Longitude

Orientation—whether in the physical, social, or metaphysical dimension—is the absolute beginning of knowledge. You’ve got to know where you are. That very word, orientation, is derived from the physical. It comes from the Latin oriri, to rise, and the rising something indicated by the word was the sun, therefore the east. East-west orientation, therefore, was relatively easy for humanity. You simply had to observe the sun. It rose in the east and set in the west.

Paradoxically, however, traveling by sea, humanity’s first effective orientations using the sun told people where they were in the north-south dimension. After we were certain that the earth was a ball, that it travelled around the sun—and at a tilt to the sun’s own rotation around its axis—we learned to use an astrolabe, thus an instrument able to measure the angle of the sun to the horizon at noon, thus at its highest point. Knowing this angle and the time of year, the astrolabe (and later the sextant) could tell us how far north or south we were of the equator at any time of year. That technique dates to 150 BC. I’ve summarized the process on this blog earlier; the link is below.

To know where we were in the east-west dimension took much, much longer. It required the development of very accurate clocks—able to operate at sea. That achievement finally came in the eighteenth century, thanks to the achievements of an English clockmaker called John Harrison (1693-1776). That story is told most eloquently by Dava Sobel in her 1995 book, Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time. It’s short, suspenseful, entertaining, has pictures, and is must reading to anyone who’d like to know this story in detail. Harrison labored to win a £20,000 prize set by the British Parliament to solve the intractable problem of longitude. That prize in today’s dollars would be $4.47 million.

Why such a huge prize? Ships, cargo, entire fleets—and all the people on them—were routinely lost at sea in those days because they had miscalculated, by so-called dead reckoning, just where they were in an east-west direction from their intended landing place. Dead reckoning used estimated measurement of speed and time, over a set course (measurable by latitude), and thus calculating distance. But especially in stormy weather, speed-measurement, indeed course calculation, was extremely chancy. Therefore dead reckoning, while it worked reasonably well, was extremely chancy. After such a period, getting a hard fix on longitude was impossible out on the open sea.

So how do clocks come into this? In brief, as observed from the earth, the sun moves 15 degrees of longitude in the space of an hour. If you know your own time accurately, and also know what time it is at another fixed point on the earth, you can use the difference in time to calculate with great accuracy how far you are from that fixed point.

The illustration shows the geometrical basis of lines of longitude calculated as angles from the Prime Meridian. The Prime Meridian here is the “fixed point”—or rather the fixed line—from which the navigator calculates his or her distance east or west. The Prime Meridian these days runs right through the Royal Observatory of Greenwich, England. That line begins at the north and ends at the south pole. If the navigator sailed west and measured time locally and it was noon, the other clock, running on universal (call it Greenwich time) said 2:00 pm, the navigator knew that he or she was 30°W longitude from Greenwich, which is at 0° longitude. Conversely, if your local time is noon, but Greenwich time is 10:00 am, where are you then? 30°E longitude. One degree is 60 nautical and 69 statue miles or 111 kilometers—that’s at the equator. More on this later.

Now more illustrations:

This one shows the longitude over the United States. Our longitudes are all west. Longitudes are further subdivided into 60 minutes, each minute into 60 seconds. My own location in Detroit is 83° 05’—although I note that not all of my sources agree about the minutes. When you see longitude or latitude figures, the fractions may also be rendered into hundreds, so that Detroit’s longitude may be shown as 83.08 and mean the same thing as above. The above courtesy of Tutapoint.com (link).

Herewith longitudes overlaying a map of the world. This graphic, of course, does not do justice to a crucial fact. The distance between longitudes is not uniform all around the world. It is greatest at the equator, 69 miles, and zero at the poles. At 40° latitude, north or south, the distance between lines of latitude shrinks to 53 miles. Therefore accurate calculations of longitude require additional lookups to adjust for the latitude where the navigator takes his or her readings. The illustration is from Jacksonville State University (link).

Herewith the big picture, showing the whole world again, as presented by Wikipedia (link).

This post is the third, and last, on the subject of the astrolabe. The others are here (first, second). The astrolabe is meaningfully connected to this subject for two reasons. A good clock measuring Greenwich time and an astrolabe would still suffice today to navigate accurately on the oceans. The astrolabe was useful for determining the exact local time, thus noon—which has always been the time for seafarers to find out where they were

Sextant: The Modern Astrolabe

An elegant, accurate instrument using mirrors became the modern version of the mariner’s astrolabe around 1730. Its first version was called the octant, thus named because its measuring surface for angles was one eighth of the circle. The sextant, the modern instrument, has a measuring arc of a sixth of a circle. The two simultaneous but independent inventors of this device, intended to measure the angle between a celestial object and the horizon, were an English mathematician, John Hadley (1682-1744) and an American glazier, Thomas Godrey (1704-1749). Isaac Newton (his name looms large) tends to be credited with inventing the principle of using mirrors in such an instruments, but while he wrote down the idea, he did not publish it.

The best way to convey the use of the sextant is by demonstration. Pictures are better than words, animations even better. Herewith the first image of an award-winning series produced by Joaquim Alves Gaspar called “Animation of the use of a marine sextant to measure the altitude of the sun”. You can see the animation by clicking on this image. It comes from Wikipedia (link).

As we can see, the user can locate the sun (in this case) and then, while watching the horizon through half of image provided by the visor, cause it to overlay the horizon line precisely by moving the index arm. Modern sextants come with filters so that looking at the sun does not endanger the eyes. When the sun is precisely on the horizon, the reading of the index arm against the arc below provides the exact altitude—exactly the same service that the astrolabe provided; in the animation that altitude is 40°. That number, can then, by means of an equation I discussed in the first post on this subject, provide the exact latitude at which the sighting was taken.

Latitude provides one of two (the horizontal, east-west) coordinate needed to locate ourselves on this planet. Longitude (the vertical coordinate, north-south) required high-precision clocks—at least before satellites rose into the sky. I’ll summarize that history in a future post.

This is the second of two posts in a series. The first is here.

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.