Normaf School KARA ARITHMETICA . a nuf nuauCb (bnaito Qytcnmiu un^oto root* 16. fififla.3rorfhr:|-^ra(iitnn nuofl raf. Crrlittmfl>ttf ftama^wmowTa^ fucac; fl cwh ttwartf tlbiintc Affonamr colfcciuo ftil) iwmh ttlto l^wtott itotcp ,=*ujtta* lerntM* < ~ ~T r^, ^ ,. jj.n. Ar. nuoai \ twc <ip\nnut ^rtkmf eft: I'l \TK I. FROM A MANUSCRIPT OF BOETHIUS, C. 1294, SHOWING FIGURATE NUMBERS RARA ARITHMETICA d^A CATALOGVE OF THE ARITHMETICS WRITTEN BEFORE THE YEAR MDCI WITH A DESCRIPTION OF THOSE IN THE LIBRARY OF GEORGE ARTHVR PLIMPTON OF NEW YORK BY DAVID EVGENE SMITH OF TEACHERS COLLEGE COLVMBIA VNIVERSITY GINN AND COMPANY PVBLISHERS BOSTON AND LONDON MDCCCCVIII COPYRIGHT, 1908, BY DAVID EUGENE SMITH ENTERED AT STATIONERS 1 HALL $rr*g GINN S: COMPANY PRO- PRIliTORS BOSTON U.S.A. VIRO DOCTISSIMO ILLVSTRISSIMO GEORGIO - ARTVRO - PLIMPTON ARTIVM MAGISTRO LIBRORVM VETVSTIORVM AMATORI AVCTORVM AMICO SOCIO ERVDITORVM LITTERARVM FAVTORI FOTORI PREFACE One of the first and most important questions for the student of mathematical history is that relating to the available sources of information. In the fields of higher mathematics scholars have been more or less successful in bringing together these sources, and in listing them in bibliographies ; but in that humbler field in which primitive mathematics first found root, only a few biblio- philes have sought to preserve the original material, and no one has seriously attempted to catalogue it. Libri, it is true, brought together two large libraries of Kara Arithmetica, but he was neither a true book-lover nor a true scholar, for he gathered his treasures purposely to see them dispersed, his commercial spirit scattering at random what should have been kept intact for the use of scholars. Prince Boncompagni, the most learned of all collectors in this domain, lived to see an unappreciative city ignore his offer to make his magnificent library permanent, and at his death it was scattered abroad, as had been the lesser ones of Kloss and De Morgan. The third great collection of early text- books which has been made in recent years is that of Mr. Plimp- ton. Of the libraries of arithmetics printed before the opening of the seventeenth century his is the largest that has ever been brought together, not excepting Boncompagni 's, and it may well be doubted if another so large will again be collected by one man. De Morgan was able to examine, in the British Museum and else- where, less than a hundred arithmetics written before 1601, including all editions ; but Mr. Plimpton has more than three hundred, a number somewhat in excess of that reached by Bon- compagni. Indeed there are few arithmetics of much importance that are not found, in one edition or another, in his library. x KARA ARITHMETICA The writers of these early printed books, not all themselves of the centuries under consideration, were by no means obscure men. Among them was Boethius, whom Gibbon called "the last of the Romans whom Cato or Tully could have acknowledged for their countryman." In the list are the names of Cassiodorus and Capella, who at least represented what there was of culture in their day, and Isidorus, the learned Bishop of Seville. There are also the names of Archimedes, who deemed it a worthy labor to improve the number system of the Greeks ; of Euclid, whose contributions were by no means confined to geometry ; of Nicoma- chus and lamblichus, who represent the declining Hellenic civiliza- tion, and of Psellus, who was a witness of its final decay. There, too, are the names of the Venerable Bede, of Sacrobosco, and of Bradwardin, all of whom testify to the culture of mediaeval Eng- land ; of that great Renaissance compiler, Paciuolo ; of Tartaglia and Cardan, who helped to make the modern algebra, and of such scholars as Ramus, Melanchthon, and Bishop Tonstall. Worthy as such a list may be, it is rendered none the less so by the names of Widman, Kobel, Borghi, Riese, and Gemma Frisius, mere arith- meticians though they were, for few who read their works can fail to recognize that they powerfully influenced education, not only in their own time but for generations after they had passed away. In view of the fact that the fifteenth and sixteenth centuries constituted the formative period in the history of printed arith- metics, I have felt it a not unpleasant duty to catalogue the volumes in Mr. Plimpton's library that belong to this period and subject, including such later editions as it may contain, to give a brief statement of their contents, and to supplement this work by a list of other arithmetics published before 1 60 1 . That a complete bibli- ography is impossible is evident to any one who considers the subject. It is a simple matter to consult the few lists of early mathematical works, and to trace the names thus secured through such catalogues as those of the British Museum and the Biblio- theque nationale (unfortunately only just begun), and through bibliographies like those of Graesse and Hain and Coppinger. It PREFACE xi is also an easy matter to examine the masterly work of Riccardi, the less accurate list of Murhard, the catalogues of Libri, and numerous other works of a similar nature. But it is manifestly impossible to read all of the published catalogues, almost invari- ably arranged only by authors. Therefore many extant books will necessarily remain undiscovered, and it is probable that some will always elude the eye of the special bibliographer. Such is the work that has been done in preparing this volume, and such is the feeling of insufficiency of achievement that remains. There is, however, a satisfaction in knowing that the bibliography is based in large measure upon an examination of the books them- selves in various libraries, and that the secondary sources are of recognized authority. Over five hundred and fifty different works are mentioned, or, including the various editions, nearly twelve hundred books in all. Of the five hundred and fifty books, about four hundred and fifty are, strictly speaking, arithmetics. Of all arithmetics known to have been printed in the sixteenth century, and to have been important enough to have two or more editions, Mr. Plimpton's library lacks less than twenty-five. Of those which were published but once, some are known only by name, while the rest are mostly ' abachetti ' or ' Rechenbiich- leins ' mere primers of a few pages and of no importance. It must also be borne in mind that it was inevitable that cer- tain arithmetics of the sixteenth century should have perished utterly, leaving not even a record of their existence. Their very commonness often caused their destruction, a law of which the unique surviving copy of more than seventy thousand New England Primers of the Franklin-Hall press is a lonely witness in our country. It is inevitable that there should be errors in such a list. Titles have been included when mentioned in even one standard bibliography, although they cannot be found in others, it being practically impossible for one person to verify every item. It is hoped, however, that a foundation has been laid upon which others may build, eliminating or otherwise as the case may be. xii KARA ARITHMETICA Of other works in this field, little need be said. De Morgan's Arithmetical Books is still one of our best single sources, although sixty years have elapsed since it first appeared. De Morgan, however, mentions altogether about one hundred and twenty editions of works originally appearing before 1601, against nearly twelve hundred listed here. Unger and Tropffke are both scholarly writers, but their bibliographies are almost exclu- sively German. Sterner was less of a student, and his list is cor- respondently less valuable. Historians like Cantor and Zeuthen have given this particular period only nominal attention, save as to a few great arithmeticians, ignoring those contributions which set forth the real work of the people's schools. The titles of the works of such writers have not been given, since any reader of a bibliography like this will know the more general histories. Mr. Plimpton's library has also a number of valuable manu- scripts on arithmetic. Since these are not available for students generally, although of great value in themselves, they have been placed after the printed works instead of being inserted in chrono- logical sequence. A study of our numeral system has justified the inclusion of books which, if printed, would hardly have place. Manuscripts written before the forms of numerals were fixed are often valuable in tracing their development, even though the books themselves are not arithmetical. Only those have been catalogued which bear with some directness upon arithmetic, and which were written before the year 1601, although numerous others, in many respects as valuable, and including several inter- esting works on the calendar, the sphere, and astrology, are in the library. One difficulty attendant upon a labor of this kind is to deter- mine what printed books to include. The number might easily have been increased by listing, as De Morgan occasionally did, works that are not at all arithmetical Peletier's algebra for example. It has been thought better to draw the line in gen- eral more closely than he did, and to depart from genuine arith- metics only in the case of the works that discuss at least some PREFACE xiii questions relating to the science or art of numbers. No effort has been made to add to the supplementary lists books that are not purely arithmetical, such as treatises on the ancient measures, although those that are in Mr. Plimpton's library, and are of value in the study of the history of arithmetic, have been included. The arrangement is chronological by first editions, but the Index allows for alphabetical and geographical reference. Although the nature of the work is usually discussed very briefly, an examina- tion of the Index will show that a fairly complete history of Renaissance arithmetic has been included in the work a his- tory which I hope to present in other form in the future. The uncertainty in the use of such symbols as 4 and 8 has led to the measuring of the page and text. The page varies owing to the binder's work, and the text is not uniform page for page, but this plan seems the most satisfactory one for giving the size of the book. The centimeter has been taken as a unit of measure, since all English and American readers of a work like this will be familiar with it, while our popular units would be unknown to most others. These measures, like the number of lines to a page, of course vary considerably in the same book. The statement ' There were no other editions ' is to be understood to mean that I have found no others that were printed before 1601. The illustrations have, in general, been selected with a view to biblio- graphical needs, although many have a marked historical interest. In copying the titles it has been the intention to follow the original as closely as possible, without attempting to imitate par- ticular forms of type or to use capitals except as initials. In the cases of misspelled words, omitted capitals, and peculiar punctua- tion, the errors have been copied as faithfully as possible. At the same time mistakes must have been made in transcribing, although it is hoped that they are not of a serious nature. DAVID EUGENE SMITH LIST OF PLATES PAGE PLATE I. FROM A MANUSCRIPT OF BOETHIUS, c. 1294, SHOW- ING FIGURATE NUMBERS Frontispiece PLATE II. FROM THE MARGARITA PHILOSOPHICA 82 A. GEOMETRY B. ARITHMETIC PLATE III. TITLE PAGE OF LAX 122 PLATE IV. FROM THE CAMPANUS MANUSCRIPT OF EUCLID, c. 1260 433 PLATE V. FROM A MANUSCRIPT OF EUCLID, c. 1294 .... 436 PLATE VI. FROM A MANUSCRIPT OF BOETHIUS, c. 1300 . . 440 PLATE VII. FROM A MANUSCRIPT OF GIOVANNI DA FIRENZE, 1422 446 PLATE VIII. FROM A MANUSCRIPT OF SACROBOSCO, c. 1442 . 450 PLATE IX. FROM AN ANONYMOUS MANUSCRIPT, c. 1460 . . 462 ABBREVIATIONS c., circa, about cm., centimeters ed. pr., edito princeps, first edition f., ff., folio, folios fol., 4, 8, . . . , folio, quarto, octavo, . . . ib., ibidem, the same place 1., 11., line, lines p., pp., page, pages r., recto, the first page of a leaf s. a., sine anno, without date of publication s. 1., sine loco, without place of publication s. 1. a., without place or date of publication v., verso, the second page of a leaf //, the end of a line of print PART I PRINTED BOOKS PRINTED BOOKS ANONYMOUS. Ed. pr. 1478. Treviso, 1478. Title. ' Incommincia vna practica molto bona et vtile // a ciafchaduno chi vuole vxare larte dela mercha-//dantia. chia- mata vulgarmente larte de labbacho.' (F. i, r. See Fig. i.) Colophon. 'A Triuifo : : A di .io. Deceb^j : : .1478.' (F. 62, r. See Fig. 4.) Description. 4, 14.6 x 20.5 cm., the text being 7.3 x 12.8 cm. 62 ff. unnumb., 32 11. Treviso, 1478. Editions. There was no other edition. So far as known this is the first practical arithmetic to appear in print, for Albert of Saxony's Tractatus (p. 9) and the Ars Numerandi (p. 23), even if earlier, and the Etymologies of Isidorus (p. 8), are not, strictly speaking, of this class. The author of the book is unknown, but from the opening lines it seems that he was a teacher of arithmetic in Treviso. The printer is also unknown, although it was probably one Manzolo, or Manzolino. The history of the work has been carefully studied by Boncompagni, his results appearing in the Atti deW Accademia Pontifitia de 1 Nuovi Lincei, vol. 16. This particular copy was in the Pinelli collection, and was sold on February 6, 1790, to a Mr. Wodhull. It afterward found its way into the library of Brayton Ives, Esq., of New York, and at the sale of that library was acquired by Mr. Plimpton. The work is commercial in character, the fundamental processes being taken up in the common order, and these being followed by the rule of three. A curious feature not at all common in early arithmetics is the rule of two. The practical applications are chiefly included under the rule of three and partnership. There is also a brief treatment of the calendar, for Church purposes. The book is lacking in applications to exchange, and probably on this account it did not appeal to the merchant class sufficiently to warrant a second edition. 3 4 RARA ARITHMETICA In considering these early works it is necessary to understand the four types of arithmetics which the Renaissance inherited from the Middle Ages. These types are as follows : i. The theoretical books. These works were based chiefly upon Jncomrmnria t>na pzactitt moltobona et roles a cufcbaduno ebi mole vjrare larte t*la mtrcba* i tniljjarnientc larte x* labbacl:o [ ttefjatp pm e'piu toh e t>a alch uni 1 3ouani a mi molto oilecnfUmi : li ill p:ctrndf uano a ttourr roU r la mercbadannatcbe per lo:o ^mojemepiaaffeafradigaitne v > puocbotitf t>argli in fmtto qualcbe f unda nic to cerca larte t* anfmctricj^biatnata rulgarmcnre laWwcbo.CTnde to conftretto per arnoa t>i kwo: rt f ttadicJ ad vttlitat)! ma cbt p:rrcnd.. no a qucl!a;fu gondola ficola intclligcntia xxrl msegno iio:bO deliberate fe non ui tntoun parte tanie fanffare a k>20.ario cbe Io:o virtuofi oefiderii tJflle fruco re ceuere poffeanp. Jn nomc tn -010 adoncba : ro&h'o per pjinpto mw d tirto e a Is onfinc coft wedo. C tlte quelle cofe:cbe x?a la p?tma oy gme bano babuco psoducimr to:per ratone e numero fono fta formadt .^ co(T come fo/ ttO'.bano'oa ftrcognofaidc.'perone la cpgiimone ce tute le cofe:quefta pzacnca e necr flaria . ger intrarnel^)pofitomio:prmio fapi lecto:e^be qn/ to fa a I p2opoftio noft ro:Ourmro e t na molnm < dine congregata ouero infemb2ada ta molte rni/ tade.er a! menow to tJnitade^ome e.a,.lquale c lo p2imo cmeno2e nunwrotcbe fe truoua.la t># nitatle e quella cofa : x>a la qiule orm cofa ft riua t>na.0eg6dario fapi:cb* fe truoua nwmcri w tre maniere^ei pzimo fe cbiama numero fimphce.lal tro namero araculo . 41 teno fe cbwma rwmc ra 4* t 11 FIG. i. FIRST PAGE OF THE TREVISO ARITHMETIC Boethius (p. 25), who wrote in the beginning of the sixth century, fol- lowing the Greek models of Nicomachus (p. 186) and Euclid (p. n). They are devoted to such matters as the theory of figurate numbers, of which the square and cube are all that now remain in elementary text- books, and the cumbersome Greek ratio-system which testified to the ancient difficulty with fractions. Books of this class were written by PRINTED BOOKS such men as Bradwardin (p. 61), Albert of Saxony (p. 9), and Jordanus (p. 62). 2. The algorisms (algorismi). These were practical arithmetics, moltipbcare per fcacbierotft quali laffaro a! fhidj o tuozmettendo It epemplt fat fotamence m fb:ma * come pozat wdere qm fotto O2 fogK w faretop2edi?tofcacbrero.5oe.3 14* fia.9 ^? 4, enocax^efarlo peril quatro modi come qoit^a fotto. 9 ^4 9 ? 4 9 3 4/i 95 /> / 9 1 Jt 9 ^r * \6 o\ 6 4 MT^ o\|o\ o\|f\ 41 ^ FIG. 2. FROM THE TREVISO ARITHMETIC, SHOWING VARIOUS FORMS OF MULTIPLICATION KARA ARITHMETICA 6 o o -i 3?o9<>y5 i I 3 16461611 I i 369679 5 3 3 565ooo 60 i 3 i 72,0$ t 6 #4$*$ I 60 P* p pop p pi 31 166 ft e compita dna^ t vmuto to impjonuflb^oe it tHrfiominsonadoIhXbf felire 146 i 6 0113? 9 1*51 5^ > valiffc duct903 gup 3 J cbe !irraooo,e $ t>a!eranno ducatia 3 o . g . li quad fonorn quarto ae t>nooucato3iclx qlie ra3one e qnefte ftano feguramente bcne* aiufendo te.cbe qnando bauerai ^a fare qaalcbe rajcone ^Da impojtann i :e cbc tu ^ubiti : non po:ai p:uouarc piu fegurametetcbe polt .ire h toa ra co^ nr.al modo cbe bat riflo ne le trc rapoe p^cditt :. Unde per quede e per k altre ra cone p^editte : le qu ale fono 111 taco namero qumdepeua puo truert FIG. 3. FROM THE TREVISO ARITHMETIC, SHOWING MULTIPICATION AND THE GALLEY FORM OF DIVISION PRINTED BOOKS 7 written to supply the mathematical knowledge necessary for business computations, and using the Hindu-Arabic numerals. These numerals were known in India, without the zero, as early as the third century B. c. They were gradually perfected, and by the time they reached Bagdad from India, in the eighth century A. D., they included the zero. An arithmetic employing these numerals was written about 800 A. D. by an Arab scholar, Mohammed ibn Musa, known by the name of al- Khowarazmi (from his birthplace, Khwarazm), and from the Latinized form of his name came the word algorism. 3. The abacus arithmetics. These were also commercial books ; but since they used the Roman numerals, which were not suited to com- putation, the actual calculations were carried on by means of calculi (Latin, pebbles), getons (French for things thrown or cast, from the Latin jacere, to throw), or counters, from which English form we have expressions like 1 cast an account.' The table on which the calculi were cast is still called a counter in our hTiw* *<&> mcir< shops, but. like the sand tables used for com- f*T /x/\/ /\/\/ puting, it was in early times called an abacus. SJJ35"*. Just at the opening of the Renaissance the con- *fcy\/ /\'A/ test was still waging between the algorists and the abacists. These arithmetics are not found, however, in Italy, because the merchants of that country abandoned the use of the counters FlG> ^ LAST pAGE QF long before this was done in other countries. THE TREVISO ARITH- In Germany the arithmetics frequently have METIC in their titles the expression ' auff der Linien und Federn ' (Riese, p. 138, 1522), or 'mit der ziffer unnd mit den zal pfenningen' (Rudolff, p. 151, 1526), ' auff der Linien ' referring to the lines on which the counters (* zal pfenningen ') were cast, and the ' Feder ' referring to the pen with which a figure (* ziffer ') was written. 4. The computi. The computus or compotus was a treatise upon the Church calendar, containing such simple directions as were neces- sary for computing the dates of Easter and the other movable feasts. The chapter on the calendar, of which there is still some trace in our arithmetics, originated in the computus. (See Anianus, p. 31, 1488.) The Treviso arithmetic is a good example of an algorism. Fig. 3 shows that multiplication was performed as it is to-day (but see Fig. 2), and that division was performed by the * galley ' method, so called because the work resembled in form an ancient galley with its sails set. 8 KARA ARITHMETICA ISIDORUS OF SEVILLE. Ed. pr. 1472. Venice, 1483. Born, probably at or near Cartagena, c. 560 or 570 ; died at Seville, April 4, 636. One of the most learned men of his time, Bishop of Seville, and writer on theology, philosophy, and the general learning of the Middle Ages. Title. 'Cjncipit epiftola Ifidori iunioris hifpalenfis epi-//fcopi ad Braulionem cefar auguftanu epifcopu.' (F. I, r.) With this is : ' Cjn chrifti nomine incipit liber primus fancti // Ifidori hifpalenfis epifcopi de fumo bono.// Qd deus fumus 2 incomutabilis fit C.Cap. I.' (F. i, r., following the first f. 105.) Colophon. ' CFinit liber etymologiarum // Cjfidori hifpalenfis epifcopi.' (F. 101, v.) Colophon of the second part. ' CFinit liber tertius : vltim9 de sumo bono // fancti Ifidori hyfpalenfis epi : Impreffus // Venetijs per Petru loflein de Langencen.// C..M.cccc.lxxxiij.C/ (F. 28, r.) Description. Fol., 19.8 x 27.7 cm., printed in double columns, each being 6.7 X 23 cm. i2Qff. numb. + 4 unnumb. = 133 ff., 58 11. Venice, 1483. Editions. Augsburg, 1472; Venice, 1483 (here described); ib., 1485 (?); ib., 1493, fol.; Basel, 1577 (mentioned below). Also two editions s. 1. a. (Strasburg, 1470 ?), and one s. 1. a. (Cologne, 1476-78 ?) This book of etymologies written by Isidorus, Bishop of Seville in the seventh century, is the standard authority upon the state of learning in Spain in that period. The subject of arithmetic is treated in book 3, beginning (f. 15) : 'CJncipit liber tertius // C.De vocabulo arithmetice // difcipline C.Cap. I.' The work consists entirely of the mediaeval theory. The treatment is very brief (5 ff.) and is followed by a few pages on the calendar. Although appearing in 1472, this cannot be called the first printed arithmetic, since it touches so briefly upon the subject. It has therefore been placed after the Treviso arithmetic. ISIDORUS OF SEVILLE. Ed. pr. 1472. Basel, 1577. Title. ' Isidori // Hispalensis // Episcopi // Originum libri viginti/ex antiquitate eruti.// . . . Basileae,//per Petrum Pernam.' (F. I, r.) Description. Fol., 20.5 x 31 cm., printed in double columns, each being 8.2 X 24.4 cm., 60 11. With an edition of Capella. Basel, 1577. PRINTED BOOKS 9 ALBERT OF SAXONY. Ed. pr. c. 1478. Venice (?), c. 1478. Born c. 1330. He lectured in the University of Paris, was Rector at Vienna in 1365, and was Bishop of Halberstadt from 1366 to 1390. He wrote several scientific works. Title. ' Eccelletiffimi magistri alberti de // faxonia tractatus pportionum incipit feliciter.' (F. i, r.) Colophon. ' Explitiunt pportiones magiftri //alberti de faxonia.' (F. 9, r.) Description. Fol., 19.7 x 28.1 cm., in double columns, each being 5.6 X 16.7 cm. 9 ff. unnumb., 39 11. S. 1. a. (Venice?, c. 1478). Editions. S. 1. a. (Venice ?, c. 1478, here described) ; Padua, 1482 (12 ff.); ib., 1484 (12 ff.); ib., s. a., but before 1487; Venice, M.cccc.xxxlvii for 1487 (10 ff.); ib., 1494 (ioff.); ib. (with another work), 1496; ib., 1496 (but no copies extant ?) ; Bologna (with another work), 1502; ib., 1506 (commentary by Vittori); Paris, s. a. An epitome by Padre Isidore Isolani Milanese was published in Pavia in 1513, and again in 1522. Prince Boncompagni, in an elaborate discussion of the various editions (Bulletino, IV, 498), mentions this rare first edition, of which he knew but one other copy, that in the Biblioteca Ambrosiana at Milan. He was of the opinion that it was printed in Venice, in a type used before 1480. Although the date is uncertain, this Tractatus may contest with the Treviso arithmetic (p. 3) and the Ars Numerandi (p. 23) the honor of being the first printed work devoted wholly to some phase of arithmetic. It treats of ratios according to the cumbersome method of Boethius as followed during the Middle Ages. It is purely theoretical and repre- sents the university treatment of scientific arithmetic in that period. Other works of 1472-1480. The mathematical activity in Italy dur- ing this period was very considerable. From 1472 to 1480 there were 38 mathematical works printed in the country. In the next decade there were 62 and in the next 100, with 13 of uncertain date between 1472 and 1500, making a total of 213 appearing in a period of less than thirty years. Before the Treviso book there were printed at least three works which touched briefly upon arithmetic. These were (i) * De re militari libri io KARA ARITHMETICA XII,' by Robertas Valturius of Rimini, printed at Verona, 1472, fol., in the second book of which the author treats ' de arithmetica & militari geometria' ; subsequent editions ib., 1483 (two editions this year); Paris, 1483 (with title changed) ; ib., 1534, fol. ; ib., 1555 (French translation); (2) 'Speculum Majus,' by Vincent de Beauvais (Vincentius Bellovacensis), the greatest mediaeval encyclopaedia, printed at Strasburg by Mentelin, i469(?)-i473, io vols., fol. The second part of this work contains one book (no. 1 6) on mathematics, in which is given a brief treatment of algorism (see p. 5), probably the first written in France (c. 1250), although M. Henry asserted that a MS. of c. 1275, which he edited, was entitled to that distinction. Incomplete editions were also pub- lished at Venice in 1484, 1493-4, 1591. The early editions of Priscian, * De figuris et nominibus numerorum,' are mentioned under the pub- lications of 1565. (3) The Etymologies of Isidorus, Augsburg, 1472 (p. 8). In 1480 an anonymous work was issued from the Caxton press in London, entitled ' The Mirrour of the World or Thymage of the same.' Chap, i o of this work began : ' And after of Arsmetrike and whereof it proceedeth,' and this was probably the first English printed matter upon the subject. There was a second edition, London, 1506, and a third s. 1. a. (London, 1527 ?), fol. (See also Boethius, p. 25, and Faber, p. 62.) About 1480 there was published at Padua a folio work by Richard Suiseth (Suicetus, Swincetus, Swinshead, Suineshevedus, the first name possibly Roger or Raymund), entitled 'Opus aureum calculationum per Johanem de Cipro emendatum et explicit.', with subsequent editions at Pavia in i488(?), 1497, 1498, fol., at Venice in 1505, 1520, and at Salamanca in 1520. I have seen an edition of this work, s. a., assigned to c. 1477. For Nicolaus Cusa, see p. 42. GIORGIO CHIARINI. Ed. pr. 1481. Florence, 1481. A Florentine arithmetician of the fifteenth century. Title. ' Qvesto e ellibro che //tracta di mercatantie //et vsan- ze de paesi.' (F. i, r. See Fig. 5.) Colophon. ' Finito ellibro di tvcti // ichostvmi : cambi : mone //te : pesi : misvre : & vsanze // di lectere di cambi : & ter// mini di decte lectere che // nepaesi sicostvmaet in // diverse terre. Per me France //fco di Dino di lacopo Kartolaio Fiore//tino Adi X di Dicembre MCCCCLXXXI. In Firenze Apreffo// almusiftero di Fuligno. ' (F. 102, r.) PRINTED BOOKS n Description. 8, 13.2 x 21.2 cm., the text being 7.1 x 14.1 cm. 3 ff. blank + 6 unnumb. +96 numb. (Roman numerals) = 105 ff., 24 11. Florence, 1481. Editions. Florence, 1481, 8 (here described); s. a. (1498), 8. The undated edition was not the first ; Coppinger has shown that it was printed in 1498. While this is not, strictly speaking, an arithmetic, it is the first printed book to give the customs relating to exchange in use among the Floren- tine merchants at the close of the fifteenth century. It is the source from which several later writers drew their material, and is particularly valu- QVESTO EELUBRO CHE TRACT A DI MERCATANTifi ETVSANZDEPAESU FIG. 5. TITLE OF CHIARINI'S WORK able in showing the nature of the practical problems of the time. Copies of this first edition are extremely rare, but the work was well enough known for Paciuolo to appropriate some of the contents. There is a question as to its authorship. EUCLID. Ed. pr. 1482. Basel, 1562. Flourished at Alexandria c. 300 B.C. He was the author of the ' Elements,' the basis of most of the textbooks on geometry. Title. 'Die Sechs Erfte Bucher // Euclidis/ // Vom anfang oder grund //der Geometrj.// In welchen der rechte grund/nitt allain der Geometrj // (verfteh alles kunftlichen/gwifen/ vnd vortailigen ge-//brauchs des Zirckels/ Linials oder Richtfcheittes vnd // andrer werckzeiige/ fo zu allerlaj abmeffen dienftlich) // fonder auch der fiirnemften ftuck vnd vortail//der Rechen- khunft/ furgefchriben vnd // dargethon ift.// Aufs Griechifcher fprach in die Teutfch gebracht/ aigene-//tlich erklart/ Auch mit verftentlichen Exempeln/ grtind-//lichen Figuren/vnd allerlaj den nutz fiirangen ftellen-//den Anhangen geziert/ Dermaffen vormals //in Teiitfcher fprach nie gefehen // worden.// Alles zu lieb vnd gebrauch den Kunftliebenden Teiitfchen/fo fych der 12 KARA ARITHMETICA Geo-//metrj vnd Rechenkunft anmaffen/ mit vilfdltiger miihe vnd arbait //zum trewlichften erarnet/vnd in Truckh ge-//geben/ Durch // Wiihelm Holtzman/genant Xylander/ //von Augfpurg. //Getruckt zu Bafel.' (P. i.) Colophon. ' Vollendet durch Jacob Kiindig/ zu Bafel/ in // Joanns Sporini koften/im jar i562.//auff den dreyfzigften tag des // Winmonats.' (P. 199.) Description. Fol., 20 X 31.3 cm., the text being 12.5 x 25.7 cm. 14 pp. unnumb. + 185 numb, -f I blank = 200 pp., 39-52 11. Basel, 1562. Editions. The editions of Euclid have not been considered in this work except in so far as they relate particularly to arithmetic. This is the first German edition. The Plimpton library contains the first edition of Euclid (Venice, 1482), but since this has no arithmetical work except Book V it has not been included in the list. Several manuscripts of Euclid are, however, included in the second part of this bibliography be- cause of their value in tracing the changes in the forms of the numerals. This edition of Euclid is mentioned because the editor has thought it necessary to add to Book II some arithmetical work. In particular he gives three forms of multiplication, first from left to right, then in the usual way, and finally for special cases in which the short processes are involved. He also considers the division of numbers in given ratios, the extraction of roots, and a few other semi-algebraic calculations. Euclid's ' Elements ' contain much work upon the Greek theory of numbers, besides what appears in Book V, and several books were pub- lished in the sixteenth century, embodying this material. These are mentioned later. Other works of 1482. In 1482 appeared the first German arithmetic, if we except the 'Ars Numerandi' (p. 23). It was written by Ulrich Wagner, a Ntirnberg Rechenmeister, and was printed by Heinrich Pet- zensteiner at Bamberg. Only nine small pieces of parchment proof sheets remain. They contain the following colophon: 'Anno dni . . . 1482 kl'i6. lunij p. Henr. peczensteiner Babenberge : finit Ulrich wagner Rechemeister zu Niirnberg.' ' Ludus Arithmomachiae,' by John Sher- wood (Shirewode), was published at Rome in the same year. (See also p. 63, and on Albert of Saxony see p. 9, c. 1478.) PRINTED BOOKS 13 PROSDOCIMO DE BELDAMANDI, AND LIVERIUS. Ed. pr. 1483. Padua, 1483. PROSDOCIMO DE BELDAMANDI was born at Padua c. 1370-1380, and died in 1428. He was educated at the University of Padua, and also taught there. He wrote on arithmetic, music, and astronomy. JOHANNES DE LIVERIUS (LIVERIIS, LINERIIS) was a Sicilian writer on astronomy who flourished c. 1300-1350. Title. ' Profdocimi de beldamandis algo-//rifmi tractatus perutilis i neceffarius // foeliciter incipit. qui de generibus cal-// culationum fpecie preteri.t nulla3.q falte // neceffaria ad h9 art^ cOgnit63 fuerat.' (F. i, r. See Fig. 6.) Colophon. ' Algorifmus. Profdocimi de beldamadis // vna cum minuciis. Johanis de liueriis. hie //felicite'finit ImprefO padue. Anno .1.4.8.3 //die zz. februarii.' (F. 21, v.) Description. 4, 15 X 20.8 cm., the text being 9 x 14.7 cm. 27 ff. numb., 32 11. Padua, 1483. Editions. Padua, 1483, fol. (here described); Venice, 1540, 8 (see p. 15). Boncompagni could learn of only three copies of the first edition, and seven of the second. This rare work was written for the Latin schools, and is a good example, the first to appear in print, of the non-commercial algorisms of the fifteenth century. It follows ' Bohectius ' (Boethius) in defining number and in considering unity as not itself a number, as is seen in the facsimile of the first page. Prosdocimo then treats of the funda- mental operations with integers, including mediation (division by 2, which the author places before duplation or multiplication by 2), pro- gressions, and the roots. The treatment of fractions is left to Liverius : ' Incipit Algorismvs de mi-//nutijs tarn vulgaribus quam phyficis magi- //ftri loannis de Liuerijs Siculi.' The vulgar minutes ' were the com- mon fractions, and the 'physical' were the sexagesimal fractions. Towards the end of the ' Algorismus de integris ' in the 1540 edition, the date of composition appears : ' ... per Profdocimum de Belda- mandis de Padua anno domini .1410. die .10. lunij compilata fufficiant.' (F. D 5, v.) The work of Prosdocimo contains the first reference that I have seen to a slate. Indigebat etiam calculator semper aliquo lapide, vel sibi conformi super quo scribere atque faciliter delere posset figuras cum quibus operabatur in calculo suo.' (See Fig. 6.) It is probable, from this statement, that computers of his time actually erased the figures in the galley form of division (see the Treviso arithmetic, p. 3), 14 RARA ARITHMETICA nlmt tractatae pcrutilto i ncaCnue foeliciter mcipit.qui dc gcnenbue caF cnladonuni fpcaie p?etcn.t noltaj.q faltc ueceflaria ad b* ait f sgmtoj foerat te.m60 area numeroe ogondi fane oaru e: atqj diuerfo0.g licet bom ejiflcrctatq} veri crdLtfi foflidiofutu .ppf ipap regulap mul' iuetui jpter am ifti modi inon faflidiofi: q> fi in akq? talculo an rolotco enc t ttigiflj: calculators o^atoj foam a capirc inctpere oponeba t: da to q; erro: fuua adbuc fatie ^pfqane cjriftcret. t boc ^pt ftgu raa in faa ogatoe deleta03ndf^ebat cna calcolato: fcmg a aq* la pidc ud fibi ^fomtufu^ quo fcnbcre atq^ faaliter delete pcffj fi^ui 00 cu gbufopabat in calculo fuo JU ga bee oia fao0 fa ; .1 diofa atq^ laDonofa mibi uifa funtrdtfpofni hteUu cdcrc in quo ota tfta abicercn^qui eaa a^ozifmu0 Hue liber de numerie de" noiart potent. Sciao & q? in boc Ubello ponef ndiiucdo nifi ca q ad colculu neceffana fmitalia q in oliis libra pracccc arifme trice tagunf ad calcnlu nouecdfaria 4>pf b:euitatqd(inuendo II doia ergo iibettue ifle de numerie tractate b$g difi'fnitone nueri ipm rcboarc udo C Tlomeros ergo F" juclide7?.fu ^5eomctHe t.f" 33obectui pf foearifmefce (IcdtfitwuTiue rue* eft mntomdo fine qnatttae difcrcta e^ untanbae ^fcfarfi t e ex nnftattbne og0regatarjt d! umta0 ifludrq? d^nno* *}>er bac g*difRnit5j nuerr b:cpofca.quof unirae no e naerneJiQ (it pnctpiu nuehxiato q? nuer^ et;a VOCOTI poPtt la go! fnrnen^nuepX^ of eo quo rcaliqua nOerare polfunma et * ocqxf nuer^in.pceflnbniflelibeUi. in q* et& v FIG. 6. FIRST PAGE OF PROSDOCIMO DE BELDAM ANDI as the Hindus had done on their sand or dust abacus, instead of cancel- ing them in the manner explained in the early printed arithmetics. The best discussion of the lives and works of Beldamandi and Liverius is in the Boncompagni Bulletino, vol. XII. PRINTED BOOKS 15 Other works of 1483. Valturius (p. 10). In this year the second German arithmetic was printed at Bamberg, only one (incomplete) copy being known. It was possibly written by Ulrich Wagner (p. 12), and, like the 1482 work, it was printed by Petzensteiner, as appears from the following colophon : ' In zale Xpi .1483. kl ' .17. des Meyen Rech- mmg // in mancherley weys in Babenberg durch henr9 // petzenfteiner begriffen : volendet.' PROSDOCIMO DE BELDAMANDI, AND LIVERIUS. Ed. pr. 1483. Venice, 1540. See p. 13. Title. ' Algorismvs de In//tegris Magistri Prosdoci-//mi Debeldamandis Pataui fimul cu algorifmo de de-//minutijs feu f ractionibus magiftri loanis de Liuerij // ficuli. Reintegratus ab erroribus comiffis a fcri-//ptoribus, a me Federico Delphino ar- tium, & // medicine doctore, mathematicarum difci//plinaru in celeberrimo gymnafio Pata//uino publico prof eff ore, additis all// quibus verbis, in aliquibus locis,//pro maiori claritate. Et da-// tus impreffioni ad infta-//tiam meoR fcholariu // nunc algorifmu //maxime de-//fideran-//tium.// Venetijs. M. D. XXXX.' (F. i, r.) Colophon. ' Venetijs per loannem Antonium de Vulpinis de Ca-//ftrogiufredo. Anno domini .M. D. XXXX. //die octauo menfis Aprilis.' (F. 52, v.) Description. 8, 9.6 X 14.5 cm., the printed part being 7.1 X 12.5 cm. 44 ff. unnumb., 30 11. Venice, 1540. (This particular copy has 8 ff. more, the sheet F appearing in duplicate.) See p. 13. RAPHAEL FRANCISCUS. Ed. pr. 1484. S. 1. a. (Florence, c. 1516). RAFFAELE FRANCESCO. A Florentine philosopher of the latter part of the fifteenth century. Title. ' Verificatio Vniversalis in // regulas Ariftotelis de motu non rece-//dens a comuni Mathema-//tico^ doctrina.' (F. i, r. See Fig. 7.) 16 KARA ARITHMETICA Description. 4, 13 X 19.9 cm., the text being 9 x 16.7 cm. 8 ff. unnumb., 41-42 11. S. 1. a. (Florence, B. Zucchetta, c. 1 5 16). Editions. Pisa, 1484, 8; this edition, s. 1. a. (Florence, c. 1516). This is a brief treatment of proportion, hardly worthy of ranking as an arithmetic. The applications relate to problems of Aristotle. VERIFICATIO VNIVERSALIS IN regulas Ariftotelis de mom non rece' densa comuni Mathema>- FIG. 7. TITLE OF THE VERIFICATIO OF FRANCISCUS PIETRO BORGHI. Ed. pr. 1484. Venice, 1484. PIERO BORGI. A Venetian arithmetician ; died after 1494. Title. ' Qui comenza la nobel opera de // arithmethica ne la qual fe tracta // tute coff e amercantia pertinente // f acta i compi- lata p Piero borgi//da veniefia.' (F. 2, numbered i, r. See Fig. 8 for the first folio.) Colophon. ' Nela inclita cita de venetia a 9orni .2.//augufto .1484. fu impofto fine ala pre-//fente opera.' (F. 118, v.) Description. 4, 14.3 X 19.3 cm., the text being 8.4 X 13. 4 cm. 2 ff. unnumb. + 116 numb. = 118 ff., 37-38 11. Venice, 1484. Editions. Venice, 1484, 4 (here described) ; ib., s. a., which Riccardi thought might be earlier than 1484, since Ratdolt, the printer, published books in Venice as early as 1476; ib., 1488, 4 (see p. 19); ib., 1491, 4 (see p. 20); ib., 1501,4; ib., 1505; ib., 1509,4; ib., 1517, 4 (see p. 20); ib., 1528, 4 (see p. 21); ib., 1534, 4 (see p. 21); ib., 1540, 4 (see p. 21); ib., 1550, 4 (seep. 22); ib., 1551; ib., 1560; ib., 1561, 4; ib., 1567; ib., 1577. It was at one time thought on the testimony of Maittaire that there was an edition of 1482, but it has been shown by sev- eral bibliographers that the first edition is that of 1484. This is the very rare first edition of Borghi's treatise, the second commercial arithmetic printed in Italy and long thought to be the first. This particular copy belonged to Count Paolo Vimercati-Sozzi. PRINTED BOOKS 17 The text of the first edition is closely followed in that of 1488 (see p. 19), except for the index, 'Tauola de li capitoli .otegnudi i qfta opa' (f. 1 1 8, r.). This does not appear in the second edition, at least in Mr. Plimpton's copy. The letters S H S U which appear twice are thought to stand for J H S U, Jesus, possibly changed on account of some conjectured pronunciation. They appear the second time on f. 1 1 8, v., in connection with a set of verses beginning as follows : 'S HSU Quanto latua memoria et alto ingegno vaglia ne larithmetica hai moftrato nel prefente volume compilato petro borgo date veneto degno.' In the verses appears the name of the printer : ' Ma limpreffor de augufta Errardo experto di lopera prefente ftampatore degno e non di mediocre laude certo.' This folio, as already stated, does not appear in the Plimpton copy of the 1488 edition, and since the later printer was not the same it prob- ably never appeared after 1484. This work is more elaborate than the Treviso arithmetic, and had far greater influence on education. More than any other book it set a standard for the arithmetics of the succeeding century, and none of the early textbooks deserves more careful study. Borghi first treats of notation (see Fig. 9), carrying his numbers as high as ' numero de mil- lion de million de million,' and making no mention whatever of the Roman numerals. In the same spirit he eliminates all of the mediaeval theory of numbers, asserting that he does this because he is preparing a practical book for the use of merchants. ' Et nota che fono // nueri de piui maniere fi cho-//me dichiara Boetio in el fuo // de arithmetiche Ma volen-//do hora tratar de quelle chof//fe che folo amerchadati apertien : pero tratado folo de quelli//che ale choffe merchadatefche fono necef- farij io laffero ogni//altra maniera de numeri.' (F. numb, i, v.) The sequence is now peculiar, for multiplication is the first operation treated. (' C.Che coffa fia moltiplichar,' f. numb. 7, r.) First comes the table, arranged in the column form, unlike the Boethian type of arithmetic, which preferred the square array. In addition to the prod- ucts through '9 uia io fa 90,' the products of 12, 16, 20, 24, 32, 36, by 2 ... io, are given, these having been necessary on account of the monetary tables of the time. The author then gives the method of i8 KARA ARITHMETICA checking by casting out 7's and 9's. (' C.Dela pruoua del .7.', ' Dela pruoua del .9.', f. numb. 8.) Then follows multiplication ' per colonna ' (i. e., by reference to the columns of the table, ' C.Del multiplicar p cholona,' f. numb. 9, r.), with its checks by 7 and 9, and ' per crocetta ' (our 'cross multiplication,' 'C.Del multiplicar per chroxeta,' f. numb. 13, v.), showing that these were the common methods in Venice. Divi- sion is then explained by the gal- **- ^ omo fe di - r 7 do batel cho lefuo P ruoue > f - numb - C opera VCdere 20 > r -)> our present method, then known as the method of giving, 'a danda,' and described by gerqucftadfcrpotraccrtificado p ac iuoio and Caiandri ( PP . S4 ssassssKgaef ** <* <** %* -" aralaprefenteoperaeafatwi Then follow addition (although ' this was used in multiplication), subtraction, denominate num- bers, common fractions (also th -HipHcatL) f three ( De la "egola del partnership, barter, alliga- FIG. 8. FIRST PAGE OF THE FIRST EDI- tion ' and false P osi tion. The rule T10N OF BORGHI'S ARITHMETIC f three had been developed many centuries earlier by Orien- tal arithmeticians. It was one of the inheritances from the Arabs, and was not improbably learned by the Venetian traders through their con- tact with the East. Partnership was to the fifteenth what the corporation is to the twentieth century, and it is only very recently that " partner- ship involving time" was thought to be a necessary subject of study. Barter, a subject until fifty years ago common in American textbooks, was necessary at a time when currency was not so plentiful as now. Alligation was a practical topic in connection with the coinage of money in the days when minting was not the monopoly of great cen- tralized governments. The rule of false position was an Oriental device which we have now replaced by the equation ; it was found in school- books in various countries until the second half of the nineteenth cen- tury. The problems are generally practical for the time, and they reveal some interesting facts concerning business customs at the close of the fifteenth century. Other works of 1484. Albert of Saxony, p. 9, c. 1478 ; Vincent de Beauvais, p. 10. PRINTED BOOKS 19 PIETRO BORGHI. Ed. pr. 1484. Venice, 1488. See p. 1 6. Title. ' Chi de arte matematiche ha piacere // Che tengon di certeza el primo grado // Auanti che di quelle tenti el vado // Vogli la prefente opera vedere//Per quefta lui potra certo CEXotnofifo:wflno trillion & million adoncba fe bie fotrxor per fcttc figure fnqnefto modo . 1000000 .pcrcbe lafepnma figura ficn cHuogo oemfara Dcmiara!:cpcrcbe 1000009 millemfara .fano vno million :eteffendoinqtiel luogo lafigura cberip:et:enta vno pero bene edieovno imlion.3&a.fqucftp modo . 1 100000 tfra'vnotntlfon e cento milia rpercbeoltraei I looooj mflUb'ffifi) foogooccentenara ocinwra:fono[afisuracberip:e jcenta vno fi cbe bene edito vno million e cento mt'Iia .32ba. in ... flmvs qucftomodo. 1 1 loooo.&iria vno mil tone cento e diccemiiia percbeoftraelmflion e cento milfa :in fuogooele oejceneoe^ mfar:fono lafigura cbe rijprtjcenta vno -.ft cbe beneedito vno rfavnmiu'oncentoevndeicemflfa per cbe oltraelmilion cento edicjcc milia:in luogo DC numcriDemfarfonoIaffgura cbe rip "icentavnoificbebenC'Cdftovnmiu'oncaitocvndcjccmilia.^a 1 1 1 1 100 inqueftomodo. 1 1 1 1 100. Diriavnmf lion cento evndecem^ iia e cetito:percbe oltra clmilion cento e vndece milfa: in luogo Oeefdrnplicecentenarfono lafiguracbe ripzacnta vno:ficbe bene editovnmilion cento e vndcfe milia e ceuto.3g>a in quefto modo. 1 1 1 1 1 io.ofriavnmilion.ccntoc vndcjccmiUa cento mill o edieye-'pcrcbeoJtra elmil/on cento evndeice miae cento :i luo^o&elefimplice&ejcene-.fono lafispracbe rip^ejcentavno. ^ainqneHomodo. 1 1 1 1 1 1 i.oiriavnmilion cento evnde I II I III JK milia cento e vndejce.per cbe ancbe in (uogo oc(e fimplice vnfta .fono lafigura cber/pzejcenta vno.ftcbe bene edirovn' milion cento e vndcjre milia cento e vndcjce.ct cbofi p:ocededo pcrfina .^^^^^.poncndo fcmpze aifuo luogiqudeflsiirc 299997* repzerentante quefinumcrf oueroocjcene ocet)tenara.cben> nominaetcetera.equcflobada ccrcba loamaiftramentoDcl nnmerar.bencbeininfinitumfipoJfa p:occder.ma cbomvna general figura mifo:cero oicbiarfrquantopoteflTeacbadcr.ct farano queflo fottopofte FIG. 9. FROM BORGHI'S ARITHMETIC, 1488 EDITION fapere // Se error fara nel calculo notado // Per quefta effer potra certificado // A formar conti di tutto maniere // A mer- chadanti molta vtilitade // Fara la prefente opera e afatori // Dara in far conti gran facilitade // Per quefta vederan tutti li errori//Ede iquaterni foi la veritade // Danari acquifterano e 20 RARA ARITHMETICA grand! honor! //In la patria e de f uori // Sapran far le rafon de tutte gente//Per le figure che fon qui depente.' (F. i, r.) ' Qui comeza la nobel opera de // arithmeticha ne laqual fe tracta // tute coffe amercantia pertinen-//te facta i compilata per Piero//borgi da Veniefia.' (F. 2, r.) Colophon. ' Stampito in Veniexia per zouane de Hall' 1488.' (F. 95, v.) Description. 4, 14.8 X 20.7 cm., the text being 12.4 x 14.9 cm. 95 ff., 44 11. Venice, 1488. Editions. See p. 16. This is the third edition, and is nearly as rare as the first. Proctor mentions three books from the press of the printer, John Leoviller, of Hall (Halle ?). The text is practically verbatim with that of the first edition (p. 16). PIETRO BORGHI. Ed. pr. 1484. Venice, 1491. See p. 1 6. Title. The title page is the same as that of 1488, except for the letters SHSU which precede the former. Colophon. 'Nela inclita citade venetia a zorni .22.//ottubero .1491. u impofto fine ala pre//fente opera.// Libro dabacho.' (F. 100, v.) Description. 4, I5-4X 20.8 cm., the text being 11.7 x 16.8 cm. 100 ff. unnumb., 4043 11. Venice, 1491. See p. 1 6. This is the fourth edition. PIETRO BORGHI. Ed. pr. 1484. Venice, 1517. See p. 1 6. Title. This is substantially the same as that of the 1484 edition already described. Colophon. ( CStampata in Venetia per lacomo pentio da Lecho ad infta//tia de Marchio Seffa & Piero di Rauani compagni // anno dni .1517. adi .25. de zugno.' (F. 100, v.) Description. 8, 15.5 X 2 1.6 cm., the text being 13.3 x 17 cm. 2 ff. blank -f 100 numb. = 102 ff., 41 11. Venice, 1517 See p. 1 6. PRINTED BOOKS 21 PIETRO BORGHI. Ed. pr. 1484. Venice, 1528. See p. 16. Title. This is substantially the same as that of the 1484 edition already described. Colophon. ' C.Stampato in Venetia per Fracefco Bindoni, & Mapheo // Pafyni compagni. Nel anno .M.D.XXVIII.// Adi .XVIII. Del mefe di Zenaro.' (F. 100, v.) Description. 8, 15.1 X 20.5 cm., the text being 13. 4 X 17.1 cm. 100 ff. numb., 41 11. Venice, 1528. See p. i 6. PIETRO BORGHI. Ed. pr. 1484. Venice, 1534. See p. 1 6. Title. This is substantially the same as that of the 1484 edition already described. Colophon. ' C.Stampato in Vinegia per Fracefco Bindoni, & Mapheo // Pafini compagni. Nel anno .M.D.XXXIIII.// Adi .25. Del mefe di Settembre.' (F. 100, v.) Description. 8, 15.4 X 20.9 cm., the text being 13.5 x 17.1 cm. 100 ff. numb., 41 11. Venice, 1534. See p. 1 6. The various editions changed but little. PIETRO BORGHI. Ed. pr. 1484. Venice, 1540. See p. 16. Title. ' Pietro Borgo // Libro de Abacho.// Chi d'arte Mathe- matice ha piacere // Che tengon di certezza il primo grado // Auanti che di quelle tenti il vado//Vogli la prefente opera vedere.// Per quefta lui potra certo fapere // Se error fara nel calculo notado // Per quefta effer potra certificado // A f ormar conti di tutte maniere.//A merchadanti molta utilitade // Fara la prefente opera e a f attori // Dara in far conti gran felicitade // Per quefta uederan tutti gli errori // E delli quaterni fuoi la ueritade // Danari acquiftaranno, e grandi honori.// In la patria e di fuori // Sapran far le raggion de tutte gente // Per le figure che fon qui depente.// Auenga che alquanto per me fu promeffo affai 22 RARA ARITHMETICA fufficientemente // alia promeffa fatisf aceff e , niente dimancho per fatisfar alle pre-//giere di qualch'uno, e maffime di alcuni Impreffori, iquali era-//no per ftampar la prefente Opera, ho uoluto alquanto ampliar la di qualche gentilezza oltra quello che prima pmiffe, benche // di quello che fe potria dir, quefto fia vna minima parte, pero cli //chi uoleffe metter pur la centifmia parte di quello che fi potria // poner, el faria molto piu la gionta di quello che fia tutta 1'opera //infi. Et pero pro nunc mi paffo con alcune cofette aggionte nel//ligar de metalli, lequal princi- piano a carte .77. & anchora in fin//de 1'opera con dieci cafi affai piaceuoli & leggiadri comincian//do a carte .98. da quello che dice. Le vno che compra tre pezze // de panno per ducati .70. &c. Et fe le niente di quelli, iquali me//hanno pregato non fuffino a fuo modo fatisfatte, prego quelli //me habbino per ifcufato.' (F. i, r.) Colophon. * Stampato in Venetia per Bernardino de Bindoni.// Ne 1'anno .M.D.XL. Del mefe di Ottober.' (F. 100, v.) Description. 4, 15.5 x 20.9 cm., the text being 13.2 x 17 cm. 2 ff. unnumb. + 98 numb. 100 ff., 39-40 11. Venice, 1540. See p. 1 6. This is the eleventh edition. The title is considerably extended and some changes are made in the text, chiefly in the way of added matter. PIETRO BORGHI. Ed. pr. 1484. Venice, 1550. See p. 16. Title. The title page is substantially identical with that of the 1540 edition already described. It bears the date, 'Anno Domini M. D. L.' Colophon. ' C.Stampato in Vinegia per Francefco Bindoni, &//Mapheo Pafmi. Nell Anno .MDL.//Adi .21. Del mefe di Nouembrio.' (F. 100, v.) Description. 4, 15.6 X 20.9 cm., the text being 12.9 x 17.9 cm. 100 ff. numb., 41 11. Venice, 1550. See p. 1 6. This is the twelfth edition, and at least five subsequent editions appeared in the sixteenth century. PRINTED BOOKS 23 ANONYMOUS. Ed. pr. c. 1485. S. 1. a. (Cologne ?, c. 1485). Title. ( Ars numerandi. Incipit copendiofus tractatul^ quin //tupliciu dclonu numealiu in quo docet' // luculet' quo ordlant' variatur pponut'//et abinuicem deriuatur dictones nume// rales.' (F. 2, r. ; see Fig. 10.) Without abbreviations this would appear as follows : ' Incipit compendiosus tractatulus quintuplicium dic- tionum numeralium in quo docetur luculenter quomodo ordinan- tur variantur componuntur et ab invicem derivantur dictiones numerales.' Description. 4, 14 X 20.6 cm., the text being 10.8 x 15.2 cm. 6 ff. unnumb. (i blank), 35 11. S. 1. a. (Cologne ?, c. 1470-1485). Editions. There was no other edition. The date of this rare book is uncertain. The style of type has led to the assertion that it was printed in Mainz by Fust and Schoeffer, about 1470. The book is not mentioned by Hain, nor is any copy known in the French libraries. Coppinger believes that it was printed in Cologne by Ulrich Zell in 1485, and in this he is followed by Zell's biographer, Merlo. The British Museum catalogue gives this date and printer, but questions each. A comparison of the water-marks in this copy with those of the fifteenth century which are described in standard treatises (e. g., Sotheby, E. L., Principia Typographica, London, 1858, vol. Ill) fails to throw any light upon the date. Riccardi attributes it to Zell, c. 1471, who had been an apprentice of Guttenberg, but had left Mainz at the sacking of the city in 1462. The book is not strictly speaking an arithmetic, but a treatise on grammatical usage as applied to numbers. A considerable portion of the text is occupied with the distinction between ordinals and cardinals, and the methods of using them. Other works 0/1485-1487. Albert of Saxony, 1487, p. 9 ; c. 1478. In 1485 there was published at Bologna, edited by Pietro Almadiano of Viterbo, a quarto work by Nicolo de Orbelli (Nicolaus Orbellis) entitled 'Compendium considerationis matematice quo ad aritmeticam et geome- triam sunt necessaria.' Orbilli's 'Cursus librorum philosophic naturalis,' which appeared in 1494, 4, and at Basel in 1503, 4, contained 2 pp. on arithmetic. See Isidorus, p. 8, 1483. RARA ARITHMETICA Ardtumtemnbt* rt opi t copm biofus tmcbtul^qum tupiiciu tmonu numcaltu m quo ft> cd* iuculct'quo (X&ian^vananjr OFonuf* ct abmtucxni txmatur tuctpiica numc mica* 3f Aiottfl numceos nrtpntSrm qfca t>ieu 9 tur cattmi ales cj fca potcatcs* qfca toiftri butiuc ftuc bi>ndc-qfca cnt>iatc6- a qfca ntultipHcatiuc ftuc d^ubialcff<t nota t>u ^ numcri t>iucrfi8 t)i<*iomb? ft^ftcaf a varia fi^n^i manccic cottabut mtf fc t>intia ct t>i lift ta fee ut pateb* ^I>c his aut fedotrfb? p ot t>mc cf> bi(5bu-*c^t puo te cavt>taii^q t>icutur io cartialc3-' cp ft cut oftiu 4tif ca catbhtc^ct tnrrinf ci: i ta t>aon* atic uualcff ^tiitut a vc^ plica* tar cca ifVas Vcl t>i cu6" carbmalcs qfi pit cipalcff * cp t>c6nc0 ahc nuaU0 ab iRi0 bttt oii gincVcl ^icu 1 qn' p riripales cp pia dp a& fi^nt tiu; ct focdu cp tJuplicc j fait t>co*3 cardma Ic0{qt>a'f'c6crctc et c)ucda abftf&c tDc cocrctt ailt carom aiibg 92 tw ipiff atic tKtiuan^pihno c t>Sm qtic t>conctJ ftcpccdut numctado Vnusr fc^fcptc- fc&cdF (efdcd - a nun^ tested p ^ teccfcptC' a non tebj mtcrpt fccc dtu^io ct ft (tnt feuc t>c5cf tibuo* a fie uf^ j ab tngmta (uo modo- ' a fie fuo modo t /i fie fuo mo ufc^ 't)u< tiduo numcralcs > eoncrcta tioianuatia fie numcta 1 FIG. 10. FIRST PAGE OF THE ANONYMOUS ARS NUMERANDI PRINTED BOOKS 25 ANICIUS MANLIUS SEVERINUS BOETHIUS. Ed. pr. 1488. Augsburg, 1488. BOETIUS. Born at Rome c. 480; died there October 25, 524. He was a Roman senator, a philosopher, and the last of the great Latin writers. Title. 'Arithmetica boetij.' (F. i, r.) Colophon. 'Finit arithmetica Boetij bene re//uifa ac fideli ftudio emendata Im//preffa per Erhardu ratdolt viri fo-// lertiffimi eximia iduftria i mira im-//primedi arte : qua nup 3fncipiuntouolibzic>e ^(ricbmeti/ fere vlla lie ciictis ibfolnM n^rn'M rr i' caam^manilMfeuerimSoety rf, m^l^XiSSSSfS& *"*' ""**"> n danfllmi * MriM ejc cofulis: nira pftoi;s:vt no cetera? quoqYar ojomaiij:patrid;:ao patriciumfim rtu a&iumentabefiberetyTlainem enoifquemuneri bnsitarecteomV ciapiecipue inter eos gfefcmagni faciunc cftimanf filtqui&ooftabit qd liber alias after re t irmciuu : i tec ab ilto vncg q6 iucun&ius benitiole tia coplecteref accepni ."Dec ipfe co ftderans : attttlt non tgnaiiaopum pooera quibua ao facinus nibil in dructins eft: cu babenbi (ttis incan t>uit:ao meritu nibil vilius cu ea fi/ bi victoi antm 9 calcata fubiccit : fed ea que c,r grecaru opulentia littera re in romane ozationis tbefaurnm fupta cducjnm. 9 .^ta eni mei quocg opens mibi ratio cotiabit :fi qttc cc fapientie ooctrinis eliati:fapienti^ fimi ittdicio copjobenf . cii&cs igif vt tarn magni laboas effect us nuij tantii ejcpectet ejcame: nee in aurea ^ire publicas nift ^octf fentetic a fKpulatioite nitaf.^n quo nibil mi ro vioeri oebet:cu i J> opus qd fapie tic iniienta perfeqitit: non auctoiia f5 a!ieno incubit arbitno jSuis $p/ pe in fhiunenris res rationts ejrpen 2>itur:cu tuoiciumcogif fubirc piti^ tetitis.Seo buic munufculo :no ea e>einqueceterts imminent artibtw numuncntacoftiruo .Tlequc enitn a&enbe molts labozeftralia fomta t>$ imaginis ratiomeceiulbe artifi^ cis mantis polfti operis nitoz ejcpe* ctat.^t pichtr^ manibns tabulf c5 mife f abzoeunr.cgre niffica obfer/ uatione t>ecerptg:cotomfuci merca toju fofertta perqiufiti:liinea opero fis elabojata tejctrinistmtiTtipliceni matena pz^ftant.Tlone ioem qwo/ que bello^ vifitur inftrumetier&ic fpidila fagiitt&erflcuif.iLli vali^ua tbozaic nigragemit incube.^lTa'i/ us rauM vmbonts teamtna ppni la bottotbi infigen^amercatur.tatn. multf ortibus ars yna_perfigit . 3llt hoftri labo:is abfblutio loge at> fa^ cilioje* ciirrit eucnt 115 . ]Cu eni fottia manuj fupzemo open impones: in quo nibil fre oecernentiu ncceite eft labojare cofenfu.Qualibet eni boc iuoiciu mttltis artibus pjobef ejrcul tu vno tamecumulat ejramine.p periare igitur licet quantn nobis in boc ftubio Ipngis tractus oci/s la^ boz a&iecerit.^nrerufubtiltum fit gas ejcercitate mentis velocitas co/ pbenoat. vtp ieiune macies o^atio nis ao ea que funt caliganribus irn peoita fentenri;s ejcpe&ienbafumci at.Quafnremibialieniquoque iu oicij lucra querunf. XTum tu vtrarii quepeririJTimus litteraprpoflis gra i^oiationisejrpertibus quantusbe, nobis lubtcare au^eant: Tola tantu pnunciationepiefcribere.^tno al/ FIG. ii. FIRST PAGE OF THE 1488 BOETHIUS 26 KARA ARITHMETICA IS |3 2-7 t* jr r* \s y t* 8 nr (To Jo 0cubavnit09. longitubo. ttauo at(& ejtpofitio bigeflg foziwt mufota (Upertf vnitate: vt btto tnu 3?i0itbuomaUte/ ra^ppofit^fpzmulfq faciut aguliuab vno ab* lo.et* lo.^pceben tiardpiciafi-rbisfub terioseeoibies copa rentiqui fcilicet a.4-angiilum incipi cntcs:invigcno8 tcrminii ponunt: buplej: ib cftpzima fpccice multipli citatie oftenbimnitavtpnmuepn' teritari 9 binariu. tcrti 9 tertm tribus: vtfmarPternariu.qrtusqrtu qter/ narij nunicrofrta tc tranfcenbat : vt 8 qternariu :z u eabe cucti fequetil Jcfemino:t8 piuratitatepzetereant. /i vcro tcrti'angulite afpiciafi^ ab 9.incboaalpgimbme la:imbincqj triceius altrinfeciis numercc]ctebit: et bit cu pn w latitubine et logitub i/ ticcoparcturariplcjcfpcciee mulct/ FIG. 12. MULTIPLICATION TABLE, 1488 BOETHIUS PRINTED BOOKS 27 venetijs nuc // augufte; excellet nominatiffimus.// Anno dni .M.cccc. Ixxxviij. Men-//fis maij die vigefima.' (F. 48, r. Fig. 14.) Description. 4, 15.1 X 20.8 cm., printed in double columns, each being 5.1 X 14.7 cm. 47 ff. unnumb. + I blank = 48 ff., 40 11. Augsburg, 1488. Editions. Augsburg, 1488, 4 (here described) ; Cologne, 1489; Leipzig, 1490; Venice, 'Opera,' 1491-92 (see p. 28); Paris, 1496, 4 ; Venice, 1497 ; ib., 1499 (bearing also the date 1497), fol. ; s. 1. a. (the Compendium of Muris, c. 1500); Paris, 1 50 1, fol. ; ib., 1503 (see p. 29) ; ib., 1507, 4; ib., 1510 (see p. 30), fol.; ib., 1511, 4; ib., 1514, fol.; Vienna, 1515 (the Muris ' Compendium,' in Tannstetter's works) ; Paris, 1521 (see p. 31), fol.; ib., 1522, fol.; 1528, fol.; Paris, 1530, 8, ' De differentiis topicis libri quatuor ' ; Basel, 1536, 8; Basel, 1546, ' Opera,' fol. ; ib. and Paris, 1549; Basel, FlG - 13- FlGUR - 1553, 'ajectis explic. per J. Scheubelium,' 8 ; Paris, 1553; Basel, 1570; Venice, 1570, 'Opera,' fol. There are undoubtedly various other editions of the Arith- metic, or the Epitome by Faber Stapulensis, these works often being bound with such treatises as the Arithmetic of Jordanus Nemorarius. Murhard (I, 160) and Rogg (p. 137) mention an edition of Faber's 'Compendium,' s. 1., 1480, but it is not given by other bibliographers. (See Hain, I, 468; Brunet, Man., I, 1059; Graesse, Tresor, I, 464; Riccardi, I, i, 159; Boncom- pagni, Bullctino, XII, 148.) The text is practically that followed by Friedlein in his standard edition of the 'Opera' of Boethius (Leipzig, 1867), except as to numerals. Here, as in the later manuscripts, the Arabic characters have replaced the Roman of the original text. (See Fig. 12.) The arithmetic of Boethius was based upon the Greek work of Nicomachus (fl. c. 100 A.D.), and related only to the theory of numbers, the 'ApiO/jirjTiKrj, as distinguished from the practical calculations, the 28 RARA ARITHMETICA y, and from the later algorismus (p. 5). Boethius gave an elabo- rate theory of ratios and devoted much attention to figurate numbers, such as the triangular, square, pentagonal, and cubic. (See Fig. 13.) The work was the standard in the Church schools throughout the Middle Ages. fut.Q*fifcrotntca m fcxfqct n.faaa.-qao fupltiBdl. gnueniaraui bic FIG. 14. LAST TWO PAGES OF THE 1488 BOETHIUS BOETHIUS. Ed. pr. of the Arithmetic, 1488. Venice, 1491-92. See p. 25. Title. See Fig. 15. Colophon. ' Impreffis venetijs per Joanne de Forli-//iiio et Gregorium fratres. Anno falutis .M//cccc.lxxxxj. die xxvj, men- fis Martij.' (F. 352, r.) On f. 256 (220 as numbered in the book), at the end of the Geometry is the following : Venetijs Impreffum Boetij opus p Joane5 i Gre//goriu de gregorijs fratres felici exitu ad fine vfqj pductu // accuratiffimeq3 emedatu Anno humane reftaurationis. // 1492. die .18. Augufti. Auguftino Barbadico Serenifli//mo Venetiarum principe Rem pu. tenete.' Description. Fol., 21.7 x 32 cm., printed in double columns, each being 6.6 X 24. i cm. 3 if. unnumb. + 345 numb. + i blank = 349 ff., 66-70 11. Venice, 1491-92 (see the colophons). PRINTED BOOKS 29 Editions. This is the editio primeps of the works of Boethius. For other editions see p. 27. j^cruntopcra JSoctii.-quf in boc,voiuminc eontfncntur. pHttoprtma. """ fditto vna. fTliogiTmoe introdiicn'o :uminXopica/Citcroiii8Ub:ire>:. 5fopfcit?lib:iquatruor. ibn^uo. tbnltiliDCr vnusj. ca*bmeitoad l^atrttinm 23c ZlDnnca lion qm ncg. Toe rcotnctriaiib:i vno. FIG. 15. TITLE PAGE, 1491-92 BOETHIUS BOETHIUS. Ed. pr. of the Arithmetic, 1488. Paris, 1503. See p. 25. Title. 'In hoc libro contenta.// Epitome/ compendiofaq3 in- troductio in libros // Arithmeticos diui Seuerini Boetij : adiecto fa-//miliari commentario dilucidata.// Praxis numerandi certis quibufdam regulis //conftricta.//Introductio in Geometriam 30 KARA ARITHMETICA breuiufculis an-//notationibus explanata. fex libris diftincta.// Primus de magnitudinibus et earu circuftan-//tiis.// Sc'dus de cofequetibus/ cotiguis/ & cotinuis.//Tertius de punctis.//Quar- tus de lineis.//Quintus de fuperficiebus.//Sextus de corporibus. // Liber de quadratura circuli.// Liber de cubicatione fphere. // Perfpectiua introductio.// Infuper Aftronomicon.' (F. i, r.) Colophon. ' Abfolutum in almo Parhifiorum ftudio/ //Anno dni qui numero definuit omnia// 1503.' (F. 48, r.) On f. 84, v. is the following: 'Geometric! introductorij:// fexti / etvltimi libri finis.// Editi anno domini // millefimo quin- //getefimo pri//mo : vicefi-//ma quin//ta no-//uem//bris.' And on f. in, v., the following: ' Id opus imprefferut Volphgangus // hopilius et Henricus ftephanus // ea in arte focii in Almo pari-//{iorum ftudio Anno Chri//fti Celorum totiufq3 // nature coditoris.// 1503. Die vice//fimafepti-//ma Iu-//nij.' Description. 8, 20 X 26.8 cm., the text being 15.6 x 22.3 cm. 48 ff. numb, in the arithmetical part, 113 in all (i blank), 54 11. Paris, 1503. The editions of Boethius differ more or less in the combinations of works which they contain. See p. 27. This is the first edition of the Jacobus Faber Stapulensis and Jodocus Clichtoveus ' Epitome.' BOETHIUS. Ed. pr. of the Arithmetic, 1488. Paris, 1510. See p. 25. Title. This is substantially identical with that of the 1503 edition already described. ColopJwn. ' CAbfolutum in almo Parifiorum ftudio/ // Anno domini qui numero defmiuit // omnia 1503. Et emiffum ex offi- //cina Henrici ftephani Anno // Christi faluatoris // omnium 1510 de-//dma quinta//die Mar-//tij.' (F. xlviii, r.) Description. Fol., 19.5 x 27.9 cm., the text being 15.4 x 25.6 cm. 48 ff. numb., 4656 11. Paris, 1510. This is one of the editions containing the commentary of Jodocus Clichtoveus on the 'Epitome' of Boethius by Jacobus Faber Stapulensis. (Seep. 27.) PRINTED BOOKS 31 BOETHIUS. Ed. pr. of the Arithmetic, 1488. Paris, 1521. See p. 25. Title. ' Divi Severi-//ni Boetii Arithmetica,// dvobvs discreta libris ; adie-//cto commentario, mysticam nvme-//rorum applica- tionem perftringente, declarata.// (Woodcut with initials of the printer : S.D.C.). Vaenundatur apud Simonem Coli-//naevm, e regione fcholae Decretorum.' (F. i, r.) Colophon. ' Excudebat Simon Colinseus, Parifijs, Anno MDXXI S-DECOLINBS FIG. 1 6. PRINTER'S DEVICE, 1521 EDITION OF BOETHIUS 1521 Quinto//Idus lulias.' (F. 1 39, v.) Description. Fol. , 19.9x28.2 cm., the text being 13.9x26 cm. 4 ff. unnumb. + 136 numb. = 140 ff., 40-52 11. Paris, 1521. This is the first edition of Boethius with the commentary of Girardus Ruffus : ' C.Girardi Rvfn, in duos arithmetical Boetii libros, commen- tarivs.' (F. 5, r.) This commentary greatly exceeds the text in extent, and as to ponderosity it leaves little to be desired. As a piece of typography, however, this is one of the best editions of Boethius. ANIANUS AND JOHANNES SACROBOSCO. Ed. pr. 1488. Strasburg, 1488. ANIANUS was a fifteenth-century astronomer and poet, of Strasburg. JOHANNES DE SACROBOSCO (SACROBUSTO, SACROBUSCHUS, HOLYWOOD, HOLYBUSH, HOJ.YWALDE, HoLYFAX, HALIFAX) was born at Halifax (Holy- wood), Yorkshire ; died at Paris in 1244 or 1256. He studied at Oxford and lectured at Paris. He wrote on astronomy and algorism. Title. ' Copotus manua//lis magri aniani. // metricus cu ;ometo//Et algorifmus.' (F. i, r. Fig. 17.) F. 45 begins, ' Incipit textus algorifmi.' 32 RARA ARITHMETICA Colophon. ' Impreffum Argii. per Johamem prytf.// Anno do- mini .1488 .i8. kail.' decembris.' (F. 44, r.) Description. 4, 13. 9 X 19.5 cm., the text being 8.8 x 14.4 cm. 53 ff. numb. + 2 unnumb. = 55 ff., 31-34 11. Strasburg, 1488. Editions. The editions of Sacrobosco's Algorismus were as follows : Strasburg, 1488, 4 (here described) ; s. 1. a. (Venice ?, 1490?), 4; Venice, 1501 (see p. 35); Paris (edited by Clich- toveus), 1498; ib., 1503; ib., 1510; Vienna, 1517; Cracow, aroanu FIG. 17. TITLE, FIRST EDITION OF ANIANUS 1504, 1509, 1521, and 1522; Paris, 1522; Venice, 1523 (p. 35); Antwerp, 1582. The editions of Anianus were as follows : Strasburg, 1488, 4 (here described); Lyons, 1489; ib., 1490, 4 ; ib., 1491, 4 ; ib., 1492 (two editions), 4 ; Rome, 1493, 4 ; Paris, 1494, 4; ib., 1498, 4; s. 1. a. (Paris?, c. 1495, 4, see p. 33); s. 1. a. (Basel, c. 1500), 4; Rouen, s. a. (1502), 8; Paris, 1501, 4; ib., 1502, 8; Lyons, 1504 (see p. 35); ib., 1509, 4; Paris, 1508; ib., 1511; ib., 1515; ib., 1519, 4; ib., 1529; ib., 1530,4; Lyons, 1540,4; Frankfort, 1549; Wittenberg, 1550; ib., 1568; Antwerp, s. a. (c. 1558); ib., 1559. There are probably others s. 1. a., and some appear PRINTED BOOKS 33 under the name of Sacrobosco. Of this first edition Boncom- pagni knew only the Munich copy. (Bulletino, XII, 126 n.) The first part of this rare book is the Compotus Manualis of Anianus, and the second is the Algorismus of Sacrobosco, described later. It is probably the first book on mathematics printed in Strasburg (but see p. 10), and it is the first edition of each of the two treatises mentioned, and the first printed work on the computus, the arithmetic of the Church calendar. In the work of Anianus appears for the first time in print the original of the rhyme beginning ' Thirty days hath September,' the English version of which is said to have been first published in the 1590 edition of Grafton's Chronicles. It also appeared in an arith- metic published anonymously in 1596. The Latin form of Anianus is as follows : ' Junius aprils feptember et ipfe nouember Dant triginta dies reliquis fuoadditur vnus, De quorum numero februarius excipiatur.' (F. B 8.) It is found in various forms in other computi, manuscript and printed. Anianus also gives for the first time in print the astronomical formula : ' Sunt Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libraque, Scorpio, Arcitenens, Caper, Amphora, Pices,' which appeared in the works of Bede under the title ' Verfus Prifciani, de Astronomia,' as ' Hinc Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Arcitenens, Capri- cornus, & Vrna, Qui tenet & Pifcis.' (1563 edn., vol. I, 517.) ANIANUS. Ed. pr. 1488. S. 1. a. (Paris ?, c. 1495). See p. 31. Title. ' Copotus cum //commento.' 'Liber qui Compotus infcribitur : vna cum figuris et ma/ //nibus neceffariis tarn in fuis locis qj in fine libri pofitis.// Incipit feliciter.' (F. i, r.) Description. 4, 13.1 X 19.4 cm., the text being 9.3 X 1 3. 1 cm. 39 ff. unnumb. + i blank = 40 ff., 35 11. S. 1. a. (Paris ?,c. 1495)- Editions. See p. 32. This rare and interesting edition was published about 1495, possibly by Mich, le Noir at Paris, although there is no date or place of publication given. It is one of the best examples of the mediaeval computus that appeared in print. Unlike the first edition (p. 31, 1488) it 34 RARA ARITHMETICA contains a number of illustrations showing the use of the hand and fingers in assisting in calendar reckoning, the title of ' Com- Zfberqoiompottd intension wz am R&xiQ etma/ nibas necclferud tarn in fuis locfe $ in fine libii r bapofluntOiiplidter confiderari . pzfmopnt Did DC oeo q eft lujc vera . loco oicebat Dauid* lat ozta eft lUito . r oe ilia luce Did? 3ba'.f . lErar IUTC vera 4 illumiar omne bomine venic/ tcinbunc mandd.^eakiopomt oefaenaa. c tXdf lut qaafi faeutc reddctc luadfuqm a fade bomuic fde/ tern efle ladda5n qUibtis verbis ad comendan'onc fcle tnio b:e ufter rangiinir.primo eft tangif fdetie^Uitudo pzedofa p boc quod Dicit lujc. Sccundo largitudo gloofa per boc qp oidt oz/ fie oicentis /^cTenSae^ fon0 indenciensJjon'tatis via .fai fal/ '5 c oatoae co^wtio Ba:ioe fic.rtlud ell valtda a pzedofam quod P' ijetniiatido^imperteaoftdtvaliduinzpfecmm.faetia modi.ergo zc. maioz dl manifcfta. minoe oeclaratur p p&m ter/ do oe ala fie tnccnte : 3ia in pncipio foe creationio e tan$ tabu la rafa in qoa nibfl oepicta dl.oepmgibilie m fdetiis ^ virturt/ bu8.*f> zmwm?pbajaaao2itate boeni ^ ratoe in pzologo artf men-icetecTenflae cop quc tera funt ^ imgmutabilto tflentie no aT" fh*aq5Compbenfione^eritarts.Batioe fic:illude!Uan$ funtti bonii quod babet largitione glonofam fcta e bmoi. ergozcma_ toz ert veramtno: ,pbamr poiflinitione fdcne q talie e.^c^eirs ^ cjda babims ate ronalie no innatus fed accjfi'tue o[m b&ma? re riunHndagatn): T tortua butnane vitc gubcrnatnjcf^ fcia fit ba/ bittis pts.qz fcia ell aliquod ejciftens in ata.fed omne illud quod ell in ala aut c bitue aut poten'a aut paflio. tboc tellaE arillo, In fecudo etbicouq^ fcta no fit paTio pt5 qz patfionea funt in va luntate fcia no e bmoi.ergo ic.cp no fit potetia pt5.q: c]l-b5 po/ ten ia fit a natura. fiat irafdbiUs.i cocjpifdbilis.z fie relmqui ear $no fitpotentia. c^fitbUu0a;eronali0pt5gpdiaa^au( a a FIG. 1 8. BEGINNING OF ANIANUS, EDITION OF c. 1495 putus manualis ' being thus justified. This differs from the 1488 edition in the notes of the commentator. The original text is, however, substantially unchanged. PRINTED BOOKS 35 ANIANUS. Ed. pr. 1488. Lyons, 1504. See p. 31. Title. 'Compotus cu //commento.' -(F. i, r.) Colophon. ' CLiber compoti cum comento finit feliciter // Im- preffus Lugduni per Claudia nour//ri. Anno domini .M.ccccc.iiij. //die .iiij. Octobris.' (F. 32, r.) Description. 8, 14.5 X 20 cm., the text being 8.9 x 14.1 cm. 32 ff. unnumb., 3440 11. Lyons, 1504. See p. 32. In this edition there is a curious misprint in the calendar verse given on p. 25 (f. 13, r.). It here begins 'Julius (instead of Junius) aprilis feptembre & ipfe nouember.' Like the edition of c. 1495 (p. 33), this is well illustrated. It differs from the 1488 edition in the notes of the commentator, but is practically identical with that of c. 1495- JOHANNES SACROBOSCO. Ed. pr. 1488. Venice, 1501. See p. 31. Title. ' Algorifmus Domini Ioa-//rris De Sacro Bufco // Nouiter Impreffi^.// Cum Gratia Et Priuilegio.' (F. i, r.) Colophon. ' Impreffum Venetijs per Bernardinum Venetum // De Vitalibus: Anno Dm .M.CCCCC.I.//Die Tertio Men. Februarij.' (F. 8, r.) Description. 4, 14.1 X 19.6 cm., the text being 10.5 X 16.8 cm. 8 ff. unnumb., 39 11. Venice, 1501. See p. 32. JOHANNES SACROBOSCO. Ed. pr. 1488. S. 1. a. (Venice, 1523). See p. 31. Title. < Algorifmus Domini Joannis // de Sacro Bufco noui- //ter impreffum.' (F. i, r.) Colophon. ' Impreffum Venetiis per Melchiorem Seffam & Petrum//de Rauanis Socios. Anno domini .M.D. XXIII. // die .XXIIII. Octobris.' (F. 8, r.) Description. 4, 14.8 X 18.6 cm., the text being 10.3 X 16.8 cm. 8 ff. unnumb., 40 11. S. 1. a. (Venice, 1523). 3 6 RARA ARITHMETICA JOHANN WIDMAN (?). Ed. pr. c. 1488. Leipzig, c. 1488. Born at Eger, Bohemia, c. 1460. He was a student at Leipzig in 1480, A.B. in 1482, bachelor of medicine in 1485, A.M. in 1486. He evidently received the doctor's degree about the same time, for he wrote a medical work in 1497 with the title, ' Tractatus clarissimi medicina ~Q doctoris Johanis widman . . . de pustulis . . . ' That he gave lectures on algebra, possibly the first at Leipzig, is proved by a passage found by Wappler in an old Dresden manuscript : ' Quare hodie hora secunda post sermonem atque Baccelaureo- rum celebrata disputatione Magister Jo. W. De. Eg. Aporismata et Regulas Algobre resumpturus pro hora atque loco conuenienti cum audeturis con- cordabit . . .' Title. ' Algorithmic Linealis.' (F. i, r.) 'Ad euitadum mul- tipli//ces Mercatorum erro-//res et alteri^ . . .' (F. 2, r.) Description. 4, 14 X 19.5 cm., the text being 7.8 x 14.5 cm. 14 ff. unnumb., 3 134 11. Initials in red, by hand. Leipzig, c. 1488. Editions. Leipzig, s. a. (c. 1488, here described); ib., possi- bly 1490, 1493; 1516; 1517, and several others. This rare treatise, the first printed work on calculation by the aid of counters ('apud noftras appellata eft calculatio '), is of unknown authorship, but was probably written by Widman. (Abhandlungen, V, 152.) At the end of the book is the device of Martin of Wiirzburg (Martinus Herbipolis), and the book was printed by him, probably c. 1488. After a brief introduction on the use of counters ('projectiles' as they are here called), the author treats of the following topics : De Additione, De Subtractione, De Duplatione, De Mediatione, De Multi- plicatione, De Diuifione, De Progreffione, De radicum extractione, De radicum extractione in Cubicis. The book closes with the words : * Et tantu de Radicum extractione et vltima huius Algorithmi fpecie Et p confequens de toto Algorithmo.' There are no applied problems, and the only computations with abstract numbers are performed 'on the line ' (i. e. by the ' projectiles ' on a line abacus), whence the name ' Algorithmus linealis.' The work is illustrated by woodcuts. Other works 0/14.88. Suiseth (p. 10, c. 1480) ; Borghi (p. 16, 1484). JOHANN WIDMAN. Ed. pr. 1489. Pforzheim, 1500. See above. Title. ' Behennd vnd hiipfch // Rechnung vff alien // kauff- manf chaff ten.' (Woodcut of a schoolroom.) (F. i, r.) Colophon. ' Gedruckt zii Pfortzheim von Thoman // anfzhelm Im Jubel Jar als man zalt 1500 //Got fey lob.' (F. 163, v.) PRINTED BOOKS 37 Description. 16, 10 x 13.2 cm., the text being 6.5 X 10.4 cm. ii ff. blank + i unnumb. + 162 numb. = 174 ff., 26 11. Pforz- heim, 1500. Editions. Leipzig, 1489, 8; Pforzheim, 1500, 8 (here described): ib., 1508, 8 (p. 39); Hagenau, 1519, 8 (p. 40); II er$im$enobet tubftetr tnfi$ bcf^Iteflen ob bie taffcl elofje taimfeie anber gefeQt i(l *b bami t?ie t>er nad; icrl id?c an ir felbft foitn tlerltc^cn befc^reiben i'(l . %ir <xl(c r e Ami n FIG. 19. FROM THE 1500 WIDMAN I Augsburg, 1526, 8 (p. 40). The title of the first edition was as follows : ' Behede vnd hubsche // Rechnung auff alien // kauffmanschafft/ and the colophon (f. 236, r.), * Gedruckt In KARA ARITHMETICA der Furstlichen Stath // Leipezick durch Conradu Kacheloffen //Im 1489 Tare.' n*cberre0clf0Fumtafot?iemxv flat QcJ^rpes 424 13 4 p:oflo:rt 305 jToret FIG. 20. EXCHANGE. FROM THE 1500 WIDMAN This is the second edition, and is even more rare than the first. It was unknown to Boncompagni when he printed his ' Intorno ad un Trattato d'Aritmetica di Giovanni Widmann di Eger ' in the Bulletino, IX, 188, the best discussion of this arithmetic that has appeared. PRINTED BOOKS 39 Widman's arithmetic was the first great German textbook on the subject, although minor works had already appeared before 1489. It is in the main a practical treatise, with good problems, and it set the standard for Germany much as Borghi's book did for Italy. Among its noteworthy features is the use of the plus and minus signs for the first time in a printed work. (See Fig. 21.) These are not used, however, as signs of operation, but as symbols of excess or deficiency in ware- house measures. The book is illustrated (see Fig. 20) with pictures showing mercantile customs, and with geometric diagrams. Widman acknowledges his indebtedness to men like Sacrobosco, 'als da lert Joannes defacrobufto vn ander mer,' although his work shows no depend- ence upon the ' Algorismus ' named. After the ' Inhalt difz buchs in einer gemein,' Widman devotes 2\ pages to Numeratio,' 2^ to 'Additio,' 2^ to ' Subtrahiren ' (including denominate numbers in these topics), i to ' Dupliren,' 2 to ' Medieren ' (i.e. multiplying and dividing by 2), u^ to ' Multipliciren/ 5 to divi- sion, 2\ to progressions, and 14 to roots (4 referring to cube root). He then takes up fractions in the same order, this work being followed by compound numbers and proportion. He then gives a large number of type problems, regulae as he calls them, although they are not stated in the form of rules as we now know them. These include the ' Regula detri' (rule of three, treated as distinct from proportion), and the regulae fusti, detri conversa, positionis, equalitatis, legis, augmenti, plurima, sentenciarum, suppositionis, residui, excessus, collectionis, quadrata, cubica, reciprocationis, lucri, pagamenti, and alligationis. Other works 0/1489. Boethius, p. 27, 1488 ; Anianus, p. 32, 1488. JOHANN WIDMAN. Ed. pr. 1489. Pforzheim, 1508. See p. 36. Title. ' Behend vnd hiipfch // Rechnung vff alien // Kauff- manfchafften.' (F. I, r.) Colophon. c C.Gedruck zu Pfborzheim von Thoman//Anfzhelm Im iar als man zalt 1508.' (F. 161.) Description. 8, 9.7 X 13 cm., the text being 6.5 X 10 cm. 7 ff. blank + 161 numb. = 168 ff., 26 11. Pforzheim, 1508. Editions. See p. 37. This is the third edition of this famous arithmetic, and is by the. same publisher as the second (1500), but is from different type. It is about as rare as the first edition. KARA ARITHMETICA JOHANN WIDMAN Ed. pr. 1489. Hagenau, 1519. See p. 36. Title. ' Behend vnd hupfch // Rechnung vff alien // Kauff- manfchafften.' (F. i, r.) Colophon. 'Getruckt zu Hagenaw durch Thoman // Anshelm. Im iar als man zalt 7/1519.' (F. 1 5 1 , r.) Description. 8, 9.7 X 14.3 cm., the text being 6.8 x n cm. i f. unnumb. + 151 numb. = 152 ff., 20-29 11. Hagenau, 1519. Editions. See p. 37. This is the fourth edition of Widman's Arithmetic. As an inscription on the fly leaf says, this copy was presented to Prince Baldasarre Boncompagni by Ludwig Kunze as a ' liber rarissimus.' See p. 39. JOHANN WIDMAN. See p. 36. Ed. pr. 1489. Augsburg, 1526. fenot>ert>e(?gley* 3+30 d)cij.6o(uiwcr 4~-~ "~'P fciejetictncrtMjb 3+44 Uixmtrt>R?<?e<mfj 3 + zi ij?/&et|l mi* 2titer 3 -i i tfc mi6t>3f<}(Je{oit* 3 -f- 50 berxmnb werfceit 4 1 6 4*3 9 & CQ 3+44 3+19 ? ""/ - . Hurt 3 Mlftf. 4 Behennde vnnd // hiibfche Rechnug auff alien // Kauffmanfchaff- ten.' (Woodcut showing two men seated at a reckoning table.) * M. D. XXVI.' (F. i, r.) Colophon. ' Getruckt zu Augfpurg durch // Haynrich Stayner// M, D, XXVI.' (F. 192, r.) Description. 9.4 X 13.7 cm., the text being 6.7 X 1 1 cm. 2 ff. unnumb. + 190 numb. = 192 ff., 25 11. Augsburg, 1 526. Editions. See p. 37. This copy has a note in the handwriting of Prince Boncompagni, together with his colla- tion of the book. It is interesting not only for its rarity but for the contem- porary coloring of the woodcuts. In the other editions described the third line of Fig. 2 1 reads 3 + 36, as it should. crosier dit 4? 3 p. Vrtt> 6fcybeit 4 1 f z tii.nim Tpn'ch I oo It ix?6i(f cin3crimcr pjO 4 ff ; nnc Dnrneit 4 1 > i tt cnb f uml FIG. 21. FROM THE WIDMAN 526 PRINTED BOOKS 41 PETRUS DE ALLIACO. Ed. pr. 1490. Augsburg, 1490. ALYACO, HELIACO, D'AILLY. Born in Compiegne in 1350; chancellor of the University of Paris, Bishop of Cambray, and Cardinal. He died in 1420. Title. ' Cocordatia aftronomie cu theologia // Cocordatia af- tronomie cu hyftorica // narratione. Et elucidariu duor^z pre- //cedentium : dm Petri de Aliaco car//dinalis Cameracenfis.' (F. i,r.) Colophon. * Explicit tractatus de cocordia aftronomice veri- tatis i narrationis hiftorice//a dno Petro cardinali Cameracen. completus in ciuitate Bafilieri. anno xpi // 1414 : menfis. Maij die decima.' (F. 33, r.) 'Opus concordantie aftronomie cum theo- logia necnon hyftorice verita : nar//ratione explicit feliciter. Magiftri Joannis angeli viri peritiffimi diligeti cor//rectione. Erhardiq3 Ratdolt mira imprimendi arte: qua nuper Venetijs nuc // Augufte vindelicorum excellit nominatiffimus. 4. nonas Januarij. 1490.' (F. 55, v.) Description. 4, 15.6 X 20.4 cm., the text being n.i x 14.7 cm. 56 ff. unnumb., 39 11. Augsburg, 1490. Editions. There was no other edition. This work has been included in the list because, while chiefly astro- nomical, it throws considerable light upon the early Computi. It was written to show the relation between theology and astronomy, and hence it has an important bearing upon the study of the mediaeval calendar. ALONSO DELATORE. Ed. pr. 1489. Seville, 1538. A Spanish savant of the fifteenth century. Title. < Vifio delectable de // la philofophia i ar//tes liberates : meta//phifica : y philofo-//phia moral .:. //M.d.xxxviij.' (This is surrounded by an elaborate woodcut.) (F. I, r.) Colophon. ' CFenefce el libro llamado vifion delectable dela Philofophia : ar//tes liberales. Es impreffo enla infigne y muy leal ciudad // de Seuilla en cafa de Juan Croberger.//Ano de .M.d.xxxviij.' (F. Ixxij, r.) Description. Fol., 19 X 26.9 cm., the text being 15.3 X 24.3 cm. 72 if. numb., 42 11. Seville, 1538. Editions. Seville, 1489, fol.; ib., 1538, fol. (here described). 4 2 KARA ARITHMETICA This rare treatise is an encyclopaedia, with chapters devoted to the various arts and sciences. The arithmetic is found in Chapter IV (see Fig. 22), and consists of only two pages of theoretical discussion. COpttulo.tui.iE)daaririnetbicart>crudtoueW' fingularcefecretoe. fTandopa'zatraueflandoeftefcndrrorvimeronenamaoel te:a oofecome^aua tn marauilloio camino: el i toe guio en t>n lugar oe cafae ? palactos mug fmgulaw: 7 ala puma oe u tnlla ballaro pna mu^fagacifftma imu^^>funda rosefla D fciericia.aqualaunqloemiebzoecubnefTecoabitofcminilrparerciaoc bap i5 acjl afcddcr co:a$6 oe mu^ penetrate i mup ingeniofo rarc.y en laoicftrateniapngrafio^bierro^enlafinieftrarnatablacbla'queada: FIG. 22. FROM THE 1538 ALONSO DELATORE NICOLAUS CUSA. Ed. pr. c. 1490. Strasburg?, c. 1490. NICOLAUS CUSANUS, NICOLAUS CHRYPFFS OR KRESS. Born at Kues on the Mosel in 1401 ; died at Todi, Umbria, August n, 1464. He held posi- tions of honor in the Church, including the bishopric of Brescia. He was made a cardinal in 1448. He wrote several other works on mathematics. Title. Usually known as the ' Opuscula.' The work begins : ' Prohemium.// [I]n hoc volumine ptinentur certi tractatus i libri altiffime ptemplato//nis et doctrine: a preclare memorie pre- ftantiflimo doctiffimoq3 viro // Nicolao de Cufa.' (F. I, v.) Description. Fol., 17.5 x 25. 4 cm., the text being 1 1.7 X i8cm. 163 ff. unnumb. + I blank = 164 ff., 45 11. S. 1. a. (Strasburg ?, c. 1490. Some bibliographers place it as early as 1480 and others assign it to Milan as late as 1505.) PRINTED BOOKS 43 Editions. Cusa's 'Opuscula varia' first appeared s. 1. a. (Stras- burg?, c. 1490 or earlier). His 'Opera' appeared at Paris in 1511 and 1514 (see below), and again at Basel in 1565. This con tains fifteen of Cusa's tractati, including 'Reparatio kalendarij,' 'De Apice theorie' (4 ff.), <De mathematicis complements,' 'De mathe- matica perfectione.' Of these mathematical chapters the first two are of some interest in the history of arithmetic, the others referring chiefly to mensuration. NICOLAUS CUSA. Ed. pr. c. 1490. Paris, 1514. See p. 42. Title. ' Hec in hoc fecudo vo//lumine contenta.// Dialogus de ignoto. 2. De tranfmutationibus Dialogus de annuncia- geometricis. 33. done. 3. De Arithmeticis com- Excitationu libri X. 7. plementis. 54. Coniectura de nouiffi- De mathematicis corn- mis diebus.// i. plementis. 59. Septem epiftolae. 3. Complementum theo- Reparatio Caledarii. 22. logicum. 92. Correctio Tabularum De prefectione mathe- Alphonfi. 29. matica. in.' (Woodcut of printing-press, with the words: 'Prelu Afcefianu.') * Venudantur cum cete//ris eius operibus in Aedibus Afcenfi- anis.' (F. I, r.) Colophon. * Emissvm est hoc librorvm Cvsae opvs // egregivm Parisiis : ex officina Ascen//siana anno Christi pientissimi om// nivm Redemptoris MDXIIII, octa//va Assvmptionis semper San//ctae semperqve Virginis// Christi Deiqve Matris//Mariae. qva patroci//nante apvd Filivm // portvm salv-//tis spera-//rnvs et//veniae.' (F. CXVI, r.) Description. Fol., 20x29 cm., the text being 12.7x26.8 cm. 114 ff. numb. (Roman numerals) + 2 unnumb. = 116 ff., 46 11. Paris, 1514. Editions. See above. 44 KARA ARITHMETICA This second volume of the Paris edition of Cusa's works, edited by Faber Stapulensis, contains the ' tractati ' already mentioned on p. 43. JOHANN WIDMAN (?) Ed. pr. c. 1490. S. 1. a. (c. 1490). See p. 36. Title. 'Algorithmic Integro^//Cum Probis annexis.' (F. i,r. See Fig. 23.) Description. 8, 14.9 X 20.6 cm., the text being 7.8 X 14 cm. 12 ff. unnumb., 29-31 11. S. 1. a. (c. 1490). Editions. The various bibliographies assign different dates, but I presume there was only this one edition. De Morgan (p. 99), whose judgment as to the dates of such early works was unreliable, estimated this as "hardly later than 1475," adding, "I think this is the oldest book in my list." Wappler, who has critically investigated the matter (Abhandlimgen, V, 158) believes that Widman wrote this work, the 'Algorithmus linealis' (p. 36), and also the ' Algorithmic Minutiarum Phisicarum.' The work opens with a quotation from Boethius, the same indeed as the opening sentence of Sacrobosco's Algorismus. After treating of numeration, addi- //tEBtroatftmus flttf eatoE ^ -1*^, du . w ^2 fc 1 ~* "* ** *- plation, mediation, multiplication, and ,> division, the author FIG. 23. TITLE OF THE ALGORITHMIC INTEGRORUM takes up progressions, roots, and the proofs of the various processes. There are no applications in the book. Other works 0/1490. Anianus, p. 32, 1488 ; Boethius, p. 27, 1488. There also appeared c. 1490, at Leipzig, an anonymous work edited by Norico, entitled ' Arithmeticae Textus communis.' ANONYMOUS. Ed. pr. c. 1491. S. 1. a. (c. 1491). Title. See Fig. 24. Colophon. 'CFinis trium Algorifmo^ cum proper //tionum vel Mercatorum regula.' (F. 10, r.) Description. 4, 13. 7 X 19. 4cm., the text being 8.6 x 15. 6 cm. 10 ff. unnumb., 34-36 11. S. 1. a. (c. 1491)- PRINTED BOOKS 45 Editions. This is not the edition described by De Morgan (p. 99), because the title has ' addita etiam regula,' instead of * addita regula.' The word 'regula' instead of ' regla ' at the intcgrts copendfofefmcfigtirarumC mo:e 3rafo Mfmttkobodl cmiclia tun bzcuiflim* cdoccne.vna cu &( gozifrme DC mmurqa vulgarfbus viddiccr ct pbiHcalibua . rjue vulga mcrcaromm rcgula^iaf Quiba&'babiris. niodicaadbibitdW'U'scnttaomncracalcuUndCmodum fa cflttmc adtpifciporcft ^***c- *K'f^y ^ ^/^^^ **<**? / ^r/t ' af*~*~*r -*~? .*** ] *^^ / / / s% */& . . a* ^ftxi^e* jt&^Zff-^-**^ ** 1 *f"^!f**~ ** ***"? FIG. 24. TITLE PAGE OF THE ALGORISMUS OF c. 1491 end shows it to be different from the one described by Brunet (edition of 1860). It is probably no. 827 of Hain, and the inter- nal evidence makes 1491 the probable year of the composition 4 6 KARA ARITHMETICA or publication. De Morgan says that three editions are known, and Giinther on the authority of Chasles gives an edition at Cologne, c. 1510. This anonymous work is of the class of the arithmetics of Muris, Peurbach, Ciruelo, and other mediaeval and early Renaissance writers. picrngoras aritbrnetriretmroducror FIG. 25. TITLE PAGE OF CALANDRI It contains a brief treatment of the ' species ' (the fundamental opera- tions) with integers, omitting ' duplatio ' and ' mediatio ' but including progression and roots. This is followed by a discussion of common fractions ('Algorifmus nouus de//minutijs vulgaribus '), a single page on sexagesimal fractions (<de minutijs Phificalibus '), and a page on proportion. It is one of the first books to identify proportion with the rule of three, or merchants' rule as it was often called. (* De regula proportionum // Sive aliter Regula Mercatorum dicta.') It is not as practical as the elaborate title would seem to indicate. PRINTED BOOKS 47 PHILIPPI CALANDRI. Ed. pr. 1491. Florence, 1491. ^ A Florentine arithmetician of the fifteenth century. Title. ( Pictagoras arithmetrice introductor.' (Woodcut of n 4,7 > per a] Uiennc LiL H4 .t* HTT ??_ ? U~ 4f o M 'Parti | g C o -parti n> g * f3>i/i -c B7r Uicnne ? i ^ inenne T? Co 480 uicnnc i C-o r - 1 fj 1 FIG. 26. FROM CALANDRI'S ARITHMETIC Pythagoras.) (F. i, v. Fig. 25.) ' Philippi Calandri ad nobi- lem et ftudiofu5 Julia//num Laurentii Medice de arimethrica opufculu.' (F. 2, r.) KARA ARITHMETICA Clu condocro cmpie u na fontc tn 4 dt T cjun do e picna non merren do ilcondocto 7 ftiir.i do iluoracoto fiuorerc beladccra feme in i i di : Uo fapere eflendo uomlafonr: tmccrcri do ilcondpcco r ftur.i doiluoraroio inqun'tt di lava plena la dccra fonte I i crdtnoi r * r * - v-*^ j^ttk'j^, i 5 dibrarrio : uo faperc ! inc]uaridt fara fnora* del deao peso I 6 ' FIG. 27. FROM CALANDRI'S ARITHMETIC Colophon. * Impreffo nella excelfa cipta di Firenze per /?// Lorenzo de Morgiani et Giouanni // Thedefco da Maganza ft// nito a di primo di // Genaio 1491.' (F. 104, v. Fig. 30.) Description. 