Geometric Proofs

as well as Fibonacci offer geometric cut-and-paste proofs for the algorithms by means of which mixed second-degree problems are solved. They are conspicuously absent from abbacus algebra, except in a few cases where they come in laterally. The first lateral entrance is found in Dardi of Pisa’s Aliabraa argibra (1344). Dardi’s proofs are similar to those of , but Dardi’s use of letters is different from anything coming before or after B he may have seen the proofs on an earlier occasion and then reproduced them from memory in his own way. His treatise was copied, but the easily recognizable proofs were not adopted by others. Next the proofs turn up in two of the three extensive “abbacus encyclopedias” that were written in Florence around 1460. They are part of sections borrowed from Guglielmo de Lunis’s translation of , and have nothing to do with the rest of the algebra of the two writers, which is in normal abbacus style. In the Summa, Luca Pacioli also offers geometric proofs B inspired by those of Fibonacci but partially of his own making. They are better integrated than in the encyclopedias, but still not very important for Pacioli’s own algebra. Their main importance appears to be that they inspired Cardano, thereby offering him a tool for his solution of the cubic equations.