Su[m]ma de arithmetica geometria proportioni et proportionalità

[Colophon:] Venice: Paganinus de Paganinis, 1494. First edition. THE MOST INFLUENTIAL WORK IN THE HISTORY OF CAPITALISM.

First edition, first issue (see below), very rare, this is a remarkable copy in entirely original condition and with a distinguished provenance, having been in the famous Giustiniani family from publication to the present day. The Summa is a work of enormous importan ce on several levels. It is the first mathematical encyclopaedia of the Renaissance, 'the first great general work on mathematics printed' (Smith, Rara arithmetica, p. 56), and the first printing of any of the works of the great thirteenth-century mathematician Leonardo of Pisa, called Fibonacci (c. 1175-c. 1250), and of Pacioli's friend, the brilliant mathematician and artist Piero della Francesca (1416-92). The first part of the Summa is the first printed comprehensive treatment of algebra and arithmetic, based largely on Fibonacci's 1202 Liber Abaci which famously introduced Arabic numbers to the West, and which was itself in part a translation of the treatises on algebra and arithmetic of the Persian mathematician and astronomer Muhammad ibn Mūsā al-Khwārizmī (c. 780-c. 850) (the word algorithm derives from his name). The second part, on geometry, is based on Fibonacci's Practica Geometriae, but includes at the end a section on stereometric geometry and regular solids taken from the Trattato d'abaco of Piero della Francesca. The first part of the Summa also contains sections illustrating the applications of arithmetic and algebra to problems in business, notably including Pacioli's original treatise Particularis de Computis et Scripturis ('Details of Accounting and Recording') (ff. 197v-210v). This is the first printed text to set out the method of double-entry bookkeeping, the single most influential work in European accounting history, which earned Pacioli the title 'Father of Accounting'; it has been called "the most influential work in the history of capitalism". De Computis introduces the 'rule of 72' for predicting an investment's future value, anticipating the development of the logarithm by more than a century. The business section of the Summa also contains the earliest discussion of mathematical probability in print (this was not in Liber abaci). "The oldest known printed source for the treatment of the problem of points is Luca Pacioli's Summa" (Schneider, p. 230) - modern probability theory is generally regarded as having begun with the exchange of letters discussing the 'problem of points' between Fermat and Pascal in the mid-1650s. In its iconic full-page woodcut of finger counting (f. 36v), from which our modern 'digital computing' took its name, the Summa contains the earliest printed representation of computation. Sangster et al. argue that the Summa was, in fact, mainly intended as a reference text for merchants - it synthesised the three major mathematical traditions - medieval European, Arab and ancient Greek - but its use of the vernacular opened it up to businessmen, students, artists, technicians and scholars alike. The Summa is also a work central to the development of Leonardo da Vinci (1452-1519). Pacioli came to Milan where he held the chair of mathematics from 1496 to 1499, during which years he lodged with Leonardo, and taught him mathematics. Leonardo owned a copy of the Summa (he paid 119 soldi for it ca. 1494, as noted in the Codex Atlanticus f. 288 recto) and refers to it in his notebooks (see below for more on Leonardo and Pacioli). Only three other copies of the first edition of the Summa are recorded at auction in the last 50 years, of which only one was in its original binding, a copy of the second issue sold at Christie's New York in 2019 for $1,215,000.

Provenance: contemporary inscription on front free endpaper recto: 'Di Giacomo Giust.[inia]no fu di m[esser] Lor.[enzo] Genovese e de soi amici'; Giacomo records the gifting of the book from his father Lorenzo Giustiniani, a member of the Genoan branch of the Giustiniani family; on verso: 'sexqui altera sexqui tertia sexqui quinta // questo sexqui, via, fia in c.[arte] 27 // algebra, algebratica vuol dire speculativa'; i.e. referring to passages concerning the superparticular ratios sesquialtera (3:2), sesquitertia (4:3) and sesquiquinta (6:5). Coincidentally, the ancestor of Lorenzo, his namesake Saint Lorenzo Giustiniani (1381-1456), is depicted in a funerary statue in the St Pietro di Castello church in Venice, holding a book with a very similar binding (thanks to Nicholas Pickwoad for pointing this out).

