Bombelli's Life and Career

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

The History of Mathematics During Bombelli's Time

By about 1500 A.D., Europe was undergoing many changes.  The Middle Ages were coming to an end and the Modern World was being created.  Great, new mathematics would be born each century leading up until modern day, but focusing on the past mathematical inventions is essential in learning what Bombelli's time was like from a mathematician's point of view.

 In 1527, Peter Aprian (1495 - 1552), a German arithmetician, published a book about the Pascal Triangle.  This version of Pascal's Triangle appears more than a century before Pascal's investigation of the properties of this triangle [3].

Around 1492, Francesco Pellos (1450 - 1500) wrote a commercial arithmetic book, Compendio de lo abaco, and he used a "dot" to indicate the division of an integer by the power of ten.  This leads to what we now know as the decimal point [3].

In 1489, Johannes Widman (1462 - 1498) wrote an arithmetic book which contained the first recorded appearances of "+" and "-" signs.  This book had more examples and became more widely known than any other book that were released around that time period [3].

There were many more ingenious mathematical processes and ideas formed around this time period, but there are also some notable, important periods in history during this time.  In 1534, the Act of Supremacy emerged making the King of England the supreme head of the Church of England.  In 1517, the Protestant Reformation occurred.  In 1492, one of the most famous discoveries were made; Columbus discovered America [3].  Of course, there are many more important points in history during this time, but the ones mentioned appear to be the most important around the time of Rafael Bombelli's birth.

References:
[3] http://www.math.wichita.edu/~richardson/timeline.html

The Life of the Bombelli Family

Rafael Bombelli was a famous mathematician most noted for his work with imaginary numbers.  He was born on January of 1526 in Bologna, Papal States (now Italy), and he passed away in 1572.  Bombelli's father was Antonio Mazzoli, but he eventually changed his name to Bombelli [1].

In 1443, the Bentivoglio family ruled over Bologna.  Sante Bentivoglio was the "lord" of Bologna from 1443, and he was succeeded by Giovanni II Bentivoglio.  Giovanni II improved the city by developing its waterways.  The Mazzoli family supported the Bentivoglio family, but in 1506 Pope Julius II took control of Bologna, and he drove the Bentivoglio family away from Bologna.  Antonio Mazzoli's grandfather continued to support the Bentivoglio family, but he, along with many other who attempted rebellion against Julius II, was executed.  For a long time, the Mazzoli family suffered and their land taken from them, but eventually Antonio Mazzoli had the property returned to him and his family [1].

Antonio Bombelli was a wool merchant, but not much information about finances was recorded [2].  He carried on his work as a wool merchant and married Diamante Scudieri, a tailor's daughter.  Rafael Bombelli was their eldest son out of six children [1].  Rafael received no university education, and all that is really known about his education is that his teacher was Pier Francesco Clementi of Corinaldo, an engineer-architect who drained swamps [2].  Rafael became a patron of Alessandro Rufini, a Roman noble later to become the Bishop of Melfi [1].

References:
[1] O'Connor, J., Robertson, E., Rafael Bombelli.  JOC/EFR, 2000.  http://www-history.mcs.st-and.ac.uk/Biographies/Bombelli.html

[2] Helden, A., Bombelli Rafael.  The Galileo Project, 1995. http://galileo.rice.edu/Catalog/NewFiles/bombelli.html