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 The Educational Significance 
 
 of Sixteenth Century 
 
 Arithmetic 
 
 FROM THE POINT OF VIEW OF THE PRESENT TIME 
 
 Lambert Lincoln Jackson, Ph.D. 
 
 Head of the Department of Mathsmatica 
 State Normal School, 3:a:k;;::, If. Y. 
 
 Published iy 
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 COLUMBIA UNIVERSITY 
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 Cornell University Library 
 QA 142.J13 
 The educational significance of sixteent 
 
 "3""l"924 001 168 511 
 
The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31 924001 1 6851 1 
 
TEACHERS COLLEGE, COLUMBIA UNIVERSITY 
 
 CONTRIBUTIONS TO EDUCATION 
 
 NO. 8 
 
 THE EDUCATIONAL SIGNIFICANCE 
 
 OF SIXTEENTH CENTURY 
 
 ARITHMETIC 
 
 FROM THE POINT OF VIEW OF THE PRESENT TIME 
 
 BY 
 
 LAMBERT LINCOLN JACKSON, PH. D. 
 
 Head of the Department of Mathematics 
 State Normal School, Brockport, N. Y. 
 
 PUBLISHED BY 
 
 Zteacbers College, Columbia TUniversits 
 
 NEW YORK 
 1906 
 
 D' 
 
& 1 1^3&4 
 
PREFACE 
 
 Although there is much material on the history of six- 
 teenth century arithmetic, comparatively little has been written 
 on the teaching of the subject at that time, and less upon the 
 educational significance of that teaching from the point of 
 view of the present. 
 
 This dissertation is the result of a research into the arith- 
 metic of the fifteenth and sixteenth centuries for the purpose 
 of showing its bearing upon the present teaching of the sub- 
 ject. The exact period chosen is that from 1478, the date of 
 the first printed arithmetic, to 1600, because between these 
 dates arithmetic took on the form which it retained for three 
 centuries, besides setting forth for the first time its educational 
 functions. So little was written at that time on the educational 
 values of studies that it is necessary to examine with unusual 
 care the treatises and text-books on arithmetic in order to de- 
 termine the principles and aims which guided their authors 
 in creating them: 
 
 No general history of mathematics can be of use in a re- 
 search Of this kind, unless it contains extensive extracts from 
 the original sources, and there is only one such work, namely, 
 that of Cantor. 1 But, if one were to depend upon Cantor for 
 a treatment of sixteenth century arithmetic, he would obtain 
 an exhaustive 'knowledge of the leading writers only, whereas 
 the great mass of minor works is necessary to supply a large 
 portion of the data needed to determine the educational signi- 
 ficance of the subject. No one has expressed more clearly 
 than DeM organ 2 the relative value of major and minor works 
 for the purpose of interpreting history. " Unfortunately, his- 
 tory must of necessity be written mostly upon those works 
 
 1 Cantor, M., "Vorlesungen uber Geschkhte der Mathematik," 3 vol. 
 (Leipsic, 3d ed., 1900). 
 
 2 DeMorgan, A., "Arithmetical Books," p. vi (London, 1847). 
 
 3 
 
4 PREFACE 
 
 wbi'ch, by being in advance of their age, 'have therefore be- 
 come well known. It ought to be otherwise, but it cannot be, 
 without better preservation 1 and classification of the minor 
 works which people actually use, and from which the great 
 mass of those who study take their habits and opinions ; — or — 
 until the historian has at his command a readier access to 
 second and third rate works in large numbers ; so that he may 
 write upon the effects as well as the causes." Other general 
 histories of mathematics like those of Kastner, 1 Suter, 2 
 Hankel, 3 Gunther, 4 and Gerhardt B contain very brief exposi- 
 tions of the arithmetic of the Renaissance, and all having been 
 written before Cantor's work furnish little, if anything, of 
 importance not covered by him. 
 
 Among those histories devoted entirely to arithmetic and 
 drawn from original sources, only two discuss to any consider- 
 able extent the works which form the basis of this disserta- 
 tion. These are Unger's, 6 chiefly on the arithmetic of the 
 Germans, and Peacock's 7 on the history of arithmetic in gen- 
 eral. Unger's treatise, because of its accuracy and its sys- 
 tematic form, has become the standard work on the history 
 of arithmetic and its teaching in Germany. Its foundation, 
 however, is limited, for, although thirty-two original works 
 of the period in question are mentioned, only twenty are 
 drawn upon, three being Italian, one French, one Flemish, and 
 fifteen German. Thus, Unger's work, excellent as it is, does 
 not furnish the historical material necessary for a study o>f six- 
 teenth century arithmetic from an international point of view. 
 Peacock's article is the most extensive exposition of the sub- 
 
 1 Kastner, " Gesdiichte der Mathematik " (1796-1800). 
 
 2 Suter, "Gesohichte der math. Wissenschaften " (1873-75). 
 
 3 Hankel, "Zur Gesichiohte d. Math, in Alterthum u. Mittelalter" (1874). 
 
 4 Giinther, " Vermischte Untersuohttngen z. Gesohichte d. math. Wissen- 
 schaften" (1876). 
 
 5 Gerhardt, "Gesohichte der Mathematik in Deutschland" (1877). 
 
 6 Unger, " Die Metholdik der praktischen Arithmetik in historisoher Ent- 
 wickelung vom Ausgange des Mittelalters bis auf die Gegenwart" (1888). 
 
 7 Peacock, " History of Arithmetic " in Encyclopedia Metropolitana, vol. 
 1, pp. 369-476 (London, 1829). 
 
PREFACE 5 
 
 ject, but, besides emphasizing the Italian works in undue 
 proportion, it omits an important link in the French contribu- 
 tion and ignores the Dutch arithmetic altogether. Had Pea- 
 cock been able to supply these important departments, and 
 had he, like Unger, added a history of the teaching of the sub- 
 ject, the historical research on which the present dissertation 
 is based would have been unnecessary. Besides the histories 
 of Unger and Peacock there are several important works 
 which contain matter germane to this subject, but they deal 
 either with a small portion of the history of the period in 
 question or with some particular phase of it. Thus : Giinther, 1 
 who probably is second only to Cantor as an authority on the 
 early history of arithmetic, closed his work with the critical 
 date, 1525. Sterner 2 covers practically the same ground as 
 Unger, but draws the bulk of his illustrative material from 
 the arithmetics of Kobel, 3 Boschenstein, and Riese only, and 
 gives a much inferior treatment of the teaching of the sub- 
 ject. Heinrich Stoy * gives very little concerning six- 
 teenth century arithmetic and confines his investigation to the 
 development of the number concept and to the various modes 
 of its expression. Grosse, 5 as indicated by the title of his 
 book, discusses those arithmetics which apply the rules and 
 processes to quantitative data which have a historical setting; 
 for example, the periods of reigning dynasties or the size of 
 armies that participated in the great battles of the past. There 
 were only a half-dozen such arithmetics of which Suevus's 
 and Meichsner's are the best types. Treutlein, 6 Villicus, 7 and 
 
 1 Giinther, " Geschichte des math. Unterrichts im deutsichen Mittelalter 
 bis 1525 (in Monumenta Germaniae Paedagogica, 1887). 
 bis 1525 (in Monumenta Germaniae Paedagogica, 1887). 
 
 3 See bibliographical list, page I of Sterner' s work. 
 
 * Stoy, H., "Zur Geschichte des Rechenunterrichts " (Diss., 1876). 
 
 B Grosse, "Historische Rechenbiicher des 16 und 17 Jahrfaunderts (1901). 
 
 6 Treutlen^ "Das Rechnen im 16. Jahrhundert " (1877). 
 
 7 Villicus, " Das Zahleruwesen der Volker im Altherthum u. die Entwick- 
 elung d. Ziff errechnens " (1880). 
 
 "Geschichte der Rechenkunst" (a slight revision of the former work) 
 (1897). 
 
6 PREFACE 
 
 Kuckuck * are other German writers who touch this period, 
 but their works are earlier and briefer than that of Unger. 
 Treutlein's work is a standard, but it treats of a limited num- 
 ber of authors, mostly German, emphasizes a few special sub- 
 jects, and neglects close comparative study. Leslie 2 deals 
 with the theory of calculation in its historical development. 
 For example, he pays much attention to the reasons for the 
 origin of the various scales of notation, and to the systems of 
 objective representation! of numbers, called Palpable arithme- 
 tic. The subject-matter of sixteenth century arithmetic found 
 in this work is very limited. It is unnecessary to mention 
 more recent philosophical treatises, like Brooks's, 5 because they 
 contain no history of value, with which this article is con- 
 cerned, not found in Peacock or Leslie. 
 
 Thus, in order to accomplish the purpose of this disserta- 
 tion, it was necessary to make a more extended and systema- 
 tic research into the arithmetic of the fifteenth and sixteenth 
 centuries than has hitherto been made. It was imperative 
 first of all to supply those departments of French and Dutch 
 arithmetic missing from the best historical treatises. Then, 
 it was necessary to consult the original sources already ex- 
 amined by others in order to find matter of educational signi- 
 ficance, much of which had not been' noted by other inves- 
 tigators, as well as to obtain a broader basis for detailed 
 comparisons. 
 
 The dissertation is divided into two chapters. Chapter I 
 contains the result of the research into' the subject-matter 
 and teaching of arithmetic in the fifteenth and sixteenth cen- 
 turies. Chapter II contains an exposition of the bearing of 
 the arithmetic of that period upon the present teaching of 
 the subject. 
 
 It is a great satisfaction to the author to acknowledge his 
 indebtedness to Professor David Eugene Smith of Teachers 
 
 1 Kuckuck, "Die Rechenkunist im 16. Jabrhundert " (1874). 
 
 2 Leslie, "Philosophy of Arithmetic," Edinburgh (1820). 
 
 3 Brooks, " Philosophy of Arithmetic " (1901). 
 
PREFACE 7 
 
 College, Columbia University, for directing the research, to 
 George A. Plimpton, Esq., of New York City, for access to his 
 extensive collection of rare mathematical works, a privilege 
 which alone made this research unique, and to Professor 
 Frank M. McMurry of Teachers College, Columbia Univers- 
 ity, for his help in making the article of practical value. 
 
 L. L. Jackson. 
 Brockport, New York, 1905. 
 
BIBLIOGRAPHY 
 
 Original Sources on Which This Research is Based 
 
 Anonymous, " Thaumaturgus Mathematicus, Id est Admirabilium effec- 
 torum e mathematicarum disciplinarum Fontibus Profluentium Sylloge." 
 
 Munich, 1651. 
 
 Anonymous. Treviso Book. (So called from the place of printing. 
 
 This is the earliest printed Arithmetic known to exist.) " Incommincio 
 
 vna practica molto bona et vtile a ciafchaduno chi vuole vxare larte dela 
 
 merchadantia chiamata vulgarmente larte de labbacho." 
 Colophon: "A Triuifo :: A di. 10. Deceb 4 :: .1478." 
 It contains 123 pages, not numbered. Size of page, 14.5x20.6 cm. 32 
 
 lines to the page. 
 
 Baker, Humphrey, " The Well spring of sciences which teacheth the per- 
 feot worke and practife of Arithmeticke, both in whole Numbers and 
 Fractions : set forthe by Humphrey Baker, Londoner, 1562. And now once 
 agayne perused augmented and amended in all the three parts, by the 
 sayde Aucthour : whereunto he hath also added certain tables of the agree- 
 ment of measures and waightes of divers places in Europe, the one with 
 the other, as by the table following it may appeare." 
 London, 1580; 1st ed., 1562. 
 
 " In spite of the date 1562 on the title page, I find no edition before 1568, 
 Indeed in the 1580 edition Baker says : ' Having fometime now twelve 
 yeres fithence (gentle reader) publifhed in print one Englifhe boke o£ 
 Arithmetick ... I have been . . requefted . . to adde fomething more 
 thereunto.'" D. E. Smith. 
 
 Belli, Silvio, " Qvattro Libri Geometrici Di Silvio Belli Vicen- 
 tino. II Primo del Mifurare con la vifta. Nel Qvale S'Insegna, Senza 
 Travagliar con numeri, a mifurar faciliffinramente le diftantie, l'altezza, e 
 le profondita con il Quadrato Geometrico, e con altri ftromenti, de' quali 
 facilmente fi pu6 prouedere con le Figure. Si mostra ancora vna belliffima 
 via di retrouare la profondita di qual fi voglia mare, & vn modo induf- 
 triofo di -mifurar il circuito di tutta la Terra. Gli Altri Tre Sono Delia 
 Proportione & Proportionality communi paffioni de Quanta. 
 
 " Vtili, & neceffarij alia vera, & facile intelligentia dell' Arithmetica, della 
 Geometria, & di tutte le fcientie & arti." 
 
 Venice, 1595. 
 
 9 
 
IO BIBLIOGRAPHY 
 
 Belli, Silvio, " Silvio Belli Vicentino Delia Proportione, et Proportion- 
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 Venice, 1573. 
 
 Boetius, Anicius Manlius Torquatus Seuerinus, "Opera." Venice, 1491. 
 
 Boetius, " Boetii Arithmetica." Augusfcae, E. Rotdolt, 1488. 
 
 Borgi, Piero, " Qui comeza la nobel opera de arithmeticha ne laquel se 
 tracta tute cosse amercantia pertinente facta t corcipilata per Piero borgi 
 da Venesia." Venice, 1488; 1st ed, 1484. 
 
 Brucaeus, Henricus, " Henrici Brucaei Belgae Mathematicarum exer- 
 
 citationum Libri Duo." „ 
 
 Rostock, 1575. 
 
 Buteo, Joan., " Logistica. Quae & Arithmetica vulgo dicitur in libros 
 quinque digesta : quorum index -summatim habetur in tergo." 
 
 Leyden, 1559. 
 
 Calanderi, Philipi, " Philippi Calandri ad nobilem et studiosum Julianum 
 Laurentdi Medicem de Aritrtwnethrica opuscu'lum." 
 
 Florence, 1491. 
 
 Cardanus, Hieronymus, " Hieronimi C. Cardani Medici Mediolanensis, 
 Practica Arithmetke, & Mensurandi singularis. In quaque preter alias 
 cotinentur, versa pagina demonstrabit." 
 
 Milan, 1539. 
 
 Cataneo, Girolamo, " Dell' Arte Del Misvrare Libri Dve, Nel Primo De' 
 Qvale S'Insegna a mifurare, & partir i Gampi. 
 
 " Nel Secondo A' Misvrar le Muraglie imbottar Grani, Vini, Fieni, & 
 Strami ; col linellar dell' Acque, & altre cose necessarie a gli Agrimenfori." 
 Brescia (no date). 
 
 Cataneo, Pietro, " Le Pratiche Delle Dve Prime Matematiche Di Pietro 
 Cataneo Senese." 
 ; Venice, 1567; 1st ed., Venice, 1546. 
 
 Champenois, Jacques Chavvet, " Les 1 Itistitvtions De L'Arithmetique De 
 Jacques Chavvet Champenois, Professeur es Matihematiques en quatre par- 
 ties : auec vn petit Traicte des fractions Astronomiques." 
 
 Paris, 1578. There is no evidence of an edition earlier than 1578. 
 
 Chiarini, Giorgio, " Qvesta e ellibro che tracta de Mercatantie et vsamze 
 de paesi." Florence, 1481. 
 
 Colophon: " Finito ellibro di tvcti ichostvmi: cambi: monete: . Per 
 
BIBLIOGRAPHY n 
 
 me Francifco di Dina di Iacopo Kartolaio Fioretino adi x di Decembre 
 
 MCCCCXXXI.'' 
 
 " I Firenze Apreffo almuniftero di Fuligno.'' 
 
 Ciacahi, Givseppe, " Regole Genarali D'Abbaco Con Le Sve Dichiara- 
 zioni, E Prove Secondo L'Vso praticato da' piu periti Arimmetici ; Con 
 Vn Breve Trattato Di Geometris, E Modi del mifurare le superficie de' 
 terreni, e corpi solidi. Descritte Da P. Givseppe Ciacchi Fiorentino.'' 
 
 Florence, 1675. 
 
 Cirvelo, Petro Sanchez, " Tractatus Arithmeticae practice qui dicitur al- 
 gorifmus. Venundantur Parrhilijs a johane Laberto eiufdem ciuitatis bib- 
 liopola in stemate diui claudij manente iuxta gymnafium coquereti." 
 
 Paris, 1513; isted., 1495 (Treutlein). 
 
 Clavius, Christopher, " Christophori Clavii Bambergensis E Societate 
 IESV Epitome Arithmeticae practica*.'' 
 
 1621 ; 1st ed., Rome, 1583. 
 
 Clicbtoveus, " Infaoductio Jacobi fabri Stapuleris in Arithmeticam Diui 
 seuerini Boetii pariter Jordani. 
 
 "Ars fupputaditam per calculos qs p notas arithmeticas fuis quidem reg- 
 ulis elegaiter expreffa Iudoci Clichtouei Neoportuenris. 
 
 " Queftio haud indigna de numeroru -, Aurelio Augustino. 
 
 " Epitome rerum geometricarum ex Geometrico introductorio Carolio 
 Bouilli, De quadratura circuli Demonstratio ex Campano." 
 Paris, c. 1506; 1st ed., Paris, 1503. 
 
 de Muris, John, 'Arithmeticae Speculativae Libri duo Joannis de Muris 
 ab innumeris erroribus quibus hactenus corrupti, & vetustate per me peri- 
 erant diligenter emendati. 
 
 " Pulcherrimis quoque exeimplis. Formisqj nouiis declarati & in usum 
 studiosae iuuentutis Moguntin&e iam recens excusi." 
 
 Moguntiae, 1538. 
 
 di Pasi, Bartholomeo, " Tariffa de i Pesi e Misure corrispondenti dal 
 Levante al Ponente e da una Terra a Luogo all' altro, quasi per tutte le 
 Parti del Monde Qvi comicia la utilissima opera chiamata taripha laq val 
 tracta de ogni sorte de pexi e misure conrispondenti per tuto il mondo 
 fata e composta per lo excelente et eximo Miser Bartholomeo di Paxi da 
 Venetia." 
 
 Venice, 1557; 1st ed., 1503. 
 
 Finaeus, Orontius, " De AritJimetica Practica libri quatuor : Ab ipsa 
 authore vigilanter recogniti multisque accessionibus recens locupletati." 
 
 Paris, ISSS ; the 1st ed. was probably printed under the title "Arithmetica 
 Practica libris quatuor absoluta" in 1525. 
 
12 BIBLIOGRAPHY 
 
 Gemma Frisius, "Arithmeticae Practicae Methodus Facilis, per Gemmam 
 Frisium Medicum ac Mathematicum, in quatuor partes divisa." 
 Leipsic, 1558; Leipsic, 1575; 1st ed., Antwerp, 1540. 
 
 Ghaligai, Francesco, " Pratica D' Arithmetica." 
 
 Florence, 1552. 
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 DeMorgan, (n.) p. 102. 
 The title of the 1521 edition is given in the Boncompagni Bulletino, 13 : 
 249. 
 
 Gio, Padre, " Elernenti Arithmetici Nelle scvole Pie." 
 
 Rome, 1689. 
 
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 schafft nach gemeinen Regeln de tre, Welschenpractic." 
 
 Frankfort, 1535. 
 
 Heer, Johann, " Compendium Arithmeticae das ist : Ein neues kurtzes 
 vfi wolgegrundtes Schul Rechenbuchlein/ von allerley Hauss: vnnd Kauff- 
 mansrechnung/ wie die taglich furfallen mogen/ so per Regulam Detri vnd 
 Practicam lieblich zu Resolvirn vnd auff zulosen sein? 
 
 " Meinen lieben Discipulis, bene=:ben alien anfahenden Rechenschulern/ 
 auoh sonsten Manniglich zu nutz/ also auffs fleissigste (So wol fur Jung- 
 fraukin als Knaben) auss rechtem grund gestellet vnd inn Druck gegeben/ 
 Dtirch Johann Heern Seniorem, Rechenmeistern vnd verordneten Visita- 
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 Nuremberg, 1617. 
 
 Huswirt, Johannes, " Enchiridion Novus Algorismi." 
 Cologne, 1501. 
 
 Jacob, Simon, " Rechenbuch auf den Linien und mit Ziffern/ sampt aller- 
 ley vortheilen/ fragweise/ Jetzt von neuem und zum neuntenmal mit vielen 
 grundtlichen anweisungen/ oder Demonstration/ sampt derselben Vnder- 
 richtung gemehret." 
 
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 quatre vacations de demie heure chacune) pratiquer les plus belles regies 
 d'icelle. Mise en lumiere, Par Alexandre lean, Arithmeticien." 
 
 Paris, 1637. 
 
 Jordanus Nemorarius, "Jordani Nemorarii Arithmetica -cum Demon- 
 
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 Kobel, Jacob, " Zwey rechenbuchlin, uff der Linien vnd Zipher/ Mit eym 
 
BIBLIOGRAPHY j, 
 
 angebenckten Visirbuoh/ so verstendlich fiir geben/ das iedem hieraus 
 on ein lerer wol zulernen. 
 
 "1 Durch den Achtbaren und wol erf amen H. Jacoben Kobel Stat- 
 schreiber zu Oppenheym.'' 
 
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 Pugliesi, "Arithmetica di Onofrio Pvgliesi Sbernia Palermitano." 
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I4 BIBLIOGRAPHY 
 
 Records, Robert, "The Ground of Artes: Teaching the woorke and 
 
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 zu recht bracht)." 
 
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 Rudolff, Christopher, " Kunstliche rechnung mit der Ziffer und mit den 
 zal pfennige/ sampt der Wellisehen Practica/ und allerley vorteil auf die 
 Regel de Tri. Item vergleighug manoherley Land ufi Stet/ gewicht/ Em- 
 mas/ Muntz ec Alles durch Christoffen Rudolff zu Wien verfertiget." 
 Wien, 1534; 1st ed., 1526. 
 
 Sacrobosco, Johann von, "Algorismus." 
 
 Venice, 1523; 1st ed., 1488. 
 
 Schonerus, Ioannes (Editor), "Algorithmus Demonstrate. " 
 Nuremberg, 1534. 
 
 Stevinus, Simon, " Les Oeuvres Mathematiques de Simon Stevin de 
 Bruges, Ou font inferees les memoires mathematiques. Le tout, correge 
 & augmente Par Albert Girard Samielois, Mathematicien." 
 
 Leyden, 1634. 
 
 Scheubel, Johann, " De numeris et diiversis rationibus." 
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 Suevus, Sigismund, "Arithmetica Historica. Die Lobliche Rechenkunst. 
 Durch alle Species vnd furnembste Regeln/ mit schonen gedenckwirdigen 
 Historien vnd Exempeln/ Auoh mit Hebraischer/ Grichischer/ vnd 
 Romischer Muntze/ Gewicht vnd Mass/ deren in Heiliger Schrifft vnd 
 gutten Gesohichte = Buchern gedacht wird/ Der lieben Jugend zu gutte er- 
 kleret Auch denen die nicht rechnen konnen/ wegen vieler schonen His- 
 torien vnd derselbigen bedeutungen lustig vnd lieblioh zu lesen. 
 
 " Aus viel gutten Buchern vnd Schrifften mit fleis zusammen getragen. 
 Durch Sigismundum Sueuum Freystadiensem, Diener der H. Gottlichen 
 Worts der Kirchen Christi zu Breslaw/ Probst zum H. Geiste/ vnd Pfarr- 
 herr zu S. Bernardin in der Newstadt." 
 
 Breslau, 1593. 
 
 Tagliente, Giovanni Antonio and Girolamo, " LiBro DABACO Che in 
 Segna a fare ogni ragione mercadantile, & pertegare le terre co l'arte dj la 
 Geometris, & altre nobilifsime ragione ftra ordinarie co la Tariffa come 
 
BIBLIOGRAPHY T e 
 
 respondeno li pefi & monede de molte terre del mondo con la inclita citta 
 di Venegia. El qual libra fe chiama Thefavro vniuersale." 
 
 Venice, 1515. 
 
 Tartaglia, Nicolo, " La Prima Parte del General Trattato di Numeri, et 
 Misure, di Nicolo Tartaglia, Nellaquale in Diecisebte Libri si Dichiara 
 Tutti Gli Atti Operativi, Pratiche, et Regole Necessarie non Solamente in 
 tutta l'arte negotiaria & mercantile, ma anchor in ogni altra arte, scientia, 
 ouer disciplina, doue interuenghi il calculo." 
 
 Venice, 1556. 
 
 Tartaglia, Nicolo, "La Seconda Parte del General Trattato Di Numeri, 
 
 et Misure Nella quale in Vndici Libri Si Notifica La piv Ellevata, et 
 
 Specvlativa Parte Delia Pratica Arithmetica, la qual e tutte regole, & oper- 
 ationi praticali delle progressioni, radici, proportioni, & quantita irrationali." 
 
 Venice, 1556. 
 
 Tartaglia, Nicolo, " Tvitte L'Opera D'Arithmetica del fatnosissimo Nicolo 
 
 Tartaglia." 
 
 Venice, 1592; 1st ed., Venice, 1556-1560. 
 
 Therfeldern, Caspar, "Arithmetica Oder Rechenbuch Auff den Linien vnd 
 Ziflfem/ mit Vortheyl vnd Behendigkeit/ auf allerley gebrauchliche Hauss/ 
 vnd Kauffmans Rechnung/ Muntzschlag/ Beschickung des 3 Thigels/ 
 Kunstrechnung/ grundlich beschrieben/ inn Frag und Antwort gestellet 
 Duroh Caspar Thierfeldern Schul und Rechenmeyster zu Steyer." 
 
 Nuremberg, 1587. 
 
 Tonstall, Cuthbert, " De Arte Supputandi. Libri Quattuor Cutheberti 
 Tcmstalli." London> ^ 
 
 Trenchant, Jan, " L'Arithmetique de Ian Trenchant, Departie en trois 
 liures. Ensemble un petit discours des Changes. Avec L'Art de calculer 
 aux Getons. Reueiie & augmentee pour la quatrieme edition, de plusiers 
 regies & articles, par L'Autheur." 
 
 Lyons, 1578; 1st ed., Lyons, 1566. 
 
 Unicorn, Giuseppe, " De L' Arithmetica vniuersale, del Sig. Ioseppo Uni- 
 corno, matihematico excellentissimo. Trattata, & amplificata con somma 
 eruditione, e connoui, &• isquisiti :modi di cihiarezza." 
 
 Venice, 1598. 
 
 Van Ceulen, Ludolf, " De Arithmetische en Geometrische fondamenten, 
 van Mr. Ludolf Van Ceulen, Met het ghebruyck van dien In veele ver- 
 scheydene constighe questien, soo Geometrice door linien, als Arithmetice 
 door irrationale ghetallen, oock door den regel Coss, ende de tafeln sinuum 
 ghesolveert." Leyden, 1615. 
 
1 6 BIBLIOGRAPHY 
 
 Van der Schuere, Jaques, "Arithmetica, Oft Reken^const/ Verchiert 
 met veel schoone Exempelen/ seer init voor alle Cooplieden/ Facteurs/ 
 Cassiers/ Ontfanghers/ etc. Gehmaeckt/ Door Jaques van der Schvere 
 Van Meenen. Nu ter tij-dt Francoysche School-meester tot Haerlem." 
 
 Haarlem, 1600. 
 
 In the voor-Reden of the 1625 edition written by his son, Denys van der 
 Schuere, it is stated that the book was published by the father " eerft in't 
 Iaer 1600. Ende dit is al de vierde mael dat het gedrukt is." 
 
 Wencelaus, Martin, T'Fondament Van Arithmetica: mette Italiaensch 
 Practijck/ midtsgaders d'aller nootwendichste stucken van den Reghel van 
 Interest. 
 
 " Beydes in Nederduyts ende in Franchois/ met redelicke ouereenstem- 
 minghe ofte Concordantien. Alles Door Martinvm VVenceslaum." 
 
 Middelburg, 1599. 
 
 Signs preface in Dutch, Marthin Wentfel; in French, Marthin Wentfle. 
 
 Widman, Johann, "Befriend und hupsch Rechnung uff alien Kauffman- 
 schafften. Johannes Widman von Eger." 
 
 Pforzheim, 1508; 1st ed., Leipsic, 1489 (Unger, p. 40). 
 
 Although there was a 1500 edition, the 1508 edition was set up anew. 
 
CONTENTS 
 
 CHAPTER I 
 
 ESSENTIAL FEATURES OF SIXTEENTH CENTURY ARITHMETIC 
 
 I . Productive activity of writers of that period 
 
 i. Number of works and editions .... 
 
 2. Causes of this awakening 
 
 a. Revival of scientific interest . . . ... 
 
 b. Commercial activity 
 
 c. Invention of printing 
 
 II. Transitions through which the subject passed 
 
 i. From the use of Roman symbols and methods to the use of 
 
 Hindu symbols and methods 
 
 2. From arithmetic in Latin to arithmetic in the vernacular . 
 
 3. From arithmetic in Mss. to arithmetic in printed books . 
 
 4. From arithmetic for the learned to arithmetic for the people. 
 
 5. From arithmetic theoretic to arithmetic practical 
 
 6. From the use of counters to the use of figures . . 
 
 III. Nature of the subject matter of sixteenth century arithmetic. 
 
 1. Definitions 
 
 a. Of numbers, unity, and zero 
 
 b. Of classes of numbers 
 
 c. Of processes . . • 
 
 2. Processes with integers 
 
 a. Notation and numeration 
 
 b. Addition ... 
 
 c. Subtraction 
 
 d. Multiplication. 
 
 e. Division 
 
 f. Doubling and Halving 
 
 3. Denominate numbers . . . 
 
 4. Fractions . . . ... 
 
 a. Definitions . . . . 
 
 Order of processes 
 Reduction. 
 
 b. 
 c. 
 d. 
 e. 
 f. 
 g- 
 
 Addition . 
 Subtraction . 
 Multiplication. 
 Division . . • 
 
 PAGE 
 23 
 23 
 
 23 
 23 
 23 
 23 
 
 24 
 
 24 
 
 24 
 
 24 
 
 24 
 
 24 
 
 24-29 
 
 2O-168 
 
 29-36 
 
 29-32 
 
 32-35 
 
 35-36 
 
 36-77 
 
 37-41 
 
 41-49 
 
 49-57 
 57-69 
 69-76 
 76-77 
 
 77-85 
 
 85-110 
 
 85-93 
 
 93-95 
 
 95-98 
 
 98-102 
 
 102-104 
 
 104-107 
 
 1 07-1 10 
 
 17 
 
!8 CONTENTS 
 
 5. Progressions 110-117 
 
 6. Ratio and Proportion 117-121 
 
 7. Involution and Evolution 121-127 
 
 8. Applied arithmetic 127-168 
 
 a. Number of writers of practical arithmetic compared with 
 
 the number of writers of theoretic arithmetic . 127-128 
 
 b. Examples of exceptional writers — Champenois, Suevus. 128-131 
 
 c. List of business rules .... 131-132 
 
 Rule of Three (Two and Five) 132-135 
 
 Welsch Practice 135-138 
 
 Inverse Rule of Three 138-139 
 
 Partnership (with and without time) I39~i4i 
 
 Factor Reckoning 142 
 
 Profit and Loss 142-143 
 
 Interest, Simple and Compound 143-145 
 
 Equation of Payments 145-146 
 
 Exchange and Banking 146-148 
 
 Chain Rule 148-150 
 
 Barter ■ • 150-151 
 
 Alligation 151-152 
 
 Regula Fusti 152-153 
 
 Virgin's Rule . . 153 
 
 Rule of False Assumption, or False Position . . . 153-156 
 
 Voyage 156 
 
 Mintage 156-157 
 
 Salaries of Servants 157 
 
 Rents 157 
 
 Assize of Bread 157 
 
 Overland Reckoning 157-158 
 
 d. Puzzles 159-163 
 
 e. Mensuration 163-168 
 
 IV. Summary 168-170 
 
 1. This was an important period in perfecting the processes of 
 
 arithmetic 168 
 
 a. The four fundamental processes with integers perfected. 168 
 
 b. The four fundamental processes with fractions perfected. 168 
 
 c. Arithmetic, geometric, and harmonic series (finite) fully 
 
 treated ... . 168 
 
 d. Involution and evolution practically complete . . . . 168 
 
 e. Tables for shortening computation were not unknown. 168 
 
 f . Decimal fractions and logarithms were the only subjects 
 
 not matured ... ... . . . . 168 
 
 2. Constructive period for applied arithmetic 168-169 
 
 a. Subject matter of applied arithmetic 169 
 
 (1) Extent 169 
 
 (2) Rich in types . . . . : 169 
 
 b. Methods of solution of applied problems 169 
 
CONTENTS I9 
 
 PAGE 
 
 (1) Unitary analysis . . 169 
 
 (2) Rule of Three 169 
 
 3. A comparison of the status of arithmetic at the beginning of 
 
 this period with that at the close 169-170 
 
 The place of arithmetic in the schools of that period 170-184 
 
 A. In the Latin Schools 170-178 
 
 1. The function of the Latin Schools 170 
 
 a. To teach the Latin language ... 170 
 
 b. To contribute to general culture 170 
 
 2. Courses of study in the Latin schools 170-172 
 
 a. Language — Latin 170-171 
 
 b. Music 172 
 
 c. Arithmetic . . . 172 
 
 d. Astronomy 172 
 
 e. Geometry 172 
 
 3. Writers of Latin School Arithmetics 172-174 
 
 a. Profession — vocation 174 
 
 b. Scholarship . . 174 
 
 4. Character of the contents of Latin School Arithmetics. 174-177 
 
 a. Prominence of pure arithmetic 174-175 
 
 (1) Definitions I74-I7S 
 
 (2) Classifications 175 
 
 (3) Plans of organization . 175 
 
 b. Applications — chiefly artificial or traditional problems 176 
 
 c. Absence of commercial arithmetic 176 
 
 5. Reasons for teaching arithmetic in the Latin Schools. 176-178 
 
 a. It was part of the classical inheritance 178 
 
 b. It was supposed to contribute to general culture or to 
 
 mental efficiency. 178 
 
 B. In the Reckoning Schools 178-184 
 
 1. The rise of the Reckoning Master . 178-179 
 
 a. General duties. 179 
 
 b. Relation to schools and education 179 
 
 2. The function of the Reckoning Schools 180 
 
 a. To teach business methods . ... . . 180 
 
 b. To teach commercial arithmetic 180 
 
 3. Courses of study in the Reckoning Schools . . . . 180 
 
 a. Practice in reading and writing the mother tongue. 180 
 
 b. Business customs and forms 180 
 
 c. Arithmetic • 180 
 
 4. Writers of Reckoning School Arithmetics 180-183 
 
 a. Profession — vocation 183 
 
 b. Scholarship 183 
 
 5. Character of the contents of the Reckoning School 
 
 Arithmetic 183-184 
 
 a. Meagre treatment of pure arithmetic 183 
 
 (1) Little emphasis on definitions 183 
 
20 CONTENTS 
 
 PAGE 
 
 (2) Little emphasis on classifications 183 
 
 (3) Processes confined to comparatively small num- 
 
 bers — those used in practical applications . 184 
 
 b. Prominence of applied arithmetic 184 
 
 (1) Applications followed closely upon the pre- 184 
 
 sentation of processes 184 
 
 (2) Concrete problems often proposed before the 
 
 required process had been developed . . 184 
 
 (3) Commercial problems and problems in men- 
 
 suration were the chief applications . . 184 
 6. Reasons for teaching arithmetic in the Reckoning 
 
 Schools . 184 
 
 a. Because of its use to artisans . . . . .... 184 
 
 b. Because of its use in commercial pursuits .... 184 
 
 CHAPTER II 
 
 EDUCATIONAL SIGNIFICANCE OF SIXTEENTH CENTURY ARITHMETIC 
 
 I. Introduction 185 
 
 II. Subject Matter 186-201 
 
 A. Kinds 186 
 
 1. Reckoning with counters and figures 186 
 
 2. Properties of numbers ... .... . . . 186 
 
 3. Denominate numbers 186 
 
 4. Business problems 186 
 
 5. Amenity and puzzle problems 186 
 
 B. Bases of selection 186-199 
 
 1. Needs of the trader 186-190 
 
 a. Commercial development tends to vitalize arithmetic 187 
 
 b. Modern conditions will not revive the Reckoning 
 
 Book 188 
 
 c. The needs of the trader lead to improved methods 
 
 of calculation 189 
 
 d. Business needs condition the selection of denomi- 
 
 nate number tables 190 
 
 2. Needs of the scholar 190-196 
 
 a. The modern disciplinary ideal demands concrete 
 
 subject matter 191-193 
 
 b. The culture ideal encourages the selection of subject 
 
 matter with a many-sided interest 194-196 
 
 c. The propaedeutics of arithmetic demand the reten- 
 
 tion of certain theoretic matter 196 
 
 3. Tradition 196-199 
 
 This tends to perpetuate obsolete material 197-199 
 
 C. Plans of arrangement ... . 199-201 
 
 1. By kinds of numbers 199 
 
 2. By kinds of processes . 199 
 
 3. Modern needs are met by a combination of (1) and (2) 200-201 
 
CONTENTS 21 
 
 PAGE 
 
 III. Method 201-222 
 
 A. Meaning 201 
 
 B . Suggestiveness of sixteenth century arithmetic 201 
 
 C. General principles of treatment of subject matter 201-202 
 
 1 . The synthetic method not adapted to elementary arith- 
 
 metic 202 
 
 2. The analytic method may improve books,[but cannot sup- 
 
 plant the' teacher 203 
 
 3. The psychological method produces the best books in 
 
 all particulars 203-204 
 
 D. Details of development 204-222 
 
 1. Definitions 204-205 
 
 2. Notation . ... 205-206 
 
 a. The use of improved notations may be hastened . . 205 
 
 b. Roman notation to thousands should be retained . 205 
 
 c. Notation for large numbers is necessary and belongs 
 
 to grammar school arithmetic 206 
 
 3. Processes with integers 207-211 
 
 a. In general, there is no best method for performing 
 
 processes .... 207-208 
 
 b. Artificial means for making number work interest- 
 
 ing should not be abandoned . 208 
 
 c. Incorrect language not an inheritance from six- 
 
 teenth century arithmetic . ... ... 208 
 
 d. Methods of testing work are derived from the early 
 
 arithmetics 208-209 
 
 e. Number combinations are not all equally important. 209-210 
 
 f. Explanations of processes should not be neglected. 210 
 
 g. Books should explain mathematical conventions . 210 
 h. Unabridged processes should precede abridged ones. 211 
 
 4. Processes with fractions . . . . 211-214 
 
 a. Common fractions are still necessary 211 
 
 b. There are three ideas necessary to the concept of 
 
 fractions . . 211-212 
 
 c. Formal multiplication should precede formal addi- 
 
 tion and subtraction 212 
 
 d. The use of the word "times " in multiplication . . 212-213 
 
 e. The two customary methods of division of fractions 
 
 are related 213 
 
 f. Fractions should be correlated with denominate 
 
 numbers . 213-214 
 
 5. Denominate numbers . 214-215 
 
 a. Operations with compound numbers should be lim- 
 
 ited to two or three denominations ... . . 214 
 
 b. There is no good reason for continuing the practice 
 
 of reduction from one table of denominate num- 
 bers to another • . ... 214-215 
 
 c. Denominate numbers should be presented under 
 
 each process with integers and with fractions . . 215 
 
22 CONTENTS 
 
 PAGE 
 
 6. Applications . . . 216-222 
 
 a. Applications may be proposed as incentives for 
 
 learning the processes ... 216-218 
 
 b. Applications should be appropriate to the different 
 
 school years . . . .... . 218-219 
 
 c. Mensuration work should be graded . . . . 219 
 
 d. Factitious problems have no place in elementary 
 
 arithmetic .... ... 219-220 
 
 e. Unitary Analysis, Rule of Three, and the Equation 
 
 are related processes of solution 220-221 
 
 f. The simple equation will become the leading 
 
 method of solution .... 221 
 
 g. Arithmetic has an interpretative function 221-222 
 
 IV. Mode 223-225 
 
 A. The heuristic mode suggests that oral work should develop 
 
 new ideas 223 
 
 B. The individual mode is apt to result in dogmatic teaching. 224 
 
 C. The recitation mode finds no precedent in sixteenth cen- 
 
 tury arithmetic 224 
 
 D. The lecture mode has no place in elementary arithmetic . 224-225 
 
 E. The spirit of the laboratory mode may be helpful in teach- 
 
 ing arithmetic 225 
 
 V. Summary . . 226-228 
 
CHAPTER I 
 
 The Essential Features of Sixteenth Century 
 Arithmetic 
 
 The arithmetic of the last quarter of the fifteenth century 
 and that of the sixteenth century show that great productive 
 activity possessed the arithmeticians of that period. 1 Ap- 
 proximately three hundred works were printed on this sub- 
 ject, some of which ran through many editions. 2 This awak- 
 ening was part of the great Renaissance and was due to the 
 same causes; those influencing arithmetic most directly were 
 the revival of scientific interest, commercial activity, and the 
 invention of printing. The first cause was the incentive which 
 led scholars to develop the science of figure reckoning; the 
 second made a knowledge of casting accounts and of reckon- 
 
 1 De Morgan (Arithmetical Books, pp. v-vi) says that from 1500 to 1750 
 probably three thousand works on arithmetic were printed in all languages. 
 He mentions fifteen hundred of them, but only seventy of the number 
 printed before 1600 had been seen by him. It is probable, according to 
 De Morgan, and by reference to Peacock's article in the Encyclopedia 
 Metropolitana, that Peacock was familiar with a still smaller number. 
 
 According to Kuckuck (Die Rechenkunst im sechTThnten Jahrhundert, 
 p. 16) over two hundred works on arithmetic were published in this period. 
 He quotes Michel Stifel (1544) as saying that a new one came out everyday. 
 
 Riccardi (Bibliotica Mathematka Italiana, Vol. II, pp. 20-22), the great 
 authority on the bibliography of Italian mathematics, gives one hundred 
 and twelve Italian works under the title : " Trattati e Compendi di Arim- 
 metica.'' Of the extant works the number in German is about equal to the 
 number in Italian. There are one-fourth as many Dutch, one-fourth as 
 many French, one-fifth as many English, and one-tenth as many Spanish. 
 Assuming that Riccardi's list is complete, and that the books lost bear a 
 constant ratio to the number extant in all languages, one may conclude that 
 there were approximately three hundred arithmetics printed before 1600. 
 
 2 Professor Smith has found that Gemma Frisius's "Arithmeticae Prac- 
 ticae Methodus Facilis " saw fifty-six editions before 1600, although Treut- 
 lein (Abhandlungen, 1 : 18) found only twenty-five. 
 
 Adam Riese's books in various combinations saw at least twelve editions 
 before 1600. Unger, pp. 49-51- 
 
 Recorde's and Baker's works in England enjoyed a similar popularity. 
 
 23 
 
24 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 ing exchange indispensable; and the third made the dissemin- 
 ation of this knowledge possible. Another circumstance 
 which encouraged both the development and the use of arith- 
 metic was the expression of the subject in the vernacular, for 
 hitherto a knowledge of theoretical arithmetic had been poss- 
 ible to scholars only. In the sixteenth century arithmetics ap- 
 peared in nearly all European languages, especially in Italian, 
 French, German, Dutch, and English. 
 
 Thus, the first century of printed arithmetics marks sev- 
 eral important transitions : The transition from the use of the 
 
 '■■ Roman symbols and methods to the use of the Hindu symbols 
 and methods; * from arithmetic expressed in Latin to arith- 
 
 ; metic expressed in the language of the reader ; 2 from arith- 
 metic in manuscript to arithmetic in the printed book; 3 from 
 arithmetic for the learned to arithmetic for the people; 4 from 
 arithmetic theoretic to arithmetic practical ; 5 and from the 
 use of counters to the use of figures. 6 Although the Hindu 
 numerals had been generally known to European mathema- 
 ticians after the twelfth century, 7 the devotion to classical 
 
 1 E. g., compare Jacob Kobel, Zwey rechenbuchlin uff der Linien und 
 Zipher/ (i537) with Robert Recorde, The Ground of Artes 
 
 (c. 1540). 
 
 2 E. g., compare Gemma Frisius, Arithmeticae Practicae Methodus Facilis 
 (1558) with Ian Trenchant, L'Arithmetique, Departie en trois livres 
 (1578). 
 
 8 The Treyiso Arithmetic (1478), so called' from the place of printing, is 
 the earliest printed arithmetic known to exist. 
 
 i E. g., compare Hieronimus Cardanus, Practica Arithmetice & Mensur- 
 andi singularis (iS39) with Willem Raets, Ein niew Cijfferboeck 
 
 - - (1580). 
 
 5 E. g., compare Joannis de Muris, Arithmeticae Speculativae Libri duo 
 (1538) with Adam Riese, Rechnung auff der Linien und Federn/ 
 (i57i). 
 
 s Adam Riese (1522) and Robert Recorde (1557) show this transition 
 by treating 'both systems of reckoning in their books. 
 
 7 M. F. Woepcke, Propogation des chiffres indiens, Journal Asiatique, 
 6 ser, t. 1, pp. 27, 234, 442. 
 
 Ch. Henry, Sur les deux plus anciens traites Francais d'Algorisme et de' 
 Geometrie, Bone. Bull., 15 :4a. 
 
 Freidlein, on John of Seville and Leonardo of Pisa, Zeitschrift der 
 Mathematik und Physik, Band 12, 1867. 
 
 A. Kuckuck, Die Rechenkunst im sechzehnten Jahrhundert (1874). "In 
 
THE ESSENTIAL FEATURES 
 
 25 
 
 study and the neglect of science tended to perpetuate the Ro- 
 man numerals. Not until the introduction of printing did a 
 full comparison of the Roman and Hindu arithmetics find ex- 
 pression, and a working knowledge of the latter sift down to 
 the common people. 
 
 Owing to the Romans' lack of appreciation of pure science 
 and to the awkwardness of their system of notation, their 
 contribution to arithmetic was small. Arithmetic, in the form 
 of the ancient Logistic, was of use to them chiefly in making 
 monetary calculation, for which they used some form of the 
 abacus. The nations of Europe received as a legacy from 
 the Romans the Roman numerals and the art of reckoning 
 with counters. This art, also called line reckoning, was wide- 
 spread in the fifteenth century, excepting in Italy. The cal- 
 culations were effected by means of parallel lines drawn on a 
 board or table, and by movable counters or disks placed upon 
 them. The lines taken in order from the bottom upward rep- 
 resented units, tens, hundreds, thousands and so on. The 
 spaces taken in the same order represented fives, fifties, five 
 hundreds, and so on, the space below units being used for 
 halves. The table on the next page shows Riese's explanation 
 of the lines and spaces : * 
 
 einer Regensburger Chronik von 1167 befinden sich die Zahlen von 1-68, 
 aber nur wie zur Uebung geschreiben. In Schlesien komtnen sie erst im 
 Jahre 1340 vor. In einem Notatenbuch des Dithmar von Meckelbach aus 
 der Zeit Kaiser Carls IV stehen die Ziffern 1-10." Pp. 4-5. 
 
 See also the following general references : 
 
 Treutlein, Geschichte unserer Zahlzeiohen (1875— Program Gymn. Karls- 
 ruhe), Bd. 12, 1867. 
 
 Gerhardt, Geschichte der Math, in Deutscbland (1877). 
 
 Freidlein, Die Zahlzeichen und das elementare Rechnen der Griechen, 
 Romer u. des christl. Abendlandes vom 7. bis 13. Jahrhundert, 1869. 
 
 Wildermuth, Rechnen, in Sohmids Encyklopadie, Bd. 6. 
 
 1 Adam Riese, Rechnung auff der Linien und Federn/ (1571 ed.), 
 
 fol. Aiij recto. 
 
26 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 1 00000 
 
 50000 
 
 10000 
 
 5000 
 1000 
 
 500 
 
 100 
 
 So 
 
 10 
 
 5 
 1 
 
 ji 
 
 ■ Hundert tausent. 
 Funfftzig tausent. 
 Zehen tausent. 
 Funf tausent. 
 Tausent. 
 Funf hundert. 
 Hundert. 
 Funfftzig. 
 Zehen. 
 Funff. 
 Eins. 
 Ein halbs. 
 
 Cross lines were drawn dividing 
 the board into sections, which 
 could be used for different de- 
 nominations of numbers, for the 
 addends in a problem of addition, 
 for the minuend and the subtra- 
 hend in subtraction, or for any 
 other sets of numbers. 
 
 The illustration * shows four 
 compound numbers arranged for 
 addition and the result expressed 
 in floren, groschen, and denarii. 
 
 Addition and subtraction with 
 counters are evidently quite easy. 
 In fact, line reckoning was often 
 recommended as preferable to the 
 processes with Hindu numerals. 2 
 
 TtVniTrotominora-uopKorTiaBmarno&jrflaerr&romfw 
 tit IDincvtricBCuptentfcoadmfnrpt'aoneq addition! con 
 rulcre licet 3Tain numcriaddcntoretreliiltabir. Sumnu. 
 
 «««*• 
 
 *-•*♦ 
 
 -®r8M»- 
 
 Aft: 
 
 •• • • eve 
 
 H-friotieampiTimm3 radon ttintptterpitmtndt $ 
 Uqaent fubtOiae compos . 
 
 ft «f ft 
 
 •♦•» 
 
 4T3rtin\t0l*tddditione'nontnaria0in0erpcrufiecoIIfCt{* 
 one m q quicqd mill* neuenano Tupril pba trpeilerar.0,0 tff 
 'ta(ntc(Iigereopoitet*SNi nunr:ri8addendie.9.rparrun colic 
 etc rocicno abicidrur quociene repetca.^llo erpedito qutc 
 QDfdnnmerifabq vlnmorCTnaTcntidod,pboin[>ruierc'ne/ 
 CcfIccflaucnaddcdotain / pba.rociuBrumn!crimib9crlCA' 
 beminne q talicngento erlincis ?fpac$0 tcducte. arte oddV 
 mitt ©in our.isnaaie tenens Sed vt [bmnte erplTt credarar 
 vnfqlcudiRBCflmbij'piobeadicianrur' 
 
 All that is necessary, after 
 
 1 From Balthazar Licht's Algorismus Linealis (1501 ed.). 
 
 2 Rudolff, Kuns'tliche rechnung mit der Ziffer und mit den zal pfennige/ 
 
 (1526), "Das die vier spezies/ auff den linien d'urch viel ringere 
 ybung auff der Ziffer gelernit werde/ mag ein yeder aus obenanzeigter vnter- 
 weisung bey jm selbst ermessen. Warlich was Fiirsten vnd Herrn 
 
 Rentkamer/ vrbarbiicher/ register/ aussgab/ empfang/ vnd ander gemeine 
 hausrechnung belangt/ dahin ist sie am bequemisten/ zu subtilen rech- 
 nungen zum dickermal seumlich." 
 
 See also: 
 
 Sterner, Geachichte der Rechenkunst, I Teil, pp. 218, 219. 
 
 Kastner, Geschichte der Mathematik, Bd. I, p. 42. 
 
 Knott, C. G., The Abacus in its Historic and Scientific Aspects, Trans- 
 
THE ESSENTIAL FEATURES o? 
 
 having expressed the numbers, is to shift or set them so as to 
 express the sum or difference of the numbers found on each 
 line. The processes of multiplication and division are more 
 complex; so much so, that, when the multiplier or divisor 
 contains more than one figure, the Hindu algorisms are far 
 superior. Several modern authorities give detailed explana- 
 tions of these processes. 1 
 
 The transition frotm line reckoning to reckoning with the 
 Hindu numerals, or pen reckoning, as it was often called, was 
 slow. So difficult was it to> abandon the old line and space 
 idea and the Romani symbols, that writers mixed the old and 
 the new symbols in calculation. This is plainly shown in 
 Toilet reckoning 2 (tabular calculation), a process used in a 
 few early works, as in the Bamberg Arithmetic and in the 
 arithmetics of Widman and Apianus. For example, the prob- 
 lem : " What is the cost of 4,367 lb. 29 lot 3 quintl of ginger 
 at 16 shillings a pound?" is solved thus by Widman : 3 
 
 actions of the Asiatic Society of Japan, vol. xiv, part i, pp. 19, 34 (Yoko- 
 hama, 1886). 
 
 1 Knott, C. G., The Abacus in its Historic and Scientific Aspects, pp. 18, 
 45-67. See preceding note. 
 
 Ku'ckuck, A., Die Rechenkunst im sechzehnten Jahrhundert, pp. 10-13 
 (Berlin, 1874)- 
 
 Villicus, Geschichte der Rechenkunst, pp. 68-76 (Wien, 1897). 
 
 Leslie, Philosophy of Arithmetic, under "Palpable Arithmetic" (Edin- 
 burgh, 1820). 
 
 A more accessible work to many is : 
 
 Brooks, Philosophy of Arithmetic, pp. lis, 160 (Lancaster, Pa., 1901). 
 
 For the historical development of the abacus see : 
 
 Cantor, M., Vorlesungen iiber Geschichte der Mathematik, Vol. I (Leip- 
 sic, 2d ed., 1894). 
 
 Also articles by Boncompagni in Atti 'dell' Accademia pontificia de 
 nuovi Lincei. 
 
 2 P. Treutlein, Abhandlungen Zur Geschichte Der Mathematik, vol. 1, 
 p. 98 (Leipsic, 1877). 
 
 3 Johann Widman, Behend und hupsch Redrawing (1508 ed.). " Es hat 
 einer kaufft 4367 lb' Ingwer 29 lot 3 qufiobl/ ye 1 lb' fur 16 /> in gold- 
 setz also.'' Fol. Ei verso, Bii recto. 
 
 (A mistake was evidently made in this edition, sfince the problem reads 
 " 13 shillings a pound " in the early editions.) 
 
28 SIXTEENTH CENTURY ARITHMETIC 
 
 4M 
 
 13000 p 
 
 4M 
 
 52000 
 
 2600 
 
 3C 
 
 1300,* 
 
 3C 
 
 3900 
 
 195 
 
 6 X 
 
 130^ 
 
 6X 
 
 780 
 
 39 floren 
 
 7 lb' 
 
 13/* 
 
 7 lb' 
 
 91 p 
 
 facit 4./? 11 ,* 
 
 2X 
 
 130/32 p 
 
 2X 
 
 260/32 
 
 8 4/32 
 
 9 lot 
 
 13/32 /» 
 
 9 lot 
 
 117/32 
 
 3 21/S2 
 
 3 quintn 
 
 13/128 / 
 
 3 quitl 
 
 39/128 
 
 39/128 
 
 The first column at the left is the multiplicand, 4 thousand, 3 
 hundred, 6 tens and 7 lb. ; 20 lot and 9 lot and 3 quintl. The 
 lot and the quintl are first expressed as fractional parts of a 
 pound. The Roman symbols, M, C, X, at the right of this 
 column are superfluous, since the figures, 4, 3, 6, placed above 
 one another designate by their positions the orders for which 
 they stand. The columns beginning 1 3000 p form the multi- 
 plier, 13, set down for each order. In modern work the 4000 
 would be multiplied by 13, but here 4 is multiplied by 13,000. 
 The fourth group represents the results of the multiplications. 
 The numbers in the last column at the right express these re- 
 sults in florins and shillings. More than half a century after 
 Widman, Robert Recorde, in the later editions of his " Ground 
 of Artes," says that before studying arithmetic proper the 
 Roman numerals must be learned. 1 
 
 A method of calculating, or a mnemonic to assist in abacus 
 work, called Finger Reckoning, was explained by a few writ- 
 ers of this period. 2 But as the method was then obsolete in 
 
 1 Robert Recorde, The Ground of Artes (1594 ed'.). "Before the intro- 
 duction of Arithmeticke, it were very good to have fome vnderftandimg 
 and knowledge of thefe figures and notes :'' 
 
 This is followed by a table of Roman numerals with the corresponding 
 Hindu numerals and the corresponding words, as : 
 
 one 
 two 
 three Fol. Bviii verso. 
 
 2 Noviomagus, De Numeris' Libri II, Cap. XIII. 
 
 Paciuolo, Suma de Arithmetica Geometria Proportioni et Proportionalita 
 (1494 ed.), fol. 36 verso, or Eiiij verso. (He does not explain, but gives 
 a page of pictures.) 
 
 Andres, Sumario breve de la practica de la arithmetica, Valencia (1515). 
 
 Tagliente, Libro de Abaco, Venetia, M. D. XV (1541 ed.), fol. Aiii verso. 
 
 Apianus, Ein newe vnd wolgegrundte vnderweysung aller Kauffmans- 
 rechnung, Ingolstadt (1527). 
 
 Aventinus, Abacvs at qve vetvstissima, vetervm latinorum per digitos 
 
 i 
 
 1 
 
 ii 
 
 2 
 
 iii 
 
 3 
 
THE ESSENTIAL FEATURES 2 Q 
 
 Western Europe, it has no significance here. Accounts of 
 this method are given in standard authorities. 1 
 
 In order to exhibit the essentials of the arithmetic of this 
 period briefly and systematically, it is best to treat it by topics, 
 as was the common practice of its authors. 
 
 DEFINITIONS 
 
 It was the common practice among Latin School writers, 
 especially among those who were influenced by the works of 
 the Greeks on theoretic arithmetic, to begin with a formidable 
 list of definitions. 
 
 Definitions of Number, Unity, and Zero 
 Number was generally defined thus : " Number is a collection 
 of units." The following is Paeiuolo's definition: 2 "Num- 
 ber is a multitude composed of units. Aristotle says, if any- 
 thing is infinite, number is, and Euclid in the third postulate 
 of the seventh book says that its series can proceed to infinity, 
 and that it can be made greater than any given number by add- 
 ing one." 
 
 The meaning of unity caused writers much concern and was 
 variously defined, as appears from the following : 
 
 i. Unity is the beginning of all number and measure, for as 
 we measure things by number, we measure number by unity. 8 
 
 manusq) mumerandi (quin etiam loquendi) cofuetudo, Ex >beda cu picturis 
 et imaginibus - Ratispone ( T 532). 
 
 Moya, Traitado de Matematiaas, Aloaba, 1573 (1703 ed., chap. ix). "Trata 
 dela orden que los antiquos timiero en catar con los dedos de las manos, 
 y otras partes del cuerpo." 
 
 1 Leslie, Philosophy of Arithmetic (Edinburgh, 1820), p. 101. 
 Villicus, Geschichte der Rechenkunst (1897), pp. 10-14. 
 
 Stoy, H., Zur GesehicMe des Reohenunterrichts, I. Teil, § 9, p. 47. See 
 plates at end of volume. 
 
 Sterner, Geschichte der Rechenkunst, I. Teil, p. 77. 
 
 Cantor, Vorlesungen fiber Geschichte der Mathematik (1900 ed.), Bd. I, 
 see index. 
 
 2 Paciuolo, Suma de Arithtnetica Geometria Proportioni et Proportion- 
 «iita (1523 ed.). "Numero: e (fecondo ciafchuno philofophante) vna 
 moltitudioe de vnita copofta : Arif totile dace : oioe. Si quid infinitum 
 eft : nuinerus eft. E per la terza petitione del f eptimo de Euclide : la fua 
 ferie in infinito potere procedere: a quocuq? numero dato: dari poteft 
 maior: vnitatem addendo." Fol. A i recto. 
 
 3 Joan Noviomagus, De Numeris Libri II (i544)- 
 
30 SIXTEENTH CENTURY ARITHMETIC 
 
 2. Unity is not a number, but the source of number. 1 
 
 3. Unity is the basis of all number, constituting the first 
 in itself. 2 
 
 4. Unity is the origin of everything. 3 
 
 This difference of opinion as to the nature of unity was not 
 new in the sixteenth century. The definition had puzzled the 
 wise men of antiquity.* Many Greek, Arabian, and Hindu 
 writers had excluded unity from the list of numbers. But, 
 perhaps, the chief reason for the general rejection of unity 
 as a number by the arithmeticians of the Renaissance was 
 the misinterpretation of Boethius's arithmetic. Nicomachus 
 (c. 100 A. D.) in his ApcB/iTjTMyc pipfaa a™ ha.d said that unity was 
 not a polygonal number and Boethius's translation was sup- 
 posed to say that unity was not a number. 5 Even as late as 
 1634 Stevinus found it necessary to correct this popular error 
 and explained it thus : 3 — 1=2, hence 1 is a number. 6 
 
 1 Gemma Frisius, Arittometlcae Practicae Melfchodus Facilis (1575 ed.). 
 " Numerum autem vocant multitudinem ex unitatibus conflatam. Itaque 
 unitas ipsa nuimerus no.n erit, sed numerorum omnium principium." Fol. 
 A, verso. 
 
 Jacques Chavvet Champenois, Institvtions De L'Arittometique (1578 ed.). 
 " Vn, qui n'est pas nombre, mais comeneement de nombre, & origine de 
 toutes chofes, - -." Page 3. 
 
 Humphrey Baker, The Well Spring of Sciences (1580 ed.). "'And' 
 Hherfore am vnitie is no number, 'but the begining and originall of number, 
 as if you doe rnultiplie or deuide a vnite by it felfe, it is refolued ikito 
 itfelfe without any inereafe. But it is in number otherwife, for there 
 can be no number, how great foeuer it bee, but that it may continually bee 
 encreafed by adding euermore one vnitie vnto the fame." Fol. Bi recto. 
 
 2 Frameiscus Mavrolycus, ArMimjeticorum Libri Duo (1575). "Unitas 
 est prmoipium & constitutrix omnium numeroriurn, .constituens autem im- 
 primis seipsaim." Page 2. 
 
 3 See definition by Champenois in note 1 above. 
 
 4 In Plato's Republic /we find : " To which class do unity and number be- 
 long?" Monroe's Source Book, p. 203. 
 
 5 Weissenbom, H., Gerbert, Beitrage zur Kenntoiss der Mathemaitik des 
 Mittelalters, p. 219 (Berlin, 18 
 
 6 After reviewing the various arguments which history has handed down, 
 Stevinus says : " Que l'unitie est nombre. II est notoire que Ton diet vul- 
 gairement que 1'unite ne soit point nombre, ains seulement son prineipe, ou 
 
THE ESSENTIAL FEATURES 3I 
 
 Zero was referred to merely as a symbol used in connection 
 with the digits to express number. When taken alone it was 
 said to have no meaning. The prevalence of nulla, nulle and 
 rein in the terms used to express it is suggestive of this 
 meaning. 1 
 
 So far were the mathematicians of that period from the 
 conception of number as a continuum; that they emphasized 
 the difference between continuous and discrete quantity and 
 limited arithmetic to the latter domain. 2 So long as they 
 
 commencement & tel en nombre come le point en la ligne; ce que nous 
 nions, & en pouvons argumenter enter en cefte forte : 
 
 La partie eft de mefme matiere qu'est fon entie/r, 
 
 Vnite eft partie de multitude d'unitez; 
 
 Ergo I'unitiS eft de mefme matiere qu'eft la multitude d'unitez; 
 
 Mais la matiere des multitude d'unitez est nombre, , 
 
 Doncques la matiere d'unite eft nombre. 
 Et qui le nie, faict comme celuy, qui nie qu' une piece de pain foit du pain. 
 Nous pourrions auffi dire ainfi : 
 
 Si du nombre donne I'on ne foubftraict mil nombre, le nombre donni 
 demeure, 
 
 Soit trois le nombre donne, & du mefme foubftrayons un, qui n'est 
 point nombre comme tu veux. 
 
 Doncques le nombre donne demeure, c'eft a dire qu'il y reftera en- 
 core trois, ce qui eft abfurd. Fol. A recto. 
 
 1 The various names for the symbol o An this period were : Zefiro and 
 nulla, Piero Borgi (1488 ed.) ; cero and nulla by Paciuolo (1523 ed.) ; 
 nulla, Rudolff (1534 ed.) ; cyphar, Recorde (1540) ; circolo, cifra, zerro, 
 nulla, Tantagiia (1556) ; cyphram, Gerruma Frisius (1558 ed.) ; circulus, 
 Ramus (1577 ed.) ; nulle or zero, Trenchant (1578 ed.) ; nul and rein, 
 Champenois (1578) ; ciphar, Baker (1580 ed.) ; and nullo, Raets (1580). 
 Buteo (1559) says that the zero could be called omicron because of its form. 
 
 The oldest manuscript actually known to have the zero bears the date 
 738 A. D., by J&ika Rashtrakuta. 
 
 2 Unicorn, De L'Arithmetioa vn.iuersale (1598 ed.). "Quantity is divided 
 into two 'Classes : continua and discreta. La quantita continua is for- 
 mally dlefinied to be that of which the terminus of every part joins the 
 terminus of another part, a c b As, for example, in the line 
 
 ab, the point c is the terminus of the part ac, and this is 
 also a terminus of the part be, and a common terminus. 
 Or in the surface abed, the line ef is the common ter- 
 minus which divides it into two parts. Of this division 
 of quantity there are five kinds : lines, surfaces, solids, 
 place and time. The treatment of these belong to geometry. 
 
 Quantita discreta is defined to be such that no part is joined to another 
 
32 SIXTEENTH CENTURY ARITHMETIC 
 
 held to this limitation, they could never think of the one-to- 
 one correspondence of numbers to points on a line. 1 The 
 Greek method of representing surds by lines was well known, 
 and it would have been, easy to arrive at the conception of 
 filling in the points of a line with numbers, had not continuity 
 been excluded. 
 
 Definitions of Classifications 
 
 The definition of number was followed by definitions of the 
 various classifications of numbers. The following taken from 
 Paciuolo will illustrate : 2 
 
 A number is prime when it is not divisible by any other in- 
 tegral number but one and the number itself. Otherwise it 
 is composite. Examples of primes: 3, 7, 11, 13, 17, etc. 
 Examples of composites : 4, 8, which is 2 X 4, 12, 14, 18, etc. 
 
 Lateral or linear numbers. Different numbers which may 
 be multiplied together, as 3 and 4, 6 and 8, compared to the 
 sides of a rectangle. 
 
 common part of another quantity, as a number. For example, in numbers 
 with periods containing three orders (the usual method of numeration), 
 the last number is the last of that group, and is not the beginning of the 
 next group. 
 
 Another difference between quantita discreta and quantita continua is 
 that the continua is divisible ad infinitum and the discreta is increasable 
 ad infinitum. 
 
 The quantita continua is divided into mobile and immobile, and by im- 
 mobile is meant the earth, and by mobile the heavens. Under the immobile 
 is included geometry, and under mobile, astrology. 
 
 Two other kinds of quantity: that which has position, as the solid, con- 
 tinuous thing; the other which has not position, as time, which is constantly 
 passing, and water and other liquids, which have not position but which 
 are limited' by other things, as by the vessel containing the water. 
 
 _ . f .mobile — cielo. TT „ . .„ 
 
 (1. Contmua j immobil e-terra. ^ Hauete P° sitl0e - 
 
 Quantita ■< ; 
 
 I t^. \ numero. , T _ , _ 
 
 ^ 2. Discreta -j orat : one 4- No hauete positione. 
 
 Fol. A 2 recto. 
 
 1 Dedekind, R., Essays on the Theory of Numbers (Beman's translationy 
 Chicago, 1901). 
 
 2 Paciuolo, Summa (1523 ed.), fol. Ai recto, Aij verso et seq. 
 
THE ESSENTIAL FEATURES 33 
 
 Superficial (plane) number. The product of two linear 
 numbers, as 12 from 3 X 4, 48 from 6X8. 
 
 Square number. The product of two similar numbers, as 
 9 from 3 X 3. l6 from 4 X 4, 25 from- 5 X 5> etc - 
 
 Solid number. The product of three linear numbers, as 12 
 from 2X3X2. 
 
