 Okay, let's start by drawing the Earth. Now that we've got the Earth, let's divide the Earth into two equal parts. We're going to call the North part of the planet the Northern Hemisphere, and we're going to call the South part of the planet the Southern Hemisphere. And that dividing line which divided the Northern Hemisphere from the Southern Hemisphere, we're going to call the equator. Equator for equa being equal. We're also going to define the equator at zero degrees of latitude. And so the term latitude is the term that we're going to use for North and South. Now let's say that we wanted to determine what the latitude was of the North Pole. The North Pole of course is the northernmost part of the planet. All we need to do to figure this out is draw an imaginary line from the North Pole to the center of the planet. And then from the center of the planet, we're going to draw a line to the edge of the equator. You'll notice that we've just made a 90 degree angle. Therefore, the North Pole is simply located at 90 degrees North latitude. Or the North Pole is 90 degrees North of the equator. We can do the same thing down here in the Southern Hemisphere. Here at the South Pole, we're going to draw an imaginary line from it to the center of the planet and we're going to draw an imaginary line from the center of the planet to the edge of the equator. Once again, we've created a 90 degree angle. Therefore we know that the South Pole is 90 degrees South latitude or 90 degrees South of the equator or South of zero. Now let's say we wanted to figure out where San Diego, California was located. Well we know that San Diego is right about here. Well what does right about here means? We're 32 degrees North latitude. So what exactly does that mean? That means if you were to draw an imaginary line from the center of the planet to San Diego and another imaginary line from the center of the planet to the edge of the equator, that angle is going to be 32 degrees. So in essence, San Diego, California is 32 degrees North of the equator which makes it 32 degrees North latitude. We can also do this in the Southern Hemisphere. If there is a point right here that's at around 32 degrees South latitude, we know that that position is 32 degrees South of the equator and a good example of that would be a place like Sydney, Australia. Now here's a question for you. Is 32 degrees North latitude San Diego only or are there other places on the planet which are also located at 32 degrees North latitude? Well the answer is obvious. Yes, I mean what about over here? This place is 32 degrees North of the equator so it too has to be 32 degrees North latitude. And what about here and here and here and here? As a matter of fact, along this entire line you're at 32 degrees North latitude. And the same is going to be true down here in the Southern Hemisphere. All along this line you're going to be at 32 degrees South latitude. Well what about somebody who's up here at maybe 60 degrees North latitude? They're not the only ones. This is 60, this is 60, this is 60. So this entire line also represents 60 degrees North latitude. And we can play this game across the entire planet and you'll notice that all of these lines that we're drawing, all of these latitudinal lines are parallel to one another. Therefore we can also call these lines parallels. So now the term latitude and parallel are used rather interchangeably. San Diego, California is located at 32 degrees North latitude and it's also located on the 32nd parallel. Now here's my question for you. How do we know that San Diego, California is located at 32 degrees North latitude? Other than me telling you it's true and other than us using for example a GPS, a global positioning system to verify it, how in essence do we know that we are at 32 degrees North latitude? Well there's a rather simple way to do this. Let's draw the Earth again. And this time that's really concentrated in the North Pole. And let's put somebody up at the North Pole, oh I don't know, let's call them Santa Claus. So here's Santa Claus up at the North Pole. If Santa Claus were to look straight up in the sky and it's nighttime, he would see a North Star. And it's called the North Star because there's going to be a star that's above North Geographic Pole and it's always going to be above North Geographic Pole. In fact that star today is called Polaris. And so Polaris is the North Celestial Pole Star. Now if Santa Claus looks straight up he'll see Polaris. And during the course of the day as the Earth rotates on its axis, Polaris is still going to be up. And Polaris isn't going to move, at least not to him. However, if there's a star here and a star here and a star here and a star here, all of those stars will appear to move in the nighttime sky. And the reason of course is, is Earth is rotating on its axis. And it's going to make one complete rotation in the time span of 24 hours. So during the course of the