atchison midterm

7/29/2019 Atchison Midterm

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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-Websters 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 Earths shape and features; collecting and storing

information about terrain, locations, and the population; modifying three-

dimensional 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 earths 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. Intermediate-

scale 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 earths 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 two-

dimensional 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. Mercators projection is conformal, but distances

are not consistent. This projection is one of the earliest projections of the entire

earth (Wolfram Map projection 2011).

M e r c a t o r P r o j e c t i o n

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).

O r t h o g o n a l P r o j e c t i o n

Trigonometry was used to derive the formulas used in deriving orthographic

projection. On the sphere, and are used to represent longitude and latitude,

respectively, with radius R, and origin (0

, 1

). The equations for the projections

onto the (x, y) tangent plane condense to: x = Rcos()sin( 0

) ,

y=R cos(

1)sin(

)

sin(

1)cos(

)cos(

1)

[ ]. Calculating the distance c from thecenter 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(1

)sin() + cos(1

)cos()cos( 0

) > 0 determines which points are to

be discarded. Inverse formulas aid in projecting a variable defined on a (, ) grid

onto a rectilinear grid (x, y). These formulas are as follows:

= arcsin cos(c)sin(1

) +ysin(c)cos(

1)

,

= 0

+ arctanxsin(c)

cos(1

)cos(c) ysin(1)sin(c)

, where = x

2 + y2 and


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c = arcsin(/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 (OConnor 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 (OConnor and Robertson 2002).

( B a b y l o n i a n c l a y t a b l e t )

( C a t a l H y u k m a p )

E r a t o s t h e n e s m a p P t o l e m y s m a p

M e r c a t o r o u t l i n e


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Eratosthenes, who lived in third century BC, made several major

contributions to cartography. He measured the Earths 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 (OConnor 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 Strabos work entitled Geographical Sketches,

written circa 23 AD. This can also be said of the work of Hipparchus (OConnor 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 Babylonians 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).

( p o r t o l a n )

( T - O m a p )

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.

( C o s m a s m a p )

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 Ptolemys work as well. A monk in Murano, Fra Mauro, created a

world map often thought of as the culmination of medieval cartography (Crone

1968).

c a t a l a n s


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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; OConnor 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. Mercators 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. Mercators 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; OConnor 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/.

OConnor, John J., and Edmund F. Robertson. The history of cartography. MacTutor.

University of St. Andrews, August 2002. Web. 21 Mar 2011. http://www-

gap.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.

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