Normaf School
KARA ARITHMETICA
.
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root* 16. fififla.3rorfhr:|-^ra(iitnn nuofl
raf. Crrlittmfl>ttf ftama^wmowTa^
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nuoai \ twc <ip\nnut ^rtkmf eft:
I'l \TK I. FROM A MANUSCRIPT OF BOETHIUS, C. 1294, SHOWING
FIGURATE NUMBERS
RARA ARITHMETICA
d^A CATALOGVE OF THE ARITHMETICS
WRITTEN BEFORE THE YEAR MDCI WITH A
DESCRIPTION OF THOSE IN THE LIBRARY OF
GEORGE ARTHVR PLIMPTON OF NEW YORK
BY DAVID EVGENE SMITH OF TEACHERS
COLLEGE COLVMBIA VNIVERSITY
GINN AND COMPANY PVBLISHERS
BOSTON AND LONDON MDCCCCVIII
COPYRIGHT, 1908, BY DAVID EUGENE SMITH
ENTERED AT STATIONERS 1 HALL
$rr*g
GINN S: COMPANY PRO-
PRIliTORS BOSTON U.S.A.
VIRO
DOCTISSIMO ILLVSTRISSIMO
GEORGIO - ARTVRO - PLIMPTON
ARTIVM MAGISTRO
LIBRORVM VETVSTIORVM AMATORI
AVCTORVM AMICO SOCIO ERVDITORVM
LITTERARVM
FAVTORI FOTORI
PREFACE
One of the first and most important questions for the student
of mathematical history is that relating to the available sources of
information. In the fields of higher mathematics scholars have
been more or less successful in bringing together these sources,
and in listing them in bibliographies ; but in that humbler field in
which primitive mathematics first found root, only a few biblio-
philes have sought to preserve the original material, and no one
has seriously attempted to catalogue it. Libri, it is true, brought
together two large libraries of Kara Arithmetica, but he was
neither a true book-lover nor a true scholar, for he gathered his
treasures purposely to see them dispersed, his commercial spirit
scattering at random what should have been kept intact for the
use of scholars. Prince Boncompagni, the most learned of all
collectors in this domain, lived to see an unappreciative city
ignore his offer to make his magnificent library permanent, and
at his death it was scattered abroad, as had been the lesser ones
of Kloss and De Morgan. The third great collection of early text-
books which has been made in recent years is that of Mr. Plimp-
ton. Of the libraries of arithmetics printed before the opening of
the seventeenth century his is the largest that has ever been
brought together, not excepting Boncompagni 's, and it may well
be doubted if another so large will again be collected by one man.
De Morgan was able to examine, in the British Museum and else-
where, less than a hundred arithmetics written before 1601,
including all editions ; but Mr. Plimpton has more than three
hundred, a number somewhat in excess of that reached by Bon-
compagni. Indeed there are few arithmetics of much importance
that are not found, in one edition or another, in his library.
x KARA ARITHMETICA
The writers of these early printed books, not all themselves
of the centuries under consideration, were by no means obscure
men. Among them was Boethius, whom Gibbon called "the last
of the Romans whom Cato or Tully could have acknowledged
for their countryman." In the list are the names of Cassiodorus
and Capella, who at least represented what there was of culture
in their day, and Isidorus, the learned Bishop of Seville. There
are also the names of Archimedes, who deemed it a worthy labor
to improve the number system of the Greeks ; of Euclid, whose
contributions were by no means confined to geometry ; of Nicoma-
chus and lamblichus, who represent the declining Hellenic civiliza-
tion, and of Psellus, who was a witness of its final decay. There,
too, are the names of the Venerable Bede, of Sacrobosco, and of
Bradwardin, all of whom testify to the culture of mediaeval Eng-
land ; of that great Renaissance compiler, Paciuolo ; of Tartaglia
and Cardan, who helped to make the modern algebra, and of such
scholars as Ramus, Melanchthon, and Bishop Tonstall. Worthy
as such a list may be, it is rendered none the less so by the names
of Widman, Kobel, Borghi, Riese, and Gemma Frisius, mere arith-
meticians though they were, for few who read their works can fail
to recognize that they powerfully influenced education, not only
in their own time but for generations after they had passed away.
In view of the fact that the fifteenth and sixteenth centuries
constituted the formative period in the history of printed arith-
metics, I have felt it a not unpleasant duty to catalogue the volumes
in Mr. Plimpton's library that belong to this period and subject,
including such later editions as it may contain, to give a brief
statement of their contents, and to supplement this work by a list
of other arithmetics published before 1 60 1 . That a complete bibli-
ography is impossible is evident to any one who considers the
subject. It is a simple matter to consult the few lists of early
mathematical works, and to trace the names thus secured through
such catalogues as those of the British Museum and the Biblio-
theque nationale (unfortunately only just begun), and through
bibliographies like those of Graesse and Hain and Coppinger. It
PREFACE xi
is also an easy matter to examine the masterly work of Riccardi,
the less accurate list of Murhard, the catalogues of Libri, and
numerous other works of a similar nature. But it is manifestly
impossible to read all of the published catalogues, almost invari-
ably arranged only by authors. Therefore many extant books
will necessarily remain undiscovered, and it is probable that some
will always elude the eye of the special bibliographer. Such is
the work that has been done in preparing this volume, and such
is the feeling of insufficiency of achievement that remains. There
is, however, a satisfaction in knowing that the bibliography is
based in large measure upon an examination of the books them-
selves in various libraries, and that the secondary sources are of
recognized authority. Over five hundred and fifty different
works are mentioned, or, including the various editions, nearly
twelve hundred books in all. Of the five hundred and fifty books,
about four hundred and fifty are, strictly speaking, arithmetics.
Of all arithmetics known to have been printed in the sixteenth
century, and to have been important enough to have two or more
editions, Mr. Plimpton's library lacks less than twenty-five. Of
those which were published but once, some are known only by
name, while the rest are mostly ' abachetti ' or ' Rechenbiich-
leins ' mere primers of a few pages and of no importance.
It must also be borne in mind that it was inevitable that cer-
tain arithmetics of the sixteenth century should have perished
utterly, leaving not even a record of their existence. Their
very commonness often caused their destruction, a law of which
the unique surviving copy of more than seventy thousand New
England Primers of the Franklin-Hall press is a lonely witness
in our country.
It is inevitable that there should be errors in such a list.
Titles have been included when mentioned in even one standard
bibliography, although they cannot be found in others, it being
practically impossible for one person to verify every item. It is
hoped, however, that a foundation has been laid upon which
others may build, eliminating or otherwise as the case may be.
xii KARA ARITHMETICA
Of other works in this field, little need be said. De Morgan's
Arithmetical Books is still one of our best single sources,
although sixty years have elapsed since it first appeared. De
Morgan, however, mentions altogether about one hundred and
twenty editions of works originally appearing before 1601, against
nearly twelve hundred listed here. Unger and Tropffke are
both scholarly writers, but their bibliographies are almost exclu-
sively German. Sterner was less of a student, and his list is cor-
respondently less valuable. Historians like Cantor and Zeuthen
have given this particular period only nominal attention, save as
to a few great arithmeticians, ignoring those contributions which
set forth the real work of the people's schools. The titles of the
works of such writers have not been given, since any reader of a
bibliography like this will know the more general histories.
Mr. Plimpton's library has also a number of valuable manu-
scripts on arithmetic. Since these are not available for students
generally, although of great value in themselves, they have been
placed after the printed works instead of being inserted in chrono-
logical sequence. A study of our numeral system has justified the
inclusion of books which, if printed, would hardly have place.
Manuscripts written before the forms of numerals were fixed are
often valuable in tracing their development, even though the
books themselves are not arithmetical. Only those have been
catalogued which bear with some directness upon arithmetic, and
which were written before the year 1601, although numerous
others, in many respects as valuable, and including several inter-
esting works on the calendar, the sphere, and astrology, are in
the library.
One difficulty attendant upon a labor of this kind is to deter-
mine what printed books to include. The number might easily
have been increased by listing, as De Morgan occasionally did,
works that are not at all arithmetical Peletier's algebra for
example. It has been thought better to draw the line in gen-
eral more closely than he did, and to depart from genuine arith-
metics only in the case of the works that discuss at least some
PREFACE xiii
questions relating to the science or art of numbers. No effort
has been made to add to the supplementary lists books that
are not purely arithmetical, such as treatises on the ancient
measures, although those that are in Mr. Plimpton's library, and
are of value in the study of the history of arithmetic, have been
included.
The arrangement is chronological by first editions, but the Index
allows for alphabetical and geographical reference. Although the
nature of the work is usually discussed very briefly, an examina-
tion of the Index will show that a fairly complete history of
Renaissance arithmetic has been included in the work a his-
tory which I hope to present in other form in the future. The
uncertainty in the use of such symbols as 4 and 8 has led to
the measuring of the page and text. The page varies owing to
the binder's work, and the text is not uniform page for page, but
this plan seems the most satisfactory one for giving the size of
the book. The centimeter has been taken as a unit of measure,
since all English and American readers of a work like this will
be familiar with it, while our popular units would be unknown to
most others. These measures, like the number of lines to a page,
of course vary considerably in the same book. The statement
' There were no other editions ' is to be understood to mean that
I have found no others that were printed before 1601. The
illustrations have, in general, been selected with a view to biblio-
graphical needs, although many have a marked historical interest.
In copying the titles it has been the intention to follow the
original as closely as possible, without attempting to imitate par-
ticular forms of type or to use capitals except as initials. In the
cases of misspelled words, omitted capitals, and peculiar punctua-
tion, the errors have been copied as faithfully as possible. At
the same time mistakes must have been made in transcribing,
although it is hoped that they are not of a serious nature.
DAVID EUGENE SMITH
LIST OF PLATES
PAGE
PLATE I. FROM A MANUSCRIPT OF BOETHIUS, c. 1294, SHOW-
ING FIGURATE NUMBERS Frontispiece
PLATE II. FROM THE MARGARITA PHILOSOPHICA 82
A. GEOMETRY B. ARITHMETIC
PLATE III. TITLE PAGE OF LAX 122
PLATE IV. FROM THE CAMPANUS MANUSCRIPT OF EUCLID,
c. 1260 433
PLATE V. FROM A MANUSCRIPT OF EUCLID, c. 1294 .... 436
PLATE VI. FROM A MANUSCRIPT OF BOETHIUS, c. 1300 . . 440
PLATE VII. FROM A MANUSCRIPT OF GIOVANNI DA FIRENZE,
1422 446
PLATE VIII. FROM A MANUSCRIPT OF SACROBOSCO, c. 1442 . 450
PLATE IX. FROM AN ANONYMOUS MANUSCRIPT, c. 1460 . . 462
ABBREVIATIONS
c., circa, about
cm., centimeters
ed. pr., edito princeps, first edition
f., ff., folio, folios
fol., 4, 8, . . . , folio, quarto, octavo, . . .
ib., ibidem, the same place
1., 11., line, lines
p., pp., page, pages
r., recto, the first page of a leaf
s. a., sine anno, without date of publication
s. 1., sine loco, without place of publication
s. 1. a., without place or date of publication
v., verso, the second page of a leaf
//, the end of a line of print
PART I
PRINTED BOOKS
PRINTED BOOKS
ANONYMOUS. Ed. pr. 1478. Treviso, 1478.
Title. ' Incommincia vna practica molto bona et vtile // a
ciafchaduno chi vuole vxare larte dela mercha-//dantia. chia-
mata vulgarmente larte de labbacho.' (F. i, r. See Fig. i.)
Colophon. 'A Triuifo : : A di .io. Deceb^j : : .1478.' (F. 62, r.
See Fig. 4.)
Description. 4, 14.6 x 20.5 cm., the text being 7.3 x 12.8
cm. 62 ff. unnumb., 32 11. Treviso, 1478.
Editions. There was no other edition.
So far as known this is the first practical arithmetic to appear in
print, for Albert of Saxony's Tractatus (p. 9) and the Ars Numerandi
(p. 23), even if earlier, and the Etymologies of Isidorus (p. 8), are not,
strictly speaking, of this class.
The author of the book is unknown, but from the opening lines it
seems that he was a teacher of arithmetic in Treviso. The printer is
also unknown, although it was probably one Manzolo, or Manzolino.
The history of the work has been carefully studied by Boncompagni,
his results appearing in the Atti deW Accademia Pontifitia de 1 Nuovi
Lincei, vol. 16. This particular copy was in the Pinelli collection, and
was sold on February 6, 1790, to a Mr. Wodhull. It afterward found
its way into the library of Brayton Ives, Esq., of New York, and at the
sale of that library was acquired by Mr. Plimpton.
The work is commercial in character, the fundamental processes
being taken up in the common order, and these being followed by the
rule of three. A curious feature not at all common in early arithmetics
is the rule of two. The practical applications are chiefly included under
the rule of three and partnership. There is also a brief treatment of the
calendar, for Church purposes. The book is lacking in applications
to exchange, and probably on this account it did not appeal to the
merchant class sufficiently to warrant a second edition.
3
4 RARA ARITHMETICA
In considering these early works it is necessary to understand the
four types of arithmetics which the Renaissance inherited from the
Middle Ages. These types are as follows :
i. The theoretical books. These works were based chiefly upon
Jncomrmnria t>na pzactitt moltobona et roles
a cufcbaduno ebi mole vjrare larte t*la mtrcba*
i tniljjarnientc larte x* labbacl:o
[ ttefjatp pm e'piu toh e t>a alch uni
1 3ouani a mi molto oilecnfUmi : li
ill p:ctrndf uano a ttourr roU r
la mercbadannatcbe per lo:o
^mojemepiaaffeafradigaitne v
> puocbotitf t>argli in fmtto qualcbe f unda nic to
cerca larte t* anfmctricj^biatnata rulgarmcnre
laWwcbo.CTnde to conftretto per arnoa t>i kwo: rt
f ttadicJ ad vttlitat)! ma cbt p:rrcnd.. no a qucl!a;fu
gondola ficola intclligcntia xxrl msegno iio:bO
deliberate fe non ui tntoun parte tanie fanffare a
k>20.ario cbe Io:o virtuofi oefiderii tJflle fruco re
ceuere poffeanp. Jn nomc tn -010 adoncba : ro&h'o
per pjinpto mw d tirto e a Is onfinc coft wedo.
C tlte quelle cofe:cbe x?a la p?tma oy gme
bano babuco psoducimr to:per ratone e
numero fono fta formadt .^ co(T come fo/
ttO'.bano'oa ftrcognofaidc.'perone la cpgiimone
ce tute le cofe:quefta pzacnca e necr flaria . ger
intrarnel^)pofitomio:prmio fapi lecto:e^be qn/
to fa a I p2opoftio noft ro:Ourmro e t na molnm <
dine congregata ouero infemb2ada ta molte rni/
tade.er a! menow to tJnitade^ome e.a,.lquale
c lo p2imo cmeno2e nunwrotcbe fe truoua.la t>#
nitatle e quella cofa : x>a la qiule orm cofa ft riua
t>na.0eg6dario fapi:cb* fe truoua nwmcri w tre
maniere^ei pzimo fe cbiama numero fimphce.lal
tro namero araculo . 41 teno fe cbwma rwmc ra
4* t
11
FIG. i. FIRST PAGE OF THE TREVISO ARITHMETIC
Boethius (p. 25), who wrote in the beginning of the sixth century, fol-
lowing the Greek models of Nicomachus (p. 186) and Euclid (p. n).
They are devoted to such matters as the theory of figurate numbers, of
which the square and cube are all that now remain in elementary text-
books, and the cumbersome Greek ratio-system which testified to the
ancient difficulty with fractions. Books of this class were written by
PRINTED BOOKS
such men as Bradwardin (p. 61), Albert of Saxony (p. 9), and
Jordanus (p. 62).
2. The algorisms (algorismi). These were practical arithmetics,
moltipbcare per fcacbierotft quali laffaro a! fhidj
o tuozmettendo It epemplt fat fotamence m fb:ma *
come pozat wdere qm fotto
O2 fogK w faretop2edi?tofcacbrero.5oe.3 14*
fia.9 ^? 4, enocax^efarlo peril quatro modi come
qoit^a fotto.
9 ^4 9 ? 4
9 3 4/i
95
/>
/ 9
1
Jt
9
^r
*
\6
o\
6
4
MT^
o\|o\
o\|f\
41 ^
FIG. 2. FROM THE TREVISO ARITHMETIC, SHOWING VARIOUS FORMS
OF MULTIPLICATION
KARA ARITHMETICA
6 o o -i
3?o9<>y5 i
I 3 16461611
I i 369679 5 3 3 565ooo
60 i 3 i 72,0$ t 6 #4$*$ I 60
P*
p pop p pi 31 166
ft e compita dna^ t vmuto to impjonuflb^oe it
tHrfiominsonadoIhXbf felire 146 i 6 0113? 9
1*51 5^ > valiffc duct903 gup
3 J
cbe !irraooo,e $ t>a!eranno ducatia 3 o . g .
li quad fonorn quarto ae t>nooucato3iclx qlie
ra3one e qnefte ftano feguramente bcne*
aiufendo te.cbe qnando bauerai ^a fare qaalcbe
rajcone ^Da impojtann i :e cbc tu ^ubiti : non po:ai
p:uouarc piu fegurametetcbe polt .ire h toa ra co^
nr.al modo cbe bat riflo ne le trc rapoe p^cditt :.
Unde per quede e per k altre ra cone p^editte : le
qu ale fono 111 taco namero qumdepeua puo truert
FIG. 3. FROM THE TREVISO ARITHMETIC, SHOWING MULTIPICATION
AND THE GALLEY FORM OF DIVISION
PRINTED BOOKS 7
written to supply the mathematical knowledge necessary for business
computations, and using the Hindu-Arabic numerals. These numerals
were known in India, without the zero, as early as the third century
B. c. They were gradually perfected, and by the time they reached
Bagdad from India, in the eighth century A. D., they included the zero.
An arithmetic employing these numerals was written about 800 A. D.
by an Arab scholar, Mohammed ibn Musa, known by the name of al-
Khowarazmi (from his birthplace, Khwarazm), and from the Latinized
form of his name came the word algorism.
3. The abacus arithmetics. These were also commercial books ; but
since they used the Roman numerals, which were not suited to com-
putation, the actual calculations were carried
on by means of calculi (Latin, pebbles), getons
(French for things thrown or cast, from the
Latin jacere, to throw), or counters, from
which English form we have expressions like
1 cast an account.' The table on which the
calculi were cast is still called a counter in our hTiw* *<&> mcir<
shops, but. like the sand tables used for com- f*T
/x/\/ /\/\/
puting, it was in early times called an abacus. SJJ35"*.
Just at the opening of the Renaissance the con- *fcy\/ /\'A/
test was still waging between the algorists and
the abacists. These arithmetics are not found,
however, in Italy, because the merchants of
that country abandoned the use of the counters FlG> ^ LAST pAGE QF
long before this was done in other countries. THE TREVISO ARITH-
In Germany the arithmetics frequently have METIC
in their titles the expression ' auff der Linien
und Federn ' (Riese, p. 138, 1522), or 'mit der ziffer unnd mit den
zal pfenningen' (Rudolff, p. 151, 1526), ' auff der Linien ' referring to
the lines on which the counters (* zal pfenningen ') were cast, and the
' Feder ' referring to the pen with which a figure (* ziffer ') was written.
4. The computi. The computus or compotus was a treatise upon
the Church calendar, containing such simple directions as were neces-
sary for computing the dates of Easter and the other movable feasts.
The chapter on the calendar, of which there is still some trace in our
arithmetics, originated in the computus. (See Anianus, p. 31, 1488.)
The Treviso arithmetic is a good example of an algorism. Fig. 3
shows that multiplication was performed as it is to-day (but see Fig. 2),
and that division was performed by the * galley ' method, so called because
the work resembled in form an ancient galley with its sails set.
8 KARA ARITHMETICA
ISIDORUS OF SEVILLE. Ed. pr. 1472. Venice, 1483.
Born, probably at or near Cartagena, c. 560 or 570 ; died at Seville, April 4,
636. One of the most learned men of his time, Bishop of Seville, and writer
on theology, philosophy, and the general learning of the Middle Ages.
Title. 'Cjncipit epiftola Ifidori iunioris hifpalenfis epi-//fcopi
ad Braulionem cefar auguftanu epifcopu.' (F. I, r.)
With this is : ' Cjn chrifti nomine incipit liber primus fancti //
Ifidori hifpalenfis epifcopi de fumo bono.// Qd deus fumus 2
incomutabilis fit C.Cap. I.' (F. i, r., following the first f. 105.)
Colophon. ' CFinit liber etymologiarum // Cjfidori hifpalenfis
epifcopi.' (F. 101, v.)
Colophon of the second part. ' CFinit liber tertius : vltim9 de
sumo bono // fancti Ifidori hyfpalenfis epi : Impreffus // Venetijs
per Petru loflein de Langencen.// C..M.cccc.lxxxiij.C/ (F. 28, r.)
Description. Fol., 19.8 x 27.7 cm., printed in double columns,
each being 6.7 X 23 cm. i2Qff. numb. + 4 unnumb. = 133 ff.,
58 11. Venice, 1483.
Editions. Augsburg, 1472; Venice, 1483 (here described);
ib., 1485 (?); ib., 1493, fol.; Basel, 1577 (mentioned below). Also
two editions s. 1. a. (Strasburg, 1470 ?), and one s. 1. a. (Cologne,
1476-78 ?)
This book of etymologies written by Isidorus, Bishop of Seville in the
seventh century, is the standard authority upon the state of learning in
Spain in that period. The subject of arithmetic is treated in book 3,
beginning (f. 15) : 'CJncipit liber tertius // C.De vocabulo arithmetice
// difcipline C.Cap. I.' The work consists entirely of the mediaeval
theory. The treatment is very brief (5 ff.) and is followed by a few
pages on the calendar. Although appearing in 1472, this cannot be
called the first printed arithmetic, since it touches so briefly upon the
subject. It has therefore been placed after the Treviso arithmetic.
ISIDORUS OF SEVILLE. Ed. pr. 1472. Basel, 1577.
Title. ' Isidori // Hispalensis // Episcopi // Originum libri
viginti/ex antiquitate eruti.// . . . Basileae,//per Petrum Pernam.'
(F. I, r.)
Description. Fol., 20.5 x 31 cm., printed in double columns,
each being 8.2 X 24.4 cm., 60 11. With an edition of Capella.
Basel, 1577.
PRINTED BOOKS 9
ALBERT OF SAXONY. Ed. pr. c. 1478. Venice (?), c. 1478.
Born c. 1330. He lectured in the University of Paris, was Rector at
Vienna in 1365, and was Bishop of Halberstadt from 1366 to 1390. He
wrote several scientific works.
Title. ' Eccelletiffimi magistri alberti de // faxonia tractatus
pportionum incipit feliciter.' (F. i, r.)
Colophon. ' Explitiunt pportiones magiftri //alberti de faxonia.'
(F. 9, r.)
Description. Fol., 19.7 x 28.1 cm., in double columns, each
being 5.6 X 16.7 cm. 9 ff. unnumb., 39 11. S. 1. a. (Venice?,
c. 1478).
Editions. S. 1. a. (Venice ?, c. 1478, here described) ; Padua,
1482 (12 ff.); ib., 1484 (12 ff.); ib., s. a., but before 1487;
Venice, M.cccc.xxxlvii for 1487 (10 ff.); ib., 1494 (ioff.); ib.
(with another work), 1496; ib., 1496 (but no copies extant ?) ;
Bologna (with another work), 1502; ib., 1506 (commentary by
Vittori); Paris, s. a. An epitome by Padre Isidore Isolani
Milanese was published in Pavia in 1513, and again in 1522.
Prince Boncompagni, in an elaborate discussion of the various
editions (Bulletino, IV, 498), mentions this rare first edition, of
which he knew but one other copy, that in the Biblioteca
Ambrosiana at Milan. He was of the opinion that it was
printed in Venice, in a type used before 1480.
Although the date is uncertain, this Tractatus may contest with the
Treviso arithmetic (p. 3) and the Ars Numerandi (p. 23) the honor of
being the first printed work devoted wholly to some phase of arithmetic.
It treats of ratios according to the cumbersome method of Boethius as
followed during the Middle Ages. It is purely theoretical and repre-
sents the university treatment of scientific arithmetic in that period.
Other works of 1472-1480. The mathematical activity in Italy dur-
ing this period was very considerable. From 1472 to 1480 there were
38 mathematical works printed in the country. In the next decade
there were 62 and in the next 100, with 13 of uncertain date between
1472 and 1500, making a total of 213 appearing in a period of less
than thirty years.
Before the Treviso book there were printed at least three works which
touched briefly upon arithmetic. These were (i) * De re militari libri
io KARA ARITHMETICA
XII,' by Robertas Valturius of Rimini, printed at Verona, 1472, fol., in
the second book of which the author treats ' de arithmetica & militari
geometria' ; subsequent editions ib., 1483 (two editions this year); Paris,
1483 (with title changed) ; ib., 1534, fol. ; ib., 1555 (French translation);
(2) 'Speculum Majus,' by Vincent de Beauvais (Vincentius Bellovacensis),
the greatest mediaeval encyclopaedia, printed at Strasburg by Mentelin,
i469(?)-i473, io vols., fol. The second part of this work contains one
book (no. 1 6) on mathematics, in which is given a brief treatment of
algorism (see p. 5), probably the first written in France (c. 1250),
although M. Henry asserted that a MS. of c. 1275, which he edited,
was entitled to that distinction. Incomplete editions were also pub-
lished at Venice in 1484, 1493-4, 1591. The early editions of Priscian,
* De figuris et nominibus numerorum,' are mentioned under the pub-
lications of 1565. (3) The Etymologies of Isidorus, Augsburg, 1472
(p. 8).
In 1480 an anonymous work was issued from the Caxton press in
London, entitled ' The Mirrour of the World or Thymage of the same.'