evening, the stars will appear to rise in the east and set in the west. And they'll be doing this type of movement. Nice circular movement in the sky, but again it's not the stars moving. Rather it's the Earth rotating on its axis giving the appearance of the stars moving. Now all the stars are going to do this except for one star. And that one star is the North Celestial Pole Star. And the reason why it does not appear to move is because the North Celestial Pole Star is on Earth's axis of rotation. So here's the axis of rotation. Earth is rotating on this axis. And because Polaris is right on the axis, it's not going to appear to move. Now because of this, we can actually use Polaris to determine our latitude and the position on planet Earth. And here's how it works. So this person looks straight up. Now according to Santa Claus here, Polaris is going to make an angle of 90 degrees with his horizon. And so your horizon is your point that extends outward that you see on the edge. So for example, if you go see a sunset and you're looking out to the Pacific Ocean and the sun appears to set and it goes below the ocean, that oceanic plane is basically our horizon. And the point directly above our head is called zenith. So for the person standing on the North Pole, Polaris is located at zenith. And Polaris therefore makes an angle of 90 degrees with horizon. Therefore we know that North Pole is located at 90 degrees North latitude. So basically, your latitude is simply how many degrees that the North Celestial Pole star makes with your horizon. In the case here of San Diego, we know we're at 32 degrees North latitude. But how do we really know that we're at 32 degrees North latitude? Well, let's draw our picture here. Here's our zenith point, which is the point directly above our head. Here is our horizon line. And here is Polaris. And we will note that Polaris makes an angle of 32 degrees North of that horizon. Make sense? Now we can play this game all over the world. Here's somebody standing at the equator. And for someone standing at the equator, Polaris is going to be right on their horizon. So basically, Polaris makes an angle of zero degrees relative to that person's horizon. And that's exactly what we're seeing here. Now here's a question for you. What if you're down here in the Southern Hemisphere? Can you use Polaris to determine your point of latitude? And the answer, of course, is no. Because the second that you go beneath the equator, or at least really close to it, Polaris is now below your horizon and you're unable to see it. So for someone standing here in the Southern Hemisphere, they can't use Polaris as their point of reference. Rather, they're going to have to use the South celestial pole star, right? And that's going to be down here beneath the South pole. Make sense? Now let's clear this out and let's summarize this one more time and try to make it clearer on the picture. So here's Earth, here's the equator at zero degrees. For someone standing right here on the North pole, here's the North celestial pole star. And their horizon makes an angle of 90 degrees relative to the North celestial pole star. As this person continues south, continuing means starts walking southbound, you'll notice that as they do, the pole star gets lower and lower in their sky. So again, you start at 90 degrees, you start walking south, you're going to smaller and smaller numbers until inevitably you hit zero. And then beyond zero, the North celestial pole star will be below your horizon, you're unable to see it, right? So here's my question for you. If I took you out in the middle of the night and I said, okay, you guys, your grade depends on this. Tell me what our latitude is. And by the way, we're not going to be in San Diego. I'm going to throw you in a car, in a bus, in a boat, on a plane. We're just going to go somewhere random. The only promise to you I'll make is that we'll stick to the northern hemisphere. Would you be able to find your latitude? So daunting question. You could say, okay, Professor Yano, all I need to do is find Polaris. And then if I find Polaris, I'm going to figure it out. And by the way, there's a really cool way to determine angle. And you can just simply use your fist. You don't, in fact, have to use a sextant. For example, let's say you're standing outside, and here's your horizon. And that said that there's a star right here. First of all, what we're going to do is the following. We're going to count knuckles, or fists, I should say. One, and start putting hand over hand. One, two, three, four, five, six, seven, eight, nine. So for me, when I go hand over hand, it takes me about nine complete fists to go from here to there. Well, I know that that angle is 90 degrees. So 90 divided by nine is 10 degrees. Therefore, if I see Polaris in the sky, and I start here from my southern horizon, and I start counting upward, then I can actually determine what the angle is of that