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PACIOLI, LUCA (b. Sansepolcro, Italy, ca. 1445;
d. Sansepolcro, 1517), mathematics, bookkeeping.
In 1500 Pacioli was appointed to teach Euclid's
Elements at the University of Pisa, which had been
transferred to Florence because of the revolt of Pisa
in 1494. The appointment was renewed annually until
1506. In 1504 he made a set of geometrical figures for
the Signoria of Florence, for which he was paid
52.9 lire. He was elected superior of his order for the
province of Romagna and shortly afterward (1505)
was accepted as a member of the monastery of Santa
Croce in Florence. During his stay in Florence, Pacioli
also held an appointment at the University of Bologna
as lector ad mathematicam (1501-1502). At this time
the University of Bologna had several lectores ad
arithmeticam, one of whom was Scipione dal Ferro,
who was to become famous for solving the cubic
equation. It has been suggested that Pacioli's presence
in Bologna may have encouraged Scipione to seek a
solution of the cubic equation, but there is no evidence
to support this apart from Pacioli's statement in the
Summa that the cubic equation could not be solved
algebraically.
Since his arrival in Florence, Pacioli had been
preparing a Latin edition of Euclid's Elements and an
Italian translation. He had also written a book on
chess and had prepared a collection of recreational
problems. On 11 August 1508 Pacioli was in Venice,
where he read to a large gathering in the Church of
San Bartolomeo in the Rialto an introduction to
book V of Euclid's Elements. A few months later, on a
supplication made by him to the doge of Venice, he
was granted the privilege that no one but he could
publish his works within the republic for fifteen years.
The works listed were the fifteen books of Euclid,
Divina proportione, “De viribus quantitatis,” “De
ludo
scachorum,” and Summa de arithmetica. The Latin
edition of Euclid and the Divina proportione were
published in 1509. Pacioli was called once more to
lecture in Perugia in 1510 and in Rome in 1514.
On several occasions Pacioli came into conflict with
the brethren of his order in Sansepolcro. In 1491, on a
complaint made to the general of the order, he was
prohibited from teaching the young men of the town;
but this did not prevent his being called to preach the
Lenten sermons there in 1493. It is likely that certain
minor privileges granted to him by the Pope had
aroused enmity or jealousy. Although a petition had
been sent to the general of the order in 1509, he was
shortly afterward elected commissioner of his convent
in Sansepolcro. A few years later Pacioli renounced
these privileges and in 1517, shortly before his death,
his fellow townsmen petitioned that he be appointed
minister of the order for the province of Assisi.
The commercial activity of Italy in the late Middle
Ages had led to the composition of a large number of
treatises on practical arithmetic to meet the needs of
merchant apprentices. Evidence of this is found in the
extant works of the maestri d'abbaco of central and
northern Italy. Some of them even contained chapters
devoted to the rules of algebra and their application,
no doubt influenced by the Liber abbaci of Leonardo
Fibonacci. The first printed commercial arithmetic was
an anonymous work that appeared at Treviso in 1478.
By the end of the sixteenth century about 200 such
works had been published in Italy. Pacioli wrote three
such treatises: one at Venice (1470), one at Perugia
(1478), and one at Zara (1481). None of them was
published and only the second has been preserved.
Pacioli's Summa de arithmetica . . . (1494) was more
comprehensive. Unlike the practical arithmetics, it was
not addressed to a particular section of the community.
An encyclopedic 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, Leonardo Fibonacci,
Prosdocimo de' Beldamandi, and others.
Although it lacked originality, the Summa was
widely circulated and studied by the mathematicians
of the sixteenth century. Cardano, while devoting a
chapter of his Practica arithmetice (1539) to correcting
the errors in the Summa, acknowledged his debt to
Pacioli. Tartaglia's General trattato de' numeri et
misure (1556-1560) was styled on Pacioli's Summa. In
the introduction to his Algebra, Bombelli says that
Pacioli was the first mathematician after Leonardo
Fibonacci to have thrown light on the science of
algebra--“primo fu che luce diede a quella
scientia.”1
This statement, however, does not mean that algebra
had been neglected in Italy for 300 years. Another
edition of Pacioli's Summa was published in 1523.
Pacioli's treatise on bookkeeping, “De computis et
scripturis,” contained in the Summa, was the first
printed work setting out the “method of Venice,” that
is, double-entry bookkeeping. Brown has said, “The
history of bookkeeping during the next century
consists of little else than registering the progress of
the De computis through the various countries of
Europe.”2
The Divina proportione, written in Italian and
published in 1509, was dedicated to Piero Soderini,
perpetual gonfalonier of Florence. It comprised
three books: “Compendio de divina proportione,”
“Tractato de l'architectura,” and “Libellus in tres
partiales tractatus divisus quinque corporum regularium.”
The first book, completed at Milan in 1497,
