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Published online by Cambridge University Press: 03 November 2016
In Mathematics crosses are not signs of illiteracy, but symbols of varied productive achievement.
It is well known that the St. Andrew’s cross (X) occurs as the symbol for multiplication in W. Oughtred’s Clavis Mathematicae, 1631, and also (in the form of the letter X) in an anonymous Appendix which appeared in E. Wright’s 1618 edition of John Napier’s Descriptio. This Appendix is very probably from the pen of Oughtred. The question has arisen, is this the earliest use of X to designate multiplication? It has been answered in the negative—incorrectly so, we think, as we shall endeavour to show.
Page 136 of note 1 Clavdii Ptolemaei Pelusiensis Alexandrini Omnia quae extant Opera. Basileae, 1551, Lib. ii. Annotationes.
Page 136 of note 2 Emil Zirkel, Zeitschr. f. math. u. naturw. Unterricht, vol. 52, 1921, p. 96. An article on the sign ×, which we have not seen and which we know only from references to it made by other writers, is N. L. W. A. Gravelaar in Wiskundig Tijdschrift, i. (1904), pp. 122-124.
Page 136 of note 3 Leonardo of Pisa, Liber Abaci, 1202. Edited by B. Boncompagni, Roma, 1857, vol. 1, p. 84.
Page 137 of note 1 Op. cit. vol. 1, p. 127.
Page 137 of note 2 Op. cit. vol. 1, p. 319.
Page 137 of note 3 Johann Widman, Behide vnd hubsche Reckenung, Leipzig, 1489. We have used J. W. L. Glaisner’s article in Messenger of Mathematics, vol. 51, 1921, p. 16.
Page 137 of note 4 Christophori Clavii Bambergensis, Opera Mathematica, Mogvntiae, 1612. Tomus secundus, Numeratio, p. 58.
Page 137 of note 5 L. Pacioli, Summa de arithmetica, geometria, etc., 1494. We have used the 1523 edition, printed at Toscolano, folio 99b, 100a, 182.
Page 137 of note 6 C. Tonstall, De arte supputandi, 1522. We have used the Strassburg edition of 1544, p. 393.
Page 138 of note 1 R. Recorde, Grovnd of Artes, 1543 (?). We have used an edition issued between 1636 and 1646 (title page missing).
Page 138 of note 2 Gall Spenlin, Arithmetica künstlicher Rechnung, 1546. We have used J. W. L. Glaisher’s article in Messenger of Mathematics, vol. 51, 1921, p. 62.
Page 138 of note 3 Juan de Hortega, Traetado subtilissimo de arismetica y de geometria, emendado por Lonçalo Busto, 1552, folio 138, 215b.
Page 138 of note 4 L’arithmétiqve de Jan Trenchant, i éd., Lyon, 1578, p. 216.
Page 138 of note 5 Samuel Jeake, AOΓI∑TIKHAOΓÍA or Arithmelick, London, 1696 (Preface 1674), p. 501.
Page 138 of note 6 Mr Wingate’s Arithmetick, enlarged by John Kersey, 11th ed., with supplement by George Shelley, London, 1704, p. 128.
Page 138 of note 7 Guido Grandi, Instituzioni di arithmetica pratica, Firenze, 1740, p. 104.
Page 138 of note 8 Daniel Adams, Scholar’s Arithmetic, 10th ed., Keene, N.H., 1816, p. 199.
Page 138 of note 9 John Preston, Lancaster’s Theory of Education, Albany [New York], 1817, p. 349.
Page 138 of note 10 Leonardo of Pisa, loc. cit. p. 132.
Page 138 of note 11 Nicolas Chuquet, Le Triparty en la Science des Nombres, edited by A. Marre, in Buttettino Boncompagni, vol. 13, 1880; reprint, Eoma, 1881, p.(115).
Page 138 of note 12 Luca Pacioli, op. cit. folio 93a.
Page 138 of note 13 The Treviso arithmetic of 1478 is described and partly given in facsimile by B. Boncompagni in Atti dell’ Accademia Pontificia dei nuovi Lincei, tome 16,1862-63, Roma 1863. See p. 568.
Page 139 of note 1 Chuquet, loc. cit. p. (84).
Page 139 of note 2 Johann Widman, op. cit.; see J. W. L. Glaisher, loc. cit. p. 6.
Page 139 of note 3 Luca Pacioli, loc. cit., 1523, folio 18, 27, 54, 58, 59, 64.
Page 139 of note 4 Christoph Rudolph, Kunstliche Rechnung, 1526. We have used one of the Augsburg editions, 1574 or 1588 (title-page missing), GV11.
