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Sarah Atchison
Dr. Pilant
MATH 646.700
4 April 2011
A Mathematical History of Cartography
Cartography, or “graph drawing”, is the science of mapmaking. For millennia,
people have been creating maps in order to aid in navigation. In early times it was
considered to be a mathematical discipline since mathematics was mainly about
measurement then. From our ancestors from the ancient world until present day,
maps have been useful tools in many aspects of everyday life. However, accuracy,
projection, and availability have been obstacles cartographers have had to battle
throughout the ages. Some of the most notable early cartographers include
Eratosthenes, Hipparchus, and Ptolemy.
According to Merriam-Webster’s dictionary, a map is “a representation
usually on a flat surface of the whole or a part of an area” (Merriam-Webster 2011).
The mathematical definition of a map is “a set of points, lines, and areas all defined
both by position with reference to a coordinate system and by their non-spatial
attributes” (Lanius 2003). They are typically created to facilitate navigation. Other
purposes for maps include, but are not limited to: locating points on Earth, showing
distribution patterns, and discovering “relationships between different phenomena
by analyzing map information” (Lanius 2003). Maps use a variety of sizes, shapes,
colors, patterns, values, orientations, and lines to convey the intended information
in an aesthetically and organizationally pleasing manner. Different thicknesses or
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colors have different meanings, even on just one map. Maps can be more of a
conceptual representation of reality that choose only the information necessary for
the purpose. Other problems and techniques cartographers must take into
consideration include: measuring Earth’s shape and features; collecting and storing
information about terrain, locations, and the population; modifying threedimensional figures to be placed on flat models; and “devising and designing
conventions for graphical representation of data” (Furuti 2009). Cartographers
must also choose an appropriate scale.
Map scale is defined as “the relationship between distances on a map and the
corresponding distances on the earth’s surface expressed as a fraction or ratio”
(Lanius 2003). Large-scale maps depict a small area in an extremely detailed
manner. On the other hand, small-scale maps show a larger area, but will little
detail. Ratios categorized as large scale include 1:24,000 and larger. Intermediatescale maps range from 1:50,000 to 1:100,000. Small-scale maps usually have a ratio
of 1:250,000 and smaller. The smaller the denominator, the larger the scale, and the
more detailed the map is (Lanius 2003).
Coordinate systems are numerical techniques for symbolizing locations on
the surface of the earth. Latitudes and longitudes form a grid on the earth’s surface
as a means of referencing locations. They are denoted by angle measures in the
form of degrees, minutes, and seconds (DMS). One degree is approximately seventy
miles, one minute measures just over a mile, and one second (which is one sixtieth
of a minute) is about one hundred feet in length. Lines of latitude, often referred to
as parallels, run east to west. Lines of longitude, also known as meridians, run north
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to south. Both sets of lines have lines of reference. The Prime Meridian, the
meridian passing through the Greenwich Observatory near London, England, is
labeled zero degrees. The other meridians are denoted by degrees to 180° east or
west halfway around the earth. All meridians intersect at the North and South Poles.
As late as 1881, fourteen different prime meridians were being used on simply
topographic survey maps. The single Prime Meridian we use today was adopted at
the International Meridian Conference of 1884 (Lanius 2003). The line of reference
for parallels is the Equator, the horizontal great circle around the center of the earth.
The equator is also labeled zero degrees, while the other parallels are denoted by
degrees to 90° North and 90° South at the poles. The actual measurement of a
degree, in regards to the meridians, can vary from roughly seventy miles near the
equator to zero degrees at the poles, since they converge at the poles. Parallels and
meridians are orthogonal to one another on a sphere.
Projection is probably the greatest obstacle cartographers face in creating a
map. Transforming locations and areas on a three-dimensional object into a twodimensional map can lead to several complications, but some level of inaccuracy is
unavoidable. Different types of projections have been formed so that distortion in
one aspect is lessened and another intensified. Examples of projections include
azimuthal orthographic, stereographic cylindrical, sinusoidal (Sanson-Flamsteed),
Mollweide, polar/equatorial azimuthal equidistant/equal-area, equidistant
cylindrical/Winkel I and II, and Aitoff, Hammer, and Winkel Triple (Furuti 2009).
Each projection type requires different grid placement (Sullivan 2000). According
to Wolfram, map projection “maps a sphere (or spheroid) onto a plane” and allows
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maps to be classified “according to common properties” such as cylindrical versus
conical or conformal, or angle preserving, versus area preserving. These schemes
are not usually mutually exclusive. Moreover, no projection can be simultaneously
conformal and area preserving. Mercator’s projection is conformal, but distances
are not consistent. This projection is one of the earliest projections of the entire
earth (Wolfram – Map projection 2011).
