. So, thank you all. It's really wonderful to be back here in Saal 1 in Hamburg.
And, yeah, I've been spending a lot of time at sea.
And this gives me a lot of opportunity to think about sort of the deep questions.
Things like, where are we? And why are we here?
Now, this talk is not going to cover that second one at all.
But, hopefully, we'll be able to answer some questions about the first one.
And people have been working on ways of navigating for millennia.
The Polynesians used the local geography that created standing
waves, and they created these really neat maps by tying sticks
together to represent the patterns of waves between their islands.
But most navigation has been based on the stars.
The Hawaiians created something they called the Star Compass that looked
at the rising and setting of the different constellations so that they
could pinpoint the location of their voyaging canoes on the open ocean.
There are some incredible Islamic astronomers who built astrolabes
that allowed them to predict the positions of the constellations
and then use that to find their position on the planet.
And it's said that the Vikings had sunstones that allowed
them to use the sun for navigation, even in the overcast
conditions that are pretty common here in Northern Europe.
So, let's try to answer a basic question first. Which hemisphere are we in?
So, if you wake up early and watch the sunrise, if it goes
up and to the right, you're in the Northern Hemisphere.
If the sun instead goes up and to the left, you're in the Southern Hemisphere.
And if the sun goes mostly straight up and down,
you're probably in the tropics or near the equator.
But who am I kidding? You all are never going to wake up in time for the sunrise.
So, you can do the same thing with the sunset.
If the sun is traveling to the right as it sets, you're in the Northern Hemisphere.
If it travels towards the left, you're in the Southern Hemisphere.
And, you know, let's say you don't get a sunset at all.
That means you're probably near one of the poles.
The easy way for that is if you see polar bears, you're in the North Pole.
And if you see penguins, you're in the South Pole.
But maybe you don't want to wait until sunrise or sunset.
So, you can just stand really still and watch your shadow.
And if it goes to the right, you're probably in the Northern Hemisphere.
And if you put some stones on the ground to
track that shadow, you've made the basic sundial.
In fact, we actually get the word clockwise from
the direction that the shadow moves on the sundial.
Now, if you're in the Southern Hemisphere, the
shadow is going to go in the opposite direction.
It moves to the left.
And unlike the toilets, it is true that sundials
in the Southern Hemisphere do run backwards.
But maybe you don't want to stand still.
So, you can put a stick in the ground and create an actual sundial.
We have an intuition that in the early morning and in the evening, our shadows are longer.
And around midday, they get short.
And if you measure the time of that shortest shadow, we call that local or solar noon.
And this happens when the sun is directly lined up with you,
either north or south, depending on which hemisphere you're in.
And we say that this is the time of the meridian
passage because the sun has passed your meridian.
However, if you're in the tropics, the sun really isn't north or south of you.
It's kind of more overhead.
And on a couple days a year, you actually get no shadow.
And on these zero shadow days, that means the sun is directly overhead.
This happened to be when we are crossing the equator on the equinox.
And I was amused by the fact that I had almost no shadow.
There's actually something serious you can do with this, serious geographical importance.
In 300 BC, Erasthenes, the Greek mathematician, had heard
that the temple at Cyene cast no shadow on the solstice.
And based on his knowledge that the earth is round, and this is how out
-of -date flat earthers are, that 2 ,000 years ago the Greeks knew it was
round, he assumed that the sun's rays coming in were effectively parallel.
And he realized that if he measured the angle of the shadow in
Alexandria, that was the same as the angle between the two cities.
So he did what any reasonable person would do.
He hired someone to walk between the two cities, counting their steps.
A little bit of 2 pi r math later, and he had computed the circumference
of the earth within a few percent of the accepted value that we use today.
He went on to write a book literally called Geography, and he
invented a lot of concepts that we use today, like the parallel
lines of latitude, as well as the north -south meridians.
He also invented a way of generating prime numbers, but
that's math, so we're not going to talk about that today.
If you wanted to recreate his experiment, or if you wanted to
use his findings, you could find a handy obelisk and look at
its shadow during local noon, the time when it's the shortest.
The angle at the top there is approximately your latitude.
It's related to your latitude.
We call this the zenith angle, because it's the angle when the sun is at its highest point.
Measuring that could be a little tricky on the obelisk.
If you can find out the height of the obelisk, you could then measure the shadow.
