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' (''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 Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accounti ...
and first published in 1494. It contains a comprehensive summary of Renaissance 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 and
accounting Accounting, also known as accountancy, is the measurement, processing, and communication of financial and non financial information about economic entities such as businesses and corporations. Accounting, which has been called the "language ...
, written for use as a textbook and reference work. Written in
vernacular A vernacular or vernacular language is in contrast with a "standard language". It refers to the language or dialect that is spoken by people that are inhabiting a particular country or region. The vernacular is typically the native language, n ...
Italian, the ''Summa'' is the first printed work on algebra, and it contains the first published description of the
double-entry bookkeeping system 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 to ...
. 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".


Contents

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 In trade, barter (derived from ''baretor'') is a system of exchange in which participants in a transaction directly exchange goods or services for other goods or services without using a medium of exchange, such as money. Economists dist ...
,
bills of exchange A negotiable instrument is a document guaranteeing the payment of a specific amount of money, either on demand, or at a set time, whose payer is usually named on the document. More specifically, it is a document contemplated by or consisting of a ...
, weights and measures and
bookkeeping Bookkeeping is the recording of financial transactions, and is part of the process of accounting in business and other organizations. It involves preparing source documents for all transactions, operations, and other events of a business. T ...
, 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 schools of contemporary northern Italy, where the children of merchants and the middle class studied arithmetic on the model established by
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
's ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describe ...
''. The emphasis of this tradition was on facility with computation, using the
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syst ...
, 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 to ...
, 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 busine ...
s and various other tools still employed by professional accountants. The business chapter also introduces the
rule of 72 In finance, the rule of 72, the rule of 70 and the rule of 69.3 are methods for estimating an investment's doubling time. The rule number (e.g., 72) is divided by the interest percentage per period (usually years) to obtain the approximate number ...
for predicting an investment's
future value Future value is the value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return; it is ...
, anticipating the development of the logarithm 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 Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays 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 Giorgio Vasari (, also , ; 30 July 1511 – 27 June 1574) was an Italian Renaissance Master, who worked as a painter, architect, engineer, writer, and historian, who is best known for his work '' The Lives of the Most Excellent Painters, Sculp ...
) to accuse Pacioli of
plagiarism Plagiarism is the fraudulent representation of another person's language, thoughts, ideas, or expressions as one's own original work.From the 1995 '' Random House Compact Unabridged Dictionary'': use or close imitation of the language and though ...
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.


History

''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 A vernacular or vernacular language is in contrast with a "standard language". It refers to the language or dialect that is spoken by people that are inhabiting a particular country or region. The vernacular is typically the native language, n ...
Italian (rather than Latin), reflecting its target audience and its purpose as a teaching text. The work was dedicated to Guidobaldo da Montefeltro, Duke of Urbino, a patron of the arts whom Pacioli had met in Rome some years earlier. It was originally published in Venice in 1494 by Paganino Paganini, 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, 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 in ...
s predating the ''Summa'' (such as the 1458 '' Della mercatura e del mercante perfetto'' by
Benedetto Cotrugli Benedetto Cotrugli ( hr, Benedikt "Beno" Kotruljević; 1416–1469) was a Ragusan merchant, economist, scientist, diplomat and humanist. Life Cotrugli was born in the city of Ragusa (modern Dubrovnik), part of the Republic of Ragusa. As a di ...
), 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 The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, result ...
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 Niccolò is an Italian male given name, derived from the Greek Nikolaos meaning "Victor of people" or "People's champion". There are several male variations of the name: Nicolò, Niccolò, Nicolas, and Nicola. The female equivalent is Nicole. The f ...
.


Commemoration

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 ''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da ...
'', another influential mathematical work by Pacioli *
List of most expensive books and manuscripts This is a list of printed books, manuscripts, letters, music scores, comic books, maps and other documents which have sold for more than US$1 million. The dates of composition of the books range from the 7th-century Quran leaf palimpsest and the ...


References


External links

{{Commonscat
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