Don't you always feel like you miss more than you see when you visit New York City? No matter how many blocks you pound, cab miles you log or subway stations you roll through, you never cover more than a fraction of a city that sprawls densely over 308.9 square miles.

Jack Eichenbaum can take care of that little problem for you. A city assessor with a PhD in urban geography, Eichenbaum devotes his spare time to giving walking tours of New York -- not the famous sites in Manhattan but the overlooked precincts of the Bronx, Brooklyn, Staten Island and Queens. Normally, he breaks the city into small bites -- a borough or a neighborhood at a time. But when I asked him if he could give me a tour of the whole city in just one day, Eichenbaum not only agreed, he insisted we use only our feet and public transit. "There's good stuff in this city that you just can't see from a car," he said. Here's how we did it.

7:50 a.m. Fortified by an early morning bagel, we meet in Manhattan at Third Avenue and 96th Street -- the boundary, reinforced by generations of real estate agents, between the Upper East Side and Harlem. The contrast remains stark. Looking south, one sees apartment buildings and brownstones for the middle class and the affluent; to the north, one sees public-housing projects. Moving two blocks west, Eichenbaum points out the spot where trains bound for Grand Central Terminal slip below Park Avenue. The tunnel begins, naturally, at 96th Street.

8:15 a.m. We board the Bronx-bound 6 subway and head north. Along the way, Eichenbaum explains that two geological features define New York City. In Manhattan and the Bronx, a spine of ridges parallels the Hudson; it's a big reason why most Manhattan subways run north-south. In Brooklyn and Queens, by contrast, the defining feature is a hilly moraine where the last glaciers stopped moving south. It runs east to west, forming a barrier of parkland in both boroughs.

8:28 a.m. We emerge at 138th Street and Third Avenue in the Bronx -- Mott Haven, a lower-middle-class area with a view of the Harlem River and the Bruckner Expressway. The strains of the 1960s -- rapid immigration, rising crime, misguided city planning and massive flight to the suburbs -- devastated the South Bronx. "Wherever you have rapid demographic turnover, you will have problems, because who's going to run the PTA and other elements of the social fabric?" Eichenbaum says.

8:38 a.m. From the elevated perch of the 4/5 subway train, we look out on rowhouses and apartments -- the fruit of the first private housing boom in the South Bronx in roughly eight decades. Twenty years ago these streets were home to burned-out hulks; a decade ago, the shells were torn down. Now "it's filling up with new homes," he says. "They're not fancy, but they're a lot better than what they replaced." We cross over the Bronx River and exit the subway near a particularly bucolic corner of Bronx Park. "The Bronx River is physically, geologically and historically significant," Eichenbaum says. "On the west side is the poorer Latino part. On the other side are white ethnics. There's even beach housing on a little peninsula called City Island -- a little New England-looking place."

9:14 a.m. We board the Q44 bus to Queens, passing big condo developments such as Parkchester on our way to the Bronx-Whitestone Bridge. After we cross the East River into Queens, we traverse affluent and middle class areas built after World War II with cars, not subways, in mind. Suddenly, Asian signs begin to dominate the commercial strips. The Korean developments tend to mirror suburban strip malls, while the ethnic Chinese and Taiwanese developments look more like traditional downtown business districts.

9:50 a.m. We step off the bus in Flushing -- once a Jewish enclave, now a place where, during the last decade, Asian business leaders have undertaken the most frenetic construction anywhere in New York City. Unlike the restaurants and stores in Manhattan's Chinatown, which are a magnet for Anglo tourists, the storefronts of Flushing cater primarily to Asians and Asian Americans, Eichenbaum says -- except in prominent locations near the subway, where rents are higher and merchants need to court a broader customer base.

10:17 a.m. We board the 7 train -- the sometimes elevated line that passes Shea Stadium, and the one that Atlanta Braves pitcher John Rocker refused to ride. "For three miles underneath us are Latin American businesses, especially Dominican and Mexican," Eichenbaum says. We get off and marvel at the entrepreneurial bustle of small stores and vendor stands -- all occurring under the hulking elevated rail lines, along a shadowy corridor never intended as a shopping district. "In the 1960s, this street was all rug stores and gas stations," he says. It took immigrant gusto to bring it back.

10:49 a.m. We walk through the Jackson Heights neighborhood, with many Indian and South Asian stores, and end up in Woodside, an old Irish enclave. We stop for a hearty lunch of beef and potato stew, served by a waitress with an authentic Irish accent.

12:12 p.m. At Woodside, we board the Long Island Rail Road. Eichenbaum notes that the LIRR was built in the 1830s, long before the postwar flowering of suburbia. The LIRR had been designed as the fastest route from New York to Boston, with an assist from a ferry link across Long Island Sound. It worked -- but only until 1850, when a Connecticut rail link was completed.

12:27 p.m. We get off the train in Jamaica, a commercial area in Queens that serves some of New York City's largest middle class African American neighborhoods. The emergence of shopping centers nearby has drawn away shoppers, but Jamaica, Eichenbaum says, "is clawing its way back."

12:36 p.m. We board the J train, which follows the edge of the glacial moraine. "The expensive areas are higher up," he says. "The flats are home to Indian and Latin American immigrants." As we cross from Queens into Brooklyn, Eichenbaum notes that Brooklyn was developed 20 years before Queens; the added age shows in the facades we can see from the elevated rails.

1:19 p.m. After transferring to the famed A train, which runs all the way to Harlem, we exit into Bedford-Stuyvesant, the heart of African American culture in Brooklyn. The neighborhood has a troubled reputation, but Eichenbaum calls it a success story. Unlike other at-risk parts of the city, Bed-Stuy has a handsome housing stock that is being restored and refurbished. Great homes here can be had for smaller sums than elsewhere in the city.

1:44 p.m. We board Brooklyn Bus 49, which takes us through Crown Heights -- a decade ago, the site of clashes between Orthodox Jews and African Americans, and now recovering, somewhat uneasily. We also pass through Flatbush, where the Brooklyn Dodgers' Ebbets Field once stood; it's now the site of an apartment building.

2:09 p.m. We exit to Bedford and Church avenues, in the middle of a vibrant Caribbean commercial strip. As we move west across 16th Street, the din immediately fades away: We're in Prospect Park South, a quiet residential neighborhood with expensive homes. These are the places that Eichenbaum loves: places where one can step across an invisible line into a seemingly different universe.

2:31 p.m. We enter the station for the Q line. It takes nine minutes for the train to come -- our longest wait of the day. Except for this, our transportation planning has gone well.

3:16 p.m. Emerging from the subway in Lower Manhattan, we head to the Staten Island ferry terminal. As pigeons fly above our heads, Eichenbaum explains that Staten Island is the borough most unlike the others. Despite a healthy rate of suburban development, its population density remains one-tenth of Manhattan's, a quarter of Brooklyn's and the Bronx's, and a third of Queens.

3:57 p.m. After sailing past Ellis Island and the Statue of Liberty, we arrive at Staten Island. By this time we have only strength enough to walk around the St. George neighborhood near the terminal.

On Staten Island's shorelines are rows of handsome old Victorian homes, but I content myself with Eichenbaum's description from memory.

After all, how much can you see in one day?