Location, Trigonometry, and Measurement of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 32

deter people from reading maps. Those who do not think about what they
are doing may get lost or get into trouble on the road from blind, thoughtless,
following of a “road routing recipe.” As with any computerized tool, “use—
not abuse!” Relative and absolute location may be already familiar in terms
of giving simple local directions. The terms are, however, more far reaching
than that. For example, absolute location can be given in hundreds of ways,
depending on the coordinate system and datum used, both of which are dis-
cussed in various parts of this topic.
Parallels and meridians were used in the previous chapter to describe the
location of a point, P , on the Earth-sphere as being at the 3rd parallel north
of the Equator and at the 4th meridian to the west of the Prime Meridian.
Recall that while this strategy might serve to locate P according to one ref-
erence system, someone else might employ a reference system with a finer
mesh. Ambiguity entered the picture, this time not in relation to orientation,
but rather in relation to partition with regard to counting parallels and merid-
ians. Many different descriptions, in how fine or coarse a mesh is created for
the graticule, can lead to one location. To convert this system to an absolute
system, which is replicable, we noted that we need to employ some com-
monly agreed-upon measurement strategy to standardize measurement—so
that different ways of partitioning data lead to different answers. The human-
constructed assumption that there are 360 degrees of angular measure in a
circle permitted such standardization. The use of standard circular measure
created a designation that was unique for P . One description leads to one loca-
tion and only to one location.
These two examples, of local navigation around town and of global navigation
on the Earth-sphere, may seem like scaled-up or scaled-down views of each
other. In some ways they are. There is, however, a critical difference. In the
local view, it was the concept of “orientation” that needed to be removed from
the picture to have accurate directions. In the global view, it was the concept
of “partition” that needed clarification. In both cases, moving from relative
to absolute strategy solved the problem. Satisfactory solution was obtained
in both cases when one description led to one location, as a “one-to-one”
transformation.
2.2 Location and measurement: From antiquity to today
One well-known classical approach to mathematical geography appears in
the work of Eratosthenes of Alexandria (c. 276-194 bc ), a Greek mathemati-
cian, geographer, and astronomer. Eratosthenes made critical discoveries
in both mathematics and geography that might seem, superficially at least,
unrelated. His prime number sieve, or algorithm, permitted complete char-
acterization of all whole numbers and offered, therefore, an understanding
of the whole number system. He worked, as well, with spherical geometry,
latitude and longitude, Earth-Sun relations, and a variety of geometric and