The four bridges that cross the East River and connect Manhattan Island with Brooklyn and Queens are the very symbol of New York: thrilling, intimidating structures that challenge the mind to wonder -- how did they build that?

A closer look shows that, like London Bridge, they are falling down.

Steel is rotting. A cable snaps on the 99-year-old Brooklyn Bridge, and a man is killed. A beam cracks on the Manhattan Bridge, and subway trains stop using it for two days because nobody wants to risk spilling 1,000 people into the river.

Chunks of concrete fall so regularly from the deck of the Queensboro Bridge that the Little League baseball field is closed beneath it on Roosevelt Island. Game postponed; watch for falling bridge.

John A. Marino, assistant commissioner in the New York State Department of Transportation whose job includes worrying about those bridges, wakes up at night sometimes, contemplating the monstrous What If?

"I believe there is the possibility of a catastrophe," he said. He ponders that, then amends it. "For me," he said, "a catastrophe is not a collapse, but to close . . . . There is no way we can operate this city without one of those bridges."

That problem is heard throughout the nation but especially in the Northeast, where cities are built along rivers, and transportation patterns established over decades make bridges as vital as gas and electricity.

In Cleveland, downtown traffic is snarled as motorists attempt to cope with the closing of one major bridge across the Cuyahoga River and find a parallel structure in dramatic disrepair.

In the Pittsburgh suburb of Duquesne, U.S. Steel trucks are taking an 18-mile, one-way detour at a cost of $12 million a year because a bridge is closed.

Across the Mississippi River between Iowa and Illinois, load-limited bridges in Keokuk, Fort Madison and Burlington are causing major delays for farmers seeking to move crops to the grain elevators on the other side. More than 1,100 of Iowa's 3,912 bridges are more than 40 years old.

Those three examples offer only a hint of the nation's bridge problems, urban and rural. The Federal Highway Administration (FHWA) says there are 557,616 bridges in the United States.

Of that number, 126,655 are listed as "structurally deficient" and 121,872 as "functionally obsolete," meaning they are too narrow or do not have enough lanes for traffic they are to carry.

Nobody really knows what it will cost to fix the bridges; the official federal guess is $47.6 billion.

The annual rate of expenditure from all sources for bridges in the current fiscal year, according to the FHWA, is about $2 billion per year.

At that rate, the problem will not be cured for 24 years.

With the advent of heavy trucks and the many highway departments' practice of pouring corrosive road salt on bridges when snow and ice begin to accumulate, the life span of the average interstate highway bridge deck is about 20 years.

The Woodrow Wilson Bridge across the Potomac in Washington, for example, was opened in December, 1961, and work has begun on a new deck because the old one has become a continuous span of temporary patches, some covering holes through which the water is visible.

"The bridge problem has reached the point that we've developed a new paving material to deal with it," joked Thomas Downs, director of the District of Columbia Department of Transportation. "We call it steel plates."

Such a plate was blamed by the National Transportation Safety Board for a spectacular accident Aug. 27, 1981, on the Calcasieu River Bridge on Interstate 10 near Lake Charles, La.

A plate in place for four months with no permanent repair having been made somehow dislodged and penetrated a truck's fuel tank. About 75 gallons of diesel fuel was spilled along the bridge, and cars began slipping and sliding. Three persons were killed and 18 were injured in a 26-vehicle collision.

New York's East River bridges are a spectacular example of what can happen when things get old. The Brooklyn, Manhattan, Williamsburg and Queensboro bridges march in stately procession north from the tip of Manhattan Island. Each day, they carry more than 395,000 cars, trucks and buses in and out of the most densely populated real estate in the country. The Queensboro and Manhattan bridges also bear six tracks that support hundreds of trains hauling thousands of commuters, lovers, school children and New York Yankees fans.

The Brooklyn Bridge will be 100 years old next March; the Williamsburg was opened in 1903 and the other two in 1909. As New York moved closer to its financial crisis of 1975, maintenance on bridges and roads was regularly cut from its budget.

So a combination of age, abuse and even an absence of paint when it would have helped have left the four bridges shabby, with chipping concrete and rusting steel. The state has assumed a major role in their rehabilitation, and a 15-year, $500 million program has been drawn to get the job done.

But in this second year of that program, federal dollars have been cut and the total federal, state and city highway and bridge effort for the New York City area has dropped from $600 million to $400 million.

"I believe this is the most critical period," Marino said. "We have a crisis out there, and the Reagan administration is saying, 'Cut, cut, cut.' "

In the meantime, Marino said, the bridges are inspected regularly for structural integrity. "We will never allow a bridge or highway to become hazardous to vehicular traffic," he said.

Bridges are expensive to build and maintain. A new deck on a major structure can cost about $30 million. The Woodrow Wilson Bridge deck will cost $27.3 million, for which 283 miles of a major rural highway could be resurfaced. In times of tight funds, state highway departments choose the 283 miles rather than the bridge deck.

Eventually, the waiting game can be played no longer and, when that happens, disruption is the rule. In Cleveland, the 7,000-foot-long Lorain-Carnegie bridge, which once carried 30,000 vehicles across the Cuyahoga daily, is closed for reconstruction, and motorists scurry to find new routes to work.

Cuyahoga County Engineer Thomas J. Neff hoped the bridge would be rebuilt by now but, after the deck was removed, he discovered unsuspected severe structural damage. The bridge will not reopen until late in the summer of 1983, and the cost has jumped from $14 million to $20 million.

Meanwhile, he is "using Band-Aids" to hold together the nearby, parallel Main Avenue bridge at cost of $1 million a year. That money, he said, "we're just throwing away. But we can't have both closed at the same time."

Does the condition scare him?

"Yes . . . it really does," he said. "We've been monitoring it weekly; it hasn't reached the hazard level yet. If it does, we'll have to start limiting traffic."

Pittsburgh and surrounding Allegheny County, Pa., have 1,765 bridges, more than half of which were built before 1930 and many of which cross the famous Allegheny, Monongahela and Ohio rivers.

Fifty-six of the bridges are posted, meaning heavy trucks cannot use them, and six are closed. "We have almost one bridge per mile of state-maintained road in Allegheny County," said Bettina Ferraro of the Pennsylvania Department of Transportation. "To bring them up to snuff, we would need about $1.8 billion right now." Total expenditures on bridges this year will be $72 million.

Many of America's bridges are getting old. Many others, particularly on the interstate system and those crossing large rivers in major northeastern cities, have worn out years before their time because of a problem little understood when their decks were poured.

That is the immense damage done by road salt to reinforcing steel in concrete bridge decks. It was known that salt contributes to corrosion but not as well understood that the type of concrete widely used in the 1950s would not protect the steel from brine formed as the salt melted ice and snow.

Federal and state highway departments have been trying new kinds of concrete and other materials to seal the steel, and drainage systems have been improved so the brine will quickly run off. But the only true test of whether engineers have solved the problem is time, and the answer is not yet available.

Loss of a bridge is more than just an inconvenience; it carries an economic penalty as well. The Federal Highway Administration, in its annual report on the status of the nation's bridges, draws this example:

"A load posted bridge carrying 2,000 autos and 200 trucks per day a relatively light traffic load must be detoured by trucks. The detour length is five miles. If the truck's running costs are 50 cents per mile, the 200 trucks per day incur additional total operating costs of $182,500 per year.

"If the bridge is closed and the 2,000 autos per day must also detour the five miles, the total additional operating costs at 20 cents per mile are $730,000 per year for the autos.

"The combined costs for autos and trucks would be $912,500 per year."