Mercator Projection
One of the most widely used projection type is azimuthal, or orthographic. In
this projection type, the earth is projected onto either a tangent or secant plane (see
below). A hemisphere is depicted as one would see it from outer space. Neither
angles nor areas are preserved, but distances tend not to be distorted along
parallels. In the second century BC, Hipparchus used orthographic projection to
determine places of star-rise and star-set. Around 14 BC, Marcus Vitrius Pollio, a
Roman engineer, used the projection to construct sundials and evaluate sun
positions. He also coined the term “orthographic”, or straight drawing, for the
projection. The earliest surviving orthographic maps are woodcut of Earth, as they
knew it from 1509. Photographs of planets from outer space have re-inspired
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interest in this type of projection in astronomy and planetary science (Wikipedia –
Orthographic Projection (cartography) 2011).
Orthogonal Projection
Trigonometry was used to derive the formulas used in deriving orthographic
projection. On the sphere, λ and φ are used to represent longitude and latitude,
respectively, with radius R, and origin (λ0, φ1). The equations for the projections
onto the (x, y) tangent plane condense to: x = R cos(φ )sin(λ − λ0 ) ,
y = R [cos(φ1 )sin(φ ) − sin(φ1 )cos(φ )cos(λ − λ1 )]. Calculating the distance c from the
center of the projection eliminates any latitudes beyond the range being depicted, so
as to not plot points on the opposite hemisphere. The equation
cos(c) = sin(φ1 )sin(φ ) + cos(φ1 )cos(φ )cos(λ − λ0 ) > 0 determines which points are to
be discarded. Inverse formulas aid in projecting a variable defined on a (λ, φ) grid
onto a rectilinear grid (x, y). These formulas are as follows:

φ = arcsin  cos(c)sin(φ1 ) +


ysin(c)cos(φ1 ) 
,
ρ


x sin(c)
2
2
 , where ρ = x + y and
ρ
cos(
φ
)cos(c)
−
ysin(
φ
)sin(c)


1
1
λ = λ0 + arctan 
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c = arcsin( ρ / R) (Wikipedia – Orthographic Projection (cartography) 2011; Wolfram
– Orthographic Projection 2011). A developable surface can be flattened without
distortion. This type of surface is common in the most common type of projection,
referred to as cylindrical projection. A cylindrical, or conic, projection is a geometric
projection onto a cylinder.
Evidence illustrates mapping developed independently in many different
parts of the world. The earliest known “map” was found in 1963 near modern-day
Ankara, Turkey. The map, believed to be from 6200 BC in Catal Hyuk in Anatolia, is
a wall painting displaying the locations of streets and houses with surrounding
features. Early map endeavors were extremely limited by ignorance of non-local
features. Native dwellers of the Marshall Islands created stick charts for navigation.
The oldest extant example of an Egyptian map is the Turin papyrus, dating around
1300 BC. Footprints represented roads on Pre-Columbian maps in Mexico. Early
Eskimos carved coastal maps out of ivory. Incas constructed relief maps of stone
and clay. As early as seventh century BC, the Chinese were making maps that were
much more detailed and accurate than those of their contemporaries. The earliest
evidence of early world maps mirror widely held religious beliefs of the times. For
instance, a map found on Babylonian clay tablets, dating around 600 BC, shows
Babylon and its surrounding areas. Babylon is represented by a rectangle and
vertical lines symbolize the Euphrates River. The surrounding area is circular and
enclosed by water, “which fits the religious image of the world in which the
Babylonians believed” (O’Connor and Robertson 2002). Anaximander is said to be
the earliest ancient Greek to have constructed a map of the world, but no details
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remain. It is believed that in sixth century BC, Pythagoras first put forth the belief
that the Earth is a sphere; moreover, Parmenides stated he believed the same the
next century. Approximately 350 BC, Aristotle gave six arguments for the purpose
of proving the Earth to be spherical. These arguments are generally accepted from
then on (O’Connor and Robertson 2002).