Page 139 of note 5 O. Schreckenfuchs, op. cit. 1551.
Page 139 of note 6 Juan de Hortega, op. cit. 1552, folio 92a.
Page 139 of note 7 Tartaglia, General Trattato di Nvmeri, la prima parte, 1556, folio lllb, 117a.
Page 139 of note 8 J. Trenchant, op. cit. p. 142.
Page 139 of note 9 Juan Perez de Moya, Arithmetica, Madrid, 1784, p. 141. This text is the same as that of the edition that appeared in Salamanca in 1562.
Page 139 of note 10 Robert Recorde, op. cit. p. 175.
Page 139 of note 11 [Leonard Digges], A Geometrical Practical Treatize named Pantometri/i, London, 1591.
Page 140 of note 1 Nicolas Chuquet, loc. cit. p. (54).
Page 140 of note 2 Luca Pacioli, loc. cit. 1523, folio 51, 52, 53.
Page 140 of note 3 Christoph Rudolph, op. cit., under addition and subtraction of fractions.
Page 140 of note 4 Petrus Apianus, Kauffmaiws Rechnung, Ingolstadt, 1527.
Page 140 of note 5 Francesco Feliciano, Libro di Aritkmetica e geometria, 1550.
Page 140 of note 6 O. Schreckenfuchs, op. cit., Annot., folio 25b.
Page 140 of note 7 Hortega, op. cit., 1552, folio 46, 55a, 63b.
Page 140 of note 8 Chr. Clavius, Opera Omnia, Tom 1, 1611 Euclid, p. 383.
Page 140 of note 9 E. Wingate, op. cit. 1704, p. 152.
Page 140 of note 10 Samuel Jeake, op. cit. p. 51.
Page 140 of note 11 J. P. de Moya, op. cit. p. 103.
Page 140 of note 12 Juan Bautista Corachan, Arithmetica demonstrada, Barcelona, 1719, p. 87.
Page 140 of note 13 George E. Crusoe, Y Mathematics (Why Mathematics?), Pittsburgh, Pa., 1921, p. 21.
Page 140 of note 14 Jacobi Peletarii Cenomani, de occvlia parte nvmerorvm, qvam Algebram vocant, Libri dito. Parisiis, 1560, folio 52 b.
Page 140 of note 15 L. Pacioli, op. cit. 1523, folio 54a.
Page 140 of note 16 Juan de Hortega, op. cit. 1552, folio 66a.
Page 140 of note 17 Samuel Jeake, op. cit. p. 58.
Page 140 of note 18 Mr. Blundevil. His Exercises contayning eight Treatises, London, 1636, p. 29.
Page 140 of note 19 Juan Perez de Moya, op. cit. p. 117.
Page 141 of note 1 Luca Pacioli, op. cit. 1523, folio 35.
Page 141 of note 2 N. Tartaglia, op. cit. 1556, folio 346.
Page 141 of note 3 Chr. Rudolph, Kunstliche Rechnung, Augsburg, 1574 or 1588 edition.
Page 141 of note 4 Chr. Clavius, Opera Omnia, 1612, tom. 1,1611, Numeratio, p. 11.
Page 142 of note 1 Juan de Hortega, op. cit. folio 42b.
Page 142 of note 2 Nicolas Hunt, Hand-Maid to Arithmetick, London, 1633.
Page 142 of note 3 Tho. York, Practical Treatise of Arithmetick, London, 1687, p. 38.
Page 142 of note 4 John Ayres, Arithmetick made Easie, by E. Hatton, London, 1730, p. 53.
Page 142 of note 5 Benjamin Workman, American Accountant, Philadelphia, 1789, p. 25.
Page 142 of note 6 Peter Barlow, Math. & Phil. Dictionary, London, 1814, art. “Multiplication.”
Page 142 of note 7 Encyclopedia Portugueza, Porto, 19—?, vol. 7, art. “Nove.”
Page 142 of note 8 Joseppo Vnicorno, De l’arithmetica universale. In Venetia, 1598, fo. 20. Quoted from C. Le Paige, “Sur l’origine de certains Signes d’Opération,” Annales de la société scientifique de Bruxelles, 16th year, 1891-92, Part IL, p. 82.
Page 142 of note 9 George E. Crusoe, op. cit. p. 6.
Page 143 of note 1 Hudalrich Regius, Vtrivsqve Arithmetics Epitome, Fribvrgi Prisgoiae, 1543, folio 61.
Page 143 of note 2 Tho. York, op. cit. p. 35.
Page 143 of note 3 Samuel Jeake, op. cit. p. 22.
Page 143 of note 4 N. Chuquet, op. cit. p. (69).