Because it's the right triangle, you can do the
tangent of these two things, but that's math.
Let's try to do this in an easier way.
If you were instead to stand at the end of the shadow and measure the angle
from the ground to the top of the obelisk, we could call that the height angle.
And then, because it's the right triangle, the zenith is 90 minus the height.
It's still a little bit of math, but it's just basic arithmetic.
I hope we can handle that.
Now, if you're at sea, maybe you don't have a handy obelisk.
You know, it's hard to find a good obelisk on board your 16th century caravel.
So, you want some sort of tools for that.
One of the early ones is called the cross staff.
And it has a fixed ruler and a sliding cross staff that you can use to measure the angle.
So, you cite it such that the sun is at the top of
that sliding part and the horizon is at the bottom.
And then, that angle can be read from the markings on the ruler section.
This has a lot of drawbacks, one of which is that you're staring directly into the sun.
It's also hard to see both ends of the staff simultaneously.
So, if you're on a more modern ship, some sort of spaceship, you might use a sextant.
And sextants do actually have a place in space navigation.
Apollo 13 used one to judge its reentry angle coming back when they had a computer failure.
And ESA has done a project to try using them on ISS as well.
But, back on the sea, you would use a sextant exactly the same way as the
cross staff to measure the height of the sun or a star or something in the sky.
And this has a lot of advantages that you can flip down
filters so that you're not staring directly into the sun.
It also has a series of mirrors so that you can bring the sun down to
the horizon and then measure the angle on the rotating scale, usually
with a vernier or some other mechanism to make it even more exact.
Now, as I said, this height angle is not directly
your latitude, but it's related to your latitude.
Oh, sorry. The way this sextant works is pretty neat.
You look through the eyepiece and see the horizon.
The sun comes through filters and then bounces off a rotating
mirror on the fixed mirror and then through the eyepiece.
And by adjusting the arm, you can rotate or pivot
that mirror and then measure the angle of the sun.
So, as I mentioned, this is related to your latitude, but not directly.
And if you were to take the height of the sun every day at
noon for the whole year, you'd find that it actually varies
almost 46 degrees between the summer and winter solstice.
This is the northern hemisphere dates.
The southern hemisphere is the other way around.
And this figure, called an analemma, results because the Earth
is tilted in its rotation relative to its orbit around the sun.
So, if we draw a diagram of the sun, [inaudible] of the Earth
-sun system, we have the Earth with its equator and its tilt.
The sun is off some distance away.
And we would say that the sun, the north -south position of the sun varies during the year.
And we call that the declination.
So, near summer in the northern hemisphere, we
expect declinations to be around 20 degrees or so.
And if we're standing here in Hamburg and we take a
height measurement, we might get, say, 57 degrees.
The zenith angle is 90 minus that, or 33 degrees.
And we can see that our latitude, which is our position relative
to the equator, not to the sun, is just the sum of those two.
So, the zenith angle plus our declination gives us
our latitude, roughly 53 degrees here in Hamburg.
Some of you might look at this diagram and say, wait, there's a symmetry here.
Wouldn't someone south of the sun also be able to measure 57 degrees?
And you're right. You can do this from both sides.
And if we make the zenith angle a signed value, so someone
looking north to the sun from the southern hemisphere, we can give
them a negative zenith angle, then the math works out the same.
That zenith plus declination gives us their latitude of 13 south.
Obviously, keeping track of that declination becomes really important.
There are almanacs that have been published. Bowditch is one of the most famous.
In fact, it's still in print 200 years later.
That's how much an epitome of navigation it is.
And included in it, what Bowditch says is the
most important part, is the table of declinations.
So this is the declination of the sun for every day of the year for several years.
So if you get the modern version, it has usually the next 10 years or so.
And keeping track of this declination is so important that Bowditch actually points
out that one of his competitors made a mistake by treating 1800 as a leap year.
And that error in position caused by that
incorrect calculation caused the loss of two ships.
So the moral here is calendar math is hard and don't write your own calendar library.
Use something a little better.
Now, the example that I gave had nice whole numbers for the degrees.
In general, most celestial navigation uses base 60 degrees, minutes, seconds.
And working with this is awful.
You have to remember to carry the 60 or borrow the 60. It's a real pain.
I make a huge number of arithmetic errors doing it.
Which is why I've designed a slide rule to help me with this.
