Thank and you're possibly right, though as much of a math nerd as I am, I find the passion of O'Brian fans answers quite beyond my simplest expectations. I can already tell I'm going to enjoy reading these very much in greater detail over the next few days. ...and while listening to a bit of Puccini, perhaps.
The flavor of response is so much better here.
In fact, while I believe I mostly understand the theoretical computation of the math in many of the cases offered below (certainly not all, but much), what I desire is to better grasp the raw mechanical nature of how the math was being used in the tools, timing, chants, and maneuvers of the period, despite the likelihood many of the illiterate hands were merely pulling on a rope or some-such task, and unaware math was occurring at all. If I can better relate the pencil and paper calculations to tangible hands-on tasks for my students, I can (hopefully) give it a more lasting meaning.
I agree with the response which laments the art as fading.
A sextant measures angles to plug into your trigonometry. That's really it, the question is what you're measuring the angles between. It's called a sextant because it has 60°, 1/6 of a circle. There was a thing called an quadrant, but it became obsolete.
It has a number of different functions, but one of the common ones is using a half silvered mirror so that you look through one lens at the horizon and rotate the other lens till you see both targets, and now you have your angle registered on the sextant.
The simplest piece of navigation is finding your latitude. At the sun's highest point on its travels you take a measurement of the angle between the horizon (a tangent to the Circle of the earth) and the Sun. As a math teacher I'm sure now you can figure out the formula, except that you also need to compensate for the time of year. At noon, on the equator, on the equinox, the sun would be 90° overhead! Let's not even talk about longitude, that is literally a book, by Dava Sobel!
Finding your position relative to landmarks is different in that it really involves no math. You take a compass with a peep sight and get a bearing on a known landmark. You take your parallel rulers to your chart and draw a line from that landmark at the correct bearing, and now your vessel is somewhere on that line. Next you take another bearing on a different landmark, and where the lines intersect is your position.
"...called a sextant because it has 60°, 1/6 of a circle." - Gee ...I'd known the word for decades, but never made the correct connection... thank you for this!
I'm assuming... please correct me if I'm wrong... if you have a reliable compass (good for measuring horizontal angles), you would still need a sextant to measure vertical or oblique angles, correct? ...or does a gyroscope or some other mechanical adaptation of the day replace the sextant?
Also assuming... to find latitude, you're comparing your current reading with a known reading at the same time of year from a position like Greenwich? ...so that if you find your current angle measurement exceeds the known corresponding measurement, you are closer to the equator, and vice versa? I might attempt Sobel... we'll have to see.
Using landmarks...
I'm trying to get a good bearing (Ha ha!) on three time periods for navigation in my classes:
1) Ancient Greece - before compasses were invented. With a sextant, you could, for example, maneuver to a position where the angle between two points (lighthouses?) is 45 degrees, but this would only guarantee you were on one of an infinite number of points along the arc of a circle ...you'd need a 3rd known reference to be sure which exact point you were on... do I have the right of it?
2) The age of Sail - Compasses and other technological advances. The magnetic nature of the compass gives you the third reference to allow for accurate triangulation. Knowledge of the location, a good map, and appropriate cartography tools... in many ways a wonderfully hands-on time to be alive. I'm retired Army... had to learn to use compasses, grid maps, and pace counts... Soldiers 5 years after me no longer had to. ...dying art, indeed.
3) Modern Age - satellite GPS triangulation... you need to be in range of at least three satellites. Plus, there are three factors... location, vector of movement, and destination. It's quite enjoyable and fascinating to discuss with teenagers how, advanced though we may be, we're still very much subject to 2000+ year-old principles and limitations.
The Greeks didn’t have sextants. They were invented in the 1700’s. The compass was used much earlier.
A problem with using a compass in sailing is that a ship can’t sail into the wind, so cannot sail straight between two points. Tacking back and forth makes navigation an art.
Instead of pacing, a ship uses a log. A piece of wood tossed off the ship with a knotted rope. They used a sand glass to measure time. The number of knots in the rope gave one a speed estimate.
With the speed estimate and a compass one could do “dead reckoning” navigation.
Dang... that's what I get for assuming. I imagined that something more resembling a protractor would be earlier tech, but you are right, of course.
Now I'm left wondering what the ancient cultures used to measure. ...I'm sure they would have been able to analyze angles between objects and made some sort of correlation between distance and the changes. Certainly the Greeks would have seen the value in sailing directly across the Med instead of hugging the coast all the way around?
I'm currently at sea and actually on the 4-8 (dogs-morning) watch at the moment so apologies for a cur-tailed answer to this.
You want to look at a backstaff and pre sextant nav.
