' (''Summary of arithmetic, geometry, proportions and proportionality'') is a book on
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
written by Luca Pacioli and first published in 1494. It contains a comprehensive summary of
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
mathematics, including practical arithmetic, basic
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, basic
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and accounting, written for use as a textbook and reference work. Written in vernacular Italian, the ''Summa'' is the first printed work on algebra, and it contains the first published description of the double-entry bookkeeping system. It set a new standard for writing and argumentation about algebra, and its impact upon the subsequent development and standardization of professional accounting methods was so great that Pacioli is sometimes referred to as the "father of accounting".


The ''Summa de arithmetica'' as originally printed consists of ten chapters on a series of mathematical topics, collectively covering essentially all of Renaissance mathematics. The first seven chapters form a summary of arithmetic in 222 pages. The eighth chapter explains contemporary algebra in 78 pages. The ninth chapter discusses various topics relevant to business and trade, including barter, bills of exchange, weights and measures and bookkeeping, in 150 pages. The tenth and final chapter describes practical geometry (including basic
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
) in 151 pages. The book's mathematical content draws heavily on the traditions of the
abacus school Abacus school is a term applied to any Italian school or tutorial after the 13th century, whose commerce-directed curriculum placed special emphasis on mathematics, such as algebra, among other subjects. These schools sprang up after the publicatio ...
s of contemporary northern Italy, where the children of merchants and the middle class studied arithmetic on the model established by Fibonacci's '' Liber Abaci''. The emphasis of this tradition was on facility with computation, using the Hindu–Arabic numeral system, developed through exposure to numerous example problems and case studies drawn principally from business and trade. Pacioli's work likewise teaches through examples, but it also develops arguments for the validity of its solutions through reference to general principles, axioms and logical proof. In this way the ''Summa'' begins to reintegrate the logical methods of classical Greek geometry into the medieval discipline of algebra.

Bookkeeping and finance

Within the chapter on business, a section entitled ' (''Details of calculation and recording'') describes the accounting methods then in use among northern-Italian merchants, including
double-entry bookkeeping Double-entry bookkeeping, also known as double-entry accounting, is a method of bookkeeping that relies on a two-sided accounting entry to maintain financial information. Every entry to an account requires a corresponding and opposite entry t ...
, trial balances,
balance sheet In financial accounting, a balance sheet (also known as statement of financial position or statement of financial condition) is a summary of the financial balances of an individual or organization, whether it be a sole proprietorship, a business ...
s and various other tools still employed by professional accountants. The business chapter also introduces the rule of 72 for predicting an investment's future value, anticipating the development of the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
by more than century. These techniques did not originate with Pacioli, who merely recorded and explained the established best practices of contemporary businesspeople in his region.

Plagiarism controversy

Pacioli explicitly states in the ''Summa'' that he contributed no original mathematical content to the work, but he also does not specifically attribute any of the material to other sources. Subsequent scholarship has found that much of the work's coverage of geometry is taken almost exactly from Piero della Francesca’s '' Trattato d’abaco'', one of the algebra sections is based on the '' Trattato di Fioretti'' of Antonio de Mazzinghi, and a portion of the business chapter is copied from a manuscript by Giorgio Chiarini. This sort of appropriation has led some historians (notably including sixteenth-century biographer Giorgio Vasari) to accuse Pacioli of plagiarism in the ''Summa'' (and other works). Many of the problems and techniques included in the book are quite directly taken from these earlier works, but the ''Summa'' generally adds original logical arguments to justify the validity of the methods.