And slide rules are portable mechanical calculators that
allow you to do easy lookups of different functions.
There are ones that are specialized for navigation.
That if you're on a spaceship, Spock here is holding the E6B student flight
computer, which is still used by beginning pilots learning to fly airplanes.
NASA has also created some custom ones back during the Apollo
era that were used for computing things related to reentry.
And mine was very much inspired by that.
I don't have a lot of manufacturing capability, so it's just printed on paper and laminated.
And you can play with an online simulator on my website if you want to.
And basically, there's an outer ring that keeps track
of the minutes of your latitude or your measurement.
There's an inner ring that can do addition, subtraction, and
then also has lookup tables to compute some different functions.
And then there's a pointer that swings around for reading out the more exact values.
Going back to Bowditch, he actually shows the four additional steps that
you need to do to correct the sextant in addition to the declination.
And he provides lookup tables, but again, that's a bunch of math that you have to do.
So, let's see how we do this with the slide rule.
The first error is called index error.
And this is a manufacturing issue with the slide rule, where when you set it at zero,
the horizon doesn't necessarily line up between the mirror and the real horizon.
There's a little discrepancy.
This is especially common with cheap plastic sextants,
like the Davis Mark III here, that swell with temperature.
So, what you do is you look out at the horizon and
you adjust the arm until the two lines line up.
And then you just make note of that number. In this case, about 1 .2 minutes or so.
And then on the slide rule, you set the minutes of your measurement.
So, 53 .6 here, where you point the index on the inner wheel at it.
And then you swing the pointer to the minus 1 .2, which is in red.
And then you can read out the corrected value once the index error is taken, 52 .4.
The second issue that you need to correct is your height of eye above the ocean.
What you see is very different if you're down in the
waves in a lifeboat or up in the mast in a tall ship.
And this is because our sextant height is measuring our view of the horizon to the sun.
But what we actually want is the angle between the tangent plane and the sun.
And this error is called the dip.
So, our apparent altitude is the sextant height minus the dip.
And the slide rule has a lookup table that will compute those values for you.
In this case, I've selected 4 .5 meters.
The third issue you need to take into account is refraction, which gives us rainbows.
But it also makes things appear to be not where they are
when light goes through different densities of air or water.
The Dutch cartographer Snellius was the one who formalized
this with Snell's law and gave us the math behind it.
And basically, the amount of refraction that you see depends on the angle of
the object to the horizon, the air temperature, and the atmospheric pressure.
So, again, there's a lookup table on the slide rule that does this for you.
So, the fourth correction, the last correction
that you need to apply, is the diameter of the sun.
When you take your sextant measurement, typically I
bring the lower limb of the sun down to the horizon.
But all of the math is based on the center of the sun.
And the sun changes diameter during the year as
we move in and out closer or further from the sun.
And then sometimes you may not have a good view of the bottom of
the sun, so you have to do an upper limb measurement from the top.
So, again, we have a little lookup table that does that computation
for us and automatically adds or subtracts that semi -diameter.
So, let's put this all together into a demo. I have one of the slide rules here.
And if I start by bringing this inner index to the 53
.6 there, I can then add my 1 .2 minutes of index error.
And then rotate the inner wheel to 4 .5 meters height above sea level.
Bring this up to the 25 degrees at 20 degrees C.
And then rotate it for the December.
And I did a lower limb measurement, so I go this way. Whoops.
And, you know, it's harder at sea, but it's hard on stage too.
And so, we now get our corrected minutes of 2 .9.
And when we swung the arm, the pointer, this way, we crossed this zero.
So, that means we need to add one to our result.
So, what we ended up with there is, after applying those four
corrections, we can read out the minutes of our sextant reading at 2 .9.
And, again, we have to carry that one, but that's not too bad.
However, we want to use the zenith angle, which, remember, is 90 minus our height.
The red numbers there give us the complement, so we don't have to subtract 2 .9 from 60.
It has done it for us. So, our zenith angle is 76 degrees, 57 .1.
We then look in today's almanac to get the declination
of the sun, and we've just passed the solstice.
So, it is a long ways down there at negative 23. So, we swing the pointer to 10 .7.
And then, we can read the minutes of latitude, 46 .4.
And then, we do a little bit of arithmetic on the zenith angle plus the declination
to get roughly the latitude of Hamburg, here at 53 and a little more than half.