star above my horizon by just simply counting the number of knuckles, or excuse me, the number of fists that I have. It's not like super, super accurate, but it's better than nothing. And I could probably round up or down to the half. So if we're at 32 degrees, I can probably guesstimate 35 or to 30 pretty easily if I can find Polaris. But the question now becomes, again, how do you find Polaris? Contrary to popular belief, Polaris is not the brightest star in the sky. In fact, it's relatively dim. And so in order for you to find it, we need to come up with a clever way to figure this out. Well, here's how we're going to do it. Polaris is the tail of a tiny asterism called the Little Dipper. Now, the Little Dipper is called the Little Dipper because it looks like a little spoon in the sky. The Little Dipper is an asterism, which is actually a component of a constellation. And this Little Dipper is connected to Ursa Minor, which is called the Little Bear. So the Little Dipper is part of the Little Bear. Now, if you can look up in the sky, and if you could find this tiny little spoon in the sky, you could probably find Polaris. But again, the problem is Polaris is very, very difficult to find. And frankly, finding the Little Dipper is relatively challenging as well. But what's not too challenging is finding the Big Dipper. And the Big Dipper actually looks like a rather big spoon in the sky. And the Big Dipper is an asterism. It's part of Ursa Major, which is called the Big Bear. So all we need to do is find this big spoon in the sky. And the very last star of this spoon, we're going to follow it to the very first star that it runs into. And that first star is going to be Polaris. And so now you can go outside and find Polaris if it's a nice, clear evening. And that is going to be the North Celestial Pole Star. OK, so that, my friends, is latitude. We have latitude. We know what latitude is. We know the latitude are also parallels. And we know how to find latitude, at least to a certain extent. Of course, nowadays everybody's got their smartphones. You whip out your smartphone. You got your GPS unit on. And voila, you're going to see San Diego, California at 32 degrees north latitude. Yes? OK, so that's latitude. What about longitude? Well, let's draw this picture again. Once again, there's the Earth. And we'll put in here the equator. And now what we're going to do is draw an imaginary line from the North Pole all the way down to the South Pole. Now, this line is relatively arbitrary. The fact of the matter is with the equator, that way is easy. The equator is simply the dividing line between the northern hemisphere and the southern hemisphere. However, with this line that we just drew, we're also going to define this at zero degrees. But again, it seems to be arbitrary. There's a history to this, which we'll talk about a little bit later on. We're going to define this line going from the North Pole to the South Pole at zero degrees longitude. So unlike latitude, longitude is going to give us east and west. Now, we're going to draw another imaginary line. And if we draw this line going to the west, we know we're in the western hemisphere. If we draw a line this way and we go to the east, we know that we're going in the eastern hemisphere. Now, this might be a little bit confusing and it's hard to see on this picture. So you know what I want to do? I want to erase this and start over again. So here's the equator at zero degrees. And here is zero degrees of longitude. I'm going to draw a sister image right here. And on this image, imagine that we're staring down on top of Earth and we're looking at the North Pole. So let's call this NP for the North Pole. Therefore, this circle basically represents the equator because, again, we're looking down on planet Earth straight from above. So this line here is pretty much the edge that we're looking down upon. Now, this line that we drew here, it's right there. So we're going from the North Pole all the way down to the South Pole. And again, we're calling this zero degrees of longitude. And if we go in this direction, we're going into the Western Hemisphere. And if we go this direction, we're going into the Eastern Hemisphere. Now, if we were to draw an imaginary line over here on the opposite side of the planet, we've just made an angle of 180 degrees. So remember with latitude, we said, OK, well, how many degrees are you going above or below the equator to find your location? We're doing the same thing with longitude. But in this case, it's how many degrees are you going east or west from zero degrees of longitude. And by the way, we're going to call this line a meridian. In fact, all these lines here are going to be called meridians. And this line here that represents zero is going to be called the prime meridian. So in this picture, we've just gone 180 degrees west of the prime meridian. So we're going to call this 180 degrees longitude. Now, you might