''Summa de arithmetica'' was composed over a period of decades through Pacioli's work as a professor of mathematics, and was probably intended as a textbook and reference work for students of mathematics and business, especially among the mercantile middle class of northern Italy. It was written in vernacular Italian (rather than Latin), reflecting its target audience and its purpose as a teaching text. The work was dedicated to
Guidobaldo da Montefeltro Guidobaldo (Guido Ubaldo) da Montefeltro (25 January 1472 – 10 April 1508), also known as Guidobaldo I, was an Italian condottiero and the Duke of Urbino from 1482 to 1508. Biography Born in Gubbio, he succeeded his father Federico da Montefe ...
, Duke of Urbino, a patron of the arts whom Pacioli had met in Rome some years earlier. It was originally published in
Venice Venice ( ; it, Venezia ; vec, Venesia or ) is a city in northeastern Italy and the capital of the Veneto Regions of Italy, region. It is built on a group of 118 small islands that are separated by canals and linked by over 400  ...
in 1494 by
Paganino Paganini Paganino Paganini (; Latin: ''Paganinus de Paganinis''; –1538), was an Italian printer and publisher from the Republic of Venice during the Renaissance. He was the original publisher of Luca Pacioli's mathematical works, ''Summa de arithmetica ...
, with an identical second edition printed in 1523 in Toscolano. About a thousand copies were originally printed, of which roughly 120 are still extant. In June 2019 an intact first edition sold at auction for .

Impact and legacy

While the ''Summa'' contained little or no original mathematical work by Pacioli, it was the most comprehensive mathematical text ever published at the time. Its thoroughness and clarity (and the lack of any other similar work available in print) generated strong and steady sales to the European merchants who were the text's intended audience. The reputation the ''Summa'' earned Pacioli as a mathematician and intellectual inspired Ludovico Sforza, Duke of
Milan Milan ( , , Lombard: ; it, Milano ) is a city in northern Italy, capital of Lombardy, and the second-most populous city proper in Italy after Rome. The city proper has a population of about 1.4 million, while its metropolitan city ...
, to invite him to serve as a mathematical lecturer in the ducal court, where Pacioli befriended and collaborated with
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested on ...
. The ''Summa'' represents the first published description of many accounting techniques, including double-entry bookkeeping. Some of the same methods were described in other
manuscript A manuscript (abbreviated MS for singular and MSS for plural) was, traditionally, any document written by hand – or, once practical typewriters became available, typewritten – as opposed to mechanically printed or reproduced i ...
s predating the ''Summa'' (such as the 1458 '' Della mercatura e del mercante perfetto'' by Benedetto Cotrugli), but none was published before Pacioli's work, and none achieved the same wide influence. The work's role in standardizing and disseminating professional bookkeeping methods has earned Pacioli a reputation as the "father of accounting". The book also marks the beginning of a movement in sixteenth-century algebra toward the use of logical argumentation and theorems in the study of algebra, following the model of classical Greek geometry established by
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
. It is thought to be the first printed work on algebra, and it includes the first printed example of a set of plus and minus signs that were to become standard in Italian Renaissance mathematics: 'p' with a tilde above (p̄) for "plus" and 'm' with a tilde (m̄) for minus. Pacioli's (incorrect) assertion in the ''Summa'' that there was no general solution to cubic equations helped to popularize the problem among contemporary mathematicians, contributing to its subsequent solution by Niccolò Tartaglia.


In 1994 Italy issued a 750- lira
postage stamp A postage stamp is a small piece of paper issued by a post office, postal administration, or other authorized vendors to customers who pay postage (the cost involved in moving, insuring, or registering mail), who then affix the stamp to the f ...
honoring the 500th anniversary of the ''Summa''s publication, depicting Pacioli surrounded by mathematical and geometric implements. The image on the stamp was inspired by the '' Portrait of Luca Pacioli'' and contains many of the same elements.

See also

* '' De divina proportione'', another influential mathematical work by Pacioli * List of most expensive books and manuscripts


External links

Full text
(1523 edition) on Google Books
Digitised edition of ''Particularis de computis et scripturis''
(First (1494) edition)
English translation of ''Particularis de computis et scripturis''
(1994) Mathematics textbooks 1494 books Medieval literature History of mathematics History of accounting History of business Accounting books