And that would have been a huge amount of, you know, columns of
addition and subtraction that we've just done in a few seconds.
So, that's our latitude.
Unfortunately, we do not have enough time to talk about longitude in real detail.
And longitude actually is really just about time.
So, the way we compute longitude is because the sun appears to be moving 360
degrees per day, which means it's moving across the Earth 15 degrees per hour,
which in human terms is almost 1700 kilometers per hour, or 463 meters per second.
The sun is really moving quite fast.
So, if we know the time that the sun passes our meridian, we can
compute the angle back to some other location, simply by taking our
local noon, subtracting the time that it passed that other meridian,
multiply by 15, and that gives us the hour angle between them.
Now, where that origin is, is a political question. The French wanted it in Paris.
The UK wanted it in Greenwich.
And in this case, the Brits won, and we use Greenwich as our prime meridian.
We also need a time source that we keep track of, or a universal time.
And UTC is what we use, which in English is neither English nor French.
So, in this case, the decision was to compromise and go with neither of them.
And keeping track of that time is a really hard problem at sea.
The ships, because they go, they swing back and forth
as they heel, means pendulums don't work very well.
And the Royal Society proposed a prize for someone who
could develop a clock that would keep time on ships at sea.
Harrison won, though Harrison completed his first clock in 1736, didn't claim
the prize, but got a stipend to continue developing it for the next 30 years.
And he actually brought it down to a pocket watch size regulator.
There's a really good book called Longitude about that development.
I, these days, just use a cheap Casio, because, you know, it keeps really good time.
Set to UTC, so I don't forget.
One last copy -out for longitude is that the sun
doesn't always pass Greenwich at noon Greenwich time.
In fact, a sundial will be fast or slow at different parts
of the year due to Kepler's Third Law, which is a lot of
math, and the fact that our orbit is a bit elliptical.
So some sundials will have this equation of time graph that you can calculate.
But it turns out that we need to think about how
accurate are we going to get with this technique.
When you're taking the noon site, the sun is following sort of a parabola up to its zenith.
And in the last few minutes, it's just barely moving.
So if we look at, like, the eight -minute span around local noon,
it's only going to be a, you know, a fraction of a degree, or
excuse me, a fraction of a minute of angle difference between them.
Which means, like, a kilometer of error, of latitude.
During those eight minutes, the sun has moved hundreds of kilometers.
So trying to pinpoint your longitude with this technique
is going to give you just absurdly large errors.
So in practice, this is not how we compute our longitude.
There's a really neat technique called the intercept method that
combines your noon site of latitude with other sites during the
day to allow you to triangulate, via geometry, your position.
It involves a lot of math.
So I've also been working on trying to develop some better tools
for this to replace the giant almanacs that you have to carry.
This is one of the ones I've done that will compute
the bearing to a celestial body, given your location.
And that replaced 90 pages of almanac with a diagram. Thank you.
The back of the slide rule has another diagram that is also used for computing longitude.
And this turns out that I reinvented a technique from one of the Islamic
astronomers, called the Universal Astrolabe, that they had very similar
design for, so they didn't have different plates for each latitude.
But in any event, this is a, like I said, longitude is a much more complicated subject.
And we are going to be having a workshop at the House of Tea pretty much after this talk.
So if you're interested and you want to try your hand at using the sextants or
the slide rules, I've got a few different ones with me that we can play with.
And they are, they're a lot of fun.
Leave your laptops and your calculators at home, because
we're going to be doing this with the old -fashioned way.
So, thanks so much for your attention and I'd love to answer any questions.
So, thank you.
Herald: Thank you very much, Trammell, for this presentation.
I'm sure some of us are now more certain about our mutiny.
Maybe there are more questions for actually doing it.
Let's see, we have microphone number one. You have a question?
Hello. What's the reason?
There is one other source of uncertainty that I can see,
namely that you need to have a horizon reference that's sharp.
And I can think of a different way to get a horizon reference, namely local vertical.
Is this a problem?
And is local vertical ever used to get a better horizon reference?
Trammell: So the question is about, do you always need to look at the horizon?
And you do not always need the horizon. You can use an artificial horizon.
Some sextants will have a bubble level for practice with.
There are some older techniques that also had a
hanging string to give you a vertical reference.
They tend to be less effective at sea, where the ship is rocking back and forth.
