Bologna, Italy, January ; d. The original family name was Mazzoli. The Mazzolis, who seem to have been small landowners, adopted the name Bombelli early in the sixteenth century. Some of them were supporters of the Bentivoglio faction.

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Bologna, Italy, January ; d. The original family name was Mazzoli. The Mazzolis, who seem to have been small landowners, adopted the name Bombelli early in the sixteenth century. Some of them were supporters of the Bentivoglio faction. One of them was Antonio Mazzoli, alias Bombelli, who later became a wool merchant and moved to Bologna.

There he married Diamante Scudieri, the daughter of a tailor. Six children were born to this marriage, of whom the eldest son was Rafael Bombelli. It has been suggested that Bombelli might have studied at the University of Bologna, but this seems unlikely when one considers his family background and the nature of his profession.

He spent the greater part of his working life as an engineer-architect in the service of his patron, Monsignor Alessandro Rufini, a Roman nobleman. The major engineering project on which Bombelli was employed was the reclamation of the marshes of the Val du Chiana.

His professional engagements seem to have delayed the completion of the book, but the more important part of it was published in His death soon afterward prevented the publishing of the remainder of the work. It was not published until It is not known whether Bombelli himself worked in Foligno; but by he had began to work for Rufini in the reclamation of the Val di Chiana marshes. Rufini began to take an interest in this project in , when the rights to take an interest in this project in , when the rights of reclamation of that part of the marches which belonged to the Papal States were obtained by his nominee.

The work of reclamation was interrupted sometime between and By Bombelli had returned to the Val di Chiana, and his work there ended in that year. In he was in Rome, where he took part in the unsuccessful attempt to repair the Ponte Santa Maria , one of the bridges over the Tiber. Rafael Bombelli grew up in an Italy that was active in the production of works on practical arithmetic. Luca Pacioli , author of the Summa di arithmetica, geometria,… , had lectured at Bologna at the beginning of the century.

So had Scipione dal Ferro, a citizen of Bologna and one of the foremost mathematicians of the time. Their successors, Cardano, Tartaglia, and Ferrari, who were attempting the solution of the cubic and biquadratic equations, lived and worked in the neighboring cities of northern Italy. Copies of the Cartelli di mathematicadisfida — , exchanged between Ferrari and Tartaglia, were circulated in the principal cities of Italy.

Such was the climate in which Bombelli conceived the idea of writing a treatise on algebra. He felt that none of his predecessors except Cardano had explored the subject in depth; but Cardano, he thought, had not been clear in his exposition.

He therefore decided to write a book that would enable anyone to master the subject without the aid of any other text. The work, written between and , was a systematic and logical exposition of the subject in five parts, or books.

In Book I, Bombelli dealt with the definitions of the elementary concepts powers, roots, binomials, trinomials and applications of the fundamental operations. In Book II he introduced algebraic powers and notation, and then went on to deal with the solution of equations of the first, second, third, and fourth degrees.

Bombelli considered only equations with positive coefficients, thus adhering to the practice of his contemporaries. He was therefore obliged to deal with a large number of cases: five types of quadratic equations, seven cubic, and fortytwo biquadratic. For each type of equation, he gave the rule for solution and illustrated the rule with examples.

Bombelli feared that the examples given in Book II would not be sufficient for a beginner who wished to master the subject, so he decided to include in Book III a series of problems by which the student would be taken, in stages, through the various operation of algebra.

For this purpose he chose problems that were common to books on practical arithmetic of his day. They were often woven into incidents that could have occurred in the marketplace or tavern.

Books IV and V formed the geometrical portion of the work. Book IV contained the application of geometrical methods to algebra, algebra linearia, and Book V was devoted to the application of algebraic methods to the solution of geometrical problems.

Unfortunately, Bombelli was unable to complete the work as he had originally planned, in particular Books IV and V. They set out to translate the manuscript, but circumstances prevented them from completing the work. The changes that Bombelli made in the first three books of his Algebra show evidence of the influence of Diophantus. His death prevented his keeping the promise. In his Algebra, Bombelli gave a comprehensive account of the existing knowledge of the subject, enriching it with his own contributions.