 Cubic number. The product of three equal numbers, as 8 
 from 2 X 2 X 2, 27 from 3 X 3 X 3, 64 from 4X4 
 X 4, etc. 
 
 Triangular numbers. Those which commence with unity, 
 and which increase upward in the form of a triangle by add- 
 ing a unit, always keeping the sides equal. 1 
 
 Besides this there are pentagonal numbers, etc. 
 
 Circular numbers, as 5 and 6 ; so called because each multi- 
 plied by itself to infinity always gives a product ending in 
 itself, as s X 5 = 25 X 5 = 125 X 5 = 625 -; 6 X 6 
 = 36 X 6 = 216 x 6 = 1296 - -. 
 
 Defective numbers are those the sum of whose factors is less 
 than the number itself, as 8 and 10. 8 = 4X2X154 + 
 2 + 1 = 7; io=5X2Xi;5 + 2 +i=8. 
 
 Superfluous numbers. Those the sum of whose factors is 
 more than the number itself, as 12, 24, etc. The factors of 
 12 are 6, 4, 3, 2, 1 ; 6 + 4 + 3 + 2 + 1 = 16: factors of 
 24 are 12, 8, 6, 4, 3, 2, 1 ; the sum of these is 36. 
 
 Perfect numbers are those the sum of whose factors equals 
 the number itself; e. g., the factors of 6 are 3, 2, 1, and their 
 sum is 6; the factors of 28 are 14, 7, 4, 2, 1, and their sum 
 is 28. 
 
 The manner of designating various ratios was also peculiar 
 and elaborate. For example, the relation of any two num- 
 bers whose ratio is 1 J4 to 1 was called sesquialteral, meaning 
 that the antecedent contains the consequent once and one-half 
 
 1 See marginal illustrations in Paciuolo. 
 
 2 The sesqudaltera stop of an organ which furnishes the perfect fifth in- 
 terval, 1 : l l A, isr named from this old Greek ratio. 
 
34 SIXTEENTH CENTURY ARITHMETIC 
 
 Similarly the ratio 
 
 iY : i, or 4 : 3 was called sesquitertiai. 
 
 1% : 1, or 5 : 4 was called sesquiquartal, and so on. 
 Of all the pairs of numbers that can result in sesqui ratios, 
 the antecedents were called superparticularis and the conse- 
 quents subsuperparticularis. 1 
 
 When the integral part of the ratio is greater than 1 the 
 above ratios were preceded by the corresponding adjectives, 
 thus the ratio 
 
 2.Y2 : 1 was called duplex sesquialteral. 
 
 2.Y2 : 1 was called duplex sesquitertiai. 
 
 3% •" 1 was called triplex sesquiquartal, and so on. 
 
 When the fraction in the ratio exceeds a unit fraction, the 
 ratio was named according to the numerator, thus the ratio 
 
 1% : 1 was called superbipartiens. 
 
 1 Y\ : 1 was called supertripartiens, and so on, meaning ex- 
 cess by two parts, by three parts, and so on. 
 
 The prefix sub was used to designate the inverse of the above 
 ratios, thus the ratio 
 
 1:1% was called subsuperbipartiens. 
 
 1 : 1 24 was called subsupertripartiens. 2 
 
 The reason for emphasizing these peculiar classifications of 
 number is not manifest. Legendre 3 explained the prominence 
 given to the subject on the ground that its study becomes a 
 sort of passion with those who take it up. This phase of 
 
 1 A discussion of the classifications of Nicomachus may be found in 
 Gow, History of Greek Mathematics (Cambridge, 1884), page 90. 
 
 2 After so much of explanation, a characteristic remark of DeMorgan 
 will be appreciated : " For some specimens of the laborious manner by 
 which the Pythagorean Greeks, in the first instance, and' afterwards Boe- 
 thius in Latin, had. endeavored to systematize the expression of numerical 
 ratios, I may refer the reader to the article Numbers, old appelations of, 
 ira the Supplement to the Penny Cyclopedia (London, 1833-43). If I were 
 to give any account of the whole system, on a scale commensurate with the 
 magnitude of the iworks written 00 it, the reader's patience would not be 
 subquatuor decupla subsuperbipartiens septimas, or, as we should now say, 
 seven per cent of what he would find wanted for the occasion." DeMor- 
 gan, Arithmetical Books, p. xx. 
 
 3 Legendre, Theories des Nombres (1798), preface. 
 
THE ESSENTIAL FEATURES 35 
 
 arithmetic had its origin in the products of the Pythagorean 
 School, was expounded by Nicomachus, 1 and was communi- 
 cated to the scholars of the Renaissance by Boethius's trans- 
 lation 2 of Nicomachus. Among the works of the sixteenth 
 •century which treated this subject, that of Maurolycus 3 is 
 noteworthy both for its exposition of the Greek classifications 4 
 and for its doctrine of incommensurables. 
 
 Writers on commercial arithmetic in the sixteenth century 
 introduced their works by definitions of arithmetic, quantity, 
 and number, but discarded the fanciful theory of numbers so 
 attractive to theoretic writers. 
 
 Definitions of Processes 
 Each process was defined when first introduced, which was 
 in connection with integers, since most of the writers treated 
 the four fundamental processes with integers before doing so 
 with fractions and denominate numbers. These definitions 
 related to integers only, and often failed to have meaning when 
 applied to fractions. Addition was generally defined as the 
 collection of several numbers into one sum, 5 and subtraction 
 as taking a smaller number front a larger one.* Certain writ- 
 ers 7 improved on this, and even recognized subtraction to be 
 the inverse of addition. 8 Multiplication was generally defined 
 
 1 See page 34, note 1. - See page 34, note 2. 
 
 3 Mavrolycus, Franciseus, Arithmeticorum LAbri Duo (1575), p. 3. 
 
 4 Jordanus (1496 ed) also is noted for its extensive treatment of the 
 Creek properties oif numbers. 
 
 5 E. g., Trenchant, UAriithme'tique (1578), "Aiouter, eft affembler plu- 
 fieur nomfores en une fomme," 'fol. B 4 recto; and Clichtoveus, Ars fuppu- 
 tadi (Paris, c. 1507). "Additio eft mrultorum numeroj fiigillatim 
 fuimptaruim in unam fumrnaim ooliectio." Fol. b iiij recto. 
 
 6 E. g., Tartaglia, General Trattato di N.umeri (1556). " Sottare non e 
 altro, che duoi proposti numeri, inequali saper trouare la loro differentia, 
 ■cioe quanto che il maggiore eccelde il menore." Fol. Bvi verso. 
 
 7 E. g., Ton'stall, De Arte Supputand'i (1522). " Subd-ucto numerorvm 
 est minoris numeri a maiore ; uel aequalis ab equale fub'tractio." Fol. E 2 
 recto. 
 
 8 Clichtoveus, Ars fupputadii (c. 1507). " Subftractio est numeri minoris 
 a majori subduotio. Et additioni ex opposito respondet." Fol. biiii verso. 
 
36 SIXTEENTH CENTURY ARITHMETIC 
 
 as repeating one number as an addend as many times as there 
 are units in another, 1 a definition not directly applicable to 
 fractions without modification. Division was generally defined 
 as finding how many times one number is contained in another* 
 the partitive phrase being often included. Its relation to sub- 
 traction was also recognized. 3 
 
 PROCESSES WITH INTEGERS 
 
 The writers of that period did mathematics a service in re- 
 ducing the number of processes in arithmetic. The processes 
 were commonly called Species, 4 due to the influence of the 
 Latin manuscripts. In 1370 Magistro Jacoba de Florentia 
 gave 9 Species, the number common in mediaeval times, viz. : 
 numeratio, additio, subtratio, duplatio, mediatio, multiplicatio, 
 divisio, progressio, et radicum extractio. The Hindus, ac- 
 cording to the Lilavati, 5 had eight processes, which were in- 
 creased to ten by the Arabs, who added Mediatio and Duplatio. 
 These latter are found in El Hassar (c. 1200), and probably, 
 according to Suter, are of Egyptian origin. Their presence 
 in mediaeval Latin manuscripts is due to the influence of Al 
 Khowarazmi. 6 ; In the sixteenth century the number ranged 
 
 1 Gemma Frisius, Arithmeticae Praoticae Methodus Facilis (1575 ed.)- 
 " Multiplicare est ex ductu vnius numeri in alterum numerum producere, 
 qui toties habeat in se vnum multiplicantium, quoties alter vnitatem, Hoc 
 est, Multiplicare est numerum quemcumqj aliquoties aut mul-toties, exag- 
 gerate." Fol. B 2 verso. 
 
 2 Trenchant, L'Arithmefrique (1578). " Parti r, eft chercher quantes foys 
 vn nombre, contient 1'autre." Fol. D 2 recto. 
 
 8 Ramus, Arithmeticae Libri Duo (1577 ed.). "Divisio est, qua divisor 
 subduotitur a dividendo quoties in eo continetur & habetur quotus." Fol. 
 A vii verso. 
 
 An interesting comparison of definitions current in the seventeenth cen- 
 tury is found in DeMorgan, Arithmetical Books, pp. 50-61. 
 
 4 The origin of the word " species " has been traced to the Greek word 
 liiog, meaning member of an equation. This word appeared in rules for 
 adding to and subtracting from the members of an equation, and was 
 translated into Latin as species, which later came to be used to designate 
 all of the fundamental processes of arithmetic. Cantor, Vorlesungen iiber 
 Geschichte der Mathematik (3d ed., 1900), Bd. I, p. 442. 
 
 6 The work of Bhaskara, a Hindu writer (c. 1200 A. D.). 
 
 6 An Arabian mathematician (c. 800 A. D.). 
 
THE ESSENTIAL FEATURES 37 
 
 from nine to five. Some writers excluded extraction of roots, 
 others progressions also. Piero di Borgi argued that the 
 number should be reduced to seven in order that it might cor- 
 respond to the number of gifts of the Holy Spirit. 1 Later in 
 the century duplatio (multiplying by 2) and mediatio (divid- 
 ing by 2) were excluded, reducing the processes to numera- 
 tion and the four fundamental operations recognized to-day. 2 
 It is probable that numeration was not always included, in 
 which case the number would be four. " The lack of agree- 
 ment with reference to the number of Species finds its ex- 
 planation in the circumstance that they fail to define the idea 
 of Species. Gemma Frisius is the only one who attempted a 
 definition : ' Moreover, we call certain kinds of operations with 
 numbers Species.' " s But this is too indefinite to give any 
 basis of selection.* 
 
 Notation and Numeration 
 The object of numeration was to teach the reading of num- 
 bers written in the Hindu notation. For a hundred years after 
 the first printed arithmetic many writers began their works 
 with the line-reckoning and the Roman numerals, and fol- 
 lowed these by the Hindu arithmetic. 5 The teaching of 
 
 1 Cardan, Practica Arithmetice (1539), Chapter II, gives seven: numer- 
 ation, addition, subtraction, multiplication, division, progression, and the 
 extraction of roots. 
 
 2 Sigismund Suevus, Arithmetice Historica (1593), fol. aii recto. 
 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1575 ed.). 
 
 " Solent nonnulli Duplationem & Mediationem assignare species distinctas 
 a Multiplicatione & Divisione. Quid vero mouerit stupidos illos nescio, 
 cu & finitio & operatio eadem sit." Fol. B 6 verso. 
 
 3 Unger, Die Methodik, page 72, § 41. 
 
 4 A few writers included the Rule of Threes — Riese for example. 
 
 5 Kobel, Z-wey reohenbuchlin (1537 ed.). After teaching the Roman 
 symbols, I, V, X, L, C, D, M, and the digits, 1, 2, 3, 9, Kobel gives 
 a comparative table entitled : " Tafel zu erkennen vnd vergleichen die zal 
 der Biichstabn aufi dem A. b. c genomen/ vnd der Figuren/ die man 
 ziferen nennet/ Underrichtung." Fol. B 6 recto and verso, or 14 recto 
 and verso. 
 
 In the table Kobel uses small letters for all numbers except ten and 
 multiples of ten, as i, ij, iij, iiij, v, - - X, xj, xij, XX -. For 500 
 he uses D c ; for 1000, j M ; 2000, ijM; for 100 he gives both C and j c ; 
 
38 SIXTEENTH CENTURY ARITHMETIC 
 
 numeration was a formidable task, since the new notation was 
 so unfamiliar to the people generally. The feeling was prev- 
 alent that one must learn the Roman system and then graft on 
 the new system. 
 
 Numbers in the Hindu system were divided into three 
 
 classes: (i) digits (digiti), i, 2, — ,9; (2) articles (articuli), 
 
 '; ten and multiples of ten, 10, 20, 30, — ; (3) composites (com- 
 
 positi), combinations of digits and articles, as 25, 37. Most 
 
 writers stated the idea of place-value simply and directly, but 
 
 some, owing to its novelty, gave it an elaborate treatment. 1 
 
 The names of the orders and the device used as a separatrix 
 
 I varied extensively. The names to hundred thousands were 
 
 ! the same as now used, but the period now called millions was 
 
 usually called thousand thousand. 2 The word million dates 
 
 200, cc and ij c ; 1100, MC, also Mjc; 1200, Mec, also Mijc; for 1300, 
 Mccc, also Miij c . 
 
 1 W'i'ckoan, Behemd und hiipsch Rechnung (1508 edl), fol. 6 recto. 
 Robert Recorde, The Ground of Antes (1594 e&). Record*, whose 
 
 book is in dialogue form, thus quaintly develops the idea of place value : 
 
 " But here must you rnarke that everie figure hath two values : One 
 alwayes certaine that it fignifietb properly, which it bath of his forme: 
 and the other vncertaine, which he taketh of his place." Fol. Cvi verso. 
 
 "M. (Master). Now then take heede, thefe certaine values euery figure 
 reprefenteth, when it is alone written without other figures joyned to him. 
 And alio when it is in the firfte place, though manie other do follow ; as 
 for example : This figure 9 is ix. standing noiw alone. 
 
 " Sc. (Scholar). How: is he alone and ftandeth in the middle of To- 
 many letters ? 
 
 " M. The letters are none of his felowes. For if you were in France in 
 the middle of a M. Frenchman, if there were none Englifh man with you, 
 you would reckon your felfe to bee alone. 
 
 " Sc. I perceiue that. And doeth not 7 that standeth in the second 
 place betaken vii? and 6 in the third place betoken vi? And so 3 in the 
 fourth place betoken three ? 
 
 " M. Their places be as you haue faid 1 , but their values are not fo. 
 For, as in the first place, euery figure betokeneth his owne value certaine 
 onely, fo in the second place euerie figure betokeneth his owne value a 
 hundretlh Dimes, fo that 6 in that place betokeneth vi. C. - -." Fol. Cvii 
 verso. 
 
 2 Raets, Een niew Cijfferboeck/ (1580). " Duyfentich duyfent 
 1000000." Fol. Aidj recto. 
 
THE ESSENTIAL FEATURES 39. 
 
 back to the thirteenth century; x but the sixteenth century 
 records the struggle of the word for its place in numeration. 
 Borgi 2 (1484) has it in "Million de million de million," 
 Chuquet ( 1484) used it in reading numbers on the six-figure 
 basis. 3 Paciuolo * (1494) used " mili one." Cirvelo 5 (1495) 
 used million for 1,000,000,000,000. La Roche (1520), like 
 Chuquet, used it on the six-figure plan. After 1 540 the word 
 appeared in many standard works. 6 The present names for 
 higher periods, though much slower to come into use, were 
 known to fifteenth century scholars. Chuquet (1484) gave 
 the remarkable list : " byllion, tryllion, quadrillion, quyllion, 
 sixlion, septyllion, ottyllion, nonyllion," using them on the 
 six-figure basis. La Roche (1520) gave billion and trillion. 
 
 Tonstall, De Arte Supputandi (1522). Tonstall reads in Latin the num- 
 ber 3210987654321 as Ter millies millena millia millies, ducenties decies 
 millies anillena millia, naningienties octvagies septies millena millia, sexcenta 
 quinquinta quattuor millia, trecenta viginti unum. Fol. C„ verso. 
 
 1 The word "million " is first found in Marco Polo (1254-1324). 
 
 2 The 1540 edition of Borgi's work gives the following names of high 
 periods : " Miar de million, million de million, miar de millio de million, 
 million de million de million." Foil. 5 verso. 
 
 The word) is also used in the Treviso arithmetic (1478). 
 
 3 By the six-figure basis is meant the use of million's to cover six orders 
 beyond hundred thousands, billions to cover the next six orders, and so on. 
 
 Thus, 18,432,750,198,246,115 would be read eighteen thousand four hundred 
 thirty-two billion, seven hundred fifty thousand one hundred ninety-eight 
 million, two hundred forty-six thousand one hundred fifteen; instead of 
 eighteen quadrillion, four hundred thirty-two trillion, seven hundred fifty 
 billion, and so on, on the three-figure plan. The six-figure grouping of 
 Chuquet and La Roche entered Germany in 1681, according to Unger (Die 
 
 Methodik . p. 71) and came into general use there in the eighteenth 
 
 century. France early adopted the three-figure system. England used the 
 old terminology at the opening of the sixteenth century, for Tonstall (1522) 
 says that millena millia (thousand thousand) is .commonly called "mil- 
 lion " by foreigners. But before the middle of the century we find Recorde 
 using million. 
 
 * Paciuolo, Sfima (1494 ed.), fol. 9 verso. 
 
 5 Cirvelo, Tractatus Ardthmetice practice (1505), uses the same notation. 
 
 6 Records, The Ground of Artes (1540) ; Gemma Fnisius (1552, Antwerp. 
 «d.) ; Cataldi (1602 ed.). 
 
4 SIXTEENTH CENTURY ARITHMETIC 
 
 Van der Schuere (1600) gave a very large number field. He 
 used millioen for million, duyset mill, (thousand million) for 
 billion, bimillioen for trillion, duyset bimill. for quadrillion, 
 and so on up to duyset quadrimill. for octillion. Trenchant 
 (1578) gave millions for million, miliar for billion, and milier 
 de miliars for trillion. 
 
 The following devices were used for separating the periods : 
 
 678935784105296 * 5678900000000000000 2 3210987654321 s 
 
 bacbacba 4 .. :: 
 
 44559886 3l554l56o 5 23456007846000305321 « 
 
 1 . 234 . 567 . 890 t 36236365463643656765656568 » 
 
 Recorde (1540) called the numbers in each period ternaries 
 and the periods denominations to assist in reading. Thus, as 
 in 222 pounds, pounds is the denomination, so in every period 
 (620,000) the last place (thousand) is the denomination. 
 
 1 Leonardo of Pisa, Liber Abaci (1202, or 1228), p. 1. 
 
 2 Borgi, Qui comeza la nobel opera (1540 ed.), fol. Av verso; Paciuolo, 
 Suma (1523 ed.), fol. 19 verso, ciii verso. 
 
 3 Tonstalll, De Arte Sivpputandd (1522), fol. C 2 verso; Kobel, Zwey 
 Rechenbuchlin (1537 ed.), fol. B 4 verso; Rudolff, Kunstliche rechnung 
 (1534 ed.), Aiij recto; Riese, Rechnung auff der Linien und Federn/ 
 (1571 ed.), fol. Aij verso; Baker, The Well Spring of Sciences (1580 ed.), 
 fol. Biiii recto. 
 
 4 Kobel, Zwey Rechenlbuchlin (1537 ed.), fol. B recto. 
 
 5 Gemma Frisius, Arithmeticae Practicae Methodus (1581 ed.), fol. Aiv. 
 
 6 Tartaglia, La Prima Parte (1556). 
 
 7 Ramus, Arithmeticae Libri Duo (1577 ed.), fol. Aii verso. 
 
 8 Unicorn, De L'Arithmetfca vniversale (1598), reads this number 
 thus: 36. millioni quatro volte, &■ ducento trenta sei millia, &■ trecen- 
 tosessanta cinque millioni tre volte, et quatro cento e sessanta tre millia 
 & sei cento e quaranta tre millioni due volte, &■ sei cento cinquanta sei 
 millia e settecento sessanta cinque millioni vna volta, & sei cento e cinquata 
 sei millia, e cinquecento sessanta otto. Fol. A^ recto. 
 
THE ESSENTIAL FEATURES 4I 
 
 The number in the period he called the numerator. Thus, in 
 .203,000,000, 203 is the numerator. 1 
 
 Addition 
 
 The treatment of addition presents much diversity, but the 
 general characteristics are the absence of tables of sums, full 
 explanations of column-adding, and tests of the work. 
 
 It would seem that the sums corresponding to the modern 
 addition' table would necessarily have received first attention 
 at a time when the Hindu numerals were so unfamiliar. But 
 the writers who used these were the exception. 2 
 
 The explanation of the processes of column-adding usually 
 
 1 Recorde, The Ground of Artes (1540). 
 
 Scholar. What call you Denominations ? 
 
 Matter. It is the lafte value or name added to any fumme. As when 
 I fay : GCxxAi. poundes : poundes is the Denomination. And likewif e in 
 faying: 25 men, men is the Denomination, and fo of other. But in this 
 place (that I fpake of before) the laft number of euery Ternarie, is the 
 Denomination of it. As of the firft Ternarie, the Denomination is Unites, 
 and of the feconde Ternarie, the Denomination is thoufandes: and of 
 the third Ternaries, thoufande thoufandes, or Millions: of the iiii, thou- 
 fande thoufande, thoufandes, or thoufande Millions : and fo foorth. 
 
 Scholer. And what fhall I call the value of the three figures that may 
 be pronounced before the Denominators : as in faying 203000000, that is 
 CCiii. millions. I perceyue by your wordes, that millions is the denomina- 
 tion: but what fhal I call the CCiii. joyned before the millions. 
 
 Mafter. That is called the Numerator ot valuer, and the whole fumme 
 that refulteth of them both, is called the fumme, value or number. Fol. 
 Dii recto (1594 ed.). 
 
 2 Tartaglia, La Prima Parte Del General Trartato (1556). Tartaglia 
 gives the addition tables as follows : 
 
 o. e 0. fa o (0 + = 0) 
 o. e 1. fa 1 (0 + 1 = 1) 
 o. e 2. fa 2 (0 + 2 = 2) 
 
 e 10. fa 10 
 
 1. e 1. 
 
 fa 2 
 
 
 1. e 10. 
 
 fa 11 
 
 and so on with all the 
 tables to 
 10. e 10. fa 20 
 
 2. e 2. 
 
 fa 4 
 
 2. e 10. 
 
 fa 12 
 
 Fol. B i verso. 
 
 That these tables were regarded as fundamental to further progress is 
 •shown by the follolwing remark from the siame folio : " Imparate adnmque 
 li soprascritti sumari necessarij di saper a menti." (Therefore to learn to 
 add, it is necessary to commit the tables written above to memory.) 
 
42 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 received first attention. Two abstract numbers were proposed 
 for addition, such that the sum of one column of figures, at 
 least, would equal or exceed ten. The first example in Een 
 mew Cijfrerboeck, by Willem Raets (1580) is: 
 
 354 
 898 
 
 1252 
 
 Cardan's (1539) first example is: 
 
 73942 
 
 4068 
 
 273 
 
 52759 
 
 131042 
 
 Noviomagus (1539) first shows the arrangement by columns, 
 as in (1) ; then adds without carrying, as in (2) ; then adds 
 with carrying, as in (3) : 
 
 (1) (2) (3) 
 
 321, not 321 321 2354 
 
 124 124 421 * 620 
 
 530 530 530 76 
 
 975 3050 
 
 Tonstall (1522) gives: 
 
 (I) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 4 
 
 309 
 
 59 
 
 389 
 
 3 
 
 204 
 
 34 
 
 204 
 
 7 513 93 93 
 
 He points out in example (2) that the ten- in the sum of the 
 first column falls under the second column, because the second 
 column is composed of zeros. Examples (3) and (4) are ex- 
 plained in order, although it would seem that (4) should pre- 
 cede (3) . The student is advised by Tonstall to learn the ele- 
 
 1 Typographical error in the original for 124. 
 
THE ESSENTIAL FEATURES 43 
 
 mentary sums, and is encouraged by the remark that it will 
 
 require only an hour. 
 
 It was usual in addition to arrange the addends in order of 
 
 size, placing the largest at the top. It is easy to state a reason 
 
 for this, although none is given 1 ; for by this plan the columns 
 
 were more easily preserved — a real difficulty for beginners, 
 
 especially for those to whom the Hindu system was unfamiliar. 
 
 It also prepared for subtraction, but this was probably not 
 
 the reason for using it, as is shown by 
 
 by the following examples : * The 3456 4602 56789 
 
 first is the only one that corresponds 9 ° 54 34 ° s 234S 
 
 . . . . . 2300 12345 
 
 to an example in subtraction, since it 6 7 g 9 
 
 has only two addends, but here the 10307 3239 
 
 smaller is written at the top. 4 
 
 1327 
 234 In the other examples, which do not corres- g6 
 
 345 pond to those of subtraction, the numbers 349 
 
 Zg are in the order of their size from the top 228 
 
 763 down. The illustration at the left shows 
 
 832 bow arrangement according to size was 3 s 
 
 3450 occasionally disregarded. 2 
 
 100 J ^> 108^2 5 
 
 63 The sums of the several columns in a J 
 
 13 problem' of addition were commonly added 
 
 as at present, by writing the first right-hand figure of 
 
 1000 ' *^ e sum °f an y column and adding the rest to the next 
 
 i column. When the columns are long, how- 
 
 T 8 ^ ever, this is not the easiest way, and occa- 3 ;r 3 
 
 sionally a writer of that period wrote the 97 gg 
 
 partial sums and added them to obtain the result. 3 
 
 A slight modification of this was the placing of the I 4 2 9° 
 
 211 
 
 numbers to be added to the next column below that 
 
 column to be added to its sum.* r 6« 
 
 A feature that is written large in arithmetic of the six- 
 
 1 Rudolff, Kunstliche redlining (1534 ed.), fol. Aidij recto. 
 
 2 Taglienite, Libro Dabaco (1541 ed.), fol. Ci recto. 
 
 3 Gemma Frisiius, Arithtneticae Practrcae Methodus facilis (1575 ed.), fol. 
 Aviii verso. 
 
 * Buteo, Logistica (1559), fol. a 7 verso. 5 Error in the original. 
 
44 SIXTEENTH CENTURY ARITHMETIC 
 
 teenth century is the matter of the so-called proofs of opera- 
 tions. These were generally not proofs, but tests more or less 
 reliable. The most common form' was that of 
 casting out nines. Thus, in the annexed case 
 
 of addition, the remainder arising from divid- 
 
 ing 354 by 9 is 3, and from dividing 898 by 1252 
 9 is 7. The excess of nines in 7 + 3 is 1 ; 
 this is the 1 above the line in \. The excess of nines in 1252 
 is 1 ; this is the 1 below the line. \ shows that the excesses 
 agree and that the work 'checks, or proves, as it was called. 
 It was customary to give a long explanation of the proof, and 
 a few writers gave a table of remainders arising from divid- 
 ing numbers from o to 90 by 9, and showed how to find the 
 remainders for large numbers. 2 
 
 That tests were given exaggerated importance is shown by 
 the fact that several writers extended them to the case of add- 
 ing denominate numbers. The excess of nines was found for 
 the highest denomination; this was expressed in terms of the 
 next lower denomination and combined with it. Then the 
 process of finding the excess was repeated. Each addend and 
 the result were similarly treated, the work of testing becoming 
 more difficult and complicated than the solution of the problem. 
 
 The test by casting out sevens was also common, but being 
 
 1 Raets, Een mew Cijfferboeck: (1580), fol. Aiiij recto. 
 
 2 Tartaglia, La Prima Parte (1556), fol. Bij verso and fol. Biij recto. 
 
 Li termini della proua del. 9. 
 
 De o. la proua e o De 10. la proua e 1 
 
 De 9. la proua e — — o De 11. la proua e 2 
 
 De 18. la proua e o De 12. la proua e 3 
 
 De 90. la proua e o De 19. la proua e 1 
 
 De o. la proua e o De 21. la proua e 3 
 
 De 1. da proua e 1 De 22. la proua e 4 
 
 De 2. la proua e 2 
 
 De 30. la proua e 
 
 De 9. la proua e o, and so on to : 
 
 De 81. la proua e 
 
 De 90. la proua e o. 
 
THE ESSENTIAL FEATURES 45 
 
 more difficult, it was placed second. Several authors mention 
 the fact that it is more accurate than that by casting out nines. 1 
 The proof by elevens was sometimes used. In addition, an 
 author occasionally used the method of adding the columns of 
 figures both upward and downward. 2 A few used subtrac- 
 tion, 3 in the case of two addends taking one addend from the 
 sum to see if the result is the other. The use of subtraction' 
 in the case of more than two addends was rare.* The reason 
 
 1 Unicorn, De LArMiimetica vniuersale (1598), " Ohe la proua del 7. sia 
 men fall&oe, che la proua del g," fol. B 2 verso. 
 
 2 Plaoiuolo, Suma (1523 ed.), fol. 20 recto, or Ciiij recto. 
 
 3 Tartaglia, La Prima Parte (1556), gives this example: 8756 
 
 a 678 b 
 
 9434 
 and says. : " Perdhe inuero il sommare e proprio in atto contrario al sot- 
 tare, & similmente il sottrare e in atto contrario al sommare,' 7 fol. Bij 
 recto (because, indeed, addition is properly the inverse of subtraction, and 
 similarly subtraction is properly the inverse of addition). 
 
 4 Ghampenois, Les Institutions De L'Ari'thmetique (1578). 
 
 4325 Preuue. 
 132 4878 Tome de l'Addition. 
 421 4325 premier* fomme. 
 
 Addition 4878 0553 premier refte. 
 
 132 feconde fomme. Page 15, or fol. Bviii recto. 
 
 421 fecond refte. 
 
 421 troifiefme fomme. 
 
 000 
 
 Also Simon Jacob, Rechenbuch auf den Linden undimiitZiffern/ (1599 ed.). 
 597-a 
 786.b 
 978.C 
 
 Summa 2361 
 597-a 
 
 1764 Fol. Cv recto. 
 
 786.b 
 
 978 
 978.C 
 
 000 
 
46 SIXTEENTH CENTURY ARITHMETIC 
 
 for the prominence of these tests is undoubtedly due to the use 
 of the various forms of the abacus. When the beads were 
 once shifted, or the counters displaced, or the symbols in the 
 sand effaced, there was no record to retrace, no possibility of 
 reviewing the work. 1 It was, therefore, very advantageous to 
 have means of testing the result by some comparison with 
 the original numbers. These means were supplied by the 
 proofs of nines and sevens. It was natural, then, that these 
 tests should appear with due emphasis in most of the first 
 printed books. 
 
 Besides the general characteristics of sixteenth century ad- 
 dition there were a few special features that have educational 
 significance; namely, the order of adding, certain short meth- 
 jods, and the use of concrete problems to introduce the process. 
 
 We have noted that Paciuolo added upward and then added 
 downward as a test of the work, but only one writer among 
 those examined confined his addition to the downward pro- 
 cess. 2 Thus, whatever virtue there may be in precedent is in 
 favor of adding upward instead of downward. 
 
 1 For some time after arithtnetioians formed the habit of writing num- 
 bers in the Hindu notation, they used line-reckoning to perforin the pro- 
 cesses, and to watch their progress they crossed the figures as they were 
 used. The influence of this is seen in the following example from Kobel 
 (Zwey Rechenbuchlin (1537 ed.), fol. m verso, 112 recto), in which 
 he crossed the figures of the addends, although using the Hindu algorism : 
 
 Zum erften wil ich zufammen thun 103 zu 966. Ich 
 
 fprich/ 6. vnd 3. ist 9. vnd fetz die 9 vnder das ftrichlin vff die 10? 
 erft ftat/ vnd durchftrekh die 6. vnd 3. fo fteht es alio. 6 ''" 
 
 ftetz ich vnd den ftrich/ vff die zweyte ftat neben die 9. zu der 9 
 
 lincke hand vnder die 6. vnnd durcfastreiehs o. vnd 6. 
 
 m 
 m 
 
 69 
 
 2 Trenchant, L'Arithmetique (1578 ed.). 
 
 "Je veux aiouter ces nombres, 581, 192, & 264. Commencant a 
 
 main droite, i'aioute toutes les figures du denier reng enfemble, 
 diftant, 1 & 2 font 3, & 4 font 7. ie pofe 7 fous celuy reng an 581 
 
 deffous du tret, & vien femblablement cueillir e precedent reng, 192 
 
 diftant 8 & 9 font 17, & 6 font 23, ie pofe le digite 3 fous ca 264 
 
 reng, & retien le nombre des dizeines qui eft 2, que i'adioute auec 1 1037 
 1'autre reng, diftant, 2 que te tien & 5 font 7, & 1 font 8, & 2 
 font 10, ie pofe o & retien 1, que ie pofe deuant o, & e'eft fet. Ainfi ces 
 troys nombres aioutez montent 1037. Fol. Biv verso. 
 
THE ESSENTIAL FEATURES 47 
 
 Another feature of sixteenth century calculation that one 
 would not expect so early in the history of European figure- 
 reckoning was the use of short methods. They were not com- 
 monly used, but there is a fair sprinkling of them through the 
 various operations. A few writers showed how equal num- 
 bers are combined while adding a column. 1 A few cases occur 
 in which the associative law is used to break up a long prob- 
 lem into shorter ones. 2 
 
 1 Recorde, The Ground of Artes (1594 ed.) : "I would) 
 adde thefe xiii fumes into one, which I let after this manner: 4599 
 then doe I begin and gather the fumme of the firft rowe of fig- 2299 
 ures which comimeth to 107, for I take 9 there x. times and that 3699 
 is 90, then 9 and 8 is 17, that is in all 107, of which fumme I 2399 
 write the 7 under the firft rowe of figures, and then for that 4090 
 100 is x. tens, I keepe x. in mind: which ten I muft adde vnto 1099 
 the nexte rowe of figures when they are added together with the 3198 
 x. that I had in my minde, make in all 125, of which fumme, I 299 
 write the digit 5 vnder the fecond rowe, t . Then for that 120 699 
 conteineth xii tens. Fol. Cii recto. 899 
 
 Cirvelo, Traotatus Arithmetice practice (1513 ed.). 499 
 
 " Et nota qj ad iftam fpeciem reducitur alia fpecies minus 389 
 
 principalis que dioitur iduplatio aut triplatio nam fi eundem 
 
 numerum bis fcripferis et addideris in vnam fummam habebis 29057 
 duplum illius : vnde pro re tam facili no oportehat dare fpeciale 
 capitulum. Exemplum." Fol. aiiij verso. 
 
 496 2 496 3 
 
 496 2 Exemplu 496 
 
 496 3 
 
 992 duplum 
 
 1488 triplu 
 An application of doubling and tripling to addition. 
 
 2 E. g., Clavius, Arithmetica Prattica (1626 ed.), fol. a 4 recto. 
 
 6008 
 5009 
 
 4009 
 
 308 108 3009 
 239 309 209 
 108 4128 308 
 
 655 4545 3526 
 
 The top row furnishes partial sums 
 of the column at the right. 
 
 6008 
 
 5009 
 
 4009 
 
 308 
 
 239 
 
 108 
 
 108 
 
 309 
 
 4128 
 
 3009 
 
 209 
 
 308 
 
 15026 
 15026 
 
 655 
 
 4545 
 3526 
 
 
 23752 
 
48 SIXTEENTH CENTURY ARITHMETIC 
 
 The plan of proposing a concrete problem in addition as a 
 motive for explaining the process occurs in several works. 
 The following will serve to illustrate : 
 
 " As if there were due to any man 223 pounds by some one 
 body, and 334 pounds by another, and 431 by another, and 
 you would know how many pounds is due to the same man 
 in all." 1 
 
 " A merchant has three purses in which there is a certain 
 number of ecus. There are known; to be 3,231 ecus in the 
 first, 2,312 in the second, and 1,213 m ^ e third. The mer- 
 chant put the contents of these purses into one. It is required 
 to know how many ecus there are in this purse." 2 
 
 " For example, if it is asked how long ago Homer lived, 
 and Gellius replies: 160 years 'before the founding of Rome, 
 which was founded 752 years before the birth of Christ. 
 Christ was born, however, 1,567 years ago. These three num- 
 bers are .added. The sum showing that Homer nourished 
 
 2,479 years ago will be as follows " : 3 160 
 
 752 
 1567 
 
 2479 
 
 It is somewhat remarkable that the idea of introducing a 
 process through concrete examples should have taken root in 
 so many countries within a period of fifty years at a time 
 when communication of ideas was so slow. Kobel in Ger- 
 
 1 Baker, The Well Spiring of Sciences (1580 ed.), foil. Bvi recto. 
 
 2 Ghamnpenois, Institutions De L'Arithimetique (1578). 
 
 "Vm nrarobant a trois bourfes ou il y a certaines fommes d'efcus,. 
 fcauoir en la premiere 3231 efcus, en la feconde 2312 efcus, & en la troi- 
 fiefme 1213 efcus. 
 
 Ce marchat vuide fes trois bourses en vne. Lon demande combien. il y 
 a d'efcus en cefte bourfe." Fol. Biiij recto, or page 7. 
 
 3 Ramus, Arithmeticae libri Duo (1577 edl). 
 
 " Ut fi quaeratur quampridem vixerit Horaerus, & respondeatur e Gellio,, 
 160 annis ante conditam Roman, quae condka fit ante natum Chriftum. 
 annis 752. Ohriftum vero natum anno abhinc 1567. addantur hi tres 
 numeri : Samma anductionis indicans Homerum annos abhinc 2479 floruif f e,. 
 erit hoc modo." Fol. Aiiij recto. 
 
THE ESSENTIAL FEATURES 4 q 
 
 many (1531), Recorde (1540) and Baker (1562) in Eng- 
 land, Ramus (1567), Trenchant (1571) and Champenois 
 (1578) in France were the pioneers in their respective coun- 
 tries. Many who began with abstract numbers introduced de- 
 nominate numbers after the first two or three problems. 
 
 Subtraction 1 
 
 Another evidence that this was the formative period in ele- 
 mentary arithmetic is seen in the treatment of the subtraction 
 of integers, for these writers were in possession of all the 
 methods of subtraction that are taught or discussed at the 
 present time. 
 
 There was little variation in their treatment where the fig- 
 ures of the minuend had greater value than the corresponding 
 ones of the subtrahend. In fact, all the writers included in 
 this investigation, with one exception, subtracted from right to 
 left the figures of the subtrahend from the figures of the min- 
 uend written above, and placed the differences below the cor- 
 responding columns. 
 
 1 A knowledge of the elementary differences required for this was pre- 
 supposed, although the tables were given in the more elaborate works only. 
 Ramus (1586 ed.) recommended learning the "alphabetum" both for addi- 
 tion and subtraction. By alphabetum he meant all the possible sums and 
 
 differences of the digits, I 9. " Subductionis mediatio in primis novem 
 
 notis eadem hie effe debet, quae fuit in additione. 
 
 " Tolle 3 de 7 manent 4, tolle 4 de 9 manent 5, & fimiliter totu alpha- 
 betum 1, 2, 3, 4, 5, 6, 7, 8, 9. Omni genere verfandiim est. Hie Pytha- 
 goreus fubductionis abacus eft." Fol. a 4 verso. 
 
 Tonstall (1522), after explaining how to subtract numbers of several 
 
 figures, states in words the differences from 1 9 and recommends that 
 
 they be learned. 
 
 " Quod f i quis ignorat : unius horae labor ; 
 Modo intentus fit animus ; if suppeditabit." Fol. F recto. 
 
5° 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 Ramus began at the left and proceeded to the right. 1 
 This is his first example, where the influence of line- 
 reckoning is again seen in the crossing of the figures. 
 Ramus was not the first to subtract from left to right, for 
 
 The following is Tonstall's table : 
 
 111 
 
 m 
 
 
 IB 
 
 18 
 
 1? 
 
 16 
 
 3.5 
 
 14 
 
 13 
 
 .JUL. 
 
 XI 
 
 19 
 
 , 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 IS 
 
 is 
 
 12 i 
 
 3 
 
 ia 
 
 _& 
 
 _a 
 
 _a 
 
 _£ 
 
 _a 
 
 _a 
 
 9 
 
 
 9 
 
 o 
 
 a 
 
 7 
 
 ft 
 
 s 
 
 4 
 
 _a__ 
 
 . ■■• 
 
 
 
 
 17 
 
 13 
 
 15 
 
 14 
 
 13 
 
 la 
 
 ll 
 
 
 8 
 
 _a 
 
 _a 
 
 _a 
 
 _a 
 
 _& 
 
 _a 
 
 _§ 
 
 
 
 , 
 
 p, 
 
 v 
 
 « 
 
 n 
 
 4, 
 
 •3 
 
 
 
 16 
 
 IV, 
 
 14 
 
 IS 
 
 is 
 
 11 
 
 
 7 
 
 
 7 
 
 7 
 
 _2. 
 
 JL 
 
 _z 
 
 -2 
 
 
 
 
 ft 
 
 S 
 
 i 
 
 « . 
 
 H, 
 
 * 
 
 
 
 IS 
 
 14 
 
 13 
 
 12 
 
 11 
 
 
 
 6 
 
 
 _fi 
 
 _& 
 
 _& 
 
 _B- 
 
 _S 
 
 
 
 
 
 _£L_ 
 
 « 
 
 7 
 
 « 
 
 F 
 
 
 
 14 
 
 13 
 
 12 
 
 11 
 
 
 
 6 
 
 
 
 
 
 _6 
 
 
 -a 
 
 
 
 13 
 
 12 
 
 11 
 
 
 
 
 4 
 
 
 
 
 -4 
 
 7 
 
 
 
 12 
 
 11 
 
 
 
 
 
 
 
 
 _a 
 
 _s 
 
 
 
 
 
 
 3 
 
 
 o 
 
 R 
 
 
 
 11 
 
 
 
 
 
 
 
 e 
 
 
 _s 
 
 
 
 
 
 
 
 
 
 
 9 
 
 Folio F, recto. 
 The following is Taxtaglia's (1556 ed.) table: 
 
 De 0. a cauarne o. resta o De 2. a cauarne 2. resta — 
 
 De 1. a cauarne o. resta 1 De 3. a cauarne 2. resta — 
 
 De 10. a cauarne o. resta • 
 
 10 De 10. a cauarne 2. resta - 
 
 De 1. a cauarne 1. resta ■ 
 De 2. a cauarne 1. resta ■ 
 
 De 3. a cauarne 3. resta ■ 
 De 4. a cauarne 3. resta ■ 
 
 De 10. a cauarne 1. resta ■ 
 and so on to the table 10 - 
 
 — 9 De 10. a cauarne 3. resta 7, 
 
 ■10 = 0, which contains this one fact only. 
 
 Fol. 
 
 C x recto. 
 
 1 Ramus, Arithmeticae Li'bri Duo (1586 ed'.). "Si dati lint plurium nota- 
 rum fubducendo infra alteram pofito, fubductio fit a finiftra dextrorfum, 
 reliquoqs ; fupernotato delentur dati, ut f i die futnma aeris lillius alieni 345 
 fubduceda fit 234 : Dispofitis ordine numeris hoc modo : 345 fubducendo 
 infra, fupra autem a quo fubdiuctio facienda, incipiam a 234 
 finiftra dextrorfum, contra qua in additione, tollo 2 de 3 manet 
 fupernoto 1 deletis 3 & 2. Deinde fubducam 3 de 4 manet 1, & 
 fupernoto 1 deletis 46-3. Deniqj fubductis 4 de 5 manet 1, & 
 fupernotabo 1 deletis 5 & 4. Vnde inveniam reliquu effe ill. cum 
 fubduxero 234 a 345. Inductio tota fie erit. Fol. A 4 verso. 
 
 1, & 
 111 
 
 w 
 m 
 
THE ESSENTIAL FEATURES 5I 
 
 the same thing was done in the Lilivati, and possibly in older 
 works. 1 Many calculators to-day recommend working from 
 left to right in both addition and subtraction, usually confining 
 the addition to two addends. But Ramus added from right to 
 left in all cases and subtracted from: left to right. He was 
 unique in his century, also, in placing the difference above the 
 minuend, as shown in the example. 
 
 The case of subtraction in which the subtrahend figure ex- 
 ceeds in value the minuend figure, received a diversity of treat- 
 ment. The three distinct methods usually taught were : ( i ) 
 Ten is added to the minuend figure before the subtrahend 
 figure is subtracted ; one is then added to the next subtrahend 
 figure. (2) The arithmetic complement of the subtrahend 
 figure is added to the minuend figure, and one is added to the 
 next subtrahend figure. (3) Ten is added to' the minuend 
 figure, and the subtrahend figure is subtracted from this sum ; 
 one is then subtracted from the next minuend figure. The 
 last is the form of solution most prevalent to-day. Lists of 
 authors who used these respective methods are given below ; 2 
 from these lists it will be noticed that the first and second 
 methods were equally popular, while the third method was 
 used very little. Ramus, subtracting from left to right, used 
 the third method, for which he gave the following example 
 
 1 See H. Siiter, Bibliotheca mathematica, VII 3 15. Gerhardt, " Etudes," 
 page 5- 
 
 2 Those who used the first kind were: Piero Borgi (1484), Tonstall 
 (1522), Paciuolo (1494), Rudolff (1526), Cardan (iS39), Noviomagus 
 (iS39), Tartaglia (1556), Van der Scheure (1600). Ton- 
 stall also gave this process in the following form : 2 9 10 10 
 
 Those who used the second ford were: Widman (1489), ** f- $ 
 Tonstall (1522), Paciuolo (1494), Tartaglia (1556), Gemma 1111 
 Frisius (1540), Riese (1522), Trenchant (157O, Baker 18 9 9 
 (1562), Unicorn (1598), Huswirt (1501), and Finaeus 
 (1525). This .method goes back to the Hindu arithmeticians. Fink, 
 Geschichte der Eleroentar-Math. (Beman and Smith's translation, Chicago, 
 1900), p. 28. 
 
 Those who used the third kind were: Paciuolo (l494)» Kobel (1531)1 
 Tartaglia (1556), Champenois (1578), Buteo (iSS9), and Raets (1580). 
 
5 2 SIXTEENTH CENTURY ARITHMETIC 
 
 and explanation : " When I take 3 from 4, I shall not 87 
 write 1, because the following subtrahend figure, 4, is W% 
 greater than the 3 placed above, but I shall keep this in ^ 
 mind and take the next figure below, which is 4, from 13. 
 This leaves 9, which for the same reason I shall not write 
 down, but shall take 1 from it and write 8 above, and keep 1 
 in mind, because the following figure to be subtracted is 
 greater; then 5 from 12 leaves 7, which will be written 
 above." 1 
 
 The proofs in subtraction, as in the case of all operations, 
 were very prominent. The three standard method's were cast- 
 ing out nines, casting out sevens, and adding the subtrahend 
 /and difference. 2 About three times as many writers used the 
 , additive method as used either of the others, which was nat- 
 ural on account of its ease and effectiveness. 3 Tartaglia, who 
 gave each of the proofs above, also subtracted the remainder 
 
 1 Ramus, Arithlmeticae Libra Duo (1586 ©d.). "Vt fi fubducenda Tint 
 345 de 432, aim fubdiicani 3 de 4 nOn fttpemofabo 1, quia 4 fequens 
 fubducenida note major est fuperapofita 3, fed il'lud mente refervabo, 4 
 fubductis a 13 rnaneret 9, quae nequaquam propter eandem caufam notabo, 
 fed uno minus 8 tentum fupernotabo, & unttm mente refervabo, quia 
 fequens fubdueeda nota major est. Jfcaqj 5 fubductis a 12 reliqua 7 fuper- 
 notabo. Vnde inveniam fubductis 34s de 432 relinqui 87. Tota inductio 
 fie erit : 87 
 
 m 
 
 w 
 
 " Trium fociorum pecunia in unum acervum congefta fit 432 : primiqj 
 fumma fit incerta, eonftet tamen focios capere 345 : ergo fuam partem is 
 per hac fubdlictionem cognofcet." Pol. A recto. 
 
 2 Piero Borgi, Aritihrnetica (1540 ed.), fol. C 6 verso. Example. Proof. 
 
 456 333 
 
 123 123 
 
 333 456 
 
 3 The proof by casting out nines was used by Rudolff ^1526), Widman 
 (1489), Cardan (1539), Tartaglia (1556), Gemma Frisius (1540), Suevus 
 (1593)- 
 
 The proof by casting out sevens was used by Widman (1489), Cardan 
 (IS39), Tartaglia (1556), Unicorn (1598). 
 
 The addition proof was used by Borgi (1484), Widman (1489), Cardan 
 (IS39), Tartaglia (1556), Riese (1522), Champenois (1578), Raets (1580), 
 Unicorn (1598), Jacob (1599), Van der Scheure (1600). 
 
THE ESSENTIAL FEATURES 
 
 53 
 
 from the minuend to find the subtrahend, as in addition he 
 subtracted the sum of all the addends but one from the result 
 to find the other addend. 
 
 The terminology and symbolism used have several points of 
 interest. In those works which used the plan of supplying 10 
 from the next order of the minuend to make subtraction pos- 
 sible, one naturally seeks to find 
 a trace of the modern vulgarism 
 *' to borrow," and recognizes it in 
 the word " entlehen " used by 
 Kobel. 1 This is suggestive, be- 
 cause Kobel was primarily an aba- 
 cist, and he would probably em- 
 ploy the same word in the algor- 
 ism that was employed to describe 
 the actual borrowing process in 
 abacus reckoning. In the tables 
 and examples of subtraction there 
 is suggested the word " rest " in 
 the sense of difference, or re- 
 mainder. 2 The book material of 
 addition and subtraction presents 
 
 4 1> |cn«t>ert}e|;glc;« 
 
 3 + Jo tfom/Sttfumier 
 
 4 1 p bt'egmetiKronb 
 
 3 + 44 Itonnft reaeaug 
 3 + » — i(J/6«oi(Jmi< 
 
 8ottner 3 1 ' K> nu6bjfeg6c(on» 
 
 3 + ?o fcerennb »er&«i 
 
 4 1 6 4TJ9& (.&« 
 
 3+44 6u 6ie jenbtncr 
 
 3 + 1 9 5ft It gemacfcetc 
 
 3 — •— i a £«fl»nnbt><i8 / 
 
 3+9 <+ bueijimtn 
 
 bar,a 2 66i'crc({ >r, > f minus, trim 
 
 f»U buftfr §»!g ai>fd)!.il;i:it allw«g fdc 
 
 0inlegdi4li. Vnosasid i 3 m«ti4. 
 
 mbmaditj 1 1 ttUarjusobicrftao — 
 
 6a«i(l>Ta ) »n6mcr6oijs> 0?cfu6» 
 
 tracer o»n49} p.Vnb BMbcn 41 fx 
 
 tt.tTiinfpnci) 1 00 H'»a6i(?eiri3ciiriKt 
 
 ps»4 ff i twte fnmm4 1 f k tt enft f umj 
 
 ' * * ff T i4H«*X)»i{lt«t)can!«iJjC 
 
 WW* 
 
 1 Kobel, Zwey Rechenbuchilin (1537 ed.), fol. P 4 recto et seq. 
 
 For discussion see Brooks, Philosophy of Arithmetic (1901 ed.), pp. 45, 
 46 ; 219, 220. Unger, Die Methoidik, pp. 73, 74. 
 
 2 Tartaglia, La Prima Parte (1556). " Sottrare nor* e altro che duoi 
 proposti Humeri, inequali saper trouare la loro differentia, cioe quanto che 
 il maggiore eccede il menore, come saria a sottrare. .4. de .9. restaria .5." 
 Fol. Bvi verso. 
 
 79374 
 5024 
 
 743SO II numero restante. 
 
 Unicom, De L'Aritihmetioa vndversale (1598), calls the remainder numero 
 ■restante, or resta dare, or residue. 
 
 Trenchant, L'Arithmetique (1578 ed.), calls the remainder restc. Fol. 
 B g recto. 
 
 Baker, The Well Spring of Sciences (1580 ed.), does not use the terms 
 minuend and subtrahend, but says, in taking 6 and 9, "there resteth 3." 
 Fol. Ciii verso. 
 
54 SIXTEENTH CENTURY ARITHMETIC 
 
 a much different appearance from that of the modern treat- 
 ment, because of the lack of symbols of operation. Although 
 ' the symbols + and — were in existence in the fifteenth cen- 
 tury, 1 and appeared for the first time in print in Widman 2 
 ( 1489) , as shown in the illustration (p. 53) , they do not appear 
 ij in the arithmetics as signs of operation until the latter part of 
 * the sixteenth century. In fact, they did not pass from algebra 
 to general use in arithmetic until the nineteenth century. They 
 were used in the sixteenth century to express excess and deficit 
 in weight, 3 as shown in note 3, where the first column is sent- 
 ners and the second pounds. These early printed examples 
 substantiate the theory that the symbols +, — , originated 
 
 1 Miiller, Historisch-Etymo'logische Studien iiber mathematische Termdn- 
 ologie. 
 
 2 Widman, Behend und hiipsch Rechnung uff alien Kauffmanschafften 
 (1508 ed.), fol. h 5 recto. 
 
 3 Wencelaus, T'Fondament Van Arithmetica (1599 ed.). 
 
 " Pijpen Olie van Oliven/ weghende alsoo hier naer volcht/ Tara op 
 elcke pijpe/ 140. tb.i. lauther 100. cost 50. ft. 6. <d. Hoe veel beloopter in 
 ghelde?" 
 
 
 Weecht. 
 
 
 
 
 cent. lb. 
 
 cent. lb. 
 
 
 
 cent. lb. 
 
 No. 1. 9 + 38. 
 
 No. 4. 9 + SO. 
 
 No. 
 
 7- 
 
 10 — 25. 
 
 No. 2. 9 -\- 44. 
 
 No. 5- 9 + 55- 
 
 No. 
 
 8. 
 
 9 + 68. 
 
 No. 3. 10 — 20. 
 
 No. 6. 9 + 56. 
 
 No. 
 
 9- 
 
 9 + 70. 
 
 " Dit + beteyckent Plus/ ende dit — Minus." 
 
 " Item 9. pipes, d'huyle d'OIives pesants comrae s'ensuit tara pour chas- 
 cune pipe 140. ft>. & couste 1. cent netto 50 ft. 6 d. Combien monte le tout 
 en argent 
 
 "Facit L. 186. ft. 4. d. io 14 /^- 
 
 " Cecy -)- signifie plus, cecy — moins." Page 59. 
 
 The problem is the same in each case, the book being printed 1 in two 
 languages in parallel columns. The translation of the problem is : 
 
 " Nine casks of olive oil have the weights given below, the tare for each 
 
 cask is 140 lb., and 1 centner net costs 50 florins 6 denarii. What is the- 
 
 entire cost?" 
 
 Weight. 
 
 cent. lb. 
 
 cent. lb. 
 
 
 
 cent. lb. 
 
 No. 1. 9 + 38. 
 
 No. 4. 9 + 50. 
 
 No. 
 
 7- 
 
 10 — 25. 
 
 No. 2. 9 -f 44. 
 
 No. 5. 9 + 55. 
 
 No. 
 
 8. 
 
 9 + 68. 
 
 No. 3. 10 — 20. 
 
 No. 6. 9 + 56. 
 
 No. 
 
 9- 
 
 9 + 70. 
 
 Ans. L. 186, ft. 4, d. 
 
 i ° 1 v 2B . 
 
 
 
 
 -f- means more, — means less. 
 
 
 
 
THE ESSENTIAL FEATURES 55 
 
 from the marks placed on packages to designate excess and 
 
 deficit with respect to listed weight. . 
 
 Van' der Scheure, in his Arithmetica (1600), fol. z x verso, 
 
 defines the symbol + and — as signs of operation thus : 
 
 + Plus Soubstraheert 1 
 — Minus Addeert. 
 
 He lapses, however, into using -=-, an old form of the minus 
 sign, in the solution of his problems. 2 The use of these signs 
 is to indicate operation, and their algebraic meaning, when em- 
 ployed in equations, is seen in Thierfeldern. 3 
 
 The lack of these symbols made tabulation in sentence form 
 impossible without the use of words. Hence, the tabular facts 
 of addition, subtraction', and also of multiplication were ex- 
 
 1 This spelling for subtract is not an accident. The title of the chapter 
 is Substractio, fol. B verso. It was quite common in the Dutch books of 
 ■that time to spell subtraction " substraction," a spelling not unheard of 
 to-day and declared -erroneous by lexicographers. 
 
 2 Van der Scheure, Arithmetica (1600), fol. z t verso, 
 "-f- Plus Soustraheert. 
 
 " — {Minus Addeert. 
 
 " Soo 9. Eyers + 2. blancken soo veel weert zijn als 12. blancken -f- 21. 
 Eyers/ Hoe veel Eyers coopt men dan om een blancke." 
 
 If 9 eyers and 2 blancken are worth as much as 12 blancken minus 21 
 eyers, how many eyers are worth as much as one blancke ? 
 
 12 -4- 21 9 + 2 : 1 
 
 2 21 
 
 10 30 Facit 3 Eyers. 
 
 3 Thierfeldern, Arithmetica (1587), page no. "Item/ 18 ft. weniger 
 85 gr. machen gleioh so vil als 25 ft. -f- 232 gr: wie vil hat 1 ft. groschen ? 
 facit 21 gr." 
 
 18 florins minus 85 groschens are equal to 25 florins minus 232 groschens, 
 how many groschens are there in 1 florin ? Ans. 21 gr. 
 
 In disen beydera Exempeln (he has given another example)/ addir das 
 Minus/ und subtrahir das Plus/ wie hie : 
 
 18 fl. -f- ?? gr. gleich 25/. -v- 232 gr. 
 + 85 
 
 18 fl. gleich 25/. 
 
 H- 147 gr. 
 
 l?fl. + 147 gr. gleich 25/. 
 18 
 
 
 147 gr. gleich 7 fl. 
 1 fl. 147 gr. 
 
 lfl. 
 
 lfl. facit 21 gr. Page 110. 
 
^6 SIXTEENTH CENTURY ARITHMETIC 
 
 pressed in words or in ruled tables according to a chosen 
 device. 1 This condition of affairs in the formative period of 
 arithmetic is responsible for the ruled tables still found in 
 modern' arithmetics, 300 years after the necessity for them has 
 disappeared. Some of them should be retained, doubtless, be- 
 cause of their suggestiveness in showing number relations, but 
 many of them might be omitted to advantage. 
 
 As in the case of addition, there are many instances of be- 
 ginning with a concrete problem. 2 Some writers who began 
 addition with abstract problems began subtraction with the 
 concrete ones. The following will illustrate: 3 "A 800347 
 man owed 800,347 livres, of which he has paid 409,- 409653 
 
 653 livres : I wish to know how much he still owes." 
 
 These problems are usually real situations, not con- 39 ° 94 
 crete merely in the sense of being subtraction of denominate 
 numbers. The problems given below from Champenois illus- 
 trate this tendency and also the care taken in grading the steps 
 in the process : 4 
 
 1 See page 50, of this article. 
 
 2 See page 48, of this article. 
 
 3 Trenchant, L'Aritbmetique (1578 ed.). "Vn homme doit 800347 l'fur 
 quoy il en paye 409653 Kures : fi de veux fcauair combien il doit de refte." 
 Fol. B g Tecto. 
 
 Dette 800347 
 Paye 409653 
 
 Refte 390694 
 
 4 'Champenois, "Les Institutions De L' Anithrnetique " (1578). 
 " Vn marobant a 58786 liures pefant de merchandife, & en vendu 35040 
 liures. On demande combien il a de refte." Page 17, or fol. C recto. 
 
 " Le 'Cotmmis general des viures a 478759 pains, & en diftribue 27000 
 pains. On demande ,comlbien 51 en de refte." Page 18, fol. C verso. 
 
 " Vn Architect a marchande faire vne murailfe qui contiet 876 toifes, en 
 a faict 374 toifes. On dlemande combien il en a enoor a faire." Page 19, 
 fol. Cij recto. 
 
 " Le Commis des viures du camp du Roy a 548 muids de ble, defquels il 
 «n a diftribue 273 muidis. On demande cobien dl en a encor' de refte." 
 Page 20, fol. Cij verso. 
 4 
 
 ?48 
 273 
 
 275 
 
THE ESSENTIAL FEATURES 57 
 
 " A merchant had 58,786 livres weight of merchandise and 
 sold 35,040 livres. It is required to find how much he had 
 left." 
 
 " The commissary-general had 478,759 loaves of bread and 
 distributed 27,000 loaves. It is required to find how much he 
 had left." 
 
 " An architect had bargained to make a wall which should 
 contain 876 toises, of which he had made 374 toises. It is re- 
 quired to know how much he had still to make." 
 
 " The steward of a royal camp had 548 measures of grain, 
 of which he had distributed 273 measures. It is required to 
 find how much still remains." 
 
 Multiplication 
 Two classes of writers may be distinguished easily by com- 
 paring their methods of treating multiplication. There were 
 those who emphasized the formal processes themselves, and 
 those who considered chiefly the applications of the processes. 
 The former class of writers made much of tabular forms and 
 devices, the latter made much of simple rules and commercial 
 problems. Both gave the multiplication tables at the outset, 
 which may be classified into three kinds : tabula per colonne, 1 
 
 1 Rudolff, " Kunstlicbe redlining mit der Ziffer und mit den zal pfen- 
 nige/ (1534 ed.), fol. Av verso. 
 
 1 1 
 
 22 24 
 
 3 3 36 3 9 
 
 44 48 4 12 
 
 1 mal 5 ist 5 2 mal 5 ist 10 3 mal 5 ist 15 and so on to 9 
 
 6 6 6 12 6 18 mal 9 ist 81. 
 
 7 7 7 14 7 21 
 
 8 8 8 16 8 24 
 99 9 18 9 27 
 
 Borgi, Arithmetica (154° ed.), fol. A 6 verso. 
 
 1 via 1 'sa 1 2 via 3 sa 6 3 via 4 sa 12 8 via 9 sa 72 
 
 2 via 2 sa 4 2 via 4 sa 8 3 via 5 sa 15 8 via 10 sa 80 
 
 3 via 3 sa 9 2 via 5 sa 10 3 via 6 sa 18 
 
 9 via 10 sa 90 
 
 10 via 10 sa 100 2 via 10 sa 20 3 via 10 sa 30 
 
 In the same way he gave tables of 16s, 20s, 24s, 32s, 36s. 
 
58 SIXTEENTH CENTURY ARITHMETIC 
 
 or column tables; the tables ruled in squares, 1 or square tables ; 
 and the tables arranged in triangles, 2 or triangular tables. 
 
 Tartaglia gave the tables — 
 
 o. fia o. fa o i. fia o. fa o 
 
 o. fia i. fa o i. fia i. fa I 
 
 o. fia 2. fa o i. fia 2. fa 2 
 
 and so on to 
 10. fia 10. fa 100 
 
 0. fia. 10. fa o 1. fia 10 fa 10 
 
 These tables were set apart to be learned. Then followed the tables of 
 
 ns, 12s, 13s, 40s for reference, and the first set of tables with the- 
 
 middle numbers ten times as large; that is, from o. fia o. fa to 10. fia 
 100. fa 1000. These latter were next combined thus : 
 
 11. fia 20. fa 220 20. fia 10. fa 200 
 
 11. fia 30. fa 330 and so on to 20. fia 20. fa 400 
 
 11. fia 100. fa 1100 20. fia 100. fa 2000 
 
 Finally he completed the tables from 11. fia 11. fa 121 to 20. fia 20. fa 
 400. Thus, 
 
 11. fia 11. fa 121 12. fia 12. fa 144 
 
 11. fia 12. fa 132 12. fia 13. fa 156 and so on to 
 
 11. fia 20. fa 220 12. fia 20. fa 240 20. fia 20. fa 400 
 
 The tables of 12s, 20s, 24s, 25s, 32s, and 36s of this list he called " Per 
 Venetia," because they were used in reckoning with Venetian money. 
 
 1 Tonstall, De Arte Supputandi (1522), fol. G„ recto. 
 
 -x 
 
 _JJ 
 
 SI 41 S! B 
 
 -AL& 
 
 p 
 
 1(1 
 
 ■4_j_a_l e 
 
 1'J IIS 
 
 .1.41 16 
 
 1* 
 
 20 
 
 3 
 
 6 | 8|1S!15!18 
 
 21124 
 
 27 
 
 30 
 
 .'* 
 
 Biiaijp,jsc'!«.j.aa l 38 
 
 Sfi 
 
 40 
 
 5 
 
 10J15 : 20 
 
 20 SO 1 35: 40 
 
 45 
 
 50 
 
 6 
 
 IS 18iK« 
 
 SO ! 36 i 42; 48 
 
 54 
 
 no 
 
 7 
 
 14 
 
 21 28IS5 42149,58 
 
 63 
 
 70 
 
 8 
 
 le 
 
 24 '32 '40 48 66164 
 
 72 
 
 80 
 
 J# 
 
 so 
 
 S0TW50lf5OI7OTn0 
 
 i81_ 
 80 
 
 _ao_ 
 
 100 
 
 2 CirveJo, Tractates Arithmetice practice (1513 ed.), fol. Avi recto. 
 
 
 
 5 
 
 .1 
 
 7 
 
 6 
 
 1 6 
 
 ^S~ 
 
 T 
 
 3 
 
 2 
 
 V 
 
 1 
 
 9 
 
 8 
 
 7 
 
 6 
 
 4 
 
 3 
 
 2 
 
 I 
 
 3 
 
 18 
 
 16114 
 
 12 
 
 3.0 
 
 ' 8 
 
 6 
 
 4 
 
 
 3 
 
 27 
 
 24 21 
 
 "28" 
 
 IB 
 
 12 
 
 9 
 
 
 
 4 
 
 sa 
 
 32 28 
 
 24 
 
 20 
 
 16 
 
 
 
 5 
 
 46 
 
 40 3fi 
 
 "So 
 
 25 
 
 ■ 
 
 
 
 6 
 
 54 48 42 
 
 36 
 
 
 
 
 7 
 
 63 
 
 66 , 49, 
 
 
 
 
 8 
 6 
 
 72| 
 81 
 
 641 
 
 
 
 
 *R 
 
 * Typographical errors in the original. 
 
THE ESSENTIAL FEATURES 
 
 59 
 
 The column tables were used by the best commercial writers, 
 and occasionally by the theoretic writers. The square arrange- 
 ment, called the Pythagorean table, was used generally by 
 authors of Latin School arithmetics. The triangular arrange- 
 ment, constructed by some from left to right and by others 
 from right to left, as shown in the notes, was the one in gen- 
 eral favor. It will be noticed in the triangular table of p. 58, 
 that the products 2 times 3 = 6, 3 times 4 = 12, and so on, 
 appear, but that 3 times 2 = 6, 4 times 3 = 12, and so on, 
 do not. Since any product, as 2 times 3, in one of these sets 
 was deemed sufficient to represent itself and the corresponding 
 product, as 3 times 2, in the other set, it is plain that writers 
 recognized the commutative law of multiplication. The rows 
 in the triangular table begin with square numbers. Gemma 
 Frisius, 1 a famous Latin School writer, called attention to this, 
 and Ramus 2 said that the pupil should first learn to multiply 
 1 Gemma Frisius, Arithmeticae Practicae Methodus FaciMs, foil. B 1 verso. 
 