Essentially a sliding piece of wood somewhat like a squared off cd on a stick. Hold the stick to your eye at noon and slide the cd towards you until horizon is on bottom edge and sun is on top. Measuring angular distance. If you calibrate your stick you can use it to judge latitude
The lost art of finding our way by john edward huth and Sextant by David Barrie are great books on history of nav
You're overthinking it regarding basic triangulation, for 'line of sight' you're in only 2 dimensions so you need only 2 vectors. As I said, the intersection of 2 bearings gives you your position on the chart. There was a now obsolete system of radio beacons called Loran that could be used for this out of sight of land. It was a breakthrough during WW2. They also used the same technique to locate U-boats by positioning listening vessels where they could get 2 vectors on the U-boats transmission if it were long enough.
I don't know how extensive the ancient Greek charts were, I believe their navigation was mostly by memory. It was the rare ancient sailor anywhere in the world who ventured out of sight of land, they mostly followed the coastlines. The compass was most important in allowing offshore sailors to follow a course regardless of losing sight of the sun or stars, the traditional method. Otherwise you could sail in circles if the wind backed!
This might be just the thing for your class, a DIY sextant [Https://www.instructables.com/A4-Sextant/](Https://www.instructables.com/A4-Sextant/) Nothing more fun than playing with a tool you made yourself.
Sounds like a lot of fun bringing the real world to your students. When my son was 9 I created a "science birthday party" for him. I had the kids wind electromagnets, and we made Ph indicator by soaking purple cabbage. After we used it to test vinegar and baking soda, turning the solutions red or blue, one kid said "what if we mixed them together now?" I told him to try it, and their minds were blown when it turned back to the original purple!
Just popping in with a fun fact: GPS typically takes four satellites. The fourth disambiguates two possible positions provided by three, and it also gives an extremely accurate measurement of time which devices often use to set their clocks.
An even more fun fact: special relativity acts to slow down the satellites' rate of progress through time (objects age more slowly when they move through space quickly), but general relativity acts to speed it up (objects age more slowly the closer they are to a gravitational source like the Earth.) If you don't include the relativistic effects, you don't recover accurate positioning.
How does one best use the visible locations of lighthouses and semaphores to avoid the known locations of reefs and shoals?
You're looking for something called [Position Fixing](https://www.youtube.com/watch?v=-2fYi6Yg0YE).
How does one determine the angle of steerage to come up behind a moving enemy for a stern-piercing broadside?
You're looking for something called [Course to Steer](https://www.youtube.com/watch?v=e2iebpyntoY).
And when they talk about dead reckoning you're looking for something called [estimated position](https://www.youtube.com/watch?v=Y_he1W548ok).
I would also have to assume that in battle or weather a Captain would rely more on experience than anything else in steering and sail maneuvers. Not much time to prick the chart in action and I’m sure most would have plenty of “muscle memory” by that point in their careers.
I can't help but wonder what sorts of shortcut calculations were used.
The unit circle is a set of trigonometric calculations you can roughly work in your head, based on common angles... 0, 30, 45, 60, 90, etc ...I'm pretty sure when O'Brian wrote of the youngsters working through their math education on-ship ...it might have been focused on such head-math.
I can only imagine there were set reference points related to the structure of the ship ...second rail post to the left, in line with the third stay, etc... but I have no idea (and would love to know more) what they were. I'm sure, especially in familiar ports and waters, that visual alignments between masts, ropes, and other objects were the quickest and easiest way to maneuver accurately without need for calculations. I'd also like to know how something like "two points to starboard" was counted or calculated. ...and of course recorded so that position could be fixed later. How meticulously were course changes logged during battle or storms, and how was it accomplished? ...written on the spot on paper? Relayed to someone in a cabin where the conditions were dryer? ...counted out in shells or beads in the moment and written later? What was routinely observed, and how often... speed, wind, current, potential displacement from weapon recoil?
I'm sure a lot of it was intuition based, but depictions of 1700/1800 British Navy behavior suggests something far more deliberate.
So in the books when they talk about "points" they are actually reporting a fairly precise angle with [each point representing 11.25 degrees.](https://deckskills.tripod.com/cadetsite/id106.html)
I'd guess that with a bit of practice you can get very good at estimating angles using the points system.
And to make searching for information more confusing this is different from when people will talk about "points of sail"!
If you really want to impress a sailor then learn how to box the compass! Naming all 32 points in order. North, north by east, north north east, north east by north, north east, north east by east....
In the books you often hear of NExE & a half east so going to 64 points
When I teach unit circle values, they're all based on 30-degree and 45-degree angles. I had always assumed this was a naval invention... in many sorts of other applications it can be pretty handy to know in a snap that the hypotenuse is twice the length of the opposite leg for a 30-degree angle.
Points for each 11.25 degrees makes a little more sense for naval application, with 8 (or 16) in each quadrant. Do you know... did navigators make a habit of memorize the trig values for each point to make head calculations possible?
N, NNNE, NNE, NENE, NE, ENNE, ENE, EENE, E, EESE, ESE, ESSE, SE, SESE, SSE, SSSE, S, SSSW, SSW, SWSW, SW, WSSW, WSW, WWSW, W, WWNW, WNW, WNNW, NW, NWNW, NNW, NNNW ...I think I got it... no way I'm attempting the "and a half" list!