8, 9.8 X 1 3.2 cm., printed in double col- umns, each 3.3 x 10.7 cm. to 11.5 cm. 104 ff. unnumb., FIG. 28. FROM CALANDRI'S ARITHMETIC 9-2611. Florence, 1491. Editions. Florence, 1491, 8 (here described) ; ib., 1518, printed by Bernardo Zuchetta, 4. PRINTED BOOKS 49 Of this rare book, the first in De Morgan's list, Mr. Plimpton pos- sesses two copies. It is beautifully printed, and is practical in its presentation of the operations, but traditional in its problems. It is Eglicunatorrcebce aim 4 o bpieda ? dap pie mpaflatmo fmme cbcctarsbo jo brae eiauofapcrcc|namo faraltwgba iwa func cbefuia,ppieaiaaUari uadclnurru: rallaei roaslclLuorre if faralunga fo brae cu glieunalbcromfufa nua dun fiumc elqua kealrofo braccia d fmmc e largbo ? o bra da 7 per fort ima di uc tofiruppcmralluogo cbclacimadeUalbcro toeeaua lariua dclftu me.Qofapereqia!trc braeaa fcnc ruppc z qtumto ncnm a fc rino too| rimafcriro \C brae eta ^ j 4 braceia fcnc ruppc FIG. 29. FROM CALANDRI'S ARITHMETIC the first printed Italian arithmetic with illustrations accompanying prob- lems, and the first to give long division in the modern form (Fig. 26) known to the Italian writers by the name 'adanda.' Indeed Calandri gives only the 4 a danda ' method, omitting the galley form, and is therefore fully a century ahead of his time. De Morgan's statement that he uses a divisor diminished by i is incor- rect, as will be seen from Fig. 26. Figs. 27 and 29 show that the problems of the cistern, the snail (serpent) in the well, the length of the hypotenuse, and the broken tree were familiar in Calandri' s time. Other works 0/1491. Anianus, p. 32, 1488 ; Boethius, p. 27, 1488 ; Borghi, p. 1 6, 1484. Xbcdcfco da ZDajanja ft nitoadiprimodi 0cnaio i +? i FIG. 30. COLOPHON OF CALANDRI 5 o KARA ARITHMETICA FRANCESCO PELLOS OR PELLIZZATI. Ed. pr. 1492. Turin, 1492. A native of Nice, living in the latter half of the fifteenth century. Title. ' Sen fegue de la art de arithme-//ticha. et femblat- ment de ieume-//tria dich ho nominat3 Copendio // de lo abaco.//i 234567890.' (F. i, r. Fig. 31.) Colophon. ' Complida es la opera, ordinada. he condida // Per noble Frances pellos. Citadin es de Nifa. . . . Impreffo in Thaurino lo prefent c5pendio de abaco per mei/ //ftro Nicolo benedeti he meiftro Jacobino fuigo de fancto ger//mano. Nel anno .1492. ad. Di .29. de feptembrio.' (F. 80, r.) Description. Sm. 4, 13.8 x 20.9 cm., the text being 9.2 x 15.2 cm. 80 ff. numb., 39 11. Some of the initials have been inserted by hand, in red. Turin, 1492. Editions. There was no other edition. This is one of the rarest arithmetics known to exist. (Brunet, IV, 475; Graesse, Tresor, V, 100 ; Riccardi, I, 2, 256.) The only good description of the work is that given by Boncompagni in the Atti del? Accad. Pontif. de 1 nuovi Lincei, XVI, 161, 332, evidently after examining this copy, since it bears a note in his handwriting. Pellos first considers the fundamental operations with integers, fol- lowing this by a treatment of proportion, square root, and cube root. He then discusses the subject of fractions in much the same order, the rule of three, certain rules relating to weights, time, money, and other measures, and such topics as partnership, barter, interest, alloys, and the rule of false position, single and double. He closes the work with a chapter on mensuration, or as he calls it, * De la art de ieumentria ' (* ieumetria ' in the title), and gives a number of interesting woodcuts. The chief interest of the book attaches, however, to the fact that Pellos came very near the invention of decimal fractions, and that he actually used the decimal point as is shown in the illustration (Fig. 32). It cannot be said, however, that he had any conception of the real value of the decimal fraction as such, the first book devoted to this subject being 'La Disme ' of Stevin (1585), hereafter described. Pellos simply uses the decimal point to indicate division by some power of ten, writing a common fraction in the quotient. Thus, to divide 425 by 70, Pellos would divide 42.5 by 7, writing the result 6. PRINTED BOOKS *"^^ feguc belaarttearitbmc/ ricba. et femblarmcnr t>cieumc/ , trm oicb bo nominal Copendio FIG. 31. TITLE PAGE OF PELLOS 52 KARA ARITHMETICA (| CTpanirper ~ 5T" 7 9 6 S 4. 8 3: 9 > 7 * 7 qaocicnf 39827419 ~ 2 O C'Partfrpcr 3 o y 8 3 6 o 4. i 3 * ; qtWrfctt 194*64 $ O C*P*rtiVp*r 7 o 9 ? ? 7 9 t *9 ^ 9 qnocfcnt I j ^ a y j 7 o i o o 9 7 6 $,87 quocicnt 400} 78 9 ^ ? 7 quotient I j 7 4 * 3000 876 S 8*7 179 I quocfent 2 p i l 9 3000 _ <L & c aqiU manfera podce fafrc obc tons fcmblls p^ru FIG. 32, FROM THE PELLOS ARITHMETIC PRINTED BOOKS 53 GEORG VON PEURBACH. Ed. pr. 1492. Wittenberg, 1534. PURBACH, PEUERBACH. Born at Peuerbach, Upper Austria, May 30, 1423; died at Vienna, April 8, 1461. He studied under Johann von Gmiinden (see p. 117), Nicolaus Cusa (see p. 42), and other great teachers, and later he became professor of mathematics at Vienna, where Regiomontanus (Johannes Miiller, of Konigsberg) was his pupil. His interests were almost entirely in astronomy. Title. ' Elemen//ta Arithmetices // Algorithmvs de // nu- meris integris auctore // Georgio Peurbachio.// De Nvmeris Practis,// Regulis communibus & // Proporcionibus.// Cum prae- fatione Philippi // Melanchthonis.// M. D. XXXIIII.' (F. I, r.) Colophon. ' Impress vm Vitebergae // per losephvm Clvg.// Anno M. D. XXXIIII.' (F. 39, v.) Description. 8, 10.8 X 15.6 cm., the text being 6.4 X 11.2 cm. 39 ff. unnumb. + I blank = 40 ff., 22-25 U- Wittenberg, 1534- Editions. S. 1., ' Explicitum est hoc opus anno Christi dom. 1492.' 4; Vienna, s. a., c. 1500; Leipzig, 1503, 4; ib., 1507, 4, and probably ib., 1510 and ib., 1511; Vienna, s. a. but ear- lier than 151 1, 4 ; ib., 1511,4; ib., 1512; Niirnberg, 1 5 1 3 ; Vienna, 1515, 4; ib., 1520, 4; Wittenberg, 1534, 8 (here described); ib., 1536, 8; ib., 1538; Venice, 1539, 8 (see below); Frankfort, 1544. The arithmetic of Peurbach went by various names, as * Opus Algo- rithmi,' Institutiones in arithmeticam,' ' Elementa arithmetices,' and * Introductorium in arithmeticam.' It is a brief treatise on the funda- mental operations with integers and fractions, and contains a few simple applications. Peurbach was too profound a mathematician to have con- sidered it a work of any importance, but it is probable that he wrote it for the benefit of students who were not yet prepared to take up his work in astronomy. GEORG VON PEURBACH. Ed. pr. 1492. Venice, 1539. See above. Title. ' Elementa // Geometriae ex Evclide // fmgulari pru- dentia collecta a loane Vo-//gelin profeffore Mathematico in 54 KARA ARITHMETICA // fchola Viennenfi.// Arithmetics practicae per Georgium // Peurbuchium Mathematicum.// Cum praefacione Philippi // Melanchthonis.' (Woodcut with motto : ' Dissimilivm. Infida. Societas.') (F. I, r.) The arithmetic of Peurbach begins on f. 32: * Elementa // Arithmetices.// Algorithmvs de nv-//rneris integris, fractis, Regulis // communibus, & de Pro-//portionibus. //Authore Georgio Peurbachio.//Omnia recens in lucem sedita fide & //diligentia fingulari.// Cum praefatione Philip. Melanth.' Colophon. ' Venetijs loan. Anto. de Nicolinis de Sabio.// Sumptu uero D. Melchioris Seffse. Anno //Domini M D XXXVIIII.//Menfe lanuario.' (F. 67, v.) Description. 8, 10.3 X 15.8 cm., the text being 8 x 12.2 cm. i f. unnumb. + 68 numb. = 69 ff., 29 11. Venice, 1539. See p. 53. Other works of 1492. Anianus, p. 32, 1488; Boethius, p. 27, 1488. Works of 1493. Anianus, p. 32, 1488; Anonymous (see Widman), p. 36, c. 1490; Isidorus, p. 8, 1483; Vincent de Beauvais, p. 10. LUCA PACIUOLO, DE BORGO SAN SEPOLCRO. Ed. pr. 1494. Venice, 1494. PACIOLUS, PATIULUS, PACIOLI. Born in Borgo San Sepolcro, Tuscany, c. 1445-1450; died soon after 1509. Not an original mathematician, but the compiler of several works. Title. See Fig. 33. Colophon to the part on arithmetic : * Et fi fequenti pti pncipali Geo e . finis decima nouembris ipofitus fuerit : huic tamen pti: die vigefi //ma eiufdem Ipofitus fuit. M?cccc.lxliiij. Per eof- dem correctorem i imprefforem vt i fine Geo e . hr.' (F. 232, numb. 224, v.) There is also the following date on f. i, v., ' M?ccc.?lxliiij?xx a . Nouembris.' Description. Fol., 21.5 x 30.5 cm., the text being 19 x 24.2 cm. 8 ff. unnumb. + 224 numb. = 232 ff. in the part on arith- metic ; 76 ff. numb, in the part on geometry ; making a total of 308 ff., 56-60 11. Venice, 1494. Editions. Venice, 1494, fol. (here described); Toscolano, 1523, fol. (p. 58). PRINTED BOOKS 55 @uma t>e Sritbmctica >co/ mctri4l>:opo:tiom i ikw porno nalita, ontincntu Oc tutta topcra. be numeric mifure in turn' modi occurrenti. t&jopojnbm'e^pomoaliraanotirfa oel.5? oeucli dc c DC turn' It aim foi lilvu fnauiouero euidenne numero.i ;.p k $nta conn/ nuc.ppojrioali oeC.6fe.7t DC uclidc crtratrc. Cuttc (e gri <xlalg;o:iTmo:abc rclcuarc . pnr. m ulri/ plicar.fi) marc^ fotrarc co rurtc fuc ,pue i (ani e rot' ri.c rad W e p:ogrcllioni. &e la regola mercanrefca Oitra od.u Toi fodamett' n'con cafi eremplariper c?m?gr.i^uadafini;^rdi te:tranrponarfoni:e mueftite. ^arti'r.muln'pf icar.fumniar.e fotrar oe k p:op0m'<> m'eDetuttefojriradicf. uidcntic general! ouer conduficmi nfc^.abfduere ognicafodxpcrrcgorc.o:dman'end fi podclTc, fuctefo^eWnomijereaTiealtrelineeuTaridafioel do: fabn*' cbeefortdamenrt. om pagnic i turn' modi.c To? pattire. ecide oe bcfliami. e lo? partire aram' i mmnnxU fcmplicucoinpofh'tc cd tempo. ambt reali-feccbtfi'tririi^ Wmt'nuriouer comunC K>criti fcmplici e a eapo oanno e aim termini* Kefti.faWi.fconti.de tempo eoenariela recare aun dipi'u partite* >?i4rgeti.elox)afftnare. ecarattare, OE)oUicafieragioniftTao?dinane variee oiuerfe 8 tutteoccurentiecommonelfafequente rauota ap' pare o^tnaMinenre oe tutte. 0:dinea fapcr tcncrognicotoefcnpture e oelqua ^tarin^a oe tuttevfantf e coftumi mercantcfcbi in tut todmoiido, [ariealtriDependennV molte altrecofed grandfflimf piacerie frutto co/ tno oifufanientc per la fcqucntc taucla apparc, FIG. 33. TITLE PAGE OF THE 1494 PACIUOLO KARA ARITHMETICA This volume, the first great general work on mathematics printed, includes treatises on arithmetic, algebra, and geometry, each being considered from a somewhat scientific rather than practical standpoint. The arithmetic, for example, gives the various methods in multiplication (see Fig. 34) and division, instead of emphasizing the one or two most prominent in business circles. In the same way Paciuolo's treatment of FIG. 34. GELOSIA MULTIPLICATION, 1494 PACIUOLO the rule of three, the rule of false ('El cataym'), partnership, pasturage, barter, exchange, and interest, while nominally practical, was too elabo- rate for the mercantile schools. His was the first printed work to illus- trate the finger symbolism of number (Fig. 35). Paciuolo copies without hesitation, practically verbatim, from the work of Chiarini (p. 10), and doubtless laid under contribution, after the manner of his time, various other works of his predecessors. In algebra he used the common sym- bolism of the time for the unknown quantities and for roots, but he made use of no symbols of operation. This part of the treatise relates chiefly to surd numbers. In geometry he follows Euclid's Book I very closely, but departs quite radically from the subsequent books. The work had a great influence on subsequent writers, including the English Tonstall (p. 132). Paciuolo's training had fitted him to write a treatise of this nature. He had been a tutor in the family of a Venetian merchant, had traveled extensively, had come in contact with practical mathe- maticians, and had studied the ancient mathematics in the cloisters ; and traces of all these influences are seen in his work. In 1497 Paciuolo wrote at Milan a work entitled ' Divina proportioned which was pub- lished at Venice in 1509 (p. 87). He also published an edition of Euclid at Venice in 1509. Other works 0/1494. Albert of Saxony, p. 9, 1478; Anianus, p. 32, 1488. PRINTED BOOKS 57 loo FIG. 35. FINGER SYMBOLISM, 1494 PACIUOLO 58 RARA ARITHMETICA LUCA PACIUOLO, DE BORGO SAN SEPOLCRO. Ed. pr. 1494. Toscolano, 1523. Title. * Summa de // Arithmetica geo//metria. Proportion! : et proportionalita : // Nouamente impresfa In Tofcolano fu la riua dil Benacenfe et // vnico carpionifta Laco : Amenisfimo Sito : de li antique i // euidenti ruine di la nobil cita Benaco ditta illuftra-//to : Cum numerofita de Impatorij epithaphij//di antique : perfette littere fculpiti do-//tato : : cu3 finisfimi i mirabil co-// lone marmorei: inumeri // f ragmenti di alaba-//Jftro porphidi i ferpentini. Cofe certo // letter mio diletto oculata fi-//de miratu digne fot-//terra fe ritro//uano.// Continentia de tutta lopera:'// (The rest of the title page is substantially identical with that of the 1494 edition.) (F. i, r.) Colophon to the part on arithmetic : ' C.Et fi fequeti pti pnci- pali Geo e . finis decima nouembris Ipofitus fuerit : huic tamen pti die vigefima // eiusde impofitus fuit .M?ccccc.xxiij. Per eosdem correctore impresforem vt in fine Geo e . hetur.' (F. 232, v.). Description, Fol., 21.1 X 30.4 cm., the text being 18.2 x 23.5 cm. 9 ff. unnumb. + 223 numb. = 232 ff. Toscolano, 1523. Bound with this is the ' Tractatus Geometria. Pars fecunda principalis huius operis i, primo eius diuifio.' This part of the book contains 75 ff., besides the index. Editions. See p. 54. De Morgan has shown (p. 2), that there are slight differences between the copies printed in 1523, proving that a second impression was necessary in that year. See p. 56, and Fig. 36. PEDRO SANCHEZ CIRUELO. Ed. pr. 1495. Paris, 1505. Born at Daroca in Aragon, c. 1470 ; died at Salamanca in r 560. One of the most learned men of his time. He was professor of philosophy at Alcala. Title. * Tractatus Arithmethice // Practice qui dicitur // Algo- rifmus.' (A large woodcut with the initials D. R. and the inscription ' Alaventvre. Tout Vient. Aponit. Qvi. Pevt. Atendre. Denis. Roce.') (F. i, r.) PRINTED BOOKS 59 fi nofcuf irnaflente ec&cfc molti pMofbpft per gudto guidzmm finno alia fubftantu para: e conct (fa octerna. Ifcerodje fconno ODgnoTtuito per- cep modo aldjana co6 in reram natora fmsa In no ptere eriltcrc. ifcer laqual cob tt In intendo Coon (adiute ee ootai d* b noftri fenft reggt J) rrgoame nm dje per altri pnTcfpi e onocf?i pfy' fofopfji nonnt fia copidfemente tractate: e in Zfroa'ca e pactua. rT*3 prrdx kxoicti gia alii tempi noftn Tonna molto obfcurt-e oa moltj male apsfi : e alle pwcndje vu[ gari mate apptoti : wldje in tow opnanortf moltD vs^ nano: e con gradi e laforioft aftanra mettano in ofrra: ft tt mmeri como V milure: vndfoi lei parlando non intendo le non quanto cJ?c alia pactica e opnare fia mdtiero: meftotodod fecondo i taogfo opairt andjoza la tfjeo nca:ecan&ataIeopnare:naiiuniericorrioaseometnVi.dNipnrnaaccio meglio qneHo dje fequiu fe fjabia appxndere : effa quantita oiuidiremo fecondo el noftro p:o pifito: e ciuidmdoU a dalcfjano fao membra aflegnaremo fiia pjopaa e tera oiffir, ^ Done e afcn'ptionf . aHoa pi feguira quello dx ^nltodle CUE in ft rondo pflcrio fantZnnfe^maj^efcimrafiqmdaunfpatmirluumguiddttc. tDBfflSnitiones/et vuffio cierrte/etcontinneqiiantkatis. jHrrj'cuuw pa'mno . tffea'me oi(h'nctjoni. D^9co adonca. ^a quantita eflfere immediate 6imemb?e^do confirms e ^ifcrctcu ^^"como foraito Itgnp km: e &ru t& jia. wTcrem oueramente numero eqnHUtecmpBrtinOTlcn^s^fc^^^fw/^mfriEcomunf.'coinoe.i.i.j.rc. ^c^rpa/na ct b ixftreffl: doeert nnmero: e pt cc b connnoa; doe 0Eometna;qua< CjDiffinitio nnmeri pjopa'idima. 2rticolu0 (ecundnff. O^mcio^ c Cfccorido daJuTiino pryloiopfjdntO ^ns mclulndinf tP tmitn co r pofta: z eflTa mia non e nnmero: ma ten pnhdpio ce cUfdjuno nnmero: e OB queUa mediante faqle ogra CD& e oicia eflere vna. fecondo feuerino )8oetjo In fiia muftca: e la mica dafdjun numero in ptenoa: z paffi j in la fua aritljmetica rf gina e fbndamento cogni nnmero lap&o. lUqual piu tnagrancandola in le cofe naot' tsae oifle in quello dje 6 ce mcate z vno. Cmnt qd eft: ideo eft: quia rnum nume^ ID eft. te ancfrna el numero in inftrafl membji oiuifo: per quello cfje cflb anftoole icE:doe.a'qmdinftninimeft:nnmeruB eft. per laterw painoneoel fepdmoce >Euc(ide: b (faa ferie in infinite ptere praadere: ? quocuqs numero pato: oari pteft nvdov wu'tatem addoido . d>a noi pi'gliaremo queue parte a noi piu note e aaomo date. Jxro oico con glialth aldjuno eflere pamo: ett quello ctje Colo ca la vnto e nn merato: e non fea altro nnmero: dje integralmente apnto to parta. aitro e oitto co^ Edlo: tt e qnttto dje oa altro numero e'mefurato: ouero numerate. jempuim p:imf Como.) .7. 1 1. i).e. i7.?c.Kplu feci3di.iC6rno.4.tljel roi to mdura e numera.-e. S. titf. J.e-4/9. i H- 1 8..e nnuUmta ronno citti numai compfh; no foto dje coftino 2 FIG. 36. FIRST PAGE OF TEXT, 1523 PACIUOLO 60 KARA ARITHMETICA Colophon. ' Arithmetice practice feu Algorifmi tractatus a Petro fanchez // Ciruelo nouiter compilatus Explicit Impref- fus Parifius In // Bellouifu. Anno dni, 1505. Die. 29. Aprilis.' (F. 14, r ) Description. 4, 13.2 X 18.4 cm., the text being 9.7 x 14 cm. 14 if. unnumb., 35-39 11. Paris, 1505. Editions. Paris, 1495, 4; ib., 1505, 4 (here described) ; ib., 1509, 4; ib., 1513, 4 (see below). Ciruelo also wrote a ' Cursus quattuor mathematicarvm artiu liberaliu,' Paris, 1516; ib., 1523; ib., 1526; ib., 1528; Alcala, 1516, fol. ; ib., 1518; ib., 1523, fol. ; ib., 1526; ib., 1528. He also edited Brad- wardin's arithmetic, Paris, 1495; ib., 1502. Ciruelo treats very briefly of the fundamental operations with integers, common fractions, and denominate numbers. Following the Spanish custom he uses cuento for million. There is little that is noteworthy in the book, and, like Peurbach, Ciruelo could not have taken his contri- bution to algorism very seriously. Other works 0/14.95. Anianus, c. 1495, p. 32, 1488. There also appeared in 1495 the treatise of Herodianus, ' De notis Graecorum Arithmeticis Grsece,' Venice. Of this work there was an edition in 1525, and a Latin edition published at Basel in 1600, but it hardly deserves to be classed as an arithmetic. PEDRO SANCHEZ CIRUELO. Ed. pr. 1495. Paris, 1513. See p. 58. Title. ' Tractatus arithmetice // Practice qui dicitur algorif- mus.// (A large woodcut with the name Jehanlambert.) Venun- dantur Parrhifijs a Johaue La//berto eiufdem ciuitatis biblio- pola in fte-//mate diui claudij manente iuxta gymna-//fium coquereti.' (F. I, r.) Colophon. ' C.Arithmetice practice feu Algorifmi tractatus a Petro fan//che Ciruelo nouiter compilatus Explicit Impreffus Parifius //per Anthonium Auffourt. Pro Johane Lamberto eiufdem ci-//iritatis bibliopola in ftemmata diui claudij manente iuxta gym-//nafium coquereti. Anno dni .1513. Dievero .21. menfis Marti j.' (F. 12, r.) PRINTED BOOKS 61 Description. 4, 13.1 x 19 cm., the text being 9.6 x 14.8 cm. 12 ff. unnumb., 43-46 11. Paris, 1513. Editions. This is the fourth edition (see p. 60), and is an exact reprint of the 1505 copy, save as to spelling and pagina- tion. As a piece of typography it is much better than its predecessors. THOMAS BRADWARDIN. Ed. pr. 1495. Paris, c. 1510. BRAGWARDINE, BRANDNARDINUS, BREDWARDYN, BRADWARDYN, DE BRADWARDINA, DE BREDWARDINA. Born at Hertfield (Hartfield) in the diocese of Chichester, c. 1290; died at Lambeth, August 26, 1349. On account of his learning he was called ' Doctor profundus.' He was pro- fessor of theology at Oxford, and died as Archbishop of Canterbury. He wrote four works on mathematics. Title. 'Arithmetica thome brauardini.// Olivier Senant// Venum exponuntur ab Oliuiario fenant in vico diui Jacobi fub si-//gno beate Barbare fedente.' (F. i, r.) Colophon. ' Explicit arithmetica fpeculatiua thoe brauardini bri re-//iiifa e t correcta a Petro fanchez Ciruelo aragonenfi ma//thematicas legete Parifius, Ipressa p Thoma anguelart.' (F. 6, v.) Description. 4, 20.3 X 27.8 cm., printed in double columns, each being 6. 8 X 19.8 cm. 6 ff. unnumb., 61 11. Paris, c. 1510. Editions. Paris, 1495, 4; ib., 1496, fol. ; ib., 1498, 4; ib., s. a. (c. 1500) ; ib., 1502, 4; Valencia, 1503, fol. ; Paris, 1504; ib., 1505,4; ib., s. a. (c. 1510, here described) ; ib., 1512; ib., 1530; Wittenberg, 1534, 8; ib., 1536, 8. His ' Tractatus de proportionibus ' appeared in several editions, as follows : Paris, 1495; Venice, 1505, fol.; Vienna, 1515 (p. 117); and a com- mentary by Vittori appeared at Bologna in 1506. Some of these editions contained two or three of his works in one volume, and it is probable that his arithmetic and his treatise on proportion appeared in other editions than those mentioned. Bradwardin was one of the earliest English mathematicians after Bseda and Alcuin. His arithmetic is of the Boethian type, relating to the theory of numbers. He gives much attention to the ancient theory of ratios and to figurate numbers. 62 KARA ARITHMETICA BOETHIUS, JORDANUS NEMORARIUS, AND FABER STAPULENSIS. Ed. pr. 1496. Paris, 1496. For the biography of BOETHIUS see p. 25. JORDANUS NEMORARIUS (JORDANUS DE SAXONIA) was born at Borgent- reich, in the diocese of Paderborn, and died in 1236. He studied at Paris and was the greatest mathematician of his time save Leonardo Fibonacci of Pisa. JACOBUS FABER STAPULENSIS (JACQUES LE FEVRE D'ESTAPLES) was born at Estaples, near Amiens, in 1455, and died at Nerac in 1536. He was a priest, vicar of the bishop of Meaux, lecturer on philosophy at the College Lemoine in Paris, and tutor to Charles, son of Francois I. He wrote on philosophy, theology, and mathematics. Title. See Fig. 37. Description. Fol., 20 x 29 cm., the text being 13.9 x 27.2 cm. 72 ff. unnumb., 6063 11. Paris, 1496. Editions. The arithmetic of Jordanus went through various editions as follows: Paris, 1496, fol. (here described); ib., 1503, fol.; ib., 1 507, fol. (p. 65); ib., 1510, fol.; ib., 15 14 (p. 65). Rogg speaks of an edition of 1480, but I do not know of it. Jordanus also wrote an * Algorithmus demonstratus,' published anony- mously at Niirnberg in 1534, 4. De Morgan, following Scho- nerus, attributed it to Regiomontanus, but the evidence shows that he only revised it and it may be due to Jordanus. His 'De Ponderibus/ edited by Apianus, was published at Niirnberg in 1533, and at Venice in 1565. An interest attaches to the 1496 edition in that it is the first printed work with which a Scotch- man's name is connected, the printer being David Lauxius of Edinburgh, then working in Paris. The greater part of this volume is devoted to the ten books on arith- metic by Jordanus Nemorarius, with the commentary of Jacobus Faber Stapulensis. The work of Jordanus is similar to that of Boethius, and is concerned only with theory of numbers. In particular, the Greek theory of ratios, as elaborated during the Middle Ages, is extensively treated. The second part consists of the work of Jacobus Faber Stapulensis on music, in four books. The third part is the Epitome of the Arithmetic of Boethius : * CJacobi Fabri Stapulenfis Epitome in duos libros Arithmeticos // diui Seuerini Boetij ad Magnificum dfium Joannem Stephanum // Ferrerium Epifcopum Verfellenfem.' PRINTED BOOKS 63 The fourth part, consisting of four and a half pages, is a description of the arithmetical game of Rithmimachia, possibly by Shirewode (John Shirwood, Bishop of Durham, who died in 1494), but usually ascribed to Faber Stapulensis. An edition appeared at Erfurt in 1577, 4. $nbocoperc contents ftritbtnetica oecem lib:is ocmonftratt (Epitome i Ubzos arttbmcncosoiui 0euerfm tfocttf lauoe Sritbmcriccs z ^fcuficco. Ccmpo:e tarn multo ooctc bcucrc fozozcc: <2iuao rennet comitco flaua mmcrua fua0 TRuc plaaoe tcrrae port tempo:a multa reuiftfflf* i6rata quoc^ ante altaa fSalUca terra placet* Ibid olmi ccleb:te futt omnto Scaia tcllu0. l7tba$o:a patriam oiffiigicntc famon* IbcIIaoa nune Imquunt: ct Ddctae pnllaoio vrbcs* Bcquaniofcp pctunt/parbtfiofcp larc^v Ibcc venit omnimooa numeroiutn dneta ca tons* 2Ucp oo cct numertd quioquio m o:be fltum eft* Bltcra oulctfono cantu/ftoibitfcp cano:ia ^oomuiflc viro0 traoimr atcp ferae: luc fua ipicrij o tenet vniea nomma mul!0 ^ niebil bae mufio $ratme clTc fplcr. 3lttamcn artiftccm ftapule mifere marine: gQui leta boc (luDtjo frontc Diarct opu0. Iboefolum (luoium atcp bee illt curan'uuarc/ Ifrriiuo et nc flt/oifperearcp labo:* FIG. 37. TITLE PAGE OF THE 1496 BOETHIUS Other works 0/1496. Albert of Saxony, p. 9, c. 1478 ; Bradwardin, p. 6 1, 1495 ; Z. Lillius, < De origine et laudibus scientiarum,' Florence, 4 (one page on arithmetic). There was also published at Paris, s. a. 6 4 KARA ARITHMETICA (c. 1496), an anonymous treatise entitled ' De arte numeradi copediu putile icipit feliciter. (Q)uoniam rogatus a plurib5 compedium artis numerandi ac breue tractalutu . . .' Works 0/1497. Boethius, p. 27, 1488 (the colophon of the 1497 no oflo FlG. 38. RlTHMIMACHIA, 1496 BOETHIUS edition has the incorrect date Mcccclxxxvii) ; Suiseth, 1497 and 1498, p. 10, c. 1480. Works, of 1498. Anianus, p. 32, 1488; Bradwardin, p. 61, 1495; Chiarini, p. n, 1481; Anonymous, 'Enchiridion sive tractatus de numeris integris, fractis,' etc., 4 (doubtless the work published at Deventer in 1499, p. 67). PRINTED BOOKS 65 BOETHIUS, JORDANUS NEMORARIUS, AND FABER STAPULENSIS. Ed. pr. 1496. Paris, 1507. See p. 62. Title. ' In hoc opere contenta.// Arithmetica decem libris demonftrata // Mufica libris demonftrata quattuor // Epitome I libros arithmeticos diui Seuerini Boetij // Rithmimachie ludus q i pugna nuero^ appellat.' (F. i, r.) Colophon. ' Cjmpreffum Parifij in officina Henrici ftephani e regione Schole decretorum fita.//Anno Chrifti fiderum con- ditoris 1507. Decimo die Nouembris.' (F. 78, r.) Description. 8, 20x26.6 cm., the text being 18.1x21.5 cm. 78 ff. unnumb. + I blank = 79 ff., 60 11. Paris, 1507. Editions. See pp. 27, 62, for the editions of Boethius and Jordanus. Like most of the works from the press of Stephanus, this book is beautifully printed. It contains the ' Elementa Arithmetica ' of Jorda- nus, with the demonstrations of Faber Stapulensis, the Epitome of Boe- thius by Faber Stapulensis, the ' Rithmimachia,' the commentary on Sacrobosco's astronomy by Faber Stapulensis, and the first four books of Euclid * a Boetio in latinum tranflate.' BOETHIUS, JORDANUS NEMORARIUS, AND FABER STAPULENSIS. Ed. pr. 1496. Paris, 1514. See p. 62. Title. ' In hoc opere contenta // Arithmetica decem libris // demonftrata.// Mufica libris demoftrata // quatuor.// Epitome in libros Arith-//rneticos diui Seuerini //Boetij. //Rithmima- chie ludus qui//et pugna numeroru ap-//pellatur.'// (Sur- rounded by an elaborate woodcut with the following wording : ' Haecsecvndariaes//tetcastigat//issimaexofficina//aemissio' // ) <C.Hec fecundaria fuperioru operum aeditio // venalis habetur Parifijs://in ofihcina Henrici Stephani e regione fchole Decre- torum.' (F. i, r.) Colophon. ' C.Has duas Quadriuij partes et artium liberalium precipuas atq3 duces cu quibufdam ammini-//cularijs adiectis: curauit ex fecunda recognitione vna formulis emedatiffime 66 KARA ARITHMETICA mandari ad ftudiorum // vtilitatem Henricus Stephanus fuo grauiffimo labore et fumptu Parhifijs Anno falutis domini : // qui omnia in numero atq5 harmonia formauit 1514. abfolutumq5 reddidit code anno: die feptima // Septembris/ fuum laborem vbicunq3 valet Temper ftudiofis deuouens.' (F. 71, v.) Description. Fol., 19.8 X 28.4 cm., the text being 17.7 x 26.8 cm. 71 ff. unnumb., 62 11. Editions. See p. 62. This edition is practically identical with that of 1496. It is the second edition of this combination of works and the fourth of Faber's Epitome. MARTIANUS MINEUS FELIX CAPELLA. Ed. pr. 1499. Vincenza, 1499. Flourished c. 475. He was probably born at Carthage, and he lived at Rome. Title. See Fig. 39. Colophon. 'Martini Capellae Liber finit: Impreffus Vincentiae Anno Salutis // M.ccccxcix. xvii. Kalendas lanuarias per Henri- cum de Sancto//Vrfo Cum gratia & priuilegio decem annorum: ne imprima-//tur neq3 cum Commentatiis: neq3 fine: & caetera: quae in ipfo pri//uilegio continentur. Laus Deo & beatae Virgini.' (F. 123, v.) Description. Fol., 20.5 X 30.3 cm., the text being 12.1 X 22.3 cm. I24ff. unnumb., 37 11. Vincenza, 1499. Editions. Vincenza, 1499, fol. (here described); Modena, 1500, fol. (p. 67); Vienna, 1516, fol.; Basel, 1532, fol.; Leyden, 1539; Basel, 1 577, fol. (p. 68); Leyden, 1592, 8 (p. 68) ; ib., I599>8; and later. An Italian translation was published at Mantua in 1578. For bibliography, see Boncompagni's Bulletino, XV, 506. This work is a medley of prose and verse, and forms a kind of ency- clopedia of the arts and sciences as known for about a thousand years. It was highly esteemed in the Middle Ages as a textbook. The seventh book is on the Greek theory of arithmetic. It treats of the various classes of numbers, such as plane and solid, and mentions the supposed mysteries of the smaller numbers, the monad suggesting one God, the dyad good and evil, the triad the Trinity, and so on. PRINTED BOOKS 67 Other works of 1499. Boethius, p. 27, 1488; Suiseth, p. 10, c. 1480 ; Anonymous, ' Enchiridion Algorismi sive tractatus de numeris integris,' Deventer, 4 (p. 64, 1498) ; George of Hungary, ' Arithmetica summa tripartita,' s. 1., reprinted at Budapesth in 1894. Opus i (E> arruni-Capdle de ^luptiie oedulectica. s- >.vc arithmetics x. Deafrronomia. t>c mafia Ubrifcprcm. FIG. 39. TITLE PAGE OF THE 1499 CAPELLA MARTIANUS MINEUS FELIX CAPELLA. Ed. pr. 1499. Modena, 1500. See p. 66. Title. ' Opvs.// Martiani Capellae // de nvptiis phi//lologiae et //mercvrii//liberi//dvo.//De gramatica. Liber. Tertius.//De di- alectica. Liber. Quartus.//DeRhetorica. Liber. Quintus.//De ge- ometria. Liber. Sextus.//De Arithmetica. Liber. Septimus. //De aftronomia. Liber. Octauus.//De mufica. Liber. Nonus.' (F. i, r.) Colophon. ' Martiani Capellse Liber finit. Impreffus Mutinae. Anno Salutis. M.//CCCCC. Die .XV. Menfis Maii. Per Dionyfiu. Berthocum.' (F. 100, r.) Description. Fol., 20 x 29 cm., the text being 13.7 x 24 cm. On f. 69, v., the part on arithmetic begins, and occupies 10 ff. 100 ff. in the entire book, unnumb., 42 11. Modena, 1500. 68 KARA ARITHMETICA MARTIANUS MINEUS FELIX CAPELLA. Ed. pr. 1499. Basel, 1577. See p. 66. Title. * Isidori // Hispalensis // Episcopi // Originum libri viginti // ex antiquitate eruti.// Et // Martian! Capellse // De nuptijs Philologiae & Mercurij // Libri nouem.// Vterque, praeter Fulgentium & Veteres Grammaticos, va-//rijs lectionibus & fcholijs illuftratus // Opera atq Industria // Bonaventvrae Vvlca- nii Brvgensis.// Cum gratia & priuilegio Caefareae Maieftatis. // Basileae,// per Petrvm Pernam.' (F. i, r.) Description. Fol., 20.5 X 31 cm., printed in double columns, each being 8.2 X 24.4 cm., 7 ff. + 240 columns + index +550 columns. The part on arithmetic in the work of Capella begins in column 155 and covers 12 pp., or as here numbered 24 col- umns, 60 11. Basel, 1577. Editions. See p. 66. This edition includes the works of both Isidorus (p. 8) and Capella, and is an excellent specimen of printing. One interesting feature of Capella' s work is the evidence that it gives of the use of the abacus in the fifth century. We are still quite uncertain as to the history of this method of calculating in the centuries following Capella. MARTIANUS MINEUS FELIX CAPELLA. Ed. pr. 1499. Leyden, 1592. See p. 66. Title. ' M. Capella.// Martiani // Minei Capellae // Carthagi- nensis // de Nvptiis Philolo-//gise, & feptem artibus // Libe- ralibus // Libri Novem // optime castigati.// Lvgdvni,// Apud Bartholomaeum Vincentium.// 1592.' (P. i.) Colophon. ' Lugduni,// Excvdebat // Stephanus Seruain. // 1592.' (P. 416.) Description. 8, 10 X 15.7 cm., the text being 7.1 X 12.8 cm. 4 pp. blank -f 18 unnumb. + 396 numb. =418 pp., 30 11. Leyden, 1592. Editions. See p. 66, PRINTED BOOKS 69 feriftft foria ficot in fcoli^ mpnibergen,0ritbmcirico:n oocct JnflozentiflTinio fltodio EipcsenfinueediVTOo minns h ttena eruditia $ ^fcercatcwbns ^ tilfe ^r majcime incipientibiw. Zectozi %urta fucdnctepateat tibt rfgtila tsctri If rangere quo valeas qaem mi ' IDuius viUfcant non t ibi oon a libn V>ii8 nurnberga niter numerar? d t infigme a b arte v iic aw mulrom contnlit i(U bout FIG. 40. TITLE PAGE OF LIGHT 70 KARA ARITHMETICA BALTHASAR LIGHT. Ed. pr. 1500. Leipzig, s. a. (1500). A German Rechenmeister of c. 1500. Title. See Fig. 40. Colophon. 'Impreffum Lipczk per Melchiar Letter.' (F. I5,v.) Description. 4, 14 X 19.9 cm., the text being 9.3 X 16.2 cm. 15 if. unnumb., 22-37 11. Leipzig, s. a. (1500). Editions. Leipzig, s. a. (1500), here described; ib., 1509; 1513; Leipzig, 1515, 4, which may be the ' Algorithmus linealis,' s. a. and 1505, by Letter, referred to by De Morgan (p. 101). There was also an * Algorithmus linealis, Impressum Lipzik per melchiorem Lotter Anno xc,' probably printed in 1490, Lotter having printed in Leipzig from 1490 to 1512. (See contra, Abhandlungen, V, 154, n., 152, n., and cf. Widman, c. 1490, p. 44.) On f. i, v., the dedicatory epistle closes with the words ' Vale ex noftra academia Lyptzen Anno 1 500,' which throws much doubt on the conjecture that Licht's work appeared earlier. This is a brief treatise on the line abacus, one of the earliest of the type represented also by Huswirt (see p. 73). LEONARDUS PORTIUS. Ed. pr. c. 1500. S. 1. a. (Venice ?, c. 1500). A Venetian jurist of the fifteenth century. Title. ' Leonardi // de Portis ivrisconsvlti Vi//centini de sestertio pe//cvniis ponderibvs et // mensvris antiqvis // libri dvo.' (F. i, r. See Fig. 41.) Description. 4, 14.4 X 19.8 cm., the text being 10 x 14.8 cm. 37 ff. unnumb., 30 11. Venice (?), c. 1500. Editions. S. 1. a. (Venice ?, c. 1500, here described); Florence, 1514 (?) ; Basel, 1520, 4; ib., 1530, 8. A work on ancient measures, using the Roman numerals throughout, except in the index. Such treatises are of value in studying the history of arithmetic, but are not, in general, included in the bibliographical lists of this work. PRINTED BOOKS 71 Other works of c. 1500. Anianus, c. 1500, p. 32, 1488; Boethius, c. 1500, p. 27, 1488 ; Bradwardin, c. 1500, p. 61, 1495 ; Capella, 1500, p. 66, 1499; Peurbach, c. 1500, p. 53, 1492. Widman, 1500, p. 37, 1489. There also appeared about this time, s. 1. a., an anonymous 4 Algorithmus minutiarum vulgarium,' blackletter, 4 (Libri, 1861 cat., 483), and an anonymous 'Ars numerandi,' 5 ff., 4, a title given to several books of this period (see p. 23), including ' De arte numerandi siue arismetice (perfectionis) summa quadripartita ' (Abhandlungen, I, 24 ; Brunet, Man., 6 (i), 458). LEONARDI DE PORTISIVRISCONSVLTI VI CENTINI DE SESTERTIO PB CVNI1S PONDERIBVS ET MENSVRIS LIBRI OVO. FIG. 41. TITLE PAGE OF LEONARDUS PORTIUS GEORGIUS VALLA. Ed. pr. 1501. Venice, 1501. Born at Piacenza in 1430 ; died at Venice in 1499. He was a physician and philologian. Title. 'Georgii Vallae Placentini viri cla-//riss. de expetendis, et fvgiendis // rebvs opvs, in qvo haec // continentvr.// De arith- metica libri .iii. ubi quaedam a Boetio praetermiffa tractantur.// De Mufica libri .v. fed primo de inuentione, & 'commodiatate eius.//De Geometria libri .vi. in quibus elementorum Euclidis difficultates omnes fere // exponuntur, ubi etia de Mechanicis fpiritalibus, Catoptricis, ac Opticis, deq; //quadrate circuli habe- tur tractatus.// De tota Aftrologia libri .iiii. in qua fabrica,ufusq; aftrolabi exaratur, & quae fi-//gnorum in exhibendis medica- minibus fit habenda obferuatio.// De Phyfiologia libri .iiii. ubi & 72 KARA ARITHMETICA Metaphyfices qda lectu q dignifs. utilifTimaq;.//De Medicina libri .vii. ubi de fimplicium natura per ordinem litterarum.// Pro- blematum liber unus.// De Grammatica libri .Hi.// De Dialectica libri .m.// De Poetica liber unus.// De Rhethorica libri .ii.// De Morali Philofophia liber unus.// De Oeconomia, fine adminiftra- tione domus libri .iii. in quibus de Architectu-/Aa, req; ruftica eft locus.// Politicon unicum uolumen, ubi de iure ciuili, ac pontificio primum, Mox de le//gibus in uniuerfum, Inde de re militari agitur.//De Corporis commodis, & incommodis libri .iii. quorum primus totus de ani-/Ana, Secudus de corpore, Tertius uero de urinis ex Hippocrate, ac Paulo aegi-//neta, deq; Galeni quaeftioni- bus in Hippocratem.// De Rebus externis liber unus, ac ultimus, ubi de Gloria, Amplitudine, & caete-/Ais huiufmodi.// Haec fum- matim, fed infunt, & alia plurima,quae legedo licet cognofcere.' (F. i, r.) Colophon. l Venetiis, in aedibus Aldi Roma-//ni impensa, ac studio Ioan-//nis Petri Vallae filii pi-//entiss. mense Decem-// bri. M.D.I.' (Not in this copy.) Description. Fol., 28.8 X 43.7 cm., the text being 18.8 x 32.7 cm. 3 ff. blank + 308 unnumb. = 311 ff., 55 11. Venice, 1501. Editions. There was no other edition. The first book consists of 23 brief chapters on the general value and nature of mathematics (18 pp.); the second book, of 18 chapters on the Greek classification of numbers (17 pp.) ; the third, of 20 chapters on figurate numbers, proportions, and the fancied properties of each number of the first decade (27 pp.); and the fourth, of 13 chapters on the operations (13 pp.)- There is nothing that is noteworthy in the treatment. Works on the value of mathematics were quite common at this time, while all university treatises on arithmetic were devoted chiefly to the Greek theory. The thirteen pages devoted to the opera- tions were a rather generous allowance for the time, especially as each page has as much matter as six or eight pages of an ordinary octavo arithmetic of that period. Valla also wrote a treatise on the astrolabe, ' Insignis philosophi Nicephori Astrolabii expositio' (Paris, 1554), and published an edition of Euclid (Venice, 1492). His collected commentaries, but without the arithmetic and other original works, appeared in Venice in 1498. PRINTED BOOKS 73 ARNALDO DE VILLA NOVA. Ed. pr. 1501. Venice, 1501. ARNAULD DE VILLENEUVE, ARNALD BACHUONE. Bom in 1248, at Villa Nova (Catalonia), or possibly Villeneuve, near Montpellier; died in 1314, shipwrecked on the Mediterranean. He is known principally for twenty works on alchemy. He lectured on philosophy and medicine at Barcelona and Paris, and was later a celebrated physician. Computus micas ditus a 2lf>agiftro naldo DC villa *flcim *#o* inter 3mp:cflum. FIG. 42. TITLE PAGE OF ARNALDO DE VILLA NOVA 74 KARA ARITHMETICA Title. ' Computus Ecclefiafticus 2 Aftrono-//micus Editus a Magiftro Ar-//naldo de villa Noua No-//iriter Impreffum.// Cum Gratia Et Priuilegio.' (F. I, r. See Fig. 42.) Colophon. ' Cjmpreffum Venetijs per Bernardino Venetu deVitalibus.//Anno Dni.M.CCCCC.J.Die .xvij.Men.Februarij.' (F. n, v.) Description. 4, 14.4 X 20.9 cm., the text being 10.5 x 16.6 cm. ii ff. unnumb., 37-39 ff. Venice, 1501. Editions. There was no other edition. This is a good example of the works on the ecclesiastical calendar in use in the Middle Ages. It employs only the Roman numerals and gives no treatment of computation. In spite of the words 'nouiter impressum,' I know of no earlier edition, and indeed these words were not infrequently used when a book was first printed. JOHANN HUSWIRT, Sanensis. Ed. pr. 1501. Cologne, 1501. A German arithmetician of c. 1500. The name Sanensis suggests his birthplace as Sayn in the Westerwald, and the problems relate to places in that vicinity. Nothing is known of his life. Title. See Fig. 43. Colophon. ' Enchiridion algorifmi fagaci cura ftudioq3 p Johane hufwirt sanefez //elaboratus. caracteri p pmiffus Colo- nie In officina felicis memorie ho//nefti viri Henrici Quentell. Anno repatoris humane feruitut3 Mccccci.' (F. 20, r.) Description. 4, 14.3 X 20.6 cm., the text being 9 x 15.4 cm. 20 ff. unnumb., 2547 11. Cologne, 1501. Editions. Cologne, 1501, 4 (here described); ib., 1503; ib., 1504, 4 (p. 77); 1507; 1554; and a French edition (Chasles). It was published with historical notes by Professor Wildermuth, at Tubingen, in 1865. Mr. Plimpton's copy has the bookplates of Dr. Kloss and Chasles. This is the earliest treatise on algorism printed at Cologne. It is divided into four * tractati,' and includes the fundamental operations through evolution (' Tractatus Primus '), a brief treatment of abacus or line reckoning (' CTractatus Secundus de proiectilibus '), common frac- tions (' Tractatus Tertius'), rule of three, partnership, and over twenty PRINTED BOOKS 75 miscellaneous rules (' Tractatus quartus de regulis mercato^ ' etc.). In the algoristic treatment of integers Huswirt places ' duplatio (doubling) after multiplication, and ' mediatio ' (halving) after division ; uition no niaftimntfuitt meneccT/ faiv tnuidc ticlatt-esflingua combrfccfurenteiti Nee nimiu rabidis garrulus efto labris. bt viuis paccm concubitarc fine FIG. 43. TITLE PAGE OF THE 1501 HUSWIRT but when he is dealing with counters and with fractions he places them before multiplication, because they are needed there in abacus calculating. It is interesting to see how these chapters on doubling and halving, of which we have traces in ancient Egypt, persisted throughout the Middle Ages and well into the sixteenth century. 7 6 KARA ARITHMETICA As in several other works of this period, there is evidence of the difficulty of finding a generally acceptable name for the character 0, a difficulty not yet removed in the English language. Huswirt gives four names to this tenth character : ' Decimo Wo theca. circul? cifra. fme figura nihili appellat'.' There is also noticeable in this work a tendency which is seen in other arithmetics of the time, to name a group of problems after some well-known type. For example, Huswirt's sixth rule is that of the flee- ing hare ( Regula Sexta de Crattatua f t)e multiplication* Hapl'm qtiartunt Itiplicatio eft numm" p:ocrcatio.p:opo:tionabC er fc I?atcnri6 ad multiplicand!} ficut multiplicanc ad rni* rem ft IjahfMjKmpU gratia tod 4 multiplicarc til mime from Pnlncrnp tn Rnmp anH .* -C rum fz p2ocreare. que fie multiplicando videlicet 4 p:o0 ) KOme, ana -, oflona ^ ur qiwmdmo d um m.ilnpacans.fcilicet j viutari cozrcfponf followed five davs later bv rcr> QUWVTMCB eft p:opojtiompla.3tem multiplicatio p:crccimru<rq ' hne mulriplicattonem Digito2um inter fefdaf.Cuiue talie oarurrctwla lepore fugiente,' f. 16, r.), although it has nothing to do with the hound and hare, but relates to a traveler going another traveler who in due time overtakes him. (' C.Ambulat qdam de Colo- nia verfus roma et ambulat 4tidie 9 miliaria. alius aut pfequit' ipfum poft 5 dies,' p:io2ipjoduao pof?ponc.vrparet W figura. ^^fl-atiarcgulaocmulripW ScnbanturCigiri fubalteme.etcuuiflittt Ciffcrcntiaa oenano vcrfue PC >:rram ponas.quje inter fe mu(nplica.erp:odu(tum inferiuc fcnhr.Dcm? Pe Diffcrcntwrn vntuaaDtgifoatteriua fubrral|jccrp:io:ip;orfucfo poll? pone, cr p :ou enicf fumma. vt paret in figura ftw plum . fepes 9 feigitus 8 x oifferenna, quoruint.mulriplc BOta. 4 MM artm'ufl fubtH[?e.ardicrunii . a. C3l iarc S llla ^ m " lr| P' tarione Ouojum numero:um infra zo quo:um quilitrr oujbua hi fcnpmseft.piopofiris itaojouobue numowpnmajinfcnc.iie eutnpni! fnperiozia multipUca^t pzotrcabitur numerus vna velouaro ntfuriE . refcruaranr adiungc.er pzoumier numcrus wa? etc.) Other works of 1501. Anianus, p. 32, 1488; Sa- crobosco, p. 32, 1488 ; Boe- thius, p. 27, 1488; Borghi, p. 16, 1484 ; H. Torrentini, ' Elucidarius carminum et hystoriarum,' Deventer, 4 (with a chapter on arith- metic), with subsequent edi- tions as follows: ib., 1503, FIG. 44. COMPLEMENTARY MULTIPLICATION, 4j Strasburg, 1505, 4; HUSWIRT (1501) Hagenau, 1507, 4; ib., 1510, 4; ib., 1512, 4; Strasburg, 1514, 4; 1515, 4; Strasburg, 1518, 4; Paris, 1530, 8; ib., 1535, 8; Cologne, 1536, 8; Paris, i55o 8. Works of 1502. Anianus, p. 32, 1488; Albert of Saxony, p. 9, c. 1478 ; Bradwardin, p. 6, 1495 ; Nicolo Calvino, a work on arithmetic and geometry, Milan, of which no extant copy is known (see Riccardi, part I, col. 213). PRINTED BOOKS 77 JOHANN HUSWIRT, Sanensis. Ed. pr. 1501. Cologne, 1504? See p. 74. Title. The same as in the edition of 1501 (see p. 74). Colophon. The last folio, with the colophon, is missing from this copy. Description. There are a few changes in type, but otherwise this edition, which is probably that of 1504, is line for line identical with that of 1501. ANTON BARTHOLOMEO DI PAXI. Ed. pr. 1503. Venice, 1503. PAST. A Venetian writer of the fifteenth and sixteenth centuries. Title. ' Tariff a de pexi e mesvre.//con gratia et privilegio.' (F. i, r.) ' Prohemio del prestantissimo miser Bartho-//lomeo di Paxi da Venetia.' (F. I, v.) ' Qvi comincia la vtilissima opera chiama-//ta taripha laqval tracta de ogni sorte // de pexi e mi- svre conrispondenti per tvto // il mondo fata e composta per lo excelen//te et eximio miser Bartholomeo di Paxi da// Venetia.' (F. 2, r. Fig. 45.) Colophon. ' Stampado in uenefia per Albertin // da lifona uer- cellefe regnante lin-//clyto principe mifer Leonardo lo//reduno. Anno domini. 1503. A di//26. del mefe de luio. Finis.' (F. k 5.) Description. 4, 15.5 x 21.1 cm., printed in double columns, each being 5.3 x 15.9 cm. I56ff. unnumb., 33-38 11. Venice, 1503- Editions. Venice, 1503, 4 (here described); ib., c. 1510, 4 (p. 79); ib., 1521, 8; ib, 1540, 8 (p. 79); ib., 1557, 8 (p. 80). The book is not a textbook on arithmetic, but a collection of infor- mation useful to merchants, relating to the measures of weight, value, length, etc., of the various cities and countries with which Venice had trade relations. It is valuable as leading to an understanding of the contemporary arithmetics of Italy, and historians could find much useful information as to the prices and the material of trade by examining this and similar works. An inspection of Fig. 45 will give some idea of the scope of Paxi's Tariffa. KARA AR1THMETICA COMINQA LA VTIL1SSIMA OPERA CHIAMA, TARIPHA LACLVAL TRACTA DE OGNI SORTE DE PEXI E MISVRE CONR1SPONDENT1 PER TVTO IL MONDO FATA E COMPOSTA PER LO EXCELEN TE ET EX1M1O MISER BARTHOLOMEO Dl PAX1DA VENETIA. AVERemoaduqjprima adechiararea uoftre excelletie tutte lerobe che fe ueda no f Venetia a pexo groflb & quelle che (c uedano a pexo Ibtile e de lordine di pe xi de li arzenti;& de le coditionc di pan> ni de lane francefche fatte in Venetian etiam dele conditione di panni de feda: e panni doro:& del ordine del uedere de le fpecie;& de le fue tare;& del ordine di pcxi dele farine e faifchoti; &del ordine de le mifure di uini:& del ordine del uen der del oio:& in che modo e pexo fe uen deno iguadi:& del ordine di frutti che fe ucndeno a nome di fter: c che pexo hano cadaun ftene come refponde el pexo groflb con tu ta la Jtalia:& tuto el leuante & ponente;& etiam come ipexi fubti^ li c pcxi groffi refpondeno con molte terre de Italia : de dalmatia e de leuante;^ come refpondeno le mefure di panni de lana con W' ta Italia;e con tuto el Ieuante;& etiam come refpondeno le mefu* re di panni de feda;e panni doro e darrcnto con tuta ltaha:& co tu to el leuante & ponenta& come refpondeno le mefure dingilterra 2oc la uirga da londrare de la taripha dalixadriat& etiam quella de damafcho di la taripha da leppo.e come torna la fporta dalixadria con molte terre de leuante e de ponete e de la ltalia;& come el can tcr forfori dalixandria refponde con alchune akre terre de leuante de ponente e de Jtalia;& etiam come refponde el canter zeroui con alchune terre del leuante del ponente e de ltalia;& come refpode cl cento de le mene con alchune terre del leuante e molte de Italia & del ponente:& come refpondeno tuti icantera de leuante e del po> nente con el pexo fubtile da Venetia: & come refpode el cnnrer da napoli de reame con moke terre del leuante del ponete e con mol- te de ltalia:& come refponde el canter de Conflatinopoli con mol' tc lenc de leuante & etiam de Italiajfrcome refponde el canter de FIG. 45. THE BEGINNING OF THE 1503 PAXI a li PRINTED BOOKS 79 ANTON BARTHOLOMEO DI PAXI. Ed. pr. 1503. S. 1. a. (Venice ?, c. 1510). See p. 77. Title. 'Tariffa De Pefi e mefure cor-//refpondenti dal Le uante al Ponete : da // vna terra a laltra : e a tutte le parte del // mondo : con la noticia delle robe // che fe trazeno da vno Paefe //per laltro. Nouamente // co diligentia Ri-//ftampata .-. *J< ' (F. I, r.) Colophon. ' C.Finiffe il prohemio de // Miffer Bartholomio di // Pafi da Venefia.// Finis.' (F. 218, v.) Description. 8, 10.2 X 15.1 cm., printed in double columns, each being 4. i X 12.5 cm. 218 ff. numb., 30 11. S. 1. a. (Venice ?, c. 1510.) See p. 77. This is the second edition of this popular ' Tariffa.' Since it was one of the first books of its kind to appear in Venice, its five editions are easily explained. ANTON BARTHOLOMEO DI PAXI. Ed. pr. 1503. Venice, 1540. See p. 77. Title. ' Tariffa // de i pesi, e misvre // corrifpondenti dal Le- uante al Ponente : // e da una terra, e luogo allaltro, qua fi p // tutte le parti dil Mondo : con la dichia-//ratione, e notificatione di tutte le robbe : // che fi tragono di uno paefe per laltro.// Composta per M. Bartholomeo di // Pafi da Vinetia. Con la fua // Tauola copiofiffima, e f aci-//liffima a trouare ogni cofa // per ordine, nuouamete // fatta : e con fomma // diligeza reuista,//e stapata.// In Vinetia. M.D. XL.' (F. i, r.) Colophon. * In Vinegia. Nelli cafe di Pietro di Nicolini da Sabbio.//Ne glianni dilla falutifera Circoncifione dil no-//ftro Signore. M. D. XL.// Dil mefe di Genaio.' (F. 212, r.) Description. 8, 10.3 x 14.7 cm., printed in double columns, each being 3.5 x 12.3 cm. 1 1 ff. unnumb. + i blank + 200 numb. = 212 ff., 30 11. Venice, 1540. See p. 77. 8o RARA ARITHMETICA ANTON BARTHOLOMEO DI PAXI. Ed. pr. 1503. Venice, 1557. See p. 77. Title. The title page is practically identical with that of the 1 540 edition, except for the date : ' In Vinegia per Paolo Gherardo. //M. D. LVII.' (F. i,r.) Colophon. ' In Vinegia per // Comin da Trino.// M. D. LVII.' (F. numbered 200, r.) Description. 8, 10.6 X 15 cm., printed in double columns, each being 3. 7 x 12. 3 cm. 1 1 ff. unnumb. + 200 numb. = 21 1 ff., 30 11. Venice, 1557 Seep. 77. BOETHIUS, JODOCUS CLICHTOVEUS, AND FABER STAPULENSIS. Ed. pr. 1503. Paris, 1503. For the biographies of BOETHIUS and FABER STAPULENSIS see pp. 25, 62. JODOCUS CLICHTOVEUS was born at Nieuport, Flanders ; died at Chartres, September 22, 1543. He was educated at the Sorbonne, and was canon of Saint-Jean, at Chartres. Like Faber Stapulensis, he was known chiefly as a commentator. Title. ' In hoc libro contenta // Epitome/ copendiofaq3 // introductio in libros // Arithmeticos diui Seuerini Boetij : adie// cto familiari cometario dilucidata.// Praxis numerandi certis quibufdam re-//gulis coftricta.// Introductio Igeometria : ex libris diftlcta // Prim^ de magnitudinib^ & earu // circuftantiis.// Secudus de cofequentibus/ conti-//giiis/ & cotinuis.// (Sur- rounded by an elaborate woodcut.) C.Tertius de puctis. C.Quar- tus de lineis. CQuItus de fuperficieb^.// C.Sextus de corporibus. . Liber de quadratura circuli. C. Liber de cubica//tione fphere. CPerfpectiua introductio. Cjnfuper aftronomicon.' (F. I, r.) Colophon. ' Id opus imprefferut Volphgangus // hopilius et Henricus ftephanus // ea in arte focii in Almo pari-//fiorum ftudio Anno Chri//fti Celorum totiufq3 //nature * coditoris.// 1503. Die vice//fimafepti-//ma Iu-//nij.' (F. cxi, v.) Description. Fol., 19x26.6 cm., the text being 16x22.6 cm. 112 ff. numb., 47-54 11. Paris, 1503. PRINTED BOOKS 81 Editions. This is the first edition of this combination of works, the second (somewhat changed) appearing at Paris c. 1 507 (see below), and the third ib., 1510. The epitome appeared as a ' Compendium arithmetices Boethii,' s. L, in 1480, and, with the arithmetic of Jordanus, at Paris, in 1496. There was an edition of Faber Stapulensis, Clichtoveus, and others, at Cologne, c. 1515, 4. Scheubel published an edition at Basel in 1553, and a work entitled ' Arithmetica Boethi epitome acced. Christiani Morisani Arithmetica ' also appeared at Basel in 1553 (pp. 27, 182, 260). The 'Praxis numerandi ' of Clichtoveus, of which this is the first edition, appeared separately at Paris in 1510, fol. The copy here described is bound with the 150910 edition of Bovillus (see p. 89). It consists of a brief introduction by Faber Stapulensis, and the arithmetic of Boethius with the commentary of Clichtoveus on Faber's epitome. This is followed by the geometry and perspective of Faber Stapulensis. BOETHIUS, JODOCUS CLICHTOVEUS, AND FABER STAPULENSIS. Ed. pr. 1503. Paris, c. 1507. See p. 80. Title. Tntroductio//Jacobi fabri Stapulefis in Arithme//cam Diui Seuerini Boetij pariter i Jordani // Ars fupputadi tarn per calcu-//los qj notas arithmeticas fuis quidem regulis elegater expreffa // Judoci Clichtouei Neoportuenfis.// Queftio haud indigna de numeroru // et p digitos i p articulos finita pgreflione ex Aurelio Auguftino // C,Epitome rerum geometrical^ ex Geometrico introductorio // Caroli Bouilli.// CDe quadratura Circuli Demonftratio ex Campano.' (F. i, r.) Description. 4, 12.6 X 17.4 cm., the text being 10.6 X 14.5 cm. 32 ff. unnumb., 45-46 11. Paris, c. 1507. The dedicatory epistle is dated * Data ano falutis // noftre Millefimo qngetefimo feptimo tercio caledas lunij,' that is, 1507. It was evidently printed at Paris, but it is without date or place. Editions. See above. 82 RARA ARITHMETICA The introduction by Faber Stapulensis to the arithmetic of Boethius and Jordanus was very popular in the university of Paris at the opening of the sixteenth century. It is, like Boethius, purely theoretical. The author begins with a dissertation <de vtilitate arithmetice difcipline,' and then gives an epitome of the two works. This is followed by the ' compendium ' of Clichtoveus, merely a set of rules for the operations. Books of this character, evidently intended as the bases of lectures to university students, show in what a hopeless state the Boethian arith- metic found itself at the end of the Middle Ages. GREGORIUS REISCH. Ed. pr. 1503. Strasburg, 1504. Born at Balingen, Wurttemberg; died at Freiburg, 1523. He was a stu- dent at Freiburg in 1487, and took his bachelor's and master's degrees there. He then entered the Carthusian order and became prior of the cloister at Freiburg, and confessor of Maximilian I. Title. ' Aepitoma omnis phylosophiae. ali-//as Margarita phy- losophica tractans // de omni genere fcibili : Cum additionibus : Qu^ in alijs non habentur.' (Large woodcut representing the liberal arts. F. 2, r. Plate II.) Colophon. ' Explicit phylofophica Margarita. Caftigatione acri // In nobili Heluecioru ciuitate Argentina Chalchogra-//phatu : Per loanne Gruninger Ciue Argetinu : I vigilia // Mathie; Anno incarnationis Saluatoris M.ccccc.iiij.// Valete & Plaudite.' (F. 289, v.) Description. 4, 15.1 x 20 cm., the text being 11.5 x 15.6 cm. 2 ff. blank + 289 unnumb. = 291 ff., 45 11. The illustra- tions are hand-colored. Strasburg, 1 504. Editions. Freiburg, 1503, 4; Strasburg, 1504, 4 (here described) ; Freiburg, 1504, 4; another edition, s. 1., by Schott (Freiburg), 1504; Strasburg, 1508; Basel, 1508, 4 (p. 83); ib., 1512, 4; Strasburg, 1512, 4; -ib., 1515, 4; Basel, 1517, 4; Paris, 1523 (first Finaeus edition) ; Basel, 1535, 4 (p. 84) ; ib., 1583,4; Venice, 1594; ib., 1599; ib., 1600. The three Venetian editions (1594, 1599, 1600) are Italian translations by Giovanni Paolo Gullucci, and contain the additions by Orontius Finaeus, and also the introduction by Faber Stapulensis to the arithmetics of Boethius and Jordanus (see p. 62), Clichtoveus A. GEOMETRY B. ARITHMETIC PLATE II. FROM THE MARGARITA PHILOSOPHICA PRINTED BOOKS 83 on arithmetic (see p. 80), and Questione di S. Agostina della progressione dei numeri per li digiti, et per li articoli.' Hart- f elder (Zeitsch. f. Gesch. des Oberrheins, II, 170) has shown that the assertion of Hain, Poggendorff , and others, that it appeared in 1496, is incorrect. This was the first modern encyclopedia to appear in print. It contains a compendium of the trivium, the quadrivium, and the natural and moral sciences. It is made up of twelve books, of which the fourth, consisting of fifteen folios in the present edition, is on arithmetic. The author first considers the definition of arithmetic, and then gives the mediaeval classification of number, including the system of ratios as set forth by Boethius and his followers. The second part of the work contains a short treatment of algorism, including the fundamental operations and roots. The third tractatus relates to common fractions and the fourth to physical or sexagesimal fractions. The arithmetic closes with a treatment of line reckoning, giving the four fundamental operations and the rule of three. The illustrations are particularly interesting. (See Plate II.) Other works 0/1503. Boethius, p. 27, 1488; Bradwardin, p. 61, 1495; Faber Stapulensis, p. 62, 1496; Huswirt, p. 74, 1501; Jor- danus, p. 62, 1496 ; Orbellis, p. 23, 1485 ; Peurbach, p. 53, 1492 ; Sacrobosco, p. 32, 1488 ; Torrentini, p. 76, 1501 ; Anonymous, 'Textus arithmetical comunis, cum Conradi Norici commentatione,' Leipzig, fol. Works 0/1^04. Anianus, p. 35, 1488; Bradwardin, p. 61, 1495 ; Johannes Carolus (see Landshut, below) ; Huswirt, p. 74, 1501 ; Reisch, p. 82, 1503; Sacrobosco, p. 32, 1488; Johann Karl von Landshut (Lanzut), ' Algorithmus integrorum,' Leipzig (see also p. 97, 1513, 1515); Henricus Stromer, 'Algorithmus linealis cum Regula de Tri,' Leipzig, 4, with other editions in 1510; 1512, 4; 1514; Leipzig, 1516, 4; ib., 1517, 4; 1520. GREGORIUS REISCH. Ed. pr. 1503. Basel, 1508. See p. 82. Title. ' Margarita philofophica // cu additionibus nouis : ab auctore fuo // ftudiofiffima reuifioe tertio fup additis. // Jo. Schottus Argen. lectori. S.// Hanc emo/ non preffam mendaci ftigmate/ Lector :// Pluribus aft auctam perlege : doctus eris.// Bafilee. 1508 ' (F. i, r.) 84 RARA ARITHMETICA Colophon. ' CTertio induftria complied Micha//elis Furterij/ et Joanis Scoti//ftudiofiifiime preffa. Ba-/Aileg.i4-Kal'.Mar// tias. Anno Chrifti.// 1508.' (F. 308, r.) Description. 4, 15.2 x 21.9 cm., the text being 12.7 x 16.7 cm. 309 ff. unnumb., 42 11. In this edition the leading initials are inserted by hand and the illustrations are colored. Basel, 1508. See p. 83. GREGORIUS REISCH. Ed. pr. 1503. Basel, 1535. See p. 82. Title. ' Marga-//rha philosophica, rati-//onalis, Moralis phi- lofophiae princi-//pia, doudecim libris dialogice cople-//ctens, olim ab ipfo autore recognita : // nuper aut ab Orontio Fineo Delphi//nate caftigata & aucta, una cum ap-//pendicibus itidem emedatis, & qua // plurimis additionibus & figuris, ab//eodem infignitis. Quoru omni-//um copiofus index, uerfa // continetur pagella.//Virefcit uulnere uirtus.//Basileae 1535.' (Surrounded by an elaborate woodcut.) (P. i.) Colophon. ' Basileae excvdebat Henricvs // Petrus, ac Con- radi Refchij impenfis. An//M. D. XXXV.' (P. 1577.) Description. 4, 15 x 20.8 cm., the text being 9.6 X 13.8 cm. 78 pp. unnumb. + 1498 numb. -f I blank = 1577 pp., 26-30 11. Basel, 1535. Editions. See p. 82. Finaeus dates the dedicatory epistle ' Parifijs ex regali collegio Nauarrae. 1523,' and his first edition appeared in that year. This edition gives only part of the elabo- rate engravings found in the earlier ones. It is, however, much better printed, being set in clear Roman type and having a more open page. See p. 83. THEODOR TZWIVEL. Ed. pr. 1505. Cologne, 1507. A German arithmet Westphalia. Title. See Fig. 46. A German arithmetician of c. 1500, from Monte Gaudio (Mongavensis), Westphalia. PRINTED BOOKS 85 Colophon. 'C. Algorithm!, qui ars dicitur numerandi. de integris // per figurarum (more Alemanorum) deletione3. Nec//no de pportionib? ingeniofi Pythagorifte Theodo//rici Tzwyuel. poft plurima praxin iam tande in hoc // opendiu3 reducti finis adeft. quod et puplica ob vti//litatem in magiftrali artis impreffori e taberna inge-//nuorum liberorum Quentell iterate diffeminari pro mtymmu opufralatmo ud t>enumcro?p:a]ci(att4a(go2itt?mL&fcunf)vnutt mtegris per ftguraru(mo:e alrniano*)t>derione. Sterufcc p^pomombws en ia vftia frcqucns m nrnficS tarrnom'ca Saitruu Bomj Dcvtilitatchuiusiibdli Tctraftichoa ioannisMurmelli] Ruremundenfis Siquis arithmetic^ optatcognofcere praxin pythagorcnumefosdifccrcfi quis amat De fophiahunc modicum cornparet ere libra FIG. 46. TITLE PAGE OF TZWIVEL //curauit. Anno a natali dominico Millefimo quin//genetefimo- feptimo.' (F. 9, v. See Fig. 47.) Description. 4, 13.9 X 20 cm., the text being 8.8 X 14.5 cm. 10 ff. (i blank), 46 11. Cologne, 1507. Editions. Giinther mentions a Miinster edition of 1505, but I have not seen it ; there was a Cologne edition in the same year; Cologne, 1507, 4 (here described). The work is divided into two parts, the first beginning as follows : ' C.Algorithmus de integris p figuraru (more alemano^z) deleti-//one 86 RARA ARITHMETICA arte numeradi enucleatim ,3pendiofeq5 edocens.' (F. 2, v.) This part contains a brief explanation of the writing of numbers and the funda- mental operations. The second part begins as follows : C.Algorithm9 de pportioib? cuius vfus freques in muficam har//monicam Seuerini Boetij ' (f. 8, v.), and two pages treat of the operations with the mediaeval ' proportiones ' or ratios. A comparison of the title pages represented on pp. 45, 75, 85, and of the works to which they belong, at least two of which were printed in Cologne, leads to the belief that the expression ' per figuraru (more alemanoqz) deletione ' (by the deletion of figures in the German way) refers to a contemporary North German custom of not actually canceling the figures in the galley division, as the Italians did. Other works of 1505. Anonymous (see Licht), p. 70, 1500 ; Borghi, p. 16, 1484; Bradwardin, p. 61, 1495 ; Ciruelo, p. 60, 1495 ; Licht, p. 70, 1500; Suiseth, p. 10, c. 1480; Tor- CBIrtO!l't|?mi'.qm'arflbfdnirnijmer9ndlKIHWgTrfl rpntini r> nf\ rcnr Anrmvmnns prfifiurarum(mo:caicmano:um)alct.o..3.-nec im P ' 7 ' I5QI ' Anonvmous n6oyjpo:tionitoingcnio(i*pytl?a0o:ift<Xbwc' ' Tractatus perutilis in arithme- ridi>W)TUel.pOftplurim3pJtmiamfandcin(?OC f - cr^nlat;,,,, ' Paric W^rTYior. pptndmjrtductiftmeadcft.ciuodetpuplicaobvtt tlca speculativa, Fans; Herman- Uuuminmafiif!raliarn8imp:efrojictalxniflin^ nus Buschius, 'Enchiridion novus nuo7umtbrre:umQiicnrdliraatoOincmmarip:o A1 . , ~ i / \ o curamt. 3nno a narali Oominico ZPilUfimo quin Algonsmi, Cologne, s. a. (i 504), 4 flemcfimofcpumoi ( see also p. 106, 1514); Nicolaus FIG. 47. COLOPHON OF TZWIVEL Horem, 'Tractatus proportion^,' Venice, in a volume with Bradwar- din and Suiseth (see also p. n 6, 1515);* Algorithmus linealis Baccalariu Wolfgangum Monacenseni,' Leipzig, 4 ; Georg Leunbach, an arithmetic. Works of 1506. Albert of Saxony, p. 9, c. 1478 ; Anonymous, p. 10, 1480; Vittori, p.-9, Albert of Saxony, c. 1478, and p. 61, Brad- wardin, 1495; Pietro Borriglione, 'Arismetices praxis,' Turin, 22 ff., with a second edition, ib., 1523 ; Raphael Maffei, ' Commentarii Urbani,' Rome, fol., an encyclopedia containing a book (no. 35) De scientiis mathematicis,' which includes a little arithmetic ; Maffei's work was also printed in 1527, and at Paris in 1511, 1515, 1526, and 1530, and at Basel in 1559. ANONYMOUS. Ed. pr. 1507. Leipzig, 1507. Title. See Fig. 48. Colophon. ' C.Impreffum Liptzck per Baccalariu Vuolfgangu // Monacenfem Anno noftre redemptionis .1507.' (F. 27, r.) Description. 4, 14.5 x 19.5 cm., the text being 9.5 X 15.3 cm. 28 if. unnumb. + i blank = 29 ff., 28-34 11. Leipzig, 1507. PRINTED BOOKS 87 Editions. Leipzig, 1507 (here described) ; ib., 1509,4; s. 1. a. (Niirnberg ?, c. 1510). This resembles several of the works on algorism appearing about this time, such as Widman's (?), Licht's, and Huswirt's. It contains a very brief treatment of the fundamental operations, including duplation and mediation. In division, only a single example is given, that of 1456 -H 1 2 . After a similarly brief treatment of fractions, the Boethian proper- FIG. 48. TITLE OF THE 1507 Algorithm** tions (ratios) are taken up : * Sequitur Algorithmus proportion^.' Then follow the Rule of Three and several other rules now entirely forgotten, such as ' Regula legis,' ' Regula augmenti,' Regula plurima,' ReguU pulchra,' and ' Regula falfi.' Such ' regulae ' were not stated like modern rules but consisted of groups of similar problems. Other works of 1507. Boethius, p. 27, 1488; Faber Stapulensis, c. 1507, p. 81, 1503 ; Huswirt, p. 74, 1501 ; Peurbach, p. 53, 1492 ; Tor- rentini, p. 76, 1501. Works 0/1508. Anianus, p. 32, 1488 ; Reisch, p. 82, 1503 ; Widman, p. 39, 1489 ; Hieronimus de Hangest, * Liber proportionum,' Paris, 4. LUCA PACIUOLO. Ed. pr. 1509. Venice, 1509. See p. 54. Title. After a vocabulary and index the work begins on f. I, r., as follows : * Excellentiffimo principi Ludouico mariae Sfor. Anglo Mediolanen//fium duci: pacis et belli ornamento fratris Lucae pacioli ex Burgo fancti // Sepulchri ordinis Minorum: Sa- crae theologiae pfefforis. De diuina pro//portione epiftola.' 88 RARA ARITHMETICA FIG. 49. FROM THE 1507 AtgoHthmus PRINTED BOOKS 89 Colophon. ' CVenetiis Impreffum per probum virum Paganinum de paganinis de // Brixia. Decreto tamen publico vt nullus ibidem totiqj dominio annorum//XV. curiculo imprimat vel iprimere faciat. Et alibi impreff um fub quouis // colore in publicum ducat fub penis in dicto priuilegio contentis. Anno Re//demptionis noftre.M.D.VIIII. Klen. lunii. Leonardo Lauretano Ve.//Rem.Pu. Gubernante. Pontificatus Iulii.II. Anno.VI.' (F. 27, r., of Part 3. A similar colophon appears at the foot of f. 35, v., of Part i.) Description. 8, 20.5 X 28.3 cm., the text being 9.1 X 21.5 (with marginal drawings). I f. blank + 2 unnumb. + 90 numb. 93 ff., 52-57 11. Venice, 1509. Editions. There was no other edition. With some hesitancy this book has been included, the only justifica- tion being the fact that there are several pages devoted to the discussion of proportion in general, including the arithmetical, geometric, and astronomic. Paciuolo excludes the other forms of ancient proportion given by ' Platone e Arifto. e yfidoro i le fue ethimologie. El feuerin Boetio in fua arithmetica.' (F. 5, r.) Most of the treatise is, however, devoted to geometry. CAROLUS BOVILLUS. Ed. pr. 1509-10. Paris, 1509-10. BOUVELLES, BOUELLES, BOUILLES, BOUVEL. Born at Saucourt, Picardy, c. 1470 ; died at Noyon, c. 1553. Canon and professor of theology at Noyon. Title. See Fig. 50. Colophon. ' Libelli De Mathematicis Svpple-//mentis Finis Anno Salutis Humane 7/1509 lanuarij Die Deci//maoctaua // C.Editum est vniversvm hoc volvmen Ambianis in edibvs Re// uerendi in Christo Patris Francifci De Hallevvin Eiufdem Loci Pontificis .-. // Et emiffum ex officina Henrici ftephani. Impenfis eiufdem et loannis parui in chalcotypa // arte fociorum Anno Christi Saluatoris omnium 1510. Primo Cal. Februarij.// Parisiis.' (F. 198, v.) Description. Fol., 19.7 X 27.4 cm., the text being 15.1 X 25.8 cm. 198 ff. numb., 53-54 11. Paris, 1509-10. Editions. Fontes has described (Toulouse Mem. (9) VI, 155- 167, for 1894) a quarto of 1510, published by Stephanus of 4[Quehoc volum/ne continetur. Liber dejntelk<2u. JLiberdefenfu, Liberdenichilo. ATS oppofitorum* Liber dcgeneratione. Liber de fapiente. Liber de duodecim num en's Epflolecomplwes. ITlnfupmathfmatfcuopus quadr^ Mathcmatic/s Rods |fDe Geometricis Corporibus d[De Gcometrids Supplementi's FIG. 50. TITLE PAGE OF BOVILLUS PRINTED BOOKS 91 Paris, entitled, ' Caroli Bovilli liber de numeris perfectis.' This book is, however, without title page, and is of course simply ff . 172-180 of this work. I know of no other editions. The first part on numbers, the Liber de duodecim numeris,' begins on f. 148, v., and ends on f. 171, r., with the colophon : ' Libri dvodecim nvmerorvm finis : editi // in domo- R. P. Francifci de Hallewin/ Pon- tificis Ambi//anenfis. Anno ab autore numerorum Incarnato.// 1 5 1 o : Maij decimafexta.' It relates solely to the mystery of numbers and to the Greek theory. The part on perfect numbers, numbers which equal one plus the sum of their factors, begins on f. 172, r., and ends on f. 1 80, r., with the following colophon : ' C.Liber perfectorvm nvmerorvm finis.// Perfecto/ trinoq5 deo laudes in gentes. Anno domi-//ni/ 1509 : lanuarij 4.' The dates of the several parts vary from October 25, 1509, to January 18, 1509 (1510 N. S.). Other works 0/1509. Anianus, p. 32, 1488; Anonymous, p. 87, 1507 ; Borghi, p. 16, 1484 ; Ciruelo, p. 60, 1495 '> Lkht, p. 70, 1500 ; Sacrobosco, p. 32, 1488. Other works of 1510. Anonymous, p. 87, 1507 ; Anonymous, c. 1510, p. 46, 1491 ; Boethius, p. 30, 1488 ; Bovillus, p. 89, 1509 ; Brad- wardin, c. 1510, p. 61, 1495 ; Clichtoveus, p. 81, 1503 ; Faber Stapu- lensis, p. 62, 1496 ; Jordanus, p. 62, 1496 ; Peurbach, p. 53, 1492 ; Sacrobosco, p. 32, 1488 ; Stromer, p. 83, 1504 ; Paxi, c. 1510, p. 79, 1503 ; Torrentini, p. 76, 1501. Works of 1511. Anianus, p. 32, 1488; Boethius, p. 27, 1488; Maffei (Maphseus), p. 86, 1506 ; Peurbach, p. 53, 1492 ; Simon Eisen- mann, Enchiridion arithmetical Leipzig, fol. JUAN DE ORTEGA. Ed. pr. 1512. Rome, 1515. JOHN DE LORTZE. A Spanish priest of the Dominican order, from Aragon. He was still living in 1567. Title. ' Svma // de Arithmetica : Geometria // Pratica vtilifTi- ma: ordina//ta per Johane de Or//tega Spagnolo//Palentino.// Cum Priuilegio.' (Surrounded by an elaborate woodcut. F. I, r.) Colophon. ' Impreffo in Roma per Maftro Stephano Guilleri de Lorena // anno del noftro Signore 1515 adi 10 de Nouebre Regnante Leo//ne Papadecimo in fuo Annotertio.' (F. 1 16, r.) Description. Fol., 20.6 X 30.2 cm., the text being 13.1 X 22.7 cm. 2 ff. unnumb. + 114 numb. = 116 ff., 32-38 11. Rome, 1515. KARA ARITHMETICA ^nelfequenre tractate fedc ~ moftraracomo feaoafa reogm barattocoiig tempo co mo fen 5a tcpo cbc prcne a lartc mcrca tilepoiuer ft modi* FIG. 51. FROM THE 1515 ORTEGA, INTRODUCTION TO BARTER PRINTED BOOKS 93 Editions. Barcelona, 1512; Lyons, 1512, 4 ; ib., 1515, 4 ; Rome, 1515, fol. (here described); Messina, 1522; 1534; Seville, 1536; ib., 1537; Paris, 1540 (?) ; Seville, 1542, 4 (see below); ib., 1552 (p. 94); s. 1. (?), 1552; Granada, 1563, 4. The Lyons edition of 1512 was the first book on commer- cial arithmetic printed in France. It differs somewhat from the Rome edition of 1515, but the latter differs only a little from the first (Barcelona) edition. Mr. Plimpton's copy of 1515 belonged to Prince Boncompagni and has his collation on the cover. It is beautifully printed and is one of the best examples of the early Italian mathematical typography. This is one of the most celebrated arithmetics written in Spain in the sixteenth century. It is a purely commercial book, beginning with notation, taking up the four processes with integers, the progressions, the roots, and the checks on operations, and the same operations in the same order with fractions, and then discussing the business rules. These last include exchange, rule of three, profit and loss, partnership, testament problems, barter, alloys, false position, and a little mensuration. Other works of 1512. Bradwardin, p. 61, 1495 ; Peurbach, p. 53, 1492 ; Reisch, p. 82, 1503 ; Stromer, p. 83, 1504; Torrentini, p. 76, 1501 ; I. Furst, ' Novus . . . algorithmus.' JUAN DE ORTEGA. Ed. pr. 1512. Seville, 1542. See p. 91. Title. ' Tratado // fubtiliffimo de Arifmeti-//ca y de Geome- tria : co-//puefto y ordenado // por el reuerendo // padre fray Jua // de Ortega d' //la orden d' // los pre//dicadores.// 1542 // 1234567890.' (The whole is surrounded by an elaborate wood- cut border.) (F. I, r.) Colophon. ' Sue impreffo el prefente libro // re Arifmetica y Geometria (agora nueuamete // corregido y emendado) en cafa d' Jacom // croberger: enla muy noble y muy leal // ciudad de Seuillara cinco dias//de deziembre de. M.d. y//quareta y dos afioz.' (F. 232, v.) Description. 4, 14.5 x 20.5 cm., the text being 11.3 X 17.2 cm. 232 ff. numb., 34 11. Seville, 1542. See above. 94 RARA ARITHMETICA JUAN DE ORTEGA. Ed. pr. 1512. Seville, 1552. See p. 91. Title. * Tractado // Subtiliflimo d'Arifmetica y de Geome// tria. Compuefto por el reueredo padre // fray Juan de Hortega, d'la orden // de los predicadores.// Elhora de nueuo emendado con mucha // diligetia por Go^alo Bufto d'muchos//errores que auia en algunas im//preffiones paffadas.// C.Van anadidas en efta impreffion las // prueuas defde reduzir hafta partir que- //brados. Y en las mas de las figuras de // geometria fus prue- uas, con ciert os aui-//fos fubjetos al Algebra. Y al fin defte tra //ctado : 13. exemplos de arte mayor.// 1552.' (Title page is printed in red, and is surrounded by an elaborate woodcut in black.) (F. i,r.) 1 Colophon. ' Hizo fin el tractado de Arifmetica Y // Geome- tria, que compufo y ordeno el reuerendo padre //fray Juan de Hortega, de la orden de los predica-//dores. Fue impreffo ela muy noble i muy leal //ciudad de Seuilla por Jua canalla, enla // collacion de fant Jua. Acabofe a diez//y feys dias del mes de Abril del // ano de nueftro criador y rede//ptor Jefu Christo de mill Hi quinientos i cin-//quenta y dos //anos.' (F. 223, r.) Description. 8, 14.6 X 20 cm., the text being 1 1.9 x 17.4 cm. 232 ff. numb. + 7 unnumb. = 239 ff., 33 11. Seville, 1552. See p. 93. The ' arte mayor ' mentioned in the title is algebra. JODOCUS CLICHTOVEUS. Ed. pr. 1513. Paris, 1513. See p. 80. Title. See Fig. 52. Colophon. ' C.Expletum eft hoc opufculum & ex officina emif- //fum/ in alma Parifiorum academia : anc domi-//ni (qui omnia numero definiuit) decimoter//tio fupra millefimu & quingente- fimu/ // decimafexta die Decembris. Per // Henricu ftephanu/ artis excu//fori libroru fedulu & indu//ftriu opifice/ e regione //fchole Decretoru //habitan-//tem.' (F. 43, v.) Description. 4, 14 x 19.5 cm., the text being 12 x 15.7 cm. 41 ff . numb. 4- 3 unnumb. + 2 blank = 46 ff., 42 11. Paris, 1513. PRINTED BOOKS 95 Editions. There was no other edition. This is, I believe, the first separate treatise on the mystery of numbers to appear in print. Paciuolo had included a good deal of such material in his Summa of 1494, and about a century later Bungus published a monumental treatise upon the subject, but Clichtoveus was a pioneer in the publication of a separate work. The result of his labors is properly |[DcmyflicanumcrorumGgnificationc opu* fculutnteorum pr gfcrtim qui in facris lit* ten's vfi tati habentur/fpirituale ipfo* rum defignationemfiiccina &e elucidans, V FIG. 52. TITLE PAGE OF THE 1513 CLICHTOVEUS included in a list of arithmetics, for, while there is nothing of compu- tation in the work, it is not unrelated to the number theories of the mediaeval writers and even of the Pythagoreans. Clichtoveus discusses, as is usual among such writers, the religious significance of one (' Quid vnitas/ numerorum fons et origio defignat. Cap. I.') and the numbers of the first decade. He also mentions several larger numbers which were supposed to have some scriptural significance, not forgetting, of course, 666, * the number of the beast.' There is also in this work a chapter, generally unrecognized by writers on the history of the subject, on finger-reckoning : ' Quomodo antiqui: numeros omnes per certas digitoru & manuum figurationes/ fignificare funt foliti.' Cap. XXVIII. JOANNES MARTINUS BLASIUS, Villagarciensis. Ed. pr. 1513. Paris, 1513. A Spanish astrologer and arithmetician of c. 1 500. In this edition the author's name appears as 'loannes Martinus Blafius diocefis Pacefis,' and in the 1519 edition as 'loannes Martinus Silecevs (and Sciliceus) Diocesis Pacefis.' Title. See Fig. 53. Colophon. 'Explicit liber Arithme//tices practice magri Joan- nis Martini Blafij Vil-//lagarciefis : Parifijs edit9 in honeftiffima Belua-//coru paleftra : impreffus vero a calcographorum ex-// pertiffimo Thoma Kees : Vvefalienfe expenfis pro//biffimorum 9 6 KARA ARITHMETICA iaronbus aDmoDum ftttlis, FIG. 53. TITLE PAGE OF THE 1513 BLASIUS PRINTED BOOKS 97 virorum : Joannis Parui et Joannis // Lambert. Anno domini. 1513. in vigilia diui Jo-//annis baptifte.' (F. 26, r.) Description. Fol., 19.5 X 28.1 cm., printed in double columns, each being 6.5 X 21.3 cm. 26ff. unnumb., 64 66 11. Paris, 1513. Editions. Paris, 1513, fol. (here described); ib., 1514, large 8; ib., 1519 (see below); ib., 1526, fol. Although an algorism, the work is mediaeval in character. The author first discusses the fundamental operations with integers, including series and roots as was the custom, but not considering duplation and media- tion as distinct topics. He is one of the earliest writers to adopt the spelling substractio, for subtraction, a custom more or less followed by the Dutch and English arithmeticians for several generations. Blasius closes his ' primus tractatus ' with a discussion of compound numbers. ' Tractatus secundus ' considers computations with counters, or ' nummi supputatorii.' The ' tertius tractatus ' is devoted to common fractions, ' fractiones vulgares ' ; the ' quartus tractatus ' to sexagesimal fractions, ' fractiones phisicae ' ; and the ' quintus tractatus ' to the rule of three (' Prima regula i fudametalis qua detri dicunt.') There are no practical applications of any value. This first edition differs greatly from the third (1519) described below. It includes only the algorism, while the latter consists of two parts, the first being on Boethian arithmetic, and the second being substantially identical with the 1513 edition. Other works of 1513. Albert of Saxony, p. 9, c. 1478; Ciruelo, p. 60, 1495 ; Gyraldus, p. 254, 1553 ; Licht, p. 70, 1500 ; Peurbach, p. 53, 1492 ; Johann Karl von Landshut (Lanzut), 'Algorithmus linealis,' Cracow, with editions, ib., 1515, 1519, 4 (see p. 83, 1504). JOANNES MARTINUS BLASIUS, Villagarciensis. Ed. pr. 1513. Paris, 1519. See p. 95. ORONTIUS FINAEUS, editor. See p. 160. Title. ' Arithmetica // loannis // Martini, Scili-//cei, i n the- oricen, et praxim // fciffa, nuper ab Orontio Fine, Del//phinate, fumma diligentia caftigata, Ion-//geq3 caftigatius q prius, ipfo cura-//te impreffa : omni hominu // condition! perq // vtilis,// & neceffaria.// Virefcit vulnere virtus.' Surrounded by an elab- orate woodcut, with the following on four sides : * Emissa ex 98 RARA ARITHMETICA officina Henrici Stephani, e regione // scholae Decretorvm Com- morantis,// vbi et vaenalis reperitvr. // Parisiis anno Christi. 1519.') (F. i, r.) Description. Large 8, 20.3 X 28.1 cm., the text being 13.3 x 20 cm. 64 ff. numb., 52 11. Paris, 1519. Editions. See p. 97. The first half of this rare work, not found in the 1513 edition, is one of the best exponents of the Boethian arithmetic of the time. Finaeus, the editor, refers to the author's work in these words : ' Hanc loanes Martinus, Sciliceus, Hifpanus, vir Mathematical peritus, noftra tepeftate Parifijs edidit.' The author shows a good knowledge of the ancient writers, mentioning particularly Pythagoras, Nicomachus, Euclid, Apuleius, and Boethius, together with Jordanus, Faber Stapulensis, and Clichtoveus. The distinctive superiority of this part of the work lies in the clearness and arrangement of the illustrations of the various classes of numbers defined. The theory of numbers ends on f. 24, v., the prac- tical part beginning on f. 25, v. Among the most noteworthy features of this half of the work is a * Tabvla mvltiplicationis et divisionis ' with all products to 50 x 50. RAGGIUS FLORENTINUS. Ed. pr. 1514. Florence, 1520. A Florentine mathematician of the fifteenth century. Title * See Ctnhocopufculohcccontinentur. 54- CCiofdfitproportio"eius fpccics Colophon. ' Um- preffum Floretiae t. . Bernardum Zuc- CQuointcl!c<aucompohtfo&diuifToproportfonum , _ ,, accjpiatur&maleopinantiumconfurationcs chetta // Anno. M.D.XX. lanua- CQ. UC ttiaf r minoruc ptoportio diccnda fit rij. X V.' (F. 1 1 , v.) Description. 4, CQuiclpa^inquftas&rcmotio I3 ' 4 x 2ai cm ' the text being 9.2 CC'onfutationesargtimentorum calculator^ x 1 5 cm. 1 1 ff. FIG. 54. TITLE PAGE OF RAGGIUS unnumb., 34 - 3 7ll. Florence, 1520. Editions. Florence, 1514, 4; ib., 1520, 4 (here described). PRINTED BOOKS 9 9 This work consists of a theoretical treatment of proportion. While partly arithmetical, this treatment relates to the fundamental theory, and is equally applicable to geometry. The book is dedicated to the illustrious Giovanni Salviati, uncle of Cosimo I, Grand Duke of Tuscany. GUILLIELMUS BUDAEUS. Ed. pr. 1514. Florence, 1562. GUILLAUME BUD. Born at Paris in 1467 ; died at Paris, August 23, 1540. Son of Jean Bude, grand audiencer of France. He became secretary to Louis XII, master of requests to Fra^ois I, royal librarian, and ambassa- dor to Leo X. He was a man of great erudition, and was instrumental in founding the College de France. Title. ( Trattato // delle Monete // e Valvta loro,// Ridotte dal coftume antico, all'vlo mo-//derno, Di M. Guglielmo//Bvdeo. //Tradotto per M. Giouan Bernardo // Gualandi Florentine.// In Fiorenza ; // Apresso I Givnti // MDLXII.// Con licenza, & Priuilegio.' (P. i.) Colophon. ' In Fiorenza apreffo gli heredi di // Bernardo Giunti//is62.' (P. 318.) Description. 8, 10.5 x 17.2 cm., the text being 6.8 X 13.1 cm. 8 pp. imnumb. + 3 blank + 309 numb. = 320 pp., uncut, 28 11. Florence, 1562. Editions. The dedicatory epistle is dated * Da. Viterbo il xxx. d'Agofto. MDLXI,' so that this is the first (as it is the only) edition of Gualandi's translation of Budaeus. I have not com- pared it with the * Libri de asse et partibus ejus,' Paris, 1514; second edition, Venice, Aldus, 1522. Although on the title page it is called a translation, the various books, six in number, begin 'Trattato delle //Monete// di M. Gio. Bernardo// Gvalandi Cittadino // Florentine,' leading to the belief that it may have been rewritten by Gualandi. This treatise is purely historical, describing in a prolix manner the ancient measures, a subject of interest to arithmeticians in the sixteenth century on account of the great number of tables of denominate numbers in use in Italy, France, and Germany. There was also a work by Budaeus entitled * Minervse Aragoniae Assis Budeani supputatio compendiaia ad monetam ponderaque et mensuras Hispanic nostre,' etc., published at Saragossa in 1536, 8- ioo KARA ARITHMETICA JOHANN BOSCHENSTEYN. Ed. pr. 1514. Augsburg, 1514. BESCHENSTEIN, BOESCHENSTAIN, BOSSENSTEIN, BOECHSENTEIN, BUCH- SENSTEIN, POSCHENSTEIN, BESENTiNUS, etc. Born at Esslingen, Swabia, in 1472; died in 1532. He taught Hebrew at the universities of Ingolstadt and Heidelberg, and also at Antwerp and NUmberg. Luther and Melanch- thon were among his pupils. Title. See Fig. 55. Colophon. 'Getruckt in der Kayferlichen ftat Augfpurg durch //Erhart 6glin Anno 1514 Jar.' (F. 24, r.) Description. 4, 14.4 X 19.2 cm., the text being 8.9 x 14.8 cm. 24 ff. unnumb., 30 11. Augsburg, \ 5 14. Editions. Augsburg, 1514, 4 (here described); ib., 1516; ib., 1518. Boschensteyn is also said to have published at Augsburg in 1514 * Ein New geordnet Rechenbiichlein auf den linien mit Rechenpfennigen,' 4, but this is doubtless Kobel's work (p. 102). This is one of the more interesting of the early German arithmetics. It is mercantile in character, and presents in condensed form the essen- tials of business arithmetic. Among the peculiarities of the book is the use of ' figures ' for 'species.' Boschensteyn gives seven of these funda- mental operations : ' Das scind nun die Siben figuren,' ' Die Erft figur Numeratio,' * Die Ander figur Additio,' etc. He includes Duplatio and Mediatio, and he checks all of his work by casting out nines. His appli- cations are chiefly in the ' Regula de Try,' partnership, and ' Regula Fufti ' (where he gives his rule in verse). JAKOB KOBEL. Ed. pr. 1514. Augsburg, 1514. KOBEL, KOBELIUS, KoBlLiNUS. Born at Heidelberg in 1470; died at Oppenheim, January 31, 1533. He studied at Cracow, where Copernicus was his fellow-student. He was a man of varied attainments, meeting with success as a Rechenmeister, printer, engraver, woodcarver, poet, and public official. Title. See Fig. 56. Colophon. ' Getruckt tzu Augfpurg durch Erhart oglin.// Anno M.D.XIIIi: (F. XXIIII, r.) Description. 4, 13.7 x 18.9 cm., the text being 9x15 cm. 6 ff. unnumb. + 24 numb, (in Roman) = 30 ff., 30-35 11. Augs- burg, 1514. PRINTED BOOKS 101 &m 1ft eav gcotfwet ift ecb enbiecbUn DenangenDdifcbtKern mw fampt fcer Kegd be Crp/v tt& fecfce regclw & rejjd joptrnft vfl artbcnt gilteti fr^ gen FIG. 55. TITLE PAGE OF THE 1514 BOSCHENSTEYN 102 KARA ARITHMETICA Editions. Kobel's Rechenbuch appeared under such varied titles and in such different combinations with his other books that it is difficult to say whether a given edition is a new work or merely a revision. It will aid the student if he recognizes in the first place that Kobel wrote three distinct books, (i) the 4 Rechenbuchlein,' (2) 'Mit der Kryden,' (3) the 'Vysierbuch.' The ' Rechenbuchlein ' first appeared at Augsburg in 1514, 4; the ' Vysierbuch,' a treatise on gauging, at Oppenheim, s. a. (1515); and * Mit der Kryden' at Oppenheim in 1520. When the 1518 edition of the Rechenbuchlein (p. 108) appeared the title was changed, and a few pages were slightly altered. The 1531 edition (p. 108), however, shows many changes, cer- tain chapters being entirely rewritten, and others considerably expanded. Although bearing a similar title, this might with some justice be called a different treatise ; and yet it is so mani- festly a revision of the 1514 work that it may more properly be classed as a new edition. The three books were sometimes published as one and some- times separately. The following list of editions is, therefore, probably incomplete, and it should be understood that any book mentioned may have been published with some other one. Editions of the Rechenbuchlein : Augsburg, 1 5 14, 4 (p. 100) ; Oppenheim, 1514, 4 (p. 106); ib., s. a. (c. 1515); Augsburg, 1516,4; 1517, 4 ; Oppenheim, 1518, 4 (p. 108; 'zumDritte male gebeffert,' and hence the third revision, although at least the sixth edition); two other editions before 1520 (Unger) ; Oppenheim, 1522, 8; ib., 1525, 12; Frankfort, 1527, 8; ib., 1531, 8 (p. 108); Oppenheim, 1531 ; ib., 1532 ; ib., 1535 ; Frankfort, 1 537, 8 (p. no); ib., 1544, 8; ib., 1549; ib., 1564 (p. in); ib., 1573; 1575; Frankfort, 1584, 8. Editions of ' Mit der Kryde od' Schreibfedern/ durch die zei- ferzal zu reche // Ein neiiw Rechepiichlein/ den angenden Schu- lern d' rechnugzu ere getruckt ' : Oppenheim, 1520 ; Frankfort, !537> 8 (p. no); probably included in various other editions of the Rechenbuchlein. PRINTED BOOKS 103 mbiecblitiauf Den lituen mtr iftecbcit pfenfogcn : oat clnfev cbchcb3it lemen mit oren nt> FIG. 56. TITLE PAGE or KOBEL'S Rechenbiechlin (1514) 1O4 KARA AR1THMETICA 55 LV LVT LVIF Lvnr LIX LX LXI &ILXII ..Lxra s4Lxnn <JJ|LXV LXVI frlLXVII LXVIII 5? I LXIX 7o|LXX LXXI 72LXXII T>> LXXtH 74ILXXHII LXXV 75 7-i 77 78 8? 8} LXXVI LXXVII LXXVJII LXXfX LXXX LXXXl LXXXfl LXXXIII .LXXXIIU 85 LXXXV 85LXXXVI 87 1 LXXX VI 83,I.XXXVII aplLXXXX! 92 97 98 99 ICO 101 102 104 105 loo 107 108 10? no in 112 2CO 5co 7CO 3co 900 I coo 2CCO xc XCI XCII xcm XCIIII XCV XCVI xcvn XCVIII XC1X c CI CII cm CHH cv CVI CVII cvni cix ex CXI CXII CXIIItc cc ccc cccc VIP Vfli 1XG oco Jcco oco 7cco Bcco cco I CCCC ZOOOO i4co V* XXX* IBCCCC (TD IIIIC V 1600 I<5|2 1700 1715 i8co 1820 It, CO O>5DCC vnc vnc 2>cc vine rtitOcr Icntcif crcfc,)cr,| FIG. 57. A PAGE FROM KOBEL'S Rechenbiechlin (1514) PRINTED BOOKS 105 bet>eut bif? figur far felbett toy! afoe I ZtfefTe figur ifl t>it benefit airt (t ertcl Wrt flfoej fill gflrtsert/fllfo mag mflit fliicfc flirt furtfftail/^$ fe4)ffatl/flw ftbeittflit o&erjwflt fec^fffli'1 2c vttb die VI 3D$ fern Scd;^ ribtattfcM fern fafcefml 6ey VIII ad^t aiit g<w8 madpert . IX ?Dif? ^tgur b ct5 ^ fcjt flttrt new ft &y i! f)ttul dflO fc^fH XI lyL *tAil/6cr XI flirt 0flrt^ tnfld)crt 4 XX XXXI ftgC tflil /t)fle feirt imbrctffigt flirt gflrt^ntfl^ert li o Iei4)tU4) lemert fl!l brSc^/t>ie bit t'n beirtert revert Commcrt/nue bu bie ver(leert/ flrtfc^retbert vrtrtb flrtf? ert etlid) Kegdrt/^rflgm/vrtrtb ^mnpel in ^crt ^fllert/wtcmflrt bi frag frfrmbt/w berert bie crjl sflle gepto^ert ift/ vn& tie mittel *mb tetfte Qflle gfln^ plcibcrt/ So mufiit bte bcrt (let iDci- gclcid^cit wiupu bit fctfiQfll fltid) bert felbert bruct) vrtb gefcbic^t bfld fl b^ccrjt Vrtrtb Ictp^flle in wto ^ Orto <rtfo gcfd?ic^(/eo FIG. 58. A PAGE FROM KOBEL'S Rechenbiechlin (1514) 106 KARA ARITHMETICA Editions of the ' Vysierbuch ' : Oppenheim, s. a. (1515), 4 (p. 113); Frankfort, 1527; ib., 1531, 8 (p. 108) ; Oppenheim, 1531 ; ib., 1532 ; s. 1., 1584; probably included in various other editions of the Rechenbuchlein (p. in, 1564). This is the first edition of this well-known arithmetic. As already stated, the title was occasionally changed, but the work was essentially but little altered. It is a purely commercial book, with all of the opera- tions performed by counters as was still the custom of the time in most parts of Germany. Kobel treats of the rule of three ('die Gulden Regel, die von dem Walen de Try genant wirt'), partnership, reduction, inher- itances, and exchange. The fundamental operations include progres- sions, and Roman numerals are used except in the section on notation (Fig. 57). Kobel makes a curious use of the Arabic method of writing fractions, the terms being written in Roman, as in the case of mfxx for % <*"*> Altogether, Kobel was a vigorous writer, and his Gothic style shows him to have been no more a follower of the Italian arithmeticians than Diirer and Holbein were of the Italian artists. Other works 0/1514. Boethius, p. 27, 1488 ; Grammateus, p. 123 ; Jordanus, p. 65, 1496 ; Stromer, p. 83, 1504 ; Torrentini, p. 76, 1501 ; Hermannus Buschius, ' Algorithmus linealis Proiectiliu : de itegris perpul- chris Arithmetice artis regulis : earundemque probationibus claris exor- natus,' Vienna, 4, 4 ff. (also catalogued as anonymous, and as written by Johannes Cusanus; see p. 43, c. 1490, for Nicolaus Cusa; see also p. 86, 1505). There was also published c. 1514 an anonymous work entitled * Arithmetics practice Tractatus qui dicitur Algorismus, cum additionibus utiliter adjunctis,' Paris, 4. JAKOB KOBEL. Ed. pr. 1514. Oppenheim, 1514. See p. 100. Title. The title page is missing. Fol. A ij begins : * Dem Ernueften Dietherichen//Remerer von Wormbsrgenant von Dal- burgk : // meinen befundern giinftigen lieben Junck-//herzen/ Eubeutich Jacob K6bel/ //dieffer zeit Statfchreiber zu Op-// penheim/mein willig dinft // allerzeyt bereydt // zu vor.' Colophon. ' Getruckt zu Oppenheym.// Anno, i c M.CCCCC. XIIII.' (F. no. XXIIII. See Fig. 59.) PRINTED BOOKS 107 tocten gelcn/t>n6 tt>$ rm fiber to ey tfbertg bkf br/ 5ae ilocl? nm fimfcn $clcn/t>fi n>ae rm ate^an aber mate iiba ctyetbabet tx>i5crumb blabr fc^ictb aud) t>ff/ 60 ^) alfo $wy$ t>olb!ac^c abcr 6ief iiberbiybat i.ilbcrtgf/ tiibctt^/ fo tegt vff 6tc luiicn ccx* ttoc^5oatf5em Qde mtr fUnffttt iibcc blteben t(? i>rt5 ale mantel? mat ate 5u one act?r/picpuleim^ernaienoc^ fybm fiberblybm iff/ fo^utaerociflfolrualwe^poj one xy. t>f6ic Uitten le^m/Bo 5<id alfo t>otnb:ac^r fo rl?u 5 u 5^ 0ale ju fanwncn An an Sum/T>n5 n>ae 5a rau^ notrt/ ^ouon joint ale 6t^f^u fanl^unbcrr x>n5 funf^ebcn 6ouort 5 vl^m/tpae ale dan am lef?e u bei-blecb r/ fo md lift fc e l?ot/t>n& tj f cm (^e cfcbeit/un fo id) cmpfm6 mic^ to mir tc^t nuo gcfc^affr l)abeu/ wd u^ fMfifttgf Uc^ mtr gottce w5 6cr oc^^elatc t^tlff fc$ merm vtA becm/aud) atc t^tlff fc$ merm vtA beflcm/aud) v>rfircit/i>ett> ftl^cn /pii5 aug fccc ^Infmctttrf \>u6 (Seomccraoffcit baren. (S FIG. 59. LAST PAGE OF KOBEL'S Rechenbiechlin (OPPENHEIM, 1514) I0 8 RARA ARITHMETICA Description. 