The Summa, the writing of which had been completed by 1487, is divided into two volumes, the first dealing with arithmetic and algebra, the second with geometry. The first volume is divided into nine chapters (distinctiones): chapters 1 to 7 on arithmetic (222 pages), chapter 8 on algebra (78 pages), and chapter 9 on business (150 pages). The second volume comprises chapters 1-8 (151 pages), on geometry; it has separate signatures and foliation and a caption title. There is a brief colophon at the end of part 1 referring to the full colophon at the end of part 2.

"In the dedication to duke Guidobaldo of Urbino [1472-1508], Pacioli gives a list of his sources. For the first part these include first and foremost Fibonacci's Liber Abaci, followed by Jordanus [fl. 13th century], Blasius of Parma [1355-1416] and Prosdocimo de' Beldomandi [1375-1428], among others ...The geometrical part is taken from Fibonacci's Practica geometriae and Archimedes ... In making this vast compilation, Pacioli availed himself of many libraries including the ducal collection at Urbino. There he consulted al-Khwārizmī's Algebra [which was translated into Italian at least by 1464, the date of a manuscript copy in the Plimpton Collection] ..., as well as Piero della Francesca's Libellus de Quinque Corporibus Regularibus. At the Venetian monastery of S. Antonio, he annotated a manuscript of Fibonacci's Liber Abaci later seen by Cardano, and at Florence he used a text of Witelo's Perspectiva [composed c. 1274]" (Rose, pp. 143-144).

Pacioli particularly emphasizes his indebtedness to Fibonacci. Fibonacci was born around 1175, the son of a Pisan merchant, and in his early life travelled widely around Italy to Genoa and Venice, but also to Barbary, Egypt, Syria, Greece and France. On these trips he learned the Arabic ways of arithmetic and computation. From 1200 he settled back in Pisa and began his mathematical writings; the first and most important work was the Liber Abaci (The Book of Calculation) written in 1202. No copy of this first version has survived, but around 1228 a second version of the text was completed, with additional chapters. "Virtually all subsequent Renaissance algebraists cite Fibonacci and Pacioli in the same breath and indeed it was difficult to cite Fibonacci independently since his works were not published until the nineteenth century" (Rose, p. 145). The first volume of the Summa follows the Liber abaci closely.

"Liber abaci is an encyclopedic work treating much of the known mathematics of the thirteenth century on arithmetic, algebra, and problem solving. It is, moreover, a theoretical as well as practical work; the methods employed in Liber abaci Leonardo firmly establishes with Euclidean geometric proofs ... General methods are established by using the geometric algebra found principally in Book II of the Elements. Leonardo turns to Book X for a foundation of a theory of quadratic irrational numbers. Throughout Liber abaci proofs are given for old methods, methods acquired from the Arabic world, and for methods that are Leonardo's original contributions. Leonardo also includes those commonplace non-algebraic methods established in the mediaeval world for problem solving, at the same time giving them mathematical legitimacy with his proofs. Among others they include checking operations by casting out nines, various rules of proportion, and methods called single and double false position" (Sigler, pp. 4-5).

"It was Leonardo's purpose to replace Roman numerals with the Hindu numerals not only among scientists, but in commerce and among the common people" (ibid.). "When Fibonacci's Liber abaci first appeared, Hindu-Arabic numerals were known to only a few European intellectuals through translations of the writings of the 9th-century Arab mathematician al-Khwārizmī. The first seven chapters dealt with the notation, explaining the principle of place value, by which the position of a figure determines whether it is a unit, 10, 100, and so forth, and demonstrating the use of the numerals in arithmetical operations" (Britannica).

In this part Pacioli gives 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 for 'plus' and 'm' with a tilde for 'minus'.

