Global Positioning System Reference
In-Depth Information
two points marked on the ship's rail. One would then multiply the speed
obtained by one hour to obtain the distance traveled in that hour. Direction
was measured with a compass, and, to use modern terminology, a trajec-
tory vector for the ship was obtained. These vectors were joined together,
tail to tip, to obtain the trajectory of the ship, as shown in figure 6.12a. In
Columbus's day the concept of vectors did not exist; mariners stored the
measured data on a
traverse board
, a circular board with the points of the
compass marked. Each hour a peg was placed on the board on a radial line
corresponding to the measured heading and at a distance from the center
that represented the calculated distance traveled in the last hour. After
24 hours, the navigator would add up all the distances and directions
(calculate the integrated path) to estimate the day's travel and would then
remove the pegs to make the board ready for the next day.
As you can imagine, dead reckoning was, especially in its early days, a
very rough-and-ready method for estimating location. The compass mea-
surement of direction was subject to error, and the estimate of time and
therefore of distance was very approximate. Worse, there may have been
systematic errors due, for example, to a current or tidal stream. Or perhaps
wind and current were sensed but compensated for incorrectly. Some
errors would be random and unbiased, which is to say that the estimate of
speed would be too large just as often as it would be too small, so that the
average estimate would be about right. More insidious were systematic
errors, which were biased in one direction and accumulated over time.
To illustrate the e√ect of dead reckoning errors, consider figure 6.12b.
Consider yourself a not-too-skillful chief navigator, heading out from Lis-
bon to the New World—say, at the beginning of the sixteenth century. You
have three ships under your command and are planning to sail due west in
order to land on North American soil at a place fairly close to what would
become New York. Each ship measures speed and direction daily, not
hourly, with a maximum random error of, let us say, 5% for distance and a
quarter degree for direction—pretty good for the period. Let us say it is
foggy the whole way; your three ships become separated and lose contact
with each other. I can simulate the paths taken by your ships on a computer
using dead reckoning, with the errors just quoted.
Two of your ships follow the more northerly courses shown in figure
6.12b. They pass through the Azores and end up reasonably close to mod-
ern New York—within a few tens of miles. The error in estimating distance
is not too serious in this instance; the sailors know they have reached the
New World when they bump into it. The unbiased direction errors result in