Chap, i o of this work began : ' And after of Arsmetrike and whereof it
proceedeth,' and this was probably the first English printed matter upon
the subject. There was a second edition, London, 1506, and a third
s. 1. a. (London, 1527 ?), fol. (See also Boethius, p. 25, and Faber, p.
62.) About 1480 there was published at Padua a folio work by Richard
Suiseth (Suicetus, Swincetus, Swinshead, Suineshevedus, the first name
possibly Roger or Raymund), entitled 'Opus aureum calculationum per
Johanem de Cipro emendatum et explicit.', with subsequent editions at
Pavia in i488(?), 1497, 1498, fol., at Venice in 1505, 1520, and at
Salamanca in 1520. I have seen an edition of this work, s. a., assigned
to c. 1477. For Nicolaus Cusa, see p. 42.
GIORGIO CHIARINI. Ed. pr. 1481. Florence, 1481.
A Florentine arithmetician of the fifteenth century.
Title. ' Qvesto e ellibro che //tracta di mercatantie //et vsan-
ze de paesi.' (F. i, r. See Fig. 5.)
Colophon. ' Finito ellibro di tvcti // ichostvmi : cambi : mone
//te : pesi : misvre : & vsanze // di lectere di cambi : & ter// mini
di decte lectere che // nepaesi sicostvmaet in // diverse terre.
Per me France //fco di Dino di lacopo Kartolaio Fiore//tino
Adi X di Dicembre MCCCCLXXXI. In Firenze Apreffo//
almusiftero di Fuligno. ' (F. 102, r.)
PRINTED BOOKS n
Description. 8, 13.2 x 21.2 cm., the text being 7.1 x 14.1 cm.
3 ff. blank + 6 unnumb. +96 numb. (Roman numerals) = 105 ff.,
24 11. Florence, 1481.
Editions. Florence, 1481, 8 (here described); s. a. (1498),
8. The undated edition was not the first ; Coppinger has
shown that it was printed in 1498.
While this is not, strictly speaking, an arithmetic, it is the first printed
book to give the customs relating to exchange in use among the Floren-
tine merchants at the close of the fifteenth century. It is the source from
which several later writers drew their material, and is particularly valu-
QVESTO EELUBRO CHE
TRACT A DI MERCATANTifi
ETVSANZDEPAESU
FIG. 5. TITLE OF CHIARINI'S WORK
able in showing the nature of the practical problems of the time. Copies
of this first edition are extremely rare, but the work was well enough
known for Paciuolo to appropriate some of the contents. There is a
question as to its authorship.
EUCLID. Ed. pr. 1482. Basel, 1562.
Flourished at Alexandria c. 300 B.C. He was the author of the ' Elements,'
the basis of most of the textbooks on geometry.
Title. 'Die Sechs Erfte Bucher // Euclidis/ // Vom anfang
oder grund //der Geometrj.// In welchen der rechte grund/nitt
allain der Geometrj // (verfteh alles kunftlichen/gwifen/ vnd
vortailigen ge-//brauchs des Zirckels/ Linials oder Richtfcheittes
vnd // andrer werckzeiige/ fo zu allerlaj abmeffen dienftlich) //
fonder auch der fiirnemften ftuck vnd vortail//der Rechen-
khunft/ furgefchriben vnd // dargethon ift.// Aufs Griechifcher
fprach in die Teutfch gebracht/ aigene-//tlich erklart/ Auch mit
verftentlichen Exempeln/ grtind-//lichen Figuren/vnd allerlaj
den nutz fiirangen ftellen-//den Anhangen geziert/ Dermaffen
vormals //in Teiitfcher fprach nie gefehen // worden.// Alles zu
lieb vnd gebrauch den Kunftliebenden Teiitfchen/fo fych der
12 KARA ARITHMETICA
Geo-//metrj vnd Rechenkunft anmaffen/ mit vilfdltiger miihe
vnd arbait //zum trewlichften erarnet/vnd in Truckh ge-//geben/
Durch // Wiihelm Holtzman/genant Xylander/ //von Augfpurg.
//Getruckt zu Bafel.' (P. i.)
Colophon. ' Vollendet durch Jacob Kiindig/ zu Bafel/ in //
Joanns Sporini koften/im jar i562.//auff den dreyfzigften tag
des // Winmonats.' (P. 199.)
Description. Fol., 20 X 31.3 cm., the text being 12.5 x 25.7
cm. 14 pp. unnumb. + 185 numb, -f I blank = 200 pp., 39-52 11.
Basel, 1562.
Editions. The editions of Euclid have not been considered
in this work except in so far as they relate particularly to
arithmetic. This is the first German edition. The Plimpton
library contains the first edition of Euclid (Venice, 1482), but
since this has no arithmetical work except Book V it has not
been included in the list. Several manuscripts of Euclid are,
however, included in the second part of this bibliography be-
cause of their value in tracing the changes in the forms of the
numerals.
This edition of Euclid is mentioned because the editor has thought
it necessary to add to Book II some arithmetical work. In particular
he gives three forms of multiplication, first from left to right, then in the
usual way, and finally for special cases in which the short processes are
involved. He also considers the division of numbers in given ratios,
the extraction of roots, and a few other semi-algebraic calculations.
Euclid's ' Elements ' contain much work upon the Greek theory of
numbers, besides what appears in Book V, and several books were pub-
lished in the sixteenth century, embodying this material. These are
mentioned later.
Other works of 1482. In 1482 appeared the first German arithmetic,
if we except the 'Ars Numerandi' (p. 23). It was written by Ulrich
Wagner, a Ntirnberg Rechenmeister, and was printed by Heinrich Pet-
zensteiner at Bamberg. Only nine small pieces of parchment proof sheets
remain. They contain the following colophon: 'Anno dni . . . 1482
kl'i6. lunij p. Henr. peczensteiner Babenberge : finit Ulrich wagner
Rechemeister zu Niirnberg.' ' Ludus Arithmomachiae,' by John Sher-
wood (Shirewode), was published at Rome in the same year. (See also
p. 63, and on Albert of Saxony see p. 9, c. 1478.)
PRINTED BOOKS 13
PROSDOCIMO DE BELDAMANDI, AND LIVERIUS.
Ed. pr. 1483. Padua, 1483.
PROSDOCIMO DE BELDAMANDI was born at Padua c. 1370-1380, and died
in 1428. He was educated at the University of Padua, and also taught there.
He wrote on arithmetic, music, and astronomy.
JOHANNES DE LIVERIUS (LIVERIIS, LINERIIS) was a Sicilian writer on
astronomy who flourished c. 1300-1350.
Title. ' Profdocimi de beldamandis algo-//rifmi tractatus
perutilis i neceffarius // foeliciter incipit. qui de generibus cal-//
culationum fpecie preteri.t nulla3.q falte // neceffaria ad h9 art^
cOgnit63 fuerat.' (F. i, r. See Fig. 6.)
Colophon. ' Algorifmus. Profdocimi de beldamadis // vna cum
minuciis. Johanis de liueriis. hie //felicite'finit ImprefO padue.
Anno .1.4.8.3 //die zz. februarii.' (F. 21, v.)
Description. 4, 15 X 20.8 cm., the text being 9 x 14.7 cm.
27 ff. numb., 32 11. Padua, 1483.
Editions. Padua, 1483, fol. (here described); Venice, 1540,
8 (see p. 15). Boncompagni could learn of only three copies
of the first edition, and seven of the second.
This rare work was written for the Latin schools, and is a good
example, the first to appear in print, of the non-commercial algorisms
of the fifteenth century. It follows ' Bohectius ' (Boethius) in defining
number and in considering unity as not itself a number, as is seen in
the facsimile of the first page. Prosdocimo then treats of the funda-
mental operations with integers, including mediation (division by 2,
which the author places before duplation or multiplication by 2), pro-
gressions, and the roots. The treatment of fractions is left to Liverius :
' Incipit Algorismvs de mi-//nutijs tarn vulgaribus quam phyficis magi-
//ftri loannis de Liuerijs Siculi.' The vulgar minutes ' were the com-
mon fractions, and the 'physical' were the sexagesimal fractions.
Towards the end of the ' Algorismus de integris ' in the 1540 edition,
the date of composition appears : ' ... per Profdocimum de Belda-
mandis de Padua anno domini .1410. die .10. lunij compilata fufficiant.'
(F. D 5, v.) The work of Prosdocimo contains the first reference that
I have seen to a slate. Indigebat etiam calculator semper aliquo lapide,
vel sibi conformi super quo scribere atque faciliter delere posset figuras
cum quibus operabatur in calculo suo.' (See Fig. 6.) It is probable,
from this statement, that computers of his time actually erased the
figures in the galley form of division (see the Treviso arithmetic, p. 3),
14 RARA ARITHMETICA
nlmt tractatae pcrutilto i ncaCnue
foeliciter mcipit.qui dc gcnenbue caF
cnladonuni fpcaie p?etcn.t noltaj.q faltc
ueceflaria ad b* ait f sgmtoj foerat
te.m60 area numeroe ogondi fane oaru e:
atqj diuerfo0.g licet bom ejiflcrctatq} veri
crdLtfi foflidiofutu .ppf ipap regulap mul'
iuetui jpter
am ifti modi inon faflidiofi: q> fi in akq? talculo an rolotco enc t
ttigiflj: calculators o^atoj foam a capirc inctpere oponeba t: da
to q; erro: fuua adbuc fatie ^pfqane cjriftcret. t boc ^pt ftgu
raa in faa ogatoe deleta03ndf^ebat cna calcolato: fcmg a aq*
la pidc ud fibi ^fomtufu^ quo fcnbcre atq^ faaliter delete pcffj
fi^ui 00 cu gbufopabat in calculo fuo JU ga bee oia fao0 fa ; .1
diofa atq^ laDonofa mibi uifa funtrdtfpofni hteUu cdcrc in quo
ota tfta abicercn^qui eaa a^ozifmu0 Hue liber de numerie de"
noiart potent. Sciao & q? in boc Ubello ponef ndiiucdo nifi ca
q ad colculu neceffana fmitalia q in oliis libra pracccc arifme
trice tagunf ad calcnlu nouecdfaria 4>pf b:euitatqd(inuendo
II doia ergo iibettue ifle de numerie tractate b$g difi'fnitone
nueri ipm rcboarc udo C Tlomeros ergo F" juclide7?.fu
^5eomctHe t.f" 33obectui pf foearifmefce (IcdtfitwuTiue
rue* eft mntomdo fine qnatttae difcrcta e^ untanbae ^fcfarfi t e
ex nnftattbne og0regatarjt d! umta0 ifludrq?
d^nno* *}>er bac g*difRnit5j nuerr b:cpofca.quof unirae no
e naerneJiQ (it pnctpiu nuehxiato q? nuer^ et;a VOCOTI poPtt la
go! fnrnen^nuepX^ of eo quo rcaliqua nOerare polfunma et
* ocqxf nuer^in.pceflnbniflelibeUi. in q* et& v
FIG. 6. FIRST PAGE OF PROSDOCIMO DE BELDAM ANDI
as the Hindus had done on their sand or dust abacus, instead of cancel-
ing them in the manner explained in the early printed arithmetics.
The best discussion of the lives and works of Beldamandi and Liverius
is in the Boncompagni Bulletino, vol. XII.
PRINTED BOOKS 15
Other works of 1483. Valturius (p. 10). In this year the second
German arithmetic was printed at Bamberg, only one (incomplete) copy
being known. It was possibly written by Ulrich Wagner (p. 12), and,
like the 1482 work, it was printed by Petzensteiner, as appears from
the following colophon : ' In zale Xpi .1483. kl ' .17. des Meyen Rech-
mmg // in mancherley weys in Babenberg durch henr9 // petzenfteiner
begriffen : volendet.'
PROSDOCIMO DE BELDAMANDI, AND LIVERIUS.
Ed. pr. 1483. Venice, 1540.
See p. 13.
Title. ' Algorismvs de In//tegris Magistri Prosdoci-//mi
Debeldamandis Pataui fimul cu algorifmo de de-//minutijs feu
f ractionibus magiftri loanis de Liuerij // ficuli. Reintegratus ab
erroribus comiffis a fcri-//ptoribus, a me Federico Delphino ar-
tium, & // medicine doctore, mathematicarum difci//plinaru in
celeberrimo gymnafio Pata//uino publico prof eff ore, additis all//
quibus verbis, in aliquibus locis,//pro maiori claritate. Et da-//
tus impreffioni ad infta-//tiam meoR fcholariu // nunc algorifmu
//maxime de-//fideran-//tium.// Venetijs. M. D. XXXX.'
(F. i, r.)
Colophon. ' Venetijs per loannem Antonium de Vulpinis de
Ca-//ftrogiufredo. Anno domini .M. D. XXXX. //die octauo
menfis Aprilis.' (F. 52, v.)
Description. 8, 9.6 X 14.5 cm., the printed part being 7.1 X
12.5 cm. 44 ff. unnumb., 30 11. Venice, 1540. (This particular
copy has 8 ff. more, the sheet F appearing in duplicate.)
See p. 13.
RAPHAEL FRANCISCUS.
Ed. pr. 1484. S. 1. a. (Florence, c. 1516).
RAFFAELE FRANCESCO. A Florentine philosopher of the latter part of
the fifteenth century.
Title. ' Verificatio Vniversalis in // regulas Ariftotelis de
motu non rece-//dens a comuni Mathema-//tico^ doctrina.'
(F. i, r. See Fig. 7.)
16 KARA ARITHMETICA
Description. 4, 13 X 19.9 cm., the text being 9 x 16.7 cm.
8 ff. unnumb., 41-42 11. S. 1. a. (Florence, B. Zucchetta, c. 1 5 16).
Editions. Pisa, 1484, 8; this edition, s. 1. a. (Florence, c.
1516).
This is a brief treatment of proportion, hardly worthy of ranking as
an arithmetic. The applications relate to problems of Aristotle.
VERIFICATIO VNIVERSALIS IN
regulas Ariftotelis de mom non rece'
densa comuni Mathema>-
FIG. 7. TITLE OF THE VERIFICATIO OF FRANCISCUS
PIETRO BORGHI. Ed. pr. 1484. Venice, 1484.
PIERO BORGI. A Venetian arithmetician ; died after 1494.
Title. ' Qui comenza la nobel opera de // arithmethica ne la
qual fe tracta // tute coff e amercantia pertinente // f acta i compi-
lata p Piero borgi//da veniefia.' (F. 2, numbered i, r. See
Fig. 8 for the first folio.)
Colophon. ' Nela inclita cita de venetia a 9orni .2.//augufto
.1484. fu impofto fine ala pre-//fente opera.' (F. 118, v.)
Description. 4, 14.3 X 19.3 cm., the text being 8.4 X 13. 4 cm.
2 ff. unnumb. + 116 numb. = 118 ff., 37-38 11. Venice, 1484.
Editions. Venice, 1484, 4 (here described) ; ib., s. a., which
Riccardi thought might be earlier than 1484, since Ratdolt, the
printer, published books in Venice as early as 1476; ib., 1488,
4 (see p. 19); ib., 1491, 4 (see p. 20); ib., 1501,4; ib., 1505;
ib., 1509,4; ib., 1517, 4 (see p. 20); ib., 1528, 4 (see p. 21);
ib., 1534, 4 (see p. 21); ib., 1540, 4 (see p. 21); ib., 1550, 4
(seep. 22); ib., 1551; ib., 1560; ib., 1561, 4; ib., 1567; ib.,
1577. It was at one time thought on the testimony of Maittaire
that there was an edition of 1482, but it has been shown by sev-
eral bibliographers that the first edition is that of 1484.
This is the very rare first edition of Borghi's treatise, the second
commercial arithmetic printed in Italy and long thought to be the first.
This particular copy belonged to Count Paolo Vimercati-Sozzi.
PRINTED BOOKS 17
The text of the first edition is closely followed in that of 1488 (see
p. 19), except for the index, 'Tauola de li capitoli .otegnudi i qfta
opa' (f. 1 1 8, r.). This does not appear in the second edition, at least
in Mr. Plimpton's copy. The letters S H S U which appear twice are
thought to stand for J H S U, Jesus, possibly changed on account of
some conjectured pronunciation. They appear the second time on
f. 1 1 8, v., in connection with a set of verses beginning as follows :
'S HSU
Quanto latua memoria et alto ingegno
vaglia ne larithmetica hai moftrato
nel prefente volume compilato
petro borgo date veneto degno.'
In the verses appears the name of the printer :
' Ma limpreffor de augufta Errardo experto
di lopera prefente ftampatore
degno e non di mediocre laude certo.'
This folio, as already stated, does not appear in the Plimpton copy of
the 1488 edition, and since the later printer was not the same it prob-
ably never appeared after 1484.
This work is more elaborate than the Treviso arithmetic, and had
far greater influence on education. More than any other book it set
a standard for the arithmetics of the succeeding century, and none of
the early textbooks deserves more careful study. Borghi first treats of
notation (see Fig. 9), carrying his numbers as high as ' numero de mil-
lion de million de million,' and making no mention whatever of the
Roman numerals. In the same spirit he eliminates all of the mediaeval
theory of numbers, asserting that he does this because he is preparing a
practical book for the use of merchants. ' Et nota che fono // nueri de
piui maniere fi cho-//me dichiara Boetio in el fuo // de arithmetiche Ma
volen-//do hora tratar de quelle chof//fe che folo amerchadati apertien :
pero tratado folo de quelli//che ale choffe merchadatefche fono necef-
farij io laffero ogni//altra maniera de numeri.' (F. numb, i, v.)
The sequence is now peculiar, for multiplication is the first operation
treated. (' C.Che coffa fia moltiplichar,' f. numb. 7, r.) First comes
the table, arranged in the column form, unlike the Boethian type of
arithmetic, which preferred the square array. In addition to the prod-
ucts through '9 uia io fa 90,' the products of 12, 16, 20, 24, 32, 36,
by 2 ... io, are given, these having been necessary on account of the
monetary tables of the time. The author then gives the method of
i8 KARA ARITHMETICA
checking by casting out 7's and 9's. (' C.Dela pruoua del .7.', ' Dela
pruoua del .9.', f. numb. 8.) Then follows multiplication ' per colonna '
(i. e., by reference to the columns of the table, ' C.Del multiplicar p
cholona,' f. numb. 9, r.), with its checks by 7 and 9, and ' per crocetta '
(our 'cross multiplication,' 'C.Del multiplicar per chroxeta,' f. numb.
13, v.), showing that these were the common methods in Venice. Divi-
sion is then explained by the gal-
**- ^ omo fe di - r 7
do batel cho lefuo P ruoue > f - numb -
C opera VCdere 20 > r -)> our present method, then
known as the method of giving,
'a danda,' and described by
gerqucftadfcrpotraccrtificado p ac iuoio and Caiandri ( PP . S4
ssassssKgaef ** <* <** %* -"
aralaprefenteoperaeafatwi Then follow addition (although
' this was used in multiplication),
subtraction, denominate num-
bers, common fractions (also
th -HipHcatL)
f three ( De la "egola del
partnership, barter, alliga-
FIG. 8. FIRST PAGE OF THE FIRST EDI- tion ' and false P osi tion. The rule
T10N OF BORGHI'S ARITHMETIC f three had been developed
many centuries earlier by Orien-
tal arithmeticians. It was one of the inheritances from the Arabs, and
was not improbably learned by the Venetian traders through their con-
tact with the East. Partnership was to the fifteenth what the corporation
is to the twentieth century, and it is only very recently that " partner-
ship involving time" was thought to be a necessary subject of study.
Barter, a subject until fifty years ago common in American textbooks,
was necessary at a time when currency was not so plentiful as now.
Alligation was a practical topic in connection with the coinage of
money in the days when minting was not the monopoly of great cen-
tralized governments. The rule of false position was an Oriental device
which we have now replaced by the equation ; it was found in school-
books in various countries until the second half of the nineteenth cen-
tury. The problems are generally practical for the time, and they reveal
some interesting facts concerning business customs at the close of the
fifteenth century.
Other works of 1484. Albert of Saxony, p. 9, c. 1478 ; Vincent
de Beauvais, p. 10.
PRINTED BOOKS 19
PIETRO BORGHI. Ed. pr. 1484. Venice, 1488.
See p. 1 6.
Title. ' Chi de arte matematiche ha piacere // Che tengon di
certeza el primo grado // Auanti che di quelle tenti el vado //
Vogli la prefente opera vedere//Per quefta lui potra certo
CEXotnofifo:wflno trillion
& million adoncba fe bie fotrxor per fcttc figure
fnqnefto modo . 1000000 .pcrcbe lafepnma
figura ficn cHuogo oemfara Dcmiara!:cpcrcbe 1000009
millemfara .fano vno million :eteffendoinqtiel
luogo lafigura cberip:et:enta vno pero bene
edieovno imlion.3&a.fqucftp modo . 1 100000
tfra'vnotntlfon e cento milia rpercbeoltraei I looooj
mflUb'ffifi) foogooccentenara ocinwra:fono[afisuracberip:e
jcenta vno fi cbe bene edito vno million e cento mt'Iia .32ba. in ... flmvs
qucftomodo. 1 1 loooo.&iria vno mil tone cento e diccemiiia
percbeoftraelmflion e cento milfa :in fuogooele oejceneoe^
mfar:fono lafigura cbe rijprtjcenta vno -.ft cbe beneedito vno
rfavnmiu'oncentoevndeicemflfa per cbe oltraelmilion cento
edicjcc milia:in luogo DC numcriDemfarfonoIaffgura cbe rip
"icentavnoificbebenC'Cdftovnmiu'oncaitocvndcjccmilia.^a 1 1 1 1 100
inqueftomodo. 1 1 1 1 100. Diriavnmf lion cento evndecem^
iia e cetito:percbe oltra clmilion cento e vndece milfa: in luogo
Oeefdrnplicecentenarfono lafiguracbe ripzacnta vno:ficbe
bene editovnmilion cento e vndcfe milia e ceuto.3g>a in quefto
modo. 1 1 1 1 1 io.ofriavnmilion.ccntoc vndcjccmiUa cento mill o
edieye-'pcrcbeoJtra elmil/on cento evndeice miae cento :i
luo^o&elefimplice&ejcene-.fono lafispracbe rip^ejcentavno.
^ainqneHomodo. 1 1 1 1 1 1 i.oiriavnmilion cento evnde I II I III
JK milia cento e vndejce.per cbe ancbe in (uogo oc(e fimplice
vnfta .fono lafigura cber/pzejcenta vno.ftcbe bene edirovn'
milion cento e vndcjre milia cento e vndcjce.ct cbofi p:ocededo
pcrfina .^^^^^.poncndo fcmpze aifuo luogiqudeflsiirc 299997*
repzerentante quefinumcrf oueroocjcene ocet)tenara.cben>
nominaetcetera.equcflobada ccrcba loamaiftramentoDcl
nnmerar.bencbeininfinitumfipoJfa p:occder.ma cbomvna
general figura mifo:cero oicbiarfrquantopoteflTeacbadcr.ct
farano queflo fottopofte
FIG. 9. FROM BORGHI'S ARITHMETIC, 1488 EDITION
fapere // Se error fara nel calculo notado // Per quefta effer
potra certificado // A formar conti di tutto maniere // A mer-
chadanti molta vtilitade // Fara la prefente opera e afatori //
Dara in far conti gran facilitade // Per quefta vederan tutti li
errori//Ede iquaterni foi la veritade // Danari acquifterano e
20 RARA ARITHMETICA
grand! honor! //In la patria e de f uori // Sapran far le rafon de
tutte gente//Per le figure che fon qui depente.' (F. i, r.)
' Qui comeza la nobel opera de // arithmeticha ne laqual fe
tracta // tute coffe amercantia pertinen-//te facta i compilata
per Piero//borgi da Veniefia.' (F. 2, r.)
Colophon. ' Stampito in Veniexia per zouane de Hall' 1488.'
(F. 95, v.)
Description. 4, 14.8 X 20.7 cm., the text being 12.4 x 14.9
cm. 95 ff., 44 11. Venice, 1488.
Editions. See p. 16. This is the third edition, and is nearly
as rare as the first. Proctor mentions three books from the
press of the printer, John Leoviller, of Hall (Halle ?).
The text is practically verbatim with that of the first edition (p. 16).
PIETRO BORGHI. Ed. pr. 1484. Venice, 1491.
See p. 1 6.
Title. The title page is the same as that of 1488, except for
the letters SHSU which precede the former.
Colophon. 'Nela inclita citade venetia a zorni .22.//ottubero
.1491. u impofto fine ala pre//fente opera.// Libro dabacho.'
(F. 100, v.)
Description. 4, I5-4X 20.8 cm., the text being 11.7 x 16.8
cm. 100 ff. unnumb., 4043 11. Venice, 1491.
See p. 1 6. This is the fourth edition.
PIETRO BORGHI. Ed. pr. 1484. Venice, 1517.
See p. 1 6.
Title. This is substantially the same as that of the 1484
edition already described.
Colophon. ( CStampata in Venetia per lacomo pentio da
Lecho ad infta//tia de Marchio Seffa & Piero di Rauani
compagni // anno dni .1517. adi .25. de zugno.' (F. 100, v.)
Description. 8, 15.5 X 2 1.6 cm., the text being 13.3 x 17 cm.
2 ff. blank -f 100 numb. = 102 ff., 41 11. Venice, 1517
See p. 1 6.
PRINTED BOOKS 21
PIETRO BORGHI. Ed. pr. 1484. Venice, 1528.
See p. 16.
Title. This is substantially the same as that of the 1484
edition already described.
Colophon. ' C.Stampato in Venetia per Fracefco Bindoni, &
Mapheo // Pafyni compagni. Nel anno .M.D.XXVIII.// Adi
.XVIII. Del mefe di Zenaro.' (F. 100, v.)
Description. 8, 15.1 X 20.5 cm., the text being 13. 4 X 17.1
cm. 100 ff. numb., 41 11. Venice, 1528.
See p. i 6.
PIETRO BORGHI. Ed. pr. 1484. Venice, 1534.
See p. 1 6.
Title. This is substantially the same as that of the 1484
edition already described.
