Abstract
In the history of mathematics a discovery is sometimes seen to have no real effect for a more or less indefinite period of time. Buried in a “relative absence”, it remains intact though outside the mathematical corpus in current use. One may speak of “absence” in so far as when the discovery occurred, it did not emerge as an active part of mathematical practice; but its “absence” is relative since the discovery did take place and was transmitted. Subsequently, even if its transmission was presented as the simple heritage of a succession of authors and not as the communication of a chapter of received mathematics, the discovery becomes an inalienable acquisition for the history of the discipline.
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Notes
See the reproduction of De Thiende and its English translation by Robert Norton (1608) in Struik (1958, pp. 386ff.). See also the French edition in Sarton (1935, pp. 230–244).
For instance, E. J. Dijksterhuis (1970, pp. 16 and 18), the best specialist on Stevin wrote: “Stevin’s main contribution to the development of mathematics being his introduction of what are usually called decimal fractions”, and further on: “Yet none of the steps taken by Regiomontanus and other writers is comparable in importance and scope with the progress achieved by Stevin in his De Thiende.” See also Sarton (1935, p. 174): “There are many examples of decimal fractions before 1585 yet no formal and complete definition of them, not to speak of a formal introduction of them into the general system of numbers”. We could multiply such references and quote numerous authors who consistently repeat this opinion. We shall simply give a recent example, that of J. F. Scott, who wrote in 1969 (p. 127): “Nevertheless, it was not until the close of the sixteenth century that we detect the first methodical approach to the system. In 1585 there appeared a short tract La Disme by Stevin… In this, the principles of the system, and the advantages which would follow from its use, are clearly set forth.”
See Gandz’s article with a preface by Sarton (1936, pp. 16–45).
Although Gandz (1936, p. 21) wrote: “The invention of Bonfils introduces two new elements; the decimal fractions and the exponential calculus”, all that may be deduced from the translation of Bonfils’ Hebrew text, reproduced by Gandz, is summarized in Juschkewitsch (1964b, p. 241) as follows: “Die kurze Skizze eines Systems von `Primen’, `Sekunden’, `Terzen’ usw. in einer Handschrift des jüdischen Mathematikers Immanuel ben Jacob Bonfils, der im 14. Jahrhundert in Tarascon gelebt hat, ist im Vergleich zur Dezimalbruchlehre al-Käsis völlig unbedeutend. — Dabei hat Bonfils keinerlei Berechnungen mit Hilfe von Dezimalbriichen vorgenommen.”
Tropfke (1930, p. 178) may be cited among many, who wrote: “Wenn noch andere Männer neben Stevin als Erfinder der Dezimalbrüche genannt werden, so ist das nicht zu verwundern. Die Erfindung der Dezimalbruchrechnung lag gleichsam in der Luft; Gelehrte aus allen Ländern beteiligten sich an ihr.” This same idea is expressed by Sarton (1935, p. 173). Lastly, see Cajori (1928, I, p. 314): “The invention of decimal fractions is usually ascribed to the Belgian Simon Stevin, in his La Disme published in 1585. But at an earlier date several other writers came so close to this invention, and at a later date other writers advanced the same ideas, more or less independently, that rival candidates for the honor of invention were bound to be advanced. The La Disme of Stevin marked a full grasp of the nature and importance of decimal fractions, but labored under the burden of a clumsy notation”.
Struik (1969, p. 7). A less eclectic but more embarrassed position is held by H. Gericke and K. Vogel, the German translators of Stevin’s La Disme. They wrote: “Al-Käschï bringt aber nicht nur die vollständige Theorie, sondern er führt auch die Rechnungen gelegentlich im einzelnen vor, einschließlich der Verwandlung von Sexagesimalzahlen und Brüchen in Dezimale und umgekehrt wobei er zur Trennung von Ganzen und Brüchen sich verschiedener Methoden bedient…” In fact, according to these authors (1965, p. 45) the only difference with Stevin is described as follows: “Was aber bei ihm im Gegensatz zu Stevin auch nicht zu finden ist and was diesem ein Hauptanliegen war, ist die konsequente Anwendung auf alle Masse, deren dezimale Einteilung von grösster praktischer Bedeutung sein musste”. On the one hand it is known that these applications had no real impact, and on the other, that al-Kâshi, as we shall see, proceeds with conversions other than certain measurements current at that time: it would therefore be incorrect to emphasize the difference. See Gericke and Vogel ( 1965, pp. 44, 45 ).
See A. Saidan’s edition of Kitab al-Fusin fi al-Hisab al-Hindi,1973.
