Abstract
Muhammad ibn Mūsā al-Khwūrizmiī1 wrote his renowned Kitāb al-jabr wa al-muqābala (Musharrafa and Ahmad, eds., 1939) at Baghdad between A.D. 813 and 833, during the reign of al-Ma’mūn. It was the first time in history that the term algebra appeared in a title to designate it as a discipline (al-Nadīm, Tajaddud, ed., 1971, pp. 338–341). Its recognition was not only insured by the title, but was confirmed by the formulation of a new technical vocabulary intended to specify its objects and procedures.
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Notes
G. Vacca (1909). Vacca was so convinced of the importance of his discovery that he published it in several other articles (1910) and (1911).
Al-Karaji (or al-Karkhi). Known since the translation of his book on Algebra by Woepcke and Hochheim on his book on Arithmetic (Kafïfra1-hisi b). Little is known about his life other than he lived in Baghdad in the late 10th century and early 11th century. For a scientific bibliography on al-Karaji, see Anbouba’s introduction (1964) and infra, I. 2.
Al-Samaw’al ibn Yahya ibn cAbbâs al-Maghribi, died in 1174. Al-Samaw’al’s autobiography is to be found in his polemical book, Ifham al-Yahúd (Perlmann,ed., 1964). A scientific bibliography is given in Rashed and Ahmad, eds. (1972). This chapter is based on the MSS 2718 Aya Sofia and 3155 Esat ef.; page numbers refer to the first manuscript.
Itard (1961, p. 73) wrote: “However, some demonstrations by recurrence or complete induction may be found. We never come across the modern rather pedantic leitmotiv: `we have checked the property for 2, we have shown that if is true for a number, it is true for the following, therefore it is general’ and those who only see complete induction in this hackneyed expression have the right to say it does not appear in the Elements.We see it in prop. VII, 3, 27 and 36; VIII, 2, 4 and 13; IX, 8 and 9”.
Al-Samaw’al, al-Bdhir,f. 43“, Rashed and Ahmad, eds. (1972), p. 104.
Ibid.,f. 43v, p. 104.
Ibid.,f. 44 p. 106.
Ibid.,ff. 44“-45`, pp. 107–108.
Ibid.,f. 45“, p. 109. Our italics.
Al-Samaw’al quoted in extenso the passage of this lost work quoted here.
To our knowledge, this is the first text, to state rules in such general terms. According to Needham (1959, p. 135), Yang Hui’s book dates back to 1261, therefore one and a half centuries after al-Karaji’s text. Al-Khayydm (1048–1131), in the wake of al-Karaji, and perhaps independently, was probably acquainted with these rules. Later in the 13th century, the same results are to be found in Nasir al-Din al-Túsi (ed. A.S. Saïdan, 1967, p. 145), with one exception: the binomial formula is always written verbally (a + b)“ — a” = Cn a“ - ”’b“’.They are also to be found in the 15th century in al-Kâshi’s Key to Arithmetic (see Luckey, 1951, p. 24 ).
In the table reproduced here, we have replaced the terms, root, square, cube... with the symbols x, x2, x3,… Al-Samaw’al, al-Bdhir,ff. 45“-47`, Rashed and Ahmad, eds. (1972), pp. 109–112.
Ibid.,f. 44“, p. 105.
Ibid.
Ibid.,ff. 45`-45“, p. 108.
From Fakhri,p. 48, reproduced in al-Bdhir,pp. 17–18.
Omnis radix multiplicata in radicem sequentem, producit duplum trianguli sibi collateralis. Demonstration: Exempli gratia, ducatur quaternarius in sequentem radicem, scilicet, quinarium: producuntur 20. Aio, quod 20. duplus est ad triangulum ipsi quaternario collateralem. Sumantur enim ab unitate ad quaternarium radices; quibus applicetur totide ordine praepostero ab unitate radices; singulae singulis: Sic enim fiet, ut crescentes cum decrescentibus singuli singulis conjuncti numeri faciant quatuor summas aequales: hoc est quatuor quinarios, quare earum aggregatum erit planus numerus, qui fit ex ductu quaternarij in quinarium: idcirco 20, erit talis planus: Duplus autem est planus ipse ad triangulum quaternarij: quandoquidem, per diff. talis triangulus est aggregatum unius dictorum ordinum: quod est dimidium plani: Igitur 20. duplus erit ad triangulum quaternarij. Et similiter in omni casu id quod proponitur demonstrabimus. Cited by Freudenthal (1953, p. 21) from Maurolico Arithmeticorum libri duo in Opuscula mathematica (1575).
I.e. instead of reasoning as usual in an almost general way for a particular n,he repeats the same reasoning for some particular numbers.
Al-Samaw’al, al-Bdhir,f. 54`, Rashed and Ahmad, eds. (1972), p. 127.
Ibid.,pp. 54`, 54“, pp. 127–128. Our translation, like other similar translations, follows the text closely, except for the substitution of signs for the terms: sum, difference, equality, etc.
Ibid.,ff. 61 “-62`, p. 143.
Ibid.,f. 62`, p. 143.