Colophon. ' C.Stampato in Vinegia per Fracefco Bindoni, &
Mapheo // Pafini compagni. Nel anno .M.D.XXXIIII.// Adi
.25. Del mefe di Settembre.' (F. 100, v.)
Description. 8, 15.4 X 20.9 cm., the text being 13.5 x 17.1
cm. 100 ff. numb., 41 11. Venice, 1534.
See p. 1 6. The various editions changed but little.
PIETRO BORGHI. Ed. pr. 1484. Venice, 1540.
See p. 16.
Title. ' Pietro Borgo // Libro de Abacho.// Chi d'arte Mathe-
matice ha piacere // Che tengon di certezza il primo grado //
Auanti che di quelle tenti il vado//Vogli la prefente opera
vedere.// Per quefta lui potra certo fapere // Se error fara nel
calculo notado // Per quefta effer potra certificado // A f ormar
conti di tutte maniere.//A merchadanti molta utilitade // Fara
la prefente opera e a f attori // Dara in far conti gran felicitade
// Per quefta uederan tutti gli errori // E delli quaterni fuoi la
ueritade // Danari acquiftaranno, e grandi honori.// In la patria e
di fuori // Sapran far le raggion de tutte gente // Per le figure che
fon qui depente.// Auenga che alquanto per me fu promeffo affai
22 RARA ARITHMETICA
fufficientemente // alia promeffa fatisf aceff e , niente dimancho
per fatisfar alle pre-//giere di qualch'uno, e maffime di alcuni
Impreffori, iquali era-//no per ftampar la prefente Opera, ho
uoluto alquanto ampliar la di qualche gentilezza oltra quello che
prima pmiffe, benche // di quello che fe potria dir, quefto fia vna
minima parte, pero cli //chi uoleffe metter pur la centifmia parte
di quello che fi potria // poner, el faria molto piu la gionta di
quello che fia tutta 1'opera //infi. Et pero pro nunc mi paffo
con alcune cofette aggionte nel//ligar de metalli, lequal princi-
piano a carte .77. & anchora in fin//de 1'opera con dieci cafi
affai piaceuoli & leggiadri comincian//do a carte .98. da quello
che dice. Le vno che compra tre pezze // de panno per ducati
.70. &c. Et fe le niente di quelli, iquali me//hanno pregato
non fuffino a fuo modo fatisfatte, prego quelli //me habbino
per ifcufato.' (F. i, r.)
Colophon. * Stampato in Venetia per Bernardino de Bindoni.//
Ne 1'anno .M.D.XL. Del mefe di Ottober.' (F. 100, v.)
Description. 4, 15.5 x 20.9 cm., the text being 13.2 x 17 cm.
2 ff. unnumb. + 98 numb. 100 ff., 39-40 11. Venice, 1540.
See p. 1 6. This is the eleventh edition. The title is considerably
extended and some changes are made in the text, chiefly in the way of
added matter.
PIETRO BORGHI. Ed. pr. 1484. Venice, 1550.
See p. 16.
Title. The title page is substantially identical with that of
the 1540 edition already described. It bears the date, 'Anno
Domini M. D. L.'
Colophon. ' C.Stampato in Vinegia per Francefco Bindoni,
&//Mapheo Pafmi. Nell Anno .MDL.//Adi .21. Del mefe
di Nouembrio.' (F. 100, v.)
Description. 4, 15.6 X 20.9 cm., the text being 12.9 x 17.9
cm. 100 ff. numb., 41 11. Venice, 1550.
See p. 1 6. This is the twelfth edition, and at least five subsequent
editions appeared in the sixteenth century.
PRINTED BOOKS 23
ANONYMOUS. Ed. pr. c. 1485. S. 1. a. (Cologne ?, c. 1485).
Title. ( Ars numerandi. Incipit copendiofus tractatul^ quin
//tupliciu dclonu numealiu in quo docet' // luculet' quo ordlant'
variatur pponut'//et abinuicem deriuatur dictones nume// rales.'
(F. 2, r. ; see Fig. 10.) Without abbreviations this would appear
as follows : ' Incipit compendiosus tractatulus quintuplicium dic-
tionum numeralium in quo docetur luculenter quomodo ordinan-
tur variantur componuntur et ab invicem derivantur dictiones
numerales.'
Description. 4, 14 X 20.6 cm., the text being 10.8 x 15.2 cm.
6 ff. unnumb. (i blank), 35 11. S. 1. a. (Cologne ?, c. 1470-1485).
Editions. There was no other edition. The date of this rare
book is uncertain. The style of type has led to the assertion
that it was printed in Mainz by Fust and Schoeffer, about 1470.
The book is not mentioned by Hain, nor is any copy known in
the French libraries. Coppinger believes that it was printed
in Cologne by Ulrich Zell in 1485, and in this he is followed by
Zell's biographer, Merlo. The British Museum catalogue gives
this date and printer, but questions each. A comparison of the
water-marks in this copy with those of the fifteenth century
which are described in standard treatises (e. g., Sotheby, E. L.,
Principia Typographica, London, 1858, vol. Ill) fails to throw
any light upon the date. Riccardi attributes it to Zell, c. 1471,
who had been an apprentice of Guttenberg, but had left Mainz
at the sacking of the city in 1462.
The book is not strictly speaking an arithmetic, but a treatise on
grammatical usage as applied to numbers. A considerable portion of
the text is occupied with the distinction between ordinals and cardinals,
and the methods of using them.
Other works 0/1485-1487. Albert of Saxony, 1487, p. 9 ; c. 1478.
In 1485 there was published at Bologna, edited by Pietro Almadiano of
Viterbo, a quarto work by Nicolo de Orbelli (Nicolaus Orbellis) entitled
'Compendium considerationis matematice quo ad aritmeticam et geome-
triam sunt necessaria.' Orbilli's 'Cursus librorum philosophic naturalis,'
which appeared in 1494, 4, and at Basel in 1503, 4, contained 2 pp.
on arithmetic. See Isidorus, p. 8, 1483.
RARA ARITHMETICA
Ardtumtemnbt*
rt opi t copm biofus tmcbtul^qum
tupiiciu tmonu numcaltu m quo ft> cd*
iuculct'quo (X&ian^vananjr OFonuf*
ct abmtucxni txmatur tuctpiica numc
mica*
3f Aiottfl numceos nrtpntSrm qfca t>ieu
9 tur cattmi ales cj fca potcatcs* qfca toiftri
butiuc ftuc bi>ndc-qfca cnt>iatc6- a qfca
ntultipHcatiuc ftuc d^ubialcff<t nota
t>u ^ numcri t>iucrfi8 t)i<*iomb? ft^ftcaf a
varia fi^n^i manccic cottabut mtf fc t>intia ct
t>i lift ta fee ut pateb* ^I>c his aut fedotrfb? p ot
t>mc cf> bi(5bu-*c^t puo te cavt>taii^q t>icutur
io cartialc3-' cp ft cut oftiu 4tif ca catbhtc^ct
tnrrinf ci: i ta t>aon* atic uualcff ^tiitut a vc^
plica* tar cca ifVas Vcl t>i cu6" carbmalcs qfi pit
cipalcff * cp t>c6nc0 ahc nuaU0 ab iRi0 bttt oii
gincVcl ^icu 1 qn' p riripales cp pia dp a& fi^nt
tiu; ct focdu cp tJuplicc j fait t>co*3 cardma
Ic0{qt>a'f'c6crctc et c)ucda abftf&c tDc cocrctt
ailt carom aiibg 92 tw ipiff atic tKtiuan^pihno c
t>Sm qtic t>conctJ ftcpccdut numctado Vnusr
fc^fcptc-
fc&cdF (efdcd - a nun^ tested p ^ teccfcptC' a
non tebj mtcrpt fccc dtu^io ct ft (tnt feuc t>c5cf
tibuo* a fie uf^ j ab tngmta (uo modo-
' a fie fuo modo t
/i fie fuo mo ufc^
't)u< tiduo
numcralcs
> eoncrcta
tioianuatia
fie numcta 1
FIG. 10. FIRST PAGE OF THE ANONYMOUS ARS NUMERANDI
PRINTED BOOKS
25
ANICIUS MANLIUS SEVERINUS BOETHIUS.
Ed. pr. 1488. Augsburg, 1488.
BOETIUS. Born at Rome c. 480; died there October 25, 524. He was a
Roman senator, a philosopher, and the last of the great Latin writers.
Title. 'Arithmetica boetij.' (F. i, r.)
Colophon. 'Finit arithmetica Boetij bene re//uifa ac fideli
ftudio emendata Im//preffa per Erhardu ratdolt viri fo-//
lertiffimi eximia iduftria i mira im-//primedi arte : qua nup
3fncipiuntouolibzic>e ^(ricbmeti/ fere vlla lie ciictis ibfolnM n^rn'M rr i'
caam^manilMfeuerimSoety rf, m^l^XiSSSSfS& *"*' ""**">
n danfllmi * MriM ejc cofulis: nira pftoi;s:vt no cetera? quoqYar
ojomaiij:patrid;:ao patriciumfim rtu a&iumentabefiberetyTlainem
enoifquemuneri
bnsitarecteomV
ciapiecipue inter
eos gfefcmagni
faciunc cftimanf
filtqui&ooftabit
qd liber alias after re t irmciuu : i tec
ab ilto vncg q6 iucun&ius benitiole
tia coplecteref accepni ."Dec ipfe co
ftderans : attttlt non tgnaiiaopum
pooera quibua ao facinus nibil in
dructins eft: cu babenbi (ttis incan
t>uit:ao meritu nibil vilius cu ea fi/
bi victoi antm 9 calcata fubiccit : fed
ea que c,r grecaru opulentia littera
re in romane ozationis tbefaurnm
fupta cducjnm. 9 .^ta eni mei quocg
opens mibi ratio cotiabit :fi qttc cc
fapientie ooctrinis eliati:fapienti^
fimi ittdicio copjobenf . cii&cs igif
vt tarn magni laboas effect us nuij
tantii ejcpectet ejcame: nee in aurea
^ire publicas nift ^octf fentetic a
fKpulatioite nitaf.^n quo nibil mi
ro vioeri oebet:cu i J> opus qd fapie
tic iniienta perfeqitit: non auctoiia
f5 a!ieno incubit arbitno jSuis $p/
pe in fhiunenris res rationts ejrpen
2>itur:cu tuoiciumcogif fubirc piti^
tetitis.Seo buic munufculo :no ea
e>einqueceterts imminent artibtw
numuncntacoftiruo .Tlequc enitn
a&enbe molts labozeftralia fomta
t>$ imaginis ratiomeceiulbe artifi^
cis mantis polfti operis nitoz ejcpe*
ctat.^t pichtr^ manibns tabulf c5
mife f abzoeunr.cgre niffica obfer/
uatione t>ecerptg:cotomfuci merca
toju fofertta perqiufiti:liinea opero
fis elabojata tejctrinistmtiTtipliceni
matena pz^ftant.Tlone ioem qwo/
que bello^ vifitur inftrumetier&ic
fpidila fagiitt&erflcuif.iLli vali^ua
tbozaic nigragemit incube.^lTa'i/
us rauM vmbonts teamtna ppni la
bottotbi infigen^amercatur.tatn.
multf ortibus ars yna_perfigit . 3llt
hoftri labo:is abfblutio loge at> fa^
cilioje* ciirrit eucnt 115 . ]Cu eni fottia
manuj fupzemo open impones: in
quo nibil fre oecernentiu ncceite eft
labojare cofenfu.Qualibet eni boc
iuoiciu mttltis artibus pjobef ejrcul
tu vno tamecumulat ejramine.p
periare igitur licet quantn nobis in
boc ftubio Ipngis tractus oci/s la^
boz a&iecerit.^nrerufubtiltum fit
gas ejcercitate mentis velocitas co/
pbenoat. vtp ieiune macies o^atio
nis ao ea que funt caliganribus irn
peoita fentenri;s ejcpe&ienbafumci
at.Quafnremibialieniquoque iu
oicij lucra querunf. XTum tu vtrarii
quepeririJTimus litteraprpoflis gra
i^oiationisejrpertibus quantusbe,
nobis lubtcare au^eant: Tola tantu
pnunciationepiefcribere.^tno al/
FIG. ii. FIRST PAGE OF THE 1488 BOETHIUS
26
KARA ARITHMETICA
IS
|3
2-7
t*
jr
r*
\s
y
t*
8
nr
(To
Jo
0cubavnit09.
longitubo.
ttauo at(& ejtpofitio bigeflg foziwt mufota (Upertf vnitate: vt btto tnu
3?i0itbuomaUte/
ra^ppofit^fpzmulfq
faciut aguliuab vno
ab* lo.et* lo.^pceben
tiardpiciafi-rbisfub
terioseeoibies copa
rentiqui fcilicet a.4-angiilum incipi
cntcs:invigcno8 tcrminii ponunt:
buplej: ib cftpzima fpccice multipli
citatie oftenbimnitavtpnmuepn'
teritari 9 binariu. tcrti 9 tertm tribus:
vtfmarPternariu.qrtusqrtu qter/
narij nunicrofrta tc tranfcenbat : vt
8 qternariu :z u eabe cucti fequetil
Jcfemino:t8 piuratitatepzetereant.
/i vcro tcrti'angulite afpiciafi^ ab
9.incboaalpgimbme la:imbincqj
triceius altrinfeciis numercc]ctebit:
et bit cu pn w latitubine et logitub i/
ticcoparcturariplcjcfpcciee mulct/
FIG. 12. MULTIPLICATION TABLE, 1488 BOETHIUS
PRINTED BOOKS 27
venetijs nuc // augufte; excellet nominatiffimus.// Anno dni
.M.cccc. Ixxxviij. Men-//fis maij die vigefima.' (F. 48, r. Fig. 14.)
Description. 4, 15.1 X 20.8 cm., printed in double columns,
each being 5.1 X 14.7 cm. 47 ff. unnumb. + I blank = 48 ff.,
40 11. Augsburg, 1488.
Editions. Augsburg, 1488, 4 (here described) ; Cologne,
1489; Leipzig, 1490; Venice, 'Opera,' 1491-92 (see p. 28);
Paris, 1496, 4 ; Venice, 1497 ; ib., 1499 (bearing also the date
1497), fol. ; s. 1. a. (the Compendium of Muris,
c. 1500); Paris, 1 50 1, fol. ; ib., 1503 (see p. 29) ;
ib., 1507, 4; ib., 1510 (see p. 30), fol.; ib.,
1511, 4; ib., 1514, fol.; Vienna, 1515 (the
Muris ' Compendium,' in Tannstetter's works) ;
Paris, 1521 (see p. 31), fol.; ib., 1522, fol.;
1528, fol.; Paris, 1530, 8, ' De differentiis
topicis libri quatuor ' ; Basel, 1536, 8; Basel,
1546, ' Opera,' fol. ; ib. and Paris, 1549; Basel, FlG - 13- FlGUR -
1553, 'ajectis explic. per J. Scheubelium,' 8 ;
Paris, 1553; Basel, 1570; Venice, 1570, 'Opera,'
fol. There are undoubtedly various other editions of the Arith-
metic, or the Epitome by Faber Stapulensis, these works often
being bound with such treatises as the Arithmetic of Jordanus
Nemorarius. Murhard (I, 160) and Rogg (p. 137) mention an
edition of Faber's 'Compendium,' s. 1., 1480, but it is not given
by other bibliographers. (See Hain, I, 468; Brunet, Man., I,
1059; Graesse, Tresor, I, 464; Riccardi, I, i, 159; Boncom-
pagni, Bullctino, XII, 148.)
The text is practically that followed by Friedlein in his
standard edition of the 'Opera' of Boethius (Leipzig, 1867),
except as to numerals. Here, as in the later manuscripts, the
Arabic characters have replaced the Roman of the original
text. (See Fig. 12.)
The arithmetic of Boethius was based upon the Greek work of
Nicomachus (fl. c. 100 A.D.), and related only to the theory of numbers,
the 'ApiO/jirjTiKrj, as distinguished from the practical calculations, the
28 RARA ARITHMETICA
y, and from the later algorismus (p. 5). Boethius gave an elabo-
rate theory of ratios and devoted much attention to figurate numbers, such
as the triangular, square, pentagonal, and cubic. (See Fig. 13.) The work
was the standard in the Church schools throughout the Middle Ages.
fut.Q*fifcrotntca m
fcxfqct n.faaa.-qao
fupltiBdl. gnueniaraui bic
FIG. 14. LAST TWO PAGES OF THE 1488 BOETHIUS
BOETHIUS.
Ed. pr. of the Arithmetic, 1488. Venice, 1491-92.
See p. 25.
Title. See Fig. 15.
Colophon. ' Impreffis venetijs per Joanne de Forli-//iiio et
Gregorium fratres. Anno falutis .M//cccc.lxxxxj. die xxvj, men-
fis Martij.' (F. 352, r.)
On f. 256 (220 as numbered in the book), at the end of the
Geometry is the following : Venetijs Impreffum Boetij opus p
Joane5 i Gre//goriu de gregorijs fratres felici exitu ad fine vfqj
pductu // accuratiffimeq3 emedatu Anno humane reftaurationis.
// 1492. die .18. Augufti. Auguftino Barbadico Serenifli//mo
Venetiarum principe Rem pu. tenete.'
Description. Fol., 21.7 x 32 cm., printed in double columns,
each being 6.6 X 24. i cm. 3 if. unnumb. + 345 numb. + i blank
= 349 ff., 66-70 11. Venice, 1491-92 (see the colophons).
PRINTED BOOKS
29
Editions. This is the editio primeps of the works of Boethius.
For other editions see p. 27.
j^cruntopcra JSoctii.-quf in boc,voiuminc eontfncntur.
pHttoprtma.
"""
fditto vna.
fTliogiTmoe introdiicn'o
:uminXopica/Citcroiii8Ub:ire>:.
5fopfcit?lib:iquatruor.
ibn^uo.
tbnltiliDCr vnusj.
ca*bmeitoad l^atrttinm
23c ZlDnnca lion qm ncg.
Toe rcotnctriaiib:i vno.
FIG. 15. TITLE PAGE, 1491-92 BOETHIUS
BOETHIUS. Ed. pr. of the Arithmetic, 1488. Paris, 1503.
See p. 25.
Title. 'In hoc libro contenta.// Epitome/ compendiofaq3 in-
troductio in libros // Arithmeticos diui Seuerini Boetij : adiecto
fa-//miliari commentario dilucidata.// Praxis numerandi certis
quibufdam regulis //conftricta.//Introductio in Geometriam
30 KARA ARITHMETICA
breuiufculis an-//notationibus explanata. fex libris diftincta.//
Primus de magnitudinibus et earu circuftan-//tiis.// Sc'dus de
cofequetibus/ cotiguis/ & cotinuis.//Tertius de punctis.//Quar-
tus de lineis.//Quintus de fuperficiebus.//Sextus de corporibus.
// Liber de quadratura circuli.// Liber de cubicatione fphere.
// Perfpectiua introductio.// Infuper Aftronomicon.' (F. i, r.)
Colophon. ' Abfolutum in almo Parhifiorum ftudio/ //Anno
dni qui numero definuit omnia// 1503.' (F. 48, r.)
On f. 84, v. is the following: 'Geometric! introductorij://
fexti / etvltimi libri finis.// Editi anno domini // millefimo quin-
//getefimo pri//mo : vicefi-//ma quin//ta no-//uem//bris.' And
on f. in, v., the following: ' Id opus imprefferut Volphgangus
// hopilius et Henricus ftephanus // ea in arte focii in Almo
pari-//{iorum ftudio Anno Chri//fti Celorum totiufq3 // nature
coditoris.// 1503. Die vice//fimafepti-//ma Iu-//nij.'
Description. 8, 20 X 26.8 cm., the text being 15.6 x 22.3
cm. 48 ff. numb, in the arithmetical part, 113 in all (i blank),
54 11. Paris, 1503.
The editions of Boethius differ more or less in the combinations of
works which they contain. See p. 27. This is the first edition of the
Jacobus Faber Stapulensis and Jodocus Clichtoveus ' Epitome.'
BOETHIUS. Ed. pr. of the Arithmetic, 1488. Paris, 1510.
See p. 25.
Title. This is substantially identical with that of the 1503
edition already described.
ColopJwn. ' CAbfolutum in almo Parifiorum ftudio/ // Anno
domini qui numero defmiuit // omnia 1503. Et emiffum ex offi-
//cina Henrici ftephani Anno // Christi faluatoris // omnium
1510 de-//dma quinta//die Mar-//tij.' (F. xlviii, r.)
Description. Fol., 19.5 x 27.9 cm., the text being 15.4 x 25.6
cm. 48 ff. numb., 4656 11. Paris, 1510.
This is one of the editions containing the commentary of Jodocus
Clichtoveus on the 'Epitome' of Boethius by Jacobus Faber Stapulensis.
(Seep. 27.)
PRINTED BOOKS 31
BOETHIUS. Ed. pr. of the Arithmetic, 1488. Paris, 1521.
See p. 25.
Title. ' Divi Severi-//ni Boetii Arithmetica,// dvobvs discreta
libris ; adie-//cto commentario, mysticam nvme-//rorum applica-
tionem perftringente, declarata.//
(Woodcut with initials of the
printer : S.D.C.). Vaenundatur
apud Simonem Coli-//naevm, e
regione fcholae Decretorum.'
(F. i, r.)
Colophon. ' Excudebat Simon
Colinseus, Parifijs, Anno
MDXXI
S-DECOLINBS
FIG. 1 6. PRINTER'S DEVICE, 1521
EDITION OF BOETHIUS
1521
Quinto//Idus lulias.' (F. 1 39, v.)
Description. Fol. , 19.9x28.2
cm., the text being 13.9x26
cm. 4 ff. unnumb. + 136 numb.
= 140 ff., 40-52 11. Paris, 1521.
This is the first edition of Boethius
with the commentary of Girardus
Ruffus : ' C.Girardi Rvfn, in duos
arithmetical Boetii libros, commen-
tarivs.' (F. 5, r.) This commentary
greatly exceeds the text in extent,
and as to ponderosity it leaves little to be desired. As a piece of
typography, however, this is one of the best editions of Boethius.
ANIANUS AND JOHANNES SACROBOSCO.
Ed. pr. 1488. Strasburg, 1488.
ANIANUS was a fifteenth-century astronomer and poet, of Strasburg.
JOHANNES DE SACROBOSCO (SACROBUSTO, SACROBUSCHUS, HOLYWOOD,
HOLYBUSH, HOJ.YWALDE, HoLYFAX, HALIFAX) was born at Halifax (Holy-
wood), Yorkshire ; died at Paris in 1244 or 1256. He studied at Oxford and
lectured at Paris. He wrote on astronomy and algorism.
Title. ' Copotus manua//lis magri aniani. // metricus cu
;ometo//Et algorifmus.' (F. i, r. Fig. 17.) F. 45 begins,
' Incipit textus algorifmi.'
32
RARA ARITHMETICA
Colophon. ' Impreffum Argii. per Johamem prytf.// Anno do-
mini .1488 .i8. kail.' decembris.' (F. 44, r.)
Description. 4, 13. 9 X 19.5 cm., the text being 8.8 x 14.4
cm. 53 ff. numb. + 2 unnumb. = 55 ff., 31-34 11. Strasburg,
1488.
Editions. The editions of Sacrobosco's Algorismus were as
follows : Strasburg, 1488, 4 (here described) ; s. 1. a. (Venice ?,
1490?), 4; Venice, 1501 (see p. 35); Paris (edited by Clich-
toveus), 1498; ib., 1503; ib., 1510; Vienna, 1517; Cracow,
aroanu
FIG. 17. TITLE, FIRST EDITION OF ANIANUS
1504, 1509, 1521, and 1522; Paris, 1522; Venice, 1523 (p.
35); Antwerp, 1582. The editions of Anianus were as follows :
Strasburg, 1488, 4 (here described); Lyons, 1489; ib., 1490,
4 ; ib., 1491, 4 ; ib., 1492 (two editions), 4 ; Rome, 1493, 4 ;
Paris, 1494, 4; ib., 1498, 4; s. 1. a. (Paris?, c. 1495, 4, see
p. 33); s. 1. a. (Basel, c. 1500), 4; Rouen, s. a. (1502), 8;
Paris, 1501, 4; ib., 1502, 8; Lyons, 1504 (see p. 35); ib.,
1509, 4; Paris, 1508; ib., 1511; ib., 1515; ib., 1519, 4;
ib., 1529; ib., 1530,4; Lyons, 1540,4; Frankfort, 1549;
Wittenberg, 1550; ib., 1568; Antwerp, s. a. (c. 1558); ib.,
1559. There are probably others s. 1. a., and some appear
PRINTED BOOKS 33
under the name of Sacrobosco. Of this first edition Boncom-
pagni knew only the Munich copy. (Bulletino, XII, 126 n.)
The first part of this rare book is the Compotus Manualis of Anianus,
and the second is the Algorismus of Sacrobosco, described later. It is
probably the first book on mathematics printed in Strasburg (but see
p. 10), and it is the first edition of each of the two treatises mentioned,
and the first printed work on the computus, the arithmetic of the
Church calendar. In the work of Anianus appears for the first time
in print the original of the rhyme beginning
' Thirty days hath September,'
the English version of which is said to have been first published in the
1590 edition of Grafton's Chronicles. It also appeared in an arith-
metic published anonymously in 1596. The Latin form of Anianus is
as follows :
' Junius aprils feptember et ipfe nouember
Dant triginta dies reliquis fuoadditur vnus,
De quorum numero februarius excipiatur.' (F. B 8.)
It is found in various forms in other computi, manuscript and printed.
Anianus also gives for the first time in print the astronomical formula :
' Sunt Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libraque, Scorpio,
Arcitenens, Caper, Amphora, Pices,' which appeared in the works of
Bede under the title ' Verfus Prifciani, de Astronomia,' as ' Hinc Aries,
Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Arcitenens, Capri-
cornus, & Vrna, Qui tenet & Pifcis.' (1563 edn., vol. I, 517.)
ANIANUS. Ed. pr. 1488. S. 1. a. (Paris ?, c. 1495).