say, well, did you forget adding the west or an east designator? And the answer is no. The fact of the matter is if you go 180, you're neither east nor west. You're right in the middle just like if you're at zero. If you were to go this way toward the west and make an angle of 90 degrees relative to zero, then you would be at 90 degrees west longitude. If you went this way and made an angle of 90 degrees east of the prime meridian, you would be at 90 degrees east longitude. And so again, it works in the same way. The only difference is when we're dealing with latitude, our numbers are going between zero and 90. When we're dealing with longitude, our numbers are going between zero and 180. Make sense? Now, here's something clever as well. You'll notice that opposite 90 degrees west is going to be 90 degrees east. Opposite zero is 180. What about something like 120? What would the opposite of 120 west be? Well, here's a very, very simple solution for you. The opposite of 120 west has to be 60 east. So this plus this always has to equal 180 degrees. By the way, San Diego, California, were right around 120 west. So that puts us over here, 32 degrees north latitude, 120 degrees west longitude. More specifically, we're at about 117. OK, so remember, all these lines here going north and south are called meridians. All these lines here going east to west, but giving us north and south, are called parallels. And you'll notice we just turned the world into a giant grid system. This is called the geographic grid, hence the name. Now, by the way, we determine how we can figure out our latitude. And we can do that by looking at for the north celestial pole star if you're in the northern hemisphere or the south celestial pole star if you're in the southern hemisphere. What about longitude? Longitude is much more quirky, and it's a little bit more complex. And in order for us to figure out longitude, what we actually have to do is use a clock, and we're going to get to that in just a second. But before we do, I'd like to look at north, south, east, and west in a little bit finer detail. Let's say that you're a navigator and you're out at sea, and you determine that you need to go from, I don't know, 32 degrees north to maybe 33 degrees north. That's not that much. That's only but one degree. But here's the caveat. One degree of longitude and one degree of latitude, if you're at the equator, equals 69 statutory miles. So imagine this. You're a navigator out at sea. You want to go from 32 to 33. You know that that's 69 miles. Let's say that you're off. Maybe you're off by a half a degree for your destination. All of a sudden, you've missed your mark by about 35 miles. That's big. That's really big. Therefore, we need to figure out a way that we can define distances a little bit more precisely, especially if we're talking about latitude and longitude and if we're out at sea, which, of course, is the early days of exploration. Well, here's how it's going to work. We know that one degree equals 69 statutory miles. And I'm using the term statutory because in a second, I'm going to use the term nautical. So statutory miles basically is distance on land. Now, let's say I want to break this down into a finer part. Well, what I can actually do is take this degree and break it into 60 equal parts. So one degree is going to equal 60 what that is an arc minute. So a little tick mark. So one degree equals 60 arc minutes. And that equals 69 miles. Well, now I can just do a little bit of math. I can divide 69 miles by 60. That equals about 1.15 statutory miles. So we know that one arc minute equals about 1.15 statutory miles. Now, that's not bad. So when you look in your atlas and you see San Diego, California, it won't say 32 degrees or won't say 117 west. It'll say like 32 degrees, 25 arc minutes. Or it'll say 117 degrees west, 33 arc minutes. So those arc minutes are just simply given us finer details. And according to this math that we've just done, we know that one arc minute equals 1.15 miles. So now if you're a navigator at sea, if you're off by only an arc minute, you're only off by about 1.15 miles, which is now not too bad. But keep in mind that 1.15 miles could be the difference between your boat hitting a big rock or your boat going into a bay. So frankly, we want to get this into even finer detail. And we can. So let's take one arc minute. And again, we know that one arc minute is the equivalent of 1.15 statutory miles. By the way, before I continue, one arc minute equals 1.15 statutory miles. That is what a nautical mile is. So one nautical mile is one arc minute. And that translates into 1.15 statutory miles. And if you're talking about a knot, a knot is simply how many nautical miles you're going per hour. So we're going to use knots and nautical miles when we're out at sea versus statutory miles when you're out at land. And again, we see why we're using arc minutes or nautical miles because that just