"In addition to teaching all of the necessary methods of arithmetic and algebra, Leonardo includes in Liber abaci a wealth of applications of mathematics to all kinds of situations in business and trade, conversion of units of money, weight, and content, methods of barter, business partnerships and allocation of profit, alloying of money, investment of money, simple and compound interest" (Sigler, p. 5). This material comprises chapter 9 of the Summa, which is divided into 12 sections (Tractati), the first ten on various items relevant to business (including barter and bills of exchange), the eleventh on bookkeeping (27 pages), and the twelfth on weights and measures and exchange rates. Pacioli interestingly observes that the term for the modern mathematics of merchants, 'abbaco', was likely derived from the phrase 'in modo Arabico' ('In the Arab manner') (see f. 19r), not from the abacus counting device (see Sangster et al., p. 116).

Within these chapters on business, the eleventh section entitled Particularis de computis et scripturis describes the accounting methods then in use among northern-Italian merchants, including double-entry bookkeeping, trial balances, balance sheets and various other tools still employed by professional accountants. "Five hundred years ago, in November 1494, one of the world's earliest printed texts included a section on accounting ... Five hundred years later, the ideas printed in that accounting manual continue to provide the guidelines for recording economic activity in all the world's great financial centers. The author of that first accounting manual is Luca Pacioli. The Summa de Arithmetica, Geometria, Proportioni at Proportionalita - which includes this first published discourse on accounting - is a work of genius" (Cripps, p. ix). Richard Brown has said [in his History of Accounting and Accountants (1905), p. 119] that "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."

"Pacioli defines double-entry bookkeeping broadly, as 'nothing else than the expression in writing of the arrangement of [a merchant's] affairs.' If a merchant follows the system Pacioli sets out, then he will always know 'all about his business and will know exactly whether his business goes well or not' ... Pacioli's formulation of Venetian double-entry bookkeeping is one of the great advances in the history of business and commerce. He recommends this method, which had been practised in Venice for two hundred years, as the best ... Rather than mingling debt and credit entries under each other down a single column or page - as did the Florentine merchants before they began keeping their books alla viniziana - Venetian ledgers separate debits and credits, dividing them into two columns, which is exactly how we organise our ledgers today ...

"In Pacioli's view, three things are needed by 'anyone who wishes to carry on business carefully. The most important of these is cash or any equivalent' ... The second thing necessary in business 'is to be a good bookkeeper and ready mathematician.' The third 'and last thing is to arrange all the transactions in such a systematic way that one may understand each one of them at a glance, i.e., by the debit and credit method.' Not much has changed today ... Pacioli does not go into detail, as he makes clear in his introduction: 'Although one cannot write out every essential detail for all cases, nevertheless a careful mind will be able, from what is given, to make the application to any particular case.' Nor does he give sample pages of worked examples, as writers on bookkeeping would begin to do in the next century. Instead, he assumes his readers are merchants and their sons, with some working knowledge of bookkeeping ...

"The first thing a merchant must do, says Pacioli, is to make an inventory of everything he owns ... Once the inventory is made, a merchant needs three books in which to record his business transactions. The first is the memoriale, or memorandum, which acted like a diary and was a temporary record of the merchant's transactions ... The second book required is the giornale, or journal. After entering his inventory into the journal, the merchant uses this book to write up in a neat and orderly fashion the details of each transaction that has been recorded in the memorandum. In Pacioli's Venetian system, every item entered into the journal must be preceded by one of two key words: per (which means 'from' and indicated that the ledger account must be debited) and a (which means 'to' and indicated that the ledger must be credited) ... The third book is the quaderno, or ledger. The ledger is made up of pages ruled in two columns (the simplest form of which is now known as a T-column) and it records - twice - every journal entry. For every entry made in the journal there will be two in the ledger: one, a debit, entered on the left side of the T-column; and the other, a credit, on the right. The ledger with its two columns marked a great advance in account-keeping. By using this system the merchant could at any moment see at a glance the precise state of his assets and his debts. It also allowed him to find mistakes in his bookkeeping relatively easily, because if his books did not balance - his debits were not equal to his credits - he had made a mistake somewhere and would have to scrutinize his books to find it. The ledger with its 'double-entries' was an innovation made by the merchants of Venice and is the reason that Venetian bookkeeping is now known as double entry" (Gleeson-White, pp. 92-100).

