Bearings Straight (I)

Mariners share two fears: bad weather and getting lost. Their deep respect for the Mercator projection reflects the map’s value for plotting an easily followed course that can be marked off with a straightedge and converted to a bearing with a protractor similar to the semi-circular plastic scales fourth graders use to measure angles. In a less direct way, the Mercator map also addresses the sailor’s fear of storms by providing a reliable base for plotting meteorological data for tropical regions. But that’s another story.

Picture yourself as a seventeenth-century navigator who knows where he is and where he wants to go. You plot both locations on a chart, join them with a straight line, and measure the angle your line makes with the map’s meridians, which run due north. If the chart is a Mercator map, all its meridians are straight lines, parallel to one another, and the course you’ve just plotted is a rhumb line, also called a loxodrome (fig. 1.1). The derivation of rhumb is obscure—possible origins include a Portuguese expression for course or direction (rumbo) and the Greek term for parallelogram (rhombos)—but math ematician Willebrord Snell (1580–1626) coined loxodromein 1624 by combining the Greek words for oblique (loxos) and course (dromos). Manuals on piloting accept rhumb as a normal part of the seaman’s language and define rhumb line as a line that intersects all meridians at the same angle. The angle between a course and a meridian is a bearing, thus a rhumb line is a line of constant bearing. Stay the course, and you’ll reach your destination.

Look down at a globe, on which the meridians meet at the North Pole, and you’ll understand why loxodromes are spirals that converge toward the pole as they wind round and round, always crossing the meridians at a constant angle (fig. 1.2). The only exceptions are rhumb lines running directly north–south, along a meridian, or directly east– west, along a parallel. The former reach the pole along the shortest possible route, whereas the latter never get any farther north or south. If a bearing is close to due north, its loxodrome approaches the pole rapidly. If a bearing is nearly due east, convergence is notably slower, with a loxodrome that originates in the tropics and circles the globe many times before crossing the Arctic Circle. Follow a loxodrome in the other direction, and it crosses the equator and starts spiraling toward the South Pole. What works in the northern hemisphere works equally well south of the equator.

Gerard Mercator (1512–94) understood loxodromes. Skilled in engraving and mathematics, he crafted globes and scientific instruments as well as maps. Like other sixteenth-century globe makers, he engraved the grid lines, coastlines, and other features on copper plates and printed the curved surface in flat sections, called gores, which were trimmed and pasted onto a ball, typically made of papier-mâché. His first experience with globe making occurred around 1537, when he engraved the lettering for a terrestrial globe designed by his mathematics tutor, Gemma Frisius (1508–55). That same year Mercator produced his first map, a six-sheet representation of the Holy Land. In 1541, he devised a navigator’s globe on which rhumb lines spiraled outward from compass roses. Intended as a navigation instrument, the globe was approximately 16.5 inches (42 cm) in diameter and consisted of the twelve gores and two polar caps pasted onto a hollow wooden ball for use at sea. According to cartographic historian Robert Karrow, this navigator’s globe was the first of its kind, and sixteen surviving copies, crafted between 1541 and 1584, attest to its success and durability.

Mercator published his celebrated world map of 1569 as a set of eighteen sheets, which form a wall-size mosaic 48 inches (124 cm) tall by 80 inches (202 cm) wide. Its projection revolutionized navigation by straightening out rhumb lines on a flat map—not just the globe’s meridians and parallels, but any rhumb line a seaman might plot. To accomplish this, Mercator progressively increased the separation of the parallels. On a grid with a constant separation of ten degrees between adjoining meridians and parallels—cartographers call this a ten-degree graticule—parallels near the equator are relatively    close, whereas those farther poleward are more widely spaced, as shown in figure 1.3. The parallels at 70°and 80°N,for instance, are much farther apart than the equator and its neighbor at 10N. And the separation between 80and 90 N cannot be shown completely because the North Pole lies at infinity. Although loxodromes converge toward the poles,on a Mercator projection they never really get there.

