When you go somewhere famous for its beauty, there’s always a bit of calculation going on: Is the place going to be all it’s cracked up to be? What if it’s just routinely wonderful?

Sometimes the unexpected, serendipitous moments are like seeing a scarlet tanager out in the open or a pretty girl on a city street.

It’s a little gift you probably didn’t deserve. But you take it and are thankful you were so lucky.

This happened to me. On July 5, I was swimming at West Hill Pond in New Hartford.

I swim slowly and inexpertly. Tugs and scows sneer at me for my gracelessness. But I do get to watch the world as I plow through the water.

Which is why I saw, first, an immature bald eagle in the sky overhead, which I thought was wonderful enough.

A few minutes later, though, a mature eagle — white head, white tail, dark brown body and wings — flew into view, circled over the pond, and circled again.

Each swing past took it lower over the water, until the world was blue sky, white clouds, blue water, green trees and a handsome bird lacing it all together.

The people on the diving platform stopped to look. The people on the beach pointed. I stopped swimming and just paddled around to better look up.

Then — hurrah! — the eagle stopped flying and perched in a tree just off the side of the beach.

Beachgoers left their towels and blankets to become cellphone Audubons, hoping to get an image to remember this moment, this bald eagle visit on the day after Independence Day.

Then a hawk dived in, mobbed the eagle, and drove it away.

I have seen lots of bald eagles in my life. On some Alaskan beaches, they seem as common as herring gulls here. But I cannot remember ever seeing one in Connecticut so clearly, for so long, or so beautifully displayed.

Because I saw both a mature and immature bird, and because the lifeguards said that, yes, they’d seen eagles other times in this summer that’s just begun, I immediately jumped to a conclusion that, until proven right, is almost certainly wrong — that there are now eagles nesting in the woods that make up much of the border of West Hill Pond.

After all, there are now eagles nesting on Candlewood Lake, which has lots more beachfront properties and noisy motorboats than quiet, untrafficked West Hill Pond.

My friends who live in Norwalk and Westport tell me there are bald eagles nesting there. Why not in my hometown?

Brian Hess, a wildlife biologist with the state Department of Energy and Environmental Protection, explained.

“With all the visibility there is on West Hill Pond, I’d assume it would have been discovered and reported,’ ” Hess said of an eagle nest.

What’s more likely is this: There are known eagle nests fairly near to West Hill Pond, Hess said. The eagles that I saw might be coming by every few days to fish, to find a meal.

“They’re nesting within 10 to 15 miles away,” Hess said. “That’s not so far if you can fly.”

There are 43 active eagle nesting sites in the state today. That may not seem overwhelming, but it’s a huge success story, a return of the national symbol to our state.

“Twenty years ago, we had one nest,” Hess said. “Now, we have between 40 and 45.”

Ornithologists had thought bald eagles could nest only in deep woods, away from human clatter. But the birds are proving more adaptable, more street-savvy than once thought.

In Windsor Locks this year, Hess said, a section of a hiking-biking trail had to be rerouted to protect a pair of eagles who built in a tree near the path.

“Eagles begin building their nests in February,” Hess said. “What seems deserted then can fill up with people and dogs by April.”

The West Hill Pond bald eagles and I may not meet up again, although I will watch for them every time I go swimming this year, and for years to come.

Lightning may not strike twice in the same place. But beauty can flare up all the time, all around.

Lucky us.

Contact Robert Miller at earthmattersrgm@gmail.com.