The section of the Summa on business contains what is almost certainly the first example of a logarithm in print. On f. 181r, Pacioli claims that the number of years necessary to double a capital placed at compound interest, is the number resulting from the division of the fixed number 72 by the rate of interest as a percentage. This was the 'rule of 72', which succeeding mathematicians including Tartaglia failed to explain. In modern notation, we want the number n such that

(1 + r/100)n = 2,

where r is the rate of interest. Our solution is n = log 2/log(1 + r/100), which if r is small is approximately (100 log 2)/r. Thus, Pacioli was giving the approximate value 0.72 of the natural logarithm of 2, more than a century before logarithms were invented, and with an error of less than 3%.

The Summa can claim to be the earliest printed book on probability, thanks to its discussion of the famous 'problem of points', or division of stakes. On ff. 197r and 198v, Pacioli states two problems. The first of these is as follows (the other is similar):

'A company [two players] plays a ball game to 60 and each goal is 10. They stake 10 ducats in all. It happens by certain incidents that they are not able to finish; and one party has 50 and the other 20. One asks what portion of the stake is due to each party.'

Pacioli gives three solutions, each of which amounts to dividing the stakes in proportion to the number of points already won. Schneider notes that although Pacioli's solution is incorrect, he had contributed to the problem of points in two significant ways. First, by simply offering a solution, Pacioli had put forward a problem that would later be brought to Cardano's attention. Secondly, Pacioli advocated the view that a definite and unique solution exists for the problem of points. In Cardano's Practica arithmetice (1539), he questions Pacioli's solution, noting that it does not take into account the number of games yet to be won by the players. In section 20, titled 'Error di Fra Luca dal Borgo,' of his General trattato di numeri (1556-60), Tartaglia notes that 'His rule seems neither agreeable nor good, since, if one player has, by chance, ten points and the other no points, then following this rule, the player who has the ten points would take all the stakes which obviously does not make sense.' These criticisms were well founded, and although Cardano and Tartaglia proposed their own solutions, it was Pascal and Fermat who, in their correspondence beginning in 1654, first correctly solved the problem of points, using a combinatorial technique that was not available to earlier authors.

The problem of points is clearly related to the division of shares in a trading company, which perhaps explains why Pacioli included it in the Summa. "Games, and especially games of chance, became a favourite pastime of the sixteenth-century merchant. Betting and economic enterprises became intertwined in a way that can explain why gaming was understood as a process which recapitulated the activities of merchant adventurers in a condensed time span. This very close connection between gaming and the economy in the sixteenth century explains very nicely why the economy was the source of inspiration for the solution of the problem of points" (Schneider, p. 220). Pacioli actually wrote a book devoted to games of chance, De ludis, but this has not survived.

