Sumario compendioso (Mexico, 1556)
The first book of mathematics printed anywhere in the Americas, issued at Mexico in 1556 — seventeen years after the press arrived and eighty-five before one reached British North America. It is not a schoolbook: it is a working table-book for the silver trade of Peru and New Spain, with assay-reduction tables, interest tables, an exchange table for transatlantic remittances, and a closing set of algebra problems.
Why this wins its question: The 1556 Sumario is usually met as a single sentence — "the first mathematics book printed in the New World" — attached to no content, because the original is among the rarest of Mexican imprints. This object reconstructs what is actually in it from the bibliographer who had the sheets in front of him: the assay tables and the maravedí rate, the interest tables, the word centavos in 1556, the printed arithmetic of the royal fifth, the peso-to-ducado table built for remittances across the Atlantic, and the algebra at the end — together with the colophon that names the press and both licensers, and the fact that the author's own surname is given two ways by the one witness.
Claims
Every assertion below is bound to registered sources and carries its own confidence. Weight them; do not treat the page as uniformly authoritative.
The book stands as entry 25, under the year 1556, of Joaquín García Icazbalceta's catalogue of books printed in Mexico between 1539 and 1600. He transcribes the title page, set in gothic type at the foot of a large coat of arms: "Sumario cōpēdioso de las quētas de plata y oro q̃ en los reynos del Piru son necessarias a los mercaderes: y todo genero de tratantes. Cō algunas reglas tocantes al Arithmetica. Fecho por Juan Diez freile." The volume is in quarto.
The licence was granted in New Spain and Icazbalceta checked it against the archive rather than taking it from the book alone. The verso of the title carries a dedication to the viceroy Don Luis de Velasco and, beginning there, a privilege for eight years dated 15 April 1556; Icazbalceta records finding that privilege entered in tomo 4 of the Mercedes of the Archivo General, folio 329 verso.
The bulk of the book is assay arithmetic done in advance. It opens by announcing tables from which one may know "el valor de cualquier barra ó tejo de plata que ensayado sea de mill y quinientos de ley hasta dos mill y cuatrocientos", set out by marcos, onzas and cuartas, reckoning fine silver at "dos mill y doscientos é cincuenta maravedís el marco". These reduction tables run from folio iij to folio xlviij, and the figures throughout the book are Roman numerals.
A second block of tables gives compound interest from 3 to 30 per cent, occupying folios xlix to lvij. The author explains that his results are carried in pesos, tomines and granos, and that a hundredth part of a peso "puédense nombrar perfectamente centavos, porque ciento valen un peso". Icazbalceta stops to note the word: here is the name centavos proposed from 1556, and only lately introduced into Mexican money.
From folio lxxxv the book turns from Peru to New Spain and becomes fiscal. The author announces that he will now set down "todo lo más necesario de las cosas tocantes á esta Nueva España, y principalmente los derechos que se deben de cualquier plata que se quintare ó dezmare, por lo cual se podrán regir todos los que quisieren saber los derechos que les han de llevar de cualquier plata con el uno por ciento". The arithmetic of the royal fifth is printed for the men who had to pay it.
The same passage sets out a currency-conversion table with its reason stated: differences between pesos, ducados and coronas are given "á causa que muchos en estas partes, y principalmente en el Perú, envían dineros á España, y por no correr el riesgo que hasta allá hay, dan acá pesos porque allá les den ducados ó coronas, sin saber la diferencia que de lo uno á lo otro hay". A printed instrument for a transatlantic remittance market, issued in Mexico in 1556.
The book closes with mathematics proper. After the tables end at folio xcj, there follows an introduction to arithmetical problems the author calls "quistiones", among which "van algunas del arte mayor, reservadas al álgebra"; they occupy folios xcij to ciij, and the volume ends at folio ciiij with a table.
The colophon, transcribed in full, names press, licensers and date: the treatise "fue impresso en la muy grande ynsigne y muy leal ciudad de Mexico, en casa de Juan pablos Bressano: con licencia del muy Illustrissimo señor Don Luys de Velasco, Visorrey y gouernador desta Nueua españa. E assi mismo cō licēcia del muy Illustre y reuerēdissimo S. dō fray Alōso de Mōtufar arçobispo de mexico: por q̃to fue visto y examinado, y se hallo ser prouecho so imprimirse. Acabose de imprimir: a veynte y nueue dias del mes de Mayo. Año dl nascimiento de nro Señor Jesu Christo d.1556 años."
What the book is for is on its own title page, and it is not disinterested. The accounts of silver and gold it teaches are declared necessary "en los reynos del Piru" to "los mercaderes: y todo genero de tratantes"; the first mathematics printed in the Americas was printed for the merchants of the silver economy and for the officials who taxed it, not for a school. The costs of that economy — the labour drafts that fed the mines — are not in view anywhere in the pages Icazbalceta prints.
The author's name is not settled by the source itself. Icazbalceta prints the title page as "Fecho por Juan Diez freile", and in his own introduction to the same volume cites the book as "DIAZ FREILE, Sumario compendioso, 1556" when listing the two Mexican imprints he reckons dated by the year of the Nativity. Both forms therefore stand in one witness, and nothing in the pages read decides between them.