The Astrolabe

Measuring and calculating instruments have always fascinated me, hence they’ve appeared on this and the earlier version of LaMarotte before (here, here, and here). During a joint vacation recently, John Magee mentioned the astrolabe. A friend of his is the navigational officer on board of an aircraft carrier. John asked him a while back if he had ever used an astrolabe. The man had not. This got us talking—and left behind an intention to look into the subject. The impression in my mind, when we were talking, was of a round sort of thing sailors had once held up toward the sky—and a vague knowledge that these babies were extraordinarily complex instruments.

Well, herewith a little introduction to the subject. It can’t be summarized in a single post, not even a dozen. The astrolabe is thought to have been invented by Hipparchus of Nicea (190-120 BC; the place is now in Turkey) around 150 BC, thus early in the Hellenistic (read the “modern, scientific”) era of Greek culture. Hipparchus was an astronomer, geographer, mathematician and also thought to be the originator of trigonometry. You now have the flavor of the thing. It turns out that the astrolabe was perhaps the earliest kind of ultra-sophisticated slide rule. The Persian astronomer, Abd al-Rahman al-Sufi (903-986) described more than a thousand uses for it, not least astronomy, time calculation, navigation, and as a trigonometric table.

Here I’ll deal narrowly with the simplest kind, known as the mariner’s astrolabe—and a single use of it, determining your latitude at sea from a single reading of the sun’s angular position. But it is well to described the actual device. It consisted of four components, a base plate known as the Mater (mother), a rotating structure above it called the Rete, a Plate that fit between the two, and a rotating ruler-pointer called the Alidade. An excellent diagram of these parts is shown on the website, The Astrolabe (link). Note that the Plate could be changed. Commercially available versions come with eight different plates one can insert depending on the application.

Herewith a picture of the front and back of a traditional astrolabe produced and sold by Norman Green (link). This one costs $180. The site shows others as well. The rete is the grayish structure on the first, the gold is the mater. The alidade, used in detecting the sun’s altitude, is shown on the second picture; it has visible sighting slits. The front has an additional pointer-ruler.

From the same page comes this simplest of astrolabes, the mariner’s ($195). It consists of a mater and an alidade and is used for obtaining one’s latitudinal position on earth.

How this instrument is used is illustrated by this cartoon taken from Wikipedia (link).

The user suspends the astrolabe (it shouldn’t actually be held in the hand as shown) and then aligns the alidade until the sun (or star) is visible through both slits. For navigation, the sighting should take place when the sun is at its highest point that day.

Latitude and longitude? Lines of latitude are horizontal lines drawn on globes and mark degrees of latitude (width—from Latin latus, wide). Why are they called degrees? The following graphic will illustrate that.

The globe, with the two poles marked as 90°, the equator as 0°, is divided into four triangles as show in blue. As the graphic shows, latitude 45 is a 45° degree elevation above or declination below the equator. If you draw a line from the 45° point of the eastern to the same point of the western triangle in the northern hemisphere, you get a line of latitude. Similarly in the southern hemisphere. By convention, therefore, latitudes are marked N or S or the southern equivalent is rendered as a negative number. The largest latitude circle is at the equator. The circles grow smaller as we go north or south and they vanish into a single point at each pole.

I show this globe at a tilt by way of emphasizing that the earth’s axis is tilted with reference to the sun’s—by 23.5°. This becomes important in finding our latitude using the astrolabe. The earth’s tilt causes our seasons; thus the sun’s altitude changes daily throughout the year. If the earth’s axis were not tilted, the angle we detect using the astrolabe would suffice, by itself, to serve as a simple indicator of our latitude; to get latitude, we would simply deduct the observed angle from 90. This becomes evident from the following graphic (courtesy of this tutorial). It shows the ecliptic, or the path of the sun, in relation to our orientation north to south. This means that in each hemisphere, the sun is beneath or above the equator depending on the time of the year:

The angle we measure using the astrolabe must be adjusted by this ever-changing declination of the sun relative to our equator. The point where the ecliptic crosses the equator twice a year is known as the equinox. At that point the declination is 0°. The sun is directly above the equator; night and day are therefore the same length. At other times the declination is positive (sun is above the equator), maxing out at 23.5° at the summer solstice, or negative (sun is below the equator), maxes out at -23.5°, at the winter solstice. In this field the word declination is used; to be sure, it is actually (as shown above) a declination followed by an inclination, but one word is used and the perceived direction of this apparent solar movement is indicated by positive or negative numbers—or zero for the equinoxes.

The navigator using an astrolabe, having correctly identified the angle of the sun, its altitude, must next calculate the declination. For this he or she will need to know the day of the year, thus have a good calendar, and use an equation. The calendar should be such that it informs the person of the number of the day. August 31 this year, for instance, was day 243. The equation to calculate the declination is the following:

declination angle in radians = 23.45 * pi/180 * sin(2*pi*((284+day)/365.25))

To render this for Excel, pi would be rendered as PI(). If we substitute 243 for the day, the result of this is 0.143834 radians. To rendered this into degrees, multiply by 180/pi. The result is 8.241088°. This is the sun’s declination on August 31.

Supposing that our astrolabe reading was 55.5. Having that and the declination of the sun for the date, we can calculate the latitude. The formula is:

latitude = 90 - (altitude - declination)

If the declination comes out negative, which it will do from the autumnal to the vernal equinox, the declination is added to altitude rather than deducted.

When we insert values for the words in the equation, in our case 55.5 and 8.24, the latitude for that sighting is 42.74° Is that correct? Well, I’ve come close. My actual latitude here is 42.4243°—but that’s not too bad when measuring the solar altitude with bits of cardboard rather than a fancy $195 astrolabe from Mr. Norman Green.

Longitude? In a word, you need a very accurate timepiece keeping Greenwich, England time—and one of the more muscular astrolabes able to calculate local time. But as for details, not this time. I all worn out with latitudinal astronomy, radians, degrees, declinations, and inclinations. My own inclination is to have lunch.