Luca Pacioli's father was Bartolomeo Pacioli, but
Pacioli does not appear to have been brought up in his parents house. He lived
as a child with the Befolci family in Sansepolcro which was the town of his
birth. This town is very much in the centre of Italy about 60 km north of the
city of Perugia. As far as Pacioli was concerned, perhaps the most important
feature of this small commercial town was the fact that Piero della Francesca had a studio and workshop in there
and della Francesca spent quite some
time there despite frequent commissions in other towns

Although we know little of Pacioli's early life, the
conjecture that he may have received at least a part of his education in the
studio of della Francesca in
Sansepolcro must at least have a strong chance of being correct. One reason
that this seems likely to be true is the extensive knowledge that Pacioli had
of the work of Piero della Francesca
and Pacioli's writings were very strongly influenced by those of Piero.

Pacioli moved away from Sansepolcro while he was still
a young lad. He moved to Venice to enter the service of the wealthy merchant
Antonio Rompiansi whose house was in the highly desirable Giudecca district of
that city. One has to assume that Pacioli was already well educated in basic
mathematics from studies in Sansepolcro and he certainly must have been well
educated generally to have been chosen as a tutor to Rompiansi's three sons.
However, Pacioli took the opportunity to continue his mathematical studies at a
higher level while in Venice, studying mathematics under Domenico Bragadino.
During this time Pacioli gained experience both in teaching, from his role as
tutor, and also in business from his role helping with Rompiansi's affairs.

It was during his time in Venice that Pacioli wrote
his first work, a book on arithmetic which he dedicated to his employer. This
was completed in 1470 probably in the year that Rompiansi died. Pacioli
certainly seemed to know all the right people for he left Venice and travelled
to Rome where he spent several months living in the house of Leone Battista Alberti who was secretary in the Papal
Chancery. As well as being an excellent scholar and mathematician, Alberti was able to provide Pacioli with
good religious connections. At this time Pacioli then studied theology and, at
some time during the next few years, he became a friar in the Franciscan Order.

In 1477 Pacioli began a life of travelling, spending
time at various universities teaching mathematics, particularly arithmetic. He
taught at the University of Perugia from 1477 to 1480 and while there he wrote
a second work on arithmetic designed for the classes that he was teaching. He
taught at Zara (now called Zadar or Jadera in Croatia but at that time in the
Venetian Empire) and there wrote a third book on arithmetic. None of the three
arithmetic texts were published, and only the one written for the students in
Perugia has survived. After Zara, Pacioli taught again at the University of
Perugia, then at the University of Naples, then at the University of Rome.
Certainly Pacioli become acquainted with the duke of Urbino at some time during
this period. Pope Sixtus IV had made Federico da Montefeltro the duke of Urbino
in 1474 and Pacioli seems to have spent some time as a tutor to Federico's son
Guidobaldo who was to become the last ruling Montefeltro when his father died
in 1482. The court at Urbino was a notable centre of culture and Pacioli must
have had close contact with it over a number of years.

In 1489, after two years in Rome, Pacioli returned to
his home town of Sansepolcro. Not all went smoothly for Pacioli in his home
town, however. He had been granted some privileges by the Pope and there was a
degree of jealousy among the men from the religious orders in Sansepolcro. In
fact Pacioli was banned from teaching there in 1491 but the jealousy seemed to be
mixed with a respect for his learning and scholarship for in 1493 he was
invited to preach the Lent sermons.

During this time in Sansepolcro, Pacioli worked on one
of his most famous books the Summa de arithmetica, geometria, proportioni et
proportionalita which he dedicated to Guidobaldo, the duke of Urbino. Pacioli
travelled to Venice in 1494 to publish the Summa. The work gives a summary of
the mathematics known at that time although it shows little in the way of
original ideas. The work studies arithmetic, algebra, geometry and trigonometry
and, despite the lack of originality, was to provide a basis for the major
progress in mathematics which took place in Europe shortly after this time. As
stated in the Summa was:-

... not addressed to a particular section of the
community. An encyclopaedic work (600 pages of close print, in folio) written
in Italian, it contains a general treatise on theoretical and practical
arithmetic; the elements of algebra; a table of moneys, weights and measures
used in the various Italian states; a treatise on double-entry bookkeeping; and
a summary of Euclid's geometry. He
admitted to having borrowed freely from
Euclid, Boethius, Sacrobosco,
Fibonacci, ...

