About 300 BC, Aristotle was the first, as far as we know, to conclude that the earth
was round. He arrived at this conclusion having observed the shadow the earth cast on the
moon during an eclipse. We know this as we wrote it down in the third of his books. A
contemporary, Erathanese, was the first to record a measurement of the radius of the
earth. Thus, that the earth is round has been known for more than two millennia. Through
the dark and middle ages, this information was of interest, perhaps, only to
intellectuals; of which there were not many during those centuries. Thus, for many
centuries, the earth remained a sphere, whose size was known only to a small degree of
precision.
As the middle ages gave way to the modern era, the need for accurate maps grew, largely
to facilitate navigation. Those involved in these efforts came to realize that the earth
was not a true sphere, but a spheroid. A spheroid? My dictionary defines a spheroid as a
"sphere like" object, implying that anything close to, but not exactly, a sphere
is a spheroid. Spheroid, therefore, is a rather general term which leaves a lot of room
for subjective interpretation. Scientists at the time, desired a more specific knowledge
of the shape of the Earth. As aptly described by John Snyder (Map Projections - A Working
Manual):
| "In the early 18th century, Isaac
Newton and others concluded that the Earth should be slightly flattened at the poles, but
the French believed the Earth to be egged-shaped as the result of measurements within
France. To settle the matter, the French Academy of Sciences, beginning in 1735, sent
expeditions to Peru and Lapland to measure meridians at widely separated latitudes. This
established the validity of Newton’s conclusions and led to numerous meridian
measurements in various locations, especially during the 19th and 20th
centuries; between 1799 and 1951 there were 26 determinations of dimensions of the
Earth." |
Of course, none of these determinations produced the same results.
However, they did all have one thing in common. They all assumed that the basic figure of
the earth was that of an ellipsoid; an oblate ellipsoid to be more specific.
An ellipsoid is defined as the solid (i.e. a three dimensional object) produced by
rotating an ellipse (a two dimensional object) about one of its axes. In the case of the
earth, the ellipse is rotated about the minor axes. An oblate ellipsoid is defined as an
ellipsoid which is flattened at the poles. Thus, the term oblate applies only to an object
which is rotating, as the Earth does, since the term pole would have no meaning otherwise.
(The term prolate ellipsoid would apply to an ellipsoid which is expanded at the poles.)
Thus, all of these measurements pretty much confirmed that the shape of the earth is that
of an oblate ellipsoid. That the Earth is an oblate, rather than prolate, ellipsoid is
assumed knowledge and the qualifying term oblate is often omitted for brevity.
Since the Earth (for the next few paragraphs anyway) is an ellipsoid of revolution, the
shape of the Earth is determined by shape the ellipse which is rotated. Ellipsoidal shape
can be defined in a variety of ways and, of course, you can expect to encounter any one of
the three most popular of these variations in the literature; or more appropriately, in
your GIS system.
Perhaps the easiest to visualize is the specification of the length of the major and
minor axes of the ellipse. However, these are terms used in mathematics text books.
Geodesists and cartographers prefer to use values which represent the major and minor
radii. (These are also referred to as the semi-major and semi-minor radii of the ellipse,
implying that both values are required to specify the shape.) In the case of the ellipsoid
we are concerned with here, the Earth, the major, or semi-major, radius is defined by the
radius of the equator; the equatorial radius. The minor axis, or semi-minor axis, is
defined bythe distance from the center of the earth to either pole, and is referred to as
the polar radius. Being an oblate ellipsoid, the polar radius is required to be somewhat
less than the equatorial radius.
One can, and several geodesists prefer, to specify the shape of the ellipse (whose
rotation produces the ellipsoid) by giving the equatorial radius, but then specifying the
polar radius as a proportion of the equatorial radius. While saying that the polar radius
of the ellipse is 99.665% that of the equatorial radius might come to us quite naturally,
this is not commonly done. Instead, the concept of flattening is used to, essentially,
accomplish the same thing. Flattening is the degree by which the equatorial radius is
reduce in order to obtain the length of the polar radius. Flattening is usually expressed
in terms of a ratio, such as 1 part in 300. That is, the polar radius is smaller that the
equatorial radius by 1/300th of the equatorial radius. More precisely, recent
precise measurements put the flattening at one part in 298.257.
If you have been exposed to some calculus and analytical geometry, you might recall
that the eccentricity of an ellipse is a measure of how much an ellipse differs from a
circle. An ellipse which is very close to being a circle will have a very small
eccentricity. On the other hand, as one stretches an ellipse to the other extreme, i.e. to
where the ellipse starts to approximate a straight line, the eccentricity approaches 1.0.
When manipulating the coordinates of an ellipse mathematically, the square of the
eccentricity is the most useful number. Therefore, it is quite common to specify an
ellipse as the equatorial radius and the square of the eccentricity. The eccentricity of
the Earth is on the order of 0.08, the square of this eccentricity is on the order of
0.0067.
|
It should be noted that given the definition of the ellipsoid in any of the above
three commonly used forms, one can compute the other two; if one has the proper formulas.
The formulas, using notation which is quite common, follow:
Hopefully we now understand the rather long list of various ellipsoids we have all seen
in books, articles, and, perhaps, in your GIS software. All lists will give the equatorial
radius (in meters). Some will then use one of the variations described above. Some may
provide all four numeric values. Take care; many authors will presume that you know that
when flattening is given as a value in the order of 300, that the reader understands that
the value given is actually the reciprocal of the true mathematical value.
Notice also, that while within a couple hundredths of a percent from each other, all of
the listed values for the size and shape of the earth are different. Even the same
individual, Clarke for example, arrived at different determinations at different times.
Thus, we have the Clarke 1866, the Clarke 1880, as well as the Kravosky, Everest, Bessel,
Helmert, Hayford, and several others. (Note, that Everest is the name of an English
geodesist, whose name was given to mountain and ellipsoid alike.) The differences in these
measurements, most agree, are due to two reasons. First, the different techniques and
equipment used produced varying degrees of accuracy. However, it eventually became obvious
that while the Earth was closer to an ellipsoid than a sphere, it is not exactly an
ellipsoid either.
We’ll complete our examination of this issue in next month’s issue of the
Casual Cartographer.