See p. 31.
Title. ' Copotus cum //commento.' 'Liber qui Compotus
infcribitur : vna cum figuris et ma/ //nibus neceffariis tarn in
fuis locis qj in fine libri pofitis.// Incipit feliciter.' (F. i, r.)
Description. 4, 13.1 X 19.4 cm., the text being 9.3 X 1 3. 1 cm.
39 ff. unnumb. + i blank = 40 ff., 35 11. S. 1. a. (Paris ?,c. 1495)-
Editions. See p. 32. This rare and interesting edition was
published about 1495, possibly by Mich, le Noir at Paris,
although there is no date or place of publication given. It is
one of the best examples of the mediaeval computus that
appeared in print. Unlike the first edition (p. 31, 1488) it
34 RARA ARITHMETICA
contains a number of illustrations showing the use of the hand
and fingers in assisting in calendar reckoning, the title of ' Com-
Zfberqoiompottd intension wz am R&xiQ etma/
nibas necclferud tarn in fuis locfe $ in fine libii
r
bapofluntOiiplidter confiderari . pzfmopnt
Did DC oeo q eft lujc vera . loco oicebat Dauid*
lat ozta eft lUito . r oe ilia luce Did? 3ba'.f .
lErar IUTC vera 4 illumiar omne bomine venic/
tcinbunc mandd.^eakiopomt oefaenaa.
c tXdf lut qaafi faeutc reddctc luadfuqm a fade bomuic fde/
tern efle ladda5n qUibtis verbis ad comendan'onc fcle tnio b:e
ufter rangiinir.primo eft tangif fdetie^Uitudo pzedofa p boc
quod Dicit lujc. Sccundo largitudo gloofa per boc qp oidt oz/
fie oicentis /^cTenSae^ fon0 indenciensJjon'tatis via .fai fal/ '5 c
oatoae co^wtio Ba:ioe fic.rtlud ell valtda a pzedofam quod P'
ijetniiatido^imperteaoftdtvaliduinzpfecmm.faetia
modi.ergo zc. maioz dl manifcfta. minoe oeclaratur p p&m ter/
do oe ala fie tnccnte : 3ia in pncipio foe creationio e tan$ tabu
la rafa in qoa nibfl oepicta dl.oepmgibilie m fdetiis ^ virturt/
bu8.*f> zmwm?pbajaaao2itate boeni ^ ratoe in pzologo artf
men-icetecTenflae cop quc tera funt ^ imgmutabilto tflentie no aT"
fh*aq5Compbenfione^eritarts.Batioe fic:illude!Uan$ funtti
bonii quod babet largitione glonofam fcta e bmoi. ergozcma_
toz ert veramtno: ,pbamr poiflinitione fdcne q talie e.^c^eirs ^
cjda babims ate ronalie no innatus fed accjfi'tue o[m b&ma? re
riunHndagatn): T tortua butnane vitc gubcrnatnjcf^ fcia fit ba/
bittis pts.qz fcia ell aliquod ejciftens in ata.fed omne illud quod
ell in ala aut c bitue aut poten'a aut paflio. tboc tellaE arillo,
In fecudo etbicouq^ fcta no fit paTio pt5 qz patfionea funt in va
luntate fcia no e bmoi.ergo ic.cp no fit potetia pt5.q: c]l-b5 po/
ten ia fit a natura. fiat irafdbiUs.i cocjpifdbilis.z fie relmqui
ear $no fitpotentia. c^fitbUu0a;eronali0pt5gpdiaa^au(
a a
FIG. 1 8. BEGINNING OF ANIANUS, EDITION OF c. 1495
putus manualis ' being thus justified. This differs from the 1488
edition in the notes of the commentator. The original text is,
however, substantially unchanged.
PRINTED BOOKS 35
ANIANUS. Ed. pr. 1488. Lyons, 1504.
See p. 31.
Title. 'Compotus cu //commento.' -(F. i, r.)
Colophon. ' CLiber compoti cum comento finit feliciter // Im-
preffus Lugduni per Claudia nour//ri. Anno domini .M.ccccc.iiij.
//die .iiij. Octobris.' (F. 32, r.)
Description. 8, 14.5 X 20 cm., the text being 8.9 x 14.1 cm.
32 ff. unnumb., 3440 11. Lyons, 1504.
See p. 32. In this edition there is a curious misprint in the calendar
verse given on p. 25 (f. 13, r.). It here begins 'Julius (instead of
Junius) aprilis feptembre & ipfe nouember.' Like the edition of c. 1495
(p. 33), this is well illustrated. It differs from the 1488 edition in the
notes of the commentator, but is practically identical with that of
c. 1495-
JOHANNES SACROBOSCO. Ed. pr. 1488. Venice, 1501.
See p. 31.
Title. ' Algorifmus Domini Ioa-//rris De Sacro Bufco //
Nouiter Impreffi^.// Cum Gratia Et Priuilegio.' (F. i, r.)
Colophon. ' Impreffum Venetijs per Bernardinum Venetum //
De Vitalibus: Anno Dm .M.CCCCC.I.//Die Tertio Men.
Februarij.' (F. 8, r.)
Description. 4, 14.1 X 19.6 cm., the text being 10.5 X 16.8
cm. 8 ff. unnumb., 39 11. Venice, 1501.
See p. 32.
JOHANNES SACROBOSCO.
Ed. pr. 1488. S. 1. a. (Venice, 1523).
See p. 31.
Title. < Algorifmus Domini Joannis // de Sacro Bufco noui-
//ter impreffum.' (F. i, r.)
Colophon. ' Impreffum Venetiis per Melchiorem Seffam &
Petrum//de Rauanis Socios. Anno domini .M.D. XXIII. //
die .XXIIII. Octobris.' (F. 8, r.)
Description. 4, 14.8 X 18.6 cm., the text being 10.3 X 16.8
cm. 8 ff. unnumb., 40 11. S. 1. a. (Venice, 1523).
3 6 RARA ARITHMETICA
JOHANN WIDMAN (?). Ed. pr. c. 1488. Leipzig, c. 1488.
Born at Eger, Bohemia, c. 1460. He was a student at Leipzig in 1480,
A.B. in 1482, bachelor of medicine in 1485, A.M. in 1486. He evidently
received the doctor's degree about the same time, for he wrote a medical
work in 1497 with the title, ' Tractatus clarissimi medicina ~Q doctoris Johanis
widman . . . de pustulis . . . ' That he gave lectures on algebra, possibly the
first at Leipzig, is proved by a passage found by Wappler in an old Dresden
manuscript : ' Quare hodie hora secunda post sermonem atque Baccelaureo-
rum celebrata disputatione Magister Jo. W. De. Eg. Aporismata et Regulas
Algobre resumpturus pro hora atque loco conuenienti cum audeturis con-
cordabit . . .'
Title. ' Algorithmic Linealis.' (F. i, r.) 'Ad euitadum mul-
tipli//ces Mercatorum erro-//res et alteri^ . . .' (F. 2, r.)
Description. 4, 14 X 19.5 cm., the text being 7.8 x 14.5 cm.
14 ff. unnumb., 3 134 11. Initials in red, by hand. Leipzig, c. 1488.
Editions. Leipzig, s. a. (c. 1488, here described); ib., possi-
bly 1490, 1493; 1516; 1517, and several others.
This rare treatise, the first printed work on calculation by the aid
of counters ('apud noftras appellata eft calculatio '), is of unknown
authorship, but was probably written by Widman. (Abhandlungen, V,
152.) At the end of the book is the device of Martin of Wiirzburg
(Martinus Herbipolis), and the book was printed by him, probably c.
1488. After a brief introduction on the use of counters ('projectiles'
as they are here called), the author treats of the following topics : De
Additione, De Subtractione, De Duplatione, De Mediatione, De Multi-
plicatione, De Diuifione, De Progreffione, De radicum extractione, De
radicum extractione in Cubicis. The book closes with the words : * Et
tantu de Radicum extractione et vltima huius Algorithmi fpecie Et p
confequens de toto Algorithmo.' There are no applied problems, and
the only computations with abstract numbers are performed 'on the
line ' (i. e. by the ' projectiles ' on a line abacus), whence the name
' Algorithmus linealis.' The work is illustrated by woodcuts.
Other works 0/14.88. Suiseth (p. 10, c. 1480) ; Borghi (p. 16, 1484).
JOHANN WIDMAN. Ed. pr. 1489. Pforzheim, 1500.
See above.
Title. ' Behennd vnd hiipfch // Rechnung vff alien // kauff-
manf chaff ten.' (Woodcut of a schoolroom.) (F. i, r.)
Colophon. ' Gedruckt zii Pfortzheim von Thoman // anfzhelm
Im Jubel Jar als man zalt 1500 //Got fey lob.' (F. 163, v.)
PRINTED BOOKS
37
Description. 16, 10 x 13.2 cm., the text being 6.5 X 10.4 cm.
ii ff. blank + i unnumb. + 162 numb. = 174 ff., 26 11. Pforz-
heim, 1500.
Editions. Leipzig, 1489, 8; Pforzheim, 1500, 8 (here
described): ib., 1508, 8 (p. 39); Hagenau, 1519, 8 (p. 40);
II
er$im$enobet tubftetr
tnfi$ bcf^Iteflen ob bie taffcl
elofje taimfeie anber gefeQt i(l
*b bami t?ie t>er nad; icrl id?c an ir felbft foitn
tlerltc^cn befc^reiben i'(l .
%ir <xl(c r e Ami n
FIG. 19. FROM THE 1500 WIDMAN
I
Augsburg, 1526, 8 (p. 40). The title of the first edition was
as follows : ' Behede vnd hubsche // Rechnung auff alien //
kauffmanschafft/ and the colophon (f. 236, r.), * Gedruckt In
KARA ARITHMETICA
der Furstlichen Stath // Leipezick durch Conradu Kacheloffen
//Im 1489 Tare.'
n*cberre0clf0Fumtafot?iemxv flat
QcJ^rpes 424 13 4
p:oflo:rt 305
jToret
FIG. 20. EXCHANGE. FROM THE 1500 WIDMAN
This is the second edition, and is even more rare than the
first. It was unknown to Boncompagni when he printed his
' Intorno ad un Trattato d'Aritmetica di Giovanni Widmann di
Eger ' in the Bulletino, IX, 188, the best discussion of this
arithmetic that has appeared.
PRINTED BOOKS 39
Widman's arithmetic was the first great German textbook on the
subject, although minor works had already appeared before 1489. It
is in the main a practical treatise, with good problems, and it set the
standard for Germany much as Borghi's book did for Italy. Among its
noteworthy features is the use of the plus and minus signs for the first
time in a printed work. (See Fig. 21.) These are not used, however,
as signs of operation, but as symbols of excess or deficiency in ware-
house measures. The book is illustrated (see Fig. 20) with pictures
showing mercantile customs, and with geometric diagrams. Widman
acknowledges his indebtedness to men like Sacrobosco, 'als da lert
Joannes defacrobufto vn ander mer,' although his work shows no depend-
ence upon the ' Algorismus ' named.
After the ' Inhalt difz buchs in einer gemein,' Widman devotes 2\
pages to Numeratio,' 2^ to 'Additio,' 2^ to ' Subtrahiren ' (including
denominate numbers in these topics), i to ' Dupliren,' 2 to ' Medieren '
(i.e. multiplying and dividing by 2), u^ to ' Multipliciren/ 5 to divi-
sion, 2\ to progressions, and 14 to roots (4 referring to cube root). He
then takes up fractions in the same order, this work being followed by
compound numbers and proportion. He then gives a large number
of type problems, regulae as he calls them, although they are not stated
in the form of rules as we now know them. These include the ' Regula
detri' (rule of three, treated as distinct from proportion), and the regulae
fusti, detri conversa, positionis, equalitatis, legis, augmenti, plurima,
sentenciarum, suppositionis, residui, excessus, collectionis, quadrata,
cubica, reciprocationis, lucri, pagamenti, and alligationis.
Other works 0/1489. Boethius, p. 27, 1488 ; Anianus, p. 32, 1488.
JOHANN WIDMAN. Ed. pr. 1489. Pforzheim, 1508.
See p. 36.
Title. ' Behend vnd hiipfch // Rechnung vff alien // Kauff-
manfchafften.' (F. I, r.)
Colophon. c C.Gedruck zu Pfborzheim von Thoman//Anfzhelm
Im iar als man zalt 1508.' (F. 161.)
Description. 8, 9.7 X 13 cm., the text being 6.5 X 10 cm.
7 ff. blank + 161 numb. = 168 ff., 26 11. Pforzheim, 1508.
Editions. See p. 37. This is the third edition of this famous
arithmetic, and is by the. same publisher as the second (1500),
but is from different type. It is about as rare as the first
edition.
KARA ARITHMETICA
JOHANN WIDMAN Ed. pr. 1489. Hagenau, 1519.
See p. 36.
Title. ' Behend vnd hupfch // Rechnung vff alien // Kauff-
manfchafften.' (F. i, r.)
Colophon. 'Getruckt zu Hagenaw durch Thoman // Anshelm.
Im iar als man zalt 7/1519.' (F. 1 5 1 , r.)
Description. 8, 9.7 X 14.3 cm., the text being 6.8 x n cm.
i f. unnumb. + 151 numb. = 152 ff., 20-29 11. Hagenau, 1519.
Editions. See p. 37. This is the fourth edition of Widman's
Arithmetic. As an inscription on the fly leaf says, this copy
was presented to Prince Baldasarre Boncompagni by Ludwig
Kunze as a ' liber rarissimus.'
See p. 39.
JOHANN WIDMAN.
See p. 36.
Ed. pr. 1489.
Augsburg, 1526.
fenot>ert>e(?gley*
3+30 d)cij.6o(uiwcr
4~-~ "~'P fciejetictncrtMjb
3+44 Uixmtrt>R?<?e<mfj
3 + zi ij?/&et|l mi*
2titer 3 -i i tfc mi6t>3f<}(Je{oit*
3 -f- 50 berxmnb werfceit
4 1 6 4*3 9 & CQ
3+44
3+19
? ""/ - .
Hurt
3 Mlftf.
4 Behennde vnnd // hiibfche
Rechnug auff alien // Kauffmanfchaff-
ten.' (Woodcut showing two men
seated at a reckoning table.) * M. D.
XXVI.' (F. i, r.)
Colophon. ' Getruckt zu Augfpurg
durch // Haynrich Stayner// M, D,
XXVI.' (F. 192, r.)
Description. 9.4 X 13.7 cm., the text
being 6.7 X 1 1 cm. 2 ff. unnumb. + 190
numb. = 192 ff., 25 11. Augsburg, 1 526.
Editions. See p. 37. This copy has
a note in the handwriting of Prince
Boncompagni, together with his colla-
tion of the book. It is interesting not
only for its rarity but for the contem-
porary coloring of the woodcuts.
In the other editions described the third line of Fig. 2 1 reads 3 + 36,
as it should.
crosier dit 4? 3 p. Vrtt> 6fcybeit 4 1 f z
tii.nim Tpn'ch I oo It ix?6i(f cin3crimcr
pjO 4 ff ; nnc Dnrneit 4 1 > i tt cnb f uml
FIG. 21. FROM THE
WIDMAN
526
PRINTED BOOKS 41
PETRUS DE ALLIACO. Ed. pr. 1490. Augsburg, 1490.
ALYACO, HELIACO, D'AILLY. Born in Compiegne in 1350; chancellor of
the University of Paris, Bishop of Cambray, and Cardinal. He died in 1420.
Title. ' Cocordatia aftronomie cu theologia // Cocordatia af-
tronomie cu hyftorica // narratione. Et elucidariu duor^z pre-
//cedentium : dm Petri de Aliaco car//dinalis Cameracenfis.'
(F. i,r.)
Colophon. * Explicit tractatus de cocordia aftronomice veri-
tatis i narrationis hiftorice//a dno Petro cardinali Cameracen.
completus in ciuitate Bafilieri. anno xpi // 1414 : menfis. Maij die
decima.' (F. 33, r.) 'Opus concordantie aftronomie cum theo-
logia necnon hyftorice verita : nar//ratione explicit feliciter.
Magiftri Joannis angeli viri peritiffimi diligeti cor//rectione.
Erhardiq3 Ratdolt mira imprimendi arte: qua nuper Venetijs
nuc // Augufte vindelicorum excellit nominatiffimus. 4. nonas
Januarij. 1490.' (F. 55, v.)
Description. 4, 15.6 X 20.4 cm., the text being n.i x 14.7
cm. 56 ff. unnumb., 39 11. Augsburg, 1490.
Editions. There was no other edition.
This work has been included in the list because, while chiefly astro-
nomical, it throws considerable light upon the early Computi. It was
written to show the relation between theology and astronomy, and hence
it has an important bearing upon the study of the mediaeval calendar.
ALONSO DELATORE. Ed. pr. 1489. Seville, 1538.
A Spanish savant of the fifteenth century.
Title. < Vifio delectable de // la philofophia i ar//tes liberates :
meta//phifica : y philofo-//phia moral .:. //M.d.xxxviij.' (This
is surrounded by an elaborate woodcut.) (F. I, r.)
Colophon. ' CFenefce el libro llamado vifion delectable dela
Philofophia : ar//tes liberales. Es impreffo enla infigne y muy
leal ciudad // de Seuilla en cafa de Juan Croberger.//Ano de
.M.d.xxxviij.' (F. Ixxij, r.)
Description. Fol., 19 X 26.9 cm., the text being 15.3 X 24.3
cm. 72 if. numb., 42 11. Seville, 1538.
Editions. Seville, 1489, fol.; ib., 1538, fol. (here described).
4 2
KARA ARITHMETICA
This rare treatise is an encyclopaedia, with chapters devoted to the
various arts and sciences. The arithmetic is found in Chapter IV
(see Fig. 22), and consists of only two pages of theoretical discussion.
COpttulo.tui.iE)daaririnetbicart>crudtoueW'
fingularcefecretoe.
fTandopa'zatraueflandoeftefcndrrorvimeronenamaoel
te:a oofecome^aua tn marauilloio camino: el i toe guio
en t>n lugar oe cafae ? palactos mug fmgulaw: 7 ala puma oe
u tnlla ballaro pna mu^fagacifftma imu^^>funda rosefla D
fciericia.aqualaunqloemiebzoecubnefTecoabitofcminilrparerciaoc
bap i5 acjl afcddcr co:a$6 oe mu^ penetrate i mup ingeniofo rarc.y en
laoicftrateniapngrafio^bierro^enlafinieftrarnatablacbla'queada:
FIG. 22. FROM THE 1538 ALONSO DELATORE
NICOLAUS CUSA. Ed. pr. c. 1490. Strasburg?, c. 1490.
NICOLAUS CUSANUS, NICOLAUS CHRYPFFS OR KRESS. Born at Kues on
the Mosel in 1401 ; died at Todi, Umbria, August n, 1464. He held posi-
tions of honor in the Church, including the bishopric of Brescia. He was
made a cardinal in 1448. He wrote several other works on mathematics.
Title. Usually known as the ' Opuscula.' The work begins :
' Prohemium.// [I]n hoc volumine ptinentur certi tractatus i libri
altiffime ptemplato//nis et doctrine: a preclare memorie pre-
ftantiflimo doctiffimoq3 viro // Nicolao de Cufa.' (F. I, v.)
Description. Fol., 17.5 x 25. 4 cm., the text being 1 1.7 X i8cm.
163 ff. unnumb. + I blank = 164 ff., 45 11. S. 1. a. (Strasburg ?,
c. 1490. Some bibliographers place it as early as 1480 and
others assign it to Milan as late as 1505.)
PRINTED BOOKS 43
Editions. Cusa's 'Opuscula varia' first appeared s. 1. a. (Stras-
burg?, c. 1490 or earlier). His 'Opera' appeared at Paris in
1511 and 1514 (see below), and again at Basel in 1565.
This con tains fifteen of Cusa's tractati, including 'Reparatio kalendarij,'
'De Apice theorie' (4 ff.), <De mathematicis complements,' 'De mathe-
matica perfectione.' Of these mathematical chapters the first two are
of some interest in the history of arithmetic, the others referring chiefly
to mensuration.
NICOLAUS CUSA. Ed. pr. c. 1490. Paris, 1514.
See p. 42.
Title. ' Hec in hoc fecudo vo//lumine contenta.//
Dialogus de ignoto. 2. De tranfmutationibus
Dialogus de annuncia- geometricis. 33.
done. 3. De Arithmeticis com-
Excitationu libri X. 7. plementis. 54.
Coniectura de nouiffi- De mathematicis corn-
mis diebus.// i. plementis. 59.
Septem epiftolae. 3. Complementum theo-
Reparatio Caledarii. 22. logicum. 92.
Correctio Tabularum De prefectione mathe-
Alphonfi. 29. matica. in.'
(Woodcut of printing-press, with the words: 'Prelu Afcefianu.')
* Venudantur cum cete//ris eius operibus in Aedibus Afcenfi-
anis.' (F. I, r.)
Colophon. * Emissvm est hoc librorvm Cvsae opvs // egregivm
Parisiis : ex officina Ascen//siana anno Christi pientissimi om//
nivm Redemptoris MDXIIII, octa//va Assvmptionis semper
San//ctae semperqve Virginis// Christi Deiqve Matris//Mariae.
qva patroci//nante apvd Filivm // portvm salv-//tis spera-//rnvs
et//veniae.' (F. CXVI, r.)
Description. Fol., 20x29 cm., the text being 12.7x26.8
cm. 114 ff. numb. (Roman numerals) + 2 unnumb. = 116 ff.,
46 11. Paris, 1514.
Editions. See above.
44 KARA ARITHMETICA
This second volume of the Paris edition of Cusa's works, edited by
Faber Stapulensis, contains the ' tractati ' already mentioned on p. 43.
JOHANN WIDMAN (?) Ed. pr. c. 1490. S. 1. a. (c. 1490).
See p. 36.
Title. 'Algorithmic Integro^//Cum Probis annexis.' (F. i,r.
See Fig. 23.)
Description. 8, 14.9 X 20.6 cm., the text being 7.8 X 14 cm.
12 ff. unnumb., 29-31 11. S. 1. a. (c. 1490).
Editions. The various bibliographies assign different dates,
but I presume there was only this one edition. De Morgan (p.
99), whose judgment as to the dates of such early works was
unreliable, estimated this as "hardly later than 1475," adding,
"I think this is the oldest book in my list." Wappler, who has
critically investigated the matter (Abhandlimgen, V, 158) believes
that Widman wrote this work, the 'Algorithmus linealis' (p. 36),
and also the ' Algorithmic Minutiarum Phisicarum.'
The work opens with a quotation from Boethius, the same indeed
as the opening sentence of Sacrobosco's Algorismus. After treating
of numeration, addi-
//tEBtroatftmus flttf eatoE ^ -1*^, du .
w ^2 fc 1 ~* "* ** *- plation, mediation,
multiplication, and
,> division, the author
FIG. 23. TITLE OF THE ALGORITHMIC INTEGRORUM takes up progressions,
roots, and the proofs
of the various processes. There are no applications in the book.
Other works 0/1490. Anianus, p. 32, 1488 ; Boethius, p. 27, 1488.
There also appeared c. 1490, at Leipzig, an anonymous work edited by
Norico, entitled ' Arithmeticae Textus communis.'
ANONYMOUS. Ed. pr. c. 1491. S. 1. a. (c. 1491).
Title. See Fig. 24.
Colophon. 'CFinis trium Algorifmo^ cum proper //tionum
vel Mercatorum regula.' (F. 10, r.)
Description. 4, 13. 7 X 19. 4cm., the text being 8.6 x 15. 6 cm.
10 ff. unnumb., 34-36 11. S. 1. a. (c. 1491)-
PRINTED BOOKS
45
Editions. This is not the edition described by De Morgan
(p. 99), because the title has ' addita etiam regula,' instead of
* addita regula.' The word 'regula' instead of ' regla ' at the
intcgrts copendfofefmcfigtirarumC mo:e 3rafo
Mfmttkobodl cmiclia tun bzcuiflim* cdoccne.vna cu &(
gozifrme DC mmurqa vulgarfbus viddiccr ct pbiHcalibua
.
rjue vulga mcrcaromm rcgula^iaf Quiba&'babiris.
niodicaadbibitdW'U'scnttaomncracalcuUndCmodum fa
cflttmc adtpifciporcft
^***c- *K'f^y ^
^/^^^ **<**? / ^r/t ' af*~*~*r -*~?
.*** ] *^^ / / / s%
*/& . . a* ^ftxi^e* jt&^Zff-^-**^
** 1 *f"^!f**~ ** ***"?
FIG. 24. TITLE PAGE OF THE ALGORISMUS OF c. 1491
end shows it to be different from the one described by Brunet
(edition of 1860). It is probably no. 827 of Hain, and the inter-
nal evidence makes 1491 the probable year of the composition
4 6
KARA ARITHMETICA
or publication. De Morgan says that three editions are known,
and Giinther on the authority of Chasles gives an edition at
Cologne, c. 1510.
This anonymous work is of the class of the arithmetics of Muris,
Peurbach, Ciruelo, and other mediaeval and early Renaissance writers.
picrngoras aritbrnetriretmroducror
FIG. 25. TITLE PAGE OF CALANDRI
It contains a brief treatment of the ' species ' (the fundamental opera-
tions) with integers, omitting ' duplatio ' and ' mediatio ' but including
progression and roots. This is followed by a discussion of common
fractions ('Algorifmus nouus de//minutijs vulgaribus '), a single page
on sexagesimal fractions (<de minutijs Phificalibus '), and a page on
proportion. It is one of the first books to identify proportion with the
rule of three, or merchants' rule as it was often called. (* De regula
proportionum // Sive aliter Regula Mercatorum dicta.') It is not as
practical as the elaborate title would seem to indicate.
PRINTED BOOKS
47
PHILIPPI CALANDRI. Ed. pr. 1491. Florence, 1491.
^ A Florentine arithmetician of the fifteenth century.
Title. ( Pictagoras arithmetrice introductor.' (Woodcut of
n 4,7 > per a]
Uiennc
LiL
H4
.t*
HTT
??_
?
