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

178 r Michael Katz

[And this (5) is a sum that is equal in squares, for if you add its

square to the square of its double, the same will be the cube of 5. And

any number preceding 5 the value of its cube to the two squares in

the sum will be as the value of the sum to 5. And above 5 this will

be in reverse.]

For the number 5 what we have here is:

53 = 5^2 + (2•5)^2

For a number, say 4, smaller than 5 we have:

43 < 4^2 + (2•4)^2 by a factor of 4 to 5

For a number, say 6, greater than 5 we have:

63 >6^2 + (2•6)^2 by a factor of 6 to 5

These are all, we note, derivatives of the identity:

N^3 = (N^2 + (2N)^2 )(N/5)

So we get the factor N to 5, which reduces to 1 if N=5. The number 5
draws a border in this formula between ratios smaller than 1 and greater
than 1. Ibn-Ezra wouldn’t have, and probably wouldn’t like, this kind of
abstract formulation. But conceivably he would be pleased to see that the
next formula in this line singles out precisely his next distinguished num-
ber, namely 10, as the next border case, for

N^3 = (N^2 + (3N)^2 )(N/10)

And indeed the number 10 draws a similar border, this time in geom-
etry, as Ibn-Ezra proceeds to show.

ואם תשים אלכסון עגול במספרו ותוציא יתר בשלישית,

יהיה המשולש שהוא שוה השוקים כמספר הקו הסובב

וכמוהו המרובע הארוך בעגול. ולפני זה המספר יהיה ערך

המשולש אל הקו כערכו אל עשרה ולמעלה ממנו הפך הדבר.

[And if you place a circle’s diameter of this number (10) and draw

a chord at the third, the triangle that is of equal sides will be of the

number of the round line, and like it the long quadrilateral in the

circle. And prior to this number the value of the triangle to the line

will be as its value to 10, and above it the reverse.]^10