The details of Mercator's plan developed over the years, but it was wide-ranging and ambitious from the start. Mercator immersed himself in the writings of the past; as his maps sought to reconcile classical wisdom with modern discoveries, so he intended that his Cosmographia would bring together the undoubted truths of Christianity with historical records and pagan authors in one great synthesis of learning. His aim, above all, was to reconcile faith in the teaching of the Bible with the growing knowledge of the physical world that seemed to call it into question. The Chronologia was to be the first of five books. In the second, he would present his view of the universe and astronomy, basing it on the celestial globe he had produced in 1551; the third and fourth would deal with astrology and the creation of the elements; and the fifth would describe the geography of the world as Mercator knew it to be, alongside the most accurate rendition yet attempted of Ptolemy's Geographia. In his dreams at least, Mercator was putting himself beside the great Alexandrian.
It was an impossible project. He was already in his fifties—a considerable age in the sixteenth century—and he tortured himself for the rest of his life with the growing realization that he would never achieve his grand design. His second volume remained unfinished, the third and fourth were never even started, and the final book was also incomplete when he died. But while working on the Cosmographia, Mercator made the discovery that would win him the immortality of which he dreamed.
THE STUDY OF GEOGRAPHY, which had taken up so much of his life, had traditionally been based on measurement and calculation, and, perhaps responding to the random, unpredictable nature of the tragedies that had struck him, he turned to the rational certainties of mathematics and the discipline of abstract thought. Central to the final part of the Cosmographia, the description of the current state of geographic knowledge, was to be a new map of the world. His map of 1538 would not do:The long, straggling shape of North America, which had represented the sum of knowledge three decades earlier, would have been laughable by the 1560s. In the Far East, there was more detail to be drawn in along the southern coast of Asia, and the English voyages to the northern seas had yielded detailed knowledge of the coastline of Scandinavia and northern Russia. More than this, the map of 1538 was useless for navigation. The double-cordiform projection that left America clinging helplessly to the edge of the design made it impossible to see how one part of the world related to another. Africa and South America were split between the two halves of the old map so that it was impossible to see their shape. Mercator needed to find a new way of reducing a three-dimensional image, the globe, to the two dimensions of a flat map— solving the problem of projection.
Artists and geometricians had been struggling with a similar problem for centuries. About 150 years before, the sculptor, architect, and engineer Filippo Brunelleschi, one of the most influential figures of the early Italian Renaissance, had calculated a mathematical way of producing the illusion of depth in a painting. Parallel lines in his picture of Florence's Piazza Duomo5 all met at a single vanishing point, and the size of an object was proportionate to its supposed distance from the observer.
This was the first modern example of a theory of linear perspective— "the rein and rudder of painting," according to Leonardo da Vinci a few years later6—and it is not too much to say that this one discovery was responsible for much of the unprecedented artistic flowering of the Renaissance over the next hundred years. Brunelleschi produced no written record of the experiments by which he established the principle of converging lines of sight—his contemporary Leon Battista Alberti some twenty years later explained the geometric principles behind it*—but his technique was essentially mathematical.
Brunelleschi demonstrated that perspective was a trick, an optical illusion. He was a showman by nature as well as an artist: He invited guests to look at the Duomo in Florence's Piazza through a small hole drilled in the back of a painting on a wooden panel. According to his biographer, Antonio di Tuccio Manetti, Brunelleschi "had made a hole in the panel . . . which was as small as a lentil on the painting side . . . and on the back it opened pyramidally, like a woman's straw hat, to the size of a ducat or a little more."7What the guest was actually seeing was not the Duomo but a small mirror reflecting Brunelleschi's painting of it; by removing and replacing the mirror so that his guest was alternately seeing the building and the painting, the artist could demonstrate the precision with which he had re-created the scene.
Brunelleschi's perspective was a theory built on rediscovered knowledge. Classical philosophers had written about it centuries before. In the first century BC, for example, the Roman architect and military engineer Marcus Vitruvius Pollio had described perspective as "a method of sketching a front with the sides withdrawing into the background, the lines all meeting in the centre of a circle."8Nobody before Brunelleschi, though, had developed the theory as a practical method of creating an apparently three-dimensional image on a flat surface, and other artists were quick to take advantage of his discovery.9The simple lead-backed mirror, a discovery of the thirteenth century, and another means of achieving the same effect, was a standard piece of equipment in an artist's studio, producing real-life examples of perspective to which the artist could refer.
Brunelleschi and the artists who followed him had a relatively simple objective: to give depth and emphasis to the scene they were painting, from the particular viewpoint they had selected. Cartographers could use similar principles—Ptolemy had done so in constructing his map projections*—but the challenge they faced was much more sophisticated. A map is a diagram, not a picture; there is no one viewpoint from which the entire globe can be seen, so perspective alone cannot solve the problem of transferring its curved surface onto a flat sheet of paper. That had been recognized for more than fifteen hundred years as a problem without a solution.