U~
4f
o M
'Parti | g C o -parti n> g
* f3>i/i -c
B7r Uicnne ? i ^
inenne T?
Co
480
uicnnc i C-o
r - 1
fj 1
FIG. 26. FROM CALANDRI'S ARITHMETIC
Pythagoras.) (F. i, v. Fig. 25.) ' Philippi Calandri ad nobi-
lem et ftudiofu5 Julia//num Laurentii Medice de arimethrica
opufculu.' (F. 2, r.)
KARA ARITHMETICA
Clu condocro cmpie u
na fontc tn 4 dt T cjun
do e picna non merren
do ilcondocto 7 ftiir.i
do iluoracoto fiuorerc
beladccra feme in i i
di : Uo fapere eflendo
uomlafonr: tmccrcri
do ilcondpcco r ftur.i
doiluoraroio inqun'tt
di lava plena la dccra
fonte
I i
crdtnoi
r * r * - v-*^ j^ttk'j^, i
5 dibrarrio : uo faperc !
inc]uaridt fara fnora*
del deao peso
I 6 '
FIG. 27. FROM CALANDRI'S ARITHMETIC
Colophon. * Impreffo nella excelfa cipta di Firenze per /?//
Lorenzo de Morgiani et Giouanni // Thedefco da Maganza ft//
nito a di primo di // Genaio
1491.' (F. 104, v. Fig. 30.)
Description. 8, 9.8 X 1 3.2
cm., printed in double col-
umns, each 3.3 x 10.7 cm. to
11.5 cm. 104 ff. unnumb.,
FIG. 28. FROM CALANDRI'S ARITHMETIC 9-2611. Florence, 1491.
Editions. Florence, 1491,
8 (here described) ; ib., 1518, printed by Bernardo Zuchetta, 4.
PRINTED BOOKS
49
Of this rare book, the first in De Morgan's list, Mr. Plimpton pos-
sesses two copies. It is beautifully printed, and is practical in its
presentation of the operations, but traditional in its problems. It is
Eglicunatorrcebce
aim 4 o bpieda ? dap
pie mpaflatmo fmme
cbcctarsbo jo brae
eiauofapcrcc|namo
faraltwgba iwa func
cbefuia,ppieaiaaUari
uadclnurru: rallaei
roaslclLuorre
if
faralunga fo brae
cu
glieunalbcromfufa
nua dun fiumc elqua
kealrofo braccia d
fmmc e largbo ? o bra
da 7 per fort ima di uc
tofiruppcmralluogo
cbclacimadeUalbcro
toeeaua lariua dclftu
me.Qofapereqia!trc
braeaa fcnc ruppc z
qtumto ncnm a fc rino
too|
rimafcriro \C brae
eta ^ j 4 braceia fcnc
ruppc
FIG. 29. FROM CALANDRI'S ARITHMETIC
the first printed Italian arithmetic with illustrations accompanying prob-
lems, and the first to give long division in the modern form (Fig. 26)
known to the Italian writers by the name
'adanda.' Indeed Calandri gives only the
4 a danda ' method, omitting the galley
form, and is therefore fully a century ahead
of his time. De Morgan's statement that
he uses a divisor diminished by i is incor-
rect, as will be seen from Fig. 26. Figs. 27
and 29 show that the problems of the cistern, the snail (serpent) in the
well, the length of the hypotenuse, and the broken tree were familiar
in Calandri' s time.
Other works 0/1491. Anianus, p. 32, 1488 ; Boethius, p. 27, 1488 ;
Borghi, p. 1 6, 1484.
Xbcdcfco da ZDajanja ft
nitoadiprimodi
0cnaio i +? i
FIG. 30. COLOPHON OF
CALANDRI
5 o KARA ARITHMETICA
FRANCESCO PELLOS OR PELLIZZATI.
Ed. pr. 1492. Turin, 1492.
A native of Nice, living in the latter half of the fifteenth century.
Title. ' Sen fegue de la art de arithme-//ticha. et femblat-
ment de ieume-//tria dich ho nominat3 Copendio // de lo
abaco.//i 234567890.' (F. i, r. Fig. 31.)
Colophon. ' Complida es la opera, ordinada. he condida // Per
noble Frances pellos. Citadin es de Nifa. . . . Impreffo in
Thaurino lo prefent c5pendio de abaco per mei/ //ftro Nicolo
benedeti he meiftro Jacobino fuigo de fancto ger//mano. Nel
anno .1492. ad. Di .29. de feptembrio.' (F. 80, r.)
Description. Sm. 4, 13.8 x 20.9 cm., the text being
9.2 x 15.2 cm. 80 ff. numb., 39 11. Some of the initials have
been inserted by hand, in red. Turin, 1492.
Editions. There was no other edition.
This is one of the rarest arithmetics known to exist. (Brunet, IV,
475; Graesse, Tresor, V, 100 ; Riccardi, I, 2, 256.) The only good
description of the work is that given by Boncompagni in the Atti
del? Accad. Pontif. de 1 nuovi Lincei, XVI, 161, 332, evidently after
examining this copy, since it bears a note in his handwriting.
Pellos first considers the fundamental operations with integers, fol-
lowing this by a treatment of proportion, square root, and cube root.
He then discusses the subject of fractions in much the same order, the
rule of three, certain rules relating to weights, time, money, and other
measures, and such topics as partnership, barter, interest, alloys, and
the rule of false position, single and double. He closes the work with
a chapter on mensuration, or as he calls it, * De la art de ieumentria '
(* ieumetria ' in the title), and gives a number of interesting woodcuts.
The chief interest of the book attaches, however, to the fact that
Pellos came very near the invention of decimal fractions, and that
he actually used the decimal point as is shown in the illustration
(Fig. 32). It cannot be said, however, that he had any conception
of the real value of the decimal fraction as such, the first book
devoted to this subject being 'La Disme ' of Stevin (1585), hereafter
described. Pellos simply uses the decimal point to indicate division
by some power of ten, writing a common fraction in the quotient.
Thus, to divide 425 by 70, Pellos would divide 42.5 by 7, writing the
result 6.
PRINTED BOOKS
*"^^
feguc belaarttearitbmc/
ricba. et femblarmcnr t>cieumc/
, trm oicb bo nominal Copendio
FIG. 31. TITLE PAGE OF PELLOS
52 KARA ARITHMETICA (|
CTpanirper ~ 5T"
7 9 6 S 4. 8 3: 9 > 7
* 7
qaocicnf 39827419 ~
2 O
C'Partfrpcr 3 o
y 8 3 6 o 4. i 3
* ;
qtWrfctt 194*64
$ O
C*P*rtiVp*r 7 o
9 ? ? 7 9 t *9
^ 9
qnocfcnt I j ^ a y j
7 o
i o o
9 7 6 $,87
quocicnt
400}
78 9 ^ ? 7
quotient I j 7 4 *
3000
876 S 8*7
179 I
quocfent 2 p i l 9
3000
_
<L & c aqiU manfera podce fafrc obc tons fcmblls p^ru
FIG. 32, FROM THE PELLOS ARITHMETIC
PRINTED BOOKS 53
GEORG VON PEURBACH.
Ed. pr. 1492. Wittenberg, 1534.
PURBACH, PEUERBACH. Born at Peuerbach, Upper Austria, May 30, 1423;
died at Vienna, April 8, 1461. He studied under Johann von Gmiinden (see
p. 117), Nicolaus Cusa (see p. 42), and other great teachers, and later
he became professor of mathematics at Vienna, where Regiomontanus
(Johannes Miiller, of Konigsberg) was his pupil. His interests were almost
entirely in astronomy.
Title. ' Elemen//ta Arithmetices // Algorithmvs de // nu-
meris integris auctore // Georgio Peurbachio.// De Nvmeris
Practis,// Regulis communibus & // Proporcionibus.// Cum prae-
fatione Philippi // Melanchthonis.// M. D. XXXIIII.' (F. I, r.)
Colophon. ' Impress vm Vitebergae // per losephvm Clvg.//
Anno M. D. XXXIIII.' (F. 39, v.)
Description. 8, 10.8 X 15.6 cm., the text being 6.4 X 11.2
cm. 39 ff. unnumb. + I blank = 40 ff., 22-25 U- Wittenberg,
1534-
Editions. S. 1., ' Explicitum est hoc opus anno Christi dom.
1492.' 4; Vienna, s. a., c. 1500; Leipzig, 1503, 4; ib., 1507,
4, and probably ib., 1510 and ib., 1511; Vienna, s. a. but ear-
lier than 151 1, 4 ; ib., 1511,4; ib., 1512; Niirnberg, 1 5 1 3 ;
Vienna, 1515, 4; ib., 1520, 4; Wittenberg, 1534, 8 (here
described); ib., 1536, 8; ib., 1538; Venice, 1539, 8 (see
below); Frankfort, 1544.
The arithmetic of Peurbach went by various names, as * Opus Algo-
rithmi,' Institutiones in arithmeticam,' ' Elementa arithmetices,' and
* Introductorium in arithmeticam.' It is a brief treatise on the funda-
mental operations with integers and fractions, and contains a few simple
applications. Peurbach was too profound a mathematician to have con-
sidered it a work of any importance, but it is probable that he wrote it
for the benefit of students who were not yet prepared to take up his
work in astronomy.
GEORG VON PEURBACH. Ed. pr. 1492. Venice, 1539.
See above.
Title. ' Elementa // Geometriae ex Evclide // fmgulari pru-
dentia collecta a loane Vo-//gelin profeffore Mathematico in
54 KARA ARITHMETICA
// fchola Viennenfi.// Arithmetics practicae per Georgium //
Peurbuchium Mathematicum.// Cum praefacione Philippi //
Melanchthonis.' (Woodcut with motto : ' Dissimilivm. Infida.
Societas.') (F. I, r.) The arithmetic of Peurbach begins on f.
32: * Elementa // Arithmetices.// Algorithmvs de nv-//rneris
integris, fractis, Regulis // communibus, & de Pro-//portionibus.
//Authore Georgio Peurbachio.//Omnia recens in lucem sedita
fide & //diligentia fingulari.// Cum praefatione Philip. Melanth.'
Colophon. ' Venetijs loan. Anto. de Nicolinis de Sabio.//
Sumptu uero D. Melchioris Seffse. Anno //Domini M D
XXXVIIII.//Menfe lanuario.' (F. 67, v.)
Description. 8, 10.3 X 15.8 cm., the text being 8 x 12.2 cm.
i f. unnumb. + 68 numb. = 69 ff., 29 11. Venice, 1539.
See p. 53.
Other works of 1492. Anianus, p. 32, 1488; Boethius, p. 27, 1488.
Works of 1493. Anianus, p. 32, 1488; Anonymous (see Widman),
p. 36, c. 1490; Isidorus, p. 8, 1483; Vincent de Beauvais, p. 10.
LUCA PACIUOLO, DE BORGO SAN SEPOLCRO.
Ed. pr. 1494. Venice, 1494.
PACIOLUS, PATIULUS, PACIOLI. Born in Borgo San Sepolcro, Tuscany, c.
1445-1450; died soon after 1509. Not an original mathematician, but the
compiler of several works.
Title. See Fig. 33.
Colophon to the part on arithmetic : * Et fi fequenti pti
pncipali Geo e . finis decima nouembris ipofitus fuerit : huic tamen
pti: die vigefi //ma eiufdem Ipofitus fuit. M?cccc.lxliiij. Per eof-
dem correctorem i imprefforem vt i fine Geo e . hr.' (F. 232,
numb. 224, v.) There is also the following date on f. i, v.,
' M?ccc.?lxliiij?xx a . Nouembris.'
Description. Fol., 21.5 x 30.5 cm., the text being 19 x 24.2
cm. 8 ff. unnumb. + 224 numb. = 232 ff. in the part on arith-
metic ; 76 ff. numb, in the part on geometry ; making a total of
308 ff., 56-60 11. Venice, 1494.
Editions. Venice, 1494, fol. (here described); Toscolano, 1523,
fol. (p. 58).
PRINTED BOOKS 55
@uma t>e Sritbmctica >co/
mctri4l>:opo:tiom i ikw
porno nalita,
ontincntu Oc tutta topcra.
be numeric mifure in turn' modi occurrenti.
t&jopojnbm'e^pomoaliraanotirfa oel.5? oeucli
dc c DC turn' It aim foi lilvu
fnauiouero euidenne numero.i ;.p k $nta conn/
nuc.ppojrioali oeC.6fe.7t DC uclidc crtratrc.
Cuttc (e gri <xlalg;o:iTmo:abc rclcuarc . pnr. m ulri/
plicar.fi) marc^ fotrarc co rurtc fuc ,pue i (ani e rot'
ri.c rad W e p:ogrcllioni.
&e la regola mercanrefca Oitra od.u Toi fodamett'
n'con cafi eremplariper c?m?gr.i^uadafini;^rdi
te:tranrponarfoni:e mueftite.
^arti'r.muln'pf icar.fumniar.e fotrar oe k p:op0m'<>
m'eDetuttefojriradicf.
uidcntic general! ouer conduficmi nfc^.abfduere
ognicafodxpcrrcgorc.o:dman'end fi podclTc,
fuctefo^eWnomijereaTiealtrelineeuTaridafioel
do: fabn*'
cbeefortdamenrt.
om pagnic i turn' modi.c To? pattire.
ecide oe bcfliami. e lo? partire
aram' i mmnnxU fcmplicucoinpofh'tc cd tempo.
ambt reali-feccbtfi'tririi^ Wmt'nuriouer comunC
K>criti fcmplici e a eapo oanno e aim termini*
Kefti.faWi.fconti.de tempo eoenariela recare aun
dipi'u partite*
>?i4rgeti.elox)afftnare. ecarattare,
OE)oUicafieragioniftTao?dinane variee oiuerfe 8
tutteoccurentiecommonelfafequente rauota ap'
pare o^tnaMinenre oe tutte.
0:dinea fapcr tcncrognicotoefcnpture e oelqua
^tarin^a oe tuttevfantf e coftumi mercantcfcbi in tut
todmoiido,
[ariealtriDependennV
molte altrecofed grandfflimf piacerie frutto co/
tno oifufanientc per la fcqucntc taucla apparc,
FIG. 33. TITLE PAGE OF THE 1494 PACIUOLO
KARA ARITHMETICA
This volume, the first great general work on mathematics printed,
includes treatises on arithmetic, algebra, and geometry, each being
considered from a somewhat scientific rather than practical standpoint.
The arithmetic, for example, gives the various methods in multiplication
(see Fig. 34) and division, instead of emphasizing the one or two most
prominent in business circles. In the same way Paciuolo's treatment of
FIG. 34. GELOSIA MULTIPLICATION, 1494 PACIUOLO
the rule of three, the rule of false ('El cataym'), partnership, pasturage,
barter, exchange, and interest, while nominally practical, was too elabo-
rate for the mercantile schools. His was the first printed work to illus-
trate the finger symbolism of number (Fig. 35). Paciuolo copies without
hesitation, practically verbatim, from the work of Chiarini (p. 10), and
doubtless laid under contribution, after the manner of his time, various
other works of his predecessors. In algebra he used the common sym-
bolism of the time for the unknown quantities and for roots, but he made
use of no symbols of operation. This part of the treatise relates chiefly
to surd numbers. In geometry he follows Euclid's Book I very closely,
but departs quite radically from the subsequent books. The work had
a great influence on subsequent writers, including the English Tonstall
(p. 132). Paciuolo's training had fitted him to write a treatise of this
nature. He had been a tutor in the family of a Venetian merchant,
had traveled extensively, had come in contact with practical mathe-
maticians, and had studied the ancient mathematics in the cloisters ; and
traces of all these influences are seen in his work. In 1497 Paciuolo
wrote at Milan a work entitled ' Divina proportioned which was pub-
lished at Venice in 1509 (p. 87). He also published an edition of
Euclid at Venice in 1509.
Other works 0/1494. Albert of Saxony, p. 9, 1478; Anianus, p.
32, 1488.
PRINTED BOOKS
57
loo
FIG. 35. FINGER SYMBOLISM, 1494 PACIUOLO
58 RARA ARITHMETICA
LUCA PACIUOLO, DE BORGO SAN SEPOLCRO.
Ed. pr. 1494. Toscolano, 1523.
Title. * Summa de // Arithmetica geo//metria. Proportion! :
et proportionalita : // Nouamente impresfa In Tofcolano fu la
riua dil Benacenfe et // vnico carpionifta Laco : Amenisfimo
Sito : de li antique i // euidenti ruine di la nobil cita Benaco ditta
illuftra-//to : Cum numerofita de Impatorij epithaphij//di antique
: perfette littere fculpiti do-//tato : : cu3 finisfimi i mirabil co-//
lone marmorei: inumeri // f ragmenti di alaba-//Jftro porphidi i
ferpentini. Cofe certo // letter mio diletto oculata fi-//de miratu
digne fot-//terra fe ritro//uano.// Continentia de tutta lopera:'//
(The rest of the title page is substantially identical with that of
the 1494 edition.) (F. i, r.)
Colophon to the part on arithmetic : ' C.Et fi fequeti pti pnci-
pali Geo e . finis decima nouembris Ipofitus fuerit : huic tamen pti
die vigefima // eiusde impofitus fuit .M?ccccc.xxiij. Per eosdem
correctore impresforem vt in fine Geo e . hetur.' (F. 232, v.).
Description, Fol., 21.1 X 30.4 cm., the text being 18.2 x 23.5
cm. 9 ff. unnumb. + 223 numb. = 232 ff. Toscolano, 1523.
Bound with this is the ' Tractatus Geometria. Pars fecunda
principalis huius operis i, primo eius diuifio.' This part of the
book contains 75 ff., besides the index.
Editions. See p. 54. De Morgan has shown (p. 2), that
there are slight differences between the copies printed in 1523,
proving that a second impression was necessary in that year.
See p. 56, and Fig. 36.
PEDRO SANCHEZ CIRUELO.
Ed. pr. 1495. Paris, 1505.
Born at Daroca in Aragon, c. 1470 ; died at Salamanca in r 560. One of the
most learned men of his time. He was professor of philosophy at Alcala.
Title. * Tractatus Arithmethice // Practice qui dicitur // Algo-
rifmus.' (A large woodcut with the initials D. R. and the
inscription ' Alaventvre. Tout Vient. Aponit. Qvi. Pevt. Atendre.
Denis. Roce.') (F. i, r.)
PRINTED BOOKS
59
fi nofcuf irnaflente ec&cfc molti pMofbpft per gudto
guidzmm finno alia fubftantu para: e conct (fa octerna.
Ifcerodje fconno ODgnoTtuito per- cep modo aldjana co6
in reram natora fmsa In no ptere eriltcrc. ifcer laqual
cob tt In intendo Coon (adiute ee ootai d* b noftri fenft
reggt J) rrgoame nm dje per altri pnTcfpi e onocf?i pfy'
fofopfji nonnt fia copidfemente tractate: e in Zfroa'ca e
pactua. rT*3 prrdx kxoicti gia alii tempi noftn Tonna
molto obfcurt-e oa moltj male apsfi : e alle pwcndje vu[
gari mate apptoti : wldje in tow opnanortf moltD vs^
nano: e con gradi e laforioft aftanra mettano in ofrra: ft
tt mmeri como V milure: vndfoi lei parlando non intendo le non quanto cJ?c alia
pactica e opnare fia mdtiero: meftotodod fecondo i taogfo opairt andjoza la tfjeo
nca:ecan&ataIeopnare:naiiuniericorrioaseometnVi.dNipnrnaaccio meglio
qneHo dje fequiu fe fjabia appxndere : effa quantita oiuidiremo fecondo el noftro p:o
pifito: e ciuidmdoU a dalcfjano fao membra aflegnaremo fiia pjopaa e tera oiffir, ^
Done e afcn'ptionf . aHoa pi feguira quello dx ^nltodle CUE in ft rondo pflcrio
fantZnnfe^maj^efcimrafiqmdaunfpatmirluumguiddttc.
tDBfflSnitiones/et vuffio cierrte/etcontinneqiiantkatis.
jHrrj'cuuw pa'mno . tffea'me oi(h'nctjoni.
D^9co adonca. ^a quantita eflfere immediate 6imemb?e^do confirms e ^ifcrctcu
^^"como foraito Itgnp km: e &ru t& jia. wTcrem oueramente numero
eqnHUtecmpBrtinOTlcn^s^fc^^^fw/^mfriEcomunf.'coinoe.i.i.j.rc.
^c^rpa/na ct b ixftreffl: doeert nnmero: e pt cc b connnoa; doe 0Eometna;qua<
CjDiffinitio nnmeri pjopa'idima. 2rticolu0 (ecundnff.
O^mcio^ c Cfccorido daJuTiino pryloiopfjdntO ^ns mclulndinf tP tmitn co r
pofta: z eflTa mia non e nnmero: ma ten pnhdpio ce cUfdjuno nnmero: e OB
queUa mediante faqle ogra CD& e oicia eflere vna. fecondo feuerino )8oetjo
In fiia muftca: e la mica dafdjun numero in ptenoa: z paffi j in la fua aritljmetica rf
gina e fbndamento cogni nnmero lap&o. lUqual piu tnagrancandola in le cofe naot'
tsae oifle in quello dje 6 ce mcate z vno. Cmnt qd eft: ideo eft: quia rnum nume^
ID eft. te ancfrna el numero in inftrafl membji oiuifo: per quello cfje cflb anftoole
icE:doe.a'qmdinftninimeft:nnmeruB eft. per laterw painoneoel fepdmoce
>Euc(ide: b (faa ferie in infinite ptere praadere: ? quocuqs numero pato: oari pteft
nvdov wu'tatem addoido . d>a noi pi'gliaremo queue parte a noi piu note e aaomo
date. Jxro oico con glialth aldjuno eflere pamo: ett quello ctje Colo ca la vnto e nn
merato: e non fea altro nnmero: dje integralmente apnto to parta. aitro e oitto co^
Edlo: tt e qnttto dje oa altro numero e'mefurato: ouero numerate. jempuim p:imf
Como.) .7. 1 1. i).e. i7.?c.Kplu feci3di.iC6rno.4.tljel roi to mdura e numera.-e. S.
titf. J.e-4/9. i H- 1 8..e nnuUmta ronno citti numai compfh; no foto dje coftino
2
FIG. 36. FIRST PAGE OF TEXT, 1523 PACIUOLO
60 KARA ARITHMETICA
Colophon. ' Arithmetice practice feu Algorifmi tractatus a
Petro fanchez // Ciruelo nouiter compilatus Explicit Impref-
fus Parifius In // Bellouifu. Anno dni, 1505. Die. 29. Aprilis.'
(F. 14, r )
Description. 4, 13.2 X 18.4 cm., the text being 9.7 x 14 cm.
14 if. unnumb., 35-39 11. Paris, 1505.
Editions. Paris, 1495, 4; ib., 1505, 4 (here described) ; ib.,
1509, 4; ib., 1513, 4 (see below). Ciruelo also wrote a
' Cursus quattuor mathematicarvm artiu liberaliu,' Paris, 1516;
ib., 1523; ib., 1526; ib., 1528; Alcala, 1516, fol. ; ib., 1518;
ib., 1523, fol. ; ib., 1526; ib., 1528. He also edited Brad-
wardin's arithmetic, Paris, 1495; ib., 1502.
Ciruelo treats very briefly of the fundamental operations with integers,
common fractions, and denominate numbers. Following the Spanish
custom he uses cuento for million. There is little that is noteworthy in
the book, and, like Peurbach, Ciruelo could not have taken his contri-
bution to algorism very seriously.
Other works 0/14.95. Anianus, c. 1495, p. 32, 1488. There also
appeared in 1495 the treatise of Herodianus, ' De notis Graecorum
Arithmeticis Grsece,' Venice. Of this work there was an edition in
1525, and a Latin edition published at Basel in 1600, but it hardly
deserves to be classed as an arithmetic.
PEDRO SANCHEZ CIRUELO. Ed. pr. 1495. Paris, 1513.
See p. 58.
Title. ' Tractatus arithmetice // Practice qui dicitur algorif-
mus.// (A large woodcut with the name Jehanlambert.) Venun-
dantur Parrhifijs a Johaue La//berto eiufdem ciuitatis biblio-
pola in fte-//mate diui claudij manente iuxta gymna-//fium
coquereti.' (F. I, r.)
Colophon. ' C.Arithmetice practice feu Algorifmi tractatus a
Petro fan//che Ciruelo nouiter compilatus Explicit Impreffus
Parifius //per Anthonium Auffourt. Pro Johane Lamberto
eiufdem ci-//iritatis bibliopola in ftemmata diui claudij manente
iuxta gym-//nafium coquereti. Anno dni .1513. Dievero .21.
menfis Marti j.' (F. 12, r.)
PRINTED BOOKS 61
Description. 4, 13.1 x 19 cm., the text being 9.6 x 14.8 cm.
12 ff. unnumb., 43-46 11. Paris, 1513.
Editions. This is the fourth edition (see p. 60), and is an
exact reprint of the 1505 copy, save as to spelling and pagina-
tion. As a piece of typography it is much better than its
predecessors.
THOMAS BRADWARDIN. Ed. pr. 1495. Paris, c. 1510.
BRAGWARDINE, BRANDNARDINUS, BREDWARDYN, BRADWARDYN, DE
BRADWARDINA, DE BREDWARDINA. Born at Hertfield (Hartfield) in the
diocese of Chichester, c. 1290; died at Lambeth, August 26, 1349. On
account of his learning he was called ' Doctor profundus.' He was pro-
fessor of theology at Oxford, and died as Archbishop of Canterbury. He
wrote four works on mathematics.
Title. 'Arithmetica thome brauardini.// Olivier Senant//
Venum exponuntur ab Oliuiario fenant in vico diui Jacobi fub
si-//gno beate Barbare fedente.' (F. i, r.)
Colophon. ' Explicit arithmetica fpeculatiua thoe brauardini
bri re-//iiifa e t correcta a Petro fanchez Ciruelo aragonenfi
ma//thematicas legete Parifius, Ipressa p Thoma anguelart.'
