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Mar 16

## Versions of Euclid's multiplication definition

Photograph of Euclid's statue by Mark A. Wilson (Public Domain)
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Pre-1570 Unary Definitions of Multiplication.

The following translations of Euclid’s definition of multiplication that pre-date Billingsley's translation of Euclid's definition of multiplication into English are all unary in nature and mention neither ‘addition’ nor the phrase ‘added to itself’.

 YEAR LANGUAGE AUTHOR DEFINITION OF MULTIPLICATION ≈888 Greek [i] Stephen the Clerk ἀριθμὸς ἀριθμὸν πολυπλασιάζειν λέγεται. ὅτ’αν ὅσαι εἰσιν ἐν αὐτῶι μονάδες τοσαυτάκις συντεθῆι ὁ πολλαπλασιαζόμενος καὶ γένηταί τις ≈950 Greek [ii] ANON MS Vat. Gr. 190 P. Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί τις ≈950 Arabic [iii] al-Karābīsī الضرب هو أن يوجد أحد العددين بعدد آحاد العدد الآخر فيكون حصة الواحد من آحاد المضروب هي المضروب فيه بعينه والمجموع هو العدد الحاصل من ضرب العدد 1482 Latin [iv] Campanus Numerous per alium multiplicari dicitur, qui totiens sibi coacervatur, quotiens in multiplicante est unitas. 1505 Latin [v] Zamberti Numerus numerum multiplicare dicitur: quando quotae sunt in ipso unitates toties componitur multiplicatus & gignitur aliquis. 1533 Greek [vi] Grynaeus Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ ὁ πολλαπλασιαζόμενος, καὶ γένηταί  τις. 1543 Italian [vii] Tartaglia Quel numero se dice esser multiplicato per un'altro, il quale si e assunato tante volte, quante unita e in lo multiplicante. 1555 German [viii] Scheubel Ain zal multiplicirt oder meret ain andere / wann die ander / als offt die erst zal ains in jr beschleüßt / genommen vnd zuesamen bracht wirdt. 1565 French [ix] Forcadel Un nombre, se dict multiplier un autre nombre, quand autant d'unitez, qu'il y a en luy, autant de fois se compose le multiplie, & en naist un autre. NOTE: The author has more instances of unary definitions that could have been cited, yet the definitions are the either the same as those presented above or have minor spelling variations only.

[i] In 888 CE the following definition of multiplication in Greek was written by ‘Stephen the Clerk’. Ἀριθμὸς ἀριθμὸν πολυπλασιάζειν λέγεται. ὅτ’αν ὅσαι εἰσιν ἐν αὐτῶι μονάδες τοσαυτάκις συντεθῆι ὁ πολλαπλασιαζόμενος καὶ γένηταί τις. Marked up text, courtesy of Henry Mason, is available online at http://www.jonathancrabtree.com/800s/888_Euclid_Multiplication_Definition_Marked_Up.jpg (Image courtesy of the Clay Mathematics Institute) The entire manuscript can be seen online courtesy of the Clay Mathematics Institute at: www.claymath.org/library/historical/euclid

[ii] Les œuvres d'Euclide, Traduites en Latin et en Français, François Peyrard, P. 383, Paris, 1814. Note: Peyrard discovered a 10th century manuscript within a collection of documents Napoleon had taken from the Vatican Library, which did not have the edits, revisions and additions of the Theon of Alexandria towards the end of the 4th century CE. Thus the manuscript (MS Gr. 190 P.) is considered by most to be the best and oldest extant source to produce the closest version of Elements to that written 300 BCE. For more information, see  P. 57 of Mathematical Thought from Ancient to Modern Times, Vol. 1., Morris Kline, Oxford University Press, New York, 1990. William Knorr has subsequently written about whether Heiberg and Heath should have relied on this manuscript as the basis for their editions of Euclid’s Elements, yet the matters raised by Knorr do not affect Euclid’s definition of multiplication.

[iii] Ahmad al-Karābīsī's commentary on Euclid's Elements, Pp. 70-71. Sonja Brentjes: In: (eds.) Folkerts, M., Lorch, R. Sic itur ad astra. Festschrift in honor of Paul Kunitzsch, Harrassowitz Verlag: Wiesbaden, 2000.