Some three-dimensional geometric forms, such as a cylinder or a cone, are capable of being laid out flat without distortion—"developable," in the mathematicians' jargon; others, such as the sphere, are not. While a mathematician can produce formulas to prove this, a layman can simply try to lay the peel of an orange flat on a table. If the spherical Earth is flattened out into two dimensions, like the orange peel on the table, it will always be distorted in some way, whether in the size or shape of the land, the distances between points, or the direction from one place to another. This means that a map projection, a system designed to lay the globe out flat, can never be perfect; all the designer can do is minimize some elements of the distortion at the expense of others.
For Ptolemy and the philosophers of classical times, this was a mathematical oddity as much as a practical problem of geography or cartography. The known world covered only a portion of the globe, and the knowledge of coastlines, the routes of rivers, and the relative positions of towns and physical features was so hazy and inexact that the distortions of projection were insignificant. In any case, there were no recorded journeys long enough to make projection a serious difficulty for sailors and navigators. Ignorance as much as mathematical law had limited the traditional worldview.
Even so, repeated attempts had been made to find the most accurate means of producing a flat map. Ptolemy, having set out his own proposals in his Geographia, had effectively thrown up his hands in despair and recommended a globe as the only truly accurate way of studying the world.
The globe, though, had disadvantages of its own, as he freely admitted: It could not be made large enough to show sufficient detail, it was awkward to handle, and one half of it would always be obscured from view. He described two different projections in detail, building on the efforts of other Greek philosophers hundreds of years before him, who had set out systems of parallels and meridians—the grid of latitude and longitude lines—to provide a basic framework for constructing a map.'10
The commonest such systems produced a network, or graticule, which was either square or rectangular. All parallels—the east-to-west lines of latitude—were straight and horizontal, re
gularly spaced according to the changes in latitude; the meridians, or north-to-south longitude lines, were similarly straight, equidistant, and parallel, crossing the lines of latitude at right angles. For the purposes of the map, each line of latitude on the globe was considered to be the same length. The projection was as straightforward to understand as it was to create, simply representing the globe as a cylinder rather than a sphere, and then unrolling it. Such a projection was ideal for maps of smaller areas, on which the inaccuracies as one moves north and south would be unnoticed."11
Although it may be easy to create, a straightforward cylindrical projection simply ignores the way that lines of longitude—the meridians—converge on the two poles. By straightening these curved and converging lines, the projection forces them apart. In addition, while every line of longitude is of equal length, running right around the world, lines of latitude are parallel circles, with only the longest one—the equator—running around the full circumference of the Earth. The rest get shorter the closer they approach to the poles—which means that each degree of longitude, being '46o of the length of a given line of latitude, will similarly represent a progressively shorter distance on the face of the Earth. A degree of latitude on the globe always represents the same distance, but the value of a degree of longitude becomes smaller as it gets farther from the equator. Those changes on the globe could not be properly reflected by a cartographer: Distances, areas, and—most important of all to navigators— directions were hopelessly distorted over much of the map. The laws of mathematics continually frustrated efforts to produce an accurate representation of the world.
Ptolemy's efforts, which survived only as mathematical descriptions rather than completed maps, were ambitious, being based on the idea of viewing the world not as a cylinder to be unrolled but as a cone. His parallels of latitude were concentric arcs rather than straight lines, and his longitudinal meridians either straight, evenly spaced radii, broken by an angle at the equator, or—in his more complicated version—a series of evenly spaced arcs. The curvature of these lines gradually reduced toward a single, straight prime meridian, producing a map shaped like a cloak spread out on the ground, gradually increasing in width from north to south.
This second projection, which Ptolemy said he preferred—"for me both here and everywhere, the better and more difficult scheme is preferable to the one which is poorer and easier"12—had the advantage of keeping different areas of the map more precisely in proportion to each other. Most cartographers who produced versions of his maps in the fifteenth and sixteenth centuries, however, were less conscientious and used the simpler version.
There were other proposals over the centuries. Globular projections, for instance, showed an entire hemisphere in a circular form and were used in the Islamic world for astronomical maps, and by medieval scholars such as the English philosopher Roger Bacon in the thirteenth century. Bacon, an English friar who studied at Paris and Oxford, quoted ancient Hebrew sources to support his view that it was possible to reach Asia by sailing west, and produced a circular map to demonstrate this. The map had parallel, straight lines of latitude crossing a single, straight, vertical central meridian at a right angle, with the curves of the other meridians gradually increasing on each side to form the outline of the globe. The world map of Franciscus Monachus, which inspired Mercator in his youth, was designed on a similar globular projection.
By Mercator's day, this mathematical theorizing had become important. The world was changing and growing, and the old guidelines for depicting it were increasingly inadequate. To the south of Europe, Africa and South America were known to stretch into areas that would be hopelessly distorted by any simple projections, while to the west, a great new continent had been discovered. Ptolemy had designed his projections to fit the world as he knew it, and cartographers had to find a way to incorporate the new knowledge that had been gained.