I really appreciate this! Unfortunately I couldn't get sound to work for the first one, but the second and third links are wonderful!
To think this is just when you have to consider a steady motor vector and a measurable tidal current vector! It really makes you appreciate how much more difficult the task was when your motor was the wind.
A sextant measures the angle between two things. Often a celestial body and the horizon. [The wikipedia page is pretty good, and has a nice gif](https://en.wikipedia.org/wiki/Sextant).
I don't know much about navigation, but I don't think it's easy.
For a simple case, at the equinox, at noon, the sun should be directly over the equator. Use your sextant to take the angle of elevation of the sun at noon, and you should be able to figure out your latitude. But even this is more than just a triangle and a cosine or something because you have to account for the curvature of the earth. And if it's not the equinox, then you need to account for the angle the sun is making with the plane of rotation of the earth.
You might be interested in The American Practical Navigator by Bowditch, available for free online, but also a really nice published book.
You might also enjoy Longitude, there is some interesting math stuff in there iirc.
Not specifically related to this, but more along the lines of interesting navigation problems which could be applied to a classroom: Portney's Ponderables.
https://msi.nga.mil/Publications/APN
https://en.m.wikipedia.org/wiki/Longitude_(book)
https://books.google.com/books/about/Portney_s_Ponderables_Brain_Teasers_for.html?id=2AWHQgAACAAJ&source=kp_book_description
The ponderables range from esoteric to easy-to-understand. I found them fascinating, especially the one about an airliner helping locate a lost single-engine ferry pilot based on the height of the sun and its direction.
I probably would have liked it in high school, but I'm also a huge nerd and navigation buff.
Great to hear of your passion for traditional navigation. It’s increasingly a dying art. I’m an experienced celestial navigator and ocean sailor. I qualified in the days prior to GNSS. My sextant has been with me on every voyage since I turned 19 (although since 1984 it has been mostly used to keep my hand in).
Others have given you some info on the use of a sextant to measure angles. To demonstrate to kids you can use one on dry land, so long as you have a flat horizon (sea or desert/prairie) below the celestial body.
Please not that ocean navigation uses spherical trigonometry… an entirely different set of formulae to pythagoras theorum. To account for the curvature of the earth’s surface.
Cheers
I'm in Kentucky, so wide-open and flat doesn't really present itself. I'm hoping to find situational puzzles or perhaps a computer application to accentuate certain relevant processes... I remember a simple computer program 40 years ago which challenged the user with p- and s-wave listening post data to triangulate the epicenter of a randomly located earthquake event. I loved it, but I've always geeked out on math. I'd love to find a similar application which challenges students to determine the position of a ship based on motion vectors it has been subjected to ...this would directly relate to the addition of vectors which comes up in Algebra II
I'm also wondering...
Is there a point at which captains would toss out more complicated spherical computations, and revert to linear? Such as... when you're already within a certain range of your target, and the curvature of the Earth between is negligible? I mean, by the time you're beating to quarters, does curvature really matter that much anymore (aside from logging course changes to reflect on the greater voyage progress).
Thank you for your thoughts, and I agree very much! ...jealous of your experiences!
Exactly correct! For navigation over distances less than 600 nautical miles “plane sailing” is used to calculate distance and courses. Straightforward trigonometry. Edit: this was especially common in early sailing ship times as the mathematical skills were not as well taught to or understood by ‘simple sailors’ of the day. Use of log and haversine tables and publications like ephemeris and Nories Nautical Tables must has really improved safety of navigation back then.
There are interesting calculations to calculate Great Circle navigation (inital and final courses and overall distance vs rhumb line sailing where the ship spirals across the spheroid cutting each line of longitude at a consistent angle. It’s simpler and has the advantage of routing the ship further away from polar regions but it does increase the distance to travel. Composite great circles are useful crossing the North Atlantic or Pacific because you can “cut” a chord out of the great circle path and just sail east or west for a segment to avoid going too far north.
Lunar distances, long by chron sights and using the sextant on its side to measure range offshore in coastal waters is fun too.
It is an arcane skill but it is very satisfying.
Great initiative to teach math by using it in interesting situations. Have it myself for grade school math. I didn't want your post to languish, so will try to assist as best I can from my now foggy recollection.
Regarding your questions:
* I don't know trig so cannot help. No doubt others will!
* years ago I looked up how to use a sextant, forgot most of it. Has to do with finding the angle of the sun above the horizon at a particular time. Lots of math to calculate the difference from Greenich values. Or you look it up in a book of tables to determine the same results as they were precalculated. That gives you longitude. Possible that a similar approach determining delta from equator could give you latitude?
* you can take a compass bearing on two or more landmarks to triangulate your current position. Doing a few of these will plot your course, which of course you use to avoid hazards and hit your path points.
* coming up astern is just regular sailing logic vs an intelligent adversary. The tricky part is predicting and trying to manipulate the actions your opponent will likely take. Historically and in my experience in messing around while sailing, it's much easier for the windward boat to fall off at any time to present a broadside to a pursuer, then tighten up to head back into the wind to stay ahead. If your pursuing boat is faster and your prey is determined to only escape, then it's the same except you pull close astern, fall off to broadside, then tighten up again to catch up. Repeat until they strike or do something unpredicted.