Because of the special nature and importance of these imaginary quantities, he took great care to make the reader familiar with them by introducing them early in his work—at the end of Book I. This was the cube root of a complex number occurring in the solution of the irreducible case of the cubic equation.

Having pointed out that the complex root is always accompanied by its conjugate, he set out the rules for operating with complex numbers and gave examples showing their application. Here he showed himself to be far ahead of his time, for his treatment was almost that followed today. Bombelli also pointed out that the problem of trisecting an angle could be reduced to that of solving the irreducible case of the cubic equation this was illustrated in Book V.

In Book III of the printed version of the Algebra one finds no trace of the influence that practical arithmetics originally had on Bombelli. In place of these applied problems he introduced a number of these applied problems he introduced a number of abstract problems, of which were taken from the Arithmetic of Diophantus. He was in fact the first to popularize the work of Diophantus in the West. Apart from the solution of the irreducible case of the cubic equation, the most significant contribution Bombelli made to algebra was in the notation he adopted.

He represented the powers of the unknown quantity by a semicircle inside which the exponent was placed: for the modern for x2, and or for 5x. In the printed work the semicircle was reduced to an arc:. The zero exponent, , was used in the manuscript, for 48, but was omitted from the published work.

The notation was used in the manuscript in applying the radical to the aggregate of two or more terms: for. He even used the radical sign as a double bracket: for. In the printed work the horizontal line was broken, and R, R3 became Rq.

He had reduced some of the arithmetical problems of Book III to an abstract from and had interpreted them geometrically. He did not feel obliged to give geometrical proofs for the correctness of the results that he had obtained by algebraic methods. In doing so, he had broken away from a long-established tradition. Bombelli was the last of the algebraists of Renaissance Italy.

While giving Bombelli due credit, he stressed the superiority of his notation to that of the Cossists. His correspondence with Huygens shows the keen interst these two men took in the work of the Italian mathematicians of the Renaissance. Original Works. Secondary Literature. For references to earlier literature, see S.

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## The Life and Work of Rafael Bombelli

It is perhaps worth giving a little family background. The Bentivoglio family ruled over Bologna from Sante Bentivoglio was "signore" meaning lord of Bologna from and he was succeeded by Giovanni II Bentivoglio who improved the city of Bologna, in particular developing its waterways. The Mazzoli family were supporters of the Bentivoglio family but their fortunes changed when Pope Julius II took control of Bologna in , driving the Bentivoglio family into exile. Antonio Mazzoli was able to return to live in Bologna.

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## Bombelli, Rafael

The solution of this kind of equation requires taking the cube root of the sum of one number and the square root of some negative number. Before Bombelli delves into using imaginary numbers practically, he goes into a detailed explanation of the properties of complex numbers. Right away, he makes it clear that the rules of arithmetic for imaginary numbers are not the same as for real numbers. This was a big accomplishment, as even numerous subsequent mathematicians were extremely confused on the topic. Bombelli avoided confusion by giving a special name to square roots of negative numbers, instead of just trying to deal with them as regular radicals like other mathematicians did. This made it clear that these numbers were neither positive nor negative.

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## Rafael Bombelli

Bombelli: Algebra In Rafael Bombelli published the first three volumes of his famous book Algebra the intended further two volumes were not completed before his death. We quote from the text where Bombelli is using fractions to approximate to square roots. We insert in square brackets the equations that Bombelli is working with in modern notation and some comments. He begins with preamble, then gives his method without justification, finally giving a detailed justification of his method:- Many methods of forming fractions have been given in the works of other authors; the one attacking and accusing the other without due cause in my opinion for they are all looking at the same end. It is indeed true that one method may be briefer than another, but it is enough that all are at hand and the one that is most easy will without doubt be accepted by men and be put in use without casting aspersions on another method. Let us first assume that if we wish to find the approximate root of 13 that this will be 3 with 4 left over.