Some of their maps simply extended the arcs of Ptolemy's meridians, while others, such as Waldseemiiller's world map, gave a much sharper break in the line of the meridians at the equator. At the start of the sixteenth century, Johannes Stabius and Johann Werner were working together in Nuremberg on the church sundial that would eventually lead them to their own heart-shaped or cordiform world map. The result of their work, the basis of Mercator's first world map in 1538, had been to increase the curve of the meridians in Ptolemy's second projection.
One factor common to all these attempts was that they were virtually useless for navigation, because the straight course along a single compass bearing, which was simplest for a sailor to follow, was effectively unplot-table on the map. Portuguese ships returning from Brazil found themselves as much as seventy leagues off course as they approached the Azores, and King Joao IPs cosmographer royal, the mathematician Pedro Nunes, who had visited Leuven while Mercator was there, had recognized the problem and made his own contribution. He had calculated the curve across the map that might represent such a straight course across the sea; but although such a course might be shown on a map by a theoretical straight line along a compass bearing between the point of departure and the destination, actually following such a bearing on the sea would send a ship increasingly off course as the curvature of the Earth distorted its direction. Nunes, in effect, had defined the problem; a practical answer to it was yet to be found.
A straight line on any existing map could not be translated into a single compass setting on the sea. Navigators were faced with the unenviable choice of either plotting their course on a globe, with all the consequent inconvenience and inaccuracy, or making constant adjustments to their calculations as they proceeded. In practice, since lines of latitude could be figured out relatively easily from the height in the sky of the Sun or the planets, they had generally minimized the problem by setting a course well to either east or west of the destination they were aiming at and then sailing along the line of latitude until they found it—a hit-or-miss technique that could add days to a long voyage, without inspiring much confidence in its outcome. What had been a theoretical challenge for cartographers was a matter of immediate practical concern. Not only were ships sailing greater distances than they had ever done before, but they were venturing farther north and south, where the problems of projection were intensified.
The task Mercator had set himself was to produce a map that sailors could use, on which a straight line was also a straight line on the sea. He spelled it out in Latin in an explanatory panel on the world map he eventually produced in 1569: The projection, he said, "spreads the surface of the globe out flat so that places are in the correct position relative to each other, both as regards direction and distance, and with the correct latitudes and longitudes." His intent was not just to update the existing picture of the world but to produce a new type of map, with a new purpose, useful to scholars and sailors alike, representing the true shape of the continents with a minimum of distortion—not just a new version but a new vision of the world.
*The Etymologiae were written sometime around AD 620, toward the end of Isidore's life, and throughout the Middle Ages were considered the most important storehouse of classical learning. They were originally circulated in manuscript copies but were printed ten times between 1470 and 1529.
†Bernardus also included Arabic astrology in his mystical and allegorical description of the way that primal matter was fused with the spirit of nature and divine wisdom to create humanity. His poem circulated throughout the Middle Ages and can still be read in translation. (The Cosmographia of Bernardus Silvestris, trans. Winthrop Wetherbee [New York: Columbia University Press, 1990].)
*Alberti's book, On Painting, first appeared in 1435. It described how an artist should frame his picture within a rectangle and then choose a single focal point within it, from which lines should be drawn to various points along the baseline of the rectangle, forming a series of triangles. The changing distances between these lines as they converged on the focal point would then show precisely how dimensions should be adjust
ed in the painting to give the illusion of depth. The book is available in a modern translation, with introduction and notes, by John R. Spencer (New Haven: Yale University Press, 1970).
*He had written explicitly about geometric optics in his book Optica, which reappeared in the West in 1154, after being translated from an Arabic version.
Chapter Eighteen
The World Hung on the Wall:
The Projection
WALTER GHIM LIKED TO CLAIM after Mercator's death that he had been in the mapmaker's studio when work started on the great map of 1569. He watched, perhaps, but Mercator worked alone."In this vast undertaking, he had no help or assistance, but engraved the whole of the map himself with the exception of the margin," Ghim declared.1
Exactly when Mercator hit on the idea of a map that would be as useful to sailors as it was to scholars is impossible to say. Twenty-three years before, he had written to Antoine de Granvelle about the failings of navigational charts: "When the courses of ships are correctly measured, the latitudes shown are often greater than they truly are, or sometimes smaller. When the latitudes shown are correct, then the distances are inaccurate." In that letter, Mercator was concentrating on the question of compass deviation that had so troubled Columbus, rather than on the specific mathematical challenge of projection. In any case, he had more pressing concerns on his mind: In 1546, he was still rebuilding his business and his life after the trauma of Rupelmonde Fort. Even so, in the same letter, he had set down his plan for the future. "There is much else to be said about the correction of voyages and marine charts . . . . If I am ever relieved of my heavier obligations, I have decided to pursue and solve this matter properly."2
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