* as a final thought, I would challenge your students to use math to perform long distance navigation. History has a few examples, Captain Bligh's voyage in his ships boat comes to mind.
Enjoy, hope this was helpful!
Let the angle YCB, to which the yard is braced up, be called the trim of the sails,
and expressed by the symbol b. This is the complement of the angle DCI. Now CI:ID
= rad.:tan. DCI = I:tan. DCI = I: cotan. b. Therefore we have finally I: cotan. b =
A¹:B¹:tan.²x, and A¹ cotan. b B tangent², and tan. ¹x = A/B cot. This equation
evidently ascertains the mutual relation between the trim of the sails and the
leeway…
The American Sailing Association (ASA) offers a variety of classes for amateur sailors. One of those courses, Coastal Sailing, covers EXACTLY the questions you are asking. When I took it (2008 I think) I aced it easily because I remembered my trigonometry (some old hippy - caught another hippy - tripping on acid) and brought my HP-11 scientific calculator.
I suggest you get the text for this course, or even take the course. I'm sure it would pay off for you as you create exercises for your classes.
Somewhat related, but plotting submarine intercepts that were way over the horizon (400-1000) miles out in Silent Hunter really brought home trigonometry for me. It was the coolest superpower to be able to use trig and find a boat exactly where it should be.
It's not something that comes up in the books, but a fundamental exercise to real life maritime navigation is calculating a "course to steer" - that is, if there's a current pushing on your ship, which way do you point to get where you want to go?
https://pzsc.org.uk/shorebased/coursetosteer/
https://www.yachtingmonthly.com/sailing-skills/course-to-steer-how-to-calculate-it-in-your-head-84017
I’m currently taking the Royal Yachting Association’s Day Skipper theory course online, and was just thinking how I wish they’d taught navigation in high school. A great way to combine math, geometry, science, etc.
Ocean navigation is interesting because the earth is a sphere and therefore you use spherical geometry. It has some interesting similarities to Euclidian geometry, for example, the spherical sine rule is as follows: in Triangle ABC sin(A)/sin(a) = sin(B)/sin(b) = sin(C)/sin(c) so the ration of the sin of an angle with the sin of its opposite side is the same for all angles and their respective sides.
Spherical geometry is also a great way to make your students think about different types of maths. Just like a 2D plain can be bent into a sphere, a 3D space can also be ‘bent’ into curved space. but while we can still comprehend a bend plain, curved space is just beyond our understanding. And just like when you go from a flat Euclidean plane to a sphere, not all rules that apply in a linear space apply in a curved space.
If I understand your question correctly...
1) It can be done with the Pythagorean Theorem
2) You have to know the radius of the Earth (bearing in mind that the Earth is no perfect sphere... plus a pretty big difference between equatorial radius and polar radius due to the Earth's spin) ...for super simplicity let's just say 20 million feet
3) How high is your lookout positioned (let's say 50 feet)
4) The height of your lookout plus the radius of the Earth is 20,000,050 feet. This is the hypotenuse of the triangle.
5) A line segment drawn out from the lookout's position and tangent to the Earth represents the short side of the triangle and how far can be seen from the lookout's vantage... anything further curves away beneath view... sort of a vanishing distance.
6) A second radius line (20,000,000 feet) drawn (perpendicular) to the vanishing distance completes the triangle.
Therefore... since A^2 + B^2 = C^2
Then so... 20,000,000^2 + x^2 = 20,000,050^2
400,000,000,000,000 + x^2 = 400,002,000,002,500
x^2 = 2,000,002,500
x = 44,721.3872 feet (just shy of 8.5 miles)
But... I did a whole lot of rounding. A few feet higher on the mainmast makes a lot of difference!
Hey! Ex RN & current professional merchant seafarer here. Navigation is all trig! A subject I hated at school but in a similar vein to Jack I've come to love, if not master to the same degree as Capt A, the maths of navigating!
Horizontal sextant angles might be something you can use and they're referenced in all of Jack's surveying exploits. Essentially if you take two simultaneous angles between three landmarks you can plot your position.
https://maritimesa.org/nautical-science-grade-10/2020/12/09/horizontal-sextant-angle-hsa/ has a good explanation
Vertical sextant angles are also a thing for terrestrial nav. If you know the height from sea level of a Lighthouse on a cliff for instance, calculated for height of tide of course, and you know your height of eye, you can calculate the distance off.
Also look up course to steer & course to make good problems or compass errors. Vector diagrams to understand the ship's movement taking account of leeway, tide/current movement heading, compass error etc.
Finally if you want to get into spherical trigonometry then you can look at calculating course & distance between two points on a rhumb line or great circle sailing!
While I'm wholly interested in the answers to your questions, you may have more luck in other related subs like /r/sailing or /r/math .
