Francesco Borromini's masterpiece, the university chapel of S. Ivo alla Sapienza in Rome, was built over three pontificates, with many radical changes after each conclave. This is a study of the first building period under Urban VIII Barberini, and in particular of the earliest known plan for the church, datable to 1642, now kept in the Archivio di Stato in Rome. The drawing shows the basic plan of the church-a triangle-plus-apses-minus-angles-being moved, manipulated, and stretched as Borromini fine-tuned his design. The plan contains echoes of centralized buildings by Baldassarre Peruzzi and Giuliano da Sangallo, and also reveals Borromini's study of ancient apsidal structures, in particular, the "funerary hexaconch" standing along-side the Via Appia, best known from Peruzzi's drawings. Neither the Barberini bee nor the star of Solomon played a role in the genesis of the plan. But the idea of a highly geometrical architecture may owe something to the mathematical culture of the Barberini court, especially to Galileo's disciple Fra Benedetto Castelli.

S. Andrea al Quirinale was Bernini's architectural masterpiece. A detailed account in the Jesuit Archives now allows a more exact chronology and shows how the design evolved over a fourteen-year period, 1658-1672. The church, which is often compared to a jewelbox, is used as the starting point for a discussion of various attitudes to wealth on the part of baroque architects and patrons. The issue of Bernini's classicism as an architect is discussed with reference to his use of geometry and his development as a designer of church façades. The famous rivalry between Bernini and Borromini is seen as a result of fundamental differences of principle, though some examples are presented from the late 17th and early 18th centuries which show attempts to reconcile the styles and approaches of the two unfriendly geniuses.