(F. 6, v.)
Description. 4, 20.3 X 27.8 cm., printed in double columns,
each being 6. 8 X 19.8 cm. 6 ff. unnumb., 61 11. Paris, c. 1510.
Editions. Paris, 1495, 4; ib., 1496, fol. ; ib., 1498, 4; ib.,
s. a. (c. 1500) ; ib., 1502, 4; Valencia, 1503, fol. ; Paris, 1504;
ib., 1505,4; ib., s. a. (c. 1510, here described) ; ib., 1512; ib.,
1530; Wittenberg, 1534, 8; ib., 1536, 8. His ' Tractatus de
proportionibus ' appeared in several editions, as follows : Paris,
1495; Venice, 1505, fol.; Vienna, 1515 (p. 117); and a com-
mentary by Vittori appeared at Bologna in 1506. Some of these
editions contained two or three of his works in one volume, and
it is probable that his arithmetic and his treatise on proportion
appeared in other editions than those mentioned.
Bradwardin was one of the earliest English mathematicians after
Bseda and Alcuin. His arithmetic is of the Boethian type, relating to
the theory of numbers. He gives much attention to the ancient theory
of ratios and to figurate numbers.
62 KARA ARITHMETICA
BOETHIUS, JORDANUS NEMORARIUS, AND FABER
STAPULENSIS. Ed. pr. 1496. Paris, 1496.
For the biography of BOETHIUS see p. 25.
JORDANUS NEMORARIUS (JORDANUS DE SAXONIA) was born at Borgent-
reich, in the diocese of Paderborn, and died in 1236. He studied at Paris
and was the greatest mathematician of his time save Leonardo Fibonacci
of Pisa.
JACOBUS FABER STAPULENSIS (JACQUES LE FEVRE D'ESTAPLES) was born
at Estaples, near Amiens, in 1455, and died at Nerac in 1536. He was a
priest, vicar of the bishop of Meaux, lecturer on philosophy at the College
Lemoine in Paris, and tutor to Charles, son of Francois I. He wrote on
philosophy, theology, and mathematics.
Title. See Fig. 37.
Description. Fol., 20 x 29 cm., the text being 13.9 x 27.2 cm.
72 ff. unnumb., 6063 11. Paris, 1496.
Editions. The arithmetic of Jordanus went through various
editions as follows: Paris, 1496, fol. (here described); ib., 1503,
fol.; ib., 1 507, fol. (p. 65); ib., 1510, fol.; ib., 15 14 (p. 65). Rogg
speaks of an edition of 1480, but I do not know of it. Jordanus
also wrote an * Algorithmus demonstratus,' published anony-
mously at Niirnberg in 1534, 4. De Morgan, following Scho-
nerus, attributed it to Regiomontanus, but the evidence shows
that he only revised it and it may be due to Jordanus. His 'De
Ponderibus/ edited by Apianus, was published at Niirnberg in
1533, and at Venice in 1565. An interest attaches to the 1496
edition in that it is the first printed work with which a Scotch-
man's name is connected, the printer being David Lauxius of
Edinburgh, then working in Paris.
The greater part of this volume is devoted to the ten books on arith-
metic by Jordanus Nemorarius, with the commentary of Jacobus Faber
Stapulensis. The work of Jordanus is similar to that of Boethius, and is
concerned only with theory of numbers. In particular, the Greek theory
of ratios, as elaborated during the Middle Ages, is extensively treated.
The second part consists of the work of Jacobus Faber Stapulensis on
music, in four books.
The third part is the Epitome of the Arithmetic of Boethius :
* CJacobi Fabri Stapulenfis Epitome in duos libros Arithmeticos // diui
Seuerini Boetij ad Magnificum dfium Joannem Stephanum // Ferrerium
Epifcopum Verfellenfem.'
PRINTED BOOKS 63
The fourth part, consisting of four and a half pages, is a description
of the arithmetical game of Rithmimachia, possibly by Shirewode (John
Shirwood, Bishop of Durham, who died in 1494), but usually ascribed
to Faber Stapulensis. An edition appeared at Erfurt in 1577, 4.
$nbocoperc contents
ftritbtnetica oecem lib:is ocmonftratt
(Epitome i Ubzos arttbmcncosoiui 0euerfm tfocttf
lauoe Sritbmcriccs z ^fcuficco.
Ccmpo:e tarn multo ooctc bcucrc fozozcc:
<2iuao rennet comitco flaua mmcrua fua0
TRuc plaaoe tcrrae port tempo:a multa reuiftfflf*
i6rata quoc^ ante altaa fSalUca terra placet*
Ibid olmi ccleb:te futt omnto Scaia tcllu0.
l7tba$o:a patriam oiffiigicntc famon*
IbcIIaoa nune Imquunt: ct Ddctae pnllaoio vrbcs*
Bcquaniofcp pctunt/parbtfiofcp larc^v
Ibcc venit omnimooa numeroiutn dneta ca tons*
2Ucp oo cct numertd quioquio m o:be fltum eft*
Bltcra oulctfono cantu/ftoibitfcp cano:ia
^oomuiflc viro0 traoimr atcp ferae:
luc fua ipicrij o tenet vniea nomma mul!0
^ niebil bae mufio $ratme clTc fplcr.
3lttamcn artiftccm ftapule mifere marine:
gQui leta boc (luDtjo frontc Diarct opu0.
Iboefolum (luoium atcp bee illt curan'uuarc/
Ifrriiuo et nc flt/oifperearcp labo:*
FIG. 37. TITLE PAGE OF THE 1496 BOETHIUS
Other works 0/1496. Albert of Saxony, p. 9, c. 1478 ; Bradwardin,
p. 6 1, 1495 ; Z. Lillius, < De origine et laudibus scientiarum,' Florence,
4 (one page on arithmetic). There was also published at Paris, s. a.
6 4
KARA ARITHMETICA
(c. 1496), an anonymous treatise entitled ' De arte numeradi copediu
putile icipit feliciter. (Q)uoniam rogatus a plurib5 compedium artis
numerandi ac breue tractalutu . . .'
Works 0/1497. Boethius, p. 27, 1488 (the colophon of the 1497
no
oflo
FlG. 38. RlTHMIMACHIA, 1496 BOETHIUS
edition has the incorrect date Mcccclxxxvii) ; Suiseth, 1497 and 1498,
p. 10, c. 1480.
Works, of 1498. Anianus, p. 32, 1488; Bradwardin, p. 61, 1495;
Chiarini, p. n, 1481; Anonymous, 'Enchiridion sive tractatus de
numeris integris, fractis,' etc., 4 (doubtless the work published at
Deventer in 1499, p. 67).
PRINTED BOOKS 65
BOETHIUS, JORDANUS NEMORARIUS, AND FABER
STAPULENSIS. Ed. pr. 1496. Paris, 1507.
See p. 62.
Title. ' In hoc opere contenta.// Arithmetica decem libris
demonftrata // Mufica libris demonftrata quattuor // Epitome I
libros arithmeticos diui Seuerini Boetij // Rithmimachie ludus
q i pugna nuero^ appellat.' (F. i, r.)
Colophon. ' Cjmpreffum Parifij in officina Henrici ftephani
e regione Schole decretorum fita.//Anno Chrifti fiderum con-
ditoris 1507. Decimo die Nouembris.' (F. 78, r.)
Description. 8, 20x26.6 cm., the text being 18.1x21.5
cm. 78 ff. unnumb. + I blank = 79 ff., 60 11. Paris, 1507.
Editions. See pp. 27, 62, for the editions of Boethius and
Jordanus.
Like most of the works from the press of Stephanus, this book is
beautifully printed. It contains the ' Elementa Arithmetica ' of Jorda-
nus, with the demonstrations of Faber Stapulensis, the Epitome of Boe-
thius by Faber Stapulensis, the ' Rithmimachia,' the commentary on
Sacrobosco's astronomy by Faber Stapulensis, and the first four books
of Euclid * a Boetio in latinum tranflate.'
BOETHIUS, JORDANUS NEMORARIUS, AND FABER
STAPULENSIS. Ed. pr. 1496. Paris, 1514.
See p. 62.
Title. ' In hoc opere contenta // Arithmetica decem libris //
demonftrata.// Mufica libris demoftrata // quatuor.// Epitome
in libros Arith-//rneticos diui Seuerini //Boetij. //Rithmima-
chie ludus qui//et pugna numeroru ap-//pellatur.'// (Sur-
rounded by an elaborate woodcut with the following wording :
' Haecsecvndariaes//tetcastigat//issimaexofficina//aemissio' // )
<C.Hec fecundaria fuperioru operum aeditio // venalis habetur
Parifijs://in ofihcina Henrici Stephani e regione fchole Decre-
torum.' (F. i, r.)
Colophon. ' C.Has duas Quadriuij partes et artium liberalium
precipuas atq3 duces cu quibufdam ammini-//cularijs adiectis:
curauit ex fecunda recognitione vna formulis emedatiffime
66 KARA ARITHMETICA
mandari ad ftudiorum // vtilitatem Henricus Stephanus fuo
grauiffimo labore et fumptu Parhifijs Anno falutis domini : //
qui omnia in numero atq5 harmonia formauit 1514. abfolutumq5
reddidit code anno: die feptima // Septembris/ fuum laborem
vbicunq3 valet Temper ftudiofis deuouens.' (F. 71, v.)
Description. Fol., 19.8 X 28.4 cm., the text being 17.7 x 26.8
cm. 71 ff. unnumb., 62 11.
Editions. See p. 62. This edition is practically identical with
that of 1496. It is the second edition of this combination of
works and the fourth of Faber's Epitome.
MARTIANUS MINEUS FELIX CAPELLA.
Ed. pr. 1499. Vincenza, 1499.
Flourished c. 475. He was probably born at Carthage, and he lived
at Rome.
Title. See Fig. 39.
Colophon. 'Martini Capellae Liber finit: Impreffus Vincentiae
Anno Salutis // M.ccccxcix. xvii. Kalendas lanuarias per Henri-
cum de Sancto//Vrfo Cum gratia & priuilegio decem annorum:
ne imprima-//tur neq3 cum Commentatiis: neq3 fine: & caetera:
quae in ipfo pri//uilegio continentur. Laus Deo & beatae
Virgini.' (F. 123, v.)
Description. Fol., 20.5 X 30.3 cm., the text being 12.1 X 22.3
cm. I24ff. unnumb., 37 11. Vincenza, 1499.
Editions. Vincenza, 1499, fol. (here described); Modena, 1500,
fol. (p. 67); Vienna, 1516, fol.; Basel, 1532, fol.; Leyden, 1539;
Basel, 1 577, fol. (p. 68); Leyden, 1592, 8 (p. 68) ; ib., I599>8;
and later. An Italian translation was published at Mantua in
1578. For bibliography, see Boncompagni's Bulletino, XV, 506.
This work is a medley of prose and verse, and forms a kind of ency-
clopedia of the arts and sciences as known for about a thousand years.
It was highly esteemed in the Middle Ages as a textbook. The seventh
book is on the Greek theory of arithmetic. It treats of the various
classes of numbers, such as plane and solid, and mentions the supposed
mysteries of the smaller numbers, the monad suggesting one God, the
dyad good and evil, the triad the Trinity, and so on.
PRINTED BOOKS 67
Other works of 1499. Boethius, p. 27, 1488; Suiseth, p. 10, c.
1480 ; Anonymous, ' Enchiridion Algorismi sive tractatus de numeris
integris,' Deventer, 4 (p. 64, 1498) ; George of Hungary, ' Arithmetica
summa tripartita,' s. 1., reprinted at Budapesth in 1894.
Opus
i (E> arruni-Capdle de ^luptiie
oedulectica.
s-
>.vc arithmetics
x. Deafrronomia.
t>c mafia Ubrifcprcm.
FIG. 39. TITLE PAGE OF THE 1499 CAPELLA
MARTIANUS MINEUS FELIX CAPELLA.
Ed. pr. 1499. Modena, 1500.
See p. 66.
Title. ' Opvs.// Martiani Capellae // de nvptiis phi//lologiae et
//mercvrii//liberi//dvo.//De gramatica. Liber. Tertius.//De di-
alectica. Liber. Quartus.//DeRhetorica. Liber. Quintus.//De ge-
ometria. Liber. Sextus.//De Arithmetica. Liber. Septimus. //De
aftronomia. Liber. Octauus.//De mufica. Liber. Nonus.' (F. i, r.)
Colophon. ' Martiani Capellse Liber finit. Impreffus Mutinae.
Anno Salutis. M.//CCCCC. Die .XV. Menfis Maii. Per Dionyfiu.
Berthocum.' (F. 100, r.)
Description. Fol., 20 x 29 cm., the text being 13.7 x 24 cm.
On f. 69, v., the part on arithmetic begins, and occupies 10 ff.
100 ff. in the entire book, unnumb., 42 11. Modena, 1500.
68 KARA ARITHMETICA
MARTIANUS MINEUS FELIX CAPELLA.
Ed. pr. 1499. Basel, 1577.
See p. 66.
Title. * Isidori // Hispalensis // Episcopi // Originum libri
viginti // ex antiquitate eruti.// Et // Martian! Capellse // De
nuptijs Philologiae & Mercurij // Libri nouem.// Vterque, praeter
Fulgentium & Veteres Grammaticos, va-//rijs lectionibus &
fcholijs illuftratus // Opera atq Industria // Bonaventvrae Vvlca-
nii Brvgensis.// Cum gratia & priuilegio Caefareae Maieftatis.
// Basileae,// per Petrvm Pernam.' (F. i, r.)
Description. Fol., 20.5 X 31 cm., printed in double columns,
each being 8.2 X 24.4 cm., 7 ff. + 240 columns + index +550
columns. The part on arithmetic in the work of Capella begins
in column 155 and covers 12 pp., or as here numbered 24 col-
umns, 60 11. Basel, 1577.
Editions. See p. 66.
This edition includes the works of both Isidorus (p. 8) and Capella,
and is an excellent specimen of printing. One interesting feature of
Capella' s work is the evidence that it gives of the use of the abacus in
the fifth century. We are still quite uncertain as to the history of this
method of calculating in the centuries following Capella.
MARTIANUS MINEUS FELIX CAPELLA.
Ed. pr. 1499. Leyden, 1592.
See p. 66.
Title. ' M. Capella.// Martiani // Minei Capellae // Carthagi-
nensis // de Nvptiis Philolo-//gise, & feptem artibus // Libe-
ralibus // Libri Novem // optime castigati.// Lvgdvni,// Apud
Bartholomaeum Vincentium.// 1592.' (P. i.)
Colophon. ' Lugduni,// Excvdebat // Stephanus Seruain. //
1592.' (P. 416.)
Description. 8, 10 X 15.7 cm., the text being 7.1 X 12.8
cm. 4 pp. blank -f 18 unnumb. + 396 numb. =418 pp., 30 11.
Leyden, 1592.
Editions. See p. 66,
PRINTED BOOKS
69
feriftft
foria
ficot in fcoli^ mpnibergen,0ritbmcirico:n oocct
JnflozentiflTinio fltodio EipcsenfinueediVTOo
minns h ttena eruditia $ ^fcercatcwbns ^ tilfe ^r
majcime incipientibiw.
Zectozi
%urta fucdnctepateat tibt rfgtila tsctri
If rangere quo valeas qaem mi
'
IDuius viUfcant non t ibi oon a libn
V>ii8 nurnberga niter numerar? d t infigme a b arte v
iic aw mulrom contnlit i(U bout
FIG. 40. TITLE PAGE OF LIGHT
70 KARA ARITHMETICA
BALTHASAR LIGHT.
Ed. pr. 1500. Leipzig, s. a. (1500).
A German Rechenmeister of c. 1500.
Title. See Fig. 40.
Colophon. 'Impreffum Lipczk per Melchiar Letter.' (F. I5,v.)
Description. 4, 14 X 19.9 cm., the text being 9.3 X 16.2 cm.
15 if. unnumb., 22-37 11. Leipzig, s. a. (1500).
Editions. Leipzig, s. a. (1500), here described; ib., 1509;
1513; Leipzig, 1515, 4, which may be the ' Algorithmus
linealis,' s. a. and 1505, by Letter, referred to by De Morgan
(p. 101). There was also an * Algorithmus linealis, Impressum
Lipzik per melchiorem Lotter Anno xc,' probably printed in
1490, Lotter having printed in Leipzig from 1490 to 1512.
(See contra, Abhandlungen, V, 154, n., 152, n., and cf. Widman,
c. 1490, p. 44.) On f. i, v., the dedicatory epistle closes with
the words ' Vale ex noftra academia Lyptzen Anno 1 500,'
which throws much doubt on the conjecture that Licht's work
appeared earlier.
This is a brief treatise on the line abacus, one of the earliest of the
type represented also by Huswirt (see p. 73).
LEONARDUS PORTIUS.
Ed. pr. c. 1500. S. 1. a. (Venice ?, c. 1500).
A Venetian jurist of the fifteenth century.
Title. ' Leonardi // de Portis ivrisconsvlti Vi//centini de
sestertio pe//cvniis ponderibvs et // mensvris antiqvis // libri
dvo.' (F. i, r. See Fig. 41.)
Description. 4, 14.4 X 19.8 cm., the text being 10 x 14.8 cm.
37 ff. unnumb., 30 11. Venice (?), c. 1500.
Editions. S. 1. a. (Venice ?, c. 1500, here described);
Florence, 1514 (?) ; Basel, 1520, 4; ib., 1530, 8.
A work on ancient measures, using the Roman numerals throughout,
except in the index. Such treatises are of value in studying the history
of arithmetic, but are not, in general, included in the bibliographical
lists of this work.
PRINTED BOOKS 71
Other works of c. 1500. Anianus, c. 1500, p. 32, 1488; Boethius,
c. 1500, p. 27, 1488 ; Bradwardin, c. 1500, p. 61, 1495 ; Capella, 1500,
p. 66, 1499; Peurbach, c. 1500, p. 53, 1492. Widman, 1500, p. 37,
1489. There also appeared about this time, s. 1. a., an anonymous
4 Algorithmus minutiarum vulgarium,' blackletter, 4 (Libri, 1861 cat.,
483), and an anonymous 'Ars numerandi,' 5 ff., 4, a title given to
several books of this period (see p. 23), including ' De arte numerandi
siue arismetice (perfectionis) summa quadripartita ' (Abhandlungen, I,
24 ; Brunet, Man., 6 (i), 458).
LEONARDI
DE PORTISIVRISCONSVLTI VI
CENTINI DE SESTERTIO PB
CVNI1S PONDERIBVS ET
MENSVRIS
LIBRI OVO.
FIG. 41. TITLE PAGE OF LEONARDUS PORTIUS
GEORGIUS VALLA. Ed. pr. 1501. Venice, 1501.
Born at Piacenza in 1430 ; died at Venice in 1499. He was a physician
and philologian.
Title. 'Georgii Vallae Placentini viri cla-//riss. de expetendis,
et fvgiendis // rebvs opvs, in qvo haec // continentvr.// De arith-
metica libri .iii. ubi quaedam a Boetio praetermiffa tractantur.//
De Mufica libri .v. fed primo de inuentione, & 'commodiatate
eius.//De Geometria libri .vi. in quibus elementorum Euclidis
difficultates omnes fere // exponuntur, ubi etia de Mechanicis
fpiritalibus, Catoptricis, ac Opticis, deq; //quadrate circuli habe-
tur tractatus.// De tota Aftrologia libri .iiii. in qua fabrica,ufusq;
aftrolabi exaratur, & quae fi-//gnorum in exhibendis medica-
minibus fit habenda obferuatio.// De Phyfiologia libri .iiii. ubi &
72 KARA ARITHMETICA
Metaphyfices qda lectu q dignifs. utilifTimaq;.//De Medicina
libri .vii. ubi de fimplicium natura per ordinem litterarum.// Pro-
blematum liber unus.// De Grammatica libri .Hi.// De Dialectica
libri .m.// De Poetica liber unus.// De Rhethorica libri .ii.// De
Morali Philofophia liber unus.// De Oeconomia, fine adminiftra-
tione domus libri .iii. in quibus de Architectu-/Aa, req; ruftica eft
locus.// Politicon unicum uolumen, ubi de iure ciuili, ac pontificio
primum, Mox de le//gibus in uniuerfum, Inde de re militari
agitur.//De Corporis commodis, & incommodis libri .iii. quorum
primus totus de ani-/Ana, Secudus de corpore, Tertius uero de
urinis ex Hippocrate, ac Paulo aegi-//neta, deq; Galeni quaeftioni-
bus in Hippocratem.// De Rebus externis liber unus, ac ultimus,
ubi de Gloria, Amplitudine, & caete-/Ais huiufmodi.// Haec fum-
matim, fed infunt, & alia plurima,quae legedo licet cognofcere.'
(F. i, r.)
Colophon. l Venetiis, in aedibus Aldi Roma-//ni impensa, ac
studio Ioan-//nis Petri Vallae filii pi-//entiss. mense Decem-//
bri. M.D.I.' (Not in this copy.)
Description. Fol., 28.8 X 43.7 cm., the text being 18.8 x 32.7
cm. 3 ff. blank + 308 unnumb. = 311 ff., 55 11. Venice, 1501.
Editions. There was no other edition.
The first book consists of 23 brief chapters on the general value and
nature of mathematics (18 pp.); the second book, of 18 chapters on
the Greek classification of numbers (17 pp.) ; the third, of 20 chapters
on figurate numbers, proportions, and the fancied properties of each
number of the first decade (27 pp.); and the fourth, of 13 chapters
on the operations (13 pp.)- There is nothing that is noteworthy in
the treatment. Works on the value of mathematics were quite common
at this time, while all university treatises on arithmetic were devoted
chiefly to the Greek theory. The thirteen pages devoted to the opera-
tions were a rather generous allowance for the time, especially as each
page has as much matter as six or eight pages of an ordinary octavo
arithmetic of that period.
Valla also wrote a treatise on the astrolabe, ' Insignis philosophi
Nicephori Astrolabii expositio' (Paris, 1554), and published an edition
of Euclid (Venice, 1492). His collected commentaries, but without the
arithmetic and other original works, appeared in Venice in 1498.
PRINTED BOOKS 73
ARNALDO DE VILLA NOVA.
Ed. pr. 1501. Venice, 1501.
ARNAULD DE VILLENEUVE, ARNALD BACHUONE. Bom in 1248, at Villa
Nova (Catalonia), or possibly Villeneuve, near Montpellier; died in 1314,
shipwrecked on the Mediterranean. He is known principally for twenty
works on alchemy. He lectured on philosophy and medicine at Barcelona
and Paris, and was later a celebrated physician.
Computus
micas ditus a 2lf>agiftro
naldo DC villa *flcim *#o*
inter 3mp:cflum.
FIG. 42. TITLE PAGE OF ARNALDO DE VILLA NOVA
74 KARA ARITHMETICA
Title. ' Computus Ecclefiafticus 2 Aftrono-//micus Editus a
Magiftro Ar-//naldo de villa Noua No-//iriter Impreffum.//
Cum Gratia Et Priuilegio.' (F. I, r. See Fig. 42.)
Colophon. ' Cjmpreffum Venetijs per Bernardino Venetu
deVitalibus.//Anno Dni.M.CCCCC.J.Die .xvij.Men.Februarij.'
(F. n, v.)
Description. 4, 14.4 X 20.9 cm., the text being 10.5 x 16.6
cm. ii ff. unnumb., 37-39 ff. Venice, 1501.
Editions. There was no other edition.
This is a good example of the works on the ecclesiastical calendar
in use in the Middle Ages. It employs only the Roman numerals and
gives no treatment of computation. In spite of the words 'nouiter
impressum,' I know of no earlier edition, and indeed these words were
not infrequently used when a book was first printed.
JOHANN HUSWIRT, Sanensis.
Ed. pr. 1501. Cologne, 1501.
A German arithmetician of c. 1500. The name Sanensis suggests his
birthplace as Sayn in the Westerwald, and the problems relate to places in
that vicinity. Nothing is known of his life.
Title. See Fig. 43.
Colophon. ' Enchiridion algorifmi fagaci cura ftudioq3 p
Johane hufwirt sanefez //elaboratus. caracteri p pmiffus Colo-
nie In officina felicis memorie ho//nefti viri Henrici Quentell.
Anno repatoris humane feruitut3 Mccccci.' (F. 20, r.)
Description. 4, 14.3 X 20.6 cm., the text being 9 x 15.4 cm.
20 ff. unnumb., 2547 11. Cologne, 1501.
Editions. Cologne, 1501, 4 (here described); ib., 1503; ib.,
1504, 4 (p. 77); 1507; 1554; and a French edition (Chasles).
It was published with historical notes by Professor Wildermuth,
at Tubingen, in 1865. Mr. Plimpton's copy has the bookplates
of Dr. Kloss and Chasles.
This is the earliest treatise on algorism printed at Cologne. It is
divided into four * tractati,' and includes the fundamental operations
through evolution (' Tractatus Primus '), a brief treatment of abacus or
line reckoning (' CTractatus Secundus de proiectilibus '), common frac-
tions (' Tractatus Tertius'), rule of three, partnership, and over twenty
PRINTED BOOKS 75
miscellaneous rules (' Tractatus quartus de regulis mercato^ ' etc.).
In the algoristic treatment of integers Huswirt places ' duplatio
(doubling) after multiplication, and ' mediatio ' (halving) after division ;
uition no
niaftimntfuitt
meneccT/
faiv
tnuidc ticlatt-esflingua combrfccfurenteiti
Nee nimiu rabidis garrulus efto labris.
bt viuis paccm concubitarc fine
FIG. 43. TITLE PAGE OF THE 1501 HUSWIRT
but when he is dealing with counters and with fractions he places them
before multiplication, because they are needed there in abacus calculating.
It is interesting to see how these chapters on doubling and halving, of
which we have traces in ancient Egypt, persisted throughout the Middle
Ages and well into the sixteenth century.
