Abstract
In the beginning was Viète. Thomas Harriot, William Oughtred, Claude-François Dechales, John Pell..., all improved the method in one way or another,1 Newton2 took it up later. Modified by Joseph Raphson, this method is known as Newton’s method in textbooks on numerical calculation. Lagrange, J. R. Mouraille and Fourier3 attempted to solve its difficulties. Independently of each other, Ruffini and Horner4 developed investigations by Viète and Newton: they proposed a more practical algorithm for extracting the root of a numerical equation of any degree.
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Notes
Harriot (1631), pp. 117–180. Herigone (1634), vol. 2, pp. 266ff. Oughtred (1652), pp. 121–196. Dechales (1647), 1, pp. 646–652. Prestet (1689), vol. 2, pp. 432–440. Wallis (1685), pp. 113–117.
In his famous letter dated 26th June 1676, Newton wrote: “Extractiones in numeris, sed methodus Vietae et Oughtredi nostri huic negotio minus idonea est, quapropter aliam excogitare adactus sum…Newton expounded his method in a letter to Oldenbourg and Leibniz dated 26th July 1677; See Gerhardt (1962), pp. 179–192. See also his letter to Collins dated 20th June 1674: Turnbull (1959), pp. 309–310. Turnbull gives the references to other letters on the same problem. We know the method is to be found in De analysi per aequationes numero terminorum infinitas (1669), reproduced in Methodus fluxionum et serierum infinitarum (1671) and only published in 1736. The first edited summary is in Wallis (1685), pp. 381–383. See also Buffon’s ”Introduction“ (1740) and Cajori (1911), pp. 29–30. Lastly, see Whiteside (1964), I, pp. 928ff.
Lagrange (1878), pp. 159ff. Mouraille (1768), Part 1. Fourier (1830). See also Cajori (1910–11), pp. 132–137.
Homer (1819), Part 1, pp. 308–335 and also Smith (1959), 1, pp. 232–252.
Burnside and Panton (1912), 1, note B: “The first attempt at a general solution by approximation of numerical equations was published in the year 1600 by Vieta. Cardan had previously applied the rule of `false position’ (called by him ‘regula aurea’) to the cubic; but the results obtained by this method were of little value”. Whittaker and Robinson (1926), see ch. 6, par. 41. Young (1843), pp. 248ff.
Sédillot (1847), pp. 69–83. Woepcke (1854a), p. xix. To calculate the value of sine he equation must be solved; B is of a higher order thanThe method exposed by Shalabi is based on an idea common to a family of approximation methods: replace the initial equation by a linear or an approached equation as often as one wants.hence, using the method of indeterminate coefficients, we havefor but then a k, b are integers, x k are not usually integers. We then take the integer part and obtainThis type of solution consists of “substituting the third-degree equation proposed by an infinity of linear equations”. A detailed description of this method is to be found in Woepcke (1854a) and Woepcke (1854b) pp. 153ff. See also Hankel (1874), pp. 290–292.
Luckey (1948), pp. 217–274. To give a general summary of al- method it should be recalled that the author solves the equation He takes as a first approximation the highest integer lower than and obtainshence Luckey showed that al-Kashused Horner’s table to calculate the coefficients for each transformed function.
Luckey (1951) analyzed the work and translated some passages. The only edition of this work is by A. S. al-Demerdash and M. H. al-Cheikh (1967). A Russian translation of al-Kashi’s work with a photographic reproduction exists, without a critical edition of the text. The translation is by Rosenfeld: al-Kash- (1956).
Luckey (1948), p. 248.
See our edition: al-Samaw’al (1972) and supra,pp. 22–33.
See Woepcke’s translation: Woepcke (1851).