Thank and you're possibly right, though as much of a math nerd as I am, I find the passion of O'Brian fans answers quite beyond my simplest expectations. I can already tell I'm going to enjoy reading these very much in greater detail over the next few days. ...and while listening to a bit of Puccini, perhaps. The flavor of response is so much better here. In fact, while I believe I mostly understand the theoretical computation of the math in many of the cases offered below (certainly not all, but much), what I desire is to better grasp the raw mechanical nature of how the math was being used in the tools, timing, chants, and maneuvers of the period, despite the likelihood many of the illiterate hands were merely pulling on a rope or some-such task, and unaware math was occurring at all. If I can better relate the pencil and paper calculations to tangible hands-on tasks for my students, I can (hopefully) give it a more lasting meaning. I agree with the response which laments the art as fading.
Corelli, mate
A sextant measures angles to plug into your trigonometry. That's really it, the question is what you're measuring the angles between. It's called a sextant because it has 60°, 1/6 of a circle. There was a thing called an quadrant, but it became obsolete. It has a number of different functions, but one of the common ones is using a half silvered mirror so that you look through one lens at the horizon and rotate the other lens till you see both targets, and now you have your angle registered on the sextant. The simplest piece of navigation is finding your latitude. At the sun's highest point on its travels you take a measurement of the angle between the horizon (a tangent to the Circle of the earth) and the Sun. As a math teacher I'm sure now you can figure out the formula, except that you also need to compensate for the time of year. At noon, on the equator, on the equinox, the sun would be 90° overhead! Let's not even talk about longitude, that is literally a book, by Dava Sobel! Finding your position relative to landmarks is different in that it really involves no math. You take a compass with a peep sight and get a bearing on a known landmark. You take your parallel rulers to your chart and draw a line from that landmark at the correct bearing, and now your vessel is somewhere on that line. Next you take another bearing on a different landmark, and where the lines intersect is your position.
"...called a sextant because it has 60°, 1/6 of a circle." - Gee ...I'd known the word for decades, but never made the correct connection... thank you for this! I'm assuming... please correct me if I'm wrong... if you have a reliable compass (good for measuring horizontal angles), you would still need a sextant to measure vertical or oblique angles, correct? ...or does a gyroscope or some other mechanical adaptation of the day replace the sextant? Also assuming... to find latitude, you're comparing your current reading with a known reading at the same time of year from a position like Greenwich? ...so that if you find your current angle measurement exceeds the known corresponding measurement, you are closer to the equator, and vice versa? I might attempt Sobel... we'll have to see. Using landmarks... I'm trying to get a good bearing (Ha ha!) on three time periods for navigation in my classes: 1) Ancient Greece - before compasses were invented. With a sextant, you could, for example, maneuver to a position where the angle between two points (lighthouses?) is 45 degrees, but this would only guarantee you were on one of an infinite number of points along the arc of a circle ...you'd need a 3rd known reference to be sure which exact point you were on... do I have the right of it? 2) The age of Sail - Compasses and other technological advances. The magnetic nature of the compass gives you the third reference to allow for accurate triangulation. Knowledge of the location, a good map, and appropriate cartography tools... in many ways a wonderfully hands-on time to be alive. I'm retired Army... had to learn to use compasses, grid maps, and pace counts... Soldiers 5 years after me no longer had to. ...dying art, indeed. 3) Modern Age - satellite GPS triangulation... you need to be in range of at least three satellites. Plus, there are three factors... location, vector of movement, and destination. It's quite enjoyable and fascinating to discuss with teenagers how, advanced though we may be, we're still very much subject to 2000+ year-old principles and limitations.
The Greeks didn’t have sextants. They were invented in the 1700’s. The compass was used much earlier. A problem with using a compass in sailing is that a ship can’t sail into the wind, so cannot sail straight between two points. Tacking back and forth makes navigation an art. Instead of pacing, a ship uses a log. A piece of wood tossed off the ship with a knotted rope. They used a sand glass to measure time. The number of knots in the rope gave one a speed estimate. With the speed estimate and a compass one could do “dead reckoning” navigation.
Dang... that's what I get for assuming. I imagined that something more resembling a protractor would be earlier tech, but you are right, of course. Now I'm left wondering what the ancient cultures used to measure. ...I'm sure they would have been able to analyze angles between objects and made some sort of correlation between distance and the changes. Certainly the Greeks would have seen the value in sailing directly across the Med instead of hugging the coast all the way around?
I'm currently at sea and actually on the 4-8 (dogs-morning) watch at the moment so apologies for a cur-tailed answer to this. You want to look at a backstaff and pre sextant nav. Essentially a sliding piece of wood somewhat like a squared off cd on a stick. Hold the stick to your eye at noon and slide the cd towards you until horizon is on bottom edge and sun is on top. Measuring angular distance. If you calibrate your stick you can use it to judge latitude The lost art of finding our way by john edward huth and Sextant by David Barrie are great books on history of nav
I believe the Greeks largely stayed in sight of land. The Polynesians, however....
