U.S. DEPARTMENT OF COMMERCE
National Oceanic and Atmospheric Administration
National Ocean Survey
Rarckyilie Md, 2Kss

Geographic Center

of the
United States

The admission of Aleske and Hawaii to the Union
es Stotes during 1959 has aroused public curiosity
as to where the geographic center of the United
States lies.

Determining the geographic center of an irregular area on
the earth's surface is < precarious business at best, There is
no unique solution and none of any seie significance.
Several methods of approach are available. The ane
employed by the National Geean Survey Is probably best
termed as the center-of-gravity method. Imagine a map of
the ares for which the geographic center is tu be determined,
has been placed on a piece of cardboard of uniform thickness
and that this is cut carefully along the outline of the map,
The center of gravity of this map outline or what might be
called the geographic center, is that point at which the map
will balance

‘The geographic center of the forty-eight conterminous states
was determined by the National Ocean Survey, formerly the
Coast and Geodetic Survey in 1918 by the method described.
This geographic center is approximately at latitude 39°50'N,
longitude S8°35°W, near Lebanon, Kansas, When Alaska
was admitted to the Union, the geographic center of the
forty-nine states shifted about 439 miles northwest to a
point at approximately latitude 44°59'N, longitude
103°38'W, about eleven miles west of Castle Rock, Butte
County, South Dakota. In arriving at this determination the
geographic center of Alaska was determined by the method

deser

bed and was found to be at latitude 62°S0'N, longitude
O0'W, with an uncertainty of about 15 or 20 miles in sny
rection. The geographic center of the combination of
Alaska and the forty-eight conterminous states |s cor
sidered to be on the great circle connecting their geograp!
centers at a point where the two areas would “balance”; j.e.,
considering each having a weight, proportional to its area,
coneentrated at the corresponding geographic center.
Later when Hawaii was admitted to the Union
geographic center was determined to be at latitude 20°13,
longitude 156°20 W, with an uncertainty set at about 3 or +
miles in any direction, The geographic center of the fifty
states was then determined as being on the great cirele
connecting the geographic centers of the forty-nine states
and Hawaii at a point where these areas would “balance.”
‘The effect of including Hawaii was rather small since its
area is unly about 1/560 of that of the forty-nine states, The
geegraphie center of the fifty states, thus determined, then
shifted about six miles west-southwest to latitude 44°58'N,
longitude 103°46'W, which is approximately 17 miles west of
Castle Rock, Butte County, South Dakota. The uncerainty
determination is set at about 101 miles in any direction.

GEOGRAPHIC
CENTER

Mit



Geographical Centers

Oscar S. ADAMS

‘Senlor Mathematician, United States Coast and Geodetic Survey

T is a characteristic of human beings that they

are curious about things of which they have no

definite knowledge. On the whole, this is a laud-
able trait but often curiosity outruns intelligence. It
is true that there are more things in heaven and
earth than are dreamed of in our philosophy, but we
should allow our intelligence to curb our curiosity in
eases where it is needed and should not insist upon
a definite statement where no exactitude exists.

A great many people are curious about what they
call the geographical center of a state or a country or
even of a whole continent. Now, let us examine into
this matter and find out, if possible, what may be the
meaning attached to this idea of geographical eenter
of any given area.

If we take a bounded plane surface we may, under
eertain cireumstances, get some reasonable meaning
into the idea of geographical center or, as it would be
in this case, the center of area, If the region under
consideration is a connected surface bounded by 3
regular system of lines such as a regular polygon or
a circle, then, what is called the ‘‘center’’ will be the
eenter of area. It will also be the center of gravity
when the surface is considered asa thin plate. The es-
sential requirement is that the area must be exactly
symmetrical with respect to this so-called ‘center of
area."’ If the boundary is irregular so that no sym-
metry is present, then, any attempt to find a center
of area will meet with difficulty. There will still be
a definite center of gravity but this point will hardly
be a satisfactory eenter of area: The suggestion has
been made to draw a straight line across the ares.
such that half of the area lies on one side of it and
the other half on the other side. Then draw a second
straight line perpendicular to this first line, also di-
viding the area into two equal parts. Then the inter-
section of these lines would be the center of area.
This is very good as far as it goes, but in what di-
rection should these lines be drawn! If the figure, as
stated before, is a symmetrical one, it would not make
any difference in what direction the lines were
drawn, for any line drawn through the center would
divide the area into two equal parts. But if sym-
metry is not present, the position of the intersection
of the two lines would depend upon their direction.
That is, the intersection of the north-south, east-west
lines would not be the same as that of the nertheast-
Southwest line with the northwest-southesst line,
Also, the eenter of gravity of the area would not be
Satisfactory, for any protuberance in any direction
would tend to pull the point in that direction.

Now, we have been diseussing merely a plane area;
but any region on the earth is a portion of a spheri-
cal surface or, if we wish to be very exact, it is a por-
tion of the surface of an ellipsoid of revolution. But
it may be objected that a plane map may be made of
the country or state and then the center of area of
this plane surface ean be determined. This is all
well and good but no map of a spherical surfaee ean
be made that is not distorted in some way. We ean
make a map that has constant area ratio, but, if so,

distortions in shape must take place, so the represen-
tation is not a true representation of the actual sur-
face. We haye already seen that, even if it were a
true representation, the center of area would have
no meaning unless the outline were symmetrical such
az a circle, an ellipse, a square, or other figure of
similar class. Unfortunately, countries and states
as political units or as physical features of the earth
are not so constituted.