 Qua- 
 dra 
 
 ti- 
 nu- 
 me- 
 
 11 
 
 S |S 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 
 
 4 1 6 
 
 a 
 
 10 
 
 12 
 
 14 
 
 18 
 
 18 
 
 2 
 
 
 1 9 
 
 12 
 
 IB 
 
 16 
 
 21 
 
 24 
 
 27 
 
 3 
 
 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 4 
 
 
 
 
 2 5 
 
 30 
 
 3b 
 
 40 
 
 45 
 
 b 
 
 
 
 
 at) 
 
 42 
 
 48 
 
 K4 
 
 rt 
 
 
 
 
 4« 
 
 56 
 
 63 
 
 i 
 
 ft 
 
 
 
 
 84 
 
 72 
 
 
 
 
 81 
 
 3 
 
 2 Ramus, Arithmeticae Libri Duo (1586 ed.), fol. A 7 verso, says that the 
 pupil should first learn to multiply single numbers by themselves, as twice 
 2 are 4, 3 times 3 are 9, 4 times 4 are 16, and so on, then the multiplica- 
 tion of each single number with other single numbers as twice 3 are 6, 
 twice 4 are 8, twice 5 are 10, and so on, but more attention should be given 
 to larger numbers, as 9 eights are 72, 9 sevens, sixes, fives are 63, S4, 45. 
 
 ,8 sevens, sixes, fives are 56, 48, 40. 
 
 " E notis autem multitudinis perdif cat primis fAngulas per fe multiplioare : 
 Bis 2 funt 4 Ter 3 funt 9 
 
 Quater 4 funt 16 Quinquies quina funt 25 
 
 Sexies 6 funt 36 Septies 7 Tunt 49 
 
 Octies 8 funt 64 Novies 9 funt Si 
 
 Huiusmodb mukiplicatio quadratura dividitur, & numero factus hoc modo 
 quadratus, factor autem latus quadrati. Fitq, ut nurnerus fecundum fuas 
 unitates pofitus & additus tantundem faciat, quantum per fe multiplicatus, 
 ut 2 & 2 faciunt 4: & bis 2 faciunt item 4. Sic 3 & 3 & 3 faciunt 9. 
 & ter 3 faciunt item 9. Analogia efciam in talibus est continua, ut 1 ad 
 factorem five latus, fie latus ad quadratum, ut in primo exemplo, ut 1 ad 
 
60 SIXTEENTH CENTURY ARITHMETIC 
 
 the single numbers by themselves, as twice 2 are 4, 3 times 3 
 are 9, and so on. This shows what the disciplinary teachers 
 of that time regarded as important. 
 
 The utilitarian writers, like Riese, Rudolff, and Kobel lim- 
 ited the elementary products to 9 X 9 or 10 X 10, occasion- 
 ally including the tables of twelves. It was more definitely 
 stated that these facts should be learned 1 than in the case of 
 the elementary sums. Tables given beyond 10 X 10, as the 
 12s, 15s, 20s, 24s, usually related to the reduction of denom- 
 inate numbers. 2 Thus, there were 12 denarii in 1 soldus, 20 
 
 2, fie 2 ad 4. Turn fingularum notarum cum fingulis multiplicatione fciat 
 quid efficiatur. 
 
 Bis 3 funt 6 : & ter 2 funt item 6. 
 
 Bis 4 funt 8: & quater 2 tantundem. 
 
 Octies 9 funt 72; & novies 8 tantundem." Fol. A„ verso. 
 
 1 Riese, Redlining auff der Linden und Federn/ (1571 ed.). 
 
 " Vndi du mult vor alien dingen das Ein imal eins wol wif fen/ und aus- 
 wendig lernen/ wie hie." Fol. Avi verso. 
 Kobel, Z<wey ireohenbuohilin (1537 ed.). 
 
 " Lern auswendig das. Ein mal ein 
 So wird dir alle Rechnung gmeyn.'' Fol. E verso. 
 
 2 Cataneo, Le Pratiche Delle Drue Prime Matematiohe (1547 ed.). 
 
 DEL MULTIPLICAEE LIRE, SOLDI ET DENARI. 
 
 " Et fe ti fuff e detto multiplica L 36. |S 12. & dena. 7 per 
 
 36.12.7 9. Segnato ohe harai le tue quantita come in margine, et tu 
 
 9 multiplica 7. uia 9. che fa 63. dena. che per effere ogni 12. 
 
 denari un foldo i detti denari 63. faranno foldi 5. & dena. 3. 
 
 329.13.3 di che fegnerai li. 3. denari & faluerai li 5. /S. dipoi multiplica 
 
 9. uie 12. & aggiugneli il 5. faluato & fara /3 113. che per effere 
 ogmi foldi 20 una L i dette /3 113. foeo L 5. & /3 13. onde feg- 
 nerai li /3 13. & faluerai le L 5. Dipoi multiplica 6. uie 9. 
 
 & aggiugneli il 5. faluato & fara 59. L dellequali fegnerai 9. 
 faluerai 5. Poi multiplica 3. uie 9. & aggiugneli il 5. faluato 
 & fara 32. ilquali fegna come da lato & harai L 329. £ 13. et 
 denari 3. per la detta multiplicatione. 
 
 DEL MULTIPLICARE, M0GGIA STAIA ET. QVARTI. 
 
 " Et dicendofi multiplica moggia 35. ftaia 11. & quarti 3. per 
 
 15. Segnato che harai le quantita come dal lato e tu multiplica 
 
 3. uie IS- che fa 45. quarti, perche ogni 4. quarti fanno uno 
 
 35.11.3 ftaio, i detti quarti 45. feranno. ftaia 11. & un quarto, onde 
 
 IS fegnerai un quarto & faluerai le 11. ftaia. Dipoi multiplica 
 
 11. uie 15 & aggiugneli lo 11. faluato & fara ftaia 176. che per 
 
 532. 8.1 effere ftaia 24. il moggio, le dette ftaia 176. fono moggia 7. 
 
THE ESSENTIAL FEATURES 6 1 
 
 soldi in i lire, 24 staia in 1 moggia, and so on. Tartaglia 
 ( x 556) ca]ls these tables "Per Venetia," because they were 
 based on the system: of Venetian measures. The theoretic 
 writers often filled in other tables, which were not of use in 
 denominate numbers, in accordance with their policy of em- 
 phasizing pure and formal arithmetic. 
 
 In the formal process of multiplication more care was taken 
 to grader the presentation than in addition and subtraction. 
 Easy-graded steps appear in some of the earliest printed arith- 
 metics. For example, Piero Borgi (1484) began with multi- 
 pliers of one figure; he next gave problems in which the 
 multiplier was a small number of two figures; then one in 
 which it was a number of two figures ending in zero; then 
 some with multipliers of three figures, and so on. It will be 
 noticed that the multiplication by multipliers of two figures, 
 as shown in examples x 2, 3 and 4 in note 1 , was often accom- 
 plished without partial products. This was done by referring 
 to the corresponding tables. See notes, pp. 57, 58, under Borgi 
 and Tartaglia. The following eight methods from- Paciuolo 
 ( 1494) show with what mastery the leading scholars of arith- 
 metic handled the Hindu algorism in the fifteenth century : 2 
 
 & ftaia 8. d'i ohe fegnerai le ftaia 6. fahierai le moggia 7. 
 
 " Dipoi multiplies. 5. uie. 15. et aggiugneli il. 7. faluato & 
 fara. 82. delquale fegnerai. 2. et faluerai. 8. Poi multiplica .3. 
 uie .15. et a quel che fa aggiugnelo .8. faluato et fara .53. 
 quale fegna come in margine & harai moggia .532, ftaia .8. et 
 un quarto per lo detto multiplicaanento per uia del quale & del 
 antedetto ti fera facile il multiplicamentO' de gli altri anoho 
 che uariati pefi mifure fuffero." Fol. Biiij verso. 
 1 Piero Borgi, Arithmetica (1540 ed.), fol. B verso, B 2 recto and verso. 
 1st 25 54 795 ' 2d 345 3d 3456 4th 3456 
 
 3 7 9 12 20 24 
 
 75 378 7155 4140 69120 82944 
 
 2 Paciuolo, Suma de Arithmetica Geometria Proportioni et Proportion- 
 alita (1523 ed.). 
 
 " Ora e da dire e mostrare in quati modi quefto acto del multiplicare 
 per la practica operatiua fe coftumi fare. Per laquel cofa dico che fimili 
 aicto de multiplicare fi coftuma fare principalmente in octo modi : di quali 
 el primo e detto multiplicare per fchachieri in vinegia ouer per altro nome 
 per bericuocolo in Firenza. El fecondo modo di multiplicare e detto caftel- 
 
62 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 i. Per schachieri, as called in Venice, meaning tesselated. 
 In Florence it was called bericuocolo. 
 2. Castellucio. 
 
 A Taveletta, or Per Colonna (by tables). 
 
 Per crocetta (crosswise). 
 
 Per quadrilatero (in form of a rectangle). 
 
 3- 
 4- 
 
 5- 
 6. 
 
 7- 
 8. 
 
 Per gelosia, 1 or graticola (lattice-work). 
 
 Per repiego (breaking up = factoring multiplier). 
 
 A scapezza (distributing = separating multiplier into 
 
 addends). 
 
 i. C Multiplicatio bricuocoli vel fchacherij. 
 
 (Multiplication bricuocoli or schacherii. Tesselated form.) 
 
 Fol. 26 recto, or Dij recto. 
 Multiplicands 9 8 7 6 
 
 Producendus 6 J $ ? 
 
 Multiplicans | 8 | 8 | £ | 8 | 4 | 
 
 fchachieri 
 bericocolo 
 4 p. I. 
 
 
 61 
 
 |7| 
 
 |9| 
 
 9 
 
 
 
 
 
 8| 
 
 \ 
 
 1 
 
 3 
 
 2| 
 
 J 
 
 15 
 
 19 
 
 12 
 
 |5 
 
 16 
 
 1 
 
 1 
 
 Sfima 6 7 4 8 16 
 
 2. C De .2°. modo multiplicand! dicto caltellucio. 
 
 (The second method of multiplication called castelluccio.) 
 
 Fol. 27 recto, or Diij recto. 
 6 Per .7. 
 
 9876 
 6789 
 
 Castelluccio. 
 
 61101000 
 
 5431200 
 
 476230 
 
 40734 
 
 Suma. 
 
 67048164 
 
 Proua 
 
 The proof in each case is that of 
 casting out sevens. 
 
 The chief feature consists in running the partial products to 
 the right and filling the vacant places with zeros. 
 
 lucio. El terzo e detto multiplicare per colona ouer a tauoletta. El quarto 
 modo de lo multiplicare e detto per crocetta : e altramente per casfelle. 
 El quinto modo e detto per quadrilatero. El fexto modo e detto per 
 gielofia : ouer gratiola. El feptimo modo e detto per repiego. Loctauo 
 modo e detto multiplicare a f capezzo." Fol. Dij reoto, or 26 recto. 
 
 1 So called because Italian ladies were protected from public view by 
 lattice-work over their windows. 
 
THE ESSENTIAL FEATURES 
 
 63 
 
 .3. C De tertio modo multiplicand.! ditto colona, 
 
 (The third method of multiplication called cotonna. By tables.) 
 
 „ Fol. 27 verso, or Diij verso. 
 
 Per .7. ' ' 
 
 4685 2 
 
 13 6 
 
 60905 
 Proua 
 
 No partial products were needed in this method, since a 
 table of 13s was presupposed. 
 
 4. C. De quarto mo multiplicandi ditto crocetta five cafella. 
 
 (The fourth method of multiplication called crocetta or casella. 
 Crosswise multiplication.) Fol. 27 verso, or Ddij verso. 
 
 _a 
 
 1 S 6 9 
 
 2 7 9 3 3 
 
 C De quinto modo multiplicandi dkto quadrilatero. 
 
 (The fifth method of multiplication called quadrilatero. 
 form of a rectangle.) Fol. 28 recto, or Diiij recto. 
 
 In the 
 
 
 5 
 
 5 
 
 i a 2 
 
 4 3 2 
 
 t 
 
 6 
 
 1 
 
 
 
 6 
 
 * 
 
 1 
 
 e 
 
 £ 
 
 a 
 
 a 
 
 ?. 
 
 i 
 
 7^ 
 
 2 
 
 « 
 
 2 
 
 7 
 
 X 
 
 6 
 
 
 
 2 9 5 6 
 
 29506624 Suma 
 
 6. C. De fexto modo multiplicandi dicto gelofia : siue graticola. 
 
 (The sixth method of multiplication called gelosia, or graticola. 
 Lattice-work.) Fol. 28 verso, or Diiij verso. 
 
 \s 
 
 8 \ 
 
 \8 
 
 \9 
 4\ 
 
 \2 
 7\ 
 
 
 \3 
 5 \ 
 
 \1 
 
 7\ 
 
 
 a 
 
 7 
 
 4 
 
 7. C. De feptimo modo multiplicandi ditto repiego. 
 
 (The seventh method of multiplication called repiego. That is, 
 multiplication by the factors of the multiplier separately.) 
 
64 SIXTEENTH CENTURY ARITHMETIC 
 
 6 1 may be broken up into the factors 2 and 3. Since 2 
 times 3 make 6, the factors of 6 are 2 and 3. The factors of 
 10 are 2 and 5, since 2 times 5 make 10. It often happens that 
 a number may be factored in different ways; for example, 12 
 has different groups of factors; it has the factors 2 and 6, 
 since 2 times 6 are 12, also the factors 3 and 4, since 3 times 4 
 make 12. And thus 24 has several groups of factors, as 12 
 and 2, 3 and 8, and 4 and 6, which multiplied together 
 make 24. 
 
 8. C De octauo cnodo multiplicandi dicto a fchapezzo. 
 
 (The eighth method of multiplication called a fchapezzo, distrib- 
 uting.) 
 
 When 2 42 is to be multiplied by 24, one of these numbers 
 (it makes no difference which) may be resolved into several 
 parts. As 24 may be resolved into four parts which added 
 together make the whole number, as 4, 6, 5, 9, then commence 
 with any one of these and multiply by 42. For instance, take 
 
 4, and 4 times 42 makes 168. Place this aside. Then 6 times 
 
 1 Paciuolo, Suma de Arithmetica Geometria Proportion! et Proportion- 
 alita (1523 ed.). 
 
 " Si commo de .6. diremmo esfer el. 2. e. 3. Perche .2. via .3. fa .6. fi 
 che el repdego de .6. e .2. e .3. El repiego de .10. e .2. e .5. perohe .2. via 
 
 5. fa 10. E acade molte volte vn. numero hauer asfai repieghi varij e 
 diuerfi: fi como .12. ane piu repieghi: poche bane el repiego de .2. e .6. 
 che .2. via .6. fa .12. Ane el repiego de .3. e .4. che .3. via .4. fa 12. E cosi 
 .24. a piu repieghi/ cioe .2. e .12. e .3. e .8. e .4. e .6. che luno elaltro mul- 
 tiplicato fa .24. : cioe per .6. e di .6. via .116. fa 696. 
 
 Fol. 28 verso, 29 recto, or Diiij verso and Dv recto. 
 
 2 Paciuolo, Suma de Arithmetica Geometria Proportioni et Proportion- 
 alita (1523 ed.). 
 
 " Si commo hauendo a multiplicare .42. via .24. dico che ne refolua vno- 
 de quefti numeri qual voli (che non fa cafo) in piu parti acio te fia piu 
 co modo el multiplicare. Or fia che tu refolua .24. in quatro parti che 
 fieno 1 luna .4. laltra .6. laltra .5. laltra .9. Dico che comenzi daqualuoli : e 
 multiplicata via .42. Or fatte idal. 4. e di .4. via 42. fa .168. Qua! metti da 
 canto. E poi .6. via .42. fa .252. e poi falua fotto .168. de ritto luno e 
 laitro: cioe numero fotto numero edicine fotto dicie^. E poi di .5. via 42. 
 fa .210. e falua fotto le altre. E poi dirai .9. via .42. fa 378. Qual simil- 
 mete falua fotto laltre e recogli mo tutte quefte .4. multiplication! infieme- 
 oioe .168. 252. 210. e 378. fanno .1008. e tanito dirai che facia .24. via .42."^ 
 Fol. 29 recto or Dv recto. 
 
THE ESSENTIAL FEATURES 65 
 
 42 makes 252. Save this with the 168, etc. Then 5 times 42 
 makes 210. Place this with the others. Finally, 9 times 42 
 makes 378. Put this also under the others in the same man- 
 ner, and then collect the four multiplications together, as 168, 
 252, 2 ix) and 378 making 1008, which is the product of 24 
 and 42. 
 
 Tartaglia (1556) and Unicorn 1 (1598) each gave seven of 
 the above methods, which shows not only Paciuolo's influence 
 upon- his countrymen, but also the tenacity with which theo- 
 retic writers held to disciplinary arithmetic. 
 
 Besides the above general processes, there are several par- 
 ticular ones of interest, such as complementary multiplication, 
 arrangement of factors, and order of multiplying. The fol- 
 lowing is an example of complementary multiplication for find- 
 ing the product of two digits, as given by Cirvelo (1513). 2 
 This was his plan of multiplying 6 by 8 : 
 
 2 (the complement of 8) X 6 = 12 6 
 
 8 
 
 60 (= 10 X 6) — 12 = 48 48 
 
 A variation of this process was as follows : To multiply 7 by 8 : 
 
 7.3 10 — 7 = 3 Multiply the complements 3 and 2 (= 6). 
 
 8.2 10 — 8 = 2 Add the numbers 8 and 7, using only units' 
 
 ■ figure in the result. 
 
 56 
 
 Such multiplication was not commonly used. Riese justified 
 its use because it was an available method for those who have 
 not learned the tables. 3 
 
 1 Unicorn, De L'Arithmetica Universale (1598 ed.), often gives credit to 
 Paciuolo. 
 
 2 Cirvelo, Tractates ArMimetice practice (1513 ed.). "Verbi gratia: 
 octies fex faciunt .48 nam ooto diftat a decez per -duas vnitates : ergo fub- 
 trahitur .6. de f exaginta q e f ua dena bis et remanebunt .48. Fol. a T recto. 
 
 According to Cantor, complementary multiplication was used by the 
 Romans. It is possible that it antedates the invention of the abacus. 
 Wefesenfoorn, Gerbert, 171-2. 
 
 5 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). "Leret 
 
66 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 432 6 
 
 18268000 
 
 1370100 
 
 91340 
 
 27402 
 
 19756842 
 
 
 
 Tonstall * said that, if the numbers are unequal, the larger 
 should be placed above as the multiplicand. In ^ , 2 s 
 
 the case of 185 times 13 bu., this arrangement 185 18s 
 
 would lead to a concrete multiplier, a form J 3 bu. 13 
 guarded against in modern teaching. But this 
 was avoided by omitting all denominations from the numbers 
 when used in calculation, as in example 2 above. The proper 
 denomination was affixed to the result when obtained. Sev- 
 eral examples of multiplication exist in which the higher 
 orders of the multiplier are used first, the 4S 6 7 3 
 
 method da Fiorentini of Tartaglia being 
 an example. The Castelluccio method of 
 Paciuolo differs from these only in having 
 the multiplier written above the multipli- 
 cand. There seems to have been no> use at 
 that time for beginning with the highest 
 order of the multiplier. But, after the decimal fraction was 
 introduced, this plan found a useful application in making ap- 
 proximations. For example, in the work in the 1.26 
 margin the part at the right of the vertical line need 
 not be calculated if the result is needed to tenths only. 
 
 Tests were prominent in multiplication, as in other 
 operations, the proof by nines and the proof by 
 seven being preferred. The example from Tartaglia at the 
 top of p. 67 illustrates the proof by casting out sevens. Divi- 
 
 2-35 
 
 2.5 1 2 
 .3I08 
 .0I630 
 
 2.9J610 
 
 viel machen/ Muft auch forne anheben/ Vnd fur alien dingen das Ein imal 
 eins/ auswendig lernen/ wie vorhin angezeiget/ oder mache es nach fol- 
 genden zweyen Regulen.'' Fol. Bv recto and verso. 
 
 8.2. 
 9.1. 
 
 
 7-3- 
 8.2. 
 
 
 6.4. 
 8.2. 
 
 
 6.4. 
 7-3- 
 
 7.2. 
 
 
 5-6. 
 
 
 4.8. 
 
 
 4.2. 
 
 The rule here suggested was known as the sluggard's rule. 
 
 Noviomagus (1544), Gemma Frisius (1540), Baker (1580), gave the 
 second method. 
 
 1 Tonstall, De Arte Supputandi (1522 ed.). "Et fi numeri lint in- 
 aequales : maior Temper supra pro multiplicado ponatur : minor infra pro 
 multiplicand." Fol. G s verso. 
 
THE ESSENTIAL FEATURES 67 
 
 sion was occasionally used to prove multiplication, S04 
 although the explanation of division constituted a 24 3 
 
 later chapter. 1 I20 9 6 ° 
 
 The short methods, although common, were con- 
 fined to three classes : (a) the use of factors in the multiplier, 
 (b) multiplication by multipliers ending in zero, and (c) 
 cross-multiplication. 
 
 (a) Multiplication by using factors of the multi- 87 
 plier. 3 
 
 This plan has been illustrated already in Paciuolo's 261 
 
 method, called repiego, number 7, page 63. The fol- 3 
 
 lowing occurs in Trenchant : 2 To multiply 87 by 9. 783 
 
 (b) Multiplication by numbers ending in zeros. 
 
 Piero Borgi, in multiplying 3456 by 20, gave the following 
 
 explanation: 3 6 X 20 = 120, then 5 X 20 = 100, 
 
 3456 
 100 -4-12 = 1 12, of which the 2 belongs to tens' place; 20 
 
 4 X 20 = 80, 80 -f- 11 = 91, of which the 1 belongs 
 
 to hundreds' place; 3 X 20 = 60, 60 + 9 = 69, the 
 whole result is 69120. In his second method of multiplying 
 by 20 he first multiplied by 2 and then by 10. Philip Calandri 
 (1491) took up multiplication by 100 as a special case, giv- 
 ing problems about 100 oranges, 100 chickens, 100 calves, 
 and various things. Tonstall directed placing at the right of 
 the multiplicand as many zeros as there are in the multiplier. 
 
 1 Riese, Rechnung auff der Linien und Federn/ (1571 ed.), fol. Bvii recto. 
 
 2 Trenchant, L'Arithmetique (1578 ed.), fol. C 4 recto. 
 
 3 Borgi, Qui comeza la nobel opera de arithmeticfaa (1540 ed.). 
 
 " E fe hauefti a moltiplicar .3456. per .20. prima metterai le due figure in 
 forma, poi cominciando dalle vnita dirai .6. via .20. fa .120. che 
 fono apunto .12. defene fenza foprauanzo de vnita, & pero in 3456 
 
 luogo delle vnita metterai .0. e dirai nulla e tien .12. defene, poi 20 
 
 alle defene .5. via .20. fa .100. e .12. che tenefti fa .112. che fono 
 
 .11. centenara e .2. defene e metterai le defene a fuo luogo. e dirai 69120 
 
 .2. e tien .11. cetenara poi alii cetenara dirai .4. via .20. fa .80. e 11. 
 che tenefti fa .91. che fono 9. rniara e vn centenaro, e metterai il centenar 
 a fuo luogo, e dirai .1. e tie .9. miara, poi alii miara dirai .3. via 20. fa .60. 
 e .9 ohe tenefti fa .69. ilql metterai a fuo luogo appreffo il. 1. fara 69120 
 adoque moltiplicato .3456. p. .20. fa 69120." Fol. B 2 recto. 
 
68 SIXTEENTH CENTURY ARITHMETIC 
 
 This method was also used by Rudolff and Car- 36 
 
 dan. 1 Gemma Frisius, 2 in multiplying two num- 7? 
 
 bers, as 3600 by 7200, rejected the zeros, multi- 72 
 
 plied as usual, and then annexed the: zeros to the 2 s2 
 
 result. This method was also used by Baker. 3 2592IOO00[ 
 
 (c) Cross-multiplication, known as per cro- 
 cetta, or per crosetta. l X l X l 
 
 The following example is taken from Tar- 456 3 
 
 taglia. 4 This method was also used by Paci- 1 48200 — 3 
 
 3 3 
 
 uolo, Unicorn, Borgi, and several others who 
 followed the Italian School. 6 
 
 Cardan gave the following methods for aiding the memory 
 in multiplication: 
 
 1. To multiply 27 by 33 : 
 
 2 7 + 33 = 60 60 -=- 2 = 30 30 2 = 900 
 
 30 — 27 = 3. 3 2 = 9 900 — 9 = 891=27X33- 
 
 2. To multiply 27 by 63 : 
 27X6=162 27X3 = 81 
 1620 -J- 81 = 1701 =27 X 63. 
 
 3. To multiply 37 by 49 : 
 
 40X50 = 2000 40—37 = 3 50 — 49=1 
 2000 + 3 = 2003 1 X 40 = 40 
 3 X 50=150 + 40=190 
 2003 — 190 = *28i3 = 37 X 49- 
 
 4. To multiply multiples of 10: 
 30 X 70 = 2 1 hundreds 
 
 700 X 800 = 56 ten thousands = 560000 
 17 X 70 =119 tens = 1 190. 
 
 Many writers of commercial arithmetic, and even some 
 Latin School writers, as Gemma Frisius, proposed a concrete 
 
 1 Cardan, Practica Arithmetice (1539 ed.), fol. Bvi verso. 
 
 2 Gemma Frisius, Arithmebicae Practicae Methodus (1581 ed.), fol. B s 
 recto. 
 
 3 Baker, The Well Spring of Sciences (1580 eld.), fol. Dvi verso. 
 
 4 TartagLia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. E 6 recto. 
 
 5 Cross multiplication is one of the six methods given by Bhaskara in 
 the Lilivati. 
 
 * Error in original. 
 
THE ESSENTIAL FEATURES 69 
 
 example in multiplication before explaining the process. 
 Calandri 1 gave as his first example : " Multiplica 9 vie 7389 
 tf 11 )8 8 d. (Multiply 9 by 7389 y 11 p 8 <£)" Gemma 
 Frisius, for his first example with a multiplier of two figures, 
 gave this example: " I wish to reduce 267 days to hours." 2 
 The following are from Champenois: "A squadron has 312 
 men in rank and 232 in file; how many men are there in the 
 squadron?" 
 
 "A wall is 1212 toises in length and 4 toises high; how 
 many toises are there in the wall ?" 3 
 
 Division 
 
 The methods of division used at that time have a peculiar 
 interest. Most of the methods of adding, subtracting, and 
 multiplying that were in general use in the sixteenth century 
 are used to some extent at the present time, but the method 
 of division most commonly used then is entirely obsolete now. 
 This was known as the scratch, or galley method. 4 
 
 A simple example from Baker will give the principles of 
 the method : G 
 To divide 860 by 4. The devidend. 860 
 
 Dividend 
 
 Deuisor. 
 
 Divisor f (2 quotient 
 
 $=(4X2) subtracting 8 from 8 leaves nothing to be placed 
 above. 
 
 1 Calandri, Arithmetica (1491 ed.), fol. 18 recto, or Cvi recto. 
 - Gemma Frisius, Arithmeticae Practicae Methodius Facilis (1581 ed.). 
 J ' Exainipli gratia, 267 dies valo redigere ad hoiras." 
 
 3 Ghaimpenois, Les Instiitvtions De L'Ardtbmetique (1578 ed). 
 
 " Vn efcadron comtient en front 312 hommes, & en flanc 232. Lon de- 
 mande combien LI y a d'hammes en l'ef cadron." Page 27, or fol. Cvi recto. 
 
 " Vne muraille contient en longueur 1212 toifes, & en hauteur 4 toifes. 
 On demande combien de toifes contiet la muraille." Page 29, or Cvii recto. 
 
 4 The galley, or .scratch., method of division is doubtless an inheritance, 
 having its origin in 'the sand-table calculation of the Hindus. Treutlein, 
 Abhandlungen, 1 : 55. 
 
 Maximus Planudes (c. 1330) also explains its origin in this way. Jour- 
 nal Asiatique, Series 6, vol. 1, p. 240. 
 
 5 Baker, The Welt Spring of Sciences (1580 ed.), fol. Dviii recto and 
 verso, Ei recto. 
 
7 o 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 In the next step the divisor is moved one place to the right. 
 
 2 subtracting 4 (= 
 W(21 
 
 4X1) from 6) 
 
 4 4 is contained in 6 once, so 1 is written in the quotient 
 
 2 The 2 placed above is the remainder after having subtracted 
 
 $S0(215 4 from 6. This 2 in tens' place with the o still left in units' 
 
 4 place forms 20. 20 divided by 4 leaves 5, the last figure of 
 
 $> the quotient. 
 
 The following is an example of the scratch method from 
 Tartaglia showing a remarkable form of galley : x 
 
 o 9 
 
 , 6 
 
 s i 
 
 o 
 
 6 i) 
 
 5 .\ 
 
 9 
 
 9 
 
 s 
 
 9 9 
 6 O 
 
 i-6 
 
 0048 
 
 6 6 6 
 
 > S S 
 0990 
 
 9 9 9 9 
 
 4 « 
 
 09 » 9 9 
 
 O I" © S 6 
 
 08^6 0SS3- 
 
 000000*9948000000019994 
 
 000000S6ff60po0o000S66666 
 000000SSSSPOO00O00SXSSSS18S 
 
 00000009^900000000099 90 9 I 
 
 00000000999000000000999 
 
 Delterzo modo dipartirc decto a danda. 
 
 4* I! rcrro modo di partire da nofiri antidii prarici t dcrto a danda < qual e" purgcneralc , fi come 3 
 parrirc per l\ucllo,oucr galca.cioe chc per cal modo fi puo partire perogni numcro ,rna in que* 
 fto non fi depe nna mai alcuna figura nd operate .come fi fa ncl parcir perbatelto , ouerg2lea, & 
 acao incglio lo apprcndi.poniamo chc ru vo^Ua partire quel medefimo 91214;. per 1 987. che 
 
 The downward method of the present day appeared 
 among those used in the earliest printed arithmetics. 
 This example is from Tartaglia : 2 
 
 hTJ- 
 
 partitore 1987 
 (Divisor) 
 
 I auenimento (Quotient) 
 9I234S I 459 
 
 ^ 
 
 M % 
 
 9123 
 
 7948 
 
 "754 
 9935 
 
 18195 
 17883 
 
 (Remainder) auanzo 312 
 
 1 Tartaglia, La Prima Parte del general Trattato (1592 ed.), fol. Gv 
 recto. 
 
 2 Tartaglia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. G 8 verso. 
 
THE ESSENTIAL FEATURES 
 
 71 
 
 Paciuolo also gave it as one of his methods, called division 
 " a danda." 
 
 — --—u 23. 
 
 JT4?>per si , \\ 
 
 l^arri 
 
 Uicnne 
 
 *T*9> — 
 
 oot44-|f 
 
 ?>> 
 312L, 
 
 o 
 
 8? 
 
 8? 
 
 + r 
 "ST 
 
 f>am 
 
 Is 
 
 a 
 
 oicnne 
 
 •J-So 
 
 Uicnne 1 1 ££ 
 
 This illustra- 
 tion is from Cal- 
 andri, 1 in whose 
 book appears so 
 far as known, the 
 first downward 
 division ever 
 printed, although 
 it is found occa- 
 sionally in manu- 
 scripts of the fif- 
 teenth century. 
 
 The lists of 
 writers given be- 
 low show rela- 
 tively the extent 
 to which the gal- 
 ley method 2 and 
 the downward 
 method 3 were 
 used. Nearly all 
 who used the 
 downward meth- 
 od also used the galley method, while many treated the 
 galley method who did not explain the downward form. 
 
 Besides these general processes there were several other 
 forms. The method a tavoletta, variously called per colona, 
 di testa, per discorso, and per toletta, was used by Paciuolo 
 
 1 Calandri, Arithmetics (1491 ed.), fol. 33 recto. 
 
 2 Among those who used the galley methods were: Borgi (1484) ; Wid- 
 man (1489) ; Cirvelo (1513) ; Tonstall (1522) ; Paciuolo (1494) ; Rudolff 
 (1526) ; Kobel (1531) ; Cardan (1539) ; Noviomagus (1539) ; Tartaglia 
 (1556) ; Gemma Frisius (1540) ; Riese (1522) ; Ramus (1567) ; Trenchant 
 (1571) ; Champenois (1578) ; Baker (1580) ; Raets (1580) ; Unicorn (1598) ; 
 Van -der Scheure (1600). 
 
 3 The dov/nward method was used by Calandri (1491) ; Paciuolo (1523. 
 ed.) ; Tartaglia (1556) ; Trenchant (1571) ; Unicorn (1598). 
 
 1 to 
 
 "Parti Co p 
 
 Go—* 
 
 4So 
 
 w'cnuc 1 Co 
 
 i? 
 
 Parti r E 
 
 3. 
 
 l> 
 
 Uteimc o 
 
 ■ ft 
 
7^ 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 (1494), Tartaglia (1556), and Unicorn (1598). This is 
 short division where the result of each part can be taken from 
 the table. Tartaglia gave as examples : 
 
 Divisor 2) 1 7953 Divisor 12) - 7630 
 
 Quotient 3976 Remainder I. Quotient 635 Remainder 10. 
 
 The method a repiego (per repiego) was used by Paciuolo * 
 (1494), Tartaglia, and Unicorn. In this method the divisor 
 was separated into factors, as in the example : * To divide 
 5867 by 48. 5867 -=- 6 = 733 with a remainder 3. 733 -=- 
 6 = 122 with a remainder 1. 
 
 Wencelaus gave a form called by him Italian division, of 
 which the following is an example : 5 
 
 To divide 11664 by 48. He first divided the divisor, 48, 
 
 „. . into halves, fourths, eighths and six- „ .. . 
 
 Divisor & Quotient 
 
 o teenths, the second group representing 
 
 24 -05 1, h 41 h iV (The first zero evidently 00625 
 
 12 °2S shows the lack of units, and the 05 x 
 
 ^^^ °J^ S figures beginning at its right repre- 
 sent respectively tenths, hundredths, 
 thousandths, and ten thousandths. It is a decimal system 
 without the use of the decimal point, a device which 
 did not appear until about 1600.) Beginning at the 
 left of the dividend, 11,664, 11 is the first number 
 that contains any one of the parts of the divisor as tabu- 
 lated. The largest part which it contains is 6. The fraction 
 which corresponds to it, 0125, is entered as part of the quo- 
 tient, as tabulated at the right. 6 is then subtracted from 11 
 and the remainder, 5, is treated similarly. Since 5 contains 
 
 1 Tartaglia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. F d recto, 
 a partir 27/7953 
 
 ne vien — 3976 — e auanza 1 
 
 2 Tartaglia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. F 7 verso, 
 a partir per 12// 7630 
 
 ne vien 635 auanza 10 
 
 * Paciuolo gave 'four methods of division : first, a Regola, or a tavoletta 
 (by table) ; second, per repiego (in parts) ; third, a danda (downward) ; 
 fourth, a Galea or per galea (galley .method). 
 
 4 Tartaglia (1592 ed.), fol. G 6 verso. 
 
 5 Wencelaus, T'Fondament Van Arithmetica (1599 ed.). 
 
THE ESSENTIAL FEATURES 
 
 73 
 
 m 
 
 012 5 
 
 0062 
 
 05 
 
 05 
 
 
 
 243 
 
 3, the next part of the quotient is the corresponding fraction, 
 tV, or 00625. 3 from.' 5 leaves 2, which is not in the list of 
 parts of the divisor, hence the next figure of the 
 dividend, 6, is annexed, making 26. This contains 
 24, so the corresponding 05 is written in the quo- 
 tient, one place farther to the right, and so on. 
 Whenever a new order of the dividend is used the 
 partial quotient is set one place farther to the right. 
 He gives as the complete form the example at the 
 Tight. 
 
 In the matter of detailed processes, Champenois was to 
 French arithmeticians what Tonstall was to English writers, 
 though less verbose. In his treatment of division Champe- 
 nois gave twelve cases : 1 
 1. To divide a digit by a digit. 2 (Exact division.) 
 
 1 Champenois, Les Institvtions De L'Arithmetique (1578 ed.), Diij rector, 
 ■or page 37 et seq. 
 
 2 The division tables were not so common as the tables of multiplication. 
 The inverse relation of division .to multiplication was generally recognized, 
 on which account one set of tables sufficed. A few writers who aimed at 
 ••completeness gave tables of division. 
 
 E. g., Tartaglia, La Prima Parte (1556), fol. Eiiij recto. 
 1 in o intra o e auanza 
 
 (1 is contained in o, o times with remainder o.) 
 1 in 1 intra 1 e auanza o 
 1 in 2 intra 2 e auanza o 
 
 1 in 9 intra 9 e auanza o 
 
 2 in o intra o e auanza o 
 2 in 1 intra o e auanza 1 
 
 9 in o intra o e auanza o 
 9 in 3 intra o e auanza 3 
 
 2 in 19 intra 9 e auanza 1 9 in 89 intra 9 e auanza 8 
 
 Tonstall, De Arte Supputandi (1522 ed.), used this inverted form of the 
 Pythagorean table, fol. Y recto. 
 
 100 1 90 1 80 1 70 1 60 1 50 1 40 1 .10 1 20 1 
 
 10 
 
 90 | 81 | 72 1 63 1 54 1 45 1 36 1 27 1 18 1 
 
 9 
 
 8 
 
 80 1 72 1 64 1 56 1 48 1 40 1 32 1 24 1 16 1 
 
 70 1 63 1 56 1 49 1 42 1 35 1 28 1 21 | 14 | 
 
 7 
 6 
 
 60 1 54 1 48 1 42 1 36 1 30 1 24 1 18 | 12 1 
 
 50 1 45 1 40 1 35 1 30 1 25 1 30 1 15 | 10 1 
 
 _5 
 4 
 
 40 1 36 1 32 1 28 | 24 | 20 | 16 1 12 | 8| 
 
 30 | 27 1 24 1 21 1 18 1 IS 1 12 1 9 1 61 
 
 .3 
 
 2 
 I 
 
 20 1 18 1 16 1 14 1 12 1 10 1 81 6 | 4 1 
 
 10 1 9l 81 7I 61 5I 4l 3l 2I 
 
74 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 2. To divide a dig-it by a digit with remainder. 
 
 3. To divide an article (a number ending in o) by a digit. 
 
 4. To divide a number whose first figure is smaller than 
 the divisor. 
 
 5. To divide an article by an article. 
 
 6. To divide a composite number (a number formed by com- 
 bining an article and a digit) by an article. 
 
 7. To divide a composite number by a composite number. 
 
 8. To divide a number when the number left after any sub- 
 traction is too small to be divided by the divisor, as 13 1328- 
 
 -*- 43 2 = 3°4- 
 
 9. Inexact division. 
 
 10. When the remainder is greater than the divisor it proves 
 that the division is incorrectly performed. 
 
 11. When the amount to be divided is less than the divisor,, 
 then fractions result. 
 
 12. To divide a number by 2. 
 
 Such development was not characteristic of the writers of 
 that period. It was customary to begin with a dividend of 
 several figures, but the methods of division were so radically 
 different from our present ones that it is not safe to say that 
 long division in the modern sense generally preceded short 
 division. It is possible, however, to find indisputable cases of 
 this plan. 1 
 
 The proofs for division were casting out nines, casting out 
 sevens, and the inverse operation. As has already been stated 
 in this article, the prevalence of proofs was due to the influ- 
 ence of abacus reckoning, and not so much to a sense of the 
 need for verification. Kobel in Germany and Baker in Eng- 
 land seemed to realize the uselessness of appending several 
 proofs to each operation, for they gave no proofs until all the 
 operations with integers had been presented. Kobel then 
 gave the proof of nines for all operations, and Baker gave 
 the inverse operations. It would be a decided improvement 
 on present teaching to introduce both the proofs by nines and 
 by the inverse operations where practicable. 
 
 1 Trenchant, L'Arithmetique (1578 ed.), fol. D 3 recto et seq. 
 
THE ESSENTIAL FEATURES 
 
 75 
 
 A few short methods were -used in division, the most com- 
 mon being those for dividing a number by 10, ioo, and iooo. 
 This example from Champenois will serve to illustrate : 1 
 " 54736 livres are to be divided among 10 men." The quo- 
 tient was formed by removing the last figure of the dividend 
 and making it the remainder. Baker also explained in the 
 same way the division by 100, 1000, and 10,000. Tagliente, 
 after his explanation of division by 10, also explained divi- 
 sion by 100 and iooo. 2 "And if you wish to divide 3497 by 
 100, do the same as above, taking off as many figures for a 
 remainder as there are zeros in the divisor, as seen in the 
 following division: 34I97." Similarly 749,745 by iooo, 
 749I745. A list of practical problems was given by Calandri 
 in which the cost of 100 things was known and the cost of 
 one required. 3 
 
 A more general case is the division of numbers by multi- 
 ples of 10. For example, 4 to divide 5,732 by 573(2 
 20, cut off the last figure of the dividend; 2 86 with 
 divide the part at the left by 2. Change this remainder 12 
 remainder 1 to 10 and add the 2 cut off. Then 5732 divided 
 by 20 gives 286 and the remainder, 12. 
 
 Finaeus ° wrote down the multiples of the divisor before 
 performing the division. 
 
 1 Champenois, "Les Institutions De L'Arithmetique (1578 ed.). 
 "Come f'il falloit diuifer 54736 liures a 10 hommes, fault trancher le 
 
 dernier nombre de la fomme a diuifer 6. Le refte 5473. donnera le Quo- 
 tient. Parquoy 54736. a partir a 10. hommes, c'eft a chacun 5473. liures, & 
 6. liures qui reftent a partir a 10 hommes." 5473(6 
 Fol. Eiij verso. I (o 
 
 2 Tagliente, Libro Dabaco (1515 ed.). "Et fe volefti partire 3497 per 
 100 farai nel modo ditto difopra itaglia tante figuare quanti .0. li a el tuo 
 parti doc come tu vedi qui fotto e fata partito. 34.I97." Similarly 749745 
 per iooo. 749 1 745. Note the bar used as a decimal point. 
 
 3 Calandri, Arithtnetica (1491 ed.). "Cento melarance choftorono 53 /3 
 4 d. che uiene luna." 100 oranges cost 53 jS 4 d., what is the cost of one? 
 "Cento pollaftre (chickens) coftorono 26 <y 10 P che uiene luna." " Cento 
 capponi (capons) coftorono 97 y 1 £ 8 d. che uiene luno." " CSto uitelle 
 (calves) coftorono 2354 yio^ che uiene luna." Fol. 29 recto and verso, 
 fol. (e) recto and' verso. 
 
 4 Tartaglia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. F ? recto. 
 B Finaeus, De Arithmetica Practica (1555 ed.), fol 9 recto. 
 
76 SIXTEENTH CENTURY ARITHMETIC 
 
 Division, like the other processes, was occasionally intro- 
 duced by concrete examples. Gemma Frisius follows his 
 definitions of dividend, divisor, and the remainder with : * 
 " 433656 aurei are to be divided among 72 men, what will 
 each receive?" and Kobel's first example is: 2 " There are 5 
 companions who must share equally 40 guldens. They would 
 like to know what part each should have." 
 
 Doubling and Halving 
 A striking example of extreme subdivision and classification 
 is the appearance of Duplatio (doubling) and Mediatio (halv- 
 ing) in the list of Species. To the student of the present 
 there seems to be no reason why doubling and halving should 
 have been treated independent of multiplication and division. 
 That the reason was not apparent to scholars of that day is 
 shown by this statement from Gemma Frisius : 3 " Some are 
 wont to regard Duplatio and Mediatio as operations separate 
 from 1 multiplication and division. I do not understand what 
 influences those stupid ones, since to double is to multiply by 
 2 and to halve is to divide by 2. If these operations are dis- 
 tinct, an indefinite number of operations will arise for con- 
 sideration, as triplatio, quadruplatio, and so on." In discuss- 
 ing the contents of the Bamberg Arithmetic (1483), Cantor 
 says : * " The treatment of doubling and halving as partic- 
 ular species or wholly unknown as such is the infallible sign 
 of whether the writer belongs to the school of Jordanus or to 
 that of Leonardo. The Bamberg Arithmetic treats of these 
 
 1 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.). 
 " Ut fi diuidendi fiat 433656 aurei 72 hominibus." Fol. B recto. 
 
 2 Ko.bel, Zwey rechenbuchlin (1537 ed.). "Es fein fiinff Gefellen/ die 
 haben zu theylen 40. gulden/ und wolten gem wiffen wie vil iedem zu 
 seinem theyl werden solt.'' Fol. F verso. 
 
 3 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.). 
 " Solent non mulli Duplationem & Mediationem a fignare species diftinctas 
 a multipllcatione &• divifione. Quid vero monuerit stupidos illos nefio, 
 cum & finitio & operatic eadem fit, Duplare endm, eft per duo multipli- 
 care. Mediare vero, per duo partiri. Quod fi hae operationes fint dif- 
 tincti, infinitae jam nobis exorientur fpeoies, triplatio, quadruplatio &c. 
 Sed fatis de illis." Fol. B 3 verso. 
 
 4 Cantor, M., Gesohichte der Mathematik (3d ed., 1900), Bd. II, page 227. 
 
THE ESSENTIAL FEATURES yy 
 
 operations as special cases under the Species and knows noth- 
 ing of them as separate species; thus it is an emanation of 
 Italian teaching spreading to southern Germany." That these 
 operations were accorded a high degree of independence is 
 shown by the fact that Widman, Kdbel, and Riese placed 
 them before multiplication and division. The following may 
 be taken as typical examples under these processes : 
 
 41232 » ;; V- 15241578570190521J 8 
 
 82464 43672136(21836068 Divisor 2 Quo. 7620789285095260 
 22222222 
 
 The first is an example in duplatio from Riese and the second 
 an example in mediatio from Gemma Frisius. Gemma Fris- 
 ius, however, treats mediatio simply as a case under division. 
 
 DENOMINATE NUMBERS 
 
 Having now discussed the four operations with integers, 
 the order of the remaining subjects must be taken arbitrarily, 
 since there was no uniformity in this matter in sixteenth cen- 
 tury works. By some writers fractions and denominate num- 
 bers * were both treated under operations with integers, and 
 
 1 Riese, Rechnung auff der Linien und Federn/ (1571 ed.), fol. Biiij 
 recto. 
 
 2 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.), 
 fol. B. verso. 
 
 8 Widman, Behead und hupsch Rechming (1508 ed.), fol. Ci verso. 
 
 4 Among those who treated the operations with integers separately were : 
 Riese (1522) ; Tartaglia (1556) ; Borgi (1484) ; Widman (1489) ; Ton- 
 stall (1522) ; Champenois (1578) ; Rudolff (1526) ; Gemma Frisius (1540) ; 
 Noviomagus (1539). 
 
 Among those who treated operations with denominate numbers under the 
 respective operations with integers were: Raets (1580) ; Unicorn (1598) ; 
 Calandri (1491) ; Baker (1562) ; Vander Scheure (1600) ; Cataneo (1546) ; 
 Cardan (iS39>- 
 
 Cardan treated under each operation all phases : e. g., under addi- 
 tion he treated addition of integers, denominate numbers, fractions, surds, 
 powers, roots, similarly under the other operations. 
 
 The expression "denominate numbers" was used by Cardan to denote 
 powers and roots of numbers. Thus, 
 
 cosa. census. cubus. census census. Relatum primum. 
 248 16 32 
 
 or the first, second, third, fourth, fifth, powers of 2 are given under the 
 last section of Chapter I on Arithmetic, fol. Avi recto. 
 
78 SIXTEENTH CENTURY ARITHMETIC 
 
 seldom were fractions and denominate numbers wholly separ- 
 ated from each other. 
 
 A thorough treatment of denominate numbers was import- 
 ant, not only because of their relation to commercial arith- 
 metic, but because the variety and complexity of the systems 
 of weights and measures then in use laid a heavy burden on 
 methods of calculation'. This is well illustrated in a work by 
 Cataneo. 1 In order to make his application of denominate 
 numbers clear, it is necessary to explain a few tables : 
 
 Measures of Length. 
 12 momementi make one minuto 
 12 minuti make one atomo 
 12 atomi make one punto 
 12 punti make one oncia 
 12 oncie make one braccia 
 (6 braccie make one cauezza). 
 The braccia was equivalent to 31 inches. 
 
 Measures of Surface. 
 I cauezza by I cauezza is an area of Y% tauole, or 3 piedi 
 1 cauezza by 1 braccia is an area of Yz piede 
 1 cauezza by 1 oncia is an area of Yz oncia 
 1 cauezza by 1 punto is an area of Yi punto 
 
 1 braccia by 1 braccia is an area of 1 oncia 
 1 braccia by 1 oncia is an area of 1 punto 
 1 braccia by 1 punto is an area of 1 atomo 
 
 1 'Cataneo, Dell' arte Del Misvrare Libri Dve. 
 
 12, mlomementi fanno vn minuto. 
 
 12, minuti, fanno vn atomo. 
 
 12, atomi, fanno vn punto. 
 
 12, punti, fanno vn oncia. 
 
 12, oncie, fanno vn piede, in fuperficie, & vn braccia in linea, 
 
 Fol. C verso. 
 Cauezzi fia cauezzi, fanno quarti di tauole, ouero piede 3 fuperficiali. 
 Cauezzi fia braccie, fanno mezi piedi fuperficiali. 
 Cauezzi fia oncie, fanno meze oncie fuperficiali. 
 Cauezzi fia punti, fanno mezi punti fuperficiali. 
 Braccia fia braccia, fanno oncie fuperficiali. 
 Braccia fia oncie, fanno punti fuperficiali. 
 Braccia fia punti, fanno atomi fuperficiali. 
 Oncie fia oncie, fanno atomi fuperficiali. 
 Oncie fia punti, fanno minuti fuperficiali. 
 Punti fia punti, fanno momenti fuperficiali. Fol. C, recto and verso. 
 
THE ESSENTIAL FEATURES yg 
 
 i oncia by i oncia is an area of I atomo 
 I oncia by I punto is an area of I minuto 
 
 i punto by I punto is an area of I momento 
 
 The ratio between the consecutive square units is 12. That is, 12 mom. 
 ■= 1 min., 12 imin. = 1 atom, 12 at. = 1 punti, etc 
 
 The following is Cataneo's method of computing the area 
 of the trapezoid whose dimensions are given in the figure : 
 
 Test 
 
 a cauezzi 17, : 
 
 " Settima Ragione, Delia (The seventh solution of the) 
 quinta Figvra (fifth figure = one above) 
 
 ■(Upper base) Tefta cau. 17, bra. 2, on. 9. 1 (Lengths of bases as given 
 
 (Lower base) Tefta cau. 19, bra. 5, on. 8. f in the figure.) 
 
 \ 
 
 (Sum) Sornma ■ 
 
 cau. 
 
 37, bra. 2, 
 
 on. 5. (See linear table.) 
 
 1 Larghezza cau. 18, 
 
 bra. 
 
 4, on, 2, pun. 6. (Half sum of bases.) 
 
 Lunghezza cau. 22, 
 
 bra 
 
 . 4, on. 9. (Altitude.) 
 
 2 Doppi cauezzi 9, bra. 4 
 
 , on. 2, pun. 6. 
 
 Doppi cauezzi 11, bra. 
 
 4, on. 9. 
 
 Tauole 
 
 99- 
 
 
 
 
 Tauole 
 
 3, 
 
 pie 8. 
 
 
 
 Tauole 
 
 0, 
 
 pie 1, 
 
 on. 
 
 10. 
 
 Tauole 
 
 0, 
 
 pie 0, 
 
 on. 
 
 S, pun- 6. 
 
 Tauole 
 
 3, 
 
 pie 0. 
 
 
 
 Tauole 
 
 0, 
 
 pie 1, 
 
 on. 
 
 4- 
 
 Tauole 
 
 0, 
 
 pie 0, 
 
 on. 
 
 0, pun. 8. 
 
 Tauole 
 
 °> 
 
 pie 0, 
 
 on. 
 
 0, pun. 2. 
 
 Tauole 
 
 O; 
 
 pie 6, 
 
 on. 
 
 9- 
 
 Tauole 
 
 0, 
 
 pie 0, 
 
 on. 
 
 3- 
 
 Tauote 
 
 0, 
 
 pie 0, 
 
 on. 
 
 0, pun. 1, at. 6. 
 
 Tauole 
 
 0, 
 
 pie 0, 
 
 on. 
 
 0, pun. 0, at. 4, m. 6. 
 
 Tauole 
 
 106. 
 
 pie 6. 
 
 on. 
 
 8. pun. 5. at. 10. m. 6. 
 
 
 
 pun 2 
 
 :|6 min. 
 
 Proua 
 
 oncie 3 
 
 16 min. Fol. G, recto. 
 4 
 
 1 Larghezza is width and Lunghezza is length. The area of the trape- 
 zoid equals that of a rectangle whose .dimensions are J /i the sum of the 
 lases of the trapezoid and its altitude. 
 
 2 Since I cau. X 1 cau. = % Tauole (see tables), y 2 of 18 and x / 2 of 
 
80 SIXTEENTH CENTURY ARITHMETIC 
 
 The vigorous commercial activity of that time demanded a 
 knowledge of weights and measures used in all the trading 
 centers of Europe. A comparison of these reveals not only 
 a great number of denominations, but also a lack of uniform- 
 ity in each denomination. An idea of what a " hundred- 
 weight " might mean in the fifteenth century may be obtained 
 from this excerpt from Chiarini : 1 " The ioo lb. of Florence 
 is 103 lb. in Siena, 102 to 104 in Perugia; in Lucca 102 lb. 
 equals 105 lb. in Pisa, and at present is the same as Floren- 
 tine weight." 
 
 A comparison of this list with corresponding data given 
 by Raets a century later shows the persistency of a condition 
 which finally led to the establishment of the International 
 System. The following is a typical problem from Raets : 2 
 " If a centner of Niirnberg weighs as much as 108 lb. at 
 Antwerp, how many centners do 11,682 lb. at Antwerp- 
 weigh?" In these problems the value of the centner of 
 Genoa, Venice, and Antwerp is compared with that of 
 Nuremberg, Aquila, Augsburg, England, Bruges, Lisbon,, 
 Sicily, and other cities and countries. 
 
 An idea of the field covered by the tables of denominate 
 numbers required in the practical arithmetic of that time may 
 be had from the following summary of Kobel's treatment : 
 
 1. A list of abbreviations of weight and money denom- 
 inations. 
 
 2. Tables of money : Rhenish, Frankfurt, Nuremberg, Aus- 
 
 22 are written down before multiplying. Then the result is 99 whole- 
 tauole. The next step is to find 22 cau. X 4 bra. This is done by finding" 
 11 cau. X 4. bra. = 3 tau. 8 pie. (See tables.) The result is 1 placed as the 
 second partial product. When all of the terms of the multiplicand have- 
 been multiplied by each term of the multiplier, all the results are added 
 as shown. 
 
 1 Giorgio Chiarini, Qvesta e ellibro che tracta de Mercatantie et vsanze- 
 de paesi (1481 ed.). 
 
 " I IBBRE cento Di Firenze fanma in Siena lib 6 ceto tire i pugiia lib^ 
 cii.I. ciiij. In Lucca lib. c. ii. I Pifa lib. c.v. & hora e tucto uno conquel 
 di Firenze -." Fol. 5. 
 
 2 Raets, Een niew Cijfferboeck (1580 ed.), "So den Centner Nuren- 
 burghs weecht tot Antwerpen 108 lb. hoe veel Centners doe 11682 lb. Ant- 
 werps?" Fol. Hiiii recto. 
 
THE ESSENTIAL FEATURES 8 1 
 
 trian, Hungarian, Meissen, Augsburg, Strassburg, Wirten- 
 berg, Venetian, Parisian, with comparisons. 1 
 
 3. Tables of common weight. 2 
 
 4. The value of a centner in Venice, Nuremberg, Frank- 
 furt, Genoa, Prussia. 
 
 5. Table of gold and silver weight. ) Worms, Oppenheim, 
 
 6. Table of wine measure. | Mainz. 
 
 7. The number of Omen in a Fuder in Heidelberg, Speier, 
 Wachenheim, Durckerm. 
 
 8. Table of grain and fruit measure. 
 
 9. Table of time: minutes, hours, days, weeks, and years. 
 He divided the minutes into 18 Puncten instead of into 60 
 seconds, and gave 364 days for a year. 
 
 10. Table of cloth measure. 
 
 11. Table of measure of Fustian. 
 
 12. Measure of salt fish. 
 
 Aliquot parts were commonly treated by commercial writ- 
 ers. Their importance has never waned, although they have 
 often been neglected, and their character changed. Baker 
 gave this definition of aliquot parts : 3 "An aliquot part 
 
 1 Kdbel, Zwey rechenbuchlin (1537 ed.). " Der Churfurften Miintz 
 am Rihein, fol. B T verso; Miintz Pranckfurter Wehrung, fol. B g recto; 
 Miintz zu Nurenbergk, fol. B g recto; Ofterreiohifch Miintz, fol. B g verso; 
 Vngerifch Miintz, fol. B g verso; Meifiwifoh Miintz, fol. B g verso; 
 Miintz zu Augfiburgk, fol. B g v«rso ; Miintz zu Strafiburgk, fol. B g verso ; 
 Miintz, in Wirtenberger land, fol. C recto; Der Venediger Miintz, fol. C 
 recto; Miintz, zu Parifi." Fol. C verso. 
 
 2 Kobel, Zwey rechenbuchlin (1537 ed.). 
 
 " Von gemeynen Gewichten. 
 
 Centner Cz 100. lb. 
 
 Pfundt lb 32. lot. 
 
 Ein Lot It hat 4. quintr/ad' ein halbe vntz. 
 
 Quint qui 4. ^9| 
 
 Mark mar *26. lot. " 
 
 ♦26 should be 16. Fol. C 2 verso. 
 
 One centner = 100 lb. 
 
 " ipfund = 32 lot 
 
 " lot = 4 quintal, or yi ounce 
 
 " Quintal = 4 pfennige 
 
 " Mark = 16 lot 
 8 Baker, The Well Spring of Sciences (1580 ed.), fol. Mvii verso. 
 
82 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 is an eue part of a shiling or of a pound or of any- 
 other thing, as %, %, %, Ys, &c, are called aliquot parts." 
 He then discussed the aliquot parts of a shilling so that in the 
 reduction the fractions of a shilling may easily be replaced by- 
 pence. 
 
 Besides the tables of weights and measures there were tables 
 to assist in the solution of problems containing denominate 
 numbers. An excellent specimen is a work compiled by Jean, 1 
 in which the author 
 showed how to 
 work problems in 
 multiplication, 
 Rule of Three, In- 
 verse Rule of 
 Three, and interest. 
 His first table com- 
 posed of multiples 
 of monetary units 
 occupies forty-six 
 octavo pages. 
 
 The numbers at 
 the top of the col- 
 umn begin at i and 
 proceed to 200,- 
 000. The numbers 
 in the first column 
 begin with 1 and 
 proceed to 25. Un- 
 der each of the 
 
 column headings there are three divisions for the livre, sou, 
 and denier respectively. 
 
 He gave a problem and explained its solution thus : " Sup- 
 pose that 29 aunes of merchandise have been bought at 7 livres 
 1 1 sous 9 deniers an aune, to obtain the cost it is necessary to 
 find column 29 and go down the column containing lira until 
 you are opposite to the 7 of the small tree. Here you will 
 
 1 Alexander Jean, Arithmetique Av Miroir (1637 ed.), fol. Aij verso. 
 
 
 
 «., 
 
 
 
 2t ~**p <i -*ar s Hflfl^" 
 
 
 
 
 Jhl* 
 
 imifrj 
 
 lui-ej 
 
 /»/* 
 
 rfcum 
 
 fc 
 
 /alx. ieaeri 
 
 1 
 
 ) 
 
 1 
 
 18 
 
 :i8. 
 
 ■-»• 4 
 
 -9 
 
 :2o. 
 
 :2. y 
 
 3" 
 
 ■■■>'■ 
 
 2. 6 
 
 I 
 
 a 
 
 S& 
 
 :,6. 
 
 4- 8 
 
 iS 
 
 :y8. 
 
 :4 . 10 
 
 00 
 
 5: 
 
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THE ESSENTIAL FEATURES 83 
 
 find 203, the number of livres for the result." 1 He finds the 
 other products in the same way and combines them to find 
 the result. 
 
 If one were to pass from the field of arithmetical text-books 
 and aim at completeness in describing the denominate number 
 systems, the result would be voluminous. An excellent idea 
 of the arithmetic of the custom-houses of that time is given 
 by Bartholomeo di Pasi. His work 2 of 200 octavo pages is a 
 compilation of tables mostly of this kind: 
 
 (The values of the lira in various cities.) 
 Bolzano L 239 
 
 Melano 
 
 L 239 
 
 Firenza 
 
 227 
 
 Genoa 
 
 247 
 
 Bologna 
 
 2i6y 2 
 
 Roma 
 
 217 
 
 Napoli 
 
 244 
 
 Piafenza 
 
 239 
 
 Mantoa 
 
 239 
 
 Ferrara 
 
 226 
 
 Parma 
 
 140 
 
 
 Nolimbergo 
 
 156 
 
 
 Geneura 
 
 164 
 
 
 Auignone 
 
 185 
 
 
 Parife 
 
 179 
 
 
 Lione 
 
 182 
 
 
 Marfiglia 
 
 193 
 
 
 Valenza 
 
 215 
 
 Fol. D s verso. 
 
 The values of the lira, the pezza (measure of length) and 
 others are given for over a hundred cities. 
 
 Writers of commercial arithmetic, on account of their ten- 
 dency to emphasize the utility of the processes, generally 
 placed denominate-number problems under each operation with 
 integers instead of deferring the whole subject to a separate 
 chapter. Baker, after having explained the addition of in- 
 
 1 Jean, Arithmetique Av Miroir (1637 ed.). " Suppofez avoir aehepte 
 29 aunes de marchandife a 7 liures 11 fols. 9 deniers l'aune, il faut trouuer 
 la Colomne 29. & defcendre dans les liures d'icelle iufques vis a vis du 7 
 du petit arbre, vous y trouuerez 203. qui sorat liures. 
 
 Pour ks 11. fols, il fault defcendre dans les fols de la dite Colomne 
 iufques vis a vis de n. dudit petit arbre, ou vous trouuerez 15 liures 19 fols. 
 
 Et pour les 9 denier, il faut defcendre dans les deniers d'icelle colomne 
 iufques vis a vis du 9. du petit arbre, ou vous trouuerez 21 fol 9 deniers. 
 
 Lesquelfes trois fommes fcauoir pour Jes liures' 203 liures, pour les fols 
 15 liures 19 fols, & pour les deniers 21 fols 9 deniers, il faut assembler & 
 vous trouuerez que 29 a.ulnes a 7 liures 11 fols 9 deniers, valient 220 liures 
 9 deniers." Fol. Aij verso. 
 
 2 Bartholomeo Di Pasi da Vinetia (1557 ed.) Tariffa de i pesi, e misure 
 corrispondenti dal Leuante al Ponente, e da una terra, e luogo all' altro, 
 quafi per tutte le parti del mondo. 
 
132 
 
 13 
 
 8 
 
 3456 
 
 16 
 
 5 
 
 789 A. 
 
 17 gr. 
 
 7 d. 
 
 67 
 
 9 
 
 6 
 
 2S2 
 
 20 
 
 3 
 
 84 SIXTEENTH CENTURY ARITHMETIC 
 
 tegers, passed at once to the addition of pounds, shilling's and 
 pence. Adam Riese, by common consent the greatest reckon- 
 ing master of his time, in his book on line-reckoning began 
 addition and subtraction with examples of denominate num- 
 bers. Thus, in addition he used the an- 
 nexed problem. 1 Even writers who were 
 chiefly concerned with traditional arith- 
 metic felt the demand for work in de- 
 nominate numbers. For instance, under 
 
 division Cirvelo says that division is ,_„„ 
 
 ■> 4729 14 5 
 
 used to reduce money of smaller denom- 
 ination to larger^ just as multiplication is used to re- 
 duce money of larger denomination to smaller. 2 Monetary 
 systems generally took precedence over all others in order 
 of treatment, as indicated by the examples above. In excep- 
 tional cases weight was placed first. The plan of introducing 
 denominate numbers under addition of integers leads at once 
 to a difficulty in sequence of processes. The reductions from 
 one denomination to another in simplifying the result occa- 
 sionally required a knowledge of division, as in these examples : 
 
 340 flo. 
 
 yp 25 d 3 
 
 
 lib 7974 
 
 />I3 
 
 7* 
 
 124 
 
 7 20 
 
 
 lib 879 
 
 fil2 
 
 6 
 
 98 
 
 6 27 
 
 
 lib 9400 
 
 P 5 
 
 7 
 
 49 
 
 12 
 
 
 lib 794 
 
 / 8 
 
 9 
 
 58 
 
 6 18 
 
 Suma 
 
 lib 19049 
 
 P 
 
 5 
 
 672 flo. S 12 d 
 
 As to the general character of the exercises given under the 
 subject of denominate numbers, it is worthy of note that they 
 
 1 Riese, Rechnung auff der Linien und Federn/ (1571 ed.), fol. Aiiij. 
 recto. 
 
 2 Cirvelo, Tractatus Arithmetice (1513 ed.). 
 
 "C Finis diuifionis eft vt fciamus quo mo pluribus debet diftribui aliqua 
 pecunia fecundu partes equales quantu debet habere e quisqj eorum, et qri 
 hafoemus aliqua. magna copia. denariorum et voluerimus videre quot folidi 
 vel quot aurei aut argentei fierent ex illis et ad plura alia valet, vnde ficut 
 multiplioationem poflutnus groffiorem reducere ad fubtiliorem : ita per 
 diuilionem poffumus ex minore moneta conftatuere maiorem." Fol. a yii 
 recto. 
 
 3 Rudolff, Kunstliche rechnung (1534 ed.), fol. Bvi verso. 
 
 4 Cardan, Practica Arithmetice (1539 ed.), fol. Avi recto. 
 
THE ESSENTIAL FEATURES 85 
 
 ■did not contain long lists of numbers to be manipulated merely 
 for practice in figuring. Concrete applications were plenti- 
 fully supplied, and ability on the part of the learner to solve 
 these practical problems was evidently the goal of instruction. 
 
 FRACTIONS 
 
 Definitions 
 Two conceptions of the fraction were prevalent among the 
 writers of that period : ( 1 ) A fraction is one or more of the 
 equal parts of a unit. (2) A fraction is the indicated quotient 
 of two integers. Among the examples of the former is the 
 treatment by Kobel, which reminds us how long the worthy 
 apple has done educational service. 1 He divides the apple into 
 twenty parts, each part of which is called a twentieth. Ten of 
 these parts make half the apple, and five of the twenty parts 
 taken together make a quarter of the apple, and so on. Kobel 
 is unique in opening the subject of fractions with a statement 
 of their utility : 2 " Since it happens that commercial questions 
 concerning measure, weight, and exchange are not always 
 asked and reckoned with in whole numbers, I shall instruct 
 you in the following pages, so that you may understand how 
 to arrange, interpret, and reckon questions involving calcula- 
 tion with fractions occurring in measure and weight. I prom- 
 
 1 Kobel, Zwey rechenbuchlin (1537 ed.). "So du ein gantzen apffel 
 hast/ vnd zerfchneideft oder tfaeyleft den felben/ in zwentzig teyl oder 
 ftuck/ so ift >der felben zwentzig theyl odder ftuck/ ieglichs ein zwentzigft 
 theyl des gantzenn .apffels genant/ rand wart inn der rechenfchafft iedes 
 ftuck ein bruch geteutfcht/ der felben zwentzigtheyl/ zahen/ fo man die 
 widerumb zufamen fetzt/ zeygen fie ein balben apffel an/ vnnd fo du der 
 zweanitzigtheyl fiinff zufamen Legft/ fiheftu ein viertheil des apffels jc. 
 vnd alfo fur vnd fur zu rechnen fein die theyl oder briich zuuenftehn." 
 Fol. H 3 verso. 
 