4, 14.1 X 19.6 cm., the text being 9. 2 x 15. 8 cm. 5 ff. unnumb. -I- 24 numb. = 29 ff., 28-34 11. Oppenheim, 1514. See p. 1 06. It is a curious fact that the first and second editions should have appeared in the same year at two different places. JAKOB KOBEL. Ed. pr. 1514. Oppenheim, 1518. See p. 100. Title. See Fig. 60. Colophon. 'Alfo Endet fich Seliglich d' Drit // Truck dif3 Rechenbuchleins/ Zu Oppenheym zufam-//men gefegt/ vnd volendt vff Dinftag des Hey-//ligen Criig Erhohungs tag.// Anno dm .1518.' (F. XLVI, r.) Description. 4, 13.7 X 18.4 cm., the text being 9.2 x 14.8 cm. 4 ff. unnumb. +46 numb. (I to XLVI) = 50 ff., 32-33 11. Oppenheim, 1518. See p. 1 06. JAKOB KOBEL. Ed. pr. 1514. Frankfort, 1531. See p. 100. Title. ' Ein new geor-//denet Kiinftlich Rechenbuchlin/ Ja-//cob Kobels/ Stattfchreiber zu Oppen-//heim/ Auff den Linien vnd Spacien/ mit Rech-//enpfeningen. Den angehn- den Schulern Rech//nens gantz leichtlich zu lernen. Vnd zu Kellerei//en Ampten/ Kauffmanfchafften/ vnd Krame-//reien dienlich vnd brduchlich. Mit vilen erkla-//rungen/ Leren/ Regeln vnd Exempln.//,g@ t3:> Mehr dan vormals ie getruckt/ //gebeffert/ vnd zugefetzt.// Im Jare M. D. XXXI.//Vifir Buchlin/ //Den Jungen/ angehnden/ // Leijfchen Vifirern gantz leichtlich zu lernen/ //verftehn vnd Rechnen. Wie man ein // Vifirrut machen/vnnd damit ein//iedes vafz vifirenn/ folle Zu //ende difes Rechenb6ch//lins angehenckt.' (F. I, r.) Colophon. 'C.Getruckt zu Franckenfurt/ am Mein. In Ver// lag vnnd Gemeynfchaff t des Ernhaffetnn // vnd Furnemen Herrn Jacoben K6-//bels/ Stattfchreiber zu Oppen-//heim. Bei Chrif- tian Ege-//nolffen. Im Mertzen/ // Nach der geburt // Chrifti. //M.D.XXXI. Jar.' (F. CXII, r.) PRINTED BOOKS 109 new ftecfe Gp acteit / nut ffmanfcl?aff t rccfyc lew? mag* / jum ID: ttte malc L ebeficrt vn ju paeer nurarbetr t^ct wnb ffifj !Da6 cv rut tt>iitb/t>etacl)t x>ewtd?t / *Dct Kawff &t |5>Bc^ \>ii leg mtt pei'h^^ i>n^ ntercf / tx^ae tc^ yrt t>n&cctt)etg SDem f cm cr noc^ nttr frctem mSt FIG. 60. TITLE PAGE OF KOBEL'S RechTepuchlein (1518). i io RARA ARITHMETICA Description. 8, 9.7 X 14.9 cm., the text being 6.9 X 11.5 cm. 112 ff. numb, in Roman, 28-29 11. Frankfort, 1531. Editions. See p. 102. It is often stated that the ' Jacob's staff ' used by surveyors was first described by Kobel in this year (1531) and that it received its name in his honor. The name was old before this time, however, as applied to some form of surveying instrument, for in the Margarita Philosophica (1503, Bk. VI, tract. II) is a conversation between a master and his disciple, in which it is mentioned : ' MAG. baculo que lacob dicunt. Dis. Qualis is eft baculus?' The master thereupon describes the instru- ment, and a picture of it is given. JAKOB KOBEL. Ed. pr. 1514. Frankfort, 1537. See p. too. Title. * Zwey rech-//enbuchlin: v ff der // Linien vnd Zipher/ Mit eym angehenck-//ten Vifirbuch/fo verftendtlich fiir//geben/ das iedem hieraufz on//ein lerer wol zulernen.// C.Durch den Achtbarn vnd wol erfarnen//H. Jacoben Kobel Statfchreiber //zu Oppenheym.// Franc. Chrift. Egen.' (F. i, r.) Colophon. ' Ende/Im lar M.D.XXXVII.' (F. numb. 144, r.) Description. 8, 9.5 X 15 cm., the text being 7.1 X 11.9 cm. 8 ff. unnumb. +9-144 numb. = 144 ff., 30 11. Frankfort, 1537. Editions. See p. 102. This is the earliest of Mr. Plimpton's copies containing the three books written by Kobel, (i) Rechenbuchlein,' (2) 'Mit der Kryde,' (3) the * Vysierbuch.' The combination of the three in a single volume formed one of the best books of the time, giving the operations both with counters and according to algorism. The latter is given in the part entitled ' Mit der kreiden // odder fchreibfederen/ durch // die ziffer- zal zu rechen/ Ein new Rechen-//buchlin/ den angehnden fchulern der//rechnung zu eren getruckt.' (F. 106.) In this work Kobel also includes the usual business problems of the period and the chapter on the calendar required by the Church schools. An unusually complete treatment of gauging is given in the part entitled 'Eyn new Vifir// Buchlin/ den Leyen/ zu // leichtem vnd begreiflichem verftandt // verordnet/ Durch H. Jacob //K6bel Stattfchreiber zu // Oppenheym.' (F. 95, r.) It is much more complete than the 1514 edition of the ' Rechenbuchlein ' (see p. 106), and is substantially the same as the PRINTED BOOKS ii i 1531 edition (see p. 108) except that it contains the third part, ' Mit der Kreiden,' which the latter does not. The Hindu-Arabic numerals were still considered difficult (' den die Zifer zal am erften zulernen fchwere,' f. 9, v.), and teachers still felt it better to begin with the common Roman characters (' wil ich zum erften die felb Teutfche zal ... hie anzeygen vn erkleren '). JAKOB KOBEL. Ed. See p. 100. Title. ' Rechenbuch/ //Auff Linien vnd Ziffern. //Mit einem Vifir buchlin/ Klar // vnd verftendtlich furgeben. // Gerechnet Buchlin/ auff alle // Wahr vnd Kauffmanfchafft / Muntz/ // Gewicht/ Elen/ vnd Mafz/ viler Land // vnd Stett verglichen.// Durch H. Jacob Kobel. // (Woodcut.) Cum Gratia & Priuilegio.// Franckfurt/ Bei Chr. Egen. Erben // M. D. LXIIII.' (F. i,r.) Colophon. ' Getruckt zu Franckfurt // am Mayn bey Chriftian E-//genolffs Erben.// M. D. LXIIII.' (F. 194, v.) Description. 8, 9. i x 15.2 cm., the text being 6.3 X 12.2 cm. 12 ff. unnumb. + 194 numb. = 206 ff., 28-2911. Frank- fort, 1564. (There were which this is the sixth.) pr. 1514. Frankfort, 1564. Von vetf tt-tttt frajfon 57 orfrt ft eft/ vims tyro af fo nad^'ompf* /fc (Tit (a*d;r Juf? f *riieit rcdjncrt/ alics o<ss Di'n'ii pfenm'njj / f<> fie betiti 0ompt / ftnOet ftc j'm forblm ' f tCbu/rtfsob^clcrt/ fo fomptDfirjcbm/ il byi'cnbnt fieVmb ctncnpfett* ni0/ VttMft wolfcyl i 3 v FIG. 6 1. FROM KOBEL'S Rechenbuch (1564) at least eight Frankfort editions, of ii2 KARA ARITHMETICA Editions. See p. 102. The first 165 ff. are the same as in the 1537 edition (see p. 1 10). The rest of the book is not found in the 1537 edition, or in any other edition in the Plimpton library. It consists of a description of foreign and domestic money, with numerous illustrations of coins. This begins S bttgctttttt/ w<ctt mi't efoanoe* gen &otit $^e/t>n&i5qfim'dj?fl8 lt/vn& mo4>t rfrtwwgtt^t mcbr t>ttmjebtndlettge> nbcym t)cit (ConQ vonCcc- bec/^lfo vat 0on ^cynrtc^ Conceit 90. Hun ifloicfr^cj //it tttt'e Vtt trt^cn Cdt$ xnt> t>tc 5tx?cn jttfamen f o inincn feint? f 5Dt'c V>nt> Derr;Icid)cn fi'itc;rj?it|ru a((b cnfc fd? cidcit. 3imt crfrcn Ic0 Me mciIcn/Mc Sort ^ann^fc meiln/t>tc Convtit emcn w0 iiKb* ^Hl FIG. 62. FROM KOBEL'S Rechenbuch (1564) (f. 165, r.): 'Von Frembden vnd Hie//landifchen Muntzen/ So difer zeit in//Teutfch vnnd Welfchen landen/inn aller // Kauffmanfchaf? t vnd Gewerb/ Hindeln/ //viler Land art im brauch/ geng/ gibig oder // ver- ruffen Muntzen/ . . .' The book closes (f. 184, v.) with a set of tables and (f. 192, v.) a ' Regifter.' PRINTED BOOKS 113 JAKOB KOBEL. Ed. pr. 1515. Oppenheim, s. a. (1515). See p. 100. Title. ' Eyn New geordet // Vyfirbuch. Helt yfi.//Wie man vff eins yden Lands Eych // vn Mafz/ ein gerecht Vyfirfit mache FIG. 63. LAST PAGE OF KOBEL'S Vysirbhch (1515) // vn do mit ein ygklich onbekant Vafz // vyfieren/ auch feynen inhalt erlernen // folle. Den anhebenden Schulern Vi//firens u 4 RARA ARITHMETICA Leichtlich/ mit Figuren vnnd//Exempeln/zu lernen/angezeigt. //Angehengt Tafeln.//Die Erften Fyer halten yn gerechet/ //fo eyn Fikter weins kaufft wirt/ vmb // Guide zu XXVI. Od' XXIIII. AT5. jc.// Was die Ome/das Fyrtel/ vnnd die Mafz gelten.//Die Andern Tafeln/Zeyge an/ Ver//anderug vn wech- felug einer Muntz//durch die ander/als p. in Ib. 2c. //Ge- druckt zu Oppenheym.' (F. i, r.) Colophon. 'Gedruckt zu Oppenheym.' (F. 32, v. See Fig. 63.) Description. 4, 14.4 X 19.7 cm., the text being 9.4 x 14.5 cm. 4 ff. unnumb. + 28 numb. = 32 ff., 30-32 11. Oppenheim, s. a. (1515). There is no date on the title page nor in the colo- phon, but the prefatory statement closes with the words, * Vol- nendet vff dornftag noch Letare. Anno & c. 1515.' Editions. See p. 106. The work is semiarithmetical, quite as much so as the chapters on mensuration in our textbooks ; chapters, indeed, which owe their origin in no small degree to these treatises on gauging so often appended to the old arithmetics. This work is illustrated with quaint woodcuts show- ing the use of the ' Vyfirftab ' or gauging measure. (See Fig. 63.) GIROLAMO AND GIANNANTONIO TAGLIENTE. Ed. pr. 1515. s. 1. (Venice), 1525. Venetian arithmeticians of c. 1500. Title. ' Opera che//insegna//A fare ogni Ragione//de Mer- catia // et a pertegare le Terre // Con arte giometrical // Intito- lata Componimeto // di arithmetica // Con grati a & preuilegio // M. D. XXV.' (F. i, r.) Description. 8, 10.3 X 15.1 cm., the text being 7.5 x 12.5 cm. 91 ff., 28-33 H- S. 1. (Venice), 1525. Editions. Venice, 1515, 8 (De Morgan having erred in say- ing 'apparently before 1500'); ib., 1520; ib., 1523, 8; ib. (s. 1.), 1525, 8 (here described); ib., 1526; ib., 1527; ib., 1528, 4; s. a. (1530?), 8; Venice, 1541 ; Milan, 1541, 8 (p. 115) ; s. 1., 1547, 8; Milan, 1548; Venice, 1548 (with probably a second Venetian edition, 1548, under the title 'Thesoro universale de abacho,' by ' lucha atonio de Uberti,' 8);ib., 1550; ib., 1554; ib., PRINTED BOOKS 115 1557; ib., 1561; ib., 1564; ib., 1567; ib., 1570; Milan, 1570; ib., 1576; ib., 1579; ib., 1586. Riccardi also mentions eleven other editions, s.a., and four such appeared in the Boncompagni sale, and four in the Fisher sale of 1906. The work also ap- peared, and is frequently catalogued, without the authors' names. These various editions have been the object of critical study by E. A. Cicogna and Prince Boncompagni. The former set forth his results in his Saggio di bibliografia Venesiana, Venice, 1847, p. 218, ascribing the work to Girolamo Tagliente 'con 1'ajuto del suo consanguineo Giannantonio Tagliente.' In the edition of 1525, here described, only the former name appears, the text beginning, 'C.A1 benigno lettore//Hieronymo Tagliente.' Boncompagni's investigations, setting forth the differences in the various editions, appeared in the Atti delf Accademia Pontif. de Nuovi Lincei, XVI, 139, 147, 155, 304. See also Riccardi, I, 2, 484, and Boncompagni's Bulletino^ XIII, 247. There was also a treatise published by the Taglientes entitled ' Regole di mercatura intitolato componimento di arithmetica,' Venice, 1524, 8, probably another edition of this work. See also the treatise on bookkeeping mentioned on p. 141, 1525. The book opens with a brief treatment of notation and finger symbols. Then follow in order the multiplication table, the proof of sevens, various methods of multiplication, division by the galley method, addition chiefly of denominate numbers, subtraction, the operations with fractions in the same order, exchange, rule of three, and applied problems. There are numerous interesting woodcuts, and such familiar problems as those of the couriers, the testament, and the sale of eggs are given with illustra- tions. In spite of the arrangement of topics, there were few textbooks so influential as this in shaping the subsequent teaching of arithmetic. GIROLAMO AND GIANNANTONIO TAGLIENTE. Ed. pr. 1515. Milan, 1541. See p. 114. Title. ' Libro // dabaco che in//segna a fare // ogni ragione mercadantile, & // pertegare le terre co 1'arte di //la Geometria, e altre no//bilifsime raginoe ftra-//ordinarie co la Ta-//riffa come n6 RARA ARITHMETICA refpon//deno li pefi & // Monede de molte terre del mon-//do con la inclita citta di Vene-//gia. Elquel Libro fe chiama //Thefauro vniuerfale.' (F. I, r.) Colophon. ' Stampato in Milano per lo. Antonio da Borgho.// Nell' anno del. M. D. XLI.' (F. 80, v.) Description. 8, 10 X 14.6 cm., the text being 7.7 X 12.3 cm. 80 ff. unnumb. + 7 blank = 87 ff., 23-30 11. Milan, 1541. Editions. Seep. 114. FIG. 64. FROM THE 1541 TAGLIENTE This differs but little from the 1525 edition, the ' Opera che insegna' (see p. 114), except in having a set of tariff tables at the end: * C.Qui comenza el terzo Libro di la fruttifera opera // chiamata la Tariffa' (f. K iiii, v.). There is some slight change in the phraseology, particu- larly at the beginning of the various sections. For two curious forms of multiplication see Fig. 64. Such arrangements of the work in multipli- cation were quite common, particularly in the early Spanish and Italian arithmetics of the first half of the sixteenth century. That they should have found place in a popular mercantile treatise is, however, rather surprising. PRINTED BOOKS 117 VARIOUS AUTHORS. Ed. pr. 1515. Vienna, 1515. JOANNES DE MURIS (JEAN DE MEURS, MURS, MURIA)- was born in Normandy, c. 1310; died after 1360. He wrote on arithmetic, astronomy, and music. THOMAS BRADWARDIN. See p. 61. NICOLAUS HOREM (NICOLAS ORESME) was born at Caen (?) c. 1323; died at Lisieux, July n, 1382. He taught in the College de Navarre at Paris, and in 1377 became Bishop of Lisieux. He wrote also an ' Algoris- mus Proportionum,' in which the idea of fractional exponents first appears. GEORG VON PEURBACH. See p. 53. JOANNES DE GMUNDEN (JOHANN VON GMUNDEN, JOHANN WISSBIER ? NYDEN ? SCHINDEL? JOHANNES DE GAMUNDIA) was born c. 1380, at Gmunden on the Traunsee, or Gemiind in Lower Austria, or Gemiind in Swabia; died at Vienna, February 23, 1442. He was educated at Vienna, and taught there, being the first professor of mathematics alone in Austria. Title. See Fig. 65. Colophon. ' Impreffum Vienne per Joannem Singrenium // Expenfis vero Leonardi i Luce Alantfe // f ratrum Anno domini .M.ccccc.xv.//Decimono die Maij.' (F. 54, r.) Description. 4, 14.3 X 18.4 cm., the text being 9.7 X 15.5 cm. (without the marginal references). 54 ff. unnumb., 26 11. Vienna, 1515. Editions. There is no other edition of this combination of works. See pp. 53, 61, 118, for the individual treatises. This interesting work consists of five parts. The first is the arith- metic of Joannes de Muris. While it is called an extract from the arithmetic of Boethius, it is merely suggested by that treatise and is really the work of Muris. This part of the work begins on f. 2 with the following title : ' Incipit Arithmetica comunis ex // diui Seuerini Boetij Arithmetica per M. Joannem // de muris compendiofe excerpta. // Prohemium.' De Morgan (p. 3) mentions a possibly earlier edition, s. 1. a., 4. (See also p. 86, 1505.) The second part of the volume begins on f. 17, v., and is the work on proportion by Thomas Bradwardin. It is a theoretical treatment of the subject, and has the following title : ' Tractatus breuis proportion u : ab-//breuiatus ex libro de Proportionibus. D. Thome // Braguardini Anglici.' The third part begins on f. 27, v., and is a treatise by Nicolaus Horem, with the following title : * Tractatus de Latitudinibus forma-// rum fcdm doctrina magiftri Nicolai Horem.' This subject attracted considerable attention in the latter part of the Middle Ages. u8 RARA ARITHMETICA The fourth part is the algorism of Peurbach, and has the following title on f. 37, v. : ' Opufculu Magiftri Georgij //Peurbachij doctifs.' As already stated (p. 53), this work takes up the four fundamental opera- tions and progressions, giving merely a theoretical discussion of the subject. The fifth part (f. 44, v.) begins : ' Incipit tractatus de Minucijs phi-//ficis : compofitus Vienne Auftrie per magiftrum //Joannem de Gmunden.' This is the treatise of Gmunden on sexagesimal fractions, or, as they were called in the Middle Ages, physical fractions. These fractions, still used by us in our degrees (or hours), minutes, and seconds, served the purposes of the later decimal fractions. They were carried much farther than is now the case, a number like 3 15' 40" 15'" 45 Iv meaning merely 3 + + + + 4 , or 3 The present symbolism (, ', ") is relatively modern. The book is particularly interesting because it combines in one volume five well-known books by mediaeval writers. In no other Stttbrncttca coiftitiunfe* pzopomonca bmics, >claritiK>imbi)9 ocmiiiucgo pbiftcfe. FIG. 65. TITLE PAGE OF THE VIENNA WORK OF 1515 single volume could the inadequacy of the mediaeval treatment of mathematics be better seen. Indeed, a manuscript of 1515, found by Gerhardt in the Wolfenbuttler Bibliothek, expressly states that the lectures on arithmetic given in the universities of that period were based on the above works of Muris, Bradwardin, Peurbach, and Joannes de Gmunden. (Monatsberichte der K. P. Akad. d. Wissensch. zu Berlin, 1867, p. 43.) PRINTED BOOKS 119 JOANNES DE MURIS. Ed. pr. 1515. Mainz, 1538. See p. 117. Title. <Arithme//ticae specvlativae // Libri duo loannis de Muris ab in-//numeris erroribus quibus hacte-//nus corrupt!, & uetuftate fer//me perierant diligen-//ter emendati,// Pvlcherri- mis qvoqve exemplis, Formisq; nouis declarati & in // ufum ftudiofae iuuentutis Mogun-//tinae iam reccens ex-//cufi.// Mo- gvntiae excvdebat // Ivo Scoffer anno.//M. D. XXXVIII.' (F. i, r.) Colophon. ' Mogvntiae excvdebat // Ivo Scheffer anno.// M. D. XXXVIII.' (F. 90, v.) Description. 8, 9.6 X 14.5 cm., the text being 6.6 X 1 1.9 cm. 4 pp. unnumb. + 3-88 numb. + 5 blank -f- I with woodcut = 96 pp. Mainz, 1538. Editions. This is the second dated edition of the arithmetic of this popular mediaeval teacher. (See p. 1 17.) It is more com- plete than the one of 1515, but it does not, like the latter, give the marginal references to Boethius, upon which it is so largely based. (See also Boethius, p. 27.) See p. 117. JOANNES FCENISECA. Ed. pr. 1515. Augsburg, 1515. An Augsburg teacher of c. 1500. Title. See Fig. 66. Colophon. * Impreffa Augufte Vindelicorum/ communibus im- penfis Io/ //annis Miller atq3 loannis fcenifece. Anno a natiuita //te domini. M.D. XV. ad. IIII. Cal.' Maias.' (F. 20, r.) Description. 4, 15.1 X 21 cm., the text being n X 13.8 cm. 20 ff. unnumb., 7-39 11. Augsburg, 1515. Editions. There was no other edition. This is an extract from a larger volume, for the folios have been numbered by hand 40-59, and the register begins with 'aa i.' Only two pages (aa ii, v., and aa iii, r.) are devoted to ' Arithmetica,' and these relate only to the Boethian system. The rest of the book is devoted chiefly to geometric figures, the mediaeval astronomy, and music. Such 120 RARA ARITHMETICA a book shows the superficiality and general emptiness of the work of the schools that were supposed to stand for culture in the period of the early Renaissance. Opera loannis Fceni&c* Augn. htciniehabent. Quadratum iapfenri^rconrinens in fefeptem a r ccs liberties vettrum* Circulos biblif .iiii.in quibus mctaphyfica mofaica. Commentaria horum. Ad Hcc/libri rubricai nfmiis fignati:nece(Tarii fu nu f^Grammatica terino Logica Monaftica Oeconomica .i.mos math cacica plulofbpliia (^ Medicina fabaltcrrm thcologia Metaphyfica Alcarithmiis fiibalrcrnus Arirfimctfca Geomccrta Pcr(pc<ftjiia fiibalrmia Mulica Aftronomia don.aIex.gua.Ia/ca.Pocri nouus bocrius boecius pcTUsiacobi boerfus boccius cphcmerid pcolcm^us Hiftonc.) JfofcoricJes bibiiUmtripTcx aai. FIG. 66. TITLE PAGE OF FCENISECA'S Opera (1515) ANONYMOUS. Ed. pr. c. 1515. Leipzig, s. a. (c. 1515). Title. ' Melchiar Lotthervs Ivnior candido lectori salutem.' (Line i.) ( Articularis Bedae prefbyteri numeroru computatio.' (Line 24.) Description. One sheet, 28 x 36.4 cm., the text being 21.6 x 26.7 cm. 42 11. Printed on one side of a single sheet. PRINTED BOOKS 121 Editions. There was no other edition, and only this copy is known of this one. This broadside was published by Letter c. 1515, and is a brief state- ment of the numerical finger-symbolism of the ancients, particularly as described by the Venerable Bede. The symbolism is practically the same as that described by such writers as Paciuolo and Aventinus (see pp. 57 and 136.) CASPAR LAX. Ed. pr. 1515. Paris, 1515. Born at Sarinena, Spain, c. 1487 ; died at Saragossa, February 23, 1560. He taught at Paris and Saragossa. His only works are the two here described. Title. See Plate III. ' Proportiones magiftri Gafparis // lax aragonenfis de farinyena . . .' (F. 101, r. Separately cata- logued, see below.) Colophon. ' Explicit Arithmetica fpeculatiua Magiftri Gafparis Lax Aragonenfis de Sarinyena duode-//cim libris demonftrata. Impreffa Parifius opera ac characteribus Magiftri Nicolai de la barre. //Expends honefti viri Hemundi le feure Bibliopole Pari- fius in vico diui Jacobi fub figno Crefce//tis albi vitam degentis. Anno Domini .1515. Die vero.i3. Menfis Decembris.' (F. ioo,v.) Description. Fol., 19.8 x 26.5 cm., the text being 14.1 X 21.8 cm. 100 ff. unnumb., 5961 11. Paris, 1515. Editions. There was no other edition. A very prolix treatment of theoretical arithmetic, based on Boethius and his mediaeval successors. As the title shows, Lax was a Spanish teacher, one of several from the southern peninsula who taught in the University of Paris in the fifteenth century. Among the others were Rollandus (originally from Lisbon, mentioned later in connection with the manuscripts) and Ciruelo (p. 58). All of the contributions of these scholars were of this general theoretical character. De Morgan face- tiously remarks, For anything that appears the author (Lax) could not count as far as 100.' CASPAR LAX. Ed. pr. 1515. Paris, 1515. See above. Title. ' Proportiones magiftri Gafparis //lax aragonenfis de fa- rinyena.// Venundatur Parifius In vico diui //Jacobi ab Emundo le feure fub fi-//gno crefcentis albi vitam degente.' (F. 101, r.) 122 KARA ARITHMETICA Colophon. ' Expliciunt proportiones Ma//giftri Gafparis Lax Aragonen//fis de Sarinyena impreffe Pa-//rifius opera Magiftri Nicolai // de la barre pro Emundo le feure // Anno dfii M. d. xv. die vo vi. me//fis Octobris.' (F. 26, 126 of the whole book, r.) Description. Fol., 19.5 X 26.6cm., printed in double columns, each being 7 x 21.5 cm. 26 ff. unnumb., 66 11. Paris, 1515. Editions. There was no other edition. Bound with the ' Arithmetica Speculativa' (p. 121). This is a prolix treatment of mediaeval ratios after the Boethian manner, and as such it ranks with works like those of Bradwardin (p. 61), Jordanus (p. 62), and Faber Stapulensis (p. 82). Other works 0/1515. Boethius, p. 27, 1488; Bradwardin, p. 61, 1495; Kobel, p. 102, 1514; Lanzut, pp.83, 97, 1504, 1513; Licht, p. 70, 1500 ; Ortega, p. 93, 1512; Peurbach, p. 53, 1492 ; Torrentini, p. 76, 1501 ; Juan Andre's, ' Sumario breve de la practica de la arithme- tica,' Valencia (from the book it appears that it was written in Saragossa in 1514; it was reprinted at Seville in 1537) ; V. Rodulphus Spoletanus, * De proportione proportionvm dispvtatio,' Rome, 4. Works 0/1516. Johann Boschensteyn, p. 100, 1514; Capella, p. 66, 1499; Ciruelo, p. 60, 1495; Kobel, p. 102, 1514; Stromer, p. 83, 1504; Widman, p. 36, c. 1488. PIETRO MARIA BONINI. Ed. pr. 1517. Florence, 1517. A Florentine writer of the first half of the sixteenth century. Title. ' Lvcidario darithmetica.' (Large woodcut. F. 3, r. Fig. 67.) Colophon. * Cjmpreffo nella excelfa cipta di Firenze per // Gianftephano di Carlo da Pauia//adi 7 di Gennaio.' (F. 18, r.) Description. 8, 10.9 x 15.9 cm., the text being 8 x 12.7 cm. 19 ff. unnumb., 24 11. Florence, 1517. Editions. There was no other edition. This interesting volume was known to De Morgan only by hearsay when he wrote his Arithmetical Books. It came into his possession, however, after that work was published, as is shown by his autograph on the title page (see Fig. 67). It is not often men- tioned by bibliographers, and is one of the rare books of the century. In the Boncompagni sale (no. 1441) there is mentioned a fteeulatiuama ansi fcefimnpen oe^^ PLATE III. TITLE PAGE OF LAX PRINTED BOOKS 123 ffLVCIDARIO DARITHMETICA. an edition of 1 547 ; but this is a misprint for 1 5 1 7, as appears from Riccardi, vol. I, col. 1534. Riccardi mentions only three copies known to him or to Boncompagni, but there was one in the Fisher sale (London, 1906). The book is a small octavo, the first two-thirds being given to mercantile prob- lems on exchange and the reduction of money. The last part treats exclusively of mensuration : ' Speculationi geometriche di piu sorte : & prima laquadratura del triangolo.' Other works of 1517. Anonymous, 'Algorithmus linealis' (see Widman), p. 36, c. 1488; Borghi, p. 16, 1484; Feliciano, p. 146, 1526; Kobel, p. 102, 1514; Reisch, p. 82, 1503; Sacrobosco, p. 32, 1488; FIG. 67. TITLE PAGE OF Widman, p. 36, c. 1488. BONINI'S Lvcidario HENRICUS GRAMMATEUS. Ed. pr. 1518. Frankfort, 1535. HEINRICH SCHREIBER; HENRICUS SCRIPTOR ; Latinized Greek, GRAM- MATEUS. Born at Erfurt, at least as early as 1496. He describes him- self as ' Henrich Grammateus // von Erffurt/ der fiben f reien kiinf ten Meyf ter.' He was a student at Cracow and at the University of Vienna (1507). The dates of his birth and death are unknown, but a record at Vienna reads : ' Anno domini millesimo quingentesimo septimo in festo sanctorum Tibureii et Valeriani martirum . . . Henricus Scriptoris de Erfordia.' He also taught at Vienna. Title. See Fig. 68. Description. 8, 9.2 x 15.2 cm., the text being 6.7 X 11.3 cm. 96 ff. unnumb., 31 11. Frankfort, 1535. Editions. Vienna, 15 18, 8; Frankfort, 1535 (here described); s. 1. (Frankfort), 1544; Frankfort, 1572. Grammateus also published an ' Algorithmus proportionum una cum monochordi generis Dyatonici compositione . . .', Cracow, 1514, 4; ' Libellus de compositione regularum pro vasorum mensuratione,' Vienna, 1518; ' Behend unnd khunst- lich Rechnung nach der Regel und welhisch practic,' Nurnberg, I24 RARA ARITHMETICA 1521, 8, an extract from the work here described ; ' Algorismus de integris Regula de tri cum exemplis/ Erfurt, 1523; ' Eyn (BemeincnKegdnbetre. ttelfcbcn mactic. Kcgcln falfr. ' J frioflu / 0eIpfci L ^rcfce zna I/ B -pd vn& fllDificr rutcn ju m^^ert Dur6 *><** t> :a t / pnb ^rtangel/mtt an o e r n Iufti0m (hic^en ber (0eomerrei. FIG. 68. TITLE PAGE OF THE 1535 GRAMMATEUS kurtz newe Rechenn unnd Visyrbuechleynn gemacht durch Heinricum Schreyber,' Erfurt, 1523. That the book was written at Vienna in 1518 appears by the dedi- cation to ' Dem Edlen fiirfichtigen weifen Johan//fen Tfchertte einer PRINTED BOOKS 125 des Senats zu Wien,' which ends: * Gebe//zu Wi//en in O-//fterreich im jar//nach der geburt vn-//fers Seligmachers. M. D. XViij.' In the chapter on bookkeeping is the date 1535, so that probably the work was revised for this edition. C&Mtfe, 31&te fern 30 abbiren bie qua witet einea n a* tnene/ald tfl. mit H: pzi m arm i t prima/fecunba mit fe cuba/tf rtia mit tertt'a K."X>nb man b:au* cjjet folcfcetticfcen aid -f- iff mefj:/t>nb /rum fcer/m welder fein 50 merrfen b:ei Ke cjel. ODann ein quan titct f>a r an beybcn o if en -f- ober fo fol man n fol^e quan ti tet abbirn ^ in 50 0efa$t bad $ey d^e n old 9ptr-f-? p:u-4-y^T> s 3ffl in ber &bcrn quantitet -]- x?nb fn ber tm br rn / vnb (- ubertriff t / fo fol bic rnber quant i te t von ber dbern fubt r a f) r t treirben/vn $ubemub:trtcn fe^ (-Soabcrbic rnber qua tiret iff 0roffer/fo fubtra j)ir bie Hn'nern uo ber 0r5(|ern/pfi 5U bem b6 bo bleitenb i(l/fecje ale So mberob0cfaRtenqtintitet iptfrr fanbe unb in ^cr vnbern +/pnb ubertriff t +/ fo |ubfra6tretn0 von bcm anbern/vnbjum ti* b:igcn fcb:cib 3f? es fiber/bae bie r nbcr qua tittttibcrrriffl bre Sbern/fojie^eemdvoi) bem anbcrn/x>nb 50 bf m cr fien fe tje -j- ab FIG. 69. FROM THE 1535 GRAMMATEUS The work is for the most part a mercantile arithmetic, the opera- tions being given according to both the abacists (with counters) and the algorists (by the Hindu- Arabic numerals), and a chapter on book- keeping being appended. Grammateus gives, however, some considera- tion to the theory of numbers, the rules of the Coss (algebra), music, 126 RARA ARITHMETICA bookkeeping, and gauging. In the treatment of the ' Regula falsi,' or rule of false position, the signs + and are first found in this connec- tion (f. Eiij). (See Widman, p. 40, 1489, who uses them for another purpose, and Vander Hoecke, p. 183, 1537.) Grammateus also uses these signs in writing algebraic binomials, as shown in Fig. 69. It is interesting to know that Rudolff (p. 150) learned algebra from Gram- mateus, as he states in the following words : ' Ich hab von meister Heinrichen so Grammateus genennt/der Cofs anfengklichen bericht emphangen. Sag im darumb danck.' Other works 0/1318. Boschensteyn, p. 100, 1514 ; Calandri, p. 48, 1491; Feliciano, p. 145, 1526; Kobel, p. 102, 1514; Riese, p. 139, 1522 ; Torrentini, p. 76, 1501 ; Perez de Oliva, ' Dialogus in laudem Arithmetics, ' Paris. ANONYMOUS. Ed. pr. 1519. Venice, 1519. Title. See Fig. 70. Colophon. ' Venetijs in Edibus Petri Liechtenftein // Anno virginei partus .1519.' (F. 12, v.) Description. 4, 15.6 X 20.4 cm., the text being 10. 1 X 16.3 cm. 12 ff. (i blank), 33-38 11. Venice, 1519. fcrcailrononnefundainc/itumpulcbcrrnnunicoB/ tinens, Clerico non minus vrilis $ necelV farms: cnmaddiriombus quibuft damnouiterapzeflis* * * SSnno. i 5 1 9 .Scncti/sin jdibus pern Kiccbtenftcin* FIG. 70. TITLE PAGE OF THE Coputus nouus Editions. There was no other edition, so far as I know, although there are several anonymous computi, and some may be the same as this. This is one of the rare works setting forth the computus as it was taught in the Church schools of the Middle Ages. (See p. 7.) PRINTED BOOKS 127 The verses on the calendar mentioned on p. 33 (Anianus, 1488) here appear as follows : 'Aprilis: Junius: September atqj Nouember Hij trigenta habent vnum reliqui fuperaddunt Februarius vigint infuper octoqj dies.' (F. 9.) Other works 0/1519. Feliciano, p. 146, 1526 ; Blasius, p. 97, 1514 ; VVidman, p. 37, 1489. JOHANNES FRANCISCUS PICUS MIRANDULA. Ed. pr. 1520. S. 1., 1520. A nephew of Pico de Mirandola, and biographer of his uncle. He was murdered in 1 533. Like his uncle he was a savant of reputation. Title. ' loannis Francisci Pici Mirandvlae domini, et // Con- cordiae comitis, examen vanitatis do//ctrinae gentivm, et veri- tatis Chri-//stianae disciplinae,//distinctvm in libros sex, qvorvm tres // omnem philosophorvm sectam vni-//versim, reliqvi Aris- toteleam //et Aristoteleis armis // particvlatim im-//pvgnant.// vbicvnqve avtem Christiana et // asseritvr et celebratvr // dis- ciplina.' (F. I, r.) Description. Fol, 20.6 X 30.6 cm., the text being 16.9 X 24 cm. 6 ff . unnumb. + 208 numb. + I blank = 2 1 5 ff . , 44 11. S. 1. , 1 5 20. Editions. There was no other edition. The dedication (f. 2, v.) bears date M. D. XX, and the privilege M. D. XIX. The book hardly deserves place in a list of this kind. It has, how- ever, been included because of the following brief chapters on the nature of arithmetic : ' Quod fuper mathematicis artibus arithmetica & geometria, fuperqj mediis Aftrologia & mufica, gentium philofophi non conueniunt. Cap. vii ' ; ' De opinione pythagoricoru, & de ratione et philolai & poftidonii . . . Cap. ix ' ; * Quid aduerfus arithmetica facultate pyrrhonii difputauerint. Cap. vii ' (of liber III). ANONYMOUS. S. 1. et a. (c. 1520). Title. ' Von dem Rachnen auff den Linien.' (Running headline.) Thirteen fragments of proof sheets of an unknown German arith- metic, three duplicates. The date is purely conjectural. The work was of at least 46 pages, since the folios were numbered and part of f. 23 i 2 8 KARA ARITHMETICA is among the fragments. The title of the book probably appears in the running headline above given, although this may be the title of only part of the work. There was at least one illustration of line reckoning in the book. The lines are 6.4 cm. in length, but there is no complete page among the fragments. Several anonymous works have already been mentioned, with some such title as 'Algorismus linealis,' and possibly this is one of them. It would probably be possible to identify it if one should examine the types and compare the fragments with possible originals. ESTIENNE DE LA ROCHE, Villefranche. Ed. pr. 1520. Lyons, 1520. Born at Lyons, c. 1480. Title. See Fig. 71. Colophon. ' C.Cy finift larifmetique de maiftre Eftienne de la roche diet ville franche natif de Lyon // fus le rofne. Imprimee par Maiftre guillaume huyon. Pour Conftantin fradin mar-//chant : libraire du diet Lyon. Et fut acheuee Ian . 1 520. le 2 e . de Juing.' (F. 234, r.). Description. Fol., 17.3 X 25.5 cm., the text being 13x21 cm. I f. blank + 4 ff . unnumb. + 230 numb. = 235 ff., 49 11. Lyons, 1520. Editions. Lyons, 1520, fol. (here described); ib., 1538, fol. (seep. 130). This is the best of the early French arithmetics. Since it is semi- mercantile in character, it was naturally printed at Lyons, then the com- mercial center of France, the theoretical books being usually printed at Paris under the influence of the Sorbonne. De la Roche gives a very complete treatment of the operations with integers, fractions, and com- pound numbers, and a large number of business applications. Perhaps no arithmetic published in France in the sixteenth century gives a more comprehensive view of the science and art of arithmetic and of the applications of the subject. Unfortunately, however, de la Roche took much of his work bodily from a manuscript of his master, Chuquet, which he had in his possession, and which has since been published. Other works 0/1520. Kobel, p. 102, 1514 ; Peurbach, p. 53, 1492 ; Raggius, p. 98, 1514; Stromer, p. 83, 1504; Suiseth, p. io,tc. 1480; Tagliente, p. 114, 1515 ; Anonymous (Tagliente ?), * Libro de Abaco,' PRINTED BOOKS 129 fus ie TRof ue oiuifec en oeujc parties pont I A p in t ere tracts DCS r>p:teres gfecn'80 rt regies oe Ufcicte fcie:come U no* toe entierae ndbze routaa regletrov5 : )La regie ouiic faulfepofition^eoeuyfauirespofitifeioappofmon em ! moti5n>e la regie oe mediatid entre le plus et Ie mofeioe la Kg4eoe!acbofe:etDelaquatiteice < pgrem60et^poi60. ; CxafecScUtractewla p:actiqu6WceHeapplicqiieeeii fait ce mono? es:en touted mar cbadifee c6iue wapperie : efpi \ cerie:mercerieetentoute0aultre0inarcbadifequifevewi dent a mefurc au poi5 ou au nobxtcn copaignies et en ttw que0:eecb5ge0 etmeritesrenfinoo: et wrgent etenlava^ ] leur oiceu>:,n arget le rev et en fincarget owe. &* oai&? ] raufc adages et eflat5.tantt>eloiqueoelarget./gten geo^ metric applicqueeauF are mecWique0c5meau):malTon0 dwpetieroetatousaultreeMongnfeaiart&mefurr. etmt FIG. 71. TITLE PAGE OF THE FIRST EDITION OF DE LA ROCHE 1 30 KARA ARITHMETICA Venice, 8; Anonymous, 'Libretto de Abaco,' s. 1., 8; Anonymous, 4 Algorismus novus de integris, de minutiis vulgaribus, de minutiis physicis,' Augsburg. To an edition of Sacrobosco's Sphere the com- mentator, Johannes Guyion, prefixed a treatise on arithmetic, 'De quantitate discreta, 1 Avignon, s. a., c. 1520. There was published, s. 1. a., possibly at Lyons in this year, ' Le liure des gectz grande- ment profitable pour messeigneurs les marchans et aultres,' 4, a work on counter reckoning (see p. 7). ESTIENNE DE LA ROCHE, Villefranche. Ed. pr. 1520. Lyons, 1538. See p. 128. Title. ' Larifmetique & Geometrie de maiftre // Eftienne de la Roche diet Ville Fran//che, Nouuellement Imprimee & des faultes corrigee,// a la qvelle font adiouftees les Tables de diuers comptes, auec leurs Ca-//nons, calculees par Gilles Huguetan natif de Lyon, Par lefquelles on pourra facil-//lement trouuer les comptes tous faictz, tant des achatz que uentes de toutes mar-// chandifes. Et principalement des marchandifes que fe uendent, ou achetent a la // mefure, come a Laulne, a la Canne, a la Toyfe, a la Palme, au Pied, & aultres fem-//blables. Au poix, come a la Liure, au Quintal, au Millier, a la Charge, au Marc,// & a Lonce, a la Piece, au Nobre, a la Douzaine, a la Groffe, au cent, & au Millier.// Auec deux Tables feruantz aulx Librayres uendeurs & acheteurs de papier. En-//femble une Table de defpence, a fcauoir a tant pour iour, combien on defped Lan // & le Moys, & a tant le moys, combien reuient Ian & le iour, & a tant pour an, co-//bien on defpend tous les moys, & a combien reuient pour chafcun iour.// Davantaige, les Tables du fin dor & dargent, pour fcauoir (fcelon que le Marc de billon tiendre//daloy, ou de fin) combien il uauldra de poix de fin or, ou dargent fin.// On les uend . . . lenfeigne de la Sphaere,//cheulx Gilles & Jacques Huguetan freres.// 1538.' (F. i, r.) Colophon. *C.Cy finift Larifmetique i Geometrie de maiftre Eftienne de la Roche diet Villefranche // Imprime a Lyon par maiftre Jacques myt Lan. 1538.' (F. 160, r.) PRINTED BOOKS 131 Description. Fol., 21.1 X 33 cm., the text being 16.3 X 30.6 cm. 2 ff. unnumb. + 158 numb. = 160 ff., 5 9-60 11. Lyons, 1538. Editions. See p. 128. The Huguetan referred to is the one mentioned on p. 188. Ed. pr. 1521. Basel, 1563. ' The Venerable Bede ' was born in England, probably near Wearmouth, Durham, c. 673, and died in 735. He was the most distinguished scholar of his time, and his works cover all the branches of learning then known. Title. ' Opera // Bedae // Venerabi-//lis Presbyte-//ri, Anglo- saxonis : vi-//ii in divinis atqve hv-//manis literis exercitatifsimi : omnia in octo to-//mos diftincta,prout ftatim poft Praefa-//tionem fuo Elencho enu-//merantur.//AdditoRerum & Verborum Indice // copiofifsimo.// Cum Caefareae Maieftatis gratia & priuile-//gio, Regisque Galliarum ad //decennium.//Basileae, per loannem // Heruagium, Anno M. D. LXIII.' (Surrounded by an elaborate woodcut with inscriptions.) (F. i, r.) Description. Fol., 24.2 x 38 cm., the text being 17.7 x 28.6 cm., printed in double columns, each 8.5 cm. wide, 61 11. 152 pp. unnumb. + 271 numbered by columns (i.e., 2 numbers to a page) = 423 pp. in vol. I. 8 vols. bound in 4. Basel, 1563. Only the first volume, containing the arithmetic, is described here. Editions. Some of his arithmetical work is said to have been published in 1521, fol. ; 1 5 2 5 (in part, see p. 140) 51529 (in part, seep. 159); Paris, 154445 (first edition of the Opera) ; ib.,1554; Basel, 1563, fol. (here described). See also c. 1515, anonymous. His ' Historia Ecclesiastica' appeared as early as 1473. The first volume contains the ' De Arithmeticis nvmeris liber ' (cols. 98-1 1 6), with little save an elaborate multiplication table and a dialogue on number, names, and symbols ; ' De Arithmeticis proportionibvs ' (cols. 133-146), with the ' Propofitiones ad acuendos iuuenes' often attributed to Alcuin, but certainly not Baeda's ; 'De ratione calcvli ' (cols. 147- 158), chiefly multiplication tables of Roman money; ' De nvmerorvm divisione ' (cols. 159-163) ; ' De loqvela per gestvm digitorvm, et tem- porvm ratione' (cols. 164-181), or, as the headline states it, 'De indigi- tatione,' giving us almost our only knowledge of the finger reckoning or symbolism of the Middle Ages in western Europe, and possibly spurious ; i 3 2 KARA ARITHMETICA ' De ratione vnciarvm ' (cols. 182-184), a treatise on Roman fractions ; an extensive treatment of the calendar and the computus, with a descrip- tion of the astrolabe. FRANCESCO GHALIGAI. Ed. pr. 1521. Florence, 1552. A Florentine arithmetician of the first part of the sixteenth century. He died February 10, 1536. Title. ' Practica // d'Arithmetica. //di // Francesco Ghaligai // Fiorentino.// Nuouamente Riuifta, & con fomma // Diligenza Riftampata.// In Firenze // Appreffo i Givnti // M. D. LII.' (F. i,r.) Colophon. ' In Firenze // Appreffo i Givnti // M. D. LII.' (F. 114, r.) Description. 4, 15.2 x 20.3 cm., the text being 12 x 17.3 cm. 2 ff. unnumb. + 112 numb. = H4ff., 37-38 11. Florence, 1552. Editions. Florence, 1521, 4 (see Boncompagni's Bulletino, VII, 486; XIII, 249) ; ib., 1 548, 4; ib., 1 5 52, 4 (here described). The 1521 edition is entitled ' Summa De Arithmetica,' but it is the same as the 1552 edition here described. Some bibli- ographers mention other editions, as of 1540, 1551, 1562, 1572, 1582, 1591, but Boncompagni's careful investigation, supported by Riccardi's, throws doubt upon all these. The book is written in the general style of the Italian works of the sixteenth century, more or less resembling Borghi (p. 16). It was intended for the use of merchants, and contains a large number of practical problems showing the conditions of trade at the time of its publication. Books 10 to 13 relate to algebra, and their chief interest attaches to the symbolism employed. Other works of 1521. Boethius, p. 31, 1488 ; Grammateus, p. 124, 1518; Paxi,p. 77, 1503 ; Sacrobosco, p. 32, 1488. CUTHBERT TONSTALL. Ed. pr. 1522. London, 1522. TUNSTALL. Born at Hackforth, Yorkshire, in 1474; died November 18, 1559. He was educated at Oxford, Cambridge, and Padua, was a man of great learning and energy, and held important positions in the Church and State. He was bishop of London, and later of Durham. Title. ' De arte svppvtandi // libri qvattvor // Cvtheberti // Tonstalli.' (Surrounded by a woodcut. F. i, r. See Fig. 72.) PRINTED BOOKS 133 SVPPVTAND BRI QVATTVOR CVTHEBERTI TONSTALLI. FIG. 72. TITLE PAGE OF THE FIRST EDITION OF TONSTALL i 3 4 KARA ARITHMETICA Colophon. ' Impress. Londini in aedibvs Ri-//chardi Pynsoni. Anno ver-//bi incarnati .M.D.XXII.// Pridie idvs octo-//t>ris. Cvm pri-//vilegio a//rege in-//dvl-//to.' (p. 2 O2, r.) Description. 4, 15.7 X 20.9 cm., the text being 10.5 x 16.4 cm. 202 ff. unnumb., 29 11. London, 1522. Editions. London, 1522, 4 (here described); Paris, 1529, 4 (p. 135); ib., 1535; ib., 1538, 4 (P- 135); Strasburg, 1543; ib. f 1544, 8 (p. 136); ib., 1548; ib., 1551. This is the first edition of the first book wholly on arithmetic that was printed in England. (But see p. 10, 1480.) In the dedicatory epistle Tonstall states that in his dealing with certain goldsmiths he suspected that their accounts were incorrect, and he therefore renewed his study of arithmetic so as to check their figures. On his appoint- ment to the See of London he bade farewell to the sciences by printing this book in order that others might have the benefit of a work which he had prepared for his own use. The treatise is in Latin, and, although it was written for the purpose of supplying a practical handbook, is very prolix and was not suited to the needs of the mercantile class. It is confessedly based upon Italian models, and it is apparent that Tonstall must have known, from his residence in Padua and his vari- ous visits to Italy, the works of the leading Italian writers. The book includes many business applications of the day, such as partnership, profit and loss, and exchange. It also includes the rule of false, the rule of three, and numerous applications of these and other rules. It is, how- ever, the work of a scholar and a classicist rather than a business man. The word ' supputandi,' in the title, was not uncommon at that time. Indeed there was some tendency to use the name ' supputation ' for arithmetic and to speak of calculations as ' supputations.' Tonstall dedicates the work to his friend Sir Thomas More, whose talented daughter Erasmus addressed as ' Margareta Ropera Britanniae tuse decus,' ornament of thine England. More speaks of Tonstall in the opening lines of his Utopia : ' I was colleague and companion to that incomparable man Cuthbert Tonstal, whom the king with such universal applause lately made Master of the Rolls ; but of whom I will say nothing ; not because I fear that the testimony of a friend will be suspected, but rather because his learning and virtues are too great for me to do them justice, and so well known, that they need not my commendation unless I would, according to the proverb, " Show the sun with a lanthorn."' The Utopia was first printed in 1516, so this sono- rous praise was written some years before Tonstall's arithmetic appeared. PRINTED BOOKS 135 Some idea of the prolixity of the treatise may be obtained from the number of closely-printed quarto pages assigned to certain topics. The chapter * De Numeratione ' fills n pages, ' De Additione ' 14 pages, ' De Svbdvctione ' 15 pages, ' De mvltiplicatione ' 14 pages, ' De parti- tione' 27 pages (the old galley method being used exclusively), and so on for the other subjects. Some 66 pages, for example, are given to the theory of ratio and proportion. The title page was engraved by Holbein, and was evidently printed after the book was completed, because in this copy the errata appear on the reverse of the first folio. The work was printed by Richard Pynson, the successor to Caxton. Following the arithmetic is an appendix : Appendix ex Bvdaei libro de as-//fe excerpta : in qua prifca Latinorum et Grecoru // fupputatio, ad aeftimationem pecunie, turn Gallicae,// turn Angli-//cae reuocatur.' Tonstall also published a work in 1518: 'In Lavdem matrimonii oratio,' second edition in 1519, now very rare. CUTHBERT TONSTALL. Ed. pr. 1522. Paris, 1529. See p. 132. Title. 'De arte svppvtandi libri qvatvor//Cvthberti Tonstalli. (Picture of a tree from which is falling a broken branch, and the words: Nolialtum fa//pere, fed time.) Parisiis ex officina Roberti Stephani//M.D.XXIX.' (P. i.) Colophon. ' Parisiis //excvdebat Robert vs Stepha-//nvs. Ann. M.D.XXIX. Prid. id. ivn.' (P. 279.) Description. 4, 13 X 18.8 cm., the text being 9.4 x 15.9 cm. 271 pp. numb. + 8 unnumb. + 2 blank = 281 pp., 36-38 11. Paris, 1529. Editions. See p. 134. The text is the same as in the first edition of 1522. There has been added, however, a second appendix with the following title : ' Gvlielmi Bvdaei Parisiensis,// secretarii regii, breviari-//vm de asse.' CUTHBERT TONSTALL. Ed. pr. 1522. Paris, 1538. See p. 132. Title. ' De arte svp-//putandi libri qua-//tuor, Cutheberti // Tonftalli.// (Large woodcut.) Parisiis.// Ex officina Roberti Ste- phani.//M.D.XXXVIII.' (P. i.) 136 KARA ARITHMETICA Colophon. * Excvdebat Robertvs Stephanvs Parisiis, // ann. M. D. XXXVIII.//xvi. cal. novemb.' (P. 259.) Description. 4, 14.4 X 20.1 cm., the text being 9.5 x 16 cm. 259 pp. numb., 39 11. Paris, 1538. See p. 134. CUTHBERT TONSTALL. Ed. pr. 1522. Strasburg, 1544. See p. 132. Title. ' De arte //svppvtan//di, libri qvatvor//Cvthberti Ton- stalli,//hactenus in Germania nus-//quam ita impreffi.// loan. Stvrmivs.//Arithmeticam Cvthbertvs // Tonftallus prae caeteris dilucide & pure tradidit : atq3 // ita tradidit, ut ars ipfa dum hie author extat, con-//tenta fcriptore, doctorem non maximopere aliquem // requirat. Non nego, poffe ex alijs quoque difci : // fed hie docet erudite, perfpicue latine, id quod non fa//ciunt caeteri: nee abest longe a perfectione, qui eius // praecepta intelligit.// Argentorati, ex offi.// Knobloch. per Georg. Machaerop.' (P. I.) Colophon. * Argentorati, ex officina//Knoblochiana, per Ge-// orgivm Machaero-//poevm, mense //febrvario//anno,//M. D. XLIIII.' (P. 478.) Description. 8, 9.5 x 15 cm., the text being 6.8 x 11.4 cm. 25 pp. blank + 453 numb. = 478 pp., 26 11. Strasburg, 1544. This copy is bound with the arithmetic of Victorius Strigelius. Editions. See p. 134. This is the same as the edition of 1 529, having the second appendix there mentioned. It is the second Strasburg edition. It is interesting to see that the classical influence on the Continent was such that seven out of the eight editions appeared in Paris or Strasburg. JOHANNES AVENTINUS. Ed. pr. 1522. Regensburg, 1532. THURNMAYER. Born at Abensberg, Bavaria, July 4, 1477 ; died at Regens- burg, January 9, 1534. He wrote on history. Title. See Fig. 73. Colophon. ' Ratifpon^ apud loannem Khol // Anno. MD- XXXII.' (Large woodcut and date, 'Io. Kol 1532.') (F. 12, r.) PRINTED BOOKS 137 Description. 4, 14.2 x 19.5 cm., the text being 9.9 x 14 cm. 12 ff. unnumb., 26 11. Regensburg, 1532. Editions. Niirnberg, 1522; Regensburg, 1532, 4 (here described). The book is primarily a treatise on numerical finger symbolism, and contains the most complete explanation of that subject extant. It gives QVE VETVSTISSIMA, VETERVM latinorumperdigitos manufcpnume* randi(quinetiam loquendi) cofue* tudo^Exbedacupiduris & ima* ginibus, inucta reginoburgij flue ractobonae r ,in biblio# thcca diui hacmerani, Atcp hoc conuc* tu auguftali Rcucrendi Atcp do&iflimi Domini Lucac bonfi] de# cani patauini fccretarfj Rcucrcndifli^ miCardinalis Laurcntrj Campcgf] zc^ Aufpkijs AIo* Aucn/ tino Edita* Gcrmania IlluflrancJa* FIG. 73. TITLE PAGE OF THE 1532 AVENTINUS illustrations showing the representation of the numbers up to one million by means of the fingers and arms (see Fig. 74). This finger symbolism is found in the works of Baeda, it was practical in both the East and 138 KARA ARITHMETICA West during the Middle Ages, and it is mentioned by several sixteenth- century arithmeticians. (Compare Fig. 74 with Fig. 35, p. 57.) r b jj FIG. 74. FINGER SYMBOLISM FROM AVENTINUS ADAM RIESE. Ed. pr. 1522. Leipzig, 1538. RYSE, Ris, RIES. Born at Staffelstein, near Bamberg, c. 1489; died at Annaberg, March 30, 1559. One of the most celebrated Rechenmeisters of the sixteenth century, and the most influential of all the Germans in replacing the counter reckoning (' auff der Linien ') by the written computa- tions (' auff Federn '). Title. ' Rechnung auff // der Linien vnd Federn //Auff aller- ley handtirung ge-//macht/ durch Adam Rifen.// (Woodcut of counting house, with reckoning on the line abacus.) Item auffs new vberfehen vnd //an viel orten gebeffert.// M.DXXXVIII.' (F. i,r.) Colophon. ' Gedruckt zu Leiptzigk // durch Valentinum // Schumann. 1538.' (F. 63, v.) Description. 8, 9.7 X 15.1 cm., the text being 7.2 x 1 1.5 cm. 63 ff. unnumb. + 2 blank (with manuscript notes) = 65 ff., 28- 33 11. Leipzig, 1538. Editions. In order to make clear the different editions of Riese's work, it is necessary to distinguish between the four arithmetics which he published. These were as follows : PRINTED BOOKS 139 1 . * Rechnung auff der linihen gemacht durch Adam Riesen vonn Staffelsteyn // in massen man es pflegt tzu lern in alien rechenschulen gruntlich begriffen anno 1518.' (Graesse, followed by Unger, p. 50, who knew of no extant copy.) A second edi- tion appeared in 1525 (' Getruckt tzu Erffordt durch Mathes Maler M. CCCCCxxv Jar/ 8, 43 ff.), and a third in 1527. This was embodied in his second arithmetic which is here described. The work is rare. 2. 'Rechnung auff //der Linien vnd Federn/ Riese's best- known work. The title of the first (i 522) edition was as follows: ' 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 vbersehen/ vnd alle gebrechen eygentlich gerechtf ertigt/ vnd zum letzten eine hiibsche vnderrichtung angehengt.' The following editions of this work are known to me: Erfurt, 1522; ib., 1525, 8; Niirnberg, 1527; 1528, 8; Erfurt, 1529, 8; ib., 1530; Leip- zig* I 533> 8 (first edition containing Helm's Visirbuch ; see p. 142); Frankfort, 1535, 12; Annaberg, 1535, 8 (p. 141); 1536, 12; Leipzig, 1538, 8 (here described); 1541; Frankfort, 1544, 8; Leipzig, 1544; s. L, 1548, 8; Leipzig, 1548,8; ib., 1550; Breslau, 1550; Frankfort, 1552, 8; Leipzig, 1554; 1556, 12; Frankfort, 1558, 8 (p. 141); Leipzig, 1562, 12; Frankfort, 1564, 8; Frankfort, 1565, 8 (p. 142); Frankfort (a. Oder ?), 1568, 8; Stettin (Frankfort?), 1570, 8; Leipzig, 1571, 8 (p. 142); Frankfort, 1574; Magdeburg, 1579, 8; Frankfort, 1581, 12; ib., 1585, 8; Leipzig, 1586; Frankfort, 1586; Wittenberg, 1587; Ntirnberg, 1592, 8; Frankfort, 1592, 8 (p. 143); Leipzig, 1598, 12. There were several editions after 1600. It is possible that some of the editions here mentioned may be of Riese's fourth book, the titles being much alike and printers varying them from time to time. 3. ' Ein Gerechent Blichlein,' first published at Leipzig in 1533; second edition in 1536. See p. 171. i 4 o RARA ARITHMETICA 4. ' Rechnung nach der lenge/auff den Linihen vnd Feder,' first published in 1550, 4. See p. 250. Kuckuck's remark, that over twenty-six editions of Riese's arithmetics appeared, greatly underestimates the number. More than forty appeared in the sixteenth century alone, and several were published in the seventeenth century. This was probably the most popular commercial arithmetic of the sixteenth century. So firmly did it impress itself upon the schools that ' nach Adam Riese ' is a common expression in Germany to-day, nearly four hundred years after the first of his books appeared. It was to Germany what Borghi's book was to Italy and Recorde's to England. It differed from Riese's first book in that it emphasized computation by the aid of the Hindu-Arabic numerals instead of the counters. There is no other book that gives as good a picture of the sixteenth- century mercantile problems of Germany, and of the methods of solving them. Other works of .1522. Albert of Saxony, p. 9, c. 1478 ; Boethius, p. 27, 1488; Budaeus, p. 99, 1514; Kobel, p. 102, 1514; Ortega, p. 93, 1512 ; Sacrobosco, p. 32, 1488; Francisco Pelacani, 'Arithmetica prattica,' Florence ; Ludovico Vincento (Vincentino) degl' Arrighi, ' La operina ... da ... bellissima Ragione di Abbacho,' Rome, 4, with editions at Venice in 1532, 1533, chiefly on chirography. Works 0/1523. Borriglione, p. 86, 1506; Ciruelo, p. 60, 1495; Grammateus, p. 124, 1518; Paciuolo, p. 54, 1494; Reisch, p. 82, 1503; Sacrobosco, p. 35, 1488; Tagliente, p. 114, 1515. Rodrigo Fer- nandez de Santaella (or Valencia, see p. 269, 1555), ' Ars coputandi,' Saragossa, fol. There was also written in this year, but published s. 1. a., a work by Vincenzo Barziza entitled ' Operetta nouamente composta,' 8, 39 ft 7 ., containing a few mercantile rules and tables. Works of 1524. Feliciano, p. 145, 1526; Tagliente, p. 115, 1515. Works 0/1525. Herodianus, p. 60, 1495 ; Kobel, p. 102, 1514; Riese, p. 139, 1522 ; Rudolff, p. 151, 1526; Tagliente, p. 114, 1515 ; Bede et al., ' Valetius Probus et Petrus Diaconus de notis Romanorum, Demettius Alabaldus de minutiis, ponderibus et mensuris, Ven. Beda de computo per gestum digitorum,' etc., Venice, 4; Angelus Mutinens (i.e. of Modena), 'Thesavro de Scrittori opera artificiosa le quale con grandissima arte, si per pratica come per geometria insegna . . .,' s. 1., M.D.XXxv, has four folios at the end relating to arithmetic, with the note 'Angelus Mutinens composuit,' and there seems to have been an edition in 1525, and another s. a. published at Rome; Giovanni PRINTED BOOKS 141 Tagliente published two editions of a work chiefly on bookkeeping, 4, 24 ff., Venice, beginning, ' Considerando io loanni Taiente quanto e necessaria cosa ali nostri magnifici getilhomeni & ad altri mercatanti.' ADAM RIESE. Ed. pr. 1522. Annaberg, 1535. See p. 138. Title. The title page is missing. Colophon. 'CNach difer vnderrichtung kanftu auffs be//hen- deft alle Exempel in der Falfi machen : W61//left folch Buch- lin vnnd kurtze erklerung ietzt/ // welches ich zum andern mal laffe aufzge-//hen/ zu danck an nemen/ wil ich ver-//dienen/ vnd dir auffs eheft ich // mag die Practica nach al-//lem fleifz heraufzftrei//chen. Datum //auff fanct // Annaberg/ // Dinftag nach // Martini. Im Iar//M. D. XXV.' (F. 55, r.) On f. 69, v., is the following colophon: ' Alfo ift ktirtzlich // befchriben vnd // begriffen // die Confection // der Vifier riiten mit // Irer iibung vnd gebrauch.//LEnd:// An. M. D. XXXV.// Im Christ- monat.' This latter is evidently the date of printing of the entire book. Description. 8, 8.8 x 13.2 cm., the text being 6.8 x 1 1.6 cm. 69 ff. unnumb., 31 11. Annaberg, 1535. Editions. See p. 139. This contains the ' Vifirbuchlin ' of Erhart Helm, as in the 1533 edition, but it does not give his name. See the 1565 edition (p. 142). ADAM RIESE. Ed. pr. 1522. Frankfort, 1558. Seep. 138. Title. The title page is missing. Page numbered 2 begins as follows : ' Vorrede in difl Rechen-//h>uch/ Adam Rifen.' Colophon. 'C.End.//Zu Franckfurt bei Chr. Egeb. erben/// Anno 1558.' (F. numb. 87, r.) Description. 8, 9.4 X 15 cm., the text being 6.5 X 12.2 cm. 87 ff. numb. + i blank = 88 ff., 31 11. Frankfort, 1558. Editions. See p. 139. This contains the * Vifirbuchlin ' of Erhart Helm, as in the 1533 edition. i 4 2 RARA ARITHMETICA ADAM RIESE. Ed. pr. 1522. t Frankfort, 1565. See p. 138. Title. ' Rechenbuch/ Vff Lini//en vnnd Ziphren/ In allerley // Handtierung/ Gefchefften vnd Kauff-//mannfchafft. Durch Adam // Rifen.// Mit new en kunftlichen Regeln vnd Ex-// emplen gemehrt/ Innhalt furge-//ftelten Regifters.//Vifier vnd Wechfelruten kunftlich vnd // gerecht zumachen/ autf dem Qua- drat/ // Durch die Arithmetic vnd Geometri. // Von Erhart Helm/ Mathema//tico zu Franckfurt/ be // fchriben.// Alles von newem jetzund widerumb erfe-//hen vnd corrigirt.//( Woodcut of counting house) Franck, Bei Chr. Egen. Erben. 1565.' (F.i,r.) Colophon. 'M. D. LXV.' (F. 113, r.) Description. 8, 9.3 X 15.5 cm., the text being 6.5 X 1 1.9 cm. 105 ff. numb. 4- 8 unnumb. = 1 13 ff., 28 11. Frankfort, 1565. Editions. See p. 139. See p. 140. The Visirbuch of Helm begins, with no separate title page, on f. 7 7, v. : * Vifirbuchlin // Hernach folget der ware // Procefz/ vnnd kurtzeft weg/ wie mann Vifir riilhen machen fol/ aufz dem // Quadraten/ Auff alle Eich.' The name of the author, 'Erhart Helmen,' appears in the headlines of each folio recto. This part of the book is strictly speaking not an arithmetic, but it includes a few explanations of those processes that are necessary in gauging. It also includes a table of square roots to the equivalent of three decimal places, and a brief explanation of roots. It was published in separate form in 1529. ADAM RIESE. Ed. pr. 1522. Leipzig, 1571. See p. 138. Title. ' Rechnung auff // der Linien vnd Federn/ // auff allerley Handtierung/ //Gemacht durch//Adam Rifen. //(Wood- cut of Adam Riese, with motto : 'Anno 1550 Adam Ries Seins Alters Im LVIII.') Auffs newe durchelefen/ vnd // zu recht bracht.// 1571.' (F. i, r.) Colophon. 'Zu Leipzig druckts // Hans Rhambaw/ //Im Jar// 1571-' (F. 94, v.) Description. 8, 9.7 x 1 5.6 cm., the text being 6.9 X 1 1.8 cm. 94 ff. unnumb., 24 11. Leipzig, 1571. PRINTED BOOKS 143 Editions. See p. 139. See p. 140. This is substantially identical with the 1538 edition except as to the title page. / ADAM RIESE. Ed. pr. 1522. Frankfort, 1592. See p. 138. Title. See Fig. 75. Description. 8, 9.5 X 15.5 cm., the text being 6. 7 X n.8 cm. 1 f. unnumb. + 79 numb. = 80 ff., 26-27 11. Frankfort, 1592. Editions. See p. 1 39. Bound with this is Helm's work of 1 592 (described later). FRANCESCO DAL SOLE. Ed. pr. 1526. Ferrara, 1546. A French arithmetician, born c. 1490, and living in Ferrara at the time of writing his books. Title. See Fig. 76. Colophon. ' In Ferrara Nella Stampa di .M. Giouanni de bugl- hat &. M. Antonio // Hucher Compagni, Ad Inftantia de .M. Rinaldo, cuoco dello Illuftriffi-//rno fignor Duca, nel mefe di zenaro 1546.' (F. 42, v.) Description. 4, 14 X 19 cm., the text being 12.1 x 16.5 cm. 2 ff. unnumb. +40 numb. =42 ff., 31 11. Ferrara, 1546. Editions. Sole published a ' Libretto di Abaco ' in Venice in 1526, 8, and this is merely a revision of that work. A third edition appeared in 1564 (see p. 146). This is little more than a primer of arithmetic. It contains the funda- mental operations, a few of the more important applications, eight pages of products and roots, and several pages on astrology. The part on astrol- ogy includes some theory of the calendar, as may be seen by the title : ' Incominciano le regoline daftrologia, p ritrouare ha quati di et minute fa la luna, la Ira dnicale, et infinite getileffe, Belli circuli, elementi, et natura, del monde.' The most distinctive feature of the arithmetic is the combination of number and space concepts. For example, in addi- tion the author considers not only abstract but compound numbers as well as geometric magnitudes. (' Regola dellae addition! in generalita, tan to geometrica, quanto arithmetica. Ca. 6.') The same idea is carried out in the other fundamental operations. 144 KARA ARITHMETICA rtu manfcDrtfft / mtt ncnwtn ffinfU FIG. 75. TITLE PAGE OF THE 1592 RIESE PRINTED BOOKS '45 lnfirnfinj.nn.llll ilflfin.tinillULDnrn-in nn luunnn nnr. On LIBRETTI NVOVI CON LE RE* co?e Di Francefco DalSok Gallo, Ne^ui, Mali* ante ww/or/, Geowefr/cz, R tociwati) Sciittorijet ludi /^ luoshifottrarmo^o Ireue ioy infcmarfi tktti:Jettefdar%e t et one tib& on adJitioni Afromwore talijDdia terragj del fuo fine^ da ritrouare la. cotjgiontione } ^y oppofit tionejdella fanajefifle mobile et ir mobile Jettere dominicale, i Jo } et de amo in anno in ^prfw, J^uoiMmente calculati etpo* ft in Tucefon gratia ft Pr/0 w/fcogio Df4^ 'Per am diece^et pea ducati if coeieffacrdficotiene M D ' XL VI FIG. 76. TITLE PAGE OF THE 1546 FRANCESCO DAL SOLE 146 RARA ARITHMETICA FRANCESCO DAL SOLE. Ed. pr. 1526. Ferrara, 1564. See p. 143. Title. ' Instrvtioni // et Regvle // di Francesco // dal Sole,// Francese.//Cittadino di Ferrara, Sopra il fon-//damento delle alme fcientie d'Abbac-//co, Arithmetica, Geometria, Cof-//mo- grafia, & Mathematica, No-//uamente riftampate, & con//par- ticolare addittioni di//effo Authore,//aggionte.//In Ferrara, Aprefso Francefco di Rossi //da Valenza.//M. D. LXIII.' (F. i , r.) Description. 4, 14.8 X 20 cm., the text being I i.i x 15.5 cm. 4 pp. unnumb. + 71 numb. = 75 pp., 38 11. Ferrara, 1564. Editions. See p. 143. Although the title of this edition is quite different from that of the first, as shown in Fig. 76, the work is essentially the same. A set of verses entitled ' II Sole ' (The Sun), a play upon the author's name, which appeared in the first edition, gives place to some Latin lines in this one. FRANCESCO FELICIANO da Lazesio. Ed. pr. 1526. Venice, 1526. Born at Lazisa, near Verona; he was living in 1563. Title. See Fig. 77. Colophon. ' <LStampato nella inclita Citta di Vinegia, apreffo // fanto Moyfe nelle cafe nuoue luftiniane: Per // Fracefco di Aleffandro Bindoni, & Ma-//pheo Pafmi, compagni. Nelli anni // del fignore, 1527. Del mefe // di Zenaro. Regnante il // Sere- niflimo Princi-//pe meffer An-//drea Gritti.// A B C D E F G HIKLMNOPQRST V.//Tutti fono duermV (F. 80, r.) Description. 4, 15.1 x 20.8 cm., the text being I3-4X 17.1 cm. 80 ff. unnumb., 41 11. Venice, 1526. Editions. Feliciano published two works, of which the first was entitled ' Libro de Abaco,' and appeared in the following editions: Venice, 1 5 1 7 (the colophon date is 1518), 8; ib., 1519, 8; ib., 1524, 8; ib., 1532, 8. His second work was a revision of his first, and is the one here described, and this appeared in the followingeditions: Venice, 1526,4 (here described); ib., 1527,4; ib., 1 5 36, 4 (P- 148) lib., 1 545, 4 (P- H9); ib., 1550, 4 (p. 149); jfraiKdco fdfdano oa Xa5tfio eronde ^ntitulato Ocala gramaUelli: Wouamence ftampato* i \ s w L 3 Chi vol aprir vna ferr agli a forte In rim a d' vna Torre ouer Caftello Bifogna hauer la Scala accio di quello Gionger f? pofla alle ferrate porte Aprcllo fa meftier Phuom feco porte CNon hauendo la Chiaue del fugello) V n ferro die apellato e grimaldello -Si non d'^prir Fimprefe Hie fien carte, Tal quefto mio libretto moftra come SagUr f? pofla aJlalte ferrature Et quclle aprir fenza fatica graue, Infinite ragion,pon ri,& m ifu re Quefto chiarilce con modo fuaue Cui fcala e grimaldello e il proprio no me, Ma non lodar mie come Co'l tuolieto biafmar,chel biafmo nafce Sol de cui nel mal dir s'alegra e pafce* M* D. XXVI* Con gratia tpziiiilegia FIG. 77. TITLE PAGE OF FELICIANO 148 KARA ARITHMETICA ib., 1560, 4 (p. 149); ib., 1560, 4 (another edition); ib., 1561 (the colophon of one 1560 edition); Verona, 1563, 8 (p. 150); s. 1. (Venice ?), 1563, 4; Venice, 1570. There were also seven- teenth-century editions extending as late as 1692. For the 1602, 1629, and 1669 editions see pp. 150, 151. This first edition was evidently begun in 1526, but completed in January 1527, as the colophon shows. It is often assigned to the latter year. Feliciano's second work was highly esteemed as a textbook for schools. It follows the lines laid down by Borghi (p. 16), and the author acknowledges his indebtedness to him and to Paciuolo (p. 54). The first part of the book is commercial in character, and in the second part the author treats of roots, rule of false, and algebra, the third part being devoted to geometry from the practical side. More complete than the Treviso book, more modern than Borghi, more condensed and practical than Paciuolo, few books had greater influence on the subse- quent teaching of elementary mathematics. The fanciful name, ' Scala grimaldelli,' is explained in the verses on the title page. Just as it is necessary in attacking a castle to have a ladder (scala) and a skeleton key (grimaldello) to open locks, so in attacking mathematics it is nec- essary to have a book that answers the same purposes. FRANCESCO FELICIANO da Lazesio. Ed. pr. 1526. Venice, 1536. See p. 145. Title. ' Libro di Arithmetica i Geometria // fpeculatiua : pra- ticale : Compofto per maeftro // Francefco feliciano da Lazifio Veronefe//Intitulato Scala Grimaldelli : // Nouamente ftampato.' (F. r, i. The rest is substantially as in the first edition, Fig. 77.) Colophon. ' CStampato nella inclita Citta di Vinegia, apreffo // fanto Moyfe nelle cafe nuoue luftiniane : Per // Fracefco di Aleffandro Bindoni, & Ma-//pheo Pafini, compagni. Nelli anni // del (ignore. 1536. Del mefe //di Zenaro. Regnante il//Sereniffi- mo Princi-//pe meffer An-//drea Grittti.// ABCDEFGHIKLM NOPQRSTV.//Tutti fono duerni.' (F. 80, r.) Description. 4, 15.6 x 20.8 cm., the text being 13.4 x 17.1 cm. 80 ff. unnumb., 41 11. Venice, 1536. See above. PRINTED BOOKS 149 FRANCESCO FELICIANO da Lazesio. Ed. pr. 1526. Venice, 1545. See p. 146. Title. This is substantially the same as in the 1526 edition (P- 147). Colophon. With the exception of the date (1545), this is sub- stantially as in the 1536 edition. Description. Substantially as in the 1 5 36 edition. Venice, 1 545. Editions. See p. 146. FRANCESCO FELICIANO da Lazesio. Ed. pr. 1526. Venice, 1550. See p. 146. Title. This is substantially the same as in the 1526 edition. Colophon. ' Stampato nella inclita Citta di Vinegia, per // Francefco Bindoni, & Mapheo Pafmi,// Nelli anni del noftro Signore.//M. D. L.// Registro.// A B CDEFGHIKLM NO//PQRST V.//Tutti fonno duerni.' (F. 80, r.) Description. 4, 15.6 x 20.8 cm., the text being 13.4 x 17.2 cm. 80 ff. unnumb., 41 11. Venice, 1550. Editions. See p. 146. FRANCESCO FELICIANO da Lazesio. Ed. pr. 1526. Venice, 1560-61. See p. 146. Title. This is substantially the same as in the 1526 edition (p. 147), but bears the date M.D LX. Colophon. 'Stampato nella Inclita Citta di Vinegia, Per Fran- //cefco de Leno. Nell' anno del N. Signore.// M. D. LXI.' (F. 79, r.) Description. 4, 15.9 x 21.1 cm., the text being 14.3 x 17.4 cm. 79 ff. unnumb., 41 11. Venice, 1560 (colophon 1561). Editions. See p. 146. Riccardi mentions two identical editions of this year, one of them, here described, with the colophon date 1561. i 5 o RARA ARITHMETICA FRANCISCO FELICIANO da Lazesio. Ed. pr. 1526. Verona, 1563. See p. 146. Title. This is substantially the same as in the 1561 edition. Description. 4, 14.9 x 19.4 cm., the text being 13.6 X 17.1 cm., 41 11., 64 if. unnumb. (part III missing). Verona, 1563. Editions. See p. 146. The third part, on geometry, is missing in this copy. FRANCESCO FELICIANO da Lazesio. Ed. pr. 1526. Verona, 1602. See p. 146. Title. This is substantially the same as in the 1526 edition except ' De nuouo riftampato, & da molti errori corretto, & ac- crefciuto di molte cofe da M.// Fillipo Marcario Veronefe Rafo- nato publico della Magnifica Citta.// Con le gionta della Regola del Catain del medefimo.//In Verona, Apreffo Dionigi Filiberi. CO 13 C II.' (P. i.) Colophon. 'In Verona,// N el la Stamparia di Angelo Tamo. 1602.' (P. 284.) Description. 4, 14.7 x 19.2 cm., the text being 11.5 x 15.7 cm. 5 pp. blank + 7 unnumb. + 276 numb. = 288 pp., 32 11. Verona, 1602. FRANCESCO FELICIANO da Lazesio. Ed. pr. 1526. Padua, 1629. See p. 146. Title. ' Scalla // Grimaldelli // Libro di // Aritmetica, et Geo- metria // Speculatiua, & Pratticale // Di M. Francefco Feliciano Veronefe.//Diviso in tre libri.// ... In Padoua, Per Donato Paf- quardi, & compagni. i629.//Con licenza de' Superiori.' (F. I, r.) Description. 4, 15 x 20.5 cm., the text being 1 1.4 X 16 cm. 4 pp. unnumb. + 276 numb. =^280 pp., 32-34 11. Padua, 1629. Editions. See p. 146. PRINTED BOOKS 151 FRANCESCO FELICIANO da Lazesio. Ed. pr. 1526. Venice, 1669. See p. 146. Title. ' Scala // Grimaldelli // libro di // aritmetica, e geo- metria // Speculatiua, e Pratticale // Di M. Francesco Feliciano // Veronese. //Diviso inTre Libri. . . . Di nuouo riftampato, e da molti errori corretto, & accrefciuto di molte cofe da // M. Filippo Macario Veronefe Rafonato publico della Magnifica // Citta. Con 1'aggionta della Regola del Catain del medefimo.// Al Molt' Illuf- tre Signor, e Padron Colendifs. il Signer Gio: Battista Sorer.// Venetia, MDCLXIX.// Preffo Gio: Giacomo Hertz.' (F. i, r.) Description. 4, 16. i X 22 cm., the text being 1 1.8 x 16 cm. 6 pp. blank + 6 unnumb. + 240 = 252 pp., 41 11. Venice, 1669. Editions. See p. 146. See p. 148. It speaks well for this work of Feliciano's that this edition should have appeared one hundred and forty-three years after the book was first published. CHRISTOFF RUDOLFF. Ed. pr. 1526. Nurnberg, 1534. Born at Jauer c. 1500, but the dates of his birth and death are unknown. Title. * Kunftliche rech//nung mit der ziffer vnnd mit // den zal pfenninge // fampt-// der Wellifchen Practica // vnd allerley vorteil // auff die Regel de Tri.// Item vergleichug mancher-// ley Land vii Stet // gewicht/ Elnmas//Muntz ec. Alles durch // Chriftoffen Rudolff zu/ Wein verfertiger.// 1534.' (F. i, r.) Colophon. ( Getrukt zu Nurmberg bey//Johan Petreio//im iar nach // der geburt Chrifti // M.D.xxx 1 1 1 1 .' (F, 1 20, r.) Description. 8, 10.2 x 14.9 cm., the text being 6.7 x n.8 cm. 1 19 ff. unnumb. + i blank = 120 ff., 31 11. Nurnberg, 1534. Editions. Rudolff published three books as follows : (i) the Coss, an algebra, in 1525 (see p. 258 for the Stifel edition of I SS3)y ( 2 ) the Kiinstliche Rechnung, here described; (3) a col- lection of problems in 1530 (see p. 159). Of the Kiinstliche Rechnung the following editions appeared in the sixteenth cen- tury : Vienna (Nurnberg ?), 1526, 8; Nurnberg, 1532, 8; ib., i 5 2 KARA ARITHMETICA 1 5 34 (here described); ib., 1537,8; ib., 1540 (below); ib., 1546; Nurnberg, 1553, 8 (below); ib., 1557 (p. 153); Vienna, 1561 ; Vienna (Augsburg ?), 1574; Augsburg, 1588, 8. This work is an extension of the first part of the Coss, and is divided into three parts : (i) Grundbuchlein, the fundamental operations with abstract and concrete numbers, integers, and fractions, with and without the abacus ; (2) Regelbiichlein, the rule of three (' Regel de Tri ') and Welsch practice (* Wellisch rechnung') ; (3) Exempelbiichlein, problems and results. It was one of the best-known of the practical arithmetics of that period. The rule of three is esteemed highly by Rudolff, for he says : * sie befchleufzt in fich die aller niitzlichfte Regel, dadurch unzeli- che rechnung in kauffen und verkauffen aufzgericht werde.' Of the Ital- ian method of solving applied problems, the ' Welsch practice,' he says : ' Dieweil die Wellifch rechnung nichts anderes ift, dan ein gefchwinder aufzug in der Regel de Tri gegriindet, wirt fie auch derhalben practica gefproche.' Other works 0/1526. Ciruelo, p. 60, 1495 ; Blasius, p. 97, 1513 ; Tagliente, p. 114, 1515; Widman, p. 37, 1489. Sterner mentions an anonymous Rechenbuchlein as printed this year at Nurnberg. CHRISTOFF RUDOLFF. Ed. pr. 1526. Nurnberg, 1540. See p. 151. Title. This is substantially the same as in the 1534 edition, but bears the date 1540. (F. i, r.) Colophon. ' C.Getruckt zu Nurmberg bey Johan//Petreo/ Anno M. D. XL.' (F. 117, r.) Description. 8, 8.8 X 13.9 cm., the text being 6.8 x 12 cm. 117 ff. unnumb., 30 11. Nurnberg, 1540. Editions. See above. CHRISTOFF RUDOLFF. Ed. pr. 1526. Nurnberg, 1553. See p. 151. Title. ' Kunftlich rech-//nung mit der ziffer vnd mit//den zal pfenningen/ fampt der // Wellifchen Practica/ vnd allerley// fortheyl auff die Regel // De Tri.// Item vergleichung manch-// erley Gewicht/ Elnmas/ MiSntz 2c. auff //etlich Landt vnd Stett. //Gemehrt mit 293 Exempeln/von man-//cherley Kauffhendeln/ PRINTED BOOKS 153 mit erklerung/ wie // die felben zu machen vnd in die // Regel zu fetzen fein.//Auffs new widerumb fleiffig vberfehen/ //vnd an vil orten gebeffert.// Alles durch Chriftoffen Rudolffzu// Wien verfertiget.// 1553.' (F. I, r.) Colophon. ' Gedruckt zu Nurmberg/ durch // Gabriel Hayn.// 1553.' (F. 206, v.) Description. 8, 10 x 15.7 cm., the text being 6.8 X 1 1.8 cm. 206 ff. unnumb. + 2 blank = 208 if., 24-26 11. Nurnberg, 1553. Editions. See p. 152. As the title states, this is a revision of the 1526 book, with some added matter and a considerable number of new examples. The new matter begins on f. T 8. The book closes with a list of gauger's char- acters, 'die vifier ziffer,' not found in the 1534 edition, the integers being represented by what are practically the mediaeval numerals, and the fractions being generally unit fractions. CHRISTOFF RUDOLFF. Ed. pr. 1526. Nurnberg, 1557. See p. 151. Description. This edition of RudolfPs arithmetic is substan- tially verbatim with that of 1553 (p. 152). 8, 9.5 x 15 cm., the text being 7X n.8 cm. 206 ff. unnumb., 26 11. Nurnberg, 1557. ANONYMOUS. Ed. pr. 1527. Cologne, 1527. Title. * Compendia-//ria ar ti s nvmerandi ratio, et // expeditif- fima practicandi uia, figuris Arithme//ticis omnes numeroru for- mulas coprehen//dens, additis quibufdam, ut raris, ita // utilibus regulis.// Radicis Cvbicae Extractio.// 13886 151 Cubus 653534 29952 Radix 40 2 I 2 I 2 O. 9648 o 8 6 38862144.' (F. I, r.) Colophon. ' Coloniae apvd Melchiorem // Nouefienfem Anno .M. .D. XXVII.//Menfe maio.' (F. 29, r.) i 5 4 KARA ARITHMETICA Description. 4, I3-4X 19.4 cm., the text being 8.9 x 14.3 cm. 29 ff. unnumb., 32 11. Cologne, 1527. Editions. There was no other edition. The dedicatory epistle is dated ' Coloniae. Anno 1527. Calendis Maijs.' This extremely rare little work, almost unknown to bibliographers, begins with a theoretical discussion of the nature of number and arith- metic. This is followed by a ' compendiaria . . . artis numerandi ratio in tres tractatus digesta.' Of these the first treats of the ' species ' and the rule of three in integers ; the second of fractions, and the third of business problems. The work is too theoretical to have had any influence on commercial arithmetic. JOHANNES WOLPHIUS. Ed. pr. 1527. Frankfort, 1534. A German mathematician, born c. 1500. Title. ' Rvdi//menta Arithmetices // Authore lohanne Vuolphio // Herfbrugienfe.// Elemen//tale Geometricvm, ex // Euclidis Geometria, a loanne Vcegelin,// Haylpronnenfi, ad omnium Mathe//matices studioforum utili-//tatem decerptum. // Franc. Chriftianus Ege-//nolphus excudebat.' (The title page is surrounded by an elaborate woodcut. F. i, r.) Colophon. At the end is the date 'M.D.XXXIIII.' (F. 56, r.) Description. 8, 9.7 X 15 cm., the text being 6.8 X 11.7 cm. 56 ff. unnumb., 2429 11. (The arithmetic occupies only 27 ff.) Frankfort, 1534. Editions. Niirnberg, 1527; Frankfort, 15 34, 8 (here described); ib., 1537; Strasburg, 1539, 8; ib., 1540; Frankfort, 1548, 8 (below) ; ib., 1561. This brief treatise on arithmetic covers the work required in some of the Latin schools of the sixteenth century. It contains little besides numeration and the fundamental operations, including duplation, media- tion, the rule of three, and fractions. There are only a few applications, coinage and partnership being the most prominent. JOHANNES WOLPHIUS. Ed. pr. 1527. Frankfort, 1548. See above. Title. ' Rvdi-//menta Arithmeti-//ces, Autore loanne Vuol-// phio Herfbru-//gienfe.// Elementa-//le Geometricvm,// Ex PRINTED BOOKS 155 Euclidis Geometria, a loanne // Voegelin, Haylpronnenfi, ad o-// mnium Mathematices ftudio-//forum utilitatem de-//cerptum.// Franc. Chri. Ege.' (F. i, r.) Colophon. At the end is the date ' M. D. XLVIII.' (F. 60, r.) Description. 8, 9.9 x 15.3 cm., the text being 8.2 x 1 1.8 cm. 60 ff. unnumb., 22-27 U- (The arithmetic occupies only 28 ff.) Frankfort, 1548. PETRUS APIANUS. Ed. pr. 1527. Ingolstadt, 1527. PETER BIENEWITZ, or BENNEWITZ. Born at Leisnig, in 1495 ; died at Ingolstadt, April 21, 1552. He wrote chiefly on astronomy, and in his Cosmographia (1524) he first showed how to determine' longitude by observ- ing the distance of the moon from certain fixed stars. He was professor of astronomy at Ingolstadt, and was one of the few university professors of his time who gave instruction in arithmetic in the German language. Title. See Fig. 78. Colophon. ' Gedriickt vnd volendt zu Ingolftadt //durch Ge- orgium Apianum von Leyfl-//nick/ jm Jar nach der geburt Chrifti // 1527. am 9. tag Augufti.' (F. 299, v.) Description. 8, 10 x 14.8 cm., the text being 7 x 11.7 cm. 299 ff. unnumb. -f- i blank = 300 ff., 27 11. Ingolstadt, 1527. Editions. Ingoldstadt, 1527, 8 (here described); Frankfort, 1537, 8 (p. 157); ib., 1544,8; ib., 1564,8; ib., 1580. Graesse mentions an ' Arithmetica/ Leipzig, 1543, 8, but questions it, and Romstock does not give it in his article on Apianus in the Astronomen, Mathematiker, und Physiker der Diocese Eichstdtt. Apianus follows Rudolff so closely as to give ground for comment. His arithmetic differs from the latter 's chiefly in the arrangement of the matter. The work is largely commercial, and includes the fundamental operations and the ordinary rules and applications of the period. There is a chapter on counters at the end of the book. Indeed, Apianus advises their use, saying : ' die Sumering der Regifter durch die rechenpfening auff der lini brauchfamer ift dan durch die federn oder kreide.' The work is also interesting on account of its quaint illustrations. The title page (p. 156) is noteworthy on account of the engraved 'Pascal triangle ' a century before Pascal studied this numerical form, and some years before Stifel mentioned it, and because of the picture of line reckoning. I know of no example of the * Pascal triangle ' in print 156 RARA ARITHMETICA before this one, although the arrangement had doubtless long been more or less familiar to mathematicians. n vn fragfliicfcu 6ettn'ffert , id) was fc:rl winfc 6e^nt>ig^it m bcr XVdfcfoe practice vii tCoIIctft geP^ucht Oesgleycfeert fnrirtrtlp ttrtt>er ut JCeo^fcbcr nod) m Wcljcber >2^d rtk riu/verfcrtigcr FIG. 78. TITLE PAGE OF THE FIRST EDITION OF APIANUS Other works 0/1527. Feliciano, p. 146, 1526 ; Riese, p. 139, 1522, the first Niirnberg edition so far as I know; Tagliente, p. 114, 1515 ; W. Peer, Ain new guet Rechenbuchlein,' Niirnberg, 8. PRINTED BOOKS 157 PETRUS APIANUS. Ed. pr. 1527. Frankfort, 1537. See p. 155. Title. ' Ein newe vnd wolge-//grundte vnderweifung aller // Kauffmans Rechnung in dreien Bu//chern/ mit fchonen Regeln vnd fragftiicken be-//griffen. Sunderlich was fortel vnnd behen- dig-//keit in der Welfchen Practica vnnd Tolle-//ten gebraucht wurt/des gleichen vor//mals weder inn Teutfcher noch in// Welifcher Spraach nie getruckt.// C.Durch Petrum Apianum von Leyfznick der // Aftronomei zu Ingolftatt Ordinarium.// (Woodcut of merchants using counters.) Franc. Chri. Egen.' (F. i, r.) Colophon. * Zu Franckfurt/ bei Chriftian Egenolff/ // Anno Domini. M. D. xxxvij.//Im Herbftmon.' (F. 183, v.) Description. 8, 9.8 X 15.3 cm., the text being 7 x 11.5 cm. 183 ff. unnumb. + i blank = 184 ff., 25-28 11. Frankfort, 1537. See p. 155. JOHANNES FERNELIUS. Ed. pr. 1528. Paris, 1528. JEAN FERNEL. Born at Clermont in 1497 ; died at Paris, April 26, 1558. He was a physician, with a taste for mathematics and astronomy. He wrote numerous works on medicine and mathematics. Title. * loannis Fer//nelii Ambianatis // de proportionibus Libri duo. //Prior, qui de fimplici proportio-//ne eft,&magnitudi- num & nu-//merorum turn fimplicium turn // fractorum rationes edocet.// Pofterior, ipfas proportiones co-//parat : earumq3 rati- ones colligit.// Parisiis // Ex aedibus Simonis Colinaei // 1528.' (F. i, r. Fig. 78.) Colophon. 'CLibellorvm de proportionibvs, loanne // Fernelio Ambianate authore, finis.' (F. 28, v.) Description. Fol., 22.2 x 31.5 cm., the text being 15.6 X 24.4 cm. 4 ff. unnumb. + 24 numb. = 28 ff., 4445 11. Paris, 1528. Editions. There was no other edition. This is one of the best of the sixteenth-century treatises on the mediaeval proportion. It follows the Boethian treatment, as seen also in the work of Bradwardin. KARA ARITHMETICA ^lOANNISFER NELII AMBIANATIS de propordonibus Libri duo. Prior, qui de umplici proportio* ne eft ,& magnicudinum & nu> merorum turn fimplicium turn fra<florum rationes edocet* Poflerior, ipfas propom'oncs co* paratreaiumcj rationes colligit. PARISIIS Ex aedibus Simonls Colinai FIG. 79. TITLE PAGE OF FERNELIUS PRINTED BOOKS 159 Other works 0/1528. Borghi, p. 16, 1484 ; Cassiodorus, p. 2 1 1, 1540 ; Ciruelo, p. 60, 1495 ; Riese, p. 139, 1522 ; Tagliente, p. 114, 1515 ; Christiernus Torchillus Morsianus, ' Arithmetica brevis et dilucida in quinque partes digesta,' Cologne, 8 (but see p. 182, 1536). Works 0/1529. Anianus, p. 32, 1488 ; Riese, p. 139, 1522 ; Tonstall, p. 135, 1522 ; Baeda, ' De natura rerum et temporum ratione libri duo,' Basel, cap. I being ' De computu vel loquela digitorum'; there was an edition by Noviomagus, Cologne, 1537 (see also pp. 131, 140, 263, 1521, 1525, 1554). CHRISTOFF RUDOLFF. Ed. pr. 1530. Augsburg, 1530. See p. 151. Title. See Fig. 80. belangenddar; t cber geffofoe We sergleycbrws 0m/ fccr C^>Myt>/ XPcyrt / 2>nrd) abfcAtertft t^er fTKing tert yrcrt n?ert>c /gegcti ein ' rurtgcn pcrfoiteti M. a xxx, FIG. 80. TITLE PAGE OF THE FIRST EDITION OF RUDOLFF'S Exempel Buchlin 160 KARA ARITHMETICA Colophon. * Getruckt in der loblichen Reychftat Aug-//fpurg/ durch Heynrichen Stayner///Volendet am 31 May im jar//M.D. XXX.' (F. 75, v.) Description. 8, 9.9 X 15.1 cm., the text being 7.4 x 12.2 cm. 75 ff. unnumb. + 2 blank = 77 ff., 28 11. Augsburg, 1530. Editions. Augsburg, 1530, 8 (here described); Nurnberg, 1538; ib., 1540. This is the third of Rudolff's works (see p. 15 1). As the name implies, it is merely a collection of problems, two hundred and ninety-two in number. Most of these problems are of a genuine business nature, and they furnish a good idea of the ordinary commercial needs of the first half of the sixteenth century in Germany. ORONTIUS FINAEUS. Ed. pr. 1530-32. Paris, 1530-32. ORONCE FINE. Born at Briar^on in 1494 ; died at Paris, October 6, 1555. He was made professor of mathematics in the (later called) College de France in 1532. He wrote extensively on astronomy and geometry, but was not a genuine scholar. Title. See Fig. 81. Colophon. ' Excvsvm est avtem ipsvm opvs Pa//rifijs in uico Sorbonico, impenfis Gerardi Morrhij, & loannis Petri. Anno // M.D.XXXII.' (F. 216, r.) Description. Fol., 24.1 x 37 cm., the text being 18.4 x 27.7 cm. Qff. unnumb. + 208 numb. = 217 ff., 48 11. Paris, 1530-32. The title page of the geometry appears on f. 49, r., with the date M. D. XXX; the cosmography on f. 101, r., with the date M. D. XXX ; the horography on f. 157, r., with the date M. D. XXXI. Editions. Paris, 15 30-32, fol. (here described) ; ib., 1535; ib., 1542, fol. (p. 163); ib., 1544, 8 (p. 163); ib., 1554; ib., 1555, 4 (p. 163); Venice, 1587, 4 (p. 164). Leslie's statement that the work appeared in 1525 is unfounded. For the ' De rebus mathematicis,' 1556, see p. 279. This is the first edition of the works of Finaeus, perhaps the most pre- tentious French mathematician of his time, and was published during the years 1530-32. The dedicatory epistle is dated ' Lutetiae Parifiorum Calendis lanuarij 1531,' or 1532 new style. The part on arithmetic is PRINTED BOOKS 161 CIT WLNERE VIRTV OR0N' FINEI DEL PHINATIS.LIBB. RALIVMDISCIPLU NARVM PtLOTtSSOi Rl( RZCII, PROTOM A T H ES IS: Opus uarium,ac fci'cu non minus mile quarn iucundum,nuncprimum iji lucem furlin'tcr emuTum. Cuius index uru'ucrfa* lis^uerfapagina ^ 4 contmctur , Cum gratia 8f priuilegioChriftianidimi Francorum Regfa, ad Decennium. FIG. 81. TITLE PAGE OF THE FIRST EDITION OF FINAEUS 162 KARA ARITHMETICA divided into four books dealing respectively with integers, common frac- tions, sexagesimal fractions, and proportion. There are no applications ORONTII FINBI DEL*H. p tt , 9 primi elementorumEuclidis facile mamfeftatur.cVangufosABH an , !mloAG f efr.*qualis(namuterqjre<!his)igrturper 4 fcxti aufdetn Eudidis, fit ficut H B ad B A, ka F C putei latitude ad C A compofitam ex G B 8C B A longi- tudinem.fiue profunditatem. Sit exempli gratia B H 10 partjum,qualium larus quadrati eft 6o:B E aute tne tiatur,& fit in exemplum 6 cubicorum,tot etiam cubirorum erit C F:fuiu enim la tera perallelogrammi B E F C oppofita,qu* per 34 eiufdem primi funt inuicera qualia)uc igitur 6 in 6o,fient j6o:quae diiudeper o, &.habebis pro quode- te iS.Tot igitur eubitoRj erit A C: a qua fi dempferis A B trium uer* bi gratia cubitorum, relinqur tur B G defyderata & in profundum depflfa putei logitudo 15 cubito'i?. IDEM Q.V O Q.V E SIC OB* tinebis. Metire H E: fitcp exempli caufa 5 cubitoru. Deinde mulnV plica y per 6o,fiem joo:hapc diui per zo,producentur Ij.uelut an- t ca.Bina nancp triangula A B H et H E F funt rurfum arquiangula. quoniam angulus A H B angulo E H F ad u err kern pofito , per JJ primi Eudidis eft xqualisa'te rex <ft us qui ad B, redo qui ad E pan' ( ter arquaf. rclfquus igitur BAH j clique H F E per $1 eiufdempri* mi eft atquah's. Vndeperfupe* riusallcgata quart! propofitione ' fexti,ficut HB ad B A.ita H E ad E F,eidem B C per hypothe(7m apqualem. Cum autem acciderit puteum rotundam habere figuram.nabenda erit cofyde* ratio diametri putealis orifici^&f reliqua omnia uelutiprfus abfoluenda. <& RELIQ.VVM ETS.VT eandem rerum in profundu de preflarum, per uulgatu quadra^ tern metiri doceamus altirudine. Sit itaqj puteus circularis E F G H.cuius diameter fit E F,aut illi a-qualis c H. Adplica igitur qua drltem ipfi putei orificio: in hue modu ,ut finis h ter is A D ad datiz punftum E conftituatur . Leua poftmodu.autdeprime quadrat tem(libero femper demuTo per* pendiculo)donec radius uifualis per ambo foramina pinnacidioj* ad inferiorcm & c diam'etrofi- gnatu rerminu u perducaf.Quo ao SC immoto quadrSte, uj'de Scrundusmo, dusnwutndi profuiKla.|)<f FIG. 82. FROM THE FIRST EDITION OF FINAEUS worthy the name, and 'the work has little to commend it. Some inter- esting illustrations showing the use of the mediaeval Quadrans are shown in Fig. 82. PRINTED BOOKS 163 ORONTIUS FINAEUS. Ed. pr. 1530-32. Paris, 1542. See p. 1 60. Title. ' Orontii // Finei Delphin. Re-//gH Mathematicarvm // Professoris: //arithmetica//practica, libris qva-//tuor abfoluta, omnibus qui Ma-//thematicas ipfas tractare volunt //perutilis, admod unique neceffa-//ria: Ex nouiffima authoris reco-//gnitione, amplior, ac emenda-//tior facta.//ydito tertia.//Parisiis.// Ex officinaSimonisColinaei.// 1 542. //Cum gratia &priuilegio Chri-// ftianiffimi Francorum Regis.' (F. i, r.) Description. FoL, 20.9 X 30.1 cm., the text being 1.6 X 28.2 cm. 2 ff. unnumb. + 66 numb. = 68 ff., 40 11. Paris, 1542. See p. 1 60. ORONTIUS FINAEUS. Ed. pr. 1530-32. Paris, 1544. See p. 1 60. Title. ' Orontii // Finaei Delphi-//natis, Regij Mathe-//mati- caru Lutetiae // Profefforis,// Arithmetica // Practica, in com- pendiu per Authorem // ipfum redacta, multifq3 acceffionibus // locupletata : Ijs qui ad liberam quauis,// nedu Mathematics adfpirant philofo-//phia perutilis, admodumqj neceffaria.// Lvte- tiae Parisiorvm // Apud Simonem Colinaeum.// I544.// Virefcit vulnere virtus.' (F. I, r.) Description. 8, n X 17 cm., the text being 8.6 X 12.8 cm. 95 ff. numb. + I unnumb. =96 ff., 31 11. Paris, 1544. See p. 1 60. ORONTIUS FINAEUS. Ed. pr. 1530-32. Paris, 1555. See p. 160. Title. 'Orontii Finaei //Delphinatis, Regii// Mathematicarum Lutetiae// prof efforis,//de arithmetica practi-//ca libri quator: Ab ipfo authore uigi-//lanter recogniti, multifque //accefsionibus recens // locupletati.// Lvtetiae Parisiorvm, //apud Michaelem Vafcofanum,// i $$$.// Ex privilegio regis.// Virefcit uulnere uirtus.' (F. i, r.) Description. 4, I5-4X 20.7 cm., the text being n.6x 17.2 cm. 4 ff. unnumb. + 72 numb. = 76 ff., 34-35 11. Paris, 1555. 164 KARA ARITHMETICA ORONTIUS FINAEUS. Ed. pr. 1530-32. Venice, 1587. See p. 160. Title. 'Opere//di//Orontio Fineo//del Delfinato : // Diuife in cinque Parti ;//Arimetica, Geometria, Cofmografia, & Oriuoli, // Traclotte // Da Cofimo Bartoli, Gentilhuome, & Academico Fiorentino : // Et gli Specchi,// Tradotti dal Caualier Ercole Bottrigaro, Gentilhuomo Bolognefe.//Nuouamente pofte in luce: //con privilegio.// In Venetia, Preffo Francefco Francefchi Senefe, 1587.' (F. i, r.) Description. 4, 14.9 X 20.9 cm., the text being 10. 1 x 16.8 cm. 8 ff. unnumb. + 8 1 numb. = 89 ff . (in the part devoted to arithmetic), 35-39 11. Venice, 1587. Editions. See p. 160. That this is the first Italian edition appears in the printer's dedicatory epistle to Guidubaldo de' Marchesi del Monte, in which he mentions * in quefta occafione dell' hauere ftampato 1'opera d'Orontio nella noftra Tofcana lingua,' with the date ' Di Venetia, il di 7. di Luglio, 1587.' See p. 1 60. ANDREAS ALCIATUS. Ed. pr. 1530. Hagenau, 1530. Born at Alzano, near Milan, May 8, 1492; died there June 12, 1550. He was an Italian jurist. Title. See Fig. 83. Description. 8, 10.3 x 15.4 cm., the text being 7.8 X n.i cm. 50 ff. and 2 plates unnumb., 28 11. Hagenau, 1530. Editions. There was no other separate edition, but the works of Alciatus appeared at Basel in 1571, 3 vol., fol. The work of Alciatus extends only to f . D i . The Oratio de legibus ' of Melanchthon then extends to f. F 5. This is followed by ' Budaei qvaedam de moneta Graeca,' etc. The work of Alciatus is not an arith- metic, but a history of weights and measures. As such it is of value for the historical development of commercial mathematics. Other works 0/1530. Boethius, p. 27, 1488; Bradwardin, p. 61, 1495; Clatovenus, p. 292, 1558; Maffei, p. 86, 1506; Riese, p. 139, 1522; Tagliente, p. 114, 1515; Torrentini, p. 76, 1501; Johann Kolross, a primer entitled ' Enchiridion : das ist Handbiichlin tutscher PRINTED BOOKS 165 Orthographi . . . Auch wie mann die Cifer vnd tiidtsche zaal version sol,' Basel, with another edition, ib., 1534,8. There was also an anonymous work entitled ' La vraye maniere pour apprendre a chiffrer et compter,' published at Lyons, s. a., 12, c. 1530. AND^AL CIATI LIBELLVS; DE PONDERIBVS BT mcnfuris* ITEM dbucnon tu ITEM Gcrmworum ufum,fentenm* kiatiqiiofyv MippiMelancbthont in Uttfam litru Cwtltf , or at tones du& \obnn. Stc Anno M. D. XXX FIG. 83. TITLE PAGE OF ALCIATUS JOACHIM FORTIUS RINGELBERGIUS. Ed. pr. 1531. Leyden, 1531. JOACHIM STERCK RINGELBERGH. Born at Antwerp, c. 1499; died c. 1536. He taught philosophy and mathematics in various cities of Germany and France. Title. ' loachimi // Fortii Ringel-//bergij Andouerpiani opera, // quae proxima pagina //enumerantur.// Virtvte dvce // comite Fortvna.//Apvd Gryphivm // Lvgdvni, //anno//M. D. XXXI.' (p. i.) 1 66 RARA ARITHMETICA Colophon. ' Lvgdvni apvd // Seb. Grypivm,// anno // M. D. XXXI.' (P. 687.) Description. 8, 10.3 X 15.7 cm., the text being 8.4 X 12.2 cm. 687 pp. numb. + 30 = 717 pp., 22-29 11. Leyden, 1531. 