Mercator’s intent is readily apparent in his map’s title, "New and More Complete Representation of the Terrestrial Globe Properly Adapted for Use in Navigation.” In 1932, the Hydrographic Review published a literal translation of the map’s numerous inscriptions, elegantly engraved in Latin. Although the chartmaker’s words reveal little about how he spaced the parallels, Mercator clearly recognized the need “to spread on a plane the surface of the sphere in such a way that ... the forms of the parts be retained,so far as is possible, such as they appear on the sphere. "Accurate bearings, he reasoned, demand a locally exact representation of angles and distances, even though “the shapes of regions are necessarily very seriously stretched.”

To compensate for the local deformation that would otherwise occur, Mercator “progressively increas[ed] the degrees of latitude toward each pole in proportion to the lengthening of the parallels with reference to the equator. «Sounds complicated, but it’s not. At 60° N ,for instance, the distance on a globe between two meridians is half the corresponding distance at the equator. Because the projection doesn’t let the meridians converge, it must stretch the sixtieth parallel to twice its true length. To compensate for this pronounced east–west stretching along the map’s parallels, the projection imposes an identical stretching in the north–south direction, along the meridians. Farther north, as east–west stretching grows progressively larger, north– south stretching increases proportionately. At the North Pole, a mere point on the globe, map scale becomes indefinitely large—the result of stretching a dimensionless spot to a measurable distance—and the pole lies “at infinity, ”or at least well off the map. That’s why Mercator world maps typically cut off northern Greenland and omit most of an otherwise humongous Antarctica.

Forcing north–south scale to equal east–west scale at all points not only preserves angles and bearings but prevents the deformation of small circles into ellipses. Modern cartographic textbooks consider this locally exact portrayal of angles and small shapes, called conformality, highly desirable for detailed, large-scale maps of small areas. In addition to depicting city blocks as rectangles, not parallelograms, a conformal map keeps squares square and circles circular. Although more than a century passed before Edmund Halley (1656–1742) recognized conformality as a mathematical property, Mercator’s 1569 world map became the first conformal projection to portray meridians and parallels as straight lines.

In addition to drawing on his experience in making globes, Mercator borrowed a concept embedded in fourteenth-century regional sailing charts. Portolan sailing charts, named after the portolani, or pilot books, that guided sailors across the Mediterranean Sea or along the coast of Europe, were distinguished by a network of straight-line sailing directions that converge at assorted compass roses. A typical portolan chart was oriented to magnetic north, covered less than one fiftieth of the earth’s surface, and lacked a consistent grid of meridians and parallels. Originally drawn to illustrate books of written sailing directions, portolan charts reduced the uncertainty of navigating across open waters. As the first whole-world sailing chart, Mercator’s map made a transatlantic journey look as straightforward as a voyage from Athens to Alexandria.

However easy to navigate, a loxodrome is rarely as direct as the great circle crudely approximated by a taunt thread stretched across a globe between a route’s origin and destination. Great circles, so called because they are the largest circles one can draw on a sphere, define the shortest path between two points. Although geometrically efficient, they are difficult to navigate because the bearing is constantly changing. The only exceptions are routes along a meridian or the equator. Because a loxodrome is not a great circle, the sailor taking its more easily followed course takes an indirect route. But if the increased distance is long enough to make a difference, the navigator can divide the route into sections and follow the rhumb line for each part. In figure 1.4,a dotted line illustrates a sectioned route from Cape Town to New York. Because the Mercator grid distorts distance, the single rhumb line marking the constant-bearing route looks deceptively shorter than either the great-circle route or its multi-rhumb approximation.

Mercator sought to reconcile the navigator’s need for a straightforward course with the trade-offs inherent in flattening a globe. These trade-offs include distortions of distance, gross shape, and area. Although all world maps distort most (if not all) distances, some projections, including Mercator’s, afford negligible distortion on large-scale detailed maps of small areas. Only a globe can preserve continental outlines, however, which cannot be flattened without noticeable stretching or compression. Relative size, which is preserved on map projections with a property called equivalence, is markedly misrepresented on Mercator charts because of the increased poleward separation of parallels required to straighten out loxodromes. Distortion of area is most apparent in the chart’s inflated portrayal of Greenland as an island roughly the size of South America. On a globe Greenland is not quite an eighth as large.