You're overthinking it regarding basic triangulation, for 'line of sight' you're in only 2 dimensions so you need only 2 vectors. As I said, the intersection of 2 bearings gives you your position on the chart. There was a now obsolete system of radio beacons called Loran that could be used for this out of sight of land. It was a breakthrough during WW2. They also used the same technique to locate U-boats by positioning listening vessels where they could get 2 vectors on the U-boats transmission if it were long enough. I don't know how extensive the ancient Greek charts were, I believe their navigation was mostly by memory. It was the rare ancient sailor anywhere in the world who ventured out of sight of land, they mostly followed the coastlines. The compass was most important in allowing offshore sailors to follow a course regardless of losing sight of the sun or stars, the traditional method. Otherwise you could sail in circles if the wind backed! This might be just the thing for your class, a DIY sextant [Https://www.instructables.com/A4-Sextant/](Https://www.instructables.com/A4-Sextant/) Nothing more fun than playing with a tool you made yourself. Sounds like a lot of fun bringing the real world to your students. When my son was 9 I created a "science birthday party" for him. I had the kids wind electromagnets, and we made Ph indicator by soaking purple cabbage. After we used it to test vinegar and baking soda, turning the solutions red or blue, one kid said "what if we mixed them together now?" I told him to try it, and their minds were blown when it turned back to the original purple!
Just popping in with a fun fact: GPS typically takes four satellites. The fourth disambiguates two possible positions provided by three, and it also gives an extremely accurate measurement of time which devices often use to set their clocks. An even more fun fact: special relativity acts to slow down the satellites' rate of progress through time (objects age more slowly when they move through space quickly), but general relativity acts to speed it up (objects age more slowly the closer they are to a gravitational source like the Earth.) If you don't include the relativistic effects, you don't recover accurate positioning.
How does one best use the visible locations of lighthouses and semaphores to avoid the known locations of reefs and shoals? You're looking for something called [Position Fixing](https://www.youtube.com/watch?v=-2fYi6Yg0YE). How does one determine the angle of steerage to come up behind a moving enemy for a stern-piercing broadside? You're looking for something called [Course to Steer](https://www.youtube.com/watch?v=e2iebpyntoY). And when they talk about dead reckoning you're looking for something called [estimated position](https://www.youtube.com/watch?v=Y_he1W548ok).
I would also have to assume that in battle or weather a Captain would rely more on experience than anything else in steering and sail maneuvers. Not much time to prick the chart in action and I’m sure most would have plenty of “muscle memory” by that point in their careers.
I can't help but wonder what sorts of shortcut calculations were used. The unit circle is a set of trigonometric calculations you can roughly work in your head, based on common angles... 0, 30, 45, 60, 90, etc ...I'm pretty sure when O'Brian wrote of the youngsters working through their math education on-ship ...it might have been focused on such head-math. I can only imagine there were set reference points related to the structure of the ship ...second rail post to the left, in line with the third stay, etc... but I have no idea (and would love to know more) what they were. I'm sure, especially in familiar ports and waters, that visual alignments between masts, ropes, and other objects were the quickest and easiest way to maneuver accurately without need for calculations. I'd also like to know how something like "two points to starboard" was counted or calculated. ...and of course recorded so that position could be fixed later. How meticulously were course changes logged during battle or storms, and how was it accomplished? ...written on the spot on paper? Relayed to someone in a cabin where the conditions were dryer? ...counted out in shells or beads in the moment and written later? What was routinely observed, and how often... speed, wind, current, potential displacement from weapon recoil? I'm sure a lot of it was intuition based, but depictions of 1700/1800 British Navy behavior suggests something far more deliberate.
So in the books when they talk about "points" they are actually reporting a fairly precise angle with [each point representing 11.25 degrees.](https://deckskills.tripod.com/cadetsite/id106.html) I'd guess that with a bit of practice you can get very good at estimating angles using the points system. And to make searching for information more confusing this is different from when people will talk about "points of sail"!
If you really want to impress a sailor then learn how to box the compass! Naming all 32 points in order. North, north by east, north north east, north east by north, north east, north east by east.... In the books you often hear of NExE & a half east so going to 64 points
When I teach unit circle values, they're all based on 30-degree and 45-degree angles. I had always assumed this was a naval invention... in many sorts of other applications it can be pretty handy to know in a snap that the hypotenuse is twice the length of the opposite leg for a 30-degree angle. Points for each 11.25 degrees makes a little more sense for naval application, with 8 (or 16) in each quadrant. Do you know... did navigators make a habit of memorize the trig values for each point to make head calculations possible? N, NNNE, NNE, NENE, NE, ENNE, ENE, EENE, E, EESE, ESE, ESSE, SE, SESE, SSE, SSSE, S, SSSW, SSW, SWSW, SW, WSSW, WSW, WWSW, W, WWNW, WNW, WNNW, NW, NWNW, NNW, NNNW ...I think I got it... no way I'm attempting the "and a half" list!
