The Birth of the Medieval Hebrew Mathematical Language as Μanifest in Ibn al-Aḥdab’s Epistle of the Number

Ilana Wartenberg

Introduction

The aim of this paper is to portray the creation of the medieval Hebrew math- ematical language, which originated in the twelfth century with the works of the two Abrahams: Abraham bar Ḥiyya and Abraham ibn ʿEzra. In part I, I will explain what mathematical language is and which attributes it should ideally possess. In part II, I will exemplify some of the rare pre-medieval mathematical traces found in the Bible and Rabbinic sources. In part III, I will define medi- eval Hebrew mathematics, and then in part IV, I will elaborate on the vari- ous linguistic processes involved in the creation of new Hebrew mathematical terms inspired by . In part V, which is the main focus of the article, I shall examine several excerpts from Ibn al-Aḥdab’s Epistle of the Number, and analyse two of the innovative algebraic Hebrew terms therein.

I The Nature of Mathematical Language

Ideally, a language that describes mathematical theories and procedures ought to be precise. Every mathematical term should be given an unambiguous defi- nition such that all its users will refer to the same concept without any doubt; at the same time, every mathematical idea should have a word to describe it. Mathematically speaking, there should be a one-to-one correspondence between a word and its mathematical referent, whether concrete or abstract. Very often, in particular in the Middle Ages, this is not the case, but even if the reality of the medieval Hebrew mathematical language is far from ideal, one discerns the birth of a scientific which aspires to have the above characteristics. Mathematical terms should also be devoid of any emotional associations and should be context-free, so that when we employ, for example, the term ‘hexagon’, we always refer to a polygon with six sides and nothing else. Similarly, the mere mentioning of ‘hexagon’ is not supposed to trigger any emotional reaction, unlike non-mathematical words, such as ‘love’, ‘sun’ or ‘sea’.

© koninklijke brill nv, leiden, ���4 | doi 10.1163/9789004277052_008 118 Wartenberg

Mathematical language is supposed to define and demarcate a new term before it is put into use, contrary to our daily language, in which we often use words that have not necessarily been officially defined, while assuming (at times, falsely) that everyone would know what the word we use means. Mathematical language often adopts a word from daily language, but only after determining its exact meaning in the new mathematical context. Very often new words are coined solely for mathematical usage.1

II Pre-Medieval Hebrew Mathematics

In his seminal work on medieval Hebrew mathematical terminology, Ṣarfatti discusses pre-medieval occurrences of mathematical language. claims that in the Bible there is no notable mathematical content, since the subject matter is too remote from any mathematical considerations. Jews of the time were not involved in complex forms of trade; neither did they hold any particular interest in astronomy or other sciences. Some arithmetical problems are pres- ent, e.g., within the discussion of the apportionment of tributes to the Priests and the Temple. One also encounters several geometrical descriptions of the tabernacle and the Temple. In the Bible we find a complete place-value decimal system which includes -seven‘ ,שבעתיים ,.both cardinals and ordinals. We also find multipliers (e.g fold’) and partial numbers (i.e. fractions of the form 1 n = 2, . . ., 10). Comple­ n n mentary fractions, (i.e. fractions of the form n + 1 ), are marked with the term -equal two thirds. And yet, it is not possi ידות i.e., parts. For example, two ,ידות ble to discern an abstract and complex mathematical system in the Bible. Even suitable nouns to denote basic geometrical figures are absent.2 In there is no sign of much mathematical development. Even words that later acquired a generic and abstract meaning designated circle’ or ‘sphere’ in‘) עיגול originally a concrete object. For example, the word a ‘rectangle’ in) מלבן .medieval Hebrew) referred to a certain part of a table both medieval and ) referred to a wooden frame of a window.3 Mišnat ha-Middot, written probably between the ninth and the twelfth cen- turies by an anonymous author, is the first known Hebrew text that deals with mathematics per se. It contains practical rules to measure areas and volumes,

1 See Ṣarfatti (1969: 2–11). 2 Ṣarfatti (1969: 14–19). 3 Ṣarfatti (1969: 20–44).