Latitude Aptitude

How did the early sailors determine their latitude position without GPS? That is the topic of today’s post.

Now first, a little background. The earth’s axis is tilted with respect to its orbit about the sun. The angle of this tilt is approximately 23.5°. This causes the northern and southern hemispheres to get more sun in summer and less in winter, which is the reason for seasons to exist. The tilted axis also causes our days to be shorter in the winter and longer in the summer. There are two times during the year when the days and nights are equal in length. The times are called the vernal and autumnal equinoxes. In the northern hemisphere, these  equinoxes occur on the first days of spring and autumn. Here in Australia in the southern hemisphere, we elected to call the start of spring on the 1st of September and the fall on the 1st of March, about 21 days short of the respective equinox. Perhaps this is because it is easier to remember. The main point here is that twice a year, at an equinox, the days and nights are equal.

At any time of the year other than an equinox, the highest height of the sun around noon is affected by the tilt of the earth’s axis. But at an equinox, the earth is in a neutral position where the axis tilt does not affect the highest sun height. At the equator (0° latitude), the sun would be directly overhead and a vertical stick in the ground would cast no shadow. As you go up or down in latitude, the highest sun height goes down and a vertical stick would cast the shortest shadow when the sun is at its highest. The below graphic shows the earth at an equinox with the sun at its maximum height. If a vertical stick is placed in the ground at your location, the sun’s rays would make an angle with it that is the same as your latitude angle.

Below is a blow-up of the vertical stick. You can see from the above picture that at the equator. The sun would be directly overhead at noon and there would be no shadow. At the poles, the sun would be at the horizon and the shadow would be very long (technically infinite). But in between, a measurable shadow would be made. 

Now you could measure the angle directly with a sextant, but I hardly know what a sextant is. let alone use one. But I am good at maths and I have a good calculator. The shadow, stick, and the line from the top of the stick to the shadow end forms a right triangle. If you remember the post on trig functions, the tangent of an angle is the length of the opposite side divided by the length of the adjacent side. We want to measure the angle 𝝀, so the adjacent side is the stick and the opposite side is the shadow: 

\[\tan\mathit{\lambda}\hspace{0.33em}{=}\hspace{0.33em}\frac{{\mathrm{length}}\hspace{0.33em}{\mathrm{of}}\hspace{0.33em}{\mathrm{stick}}}{{\mathrm{length}}\hspace{0.33em}{\mathrm{of}}\hspace{0.33em}{\mathrm{shadow}}}\]

On your calculator, if you have the trig functions, you would also have keys labelled “arctan” or “tan-1“. These keys mean “what is the angle that has what you entered as its tangent”. So if you enter the results of the division and then hit this key (making sure that your calculator is in “degrees” mode), you will get your latitude.

Now this method will not tell you if the latitude is positive (North) or negative (South). But if you are so lost that you don’t even know what hemisphere you are in, finding your latitude is probably the least of your troubles.

Also, waiting for noon to find your latitude is not too bad, but waiting for an equinox is fairly restrictive. Fortunately, our early sailors had tables to correct the angle found depending on the time of the year.