(Babylonian clay tablet)
Eratosthenes’ map
(Catal Hyuk map)
Ptolemy’s map
Mercator outline
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Eratosthenes, who lived in third century BC, made several major
contributions to cartography. He measured the Earth’s circumference very
accurately using angle measures. Eratosthenes was able to precisely sketch the
course from the Nile to the Khartoum, showing the two Ethiopian rivers. He also
used a grid to “locate positions of places on the Earth”; however, Dicaearchus, a
follower of Aristotle, had already been the first to devise a grid some fifty years
earlier (Lanius 2003). Using these positional grids was an early form of Cartesian
geometry. The grid Eratosthenes used was similar to the one we use today. Using
Dicaearchus’ methods, he chose a line through Rhodes and the Pillars of Hercules to
produce one of the lines of reference of his grid, known as 36 degrees North. This
grid system was highly accurate. The principal line of his grid sliced the world as he
knew it into two relatively equal halves and “defined the longest east-west extent
known” (O’Connor and Robertson 2002). He also selected a defining north-to-south
line through Rhodes and drew seven parallels to each of his defining lines, forming a
rectangular grid. Eratosthenes believed that two locations with similar climates and
environmental byproducts must also lie on the same parallel. This, of course, was
not the case. Eratosthenes formulated meridians by “transforming distances into
their angular values in relation to the circumference of the globe “ (Crone 1968).
His work inspired the most dominant of the projections devised before the
Renaissance, equirectangular projection. None of Eratosthenes’ works remain, but
we know of its existence through Strabo’s work entitled “Geographical Sketches”,
written circa 23 AD. This can also be said of the work of Hipparchus (O’Connor and
Robertson 2002; Sullivan 2000).
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Hipparchus was more of an astronomer, as he never constructed a map.
Hipparchus was essentially the founder of the coordinate system used in
cartography today. His system follows the Babylonian’s sexagesimal system,
involving latitudinal and longitudinal geodesics, which divide the Earth into 360
degrees. Each degree is comprised of sixty minutes and each minute is comprised of
sixty seconds. Hipparchus’ astronomical observations described eleven of the
parallels.
With great gratitude towards the conquests of Alexander the Great and the
Romans, the world as everyone knew it expanded, lending an enormous amount of
detail to future cartographers, who would then be able to the job put forth by
Eratosthenes and Hipparchus very confident in their ability to succeed. Claudius
Ptolemy was the last of the ancient Greeks to make a major contribution to
cartography. According to Snyder, “Ptolemy was possibly the single most influential
individual in the development of cartography in Europe and the Middle East at the
dawn of the Renaissance, although he lived 1300 years earlier” (Snyder 1993). He
was a famous mathematician who lived around 140 AD. He gave elaborate
instructions regarding a few methods of map projection. He wrote an eight-book
long “Guide to Geography”. This work was basically an extensive list of coordinates
“based on his study of itineraries, sailing directions, and topological descriptions”
(Sullivan 2000).
Several popular map types dating before the Renaissance were centered
around philosophy rather than mathematics. One example of this is the T-O map. In
the T-O map, all landmasses are contained within a circle. The circle, or O, is the
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limit of the known world. The horizontal segment of the T is “the approximate
meridian running from the Don to the Nile, and the perpendicular stroke the axis of
the Mediterranean" (Crone 1968).
(T-O map)
(portolan)
During the Middle Ages, Cosmas made a map that was the epitome of the
eclectic maps of the times by integrating in religious motifs and allusions.
(Cosmas’ map)
On the other hand, maps from Arab cartographers, namely Al-Idrisi, held true to the
earlier Greek techniques, even enhancing them. During this time, knowledge of
geography was lacking, so much of cartography was simply repetitive copying of
information; thus, inaccuracies were also being copied from one map to another.
Cartography really began emerging from its lull by 1300 AD. Progressions in
astronomy and mathematics stimulated the work cartographers were doing. Near
the end of the thirteenth century, maps known as portolans came into use in
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Western Europe. Portolans depicted coastlines and ports for sailors and were based
on observations made with compasses. The few portolans that survive have a
couple of features in common: they cover the Mediterranean and Black Seas along
with portions of Europe coastlines along the Atlantic Ocean; and the also include a
system of sixteen to thirty-six lines that cover the entire map (Sullivan 2000).