26 I O. F O R T I I RING. Pyntmidm nmm hoc pufto digeruntur. J 6 10 FIG. 84. FROM THE FIRST EDITION OF RINGELBERGIUS Editions. Leyden, 1531, 8 (here described); ib. (at least the arithmetic part), 1539,8; Basel, 1541, 8 (see p. 167); Leyden, PRINTED BOOKS 167 1556, 8 (see p. 168). The ' Epistola ad Lectorem ' is dated ' Louanij Idib. August!, Anno M. D. XXIX.' This work of Ringelbergius is somewhat encyclopedic in charac- ter. The 'Liber de Ratione ftudij,' with Annotationes ' thereon and a ' Horoscopus libri ratione ftudij ' to show that it was written in an auspi- cious time, is followed by six other books. These relate to Grammar, Dialectics, Rhetoric, Mathematics, and Divination, closing with a book entitled ' Commvnis cvivsdam Naturae funt.' The book on mathematics includes a chapter on arithmetic in which, in 1 7 pages (about i o pages excluding the illustrations), the author treats of the Boethian ratios, figurate numbers (see Fig. 84), and the fundamental operations with figures and upon the line abacus. The part relating to astronomy had already been published at Basel in 1528, and the cosmography in Paris in 1529. Other works 0/1531. Kb'bel, p. 101,1514; Juan Gutierrez de Gualda, ' Arte breue y muy prouechos de cuenta castellana y arismetica,' Toledo, 4; ib., 1539; Saragossa, 1557, 1564; Alcala, 1570; H. C. Agrippa, ' De occulta philosophia libri tres,' s. 1., with later editions, Cologne, 1533, fol. ; 1541; Lugduni, 1550, 8; s. 1., 1565; Basel, 1567, 8; s. 1. a. (Paris, 1567 ?), included by De Morgan without much reason. JOACHIM FORTIUS RINGELBERGIUS. Ed. pr. 1531. Basel, 1541. See p. 165. Title. ' loachimi // Fortii Ringelber//gii Andoverpiani Ivcvbra- //tiones, uel potius abfolutifsima KVK\o7rai$t8<z : nem-//pe liber de Ratione ftudij, utriusq; linguae, Gramatice,// Dialectice, Rhe- torice, Mathematice, & fublimioris // Philofophiae multa. Quoru fXrywv fub fequenti pa//gina enumeratur. Atq; haec omnia eo iudicio & //ordine funt tradita, ut uel fola cuiq; //meliorum lite- rarum ftudiofo // fatis ad fummum inge-//nij cultum effe // pof- sint.//Basileae.//Anno M. D. XLI.' (P. i.) Colophon. ' Basileae apvd Bartholo-//mevm Vvesthemervm // anno M. D. XLI.' (P. 797.) Description. 8, 10.3 X 15.7 cm., the text being 8.2 x n.8 cm. 796 pp. numb. + 2 unnumb. = 798 pp., 25 11. Basel, 1541. See p. 1 66. Although the title differs from that of 1531, the work is the same. 168 KARA ARITHMETICA JOACHIM FORTIUS RINGELBERGIUS. Ed. pr. 1531. Leyden, 1556. See p. 165. Title. ' loachimi // Fortii // Rin-//gelbergii // Andoverpiani // Opera,// Quae proxima pagina enumerantur.// Lvgdvni,// Apud loannem Frellonium.' (P. i. The rest of the page is torn off.) Colophon. ' Lvgdvni,// ex officina typogra-//phica Michaelis // Sylvii,//M. D. LVL' (P. 663.) Description. 8, 10.5 x 15.2 cm., the text being 9X 13 cm. 4 pp. unnumb. + 4 blank + 660 numb. = 668 pp., 26-31 11. Leyden, 1556. See p. 1 66. Although the title differs from that of 1531, the work is the same. MICHAEL PSELLUS. Ed. pr. 1532. Venice, 1532. Called the Younger, to distinguish him from a philosopher of the same name who lived about 870 A.D. Born at Constantinople in 1020; died in a cloister in mo. He studied at Athens and taught philosophy at Con- stantinople. Title. See Fig. 85. Description. 8, 9.9 x 14.9 cm., the text being 7.4 X 1 1.4 cm. i f. blank +104 unnumb. = 105 ff., 24-25 11. Greek, Venice, 1532. Editions. Venice, 1532,8, Greek (here described); Paris, 1538, Greek, 4; Paris, 1545, Greek and Latin; Augsburg, 1554, 8, Greek and Latin; Wittenberg, 1556, Latin; Basel, 1554 and l $$6> 8, Greek and Latin; Paris, 1557, 8, Latin (p. 170); Wittenberg, 1560, Latin; Paris, 1585, Latin; Leipzig, 1590, 8, Greek and Latin (p. 170); Heidelberg, 1591, Latin; Tours, 1592, Latin. Psellus was one of the last of the Greek writers on arithmetic. This part of his work is devoted solely to the theory of numbers, and it represents the arithmetical inheritance derived from the older Hellenic civilization. The treatise covers the mediaeval Quadrivium arithmetic, music, geometry, and astronomy and is the only late Greek work on arithmetic that attracted attention in the Renaissance period. The arith- metic is merely a primer for the study of Nicomachus. PRINTED BOOKS 169 Other works of 1532. Aventinus, p. 136, 1522; Capella, p. 66, 1499 ; Feliciano, p. 146, 1526; Kobel, p. 102, 1514; Rudolff, p. 151, 1526; Stifel, p. 223, 1544; Vincento, p. 140, 1522; Johann Brandt, 'Kunst- Trfo/ eig T TI SAPIENTISSIMI PSEL li opus dilucidum in quattuor Ma; thematicas difciplmas, Arith/ iticticam a Muficam t Geomc triam,K Aftronomiam. Numcrorum hie contraftior explicatfo* Elaboratum Mufices Compendium* Copendiu rurftis Geometri^ rationu* .Aftronomig coadio pcrfpicua* VENETJ 4 IS.MDXXXIL C Cum gratia* FIG. 85. TITLE PAGE OF THE FIRST EDITION OF PSELLUS liche Rechnimg mit der Zyffern vnd Pfennigen, Auff allerley handt- tierung,' Cologne, 8, 39 ff. ; Georg Reichelstain, ' Kauffmans hand- biichlin. Aller Rechennschafft behendigkeyt, auff Linien vnd Zififern,' Frankfort, sm. 8, with another edition in 1534. 170 KARA ARITHMETICA MICHAEL PSELLUS. Ed. pr. 1532. Paris, 1557. See p. 1 68. Title. ' Michael // Psellvs de // Arithmetica,// Mvsica, Geo- metria://& Proclus de // Sphaera,// Elia Vineto Santone inter- prete.// (Woodcut with motto : Inpingvi Gallina.) Parisiis,// Apud Gulielmum Cauellat, in pingui gallina,//ex aduerfo collegij Cameracenfis.// 1557.' (F. i, r.) Description. 8, 10.6 x 16.2 cm., the text being 6.6 X 13.3 cm. 2 ff. unnumb. + 76 -numb. + 3 blank = 81 ff. (18 ff. on arithmetic), 22 11. Paris, 1557. Editions. See p. 168. This is one of the Latin editions. See p. 1 6 8. MICHAEL PSELLUS. Ed. pr. 1532. Leipzig, 1590. See p. 168. Title. ' Pselli // Philofophi & Mathemati-//ci clarifsimi // Arithmetica // Edita ftudio//M. Chriftophori Meureri,//Mathe- matum Profefforis public! //in Academia// Lipfienfi.// 1590.77 Plato interrogatus, cur homo fit a-//nimal fapientifsimum : on apid-/ /pew eTTigarai, refpondit.// Lipsiae.' (F. I, r.) Colophon. ' Lipfiae, imprimebat Michael // Lantzenberger.// Anno M. D. XC.' (F. 24, v.) Description. 8, 9.5 x 15.8 cm., the text being 6.6 X 12.3 cm. 24 ff. unnumb., 30 11. Leipzig, 1590. Editions. See p. 168. The Latin dedicatory epistle is dated ' Lipfiae XV. Cal. Nouemb. Anno poft Chriftum natum 1590.' The text is in Greek and Latin. See p. 1 68. MICHAEL PSELLUS. Ed. pr. 1532. Leipzig, 1616. See p. 168. Title. ' Pselli // arithmetica // Guilhelmo Xylandro // inter- prete // Cum Praefatione // Christophori Meureri D.// Mathe- matum Profefforis //in Academia Lipfienfi.// i6i6.// Lipsiae // Typis Abrahami Lambergi.' (F. I, r.) PRINTED BOOKS 171 Description. 8, 9 x 14.4 cm., the text being 6.6 X 12.1 cm. 39 pp., 1-8 unnumb., then numb. 1-31 ; 24-25 11. Leipzig, 1616. See p. 1 68. ADAM RIESE. Ed. pr. 1533. Leipzig, 1536. See p. 138. Title. See Fig. 86. Description. 4, 14.6 x 19 cm., the text being 9.6 x 15.1 cm., and the tables 9.5 X 12 cm. 79 ff. unnumb. + i blank = 80 ff., 17-24 11. in the Introduction (3 ff.), the rest of the book con- sisting of tables. Leipzig, 1536. Editions. Leipzig, 1533; ib., 1536, 4 (here described). This is a set of mercantile tables for the multiplication and division of denominate numbers. GEORGE AGRICOLA. Ed. pr. 1533. Paris, 1533. Born March 24, 1490, at Glauchau, Saxony; died November 21, 1555, at Chemnitz. He was rector of a school at Zwickau (1518-1522), and in later life a physician. He wrote a number of scientific works. Title. See Fig. 87. Description. 8, 10.2 x 15 cm., the text being 6.8 x 12.4 cm. 7 pp. unnumb. + 3-261 numb. +6 blank -f- 1 with woodcut = 273 pp., 28 11. Paris, 1533. Editions. Paris, 1533, 8 (here described); Venice, 1533, 8; ib., 1535, fol. ; Basel, 1549, 8; ib., 1550, fol. There was also an ' Epitome omnium Georgii Agricolae de mensuris et ponderibus per G. Philandrum' published at Lyons in 1552, 8. The work can hardly be called an arithmetic, but, like a few others included in this list, it is a valuable book of reference on the history of ancient measures. It consists of five books as follows : ' Liber primus, de menfuris Romanis ' (p. 9) ; * Liber secundus de Mensuris Graecis ' (p. 75); 'Liber tertius, de Pondere rerum quas metimur ' (p. 144); 'Liber quartus, de Ponderibus Romanis' (p. 188) ; 'Liber quintus, de Ponderibus Graecis' (p. 219). The book is also valuable to the student of Roman and Greek numerals, and of the various symbols of measures. Such works explain the origin of certain systems of 172 RARA ARITHMETICA btfe XJolmbet vnb af0on0eitant FIG. 86. TITLE PAGE OF THE 1536 RIESE PRINTED BOOKS 173 measures employed before the metric system was developed, and of such symbols as are still used by apothecaries. GEORGII AQRJCOLAB MEDICI LI BUI quinquede Menfuris CT Ponderibusjnffa bits pltraqtie* B V D AE O QT P ORT f O pa rum aninuducrft Ofw twnc prinwm in lucent tfitm* P A R I 3 I I 5, Etttfcfcbti Cbn/lwfws Wrcbrftisjfi BKOttcc* JH.D. XXXIIK FIG. 87. TITLE PAGE OF THE FIRST EDITION OF AGRICOLA Other works 0/1533. Agrippa, p. 167, 1531 ; Apianus, p. 62 ; Jorda- nus, p. 62, 1496; Riese, p. 139, 1522 ; Schonerus (editor), p. 178, 1534; Vincento, p. 140, 1522; Anonymous, 'Libretto de Abaco,' Venice; Anonymous (sometimes attributed to Regiomontanus), p. 178, 1534. i 7 4 KARA ARITHMETICA GIOVANNI MANENTI. Ed. pr. 1534. Venice, 1534. ZUAN MANENTI. A Venetian mathematician of the sixteenth century. Title. See Fig. 88. Colophon. ' In Vinegia per Giouan' Antonio di Ni-//colini da Sabio A Inftantia de. M.// Zuan Manenti. Nelli Anni del // figno- re. MDXXXIIII.//del Mefe di Genaro.//Neffuno ardifca Stam- par quefte Tariffe// de cambii & de diuerfe cofe fotto pe//na de excommunicatione late // fententie come nel Priui-//legio fe con- tiene.// M D XXXIIII.' (Large woodcut.) (F. 400, v.) Description. 12, 7.5 x 13.9 cm., the text being 5.4 X 10.6 cm. 402 ff. (2 blank) unnumb., 21-26 11. Venice, 1534. Editions. There was no other edition. As the title indicates, this is a set of tables of exchange, and it was intended for the use of Venetian bankers and merchants. GIOVANNI SFORTUNATI. Ed. pr. 1534. Venice, 1534. JOHANNES INFORTUNATUS. An Italian arithmetician, born at Siena c. 1 500. Title. See Fig. 89. Colophon. ' Stampata in Vinegia per Nicolo di Ariftotile // detto Zoppino.//M.D. XXXIIII.' (F. 129, r.) Description. 4, 15.4 x 21 cm., the text being 13.2 x 16.9 cm. 129 ff., 40-41 11. Venice, 1534. Editions. Venice, 1534, 4 (here described); ib., 1543, 4; ib., 1544 (colophon 1545, see p. 17?), 4 5 ib., I545,4(p. 177); s. 1., Venice, c. 1550; ib., 1561, 4 (p. 177); ib., 1568. The privilege is dated 1532. Sfortunati wrote his treatise along the lines followed by Borghi and Feliciano, and in his preface he acknowledges his indebtedness to them and to Maeftro Luca dal Borgo dell' ordine di fanto Francefco ' (p. 54) and to the 'operetta di Filippo Caladri Cittadino Fiorentino ' (p. 47). Like these authors, he was a popular writer, as the seven editions of his book go to prove. His work is fairly complete as to the opera- tions with integers and fractions, and is satisfactory as to the examples illustrating the Italian business life of the sixteenth century. The treatise closes with some work in practical mensuration and some mer- cantile tables. PRINTED BOOKS 175 DF C AMBI E ALTRO Copofia per ZuaMsnenti. Ccn priuilcgfo del Uluft.Senatcurenero ch'altrichcZ.Manoti infra anni X fta far re far ftapar no la pofTi, (b*ro Ie pc ne cotenute in qusllo.M D X X XIII I. FIG. 88. TITLE PAGE OF MANENTI 7 6 KARA ARITHMETICA $> INVOVO ,IBRO DI ARITHMETICS Intirahto: Naouotnme impoche molte .ppofitfo tii cheperaltn autorufono falfamete coclufe: in qucfto fi emldano:& caftiganorcon chiate: In cide:& aperte dimoftrationitmolto betie di fciiCTe,& ventillateC6 y no breae rrattato di Geometria:per quato d vnopratico Agrimenfoie fi conucga,con ta^ uole da rcmporre le corde:da mj furarela tenuta di ciafchinu botte.& eriam liftaggiuox Ii da mifuiare gli fee mi diquelle* Compofto per lo acutiffimo prefcratato're,delle Archimcdfane&Euclidiane dottrine GIOVANNI SFORTVNATI DA SIENA. CVM GRATIA ET PRIVILEGIO* M* D. XXXllll* QVISEHVMILI AT-IXALTABni FIG. 89. TITLE PAGE OF THE FIRST EDITION OF SFORTUNATI PRINTED BOOKS 177 GIOVANNI SFORTUNATI. Ed. pr. 1534. Venice, 1544-45. Seep. 174. Title. The title page is practically identical with that of 1534, except for the date : M. D. XLIIII. Colophon. ( Stampata in Vinegia per Bernardino de Bindoni //Milanefe Anno domini. M. D. XLV.' (F. 129, r.) Description. 4, 14.8 X 20 cm., the text being 12.2 x 16.7 cm. 2 ff. unnumb. + 127 numb. = 129 ff., 40 11. Venice, 1544-45. Editions. See p. 174. It will be noticed that the dates in the colophon and on the title page do not agree. See p. 174. GIOVANNI SFORTUNATI. Ed. pr. 1534. Venice, 1545. See p. 174. Title. The title page is practically identical with that of the 1534 edition. Colophon. ' In Vinegia per Giouan' Antonio & Pietro fratel-//li de Nicolini da Sabio. Ad inftantia di // Giacomo da Coneano libraro a fan // Fantin. M: D. XLV.' (F. 129, r.) Description. 4, 15.4 x 21.2 cm., the text being 12.9 x 16.8 cm. The text is practically identical with that of the 1534 edition. See p. 174. GIOVANNI SFORTUNATI. Ed. pr. 1534. Venice, 1561. See p. 174. Title. The title page is practically identical with that of the 1534 edition. Colophon. * In Venetia per Francefco del Leno,//M D LXI.' (F. 129, r.) Description. 4, 15. 4 X 21.6 cm., the text being 12.9 X 17.3 cm. 5 ff. blank + 129 numb. = 134 ff., 41-43 11. Venice, 1561. See p. 174. i;8 KARA ARITHMETICA ANONYMOUS. (Schonerus editor.) Ed. pr. 1534. Nurnberg, 1534. Johannes Schonerus (Schoner) was born at Karlstadt, near Wiirzburg, January 16, 1477, and died at Nurnberg January 16, 1547. He was a preacher at Bamberg, and later (1526-1546) a teacher of mathematics in the Aegidiengymnasium at Nurnberg, in which Melanchthon took such interest. Title. See Fig. 90. Colophon. ' Norimbergse apud lo. Petreium,// Anno M. D. XXXIIII.' (F. 32, r.) Description. 4, 15 X 20.9 cm., the text being 1 1.6 X 15.6 cm. 32 ff. unnumb., 29-43 11. Nurnberg, 1534. Editions. This is merely an edition of the anonymous medi- aeval 'Algorithmus Demonstratus,' with notes by Schonerus. As might be expected, therefore, it is purely theoretical, being a late variation of the Boethian works. In the preface Schonerus speaks of it as the ' Algorithmus Demonftratus incerti autoris.' De Morgan thought that it might have been written by Regio- montanus, but he was wrong in asserting that Schonerus attrib- uted it unquestionably to him. As a matter of fact the authorship goes back at least to the fourteenth century. There is said to have been an edition published at Nurnberg in 1533, attributed to Regiomontanus, but I have not seen it. JOHANN ALBERT. Ed. pr. 1534. Wittenberg, 1561. A Wittenberg Rechenmeister of c. 1500-1565. Title. ' Rechenbdchlin // Auff der Federn/ Gantz // leicht/ aus rechtem Grund/ In // Gantzen vnd Gebrochen/ Neben // an- gehefftem vnlangft ausgelaffnem Buch-//lin/ Avff den Linien/ Dem einfel-//tigen gemeinem Man/ vnd anhe-//benden der Arith- metica//zu gut.//Durch Johann. Albert/ // Rechenmeifter zu Wittembergk/zufamen//bracht. Auffsnew/mit allemvleis vber- //fehen/ gemehrt vnd gebeffert/ //zum dritten mal.// Wittem- berg.//i 5 6i.' (F. i, r.) Colophon. ' Gedruckt zu // Wittemberg/ // durch Geor-//gen Lawen // Erben.// 1561.' (F. 118, v.) PRINTED BOOKS 179 ^'-^^^'^ ALGO RITHMVS DE MONSTRATVS. rhemaucas dcmonftrarionc^in earn calcu landi artem^uam uulgus Algorithmum uocat,quibusfons& origo,irem caufije& certitudo eiuSjdarifnmeCutin Madiemati* cis omnibus fieri folet)tibi ob oculos pos mint u r. Etquamuisetiam citra hanc cogni tionc,&diici &excrceri poflit, tamen,auan turn intcreft inter caecum, alienarrepide cu ndanter ucfligia fcqucn tern , & inter ocu latum.fecure&expcdjceinccdentemjtantu intereA inter harum demonftrarionum na 4m & peritvi calcu tatorrm . Quarc lege,&iuuaberis. L^ V J^ FIG. 90. TITLE PAGE OF THE 1534 Algorithmvs Demonstratvs i8o KARA ARITHMETICA Description. 8, 9.4 X 15.1 cm., the text being 6.7 x 10.8 cm. 1 20 ff. unnumb. (two blank), 7-24 11. Wittenberg, 1561. Editions. Wittenberg, 1534; ib., 1541, 12; Frankfort, 1541 (the colophon is dated 1542), 8; Wittenberg, 1553,8; ib., 1554; Frankfort, 1558, 8; Magdeburg, 1559, 8; Wittenberg, 1561, 8 (here described); ib., 1564, 8; Magdeburg, 1579, 8; Witten- berg, 1586, 8. That this edition was revised in 1541 appears from the dedication, which is dated 'im taufent/ f tinff-//hundert/ ein vnd vier-//tzigften Jar.' Murhard also mentions an ' Intro- ductio Arithmetices,' Cologne, 1542, 8. Although from the title it would seem that algorism ( Auff der Federn ') is emphasized, counter reckoning (' die Species auff den Linien ') is first described (ff. A 3~F 2). This is followed by the second part, the algorism: 'Das Ander Rechenbfichlein/ auff der//Feder/ auffs aller kiirtzeft // vnd leichteft ver-//faffet ' (f . F 3). In each part there are many commercial problems, and the book ranks as one of the most practical of its day. It is a valuable source of information as to the commercial activities of its time. Other works 0/1534. Borghi, p. 16, 1484; Bradwardin, p. 61, M95 ; Jprdanus, p. 62, 1496; Kolross, p. 164, 1530; Ortega, p. 93, 1512; Peurbach, p. 53, 1492; Reichelstain, p. 169, 1532; Rudolff, p. 151, 1526 ; Wolphius, p. 154, 1527 ; Rabbi Elias Misrachi, nrtf^Q "1CDDH or IBDOn 1ED (perhaps 1533? 1532?), Constantinople, 4 (another edition, with a Latin translation by Schreckfuchs, and a com- mentary entitled "1CDDH H-N^D "lisp by Minister, Basel, 1546, 4). GIOVANNI MARIANI. Ed. pr. 1535. Venice, 1580. ZUANE MARIANI. A Venetian arithmetician of the sixteenth century. Title. ' Tariffa perpetva // Con le ragion fatte per fcontro di // qualunque Mercadante fi voglia,// che dimoftra quanto monta ogni // quantita de cadauna mercantia ad // ogni pretio, si a pefo come a nume//ro. Buona per ogniuno in Venetia,// Dalmatia, & altri luoghi; nelli quali // fi ragiona, & fi fpende a moneda// Vene- tiana. Et e buona per Verona,// Breffa, Bergamo, Milan, Cre- mona,// Mantoua, & altri luoghi ddue fi // ragiona, & fi fpende a moneda Im-//periale, & Breffana : Con la redutio//di moneda Venetiana in mone-//da Imperiale, & della imperiale in // PRINTED BOOKS 181 Venetiana. Et e buona a ridurre // ogni forte de ori in moneda cor-//rente, si Venetiana come Im-//periale : & Breffana ad // ogni precio.' (F. i, r.) Colophon. ' Stampata in Venetia per gli Heredi di // Francefco Rampazetto. // Ad inftantia de lAutore Zuane // Mariani.// L'Anno .M. D. LXXX.' (F. 299, r.) Description. 12, 8.5 X 15.2 cm., printed in double columns, each being 2.6 X 13.3 cm. 2791!. numb. + 20 unnumb. = 2991!., 32 11. Venice, 1580. Editions. This is apparently the first of two books by Mariani, the various editions appearing as follows : Venice, 1535; ib. , 1 5 5 3 ; ib., 1559; ib., 1564, 8; ib., 1567; ib., 1569, 16; ib., 1572; ib., 1575; ib., 1579; ib., 1580, 12 (this edition); ib., 1591, 16. The second book, also a Tariffa, appeared three times at Venice, viz. in 1538, 1555, and 1558. That these were different works I know only from such bibliographers as Riccardi and Libri. Like other books with the same title, this is simply a set of tables for the use of merchants. It includes both interest and exchange tables, and is adapted to the needs of Northern Italy. Other works 0/1535. Agricola, p. 171, 1533; Angelus Mutinens, p. 140, 1525 ; Finaeus, p. 160, 1530-32 ; Grammateus, p. 123, 1518; Kobel, p. 102, 1514 ; Reisch, p. 82, 1503; Riese, p. 141^522 ; Ton- stall, p. 134, 1522 ; Torrentini, p. 76, 1501 ; Pedro Melero, 'Compendio de los niimeros y proporciones,' Saragossa, 4. HUDALRICH REGIUS. Ed. pr. 1536. Freiburg, 1550. A German teacher of the first half of the sixteenth century. Title. ' Vtrivs-//qve arithme-//tices epitome, ex variis // authoribus concinnata, per // Hvdalrichvm // Regium.// Nvnc Tertio omnia // diligenter reuifa & emendata.// Friburgi Brif- goiae,// Stephanus Grauius excu-//debat, Anno // M. D. L.' (F. i, r.) Colophon. ' Fribvrgi Brisgoiae, // Stephanus Grauius // excu- debat,//Anno M. D. L.' (F. 104, v.) Description. 8, 10 x 15.3 cm., the text being 7 x n.i cm. i f. unnumb. + 103 numb. = 104 ff., 17-22 11. Freiburg, 1550. 182 RARA ARITHMETICA Editions. Strasburg, 1536, 8; Freiburg, 1543, 8; ib., 1550, 8 (here described). This work was intended for the Latin schools. It is only slightly practical, and as compared with a book like that of Gemma Frisius it is reactionary. The first part (to f. 48) treats only of Boethian arith- metic, the theory of numbers, closing with the words : * Hactenus de numerorum Theo-//rijs, nunc de eorundem // Praxi.' The practical part gives the operations in the usual style of the Latin writers of the time, and closes with several pages on the use of counters. Other works of 1536. Boethius, p. 27, 1488; Bradwardin, p. 61, 1495 ; Budaeus, p. 99, 1514; Feliciano, p. 148, 1526; Ortega, p. 93, 1512; Peurbach, p. 53, 1492 ; Riese, p. 139, 1522 ; Torrentini, p. 76, 1501 ; Christiernus Torchillus Morsianus, ' Arithmetica practica,' Basel, 8, with subsequent editions, ib., 1538, 1553, 8 (but see p. 159, 1528) ; Georg Walckl, ' Die Walfch practica/ gezoge auf3 der kunft der Proportion,' Strasburg (Niirnberg ?), 8; Rycharde Benese, 'This boke sheweth the maner of measurynge of all maner of lande, as well of woodlande, as of lande in the felde, and comptynge the true nombre of acres of the same,' London, 4. (De Morgan includes this book because of its com- putations and early mathematical tables. Subsequent editions appeared in 1537, 1540, c. 1558, 1562, and 1564.) L. Culman, ' Wie iunge und alte Leut recht petten sollen,' Niirnberg, 8; ib., 1537. ABRAHAM BOSCHENSTEYN. Ed. pr 1536. 8.1,1536. The son of Johann Boschensteyn (see p. 99). Title. ' Ein niitzlich // Rechenbuch-//lin der Zyffer/ daraufl ein // yeder/ durch fein aygen fleyfz mit // kleyner huff/ lernen mag anfeng//klich rechenen/ Aufzgange durch // Abraham Bofchenfteyn/ Vnnd // yetzo ztim dritten mal mit fleyfz // vber- fehen vnnd Corrigiert/ mit // erlichen ziigethanen Exem//plen/ Durch Johann // Bofchenfteyn/ den alte.// M. D. XXXVI.' (F. i, r.) Description. 8, 9.5 x 14.5 cm., the text being 6.8 X 1 1.9 cm. 40 ff. imnumb., 20-23 H- S. 1., 1536. Editions. There was no other edition, but the book seems to have been written in 1530, for the dedicatory epistle is dated 'Gebe am i9.//tag Aprilis/ An-//no K. im 30 //jar d' min// derenn //zal.' The work is very rare. PRINTED BOOKS 183 In the epistle the author mentions his father's work (p. 99) : ' Wie- wol meyn Herr vatter/ herr Johan Bofchenfteyn vor 17. jaren auch der gleych zfi Augfpurg inn den Truck mitgetheylt hat/ vnnd zum drittenn mal getruckt worden.' Seventeen years before 1530 was 1513, when Johann's book was probably written, since it was published in 1514. Abraham's work is not much of an improvement on his father's, and resembles it in many respects. It gives seven ' Species,' including * Duplicatio ' and ' Mediatio,' as Johann's work had done. The prin- cipal additions are in the applied problems. GIEL VANDER HOECKE. Ed. pr. 1537. Antwerp, 1537. A Dutch arithmetician of the first half of the sixteenth century. Title. See Fig. 91. Colophon. ' Gheprent Thantwerpen op die Lombaerden vefte //teghen die gulden hant ouer by mi Symon Cock.//Int Jaer ons Heeren ; M.CCCCC. ende // XXXVII. den. ix. dach Febru- arij.' (F. 180, v.) Description. 8, 9.4 X 14.7 cm., the text being 7.6 x 12.8 cm. 5 ff. unnumb. + 176 numb. = 181 ff., 26-32 11. Blackletter, as shown in Figs. 91, 92. Antwerp, 1537. Editions. So far as I know this edition of 1537 is the first, the date 1514 in the British Museum catalogue being evidently an error for 1 544. The book was again published by the same printer in 1 544, and there was an edition in 1 548. (^eeBibliotheca Mathematica^ 1906, p. 211.) The author begins with the fundamental processes with integers, con- sidering the subject in a practical way. He then considers the same processes with counters on the line abacus. This preliminary work is followed by chapters on denominate numbers, fractions, the rule of three, roots, and the mediaeval proportion. The second part of the treatise is devoted to algebra and the applications of arithmetic. The work closes with a brief treatment of mensuration. It is especially noteworthy on account of the early use of the plus and minus signs, not heretofore noticed by writers on the history of the subject. There is no other Dutch book of this period that makes as much use of these signs (see Fig. 92), and Vander Hoecke should be recognized as among the pioneers in appre- ciating their value in connection with algebraic quantities. (See Gram- mateus, p. 125, 1518, who used them in a similar way.) 1 84 RARA ARITHMETICA CMonbetltngfte perfect? reflukn/alo metten fipectl mt gftebeele/cn mt gtjcbjofcett. C$ ie reffule oan tupen wt frtcljeclc en nit gtobjofte* 4)tf re0wlc uan ecnoalfctje tneratio/aDbttto/fubtracrto/mulrtpHcatfo/enDOnfio/ tit met fjacr e0a!iane oft glj eliwabc/cfi met Die rc0t^ le fter quantitept/annejc Oer re0u(en<ro0 en DatfdiK ooc tc wctchc tioct bfei^ulc oanp^ctiKe COM* re0u!c ban g^cfclfrap/ntet b C^icrc0u!cuanfmaeltipelm0c o/tom ^cUingcn btdc traufpoozttnuaittienl tocr allcontfang^croua fubucncicn, en oan 6Iuer/ett nan f nannen Dan ttiapf nen, ^)c ^b;iHe Dana wtjaroefce oft otfier roiOc/en bet tifg X>e pzactme oan eenen ff tche lante te tne (rf ba Die ten alfo tud Dat onb c0an0e!ijc 10 f nttJto Den kuater/ofiS andcr o/alo bat begangtjdtjc io. Cibefalculeertefioetfaemt met gtooter ita^ FIG. 91. TITLE PAGE OF VANDER HOECKE PRINTED BOOKS 185 CJtfntwflHtaftrccteit.fi: fcacrnamuitf wm re/j -, $[;irrationalc. Met op mce+itacr Sen i _i^| CJtrmiml&iaftrccltcH& -^ anfi; 5 1 tanfic} fCi tern Hh uan -j- ofif Dan fubtraijcm cube + uan oft Dan + attor crt iiiHcfubtracttc fo ucr* reals fi adDccrlicfun eft fubtraljcrritf. C l^tiltipliraac mtjrn (^ onnraticttalcn. JJTTT IDtiDi multtpltfftcn int)f n IV: foctucrt dat 0hp 'VAinnccfTtHmalleOcnomnicrovjaii cnidu iiaru realoix temulttpitcrrcn met fimpctcn nomtmrfcc mocr (jl-i Den nomincr multi'plucrcn ttac tic qualit cyt Sco i%: Ml o tm'Ifct multiplir tr en IV 9 met 4 fo frt 4 in Cuen l%: nuiltipliccert 4 mtiacr fduen cocnif tv 10 I:Q tmilttpticccvtomctio coft 144 liicr tut rrctt&toct iz, foe uccl io CN 9 flf)f muIripUf cc rt met 41 want l\ 9 i o ; Dit inulf ip! tcccrt met 4 coemt i i a!o oo ;n u w iiDi muitipitr ae IS;'* 8 met 5 fo inuttipiteccrtf cu FIG. 92. FROM VANDER HOECKE'S Arithmetica 1 86 KARA ARITHMETICA Other works of 1537. Apianus, p. 155, 1527; Baeda, p. 159, 1529; Benese, p. 182, 1536; Kobel, p. no, 1514; Rudolff, p. 152, 1526; Wolphius, p. 154, 1527 ; Andres, p. 122, 1515 ; B. C. Symphorien Cham- pier, * Libri VII,' Basel, 8 (one chapter De Arithmetica ') ; Culman, p. 182, 1536 ; Anonymous, * An Introduction for to lerne to reckon with the Pen and with the Counters after the true cast of Arsmetyke, or Awgrym,' St. Albans. NICOMACHUS. Ed. pr. 1538. Paris, 1538. Born at Gerasa; flourished c. 100 A.D. He was a neo-Pythagorean phi- losopher and mathematician, and attempted, unsuccessfully, to do for the Greek theory of numbers what Euclid had done for geometry. Two works of his are extant, this treatise and a ' Harmonices Manuale.' Title. See Fig. 93. Description. 4, 15.7x23.1 cm., the text being 9.4 X 17.7 cm. 77 pp. numb., 32 11., Greek. Paris, 1538. Editions. This is the first edition of the arithmetic of Nicoma- chus in Greek. A second edition appeared, * Explicata per Joach. Camerarium,' at Augsburg in 1554 (p. 263). For the commentary of lamblichus, see p. 188. The arithmetic of Nicomachus is the most celebrated of the few Greek treatises upon the subject. It was written during the decline of Greek learning, and is not a work of great merit, being chiefly a com- pilation of the general number theory of the Pythagoreans. There are several commentaries upon the 'Introductio,' that of lamblichus (c. 325 A.D., see p. 1 88) being the best known of the ancient ones, and that of Camerarius (see p. 262) being the most important one of the Renaissance. After a philosophical introduction, Nicomachus classifies numbers as even and odd, and the odd as prime and composite. Perfect, excessive, and defective numbers are also considered, and the elaborate system of ratios which later characterized the work of Boethius and the mediaeval writers is given. Polygonal and solid numbers and proportions are treated in the second part, a ratio being loosely defined as ' the relation between two terms,' and proportion as ' the composition of ratios.' The work differs essentially from Euclid in its presentation, being inductive instead of deductive in treatment. It is also a matter of interest that the first multiplication table, the ' mensa Pythagorica ' of mediaeval writers, to be found in any treatise appears here, although Hilprecht found them on the Babylonian cylinders of about 2000 B.C. The best edition of the works of Nicomachus is that of Hoche (Leipzig, 1866). PRINTED BOOKS NIKOMAXOY FEPA: A'pieMHTiicHV BIBAI'A Ar'o. 187 NICOMACHI GERA: SINJ ARITHMETl- cae iibri duo. Kmcfrimim tyfis awpjn Imn edmtor. P A R I S 1 1 S. In officina Chriftiani Wechcli. M. D. XXXVIIL FIG. 93. TITLE PAGE OF NICOMACHUS 1 88 KARA ARITHMETICA GILLES HUGUETAN. Ed. pr. 1538. Lyons, 1538. A Lyons arithmetician, born c. 1500. Title. See Fig. 94. Colophon. ' Icy finiffent les tables des comptes compofees et calculees // par Gilles huguetan : Et imprimees cheux // ledict Gilles et Jacques huguetan // freres. Lan 7/1538.' (P. 115.) Description. Fol., 21.3 x 32.9 cm., the text being 18.1 X 27.3 cm. 25 pp. unnumb. + 90 numb. = 115 pp., 5964 11. Lyons, 1538. Editions. There was no other edition. Because it is composed largely of multiplication and division tables, and of other tables of use to stationers and merchants, this work is not often included among the arithmetics of the century. It should be so classed, however, since the first eleven folios are devoted to the expla- nation of the fundamental operations both with written numbers and with counters. The illustrations of counter reckoning are striking, the 'gectz' (counters) being represented full size. The book is one of the earliest Lyons arithmetics in which the line abacus is mentioned. IAMBLICHUS. Ed. pr. 1538. Arnheim-Deventer, 1668. Born at Chalcis, in Coele-Syria, c. 283 ; died at Alexandria, c. 330. He was a neo-Platonic philosopher and a voluminous writer. Four of his works are extant, this introduction to the arithmetic of Nicomachus being one. Title. See Fig. 95. Description. 4, 15 X 19.8 cm., printed in double columns, Greek on the left and Latin on the right, each column being 5 X 14.8 cm. 12 pp. unnumb. + 181 numb. = 193 pp., 34 11. Bound with the Camerarius edition of Nicomachus (p. 262, 1554). Arnheim-Deventer, 1668. Editions. There was no other edition in the sixteenth century than that of 1538. See Nicomachus, p. 186. This commentary by lamblichus forms the fourth part of his treatise on the Pythagorean philosophy, the greater part of which is still extant. Other works of 1538. De la Torre, p. 41, 1489 ; Glareanus, p. 191, 1539; Mariani, p. 181, 1535 ; Morsianus, p. 182, 1536 ; Peurbach, p. 53, PRINTED BOOKS 189 ^LESTABLESDEDIcte VERS COMPTES, AVEC LEVRS CANONS, cakukcsparGlLLES H VG VETAN,natif deLyon, Par kfquelks on pourra facilcment trouuer ks Comptcs tous faictz, tant dcs achat2 que uentcs dc routes marchandi{es,foit en gros,ouen detail, a la Mefurc, ou au Poix, a la Charge, ou au Nombre. JLes Tables aufli du fin Dor & Dargent, pour fcauoir, fcelon que k Marc de billon tiendra de fin,ou daloy,combien il uauldradepoixde finOr,ouDargent fin. Deux Tables feruantz aux Libraires.Et une Table de Defpence, a fcauoir a tant pour Iour,combienonde(pend Ian 8tk Moys, S^arayfondu Moys combi'cnrcuientpour an 8^ pour chafcunlour,8ia tant pour Ancombicn ondelpend leMoys,S^ chafcun lour. La maniere de Aualuer,ouRcduyre par icelles Tables toutes Monnoycs,en liurcs, folz.&dcniers. $ L A R T S^fcience deNombrer,Adioufter,Souftraire,MuItJplier, 8^Partir,par le compte desGectz, On les uend aLyon,a kn(eigne de laSpnccrc,cheux Gilk$,8C Jaques Huguetan,freres. FIG. 94. TITLE PAGE OF HUGUETAN i 9 o KARA ARITHMETICA 1492 ; Psellus, p. 168, 1532 ; Riese, p. 138, 1522 ; Roche, p. 130, 1520; Rudolff, p. 1 60, 1530 ; Tonstall, p. 135, 1522 ; Tomas Klos, ' Algoritmus : JAMBLICHUS CHALCIDENSIS Ex Coele- Syria I N NICOMACHI GERASENl Arithmeticam introdu&ionem, T D E F A T O. 3\(uncprimum editu*> in Latinumfermonem converfus, notis perpetuis ittuftratus \ SAMUELE TENNULIO, Accedic JOACHIMI CAMERARII Explicatio in duos Libros Nicomachi, cum Indice rerum & verborum locupleudimo. A R N H E M I Troftancapud jutt. FRIDERICUM HAGIUM, Daventfiz typis defcripfir WILHELMUS WUR, clo Io c LXVIII. FIG. 95. TITLE PAGE OF THE 1668 IAMBLICHUS to iesth nauka Liczby, Polska rzecza wydana : Przez Ksiedza Tomasza Klosa,' Cracow (reprinted at Cracow in 1889) ; Eysenhut, 'Ein klinstlich Rechenbuch,' Augsburg, 8. PRINTED BOOKS 191 HENRICUS LORITUS GLAREANUS. Ed. pr. 1539. Paris, 1543. LORITI, LORETI. Born at Mollis, canton of Glarus, Switzerland, in June, 1488 ; died at Freiburg, Breisgau, May 28, 1563. He was professor of mathe- matics and philosophy at Basel (1515-1521), and professor in the College de France, Paris (1521-1524), and later taught at Basel and Freiburg. He wrote on arithmetic, music, and geometry. Title. See Fig. 96. DcVi.Arith- METICAE PRACTI- CAE SPECIEBVS > HEN- R ICI G L AR E ANI Epitome. P ARISIIS Ex officina lacoli Ga^elli^fub in- figni Inuidi<,e regionegym- nafri Cameracenfis. 1 5 4 FIG. 96. TITLE PAGE OF THE 1543 GLAREANUS i 9 2 KARA ARITHMETICA Description. 8, 10.6 x 17 cm., the text being 6.7 x 12.9 cm. 2 ff. unnumb. + 21 numb. = 23 ff., 29 11. Paris, 1543. Editions. Glareanus is sometimes mentioned as the author of two works on arithmetic, the ' Isagoge Arithmetica ' and the work here described. Under the former title the following edi- tions are mentioned by various writers: Freiburg, 1 539, 8 (unless Tropfke is correct in saying there was an edition of 1538); Paris, I 554> 8; Lyons, 1554, 8. Under the above title the following editions appeared: Paris, 1543, 8 (here described. The dedica- tion is dated 'Friburgi Brisgoiae . . . M. D. XXXVIII,' so there may have been an edition as early as 1538); Freiburg, 1543, 8; Cracow, 1549; Freiburg, 1550, 8 (below); Paris, 1551, 8 (p. 193); Freiburg, 1555; ib., 1558, 8; Paris, 1558,8. The biog- rapher of Glareanus, Schreiber (Freiburg, 1837), gives these as the same work. Besides this book, Glareanus also published an ' Arithmetica et musica operum Boethii demonstrationibus et figuris auctior,' Basel, 1546, fol., and a ' Commentarius in Arithmeticam et Musicam Boethii,' Basel, 1546, fol. ; ib., 1570, 4. A handbook for the Latin schools. In it Glareanus first treats of notation, including the Greek, Roman, and Arabic systems; then of the elementary operations with integers ; then, briefly, of progressions and proportion. There is nothing in the little book to commend it. HENRICUS LORITUS GLAREANUS. Ed. pr. 1539. Freiburg, 1550. See p. 191. Title. De. VI. Ari//thmeticae // Practicae Speciebvs // Henrici Glareani//P. L. Epitome.// Fribvrgi Brisgoiae. //Cum gratia ac Priuilegio Regio,//ad annos fex.' (P.) Colophon. 'Apud Friburgum Brifgoicum // Anno M. D. L.// Stephanus Grauius //excudebat.' (P. 77.) Description. 8, 10 x 15.8 cm., the text being 6.4 X 1 1.7 cm. 2 pp. unnumb. -f 2 blank + 74 numb. = 78 pp., 22 11. Freiburg, 1550. PRINTED BOOKS 193 HENRICUS LORITUS GLAREANUS. Ed. pr. 1539. Paris, 1551. See p. 191. Title. 'De fex Arith-//meticae Practi-//cae Speciebvs,//Hen- rici Glareani // Epitome.// Parisiis,// Apud Gulielmum Cauel- lat, in pingui Gallina,// e regione colleij Cameracenfis. // 1551.' (F. I, r.) Colophon. ' Excudebat Lutetiae Parifiorum Benedictus // Pre- uotius Typographus in vico Fre-//mentello, fub Stella Aurea:// III. Non. lanuarij,// 1551.' (F. 23, r.) Description. 8, 10.4 X 16. 1 cm., the text being 6.4 x 11.9 cm. 2 ff. unnumb. + 21 numb. + I blank = 24 ff., 29 11. Paris, I55L See p. 192. HIERONYMUS CARDANUS. Ed. pr. 1539. Milan, 1539. GERONIMO or GIROLAMO CARDANO, JEROME CARDAN. Born at Pavia, September 24, 1501 ; died at Rome, September 21, 1576. He was a physi- cian and professor of mathematics at Milan (1534-1559) and professor of medicine at Pavia and (1562-1570) Bologna. Later he was a papal pen- sioner at Rome. He was one of the most acute mathematicians of his century, and wrote numerous treatises on mathematics and natural science. Title. See Fig. 97. Colophon. 'Anno a Virgineo partu.// M. D. XXXIX. // lo. Antonins Caftellioneus Me//diolani Imprimebat Im-//penfis Bernardini // Calufci.' (Printer's mark, with ' B.C.') (F. 304, v.) Description. 8, 9.9 X 15.3 cm., the text being 8.2 x 12.7 cm. 304 ff. unnumb., 33 11. Milan, 1539. Editions. Milan, 1539, 8 (here described); Niirnberg, 1541; ib., 1542. See also p. 338, 1570. A 1537 edition is given by Villicus, and I have seen it mentioned in a dealer's catalogue ; but I think the date a misprint, or that some one has taken the date of the preface instead of looking at the colophon. This is one of the most pretentious arithmetics of the sixteenth cen- tury, and it did much to influence the advanced teaching of the subject. It is in no sense a practical book, having been written by a mathematician i 9 4 KARA ARITHMETICA for the use of scholars. It opens with a discussion of the kinds of num- bers considered in arithmetic, such as integers, fractions, surds, and denominate numbers. This is followed by the fundamental operations HIERONIMI CCARDANI MEDICI MEDIOLA NENSIS,PRACTICA ARITFL *netice,& MenfurandifingularisJnqua que pretcr alia s c6tincntur,vcrfa pagina oemonftrabit* FIG. 97. TITLE PAGE OF CARDAN with these numbers and a treatment of proportion. The properties of numbers occupies a considerable space and includes much of the ancient theory. The work then runs into algebra, combining this with arithmetic. PRINTED BOOKS 195 There are numerous business applications in the treatise, such as partner- ship, exchange, profit and loss, and mensuration, but these are treated from the theoretical standpoint rather than from that of the practical needs of the merchant class. The great prominence of the author and the scholarly nature of the work account for the various editions of the book. His well-known ' Ars Magna ' (algebra) appeared in 1545. JOHANN NOVIOMAGUS. Ed. pr. 1539. Paris, 1539. NEOMAGUS, JAN BRONCKHORST. Born at Nimwegen in 1494; died at Cologne in 1570. He was for a time professor of mathematics at Rostock. He not only wrote on numbers, but edited works of Basda and Ptolemy. Title. See Fig. 98. Description. 8, 9.7 X 15.4 cm., the text being 6.7 x 12.3 cm. 117 pp. numb. + 2 unnumb. = 119 pp., 26-28 11. Paris, 1539. Editions. Paris, 1539, 8 (here described); Cologne, 1544, 8 (below) ; Deventer, 1551 (p. 197). The book was intended for the classical schools. It sets forth the Roman and Greek notations, the fundamental operations both with the Hindu numerals and upon the line abacus, the finger notation as found in the works of Baeda, the astrological numerals of the Middle Ages, and the Boethian theory of numbers. Other works 0/1539. Capella, p. 66, 1499 ; Peurbach, p. 53, 1492 ; Ringelbergius, p. 166, 1531 ; Vogelin (see Peurbach, p. 53, 1492); Wol- phius, p. 154,1527; Anonymous, ' Abacho novo con il quale ogni persona puol imparar Abacho senza che alcuno li insegni,' Venice ; Anonymous, ' An introduction to algorisme, to learne to reckon with the penne,' Lon- don, 8, with another edition ib., 1581, 8; Juan Gutierrez de Gualda, p. 167, 1531. JOHANN NOVIOMAGUS. Ed. pr. 1539. Cologne, 1544. See above. Title. ' De Nvme//ris Libri II. Qvo-//rum prior Logifticen, & ueterum nu//merandi confuetudinem : pofterior//Theoremata numerorum complecti-//tur, autore loan. Nouiomago. // Nunc rece"ns ab ipfo autore recogniti.//( Woodcut with motto: 'Discite Ivsticiam moniti.') Colonise loan. Gymnicus excudebat,// Anno M.D.XLIIII.' (F. i, r.) 196 KARA ARITHMETICA DcNumensli^ BRI DVO, CLVORVM PRIOR Logifticen & vecerum numerandi confuetudf- nem , pottcrior Theoremata numerorum com-* plcditur,ad do&ifsimum'virtim Andre* am Eggerdem profcflbrcm Roftochienfem. Nunc recens in lucem emiTsi authore loanne Nouiomago. PARISIIS Ex ofRcfna Chriftiani Wechcli,fub fcuto Bafilknfyn vico Iacobeo:& iub Pegafojin vico BelloiucenfL M, D. XXXIX. FIG. 98. TITLE PAGE OF NOVIOMAGUS PRINTED BOOKS 197 Description. 8, 10 x 14.8 cm., the text being 6.7 x 11.5 cm. 59 ff., 25 11. Cologne, 1544. Editions. See p. 195. This is merely a reprint of the first edition (p. 195, 1539). JOHANN NOVIOMAGUS. Ed. pr. 1539. Deventer, 1551. See p. 195. Title. ' De nvme//ris libri II. qvo-//rvm prior logisticen, et // ueterum numerandi confuetudinem : pofte-//rior Theoremata numerorum com//plectitur, Autore loan.// Nouiomago.// Nvnc recens ab ipso//autore recogniti.// (Woodcut, and ' T. B. Fons lovis.') Daventriae,// Theodoricus Bornius excudebat.// Anno M. D. LI.' (F. i, r.) Description. 8, 9.8 x 14.9 cm., the text being 6.8 X 10.7 cm. 50 ff. unnumb., 25 11. Deventer, 1551. Editions. See p. 195. Like the 1544 edition, this is a reprint of that of 1539. JODOCUS WILLICHIUS. Ed. pr. 1540. Strasburg, 1540. WILKE, WILCKE, WILD. Bom at Resel, East Prussia ; died at Lebus, November 12, 1552. He was professor of Greek (1540), and then of medicine, in the university of Frankfort an der Oder. Title. See Fig. 99. Colophon. 'Argentorati ex officina // Cratonis Mylii,// mense Sept.// anno //M. D. XL.' (P. 125.) Description. 8, 10.2 x 1 5.4 cm., the text being 8.2 x 1 1.4 cm. 2 pp. unnumb. + 123 numb. = 125 pp., 26 11. Strasburg, 1540. Editions. Strasburg, 1540, 8 (here described); ib., 1545. This is a book intended for the classical schools. It is written chiefly in Latin, but contains numerous extracts from the Greek. It is based upon Greek models, and contains several quotations from Nicomachus. The plan of treatment is catechetical (see Fig. 100), and it is interesting to note that this work appeared in the same year in which Recorde may have published his * Ground of Artes ' in England (see p. 213), a book in which the author also adopted the catechism form. It is manifestly inspired by Boethius, and is hair-splitting in theory and useless in prac- tice. Willichius begins in a grandiloquent style, ' De Arithmeticae, quae 198 RARA ARITHMETICA Mathefeos mater eft, finitione.' The history of arithmetic has few more curious examples than the first chapter of this work, with its learned references to Pythagoras, Augustine, the Platonists, and ' an Arab phi- losopher named Algebras.' (' Eadem Autem hodie ab authore quodam Arabe Philofopho, cui nomen erat Algebras, nomine regularum Algebra IODOCI VVILLICHII fcBSBLLIANI, Arithmetic* libri tret. ARGENTORATI 'JM t D. XL. FIG. 99. TITLE PAGE OF WILLICHIUS explicatur.' P. 19.) Willichius follows the ancient Greek plan of divid- ing arithmetic into two parts, the first being the practical, the logistica of the classical civilization (' vna TiyxxKTi/oy, qua fupputatio domeftica fit ... apud ueteres a ratiocinando Aoyi^i/d; dicta eft,' p. 19), and the second being theoretical, the ancient arithmetica. ( Eft tfewp^TiKT/, qua velut fanctiora myfteria continentur, & haec fola intelligentia animi PRINTED BOOKS 199 conftat,' p. 20). In his number mysticism he calls unity Jupiter (' Vnitas eft lupiter,' p. 22), saying that others call it Cupid, others Amicitia, and others Concordia, and quotes ' Zarathas the teacher of Pythagoras ' as calling one the father and two the mother of numbers. ('Proinde apte Zarathas Pythagorae praeceptor dixit, //.oraSa effe numerorum patrem, $ o ARITHMETICAE NICOLAV5. /~\VkeJttdtcrnmeruspdr* I VST. E# V-x pariterunparyuelAparibui impar^GrMK Pariter tnt* ***^^**i^&fai&t*+i {wr gtKiof.Eft autem, cum primumdiuiditur^mox ft indiuipbilii , t 1 4 . 1 8 . as , N I C O L. Quomodo eum finit borum numerorum txquiji* ties magifer Euclides * IV ST. Sic: fyrt / a V7TO diuidens par eft, fed diuiforiM mox impar exur get. NICOL. CuridnominKiUiinditum&* I V S T . Idea, quod quilibet eiw ordinis numeri paresfifttfuntperimparem multiplicationtmiut Barker vaifwr fa ter y [enttrium^ bit quin% denarium conficiunt. VerumfictriaUiutcontempiarilibet,cundem uo* cabit imparem infud qnxntitate 9 fed parem in dFeno wfmtfwne.E/fe exempUgratia^demtrii^ cum <&* teu pars cftquinariMtfui quantitatejioc eft, mo* nadum congregations eft imparted quid .1 binario denomintttur,par iudicabitur . Que ratio nomiws x Tloetbio cotUgitur : Alia auteni Euclidi efft uide tor* N 1 C O L. sum ne huic deitto aliquot tbeo Symbol*, ex rtmat* f IV ST. Quid ni ? Vnum eft . Si nuts, p<tr tier imp fa merus dimidium impar babuerit, pariter impdf vi e$ tgntttm * N^m hie dimtaxa extrfmm , quod FIG. ioo. FROM THE ARITHMETIC OF WILLICHIUS autem matrem,' p. 22.) The contrast between this work and that of Gemma, which appeared in the same year, is very marked. It must be admitted, however, that the book has some value in interpreting the ideas of such followers of Nicomachus as Boethius, Jordanus, and Faber Stapulensis, as is shown in Fig. ioo. It has at least the merit of having been written for beginners. 200 RARA ARITHMETICA GEMMA FRISIUS. Ed. pr. 1540. Wittenberg, 1542. More properly GEMMA RAINER or REGNIER, the Frisian. Born at Dockum, in East Friesland December 8, 1508; died at Louvain, May 25, 1555. He received the degree of Doctor of Medicine in 1541, when he abandoned his mathematical studies. He wrote upon astronomy and arithmetic, and his son Cornells was a contributor to the former science. Title. See Fig. 101. Colophon. ' Impreffum Vitebergae apud // Georgium Rhau.// Anno M.D.XLII.' (F. 80, r.) Description. 8, 10 X 15.3 cm., the text being 7.8 x 15.3 cm. 80 ff. unnumb., 23 11. Wittenberg, 1542. Editions. Antwerp, 1540, 8; Wittenberg, 1542, 8 (here described); ib., 1543, 12; Paris, 1543 (?); Wittenberg, 1544, 8 (p. 202); Paris, 1545 (Peletarius edition); Antwerp, 1547; Wittenberg, 1548; Paris, 1549, 8 (p. 202); Paris, 1550, 8 (p. 203); Wittenberg, 1550, 12; Antwerp, 1550; Wittenberg, 1551, 8 (p. 203); Paris, 1551; Antwerp, 1552, 8 (p. 203); Paris, 1553, 8 (p. 204); Wittenberg, 1553; ib., 1555; Lug- duni, 1556, 8; Paris, 1557; Leipzig, 1558, 8 (p. 204); s. a., but c. 1558; Paris, 1559; Leipzig, 1559; Paris, 1561,8; Wit- tenberg, 1561, 8 (p. 204); Leipzig, 1562; Antwerp, 1562; Paris, 1562; ib., 1563 (p. 205); Wittenberg, 1563, 8 (p. 205); Cologne, 1565; Leipzig, 1565, 8; Lugduni, 1566, 8; Witten- berg, 1567, 4; Paris, 1567; Venice, 1567, 4 (p. 205) ; Leipzig, 1568; Paris, 1569, 8; Wittenberg, 1570, 12; Cologne, 1571, 8 (p. 206) ; Paris, 1572 ; Leipzig, 1572 ; ib., 1575, 8 (p. 206) ; Cologne, 1576, 8; Paris, 1578, 8 (p. 207); Wittenberg, 1579; Leipzig, 1580; Antwerp, 1581, 8 (p. 207); ib., 1582, 8 (the first with Forcadel's notes?); Wittenberg, 1583, 8 (p. 208); Paris, 1585, 8; s. 1., 1588; Leipzig, 1588, 8 (p. 208) ; ib., 1591; ib, 1592, 8 (p. 208); Cologne, 1592, 8; Wittenberg, 1593; Frank- fort, 1 597 (the only German translation ?). There were numerous editions after 1600. Treutlein (Abhandlungen, I, 18), following Murhard, says there were at least twenty-five editions in the sixteenth century. In reality there were more than twice as many ; the above list, probably incomplete, mentions fifty-nine. PRINTED BOOKS 201 The editions of Gemma varied but little until Peletarius, several years before the former's death, added his notes. These amplified the text, but they made no changes of importance. This was the most popular arithmetic of the sixteenth century, at least among those intended for the Latin schools. It combined the older ARITHME, TICAE PRACTI* CAB METHODVS FACILE, per Gemmam.Frifium MediV cum ac Mathema ti'cum. VITEBERGAE, M. D.-XLIf. FIG. 10 1. TITLE PAGE OF THE 1542 GEMMA FRISIUS 202 RARA ARITHMETICA science of numbers with the commercial arithmetic of the Italian writers in such way as to appeal in a remarkable degree to the teachers of the period. The book opens with a discussion of the various fundamental operations, presented without much explanation and with numbers of relatively small size. These operations include the subjects of dupla- tion and mediation as was customary in the Latin books of that time. The author closes the first part of his work with a treatment of progres- sions and the rule of three. The second part is devoted to fractions, the sequence being the same as with integers. The third part includes such common rules of business as partnership, alligation, and rule of false, together with roots and a little algebra. The fourth part treats of proportion, and has a few pages on arithmetical recreations. GEMMA FRISIUS. Ed. pr. 1540. Wittenberg, 1544. See p. 200. Title. This edition is substantially the same as that of 1542 with the following exceptions : F. i, r., title page : ' Vitebergae Anno M,D,XLIIII.' Colophon. ' Anno M,D,XLIIII.' (F. 88, r.) Description. 8, 9.5 X 15 cm., the text being 7.9 X 11.3 cm. 88 ff. unnumb., 23 11. Wittenberg, 1544. See p. 20 1. GEMMA FRISIUS. Ed. pr. 1540. Paris, 1549. See p. 200. Title. ' Arithmeticae practicae // Methodvs facilis per // Gem- mam Frisivm, Medicvm // ac Mathematicum, lam recens ab ipfo //authore emendata & multis in lo-//cis infigniter aucta.// Hvc accesservnt lacobi Pe-//letarii Cenomani annotationes : Eiufdem item de Fractio-//nibus Aftronomicis compendium : Et de cogno- fcedis per // memoriam Calendis, Idib. Nonis, Feftis mobilibus, & loco // Solis & Lunae in zodiaco.// (Woodcut with motto : ' In pingvi. gallina.') Parish's. //Apud Gulielmum Cauellat, in pingui gallina, ex // aduerfo collegij Cameracenfis.// 1549.' (F. i, r.) Description. 8, 10.5 X 16.6 cm., the text being 7 x 12.7 cm. i f. unnumb. +95 numb. =96 ff., 26 11. Paris, 1549. PRINTED BOOKS 203 Editions. This is the ninth edition (see p. 200), and the first which I have seen with the notes of Peletarius, although these notes bear the date 1545 (f. 77, r.) They are here given in an appendix (Tacobvs Peletarius Lectori,' f. 77, r.), but later, as in the 1571 edition (see p. 206), they are introduced in the body of the text. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Paris, 1550. See p. 200. Title. Except for the date (1550) the title page is the same as that of the 1549 edition (p. 202). Description. 8, 10.3 X 16.1 cm., the text being 7.3 x 13 cm. i f. unnumb. + 95 numb. =96 ff., 26 11. Paris, 1550. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Wittenberg, 1551. See p. 200. Title. ' Arithme//ticae practicae // method vs facilis, per // Gemmam Frifium Medi-//cum ac Mathe-//rnaticum.// (Wood- cut showing computing with counters.) Vvitebergae.// Anno M. D. LI.' (F. i, r.) Description. 8, 9 X 14.2 cm., the text being 6.4 X 11.2 cm. 87 ff. unnumb. + i blank = 88 ff., 23 11. Wittenberg, 1551. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Antwerp, 1552. See p. 200. Title. < Arithme-//ticae practicae metho-//dus facilis, per Gem- mam Frifium Me-//dicum ac Mathematicum,iam re-//cens ab ipfo auctore emen-//data, & multis in locis // infigniter aucta.// Cor. Graphevs.// Si numerandi artem, cunctis ex artibus illam //Vel primam, exacte difcere lector amas,// Hanc gemma, ingenio fum- mus quam Gemma libello // Hoc paruo includit, carpito, doctus eris.// Sunt conati alij prolixis tradere chartis // Hanc artem, at 204 KARA ARITHMETICA multis non placet ille labor.// Porro hie Gemma fuam gemmam fie temperat, ipfa // Vt placeat cunctis commoditate breui.// C.Antuerpiae, apud Gregorium Bontium//a Caef. Maieft. libra- rium admiffum.// Cum gratia & priuilegio.// I55 2 -' ( F - i, r -) Description. 8, 9.5 X 13.9 cm., the text being 7.4 X 1 1.8 cm. 4 ff. unnumb. + 76 numb. = 80 ff., 28 11. Antwerp, 1552. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Paris, 1553. See p. 200. Title. Except for the date (1553) the title page is the same as that of the 1549 edition (p. 202). Description. 8, 9.8 x 14.8 cm., the text being 6.8 X 12.7 cm. i f. unnumb. + 95 numb. = 96 ff., 26 11. Paris, 1553. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. S. 1. (Leipzig), 1558. See p. 200. Title. ' Arithme-//ticae practice // methodvs facilis, per // Gemmam Frifmm Medicum ac // Mathematicum // (Woodcut as in the 1542 edition.) Anno M. D. LVIII.' (F. i,r.) Description. 8, 10. i x 15.4 cm., the text being 7.8 X 1 1 cm. 87 ff. unnumb., 22-23 H- S. 1- (Leipzig), 1558. Editions. See p. 200. See p. 201. A set of manuscript notes in the back, in a sixteenth- century hand, gives an interesting synopsis of arithmetic as taught in the universities of that time. GEMMA FRISIUS. Ed. pr. 1540. Wittenberg, 1561. See p. 200. Title. ' Arithme-//ticae practicae me-//thodvs facilis, per Gem- //mam Frisivm Medi-//cum ac Mathematicum.// (Woodcut as in the 1542 edition.) Vvitebergae // ex officina Haeredvm // Georgii Rhavv//M.D.LXI.' (F. i, r.) Colophon. * Vvitebergae // ex officina Haeredvm // Georgii Rhavv//M.D.LXI.' (F. 88, v.) PRINTED BOOKS 205 Description. 8, 9.8 X 14.9 cm., the text being 7.7 x 1 1.2 cm. 88 ff. unnumb., 23 11. Wittenberg, 1561. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Paris, 1563. See p. 200. Title. The title page is substantially the same as that of 1549 (p. 202), with the following addition: loth line, ' Quibus demum ab eodem Peletario additae funt Radicis // vtriufque demonftrationes.' Description. 8, 10.4 X 16.3 cm., the text being 7.1 x 12.5 cm. 2 ff. unnumb. + 102 numb. = 104 ff., 26 11. Paris, 1563. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Wittenberg, 1563. See p. 200. Title. ' Arithme-//ticae practicae me-//thodvs facilis, per // Gemmam Frifium Medicum ac // Mathematicum.// VVitebergas //M.D.LXIII.' (F. i,r.) Colophon. ' VVitebergae // ex officina Haeredvm.// Georgii Rhavv.// M.D.LXIII.' (F. 88, r.) Description. 8, 10.5 x 15.6 cm., the text being 8.1 X 1 1.2 cm. 88 ff. unnumb., 23 11. Wittenberg, 1563. Editions. See p. 200. v GEMMA FRISIUS. Ed. pr. 1540. Venice, 1567. See p. 200. Title. ' Aritmetica//prattica facilissima,//composta da Gem- ma Frisio Medico, //et Matematico ; //Con 1'aggiunta dell' Ab- breuiamento de i Rotti Aftronomici di // Giacomo Pelletario; & del conofcere a mente le Calende,// gl' Idi, le None, le Fefte Mobili, il luoco del Sole, & della Luna nel // Zodiaco; & la dimo- ftratione della Radice Cubica : lequali tutte // cofe dal latino, ha in quefta lingua ridotte Oratio Tosca-//nella della famiglia di Maeftro Luca Fiorentino ; & halle de-//dicate // allo illvstre 206 KARA ARITHMETICA fignore, il fignor//Ettore Podocataro.// (Woodcut.) InVenetia, Apreffo Giouanni Bariletto.//MDLXVII.' (F. i, r.) Colophon. < Registro.// *ABCDEFGHIKL M.// Tutti fono Quaderni ; eccetto M,//che e Quinterno //In Venetia,// Apreffo Giouanni Bariletto.//MDLXVII.' Description. 4, 14.9 X 20.7 cm., the text being 10.6 X 17.4 cm. 5 ff. unnumb. + 51 numb. = 56 ff., 4042 11. Venice, 1567. Editions. See p. 200. This is the only Italian edition of the sixteenth century. See p. 201. The ' Maeftro Luca Florentine' mentioned in the title page was a well-known arithmetician of Florence. A manuscript of his of c. 1425 is described in the second part of this work. GEMMA FRISIUS. Ed. pr. 1540. Cologne, 1571. See p. 200. Title. ' Arithme-//ticae practicae //methodvs facilis, per Gem- mam // Frifium, Medicum ac Mathematicum, iam re-//cens ab ipfo authore emendata, & multis // in locis infigniter aucta.// Hvc accesservnt lacobi Pe-//letarij Cenomani annotationes : Eiufdem item de // Fractionibus Aftronomicis compendium : Et // de cognofcendis per memoriam Calendis,// Idibus, Nonis, Feftis mobilibus, // & loco Solis & Lunas in// Zodiaco.//Nunc ver6 a loanne Stein recognita, & no-//uis aucta additionibus.// (Woodcut with motto : Benedices // Coronae Anni //Benignitatis //Tvae. Psal. 64 //) Coloniae,// Apud Maternum Cholinum.// M. D. LXXI.//Cum gratia & priuilegio Caef. Maieft.' (P. i.) Description. 8, 9.5 x 15.8 cm., the text being 6.5 X 12.1 cm. 3 pp. unnumb. + 281 numb. = 284 pp., 28-30 11. Cologne, 1571. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Leipzig, 1575. See p. 200. Title. 'Arithme-//ticae practicae // methodvs facilis, //per// Gemmam Frisivm // Medicum ac Mathematicum.// (Woodcut representing a counting house, with line reckoning.) Lipsiae // lohannes Rhamba excudebat.// M. D. LXXW (F. i, r.) PRINTED BOOKS 207 Description. 8, 9.5 X 15.3 cm., the text being 7.8 X n cm. 87 ff. unnumb., 23-24 11. Leipzig, 1575. See p. 201. The book has been rebound with added pages which have been used for manuscript. These notes are in Latin, in a Ger- man sixteenth-century hand, and include both work on the fundamental operations and numerous commercial problems. GEMMA FRISIUS. Ed. pr. 1540. Paris, 1578. See p. 200. Title. The title page is substantially the same as that of 1542 (p. 201), except for the following: ' In me Mors,//In me Vita.//Parisiis,//Apud Hieronymum de Marnef, & viduam // Guliel. Cauellat, fub Pelicano monte D.Hilarij. 7/1578.' (F. i,r.) Description. 8, 10.9 X 17.3 cm., the text being 7.4 x 14.1 cm. 3 ff. unnumb. + 93 numb. = 96 ff., 29 11. Paris, 1578. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Antwerp, 1581. See p. 200. Title. ' Arithmetics //practicae methodvs fa-//cilis, per Gem- mam Frisivm, Medi-//cum, ac Mathematicum confcripta : iam recens ab Auctore // pluribus locis aucta & recognita.// In ean- dem loannis Steinii & Iacobi//Peletarii Annotationes.//Eiufdem de Fractionibus Aftronomicis Compendium,// & de cognofcendis per memoria Kalendis, Idibus, No-//nis, Feftis mobilibus, Ioco'q5 Solis & Lun< in Zodiaco. // (Woodcut of astronomer.) // Ant- verpiae.// Apud loannem Bellerum ad infigne Aquilae // aureae. Anno 1581.' (P. i.) Description. 8, 9.7 x 15.7 cm., the text being 6.5 X 13 cm. 5 pp. unnumb. -f 178 numb. + I blank = 184 pp., 31 11. Antwerp, 1581. With this copy is bound: 'Ivl. Pacii//a Beriga // Insti- tvtiones // Logicae,// in vsvm scholarvm//Bernensivm //editae. //Bernae. M.D.C.' See p. 201. 2 o8 KARA ARITHMETICA GEMMA FRISIUS. Ed. pr. 1540. Wittenberg, 1583. See p. 200. Title. ' Arithme-//ticae Practicae // Methodus facilis,// per // Gemmam Frisivm // Medicum ac Mathematicum.// (Woodcut as in the 1 542 edition.) Vvitebergae//Ex officina Matthaei VVelacij. //M. D. LXXXIII.' (F. i, r.) Description. 8, 9.5 X 15 cm., the text being 7.9 x 11.2 cm. 87 ff. unnumb., 23 11. Wittenberg, 1583. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Leipzig, 1588. See p. 200. Title. * Arithme-//ticae practicae // methodvs facilis, //per// Gemmam Frisivm // Medicum ac Mathematicum.// Lipsiae,// Anno M. D. LXXXVIII.' (F. i, r.) Colophon. ' Lipsiae, //ex officina typo-//graphica Abrahami // Lambergi.//Anno//M. D. LXXXVIII.' (F. 87, r.) Description. 8, 9.4 X 15 cm., the text being 7.8 X 11.5 cm. 87 ff. unnumb., 24 11. Leipzig, 1588. See p. 201. GEMMA FRISIUS. Ed. pr. 1540. Leipzig, 1592. See p. 200. Title. ' Arithme//ticae Practicae // Methodvs facilis,// per // Gemmam Frifium Medi-//cum ac Mathematicum.// (Woodcut as in the 1 542 edition.) Lipsiae,// Anno // M. D. XCII.' (F. i , r.) Colophon. (Woodcut of Pegasus.) ' Lipsiae, //ex officina typo-// graphia Abrahami // Lambergi.// Anno //M.D. XCII.' (F.88,r.) Description. 8, 9.2 x 15.5 cm., the text being 8.1 X 11.4 cm. 88 ff. unnumb., 24 11. Leipzig, 1592. See p. 201. HENRICUS URANIUS. Ed. pr. 1540. Solingen, 1540. A German classicist, born at Reesz, Prussia. He lived at Emmerich (Em- rich) when he wrote this work. Title. See Fig. 102. PRINTED BOOKS 209 Description. 8, lox 15 cm., the text being 7.9 X 11.7 cm. 20 ff. unnumb., 28 11. Solingen, 1540. Editions. There was no other edition. This is a semi-historical discussion of the various measures which the sixteenth century received from the Roman civilization, or which were JNVMARIA, MENSVRIS ET P O N- deribus Epitome ex Budaeo, Porcio, Alciato O" Georgio Agricola concinnata per Henncuro Vranium Reffenfem. IN ZOILVM- Zotlr qttttfpropfrdrfwM We <juod rode re poJ?iV, Qmnitfunt doftit ante probatA mis. Qcyw bincfugitoji ncfcis quo fit ewdm: Enmonftrdo, mlm Zoik adito cm cm* IALING1ACI, loannes Sorer cxcudebar. AN. M.D.XL. FIG. 102. TITLE PAGE OF URANIUS mentioned in the most commonly read classics of the Renaissance period. No arithmetical operations are given in the book. It begins with the fractional parts of the as, the twelfth being called the uncia (the Troy ounce), the sixth the sextans, the fourth the quadrans, and so on. 210 KARA ARITHMETICA PHILIP MELANCHTHON. Ed. pr. 1540. Leyden, 1540. Horn at Bretten, Baden, February 6, 1497 ; died April 19, 1560. His family name was Schwarzerd; he assumed the Greek equivalent, Melanchthon, when he entered the university of Heidelberg (1509). He was one of the most famous classicists of the Renaissance, and a leader in the Reformation. Title. See Fig. 103. M A T H E M A* TICARVM DISCI- PLINARY**, TVM ETI A M A S TRO LO- G I A E E N C fl- irt I A> * PER PHIL. ME LAN CHT. ITEM Phenomena loachimi Camerarij, clegantifsimo carmine dcfcripta* APVD SEE. GRVPHIVM L V G D V N J, 1540. FIG. 103. TITLE PAGE OF MELANCHTHON PRINTED BOOKS 211 Description. 8, 9.9 x 15.2 cm., the text being 7.2 x 12.2 cm. 40 pp. numb., 30 11. Leyden, 1540. Editions. There was no other separate edition. The work of Melanchthon consists of three letters: (i) to Simon Grynaeus, dated Vitebergae, mefe Augufto. M. D. XXXI ' ; (2 ) to Johann Reiffenstein, dated ' Menfe Augusto. Anno M. D. XXXVI ' ; (3) to Johann Schonerus (see p. 178), dated * Vuittebergse, Mefe Augusto, Anno M. D. XXXVI.' These epistles are all upon the value and the nature of mathe- matical thought, and are replete with classical and religious references. As mathematics they have no value. The verses of Camerarius, ' loachimi Camerarii Phaenomena ad clarissimum ivvenem Danielum Stibarum,' relate chiefly to astronomy. MAGNUS AURELIUS CASSIODORUS. Ed. pr. 1540. Paris, 1550. CASSIODORIUS. Born at Scylaceum, c. 470 ; died, probably at Rome, c. 564. A Roman statesman and historian. Title. 'Aurelij Cafsiodori Se-//natoris Cos.qve Romani//de quatuor Mathematicis difciplinis// Compendium. //Parisiis//Apud Vafcofanum, uia lacobaea ad infigne Fontis.// M. D. L.' (F. i, r.) Description. 4, 15.9 x 20.8 cm., the text being 9.7 x 17.2 cm. i f. unnumb. + 7 numb. = 8 ff., 25-29 11. Paris, 1550. Editions. Paris, 1540; ib., 1550, 4 (here described); ib., 1580. The ' Opera ' appeared in Paris in 1598 (and 1584?). The Compen- dium is embodied in the * Disciplinarum liberalium orbis,exP. Con- sentio et Magno Aurelio Cassiodoro/ published at Basel in 1528. This brief treatise on the nature of arithmetic, music, geometry, and astronomy, the four mathematical disciplines, was held in high esteem in the Middle Ages. Other works of 154.0. Anianus, p. 32, 1488 ; Beldamandi, p. 15, 1483 ; Benese, p. 182, 1536; Borghi, p. 21, 1484; Ortega, p. 93,1512; Paxi, p. 79, 1503 ; Rudolff, p. 152, 1526 ; p. 160, 1530; Scheubel, p. 233, T 545 > Wolphius, p. 154, 1527; Anonymous (Ortega?, p. 91, 1512), * CEuvre tres subtile y profitable de Tart y science de arismeticque y geometric translate nouvellement d'Espaignol en Francoys,' Paris, 8. Works of 1541. Agrippa, p. 167,1531; Albert, p. 180, 1534 ; Cardan, P- J 93 J539; Riese, p. 139, 1522; Ringelbergius, p. 167, 1531; Ta- gliente, p. 115, 1515 ; Georg Rheticus, Arithmetic, Strasburg. 212 KARA ARITHMETICA ANONYMOUS. Various authors. Ed. pr. 1542. Cologne, 1542. Title. See Fig. 104. ARITHME TICES ftio ex uarijs ctuthoribu* cow cinnata* <iiMt/ Coloniaexcudebat loannesGytnnicw <-Snno M+ D. XLIL $,<i*'J> , A- 5 f i'Wif/F , ^v./^r S'</ i'viivvwt- ^lufi-i't- >X* fa.<i^ vc/^i_ FIG. 104. TITLE PAGE OF THE 1542 ANONYMOUS WORK Description. 8, 9.7 X 15.1 cm., the text being 6.9 x 12.3 cm. 20 ff. unnumb., 25-29 11. Cologne, 1542. PRINTED BOOKS 213 Editions. Cologne, 1542, 8 (here described); ib.,1546; Dort- mund, 1549, 8 (see below). This is one of several anonymous compilations made in the sixteenth century for use in the Latin schools. It has no merit, save that of brevity. It contains a brief treatment of the fundamental operations, followed by a chapter ' De Progrefione,' the ' Regvla mercatorvm feu de tribus,' and 7 pages * De minutijs.' I notice that folio B 6, v., is an exact copy of Wolphius (see p. 154) folio B i, v. This is the same as the work next mentioned, published at Dortmund in 1549. ANONYMOUS. Ed pr. 1542. Dortmund, 1549. Title. ' Brevis // Arithme//tices Intro-//dvctio ex Variis // Authoribus con-//cinnata. Tremoniae excud. Melch. Soter./ Anno M.D.LXIX.' (P. i.) Description. 8, 9.8 x 14.9 cm., the text being 6. i x 12.3 cm. 48 pp. unnumb. (i blank), 25 11. Dortmund, 1549. See above. ROBERT RECORDE. Ed. pr. c. 1542. London, 1558. Born at Tenby, Pembroke, c. 1510; died in Southwark prison, probably soon after June 28, 1558 (the date of his will). He was educated at Oxford and Cambridge, and taught mathematics at the former and probably at the latter university. He became royal physician and wrote on medicine as well as mathematics. Title. See Fig. 105. Colophon. ' Imprinted at London in Paules churchyard // at the figne of the Brafen Serpent //by Reginalde Wolfe.// Anno Domini M. D. LVIII.' (F. 205, v.) Description. 8, 8.9 X 13. i cm., the text being 8 X n.8 cm. 205 ff. unnumb., 31 11. London, 1558. Editions. There is considerable uncertainty as to the date of the first edition of this work of Recorde's. It appeared, however, between 1540 and 1542. For a discussion of the question see the Dictionary of National Biography, and De Morgan, p. 22. The former says there were twenty-seven editions of the book, but there were at least twenty-eight (see p. 214). On account of the influence of the work on English education, the bibliography 214 RARA ARITHMETICA has been extended through the seventeenth century. London, c. 1542; ib., 1543,8; ib., 1549,8; ib., 1551. 8; ib., 1556; 1552; London, 1558, 8 (here described); ib.,i56i, 8 (the earliest seen by De Morgan); 1570; London, 1571, 8; ib., 1573; ib., 1577; ib. 1579, 8 (p. 217); ib., 1582, 8 (the Mellis edition); THE GROVN D OF A R T E S: pzactifc of 0rftt)mcnlir, bott) in \Wjolctwmbies iwo factions , eftcc a mote fafpcc anD cractec fozte, tben anye lyftc Ijnttjljptlwrobeene ueva ueib f,o o B ER T B R E C O R D E of FIG. 105. TITLE PAGE OF THE 1558 RECORDE ib., 1586, 8; ib., 1590, 8 (the Dee and Mellis edition); ib., 1594, 8 (p. 207); ib., 1596,8 (p. 219); ib., 1618, 8;ib., 1623, 8; 1636; London, 1646, 8 (p. 219); ib.,c. 1646 (p. 220); 1652; 1654; London, 1662, 8 (p. 220); ib., 1668, 8 (p. 221); 1673; 1699. Presumably all of these were published at London. PRINTED BOOKS 215 ADDITION. none otbcr tramples fo? to learnet&f nume tattoncf tftfsfo:mc- But tljig fljall p cii mrhe,tftat as pou fcpD In t!)c ot&et hpnfc of arW;meti& fc? a pncfcc tntDeplftccs of tUoufaufccs, w tijtstoooifte pou QjaU Cet a ttarrc,a0 pou fee bcfozc* S* JTftcn 7 pcrccauc ^umcrattcn : but 3 pia?c pou, Doto J^aii 3 &o in tDm art to rtft tteoo Dimmer o; mo:c togttfter? ADDITION. be caficft toap tn tftts artels to ante but ttoo futttittcs at ones togptfjcr: ', pou wape afc&e meje,as J tori re [ je* tl)crcfoje tbljcnnc pou topll** ttoo fummes,pou$aH fp^ae fet botbnc one of meuut tocctft not tbfttc^r, <?nt> then b? ft D^ato a Ipne crolTe tf>c otDcr Ipnts. vino if pou ttxwlfee awit ^^- $- **!9 tO ^341 > POU &<&% ^ mud fct pour famca j ^ -a-^^ 1 ?cn<f pou -^,_L^^ Ipft, pou map*: aaar tftc oic to t*K otfter tn tlje fame place, 02 els pou map am tl):mbor&rro0itl)crui a trcto U 3 FIG. 1 06. COUNTER RECKONING FROM THE 1558 RECORDE 2i6 KARA ARITHMETICA The first arithmetic to be published in England was that of Tonstall (p. 132). There must have been several other books published between that date and the first appearance of Recorde's, because in the preface to the latter the author says : ' And if any man obiect, that other books haue bene written of Arithmetike alreadie fo fufriciently, that I needed not now to put penne to the booke, except I wil codemne other mens writings : to them I anfwere. That as I codeme no mans diligence, fo I know that no man can fatisfie euery man, and therefore like as many do efteeme greatly other bookes, fo I doubt not but fome will like this my booke aboue any other Englif h Arithmetike hitherto written, & namely fuch as fhal lacke inftructers, for whofe fake I haue plain-ly fet forth the exaples, as no book (that I haue feene) hath hitherto : which thing fhall be great cafe to the rude readers.' (From the 1594 edition.) This is, however, the first commercial arithmetic of any note used in the English schools. It is written in the form of a dialogue between the master and his pupil, and the language is so formal that it seems strange that the book should have been so successful. The first part is devoted to integers, the fundamental operations being followed by a section on denominate numbers. This is followed by proportion and the 'golden rule ' of three, the backer rule of three (inverse proportion), the double rule of three (compound proportion), the rule of three com- posed of five numbers, and the rule of fellowship (partnership). The second part relates to fractions, and includes the same general topics as the first, together with alligation and the rule of false. Counter reck- oning is given (Fig. 106) as well as computation with Arabic numerals. Among other works, Recorde wrote ' The Castle of Knowledge ' (p. 253) and 'The Whetstone of Witte ' (p. 286), the latter being chiefly on algebra. His geometry, ' The Pathway to Knowledge,' appeared in London in 1551, 4; ib., 1574, 4. Recorde's works were the most influential English mathematical pub- lications of the sixteenth century. Thomas Willsford, in his 1662 edition of the ' Ground of Artes,' was able to say with much truth that this book was ' entail'd upon the People, ratified and fign'd by the approbation of Time.' Other works 0/1542. Albert, p. 180, 1534; Cardan, p. 193, 1539 ; Finaeus, p. 163, 1530-32 ; Gemma, p. 200, 1540; Ortega, p. 93, 1512 ; Diego el Castillo, 'Tratado de quentos,' Salamanca, 4; Giambattista Verini, Spechio del mercatanti,' Milan, 8 (Brunet says, ' Libro de Abaco e gioco de memorie,' Milan, sm. 8); Han vander Wehn, ' Exem- pelrechenschaft der Regel de Tri, die man nennt die Kaufmanns guldene Regel ganz und gebrochen,' s. 1., 8. PRINTED BOOKS 217 ROBERT RECORDE. Ed. pr. c. 1542. London, 1579. See p. 213. Title. ' The // Grounde of Artes :// teaching the work and pra-//ctife of Arithmetike, bothe in whole numbers // and Frac- tions, after a more eafyer and //exacter forte than any like hath hither-//to bin fet foorthe :// Made by Mayfter Roberte Re-// cord, Doctor in Phyfike, and now of late// diligently ouerfeene and augmented // with new and neceffarie //Additions.// I. D. That which my freende hath well begonne, For very loue to common weale, Neede not all whole to be new done, But new encreafe I do reueale. Some thyng heerein, I once redreft, And nowe agayne for thy behoofe, Of zeale I doe, and at requeft, Both mend and adde, fitte for all proofe. Of Numbers vfe, the endleffe might, No witte nor language can expreffe, Applie and Trie, both day and night, And then this truth thou wilt confeffe. Printed at London by H. Bynneman.// Anno Domini. 1579.' (F. i,r.) Colophon. ' Imprinted at London by Henry // Binneman, and John Harifon.//Anno Domini M.D.LXXVII.' (F. 261, r.) Description. 8, 8.7 x 13.9 cm., the text being 6.5 X 1 1.7 cm. 261 ff. unnumb., 27 11. London, 1579. It will be noticed that the colophon is dated 1577. This is therefore one of the cases where a large edition was printed, and a new title page was added from year to year as necessary. The edition is more rare than its date would suggest. ROBERT RECORDE. Ed. pr. c. 1542. London, 1594. See p. 213. Title. ' The // Grovnd of // Artes, teaching // the perfect worke and practife //of Arithmeticke, both in whole numbers // and Fractions, after a more eafie and exact // fort, than hitherto 2i8 KARA ARITHMETICA hath been fet foorthe.// Made by M. Robert Record, // D. in Phificke.// And now lately diligently corrected and beauti-//fied with fundry new Rules and neceffary Addi-//tions : And further endowed with a third part, of // Rules of Practife, abridged into a briefer method // than hitherto hath bene publifhed : with di-//uerfe fuch neceffairie Rules as are // incident to the trade of // Merchandife.// Whereunto are alfo added diuerfe Tables and In-//ftructions that will bring great profite and delight vnto Merchants, Gentlemen, and others,// as by the Contents of this Trea-//tife fhall appeare.// By lohn Mellis.// At London,// Imprinted by T.D. for lohn Harifon, at the Greyhound in// Paules Churchyard.// 1594.77 (P. I.) Colophon. * Imprinted at London by Thomas Dawfon, for lohn // Harrifon, dwelling in Paules //Churchyard, at the figne of //the Greyhound,// 1594.' (P. 558.) Description. 8, 10. i x 15.5 cm., the text being 7.6 X 13.1 cm. 34 pp. unnumb. + 493 numb. 33-525 4- i blank = 528 pp., 30-31 11. London, 1594. Editions. See p. 213. This is probably the third edition by Mellis. In the dedication he says : ' And feeing that within this 8. yeares, two impreffions of thefe my labors dedicated to your Worfhip are already worne out . . .', although not much reliance can be placed on the statement, since it also appears in the 1596 edition. See p. 216. The first two parts, covering 404 pp. of this book, are substantially identical with the 1558 edition (p. 213). The third part (pp. 405-5 5 7), the work of John Mellis, also appeared in the 1591 edition. It includes the ' Rules of Practife,' ' The order & worke of the Rule of three in broken Numbers, after the trade of Marchants, digreff- ing fomething from M. Recordes,' * Loffe and Gaine,' ' Rules of Payment' (equation of payments), barter, exchange, interest, and other business applications, together with a chapter on ' Sportes and Paftimes done by Number.' These mathematical recreations had already appeared in printed textbooks, and they played an interesting role until the latter half of the nineteenth century. Those in this treatise related to number guessing, the rules being easily developed by our present algebra, but rather mysterious by sixteenth-century arithmetic. PRINTED BOOKS 219 ROBERT RECORDE. Ed. pr. c. 1542. London, 1596. See p. 213. Title. The title page is missing. Colophon. ' Imprinted at London by Richard // Field, for lohn Harrifon, dwelling // in Pater nofter Row at the // figne of the Greyhound.// 1596.' (P. 559.) Description. 8, 9.8 x 15.5 cm., the text being 8 x 13.3 cm. 559 PP-> 30-3 1 11- London, 1596. Editions. See p. 213. A note on the last page says that this copy was bought in 1686 for is. 6d., not a very low price at that time for a book only ninety years old. This edition is practically identical with that of 1594 (p. 217). ROBERT RECORDE. Ed. pr. c. 1542. London, 1646. See p. 213. Title. ' Records Arithmetick :// or,// The Grovnd // of Arts : // Teaching // The perfect work and Practice of Arithmetick,// both in whole Numbers and Fractions, after a more//eafie and exact form than in former time hath been fet forth ://Made by M. Robert Record, D. in Phyfick. //Afterward, augmented by M. John Dee.// And fince enlarged with a third part of Rules of Pra-//ctife, abridged into a briefer method than hitherto hath been//publifhed, with divers neceffary Rules incident to the Trade // of Merchandife : with Tables of the valuation of all Coyns,//as they are currant at this prefent time.// By John Mellis.//And now diligently perufed, corrected, illuftrated and en-// larged ; with an Appendix of figurate Numbers, and the Extraction // of their Roots, according to the method of Chriftian Vrftitius : with // Tables of Board and Timber meafure ; and new Tables of Intereft //upon Intereft, after 10 and 8 per 100 ; with the true value of // Annuities to be bought or fold prefent, Refpited, or in Rever-//fion : the firft calculated by R. C. but corrected, and the // latter diligently calculated by Rob : Hart- well, Philomathemat.// Scientia non habet inimicum nifi ignoran- tem.// Fide. fed - Vide.// London,// Printed by 220 KARA ARITHMETICA M. F. for John Harifon, and are to be fold by //Geo: Whitting- ton, and Nath: Brooks, at trie fignof//the Angell in Corn-hill. 1646.' (P. i.) Description. 8, 10 x 15.5 cm., the text being 8.6 x 13.7 cm. 27 pp. unnumb. + 629 numb. = 656 pp., 31 11. London, 1646. Editions. See p. 213. Like the 1594 edition this has the additional 'third part' by John Mellis. The Hartwell chapter on roots begins on p. 573, and is based, as the title says, on the work of Urstisius. ROBERT RECORDE. Ed. pr. c. 1542. London, 1646 (?). See p. 213. Title. This edition is said to have been published in 1646, but no date appears in the book itself. It has the same title page as the one dated 1646, except as follows : The words ' Records Arithmetick' (I.I) are here omitted ; the last lines read ' Printed by M. F. for John Harifon, and are to be // fold at his Shop in Pauls-Church-yard.' The number of words on some of the lines in the title page is different, but with the above exceptions the page is the same. The body of the book is from the same setting of type as in the other edition of 1646. Description. 8, 10.2 x 15.7 cm., the text being 8.6 X 13.7 cm. 27 pp. unnumb. + 629 numb. = 656 pp., 32 11. London, s. a. (1646?). See p. 216. ROBERT RECORDE. Ed. pr. c. 1542. London, 1662. See p. 213. Title. The title page is practically the same as that of 1646, except as to the imprint : ' Printed by James Flefher, and are to be fold by Jofeph // Cranford, at the figne of the Gunn in St. Pauls // Church-yard. 1662.' Description. 8, 10.7 x 16.5 cm., the text being 9.2 x 13.9 cm. 22 pp. unnumb. + 536 numb. = 558 pp., 33 11. London, 1662. See p. 216. PRINTED BOOKS 221 ROBERT RECORPE. Ed. pr. c. 1542. London, 1668. See p. 213. Title. The title page of this edition is practically the same as that of 1646, except as to the imprint : 'Printed by James Flefher, and are to be fold by // Robert Boulter, at the Turks-head in Bishopsgate-//ftreet, next the great James. 1668.' Description. 8, 10.4 X 16.4 cm., the text being 9.2 X 13.7 cm. 22 pp. unnumb. + 536 numb. = 558 pp., 33 11. London, 1668. JOHANN FREY. Ed. pr. 1543. Niirnberg, s. a. (1543). A Niirnberg gauger of the middle of the sixteenth century. Title. See Fig. 107. Colophon. ' Gedruckt zu Nurnberg durch//Georg Wachter.' (F. 36, r.) Description. 8, 9.5 x 15.5 cm., the text being 7.1 x 10.9 cm. 36 if. unnumb., 21-26 11. Nurnberg, s. a. (1543). Editions. There was no other edition. The book bears no date except in the dedicatory epistle to the reader, which closes with the words, 'Anno 1543.' This epistle gives, also, the only reference to the author, who there speaks of himself as ' ich Johan Frey/ burger zu Nurmberg.' The subject of gauging occupied a great deal of attention on the part of German writers on arithmetic in the sixteenth century, and occasionally, as in this instance, separate books were prepared. It was not so common in England as on the Continent, not appearing, for example, in as extensive a work as the Dee and Mellis edition of Recorde mentioned above. It had, however, some standing in the early Amer- ican arithmetics, and is found as a separate chapter as late as the mid- dle of the nineteenth century. The title page gives some idea of the work of the gauger before the days of standardization of casks. The American rule for gauging a cask was substantially as follows : Add to the head diameter 0.7, 0.65, 0.6, or 0.55 of the difference between the head and bung diameters (according to the degree of curvature of the staves), and multiply the square of this sum by the length ; divide by 359 for ale or beer gallons, and by 294 for wine. Thus a cask with bung diameter 36 in., head diameter 30 in., and length 48 in., contains 153.65 ale gallons. 222 RARA ARITHMETICA This particular work is interesting because, although it was printed in 1543, some of the numerals are quite like those of a hundred years earlier. The mediaeval 4, 5, and 7 are exclusively used in the engraved wclcbe* inn^elt/wte man fcurcb ten <D,it a&Mtcrt 4uff e^nes ^efcen (onto UEidvctw JWpen jftbe revten/vn feamft ^eth'cbe tmbef *m0 vag wen w gebeflert vnfegemcrt* FIG. 107. TITLE PAGE OF FREY figures, although the types used in the body of the book give the later forms. These mediaeval figures may be seen in several illustrations in the second part of this bibliography. PRINTED BOOKS 223 ANONYMOUS. Ed. pr. 1543. Paris, 1543. The author of the Theologoumena is unknown. He lived after Anatolius (Bishop of Laodicea, 270 A.D.) and probably after lamblichus (fourth century). Title. See Fig. 108. Description. 4, 15.2 X 23 cm., the text being 9.1 x 18.1 cm. 7 pp. unnumb. +61 numb. (5-65)4-2 blank = 70 pp., 30 11. Paris, 1543. Greek text, except the dedicatory epistle, which, being dated 'Lutitiae Parifiorum 1543. 6. Calendas lulias,' shows this to be the first edition. Editions. This edition was unknown to De Morgan. The best edition is that of Fr. Ast, Leipzig, 1817. A work of little importance, on the Greek theory of numbers. Gow (p. 88) describes it as a ' curious farrago.' Cantor (Geschichte der Mathe- matik, Kap. 22) says that the author may have drawn on lamblichus. The most valuable feature of the work is the light which it throws on an earlier work by Speusippus, nephew of Plato. Other works of 1543. Archimedes, p. 228, 1544; Gemma, p. 200, 1540; Glareanus, p. 192, 1539; Recorde, p. 214, c. 1542; Regius, p. 182, 1536 ; Sfortunati, p. 174, 1534; Tonstall, p. 134, 1522 ; Nicolaus Medlerus, * Rudimenta Arithmetics practice, ' s. 1., 8 (with subsequent editions: Wittenberg, 1550; ib., 1558; Leipzig, 1556; Weissenfels, 1564, 8). MICHAEL STIFEL. Ed. pr. 1544. Nurnberg, 1544. STIEFEL, STYFEL. Born at Esslingen, April 19, 1487 ; died at Jena, April 19, 1 567. He was a priest, a reformer, and a fanatic, but was one of the most skillful arithmeticians of his time. Title. See Fig. 109. Colophon. 'Excudebatur Norimbergae // apud loh. Petreium.' (F. 325, r.) Description. 4, 15.5 x 20.2 cm., the text being 10.2 X 15.1 cm. 326 ff. (i blank, 6 unnumb.), 33 11. Nurnberg, 1544. Editions. Stifel wrote five works on mathematics, all but one appearing after his period of religious fanaticism. These works, with their various editions, are as follows : i. ' Ein Rechen Biichlein vom End Christ. Apocalysis in Apocalysim,' Wittenberg, 1532. A little-known work on the theory and mysticism of numbers. OAO- TorMENA TH2 A*P I M H T I K H 2. Habes hie o ftudio- SE LECTOR, NOVVM OPVS- culum antehac nufcjuam cxcufum , in quo ita Numerorum ratio explicatur,vt non fit obfcu- rum intclligere hanc arithmetica ad interiore ilia de philofophia diiputationem,quam Theologian! veteres vocabant, conferre piurimum. PARIS IIS. ApudChriftianumwechelum fubfcuto Bad- lienfi,in vico Iacobceo:& fub Pegafo^'n vi- coBellouacenfi. M. D. XLIIL FIG. 1 08. TITLE PAGE OF THE Theologoumena PRINTED BOOKS 225 ARITHMETI CA INTEGRA. Authorc Michaclc Stifelio* Norimbergae apud loham Petreitim, Anno Chrifti JMU D. x LIIIU Cum gratia & priuilcgio Carfareo atcp Regio ad Scxennium. FIG. 109. TITLE PAGE OF STIFEL'S SECOND ARITHMETIC 226 KARA ARITHMETICA 2. 'Arithmetica Integra/ Niirnberg, 1544, 4 (here described); ib., 1545, 4; 1546; Nurnberg, 1548; ib., 1586, 4. 3. 'Deutsche Arithmetica ' (p. 231), Nurnberg, 1545, 4. 4. ' Rechenbuch von der Welfchen vnd Deutfchen Practick/ auff allerley vorteyl vnd behendigkeit/ mit erklerung viler Exem- peln/ . . .', Nurnberg, 1546. 5. <DieCoss',K6nigsberg, 1553-54, 4(p. 259); 1571; 1615. The dedication and Melanchthon's preface, both of which have been removed from this copy, are dated 'Vuitebergse 1543.' This copy has evidently been owned by some one unsympathetic with Melanchthon, because not only has the preface been removed but the reformer's name has been crossed out of the title page (Fig. 109). The work is one of the most scholarly arithmetics and algebras that came out in Germany in the sixteenth century. It is divided into four books, the first being an arithmetic and giving not only the theory of the subject, but a con- siderable amount of practical work. The second book is on irrational numbers, a chapter which we now insert in algebra, and the third is on algebra itself, the name of which subject is said to be 'a Gebro Aftronomo, auctore eius,' a common opinion at that time. This work did for Germany what Cardan's and Tartaglia's did for Italy. It was a storehouse from which subsequent writers drew, and, although not a practical mercantile book, it materially influenced even the elementary textbook makers. Stifel himself recognized the demand for such a work, for he says : * Quanqj autem plurimi de Arithmetica libelli extent, & quotidie plures noui gignunter, ego tamen adhuc nullum uidi qui Integra artem traderet.' Stifel makes much use of the plus and minus signs. ARCHIMEDES. Ed. pr. 1544. Basel, 1544. Born at Syracuse, c. 287 B.C. ; died there in 212 B.C. The greatest of Greek mathematical physicists. Title. See Fig. 1 10. Colophon. 'Basileae, per loannem // Hervagivm, anno ab orbe re-//dempto,M.D.XLIIII. menfeMartio.' (P. 69 of the last part.) Description. Fol., 21.8 X 31.7 cm., the text being 13.2 X 23.5 cm. 455 pp. (7 blank and 12 unnumb.), 51 11. The work is made up of four parts, all of the same date, separately paged, and bound together. The '^Pa/u/uT?;?, De harenae numero,' is the sixth work in the collection. Basel, 1 544. PRINTED BOOKS 227 APXIMHAOYS TOY 2YPAKOYSIOV, TA MEXP1 ARCHIMEDIS SYRACVSANI PHI LOS OP HI ^AQ GEOMETR-t-^E E JT, cellentiffimi Opera , qusequidem extant,omnia 3 multis iam feculisdefi' derata,atqp a quam pauciflimishadenus uiTa,nunc9 primum & Grace 8^ Latine in lu* cem cdita. Quorum Catalogum uerfa pagina reperies. a (Juofifunt IN EOSDEM ARCHIMEDIS^ LI* bros Commentaria,item Grace &Latine, nunquam antea excufa. Cum C&J. Mate fl. gratia <ty priuiltgio ad quinquennium. j B ^f S J L loftnnes Htruagiiis cxciidificit. An. M D XL i MI. FIG. no. TITLE PAGE OF THE 1544 ARCHIMEDES 228 RARA ARITHMETICA Editions. This seems to be the first edition of the works of Archimedes to contain the ' De arenae numero.' This chapter also appears in the 1558 edition (below), and separately in Paris in 1557, 8 (the Hamellius edition, below). It was not in the Tartaglia edition of 1543. It appeared in the elaborate editions of Barrow (1675), Torelli (1792), Peyrard (1807-8), and Hei- berg (1880). Riccardi says that it * f u illustrato dal Clavio ne' suoi commenti alia sfera del Sacrobosco.' There are several works of Archimedes extant. The ' De arenae numero,' included in this edition, is not an arithmetic, but it treats of the numeration of large numbers. It is addressed to Gelon, King of Syracuse, and proposes to show, 'by geometric proofs which you can follow, that the numbers which have been named by us and are included in my letter to Zeuxippus are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe.' Archimedes then proceeds to develop a system of numera- tion by octads. In this work he incidentally refers to a fact which would now be expressed by the symbols x m x n = x m + n . ARCHIMEDES. Ed. pr. 1544. Paris, 1557. See p. 226. Title. ( Paschasii // Hamellii Regii // Mathematici // Commen- tarius // in // Archimedis // Syracufani praeclari Mathematici libru // de numero arenae, multis locis per // eundem Hamellium //emendatum.//Lvtetiae// Apud Gulielmum Cauellat, fub pin- gui Gallina,//ex aduerfo collegij Cameracenfis.//i557/ (F. i, r.) Description. 8, n x 17.1 cm., the text being 7.1 x 12.6 cm. 48 ff. numb., 23-30 11. Paris, 1557. Editions. See above. This commentary of Hamellius is quite as satisfactory as any of the earlier ones. ARCHIMEDES. Ed. pr. 1544. Venice, 1558. See p. 226. Title. ' Archimedis // opera non nvlla // a Federico Comman- dino // Vrbinate // nvper in Latinvm conversa, // et commenta- riis //illvstrata.// Quorum nomina in fequenti pagina leguntur. PRINTED BOOKS 229 //Cvm privilegio in annos X.// Venetiis,//apud Paulum Manu- tium, Aldi F.//M D LVIII.' (F. 2, r.) Description. Fol., 21.1 X 30 cm., the text being 13.2 x 22.8 cm. 128 ff. (2 blank, 8 unnumb.), 39 11. Venice, 1558. With this is bound the * Commentarii // in opera non Nvlla // Archi- medis.// Venetiis,// apud Paulum Manutium, Aldi F.// M D LVIII.' (F. 60, r.) It contains the book entitled ' Archimedis // Liber de Arense //nvmero.' See p. 228. JUAN SARAVIA, de la calle Beronese. Ed. pr. 1544. Medina, 1544. A Spanish arithmetician, of Medina, of the middle of the sixteenth century. Title. See Fig. 1 1 1 . Colophon. ' C.Fue impreffa la prefente: // obrallamada Inftrucio de mercaderes enla muy // noble villa de medina d'l campo por Pedro de // Caftro Ipreffor. Acofta d'Antono de vruena // mer- cader d'libros. Acabofe atreynta dias // del mes de Julio. Ano de mil \ quinie-//tos i quarenta i quatro anos.' (F. cvij, i.) Description. 8, 14x20.1 cm., the text being 12.1 x 16.9 cm. 2 ff. unnumb. + 105 numb. (Roman numerals) = 107 ff., 30 11. Medina, 1544. Editions. Medina, 1544, 4 (here described); ib., 1547, 4; Venice, 1561, 8 (p. 231.) This rare work was evidently written about 1542 (see f. xcviij, r.). It is not a textbook on arithmetic, but it relates to commercial problems, the topics being curiously interspersed with biblical illustrations to show the justice of the customs involved. Other works 0/1544. Apianus, p. 155, 1527; Baeda, p. 131, 1521; Finaeus, p. 163, 1530-32 ; Gemma, p. 200, 1540; Grammateus, p. 123, 1518; Kobel, p. 102, 1514; Noviomagus, p. 195, 1539 ; Peurbach, p. 53, 1492 ; Riese, p. 139, 1522 ; Sfortunati, p. 177, 1534; Tonstall, p. 136, 1522 ; Vander Hoecke, p. 183, 1537 ; H. Bock, Ein new Rechenbuch- lein,' Niirnberg, 8; Johannes Bogardus, a work on finger reckoning based on Aventinus (p. 136), Paris; Leonard Hegelin, 'Em kiinstlich Rechenbuchlin auff Zyffer vnnd andern hiipschen Regeln,' Ulm(?); ^^nla qual ft enfen^como bcuen fo> ctar^yOcqnemiiiicr*ifcbant)eeui- tar lasvfuraa&e todoofostractos bcventss^ com do otrotractadobccambios* ncl qu^l fe trscta ^clos cambfos licitos yrep:ouadcs ' FIG. in. TITLE PAGE OF SARAVIA PRINTED BOOKS 231 Antonio Martin, 'Tractado de Arithmetica y Geometria,' Alcala, 4; H. Vuelpius, ' De minutiis phyficis et practicis aftronomicae arithmeticae regulis,' Cologne, 4, and ' Libellus de communibus et ufitatis arithme- ticae practicae regulis,' ib., 4. JUAN SARAVIA. Ed. pr. 1544. Venice, 1561. See p. 229. Title. < Institvtione//de' Mercanti//che tratta del comparare // et vendere,// et della vsvra chepvo // occorrere nella Mercan- tia//insieme con vn trattato // de' Cambi.// Et in somma fi ra- giona di // tutto quello che al Mercante Christiano // fi conuiene. // Composta per il Dottor Sarava,// & nuoamente tradotta di lin- gua spagnuola // dal S. Alfonso D' Vlloa.