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Bearings Straight (V)

 

The Wright Approach


Edward Wright was a mathematician, not a mapmaker. Born in 1561 in the village of Garveston, about one hundred miles northeast of London, he attended Gonville and Caius College at Cambridge, where he received a bachelor of arts degree in 1581 and a master of arts three years later. In 1587 a research fellowship allowed him to focus on mathematical cosmography and its use in navigation. In 1589 a seven-month leave to help the Earl of Cumberland plunder Spanish shipping in the Azores provided practical experience at sea. Appalled by mariners’ misuse of almanacs, charts, and navigation instruments, Wright undertook a mathematical critique of contemporary navigation. His search for new solutions to old problems included a sea chart with straight-line loxodromes: the map Mercator had demonstrated but never explained.

Wright compared the projection to an inflatable globe inside a glass cylinder. Imagine a spherical bladder, he suggested, with meridians, parallels, and a selection of rhumb lines inscribed on its surface. Inflate the sphere initially so that its axis aligns with the axis of the cylinder and its equator just touches the glass. This step establishes the equator as the standard line, with the same scale on both globe and cylinder. Then slowly inflate the bladder so that all rhumb lines remain straight and the stretching at every point is the same in all directions—the angle-preserving condition now known as conformality. Although Wright’s mythical model requires an infinitely expansive bladder of extreme flexibility, it describes perfectly the transformation of a globe into Mercator’s conformal cylindrical projection: the parallels grow farther and farther apart as the bladder inflates, but because the cylinder is open ended, the poles never touch the map.

To describe the growing separation of the map’s successive parallels Wright worked up a table with three columns. The first two list the degrees and minutes of latitude for parallels spaced ten minutes apart on the sphere, and the third reports the parallel’s projected distance from the equator. Because the northern and southern hemispheres have identical grids, the table runs from the equator at 0° to a generic pole at 90°, and because a degree contains sixty minutes, an interval of ten minutes divides each half meridian into 540 (90 X 6) “meridional parts.” To simplify the calculations, Wright set to 100 the distance encompassed by an arc of ten minutes at the equator. With minimal distortion near the equator, the parallels for 0° 10' and 0° 20' plot at 100 and 200 distance units, respectively. Because the table’s third column has no decimal places, the slowly growing separation of parallels is not apparent until the sixteenth meridional part positions the parallel for 2° 40' at 1,601—up 101 units (rather than 100) from the parallel for 2° 30' at 1,500. Vertical stretching becomes only slightly more apparent at 15° 00', which plots at 9,104—only 103 distance units away from 14° 50', which plots at 9,001. Separations increase, and in its final rows the table locates the parallels for 89° 40' and 89° 50' at 201,513 and 226,223, respectively, and describes the polar parallel of 90°00' as “Infinite.”With Wright’s “Table for the true dividing of the meridians in the Sea Chart,” any mapmaker or sailor could easily lay out a Mercator grid.

Figure 5.1 The title page of the first edition of Wright’s treatiseWright used at least three decimal places in his calculations, but omitted them from the abridged table in the first edition of Certaine Errors in Navigation (fig. 5. 1), published in 1599, in order “not at this time to trouble [chartmakers and navigators] with more than thought to be of use.”Another concern might have been his publisher’s bottom line: the condensed table with a ten-minute interval occupies a mere six pages in the 1599 edition, whereas the complete table, with a one-minute interval and smaller type, consumes twenty-three pages in the second edition, published in 1610. Of little direct use to most readers, the added precision of 5,400 (90 X 60) small meridional parts minimized cumulative error.

However tedious, Wright’s calculations are straightforward. The map’s rectangular grid, which stretches the parallels to equal the equator in length, compensates for this increasing horizontal exaggeration by shifting the parallels farther apart vertically. The left part of figure 5.2 describes key elements in the calculation: a pair of meridians divide the equator and a parallel at latitude Ǿ into sections with lengths c and c', respectively. Note that on the globe c' becomes progressively smaller than c with increasing latitude. Because the meridians on the map (right side of fig. 5.2) cannot converge, the mapped arc at latitude Ǿ is stretched horizontally by a factor of c/c'. At 60°, where the full circumference of the parallel on the globe is half the length of the equator, the stretching ratio c/c' equals 2.0. Farther poleward, as the latitude approaches 90°, c' shrinks to zero and the stretching factor approaches infinity. Near the equator, though, east–west stretching is comparatively minor and the ratio is only marginally greater than 1.

