De Waal's Chimpanzee Politics was one of 25 books on a recommended reading list Mr Gingrich put out earlier this year for the benefit of young congressional Republicans newly arrived in Washington.
Chimpanzee Politics is a study of the struggle for power between apes striving to acquire the coveted status of "alpha male". De Waal's conclusion is that "the roots of politics are older than humanity", that when it comes to the tactics employed for the pursuit of power the main difference between humans and chimpanzees is that one species talks and the other doesn't.
De Waal's book centres on a protracted leadership battle between Yeroen, the incumbent alpha male of a chimpanzee colony in Arnhem, and Luit, his ambitious challenger. Luit wages a campaign to dethrone the autocratic Yeroen. His eventual success rests on a strategy that combines populism, coalition-building and naked aggression.
For a simple understanding of the secrets of power Mr Gingrich found in the book, all one need do is substitute Newt for Luit and Bill Clinton for Yeroen.
Luit, like Newt, "has a playful, almost mischievous character", De Waal writes. He engages in "bluff displays": screaming, jumping-up-and-down performances designed simultaneously to intimidate Yeroen and trumpet the impending challenge to his leadership. Yeroen responds in kind but tires more easily.
Luit "mocks Yeroen by openly usurping his position in the early stages of the struggle". Not 10 metres from Yeroen he proceeds to mate with Spin, the most attractive female in the colony. Yeroen pretends not to notice but "unobtrusively watches Luit through the hair on his shoulders".
An analogy might be found with Mr Clinton's response - peeping, as it were, from behind the White House curtains - to Mr Gingrich's flauntingly presidential demeanour following the Republicans' electoral seizure of Congress.
Mr Clinton's first counter-tactic uncannily aped Yeroen's. Mr Clinton went into a huddle with his most loyal White House supporters. Yeroen, "calculating by nature", lobbies among his trusted allies, most of whom happen to be female. (Mr Clinton's support base is much more solid, according to the statistics, among women than men.)
Yeroen, his self-esteem beginning to grow again, regroups and charges at Luit (Mr Clinton's successful State of the Union address in January?) with his female supporters in tow.
But Luit learns from Yeroen's example. He goes to the females, plays with their children. (Mr Gingrich's response to Mr Clinton's charge that the Republicans were not sensitive to children's needs was to hand out free books at a Washington school.) Luit builds up his alliances, grows stronger.
"Luit displays forcefully at Yeroen and never bares his teeth. Yeroen, on the other hand, does bare his teeth, a sign of uncertainty. Yeroen was, as it were, being weaned from power ... He looked a pitiful sight. He had lost all his former self confidence and the look in his eyes reflected the psychological beating he had taken."
Sensing victory, Luit changes tack. He offers a truce. (Mr Gingrich has made a point recently of praising Mr Clinton, pledging to work with him for the common good.)
But Luit feels he has won, as Mr Gingrich feels he has, having successfully cajoled the House of Representatives into voting on all the items on his Contract with America during the first 100 days of Congress.
In the last phase of the struggle Yeroen greets Luit with a grunt: the chimpanzee equivalent of the white flag of surrender.
But, there is a tragic postscript. After Chimpanzee Politics was published, the deposed Yeroen hatched a cruel plot with a new young ally. One night they caught Luit unawares and brutally murdered him, tearing off his toes and yanking out his testicles.
The lesson for Mr Gingrich? That he might usefully consider paying more attention to peace-making, less to brash acts of bluff self-display. If he doesn't know, he should be told.