It may be suggested that we could make such 2 de-
termination on the surface of a globe and so attain
some definiteness. But here we run into new diffi-
culties. If we wish to define the center of area as the
eenter of gravity of the surface, such center would
not even lie in the surface but would be some dis-
tance beneath the surface of the globe. On the other
hand, if we wish to make the determination by the
intersection of two lines mutually perpendicular,
such that each of them divides the area into two
equal parts, what lines should be adopted for the
purpose? If we take one of them to be the meridian,
should the other be an east and west great cirele or
should it be a parallel? Two mutually perpendicular
great circles in any other directions would necessarily
give a different point due to the lack of symmetry
in the region considered. When the matter is looked
at in any way, elements ¢f uncertainty arise and we
must conclude that there is no definite center of area
or geographical center of any political or physieal di-
visions of the earth’s surface.

In the United States Coast and Geodetic Survey
we reeeive many requests for information regarding
the geographical center of the United States or even
sometimes regarding that of the whole of North
America. Different results have been given by differ-
ent investigators and at times we are requested to ar-
bitrate the rival claims. From the preceding diseus-
sion it is seen that this discrepancy of results is to be
expected. As a matter of fact, the conclusion is
forced upon us that there is no such thing as the
geographical center of any state, country or conti-
nent. The point determined will depend entirely
upon the definition given by the one making the com-
putation. No aceount should be taken of a character-
istic of an area that will depend entirely upon the
supposition made in its determination. If there were
some rational definition so that all computations
would necessarily lead to the same result, then some
meaning might be attached to sach a result. How-
ever, in most cases it is nothing but idle curiosity that
leads to an interest in such a point, or some man may
want it definitely located in his front yard. To all
such the warning should be given that the point is
much like the proverbial flea; when you think you
have it you have it not.

A method that was used in the Coast and Geo-
detic Survey a number of years ago was the follow-
ing: An equal-area map of the United States was
constructed on thin cardboard and then the outline
map was cut out along the various boundaries. The
center of gravity of this outline map was then deter-



mined. This could easily be done if it is assumed
that the cardboard is perfectly homogeneons and
everywhere of the same thickness. These are ideal
conditions that are probably never found actually to
exist. However, assuming this to be true, we need
only to suspend the Sgure t from some point and draw
@ vertical line on the map from the point of sus-
pension. Then the same is done from another point,
preferably so that the second vertical line is approxi-
mately at right angles to the first, The intersection
of these two lines will be the center of gravity of the
ares in question under the’assumption stated above.
This point ean then be considered the geographical
center if it is satisfactory so to define it.

Let us give a little further consideration to this
procedure. As already stated, an equal-area map has
the same ares ratio at all points but, in order that
this may be true, it is necessary to distort the shapes
and this distortion becomes greater as the area map-
ped imereases. This fact would have an influence
upon the shape of the outline of the country as it is
eut from the cardboard. Hqual-area maps of the
same country constructed on two different projections
would give different results for the center of gravity
of the outline. In each case a perfectly definite cen-
ter of gravity for each outline would be determined.
but when sealed in terms of latitude and longitude
from the different maps the results would not. agree.
If 4 similar attempt were made with the outline cut
from the spherical surface, a result not in agreement
with any of the plane maps would be found. Also
projecting regions such as Florida and Maine in the
United States would have an undue influence upon
the location of the point to be determined. In view
of ail these considerations it is clearly seen that the
definition of the geographical center as a center of
gravity is far from satisfactory.

We shall now examine the method of two perpen-
dicular lines, each of them dividing the area in
halves. We might attempt this on an equal-area map
with straight lines, one north-south and the other
east-west. In the first place the practical difficulty
in the way ef such a method is very great. We could
quite easily determine an approximate area for the
whole map with the use of a planimeter but no great
exactness could be obtained. However, the attempt
to determine a north-south line that accurately bi-
sected the area would meet with serious difficulties.
After a number of trials one might finally arrive at
an approximate result. Also by cutting the map in
two and weighing the parts against each other after

several trials an approximation might be found.
Let us suppose the two required lines have been
found. Then their intersection may be assumed as
the geographie center of the map. Here we meet
with the same objection as before. The location of
the point in terms of latitude and longitude would
depend upon the kind of projection used in construet-
ing the map; also any two lines similarly drawn
in different directions would give a yet different re-
sult. Again, if we resort to the globe the same ob-
jections apply with still greater practical difficulties
to contend with,

In regard to the United States, almost any of these
methods would probably locate the point somewhere
in the State of Kansas. but at just what point in
terms of latitude and longitude one would be rash to
state. It would seem then, in view of these uneer-
tainties, that it is not an advisable thing te do to
make any detinite statement regarding the matter.
Since there is no definite way to locate such a point,
it would be best to ignore it entirely since any state-
ment regarding the position of the point may be
called in question by one who considers the matter
from another point of view. If it were only possible
by any analytical process to determine a definite
point that could be reproduced by any other person
with the same assumption, it might be well to take
some stand in the matter, But as things are consti-
tuted, the resulting point would not only depend
upon the definition adopted but would slso depend
upon the investigator and upon the tools employed
in the investigation. The conelusion seems to be
that it is hazardous to make any definite statement
about a matter that can be viewed in so many differ-
ent ways that inevitably lead to discordant results.

Tf the fact is recognized that no method of deter-
mination of a geographical center of a country is en-
tirely satisfactory, and if it is admitted that such a
point can not be uniquely determined, then it is true
that almost any one of the methods already outlined
will give a point that is accurate enough for all prac-
tical purposes. As a matter of fact, it is hardly con-
ceivable that such a point should meet any ‘*practi-
cal purpose’ in any case. It is a coneeption that de-
pends almost entirely for its existence upon the curi-
osity of mankind. It is inevitable that there are as
many geographical centers of a state or country or
other area as there are persons determining them.
‘Any reasonable method employed will give a center
as satisfactory as any other one. This is a case in
which all may differ but all be right. Moser