Whatever else a summer day's trip to the Mall may include, for my kids it always means a ride on the carousal in front of the Smithsonian castle.

Heads thrown back, eyes closed to the wind in their faces, they move up and down astride their mounts, primally soothed by the rocking motion of the horses and the heavy beat of the music that bursts from the Wurlitzer 153-band organ nearby.

Juding from the crowds of riders who line up to share in this 40-cent experience, there are plenty of people besides us who use their summers to horse around. The carousel on the Mall is only one place to do it -- and a good place to begin.

Here are three optional warming-up exercises:

Read through Frederick Fried's book A Pictorial History of the Carousel , available at libraries and on sale at the Glen Echo gift shop.

View the Dentzel lion and giraffe, and the Stein and Goldstein horse on display at the Eleanor and Mabel Van Alstyne American Folk Art Collection in the Museum of History and Technology.

Touch your toes, do two deep knee bends, and breathe deeply to the count of eight.

For what to do next, here's a guide to carousel-hopping in the area. THE BALTIMORE ZOO CAROUSEL, Druid Hill Park, Balitmore. 301/396-7102. 10 to 4 daily; 35 cents. This thoroughly modern Merry was built with an intentional zoo motif in 1965 by a company in Kansas that manufactures carnival equipment. Its 36 creatures -- including an elephant -- are made out of aluminum and spin round to the sounds of an eight-track tape system.

It's a slick, contemporary participant in a tradition that, according to Frederick Fried, is at least 1,500 years old. There exists visual record of a carousel in a Byzantine bas-relief, and numerous pictures and written descriptions of carousels in Turkey and Bulgaria in the 1600s and in England, France and Germany by the 1700s.

The carousel in America was indtroduced and developed by European woodworkers and machinists who immigrated in the 19th century. Early carousels were made of light wood and had horses or seats that hung from overhead beams. The platform design we know appeared after 1876. With the advent of steam power, carousels were made more elaborately and could carry heavier weights. What may have been the largest carousel ever constructed was built in 1914 by the Philadelphia Toboggan Company for Olympic Park, New York. Its platform was 60 feet in diameter, there were 80 animals on it and four charioits -- and seats enough for 99 passengers. THE LAKESIDE HERSCHELLS: Lake Fairfax, Vienna. 703/471-5414. 10 'til dark daily. 40 cents. Burke Lake, Route 123, Virginia. 703/323-6600. 10 'til dark daily -- the only carousel open year-round. 40 cents (2 and under free). Lake Accotink Park Virginia. 703/569-3464. 10 'til dark "as long as the weather permits." 40 cents. These are Herschell machines built approximately 50 years ago in North Tonawanda, New York. None is original to its present site. Their builder was Allan Herschell, a Scotch immigrant who came to America in 1870 and was running a carousel business by 1890. He's remembered for his moderately sized, minimally ornate rides. A Herschell-Spillman catalogue of 1913 advertises a steam-powered carousel for less than $2,000.

Herschell carousels were -- and are -- numerous. Both the carousel on the Mall and the one in the mall at Bethesda Square Shopping Center are also of this make. The one on the Mall is the oldest and of most historic interest. It was built about 1910 and features among its animals a dog, a pig and two zebras; more recent models carry all horses. The figures are small, carved wooden ones, not at all threatening to young riders who sit them more comfortably than adults do.

North Tonawanda was also the birthpalce of the American band organ, the work of Eugene DeKleist in 1891. Wurlitizer bought him out in 1908 and successfully marketed band organs to provide carousel music until 1928, when the radio and the amplified phonograph proved too competive. The Wurlitizer band organ is now a vanishing instrument. GLENECHO, Glen Echo Park, Maryland. 301/492-6282. Weekends, noon to 6.25 cents. The most ride for the least money: For two bits, you get it all at Glen Echo -- the color, the music and the movement of the big carousel ear. Among the 50-odd beautifully carved and painted animals on the brightly lighted platform are an ostrich, a rabbit, a deer, a tiger, a lion and two giraffes. And there's a Wurlitizer 165-band organ with bass and snare drums, triangle, castanets, bells and pipes continuously playing good oldies like "Papa Won't You Dance With Me" and "Too Fat Polka."

The carousel is a Dentzel, made in Germantown, Pennsylvania, and shipped new to Glen Echo in 1922. It has remained at its original site thanks to efforts of Glen Echo residents who in 1970 raised $80,000 to buy the ride for the National Park Service.

The Dentzel Corporation was founded by Gustav A. Dentzel, a German immigrant, more than a hundred years ago. Gustav's father had been a wheelwright in Kreuznach, Germany, and the boy grew up watching his father experiment with carousel design. His own rides are grand, imaginative creations with huge, artfully detailed characters. In 1890, they were selling for as much as $20,000.

When Gustav died in 1909, the business passed to his son. After William's death in 1928, it was auctioned off.

The Wurlitizer that operates with the carousel was installed in the late 1920s and then cut down to take Caliola rolls some time in the 1950s. Durward Center of Baltimore restored it to is original sound about two years ago. CHESAPEAKE CAROUSEL, Watkins Regional Park, Maryland. 301/249-9220. 10 to 5 daily. 50 cents. You can smell the salt spray in the wood of this one -- it was bought in 1975 and moved to Watkins Regional Park from Chesapeake Beach, where it's believed to have operated in the amusement park near the steamboat landing since the turn of the century. Oldtimers in Calvert County can remember going by wagon to Chesapeake Beach and riding the carousel there. Now their grandchildren drive to the new site by car and make memories of their own.

The carousel has three bands of animals on it. Those on the outer ring are believed to be from Gustav Dentzelhs workshop, and some may date back to the last quarter of the 19th century.

Among the creatures on the inner rings are a seahorse, a goat, burros and buffalos -- and a one-of-a-kind kangaroo with hinged hind legs that produce a jumbing effect. Had the carousel not been purchased whole by Prince George's County, the pieces might well have been bought individually by collectors.

The carousel is fully functional and an interesting example of early-20th-century folk art. LEE DISTRICT PARK CAROUSEL, Telegraph Road and Roosevelt Drive, Virginia. 703/971-0294. Monday, Wednesday, and Friday, 4 to 8; Tuesday, Thursday and Saturday, 10 to 8; Sunday 1 to 8. 30 cents. This carousel was built early in this century by the Chance company of Wichita, Kansas, but you won't be taking a chance when you ride it -- because the park people redid its wiring after they bought it, before they hooked its lights up. That's when they were able to date its manufacture. Its 48 wooden horses are arranged in two rings.