How exactly did Brunelleschi build the dome of Florence Cathedral without conventional centering?

Brunelleschi left no treatise and guarded his methods in his own lifetime, so the surviving evidence is the dome itself, the account books, and machines described secondhand. How the herringbone brickwork and the double shell actually stabilized each fresh course of the rising vault, and how the courses were laid out in space without centering to guide them, is still argued among historians of construction, with competing geometric reconstructions defended in the literature and none established.