Like many innovations, the new projection did not catch on right away. One impediment to a wider, swifter adoption was the lack of a detailed procedure for progressively separating the parallels. Wordy inscriptions explained the map’s purpose but offered no instructions for constructing or refining its grid. That Mercator produced a generally accurate solution for the lower and middle latitudes was quite an accomplishment in an era with neither logarithms to expedite calculation nor integral calculus to derive a concise mathematical formula. Trigonometric tables of secants and tangents, which might have been especially useful, were also lacking. Some scholars think Mercator used a mathematical approximation to lay out parallels ten degrees apart; a few suggest that he developed the separations graphically by copying loxodromes from a globe to a map. Whatever his approach, Mercator’s map stimulated further work by English mathematicians Edward Wright (1561–1615) and Henry Bond (1600–1678), discussed in chapter 5. In 1599, in a treatise with a long title that begins Certaine Errors in Navigation, Wright included a table of “meridional parts,” with which a chartmaker or seaman could efficiently lay down a Mercator grid. And in 1645, Bond suggested a mathematical formula after discovering a similarity between Wright’s table and a table of logarithmic tangents.
Another obstacle was the primitive technology for taking compass readings at sea and correcting for magnetic declination. An inscription on the 1569 world map discusses the vexing discrepancy between the poles that anchor the earth’s grid and the poles believed to attract compass needles. Eager to include a north magnetic pole on his map, Mercator consulted “a great number of testimonies, "which suggested diverse positions for a magnetic meridian aimed at the magnetic pole. Some observations placed this magnetic meridian in the Cape Verde Islands, where magnetic north coincided with true north; others placed it at Corvo,in the Azores. Equally suggestive was Marco Polo’s report that “in the northern parts of Bargu [in northeast Asia] there are islands, which are so far north that the Arctic pole appears to them to deviate to the southward.” Without marking the Corvo meridian explicitly on his map, Mercator extended it up over the pole and then south toward Asia. In doing so, he wrongly assumed that compass needles point along great circles that converge at the magnetic poles.

Aware that, because of this uncertainty, the location didn’t warrant an X or a compass rose, Mercator marked the spot with what looks like a fried egg with a very small yoke (fig. 1.5). An adjacent inscription restates the premise: “It is here that the magnetic pole lies if the meridian which passes through the Isle of Corvo be considered at the first. To hedge his bets, the chartmaker placed a second magnetic north pole farther south and a bit to the east, where a larger symbol that cartographic historians Helen Wallis and Arthur Robinson describe as “a high rocky island” carries a more confident explanation: “From sure calculations it is here that lies the magnetic pole and the very perfect magnet which draws to itself all others, it being assumed that the prime meridian be where I have placed it.” Confronting uncertainty, Mercator used a pair of “extreme positions” to bracket the magnetic pole’s true location “until the observations made by seamen have provided more certain information. "Too few present-day cartographers, sad to say, are as frank about geographic ambiguity.

 Ships carried magnetic compasses as early as the twelfth century, but seamen seldom used them because of an innate mistrust of innovations as well as quirky needles that didn’t point directly north. Magnetic declination was not discovered until the fifteenth century, and as Mercator’s experience illustrates, geomagnetism proved less wellbehaved than sixteenth-century mapmakers had originally believed. Adjustment for geomagnetic distraction was not possible until 1701, when Edmund Halley published a pioneering but simplistic map of isogons (lines of equal magnetic declination) for the Atlantic Ocean (fig. 1.6). Determining a ship’s location at sea was equally troublesome. Latitude could be figured simply by sighting on the northern star at night or by measuring the sun’s noontime elevation above the horizon, but longitude, calculated from the difference between local time and time at the prime meridian, required a highly accurate chronometer, not available until the mid-eighteenth century, when John Harrison (1693–1776) devised a clock that lost only fifteen seconds in 156 days. The ship’s compass, magnetic adjustment, and an accurate chronometer were parts of a puzzle that included Mercator’s projection. Not until all the pieces were in place could mariners fully appreciate Mercator charts.