Trigonometry conveniently enters the picture at this point because c/c' is the secant of angle Ǿ. (In trigonometry the secant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the adjacent side.) By consulting a table of secants, readily available to any late sixteenth-century university mathematician, Wright could look up the stretching factor for any latitude.
The crux of Wright’s method is a cumulative vertical adjustment for horizontal exaggeration. Figure 5.3 describes the process. A pair of meridians one minute apart define the east and west sides of a series of quadrangles covering one minute of latitude or longitude on all sides and extending upward from the equator. Stacked vertically, the quadrangles have curved sides on the sphere but plot as rectangles on the projection. Because the map’s meridians cannot converge, the first quadrangle, which covers distance d along the equator, must be slightly taller to compensate for east–west stretching along its upper edge, which is proportional to the secant of its latitude.

Thus its upper edge, defined by the parallel at 0° 1', is d times the secant of 1', which is written as d sec 1'. Similarly, the height on the map of the second quadrangle, ever so slightly taller, is d sec 2', and the height of the quadrangle immediately above is d sec 3'. As the diagram shows, the vertical distance from the equator to the parallel at 0° 3' is the sum of the heights of these three rectangles. More generally, the map distance y from the equator to the parallel at latitude Ǿ can be computed as How accurate is Wright’s table? To find out, I wrote a short computer program—something cartographers rarely do these days, now that commercial software handles most mapping tasks. Table 5.1 compares my results with Wright’s abridged and complete tables, published in 1599 and 1610, respectively. Although the differences are more apparent for higher latitudes, where cumulative error should be most noticeable, the numbers are remarkably close. To put these discrepancies in perspective, I calculated the error (assuming my computer is reliable) for a world Mercator map three feet wide. At this scale the greatest discrepancy, a mere 2.366 at 80°, represents a nearly infinitesimal 0.00039 inches on the map—well within the tolerance of the most precise automatic plotters. It’s hard not to be both amazed and impressed.

The calculations no doubt impressed mapmaker-engraver Jodocus Hondius, who was in London in the early 1590s, sitting out the Netherlands’version of the Spanish Inquisition. Hondius heard of Wright’s work and borrowed a draft manuscript for a brief period after agreeing not to publish any of its contents without permission. But an accurate table of meridional parts was too great a temptation for the Dutch mapmaker, who drew on the English mathematician’s labors for several regional maps as well as a world map he published in Amsterdam in 1597 (fig. 5.4). No less apparent than the progressive pole-ward spacing of the map’s parallels is the allegorical engraving of a Christian knight battling Sin, the Flesh, and the Devil. Hondius was mute about how he laid out the grid but dedicated the map, in Latin, to “Ed. Wrichto” and two other Englishmen.

Wright was outraged. In the preface to Certaine Errors, he quoted a letter in which Hondius had offered a vague apology: “I hear that you are somewhat offended with me because I have taken those few things out of your hand-written book.... Truly I told all my friends plainly that you are the author thereof, and I tell them so still.” In what historian Lawrence Wroth termed “the most inept rationalization a plagiarist ever made,” Hondius pleaded, “I was purposed to have set this forth under your name, but I feared that you would be displeased therewith because I have but rudely translated it into Latin.” Neither moved nor mollified, Wright spared no sarcasm in condemning his former friend’s deceit and greed: “But how well and honestly he [honored his agreement], grounded upon faith and credit, the world may now see: and how thankful he hath been to me for that which hath been so profitable and gainful unto himself, as may appear by so com­mon sale of his maps of the world, and of Europe, Asia, Africa, and America (all which had been yet unhatched, had he not learned the right way to lay the groundwork of them out of this book) I myself know too well. But let him go as he is.”

The Christian-Knight map (as map historians call it) was not the only premature publication of Wright’s results. His table of meridional parts appeared in print in 1594, in mathematician-navigator Thomas Blundeville’s Exercises for Young Gentlemen, and again, three years later, in Sir William Barlow’s The Navigator’s Supply. Although both authors had obtained Wright’s permission, only Blundeville acknowledged him by name. Barlow vaguely credited “a friend of mine of like profession.” More devious was Abraham Kendall, a navigator with Sir Robert Dudley’s 1594 expedition to Guiana and Trinidad. Kendall borrowed a draft of Wright’s manuscript, made a longhand copy without permission, and carried it with him on Sir Francis Drake’s 1595 expedition to the West Indies. Whatever his intentions, Kendall died off Porto Bello, and the manuscript found its way back to London, where someone, thinking it original scholarship, sent it to the Earl of Cumberland, who immediately recognized the work of his former hydrographer. According to maritime historian David Waters, two brushes with plagiarism—at the hands of Kendall and Hondius—convinced Wright to publish his book, which he gratefully dedicated to the Earl.