The second volume of the Summa is largely, though not entirely, a version of Fibonacci's Practica geometriae, composed in 1220 or 1221. 'Practical Geometry' is the name of the craft of medieval land-measurers, known as agrimensores in Roman times, and as surveyors in modern times. Fibonacci wrote Practica geometriae for these artisans, a fitting complement to Liber abaci. The first chapter of the second volume of the Summa contains a summary of the books of Euclid's Elements on fundamental geometric constructions, calculations of areas, and similarity theory.
The second is concerned with special lines in a triangle.
The third treats right triangles and the associated solution of quadratic equations (theorem of Pythagoras). In contrast to the writings of al-KhwārizmÄ«, here the solution of quadratic equations is presented theoretically, not by means of examples. In addition, Pacioli dealt with equations of the third and fourth degree, which he held to be as unsolvable (impossibile) as the quadrature of the circle, an assertion that not long afterwards was refuted by Scipione del Ferro (1515) (Rose points out that Pacioli may have stimulated Scipio's discovery - he was one of Pacioli's colleagues when Pacioli was lecturing Bologna in 1501-1502). The fourth chapter concerns the theory of the circle. Tables of chords give information on the lengths of chords and their associated arcs. For π, Pacioli gives the approximation 3 33/229. In the fifth chapter, the division of geometric figures is discussed (theory of ratios). The sixth chapter explains how to calculate the surface area and volume of geometric solids. The seventh chapter introduces apparatus and methods of measurement. The eighth chapter contains a number of applications of different types: calculation of the volume of a barrel (approximately described as two frusta of cones), calculations on regular solids, and the inscribing of several equal circles of maximal size in a triangle and in a circle. Finally, Pacioli provides an overview of the coinage and weights and measures of the various Italian city-states.

In both the algebraic and geometric volumes of the Summa, Pacioli includes material from the Trattato d'abaco of Piero della Francesca (which was not published until the 20th century). The Trattato, composed between 1460 and 1470, building on the work of Fibonacci, "belongs to the so-called abacus books, which, despite their name, presented not the abacus but mathematics on an elementary level for future merchants, bank clerks, artisans, and artists. The books dealt primarily with arithmetic, but often included some algebra and practical geometry as well - which is also true of Piero's Trattato. In addition, Piero treated some more advanced geometrical objects, such as the regular polyhedra. He returned to these solids in the last of his books Libellus de quinque corporibus regularibus, in which he also included five semi-regular, also called Archimedean, polyhedral [convex polyhedra that have regular polygons of more than one kind as faces] ... In Trattato [Piero] described two of these, one - later called a cuboctahedron - having eight equilateral triangles and six squares as faces, obtained by cutting off the corners of a cube through the midpoints of the edges. The second was a truncated tetrahedron that Piero constructed by cutting off the vertices of a tetrahedron through points situated one third of the edge length from the corners. In Libellus he came back to the truncated tetrahedron and then added the truncated versions of the remaining four regular polyhedra. Pacioli later incorporated the Libellus into his Divina proportione (1509). As to the rest of the contents of Trattato and Libellus, Piero benefited a great deal from earlier treatises - the material for which mainly dates back to Euclid, al-Khwārizmī, and Leonardo da Pisa" (DSB, under Piero della Francesca). Pacioli included more than a hundred problems of algebra and arithmetic from the Trattato in the first volume of the Summa; much of the geometrical content of the Trattato appears at the end of the second volume, including new theorems of Euclidean geometry, stereometric problems (e.g., the formula for the volume of a tetrahedron in terms of the lengths of its sides), as well as the material on polyhedra.

"The Summa was widely circulated and studied by the mathematicians of the sixteenth century. Cardano, while devoting a chapter of his Practica arithmetice to correcting the errors in the Summa, acknowledged his debt to Pacioli. Tartaglia's General trattato de' numeri et misure was styled on Pacioli's Summa. In the introduction to his Algebra [1570], Bombelli says that Pacioli was the first mathematician after Leonardo Fibonacci to have thrown light on the science of algebra" (DSB, under Pacioli).

Born 1446-1448 in Borgo Sansepolcro, Pacioli was first schooled in Venice in the abbaco system, an applied, commercial education focusing on mercantile mathematics. He later studied mathematics under the great Florentine polymath mathematician and architect Leon Battista Alberti. Pacioli became a Franciscan friar in the early 1470s and shortly thereafter was named Perugia's first public lecturer in abbaco arithmetic. Pacioli soon rose to teach mathematics at the university level and so was highly unusual in mastering both practical and theoretical mathematics. Prior to its publication in 1494, Pacioli had been working on the Summa for a period of thirty years. Pacioli regretted the low ebb to which teaching had fallen and he thought that the fault lay in the use of improper methods and in the scarcity of available subject matter. He sought to correct these faults in the Summa.