We know through ancient Arabic bibliographers that al-Samaw’al wrote a Treatise on Arithmetic entitled al-Qiwami ft al-Hisab al-Hindi. The thirteenth-century bibliographer Ibn AbT Usaybica wrote in his Tabaqat (1965, p. 462) that al-Samaw’al had completed this treatise in 568 of the Hegira (1172–1173). See also Suter (1900) and Sezgin (1974, p. 197). The entire work is still lost. But a work by al-Samaw’al entitled al-Maqala al-thalitha ‘ilm al-misaha alhindiyya (“The third volume of the work on Indian mensuration”) is to be found in the Biblioteca Medicea Laurenziana, Orient. ms. 238. This manuscript comprises 115 in-folio pages and the copy, dated 751 of the Hegira, is in poor condition. The above title is obviously incorrect. I take this opportunity to thank Sezgin (1974, p. 197), who sent me a microfilm copy of this manuscript, whose existence he mentioned in the above work, observed quite correctly that the manuscript “hat trort ihres Titels mit der indischen Ausmessung nicht direkt zu tun”. We can effectively show that it is a change with 115 in-folio pages from al-Qiwami (“Treatise on Arithmetic”). The copist wrote at the end (ff. 114`, 114“): ”We have completed the book al-Samaw’al composed at Baku and finished on 29th of the month of Ramadan the year five hundred and sixty-eight“. He also mentioned possessing a copy written by al-Samaw’al himself. The subject, dates and attribution leave no doubt as to the identity of the manuscript. We set forth the results of this discovery for the first time at the Congress of the History of Arabic Sciences at Aleppo and we have undertaken a critical edition of this difficult text.
Al-Samaw’al, al-Qiwami,f. 32’.
Juschkewitsch (1964b) where the author apparently repeats the conclusions of the analysis in the introduction of the Russian translation of al-Kâshï’s work (Al-Kâshi 1956). A. S. al-Demerdash and M. H. al-Cheikh, the editors of al-Kâshi’s work (1967) apparently shared Luckey’s opinion. Lastly, in a detailed study by Dakhel (1960), Luckey’s analysis and point of view is repeated.
Al-Qiwami,f. 108’. Al-Samaw’al used the Jummal system to express numbers. It consists of the 28 letters of the Arabic alphabet arranged in an ancient Semitic order for expressing numbers. For typographical reasons we have used the corresponding numbers.
Ibid.,f. 108`.
The literal translation of al-mu’tiya is “indicator”, which means “which gives one of the figures of the root”. The text would have been incomprehensible if we had retained this translation.
Al-Qiwami,ff. 108`-108“.
Al-mu`tiya (see note 13).
Al-maraf ï`.
Al-Qiwâmi; ff. 108`-109`. In the ms. the left-hand column headings are omitted both in this and the following tables.
Let f be a real function with a real continuous variable: a a real number strictly positive, we call the dilation of f of ratio a the application (p,,; cp,.(x) = f(a lx) for any x.
Al-Qiwâmi,f. 109’.
Ibid.,f. 104“.
Ibid.,ff. 104`-105’.
Ibid.,f. 109’.
Ibid.,f. 109“.
Ibid.
Ibid.
Ibid.
Ibid. There is obviously an important break in the manuscript. We have been able to establish that f. 110` follows f. 69’.
Ibid.,f. 109’.
Infra,Chap. III, pp. 153ff.
Infra,Chap. III, pp. 201–202, note 12.
Infra,p. 158. See also al-Tosi (1986), I, pp. 32ff.
I.e. expansion of the relation 0 a 1.
Infra,p. 164.
Ibid.
Infra,pp. 162ff.
Al-Qiwâmi,f. 110r.
Infra,pp. 202–203, note 15.
Al-Qiwâmi,f. 111’.
Ibid.
Al-Qiwâmi,ff. 68’ and 68“.
Ibid.,ff. 68“, 69’, 69”.
This point was brought to our attention after publication of this article in a correspondence with M. Bruins and also independently by Waterhouse (1978).
Al-Takmila fi al-Hisab,MS 2708, Lâleli, Istanbul, 22’ff. and 29’ff.
Al-Qiwâmi,ff. 58’ff.; 62’ff.; 64’ff.
Al-Nasawi, al-Mugni` fï al-hisâb al-hindi,MS Leiden arabe, n°556, ff. 21, 22.
Al-Samaw’al, al-Tabsira fï ‘Um al-hisâb,MS Oxford Bod. Hunt. 194, f. 18’.
See Al-Qiwâmi,f. 27“.
Ibid.,f. 111’.
Ibid.,ff. 111’, 111“.
Al-Samaw’al, Al-Bahir,1972, pp. 21 and 22 (Arabic text) and pp. 18 and 19 (French introduction).
Ibid.
Al-Qiwâmi,f. 112“.
Ibid.,f. 113’.
Ibid.,f. 114’.
Ibid.
Saidan (1966). The same idea was expressed on several occasions in his edition of al-Uglidisi’s book. He writes for example: “What inclines us to be proud of al-Ug1Idisi is that he was the first to deal with decimal fractions, suggest a sign for separating integers from fractions, and treat fractions as he treats integers. Before al-Uglidisi was known, common opinion held that the first to deal with decimal fractions was Gamnd ben MasTid al-Käshi’, p. 524.
