There is a president named Bush with high public-approval ratings. There is an economy that is sputtering, even on the best of days. There is a military triumph in Iraq that suddenly doesn’t look all that triumphal. There is, according to a theory the Democrats have developed in the tiny spot where history and hope collide, reason to believe that all those factors will combine to invite them back into the White House.

It happened once, and in theory it can happen again. History doesn’t repeat itself, but it sometimes can seem repetitive. And history does present an intriguing juxtaposition of facts: Only one other presidential son has been elected to the White House. He did so without winning the popular vote. Four years later, John Quincy Adams was not re-elected.

None of that means that George W. Bush is predestined to be defeated next year, and the danger is that tantalizing historical circumstances can obscure rather than enhance our understanding of the president — and of our politics. The next election is being held in circumstances that could not be more different from those of 1828, when John Quincy Adams was defeated by Andrew Jackson. But more important, the George W. Bush presidency is not a sequel to his father’s, just as the 43rd president is not a modern version of the 41st.

Let us count the ways: The first President Bush was a moderate who became a late, reluctant and occasionally unconvincing convert to the new conservatism and the new conservative economics. The second President Bush is a genuine conservative with supply-side sentiments coursing through his blood. The first President Bush worried endlessly about the budget deficit and resorted to a tax hike to fight it. The second President Bush is sanguine about the growing deficit. Last month he signed a big tax cut.

The first President Bush believed deeply in international institutions such as the United Nations, where he once served as the chief American delegate. The second President Bush is impatient with all international institutions, even the G-8, whose meetings he endured dutifully but not enthusiastically this week.

The first President Bush was a diplomat by instinct, training and conviction. The second President Bush may be a conciliator by instinct, but he has no diplomatic training and no conviction that the traditional avenues of diplomacy have any relevance to the new challenges the United States faces around the world.

The father saw the United States as having strategic interests that changed with time. The son sees the United States as having permanent friends and permanent enemies — which is why Jacques Chirac shouldn’t be dreaming of any Lincoln Bedroom soirees anytime soon.

Though John Adams and his son were both aloof and difficult — the younger Adams called himself “a man of reserved, cold, austere and forbidding manners” — blood lines do not necessarily determine personality types, particularly in political families. John F. Kennedy was a cautious politician; Robert F. Kennedy, who was assassinated 35 years ago this week, was far more daring. Both had executive personalities. Their brother, Sen. Edward M. Kennedy, the Massachusetts Democrat, is the only member of his family to succeed in a legislative role.

So it should not be surprising, for example, that the younger Bush prides himself on his sharp edges while the father was proud of the fact that his edges were rounded off. The son sees the world in black and white — the terrorist attack of September 2001 affirmed his worldview — while the father saw a landscape with many grays. The son grew up in Texas, the father in Connecticut. The son believes that being characterized as a “cowboy” is a compliment; the father saw it as a slur. The son doesn’t mind seeming a little cocky. For the father, reared by an old-school mother who instilled in him a lifelong fear of being seen a braggart, humility was a telling mark of character.

In the eternal struggle between the State Department and the Defense Department, the first President Bush leaned toward Foggy Bottom. The second President Bush leans toward the Pentagon.

A central player in the Bush drama is Colin L. Powell. The retired general and the first President Bush were almost like father and son — and even today the elder Bush inquires regularly and discreetly about Powell’s position in his son’s inner circle.

Powell and the second President Bush are like brothers — but from the start it was clear that they were brothers with a bit of a sibling rivalry. The father was a natural businessman and an awkward politician, the son an awkward businessman and a natural politician. This distinction explains the two Bushes’ view of politics and policy.

“The father thought that if you had a good policy, you wouldn’t have to worry about the politics,” Peter B. Teeley, who was the elder Bush’s press secretary between 1979 and 1985, said in an interview this week. “The son doesn’t take any chances on that.”

That’s because, more than anything, the president is his mother’s son. Barbara Bush is opinionated, and she’s not reluctant to share her opinions. Barbara Bush holds grudges, and she’s not above acting on them. With her, everything is visceral, little is internalized. All that can be said of the son. She is a warrior, and proud of it. Her husband is a statesman, and proud of that.

But all those years when George W. Bush was growing up, George H.W. Bush wasn’t really around. The 43rd president was reared by Barbara Bush. He is, to be sure, his father’s son, but he doesn’t necessarily share his father’s perceptions, instinct — or destiny.