Ibid.,f. 62“, p. 143.
Ibid.,ff. 62`-62“, pp. 143–144.
Ibid.,f. 62“, p. 144.
Ibid.,f. 53`, p. 125.
Ibid.,f. 53“, p. 125.
This is basically the proof of a formula equivalent to P„.,i = (n + 1)P„ where P„ is the set of permutations of n distinct elements. For Levi ben Gerson, see (1909, pp. 48ff.). See Carlebach (1910) and Rabinovitch (1970, p. 242).
Hara (1962, p. 287) wrote: “For the first time in Pascal, we encounter not only a systematic application but an almost completely abstract formulation of the method rigorously intended”. Hara thought he conveyed not only his point of view but that of M. Freudenthal. However, the latter is apparently more reserved on this point as he wrote (1953, p. 33): “Nicht die Anwendung, auch nicht die systematische Anwendung ist das Auffallende, sondern die fast vollständig abstrakte Formulierung, die übrigens später nocheinmal, an anderen Objeckten, wiederholt wird”.
This involves the application of arithmetic to algebra or the extension of algebra to elementary arithmetical operations, so these operations may be applied to [0, co]. The obvious result of this project was to define the frontiers between algebra and geometry and achieve autonomy and specificity for algebra. The principal means was to extend abstract algebraic calculation. As this renewal progressed, the following discoveries were made for the first time in history:(1) the multiplication and division of algebraic powers;(2) the theory of the division of polynomials;(3) the calculation of signs.Concurrently, we encounter the calculation of binomial coefficients and the binomial formula, including various enumeration problems, later to be known as combinatorial analysis. See Rashed (1974) and our introduction to al-Samaw’al (1972).
For Frénicle and for Levi ben Gerson earlier, this concerns permutation problems. In fact Frénicle proves a formula equivalent to P„.. 1 = (n + 1)P„. For instance, having shown that P3 = 3P2 = 6; P4= 4P3 = 24; P5 = 5P4= 120, he writes (1729, p. 92), “and so on, we must multiply the preceding combination (read permutation) by the number of the given multitude; and that is clear proof which serves to demonstrate the construction of the table”. For studies on Frénicle, see Coumet (1968, pp. 209ff.).
Bachet ( 1624, p. 2): Bachet’s proof is based principally on type R,. Here Bachet writes as follows (pp. 3ff.): “Euclid proved in VII-16, that for two numbers, whether the first is multiplied by the second, or the second by the first, the product is always the same. Here I want to prove that the same applies to three or more numbers. However, two or more numbers are said to be multiplied together when we multiply two together and the product of the one and the product of the other, and so on or as many numbers as there are.Firstly, let us take three numbers A, B, C, and by multiplying A by B we have D, which multiplied by C produces E. Let us change the order and multiply B by C which gives F which, multiplied by A gives G. Let us change the order once again and multiply A by C, which gives H which, multiplied by B gives K (these are all the different ways for multiplying three numbers together). I say that three products E•K-G have the same number. Since B multiplied by A • C. gives D•F, there is a ratio of A to C and D to F therefore the same number is given by multiplying A by F and C by D in 19 in 7. Let us start with E and G the same number. Similarly, since C multiplied by A and B gives H and F, there is a ratio between A and B and between H and F. Therefore, the same number is given multiplying A by F and B by H. Therefore, the same number is given multiplying A by F and B by H, therefore KG is the same number. Consequently, all three E•K•G. are the same number which it what had to be proved. Now let us take four numbers A•B•C•D and multiply A by B and the product of C which gives E, which multiplied by D gives K. Then let us change the order and multiply D by C and the product by B which gives F which, multiplied by A gives H. I say that K•H is the same number and that the same number will always be produced in any way when we multiply the four numbers A•B• C•D together. Since by multiplying together on one side the three numbers A • B - C and on the other side D•C•B, we find B•C on both sides, multiply B•C together, which gives G. However, by what was proved for three numbers the same E, I say, which is made by multiplying A by B and the product by C, the same E is reached as well by multiplying B by C and the product (i.e. G) by A. Similarly, we will prove that F is made by multiplying D by G, therefore as the same G multiplies the two A•D produces E.F. There is the same ratio in A•D between E.F. Consequently, the same number is reached by multiplying A by F and D by E. Therefore, K•H are the same number. However, using the same means we always prove the same. Since of four numbers, by multiplying three together on one side and three on the other, there will always be two the same between three taken from one side and the other and consequently the same proof occurs”.
Pascal (ed. Seuil, 1963), p. 53. This is the twelfth consequence which states: “in every arithmetical triangle, two contiguous cells having the same base, the higher is to the inferior like the multitude of the cells from the higher to the top of the base to the multitude of those from the inferior to the bottom inclusive”.
For instance, after criticizing Wallis, J. Bernoulli proceeds by complete induction to find the general rule for the sum of squares, cubes, etc. and n first natural integers, i.e.k=1with c = 1, 2, 3,… See Bernoulli (1713, pp. 96ff.).
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Rashed, R. (1994). The Beginnings of Algebra. 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_2
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