// Cvm Privilegio.// In Venetia//Apreffo Bolognino Zaltieri.// M D LXI.' (F. i, r.) Colophon. 'In Venetia apreffo Bolognino Zaltieri // 1561.' (F. 135, v.) Description. 8, IO.2X 15.1 cm., the text being 7.6 X 12.2 cm. 135 ff. (3 unnumb.), 29 11. Venice, 1561. Editions. See p. 229. This is an Italian translation of the Spanish edition of 1544 (p. 229). The book was evidently written in 1542 as already stated, for the author speaks (f. 109) * del capo quefto anno M D XLII.' MICHAEL STIFEL. Ed. pr. 1545. Niirnberg, 1545. See p. 223. Title. See Fig. 112. Colophon. * Zu Nurnberg Truckts Johan Petreius.// 1545.' (F. 96, r.) Description. 4, 15.5 X 20.7 cm., the text being 10.8 x 15.5 cm. 4 ff. unnumb. + 92 numb. = 96 ff., 32-34 11. Nurnberg, 1545- Editions. See p. 226. The ' Deutsche Arithmetica ' is divided into three parts. In the ' Haufsrechnung ' there are 1 2 chapters, all relating to the arithmetic of the common people, there being ' no household so narrow and poor that common arithmetic is not both serviceable and necessary to its welfare.' ('Es ift kein Haufzhaltung/ niendart fo gering vnnd fchlecht/ 232 RARA ARITHMETICA c ficmb^crl troucti/vcrmertt nD verMenb/fcfewer tftgcmefen/So trt fie ^te mt f amUcfcen tvoif en vft ^jrcmpcth erwpfVf. * ae anDer fo^tertn ^elert win toon fcer -Oaufjrec^nun^ t>nb ^I'rcfjenrednu feirten l>m i--f 4ft Cum Pnuflcgio ad Qiifnqueniifam, FIG. 112. TITLE PAGE OF STIFEL'S Deutfche Arithmetica PRINTED BOOKS 233 das jr die Haufzrechnung oder gemeine rechnung/ nicht niitzlich vnnd dieftlich sey.' ) Stifel first treats * Vom Algorithmo der Rechenpfenning,' recommending the operations as being ' wunderleichtlich durch die Rechenpfenning gelernet vnd gelehret.' The coss (algebra) relates largely to arithmetic, although touching upon the most common alge- braic operations (Fig. 113). In it Stifel gives the rule for dividing one fraction by another by using the inverted divisor as a multiplier. The third part relates to the ecclesiastical calendar ('Yonder Kirchenrech- nung/die man nennet Computum Ecclefiafticum ' ) , a subject in which he acknowledges his indebtedness to ' Johannes de Sacro bufto.' With his usual commendable but effusive piety Stifel closes with praise not only to the Prince but to * vnferm Vater im Hymel vnd feinem eynigen naturlichen Son/ vnferm Herren lefu Chrifto.' JOHANN SCHEUBEL. Ed. pr. 1545. Leipzig, 1545. SCHEYBL, SCHEUBELIUS. Born at Kirchheim, Wiirttemberg, August 18, 1494; died February 20, 1570. Professor of Mathematics in the University of Tubingen, to which institution he bequeathed most of his manuscripts. He wrote on arithmetic and algebra, and edited part of Euclid. Title. See Fig. 114. Colophon. ' Lipfiae ex Officina Michaelis//Blum, a reftituta falute.// Anno M. D. XLV.// Idib: Maij.' (F. 255, r.) Description. 8, 10 X 15.9 cm., the text being 6.9 x 12 cm. 255 ff. unnumb. + i blank = 256 ff., 20-26 11. Leipzig, 1545. Editions. It is sometimes stated that this work appeared at Strasburg in 1540, but I cannot verify the statement. I know of no edition other than this of 1545. Scheubel also wrote a 4 Compendium arithmeticae ' (p. 246, 1549). Murhard mentions an ' Arithmetica sive de Arte supputandi Liber,' Lipsiae, 1545, 8, but he may refer to the ' De Numeris.' Scheubel also pub- lished an algebra (Paris, 1551), and the seventh, eighth, and ninth books of Euclid (1558). This work is the production of a scholar rather than a man con- versant with the demands of business. While Scheubel tried to write a mercantile arithmetic, the result was far removed from the needs of the common people. It carries the work in subjects like the roots so far that the ordinary Rechenmeister could not have used it. Moreover, it is written in Latin and is much more extended than the work of Gemma Frisius, so that it appealed neither to the business school nor 234 KARA ARITHMETICA *mt> . VII. ^ tct> t>on icpdjen reDen ftxrDe/foltu mtclj t>c r jfcjjn t>on Wfen ieicfem -f- twD nung Dcr tafcn.533a icfe nu rcDe von gWcfem tmfcc/ foltu w verjle^n toon -f- t>n& -f- / oDer toon twb attc^/wa fcft von vngW^en Wdfccnrt&e / fo tocrjt^e wvon @o Ijafcw nit t)ift twep Jctc^en -f-t^ / ein nv ^Dcnn er too^m^efagtifttoonDiTcnnamen fum: fum:X :?. Dae B^ dzct atttt |ic bcr / a(e tmtar cm cmigcn 2Hg o* rit^mum. $fc erp ^egct t>on tern 2(bt>tre VIII. setd)eit/ma^ert eben ba* fcl* im 21&6iren i?ii ol>n rtllcm fb 6w im fubr^I)ircn bic jal / bit 6n foltefl fttbtra^trc/iiid^t tonfi fttbtm^irc. 8 urn: -t- 11 urn: -h 7* it* $ (Sum: rS. *o (Sum: -f- IS. H <Sum; 24. FIG. 113. FROM STIFEL'S Deutfche Arithmetica nnpto PRINTED BOOKS 235 to the ordinary classical school. A great deal of attention is given to exchange, the rule of three, and the extracting of roots of high order. Attention is also given to problems which would now form part of algebra, and there is a little treatment of geometry from the standpoint of mensuration. The nature of the work can be somewhat understood DENVME. RIS ET DIVERSTS RATIONIBVS feu regulfs computationum opuiculum, a loanne Scheubelio compofi'tum, Non folum ad ufurn quondam uul garem,fed etiam cognftfonem & fcfentfam exquni tiorcm arithmetics acco* modatum, s* syr v M, D. XLV, FIG. 114. TITLE PAGE OF SCHEUBEL from the titles of the five ' tractati ' : i. ' De numeris integris ' ; 2. De proportionibus, proportionalitatibus & alijs ' ; 3. 'De minutijs uulgari- bus ' ; 4. De minutijs phyficalibus ' ; 5. ' Aliquot regulae.' While Scheubel is not much appreciated to-day, he was really ahead of his time. He tried to banish the expression ' rule of three ' and to substitute 'rule of proportion.' His explanation of square root is in 236 KARA ARITHMETICA some respects the best of the century, and he dismisses with mere men- tion the ' duplatio ' and ' mediatio ' of his contemporaries. He extracts various roots as far as the 24th, finding the binomial coefficients by means of the Pascal triangle a century before Pascal made the device famous. SACROBOSCO. Ed. pr. 1545. Wittenberg, 1550. See p. 31. Title. * loannis // de Sacrobvsto // Libellus de Sphaera. // Accessit eivsdem // avctoris Compvtvs // Ecclefiafticus, Et alia quaedam // in ftudioforum gra-//tiam edita.// Cum Praefatione Philippi // Melanthonis.' On f. 67, r., begins ' Libellvs // loannis de Sacro//bvsto, de Anni Ratione,//sev vt vocatvr vvl-//go Com- pvtvs Ec-//clesiasti-//cvs.// Cvm Praefatione // Philippi Melan- //thonis.//Anno M. D. XLV.' Colophon. ' Impreffum Vuitebergae apud // lohannem Crato- nem.//Anno//M. D. L.' (F. 134, r.) Description. 8, 10.1 x 15.5 cm., the text being 6.5 x 10.8 cm. 1 34 ff. unnumb. + 2 blank (2 plans) = 1 36 ff ., 26 11. Witten- berg, 1550. Editions. The preface by Melanchthon is dated 'Menfe Auguf- to.// Anno // M. D. XXXVIII,' so that his edition of the book could not have appeared earlier than that year. The fact that this volume was printed in 1550, as shown by the colophon, makes it probable that the date 1 545 on the title page is that of the first edi- tion. There are several editions of the * Sphaera' in Mr. Plimpton's library, but this is the only one containing the Computus. This is less properly included in a list of arithmetics than many of the other computi. Since several others have been included, this, which is one of the most celebrated, is given place ; but the arithmetical work is practically nil. EUCLID. Ed. pr. (arith. books) 1545. Rome, 1545. See p. ii. Title. See Fig. 115. Colophon. ' Stampata in Roma per Antonio Blado Afolano.// M D XLV.' (P. 12.) PRINTED BOOKS 237 Description. 8, 9.8 X 15.1 cm., the text being 6.7 X 10.9 cm. 112 pp. (3 blank, 6 unnumb.), 24-26 11. Rome, 1545. Editions. There were several editions of one or more of those books of Euclid that relate to some part of the theory of arith- I Q.V1NDICI LIBRI DEGLI ELEMEN TI DI EVCLIDB, DI GRE CO TKADOTTI IN LINGVA THO. SCANA* IN ROMA. M DXXXXV. S. N. S, PattloTerfa &dfllt Sertnisfima republic* Venetian* (er cinque tnni. FIG. 115. TITLE PAGE OF THE 1545 EUCLID metic. Of these separate books, this is the rare first Italian edition. Among others may be mentioned those of Wittenberg, 1546; 1549; Paris, 1551,4 (p- 238); ib., 1554, 4 (P- 2 38); I 555 ( b Y Scheubel) ; Paris, 1557, 8 (p. 240) ; Wittenberg, 1564, 238 KARA ARITHMETICA 8 (p. 240). There were very many editions of Euclid's work published before 1600, practically all including Book V, on pro- portion, and some including the other arithmetical books. Euclid's treatment of arithmetic was purely theoretical, no work on computation being included. This copy has the following note by De Morgan, who once owned it : " This book was also printed in Greek same year, place, size, printer, and dedication. There is a Greek copy in the British Museum. A. De Morgan, Feb. 29, 1852." Other works 0/1545. Baeda, p. 131, 1521 ; Feliciano, p. 149, 1526 ; Psellus, p. 168, 1532; Stifel, p. 226, 1544; Sfortunati, p. 177, 1534; Willichius, p. 197, 1540; Antonius de Barres, ' Arithmeticae practicae libri IV,' Louvain, 4 ; Johann Obers, ' Newgestelt Rechenpuchlin,' Augsburg ; Pedro Espinosa, ' Tractatus proportionum,' Salamanca, fol. EUCLID. Ed. pr. (arith. books) 1545. Paris, 1551. See p. n. Title. See Fig. 1 16. (Note the signature of Giuliano de Medici.) Description. 4, 16.3 x 23.4 cm., the text being 9.5 X 17.3 cm. 162 ff. (2 blank, 20 unnumb.), 29 11. Paris, 1551. Editions. See p. 237. This is not the same as the 1545 edition already described, but, like it, this relates to the numerical side of mathematics. EUCLID. Ed. pr. (arith. books) 1545. Paris, 1554. See p. ii. Title. ' Evclidis // Elementa qvaedam // Arithmetica.// Lvte- tiae,//Apud Vafcofanum, uia lacobaea, ad infigne Fontis.// M. D. LIIII.//Cvm privilegio regis.' (F. i, r.) Description. 4, 15. 4 X 19.4 cm., the text being 11.2 x 17.8 cm. 1 8 ff. numb., 29 11. Paris, 1554. Editions. See p. 237. This work is made up of certain extracts from the various books of Euclid relating to Arithmetic. It is in Greek with a Latin translation following each definition or theorem. It consists of such standard old definitions as * Numerus autem, ex unitatibus composita multitude,' and such theorems as ' Omnis primus numerus, ad omnem numerum quern non metitur, primus est.' There are no discussions, illustrations, or proofs of the propositions. PRINTED BOOKS 239 Euclidiselcmentoruni LIBER DECIMVS, PETRO Montaurco interprete. Ad loannem Bellaium Cardinalem. L VTETIAE, d Vafcofanum^ma lacobda ad inpgne fontit. M. D. LL CVM PRIVILEGIO FIG. 1 1 6. TITLE PAGE OF THE 1551 EUCLID 240 KARA ARITHMETICA EUCLID. Ed. pr. (arith. books) 1545. Paris, 1557. See p. ii. Title. ' Evclidis // Elementorvm // Libri XV. Grae-//ce & Latine,// Quibus, cum ad omnem Mathematics fcientiae // par- tern, turn ad quamlibet Geometriae tra-//ctationem, facilis com- paratur aditus.// ^Tri^pa^fia Trakaiov.// 2^/Aara TreWe riXara)- 1/05, a Hvdaydpas o-o-//<o? evpe.// HvOaydpas cro<o? evpe, TlXdrcov B* aptSrjX eSi-f/Sagev / / Eu/cXetS??? eirl rolai /eXe'o? Tre/ot/caXXe? Tvj;v. Lvtetiae,// Apud Gulielmum Cauellat, in pingui Gallina, //ex aduerfo collegij Cameracenfis.// 1557.' (F. i, r.) Description. 8, 10.5 x 16.9 cm., the text being 7 x 12.8 cm. 1 6 ff. unnumb. + 130 numb. = 146 ff., 25 11. Paris, 1557. See p. 238. EUCLID. Ed. pr. (arith. books) 1545. Wittenberg, 1564. See p. ii. Title. ' Arithmetices // Evclideae // Liber Primvs.// Alias in ordine reliquorum // Septimvs : Qui citra // praecedentium Sex librorum // Geometricorum opem erudite //perfequitur, cum reli- quis duobus // fequentibus, uera principia ac // folidiora funda- menta Logi-//itices,id eft,ut uocant,//ArithmeticesPra-//ctic3e. // Per // loan. Sthen. Luneb.// In scholarvm vsvm Kara TO on tractatus epcorrj^arLKw^, difquifitione nimi-//rum Dialectica quae Dialogorum //est propria.// 1564.' (F. i, r.) Colophon. 'VVittenbergae.// Anno// 1564.' (F. 106, v.) Description. 8, 9.2 x 14.8 cm., the text being 6.3 X 11.5 cm. 107 ff. unnumb., 22-24 U- Wittenberg, 1564. Editions. See p. 237. GASPARD DE TEXEDA. Ed. pr. 1546. Valladolid, 1546. A Valladolid arithmetician of the first half of the sixteenth century. Title. See Fig. 117. Colophon. ' C.Fue impreffala prefente // obra d'Arithmetica En la muy noble // y felice villa de Valladolid (Pincia // otro tiempollamada) En la offici-//na de Francifco Fernandez // de PRINTED BOOKS 241 FIG. 117. TITLE PAGE OF TEXEDA 242 KARA ARITHMETICA cordoua/ junto alas // efcuelas ma//yores // Acabofe a quatro dias del mes // de Henero defte ano del // fenor de mill i quini- //entos i quaren//ta 2 feys // Anos.' (F. Ixiiij, r.) Description. 8, 13.5 X 19.5 cm., the text being 10.3 X 16.5 cm. 64 ff., numb, in Roman, 32-34 11. Title page engraved on wood. Valladolid, 1546. Editions. There was no other edition. De Morgan (p. 103) gives the date as 1545, which is that of the privilege, the colo- phon of his copy having been torn out. This rare Spanish arithmetic gives the fundamental processes with integers, fractions, and denominate numbers, introduces some practical mensuration under the title ' De Geometria,' and gives a rather extended treatment of the business rules. It is interesting because of the treat- ment of Spanish and Arab (algoristic) notation, or, as the author says, 'de numerar en caftellano y en guarifmo.' For example, his two methods of writing 160,462,009,621, are : c. Ix. 11462 qs. . ix U62i 1 60 11462 qs 009 U 621. (F. iiij, r.) PIETRO CATANEO. Ed. pr. 1546. Venice, 1546. A sixteenth-century arithmetician, of Siena. Title. See Fig. 118. Colophon. ' Stampato in Venetia per Niccolo Bafcarini.// M D XLVI.' (F. 64, r.) Description. 4, 15.3 X 20.8 cm., the text being 12 x 16 cm. i f. unnumb. + 63 numb. = 64 ff., 32-35 11. Venice, 1546. Editions. A very rare edition, and probably the first in spite of the words ' nvovamente stampate,' for the dedication is dated M. D. XLVI. (F. i, v.) There were two later editions, Venice, 1559, 4 (P. 244), and Venice, 1567, 4 (p. 244). The work is fairly practical, and in many respects is in advance of its time. Unlike most Venetian books it uses the Florentine name biri- cvocolo ' for the common form of multiplication, and gives the 'a danda ' division before the galley form, recommending it as ' molto neceffario.' The applications, while not numerous, are practical, and throw some light upon the business customs of Siena and Venice. Cataneo was not, however, an original writer. His arithmetic is composed quite largely PRINTED BOOKS 243 of didactic statements to be found in the works of his predecessors, and the fact that he gives four methods of multiplication shows that he could not escape the influence of writers like Paciuolo. LE PRATICHE DELLE DVB PRIME MATHEMATICHE Dl PI ETRODE CATANI DA SIENA LIBRO D'ALBACO E GEOMETRiA STAMPATE, INVENET1A W P XLVi FIG. 1 1 8. TITLE PAGE OF THE 1546 CATANEO Other works of 1546. Anonymous, p. 213, 1542 ; Boethius, p. 27, 1488; Euclid, p. 237, 1545; Glareanus, p. 192, 1539; Helmreich, p. 303, 1561; Manzoni, p. 257, 1553; Misrachi, p. 180, 1534; 244 KARA ARITHMETICA Rudolff, p. 152, 1526; Sole, p. 143, 1526 ; Stifel, p. 226, 1544 ; Anony- mous, * An introduction for to lerne to recken with the pen, or with the counters accordyng to the trewe cast of Algorisme, in hole numbers, or in broken, newly corrected. And certayne notable and goodly rules of false positions thereunto added, not before sene in our Englysche Tonge,' London, 8, with another edition at London in 1574, sm. 8; Anony- mous, * Ein new kurtz Rechenbtichlein auff der Linien und Federn,' Frankfort, 8, possibly by Giilfferich (p. 269, 1555); Alfonzo Lopez de Corella, ' Secretos de las cuatro mathematicas ciencias,' Valladolid. Works of 1547. Gemma, p. 200, 1540; Saravia, p. 229, 1544; Tagliente, p. 114, 1515. Works 0/1548. Gemma, p. 200, 1540; Ghaligai, p. 132, 1521; Riese, p. 139, 1522; Stifel, p. 226, 1544; Tagliente, p. 114, 1515; Uberti, see Tagliente, p. 114, 1515 ; Wolphius, p. 154, 1527; Anony- mous, * Specie principali, et primi principii del' Arithmetica di C. de C. P.,' Bologna. PIETRO CATANEO. Ed. pr. 1546. Venice, 1559. See p. 242. Title. * Le // pratiche // delle dve prime // Matematiche // di Pietro Cataneo // con la aggionta,// libro d'Albaco e Geome- tria con il //pratico e uero modo di mifurar la Terra.// Non piv mostra da altri.// (Woodcut of griffin with motto : ' Virtute dvce //comite Fortvna.') In Venetia, apreffo Giouanni Griffio, MD LIX.' (F. i, r.) Colophon. ' In Venetia, apreffo Giouan Griffio, ad inftantia // di M. Pietro Cataneo, M D LIX.' (F. 83, v.) Description. 4, 15.3 X 20.3 cm., the text being n.4X 15.1 cm. i f. unnumb. + 82 numb. + i blank = 84 ff., 32-35 11. Venice, 1559. See p. 242. PIETRO CATANEO. Ed. pr. 1546. Venice, 1567. See p. 242. Title. ' Le // pratiche // delle dve prime // Matematiche // di Pietro Cataneo Senese,// ricorrette, & meglio ordinate, con alcu- ne agV/giontioni de lo steffo Autore.// Diuife in libri quattro.// PRINTED BOOKS 245 (Woodcut of griffin, with motto : ' Virtvte dvce,// comite For- tvna.') In Venetia, apreffo Giouanni Griffio,//M D LXVII.' (F. i, r.) Colophon. ' In Venetia, apreffo Giouan Griffo, ad inftantia di //M. Pietro Cataneo, M D LXVII.' (F. 88, r.) Description. 4, 15.5 x 20.7 cm., the text being 11.7 x 15.7 cm. 88 ff. numb., 32-35 11. Venice, 1567. Editions. See p. 242. This differs from the first (1546) edi- tion only in the fact that the part relating to geometry contains considerable additional matter. See p. 242. JACQUES PELETIER. Ed. pr. 1549. S. L, 1607. Bom at Mans in 1517 ; died at Paris in July, 1582. He became principal of a college, traveled extensively, and contributed both to literature and to elementary mathematics. Title. ' L'Arithmeti-//qve de lacqves // Peletier dv // Mans, //Departie en quatre liures.//Troifieme edition, reucue et aug- mentee.//Par lean de Tovrnes.//M. DC. VII.' (P. 3.) Description. 8, 10.3 x 16 cm., the text being 7.7 X 13 cm. 297 pp. (6 blank, I unnumb.), 26 11. S. 1., 1607. Editions. Poitiers, 1549, 4 (Graesse, Sup.); ib., 1552, 8 (Graesse says 1551); Lyons, 1554, 8. Graesse mentions all of these, but the above title shows this edition of 1607 to be the third, possibly the third revision. I have also seen mentioned an edition of 1567, 4, a Lyons edition of 1570, and a Latin edi- tion at Paris in 1563 and 1578. The work is quite practical, although it contains a considerable amount of mediaeval matter. The first book treats of the fundamental operations with integers, the second of fractions, the third of roots and proportion, and the fourth of the applications of arithmetic. It contains a number of such traditional problems as the hare and hound. Peletier also wrote a chapter ' De fractionibus astronomicis compendium de cognoscendis per memoriam calendis,' that was published in his editions of Gemma Frisius. Graesse mentions an * Arithmeticae modus,' Paris, 1563, 8, probably the Paris edition referred to above. Peletier also wrote one of the first practical textbooks on algebra. 246 KARA ARITHMETICA JOHANN SCHEUBEL. Ed. pr. 1549. Basel, 1549. See p. 233. Title. The title page is missing in this copy, but the running headline is ' Compendium Arithmetics. ' (See the 1560 edition.) Colophon. ' Basileae, per lacobvm // Parcvm, expensis // loan- nis Oporini,// Anno 1549.' (F. 87, r.) Description. Sm. 8, 8.7 X 13.7 cm., the text being 6.3 X 1 1.8 cm. 87 ff. unnumb., 27 11. Basel, 1549. Editions. Basel, 1549, sm. 8 (here described); ib., 1560, 8 (below). That this is the first edition is seen in the ' Epiftola Dedicatoria,' which bears date ' Tubingae, idibus Martij //annni fefqui millefimi // quadragefimi noni.' (P. 7.) < While open to some of the criticism mentioned in connection with ScheubeFs De numeris ' ( p. 233, 1545), this book is more practical than his earlier one, and was enough in demand to warrant two editions. It is not, however, a commercial textbook. JOHANN SCHEUBEL. Ed. pr. 1549. Basel, 1560. See p. 233. Title. ' Compen-//divm Arithme-//ticae Artis, vt bre-//uif- sum ita longe utilifsimum eru//diendis tyronibus, non folum pro- //pter ordinem, quo paucis perftrin-//guntur omnia huius artis capita : fed // etiam caufa perfpicuitatis, quae plu-//nmum de- lectat & iuuat difcentes,// fummopere expetedum : per loan-// nem Scheubeliu adornatum//& confcriptum.//Iam denuo ab ipfo autore recognitum // & emandatum.// Continent autem utrunq3 hoc Compendia, //numerorum fcilicet & calculorum, feu//pro- iectilium (ut uocant) ra-//tiocincationem.// Bafilae, anno 1560.' (P. i.) Colophon. ' Basiliae // excudebat lacobus Parcus,// expenfis loannis Opo-//rini, anno M.D.LX.//menfe Martio.' (P. 205.) Description. 8, 9.5 X 15 cm., the text being 6.1 x 12 cm. 14 pp. unnumb. + 191 numb. (3-193) + I blank = 206 pp., 23- 24 11. Basel, 1560. See above. PRINTED BOOKS 247 JOHANN FISCHER (Piscator). Ed. pr. 1549. Stettin, s. a. (1565 ?). A German Rechenmeister of the second half of the sixteenth century. Title. ' Ein kurtz Rechenbuchlein fur die anfahende Schuler gemacht //Durch Johann Fifcher. Gedruckt zu Alten Stettin// in Johan Eichorns Druckerey.' (F. i, r.) Description. 8, 9.8 X 15.4 cm., the text being 6.7 x 1 1.2 cm. 1 6 ff. unnumb., Stettin, s. a. (1565 ?). Editions. This work appeared first in Latin under the title * Arithmeticae Compendium, pro Studiosis hujus artis tyronibus recognitum,' Leipzig, 1549, with subsequent Latin editions, ib., 1554, 1559, 1582, 1592, 1598, and Wittenberg, 1592, all 8. Of the German translation there were editions as follows : Stettin, s. a. (1565 ?), 8 (here described) ; Frankfort an der Oder, 1566 ; Leipzig, 1581,8; 1 592, 8. Fischer also published a work, said to be different from the ' Compendium,' entitled ' Ein kiinstlich Rechenbuchlein, ' Wittenberg, 1559, with four editions from 1559 to 1592. As the title suggests, this is merely a compendium, designed to serve as an introduction to practical arithmetic. It has no more merit than any brief primer. JUAN DE YCIAR, Vizcayno. Ed. pr. 1549. Saragossa, 1549. ICIAR. A Basque arithmetician, born at Durango in 1525. On f. 3, r., is a large portrait with the inscription ' loannes de Yciar aetatis sve anno xxv.' He lived in Saragossa, and was well known as a calligrapher. Title. See Fig. 119. Colophon. ( C.Fue impreffo el prefente libro en la muy noble //y leal ciudad //d'c^aragoga en cafe de Pedro Bernuz/a cofta //del auctor y de Miguel de gapila mercader d'libros,// Acabo fe a .xvj. de Febrero del ano de mil y // quinientos y quarenta y //nueue.' (F. 61, v.) Description. Fol., 19 x 28.9 cm., the text being 14.3 X 21.5 cm. 4 ff. unnumb. + 56 numb. + 3 blank = 63 ff., 27-38 11. Saragossa, 1549. /v,.,, &RY '*/& Wool 248 RARA ARITHMETICA Cxfarauguflx. M.D.XLIX. FIG. 119. TITLE PAGE OF YCIAR PRINTED BOOKS 249 Editions. Saragossa, 1549, fol. (here described) ; ib. 1555, 4; ib., 1564, 4. The book probably had several other editions, for Heredia (I, 154) says that this is ' une des plus anciennes Edi- tions de ce traite.' There are numerous interesting features in this book. Among these is Yciar's fanciful explanation of the origin of the Roman numerals, part of it traditional, as that V stands for five because it was the fifth Latin vowel, and part more recent, as that L was half of the old form for C. U is used instead of M, as with several Spanish writers, and cuento is used for million as was their general custom. (See p. 60, Ciruelo.) The treatment of the fundamental operations is followed by progressions, compound numbers, roots, mensuration, and such common applications as ' las companias fin tiempo ' and ' con tiempo,' and ' las reglas de teftamentos.' Other works of 1549. Anianus, p. 32, 1488 ; Anonymous, p. 213, 1542 ; Boethius, p. 27, 1488 ; Euclid, p. 237, 1545 ; Gemma, p. 202, 1540; Glareanus, p. 192, 1539 ; Kobel, p. 102, 1514 ; Recorde, p. 214, c. 1542 ; Hans Bock, * Ein new Rechenbiichlein auff der Linien und Federn,' Niirnberg (probably a second edition of the 1544 book, p. 229) ; Joannes Stigelius, ' Arithmetica,' Leipzig, 8, with a second edition, s. 1., 1554 (Victorinus Strigelius of 1563 ? see p. 311) ; Juan Vejar, 'Arith- metica practica,' Saragossa, 4. (Yciar's work ?) VALENTIN MENHER de Kempten. Ed. pr. 1550. Antwerp, 1565. MENNHER. A German-Dutch arithmetician of the sixteenth century. See also p. 281, 1556. Title. ' Practicqve // pour brievement // apprendre a Ciffrer, & tenir Liure // de Comptes, auec la Regie de // Cofs, & Geome- tric. //Par M. V. Menher Alleman.// (Woodcut of counting house.) A Anvers, 1'an M. D. LXV.//Auec priuilege du Roy pour 4 ans.' (F. i, r.) Bound with this in the third part is 'La Regie d' Algebra,' or 'Cofs.' Also the ' Practicqve // des Triangles //Spheriqves.//. . . Anvers . . . M. D. LXIIII.' Colophon. ' Imprime en Anuers par ^Egidius Dieft,//l'An de noftre Seigneur lefu Chrift.//M. D. LXV.// 19. lanuarij.' (F. 113, v.) 250 RARA ARITHMETICA Description. 8, 9.6 X 15.6 cm., the text being 7.3 X 12.6 cm. 113 ff., 26 11. The algebra contains 120 ff. unnumb. (part 3); the geometry, 102 ff. (part 4), besides the ' Practique des tri- angles spheriques.' Antwerp, 1565. Editions. Menher wrote three or four arithmetics, as follows : 1. ' Practique briesve pour cyfrer et tenir Livres de compte,' Antwerp, 1550,8; 1556 (probably the one mentioned on p. 281); Antwerp, 1565, 8 (here described). Unlike the De Morgan copy this does not have 1564 for 1565 in the colophon. Indeed I think De Morgan probably looked at the colophon of the geome- try instead of the arithmetic. 2. ' Arithmetique seconde,' Antwerp, 1556 (p. 281). A com- parison of this with the ' Practique ' shows it to be substantially the same work. In his epistle to the reader, Menher speaks of the 1565 edition of the 'Practique' as merely a revision of * noftre feconde Arithmetique de Tan M. D. LVI.' 3. 'Livre d'Arithmetiqve,' Antwerp, 8, 1573 (p. 347). I know of no other sixteenth-century edition of this work, although there was a Rotterdam edition in 1609, 8. The work is entirely differ- ent from the * Practique.' 4. ' Arithmetica Practice,' Autorff, 1560, 8. I know nothing of this work. The ' Practique ' and the ' Arithmetiqve seconde,' essentially the same work, are mercantile textbooks, possessed of the spirit of the 1 Livre d'Arithmetiqve ' (p. 347 ), but not as successfully written. Menher was one of the pioneers among the Dutch arithmeticians, and his successors, particularly in the period from 1600 to 1650, produced some very practical textbooks. ADAM RIESE. Ed. pr. 1550. Leipzig, 1550. See p. 138. Title. See Fig. 120. Colophon. ' Gedruckt zu Leipzig durch// Jacobum Berwalt.' (F. 196, r.) Description. 4, 15.5 x 17.8 cm., the text being 10.6 X 14.4 cm. 4 ff. unnumb. + 1 96 numb. = 200 ff., 29-31 11. Leipzig, 1550. PRINTED BOOKS 251 auff t>w i 7 9 o 3*?' X ms^>^ ^^^ Cum gratia & priuilcgfo C^farco* FIG. 120. TITLE PAGE OF THE 1550 RIESE 252 RARA ARITHMETICA Editions. See p. 140. This is the fourth of Riese's books (p. 139). Bound with it is Isaac Riese's arithmetic of 1580 (p. 365). The date is also given in the dedicatory epistle, ' im 1550 jhar.' The first forty-six folios contain the treatise ' auff den Linihen,' the counter reckoning. This is followed (ff. 47-105) by that 'auff der Feder,' the common algorism. The third part is the ' Practica,' and the fourth the * Vifieren ' or gauging. The book represents the cul- mination of Riese's work, and is the best exponent of the practical arithmetic of the middle of the century in Germany. Other works 0/1550. Agricola, p. 171, 1533; Agrippa, p. 167, 1531; Anianus, p. 32, 1488; Borghi, p. 22, 1484 ; Cassiodorus, p. 2 1 1 , 1540; Feliciano, p. 149, 1526; Gemma, p. 200, 1540; Glareanus, p. 192, 1539; Lonicerus, p. 253, 1551; Medlerus, p. 223, 1543; Regius, p. 181, 1536; Riese, p. 139, 1522; Tagliente, p. 114, 1515; Torrentini, p. 76, 1501. There was also an edition of Sfortunati, c. 1550 (p. 174, 1534). Two other works published c. 1550 should be men- tioned : Anonymous, ' Opera che insegna a tener conto de libro secondo lo cosueto di tutti li lochi della Italia al modo mercantile,' s. 1. a., with some mercantile arithmetic ; William Buckley, ' Arithmetica memorativa sive compendaria Arithmeticae tractatio,' 8, s. 1. a.; but later in Seton's Logic, London, 1572, 1574, 1577, 1584, 8. JOHANN SCHEUBEL. Ed. pr. 1551. Paris, 1551. See p. 223. Title. ' Algebrae//compendiosa//facilisqve descri-//ptio, qua depromuntur magna // Arithmetices miracula.// Authore loanne Scheubelio Mathematicarum //prof eff ore in academia Tubingenfi. // Parisiis, // Apud Gulielmum Cauellat, in Pingui Gallina,//ex aduerfo Collegii Cameracenfis.// 1551 .//C vm privilegio. ' ' (F. i , r.) Colophon. ' Excudebat Lutetiae Parifiorum, Benedictus Preuo- tius Typo-//graphus,invico Frementello, fub infigni ftellae aureae. //I55I.' ( F - numb - 52, v.) Description. 4, 13.2 x 18.5 cm., the text being 9.4 X 15.2 cm. 52 ff. numb., 32-37 11. Paris, 1551. Editions. There was no other edition. I have included this algebra because it contains some work in the extracting of roots by the galley method, and therefore shows the per- sistence of this mediaeval plan. PRINTED BOOKS 253 ROBERT RECORDE. Ed. pr. 1551. London, 1596. See p. 213. Title. 'The Castle //of // Knowledge.// To Knowledge is this Castle fet,// All Learnings friends wil it fupport,// So fhall their name great honour get,// And gaine great fame with good report. // Though fpitefull Fortune turn'd her wheele,// To ftay the Sphere of Vranie,//Yet doth the fphere refift that wheele,// And flee'th all Fortunes villanie // Though earth do honour Fortunes ball,// And beetles blinde her wheele aduance, //The heauens to Fortune are not thrall, //The fpheres furmount all Fortunes chance.// London // printed by Valentine Sims, afsigned // by Bonham Norton.// 1596.' (P. i.) Colophon. ' Imprinted at London by Valentine // Simmes. 1596.' (P. 236.) Description. 4, 14.4 x 19.2 cm., the text being 12.5 x 17.5 cm. 3 pp. unnumb. + i blank + 232 numb. = 236 pp., 41 11. London, 1596. Editions. London, 1551 ; ib., 1556, fol. ; ib., 1596, 4 (here described). Recorde's name does not appear on the title page, but he signs the letter of dedication to ' Princesse Marie,' ' Robert Record Phyficion.' The work is on astronomy, and is of interest in the history of arith- metic only in the operations involving sexagesimal fractions. Division is performed by the galley method, and there are no symbols for degrees, minutes, and seconds. The arithmetical part includes the rule of three. Like the author's other works (pp. 213, 286), this is in the catechism form. Other works 0/1551. Borghi, p. 16, 1484 ; Euclid, p. 238, 1545 ; Gemma, p. 200, 1540; Glareanus, p. 193, 1539 ; Noviomagus, p. 197, 1539; Peletier, p. 245, 1549; Recorde, p. 214, c. 1542; Tonstall, p. 134, 1522; Adam Lonicerus, Arithmeticae brevis Introductio,' Frankfort, 8, with subsequent editions, ib., 1568, 8 ; 1570, 12; 1581 ; 1585 ; 1600, 8 (Tropfke puts the first edition as 1550) ; Andres Garcia de Lovas, ' Tratado del compute,' Salamanca, 8 ; Innocenzo Ringhieri, ' Centi givochi liberali ... in dieci libri descritti,' Bologna, 4, with subsequent editions, Venice, 1553, 4 ; Bologna, 1580 ; Lyons (French translation), 1555, 4 (contains some number games). 254 KARA ARITHMETICA MARCO AUREL. Ed. pr. 1552. Valencia, 1552. Aurel was, as he states, a German. He lived, however, for several years in Valencia, and published a work there in 1541. Title. See Fig. 121. Description. 4, 14.5 x 19.3 cm., the text being 9.9 x i6.6cm. 4 ff. unnumb. + 140 numb. = 144 ff., 31-36 11. Valencia, 1552. Editions. There was no other edition. Aurel, in his letter to the reader, dwells upon the unfortunate state of mathematics in Spain, and says that he feels called upon to assist in making known a science so necessary to humanity. Of the twenty-four chapters in the book, the first six may be said to relate to arithmetic as we ordinarily consider it, the rest referring entirely to algebra. The arithmetical chapters present the subject in a fairly practical way, but are deficient in genuine problems. To subtract is called * Restar,' as at present in Spanish, the same root appearing occasionally in other lan- guages, and our word * rest ' (for remainder) being a relic of this name. Division is performed entirely by the galley method, but the figures are not canceled as is generally the case. ' Proporcion ' is used for ratio, and * proporcionalidad ' for proportion, as was generally the custom in the early arithmetics of all Latin countries, a custom derived from the Boethian books. The ' proporciones ' (ratios) are treated at consider- able length after the fashion set by the mediaeval writers. The applica- tions are almost entirely under the ' Regla de tres ' (rule of three). In the part devoted to algebra, surd numbers are first treated, the root symbols showing the German influence. The plus and minus signs are also used as extensively as in the works of writers like Stifel and Scheubel, and the symbols for the various powers of the unknown quantity are such as are found in the works of contemporary writers in other countries. LILIUS GREGORIUS GYRALDUS. Ed. pr. 1552. Venice, 1553. A philosopher of Ferrara, of the middle of the sixteenth century. Title. See Fig. 122. Description. 8, 9.3 x 15.5 cm., the text being 6.1 X 12.4 cm. 184 pp. (2 blank, 16 unnumb.), 29 11. Venice, 1553. Editions. Venice, 1552, 8; ib., 1553, 8 (here described). This treatise also appeared in his ' Opera,' Basel, 1580, sm. fol., and Leyden, 1696. 2*>LIBRO PR I- T>E Mgebratica , enel qualfe contiene el arte Afercanttuol, con otras muchas Qfyglas del arte menor^yla^gla del jflgtbra , Vulgarmente llama Ja .Arte mayor 9 o fygla de la co fa tjin la qualnofe fodra entenderel dedmo de Eu+ elides y ni otros muchos primom , affi en jfrithmetica ro wo en Geometria:compuefto 9 ordenado,y becho Impri* ww for Afarco Aurel, natural Meman : Intitulado, De* fyertador dcingenios. Vadirigido at muy magni- foofenormoffen ^Bernardo Cimon, dadano dela muy infigney co ronada Ciudad de J vVvfvwVVVt ^ Con Priuilcgio de fu Magcflad^ por ticmpo dc diez anos EN r En ofa dc loan de Me?, Handrof FIG. 121. TITLE PAGE OF AUREL 2 5 6 KARA ARITHMETICA LILII GREGORII GYRALDI FERRARIEN. SVA* R V M Q.V ARVNDAM ANNOc tationum DialogifmiXXX. ad Ants plifi. Cftrd* Sduiitum. Item Lturcntij frizzolij Solitncnfls Dialogifinw unicm de ipflm Lilij uitct cr opmbus. ApudGutltcrumScottum M D L I I I. j FIG. 122. TITLE PAGE OF THE 1553 GYRALDUS PRINTED BOOKS 257 This set of dialogues is of interest in the history of mathematics in that the second and third parts deal with notation and finger symbolism. ' Dialogismvs secvndvs de manus & digitorum nominibus de"q; nume- randi per eos antiquorum ratione ' (p. 10) is an almost unknown sketch of finger symbolism. ' Dialogismvs tertivs ad Baptistam Lucarinum FR. filium optimse fpei ac indolis puerum, de notis & finguris numerorum, quibus Latini ac Graeci utebantur ' (p. 20) is an equally interesting sketch of the Greek and Latin numerals. Gyraldus also published a 'Brevis instructio de Grecor^z numerali supputatione ' in 1513, fol. Other works 0/1552. Agricola, p. 171, 1533; Gemma, p. 203, 1540; Ghaligai, p. 132, 1521; Herman Giilfferich, < Ein new kurtz Rechenbiichlein,' Frankfort, 8, with editions ib., 1555, 12, 1568, 8 (see also p. 244, 1546, and p. 292, 1559); Ortega, p. 94, 1512 ; Peletier, p. 245, 1549 ; Recorde, p. 214, c. 1542 ; Riese, p. 139, 1522 ; Joachim Camerarius, ' Arithmologia,' 1 2 (there was also an edition at Basel, s. a., 1 6); Dunkel, ' Arithmetical Leipzig, 8; Christopher Falconius, ' Re- chenbuch,' Konigsberg in Preuften, 4 (Murhard cites as a different work of the same date ' Rechenbuch auff die Preusche miintz mass und Gewicht,' ib., 4). DOMENICO MANZONI. Ed. pr. 1553. S. 1. (Venice), 1553. A sixteenth-century arithmetician, born at Oderzo. Title. See Fig. 123. Description. 8, 10.5 x 15.8 cm., the text being 8.1 x 13. 4 cm. 1 6 ff. unnumb., 26 11. except where arranged in sections. S. 1. (Venice), 1553. Editions. There was no other edition of this work. Manzoni had already published a textbook entitled ' Libretto molto utile per imparar a leggere, scrivere et Abaco, con alcuni Fondamenti della Dottrina Christiana,' Venice, 1546, 8, 31 pp. of which are devoted to arithmetic. He also published in 1553 a more elabo- rate treatise, < La Brieve Risolvtione di Aritmetica universale in qualsi voglia negotio, doue interuenga numero, peso, & misura,' Venice, 1553, 8, 246 ff. A beautifully printed little manual of the fundamental operations of arithmetic. The Abbreviatvre delle monete, Pefi, & mifure, chi fi ufano in Vinegia' (f. 2) is helpful to students of the history of sixteenth- century arithmetic and to all who are interested in the metrology of the countries with which Venice traded. 258 KARA ARITHMETICA ABACHETTO NVOV MOLTO COPIOSO , ET AKTI- ficiofamente ordinaco a modo di tariffa d'alcune moltipli* cationi <5 partimenti di numcri rapprefentantt diuerfe nature, $C qualita di mo* nete,peil,ec mifurc, OPERETTA A CIASCVNO Vtilisfima , & uniuerfalmente in tutce le Citta & Prouintie molco neceiTaru . FIG. 123. TITLE PAGE OF MANZONI MICHAEL STIFEL. Ed. pr. 1553. Konigsberg i. Pr., 1553. See p. 223. Title. See Fig. 124. Colophon. 'Gedruckt zu K6nigs-//berg in Preufse durch Alex- andrum // Behm von Luthomifl/ Voll // endet am dritten tag PRINTED BOOKS 259 defs Herbft-//monats/ Als mann zalt nach // der geburt vnfers lieben // herrn Jefu Chrifti.// 1554.' (P. 505.) Woff* far Ccf* Stfcl f r r ? FIG. 124. TITLE PAGE OF STIFEL'S RUDOLFF Description. 4, 14.7 X 19.6 cm., the text being 9.9 X 16 cm. 507 pp. (4 blank, 12 unnumb.), 2 1-26 11. Konigsberg in Preufien, 1553- 2 6o KARA ARITHMETICA Editions. Konigsberg in Preufien, 1553, 4 (here described); 1571. See also p. 226. Although the book is nominally an algebra (the word ' cofs ' coming from the Italian cosa, ' thing,' meaning the unknown quantity), the first part of the book is devoted entirely to arithmetic. This part is intended as an introduction to the algebra that comes later, and gives the funda- mental operations as treated by Rudolff, with Stifel's commentary. It is entirely theoretical, and it naturally leads into the theory of irrational numbers, which constitutes the first part of the algebra. Other works 0/1533. Albert, p. 180, 1534 ; Boethius, p. 27, 1488 ; Gemma, p. 204, 1540 : Gyraldus, p. 254, 1552 ; Mariani, p. 181, 1535 ; Morsianus, p. 182, 1536; Ringhieri, p. 253, 155 i ; Rudolff, p. 152, 1526 ; Bernardu Wojewodki, ' Algorithm, to jest nauka liczby, po polsku na linijoch uczyniony,' Cracow. CLAUDE DE BOISSIERE. Ed. pr. 1554. Paris, 1554. CLAUDIUS BUXERIUS. Born in the diocese of Grenoble, probably c. 1500. He also wrote on poetry, music, and astronomy. For his Rythmomachia see p. 271. Title. See Fig. 125. Colophon. 'Acheue d'Imprimer le xiij. // iour d'Octobre,// 1554.' (F. 74, i.) Description. 4, 10 x 14.8 cm., the text being 7.2 x 12.9 cm. 2 ff. unnumb. + 73 numb. = 75 ff., 32 11. Paris, 1554. Editions. Paris, 1554, 8 (here described); ib., 1563, 8 (p. 262). This is a theoretical work in two ' livres,' probably written for students in the University of Paris. It is the second printed work which I have noticed that carries the system of numeration as high as thousands of quintillions, ' Mille de Quintillions,' although Chuquet, in his manu- script of 1484, carries it to ' nonyllions.' De la Roche (p. 128) followed Chuquet in this as in other respects. Boissiere's plan starts with the names of millions, bimillions (for million millions), trimillions, etc. He then says that to avoid ambiguity, as of bimillions for two millions, these names are abridged to billion, trillion, etc. He moreover numerates in periods of six figures each, as in England at present, and not in periods of three figures as is now the custom in France and America. He first treats of the classes of number according to the old plan, distinguishing between digits (1-9), articles (multiples of 10), and com- posites (articles -f- digits), of which names we still use the digit. Instead of extending the fundamental operations to six, or even nine, as his PRINTED BOOKS 261 contemporaries so often did, he limits them to four, as we do. His work in addition and multiplication is substantially like ours, but his ' sub- straction ' is as follows (1563 edition) : Refte 9399739 01 C. Debte ou fomme 1840006503 A. Paye ou a Subftraire 900032602 B. I/ART D'ARYTHME- TICLVE CONTENANT TOVTE DIMENTION, TRES- SINGVLIER ET COMMODE, tant pout 1'art militaire o,ue autres calculations. Auec priuilege du Roy. fmprlme'a parK,p<tr^nncttnere,A I'cnfci jclalian , rue des Porfa . FIG. 125. TITLE PAGE OF BOISSIERE 262 KARA ARITHMETICA In division he uses the galley method only. Boissiere is one of the first writers, I believe the first in France, to invert the divisor in the division of fractions, as Stifel had done a few years before in Germany. Book I closes with a somewhat extended treatment of the rule of three. The second book relates largely to mensuration, figurate numbers, roots, and the mediaeval proportions. Some of the rules are in verse. Of military matters, referred to in the title page, there is little mention. CLAUDE DE BOISSIERE. Ed. pr. 1554. Paris, 1563. See p. 260. Title. This is practically identical with that of the 1554 edi- tion, except for the following : ' Reueu & augmente par Lucas Trembley Parifien,// profeffeur des Mathematiques.// A Paris,// Pour Guillaume Cauellat, a 1'enfeigne de la Poulle//graffe, deuant le college de Cambray.// 1563.' (F. I, r.) Description. 8, 10.2 x 1 6. 8 cm., the text being 6. 8 X 13 cm. 3ff. blank + I unnumb. -f 71 numb. = 75 ff., 25-30 11. Paris, 1563. See p. 260. Two other books are bound with this : La Declara// tion et Vsage de // L'inftrument nomm Canomettre,// Par G. des Bordes, Gentilho-// me bordelois, profeffeur ez // Mathematiques.// . . . 1570.' and ' Vsage // Dv // Compas // de // Proportion.// Par D. Hen- rion, Mathem.// . . . M. DC. XVIII.' JOACHIM CAMERARIUS. Ed. pr. 1554. Deventer, 1667. Born at Bamberg, April 12, 1500; died at Leipzig, April 17, 1574. The office of chamberlain (Kammermeister) to the Prince-Bishop of Bamberg being hereditary in the family of Liebhard, he took the Latin name of Camerarius. He was a distinguished classicist, a friend of Melanchthon, and a professor at Tubingen and Leipzig. Of the various commentaries on Nicomachus, his was the most important of the Renaissance. Title. ' Explicatio // loachimi Camerarii // Papebergenfis// in dvos libros//Nicomachi Geraseni//Pythagorei//Deductionis// Ad Scientiam Numerorum.// Et Notae // Samuelis Tennulii // in //Arithmeticam // Jamblichi Chalcidensis.//Daventriae.//Typis Wilhelmi Wier, CI3 IOCLXVII.' (P. i.) Description. 4, 14.7 x 19.8 cm., the text being 10.3 X 14 cm. 2 pp. unnumb. + 239 numb. = 241 pp., 34 11. Bound with PRINTED BOOKS 263 the commentary of lamblichus on Nicomachus (p. 188, 1538). Deventer, 1667. Editions. The first edition was Augsburg, 1554, 8. There was also an edition published in 1569. Camerarius also wrote a work entitled ' De logistica,' published at Augsburg in 1554; ib., 1557, 8; Leipzig (?), 1569, 8. See also p. 257, 1552. A commentary on the theoretical work of Nicomachus (p. 186). Other works 0/1554. Albert, p. 180, 1534; Baeda, p. 131, 1521; Buteo, p. 292, 1559 ; Euclid, p. 238, 1545 ; Finaeus, p. 160, 1530-32 ; Fischer (Piscator) p. 247, 1549; Glareanus, p. 192, 1539; Huswirt, p. 74, 1501 ; Nicomachus, p. 1 86, 1538 ; Peletier, p. 245, 1549 ; Psellus, p. 168, 1532 ; Riese, p. 139, 1522 ; Stigelius, p. 249, 1549 ; Tagliente, p. 114, 1515 ; Earth. Barchi, 'Tariffe della valuta di tutte le monete,' Mantua, 4 (hardly an arithmetic) ; Claudio Bertholio, ' De numerandi ratione aphorismi,' Paris, 8 ; Ian Gentil (Vander Schuere, in his 1634 edition, f. 201 see p. 424 refers to his arithmetic as published at Paris, 1554); Caspar Hiitzler, < Eyn behende und Kunstrik Rekens- boeck op. der Linien und Tziferen,' Liibeck, 8. PETRUS RAMUS. Ed. pr. 1555. Paris, 1555. PIERRE DE LA RAMEE. Born at Cuth, Vermandois, 1515; killed at Paris the night of August 24-25, 1572, in the Massacre of St. Bartholomew. He lectured on philosophy at Paris, and for a short time at Heidelberg. He wrote on arithmetic, geometry, optics, and mathematics in general. Title. See Fig. 126. Colophon. ' P. Rami Eloqventiae et Phi-//losophiae professoris //Regij Arithmeticae,// Finis.' (P. in.) Description. 4, 13.8 X 19.1 cm., the text being 9.5 x 17.2 cm. 128 pp. (2 blank, 16 unnumb.), 26-27 11. Paris, 1555. Editions. Paris, 1555, 4 (here described); ib.,1557, 8; Basel, 1567; Paris, 1584. See also Gleitsman, 1600 (p. 427), and the ' Libri Duo' of Ramus, 1569 (p. 330). This arithmetic was popular in the Latin schools for half a century. It is theoretical, consisting largely of definitions, extracts from the Greek writers, a little work on the fundamental operations, and the mediaeval theory of ratios. Ramus had not the faculty of putting together a text- book that should be a rival to that of Gemma Frisius. 264 KARA ARITHMETICA P. Kami, eloquentix ET PHILOSOPHIC PROFES- SORIS REGII,ARITH- metics libri tres, Carolum Lotbaringum Cardinatem, PARTSIIS, Apud Andream Wechelum,fub Pcgafo,in vicoBellouaco, ^4nno Salfttif* 1 5 5 5- Cum priuilegio Regis. FIG. 126. TITLE PAGE OF RAMUS PRINTED BOOKS 265 JACOBUS MICYLLUS. Ed. pr. 1555. Basel, 1555. MOLTZER. Born at Strasburg, April 6, 1 503 ; died at Heidelberg (?), January 28, 1558. He was a well-known classical scholar. Title. See Fig. 127. ; Arithmetics LOGISTIC AE DIBRI duo , ex diuerfis dus artis fcnV ptoribus colle&ij & exemplis plurimis, ijfdem'cp utilifs, nuperilluftrati; IACOBVM Cum gratia &priuilfgio Imperiali ad deceniuum. BASILEAE, PER I O A N* FIG. 127. TITLE PAGE OF MICYLLUS Colophon. s Basileae, ex officina//IoannisOporini, Anno Salutis humanse//M. D. LV. Menfe Martio.' (P. 319.) Description. 8, 9.3 x 15.2 cm., the text being 7.8 X 11.3 cm. 320 pp. (3 blank, 23 unnumb.), 25-27 11. Basel, 1555. 266 KARA ARITHMETICA Editions. There was no other edition. Treutlein's statement (Abhandlungen, I, 15) that the book was published in Heidelberg is incorrect. It probably came from the fact that the ' Epistola Nvncvpatoria' is dated ' Heidelbergae, 10 Calend. Nouemb. Anno Domini 1553.' The book was written for the classical schools. Although Micyllus gives the fundamental operations in a practical manner, the latter part of his book is theoretical, presenting some of the ancient arithmetic in the Greek language. The work includes an unusually complete treat- ment of sexagesimals, ' De partibvs Aftronomicis, & earum fupputa- tione ' (p. 201), and a chapter on the computus (see p. 7), ' De Temporvm svppvtatione, qva Ecclefiaftici utuntur.' The latter shows that the name ' Computus ' had become unpopular, for it begins with the following statement : ' Est & temporum fupputatio quaedam, quern Com- putum Ecclefiafticum, barbaro nomine, uocant.' (P. 269.) The follow- ing is an example of his applied problems : ' Scribit Plinius, Alexandriam a Rhodo diftare 583000 pafsuum. Cupio autem fcire, quot ftadia ijdem pafsus efficiant. Diuido igitur 583000 per 125.' MICHAEL NEANDER. Ed. pr. 1555. Basel, 1555. Born in the Joachimsthal, April 3, 1529; died at Jena, October 23, 1581. He was professor of mathematics, Greek, and medicine, in the university of Jena. He wrote on physics and cosmography. Title. See Fig. 128. Colophon. ' Basileae, ex Officina // loannis Oporini, Anno // Salutis humanae M. D. LV. Men-//fe lanuario.' (P. 119.) Description. 4, 13.9 X 19.5 cm., the text being 9.1 x 15.7 cm. 104 pp. numb. + 15 unnumb. -f I blank =120 pp., 2529 11. Basel, 1555. Editions. There was no other edition. The * Epistola Nvncv- patoria ' is dated * Die Pafchalis, Anno LIIII,' but the book was printed, as the colophon shows, in 1555. This is a historical treatise on Greek, Egyptian, Roman, Arabic, and mediaeval European weights and measures, and is particularly inter- esting because of the symbolism which it contains. The origin of our present apothecary's symbols of measure is seen in the ancient Roman abbreviations. The text is mostly in Latin, but eight pages of ' Merpcov KCU ^TaO/jiotv ' are in Greek. 2YNO MENSVRARVM ET PON- DER VM, PONDERATIONIS'* que menfurabilium fecundum Romanos, Athcnienfes , y^f^ov^, H9u iTmot^cvsy ex pracftantifsimis authoribus huius generis cotraAaopera MICHAELIS DRi ex Vallc Anno M D ACCE55ERVNT ETIAM, Q^VAE APVD plenum hdttemif cxtdbdntdcponderumzrmenfurarumrdtionente bcmcntcr dcpr<mtfd,mmc Greece cr UW m/^o corr^wrrf, Item rcrum uerborum in his omnibus mcmorabilium Index, LEVITICI XIX. BASILEAE, PER 10 w^w Oporinum. FIG. 128. TITLE PAGE OF NEANDER 268 KARA ARITHMETICA ANTOINE CATHALAN (?). Ed. pr. 1555. Paris, 1556. A French arithmetician. The author's name does not appear in this edition. A work with the same title is assigned to Cathalan, Lyons, 1555. Title. See Fig. 129. L'ARITHMETIQJE ET MANIERB DE APPKBNDIIE a Chijfrfr & comfter far la flume & fat lesgeftz en nombre entier & yam* t>u facile a appeudre& tref- utileatoutesgenSi Denouueau reueuc &corrigee. ^4 tyuelle font adiouftees fhfottrs tjue &. txftnptes pour f air e lafcienrtfltt* facile t &- flus Ifgttn a com- frendre. A PARIS. Par leJian Ruclle,Jemournnt en la Rue fainft Iacquts,a TeDfcignc de Ja queue de Regard. 1 T 5 * FIG. 129. TITLE PAGE OF THE 1556 CATHALAN Description. Small 8, 7.3 x 11.4 cm., the text being 5.3 X 9.5 cm. 79 ff. numb. + i unnumb. = 80 ff., 27 11. Paris, 1556. PRINTED BOOKS 269 Editions. Lyons, 1555, 16; Paris, 1556, sm. 8 (here described). This is a very good little primer of algorism for the time. Concern- ing this work the author says : ' Lequel art trouua premierement vn philofophe d'Arabie, nomme Algus. Dont cefte fcience prent fon nom d'Algorifme ' (f. 3). In multiplication the gelosia arrangement is given as a third method, under the name ' Mvltiplication per quarreaux.' In division only the galley form appears. The ' premier liure ' covers the four fundamental operations with integers, and a brief treatment of pro- gressions. The * fecond liure' relates to the use of counters ('gectz'), which the author esteems so highly as to say : ' Et note que cefte efpece de addition eft plus vtile & facile aux gectz que aux chifres ; ' ' il eft facile par les gectz, & difficile par les chifres.' The last part of the book is devoted to such standard problems as the testament, the pipes filling the cistern, and the broken eggs, and to applications like ex- change and pasturage. LODOICO BAEZA. Ed. pr. 1555. Paris, 1555. A Spanish scholar of the sixteenth century. Title. See Fig. 130. Colophon. ' Excudebat Benedictus Preuotius, via // Fremen- tella, ad infigne ftellae aureae.' (F. 68, r.) Description. 8, 10.3 X 15.6 cm., the text being 7.8 X 13 cm. 2 ff. unnumb. + 66 numb. = 68 ff., 26-27 N- Paris, 1555. Editions. This rare first edition of Baeza's arithmetic is sel- dom found in libraries. The second edition (1556), 8, is, how- ever, the same impression with the title page changed. The book is entirely theoretical, making much of the classification of numbers and of the ancient ratio systems. It is in Latin with nu- merous Greek quotations. Other works of 1555. Euclid, p. 237, 1545 ; Finaeus, p. 163, 1530-32; Gemma, p. 200, 1540; Glareanus, p. 192, 1539; Herman Giilfferich, p. 257, 1552; Mariani, p. 181, 1535; Ringhieri, p. 253, 1551 j Valturius, p. 10, 1472 ; Yciar, p. 249, 1549 ; Jacob Cuno, 'Arith- metica,' Wittenberg, 8 ; Taddeo Duni, ' Liber de arithmetical Basel, 4 ; Melchior Goldammer, 'Arithmetica Pratica,' Wittenberg, 8 ; Jero- nimo de Valencia, 'Arte de compute,' in Santaella's 'Vocabularium ecclesiasticum,' a separate edition appearing at Saragossain 1601. (See also p. 140, 1523.) 2 7 RARA ARITHMETICA N V M E R A N- DI DOCTRINA PR AE CLARA METHODO expofita, in qua brcuitcr continents, cxponuntur aperte ea 3 qu& ex yn rithmetieajuntadyfitm potiora. futhorc Lodoico B<te%4* L V T E T I AF, liclmum Cauetlat 9 /ub pingui Galli- n<t,ex aJuerfe cottegij Cameracenfis. 1 5 5 5- FIG. 130. TITLE PAGE OF BAEZA PRINTED BOOKS 271 CLAUDE DE BOISSIERE. Ed. pr. 1556. Paris, 1556. See p. 260. Title. See Fig. 131. Description, 8, 10.4 x 16.6 cm., the text being 6.7 X 13.1 cm. i f. unnumb. + 52 numb. + i blank = 54 ff., 26-27 ^- Paris, 1556. Editions. Paris, 1556, 8 (here described); French edition, ib., 1556, 8. Of the three standard treatises on the ancient number game of Rythmomachia mentioned in this list, the others being the one of 1496 of uncertain authorship and Barozzi's work of 1572, this is the clearest. It describes very carefully the checkerboard on which the game is played, the nature of the calculi used, and the general mode of procedure. Moreover, it is profusely illustrated (Fig. 132), which adds much to the value of the book. The game was connected with the mediaeval number classifications and ratios, and could never have been understood by any save those who were well educated in the ancient theoretical arithmetic. GALLUS SPANLIN. Ed. pr. 1556 (?). Niirnberg, 1566. An Ulm Rechenmeister, as he describes himself on the title page. Title. See Fig. 133. Colophon. 'Gedruckt zu Nurmberg//durch Chriftoff // Heufz- ler.' (P. 382.) Description. 8, 9.2 x 14.7 cm., the text being 7.1 x 1 1.5 cm. 15 pp. unnumb. + 4 blank + 365 numb. = 384 pp., 21-25 U- Niirnberg, 1566. Editions. The dedication to ' Den Edlen Ehrnveften/ Furfich- tigen/ Erfamen vnd Weyfen Herren/Eltern Burgermeiftern vnd Rath/des heyligen Reichs Stadt Ulm' is dated < 3 Julij/Anno Chrifti/ 1556,' so that this is possibly the date of the first edi- tion. I know of no other edition except this of 1566. The author devotes twenty-one pages to ' Rechnung auff der Linien,' this counter reckoning being still the popular method at the time he wrote. He then (p. 24) takes up the processes with Arabic numerals, at first using a few abstract numbers, but soon, as was the custom, introducing many practical applications. On the whole the book may be said to be a rather good exponent of Riese's school. 272 KARA ARITHMETICA NOBILISSI- MVS ET ANTIQVISSI- mus ludusPythagoreus Cqui Rythmo machia nominator) in vtilitatem & re- laxationem ftudioforum comparatus ad veram & facilem proprietatem & rationem numcrorum affcquendam, nunc tadem per Claudium Buxcrium Delphinatcm illuftratus. L V T B T I AB, ngut Gti- ge x'adtterfo collegij Cameracfnfis in apttd lottnncm Gentil. M 5 6 CVM PRIVILEGIO REGIS. FIG. 131. TITLE PAGE OF THE 1556 BOISSIERE PRINTED BOOKS 273 T8 V V FIG. 132. FROM BOISSIERE'S Rythmomachia 274 KARA ARITHMETICA miff Jltirii alf urn epnK ' FIG. 133. TITLE PAGE OF SPA'NLIN PRINTED BOOKS 275 DOMENICO DELFINO. Ed. pr. 1556. Venice, 1565. A sixteenth-century Venetian. He was born of a noble family, which had produced a doge, a general of the Camaldoli, and other men of prominence, and which was later to produce a cardinal and a captain-general of the Venetian fleet. Title. See Fig. 134. Description. 8, 9.8 x 14.4 cm., the text being 8. 5 x 12.7 cm. 416 pp. (56 unnumb.), 26 11. Venice, 1565. Editions. Venice, 1556, 4; ib., 1565, 8 (here described); ib., 1568, 8; ib., 1584, 8, and later. The title page bears the date 1565 as stated, and the dedicatory letter ' Al Mag. co et Ecc. mo Sig. Nicolo Crasso,' signed by Ludouico Dolce, is dated ' In Venegia a VII. di Febraio. M D LXIIII,' 1565 new style. The author's preface, ' Intentione dell' avtore,' which follows, bears no date, but succeeding this there is a letter ' AH'Illvstris- simo et Reverendissimo Signor, II S. Cristoforo Madrvccio, Cardinal di Trento, Governator Meritissimo di Milano,' signed by F. Nicolo Croce, and dated ' Di Vinetia il di primo di Settem- bre. M D L VI.' There is no colophon to fix the date more accurately, unless it appeared on some page after 360, with which this copy terminates. The book, a ' summary of all the sciences,' devotes a brief chapter to the Boethian arithmetic (pp. 43-48) : ' Dell' Aritmetica, de' svio // inuentori, utilita, modo, & altri fecre // ti Cap. III.' The author is highly lauded in Croce 's letter as of noble family, ' Illuftre per lo splendor del fangue, non dimeno uia piu per Pornamento e per la gloria della dottrina.' The Delfino family was at that period very prominent in Venice. NICOLO TARTAGLIA. Ed. pr. 1556. Venice, 1556. TARTALEA. Born at Brescia, c. 1506; died at Venice in 1559. He was one of the best mathematicians of his time, and was the first to give a gen- eral solution of the cubic equation. Title. See Fig. 135. Description. Fol., 21.1 X 29. 1 cm., the text being 13.8 X 23.6 cm. 285 ff. (3 blank, 6 unnumb.), 52-56 11. Venice, 1556. Editions. Tartaglia's works include the ' Nova Scientia,' chiefly algebraic, editions of Euclid, Archimedes, and Jordanus, 276 KARA ARITHMETICA S C I E N Z E, DEL M^f G TV I F I CO M E S S E 1 Domenico Deljino y nobile Vmitia.no> DALQVALE SI POSSONO IMPARAB molte colc_appartenenti ai umere humano , & alia cognmon di D i o Ccn U "Tauola, & le FojMe deiie cofe ptu notattili, IN VlN^GIA_jjP PRESS (T~GATB 1C ~L G10LITO DF FERRARI. M n f T v FIG. 134. TITLE PAGE OF DELFINO 7 LA PRIMA PARTE DEL GENERAL TRATTATO DI N V* MERI, ET HISVRE DI NICOLO TARTAGLIA, NELLACLVALE IN DIECISETTE LZBRI SI DICHIARA TVTTI GLI ATTI OPEI^ATIV/, PRATICHE, ET REGOLE NECESSARIE NON SOL uicntc in tucta 1'arte ncgor ian'a,& mcrcan tile,ma anchor in ognialtra attc/dcnciajOucr difcip!ina,doue intcrucngliiil calculo. WALIGNITA* CON LI SVOI PRIVILEGII. In l^intgid per Cttrtio Troia.no deiNauo, M D Ll^I. FIG. 135. TITLE PAGE OF TARTAGLIA 278 KARA ARITHMETICA the * Quesiti et invention! ' and ' Regola generate,' chiefly phys- ical, and the ' General Trattato ' containing the arithmetic. Of the arithmetic there were editions in Venice, 1556 and 1560, 4 (?) ; ib., 1592-93, 4 (p. 279) ; Paris, 1578, 8 (below) ; Ant- werp, 1578, 8. There was also a ' Scelta di Abbaco ridotto dal famosissimo Nicolo Tartaglia,' published at Venice in 1596. The entire work consists of six volumes, bound usually in three, and the publication extended over a period of five years, from 1556 to 1560 inclusive. This volume contains the arithmetic, the most scholarly con- tribution to the subject that appeared in the sixteenth century. It is more elaborate than the treatise of Paciuolo (p. 54, 1494), and like that work it enters into the various minute details of the operations and commercial rules of the Italian arithmeticians. For example, Tar- taglia gives seven methods for the multiplication of integers, and four for division ; he enters very fully into the discussion of denominate numbers and exchange ; and his treatment of such rules as ' Regula de tri ' is unusually elaborate. Indeed, there is no other treatise that gives as much information concerning the arithmetic of the sixteenth cen- tury, either as to theory or application. The life of the people, the customs of the merchants, the struggles to improve arithmetic, are all set forth by Tartaglia in an extended but interesting fashion. NICOLO TARTAGLIA. Ed. pr. 1556. Paris, 1578. See p. 275. Title. L' Arithmetiqve // de Nicolas // Tartaglia Brescian,// grand mathematicien,// et prince des practiciens.// Diuifee en deux parties.// La .declaration fe verra en la page fuyuante.// Recueillie, & traduite d'ltalien en Francois, par // Gvillavme Gosselin de Caen.//Auec toutes les demonftrations Mathema- tiques : & plufieurs in-//uentions dudit Gosselin, efparfes chacune //en fon lieu. //A tres-Illustre & Vertueufe Princeffe Mar-// gverite de France, Royne de Nauarre.// Premiere Partie.//A Paris, //Chez Gilles Beys, rue S. lacques, au Lis blanc.// 1578. //Avec Privilege dv Roy.' (F. I, r.) Description. 8, 10.6 X 16.6 cm., the text being 7.5 x 13.4 cm. 286 ff. (152 in part I, 134 in part II, 28 being unnumb.), 27-32 11. Paris, 1578. PRINTED BOOKS 279 See p. 278. The * Seconde Partie ' follows f. numb. 136 of the first part, and is of the same date. The Privilege follows f. numb. 122 of the second part, and is dated 'Paris le 17. Septembre. 1577.' This is merely a French translation of the work already described with notes by Gosselin. These notes are, rather naively, printed in more prominent type than the original text, but are generally of little value. They cover such points as multiplying by or dividing numbers ending in zero, they amplify certain demonstrations (as in the division of fractions), and they adapt the commercial chapters to French usage. NICOL6 TARTAGLIA. Ed. pr. 1556. Venice, 1592. See p. 275. Title. ' Tvtte Fopere // d'arithmetica // del famosissimo // Nicolo Tartaglia.// Nelle qvale in XVII. libri con varie prove,// & ragioni, moftrafi ogni prattica naturale, & artificiale; i modo, & le regole da//gli Antichi, & Moderni vfate nell' arte mercantile; & oue interuiene calcolo,//pefi, denari, tariffe, calmeri, baratti, cambi di banchieri, e di fiere, faldi, fconti,//giuochi, traffico di compagnie, compre, vendite, portar mercantie da un paefe // all'altro, conuertir monete, congiungimento di metalli, & opere de'zecchieri.//Sopra le qual cofe tutte, formanfi belliffimi que- fiti, & fi fciolgono le diffi-//colta, CO n vgual chiarezza, & dili- genza, per vtile rileuato de i mercanti, & te-//forieri, a Capitani, e Matematici, & Aftrologhi, &c. // Parte Prima.// Con Privilegio. // In Venetia,//All'Infegna del Leone. M. D. XCII.' (F. i, r.) Description. 4, 13.6 X 18.8 cm., the text being 11.5 X 18 cm. 479 ff. (204 in part I, 275 in part II, 8 unnumb.), 40 11. Venice, 1592. Riccardi says that it was not completed until 1593. Editions. See p. 278. This is substantially the same as the first volume of the ' General Trattato ' of 1556 (p. 275), except that it is in quarto instead of folio. ORONTIUS FINAEUS. Ed. pr. 1556. Paris, 1556. See p. 160. Title. See Fig. 136. Description. Fol., 19.6 X 28.5 cm., the text being 13 X 21.4 cm. 6 ff. unnumb. + 1 36 numb. = 142 ff., 1 5-34 11. Paris, 1 5 56. ORONTIIFINAEI, DELPHINATIS.RE- GII MATHEMATI- CARVM PRO- FESSORIS, De rebus mathematicis, ha<5tenus defiderans, Libri IIII. ^ Qwbus inter cetera, Circuli quadratura Centum modis,& fupra, per cundcm Orontium rcccntcr excogitatis,dcmonftratur. LVTETIAE PARISIORVM, Anno Chrifti Scruatoris, M. D. LVI. ExofficinaMichaelis Vafco(ani,uia lacob^a adinfigncFontis. Cum Pnuilegio Regis. FIG. 136. TITLE PAGE OF THE 1556 FINAEUS PRINTED BOOKS 281 Editions. There was no other edition. For his arithmetic, see p. 1 60. Although chiefly on geometry, this work is included in the list of arithmetics because its treatment of proportion is more arithmetical than that of Euclid and his followers. (Ff. 25-29.) There was, how- ever, nothing original in the work of Finaeus. VALENTIN NABOB. Ed. pr. 1556. Cologne, 1556. NAIBOD, NAIBODA. Born at Cologne; died at Padua, March 3, 1593. He was for a time professor of mathematics at Cologne. He wrote on astronomy as well as arithmetic. Title. See Fig. 137. Colophon. ' Coloniae, Typis Iohannis//Bathenij.' (F. 100, r.) Description. 8, 9.5 X 14.9 cm., the text being 6.3 x 1 1.9 cm. 8 ff. unnumb. + 92 numb. = 100 ff., 28 11. Cologne, 1556. Editions. There was no other edition. This is a Latin work, with occasional Greek passages, and was written for the classical schools of Germany. Although assuming to be a prac- tical treatise on calculating, it so lacks the merit of brevity as to be unfitted for use as a textbook. It has few problems, and these are of no commercial value. In his desire to exalt the classical learning, Nabod, like other Latin writers of his time, assigns the Arab-Hindu numerals to the Pythagoreans. VALENTIN MENHER de Kempten. Ed. pr. 1556. Antwerp, 1556. See p. 249. Title. See Fig. 138. Colophon. ' Imprime en Anuers par Ian Loe //1'An de noftre Seigneur i556//le 20 iour d'Auril.' (F. 184, v.) Description. 8, 9.8 x 14.9 cm., the text being 7.8 x 13.2 cm. 184 ff. unnumb., 27-28 11. Antwerp, 1556. Editions. See p. 250. This is usually mentioned as the second of Menher's arithmetics. It is, however, the same as the work described on p. 249, 1550, save for a few minor changes. 282 KARA ARITHMETICA VALENTI- NI NABOD I D E CALCVLATORIA NV- merorumque natura Seftio- nesquatuor. CLARIS SIMVM Vl^fU GASPARVM DOVCIVM JJLORENTINVM, CAESAREAE Maieftatis Coniiliarium. COLONIAfi AGRIPPINAE, Apud hseredes Arnold! Birck- manni 1556. FIG. 137. TITLE PAGE OF NABOB PRINTED BOOKS 283 ARITHMETI QJ/E SECONDE PAR ~M. VALENTIN MENNHER deKempten. Auec grace & priuilege de PEm- pereur pour quaere ans. FIG. 138. TITLE PAGE OF MENHER 284 KARA ARITHMETICA PIERRE FORCADEL. Ed. pr. 1556-57. Paris, 1556-57. Born at Beziers ; died at Paris in 1574. He was (1560) professor of mathe- matics in the College Royale, Paris. He also wrote on astronomy, and trans- lated the works of several Greek mathematicians. Title. The title page of this copy of Book I is missing. The title page of Book II reads as follows : ' Le // Second Livre de // T Arithmetiqve de P. For-//cadel de Beziers.// Av qvel seront declarees les// fractions vulgaires, auec leurs demonftrations, par les // quantitez continues, & premieres caufes des egaliffemens // de F Algebre.// Le tout nouuellement inuente par 1'auteur.// (Woodcut with motto : In Pingvi Gallina.) A Paris,// Chez Guillaume Cauellat, a 1'enfeigne de la // poulle graffe, deuant le college de Cambray.// i556.//Avec Privilege.' (F- 94-) The title page of Book III reads as follows : * Le // Troysieme Li-//vre de 1' Arithmetiqve // de P. Forca- del de Beziers.// Avqvel sont traictees les de-//rnonftrations de toutes fortes de racines, auecques 1'entiere pra-//ctique de 1'extraction d'icelles, enfemble plufieurs queftions, reigles,//& demonftrations Mathematiques, auecques le propre fubiect de 1' Algebre.// Le tout de 1'inuention de l'Autheur.//A Paris,// Chez Guillaume Cauellat, a 1'enfeigne de la // Poulle graffe, de- uant le college de Cambray.// i $$?.// Avec Privilege.' (F. i , r., of Part III.) Description. 4, 13.2 x 19.2 cm., the text being 10. i x 16 cm. 323 ff. (5+93, 1 10, 4+ 1 1 1, in the above three books); 25-31 11. Paris 1556-57. Editions. Forcadel wrote four arithmetics, as follows : 1. * L'arithmetiqve,' Paris, 1556 57, here described. 2. ' L'Arithmetique par les gects,' Paris, 1558, 8. 3. 'Arithmetiqve entiere et abregee,' Paris, 1565 (p. 316). 4. ' Arithmetique demonftree,' Paris, 1570,4. This is perhaps the most elaborate French treatise on arithmetic published in the sixteenth century. Its three books of about one hun- dred pages each form a work of the nature of the great contemporary Italian arithmetic of Tartaglia, or rather of Tonstall's Latin treatise. Of PRINTED BOOKS 285 L'ARITHMETICQVE DE P Forcadel -deBczicn. / Vela cognoijfince & dftofiicndfs figures ><Mtrcment de* nombresjimplft. o v T E 1'A rithmcticque confiftc a fjauoir cognoi- ftredix figures, oucarafteres,propres guidons de toutc computation Mathematicque;qui font cy dcf (bus rcngcz felon leur propre nature. DA x. a. 3. 4. 5. 6. 7. 8. $>. o. zero. 3. 4. 5. tf. 7. 8. p. o. i. premiere f. 4. IT. 7. 8. p. o. j.^2. (econdes. 5. 6. 7. 8. $. o. i. a. 3. tierces. 6. 7. 8. p. o. i. a. 3. 4. quartcs. 7. 8. 9. o. i. a. 3. 4. j. quintes. 7. 8. p. o. i. i. 3. 4. 5. 6. fixiefmes. p, o. i, i. 3, 4. 5. (T. 7. feptiefmes. o. i. 2. 3. 4. 5. 6. 7. 8. huiticfmes. 'a^jnu no ojaz o. i. 2. 3. 4.5. 6.7.8. 9, ncufielmcs, 3 *3 ? '* 3* S*^ S" FIG. 139. FIRST PAGE OF TEXT OF THE 1556-57 FORCADEL 286 KARA ARITHMETICA practical problems it has very few, and it is equally deficient in the theories of the ancient arithmeticians. It is simply a ponderous work on the theory of arithmetical calculations and rules, valuable for a scholar but useless as a practical textbook. For a biographical study of Forcadel see Boncompagni's Bullet ino, II, 424. Other works of 1556. Bae'za, p. 269, 1555 ; Cathalan (Anonymous), p. 269, 1555 ; Gemma, p. 200, 1540 ; Medlerus, p. 223, 1543 ; Medicus, p. 290, 1557 ; Psellus, p. 168, 1532 ; Recorde, p. 214, c. 1542 ; Riese, p. 139, 1522 ; Ringelbergius, p. 168, 1531 ; Xylander, p. 356, 1577 ; Anonymous, ' Arithmetices Epitome,' Freiburg, 12; A. Lottini, ' Calculi e conti per quelli che hanno denari,' Lyons, 8 ; Casp. Pauerus, ' Logis- tice astronomica,' Wittenberg, 8 ; Joannes Pierius Valerianus, ' Hiero- glyphica sive de Sacris Aegyptiorvm literis Commentarii ' (ancient finger reckoning), Basel, fol., with editions ib., 1567, fol. ; ib., 1575, fol. ; Lug- duni, 1579, fol. ; Juan Diaz Freyle, ' Sumario compendioso . . . con algunas reglas tocantes al Aritmetica,' Mexico (the first arithmetic printed in America). ROBERT RECORDE. Ed. pr. 1557. London, 1557. See p. 213. Title. See Fig. 140. Colophon. * Cjmprinted at London, //by Jhon Kyngfton.// Anno domini. 1557.' (F. 164, v.) Description. 4, 13 X 18 cm., the text being 8.1 x 14.7 cm. 164 ff. unnumb., 36 11. London, 1557. Editions. There was no other edition. Recorde speaks of this work as ' The feconde parte of Arithmetike, containyng the extraction of Rootes in diuerfe kindes, with the Arte of Coffike nombers, and of Surdes nombers also, in fondrie fortes.' It is not, however, purely algebraic, for the first half of the book is a treatise on Boethian arithmetic. For example, the following is one of the defi- nitions : ' A Diametralle nomber, is fuche a number as hath twoo partes of that nature : that if thei bee multiplied together, thei will make the faied diametralle nomber.' The treatise on ' Cossike nombers ' begins on f. S i, and 'The rule of equation, commonly called Algebers Rule' on f. Gg 4. The sign of equality, ' a paire of paralleles, or Gemowe lines of one lengthe, thus ; , bicaufe noe .2. thynges, can be moare equalle,' is found for the first time in print on f. Ff i (see Fig. 141). PRINTED BOOKS 287 Cljctoljetftone oftoftte, to&tc&e is tl)e fcconor parts of Arithmetiketconf flfnpn tfon of ttocte*: HS&c&J?/ fottlj tfje rule of ? fuffo T^oag^ way/tones doe learegrtate price, The ^hctftone is for eterfce jfs neadefuttjmdm Tworkj atftrattnge: Quite t&nges andkardeiftoillfo dating jfndmdy tbem/karfaio right good* ft: Mlartefintn 1&oW,tbei can mt cbufe, $ut *fe bit belpeyet at men fee, Hoefbarptneffefemetb in it to bee. TAegroundeofartes dtdlrede tbttftone: His \>fi isgreatfjtmlmoare then one, Here 1/1 ou lift your %ittts to Ttbttte, toulle toitte thereby doegrettely mtnde, Sbarpe Tutte* are fined to tbeirfulle tvde. jfndtojottrfel/be not +nl(inde CM cfe 250ofte* are to bee folDe, at tie OTette ooo;e of pouiw, FIG. 140. TITLE PAGE OF RECORDE'S Whetftone of witte 288 KARA ARITHMETICA as tf)ct r too;fer0 Doe ctfcnfic ) to Diaincte it onety into tfooo parte*. SflJfcereof tbc firfte is, v^f OB* nomberu cqutllc vnto one other. 3nO ttje feconDe is , *k onr now; if r u compared as e^aalle Vnto.^otbcr nombers* ai toaics lulling ^ou to rcmclicr, tljat f o u reduce pournombew, totljeirlcaffc Denominations, anu Onallelte fo;me0^efo;e ?ou p^oceDe anp farther* 0nD again, if pour equation be for tjc, tbat tljc grca- tefte Denomination G/?/^ ? be i oineD to an? parte of 4 compounDc nomfaer , T?OU Qall ton rnc it fo , ti>at tlje nombcrof tljegreatcftc 0gne alone , inaicttanDcaa cquallctotticrcttc. anD tins is all tljat neaDetb to be taugljte , conccc- npngtbustooojUe* !i^o tubcit, fo^ caOfc alteratio of eftutiont.% tuf II p^o^ pounDe a fetoe craples, bicaufc t^e extraction of tbeic rootes,maie toe mo^e aptlp bee tojo ugtjte. anD to a- uotDe tlje teDioufe repetition of tljefe tuoojoe* : is c^ rjuallc to : 31 foill fette as 31 Doe often m luo o:Uc ijfc, a paire of paralleles,o^ >eniotne lines of one lengtbc, rqualle. 3nD noto marUe tljcfc nombers* 1. 1 4.t 2. 2 o, t v 26. > I 4. T 1 8.2^ f -2 4* 6. 545- -- I2^=4o2e > ~-f 48of 9,5- i. 3in tbc firftc there nppearctlj, 2 , nomfaf rs , tbat ts 14-%. FIG. 141. FROM RECORDE'S Whetftone of witte PRINTED BOOKS 289 LUCAS LOSSIUS. Ed. pr. 1557. Frankfort a. d. Oder, 1557. A German arithmetician of the sixteenth century, born at Liineberg. Title. See Fig. 142. ARITHME' TICES EROTE* MAT A PVE- R I L I A. IN QVIBVS SEX SPECIES huius vtilifsimae artis, & Regula>quam vocant, Detrijbreuiter & per- fpicue traduntun IN GRATtAM ET VSVM SCHOLAa rum puerilim Ldtmnvm coUettt , cr m lucem iani recent cdita* - A. ; - LFCA LOSS 10 Luneburgenfi. FRANCOFORDI^ AD ODERAM IN OFFICINA IOHANNIS EICHORNL FIG. 142. TITLE PAGE OF LOSSIUS Description. 8, 9.2 x 14 cm., the text being 6.8 x 11.9 cm. 4 ff. unnumb. + 27 numb. = 31 if., 22-24 U- Frankfort. For date see f. 4, v. : ' Datae Lunebar, Anno 1557.7/7. Februarij.' 2 9 o KARA ARITHMETICA Editions. Frankfort an der Oder, 1557, 8 (here described); s. 1., 1562, 8; Leipzig, 1568, 8; Frankfort, 1569; Magdeburg, 1585, 8. A small book, intended, as the title states, for beginners in the Latin schools. It is arranged on the catechism plan, a feature not common with arithmetics printed in Germany at this time, although extensively used by Recorde in England (see p. 210). It begins : * Quid eft Arith- metica? Est bene & artificiofe numerandi & computandi fcientia.' (F. 5.) The ' species ' are treated in a practical way, but the problems are all traditional, part of them being based upon biblical incidents. Other works of 1557. Archimedes, p. 228, 1544; Camerarius, p. 263, 1554; Gemma, p. 200, 1540; Gutierrez de Gualda, p. 167, 1531 ; Jacob, p. 298, 1560; Paxi, p. 80, 1503; Psellus, p. 170, 1532 ; Ramus, p. 263, 1555; Rudolff, p. 153, 1526; Tagliente, p. 115, 1515; Martinus Carolus Cressfelt, Arithmetica, Reeckeninge op den Linien end Cyfferen na allerley Hantieringe,' Deventer (second edition 1577) ; Sixtus Medicus, ' De Latinis numerorum notis,' Venice, 4 (colophon date, 1556; hardly an arithmetic). GIOVANNI FRANCESCO PEVERONE. Ed. pr. 1558. Lyons, 1581. An Italian arithmetician of the sixteenth century, bom at Cuneo, in Piedmont. Title. See Fig. 143. Description. 4, 15.3 x 21.7 cm., the text being 10.2 x 17.6 cm. 136 pp. (60 on arithmetic), 35-37 11. The dedicatory epistle is dated 1556. Bound with this is 'II breve Trattato// di Geometria.' Lyons, 1581. Editions. Although the dedicatory epistle is dated 1556, and the portrait of the author bears the date 1550, I know of no edition before 1558, Lyons, 8. This edition of 1581 seems to have been the second. The first part of the work treats of the operations with integers. It is not a particularly progressive textbook, as is seen in the fact that it includes the ' gelosia' multiplication of the early Venetian writers. The second book relates to fractions, the third to business operations, and the fourth to roots. The work is in no sense a scholarly production. ARITHMETICA E GEOMETRIA DEL SIC.* Gio. Francefco Peuero/te di Canto. IN LTONE* P1K CJO* DI TORNES. M. D. LXXXI. Con Priuilcgio del Re. FIG. 143. TITLE PAGE OF PEVERONE 292 KARA ARITHMETICA ALVISE CASANOVA. Ed. pr. 1558. Venice, 1558. A Venetian teacher of the sixteenth century. Title. See Fig. 144. Description. 4, 15.5 X 20.8 cm., the text being 12.5 x 16. 1 cm. 136 ff. (partly numbered), 33-38 11. Venice, 1558. Editions. There was no other edition. Although this rare work is usually classed as a commercial arithmetic, it is rather a collection of bookkeeping problems. As such it was one of the most prominent of the century, and it gives an idea of the busi- ness questions of the Venetian merchants of its time. Other works 0/1558. Anianus, p. 32, 1488; Archimedes, p. 228, 1544 ; Forcadel, p. 284, 1556-57 ; Gemma, p. 204, 1540 ; Glareanus, p. 192, 1539 ; Mariani, p. 181, 1535 ; Medlerus, p. 223, 1543 ; Recorde, p. 213, c. 1542 ; Riese, p. 141, 1522; Andreas Clatovenus, 'Arith- metica Bohemice,' Prag, 8 (Wydra, in his Historia Matheseos in Bohemia, p. 18, mentions it as printed in Niirnberg, 1530). There were also two arithmetics published s. a., but c. 1558, viz. : Benese, p. 182, c. 1536 ; Gemma, p. 200, 1540. JOHANNES BUTEO. Ed. pr. 1559. Lyons, 1559. BOTEO, BUTTON, BATEON. Born in Dauphine, c. 1485-1489; died in a cloister in 1560 or 1564. He belonged to the order of St. Anthony, and wrote chiefly on geometry, exposing the pretenses of Finaeus. Title. See Fig. 145. Description. 8, 10.4 X 16.9 cm., the text being 7.2 X 13.2 cm. 400 pp., 23-27 11. Lyons, 1559. Editions. Lyons, 1559, 8 (here described); ib., 1560,8. An edition of his 'Opera' appeared in 1554. The work is divided into five books, the first treating of the funda- mental operations with integers, the second of fractions and the rule of position, the third of algebra, the fourth of arithmetical problems, and the fifth of algebraic problems. The problems are not of practical value, and hence the arithmetic never attained any popularity. Other works 0/1559. Albert, p. 180, 1534 ; Anianus, p. 32, 1488; Cataneo, p. 244, 1546 ; Gemma, p. 200, 1540 ; Mariani, p. 181, 1535 ; Johann Fischer (Piscator), p. 247, 1549 ; Herman Giilfferich, ' Ein new Rechenbuchlin auff der Linien und Federn' (perhaps the work men- tioned on p. 257, 1552); Pedro Juan Monzd (Monzoni), ' Elementa Arithmeticae,' Valencia, 8, and s. 1., 1566, 1569. PRINTED BOOKS 293 SPECCHIO LVCIDJSSIMO, nelqjuale.fiucdeno. effere dlfFmito _ tutti i modi , & ordini dc fcrittura, che fi.dcuc inenare nelli nego- tianienti della Mcrean- J(EC./MB!1 It loro corriftondeBtie > difearba- gliando , jjr illuminaitdo Citttettetto a nego- tianti . OPERA NON piv VEDVTA, compolta per Aluifc Cafanoua , Cittadiil.Venetiano . Annc Dommi. O ON O R A TT A ET FIG. 144. TITLE PAGE OF CASANOVA 294 KARA ARITHMETICA B V TE O N IS LOG IS TIC A, QV & & Arithmetica vulgo dicitur in h- bros quinqne digefla:quo- rum index fummatim habetur inter go. E 1 V S 7> E M , i/td locumUitruuii corruftum '&stitutio,qui eft dcfroponme lafidum mitten Jorum ad baliftt Dccimo, I V G D V N 1, JIPVD SVB SCVTO VENETO. JA. 7). LIX. Cum FIG. 145. TITLE PAGE OF BUTEO PRINTED BOOKS 295 FRANCISCUS BAROCIUS. Ed. pr. 1560. Pavia, 1560. FRANCESCO BAROZZI. Born at Venice, c. 1538 ; died after 1587. He wrote on cosmography, and edited Proclus. Title. ' Francisci Barocii // patritii Veneti // opvscvlvm,// in quo vna Oratio, & cluaeQueftiones://altera de certitudine, & altera // de medietate // Mathematicarum continentur.// Ad Reueren- disfimum Danielem Barbarum Patriarcham//Aquileienfem defig- natum Virum Clariff.// Patavii, E. G. P.//M. D. LX.' (F. I, r.) Description. 4, 14.1 X 19.6 cm., the text being 10.1 x 15.3 cm. 40 ff. numb., 26 11. Pavia, 1560. Editions. There was no other edition. This philosophical discussion has been mentioned because it includes some reference to the old Boethian arithmetic. No arithmetical pro- cesses are discussed. OLIVIERO FONDULI. Ed. pr. 1560. Bologna, 1560. A Bolognese teacher of the middle of the sixteenth century. Title. See Fig. 146. Description. 8, 9.9 X 14.6 cm., the text being 6.9 X I i.i cm. 23 ff. unnumb. (possibly one missing at the end), 24 11. The date is given on f. I, v.: 'Dat. Bonoiae . die xxv. Septemb. 1560.' Bologna, 1560. Editions. There was no other edition. This rare little handbook of commercial customs, from the press of Pellegrino Bonardo of Bologna, is hardly an arithmetic, although it explains certain arithmetical processes, and gives a considerable number of business problems. SIMON JACOB. Ed. pr. 1560. Frankfort, 1565. Born at Coburg; died at Frankfort am Main, June 24, 1564. He was one of the best-known Rechenmeisters of his time. Title. Sec Fig. 147. Colophon. ' Getruckt zu Franck-//furt am Main/ bey Georg Raben///in verlegung Sigmund Feyerabends/ //vnd Simon Huters. Sigmund Feyerabent // Simon Hutter // M. D. LXV.' (F. 363, v.) 296 KARA ARITHMETICA PRATICHE DE PIORETTI MERCHANTILI. vtilifsime a ciafcheduna perfbna 3 di mandare a memorialebreuelnuetioni fabrichate fopr* al Valutar de pefi 5 & mifiire, & altre in ftruttioni necceflarie da fapere . Et inchora 3 quadrare Muraglie , Taflelli , & Coperti *- Tuade , Fieno , & Legne.-Con la decchiaratrone, & Exempli loro come legendo intenderai. FIG. 146. TITLE PAGE OF FONDULI PRINTED BOOKS 297 m&mfttenVwtnvfiMem/ extraaian Radicum,t>it t> on ton <propom'onm/mtf tnfen fuftigm Sra^nvft ^tcic^en noc^ nie an tag fommm. CDii6 &ann i?on t> Ocomct via/ tvic man raaitc^ erle;? ^eU iD e&itf /ancf) atterlep (Totpota?7vegufarla bnb 3rr(0ulatrta/meffen/2(ream finteen t>fi reefy* ttcnfo( 2iflc8t>ur^0imoit3aco& t>0n$o(urav^ih-acri>ut>3fccJ)enmcifteritt FIG. 147. TITLE PAGE OF JACOB 29 8 KARA ARITHMETICA Description. 4, 15.2 x 18.6 cm., the text being 10.3 x 15 cm. 354 ff. (349 numb.), 30 11. Frankfort, 1565. Editions. Jacob published two works, as follows : 1. ' Rechenbiichlein auf den Linien und mit Ziffern,' Frank- fort, 1557, 12; 1574, 12; 1589; 1590; 1599. 2. ' Ein new vnd wolgegrundt Rechenbuch,' 1560 ; Frankfort, 1565, 4 (here described); 1569; 1600, 4 (below), and in the seventeenth century. Jacob's arithmetics followed the general plan of the popular books of Riese, and were deservedly well received in the second half of the century. They were commercial textbooks, and although they do not show any mathematical advance they are historically valuable for their applied problems. The title page (fig. 147) is interesting because of the variety of mathematical instruments illustrated. Other works of 1560. Borghi, p. 16, 1484 ; Buteo, p. 292, 1559; Feliciano, p. 149, 1526 ; Psellus, p. 168, 1532 ; Scheubel, p. 246, 1549 ; Tartaglia, p. 278, 1556 ; Gasparo Rizzo, ' Abbaco nvovo molto copioso et artificiosamente ordinato,' Venice, 8 ; Juan Ventallol, * Aritme'tica,' a rare work, s. 1. a., which appeared in Spain about this time. SIMON JACOB. Ed. pr. 1560. Frankfort, 1600. See p. 295. Title. This is practically the same as in the 1565 edition described above. Colophon. ' Getruckt zu Franckf urt am Mayn/ bey // Matthes Beckern/ In Verlegung Chris-//ftian Egenolphs Erben.// Anno 1600.' (F. 360, v.) Description. 4, 15.5 x 19.2 cm., the text being 10.5 XI4.8 cm. 1 1 ff. unnumb. + 349 numb. = 360 ff., 28 11. Frankfort, 1600. See above. JOSEPHUS UNICORNUS. Ed. pr. 1561. Venice, 1561. GIUSEPPE UNICORNO. A Bergomese arithmetician, born in 1523; died in 1610. Title. See Fig. 148. Description. 8, 9.4 x 13.9 cm., the text being 7.4 x 12.7 cm. i f. unnumb. + 78 numb. = 79 ff., 30 11. Venice, 1561. PRINTED BOOKS 299 10SEPHI VNICORNI BERGOMATIS LIBER DE VTILITATE MATHEMATICAL R V. M A JR T I V flU In quo candide Ltftor mulu quidem fcituperiucw* d<t , admiratione digna, cr bmano ufid * neceffirit pafsim reptriet. V E K s T 1 1 s , ApudDominicumde Nicolinis, AnnoDomim, M D L x I- FIG. 148. TITLE PAGE OF UNICORNUS 300 KARA ARITHMETICA Editions. There was no other edition. This work is not an arithmetic, Unicorn's textbook on this subject appearing much later (p. 412, 1598). It is a prolix dissertation on the uses of mathematics, and of arithmetic in particular, with extracts from the ancient and mediaeval writers. Like Agrippa's ' De vanitate scien- tiarum,' it is interesting but profitless. NICOLAUS WERNER. Ed. pr. 1561. Niirnberg, 1561. A Niirnberg Rechenmeister, born c. 1520. Title. See Fig. 149. Colophon. ' Gedriickt zu Niirnberg/ //durch Johann vom Berg/ //vnd Ulrich Newber.' (F. 156, r.) Description. 4, 14.2 X 19.3 cm., the text being 10.1 X 15.5 cm. 156 ff. unnumb., 23-31 11. Niirnberg, 1561. Editions. There was no other edition. This is a purely mercantile arithmetic, and like many of the Italian textbooks it assumes some preliminary knowledge of the fundamental operations. It begins with a treatment of Welsch (Italian) practice, and is composed almost entirely of practical problems of the day, solved by this method. Welsch practice differs from the rule of three only in having, ordinarily, unity for the first term, as in the following example : If i book cost $2, how much will 7 books cost? This method was very popular with all German arithmeticians of the sixteenth century, and survived in the chapter on Practice still to be found in English arith- metics. It appeared also in the early American textbooks. The book gives a very good idea of the business requirements of the second half of the sixteenth century. The value placed upon exchange and * profit and loss ' may be seen in the fact that twenty-six pages are devoted to the former and forty-seven to the latter, while partnership also has twenty-six pages for its share. JOHANNES MONHEMIUS. Ed. pr. 1561. Diisseldorf, 1561. A German teacher of the second half of the sixteenth century. Title. See Fig. 1 50. Description. 8, 9.4 X 15 cm., the text being 6.5 x 12 cm. 22 ff. unnumb., 25-27 11. Diisseldorf, 1561. Editions. There was no other edition. PRINTED BOOKS 301 nxlot fottbetlidj foui'e! beeHfcitbei;0ifcl)ettS.<mb9 mt/twb gcbjtiucbbdangr/mi't fonbetm vwteyl/nib be= igfci f mjff Jncl;criey art pradtce $Kicol(ttim vubfoiiflinenigHiebenju in8 in FIG. 149. TITLE PAGE OF WERNER 3 02 KARA ARITHMETICA This work is almost unknown to students of arithmetic. It seems to be the only edition, although the introduction bears the date 1542. It is a small work and is based somewhat on the mediaeval texts. It gives M E T H COMPVTATORlAE,OMNEM puttnti Arttm trddcm:t41tl4m,<i*mti$ numcroruni,cjtuni qii olimcalculis, nunc numwisfuper <eque diftdn tcslinets (itrfurn <tc dc* orfurtt poff/w, per fie it ur. Auttore lo. Mbnbemioi DVSSELDORPIl Anno t | * i- FIG. 150. TITLE PAGE OF MONHEMIUS the fundamental operations with integers and a brief treatment of the rules of three, partnership, and false. There is also a brief explanation of counter reckoning. PRINTED BOOKS 303 BENEDICTUS HERBESTUS. Ed. pr. 1561. Cracow, 1577. Born at Novomiasti, Poland, in 1531 ; died at Jaroslaw, March 4, 1593. He was a Jesuit priest at Cracow. Title. See Fig. 151. Description. 8, 9.5 X 15.3 cm., the text being 8 X 11.5 cm. 42 ff. unnumb., 27-28 11. Cracow, 1577. Editions. 1561; Cracow, 1564,8; ib., 1566; ib., 1569; ib., 1577, 8 (here described). A Latin book intended for the Church schools, and revised, as the ' Proaemivm auctoris ' states, ' Mariaeburgi in Prufsia, fub finem Anni Domini 1576.' It consists of two parts, the first being devoted to the arithmetical operations, chiefly with counters (and this as late as 1577), and the second to the calendar. Besides the operations, the first part also includes a chapter on progressions (' De Progressione. Capvt VII.'), and one on the rule of three (' De tribvs nvmeris integris. Capvt VIII.'). ANDREAS HELMREICH. Ed. pr. 1561. Eisleben, 1561. A Halle Rechenmeister of the latter half of the sixteenth century. The dedication of the 1561 edition is signed by 'Andreas Helmreich Rechen- meifter vnd Vifierer zu Halle,' and the 1588 edition by ' Andreas Helmreich von Eifzfeldt/ Notarius publicus/ Rechenmeifter vnd Vifierer zu Halle.' Title. See Fig. 152. Colophon, ' Gedruckt zu Eisleben/ bey // Vrban Ganbifch.' (F. in,r.) Description. 4, 14.2 x 19.3 cm., the text being 9.3 x 14.9 cm. in ff. unnumb. + I blank = 112 ff., 27-33 H- Eisleben, 1561. Editions. This is probably the first edition of Helmreich' s arith- metic, the dedicatory epistle being dated 'zu Halle inSachffen den Sontag Oculi/ welcher war der 9. tag des Monats Marcij/ nach Chrifti Ihefu vnfers lieben Herrn vnd Seligmachers Geburt 1561.' Murhard (1, 1 5 5) mentions, however, an edition appearing at Halle in 1546, 4. The book was republished at Leipzig in 1588 (p. 306), and again in 1595, 4 (p. 306), and possibly ib., 1596, 4. The work is an unsuccessful attempt to combine the old and new arithmetics. The author begins with a semi-Boethian treatment of ratios, assuming a knowledge of the fundamental operations with integers, and then introduces a course in mercantile arithmetic. He closes with a considerable amount of work on mensuration, including 3 o 4 KARA ARITHMETICA ARITHM TICA LINEA* RIS, eicj adiun&a FIGVR.ATA, cumquibufdamex COMPVTO necdTarijs; IESV Trejbytero, cum facilitate Superiorum. CVH GRATIA ET PRtoifego S. R. M. In Officina Mattbai Sicktuy J 5 7 7. "UJcf U?voV^(|P FIG. 151. TITLE PAGE OF THE 1577 HERBESTUS PRINTED BOOKS 305 ttortci tuft Spractica/wit jtm (fgcnttff &en/ OeutUc&m md pnterfc tyieblkfen proportsen/ fampt gewlffer rnt> be Denser art /mancferlej? ^ifler* FIG. 152. TITLE PAGE OF THE 1561 HELMREICH 3 o6 KARA ARITHMETICA gauging (Visierrechnung). It is somewhat surprising that such a weari- some treatise ever went through three editions. Other works of 1561. Albert, p. 178, 1534; Borghi, p. 16, 1484; Feliciano, p. 148, 1526; Gemma, p. 204, 1540; Recorde, p. 214, c. 1542 ; Rudolff, p. 152, 1526; Sfortunati, p. 177, 1534 ; Tagliente, p. 115, 151$; Wolphius, p. 154, 1527; A. Citolini, Tipocosmia,' Venice, 8 (a synopsis of science, including arithmetic) ; Ognibene de Castellano, * Opera,' Venice, 4 (principally on geometry, but including some arithmetic ; see also p. 375, 1582) ; Antonio Maria Venusti, ' Com- pendio utilissimo,' Milan, 8 (includes some treatment of exchange). ANDREAS HELMREICH. Ed. pr. 1561. Leipzig, 1588. See p. 303. Title. ' Rechenbiich //Von vortheil vnd behendig-//keit/ nach der Welfchen Practica/ mit // jhren vnterfchiedlichen Propor- tionen.//Grund vnd vrfach der Regel Detri/ //Sampt gewiffer vnd behender art/ mancherley Vi-//fier Ruthen vnd Schnure/ nach dem Quadrat vnd // Cubo, auff alle Ohme vnd Eiche zu machen/ alles mit war-//hafftigen perfpectiuifchen/ Geometri- fchen vnd Arithmetifchen demon-//ftrationen, aufz den Buchern Euclidis fundirt/mit fleifz //zufammen gebracht vnd in Druck aufzgangen///Durch//Andream Helmreich/Rechenmeifter vnd Vifierer//zu Hall in Sachffen.// (Cut showing dimensions of a cask.) Im Jahr 1588.' (P. 3.) Colophon. 'Gedruckt zu Leipzig/ //Durch Abraham Lamberg //Anno//M. D. LXXXVIII.' (P. 336.) Description. 4, 15.5 X 19.2 cm., the text being 10. i X 14.5 cm. 9 pp. blank + 1 6 unnumb. + 317 numb. = 342 pp. , 31 11. Leipzig, 1588.. See p. 303. ANDREAS HELMREICH. Ed. pr. 1561. Leipzig, 1595. See p. 303. Title. 'Rechenbuch/ //Erftlich/Von // Vortheil vnnd Behen- // digkeit/ nach der Welfchen Practica/ mit ih-//ren vnterfchied- lichen Proportionibus, Grund vnd vr-//fach der Regel Detri oder proportionum.//!!. Von zubereitung mancherley Vifier Ruthen/ PRINTED BOOKS 307 // vnnd Schnuren/ damit alle Vahffer vnnd andere Corpora nach dem // Quadrat vnd Cubo auff alle Ohme vnd Eiche-Vifiere wer- den.// III. Wie man kunftlich das Feld vnd Erdreich/auff man- //cherley art/ mit gewiffer Meflruthen vnd Schnuren/ nach eines jeden Landes//oder Stadt gebrauch/ Geometrifcher vnd Jdio- tifcher weifl/ vnd was hier-//innen der Vnterfcheid fey/ recht meffen fol.// HII. De Diftantijs Locorum, das i ft/ Wie man wunderbar-//licher Geographifcher vnd Cofmographifcher/ nach Geometrifcher weife/ durch // Arithmeticam, in der gantzen Welt/zweyer Stedte oder Wonung Diftantiam, oder//wie weit die von einander gelegen/ nach ihren Longitudinibus, vnd Lati- tudinibus, // fol Rechnen vnd finden/ Sampt einer Landtaffel daraus zu //machen vnd zu befchreiben.