His face adorns the $10 bill, but as Richard Brookhiser, host of “Rediscovering Alexander Hamilton” (airing on PBS April 11), finds when conducting a quick street canvas – many Americans cannot identify him.

“Washington has a monument,” Brookhiser intones. “Jefferson has a memorial. It’s often said that New York City is Hamilton’s monument.”

That would be more than enough for any man, yet, as this engrossing film from producer Michael Pack makes clear, it doesn’t quite do justice to the genius of Hamilton. First secretary of the Treasury, a drafter of the Constitution, author of two-thirds of the Federalist Papers, and father of the U.S. economy, Hamilton was also the prototype of the self-made American success – the original Horatio Alger hero, and then some.

Unlike the planters, wealthy merchants, and successful lawyers from established families who comprised the other founders, Hamilton was born in the Virgin Islands, “the bastard brat of a Scotch peddler,” as John Adams sneered in one of his less charitable moments. (To be fair, Hamilton could be lacerating about Adams, too.)

He was a bastard – but some brat. At age 11, orphaned and penniless, Hamilton found work in a St. Croix counting house. There he learned that strong application could yield advancement. He was so gifted at administration that his boss was willing to leave the 14-year-old Alexander in full charge of the business when he left for four months.

Also in St. Croix, Hamilton saw the suffering of slaves, forced to work endless hours in the scorching sun harvesting sugar cane. The camera lingers on the lanky, bamboo-shaped stalks. Most slaves, Brookhiser notes, “died within seven years.” Hamilton became a fervent and lifelong opponent of slavery.

So prodigious were his talents that a few of the merchants on St. Croix sponsored his emigration to the colonies to further his education. He was 16. Within the next two decades, he would serve as deputy to Gen. George Washington, achieve glory in battle himself, excel at the law, and, from nothing, create for his adopted country its first monetary system, its first fiscal system, its first accounting system, and its first central bank. He also founded the Coast Guard, the Customs Service, and the New York Post. In a touching moment, the film captures the ritual in which newly minted Coast Guard officers – to this day – salute the grave of the service’s founder.

Historical documentarians face a problem – no footage. Most resort to long pans of period paintings, or linger over photographs and sunsets, or throw in the occasional actors in period costume marching off to battle, along with talking heads. There’s nothing wrong with that style (Ken Burns, the master of the genre, has a great new film on Prohibition coming in October). But this film takes a different approach, setting itself firmly in the contemporary world – the world that Hamilton did so much to create.

Brookhiser travels from a prison in the Virgin Islands, where he chats with women who, like Hamilton’s mother, are behind bars, to the People’s Court for a re-enactment of one of Hamilton’s famous law cases, to the hectic streets of New York City, pulsing with business. He and Bernard-Henri Levi play-act the meeting between Hamilton and Talleyrand. He chats with Larry Flynt about the sex scandal that nearly ended Hamilton’s career, and with former gang members about the touchy matter of honor, which did end his life.

To appreciate Hamilton fully, it’s necessary to set the stage, as Brookhiser and historian Ron Chernow do, explaining that after the Revolution, the United States was an economic cripple, deeply in debt, its currencies nearly worthless as a result of inflation.

“We were,” says Chernow, “the deadbeat of world finance. We were like a Third World country.” Hamilton steered the new republic toward solvency. (We could use him now!)

Unlike the other founders, Chernow notes, who had mainly “pre-capitalist worldviews” with a strong bias toward agriculture, and who tended to see commerce and manufacturing as “corrupting influences,” Hamilton foresaw that the United States could become a great trading nation. From his earliest days in the St. Croix counting house, doing business with people from around the world speaking many languages, Hamilton understood that wealth is created by trade and commerce, not just from the soil.

Hamilton was an economic wizard, but also a profound political philosopher, a deep-dyed patriot, a gifted administrator who served as Washington’s informal “prime minister” during the first president’s term – and also a human being with weaknesses and foibles. He spoke brilliantly, the film reminds us, but sometimes too much. He might have bitten his tongue a bit more on the subject of Vice President Aaron Burr. But he did not, and the film takes us, reluctantly but inexorably, to the dueling ground at Weehawken, N.J., where we feel anew that day’s terrible toll.

“Rediscovering Alexander Hamilton” has depth, humor, original techniques, and insight. Highly recommended.

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