Final answer:
Filippo Brunelleschi's design triumphed over the challenge of building a large dome on the Florence Cathedral by innovating a double-walled brick design without scaffolding, utilizing a herringbone pattern for stability, and inventing hoisting machines.
Step-by-step explanation:
Filippo Brunelleschi's design for the dome of Florence Cathedral was a triumph over adversity because it resolved the daunting engineering challenge of constructing a massive dome over the cathedral's wide span without the use of any supporting framework. By introducing a double-walled design made from lightweight bricks laid in a herringbone pattern and reinforced with horizontal chains, Brunelleschi managed to create a structure that could support itself during construction. Additionally, his innovative use of hoisting machines to lift materials up to the great heights of the dome exemplified a combination of mathematical precision and architectural innovation. This monumental achievement not only covered the cathedral, protecting it from the elements but also set a new precedent in the use of classical architectural principles, which became central to Renaissance architecture.