APIAN

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      A very attractive copy of the first edition of Apianus's Kauffmans Rechnung. This is the second arithmetic textbook written in German (preceded only by the 1518 work of Grammateus/Heinrich Schreiber), notable both for the rendering of Pascal's triangle on the title page-a century before Pascal studied it himself-and for its inclusion by Holbein in his famous double portrait "The Ambassadors." Clearly a much-used resource of a sixteenth-century merchant, this copy is bound with two early German-Latin dictionaries, and offered in its original blind-tooled binding, dated 1531.In addition to containing the first known appearance of the Pascal triangle ("some years before Stifel mentioned it"), Smith heralds the Kauffmans Rechnung for its depiction of line reckoning, also on the title (Rara Arithmetica, p. 156). John Glenn notes that Apianus's "method of division introduces... decimal fractions, nearly a century before Stevinus produced his treatise on them"; admitting however that the notation, where halves, quarters and eighths are written as 05, 025 and multiples of 0125, seems "rather clumsy" (Glenn, LRB Letters vol. 25, no. 6).The Kauffmans Rechnung covers the six basic mathematical operations (including roots up to eight, "casting out nines," and several contemporary methods of division), interest calculations, rates of exchange, counters, and other mercantile applications. Apianus, who drew upon Christoff Rudolff's 1525 algebraic work, also touches on all manner of commerce, FROM MINING TO THE SPICE TRADE, and discusses the "foreign" techniques employed in Florence. He includes many sample questions for practice. Part Two contains a chapter on the "Regula virginum," explaining how to calculate the varying wages of men, women, maidens and young children. Hans Holbein included the first edition of the Rechnung in his 1533 painting "The Ambassadors"; the small, partly open book appears on the lower shelf to the left of the famous anamorphic skull, open to a page that begins with the word dividirt (f. [Q8]v). OCLC lists only two copies of this first edition: Columbia and U. Chicago. Bound with Apianus's work are two bilingual dictionaries, the first an anonymous work directed at children that begins with a list of Latin and German syllables, followed by the Ten Commandments, Lord's Prayer and other devout passages. It contains 55 pages of vocabulary, where each German term is accompanied by a group of three letters and a number (g m d 2, g f d 3, etc.)-a type of shorthand, perhaps identifying the relevant parts of speech? (DigiBib lists a 1536 edition also from Wittemberg.) The second dictionary in this sammelband is an early edition of Sebald Heyden's popular Nomenclatura rerum domesticarum. Although this boasts more vocabulary words-each page features two double-columned lists-it lacks the handy bilingual subheadings. * Smith, Rara p. 155; Ortroy 67; VD16 A3094; Cantor pp. 401-5; John Glenn, London Review of Books, vol. 25 no. 6 (20 March 2003) http://www.lrb.co.uk/v25/n06/letters.html; http:// employees. oneonta.edu/farberas/arth/ARTH214/Ambassadors_Home.html