Fibonacci had not really understood Archimedes’ proof. Archimedes knew that $256/153 < \sqrt{3} < 151/78$ and used the appropriate bound as an approximation when calculating the perimeters of the circumscribed and inscribed polygons, so he knew that he would obtain upper and lower bounds for the actual perimeters. But in his own account of Archimedes' work, Fibonacci used the same approximation for both and rounded up or down to the nearest number at each stage.
Later, Luca Pacioli (c.1445–1517) called Archimedes' result a ‘beautiful and subtle [bella e sotile]’ discovery. He followed Fibonacci's exposition, but mangled the line of Archimedes' thought even more than Fibonacci had, ignoring the whole idea of finding bounds.
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