Thursday, October 18, 2012

Dead reckoning and the exploration explosion

Navigation is the art or science of combining information and reducing error to keep oneself on, or return oneself to, a route that will get you where you want to go. Note what I did not say here. Navigation is not necessarily the art or science of locating where you are. While answering the latter question – i.e. locating oneself in a Euclidean space, or a space reasonably projectable onto a Euclidean space – can usefully solve the navigation problem, figuring out such a location often requires different, and often more, information than you need to answer the questions of how to stay on or return to your desired route. And indeed this is what dead reckoning does – it gets you where you want to go with different information than what you would need to draw or find yourself on a normal map. I hope to explain more about this important incompatibility between the pilots’ and cosmographers’ systems during most of the age of exploration in a future post, but for now I will give an overview of the historical development of dead reckoning.

Between Italy of the late 13th century and the advent of GPS, dead reckoning formed the basis of most modern navigation. Dead reckoning was in particular the primary method of navigation used during the exploration explosion of the late 15th and early 16th centuries – the startlingly unprecedented voyages across unknown oceans of Dias, da Gama, Columbus, Magellan, and so on.

Dead reckoning is based on a sequence of vectors. Each vector consists of two essential pieces of information: direction and distance. Distance is typically calculated from time and speed, so each vector typically consists of the tuple {direction, time, speed}. With only speed and time, we have only a scalar distance value – it could be in any direction. With time but not speed, or speed but not time, we don’t have enough information to determine the distance covered.

From the start of a voyage to the last docking at the home port, dead reckoning was a strict regimen that never stopped: day and night, in calm and in storm, its measurement, recording, and diagramming protocols were rigorously followed.

Measuring or estimating the speed of a ship was a craft mystery the nature of which is still debated today, so I’ll skip over that and focus on the two more straightforward innovations in measurement, both of which occurred in or reached Italy and were first combined there in the 13th century: in measuring direction and in measuring time.

For measuring time mariners used the sand glass, invented in Western Europe during that same century. I have discussed this invention here. A strict regimen of turning the glasses was kept non-stop throughout a voyage.

For measuring direction, the ships of the exploration explosion typically had at least two magnetic compasses, usually built into the ship to maintain a fixed orientation with the ship. Typically one compass was used by the helmsman, in charge of steering the ship, and the other by the pilot, in charge of ordering both the sail rigging and the general direction for the helmsman to keep.

The magnetic compass was probably first invented in China, used first for feng shui and then for navigation by the early 12th century. Somehow, without any recorded intermediaries, it appears in the writings of authors in the region of the English Channel in the late 12th century where it was quite likely being used for navigation in that often cloudy region. Its first use in Italy was associated with the then-thriving port city of Amalfi. As both Amalfi and the English Channel were at the time controlled by the Normans, this suggests to me either a Norman innovation, or arrival via Norse trade connections to the Orient via Russia combined with now unknown Chinese trade routes. This is conjectural. Neither the Norse sagas nor writings about the Normans during earlier periods mention a magnetic compass, nor do Arab sources mention it until the late 13th century in the Mediterranean. In any case, it is the Italians who made the magnetic compass part of a rigorous system of dead reckoning.

Green dots indicate, in the case of northern Europe, the location of authors who mention use of the magnetic compass for navigation in the late 12th and 13th centuries, and for Italy, the traditional Italian association of the invention of the compass with Amalfi in the 13th century. Red indicates areas controlled by the Normans.

A dead reckoning itinerary can be specified as a sequence of tuples { direction, speed, time }. It can be drawn as a diagram of vectors laid down head-to-tail. However, as mentioned above, this diagram by itself, for nontrivial sea and ocean voyages, contains insufficient information to map the arrows accurately onto a Ptolemaic map (i.e. maps as we commonly understand them, based on celestial latitudes and longitudes), yet sufficient at least in theory to guide a pilot following such directions to their destination.

For recording speed and direction for each sand glass time interval (e.g. half hour), pilots used some variation of the traverse board, in which these values were specified by the locations of pegs in the board.

Traverse board. Pins on the upper (circular) portion indicate compass heading and (via distance from the center) for each half hour. Pins on the lower (rectangular) portion indicate estimated speed during each hour. The board thus allows an a pilot on a wet deck unsuitable for a paper log to record an equivalent of a sequence of tuples { direction, speed, time } over four hours, after which time this information is transferred to the ship’s written log(normally kept indoors), the progress is plotted as a head-to-tail diagram on a chart (also kept indoors), and the traverse board is reset. Note that the direction is read directly off the magnetic compass: thus north (the fleur-de-lis) is magnetic north, not geographic (celestial) north.
In a future post I hope to discuss more about dead reckoning directions and explain how the errors that can accumulate in such directions over long distances were corrected. I will also explain why neither the directions nor even the corrections could be accurately drawn on a normal (Ptolemaic or celestial coordinate) map, and yet such dead reckoning directions are sufficient at least in theory for the pilot to guide his ship from the starting port to the intended destination port. In practice, pilots "fixed" errors in their dead reckoning using landmarks and sounding, which I will also describe. And I hope to describe how this resulted in two incompatible systems of “navigation” (broadly speaking) during exploration explosion -- the pilot’s dead reckoning methods versus the cosmographers’ maps and globes based on latitude and longitude.

I also hope to someday figure out just why the exploration explosion occurred when it did. The advent of rigorous dead reckoning -- combining the compass, the sand glass, and decent estimates of speed with rigorous log-keeping -- did not occur in Asia (where the Chinese, lacking the sand glass at least, made a less systematic use of the compass), nor with the Arabs (who seldom used either sand glass or compass), which along with naval superiority explains why the exploration explosion occurred from western Europe. The puzzle of why the explosion started specifically in the 1480s, and not sooner or later, however, remains a mystery to be solved.


Blogger gwern said...

I don't understand the traverse board: there are 4 pegs placed in the top half giving compass headings, OK, but if there are 4 there, why are there only 2 pegs placed in the rectangular bottom-left section? Where are the speeds for the other 2 time periods? (And what's that 1 peg in the small square section bottom-right?)

7:41 AM  
Anonymous nick said...

Each half covers four hours. The top half covers eight half-hour intervals and the bottom half covers four one-hour intervals. So far two hours have elapsed.

The bottom half shows the speed was six knots the first hour and eight and one-half knots the second hour. (The bottom right peg shows the fraction).

12:47 PM  

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