Chapter 9. Measurement
The word geometry comes from the Greek words ge, meaning earth, and metrein,
meaning measure. It seems that all human societies have had to measure fields
for agriculture or compute the amount of work involved in excavating a building
site. That there is a basic similarity in approaches to these problems is attested
by the presence of words for circles, rectangles, squares, and triangles in every
language. Geometric intuition seems to be innate to human beings. Many different
societies independently discovered the Pythagorean theorem, for example. These
external similarities conceal certain differences in outlook, however. For example,
we are taught to think of a line as having no thickness. But did the Hindus,
Egyptians, and others think of it that way? The word line comes from the Greek
linon (Latin linea) meaning string; the physical object (a stretched string) on which
the abstraction is based is clear. Early Hindu work on geometry also uses Sanskrit
words for ropes and cords. It is very likely that ancient engineers thought of a line
as a physical object: a rope, stretched taut. The quantity of rope was given as a
number (length). Geometry at this stage of development was a matter of relating
lengths to other quantities of geometric interest, such as areas, slopes, and volumes.
It was an application of arithmetic and useful in planning public works projects,
for example, since it provided an estimate of the size of a job and hence the number
of workers and the amount of materials and time required to excavate and build a
structure. It also proved useful in surveying, since geometric relationships could be
used to compute inacessible distances from accessible ones. The fact that similar
triangles are the basic tool in this science has caused it to be named trigonometry.
The elementary rules for measuring regular geometric figures persist in treatises
for many centuries. Nearly always the author begins by describing the standard
units of length, area, volume, and weight, then presents a variety of procedures
that have a great resemblance to the procedures described in all other treatises of
the same kind. This geometry, though elementary, should not be thought of as
primitive. Textbooks of "practical mathematics" containing exactly this material
are still being written and published today.
Although geometry looks very much the same across cultures, there is one
place where we must exercise a little care in order to understand it from the point
of view of the people we are studying. Textbooks often give approximate values of
the number ð allegedly used in different cultures without being clear about which
constant they mean. When calculating the circumference C of a circle in terms of
its diameter, we use the formula C = ðÜ. When calculating the area of a circle
(disk) in terms of its radius, we use the familiar formula A — ðí^2. When calculating
the area of a sphere in terms of its radius, we use A = 4nr^2 , and for the volume, the
formula V — f ðô·^3. There are really four different values of ð here, depending on
the dimension and flatness or curvature. The first formula reflects the fact that the
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