Although Wright was neither the engraver nor the publisher, historians credit him with the two-sheet world map (fig. 5.5) prepared for the second volume of geographer-navigator Richard Hakluyt’s Principal Navigations, Voyages, Trafiques and Discoveries of the English Nation, published in 1599. Sometimes called the Wright-Molyneux map because Wright laid out the graticule and transferred features from Emery Molyneux’s 1592 terrestrial globe, the chart reflects recent discoveries by explorers like John Davis. Conspicuously absent are the northern Pacific coasts, the vast southern continent, and other questionable features that populate unexplored regions on most late sixteenth-century world maps. Hungry for accurate information, Wright and his co-compilers consulted the latest Dutch, Portuguese, and Spanish charts and amassed a total of 1,209 place names, mostly coastal. Acclaimed by the English intelligentsia for correcting numerous inaccuracies on existing charts, the map’s fame is affirmed in act 3, scene 2 of William Shakespeare’s Twelfth Night, in the line “He does smile his face into more lines than is in the new map with the augmentation of the Indies.”

Wright died in 1615, knowing he had made an important contribution to navigation and cartography. Although his principles and calculations promoted a wider use of Mercator’s projection—which a few logrolling British historians proposed calling the Wright-Mercator projection—it’s now clear that another English mathematician, Thomas Harriot (1560–1621), had begun to address the problem of meridional parts around 1589, about the same time as Wright. What’s more, Harriot’s solution is cleaner and more mathematically elegant insofar as he had progressed from merely adding up secants, as Wright had done, to a logarithmic tangents formula that affords a more exact and direct solution. We know this because Harriot left behind a massive collection of unpublished drawings, tables, notes, and manuscripts— over ten thousand pages worth, according to the Dictionary of Scientific Biography, which attributes his aversion to publishing to “adverse external circumstances, procrastination, and his reluctance to publish a tract when he thought that further work might improve it.” A brilliant scholar with a profound understanding of astronomy and physics as well as mathematics, Harriot is the epitome of the perfectionist academic who rarely publishes.
Harriot’s solution anticipated the serendipitous discovery of another English mathematician, Henry Bond (ca. 1600–1678), who around 1645 noticed a surprising correspondence between Wright’s table of meridional parts and a table of logarithms of tangents published in 1620 by Edmund Gunter (1581–1626). It wasn’t a direct correspondence—Bond had to reorganize Gunter’s table to show logarithmic tangents of (450 + Ǿ/2), where Ǿ is latitude—but once the numbers were rearranged, an exact match suggested strongly that the distance y from the equator of the parallel at latitude Ǿ on a Mercator projection could be computed as where R is the radius of a globe that defines the projection’s scale and in specifies a natural (or Napierian) logarithm. Bond’s insight is important for two reasons. First, because the equation is not based on a succession of sums, it promotes a more straightforward, less error-prone calculation of projected coordinates using either a computer or a table of logarithmic tangents. (A moot point, perhaps, if Wright’s table is at hand.) Second, and more important, as an equation readily manipulated using algebra and calculus, Bond’s formula fosters a detailed mathematical examination of the projection’s geometric distortion.

I looked in vain for a copy of the 1645 edition of Richard Norwood’s Epitome of Navigation, in which Bond, who was its editor at the time, first published his observation. But no less than the eminent Edmund Halley confirmed Bond’s discovery in a 1696 essay in the Philosophical Transactions of the Royal Society of London. Halley titled his article “An Early Demonstration of the Analogy of the Logarithmick [sic] Tangents to the Meridian Line or Sum of the Secants.” After crediting “our Worthy Countryman Mr. Edward Wright” with a valuable table “to be met with in most Books treating of Navigation, computed with sufficient exactness for the purpose,” he turned to the subject of his essay in noting, “It was first discovered by chance, and as far as I can learn, first published by Mr. Henry Bond, as an addition to Norwoods Epitome of Navigation, about 50 Years since, that the Meridian Line was Analogous to a Scale of Logarithmick Tangents of half the Complements of the Latitudes.” Halley’s article is important because he not only validates the Bond legend but also substantiates Wright’s claim to priority. Like Wright, Halley was innocently ignorant of Harriot’s unpublished solution.

Like most mathematicians I know, Halley was less concerned with the proposition’s history than with its proof. An earlier proof, by James Gregory (1638–75), was hardly elegant, or as Halley saw it, “not without a long train of Consequences and Complications of Proportions, whereby the evidence of the Demonstration is in a great measure lost, and the Reader wearied before attaining it.” And while subsequent attempts strayed from the point of Bond’s discovery, Halley’s own demonstration, the focus of his essay, was simple, on target, and probably original, as he boldly asserts in a remarkably candid and irresistibly quotable disclaimer:

Wherefore having attained, as I conceive, a very facile and natural demonstration of the said Analogy, and having found out the Rule for exhibiting the difference of Meridional parts, between any two parallels of Latitude, without finding both the Numbers whereof they are the difference: I hope I may be entitled to share in the improvements of this useful part of Geometry. Desiring no other favour of some Mathematical Pretenders, than that they think fit to be so just, as neither to attribute my desire to please the Honourable Royal Society in these Exercises, to any kind of Vanity or Love of Applause in me, (who too well know how very few these things oblige, and how small reward they procure) nor yet to complain, coram non judice, that I arrogate to my self the Inventions of others, and upon that pretext to depreciate what I do, unless at the same time, they can produce the Author I wrong, to prove their assertions. Such disingenuity as I have always most carefully avoided, so I with not too much experience of it in the very same persons, who make it their business to detract from that little share of Reputation I have in these things.