In his English translation of this passage, A. Saidan (1966, p. 485) integrated the sign in the text. We may therefore read: “… and mark the unit place with the mark ’ over it…”. However, if we consult the manuscript of al-Fusel,this sign is missing. Nor is it to be found in Saidan’s edition, only in his English translation. For this reason we prefer to use the manuscript although we always refer to Saidan’s edition, easier to consult.
Al-Uglldisi gives the following example: “we want to separate 19 into two halves five times, we say: half of 9 is four and a half, we write half 5 in front of 4, we divide ten into two halves, and we mark the unit position, we obtain 95; we separate five and nine into two halves, we obtain 475, which we divide into two halves, we obtain 2375 and the unit position is (the position) of thousands for what precedes it. If we want to pronounce what we have obtained, we say that the separation into two halves makes two, plus 375 of a thousand. We separate this into two halves, we obtain 11875, we separate into two halves a fifth time, we obtain 059375, we find 59375 in Saidan’s edition which is 59375 of a hundred thousand. Its ratio is said to be a half plus a half eighth plus a quarter of an eighth”. Al-Fustil,MS 802, Yeni Carni, Istanbul, 58`. See al-Uglldisi, Al-Fusúl,pp. 145–146.
In the text one reads “the calculation of stars”, this is probably a copyist’s error. The accepted term is the calculation of astronomers.
Luckey ( 1951, p. 103) wrote: “Während also K. die ganzen wie die gebrochenen Sechzigerzahlen von Vorgängern übernahm, schreibt er sich wiederholt ausdrücklich die Einführung der Dezimalbrüche zu. Meines Wissens fand man bisher zwar in keinem älteren arabischen Texte, wohl aber in Schriften, die arabisches Gut wiedergeben oder auf solchem fussen, den Gedanken ausgesprochen, daß an die Stelle der Grundzahl 60 der Sexagesimalbrüche eine andere Grundzahl treten könne, als welche im (Algorismus de minutiis) von Seitenstetten aus dem 14. Jahrhundert neben 12 auch 10 genannt sein soll. Auf das, was Immanuel Bonfils aus Tarascon über Dezimalbrüche sagt, soll später eingegangen werden. Der Gedanke der Dezimalbrüche mag also im Mittelalter in der Luft gelegen haben. Wie andere vor und nach ihm, so kann auch K. sehr wohl selbaständig den Einfall gehabt haben, nach dem Vorbild der Sechzigerbrüche Dezimalbrüche einzuführen. Jedenfalls aber hat man bisher in keiner vor seine Zeit fallenden Schrift eine ausführliche praktische Durchführung der Methode der Dezimalbrüche im Positionsystem, wie er eine solche bringt, nachgewiesen.”
Luckey (1951, pp. 115ff.). Let us remark moreover that the problem of the periodicity of the fraction may occur during the solution to this problem. We know that it is always possible to express a decimal fraction exactly by a sexagesimal fraction; but it is not always possible to express a sexagesimal number by a limited decimal fraction. In the French translation of the section on the history of Arabic mathematics, Youschkevitch (1976) wrote: “Note that al-Käsh7 neither mentioned nor remarked on the obvious periodicity of the fraction 0,141 (592) he obtained”. In a note on the French translation (p. 168), he recalled a remark made by Carra de Vaux: the periodicity of a sexagesimal fraction is indicated by the thirteenth century mathematician al-Mârdlnl. We are now able to show that the periodicity of a sexagesimal fraction was already known in the twelfth century. For instance, to convert a fraction — e.g. 4/11 — into a sexagesimal fraction, al-Samaw’al obtains;21,49,5,27,16. He then writes (al-Qiwâmi,f. 89`): “And these five figures repeat themselves indefinitely. If we are satisfied with limiting ourselves to a tenth of a tenth [60-20] for example, the answer is;21,49,5,27,16,21,49,5,27,16,21,49,5,27,16,21,49,5,27,16 and if we want a more correct [number] than this we repeat these five figures up to a higher [place] of these places”. These computations demonstrate that the problem of periodicity at least was already known in the twelfth century.
Hunger and Vogel (1963). The example given is the following: to calculate the price of 15312 measures of salt, the price of each being 161/4 aspra, i.e. 15312 161/4. The author wrote that the Turks put 5 instead of half, and 25 instead of a quarter. So we obtain 2494375, from which we separate the last three figures which are in the three last positions (tyriOt x). The calculation was made as follows: We note that the zero (oOOSéu) is indicated by a point, to what digits the Greek letters correspond in the positional number position and that the fractional part is separated by a vertical line.
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Rashed, R. (1994). Numerical Analysis. 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_3
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