[iv] Preclarissimus liber elementorum Euclidis perspicacissimi in artem Geometrie, Erhard Ratdolt, Venice, 1482. This Latin edition of Elements is the first printed. It is based on an edition by the Italian Campanus of Novara in the 13th century, which was based on a translation from Arabic to Latin by Englishman Adelard of Bath, in the 12th century. The author has been able to explore two first edition Ratdolt printings of Euclid’s Elements.

[v] Euclidis Megaresis, Philosophi Platonici Mathematicarum, Bartolomeo Zamberti, Venice, 1505. This was the first Latin edition of Elements direct from Greek sources rather than via Arabic origins.

[vi] Eukleidou Stoicheion biblon, Simon Grynäus (Grynaeus), Basel, 1533. This edition is the first printed edition of Elements in Greek.

[vii] Euclide Megarense ... solo introduttore delle scientie mathematice, (Euclid of Megara (wrong should be 'of Alexandria' - a common error) ... only introducer of scientific mathematics.) Niccolo Tartaglia, Venturino Ruffinelli, Venice, 1543.

[viii] Das sibend, acht und neünt Büch, des hochberühmbten Mathematici Euclidis, (The seventh, eighth and ninth book by the renowned Mathematician Euclid), Johann Scheubel, Valentine Ottmar, Augsburg, 1555.

[ix] Les septieme huictieme et neufieme livres des Elemens d'Euclide, (The seventh, eighth and ninth book of the Elements of Euclid), Pierre Forcadel, Charles Perier, Paris, 1565.

Unary non-English multiplication definitions post 1800 that don't have repeated binary addition.

 1802 Latin [i] Horsley Numerus numerum multiplicare dicitur, quando quot unitates sunt in ipso, toties componitur (put together) multiplicatus, & aliquis gignitur. 1826 Greek [ii] August Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ (put together) ὁ πολλαπλασιαζόμενος, καὶ γένηταί  τις. 1855 Swedish [iii] Rundback Ett tal säges multiplicera ett tal, när det sednare talet tages (taken) så många gånger, som enheter finnas i det förra, och ett annat tal (produkten) deraf uppkommer 1857 Chinese [iv] LI and Wylie 乘數者，數有若干倍，即若干為乘數。Addition does not appear in the multiplication translation of LI Shan-lan and Alexander Wylie. 1865 Hungarian [v] Brassai Szám számot szorozni mondatik, midon a hány egység van benne, annyiszor rakatik (put) a szorzandó, és igy származik szám. 1884 Greek & Latin [vi] Heiberg Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ (put together) ὁ πολλαπλασιαζόμενος, καὶ γένηταί  τις. | Numerus numerum multiplicare dicitur, ubi quot sunt in eo unitates, toties componitur (put together) numerous multiplicatus, et oritur aliquis numerus. 1907 Czech [vii] Servit Pravíme, že číslo číslem se násobí, když násobené (násobenec) tolikrát se složí, (put together)  kolik v druhém jest jednotek, a nějaké vznikne. 1935 German [viii] Thaer Man sagt, daß eine Zahl eine Zahl vervielfältige, wenn die zu vervielfältigende so oft zusammengesetzt (put together) wird, wieviel Einheiten jene enthält, und so eine Zahl entsteht. 1949 Russian [ix] Morduhai-Boltovskii Говорят, что число умножает число, когда сколько в нем единиц, столько раз составляется (put together) умножаемое и что-то возникает. 1953 Greek [x] Stamatis Ἀριθμὸς ἀριθμὸν πολλαπλασιάζειν λέγεται, ὅταν, ὅσαι εἰσὶν ἐν αὐτῷ μονάδες, τοσαυτάκις συντεθῇ (put together) ὁ πολλαπλασιαζόμενος, καὶ γένηταί  τις. 1987 Latin [xi] Busard Numerus numerum multiplicare dicitur quando quot sunt in ipso unitates tociens componitur (put together) multiplicatus et sit aliquis. 2007 Italian [xii] Acerbi Un numero è detto moltiplicare un numero, quando, quante unità siano in esso, tante volte sia composto  (compounded) quello che è moltiplicato, e risulti un certo .

[i] Componitur means put together, place together, compose           www.perseus.tufts.edu/hopper/morph?l=componitur&la=latin

[ii] Eukleidou Stoicheia (Euclidis Elementa) Ex optimis libris in usum tironum, P. 193, Ernesto August, Trautwein, Berlin, 1826.