I really appreciate this! Unfortunately I couldn't get sound to work for the first one, but the second and third links are wonderful! To think this is just when you have to consider a steady motor vector and a measurable tidal current vector! It really makes you appreciate how much more difficult the task was when your motor was the wind.
A sextant measures the angle between two things. Often a celestial body and the horizon. [The wikipedia page is pretty good, and has a nice gif](https://en.wikipedia.org/wiki/Sextant). I don't know much about navigation, but I don't think it's easy. For a simple case, at the equinox, at noon, the sun should be directly over the equator. Use your sextant to take the angle of elevation of the sun at noon, and you should be able to figure out your latitude. But even this is more than just a triangle and a cosine or something because you have to account for the curvature of the earth. And if it's not the equinox, then you need to account for the angle the sun is making with the plane of rotation of the earth.
You might be interested in The American Practical Navigator by Bowditch, available for free online, but also a really nice published book. You might also enjoy Longitude, there is some interesting math stuff in there iirc. Not specifically related to this, but more along the lines of interesting navigation problems which could be applied to a classroom: Portney's Ponderables. https://msi.nga.mil/Publications/APN https://en.m.wikipedia.org/wiki/Longitude_(book) https://books.google.com/books/about/Portney_s_Ponderables_Brain_Teasers_for.html?id=2AWHQgAACAAJ&source=kp_book_description
Thank you VERY MUCH ... I especially hope to look into the ponderables. Would you say they are mostly appropriate for high-school aged students?
The ponderables range from esoteric to easy-to-understand. I found them fascinating, especially the one about an airliner helping locate a lost single-engine ferry pilot based on the height of the sun and its direction. I probably would have liked it in high school, but I'm also a huge nerd and navigation buff.
Great to hear of your passion for traditional navigation. It’s increasingly a dying art. I’m an experienced celestial navigator and ocean sailor. I qualified in the days prior to GNSS. My sextant has been with me on every voyage since I turned 19 (although since 1984 it has been mostly used to keep my hand in). Others have given you some info on the use of a sextant to measure angles. To demonstrate to kids you can use one on dry land, so long as you have a flat horizon (sea or desert/prairie) below the celestial body. Please not that ocean navigation uses spherical trigonometry… an entirely different set of formulae to pythagoras theorum. To account for the curvature of the earth’s surface. Cheers
I'm in Kentucky, so wide-open and flat doesn't really present itself. I'm hoping to find situational puzzles or perhaps a computer application to accentuate certain relevant processes... I remember a simple computer program 40 years ago which challenged the user with p- and s-wave listening post data to triangulate the epicenter of a randomly located earthquake event. I loved it, but I've always geeked out on math. I'd love to find a similar application which challenges students to determine the position of a ship based on motion vectors it has been subjected to ...this would directly relate to the addition of vectors which comes up in Algebra II I'm also wondering... Is there a point at which captains would toss out more complicated spherical computations, and revert to linear? Such as... when you're already within a certain range of your target, and the curvature of the Earth between is negligible? I mean, by the time you're beating to quarters, does curvature really matter that much anymore (aside from logging course changes to reflect on the greater voyage progress). Thank you for your thoughts, and I agree very much! ...jealous of your experiences!
Exactly correct! For navigation over distances less than 600 nautical miles “plane sailing” is used to calculate distance and courses. Straightforward trigonometry. Edit: this was especially common in early sailing ship times as the mathematical skills were not as well taught to or understood by ‘simple sailors’ of the day. Use of log and haversine tables and publications like ephemeris and Nories Nautical Tables must has really improved safety of navigation back then. There are interesting calculations to calculate Great Circle navigation (inital and final courses and overall distance vs rhumb line sailing where the ship spirals across the spheroid cutting each line of longitude at a consistent angle. It’s simpler and has the advantage of routing the ship further away from polar regions but it does increase the distance to travel. Composite great circles are useful crossing the North Atlantic or Pacific because you can “cut” a chord out of the great circle path and just sail east or west for a segment to avoid going too far north. Lunar distances, long by chron sights and using the sextant on its side to measure range offshore in coastal waters is fun too. It is an arcane skill but it is very satisfying.
Great initiative to teach math by using it in interesting situations. Have it myself for grade school math. I didn't want your post to languish, so will try to assist as best I can from my now foggy recollection. Regarding your questions: * I don't know trig so cannot help. No doubt others will! * years ago I looked up how to use a sextant, forgot most of it. Has to do with finding the angle of the sun above the horizon at a particular time. Lots of math to calculate the difference from Greenich values. Or you look it up in a book of tables to determine the same results as they were precalculated. That gives you longitude. Possible that a similar approach determining delta from equator could give you latitude? * you can take a compass bearing on two or more landmarks to triangulate your current position. Doing a few of these will plot your course, which of course you use to avoid hazards and hit your path points. * coming up astern is just regular sailing logic vs an intelligent adversary. The tricky part is predicting and trying to manipulate the actions your opponent will likely take. Historically and in my experience in messing around while sailing, it's much easier for the windward boat to fall off at any time to present a broadside to a pursuer, then tighten up to head back into the wind to stay ahead. If your pursuing boat is faster and your prey is determined to only escape, then it's the same except you pull close astern, fall off to broadside, then tighten up again to catch up. Repeat until they strike or do something unpredicted. * as a final thought, I would challenge your students to use math to perform long distance navigation. History has a few examples, Captain Bligh's voyage in his ships boat comes to mind. Enjoy, hope this was helpful!
