1914 ALGORISM OF JOHN K1LUNGW0RTH 707 The Algorism of John Killingworth THE development of the system of arithmetic which we use extends over many centuries and includes contributions from many peoples. The decimal basis of the system, resulting from the common practice of reckoning on the fingers, is wellnigh universal. Associated with this system in many lands is the abacus, which Downloaded from was possibly invented by the Egyptians. At any rate, their use of the instrument is attested by Herodotus,1 and they contrived a number of games and other devices to make the study of arithmetic attractive.2 Of the use of the abacus in ancient Greece abundant evidence exists, such as the statement of the http://ehr.oxfordjournals.org/ orator Lysias that the office furniture of a banker consisted of a table and an abacus.3 In 1846 there was found on the island of Salamis a well-preserved specimen of a Greek abacus, a marble tablet (1*5 m. by 0-75 m.) with a series of five columns for integers together with four smaller columns at the right for fractions.4 The column of integers is divided by a horizontal line into equal parts, the upper section doubtless being intended, as in the Roman at Carleton University on June 23, 2015 abacus, to represent fivefold the lower; but unfortunately no explanation of the table has been transmitted to us. For cen- turies not only in Greece and Rome but in all Europe the common operations of arithmetic were performed upon such an instru- ment, and even to-day no village merchant in China or Russia makes a sale without his abacus at hand. Towards the end of the tenth century an innovation was effected in methods of calculation on the abacus by the introduction of separate disks, apices, or counters to represent each of the first nine integers. These High-a were marked either with the Greek numeral letters or with the Roman numerals, and later, after the introduction of the Hindu system, such disks or stones were marked with the numbers of algorism.* Thus 1 ii. 36. * Plato refers frequently to Egyptian method* of instruction and says particularly of arithmetic, Lava, vii, p. 819 (Jowett's translation): ' In that country systems of calculation have been actually invented for the use of children, which they learn as a pleasure and amusement.' 1 Quoted by Julius Pollux, Onomtuticon, x. 105. * A. Nagel, ' Die Bechenmethoden auf dem griechischen Abakus,' in AbhandL nr Ouch, der mathim. WisstnschafUn, ir (Leipzig, 1899), 337-37, -with plate; Cantor, Voriesungen Qbtr Qtach. der Mathtm. i, ed. 3, 130-4. * Down to the sixteenth century any work pi-plaining the Hindu art of reckoning with nine digits and zero was commonly called Algorism, from the surname of Mohammed ibn Musa al-Khowarizmi. whose Arabio treatise on arithmetic, written Z Z 2 708 ALGORISM OF JOHN KILLINGWORTM October Chaucer 'uses the expression avgrimstones, and even in the six- teenth century Palsgrave7 says : ' I caste an accomptes with coun- ters after the aulgorisme maner,' although the algorism (avgrim) method is fundamentally opposed to the use of stones or counters. Wide circulation was given to this new type of abacus through its adoption and discussion by Gerbert, afterwards Pope Silvester II, and his pupils. One of them, Richer,8 informs us that Gerbert secured a thousand of these disks of horn to use in calculating, upon an abacus of twenty-seven columns. That these ' novem ntunero notae omnem numerum significantes', as Richer terms- them, were the Hindu (vulgo Arabic) numerals is now generally Downloaded from accepted.* Not the least advantage of this innovation is that it lent itself to calculation with the pen upon ruled columns. This column-reckoning continued in use down to the fifteenth century, and we find it employed in combination with the ordinary methods of John KiUingworth in his Algorism of 1444. But this http://ehr.oxfordjournals.org/ system was replaced in the fifteenth and sixteenth centuries by reckoning upon horizontal lines u with ' projectiles ' or counters, a system which was widely used in Germany especially by bankers. Here the decimal powers from one to a million were represented by counters (Rechenpfennige) on the lines, increasing upwards, units being placed on the lowest line, while a counter in any space represented five of the units corresponding to a counter at Carleton University on June 23, 2015 on the line below the space. The rise and prevalence of these various types of tangible arithmetic indicate the difficulties of popular the fundamentally abstract, although simple. about A- D. 825, wu translated into Latin early in the twelfth centaiy. Thif treatise wma the source of many subsequent works on the subject. The texts which contributed most to the spread of the Hindu system were the Algorisvau vaigarit (c A.D. 1226), by John Halifax (or Holywood, de Sacrobosco) and the contemporary Carmen de