// V. Vnnd wie man die Funff Horologia communia, oder ge-//meine Sonnen Vhrn/ als Horizontale, Meridionale, Septentrionale, Ori-//entale, vnd Occi- dentale, auff einen Cubum, aufzwendig/ oder fonften an die Wen- //de/ Mawer vnd ebenen/ Vnd das Fundamentum Horologiorum abreiffen. Item/ // der Sonnen vnd des Monds am Himmel ge- fchwinden Lauff rechnen. Vnd auch eine fonderliche gar kunft- //liche Sonnen Vhr/ der Cylinder genandt/ darinne alle Stunde nach der Sonnen fchein de Tages grtmd-//lich zu erfehen vnd zu befinden/ machen/ vnd fampt andern mehr fchonen Kunften vnd Vbun-//gen/ nutzlich gebrauchen fol.//Alles mit warhaff- tigen Demonftrationibus vnd Figuren/ aus den Buchern Euclidis vnd andern fundirt. Durch // Andream Helmreich von Eiflfeldt/ Rechenmeifter vnd // Vifierer zu Halle in Sachfen an der Sahle. //Cum Priuilegio.' (P. i.) Colophon. ' Gedruckt zu Leipzig/durch Zacha-//riam Berwald. // Im lahr/ // M. D. XCV.' (P. 645.) Description. 4, 14.5 X 18.7 cm., the text being 10.9 X 15.4 cm. 17 pp. unnumb. + 3 blank + 627 numb. =647 pp., 26-34 11. Leipzig, 1595. Editions. See p. 303. This edition contains the Rechenbuch of 1561, together with much additional matter as summarized on the title page, viz. books III, IV 3 o8 KARA ARITHMETICA (pp. 306-631). This includes a considerable amount of work on gaug- ing, surveying, cosmography, and the ' Sphere.' The first 304 pages are practically identical with the first edition, except for the addition of numerous woodcuts. GUGLIELMO PAGNINI. Ed. pr. 1562. Lucca, 1562. A Lucca mathematician of the sixteenth century. Title. ' Practica// Mercantile // Moderna.// Di Guglielmo Pa- gini // Lucchefe. // In Lucca per il Bus-//dragho. MDLXII.' (P. 5-) Description. 4, 14.5 x 20.6 cm., the text being 9. 1x15.3 cm. 14 pp. unnumb. + 161 numb. = 175 pp., 19-28 11. Lucca, 1562. Editions. There was no other edition. This is a commercial arithmetic, the only one I have seen that appeared in Lucca in the sixteenth century. The author begins with the operations on denominate numbers, but places division immediately after addition. The galley method of dividing is not given at all, in which Pagnini shows a more advanced spirit than most of his contem- poraries. There is first the ' Modo di partire ditti partiri piccoli,' or short division ; then the ' Modo di partir per ripiegho,' ' Modo di partire per colonna,' and finally the ' Modo di partire, a danda per altro nome partire grande.' Following division is the work in multipli- cation, including ' regoletto,' ' biricuocholo,' 'crocetta,' * per colonna,' and ' per ripiegho.' Subtraction or traction ' (* Modo di trare, o fottrare ') follows multiplication. The author then takes up fractions in the same order. This work in the fundamental operations is followed by chapters on exchange and mercantile problems, the ' Regola del tre,' profit and loss (* Gvadagni e perdite '), partnership (' Compagnie '), barter, interest and discount (' Meriti e sconti simplici e capo d'anno'), and a further treat- ment of exchange. Altogether the arithmetic is one of the most interest- ing of the smaller books of the time issued under Florentine influence. JUAN PEREZ DE MOYA. Ed. pr. 1562. Salamanca, 1562. Born in San Stefano (Santisteban del Puerto), in the Sierra Morena, in the first third of the sixteenth century. He studied at Alcala and Salamanca. He was canon at Granada in the latter half of the century. Title. See Fig. 153. Colophon. ' En Salamanca. //Por Mathias Gaft.// Ano de 1 562.' (p. 765.) PRINTED BOOKS 309 ARITHMETIC A PRACTICA,Y SPECV- latiua delBachiller luan PereideMoya. Agora nucuainentc corregida,y anadidas por el mifmo author muchas cofas,cor* otros dos libros,y vna Tabla muy copio fade las cofas mas notables de todo lo que en efte libro fe contiene. irifffa almuy alloy ntuypodtro* Jofenor don Carlos Principe deEfpananueftro fenor* EN SALAJMJINCA, PorWathiasGafU t y 6 * FIG. 153. TITLE PAGE OF THE 1562 MOYA 3 io RARA ARITHMETICA Description. 8, 9.3 X 14.9 cm., the text being 7 x 12.1 cm. 811 pp. (765 numb.), 29 11. Salamanca, 1562. Editions. This is the first edition of this work that I have been able to find, in spite of the words ' agora nueuamente corre- gida' in the title. It appeared again at Alcala in 1573, and at Madrid in -1598, 8, and there were at least thirteen editions between 1609 and 1706. For the 1703 edition see below. Moya also published three other works : ' Reglas para cotar sin pluma y de reduzir unas monedas castellanas en otras,' in 1563, 4 ; ' Manval de contadores,' Alcala, 1582, 8, and Madrid, 1589, 8; and ' Tratado de matematicas,' Alcala, 1573, fol., containing a section on arithmetic less complete than the ' Arith- metica practica.' This is an elaborate treatise of 765 pages of text, covering the ordinary calculating by algorism, the use of counters, business arith- metic, the elements of algebra (of which word he gives the etymological meaning, adding the rest of the ancient title, ' almucabala '), practical geometry (' Trata algvnas reglas de Geometria pratica neceff arias para el medir de las heredades,' p. 304), and the calendar. Besides all this, Moya gives a large amount of information concerning matters of his- torical interest. For example, he treats the subject of notation very fully, giving the Greek, Roman, Hebrew, digital, and astrological systems, together with a brief mention of other systems. Altogether it is the most noteworthy book on mathematics published in Spain in the sixteenth century. JUAN PEREZ DE MOYA. Ed. pr. 1562. Barcelona, 1703. See p. 308. Title. ' Arithmetica // Practica,// y Especvlativa,// del Ba- chiller // Jvan Perez //de Moya.//Aora nvevamente corregida, //y aftadidas por el mifmo Autor mu-//chas cofas.//Con otros dos Libros,//y vna Tabla muy copiofa de las cofas mas //nota- bles de todo lo que en efte Libro//fe contiene.// Afio 1703. Con Licencia : En Barcelona, en la Imprenta // de Rafael Fi- guero.' (F. i, r.) PRINTED BOOKS 311 Description. 8, 15 x 20 cm., the text being 10.7 x 17.9 cm. 396 pp. (380 numb.), 39-41 11. Barcelona, 1703. Editions. See p. 310. See p. 310. It shows the popularity of this Spanish treatise that this edition should appear 141 years after the first one, and that another edition was published as late as 1761. ANONYMOUS. Ed. pr. 1562. Paris, 1562. Title. See Fig. 154. Description. 8, 10.5 X 15.9 cm., the text being 6.9 X 12.5 cm. 98 pp. numb., 24-30 11. Paris, 1562. Editions. There was no other edition. The work is largely theoretical, but it contains a few applied prob- lems. It was without merit. Other works 0/1562. Benese, p. 182, 1536 ; Budaeus, p. 99, 1514 ; Euclid, p. n, 1482; Gemma, p. 200, 1540; Ghaligai, p. 132, 1521; Lossius, p. 290, 1557; Ramus, p. 330, 1569; Riese, p. 139, 1522; Xylander, p. 356, 1577; Francesco Spinola, ' De Intercalandi ratione corrigenda, & tabellis quadratorum numerorum, a Pythagoreis disposi- torum,' Venice, 8. VICTORINUS STRIGELIUS. Ed. pr. 1563. Leipzig, 1563. STRIGEL. Born at Kaufbeurn, December 26, 1524; died at Heidelberg, June 26, 1569. He was professor of theology at Jena, Leipzig, and Heidel- berg, and, aside from the work here mentioned, wrote entirely on that subject. Title. See Fig. 155. Description. 8, 9. i X 14.9 cm., the text being 6.3 X 1 1.8 cm. 83 ff. unnumb. + i blank = 84 ff., 2127 M- Leipzig, 1563. Editions. There was no other edition. It seems by the pref- ace that the work was written in 1551. See p. 249, 1549. For the date of the printing of this work, 1563, see f. 6, v., of the preface. F. 10, v., gives the year in which it was written, MDLI. This comparatively unknown work presents the subject of arithmetic from the classical standpoint. The author speaks of the dignity of arithmetic (' De dignitate arithmeticae '), and follows this discussion by 3 I2 KARA ARITHMETICA ARITHME- TicA. * A & X S I X 5, Apud Andrcam Wechelum. 25*2. Cum privilegio Regis. FIG. 154. TITLE PAGE OF THE ANONYMOUS 1562 ARITHMETIC PRINTED BOOKS y Arithmcticus LIBELLVS CONTINEKS MO DO T K52T^4T^f, SED etiam demonftrationes pneceptorum > E D I T p S IN F F I C I N Y OE.CL\l X/tfv^. FIG. 155. TITLE PAGE OF STRIGELIUS 314 KARA ARITHMETICA a series of definitions from Euclid, a brief treatment of the operations, the theory of proportion, the operations with fractions, and some of the Greek theory of numbers. One curious feature of the book is the notation of Greek fractions after the Arabic manner. PIERRE SAVONNE. Ed. pr. 1563. Lyons, 1571. A French arithmetician, born at Avignon c. 1525. Titl$. ' L'Arithmetiqve // de Pierre Savonne,// diet Talon, natif d' Auignon // comte" de Veniffe. // En laquelle font conte- nues plufieurs reigles briefues & fubtiles, pour les traffiques de plufieurs pays, mentionnez // en la table dudit liure : avec la dif- ference des poids, aunages // & monnoyes de chacun defdits lieux, alliage de metaux :// neceff aire pourtous Maistres de mon- noyes, Orfeures & // Changeurs, avec le fait & maniement des Changes & Ban-//ques qui fe font iournellement a Lyon, & par les places ac-//coustumees : comme Flandres, Angleterre, Hef- pagne,// Italic, & autres lieux. //A Lyon, par Benoist Rigavd. //M. D. LXXI.' (F. i, r.) Colophon. 'A Lyon, de I'lmprimerie de // Pierre Roufsin.// 1571.' (F. 148, r.) Description. 8, 10.8 X 17 cm., the text being 7.4 x 13.6 cm. 156 ff. (147 numb.), 28-30 11. Lyons, 1571. Editions. Paris, 1563, 4; ib., 1565, 4; Lyons, 1571, 8 (here described) ; ib., 1585; ib., 1588, 8. This is one of the early French commercial arithmetics, well arranged but with no marked peculiarities. It was so popular that an edition appeared as late as 1672. Like all arithmetics appearing in Lyons it devotes much attention to banking and exchange, this city being at that time the commercial center of France, and the seat of one of the great international fairs. Other works 0/1563. Bseda, p. 131, 1521 ; Boissiere, p. 262, 1554 ; Feliciano, p. 148, 1526; Gemma, p. 205, 1540; Moya, p. 310, 1562; Ortega, p. 93, 1512; Peletier, p. 245, 1549; Wolffgang Hobel, ' Ein niitzlich Rechenbiichlein mit viel fchonen Regeln und Fragftiicken,' Niirnberg, with editions s. 1. (Niirnberg?), 1565 and 1577, 8; Simon Schweder, < Rechenbuch von alles kauffmanschaft der Landt auff der Feder und Linien,' Konigsberg, 8. PRINTED BOOKS 315 COSIMO BARTOLI. Ed. pr. 1564. Venice, 1589. A Florentine geometer, born in 1503; died in 1572. He also translated the works of Finaeus (see pp. 160, 164). Title. ' Cosimo Bartoli //gentil' hvomo, et // Accademico Flo- rentine,// Del Modo di Misvrare // le diftantie, le fuperficie, i corpi, le//piante, le prouincie, le profpettiue, & //tutte le altre cofe terrene, che poffo-//no occorrere a gli huomini,// Secondo le vere regole d' Euclide, & de gli altri//piu lodato fcrittori.// (Elab- orate woodcut.) In Venetia, Per Francefco Francefchi Sanefe. 1589.' (F. i,r.) Colophon. ' In Venetia,// Per Francefco Francefchi Sanefe. //M. D. LXXXIX.' (F. 148, r.) Description. 4, 15.5 x 21.6 cm., the text being 11.2 x 15.5 cm. 148 ff. (6 unnumb.), 29 11. Venice, 1589. Editions. Venice, 1564, 4; ib., 1589, 4 (here described), and one edition after 1601. The dedicatory epistle is dated ' il di 10. di Agofto del 1559,' so there may have been an earlier edi- tion than that of 1564. Although the book is on practical mensuration, the ' libro sesto ' (f. 130, r.) is upon square and cube root. The galley method is used, and the common sixteenth-century device of annexing 2 n ciphers in square root and dividing the root by 10" (and similarly for cube root) is em- ployed. Bartoli also gives a table of squares to 662 2 . The chapter on roots is followed by one on the ' Regola delle tre cofe, ouero quattro proportional!.' Works of 1564. Albert, p. 180, 1534; Apianus, p. 155, 1527; Benese, p. 182, 1536; Euclid, p. 240, 1545; Gutierrez de Gualda, p. 167, 1531 ; Herbestus, p. 303, 1561 ; Kobel, p. 1 1 1, 1514 ; Mariani, p. 181, 1535; Medlerus, p. 223, 1543; Riese, p. 139, 1522; Sole, p. 146, 1526; Tagliente, p. 115, 1515; Thierfelder, p. 391, 1587; Ulman, p. 391, 1587 (Thierfelder) ; Yciar, p. 249, 1549 ; Barlaamo, ' Arithmetica demonstratio eorum quse in secundo libro Elementorum (Euclidis) sunt,' Strasburg (see also p. 343, 1572) ; Giovanni Camilla, ' Enthosiasmo,' Venice, 8 (containing a little work on arithmetic) ; Manuel Fernandez Lagasa, ' Libro de quentas,' Salamanca, 4, with a little work on arithmetic ; Petrus Nonius (Nunez), ' Libro de Algebra en Arithmetica y Geometria,' with two editions at Antwerp in 1567, 8 ; the 'Opera' of Nonius appeared at Basel in 1592. 316 KARA ARITHMETICA PIERRE FORCADEL. Eel. pr. 1565. Paris, 1565. See p. 284. Title. See Fig. 156. Description. 4, 14.8 X 20 cm., the text being 9.9 x 15.9 cm. 192 pp. numb., 32-39 11. Paris, 1565. Editions. See p. 284. This is the third work on arithmetic published by Forcadel. The dedicatory epistle is dated 'De Paris, ce 19. de May. 1565.' There was only one edition, and this is not often found in dealers' catalogues. In this book Forcadel does not, as in his other works, take up the subject of counter reckoning, but he gives a very satisfactory treatment of the fundamental operations, the rule of three, partnership, alligation, and the common applications of the day. It is not, however, as practical as the arithmetics of Savonne and Trenchant. ANTICH ROCHA de Gerona. Ed. pr. 1565. Barcelona, 1565. A Spanish arithmetician of the second half of the sixteenth century, born in Gerona, lecturer at Barcelona. Title. See Fig. 157. Description. 8, 9.7 X 14.7 cm., the text being 7.5 x 12 cm. 314 ff. (267 numb.), 28 11. The dedication by Rocha bears the date * Hechaen Barcelona, a. 23. de Nouiembre. 1564.' Barce- lona, 1565. Editions. There was no other edition. This rare compilation is based upon several Italian books, and the writer claims to have consulted a large number of authors. His list includes the names of Feliciano, Faber Stapulensis, Buteo, Scheubel, Finaeus, Ramus, Gemma, and various other writers of the time. Rocha has, however, omitted some of the best textbook-makers who preceded him. The book is a fairly complete elementary treatise, the writer hav- ing taken up the fundamental operations with various kinds of numbers, and treated each rule in a rather scientific way. Although a consider- able number of practical problems relating to mercantile affairs appear in the last half of the work, the style of the writer is so prolix that the book could never have been well received by the mercantile classes. Bound with this work is another with the following title : ' Compendio ARITHMETIQyE ENTIERE ET ABREGEE DE PIERRE FORCADEL, LECTEVR D V ROY ES JUATHtMATI QJf S. A PARIS. Chez Charles Pcrier, rue S.Iean de Bcauuais, au Bcllerophon. i j tf y. FIG. 156. TITLE PAGE OF THE 1565 FORCADEL 3 i8 KARA ARITHMETICA y breue // instruction por tener Libros de Cuen//ta, Deudas, y de Mer- caduria : muy prouechofo // para Mercaderes, y toda gente de negocio. //traduzido de Frances en Caftellano. En Barcelona.// En cafe de ?3ARITHMETI- ca por Antich Rocha dc Gerona compucfta , y de varies Au^ores recopilada:pr6uechof;i para todos eftados de gentes. Vaanadido vn Compendio,para ttner y re. gir los libros de Cuenta-.traduzido de lea* gua Francefa en Romance Caftellano. EN BARCELONA En cafa dc Claudio Bornat, a la Aguila fuertfc 1S6 ^ Con priuilcgio por dkz anos. FIG. 157. TITLE PAGE OF ROCHA Claudio Bornat//al Aguila fuerte. i565.//Con priuilegio por diez anos.' (F. 286.) This relates, as the title suggests, entirely to bookkeep- ing, and is one of the earliest treatises upon the subject in the Spanish PRINTED BOOKS 319 language. In the fourth book of this second part is a treatise on alge- bra, one of the first to appear in Spain. Rocha speaks of an Aritmetica by Juan Ventallol, of which we do not know the date, but which must have appeared before 1565 (p. 298). ERHART HELM. Ed. pr. 1565. Frankfort, 1592. A Frankfort arithmetician and gauger of the middle of the sixteenth century. Title. '1592. Erhart Helm/ // Mathematicus // zu Franck- furt am Mayn/ // von // Geometrifcher Abmeffung der Erden.// Item :// Kunftliche Vifier vnnd // Wechfelruthen/ aufl dem Qua- drat/ //durch die Arithmeticam vnnd Geometri-//am/ gerecht zumachen/ fampt einer luftigen behen-//digkeit in Weinrech- nung/ // alles durch obgedachten // Authorem befchrieben/ vnd jetzt von neuwem // widerumb fleiffig erfehen vnd cor-//rigiert.// Franckf. bey Chrift. Egen. Erben. 1592.' (F.i, r.) Description. 8, 9.5 X 15.4 cm., the text being 6.5 X 12.2 cm. 23 ff., 26-27 11. Bound with Adam Riese's arithmetic of 1592 (p. 143). Frankfort, 1592.- Editions. The work appears in the 1565 edition of Adam Riese (p. 142), but not with a separate title page. I have seen no separate edition before this one of 1592. This is a brief treatise on mensuration, and in particular on gauging, but it contains some explanation of arithmetical processes, including the extraction of roots, and a table of square roots to the equivalent of three decimal places. Other works 0/1565. Agrippa, p. 167, 1531 ; Apianus, p. 62, 1496 (Jordanus) ; Belli, p. 343, 1573 ; Cusa, p. 43, c. 1490 ; Delfino, p. 275, 1556 ; Fischer (Piscator), p. 247, 1549 ; Gemma, p. 200, 1540 ; Hobel, p. 314, 1563 ; Jacob, p. 295, 1560 ; Jordanus, p. 62, 1496 ; Menher, p. 249, 1550 ; Padovanius, p. 389, 1587 ; Riese, p. 142, 1522 ; Savonne, p. 314, 1563 ; Priscian, Rhemnius, Fanius, Baeda, Metianus, 'Liber de nummis, ponderibus, mensuris, numeris, eorumque notis, et de vetere computandi ratione, ab Elia Vineto emendati,' Paris, 8. (Priscian's ' De figuris et nominibus numerorum ' had already appeared in his works published at Venice in 1470, fol., with later editions in 1488, 1492, 1495, J 496, 1519, 1525. Such books, of which Mr. Plimpton has several, are not generally included in this list.) 320 KARA ARITHMETICA HIERONYMUS MUNYOS. Ed. pr. 1566. Valencia, 1566. Born at Valencia; died in 1584. He was professor of mathematics and Hebrew at Ancona, and later at Valencia. He also wrote an astronomical work. The family name is more strictly Munoz. Title. See Fig. 158. Description. 4, 15 X 20.4 cm., the text being 9.3 x 14.6 cm. 4 ff. unnumb. + 77 numb. + i blank = 82 ff., 27-31 11. Valencia, 1566. Editions. There was no other edition. The ' epistola' is dated ' Calendis Aprilis, anni M. D. Lxvj.' The work consists of three books. The first treats of the funda- mental operations, including proportion and some work in the Greek theory of numbers. The second book treats of fractions, including sexagesimals, these being needed by the astronomers for whom Munyos was writing. The third book relates to ratio and proportion. Altogether the work is too theoretical to have much influence upon the develop- ment of arithmetic. IAN TRENCHANT. Ed. pr. 1566. Lyons, 1578. A Lyons arithmetician, born c. 1525. Title. ' L' Arithme-//tiqve de Ian // Trenchant, //Departie en trois //liures.//Enfemble vn petit difcours des Changes.// Avec // L'art de calculer aux Getons.// Reueiie & augmentee pour la quatrieme edition, //de plufieurs regies & articles,// par 1'Autheur. // A Lyon,// par Michel love,// et lean Pillehotte.// a 1'enfeigne du lefus.// 1578. // Auec priuilege du Roy.' (P. I.) Description. 8, 10.4 x 15.5 cm., the text being 7 x 13.7 cm. 375 pp. numb. + 5 unnumb. = 380 pp., 3031 11. Lyons, 1578. Editions. Lyons, 1566; ib.,i57i; ib., 15 78, 8 (here described). The dedication of the 1578 edition bears the date * De Lyon ce 9. de luillet 1571,' and this is often given as the date of the first edition. Although this is described on the title page as the fourth edition, I know of only two earlier. Cantor says that not less than six editions were published at Lyons from 1588 to 1602, and I have seen mentioned editions of 1608, 1610, 1632, and 1643. Trenchant was one of the best of the sixteenth-century textbook- makers of commercial arithmetic in France. His work is divided into PRINTED BOOKS INSTITVTIONES ARITHMETICAE AD JPER- CIPIENDAM ASTROLOGIAM ET Mathem aclcas facultates neceflariac. mo Jxtunyos tJalt ntin 9 Hebrtit* lip? gueftriter at^ Mathernatumi mnafio 1J tlentino pullico t- r 'f*8 r -<. V A i E N T I AE. Ex typograpbia Joannis Mey. Aono 1 FIG. 158. TITLE PAGE OF MUNYOS 322 KARA ARITHMETICA three books, the first dealing with the fundamental operations with integers and fractions, and containing a considerable number of applied problems. The second book treats of the rule of three in its various forms together with such applications as barter, partnership, commis- sion, and alloys. The third book treats of the properties of numbers, including figurate numbers, roots, and progressions, and has some work on discount, together with a few recreations. In the 1578 edition the third book is followed by a chapter on exchange, and an explanation of the method of calculating with counters. GEORGIO LAPAZZAIA. Ed. pr. 1566. Naples, 1569. LAPEZAJA, LAPIZAYA, LAPAZAIA. Probably a resident of Naples, but born at Monopoli, a town in the province of Bari, on the Adriatic. He was a priest, and wrote only the work here described. Title. See Fig. 159. Colophon. ' In Napoli // Apreffo Mattio Cancer. M.D.LXIX. //Con Priuilegio per anni diece.//Marius Carrafa Archiepifcopus Neapolitanus.' (P. 262.) Description. 4, 14.1 x 21 cm., the text being 10.8 X 16.9 cm. 13 pp. unnumb. + 250 numb. = 263 pp., 29 11. Naples, 1569. Editions. Naples, 1566, Latin edition with the title 'Defami- liarite arithmeticae et geometriae,' this date appearing also in the privilege, 'Datum Neapoli die vltimo lulii M.D.LXVI;' ib.,1566, Italian edition; ib., 1569, 4 (here described); 1575; Naples, 1590, 4 (p. 324). There were several editions after 1600, one appearing as late as 1784. For the 1601 edition see p. 324. The book has nothing to commend it except its popular style. Lapazzaia begins with the fundamental operations with integers, and then treats of ratio, fractions, and progressions. He considers also the rule of three, the rule of five, interest, exchange, partnership, alligation, rule of false, and the extraction of roots. The last part of the work is on mensuration. Other works of 1566. Belli, p. 343, 1573 ; Fischer (Piscator), p. 247, 1549 ; Gemma, p. 200, 1540 ; Herbestus, p. 303, 1561 ; Monzd, p. 292, J 559 5 Georg Meyer, * Rechenbiichlein defs Silberkauffs und gemachter Arbeit,' Augsburg, 16; Mathaus Nessen (Nesse), 'Zwei neue Rechen- biicher,' Breslau, 8 ; Johannes de Segura, ' Mathematicae quaedam selectae propositions, ' Alcald, 4, and ' Compendium Arithmeticae et Geographiae partis,' Alcala, 4. PRINTED BOOKS 323 DARITMETTCA E GFO METRIA DELL'ABBATE GEOR, GlO IAPAZZAIA MONOPOLITANO. FIG. 159. TITLE PAGE OF THE 1569 LAPAZZAIA 3 2 4 KARA ARITHMETICA GEORGIO LAPAZZAIA. Ed. pr. 1566. Naples, 1590. See p. 322. Title. ' Opera // terza // de Aritmetica // et Geometria. // Dell'abbate Georgio Lapazaia//da Monopoli.//Intitolata il Ra- maglietto. // In Napoli,//Apreffo gli Eredi di Mattio Cancer.// M. D. LXXXX.' (P. i.) Colophon. 'In Napoli // Apreffo gl' Eredi di Mattio Cancer.// M. D. LXXXX.' (P. 176.) Description. 4, 13.9 x 18.7 cm., the text being 10.8 X 16.5 cm. i p. unnumb. -f- I blank+ 174 numb. = 176 pp., 29 11. Naples, 1590. Editions. See p. 322. Although bearing a different title, this is merely a revision of the 1566 work, with a slight variation in the problems. The preface is dated 1569, when the second edition of the Italian version appeared, but it is not found in that edition. The title * Opera terza' means simply the second revision, or the third writing of the book. Lapazzaia's work shows the increasing attention given by the Church schools to the business needs of the people. GEORGIO LAPAZZAIA. Ed. pr. 1566. Naples, 1601. See p. 322. Title. ' Libro // d Aritmetica // e Geometria, // dell' Abbate Giorgio Lapazzaia // Canonico Monopolitano, e Protonotario Apoftolico.// Nouamente in queft'vltima imprefsione efpurgato da molti errori, & arric-//chite d'vna Prattica d'Abbaco, non meno vtilifsima, che neceffaria.// Al Signer Diego d'Aldana,// Prefidente della Regia Camera della Summ. per Sua Maiefta.// (Large woodcut, coat of arms.) In Napoli, //Apreffo Tarquinio Longo. MDCI.' (P. i.) Colophon. 'In Napoli, //Apreffo Tarquinio Longo. MDCI.' (P. 215.) Description. 4, 14.9 x 20 cm., the text being 10.7 x 15.8 cm. 216 pp. (3 blank, 8 unnumb.), 30-38 11. Naples, 1601. See p. 322. PRINTED BOOKS 325 NICOLAUS PETRL Ed. pr. 1567. Amsterdam, 1635. Born at Deventer. He taught at Amsterdam from 1567 to 1588. He also wrote on algebra and astronomy. Title. 'Practicque //Om te leeren//Reeckenen/ Cypheren // ende Boeckhouwen/ met die regel Cofs/ // ende Geometric/ feer profijtelijcken voor alien // koop-luyden. Van nieus gecorrigeert //ende vermeerdert/ //Deur Nicolaum Petri Daventrienfem.// L'homme propofe, Et dieu dispofe. A 1603. // (Woodcut of author.) t'Amstelredam,//Voor Hendrick Laurentfz. Boeckver- cooper op het // water int Schrijf-boeck, Anno 1635.' (F. i, r.) Description. 8, 10.3 x 16.3 cm., the text being 8.9 x 14.4 cm. 6 ff. unnumb. + 281 numb. = 287 ff., 23-32 11. Bound with this is ' i596.//Iournael-Boeck//gheteeckent met die // Letter // Anno M. DC. XXXV.,' 10 ff. Amsterdam, 1635. Editions. From the preface it appears that the work was first published in 1567. The other sixteenth-century editions were Amsterdam, 1576, 8; 1583; 1591; and Alkmaar, 1596, 8. The arithmetic is of the ordinary Dutch type. It uses only the galley form of division, and its chief value to the student of history lies in its business problems. Other works 0/1567. Agrippa, p. 167, 1531 ; Borghi, p. 16, 1484 ; Cataneo, p. 244, 1546 ; Gemma, p. 205, 1540 ; Mariani, p. 181, 1567 ; Nonius (Nunez), p. 315, 1564; Ramus, p. 263, 1555; Tagliente, p. 115, 1515 ; Conradus Dasypodius, ' Logistica,' Strasburg, 8; Guill. de la Toissoniere, ' Compost arithmetical,' Lyons. STEFANO GHEBELINO. Ed. pr. 1568. Brescia, 1568. An arithmetician of Brescia, of the second half of the sixteenth century. Title. See Fig. 160. Colophon. ' In Brescia,// Apresso Vincenzo // di Sabbio.// M. D. LXVIII.' (F. 34, r.) Description. 4, 13.5 x 19.5 cm., the text being 10. i x 16.5 cm. ii ff. unnumb. + 33 numb. =44 ff., 37 11. Brescia, 1568- Editions. There was no other edition. The dedicatory epistle shows that this work was written at Brescia in October, 1568. It is composed almost entirely of tables for the use 326 KARA ARITHMETICA IEKZ T A V O L E BREVISSIME A R 1 T I! M E- TICHE, Cow lequdi ciafcuno ageuolipim&- mente potr.i /sr % in un tratto. quafi c^ni forte de contij Corapofte per Stefano Ghebelino Btefciano, & della Scol.i di M. Hieronimo Catanco Nouarefe M Jtematico raro. FIG. 1 60. TITLE PAGE OF GHEBELINO PRINTED BOOKS 327 of merchants and bankers in the north of Italy. There is some explan- atory matter in the beginning relating to the arithmetic of exchange, but the work can hardly be called a school textbook. Riccardi speaks of it as * uno dei primi esempi di tavole di conti fatti.' HUMPHREY BAKER. Ed. pr. 1568. London, 1580. Bom at London ; died after 1587. Title. 'The Well fpring of //Sciences.// Which teacheth the perfect // worke and practife of Arith-//rneticke, both in whole Num-//bers and Fractions : fet // f orthe by // Humf rey Baker // Londoner, 1562. //And nowe once agayne perufed //augmented and amended in all //the three partes, by the sayde //Aucthour: where unto he // hath alfo added certein // tables of the agree-// ment of meafures // and waightes // of diuers places in Europe, // the one with the other, as // by the table following // it may appeare.// 1580.' (F. i, r.) Description. 8, 9.9 x 14 cm., the text being 5.6 x 11.2 cm. 227 ff. (28 unnumb., last folio missing), 24-26 11. London, 1580. Editions. London, 1568, 8 (written in 1562, which explains the date on the title page); ib., 1574, 8; ib., 1580, 8 (here described); ib., 1583, 8; ib., 1591, 8. There were several editions after 1601 (see pp. 328, 329). For a long time Baker's arithmetic was the only English rival to Recorde's ' Ground of Artes ' (see p. 213), and it was in many respects better than that popular work. This edition is more complete than that of 1568, the book having, as the author states, been rewritten. In 1 The Prologue to the gentle Rerder ' he says : ' Hauing fometime now twelue yeres fithence (gentle Reader) publifhed in print one Englifhe boke of Arithmetick, conteyning as I suppose, fundry necfsarie and profitable documentes for fuch as are willing to attayne any knowlege therein. I have bene often fmce that time, and of very late alfo, requefted by fundry of my friendes to perufe the fame worke, and as I fhold nowe iudge it expedient, to adde fomething more therevnto, and to amplifye the fame.' He complains of the criticism of foreigners that English arithmetic is not as advanced as that on the continent : For when I perceyued the importunitie of certayne ftraungers not borne within this lande, at this prefent, and of late dayes fo farre proceeding, that they aduaunced and extolied them felues in open talke and wri tinges, 3 28 KARA ARITHMETICA that they had attayned fuch knowledge and perfection in Arithmetike, as no englifh man the like : Truly me thought that the fame reporte not only tended to the (disprayfe) difpraife our Countreymen in gen- eral : But touched efpetially fome others & me, that had trauailed & written publiquely in the fame facultie. For vnto this fame effecte they haue of late paynted the corners and poftes in euery place within this citie with their peeuifhe billes, making promife and bearinge men in hande that they coulde teache the fumme of that Science in breefe Methode and compendious rules fuch as before their arriuall hath not bene taughte within this Realme.' These words, and others in the same strain, give an interesting picture of English arithmetic in 1580, and of the work of the teacher at that time. The criticism was a just one, for the Dutch, French, Germans, Spanish, and Italians were much ahead of the English at that period in the matter of arithmetic. Baker follows the continental models, giving the usual operations and the applications to * Marchandife,' ' Felowfhip,' barter, alligation, false position, and the like. He closes his text with ' Quftions of Paftime.' He still uses duplation, generally uses the form * substraction ' (following the Dutch books of the time), and makes relatively little of ' Deuifion,' which he treats by the usual galley method, but he succeeds in producing a fairly practical mercantile book. Other works 0/1568. Anianus, p. 32, 1488 ; Delfino, p. 275, 1556; Gemma, p. 200, 1540; Giilfferich, p. 257, 1552; Lonicerus, p. 253, 1551 ; Lossius, p. 290, 1557; Riese, p. 139, 1522; Sfortunati, p. 174, 1534; Mauritius Steinmetz, 'Aritmeticae praecepta in quaeftiones redacta,' Leipzig, sm. 8. HUMPHREY BAKER. Ed. pr. 1568. London, 1659. See p. 327. Title. ' The // Wei-spring // of // Sciences : // teaching The perfect Work and Practice of // Arithmetick,// both in Numbers and Fractions.// Set forth by// Humphrey Baker // Londoner.// And now again Perufed, Augmented, and //Amended in all three Parts, by // the faid Authour.// Whereunto are added certain Tables of //the agreement of Meafures & Weights //of divers places in Europe, //the one with the other, as by the Table appear- eth.// London,// Printed for A. Kemb, at St. Margarets Hill in //Southwark, to bee fold by Tho. Brewfter,// at the three Bibles in Pauls Church-yard : 1659.' (P. i.) PRINTED BOOKS 329 Description. 8, 8.4 x 13.6 cm., the text being 6.3 x 1 1.2 cm. 366 pp. (54 unnumb.), 26-27 H- London, 1659. Editions. See p. 327. This is not materially different from the 1580 edition (p. 327). The publisher says that ' The friendly Reader may pleafe to take notice that in this Impreffion of 1659, the whole Book hath been revifed, every Queftion therein examined, the Faults that were committed in former Impreffions, Corrected, the whole reftored to its first integrity.' In spite of this statement, there is little improvement in the book either in methods of operating or in symbolism. It is interesting to read that ' The Fractions [in the tables at the end of the book] which before were in the common way (and fo the figure being fmall in many not difcerned) are put into the decimal parts, and fo the fame with the integral, but farre more true than the Common Fractions can exprefs it in one figure, and if in the common it be expreft in many, (as it must be, if true) then the decimal is far more eafie, becaufe the Denominator is one and the fame to all, whereas the other is differing.' In fact, very slight knowledge of decimals is shown, and when they are employed the bar is generally used instead of the point. HUMPHREY BAKER. Ed. pr. 1568. London, 1687. See p. 327. Title. <Licenfed,//Feb. 28, i68f,//Rob. Midgley.' (P. 4, first page of print.) ' Baker's // Arithmetick :// Teaching // The perfect Work and Practice of //Arithmetick both in //Whole Numbers & Fractions.// Whereunto are Added // Many Rules and Tables of // Intereft, Rebate, and Purchafes, & c.// Also// The Art of Decimal Fractions, //intermixed with Common Frac- tions, for the // better Underftanding thereof.// Newly Corrected and Contracted, and //made more plain and eafie //By Henry Phillippes.// London.// Printed by J. Richardfon for William Thackery at the //Angel in Duck-Lane, and Matthew Wotton at the Three // Daggers in Fleet ftreet, and George Conyers at the // Ring without Ludgate, 1687.' (P. 5.) Description. 12, 8.3 X 14.3 cm., the text being 6.8 X 12.8 cm. 8 pp. blank + 10 unnumb. + 228 numb. = 246 pp., 32-39 11. Lon- don, 1687. 330 KARA ARITHMETICA Editions. See p. 327. Phillippes, the editor, pays a deserved tribute to Baker in his letter ' To the Reader.' He begins as follows : ' This little Book, as it was one of the firft, fo it is one of the beft of this Subject, and hath had as good Acceptance, as any other ; which may appear by the often Impref- fions of it. Indeed as long as the Author lived, he was careful to be ftill adding and correcting it : and though he be dead, 'yet his Book is thought worthy to live, and not only to live, but to flourifh.' It certainly speaks well of the book that this edition should have been published 121 years after the first one appeared. The treatment of decimal frac- tions is very satisfactory, and, of course, is not found in the original edition. These fractions, the necessity for which became apparent in the sixteenth century, were first scientifically treated at any length in a work by Stevin, published in 1585. (See p. 386.) PETRUS RAMUS. Ed. pr. 1569. Basel, 1569. See p. 263. Title. ' P. Rami Arith-//meticae libri // dvo : Geometriae // septem et viginti.// (Woodcut.) Basiliae, per Evsebivm // Epif- copium, & Nicolai fratris haeredes.// Anno M. D. LXIX.' (P. i.) Description. 4, 17.5 X 23.3 cm., the text being 12.3 x 18.2 cm. 198 pp. (2 blank, 6 unnumb.), 42 11. Basel, 1569. Editions. Basel, 1569,4 (here described); Paris, 1577, 8 (p. 331) ; Basel, 1580, 4 (p. 331) ; Paris (Stadius edition), 1581, 12; Frankfort (Schonerus edition), 1586, 8 (p. 331) ; ib., 1591, 8 (Stegerua edition) ; ib., 1 592, 8 (Schonerus) ; ib., 1 596, 8 (the Snellius and Schonerus * Explications, ' p. 333); ib., 1599, fol. (Schonerus, p. 333) ; Lemgo, 1 599, 4. There was also published at Paris in 1562, and in two editions the same year, an Arith- metica,' in two books, without the author's name, attributed to Ramus, but I do not know whether it is the same as this work. There was also an English edition of ' The Art of Arithmeticke in whole numbers and fractions ... by P. Ramus . . . translated . . . by William Kempe,' London, 1592, 8. This is a better book than the ' Libri Tres' of 1555. Although it is too theoretical to have met the commercial needs, it is a nearer approach to a practical work than its predecessor. PRINTED BOOKS 331 PETRUS RAMUS. Ed. pr. 1569. Paris, 1577. See p. 263. Title. 'Petri Rami // Professoris Regii,// Arithmeticce //li- bri dvo.// Parisiis,// Apud Dionyfmm Vallenfem, fub // Pegafo, in vico Bellouaco.// 1577.' (F. I, r.) Description. 8, 10.3 x 16 cm., the text being 7.1 x 13.1 cm. 97 ff. (2 blank, I unnumb.) with chart; 32 11. Paris, 1577. See p. 330. PETRUS RAMUS. Ed. pr. 1569. Basel, 1580. See p. 263. Title. ' P. Rami // Arithmeticae // libro dvo // Geometriae // septem et viginti.// (Woodcut.) Basileae, per Evsebivm // Epifco- pium, & Nicolai fratris haeredes.// M D LXXX.' (P. i.) Colophon. ' Basileae, per Evsebivm Episco-//pium, & Nicolai fratris haeredes. Anno//M. D. LXXX.' (P. 192.) Description. 4, 15.9 x 21.2 cm., the text being i i.i x 17 cm. 200 pp. (9 unnumb.), 28-35 N- Basel, 1580. See p. 330. PETRUS RAMUS and LAZARUS SCHONERUS. Ed. pr. 1569. Frankfort, 1586. See p. 263. Title. See Fig. 161. Description. 8, 10.5 x 17.3 cm., the text being 7 x 13.1 cm. 16 pp. unnumb. + 406 numb. = 422 pp., 2632 11. Frankfort, 1586. Bound with this is * P. Rami, Regii // Eloquentiae et // Philo- sophiae Pro-//fefforis, liber de moribus //veterum Gallorum,// ad // Carolum Lotharingum // Cardinalem.// Parisiis,// Apud An- dream Wechelum.// 15 62. //Cum privilegio Regis.' Editions. See p. 330. The commentary of Schonerus also appeared with that of Snellius in 1596 (p. 333), and without the latter in 1599 (p. 333). There were also editions by Steger pub- lished at Leipzig in 1591 and at Frankfort in 1592. 332 KARA ARITHMETICS PETRI KAMI ARITHMETICES LI- BRI DVO, ET ALGEBRA totidem:aLAZARO SCHONERO cmendati &explicati. Eiufdem SCHONERI l&riduo: alt<r,De alter, De Logiftic* FRANCOFURDI Apud hcredes Andreas Wecheli, MDLXXXVr. (um S.CtpsJfyCaicftatuprwilegio adfcxcnniitm . FIG. 1 61. TITLE PAGE OF THE 1586 RAMUS PRINTED BOOKS 333 The first of these works is one of several commentaries on the theo- retical arithmetic of Ramus (p. 263). It is more practical than that of Snellius (mentioned below), giving the various operations and making an attempt at introducing some commercial problems. In the first part are included two works by Schonerus, ' De nume- ris figuratis Lazari Schoneri liber ' and ' Lazari Schoneri De logistica sexagenaria liber.' The former is, as the title suggests, a treatise on the Greek theory of numbers, and the second is on the sexagesimal fractions used by the astronomers. Schonerus writes his sexagesimals Ilae lae o I II , 2 20 40 thus: for 3'6o 2 + 39'6o + 40 + + . 3. 39. 40. 20. 40., 60 6o 2 This is one of the early approaches to our symbols , ', ". PETRUS RAMUS and RUDOLPHUS SNELLIUS. Ed. pr. 1569. Frankfort, 1596. See p. 263. SNELLIUS, born at Oudewater, October 8, 1546; died at Leyden, March 2, 1613. Title. See Fig. 162. Description. 8, 9.6 X 16 cm., the text being 6.6 x 12.8 cm. 3 pp. unnumb. + 154 numb. = 157 pp., 29-30 11. Frankfort, 1596. Bound with this are ' Rvdolphi // Snellii in //P. Rami Geome-//triam Pr3ele-//ctiones,' and ' Rudolphi // Snellii in Sphaeram Cor-//nelii Valerii //praelectiones,' both of 1596. Editions. See p. 330. The first edition of Snell's commentary. Like the arithmetic of Ramus, this work is theoretical rather than practical. Only the prominence of Ramus could have justified such efforts as these of Snellius, Salignacus, and Urstisius. PETRUS RAMUS. Ed. pr. 1569. Frankfort, 1599. See p. 263. Title. ' Petri Rami // Arithmeticae // libri dvo : Geometriae // septem et viginti.// A Lazaro Schonero recogniti & aucti.// Fran- cofvrti,// Apud Andreae Wecheli heredes,// Claudium Marnium, & loannem Aubrium.//M. D. XCIX.' (P. 3.) Description. Fol., 17 X 22.5 cm., the text being II.QX 17.6 cm. 244 pp. in the arithmetic, 184 pp. in the geometry, 39 11. Frankfort, 1599. Bound with this is the geometry of Ramus. 334 KARA ARITHMETICA IN P. KAMI ARITH- METIC A M RVDOLPHI S.N E L L I I Explicationes le&iflimae : LAZARI SCHO^ERIy BERT^ff. S Aligned , & Chriftiani Vrfttpi , co m- went cuftetat*. FRANC OFVRTI Ex Officin a Ty pograph ica loan nis Saur i i, impenfis haeredum Pecri Pitched. M. D. xcyi. FIG. 162. TITLE PAGE OF THE 1596 RAMUS AND SNELLIUS PRINTED BOOKS 335 PETRUS RAMUS. Ed. pr. 1569. Basel, 1569. See p. 263. Title. See Fig. 163. Colophon. ' Basileae, per Evsebivm Episco-//pium, & Nicolai fratris haeredes. Anno Salutis humanse//M. D. LXIX.' (P. 190 of the geometry, bound with the above, or 5 34 of the entire book.) Description. 4, 17.5 X 23.3 cm., the text being 12.3 x 18.2 cm. 1 6 pp. unnumb. + 320 numb. = 336 pp. of the above (not including the rest of the work), 42 11. Basel, 1569. Editions. Basel, 1569, 4 (here described); ib., 1578 (Schone- rus edition); Frankfort (also Schonerus edition), 1599, 4. The first three books also appeared at Paris in 1567, 8, under the title ' Praemium Mathematicarum.' An extensive and tiresome treatise on the philosophy of elementary mathematics in general. THOMAS DE MERCADO. Ed. pr. 1569. Salamanca, 1569. A Spanish priest of the middle of the sixteenth century. Title. See Fig. 164. Description. 8, 13.5 X 19.4 cm., the text being 11.5 x i6cm. 277 ff. (29 unnumb.), 32 11. Salamanca, 1569. Editions. There was no other edition. This is not a textbook on arithmetic, but a treatise on the applica- tions of the subject to mercantile affairs. It is so prolix and theoret- ical that it was never republished. That it is the first edition appears from ' La Tassa ' and from the dedication to the king, the former being dated October 6, 1569, and the latter May 6 of the same year. The license is, however, dated August 13, 1568, and one of the decrees May 9, 1568. Although the work professes to be of a mercantile char- acter, it is too ponderous in style for the purpose for which it was intended. It is interesting historically because in several chapters the author has considered the development of arithmetic and of mercantile customs. It is also interesting because of its reference to the recently awakened commerce. For example, chapter 13 has the title ' De los tratos de Indias, y tratantes en ellos.' Chapter 16 is also suggestive of the methods of trade of the period, the title being 'De los baratas y 336 RARA ARITHMETICA J, J m ^ KAMI SCHOJL_i LARVM MA THEMATIC TRIGINTA BASILEAE, PER EVSEBIVM EPISCOPIVM, CT RUoUi Fr4frw heredtt. ANNO M. D. LXIX W- J~ PIG. 163. TITLE PAGE OF THE Libri vnvs et triginta OF RAMUS TRATOSYCON- TRATOS DE MERCADERES y tratantes difcididos y determinados,por elPadre Prefentado Fray Thomas de Mercado,de la orden de los Predicadores. Conlicenciay priuilegio real. EN SALAMANCA Por <&4athia$ (jjaft. $Ano I 5 6 9. Efla taflado en cinco realcs. FIG. 164. TITLE PAGE OF MERCADO 33 8 KARA ARITHMETICA de la nauegacion de las Indias.' The treatment of interest and exchange is more extensive than usual, the former having been by no means popular in Spain. The last part of the book is devoted entirely to legal questions. Other works 0/1569. Belli, p. 343, 1573; Camerarius, p. 263, 1554 ; Gemma, p. 200, 1540 ; Herbestus, p. 303, 1561 ; Jacob, p. 298, 1560 ; Lapazzaia, p. 322, 1566 ; Lossius, p. 290, 1557 ; Mariani, p. 181, 1535; Monzo, p. 292, 1559; Urstisius, p. 361, 1579; Jacob Frey, * Exempelbiichlein allerley Kaufmannshandel,' Niirnberg (there was also an Augsburg edition of 1 603, 1 6) ; Adriaen van der Gucht, Cyferbouck,' Bruges, 4 ; James Peele, The pathewaye to perfectnes in th' accomptes of debitour and creditour,' London, fol. (second edition). HIERONYMUS CARDANUS. Ed. pr. 1570. Basel, 1570. See p. 193. Title. See Fig. 165. Colophon. * Basileae,// ex officina Henricpetrina, Anno // Salv- tis M. D. LXX. Mense // Martio.' (P. 1 1 1 of the third part.) Description. Fol., 20.4 X 30 cm., the text being 13.2 x 24 cm. 291 pp. (4 blank, 16 unnumb.) in this book. Bound with this is the 'Ars Magna' (second edition), 163 pp. numb., and the * De aliza regvla liber, hoc est, algebraicse logifticae fuse . . .,' 120 pp. (in numb.), 41 11. Basel, 1570. Editions. There was no other separate edition of the * Opus Novum.' This work is particularly interesting in its application to physical problems, these being well illustrated. The only reason for including it in a list of arithmetics is that it contains some work on proportion less geometric than that given in Euclid. See also p. 193, 1539. Other works of 1570. Belli, p. 343, 1573; Boethius, p. 27, 1488; Feliciano, p. 148, 1526; Forcadel, p. 284, 1556-57; Gemma, p. 200, 1540; Glareanus, p. 192, 1539; Lonicerus, p. 253, 1551 ; Recorde, p. 214, c. 1542 ; Riese, p. 139, 1522 ; Tagliente, p. 115, 1515 ; Anony- mous, ' Briefue arithmetique fort facile a comprendre ' ; Johann Weber, ' Gerechnet Rechenbiichlein auf Erfurtischen Wein- und Tranks-Kauff,' Erfurt, with a second edition in 1583. Works 0/1571. Digges, p. 343, 1572; Gemma, p. 206, 1540; Riese, p. 139, 1522; Savonne, p. 314, 1563; Stifel, p. 260, 1553; HIERONYMJ CARDANI MEDIO LANRNSIS, CIVISCVV'E BONO* NJfcNSIS, PHILOSOP.HI, MEDICI ET Mailicmatici clanfsimi, OPVS NOVVM DE PROPORTIONIBVS NVMERORVM, MO Y V V M, P ON D E R. V M , S O N O R V M, A L I A R V M Q_v' U R E R V M menfurandafum^ion foliim Geomctrico more ftabilitum,fcd ctiain uarrjs experimcntis 6^ obfcruan'onibusrerum fnnatura,foleni dcmonfl;rationciilun:ratum,acl multiplices ufus ac* commodatum^in Vlibros digeftum. P R AE T E R E A. ARTIS MAGN^, SIVE DE REGVLTS ALGEBRAICI5, LIBER VNV'S, AB $ T R V S I S S I M V $ 8^inexhauftuspIanetotiasAnthmet!ca:tIieraurus,ab authorc recens mulcis in loci's recogni- i T n M. DE ALIZA REGVLA LIBBR, ftOC EST, ALGEBRAIC AB logiftics-' ruae,numeros recondit.i numcrand i fubtilitate,fcamdum Gco^ mftricas quantitates inqiiircnrfs , ijccolliru Coron/s, nunc deni u rain lucem c4ita. Opwj thy/tcii cr Mdthematick inprimis utilecrnccefftrim. Cum Caef.Maieft. Gratia & Priuilegio. B A S I L B M. FIG. 165. TITLE PAGE OF THE 1570 CARDAN 340 KARA ARITHMETICA Trenchant, p. 320, 1566; Anonymous, ' Les principaux fondemens d'arithmetique ' ; Nicolaus Eschenburg, ' Arithmetica logiftica,' Frank- fort ; Alex. Vandenbussche, ' Arithme"tique militaire,' Paris, 4. FRANCISCUS BAROCIUS. Ed. pr. 1572. Venice, 1572. See p. 295. Title. See Fig. 166. Description. 4, 13.9 X 19.3 cm., the text being 10.3 x 17 cm. 3 ff. unnumb. + 23 numb. = 26 ff., 17-36 11. Venice, 1572. Edition. There was no other edition. A German translation was published in Leipzig in 1616. This is an attempt to popularize the mediaeval number game of Rithmomachia (Rithmimachia, Rythmomachia), set forth in Latin pos- sibly by Shirwode or by Faber Stapulensis, in an edition of Boethius in 1496 (see p. 63), and afterwards amplified by Claude Boissiere (see p. 271, 1556). The game was often, with no authority, attributed to Pythagoras. Barocius (or Barozzi) amplified the treatment attributed to Faber Stapulensis, and his discussion of the subject is clearer than that of the latter, although hardly equal to that of Boissiere already described. He had already published a philosophical discussion of arithmetic as stated on p. 295. LEONARD AND THOMAS DIGGES. Ed. pr. 1572. London, 1579. LEONARD DIGGES came of an ancient family whose seat was Digges Court, Barham, Kent. He studied at Oxford, and was an expert mathematician for the time. He died c. 1571. THOMAS was a son of Leonard, and was born in Kent. He was educated at Oxford, and died in London, August 24, 1595. Title. See Fig. 167. Colophon. ' Imprinted at Lon-//don, by Henrie Bynneman, dwel-//Hng in Thames Street, neere vnto // Baynardes Caftle.// Anno 1579.' (P. 192.) Description. 4, 12.8 x 18 cm., the text being 8.9 X 14.7 cm. 1 6 pp. unnumb. + 191 numb. = 207 pp., and one plan (p. 176), 35 11. London, 1579. Editions. London, 1572, 4; ib., 1579, 4 (here described) ; ib., 1585, 4; ib., 1590, 4. The 1579 edition was a revision, IL NOBILISSIMO ET ANTIQVISS1MO GIVOCO PYTHAGOREO NO MI NATO Rythmomachia CIOE BATTAGLIA DE CONSONANTIE DE NVMERI. RilroH4te fer < vtitsti,& folftgo delli Studio/!. Etal prefente per Francefco Barozzi Gentil'hiiomo Yenetiano in lingua volgare in mododi Paraphrafi compofto . IN VENETIA. ratiofo Perchacino* 1 5 7 i .' FIG. 1 66. TITLE PAGE OF BAROZZI 342 KARA ARITHMETICA if An Arithmetical! Mill tare Treatile^iamed ST^A^IOTICOS: Compendioujly teaching the Science of3\ubers, as well in Fradions as Integers, and fo much of the Ru> Jesand /Equations Algebraical! and Arte of Numbers Cofsicall,as are requifice for the Profefsion of a Soldiour* Together with the Modcrne Militare Discipline, Offices,! awes and Ducties in cucr}' vvel goucrncd Cainpc ami Annie to be obferued : Long fince attested by LEONARD DIOGES Gentleman, Augmented, digcftcd, and lately (iniflied, by THOMAS D i c o E s, his Sonne. W 'hereto he hath ttlfoivkoyr.edccnMiic Questions of gr rejoined hi* other TreAt:z*of Pyrotechny and great j4rttttene> hereafter lobee publifted. VIVET POST FVNERA VIRTVS. LONDON; Printed by HenneBynncmaiu FIG. 167. TITLE PAGE OF THE 1579 DIGGES PRINTED BOOKS 343 for the preface ' To the Reader ' (f . a 2) states that it was ' fin- ifhed the 13. of October. 1579.' Considering its date, this work is a very good introduction to the study of arithmetic. The arithmetic proper extends, however, only to page 32. Then follows a brief treatment of algebra (pp. 33-51), after which are certain problems (to p. 70) relating to military matters. Pp. 81-191 are devoted entirely to military affairs. The father and son wrote several mathematical treatises, but none directly on arithmetic. One was the ' Pantometria ' of 1571, which De Morgan includes, and which is in Mr. Plimpton's library, but which I have omitted because it is in no sense an arithmetic. Other works of 1572. Buckley, p. 252, 1550 ; Gemma, p. 200, 1540 ; Grammateus, p. 123,1518; Mariani, p. 181, 1535 ; Barlaamo, ' Aoyicrri/o;, sive arithmetics, algebraicse libri VI ' (in a work on spherics), Strasburg, with later editions at Paris, 1594, 4; 1599, 4; 1600, 4 (see also p. 315, 1564); John Seton (see Buckley, p. 252, c. 1550). SILVIO BELLI. Ed. pr. 1573. Venice, 1573. Born at Vincenza. He died in 1575. He was an architect at Rome and Ferrara, and wrote on practical geometry. Title. See Fig. 168. Description. 4, 15 x 20.6 cm., the text being 9 x 15.1 cm. 46 ff. (40 numb.), 19-21 11. Venice, 1573. Editions. This is the only separate edition of this semi-geo- metric work. Belli published a ' Libro del misurar con la vista,' which passed through the following editions : Venice, 1565, 4; ib., 1566,4; 1569,4; Venice, 1570,4; ib., 1573,4; ib., 1595, 4. This was united with the ' Delia Proportione 'in 1595 (Venice) to form the ' Quattro libri geometrici.' This work is included in this list because of the treatment of mediaeval proportion which it contains. LUCAS PAETUS. Ed. pr. 1573. Venice, 1573. A Venetian jurist of the sixteenth century. Title. See Fig. 169. Description. Fol., 20.5 X 29.5 cm., the text being 15 X 24.5 cm. 8 pp. unnumb + 93 numb. + i blank = 1 02 pp., 50 11. Venice, 1573- 344 KARA ARITHMETICA SILVIO BELLI V I C E N T I N O *D E L. Z, A PROPORTIONE, ET PROPORTIONALITY Communi Paflioni del Quanto . L I B RI TR E. Vtifi f $ necejptrij alia. <-ueu , ft) '/actle intelligentia^ tutte le fcicntie & arti. Al Magnanimo Alejandro Farnefo Card. I H V N E TI A, ^ppreffb Francefio de rrancefibi Sanefc^. 1/75. FIG. 1 6 8. TITLE PAGE OF BELLI PRINTED BOOKS 345 LVCAE PAETI IVRISCONSVL DEMENSVRIS, ET PONDERIBVS ROMANIS, ET GRAECIS, CVM HIS OVAE HODIE ROMAE SVNT COLtATIS LIBRI LECTlONriH HJIEJt AD SANCTISSIMVM OPTIMVMOVE PRINCIPEM PIVM QJINCTVM PONT. MAX. MAXIMILIANI.il O > > * (X VENETIIS MD LXXIII FIG. 169. TITLE PAGE OF PAETUS 346 KARA ARITHMETICA Editions. There were two editions of this work published at Venice in 1573. (See next title.) Although not an arithmetic, this work is a scholarly and interesting contribution to the history of the weights and measures of Greece and Rome, and the symbols inherited by the Middle Ages. It also contains several illustrations of ancient measures. LUCAS PAETUS. Ed. pr. 1573. Venice, 1573. See p. 343. This is a different edition from that just described. The title page is, however, substantially the same. Description. Fol., 17.5 X 23.2 cm., the text being 1 1.6 x 18.5 cm. 144 pp. (127 numb.), 38 11. Venice, 1573. See above. VALENTIN MENHER. Ed. pr. 1573. Antwerp, 1573. See p. 249. Edited by Michiel Cognet, born c. 1549 at Antwerp; died at Antwerp. Title. See Fig. 170. Colophon. ' Antverpiae // Typis Ant. Dieft. 1573.' Description. 8, 9.3 X 13.6 cm., the text being 6.7 X n.8 cm. 141 ff. unnumb., 24 11. Antwerp, 1573. Editions. I have no doubt there was an earlier edition, although Cognet may have edited this from a manuscript left by Menher. This is one of the best of the purely business arithmetics of its time. It shows, better than most works of the kind, the state of commerce in the second half of the sixteenth century in Antwerp, then the most pro- gressive of the mercantile cities of the North. In it may be studied the merchandise, the trade routes, the customs of merchants and bankers, and the prices prevailing in that period. It was to the Netherlands what Riese's book had been to Germany and Borghi's to Italy. As the title page shows, it also took a progressive attitude with reference to practical geometry. (On Cognet see p. 365.) Other works 0/1573. Kobel, p. 102, 1514; Moya, p. 310, 1562; Recorde, p. 214, c. 1542; Peter Beausard, ' Arithmetices praxis,' Louvain, 8 ; Simon Kopfer, ' Grundbiichlein der Regel Detri,' Ntirn- berg ; Bartolomeo Piccini, * Trattato de' Cambi,' Florence, 4. PRINTED BOOKS 347 Works of 1574. Anonymous, p. 244, 1546; Baker, p. 327, 1568; Buckley, p. 252, c. 1550; Jacob, p. 298, 1560; Riese, p. 139, 1522; Rudolff, p. 152, 1526 ; Seton (see Buckley, p. 252, c. 1550) ; Lorenzo J. I V RE D'ARITHMETIQVE, contenant plufieurs belles queftions& de- mand es^propres & vtiles a tons ceuxxjui hantentla Trafiquede Marchandifc. Compojgparfiu Falensm Mennber<s4llc mand; rGueu,corrige,& augments enpluficurs endrom par Michiel Cognet. ENSEMBLE Vm ample declaration fur le fait Jes Changes. ITEM totot difcours dt lien &deu'ementdifconterjiMC la Solution fur dtuerfes opinions y propofies* AVEC La Solution des <jueftions Mathemat'iques par la fiippu ta tion de Sinus^illuftrees & am- plificcs paries demonftrations Geome- triques neceflaires Micelles. A ANVERS ean \\aeslergbe j, I'cfcu de FUndres. AVEC PRIVILEGE. FIG. 170. TITLE PAGE OF THE 1573 MENHER Bonocchio, ' Breve et universale risolutione d'aritmetica,' Brescia (Ven- ice ?), 4; ib., 1597 ; Milles de Norry, * Arithmetique,' Paris, 4; Johann Sekgerwitz, Rechenbiichlein auff allerley Handthierung,' Breslau. 348 KARA ARITHMETICA DIOPHANTUS. Ed. pr. 1575. Toulouse, 1670. A Greek mathematician, c. 300 A.D. He was the first great writer upon algebra. Title. ' Diophanti // Alexandrini // Arithmeticorvm // libri sex,// et de nvmeris mvltangvlis // liber vnvs.// Cvm commen- tariis C. G. Bacheti V.C.//& obferuationibus D. P. de Fermat senatoris Tolofani // Acceflit Doctrinse Analyticae inuentum nouum, collectum //ex varijs eiufdem D. de Fermat Epiftolis.// (Engraving ' Rabault Facit,' with motto : ' Obloqvitvr nvmeris septem discrimina vocvm.') Tolosse, // Excudebat Bernardvs Bosc, e Regione Collegij Societatis Iefu.//M. DC. LXX.' (P. i.) Description. FoL, 23.4 x 36.3 cm., the text being 15.1 X 23.9 cm. 5 pp. blank + 6 unnumb. + 341 numb. + 48 of notes = 400 pp., 5055 11. Toulouse, 1670. Editions. Basel, 1575. There was no other sixteenth-century edition. Athough entitled an arithmetic this is really a treatise on algebra, the first systematic one ever written. It contains, however, a good deal of matter upon the Greek theory of numbers, notably the ' Clavdii Gasparis Bacheti Sebusiani, in Diophantvm Porismatvm, Liber Primus,' ' Liber Se- cundus,' and * Liber Tertius.' A certain amount of this work also enters into the treatise itself, but this is generally algebraic in character, the standard problem requiring the finding of a number satisfying given conditions. This leads to numerous indeterminate (Diophantine) equations. This edition, by Bachet and Fermat, is one of the best that has been published. FRANCISCUS MAUROLYCUS. Ed. pr. 1575. Venice, 1575. FRANCESCO MAUROLICO. Born at Messina, September 16, 1494; died near there, July 21, 1575. He entered the priesthood and later became professor of mathematics at Messina. He wrote chiefly on astronomy, and edited several works of the Greek mathematicians. Title. See Fig. 171. Colophon. 'In monafterio S. Maria //a parte 19. lulij die,// $. ii. Indictionis,// 1553.' (P. 305.) Description. 8, 15.7 x 20.9 cm., the text being 10.6 x 17.5 cm. 20 pp. unnumb. + 285 numb.+ I blank = 306 pp., 39-40 11. Venice, 1575. PRINTED BOOKS 349 D. FRANCISCI MAVROLYCL ABBATIS MESSANENSIS, Opufcula Mathematica , nc frimiim in lucem &ditn , cum rerum omnium notfttu dignarwrru . INDICE LOCYPJLETISSIMO. *PAGELLA HriC PROXIME CONTlGVA* corum Catalogus ett . CVM PRIV1LEGIO. Vcnetijs, Apud Francifcum Fraacifcium Senenfcm M D L X X V. FIG. 171. TITLE PAGE OF THE Opuscula OF MAUROLYCUS 350 KARA ARITHMETICA Editions. There was no other edition. (See below.) It appears from the colophon that this work was composed in 1553, although it was not published until 1575. It includes (pp. 26-47) a ' Compvtvs ecclesiasticvs in svmmam collectvs.' The rest of the treatise is chiefly astronomical. It forms the first of two volumes on mathe- matics, the second being the * Arithmeticorum libro duo ' (see below). FRANCISCUS MAUROLYCUS. Ed. pr. 1575. Venice, 1575. See p. 348. Title. See Fig. 172. Colophon. ' Libri fecundi Arithmeticorum Maurolyci finis: hora //decimaoctaua, diei Sabbati, qui fuit lulij 249. Cum // Meffanae cum multo pontis & arcus // apparatu expectaretur lo. Cerda,// MethynenfiumDux,//Prorex. Indict. I5.//M. D. LVII.// Vene- tiis, M D LXXV.//Apud Francifcum Francifcium Senenfem.' (p. i8 3 .) Description. 4, 15.9 X 20.9 cm., the text being 13.3 x 17.2 cm. 200 pp. (175 numb.), 40 11. Venice, 1575. Editions. This is the first edition, and from the colophon it appears that it lay in manuscript from 1557 to 1575. A second edition appeared in Venice in 1580. The 'Arithmeticorum libri dvo' formed the second volume of the 'Opuscula Mathematica' (Venice, 1575; p. 348). The work is mediaeval, dealing solely with the Boethian theory of numbers. It was one of the last of the extensive sixteenth-century Italian works of this nature, and shows considerable originality in the treatment of figurate numbers. Maurolycus was by no means a mere compiler, but a man of creative power. HENRICUS BRUC^US. Ed. pr. 1575. Rostock, 1575. Born at Alost, Flanders, c. 1531 ; died at Rostock, December 31, 1593. He was professor of mathematics at Rome, and later professor of medicine at Rostock. Title. See Fig. 173. Description. 8, 9.5 X 15.1 cm., the text being 6.5 X u.6 cm. 76 ff. (i blank), 25 11. Rostock, 1575. D. FRANCISC1 MAVROLYCL ABBATIS MESSANENSIS, Mathematici celeberrimi > L1BR1 NVNC PRIMVM INLVCEM (urn rerwn omnium notnbilmmj . INDICE COPIOSISSIMO. EDIT I, CV M PRIVILEGIO. Venetijs> Apud FrancifcumFrancifcium Sencnfcm M D LX X V. FIG. 172. TITLE PAGE OF THE Lib ri duo OF MAUROLYCUS 352 RARA ARITHMETICA Editions. There was no other edition. This is a Latin-school manual, in two ' books.' The first book treats of arithmetic, and chiefly of mediaeval ratios. The second treats of algebra, the equations being considered from the standpoint of geometry, HENRICI B R V C R. I MATHEMATICA^M EXERCITATIONr M LIBRI ROSTOCHX! JXCYDEIAT IACOJJVS TRANSYLVANVS. M. D. FIG. 173. TITLE PAGE OF BRUC/EUS and some attention being given to surd numbers, roots, and the rule of false. Other works of 1575. Gemma, p. 206, 1540; Kobel, p. 102, 1514; Lapazzaia, p. 322, 1566; Mariani, p. 181, 1535; Salignac, p. 359, PRINTED BOOKS 353 1577: Xylander, p. 356, 1577: Mauricius Steinmetz Gersbach, ' Arith- metices praecepta,' Leipzig, 8. Works of 1576. Gemma, p. 200, 1540; Petri, p. 325, 1567 ; Ta- gliente, p. 115, 1515 ; Joseph Lange, ' Arithmetical Copenhagen, 8. GIRJKA GORLA Z GORLSSTEYNA. Ed. pr. 1577. Czerny, 1577. A Polish Rechenmeister of the latter half of the sixteenth century. Title. See Fig. 174. Colophon. ' Wytifftenow // Starem Hefte Pra5 ke"m//v Girijka g3erneho.//Letha Pane///M. D. LXXVII.' (F. XC, r.) Description. 8, 9.5 X 15.9 cm., the text being 7.2 x 11.7 cm. 9 ff. unnumb. + 89 numb. (Roman) = 98 ff., 2 1 11. Czerny, 1 577. Editions. There was no other edition. This is one of the few arithmetics in the Polish language published in the sixteenth century, and is very rare. There was a copy in the Boncompagni sale, but the book is seldom mentioned by bibliographers. It consists of five parts, the first dealing chiefly with the fundamental operations with counters, the second with written operations, the third with fractions, the fourth with business arithmetic, and the fifth with the rule of false and allied topics. DIONIS GRAY. Ed. pr. 1577. London, 1577. A London goldsmith of the second half of the sixteenth century. Title. See Fig. 175. Description. Sm. 8, 8.8 x 13.8 cm., the text being 6.5 x 12 cm. 126 ff. (8 unnumb.). London, 1577. Editions. London, 1577, 8 (here described); ib., 1586, 8. This is a practical arithmetic, consisting of four parts. ' The firft containeth foundrie partes of Arithmetique, that is to fay,' the funda- mental operations with integers, including progressions. ' The feconde parte, containeth the faid partes feruyng for practife of broken numbers or fraccions.' ' The third part containeth the fondrie Rules of pro- portion, furthered by vfe of the fore faid partes,' and includes alligation and the rule of false. ' The fourth parte containeth fondrie Rules of breuetie,' or short processes. It is one of the earliest English arith- metics to contain rules and definitions in rhyme. For example, in 354 KARA ARITHMETICA ARITHMETIC A ctSiv/iw tintib 4 Cy flfw'd> wcimt vstrccn * 4 prop / w Scatr'm ttTli^|f^ domuob (laro 04 fec'Ciiem \>2ftefrc!;0 tir^ M. D. LXXVII. oat aBvs&Mi FIG. 174. TITLE PAGE OF GIRJKA GORLA z GORLSSTEYNA PRINTED BOOKS 355 speaking of addition Gray says : ' And for to amplifie the effecte, take here a fewe lines in verfe : ftore-houfeof "Breuitie in njnuoorkes ofo/1- rithcmctike , containyng afvvcll the fottndrie panes of the Science in nbote ana broken numbers, tui tb t& c Uu= leg of p^opouf on,fattbcren to p?ofi= table life : as alfo ftinoerie rules of T^ewrie of io^e,of rdre,plea' fnunte, ana commo&iou* effecre,fetfo?tljebp lonbon 6olormitf;. '577- ff Imprinted at LVdonfor Norton y indlhon Hanfo.t, dwefarrg' Church-yard. FIG. 175. TITLE PAGE OF GRAY 1 C.Of fondrie fommes perticulars, one totall for to frame, Set them doune right orderly, as worke doeth beft require : What place ye giue to any one, the reft let haue the fame, So maie you well performe the 'ffecte, of what you doe defire. The rule is then continued in a series of verses. 35 6 KARA ARITHMETICA GUILIELMUS XYLANDER. Ed. pr. 1577. Heidelberg, 1577. WILHELM HOLTZMANN. Born at Augsburg, December 26, 1532 ; died at Heidelberg, February 10, 1576. He was professor of Greek at Heidelberg. Title. See Fig. 176. Colophon. ' Excudebat lacobus Mylius, impenfis // Matthaej Harnifch.//M. D. LXXVII.' (F. 52, v.) Description. 4, 14.9 X 19.2 cm., the text being 10.5 X 16. 1 cm. 2 ff. unnumb. + 50 numb. = 5 2 ff . , 3 1 11. Heidelberg, 1577. Editions. There was no other edition of this work. Xylander translated several Greek works into Latin, among them Euclid (Basel, 1 562) and Diophantus (Basel, 1575, fol.). The 'Opuscula' appeared a year after his death. Xylander also edited Psellus (Basel, 1556). As the title states, this work is divided into four parts. The first and most extensive (ff. 1-22) relates to astronomy. The second is purely arithmetical (ff. 22-36), and treats of common fractions, giving the usual operations and the rule of three. The arrangement of this part is peculiar, addition and subtraction being treated together, after which division is explained, multiplication coming last. This order would be justified if Xylander had reduced his fractions to fractions having a common denominator, but he merely follows the usual rule of cross multiplication (f. 29, v.). The third section is ' De svrdis, qvos vocant, nvmeriis iis, qvi a qvadratis primo nafcuntur, Inftitutio do- cendo explicanda,' a chapter now conventionally placed in our alge- bras. The fourth section relates to the celestial globe and the astrolabe (ff. 46-50). PIETRO ANTONIO CATTALDI. Ed. pr. 1577. Bologna, 1577. CATALDI, CATALDO. Born at Bologna in 1548 ; died at Bologna, February i r, 1626. Professor of mathematics and astronomy at Florence (1563), Peru- gia (1572), and Bologna (1584). He wrote several mathematical works, and to him is due the beginning of the theory of continued fractions (1613). Title. See Fig. 177. Description. 4, 14.3 x 19.6 cm., the text being 10.9 x 16.7 cm. 8 ff. unnumb., 40 11. Bologna, 1577. PRINTED BOOKS 357 OPVSCVL A MATHEMATICA DOMINI IELM1 XrL AVG VSTAKJL i Cofm ogntf bid liter L 2)e%finu(itf tiler L uriforum Nutnerorum natura & tmSathne tiler* HE 1 D E L 3 E Excudeba tlacobus My li us > impends MatthjHarni(ch. M, D. LXXVI1. FIG. 176. TITLE PAGE OF XYLANDER 358 KARA ARITHMETICA DVE L E T T I O N I DI PIETR' ANTONIO CATTALDI BOLOGNESE to Diffegno dt Perugia, ALLI GENEROSI.ET VIRTVOSISSIMI Signon Academici , il Signor Caualier Paciotto & il Signor Caualiero Anaitagi . IN BOLOGNA. PerGiouanni RoGi MDLXXVII. Cm licentia de Superior! . FIG. 177. TITLE PAGE OF CATTALDI PRINTED BOOKS 359 Editions. There was no other edition. Cattaldi wrote, under the pseudonym Perito Annotio, a ' Prima Parte della Pratica Aritmetica' (Bologna, 1602) and a ' Trattato dei numeri per- fetti' (Bologna, 1603), but like all of his strictly mathemati- cal works they appeared after 1600. The second part of the 'Pratica' appeared under his own name in 1606. This book hardly deserves to be ranked in a list of arithmetics. I have included it, however, because in his first address Cattaldi treats somewhat of numbers, and in the second address he applies arithmetic to mensuration. The treatment is mediaeval, and the chief interest in the book is a typographical one ; for, the printer not being able to set such numbers, all of the fractions have been written in by hand. Other works 0/1577. Borghi, p. 16, 1484 ; Buckley, p. 252, c. 1550; Capella, p. 68, 1499 ; Cressfelt, p. 290, 1557 ; Herbestus, p. 303, 1561 ; Hobel, p. 314, 1563 ; Ramus, p. 331, 1569 ; Seton, p. 252 (see Buckley, c. 1552); Anonymous, ' Arithmetica,' Debreczin ; Miguel Berenguer, ' De numerorum antiquorum nods,' Saragossa ; Georg Gehrl, ' Ein nutzlich und kiinftlich Rechenbuch auff der Federn,' Prag, 8 ; Johann Jung, an arithmetic, Llibeck, c. 1557 (no copy extant?); Bernhard Salignacus, * Tractatus de Arithmetica Partium et Alligationis,' Frankfort (Peacock says 1575); also ' Regula veri,' Heidelberg, 1578, and * Arithmeticae Libri II, et Algebrae totidem,' Frankfort, 1580, 4; ib., 1593, 4. JACQUES CHAUVET CHAMPENOIS. Ed. pr. 1578, Paris, 1578. Professor of mathematics in the University of Paris, in the second half of the sixteenth century. Title. See Fig. 178. Description. 8, 10. i x 17 cm., the text being 7.5 x 14 cm. 9 pp. unnumb. + 383 numb. = 392 pp., 28-29 N- Paris, 1578. Editions. There was no other edition. The privilege is dated ' a Paris le huictiefme iour de Septembre, M. D. LXXVII.', and the dedicatory epistle * De Paris ce 28. de Nouembre 1577.' The chief interest in the book lies in the fact that it was written by a man who was so interested in military matters as to take a large num- ber of his applied problems from army life. The arrangement of the book is not peculiar, but the great array of military problems is unique, and, having been prepared especially for this work, would form an 360 KARA ARITHMETICA LES INSTITVTIONS DEL'ARITHMETIQJ/E DB I A C QJT ES CHAVVET C H A M P- E- nois,ProfefleUr es Mathetnatiques en rVniuerfite de Paris,diuifecs ea quatre partics.-aucc vn pe titTraid^ des fraftions Altronomiques. A PARIS, Chefc Hierofme de Marnef,au mont S.Hilaire,arenfeigne du Pelican. 1578. AVHC PRIVILEGE DV ROY.. FIG. 178. TITLE PAGE OF CHAUVET CHAMPENOIS PRINTED BOOKS 361 interesting source for the study of army conditions in France in the sixteenth century. Other works 0/1578. Capella, p. 66, 1499 ; Gemma, p. 200, 1540 ; Ramus, p. 355, 1569; Salignacus, p. 359, 1577; Tartaglia, p. 278, 1556 ; Trenchant, p. 320, 1566. JOHANN OTTO. Ed. pr. 1579. Leipzig, 1579. A Freiburg Rechenmeister, born c. 1529. The dedicatory epistle is dated 1579, and is signed ' Johan Otto. ^Etatis fuse 50,' which approximately fixes the date of Otto's birth. Title. See Fig. 179. Colophon. ' Getruckt zu Leipzig/ bey Johan // Rhambawes Seligers hinderlaf-//rien Erben/ 1579.' (P. 423.) Description. 4, 14.5 X 19.1 cm., the text being 11.7 X 17.5 cm. 424 pp. (381 numb.), 28-38 11. Leipzig, 1579. Editions. There was no other edition. The book is composed almost entirely of tables, although the first few pages give a brief treatment of counter reckoning. CHRISTIAN URSTISIUS. Ed. pr. 1579. Basel, 1579. ALLASSIDERUS, ALLASSISIDERUS, WURSTEISEN, URSTIS. Born at Basel in 1544 ; died at Basel March 30, 1588. He was educated at Basel and became professor of mathematics (1565) and afterwards (1585) of theology in that university. Title. See Fig. 180. Colophon. ' Basileae Helve-//tiorvm,//per Sebastianvm Hen- //ricpetri, An. hvmanitatis // Filii Dei, CI3. 13. LXXIX.// Menfe Augufto.' (P. 192.) Description. 8, 10.3 X 16.6 cm., the text being 6.4 X 1 1.2 cm. 192 pp., 25-28 11. Basel, 1579. Editions. On p. 192 is a woodcut with the date 1569, evi- dently used from some earlier work of the printer's. Murhard (vol. I, p. 173) mentions a work by Urstisius, < Zwey Bucher von der Rechenkunft, defsgleichen in der deutfchen Sprache nie aus- gegangen,' Basel, 1569, 4, in which this woodcut may have been used. He also mentions an edition at Basel in 1595. An English translation by T. Hood appeared in 1 596. Calculator. news / afiff We (left; 2rcfjan6crnBte6tm 0(e^ofaCo bet (EfliUicr atlff no tcuff tfttff (Ten tiicr gttt/ i?n{) (jnbetcr fdfftcttttyicft ma 0ar Wen5 one Multipliciren t)iU) Diuidircu 3 fpeacs 5nr Addition ^u $rancffcrtan^cr vrib <n fumma al(e toitffol 1579* FIG. 179. TITLE PAGE OF OTTO PRINTED BOOKS 363 This is a book written by a gymnasium teacher, who was filled with a love of the classical learning, and yet who recognized that the old Boethian arithmetic must give way to the practical treatment demanded ELEMENTA A R I T H- MEXICO, LOG I* C IS. LE G IB VS DED VCT A, In ufum Academise Bafil. CHRISTIAN i Mathtmat tcdruni frofefc for'u. BASILED, PER. SEBASTIANVM HEN- RI CPE Till. FIG. 1 80. TITLE PAGE OF URSTISIUS by modern conditions. As a result it is a somewhat heavy treatment of part of the old theory of numbers, together with the fundamental oper- ations, roots, and certain commercial applications of the rule of three, of partnership, and of alligation. Urstisius expresses his indebtedness 364 RARA ARITHMETICA chiefly to Euclid, Ramus, Salignacus, Gemma Frisius, and Scheubel, not a very extensive list for 1579. Other works 0/1579. Digges, p. 340, 1572 ; Gemma, p. 200, 1540; Mariani, p. 181, 1535; Riese, p. 139, 1522; Tagliente, p. 115, 1515; Anonymous, ' Nouvelle et facile methode d'arithmetique,' Lyons, 16; Miguel de Eleyzalde, ' Guia de contadores,' Madrid, 4. GIAMBATTISTA BENEDETTI. Ed. pr. 1580. Turin, 1585. JOHANNES BAPTISTA BENEDICTUS. Born at Venice, August 14, 1530; died at Turin, January 20, 1590. Philosopher and mathematician of the Duke of Savoy. Title. ' lo. Baptistae // Benedicti // Patritij Veneti Philofophi. //Diversarvm Specvlationvm // Mathematicarum, & Phyficarum // Liber. //Quarum feriem fequens pagina indicabit.// Ad Sere- nissimvm Carolvm Emanvelem // Allobrogvm, et Svbalpinorvm // D vcem Invictissimvm.// (Large woodcut.) Tavrini, Apud Hsere- demNicolaiBeuilaquae,MDLXXXV.//Superioribuspermiffum.' (F- 3.) Description. Fol., 20.7 X 30 cm., the text being 14.5 x 23 cm. 425 pp. numb. + 10 unnumb. + I blank = 436 pp. (118 in the part devoted to arithmetic), 42-46 11. Turin, 1585. Editions. Turin, 1580, fol.; ib., 1585, fol. (here described); Venice, 1599, fol. This work is composed of six parts, of which the first is entitled ' Theoremata Arithmetical The other five parts relate respectively to perspective, mechanics, Aristotle, Euclid's book on proportion, and physics. The arithmetic is a scientific consideration of various matters of theory, and is best illustrated by the following theorems, which, as is usually the case, are stated in the form of questions : ' Theorema IIII. Cvr multiplicaturi fractos cum integris, recte multiplicent numerantem fracti per numerum integrorum, partianturque productum per denomi- nante fracti, ex quo numerus quaefitus colligitur.' 'Theorema XXIX. Qvid caufae eft, cur fubtracto duplo producti duorum numerorum ad inuicem multiplicatoru ex fumma fuorum quadratorum, femper quod fuper eft duorum numerorum quadratum differentiae fit? ' All such questions are answered by the aid of diagrams, quite as Euclid would have done. The graphic treatment is even applied to such problems as that of the couriers ('Theorema CXIIII'). There is an interesting PRINTED BOOKS 365 ' Appendix de specvlatione regvlae falsi,' which closes the arithmetic, and sets forth an elaborate explanation, with graphic aids, of the rule of false which was then so common. As a specimen of graphic arithmetic, combining the Euclidean the- ory with the Renaissance practice, Benedetti's book is worthy of more .attention than it has received. It may be inferred from some of his statements that, although purely a theorist himself, he recognized the obsolete nature of much that the practical arithmetics had to offer. It is possible that the common partnership problems were already con- sidered too traditional, for he says : ' Svpponunt antiqui aliquot merca- tores dantes pecunias lucro in diuerfis vnius anni temporibus,' etc. ISAAC RIESE. Ed. pr. 1580. Leipzig, 1580. One of the five sons of Adam Riese (p. 138). Title. See Fig. 181. Colophon. ' Gedruckt zu Leipzig // ben Hans Rhambaw/ //im Jar//M. D. LXXX.' (P. 402.) Description. 4, 14.1 x 17.9 cm., the text being 1 1 x 16.5 cm. 36 pp. unnumb. + 366 numb. =402 pp. Leipzig, 1580. Editions. There was no other sixteenth-century edition. Although this book contains a brief treatment of the operations, it is largely made up of tables for the use of merchants. It is therefore not a textbook, and I believe it went through only one other edition (Leipzig, 1619). WILLEM RAETS. Ed. pr. 1580. Antwerp, 1580. A Dutch arithmetician of Maastricht. The privilege is dated May 22, 1576, and mentions only the name of Raets. Coignet (Cognet) in his preface, how- ever, speaks of his particular friend ('mijn zoderlinge goet vrient') Raets as dead ('meynen ouerlede vrient '), so that he very likely died between 1 576 and 1580. Coignet was also the editor of Menher's arithmetic (see p. 346). Title. See Fig. 182. Description. 8, 9.8 X 14.7 cm., the text being 7.4 X 1 1.9 cm. 88 ff. unnumb., 27-31 11. Antwerp, 1580. Editions. There was no other edition. This work is one of several elementary business arithmetics appearing in the Low Countries about this time. It is a small book of no special merit save as it shows the style of commercial problems of the period. The similarity of the title page to that of Menher's ' Arithmetiqve seconde' of 1556 (p. 283) is interesting. 3 66 KARA ARITHMETICA fftrn o.funt>t3<ttrtrW<jet/ Dartnmn We in cmf anffcn tonD t>crf auffcn/ one fonDm wcttte ufft tgc ^ccljnuna f auff . amp rcn'altftye SjJecercicn / tR^en vcrgln'c^un^ * th'djc r COJiln^c nl> fwfcfcf gcri^tcf. Se^ctc^n dne ^cc^feJ SKccfenung auff Jrctfjett irt sc&cii jtotttt n j cf) t ad)$tt&rttcf en- i f o* FIG. 181. TITLE PAGE OF ISAAC RIESE PRINTED BOOKS 367 Other works 0/1580. Apianus, p. 155, 1527 ; Baker, p. 327, 1568 ; Cassiodorus, p. 211, 1540 ; Gemma, p. 200, 1540; Gyraldus, p. 254, 1552 ; Mariani, p. 180, 1535 ; Maurolycus, p. 350, 1575; Ramus, p. 331, 1569; Ringhieri, p. 253, 1551; Salignacus, p. 359, 1577; J. Ammo- oft (Sen meib Cifffetboecit/ .ban ftaets/ 7 Vaer in die Fondamenten ieer grondeiijck verclaert efi met veel fchoone quefticn gheilluftreert vvor- den,tot nut endeoorbaer vanalle CoopUede endc lief hcbbers der feluer Conftcn, Met nocjj een Trattaet vandt Wijfetroede, met Ann*. Mien vercitrt t <toffr Mich'iel C tcliebiocme, 1580. Met Priuilcgtc van thtcn lacren. FIG. 182. TITLE PAGE OF RAETS nius, < Isagoge Arithmetical Wittenberg, 8 ; H. Flicker, Arithmetices introductio,' Cologne, 8, and * Compendium calculorum, seu projectili- um ratiocinations,' ib., 8; C. Zuccantini, ' Libro d'Albaco,' Siena, 12. 3 68 KARA ARITHMETICA ANTONIO MARIA VISCONTI. Ed. pr. 1581. Brescia, 1581. A mathematician of Piacenza, of the latter part of the sixteenth century. Title. See Fig. 183. Colophon. ' Brixiae.// apud lacob. & Policretum de Turlinis.// M D L XXXI.//Svperiorvm Premissv.' (P. 301.) Description. 4, 14.8 X 20.2 cm., the text being 9.4 x 15.6 cm. 304 pp. (289 numb.), 37 11. Brescia, 1581. Editions. There was no other edition. Riccardi believes the date 1551 in Murhard is a misprint. I find no such edition. This rare and curious book is a combination of algebra, advanced arithmetic, geometry, and the mensuration of river lands. The arith- metic is designed to be an application of the algebra, and includes roots, equation of payments, proportion, and a little bookkeeping. Other works of 1581. Anonymous, p. 195, 1539 ; Fischer (Piscator), p. 247, 1549; Gemma, p. 200, 1540; Lonicerus, p. 253, 1551; Peverone, p. 290, 1558; Ramus, p. 330, 1569; Riese, p. 139, 1522 ; Johann Kandleon, ' Arithmetical Regensburg (referred to in the care- lessly prepared Boncompagni sale catalogue, but probably the Kaudler book of 1591). There was also 'A short Introduction to Arithmetic' published anonymously in London c. 1581-90, 8. JULIUS CAESAR of Padua. Ed. pr. 1582. Frankfort, 1678. A German-Italian teacher of the sixteenth century. Title. ' Julii Caesaris // von Padua // Arithmetifche // Prac- tick/ // Welche in alien LSndern // fehr nutzlich kan gebraucht // werden/ bey Kauff- und Ver-//kauffung allerley Wahren/ auch//die groffen Muntz-Sorten in//kleine/und die kleine in groffe // zu verwandeln :// Samt der Erklarung/ wor-//inn ein Jedweder/ der nur die // Ziffern kenet/ alfobald fehen kan/ // wie dip Buchlein zu verftehen ift.// Nebenft Morgen- und Abend- //Gebehtenund Gefdngen/den//reyfenden Perfonen gar bequem // bey fich zu fuhren.// Und dann letzlich/ eine kurtze Be-// fchreibung/ der denckwurdigften // Sachen/ fo von Anfang der Welt/ bitf // zu diefer Zeit/ vorgangen.// Franckfurt am Mayn/ //Druckts Blafius Jlflner/im Jahr 1678.' (P. i.) PRINTED BOOKS 369 ANTONII MARIAE VICECOMITIS CIVIS PLACENTINI, Pra&ica Numerorum, & Menfurarum , ac Alluuionis particionem, inueftigandi , & vt in Indice fequenci . B R I X I AE. APVD IACOBVM, ET POLYCHETVM dc Turlinis Fratrcs . 158** FIG. 183. TITLE PAGE OF VISCONTI 370 KARA ARITHMETICA Description. 12, 4.7 x 11 cm., the text being 3.9 x 9.8 cm. 504 pp. (239 and 258 numb.) ; tables 28 11., other pages 22 11. Frankfort, 1678. Editions. Strasburg, 1582, 16; ib., 1583; ib., 1585; ib., 1592, and after 1600 as late as 1679. The first part of the work is devoted to multiplication tables. This is followed by a chapter on chronology. The last part of this edition is a separate book of prayer, ' Chriftliche Morgen- Und Abend Gebeht,' of the same date (1678). There is nothing arithmetical in the work except the tables. GASPARO SCARUFFI. Ed. pr. 1582. Reggio, 1582. An Italian jurist of the sixteenth century. Title. See Fig. 184. Colophon. 'In Reggio,// Per Hercoliano Bartoli. // M.D.- LXXXII.' (F. 65, v.) Description. Fol., 21 x 30.2 cm., the text being 12,7 x 19.5 cm. 65 ff., 32 11. Reggio, 1582. Editions. This is the only sixteenth-century edition of Sca- ruffi's work, the privilege being dated July 15, 1582. The dedi- catory epistle is, however, dated at Reggio, May 16, 1579. The work is called in the running headlines a ' Discorso sopra le monete,' and is a historical treatise on money and coinage, touching slightly on exchange. GASPARO SCARUFFI. Ed. pr. 1582. Reggio, 1582. See above. Title. * Breve Instrvttione // sopra il discorso // fatto dal Mag. M.//Gasparo Scarvffi,// per regolare le cose delli // danari.// (Woodcut representing a bishop, surrounded by these words: S. Prosper // Episcopvs // Regii //.) In Reggio, //per Hercoliano Bartoli.// M. D. LXXXII.' (F. I, r.) Colophon. <Di Reggio il xvij. Aprile. M.D.LXXXI.' (F. 9, r.) The colophon and title page do not agree as to date. Description. Fol., 21 X 30.2 cm., the text being 12.8 x 20 cm. 9 ff., 27-30 11. Bound with the preceding work. Reggio, 1582. PRINTED BOOKS L ALITINONFO DIM.GASPARO SCARNfTFI REGIANQ PER FARE RAGIONE. ET CONCORDANZA D'ORO E ttARGENTO; CHE SERVIRA IN VNIVERSALE; TANTO PER PROVEDERE A GLI INFINITI ABVSI DEL TOSARE, ET GV^STARE MONFTE; Q^ANTX) PER REGOLARE .OGNI SORTE DI PAGAMENTL ZT R1DVRRE ANOO TVTTO IL MONDO ADVNA SOLA MONETA. RECEDANT TENEBRE. SEMPER. ICANDORMEVS IRRADIET: IN REGGIQ PER HERCOLIANO fiARTOLI. M.D.LXXXO. CON LICEN2A DI' SVPERIORI. i 1 FIG. 184. TITLE PAGE OF SCARUFFI 372 KARA ARITHMETICA Editions. There was no other edition. A verbose commentary on certain parts of Scaruffi's work. See P- 370. JOANNES THOMAS FREIGHTS. Ed. pr. 1582. Basel, 1582. A Swiss educator of the sixteenth century. Title. ' loan. Thomse // Freigii I. V. D.// Psedagogvs.// Hoc est, libellvs //ostendens, qva ratio-//ne prima artivm ini-//tia pueris quam facilU-//mi tradi pof-//fint.// Basileae,// per Sebas- tianvm//Henricipetri.' (P. i.) Colophon. ' Basileae,// per Sebastianvm Hen-//ricpetri, anno salvtis//noftrae inftauratae CIO. ID. XXCII.//Menfe Septem- bri.' (P. 383.) Description. 8, 10. 1 x 15.8 cm., the text being 6.8 X 12.4 cm. 17 pp. unnumb. + i blank -f 366 numb. = 384 pp., 303 1 11. Basel, 1582. Editions. There was no other edition. This is a general summary of the subject-matter of education, pub- lished, after the author's death, by his two sons John Thomas and John Oswald (Joannes Osualdus) Freigius. The section devoted to arith- metic begins on p. 144 and ends on p. 156, and p. 145 is reproduced in Fig. 185. Only the fundamental operations with integers, fractions, and compound numbers are given, save for nine lines on the 'Aurea Regula . . . uulgo uocatur regula Detri.' MATTHEW HOSTUS. Ed. pr. 1582. Antwerp, 1582. A German educator of the sixteenth century. Title. See Fig. 186. Colophon. ( Matthaeus Hoftus Francofordiae ad Oderam haec obferuata congerebat & edebat elegantioris literaturae // ftudiofis gratificaturus, Anno Chrifto nato CI3. ID. LXXXI.' (On map at end.) Description. 8, 10 x 14.6 cm., the text being 7.6 X 13.3 cm. 61 pp. numb. + 3 blank = 64 pp., 20-32 11. Antwerp, 1582 (colophon 1581). Editions. There was no other edition. PRINTED BOOKS 373 ARITHMETIC A. 14. f quid' eft A ti thine tic a.? Eft An bcne nwnier*ndi.SubitRwn igitur Arithmetic* Qnpt (tint confidcranda in numero? "Duo; Wotauoo-numeratio. Qiiacnaut eft Notatio > tfimeri fa alacofcribtndi & notandidec-emnottfimt* i .x. $ . 4- 5 $*7' 8 9* Circ ?r pcr/e niktlfigwficat, wd- Iff M/ncn tf i 4/J4i no f dr antplificddMn pro U4r ^ ^ru di^Mf |rt I O.I 00.10 00. 1 OO.OO. Qiisenam fuerunt no tee Koma* noriini? 1. t. r- 1. X. 10. ^. ?o. C. 100. j. D. lo. 500. Qpmgfn&f. CXO. CD, CID. looo. X/Mtf. Mlffe. (\-. IDD. 5000. Qjim^tfrmf/fof. CMo. ^ - CCiDD-.toood* UJ&oc.Deccmmiiridt ^ 1333-5 oooo. Qp M^tMigmftf W/ff r . CCCIO33- won pogreiltmtw ultrd dfcies cement QO . CO.. ZOOO. Clo. CI3. CIO. 3000* CI3, 13, 1500. CD fc Fie. 185. FROM THE Pczdagogvs OF FREIGIUS 374 KARA ARITHMETICA D E NVMERATIONE EMENDATA, VETERIB VS LATINIS E/r' GRwtClS VS1TATA, <?Matth<to Hotfo auffort. ANTVERP I/E, Ex officina Chriftoplion Plantini M. O. L X X X I I. Fig. 1 86. TITLE PAGE OF HOSTUS PRINTED BOOKS 375 This is a semi-historical treatise on the various numeral systems found in Renaissance literature. It includes the Arabic system, and the Greek, Latin, and Hebrew (' gens ludaica,' as Hostus speaks of it) systems, and a chapter * De notis Numerorum Aftronomicis quibufdam vfitatis,' a set of mediaeval astrological numerals also fully described by Noviomagus. MAFFEO POVEIANO. Ed. pr. 1582. Bergamo, 1582. A Veronese arithmetician of the sixteenth century. Title. See Fig. 187. Description. 4, 15.1 X 16.5 cm., the text being 10.3 x 15.6 cm. 92 ff. (83 numb.), 22-26 11. Bergamo, 1582. Editions. There was no other edition. An ordinary treatment of the fundamental operations, with a few applications to mercantile affairs. The book had not enough merit to warrant a second edition. The second part of the work treats of ele- mentary mensuration. The book is little known, and, like many others in this list, is not mentioned by De Morgan. Other works 0/1582. Gemma, p. 200, 1540; Moya, p. 310, 1562 ; Sacrobosco, p. 32, 1488 ; Stevin, p. 386, 1585 ; Clement, * Summa del arte arithmetica, de Fr. de Sant Clement,' Barcelona, 4 ; Ognibene de Castellano, ' II lineamento pertinente all' intendere facilmente quello, che Euclide & altri Eccellentiss. Mathematici ha trattato oscuramente/ Vincenza, 8 (contains some theory of numbers ; see also p. 306, 1561) ; Mellema, ' Arithme'tique composed de plusieurs inventions et problemes nouveaux,' Antwerp, 2 vol., 1582 and 1586. CHRISTOPHER CLAVIUS. Ed. pr. 1583. Rome, 1583. CHRISTOPH KLAU. Born at Bamberg in 1537 ; died at Rome, February 6, 1612. He was a Jesuit priest, and taught mathematics in the Jesuit college at Rome. He wrote a number of treatises on mathematics. Title. See Fig. 188. Description. 8, 10.5 X 16.6 cm., the text being 8 x 13.3 cm. 219 pp. numb, -h 13 unnumb. = 232 pp., 38 11. Rome, 1583. Editions. Pome, 1583, 8 (here described) ; Cologne, 1584; Rome, 1585, 8 (p. 378); Cologne, 1592. There were also edi- tions after 1600 (see p. 378 for the 1602 edition). The Italian translation of 1586 is mentioned on p. 378. The collected works of Clavius in five volumes appeared at Basel in 1612, fol. 376 KARA ARITHMETICA FATTOKE LIBRO D'ARITHMETICA, ET GEOMETRIA PRATTICALE. DI M. MAFFEO POVEIANQ VERONESE. Opera noua , & vtili Ddlepit bmi, & general} prattlcbe , cbc ufar ftp oflano ntctfpnrieadogrfuno. Con licenza de' Supcrion. IN BERGAMO L'ANNO DI N. SIG. M D L X.X XII. Per Comin Ventura , Scampatorein c/Ta Citcn. FIG. 187. TITLE PAGE OF POVEIANO PRINTED BOOKS CHRISTOPHORI C L A V I I BAMBERGENSIS E SOCIETATE i E S v EPITOME 377 PER.MISSV SVPRIORVM. 1(0 M **E Ex Typcgrapbiz Dortawcl .{/& 1 5$ j , FIG. 1 88. TITLE PAGE OF CLAVIUS 378 KARA ARITHMETICA Clavius was an excellent teacher of mathematics, and his textbooks were models of good arrangement. This work is an attempt at a prac- tical arithmetic. It is conservative in treatment, the applications being confined, as was the custom, largely to the rule of three. It was too scholarly to be popular in schools under the mercantile influence, but it was influential in the classical schools. CHRISTOPHER CLAVIUS. Ed. pr. 1583. Rome, 1585. See p. 375. Title. ' Christophori // Clavii // Bambergensis // e Societate // lesv // Epitome Arithmetics // Practicae nunc denuo ab ipfo auctore//recognita.// (Woodcut with I. H. S. in center.) Permis- sv Svperiorvm // Romae Ex Typographia Dominici Bafae. 1585.' (P. i.) Description. 8, 10.3 X 16 cm., the text being 8x13 cm. 337 pp. (321 numb.), 31 11. Rome, 1585. See above. CHRISTOPHER CLAVIUS. Ed. pr. 1583. Rome, 1586. See p. 375. Title. ' Aritmetica // Prattica // composta dal Molto // Reuer. Padre Chriftoforo Clauio // Bambergenfe della Com-//pagnia di I E S V.//Et tradotta da Latino in Italiano dal Signor// Lorenzo Caftellano Patritio// Romano. //Con Licentia del Svperiori.// (Woodcut with I H S in center.) In Roma,// Nella Stamperia di Domenico Bafa.//M. D. LXXXVI.' (P. i.) Description. 8, 10.3 X 1 5.6 cm., the text being 8.2 x 13.3 cm. 302 pp. (275 numb.), 39 11. Rome, 1586. Editions. See p. 375. This is merely an Italian translation of the 1583 edition (see p. 375). Clavius was unable, however, to popularize the book in the mercantile schools of Italy, although several editions appeared after 1600. CHRISTOPHER CLAVIUS. Ed. pr. 1583. Rome, 1602. See p. 375. Title. ' Arithmetica // Prattica //composta dal Molto // Reuer. Padre Chriftoforo Clauio // Bambergenfe della Com-//pagnia di PRINTED BOOKS 379 IESV.// Et tradotta da Latino in Italiano dal Signer Lorenzo // Caftellano Patritio Romano.// Reuifta dal medemo Padre Clauio //con alcune aggiunte.//Con Licentia de i Svperiori.//In Roma, //(woodcut with I. H. S. in center.) Per li Heredi di Nicolo Mutij//M. DC. II.' (P. i.) Colophon. ' In Roma,// Per li Heredi di Nicolo Mutij M D C II.' (P. 312.) Description. 4, 10.4 X 1 5.7 cm., the text being 8.3 x 13.3 cm. 312 pp. (281 numb.), 39!!. Rome, 1602. Editions. This is the second Italian edition (p. 375). See p. 378. NICOLAUS REYMERS. Ed. pr. 1583. Leipzig, 1583. A German surveyor, born at ' Henstede in Dietmarschen.' The ' Be- schluss' is dated 'zu Hattftede in Diethmarchen,' September 14, 1583. Title. < Geodaesia//Ranzoviana.// LandtRechnen///vnd Feld- meffen/fampt meffen aller-//hand gr<5ffe. Alles auff eine leichte/ behende/ // vnd vormals vnbekandte newe art/ kunft-//lich/ grundlich vnd deutlich // befchrieben/ // Zu Ehren // Dem Edlen/ Beftrengen // vnd Ehrnuehften Herrn/ Heinrichen // Rantzouen/ Herrn Johans feligen Sohne/ der // Kon. Mayft. zu Dennemarcken/ etc. In den // Furftenthumben Schlefewick/ Holftein/vnd Diethmar-//fchen/Stadthaltern/Rhat vnd Ambt- man auff // Segeberge/ Erbgefeffen zum // Breitenberge/ etc.// Durch// Nicolaum Reymers/ von Henftede/ //in Dietmarfchen. //Cvm Privilegio.' (F. i, r.) Colophon. ( Gedruckt zu Leipzig bey // Georg Defner/ // Im Iahr//M. D. LXXXIII.' (F. 44, v.) Description. 4, 15.1 x 1 8. 6 cm., the text being 9.7 X 14.5 cm. 44 ff. unnumb., 25 11. Leipzig, 1583. Editions. There was no other edition. Although nominally a book on surveying, this work may properly rank as an arithmetic, the ' Erfte Buch ' being entirely devoted to that sub- ject. Of this book the first chapter is entitled ' von zahlen ' ; the second, 'von bruchen ' ; the third, 'von fummieren ' ; the fourth, 'von viel- feltigen,' multiplication thus directly following addition ; the fifth, 'von 3 8o KARA ARITHMETICA abziehen ' ; the sixth, ' von Theilen ' (division) ; the seventh, ' von den Wurtzel ' ; the eighth, ' von der gevierten Wurtzel.' The rest of the work is devoted to mensuration, and in particular to surveying. The chief interest in the first book is in the use made of the compound numbers then needed in surveying. Like most such manuals, it shows no insight into educational problems, and the treatment is very unsatis- factory. Other works of 1583. Baker, p. 327, 1568 ; Caesar, p. 370, 1582 ; Gemma, p. 200, 1540; Petri, p. 325, 1567; Reisch, p. 82, 1503; Anonymous, ' Rechenbiichlein auf Erfurtifche weifs,' Erfurt, 8 ; Johnn Weber, * Ein new Kunstlich Rechenbuch auff den linien und ziffern,' Leipzig, 4 (see also p. 338, 1570). PETRUS BUNGUS. Ed. pr. 1583-84. Bergamo, 1584-85. Born at Bergamo ; died September 24, 1601, at Bergamo. He was a canon of the cathedral in that city. He wrote only on the mystery of numbers. Title. See Fig. 189, which gives the title page of the first part. Bound with this is the second part with the title : ' Mys- ticae // nvmerorvm // significationis // pars altera, // lo. Petro Bongo Canonico Bergomate //avctore,//In qua de Numeris in Sacris libris potiffimum repertis, ex Theo-//logorum maxime fen- tentia, & probatorum aliorum cuiufuis // f acultatis Scriptorum, ita exacte, dilucide, & accurate difTeri-//tur, vt ferme nil addi, aut detrahi poffe videatur://Opus varia fane, et multiiuga adeo re- fertum doctrina, vt non Theo-//logis folum; fed etiam Philofo- phis, Mathematicis, atque alijs ftu-//diofis omnibus, tarn vtile, quam iucundum fit futurum.