And the algebraists or arithmeticians of the 16th century, such as Luca Pacioli (Lucas de Borgo), Geronimo or Girolamo Cardano (1501-1576), and Niccola Tartaglia (1506-1559), had used geometrical constructions to throw light on the solution of particular equations.
During the 11th, 12th and 13th centuries Parma had its full share of the Guelph and Ghibelline struggles, in which it mainly took the part of the former, and also carried on repeated hostilities with Borgo San Donnino and Piacenza.
The incident led to a feud with the supporters of Morati, among whom was Pozzo di Borgo (destined to be his life-long enemy), and opened a breach between the Bonapartes and Paoli.
Pseudo-Joachimite treatises sprang up on every hand, and, finally, in 1254, there appeared in Paris the Liber introductorius ad Evangelium aeternum, the work of a Spiritual Franciscan, Gherardo da Borgo San Donnino.
This book, probably published after the death of its author and probably interpolated by his disciples, contains, besides Joachimite principles, an affirmation even clearer than that of Gherardo da Borgo of the elect character of the Franciscan order, as well as extremely violent attacks on the papacy.