Mercator's Magic Map

Nautical history provides some fascinating insights into our ancient need for geographical information.  The sea has always been a path of exploration and a challenging adversary in our quest to comprehend the globe.  The haunting sirens, horrible monsters and perilous Edge of the Earth may have been real enough to some, but naivety was never the real issue.  Intelligence and reason have been abundant in every age.  Those apparitions of the deep are more likely to be symbolic of the profound conundrums that perplexed ancient scholars and sailors alike.

The predictable behavior of the sun and stars and even the lodestone had been known for millennia.  Unfortunately, these natural tools of navigation could only do so much.  While latitude was easy, longitude was perpetually elusive.  It was common knowledge that position east or west was a fraction of rotation, or a day, and could be measured in time.  Time was completely familiar, too.  Even before recorded history, men knew the difference between solar and sidereal time, understood the analemma and could compute the very moment of equinox.  But time was also a paradox.  Although men could easily tell time, they could not keep it.

Bearing was the only remedy for navigators.  True north was easily determined from the sun or stars.  A compass worked day and night and could be calibrated for declination.  You could measure speed at sea with a chip log and hourglass.  So, with constant vigilance, sailors could approximate their global position with dead reckoning.  (NOTE: Although mistakes could be deadly, that term derives from deduced reckoning — the use of math and logic.)

Bearing, speed and elapsed time were meticulously recorded in ship's logs for centuries.  It was a mariner's legal obligation to do so, and to preserve the log for future reference at all costs.  Early cartographers poured over these accounts to verify and correlate positions for new charts and maps.  While their work may seem crude and even fanciful today, these were serious endeavors with lives at stake.  Uncountable man hours were involved.  Coordinates and graticules were known and sometimes used, especially on maps of the entire world where spatial accuracy was clearly respected. 

Detail from Francesco Berlinghieri's pre-Columbian world map of 1482.  The lack of precise locations does not diminish his obvious understanding of spherical coordinates.
Giacomo Gastaldi's World Map of 1546 employs a graticule quite similar to the modern Robinson projection.  It also shows how dramatically the understanding of world geography had improved in just a few decades!

Nautical charts, however, were far more important than maps.  Land travel was difficult, expensive and provided little commercial value.  What might appear to us today to be naive and distorted attempts to draw the land, were nothing of the kind.  They were instructions for sailors.  The shaped of land areas were irrelevant and only coastal detail was usually noted.  In that respect, most sea charts from the age of exploration are not unlike a modern subway map that shows only a schematic diagram of routes and stations.

Giovanni Camocio's 1558 chart of the North and Baltic Seas.  Notice the lack of graticule but generous display of rhumb lines radiating from six rosettes at 11.25º increments (360/32).

This was the prevailing technology in 1569 when Gerardus Mercator introduced a new world map with an almost magical quality.  It was conformal.  That means that all shapes were proportionally correct and the direction from any point to any other point was accurate.  Size and distance were dramatically distorted, but that was not the point.  How to get there was the key.  Now you could simply draw a line to your destination, maintain a bearing equal to the slope of that line and off you go!

This works because Mercator had figured out how to make a straight loxodrome.  A loxodrome is a line that crosses every meridian at the same angle.  These point-to-point paths often appeared as a spider web of rhumb lines on the sea charts of the day.  They are not the shortest path between two points, but they were the easiest path to follow in a fog of uncertainty.

A loxodrome on a globe is a spiral.  It curls around the sphere like the swirl on an ice cream cone.  On a map with uniform latitudes, such a line would be curved.  To make a straight loxodrome, that dollop of ice cream needs to be figuratively stretched upward at an increasingly disproportionate rate.  Think of the meridians on a globe; they converge at the poles.  On Mercator's map, they had to be straight and parallel so that a loxodrome would cross each one at the same angle.  This causes an increasing horizontal distortion at extreme latitudes.  To compensate, Mercator simply increased the vertical distance to match.  His inspiration was brilliant.  Now every line was a rhumb line!  What a glorious Aha! moment that must have been.

Some non-trivial math is required to perform Mercator's transformation and his original method was probably geometric construction.  A formal projection didn't exist until 1599 when the English mathematician Edward Wright devised a workable solution.  This was before the invention of logarithms (by John Napier in 1614) which would certainly have helped.  The first mathematical proofs were not published until 1668 and 1670 and they predate the invention of calculus (by Isaac Newton in 1687).  The cartographers of old were apparently well ahead of their time!

Mercator's concept is still used in marine navigation and is, in fact, the basis for Google and all other web maps.  The advent of GPS and the ubiquity of hand-held and in-vehicle routing apps exploit its fantastic utility.  At every scale, straight is straight and all angles are correct.  Sure, sizes are distorted over large areas and the poles extend to infinity (and are therefore excluded) — but Mercator clearly understood this.  So did all the mariners of his time.  It should be noted that the title for his map was Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata ("New and Augmented Description of Earth Corrected for the Use of Sailors").  That could not have been more clearly stated.

Sadly, the contemporary use of Mercator is being increasingly criticized.  Its distortion of size is blamed for creating false impressions about the relative importance of various places in the world.  That seems unfair.  Perhaps such critics were so annoyed by the Mercator map in their elementary classrooms that they failed to notice the globe in the corner.  The 16th century seems quite enlightened by comparison.

The familiar Mercator map showing how latitude is increasingly distorted towards the poles.


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Posted in Geography, History on Oct 10, 2016