The Convergence of Judaism and Islam. Religious, Scientific, and Cultural Dimensions

Al-Khwarizmi’s Mathematical Doctrines in Ibn-Ezra’s Biblical Commentary r 179

To understand what Ibn-Ezra tells us in this paragraph we need the
following plot, where the two chords cut the diameter at one-third and
two-thirds of its length.

The area of the rectangle is equal to the area of the big triangle, Ibn-
Ezra says. This is easily seen to be true, since the height of the triangle is
twice the width of the rectangle while the triangle’s base coincides with
the rectangle’s long side. (And also since the two small triangles cut by the
big triangle from the two corners of the rectangle are congruent to the two
small triangles cut from the big one above the rectangle.)
Now, if the length of the diameter is 10, Ibn-Ezra maintains, then the
area of the big triangle (and hence also the area of the rectangle) is equal
to the length of the circle’s perimeter. While if the length of the diameter
is smaller than (or greater than) 10, then the ratio of the triangle’s area to
the circle’s perimeter (or the reverse of this ratio) is the same as the ratio
of the diameter length to 10.^11
These comments of Ibn-Ezra are captured by the formulas below,
where A denotes area, T is for triangle, R for rectangle, P for perimeter, C
for circle, and D for diameter.

A(T) = A(R) = P(C)•D(C)/10

Hence, clearly—

A(T) = A(R) = P(C) if D(C) = 10

A(T) = A(R) < P(C) by a factor of D(C) to 10 if D(C) < 10

A(T) = A(R) >P(C) by a factor of D(C) to 10 if D(C) >10

General Principles

So much for specific properties of the four distinguished numbers. And
now to three fundamental principles of arithmetic. We see the follow-
ing lines as the truly exciting part in Ibn-Ezra’s commentary on the Holy