In 1510 Pacioli returned to Perugia to lecture there again. He also lectured again in Rome in 1514 but by this time Pacioli was 70 years of age and nearing the end of his active life of scholarship and teaching. He returned to Sansepolcro where he died in 1517 leaving unpublished a major work De viribus amanuensis on recreational problems, geometrical problems and proverbs. This work makes frequent reference to Leonardo da Vinci who worked with him on the project, and many of the problems in this treatise are also in Leonardo's notebooks. Again it is a work for which Pacioli claimed no originality, describing it as a compendium.
Despite the lack of originality in Pacioli's work, his contributions to mathematics are important, particularly because of the influence which his book were to have over a long period. In the importance of Pacioli's work is discussed, in particular his computation of approximate values of a square root (using a special case of Newton's method), his incorrect analysis of certain games of chance (similar to those studied by Pascal which gave rise to the theory of probability), his problems involving number theory (similar problems appeared in Bachet's compilation), and his collection of many magic squares.
In 1550 there appeared a biography of Piero della Francesca written by Giorgio Vasari. This biography accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry. This is an unfair accusation, for although there is truth that Pacioli relied heavily on the work of others, and certainly on that of della Francesca in particular, he never attempted to claim the work as his own but acknowledged the sources which he used.
J J O'Connor and E F Robertson
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