Algorismo by Alnxander de Villa Dei. Manuscript copies of both are common and both were printed by Halliwell, Bora mathematka (London, 1839), pp. 1-26 and 73-83 respectively. No lees than fourteen editions of the former appeared daring the fifteenth and •rirt^^fm+.ti "*"*""— The best and most recent edition is printed with the thir- teenth-century commentary by Petrus de Dacia (Petri PhUtmeni dt Dacia inalgori- rmum vuigarem Iohamtit de Socrobotco commentarivs. Una cum algoritmo ipto edidit et praefatus tit MaxinUianyu Curtie, Copenhagen, 1897). The statement in the Did. of Nat. Biogr. xrviL 217, that John de Sacroboaoo's fame rested entirely on his Tractates de Spkaera needs correction. • In his description of the clerk in the Canterbury Tale*, A. -3210, ffin augrim^tones layen faire apart On shelves couched at his beddes heed. See my paper on ' Augrim-stones', in Modern Language Nottt, xxvii 206-9. ' Quoted in the Nae English Dictionary under Algorim. • tfiit.iii.54. • Bubnov, OtrberH Opera ma&ematica (Berlin, 1899), p. 381, n. 23; Smith and TCurpinnlri, Tht Hindu-Arabic Numerals (Boston, 1911), pp. 112-13, 116. >• Adam Biese's Rtchrtung auff dtr Linihen (Erfurt, 1S22) passed through three editions, and his Rechnung auff der Linien urtd Ftdern (Erfurt, 1522) through more than forty editions ; see Smith, Bora Arithmetica (Boston, 1908), pp. 138-9. 1914 ALGORISM OF JOHN K1LL1NGW0RTH 709 system of numbers which we use, while it also illustrates the hold which the abacus had attained upon people's minds. Not only did the reckoning on the abacus flourish long after the introduction of the Hindu numerals, but its terminology and methods survive also in the mannala explaining the new system. Thus, the size of the ordinary abacus permitted the representa- tion upon it of only two numbers, in general, at one time. In the process of adding such numbers it is convenient to place the larger number above and to combine the lower number with it by moving up the counters of the lower number so as to Downloaded from leave the sum in the position of the larger number. Ten counters combined in any column are replaced, of course, by one in the next column to the left. In the two Latin versionB of Al-Khowarizmi's arithmetic11 exactly this procedure is fol- lowed, and the words pono, lew, erigo are used quite as though http://ehr.oxfordjournals.org/ counters or pebbles were actually being moved about. ' Cum uolueria addere numerum super numerum . pone utrosque numeros in duohus ordinihus.... Cum collects fuerint in aliqua differentia- rum, id est in differentia unitatum uel decenorum, due in aha aliqua decem, pones pro eis "Tinny et eriges enm ad superiorem differentiam id est si habneris in prima differentia, que est differentia unitatum, decem, fades de eis unum, et subleuabis enm ad differentiam decenorum. .u Cum igitur uolueris numerum numero agregare, numerum cui agregan- at Carleton University on June 23, 2015 dum est ordinabiliter per differentias suas pones. Deinde numerum agre- gandom sub eo per suas etiam differentias consimiliter dispones. Et si ex eorum agregatione collectus fuerit articulus et digitus, articulum numero precedentis differentie superpones ; sed in loco numeri cui agre- gaueras iam deleti digitum scribes.' u Here express mention is not made as it is later by John de Sacrobosco14 that the larger number should be placed above, but this practice is followed in the different problems presented. It is evident that in the new arithmetic several numbers could be added as easily as two, and there is no necessity for deleting the one number during the process as is indicated in the second passage cited above. A study of the process of multiplication as taught in these early algo- risms shows the retention of a procedure adapted to a board with movable counters rather than the intelligent application of a process suited to a written system which includes a rero to give value of position. First the multiplicand is written and below it the multiplier, so placed that the unit figure of the multiplier stands below the highest order of the " Both of uncertain authorship and both, doubtless, of the twelfth century; published by Boneompagni, Trattati d'Aritmetiea, Rome, 1857. a ' Algoritmi de Numero Indorum,' in Tratiati, pp. 7-8. u ' Liber Algorismi de Prmctic* Arismetric*,' in Tnxttati, p. 30. " AJgoTumtu vtUgarvs, ed Curtie, I.e., p. 3: ' Comprtentius ett, ut minor numerra gubecribatar et maiori addatur.' 710 ALGORISM OF JOHN KILLINGWORTH October multiplicand. In the problem 2,326 multiplied by 214, which is taken from the first of the Trattati, the numbers are placed as here indicated : 2 3 2 6 2 14 Each digit of the multiplier beginning with the highest is then multiplied by the highest digit of the multiplicand and the product is written in the line of the multiplicand above the corresponding figures of the multiplier.