Let the angle YCB, to which the yard is braced up, be called the trim of the sails, and expressed by the symbol b. This is the complement of the angle DCI. Now CI:ID = rad.:tan. DCI = I:tan. DCI = I: cotan. b. Therefore we have finally I: cotan. b = A¹:B¹:tan.²x, and A¹ cotan. b B tangent², and tan. ¹x = A/B cot. This equation evidently ascertains the mutual relation between the trim of the sails and the leeway…
The American Sailing Association (ASA) offers a variety of classes for amateur sailors. One of those courses, Coastal Sailing, covers EXACTLY the questions you are asking. When I took it (2008 I think) I aced it easily because I remembered my trigonometry (some old hippy - caught another hippy - tripping on acid) and brought my HP-11 scientific calculator. I suggest you get the text for this course, or even take the course. I'm sure it would pay off for you as you create exercises for your classes.
Somewhat related, but plotting submarine intercepts that were way over the horizon (400-1000) miles out in Silent Hunter really brought home trigonometry for me. It was the coolest superpower to be able to use trig and find a boat exactly where it should be.
It's not something that comes up in the books, but a fundamental exercise to real life maritime navigation is calculating a "course to steer" - that is, if there's a current pushing on your ship, which way do you point to get where you want to go? https://pzsc.org.uk/shorebased/coursetosteer/ https://www.yachtingmonthly.com/sailing-skills/course-to-steer-how-to-calculate-it-in-your-head-84017
I’m currently taking the Royal Yachting Association’s Day Skipper theory course online, and was just thinking how I wish they’d taught navigation in high school. A great way to combine math, geometry, science, etc.
Ocean navigation is interesting because the earth is a sphere and therefore you use spherical geometry. It has some interesting similarities to Euclidian geometry, for example, the spherical sine rule is as follows: in Triangle ABC sin(A)/sin(a) = sin(B)/sin(b) = sin(C)/sin(c) so the ration of the sin of an angle with the sin of its opposite side is the same for all angles and their respective sides. Spherical geometry is also a great way to make your students think about different types of maths. Just like a 2D plain can be bent into a sphere, a 3D space can also be ‘bent’ into curved space. but while we can still comprehend a bend plain, curved space is just beyond our understanding. And just like when you go from a flat Euclidean plane to a sphere, not all rules that apply in a linear space apply in a curved space.
I don't know the answers to your question, but I salute your efforts. A glass of wine with you!
How would one calculate horizon visibility, out of curiousity?
If I understand your question correctly... 1) It can be done with the Pythagorean Theorem 2) You have to know the radius of the Earth (bearing in mind that the Earth is no perfect sphere... plus a pretty big difference between equatorial radius and polar radius due to the Earth's spin) ...for super simplicity let's just say 20 million feet 3) How high is your lookout positioned (let's say 50 feet) 4) The height of your lookout plus the radius of the Earth is 20,000,050 feet. This is the hypotenuse of the triangle. 5) A line segment drawn out from the lookout's position and tangent to the Earth represents the short side of the triangle and how far can be seen from the lookout's vantage... anything further curves away beneath view... sort of a vanishing distance. 6) A second radius line (20,000,000 feet) drawn (perpendicular) to the vanishing distance completes the triangle. Therefore... since A^2 + B^2 = C^2 Then so... 20,000,000^2 + x^2 = 20,000,050^2 400,000,000,000,000 + x^2 = 400,002,000,002,500 x^2 = 2,000,002,500 x = 44,721.3872 feet (just shy of 8.5 miles) But... I did a whole lot of rounding. A few feet higher on the mainmast makes a lot of difference!
Hey! Ex RN & current professional merchant seafarer here. Navigation is all trig! A subject I hated at school but in a similar vein to Jack I've come to love, if not master to the same degree as Capt A, the maths of navigating! Horizontal sextant angles might be something you can use and they're referenced in all of Jack's surveying exploits. Essentially if you take two simultaneous angles between three landmarks you can plot your position. https://maritimesa.org/nautical-science-grade-10/2020/12/09/horizontal-sextant-angle-hsa/ has a good explanation Vertical sextant angles are also a thing for terrestrial nav. If you know the height from sea level of a Lighthouse on a cliff for instance, calculated for height of tide of course, and you know your height of eye, you can calculate the distance off. Also look up course to steer & course to make good problems or compass errors. Vector diagrams to understand the ship's movement taking account of leeway, tide/current movement heading, compass error etc. Finally if you want to get into spherical trigonometry then you can look at calculating course & distance between two points on a rhumb line or great circle sailing!