The Casbah. Most Americans know about it from the movies. ''Come with me to the Casbah'' is an enticement Charles Boyer is widely thought to have offered to Hedy Lamarr in the 1938 motion picture ''Algiers.'' In fact, Boyer never spoke those words in the film, but the phrase has, nonetheless, become a symbol of romance and intrigue in the French colonial empire.
Director Gillo Pontecorvo set his classic 1960s ''Battle of Algiers'' in the Casbah's maze of narrow staircase alleys that substitute for streets in the old Arab quarter of the North African capital. It represented the heart of the Algerian revolution.
The original ''Casbah'' is once again a symbol, this time for one of the worst population problems in the world.
This medieval quarter, an architectural labyrinth spilling down a steep hill overlooking the Mediterranean, is also now a labyrinth of humanity. European diplomats claim that the Casbah population is ''incalculable.'' They say it is many times worse than the average Algerian home, which has nine children living in three rooms or less.
The situation is the result of political, religious, and geographic ''encouragement.'' In the mid-1960s, former President Houari Boumedienne encouraged large families to make up for the massive war losses and because of a demographic boom in Morocco, Algeria's neighboring rival.
Islam, which is practiced by 99 percent of Algerians, does not encourage the use of birth control or family planning.
And geographically, the 10th largest nation in the world is almost 80 percent desert. The vast majority of the 20 million Algerians live along the fertile northern coastal strip.
A severe drought in the Sahara Desert in the 1970s led thousands to flee to the cities, where many have stayed despite official efforts to repopulate the rural areas and oases.
The Casbah is the ideal hiding place for the migrants, who have set up ''temporary'' quarters on rooftops of the shabby old stone buildings, or tents in corners of the alleys. As proven by the resistance fighters during the war of independence against France, one can ''disappear'' into the crevices of the Casbah.
Algeria has one of the world's highest birth rates - 3.2 percent - and officials admit the situation will probably get worse. A US economic brief predicts another 7 million people over the next seven years.
The government has taken two steps to alleviate the problem. Last month banner headlines in Algiers papers announced a ''spacing of birth program,'' which encourages a greater span between births.
The government's five-year development plan also calls for construction of 450,000 housing units to cope with expanding numbers. Much of the new housing is to be built far from the major cities. This includes what a US economic report descibes as ''probably the world's largest program of prefabricated construction.''
Meanwhile a government agency, Comeder, has laid out plans to beautify and revitalize the Casbah. But few residents feel the program can be successful until the number of residents is at least halved.
Space is so cramped that a single room in an old Moorish building often serves as both home and shop for two generations - or more. Many merchants peddle their wares - from homemade mattresses to vegetables - on the streets.
Gone forever is the glorious period when the Turkish Deys ruled from palaces perched at the summit of the Casbah. It was from here that the ruler, Husayn, slapped the French consul in the face with a flywhisk - and triggered French retaliation that led to 132 years of domination, and development of the area that now surrounds and overwhelms the Casbah.
The Casbah's character is as rundown and chipped as the famous Delft ceramic tiles that decorate the walls and floors of homes and inner courtyards and were once used to pay as ransom to Barbary pirates.
Sadly, it is no longer a place where an invitation to ''come with me to the Casbah'' would be accepted.