 2 Kobel, Zwey rechenbuchlin (1537 ed.). " Dieweil fich nit allweg 
 begibt/ das die handel/ kauff und fragen/ in gantzen zalen/ inaffen/ 
 gewichten/ oder verwechfilungenn gefchehen/ gefragt vnnd gerechnet wer- 
 den/ wil ich dich hernach leren/ fo dir in fragenn/ oder rechnungen 
 gebrocbne zalen/ ungerade gelt/ tnafi oder gewicht fiirkumpt/ wie du das 
 ordnen/ verftehn und rechnen folt/ fo vil zu difem gmeyneim heufilichem 
 gebraucih und Rechnen/ ich dir verheyffen un einem angenden Rechner am 
 erften zu wiffen not ift offenbaren." Fol. H recto. 
 
86 SIXTEENTH CENTURY ARITHMETIC 
 
 ise to teach you as much as is evidently necessary for every- 
 day use for a beginner in the art of calculation." 
 
 Another example of the first definition, and one in which 
 the measured units of denominate numbers serve to define the 
 fraction, -is given by Champenois as follows : * "A fraction is 
 part of an integral whole. As a livre is an integral whole, 
 and its parts are 20 sous; and one sou is an integral whole 
 whose parts are 12 deniers; an aune is an integral whole, 
 and its parts are three tiers, four quarts, and other parts, 
 a c d e b 
 
 " If the aune ab be divided into four equal parts at the 
 points c, d, and e, acde will be the three-fourths, which the 
 purchaser took, and the other fourth, eb, will be kept by the 
 merchant." 2 The graphical method of explaining fractions 
 was very rare in that period. 
 
 The definition of Gemma Frisius is of the same kind and 
 contains an explanation of the terms numerator and denomi- 
 nator. 3 " We call the numbers showing the parts of an in- 
 tegral thing fractions,, or parts, as J4 signifies one-half; Ya, a 
 
 1 Champenois, Les Institutions De L'Arithmetique (1578 ed.). "Frac- 
 tion eft partie d'vn entier. Comme vne liure eft vn entier, & fes par- 
 ties .font 20. fols. & vn fols eft vn entier, & fes parties font 12 deniers, 
 vne aulne eft vn entier, & fes parties fot trois tiers, quatre quarts, & autres 
 parties." Page 85, fol. Giij reoto. 
 
 2 Champenois, Les Institutions De L'Arithmetique (1578 ed.). " Soit 
 l'aulne .a.h. diuifee en quatre parties egales, au poinct .c.d.e. les trois quarts 
 feront a.e.d.e. que 1'achepteur predra, & reftera l'autre quart au marchant 
 .e.b." Page 86, Giij verso. 
 
 3 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.). 
 " Fractiones, rninutias, aut partes, appellamus numeros integrae rei par- 
 tes significantes, vt -| semissem significat, A quadrantem siue quartam 
 partem, | dodrantem, aut tres quadrantes. Scribuntur duobus numeris, 
 superiorem numeratorem, inferiorem denominatorem appellant : hunc quod 
 denotet, quot in partes integrum secari oporteat : ilium, quia quot huius- 
 rnodi sumendae sint particulae, numeret. Veluti -|, hie inferior denotat 
 integrum dividendum in 7, sumendas tamen tantu itres septimas innuit 
 superior. Cum igitur duo hi fuerint aequales, semper integrum fcawtum de- 
 notatur, vt ^. Cum superior maior est, plus integro: cum minor est, 
 minus integro significat." Fol. C verso. 
 
THE ESSENTIAL FEATURES 87 
 
 fourth; and $4, three-fourths. They are written with two 
 numbers ; the upper one is called 1 the numerator, the lower one 
 the denominator, the latter of which denotes into how many 
 parts the integer must be divided, the former shows how many 
 of these parts are to be taken. For example, in f the lower 
 number denotes that the integer is to be divided into seven 
 parts ; the upper one shows, however, that only three-sevenths 
 are to be taken. Therefore, when these two numbers arc 
 equal, a whole number is designated, as 11- When the upper 
 number is greater, it signifies more than a whole number; 
 when it is less, it signifies less than a whole number." 
 
 The second conception of the; fraction is well illustrated 
 from Trenchant : 1 " The teaching of fractions, of which we 
 have given the definition in Chapter 2, should follow division, 
 so as to follow the proper source from which it originates. 
 For this will happen most often when a smaller number is to 
 be divided by a larger ; or when it results from a division, as 
 24 divided by 60 makes f-f; or when it results from a divi- 
 sion, as 24 resulting from a division by 60 makes f^. The 
 24 is the numerator and 60 the denominator, and the fraction 
 is called twenty-four sixtieths." 
 
 The second form- of definition was used by Raets : 2 " Frac- 
 tions arise (as has been explained) from division of a number 
 by a greater number. For instance, when 2 is divided by 3, 
 then § results," and also by Tartaglia, who gave this illus- 
 tration : 3 " To divide 15 by 2. It will be impossible to divide 
 15 into two equal parts. After dividing there will be one 
 
 1 Trenchant, L'Arifchmetique (1578 ed.). "La doctrine <lu nobre rompu, 
 duquel auons donne la diffinition au 2 chap, doit fucceder a la diui- 
 f>ion, come fuyuant sa .propre fourfe dont il prent origine. Car iceluy 
 auient le plus fouuent quand Ion diuife vn moindre nombre, par vn maieur : 
 icomme 24 par 60, fet 2-i: ou quand il relte d'vne partition: comme 24 
 reftant d'vne partition par 60 fet |.£: le 24 eft numerateur: & le 60, 
 denominateur : & s'exprkne vint & quatre foixantiemes." Fol. G recto. 
 
 2 Raets, Arithmetica Oft Een niew cijfferboeck/ (1580). "Die ghebroken 
 ghetalen spruyten (als verclart is) wuter Diuision/ alsmen een ghetal 
 divideert met een grooter. Ghrlijck alsmen divideert 2. met 3. foo comen- 
 der %." Fol. Biiij verso. 
 
 3 Tartaglia, La Prima Parte Del General Trattato (1556 ed.). 
 
88 SIXTEENTH CENTURY ARITHMETIC 
 
 part left, which is still to be divided by the divisor. Then i 
 is taken for the numerator and 2 for the denominator of the 
 fraction." 
 
 Thus, the proper relation of the fraction to the process of 
 division was recognized, and the fraction was taught as a 
 necessary step in the growth of the number system. 
 
 The second method of approach to the fraction, though less 
 common than the first, persisted for two centuries. It is in- 
 teresting to compare Tartaglia's treatment quoted above with 
 the following from Gio (1689) more than a century and a 
 quarter later : 1 " When a divisor is greater than a dividend. 
 When it is necessary to divide a smaller dividend by a divisor, 
 place the divisor under the dividend, and, as no quotient is 
 obtained, a fraction is formed. Thus, if the divisor is 40 and 
 the dividend 20, place 40 below the 20, then the quotient |$ 
 results." 
 
 The degree to which these conceptions of the fraction were 
 held to be incompatible by some is exemplified in the work of 
 Unicorn, 2 who began with the first definition and ended a long 
 treatment with these cases of division, the last and least of 
 which is the second definition. 
 Division of fractions : 
 
 A fraction by a fraction. 
 
 An integer by a fraction. 
 
 A mixed number by a fraction. 
 
 A mixed number by a mixed number. 
 
 A fraction by an integer. 
 
 An integer by a mixed number. 
 
 A fraction by a mixed number. 
 
 A mixed number by an integer. 
 
 1 Gio, Padre, Elementi Arithmetic! (1689 ed.). Quando un Partitore 
 foffe moggiore de composto. Quanto s'haueffe da partire vn Compofto 
 minore del Partitore ; si mette il Partitore lotto il compolto, e ne viene di 
 Quotiente quel rotto che fi forma. Gome se foffe Partitore 40, Composto 
 20. Perche il 40. non puo entrare in 20., percio neffo 40. fotto il 20, resta 
 di Quotiente |^. 4o)|~g- Quotiente." Fol. Cviij verso. 
 
 2 Unicorn, De L'Arithmetica universale (1598 ed.). 
 
THE ESSENTIAL FEATURES 89 
 
 One integer by another; that is, in case the dividend is 
 smaller than' the divisor, as 48 by 64 gives $f . 
 
 A curious mixture of the two definitions is found in 
 Ramus : x " If 8 is divided by 3, the quotient is 2, and % are 
 left. 2 is the number of parts, 3 the name. 
 
 If I wish to divide 11 asses by 3, this division ty (3I) 
 will indicate 3 asses and % of 1 ass. 
 
 If the numerator and the denominator are equal, as ff£ 
 the number is an integer; if larger, as J-ff , it will be more 
 than an integer." 
 
 Both were happily combined by Tonstall, whose definition 
 is : 2 "Any integer may be divided into as many parts as one 
 wishes from 1 1 to infinity." Although Tonstall often goes to 
 extremes in details, his treatment of the relative size of frac- 
 tions should have a mission. One of the chief reasons why 
 ■children have difficulty in mastering fractions is that they do 
 not make a sufficient number of comparisons. They do not 
 ■observe the change in the fraction by varying its terms. Ton- 
 stall's comparisons are suggestive of good method : 3 " The 
 greater the denominator and the fewer of these parts there are, 
 the farther is the fraction removed from the integer; the 
 smaller the denominator and the more of these parts there are, 
 
 1 Ramus, Arifchmeticae Libri duo (1586 ed.). "Ut efto 8. dividendus 
 per 3. quotus integer est 2, 6* fuperfunt 2, que interpofita linea fupernotata 
 divisori, tandem ipfa quoqj divifa flint, inventaqi fraotio % quoto priori 2 
 ad <texitram est affcribenda fie: 
 
 2 
 
 3 " Fol. D 7 recto. 
 
 * Tonstall, De Arte Supputandi (1522 ed.). " Omne integrum in Partes, 
 quotcuq? velis: folui per intellects poteft. et quemadmodum integrorum 
 numeratio ab uno incipit : atq; in infinitum poteft extendi : fie integrorum 
 fectio a fecfldis orditur partibus." Fol. 4 recto. 
 
 3 Tonstall, De Arte Supputandi (1522 ed.). " Semper autem in omnibus 
 diffectorum minutijs, quo maior denominator fuerit: eo minores erunt 
 partes, remotioresq; ab integris. et quo fuerit minor: eo partes maiores 
 •erunt. atoj ad Integra propius accedet. Nam due partes fecundae maiores 
 funt : q duae tertif et duae tertiae maiores, q duae quartae et duae 
 .quarte maiores, q duae quintae. et fie in uniuerfum, quanto magis numer- 
 ado crefcit denominator : tanto magis quantitate partes diminuunter." Fol. 
 P recto. 
 
9 o SIXTEENTH CENTURY ARITHMETIC 
 
 the more nearly a fraction approaches an integer. For z 
 halves are greater than f, f than f , f than f, and so on. 
 
 " In dealing with fractions these things should be noticed : 
 
 First. Whenever the numerator and denominator are 
 equal, then the fraction equals an integer. Thus, f, i, \ are 
 equal to i. 
 
 Second. Whenever the numerator is greater than the 
 denominator, by as many units as the numerator exceeds the 
 denominator, the value of the fraction exceeds an integer, as in 
 i> h f > f is an integer and J /& over, and so on. 
 
 Third. By as many parts as the numerator is less than 
 the denominator in units, by so much is the fraction less than 
 an integer, J4 is one-fourth less than an integer." 
 
 Classes of Fractions 
 
 Two classes of fractions were recognized : 
 
 Common fractions, variously called numeri rotti, fragmenta,. 
 nombres rouptz, fractiones, 1 minutiae vulgares seu mercatoriae. 
 
 Sexagesimal fractions, called fractiones astronomicae, or 
 minutae phisicae. 2 
 
 Cirvelo, in the third part of his Practica Arithmetica (1555. 
 ed.), explained the operations with the sexagesimal fractions 
 by reference to those with denominations of weight and value. 
 The following shows his method of associating the addition 
 of signs, degrees, minutes, etc., by reference to the addition of 
 ducats, soldi, denarii, etc. 
 
 signa 
 
 gradus 
 
 minuta 
 
 secunda 
 
 tertia 
 
 5 
 
 47 
 
 39 
 
 S3 
 
 15 
 
 3 
 
 26 
 
 54 
 
 18 
 
 34 
 
 2 
 
 17 
 
 23 
 
 45 
 
 45 
 
 11 31 57 57 45 fua 
 
 1 The word "fractio" is as old as Hispalensis (c. 1150). Treutlein, Ab- 
 handlungen, 3 : 112. 
 
 a Planudes introduced sexagesimal fractions under the title " Zodiac " 
 and showed how to use them in the four operations. According to Sayce 
 and Bosanquet, the origin of these fractions is sometimes incorrectly attrib- 
 uted to the Assyrians. Publications of the Royal Asiatic Society, 1880, vol. 
 xl, no. 3. 
 
Auri 
 
 argenti 
 
 den. 
 
 ducati 
 
 solidi 
 
 
 12 
 
 23 
 
 9 
 
 8 
 
 16 
 
 7 
 
 23 
 
 14 
 
 11 
 
 THE ESSENTIAL FEATURES 91 
 
 Practica in Monetis. 
 oboli 
 
 4 
 3 
 5 
 
 53 25 40 fua Fol. b alj recto. 
 
 6 oboli = 1 denarius. 12 denarii = 1 solidus. 30 solidi = 1 ducatum. 
 
 A double entry multiplication table is also given by Cirvelo, 
 page 38, but it was not an invention of the sixteenth century, 
 for the Arabs had used this means of calculation 1 much earlier. 
 
 In German works that treated of line-reckoning, Roman 
 notation was occasionally used to express fractions. An ex- 
 cellent illustration showing the struggle of the Hindu and 
 Roman systems is the following from Kobel : 1 
 
 " The numerator 1 This symbol is one-oneth, 
 The separatrix — that is, the integer 1. 
 
 The denominator 1 
 
 " Then, whatever equal numbers are found in the numerator 
 and denominator, they always mean in such a symbol the in- 
 teger 1 ; for example, 4 fourths are 1 and written j-ja i n th e 
 German numerals and £ in figures. Similarly, 6 sixths is 
 also 1 and written ^j- or f . " The illustration on the next page 
 shows a part of a page from Kobel's arithmetic (1544). 
 
 1 Kobel, Zwey rechenbuchlin (1537 ed.). 
 
 " Der zeler I Difi figur ift vnd bedeut ein eintheyl/ das 
 
 Das ftrichlin — ift ein gantzs. 
 
 Der Nenner I 
 Damn in welcher zal du den Zeler vnd Nener gleioh findeft/ fo bedeutten 
 die felbigen figuren alwegen. ein gantze zal/ als vier viertheil/ die tnaohen 
 ein gantz/ vnd werden alio mit teutfoher zal gefiOhriben j^/ aber mit 
 den ziffern alio £ Dergleicheni ift 6. fechfteil audi ein gantzs/ vfi fctorei'b 
 es alio fj/ vnd mit den ziffem |.." Fol. H 4 recto. For examples from 
 other of Kobel's works see linger, pp. 15-16. 
 
9 2 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 fce&eifc bi$fw* fccr felbett ttyl aim • 
 
 I fcteffe fi'gur tfl vn bebetft <u'it (fertel %>Oit tint* 
 iflT B<*«0en/rtlfo mag itiait and) aftt fififfffofl/ffpt 
 
 fcd)(|rtO/rt(rt ftbcitwil CDerjtvat fcd}jftu'l2c- jwb «U* 
 
 VF 5D# few ©cd;6 aM/tae few fedjetoil tar 
 VHT «d;t ain qm% mcvfym . 
 
 IX fci£ #g»r bc^atgt <wit wewrt ayflfjtdfl fo*e fcyn 
 KT IX*tfltl/bcrX[.rttitgrtrtom<K^eii» 
 
 XX SD# #g»r bet5fltc|)et/$wert«3tg£dwuil&rey* 
 XXXI ft0* Mil /fo»e few 3weij$fg£ Mil *fcer dine* 
 iwt>rciffig£ attt g<m$ mac^eit * 
 
 II C 3D$ few nvocdfymbtxt idfl/fcer ^ter^nn# 
 IIllc.LX ben vnb fcdjBig? tin g<WB mri&ai ' 
 
 Many -writers followed their definition of fractions by an 
 explanation of numeration of fractions. This table from Van 
 der Schuere was designed to teach numeration to iV- * 
 
 1 
 1 
 
 a 
 
 2 
 
 a 
 
 3 
 
 4 
 4 
 
 6 
 
 6 
 
 7 
 
 7 
 
 a 
 
 8 
 
 2. 
 9 
 
 
 2 
 
 2 
 3 
 
 a 
 
 4 
 
 6 
 
 B 
 6 
 
 6 
 7 
 
 7 
 8 
 
 8 
 9 
 
 
 
 1 
 3 
 
 2 
 
 4 
 
 a 
 
 5 
 
 4 
 
 6 
 
 7 
 
 8 
 
 7 
 
 9 
 
 
 
 1 
 4 
 
 2 
 5 
 
 a 
 
 6 
 
 4 
 7 
 
 6 
 8 
 
 9 
 
 
 
 1 
 
 6 
 
 s. 
 
 6 
 
 a 
 
 7 
 
 8 
 
 •a 
 
 9 
 
 
 
 1 
 6 
 
 2 
 
 7 
 
 3 
 8 
 
 1 
 9 
 
 
 
 1 
 7 
 
 2 
 
 8 
 
 a 
 
 9 
 
 
 
 1 
 
 8 
 
 & 
 9 
 
 
 
 9 
 
 1 Van, der Scheure, Arithmetica, Oft Reken=const (1600 ed.)- Fol. D g 
 yerso. 
 
THE ESSENTIAL FEATURES 
 
 93 
 
 Order of Processes 
 
 The order of the processes with integers seems to have im- 
 pressed itself so generally upon' writers that it was usually fol- 
 lowed blindly in the case of fractions. The order of presenta- 
 tion) in fractions now prevalent in formal arithmetic was the 
 one commonly used in the sixteenth century, namely : i . De- 
 finition. 2. Reduction to lower terms. 3. Reduction to the 
 same denominator. 4. Aldd'ition. 5. Subtraction'. 6. Mul- 
 tiplication. 7. Division. 
 
 The following outline from Tartaglia is typical as to the or- 
 der of the main topics ; 1 the order and number of minor topics 
 differed with different authors : 
 
 ■Numeration, or representation of fractions. 
 Reduction of fractions to lower terms. 
 Changing mixed numbers to fractions. 
 Reduction to the same denominator. 
 Addition. 
 
 Of fractions. 
 
 Of fractions and mixed numbers. 
 
 Of fractions, whole and mixed number. 
 Subtraction. 
 
 Of fractions. 
 
 Of fractions from mixed numbers. 
 
 Of fractions from whole numbers. 
 
 Of whole numbers from mixed numbers. 
 
 Of mixed numbers from mixed numbers. 
 Multiplication. : 
 
 Of a fraction by a fraction. 
 
 Of a mixed number by a fraction, and conversely. 
 
 Of a fraction by a whole number. 
 
 Of mixed numbers by mixed numbers. 
 
 Of whole numbers by fractions. 
 
 Of a fraction by a fraction by a fraction. 
 
 Of a whole number by a mixed number by a mixed number. 
 Division. 
 
 Of a fraction by a fraction of the same denominator. 
 
 Of a fraction by a fraction of different denominator. 
 
 Of a whole number by a fraction. 
 
 Of a mixed number by a fraction. 
 
 Of a mixed number by a mixed number. 
 
 Of a whole number by a mixed number. 
 
 Of ia mixed number by a whole number. 
 
 1 Tartaglia, La Prima Parte Del General Trattato di numeri, et misure 
 (1556 ed.). 
 
94 SIXTEENTH CENTURY ARITHMETIC 
 
 But should the order of processes with fractions in formal 
 arithmetic necessarily be the same as that with integers ? It 
 is not followed in primary arithmetic, it does not conform 
 readily to spiral treatment, and, furthermore, historical pre- 
 cedent is not unanimous in its favor. Calandri ( 1491), Borgi 
 (1484), Paciuolo (1494), and Rudolff (1526) each gave 
 the following order : multiplication, addition, subtraction, and 
 division. Gemma Prisms introduced multiplication first and 
 repeated it after subtraction. The subject was introduced by 
 the definition of a fraction and its terms, and followed by the 
 explanation of a fraction 1 of a fraction. 1 "As f of $ of f, 
 that is, the integer is divided into 7 parts of which 6 are taken, 
 this is again divided into 3 parts, of which 2 are taken, and the 
 resulting fraction is divided into 4 parts, of which 3 are taken. 
 The fractions are most easily multiplied by multiplying the nu- 
 merators together for a new numerator and the denominators 
 together for a new denominator." Francesco Ghaligai pre- 
 ceded addition and subtraction by both multiplication and 
 division. 2 These presentations cannot be regarded as experi- 
 ments of minor authors, for Paciuolo, Gemma Frisius, and 
 Calandri were representatives of the highest scholarship and 
 were leading spirits of both the Latin and the Commercial 
 Schools. Neither can they be regarded as entirely arbitrary or 
 accidental plans, for Paciuolo-, who did more than any other 
 writer (save possibly Borgi) to formulate the arithmetic of 
 that period, actually expressed his purpose thus : 3 "In the ex- 
 
 1 Gemma Frisius, Arifchmetioae Practicae Methodus Facilis (1581 ed.). 
 " Item 1. 1 -| I -jj-, hoc est, tres quare a duarum tertiaru ex Tex feptimis : 
 hoc eft integri dduif i in 7, cape f ex particulas : quas rurf us f eca in tres : 
 faarum aocipe duos : quas diuide in quaituor : tandem tres huiufmodi fig- 
 nificatur particule." Fol. C recto. 
 
 2 'Ghaligai, Practica D'Arithmetica (1552 ed.). 
 
 " Paciuolo, Suma de Aritfametica Geometria Proportioni et Proportion- 
 alita (1523 ed.). 
 
 " De imuhiplicatione fractorum inter se. 
 
 Nelli fani el f ummare : fe ben notafti fo piu facile che nullaltra a 
 parte: el multiplicare fo piu difficile die lui. Qui nelli rotti asfai piu 
 facile : e <k> multiplicare ehe non e k> f ummare. E nelli fani la piu difficile 
 parte fo ed partire. E qui nelli rotti e la piu facile : che quella intendi di 
 
THE ESSENTIAL FEATURES 95 
 
 planation of reckoning with fractions, multiplication precedes 
 addition, for fractions and whole numbers are so different that 
 in whole numbers addition is more easily .treated, while in 
 fractions it is more difficult than multiplication and should 
 follow." 
 
 Reduction 
 The reduction 1 of fractions to lowest terms was effected in 
 simple cases by inspection'. Ini other cases it was necessary to 
 apply tests of divisibility to find the common factors of the 
 numerator and the denominator. Reduction to lowest terms 
 was the only method of simplifying fractions, since the decimal 
 fractions had not yet come into use ; consequently, the tests of 
 ■divisibility, and occasionally the Euclidean method of Great- 
 est Common Divisor, were given by theoretic writers before 
 fractions. For example, Ramus preceded his treatment of 
 fractions by five chapters, as follows : 1 
 
 Chapter VI. Concerning odd and even numbers. 
 
 From division arise different kinds of num- 
 bers : odd and even', prime and composite. 
 Chapter VII. Concerning prime and composite numbers, 
 in which the sieve of Eratosthenes is used. 
 Chapter VIII. Concerning numbers prime to one another. 
 Chapter IX. Concerning composite numbers and their 
 
 Greatest Common Divisor. 
 Chapter X. Concerning the Least Common Denomin- 
 
 ator. 
 
 fani. El fotrare in quelli ene asfai piu facile: .che non e el fotrare in li 
 rotti. E pero dal multiplicare me pare ben comenzare." Fol. 50 recto, Gij 
 xecto. 
 
 Reconde (1558 ed.) makes the same explanation. Fol. Riiij verso. 
 
 1 Ramus, Arithmeticae Libri duo (1586 ed.). 
 " Caput VI De Numero Impari et Pari. 
 
 E divilione oritur numeri differentia duplex, imparis & paris, primis & 
 <compofiti. Fol. C 5 recto. 
 
 Caput VII. De Numero Primo et Composito. Fol. C recto. 
 
 Caput VIII. De Numeris Primis inter Se. 
 
 Cap. IX. De Numeris Inter Se compofitis, eorumq; communi divi- 
 
 fore maximo. Fol. D 3 verso. 
 •Cap. X. De mirairno Communi Dividuo." Fol. D verso. 
 
96 SIXTEENTH CENTURY ARITHMETIC 
 
 Stevinus under division of integers states the use to be made 
 of the Greatest Common' Divisor ; namely, as an aid in the re- 
 duction of fractions. 1 It was more common, especially among- 
 commercial writers, to give the tests of divisibility under frac- 
 tions or in connection with the actual problem solved. Thus, 
 in Rudolff's second section in the treatment of fractions Re- 
 duction has this treatment : 2 The first example, solved by in- 
 spection, — both terms of 4i~i divided by 2 = T 4 ^-, which 
 divided by 3 = if, which divided by 7 = 7, — is followed 
 by the rules of divisibility : 
 
 " In order to be divisible by 2 a number must be even ; 
 
 " In order to be divisible by 3 it must have a remainder 
 of o, 3, or 6 after the nines have been cast out ; 
 
 " A number is divisible by 5, when the number ends in 
 5 or o; 
 
 " A number is divisible by 10, when the number ends in o."" 
 
 As an example of these tests he gives the one in the all ? 
 margin. Then follows the Euclidean method : 
 
 " How we may easily find whether a fraction may be 
 made smaller or not. jf^ 
 
 " Divide the larger number by the smaller. If any- 2 7 
 
 thing is left, divide the former divisor (the smaller 
 number of the fraction) by this, and so on. Then -^ 
 
 divide the larger number of the fraction, and, if the 
 division comes out even, the fraction is reducible. As, 
 
 243 
 7T5~ 
 
 715 
 
 *|4- 286 
 
 7 15 
 
 715 
 
 286 = 2, remainder 143. 
 143 = 2, no remainder. 
 143 = 5." 
 
 1 Stevin, Les Oeuvres Mathematiques (1634 ed.). "Eftant doncques. 
 donnez nombres Arithmeticques entiers, nous avons trouve leur plus grande 
 ■commune inefure; ce qu'il falloit faire. 
 
 Problem VI. 
 
 Estant donae nombre Arithmetique rompu : Trouver Ton premier rompu. 
 
 Explication du donne. Soit donne 'rompu yU^. Explication du requis. 
 II faut trouver fon premier rompu. Conftruction. On trouvera la plus 
 grande commune mefure de 91 & 117." Fol. B 5 recto. 
 
 2 Rudolff, Kunstliche rechnumg mitder Ziffer und Zalpfennige/ (1534 ed.). 
 
THE ESSENTIAL FEATURES gy 
 
 Another interesting example is found in Raets. 1 
 The term, " reduction of fractions " meant several things : 
 with some writers it meant reduction to lowest terms, with 
 others it meant changing fractions to others having a common 
 denominator, and with still others it meant expressing a frac- 
 tional part of a denominate number unit in terms of lower 
 orders. It is now customary for modern writers to precede 
 the treatment of common denominators by that of reduction 
 to lowest terms, but the influence of applied arithmetic, es- 
 pecially of denominate numbers, and the freedom 1 accompany- 
 ing dogmatic treatment sometimes led sixteenth century writ- 
 ers to reverse this order. Thus, Baker's second chapter is 
 Reduction of Fractions to Common Denominator, and his 
 third chapter is Reduction of Fractions to Lowest Terms. The 
 influence of denominate numbers upon the treatment of frac- 
 tions is apparent among both Commercial and Latin School 
 writers. Thus, Adam Riese follows his definition of a frac- 
 tion by : 2 
 
 To reduce Yn floren. 
 
 Solution. Multiply 3 by 21 and divide by 4, the result is 15 gr. and 9 <*. 
 
 1 Raets, Aribhmetioa Oft Eem niew Cijfferboeck/ (1580 ed.). 
 " Abbreviatie int ghebroken. 
 
 " Leert/ hoemen die gebroken ghetalen reduceren sal in een min- 144 
 
 der proportie. Het welcke gheschiet nae der leeringhe Euclidis/ 108 
 
 In der tweeder propohtien des feuenden boers der Elementen/ in 36 
 
 deser voegen. Ten exemplel: Om habhbeuieren 1^| fuibshaheert 72 
 
 den Felder 108. va den Kooimer 144. ende daer fullen .resteren 36. 36 
 
 die suibtrahert van 108. , 36 
 
 o 
 
 "Das voor J.-SJ unde pla-et se van 108. ledt 3. ende voor 144. fedt 
 
 4. Too comender # vor ylr'te meten/ die ^ doen loo veel als i||." Fol. 
 Bv recto. 
 
 2 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). 
 
 " Wil'tu -wiffen/ wie viel ein jglicher Bruch in fich behelt/ To refoluk 
 den Zeler in fenien werdt/ vnd teil ab mit dem Nenner. Alsi J4 floren/ 
 multiplicir 3 mit 21 gr./ vnd tiheil ab mit dem Nenner/ als 4/ kommen 15 
 gr./ vnd 9 g. Alfo dergleichen von Gewichlten vnd andern." Fol. D verso. 
 
q8 SIXTEENTH CENTURY ARITHMETIC 
 
 Gemma Frisius follows Reduction of Fractions to Lowest 
 Terms by : 
 
 " $ Joachimi or Thaleri are worth how many grossi?" 
 124 grossi = 1 Thaler. - = 10% grossi. 
 
 5= 8 numuli. 
 
 3 
 
 Ramus precedes his Chapter XIIII, 1 Operations with Frac- 
 tions, by a chapter, Reduction of Integers and Fractions, in 
 Which examples of this type occur : 
 
 To reduce 12 asses to uncias Jj^*. 
 
 To reduce -^ to an integer, divide 144 by 12. 
 
 Addition 
 The usual order in the addition of fractions was : ( 1 ) The 
 addition of fractions with denominators alike, (2) Of those 
 with denominators unlike. For example, the following prob- 
 lems are given by Noviomagus in this order : 2 
 
 (1) i and l and J = V 5 -. 
 
 (2) f . |. Multiply 7 by 3 = 2i, 5 by 4 = 20. 
 
 The sum = 41 for numerator and 7 X 4 = 28 for denominator. 
 
 5 3 41 T 1S 
 
 The second kind required reduction to a common denominator, 
 which process was treated either in a section previous to addi- 
 tion or in connection with the actual problems to be added. 
 The number of writers who followed each plan was about 
 the same. 
 
 A typical case of reducing to a common denominator is the 
 following from Trenchant : 3 To reduce f , f , f, | to a 
 common denominator: 
 16 18 20 21 Tb e common denominator is written 
 
 2 . . 7 below, and the new numerators are 
 
 3 4 6 8~ written above the corresponding frac- 
 
 2 4 tions. 
 
 1 Ramus, ArMimeticae Libri duo (1577 ed.), fol. 4 recto; fol. D g recto. 
 
 s Noviomagus, De Nutneris Libri II (1544 ed.). 
 
 3 Trenchant, L'Airithmetique (1578 ed.), fol. G verso. 
 
THE ESSENTIAL FEATURES 
 
 99 
 
 The usual form of expressing the work of addition is shown 
 in this example from Rudolff : 1 
 8 g The fractions to be added, f and |, 
 
 2 3 I7 appear in the center, the common de- 
 
 3 4 ' " 12 nominator, 12, below, and the new 
 12 numerators, 8 and 9, above. 
 
 A clever formi of adding fractions, considering the fact that 
 signs of operations were not in use, is illustrated by the fol- 
 lowing from Tartaglia : 2 
 
 2 3 g The cross indicates how the terms 
 
 34 8 are multiplied, but it probably has 
 
 fanno ^- die saria 1— n ° coinnec,t ioo witn the symbol of 
 
 12 I2 multiplication. 
 
 So rarely were reasons given for processes that the follow- 
 ing explanation by Rudolff of why fractions must be reduced 
 before adding is noteworthy : 3 "It is impossible to add 4 
 florins and 3 soldi to make either 7 florins or 7 soldi. It is 
 also impossible to add f and \ and to get either £ or f. 
 The florins must be reduced to lower denomination', and the 
 fractions must also be reduced to a common denominator. 
 
 ■8 1 ftr li 8 m „U~ 2 3 " 
 
 "8" t or tit ¥ir make ^7. 
 
 The order, 4 or the method of grouping, used when several 
 fractions were added was not uniform. The prevailing usage 
 was to add the first fraction to the second, then to add this re- 
 sult to the third and this result to the fourth, and so on. 5 
 
 1 Rudolff, Kunstliche Redlining (1534 ed.), fol. Cvii verso. 
 
 2 Tartaglia, La Poma Parte Del General Trattato (1556 ed.), fol. Tiij 
 verso. 
 
 3 Rudolff, Kunstliohe Rechnung, fol. Di recto. 
 
 4 Baker, The Well Spring of Sciences (1580 ed.), adds -|, f, £, f by 
 adding -J and £, then -| and ^, combining the results. Fol. l v verso. 
 
 He also direots haw to add a fraction of a fraction to a fraction of a 
 fraction' by making the multiplications first and then adding the results. 
 This has significance in showing that Baker gave multiplication precedence 
 over addition, as is done in the modern convention concerning a series of 
 operations. 
 
 5 Riese, Rechnung auff der Linien und Federn/ (1571 ed.), adds J, |, 
 and A thus : §• and |- are i| and -| = 2l|-. " 'Summir die erfte zween 
 brtiche/ als nemlich/ g vnd f/ werden i-|/ darzu A/ kommen 2i| teil." 
 Fol. Dij recto. 
 
IOO SIXTEENTH CENTURY ARITHMETIC 
 
 Some writers, as Cardan, 1 added in groups and combined the 
 results according - to the associative law. The modern method 
 of reducing 1 all of the fractions to a common' denominator and 
 adding their numerators was also recognized. 2 Tonstall ft 
 combined the first and the third methods. In adding £, f, 
 and | he suggested that the first two be added and that the 
 third be added to this result, as : 
 
 | + £ = H and H X 4 = Vtr-. or that they may be taken as 
 
 i = i$,i = H,i + U- The sum is W- 
 When the modern method was used, the product of the sev- 
 eral denominators was often taken instead of their least com- 
 mon multiple. The following problem from Raets shows this : 4 
 To reduce f , $, f . 
 
 4 
 
 160 
 
 8 
 
 7 
 
 32 
 
 
 5 
 
 3 
 
 
 W0(14O 
 
 160 
 
 ffl$ 
 
 3 
 
 
 
 160 
 
 #0(120 
 
 4 
 
 m 
 
 
 
 
 1 
 
 
 W128 
 
 1 Cardan, Practica Arithmetice (1539 ed.). "Exemplu vok> agregare J, 
 ■|, -J, -J agrego p modu dictu Prima duo & fatiut i-| & reliq duo & fatiut 
 ||, deinde agrego .11 & ♦» & fifit 1 a ° | S 9 8 - & sut integri tres & -^ & hoc 
 eft facile.'' Fol. Bii recto. 
 
 2 It was not customary, however, to use the least common -denominator. 
 
 3 Tonstall, De Arte Supputandi (1522 ed.). "Si vero plura fuerunt 
 fffagrneta: uti duae tertie. tres quartae. quatuor quintae. poft duo priora 
 fragmemta, ficuti diximus, redticta, iterum denominator comunis prius in- 
 ueftigatus per tertij fragmenti denominaforem multiplicetur : et f urgent 
 fexaginta, omnium denominator communis. % i i *TT * 6 
 
 * Should be 17 in the original, which shows that the fractions were added. 
 
 "Qj si fcire oupis : quot partes fexagefirnae Tint in quouis fragmento- 
 numeratoram ipfius fragmenti in denoaninatorem comunemi multiplies: 
 nempe fexaginta: numerumoj iprocrea'tu diuide per eiufdem fragmenti de- 
 nominatorem. Ita deprehendes in duabus tertij s quadraginta fexagefimas 
 Q |_o et in tribus quartis quadraginta quinq; . fexagefimas | M- 
 et in quatuor quitis quadraginta octo fexigefimas." Fol. P 3 verso. 
 
 * Raets, Arithmetica Oft Een niew Cijfferboeck (1380 ed.), fol. Bvi verso, 
 to Bviii reoto. 
 
W(40 
 & 3 
 
 W(20 
 
 W 7 
 
 W(82 
 «» 4 
 
 120 
 5 
 8 
 
 140 
 4 
 5 
 
 128 
 4 
 8 
 
 40 
 3 
 
 20 
 
 7 
 
 32 
 
 4 
 
 THE ESSENTIAL FEATURES ioi 
 
 Therefore, the 'fractions become f£#, ££■$, f|f • 
 2. 
 
 3. 
 
 120 140 128 
 
 This problem has an additional interest on account of its three 
 solutions. In the first the common denominator is multiplied 
 by each numerator, and the result is divided by the correspond- 
 ing denominator. This is evidently the longest method. In 
 the second the common denominator is divided by each de- 
 nominator, and the results multiplied by the corresponding nu- 
 merators. In the third the product of two denominators is 
 multiplied by the numerator of the third fraction; this is the 
 shortest process. 
 
 Mixed numbers were added in two ways : ( i ) By adding the 
 integers and fractions separately and combining the results. 
 (2) By reducing the mixed numbers to improper fractions and 
 adding. The following example from Rudolff * shows a com- 
 mon form for arranging the work in adding by the first 
 method : 
 
 13J 3 8 10 6 
 
 n 
 12? iif* 
 
 19} 
 
 53£ 12 12 
 
 The following shows the work of adding mixed numbers by 
 the use of the second method : 2 
 
 1 Rudolff, Kunstliche reohnung (1534 ed.), fol. Dii recto. 
 
 2 Ciaccihi, Regole Generali D'Abbaco (1675 ed.). " Del secondo modo di 
 sommare interi, e rotti. Supponga fi per efempio, ohe vno fi trouaffe debi- 
 tx>re d'vn altro di diuerfe fomme di danari, cioe di lire 16=-, di lir. 4^, 
 di lir. Us-, di lir. 25^, di lir. 6£-, di lir. 15.7.12 firm, e di lir. 4.5.12 fimi, 
 si domando di quanta f i douera afcriuere in vna fola partita ; Si ponghino 
 
102 
 
 iixian^in 
 
 cc/Vi uki 
 
 ' AKlit 
 
 iM2.UL 
 
 
 16* 
 
 4| Hi 
 
 25f 
 
 6$ 
 
 IS* 
 
 ^ 
 
 il 
 
 1 2 9. 
 
 4 
 
 A 
 
 A 
 
 £ 
 
 ¥ 
 
 ¥ ¥ 
 
 Somma prima 
 
 seconda 
 
 terza 
 
 quarta 
 
 quinta 
 
 seJJta 
 
 settima 
 
 594 
 168 
 405 
 930 
 
 232 
 561 
 159 
 
 ¥ 
 
 W 
 
 H 
 
 3049 36] 84.13.10.% 
 
 The top row contains the numbers to be added. The third 
 row is composed of the same numbers reduced to improper 
 fractions. The middle row contains numbers of which the 
 numerator of each fraction is to be multiplied to give an equal 
 fraction with the denominator 36. The column which fol- 
 lows is the sum of these numerators. The result is expressed 
 in lira, soldi, and denarii. 
 
 Under addition and other processes with fractions there 
 generally were included problems in denominate numbers. 1 
 
 Subtraction 
 
 The order in subtraction was naturally the same as that in 
 addition. That is, the subtraction of fractions ( 1 ) with the 
 
 le fomme per ordine, e fi reduehino gl' interi a quella parte, con la quale fi 
 trouano copulati, e s'offerui il modo notato nell' antecedente." Fol. D 
 recto. 
 
 1 This problem from Trenchant (L'Arithmetique, 1578 ed.) will illus- 
 trate: 
 
 The fractions represent parts of a lira, 
 a money denomination, the second col- 
 umn contains the equivalent amounts 
 expressed- in the lower denominations, 
 soldi and denarii. 
 
 }, 
 
 10 f s 
 
 § 
 
 13 4 
 
 i 
 
 5 
 
 * 
 
 16 8 
 
 ft 
 
 7 6 
 
 & 
 
 11 8 
 
 3& 
 
 3 1. 4 r. 2 s 
 
THE ESSENTIAL FEATURES 103. 
 
 like denominators, (2) with different denominators. Baker, 1 
 Gemma Frisius, 2 and Tonstall 3 emphasized the subtraction of 
 a fraction' from an integer, performing the process in two 
 ways : By detaching one from the integer and subtracting the 
 fraction from that, and by reducing the integer to an improper 
 fraction of the same denominator as the given fraction and 
 then subtracting. 
 
 The subtraction of mixed numbers was accomplished in 
 two ways : by subtracting the fractions separately, adding one 
 to the minuend when necessary, and by reducing both minuend 
 and subtrahend to improper fractions. Theorists like Ton- 
 stall * who were fond of extreme classification gave three cases 
 under mixed numbers : 
 
 1. Subtraction of a fraction from a mixed number. 
 
 2. Subtraction of an integer from a mixed number. • 
 
 3. Subtraction of a mixed number from a mixed number. 
 The placing of ( 1 ) before (2) in this list is an example of the 
 illogical order that characterized even the work of careful 
 writers of this period. 
 
 1 Baker, The Well Spring of Sciences (1580 ed.), fol. Ki recto, Kii recto. 
 
 2 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.), 
 fol. Cv verso. 
 
 » Tonstall, De Arte Supputandi (1522 ed.). To subtract f from 12, 
 take 1 from 12 which leaves 11. 1 = -f and fy — A = f. Hence, 12 — 
 
 * = "f 
 
 Or, 12 may be reduced to sevenths, making 4*.. 3*. — * = ^ lit. 
 
 " Quando minutiae fubducentur ab irategris : f uffecerit eas ab uno integ.ro, 
 
 in minutias foluto, fubducere. et quod tarn de integris q de minutijs ref- 
 
 tabit: totius fubductionis erit reliquum. Veluti fi | fubtrahendae funt 
 
 a 12. fumamus .1. de .12. et reftabunt .11. ab illo autem uno demptis £, 
 
 reliquentur |. que copulatae cum. 11. reftare faciunt .nf. tantum 
 
 fupereft: fi A a. 12. fubducimus. Alij integra minutiarum more, fupra 
 
 lineam notant: cui unitatem subijciunt, ad integra defignanda. Deinde 
 
 quafi minutiae a minutijs subducendae ef f ent : pofit obliquam numeratorum 
 
 in denominatores multiplkaitionem, minoram productum a maiore fub- 
 
 doicunt: et fupra. lineami notant. cui denotninatorem fubdunt. ita. 7. in 12. 
 
 ducta creant .84. et. 4 in .1. ducta faciunt .4. que fubduota ab .84. reliquunt 
 
 80 Ea, fi reducas ad integra: fient .11$. ita res ad idem recidet. | X 
 
 8. 8 0" Fol. S recto. Riese (1571 ed.) used this plan, fol. Dii verso. 
 
 * Tonstall, De Arte Supputandi (1522), fol. S 2 recto. 
 
104 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 Among the works which correlate fractions and denominate 
 numbers, it would be difficult to find a better specimen than that 
 of Wencelaus. This is an example under subtraction of frac- 
 tions : 1 "A silversmith had an amount of silver weighing 1 5 
 marc if once, he wished to make from it an article weighing 
 8 marc 5$ once. The question is: How much silver is left?" 
 
 Multiplication 
 
 The multiplication of fractions was generally based upon the 
 definition of a fraction'. This statement, often included in the 
 general definition of a fraction, gave the common rule for 
 forming the product of two fractions. That the multiplication 
 of one fraction by another does not require special emphasis 
 was observed by Champenois : 2 " Fractions of fractions occur 
 more rarely than the others (simple fractions), as two-thirds 
 of three-fourths is written thus: f of |. Likewise, three- 
 fifths of a half, f of I. Also, a half of two-thirds is \ of f . 
 Two-thirds of three-fourths of five-sevenths is f of f of f." 
 Calandri, who placed multiplication of fractions before addi- 
 tion, introduced the subject by finding the product of an in- 
 teger and a fraction. 3 Though complicated by the use of 
 
 1 Wencelaus, T'Fondament Van Arithmetica (1599 ed.). 
 
 Item >eenen Silversmit faeeft een Item vn Orsebure a vn masse 
 
 Masse Silvers van 15. Marc/ inde dargent de 15. marc. & IJ|. d'once, 
 
 1. &. once/ daer wt wil hy een il en veut faire vn ouurage de 8. 
 
 ■weeck toerichten van 8. marc, ende marc. 5%. once, la demande est, 
 
 5. i. oncen. De vraghe is : Hoe marc. Sj. once, la demande est, 
 
 veel Silvers restar nooh? P. 82. 82. 
 
 2 Champenois, Les Institvtions De L'Arithmetique (1578 ed.). "Les 
 Fractions de Fractions aduinnet plus xarement que les autres, & f efcriuent 
 par plufieurs fimples minutes, comme deux tiers de troisi quarts ainfi fig- 
 ure, |- de £. Item trois einquiefmes d'vn demy, -| de A. Plus vn demy de 
 deux tiers ^ de %. Dauantage deux tiers de trois quarts de cinq feptiemes 
 f de I de f " P. 88, fol. Giiij verso. 
 
 3l Calandri, Arithmetica (1491 ed.). The problem is: "A man gained 
 45 y IS P 4 6 in one year, what will he gain in 23 years 3-| months ?" 
 
 Original. "Lhuomo guadagna lanno 45 ^ 15 /8 4 $ che ghuadagnera 
 in 23 anni 3 mefi £." (See next page for the solution.) 
 
THE ESSENTIAL FEATURES 
 
 I05 
 
 denominate numbers it amounts to finding 3 /i of 45, 15, and 4. 
 This is the appropriate phase with which to begin the multipli- 
 cation of fractions and, as such, should precede formal work 
 in addition and subtraction. Such a plan would be an im- 
 provement in modern arithmetic. The directions for forming 
 the products of two fractions were the same as those in present 
 use : Take the product of the numerators for a new numerator 
 and the product of the denominators for a new denomina- 
 tor. 1 Thus, f met f comt ■&. Mixed numbers were usu- 
 ally reduced to improper fractions, as in : 2 
 
 3* 6 i 
 
 met 
 
 19 55 
 
 A rare representation of the product of three fractions was 
 given by Champenois : 3 I wish to reduce a half of two-thirds 
 
 23 3f 
 
 He solves the prob 
 
 lern thus : 
 
 T r* 1 
 
 45 ■ 
 
 *5 4 
 
 
 3 16 3i 
 
 
 IS 3 
 
 
 920 | 12 
 
 
 "5 1 5 
 
 
 17 5 
 
 
 7 8 
 
 
 9 
 
 
 2 8 
 
 
 9 
 
 
 1 
 
 
 2 s — 
 
 
 9 
 
 1066 y. 70 3d 
 
 He •will gain in the time stated above if 1066 P 7 s 3- 
 "Guadagnera nel fopra decto tempo difopra if. 1066 P > d 3 a pl'd." 
 Pol. dvi recto, fol. 26 r. Another example from Calandri is: "El cogno 
 de uino vale 37 if 15 P 8 <S die uarranno 1 6 'cognia 5 barili 2 / 3 . Fol. d rt 
 recto. 
 
 iVan der Scheure, Arithmetica (1600 ed.), fol. E ? recto. 
 
 2 Van der Scheure, Arithmetica (1600 ed.), fol. E 7 verso. 
 
 3 Champenois, Les Institutions De L'Arithmetique (1578 ed.). "Ie veux 
 4. Ie multiplie le premier Numerateur 1. par le secod Numerateur 2. Ie 
 reduire vn demy de deux tiers de fix feptiefmes ainfi figure £ de § de 
 jiduicl done 2. que ie nuultiplie par le troifiefme Numerateur 6. le prodoiicT: 
 
io6 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 of six-sevenths, represented in figures thus : \ of f of f . I 
 multiply the first numerator, i, by the second numerator, 2, the 
 product is 2, which I multiply by the third numerator, 6. This 
 gives the product, 12, for the numerator of the reduction. 
 
 " I then multiply the first denominator, 2, by the second, 3,. 
 and obtain 6 as the product, which I multiply by the third 
 denominator, 7, giving a product 42 for the denominator of the 
 reduction. Then \ of f of f is reduced to one fraction,, 
 making -Jf, or f." 
 
 The meaning of the above multiplication was shown gra- 
 phically by the following diagram : 1 
 
 p 
 
 
 Q 
 
 
 
 
 
 
 M 
 
 
 N 
 
 O 
 
 
 m 
 
 
 n 
 
 
 
 
 i k 
 
 I 
 
 
 a 
 
 c 
 
 d 
 
 e 
 
 f 
 
 B 
 
 h 
 
 
 a 
 
 c 
 
 d 
 
 e 
 
 f 
 
 E 
 
 h 
 
 b 
 
 The base line, ab, represents a unit and ah is f of it. Then 
 af is § of f , and ad is \ of f of $ . 
 
 Trenchant 2 gave a diagram, which actually demonstrated 
 
 dbime 12. pour le N-umenateur de la reduction. Apres ie multiplie le pre- 
 mier Denominateur 2. par le fecoind 3. le produict donne 6. que ie multiplie 
 par le troifiefme Denominateur 7. le produict done 42. pour le Denominateur 
 de la reduction. Parquoy £ de § de & font reduicts en Vne Fractio fcauoir 
 J|-iemes, ou -|ieme." Page 89, fol. Gv recto. 
 
 1 Champenois, Les Institutions De L'Arithmietique (1578 ed.), p. 90, fol. 
 Gv verso. 
 
 2 Trenchant, L'Arithmetique (1578 ed.). 
 
 MULTIPLICATION OF FRACTIONS. 
 
 " Pour en auoir demonstration plus fuffifante : veyez 
 cete jfuperfice ou quarre A, B, qui eft le produit die la 
 ligne A, C : multiplie en Toy meme, laquelle faut entendre 
 vn entier diuife en 5. Or eft il certain que 1 qu'elle rep- 
 refente multiplie em foy fet 1, denotaot le total quarre A, 
 B : mais le 1 d'icelle, fcauoir eft, D, C, en foy multiplee, 
 
 
THE ESSENTIAL FEATURES 107 
 
 the truth of the multiplication, a thing unique among the treat- 
 ments examined. Thus he demonstrated that £ X i = tV> 
 t X 5 = tV> tV X tV = ts, and so on. It is evident that 
 the products of other fractions may be found from similar dia- 
 grams. The applications of multiplication involving denomin- 
 ate numbers were of the type, — the product of an integer and 
 a fraction, — as illustrated by this problem from Kobel. 1 " 76 
 persons have 4698 gulden to be divided among them. The 
 question is: 'How much belongs to each?' Solving by the 
 Rule of Three, one finds that the share of each is 71 If 
 guldens. Reduce further this fraction and all resulting frac- 
 tions according to the method described in the rule above and 
 according to the example which follows, until the lowest de- 
 nomination and the lowest fraction: of that denomination is 
 reached." 
 
 Division 
 The prevalent order in the treatment of the division of frac- 
 tions was ( 1 ) to divide the numerators of fractions having a 
 common denominator, (2) to reduce to a common denomin- 
 ator and divide the numerators, (3) to multiply crosswise the 
 numerator of each fraction by the denominator of the other, 
 (4) to invert the divisor and multiply. The third method is 
 not so frequently used as the first and second, and the fourth 
 is very rare. An excellent example of the fourth method is 
 From Thierf eldern : 2 " When the denominators are different, 
 
 rae fet que le quarreau E : qui n'eft que le -^ du total & enltier quarre 
 A, B, qui en contient 25 femblables. Parquoy apert que 1 par A multi- 
 plie, ne fet que -fo d'entier : comme auffi § par £ ne fet que -^ ; & § par 
 
 I font TS- 
 
 " Semblablemeot fe peut .demontrer par vne autre fuperfice, ayant 4 de 
 
 long, & 3 de large: que £ par \, fet ^; & § par f, fet ^: & ainfi des 
 
 autres, ce qui nous a femble bon de declarer en paffant" Fol. H 3 recto. 
 
 1 Kobel, Zwey rechenbuchlin (iS37 ed.). "Auff das nim difi Exempel. 
 76. perfonen haben vnder ficb zu teylen 4698. gulden. 1st die frag/ 
 ■was vfi wie vil geburt jr jeglichem. Machs nach der Regel de Tri/ fo 
 findet rich das einem geburt 71. guide/ und -|-| eines guldens. Dife vnd 1 
 all ander ibriich reducier/ minder/ und bring fie alfo nach aufi weifung 
 obgefchribner regel vnd Exempel/ wie naohuolgt/ in den kleyneren bruch 
 Oder in das kleyner theyl." Fol. H g verso. 
 
 2 Thierfeldern, Arithmetica Oder Rechenbuch Auff den Linien vnd Ziffer/ 
 
108 SIXTEENTH CENTURY ARITHMETIC 
 
 invert the divisor (which you are to place at the right) and 
 multiply the numbers above (new numerators) together and 
 the numbers below (new denominators) together, then you 
 have the correct result. As, to divide f by f, invert thus 
 I X f = If = ii" Thierfeldern used cancellation to 
 simplify the work, as shown in the ex- 1 3 
 
 7 go 
 
 ample x at the right. This was rarely ^ mit Ti facit lf 
 
 done in the works of that time. (4) (2) 
 
 Teachers often complain that their pupils do not readily pass 
 to the multiplication 1 by the reciprocal of the divisor after they 
 have begun by using the common denominator method. The 
 crosswise method of the sixteenth century forms a connecting 
 link between these two plans, and it is possible that it could 
 be used to advantage in making the transition in teaching. 
 The relation is shown in this example from Baker. 2 The 
 divisor was generally written first (which would 9 
 
 not be done now) and the terms of the result above ixf 
 and below. If these fractions were changed to 
 fractions with a common denominator, 12, the numerator 
 would be found by multiplying the same numbers that are 
 multiplied in the above work ; and, since the result is the quo- 
 tient of these numerators, Baker's process is the same as the 
 one in which the common denominator is used, only the 
 changed fractions are not in evidence. 
 
 The denominate number problems following division were 
 of this type : 3 "At Breslau a man buys 3 sacks of wool weigh- 
 ing 14 stein, 12 stein, and 15 stein. How many centners is 
 this, if one centner is equal to 5^4 stein, and a stein is equal 
 to 24 lb. ? Ans. 7 ceh 2. stein 2 lb. 
 
 (1578 ed.). "Da aber die Nenner vngleich/ £o kehre allzeit den Theyler 
 
 (welchen :du zur rechten Hand fetzen folt) vmb/ vnd multiplicir darnacb 
 
 die obern vnd vndern mit einander/ fo haft du es verricht/ Als : 4 in 4. 
 
 stehet vmb gekehrt also : 
 
 24 I 
 4. in -| r il das Facit." Pages 64 and 65. 
 
 1 Thierfeldern, Arithmetica (1578 ed.), page 63. 
 
 2 Baker, The Well Spring of Sciences (1580 ed.), fol. Kviii recto. 
 
 3 Rudolff, Kunstliche rechnung mit der Ziffern und mit den Zalpfennige/ 
 (1534 ed.). 
 
THE ESSENTIAL FEATURES IOO 
 
 " 12 Nuremberg pfennings equal what part of a pound? 
 Ans. f ." 
 
 " i A f 6 is what part of i L ? Ans. -^ L." 
 Doubling and halving were often included under fractions 
 with the same meaning as under integers and with as little 
 use for independent existence. Baker, whose work is char- 
 acterized by minute classification, devotes the eighth chapter 
 in his book to duplation (doubling), triplation, and quadrupla- 
 tion of fractions. He gives these examples : 1 
 
 (i) Duplation. To double any fraction, divide by 4. 
 
 6 
 To double f . | X f . 
 
 8 
 (2) Triplation. To triple |, divide £ by $=■§■. 
 
 Proofs for the operations with fractions were far less nu- 
 merous than for those with integers. One would expect this 
 to be so. The work with fractions, having to do with small 
 numbers, had usually been done mentally, without the use of 
 the abacus, and so without proofs. Hence there were no 
 proofs to carry over into figure reckoning, as there were in the 
 case of integers. Only one-eighth of the writers gave proofs 
 in fractions and commonly placed them at the close of the 
 treatment of each operation, or in a list at the end of the 
 chapter. 
 
 This table from Van der Scheure shows that each operation 
 may be proved by applying the inverse operation. 2 
 
 Soubstractio 
 Additio 
 Divisio 
 Multiplicatio 
 
 f Additio 
 De proeve J Substractio 
 van I Multiplicatio 
 
 (_ Divisio 
 
 Commercial arithmetics, especially those which did not have 
 denominate numbers under each operation with fractions, con- 
 tained a section following the operations called the " Rule of 
 Three with Fractions." Van der Scheure opens this section 
 
 1 Baker, The Well Spring of Sciences (1580 ed.), fol. Liii verso, Liiii 
 verso. 
 
 2 Van 'der Scheure, Arithmetica (1600 ed.), fol. E 8 recto. 
 
HO SIXTEENTH CENTURY ARITHMETIC 
 
 containing forty-one problems thus : * Having learned to use 
 the species understandingly, do not fail to heed the advice to 
 turn your thoughts quickly to the Rule of Three with Fractions. 
 
 The following serve to illustrate (the application of this 
 rule : 2 
 
 " If a centner of anything costs 9% florins, how much does 
 a pound cost?" 
 
 " If 45 ells of cloth cost 13 H 17 gr., how much do 7 ells 
 cost ?" Ans. 2 U 3 gr. 1 s o % hi'. 
 
 " 4^2 braza are worth 17 sol., what will 8 braza be worth? 3 
 
 " If a centner of wax is worth 13% florins, how much are 
 I7J4 ft. worth?" 
 
 PROGRESSIONS 
 
 Among those arithmeticians on whom tradition worked its 
 spell, none failed to give progressions (Arithmetic, Geometric 
 and often Harmonic) a place. This subject appealed to both 
 the classical and the modern scholars of that time, since it was 
 persistent in the few classical works that survived and in the 
 more recent acquisitions from the Hindus. Theoretical writ- 
 ers did not attempt to justify the presence of Progressions in 
 their text-books. That they were in existence was deemed 
 sufficient reason why all educated persons should know of 
 them. 
 
 Although their position in the list of species was variable, 
 the favorite place for them was after the division of whole 
 
 1 Van der Scheure, Arithmetica (1600 ed.). 
 
 " Die Special gheleert. 
 Zijn nu met claer bediea, 
 Das laet den moet niet v&llen, 
 Maer snel u sinnen keert 
 Toto den Reghel van Drien, 
 Ghebroken in ghetallen." Fol. E 8 verso. 
 
 2 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). 
 
 "Item/ Ein Centener fur. 9 floren/ vnd 1 orth/ Wie kom.pt ein pfund?" 
 Fol. Dv recto. 
 
 "Item/ 45 Ella tuchs fur 13 H/ 17 gr/ "Wis kommen 7 elln? Facit 2 A/ 
 3 gr/ 1 8/ o hi'/ y 3 ." Fol. Dvi recto. 
 
 3 Borgi, Arithmetica (1540 ed.). 
 
 C E Xel te fuffe detto fe braza .4^. de tela val. fol. 17. che vaJera braza 
 .8. Fol. E. recto. 
 
THE ESSENTIAL FEA TURES t z j 
 
 numbers. 1 In some cases they followed proportion, in others 
 roots. 2 Kobel and Van der Scheure reserved the subject for 
 the latter part of their works, after practical arithmetic had 
 been completed, and Noviomagus deferred it to his second 
 book, entitled Liber Secundus Arithmeticae, qui est de numer- 
 orurn Theorematis. 3 
 
 The usual treatment consists in defining the series and in 
 giving the rules for the last term and for the sum of a specific 
 series. No proofs for these rules were attempted in arithmetic. 
 This treatment of arithmetical progression from Tonstall * is 
 more elaborate than those commonly given: 
 
 "Arithmetic progression is the collection, into one whole, of 
 1 Riese, Baker, Tonstall, and Buteo. 2 Finaeus. 
 
 3 Noviomagus, De Numeris Lrbri II. 
 
 4 Tonstall, De Arte Supputandi (1522 ed.). " Progressio Arithmetica 
 ■eft numeroru inter fe equoliter difbantium in unain fummam collectio: 
 
 - Eius autem duae funt fpecies. Altera eft: in qua naturali numer- 
 orum ferie feruata, numerus quilibet fequens Tola unitaite praecedentem 
 fuperat: ficut in 1 hoc exemplo. 1.2.3.4.5. 6.7.8.9. Altera in qua numeros 
 •quoflibet omittenjtes, et pari'a feruantes interuialla, longam numeroruim 
 feriemi connectimus. velutei. 1.3.5.7.9.11.13. -." Fol. M ± verso. 
 
 " Ita net : vt numerus ex hoc productus fummam omnifi commonftret. 
 veluti in hoc exemplo .1.2.3.4.5.6.7.8. primus numerus .1. ad poftremum .8. 
 addatur at fleet .9. Cumqj in tota ferie lint .8. loca. ducamus .9. in 4, 
 corum dimidiu. et prodibunt .36. quae omnium eft f unima. -." Fol. M. x 
 recto. 
 
 "Q7 -fi numeroru a fe equaliter diftantium atqj ordine continuo difpofi- 
 torum feries erit impair: tunc numerus indicans, quot loca funt in ferie, 
 non in eum numerum ducatur : qui indicat quotus locus eft in ferie medius, 
 fed in eum numerum qui in ferie medius reperitur: atqj ab utroqj extremo 
 aequaliter diftat Ita numerus procreatus omnium fumma patefaciet. ficuti 
 in hoc exemplo .1.2.3.4.5.6.7. quia loca feriei funt .7. et medius numerus eft 
 .4.7. in .4. duoamus: et fient .28. quae fuma eft uniuerforum. Itidem fi 
 exempli caufa fumantur. 1.4.7.10.13. quia loci ferie funt .5. et medius niuimer' 
 eft .7. 5. in .7. ducamus: et fient 35. quae fumma eft omnium." Fol. M s 
 recto. 
 
 " In omini progref fione Arithmetica, fine feries par., f iue impar fuerit : 
 numerus ab extremorum additione collectus in numerfi indicatem, quot loca 
 funt in ferie, imultiplicetur. numerusq) productus postea dimidietur. et 
 fumma progreffionis' habebitur. Exemplum in ferie pari. 1.3.5.7.9.11. pri- 
 mus numerus additus ad postremu facit .12. et quia .6. loca feriei funt .12. 
 per .6. multiplicemus. et f urgent .72. quae fi dknidientur : fient. 36. quae 
 fumma eft progreffionis." Fol. M 2 verso. 
 
112 SIXTEENTH CENTURY ARITHMETIC 
 
 numbers equally distant from one another. There are two 
 kinds, one is the natural number series, i, 2, 3, 4, 5, 6, 7, 
 where the numbers differ only by 1 . The other in which any 
 number of terms of natural series is regularly omitted, as 
 
 x > 3. 5. 7, 9> l x > x 3- The sum of the serie s T , 2, 3, 4, 5> 6, 7, 8 
 is found by adding 1 and 8, the first and last terms, and mul- 
 tiplying this result by half the number of terms, ■§, or 4. It 
 gives 36, which is the sum of the series. Similarly in 1, 3, 5, 
 7, 9, 11; (1 + 11) f = 123 = 36. In the case of an odd 
 number in the series, the sum of the series is equal to the 
 middle term of the series multiplied by the number of terms. 
 Thus, in 1, 2, 3, 4, 5, 6, 7; 74 = 28, the sum of the series: 
 and in 1, 4, 7, 10, 13, there are 5 terms and 7 is the middle 
 one. Then 7-5 = 35, the sum of the series. 
 
 " In all cases of arithmetic progression, whether the num- 
 ber of terms be odd or even, the sum of the series may be 
 found by adding the first and last terms, multiplying the result 
 by the number of terms and dividing this result by 2. Thus, 
 !. 3> 5> 7. 9, IJ ; ( IX + 1) 6= 126 = 72. 72 -r-2 = 36, 
 the sum of the series." 
 
 The following treatment of geometric progression from 
 Adam Riese is a typical one from commercial arithmetic : 1 
 
 " When numbers follow each other in twofold, threefold, 
 or fourfold ratio, and so on, and you wish to find the sum; 
 multiply the last number by the rate of progression, subtract,, 
 the first term and divide this result by the rate of progression 
 minus 1. 
 
 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048. 
 
 2048 X 2 = 4096 4096 — 2 = 4094. 
 
 -|4i*_ = 4094, sum." 
 
 1 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). 
 
 " So aber eine zal die an der vbertrit/ zweyfeltig/ dreyfeltig/ vierfeltig/" 
 vc. vnd wolteft die Summa wiffen/ To multiplicire die letzte zal mit der 
 vbertrettung/ nim von folchen die erfte/ was bleibt/ theil ab mit der Vber 
 treittung/ weniger 1/ Als hie in folgenden Exempeln. 
 
 " Item/ 2.4.8.16.32.64.128.256.512.1024.2048. duplir 2048/ komen 4096/ nim 
 ab 2/ bleiben 4094/ die teil ab mit 2/ weniger 1/ als/ 1/ bleibt die zal an 
 jr felbs.'' Fol. Cij recto. 
 
THE ESSENTIAL FEATURES j 13 
 
 Thierfeldern combines geometric progression with the ex- 
 traction of roots. 1 " Hiernacht folget die Tafel/ dadurch 
 ausgezogen werden alle Wurtzeln Geometrischer Progres." 
 
 tf 
 
 X 
 
 
 
 
 
 } 
 
 2 
 
 i 
 
 
 
 Cf 
 
 3 
 
 3 
 
 t 
 
 
 in 
 
 4 
 
 6 
 
 4 
 
 t 
 
 
 
 (3 
 
 5 
 
 10 
 
 10 
 
 6 
 
 i 
 
 
 
 ft 
 
 8 
 
 IE 
 
 20 
 
 15 
 
 8 
 
 t 
 
 
 
 66 
 
 7 
 
 21 
 
 36 
 
 36 
 
 21 
 
 7 
 
 t 
 
 
 
 w 
 
 8 
 
 28 
 
 68 
 
 70 
 
 56 
 
 28 
 
 8 
 
 y 
 
 
 cq 
 
 9 
 
 36 
 
 84 
 
 128 
 
 128 
 
 84 
 
 36 
 
 9 
 
 % 
 
 The characters in the left hand column represent the succes- 
 sive powers of a number from the first to the ninth ; that is, 
 these were algebraic symbols for x, x 2 , and so on. The vari- 
 ous lines are seen to be the binomial coefficients, and the form 
 is essentially that of Pascal's triangle. 
 
 The traditional applications of the progressions were com- 
 monly given, but when an attempt was made to supply vital 
 applications, the results were curious. The following are 
 from Baker : 2 
 
 "A marchant hath sold ioo kerfies after this manner fol- 
 lowing, that is to fay, the first peece for i s, the second peece 
 for 2 s, the thirde for 3 s, and so foorth, rifing 1 s in every 
 peece of kerfey unto the 100 peeces. The queftion is to know 
 how much he fhall receiue for the fayd 100 peeces of kerfeys." 
 This is evidently an attempt at giving a practical problem. 
 
 " I woulde laye 100 f tones or other thinges in a right line, 
 of every of the fayde f tones to be a juft pace one from an 
 other, and one pace of from the firft ftone, there ftandeth a 
 bafket. I demaunde howe manye paces a man fhall goe in 
 gathering up the fayde ftones, and bearing them unto the 
 
 1 Thierfeldern, Arithmetica Oder Rechenfouch Auff den Linien vnd Zif- 
 ferm/ (1587 «*•), page 331. 
 
 2 Baker, The Well Spring of Sciences (1580 ed). Fol. Fiii recto. 
 
ii4 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 bafket, the [tone after the other." This was a puzzle prob- 
 lem of long standing. 
 