makes navigation more precise when we're in fact looking for more precision. Of course, nowadays, you're going to use a lot of decimal degrees. And in decimal degrees, you're just going to have decimal point representation of a degree. But for now, we'll stick with this arc minutes. OK, so we know that one arc minute equals 1.15 statutory miles, which of course equals a nautical mile. We can break down that one arc minute into 60 equal parts. So one arc minute equals 60 arc seconds. And we know that that equals 1.15 miles. So again, we're going to do just a little bit of math. We're going to divide both sides by 60 arc seconds. Therefore, one arc second equals 1.15 miles divided by 60, which is about 2 tenths of a mile. So now we've got really nice, fine detail. When you break open your Atlas and you find San Diego, California, it'll give you a little bit more precision. You'll have an answer in degrees, arc minutes, arc seconds. And again, the more detail you want, then you're going to start to see arc seconds, because that gives you distance up to 2 tenths of a mile, which is really quite precise or accurate, especially if you're a navigator. OK, let's do a little bit of trivia. Let's draw Earth here. Here's the equator. Here's the North Pole. And again, let's take a look at the perspective from looking straight down on the planet. So here's the North Pole. And let's draw all these meridians. You'll notice that as we approach the North Pole, meridians are getting tighter and tighter and tighter together until inevitably all the meridians converge at the North Pole. And then the same is true with the South Pole. They'll converge there as well. Let's say that at the equator, this distance here represents one degree or about 69 miles. As you go higher and higher in latitude, clearly the distance between one degrees gets less and less and less until eventually you're at zero. So if you are standing right here, one degree of longitude is zero miles. If you're standing right here, one degree of longitude is 69 miles, right? It's quirky, but that's what's going on. So the fact of the matter is we know that one degree of longitude equals 69 times cosine of whatever your latitude is. So if your latitude is at zero degrees, then we know that one degree longitude equals 69 times cosine zero equals 69. If you're at 90 degrees latitude, then we know that one degree of longitude equals 69 times cosine 90, which equals zero. So we can actually use this little equation here to determine what a degree of longitude is as you go higher and higher and higher North or higher and higher and higher South, if you want to use the term higher. Now this will become important for us a little bit later on, especially when we start to talk about great circles and small circles. In fact, let me give you a quick, quick example. Let's say we're here in San Diego. And so let's draw the United States of America. And here we are. And let's say that you want to fly all the way over here to England. So let's say we want to go to London Heathrow, which is way over here. Now, let's say that you take this flight. It's a long flight. And let's say you're flying in a Boeing 777 and right in front of you in the seat in front of you, you've got a little TV screen. And the little TV screen shows you a map. And this map is going to show you the path with which the pilot is going to take to get you to your destination. Now, you and I, we're going to look at this map and we'll go, OK, well, let's go the shortest distance. And we would think that the shortest distance is going to be like this, straight line. Well, as it turns out, that's not the case. The shortest distance on a sphere is going to actually look like this. And that line is called a great circle. And so a great circle root on a sphere basically means that the center of the sphere is the center of that arc or the equal distance point of that arc. That's the great circle. So here around the equator, that's a great circle. Here is not a great circle. That would actually be called a small circle. On a sphere, the shortest distance is always the great circle. And part of that we can actually understand by, again, referring to this whole idea of longitude. As you go to higher latitude, your distance of a degree of longitude gets smaller and smaller and smaller until it converges at the north pole and it turns to zero. So here, you're flying this root. You're actually flying a shorter distance on the sphere. And that's what a great circle is. OK, let's go back now. Let's go back now. We know how we can determine latitude. And again, determining latitude is all about finding the north celestial pole star if you're in the northern hemisphere or the south celestial pole star if you're in the southern hemisphere. But what about longitude? Well, in order for us to figure out longitude, the first thing we actually need to do is talk about time zones. So let's do time zones really quickly.