Navigators began to use the Mercator map in the early 1600s, after British geographer Richard Hakluyt (1552–1616) included a world map drawn by Wright in the second edition of his Principall Navigations, Voiages, Traffiques and Discoveries of the English Nation, published in 1599. Wright not only corrected inaccuracies in Mercator’s grid but provided an updated view of world geography, taken from a 1592 globe by Emery Molyneux (d. 1598/9). Measuring 17 inches (43 cm) tall by 25 inches (64 cm) wide and printed in two sections, the Wright-Molyneux map, as it’s often called, is smaller and more readily reproduced, displayed, and archived than Mercator’s eighteen-sheet mosaic. According to Robert Karrow, nineteen copies of the Wright Molyneux map exist, in contrast to only three copies of Mercator’s, which is seldom reproduced in one piece because of its size. Despite suggestions that the grid be called the Wright projection, Mercator’s name stuck, reinforced no doubt by his impressive contribution as an atlas publisher. Cartographic historians celebrate Gerard Mercator for two epic achievements: his world map of 1569 and his monumental three-volume world atlas, completed in 1595.

Mercator might not have been the first to use the projection that bears his name. In 1511, Erhard Etzlaub (ca. 1460–1532),a Nuremberg compass maker, crafted a portable sundial with a map on its lid. A mere 3.1 inches (80 mm) wide and 4.3 inches (108 mm) tall, Etzlaub’s map puts south at the top and covers only Europe and North Africa (fig. 1.7). It lacks a graticule ,but latitude gradations at one-degree intervals along the sides and numerical labels every five degrees reflect the progressively spaced parallels of the Mercator grid. This similarity is hardly an accident. Etzlaub produced a similar but slightly larger sundial map two years later and presumably made others that didn’t survive. An instrument maker and physician with an active interest in astronomy and cartography, he produced several other maps, principally woodcuts with south at the top. Especially note-worthy is his 1500 road map of central Europe, cast on a stereographic projection—also conformal—to promote the accurate alignment of compass points with travel directions. According to cartographic historian Brigitte Englisch, his 1511 “compass map” not only was the earliest rectangular conformal projection but also accords exceptionally well with modern versions of the Mercator projection. Englisch argues that Mercator no doubt knew of Etzlaub’s invention and that “the projection of varying latitudes should be known as the Etzlaub-Mercator projection.”

Wright and Etzlaub are not the only mapmakers in line to share Mercator’s fame. Another contender is the unidentified Chinese scholar who drafted the tenth-century Dunhuang star map. According to The Timetables of Science, a chronology published in 1988 and cited on several Web sites, the star chart “uses a Mercator projection [and is] the first known use of this kind of map projection. "I tracked this assertion no further than the multivolume History of Cartography, which includes a black-and-white photo of the narrow, scroll-like map. How the claim arose is a puzzle insofar as the chart contains neither a grid nor marginal tick marks. As a key sentence in its caption tellingly observes: “There is no attempt at a projection on this rather crude chart.” Projection guru John Snyder wholly ignored the Dunhuang star chart in his epic history of map projection, in which he noted Etzlaub’s “similar projection "but concluded that “the principle remained obscure until Mercator’s independent invention.”