The geometrical part of Pacioli's Summa is discussed
in detail in [6]. The authors write:-

The geometrical part of L Pacioli's Summa [Venice,
1494] in Italian is one of the earliest printed mathematical books. Pacioli
broadly used Euclid's Elements,
retelling some parts of it. He referred also to Leonardo of Pisa ( Fibonacci).

Other interesting aspects of the Summa was the fact
that it studied games of chance. Pacioli studied the problem of points, see,
although the solution he gave is incorrect.

Ludovico Sforza was the second son of Francesco
Sforza, who had made himself duke of Milan. When Francesco died in 1466,
Ludovico's elder brother Galeazzo Sforza became duke of Milan. However,
Galeazzo was murdered in 1476 and his seven year old son became duke of Milan.
Ludovico, after some political intrigue, became regent to the young man in
1480. With very generous patronage of artists and scholars, Ludovico Sforza set
about making his court in Milan the finest in the whole of Europe. In 1482 Leonardo da Vinci entered Ludovico's service
as a court painter and engineer. In 1494 Ludovico became the duke of Milan and,
around 1496, Pacioli was invited by Ludovico to go to Milan to teach
mathematics at Ludovico Sforza's court. This invitation may have been made at
the prompting of Leonardo da Vinci who
had an enthusiastic interest in mathematics.

At Milan Pacioli and
Leonardo quickly became close friends. Mathematics and art were topics
which they discussed at length, both gaining greatly from the other. At this
time Pacioli began work on the second of his two famous works, Divina
proportione and the figures for the text were drawn by Leonardo. Few mathematicians can have had a
more talented illustrator for their book! The book which Pacioli worked on
during 1497 would eventually form the first of three books which he published
in 1509 under the title Divina proportione (see for example). This was the
first of the three books which finally made up this treatise, and it studied
the 'Divine Proportion' or ' golden ratio' which is the ratio a : b = b : (a +
b). It contains the theorems of Euclid
which relate to this ratio, and it also studies regular and semiregular
polygons (see in particular for a discussion of Pacioli's work on regular
polygons). Clearly the interest of
Leonardo in this aesthetically satisfying ratio both from a mathematical
and artistic point of view was an important influence on the work. The golden
ratio was also of importance in architectural design and this topic was to form
the second part of the treatise which Pacioli wrote later. The third book in the
treatise was a translation into Italian of one of della Francesca's works.

Louis XII became king of France in 1498 and, being a
descendant of the first duke of Milan, he claimed the duchy. Venice supported
Louis against Milan and in 1499 the French armies entered Milan In the
following year Ludovico Sforza was captured when he attempted to retake the
city. Pacioli and Leonardo fled
together in December 1499, three months after the French captured Milan. They
stopped first at Mantua, where they were the guests of Marchioness Isabella
d'Este, and then in March 1500 they continued to Venice. From Venice they
returned to Florence, where Pacioli and
Leonardo shared a house.

The University of Pisa had suffered a revolt in 1494
and had moved to Florence. Pacioli was appointed to teach geometry at the
University of Pisa in Florence in 1500. He remained in Florence, teaching
geometry at the university, until 1506.
Leonardo, although spending ten months away working for Cesare Borgia,
also remained in Florence until 1506. Pacioli, like Leonardo, had a spell away from Florence when he taught at the
University of Bologna during 1501-02. During this time Pacioli worked with
Scipione del Ferro and there has been
much conjecture as to whether the two discussed the algebraic solution of cubic equations. Certainly Pacioli discussed
this topic in the Summa and some time after Pacioli's visit to Bologna,
del Ferro solved one of the two cases
of this classic problem.

During his time in Florence Pacioli was involved with
Church affairs as well as with mathematics. He was elected the superior of his
Order in Romagna and then, in 1506, he entered the monastery of Santa Croce in
Florence. After leaving Florence, Pacioli went to Venice where he was given the
sole rights to publish his works there for the following fifteen years. In 1509
he published the three volume work Divina proportione and also a Latin
translation of Euclid's Elements. The
first printed edition of Euclid's
Elements was the thirteenth century translation by Campanus which had been published in printed form in Venice in
1482. Pacioli's edition was based on that of
Campanus but it contained much in the way of annotation by Pacioli
himself.

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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