It's unclear as to exactly how Bombelli learned the leading mathematical works of that time period, but he lived in the right part of Italy to be involved in major events surrounding the solution of cubic and quartic equations. Scipione Del Ferro was the first to solve the cubic equation, and he was a professor at Bologna, but he died the year Bombelli was born. Cardan's major work on the topic
Ars Magna was published in 1545, and it is known that Bombelli studied Cardan's work [1].
From about 1548, Pier Francesco Clementi, Bombelli's teacher, worked for the Apostolic Camera, which was a specialized department of the papacy in Rome formed to deal with financial and legal matters. The Apostolic Camera employed Clementi to reclaim the marshes near Foligno, and it was probably that Bombelli assisted his teacher in this project. However, there is no confirmation that this actually occurred [1].
In 1549, Bombelli was given the opportunity by Rufini to work on a major engineering project to reclaim the marshlands of the Val di Chiana, which belonged to the Papal States [4]. By 1551, Bombelli was in the Val di Chiana recording the boundaries to the land that was to be reclaimed [1]. In 1555, the project was put on hold [4].
While Bombelli waited for his work on the Val di Chiana project to continue, he decided to write an algebra book. He felt there were too many arguments between leading mathematicians at the time, and he wanted to clarify certain aspects of it. He believed there was a lack of careful exposition of the subject of algebra. Bombelli believed that only Cardan truly delve into the topic in great depth, but his work was not entirely readable to those without great knowledge with mathematics. The preface of his book goes as follows:
"
I began by reviewing the majority of those authors who have written on up to the present, in order to be able to serve instead of them on the matter, since there are a great many of them" [1].
Of course, he was referring to the subject and matter of algebra. He believed that he could communicate the ideas of algebra to
any reader, regardless of any previous skills with algebra or any understanding of the subject.
In 1560, work on the Val di Chiana continued, but Bombelli had not completed his algebra book yet. However, there must not have been much work to finish because the project was completed before the end of 1560. Bombelli gained much recognition as a reputable hydraulic engineer, and he was then tasked to do many other projects [4]. In 1561, he visited Rome to repair the Santa Maria bridge over the Tiber, but he failed. This did not dampen his reputation, though, because he was taken as a consultant for a project to drain the Pontine Marshes. Many attempts to drain these marshes had previously failed, and the project in which Bombelli, as a consultant for Pope Pius IV, attempted had also failed [1].
On one of Bombelli's visits to Rome, he made an important mathematical discovery. He learned of Diophantus's Arithmetica, and since he had not completed his algebra book there was great influence of Diophantus on his math project [4]. He and Antonio Maria Pazzi, who taught mathematics at the University of Rome, examined Diophantus's manuscript. They made a translation, and Bombelli wrote:
"... [
we]
, in order to enrich the world with a work so finely made, decided to translate it and we have translated five of the books (
there being seven in all)
; the remainder we were not able to finish because of pressure of work on one or other." [1].
Bombelli then published three separate books of his work, and many of the problems that arise in Book III (143 of the 272 problems), are taken directly from Diophantus. Bombelli gives full credit to Diophantus, acknowledging that he had taken many of Diophantus's problems from the
Arithmetica. Bombelli intended to write five books, but he died shortly after writing his third [1].
Although Bombelli never finshed his last two books, his manuscript was discovered in a library in Bologna by Bortolotti in 1923. There was an unfinished manuscript of the last two books, and Bortolotti published the incomplete geometrical part of Bombelli's work in 1929. It is noted that many of Bombelli's methods relate to the geometrical procedures of Omar Khayyam.
References:
[1] O'Connor, J., Robertson, E.,
Rafael Bombelli. JOC/EFR, 2000.
http://www-history.mcs.st-and.ac.uk/Biographies/Bombelli.html
[4] Freeman, L.,
Rafael Bombelli. Fermat's Last Theorem, 2006.
http://fermatslasttheorem.blogspot.com/2006/11/rafael-bombelli.html
