Leonardo contributed to the understanding of the volume of solids and their projected heights. In â€œDivine Proportionsâ€, which he co-authored with Luca Pacioli, Leonardo illustrated such shapes as polygons, icosahedrons, and dodecahedrons, contributing to the development of the concept of volume and projections of such shapes (Stakhov and Olsen 172; Frere 110). Although earlier biographers tended to portray Leonardoâ€™s understanding of geometrical forms and figures as relatively limited, Dauben and Scriba believe that the basis of such view was on the limited study of his works, and the latter-day understanding of Leonardoâ€™s works reveal a deeper knowledge of this subject (87). Leonardoâ€™s method of operation in representing these mathematical figures, according to Moon, does not involve equations (162). Instead, Leonardo â€˜drawsâ€™ his explanations. For instance, Leonardo is credited with having proved Pythagorasâ€™ theory, a proof that he illustrated by adding triangles and squares on an original triangle to provide the proof of the theory. Leonardo was very keen on studying available Greek manuscripts on various mathematical principles, and he proved the Pythagorean Theorem in his unique way, before delving into Archimedesâ€™ principles (Nelsen and Alsina 28). The Renaissance period did not only involve a passive study of the Greek texts, but also active analysis and development of such texts.
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