Piero della Francesca

Piero della Francesa (1412 – 1492) is acknowledged as one of the most important painters of the fifteenth century.  Besides being a talented painter, he was also a highly competent mathematician.  Piero wrote three treatises: “Abacus treatise” (Trattato d’abaco), “Short book on the five regular solids” (Libellus de quinque corporibus regularibus), and “On perspective for painting” (De prospectiva pingendi).

Piero’s perspective treatise is believed to be the first of its kind.  It was concerned not with ordinary natural optics, but with what was known as “common perspective” (perspectiva communis), the special kind used by painters.  Piero felt that this new part of perspective should be seen as a legitimate extension of the older established science.  He divided his treatise into three books, and his first original theorem appeared in one of them (Fig. 20).

Fig. 20 – Proposition 8, Book 1 of
Piero della Francesca’s De prospectiva pingendi

A given straight line BC is divided into several parts.  Another line HI is drawn parallel to the first.  The theorem states that if lines from the points dividing BC converge at point A, HI will be divided in the same proportion as BC

The proof uses similar triangles.  Since the two lines are parallel, AK/AD = HK/BD.  Also, AK/AD = KL/DE.  It follows that HK/BD = KL/DE.  Since BD = DE, this means HK = KL.  The same reasoning can be extended to LM, MN and NI.

Indeed, so influential was Piero’s treatise that many following it were merely simplifications of it.  It was found that examples used in many treatises on perspective were derived from Piero’s work.  Furthermore, it is possible Piero had shown that Alberti’s and the distance point constructions are equivalent.  Nevertheless, we will attempt to prove this in the next section.