Balancing the Books in Rare Books

by Julia Blakely

As the old Sam Cooke song goes:

Don’t know much trigonometry
Don’t know much about algebra,
Don’t know what a slide rule is for
But I do know that one and one is two,

And I do know something about early printing and history. And books of firsts are always fun. So there is much to love about Luca Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalita (Summary of arithmetic, geometry, proportions and proportionality), printed in Venice in 1494.

This title is one of the more famous works in the history of mathematics. It is the first printed book on algebra written not in scholarly Latin but in a vernacular language (Italian, except for the title and headings). It draws on the writings of the great Renaissance painter, Piero della Francesca, as well as Euclid’s geometry. A compendium of collected knowledge, the Summa nevertheless presents the revolutionary concept of the mathematician providing proportion and scale to art.

Fueling the cultural rebirth of art in the fifteenth century was a renaissance in business and economics. Although Pacioli did not invent the system, his Summa has the first description on double-entry bookkeeping (in folios 198-210), a short but tremendously influential section. This is the accounting equation of assets equals liabilities plus owner’s (or stockholder’s) equity. Written in the common tongue, the mechanics of transactions in the marketplace were made accessible to the mercantile class in this work. The close connection between gaming and economics is also pointed out in the pages of the Summa. The author mentions the financial “Rule of 72,” a method for estimating an investment’s doubling time. Included are tables of currencies and the weights and measures used in the various Italian states. The volume is very much about business, of great use to the Venetian merchants. Pacioli declared “a merchant must be a good bookkeeper.”

Luca Pacioli (1445-1517) was a tutor, Franciscan friar, and, later, a university lecturer. His Summa de arithmetica was a summary of mathematical studies of the time, a textbook that is still admired for its approach to teaching. It is an example of one of the Dibner Library’s incunabula (literally, “things in the cradle”), a book produced before 1501, the earliest years of printing. It is a fine specimen, with some printing in red and black, a lovely woodcut strapwork border (white on black background), large ornamental initials, full-page images of finger counting and a tree of proportion, and illustrations of weighing instruments and barrels. This copy was read carefully by a previous owner as there are early manuscript annotations.

Pacioli must have been a great teacher. He works in proverbs in his discourses, including “Who does nothing, makes no mistakes; who makes no mistakes learns nothing.” The author lays out in his Summa the correct use of ledgers and admonished that no one should end the work day without balancing the books, making sure that the debit and credit columns agree. Oh, if I could achieve that with my accounts, and understand his method of determining economic returns, what a wonderful world this would be!

The famous woodcut illustration of dactylonomy, or the art of expressing numbers by the position of the fingers. For example, the number one is indicated by folding down the left hand’s little finger, 100 by the right hand’s.

The Dibner Library also holds Pacioli’s 1509 edition of Euclid and a later reprinting of his Summa, in 1523. Many editions followed the 1494 Summa; it was translated into Dutch, German, Russian, French, and English.

Portrait of a Renaissance Man: Luca Pacioli and Disciple. In the habit of the Franciscan order, the author is shown presenting Euclidean geometry to a pupil and to us, the viewers. This painting, now in the Capodimonte Museum in Naples, is often attributed to Jacopo de’ Barbari but the artist is much debated. The friar draws on a slate board with the inscription Euclides while his other hand rests upon an open book, surely an early edition of Euclid, perhaps a copy of the Elements printed by Erhard Ratdolt in Venice in 1482 (also in the Dibner Library). The closed volume is his Summa de Arithmetica with the initials on the fore-edge of LI. RI. LUC. BUC (Liber Reverendi Luca Burgensis; Pacioli was also known as Luca di Borgo, after his birthplace in Tuscany). Upon the leather-covered boards, with metal bosses and clasps, of the Summa, sits a dodecahedron (The Dibner Library’s copy was rebound sometime in the 18th century when its leaves were trimmed, sadly cutting off some of the manuscript notes). Also on the table are other tools of the mathematician: a protractor, a pen, chalk, compass. A rhombicuboctahedron, half filled with water, balances the careful composition (illustration from Wikimedia Commons).




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