7 6
KARA ARITHMETICA
As in several other works of this period, there is evidence of the
difficulty of finding a generally acceptable name for the character 0,
a difficulty not yet removed in the English language. Huswirt gives
four names to this tenth character : ' Decimo Wo theca. circul? cifra.
fme figura nihili appellat'.'
There is also noticeable in this work a tendency which is seen in
other arithmetics of the time, to name a group of problems after some
well-known type. For example, Huswirt's sixth rule is that of the flee-
ing hare ( Regula Sexta de
Crattatua
f t)e multiplication* Hapl'm qtiartunt
Itiplicatio eft numm" p:ocrcatio.p:opo:tionabC
er fc I?atcnri6 ad multiplicand!} ficut multiplicanc ad rni*
rem ft IjahfMjKmpU gratia tod 4 multiplicarc til mime
from Pnlncrnp tn Rnmp anH .* -C rum fz p2ocreare. que fie multiplicando videlicet 4 p:o0
) KOme, ana -, oflona ^ ur qiwmdmo d um m.ilnpacans.fcilicet j viutari cozrcfponf
followed five davs later bv rcr> QUWVTMCB eft p:opojtiompla.3tem multiplicatio p:crccimru<rq
' hne mulriplicattonem Digito2um inter fefdaf.Cuiue talie oarurrctwla
lepore fugiente,' f. 16, r.),
although it has nothing to
do with the hound and hare,
but relates to a traveler going
another traveler who in
due time overtakes him.
(' C.Ambulat qdam de Colo-
nia verfus roma et ambulat
4tidie 9 miliaria. alius aut
pfequit' ipfum poft 5 dies,' p:io2ipjoduao pof?ponc.vrparet W figura. ^^fl-atiarcgulaocmulripW
ScnbanturCigiri fubalteme.etcuuiflittt Ciffcrcntiaa oenano vcrfue PC
>:rram ponas.quje inter fe mu(nplica.erp:odu(tum inferiuc fcnhr.Dcm?
Pe Diffcrcntwrn vntuaaDtgifoatteriua fubrral|jccrp:io:ip;orfucfo poll?
pone, cr p :ou enicf fumma. vt paret in figura ftw plum . fepes 9
feigitus 8 x oifferenna, quoruint.mulriplc
BOta.
4 MM
artm'ufl fubtH[?e.ardicrunii
. a. C3l iarc S llla ^ m " lr| P'
tarione Ouojum numero:um infra zo quo:um quilitrr oujbua hi
fcnpmseft.piopofiris itaojouobue numowpnmajinfcnc.iie eutnpni!
fnperiozia multipUca^t pzotrcabitur numerus vna velouaro ntfuriE
.
refcruaranr adiungc.er pzoumier numcrus wa?
etc.)
Other works of 1501.
Anianus, p. 32, 1488; Sa-
crobosco, p. 32, 1488 ; Boe-
thius, p. 27, 1488; Borghi,
p. 16, 1484 ; H. Torrentini,
' Elucidarius carminum et
hystoriarum,' Deventer, 4
(with a chapter on arith-
metic), with subsequent edi-
tions as follows: ib., 1503, FIG. 44. COMPLEMENTARY MULTIPLICATION,
4j Strasburg, 1505, 4; HUSWIRT (1501)
Hagenau, 1507, 4; ib.,
1510, 4; ib., 1512, 4; Strasburg, 1514, 4; 1515, 4; Strasburg,
1518, 4; Paris, 1530, 8; ib., 1535, 8; Cologne, 1536, 8; Paris,
i55o 8.
Works of 1502. Anianus, p. 32, 1488; Albert of Saxony, p. 9, c.
1478 ; Bradwardin, p. 6, 1495 ; Nicolo Calvino, a work on arithmetic
and geometry, Milan, of which no extant copy is known (see Riccardi,
part I, col. 213).
PRINTED BOOKS 77
JOHANN HUSWIRT, Sanensis.
Ed. pr. 1501. Cologne, 1504?
See p. 74.
Title. The same as in the edition of 1501 (see p. 74).
Colophon. The last folio, with the colophon, is missing from
this copy.
Description. There are a few changes in type, but otherwise
this edition, which is probably that of 1504, is line for line
identical with that of 1501.
ANTON BARTHOLOMEO DI PAXI.
Ed. pr. 1503. Venice, 1503.
PAST. A Venetian writer of the fifteenth and sixteenth centuries.
Title. ' Tariff a de pexi e mesvre.//con gratia et privilegio.'
(F. i, r.) ' Prohemio del prestantissimo miser Bartho-//lomeo di
Paxi da Venetia.' (F. I, v.) ' Qvi comincia la vtilissima opera
chiama-//ta taripha laqval tracta de ogni sorte // de pexi e mi-
svre conrispondenti per tvto // il mondo fata e composta per lo
excelen//te et eximio miser Bartholomeo di Paxi da// Venetia.'
(F. 2, r. Fig. 45.)
Colophon. ' Stampado in uenefia per Albertin // da lifona uer-
cellefe regnante lin-//clyto principe mifer Leonardo lo//reduno.
Anno domini. 1503. A di//26. del mefe de luio. Finis.' (F. k 5.)
Description. 4, 15.5 x 21.1 cm., printed in double columns,
each being 5.3 x 15.9 cm. I56ff. unnumb., 33-38 11. Venice,
1503-
Editions. Venice, 1503, 4 (here described); ib., c. 1510, 4
(p. 79); ib., 1521, 8; ib, 1540, 8 (p. 79); ib., 1557, 8 (p. 80).
The book is not a textbook on arithmetic, but a collection of infor-
mation useful to merchants, relating to the measures of weight, value,
length, etc., of the various cities and countries with which Venice had
trade relations. It is valuable as leading to an understanding of the
contemporary arithmetics of Italy, and historians could find much useful
information as to the prices and the material of trade by examining
this and similar works. An inspection of Fig. 45 will give some idea of
the scope of Paxi's Tariffa.
KARA AR1THMETICA
COMINQA LA VTIL1SSIMA OPERA CHIAMA,
TARIPHA LACLVAL TRACTA DE OGNI SORTE
DE PEXI E MISVRE CONR1SPONDENT1 PER TVTO
IL MONDO FATA E COMPOSTA PER LO EXCELEN
TE ET EX1M1O MISER BARTHOLOMEO Dl PAX1DA
VENETIA.
AVERemoaduqjprima adechiararea
uoftre excelletie tutte lerobe che fe ueda
no f Venetia a pexo groflb & quelle che
(c uedano a pexo Ibtile e de lordine di pe
xi de li arzenti;& de le coditionc di pan>
ni de lane francefche fatte in Venetian
etiam dele conditione di panni de feda:
e panni doro:& del ordine del uedere de
le fpecie;& de le fue tare;& del ordine di
pcxi dele farine e faifchoti; &del ordine
de le mifure di uini:& del ordine del uen
der del oio:& in che modo e pexo fe uen
deno iguadi:& del ordine di frutti che fe ucndeno a nome di fter: c
che pexo hano cadaun ftene come refponde el pexo groflb con tu
ta la Jtalia:& tuto el leuante & ponente;& etiam come ipexi fubti^
li c pcxi groffi refpondeno con molte terre de Italia : de dalmatia e
de leuante;^ come refpondeno le mefure di panni de lana con W'
ta Italia;e con tuto el Ieuante;& etiam come refpondeno le mefu*
re di panni de feda;e panni doro e darrcnto con tuta ltaha:& co tu
to el leuante & ponenta& come refpondeno le mefure dingilterra
2oc la uirga da londrare de la taripha dalixadriat& etiam quella de
damafcho di la taripha da leppo.e come torna la fporta dalixadria
con molte terre de leuante e de ponete e de la ltalia;& come el can
tcr forfori dalixandria refponde con alchune akre terre de leuante
de ponente e de Jtalia;& etiam come refponde el canter zeroui con
alchune terre del leuante del ponente e de ltalia;& come refpode cl
cento de le mene con alchune terre del leuante e molte de Italia &
del ponente:& come refpondeno tuti icantera de leuante e del po>
nente con el pexo fubtile da Venetia: & come refpode el cnnrer da
napoli de reame con moke terre del leuante del ponete e con mol-
te de ltalia:& come refponde el canter de Conflatinopoli con mol'
tc lenc de leuante & etiam de Italiajfrcome refponde el canter de
FIG. 45. THE BEGINNING OF THE 1503 PAXI a li
PRINTED BOOKS 79
ANTON BARTHOLOMEO DI PAXI.
Ed. pr. 1503. S. 1. a. (Venice ?, c. 1510).
See p. 77.
Title. 'Tariffa De Pefi e mefure cor-//refpondenti dal Le
uante al Ponete : da // vna terra a laltra : e a tutte le parte del //
mondo : con la noticia delle robe // che fe trazeno da vno Paefe
//per laltro. Nouamente // co diligentia Ri-//ftampata .-. *J< '
(F. I, r.)
Colophon. ' C.Finiffe il prohemio de // Miffer Bartholomio di //
Pafi da Venefia.// Finis.' (F. 218, v.)
Description. 8, 10.2 X 15.1 cm., printed in double columns,
each being 4. i X 12.5 cm. 218 ff. numb., 30 11. S. 1. a. (Venice ?,
c. 1510.)
See p. 77. This is the second edition of this popular ' Tariffa.' Since
it was one of the first books of its kind to appear in Venice, its five
editions are easily explained.
ANTON BARTHOLOMEO DI PAXI.
Ed. pr. 1503. Venice, 1540.
See p. 77.
Title. ' Tariffa // de i pesi, e misvre // corrifpondenti dal Le-
uante al Ponente : // e da una terra, e luogo allaltro, qua fi p //
tutte le parti dil Mondo : con la dichia-//ratione, e notificatione
di tutte le robbe : // che fi tragono di uno paefe per laltro.//
Composta per M. Bartholomeo di // Pafi da Vinetia. Con la fua
// Tauola copiofiffima, e f aci-//liffima a trouare ogni cofa // per
ordine, nuouamete // fatta : e con fomma // diligeza reuista,//e
stapata.// In Vinetia. M.D. XL.' (F. i, r.)
Colophon. * In Vinegia. Nelli cafe di Pietro di Nicolini da
Sabbio.//Ne glianni dilla falutifera Circoncifione dil no-//ftro
Signore. M. D. XL.// Dil mefe di Genaio.' (F. 212, r.)
Description. 8, 10.3 x 14.7 cm., printed in double columns,
each being 3.5 x 12.3 cm. 1 1 ff. unnumb. + i blank + 200
numb. = 212 ff., 30 11. Venice, 1540.
See p. 77.
8o RARA ARITHMETICA
ANTON BARTHOLOMEO DI PAXI.
Ed. pr. 1503. Venice, 1557.
See p. 77.
Title. The title page is practically identical with that of the
1 540 edition, except for the date : ' In Vinegia per Paolo Gherardo.
//M. D. LVII.' (F. i,r.)
Colophon. ' In Vinegia per // Comin da Trino.// M. D. LVII.'
(F. numbered 200, r.)
Description. 8, 10.6 X 15 cm., printed in double columns,
each being 3. 7 x 12. 3 cm. 1 1 ff. unnumb. + 200 numb. = 21 1 ff.,
30 11. Venice, 1557
Seep. 77.
BOETHIUS, JODOCUS CLICHTOVEUS, AND FABER
STAPULENSIS. Ed. pr. 1503. Paris, 1503.
For the biographies of BOETHIUS and FABER STAPULENSIS see pp. 25, 62.
JODOCUS CLICHTOVEUS was born at Nieuport, Flanders ; died at Chartres,
September 22, 1543. He was educated at the Sorbonne, and was canon
of Saint-Jean, at Chartres. Like Faber Stapulensis, he was known chiefly
as a commentator.
Title. ' In hoc libro contenta // Epitome/ copendiofaq3 //
introductio in libros // Arithmeticos diui Seuerini Boetij : adie//
cto familiari cometario dilucidata.// Praxis numerandi certis
quibufdam re-//gulis coftricta.// Introductio Igeometria : ex
libris diftlcta // Prim^ de magnitudinib^ & earu // circuftantiis.//
Secudus de cofequentibus/ conti-//giiis/ & cotinuis.// (Sur-
rounded by an elaborate woodcut.) C.Tertius de puctis. C.Quar-
tus de lineis. CQuItus de fuperficieb^.// C.Sextus de corporibus.
. Liber de quadratura circuli. C. Liber de cubica//tione fphere.
CPerfpectiua introductio. Cjnfuper aftronomicon.' (F. I, r.)
Colophon. ' Id opus imprefferut Volphgangus // hopilius et
Henricus ftephanus // ea in arte focii in Almo pari-//fiorum
ftudio Anno Chri//fti Celorum totiufq3 //nature * coditoris.//
1503. Die vice//fimafepti-//ma Iu-//nij.' (F. cxi, v.)
Description. Fol., 19x26.6 cm., the text being 16x22.6
cm. 112 ff. numb., 47-54 11. Paris, 1503.
PRINTED BOOKS 81
Editions. This is the first edition of this combination of
works, the second (somewhat changed) appearing at Paris c.
1 507 (see below), and the third ib., 1510. The epitome appeared
as a ' Compendium arithmetices Boethii,' s. L, in 1480, and,
with the arithmetic of Jordanus, at Paris, in 1496. There was
an edition of Faber Stapulensis, Clichtoveus, and others, at
Cologne, c. 1515, 4. Scheubel published an edition at Basel
in 1553, and a work entitled ' Arithmetica Boethi epitome acced.
Christiani Morisani Arithmetica ' also appeared at Basel in 1553
(pp. 27, 182, 260). The 'Praxis numerandi ' of Clichtoveus,
of which this is the first edition, appeared separately at Paris in
1510, fol.
The copy here described is bound with the 150910 edition of
Bovillus (see p. 89). It consists of a brief introduction by Faber
Stapulensis, and the arithmetic of Boethius with the commentary of
Clichtoveus on Faber's epitome. This is followed by the geometry and
perspective of Faber Stapulensis.
BOETHIUS, JODOCUS CLICHTOVEUS, AND FABER
STAPULENSIS. Ed. pr. 1503. Paris, c. 1507.
See p. 80.
Title. Tntroductio//Jacobi fabri Stapulefis in Arithme//cam
Diui Seuerini Boetij pariter i Jordani // Ars fupputadi tarn per
calcu-//los qj notas arithmeticas fuis quidem regulis elegater
expreffa // Judoci Clichtouei Neoportuenfis.// Queftio haud
indigna de numeroru // et p digitos i p articulos finita pgreflione
ex Aurelio Auguftino // C,Epitome rerum geometrical^ ex
Geometrico introductorio // Caroli Bouilli.// CDe quadratura
Circuli Demonftratio ex Campano.' (F. i, r.)
Description. 4, 12.6 X 17.4 cm., the text being 10.6 X 14.5
cm. 32 ff. unnumb., 45-46 11. Paris, c. 1507. The dedicatory
epistle is dated * Data ano falutis // noftre Millefimo qngetefimo
feptimo tercio caledas lunij,' that is, 1507. It was evidently
printed at Paris, but it is without date or place.
Editions. See above.
82 RARA ARITHMETICA
The introduction by Faber Stapulensis to the arithmetic of Boethius
and Jordanus was very popular in the university of Paris at the opening
of the sixteenth century. It is, like Boethius, purely theoretical. The
author begins with a dissertation <de vtilitate arithmetice difcipline,'
and then gives an epitome of the two works. This is followed by the
' compendium ' of Clichtoveus, merely a set of rules for the operations.
Books of this character, evidently intended as the bases of lectures to
university students, show in what a hopeless state the Boethian arith-
metic found itself at the end of the Middle Ages.
GREGORIUS REISCH. Ed. pr. 1503. Strasburg, 1504.
Born at Balingen, Wurttemberg; died at Freiburg, 1523. He was a stu-
dent at Freiburg in 1487, and took his bachelor's and master's degrees there.
He then entered the Carthusian order and became prior of the cloister at
Freiburg, and confessor of Maximilian I.
Title. ' Aepitoma omnis phylosophiae. ali-//as Margarita phy-
losophica tractans // de omni genere fcibili : Cum additionibus :
Qu^ in alijs non habentur.' (Large woodcut representing the
liberal arts. F. 2, r. Plate II.)
Colophon. ' Explicit phylofophica Margarita. Caftigatione acri
// In nobili Heluecioru ciuitate Argentina Chalchogra-//phatu :
Per loanne Gruninger Ciue Argetinu : I vigilia // Mathie;
Anno incarnationis Saluatoris M.ccccc.iiij.// Valete & Plaudite.'
(F. 289, v.)
Description. 4, 15.1 x 20 cm., the text being 11.5 x 15.6
cm. 2 ff. blank + 289 unnumb. = 291 ff., 45 11. The illustra-
tions are hand-colored. Strasburg, 1 504.
Editions. Freiburg, 1503, 4; Strasburg, 1504, 4 (here
described) ; Freiburg, 1504, 4; another edition, s. 1., by Schott
(Freiburg), 1504; Strasburg, 1508; Basel, 1508, 4 (p. 83);
ib., 1512, 4; Strasburg, 1512, 4; -ib., 1515, 4; Basel, 1517,
4; Paris, 1523 (first Finaeus edition) ; Basel, 1535, 4 (p. 84) ;
ib., 1583,4; Venice, 1594; ib., 1599; ib., 1600. The three
Venetian editions (1594, 1599, 1600) are Italian translations by
Giovanni Paolo Gullucci, and contain the additions by Orontius
Finaeus, and also the introduction by Faber Stapulensis to the
arithmetics of Boethius and Jordanus (see p. 62), Clichtoveus
A. GEOMETRY
B. ARITHMETIC
PLATE II. FROM THE MARGARITA PHILOSOPHICA
PRINTED BOOKS 83
on arithmetic (see p. 80), and Questione di S. Agostina della
progressione dei numeri per li digiti, et per li articoli.' Hart-
f elder (Zeitsch. f. Gesch. des Oberrheins, II, 170) has shown that
the assertion of Hain, Poggendorff , and others, that it appeared
in 1496, is incorrect.
This was the first modern encyclopedia to appear in print. It contains
a compendium of the trivium, the quadrivium, and the natural and moral
sciences. It is made up of twelve books, of which the fourth, consisting
of fifteen folios in the present edition, is on arithmetic. The author
first considers the definition of arithmetic, and then gives the mediaeval
classification of number, including the system of ratios as set forth by
Boethius and his followers. The second part of the work contains a
short treatment of algorism, including the fundamental operations and
roots. The third tractatus relates to common fractions and the fourth
to physical or sexagesimal fractions. The arithmetic closes with a
treatment of line reckoning, giving the four fundamental operations
and the rule of three. The illustrations are particularly interesting.
(See Plate II.)
Other works 0/1503. Boethius, p. 27, 1488; Bradwardin, p. 61,
1495; Faber Stapulensis, p. 62, 1496; Huswirt, p. 74, 1501; Jor-
danus, p. 62, 1496 ; Orbellis, p. 23, 1485 ; Peurbach, p. 53, 1492 ;
Sacrobosco, p. 32, 1488 ; Torrentini, p. 76, 1501 ; Anonymous,
'Textus arithmetical comunis, cum Conradi Norici commentatione,'
Leipzig, fol.
Works 0/1^04. Anianus, p. 35, 1488; Bradwardin, p. 61, 1495 ;
Johannes Carolus (see Landshut, below) ; Huswirt, p. 74, 1501 ; Reisch,
p. 82, 1503; Sacrobosco, p. 32, 1488; Johann Karl von Landshut
(Lanzut), ' Algorithmus integrorum,' Leipzig (see also p. 97, 1513,
1515); Henricus Stromer, 'Algorithmus linealis cum Regula de Tri,'
Leipzig, 4, with other editions in 1510; 1512, 4; 1514; Leipzig,
1516, 4; ib., 1517, 4; 1520.
GREGORIUS REISCH. Ed. pr. 1503. Basel, 1508.
See p. 82.
Title. ' Margarita philofophica // cu additionibus nouis : ab
auctore fuo // ftudiofiffima reuifioe tertio fup additis. // Jo.
Schottus Argen. lectori. S.// Hanc emo/ non preffam mendaci
ftigmate/ Lector :// Pluribus aft auctam perlege : doctus eris.//
Bafilee. 1508 ' (F. i, r.)
84 RARA ARITHMETICA
Colophon. ' CTertio induftria complied Micha//elis Furterij/
et Joanis Scoti//ftudiofiifiime preffa. Ba-/Aileg.i4-Kal'.Mar//
tias. Anno Chrifti.// 1508.' (F. 308, r.)
Description. 4, 15.2 x 21.9 cm., the text being 12.7 x 16.7
cm. 309 ff. unnumb., 42 11. In this edition the leading initials
are inserted by hand and the illustrations are colored. Basel,
1508.
See p. 83.
GREGORIUS REISCH. Ed. pr. 1503. Basel, 1535.
See p. 82.
Title. ' Marga-//rha philosophica, rati-//onalis, Moralis phi-
lofophiae princi-//pia, doudecim libris dialogice cople-//ctens,
olim ab ipfo autore recognita : // nuper aut ab Orontio Fineo
Delphi//nate caftigata & aucta, una cum ap-//pendicibus itidem
emedatis, & qua // plurimis additionibus & figuris, ab//eodem
infignitis. Quoru omni-//um copiofus index, uerfa // continetur
pagella.//Virefcit uulnere uirtus.//Basileae 1535.' (Surrounded
by an elaborate woodcut.) (P. i.)
Colophon. ' Basileae excvdebat Henricvs // Petrus, ac Con-
radi Refchij impenfis. An//M. D. XXXV.' (P. 1577.)
Description. 4, 15 x 20.8 cm., the text being 9.6 X 13.8 cm.
78 pp. unnumb. + 1498 numb. -f I blank = 1577 pp., 26-30 11.
Basel, 1535.
Editions. See p. 82. Finaeus dates the dedicatory epistle
' Parifijs ex regali collegio Nauarrae. 1523,' and his first edition
appeared in that year. This edition gives only part of the elabo-
rate engravings found in the earlier ones. It is, however, much
better printed, being set in clear Roman type and having a more
open page.
See p. 83.
THEODOR TZWIVEL. Ed. pr. 1505. Cologne, 1507.
A German arithmet
Westphalia.
Title. See Fig. 46.
A German arithmetician of c. 1500, from Monte Gaudio (Mongavensis),
Westphalia.
PRINTED BOOKS 85
Colophon. 'C. Algorithm!, qui ars dicitur numerandi. de integris
// per figurarum (more Alemanorum) deletione3. Nec//no de
pportionib? ingeniofi Pythagorifte Theodo//rici Tzwyuel. poft
plurima praxin iam tande in hoc // opendiu3 reducti finis adeft.
quod et puplica ob vti//litatem in magiftrali artis impreffori e
taberna inge-//nuorum liberorum Quentell iterate diffeminari pro
mtymmu
opufralatmo
ud t>enumcro?p:a]ci(att4a(go2itt?mL&fcunf)vnutt mtegris per
ftguraru(mo:e alrniano*)t>derione. Sterufcc p^pomombws en
ia vftia frcqucns m nrnficS tarrnom'ca Saitruu Bomj
Dcvtilitatchuiusiibdli Tctraftichoa
ioannisMurmelli] Ruremundenfis
Siquis arithmetic^ optatcognofcere praxin
pythagorcnumefosdifccrcfi quis amat
De fophiahunc modicum cornparet ere libra
FIG. 46. TITLE PAGE OF TZWIVEL
//curauit. Anno a natali dominico Millefimo quin//genetefimo-
feptimo.' (F. 9, v. See Fig. 47.)
Description. 4, 13.9 X 20 cm., the text being 8.8 X 14.5 cm.
10 ff. (i blank), 46 11. Cologne, 1507.
Editions. Giinther mentions a Miinster edition of 1505, but
I have not seen it ; there was a Cologne edition in the same
year; Cologne, 1507, 4 (here described).
The work is divided into two parts, the first beginning as follows :
' C.Algorithmus de integris p figuraru (more alemano^z) deleti-//one
86 RARA ARITHMETICA
arte numeradi enucleatim ,3pendiofeq5 edocens.' (F. 2, v.) This part
contains a brief explanation of the writing of numbers and the funda-
mental operations. The second part begins as follows : C.Algorithm9
de pportioib? cuius vfus freques in muficam har//monicam Seuerini
Boetij ' (f. 8, v.), and two pages treat of the operations with the
mediaeval ' proportiones ' or ratios. A comparison of the title pages
represented on pp. 45, 75, 85, and of the works to which they belong, at
least two of which were printed in Cologne, leads to the belief that the
expression ' per figuraru (more alemanoqz) deletione ' (by the deletion
of figures in the German way) refers to a contemporary North German
custom of not actually canceling the figures in the galley division, as
the Italians did.
Other works of 1505. Anonymous (see Licht), p. 70, 1500 ; Borghi,
p. 16, 1484; Bradwardin, p. 61, 1495 ; Ciruelo, p. 60, 1495 ; Licht, p. 70,
1500; Suiseth, p. 10, c. 1480; Tor-
CBIrtO!l't|?mi'.qm'arflbfdnirnijmer9ndlKIHWgTrfl rpntini r> nf\ rcnr Anrmvmnns
prfifiurarum(mo:caicmano:um)alct.o..3.-nec im P ' 7 ' I5QI ' Anonvmous
n6oyjpo:tionitoingcnio(i*pytl?a0o:ift<Xbwc' ' Tractatus perutilis in arithme-
ridi>W)TUel.pOftplurim3pJtmiamfandcin(?OC f - cr^nlat;,,,, ' Paric W^rTYior.
pptndmjrtductiftmeadcft.ciuodetpuplicaobvtt tlca speculativa, Fans; Herman-
Uuuminmafiif!raliarn8imp:efrojictalxniflin^ nus Buschius, 'Enchiridion novus
nuo7umtbrre:umQiicnrdliraatoOincmmarip:o A1 . , ~ i / \ o
curamt. 3nno a narali Oominico ZPilUfimo quin Algonsmi, Cologne, s. a. (i 504), 4
flemcfimofcpumoi ( see also p. 106, 1514); Nicolaus
FIG. 47. COLOPHON OF TZWIVEL Horem, 'Tractatus proportion^,'
Venice, in a volume with Bradwar-
din and Suiseth (see also p. n 6, 1515);* Algorithmus linealis Baccalariu
Wolfgangum Monacenseni,' Leipzig, 4 ; Georg Leunbach, an arithmetic.
