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**L’extraction de la racine \(n\)-ième et l’invention des fractions décimales (XI\(^e\) – XII\(^e\) siècles).**
*(French)*
Zbl 0389.01003

The Sumerians discovered the (sexagesimal) positional system and developed it perfectly. The decimal positional system is much inferior to the sexagesimal system and it is more a concession to the arithmetically lower level of the greater and greater becoming classes which had to compute that the decimal system arose, regrettably. Astronomers never forgot the excellent qualities of the basis 60 and continued to use the sexagesimal system till modern times. We should not forget that the “invention” of the decimal system is “a step backwards”. Simon Stevin was a propagandist of the decimal system, never called himself the inventor of this system. Indeed many people as Rudolff, Apianus, Bonfils have been indicated as using the decimal system before Stevin. Al-Kâshî explains both the sexagesimal and the decimal system in his “Key of Arithmetics” and the present author emphasizes that not al-Kâshî (\(\dagger\) 1437) or Bonfils (1335) following A. Saidan but al-Uqlîdisi in the Xth century is the first who is known to write on the decimal system.

The author considers the works of the school of al-Karaji and in particular of as-Samaw’al (1172) for the application of the decimal system in computing \(n\)-th roots. Also al-Tûsi’s work on more general equations is elucidated. The author discusses in detail the extraction of a fifth root by as-Samaw’al and sees – as is en vogue during some decades – “Ruffini-Horner”-methods, and a “Horner-triangle”. Horner’s paper (Phil. Trans. 1819) is mirabile dictu, concerned with the remark that one should never use more than two lines of the algorithm named nowadays after him, that one should not compute one more digit at each step, that one should not use more digits than necessary, i.e. for an equation of degree\(n\) never more than \(1/n\)-th of the digits wished in the result.

Al-Kâshî and as-Samaw’al are therefore doing just what Horner combats: computing too much and too many numbers without importance. Indeed when as-Samaw’al computes sexagesimally the 5-th root of the number \[ 2.33.43;3.43.3.43.36.48.8.16.52.30 \] at three places only \(2.33.43\) do play a role, all the others can be disregarded, and when al-Kâshî computes the 5-th root of \(44240899506197\dots\) the only digits playing a role for the three digits result are \(4424\). All the other computations are meant for “unskilled people” who have to follow blindly an algorithm.

Already in ancient Babylonian times by computing the powers of 1.1 – shifting and adding continuously – the “binomial coefficients” arose and the arithmetical series of higher order on the \(45^\circ\) lines became evident. Horner used these directly in computing a value of a polynomial, not following \(\{[a_0x+a_1)x+a_2]x + \dots\}\).

In addition to this we have to remark that the pre-Greek and Greek computational schemes for square and cubic roots – easily generalised for \(n\)-th roots – are far superior to the later developed algorithms …and independent of the representation of numbers used. The procedure developed by as-Samaw’al “dividing the difference by the double of the \((n-1)\)-th power of the integral part of the root to which one adds the lower powers” is not only inferior to other ancient procedures …but in many cases not even convergent in iteration: one has only to try the cubic root of 5 in \(x + (5 -x^3)/3\), \(3 =2a^2 +a\), \(a = 1\). The doubling of the \((n-1)\)-th power was undoubtedly suggested to as-Samaw’al by the factor 2 occurring for square and cubic roots, but it should have been “\(n\)-times the \((n-1)\)-th power”.

The system of al-Tûsi, concerning the general equation uses the old Babylonian KI-GUB, putting the unknown \(x+y\) and computing the coefficient of \(y\) as a function of \(x\), i.e. the first derivative in modern terminology. His scheme is therefore equivalent to Newton-approximation, which latter was in Newton’s time ameliorated by E. Halley considering third order approximations and later by Chebyshev …whose results can be obtained not using calculus but simple algebraical tools for the changing of \(y=f(x)\) to \(x=F(y)\), the convergence of the latter series requiring in general “small values” of \(y\) …which is obtained by a shift as al-Tûsi effectuates.

In an Appendix the Arabic text of a section of as-Samaw’al treatises is given. Here the computation of square, cubic, fourth and fifth roots of \(60,10,40,650\) is given according to \(a + (D - a^n)/D\), \(D= (a+1)^n - a^n\), which value of \(D\) is obtained using the proper coefficients of the difference polynomial.

The author considers the works of the school of al-Karaji and in particular of as-Samaw’al (1172) for the application of the decimal system in computing \(n\)-th roots. Also al-Tûsi’s work on more general equations is elucidated. The author discusses in detail the extraction of a fifth root by as-Samaw’al and sees – as is en vogue during some decades – “Ruffini-Horner”-methods, and a “Horner-triangle”. Horner’s paper (Phil. Trans. 1819) is mirabile dictu, concerned with the remark that one should never use more than two lines of the algorithm named nowadays after him, that one should not compute one more digit at each step, that one should not use more digits than necessary, i.e. for an equation of degree\(n\) never more than \(1/n\)-th of the digits wished in the result.

Al-Kâshî and as-Samaw’al are therefore doing just what Horner combats: computing too much and too many numbers without importance. Indeed when as-Samaw’al computes sexagesimally the 5-th root of the number \[ 2.33.43;3.43.3.43.36.48.8.16.52.30 \] at three places only \(2.33.43\) do play a role, all the others can be disregarded, and when al-Kâshî computes the 5-th root of \(44240899506197\dots\) the only digits playing a role for the three digits result are \(4424\). All the other computations are meant for “unskilled people” who have to follow blindly an algorithm.

Already in ancient Babylonian times by computing the powers of 1.1 – shifting and adding continuously – the “binomial coefficients” arose and the arithmetical series of higher order on the \(45^\circ\) lines became evident. Horner used these directly in computing a value of a polynomial, not following \(\{[a_0x+a_1)x+a_2]x + \dots\}\).

In addition to this we have to remark that the pre-Greek and Greek computational schemes for square and cubic roots – easily generalised for \(n\)-th roots – are far superior to the later developed algorithms …and independent of the representation of numbers used. The procedure developed by as-Samaw’al “dividing the difference by the double of the \((n-1)\)-th power of the integral part of the root to which one adds the lower powers” is not only inferior to other ancient procedures …but in many cases not even convergent in iteration: one has only to try the cubic root of 5 in \(x + (5 -x^3)/3\), \(3 =2a^2 +a\), \(a = 1\). The doubling of the \((n-1)\)-th power was undoubtedly suggested to as-Samaw’al by the factor 2 occurring for square and cubic roots, but it should have been “\(n\)-times the \((n-1)\)-th power”.

The system of al-Tûsi, concerning the general equation uses the old Babylonian KI-GUB, putting the unknown \(x+y\) and computing the coefficient of \(y\) as a function of \(x\), i.e. the first derivative in modern terminology. His scheme is therefore equivalent to Newton-approximation, which latter was in Newton’s time ameliorated by E. Halley considering third order approximations and later by Chebyshev …whose results can be obtained not using calculus but simple algebraical tools for the changing of \(y=f(x)\) to \(x=F(y)\), the convergence of the latter series requiring in general “small values” of \(y\) …which is obtained by a shift as al-Tûsi effectuates.

In an Appendix the Arabic text of a section of as-Samaw’al treatises is given. Here the computation of square, cubic, fourth and fifth roots of \(60,10,40,650\) is given according to \(a + (D - a^n)/D\), \(D= (a+1)^n - a^n\), which value of \(D\) is obtained using the proper coefficients of the difference polynomial.

Reviewer: E. M. Bruins (Amsterdam)

### MSC:

01A30 | History of mathematics in the Golden Age of Islam |