The Hindu-Arabic Numerals Part 24

[567] Ibid., p. 79.

[568] _Liber de Remediis utriusque fortunae Coloniae._

[569] Fr. Walthern et Hans Hurning, Nordlingen.

[570] _Ars Memorandi_, one of the oldest European block-books.

[571] Eusebius Caesariensis, _De praeparatione evangelica_, Venice, Jenson, 1470. The above statement holds for copies in the Astor Library and in the Harvard University Library.

[572] Francisco de Retza, _Comestorium vitiorum_, Nurnberg, 1470. The copy referred to is in the Astor Library.

[573] See Mauch, "Ueber den Gebrauch arabischer Ziffern und die Veranderungen derselben," _Anzeiger fur Kunde der deutschen Vorzeit_, 1861, columns 46, 81, 116, 151, 189, 229, and 268; Calmet, _Recherches sur l'origine des chiffres d'arithmetique_, plate, loc. cit.

[574] Gunther, _Geschichte_, p. 175, n.; Mauch, loc. cit.

[575] These are given by W. R. Lethaby, from drawings by J. T. Irvine, in the _Proceedings of the Society of Antiquaries_, 1906, p. 200.

[576] There are some ill-tabulated forms to be found in J. Bowring, _The Decimal System_, London, 1854, pp. 23, 25, and in L. A. Cha.s.sant, _Dictionnaire des abreviations latines et francaises ... du moyen age_, Paris, MDCCCLXVI, p. 113. The best sources we have at present, aside from the Hill monograph, are P. Treutlein, _Geschichte unserer Zahlzeichen_, Karlsruhe, 1875; Cantor's _Geschichte_, Vol. I, table; M. Prou, _Manuel de paleographie latine et francaise_, 2d ed., Paris, 1892, p. 164; A.

Cappelli, _Dizionario di abbreviature latine ed italiane_, Milan, 1899. An interesting early source is found in the rare Caxton work of 1480, _The Myrrour of the World_. In Chap. X is a cut with the various numerals, the chapter beginning "The fourth scyence is called arsmetrique." Two of the fifteen extant copies of this work are at present in the library of Mr. J.

P. Morgan, in New York.

[577] From the twelfth-century ma.n.u.script on arithmetic, Curtze, loc. cit., _Abhandlungen_, and Nagl, loc. cit. The forms are copied from Plate VII in _Zeitschrift fur Mathematik und Physik_, Vol. x.x.xIV.

[578] From the Regensburg chronicle. Plate containing some of these numerals in _Monumenta Germaniae historica_, "Scriptores" Vol. XVII, plate to p. 184; Wattenbach, _Anleitung zur lateinischen Palaeographie_, Leipzig, 1886, p. 102; Boehmer, _Fontes rerum Germanicarum_, Vol. III, Stuttgart, 1852, p. lxv.

[579] French Algorismus of 1275; from an unpublished photograph of the original, in the possession of D. E. Smith. See also p. 135.

[580] From a ma.n.u.script of Boethius c. 1294, in Mr. Plimpton's library.

Smith, _Rara Arithmetica_, Plate I.

[581] Numerals in a 1303 ma.n.u.script in Sigmaringen, copied from Wattenbach, loc. cit., p. 102.

[582] From a ma.n.u.script, Add. Ma.n.u.script 27,589, British Museum, 1360 A.D.

The work is a computus in which the date 1360 appears, a.s.signed in the British Museum catalogue to the thirteenth century.

[583] From the copy of Sacrabosco's _Algorismus_ in Mr. Plimpton's library.

Date c. 1442. See Smith, _Rara Arithmetica_, p. 450.

[584] See _Rara Arithmetica_, pp. 446-447.

[585] Ibid., pp. 469-470.

[586] Ibid., pp. 477-478.

[587] The i is used for "one" in the Treviso arithmetic (1478), Clichtoveus (c. 1507 ed., where both i and j are so used), Chiarini (1481), Sacrobosco (1488 ed.), and Tzwivel (1507 ed., where jj and jz are used for 11 and 12).

This was not universal, however, for the _Algorithmus linealis_ of c. 1488 has a special type for 1. In a student's notebook of lectures taken at the University of Wurzburg in 1660, in Mr. Plimpton's library, the ones are all in the form of i.

