I have read that a lost book by Fibonacci (a commentary on Book X of Euclid's Elements) gives a numerical treatment of incommensurable magnitudes.

Given that Fibonacci grew up in North Africa and travelled extensively throughout the Middle East where he learned the Hindu-Arabic number system and became familiar with the contemporary developments in mathematics, it seems reasonable to wonder if he may have encountered a numerical treatment of incommensurable magnitudes during these travels.

Q: Is there any record of Arab or Persian mathematicians developing a numerical treatment of incommensurables, or is Fibonacci’s lost commentary the first known treatment?

(N.B. A number of questions have already asked here regarding the Greek's discovery of incommensurable magintudes, but none appear to address this question.)


From the MacTutor page on Fibonacci

His book on commercial arithmetic Di minor guisa is lost as is his commentary on Book X of Euclid's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view.

We know that the fourth section of Liber Abaci has Fibonacci giving numerical approximations of surds, thereby removing them from geometry. Thus, the missing text must go further than simple numerical approximations.

Further, in the 1225 work, Flos, Fibonacci details his solutions to the three questions posed to him by Johannes of Palermo during a meeting at the court of Fredrick II. In the second of these problems he finds the only real root of the equation $x^3 + 2x^2 + 10x = 20$. His argument establishes that such a root cannot be integral or rational, and that it cannot take any of the forms from Book X of Elements. As such, this represented a new type of irrational number - one not capable of construction by straight edge and compass.

And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.

Then, without justification, Fibonacci states that root is approximately

$1; 22, 07, 42, 33, 04, 40$

in sexidecimal notation.

All of these know facts suggest that the missing commentary of Book X represented a significant advance in our understanding of irrational numbers. I was hoping that any existing Middle Eastern sources might give a hint as to what this treatment may have included.

  • $\begingroup$ @MauroALLEGRANZA Please see my edit to the question. $\endgroup$
    – nwr
    Jan 16, 2017 at 18:29

2 Answers 2


Transition from Eudoxian ratios of magnitudes to irrational numbers was a long incremental process with no sharp thresholds. Some argue that already Archimedes deviates from Euclidean strictures about magnitudes when he "approximates" π in the Measurement of the Circle (c. 250 BC), and that his works show some "intuitive idea" of irrational numbers. Heron described a popular iteration process for numerically approximating square roots in his Metrica, he also had a version for cube roots. Ptolemy in Almagest (and likely already Hipparchus c. 150 BC), despite heavily Euclidean language, has no problem with expressing chords and angles in sexagesimals and even describes how to interpolate their tabulated values.

This was somewhat opportunistic and sporadic. Medieval Islamic mathematicians became more systematic after the introduction of the decimal system and algebra. According to Katz's History of Mathematics:

"The process of relating arithmetic to algebra, begun by al-Khwarizmi and Abu Kamil, continued in the Islamic world with the work of Abu Bakr al-Karaji (d. 1019) and al-Samaw’al over the next two centuries. These latter mathematicians were instrumental in showing that the techniques of arithmetic could be fruitfully applied in algebra and, reciprocally, that ideas originally developed in algebra could also be important in dealing with numbers...

[...]Al-Karaji was more successful in continuing the work of Abu Kamil in applying arithmetic operations to irrational quantities. In particular, he explicitly interpreted the various classes of incommensurables in Elements X as classes of “numbers” on which the various operations of arithmetic were defined, but then noted that there were indefinitely many other classes composed of three or more surds. Like Abu Kamil, he gave no definition of “number,” but just dealt with the various surd quantities using numerical rather than geometrical techniques. As part of this process, he developed various formulas involving surds..."

[...]Given that al-Samaw’al thought of extending division of polynomials into polynomials in $1/x$, and thought of partial results as approximations, it is not surprising that he would divide whole numbers by simply replacing $x$ by $10$. As already noted, al-Samaw’al was the first to explicitly recognize that one could approximate fractions more and more closely by calculating more and more decimal places."

For more on al-Samawal see What is the origin of polynomials and notation for them? By the way, not taking a form of Book X irrationals does not amount to being non-constructible by straightedge and compass, Book X irrationals are quadratic surds, they do not cover iterated square roots. Not even Gauss could prove roots of cubics non-constructible by straightedge and compass many centuries later, it had to wait until Wantzel, see Suzuki's Brief History of Impossibility. Just how much trust medieval mathematicians put into irrationals as "numbers" is attested to by the etymology of "surd": surdus "deaf, dumb" is a Latin translation of Arabic aṣamm "deaf", itself apparently derived from Euclid's Greek alogos "speechless, without reason". Latin surd shares the root with "absurd". Even Cardano in Ars Magna (1545) treats them with gloves.

But even if we are charitable, calling surds "numbers" does not amount to treating them as "irrational numbers" as understood today. That assumes familiarity with a whole class of "numbers" (say our real numbers) in which they belong to the complement of rationals. Neither Heron, nor Al-Karaji, nor Fibonacci, nor Cardano came close to that. We had a long debate in Who gets credit for the real numbers? thread on whether Stevin's treatment of arbitrary unending decimals in Arithmetic (1585), amounts to such an insight. For subsequent developments up to Cantor and Dedekind see When did it become understood that irrational numbers have non-repeating decimal representations?

  • $\begingroup$ Yes, I should have known better than to have posed the OP in terms of "the first". Indeed, even when history credits an individual as "the first", we can usually safely assume that they were not in fact the first. It seems reasonable to assume that Fibonacci was familiar with the work of Abu Kamil and its development by Al-Karaji. It is interesting to see that he did work with arithmetic operations on irrationals. I take your point that not being treated in BookX does not equate to Euclidean non-constructability. Thx! Very well informed as always. $\endgroup$
    – nwr
    Jan 17, 2017 at 2:46

It is highly unlikely that Fibonacci could give a rigorous treatment of incommensurables. Decimal fractions for representation of what we call now real numbers were introduced by John Napier in the end of 16th century, but the rigorous theory of incommensurable magnitudes (on the same level of rigor as the Greek treatment of them) had to wait until Dedekind and Cantor (second half of 19th century).

So the answer to your question "Who was to replace" is this: Napier for practical purposes, and Dedekind/Cantor rigorously.

EDIT. Of course it was understood since the times immemorial that incommesurable quantities can be APPROXIMATED by rationals. But approximate is not the same as represent. As I understand, nobody was talking about infinite (decimal) expansions before Napier, which indeed represent the irrational numbers. So there was no novelty when Fibonacci approximated his root with a sexagesimal fraction. Archimedes approximated pi with an ordinary fraction, and perfectly understood that such approximation can be obtained with any precision.

  • $\begingroup$ No, obviously not rigorous, at least my today's standards. Please see the edit to my original question for further background and detail. $\endgroup$
    – nwr
    Jan 16, 2017 at 18:31

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