The Institut de France is one of the most prestigious French institutions. It brings together five Académies, among which is the Académie Française, founded in 1635 by Cardinal Richelieu. It is the strict guardian of the French language - especially, today, against such dreaded Anglicisms as "fast food" (which the honorable academicians render as la restauration surrapide). 

The building itself was the brainchild of another cardinal, Mazarin, who in his will devoted part of his immense fortune to establish the Collège des Quatre-Nations and give it appropriate headquarters. The building, designed by King Lous XIV's architect Louis le Vau, was later assigned by Napoleon to host the Institut de France and its academies. For further details about the building's very interesting architecture, history and traditions, see this article from wikipedia (also, in French). 

The most important academy for our tour is the Académie des Sciences. John Adams was passionately interested in science. He was also rather jealous of Benjamin Franklin's popularity with members of both sexes and all classes. Hence the slightly snide tone to his journal entry for April 29, 1778 (given here with his spelling, punctuation and capitalization, and with old French spelling): After dinner We went to the Accademy of Sciences, and heard Mr. D'Alembert as Secretary perpetual, pronounce Eulogies on several of their Members lately deceased. Voltaire and Franklin were both present, and there presently arose a general Cry that Monsieur Voltaire and Monsieur Franklin should be introduced to each other. (They had in fact already met.) This was done and they bowed and spoke to each other. This was no Satisfaction. There must be something more. Neither of our Philosophers seemed to divine what was wished or expected. They however took each other by the hand.... But this was not enough. The Clamour continued, untill the explanation came out 'Il faut s'embrasser, a la francoise' [They have to hug each other the French way]. The two Aged Actors upon this great Theatre of Philosophy and frivolity then embraced each other by hugging one another in their Arms and kissing each others cheeks, and then the tumult subsided. And the Cry immediately spread through the whole Kingdom and I suppose over all Europe Qu'il etoit charmant. Oh! il etoit enchantant, de voir Solon et Sophocle embrassan[t]. How charming it was! Oh! it was enchanting to see Solon and Sophocles embracing!

In Adams's cynical rendering of the scene, Franklin, of course, was the Greek statesman Solon and Voltaire the Greek playwright Sophocles.

About a year later, at a Boston dinner in honor of the French ambassador to the U.S., Adams rhapsodized about the science collections he had seen in France, and suggested that his own fledgling nation - already a storehouse of scientific talent - should create an Academy of Arts and Sciences. The legislature of his home state, Massachusetts, did this the next year, 1780, and the American Academy of Arts and Science has being going strong ever since.