 " There is one man departeth from London to Chefter, and 
 fo to Carnaruan, the diftaunce beeing about 200 myles : He 
 goeth the fyrfte day 1 mile, the fecond day 2 myles, the thirde 
 day 3 : and so orderlye by natural progreffion. An other man 
 departeth at the fame inftante from Carnaruan to London, and 
 goeth the fyrfte day 2 myles: the fecond day 4 myles, the 
 thyrde day 6 miles, and so encreafing every day 2 miles. 
 The queftyon is to know in howe manye dayes they two per- 
 sons shall meete togither." This is a courier problem. 
 
 The following, though an artificial business problem, seems 
 more plausible: "A man oweth me 400 l'i, to be payd in 10 
 yeares, by progref fyon Arithmeticall, that is to fay 40 l'i at the 
 end of the fyrfte yeare, and euery yeare following 40 l'i, to> the 
 end of 10 yeares, hee offereth to pay me the fayd 400 pound 
 al at one paiment. The question is to know at what time hee 
 ought to paye mee the fame at one paimente, that I be not 
 intereffed in the time." 
 
 Only one problem in geometric progression was given. It 
 is this : 
 
 "A marchante hath folde 15 yeardes of Satten, the firfte 
 yarde for 1 s, the fecond 2 s, the thyrde 4 s, the fourth 8 s, 
 and so increafing by double progreffion Geometricall. The 
 queftyon is to know, how muche the fayde marchant fhal re- 
 ceiue for y fayd 15 yardes of fatten." 
 
 Although the French writers were chiefly '-theorists, Cham- 
 penois wrote to meet business needs, and his introduction to 
 this chapter reads :* " Progression is a series of numbers in 
 which there is a certain excess between the numbers, and the 
 first is always exceeded by the second as much as the second 
 
 1 Qhampenois, Les Institutions De L'Arithmetique (1578 ©d.). 
 
 " Pirogreffion eft vne Irate de notnbre, qui out vn certain excez les vns 
 entre les auitres, & toufiours le premier eft autant excede du fecond, que 
 le fecond du troifiefme, & ainfi des autres. L'vfage de la Progreffion eft 
 vn compendium d' Addition, & eft fort vtile, tant en diuerfes questios 
 d'Aritlimetique, Geometrie, Mufique, qu' Aftrologie, la ou plufieurs regies 
 font faiotes par la nature de la Progreffion." Fol. Eviij verso. 
 
THE ESSENTIAL FEA TURES I x 5 
 
 is by the third, and so on with the others. The practice of 
 progression is a summary of addition and is very useful in 
 questions of arithmetic, geometry, music, and astrology, where 
 many rules are made in the nature of progression." 
 
 Among his problems in progression are the following : 
 
 "A merchant has 60 horses and sells them to another mer- 
 chant; he is to have 4 ecus for the first, 8 for the second, 12 for 
 the third, and so on for the others. The question is : ' How- 
 much should he receive for the last, and how much should 
 he receive for all ?' " x 
 
 "A man about to die gives all his money to his relatives on 
 the condition that the first should have 4 ecus, the second 6 
 more than the first, and the amount for the third and the 
 others should increase at the same rate to the last. After 
 the distribution was made, it was found that the last had 
 118 ecus for his part. The question is: 'How many rela- 
 tives had he and how many ecus ?' " 2 
 
 "A merchant sells a piece of velvet containing 16 aunes to 
 another merchant for which he shall give 2 sous for the first 
 aune, 6 sous for the second, and for the others he shall give 
 at the same rate up to the sixteenth aune. The question is : 
 * How much should he give for the 16 aunes?' " 3 
 
 "A gentleman has 12 horses, which he wishes to sell, and 
 
 1 " Vm marchant a 60 eheuailx, & le's vend a vn autre tnarcbant, par tel 
 fa qu'il doit auoir 4 efcus du premier, 8 du fecond, 12 du troifiefme, & ' 
 ainfi 'dies autres. On. demande combien il doit Teceuoir du dernier, & com- 
 bien il doit reoeuoAr du tout." Page 63, fol. Eviii recto. 
 
 2 " Vn. foomime allamt de vie a trefpas, donne tous- fes efcus a fes parents, 
 par telle icoditiom que le premier doit auoir 4 efcus : le fecond 6. plus que 
 le premier, & ainfi du troifiefme, & autres iusques au dernier. Et apres le 
 paittage faict Ion a trouue que le dernier a eu 118. efcus pour fa part. Lon 
 demande combien il auoit de parents, & combien d'efcus." Page 64, Eviij 
 verso. 
 
 3 " Vn marchant vend vne piece de velours, qui comtient 16. autoes, a vn 
 autre marchant, par tel fi qu'il donnera 2. fols pour la premiere aulne, & 
 6. fols de la feconde, & ainfi pourfuiuant la Progreffio >des autres iufques 
 a la feiziefme aulrae. Lon demande combien il doit des 16. aulnes." Page 
 66, fol. F verso. 
 
n6 SIXTEENTH CENTURY ARITHMETIC 
 
 gives the first for 3 sous, the second for 12, the third for 48, 
 and so on with the others up to the last." 1 
 
 "A dealer wishes to sell a robe to a Master of Arts, who 
 has little money. The Master of Arts, however, seeing him- 
 self without a robe, and that this one for which he was bar- 
 gaining, seems to be well made, says to the dealer that he will 
 give 4888 livres for it, reduced twenty times by half. The 
 dealer is well satisfied and gives the robe to the Master of 
 Arts, who does not refuse it, but joyfully putting it on his 
 shoulders makes his reckoning with the dealer. At the end 
 of the calculation he finds due to the dealer 2^1^ deniers, 
 which is worth f of a quarter of a pite, and ffths of yg- of 
 a pite. To pay this large sum he takes from his purse a piece 
 worth 3 deniers, gives it to the dealer, and demands his 
 change. The dealer driven to despair asks if he will put it at 
 stake. The Master of Arts agrees. The dealer loses, the 
 Master of Arts gains, and taking his 3 deniers he joyfully 
 takes leave of the dealer. The dealer thanks him, saying that 
 he is at his service. Thus, the Master finds himself well 
 dressed at little expense by means of the cunning of an arith- 
 metical rule." 2 
 
 1 " Vn Seigneur a 12 cheuaux, qu'il veut vedre, & donne le premier pour 
 3. fols, le fecod pour 12. le troifiefine pour 48. & ainfi des autres iufques 
 au dernier." Page 67, Fii recto. 
 
 2 Champenois, Les Institutions (1578 ed.). "Vn frippier veut vendre 
 vne robbe 58. lires a vn maiftre es arts, qui n'auoit pas beaucoup de pecune. 
 Toutefois fe voyant fans robbe, & que cellef qu'il marchadoit, luy fembloit 
 bien faicte, diet au frippier qu'il en doneroit 4888 liures, en rabatant de 
 moitie iufques a 20 fois. Le frippier fut content, & dona la robbe au 
 maiftre es arts : lequel ne la refufa pas, tnais d'vne gayete de coeur la mit 
 deffus fes efpauks, & fit copte auec le frippier. Et a la fin du copte fe 
 trouua redeuable au frippier de 2. deniers & ?l?, ieme d'vn denier qui 
 vallet Y$ d'vn quart de pite, & -||.ieme d'vn quart de quart de pite. Et pour 
 faire le payement de cefte grande fonime, preden fa bourfe vne grande 
 piece de trois deniers, & la done au frippier, & demande fon refte. Le 
 frippier tout defefpere luy demanda f'il vouloit iouer fon refte a fa grande 
 piece de trois deniers. Le maiftre es arts diet qu'il eftoit cotet. Le frip- 
 pier perd : le maiftre es arts gaigne, & pred fa piece de trois deniers, & 
 ioyeufemet prend conge du frippier. Et le frippier le remercia, & luy diet 
 que luy & fon bie eftoiet a fon comarudemen't. Monfieur le Maiftre se 
 trouua bien pare a peu de frais, par le Tnoyen de la fubtilite d'vne regie 
 d'Arithmetique." P. 70, fol. Fiij verso. 
 
THE ESSENTIAL FEATURES uj 
 
 RATIO AND PROPORTION 
 
 The treatment of this subject, more than that of progres- 
 sions, followed the traditional lines of Greek and Hindu 
 works x and occurred less often in the practical arithmetics of 
 the period. The most useful tool in the solution of commer- 
 cial problems, the Rule of Three, was, of course, proportion, 
 but it was seldom connected with that subject. 
 
 It was the universal custom borrowed from' the Greeks to 
 use the terms, proportion and proportionality, for ratio and 
 proportion until the latter part of the sixteenth century. Ton- 
 stall stated the Greek meaning of proportion in such a way as 
 to enable one to recognize at once the corresponding modern 
 ideas of Ratio and Proportion and gave an explanation that is 
 more than a mere category. His treatment was as follows : 2 
 
 " Proportion is nothing else than a comparison of things 
 
 1 Unicorn, De L'Arithmetica vinversale (1598 ed.). " Che contien li altri 
 4. Algorismi della prattica di Arithmetics, cioe de radici, del piu, & men, 
 de binomii, & recisi, & de proportioni, de radice uniuersali, & estrattioni 
 de radici delli 'binomii, & recisi, quali sono il neruo del decimo libro de 
 Euclide, & della proportione hauente il mezzo, & doi estremi, della qual il 
 decimoterzo libro di Euclide.'' Fol. Fi recto. (Besides the four algorisms 
 of practical arithmetic, this contains roots, positive and negative number, 
 binomials, surds, proportion, roots in general, the extraction of roots of 
 binomials and surds, which are contained in the tenth book of Euclid, and 
 the extremes and means of proportion from the thirteenth book of Euclid) 
 
 2 Tonstall, De Arte Supputandi (1522 ed.). "Ilia itaqj habitudo, qua 
 fefe uel aequaliter, quando funt equales : -mutuo refpiciunt. uel inequaliter, 
 quando earum altera maior reliqua, aut minor eft : appellatur proportio. que 
 nihil aliud eft : q earum inter fe comparatio eiuf dem quoqj generis quan- 
 titates effe deibent: inter quas cadit proportio. Veluti duo numeri, duae 
 lineae, due fuperficies, duo corpora duo loca, duo tempora. neqj enim linea 
 maior aut minor f uperficie eft, aut corpora : nee tempus loco maius eft, aut 
 minus, fed linea linea: fuperficies fuperficie: corpus corpore. fola enim, 
 quae unius funt generis: inter fe comparabilia funt." Fol. ij verso. 
 
 " Quippe proportio apud ueteres in tria fecatur genera, quorum unum eft 
 difcretorum, uidelicet numerorum: quod uocant Arithmeticum. Alterum 
 oontinuorum: quod geometricum appellant. Tertium fonorum et concen- 
 tuum: quod armonicu nuncupant, ex illoru utroq) mixtu: qj mufica in 
 pauf is <et prolaitionibus tepus fpectet : in uibrante nocii notarumqj diuif ione, 
 numeros." Fol. i l verso. 
 
 " Vnde fit : ut quecumqj proportio occurrit in numeris : eadem reperiatur 
 in omni genere cocitinuoruin. puta in lineis, fuperficiebus, corporibus, et 
 temporibus." Fol. i 2 recto. 
 
H8 SIXTEENTH CENTURY ARITHMETIC 
 
 among themselves. Quantities which form a proportion must 
 be of the same kind, as two numbers, two lines, two surfaces,, 
 two solids, two like orders, and not, a line is greater than a 
 surface, or solid, or time is greater or less than a space, but 
 line is comparable with line, surface with surface, solid with 
 solid, and so on. 
 
 "Among the ancients proportion is divided into three 
 classes, arithmetic, geometric and harmonic. 
 
 " Just as proportion occurs in numbers, so it is found in 
 all continua, as in lines, surfaces, solids, and time." 
 
 Kinds of Rational Proportion 1 
 "Any quantity compared with another is equal or unequal to 
 it. A quantity is equal to another when it does not exceed 
 that quantity, or is not exceeded by it, as a cubit to a cubit, or 
 a foot to a foot, and so on. Unequal quantities are those in 
 Which one quantity exceeds the other. A series of numbers 
 having a common difference such that the difference is an 
 integral part of the first number is said to be composed of 
 proportional numbers, and the series is classified according to 
 the name of the part. They are called sequialteri, sequitertia, 
 and so on. This is the origin of arithmetic progression." 
 
 Tonstall here gave the Pythagorean table again, showing 
 ratio, proportion, multiplication, and division in their inter- 
 dependence. 2 " Proportionality is the similarity of propor- 
 tions among themselves." In this chapter Tonstall gave 
 arithmetic and geometric proportionality and referred to 
 
 1 Tonstall, De Arte Supputandi (1522). 
 
 " PROP0RTI0NIS EATIONALIS SPECIES.'' 
 
 " Oinnis quatitas ad' alia, coparata aut ei gqualis : auit inecnialis reperitur. 
 Quantitas equal-is ef<t: quae nee fibi comparatarn excedit: nee ab ea exce- 
 ditur. Veluti cubitus ad cubitum collaitus. pes ad' pedem. numerus quater- 
 narius ad quatemarium.'' Fol. i 2 verso. 
 
 " Inequalis autem quantitas, quae fibi comparatarn. excedit : propontionem 
 ad ilia habet ingqualitatis <maioiris : ueluti cubitus ad pedem. numerus quater- 
 oarius ad binarium." Fol. i 2 verso. 
 
 2 Tonstall, De Arte Supputandi (1522). " Proportionalitas eft propor- 
 tiomim inter fe limilitudo." Fol. K, Tecto. 
 
THE ESSENTIAL FEATURES 119 
 
 Jordanus as having - given eight kinds. Vicentino gave four : 
 arithmetic, geometric, harmonic, and contraharmonic. 1 This 
 subject was given a more rigorous treatment by algebraists. 3 
 Ratio and proportion in the modern sense came into arith- 
 metic about 1550. Orontius Finaeus under Ratio, Propor- 
 tion, and Rules of Proportion gave this definition : s " Ratio 
 (as we shall use it) is the method of comparing two num- 
 bers of the same kind." The same conception of ratio will 
 be recognized in this example from Buteo : 4 " When two 
 numbers are proposed, as for instance, 12400 and 124, the 
 ratio between them will be immediately shown by dividing 
 the greater by the less; since it is 100 in this case, we shall 
 call the ratio between the given numbers a hundredfold ratio." 
 
 1 Viracentino, Proportione, et Proportionalita (iS73 «d-)- 
 " Proportionality, fi dice la ugualita delle proportions 
 ' La proportionalita, I i divide in Arithmetica, Geometrica, Harmonica, et 
 Contraharmonica. 
 
 2 2 
 " La Aribhimetica 5 7 9 
 
 "La Geometrica 4 6 
 
 2 1 
 " La Harmonica 643 
 
 "L'una ch'faa la differentia dello antecedente fopra il confeguente, alia 
 differentia del confeguente fopra il terzo quanto, l'altra la proportione dell' 
 antecedente al terzo quanto. 
 1 2 
 
 "La contraharmonic 653 
 
 " L'una die ha la differentia del confeguente fopra il terzo quanto, alia 
 differentia dell' antecedente, fopra il confeguente, l'altra la proportione dell' 
 antecedente, l'altra la proportione dell' antecedente al terzo quanto." Fol. 
 G, verso. 
 
 2 Tartaglia, La Prima Parte Del General Trattato (1SS6), Book 7. 
 
 s Finaeus, De Arithmetica Practica (iSSS ed.). "Ratio igitur (ut ad 
 remipfarn deueniamus) eft duarum quantitatum eiufdem fpeciei adinuicem 
 comparataru habitudo." Fol. Q 1 recto. 
 
 * Buteo, Logistica, Quae Arithmetica vulgo dicitur (iSS9 ed.). "Pro- 
 pofitis duobis numeris, vt puta 12400, et 124, quaena fit inter eos ratio, 
 partitione maioris in minorem, ipfum prouenies ftatim oftendet, quod cum 
 fit in hoc loco 100, dicemus inter datos huiufmodi numeros rationem effe 
 centuplam." Fol. f B recto. 
 
120 SIXTEENTH CENTURY ARITHMETIC 
 
 Ramus defined ratio as division. 1 He then added the old 
 conceptions, as : " The ratio of 3 to 2 is sequialter, because 3 
 contains 2 one and a half times. The ratio of 5 to 3 is super 
 biteria, because 5 contains 3 one and f times." He defined 
 proportion as the equality of two ratios. 2 Thus, proportion 
 came to replace the old term proportionality, and at the be- 
 ginning of the seventeenth century, the title, Ratio and Pro- 
 portion', had taken the place of Proportion and Pro- 
 portionality. 3 
 
 Ludolf van Ceulen (161 5) gave a chapter on proportion as 
 applied directly to practical problems. He says that pro- 
 portion is necessary in comparison of weights and measures," * 
 as when one says that a pound is heavier than a mark in com- 
 paring weight, or that the distance from Utrecht to Delft is 
 farther than that to Leyden, in comparing length." 
 
 Ratio of fractions 5 and the Rule of Three s of fractions are 
 
 1 Ramus, Arithmeticae Libri Duo (1577 ed.). "Ratio est comparatio, 
 quoties terminus in termino continetur." Page 41. 
 
 2 Ramus, Arithmeticae Libri Duo (1586 ed.)- " Proportiono arithmetica 
 fie est, geometrica fequitur in rationem aequalitate." 
 
 3 Stevin, Les Oeuvres Mathematiques (1634 ed.). " Des definitions de 
 la Raison et Proportion." " La proportion, pour en parler un peu en gen- 
 eral, avant que parvenir au particulier, eft la fimilitude de deux raifons 
 egales. Raifon eft comparaifon de deux termes d'une mefme efpece de 
 quamtite." Fol. Bi verso. 
 
 4 Van Ceulen, De Arithmetische en Geometrische fondamenten (1615 
 ed.). "Alsmen spreeckt een <pondt is -swaerder als een marcq dan ver- 
 ghelijcktmen gfaevviohte, ofte van Delfft is Vtrecht verder als Leyden, hier 
 vvert de lenghte vergheleecken." Fol. Biij verso. 
 
 K Buteo, Logistica (1559 ed.). 
 
 To find the ratio of £ and A : 
 
 |- X ^ ( The ratio is 21 to 20, or sesquivigesima. (See pages 
 21 20 I 33, 34 of this article.) 
 
 " De particularum fragmentorumque rationibus, quomodo dignofcantur." 
 " Non folum numeri conferuntur numeris, fed etiam particulis, atque frag- 
 ments, &■ ipfae etiam inter fe particular Quarum rationem inuestigabis 
 hoc modo. Sit propofitum dare rationem quam habet & ad £. Difpo- 
 natur datae particulae atque nmltiplicetur in decufiim, 
 fientque producta 21 & 20. Quae eft igitur ratio 21 ad 20, 21 20 
 
 •eadem eft 3 ad 4, hoc eft fefquiuigefima.'' Fol. f verso. 4 X % 
 
 6 Ramus, Arithmetice Libri Duo (1577 ed.), Liber II, Chap. VI. 
 
THE ESSENTIAL FEATURES I2I 
 
 subjects sometimes met. The latter is generally characteristic 
 of the commercial writers. 1 
 
 INVOLUTION AND EVOLUTION 
 
 The commercial arithmetics naturally contain little concern- 
 ing powers of numbers, and one might expect to find much less 
 on evolution. But there was a large number of problems in 
 mensuration, a subject of real interest at that time, which re- 
 quired a knowledge of roots. Consequently, most works of 
 any pretension gave square and cube root. Theorists like 
 Tartaglia 2 and Unicorn 3 explained evolution of orders higher 
 than the third. It was quite common to preface the work on 
 roots by a table of powers. 4 
 
 1 Thierfeldern, Arithmetica Oder Rechenbuch Auff den Linien vnd 
 Ziffem/ (1587 ed.), Chap. IIII. 
 
 Baker, The Well Spring of Sciences (1580 ed.). "If one elle coft me 
 17 s" what fhall 15 elles % part cofte? which Y% is halfe a quarter of an 
 «Lle." Fol. Qvii recto. 
 
 Kobel, Zwey rechenbuchlin (1537 ed.). "So einer dich fragt/ wan einer 
 9. elen vnnd Ya- Tuchs vmb 7. gulden kaufft het/ was in 6. Elen koften." 
 
 " If a person bought g% ells of cloth for 7 gulden, what will 6 ells 
 cost?" Fol. I 2 recto. 
 
 2 Tartaglia, La Seconda Parte Del General Trattato (1556), Book II, 
 fol. Cvi verso. 
 
 3 Unicorn, De L' Arithmetica vniversale (1598), Book 2. 
 
 * Champenois, Les Institvtions De L'Arithmetique (1578). Ohampenois 
 
 gives the squares of numbers from 1 9, inclusive, illustrating each by 
 
 little squares beneath, thus : 
 
 Racines 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 ? 
 
 8 
 
 9 
 
 Simples 
 Quarrez 
 
 1 
 
 4 
 
 9 
 
 16 
 
 2S 
 
 36 
 
 49 
 
 .64 
 
 81 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Page 258, fol. S verso. 
 
 The subjects of powers and' roots were more generally treated in 
 algebra or in algebraic chapters of the arithmetical works. The following 
 
122 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 At that time roots were extracted by at least four methods, 
 each subject to some variation. 
 i. Method of line reckoning. 
 
 2. Geometric method. 
 
 3. Scratch method. 
 
 4. Downward method. 
 
 Although the methods by the use of figures were not sim- 
 ple, the method by counters was still more difficult, and was 
 rapidly superseded by the former. Examples solved by line 
 reckoning were given by Kobel, Riese, and a few others. Geo- 
 metric illustrations of square and cubic numbers were not un- 
 common, but graphical methods of extracting roots were rare. 
 Trenchant in the third book of his Arithmetic gave the geo- 
 metric demonstration commonly 
 found in American arithmetics 
 of the nineteenth century. In 
 this he explained the construc- 
 tion of the diagram as here 
 given and proved the equality of 
 the rectangles, AF and FD, by 
 reference to Euclid, Bk. II, Prop. 
 4. He then said : " From the 
 above it is easy to see clearly 
 and to demonstrate the logic of 
 
 is a table from Ghaligai, Practica D'Arithmetica (1552 ed.), explaining his 
 power symbols. 
 
 40 
 
 c 
 
 7 
 
 280 
 
 
 49 
 
 
 F 
 
 
 2209. 
 1600. 
 
 
 280 
 
 n° Humero 
 
 O o Coea — 
 
 a Censo — 
 
 ay Cubo — 
 
 a ete a a di o 
 
 B Relato 
 
 S 
 m eU £D 
 
 s 
 
 mu^adia ro di o di nil 
 
 SJ DTomico 
 
 E3M. a ff di 
 
 m • B ™ B 
 
 1 
 
 2 
 
 4 
 
 8 
 
 16 
 
 82 
 
 ID Jla -64 
 
 Pronico 128 
 
 o di o di D 256 
 
 en di m 612 
 
 di n 1024 
 
 Tromico 2048 
 
 4096 
 
 8102 
 
 16384 
 
 32768 
 
 Fol. Kii recto. 
 
 * Error in original. It should be turned horizontally, "di" indicates "to the power. 
 Thus, D di D means 4, DZ] di D means 8 to the second power, l"T~l di OO means 8 to the 
 third power, and so on. 
 
THE ESSENTIAL FEATURES 12 t > 
 
 extracting square root : for let us take the same number, 2209 
 (which is the square on the whole line AB), to find the square 
 root, the extraction of which we shall demonstrate from the 
 figure and maintain the logic. First having separated the 
 number into two periods (according to Article 4 of this chap- 
 ter), I seek, according to Article 5, the root of the first period, 
 22 : this is 4, which (because there is another root figure, or 
 because it is of the tens' period) must be 40 and denotes the 
 longer segment of the line, i.e., AC, and consequently, each of 
 the sides of the square, EF, and also the longer sides of the 
 supplementary rectangles, AF and DF. From 22 I subtract 
 the square of 4, which is 16, and because 4 denotes 40, then 
 its square 16 denotes 1600. This is the square EF. Then 
 the remainder from 22 is 6, or from 2209 will leave 609, 
 Which is the area of the two rectangles, AF, DF, and of the 
 small square, BF. But, if I divide the area of a rectangle by 
 one of its sides, I obtain the other. That is, if I divide the 
 product of two numbers by one of them, I obtain the other. 
 So if I divide 280 (the product of 7 by 40) by 40, or 560 
 (the product of 7 by 80) by 80, I shall get 7. Therefore, 
 knowing that 609 is the area of the two rectangles, AF, DF, 
 and of the small square BF, and that 40 is the side of one of 
 these rectangles, I double 40 making 80, denoting the side of 
 the rectangles joined. Then I divide 609 by 80, so that I can 
 also take away the square of the quotient, and obtain 7, which 
 denotes the shorter sides of the rectangles, AF, DF, and con- 
 sequently the side of the small square BF, which is BC. Fin- 
 ally, I multiply 80 by 7, and then 7 by 7, producing 560 and 
 49, which make 609; then I subtract the 609, and the re- 
 mainder is zero. Thus, the square root of 2209 is 47." 2 
 
 1 It is well known that the Western Arabs used the sand-board in ex- 
 tracting roots-; hence it is reasonable to suppose that they used the scratch 
 method, whence the Italians obtained it. Cantor, Bd. I, p. 767 (igoo ed.). 
 
 2 Trenchant, L'Arithmetique (1578 ed.), Book 3. "Par ce que deffus 
 fe peut clerement veoir, & demontrer la raifon des extractions quarees : 
 pourquoy fere prendrons ce meme nombre 2209 (qui eft le quarre de la 
 totale ligne A, B) pour en tirer la racine quanree: a 1' extraction da laquelle 
 -monftrerons fur la figure la raifon pretendue. 
 
 " Premierement ayant coupe iceluy nombre en deux lections (felon le 4 
 
124 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 The accompanying diagrams from Trenchant's pages, for 
 extracting the cube root of 103,823, show that he used the 
 modern block method. 
 
 Fol. Q 2 verso. 
 
 The scratch method was the common algorism for this pro- 
 cess. The following examples from Tonstall will illustrate 
 it : * " The extraction of the square root is nothing else than 
 the finding of a number which multiplied by itself will pro- 
 duce the proposed number, if it be a square, or the greatest 
 square number contained in it, if it be not a square." 
 
 " Example in square root : 
 
 antic, de ce chap.). Ie ohercfae, felon le 5 art. la ratine de la premiere 
 fecitiom 22 : c'eft 4, lequel (a caufe de la figure radicale future, ou qu'il eft 
 de la fections des dizeines) vaut 40 dieniotat la maieure fection de la ligne 
 A, C, & par confequent, chacun des cotez du quarre E, F, & auffi les plus 
 grans cotez des fupplemens A, F, & D, F. En apres de 22, ie leue le 
 quarre de 4, c'eft 16, lequel comme 4 denote 40, auffi fon quarre 16 deno- 
 tera 1600, c'eft le quarre E, F. Par ainfi de 22 refte 6: ou de 2209, ref- 
 tera 609 qui eft la fuperficie de 2 fupleimens A, F, & D, F, & du petit 
 quarre B, F. Or qui diuife la fuperfice d'vn rectangle, par l'vn de fes 
 cotez vient l'autre : cef t a dire, qui diuife le produit de deux nombres par 
 ■l'vn d'iceux, vient l'autre : comme fi ie diuife 280 (prooienu de 7 foys 40) 
 par 40, ou 560 (prouenu de foys 80) par 80 viendra 7. Parquoy fachant 
 que 609 eft la fuperficie des deux fuplemens A, F, & D, F, & du petit 
 quarre B, F, & que 40 eft le cote de l'vn dfaceux fuplemens, ie double 40 
 fet 80, denotant le cote des deux fuplemens affemblez. Donques ie diuife 
 609 par 80, de forte que i'en puif fe auffi leuer le quarre du quotient, vient 
 7 qui denote les moindres cotez d'iceux fuplemens A, F, & D, F, & par 
 confequent ceux du petit quarre B, F, dont la fection B, C, en eft l'vn. 
 Finalblement ie multiplie 80 par 7, & encores 7 par 7, pirouient 560, & 49, 
 qui font 609, que ie leue de 609, & n'y refte rien. Ainfi la racine quarree 
 de 2209 eft 47." Fol. P e recto. 
 
 1 Tonstall, De Arte Supputandi (1522), fol. N, recto et seq. 
 
THE ESSENTIAL FEATURES I2 c 
 
 " The number given, 57836029, should 2 
 
 be properly pointed off. Beginning at ? * 
 
 the left, we should search for the first •*■'■ 4 Re]i< l uum ' 
 
 number which multiplied by itself will Ve 5 Eadix 
 give the first number marked off at the. Ifflffl 
 left, or, if it is not a perfect square, the . W 
 number which multiplied by itself will give the nearest square 
 which it can contain. The number is 7, and its square is 49, 
 which subtracted from 57 leaves 8, and so 7 is placed between 
 the parallel lines first made, 8 is put above the first „ 
 point and crossed out. Then the 7 between the . . . 
 parallel lines is doubled, which makes 14, of ^7836029 
 
 which 4 is placed below and at the right, and the 
 
 10 left should be placed below the lines directly below the 8, 
 which was the result of the former subtraction. Then again 
 another number is to be found which, multiplied by 
 14, will give the number nearest to 88 (the 8 left .4. . 
 from subtracting and the next figure of the given 57836029 
 
 number.) This figure is 6, which multiplied by -r 
 
 14 gives 84; this taken from 88 leaves 4; then 
 this number 6 multiplied by itself gives 36, which subtracted 
 from 43 (the 4 left from the last subtraction and the 
 next figure in the given number) leaves the num- 
 ber 7, which is written above the next point in .$. . . 
 the number given. The 6 is placed between the 57836029 
 
 parallel lines as the next figure of the root, and -= 
 
 so on." 
 
 " If one wishes to find out 
 whether the number found is 
 right, multiply the number found 
 by itself and add the remainder, if 
 there is one. If the work is cor- 
 rect, this result will be the given 
 number." x 
 
 The example in cube root 
 shown in the illustration is a re- 
 production from Tonstall's arith- 
 metic : 2 
 
 exempla auncaffuamus : quffinguUmaniMlrt. 
 At^cxduecjitiesqm'nquagi'cs milli'esmillcim miUibm, 
 quingenuuutautamfllenfsmi1bbus,qm'ngentisoAcM 
 £int;iduobusmfl!ibus,quadringtntis focaginu quatU 
 o^iadieemcubicametuanuis. 
 .*■ ii a 
 9 . .#* • . 
 y X M Jt * r X \* * # & # Nutna'cnb' 
 s ) o 4 Radixtubl 
 
 * jr. M * m jt 
 
 * * Xjf * ft 
 
 Jirjtfi # * # * 
 
 STATIHQ.VE poft^Rumcrfftioordinepnferfpri; 
 du&xq} fubtus parallel x : u miUcnarionun fedcs puit 11 
 cnint fi'gnatc : Tub poftrano ad (uu'ftram m illenar io pum 
 {to oouionumenis aliqu f s prfmarius exqui'ratur : quifo 
 
 tncl in fe.etucrum in DUBicriJ produftum muldplicaou. 
 
 iToostall, De Arte Supputandi (1522), fol. N 2 verso. 
 2 Tonstall, De Arte Supputandi (1522), N 4 verso. 
 
126 SIXTEENTH CENTURY ARITHMETIC 
 
 A peculiar form in which the process is the same as the 
 scratch method, only the numbers are not crossed out, is found 
 in Cirvelo : x 
 
 (a) oo 1 
 i6| 
 
 (b) 00 1 
 
 25 1 
 
 (c) o oo| 
 i 1 44 
 
 
 
 (d) 03 1 oo 
 67I24 | 
 
 12 
 (e) 1 28 
 
 5! 47 56 
 
 4l 
 
 Si 
 
 I | 22 
 
 8 162 | 
 
 2 1 43 1 94 1 
 
 4| 
 
 Radix. s | 
 
 Ra. *i | Ra. 
 
 8|2| 
 
 U 1 
 
 2 I 3 I 4 |Ra. 
 * The 2 is missing in the original. 
 
 Problem (a) is the process of finding the square root of 16, 
 (b) of 25, (c) of 144, in this, however, the root is incom- 
 plete, (d) of 6724, and (e) of 54, 74, 756. The powers are 
 written above the double lines and the roots below them. 
 
 A downward process resembling in many respects our pres- 
 ent form is found in Widman : 2 
 
 To extract the square root of 207936 : 
 
 (a) (b) (c) 
 
 47936 47936 5436 
 
 8 8 90 
 
 4 45 45 
 
 The figures in the top row are the successive remainders with 
 all periods brought down at each step. The last row is the 
 root, the numbers in the root being repeated each time. The 
 middle row is the trial divisor, not always having the zero 
 added. Step (b) is really step (a) with the next figure, 5, 
 of the root written down. 
 
 An interesting plan is given by Paciuolo, which has the 
 principle common to all the other methods, but which differs 
 from them in arrangement of calculation : 3 
 
 1 Cirvelo, Arithmetice practice seu Algorismi (1513 ed.), fol. b; recto. 
 
 2 Widman, Behend und hiipsch Rechnung (1508 ed.), fol. di recto. 
 
 3 Paciuolo, Suma de Arithmetica (1523 ed.). 
 
 "De aproximatione .R. in furdis." 
 "Apresfo per le. R . forde : cioe qlle che no fonno dif crcta : qui f equente 
 mettaro vna re", p laquale f emp tu piu ohel copagno te porrai aproximare : 
 vnde a voler trouare ditt?. R . fempre troua prima la fua. R . derita de 
 poto: conio fai laltre d fopra. E qn tu hai itrouato la prima. R. fanne 
 pua. e vedi quanto la pasfa el detto n°. Alora torrai quel piu: cioe la 
 dria e ptirala per lo doppia de qfta prime. R. che te la data: e quello 
 dhe virra de ditto ptimento cauaralo de ditta pma . R . el remanente f era la 
 
THE ESSENTIAL FEATURES I2 - 
 
 To find the square root of 6 : 
 
 ^= ' (^6- -W^-iV 
 
 5 
 
 = %¥irV» and so on. 
 
 4liT 
 
 This agrees with the general formula : 
 
 a - A ~ a ' _ , , A - a\ 
 
 d ^ — a i> aH — — = aj, and so on. 
 
 It is easy to recognize in every method the plan of dividing 
 the remainder by twice the part of the root found and adding 
 the result to the root for the next figure. 
 
 Square and cube roots of fractions were common. Wid- 
 man gives these examples : Find the square root of ff and 
 the cube root of ^." " Give a number, i of i of i of 
 £ of which is its own quare root." Ans. AVAV 1 
 
 APPLIED ARITHMETIC 
 
 The authors of arithmetic of the sixteenth century may be 
 classified into three groups : writers of theoretic arithmetic, 2 
 
 .R. fcda. de ditto n . afai piu p fimana che la p*. .poi p aproximarte piu: 
 farai la pua ancora di qfta. e vederai quato e la fupchia ditto n°. E anche 
 quel piu che te dara qfta feconda. R. ptiralo pure per lo dopio de esfa 
 .R. 2*. che te la dato : e qllo auenimento caualo de ditta .2". R. el remanete 
 fia. . 3"- piu pximana de ditto ai°. Poi a mo ditto: farane proua. e 
 vederai quanto la pasfi ditto n°. e pigliarai anche quella dria e ptira la per 
 lo doppio pure di quefta. R. 3". che tal dria te dette: e lauenimento de lei: 
 caua el rimanente fera. R. quarta piu pfimana de ditto n°. E cofi fempre 
 in infinitu andarai facedo: e guarda fempre de cauare li ditti auenimeti de 
 le R. fchiette: e no delle duplate verbi gra. R. p a . di 6. ene. 2 z /i. E qfta 
 pasfa de. %. pche. 2]/ 2 . via. 2 l / 2 . fa. 6J4- parti quel. J4. quale e la dria 
 p lo doppio di la R . prima che la dato : cioe p .5. neue J_ . qual caua de 
 .2j4. Che e la. R fohietta resta -2-^ e quefto di co die. 2". R. di 6 piu 
 pximana che la prima, cioe. 2}A. E cofi andarai fequitando a modo ditto 
 e trouerai che qfta 2*. R.- pasfa .6. de • T ^ Tr la terza R fera .2^^. 
 E quefta pafsara .6. de iti | i()|> . la quarta .R. fera fecondo che vedi q 
 de fcripto in margine. e cofi la quinta al medifimo modo trouato. E 
 quefto fin qua e detto fe habia intendere de li numeri fani. Fol. 45 recto, 
 F T xecto. 
 
 r Widman, Behend und hupsch Rechnung (1508 ed.). "Gyb ein zal 
 welch mit irm £, }, \, 7 sey ir selbst qdrata radix. Ans. 
 recto. 
 
 2 De Muris (1538), Maurolycus (lS7S)- 
 
128 SIXTEENTH CENTURY ARITHMETIC 
 
 writers of arithmetic primarily theoretic and secondarily prac- 
 tical, 1 and writers of arithmetic primarily practical and sec- 
 ondarily theoretic. 2 The first class is the smallest, and the 
 second is the largest. Taking the three classes in order of 
 their size, their numbers are approximately proportional to i , 
 4, 6, as the lists given below suggest. It was common usage 
 to divide arithmetic into theoretic and practical. For example, 
 Trenchant, in the beginning of his arithmetic, denned these 
 divisions thus : 3 " Theory is the speculation through which 
 one becomes acquainted with the property of numbers. Prac- 
 tice is the performing of the operations which arise from such 
 knowledge and speculation." 
 
 As would be expected, practical arithmetic was about the 
 same in its general features wherever found. The exceptions 
 are due to local influence or to eccentricity of authorship. For 
 example, Champenois drew his problems mainly from military 
 affairs. Thus : 
 
 "The Commissary-General had four stewards; the first 
 had 7,836 loaves of bread, the second 6,342 loaves, the third 
 5,424 loaves, and the fourth 6,398 loaves. The question is : 
 How much bread did they have altogether ?' " 4 
 
 "A squadron contains on the front 312 men and on the 
 side 232. The question is : ' How many men are there in a 
 squadron ?' " B 
 
 1 Tonstall (1522), Paciuolo (1494), Cardan (1539), Noviomagus (1539),. 
 Tartaglia (1556), Gemma Frisius (1540), Riese (1522), Ramus (1567), 
 Trenchant (1571), Champenois (1578), Undcom (1598). 
 
 2 Borgi (1484), Calandri (1491), Widman (1489), Cirvelo (1505), Ru- 
 dolf! (1526), Kobel (1531), Baker (1562), Raets (1580), Van der Schuere 
 (1600). 
 
 3 Trenchant, L'Arithmetique (1578). "La Theoreque eft la f peculation 
 par laquelle l'on vient a connoetre la propriete de leur fuget. Et la pra- 
 tique, eft l'operatJon & effet qui prouient de telle connoefanoe & fpecnv 
 lation.'' Fol. A^ verso. 
 
 * Champenois, Les Institvtions De L'Arithmetique (1578). "La Commis 
 general des viures a quatre prouifeurs de pain. Le premier a 7838 pains : 
 le f econd, 6342 : le troif iefine, 5424 : & le quatriefme, 6398. Lorn demand* 
 comhien il y de pain en tout." Page 10, fol. Bv verso. 
 
 Champenois, Les Institvtions De L'Arithmetique (1578 ed.). "Vn efca- 
 
THE ESSENTIAL FEATURES I2 o, 
 
 "A captain with 4,000 soldiers is besieged in a fortress by 
 the enemy for seven months ; they have food for five months 
 and are without hope of obtaining any during the period of 
 the siege, which is seven months. The question is : ' How 
 much should the captain diminish the rations of the soldiers 
 that the food may last through the time of the siege, which 
 is 7 months ?' " 
 
 Another writer who used peculiar problems was Suevus. 
 His interest inclined to history. In the dedication of his 
 arithmetic he named the following applications : x "To reckon 
 feast days of the church, also many mysteries and secrets of 
 the church. For use in schools, which Cicero, that learned 
 pagan, called the foundation of the whole republic, as all other 
 arts are learned so much better through arithmetic. Also in 
 the army; among merchants; among manual workers, as 
 artists, goldsmiths, mint masters, watchmakers, painters,, 
 builders, masons and others ; and in housekeeping." 
 
 The subject-matter of the book presents a most remark- 
 able collection of historical material. Some of the problems 
 illustrate a tendency not uncommon at that time to correlate 
 religious teaching with school instruction. Under Numera- 
 tion the first example is : " The number of years from the be- 
 ginning of the world to the birth of Christ, our Saviour, and 
 the Incarnation was : 
 
 3970. 
 
 " That is, Three thousand nine hundred seventy years. 
 
 " That was the time decided upon when God promised to 
 send his Son. This promise he had fulfilled (Galat. 3), by 
 which we may know his truth and uprightness, and putting 
 aside all sorrow and doubt, we may sing cheerfully with the 
 dear David from the 33d Psalm, and we may say : ' The 
 
 dron contient en front 312 homines, & en flanc 232. Lon deman.de combdcn 
 il y a d'hommes en l'ercadron." Page 27, fol. Ovi recto. 
 
 "Vn Captaine auec 4000. foldats eft affiege en vne fortereffe, de 
 l'ennemy pour 7. mois, & n'ont de viures que pour 5. mois, & sans efper- 
 atiee d'en pouuoir Tecouuir durant le temps de 1'affiegement qui eft 7. 
 mois? Lon demande combien le Captaine doit apetiffer la penfion du 
 foldat, afin que le viure puiffe durer le temps de l'affiegement, qui eft 7- 
 mois." Page 83, fol. Gij recto. 
 
 1 Suevus, Arithmetica Historica (1593 ed.). 
 
130 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 word of the Lord is true, and that he hath promised will he 
 surely fulfil." x 
 
 Other examples under Numeration are : 
 
 "According to the statements of Theodore Bibliander, the 
 cost of building Solomon's temple was 13,695,380,050 
 crowns." 2 
 
 " The yearly cost of maintaining the wars of the Emperor 
 Augustus, especially in holding the Roman borders, was 12,- 
 000,000 crowns." 
 
 " The annual income of King Ptolemy Auletes was 7,500,- 
 000 crowns." 3 
 
 Addition finds application in determining the age of Methu- 
 selah: 
 
 "Methuselah was 187 years old when he begot Lamech; 
 after that he lived 782 years. What was his age? Ans. 969 
 years." * 
 
 Division is applied thus : 
 
 " In his thirty-fourth Book, Livy informs us that 1,200 
 Gallic prisoners were released with 100 talents. The question 
 is : ' How much should each receive?' " 6 
 
 1 Suevus, Arithmetica Historica (1593). "Die Jarzal von anfang dex 
 Welt/ hiss auff C'hristi vnsers Heylandes Geburth vnd Menschwerdung. 
 
 3970. 
 
 " Das sind : Drey tausend/ neun hundert vnd siebentzig Jar. 
 
 " Das ist die bestimpte zeit/ darin Gott seinen Son zu senden verheissen/ 
 auch seine zusage kreffigerfullet hat/ Galat. 3. daraus wir seine Trew vnd 
 Wartieit kennen lernen/ vnd wir alien kurnmer vnd' zweiffel/ mit dem 
 lieben David aus dem 33. Psalm getrost singen und sagen mugen : Des 
 Herrn Wort ist warhafftig/ vnd was er zusagt/ das helt er gewiss." Fol. 
 Aij verso. 
 
 2 " Des Tempels Salomonis vnkosten zu bawen/ nach des Theodori Bib- 
 liandri verzeichrtis. 13695380050 Cronen." Fol. Aij verso. 
 
 " Des Keysers Augusti Jarlich Kriegs vnkosten/ sonderMch des Rom- 
 ischen Reichs Gretzen zu halten. 12006000 Cronen." Fol. Aiij recto. 
 
 3 " Des Konigs Ptolomei Auletis Jahrliohs Einkommen. 7500000 Cro- 
 nen." Fol. Aiij recto. 
 
 4 Methusalem war hundert vnd sieben vnd achtzig Jahr alt/ vnd zeugete 
 Lamech/ vnd lebete darnach sieben hundert vnd zwey vnd achtzig Jahr. 
 Wie gros ist denn sein gantzes Alter geworden? Antwort, neun hundert 
 und neun vnd sechzig Jahr." Page 22. 
 
 5 "Liuius Lib. 34 meldet: 'Das zwolff hundert Welche gefangene Kriegs- 
 leute mit hundert Talentis sind aiusgeloset ,worden.' " 
 
 " Ist die Frage : ' Wie viel fur eine Person gegeben sey ?' " 
 
THE ESSENTIAL FEATURES I3I 
 
 The following problems occur under the Rule of Three : 
 
 " In the 7th Chapter of his 12th Book, Pliny tells us that a 
 pound of black pepper was bought for 4 denarii— that is, for 
 a half Taler. Here is the question : ' If 3I pounds cost 12! 
 denarii, what will 14I pounds cost ?' " 1 
 
 " Martial (the poet) informs us that an amphora of wine 
 was sold for 20 asses, that is, for 2 denarii, which is as much 
 as a quarter of a Taler. The question is : 'At this rate how 
 much should a Roman sextarius cost, if it takes 64 sextarii to 
 fill a Greek amphora?' " 2 
 
 In the early printed arithmetics, as in those of Borgi 3 and 
 Calandri,* the number of problems is meagre, each problem 
 usually being followed by its solution. In the arithmetics of 
 a date later than 1525 problems for practice are numerous.. 
 The explanation is probably to be found in the fact that the 
 cost of paper and printer's composition rapidly decreased. 
 
 Although, as we have said, applied arithmetic was gener- 
 ally thought of as a department by itself, certain applications 
 were distributed among the simple operations. The most im- 
 portant of these are the problems of denominate numbers (pp. 
 77-85 of this article), which were real applications and not 
 mere exercises in manipulating symbols often found in modern 
 arithmetics. But, passing to that department commonly called 
 practical arithmetic by sixteenth century authors, we find that 
 it was generally composed of a list of rules of operations under 
 
 1 Suevus, Arithmetica Historica (1593). " Plinius Lib. 12. Cap. 7. mel- 
 det: Das man ein Pfund sehwartzen Pfeffer v-mb vier Denarios gekaufft 
 habe/ das ist vmb einen halben Taler. 
 
 " Hier ist die Frage : Wenn drey Pfundt/ vnd drey viertel eines Pfundes 
 vmb zwolff Denar/ vnd vier Funmel eines Denarij gekaufft wurden : Wie 
 tewr vierzehen Pfund vnd zwey Drittel eines Pfundes im Kauff sein wur- 
 den?" Page 250. 
 
 2 Suevus, " Martialis meldet, Das ein Amphora Wein sey vmb 20. Asses 
 verkaufft worden/ das ist vmb 2 Denar/ so viel als ein Ort eines Talers. 
 Ist die Frage : Wie thewr ein Romisch Sextarius oder Nossel/ deren vier 
 vnd sechtzig auff ein Griechische Ampihoram gehen/ zu .rechnen sey?" 
 Page 256. 
 
 s Borgi, Arithmetica (1488 ed.). 
 •*Calandri, Arithmetica (1491), 
 
132 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 which are grouped the corresponding problems of business 
 concern. 
 
 The following may be taken as a typical category : 
 
 Rule of Three (Two and Five). Exchange and Banking. 
 
 Welsch Practice. Chain Rule. 
 
 Inverse Rule of Three. Barter. 
 
 Partnership (with and without time). Alligation. 
 
 Factor Reckoning. Regula Fusti. 
 
 Profit and Loss. Virgin's Rule. 
 
 Interest, Simple and Compound. Rule of False Assumption, or False 
 
 Equation of Payments. Position. 
 
 Besides these more general rules, the following were often 
 added : 
 
 Voyage. Rents. 
 
 Mintage. Assize of Bread. 
 
 Salaries of Servants. Overland Reckoning. 
 
 Gemma Frisius recognized the dependence of most of the 
 rules given in the above list upon the Rule of Three : 1 " From 
 this one rule, which in fact may be called ' The Golden Rule,' - 
 grow many different rules or methods of work, as the 
 branches of a tree grow from its trunk, so much so that it 
 has place in nearly all questions, and all canons lean upon it 
 as a foundation, or base, one of which is the Double Rule, 
 which you will understand from the following example." 
 
 The Rule of Three 3 was the method of simple proportion. 
 
 1 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1575 «&)■ 
 
 De Regulis vulgaribus. 
 " Ex una hac regula (quam vere auream licet appellane) multae diuer- 
 saeqs regulae, siue Canones operandi tanquam rami ex trunco oriuntur, 
 adeo vt in omnibus fere quaestionibus locum habeat ac omnes Canones 
 hinc innitantur, tanquam fundarnento seu basi, quarum vna est regula 
 duplex, quam ex tali exemplo intelliges.'' Fol. D g recto. 
 
 2 Jacob, Rechembuch auf den Ldnien und mit Zififern/ (1599 ed.). "So 
 viel hat ich von den progreffionen erzehlen und fetzen wollen/ Folgt/ 
 ferrner die Regel De tri/ von etlichen/ Proportionum, aiuoh fonften Aurea 
 Mercatorum genannt." Fol. D 6 recto. 
 
 3 Aryabhatta (c. 500 A. D.) used Rule of Three. See Rodet's Lecpns 
 de Calcul d'Aryabhata. 
 
THE ESSENTIAL FEATURES 133 
 
 Its explanation was sometimes expressed in general terms by 
 the writers of arithmetic, as in Heer's Arithmetic : * 
 
 " I am composed of three parts ; always place the question 
 last; whatever number is like the question put in the first 
 place. Multiply together the last and middle numbers and 
 divide the result by the first number. The quotient will be of 
 the same kind, or denomination, as the middle number. Thus 
 is the question solved." 2 Such generality, however, was un- 
 usual, the solution of a typical example ordinarily served as a 
 guide without the aid of a general theory or formula. 
 
 This statement of the rule was given by Borgi : 3 
 "A Rule Pertaining to Trading." 
 
 " Three quantities are known to find the other. Multiply 
 the second by the third and divide the result by the first." 
 
 " Example. The three numbers are 2, 3, 4. 
 
 3 X 4=12 12 -j- 2 = 6." 
 
 1 Heer, Compendium Arithmeticae (1617 ed.). 
 
 " Von dreyen bin ich zufamni gefetz 
 Die Frag fetz alle mal zu letzt 
 Vnd was die Frag fur Namen hat 
 Das ordne an die voider ftatt 
 Das hinder vnd rrmtler Multiplioir. 
 Was komt durchs vorder Dividir 
 Der Quotient bringt dir zur frift 
 Den Nam/ fo mitten geftanden aft 
 Damit if t der Frag auff gelof t du wift ?" 
 
 Fol. Avii verso and Aviii recto. 
 
 2 It is clear from this rule that, if the unknown term had been written, 
 it would have taken the fourth place in. the proportion, whereas in 1 the 
 modern form it takes ithe first place. 
 
 3 Borgi, Arithmetica (1540 ed.). 
 
 "■Como fi procede in tutte rafon merchadantefche per ditta regola." 
 
 " Altro non ci refta fe non a ueder in die modo per la precedete regola 
 fe die prooeder in el far delle rafon merchadantefche, cominciando in 
 quefto modo, fel te fuffe detto, fe .2. val .3. che valera .4. prima metti 
 quefte tre cofe vna drieto a laltra, cioe 2. val .3. che valera .4. prima metti 
 quefte tre cofe vna dnieto a laltra, cioe .2. 3. & .4. fi come tu vedi, poi 
 moltiplica la feconda in la terza, cioe .3. via .4. e fara .12. el qual .12. 
 parti per ia prima, cioe per .2. & in fira .6. e tanta> val el .4. adonque achi 
 ti diceffe fe .2. val. 3. clue valera 4. tu hai a rifponder che 1 val. 6. 
 
 "Se braza .3. de tela val fol .15. ohe valera el brazo. 
 
 " Se braza .41. de tela val fol. 17. che valera braza. 8." Fol. E 5 recto. 
 
I3 4 SIXTEENTH CENTURY ARITHMETIC 
 
 Problems: i. If 3 braza cost i$p. what does 1 braza cost? 
 
 6|3 PIS H 1 
 
 2. If 4>4 braza are worth /> 15, what will 8 braza be worth? 
 b I 4 J / 2 pij 6 | 8 
 
 Problem 1, of course, is a direct case of division. Problem 
 2 is the usual type of proportion. It was not unusual to pre- 
 face problems like the second by some like the first. 
 
 It has been explained on pp. 1 19-120 of this chapter that 
 proportion came to have its present meaning- in the sixteenth 
 century. Consequently one would expect to find some use of 
 proportion in solving problems and its relation to the Rule of 
 Three. This connection is supplied by Buteo in the following 
 treatment : 
 
 " Three numbers being given, to find a fourth proportional 
 number; called the Rule of Three." 
 
 " By many, indeed, it is called the Regula trium, or, as a 
 certain contemporary foreigner wrote it, regula de tri. By 
 others it is called ' The Rule of Four Proportional Num- 
 bers.' " 1 
 
 Ramus also combines the rule with proportion, for after 
 applying it to problems involving integers and fractions in the 
 usual way, he gives the following chapters : 
 
 Chap. VII. Golden Rule requiring Antecedent Proportion. 
 
 A typical problem is : " The lion on a fountain had 4 pipes, 
 of which the first fills the pool below in 24 hours, the second 
 in 36 hours, the third in 48 hours, and the fourth in 6 hours. 
 If they flow simultaneously, in how many hours will they fill 
 it? Add in turn the four ratios, 1 pool to the 4 periods of 
 time, and the total will be 37 pools and 144 hours, antece- 
 dents of the proposition. Since 37 pools are filled in 144 
 hours, therefore 1 pool is filled in 3-fr hours." 2 
 
 1 Buteo, Logistica, Quae & Arithmetica vulgo dicitur (1559). "A multis 
 fiquidem dicitur regula trium, vel ficut quidam Barbarus Icripfit tempore 
 no-ftro, regula de tri. Ab aliis regula quatuor proportionalium." Fol. g 4 
 recto, p. 104. 
 
 2 Ramus, Arithrneticae Libri duo. Liber II, Cap. VII. 
 
 "Leo fontis 4 fistulas 'habet, quarum prima knplet fubjectum lacum 24 
 horis, fecunda 36, tertia 48 quarta 6: fi fimul fluant, quot horis implebuntr 
 
THE ESSENTIAL FEATURES 1 ^e ) 
 
 Chapter VIII. Concerning Reciprocation. 
 
 A typical problem is : " When a measure of wheat is sold 
 for 5 aurei, a loaf of bread weighs 4 unciae; when, however, 
 it is sold for 3 aurei, a loaf of bread will weigh 6f unciae." 1 
 
 Chapter IX. Composite Proportion through Addition. 
 
 Chapter X. Alligation. 
 
 Chapter XL Composite Proportion through Multiplica- 
 tion only. 
 
 "A piece of tapestry 2\ ells long and 2 ells wide was sold 
 for 50 libelli, therefore a piece of tapestry of the same quality 
 1 ell long and f of an ell wide will sell for 1. 8 s. 6 d. 8." 2 
 
 Chapter XII. Composite Proportion through Multiplica- 
 tion and Addition. 
 
 Chapter XIII. Continuous Proportion for Finding the 
 Smallest Term in a Given Series. 
 
 In the fifteenth century the Italians originated a modifica- 
 tion of the Rule of Three for certain problems involving de- 
 nominate numbers. This method was first published in Ger- 
 many by Schreiber (Grammateus) in 15 18 3 under the name 
 of Welsch Practice. At that time the people of southern 
 France and northern Italy were often called Welsch by the 
 Germans, whence the name given to the process. Italian 
 writers refined the processes in arithmetic in many ways, 
 
 Adde rurfum quatuor rationes 1. lacus ad quadruplex tempus, tota ratio 
 erit 37 lacuum 144 horas antecedent proposition's. Die igitur 37 lacus ira- 
 plenitur 144 lioris, ergo 1. lacus impletur horis 344- -Tie: 
 
 1. 24 
 
 1. 36 
 
 1. 48 
 
 1. 6 
 
 37. 144. 1. 3ff" 
 
 1 Ramus, Arithmeticae Libri duo (1577 ed.). " Cum modiius tritici vaenit 
 5 aureis, turn panis est 4 unciarum : ergo cum vaenit 3, panis erit unciarum 
 6y 3 ." Page 59. 
 
 2 Ramus, "Aulaeum longum ulnas 2 & £, latu 2 emitur 50 libellis : ergo 
 tapetum ejusdem generis alteram longum ulnam 1, latum % emetur 1. 8. 
 s. 6. d. 8." Page 73. 
 
 3 Schreiber, Ayn new Kiinstlich Buch welcher gar gewifs vnd behend/ 
 lernet nach gemadnen regel Detre/ welschen practic/ (1518 ed.), fol. D x 
 verso. 
 
136 SIXTEENTH CENTURY ARITHMETIC 
 
 which led to the custom of designating- any ingenious opera- 
 tion originating with them as Welsch Practice. Thus, the 
 name became widely used in the broader sense, occurring in 
 the titles of many arithmetics of the sixteenth and seventeenth 
 centuries. 
 
 The technical difference between the Rule of Three and 
 Welsch Practice in the original narrow sense may be seen 
 from two examples from Riese. 
 
 1. Problem. " If 1 pound costs 3 groschen 9 denarii, how 
 much will 3 centner 2 stein 7 pounds cost ?" 1 
 
 Solution by the Rule of Three : 
 
 1 : 3 groschen 9 denarii = 3 centner 2 steins 7 pounds : 
 ( ). Since 1 centner = no lb., 1 stein = 22 lb., then 3 
 cent. 2 stein 7 lb. = 381 lb. Reduce the 3 gr. 9 d. to denarii. 
 The result is 45 d. 
 
 Then 1 145 = 381 lb. : ( ) 
 
 Reduce 381 X 45 d. to florins, groschen, and denarii. 
 
 2. Problem. If 5! lb. cost 32 fl. 13 gr. 12 d., what will 
 47 lb. 25 lot cost? 
 
 Then 5I- : 32 fl. 13 gr. 12 d. = 47 lb. 15 lot: ( ). 
 
 He now has to multiply the means together as in the Rule 
 of Three, but instead of reducing each compound number to 
 one denomination he performs the work as follows : 
 
 Sf lb. kosten 
 
 32 fl. 
 
 13 n gr. 12 <!. was 47 lb. 25 
 
 lot? 
 
 6 
 
 
 16 
 
 i\b. 
 
 35 
 
 194 fl. 
 
 22 n gr. 8 
 
 1 
 i 
 
 
 1358 fl. 
 
 lojifl. 1 
 
 1 
 
 T 
 
 
 776 
 
 10 li 
 2 U aus -J- 
 
 
 15 A- 
 
 20 n gr.T 
 
 
 15" 
 
 20 " j = 22 gr., mal. 47 
 
 
 3" 
 
 4 ■* J 
 
 
 97" 
 
 11 " -1 
 
 
 48" 
 
 20 " 9 (5.1= Preis fur 25 Lot 
 
 
 6 " 
 
 2 " io£ "J 
 
 
 5) 
 7)- 
 
 9304 fl. 
 i860 " 
 
 18 n gr. i\ (5. Summa 
 27 " "4V " 
 
 265 fl. 25 n gr. 6|f J 6. 
 1 Riese, Rechnung auff der Linien vnd Federn/ (1571 ed.), Unger, p. 92. 
 
THE ESSENTIAL FEATURES X yj 
 
 Explanation, i. Multiply the terms of the first ratio by 6 
 to avoid fractions. 
 
 Then, 35 : 194 fl. 22 gr. = 47 lb. 15 lot: ( ). 
 
 The equivalents here are 1 fl. = 30 gr., 1 gr. = 18 d., and 
 
 I lb. = 32 lot. 
 
 2. Multiply 194 fl. by 47. The result is 1358 
 
 776 
 
 3. Multiply 22 gr. by 47. To do this separate 22 gr. into 
 
 10 gr. -+- 10 gr. + 2 gr., which equals j/3 fl. + Yz fl. + H 
 of Y$ fl., and multiply each by 47. The result is 15 fl. 20 gr. 
 -f- 15 fl. 20 gr. + 3 A- 4 g r - (Note that 10, 10, 2 are parts 
 O'f 30 gr. (= 1 fl.) expressed by unit fractions.) 
 
 4. Multiply 194 fl. 22 gr. by 25 lot. To do this separate 
 25 lot into 16 1. + 8 1. + 1 1. = J4 lb. + J4 of J£ lb. + # 
 of Y* of Y* lb. and multiply 194 fl. 22 gr. by each term. The 
 result is 97 fl. 11 gr., 48 fl. 20 gr. 9 d. by taking Y* of the last 
 result, and 6 fl. 2 gr. 10% by taking Y& of the last result. 
 (Note again the unit fractions.) 
 
 5. The partial products are now added and divided by 5 
 and 7, or 35. 
 
 It is the ingenious method of separating the multipliers in 
 order to avoid reduction to one denominator, as in the solu- 
 tion by the Rule of Three, that constitutes the characteristic 
 feature of Welsch Practice; a good example of how the short- 
 est process may become the longest. 
 
 Problems then solved by the Rule of Three are now solved 
 by unitary analysis or by the equation, and Welsch Practice 
 has lost its virtue on account of the comparative simplicity of 
 the modern denominate number systems. It is interesting to 
 note that Welsch Practice in its early. form exactly coincides 
 with our present unitary analysis. This may be seen by an 
 example from Calandri : 1 " If a cogno of wine is worth 33 tf 
 
 II |8 4 s, what are 5 barili worth?" 
 
 1 Calandri, Arithmetica (1491 ed.). "El cogno del uino uale 33 if 
 
 11 p 4 s ehe uarranno 5 barili." Fol. 28 verso. 
 
33 — 
 
 II 
 
 — 4 
 
 S 
 
 3 — 
 
 7 ■ 
 
 
 - 1 ! 
 
 IS 
 
 
 
 
 i - 
 
 
 IS - 
 
 — o 
 
 o - 
 
 
 o - 
 
 — 5 
 
 - 
 
 
 - 
 
 — 3 
 
 138 SIXTEENTH CENTURY ARITHMETIC 
 
 EXPLANATION. 
 
 10) 33 — 11 — 4 S The top line is the statement 
 
 of the problem, namely, 10 barili 
 
 : 33 y 11 P 4 <? = 5 ban" : 
 
 ( ). This is evident, since 1 
 
 cogno =10 barili. The second 
 
 line is found by dividing the first 
 
 16 y 15 /3 8,5 two terms by 10. This gives the 
 
 cost of 1 barili. The other lines 
 
 are the results of multiplying the denominators of the second 
 
 line by 5, the relations being 20 P = 1 #, 12 s = 1 p. 
 
 Thus, the whole solution is briefly, if 10 barili cost 13 R 
 it P 4 6, 1 barili costs ■& of this. 5 barili cost 5 times the 
 latter result. This process we call unitary analysis. 
 
 In the Double Rule of Three, also called the Rule of Five, 
 are recognized the problems of compound proportion. Five 
 quantities are given to find a sixth. A typical example from 
 Gemma Frisius is: 
 
 " If 4 aurei must be paid for transporting 20 pounds of 
 merchandise 30 miles, how much must be paid for transport- 
 ing 50 pounds 40 miles ?" x 
 
 The Rule of Two was the process of dividing the product 
 of two numbers by the sum, and was applied to problems 
 whose solution could thus be obtained. In the Treviso book 2 
 it is applied to certain courier problems. 
 
 Inverse Rule of Three 
 The Inverse Rule of Three is explained by Baker thus : 
 
 *' Of the Backer Rule of Three." 
 " The backer rule of three is so called because it requireth a 
 contrarye working to that, which the Rule of three direct 
 doeth teache, whereof I have nowe treated." s 
 
 1 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.). 
 " Pro 20 Libris cuiufuis mercis, aductis per 30 milliaria, foluendi fumt 4 
 aurei, quantum pro 50 lib. aduoctis per 40 milliaria?" Pol. E verso. 
 
 2 See p. 9, bibliographical note. 
 
 3 Baker, The Well Spring of Sciences (1580 ed.), fol. Gvi recto. 
 
THE ESSENTIAL FEATURES I39 
 
 Example. "If 15 shillinges worth of Wyne will serue for 
 the Ordinary of 46 men when the Tonne of wyne is woorth 
 12 pounds: for howe many men will the same 15 shillinges 
 worth of wine suffice; when the ton of wine is woorth but 8 
 pounds." 
 
 Partnership 
 
 The first kind, simple partnership, was concerned with 
 dividing a gain or loss proportional to a given set of numbers. 
 The following is an example from Suevus prefaced by an in- 
 troduction resembling a Herbartian Preparation : 1 
 
 Introduction. " In the first chapter of the Prophet Jonah 
 we read how God enjoined upon the Prophet to proclaim to 
 the Ninevites his rig'hteous anger against their sins; this 
 Jonah refused to do, and for this reason betook himself to a 
 ship and departed with the sailors. But Almighty God ar- 
 rested the progress of the Prophet Jonah by a mighty tempest 
 which greatly frightened the sailors. As soon as they had 
 
 1 Suevus, Arithmetica Historica (1593). "Im Propheten Jona Cap. 1. 
 •lesen wir/ wie gott der herr dem Propheten auff erkget hat/ den Niniuiten 
 seineri gerechten Zorn wider ihre Sunde zuuerkundiigen/ des sich der 
 Prophete gewegert/ vnd sich derhalben auff ein Schiff begeben hat/ vnd 
 mit den Schiffleuten dauon gefahren isit/ aber der Allmaohtige gott hat 
 den Propheten Jonam, duroh einen grossen Sturmwind auff dem Meer 
 arestiret vnnd auffgehalten/ daruber die Schiffieute sehr ersohrocken sind/ 
 als bald das Schiff zu/ leichtern/ etlich Gerethe aussgeworffen/ audi 
 darumb (sender zweiffel aus sender schickung gottes) das Loss geworffen 
 haben/ zu erkundigen/ umb wen es doch inusse zu thun sein/ und weil 
 das Loss den Propheten lonam getroffen/ hat er sich willig darein begeben/ 
 das sae ihn aus dem Schiffe ins Meer gesturtzet haben/ welchen als bald 
 ein grosser Wallfisch auffgefangen vnd verschlungen/aber noch dreyen 
 tagen vnd nachten wieder zu Rande vnd Lande gebracht hat/ das er nach 
 dem befehl das Herrn den Niniuiten die Busse gepredigt hat. _ 
 
 " Dauon wollen wir aiuch ein nutzlich Exempel nemen. 
 
 " Wafi vier Kauffleuthe ein Schieff jnik Guttern beladen hatten/ 
 
 " 1. Einer mat vier und funfftzig Lasten : 
 
 " 2. Der ander miit zwey vnd Siebentzig Lasten : 
 
 " 3. Der dritte mit Hundert vnd vier vnd zwantzig Lasten : 
 
 " 4. Vnd der vierde mit Hundert vnnv Funfftzig Lasten : Jeder Last 
 auff 12 Tonnen zu rechnen/ dauon die Sohieffleute inn grossem vngewitter 
 das Schiff zu Leichtern/ haben Sechs Last vnd vier Tonnen auswerffen 
 mus'sen. 1st die Frage: Wie viel ein jeder Kauffman in sonderheit habe 
 schaden Leiden mussen." Pages 339-340. 
 
