Finding your latitude by the stars. Image courtesy NASA.

How do you find your way around the world? GPS? Well what if you
haven't got reception? A map? Good idea, but what if you're travelling
on the high seas with no land mark in sight? That's a situation that
many, many people have found themselves in over the millennia. These
brave seafarers used the Sun and the stars to navigate instead. And to do
this, they needed a fair bit of geometry, in particular
trigonometry.

Suppose you are on the open ocean and you want to work out your
latitude (see here for a definition of latitude). The Sun and most of the stars change their position
in the sky over time. But some stars always appear to be in the same
place. An example is *Polaris*, also called the *North
star*, which always appears to be sitting directly overhead the
North pole. It turns out that your latitude is the angle at which
Polaris appears to sit above the horizon.

To see why, let's look at a two-dimensional picture. Consider
the plane that contains the North pole, the point $X$ you are sitting
at and the centre $O$ of the Earth. Strictly speaking, Polaris
doesn't sit vertically above $X$, as shown in the picture, but it is
so far away that the line of sight from $X$ to Polaris is very nearly
vertical, so we can pretend that it does.

The angle that "Polaris sits above the horizon" is the angle
$\theta$ indicated in the figure. It's the angle our line of sight to
Polaris, call it $l$, forms with the line $t$ that is tangent to the
Earth at the point $X$ (that's our line of sight towards the
horizon).
Extending $t$ and $l$, we see the angle $\theta$ again on the other
side of the crossing point $X$:

The latitude of $X$ is defined to be the angle $\phi$ that the
line $r$ from $O$ to $X$ makes with the plane containing the
equator. In our two-dimensional picture, that equatorial plane is just
a horizontal line $e,$ which passes through $O.$ It meets the vertical
line $l$ in a point $L$ and the tangent line $t$ in a point $T.$

Because $r$ is the radius of the circle and $t$ a tangent, we
know that $r$ and $t$ form a right angle at $X.$ And since $t$ and $l$
form an angle $\theta,$ we know that the angle between $l$ and $r$ is
$90-\theta.$

Now consider the triangle with vertices $O$, $X$ and $L.$ As we have just seen, the angle at $X$ is $90-\theta.$ Since $l$ is vertical and $e$ horizontal, the angle at $L$ is $90^\circ$. And because angles in a triangle always add up to $180^\circ$, the angle $\phi$, which is our latitude, is
$$\phi = 180 - 90 - (90-\theta) = \theta.$$

Et voilà — your latitude is given by the angle $\theta$ Polaris
sits above the horizon. The Greek astronomer Hipparchus defined
latitude in this way over 2000 years ago. He didn't even know that the
Earth was round, but our little picture here should explain why
Hipparchus' definition agrees with the modern one.

There is no equivalent of Polaris in the South, but to find your latitude if you are in the Southern hemisphere you can use a constellation called the *Southern cross* (illustrated on the flag of Australia) and two stars called the *Southern Pointers.*

Over the millennia navigators have used different devices to measure the angle at which a star appears above the horizon. These include beautiful astrolabes and sextants, which you often find in antique shops and museums.

That sorts out latitude, but how do you work out your longitude?
That's another story.