Al-Khwarizmi’s Mathematical Doctrines in Ibn-Ezra’s Biblical Commentary r 177כל מספר הוא באחד בכח, והוא בכל מספר במעשה. והוא
יעשה בפאה אחת מה שיעשה כל מספר בשתי פאות.
והנה כח האחד, ובחברך מרובעו אל מרובע כפלו אז יהיו
חמשה... וכאשר תחבר מרובע האחד אל מרובע ראש הנפרדים
יהיה המחובר הוא הדומה.[Every number is in 1 potentially, and 1 is in every number actually.
And it will do on one side what every number will do on two sides.
And here is the power of the 1, and when you add its square to the
square of its double, then there will be 5... and when you add the
square of 1 to the square of the head of separates, the sum will be
that which is the like.]The first sentence here refers to the most basic idea that the number 1 can
give rise to any potential number, and any existing number is the sum of
several repetitions of 1.^9 This entails a certain specific property presented
in the next sentence. The number 1 is half of its successor, 2. Any other
number is half of the sum of its successor and predecessor. That is, in
present-day mathematical formulation:
1 = ½(2) 2 = ½(1+3) 3 = ½(2+4) and so on.(As we shall explain below, Ibn-Ezra doesn’t have here 0 as a number.
If he had, he would have had to notice that the number 1 is also the mean
of its successor and predecessor, i.e. 1 = ½(0+2).)
I translate the word Hineh into Here, though there are other equally
good potential translations. And “the head of separates” is the first odd
number (3). Also “the like” is the number 10, so called as it is “like 1,” as we
shall soon see. Thus, in present-day mathematical formulations, the last
two sentences in the quotation above simply say:
12 + 2^2 = 5 and 1^2 + 3^2 = 10.And these two sentences are meant to lead us to properties of the
numbers 5 and 10.וזהו חשבון השוה במרובעים, כי אם תחבר מרובעו אל
מרובע כפלו כה יהיה מעוקב חמשה. וכל מספר לפני חמשה
יהיה ערך המעוקב אל השנים המרובעים בחשבון בערך
החשבון אל חמשה. ולמעלה מחמשה יהיה הדבר בהיפוך.