7. arithmetic & number theoretic recreations a. Fibonacci numbers

First Book, Ninth Section: On arithmetic checks, f. 274a, pp. 70-71. Brief description of casting out nines.

Second Book, Eleventh Section: On checks, ff. 280a-280b, pp. 94-97. Brief description of casting out nines in base 60.

Introduction, pp. 32-33, discusses the above, noting that use of 9 in base 60 is unreasonable, but others also did it. Says Sibţ [NOTE: ţ denotes t with an underdot.] al-Māridīnī (16C) used casting out 8s and 7s in base 60, but that al-Kāshī (15C) used casting out 59s in base 60. Kūshyār does not state that the check proves the correctness of the result, though this was commonly believed, e.g. by Fibonacci and Sibţ [NOTE: ţ denotes t with an underdot.], though al-Kāshī clearly discusses the question.

Ibn Sina = Avicenna. Treatise on Arithmetic. c1020. ??NYS. Complete rules for checking operations by casting out 9s, attributed to the Hindus, (Smith, Isis 6 (1924) 319). (See also Cammann -- 3 (cited in 7.N); Datta & Singh, I, 184; and Kaye, above, who cite F. Woepcke; Mémoire sur la propagation des chiffres indiens; J. Asiatique (6) 1 (1863) 27-529; p. 502, ??NYS.) The DSB entry indicates that the material is in Ibn Sina's Al Shifâ' (The Healing) and there doesn't appear to be a translation. Suter 89 mentions some Latin translations but I'm not clear whether they are this book or a related book.

Saidan's discussion says Woepcke (p. 550 [sic]) construes ibn Sina as saying that the method is Indian, but this is a contentious interpretation. Kaye, above, says Woepcke is wrong. Smith, History II 151, says the expression "has been variously interpreted".

Bhaskara II. Lilivati. 1150. Smith, History II 152, cites this in Taylor's edition, p. 7, but the method is not in Colebrooke and neither Dickson nor Datta & Singh cite it, so perhaps it is an addition in the text Taylor used??

Fibonacci. 1202. Pp. 8-9, 20, 39, 45 (S: 24-26, 41, 67, 74) uses checks (mod 7, 9 and 11). On p. 8 (S: 24), he implies that if the 'proof' is right, then the calculation is correct -- see comments at Kūshyār above.

Maximus Planudes. Ψηφηφoρια κατ' Ivδoυσ η Λεγoμεvη Μεγαλη (Psephephoria kat' Indous e Legomene Megale) (Arithmetic after the Indian method). c1300. (Greek ed. by Gerhardt, Das Rechenbuch des Maximus Planudes, Halle, 1865, ??NYS [Allard, below, pp. 20-22, says this is not very good]. German trans. by H. Waeschke, Halle, 1878, ??NYS [See HGM II 549; not mentioned by Allard].) Greek ed., with French translation by A. Allard; Maxime Planude -- Le Grand Calcul selon des Indiens; Travaux de la Faculté de Philosophie et Lettres de l'Univ. Cath. de Louvain -- XXVII, Louvain la Neuve, 1981. Proofs by casting out 9s are given in the material on the operations of arithmetic.

Narayana Pandita (= Nārāyaņa Paņdita [NOTE: ņ denotes n with an overdot and the d should have an underdot.]). Gaņita[NOTE: ņ denotes n with an underdot.] Kaumudī (1356). Edited by P. Dvivedi, Indian Press, Benares, 1942. Introduction in English, p. xv, discusses the material. Allows any modulus. (English in Datta & Singh I 183.)

The Treviso Arithmetic = Larte de labbacho. Op. cit. in 7.H. 1478. F. 4v onward (p. 46 in Swetz) uses casting out 9s as a check on many examples. Swetz (p. 189) refers to Avicenna and the Hindus. On f. 10v (Swetz p. 59), the anonymous author says that proving a subtraction by addition "is more rapid and also more certain than the proofs by 9s" and he makes similar statements regarding multiplication and division.

On p. 323 of his Isis article, Smith says the author "gives a proof by casting out sevens". This would be on or near f. 17v (Swetz p. 73). I can find nothing of the sort -- the author has an example of multiplication by 7, but he checks it by casting out 9s.

Borghi. Arithmetica. 1484. Ff. 8r-9r (1509: ff. 9r-9v). Casting out 7s and 9s. This is applied over the next few sections, but I don't see any indication that casting out 9s is not a certain test. However he uses casting out 7s more often than 9s which may indicate that he was aware that 7s is a more secure test than 9s.

Chuquet. 1484. Triparty, part 1. English in FHM 41-42. "There are several kinds of proofs such as the proof by 9, by 8, by 7, and so on by other individual figures down to 2, .... Of these only the proof by 9, because it is easy to do, and the proof by 7, because it is even more certain than that by 9 are treated here." He then notes that these proofs are not always certain.

Pacioli. Summa. 1494. Ff. 20v-23v. Discusses casting out 9s and 7s and notes that these tests are not sufficient.

Apianus. Kauffmanss Rechnung. 1527. Gives numerous examples of testing by 9s, and also by 8s, 7s and 6s, in his sections on the four arithmetic operations and also under arithmetic progressions.

Recorde. First Part. 1543. Discusses the proof by nines in his chapters on: addition, ff. D.i.r - D.iii.v (1668: 29-32: The proof of Addition); subtraction, ff. F.iii.r - F.iiii.r (omitted in 1668); multiplication, ff. G.vi.r - G.vi.v (1668: 70-72: Proof of Multiplication); and division, ff. H.iii.v - H.iiii.v (1668: 82-84: Proof of Division).

Hutton. A Course of Mathematics. 1798? 1833 & 1857: 6-12. In his discussion of the basic arithmetic operations, we find on p. 7 under To Prove Addition, "Then, if the excess of 9's in this sum, ...., be equal to the excess of 9's in the total sum ..., the work is right." A footnote explains the idea and is less clear as to the direction of implication being asserted: "it is plain that this last excess must be equal to the excess of 9's contained in the total sum". The note concludes: "This rule was first given by Dr Wallis in his Arithmetic, published in the year 1657." However, Hutton does not mention the rule under subtraction and under multiplication on pp. 10-11, he says the "remainders must be equal when the work is right." All in all, it seems that he is surprisingly unclear for his time.

Boy's Own Book.

The number nine: "To add a figure to any given Number, which shall render it divisible by Nine". 1828: 179-180; 1828-2: 235; 1829 (US) & 1881 (NY): 103; 1855 & 1859: 389; 1868: 429; 1880: 459. Here the digit is actually added, but then he indicates that it can be inserted. [Incidentally, this avoids the hazard of discovering the missing digit might be either 0 or 9.] This section is extended and combined into Properties of certain numbers from 1868. Cf 1843 (Paris): 342