I4 o SIXTEENTH CENTURY ARITHMETIC 
 
 thrown over some of their wares to lighten the ship, they cast 
 lots (without doubt by the special Providential arrangement 
 of God) to find for whose sake the calamity had to happen, 
 and because the lot fell to the Prophet Jonah, he voluntarily 
 gave himself up that they might cast him from the ship into 
 the sea : as soon as they did this a great fish seized and swal- 
 lowed him. After three days and nights, however, he was 
 brought to the shore again that he might preach of repentance 
 to the Ninevites." 
 
 Problem. " From this we also wish to take a useful ex- 
 ample : 
 
 " Four merchants loaded a ship with goods ; the first fur- 
 nished 54 Lasten, the second 72 Lasten, the third 124 Lasten, 
 and the fourth 1 50 Lasten. Each Last is to be considered as 
 12 tons. In a violent storm the sailors had to throw over- 
 board 6 Lasten and 4 tons to lighten the ship. The question 
 is : ' How much must each mercant share in the loss ?' ' 
 
 The second kind, Partnership with Time, may be illustrated 
 by the following from Riese : 1 
 
 " Three men formed a partnership. The first gave 20 
 
 florins for 4 months, the second 24 florins for 3 months, and 
 
 the third 40 florins for 1 month. They made 101 florins; 
 
 how much belonged to each?" 
 
 Ans. 1st. 42 fl. 1 & 8 heller. 
 2d. 37 fl. 17 fi 6 heller. 
 3d. 21 fl. 10 heller. 
 
 80 
 
 192 101 fl. 72 
 
 40 
 
 1 Riese, Redlining auff ,der Linden und Federn/ (1571 ed.). "Item/ 
 Drey machen ein Gefelfohafft alio/ Der erft legt 20 floren/ 4 monat/ Der 
 ander 24 floren drey monat/ Vnd der dritte 40 floren ein monat/ Haben 
 101 fl gewunnen/ wie viel geburt jglichem? Facit dem eriten 42 fl/ I £/ 
 8 heller/ Dem andern 37 fl/ 17 fi/ 6 heller/ Vnd dem dritten 21 fl/ 10 hi'. 
 " Machs alio/ multiplioir jglichs Geld mit fainer zeit/ Summir/ wlrd 
 dem Theiler/ Vnd fetz darnach in maffan/ wie du oben gethan haft/ 
 Stehet alio." Fol. Hviiii recto. 
 
 80 
 
 192 101 fl 72 
 
 40 
 
THE ESSENTIAL FEATURES I4I 
 
 The problems in partnership reflect real conditions of that 
 time, although they seem artificial now, for there was a pop- 
 ular prejudice among Christians against taking interest, and 
 usury laws were made to prevent it. Hence, merchants 
 pooled their money in enterprises for various periods, often 
 very brief, and shared the profits in proportion to the amounts 
 invested and the times for which they were furnished. 1 
 
 The problems concerning pasturage (pasturing partner- 
 ship 2 ) are relics of the days of commons and shepherds. 
 Contractors rented large sections of the estates, which had 
 been farmed out to them by the state, to the owners of stock, 
 and these paid in proportion to the numbers in their herds.* 
 Such problems may be found in a few arithmetics at the 
 present time. 
 
 Tonstall gives a third case of partnership in which, while 
 different times intervene, money is drawn out also. 
 
 Example. " Four merchants formed a partnership for two 
 years. The first contributed 30 aurei at the beginning, and 
 after 8 months drew out 10 of them. At the beginning of the 
 twentieth month he contributed 12 aurei to the partnership. 
 At the beginning the second contributed 24 aurei, and at the 
 beginning of the sixteenth month he contributed 14 more 
 aurei. The third contributed at the beginning 20 aurei, at the 
 beginning of the seventh month he withdrew all his money, 
 and at the beginning of the eighteenth month contributed 16 
 aurei. The fourth at the beginning of the seventh month con- 
 tributed 18 aurei,. and at the beginning of the fourth month 
 after drew out 9 of these; again in the seventeenth month he 
 added 15 aurei to the business. How should a gain of 100 
 aurei be divided among them?" * 
 
 He gives the following solution with a careful explanation : 
 
 1 W. Cunningham, The Growth of English Industry and Commerce 
 during the Middle Ages (London, 1896), p. 364. 
 
 2 Ortega's Arithmetic (1515 ed.), fol. 762, has the expression "' com- 
 pagnia pecovaria." 
 
 " Ramsey, Manual of Roman Antiquities (London, 1901), p. 548. 
 4 Tonstall, De Arte Supputandi (1522), fol. Z 2 verso. 
 
620 
 
 
 -, c 205 
 
 558 
 
 100 
 
 -> T 4 03 
 3!«T 
 
 252 
 
 
 I4H» 
 
 3i8 
 
 
 i8ftV 
 
 1 4 2 SIXTEENTH CENT CRY ARITHMETIC 
 
 17480 
 
 Factor Reckoning corresponds to the modern topic of com- 
 mission, as shown by the following problems from Baker : 
 
 "A marchant hath delivered to his Factor 1200 li. to gouerne 
 them in the trade of marchandife, upon fuch condition, that 
 he for his feruice fhall have the i of the gaine, yf anything 
 be gained, and he fhall beare the i of the loss if any thinge be 
 loste : I demaunde for how much his person was esteemed." * 
 
 "A marchante hath delivered unto his Factor 1200 l'i and 
 the Factor layeth 500 l'i and his person. Nowe, because hee 
 layeth in 500 li. and his person, it is agreed between them y 
 he shall take § of the gaine : I demaunde, for how much his 
 person was esteemed ?" 2 
 
 Profit and Loss 
 
 Problems of this kind were often unclassified and used as 
 applications of the Rule of Three. But some writers grouped 
 them under a separate title, thus setting the precedent fol- 
 lowed until the present time. 3 
 
 A type example and the plan of solution is seen in the fol- 
 lowing from Riese: 
 
 "A man bought a centner of wax for i6| florins. How 
 many pounds will he sell for 1 /?. if he wishes to make 7 H. on 
 100 florins? Answer, 5 pfund 29 loth 3 quintle 2 pf. gewicht 
 otfif- heller." 4 
 
 (Reckon^ first how much wax. was bought for 100 florins, 
 
 1 Baker, The Well Spring of Sciences (1580 ed.), fol. Wii recto. 
 
 2 Baker, The Well Spring of Sciences (1580 ed.), fol. Wv recto. 
 
 3 Sfortunati's Arithmetic (1545 ed.), fol. 47 recto, has "di guadagni e 
 perdite.'' 
 
 4 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). "Item/ Ein 
 Centner Wachs fur 16 floren/ 3 ort/ Wie viel pfund komen fur 1 fl/ fo 
 man an 100 gewinnen wil 7 floren? Facit 5 pfund/ 29 loth/ 3 quintle/ 
 2 gewicht/ o heller/ vnd •g-S-f-j- teil. Machs alfo rechne zum erften 
 wie viel Wachs fur 100 floren kompt/ Als denn addir die 7 floren zu 100/ 
 vnd fprich/ 107 floren geben fo viel Wachs/ als hierin 63444 lb/ Was 
 gibt 1 fl? Brichs/ ftehet alfo." 
 
 6741 40000 lb 1 fl. Fol. Evii verso. 
 
THE ESSENTIAL FEATURES I43 
 
 then add 7 florins to 100, for which he will sell the same 
 amount of wax, or 634 £f lb.) How much does he sell for 
 1 A? 
 
 6741 — 40,000 lb. — 1 a. 
 
 It was common to speak of the gain per hundred, or per 
 cent, as it is now, for per cent always meant a rate in that 
 period. Thus, in Rudolff: 
 
 "A piece of velvet cost 36 no. ; it contained 15$ ells, for how 
 much should 6 ells be sold so as to gain 10 florins on a 
 hundred ? 
 
 Gain was also reckoned with time. The first example in 
 Tonstall is : 
 
 "A merchant gained 5 aurei in 3 months from. 70 aurei. 
 At this rate what would he gain from 70 aurei in 13 
 months?" 1 
 
 The other cases treated by Tonstall were : To find the time 
 when the gain is given. To find the gain from a larger 
 amount when the gain on a small amount is given. To find 
 the time in which a gain greater than: the money invested can 
 be found. 
 
 Simple- Interest 
 
 Van der Schuere began the subject thus: 2 Simple interest 
 is much like gain and loss with time, so that if you can work 
 one subject, you can easily understand the other. 
 
 "A man placed 100 L at simple interest for 4 years at 6%%; 
 how much did he have at the end of the time?" 3 
 
 The rates varied from 6 per cent to 12 per cent, although 
 there were exceptional extremes. 4 
 
 1 Tonstall, De Arte Supputandi (1522). "Mercator ex avreis septva- 
 ginta per menfes tres lucri fecit quincfc. quatum lucri tredecim menfibus ex 
 aureis feptuaginta obueniet?" Fol. a 1 verso. 
 
 2 Van der Schuere, Arithmetica, Oft Reken = const/ (1624 ed.). 
 
 "Den simp'len Int'rest, is, Winst end Verlies met Tijdt, 
 ■Ghelij ckend' een groot deel, dus van het werck subijt 
 Suldy veel haest verstant ghecrdjgh'en door u vlijt." 
 
 Fol. Pv recto. 
 
 3 Van der Schuere, Arithmetica (1624 ed.). "Eenen gheeft op Interest 
 100 L voor 4. Jaer/ om daer voor te hebben simpeleln Interest/ teghen 
 6% ten 100 t's Jaers/ Hoe veel ontfang hy dan ten eynde des tijdts." Fol. 
 Pv recto. 
 
 * Raets mentions 14%, and Trenchant 10% (1578 ed.), p. 300. 
 
144 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 Jean solved problems of interest from a table. This ex- 
 ample will illustrate: 
 
 " I wish to find the interest on 720 livres at 16 deniers per 
 livre. On line 16 I search for the sum, and when I find it I 
 refer to the number at the top of the column, where I find 45 
 which is the interest on 720 livres." * 
 
 Trenchant gives the following- interest table with interest at 
 12 .per cent on sums from 10,000 livres to 1 sou. 2 
 
 Interest. 
 I V. 
 
 Principal. 
 
 Interest. 
 
 
 Principal. 
 
 10000 V. 
 
 1200 1'. 
 
 
 9 1'. 
 
 
 9000 
 
 1080 
 
 
 8 
 
 
 8000 
 
 960 
 
 
 7 
 
 
 7000 
 
 840 
 
 
 6 
 
 
 6000 
 
 720 
 
 
 5 
 
 
 5000 
 
 600 
 
 
 4 
 
 
 4000 
 
 480 
 
 
 3 
 
 
 3000 
 
 360 
 
 
 2 
 
 
 2000 
 
 240 
 
 
 1 r 
 
 
 1000 
 
 120 
 
 
 
 19 
 
 900 
 
 108 
 
 
 
 18 
 
 800 
 
 96 
 
 
 
 17 
 
 700 
 
 84 
 
 
 
 16 
 
 600 
 
 72 
 
 
 
 15 
 
 500 
 
 60 
 
 
 
 14 
 
 400 
 
 48 
 
 
 
 13 
 
 300 
 
 36 
 
 
 
 12 
 
 200 
 
 24 
 
 
 
 11 
 
 100 
 
 12 1'. 
 
 r. 
 
 
 10 
 
 90 
 
 10 — 
 
 16 
 
 
 9 
 
 80 
 
 9 — 
 
 12 
 
 
 8 
 
 70 
 
 8 — 
 
 8 
 
 
 7 
 
 60 
 
 7 — 
 
 4 
 
 
 6 
 
 So 
 
 6 — 
 
 
 
 5 
 
 40 
 
 4 — 
 
 16 
 
 
 4 
 
 30 
 
 3 — 
 
 12 
 
 
 3 
 
 20 
 
 2 — 
 
 8 
 
 
 2 
 
 10 
 
 1 — 
 
 4 
 
 
 1 
 
 1 
 
 r. 
 
 H d - 
 
 19 
 
 — 
 
 2 t 
 
 16 
 
 — 
 
 9f 
 
 14 
 
 — 
 
 4* 
 
 12 
 
 — 
 
 
 9 
 
 — 
 
 n 
 
 7 
 
 — 
 
 H 
 
 4 
 
 — 
 
 9* 
 
 2 
 
 — 
 
 4* 
 
 2 
 
 — 
 
 1 9 
 3"5T 
 
 2 
 
 — 
 
 *n 
 
 2 
 
 — 
 
 °ii 
 
 
 — 
 
 "A 
 
 
 — 
 
 9f 
 
 
 — 
 
 8 A 
 
 
 — 
 
 6 it 
 
 
 
 
 
 
 
 
 
 •>21 
 2 § 
 
 
 — 
 
 °tt 
 
 
 — 
 
 "it 
 
 
 — 
 
 I0 ^5- 
 
 
 — 
 
 m 
 
 
 — 
 
 7i 
 
 
 
 cl 9 
 5 T7 
 
 
 
 
 — 
 
 A 8 
 
 
 — 
 
 2 M 
 
 The following is an example worked from the above table : 
 " To find the interest on 16,097 livres 8 sous." 
 
 1 Jean, Arithmetique (1637 ed.). " Ie veux tirer l'interest au denier 16 
 de la somme de 720 liures : le cherche done" ladite somme dans la ligne 16, 
 & l'ayant trouuee, ie regarde directement au dessus en la ligne capitale, ou 
 ie tro-uue 45, qui est 45 liures de rente que donnent lesdites 720 liures.'" 
 Fol. Aiiij recto. 
 
 2 Trenchant, L' Arithmetique (1578 ed.), fol. M recto. 
 
THE ESSENTIAL FEATURES I4 g 
 
 (Soit maintenant qu'il faille feauoir les interests de 16097 
 liures, 8 fouz de principal.) 
 
 10000 1' 1200 1' of den. 
 6000 720 
 
 go 10 16 
 
 7 
 
 16 9f 
 
 0-8 J "M 
 
 1931 1' 13 f 9^3- 6 
 
 Compound Interest 
 
 Compound interest was commonly called Jewish interest, 
 or profit, as is shown by the following from Van der Schuere : x 
 
 " When one wishes to gain money more quickly than can 
 be gained in the usual time, then one must learn to reckon well 
 what is his just due according to the Jewish profit." 
 
 The same tendency to associate compound interest with 
 Jewish practice is seen in Riese : 2 
 
 "A Jew lent a man 20 florins for 4 years, every half-year 
 he added the interest to the principal. Now I ask, how much 
 will the 20 florins amount to in 4 years, if every week the 
 interest on 1 floren is 2 denarii? Answer, 69 florins 15 gr. 
 
 ,-. 81256480 •>. 8 4 5 . " 
 9 39388806 86167 <>■ 
 
 Equation of Payments 
 
 This topic was not commonly given, A few authors gave 
 
 it separate treatment, but most of them condensed it into a 
 
 few problems and placed them under other rules, as that of 
 
 interest. Trenchant states the object of the process thus : To 
 
 1 Van der Scheure, Arithmetica, Oft Reken=const/ (1600 ed.). 
 
 "Soo yemandt van t'ghevvin oock vvinst vvil faeben snel, 
 Als vvinste niet ibetaelt en vvordt ter rechter tijdt, 
 So moet hy leeren hier berekenen seer vvel, 
 Wat hem met recht toecomt, al ist een Ioodtsch profijt." 
 
 Fol. Q 8 recto. 
 
 2 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). "Item/ ein 
 Jude leihet einem 20 floren 4 Jar/ vnd alle halbe Jar rechent er den gewin 
 zum hauptgut/ Nu frage ich/ wie viel die 20 floren angezeigte vier Jar 
 bringen mugen/ fo alle wochen 2^ von einem A gegeben werden? Facit — 
 gewin vnd gewins/ t z 69 floren/ 14 gr/ 9 <;/ vnd |i ||6480 2||45 te ;i » 
 Fol. Gv verso. 
 
146 SIXTEENTH CENTURY ARITHMETIC 
 
 reduce to a single payment at one time several items payable 
 at different times. 1 
 
 Exchange and Banking 
 
 Exchange as a business custom existed among the Greeks, 
 from whom it was communicated to the Romans. It is known 
 with certainty that Bills of Exchange existed about 309 B. C. 
 and were introduced into Italy from Greece. In Rome private 
 bankers were known as "Argentarii," and the practice of Ex- 
 change as " Permuta.tio." 2 The subject as it has appeared in 
 arithmetics was developed by the Italians in the sixteenth cen- 
 tury. According to Unger(Die Methodik, p. 90) the earliest 
 appearance of a bill of exchange was in Borgo's (Paciuolo's) 
 Summa (1494), fol. 167. The earliest Italian bank was 
 a kind of subtreasury of the Mint and was located at Venice. 
 Public banking took its rise in that city in 1587. 3 Tartaglia 
 gave four kinds of Exchange 4 and explained the conditions 
 for acceptance, protestation, and return. The problems of ex- 
 change are chiefly concerned with the translation of money 
 units, weights and other denominate number tables from one 
 system to another, as shown in Adam Riese : 5 
 
 " 894 Hungarian florins are equal to how many Rhenish 
 florins, when 100 Hungarian florins equal 129 Rhenish? Ans. 
 11 S3 ^- 5 P 2 t heller. Proceed thus: Add the exchange 
 to 100 Rhenish and say that 100 Hungarian florins make 129 
 
 1 Trenchant, Arithmetique (1578 ed.). " Remettre a vn iour de payment 
 vne ©u plufieurs parties payables a diuers termes." Page 316. 
 
 2 See " Exchange, Roman," in Harper's Diet. Classical Lit. and Antiq. 
 (N. Y., 1897), 2:1597. 
 
 3 C. A. Conant, A History of Modern Banks of Issue (N. Y., 1896). 
 
 4 Tartaglia, Tvtte l'Opera (1592 ed.), II, fol. 174 recto. 
 Catnbio (Exchange). 
 
 1. Minuto = common, meant changing money from one system to another. 
 
 2. Reale = chief, meant expressing the value of a sum of money in dif- 
 ferent places and covered remittances. 
 
 3. Secco=:dry, treated of drafts drawn on the maker. 
 
 4. Fittitio = special kind of secco, meant bills drawn with various de- 
 vices to prevent fraud. 
 
 K Riese, Rechnung auff der Linien und Federn/ (1571 ed.). "Item/ 894 
 Vngerifch fioren/ wie viel machen die Reinifch/ 29 auff? Facit 1153 
 Reinifch/ 5/82 heller/ und | teil. Thue jm alfo/ Addir den Auffwechffel 
 zu 100 Reinifch/ und fprich/ 100 Vngerifch thun 129 Reinifch/ wie viel 
 894 Vngerifch ? Facit wie oben." Fol. F verso. 
 
THE ESSENTIAL FEATURES 
 
 147 
 
 Rhenish. Haw many Rhenish florins will 894 Hungarian 
 florins make? Ans. Same as above." 
 
 The questions often included the matter of remittances also. 
 Thus, a person in Paris wishes to order the payment of 1200 
 crowns in Augsburg;, how many florins must be paid in 
 Augsburg ? 
 
 Since in the sixteenth century nearly every principality had 
 its own mint and its own system of coinage, a treatment of 
 exchange required a statement of the equivalents of many 
 systems. Thus, Tartaglia treats of exchange between these 
 places : 
 
 
 Rome 
 
 
 Venice 
 
 
 Naples 
 
 
 Pisa 
 
 
 Lyons 
 
 
 Siena 
 
 
 Antwerp 
 
 
 Naples 
 
 
 London 
 
 
 Bologna 
 
 
 Paris 
 
 
 Milan 
 
 Venice 
 
 Milan 
 
 
 Barcelona 
 
 and 
 
 Pisa 
 
 
 Provence 
 
 
 Perugia 
 
 
 Aquila 
 
 
 Bologna 
 
 Florence 
 
 Sicily 
 
 
 Genoa 
 
 and 
 
 Perugia 
 
 
 Florence 
 
 
 Rome 
 
 
 Valencia 
 
 
 Geta 
 
 
 Palermo 
 
 
 Avignon 
 London 
 
 
 Barcelona 
 
 
 Genoa 
 
 Avignon 
 
 Paris 
 
 
 Flanders 
 
 and 
 
 Florence 
 
 
 Majolica 
 Apulia 
 
 
 Venice 
 
 
 Rhodes 
 
 Pisa 
 
 Perugia 
 
 
 Constantinople 
 
 and 
 
 Rome 
 
 
 
 
 Barcelona 
 
 
 Venice 
 Genoa 
 
 
 Venice 
 
 Milan 
 
 Avignon 
 
 
 Milan 
 
 and 
 
 Pisa 
 
 
 Genoa 
 
 
 Paris 
 
 
 Paris 
 
 
 
 Bologna 
 
 Pisa 
 
 
 Venice 
 
 and 
 
 Rome 
 
 
 Pisa 
 
 
 Perugia 
 
 Genoa 
 
 Rome 
 
 
 Ferrara 
 
 and 
 
 Palermo 
 
 
 Siena 
 
 
 Barcelona 
 Paris 
 
 Paris 
 
 Bruges 
 
 
 
 and 
 
 Pisa 
 
 
 
148 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 It is easy to note from the problems of exchange the various 
 articles of trade. A few are : saffron, wax, wool, soap, tin, 
 sable, tallow, pepper, skins, furs, grain, ginger, cloves, camlet, 
 caps, fustian, tapestry, taffeta, worsteds, musk, linen, satin, 
 velvet, lead, iron, steel. 
 
 The following is a partial list of articles which were items 
 of exchange between the cities named : 1 
 
 Place sent from. 
 
 Article. 
 
 Place sent to. 
 
 Breslau 
 
 garments 
 
 Vienna 
 
 Bohemia 
 
 wool 
 
 Breslau 
 
 Prague 
 
 cloth 
 
 Ofen 
 
 Venice 
 
 cloves 
 
 Nuremberg 
 
 Eger 
 
 tin 
 
 Nuremberg 
 
 Venice 
 
 saffron 
 
 Nuremberg 
 
 Nuremberg 
 
 pepper 
 
 Vienna 
 
 Nuremberg 
 
 pepper 
 
 Breslau 
 
 Basel 
 
 paper 
 
 Nuremberg 
 
 Breslau 
 
 wax 
 
 Nuremberg 
 
 Posen 
 
 wax 
 
 Nuremberg 
 
 Augsburg 
 
 almonds 
 
 Vienna 
 
 Nuremberg 
 
 tin 
 
 Augsburg 
 
 Venice 
 
 soap 
 
 Augsburg 
 
 Venice 
 
 cloves 
 
 Vienna) 
 
 Vienna 
 
 wine 
 
 Nuremberg 
 
 Augsburg 
 
 silver 
 
 Vienna 
 
 No adequate idea of the subject of Exchange can be given 
 in a brief general article, for when the equivalents for weights, 
 measures, and moneys of different systems are considered the 
 matter increases to volumes. For a work of two hundred 
 pages on this subject see Pasi, page 83 of this monograph. 
 
 Chain Rule 
 
 Although the name of this rule had a curious origin, 2 the 
 
 1 Rudoiff, Kiinstliche rechnung mit der Ziffer und mit den zalpfennige/ 
 (1534 ed.). Under Chapter on Wechfel. 
 
 See also Exempel Buchlin (1530 ed.), fol. d 8 recto. 
 
 2 According to Cantor this term has its origin thus : In Menelaus's pro- 
 position in whioh a line divides the sides of a triangle into six segments, 
 the transversal was called sector (cutter). The Arabs translated this 
 
THE ESSENTIAL FEATURES 149 
 
 meaning given to it in the sixteenth century was peculiarly 
 appropriate. In the Arithmetics of that period it meant a rule 
 to find the relation between two' denominate numbers meas- 
 ured in different units by means of a series of intermediate de- 
 nominate numbers. The following example from Riese will 
 illustrate: 7 pounds at Padua make 5 pounds at Venice, 10 lb. 
 at Venice make 6 lb. at Nuremberg, and 100 lb. at Nuremberg 
 make 73 lb. at Cologne; how many pounds at Cologne do 
 1000 lb. at Padua make? 
 
 The work is arranged thus : 
 
 7 lb. Padua = 5 lb. Venice 
 
 10 lb. Venice — 6 " Nuremberg 1000 Padua 
 
 100 lb. Nuremberg = 73 " Cologne 
 Then, 7,000 lb. Padua = 2190 lb. Cologne, multiplying in 
 columns. Therefore, 1000 lb. Padua = 1000 lb. X rrir = 3 I2 # 
 lb. Cologne. 
 
 The Italian plan of arranging the work brought out more 
 clearly the significance of the name, Chain Rule. By their 
 method the above problem would be solved thus : 
 
 7 ^«u(t**_ 
 
 j~YZnice. 
 
 J 00 touutAiU > 
 
 <73 C^k^. 
 
 
 /0 lrtAA**jL 
 
 6 UuA€**+t*ULlf 
 
 "* (/CV-Q^r>r/<.cttJ 
 
 Divide the product of the numbers on the broken line from 
 Padua to Padua by the product of the numbers on the line 
 from Cologne to Venice. Then 1000 lb. Padua= ^-^".fF— 
 lb. Cologne = 312! lb. 
 
 Examples of this nature were given by Brahmagupta 1 
 (c. 700 A. D.) and by Leonardo of Pisa 2 (1202). In Ger- 
 many they were usually solved until 1550 by repeated ap- 
 plication of the Rule of Three. Widman, 3 however, gave the 
 
 al-katta, which appeared in the Latin of Leonardo of Pisa's Liber Abaci, 
 as ifigua cata. Cantor, Geschichte der Mathematik, 2 : 15. 
 
 1 Unger, Die Methodik der praktischen Arithmetik, p. 91. 
 
 2 Scritti di Leonardo Pisano, I, pp. 126, 127. 
 
 8 Widman, Behede Rechnung (1489 ed.), fol. 152. 
 
15° 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 Italian form, Apianus (1527) explained the difference be- 
 tween the Chain Rule and the Rule of Three, Rudolff ( 1 540) 
 gives some examples, and Stifel (1544) made a clear and 
 formal explanation. The method reached England in the 
 seventeenth century, for it appears in Wingate's Arithmetic 1 
 (1668). 
 
 The rule has several names, " Vom Wechsel," because of 
 its connection with exchange, " Vergleichung von Mass und 
 Gewicht," "Verwechselung von Mass und Gewicht," 2 " figua 
 cata," s " regula del chatain," 4 " Regula pagamenti, 6 and 
 " Kettensatz," the name that became general in Germany in 
 the eighteenth century. 
 
 Barter 6 
 Although Barter was an extensive custom among primitive 
 peoples, 7 it may seem strange that it should find place as a 
 subject of instruction up to the last century. 8 There are 
 two reasons why the subject was of sufficient importance in 
 the sixteenth century to have given rise to a chapter in the 
 arithmetics. 9 First, the scarcity of coined money, 10 and second, 
 the custom of holding interstate fairs. 11 Not until the dis- 
 covery of large quantities of gold in the New World was 
 there a suitable metal in sufficient quantities to supply the de- 
 mands of trade; hence, the direct exchange of goods was 
 
 1 Villicus, Geschichte der Rechenkunst, p. 101. 
 
 2 Widman. 
 
 8 Leonardo of Pisa. 
 
 * Ghaligai, Practica D'Arithmetica. 
 
 " Widman. 
 
 8 Often called Stich Rechnung. Heer, Compendium Arithmeticae (1617 
 ed.). Fol. Giij recto. 
 
 7 W. Cunningham, The Growth of English Industry and Commerce (Lon- 
 don, 1896), p. 114. 
 
 " It persisted in holding a place in the Arithmetics of the nineteenth 
 century. See Pike's Arithmetic, 8th ed. (N. Y., 1816), p. 221. 
 
 9 Ciacchi, Regole generali d'abbaco (Florence, 1675), p. 114. 
 
 10 See "Barter," New International Encyclopedia (New York, 1901-4). 
 Cunningham, Cambridge Modern History, I, Chap. XV (London, 1902). 
 
 11 Cataneo, Le Pratiche (1567 ed.), fol. 49 verso. 
 
THE ESSENTIAL FEATURES 
 
 ISI 
 
 essential to commercial progress. The great fairs which cor- 
 responded to the International Expositions of the present time 
 served to encourage this form of trade. Thus, the many 
 technical questions about the expressions of values of goods in 
 different systems and the methods of calculating the amount 
 of one product to be exchanged for another necessitated a 
 treatment of the subject in the arithmetics of that time. In 
 barter the prices of articles were usually placed higher than in 
 selling for cash. An example in barter from Baker reads : * 
 " Two marchants will change their marcadise, the one with 
 the, other. The one of them hath cloth of 7 s 1 d. the yard 
 to sell for readye money, but in banter he will sell it for 8 s 4 d. 
 The other hath Sinamon of 4 s 7 d' the li. to sell for readye 
 moneye. I demaunde how he shall sell it in barter that he be 
 no loser." 
 
 Alligation 
 
 In the sixteenth century alligation found application chiefly 
 in problems of the mint. Among others Rudolff gave these 
 two problems : 2 
 
 "A man has refined silver containing 14^ lot per marck and 
 coins containing 4i lot per marck. How much of each will he 
 need to make 40 marcks in which each marck will be 9 lot 
 fine? Ans. 18 m of silver and 22 m of coins." 
 
 "A mint-master has some refined silver containing 14^ lot 
 per marck. How much silver and how much copper must he 
 take in order to have 45 fh, each marck being 9 lot fine? Ans. 
 Pure silver, 27 fh 14 lot 3 qh 2 ^ ', copper, 17 in 1 lot o qn 
 
 1 9 a 
 
 The following is from Thierf eldern : 3 
 
 1 Baker, The Well Spring of Sciences (1580 ed.), fol. Wv verso. 
 
 2 Rudolff, Kiinstliche rechnung mit der Ziffer und mit den zalpfennige/ 
 (1534 ed.). 
 
 Similar problems are found in Rudolff 's Exempel Buchlein (1530), fol. 
 F g verso and G recto. 
 
 3 Thierfeldern, Arithmetica (1587 ed.). "Item/ ein Herr hat dreyerley 
 Gold/ wegen/ das erste 15 marck/ helt ein marck 15 karat/ 3 gran/ das 
 ander 21 marck/ helt die marck 17 karat 2 gran/ das dritte 48 marck 
 
152 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 "A man had three qualities of gold, the first contained 15 
 marcks, each marck containing 15 karats 3 grains, the second 
 contained 21 marcks, each marck containing 17 karats 2 
 grains, the third contained 48 marcks, each marck containing 
 12 karats 1 grain. What is the greatest weight the metal 
 resulting from a mixture of these can have so that each marck 
 may contain 14 karats 3 grains? Ans. The resulting metal 
 will weigh 56^ marcks, for which he takes the first two and 
 from the third takes 29^5- marcks." 
 
 Regula Fusti 
 
 The Regula Fusti is an application of the Rule of Three to 
 problems involving a reduction for impure or damaged goods. 
 The problems refer to such commodities as spices, gold, silver, 
 honey, and oil. Thus : x 
 
 "A sack of pepper weighs 3 centners 50 lb. The tare for 
 the sack is 3I lb., each centner contains 11 lb. of fusti. One 
 pound of fusti cost 4 gr. and a centner of pure pepper 724 fl. 
 The question is : What is the pepper worth ?" 
 
 Another example is from Simon Jacob : 2 
 
 "A merchant bought at Frankfurt a sack of cloves weighing 
 
 2 centners 45^ lb. The tare was 8i lb., and each centner con- 
 tained 16 pounds of fusti. A pound of pure cloves cost 21 /?, 
 and a pound of fusti 6 /3. He then went to Nuremberg. His 
 expenses were 5! fl. For how much must he sell the cloves 
 
 helt 1 marck 12 karat/ 1 gran/ wie vil mag er von difen am meyften 
 ■befchicken/ das ein marck hake 14 karat/ 3 gran? facit/ das Werck wird 
 56-jij. ms. dar zu nimpt er die erften zwey/ vnd von dritten 29 1 marck." 
 Page 193. 
 
 1 Thierfeldern, Arithmetiica (1587 ed.). "Item/ ein Sack Pfeffer .wigt 
 
 3 cr. 50 lb. Thara fur den Sack/ 3| lb. helt der cr. 11 lb. Fufti/ kost 
 1 lb. Fufti 4 gr. vnd ein cr. lauter 72^ fl. Ift die Frag/ was der Pfeffer 
 geftehe? Fa. 231 ft. o gr. 8 8 o|| hr." Page 119. 
 
 2 Jacob, Reohenbuch auf den Linden und mit Ziffern (1599 ed.). "Item/ 
 einer kaufft zu Franckfurt einen Sack mit Naglin/ der wigt 2 centner 454 
 lb. tara 8-J pfundt/ helt der centner 16 pfund Fusti/ .das lb. lauter vmb 
 21 /3. das lb. Fusti vmb 6. /3. die bringet er gehn Nurnberg/ gestehen mit 
 vnkosten dahin s-J fl. wie soil er da selbst 1 lb. durch einander verkauffen/ 
 das er vber alien kosten 30 gulden gewine. Vnd ich «etze das Frankforter 
 gewicht gleich dem Nurntoerger ?" Fol. Mv verso. 
 
THE ESSENTIAL FEATURES 
 
 153 
 
 a pound that he may gain 30 guldens above all costs? And I 
 reckon the Frankfurt weight equal to that of Nuremberg." 
 
 Virgin's Rule, also called Rule of Drinks. 
 
 This rule (Regula Virginum, or Regula Cecis) 1 grew out 
 of the custom of charging men, women and maidens different 
 prices for their drinks. Riese explains it thus : 2 "At times it 
 chances that many people of different kinds are included in one 
 bill and the reckoning is obscure as when men, women, and 
 maidens are included in a reckoning for money spent in drink- 
 ing and they are not to pay equally. To make such a reckon- 
 ing you must study industriously this excellent rule, called the 
 Rule of Drinks." 
 
 Thierfeldern gives this example: 3 
 
 " 47 people, men, women, and maidens together spent 47 gr., 
 each man gave 5 gr., each woman 3 gr., and each maiden 1 hr. 
 How many persons of each kind were there? Ans. 3 men, 4 
 women, and 40 maidens." 
 
 Rule of False Position (Single; Double) 
 
 The Rule of False Position (Regula Falsi), essentially an 
 
 algebraic process, is as old as Egyptian mathematics. It was 
 
 used to solve various indeterminate problems. 4 Gemma 
 
 Frisius explains the name thus : 5 " This rule which we are 
 
 1 This name is derived from the Arabic cintu Sekes, according to Zeu- 
 then in L'Interm., 1896, p. 152 (quoted by Enestrom B. M., 10(2) : 96). 
 
 2 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). " Es begeben 
 fich zu zeiten viel und mancherley rede unter den Leyen/ und unverften- 
 digen der Rechnung/ Als wenn Menner/ Frawen/ und Jungfrawen in 
 einer Zeche verfamlet/ ein anzal gelds vertrincken/ und nicht zu gleich 
 bezahlen/ Solches zu machen/ foltu mit fleis diefe hiibfche Regel mercken/ 
 welche Cecis genant wird." Fol. Lvii recto. 
 
 3 Thierfeldern, Arithmetica Oder Rechenbuch (1587 ed.). "Item/ 47 
 Perfonen/ Mann/ Frawen und Jungfrawen/ haben verzehrt 47 gr. ein Mann 
 gibt s gr. ein Fraw 3 gr. ein Jungfraw 1 hr. Wie vil find jeder Perfon 
 dn fonderheit. facit/ 3 Man/ 4 Frauwen/ vnd 40 Jungfrawen." Page 215. 
 
 * The Arabs called it the operation with scales, because of the figure, 
 ZZXZZ, used in the method. Steinschneider, Abhandlungen, 3:120. 
 
 6 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1575 ed.). 
 " Vocatur autem regula quam iam docemus, Falsi, non quod falsum doceat, 
 sed ex falso verum elicere, fit qj in hunc modum." Fol. F recto. 
 
154 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 now teaching is called the Regula Falsi, not because it teaches 
 what is false, but because it teaches to find the true through 
 the false." 
 
 Tonstall * says of the name that the Arabs and Phoenicians, 
 celebrated merchants, from whom arithmetic is thought to 
 have originated, called this method of finding the truth by the 
 foreign word, cathaym. The Latin races called it either the 
 Rule of False Position or the Rule of False Assumption. 
 
 Variations of this word are Kataim used by Cardan 2 and 
 Helcataym used by Tartaglia. 3 Baker 4 speaks of this rule 
 as the " Rule of Falsehoode, or false positions." 
 
 Under applications of this rule Widman gives many number 
 puzzles, as : 5 " You are to find for me a number to which if 
 I add | of itself and divide the result by 4^, the answer will 
 be 12." 
 
 "Divide for me 15 into two unequal parts SO' that if I 
 divide the larger by the smaller, the result will be 19." 
 
 The following example from Onofrio will illustrate the 
 method of working : 8 " This principle is illustrated by several 
 
 1 Tonstall, De Arte Supputandi (1322 ed.). "Arabes et Phoenices mer- 
 catura celebres, et a quibus Arithmetica profecta primum putatur : artem 
 illam ueritatis inueniende barbaro uocabulo Cathaym appelant. Latini fiue 
 falfaru pofitionu, fiue falfaru coiecturaru regulas uocat. Fol. r recto. 
 
 2 Cardan, Practica Arithmetica (1539 ed.), Chap. 47, Lii verso. 
 
 3 Tartaglia, La Prima Parte Del General Trattato (1556 ed.), Book 16. 
 * Baker, The Well Spring of Sciences (1580 ed.), fol. Zv verso. 
 "Widman, Behend und hupsch Rechnung (1508 ed.). " Du solt mir 
 
 sflche ein zal wen ich J der selben zal dar zu addir/ vn darnach das aggre- 
 gat in 4-J. partir/ das mir 12 kumen." Fol. e 3 recto. 
 
 "Diuidir mir 15 in 2 teil die vngleich sein/ vnd we ich dz grost diuidir 
 durch dz kleinst das 19 kume/. 
 
 Onofrio, Arithmetica (1670 ed.). "E per dar principio a gl' effempij 
 f ia quefto il primo. Ottauiano Semproni compro tre diamanti ; il fecondo 
 li cofto on. 4. piu del primo, & il terzo quanto il primo, e fecondo, & on. 
 5 piu; in tutto fpefe on. 81. quanto dunque li cofto ciafcun diamante? 
 Per folutione della prefente domanda, fupponi il primo diamante efferli 
 coftato on. 24. il fecondo, perche dice la domanda, che li cofto on. 4. piu 
 del primo; li fara coftato on. 28. & il terzo, perche cofto quanto il primo, 
 e fecondo, & on. 5. piu, dunque cofto on. 57. la fomma delli quali tre 
 numeri s'e on. 109. & eglino coftarono on. 81. dunque la noftra pofitione 
 
THE ESSENTIAL FEATURES 
 
 155 
 
 examples, of which this is the first : Ottaviano Semproni 
 bought three jewels, the second of which cost 4 on. more 
 than the first, the third cost 5 on. more than the first, and the 
 third cost 5 on. more than the first and second together, and 
 all three cost 81 on. Required the cost of each. In order to 
 solve, suppose for the present that the first jewel cost 24 on., 
 the second, because it was to cost 4 on. more, cost 28 on., the 
 third, because it was to cost as much as the first and second 
 together and 5 more, cost 57 on. The sum of these make 
 109, and since they are to cost 81 on., then our position is 
 false by excess; this excess (since 28 is the result of taking 
 81 from 109) will be designated by the letter, P, in this man- 
 ner 24 P. 28. 
 
 fu falfa per ecceffo, quale ecceffo (che fono on. 28. perdhe tanto auanza 
 il numero 109. al numero 81.) fi notera con la lettera P, in quefta maniera. 
 24. P. 28. 
 
 " Faccifi vn' altra nuoua pof itione e fuppongafi il primo diamante hauer 
 coftato on. 20. il fecondo, perche cofto onze 4. piu, fara conftato on. 24. 
 & il terzo, perche cofto quanto il primo, e fecondo, e 5 piu, hauera coftato 
 on. 49. fommati quefti tre numeri fanno on. 93. & eglino 
 doueano fare on. 81. dunque habbiamo di nuouo auanzato dalla 24. P. 28 
 verita per on. 12. e pero noteremo quest' errore parimente con 20. P. 12 
 la lettera P, cosi dunque ftara l'effempio. 
 
 " Hor per trouare la verita mediante la proportionalita della pofitioni 
 con quella degl' errori, cosi s'operira. Perche l'vna, e l'altra pofitioni haue 
 auanzato la verita, fi fottrarra il minore errore dal maggiore, cioe 12. da 
 28. e rimarra 16. quale fi notera fotto per partitore : doppo fi moltiplice in 
 croce la prima pofitione, cioe 24. per il fecondo errore 12. & il prodotto 
 288. fi fcriuera alia parte deftra del medefimo errore 12. come in queft' 
 ef fempio appare : parimente f i moltiplichera la f econda pofitione 20. per 
 il primo 24. P. 28 560 
 
 >< 
 
 20. P. 12 288 
 
 Partitore 16. 272 Partitione 
 
 Quotiente 17. 1 12 
 
 — o 
 errore 28. & il prodotto 560. fi fcriuera dalla parte deftra del medefimo 
 errore 28. delli quali due prodotti fottraito il minore dal maggiore, cioe 
 288. da 560. reftera 272. da partirfi al partitore 16. fi che partendo 272. a 
 16. il quotiente fara 17. & onze 17. cofto il primo diamente, il fecondo on. 
 21. cioe on. 4. piu, che il primo, & il terzo on. 43. cioe quanto il primo, e 
 fecondo, e 5 piu, quali tre numeri infieme vniti fanno on. 81. come nella 
 domanda fi cercaua." Fol. Ee verso. 
 
156 SIXTEENTH CENTURY ARITHMETIC 
 
 " Take a new position and suppose that the first cost 20 
 
 on., the second, since it cost 4 on. more, would cost 24, and 
 
 the third, because it was to cost 5 more than the first and 
 
 second together, would cost 49 on. The sum of these would 
 
 then be Q3, where it should be 81, then we have 
 
 yj 24. P. 28 
 
 the new variation from the truth, 12, which error 20 p I2 
 
 we designate by the letter, P, as in the example. 
 
 " The truth is found by finding the mean proportionals be- 
 tween these positions together with their errors. 
 
 " Since the former and the latter positions vary from the 
 truth, if the smaller group is subtracted from the larger, as 
 12 from 28, there remains 16, which we place below for a 
 divisor. Then we multiply crosswise the first position, 24, 
 by the second error, 12, which gives the product, 288, which 
 we place at the right of the error, 12, as shown in this example: 
 24. P. 28 560 
 
 >< 
 
 20. P. 12 288 
 
 Partitore 16. 272 Partitione 
 
 Quotiente 17. 112 
 
 — o 
 
 " Then multiply the second assumption, 20, by the first 
 error, 28, and the product is 560, which is placed at the 
 right of the error, 28, then the difference between these pro- 
 ducts is found which is 272. When this remainder is divided 
 by 16, the quotient will be 17, therefore the first jewel cost 
 17 on., the second 21,4 on. more than the first, and the third, 
 43, equal to 5 more than the first and second; the three to- 
 gether make 81, as the problem required." 
 
 Most of the minor rules, mentioned p. 132, are special cases 
 of other rules. That is, they designate more minute divisions 
 used by a few authors for particular problems and included by 
 most writers under more general rules. Thus, voyage J was 
 commonly used to stand for courier problems. 
 
 Problems of the mint were very important at that time, be- 
 
 1 Van der Schuere, Arithmetica, Oft Reken = const (1600 ed.), fol. 
 Ziiii recto, gives the Hound and Hare problem, the Mule problem, and 
 other courier problems. 
 
THE ESSENTIAL FEATURES 1 ry 
 
 cause the coinage of money was delegated to local autl. rities 
 and on account of the multiplicity of standards. Although 
 these questions were often treated under Alligation, many 
 authors grouped them under the title Mintage. 1 
 
 Certain practical arithmetics emphasized solutions of ques- 
 tions of householders and landlords and designated the prob- 
 lems by such titles as Salaries of Servants and Rents. 2 
 
 Among the many safeguards which European nations have 
 thrown about general public interests for centuries is the legal 
 standardizing of bread. The weight of a loaf of bread which 
 sold for a fixed price was regulated according to the price of 
 the grain from which it was made. The earliest regulation 
 yet found is the Frankfurt Capitulare (794 A. D.). London 
 regulations are found as early as the twelfth century. An- 
 other good specimen is the "Assize of Bread " of the time of 
 Henry II. The general law which was practically followed 
 was : The weight of the loaf varies inversely as the price of 
 wheat. 3 The following from Finaeus is a typical bread prob- 
 lem, as found in the arithmetics of the sixteenth century : * 
 "When a bushel of wheat is sold for 34 shillings (for ex- 
 ample), and the bread made from it is sold at 6 denarii per 
 loaf, one observes that the weight is 12 ounces; if the same 
 bushel of wheat is sold at 28 shillings, how many ounces must 
 be put into each loaf to sell for 6 denarii?" 
 
 Overland Reckoning 
 The title, Rechnung Uber Land, really a synonym for Ex- 
 
 1 Van der Schuere, Arithmetica Oft Reken = const (1600 ed.), fol. Yiiii 
 recto. 
 
 2 Unicorn, De L' Arithmetica universali (1558 ed.), fol. Ccccc 4 verso. 
 
 a W. Cunningham, The Growth of Industry and Commerce during the 
 Early Middle Ages (London, 1896), p. 68. 
 
 An excellent explanation of the English Law in 1800 is found in Nasmith, 
 An Examination of the Statutes now in Force Relating to the Assize of 
 Bread (Wisbech, 1800). 
 
 4 Finaeus, De Arithmetica Practica (1555 ed.). "Cum medimnus tritici, 
 uaenit (exempli gratia) duodenis 34, & confectus ex illo panis 6 denarioru 
 turonen obferuat podus 12. unciarum: fi idem medimnus tritici, uenerit 
 ad pretium 28 duodenorum, queritur quot unciaru formandus erit idem 
 panis 6 denorioru?" Fol. Sij recto. 
 
158 SIXTEENTH CENTURY ARITHMETIC 
 
 change, was used by some writers in a broader sense, namely, 
 to include problems concerning the purchase of foreign goods 
 as well as the methods of remitting money. Trenchant * 
 gave twenty-three pages to the subject as well as several ap- 
 pendices and included in it the treatment of Exchange, explain- 
 ing four cases similar to those of Tartaglia. See page 146, 
 note 4 of this monograph. 
 
 Because problems on gain and loss, interest and discount 
 were abundant, one naturally seeks for a treatment of per- 
 centage. But percentage as a separate subject did not appear 
 until the end of the sixteenth century. There appeared, how- 
 ever, under the various subjects, problems of that nature. The 
 symbol, %, although not in the text-books of that period, 
 originated about the beginning of the fifteenth century. 2 
 
 The following are among the various expressions used for 
 
 per cent: 
 
 p cr. 3 von 100 * pour ioo D ten 100 ° 
 
 mt hundert 7 int hundert s met 100 ° .mit IOO 10 
 
 per cento 11 pro 100 12 p 100 13 uppon the 100 1 * 
 
 an ioo 15 P 2 1G auff 100 17 per C. 18 
 
 1 Trenchant, L'Arithmetique (1578 ed.), p. 340. 
 
 2 The origin of the present sign, %, as an abbreviation for per cento 
 has recently been traced by Dr. David Eugene Smith, Columbia Univer- 
 sity, New York, to a manuscript of the first half of the fifteenth century. 
 
 3 E. g., Heer, Compendium Arithmeticae (1617), fol. F recto et al. 
 
 i E. g., Heer, fol. Fv verso. Jacob, Rechenbuch auf den Linien und mit 
 Ziffern (1599 ed.), fol. Mv recto. 
 
 6 E. g., Wencelaus, T'Fondament Van Arithmetica (1599 ed.), under In- 
 terest. 
 
 c Ibid. 
 
 7 E. g., Van der Schuere, Arithmetica (1600 ed.), under Gain and Loss. 
 
 8 Ibid. "Ibid. 10 E. g., Jacob, under Gain and Loss. 
 11 E. g., Chiarini and Jacob, under Gain and Loss. 
 
 12 E. g., Finaeus, De Arithmetica Practica (1555 ed.), fol. Si verso. 
 
 13 E. g., Unicorn, De L' Arithmetica vniuersali (1598 ed.), under Interest. 
 14 E. g., Baker, The Well Spring of Sciences (1580 ed.), fol. Riiii verso. 
 
 16 E. g., Heer, fol. Fiij recto. 
 
 10 E. g., Giocomo Filippi Biordi, Arithmetica et Prattica (MS.) (1684). 
 He uses "12 p |-|- i'' for I2i%. 
 
 17 Jacob, " Setz 93 fl. Iosung (verftehe aufi 100 fl.) gehen 3 J fl. wie viel 
 117 fl." Fol. Mv recto. 
 
 18 Chiarini, Qvesta e ellibro che tracta de Mercatanti et vsage de paesi." 
 
THE ESSENTIAL FEATURES 
 
 159 
 
 Puzzles 
 
 Besides the practical problems which were classified under 
 the general rules, there were many problems famous as 
 puzzles or amenities. The work of Bachet de Meziriac 1 
 (1624), a book now generally accessible through modern 
 editors, is a collection of such problems known in his time. 
 The following list is incomplete, but it contains the most in- 
 teresting of those contained in the arithmetics consulted in 
 preparing this monograph ; the writers mentioned are those in 
 whose arithmetics the problems appeared, but not the origin- 
 ators of the problems. Only the essential feature of each 
 problem is stated : 
 
 Potato Race. One hundred stones (or potatoes) are placed 
 in a row, the adjacent stones being 1 yard apart; how many 
 yards will one have to travel in starting with the first and 
 bringing each stone separately to the position of the first? 
 (Trenchant). 
 
 Snail in the Well. There is a well 20 fathoms deep. Every 
 day a snail climbs 7 fathoms and at night falls back two 
 fathoms. In how many days will he come from the well? 
 Ans. 3f days. (Rudolff, Riese, and others). 
 
 Chess-board Problem. Required the number of kernels of 
 wheat needed in order to place 1 kernel on the first square of 
 a chess-board, 2 on the second, 4 on the third, and so on for 
 the 64 squares. Given by Masudi, (Cairo, 950) in " Meadows 
 of Gold." See Bone. Bull., 13 :274. 
 
 Eating and Drinking Problems. Some hunters with 
 loaves of bread and bottles of wine meet at a spring; they 
 seek to divide the refreshments so that each shall share ac- 
 cording to what he brought. (Ghaligai, fol. 66 recto). 
 
 Similar problems about drinking wine were given by Re- 
 corde. In the Dutch arithmetics this problem takes the form 
 of a dispute between a lion, a wolf, and a dog over their prey. 
 
 Horseshoe Problem. A man agrees to pay one penny for 
 the first nail, 2 pence for the second, 4 for the third and so on 
 
 1 Claude Gaspard Bachet de Mezeriac, Problemes plaisants (1624). 
 
160 SIXTEENTH CENTURY ARITHMETIC 
 
 for all the nails used in shoeing his horse; how much does 
 he pay? This is similar to the Chessboard problem. 
 
 Courier Problems. The name is now used to represent the 
 whole class of problems that concern the movement of bodies 
 at given rates in which some position of these bodies is given 
 and the time required before they will assume another given 
 position. The name originated with the French to designate 
 problems about messengers delivering despatches in connection 
 with military service. 
 
 The problems of the clock hands and the times of conjunc- 
 tion of the planets fall into this general class. 
 
 Courier problems appear in the Bamberg Arithmetic 
 (1483) under the title, "Van Wandern." They are found 
 in Calandri, Tonstall, Kobel, Cardan, and Trenchant. 
 
 Three Casks. Three casks together con- 
 tain 79 gal. ; the second contains 3 gallons 
 more than § as much as the first, and the 
 third contains 7 gallons less than the 
 second ; how many gallons are there in each ? three jugs. 1 
 (Trenchant, Baker, Kobel.) 
 
 God Greet You Problem. God greet you with your 100 
 scholars! We are not 100 scholars; but our number and the 
 number again and its half and its fourth are 100; how 
 many are we? (Rabbi Ben Ezra, Alcuin, Leonardo, Gram- 
 mateus, Riese, Kobel and Van der Schuere.) 
 
 Thief Problem. A thief having robbed a castle met a guard 
 in trying to escape whom he bribed with i of his plunder; 
 at the next gate he met a guard whom he bribed with i of 
 what he has left; he escaped with 15 lb. How much did the 
 owner of the castle lose? (Tonstall and Kobel.) 
 
 This problem ran through many variations as the plunder- 
 ing of gardens and the stealing of apples. 
 
 Reed Problem. A reed standing in the center of a circu- 
 lar pond 12 feet deep projects 3 ft. out of water; when the 
 wind blows it over to the side of the pond, it just reaches the 
 surface; what is the distance across the pond? 
 
 1 This illustration is from a fourteenth century manuscript. 
 
THE ESSENTIAL FEATURES x 6i 
 
 Tree Problem. A tree 50 ft. high was broken so that its 
 top touched the ground 30 ft. away from its base. How much 
 was broken off and how much remained standing? (Calandri). 
 
 Mill- Wheel Problem. A mill has 5 wheels, the first wheel 
 grinds 7 staria of wheat in 1 hour, the second 5 staria, the 
 third 3 staria, the fourth 2, and the other 1. In how many- 
 hours will they altogether grind 50 staria? 
 
 Jealous Husband Problem. A boatman 
 has his wife, two strangers, and their 
 wives to ferry across a stream. His boat 
 will carry only two persons. Being jealous 
 of his wife he is not willing to leave her boat and dock. 1 
 with either of the strangers. How many trips must he make 
 to ferry the party across and not leave his wife with a stranger? 
 
 A familiar form of this problem is that of the boatman with 
 a fox, a goose and some corn, or with a wolf, a goat and 
 some cabbage. 
 
 Cistern Problem. See page 134. Many variations of this 
 problem are still found in text-books concerning the building 
 of walls, digging of trenches, and the famous " If A can do a 
 piece of work in 5 days, and B can do it in 8 days, how long 
 will it take both working together to do it?" 
 
 Market Problem. A woman going to market with a basket 
 of eggs found that when she counted them by twos there 
 was one over, but when she counted them by threes there were 
 two over. The whole number was between 50 and 60; how 
 many were there in the basket? (Baker.) 
 
 Mule and Ass Problem. A mule asked an ass whether 
 his load was heavy; the ass replied: "My load is thrice as 
 heavy as yours, but, if I had yours and mine together, it would 
 be only half a ton; find the result yourself." (Gemma 
 Frisius.) 
 
 Servant Problem. A master bargained with a servant to 
 give him 10 guldens a year and a coat. The servant re- 
 mained only 7 months. At that time the master said : " Leave 
 my house and take the coat that I gave you ; I owe you noth- 
 1 This illustration is from a fourteenth century manuscript. 
 
162 SIXTEENTH CENTURY ARITHMETIC 
 
 ing more. How many guldens was the coat worth ?" (Uni- 
 corn and Kobel.) 
 
 Casket Problem. A jewel casket and lid weigh 27 oz. ; the 
 lid is I as heavy as the casket; what is the weight of the lid?" 
 
 Striking of Clock. Venetian clocks strike from 1 to 24; 
 how many days and nights go by for 300 strokes of the clock ? 
 
 Tower Problem. One-third of a tower is hidden under 
 the earth, a fourth is submerged under water. 60 cubits rise 
 above the water. It is desired to know how many cubits are 
 under the earth and how many are submerged under water. 
 (Tonstall.) 
 
 Garrison Problem. A captain with 4000 soldiers was be- 
 sieged in a fortress by the enemy for 7 months. They had 
 only provisions for 5 months and were without hope of re- 
 ceiving any during the siege of 7 months. It is required to 
 know how the captain shall apportion the rations that they 
 may last during the siege. (Champenois.) 
 
 Will Problem. A man on his death-bed made his will thus : 
 If his wife (about to be confined) should bear a son, he 
 should receive \ of the property valued at 3600 aurei, if she 
 should bear a daughter, the daughter should receive a third. 
 She gave birth to a son and a daughter; it is required to know 
 how much each should receive. This question was also known 
 as the Widow Problem and the Problem of Inheritance and 
 dates back to the Greeks and Romans. It was stated by 
 Salvianus Julianus in the reign of Hadrian. (Given by 
 Gemma Frisius, Tartaglia, Rudolff, Trenchant, Widman.) 
 
 Hiero's Crown. Vitruvius relates in Book 9, Chapter 3, 
 that when Hiero, the king, had decided to make an offering 
 to the gods of a crown of pure gold, he entrusted it to a work- 
 man, who (as they are always wont to do) mixed a portion 
 of silver with the gold. A fraud was suspected when the 
 crown was finished and Archimedes of Syracuse detected it 
 thus : He obtained a mass of pure gold of the same weight 
 as the finished crown and another mass of pure silver of the 
 same weight. He placed each separately into a vessel filled 
 with water saving the water which flowed out each time from 
 
THE ESSENTIAL FEATURES ^3 
 
 the vessel and thus found the amount of gold and silver. 
 Let us suppose the weight of the crown and the two pieces 
 of metal to have been 5 lb. each; 3 lb. of water overflowed 
 from the immersion of the gold, 3^ lb. from that of the crown, 
 and \\ from that of the silver. Therefore the question is : 
 how much gold and how much silver were there in the crown ? 
 (Gemma Frisius.) 
 
 Statue of Minerva Problem. I am the statue of Minerva; 
 my gold, however, is the gift of the youthful poets. Charisius 
 furnished £, Thespis -J; Solon, ■£$, Themison, ^V ; the re- 
 mainder, nine talents, was the gift of Aristodicus. (Ramus.) 
 
 Herds of Alcides. To one asking the number in the herds 
 of Alcides it was replied that $ were near the gently flowing 
 Alpheus, $ grazed on the hill of Saturn, ^ on the mountain 
 of Tarixippus, ^V near the divine Elides, and £$ in Arcadia. 
 The rest of the herd is 50. 
 
 Beggar Problem. Three mendicants approached a priest 
 holding a purse of money to be distributed among the poor. 
 Having compassion for their poverty he gave to the first one- 
 half of what he had in the purse and 2 nummi ; to the second 
 he gave half of what was left and 3 nummi besides; to the 
 third he gave 4 nummi more than half of what was left. Only 
 one nummus remained in the purse. It is required to find 
 how many nummi were in the purse at first. (Tonstall.) 
 
 Ring problem. To find who has a ring in a company. If 
 one person in a company standing in line has a ring on a cer- 
 tain finger and you wish to know which one has it and on 
 which finger it is, have one of the company silently double 
 the number denoting the order of the one who has the ring, 
 add 5, multiply this by 5, add the number of the finger on 
 which the person has the ring, and tell the result. Take 
 away 25 and the tens' digit will be the number of the person 
 and the units' digit the number of the finger. (Trenchant.) 
 
 Mensuration 
 Besides the commercial applications, arithmetic of the six- 
 
164 SIXTEENTH CENTURY ARITHMETIC 
 
 teenth century was very serviceable in the field of mensura- 
 tion. Practical arithmetics commonly contained a section de- 
 voted to mensuration. The combination of the Reckoning 
 Book (Rechenbuch) and the Mensuration book (Visirbiich) 
 by Kobel * represents the two forms of arithmetic at that time. 
 The Visirbiich was better illustrated than any of its contem- 
 poraries. The name, Visir, means gauge and Visirbiich, 
 technically gauge book, means a book to teach gauging of 
 casks. Its contents, however, were concerned with all forms 
 of practical mensuration. 
 
 The following list given by Cataneo furnishes an idea of 
 the extent of the subject of mensuration: 2 
 
 The Measurement of Wood. 3 
 
 The Measurement of Solid Bodies.* 
 
 The Measurement of Triangular Prisms. 5 
 
 The Measurement of Square Prisms. 
 
 The Measurement of Square Pyramids. 7 
 
 The Measurement of Parallelepipeds. 8 
 
 The Measurement of Walls. 
 
 Another Method of Measuring Walls, Floors, Surface to 
 be Whitewashed. l ° 
 
 The Measurement of Casements. J 1 
 
 The Measurement of the Scarp of a Wall. 12 
 
 'Kobel, Zwey rechenbuchlin. " Uff der Linien vnd Zipher/ Mit eym 
 angehenckten Visirbiich/ so verstendlich fur geben/ das iedem hierauS on 
 ein lerer wol zulernen " (1531). 
 
 2 Cataneo, Le Pratiche Delle Due Prime Matematiche (1567 ed.). 
 
 s Del Misvrar De I Boschi, fol. W. verso. 
 
 *■ Del Qvadrar Le Cose Corporee, fol. Xi recto. 
 
 5 Del Qvadrar Le Colonne Triangulari, fol. Xi recto. 
 
 Del Qvadrar Le Colonne Qvadrangulari, fol. Xi verso. 
 
 T Del Qvadrar Le Piramide Qvadrangulari, fol. Xi verso. 
 
 " Del Riqvadrar I Vasi Qvadrangulari, fol. Xi verso. 
 
 Del Riqvadrar Le Muraglie, fol. Xij recto. 
 
 10 Altro Modo Di Riquadrar Muraglie, palchi, sciabli, e legnami, fol. Xij 
 recto. 
 
 11 Del Misvrar I Casamenti, fol. Xij verso. 
 
 "Del Riqvadrar Le Scarpe De I muri, fol. Xij verso. 
 
THE ESSENTIAL FEATURES ^5 
 
 Further subjects are: 
 
 The Measurement of Bodies with Square Surfaces. 
 
 The Measurement of Floors and the Number of Bricks 
 Necessary for a Square Surface. 
 
 The Method of Finding the Number of Bricks in a Wall. 
 
 The Measurement of Round Bodies, first the Measurement 
 of a Ball. 
 
 The Measurement of a Cistern. 
 
 The Measurement of a Cylinder. 
 
 The Measurement of a Cone. 
 
 The Measurement of a Pile of Grain. 
 
 Another Method of Finding the Contents of a Pile of Grain. 
 
 A still different Method of Finding the Contents of a Pile 
 of Grain. 
 
 The Method of Finding the Capacity of a Cask. 
 
 The Method of Finding the Capacity of a Barrel. 
 
 Another Method of Finding the Capacity of a Barrel. 
 
 The Method of Finding the Capacity of a Bin of Grain. 
 
 The Measurement of a Small Cylinder. 
 
 The Measurement of Cylindrical Walls. 
 
 Besides the guage which served to measure the capacities of 
 castes and small receptacles, another instrument, called the 
 quadrans, or quadrant, was used to measure cisterns, walls, and 
 distances. There were two forms of the latter instrument, one 
 a square with graduated sides and the other a quarter of a 
 circle with a graduated arc. Although the proportionality of 
 the corresponding sides of similar triangles was the chief prin- 
 ciple used in the solution of problems, the graduated arc in 
 connection with the plumb-line made possible the measurement 
 of angles. In this form of the quadrant we recognize the be- 
 ginnings of the modern theodolite. 
 
 The following page from Finaeus (1532) illustrates the 
 method of finding heights and distances by use of each form 
 of the Quadrans. 
 
1 66 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 tiaasd m . 
 SaonAn m 
 
 ORO NT XI PINEI DEL FH. 
 
 „* orimi dementorumEudidis facile maniWhtur.&angulus ABklan- 
 ^ AG F cftxquaKs(jumucer(^re&is)ig£turper 4 i«ti auTdrra Eudidis, 
 
 icffcnt k b ad s a, ica * C putd laurudoad C A corapoGtam aGB&iSA lon£i. 
 
 tudincm.due prcfijiditsccm. tt ^«.» - 
 
 Sic exempli gratia BH20 partnrm/jualiuci fetus quadrati eft 6o:b E aute me* 
 
 tiitur,& fit in exeroplum 6* cubiconinvot etiara cnbicorum erit G Fdunt enira la 
 
 cent prralldogranfcm BEPC oppoGa,qux per J4 dsf&cm primi funt inuicem 
 
 aec]uaIia,Duc igicur 6m 6Q&nt }6o:(px diuidepcr 20, &iabehis pro quoded 
 
 tciS-Totigirur cubitojfc crii A G: 
 
 & quaG dempferis A b mum uer* 
 
 hi gratia cubiiorum, rdfnqtrrtnr 
 
 BGdefyderata & in profundom 
 
 depfla patcHogimdoif cubitoR:. 
 
 IDEM Q_V O CLV 5 SIC OB* 
 
 tinebis. Metire H e : Gtcp exempli; 
 
 caufa 5 cub i :cr lj . Deindc inula* 
 
 plica y per 6o,Sent joojiarc diul 
 
 per zOjproducentur ly.udnt an* 
 
 tea.Binanancjrtriangula A BH et 
 
 HEf funt rurfum equiangular^ 
 
 quoniam angulus ahb angulo 
 
 E H f ad ucrricem pcfito , per ic 
 
 primi Eudidis. eft scquaji5.]t5 re* 
 
 crus qui ad B, reSoqui ad e pari 
 
 ter acquac. rcliquus igicur BAH 
 
 reliquo H F E per J2 eiufdem pri* 
 
 nueftarqualis. Vndc per fupc* 
 
 rius allcgara quarca propoGcionc 
 
 fexri,Gcuc hb ad e A,ira h E ad E F.eidem B G per aypotbefim aequalem. 
 
 Cum autem accident puceum rotundam habere figuram,habenda eric cofy dc* 
 ratio diamem putcalis oriGcij J & reliqua omnia ueluri prius aifoluenda, 
 
 $fl* B.ELIQ.VVUETS,VT 
 
 eandemrerumin profundi! dc* 
 pnaTarum, per uulgatu qusdra* 
 tem mean doceamus alticudine. 
 Ei't icacp puteus circularis efg 
 H^uius diamerer Gc e F,aut illi 
 scqualis G H. Adplica igicur qua* 
 dratem ipG pucei orifioo: in Hijc 
 mod^ut finis lateris A D ad datu 
 punctumEconfticuatur. Leua 
 poftmodu,aut deprime quadra* 
 
 tern(Itbero femperdemiflb per 
 pendiculo)donec radius uifualis 
 per ambo foramina pinnacidioRE 
 ad inferiorem & e diametroG- 
 |™u termini! Hperdueac.Quo 
 fa<fb & immoto quadrate, uide 
 in qui 
 
 A typical problem of this kind is the following - from 
 Belli : 1 
 
 1 Belli, Silvio, Qvattro Libri Geometrici (iS95 ed.). "La ragione 
 e qvefta, l'angolo del triangolo AEB, euguale al l'angolo B del triangolo 
 CEB, perche 1'vno, e l'altro d'effi e retto, & l'angolo E e commune ad 
 amendue i detti triangoli : Onde per la trigefima feconda del primo libro 
 de gli elementi d'Euclide, il reftante angolo -del l'altro. Et per la quarta 
 del fefto i lati, che riguardano gli angoli vguali fono proportionali. Adun- 
 
THE ESSENTIAL FEATURES 
 
 167 
 
 Required to find the distance from point E to point A 
 and the distance from B to A as shown in the figure. The ex- 
 
 planation is as follows: The angle (B) of triangle AEB equals 
 the angle B of the triangle CEB, because they are right angles ; 
 and the angle E is common to both of these triangles. Then 
 by the thirty-second proposition of the First Book of Euclid's 
 Elements the remaining angles are equal, and by the fourth 
 proposition of the Sixth Book, the sides opposite to the equal 
 angles are proportional. Then BC is to BA as EC is to EA, 
 as the side BE of the small triangle is to the side BE of the 
 large triangle. And the side BE of the small triangle, as was 
 presupposed, occupies as many small divisions of the square as 
 there are paces in the side BE of the large triangle, where- 
 fore the small divisions of the side BC of the small triangle 
 are as many as the paces of the side BA of the large triangle 
 which was the first thing sought. And in the same way the 
 
 que la proportione del lato BC al lato BA, & dello EC alio EA, fi come 
 del lato BE del picciolo, al lato BE del grande, & il lato BE del picciolo, 
 dal prefuppofito ha tante delle particelle del lato del Quadrato quante 
 fono le paffa del lato BE del grande: per la qual cofa ancor le particelle 
 del lato BC del picciolo fono quante le paffa del lato BA del grande, che 
 e il primo intento. Et per lo medefimo modo le particelle del lato CE 
 del picciolo triangolo fono vguali per numero alle paffa del lato AE del 
 grande, che e il fecondo." Fol. A f verso. 
 