Anyone who thinks cartographic folklore inflates Mercator’s contribution should be mollified if not amused by an offhand comment in the U.S. Coast and Geodetic Survey’s bible on map projection, introduced in 1921 and shepherded through numerous revisions by Charles Deetz and Oscar Adams. In discussing the sinusoidal projection, on which converging meridians yield a world map shaped like an antique Christmas ornament, Deetz and Adams noted the occasional use of an alternative name, Sanson-Flamsteed projection, commemorating Nicolas Sanson and John Flamsteed, who used it around 1650 and 1729, respectively. In their opinion, the projection “might well have been termed the ‘Mercator equal-area projection’ in the first place, from the fact that the early atlases bearing his name gave us the first substantial maps in which it is employed. Mercator’s name has, however, been so clearly linked with his nautical conformal projection that it becomes necessary to include with his name the words equal area if we wish to disregard the later claimants of its invention, and call it the Mercator equal-area projection.” To underscore the point, they titled the section “Sinusoidal or Mercator Equal-Area Projection.”
Whatever its authorship, the better-known Mercator conformal projection gathered adherents among scientists as well as navigators. Noteworthy adoptions include Robert Dudley’s pioneering sea atlas of the world, published in 1647, and Edmund Halley’s revolutionary maps of the trade winds and magnetic declination, published in 1686 and 1701, respectively. In 1769 the Mercator grid provided the geographic framework for a ground-breaking Gulf Stream chart by Benjamin Franklin and whaling captain Timothy Folger, and in the early nineteenth century it gained wider exposure in the influential line of geography textbooks written and published by Jedidiah Morse, father of portrait painter and telegraphic experimenter Samuel F. B. Morse. In 1919 Vilhelm Bjerknes, the Norwegian meteorologist who discovered fronts and air masses, proposed the Mercator projection as the world standard for weather maps of the tropics, and in 1937 the World Meteorological Committee recognized the importance of conformality on atmospheric maps by endorsing Bjerknes’s recommendation. In the commercial sphere, publishers of reference atlases and wall maps adopted the Mercator grid for regional maps of Australia, the Pacific islands, and the world’s oceans.

To the disgust of geographic educators, Mercator’s grid framed many whole-world maps with no bearing on navigation, weather, or geophysics. As I show in chapter 9, geopolitical motives were apparent in a few cases, but much of the projection’s misuse reflects a mix of comfortable familiarity, public ignorance, and institutional inertia. No one was hawking the Mercator brand, at least not overtly, but no one had to—many people who grew up with the map apparently believed this was how a flattened earth should look. How else to explain the ascendancy of an utterly inappropriate perspective and widespread resistance to superior substitutes?

If there is a villain here, it's not Gerard Mercator, who used equal area maps in his atlases and was quite clear about why he devised a rectangular conformal projection. Wary of wrongheaded finger pointing, Deetz and Adams chided the chartmaker’s critics in verse:
Let none dare to attribute the shame
Of misuse of projections to Mercator’s name;
But smother quite, and let infamy light
Upon those who do misuse, Publish or recite.
Although educators and scientists understood the problem, few seemed willing to challenge the conventional stupidity.

The most famous attack on the Mercator map’s undeserved prominence came well after the tide had turned. In the 1970s German historian Arno Peters (1916–2002) proposed a ludicrously inapt solution now known as the Peters projection. As chapter 11 explains, the Peters map is not only an equal-area map but an exceptionally bad equal-area map that severely distorts the shapes of tropical nations its proponents profess to support. Its popularity among Third World advocacy groups like Oxfam and the World Council of Churches is hard to explain. Perhaps it’s a reflection of what I call the Monty Python Effect, named for the parody troupe’s well-known transition line, "And now for something completely different." To most people who see it for the first time, the Peters map is indeed different: as figure 1.8 illustrates, Africa and South America look like land masses stretched into submission on a medieval torture rack. In asserting a new solution to an old problem, Peters ignored other, demonstrably better equivalent projections. And in claiming his projection was original, he overlooked an identical map presented in 1855 by James Gall (1808–95),a Scottish clergyman. Dare I say it? Peters had a lot of Gall in as many ways as possible. Mercator’s legacy is much more than the life and works of a Flemish chartmaker. As the remaining chapters illustrate, the Mercator projection lies at the intersection of a diverse collection of intriguing tales about navigation, cartographic innovation, military precision, media mischief, and political propaganda.


To be followed next week




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