[iii] Öfversättning och bearbetning af sjunde, åttonde, nionde och tionde böckerna af Euclids Elementer. (Translation and adaptation of the seventh, eighth, ninth and tenth books of Euclid's Elements), Abraham Rundbäck, P. 2,  1855. “A number is said to multiply a number, when the latter number is taken as many times, as there are units in the former, and another number (the product) thereby arises.”  The author is grateful to Dr. Rikard Bögvad, for the analysis of the Swedish text. Dr. Bögvad is a Professor of Mathematics at Stockholm University, Sweden.

[iv] Christopher Cullen.

[v] EUKLIDES ELEMEI, XV KÖNYV, (Euclid’s Elements, 15 Books) P. 210, Sámuel Brassai, Magyar Tudományos Akadémia Publisher: Pest, 1865.  “A number is said to multiply a number when, as many times as units in it, so many times the multiplicand is put, and thus number originates.” The author is grateful to Dr. Gábor Kutrovátz  for the analysis of the Hungarian text. Dr. Kutrovátz  is Assistant Professor, Eötvös Loránd University, Department of History and Philosophy of Science, Budapest.

[vi]

[vii]

[viii]

[ix]

[x]

[xi]

[xii] Euclide, Tutte le opere, Fabio Acerbi, P. 1091, Bompiani, Milan, 2007. Composto equates to compounded/composed. The word compounded was used by Leeke and Serle within their translation of Euclid’s definition of multiplication from their 1661 English edition of Elements, as did Edmund Stone in his 1745 English edition. See also: www.collinsdictionary.com/dictionary/italian-english/composto
Translations of Euclid's multiplication  definition with either the binary word 'added' or phrase 'added to itself'.

[i] Les œuvres d'Euclide, Traduites en Latin et en Français, (The Books of Euclid, Translated in Latin and in French), Francois Peyrard, P. 383, Paris, 1814.

[ii] The Thirteen Books of Euclid’s Elements, Volume 2, Thomas L. Heath, P. 278, Cambridge, at the University Press, 1908.

[iii] Euklids Elementer VII - IX, Thyra Eibe, P. 4, København : Gyldendalske Boghandels, 1912. “A number is said to multiply a number when the one that is multiplied is added together as many times as there are units in the first and some [literally: one or other] number is produced” Translated with the assistance of Professor Jesper Lützen from the Department of Mathematical Sciences, University of Copenhagen, Denmark.

[iv] De elementen van Euclides, Eduard J Dijksterhuis, P. XXX, Historische bibliotheek voor de exacte wetenschappen, 1930.

[v] Euklidész Elemek, P. 206, Gyular Mayer, Gondolat, 1983. “We say that we multiply with a number another one when we construct a number so that we add the multiplied number as many times as there are units in the

multiplier.” Translation courtesy of Professor Michael Szalay,  Faculty of Science, Institute of Mathematics, Department of Algebra and Number Theory, Eötvös Loránd University, Hungary.

[vi] French, Itard, 1962.

[vii] Oujilide Jihe yuanben, (Euclid’s Elements) Lan Jizheng, Zhu Enkuan, Trans., Xi'an, Shaanxi Kexue Jishu Chubanshe, [in Chinese] 1990.

[viii] Vitrac 1994.

[ix] In 1997 Dr. David Joyce’s definition of multiplication attributed to Euclid was “A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other.”  After the author contacted Dr. Joyce in 2012 with some research that led to this essay, Dr. Joyce changed Billingsley’s definition of multiplication to read, “A number is said to multiply a number when the latter is added as many times as there are units in the former.” This is Dr. Joyce’s explanation, not Euclid’s definition. Euclid’s Greek neither has added nor the words latter and former. 1997: https://web.archive.org/web/19970111012516/http://aleph0.clarku.edu/~djoyce/java/elements/bookVII.html#defs

[x]

[xi]

[xii] At https://web.archive.org/web/20131229191827/http://scienzaatscuola.it/euclide/index.html Professor Gianluigi Trivia says he used the Italian translation of Fabio Acerbi, yet his definition of multiplication is a translation from the English on David Joyce’s website, which at the time, cited Billingsley’s definition of multiplication as it appeared in Thomas Heath’s edition. The definition of multiplication with the ‘added to itself’ phrase in Italian, ‘aggiunto a se stesso’, is at
https://web.archive.org/web/20140609062955/http://www.scienzaatscuola.it/euclide/libro7.html

[xiii]

[xiv]

[xv] Source: David Joyce, http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII15.html