Accounts by Brunelleschi's biographers of the discovery of linear perspective are briefly reexamined. It is suggested that Brunelleschi's familiarity, as a clockmaker, with sundials may have suggested to him a method of drawing perspectives without recourse to geometrical construction from plans and elevations. This would explain several puzzling features of the first perspective, including the choice of subject (the Baptistery of San Giovanni), the lateral inversion of the painting, the necessity for a hole “as tiny as a lentil bean”, and the use of mirrors. The second perspective is then shown to embody lessons learned from the shortcomings of the first. It is concluded that the descriptions written by Vasari and Manetti are essentially sound, but their usual interpretation is thrown into question. In particular they can no longer sustain the centuries-old claim, for which they formed the basis, that the geometrical construction of linear perspective from plans and elevations was discovered by Brunelleschi.