168 SIXTEENTH CENTURY ARITHMETIC 
 
 number of small divisions of the side CE of the small triangle 
 is equal to the number of paces of the side AE of the large 
 triangle, which was the second thing sought. 
 
 Thus applied arithmetic represented the vital commercial 
 interests of that period; it naturally contained some traditional 
 and artificial problems, but in the main it expressed the quan- 
 titative side of contemporary trade and industry. 
 
 Summary 
 
 The foregoing exposition of the subject matter of arith- 
 metic belonging to the first century of printed books shows 
 | that this was an important period in perfecting the algorisms, 
 or the processes of present day arithmetic with the Hindu nu- 
 merals. From the treatment of processes (pp. 36-37) it ap- 
 pears that the addition, subtraction, multiplication, and divis- 
 ion of integers reached their highest development in the six- 
 teenth century. Some forms of these processes have since 
 been discarded, 1 but all that are in current use to-day are an 
 inheritance from that period. All processes with common 
 fractions now used, including some graphical methods, were 
 practiced in the sixteenth century (pp. 85-110). Some meth- 
 ods, however, were rare, as multiplying by the reciprocal of 
 the divisor in division (p. 107). Arithmetic, geometric and 
 harmonic progressions (pp. 1 10-1 16) as finite series, were com- 
 monly explained by particular examples. The processes of in- 
 volution and evolution were practically complete (pp. 121-127). 
 Even the use of tables to save calculation was a common prac- 
 tice. (Tartaglia, p. 58; Trenchant, p. 144, and Jean, p. 82.) 
 Thus the elementary processes reached their maturity except 
 in the case of decimals and logarithms. The decimal frac- 
 tion, the prefection of the processes with decimals, and 
 logarithms were contributions of the seventeenth century. 
 
 The constructive period of applied arithmetic began about 
 1400 and lasted for more than a century. Mensuration 
 
 1 For example, addition and subtraction in which the result was placed 
 above the addends or above the minuend; multiplication by complements; 
 multiplication by tables above 12 X 12; multiplication by fancy geometric 
 forms ; division by galley and other methods. 
 
THE ESSENTIAL FEATURES rfg 
 
 (p. 163) and commercial problems (p. 132) received exhaustive 
 treatment at the hands of Paciuolo, Borgi, Tartaglia, Rudolff 
 and Riese. Little has been added to the work of these writ- 
 ers from that time to the present, except in such particulars as 
 industrial problems, insurance, compound interest and ex- 
 change. There are few problems found in present school 
 arithmetics whose types were not foreshadowed in the 
 arithmetic of that century. The sixteenth century problems 
 in accounts, exchange, transportation, gain, loss, percentage, 
 discount, salaries, rents, taxes, interest, tare, duties and part- 
 nership testify to this fact (pp. 132-139). 
 
 Even the methods of solution were more like those now in 
 use than is generally supposed. Besides the solution of in- 
 determinate problems, which were really solved by algebra in 
 the form of rules as now, there were two methods of solving 
 business problems: the method of unitary analysis (pp. 137- 
 138) and the Rule of Three (pp. 132-139). 
 
 The actual contributions to the theory of arithmetic made 
 in the fifteenth and sixteenth centuries were few. The first 
 great arithmetic printed in Europe, that of Paciuolo (1494), 
 doubtless possessed some originality, but its chief merit con- 
 sisted in its being a systematic and exhaustive summary of all 
 the pure mathematics known at that time, and little was added 
 to Paciuolo's arithmetic by way of theory in the sixteenth 
 century. There were certain variations in processes, as the 
 reduction of the number of species (pp. 36-37), different or- 
 ders of presentation, and graphical illustrations (pp. 104-106) 
 which may be called improvements, but the processes with in- 
 tegers and common fractions were practically complete before 
 the era of printed books. With symbolism the case was dif- 
 ferent, for the improvements in methods of notation of whole 
 numbers (pp. 37-39) and of decimal fractions (p. 72) and in 
 the symbols of operation (p. 53) show that arithmetic was 
 in a rapid state of development. Besides the advance in no- 
 tation and symbolism, great progress was made in applied 
 arithmetic; the cumbersome form of unitary analysis (as used 
 by Calandri, p. 138) developed into the equational form seen 
 
170 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 in Thierfeldern ; the arbitrary Rule of Three came to be sim- 
 ple proportion; exchange came to embrace a comparison of 
 the denominate number systems of all Europe and the Levant ; 
 and finally, many processes were superceded by tables of re- 
 ference (pp. 8a, 144). 
 
 The Place of Arithmetic in the Schools of that 
 
 Period 
 
 Arithmetic in the Latin Schools 
 The Latin Schools were the chief agency through which 
 the great reformers and teachers of the Renaissance recon- 
 structed and extended elementary education. Seventy 
 gymnasia 1 were founded in the kingdom of Prussia during 
 the sixteenth century. These were not everywhere known, as 
 in Germany, by the name of " Latin Schools," for in England 
 they received the name, " Grammar Schools," and in France 
 and Italy they were known as the " Schools of the Teaching 
 Orders." 
 
 The chief functions of these schools were to teach the Latin 
 language and to furnish general culture. A knowledge of 
 Latin was necessary at that time for one who wished to be 
 educated, for the standard works of literature, science, philo- 
 sophy, law and theology were all expressed in the classical 
 languages. 
 
 The courses of instruction in these schools were not uni- 
 form. Many of the greatest scholars and teachers of that 
 period advocated the study of philology to the exclusion of 
 history, geography, physics and mathematics. In his school 
 laws Trotzendorf prescribed : " The students are never to use 
 their mother tongue with teachers, fellow students or learned 
 people." It was boasted of Trotzendorf's teaching : 2 " He 
 has so thoroughly instilled the Latin language in all, that it is 
 considered a disgrace to speak in the German language; to hear 
 the servants speak Latin, one would believe that Goldberg 
 lay in Latium." The school regulations 3 for Stuttgart in 
 
 1 Unger, Die Methodik der praktischen Arithmetik, p. 1. Leipzig, 1888. 
 
 2 Ibid., p. 5. 3 Ibid. 
 
THE ESSENTIAL FEATURES j 7i 
 
 1 501 commanded the school master to punish with scanty- 
 food the students who spoke German to one another. Even 
 Melancthon, who did much through lectures to encourage the 
 study of mathematics, laid special stress upon language. 
 " The school master, so far as possible, is to speak nothing but 
 Latin with the boys in order that they may become accus- 
 tomed to it." The early Jesuit schools in France were like- 
 wise closely confined to the Latin language. In England 
 Roger Ascham, though at one time a lecturer in mathematics 
 at Cambridge, expressed his views thus : 1 " Some wits, mod- 
 erate enough by nature, be many times marred by overmuch 
 study and use of some sciences, namely, music, arithmetic 
 and geometry. These sciences, as they sharpen men's wits 
 overmuch, so they change men's manners oversore, if they 
 be not modestly mingled and wisely applied to' some good use 
 in life." But in all of these countries there were men who 
 took a broader view of education. Martin Luther encour- 
 aged the extension of the course of study. He said : 2 "If 
 I had children, and if it were possible, they would not only 
 learn language and history, but also music, singing and 
 mathematics." Michael Neander, head of the Latin School 
 of Ilfield, contrary to the practice of Sturm and Trotzendorf, 
 taught geography, history and the natural sciences. 3 Pere 
 Lamy, a leader of the Oratorians, exerted a similar influence 
 in France. He said : 4 " I know of nothing of greater use 
 than algebra and arithmetic." And Rabelais, 6 the great 
 Benedictine, advocated the learning of mathematics " through 
 recreation and amusement." 
 
 In all of the Latin Schools the study of Latin consumed the 
 most time. Mathematics seldom extended over more than 
 half of the course, and in some schools it was not taught 
 until the fifth or sixth year, and then only an hour weekly. 
 
 1 The Scholemaster. 
 
 2 Unger, Die Methodik, p. 5. 
 
 3 Seeley, History of Education, p. 179. 
 
 * Compayre, History of Pedagogy, p. 151. 
 6 Ibid., p. 98. 
 
172 SIXTEENTH CENTURY ARITHMETIC 
 
 The plan of individual instruction used in these schools tended 
 to discourage the teaching of arithmetic in the first years of the 
 course. For each pupils had to prepare his lessons from a 
 text-book in Latin, hence he had to master the language be- 
 fore he could read his book. Classes which read Terence 
 and Cicero and studied syntax and prosody, learned only the 
 four operations with whole numbers. The full course sel- 
 dom advanced beyond fractions and the Rule of Three. The 
 following extract from a curriculum of a six-class Latin 
 School of the sixteenth century gives some idea of the studies 
 pursued : 1 
 
 " The practice of the sixth class ; figures and numbers ; of 
 the fifth, common reckoning ; of the fourth, not given ; of the 
 third, music, arithmetic and astronomy; of the two highest 
 classes, second and first, not given." 
 
 The following is from the curriculum of a five-class Latin 
 School of the same period : 
 
 " On Friday from 12 to i the arithmetic shall be read. The 
 preceptors shall use no other arithmetic than that of Piscator 
 (John Fischer), and in the fourth class (the next to the high- 
 est in which Terence and Cicero are read) the species alone, 
 in the fifth (highest class) the whole arithmetic is read.'" 
 
 The courses of the early English Grammar Schools were 
 very similar to those of the Latin Schools. About three- 
 fourths of the time was given to the study of the classics. 
 Few head masters paid any attention to mathematics. The 
 Writing Master usually taught some arithmetic in the forms 
 below the fifth and sixth. 2 
 
 Since Latin was prescribed as the language of instruction 
 in the Latin Schools, it was necessary to provide reckoning 
 books in that language. Among the most important writers 
 of these text books were: Michel Stifel, Hieronymus Car- 
 danus, Gemma Frisius, Peter Ramus, Christopher Clavius, 
 Simon Stevin, Piscator (John Fischer). 
 
 1 Unger, Die Methodik, pp. 24-25. 
 
 2 W. H. D. Rouse, History of Rugby School, p. 130. 
 
THE ESSENTIAL FEATURES jy^ 
 
 Michel Stifel (1487-1567), an Augustinian monk, escaped 
 from a monastery and went to Wittenberg in 1523, where he 
 became a friend and follower of Luther. The last years of 
 his life were spent at Jena as a private teacher of mathe- 
 matics. Not only was Stifel interested in serious theological 
 and mathematical subjects, but also in visionary fancies 
 about the secrets of number. These fancies led him to write 
 " Ein Rechen Biichlein von End. Christ. ' Apocalysis in 
 Apocalysim, Wittenberg, 1532." He reckoned the last day 
 of the world to be October 19, 1533, and imparted this know- 
 ledge to his parishioners at Lochau, who consumed their for- 
 tunes and goods. Stifel's life was endangered by this false 
 prophecy and was saved only by the personal intervention 
 of Luther. He then gave himself up to serious study and in 
 1544 wrote "Arithmetica integra," a work which gave him 
 a place among the leading mathematicians of the sixteenth 
 century. 
 
 Hieronymus Cardanus (1501-1576) taught mathematics in 
 many Italian towns. He wrote " Practica arithmeticae gen- 
 eralis et mensurandi singularis 1537," and "Ars magna arith- 
 meticae. Artis magnae sive de regulis algebraicis liber unus, 
 
 1 545-" 
 
 Gemma Frisius (1508-1558) was Professor of Medicine at 
 the University of Lourain and the author of "Arithmeticae 
 Practicae Methodus Facilis, 1540." This book, small in com- 
 pass, but rich in contents, was as popular in the Latin Schools 
 as Adam Riese's was in the Reckoning Schools. It is com- 
 posed of rules with an example to illustrate each. 
 
 Peter Ramus (1 515-1572), an anti-scholastic, was a pro- 
 fessor at the University of Paris; but on account of his af- 
 filiation with the Huguenots, he was obliged to leave France 
 in 1560. He returned to Paris ( 1571) and was slain in the 
 St. Bartholomew massacre (1572). Ramus was more im- 
 portant as a philosopher than as a mathematician. He was 
 the author of " Scholarum mathematicarum libri XXXI, 
 Basel, 1569" and "Arithmeticae libri duo" (1567.) 
 
 Christopher Clavius (1537-1612), a Jesuit, became a teacher 
 
174 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 of mathematics in the Jesuit College at Rome. He assisted 
 in a revision of the calendar under Gregory XIII (1582), 
 and wrote Opera Mathematica, 161 1," 5 vols., and " Chris- 
 tophori Clavii Bambergensis e societate Jesu Epitome 
 Arithmeticae Practicae, Rome, 1583." This was to be only 
 the predecessor of a complete arithmetic, which, unfortun- 
 ately, did not appear. In the subject matter of his works 
 Clavius did not approach Tartaglia, but in methodical treat- 
 ment was his peer. 
 
 Simon Stevin (1548-1620) was a book-keeper at Antwerp, 
 a revenue officer at Bruges, and a teacher and favorite of 
 Prince Maurice of Nassau, by whom he was appointed in- 
 spector of the dikes in Holland. He discovered decimal frac- 
 tions and published the oldest discount table. His works 
 were edited by Girard : " Les oeuvres mathematiques de 
 Simon Stevin de Bruges. Le tout revu, corrige et augmente 
 par Albert Girard, Leyden, 1634." The arithmetical part, 
 written in the form of Latin compendia, is characterized by 
 complete definitions and the development of processes from 
 concrete problems. 
 
 Piscator (John Fischer) was the author of Arithmeticae 
 Compendium (1545). The school regulation of Kursachen 
 ( 1 580) prescribed this arithmetic of Piscator to be used in the 
 school. He also wrote reckoning books in the German 
 language. 
 
 Thus, the writers of Latin School arithmetics, with few ex- 
 ceptions, were instructors in private schools, in Latin Schools, 
 or in Universities. Their activities, however, were not con- 
 fined to mathematics ; for example, Ramus achieved fame as a 
 philosopher; Frisius as a physician, and Stifel as a theologian. 
 Hence, although the arithmetic of the Latin Schools was 
 meagre, the text-books were written by the foremost schol- 
 ars in medicine, science, engineering and philosophy. 
 
 Since the essential features of the arithmetic produced by 
 the above writers have been noted (pp. 23-170) it is necessary 
 only to summarize them here. These works are characterized 
 by a prominence of pure arithmetic, somewhat precise 
 
THE ESSENTIAL FEATURES jy^ 
 
 definitions (pp. 29-35), rules and explanations of processes 
 with examples. Although one can recognize few principles 
 of method consciously applied in mathematical instruction at 
 that time, the arithmetics of certain Latin School writers con- 
 tain material classified and organized with a view to facilitat- 
 ing the educational process; for example, the graded treat- 
 ment of the cases in the multiplication of integers (p. 61), in 
 the validity and sufficiency of the proofs of sevens, nines and 
 elevens (pp. 44, 66), and the recognition of the relation of 
 inverse processes (p. 109). The subject matter was usu- 
 ally organized on the following plan : The whole realm of in- 
 tegers was presented at the outset (pp. 36-77) and repre- 
 sented under each operation. The four processes were re- 
 peated with fractions and often with denominate numbers. 
 There were no stages based on the size of numbers, no con- 
 centric extension of operations, and no limits corresponding 
 to the different ages of pupils. The following examples il- 
 lustrate the rapid plunge into large numbers in the addition 
 of integers : 1 
 
 4 
 
 3 
 
 3°9 
 204 
 
 13 
 
 59 
 34 
 
 93 
 
 389 
 204 
 
 93 
 
 7389 
 1264 
 
 8653 
 
 7389 20 
 6264 10 
 
 7 
 
 I36S3 30 
 
 402 
 301 
 
 4052 
 3601 
 
 
 4321 
 10 
 
 7300S804 
 60203643 
 
 69001303 
 69000000 
 69008000 
 
 
 
 
 
 
 
 
 79017100 
 
 
 
 
 
 
 79000003 
 79006000 
 89004000 
 89026200 
 89008000 
 89005000 
 
 
 
 
 
 
 99002400 
 
 
 
 
 
 
 99017002 
 
 
 1008095008 
 
 1 Cuthbert Tonstall, De Arte Supputandi (1522), fol. C 3 recto, et seq. 
 
176 SIXTEENTH CENTURY ARITHMETIC 
 
 The applications of the processes given in the Latin School 
 books were not numerous. The tendency was towards puz- 
 zles, and factitious and traditional problems. For example, 
 Ramus 1 gave the courier, the tower, statue of Pallas, the 
 herds of Alcides, the mill-wheel, the fountain, the architect, and 
 the widow problems (pp. 138-9). In all of the Latin School 
 arithmetics there was a dearth of business problems. The 
 rule of three, partnership, alligation, rule of false position and 
 a few others were generally included, but the practical prob- 
 lems characteristic of the commercial world of that time were 
 generally ignored. The arithmetic of Gemma Frisius, which 
 ran through some fifty-five editions, and which is probably the 
 best example of Latin School text-books, contains for its ap- 
 plications artificial problems of this kind : 2 
 
 "A man having a certain number of aurei bought for each 
 aureus as many pounds of pepper as equaled half of the whole 
 number of aurei. Then upon selling the pepper he received 
 for each 25 pounds as many aurei as he had at the beginning. 
 Finally, he had 20 times as many aurei as he had at first. 
 The number of aurei and the quantity of pepper are required." 
 
 " Three men together have a certain amount of silver, but 
 each one is ignorant of the amount he has. The first and 
 second together have 50 aurei, the second and third, 70 aurei, 
 the third and first, 60 aurei. It is required to know how 
 much each one has." 
 
 The reasons for teaching arithmetic in the Latin Schools 
 must be deduced from indirect evidence, for discussions of 
 courses of study or of educational values were so rare in 
 that period that history gives little direct evidence on the sub- 
 ject. One must look to the scholars and leading educators of 
 that age, as of every age, and not to the directors and teach- 
 ers of schools, to learn of the ideals and purposes of education. 
 The scholars and educators of the sixteenth century were 
 
 1 Peter Ramus, Arithmeticis Libri Duo (1586 ed.), fols. M 4 recto, M 5 
 recto. 
 
 2 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1540). 
 
THE ESSENTIAL FEATURES iyj 
 
 enthusiasts for classical studies and the schools reflected their 
 tastes. There was much of grammar, rhetoric, and literature 
 in the Latin language, hence there was much of grammar, rhe- 
 toric and literature in the schools. There was little of physi- 
 cal science and mathematics available in Latin, hence there 
 was little of these subjects in the schools. Greek mathema- 
 tics, however, including the works of Euclid, Ptolemy, Archi- 
 medes and Diophantus were of great interest to scholars, and, 
 being early made available by translation, gradually found 
 place among school studies. 
 
 That the arithmetic of the classical languages did leave its 
 stamp upon the works of the sixteenth century writers is evi- 
 dent from the Greek classifications of numbers found in the 
 extended treatments of proportion of the Latin School Arith- 
 metics. Gemma Frisius treats at length harmonic and arith- 
 metic proportion, proportion of equal and unequal numbers, 
 multiplex, superparticular, superpartiens, multiplex superpar- 
 ticulare and multiplex superpartiens proportion, proportion of 
 fractions, mean proportional, and addition and subtraction of 
 proportion. The extent to which the classics influenced Maur- 
 olycus, a contemporary, is shown by this partial bibliographi- 
 cal list : 1 
 
 Euclidis elementa. 
 
 Theodosij Sphaerica elementa. 
 
 Menelai Sphaerica. 
 
 Apollonij Conica elementa. 
 
 Sereni Cylindrica. 
 
 Archimedis opera. 
 
 Jordanj Arithmetica. 
 
 Theonis Data geometrica. 
 
 Rogerii Baoconis. & Io. Petsan Perspectivae breuiatae cum adnotation- 
 
 ibus errorum. 
 Ptolemei Specula. 
 Autolyci de sphera. 
 Theodosii de habitationibus. 
 Euclidis Phaenomena brevissime demonstrata. 
 Aristotelis problemata mechanica, cum additionibus complurimis, & iis, 
 
 quae ad pyxidem nauticarn, & quae ad Iridem spectant. 
 
 1 Franciscus Maurolycus, Arithmeticorum Libri Duo (i57S)» fol. Hh 
 recto. 
 
178 SIXTEENTH CENTURY ARITHMETIC 
 
 Therefore, one reason for the teaching of arithmetic in the 
 Latin Schools was that it formed at least a small part of the 
 classical inheritance. Another reason why arithmetic was 
 taught in the Latin School was for its culture value. It was 
 believed by the writers of arithmetical text-books to be an es- 
 sential part of knowledge and of an education. The preface 
 to nearly every arithmetic contains a eulogy on the importance 
 of the science and its functions. The authors referred to the 
 opinions of Greek philosophers, as Plato, who held that arith- 
 metic alone distinguished men from the unreasoning beasts. 
 They quoted from the Church Fathers and Holy Writ. They 
 declared arithmetic to be valuable not only as an art, but as an 
 essential to philosophical attainment and to an understanding 
 of the mysteries of the Scriptures. That arithmetic was not 
 taught in the Latin Schools in order to make proficient reck- 
 oners is shown by the lack of practice problems in their text- 
 books ; and, likewise, the lack of vital commercial problems of 
 that day show that it was not taught in order to prepare for 
 a business life. It is evident from such considerations that 
 these scholars and writers neglected the applications of the sub- 
 ject and believed theoretic arithmetic to be an essential in pro- 
 ducing mental efficiency. 
 
 Arithmetic in the Reckoning Schools 
 
 The Reckoning School was born of necessity and owed its 
 origin to the commercial development of the thirteenth cen- 
 tury. The schools conducted by the clergy taught chiefly 
 reading and writing. Instruction in book-keeping and reck- 
 oning, the necessary equipment for a business career, was not 
 found in the schools. Hence, merchants were obliged to in- 
 struct their sons and to take others as apprentices who wished 
 to enter commercial life. Under the direction of the Han- 
 seatic League commerce grew so rapidly that merchants had 
 to employ assistants to give instruction in reckoning. These 
 teachers were the forerunners of the Reckoning Masters. 
 As the merchant leagues increased in importance, municipal 
 governments conferred upon them privileges relating to the 
 
THE ESSENTIAL FEATURES I7 g 
 
 management of trade; and this connection between govern- 
 ment and private enterprise accounts for the official position 
 and title, " Privileged Town Reckoning Master," to which 
 the teacher of commercial arithmetic arose. As a public offi- 
 cer he usually acted as town clerk, inspector of weights and 
 measures, shipping master, notary, and sometimes as surveyor. 
 In his capacity as teacher he often monopolized the secular in- 
 struction of his town. Thus, were founded the Reckoning 
 Schools, which sprang up in the towns along the trade routes 
 of the Hanseatic League, and, with certain modifications, throve 
 in Italy, Germany, France, the Netherlands, and possibly in 
 England. The Reckoning Master became such an important 
 factor in education toward the end of the sixteenth century 
 that he was even called upon to supplement the work of the 
 Latin Schools. A record of appointment of a Reckoning Mas- 
 ter at Rostock in 1627 contains the following: 1 "We, the 
 Burgermeisters and members of the council at Rostock, an- 
 nounce hereby that we have appointed the honorable and well 
 educated Jeremias Bernsterz for our common Town Writing 
 and Reckoning Master, until one party or the other shall give 
 notice a half-year in advance. In this document we also com- 
 mand him to attend the Latin School every week an hour each 
 on Mondays, Tuesdays, Thursdays and Fridays and there to 
 teach the youths without discrimination, and for a moderate 
 monthly or weekly salary to teach others outside of the school, 
 whether they be boys or girls who desire instruction, Latin and 
 German writing, reckoning, book-keeping and other useful 
 arts and good manners ; he is also to do with greatest industry 
 and according to the best of his knowledge and ability all 
 other things which are properly the duties of an industrious 
 and honest Writing and Reckoning Master. That his honest 
 service may be properly rewarded, we shall pay him 400 
 marks a year from the common treasury at four quarterly 
 payments; we also promise him exemption from taxes, ex- 
 cises, hundred penny tax, soldier money and all other con- 
 
 1 Unger, Die Methodik der praktischen Arithmetik, pp. 26-27. 
 
180 SIXTEENTH CENTURY ARITHMETIC 
 
 tributions, whatever they may be; we grant him the freedom 
 of the city and a free dwelling. We promise all in honor and 
 good faith." 
 
 As the number of Reckoning Masters increased, especially 
 in the larger commercial centers, guilds, regulated by a con- 
 stitution and by-laws, were formed. Boys who wished to be- 
 come Reckoning Masters served an apprenticeship of six 
 years, after which, by successfully passing a prescribed ex- 
 amination and signifying their allegiance to the constitution 
 of the guild, they were admitted to membership. There is 
 preserved in Nuremburg school history the record of a diploma 
 certifying that a candidate had passed the required examina- 
 tion, July 12, 1620. 1 Among the signatures is that of Johann 
 Heer, Reckoning Master and author of an excellent reckon- 
 ing book (p. 12 Bibl.). Although these guilds were of great 
 service in the sixteenth century in furnishing the common 
 people a practical education, they later became a serious hin- 
 drance to public instruction, because they were not obliged to 
 follow the advances in educational methods. 2 
 
 The function of the Reckoning Schools was to teach reckon- 
 ing, business forms and the solution of commercial problems. 
 But, since individual instruction from a text-book was the pre- 
 vailing method of teaching, the pupil was obliged to read and 
 write in his native tongue before he could study arithmetic. 
 
 The courses of study, as verified by the above record of 
 appointment consisted of the vernacular, writing, business 
 forms, arithmetic, and sometimes Latin. Arithmetic was the 
 chief subject, and language a secondary one, in contrast with 
 ■the Latin Schools, where language held the major place and 
 arithmetic a minor one. 
 
 The authors of arithmetical text-books for Reckoning 
 Schools wrote in German, Dutch or Italian to correspond to 
 the language of their readers; an occasional one, as Piscator 
 (John Fischer) also wrote books for the Latin Schools. 
 Similar books for the use of merchants appeared in other coun- 
 
 1 Unger, Die Methodik der praktischen Arithmetik, pp. 31-32, 33. 
 
 2 Ibid. 
 
THE ESSENTIAL FEATURES x 8i 
 
 tries, as in France and England. The most important writers 
 of reckoning books were : Ulrich Wagner, Johann Widman, 
 Jacob Kobel, Adam Riese, Christopher Rudolff, Peter Apianus 
 (Bienewitz), and Simon Jacob. 
 
 Ulrich Wagner, a Nuremberg Reckoning Master, was the 
 author of the first arithmetic published in Germany 1 (14S2). 
 
 Johann Widman entered the University of Leipzig with 
 a certificate of poverty and took the degree of Bachelor of 
 Arts in 1482 and that of Master of Arts in 1485 without cost. 
 He became a teacher, lectured at Leipzig, and probably held 
 a professorship. His chief work was his reckoning book : 
 " Behend und hupsch Rechnung uff alien Kauffmanschaff ten," 
 Leipzig, 1489. Five other editions are known, Pforzheim, 
 1500, 1508; Hagenau, 1519, 1521; Augsburg, 1526. 
 
 Jacob Kobel (1470-1533) studied jurisprudence, mathe- 
 matics and astronomy at Heidelberg. He made a special 
 study of mathematics at Krakau (1490), where he was a 
 fellow student of Copernicus. After his return to southern 
 Germany, Kobel became town clerk at Oppenheim (1511) 
 where he passed the remainder of his life. He is best known 
 as a poet, designer, wood carver, printer, publisher and mathe- 
 matician. His fame as a reckoning book writer rests upon 
 these works : " Eynn newe geordent Rechebiichlein vf den 
 Linien mit Rechepfenigen (1514); Das new Rechepuchlein 
 Wie mann vff den Linien und Spacien/ mit Rechepfenning/ 
 kauffmanschaft (1518)"; "Mit der kryde od' schreib- 
 federn/ durch die Zieferzal (1520) ;" " Ein new geordet Visir- 
 biich (151 5)." In 1 53 1 Kobel combined the last three books 
 into one which was designed for self-instruction. 
 
 Adam Riese, also Ries, Rys, Ryse (1492-1559), Germany's 
 most famous Reckoning Master, taught at Erfurt (1522) and 
 at Annaberg (1525). His reckoning books are: I. "Rech- 
 nung auff der linihen gemacht durch Adam Riesen vonn Staf- 
 felsteyn/ in massen man es pfiegt tzu lern in alien rechen- 
 
 1 A book published by the same printer, Petzensteiner (1483), probably 
 •was also a work of Wagner. This is known as the Bamberg Reckoning 
 Book. 
 
1 82 SIXTEENTH CENTURY ARITHMETIC 
 
 schulen gruntlich begriffen anno 1518," Colophon : " Getruckt 
 tzu Erffordt durch Mathes Maler M. CCCCCXXV Jar." ; II. 
 Rechenung auff der linihen vnd federn in zal/ mafs, vnd 
 gewicht auff allerley handierung/ gemacht vnd zusamen 
 gelesen durch Adam Riesen von Staffelstein Rechenmeister 
 zu Erffurdt im 1522 Jar. Itzt vff sant Annabergk durch in 
 fleyssig vibersehen/ vnd alle gebrechen eygentlich gerecht- 
 fertigt/ vnd zum letzten eine hiibsche vnderrichtung ange- 
 hengt." III. " Rechnung nach der lenge/ auff den Linihen 
 vnd Feder. Darzu forteil vnd behendigkeit durch die Propor- 
 tiones Practica genant/ mit griintlichem vnterricht des visi- 
 rens. Durch Adam Riesen im 1550 Jar." Colophon: 
 " Gedruckt zu Leipzig durch Jacobum Berwalt." IV. Ein 
 Gerechent Biichlein/ auff den Schoffel/ Eimer vnd Pfundt 
 gewicht/ zu ehren einem Erbarn/ Weisen Rathe auff Sanct 
 Anr.aberg durch Adam Riesen, 1533. Zu Leiptzick hatt 
 gedruckt diss gerechent Biichlein Melchior Lotter. Volendet 
 vnd aufgangen am abendt des Newen Jars. 1536." The pic- 
 ture of the author appears in his third book with the legend, 
 "Adam Riese in the year 1550 at the age of 58," which fur- 
 nishes the date of his birth. Riese's reckoning books were 
 for a century the most popular books of the people and were 
 reprinted many times in several combinations. 
 
 Christopher Rudolff was educated at the Vienna Hoch- 
 schule. His writings were a Coss (1525), Kiinstliche Rech- 
 nung mit der Ziffer vnd mit den Zalpfennigen sampt der 
 Wellischen Practica/ vnd allerley vortheyl auff die Regel de 
 Tri/ alien Liebhabern der Rechnung vnd sonderlich dersel- 
 bigen kunst anfahenden Schiilern zu nutz/ Wien 1526," and 
 "Exempel Buchlin" (1530). The second, a reckoning book, 
 was his most popular work. Three later editions were pub- 
 lished (1546, 1574, 1588). Rudolff was one of the few 
 writers of that time who did not copy Riese. 
 
 Peter Apianus (Bienewitz) (1495-1 552), a scholar of great 
 breadth, was educated at Leipzig and became the professor 
 of Astronomy at the University of Ingolstadt and teacher of 
 Kaiser Karl V. His interest included astronomy, geography, 
 
THE ESSENTIAL FEATURES jg^ 
 
 philology and mathematics. Besides a work on the Coss and 
 one on astronomy, for which the emperor presented him 
 with 3,000 gulden, Apianus wrote " Eyn newe vnd wolge- 
 griindte vnderweysung aller Kauffmannfsrechnung in dreyen 
 Biichern mit schonen Regeln vnd fragstucken begriffen, Sun- 
 derlich was fortl vnd behendigkeit in der Welschen Practica 
 vnd Tolleten gebraucht wirdt. Desgleichen fiirmalfs weder in 
 Teutzcher noch in Welscher sprach nie gedruck. Durch Pe- 
 trum Apianum von Leylsnick d'Astronomie zu Ingolstadt 
 Ordinariu verfertigt." Colophon : " Gedruckt vnd vo- 
 lendt zu Inglostadt durch Georgium Apium von Leylsnick 
 im Jar. nach der geburt Christi 1527 am 9. tag Augusti." 
 Thus he deprived Grammateus of the honor of being the only 
 University teacher who had written a German reckoning book. 
 
 Simon Jacob of Coburg (died, 1564), was the most import- 
 ant reckoning master of the last half of the sixteenth century. 
 He wrote a reckoning book as well as a work on geometry. 
 
 The first German work on book-keeping was written by 
 Heinrich Schreiber, (Grammateus). 
 
 Corresponding to the German writers there was a school 
 of writers of practical and commercial arithmetics in other 
 countries. Among the Italians were the author of the Tre- 
 viso book (1478), the first printed arithmetic, Philip Calandri 
 and Piero Borgi ; among the Dutch a century later, Willem 
 Raets, Jaques Van der Schuere, Martin Wencelaus and 
 Ludolf van Ceulen; among the French, Savonne and Tren- 
 chant ; among the Spanish, Juan Perez de Moya, Ortega, and 
 among the English, Robert Recorde and Humphrey Baker. 
 Most of the authors of reckoning school arithmetics, like the 
 writers of Latin School arithmetics, were scholars of high 
 rank. But they were in closer touch with the needs of the 
 schools for which they wrote, for many of them served as 
 Reckoning Masters, while the Latin writers more often were 
 professors in the Universities. 
 
 It is noticeable that the contents of the reckoning school 
 arithmetics were deficient in pure arithmetic. Little attention 
 was given to definitions; the operations were stated dogma- 
 
184 SIXTEENTH CENTURY ARITHMETIC 
 
 tically; few attempts were made to grade any operation into 
 steps; and numbers larger than those commonly used in 
 business were avoided. Numerous examples for practice were 
 given in connection with the processes or in separate chap- 
 ters. Rudolff's book, for example, was divided into three 
 parts, Book of Elements, Book of Rules, and Book of Ex- 
 amples. Much importance was attached to rules, and several 
 writers resorted to verse in order to render them more effec- 
 tive (p. 133). The following subtraction rule by Reichel- 
 stain (1532) is a popular illustration: 
 
 So du magst von der obern nit 
 Ein ziffer subtrahirn mit sitt, 
 Von zehen sollt sie ziehen ab, 
 Der nechst under addir eins knab. 
 
 Thus, the theory of arithmetic was treated less scientifically 
 in these books than in the Latin School books. 
 
 The reckoning book, in harmony with its purpose, was rich 
 in applied problems. Problems involving denominate num- 
 bers, exchange and merchants rules occupied the greater part 
 (pp. 77-85, 132, 158). Besides commercial problems, men- 
 suration received much attention, because the Reckoning Mas- 
 ter, who often performed the duties of surveyor in his town, 
 included in his reckoning book the methods for finding heights 
 and distances (pp. 163-7) ; and, since as inspector of im- 
 ports he had to measure the bales and casks, he gave in his 
 books the theory and practice of the gauge. 1 It was usual 
 for the applications to follow directly the explanation of the 
 processes, although in several works a problem 1 was pro- 
 posed before the process required had been taught (p. 48), 
 a plan revived in the eighteenth century by Sturm and Wolf. 
 
 The reasons for teaching arithmetic in the reckoning schools 
 scarcely need to be stated. The origin and function of these 
 schools and the interests of their teachers leave no doubt as 
 to why arithmetic was taught. Utility was their shibboleth. 
 They sought to make good computers and to give a thorough 
 training in industrial and commercial arithmetic. 
 
 1 Jacob Kobel, Ein new geordet Visirbiich (1515). 
 
CHAPTER II 
 
 Educational Significance of Sixteenth Century 
 Arithmetic 
 
 We have set forth in Chapter I the contents of sixteenth 
 century arithmetic. We have seen that it was produced by 
 the leading scholars and teachers of that period, and that the 
 chief influences responsible for this product were : the demands 
 of the commercial and industrial world, the educational ideals 
 and practices, the restraint of traditional customs, and the de- 
 mands of science. We have seen that the purposes or aims 
 of the writers of arithmetic were : to furnish useful informa- 
 tion', to provide material for mind training, and to advance 
 human knowledge for its own sake. And, furthermore, since 
 the leading spirits among the authors of arithmetic in that 
 period were also the leading spirits among the teachers of the 
 subject, the teacher's aims corresponded to those of the author; 
 namely, to furnish the mind with useful information, to exer- 
 cise it in mathematical thinking, and to interest it in arith- 
 metic for the sake of knowledge getting. 
 
 We recognize in these influences and aims the same forces 
 and purposes that control the present day production and 
 teaching of arithmetic. Hence, a study of the questions which 
 arose and were solved at that time in relation to the corres- 
 ponding questions now pressing for an answer will show to 
 what extent we may profit from this inheritance; in other 
 words, it will show the educational significance of sixteenth 
 century arithmetic from the point of view of the present time. 
 
 Both the problems of that period and those of to-day con- 
 cerning the teaching of arithmetic may be classified under 
 the heads: subject-matter, method, and mode. 
 
 185 
 
1 86 SIXTEENTH CENTURY ARITHMETIC 
 
 Subject-Matter 
 We have seen that the subject-matter of arithmetic con- 
 sisted of rules for reckoning with counters (line reckoning, 
 pp. 25-26), rules for reckoning with the Hindu numerals 
 (figure reckoning, pp. 27-29), a body of properties and rela- 
 tion of numbers (pp. 29-77), a system of denominate num- 
 bers (pp. 77-85), and a heritage of amenity or puzzle problems 
 
 (P- 159)- 
 
 The chief question relating to subject-matter which con- 
 fronted sixteenth century writers was the basis of its selec- 
 tion. The few who wrote in an encyclopedic fashion, Paciuolo 
 and Tartaglia for example, tried to furnish arithmetic of all 
 kinds and suited to all purposes; but most writers grasped a 
 special need or set up an ideal and formulated their works in 
 harmony with it. 
 
 We have seen in the cases of Riese and Kobel and other 
 reckoning masters that the basis of the selection of subject- 
 matter was the need of the trader. 1 By them, processes and 
 number relations were reduced to those only which were of 
 use in solving commercial problems. The simple operations 
 with comparatively small integers and fractions were briefly 
 explained but elaborately applied to business questions; and 
 denominate numbers, because of their variety and complexity, 
 were made a prominent feature. In the cases of Cardan, Uni- 
 corn, Ramus, and other Latin School writers the basis of selec- 
 tion of subject-matter was the need of the scholar. Conse- 
 quently, these authors gave a more elaborate treatment of 
 processes covering a larger number field, emphasized theoretic 
 subjects, like proportion, progressions, and roots, extended 
 arithmetic to the solution of equations, retained the puzzle 
 problems, but neglected the business applications. In the 
 
 1 This is seen from such dedications as : " Prepared for merchants " 
 (Borgi), "For business purposes" (Riese), "For merchants, bookkeepers 
 and beginners" (Van der Schuere), "To the right worshipfull, the gov- 
 ernoure assistentes, and the rest of the companye of Marohants adven- 
 turers : Humfrey Baker Lodoner wisheth helth with continuall increase of 
 comodity by their worthy travail." Baker. 
 
EDUCATIONAL SIGNIFICANCE 
 
 I8 7 
 
 cases of still others the basis of selection seemed to be a pre- 
 ference for the traditional. What had attracted Nicomachus 
 and other Greeks, what had seemed to Boethius worth while, 
 appealed to these authors and caused them to perpetuate and 
 extend the fanciful classifications of numbers handed down 
 from antiquity (pp. 32-35). 
 
 The same ideas control the selection of material in the pre- 
 paration of arithmetics to-day. The needs of the trader in- 
 spire the " commercial arithmetic," the needs of the scholar 
 inspire the " disciplinary school arithmetic," and the influence 
 of tradition is responsible for the retention in both of these 
 books of much formal arithmetic long since obsolete. 
 
 A comparison of the conditions which made the " Rechen- 
 buch " of the sixteenth century, with the corresponding con- 
 ditions of to-day, indicates that we are entering upon a new era 
 of commercial arithmetic ; for, when a nation passes from pas- 
 toral and agricultural pursuits to those of manufacture and 
 trade, the transition seems to be reflected in the subject-mat- 
 ter of its arithmetic. It was true of the European nations 
 of the fifteenth and sixteenth centuries. The fall of Con- 
 stantinople (1453) and the consequent westward movement of 
 Christian civilization initiated the educational and industrial 
 awakening of Italy. This movement was reflected in the first 
 printed arithmetic (Treviso, 1478) and in the subsequent 
 Italian works of Calandri (1491), Borgi (1484), Paciuolo 
 (1494), and Tartaglia (1556). The Hanseatic League by 
 opening trade routes developed the commercial possibilities 
 of Germany. The establishment of reckoning schools along 
 these routes provided the opportunities by which the great 
 German Reckoning Masters, Ulrich Wagner, Adam Riese, 
 and Christopher Rudolf? rose to fame. The spirit of enter- 
 prise invaded France in 1553, and Savonne ushered in com- 
 mercial arithmetic. The Netherlands awoke to her maritime 
 possibilities under the Inquisition through the oppression of 
 the Spanish king, and practical arithmetic immediately found 
 expression in the works of Van der Schuere and Raets. Eng- 
 land, feeling the pulse of continental commerce, increased her 
 
188 SIXTEENTH CENTURY ARITHMETIC 
 
 trade, and Recorde and Baker founded her practical arith- 
 metic. The conditions in America to-day are somewhat 
 similar to those of the European countries of the sixteenth 
 century. The United States has built up a vast industrial sys- 
 tem on the basis of its natural resources. It has recently 
 taken the position of a world power through commercial and 
 political expansion, and the teaching of arithmetic is just be- 
 ginning to respond to this movement. Teachers and educa- 
 tors are demanding that the subject-matter of arithmetic shall 
 meet the needs of the present. The school cannot prepare for 
 life unless it reflects the interests of the people. It must re- 
 cognize in its teaching the quantitative aspect of such far- 
 reaching enterprises and industries as our railways, steam- 
 ships, telegraphs, telephones, mines, quarries, ranches, plan- 
 tations, and factories. The following is a broad classification 
 of the type of problems that are beginning to appear in the 
 arithmetics of the twentieth century : 
 
 Problems of Resources : Products of soil and mine ; supply 
 of timber and water power; capacity of development and 
 consumption. 
 
 Problems of Industries : Labor and capital, production of 
 food products, clothing, tools, and luxuries. 
 
 Problems of Transportation: Express rates, freight rates 
 by railroad and steamships, capacities of carriers, and storage. 
 
 Problems of Communication: Postal rates, telephone, tele- 
 graph, and cable rates. 
 
 Problems of Government: Expenses, revenues, enterprises, 
 imports and exports, immigration, and employees. 
 
 Problems relating to Education: Schools, libraries, print- 
 ing, and cost of public instruction. 
 
 Problems of Science and Art : Domestic economy and nature 
 study. 
 
 This, however, does not mean that twentieth century arith- 
 metics are to be a revival of the Reckoning Books of the six- 
 teenth century. The Reckoning Book owed its existence to 
 the exclusion of commercial arithmetic from; the schools of the 
 Church, the Municipality, and the Teaching Order. No com- 
 
EDUCATIONAL SIGNIFICANCE r gg 
 
 promise was possible, the educational ideal of the schorl men 
 of that day was too limited to appreciate the needs of the 
 merchant. Modern systems of education on the other hand 
 welcome utilitarian arithmetic. The business men of to-day 
 are not obliged to found special schools and seek special books 
 in order to promote elementary business knowledge. The 
 business college is a convenience and short cut to certain vo- 
 cations rather than a substitute for public school instruction. 
 Modern education tends toward uniformity and toward a cor- 
 relation of interests, hence twentieth century arithmetic will 
 not be wholly utilitarian like that of the Reckoning Books, 
 but will be a composite of materials chosen on all three bases, 
 the needs of the business man, the needs of the scholar, and 
 the influence of tradition. 
 
 Another modification of the subject matter of arithmetic in 
 the sixteenth century due to the commercial ideal was the 
 transition from line (abacus) reckoning to figure reckoning 
 (algorism). This change in the subject matter of arithmetic 
 was hastened by the conviction that the Hindu reckoning was 
 a more efficient system of calculation than the abacus in solv- 
 ing business problems. The abacus was not discarded because 
 at was a machine, but because a better method was found to 
 | do its work. A similar tendency exists to-day through the 
 introduction of the calculating machine. Although the mod- 
 ern tendency is to change from head work to automatic ma- 
 chinery, which seems to be exactly opposite to the change in 
 the sixteenth century, it is, nevertheless, similar • in being a 
 change from the difficult to the more easy method of practical 
 calculation. If the cost of a reliable machine for performing 
 the four fundamental operations did not prevent its use from 
 becoming universal, one might reasonably predict the dis- 
 appearance of the formal processes from business arithmetic 
 before the end of the present century. But this is certain, 
 that the mechanical calculator by decreasing the need for 
 expert accountants will lessen the amount of abstract drill 
 matter in arithmetic and will limit the treatment of the oper- 
 ations to the field of small numbers. 
 
igo SIXTEENTH CENTURY ARITHMETIC 
 
 The needs of the trader affected the selection of subject 
 matter in another way. No business arithmetic could be effi- 
 cient without a thorough treatment of contemporary denom- 
 inate numbers, and a glance at the practical arithmetics of the 
 sixteenth century conveys the impression that an elaborate and 
 exhaustive treatment of such numbers was essential. But a 
 careful inquiry into the weights, measures, and moneys of that 
 time shows that the variety in use was so great and the 
 awkwardness of the systems so extreme that even a con- 
 servative exposition of them occupied a large share of the 
 text-book (pp. 83, 147). This necessary prominence given to 
 denominate numbers in the first printed books initiated a cus- 
 tom which for centuries led writers to compile tables of weights 
 and measures long after they had ceased to be useful. The les- 
 son which the sixteenth century has for modern writers on this 
 point is that they should present only the contemporary sys- 
 tems, taking full advantage of their simplicity and uniformity. 
 
 We have shown what lessons concerning the selection of the 
 subject matter of arithmetic may be drawn by comparing the 
 influence of the sixteenth century commercial renaissance with 
 that of the present industrial development. Similarly, we may 
 obtain further suggestions by comparing the effects of the 
 educational ideal then prevalent with the effects of our own. 
 The scholar's need of arithmetic at that time was thought to de- 
 pend upon three things, mental discipline in the narrow sense 
 of logical thinking, culture in the broad sense of information 
 and pleasure in knowledge getting, and propaedeutics, or pre- 
 paration for scientific research. 
 
 The teaching of arithmetic at the opening of the twentieth 
 century is in much the same condition that it was at the be- 
 ginning of the sixteenth. The nineteenth century was an 
 epoch of mental discipline, and just as the Latin Schools in- 
 herited their ideal of formal studies, like astronomy and arith- 
 metic from the Ouadrivium of the Middle Ages, so the men- 
 tal disciplinarians of the nineteenth century inherited their 
 ideal from the Latin school of the Renaissance. 
 
 It is true that the functions of school studies had not been 
 
EDUCATIONAL SIGNIFICANCE lgl 
 
 so thoroughly differentiated then as now, the time for begin- 
 ning the study of the classics had not been advanced to the 
 period of secondary instruction, and therefore, arithmetic re- 
 ceived a comparatively small part of the time given to school 
 work. But the books inspired by the disciplinary idea had a 
 distinctive content (pp. 174-176), operations extended over a 
 large number field, lack of commercial problems, prominence 
 of definitions and principles, and extreme classifications. 
 These same features characterize the disciplinary arithmetics 
 of to-day; and, in the light of history, should it not be so? 
 It should, other things being equal, but they are not. We 
 have a clearer and more correct idea of the disciplinary value 
 of arithmetic now than schoolmen had then. We know that 
 this value does not depend so much upon complex processes, 
 definitions, principles, and minute classifications as upon the 
 mathematical reasoning found in the applications of the sub- 
 ject. " Mathematics is the science which draws necessary 
 conclusions." x "According to this view, there is a mathema- 
 tical element involved in every inquiry in which exact reason- 
 ing is used, and one is not justified in calling reasoning 
 mathematical unless it is exact." 2 An appreciation of the 
 mathematical element in elementary arithmetic is chiefly gained 
 by using the syllogism in solving concrete problems. In fact, 
 the simple applications of arithmetic are the most available and 
 sufficiently simple material for children to use as exercises in 
 logic. For example, the second grade pupil who solves the 
 problem, If one orange costs 2 cents what do five oranges cost ?, 
 traverses the steps of the syllogism. For the minor premise 
 is, one orange costs 2 cents; the major premise is, 5 oranges 
 cost 5 times as much as 1 orange; and the conclusion is 5 
 oranges cost 5 times 2 cents, or 10 cents. The complex prob- 
 lems of more advanced arithmetic are generally composed of 
 
 1 Benjamin Peirce, Linear associative algebra (1870) ; also American 
 Journal of Mathematics, vol. 4. 
 
 2 Maxime Bocher, Bulletin of the American Mathematical Society, Vol. 
 XI, Number 3 (1904), p. 117 '. 
 
192 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 a series of steps, each of which is solved by the application of 
 the syllogism, as in the simplest case. 
 
 For mind training- the concrete side of arithmetic is more 
 efficient than the abstract, because there is an unlimited amount 
 and variety of the former, sample in point of logic, while there 
 is little of the latter, whose logic is not beyond the elementary 
 pupil's powers. Besides having variety the concrete side pos- 
 sesses interest gained through the correlation of arithmetic 
 with other school subjects and through the interpretation of all 
 environment "in its quantitative aspect,, whereas the abstract is 
 usually uninteresting to children. 
 
 It has still another advantage over the abstract in that it 
 deals with the same material which is used in other school 
 subjects where the reasoning does not relate to quantity, al- 
 though it is till mathematical. We are accustomed to say 
 that arithmetic is the only subject from which the elementary 
 school pupil can obtain practice in mathematical reasoning. 
 But, " Mathematics does not necessarily concern itself with 
 quantitative relations, any subject becomes capable of mathe- 
 matical treatment as soon as it has secured data from which 
 important consequences can be drawn by exact reasoning." * 
 Thus, all of the following, though drawn from different sub- 
 jects, are examples of mathematical reasoning: 
 
 1. Geography, a. There is cheap water power wherever 
 there are rapidly flowing rivers. 
 
 b. There are rapidly flowing rivers in New England. 
 
 c. Therefore, there is cheap water power in New England. 
 
 2. Applied Arithmetic, a. The ratio of the circumference 
 to the diameter of any circle is *. 
 
 b. The diameter of a circle is 3 ft. 
 
 c. Therefore, the circumference of the circle is w -3 ft. 
 
 3. Pure Arithmetic, a. A fraction whose numerator is 
 greater than its denominator is an improper fraction. 
 
 b. In -f the numerator is greater than the denominator. 
 
 c. Therefore, | is an improper fraction. 
 
 1 M. Bocher, Bulletin American Mathematical Society, Vol. XI, No. 3, 
 p. 118. 
 
EDUCATIONAL SIGNIFICANCE IO , 
 
 4. History, a. Oppression of a people means revolution, 
 and revolution means democratic institutions. 
 
 b. There is oppression in country A. 
 
 c. Therefore, there will be democratic institutions in coun- 
 try A. 
 
 5. Pure Mathematics, a. An A-object is a necessary and 
 sufficient condition for a B-object, and a B-object is a neces- 
 sary and sufficient condition for a C-object. 
 
 b. a is an A-object. 
 
 c. Therefore, there is a C-object. 
 
 It is sometimes claimed that arithmetic affords better dis- 
 cipline in reasoning than does history, because the data in his- 
 tory are subject to exceptions and lacking in generality. 
 Thus, examples 2 and 3 above would be regarded as having 
 greater educational value than example 4, because oppression 
 has not always led to revolution, nor revolution ito democratic 
 institutions. But mathematical reasoning is not concerned 
 with the validity of its data; therefore, as an exercise in ■mathe- 
 matical logic, example 4 is in no way inferior to the others. 
 If the soundness of the data be considered, the study of his- 
 tory would seem to be more disciplinary than that of arith- 
 metic. For, in arithmetic no forethought can be exercised in 
 selecting true data, since the data are determined a priori. 
 From this point of view modern grammar school teaching, 
 which encourages exact reasoning from cause to effect in re- 
 gard to data drawn from history, from geography, and from 
 industrial and domestic science is really securing the essence of 
 mathematical discipline through varied symbols of expression. 
 
 Thus it appears in selecting the subject-matter of arithmetic 
 for disciplinary purposes that the concrete side should have the 
 preference, because it furnishes material suited to easy logical 
 thinking; material which is varied, abundant, interesting, and 
 related to other school subjects. 
 
 The conviction that arithmetic supplies a certain kind of in- 
 formation needed in non-technical daily life has never in a 
 large way influenced writers in the selection of subject matter. 
 The probable reason for this is that the average citizen, not 
 
194 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 engaged in a trade or business especially requiring a know- 
 ledge of arithmetic, uses very little of the subject. 1 Ancient 
 peoples like the Chinese, Hindus, and Hebrews, who con- 
 ceived of no special reason for teaching arithmetic used the 
 merest elements; practically all of the arithmetics of the six- 
 teenth century agreed in the material essential to the needs of 
 the common people, and similarly, all arithmetics of to-day 
 agree in the subject matter pertaining to this purpose. But, 
 the broader idea of information giving, that of presenting the 
 subject for its own sake, and the broader idea of knowledge 
 getting, that of learning for the pleasure of knowing, have 
 greatly influenced the selection of the subject matter of arith- 
 metic. The Latin School books (pp. 174-176) of the six- 
 teenth century shew the effect of this influence. As a rule 
 they contain a more complete list of topics than do the Reck- 
 oning School books (pp. 183-184). They often contain, in 
 addition to a rounded treatment of the theory of the subject, 
 various practical rules, much material of historical interest 
 (pp. 129-131), and recognize the amenities of the subject 
 (p. 159). It is the idea contained in the last two particulars 
 which throws light on the selection of the materials of present 
 day arithmetic. 
 
 Although it is customary to place the educational doctrine 
 of interest at a very recent date, and correctly so in the sense 
 of a formally stated theory, the essence of this idea has long 
 been appreciated. It was this central thought of education 
 that led the Latin School writers of the sixteenth century to 
 humanize arithmetic by applying it to historical data, and by 
 emphasizing the amenities of the subject. There is reason to be- 
 lieve that this idea was strong enough to have thoroughly vital- 
 ized the Latin School arithmetic had there been such subjects 
 as Commercial Geography, Domestic Art, and Nature Study 
 in the schools, available for correlation. Furthermore, that 
 these authors, although necessarily unaware of the results of 
 modern researches into child nature, grasped the importance 
 
 1 David Eugene Smith, The Teaching of Elementary Mathematics, page 
 21, New York, 1900. 
 
EDUCATIONAL SIGNIFICANCE I95 
 
 of the play or game element in education, appealed to curiosity 
 through magic squares and other remarkable properties of 
 number, 1 and recognized the sociological phase through prob- 
 lems touching social usages and conditions. One would seem 
 justified in concluding that, since men while groping after edu- 
 cational doctrine selected a part of the subject-matter of arith- 
 metic on the basis of interest, writers to-day under the present 
 flood of educational light should consult the natural instincts 
 of the individual in selecting such material. 
 
 There is evidence of a basis for number in the sense of 
 rhythm and the tendency to repeated action early manifested in 
 the child. The tendency to imitate the doings of others cor- 
 relates the simple muscular activities into more complex move- 
 ments and enables the child to participate in games and occu- 
 pations which have a quantitative side. The occupations and 
 industries of the neighborhood, facts about food, clothing, and 
 shelter, besides furnishing an abundance of material, make a 
 basis for instruction in larger interests. In passing beyond 
 the simple facts of home life, as the pupil's experience grows, 
 the whole range of production, transportation, communication, 
 and consumption, both local and national, are available as in- 
 teresting applications of arithmetic. After leaving the prim- 
 ary school the pupil enlarges his social interests. His ten- 
 dency to hero worship influences his intellectual tastes, and 
 the time is right for problems about fire and police protection, 
 
 1 E. g., Unicorn, De L'Arithmetica vniversale (1558), mentions this curi- 
 osity as taken from Frate Luca (Paciuolo), the point of interest being the 
 repetition of the same digit in the product. 
 
 777 777 
 
 2 143 
 
 1554 
 143 
 
 2331 
 3108 
 777 
 
 4662 
 6216 
 
 1554 
 
 nun 
 
 222222 
 
196 SIXTEENTH CENTURY ARITHMETIC 
 
 about the army, the navy, the life-saving service and similar 
 things of local or national concern. Even the ethical phase 
 of instruction is not unrelated to arithmetic, for acts of charity 
 and self-sacrifice in the school, the neighborhood, and the 
 state dignify the problems of economics. 
 
 Thus we have seen that the early printed arithmetics in their 
 culture phase were prophetic of the important role which the 
 subject is to play in education. 
 
 The selection of arithmetic according to its propaedeutic 
 value relates to the scholar's need of a basis for higher mathe- 
 matical study. The sixteenth century possessed some of the 
 greatest mathematicians of all time, some who did the world 
 an inestimable service by advancing the science of mathema- 
 tics. It is notable that such men as Tartaglia, Cardan, and 
 Rudolff placed at the foundation of their works a well rounded 
 treatment of theoretic arithmetic. It is probably true that 
 mathematics for the sake of its presentation and advancement 
 as a science requires a theoretic basis in arithmetic which 
 should be considered in the selection of material for the pur- 
 pose of its study. 
 
 Finally, there should be considered in the selection of the 
 subject-matter of arithmetic the influence of tradition. The 
 effect of this influence may not be always entirely separable 
 from that of either the needs of the business man or from 
 those of the scholar, for many business customs reach far 
 back into history, and the scholar's interest pertains to all 
 ages, but it has one characteristic, namely, the tendency to per- 
 petuate things regardless of their usefulness or interest. Just 
 as certain arithmetics of the sixteenth century contained ma- 
 terial which was there simply because it had long been in 
 arithmetics (pp. 32-34, 159), so some text-books of to-day con- 
 tain matter whose presence has no other justification. If we 
 adopt the theory that a sufficient amount of mental discipline 
 can be obtained from utilitarian subject-matter, then the fol- 
 lowing items may well be omitted from twentieth century 
 arithmetic. 
 
EDUCATIONAL SIGNIFICANCE ^y 
 
 Concerning Integers: 
 
 Greatest common- divisor of large numbers and the 
 
 Euclidean method. 
 
 The need for this process disappeared in the seven- 
 teenth century with the introduction of the notation of decimal 
 fractions. Before that time it was necessary to find the great- 
 est common divisor of the terms of a fraction in order to re- 
 duce it to its lowest terms (pp. 95-96). 
 
 Least common multiple of large numbers, or of those 
 
 not readily factored. 
 
 The need for this process is confined to reducing frac- 
 tions to a common denominator, but there is no actual demand 
 for the treatment of fractions whose denominators are large 
 numbers not readily factored. 
 
 Cube root, especially o>f large numbers not readily 
 
 factored. 
 
 There is scarcely any excuse for teaching the general 
 process of extracting cube root in the grammar school, or, in 
 fact, square root, although the latter is sometimes met in 
 practical measurement. The elementary school pupil seldom 
 understands the reasoning involved in the process and there- 
 fore loses even' the supposed mental training. 
 
 Progressions. 
 
 These subjects have no vital applications, and their 
 theory belongs to algebraic analysis. 
 Concerning Fractions: 
 
 Complex fractions. 
 
 Drill work in fractions above thousandths. 
 
 All fractions not common in business practice, as 39ths, 
 
 47ths, 6ists, and the like. In such matters the ques- 
 tion is not so much one of omission as of emphasis. 
 
 The presence of such fractions in arithmetic may even 
 
 be desirable, but continuous drill work with them is 
 
 deadening. 
 Concerning Decimals: 
 
 Decimals beyond thousandths should not receive 
 
 emphasis. 
 
198 SIXTEENTH CENTURY ARITHMETIC 
 
 All practice in circulating decimals. 
 Reduction of decimals to common fractions should re- 
 ceive little emphasis. 
 
 Concerning Denominate Numbers : 
 Troy zv eight. 
 Apothecary's zveight. 
 Surveyor's measure. 
 Duodecimals. 
 
 Not only should all obsolete tables be eliminated, but 
 also those which are oi use to specialists only. 
 Gill, perch, mill, and rod. The last is still spoken of 
 in rural districts, but is seldom met in practical 
 calculation. There is no defence for its great promi- 
 nence in text-book exercises. 
 
 Tables of English Money should receive less attention 
 than the exchange value of the pound sterling, the 
 mark, and the franc. 
 
 Compound numbers of more than tzvo or three de- 
 nominations are not justifiable in operations. 
 
 Concerning the applications of arithmetic : 
 
 Profit and Loss as a separate subject. A few prob- 
 lems to illustrate percentage of gain and loss are 
 sufficient. 
 
 True discount. Bank discount has taken its place en- 
 tirely. 
 
 Partial payments in the form O'f state rules and irre- 
 gular indorsements. Promissory notes which provide 
 for payment before maturity are drawn with the privi- 
 lege of paying stated amounts on interest days. 
 Annual interest, except in the form of bond coupons. 
 Equation of payments. 
 
 Partnership zvith time. Partnership without time may 
 serve as an introduction to the explanation of the mod- 
 ern Stock Company. 
 Alligation. 
 
 Compound Proportion. Simple proportion is import- 
 ant, but the old form and the traditional processes of 
 
EDUCATIONAL SIGNIFICANCE Ig9 
 
 inversion, alternation, composition, and so on, are be- 
 ing replaced by the simple equation. 
 Besides these whole topics there are many types of problems 
 which have become obsolete. The chief ones are: 
 
 Problems in commission that represent the agent as re- 
 ceiving money from his principal after deducting his 
 commission. 
 
 Problems in stocks that represent the purchase and sale 
 of fractional shares of stock. 
 
 Problems in exchange that represent all rates of ex- 
 change at par and that do not conform to modern 
 banking customs. 
 
 Problems in compound interest, apart from some bank 
 or life insurance reckoning in which tables are 
 employed. 
 
 Then there are the objectionable inverse problems 
 found everywhere, as : Given the proceeds, rate, and 
 time to find the face. Given the rate and amount to 
 find the base. Given the rate, time, and interest to find 
 the principal. Such problems may occasionally be per- 
 mitted for the purpose of jogging the intellect, but 
 seldom on the ground of utility. 
 Having discussed the bases of selection of subject-matter, 
 the organization of materials may next be considered. Al- 
 though any arrangement of subject-matter must have a bear- 
 ing upon method, it may be made for logical rather than for 
 educational reasons. In the making of sixteenth century arith- 
 metics there were three plans of construction. The first and 
 most common was the prevailing nineteenth century type. 
 The whole realm of numbers was presented at the outset 
 (pp. 36-77) and was represented under each operation. The 
 four processes were repeated with fractions and often with de- 
 nominate numbers. Another plan, best illustrated by Cardan, 1 
 was based upon the idea of teaching all the operations with 
 each kind of number. The following outline of the first few 
 chapters will illustrate: 
 
 1 Cardan Practica Arithmetice (1539). 
 
200 SIXTEENTH CENTURY ARITHMETIC 
 
 Chapter I On the Subjects of Arithmetic. 
 
 The subjects defined are: integral numbers, 
 fractions, surds, and denominate numbers. 
 
 Chapter II On Operations. 
 
 There are seven operations : numeration, ad- 
 dition, subtraction, multiplication, progres- 
 sion, division, and the extraction of roots. 
 
 Chapter III Numeration of .Integers. 
 
 Chapter IV Numeration of Fractions. 
 
 Chapter VI Numeration of Denominate Numbers. 
 
 Chapter VII Addition of Integers, including denomin- 
 ate numbers. 
 
 Chapter VIII Addition of Fractions. 
 
 Chapter IX Addition of Surds. 
 
 Chapter X Addition of Powers. 
 
 Chapter XI Subtraction of Integers and Denominate 
 Numbers. 
 The third plan was a modification of these two in which a sub- 
 ject like denominate numbers occurred under two or three 
 operations. 
 
 Although these schemes of arrangement were not based on 
 avowed educational principles, they are none the less sugges- 
 tive ait the present time. Educators are generally agreed that 
 the extreme topical plan of nineteenth century arithmetics is 
 defective, and are seriously questioning the so called spiral 
 plan, hence it may be useful to know by reference to six- 
 teenth century arithmetic that there is no practical arrange- 
 ment except a combination of the first and second plans de- 
 scribed above. The scheme of grading the subject according 
 to the size of the numbers and complexity of processes, instead 
 of by kinds of numbers or kinds of processes seems to be an 
 essential for primary teaching, but its value depends upon the 
 size of the groups and upon the systematic treatment within 
 each group. Some writers have chosen a group of topics, 
 reproducing them every ten pages and adding slowly to the 
 content presented. Some have chosen a group to represent a 
 term's or a year's work, reproducing and extending the con- 
 
EDUCATIONAL SIGNIFICANCE 20I 
 
 tent in each succeeding period. Others have reduced the 
 amount of repetition to a minimum by giving only two cycles, 
 one for the primary school and one for the grammar school. 
 The proper number of periods lies between these extremes and 
 probably approaches the larger rather than the smaller cycle. 
 The minimum cycle leads to confusion and scrappiness, while 
 the larger cycle makes possible the systematic treatment of ad- 
 dition, subtraction, multiplication, and division separately un- 
 der each kind of number within each group. It not only 
 makes possible the appearance of these processes, but it in- 
 sures to each an extent of application sufficient to leave an 
 impression of its importance upon the pupil. 
 
 Method 
 
 Method as used in this article relates to an author's manner 
 of treatment of subject-matter and not to the teacher's mode 
 of presentation of it in the class room. It is by considering 
 method in this aspect that one obtains most light from sixteenth 
 century sources on the teaching of arithmetic, because the text- 
 books constitute the chief source o>f information concerning 
 the mathematical instruction o-f that time. Even from this 
 narrower point of view one cannot expect to gain informa- 
 tion on all present questions of method, since the text-books 
 of that time were written before much of modern educational 
 doctrine saw the light. But there is enough of suggestion to 
 justify a comparison of the methods of development of subject- 
 matter at that time with those prevalent to-day, and for this 
 purpose we may consider three plans, the synthetic, the analy- 
 tic, and the psychologic. 
 
 The synthetic method is the form in which the mathema- 
 tician casts the finished product of his reasoning. It rarely is 
 the way by which he reaches the truth, but having reached it, 
 the synthetic development of the steps in the process makes 
 an elegant .and direct exposition. 
 
 The analytic method is the process of experimentation and 
 discovery. It is the pulling to pieces and comparison of parts 
 which show on what principles the system may be built up. 
 
202 SIXTEENTH CENTURY ARITHMETIC 
 
 The psychologic method seeks to develop the subject along 
 the path of least resistance by taking into account all accepted 
 educational principles. This method is not wholly separate 
 from the other two any more than logic is separate from psy- 
 chology. In general, it conforms now to one and now to the 
 other; but its characteristic feature consists in this, that it fol- 
 lows psychological principles, whereas the other methods fol- 
 low the canons of logic. 
 