During the fourteenth century, the first attempt, since ancient times, was
made to include an accurate representation of Asia in world maps. This was seen in
Catalan world maps produced by the Catalan school. The Catalan world map of
1375 was constructed with the use of three resources: “(1) elements derived from
the circular world map of medieval times; (2) outlines of the coasts of western
Europe based on the normal portolan chart; (3) details drawn from the narratives of
the thirteenth and fourteenth century travelers in Asia” (Sullivan 2000). The maps
produced in the fifteenth century, were in line with the later Catalan maps, except
they reflected Ptolemy’s work as well. A monk in Murano, Fra Mauro, created a
world map often thought of as “the culmination of medieval cartography” (Crone
1968).
catalans
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The introduction of “the magnetic compass, telescope, and sextant enabled
increasing accuracy” in mapping (Wikipedia – Cartography 2011). Renowned
cosmographer Martin Behaim paved the way for a new wave of cartography by
making the first globe in 1492. This development coupled with vast amounts of data
resulting from new overseas explorations lead to a plethora of maps being produced
in the sixteenth century (Sullivan 2000; O’Connor and Robertson 2002; Wikipedia -Cartography). It was during this time that the Mercator projection, created by
Gerardus Mercator, made its debut, allowing seamen to navigate to their
destinations by following a rhumb line. Mercator’s vision was to provide a
comprehensive and up-to-date map. Highlights of the sixteenth century in regards
to cartography include the globe of 1541 and the world map of 1569. The globe of
1541 was the first to incorporate loxodromes, or “lines of constant bearing”, while
the world map of 1569 was the first to lay the loxodromes onto a two-dimensional
map (Sullivan 2000). This chart was intended to depict landmasses as precisely as
possible as well as for navigation. Mercator’s projection was a “regular cylindrical
projection, with equidistant, straight meridians, and with parallels of latitude that
are straight, parallel, and perpendicular to the meridians” (Snyder 1993). As seen in
the map above, Greenland appears to be much larger in surface area than South
America, which is actually nine times larger than Greenland (Lanius 2003; Sullivan
2000; O’Connor and Robertson 2002).
In the seventeenth century, the inventions of the pendulum clock, the
telescope, tables of logarithms, differential and integral calculus, and the law of
gravity all aided in scientists’ ability to make new observations of Earth and its
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characteristics. The development of measuring an arc on the surface of the Earth
also furthered the progresses in cartography. Isaac Newton theorized that, “due to
the centrifugal force of the spinning Earth, strongest at its equator, the Earth bulges
at the equator and flattens at the poles” (Lanius 2003). Newton revealed that the
Earth is actually not a true sphere, but is in fact a spheroid. During the eighteenth
century, Newton also assisted in perfecting the technique of evaluating longitudes
within one degree of accuracy. This allowed outlines of landmasses and positions of
locations to be more precise and exact. The settling of North American colonies and
ongoing rivalries between the Anglos and the French spawned a high demand form
more reliable, accurate, and up-to-date maps (Sullivan 2000; Lanius 2003). In the
nineteenth century, Europe executed the metric system, which presented “a simpler
and more universal language for map scale” (Lanius 2003). The Greenwich Prime
Meridian was also dubbed the sole Prime Meridian during this century, as
mentioned earlier.
In conclusion, cartography is more about mathematics than geography. This
science has affected and been effected by mathematical developments throughout
the ages. In times of mathematical downtimes, cartography also experienced lulls,
and vice versa. Cartographical developments are just beginning and only time can
tell just how much further we can advance in this area of mathematics and
geography.
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References
Crone, G.R. Maps and their Makers, London, England: Hutchinson & Co. Ltd, 1968.
Furuti, Carlos A. “Cartographical Map Projections.” Progonos, July 2009. Web. 24
Mar 2011.
http://www.progonos.com/furuti/MapProj/Normal/TOC/cartTOC.html.
Lanius, Cynthia. Rice University, Houston. 2003. Web. Accessed 20 Mar. 2011.
http://math.rice.edu/~lanius/pres/map/.
O’Connor, John J., and Edmund F. Robertson. “The history of cartography.” MacTutor.
University of St. Andrews, August 2002. Web. 21 Mar 2011. http://wwwgap.dcs.st-and.ac.uk/~history/HistTopics/Cartography.html.
Snyder, John P. Flattening the Earth, Chicago and London, England: The University of
Chicago Press, 1993.
Sullivan, John. “Mapmaking and its History.” Rutgers University, 2002. Web. 24 Mar
2011.
http://www.math.rutgers.edu/~cherlin/History/Papers2000/sullivan.html.
Weisstein, Eric W. "Map Projection." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/MapProjection.html
Weisstein, Eric W. "Orthographic Projection." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/OrthographicProjection.html
Wikipedia contributors. “Cartography.” Wikipedia, The Free Encyclopedia. Wikipedia,
The Free Encyclopedia, 4 Apr. 2011. Web. 4 Apr. 2011.
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Wikipedia contributors. "Orthographic projection (cartography)." Wikipedia, The
Free Encyclopedia. Wikipedia, The Free Encyclopedia, 17 Feb. 2011. Web. 24
Mar. 2011.