Pacioli, who had studied chiefly in Venice, published his Summa there in 1494, two years before his extensive travels as a teacher brought him to Milan, summoned by the Duke Lodovico Sforza. There he held the chair of mathematics from 1497-99. In Milan he met Leonardo da Vinci, also in the service of Duke Lodovico Sforza 'il Moro'. They became friends and immersed themselves in common intellectual pursuits: 'the remarkable convergence of their thought after 1496 was such that many of Luca's published opinions would sit easily in Leonardo's notebooks' (M. Kemp, Leonardo da Vinci: the marvellous works of nature and of man, p 148). A note in the Codex Atlanticus reads: "Learn the multiplication of roots from Maestro Luca" (f. 331r), and both the Madrid and Forster codices contain notes on the Somma - in particular, sections on proportions and proportionality, including a mirror-version of the arbor proportionis et proportionalitatis on f. 82r of Pacioli's work. Pacioli familiarized Leonardo with Euclidean mathematics and "Luca's presence in Milan served to encourage Leonardo to pursue far more explicit and fundamental investigations into the mathematical order which had formed the implicit basis of much of his earlier art and science" (ibid). On the 8th of February 1498 Pacioli argued in a court disputation for the status of painting as a liberal art involving mathematical knowledge.

Unemployed as a result of the fall of the Sforza regime in 1499, Pacioli and Leonardo resumed their peripatetic careers, ending up together in Florence where they found protection under its republican ruler Piero Soderini. There Pacioli embarked upon compiling his De divina proportione, illustrated by Leonardo, and a work best understood as the product of their collaboration and intellectual friendship. Pacioli's acknowledgement of Leonardo in the introduction is fulsome, referring as he does to the supreme beauty of his 'figures of the platonic and mathematical regular solids ... made and formed by that ineffable left hand accommodated to all the disciplines', comparing Leonardo to the greatest of the ancient artists, and lauding the happy time they spent together at the Sforza court in Milan. He also refers to Leonardo's never-to-be-completed treatises on painting, on local motion, on weight and on percussion, and praises his recently painted Last Supper in the refectory of Sta. Maria delle Grazie in Milan and the colossal clay model of the unexecuted equestrian statue of Lodovico il Moro, the bronze for the casting of which was used by the Duke to make cannon in his last-ditch (and unsuccessful) attempt to defend his city against the invading armies of Louis XII of France.

One large repeated initial depicts the author standing over an illustrated mathematical text, presumably the Summa, with a pair of dividers in his hand. In the portrait of Pacioli by Jacopo de' Barbari (1495; also attributed to Jacometto Veneziano) in the Capodimonte, Naples, Pacioli has his left hand on an open copy of Euclid, as "he demonstrates the eighth proposition from book XIII of the Elements of Euclid to a disciple, dressed according to the aristocratic fashion of the time, and identified as the duke Guidubaldo da Montefeltro, who instead turned his eyes towards the viewer. The friar had dedicated to the young duke the Summa de Arithmetica, geometry, proportions and proportionality, printed in Venice in 1494 and depicted in the painting right in front of the gentleman, with the inscription Li[ber] R[egularum] Luc[ae] Bur[gensis]" (Il ritratto di Luca Pacioli, Capodimonte website). Placed on top of the Summa's red leather binding is a pentagonal dodecahedron, while a large crystal rhombicuboctahedron, half filled with water, is suspended beside him; this polyhedron has been attributed to the hand of Leonardo by some scholars. Pacioli also appears in the Brera Madonna by Piero della Francesca, who has used his features in the depiction of Peter Martyr.

