Al-Khwarizmi’s Mathematical Doctrines in Ibn-Ezra’s Biblical Commentary r 189text, just like the Hebrew original, is not easy to read and understand. I hope the added
explanations and remarks will be of some help.
- Here and in the sequel I use the term Sum to translate the word Heshbon. In the
present context this seems more appropriate than calculus or calculation or computation
(the terms to which the word Heshbon usually refers in modern Hebrew). - This is another place where al-Khwarizmi’s influence is evident, as we show at the
end of the section on the Ten Commandments. - Ibn-Ezra uses the term Alakhson for the diameter of a circle, while in today’s
Hebrew this term refers to the diagonal of, say, a rectangle or a parallelogram, and the
word for diameter is Kotter. Similarly, he uses here the term Yetter for a circle’s chord,
while in modern Hebrew Yetter is the hypotenuse of a right triangle and the word for
chord is Meitar—quite similar to Yetter and from the same root. We find here, perhaps
for the first time, the term Shveh Shokayim for an isosceles triangle, just as in present-day
Hebrew. - Two remarks are needed regarding this point. First, Ibn-Ezra did not mean to
equate area and length conceptually, but only the numerical values of these two entities
in the present context. This almost goes without saying. Second, the “equality” here is
in fact only an approximation (though a very close one). It’s hard to know whether Ibn-
Ezra was aware of this. He would have been, had he tried to provide a formal proof of
his claim. However, in the spirit of al-Khwarizmi and other Moslem mathematicians of
the time, Ibn-Ezra showed little interest in formal “Greek type” proofs. - It is worth noting that for similar reasons the number 9 is held in awe by various
traditional cultures. Here are two examples. In Chinese culture this awe is attested, e.g.,
by the nine concentric circles in Beijing’s Temple of Heaven, where there are nine stones
in the first circle, eighteen in the second, and so on up to eighty-one in the last circle.
And in Buddhist temples, in India and elsewhere, we often find nine stairs leading to the
Buddha. - For example, Ramban (Rabbi Moshe Ben Nahman) and Hezkoni (Rabbi Hezekiah
Ben Manoah). - See André Allard, “The Arabic Origins and Development of Latin Algorithms in
the Twelfth Century,” Arabic Sciences and Philosophy 1, no. 2 (1991): 233–83.