[588] Thus the date [Numerals 1580], for 1580, appears in a MS. in the Laurentian library at Florence. The second and the following five characters are taken from Cappelli's _Dizionario_, p. 380, and are from ma.n.u.scripts of the twelfth, thirteenth, fourteenth, sixteenth, seventeenth, and eighteenth centuries, respectively.

[589] E.g. Chiarini's work of 1481; Clichtoveus (c. 1507).

[590] The first is from an algorismus of the thirteenth century, in the Hannover Library. [See Gerhardt, "Ueber die Entstehung und Ausbreitung des dekadischen Zahlensystems," loc. cit., p. 28.] The second character is from a French algorismus, c. 1275. [Boncompagni _Bulletino_, Vol. XV, p. 51.]

The third and the following sixteen characters are given by Cappelli, loc.

cit., and are from ma.n.u.scripts of the twelfth (1), thirteenth (2), fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and eighteenth (1) centuries, respectively.

[591] Thus Chiarini (1481) has [Symbol] for 23.

[592] The first of these is from a French algorismus, c. 1275. The second and the following eight characters are given by Cappelli, loc. cit., and are from ma.n.u.scripts of the twelfth (2), thirteenth, fourteenth, fifteenth (3), seventeenth, and eighteenth centuries, respectively.

[593] See Nagl, loc. cit.

[594] Hannover algorismus, thirteenth century.

[595] See the Dagomari ma.n.u.script, in _Rara Arithmetica_, pp. 435, 437-440.

[596] But in the woodcuts of the _Margarita Philosophica_ (1503) the old forms are used, although the new ones appear in the text. In Caxton's _Myrrour of the World_ (1480) the old form is used.

[597] Cappelli, loc. cit. They are partly from ma.n.u.scripts of the tenth, twelfth, thirteenth (3), fourteenth (7), fifteenth (6), and eighteenth centuries, respectively. Those in the third line are from Cha.s.sant's _Dictionnaire_, p. 113, without mention of dates.

[598] The first is from the Hannover algorismus, thirteenth century. The second is taken from the Rollandus ma.n.u.script, 1424. The others in the first two lines are from Cappelli, twelfth (3), fourteenth (6), fifteenth (13) centuries, respectively. The third line is from Cha.s.sant, loc. cit., p. 113, no mention of dates.

[599] The first of these forms is from the Hannover algorismus, thirteenth century. The following are from Cappelli, fourteenth (3), fifteenth, sixteenth (2), and eighteenth centuries, respectively.

[600] The first of these is taken from the Hannover algorismus, thirteenth century. The following forms are from Cappelli, twelfth, thirteenth, fourteenth (5), fifteenth (2), seventeenth, and eighteenth centuries, respectively.

[601] All of these are given by Cappelli, thirteenth, fourteenth, fifteenth (2), and sixteenth centuries, respectively.

[602] Smith, _Rara Arithmetica_, p. 489. This is also seen in several of the Plimpton ma.n.u.scripts, as in one written at Ancona in 1684. See also Cappelli, loc. cit.

[603] French algorismus, c. 1275, for the first of these forms. Cappelli, thirteenth, fourteenth, fifteenth (3), and seventeenth centuries, respectively. The last three are taken from _Byzantinische a.n.a.lekten_, J.

L. Heiberg, being forms of the fifteenth century, but not at all common.

[Symbol: Qoppa] was the old Greek symbol for 90.

[604] For the first of these the reader is referred to the forms ascribed to Boethius, in the ill.u.s.tration on p. 88; for the second, to Radulph of Laon, see p. 60. The third is used occasionally in the Rollandus (1424) ma.n.u.script, in Mr. Plimpton's library. The remaining three are from Cappelli, fourteenth (2) and seventeenth centuries.

[605] Smith, _An Early English Algorism_.

[606] Kuckuck, p. 5.

[607] A. Cappelli, loc. cit., p. 372.

[608] Smith, _Rara Arithmetica_, p. 443.

[609] Curtze, _Petri Philomeni de Dacia_ etc., p. IX.

[610] Cappelli, loc. cit., p. 376.

[611] Curtze, loc. cit., pp. VIII-IX, note.