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Full text of "Rara arithmetica; a catalogue of the arithmetics written before the year MDCI, with description of those in the library of George Arthur Plimpton, of New York"

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 



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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&quoteius 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&ltctt 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 



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FIG. 75. TITLE PAGE OF THE 1592 RIESE 



PRINTED BOOKS 



'45 




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LIBRETTI NVOVI CON LE RE* 
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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, 

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J vVvfvwVVVt 




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por ticmpo dc diez anos 



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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* 

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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. 




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ir tcnitl JwW . ] b ucrwr t'.u : fi tutl avf 
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cy^lty^ttcf tint*'ca[tui-v. (^ ^ fjf ^i- <j<ocf . tunico'). 

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-- u ^, t ,,;;t^;-!! 



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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 



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