air Navigator in the Canadian Forces on C-130 Hercules. We did 3-Star Fixes with a Periscopic Sextant using the INTERCEPT METHOD of plotting whilst travelling at 330 Knots to HOPEFULLY obtain a small “Cocked Hat” from 3 LOPs! My $0.02 worth.

According to your example. The height of the star is about 1073 miles? If you can use trigonometry to find your distance from the 90 degree point you can also use it to find the distance to the star.

Thank you! I enjoyed this video. You taught me some thing very important on Astro Navigation. I has got me closer to my goal. Thanks once again and God bless you.

GP of stars gives latitude but you have to consider GMT and time zones to figure longitude. Great starter video.

FYI I have determined from history that 360° 60' 60" DMS latitude and longitude was a prehistorical invention. It ivented hexidecimals for a pragmatic, angular ratio grid on the the stars that also made a standard radian Nautical Miles each minute of latitude (of 1000 standard fathoms or 6000 original feet) at sea level. Fathoms being the familly set of measurements including 2 yards, 4 cubits, 6 feet, and inches used since sailors started using boats instead of everyone walking. That star grid also creates an angular grid on the earth. As a Radian distance, spherical Nautical Mile calculation in a quadrant are greatly simplified by reducing converging longitude distance to true distance multiplied with the ratio cosine of latitude midpoint. The resulting true longitude "distance^ can be used with the direct true latitude distance simply as a flat right angle triangle to have an adequate course forecast. The original ancient foot, in the ancient sailors fathom familly set of 1fathom/2 yards/4 cubits/6 feet/inches, are longer than the yard/foot introduced by English King Edward 1st after 1275 CE used with a 5280 foot Statute mile in a failed attempt to create a 24,000 Mile Equator for a 24 hour day. Typing 1 fathom into ConverPad app shows that the ancient standard fathom was the basis for over Twenty ancient measurements including cubits (Ancient Sumariansand Egyptians), 10 Stadia (Ancient Greeks) per Nautical Mile etc. An English scientist commented after the Edward foot was introduced that there was 6068 of them in a Natical Mile. (About 1.15% different in length.). This DMS ratio is inherently future proofed by providing a 6000 foot Nautical Mile giving an arc second (") of100 feet, a "ten"th if a second of "ten" feet and a hundredth of a second of one foot. Pragmatically, either (foot) distance still works for navigation distance calculations and provides simplistic mathematical navigation - between angular latitude and longitude positions determined without distance - on a line between the centre of the Earth and a DMS grid on the stars. GPS, Immortalising latitude and longitude used on all maps and charts, is accurate to 6 metres or 20 feet (0° 00' 00".20 DMS) or DDD 0.000056° (HiPER Calc app) Few people understand that before the ancient Sumarians and Egyptians (used cubits) latitude and longitude positions provided a control base for bearing triangle (to scale) for mapping and charting without distance measurement of coasts and interiors by navigators and explorers. (Triangulation) This mapping method gained the name Plane Table mapping in the 1400's and was the method of accurate world mapping until about World War 2. Complex 3D spherical math for navigation is greatly simplified when you understand the pragmatism of forecasting navigation from one latitude a longitude point to another is altered along the course by crosswinds, currents, storms and steering ability changing where you end up in distance and/or time. No long course, especially on land or in the air is an undeviating curved line. Since Nautical Miles are radian distance no earth radius calculations are needed. Simply put -distance between latitudes is true Nautical Miles on one side of a right angle triangle - and distance between longitudes on the other side changes by the factor of the cosine of the latitude at the midpoint of the line midpoint in a quadrant. The cosine of zero degrees at the equator is 0ne (1), the cosine at the poles of 90° is zero (0) and between these values will multiply the length between longitudes to provide an adequate true length for the other side of the right angle spherical triangle. Since the curvature of the earth can be considered a constant ratio, flattening the spherical triangle to a plane 90° triangle is adequate to calculate a forecast plane hypotenuse course distance. Lines crossing the equator or longer than 90° outside a quadrant can need calculation as more than one line. Sometimes latitude and longitude might be measured as accurately as one Nautical Mile., but you will never know when.

If you’re using right triangles, your baseline is a horizontal flat plane. Let that sink in.

hello; I'm not sure if there was one single person that constructed the book; I suspect the data found in this book was not developed until 1600s as navigators needed accurate time insturments to find Long.....

Do you remember how good the teacher was when you were taught celestial navigation in officer training? I do. it was bad. Your approach is refreshing. 4.0.

Thank you for this incredible "Navigation" documentation. I've just been "getting into it"; so again TY!

great questions; the more LOP's the more accurate the fix becomes- it's still just one "fix" or position; as to the hemispheres this is determined by which starts you see- i.e. in the S. Hemi you can't see the North Start (Polaris)..final thought: the video was very general but the actual procedure is rather involved and would answer your questions in more detail...

thanks for watching

Thank you! I so happy you enjoyed the video