So, that can be challenging.
Some of the aircraft ones were fed from the artificial horizon and were able to use that.
I didn't include this in the slides, but aircraft up until World War II had
bubbles on the roofs where the navigator could stand and take star sites.
And that typically used the artificial horizon for that.
Guest: Thank you.
Thank you. Let's see. We have some questions from the internet. Signal Angel.
Signal: Yeah, thank you.
The question is, as you have to account for your height over the
horizon, do you also have to account for the Earth being non -spherical?
Trammell: So, the fact that the Earth is not a perfect sphere will introduce errors, but they
are much smaller than the mechanical and the other errors that you're dealing with.
If you can get within a few kilometers of your position, that's considered really good.
And the non -spherical parts are, I think, significantly smaller errors than that.
Thank you. Microphone number eight.
Guest: Hi.
So, if you were planning for being a shipwreck and having no navigational equipment
with you, is there anything you would tattoo onto your body to help you navigate?
Trammell: I've considered getting logarithmic scales tattooed so I could make my own slide rule.
Herald: So, that's probably another question next talk then?
Trammell: Yes, yes.
Herald: Are the tattoo sessions at House of Tea?
Trammell: Yes, stick and poke tattoos are probably not, I don't know
if they're legal in the EU, they're not very hygienic.
So, basically, you're still looking for volunteers, right? Number two, please.
Guest: At the start, you showed an old tool using standing waves.
Do I understand correctly that they were using the
standing waves of the ocean generated by the landscape?
Trammell: That's correct.
So, the Polynesian wave navigation is specialized for that particular area.
The prevailing winds create almost standing waves and the
intersection angle between the waves actually identifies the location.
There's a really wonderful exhibit in a museum in
Leiden where that photo came from on the technique.
Guest: Thank you.
Thank you. And I think number six over there.
Guest: Thank you for the talk.
What do you do if you miss the zenith or when you are maybe
at noon or in the evening and want to know your position?
Trammell: That's a great question.
So, you typically do want your position much more often than once
per day and the intercept method works at any time of the day.
So, it's a lot more complex, but it involves computing the
great circle based on not the due north -south line of the
meridian that you're on, but an arbitrary bearing to the sun.
So, it happens to be that for the noon passage, basically
you end up with a bunch of cosines of 90 and sines of zero.
So, all these extra terms drop out and you get a much more direct measurement.
But you can, with all that extra trig, you can actually use
any measurement of the sun or a star at any time of the day.
Guest: So, that means you need more math.
Trammell: You need a lot more math.
And so, this diagram on the back of the side rule solves the spherical triangles
that are necessary for computing position during non -meridian passages.
Okay. Was there a question from the internet or was it already answered?
Signal: Yes.
Someone would like to know the name of the ship you showed in the earlier slides.
Trammell: There were many ships in the slides.
Signal: I think they meant the first one.
Trammell: Oh, the first one, that was probably Stad Amsterdam, which is a really
nice three -masted clipper, turn -of -the -century design from 1995.
Other ships were Tres Hombres, which is a cargo ship.
Draken, which is a replica Viking ship.
I don't remember what other ones are on there. Some of the Dutch Brown fleet.
So, lots of different ships. I've been spending a lot of time on ships.
Thank you. Number one, please.
Hello. If I understood correctly, you still need the declination at the beginning.
With an Almanac, have you thought about replacing
that with an approximation for a diagram as well?
Trammell: There is a declination in the center.
That's one of the functions we didn't cover on the slide roll.
There's actually six functions that the front of it does,
and one of them is an approximation of the declination.
I see. Thank you.
Herald: And number four, please.
Guest: In case you have bad weather and your interception method is not
working anymore, how long do you need to wait until you get eligible
data again and you can continue your navigation with this method?
And do you carry a backup GPS or something like that?
Trammell: I've never been on a ship that didn't have a GPS.
So, yes, we've always had that modern backup. And this is more for the fun of it.
With the intercept method, you typically want
measurements at least 30 minutes or an hour apart.
So, the time of the intercept method, or the time of the
measurements of the intercept method doesn't really matter.
That whenever you can get a clear sight of something in the
sky and the horizon, you can use that to update your position.
Guest: Thank you.
Herald: So, it seems we don't have more questions for now.
In that case, Trammell, thank you very much for your talk.
Give him a big round of applause. Thank you.