The case of Sharaf al-Din al-Tus! is not uncommon in the history of mathematics. The importance of his work, frequently reaffirmed by algebraists, contrasts with a total lack of historical studies on his work. And if the same situation is almost true for al-Samaw’al, the case of al-Túsi is more surprising, it reveals the inadequacy of historical research in this field and renders our knowledge of Arabic algebra and the Renaissance precarious. Al-Túsi was not only cited by Arab algebraists but by historians as well. For instance, al-Kash! cited him in The Key to Arithmetic on the solution of cubic equations, p. 198. He was also cited by the famous Kamal al-Din al-Fârisi in MS Shehid Ali Pasha 1972, Asas al-Qawa’id fi usúl al fawa=id (“The foundation of rules…”). Several references show that some algebraists recognized the importance of al-Túsi’s contribution. For instance, the early thirteenth-century mathematician Shams al-Din Ismril ibn Ibrâhim al-Mardini attributed the invention of the “table method” to him, i.e. the numerical solution of cubic equations. See Nisab al-habr,MS Istanbul, Feyzullah, 1366. Al-Túsi was no less unknown to ancient or modern biographers. Sarton wrote his biography and recalled that he composed “a treatise on algebra… in 1209–10 [which] is only known through a commentary [talkhis] by an unknown author”. See Sarton (1950), II, pp. 622–623. This fact was mentioned in Suter’s bibliography earlier and in the MS in the India Office. See Suter (1900, p. 134). See also Brockelmann (1898), 1, p. 472. Al-Túsi is also known as the inventor of the Linear Astrolabe since the translations by Suter and Carra de Vaux. See Carra de Vaux (1895), pp. 464–516. Also Suter (1895), pp. 13–18 and (1896), pp. 13–15. Ancient biographers, at least those we were able to consult, cites al-TUsí but without giving any important biographical details. See al-Qifti (ed., 1903), p. 426. Ibn Khallikan (ed., 1978), p. 314 where we may read: “Shaykh Sharaf al-Din al-Muzaffar ibn Muhammad ibn al-Muzaffar al-TUsí is the inventor of the linear astrolabe called the stick”. Tashkupri-Zadeh (1968), p. 392: “Sharaf al-Din, Muhammad Masud ibn Masud al-Masud! [composed] a ‘development’ [the opposite of a summary] in this discipline [Algebra]”. As yet virtually nothing is known about his life. He lived in the twelfth century, taught in Damascus where the famous Muhadhdhab al-Din ibn al-Din al-Hajib was his pupil, and in Mosul where his pupils were Kamal al-Din ibn Yunus and Muhammad ibn `Abd al-Karim al-Harithi, and lastly, in Baghdad. From Tus in Khurasan, he probably died in the last quarter of the 12th century.Of his works, the following are known: Risâla fi sanc al-asturlab al-musattah,MS Leiden 591 (“A Discourse on the linear astrolabe”); “A Reply to a geometrical question asked by his friend Shams al-Din”, cited by Suter; Risola fi al-hattayn alladhayni yaqrubani wa la yatagabalani,cited by Brockelmann (1898), I, supp. Bd. p. 850, probably a treatise on asymptotes; lastly, the Algebra. The only surviving copy of his treatise on algebra entitled On equations (al-Túsi, ed., 1986) is in the India office, London. For a translation and complete study of this work including a new translation of a1-Khayyâm’s work see al-Túsi (ed., 1986). The text is not a “commentary” as Sarton wrote, but really a summary as the unknown author explained: “In this work I wanted to summarize the art of algebra and al-mugabala,adapt what has survived from the great philosopher Sharaf al-Din al-Muzaffar ibn Muhammad al-Túsi, and reduce his overlengthy exposition to a moderate size; I eliminated the tables he drew up to make his computations and solve his problems”. The manuscript is therefore an adaptation of al-Tusi’s algebraic work minus the tables and some figures; if we include the copyist’s errors, it is understandable why it is difficult to read. This probably explains why it has never aroused the curiosity of historians. The manuscript comprises 154 in-folio pages.
We now know that this work on algebra was completed by Karaji and his successors such as al-Samaw’al. See al-Samaw’al (ed., 1972), and Rashed (1971).
Infra, I.4, pp. 64–68.