//De Svperiorvm licentia.// Bergo- mi CIO ID XXCIV.//Typis Commini Venture, eiufdem Vrbis typographi. 1 Description. Fol., 20.6 x 30.4 cm., the text being 17 x 28.9 cm. 276 pp. (245 numb.) in the first part ; 198 pp. (177 numb.) in the second part; 36 11. Bergamo, first part 1585, second part 1584. Editions. The first edition appeared in 1583-84, ' Bergomi, typis Comini Venturae.' The second part of this work, as the title page shows, is therefore of the first edition. The first edi- tion of the first part was evidently exhausted before that of the M Y S T I C AE NVMERORVM SIGNIFICATIONS LIBER /AT DVAS DIVISVS TARTES, R. D, P E T R O BONGO CANONICO BERGOMATE A V C T O R E: Opus maximarum rerum , & plurimarum doftrina,, fua- uitato,copia*> & uarictate refcrtuiiL,, u, JMathematicis , atque ali/s fludiofis omnibus, tam vtilitatem, quam iucwditat attatumrru. DE SVTERIORVM LICENTIA. BERGOMI CD D X X C V- Typis Comini Vcncurae , &Sodj. FIG. 189. TITLE PAGE OF THE 1585 BUNGUS 382 RARA ARITHMETICA second part was sold, and hence the first part of the present work is the second edition. There were also editions as follows: Venice, 1585, 8; Ber- gamo, 1590; ib., 1591, 4 (which De Morgan incorrectly calls the second edition) ; ib., 1614 (p. 384). It appeared under the Romanorum CO m c oo H m HIM CO 1*> +099. 00 m fOOO. CIO OO OO ca FIG. 190. FROM THE 1614 EDITION OF BUNGUS title 'Numerorum mysteria,' Bergamo, 1599, 4 (p. 384). I have an edition under this title, published at Paris in 1618 (1617 in the colophon), 4. For the 'Praecipuae numerorum notae et earum valor,' Parma, 1689, see p. 384. This is a mass of erudition, prolix and unscientific, relating to the mystery of numbers. It was written by a priest for the use of preachers, PRINTED BOOKS 383 and it includes all of the allusions to such matters as the mystic three that Bungus could find in ancient literature. He takes up the various numbers from one to ten in the same way, together with a few of the more interesting larger numbers. For students interested in popular number mysticism the book still remains the classic in its way. It is also of much value in showing the nature of the Roman numerals in use in the sixteenth century. (See Figs. 190, 191.) Numcratio. CD CO CC1W CCIOO GGIOO X CGLCC DMC OMD IMI CCDD COI03 00 fYYY") co CD VA-d^y^ 12000. COIO3 * <* CCI03 CCDD "^> CCIDD 00 m '* 000 ' CCDD JfOOO. FIG. 191. FROM THE 1614 EDITION OF BUNGUS Other works 0/1584. Buckley, p. 252, 1550; Cassiodorus, p. 211, 1540; Clavius, p. 375, 1583; Delfino, p. 275, 1556; Kobel, p. 102, 1514; Ramus, p. 263, 1555 ; Anton Schulze, ' Arithmetica oder Rech- enbiichlein,' ( * Arithmetica oder Rechenbuch neben einer dienlichen Anleitung zum Buchhalten'?) s. 1., with subsequent editions at Liegnitz, 1600, 4, and Frankfort, 1600, 4. 384 RARA ARITHMETICA PETRUS BUNGUS. Ed. pr. 1583-84. Bergamo, 1599. See p. 380. Title. * Petri Bongi // Bergomatis // Numerorum myfteria.// Opvs maximarvm rervm // Doctrina, et copia refertvm,// In quo mirus in primis, idemq perpetuus Arithmeticae Pythagoricae cum // Diuinae Paginae Nvmeris confenfis, multiplici ratione pro- batur.// Poftrema hac editione ab Auctore ipfo copiofo Indice, & ingenti//Appendice avctvm.//Cum Superiorum approbatione.// Bergomi, Typis Comini Venturae, eiufdem vrbis Typographi.// ooIOXCIX.' (P. i.) Description. 4, 17.5 x 24.3 cm., the text being 12. 6 X 18.8 cm. 770 pp. (676 numb.), 32 11. Bergamo, 1599. See p. 382. Although the title is slightly changed, this is the work already described. PETRUS BUNGUS. Ed. pr. 1583-84. Bergamo, 1614. See p. 380. Title. ' Petri Bvngi // Bergomatis // Nvmerorvm myfteria // Ex abditis plurimarum difciplinaru fontibus haufta://Opvs maxi- marvm rervm //Doctrina, & copia refertum: In quo mirus in primis, idemq ; perpe-//tuus Arithmetica&Pythagorice cum Diuinae Paginae Nu-//meris confenfus, multiplici ratione probatur.// Poftrema hac editione ab Auctore ipfo copiofo Indice, & ingenti // Appendice avctvm.// Illuftriffimo viro, Virtutum omnium, ac difciplinarum //genere ornatiffimo //Ranvtio Gambarae // Comiti Virolae. & c. //Bergomi, Typis Comini Venturae. 1614.' (P. I.) Description. 4, 17.5 X 23.4 cm., the text being 12.6 X 18.8 cm. 970 pp. (753 numb.), 32 11. Bergamo, 1614. See p. 382. PETRUS BUNGUS. Ed. pr. 1583-84. Parma, 1689. See p. 386. Title. ' Praecipuae // Numerorum // Notae,// Et earum valor,// Secundum // Petrum // Bungum.//Parmae, Ex Typographia Du- cali.//CLD. IDC. LXXXIX.' (P. i.) PRINTED BOOKS 385 Description. 4, I i.i X 17.7 cm., the text being 7.9 x 12.6 cm. 16 pp. (12 numb.), 13-19 11. Parma, 1689. Editions. See p. 380. This is simply an extract from the later editions of the ' Numerorum Hysteria.' The treatment of Roman numerals by Bungus is the most elaborate and interesting to be found in any of the works of Renaissance writers. MONTE REGAL PIEDMONTOIS. Ed. pr. 1585. Lyons, 1585. The title states that he was professor of mathematics in the University of Paris ; evidently in the second half of the sixteenth century. Title. See Fig. 192. Description. 32, 5.3 X 9.8 cm., printed on vellum in double col- umns, each 1.8x9.5 cm. 144 pp. (100 numb.), 27-28 11. Lyons, 1585. Editions. There was no other edition. The privilege is dated August 6, 1581, but the work does not seem to have been pub- lished until 1585. The book is ex- ceedingly rare, and is unknown to most bibliographers. The author speaks of having published part of the tables in Venice in 1575. This is a collection of tables, largely for multiplication, beautifully printed on split vellum. These tables give the products of numbers to 100 times 1000. The last ten pages contain a table of arrangement of soldiers in order of battle : ' Le moyen et ordre qui fe doit tenir'pour mettre en or- donnance les batailles de dix soldats iufques au nobre de quarante mille.' This is dedicated * A pvissant et illustre Seigneur le Baron de Mont-clar.' INVENTION NOVVELLE ET admirable, poor faire route forte de coptc , tant de mar- chandife , com me de chager monnoYes,poids,mefures,de dinerfe iruuiere, pour vend re & acheter,laqueHc feruira en tout Je monde auec grande ficilite,fans gecosne plume Et A Ufn le moyen tie meilre vn exercit en lattttlte de cent ( Le tout nouudlcnif tit compote & | mis en luuvieie par le MOD te Re : gal Pirdmomois , Profetfeur d Mathenurique en I'Vniuerfuid Palis. A L Y O N> Ik ie vend en rue Merciere a Tcnfemne de la Sphere. FIG. 192. TITLE PAGE OF MONTE REGAL PIEDMONTOIS 386 KARA ARITHMETICA SIMON STEVIN. Ed. pr. 1585. Leyden, 1585. Born at Bruges in 1548; died at the Hague in 1620. He was a merchant, a soldier, and an officer in the civil service. Title. See Fig. 193. Description. 8, 10.5 X 16 cm., the text being 7.6 X 12.7 cm. 3 1 pp. unnumb. + I blank + 642 numb. + (203 numb. + 1 2 unnumb. + i blank in * La Pratiqve d'Arithmetiqve ') = 890 pp., 29 11. Leyden, 1585. Editions. Stevin's first work was an interest table, Antwerp, 1582. His arithmetic first appeared in Flemish, at Leyden, 1585, and was reprinted in that language at Gouda in 1626 and in 1630. The French translation, which is here described, appeared at Leyden in 1585 and again in 1586. The first edition under the editorship of Girard appeared at Leyden in 1625. For the edi- tion of 1634 see below. This work consists of three distinct parts: (i) L'Arithmetiqve,' in two books, the first treating of powers and roots, and particularly of surds, and the second of operations on numerical and algebraic expres- sions and of the solution of equations ; (2) ' Les qvatre premiers Livres d'Algebre de Diophante d'Alexandrie,' translated by Stevin, apparently from Xylander's text ; (3) ' La Pratiqve d'Arithmetiqve,' an attempt at a practical textbook, but too scholarly for its purposes. The Pratiqve contains 'La Regie d'Interest avec ses tables,' the 1582 work above mentioned, ' La Disme. Enfeignant facilement expedier par nom- bres entiers fans rompuz, tous comptes fe rencontrans aux affaires des Hommes. Premierement defcripte en Flameng, & maintenant conuertie en Francois,' and a 'Traicte des incommensvrables Grandevrs.' The interest centers in ' La Disme,' in which decimal fractions are for the first time treated in any elaborate way (see Fig. 194). SIMON STEVIN. Ed. pr. of the arithmetic, 1585. Leyden, 1634. See above. Title. ' Les //CEuvres//Mathematiques//de //Simon Stevin, // Augmentees // Par Albert Girard.' (P. i .) Page 3 reads : Les // CEuvres // Mathematiques //de Simon Stevin de Bruges. //Ou font infere"es les // Memoires Mathema- tiqves,// Efquelles s'eft exerce" le Tres-haut & Tres-illuftre Prince PRINTED BOOKS 387 1'ARITHMETIQJE DE SIMON STEVIN DE BRVGES: Conccnanc les computations des nombres Arithmetiques ou vulgaircs : Aufi l'Al?ebT?,aueclesequaiwnsdeciM qtidmittt*. Enfemblc les quatrc premiers liures d'Algebre dc Diophante d' Alcxandrie,maintenanc pre- miercraenc craduifts en Franfois. Em ore vn littreparticttlier de la Pratique conteaxx cmre autres,Les Tables d' Inter e ft > Et vn tr.iicle dc$ Incommenfiirables grandeurs : Aucc I' Explication dti Dixiefmc Linn d'Eudide. A L E Y D E, De nmprimeriedeChviftophlc Plantiii* C!D. ID. LXXXV. FIG. 193. TITLE PAGE OF THE 1585 STEVIN 3 88 KARA ARITHMETICA SECONDE PARTIE DE JLA DISME DE L'OPE, RATION, PRpPOSJTION I, DE L*A D D I T I O N. ESrant donnez, nombfes de Difne a ajoufter : Trmer leur fomme : Explication du donns. II y a trois ordres de nombrcs de, Difinc, dcfcjuels le premier 17 @8 47(|Xk deux- iefme 37 8 7 (2)5@ 5 !e troifiefme 875 7 8@tg) f "Explication du requis. II nous faut trouver Jeur fomme . CwjJrffw. On mcrtra les nombrcs donncz en ordre comme ci joignanc , les aiouftant felon la vulgaire manierc d'aiouftcrnombrcscntier^encefte forte: Donne {bmme (par Ic i probleme de rArithmeti- quc) 941304, qui font fee que dcmoDftreut les (igncs deflus les nombres) 941 @5o@4-I cc ^> 4 UC les mefrnes font h fomme requife. Demonstration, tes 2784(|)7(!) donnez, font (par la jc definition^ *7 A. T| Tfer- enfemble 27 -, &parmefmc raifon les $7 6 7 5 valient 3 7 rl^ & les 8 ,7 5 7 8 4 feront 8 7 f ^fe, lefquels trois nombres,comme 27 ~ 4 ^, 37 T^fe, 8 75 ^ cnfemble (par le io e probleme de 1 Arith.) 941 mais autant raut aufll la fomme 9io cefF FIG. 194. FROM THE 1634 EDITION OF STEVIN PRINTED BOOKS 389 Maurice // de Nassau, Prince d'Aurenge, Gouverneur des Pro- vinces des // Pai's-bas unis, General par Mer & par Terre, &c.// Le tout reveu, corrige, & augmente // Par Albert Girard Samie- lois, Mathematicien.// A Leyde // Chez Bonaventure & Abraham Elfevier, Imprimeurs ordinaires //de 1'Univerfite, Anno clo ID c xxxiv.' Description. Fol., 21.7 x 34 cm., printed in double columns, each being 8. i x 28 cm. 910 pp. (232 on arithmetic, lounnumb.), 63 11. Leyden, 1634. Editions. See p. 386. Other works of 1585. Benedetti, p. 364, 1580; Bungus, p. 382, 1583 ; Caesar, p. 370, 1582 ; Clavius, p. 378, 1583 ; Digges, p. 340, 1572; Gemma, p. 200, 1540; Lonicerus, p. 253, 1551; Lossius, p. 289, 1557 ; Psellus, p. 168, 1532 ; Riese, p. 139, 1522 ; Savonne, p. 314, 1563 ; Io. Frans. Fulconis, ' Cisterna Fulconica, libro d'abaco in lingua provenzale,' s. 1. (Lyons?), 8; Thilman Ofenlach, ' Rechen- biichlein mit der Ziffer imd auf den Linien mit Zahlpfennigen,' Basel, 8 ; Johann Schreckenberger, Rechenbuchlein auff den Linien imd der Federn,' Strasburg, 8; Samuel Eisenmenger (pseud. Siderocrates), Cyclopaedia Paracelsica Christiana,' s. 1., with a section on arithmetic. Works of 1586. Clavius, p. 378, 1583 ; Gray, p. 353, 1577 ; Mellema, p. 375, 1582 ; Ramus, p. 331, 1569 ; Riese, p. 139, 1522; Schonerus, p. 331 (Ramus, 1569); Stevin, p. 386, 1585; Stifel, p. 226, 1544 ; Tagliente, p. 115, 1515 ; Anonymous (Sterner mentions a Rechenbuchlein without title page, Magdeburg) ; Georg Hoflein, 'Rechenbuchlein mit der Ziffer und mit den Zahlpfennigen auf der Linie,' Strasburg, 8 ; Paulus Alexandrinus, Rudimenta,' Greek and Latin, Wittenberg, 4 (with some notes on Jewish arithmetic). JOHANNES PADOVANIUS. Ed. pr. 1587. Verona, 1587. GIOVANNI PADOVANI. A Veronese mathematician of the second half of the sixteenth century. Title. See Fig. 195. Description. /t, 14.4 X 20.4 cm., the text being 9.6 X 15.7 cm. 7 pp. unnumb. + 73 numb. = 80 pp., 29-32 11. Verona, 1587. Editions. Padovani published a work in Venice in 1565 con- sisting of six parts, the fourth of which was * De Arithmetical 390 KARA ARITHMETICA IOANNIS PADOVANII VERONENSIS, DE ARITHMETICA OPVS,IN QVO NON folum omnis generis numerandi ars ram Latino fermone,qudm Greco pcrdifcitur: ucruin etiam quicquid ad quafcunque ra- tiocinationes pertinet , faciii do&rina apcritur, &ln fr'wisreiputiicAltterarM ftudtofts perneceffarium* V E R O N AE, ExTypographiaSebaftiani a Donnis. jjsz Delicentia Superiorum. FIG. 195. TITLE PAGE OF PADOVANTUS PRINTED BOOKS 391 That work is probably the same as this. This seems to be the first and only separate edition, and is very rare. This is in no sense a practical treatise, but it discusses the subject of arithmetic in a learned way, at first rather on the Greek plan. It then considers the Arabic arithmetic, taking up the four processes, roots, pro- portion, partnership, and two or three other applications. The style in which it is written is such that it would hardly have appealed to the practical merchant even if it had not been in Latin. For Padovani's other works see Riccardi, I, i, 251. CASPAR THIERFELDER. Ed. pr. 1587. Niirnberg, 1587. A German Rechenmeister of Steyer, born c. 1525. Title. See Fig. 196. Colophon. * Gedruckt zu Nurnberg/ // durch Leonhardt // Heufzler.' (P. 369.) Description. 8, 9.4 X 1 5.2 cm., the text being 7 x 13.3 cm. 30 pp. unnumb. + 369 numb. = 399 pp., 25-26 11. Nurnberg, 1587. Editions. There was no other edition. Thierfelder also pub- lished a book with Ulman at Freiburg in 1 564, 8, entitled ' Neues Kunft-Rechenbuch auf der Linie und Feder ; dergleichen weder in lateinifchen noch deutfcher Sprache ausgegangen.' This is a commercial arithmetic, based on Rudolff and other Ger- man writers, and with no particular individuality. It contains eighteen chapters, the last one being upon mathematical recreations, ' Von der Schimpff Rechnung/vnd Erfindung derfelben Regeln,' a subject that began to come into prominence about this time. Other works of 1587. Finaeus, p. 160, 1530-32 ; Riese, p. 139, 1522; Michael Gempelius, ' Arithmetik,' s. 1., 8; Aurelio Marinati, ' Delia prima parte della somma di tutte le scienze,' Rome, 4 (with a brief treatment of arithmetic) ; Conr. Poeppingius, < Neues Rechen- biichlein auf Linien und Federn,' Braunschweig, 8 (four editions after 1600). Works of ij&S. Gemma, p. 208, 1540 ; Helmreich, p. 306, 1561 ; Rudolff, p. 152, 1526; John Mellis, 'A brief instruction and maner how to keepe bookes of Accompts,' London, 8 (containing a short chapter on arithmetic) ; Heinrich Striibe, Arithmetica oder new kiinft- liches Rechenbiichlein,' Zurich, 8, with an edition ib., 1599, 12. 392 KARA ARITHMETICA (gentili $<($<miK9fto iu ^t^eF, H. D, LXXX.VII, FIG. 196. TITLE PAGE OF THIERFELDER BERNAERT STOCKMANS. Ed. pr. 1589. Dordrecht, 1609 A French schoolmaster at Dordrecht (Dort), in the second half of the six- teenth century. Title. ' Een corte ende een-//vuldige Inftructie/om lichtelij- cke // en by hemfelve/ fonder eenige meefter oft onderwij-//fer PRINTED BOOKS 393 te leeren cijfferen. Seer nut en profijtelijcken //alle menfchen/ die in de Confte van Arith-//metica heel ftecht/ onervaren // ende eenvuldich zijn.// Geftelt ende by een vergadert, door Bernaert // Stockmans lanfz. Franc.oyfche Schoolmeefter // inde vermaerde Coop-ftadt Dordrecht.// Hier zijn ooc bygevoecht de Differentien van de co-//ren-Mate der voornaefte ftede in Hol- lant/ t'Sticht/ // Zeelant/ Brabant/ Vlaenderen/ Gelderlant/ En-//gelant/ Vrancrijck/ en Ooftlant/ tot dienft // van alien Graen-coopers.// Van nieus overfien ende verbetert door C. P. Boeye.// Item/ noch van nieus bygevoecht een clare onder-// wyfmge om de tafelen van Intereft te leeren maken/ //alles tot dienft vanden onervarenen.// Sapiente 1 1. verf. 22. // Maer ghy hebbet alles in Mate/ Tellen/ // ende Ghewichte gheordineert.// Tot Dordrecht// By my Pieter Verhagen/^woonende inde Drue // kerije/ teghen over de Wijnbrugge. 1609.' (F. i, r.) Colophon. 'Tot Dordrecht.// By Adrien lanfz Bot.//Anno 1609.' (F. 214, v.) Description. 8, 9. i x 14 cm., the text being 7.4 x 11.9 cm. 4 ff. blank + 211 unnumb. = 215 ff., 3 1-33 11. Dordrecht, 1609. Editions. This book was first printed in 1589, and the dedi- catory epistle is dated * den 20. defer Maent Julij / 1 589.' There was no other edition in the sixteenth century. This is one of the noteworthy arithmetics of Holland, and it went through a number of editions after 1601 (see p. 394). It is a commer- cial work, and like Vander Schuere's book it gives an excellent view of the mercantile life of Holland in this period. Other works of 1589. Jacob, p. 298, 1560; Moya, p. 310, 1562; Fabio Paolini (Paulinus), ' Hebdomades, sive septem de septenario libri,' Venice, 4, containing a little arithmetic, and ib., 1598 (?). Works 0/1590. Bungus, p. 382, 1583; Digges, p. 340, 1572; Jacob, p. 298, 1560; Lapazzaia, p. 324, 1566; Psellus, p. 170, 1532; Recorde, p. 214, c. .1542; Heizo Buscher (Boscherus), ' Arithmetics libri duo,' Helmstadt, sm. 8, with editions, ib., 1591, 8; Hamburg, 1592, 8; ib., 1597, 8; Frankfort, 1600, 8, and later; Franciscus Brasser, an arithmetic, Liibeck, with later editions ; Cyprian Lucar, ' A treatise named Lucarsolace,' London, 4; Bartolome" Solorzano, ' Librode caxa y Manual de cuentas de Mercaderes,' Madrid. 394 KARA ARITHMETICA BERNAERT STOCKMANS. Ed. pr. 1589. Gouda, 1644. See p. 392. Title. ' Aritmetica,// Door Bernardus Stockma // eertyts Franfoyfche fchool-//meefter inde vermaerde coopftadt // Dor- derecht nu van nieu 8 curieus // gecorigert ende verbetert noch is // hier by gevoecht een tafelken om te // rabatteeren op fulcken tyt of te intereft // men begeert door // Abel. W. Waesenaer // Rekenmeefter tot Vtrecht // (Portrait of author.) Gedruckt tot Vtrecht by//Efdras Willemfsen Snellaert.//boeckvercooper // Anno. 1637.' (P. i.) Colophon. ' Ter Govde,// Gedruckt by Pieter Rammazeyn, Boeck-// drucker in't vergulde A B C.//Door Efdras Willemfz Snellaert, Boeck-//vercooper tot Vtrecht by de waert-poort 1644.' (P. 421.) Description. 8, 9.3 x 13.9 cm., the text being 7.5 x 12.4 cm. 429 pp. (410 numb.), 32-33 11. Gouda, 1644. Editions. See p. 393. The engraved title page of the 1637 edition has been used with this edition, which, as the colophon shows, was printed in 1644. See p. 393. FRANCESCO PAGANI. Ed. pr. 1591. Ferrara, 1591. An Italian arithmetician of the sixteenth century, born at Bagnacavallo. Title. See Fig. 197. Description. 4, 14.2 x 17.9 cm., the text being 10.1 X 15.9 cm. 210 pp. (200 numb.), 38 11. Ferrara, 1591. Editions. There was no other edition. This rare and almost unknown work is based upon the Borghi model (see p. 1 6) which served for so many of the best Italian writers. It was written, as the dedicatory epistle states, at Bagnacavallo, and is one of the few books on mathematics published in the sixteenth century at Ferrara. It has no merit save as its applied problems give a view of the business life of the time. In its numerical work it is reactionary, making, for example, a strong plea for the galley as opposed to the ' a danda ' or modern method of division. PRINTED BOOKS 395 ARITHMETICA PRATT1CA VTILISSIMA, ARTIFICIOSAMENTE ORDINATA Da M. FRANCESCO PAGANI da Bagnacauallo, J^ella qualeficonticne il<vero> <& facile modo di conteggiart^>. Conmolti Qucfiti importantf^&neccflanj a Ragionicri,a Mercanti,& ad ogni perfona^in tutti i Pacfi. *AL &IOLTO JLLVSTRE S1GNOR Appreffb Vittorio Baldini Con liccnza dc' Supcriori. M. D. xci. FIG. 197. TITLE PAGE OF PAGANI 396 KARA ARITHMETICA RENE BUDEL, et al. Ed. pr. 1591. Cologne, 1591. For biographies, see below. Title. See Fig. 198. Description. 4, 18.2 x 23.6 cm., printed in double columns, each being 6.4x18.6 cm. 38 pp. unnumb. -f 798 numb. = 836 pp. (PP. 271-350 are not in the volume, and if they were ever bound in any copies they must have constituted a section by themselves.) 46-49 11. Cologne, 1591. Editions. There was no other edition. This rather massive treatise on the history of monetary measures con- sists of two books by Budel (Budelius), director of the Bavarian mint, and several appended chapters by the following writers : Albertus Brunus (1461-1541), counselor to Louis of France, and ambassador; Johannes Aquila, friend of the astrologer Stoffler who died in 1531 ; Bilibaldus Pirkheymer (1470-1530), a celebrated humanist; Martinus Garatus Laudensis, who writes a chapter ' De monetis ' ; Franciscus Curtius, and Joannes Regnaudus of Avignon, who write on the same topic; Carolus Molinseus (1500-1566); Didacus Covarrubias (1512-1577), bishop of Ciudad Rodrigo; Henricus Mameranus, a Belgian printer; Henricus Hornmannus; Franciscus de Arctic (1418-1483), a cele- brated lawyer of Arezzo; Nicolas Everardus (1473-1532), a celebrated Dutch lawyer, of Middelburg; Jacobus Menochius (1531-1607), an Italian lawyer, and various others. It is a monumental work, and is helpful in the investigation of the history of monetary tables. Other works of 1591. Baker, p. 327, 1568 ; Bungus, p. 382, 1583 ; Buscher, p. 393, 1590; Gemma, p. 200, 1540; Mariani, p. 181, 1535 ; Petri, p. 325, 1567; Psellus, p. 168, 1532; Ramus, p. 331, 1569; Vincent de Beauvais, p. 10, 1473; Jordanus Bruno, ' De Monade numero et Figura liber,' Frankfort, 8 ; Johann Kaudler, * Arithmetica oder Rechnung auf der Linien und mit Ziffern,' Regensburg, 8 ; Hans Jacob Mewrer, * Bericht von dem Rechnen mit den Zahlpfennigen oder auf der Linien,' Ziirich, 8. THOMAS HYLLES. Ed. pr. 1592. London, 1600. An English mathematician of the latter part of the sixteenth century. Title. See Fig. 199. Description. 4, 14.2 X 19.6 cm., the text being 10.8 X 16.9 cm. 1 5 ff. unnumb. + 270 numb. = 285 if., with 2 charts at the end; 40 11. London, 1600. PRINTED BOOKS 397 MONETIS, ET RE NVMA- RIA, LIBRI DVO: QJORVM PRiMVS ARTEM CVDENDAI MO NETAE: SECVNDVS VER.O QVAESTIONVM MO- NETARIARYM DECISIONS* CoNTINET. HIS ACCESSERVNT TRACTATVS VARII ATQVB YT1LES, NECNON CONSILIA, S I N G V L A RES QJ E A D D I- tionc tam vcteriim.quamNeotericorum Authorum/qui deMonctis,carundcmquevaIore,liga, pondere, poteftate,tnutationc,variatione,faliitate, aciimilibusfcripferunc. Quorum omnium Catalogumpagina duodecimo indicat. AVTHORE ET COLLECTORS CLARIS5. VIRO RENERO BVDELIO RVREMVNDANO, 1C. NECNON REVE- readifs.atque Illiiftrifs.Principisac D.Domini Ernefti Elc^tods Colonien Bauaciae Duc.&c. Monctarum,tam Rhe. nenlium, quam Vveftphaticarum Archie. pifcopaliura Przfedo. CVM SVMMARIIS ET INDFCE COPIOSO. COLONIAE AGRIPP1NAE, APVD IOANNEM GYMNICVM, SVB MONOCEROTE. A NNO M. D. LXXXXI. CVM PRIVILEO. CAES. MAIEST. AD SEXENNIVM. FIG. 198. TITLE PAGE OF BUDEL ET AL. 398 KARA ARITHMETICA The Arte of vulgar arith- meticke , both in Integers and Fractions, deuidcdinto typo 'Boofa'.wbereof the first it called WomodidaclttiNujneroruinj3A\d( the fecond PorttisProporthnttm. with ceiceine Demonftrations,t educed into fo plaine and perfect Me- thodjti the Iffy huth not hitherto beene pHbhfked in English Whtrtwti i i added a third Booke s cnutulcd zJMttfft Merc-itomm \ Com- prehending all the moil ncceflhrie and profiiablei^ulcs vfed in the trade tfidtr cba'iJiff^ In all which thrceBookes : theRules^Precepts^andMaxims^renoc cnely composed inmeeter for the better retaining oj them in mcmorie^ butalfothe operations, examples.demonHrations, and queftions, are in mo ft e*ft( wift ex founded and explanfd, in the forme ofa Diahguc,foi flse Readers more dcerc v&derttar ding. t^d htcwhdgefleAfwtfor Gentlemen^ commendable for CapttMfj and Soldiers, profitable for Merchants^ and generally tte c eflkriefor alleflata and degees. Newly colleftedjdigefted, and in fome part deuifed by a V?cl\\tl!cr to the Mathematical?. Ecdcfiafticus. cap, ip, Learning vnto fooles is as fetters on their feete and Manicles vpcn their right hand: but to the vvifeit isalcwcllof ^oldi^aiidljke a Bracelet vpon his right arme Bottiuslibr. i. Arith, rap. Atura con Hgc Wtmfaitfrmttptb in (wimo conditeru txawflar* Imprinted at Londo FIG. 199. TITLE PAGE OF THE 1600 HYLLES PRINTED BOOKS 399 Editions. London, 1592; ib., 1600, 4 (here described). This rather ponderous work of 570 pages is written in the form of a dialogue, following the popular textbooks of Recorde. Hylles intro- duces his rules and definitions in verse, presumably with the idea that they can be more easily memorized in this form. An illustration of this feature is seen in the following description of the first case of * Barters or Trucques,' a chapter now obsolete, but one of which we have a reminder in the words ' barter ' and ' truck ' : Of Barters or trucques, there are diuers kindes, Whereof the firft, is when the Trucquers take, But ware for ware, by agreement of mindes, No partie grating, greater gaines to make, Thequalitie of which exchange of wares, The compound rule afcending fole declares.' (F. 255.) The author seems to take up every rule known to the English arith- meticians of the time, and his book is a good source of information concerning British commerce of the period. It was not a popular work, probably because it elaborated its rules too much to be usable. JOANNES ANTONIUS MAGINUS. Ed. pr. 1592. Venice, 1592. GIOVANNI ANTONIO MAGINI. Born at Padua, June 13, 1555; died at Bologna, February n, 1617. He was professor of astronomy and mathe- matics at Bologna, and wrote numerous works on these subjects. Title. * Io.// Antonii // Magini Patavini // Mathematicarvm in almo // Bononienfi Gymnafio profefforis.// De Planis Triangv- lis // Liber Vnicus.// Eiufdem // de Dimetiendi ratione //per Quadrantem, & Geometricum Quadratum,// Libri Qvinqve.// Opus valde vtile Geometris, Aftronomis, Geographis, Mechanicis, Ar-//chitectis, Militibus, Agrorum menforibus, & denique //om- nibus Matriematicarum profefforibus.// Cvm Privilegio.// Vene- tiis, apvd Robertvm Meiettvm.//M. D. XCII.' (F. i, r.) Colophon. * Bononiae,// apud lannem Baptiftam Ciottum,// Typis Victorij Benacij,// Anno Domini, M. D. XCII.//Superi- orum permiffu.' (F. 114, v., and 132, v.) Description. 4, 16 X 21.6 cm., the text being 1 1.8 X 18.2 cm. 4 ff. unnumb. + 1 10 numb, (in the first book) + 124 numb. 4- 4 400 RARA ARITHMETICA unnumb. (in the second book) = 242 ff., 34-38 11. Venice (but printed in Bologna, as the colophon states), 1592. Editions. There was no other edition. Following some other bibliographers, I have included this treatise on plane triangles, because of the ' Expositio, ac vsvs tabvlae tetragonicae, feu Quadratorum numerorum cum fuis radicibus iuxta fequentes octo Canones ' (f. 5, r.) which it contains. The ' Tabvla nvmerorvm quadra- torum ' begins on f. 41 and extends to f. 64, inclusive. It is the most extensive table of squares and roots that had appeared up to 1592. THOMAS MASTERSON. Ed. pr. 1592-95. London, 1592-95. An English mathematician of the latter part of the sixteenth century. Title. See Fig. 200. Description. 4, 12.8 X 18.1 cm., the text being 10. i x 15.6 cm. 255 pp. (230 numb.), 3435 11. London, 159295. Book II (148 pp.) bears the date 1592 ; Book III (78 pp.) bears the date 1595. Editions. There was no other edition. There was, however, an addition to the * First Booke ' published at London in 1594. The author says in his dedicatory epistle, dated * London this 20 Auguft Anno. 1592,' that he has ' vnder taken to write and publifhe fixe bookes of the Art of Arithmeticke : with this order and methode, that the firft, third, and fift bookes, fhalbe as a fummarie and ground, teaching, the true ingenious, inuentions, and the perfect figuratiue and caractericall operations of the fame Art . . . Then the feconde, fourth, and fixt bookes, fhalbe of Arithmetical! queftions and demaundes, with the application of the definitions, common fenteces and inftructions of the firft, third, and fift bookes.' The plan was not carried out beyond the publication of the three books and the supplement mentioned above. Book I is on the fundamental operations with integers and fractions, and makes no advance in the ordinary textbooks of the period. Book II is a collection of practical problems representing the mer- cantile activities of London at the close of the sixteenth century (see Fig. 201). Book III would now be classed as part of algebra (see Fig. 202), since it refers chiefly to irrational numbers. Other works 0/1592. Buscher, p. 393, 1590 ; Capella, p. 68, 1499 ', Clavius, p. 375, 1583; Fischer (Piscator), p. 247, 1549; Gemma, PRINTED BOOKS THOMAS MASTERSON HIS FIRST BOOKE OF ARITHMETICKE. Shewing the ingenious inuentions , and figuratiue opera- tions , by which to calculate the true folution or anfwcres of Arithmcticallqueftions: after a more perfect, plainc, bricfc, well ordered Anthmeticall way , then any other heretofore publifhed ivcric neccflarie for all men. Nothing without labour. \^tll things uvitbrenfon. 401 Imprinted at London by Richard Ficld,d\velling in the Blacke friers ncare Ludgate. o 03 FIG. 200. TITLE PAGE OF MASTERSON 402 KARA ARITHMETICA p. 200, 1540; Nonius, p. 315, 1564 ; Psellus, p. 168, 1532 ; Ramus, P- 33 1 , !5 6 9; Riese, p. 139, 1522 ; Tartaglia, p. 279, 1556; Rutilio Cosentino Benicansa, ' Corona di tutte le scientie de Abaco,' Naples, m THOMAS MASTERSON HIS 5 . . 200 . . 3 Facit 120 . 4 . . 800 . 4 120,0 80 . . 2600 . . 12O Fuck 3poo. 400 ., 5- . . 2000 15)00 Tacit B tooke out 1 2 8 pound. j j8 Two marchants made a companie, A put in 300 pound for z monethes,and then putteth yet in 100 pound , and 6moncthes after that taketh out 200 poud,and with the reft remaincth vn- till the yeares end.B put in i oo pound for one moneth,and then putteth yet in 700 pound , and 6 monethes after that taketh out acertaine fnmme of money , and with the reft remaineth vntill the yeaVes endand then finde to haue gained together 400 poiid, whereof B muft haue 80 pound more then A , the queftion is how much money B tooke out of the companie, without recko- ning intereft vpon intercft. 3 oo . . 2 . . 600 400 i oo 80 400 . . 6 . . 2400 3 20 200 i 60 200 . . 4 , . 800 160 ' 38o~o . . 240 19 I 00 . .. J . . 100 Z. 00 800- 6 _. . 4 800 Facit B tookc out 6 4 o pound. 4900 800 i v FIG. 201. FROM MASTERSON 8 ; Julius Caesar of Padua, ' Gewiffe Erinnerung einer allgememen arithmetifchen Practic,' Cologne, 16 (see also p. 368, 1582) ; Johann Krafft, ' Ein neues vnd wohlgegriindtes Rechenbuch,' Ulm. PRINTED BOOKS 403 $ THOMAS MASTERSON HIS hauing here their number written oiler them. i. 2. 3. 4 ? & 7* 8. 9. 10. The firft caracter written thus ?, doeth fignifie any num- ber before which it is written, to be the firft number giuen, taken, or imagincdiand is called radix, or roote.for that all the other caractershaue theiroriginallorof-ipringof k. Thefe- cond written thus *, is called zenje or fquare, and doth fignifie any number before which it is written, to be the product of the firft multiplication of the roote by it fclfe : that is of the roote two times taken and multiplied. The third written thus c, is called cttfo, and doth fignific the number following the fame to be the product of the fecond multiplication of the roote,threc times taked and multiplied , that is of the ^ multiplied by the *. The fourth is called ^enfe^enfe t and dothfignifie the num- ber following the fame,to be the product of the third multiph- cation,that is the product of thzp foure times taken and mul- tiplied. The fift is called furjbluk, and doth fignific the number following the fame, to be the product of the fourth multiplica- tion. The fixe is called ztnfecube , and doth fignifie the number following the fame>to be the product of the fift multiplication, The (euenth is called bfarfolide ,. or fecond furfolide , and doeth fignifie the number folio wing the fame,to be the product of the fixt multiplication. The eight is te\\c&&nfczenfe*nfe t 8< doth fignific the number following thcfamc,to be the product of the feuenth multiplication. The ninth is called cubecubc , and doth fignifie, the number following the fame,to be the product of the eight multiplication. The tenth is called *,enfefurfobde, & doth fignifie the number following the fame , to be the product c. The eleuenth, is called cfurfolidejx third furfblide,The twelfth is called *jenfe*.cnfecube. The thirteenth is called dfurfolide^Qr fourth furfolide. You may proceede further at your pleafure, if you markc-that the fift caraftcr is the firft furfolide, the feuenrh the fecond furfolide, and the next vncompound number fol- lowingjthcnextfuriblidcjo'c fo infinitely cucr the next vncom- pound FIG. 202. FROM MASTERSON, SHOWING ALGEBRAIC SYMBOLS 404 KARA ARITHMETICA SIGISMUNDUS SUEVUS. Ed. pr. 1593. Breslau, 1593. A German priest, living at Breslau. Born c. 1550. Title. See Fig. 203. Colophon. ( Gedruckt zu Brefslaw/ durch // Georgium Baw- mann/J. Jn Mitvorle-//gung Andreae Wolcken. Jm lahre :// M. D. XCIII.' (P. 523.) Description. 4, 14.8 x 19.5 cm., the text being 11.2 x 15.8 cm. 524 pp. (455 numb.), 28-31 11. Breslau, 1593. Editions. There was no other edition. This work was probably intended to be a practical arithmetic, but the author's theological interests unfitted him for the task of writing such a book. Although the fundamental operations and the common rules of the day are treated in somewhat the usual way, the problems are largely biblical, or mystical. The size of Goliath's armor and Gematria used to foretell the famine of Poland represent the applications. ALESSANDRO ALAMAGNI. Ed. pr. 1593. Venice, 1593. A Venetian arithmetician of the sixteenth century. Title. See Fig. 204. Description. 12, 8.7 X 15.4 cm., the text being 6. 3 X 13.2 cm. I f. unnumb. + 96 numb. = 97 ff., printed in double columns, 30 11. Venice, 1593. Editions. There was no other edition. The word * tariffa ' in these early books means a table used by mer- chants to assist in their computations involving the measures and awk- ward monetary systems of the time. See Paxi, p. 77 ; Mariani, p. 180. Other works of 1593. Gemma, p. 200, 1540; Tartaglia, p. 279, 1556 ; Horatio Galasso, * Giochi di Carte,' Venice, 8, with an edition at Verona in 1597, 12, and a French translation in 1603 (containing several ' giochi d'abbaco '). MIGUEL GERONIMO SANTA CRUZ. Ed. pr. 1594 Madrid, 1643. A Spanish merchant and arithmetician of the second half of the sixteenth century, born at Valencia. He lived at Seville. Title. * Libro de // Arithmetica // especvlativa, y pra-//tica, intitvlado, el Dorado // Contador, contiene la fineza y reglas de PRINTED BOOKS 405 RITHMETICA HISTORICA, firnmtffe $Kr$f In/ k fcfcottw artwcf ftrfrtfem iftotfm tmb tcmgefo / Xucfc imt JJefrraifdjcr/ ridjifdjrr/ wib Otfmifcfcrr $j?an$f/r* Dnt c^ Drnrn Die nu(e mfcnm fdtmm/ trrgrn fc^ncn JJ)ff suammcn Sigifmundum Sueuum Freyftadicnfem* }iVnrr to* ft* oHlicfef n 3Bort$ OT&fafrn m' PUtt aunt Bcrnard ott tnft CVM GRATIA ET PRIVILEGJO. FIG. 203. TITLE PAGE OF SUEVUS 406 RARA ARITHMETICA TA R I F F A N V O V A, detli Cecckini , d* Lire died , e foldi Fino a # iz / 8. Laqual fcrie anco per ogni lorttdi Mcrcantie, 4 F A T T A DA M. Alcflandro Alamagni. IN V E N E T I A, ApprciTo Gio. Am. Raropazetto. 1 5 p j. FIG. 204. TITLE PAGE OF ALAMAGNI PRINTED BOOKS 407 contar//oroy plata, y los Aneages de //Flandes.//Por moder- no y com-//pendiofo eftilo.//Compvesto por Migvel Gero-//nimo de Santa-Cruz, natural de la Ciudad, y//Reino de Valencia, y vezino de // Seuilla.// Al L do Don Pero de Baraiz,//Tenience mayor de Corregidor de la infigne Villa de //Madrid, Corte del Rei nueftro Sefior.//Ano 1643 //Con licencia.// En Madrid, Por Francifco Martinez. //A cofta de luan BautiftaTabano, Mercader de libros, vendefe al//lado del Colegio de Atocha.' (F. i, r.) Description. 4, 13.8 x 20 cm., the text being 9.9 x 16.8 cm. 6 ff. unnumb. + 238 numb. = 244 ff., 32-34 11. Madrid, 1643. Editions. I know of no sixteenth-century edition, but the orig- inal privilege is dated May 8, 1594. This edition of 1643 seems by the dedication of Tabano to have been published after the author's death. There was an edition as late as 1794. The work follows the general plan of the Spanish and Italian mer- cantile arithmetics of the sixteenth century. It is not, however, a very practical book, the author having been too much influenced by the theoretical side of works like Tartaglia's. Other works of 1594. Barlaamo, p. 343, 1572 ; Masterson, p. 400, 1592 ; Recorde, p. 217, c. 1542 ; Reisch, p. 82, 1503 ; Wenceslaus, p. 421, 1599 ; Thomas Blundevile, 'His Exercises, containing sixe Trea- tises,' London, 4 (the first being on arithmetic ; there was an edition ib., 1597, 4, and others appeared after 1601) ; Smiraldo Borghetti, 'Opera d'abbaco,' Venice, 8 ; Jerdnimo Cortes, ' Arithmetica practica,' Valencia. Works 0/1595. Belli > P- 343> J 573 ; Helmreich, p. 306, 1561 ; Masterson, p. 400, 1592 ; Urstisius, p. 361, 1579. JOANNES BILSTENIUS. Ed. pr. 1596. Basel, 1596. A sixteenth-century German educator. Title. Syntagma // Philippo-//rameum // Artium Li-//bera- lium, Methodo brevi ac per-//fpicua concinnatum // per // loan. Bilstenium // Marfbergianum.// In // Gratiam Tyronum partim difficilibus // vocibus Germanica adiecta // est explicatio.// Basileae,//Typis Conr. Waldkirch.//CID 10 XCVI.' (F. i, r.) Description. 8, 10 X 16.1 cm., the text being 6.8 X 12 cm. 16 pp. unnumb. + 586 numb. = 602 pp., 26-28 11. Basel, 1596. Editions, There was no other edition. 4 o8 KARA ARITHMETICA This work consists of twenty parts, treating respectively of the lead- ing branches of knowledge as considered in the sixteenth century. Of these the eleventh is ' De Arithmetical and gives in 43 pages a succinct account of algorism. It begins with the catechism form : ' Quid est Arithmetica? Arithmetica, die Rechenkuft/ est ars bene numerandi. Subjectum Arithmeticae eft Numerus.' There is nothing progressive in the treatment, division, for example, being performed by the galley method only, and the applications being confined largely to the ' Aurea Regula' (the 'Golden Rule' of three). Other works of 1596. Helmreich, p. 303, 1561 ; Hood (translator ; see Urstisius), p. 361, 1579; Petri, p. 325, 1567; Ramus, p. 330, 1569 ; Recorde, p. 219, c. 1542 ; Snellius (see Ramus), p. 333, 1569 ; Tartaglia, p. 278, 1556 ; Urstisius, p. 361, 1579 ; Anonymous (William Parley (?), translator), ' The Pathway to Knowledge,' London, 4 (con- taining Thirty days hath September,' see p. 33) ; Anonymous, ' Arith- metica,' Frankfort, 8 ; Sebastian Brandt, ' Plenaria artis Arithmeticae refolutio,' Frankfort, 8 ; C. M. Glysonius, ' Arithmetica practica,' Venice, 4, with editions as late as 1783 ; Giacomo Trevisano, ' Me- moriale di abbaco,' Venice, 8 (title page 1597, colophon 1596) ; Bernardo Vila, ' Reglas breves de Arithmetica,' Barcelona, sm. 8 ; Antonio Rodriguez, ' Aritmetica pratica y theoretica,' Salamanca, 8. ANONYMOUS. Ed. pr. 1597. Luyck, 1597. Title. * Cleynen // Catholyken // Catechismvs.// . . . Tot Luyck by Jan Voes.// M.D.XCVII.// Met priuilegie van fes laren.' (F. i, r.) Colophon. 'Tot Luyck by Ian Voes, 1597.' (F. 12, v.) Description. 4, 9 X 14 cm., the text being 6.2 x n.8 cm. 12 ff. unnumb., 16-21 11. Luyck, 1597. This is one of many similar children's books in Mr. Plimpton's library. It is introduced here merely as a type of those primers that taught some work in number in connection with grammar or the catechism. In this book the child learns the Roman and Arabic numerals to 84. There are also in the library numerous Latin and Greek grammars of the six- teenth century in which numeration is taught, but these have been omitted from this bibliography. Other works 0/1597. Blundevile, p. 407, 1594 ; Bonocchio, p. 346, 1574 ; Buscher, p. 393, 1590 ; Galasso, p. 404, 1593 ; Gemma, p. 200, 1540; Trevisano, above, 1596 ; Francis Meres, 'God's Arithmetique,' London, 8. PRINTED BOOKS 409 HENRY DE SUBERVILLE. Ed. pr. 1598. Paris, 1598. A French scholar of the close of the sixteenth century. He describes himself as canon of the cathedral at ' Xaintes,' and ' Aduocat en la Cour de Parlement de Bourdeaux,' and signs his name with the birthplace ' Breton- Bearnois.' The dedicatory epistle is dated at ' Kimpercorentin.' Title. ' L' Henry-metre, // instrvment royal, et // vniversel, avec sa theoriqve,// vsage, et pratiqve demonstree par // les Pro- pofitions Elementaires d'Euclide, & regies familieres // d'Arith- metique : & auffi fans Arithmetique : Lequel prend toutes mefures Geometriques, & Aftronomiques, qui luy // font circu- lairement oppofees tant au Ciel, qu'en la Terre,//svr vne sevle station, par vn // feul triangle Orthogone, fans le bouger de fa place, ny aller // mefurer aucune diftance de ftation, ainfi qu'on eft con-//trainct de faire auec les autres Inftruments Geome- triques. //De 1'inuention// D'Henry de Suberuille Breton, Cha- noine en 1'Eglife Cathedrale S. Pierre //de Xaintes : & Aduocat en la Cour de Parlement de Bourdeaux.// Item,// Vn petit traicte fur la Theorique, & Pratique que de T Extraction des racines quarrees,//pour dreffer les Scadrons, & Bataillons quarres.// Dedies au Roy.// Diev a difpofe toutes chofes en Poids, Nombre, & // Mesvre. Sap. 1 1 . 2 1 .// A Paris,// Chez, Adrien Perier, rue fainct laques en la //boutique de Plantin au Compas.// I598.// Auec Priuilege du Roy.' (F. I, r.) Description. 4, 17 x 22.1 cm., the text being 1 1.8 x 17.8 cm. 39 pp. unnumb. + 225 numb. = 264 pp., 39-42 11. Paris, 1598. Editions. There was no other edition. Although this is primarily a treatise on mensuration by the use of a trigonometric instrument somewhat resembling a quadrans, it is properly included in this list because it contains several chapters on fractions and denominate numbers. These are of no special merit, and are introduced as a preliminary to the calculations involved in the use of the ' Henry-metre ' which the author invented. The book is an interesting effort to perpetuate an inventor's name by a work of no special scholar- ship describing an instrument of no particular value. There is, however, a value in the general study of all of these early instruments. Many of them are easily constructed, the quadrans for example, and their more extensive use in the teaching of trigonometry would be very helpful. 410 KARA ARITHMETICA JOHANN FRIDOLIN LAUTENSCHLAGER. Ed. pr. 1598. Freiburg in Uchtland, 1598. A Freiburg Rechenmeister of the latter half of the sixteenth century. Title. See Fig. 205. m un|i Dm <wfaf)cttDM fcgriffw/ o^annem SrtDoIittiim iaitfen? Magiftro VVilbelmo Mtft. Anno FIG. 205. TITLE PAGE OF LAUTENSCHLAGER PRINTED BOOKS 411 Description. 8, 9.4 X 14.7 cm., the text being 6.9 X 12.3 cm. 4 pp. unnumb. + 58 numb. = 62 pp., 21-30 11. Freiburg in Uchtland, 1598. 19 <?crunnt anea prof C2ttonum fie gat fcecfuenrticfj Rauff/begett fcref fctti<j /ftie ^rag jum Dtictcn; >as 2Bcrtb foQ ^t l^cn in ber mitten / ^inbeii/uonien gl<id> 'ttameit rtcfct / ^)ie Itnn &=e grofj all ?77u(t pltcter Die btn&cr Oev niittlcrcn in pro* ^ifar it mu|r, in We mitten gc ^en/ nutler oa&lra fommenfcU/ / fo fcajte getroffm w^ ? FIG. 206. FROM LAUTENSCHLAGER Editions. There was no other edition. This is the first arithmetic that I have seen composed entirely in rhyme. Various others had already contained verses, but this is 4 i2 KARA ARITHMETICA substantially all in meter. Like all such educational vagaries, it is weak in subject matter, the most difficult example in division being that of 65443 by 6. The illustration (Fig. 206) shows the rule of three in verse. CASPAR SCHLEUPNER. Ed. pr. 1598 Leipzig, 1598. A Breslau Rechenmeister, born at Niirnberg c. 1535. Title. See Fig. 207. Colophon. ' Gedruckt zu // Leipzig/ bey Frantz // Schnellboltz. //Typis haeredum Beyeri.// (Woodcut.) Im lahr// M. D. XCviij.' (F. 99, v.) Description. 8, 9.3 X 14.7 cm., the text being 6.9 X 12.2 cm. 99 ff. unnumb., 22-26 11. Leipzig, 1598. Editions. Leipzig, 1598, 8 (here described) ; Breslau, 1599, 8. Schleupner was one of the last serious advocates of the old ' line reckoning' with counters. He was a disciple of the Newdorffer school of Niirnberg Rechenmeisters, as he states in his preface, and had him- self taught at ' Neyfe ' and ' Brefzlaw ' for many years when he decided to write this book. He makes an attempt at an easy method of pre- senting the fundamental operations, hoping, as he says, to set forth the doctrines of Adam Riese and Johan Seckerwitz (another Niirnberg Rechenmeister) in simple fashion. The work is made up of a series of impossible questions and answers between a father and his son. The latter always begins with 'Geliebter Vater,' and the father's replies, while always clear, are often very extended, the entire 99 pages cover- ing little more than the four operations, together with a few insignifi- cant problems. As a simple presentation 1 of line reckoning, however, the book has few equals. JOSEPHUS UNICORNUS. Ed. pr. 1598. Venice, 1598. See p. 298. Title. See Fig. 208. * Parte Seconda,' of the same date, follows f. 204, the two being bound in one, and the pages being numbered consecutively. Colophon. * In Venetia,// Apreffo Francefco de' Francefchi Senefe.//M D XCVIII.' (F. 413, v.) Description. 4, 15.5x21.6 cm., the text being 10.4 X 1 7.6 cm. 12 ff. unnumb. + 395 numb. = 407 ff., 29-39 11. Venice, 1598. PRINTED BOOKS 413 Maifetit/ Lttfftwgwtfcrt, 3nn gcfcrawfyt, SDoraus rtt fleffctar &fer / ct ' aaff i?ter ma( CQeune/autt)ai&ig fan/ obne eini^cc CO^flnblic^w fre* Irictt/ f wv fic|) felbp / bie ^inien Ovec^nuitg Ictnen Ian/ feme (^peges aoffbcr Jebcw ' Icrnet Z)croglcict>cn ^leuor/ Jn 'Ccutfc^er ^)urcfc ff afpar cf j Don ^iJrmberg/Scubfclj . 3)?cnifigltct)cn/ n& toctfettiger. Saui 5llbrc*ts, FIG. 207. TITLE PAGE OF SCHLEUPNER L'ARITHMETICA VNIVERSALE . DEL SIG. IOSEPPO VNICORNO. MATHEMATICO ECCELLENTISSIMO. PartePrima: Ndlaquale fi contiene non folo la Theorica di tu tti i Hum eri. ma ancoralaPrattica appartcticntc a tuttti negocij humani. Trattata , d* amplifcatacon fimma erudition* ,econ ou'h & tfyui/tti modi di chiare%$4* CON PRIVILEGIO. FIG. 208. TITLE PAGE OF UNICORNUS PRINTED BOOKS 415 Editions. There was no other edition. This is one of the most elaborate treatises on arithmetic published in Italy in the sixteenth century. It consists of six books, the first four making up Part I. The first book treats in a detailed fashion of the fundamental operations. Unicornus, for example, gives six methods of multiplication, a treatment that recalls those of Paciuolo and Tarta- glia. There is a good discussion of the two general methods of dividing, the downward ('a danda') method having as much attention as the galley plan. Fractions are also treated in Book I. Book II deals with the theory of numbers after the Boethian method. Book III treats of roots, surds, and proportion ; Book IV, of the rules of three and false ; and Book V, of business arithmetic, including exchange, interest, and alligation. The work was too theoretical to be popular, but it is an excellent source for the study of the development of elementary mathematics. Unicornus gives a number of interesting historical references. Other works 0/1598. Cassiodorus, p. 2 1 1 , 1 5 40 ; Fischer (Piscator), p. 247, 1549 ; Moya, p. 310, 1562 ; Paolini, p. 393, 1589 ; Raymundus Lullius, * Opera ea quae ad adinventam ab ipso artem universalem . . . pertinent cum diversorum commentariis,' Strasburg, 1598, 8 (see also p. 457); Fabricio Mordente, ' Le propositioni di Mordenti,' Rome, 4 (geometric, but with a little arithmetic). ANTHON NEWDORFFER. Ed. pr. 1599. Nurnberg, 1599- One of the famous Newdorffer family of Niirnberg Rechenmeisters. Title. ' Ktinftliche vnd Ordentliche An//weyfzung der gantzen Practic vff // den Jetzigen fchlag vnd derfelbenn // herlichen gefchwinden Exempel // vff s kiirtzt zufammen getzogen &c.// Meinen lieben Difcipeln zu fon-//derlichem Nutzen geftelt. Durch // mich Anthonium Newdorffer //Rechenmaifter vnd Mo- dift der//Statt Nurnberg// Anno .M.D.IC.' (F. 2, r.) On f. i, r., is an engraved frontispiece with figures of Euclid and Py- thagoras, and a triangular multiplication table. On f. i, v., are 14 11. of verse ' Ad Stvdiosos Artis Nvmerandi.' Colophon. ' Gedruckt zu Nurnberg/ // durch Paulum Kauff- mann.//M. D. XCIX.' (F. 65, r.) Description. 4, 15 x 19.3 cm., the text being 11.4 x 15. 5 cm. 65 ff. unnumb., 22-32 11. Nurnberg, 1599. 416 RARA ARITHMETICA Editions. There was no other edition. This is an excellent illustration of the work of the celebrated Niirn- berg Rechenmeisters. It is divided into twelve books, of which Das erfte Buchlein handelt von der Venetianifchen oder Kauffmennifchen Practic,' usually called Welsch practice by the German writers. It is composed chiefly of problems, the rules and explanations being left for the master. Directions are given, however, in the case of fractions. Book II relates to denominate numbers and the ' Regula de Tribus.' Book III treats of the weighing of commercial products. Book IV ' Handelt von der Rech- nung eines Caffierers,' including exchange. Book V is entitled ' lornates, Das ift/ Rechnung von allerley Handtierung,' and consists of practical problems relating to the purchase of goods. Book VI ' Tractiert von der Regel Converfa vnd Quinque,' inverse and compound proportion. The rest of the twelve books are devoted to practical business questions of the day. OBERTO CANTONE. Ed. pr. 1599. Naples, 1599. A Genoese arithmetician of the sixteenth century, residing in Naples. Title. See Fig. 209. Description, Fol., 14.5 x 20.1 cm., the text being 10.4 X 15.7 cm. 304 pp. (292 numb.), 34~37 n - Naples, 1599. Editions. This is the first edition of this work, three other editions appearing in the seventeenth century. Arithmetics written by Genoese masters in the sixteenth century are rare, even in manuscript form, and this is probably the only such work printed in Naples. Genoa was a mercantile center, but its dialect was not conducive to the success of a textbook, as is seen in the case of Zucchetti's treatise (p. 425). Naples was too far from the path of international commerce to produce many mercantile works. But although Oberto Cantone was a Genoese, he was a ' professor delle dis- cipline matematiche ' in Naples, and he dates his dedicatory epistle ' Di Napoli li 15. luglio 1599.' Naples, however, had a commerce of its own, even if not as extensive as that of Venice or Florence, and by the close of the century it was natural to expect works of this kind. The book is mercantile and is based upon Borghi and similar writers of the North. It gives our present method of multiplication, but makes no use of the Venetian or Florentine names, with the exception of * per colonna.' Two or three short methods are given, but no such extended treatment of varied forms appears as in Tartaglia, or his great prede- cessor Paciuolo. Division, except in simple cases, is postponed to p. 142, L VSO PRATTICO DELL'ARITMETICA OBERTO CANTONE DA GENOVA> PROFESSOR DELLE DISCIPLINE Maecmatiche . QTALE CON NVQVA INVENTIONS sinfegna in mat ma di conti , I'vfo tanto delta Regia Camera delta Sommaria, quanta di Negotiant^ Mercadanti&Artegiani. come Nafolicamby, f recambijin ciafcuna piazza. CON PRIVILEGIO. o w P TARQJTINIO LONG HO. 2^. D. Si vendono dal ir.edcilir6 Aucorc a Banchi nuoui. FIG. 209. TITLE PAGE OF CANTONE 418 KARA ARITHMETICA and is there treated in the modern form, ' a danda.' Most of the applied problems are in exchange, although a few other types are given. The book is poorly constructed, being too prolix in the treatment of the operations, and too narrow in its applications. JOANNES MARIANA. Ed. pr. 1599. Toledo, 1599. Born at Talaver^ de la Reina, in 1536; died at Toledo, February 17, 1624. He was a Spanish Jesuit and a famous historian. Title. See Fig. 210. Colophon. 'Toleti, Apud Thomam Guf-//manium, Anno . 1 599.' (P. 206.) Description. 4, 1 4 x 1 8.6 cm. , the text being 9.3 X 1 5 .6 cm. 6 pp. blank + 8 unnumb. + 192 numb. = 206 pp., 26 11. Toledo, 1599. Editions. There was no other edition. The author is not the same as the Giovanni Mariani mentioned on p. 1 80, but a Spanish Jesuit of some fifty years later. The work is on the history of the weights and measures used in Spain in the sixteenth century. It traces these measures from the Roman, Greek, and Hebrew sources, and is valuable for the study of the history of the subject. It closes with a table of comparative measures. MARTIN WENCESLAUS. Ed. pr. 1599. Middelburg, 1599. A Dutch arithmetician of the sixteenth century. Title. See Fig. 211. Colophon. ' Eynde defes onfes eerften Vo-//lumens ; Ghedruct int Jaer ons Hee-//ren 1599. ^ n defe vermaerde Coop-//ftadt van Middleburgh in // Zeelandt.'// Also in French : ' Fin de ce noftre premier Volu-//irien: Imprime 1'An de noftre Seig-//neur 1 599. A la tref-renommee // ville Marchande de Middel-//b>ourg en Zeelande.' (F. 157, r.) Description. 4, 15.2 X 20.1 cm., printed in double columns, one in Dutch and the other in French, the Dutch being 6.7 X 15.7 cm., the French 5 X 15 cm. 4 ff. blank + 18 unnumb. + 137 numb. = 159 ff. Dutch: 30-41 11.; French: 16-26 11. Middelburg, 1599. Editions. This is evidently the first edition, since the dedica- tion is dated 'Den 10. dach van Decem. ftijlo nouo. Anno Chrifti. PRINTED BOOKS 419 JO ^KKIS MARIANAE Hifpani, E SOCIE* IESV, DE P O N<DE Rl%rs ET menfaris. Anno CVM PRIVILEGIO. Volet i, lAfudThornxm Gufmanium. FIG. 210. TITLE PAGE OF MARIANA 420 KARA ARITHMETICA T'FONDAMENT Ueicnfc^e ^actijcb / mtDtfgaDeta bailee nbotturnatrtjftc (luchm tian&cnUrgljdtian met r tDdicUe ow r tcnft cmmmglje of re Conto^Dantiett* MART1NVM VVENCESLAVM, AQVISGRANENSEM. LE FONDEMENT DE DARITHMETIQVE AVEC LAPRACTICQVE ITAL1ENNE ENSEM, ble Ics pieces , les plus necefsairesde la BsCglc d'Imercft. ENSEMBLE EN BAS ALLEMAN ET EN !ran$ois >avec raifonnable Harmonie on Concordance. tl TOVT Par <JfA MIDDELBVRGH. Cn^e men tomtfe te c oope/ bp Adriaen uan be Vivere . 25oer febercooper/ UiooncuDc bp be nieutot ^urf^/inDen bergulom 23plicl, FIG. 211. TITLE PAGE OF WENCESLAUS PRINTED BOOKS 421 1598,' 'le 10. lour de Decembre ftylo nouo. Anno. Christi- 1598.' The long and stupid preface is dated November 30 of the same year. Wenceslaus had already published two works before this one, the ' Proportionale ghesolveerde Tafelen van Interest,' 1 594, 8, and the ' Boukhoudens Instruction,' 1595. The book is interesting, both because of its arrangement of the Dutch and French texts in parallel columns, thus serving a purpose in lan- guage teaching in a bilingual country, and because of the mercantile problems which reveal, as is particularly true of the Dutch books, the contemporary life of the people. Other works of 1599. St. Augustine (see Reisch), p. 82, 1503 ; Barlaamo, p. 343, 1572; Benedetti, p. 364, 1580; Boethius (see Reisch), p. 82, 1503; Bungus, p. 384, 1583-84; Capella, p. 66, 1499 ; Clichtoveus (see Reisch), p. 82, 1503; Faber Stapulensis (see Reisch), p. 82, 1503; Finaeus (see Reisch), p. 82, 1503; Gallucci (see Reisch), p. 82, 1503; Jacob, p. 298, 1560; Jordanus (see Reisch), p. 82, 1503; Ramus, p. 330, 1569; Reisch, p. 82, 1503; Schleupner, p. 412, 1598; Striibe, p. 391, 1588; Johann Heere, Rechenbuchlein von allerhand gebraiichlichen Fragen,' Niirnberg, 8 ; Andreas Reinhard, Drey Regifter Arithmetifcher Anfang zur Practik,' Leipzig, 8, with a second edition in 1600. JACOB VANDER SCHUERE. Ed. pr. 1600. Haarlem, 1600. A Dutch arithmetician of Meenen, c. 1550-1620. Title. See Fig. 212. Description. 8, 9.4 X 14.4 cm., the text being 7.2 x 12.6 cm. 2 ff. unnumb. + 202 numb. = 204 ff., 22-23 11. Haarlem, 1600. Editions. That the date of the first edition is 1600 appears in the Voor-Reden of the 1625 edition, where his son, Denys, says that the book was published by the father ' eerst in't laer 1600. ende dit is al de vierde mael dat het gedrukt is.' There were various editions in the seventeenth century, including the following: Haarlem, 1611 ; Rotterdam-Schiedam, 1624, 8 (p. 423) ; Haarlem, 1625 ; c. 1630, 8 (p. 423) ; Gouda, 1634, 8 (p. 424); Amsterdam, 1643, 8 (p. 424) I Rotterdam, 1653, 8 (p. 425); Amsterdam, 1675. 422 KARA ARITHMETICA ARITHMETICA, l^eten tonff/ toed fcljoone C'jrempelen/feer nut boo? alle Coop* Uefen/ f arteurjs/ Caflter^ / <8ntfan- fitters/etc. IAQVES VAN DER SCHVERE VAN MEENEN. Ku ter tijJtFrancoyfche School-meefter tot HAERLIM. FIG. 212. TITLE PAGE OF VANDER SCHUERE PRINTED BOOKS 423 Vander Schuere's work on bookkeeping is mentioned under the later editions (p. 424), although it was not published in the sixteenth century. No effort has here been made to complete the list of seventeenth-century editions. This is one of the many practical arithmetics that appeared in Holland about this time. It takes up the fundamental processes, rule of three, fractions, the rule of practice, partnership, commissions, in- heritance problems, profit and loss, interest, exchange, barter, alliga- tion, and the various other rules in use at that time. It was one of the most successful Dutch textbooks. JACOB VANDER SCHUERE. Ed. pr. 1600. Rotterdam-Schiedam, 1624. See p. 421. Title. ' Arithmetica//Oft// Reken-konft;//Verciert met veel fchoone // Exempelen/ zeer nut voor alle vlijtighe // Oeffenaers ende leer-ghierighe Aenvanghers// defer Konft/ etc. Ghemaeckt door // lacob Vander Schuere, Meenenaer,// Nu ter tijdt Fran- foyfche. School-meefter//tot Haerlem.//Ende nu int herdrucken overfien ende ghebetert. // (Woodcut of Vander Schuere with motto: Door siet den Grond.)//Tot Rotterdam,// Voor Pieter van Waefberghe/ //Anno 1642.' (F. i, r.) Colophon. 'Tot Schiedam, //Ghedruckt by Adriaen van Delf.// Anno 1624.' (F. 208, r.) Description. 8, 8.7 X 14.7 cm., the text being 7.6 x 12.9 cm. 2 ff. unnumb. + 2o6 numb. = 208 ff., 22-27 H- Rotterdam- Schiedam, 1624. See above. JACOB VANDER SCHUERE. Ed. pr. 1600. S. 1. a. (?), c. 1630. See p. 421. Title. ( Arithmetica oft Reken-konst.' This edition is without date, the title page being missing, but is probably c. 1630. F. I is missing. 9 X 1 5.6 cm., the text being 7.3 X 12.2 cm. 5 ff. unnumb. + 252 numb. = 257 ff., 35 11. 4 2 4 KARA ARITHMETICA JACOB VANDER SCHUERE. Ed. pr. 1600. Gouda, 1634. See p. 421. Title. ' Arithmetica // oft // Reken-konf t . // Door lakob vander Schuere, Meenenaer // Eertyts Frangoysche Schoolmeefter//tot Haerlem,//En in't herdrucken by hem ouerfien // verbetert en vermeerdert, en noch by-//geuoecht een kort onderricht van // 'tltaliaens Boeck-houden // ter Goude//By Pieter Rammefeyn, Boeck-verkooper // inde Korte Groenen-dal, in't Vergult A, B, C. //a 1634. W Akerfl : fee.' (The title page is elaborately en- graved, with a portrait of the author surrounded by the following: ' Doorseit den Grond lakob Vander Schvere aet 50.') (F. i, r.) Description. 8, 8.8 x 14.2 cm., the text being 7 x 12.7 cm. 8 ff. unnumb. + 208 numb. = 216 ff. Bound with this is ' Kort onderricht //over het // Italiaens // Boek-houden ; // Nu int licht ghebracht // Door lakob vander Schvere,' etc., with I f. unnumb. + 37 numb. = 38 ff., making a total of 254 ff. in the book, 28-33 11- Gouda, 1634. See p. 423. JACOB VANDER SCHUERE. Ed. pr. 1600. Amsterdam, 1643. See p. 421. Title. 'Arithmetica //oft // Reken-konf t.// En een kort onder- richt van't Italiaens Boeckhoude // Door lacob vander Schvere Meenenaer. // By den Autheur overfien, verbeetert en vermeer- dert. //Tot Amsterdam, //voor Michiel de Groot Boeckverkooper // op de nieuwendyk inde Bieftkens Bybel. 1643.' (Engraving of author with the following wording : ' lacob vander Schvere Mee- nenaer. Out67.Iaer. Doorsiet denGrondt. Anno 1643.') (F. i,r.) Description. 8, 9.5 X 14.7 cm., the text being 7.2 X 12.5 cm. 272 ff. (216 numb -f 8 unnumb. in the arithmetic), 26-31 11. Amsterdam, 1643. Bound with this is ' Kort onder-richt //Over het //Italiaens // Boeck-houden.// In't Licht gebracht // door // Jacob van der Schuere. //t'Amsteldam,// By Michiel de Groot, PRINTED BOOKS 425 Boeckverkooper // op de Nieuwendijck/tuffchen de twee Haer-// lemmer Sluyfen. A. 1675.' (F. i, r., of the bookkeeping.) See p. 423. JACOB VANDER SCHUERE. Ed pr. 1600. Rotterdam, 1653. See p. 421. Title. * Arithmetica // ofte // Reken-konft, // En een kort onder-richt van't Italiaens // Boeck-houden.// Door // lacob van der Schuere Meenenaer.//By den Autheur over-fien, verbetert en // vermeerdert.//Tot Rotterdam, //Gedruckt by Pieter Waef- berge, woo-//nende op't Steyger/ in de gekroonde // Leeuw/ Anno 1653.' (F. J > r -) Description. 8, 8.7 x 14.1 cm., the text being 7 x 12.1 cm. 275 pp. (219 numb. + 8 unnumb. in the arithmetic), 26-33 H- Rotterdam, 1653. Bound with this is 'Kort onder-richt // over het // Italiaens Boeck-houden. // In't Licht gebracht // door // Jacob van der Schuere.// Tot Rotterdam, //Gedruckt by Pieter van Waefberge,//Ordinaris Drucker/ woonende op't Steyger// in de gekroonde Leeuw/ Anno 1653.' (F- J > r -> f tne book- keeping.) See p. 423. GIOVANNI BATTISTA ZUCHETTA. Ed. pr. 1600. Brescia, 1600. Born April 21, 1550. A Genoese arithmetician. Title. See Fig. 213. The privilege is dated ' In Genoua nel di 30. di Genaro. M. CCCCCC.' Description. Fol., 24.2 x 33.4 cm., the text being 17.1 X 26 cm. 444 pp. (412 numb.), 33-37 11. Brescia, 1600. Editions. There was no other edition. Brunet says this ' Prima Parte ' is the only one that appeared. The work has several interesting features, not the least one being the apology ' Al generoso lettore,' in which the author speaks of the criticism liable to be directed against a Genoese author on account of his pro- vincial Italian. The ' Prologo ' is a curious dissertation on the Arti, 426 KARA ARITHMETICA ato che rifolue qualunque qneCto bifogaofo a ZecchierijOrefici,* ArgentarL Cvi CO 7^ FIG. 213. TITLE PAGE OF ZUCHETTA PRINTED BOOKS 427 Scienze, & altro,' with some ninety-eight arguments to show the need for arithmetic on the part of all classes of humanity. The farmer, the musician, the thief, the cook, the prelate, all are shown to have need of number ; and Nature, Intelligence, and even God himself make use of it. The book presupposes a knowledge of the arithmetic of integers, and opens with a treatment of fractions. The rule of three, in all of its forms, and with most unbusinesslike numbers, is then discussed at great length, and this is followed by various complications of the Regola del Cattaino, ' cosi detta da gli Arabi inuentori di quello, ch' in lingua nostra fignifica falfa pofizione.' The latter part of the book treats of such topics as partnership, barter, and alligation. The work was not of a nature to have any influence on Italian arithmetic. WILHELM SCHEY. Ed. pr. 1600. Basel, 1600. A German Rechenmeister at Solothurn. Born c. 1560. Title. See Fig. 214. Description. 4, 15.1 X 19.4 cm., the text being 10 x 15 cm. 486 pp. (470 numb.), 36-37 11. Basel, 1600. Editions. That this is the first edition appears from the dedi- cation : 'Datum den 12. Septembris. Anno Domini 1600.' The book is an attempt at a complete commercial arithmetic, but is not well constructed. The author likes to arrange his computations in a bizarre fashion, for effect. He extends his explanations unduly, and for so large a book the mercantile information is not as complete as it should be. There is a curious arrangement of the figures in the proofs by casting out nines, as here shown, although whether this is a typographical matter or a notion of Schey's there is nothing to indicate. In the ' Regula Falsi ' there is a rather early use of -f- for the minus sign. In general the book is reactionary, giving only the galley division (with much attempt at effect), and mentioning, although not treating, ' Duplatio ' and ' Mediatio.' Other works 0/1600. Barlaamo, p. 343, 1572 ; Buscher, p. 393, 1590; Chambers (see Barlaamo), p. 343, 1572; Herodianus, p. 60, 1495 ; Hylles, p. 396, 1592 ; Jacob, p. 298, 1560 ; Lonicerus, p. 253, 1551 ; Ramus, p. 263, 1555 ; Reisch, p. 82, 1503 ; Reinhard, p. 421, 1599; Schulze, p. 383, 1584; M. van den Dycke, Chyfer-Boeck ' (second edition, I do not know the first), Antwerp, 8, and ' La vraye reigle d'Arithmetique ' (a translation of the other?), ib., 8; Georg Gleitsmann, Kiinftliches Rechenbuch fowohl auf Linien als mit Ziffern 428 KARA ARITHMETICA ARITMETICA Att^ ttd^ttttt titttb ^itt^di *>&cr izoo. (^jrcmpl<i ri&cit/bermafrm/l>a8Ctn ^M ^ W Wuntttic^en ferid^ t/bon im fcl&tj/ WPl ncn anfafxntm ^cfyuUrn/imft cimm K^n ^cgimg Anno M DC FIG. 214. TITLE PAGE OF SCHEY PRINTED BOOKS 429 nach defs Rami Arithmetica geftellt,' Frankfort, 8 ; M. Johann Taf, ' Schones neues . . . Rechenbuch,' Cologne, 4. There was also pub- lished, s. a., but c. 1600, a work by Vincenzo da Bergamo, ' Arith- metiche instruttioni.' There were many other arithmetics published in the sixteenth cen- tury without date, including the following : Anonymous, ' Art et Science de Arismetique,' Paris, 12, 96 ff. (Boncompagni sale); Anonymous, * Livre des gects,' s. 1., 4 (see p. 130, 1520) ; Anonymous, ' Abbaco di succincte dimostrazioni,' s. 1. (Milan?) (Brunet) ; Anonymous, ' (Q)ui apresso e inaci col nome di dio intedo di tractare e scriuere alquatimo di e regole sopra larte del numero altrimeti chiamato algurismo,' s. 1., 33 ff. (Riccardi) ; Joachim Ammonius, ' Isagoge Arithmetices . . . cum praefatione P. Melanchthoni,' Wittenberg ; Angelus Mutinens, p. 140, 1525 ; Sarafino da Campora, * Delia ragione dell' Abbaco,' and a work on the calendar (Messina, 1559, and Rome, 1560) (Riccardi); Lauro Quirini, ' De mysterio numerorum ; ' Matteo Ricci, ' T'ung-wen suan- chih ' (Practical arithmetic in 1 1 books, the work of a Jesuit missionary in China, b. 1552, d. 1610; possibly not published during his lifetime). PART II MANUSCRIPTS ^tmar cnic Uncc tatt futt;"ttcttlt PLATE IV. FROM THE CAMPANUS MANUSCRIPF OF EUCLID, c. 1260 MANUSCRIPTS EUCLID. Latin MS., c. 1260. See p. n. Title. ' In hoc libro otinet' geomet'a euclidis // cQ omento magri campani.' (F. i, v.) Colophon. ' C.Explicit geometria euclidis cu comen//to magiftri campani.' (F. 165, v.) Description. Fol., 17.8 x 25.8 cm., the text being 8.5 x 17.2 cm. besides the marginal figures. 165 ff., 23-44 11. Written on vellum c. 1260. Editions. This work being primarily a geometry, I have not given a list of the editions. For the arithmetical books, which (except for Book V, on proportion) are not included in this manuscript, see p. 237. This is a Latin manuscript of Euclid, with the commentary (proofs) of Campanus, written on vellum about 1260. The translation of the theorems is that made by Adelard (^thelhard) of Bath, c. 1 120, but in the early printed editions of the ' Elements ' it is generally referred to as that of Campanus. Of Campanus himself not very much is known. His first name was probably Johannes, and he is known to have prepared a set of planet- ary tables, and to have been chaplain to Urban IV. (See Boncompagni's Bulletino, I, 5, and XIX, 591.) This manuscript has been studied by Mr. C. S. Peirce (Science, N.S., XIII, 809), who believes it to be the copy given by Campanus himself to Jacques Pantaleon when the latter was Patriarch of Jerusalem, hence before August 29, 1261, when he became Pope Urban IV. He bases his belief on a sentence written in a cursive hand just below the colo- phon, containing the words ' Jacobus Dei gratia Patriarcha Jerusalemi- torum.' The complete sentence is : 'In noTe dm ame Jacobus dei g ia 433 434 KARA ARITHMETICA patriarcha Jerufalemitar omibus xpi fidelibus falutem defiderium,' ' In the name of the Lord, amen; Jacob by the grace of God patriarch of the Jerusalemites to all the faithful of Christ, greeting and love.' While this seems more of a blessing or quasi imprimatur than a mark of ownership, it is equally valuable in serving to fix the date. On f . i , r., is the inscription in a fourteenth-century hand, ' mgri adolphi di Werda,' and a statement that the manuscript belonged to the Phillips collection, no. 4633. On f. 165, v., is an inscription in an English hand of c. 1400, ' lib 6 ifte fuit Di armachani,' ' This book belonged to Dominus Arma- chanus.' (For a page of this MS. see Plate IV.) ANICIUS MANLIUS SEVERINUS BOETHIUS. Latin MS., c. 1294. See p. 25. Title. The manuscript begins : ' Incipit prolog in arithmetics boetii.' (F. i, v.) Colophon. ' Explic* arifme ca // boetij ad fymacu^ patriciu c .' (F. 28, r.) Description. Fol., 19.8 X 27 cm., in double columns, each being 7 x 20. i cm. 28 ff., 41 11. Written on vellum c. 1294. Bound with the Euclid described below but in a different hand. It is beautifully written and illuminated. The contemporary pigskin binding has the inscription, 'lib' arifmetice boecij.' The text is practically that followed by the Friedlein edition (Leipzig, 1867). (See Plate I.) Editions. For printed editions see p. 27. See p. 27. EUCLID. Latin MS., c. 1294. See p. ii. Title. The work begins without title : ' Punct^ eft cui9 ps He.' (F. 29, v.) Colophon. ' Explicit geometri euclidis cum comto campani.' (F. in, v.) Description. Fol., 19.8 x 27 cm., the text being 17.5 X 19.5 cm. in ff. (the first 28 ff. being the arithmetic of Boethius MANUSCRIPTS 435 described on p. 434), 44 11. Written on vellum, c. 1294. It has beautifully executed figures and is a fine specimen of the work of the mediaeval scribe. It is written in a different hand from that of the Boethius with which it is bound. The cover, which seems contemporary with the manuscript, has the number cclxxxxiiij, possibly for the date 1 294, the M being omitted as is often the case. (See Fig. 215 and Plate V.) ANICIUS MANLIUS SEVERINUS BOETHIUS. Latin MS., c. 1300. See p: 25. Title. The manuscript begins: Tncipit libr arismetice art . . .' Description. Fol., 13 x 18.6 cm., the text being 8.9 x 14.1 cm. 37 ff., 32-36 11. Beautifully written on vellum. The work is complete, and, like the manuscript described on p. 434, this shows a text very similar to that followed by the Friedlein edition. At the end of the manuscript are two folios of com- mentary, closing with two almost illegible lines containing the words ' Com . . . campani . . . ,' referring to the commentary of Campanus. Roman numerals are used throughout the text, which was not always the case in manuscripts of this date. (See Plate VI.) The commentary, which seems to have been added about a century later, has some Hindu-Arabic numerals. PAOLO DAGOMARI. Italian MS., c. 1339. PAOLO DELL'ABACO, PAOLO ASTROLOGO, PAOLO GEOMETRA, PAOLO ARIS- METRA, PAUL OF THE ABACUS. Bom in Prato, c. 1281 ; died at Florence in 1374, or, according to some writers, in 1365. He was a celebrated Flor- entine arithmetician, 'geometra grandissimo, e peritissimo aritmetico . . . diligentissimo osservatore delle Stelle, e del movimento de'cieli,' as Villani (Le Vite cT Uomini ilustri Fiorentini) calls him. Title. ' Trattato d' Abbaco, d' Astrono-//mia, e di segreti natu- rali //e medicinali.' (F. i, r.) Description. Fol., 21.7 X 29.3 cm., the text being 15 X 21.4 cm. 138 ff. (7 blank), 32-35 11., clearly written on paper, c. 1339 (possibly copied later). 436 RARA ARITHMETICA " UJTO mCOn Hat eft' ^a biojor^ cw > ftt^mtiftfo.l lm"7 Vim aranim ^V o <f Ht <n 4 fovlar- m Jnmft vV awufl f&rquoufip nu jiff \^tt* x ? ! {~. *~nt - ' " ^w 11 * nn mcMctaa c i yar / V * i -1 ' * t wyd.* A ? l>-w < FIG. 215. FROM THE 1294 MANUSCRIPT OF EUCLID Bernardino Baldi testifies to the esteem in which the author of this work was held, in the following words : ' Di patria Fiorentino fu Pauolo ; il quale, per 1' eccellenza ch' egli hebbe ne le Matematiche, lasciato il proprio cognome, fu chiamato da tutti il Geometra. Come apunto fra gli antichi auenne ad Apollonio Pergeo.' This manuscript is primarily a treatise on arithmetic. The writer, however, left a number of blank pages at the end, and these have been filled in from time to time by various owners. \vdWf-i?.iu ft-'wruifiTT -Ouf . U.TiuiijT 4 - P" cqleui ne.< f.lun-nuicr >M ,leiHnc.< f.UinTiiug-^m O.j,^. .^ihniOTnu/intJJ aatmwnn.mouqaw)umiU'Utq7^'-<:nrq,-^Y lvts -^ 1|! lt > t I Mni<qbpa.vq ^rtnt^tnll)^<^irtm^.^f-tq^ui4.nuyAdt|.v't^l 1 an(f.i.ap.5v5.bujlciiKi Ct sf'fr u'innlF-7nirli3io co^ithfmi n^nni(.l>f'j|hntnjtF\'>tt^'f8^'i icUoawji rocfctucf U^tnu '^* V/ Ki ^, fpaUflDyu.-!^ tqdrthtaau; ""Sileto Pw^W^-lSoSKSfah^^Sa^ttS^ PLATE V. FROM A MANUSCRIPT OF EUCLID, c. 1294 MANUSCRIPTS 437 The arithmetic begins (f . i , r.) : ' El nome sia di Dio et a reverentia //della fuo potentia et della fanta trinitade. Et dello fuo madre// uirgino fempre fanta maria Et del beato Sco Giouani batifto // . . . Al chominiciamento del noftro trattato . . .' On f. i, r., is a table of V T FIG. 216. FROM THE DAGOMARI MANUSCRIPT money. F. 2, r., f. 3, v., is a table of contents : ' Quefti sono echapitoli del noftro tractato,' and this, states that the work includes the * Re- gholvzze del Maestro Pagolo Astrologo.' (ff. 121, r., 131, r.). This part of the volume is an ordinary commercial arithmetic such as the Florentine teachers pro- duced in the fourteenth and fifteenth centuries. The writing and the forms of the numerals indi- %' 1 I -\ 9 cate this period, and are not unlike that of a fourteenth-century computus (see p. 445) in this library. This may, however, be a fifteenth-century copy. The exam- ples in division are unique, since they follow neither the galley nor the 'a danda' method, as is here shown in the case of 49289^-23 =2143. The peculiar position of the remainders should be noted. The o indi- cates no remainder, and the 2,2, and i are excesses of y's in the proof. 438 KARA ARITHMETICA That the author considers this the ' a danda ' plan (the forerunner of our present long division) appears in his use of this name immediately after. Another odd feature is the placing of the divisor second in the FIG. 217. FROM THE DAGOMARI MANUSCRIPT division of fractions, it usually appearing first in that period. The problems are of the usual Florentine mercantile character. The first few are followed by a treatise on 'Nvmeri perfetti ' (f. 67, r.), and this MANUSCRIPTS (f. 73, r.) by further business problems. The work curious illustrations (Figs. 217, 218). The most interesting feature of the treatise is, however, the internal evidence as to its date. It is usually possible to determine quite accurately the date of a Florentine manuscript on arithmetic by the examples in equation of payments, a favorite application with the Tuscan arithmeticians, and one requiring the year to be stated. This is the case here, where the dates in these examples are all, save two, 1339. The writer has also used approximately these dates in other examples, in part as follows: 1329- 1332 (a problem on the calendar, f. 27, r.) ; 1310- 1404 (an astronomical table, which would naturally extend well into the future, f . 79, r.) ; 1 330 (f. 1 1 6, r.), and 1339 frequently. A table of the ig-year cycle (f. 123, v.) begins with 1337, as is easily computed from a marginal note in a different hand bearing the dates 1394 and 1412. This part of the work ends on f. 131, r., and there can be no doubt that it was written about 1339, the date so frequently used in the problems. F. 1 2 1, r., begins ' Regholvzze del Maestro Pagolo Astrologo,' Dagomari being referred to in the table of contents as Paul of the Abacus, ' Regholuze del Maeftro pagholo delabacho . . .' The ' Regholvzze ' was first published in Libri's Histoire, vol. Ill, p. 296. Dagomari is included in the list of Bernardino Baldi's (1589) biographies, published in the Bon- compagni Bulletino, XIX, 600. Ff. 131, v., and 132, r., are in a different hand. These and the following leaves were originally blank, and after ff. 132, v., 133, v., had been written upon, some owner used these two blank pages. He has also left his dates, viz. 1400, 1402, 1406, 1412, 1435, the 1402 being used several times. This was therefore written about 1400-1435. In this occurs the first per-cent sign I have met, other than p. i oo, which is in the earlier part of this volume. This unknown writer of about 1425 uses a symbol which, by natural stages, developed into our present %. 439 contains several FIG. 218. FROM THE DAGOMARI MANUSCRIPT 440 KARA ARITHMETICA Instead of writing 'per ioo/ 'p 100,' or 'p cento,' as had commonly been done before him, he wrote ' p <_? ' for ' p c,' just as the Italians wrote !,,... and i, 2, . . . for primo, secundo, etc. In the manu- scripts which I have examined the evolution is easily traced, the <_? becoming g about 1650, the original meaning having even then been FlG. 219. FROM THE ADDITION (c. 1400-1435) TO DAGOMARI lost. Of late the ' per ' has been dropped, leaving only % or %. See Figs. 219, 220. Ff. 132, v., and 132, r., are in a different hand. They refer to the calendar, and contain the dates 1380 and 1382, each twice. F. 133, v., is in a still different hand, although also on the calendar. The date of this part is fixed by the expression, ' et i ifto prefeti aho. f. 1447 finit circuluf anni.' EUCLID. Latin MS, c. 1350. See p. ii. Title. None. Fragments of the ' Elements.' Description. 4, 16.5 x 21.4 cm., the text being 10.5 x 15.8 cm, with marginal drawings. 6 ff. unnumb, 32-41 11. A Latin manuscript, written on vellum in a hand of c. 1350. The manuscript is a fragment of 6 folios, and includes part of Books III and IV, and a list of propositions evidently based upon, but not identical with, Euclid's sequence. ^viiKTc.1 tucdxof cvpUcjr ; -^ ^^ ^r ^oU W^l- tf:ft ^-^^r *** #w* ^p^,-,^,,^ &<, &rtv *t* Ana m-ou - utniT. $$r.H.ru\fcnr cv ur?a tif w.tn-.-u.u .o-Ktf ir tcnitl JwW . ] b ucrwr t'.u : fi tutl avf c^i.w f x \ M /^ntriwfllbtf 4tn*Sli'*utti f.]:il -.f i- ^.Mnc uicmr.ft..r.ab.viu.t .^ ici k . C -r cf XWMC fxib mmtracib 4 idmi . c. ct ^oi ftr ?r cy^lty^ttcf tint*'ca[tui-v. (^ ^ fjf ^i- <j<ocf . tunico'). N 4 * I r "0n C4 . 1 itnc^t^T 1 -- u ^, t ,,;;t^;-!! Au fcjjfl tit ii , f . " ! i^Hii j^^j^^^r^TTsrrdSH ^tesc^sSIvJ , ^ ^ " ltnrfcl ^.JT^^^^feAnd^wcA . .w.?^R ^t^mril.^.^tH^S^. V7 m, w Aii ' m U ^;U x ^|^u r i iftu.^.- muck cvunnacvf tndi.. i PLATE VI. FROM A MANUSCRIPT OF BOETHIUS, c. 1300 MANUSCRIPTS 44I Ji ^ fl <n Ortfl *s 610 \4~* i >? I iSi. ^ FIG. 220. FROM A 1684 MANUSCRIPT, NOT HERE CATALOGUED 442 KARA ARITHMETICA ALBERTUS MAGNUS. Latin MS., c. 1350. ALBERTUS TEUTONICUS, DE COLONIA, or RATISBONENSIS. Born at Lau- ingen, Swabia, c. 1 193 ; died at Cologne, November 25, 1280. He was Count of Bollstadt, a Dominican priest, and Bishop of Regensburg. He studied at Padua and taught at Bologna, Strasburg, Freiburg, Cologne, and Paris. He was so prominent as a philosopher that he was known as ' Doctor Universalis.' Title. 'De C^lo et Mundo.' (F. i, r.) Description. 22.4 x 32 cm., the text being 14 x 21.3 cm. with- out the marginal notes, and arranged in double columns, each 6.5 cm. wide. 90 ff., 50 11. This is a beautifully written manuscript, in a fourteenth-century hand, with fine initials in red, black, and blue. The first folio contains part of the calendar, and a few random memoranda, including an old price mark of three ducats. The twenty-ninth folio is blank except for some crude circles, and one folio has been cut out. There is a break here in the manuscript, f. 28, v., closing with these words from the fourth tractatus of liber I : * Rei a' gnate e ultim i finis in tpr aut actu aut po. quia si e corrupta tuc ht actu fine 2 si e adhuc corrupt ' (* Rei autem generatae est ultimum et finis in tempore aut actu aut potentia : quia si est corrupta, tune habet actu finem : et si est adhuc corruptabilis ' - the rest of the sentence being * habet finem potentia.') After the blank folio, f. 31, r., opens with these words from tractatus I of liber II : ' que eft f b'a sepata ' (' quse est substantia separata,' the preceding miss- ing words of the sentence being ' quse tamen non limitant operationem formae ejus ') Aside from this the manuscript is complete. F. 89 closes in col. i with the words ' Explicit liber de celo et mudo fratris alberti deo agamus gratias.' F. 90, r., is blank, but 90, v., has the zodiac and planets. There is a brief reference to Pythagorean arithmetic in liber I, tractatus I, caput II. The best edition of this work is in ' B. Alberti Magni // Ratisbonensis Episcopi, ordinis Prsedicatorum,// opera omnia,// . . . cura ac labore // Augusti Borgnet.// Volumen quartum // Parisiis . . . MDCCCXC.' The first edition of the Opera Omnia appeared in Leyden in 1651. EUCLID. Latin MS, 1375. See p. 1 1. Title. The work begins : ' Punctus eft cui9 ps no e".' Description. Fol, 20.9 X 30.1 cm., the text being 11.9 x 19.5 cm. 39 ff. unnumb. + i blank = 40 ff, 44-47 11. A Latin man- uscript, written on paper, c. 1375. MANUSCRIPTS 443 This manuscript includes the first five books of Euclid. This is fol- lowed by a treatise upon astronomy and mensuration, also in Latin, in a different but contemporary hand. The mensuration includes some work on areas and volumes. ANONYMOUS. Latin MS., 1384. Title. None. A Computus manualis. Description. 4, 13.3 x 18.8 cm., the text being 9 x 13.3 cm. 33 ff. (3 blank), 40-44 11., written on paper. The first written folio has been torn, and the opening lines are miss- ing. The work is that particular kind of mediaeval computus (see p. 7) in which a finger mnemonic system is used (see p. 34). It is a copy of an older treatise, the text of which is here written in Gothic characters, the copyist's notes appearing in a smaller hand. As in most computi, the numerals are generally in Roman, both in the text and in the commentary, but the date is twice given, as follows (f. 19, r.) : 'ano dnj 1000.300.80.4'; 'anno dnj 1000.300.80.4' (see Fig. 221). The text is iri Latin, except for one page which is in French. A later owner has written the date of his ownership, 1600 (f. 2, r.). ANONYMOUS. Italian MS., 1393. Title. ' CQui chomincia ilpologho d-1 conpoto //d-lcorfo d-1 folo i d-lla luno// Prolagho.' (F. 3, r., Fig. 222.) Description. Fol., 22 x 29.7 cm., written in double columns, each being 7.5 x 20.5 cm. 70 ff. (6 blank), 35 11. This is an excellent example of a computus (see p. 7). The verse ' Thirty days hath September,' the only relic of the old computi now familiar to most people, appears in this manuscript in the following Italian form: * Trenta di a noumbre apile // giugno & fettenbre di uentotto ( ) vno tutti glialtrj fono trentuno.' (F. 12, v.) (See also p. 33.) GIOVANNI, the son of Luca da Firenze. Italian MS., 1422. The son of Maestro Luca, a celebrated Florentine arithmetician, mentioned on p. 468. Title. ' Trattato di aritmetica.' (F. I, r.) Description. 4, 14.9 X 2 1.9 cm., the text being 9. 3 X 16.3 cm. 145 ff. 27-30 11. 444 KARA ARITHMETICA .* *o a* **- -ok *~x* frov~S> -|H * ^^r r 4 "*~ t * ^^T" Jjftna *%3*Sg!^sfesi3 5JJT^ S, r.-^u^*- c^^*-^ 5 ?*- <^1; "A fi^V'^ j*- ^1 \C<^ &*~n biW^ril^S^^P gS^-^J? ^.JHSZJ F-T ? ia^ itlii tftu*) 'qii^J w tit Ufg?t0 flmuV; Vmntt FIG. 221. FROM THE 1384 COMPUTUS MANUSCRIPTS 445 This is one of the best examples of an early fifteenth-century com- mercial arithmetic known. Florence was at this time an important financial center, and the arithmeticians of the city were highly esteemed. Several of the applications found in arithmetics for the next three cen- turies had their origin here. Subjects like equation of payments and partnership involving time, customs like ' days of grace ' in exchange, and forms like time drafts in sets of two or three can be studied to good advantage in the arithmetical manuscripts of this period. ft* Srtttu rrc<lv*m ^ftx. prft 7 p* FIG. 222. FIRST PAGE OF THE 1393 COMPUTUS 446 KARA ARITHMETICA On f . i , v., the writer says he proposes to treat of arithmetic or the abacus : ' arifmetricho volgharemente e chiamata abacho.' The word 1 abacus ' had come at this time, in Italy, to mean simply arithmetic, the original meaning having been lost. Following a custom of Floren- tine arithmeticians, Giovanni gives an extensive multiplication table, for purposes of reference, and then begins at once with examples in com- pound numbers and fractions, thus presupposing a knowledge of the fundamental operations. These examples are of a mercantile character and constitute the entire portion devoted to arithmetic. The examples in the equation of payments serve, as usual, to fix the date of the manu- script. They all refer to the years 1418-1426 (ff. 113-122). The date is, however, fixed exactly by the closing lines of a folio near the end : ' quefto libro Ifcriffe Giouannj del maeftro lucho dellabacho e finillo quefto d dottobre 1422 - ' (f. 136, v.). In the first page is the date 'ad 28 dottobre 1422.' The book also contains a section on mensuration and the calendar, with curious illustrations of the months (PL VII). The last five folios, originally blank, seem to have been written by a different hand about the same period. ROLLANDUS. Latin MS., 1424. A native of Lisbon, canon of Sainte-Chapelle, Paris, c. 1425. Title. l Scientia de numero ac virtute numeri.' (F. 3, r.) Description. Fol., 21 X 29.8 cm., the text being 15.5 X 21.5 cm. 168 ff., 28-31 11. This is an exceedingly interesting manuscript, written in the year 1424. It was prepared at the command of John of Lancaster, Duke of Bedford, son of Henry IV of England, at one time Protector of England and Regent of France. To him Rollandus dedicates the treatise : ' Illuf- triffimo ac fereniffimo principi metuendiffimo domino domino lohanni patruo domini noftri regis ffrancie et anglie regent! Regnum ffrancie. duci bethfordie Rollandus fcriptoris veftre celfitudinis phyficus vlex- bonenfis fe ipfum ex debito iuramenti.' In 1423 Lancaster issued an ordinance for the restoration of studies in the University of Paris, and it was probably as a result of this that this textbook was written by a Portuguese physician, Rollandus, who was then a canon of Sainte- Chapelle in Paris. The dedication sets forth Lancaster's interest in learning in France and the status of mathematics at that time. Rol- landus covers all of theoretical arithmetic as then known, but takes up no practical problems. He also treats of irrational numbers, a topic MANUSCRIPTS 447 which is now considered part of algebra. It is doubtful if there is a manuscript extant which throws more light upon the nature of French university mathematics at the time this was written. Rollandus also wrote a work on surgery and one on physiognomy. He may possibly be the Rolland who in 1410 was rector of the University. Since this manuscript is evidently a copy, others must have existed, but I have found no reference to them. Fig. 223 shows the forms of the numerals used at this time. It should be noticed that the numerals that have changed materially in form since the twelfth century are 4, 5, and 7. These are shown opposite the letters b, /, and /, re- spectively, in Fig. 223. The changes in the other forms have been more evidently due to the fashion in handwriting. All of the forms are, however, quite different from the primitive ones found in the cave inscrip- tions of India. ANONYMOUS. Italian and Latin MSS., c. 1430 and 1478. Title. The first folio is missing. Description. Fol., 17 X 22.1 cm., the text being 11.2X14.4 cm. 183 ff., 25-28 11. This volume consists of two Florentine manuscripts, one on commercial arithmetic and the other on the computus. The por- tion on arithmetic was probably written between 1420 and 1444, these being the extremes of the dates in the examples in equation of payments. It is not unlike the other contemporary Florentine arithmetics described on pp. 443 and 464. The author assumes the student's ability to perform the fundamental operations with integers, although, after numerous multiplication tables, he gives (see Fig. 224) an example under Multiplicha p modo de barichocholo ' (the Florentine name for our present method), and one under ' Multi- plicha p modo de Quadrate' (the 'gelosia' method of the Venetians). FIG. 223. FROM THE 1424 ROLLANDUS 448 KARA ARITHMETICA FIG. 224. FROM THE ANONYMOUS MANUSCRIPT OF c. 1430 MANUSCRIPTS 449 In the latter part of the work, as if it were a new topic, the author has introduced (f. 126, v.) a chapter entitled ' Partire a danda.' He closes with several folios on mensuration. The second part (beginning on f. 154, v.) is in a different hand. It is a computus, and from the dates it would seem that it was written in 1475. There is a third part (beginning on f. 172, r.) consisting of a set of religious verses, in a still different hand, bearing the date 1478. The language is Italian, excepting for the computus which is Latin. ANONYMOUS. Latin MS., c. 1435. Title. ' De tempore de compositione quadratis et compositione astrolabii plani,' etc. Description. Bound with the 'Computus cyrometralis ' of 1476. Written on paper, in a German hand. Some dates which it con- tains indicate that the book was written about 1435. It is related to the history of arithmetic only in the forms of numerals used. JOANNES DE GMUNDEN. Latin MS., c. 1439. See p. 117. Title. None. A treatise on the computus. Colophon. On f. 17, v., are the words, ' Explicit kalendariu mgri Joh'is gmund.' Description. Fol., 19. i X 25.7 cm., the written part being about 14. IX 18.5 cm. (The tables vary and are larger.) 20 ff., 32-38 11. Latin manuscript. Written on vellum, in red and black, about 1439, as shown by the dates on f. 16. Although entitled a Kalendarium,' it is so much like the mediaeval computi as to have a place in this list. It is really a semiarithmetical treatise on the Church calendar. ANONYMOUS. Latin MS., c. 1441. Title. None. A computus. Description. Fol., 21.5 X 28.5 cm. (varies), 10 ff. (i blank); the number of lines to each folio varies considerably. Latin manu, script, c. 1441. The first part of this manuscript is a computus, not very extended. It relates rather to the calendar itself than to the computations upon 450 RARA ARITHMETICA which it is founded. The last folio is in a later hand, and from the tables which it contains it seems to have been written c. 1524. ANONYMOUS. Latin MS., c. 1442. Title. None. A treatise on the planets. Description. Bound with the ' Computus cyrometralis ' of 1476. Written on paper, in a German hand. It contains the date 1442 in two places (f. 102). Its value in connection with arithmetic is confined to the forms of the numerals used. JOHANNES SACROBOSCO. Latin MS., c. 1442. See p. 31. Title. An algorismus, beginning with the words, ' Omnia que a primeua rerum origine processerunt.' (F. 3. r.) Description. 