 All of these types of development are represented among 
 sixteenth century arithmetics. The synthetic type, however, 
 was the common method, the others being exceptional. Al- 
 though its use was not confined to one school of writers, it pre- 
 vailed almost exclusively among those of the Latin School. It 
 was the traditional method of the ancient mathematicians, and 
 the great classical scholars of the sixteenth century, as 
 Paciuolo, Cardan, Ramus, Tartaglia, and Unicorn, preferred 
 its elegant form of expression. Of course, arithmetic does 
 not furnish opportunities for demonstrations to the same ex- 
 tent that geometry does ; there is more of definition and me- 
 chanical process; but the spirit of the synthetic method is 
 basal in all dogmatic treatments and in all topical arithmetics 
 in which abstract theory precedes the concrete matter. This 
 is the type represented by the old style books in use to-day. 
 It is a relic of the old education in which adult psychology 
 stood in lieu of child psychology and information giving in 
 place of self-realization. 
 
 The sixteenth century arithmetics which may be called 
 analytic were cast in the catechetical form,, a style somewhat 
 prevalent for half a century. Among the first of these books 
 was one by Willichius ( 1 540) , but it was not a very success- 
 ful arithmetic. Perhaps the best extant example of this 
 method is Recorde's Ground of Artes. The following is an 
 illustration taken from his clear method of teaching the signi- 
 ficance of place value in notation : 
 
 " M. (Master) Now then take heede, these certayne values 
 every figure representeth, when it is alone written without 
 other figures joyned to him. And also when it is in the first 
 
EDUCATIONAL SIGNIFICANCE 203 
 
 place though many others doo follow : as for example. This 
 figure 9 is IX. standing now alone. 
 
 " Sc. (Scholar) How, is he alone and standeth in themydle 
 of so many letters ? 
 
 " M. The letters are none of his felowes. And if you 
 were in Fraunce in the middle of M. Frenchmen, if there were 
 none Englysh man with you, you wolde recken your selfe to 
 be alone. 
 
 " Sc. I perce'aue that." fol. Bvii verso. 
 The catechetical method is one of those extreme forms of pre- 
 senting arithmetic which has been tried and found wanting. 
 But the lesson for present purposes is not that arithmetics 
 should not be analytic in form, but that in making them 
 analytic, there should be no attempt to usurp the function of 
 the teacher. The ideal arithmetic must be inductive and must 
 suggest ways of presenting subjects in the class-room, but all 
 arithmetics which may be designed to serve both as text-book 
 and teacher will prove unsuccessful as were the catechetical 
 ones of the sixteenth century. 
 
 Among the sixteenth century writers who were less en- 
 slaved to the classical tradition, that is, the authors of the 
 practical arithmetics of that period, are found those whose 
 works represent the third or psychological method of develop- 
 ment. For, although these writers were practically bound by 
 prevailing conditions to dogmatic instruction, and, although 
 they lived before the time of genetic pschology, they were not 
 without educational sense. 
 
 A noted example is that of Riese (p. 181) who recognized 
 the following principles: 
 
 1. From the concrete to the abstract — by placing reckoning 
 with counters before reckoning with figures. 
 
 2. From the simple to the complex — by presenting the full 
 form of processes before the abridged form. 
 
 3. Repetition, practice makes perfect — by working over the 
 same material in different forms. 
 
 It is significant that the authors who possessed this psycho- 
 logical insight belonged to the practical school rather than to 
 
204 
 
 SIXTEENTH CENTURY ARITHMETIC 
 
 the disciplinary school, and that the most useful and popular 
 books of that century were those conforming to their method. 
 The man with the greatest educational grasp, Adam' Riese, 
 was the greatest reckoning master and wrote the greatest reck- 
 oning book of his time. This is exactly in accord with what 
 promises to be true in the twentieth century, namely, that the 
 ideal text-book will be the utilitarian arithmetic constructed 
 according to the psychological method. 
 
 Besides the results of these general comparisons relating to 
 the organization of subject-matter, sixteenth century arithmetic 
 set various precedents and is suggestive in respect to many de- 
 tails. The educational significance of these particulars will 
 best be expressed by grouping them in four classes : those 
 pertaining to definitions, to> symbolism,, to processes, and to 
 applications. 
 
 We have already noted that formal definitions, although not 
 abandoned by any class of writers, were especially emphasized 
 by the Latin School men. It was their text-book that set the 
 practice, -characteristic of all disciplinary arithmetics, of defin- 
 ing terms and processes at the outset. But the style of the 
 definition is significant. Thus, addition was generally defined 
 as the collection of several numbers into one sum (pp. 35-36) 
 in place of the process of finding the sum of tzvo numbers. 
 Subtraction was generally defined as taking a smaller number 
 from a larger one, which is more suggestive than the defini- 
 tion, the process of finding the difference between two numbers. 
 Certain writers improved on this and even recognized sub- 
 traction to be the inverse of addition. Multiplication was 
 generally defined as repeating one number as an addend as 
 many times as there are units in another, a better definition 
 than the process of finding the product of two numbers, al- 
 though it is not generally applicable to fractions without modi- 
 fication. Division was generally defined as finding how many 
 times one number was contained in another, the partitive phase 
 often being included, a more definite form than the process of 
 finding the quotient. 
 
 It is true that sixteenth century definitions were not general, 
 
EDUCATIONAL SIGNIFICANCE 205 
 
 nor can such definitions be used in elementary work. A 
 statement which tells nothing and one which states an im- 
 portant truth unintelligibly are equally useless. It is this fact 
 which has led modern teachers to the extreme of wholly ne- 
 glecting definitions. But the power of expressing thought is 
 too significant to be crippled. Descriptions of a process or of 
 the characteristic property of a term leads to clearness and 
 efficiency. Sixteenth century arithmetic points to the conclu- 
 sion that all definitions used in teaching should be working de- 
 finitions, that is, they should state how the process is per- 
 formed. It is possible to formulate statements of this kind that 
 are not beyond the experience of the elementary school pupil, 
 and which are sufficiently general to admit of extension as new 
 fields of mathematics are entered. 
 
 Any phase of the growth of mathematical notation is an in- 
 teresting study, but the chief educational lesson to be derived 
 is that notation always grows too slowly. Older and in- 
 ferior forms possess remarkable longevity, and the newer and 
 superior forms appear feeble and backward. We have noted 
 the state of transition in the sixteenth century from the Ro* 
 man to the Hindu system of characters (pp. 24-26), the intro- 
 duction of the symbols of operation, +, — , (pp. 53-55) and 
 the slow growth toward the decimal notation (pp. 72, 75). 
 The moral which this points for twentieth century teachers 
 is that they should not encourage history to repeat itself, but 
 should assist in hastening new improvements. For example, 
 the use of x for (?) in equations, singular abbreviations, as lb. 
 for lbs. ; and the use of %o for per thousand; 2 /z for f; $ for %; 
 and a convenient abbreviation of denominations of the metric 
 system. 
 
 No mention of Roman notation can pass without reference 
 to the question, Why are we teaching it to-day ? Less than a 
 decade ago, it appeared that the traditional clock-face and the 
 numbering of introductory pages and chapters with Roman 
 numerals were doomed. But the recent craving for antiques 
 and its influence on certain crafts are making a knowledge of 
 Roman notation to thousands still desirable. 
 
206 SIXTEENTH CENTURY ARITHMETIC 
 
 The practice in sixteenth century arithmetics of treating very 
 large numbers under the natation of integers was a prominent 
 feature (pp. 34-40). Tonstall, for example, taught the reading 
 of integers to five periods and remarked that scarcely ever in 
 human experience do larger numbers occur. The treatment of 
 large numbers so early was chiefly due to the topical system 
 which required the explanation of reading and writing num- 
 bers, as large as the wisest might ever encounter, to be placed 
 in the first chapter. Then too, for the purpose of exhibiting 
 the idea of periods, a number less than one million was scarcely 
 ettective. Pestalozzi and his precursors changed all this by 
 grading arithmetic, and we might say that large numbers have 
 properly disappeared were it not for the gigantic enterprises 
 of our 'century. Let no one endeavor to confine arithmetic 
 to thousands when the daily press, in describing the improve- 
 ments in our own metropolis, deals with numbers like the fol- 
 lowing : " Plans accepted and plans that are certain of accept- 
 ance provide for an expenditure of five hundred million, dol- 
 lars within the next few years." " Two separate plans for 
 the extension of the subway system of New York City rep- 
 resenting a cost of two hundred thirty million dollars are now 
 under consideration by the rapid-transit commission." " In 
 improving the Grand Central Station one million five hundred 
 thousand cubic yards of earth will be removed, and thirty 
 thousand tons of structural steel be Used." '" In all two 
 million cubic yards of earth and rock will be carted away before 
 the work of building the Pennsylvania Station can be begun." 
 Let no one attempt to limit arithmetic to millions when our 
 statistical reports contain data like these : " The annual num- 
 ber of letters transmitted through our post-offices is 20,000,- 
 000,000 and of newspapers 12,000,000,000." " The amount 
 of deposits in the savings banks of the United States in one 
 year was $2,769,839,546." " The total amount of life in- 
 surance policies in force is $9,593,846,155, and the value of 
 our railroads and equipment is $10,717,752,155." 
 
 We cannot deny the necessity for these numbers, but we 
 can defer them to the higher grades of the grammar school. 
 
EDUCATIONAL SIGNIFICANCE 2 OJ 
 
 It is only in connection with the larger interests that these 
 numbers are met, hence they should be reserved for those pupils 
 who are able to appreciate their uses. 
 
 Another feature of the arithmetic of the Renaissance was 
 the variety of ways used to perform the processes. E. g., there 
 were three methods of subtraction (p. 51), eight methods of 
 multiplication (p. 62), and seven methods of division (pp. 
 69-72). This excess of variety was characteristic of the en- 
 cyclopedic and Latin School writers, and probably was due to 
 the tendency to follow Boethius and Hindu-Arabian classics. 
 It is quite impossible by reference to sixteenth century arith- 
 metic to prove any method to be the preferred one. There 
 is precedent for almost every form-. Some writers added up- 
 ward, and others added downward, (pp. 45-46) ; some added, 
 I subtracted, and multiplied from left to right and others from 
 right to left (pp. 51, 66) ; some placed the difference above the 
 minuend and others below the subtrahend (p. 52). But the 
 practical and commercial arithmetics generally presented one 
 or two methods, although not always the same ones. The lat- 
 ter custom is followed to-day in most cases; but there are 
 exceptions, as in the case of subtraction and division. Teach- 
 ers are asking : " Shall we teach the plan of taking a unit from 
 the next higher order of the subtrahend, or increase the minu- 
 end by ten and add one to the next order of the subtrahend ?" 
 The following from a New York educational journal shows the 
 importance attached to this subject by a superintendent of 
 schools : 1 " Letters received by the committee appointed at 
 the Teachers' Institute show that the educators do not agree 
 ; upon any best method, and I feel that it is criminal thus to 
 j: lead the millions of school children into ways that are not the 
 best and most practical at an annual expense of millions of 
 dollars to the taxpayers." After investigating the subject by 
 soliciting opinions from school men and business men his 
 , conclusion was: "What shall be done when 1 doctors dis- 
 i agree?" It is safe to say that it is not the author's business 
 to determine the only method, but to explain the advantages 
 
 1 Educational Gazette, Vol. 22. No. 5, p. 186. 
 
208 SIXTEENTH CENTURY ARITHMETIC 
 
 of the leading ones and to present one thoroughly, preferably 
 the one personally believed in. This conclusion applies to 
 division in the matter of making the divisor a whole number, 
 or of writing the quotient above the dividend or at the right 
 or under the divisor. 
 
 Something of educational significance attaches itself to the 
 old galley method of division (p. 70), for the Italian, or down- 
 ward method, to-day recognized to be by far the best of the 
 half-dozen ways then in use, scarcely gained a foothold in the 
 sixteenth century, although it was known at that time. The 
 significance lies in the reason for the persistence of the galley 
 method. I It possessed the pictorial feature, it aroused curios- 
 ity, and it held interest by association with the fascinating 
 concept of the ship. 1 There is now a tendency to abandon 
 artificial means such as numbers arranged in squares, triangles, 
 and fanciful forms for the purposes of drill on their combin- 
 ations. But history tells us that the mind clings to such de- 
 vices, and that they are a valuable element in teaching number 
 when properly chosen and not used to excess. 
 
 Much is said in discussions of the details of grade work 
 » about the proper use of language, especially about such words 
 las "borrow" and "carry." It would be difficult to estab- 
 lish general usage for these in sixteenth century arithmetic, 
 because a translation of a foreign word is seldom unique. But 
 there is no mistaking such a statement as Recorde's " keepe in 
 mind " (p. 47, n. 1 ) for carry. 
 
 Probably no teacher of arithmetic would advocate the total 
 neglect of short processes, but some prefer to regard them as 
 a high polish to be added to the basal attainments of the ele- 
 mentary school, while others believe that the shorter forms of 
 ' calculation should be taught in immediate sequence to the cor- 
 responding unabridged forms. Sixteenth century custom was 
 in accord with the latter plan (p. 75). 
 jThe origin of proofs, or tests of the work of calculation, 
 
 1 The old copy-books contain examples solved by pupils in order to shape 
 an attractive form of galley. The same tendency is seen in the old cro- 
 cetta and gelosia methods of multiplication (p. 63). 
 
EDUCATIONAL SIGNIFICANCE 2 oa 
 
 has 'been explained (pp. 44, 66, 75). It was seen that the 
 original reason for their use vanished with the transition from 
 the abacus reckoning to figure reckoning and that their use in 
 the latter practice decreased through the century. This neg- 
 lect increased with the spread of the Hindu system until, with 
 the exception of the schools in certain European states, proofs 
 of operations practically disappeared from arithmetic. In the 
 latter half of the nineteenth century it is safe to say that in 
 general the rank and file of the teaching body in America had 
 no knowledge of the existence of such proofs. The answer 
 book, or the teacher's results,, or the answer found by the ma- 
 jority of the class under instruction constituted the only court 
 of appeal in determining the correctness of a solution, j But 
 the methods of testing operations in the form of casting out 
 nines or by reverse processes or by substitution begin to ap- 
 pear in recent arithmetics, not because they are an inheritance, 
 but because they supply an educational need. For it is now 
 generally held that convenient tests are a valuable means of 
 developing the self-confidence and independence of the pupil 
 in the matter of calculation. Thus, we may not derive our 
 present reasons for teaching proofs from sixteenth century 
 arithmetic but rather the methods themselves. 
 
 Faith in the Grube plan for teaching the elementary number 
 facts is rapidly waning, 1 and sixteenth century arithmetic 
 throws some light on this movement. The chief claim for the 
 Grube method is the thoroughness secured by successively 
 treating each integer in relation to all the integers preceding 
 it. But this does not take into account that some combina- 
 tions of integers are less useful than others and, therefore, 
 should receive less emphasis. This is now recognized as a 
 vital defect and one which is not inherited from the masters, 
 for, although the early printed arithmetics often contained mul- 
 tiplication tables to 36x36 (p. 58), their authors took pains 
 to designate the most useful ones. E. g., Tartaglia called the 
 
 1 David Eugene Smith, The Teaching of Elementary Mathematics, p. 
 118. McLellan and Dewey, Psychology of Number, pp. 85-92. McMuriry, 
 Special Method in Arithmetic, p. 46 et seq. 
 
2io SIXTEENTH CENTURY ARITHMETIC 
 
 tables of 12s, 15s, 20s, 24s, — Venetian, because they were 
 of special use in computing with Venetian denominations 
 (p. 61). Then the Reckoning Books or practical arith- 
 metics of that time devoted little attention to drill on all com- 
 binations of numbers in a given field; the products of 10 x 10 
 were usually recommended (p. 60, n. 1), but formal lists of all 
 these facts were often omitted (pp. 65, n. 3, 66, Sluggard's 
 Rule). 
 
 We are now passing through a period of neglect of detailed 
 explanation of processes with abstract numbers. Writers have 
 reached the extreme in this particular, and fuller treatments 
 are again appearing in recent books. No doubt this feature 
 of arithmetic was exaggerated in the old style books of the 
 nineteenth century, but even in them the practice was not com- 
 parable with that of the sixteenth century. The early text- 
 books cannot be taken as a guide in this matter, because the 
 prominence given to processes by them was largely due to the 
 novelty of the Hindu algorism. 
 
 Teachers sometimes fail to distinguish practices which are 
 merely conventional from those which should be followed for 
 the sake of logic or for educational reasons. This failure is 
 sufficiently prevalent to command attention at state institutes 
 for teachers. Among these practices is the one relating to the 
 performing of processes in a series. E. g, 2 + 3 X 5 — 6-7-2. 
 This has the value 14 and not g$ or some other value, because 
 mathematicians have established this convention for a series of 
 operations : the processes of multiplication and division take 
 precedence of addition and subtraction. The convention might 
 have been to perform the operations in order from left to right, 
 and some teachers proceed as if that were the case. The 
 trouble with these teachers is that they are not aware that it 
 is a matter of convention, consequently they do not seek to 
 know what the custom is. There is precedent in sixteenth 
 century for the present order (p. 99, n. 4), and when such ex- 
 pressions occur in text-books the customary method of sim- 
 plification should be given. Another question which receives 
 some attention in discussions on the teaching of arithmetic is : 
 
EDUCATIONAL SIGNIFICANCE 2 I1 
 
 Which shall be taught first, long division or short division r 
 This is not an arbitrary matter, like the one above referred to, 
 but depends upon the educational axiom: Proceed from the 
 simple to the complex. In the early printed arithmetics the 
 unabridged processes usually preceded the abridged (p. 74). 
 
 The first printed arithmetics also throw some light on the 
 teaching of fractions. The fraction from the earliest times 
 has been much more difficult to master than the integer. The 
 Egyptian tables- of unit fractions found in the papyrus of 
 A'hmes * exhibit the meagre knowledge of fractions which the 
 early civilizations possessed. The Greeks and Romans made 
 little progress; e. g., the Greeks 2 wrote «5' w Xe'' for if, and 
 the Romans used a still more clumsy notation. Even the in- 
 troduction of the Hindu numerals did not sufficiently simplify 
 the work with fractions. The decimal form is the only one in 
 which the operations are practically as simple as those with 
 integers, but we still have large use for the common frac- 
 tions, and their teaching gives rise to several educational ques- 
 tions. 
 
 Since the spread of Pestalozzian ideas, the method of pre- 
 senting fractions has been through sectioned objects and dia- 
 grams. In this plan the unit is made prominent. A plan re- 
 cently recommended is that of representing the fraction as a 
 ratio, which, in the concrete aspect, depends upon the act of 
 measuring, and which emphasizes the collection or group of 
 units in place o>f the absolute s unit. A third plan suggested 
 by the evolution of our number system is to define the frac- 
 tion as an indicated quotient as soon as inexact division is en- 
 countered, for the disposition of the remainder gives the logical 
 opportunity for introducing the fractional notation. Of 
 course, if this were done in primary teaching, concrete illustra- 
 tions for the purpose of giving content to the fractional sym- 
 bol would not be excluded. It is quite probable that no one of 
 
 1 Eisenlohr, Ein mathematisches Handbuch der alten .^Egypter (Leipzig, 
 
 1877). 
 
 2 Gow, History of Greek Mathematics, pp. 42, 48. 
 
 8 McLellan and Dewey, Psychology of Number, pp. 157-162. 
 
2I2 SIXTEENTH CENTURY ARITHMETIC 
 
 these methods is best, and that all three should find place in 
 the teaching of fractions. The partition of the single thing, 
 the partition of the group, or measured unit, and the quotient 
 of two integers all appear to be essential to a general concept 
 of the fraction. Furthermore, this is in accord with the treat- 
 ments in sixteenth century arithmetics, for Kobel used the first 
 plan 1 and called into service the now illustrious apple (p. 85). 
 Writers who combined fractions and denominate numbers used 
 the second plan, and others, as Ramus and Raets, used the 
 third plan (p. 87). Tonstall suggested an exercise which, 
 if used in connection with the last form would assist in clarify- 
 ing the pupil's notion of the fraction, namely, the consideration 
 of the effect upon the value of the fraction produced by vary- 
 ing its terms (p. 89). 
 
 No one seriously questions the logical order of addition, 
 subtraction, multiplication, and division in the field of integers 
 (the Grube simultaneous plan being a misnomer), but in the 
 case of fractions this order may well be violated, even to &. 
 greater degree than at present. 
 
 The teaching of fractions by sectioning objects leads early 
 to such products as 2 X s = i and i of f = f , not necessarily 
 before the presentation of sums like \ + f = f , but generally 
 before those like i + s = TT- In other words, teachers of 
 primary arithmetic recognize that certain products in the field 
 of fractions are more easily found than the sums of fractions 
 having different denominators. But writers of text-books, es- 
 pecially in their formal treatments, have given little heed to this 
 fact, although much was made of this feature during the first 
 century of printed books. The Bamberg Arithmetic (1483) 
 (p. 181) placed the processes with fractions in the following 
 order: multiplication, addition, subtraction, and division. 
 Calandri (p. 94) and Paciuolo adopted the same order, and the 
 latter explained his position on educational grounds (p. 94). 
 
 The multiplication of a fraction by a fraction gives rise to 
 a special difficulty, namely, the explanation of " times " in cases 
 
 1 See also 'Champenois, p. 81 of this monograph. 
 
EDUCATIONAL SIGNIFICANCE 
 
 213 
 
 likeiX£ = 
 
 1 
 
 T5- 
 
 For lack of effective means of illustration 
 
 1 
 
 * 
 
 most writers banish the usual symbol, X, and substitute the 
 word " of." It would seem to be a dangerous practice to set 
 aside the general notation whenever a difficulty is encountered; 
 and one may reasonably claim Tren- 
 chant's plan (1571) to be prefer- 
 able, for he showed that, since the 
 area of the whole square here 
 shown is 1 X 1, or 1, the area of 
 the small square should be J X i or 
 -jrV (p. 106). 
 
 There are two methods in com- 
 mon use for dividing one fraction 
 by another. First, reduce the frac- 1 
 
 tions to a common denominator 
 
 and divide the numerators. Second, invert the divisor and 
 multiply the fractions. The first method is commonly taught 
 to beginners, because it is more easily explained; but, since 
 the second method is simpler in practice, a transition from the 
 first to the second is necessary. It is not the custom of mod- 
 ern arithmetics to show the connection between these two 
 plans, but sixteenth century arithmetic suggests that we might 
 well make the connection in the following way : Divide f by f . 
 
 By the first plan 
 
 
 
 
 
 Iff 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 . 
 
 
 
 1 ^ 1 = 3x3 _J_ 4X2 _ 3X3 
 
 4'3 3X44X3 4X2 
 
 By the second plan 
 
 x 3 = 3x3 
 2 4x2 
 
 The second plan is the same as the first only the fractions with 
 the common denominators are not written. Before writers 
 began to invert the divisor they multiplied crosswise (p. 108) 
 which is the very thing done in finding the fractions with a 
 common denominator, and thus the connection between the 
 two methods is made apparent. 
 
 In the problems arising from actual experience, fractions 
 
2i 4 SIXTEENTH CENTURY ARITHMETIC 
 
 usually occur in connection with denominate numbers, and in 
 business problems the aliquot parts of one hundred furnish 
 the fractions commonly met. This suggests that writers of 
 text-books should place emphasis mainly on fractions whose 
 denominators are not greater than one hundred and that little 
 attention be paid to fractions whose denominators are prime 
 numbers greater than five. This would be in accord with the 
 custom of early arithmeticians, for, although they often in- 
 dulged in large fractions, the decimal form not having been in- 
 vented, they correlated denominate numbers with fractions 
 (p. 102) and emphasized the aliquot parts of one hundred 
 (p. 81). 
 
 We have already mentioned the importance of denominate 
 numbers and the prominent place given to them in the first 
 practical arithmetics, but there remain a few points of inter- 
 est in relation to the method of presenting them for teach- 
 ing purposes. 
 
 The money system of a nation enters so largely into the 
 data of its arithmetic that this system becomes a controlling 
 factor both in the matter of material and of method. The 
 money systems of Europe in the sixteenth century were not 
 decimal systems, consequently operations involving them were 
 comparatively complex. This circumstance coupled with the 
 fact that the variety of weights and measures was great 
 (p. 83) necessarily led to much adding, subtracting, mul- 
 tiplying, and dividing with compound numbers having several 
 denominations (p. 84). This practice encouraged by the sur- 
 vival of many awkward systems has tended to perpetuate the 
 processes with compound numbers to the present time. But 
 the increasing popularity of the decimal system and the prac- 
 tice of expressing compound numbers in terms of the larger 
 units or fractions of the same are sufficient reasons for limit- 
 ing the operations with compound numbers to those having 
 only two or three denominations. 
 
 Another form of exercise that has not wholly disappeared 
 from arithmetic is the reduction from one table of denominate 
 numbers to another. E. g.„ find how many pounds Troy there 
 
EDUCATIONAL SIGNIFICANCE 215 
 
 are in 17J lb. avoirdupois. This unquestionably is a type de- 
 rived from sixteenth century problems of reduction made 
 necessary at that time by the lack of uniformity in the value 
 of current units. An example of this diversity may be seen 
 in the following from Riese : x 
 
 10 pounds at Venice = 6 pounds at Nuremberg 
 
 3 centner at Eger = 4 centner at Nuremberg 
 
 10 pounds at Nuremberg = 11 pounds at Leipzig 
 
 100 pounds at Nuremberg = 128 pounds at Breslau 
 
 7 pounds at Padua = 5 pounds at Venice 
 
 10 pounds at Venice = 6 pounds at Nuremberg 
 
 100 pounds at Nuremberg = 73 pounds at Cologne 
 
 But the day of diversities of this kind having passed so far as 
 practical calculation is concerned, it is difficult to find a reason 
 for continuing the practice in our arithmetic of reduction from 
 one table of denominate numbers to another. 
 
 Another matter in connection with the presentation of de- 
 nominate numbers is worthy of notice. The careful grading 
 of primary arithmetic and especially the spiral plan (p. 200) 
 have tended to distribute the subject-matter of weights and 
 measures throughout the whole course in arithmetic. Some 
 educators scent danger here, fearing that the subordination of 
 so important a subject as denominate numbers may lead to a 
 scrappy presentation and unsystematic knowledge. A recent 
 remedy is found in the plan of placing a thoroughly organized 
 review chapter in grammar school arithmetic. Sixteenth cen- 
 tury arithmetic suggests another solution, which consists of 
 placing a section on denominate numbers under each process 
 with integers and fractions and a final treatment under applica- 
 tions. For example, under addition of integers would be 
 placed the addition of compound numbers, only such numbers 
 and reductions being chosen as could be manipulated by pupils 
 at that stage; and similarly for other processes. This plan is 
 imperfectly shown in the Bamberg Arithmetic which presents 
 the topics thus : Addition, Subtraction of Integers ; Addition 
 and Subtraction of Denominate Numbers. Mention has been 
 made of more detailed examples (p. 94). 
 
 1 Riese, Rechnung auff der Linien und Federn/ (1571 ied.). 
 
216 SIXTEENTH CENTURY ARITHMETIC 
 
 Besides the questions concerning the proper selection of ap- 
 plications of arithmetic (pp. 187-199), there are several im- 
 portant considerations that relate to their form of presenta- 
 tion. The greatest question of method in this connection is : 
 Shall the applications precede or follow the process to be ap- 
 plied ? This query seems to imply the absurdity that the effect 
 may precede the cause, until one determines which is the cause 
 and which the effect. Since every unit of instruction should 
 have its aim, and since the consciousness of this aim on the part 
 of the pupil is a factor in his interest in the subject, a know- 
 ledge of the end for which he studies may well be outlined 
 before he is set to acquire the means. In this sense the ap- 
 plication may precede the arithmetical process. That is, the 
 use to which the process is to be put may be proposed as an 
 incentive for learning the process. For ex- 
 ample, formal multiplication may be intro- 
 duced in this way. A bushel of oats weighs 
 32 lb. Find the weight of 5 bu. by a shorter 
 process than addition. The pupil who 
 knows only formal addition uses the first 
 process. But when his attention is called 
 to the fact that his knowledge of the mul- 
 tiplication tables furnishes the sums of the l6o lb l6o Ib 
 columns he is ready to appreciate the 
 second, or shorter form. This is not a new theory for it had 
 a large number of adherents in the sixteenth century (pp. 48, 
 56, 76) . Although arithmetical writers of that time were not 
 concerned with primary instruction or method in the modern 
 sense, nevertheless they often proposed concrete problems be- 
 fore explaining the processes which entered into their solu- 
 tion. For example, Champenois introduced the subjects of 
 addition and subtraction by proposing military problems 
 (p. 128), Baker took commercial problems for his method of 
 approach, and Suevus employed facts of ancient history 
 and fancies quite his own for his center of interest 1 (pp. 
 
 1 That the attempt to stimulate interest through concrete situations may 
 
 (1) 
 
 (2) 
 
 32 lb. 
 
 
 32 
 
 
 32 
 
 
 32 
 
 32 lb. 
 
 32 
 
 5 
 
 10 
 
 10 
 
 IS 
 
 IS 
 
EDUCATIONAL SIGNIFICANCE 2 j 7 
 
 129-130. There was one writer, Robert Recorde, who 
 employed the motive of utility in a masterly way. His plan 
 was the precursor of the modern principle of appealing to the 
 pupil's vital interests. It would be difficult to find a better 
 type of instruction than his method of approach to the subject 
 of addition by reference to his Oxford pupil's expense account : 
 
 " S. (Scholar) Then wyll I caste the whole charge of one 
 monethes comons at Oxford with battelyng also. 
 
 " Master. Go to, let me see how you can doo. 
 
 " S. One wekes comons was 11 d. ob, q. and my battelyng 
 that weke was 2 d. q. q. The seconde weekes comens was 
 12 d. and my batlyng 3d. The third wekes cornos 10 d. ob. & 
 my batling 1 d. ob, c. The fourth wekes comos 11 d, q, & 
 my batling 1 d, ob, c. These 8 sumes wold I adde into one 
 whole summe." 
 
 But this method has fallen into complete neglect with the 
 passing of the centuries between that time and the present. 
 The customary practice is not sufficiently inductive. It pre- 
 sents the processes in the fields of integers, of fractions, and of 
 denominate numbers as ends in themselves, and treats the ap- 
 
 also lead to artificial means is illustrated 'by the ingenuity of Suevus, who 
 introduced Roman notation by the story of a famine. 
 
 ON THE GREAT FAMINE IN POLAND AND SILESIA. 
 
 "That the time of the famine may not be concealed, behold cvcvllvm. 
 That is, in order that the time of the famine and distress, which long ago 
 took place in Poland and Silesia, shall remain concealed from no one, but 
 shall be known by all, the year is to be reckoned from the little word 
 cvcvllvm, which here means a cap of sorrow.' - 
 
 m = 1000 
 
 ll = 100 
 
 cc = 200 
 
 vw = 15 
 
 CVCLLVM = 1315 
 
 Sigismund Sueuus, Arithmetica Historica. Die Lobliche Rechenkunst 
 (page 64). "Vt lateat nullum tempus famis Ecce cvcvllvm. Das ist: 
 Auff das die grosse Thewrung vnd Hungersnot/ die vor zeiten in Polen 
 vnd Schlesien gewesen ist/ niemande verborgen bleibe/ sondern von men- 
 niglichen wol in acht genommen werde/ so sol man durch das Wortlein 
 cvcvllvm, welches hier eine Trawerkappe heist/ die Jahrrechnung machen." 
 Page 64. 
 
218 SIXTEENTH CENTURY ARITHMETIC 
 
 . plications as convenient drill work for fixing the methods oi 
 calculation. Applications, when valued for their own sake, are 
 set apart from, the processes so as to form an independent 
 center of interest. 1 The true course doubtless lies between 
 the usage of the present and that of the past. The present 
 custom of process worship has led to poverty of ideas, and to 
 approach every variety of calculation through a problem in- 
 volving its use might lead to a confusion of ideas. 
 
 Another question of method is this : Is the great prom- 
 inence now given to commercial arithmetic in all grades 
 justifiable on educational grounds? The present place of 
 the subject in instruction is indirectly due to sixteenth 
 century teaching; for, when Pes'talozzi founded primary 
 arithmetic, he naturally drew his material from the books of 
 the Reckoning Schools which had up to that time monop- 
 olized instruction in arithmetic. These schools had little 
 to offer except commercial arithmetic plus some mensura- 
 tion, the two phases of applied arithmetic united by Kobel 
 (p. 164). Thus, the arithmetics of the nineteenth century 
 became saturated with commercial arithmetic in a narrow 
 sense. Money value was emphasized everywhere,. " bought 
 at " and " sold at " being the usual data. Subjects like part- 
 nership were introduced before the pupil had any feeling of 
 interest in them, but modern educational research is finding 
 more appropriate material for the applications of arithmetic 
 in the lower grades, — things which come within the pupil's 
 experience and for which he is willing to study the subject. 
 For example, his games,, purchases, and possessions. There 
 is a quantitative side also to manual training, domestic art, 
 geography, nature study, and drawing. If a boy is making a 
 box, a model, or an iron ornament in his manual training work, 
 there are related problems about size, amount, and cost of 
 materials. If a girl is making an apron, or a book-bag, or 
 cooking in domestic art, there are related problems about 
 amount and cost of materials. Likewise, the geography reci- 
 
 1 McMurry, Special Method in Arithmetic, p. 113 (New York, 1905). 
 
EDUCATIONAL SIGNIFICANCE 
 
 219 
 
 tation suggests problems about distances, areas, population, 
 and production. Nature study suggests problems about 
 weight, time, and motion, and drawing is full of scale meas- 
 urements and proportion. If the concrete material of the arith- 
 metic hour is drawn in part from recitations in other subjects, 
 not only is time saved and review secured, but the ideas of 
 arithmetic are enriched by association with varied and vital 
 interests. 
 
 Consequently, although sixteenth century arithmetic may 
 have led writers to put too much commercial arithmetic into 
 the primary course, it set a step in the direction of making 
 arithmetic concrete and paved the way for a better and more 
 teachable system of subject matter. 
 
 In the early arithmetics mensuration was presented in a 
 separate chapter usually placed at the end of the book, which 
 place it has since occupied. But we are now coming to a 
 more efficient use of this subject in the teaching of arithmetic. 
 Fractions are given concreteness by the handling of graduated 
 rulers, scale relations are learned by drawing lines and figures, 
 decimal fractions are illustrated and made familiar to the stu- 
 dent by the use of the meter stick, and so on. 1 That is, the 
 facts of measurement and related properties of geometric fig- 
 ures are being graded and correlated with number work from 
 the fourth year to the eighth. This plan has the further ad- 
 vantage of preparing for the geometry of the high school, and 
 of offering the opportunity for practical field mathematics, 
 through determining heights, distances, and areas. 
 
 There is a class of problems that may be called factitious or 
 artificial. For example, 100 potatoes are placed in a row at 
 intervals of 10 yd. A basket is placed at one end of the row, 
 how long will it take a person, who can walk 20 yd. a minute, 
 to bring all the potatoes to the basket ? In the sixteenth cen- 
 tury such problems served as applications to the progressions 
 (p. no), and similar ones to proportion and evolution. The 
 
 1 F. T. Jones, in School Science and Mathematics (Chicago), Vol. 5, 
 No. 6, p. 408 (190s). 
 2 W. T. Campbell, Observational Geometry, Boston (1901). 
 
220 SIXTEENTH CENTURY ARITHMETIC 
 
 early writers understood the function of these exercises to be 
 drill in processes which had no real applications, but which 
 were valuable because of their importance in other branches 
 of mathematics. But now, since the processes themselves have 
 been promoted to more advanced mathematics, there is no good 
 reason for retaining these artificial problems in elementary 
 arithmetic. 
 
 Although there is no single rule that will solve all of the 
 problems of arithmetic, there are two general methods which 
 have wide application, unitary analysis and the equation. All 
 grades of problems from " If one orange costs 3 cents what 
 would 5 oranges cost?" to " If 5 men working 8 hours a day 
 can dig a trench 3 ft. wide, 12 ft. deep and 300 ft. long in 
 ro days., how many men will it take working 10 hours a day 
 to dig a trench 3^ ft. wide, 18 ft. deep, and 450 ft. long?" 
 may be solved by either method. Unitary analysis, although 
 known in the sixteenth century (p. 137) was commonly modi- 
 fied into the Rule of Three (p. 132). This rule, a great 
 favorite for centuries, is still preserved in the subject of Pro- 
 portion (p. 134), a method now yielding to the equation. The 
 last method, too, should not be thought of as a modern in- 
 vention, but rather as a modern form of the sixteenth cen- 
 tury Rule of Three. The solution of the following example 
 from Thierfeldern by the Rule of Three shows a marked 
 similarity to the present method by the simple equation : " If 
 18 florins minus 85 groschens are equal to 25 florins minus 
 232 groschens, how many groschens are there in 1 florin ?" * 
 
 1 Caspar Thierfeldern, Arithmetica Oder Rechenbuch Auff den Linien 
 vnd Ziffern/ (1587). 
 
 " Item/ 18 A. weniger 85 gr. machen gleich so vil gr. als 25 A. -~- 232 gr. 
 wie vil hat 1 A. groschen? Facit 21 gr." 
 
 " In disem beyden Exempeln/ addir das Minus/ und subtrahir das Plus/ 
 wie hie." Page no. 
 
 I. 18 A. -h 85 gr. gleich 25 A. -e- 232 gr. 
 
 85 gr. 
 
 18 A. gleich 25 A. -H 147 gr. 
 
 18 A. -H 147 gr. gleich 25 A. 
 18 
 
 147 gr. gleich 7 A. 
 
EDUCATIONAL SIGNIFICANCE 2 2I 
 
 These solutions I and II depend upon the laws of transpo- 
 sition and are the same except that the unknown quantity, x, 
 is missing from the first. But the ease with which the second 
 form can be followed shows the advantage of the modern plan 
 with x. Thus, although changed to a more convenient form 
 by improved symbollism, the great method of the sixteenth 
 century will become the favorite one of the twentieth. 
 
 Undoubtedly, one of the chief reasons why the equation did 
 not take its place in arithmetic much earlier is the prominence 
 given to the Rule of False Position (p. 153). The use of 
 this method of approximation having been introduced from 
 algebra delayed the more definite process of the equation. 
 
 In considering the form of presentation best suited to the 
 more advanced applications of arithmetic one must take into 
 account that arithmetic is an important factor in interpreting 
 the world about us. " It is a standpoint from which the bet- 
 ter to see through and around a great many important topics. 
 
 Without the illumination from mathematics a great many 
 important facts and bodies of knowledge in geography, his- 
 tory, natural science, and practical life remain hazy and not 
 clearly intelligible." 1 Thus, the concrete material from which 
 the processes of arithmetic may be taught should do more than 
 furnish the center of interest. It should furnish types of 
 quantitative experience of use in appreciating the larger inter- 
 ests of life. " It is now thought proper to take a class to a 
 saw-mill, a stone quarry, a cotton factory, or a foundry as to 
 a laboratory or a recitation room. The industries of the 
 
 7 ft. 147 gr. 1 «. 
 
 facit 21 
 
 II. 1. Let x = the number of groschens in 1 florin 
 
 2. Then i8x — 85 = 25X — 232 
 
 3. Therefore, i8x = 25X — 147 
 
 4. " i8x + 147 = 25X 
 5- " 147 = 7x 
 
 6. " 7x = 147, and x = 21. 
 
 1 McMurry, Special Method in Arithmetic (New York, 1905), pp. 113- 
 114. 
 
222 SIXTEENTH CENTURY ARITHMETIC 
 
 neighborhood become standards by which the world's work of 
 various sorts is estimated and judged." 1 
 
 Arithmeticians have been very slow to grasp this inter- 
 pretative function of arithmetic while writers of text-books in 
 other subjects have made marked progress in this direction. 
 Geography, for example, no longer consists merely of unre- 
 lated facts about the political divisions and physical features 
 of the earth, but treats of the influence of heat, moisture, soil, 
 climate, and other factors on resources and industries together 
 with the significance of the latter in determining national con- 
 ditions and peculiarities. One cause which has helped to keep 
 arithmetic in this backward condition is the disciplinary ideal 
 inherited from the Latin Schools of the sixteenth century 
 (pp. 174-178). This ideal has led writers and teachers to 
 look upon the applications of arithmetic as an instrument of 
 discipline to the exclusion of its larger function of informa- 
 tion giving. For example, the chapter on percentage was 
 given over by some to the consideration of the mechanics of 
 its nine cases : 
 
 Base X rate = percentage Percentage -=- rate =base 
 
 Base X (1 + rate) = amount Amount-:- (1 + rate) =base 
 
 Base X (1 — rate) = difference Difference h- (1 — rate) = base 
 
 Percentage ■+■ base = rate Amount -*- base = 1 + rate 
 
 Difference -s- base = 1 — rate 
 
 instead of to its use in answering the quantitative ques- 
 tions of commerce and of the crafts and sciences. Another 
 cause is the recent reaction against the topical plan. Six- 
 teenth century arithmetic presented all its subject-matter both 
 theoretic and applied arranged by topics (pp. 29-170), which 
 remained the standard form until the closing decade of 
 the nineteenth century when a school of writers favoring ex- 
 treme gradation reduced the applications of arithmetic to a 
 mass of unrelated questions. It is now necessary to reorgan- 
 ize the problems of arithmetic into groups and vitalize them 
 so that they may shed light upon the various topics which 
 school subjects should illuminate. 2 
 
 1 Dutton, School Supervision, p. 208 (New York, 1905). 
 
 2 Smith and McMurry, Mathematics in the Elementary School (New 
 York, 1903), pp. 58, 59. 
 
EDUCATIONAL SIGNIFICANCE 223 
 
 Mode 
 
 In addition to the questions which relate to the selection of 
 subject-matter and to its organization for teaching purposes, 
 there are questions concerning the mode of class-room in- 
 struction. By mode is here meant the form of class exercise. 
 
 Although, directly, sixteenth century arithmetic throws little 
 light upon this department of teaching, indirectly it has im- 
 portant bearings. For, among the modes in use to a greater 
 or less extent at the present time, the early printed arithmetics 
 touch in a significant way the heuristic, the individual, the lec- 
 ture, the recitation, and the laboratory modes. The bearing 
 is an indirect one, because it is largely through the method of 
 the subject-matter that one must determine the mode of teach- 
 ing at that time. This plan of investigation, which would, 
 in general, be misleading, is quite safe in the present instance, 
 because the authors of many of the most significant arithmetics 
 were prominent teachers; and besides formulating the subject- 
 matter in harmony with their favorite mode, they often named 
 in the prefaces or introductions to the works the kind of teach- 
 ing for which their arithmetics were adapted (pp. 178, 186). 
 
 The heuristic mode finds its prototype in the catechetical 
 books (p. 202). There can be no mistaking the form which 
 teaching assumed that drew its material from these books. 
 The purely catechetical mode, of course, was not heuristic, 
 but sudh a treatment as Recorde's (p. 202) contains the de- 
 veloping process, the real unfolding of new ideas by means of 
 suggestion and skillful questioning. Perhaps the most im- 
 ) portant lesson to be drawn from this is that the heuristic mode 
 ! of teaching should not be carried to an extreme, that is, should 
 \ not reduce all instruction to a system of interrogations lest it 
 I suffer the fate of the purely catechetical form (p. 203). A 
 lesson of secondary importance concerns the function of oral 
 arithmetic. If the historical precedent be followed, we may 
 infer from Fischer, Suevus, and Recorde that the oral work of 
 the recitation should be given more to the development of new 
 ideas than to drill upon those already taught. 
 
 Although on account of the vastness of public education in 
 
224 SIXTEENTH CENTURY ARITHMETIC 
 
 this country, mass instruction! will be inevitable for a long time 
 to come, much is said about the advantages of individual in- 
 struction, and many devices are being employed to obviate the 
 defects of the class system. Among these is the laboratory 
 mode discussed on page 225. The common plan of teaching in 
 the sixteenth century was the individual one (p. 180), and 
 many arithmetics were written in conformity with this practice. 
 It was often stated in the prefaces of these books that they 
 were also adapted to self-instruction. This was true, for ex- 
 ample, of the Bamberg Arithmetic, of Kobel's Zwey Rechen- 
 buchlin, and of Recorde's The Ground of Artes. The char- 
 acteristic feature of these books, excluding an occasional 
 work like Recorde's, was the direct presentation of processes. 
 The formal rule, the "Thue ihm also" of Riese, was the sign- 
 board over every new path. Thus, history suggests that the 
 present danger to arithmetic in individual instruction may 
 
 , consist iru this, that the teacher who attempts to instruct even 
 
 1 a small class on this plan is liable to fall into the error of dic- 
 
 1 tating method instead of directing it. 
 
 The practice of merely assigning lessons to be learned out- 
 side of school hours and recited at the next meeting of the 
 class, commonly called the recitation mode, is an outgrowth of 
 modern machine teaching. There is no evidence that arith- 
 metic was taught in this fashion in the sixteenth century; on 
 the contrary, the contents of the text-books and the use of the 
 separate problem book 1 indicate that the time spent in school 
 was given to instruction, and that the work assigned was in 
 the form of applications. It is safe to say that sixteenth 
 century teaching of arithmetic with all its faults was better 
 than the so-called recitation mode. 
 
 There is still an occasional educator who believes in the 
 lecture mode of instruction, who holds that text-books have 
 disadavantages which outweigh their usefulness; but, since 
 there is no likelihood of this mode's becoming common in our 
 
 elementary schools,, it may be passed over briefly. In the six- 
 
 1 As Rudolff's Exempel Buchlin (p. 182, this monograph). 
 
EDUCATIONAL SIGNIFICANCE . :25 
 
 teenth century arithmetic was taught in the universities. In 
 fact, several authors, as Ramus, Widman, and Apianus were 
 university professors. It is undoubtedly true that arithmetic, 
 like other university subjects, was often taught by lecture, but 
 there is nothing in the early teaching of arithmetic that sug- 
 gests the advisability of using the lecture mode in the modern 
 elementary school. 
 
 Among the ways most discussed at present for presenting 
 subject matter in the class room is one known as the laboratory 
 mode. The characteristic feature of this mode consists in the 
 teacher directing the pupil in discovering and verifying im- 
 portant truths by the aid of some form of laboratory equip- 
 ment. In mathematics the equipment consists of books, maps, 
 charts, blanks, legal forms, drawing materials, physical appar- 
 atus, and other materials suited to the subject in hand. So 
 far, the experiments in the use of this plan have practically 
 been confined to secondary schools, but there is room for at 
 least an adaptation of it in the elementary school. The his- 
 tory of the formative period of commercial arithmetic shows 
 that its teaching began with this mode (p. 178 et seq.). The 
 precursor of the Reckoning School was a system of apprentice- 
 ship. A knowledge of arithmetic was obtained by working 
 in the ledgers of the counting-house, by keeping the public 
 records of the municipality, by listing, weighing, and measur- 
 ing in the warehouse and by serving the surveyor in the field. 
 The modern business college mathematics in which nominal 
 sales, shipping, and banking departments lend a semblance of 
 reality to the work are in harmony with the historical de- 
 velopment of business arithmetic. And one may reasonably 
 ask, in view of the present movement toward the vitalization 
 of arithmetic, toward its application to manual and domestic 
 arts, and toward the recognition of its interpretative function in 
 the economic questions of our people, if the spirit of the labor- 
 atory mode may not be a needed element in teaching element- 
 ary arithmetic in our public schools? 
 
226 SIXTEENTH CENTURY ARITHMETIC 
 
 SUMMARY 
 
 From the foregoing we may conclude that the teaching 
 of arithmetic in the sixteenth century supports the following 
 general theses : 
 
 Concerning Subject-Matter 
 
 i . Rapid commercial and industrial development has a vital- 
 izing effect upon the subject-matter of arithmetic. It tends 
 to enrich its problems; it encourages improved methods of 
 calculation and conditions the selection of denominate num- 
 bers. Commercial development had this controlling tendency 
 in the sixteenth century, and it is exerting the same influence 
 in the twentieth century. 
 
 2. The disciplinary function of arithmetic reaches its great- 
 est efficiency through the uses of number rather than through 
 the properties of numbers, a principle not generally recognized 
 in the Latin School books. The culture ideal has always led 
 to the selection of subject-matter having a many-sided inter- 
 est, and the propaedeutics of arithmetic at present, as in the 
 past, require the retention of certain theoretic matter. 
 
 3. Traditional custom is not a safe guide in the selection of 
 subject-matter, since it tends to perpetuate obsolete material. 
 Present commercial, industrial, and educational needs are the 
 true basis of such selection. 
 
 4. The two sixteenth century schemes of arranging subject- 
 matter, namely, by kinds of numbers and by kinds of processes 
 have proved failures in modern graded curricula. Present 
 needs can be met only by a combination of the two plans. 
 
 Concerning Method 
 
 1. The psychologic method produces ideal modern books. 
 The arithmetic of the Renaissance furnishes excellent speci- 
 mens oi the synthetic and analytic methods, and marks the 
 birth of the psychologic. The last method is eclectic, taking 
 the best features from the other two, and, besides, pursues the 
 path of least resistance in accord with modern educational 
 principles. 
 
EDUCATIONAL SIGNIFICANCE 22? 
 
 The following details of method find support in the first 
 century of printed arithmetics : 
 
 1. Artificial means for making number work interesting 
 should not be abandoned. 
 
 2. Number combinations are not equally important. 
 
 3. Unabridged processes should precede abridged ones. 
 
 4. Methods of testing the work of calculation should be 
 given. 
 
 5. Definitions are desirable in arithmetic. In cases of pro- 
 cesses they should tell how the operations are performed. They 
 should also admit of easy extension to cover the processes 
 when new fields of numbers are entered. Sixteenth century- 
 definitions are superior to most modern ones in this respect. 
 
 6. In regard to notation we may learn from the results of 
 conservatism in the past how necessary it is to welcome im- 
 proved symbolism. 
 
 7. Our books should explain the processes by the generally 
 preferred methods. The best text-books among the early 
 arithmetics did this. Encyclopedic works, now as well as 
 then, are valuable only for reference. 
 
 8. The partitive idea, the measuring idea, and the ratio 1 idea 
 are necessary to the concept of the fraction. All of these 
 ideas were recognized in sixteenth century arithmetic. 
 
 9. In the formal treatment of fractions the logical order, 
 addition, subtraction, multiplication, and division should be 
 replaced by the psychological order, multiplication, addition, 
 subtraction, and division. 
 
 10. Fractions should be correlated with denominate num- 
 bers, 
 
 11. The presentation of denominate numbers should be dis- 
 tributed under the processes with integers and fractions. In 
 compound numbers only those of two 1 or three denominations 
 require emphasis, and reductions from one table to another 
 are no longer necessary. 
 
 12. The applications of arithmetic may be proposed as in- 
 centives for learning the processes. They should be classified 
 in accordance with the aims and motives of the different 
 
228 SIXTEENTH CENTURY ARITHMETIC 
 
 periods of school instruction and should perform an inter- 
 pretative function. 
 
 13. The simple equation is an improved form of sixteenth 
 century analysis and promises to become the favorite method 
 for solving the problems of arithmetic. 
 
 Concerning Mode 
 
 1 . The heuristic mode as used in the sixteenth century sug- 
 gests that oral work should be used chiefly to develop new 
 ideas. 
 
 2. The individual mode is apt to result in dogmatic in- 
 struction. 
 
 3. The recitation mode finds no precedent in early teaching. 
 
 4. The lecture mode has no place in elementary arithmetic. 
 
 5. Renaissance arithmetic suggests the use of the laboratory 
 mode. 
 
 Thus, we conclude that the history of the formative period 
 of arithmetic supplements in many ways the conclusions of 
 educational theory in regard to the subject-matter, method, 
 and mode of modern arithmetic. 
 
INDEX 
 
 Abacist, 53 
 
 Abacus, 25, 46, 74, 209 
 
 Abstract Problems, 56, 192 
 
 Addends, 51 
 
 Addition, 26, 35, 201 
 
 of integers, 41, 61, 175 
 of fractions, 98 
 Ahmes, 211 
 Alcuin, 160 
 Algorism, 27, 61, 168 
 Al Khowarazmi, 36 
 Aliquot parts, 81, 82 
 Alligation, 132, 151, 157, 198 
 Amenities, 194 
 Andres, 28 
 Apianus (Bienewitz), 27, 28, 150, 
 
 181, 182, 225 
 Applications, 216, 218, 220, 222, 227 
 Applied Arithmetic, 97, 127, 129, 
 
 131, 169, 176, 198 
 Approximations, 66 
 Arabs, 36 
 Archimedes, 177 
 Area, 79 
 Aristotle, 29 
 Arithmetic, 35 
 Articles, 38 
 Aryabhatta, 132 
 Ascham, 171 
 Assize of Bread, 132, 157 
 Associative Law, 47, 100 
 Aventinus, 28 
 Bachet de Mezeriac, 159 
 Baker, 23, 30, 48, S3, 69, 74, 75, 81, 
 
 97, 99, 103, 108, 109, 113, 121, 138, 
 
 142, 151, 183, 186, 188, 216 
 Bamberg Arithmetic, 27, 76, 212, 
 
 215, 224 
 Banking, 146 
 Barter, 132, 150 
 Bhaskara, 36 
 Binomial Coefficients, 113 
 Biordi, 158 
 Bocher, 191, 192 
 Boethius, 30, 35, 187, 207 
 Bookkeeping, 183 
 Borgi, 37, 39, 52, 57, 61, 67, 94, 110, 
 
 133 
 Borgo. See Paciuolo. 
 Bosanquet, 90 
 
 Brahmagupta, 149 
 
 Brooks, 27, 53 
 
 Business Applications, 186 
 
 Buteo, 31, 119, 134 
 
 Calandri, 67, 69, 71, 75, 94, 104, 131, 
 
 137, 160, 169, 183, 187, 212 
 Calculation, 66, 78, 85, 85 
 Campbell, 219 
 
 Cantor, 27, 29, 36, 65, 76, 123, 148 
 Cardanus, 37, 42, 84, 100, 154, 173, 
 
 186, 196, 199, 202 
 Casting Accounts, 23 
 Casting out Nines, 45, 52, 74 
 Casting out Sevens, 66, 74 
 Cataldi, 39 
 
 Cataneo, Girolamo, 78, 79 
 Cataneo, Pietro, 60, 150, 164 
 Catechetical Arithmetic, 202 
 Chain Rule, 132, 148 
 Champenois, 30, 45, 48, 49, 56, 69, 
 
 73, 75, 86, 104, 105, 106, 114, 116, 
 
 121, 128, 216 
 Chiarini, 80, 158 
 Chuquet, 39 
 Ciacchi, 101, 150 
 Cicero, 129, 172 
 Circular Numbers, 33 
 Cirvelo, 39, 47, 58,65,84, 90, 91, 126 
 Classification, 32-34, 187 
 Clavius, 47, 173 
 Clichtoveus, 35 
 Commercial Arithmetic, 35, 83, 179, 
 
 187, 207, 219 
 
 Commercial Problems, 57, 169, 178, 
 
 184, 191 
 Commission, 199 
 Common Denominator, 97, 98, ioi, 
 
 108 
 Compayre, 171 
 
 Complementary Multiplication, 65 
 Complex Fractions, 197 
 Composite Numbers, 32 
 Conant, 146 
 Concrete Problems, 46, 48, 56, 68, 
 
 76, 216 
 Correlation, 194 
 Counters, 24, 46 
 Courses of Study: 
 
 in Latin Schools, 172 
 in Reckoning Schools, 180 
 229 
 
230 
 
 INDEX 
 
 Cube Root, 125, 197 
 Cubic Numbers, 33 
 Culture Value, 178 
 Cunningham, 141, 150, 157 
 Custom House, 83 
 Decimals, 66, 95, 197, 198, 214 
 Dedekind, 32 
 Defective Numbers, 33 
 Definitions, 191, 204, 227 
 
 of numbers, 29 
 
 of processes, 35 
 
 of fractions, 85 
 De Morgan, 23, 34, 36 
 De Moya, 183 
 De Muris, 127 
 
 Denominate Numbers, 49, 56, 60, 
 77-85. 97. 135. 184, 186, 190, 198, 
 214, 215, 217, 226 
 Dewey, 211 
 Diagrams, 106, 124 
 Digits, 38 
 Diophantus, 177 
 Di Pasi, 83, 148 
 Discount, 169, 198 
 Division, 27, 36, 201 
 
 of integers, 69-76 
 
 downward method, 70, 71 
 
 twelve cases, 73 
 
 of fractions, 107 
 Doubling, 36, 47, 76, 109 
 Duplatio. See Doubling. 
 Duties, 169 
 Dutton, 222 
 Eisenlohr, 211 
 El Hassar, 36 
 
 Equation of Payments, 132, 145 
 Equations, 55 
 
 Euclidean Method of G. C. D., 96 
 Evolution, 121-127 
 Exchange, 24, 132, 146, 158, 199 
 
 Bills of, 146 
 Factor Reckoning, 132, 142 
 Finasus, 75, 119, 157, 165 
 Finger Reckoning, 28 
 Fischer, 172, 174, 180, 223 
 Fractions, 85-110, 227 
 
 Classes of, 90-92 
 
 Definitions of, 85-90 
 
 Order of processes in, 93-95 
 
 Reduction of, 95-98 
 
 Teaching of, 212-214 
 Freidlein, 24 
 
 Galley Method, 69, 70, 208 
 Gauge, 145, 165, 184 
 Gemma Frisius, 30, 36, 37, 59, 69, 
 76, 77, 86, 94, 103, 132, 138, 153, 
 173, 176, 177 
 Gerhardt, 51 
 
 Ghaligai, 94, 122, 150, 159 
 Gio, 88 
 
 Girard, 174 
 
 Golden Rule, 132 
 
 Gow, 34, 211 
 
 Grammateus. See Schreiber. 
 
 Graphical Methods, 106, 213 
 
 Greatest Common Divisor, 95,96, 197 
 
 Grube, 209, 212 
 
 Halving, 36, 76 
 
 Hanseatic League, 178, 179 
 
 Heer, 133, 150, 180 
 
 Henry, 24 
 
 Herbartian Preparation, 139 
 
 Hindu Numerals, Symbols, 24-26, 
 
 37, 41, 186, 209 
 Hispalensis, 90 
 Incommensurables, 35 
 Integers, 197 
 Interest, 132 
 Annual, 198 
 Compound, 145, 199 
 Simple, 143 
 Tables of, 144 
 International System, 80 
 Inverse Operations, 74, 199 
 Inverse Rule of Three, 132, 138 
 Involution, 121-127 
 Jacob, 45, 132, 152, 158, 181, 183 
 Jacoba, 36 
 
 Jean, 82, 83, 144, 168 
 Jewish Profit, 145 
 Jones, 219 
 
 Jordanus, 35, 76, 119 
 Kastner, 26 
 
 Kettensatz. See Chain Rule. 
 Knott, 26, 27 
 Kobel, 37, 46, 48, 53, 60, 74, 76,81, 
 
 85, 91, 107, i2i, 164, 181, 184, 186, 
 
 212, 218, 224 
 Kuckuck, 23, 24, 27 
 Lamy, 171 
 La Roche, 39 
 Latin Schools, 170, 178, 179, 190, 
 
 204, 222 
 Least Common Multiple, 197 
 Legendre, 34 
 Leonardo, 76, I4g, 150 
 Leslie, 27, 29 
 Licht, 26 
 Lilivati, 36, 51 
 Linear Numbers, 32 
 Logarithms, 168 
 Luther, 171, 173 
 Magic Squares, 195 
 Masudi, 159 
 Maurolycus, 30, 35, 177 
 McLellan, 211 
 McMurry, C, 218, 221 
 McMurry, F., 222 
 Mediatio. See Halving. 
 Melancthon, 171 
 
INDEX 
 
 231 
 
 Mensuration, 163, 184, 218 
 Method, 185, 201 
 
 Analytic, 226 
 
 Psychologic, 226 
 
 Synthetic, 226 
 Million, 38, 39 
 
 Mint and Mintage, 132, 147, 151 
 Mixed Numbers, 101, 103, 105 
 Mode, 185, 223, 228 
 
 Heuristic, 223, 228 
 
 Individual, 224, 228 
 
 Laboratory, 225, 228 
 
 Lecture, 224, 228 
 
 Recitation, 224, 228 
 Monroe, 30 
 Moya, 29 
 Muller, 54 
 Multiplication, 27, 35, 201 
 
 of integers, 57-69 
 
 Eight methods of, 62-65 
 
 of fractions, 104 
 Nasmith, 157 
 Neander, 171 
 Nicomachus, 30, 35, 187 
 Notation, 37, 205, 206 
 Noviomagus, 28, 29, 42, 98, 111 
 Numeration, 37, 129 
 One-to-one Correspondence, 32 
 Onofrio, 154 
 Operations, 76, 77, 191 
 Order of Processes, 212 
 Orders, 38 
 Ortega, 141, 183 
 Overland Reckoning, 132, 157 
 Paciuolo, 28, 29, 32, 39, 45, 61, 64, 
 71, 72, 94, 126, 146, 169, 186, 187, 
 195, 202 
 Partial Payments, 198 
 Partnership, 132, 139, 140, 198 
 Peacock, 23 
 Peirce, 191 
 Percentage, 158, 212 
 Per cent Sign, 158 
 Perfect Numbers, 33 
 Pestalozzi, 206, 218 
 Piscator. See Fischer. 
 Place Value, 38 
 Planudes, 69, 90 
 Plato, 30, 178 
 Polygonal Numbers, 30 
 Powers, 121 
 
 Practical Arithmetic, 24, 80, 117, 128 
 Primary Arithmetic, 218 
 Processes, 27, 73, 175, 184, 207, 226, 
 227 
 
 with fractions, 93 
 
 with integers, 36, 83 
 Profit and Loss, 132, 198 
 Progressions, 110-116, 186, 219 
 Proofs, or Tests, 44-46, 52, 66, 74, 
 109, in, 208 
 
 Proportion, 117-120, 132, 134-135, 
 
 18S, 220 
 Ptolemy, 177 
 Puzzles, 159-163, 186 
 Quadrans, or Quadrant, 165 
 Quadrivium, 190 
 Rabelais, 171 
 Raets, 38, 44, 80, 87, 97, 100, 143, 
 
 183, 187, 212 
 Ramsey, 141 
 Ramus, 36, 48, 50, 52, 59, 89, 95, 58, 
 
 120, 134, 172, 173, 174, 176, 186, 
 
 202, 225 
 Rashtrakuta, 31 
 Ratio, 1 1 7-1 20 
 
 Reckoning. See Abacus, Finger, 
 
 Hindu, Counters. 
 Reckoning Book, 164, 174, 181, 
 
 187-9, 210 
 Reckoning Master, 178, 179, 180, 184 
 Reckoning Schools, 178, 179, 180, 
 
 218, 225 
 Recorde, 22, 28, 38, 39-41, 47, 49, 
 
 183, 188, 202, 208, 217, 223, 224 
 Regula del chatain, 150 
 Regula Fusti, 132, 152 
 Reichelstain, 184 
 
 Renaissance, 23, 30, 170, 190,207, 226 
 Rents, 132, 157 
 Riccardi, 23 
 Riese, 23, 60, 65, 67, 77, 84, 97, 99, 
 
 no, 112, 136, 140, 142, 145, 146, 
 
 149, 153, 169, 173, 181, 182, 187, 
 
 203, 204, 215, 224 
 Rodet, 132 
 
 Roman Symbols, 24, 27, 28, 37, 91, 
 
 205 
 Roots, 37, in, 113, 121-127, 186 
 Rouse, 172 
 
 Rudolff, 26, 57, 60, 84, 94, 96, 99, 
 101, 108, 143, 148, 150, 151, 169, 
 181, 182, 184, 187, 196, 224 
 Rule of Drinks. See Virgin's Rule. 
 
 of False Position, 132, 153, 221 
 
 of Three, 109, 131, 132, 137-139, 
 142, 149, 220 
 Rule of Two, 138 
 Rules, Minor, 156 
 
 in verse, 184 
 Salaries of Servants, 132, 157 
 Sayce, 90 
 Savonne, 183 
 
 Schools of Teaching Orders, 170 
 Schreiber (Grammateus), 183 
 Scratch Method, 69, 122 
 Seeley, 171 
 Series, no, in 
 
 Arithmetic, no 
 
 Geometric, no 
 
 Harmonic, no 
 
 Sfortunati, 142 
 
232 
 
 INDEX 
 
 Short Division, 72, 74 
 
 Short Methods, 46, 47, 67, 75 
 
 Sluggard's Rule, 65, 66 
 
 Smith, D. E., 23, 158, 194, 209 
 
 Solid Numbers, 33 
 
 Species, 36, 77, no 
 
 Square Numbers, 33 
 
 Steinschneider, 153 
 
 Sterner, 26, 29 
 
 Stevinus (Stevin), 30, 96, 120, 172 
 
 Stifel, 23, 150, 172, 173, 174 
 
 Stocks, 199 
 
 Stoy, 29 
 
 Sturm, 171, 184 
 
 Subject-Matter, 185, 186-201, 226 
 
 Subtraction, 26, 35, 45 
 of fractions, 102 
 of integers, 49, 53, 55, 61 
 
 Suevus, 37, 129, 130, 131, 139, 217, 
 223 
 
 Superficial Numbers, 33 
 
 Superfluous Numbers, 33 
 
 Suter, 36, 51 
 
 Syllogism, 191 
 
 Symbolism, 53-55 
 
 Tables, 41, 44, 50, 56, 57. 59. 82, 92, 
 144 
 
 Tagliente, 28, 75 
 
 Tare, 169 
 
 Tartaglia, 35- 41, 44, 45. 5°. S3, 58, 
 61, 65, 66, 70, 72, 73, 75, 87, 88, 
 93, 99, 119, 121, 146, 147, 154, 168, 
 169, 186, 187, 196, 202, 209 
 
 Taxes, 169 
 
 Terence, 172 
 
 Theoretic Arithmetic, 24, 128, 196 
 
 Thierfeldern, 55, 107, 108, 113, 121, 
 151, 153, 170, 220 
 
 Toilet Reckoning, 27 
 
 Tonstall, 35, 39, 42, 49, 50, 58, 65, 
 
 73, 89, 100, 103, in, 117, 118, 124. 
 125. 141, 143, 154, 175, 206, 212 
 
 Trenchant, 35, 36, 46, 49, 53, 56, 67, 
 
 74, 87, 98, 102, 106, 122, 123, 124, 
 128, 142, 146, 158, 183, 213 
 
 Treutlein, 23, 27, 69, 90 
 
 Treviso Arithmetic, 24, 39, 138, 183, 
 
 187 
 Triangular Numbers, 33 
 Trotzendorf, 170 
 Unger, 37, 39, 53, 146, 149, 170, 171, 
 
 172, 179, 180 
 Unicorn, 31, 45, 53, 65, 72, 88, 117, 
 
 121, 186, 195, 202 
 Unitary Analysis, 137, 138, 169, 220 
 Unity, 29, 30 
 Usury, 141 
 Van Ceulen, 120, 183 
 Van der Scheure, 55, 92, 105, 109, 
 
 no, in, 143, 145, 156, 186, 187 
 Villicus, 27, 29, 150 
 Vincentino, 119 
 
 Virgin's Rule, Regula Cecis, 132, 153 
 Visirbiich, 164 
 Voyage, 132 
 Wagner, 181, 187 
 
 Weights and Measures, 80, 85, 120 
 Weissenborn, 30, 65 
 Welsch Practice, 132, 135-137 
 Wencelaus, 54, 72, 104, 158, 183 
 Widman, 27, 28, 38, 54, 77, 126, 127, 
 
 149, 150, 154, 162, 181, 225 
 Willichius, 202 
 Woepcke, 24 
 Wolf, 184 
 Zero, 29, 31, 42 
 
y