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PACIOLI, LUCA (b. Sansepolcro, Italy, ca. 1445;
d. Sansepolcro, 1517), mathematics, bookkeeping.

is dedicated to Ludovico Sforza. Its subject is the
golden section or divine proportion, as Pacioli called
it, the ratio obtained by dividing a line in extreme and
mean ratio. It contains a summary of Euclid's
propositions (including those in Campanus' version)
relating to the golden section, a study of the properties
of regular polyhedrons, and a description of semi-regular
polyhedrons obtained by truncation or stellation
of regular polyhedrons. Book 2 is a treatise on
architecture, based on Vitruvius, dedicated to Pacioli's
pupils at Sansepolcro. To this he added a treatise on
the right proportions of roman lettering. The third
book is an Italian translation, dedicated to Soderini,
of Piero della Francesca's De corporibus regularibus.

Also in 1509 Pacioli published his Latin translation
of Euclid's Elements. The first printed edition of Euclid
(a Latin translation made in the thirteenth century by
Campanus of Novara from an Arabic text) had
appeared at Venice in 1482. It was severely criticized
by Bartolomeo Zamberti in 1505 when he was
publishing a Latin translation from the Greek.
Pacioli's edition is based on Campanus but contains
his own emendations and annotations. It was published
in order to vindicate Campanus, apparently at
the expense of Ratdolt, the publisher of Campanus'
translation.

Among the works that Pacioli had intended to
publish is “De viribus quantitatis,” a copy of which,
in the hand of an amanuensis, is in the University
Library of Bologna.3 The name of the person to whom
the work was dedicated has been left blank. It is an
extensive work (309 folios) divided into three parts:
the first is a collection of eighty-one mathematical
recreational problems, a collection larger than those
published a century later by Bachet de Méziriac and
others; the second is a collection of geometrical
problems and games; the third is a collection of
proverbs and verses. No originality attaches to this
work, for the problems are found scattered among
earlier arithmetics and, in fact, a collection is attributed
to Alcuin of York. Pacioli himself called the work a
compendium. Some of the problems are found in the
notebooks of Leonardo da Vinci, and the work itself
contains frequent allusions to him.

Pacioli's Italian translation of Euclid's Elements and
his work on chess, “De ludo scachorum,” dedicated to
the marquis of Mantua, Francesco Gonzaga, and his
wife, Isabella d'Este, were not published and there is
no trace of the manuscripts.

Vasari, in writing the biography of Piero della
Francesca, accused Pacioli of having plagiarized the
work of his compatriot on perspective, arithmetic, and
geometry.4 The accusations relate to three works by
Piero--De prospectiva pingendi, “Libellus de quinque
corporibus regularibus,” and Trattato d'abaco, all of
which have been published only since the turn of the
twentieth century.5 In 1908 Pittarelli came to the
defense of Pacioli, pointing out that any accusation of
plagiarism in regard to De prospectiva was unjust,
since Pacioli had acknowledged Piero's work in both
the Summa and the Divina proportione.6 As for the
Libellus, it has been established by Mancini that
Pacioli's work is a translation of it that lacks the
clarity of the original.7 In the case of the Trattato,
although Piero can claim no originality for it, it has
been possible to find in it at least 105 problems of the
Summa.8

The writings of Pacioli have provided historians of
the Renaissance with important source material for
the study of Leonardo da Vinci. The numerous
editions and translations of the De computis et
scripturis are evidence of the worldwide esteem in
which Pacioli is held by the accounting profession.
Pacioli made no original contribution to mathematics;
but his Summa, written in the vulgar tongue, provided
his countrymen, especially those not schooled in Latin,
with an encyclopedia of the existing knowledge of the
subject and enabled them to contribute to the advancement
of algebra in the sixteenth century.

NOTES

1. Rafael Bombelli, Algebra (Bologna, 1572), d 2v.
2. Brown, History of Accounting, p. 119.
3. An ed. of the MS by Paul Lawrence Rose of New York
University is in press.
4. Vasari, Vite, pp. 360, 361, 365.
5. Codex Palat. Parma, published by C. Winterberg (1899);
Codex Vat. Urb. lat. 632, published in 1915 by Mancini;
Codex Ash. 280, published in 1971 by Arrighi.
6. Pittarelli, “Luca Pacioli . . ..”
7. Mancini, “L'opera ‘De corporibus regularibus’ . .
..”
8. Jayawardene, “The Trattato d'abaco of Piero della
Francesca.”

BIBLIOGRAPHY

I. ORIGINAL WORKS.

Pacioli's writings include Summa
de arithmetica, geometria, proportioni et proportionalita
(Venice, 1494; 2nd ed. Toscolano, 1523)--there are
several eds. of the treatise on bookkeeping, “De computis
et scripturis,” contained in the Summa, fols. 197v-210v,
in the original Italian and in trans.; Divina proportione
(Venice, 1509)--there are two extant MSS containing the
“Compendio de divina proportione,” one in the University
of Geneva Library (Codex 250) and the other
in the Biblioteca Ambrosiana, Milan (Codex 170, parte
superiore), the second published as no. XXXI of Fontes
Ambrosiani, Giuseppina Masotti Biggiogero, ed. (Verona,
1956); Euclid megarensis opera . . . a Campano . . . tralata.
Lucas Paciolus emendavit (Venice, 1509); “De viribus