If Thomas Harriot had been as eager to publish, Edward Wright might be no better known today than Abraham Kendall or Henry Bond.

 

 

 

In a self-esteem contest, Halley could not hold a candle to Johann Heinrich Lambert (1728–77). During aninterview for membership in the Prussian Academy of Sciences, Frederick the Great asked Lambert to name the science in which he was most proficient. Without hesitation, the candidate calmly answered, “All.” Hardly an overstatement, though, for a genius whose contributions encompass mathematics, physics, astronomy, philosophy, and artography. Born in Alsace to poor German parents and largely self-educated, Lambert worked as a clerk, secretary, and tutor before moving to Berlin in 1764. According to the Dictionary of Scientific Biography, his appointment to the Academy was delayed a year because of “his strange appearance and behavior.” Lambert was openly religious, perhaps obnoxiously so, and he had an exceptionally high forehead, highlighted in the intriguing portrait (fig. 5.6) that decorates nearly every account of his life and work.

I suspect, though, that the engraver, working from sketches decades after his eminent subject’s demise, exercised a bit of artistic license in endorsing popular ideas about superior intelligence and cranial capacity.

Lambert’s contributions to cartography include seven different map projections as well as an illuminating mathematical analysis of conformality. In addition to using calculus to derive Bond’s analytical formula for the Mercator projection, he demonstrated that the Mercator map is a “special case” in a family of conformal projections with polar and conic versions. As figure 5.7 illustrates, the cylinder and the plane are extreme forms of a cone tangent to the sphere along a “standard parallel.” Positioning the apex at infinity converts the cone to a cylinder, with the standard parallel at the equator. Putting the apex on the North Pole flattens the cone to a plane and shrinks the standard parallel (at 90°) to a point. If the projections are conformal, the cylindrical case is the Mercator, the planar case is the polar stereographic (in use since about 150 BC), and all intermediate cases are instances of the Lambert conformal conic projection, presented in 1772.

Lambert’s insight stimulated further work on map projection by three of the era’s greatest mathematicians, Euler, Lagrange, and Gauss. For me, though, the next most decisive contributor is an otherwise obscure Paris mathematics teacher, Nicolas Auguste Tissot (pronounced “tea-so”), who devised an analytical description of map distortion. (I searched for a biography or obituary, but found nothing.) Tissot’s monograph Mémoire sur la représentation des surfaces et les projections des cartes géographiques, published in 1881, focuses on “the indicatrix,” a simple device for describing distortion of angles and shape. Picture a globe with many small circles—infinitesimally small, in theory—all the same size. On conformal projections, which do not distort angles, the tiny circles remain circles but vary in area—as Mercator’s map demonstrates, conformal projections suffer severe areal distortion in zones far from a standard line. By contrast, on projections that are not conformal, compression and stretching deform most circles into ellipses as shown in the indicatrix in figure 5.8.

In this example, point M on the circle corresponds to point M' on the ellipse, which reduces the angle ROR1 on the globe by an amount equal to twice the angle MOM'. Using calculus and his indicatrix, Tissot calculated areal distortion or maximum angular distortion at grid intersections for a variety of projections, including Mercator’s. In the next century his formulas helped analytical cartographers design customized projections that minimize distortion for specific regions.

As a graphic device for evaluating map projections, Tissot’s indicatrix is unrivaled. Anyone who grasps the notion of a network of small, uniform circles on the globe can easily compare areal distortion on the Mercator projection with angular distortion on the Peters map. On the Mercator map (fig. 5.9, left) small circles grow ever larger with increasing distance from the equator—the price of preserving angles and loxodromes on a rectangular projection. An altogether different trade-off arises with the Peters projection (fig. 5.9, right), on which perfect shape at 45° N and S gives way to severe north–south stretch­ing in the tropics and an equally troubling east–west wrenching around 75°, where severely deformed ellipses overlap. In neither case would the indicatrix be plotted at the poles, where east–west scale is indefinitely large. In chapter 9, Tissot’s clever device supports an in­sightful appraisal of promising substitutes for the Peters projection, such as those of Robinson and Goode.
Given Tissot’s contribution to the visual evaluation of map projections, it’s ironic that his treatise contains very few diagrams and no maps. Hardly surprising, though: mathematicians like Lambert and Tissot were numerical theoreticians, not mapmakers. Proficient in successfully attacking important cartographic problems analytically, they had little concern for the practical implications of their work.

 

 

 

 

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