4, 14.1 X 20 cm., the text being 10.5 X 17.6 cm. 21 ff., 25-3911. The first folio has a picture of an astronomer with a celestial sphere, and the name of the student who copied the MS., ' hainricus muglinchk ' (f, i, r.). On the next page (f. i, v.) Mugling's name again appears, together with three dates: 'Item hainric^ mugling aftronimV 1442, 1443, 1444. The ' Algorismus ' begins on folio 2, r. : ' Omnia que a pri- meua //reru orignj,' and ends on f. 10, v. : ' Explicat algorafmus.' Then follows a picture (PL VIII) in colors, representing a master teaching his pupil the Hindu numerals from a kind of large hornbook, with a motto : 'Ich pin algorifm9 genant//Das ... (?)... hau ich in mein//nes hant.' A table explaining place value is given on f. n, r. Beginning on f. n, v., is another treatise on arithmetic, giving the fundamen- tal operations and some work in progression, and ending with a multi- plication table (ff. 1 8, v. ; 19, r.). A later hand has added three pages on progressions, rule of three, partnership, and interest. (Ff. 19, v. ; 20.) Sacrobosco's algorismus was the first arithmetic, based on the new numerals, written by an English scholar. It consists of eleven chapters, viz. Numeratio, Additio, Subtractio, Mediatio, Duplatio, Multiplicatio, Divisio, Progressio, Perambulum ad radicum extractionem, Extractio radicum in cubicis. For the various editions of this work see p. 32. The title of the Paris edition of 15 10 is ' Opvscvlvm de praxi numerorum quod algorifmum vocant,' and the work consists merely of four folios containing the chapter ' De arte numerandi.' There was also published at Antwerp in 1547 (with later editions, Paris 1550, Venice 1564, MANUSCRIPTS 451 Wittenberg 1578), Sacrobosco's * Libellvs, de anni ratione: seu vt voca- tur vulgo, Computus ecclesiasticus.' His work on the Sphere was pub- lished in 1488 at Venice. For a discussion of the authenticity of the Algorismus, see De Morgan, p. 13. As stated on p. 31, the date of the death of Sacrobosco is uncertain. It is either 1244 or 1256, according as we interpret certain lines on his tomb: ' M. Christi bis C quarto deno quater anno De Sacro Bosco discrevit tempora ramus, Gratia cui dederat nomen divina Johannes.' MARO ANTONIO ROZINO. Latin MS., 1447. Title. ' Qories Marci Antonij rozoni artiu // doctoris sacre the- ologie magn et papie phiam legends.' (F. i, r.) Description. Fol., 20.3 X 28.1 cm., the text being 13.7 X 24 cm. 99 ff., 32-33 11. This is not an arithmetic, and it has been included in this list only because it is semi-mathematical and is bound with Bradwardin's treatise on the theory of proportion. It is a Latin treatise on the theory of per- spective. It is written in a clear Italian hand, and was part of a volume numbered 493 in the Boncompagni sale, containing four manuscripts. Some dealer has removed the first of these manuscripts, the Perspectiva communis ' of John Peckham, archbishop of Canterbury. A memo- randum in the Boncompagni catalogue, probably from a leaf removed with the first treatise, shows that the manuscript was copied in 1447 : ' scripte per me antonium confaronesium ut (vocatur?) de lavilata Anno drii M"ccccxlvij.' The other manuscripts bound with this are described below and on p. 452. THOMAS BRADWARDIN. Latin MS., 1447. See p. 61. Title. None. A treatise on proportion. Colophon. ' Expliciiit propor nes thomi brardi //scripte p me.' (F. 115, r.) Description. Fol., 20.3 X 28.1 cm., the text being 13.7 X 22.8 cm. 15 ff., 33 11. This forms part of the volume last mentioned, and is written in the same hand. Bradwardin's treatise on proportion was published in Paris, 1495 (p. 61). 452 KARA ARITHMETlCA ANONYMOUS. Latin MS, 1447. Title. A treatise on lenses. Description. Fol., 20.3 X 28.1 cm., written in two columns, each being 6.6 X 22.9 cm. 7 ff, 24-33 H- This rather early work on optics forms part of the volume mentioned on p. 451, and is written in the same hand. ST. BERNARD OF SIENA. Latin MS., c. 1450. Born at Massa, Tuscany, in 1383; died at Aquila in 1444. He was a zealous founder of monasteries, and wrote various religious treatises. Title. This book of sermons begins as follows: ' S mo .33.9 .de reftitutioe.// Dorii .q. in quadrageffla ordo dicendo^ p feptimana fequete3 I li de x'ana re//ligione .a. R do p. S. B. no de fenif ordonif mino^ edito.' (F. i, r.) Colophon. ( Explicit tra//ctatuf de uf3if 2 pctib3 303 Bntu3 // Bnardim'3 (?) de fenif. ordif minor^.' (F. 129, r.) Description. 4, 14.8 x 19.8 cm., the text being 10 x 13.8 cm. 129 ff., part vellum and part paper. After f. 60 the pages are arranged in double columns, each being 5 x 14.5 cm, 44-46 11. This beautifully written manuscript of c. 1450 is included in this list because it contains several sermons bearing upon the mercantile customs of the time, including ' De usuris,' * De cambiis,' ' De con- tractibus,' and ' De mercatoribus.' Part of this work was translated into Italian in the fifteenth century, as appears from a codex in the Biblioteca Comunale at Siena. One of the sermons was published by Riccomanni, in the Scelta di Curiosita Let- terarie inedite o rare (no. 13) of Romagnoli, Bologna, 1862. This ser- mon, ' Sulle Soccite di Bestiami,' contains considerable information as to the business problems of monastic institutions of the fifteenth cen- tury. A copy of several of these sermons, made by one Eustachio da Feltre in 1469, is mentioned on p. 466. ANONYMOUS. Latin MS, c. 1450. Title. None. On the Quadrivium. Description. 4, 14.7 x 21 cm, the text being 14.5 x 17 cm, 47 pp., 18-40 11. This is a Latin manuscript on the Quadrivium. It is written on paper, in a German hand of about 1450. It includes a brief treatment MANUSCRIPTS 453 of arithmetic (5 ff. + i blank, the text occupying ff. 18 - 22, v.), and a brief treatment of the calendar and the zodiac (ff. 15, r . 1 7, v.). The geometry begins with modifications of Euclid's definitions : ' Pvnct9 eft c9 pars no eft/Linea e longido fn latitu"' 2 pfuditate.' It closes : C.Explicat prim9 liber euclidis cu <ometo capani.' Latin MS., c. 1450. JOHANNES SACROBOSCO. See p. 31. Title. ' Spera mundi secundum Joha//nnem de facrobofco ' (F. i, r.) Colophon. < Explicit tractat9 de fpa fm // Johem de facrobofco am/ (F. 35, v.) FIG. 225. FROM SACROBOSCO'S Spera, c. 1450 Description. Fol., 35 ff., written on vellum. Printed works of this kind have not been included in this catalogue. This manuscript, however, shows the state of the numerals about 1450, and therefore is mentioned. The treatise begins (f. i, r.) : ' De fpera in quatuor capitula diftin- gui9 // dicentes p'mo quid fit fpera/ quid eius // centrum/ quid axis fpere /et quid fit po//lus.' The first four folios have a marginal commentary, closely written in a later hand. The figures are carefully drawn throughout (see Fig. 225). 454 KARA ARITHMETICA There is nothing to show the exact date of the MS., but the handwriting and numerals indicate the last half of the fifteenth century. Sacrobosco's * Sphere ' was published in 1488, and often thereafter. It was the great mediaeval work on astronomy. ANONYMOUS. Latin MS., c. 1450. Title. ' Incipiut floref Arifmetrice.' (F. i, r.) Description. 4, 15 x 21.8 cm., the text being 8.4 X 15.5 cm. 70 ff. unnumb. + 2 blank = 72 ff., 45 11. Latin MS., in a Ger- man hand, c. 1450. This is a theoretical treatise on arithmetic and algebra, written respectively on the Boethian and Al-Khowarazmian models. Only a little elementary treatment of the fundamental operations (chiefly mul- tiplication) is given, the writer devoting most of his attention in the first part of the work to subjects like progressions, ratios, and propor- tions. The latter part of the book is algebraic and may prove to be a copy of some mediaeval work of importance. It resembles in some places the work of al-Khowarazmi, a manuscript of which follows this one in the same volume. MOHAMMED IBN MUSA AL-KHOWARAZMI. Latin MS., 1456. ABU 'ABD ALLAH MOHAMMED IBN MUSA AL-KHUWARIZMI. Born in the province of Khwarazm (whence his name), died c. 831. The most cele- brated algebraist of his time, and the first to write a book bearing the title Algebra. From his name comes the word algorism (see p. 7). Title. ' Liber mabometi de Algebra et almuchabilax comparif et oppof.' (See Fig. 226.) Colophon. There is none, but f. numb. 85 bears the date 1456, and the forms of the numerals and letters are of that period. Description. 4, 15 x 21.8 cm., the text being 12 x 17.2 cm. 23 ff. unnumb. + I blank = 24 ff., 44-48 11. Latin MS. in Ger- man hand, 1456. This interesting manuscript of the first book bearing the name alge- bra is more complete than the one found by Libri in the Bibliotheque Nationale (Histoire des sciences mathematiques, I, note XII). It more nearly resembles the one which I found in the Columbia University Library in 1904, and showed to be in the handwriting of Scheubel. The two deserve to be edited and compared with the Rosen translation MANUSCRIPTS 455 FIG. 226. FIRST PAGE OF THE 1456 AL-KHOWARAZMI 456 KARA ARITHMETICA (London, 1831 ). This manuscript is particularly valuable because, unlike the one published by Libri, it has the Arabic numerals and the mediaeval algebraic symbols. The et used for plus so closely resembles the + as to leave little doubt that the latter was derived from this Latin word. Like Regiomontanus, the writer uses ^> for minus. This is followed (ff. numb. 97) by a brief treatise on rhetoric, and by three manuscripts on mathematics of 1501, c. 1475, and c. 1550, described on pp. 480, 468, 486. ANONYMOUS. Italian MS., c. 1450. Title. A treatise on mensuration, roots, and algorism. Description. Fol., 21.6 X 30.3 cm., the text being 10.5 x 15.8 cm. 177 pp. (2 blank), 20-25 H- Written on paper, in a Flor- entine hand, c. 1450. This is a general treatise on mathematics, with divisions as follows : A 'praticha de Geometria' (ff. 2-11); fractions (ff. 13-16); square root (ff. 17-19, r.) ; cube root (ff. 19, v., 21); ratios (ff. 22-26); algebra (ff. 26-30); mensuration (ff. 31-46); algorism (ff. 47-177). The algorism is evidently the work of some Florentine teacher, and the handwriting is that of the middle of the fifteenth century. It includes the usual applications of the period, such as profit and loss, partnership, and interest, and it makes prominent the rule of three and the rule of false position. PETER PAUL VERGERIUS. Latin MS., c. 1450. Born at Capo d'Istria, Venice, c. 1350; died in Hungary c. 1420, or pos- sibly in 1444. Bishop of Capo d'Istria. Title. ' De Ingeniis Moribus et liberalibus Studiis. Francis- cvs//fenior auus tuus cui9 ut extant //plurime res magnifice gefte ita//et multa paffim ab eo fapieter i// dicta memorantur. . . .' (F. i,r.) Colophon. ' Petrj pauli uergerij de uiftinopoli ad uber-//tinu3 Cararienfem de ingenijs moribs i / / liberalibs ftudijs adolefcentie liber felici-//ter. Explicit.// Amen.' (F. 23, v.) Description. 4, 14 X 20.3 cm., the text being 8.8 X 13.6 cm. 23 ff. (10 vellum, 13 paper), 26-36 11. The first 10 (vellum) folios are in Gothic script; the last 14 are plainly written in cursive characters. MANUSCRIPTS 457 The treatise refers to ' arifmeticha ' and * geumetria '(.15, r.) among the liberal arts. It is bound with several other manuscripts. RAYMUNDUS LULLIUS. Latin MS., c. 1450. RAYMOND LULLY, RAMON LULL. Born in Palma, Majorca, c. 1234; died in 1315. He was a Catalan alchemist, philosopher, and missionary, and was known as ' Doctor illuminatus.' Title. 'Ars Brevis.' A note by a somewhat later hand, written on f. i, r., reads : 'Ars breuis Raymundi Luli quam scripsit Pisis //in monasterio Sancti Dominici anno 1307. a. 0.23.' The Ars brevis begins on f. 5, r. : 'Deus cu tua gra fapia et amore Incipit ars breuis que eft //ymago artis gnalis. Na ifta fata ab intelectu fb'tili et fun//dato ipe pot fcire gnale arte.' Colophon. 'Finiuit Raymundus arte breuen pifis in monaft'io fac//ti dominici Anno ab incarnatione dni. 1 307. // Explicit feli- citer.' (F. 23.) Description. 4, 15.4x20.8 cm., the text being 9.9 X 13.2 cm. 24 ff., 28 11. This is the second part of a manuscript of 59 folios, of which the first is the Sensuale of Lullius. This and several other man- uscripts are bound with the Vergerius already described (p. 456). The Ars brevis was originally written in 1307 ; this copy was made about 1450. Although not an arithmetic, the work contains several mathematical definitions. JOHANNES ROS. Latin MS., 1450. A Valencian priest. Title. ' Artificium artis arithmeticae.' (F. i, r.) '[DJEus qui es unus . . . Incip artifi" 1 ? atis alfmet'ce De alphabeto.' (F. 24, r.) Colophon. * Ad laudem // oipotentis dei et uirginis marie . . . fi//niuit frat' Johanis ros de Valencia puincie aragonii hoc // artificiu arifmetice pad ... I loco fratrum rnino^ de ofpitali // 1450 die .5. Januarij I uigilia epiphanie domini.' (F. 14 = 37, r.) Description. 4, 15.1 X 19 cm., the text being 9.2 X 13.7 cm. 36 ff., 28 11. Bound with the Lullius and Vergerius described above and on p. 456. 458 KARA ARITHMETICA This rather early Spanish monastic treatise on algorism is the third of three manuscripts in the same hand, written c. 1450, of which the first two are mentioned on pp. 456, 457. It begins on f. 24, r., and ends on f. 37, r. After some definitions, tables, and computus figures, the author takes up the ' nine subjects ' : ' Noue f'biecta ponut I arif- metica ' (f. 3 = 26, v.). These are treated very briefly. This is followed by a religious work, in the same hand (to f. 57). ANONYMOUS. Latin MS., 1442. Title. ' De ymagine mundi.' (F. 2, r.) Colophon. * <Ldeo grat. anj. (amen ?) Ano. dnj. 1442 . . . C.Ex- plicit lib. 9 de ymagine mundi deo grat . .' .' Description. 8, 14.5 x 21.5 cm., the text being 11.5 x 19 cm. 1 58 ff. in the entire manuscript, n ff. in this portion. The other portions are described on pp. 477, 478. This is the first manuscript in a collection of 156 folios, and begins with the lines ' I. H. S.//C.de ymagine Mundi ' (f . 2, r.). It is written in Latin and treats of physical and descriptive geography. It has some interest in the history of arithmetic through the Roman numerals which are generally employed except in the case of large numbers. This manu- script and the others bound with it are described in Narducci's catalogue of the Boncompagni manuscripts (Rome, 1862, no. 81 ). As there stated, at one time it belonged to Alessandro Padovani, a celebrated collector of the sixteenth century. ANONYMOUS. Italian MS., c. 1456. Title. The title is missing. Description. 4, 14.6 x 21 cm., the text being n x 15.6 cm. in ff., (12 with drawings), 30 11. No. 168 in the Boncompagni sale catalogue. This is a business arithmetic, written in northern Italy, and com- pleted, as appears from a note, July 15, 1456. As in most of the busi- ness arithmetics of the period, the first pages contain a set of tables, these being followed by a discussion of the fundamental operations with denominate numbers. Although written in 1456 it is probably a copy of an earlier work of about 1420, for the examples in the equation of payments involve dates from 1418 to 1425 (ff. 78-84). The work is also interesting because it contains the early form of the sign % (see Fig. 227) already mentioned on p. 439. The column tables used by MANUSCRIPTS 459 merchants in their multiplication < per colonna,' and common in the Italian manuscripts of this nature, are shown in Fig. 228. FIG. 227. FROM AN ITALIAN ARITHMETIC OF c. 1456 ANONYMOUS. Italian MS., c. 1460. Possibly by Raffaele Canacci, a Florentine mathematician. Title. There is none given, but the work is a general treatise on mathematics. Description. Fol., 28.2 x 39 cm., the text being 16.5 x 28 cm. 2 blank + 322 numbered ff. = 324 ff., 5 1 11. Italian manuscript, c. 1460. This is an Italian manuscript, beautifully written on vellum, with finely executed initials in colors and gold at the beginning of each of its sixteen books. It belonged at one time to Libri, and later to Bon- compagni. Narducci describes it in the catalogue of the latter's man- uscripts (no. 14). The author begins (f. i) with a description of the work : Come e in che modo eldetto trattato e diuifo/ cioe cio che lopa cotiene.' The successive chapters are as follows : 460 KARA ARITHMETICA r: " - <M - i.. i -*.n ** i 7 * 41 . >o . -, - *"' * > 4- I -*>S> *" 1 * ^ . 4-| |^ <M | -f I 1 - *-| | -UJ, ^. ^ . ^ o x's <e> " K 11~+ 4/ ^ - 4- . ^l^J e "V O ,00 r ? FIG. 228. FROM AN ITALIAN ARITHMETIC OF c. 1456 MANUSCRIPTS 461 I. ' Qui chomincia el pimo libro del detto trattato, & pima pone la diuifione del detto primo libro/ laquale e achapitoli/ cioe e diuifo i .4. capitolj.' (F. i, r.) The four ' capitoli ' are as follows : 1 . 'El primo capitolo del pimo libro/ doue fimoftra lordine e modo del numerare le fighure chefufano afcriuere enumeri.' (F. i, v.) In this are explained the Hindu-Arabic notation, the nine ' figure fignifichatiue,' and the 0, ' che i arabia fidice cero.' 2. The title of chapter 2 is wanting, f. 3 having for some reason been left blank. It related to the addition of integers and compound numbers. 3. 'Qui chomincia el terzco capitolo del pimo libro/ doue fitratta del modo & hordine del trarre el numero minore del numero maggiore.' (F. 5 , r.) The method is that of borrowing and repaying. 4. ' El quarto capitolo del primo libro di quefto trattato/ Doue fitratta del modo & hordine del @chare e numerj.' (F. 6, v.) The column form of the multipli- cation table is first given, XlX K>- FIG. 229. MULTIPLICATION QUADRATO,' C. 1460 PER and is followed by various methods of multiplying. The names ' El berichuocholo ' (f . 10, v.) and <p quadrato' (f . u, r.) show the work to be Florentine rather than Venetian, and the handwrit- ing and the numerous refer- ences to Florence confirm this fact. The method * per quadrato ' is shown in Fig. 229. Five pages of multiplication tables are given, such an elaborate treatment having been rather common in the arithmetics of that city. Division is not treated in Book I. II. 'El fecondo libro del detto trattato. Nelquale fi contiene la natura & proprieta De numerj. & prima come e diuifo & aprj lontel- letto.' (F. 17, r.) This is divided into two chapters : 1. 'El pimo capitolo del fjo libro/ Nelqual fitratta la natura che a ife el nuo con diffinitionj fapute.' (F. 1 7, r.) This relates to such ancient classifications of number as odd and even, prime and composite. 2. 'El fecondo capitolo del fjo libro/ Doue fitratta de numerj nomi- nati per nomj apropiati alle fighure geometre.' (Figurate numbers.) 462 KARA ARITHMETICA III. * El terzco libro della praticha darifmetricha. Nelquale fitratta el modo di partire p nuj. & pima I che forma e modo el detto libro e diuifo.' (F. 23, r.) The subject of division is now taken up in three chapters : i. By the use of the table, el modo di partire per gli numerj fcripti I fulle librettinj.' (F. 23, v.) 2. The ' ripieghi ' method (f. 28, v.), by the successive factors of the divisor. 3. Long division. (F. 32, r.) The author gives not only the galley method, but an early description of the method ' a danda,' substan- tially our present plan (see Fig. 230). IV. This treats of common fractions. (F. 33. v -) V. Ratios, ' quantita proportional].' (F. 46, v.) VI. Mercantile arithmetic (f. 58, v.), with numerous references to the customs of Floren- tine merchants. Sixty-six large folios (132 pp.) are given to this book. VII. The rule of false position : ' El feptimo libro diquefto trattato nelquale fitratta del modo delafoluere de chafi p lofemplice modo delchatain che p moltj fidicono principij del chatain.' (F. 124, v.) VIII. Simple and compound interest (f. 134, v.), with 2 pages (4 incomplete) of tables giving the interest on ^100 for 1-21 yrs. at rates varying from ' 5 p TOO. lano ' (f. 140, v.) to ' 40 p 100. lano ' (f. 150, v.). Equation of payments is also presented in this book (f. 152, v.), the dates ranging from 1458 to 1464. IX. False position as treated by Leonardo of Pisa. (F. 170, v.) Leonardo is mentioned in PI. IX. X. Miscellaneous problems. (F. 176, r.) These are largely tradi- tional examples, and include the hare and hound, finding numbers satis- fying given conditions, problems about eating, the jealous husbands, and the testament complication. XI. Proportion, based on Euclid V. (F. 225, r.) XII. Algebra, 'della regola della algebra.' (F. 233, v.) Unfortunately few of the figures for this book were drawn. The treatment is rhetor- ical, practically no symbolism being used. FIG. 230. FROM AN ITALIAN MANUSCRIPT OF C. 1460 ^^rcu^^u^^.^ ^"X.mpUxovu^o ^ CUA ' \;sh to -5SftS,Sit;s J^ll ^ P( Vv ' Pn, crto dbbtno ^Ub(lKft / - >^ -NT - ^^ T3TIVI . S^s^^^-^^^^^ ^Sffi^asS PLATE IX. FROM AN ANONYMOUS MANUSCRIPT, c. 1460 MANUSCRIPTS 463 XIII. Algebra continued, < la regola de Algebra amucabale.' (F. 279, r.) This is a very interesting treatment of the subject fecondo ghugliel- mo de lunis ' and ' Lionardo pifano,' and it throws some light upon alge- bra as studied in the fifteenth century. (See Plate IX.) XIV. Algebra continued (f. 295, v.), according to Master Biagio (' certj cafi che ferine m? biagio nel fuo trattato di pratica') of 1340, Master Gratia de Castellani (' fecodo che fcriue m? gratia de caftellani), and Leonardo of Pisa. XV. Algebra continued (f.- 312, r.), according to certain 'maeftrj antichi,' viz. : ' Maeftro paolo,' ' m? Antonio,' ' m? giouanj,' Leonardo pifano,' * m? biagio che circha al. 1340. anj morj,' m? paolo fiorj che circha al. 1360. duro,' ' m? michele padre di m? mariano,' ' m? lucha,' and ' un altro m biagio.' XVI. This is missing, the manuscript ending with Book XV not quite completed. On f. i, r., there is mentioned the ' fedecimo e ultimo' chapter, and this might have contained the name of the author had the work been finished. EUCLID. Latin MS., c. 1460. See p. n. Title. None. The first book of the Elements. Colophon. ' CExplicat prim^ liber euclidis cu ometo capani.' (F. 14, v.) Description. 4, 14.7 x 2 1.2 cm., the text being 9.9 X 1 5.8 cm. 22 ff., 18-40 11. This manuscript of the first book of Euclid, with the commentary by Campanus, has been included on account of the forms of the numer- als used. It is written in a German hand of c. 1460, and is bound (ff. 1-14) with the two manuscripts next described. ANONYMOUS. Latin MS., c. 1460. Title. None. A treatise on the calendar. Description. See the preceding manuscript. This is ff. 1 5-17 of the volume. This manuscript, bound with the Euclid just described, was probably written by the same hand. It is a brief treatise on the calendar, and was intended, as usual, for the Church schools. In it occurs the date 1460. 464 KARA ARITHMETICA ANONYMOUS. Latin MS, c. 1460. Title. None. On the Quadrivium. Description. See p. 463. This is ff. 18-22 of the volume. This is a general treatise on the quadrivium, and therefore contains a chapter on arithmetic. BENEDETTO DA FIRENZE. Italian MS, c. 1460. A Florentine arithmetician of the first half of the fifteenth century. Title. ' Inchomincia el trattato darifme//tricha efpelialmete quella pte // che e fotto pofta alia mercatatia // e comminciando alnome didio.' (F. n, r.) q.q.1 7 t 7 f T*-9 78<?- T.-H; 400 9 ^ 9 u 9 90 FIG. 231. TABLES FROM BENEDETTO DA FIRENZE Description. 8, 8.3 x 11.7 cm, the text measurement vary- ing. 2 ff. missing + 348 ff. unnumb. = 350 ff, 20 11. Manuscript on parchment. This is a parchment codex written about the middle of the fifteenth century. It lacks the first two folios and possibly also a few at the end. It is one of the best examples of the mercantile arithmetics in Italy MANUSCRIPTS 465 preceding the printed works. The author begins, as usual, with several pages of tables (see Fig. 231), the multiplication table including the prime numbers below fifty. There is also a table of squares and one for the multiplication of compound numbers. Benedetto presupposes that the reader is able to perform the fundamental operations with integers, and he begins at once with operations on compound numbers and frac- tions. The applications include exchange, partnership, and equation of P'lG. 232. FROM BENEDETTO DA FIRENZE payments, the dates in the problems including the years from 1460 to 1464. The latter part of the book contains a number of such traditional problems as those of the grains of wheat on a chess-board, the couriers, the hare and hound (see Fig. 232), the jealous husbands, and the testa- ment of the dying man. The author closes with a brief treatment of mensuration. ALBERT OF SAXONY. Latin MS., 1462. See p. 9. Title. ' Tractatus proportionum. IncipiGt pportioef copo te a dno albertutio.// [ ] Roportio coitr da e ducxp .operito^ in a co t'rio uni co //ad inuice hltudo.' (F. i, v.) Colophon. ' Explicat pportioef 2po te // p reuedo mro alberto defifona // Finis //' (F. 12, v.) 4 66 KARA ARITHMETICA Description. 4, 14.5 X 21.8 cm., the text being n.8 x 13.8 cm. (varies). 19 ff., 23 11. Bound with the Vergerius and Lullius mentioned on pp. 456, 457. This is the treatise on proportion that was printed without date, prob- ably at Venice c. 1478 (p. 9). It is followed by another treatise in the same hand, 'De latitudinibus formarum,' which bears the date 1462. ST. BERNARD OF SIENA. Latin MS., 1469. See p. 452. Title. None. Sermons of St. Bernard (Bernardinus) of Siena. Description. 4, 15 X 20.6 cm., the text being 11.5 x 14 cm. 130 ff. (12 blank), 28 11. Editions. The works of St. Bernard were first printed in Venice in 1591, 4, and again in Paris in 1636, 2 vol., fol. Some of his sermons were, however, printed in Florence, and some in Venice, in 1495. This manuscript, containing several of the sermons of St. Bernard, was copied by one Eustachio de Feltre in 1469. Two dates are given showing the completion of parts of the work, September 22, 1469, and October 6, 1469. For the reasons for including this manuscript, see p. 452. ANONYMOUS. Italian MS., 1473. Title. 'Trattato della Arithmetica.' (F. i, r.) Description. Fol., 16.7 x 23.7 cm., the text being 10 x 16.2 cm. 190 ff., 35 11. This treatise was composed, as the first folio states (Fig. 233), in the year 1473. I n tne examples in partial payments the dates given are about 1490, and this particular manuscript may have been copied about that time. There is, however, one example with the date 1392, which would seem to indicate that at least part of the book was copied from some earlier writer. There is also (f. 183) a brief treatment of the calendar with two dates 1443. On f. 180 there are also the dates 1452 and 1453. In the margins some sixteenth-century dates have been added by a later owner. The work is beautifully written on vellum. The first part consists entirely of arithmetic, treated from the mercantile standpoint. That it is a copy of an earlier work also appears from the fact that a few pages MANUSCRIPTS 467 A-emo pvtnetpio Alnornc Jt du> <^d riQ/rxiTno Jel Jo _ __ Sro CrtcE.ntio.>Tv<ano _ Jovtofa wazHrc. n-u^ S CD LrittL-ertHo cc ov ccitftuale. cot-hz. cltjocivcidifo c^ndli daxtoHximATTtcrtfu ^J> mo ttJjbnn>o m^o^finc. acD7anidhio giv*X dihmac- i (jiuJi* naoai tB feno dllovo |aticl<z.Q^ve.xte.renj oni-a (ui. cUUmtX f \dt a.^didi'cB.Tntrudt di S'cu" GorUiani s 'on^tt miriifi _ _ _ nclx p/ente.'^ihv. atmo c-ltx^q^fD JT(O j to ctcJA. ndlui'Xi-at'tiotci/c ifri'ibuivc u atlino (tcwz. irii. oikc. at fUttut^tlc. cozfo mo rvactt V)in-ViA filLtcu, nie.nt. cttmeno (aiiioiiiYt\.^p^. at cjitGilj e.-pnv Additive, audio ^/^ lovo J^4o dA UnAhirA coaJu UJA wlonmA. t>Oti2jvhX-*(W<|udL. fcrnvji-G. i-i"a)ra_iajiitzllx 'pv^ <^ fwivi lorm'o jL ^loiictT'hL ciAi* jYyyp^TJx <><-"* >^^ FIG. 233. FIRST PAGE OF THE 1473 Trattato 468 KARA ARITHMETICA are left blank for subsequent insertions, the scribe having written : ' Hie dificiunt quatuo // chartse q non funt//i exemplario ' (f. 65, v.) The second part of the work consists of practical geometry and mensuration (ff. 103 to 128). The third part (f. 129) relates to irrational numbers. On f. 135 begins the ' Regole de la Arcibra,' a chapter on algebra. There is nothing to indicate the name of the author or the copyist. ANONYMOUS. Latin MS., c. 1475. Title. <De duplirj Arti Viforir.' (F. I, r.) Description. 4, 15 x 21.8 cm., the text being 10 x 17 cm. (varies). 1 1 ff . + I blank = 12 ff., 40-45 11. Written in a Ger- man hand of c. 1450-1475. Bound with the manuscript of Mohammed ibn Musa (p. 454). This manuscript gives some account of gauging, and closes with a brief treatment of trigonometry. The mediaeval numeral forms are used throughout. A copy also appears in the 1501 manuscript described on p. 480. LUCA DA FIRENZE. Italian MS., c. 1475. Maestro Luca da Firenze lived in the fifteenth century, and was the son of the celebrated Florentine arithmetician Matteo, who was born in the fourteenth century. Title. ' Inprencipio darte dabaco.' (F. i, r.) F. 2 begins: ' Inprencipio darte dabacho fecondo loftile dinfegniare del ma-// eftro luca di Matteo da fflrenc.e.' (Fig. 234.) Description. 4, 16.8 x 23.3 cm., the text being 11.7 x 13.8 cm. 46 ff., 29 11. Although Fabbroni's Storia delF Universita di Pisa (I, 97) says that Luca's son Giovanni went with Lorenzo dei Medici to the University of Pisa in 1515, thus putting his birth about 1495 and Luca's birth perhaps about 1450, I feel that either this is incorrect, or it is to some other Giovanni and Luca that he refers. One of the best evidences of the date of an arithmetic is found in the dates given in its problems. Authors usually mention years that are not remote from the time when they write, and in the examples in equation of payments (f. 29, v.) Luca uses dates from 1410 to 1441. I therefore think that either he copied a problem from his father (Matteo), or, what is more probable, he him- self wrote about that time, say c. 1425, and that in either case this Gio- vanni was a descendant but not a son. MANUSCRIPTS 469 The arithmetic resembles numerous others written in Florence about this time, such as those described on pp. 443 and 464. The author begins with the fundamental operations and follows these by a treatment zengwg.me.ta. fVlH dvc fWuc-ra fufte -ZCTqKmi' firtc cfjo-n-ptjvrtc. ft-cttf&zaTno pi-mn^ le >Mio ''---. -' FIG. 234. FIRST PAGE OF LUCA DA FIRENZE of fractions and denominate numbers, closing with a series of applica- tions to the business problems of the time. Subjects like equation of payments, partnership, and exchange are given the most attention. The handwriting indicates that the manuscript is a copy made about 470 KARA ARITHMETICA 1475-1500. The symbols in Fig. 235 may also throw some light on the disputed origin of our symbol for dollars, $. No one seriously considers such fanciful theories as the combination of U and S, or the Spanish banner about the Pillars of Hercules. The symbol first appears in print in The American Accomptant, by Chauncey Lee (Lansingburgh, 1797), but in a very different form from that now used. It was common among merchants for some time before it was cast in type form, for a note in one of the early American arithmetics says that the symbol was in use, but that there was no type for it. The third edition of Pike's arithmetic (1798) uses m., c., d., D., and E., for mills, cents, dimes, dollars, and eagles, but Daboll's School maffer's Afflftant (4th ed., 1799, p. 20) gives the symbol $ very nearly in its present form. Now whence come symbols like this ? If they are invented de novo it is usually easy to find their first appearance, as in the case of symbols like TT, e, and / (for V i). But mercantile symbols usually develop slowly, like from libra t /- from the old s (/) from soldi or solidi, and d from denarii. So it is probable that $ was simply developed from some earlier symbol of value, such as that for pounds or reales. Now the symbol for pounds (libra, lire) has various forms, appearing in England as or lb., but generally in Italy as L, or Ib. with two bars across. The former is seen in Fig. 243, from a manuscript of 1545, and the latter in Fig. 216 from Dagomari's work. The latter form, in the fifteenth and sixteenth century manuscripts, appears as practically our dollar sign, as shown in Figs. 235, 237, 246, and it is not improbable that our early American merchants used it for the new unit of value, the dollar, just as the /- for shilling is still used in many parts of our country for \2\ c., although the original meaning is entirely lost. The symbol came into general use in printed books between 1800 and 1825. ANONYMOUS. Latin MS., 1476. Title. ' Computus cyrometralis.' Description. 4, 15.5 x 20.9 cm. A volume of 180 ff., con- sisting of four treatises, the other three being elsewhere described. The first of the four manuscripts is a computus of the usual fifteenth- century type. (See Fig. 236.) In it the Latin original of the verses beginning ' Thirty days hath September ' appears (f . 6) as follows : ' Ja. mar ma. iul. aug. oc. de. deca- trib9. et. vno. alij. trigeta. f$. februus. octo. viginte.' MANUSCRIPTS 18 13 '? n-O iO v r rana V 7 T */rto-rx- / fo |o/>n.vvi'n4-/Jp t Sn*5\|;? >&/* 4*" 11$' '</$* *<v FIG. 235. TABLES FROM LUCA DA FIRENZE 472 KARA ARITHMETICA JWKC *S. WMk<Mr*.-MM* n f^HM-aft <j* 4pfe* v BMtCgfe. ..aaMap^a'^ayttgi 'jr*~&***nr***r* 1 " ' i"^~^ ' T~- -. -- -i . -. &**+*>+&.& $4 * ^^9-teI^^ Bi&^-^ ^^>^^^^tfi&j^ <-~TPCH ^iv^eS^f^fiBVp^^ ^J^^^i *?<3a ^r pr f^.te^ a^fe t*r As^^Si 01 -.^ % I ' f^V L ._ C _^ _ -v xv* . .-** FIG. 236. FROM THE 1476 Computus cyrometralis MANUSCRIPTS 473 (See also p. 33.) It does not start here, however, but goes back at least as far as Sacrobosco (i3th cent., see p. 31), in whose Computus (see p. 45 1 ) it appears as follows : ' Sep. No. lun. Ap. triginta dato, reliquis magis uno. Ni fit Biffextus, Februus minor efto duobus.' (1545 edn., f. K 6.) JACOBUS OBERNHEYM. Latin MS., 1476. A Nurnberg computist of c. 1431. Title. ' Computus norembergensis.' Colophon. ' Anno dm 1476 complete. ' Description. See the preceding manuscript, with which it is bound. This is a German computus written by the same hand as the preced- ing. It is evidently a copy of a work written in 1431, for the following statement appears on f. 65 : 'Anno dm mccccxxxi quarta feria p9 judica finitg e liber ifte per me Jacobu Obernheym.' ANONYMOUS. Latin MS., 1477. Title. None. A calendarium. The first folio begins 'Albeto belo caro decagolo ca9 nobilis.' Description. 4, 15.4 x 20.4 cm., the text being 10.5 x 16.4 cm. 14 ff., 17 11. Bound with the Vergerius and Lullius de- scribed on pp. 456, 457. This is a manuscript of 14 folios on the calendar, and has two dates, M.cccc.lviij and M.cccc.lxxvij, on the first folio. It also has some lunar tables for 1364-1381, the treatise having probably been copied about 1458-1477 from another one of a century earlier. NICOLO DE ORBELLI. Latin MS., 1478. Title. 'Incipit op9 fratris (?) dorbelli fup . . .' (F. I, r.). ' In- cipit. Matematica' (F. 132, r.). Colophon. On f . 1 26, v. : ' Explicunt Scripta f fs nicolai de orbel- lis // doctoris eximini fup ... // Deo gratias et xpo ihu Amen //M? 4 78.' Description. 6, 10.5 X 15.6 cm., the text being 9.2 X 1 1.9 cm. 270 ff. (6 blank), 41 11. Latin MS. on paper, 1478. 474 KARA ARITHMETICA The first part of the book is devoted to dialectics and logic. The mathematics begins on f. 132, r., and consists of two folios on the theory of numbers and three on geometry. On f. 140, r., begins a treatise on philosophy, containing (f. 184, v.) a drawing of 'Johs fcotus docts fubtilis.' On f. 193, r., begins a treatise on astronomy: ' Hie // Incipit liber de // celo & mundo.' The rest of the work relates to science in general. (See p. 23, 1485.) LEONARDUS MAYNARDUS. Latin MS., c. 1488. A fifteenth-century mathematician of Cremona, Italy. Title. ' Leonardi Cremonenfis artis metri//ce pratice compi- latio. Primus tractatus.' (F. i, r.) Description. 4, 17.2 x 24.4 cm., the text being 7 x 10.8 cm. 24 ff., 25 11., figures on the margins. This is a treatise on trigonometry, and has been included in this catalogue because of the arithmetical nature of some of the problems. Favaro's careful investigation of the time when Maynard lived still leaves the matter in doubt. He may have lived in the latter part of the fifteenth century, or he may possibly have been the 'Leonardus de Antoniis de Cremona, ordinis minorum, bacalarius ' who lived early in the fifteenth century. In the catalogue of the Boncompagni sale another Latin manuscript of Leonardo is given as 'in pelle di 33 carte membranacie del secolo XIV.' If this is correct, which is doubtful, Leonardo must have lived before 1400. See also Enestrom in Bibliotheca Mathematica IV (3), p. 290, and Favaro in the same journal IV (3), p. 334. The latter gives a bibliography, with some quotations. In 1902 M. Curtze translated and edited one of the three manu- scripts of this work known to him. This is in the Venetian dialect, and belongs to the University library at Gottingen. He also consulted the two Latin manuscripts formerly belonging to the Boncompagni library, of which this is the later by a few years. This manuscript belonged to the advocate Cav. Carlo Morbio, in Milan, before Bon- compagni secured it. A note on f . 15, r., shows that in 1655 it belonged to Bonifacio or Joseffo Aliprandi. Another note, on f . i , refers to a pas- sage in a work by Franciscus Arisius, printed at Parma in 1702. This passage is as follows: ' LXXXVIII (i.e. 1488) Leonardus Maynardus Insignis Astronomus, Physicus et Mathematicus, cuius opusculum M. S. Mediolani servatur, mihi indicatum ab eruditissimo Viro Lazaro Augstino Cotta I. C. amico meo nequaquam satis laudato, cui est initium : (Here MANUSCRIPTS 475 follow the first few lines of this work, so nearly identical with the manuscript as to show that Arisius probably had this very one at hand.) . . . Fuit ante Blasium Leonardos Maynardus, qui suo tempore non tan- turn inter nos, sed etiam inter omnes in iis studiis tenuit principatum.' A similar passage appears in the manuscript Biografia Cremonese of Vincenzo Lancetti, now in the civic library at Cremona, from which we may infer that he lived before Battista Piasio, of a noble Cremonese family, a philosopher, physician, and astrologer, who flourished about 1500. This manuscript of Maynardus is described (no. 254) in Nar- ducci's catalogue of the Boncompagni manuscripts (Rome, 1862). ANONYMOUS. Italian MS., c. 1490. Title. None. A manuscript on elementary mathematics. 'Choncio fia chofa che fono noue fighure nellaba//cho.' (F. i, r.) Description. 4, 14.5 X 21.5 cm., the text being 9 x 15.2 cm. 158 ff. (5 blank), 26 11. The arithmetic is of the ordinary commercial type, and includes the fundamental operations together with a considerable range of business applications. It follows the general style of the Florentine arithmetics and uses the Tuscan berichuocholo instead of the Venetian scachero to designate our present multiplication : * Volendo multiprichare .2. numeri p berichuocolo ' (f. 17). The dates in the problems in equation of pay- ments indicate that it was written about 1490 (ff. 83-86-129). The per-cent symbol % here appears as p c, p c, p c- , as well as p 100 (ff. 84, 86, 134, v. See Fig. 237). The second part of the work (f. 91) relates to algebra : ' Qui aprffo fcriuerio lareghola dellarabre (dell' algebra). mochabiln (e muqabala).' The work is rhetorical, there being practically.no symbolism employed. This part of the work closes with the words, 'Voglia hora fare fine enondire piu fopa a que//fta reghola delagebremghabile . . . (f. 109, r.). The geometry begins on f. 109, r., and is confined to simple men- suration. The work closes (ff. 123-153) with a series of miscellaneous problems. ANDREA DI GIOVANNI BATTISTA LANFREDUCCI. Italian MS., c. 1490. An officer of the Republic of Pisa in 1 505. Title. None. A treatise on arithmetic. 'Choncio fia chofa che fono noue figure nellabacho.' (F. 3, r.) 476 KARA ARITHMETICA tl ea l l cw AI ucdutu tjiigo ^wrgrmaafio it too cmnfraU'Uria <nlrncfc etUi u*4i cntcfl Sucllc thm c^vurlU Jrflri ck>fi vitcnc a^cUtta tctim outlet te (vr^i (eticnac . U c ootbuinio ttcJcrc i c vi nclU g^ vtiiVtid ^Jj y Hvfcffenbrc nc(L C tU Ud mClC C ^ at A- |<3Jfr C CdC-A- T-&VA& t(t1C Ce wct:ttn ^ &$<$' ^Ttwdaaj-nuwtf in/ j c. Ji ccc nc nc 5. wclr 41 ^ra del ntmmrc , vw f tneft ^li 4-iO EC tweu^cL^wti^^^d^i ucacre uWh? *wi cci^llavm^ snttttn nclu) Qt ctu' -> tncfi e- i.ora aci .^ ^^ ej ) ucacre a AAI- -net d-^u4-ti wcfM^cli wt^i dci FIG. 237. FROM THE ANONYMOUS MANUSCRIPT OF c. 1490 MANUSCRIPTS 477 Colophon. ' Qvefto Libro. afcritto. difua. popia ajano. andrea. Lanf reduce].' (F. 96, v.) Description. 4, 16.8 x 23.4 cm., the text being 12.1 x 16.5 cm. 96 ff. (15 blank), 29 11. This book is an Italian commercial arithmetic of the Florentine type written about 1490. After a brief treatment of notation the author takes up the ' Librettine minore,' or the smaller multiplication table to 10 x 10, following this by several examples. He then gives (f. 9, r.) 1 le libretine maggiore ' or larger table, with multiplication ' p quadrato ' and ' p berichuochulo,' these names showing the work to be Tuscan. Division is followed by a large number of problems, per cent (il c 2 and p 100) playing a large part. The problems in equation of payments show the book to have been written between 1489 and 1491 (ff. 72-74). JOHANN NEWDORFFER. German MS., 1492. A Niirnberg Rechenmeister, c. 1450-1500. Title. ' Hanns Dimpf el // (Multiplication table) // Johann New- dorffer Rechen-//maifter V nd Modift zu N.// 1492.' (F. I, r.) Description. 8, 10.5 X 13.9 cm. n ff., 20 11. Written on vellum. This is a beautifully written primer apparently done under the guid- ance of one of the celebrated Rechenmeisters of the Newdorffer family of Niirnberg. It was written in 1492 and is evidently the work of a beginner in commercial arithmetic. It opens with the addition and multiplication of compound numbers, and this is followed by the. rule of three, with some applied problems. The name Hanns Dimpfel is doubtless that of the pupil whose work was done under the direction of Newdorffer. As to the latter, it is not improbable that he was the father of the celebrated Schreib- and Rechen- meister Johann Newdorffer, the founder of the German calligraphy, who was born at Niirnberg in 1497, and died there November 12, 1563, and whose son Johann (b. February 22, 1543, d. October 28, 1581) was also a well-known Rechenmeister. ANONYMOUS. Latin MS., c. 1500. Title. ' De comutata pporne.' (F. 14.) Description. Fol., 14.5 X 21.5 cm., the text being 1 1.5 X 19011. This is the second part of the 1442 manuscript, ff. 14-32 (p. 458). KARA ARITHMETICA This is a Latin treatise on arithmetic, apparently written about 1450- 1500. It relates almost exclusively to Boethian ratios and figurate numbers. It closes with a few pages on the circle, including the follow- ing : ' Rem noua mirabilej. quadra//tura ; circuli velut ifcritabile^.// apud doctof ppli' olim. f. fabile.//pure ceriiut oculi vere demra//bilem nuc I fine feculi ' (f. 30, v.) ; C.de quadratura circuli ' (f. 32, r.). The table in Fig. 238 shows some work in series such as is common in arith- 4- I .3.- 8- .?. \t~ '+' .\crr J .2-O- SEZI [77\- HE -^ZTT 9 .<^ 40- *& .3 cr? 0' .(2-. .1^. -0. .*.( * .2-A r U FIG. 238. FROM ./?<? comutata pporne\ c. 1500 metics of this period. Fig. 239 shows the multiplication table as com- monly seen in the Boethian arithmetics (compare p. 26). It also shows (on line i) the use of the Roman numerals in connection with the Arabic, and (on lines 9 and 10) the absurdly long Latin names for ratios. ANONYMOUS. c. 1500. Description. Fol., 14.5 X 21.5 cm. It is bound with the pre- ceding manuscript, but is in a later hand. It consists of ff. 33-38 of the volume. The manuscript consists of an interesting set of drawings, including one of an astrolabe (f. 33) and several horoscopes (f. 35). The latter serve to fix the date of this portion (c. 1500) and the country (Hungary) in which it was written. They include (f. 35, r.) a horoscope cast at the birth (1456) of Ladislas or Uladislas VII, possibly by Peurbach (see p. 53), and one at his coronation as king of Hungary on September 18, 1490, mentioning his coronation as king of Bohemia on August 25, 1471, and his election as king of Hungary on July 15, 1490. ANONYMOUS. Latin MS., c. 1500. Description. Fol., 14.5 x 21.5 cm. It is bound with the pre- ceding manuscript, is possibly in the same hand as the horo- scopes, and is certainly on the same paper. It occupies ff. 3944 of the volume. This is a treatise on the mensuration of the circle and dates from c. 1500. It has some bearing upon the metrical computations of the time. MANUSCRIPTS 479 ^ XHi| ttd -^ \,f /' r ' ft./**!/ vr-rxUt^N,,, <; " ' of ci'm.. t-Il<l xl ^' FIG. 239. FROM Zte cdmutata pporhe, c. 1500 480 KARA ARITHMETICA The fifth portion of the volume (ff. 45-52) consists of a set of Latin, and the sixth (ff. 53-58) of a set of Italian verses. The seventh (ff. 59- 60) contains a brief reference to astrology. The rest of the book is of a literary or astrological character. There are, in all, thirteen different manuscripts in the volume. For details concerning the non-mathematical portions, see Narducci, 1. c., p, 31. ANONYMOUS. Italian MS, c. 1500. Title. ' Regole per far Orologi da sole //con le sue Figure.' Description. 4, 16.8 X 22.3 cm. 22 ff. Bound with the Verge - rius manuscript described on p. 456, but written in a later hand. This is an Italian treatise on dialing, written in a fine hand of the sixteenth century, with well-executed figures. Several other manu- scripts, not of a mathematical character, are also bound with it. ANONYMOUS. Latin MS., 1501. Title. None. A treatise on mensuration. Colophon. There is none, but one of the folios bears the date '1501 adj feptembr In Niirnberg.' Description. 4, 15 x 21.8 cm, the text being 9.8 x 17.1 cm. 89 ff. unnumb. + 7 blank = 96 ff, 27-30 11. Latin MS, written in a German hand. Bound with the manuscript of Mohammed ibn Musa (p. 454). This is a treatise on mensuration, of no particular merit save as it shows the nature of the work at the opening of the sixteenth century. It contains a copy of the c. 1475 manuscript described on p. 468. STEPHANO DI BAPTISTA BELLI STEPHANI DA MERCATELLO. Italian MS, Mercatello, c. 1522. An Italian teacher, born at Mercatello, and living there in 1522. He was a pupil of Paciuolo. Title. <Svmme//Arismetice.' (F. i, r.) ' Stephano.D.B.Del- listpha//ni . damercatello . atvtti . qve//ili . liqvali . in arte . mercan// tile.exercitare.sidilectano.' (F. i, v.) Description. 8, 13.5 x 20.7 cm, the written part being 9.1 X 14.3 cm. 153 ff. numbered (5 blank), 29 11. Italian MS. on paper, except f. i, which is on vellum. Written, as the problems show, at Mercatello, c. 1522. MANUSCRIPTS 481 This is a manuscript on commercial arithmetic, unusually complete in its applications, and also unusually well written. It is of the general Florentine type, but, as appears from a date on f. 101, r., was written at Mercatello, a town south of Ferrara and east of Florence. The name of the author appears not only in the dedicatory epistle, but also at the end of a bill of exchange dated '1522 A di./25/maggio in M n ' (Mercatello), in the latter case as < Stefano di Bap to Stefan j 13.' The examples in equation of payments are dated 1371-72, showing that these were copied from some earlier work, and in fact they were taken from Paciuolo (' Diftinctio nona, tractatus quintus '), who in turn borrowed them from some predecessor. Stephano states in his dedicatory epistle that he was a pupil of Paciuolo, and that he is chiefly indebted to him for his material : ' et maxime dal mio R do . et ex te affme & pceptor'. M. Luca dal Borgo.' This is quite evident on comparing certain passages ; for, while Stephano does not usually copy his master verbatim, there is often a great simi- larity between them, and sometimes (as in the chapter 'De le. 2. falfe pofitioni ') there is evident plagiarism. Stephano, however, omits most of the theory of numbers to be found in Paciuolo, and confines himself to mercantile applications. These cover barter, partnership, various forms of discount and exchange, and other similar topics. There are also given a number of mediaeval puzzles, including the testament problem, the sale of the eggs, the hound and hare, and the guessing of numbers. A little work in mensuration and the calendar is given at the end of the book. BARTOLOMEO ZAMBERTO, editor. Latin MS., c. 1525. A Venetian scholar of c. 1 500. He was born c. 1473. Title. ' Evclidis // Megarenfis graeci philofo-//phi ex Theone graeco com-//metare Interprete Zaber-//to veneto triplex prlci- pioru // genus primu diffinitiones : // Signv//est cuius pars nulla : // Linea vero, lon-//gitudo ilia tabilis.//Lineae autemlimites,funt figna. // Recta linea, est quae ex aequa-//li fua interiacet figna. Superfi-//cies, est quae longitudinem Iatitu//dinemq3 tantum habet. Super-//ficiei extrema, funt lineae. Pla-//na fuperficies, est quae ex aequa-//li, fuas interiacet lineas.' (F. 2, r.) Description. 12, 8 X 11.7 cm. 152 ff. (2 blank), 16-20 11. Written on parchment. This beautifully written manuscript has been included in this cat- alogue of arithmetics without much justification, since it does not 482 KARA ARITHM^TICA contain even Book V. It is, however, valuable in showing the influence of printing upon written numerals. It is a copy of a translation of the first three books of Euclid made by Bartolomeo Zamberti of Venice in 1513. The complete translation was first printed in Venice in 1513, the statement of some bibliographers that it appeared in Paris in 1505 being unsupported by any evidence. It also appeared in Venice in 1517, in Paris in 1516, and in Basel in 1537, 1546, and 1558. See Riccardi, II, i, 644; Weissenborn, Die Uebersetzungcn des Euklid durch Campano und Zamberti, Halle, 1882. This manuscript was evi- dently written about 1525 for some noble family, for it has (f. 2, r.) three illuminated coats of arms. ANONYMOUS. Latin MS., 1533. Title. * Declaratio Calendarii et // Almanach huius Cifte.' (F. i, r.) ' Ars supputandi cum Denariis.' (F. 66, r.) Description. 4, i6.6x 23.3 cm., the text being 1 1.8 X 16.3 cm. 8 1 ff. unnumb., 18 11. Latin MS., written on vellum, in 1533. This beautifully written Latin manuscript consists of two distinct works. Of these the first is a computus in twenty-three chapters, written apparently in Salisbury cathedral in 1533. The second part is a treatise on counter reckoning, and consisted originally of six chapters, ' De Numeratione, Additione, Subftractione, Multiplicatione, Diuifione, Frac- tione minutiis.' The last of these chapters is missing. The manuscript is particularly interesting because it gives the counter reckoning as it was used in England early in the sixteenth century, the numbers all being written in Roman when they are not represented * on the line.' It is illustrated by numerous diagrams representing the line abacus. The manuscript closes with five pages * De proportione vel regula Detri,' and De Proba regule Detri.' ANONYMOUS. Italian MS., c. 1535. Title. ' Trattato d'Aritmetica, e del Misure.' (F. i, r.) Description. Fol., i8x 23.8 cm., the text being 14.5 X 21 cm. 148 ff., 23-27 11. This is an Italian manuscript, written, as the dates on folios 67, 68 show, about 1535. It is a commercial treatise, beginning, as was fre- quently the case, with the fundamental operations with compound numbers. The method of division ' a danda ' is preferred to that ' per galea,' although both are given : * II partir a galea e molto legiadro et MANUSCRIPTS 43 fpeditiuo, ma non tanto ficuro per un principiante quanto il partir a danda' (f. 19, v.). The applied problems are generally of a practical type useful to merchants' apprentices in the north of Italy. (Fig. 240.) FIG. 240. FROM THE ANONYMOUS Trattato OF c. 1535 Much of the latter part of the treatise relates to mensuration. A set of tables and some notes are given at the end. Among the notes is the celebrated problem of the horseshoe. (Fig. 241.) 484 KARA ARITHMETICA U ' -*. 4. 1. f *~ * tf . J * f fC > / r T. FIG. 241. NOTES IN THE ANONYMOUS MANUSCRIPT OF c. 1535 The manuscript is no. 23 in Narducci's catalogue of the Boncom- pagni manuscripts (Rome, 1862, p. 16). LUDOVICO ALT DI SALISPURGO. Italian MS., 1545. A sixteenth-century student. Title. '.M .D. XLV.// Ludouichoalt de Salifpurga.' (F. i, r.) Description. 8, I2.6X 17.8 cm. 92 ff. MANUSCRIPTS 485 This is a business arithmetic written by some student in 1545. The author first treats of the fundamental operations with denominate num- bers, following this by a treatment of fractions. (Fig. 242.) The last part FIG. 242. FROM THE LUDOVICO ALT of the work is a * praticha ' (f. 48), and considers the ordinary business arithmetic of the time. The author uses almost the same form of the per- cent symbol as the writer of 1456 referred to on p. 458, viz. p ^ (see Fig. 243). The manuscript is described in Narducci's catalogue, p. 152. 486 G RARA ARITHMETICA # A J too : 1 FIG. 243. FROM THE LUDOVICO ALT ANONYMOUS. Latin MS., c. 1550. Title. ' Introducoriu breue fup elementa Euclidis.' Description. 4, 15 x 21.8 cm., the text being 14.2 x 19 cm. 23 ff. unnumb. + 2 blank = 25 ff., 28-33 H- Written in a Ger- man hand, c. 1550. MANUSCRIPTS 487 This manuscript is the last one in the volume containing the algebra of Mohammed ibn Musa (p. 454). It consists of introductions, book by book, to books I-XV of Euclid. The symbols +, -=- (for ), V , and W~ (for v^) are used. HONORATUS. Latin MS., c. 1550-1600. A Venetian monk of the sixteenth century. Title. ' Opus Arithmetica D. Honorati vene-//ti monachj coenobij S. Lauretij.' (F. i.) FIG. 244. FROM HONORATUS Description. 8, 10.9 X 16 cm., the text being about 9.5 x 15.8 cm. i ii ff. This manuscript was written by a pupil of a Venetian monk named Honoratus evidently about 1550-1600. It is a practical arithmetic, 488 RARA ARITHMETICA the author taking up rather fully the fundamental operations with inte- gers and denominate numbers. He also treats of such common applica- tions as partnership and barter. The illustrations are those which a pupil would be likely to make, and prove the manuscript to be the work of an immature hand. The common galley method of division is given, with the ship in full form (Fig. 244), and the sign % appears as per cr- (see p. 439). ANONYMOUS. Italian MS., c. 1560. Title. ' Dichiarazione per intelligentia de Principiantj // del vso, che merchantilmente tiene la Citta dj Firenze // sopra le monete, pesi, e Misure.' (F. 2, r.) On f. i, r., the coat of arms of the Angelotti (?) family is painted on parchment. Description. Fol., 18.7 x 25.5 cm., the text being 12.5 x 20 cm. 1 66 ff. (3 blank), 8-42 11. This is a very clearly written Florentine manuscript on commercial arithmetic. The 8's are all made like S on its side, thus (CO), a feature not infrequently seen in the second half of the sixteenth and even in the seventeenth century. As was often the case, the author presupposed a knowledge of the fundamental operations, reviewing them only briefly with compound numbers. The problems are of the common mercantile type of the period. The manuscript seems to have been written about 1560. (See Fig. 245.) ANONYMOUS. Latin MS., c. 1565. Title. ' Trattato d'aritmetica mercantile.' (Cover.) Description. Sm. 4, 11.9 x 16.8 cm., 279 ff., 20-25 li- lt appears by the problems that this mercantile arithmetic was written at Bologna about 1565. Like so many similar treatises, it opens with a set of column multiplication tables. The first operation is * par- tiri piccoli,' or short division, this being necessary for the simple reduc- tions in the addition of denominate numbers. This is followed by the addition of pounds, shillings, and pence (' Sommare di lire e fs e de '). The next operation is multiplication, several methods being given. The first is cross multiplication (' modo di mcare p + ') ; the second, from left to right (' modo di multrip re , per la dirieto,' multiply always appear- ing as ' multriply,' ' multripich,' in this manuscript) ; the third, by ref- erence to the tables (' p colonna ') ; the fourth, our common form, the ' bericuocolo ' of the Florentines, but here called ' modo per bilicuocolo ' MANUSCRIPTS Vno >ia viceu.ro d* jwo a ragume dt^ da dungue quanta /ii il XO O O -------- xooo --------- 00* c^o ooo gr xoo coo Az. " ^ FIG. 245. FROM AN ITALIAN MANUSCRIPT OF c. 1560 490 KARA ARITHMETICA (also 'bilicocolo '). This is followed by the subtraction of denominate numbers, and this by a more extended treatment of multiplication. Division follows, at first ' per ripiegho,' and then ' per danda' (Fig. 246), the galley method not being given. This is followed by a treatment of fractions, percentage (' Commicano e centi dalcuna merchantia '), the rule of three (' Commica lareghola del tre '), profit and loss, equation of payments, exchange, and the other mercantile rules of the period. Alto- gether it is one of the best of the sixteenth-century manuscripts on the commercial arithmetic of Northern Italy. GAUDIOSO FRASCADA. Italian MS., 1568. A schoolmaster of Brescia, about the middle of the sixteenth century. Title. ' Libro di Arithmetica // et Geometria.' (F. i, r.) Colophon. ( II pnte libro cioue opera di abaco et geometria scrit- to ad // instantia de figliolo di m Bertholameo sachetto // habitante nella terra de' 1'orala algise' scritto per mi // Camillo sachetto 1'anno 1 568 a honor de dio et della // uergine maria laus deo ' (f. 43, v.). But f. 5, v., has the following : ' Questo libbro e ftato fcritto p Gaudioso frafcada cittadino di // Bressa a di 4 Nouembrio lanno 1555 laus dei.' Description. Fol., 20.2 x 28.6cm., the text being 14.7 x 24.2 cm. 46 ff. (3 blank), 17-29 11. This is a copy, made in 1568, of a treatise composed by Frascada, a schoolmaster of Brescia, in 1555. The author uses both the Floren- tine and the Venetian forms and names for the operations, as in the expression ' Multiplicar per fchacchiere feu baricocolo.' Several forms of multiplication are given, as in the work of Paciuolo, but in division only the galley method appears. The examples are generally of the ordinary business nature, and the rules include ' Raggioni p lo cattayno, cioe position false.' The last few folios refer to practical geometry. ANONYMOUS. German MS., c. 1575. Title. ' Von Kiiftlicher Abmeffung aller groffe,//ebene, oder nidere, in die lenge, hohe, breite vnd//tiefte als graben Cifternen, vnd Brunnen.' (F. i, r.) Description. 12, 10.2 X 15 cm., the text being 8.6 x 13 cm. 30 ff. unnumb. + 4 blank = 34 ff., 24 11. Written in a German hand and in the German language, c. 1575. MANUSCRIPTS 491 \f** isjiB ^. a/L r-g; '"*-- 1*7 *6? + -s><s z%&-< -3*1 ZQ^^+J \^** LZL FIG. 246. FROM AN ITALIAN MANUSCRIPT OF c. 1565 492 RARA ARITHMETICA This manuscript contains a few numerical problems in connection with mensuration. The work is of no merit save as it throws light upon the mensuration of the sixteenth century in Germany. Like most books of this kind, written at that time, it has little or no explanation of the rules used. DOMENICHO DA BIEN DE VALSUGANA. Italian MS., 1579. A Venetian student of 1579. Title. None. An elementary algorism. Colophon. On p. 21 there is this statement: * L'anno. 1579. a di .9. del // mese di marzo in liedolo // Per Domenicho da Bien de Valsugana.' Description. 12, 9.5 X 14.5 cm., the text being about 7.5 x 12.5 cm. 48 pp., varying number of lines. Probably written at or near Venice. 1579. This is a student's manuscript on commercial arithmetic. It opens with the fundamental operations, the treatment of * Multipticare Per scachiero ' (p. 12) showing the Venetian instead of the Florentine influ- ence. Division is performed only ' per galia, ouero per battello ' (* p Galera' Domenicho calls it elsewhere), or 'per Collona,' the modern form not appearing. The examples are wholly mercantile, most of the applications being in the rule of three. The work also contains some simple examples in mensuration. FRANCESCO GIUNTINI. Italian MS., Lyons, 1579. An Italian student or teacher of astronomy. I judge from his horoscope that he was born November 14, 1522, at Florence. Title. None. Colophon. ( Di Lione adi 13. di Maggio 1579: Francesco Giuntinj.' Description. 4, 15.5 x 21.8 cm., 32 ff. (2 blank), 26-30 11. Written on paper. This manuscript is clearly written in Italian, and relates to astrology, in particular to Giuntini's own horoscope. Its interest in the history of arithmetic lies wholly in such symbols as that for degrees, and in the forms of numerals. MANUSCRIPTS 493 JOHANNES KLUMPIUS. Latin MS., 1598-99. Title. * Varij Tractatus Mathematices // a Joanne Klumpio phili2e//ftudiofo excpt Ingol-//ftadij Anno 7/1598 et fequentj.' (P. i.) Colophon. At the end of the ' Arithmetica practica ' are the words <Abfoluim9 15 Januarij // Anno 1599.' Description. 4, 15.3 x 20 cm., the text being 10.5 x 16.5 cm. (varying). 70 pp. blank + 442 pp. written = 5 12 pp., 19-21 11. Latin MS., written in a German hand, Ingolstadt, 1598-99. This is a set of lecture notes on general mathematics, including geom- etry, trigonometry, arithmetic, and astronomy. The arithmetic consists of two distinct parts, probably the result of two courses of lectures. The first treats of the four fundamental operations with integers and common fractions ; the second of practical arithmetic, as Klumpius calls it, although it simply gives the fundamental operations, progressions, roots, and a few rules like the rule of three, partnership, alligation, and the rule of false. B. ROTH. German MS., 1599. A German student of c. 1600. Title. 'Das Fiinfft Capital.// Item Im Funfften Capitel ift von dem Algorifmo // oder Cofs.' (F. 4, r.) Colophon. Not entirely legible. It contains the date, August 4, 1599. Description. 4, 18.8 X 24.6 cm., the text being 13.5 x 19.8 cm. 321 ff. (13 blank), the lines varying. This is a German manuscript containing the solutions of the problems in Stifel's edition of Rudolff's Coss, beginning with Chapter 5. (See p. 258.) This manuscript was written, as the colophon shows, in 1599. The name of the writer is not entirely legible, but it seems to be Brite- nus Roth. The solutions are written in a very clear hand, and furnish an excellent example of the symbolism of that period. ANONYMOUS. Dutch MS., Louvain, c. 1600. Title. None. A treatise on mensuration. Description. Sm. 8, 10.5 X 16 cm., the text being about 8 X 13 cm. 80 pp., 33-46 11. Written on vellum. 494 RARA ARITHMETICA This is a Dutch manuscript on gauging and general mensuration. The ganger's tables are as clearly written as in the best Florentine manuscripts. ANONYMOUS. Latin MS., c. 1600. Title. None. A treatise on arithmetic. Description. Sm. 4, 10.5 x 14.1 cm., the text being about 6x12 cm. 197 ff. numb, (there have been added 8 ff. in the back, written on vellum, making a total of 205 ff.), 22 11. (varies). Latin MS., written in a German hand. This manuscript begins with the fundamental operations after the manner of algorism, and then takes up the theory of numbers according to the Boethian system. Figurate numbers and proportions (ratios) are treated at great length, as in the arithmetic of Boethius, although this is not a copy of that work. It is not common to find manuscripts written as late as this that go so fully into the ancient theory of pure arithmetic. At the end of the book the author has placed eight folios of ' Arithmetica tabulals et formularis,' clearly written on vellum, con- taining multiplication (or ' area ') tables of primes, of * oddly even ' and ' evenly odd ' numbers, of i solid numbers,' and the like. ANONYMOUS. German MS., c. 1600. Title. 'Nutzlicher Gebrauch // Der Weldt Kiigel.' (P. 3.) Description. Fol., 20.5 x 33 cm., the text being 19.5 x 27 cm. (varies), 160 pp. (several blank), 24-29 11. (varies). Written on paper, in German, c. 1600. Although the first part of this manuscript is on the terrestrial sphere, the second part, beginning on p. 83, is on arithmetic as needed by cos- mographers. This includes the fundamental operations, including square and cube roots. The galley method of division is used exclusively. ADDENDA Since the completion of the manuscript for the edition de luxe of this work, two years ago, numerous additions have been made to Mr. Plimpton's collection of early arithmetics. Excluding a number of early Arabic manuscripts, these acquisitions, with references to the pages on which they would naturally appear, are as follows : PAGE 16. The 1561 edition of Borghi has been acquired. PAGE 23. The 1485 edition of Nicolo de Orbelli has been acquired. See also page 473. PAGE 36. There has been acquired a work on the calendar by Regiomontanus : 'Almanach magiftri Johanis//de monteregio ad anos. xviij.//acuratifiime calculata.' Colophon .-...< Erhardi Ratdolt Auguf- ten // Vindelico . . . M.cccc.lxxxviij.' This is not an arithmetic, but it is interesting on account of its mathematical treatment of the calendar. PAGE 62. There has been acquired a Cracow edition of a work by Faber Stapulensis, containing a little arithmetic : ' Jacobi Fa-//bri Stapv-//len I Artiu diuifione itroductio . . .' Colophon : * Cracouig . . . 1534.' 4, 13.8 x 18.7 cm., the text being 10.2 x 16 cm. PAGE 82. The first (1503) edition of the Margarita Philosophica has been acquired. The colophon is as follows : ' Chalchographatum primi- ciali hac // peffura/ Friburgi p loanne Scho//ttu Argen. citra feftu Margarethg //anno gratise M:CCCCC.III.' PAGE 1 14. An edition of Tagliente's ' Libro //de abaco,' ' Vinegia . . . M.D.XLIIII ' has been acquired. Also the Venice edition of 1570, and an edition s. a. PAGE 115. The 1570 Milan edition of Tagliente has been acquired : ' Libro //de Abbaco che inse-//gria a fare ogni ragione // mercadantile . . . Milano 1570.' PAGE 139. The 1525 edition of Riese's first book has been acquired : ' Rechnung auff der linihen //gemacht durch Adam Riefen vonn Staffel- //fteyn/in maffen man es pflegt tzu lern in alien // rechenfchulen RARA ARITHMETICA gruntlich begriffen anno I5i8.//vleyfigklich vberlefen/ vnd zum andern mall //in trugk vorfertiget.//C.Getruckt zu Erffordt zcum//Schwartzen Horn.// 1525.' Colophon: 'C.Gedruckt tzu Erffordt/ durch//Mathes Maler. M.//CCCCC.xxv. Jar.' 8, 8.8 x 14.6 cm., the text being 7.1 x lo.ycm. 44ff., 17-20 11. PAGE 139. Another Erfurt edition of Riese's second work has been acquired : * Rechnung auff // der Lynihen vn Federn/ // AurT allerley handthirung/ gemacht durch//Adam Ryfen.// Zum andern mal vber- fehen///vnd gemehrt.// Anno M. D.XXvij.' Colophon: ' Gedruckt zu Erffurdt zum Schwartzen Horn.' PAGE 148. The 1570 Venice edition of Feliciano has been acquired. PAGE 167. 'Ulrich Kern von Freysing Eyn new Kunftlichs wolge-// grtindts Vifierbuch/ gar grvifi vnnd behend // aufz rechter art der Geometria/. . . M.D.XXXI.' Fol., 18 x 27.8 cm., the text being 12.7 x 23 cm. 57 ff., 46 11. Strasburg, 1531. A work on guaging. PAGE 181. Two more editions of Mariani's 'Tariffa' have been acquired, Venice 1564 and 1572. PAGE 223. There is mentioned an arithmetic by Medlerus, 1543, with a second edition in 1550. This work, recently acquired, may be all that there is of the 1550 edition: ' Facili-//ma et exactis-//sima ratio extra//hendi radicem Quadratam // & Cubicam, a Doctore // Nicolao Medlero in // gratiam ftudiofae // iuuentutis // sedita. // Anno Domini. // M.D.L.' Colophon: ' Impressvm VVitem- // bergae, per Vi-//tum Creutzer.' 8, 9 x 14.8 cm., the text being 7x11. 2 cm. 7 ff., 2 7 H. PAGE 322. The 1575 edition of Lapazzaia has been acquired. It is not the same, however, as the work described on page 322, either in title or in contents. ' Opera //terza//de aritmeti-//ca et geo-//metria. // Dell' Abbate Georgio Lapazaia // da Monopoli. // Intitolata il Rama- glietto.//In Napoli // Apreffo Mattio Cancer. M.D.LXXV.' 14.2 x 20 cm., the text being n x i6cm. 4 + 169 pp., 2 9 11. PAGE 322. The 1566 edition of Lapazzaia has been acquired : ' Fami- liarita // d'arithmetica, e geometria // con 1'vsitata prattica Napo-// litana, Compofta & ordinata per Abbate Geor-//gio Lapizzaya Canonico Monopolitano // Nuouamente con fomma diligentia // Riftampata, e corretta.// ... In Napoli //Appreffo Horatio Saluiani //MDLXVI.' 4, 14 x 22 cm., the text being 11.2 x 16.2 cm. 2 + 62 ff., 28 11. ADDENDA PAGE 340. L'Arithmetiqve//militaire d'Alexandre//Vandenbussche Flandrois//departie en deux liures.//. . . A Paris . . .' (s. a.). 8, 14.8 x 2o.8cm., the text being 9.8 x i6cm. 35 ff., 34 11. Paris 1571. The Proeme is dated " De la groffe tour de Bourges le douziefme d'o-// ctobre. 1571." An arithmetic for instruction in military circles. PAGE 347. A 1590 Antwerp edition of Menher has been acquired: Livre // d'Arithme-//tique contenant plufieurs belles que-//ftions & demandes, bien propres & // vtiles a tous Marchans, // Par M. Valentin Mennher // de Kempten. // Reueu par M. Melchior van Elftaer.// . . .' PAGE 375. The 1584 Cologne edition of Clavius has been acquired. PAGE 383. < Arithmetica // oder // Rechenbuch // . . . durch // An- thonium Schultzen/. . . Zur Liegnitz . . . 1600.' Fol., 13.6 x 18.5 cm., the text being 10.2 x 13.5 cm. 4 + 259!?., besides an appendix on Buchhalten,' 30 11. The first edition appeared in 1583. A book of no particular merit. PAGE 389. In 1585 Fr. Barocio published a Cosmographia, at Venice. It contains 5 pp. on arithmetic. PAGE 391. The work of Aurelio Marinati has been acquired: ' La prima parte // della // sommadi tvtte// le scienze // nella qvale si tratta delle // sette arti liberali // . . . Roma ...1587.' 4, 1 6 x 2 1 .5 cm., the text being n x 17.7 cm. 8 + 156 pp. The chapter ' Dell' aritmetica' begins on p. 99 and ends on p. 114. PAGE 408. Vila's arithmetic has been acquired: ' Reglas // brevs de Arith-//metica . . . per Bernat //Vila, . . . Barcelona . . . Any. M.D.- LXXXXVI.' 8, 9.8 x 14.3 cm., the text being 7.5 x 12 cm. 8+136*!. PAGE 418. The 1611 edition of Mariana's work has been acquired. * loannis // Marianse // Hifpani //, e socie. lesv, // De Ponderibvs // et Mensvris.//Typis Wechelianis. // Anno M.DC.XI.' PAGE 425. A manuscript of Zuchetta's work, copied in 1692, has been acquired. PAGE 440. There has been acquired one folio of manuscript on parchment, in a fourteenth-century hand, containing part of the Ety- mologies of Isidorus. 31 X 45.7 cm. See also p. 8. PAGE 456. There has been acquired an anonymous manuscript on astrology, with some directions for arithmetical computations. It bears no date but was written c. 1450. 5 ff., on paper, in Latin, in a German hand. PAGE 480. A manuscript of Bede, ' De Scientia computandi,' written in Latin, c. 1520, has been acquired. Fol., 22. 7x33 cm., the text KARA ARITHMETICA being 13 x 22.5 cm. In the same volume and written by the same hand is ' Franconis De Quadratura Circuli lib. V.' PAGE 487. A manuscript of Boethius and Gerbert, on geometry, written in Latin, c. 1550, has been acquired. The manuscript of Ger- bert begins ' Incipit liber geometri^ artis aeditvs a Dno // Gerberto Papa et Philosopho. Qvi et Silvester // secvndvs est nominatvs.' Each manuscript contains some interesting number work. Fol. on paper, 22 X 33-5 cm -> tne text being 15.5 x 23.5 cm. As stated in the preface, there will naturally be found from time to time numerous additions to the bibliography contained in this work. The following have recently come to the attention of the author : PAGE 70. Portius. There was also an edition, Rome, 1524, 4. PAGE 81. Boethius, Clichtoveus, and Stapulensis. There was also an edition, Paris, 1514, fol. PAGE 97. There is an anonymous 'Algorithmus linealis,' 1513 at Gottingen, probably Licht. PAGE 106. Kobel. There was an edition of the ' Vysierbuch,' Oppen- heim, 1519. PAGE 123. Grammateus. There was also an edition, Frankfort, 1554. PAGE 139. Riese. There was also an edition, 1559, and a Frankfort edition, 1563, of No. 2. PAGE 180. Albert. There was also an edition, Magdeburg, 1588. PAGE 195. Noviomagus. There was also an edition, Cologne, 1539. PAGE 214. Recorde. There was also an edition, London, 1658. PAGE 257. Giilfferich. There was also an edition, Frankfort, 1561. PAGE 286. There was published a small Tariffa by Marcello in 1566. PAGE 359. There may have been a Rechenbuch by Junge published in 1577. PAGE 396. Lindebergius published at Rostock, in 1591, a work ' De prsecipuorum . . .,' containing a little arithmetic. PAGE 429. There is also a work by Lachar, ' Algorithmus mercatorum,' s. 1. a. The statement in the preface, page ix, relating to the De Morgan library should be modified. While some of the books were sold (for example, see page 122), most of them were purchased by Lord Over- stone and presented to the University of London, where they may now be seen. INDEXES INDEX OF DATES DATES OF PRINTED BOOKS 1472-80 8, 9, r o M78 3 1481 10 1482 II, 12 M83 13, IS 1484 15, 18 1485-7 23 1488 25, 36 1489 36, 39 1490 41, 44 1491 47, 49 1492 50, 54 T 493 54 1494 54, 56 1495 S 8 , 60 1496 62, 63 1497 64 1498 64 1499 66, 67 1500 70,71 1501 71, 76 1502 76 i53 77,83 1504 83 1505 84,86 1 506 86 1507 86,87 1508 87 1509 87,91 1510 89, 91 1511 91 1512 91, 93 r 5 J 3 94,97 1514 98, 106 1515 114, 122 '549 245, 249 1516 122 '550 249, 252 1 5 1 7 122, 123 '55i 252, 253 1518 123, 126 254, 257 1 5 I 9 126, 127 '553 257, 260 1520 127, 128 1554 260, 263 1521 131, 132 '555 263, 269 1522 132, 140 1556 271, 286 1523 140 1557 286, 290 1524 140 1558 290, 292 1525 140 '559 292 1526 143, '52 1560 295, 298 1527 153, 156 1561 298, 306 1528 '57. 159 1562 308,311 1529 '59 1563 3", 3H '53 159, 164 1564 3'5 1531 165, 167 ! 5 6 5 316, 319 1532 1 68 1566 320, 322 '533 i?if '73 1567 325 '534 174, 180 1568 325, 328 '535 180, 181 1569 33, 338 1536 181, 182 1570 338 1537 183, i 86 1571 338 1538 186, 188 1572 340, 343 '539 191, 195 '573 343, 346 1540 197, 211 1574 346 1541 211 1575 348, 352 1542 212, 2l6 1576 353 1543 221, 223 1577 353' 359 1544 223, 229 1578 359, 3 6 ' 1545 2 3 I, 2 3 8 1579 361, 364 1546 240, 243 1580 364, 3 6 7 1547 244 1581 368 1548 244 1582 368, 375 497 498 RARA ARITHMETICA 1583 375' 38o 1589 392, 393 1595 407 1584 380, 383 1590 393 1596 407, 408 1585 385, 389 J 59' 394, 396 1597 408 1 5 86 389 1592 396, 400 1598 409, 415 1587 389, 391 1593 404 '599 415 1588 39i 1594 404, 407 1600 425, 427 DATES OF MANUSCRIPTS 1260 433 M47 45 1 1501 480 1294 434 145 452 1522 k 480 1300 435 M5 6 454, 458 1525 481 1339 435 1460 459 1533 482 1 3S 440 1462 465 '535 482 1375 442 1469 466 1545 484 1384 443 M73 466 1550 487 '393 443 M75 468 1560 488 1400-35 439 1476 470 1565 488 1422 443 M77 473 1568 49 1424 446 1478 473 1575 49 M3 447 1488 474 1579 492 M35 449 1490 475 1598 493 1441 449 1492 477 '599 493 1442 458 1500 477 1600 493 INDEX OF NAMES, PLACES, AND SUBJECTS Abacus, 7, 70. See Counters Adriaen van der Gucht, 338 Agricola, 171 Alamagni, 404 Albert, J., 178 Albert of Saxony, 9, 3, 465 Albertus Magnus, 442 Alcala, 60, 167, 231, 310,322 Alciatus, 164 Alexandrinus, 389 Algebra, 56, 125, 132, 226, 233, 252, 254, 260, 286, 400, 403, 454 Algebra (facsimile), 455 Algebraic symbols (facsimile), 403 Algorism, 5, 7, 10, 13, 15, 28, 33, 74, 106, 45> 454 Al-Khowarazmi, 7, 454 Alkmaar, 325 Alligation, 18 Almadiano, 23 Alonso Delatore, 41 America, 286 Ammonius, 367, 429 Amsterdam, 325, 424 Andrea Lanfreducci, 475 Andres, 122 Andres Garcia de Lovas, 253 Angelus Mutinens, 140 Anianus, 31,7 Annaberg, 139 Anonymous, 3, 23, 64, 67, 71, 83, 86, 106, 120, 126, 127, 128, 130, 153, 173, 178, 186, 195, 211, 212, 223, 243, 244, 252, 286, 311, 338, 340, 359, 368, 389, 408, 429, 443, 447, 449, 45' 45 2 454, 456, 458, 459, 463, 464, 466, 468, 470, 473 475. 477, 47^, 480, 482, 486, 488, 493, 494 Antonius de Barres, 238 Antwerp, 32, 183, 200, 249, 250, 278, 281, 3 1 S' 345' 3 6 5> 372, 375, 3 8 <>, 427 Apianus, 155, 62 Aquila, 396 Arabic numerals, 1 1 1, 447 Arabic numerals (facsimiles), 447, PI. VIII, et passim Archimedes, 226 Aristotle, 15 Arithmomachia, 12. See Rithmomachia Arnheim, 188 Astrolabe, 72 Astrological numerals, 375 Astronomical fractions. See Sexagesi- mals Augsburg, 8, 10, 25, 27, 37, 41, 100, 102, 119, 130, 152, 1 60, 1 68, 1 86, 190, 238, 263, 322, 338 Augustine, 83 Aurel, 254 Austria. See Cracow, Vienna, etc. Autorff, 250 Aventinus, 136, 229 Avignon, 130 Bachet, 348 Baeda, 131, 140, 159 Baeza, 269 Baker, 327 Bamberg, 12, 15 Barcelona, 93, 310, 316, 375, 408 Barchi, 263 Barlaamo, 315, 343 Barocius (Barozzi), 295, 340 Barres, 238 Barter, 18, 399 Bartoli, 315 499 500 KARA ARITHMETICA Barziza, 140 Basel, 8, n, 27, 43, 60, 66, 70, 82, 86, 131, 164, 165, 166, 167, 168, 171, 180, 182, 186, 192, 211, 226, 246, 265, 266, 269, 286, 315, 330, 335, 338, 348, 356, 361, 372, 375, 389, 407, 427 Beausard, 346 Bede. See Bxda Beldamandi, 13 Belgium. See Antwerp, Bruges, etc. Belli, 343 Benedetti, 364 Benedetto da Firenze, 464 Benese, 182 Benicansa, 402 Berenguer, 359 Bergamo, 375, 380, 382, 384 Bernard, 452 Bertholio, 263 Bilstenius, 407 Blasius, 95 Blundevile, 407 Bock, 229, 249 Boethius, 25, 62, 80, 4, 13, 82, 434 Bogardus, 229 Boissiere, 260, 271 Bologna, 61, 244, 253, 295, 356, 359 Bonini, 122 Bonocchio, 347 Borghetti, 407 Borghi, 1 6 Borriglione, 86 Boschensteyn, A., 182 Boschensteyn, J., 100 Boscherus, 393 Bouvelles (Bovillus), 89 Bradwardin, 61, 5, 86, 117, 451 Brandt, 169, 408 Brasser, 393 Braunschweig, 391 Brescia, 325, 347, 368, 425 Breslau, 139, 322, 347, 404, 412 Brucaeus, 350 Bruges, 338 Bruno, 396 Brunus, 396 Buckley, 252 Budaeus, 99, 35, 164 Budapesth, 67 Bud<?. See Budaeus Budel, 396 Bungus, 380 Buscher, 393 Buschius, 86, 106 Buteo, 292 Caesar of Padua, 368, 402 Calandri, 47 Calculi. See Counters Calendar. See Computus Calvino, 76 Camerarius, 262, 186, 211 Camilla, 315 Campanus, 433 Canacci, 459 Cantone, 416 Capella, 66 Cardan us, 193, 338 Cardanus (portrait), 194 Casanova, 292 Cassiodorus, 211 Cataldi, 356 Cataneo, 242 Catechism method, 197, 210, 290 Catechism method (facsimile), 199 Cathalan, 268 Cattaldi, 356 Champenois, 359 Champier, 186 Chauvet, 359 Chiarini, 10, 56 China, 429 Chuquet, 128 Cipro, 10 Ciruelo, 58, 61 Clatovenus, 292 Clavius, 375 Clement, 375 Clichtoveus, 30, 80, 82, 94 Cognet, 346, 365 Coignet. See Cognet Cologne, 23, 27, 46,74, 76, 85, 153, 154, 159, 167, 169, 1 80, 195, 2OO, 212, 231, 281, 367, 375, 396, 402, 429 Computus, 7, 31, 41, 73, 126, 443, 449, 463, 473 Computus (facsimiles), 444, 445, 472 Constantinople, 180 Copenhagen, 353 Corella, 244 INDEX SGI Cortes, 407 Coss, 125, 226, 233, 260, 286. See Algebra Counters, 7, 36, 155, 271, 412, 482 Counters (facsimiles), 69, 103, 156, 166, 201, 214, 215 Covarrubias, 396 Cracow, 32, 97, 123, 190, 260, 303 Cuento, 60, 249 Culman, 182 Cuno, 269 Curtius, 396 Cusa, J., 106. See also Nicolaus Cusa Czerny, 353 Dagomari, 435 Debreczin, 359 Decimal point (facsimiles), 52, 388 Decimals, 50, 329, 330, 386, 388 Dee, 214 De la Roche, 128 Delfino, 275 Denmark. See Copenhagen De Suberville, 409 Deventer, 64, 67, 76, 188, 195, 262 Dialing, 480 Diego el Castillo, 46 Digges, 340 Diophantus, 348 Division, 18, 49, 449 Division (facsimiles) A danda, 47, 462, 489 Galley, 6, 100, 483, 486, 487 Of fractions, 485 Division of fractions, 233, 262 Dollar sign, 470 Dollar sign (facsimiles), 471, 491 Domenicho de Valsugana, 492 Dordrecht (Dort), 392 Dortmund, 213 Duni, 269 Duplation, 75 Diisseldorf, 300 Dycke, 427 Eclipse (facsimile), 453 Eisenmann, 91 Eisenmenger, 389 Eisleben, 303 Elias Misrachi, 180 England, 134,216,327,450. See London Equality sign (facsimile), 288 Equation of payments, 439, 446 Erfurt, 124, 139, 338 Eschenburg, 340 Espinosa, 238 Euclid, ii, 236, 4, 56, 433, 435. 44<>, 442, 463, 481, 486 Euclid (facsimiles), 436, PI. IV, VI Everardus, 396 Exchange, 38 Exchange (facsimile), 38 Eysenhut, 190 Faber Stapulensis, 62, 80, 27, 30, 82 False position, 18 Feliciano, 146 Fermat, 348 Fernandez de Santaella, 140, 269 Fernelius, 157 Ferrara, 143, 146, 394 Figurate numbers (facsimiles), 27, PL I Finaeus, 160, 279, 82, 97, 106 Finger symbols, 34, 57, 95, 115, 121, I3'> 137. 257 Finger symbols (facsimiles), 57, 138 Fischer, 247 Flicker, 367 Florence, 10, 15, 16, 47, 63, 70, 98, 99, 122, 132, 140, 346 Foeniseca, 119 Fonduli, 295 Fonduli (portrait), 296 Forcadel, 284, 316 Fractions, 106. See Decimals, Sexa- gesimals, Division France, 284. See Paris, Lyons, Avignon Francesco dal Sole, 143 Franciscus, 15 Franciscus de Oretio, 396 Frankfort a. M., 32, 53, 102, 106, 123, 139, 154, 155, 169, 1 80, 200, 244, 253, 269, 290, 295, 298, 319, 330, 331, 340, 359. 3 68 > 3 8 3> 393' 396, 408, 429 Frankfort a. d. Oder, 139, 247, 289 Frascada, 490 Freiburg, 82, 181, 182, 192, 391, 410 Freigius, 372 Frey, Jacob, 338 Frey, Johann, 221 Freyle, 286 502 Fulconis, 389 Furst, 93 KARA ARITHMETICA Galasso, 404 Gauging, 114, 142, 221, 468 Gauging (facsimiles) 113, 222 Gehrl, 359 Gematria, 404 Gemma Frisius, 200 Gempelius, 391 Genoa, 416, 425 Gentil, 263 George of Hungary, 67 Germany. See Augsburg, Bamberg, Heidelberg, etc. Gersbach, 353 Getons. See Counters Ghaligai, 132 Ghebelino, 325 Giovanni da Firenze, 443 Girard, 386 Gifjka Gorla z Gorlssteyna, 353 Giuntini, 492 Glareanus, 191 Gleitsmann, 427 Glysonius, 408 Gmunden, 117, 449 Goldammer, 269 Gosselin, 279 Gouda, 386, 394, 424 Grammateus, 123 Granada, 93 G^y, 353 Gualda, 167 Gucht, 338 Gulfferich, 269, 292 Gullucci, 82 Gutierrez de Gualda, 167 Guyion, 130 Gyraldus, 254 Haarlem, 421 Hagenau, 37, 76, 164 Halle, 303 Hamburg, 393 Hanaellius, 228 Hangest, 87 Hartwell, 220 Heere, 421 Hegelin, 229 Heidelberg, 168, 266, 356, 359 Heller, 67 Helm, 141, 142, 319 Helmreich, 303 Helmstadt, 393 Henry-metre, 409 Herbestus, 303 Herodianus, 60 Hobel, 314 Hoflein, 389 Holland. See Amsterdam, Deventer, etc. Honoratus, 487 Hood, 361 Horem, 86, 117 Hornmannus, 396 Horoscope, 478, 492 Hostus, 372 Huguetan, 188, 131 Hungary. See Budapesth, Debreczin Huswirt, 74 Hiitzler, 263 Hylles, 396 lamblichus, 188, 186 Ingolstadt, 155 Instruments (facsimiles), 124, 283, 297, 367 Isidorus, 8, 3, 10 Italy. See Florence, Venice, etc. Jacob, 295 Jacob's staff, 1 10 Jeronimo de Valencia, 269 Joannes de Gmunden, 117, 449 Joannes de Muris, 117 Johann von Gmunden, 117, 449 Jordanus, 62, 5, 27, 82 Kandleon, 368 Kaudler, 368, 396 Klos, 190 Klumpius, 493 Kobel, 100 Kolross, 164 Konigsberg, 226, 258, 260, 314 Kopfer, 346 Krafft, 402 Lagasa, 315 Landshut, 83, 97 INDEX 503 Lanf reducci, 475 Lange, 353 Lanzut. See Landshut Lapazzaia, 322 Lapazzaia (portrait), 323 Laudensis, 396 Lautenschlager, 410 Lax, 121 Lazesio, 146 Leipzig, 27, 36, 37, 44, 53, 70, 83, 86, 91, 120, 138, 139, 155, 1 68, 171, 200, 223, 233. 2 47, 249, 250, 263, 290, 303, 311, 33 r 340, 353, 361, 365, 379, 412, 421 Lemgo, 330 Leonardus Maynardus, 474 Leunbach. 86 Leyden, 66, 165, 166, 210, 254, 386. See Lugduni Licht, 70 Liegnitz, 383 Lillius, 63 Liverius, 13 London, 10, 132, 134, 182, 195, 213, 214, 244, 252, 253, 286, 327, 330, 338, 340, 353' 3 68 39 i> 393' 39 6 > 4o, 407, 408 Lonicerus, 253 Lopez de Corella, 244 Loritus, 191 Lortze, 91 Lossius, 289 Letter, 70 Lottini, 286 Louvain, 238, 346 Lovas, 253 Liibeck, 263, 393 Luca da Firenze, 468 Luca de Borgo, 54, 87 Lucar, 393 Ludovico Alt, 484 Lugduni (Lyons or Leyden), 167, 200, 286 Lullius, 415, 457 Luyck, 408 Lyons, 32, 93, i S, 130, 165, 171, 192, 245, 253, 269, 286, 290, 292, 314, 320, 385, 389. See Lugduni Madrid, 310, 393, 404 Maffei, 86 Magdeburg, 139, 180, 290, 389 Maginus, 399 Mainz, 23, 118 Mameranus, 396 Manenti, 174 Mantua, 66, 263 Manzoni, 257 Mariana, 418 Mariani, 180 Marinati, 391 Martin, 231 Masterson, 400 Mathematics, value of, 72, 127, 231, 300 Maurolycus, 348, 350 Maynardus, 474 Measures, 70, 99, 164, 171, 209, 266, 319 Mediate, 75 Medina, 229 Medlerus, 223 Melanchthon, 210, 53, 164, 226, 236 Melero, 181 Mellema, 375 Mellis, 218, 214, 391 Menher, 249, 281, 346 Menochius, 396 Mensuration, 478 Mercado, 335 Mercatello, 480 Meres, 408 Messina, 93, 429 Mewrer, 396 Mexico, 286 Meyer, 322 Micyllus, 265 Middelburg, 418 Milan, 76, 114, 115, 193, 216 Military arithmetic, 343, 359, 385 Million, 17 Million (facsimile), 19 Mirandula, 127 Misrachi, 180 Modena, 66 Mohammed ibn Musa, 7, 454 Molinaeus, 396 Monhemius, 300 Monte Regal Piedmontois, 385 Monzo, 292 Mordente, 415 Morsianus, 159, 182 Moya, 308 Mugling, 450 504 KARA ARITHMETICA Multiplication, 56, 115, 447 Multiplication (facsimiles) Complementary, 76 Fanciful, 1 1 6 Gelosia, 5, 448, 461 Per quadrato, 5, 448, 461 Per scachiero, 5, 6, 448, 483 Table, 26, 37, 446, 460, 464, 479 Multiplication table, 37, 446 Miinster, 85 Munster, Seb., 180 Munyos, 320 Muris, 117 Mutinens, 140 Mystery of numbers, 95, 91, 199, 383 Nabod, 281 Naples, 322, 324, 402, 416 Neander, 266 Nessen, 322 Newdorffer, 415, 477 Nicolaus Cusa, 42 Nicolo de Orbelli, 23, 473 Nicomachus, 186, 4, 27 Nonius, 315 Norico, 44 Norry, 347 Noviomagus, 195, 159 Numerals. See Arabic, Astrological, Roman Numeration, 260 Nunez, 315 Nurnberg, 53, 62, 86, 123, 137, 139, 151, 152, 154, 156, 160, 178, 182, 193, 221, 223, 226, 231, 249, 271, 292, 300, 314, 33 8 346, 39 1 ' 412, 415, 421 Obernheym, 473 Obers, 238 Ofenlach, 389 Oliva, 126 Oppenheim, 102, 106 Oresme. See Horem Ortega, 91 Otto, 361 Paciuolo, 54, 87 Padovanius, 389 Padua, 9, 10, 13, 150 Paetus, 343 Pagani, 394 Pagnini, 308 Paolini, 393 Paolo dell' Abaco, 435 Paris, 9, 27, 32, 43, 58, 60, 61, 62, 63, 76, 82, 86, 87, 89, 93, 95, 97; 99, 1 06, 121, 126, 131, 134, 157, 160, 167, 168, 171, 186, 191, 192, 195, 200, 211, 223, 228, 229, 233, 237, 245, 252, 260, 263, 268, 269, 271, 278, 279, 284, 311, 314, 316, 330, 34. 343 347. 359. 32, 409. 446, 450 Parley, 408 Parma, 382, 384 Partnership, 18 Pascal triangle, 155, 236 Pascal triangle (facsimile), 156 Pauerus, 286 Paul. See Paolo, Paulus Paulinus, 393 Paulus Alexandrinus, 389 Pavia, 9, 10, 295 Paxi, 77 Peele, 338 Peer, 156 Pelacani, 140 Peletier, 245 Pellos, 50 Per-cent sign, 439, 441, 458 Per-cent sign (facsimiles), 437, 440, 441, 459 476 Perez de Oliva, 126 Perfect number, 91 Petri, 325 Petrus de Alliaco, 41 Peurbach, 53, 117 Peverone, 290 Peverone (portrait), 291 Pforzheim, 36 Piccini, 346 Picus Mirandula, 127 Piedmontois, 385 Pirkheymer, 396 Pisa, 1 6 Piscator, 247 Plus and minus, 40, 126, 183, 254, 456 Plus and minus (facsimiles), 40, 125, 185, 234 Poeppingius, 391 Poitiers, 245 INDEX 505 Poland, 353. See Cracow Portius, 70 Poveiano, 375 Prag, 292, 359 Priscian, 319 Problems (facsimiles) Cistern, 48 Couriers, 112 Fallen tree, 49 Fish, 438 Gauging, 113, 222 Hare and hound, 465 Horseshoe nails, 484 Market women, 1 1 1 Partnership, 402 Snake and well (tree), Thief, 438 Proportion, 46, 89, 99, 235 Proportion (facsimile), 28 Prosdocimo, 13 Psellus, 168 Pythagoras (portrait), 46 Quadrans (facsimile), 162 Quirini, 429 Raets, 365 Raggius, 98 Rainer, 200 Ramus, 263, 330, 335 Raphael Franciscus, 15 Raymundus Lullius, 457 Recorde, 213, 253, 286 Recreations, 391 Regensburg, 136, 137, 396, Reggio, 370 Regiomontanus, 62 Regius, 181 Regnaudus, 396 Regnier, 200 Reich elstain, 169 Reinhard, 421 Reisch, 82 Reymers, 379 Rheticus, 211 Ricci, 429 Riese, A., 138, 171, 250, 7 Riese, A. (portrait), 251 Riese, I., 252, 365 Ringelbergius, 165 Ringhieri, 253 Rithmomachia (Rithmimachia, Ryth- momachia), 12, 63, 271, 340 Rithmomachia (facsimiles), 64, 273 Rizzo, 298 Rocha, 316 Roche, 128 Rodolphus Spoletanus, 122 Rodriguez, 408 Rollandus, 446 Roman numerals, 104, 106, 249, 382, 383, 385 Roman numerals (facsimiles), 104, 105, !7> 373. 382, 383 Rome, 32, 86, 91, 93, 122, 140, 236, 375, 48 439 378, 39 1 . 415. 429 Ros, 457 Rostock, 350 Roth, 493 Rotterdam, 250, 423, 425 Rouen, 32 Rozino, 451 Rudolff, 151, 159, 126, 7, 260, 493 Ruff us, 31 Rule of three, 18, 46 Rules, 76 Sacrobosco, 31, 236, 450, 453 Salamanca, 10, 216, 238, 253, 308, 315, 335. 408 Salignacus, 359 Salispurgo, 484 Santa Cruz, 404 Santaella, 140, 269 368 Sarafino da Campora, 429 Saragossa, 99, 122, 140, 167, 181, 247, 249, 269, 359 Saravia, 229 Savonne, 314 Scaruffi, 370 Scheubel, 233, 246, 252, 454 Schey, 427 Schiedam, 423 Schleupner, 412 Schonerus, J., 178 Schonerus, L., 330, 333 Schreckenberger, 389 Schreckenfuchsius (Schreckfuchs), 180 Schreiber. See Grammateus Schuere, 424 KARA ARITHMETICA Schulze, 383 Schweder, 314 Segura, 322 Sekgerwitz, 347 Series (facsimiles), 88, 478 Seville, 41, 122, 93 Sexagesimals, 333 Sfortunati, 174 Sherwood, 12 Siderocrates, 389 Siena, 367 Silicius, 106 Slate, 13 Snellius, 330, 333 Sole, 143 Solingen, 208 Solorzano, 393 Spain, 254. See Alcala, Barcelona, etc. Spanlin, 271 Sphere, 453 Spinola, 311 Spoletanus, 122 St. Albans, 186 St. Bernard, 452, 466 Stephano da Mercatello, 480 Stettin, 139, 247 Stevin, 386 Stifel, 223, 231, 258, 493 Stigelius, 249 Stockmans, 392 Strasburg, 10, 32, 33, 42, 76, 82, 134, 154, 182, 197, 211, 233,315, 343.370, 389. 415 Strigelius, 311 Striibe, 391 Suberville, 409 Substractio, 97 Suevus, 404 Suiseth, 10, 86 Supputandi, De Arte, 134 Swinshead. See Suiseth Switzerland. See Basel Tables, 385, 400. See Tariffa, Multi- plication Taf, 429 Tagliente, 114, 141 Tariffa, 77, 180, 404, 175, 181 Tartaglia, 275 Tartaglia (portrait), 277 Texeda, 240 Theologoumena, 223 Theoretical books, 4 Thierf elder, 391 Toledo, 167, 418 Tonstall, 132 Torrentini, 76 Toscolano, 54 Toulouse, 348 Tours, 1 68 Trenchant, 320 Trevisano, 408 Treviso, 3 Trigonometry, 474 TUbingen, 74 Turin, 50, 86, 364 Turkey. See Constantinople Tzwivel, 84 Uberti, 114 Ulm, 229, 402 Ulman, 391 Unicornus, 298, 412 Uranius, 208 Urban IV, 433 Urstisius, 220, 361 Valencia, 61, 122, 254, 292, 320, 407 Valencia, J. de, 269 Valerianus, 286 Valla, 71 Valladolid, 240, 244 Valturius, 10 Vandenbussche, 340 Van den Dycke, 427 Vander Hoecke, 183 Vander Schuere, 424 Vander Schuere (portrait), 422 Vander Wehn, 216 Vejar, 249 Venice, 8, 9, 10, 12, 13, 15, 16, 19, 20, 21, 22, 27, 28, 32, 54, 60, 61, 62, 70, 7i> 73' 77, 82, 87, 99, 114, 115, 130, 140, 141, 146, 148, 160, 168, 171, 173, 174, 180, 181, 195, 200, 229, 242, 253, 254, 257, 275, 278, 292, 298, 315, 340, 343' 346, 347, 348, 35' 364, 382, 389, 393, 399, 404, 407, 408, 412 Ventallol, 298, 319 Vergerius, 456 INDEX 507 Verini, 216 Verona, 10, 148, 389, 404 Verse, 399, 411, 100, 262 Verse (facsimile), 411 Vienna, 27, 32, 53, 61, 66, 106, 117, 123, 151, 152 Vila, 408 Vincent de Beauvais, 10 Vincento, 140 Vincenza, 66, 375 Vincenzo da Bergamo, 429 Visconti, 368 Vittori, 6 1 Vogelin, 195 Von Szily, 67 Vuelpius, 231 Wagner, 12, 15 Walckl, 182 Weber, Johann, 338 Weights. See Measures Weissenfels, 223 Welsch practice, 152, 416 Wenceslaus, 418 Werner, 300 Widman, 36, 40, 44 Willichius, 197 Willsford, 216 Wittenberg, 32, 53, 61, 139, 168, 178, 180, 200, 223, 236, 237, 247, 269, 286, 367, 389, 429 Wojewodki, 260 Wolphius, 154 Xylander, 356 Yciar, 247 Zamberto, 481 Zero, 76 Zuccantini, 367 Zuchetta, 425 Zuchetta (portrait), 426 Zurich, 391, 396 Z Smith, David Eugene 6654 Kara arithmetica A7S7 PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY