{ "id": "caramuel-mathesis-biceps-1670", "type": "entity", "name": "Caramuel's arithmetics of any base (Mathesis biceps, 1670): a printed binary table, and a Guarani counting system defended against its own missionary", "summary": "A Spanish mathematician printed, on an episcopal press in 1670, a table of the numbers 0 to 32 written with two characters, alongside articles on arithmetic by threes, fours, fives, nines and sixties. His argument was that the base is arbitrary - chosen by the inventor, not by nature. He then applied it to a missionary's report that the Guarani could not count past four, and answered that having a different arithmetic is not the same as having none.", "locale": "en", "tags": [ "mathematics", "computing", "science", "spain", "seventeenth-century", "paraguay", "guarani", "indigenous-knowledge", "americas", "jesuit-missions", "language", "ideas" ], "relations": [ { "rel": "related", "target": "montoya-conquista-espiritual-1639" }, { "rel": "related", "target": "montoya-tesoro-lengua-guarani-1639" }, { "rel": "related", "target": "diez-sumario-compendioso-1556" }, { "rel": "related", "target": "imprenta-de-las-misiones-guaranies" }, { "rel": "related", "target": "feijoo-mapa-intelectual-1728" }, { "rel": "related", "target": "hervas-catalogo-lenguas-americanas-1800" } ], "questions": [ "Who printed a binary numeral table before Leibniz?", "What is Caramuel's Mathesis biceps and what does it say about number bases?", "Did any early modern European argue that American peoples had a base-four arithmetic?", "When did the Paris Academy publish Leibniz's binary arithmetic?" ], "claims": [ { "id": "c1", "text": "The title page reads \"IOANNIS CARAMVELIS MATHESIS BICEPS. VETVS, ET NOVA\", and gives the imprint \"CAMPANIAE, In Officina Episcopali Anno M.DC.LXX. SVPERIORVM PERMISSV. Prostant Lugduni apud Laurentium Anisson\" - printed at Campagna, in the Kingdom of Naples, on an episcopal press, in 1670, by permission of superiors, and sold at Lyon by Laurent Anisson. The page lists forty numbered subjects, beginning with Arithmetica and Algebra and running through Cosmographia, Combinatoria, Musica and Meteorologia.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.9 }, { "id": "c2", "text": "The opening essay of the volume, the \"Meditatio Prooemialis\", asks whether arithmetic is one or many, and carries the running head \"De Arithmeticarum numero, & varietate\" - on the number and variety of arithmetics. The volume's own printed contents give it twelve articles, eleven of them one base each: Binaria at folio XLV, Ternaria XLVIII, Quaternaria L, Quinaria LIII, Senaria at the same folio, Septenaria LIV, Octonaria LV, Novenaria LVI, Denaria LVII, Duodenaria LX, Sexagenaria LXI, and then Articulus XII, \"Quaenam ex his Arithmeticis admitti debeat?\" - which of these arithmetics ought to be adopted - at LXV. Base eleven is refused a place, with the reason printed in the contents: the Spaniards may make up their ducat out of eleven silver pieces, but they do not count elevens of elevens.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.9 }, { "id": "c3", "text": "Folio XLVI prints a positional table of the integers from 0 to 32 written in two characters only, a and 0, set in a column beside the ordinary decimal numerals: 2 is a0, 3 is aa, 4 is a00, 8 is a000, 16 is a0000 and 32 is a00000, with every intermediate value given. A second table on the same page, headed \"Revolutiones Arithmeticae. Binariae | Communis\", compares the doubling sequence 1, 2, 4, 8, 16, 32, 64 against the decimal sequence 1, 10, 100, 1,000 and upwards.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.9 }, { "id": "c4", "text": "The reasoning behind the tables is stated at folio XLV: how many units are to be put in the first period \"non a natura rei, sed ab arbitrio Inventoris dependet\" - depends not on the nature of the thing but on the choice of the inventor - and, in the same paragraph, that \"just as different peoples have different languages, so there can be different arithmetics\", one people proceeding by fours and fours of fours, another by sixes and sixes of sixes.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.9 }, { "id": "c5", "text": "He applies the same reasoning to base ten and refuses to privilege it: at folio XLVI he writes that he will admit this number could, or should, have been chosen rather than another, and that it was adopted \"non quia sic licuit, sed quia sic placuit\" - not because it had to be, but because it pleased. On the same page he argues that the binary progression is naturally seated in music, where all duple proportions are consonant.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.85 }, { "id": "c6", "text": "Folio L, Articulus III, number VIII, opens the American question: that the Paraguayans and many other peoples of the West Indies are ignorant of arithmetic is reported, Caramuel writes, by those who have travelled Peru, Brazil and other American regions, and he names his source - Father Antonio Ruiz [de Montoya], in the book entitled \"Conquista Espiritual, hecha por los Religiosos de la Compania de Iesus, en las Provincias del Paraguay, Parana, Vruguay\" and a fourth province whose name the page carries in a damaged setting - and quotes it in Spanish at article 10: \"Cuentan los anos por los inbiernos, que llaman Roy. Su numerar no llega mas, que a quatro: y de alli con confusion alguna asta diez: y assi les vamos ensenando nuestra cuenta, importante para las confessiones.\"", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.9 }, { "id": "c7", "text": "Caramuel declines the inference. At folio LI he writes: \"At assentiri non audeo Patri Ruizio, aliud est enim hos populos Arithmeticam nullam habuisse, & aliud toto coelo diversum, habuisse aliam, quae non per decades, ut nostra, sed per tetrades, suas periodos involvit\" - it is one thing for these peoples to have had no arithmetic, and a wholly different thing for them to have had another one, which closes its periods by fours and not, as ours does, by tens.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.9 }, { "id": "c8", "text": "His stated ground for suspecting the missionary was mistaken is linguistic: what compels him, he writes on the same folio, is \"illorum hominum ingeniosa Grammatica\" - the ingenious grammar of those people, who have very many properties and elegances that Europeans lack - and he instances their naming, deriving Paraguay as \"a crown of feathers\" and the name of God, Tu-pa, from a particle of wonder and a particle of question. He adds a practical argument: if they truly could not count beyond four, neither a chief robbed of one wife out of thirty nor a countryman robbed of one sheep out of a hundred would notice the theft. He closes with a thrust at the missionary's method - Ruiz taught them the European count, he suggests, because he did not wish to undergo the trouble of learning theirs.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.8 }, { "id": "c9", "text": "The passage is not a modern ethnography and this object does not present it as one. Caramuel never saw the Guarani; his base-four reading is an inference drawn from a printed missionary report, and he reproduces from the same report material of a very different kind on the facing column - that the Paraguayans knew God and his unity, that what remained of it came from the preaching of the Apostle Saint Thomas, that a demon had led them to venerate the bones of famous Indian magicians, together with notices of polygamy and of a flood tradition called Yporu. The purpose Ruiz himself gives for teaching European counting, in the sentence Caramuel quotes, is that it is necessary for confession. Whether the Guarani numeral system was in fact quaternary is not asserted here.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.85 }, { "id": "c10", "text": "Articulus XI, at folio LXI, sets out sexagesimal arithmetic with its ladder of apices - 6, 36, 216, 1,296, 7,776, 46,656, 279,936 - noting that these are born of senary arithmetic if zeros are added successively, and records that reckoning by sixties was not confined to astronomers: in Bohemia, he writes, civil accounts were still settled by sixties in his own day, small goods being sold not by tens but by sexagenae, so that a hundredweight there meant 120 pounds.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.85 }, { "id": "c11", "text": "The Academie Royale des Sciences in Paris registered binary arithmetic as new in 1703. Its own alphabetical table for 1699-1710 carries the entry \"ARITHMETIQUE Binaire (Nouvelle) H. 1703. p. 58\", followed by \"Explication de l'Arithmetique binaire, qui se sert des seuls Caracteres 0. & 1. avec des Remarques sur son utilite, & sur ce qu'elle donne le sens des anciennes Figures Chinoises de Fohi. Par M. LEIBNITS. M. 1703. p. 85\", and adds \"Inventee par M. LEIBNITS, & vers le meme tems par M. DE LAGNY\". Caramuel's printing of a binary table therefore precedes that publication by thirty-three years.", "sources": [ "academie-royale-table-des-matieres-1699-1710" ], "confidence": 0.9 }, { "id": "c12", "text": "No claim of influence is made here. The Paris table gives no indication that Leibniz or Lagny knew Mathesis biceps, the 1703 memoir itself was not opened for this object, and priority in binary notation is contested by earlier unpublished and semi-published work that is not examined here. What the two documents together support is narrower and firmer: a printed binary table and a general theory of arbitrary bases existed in a Spanish author's book in 1670, and the Paris Academy was announcing the same notation as a novelty in 1703.