This painting is The Astronomy Lesson of the Duchesse du Maine, by François de Troy. It was done in the period 1705 - 1710. The Duc and Duchesse were favorites of King Louis XIV (the Duc was illegitimate and Louis was his father). The Duchesse was interested in science, and is portrayed being lectured in astronomy. The armillary sphere in the gallery is similar to the one shown in the painting.
The painting on the right is The Duchesse du Polignac Wearing a Straw Hat, a 1782 painting by Élisabeth Louise Vigée Le Brun. Vigée Le Brun was one of the foremost portrait painters of 18th century France. Marie Antoinette was her most important supporter. The Duchesse was considered one of the great beauties of the age. She had humble origins but married a count, who was elevated to a dukedom. The duchess became the governess of the royal children, a high honor and a lot of work! Vigée Le Brun and the duchess both fled France when the monarchy fell. The Atheneum's painting is a duplicate; the original is in Versailles. (Hi, Cieldequimper.)