In this respect, al-Tús1’s work may only be understood if sufficient emphasis is laid on the development of method invented to extract roots from a number. Two movements have marked the history of this problem: the first was dominated by the first algebraist, al-Khwúrizmi, while the second was almost achieved by the renovator of algebra: al-Karaji. If the Arabic text of al-Khwúrizmi’s arithmetic remains lost, its Latin translation has, on the other hand, survived, see Vogel (1963). This text teaches us that al-Khwíirizmi saw the problem of the extraction of the square root as a step towards the systematic study of arithmetical operations. He gave the rule of approximation for the square root of a numberThis fact was confirmed by al-Khwârizmi’s Arab successors. For instance, al-Baghdâdi (d. 1037), in his work al-Takmila (MS Laleli 2708/1 Istanbul), attributed this approximation to al-Khwürizmi, and recalled that later Arab mathematicians deliberately abandoned it as unsatisfactory for values such as q2, 3. Much more important than the rule of approximation are al-Khwârizmi’s fundamental ideas on the subject and which may be Hindu in origin. He uses both the development His method consists of:differentiating between two-digit groups, i.e. the positions 10the set of digits of the number from which one wants to extract the root, working from right to left.(2)hen finding a number whose square is the highest square included in the last group (on the left) of two digits. This number will be the first digit of the root, that is a, written in its decimal order.(3)subtracting a to obtain the first remainder, and determining the second digit of the root with its decimal order, that is b, subtract tab, b 2 from the first remainder and so on. Al-Khwârizmì’s computation was not direct and his exposition remained imperfect. Later on, Arab mathematicians attempted to improve the approximation, and perfected the representation of the method, and finally extended it to the extraction of higher order roots: these were the three goals al-Khw.rizmi’s successors attempted to reach.by default and Kúshyar ibn Labban (1000 c.) wanted to improve the result and representation. For instance, in the example N = 65342, he proposes the following figures: In this representation, he endeavours, as we see, to give the figure of the roots higher than N written out in full, and explicitly marks the positions and decimal order of each number in order to make the procedure uniform or “standard”. See Saidan (1967), pp. 65–66 and the English translation with an historical introduction: Levey and Petruck (1965). Ibn Labban’s pupil, al-Nasawi went further, at least as far as the roots of fractional numbers are concerned. Arab mathematicians were later to improve on the representation and indicate the group of two digits by small circles, like Sharaf al-Din al-Túsl. Kúshyàr ibn Labban and his pupil, al-Nasawi, did not stop there, they extended the same method to the extraction of cubic roots. They therefore used the developmentnd always the decimal decomposition. their own formula,while other Arabic mathematicians used what Nasir al-Din al-Tilsi was to call “conventional approximation n approximation that is to be found later in Leonardo of Pisa. See al-Túsi (ed., 1967), pp. 141ff. This method of extracting the roots of ”pure powers“ according to sixteenth-century terminology, was to be found together with some inessential variations by mathematicians before al-Tiisi. So we have attempted to show the result rather than a real history of the problem. See also Suter (1966), pp. 113–119; Luckey (1948).
Woepcke (1851), p. 13.
See Schau (1923), 8, p. xxxii; Wiedemann and Suter (1920–21); Boilot (1955), 2, p. 187.
There is evidence of a generalized usage of tables in al-Samaw=al (ed., 1972).
Only al-Mârdinï expressly attributed the invention of this method to al-Túsi (ed., 1986 ), I. But in absence of further confirmation, this evidence is not decisive.
Viète (ed., 1970): “Numerosam resolutionem potestatum purarum imitatur proxime resolutio adfectarum potestatum see also p. 224.
Viète (ed., 1970), p. 173: “Intelliguntur videlicet componi adfectae potestates à duobus quoque lateribus, immiscentibus se subgradualibus magnitudinibus, una vel pluribus, and in eadem resolvuntur contraria compositionis via, observato coefficientium subgradualium, sicut potestatis, and parodicorum graduum, congruente situ, ordine, lege, and progressu.”
Al-Túsi (1986), I, pp. 47–48; ff. 58–59“.
Al-Túsi (1986), II, pp. 19–32; ff. 113–120`.
The conclusion HL.HB is not valid for any value of a; it is not indispensable here as it may be shown directly that HL
Al-Túsi (1986), II, pp. 71–72; f. 143’.
Al-Túsi (1986), II, p. 73; f. 144’; the expression for the higher limit is: nihâya fï al-izam.
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Rashed, R. (1994). Numerical Equations. In: The Development of Arabic Mathematics: Between Arithmetic and Algebra. Boston Studies in the Philosophy of Science, vol 156. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3274-1_4
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