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio", "academie-royale-table-des-matieres-1699-1710" ], "confidence": 0.8 }, { "id": "c13", "text": "The Academy reached Caramuel's conclusion about base ten independently and thirty-three years later. The same table records, under ARITHMETIQUE, that counting by the decuple progression \"ne paroit pas fort ancienne\", that it seems to have been introduced into Europe by Pope Sylvester II, and that \"son fondement est purement arbitraire\" - its foundation is purely arbitrary.", "sources": [ "academie-royale-table-des-matieres-1699-1710" ], "confidence": 0.8 }, { "id": "c14", "text": "What the folios read contain, and what they do not, is worth stating. The binary pages give a notation, a comparative table of periods and an argument about the arbitrariness of the base; they do not work out rules for operating in base two, and they do not connect the notation to the Chinese figures that the title of Leibniz's memoir advertises. Caramuel does reach the operations for another base: the printed contents record that duodecimal arithmetic is treated at greater length from page 90, with three tables and rules for adding, subtracting, multiplying and dividing at pages 94-95. Those pages were not opened for this object, and nothing here rests on them beyond the contents entry itself.", "sources": [ "caramuel-mathesis-biceps-1670-meditatio" ], "confidence": 0.85 } ], "takeaways": [], "faqs": [], "evidence_tier": "primary", "evidence": { "level": "industry_observation", "source_types": [ "paper" ] }, "moat_flag": false, "winning_edge": "Caramuel appears in English mostly as a footnote to Leibniz, asserted without a page: \"a Spanish bishop had binary first.\" This object opens the book. It quotes the folio where the table of 0 to 32 in two characters is printed, the sentence in which the base is declared to depend on the inventor's choice and not on nature, and the Paris Academy's own index announcing the same notation as new in 1703 - and then declines to claim influence, which the loose versions of the story do claim. It also carries the half of the Meditatio that nobody repeats: the article in which a European mathematician reads a Jesuit's report that the Guarani \"could not count past four\" and answers that a different arithmetic is not an absent one, with his reason - their grammar - and the missionary purpose, confession, left on the page beside it.", "confidence": 0.85, "last_verified": "2026-08-14", "canonical_url": "https://hispanic-legacy.com/k/caramuel-mathesis-biceps-1670", "api_url": "https://hispanic-legacy.com/api/k/caramuel-mathesis-biceps-1670.json", "jsonld": { "@context": "https://schema.org", "name": "Caramuel's arithmetics of any base (Mathesis biceps, 1670): a printed binary table, and a Guarani counting system defended against its own missionary", "description": "A Spanish mathematician printed, on an episcopal press in 1670, a table of the numbers 0 to 32 written with two characters, alongside articles on arithmetic by threes, fours, fives, nines and sixties. His argument was that the base is arbitrary - chosen by the inventor, not by nature. He then applied it to a missionary's report that the Guarani could not count past four, and answered that having a different arithmetic is not the same as having none.", "url": "https://hispanic-legacy.com/k/caramuel-mathesis-biceps-1670", "license": "https://spdx.org/licenses/CC-BY-4.0.html", "dateModified": "2026-08-14", "citation": [ { "@type": "CreativeWork", "name": "Ioannis Caramuelis Mathesis biceps, vetus et nova - 'Meditatio Prooemialis. De Arithmeticarum numero, & varietate' (tomus I, foliation XLV-LXI)", "url": "https://books.google.com/books?id=5cmBsjnRQzwC&pg=PR45" }, { "@type": "CreativeWork", "name": "Table alphabetique des matieres contenues dans l'Histoire et les Memoires de l'Academie Royale des Sciences, 1699-1710 (entries ARITHMETIQUE and ARITHMETIQUE Binaire, pp. 57-58)", "url": "https://books.google.com/books?id=l79eAAAAcAAJ&pg=PA57" } ], "@type": "Article", "headline": "Caramuel's arithmetics of any base (Mathesis biceps, 1670): a printed binary table, and a Guarani counting system defended against its own missionary" } }