What is small, round, flat, and moves 75 feet a day?

Don't look now, but the 1980 center of US population is in a pile of leaves in a Missouri wood, 40 miles southwest of the Gateway Arch in St. Louis. That's according to the delayed results of the 1980 census. Trouble is, by the time you find it, the center will have shifted farther south and west.

In 1790, the center of the country's populace was near Baltimore. Its present rate of movement is about 751/2 feet a day. Its latest known address is latitude 38 degrees and 8 minutes and 13 seconds north; and longitude 90 degrees 34 minutes and 26 seconds west.

It's headed for the sunset.

The center of population is defined by the Census Bureau as the point at which ''an imaginary flat, weightless, and rigid map of the United States would balance if weights of identical size were placed on it so that each represented the location of one person.'' That spot has more than symbolic meaning. In keeping with time-honored tradition, a permanent marker is set on each spot - the latest by an official representing Commerce Secretary Malcolm Baldrige for the Census Bureau.

(As a matter of note, the woods where the marker will go are the property of Henry Koch, an assistant to Missouri's secretary of state. He is proposing to develop it as the ''Center of Population Park.'' It gets a copper disk, commemorative plaque, and flagpole).

The US center of population started out, calmly enough, on Maryland's Eastern Shore, with enough homespun Americans living north, south, east, and west around it to balance the mythical ''flat, weightless, and rigid map'' of the new nation. The center moved out from there and, in a series of decennial hops, left a trail of markers heading for the present location in Missouri.

The center took about 50 years crossing West Virginia and reached a spot near Waverly, Ohio, in 1860. During the 1970s the center of population crossed the Mississippi River. A millenium hence, if prospectors dig up a four-inch copper disk set in concrete, they will have uncovered the current abiding place of the population center.

Since it started out in Maryland 190 years ago, the center has moved about 750 miles west and 80 miles south, reflecting the enlargement of the country (including Alaska and Hawaii) and the internal migration toward the West and South.

By one calculation, the center in the 1970s moved about 751/2 feet a day in its inexorable course toward the Sunbelt. Instead of being a fixed point, the center is a transitory concept, of deep interest to historians, demographers, and geographers, but with no physical reality save a copper disk.

By jumping the Mississippi, demographers say, the average American is no longer a Northerner. More Americans now live in the West and South than in the East and North.

Being at the center of population has some commercial advantage for local inhabitants. The last stop of the suppositious point was at Mascoutah, Ill., in 1970 - a town now dethroned by the new center. It became a minor tourist attraction after its designation.

The new center is near the small town of De Soto, named after a Spanish explorer. But the adventurer De Soto apparently never got here and his supposed visit, according to historians, is as insubstantial as the concept of the population center itself.

Census Bureau analysts, in a study in the American Demographic Magazine, say that the center, now that it has arrived in Missouri, will stay there for a long time.

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