This copy is the first issue; for reasons still unexplained the work was reissued twice (the second reissue can be dated to 1507-1509); for example, the copy in the British Library has some leaves in Boncompagni's setting B (IGI 7134). For details see Clarke p 91. Of the first issue, Volume 2 was completed on November 10, 1494, and the introduction to Volume 1 was completed on November 20, 1494. Pacioli tells us that he was present during 1493 and 1494 to oversee the printing, but it is clear that this did not include proof-reading as there are numerous typographical errors. The Venice printer, Paganino de' Paganini (c. 1450-1538), ran a fairly small print shop at the time the Summa was published, although his business grew in the early 16th century when he was joined by his son Alessandro. Paganini also published Pacioli's Divina proportione and his Italian translation of Euclid's Elements, both in 1509.

The Summa was an extremely large book for the period, and sold for 119 soldi, about one week's wages for a university teacher at the time. Sangster et al. estimate that between 1000 and 2000 copies of the Summa were printed, of which about 160 have survived. Olschki (Geschichte der Neusprachlichen Wissenschaftlichen Literatur, 1918) wrote that, for fifty years after its publication, the Summa was the most widely read mathematics work in Italy, and Favier [Gold and Spices: The Rise of Commerce in the Middle Ages, 1998], then president of the French Bibliothèque Nationale and author of many books on the Middle Ages, averred that the Summa was "an instant success and [was] for many years used by the business world" and that "merchants from every country rushed to buy this guide to accountancy" (Sangster et al., p. 142).

ISTC il00315000 (all three issues); IGI 7132; GM 18913; BMC V pp. 457-8; Essling 779; Goff L315; Goldsmiths 5; HC 4105; ICA p. 1; Klebs 718.1; Riccardi ii, 226; Smith, RaraArithmetica p. 54; Essling 779; Mortimer, Italian 16th Century Books II 2358 (second issue); Sander 5367; Stillwell 203. Jeremy Cripps, Particularis de Computis et Scripturis ... A Contemporary Interpretation (Seattle: Pacioli Society, 1994); Alan Sangster, et al., 'The Market for Luca Pacioli's Summa arithmetica', Accounting Historians Journal, vol. 35, 2008, pp. 111-134; P. L. Rose, The Italian Renaissance of Mathematics (Geneva: Librairie Droz, 1975), pp. 143-50; Derek Ashdown Clarke, 'The First Edition of Pacioli's "Summa de Arithmetica" (Venice, Paganinus de Paganinis, 1494)', Gutenberg Jahrbuch (1974), pp. 90-92; J. Gleeson-White, Double Entry: How the Merchants of Venice Created Modern Finance (New York: W. W. Norton, 2012); Jayawardene, 'The 'Trattato d'abaco' of Piero della Francesca', in C. Clough, ed., Cultural aspects of the Italian Renaissance, essays in honour of P.O. Kristeller (Manchester, 1979); Ivo Schneider, 'The market place and games of chance in the fifteenth and sixteenth centuries,' in Mathematics from Manuscript to Print, ed. Cynthia Hay (New York: Oxford University Press, 1988); L. E. Sigler, Fibonacci's Liber Abaci (New York: Springer, 2002); A. Ciocci, Luca Pacioli tra Piero della Francesca e Leonardo, (Sansepolcro: Aboca Museum, 2009); A. Ciocci, Luca Pacioli e la mathematizzazione del sapere nel Rinascimento (Bari: Cacucci, 2003); E. Giusti and C. Maccagni, eds, Luca Pacioli e la mathematica del Rinascimento (Florence: Giunti, 1994).

Two vols. in one, folio (319 x 217 mm), ff. [8], 224; 76, with large white-on-black woodcut initials (including a repeated depiction of Pacioli with a pair of dividers and a copy of the Summa before him), full-page woodcut 'tree of proportion' printed in red and black, full-page woodcut showing finger symbolism for numbering, numerous woodcut mathematical and geometrical diagrams, and illustrations showing instruments and methods of measuring, in margins; a few minute wormholes in a few gatherings, not affecting text, a fine, large copy, with numerous deckle edges, in its original Italian [Venetian?] binding of quarter leather with blind-ruled panels on sides, over bevelled wooden boards, with four embossed brass catchplates and remains of clasps.