Works of 1506. Albert of Saxony, p. 9, c. 1478 ; Anonymous, p.
10, 1480; Vittori, p.-9, Albert of Saxony, c. 1478, and p. 61, Brad-
wardin, 1495; Pietro Borriglione, 'Arismetices praxis,' Turin, 22 ff.,
with a second edition, ib., 1523 ; Raphael Maffei, ' Commentarii Urbani,'
Rome, fol., an encyclopedia containing a book (no. 35) De scientiis
mathematicis,' which includes a little arithmetic ; Maffei's work was
also printed in 1527, and at Paris in 1511, 1515, 1526, and 1530, and
at Basel in 1559.
ANONYMOUS. Ed. pr. 1507. Leipzig, 1507.
Title. See Fig. 48.
Colophon. ' C.Impreffum Liptzck per Baccalariu Vuolfgangu
// Monacenfem Anno noftre redemptionis .1507.' (F. 27, r.)
Description. 4, 14.5 x 19.5 cm., the text being 9.5 X 15.3
cm. 28 if. unnumb. + i blank = 29 ff., 28-34 11. Leipzig, 1507.
PRINTED BOOKS 87
Editions. Leipzig, 1507 (here described) ; ib., 1509,4; s. 1. a.
(Niirnberg ?, c. 1510).
This resembles several of the works on algorism appearing about this
time, such as Widman's (?), Licht's, and Huswirt's. It contains a very
brief treatment of the fundamental operations, including duplation and
mediation. In division, only a single example is given, that of 1456 -H
1 2 . After a similarly brief treatment of fractions, the Boethian proper-
FIG. 48. TITLE OF THE 1507 Algorithm**
tions (ratios) are taken up : * Sequitur Algorithmus proportion^.' Then
follow the Rule of Three and several other rules now entirely forgotten,
such as ' Regula legis,' ' Regula augmenti,' Regula plurima,' ReguU
pulchra,' and ' Regula falfi.' Such ' regulae ' were not stated like modern
rules but consisted of groups of similar problems.
Other works of 1507. Boethius, p. 27, 1488; Faber Stapulensis, c.
1507, p. 81, 1503 ; Huswirt, p. 74, 1501 ; Peurbach, p. 53, 1492 ; Tor-
rentini, p. 76, 1501.
Works 0/1508. Anianus, p. 32, 1488 ; Reisch, p. 82, 1503 ; Widman,
p. 39, 1489 ; Hieronimus de Hangest, * Liber proportionum,' Paris, 4.
LUCA PACIUOLO. Ed. pr. 1509. Venice, 1509.
See p. 54.
Title. After a vocabulary and index the work begins on f.
I, r., as follows : * Excellentiffimo principi Ludouico mariae Sfor.
Anglo Mediolanen//fium duci: pacis et belli ornamento fratris
Lucae pacioli ex Burgo fancti // Sepulchri ordinis Minorum: Sa-
crae theologiae pfefforis. De diuina pro//portione epiftola.'
88
RARA ARITHMETICA
FIG. 49. FROM THE 1507 AtgoHthmus
PRINTED BOOKS 89
Colophon. ' CVenetiis Impreffum per probum virum Paganinum
de paganinis de // Brixia. Decreto tamen publico vt nullus ibidem
totiqj dominio annorum//XV. curiculo imprimat vel iprimere
faciat. Et alibi impreff um fub quouis // colore in publicum ducat
fub penis in dicto priuilegio contentis. Anno Re//demptionis
noftre.M.D.VIIII. Klen. lunii. Leonardo Lauretano Ve.//Rem.Pu.
Gubernante. Pontificatus Iulii.II. Anno.VI.' (F. 27, r., of Part 3.
A similar colophon appears at the foot of f. 35, v., of Part i.)
Description. 8, 20.5 X 28.3 cm., the text being 9.1 X 21.5
(with marginal drawings). I f. blank + 2 unnumb. + 90 numb.
93 ff., 52-57 11. Venice, 1509.
Editions. There was no other edition.
With some hesitancy this book has been included, the only justifica-
tion being the fact that there are several pages devoted to the discussion
of proportion in general, including the arithmetical, geometric, and
astronomic. Paciuolo excludes the other forms of ancient proportion
given by ' Platone e Arifto. e yfidoro i le fue ethimologie. El feuerin
Boetio in fua arithmetica.' (F. 5, r.) Most of the treatise is, however,
devoted to geometry.
CAROLUS BOVILLUS. Ed. pr. 1509-10. Paris, 1509-10.
BOUVELLES, BOUELLES, BOUILLES, BOUVEL. Born at Saucourt, Picardy,
c. 1470 ; died at Noyon, c. 1553. Canon and professor of theology at Noyon.
Title. See Fig. 50.
Colophon. ' Libelli De Mathematicis Svpple-//mentis Finis
Anno Salutis Humane 7/1509 lanuarij Die Deci//maoctaua //
C.Editum est vniversvm hoc volvmen Ambianis in edibvs Re//
uerendi in Christo Patris Francifci De Hallevvin Eiufdem Loci
Pontificis .-. // Et emiffum ex officina Henrici ftephani. Impenfis
eiufdem et loannis parui in chalcotypa // arte fociorum Anno
Christi Saluatoris omnium 1510. Primo Cal. Februarij.// Parisiis.'
(F. 198, v.)
Description. Fol., 19.7 X 27.4 cm., the text being 15.1 X 25.8
cm. 198 ff. numb., 53-54 11. Paris, 1509-10.
Editions. Fontes has described (Toulouse Mem. (9) VI, 155-
167, for 1894) a quarto of 1510, published by Stephanus of
4[Quehoc volum/ne
continetur.
Liber dejntelk<2u.
JLiberdefenfu,
Liberdenichilo.
ATS oppofitorum*
Liber dcgeneratione.
Liber de fapiente.
Liber de duodecim num en's
Epflolecomplwes.
ITlnfupmathfmatfcuopus quadr^
Mathcmatic/s Rods |fDe Geometricis Corporibus
d[De Gcometrids Supplementi's
FIG. 50. TITLE PAGE OF BOVILLUS
PRINTED BOOKS 91
Paris, entitled, ' Caroli Bovilli liber de numeris perfectis.' This
book is, however, without title page, and is of course simply ff .
172-180 of this work. I know of no other editions.
The first part on numbers, the Liber de duodecim numeris,' begins
on f. 148, v., and ends on f. 171, r., with the colophon : ' Libri dvodecim
nvmerorvm finis : editi // in domo- R. P. Francifci de Hallewin/ Pon-
tificis Ambi//anenfis. Anno ab autore numerorum Incarnato.// 1 5 1 o :
Maij decimafexta.' It relates solely to the mystery of numbers and to
the Greek theory. The part on perfect numbers, numbers which equal
one plus the sum of their factors, begins on f. 172, r., and ends on f.
1 80, r., with the following colophon : ' C.Liber perfectorvm nvmerorvm
finis.// Perfecto/ trinoq5 deo laudes in gentes. Anno domi-//ni/ 1509 :
lanuarij 4.' The dates of the several parts vary from October 25, 1509,
to January 18, 1509 (1510 N. S.).
Other works 0/1509. Anianus, p. 32, 1488; Anonymous, p. 87,
1507 ; Borghi, p. 16, 1484 ; Ciruelo, p. 60, 1495 '> Lkht, p. 70, 1500 ;
Sacrobosco, p. 32, 1488.
Other works of 1510. Anonymous, p. 87, 1507 ; Anonymous, c.
1510, p. 46, 1491 ; Boethius, p. 30, 1488 ; Bovillus, p. 89, 1509 ; Brad-
wardin, c. 1510, p. 61, 1495 ; Clichtoveus, p. 81, 1503 ; Faber Stapu-
lensis, p. 62, 1496 ; Jordanus, p. 62, 1496 ; Peurbach, p. 53, 1492 ;
Sacrobosco, p. 32, 1488 ; Stromer, p. 83, 1504 ; Paxi, c. 1510, p. 79,
1503 ; Torrentini, p. 76, 1501.
Works of 1511. Anianus, p. 32, 1488; Boethius, p. 27, 1488;
Maffei (Maphseus), p. 86, 1506 ; Peurbach, p. 53, 1492 ; Simon Eisen-
mann, Enchiridion arithmetical Leipzig, fol.
JUAN DE ORTEGA. Ed. pr. 1512. Rome, 1515.
JOHN DE LORTZE. A Spanish priest of the Dominican order, from
Aragon. He was still living in 1567.
Title. ' Svma // de Arithmetica : Geometria // Pratica vtilifTi-
ma: ordina//ta per Johane de Or//tega Spagnolo//Palentino.//
Cum Priuilegio.' (Surrounded by an elaborate woodcut. F. I, r.)
Colophon. ' Impreffo in Roma per Maftro Stephano Guilleri
de Lorena // anno del noftro Signore 1515 adi 10 de Nouebre
Regnante Leo//ne Papadecimo in fuo Annotertio.' (F. 1 16, r.)
Description. Fol., 20.6 X 30.2 cm., the text being 13.1 X 22.7
cm. 2 ff. unnumb. + 114 numb. = 116 ff., 32-38 11. Rome,
1515.
KARA ARITHMETICA
^nelfequenre tractate fedc
~ moftraracomo feaoafa
reogm barattocoiig
tempo co mo fen 5a
tcpo cbc prcne
a lartc mcrca
tilepoiuer
ft modi*
FIG. 51. FROM THE 1515 ORTEGA, INTRODUCTION TO BARTER
PRINTED BOOKS 93
Editions. Barcelona, 1512; Lyons, 1512, 4 ; ib., 1515, 4 ;
Rome, 1515, fol. (here described); Messina, 1522; 1534;
Seville, 1536; ib., 1537; Paris, 1540 (?) ; Seville, 1542, 4
(see below); ib., 1552 (p. 94); s. 1. (?), 1552; Granada, 1563,
4. The Lyons edition of 1512 was the first book on commer-
cial arithmetic printed in France. It differs somewhat from
the Rome edition of 1515, but the latter differs only a little
from the first (Barcelona) edition. Mr. Plimpton's copy of 1515
belonged to Prince Boncompagni and has his collation on the
cover. It is beautifully printed and is one of the best examples
of the early Italian mathematical typography.
This is one of the most celebrated arithmetics written in Spain in
the sixteenth century. It is a purely commercial book, beginning with
notation, taking up the four processes with integers, the progressions,
the roots, and the checks on operations, and the same operations in
the same order with fractions, and then discussing the business rules.
These last include exchange, rule of three, profit and loss, partnership,
testament problems, barter, alloys, false position, and a little mensuration.
Other works of 1512. Bradwardin, p. 61, 1495 ; Peurbach, p. 53,
1492 ; Reisch, p. 82, 1503 ; Stromer, p. 83, 1504; Torrentini, p. 76,
1501 ; I. Furst, ' Novus . . . algorithmus.'
JUAN DE ORTEGA. Ed. pr. 1512. Seville, 1542.
See p. 91.
Title. ' Tratado // fubtiliffimo de Arifmeti-//ca y de Geome-
tria : co-//puefto y ordenado // por el reuerendo // padre fray Jua
// de Ortega d' //la orden d' // los pre//dicadores.// 1542 //
1234567890.' (The whole is surrounded by an elaborate wood-
cut border.) (F. I, r.)
Colophon. ' Sue impreffo el prefente libro // re Arifmetica y
Geometria (agora nueuamete // corregido y emendado) en cafa
d' Jacom // croberger: enla muy noble y muy leal // ciudad de
Seuillara cinco dias//de deziembre de. M.d. y//quareta y dos
afioz.' (F. 232, v.)
Description. 4, 14.5 x 20.5 cm., the text being 11.3 X 17.2
cm. 232 ff. numb., 34 11. Seville, 1542.
See above.
94 RARA ARITHMETICA
JUAN DE ORTEGA. Ed. pr. 1512. Seville, 1552.
See p. 91.
Title. * Tractado // Subtiliflimo d'Arifmetica y de Geome//
tria. Compuefto por el reueredo padre // fray Juan de Hortega,
d'la orden // de los predicadores.// Elhora de nueuo emendado
con mucha // diligetia por Go^alo Bufto d'muchos//errores
que auia en algunas im//preffiones paffadas.// C.Van anadidas
en efta impreffion las // prueuas defde reduzir hafta partir que-
//brados. Y en las mas de las figuras de // geometria fus prue-
uas, con ciert os aui-//fos fubjetos al Algebra. Y al fin defte tra
//ctado : 13. exemplos de arte mayor.// 1552.' (Title page is
printed in red, and is surrounded by an elaborate woodcut in
black.) (F. i,r.)
1 Colophon. ' Hizo fin el tractado de Arifmetica Y // Geome-
tria, que compufo y ordeno el reuerendo padre //fray Juan de
Hortega, de la orden de los predica-//dores. Fue impreffo ela
muy noble i muy leal //ciudad de Seuilla por Jua canalla, enla //
collacion de fant Jua. Acabofe a diez//y feys dias del mes de
Abril del // ano de nueftro criador y rede//ptor Jefu Christo de
mill Hi quinientos i cin-//quenta y dos //anos.' (F. 223, r.)
Description. 8, 14.6 X 20 cm., the text being 1 1.9 x 17.4 cm.
232 ff. numb. + 7 unnumb. = 239 ff., 33 11. Seville, 1552.
See p. 93. The ' arte mayor ' mentioned in the title is algebra.
JODOCUS CLICHTOVEUS. Ed. pr. 1513. Paris, 1513.
See p. 80.
Title. See Fig. 52.
Colophon. ' C.Expletum eft hoc opufculum & ex officina emif-
//fum/ in alma Parifiorum academia : anc domi-//ni (qui omnia
numero definiuit) decimoter//tio fupra millefimu & quingente-
fimu/ // decimafexta die Decembris. Per // Henricu ftephanu/
artis excu//fori libroru fedulu & indu//ftriu opifice/ e regione
//fchole Decretoru //habitan-//tem.' (F. 43, v.)
Description. 4, 14 x 19.5 cm., the text being 12 x 15.7 cm.
41 ff . numb. 4- 3 unnumb. + 2 blank = 46 ff., 42 11. Paris, 1513.
PRINTED BOOKS 95
Editions. There was no other edition.
This is, I believe, the first separate treatise on the mystery of numbers
to appear in print. Paciuolo had included a good deal of such material
in his Summa of 1494, and about a century later Bungus published a
monumental treatise upon the subject, but Clichtoveus was a pioneer
in the publication of a separate work. The result of his labors is properly
|[DcmyflicanumcrorumGgnificationc opu*
fculutnteorum pr gfcrtim qui in facris lit*
ten's vfi tati habentur/fpirituale ipfo*
rum defignationemfiiccina
&e elucidans,
V
FIG. 52. TITLE PAGE OF THE 1513 CLICHTOVEUS
included in a list of arithmetics, for, while there is nothing of compu-
tation in the work, it is not unrelated to the number theories of the
mediaeval writers and even of the Pythagoreans.
Clichtoveus discusses, as is usual among such writers, the religious
significance of one (' Quid vnitas/ numerorum fons et origio defignat.
Cap. I.') and the numbers of the first decade. He also mentions several
larger numbers which were supposed to have some scriptural significance,
not forgetting, of course, 666, * the number of the beast.'
There is also in this work a chapter, generally unrecognized by writers
on the history of the subject, on finger-reckoning : ' Quomodo antiqui:
numeros omnes per certas digitoru & manuum figurationes/ fignificare
funt foliti.' Cap. XXVIII.
JOANNES MARTINUS BLASIUS, Villagarciensis.
Ed. pr. 1513. Paris, 1513.
A Spanish astrologer and arithmetician of c. 1 500. In this edition the
author's name appears as 'loannes Martinus Blafius diocefis Pacefis,' and
in the 1519 edition as 'loannes Martinus Silecevs (and Sciliceus) Diocesis
Pacefis.'
Title. See Fig. 53.
Colophon. 'Explicit liber Arithme//tices practice magri Joan-
nis Martini Blafij Vil-//lagarciefis : Parifijs edit9 in honeftiffima
Belua-//coru paleftra : impreffus vero a calcographorum ex-//
pertiffimo Thoma Kees : Vvefalienfe expenfis pro//biffimorum
9 6
KARA ARITHMETICA
iaronbus aDmoDum ftttlis,
FIG. 53. TITLE PAGE OF THE 1513 BLASIUS
PRINTED BOOKS 97
virorum : Joannis Parui et Joannis // Lambert. Anno domini.
1513. in vigilia diui Jo-//annis baptifte.' (F. 26, r.)
Description. Fol., 19.5 X 28.1 cm., printed in double columns,
each being 6.5 X 21.3 cm. 26ff. unnumb., 64 66 11. Paris, 1513.
Editions. Paris, 1513, fol. (here described); ib., 1514, large
8; ib., 1519 (see below); ib., 1526, fol.
Although an algorism, the work is mediaeval in character. The author
first discusses the fundamental operations with integers, including series
and roots as was the custom, but not considering duplation and media-
tion as distinct topics. He is one of the earliest writers to adopt the
spelling substractio, for subtraction, a custom more or less followed by
the Dutch and English arithmeticians for several generations. Blasius
closes his ' primus tractatus ' with a discussion of compound numbers.
' Tractatus secundus ' considers computations with counters, or ' nummi
supputatorii.' The ' tertius tractatus ' is devoted to common fractions,
' fractiones vulgares ' ; the ' quartus tractatus ' to sexagesimal fractions,
' fractiones phisicae ' ; and the ' quintus tractatus ' to the rule of three
(' Prima regula i fudametalis qua detri dicunt.') There are no practical
applications of any value.
This first edition differs greatly from the third (1519) described
below. It includes only the algorism, while the latter consists of two
parts, the first being on Boethian arithmetic, and the second being
substantially identical with the 1513 edition.
Other works of 1513. Albert of Saxony, p. 9, c. 1478; Ciruelo,
p. 60, 1495 ; Gyraldus, p. 254, 1553 ; Licht, p. 70, 1500 ; Peurbach, p.
53, 1492 ; Johann Karl von Landshut (Lanzut), 'Algorithmus linealis,'
Cracow, with editions, ib., 1515, 1519, 4 (see p. 83, 1504).
JOANNES MARTINUS BLASIUS, Villagarciensis.
Ed. pr. 1513. Paris, 1519.
See p. 95.
ORONTIUS FINAEUS, editor. See p. 160.
Title. ' Arithmetica // loannis // Martini, Scili-//cei, i n the-
oricen, et praxim // fciffa, nuper ab Orontio Fine, Del//phinate,
fumma diligentia caftigata, Ion-//geq3 caftigatius q prius, ipfo
cura-//te impreffa : omni hominu // condition! perq // vtilis,//
& neceffaria.// Virefcit vulnere virtus.' Surrounded by an elab-
orate woodcut, with the following on four sides : * Emissa ex
98 RARA ARITHMETICA
officina Henrici Stephani, e regione // scholae Decretorvm Com-
morantis,// vbi et vaenalis reperitvr. // Parisiis anno Christi.
1519.') (F. i, r.)
Description. Large 8, 20.3 X 28.1 cm., the text being 13.3 x
20 cm. 64 ff. numb., 52 11. Paris, 1519.
Editions. See p. 97.
The first half of this rare work, not found in the 1513 edition, is
one of the best exponents of the Boethian arithmetic of the time.
Finaeus, the editor, refers to the author's work in these words : ' Hanc
loanes Martinus, Sciliceus, Hifpanus, vir Mathematical peritus, noftra
tepeftate Parifijs edidit.' The author shows a good knowledge of the
ancient writers, mentioning particularly Pythagoras, Nicomachus, Euclid,
Apuleius, and Boethius, together with Jordanus, Faber Stapulensis, and
Clichtoveus. The distinctive superiority of this part of the work lies in
the clearness and arrangement of the illustrations of the various classes
of numbers defined. The theory of numbers ends on f. 24, v., the prac-
tical part beginning on f. 25, v. Among the most noteworthy features
of this half of the work is a * Tabvla mvltiplicationis et divisionis ' with
all products to 50 x 50.
RAGGIUS FLORENTINUS. Ed. pr. 1514. Florence, 1520.
A Florentine mathematician of the fifteenth century.
Title * See
Ctnhocopufculohcccontinentur.
54-
CCiofdfitproportio"eius fpccics Colophon. ' Um-
preffum Floretiae
t. . Bernardum Zuc-
CQuointcl!c<aucompohtfo&diuifToproportfonum , _ ,,
accjpiatur&maleopinantiumconfurationcs chetta // Anno.
M.D.XX. lanua-
CQ. UC ttiaf r minoruc ptoportio diccnda fit rij. X V.' (F. 1 1 , v.)
Description. 4,
CQuiclpa^inquftas&rcmotio I3 ' 4 x 2ai cm '
the text being 9.2
CC'onfutationesargtimentorum calculator^ x 1 5 cm. 1 1 ff.
FIG. 54. TITLE PAGE OF RAGGIUS unnumb., 34 - 3 7ll.
Florence, 1520.
Editions. Florence, 1514, 4; ib., 1520, 4 (here described).
PRINTED BOOKS 9 9
This work consists of a theoretical treatment of proportion. While
partly arithmetical, this treatment relates to the fundamental theory,
and is equally applicable to geometry. The book is dedicated to the
illustrious Giovanni Salviati, uncle of Cosimo I, Grand Duke of Tuscany.
GUILLIELMUS BUDAEUS. Ed. pr. 1514. Florence, 1562.
GUILLAUME BUD. Born at Paris in 1467 ; died at Paris, August 23, 1540.
Son of Jean Bude, grand audiencer of France. He became secretary to
Louis XII, master of requests to Fra^ois I, royal librarian, and ambassa-
dor to Leo X. He was a man of great erudition, and was instrumental in
founding the College de France.
Title. ( Trattato // delle Monete // e Valvta loro,// Ridotte
dal coftume antico, all'vlo mo-//derno, Di M. Guglielmo//Bvdeo.
//Tradotto per M. Giouan Bernardo // Gualandi Florentine.//
In Fiorenza ; // Apresso I Givnti // MDLXII.// Con licenza, &
Priuilegio.' (P. i.)
Colophon. ' In Fiorenza apreffo gli heredi di // Bernardo
Giunti//is62.' (P. 318.)
Description. 8, 10.5 x 17.2 cm., the text being 6.8 X 13.1
cm. 8 pp. imnumb. + 3 blank + 309 numb. = 320 pp., uncut,
28 11. Florence, 1562.
Editions. The dedicatory epistle is dated * Da. Viterbo il xxx.
d'Agofto. MDLXI,' so that this is the first (as it is the only)
edition of Gualandi's translation of Budaeus. I have not com-
pared it with the * Libri de asse et partibus ejus,' Paris, 1514;
second edition, Venice, Aldus, 1522. Although on the title
page it is called a translation, the various books, six in number,
begin 'Trattato delle //Monete// di M. Gio. Bernardo// Gvalandi
Cittadino // Florentine,' leading to the belief that it may have
been rewritten by Gualandi.
This treatise is purely historical, describing in a prolix manner the
ancient measures, a subject of interest to arithmeticians in the sixteenth
century on account of the great number of tables of denominate numbers
in use in Italy, France, and Germany.
There was also a work by Budaeus entitled * Minervse Aragoniae Assis
Budeani supputatio compendiaia ad monetam ponderaque et mensuras
Hispanic nostre,' etc., published at Saragossa in 1536, 8-
ioo KARA ARITHMETICA
JOHANN BOSCHENSTEYN.
Ed. pr. 1514. Augsburg, 1514.
BESCHENSTEIN, BOESCHENSTAIN, BOSSENSTEIN, BOECHSENTEIN, BUCH-
SENSTEIN, POSCHENSTEIN, BESENTiNUS, etc. Born at Esslingen, Swabia, in
1472; died in 1532. He taught Hebrew at the universities of Ingolstadt
and Heidelberg, and also at Antwerp and NUmberg. Luther and Melanch-
thon were among his pupils.
Title. See Fig. 55.
Colophon. 'Getruckt in der Kayferlichen ftat Augfpurg durch
//Erhart 6glin Anno 1514 Jar.' (F. 24, r.)
Description. 4, 14.4 X 19.2 cm., the text being 8.9 x 14.8
cm. 24 ff. unnumb., 30 11. Augsburg, \ 5 14.
Editions. Augsburg, 1514, 4 (here described); ib., 1516;
ib., 1518. Boschensteyn is also said to have published at
Augsburg in 1514 * Ein New geordnet Rechenbiichlein auf den
linien mit Rechenpfennigen,' 4, but this is doubtless Kobel's
work (p. 102).
This is one of the more interesting of the early German arithmetics.
It is mercantile in character, and presents in condensed form the essen-
tials of business arithmetic. Among the peculiarities of the book is the
use of ' figures ' for 'species.' Boschensteyn gives seven of these funda-
mental operations : ' Das scind nun die Siben figuren,' ' Die Erft figur
Numeratio,' * Die Ander figur Additio,' etc. He includes Duplatio and
Mediatio, and he checks all of his work by casting out nines. His appli-
cations are chiefly in the ' Regula de Try,' partnership, and ' Regula
Fufti ' (where he gives his rule in verse).
JAKOB KOBEL. Ed. pr. 1514. Augsburg, 1514.
KOBEL, KOBELIUS, KoBlLiNUS. Born at Heidelberg in 1470; died at
Oppenheim, January 31, 1533. He studied at Cracow, where Copernicus
was his fellow-student. He was a man of varied attainments, meeting with
success as a Rechenmeister, printer, engraver, woodcarver, poet, and public
official.
Title. See Fig. 56.
Colophon. ' Getruckt tzu Augfpurg durch Erhart oglin.// Anno
M.D.XIIIi: (F. XXIIII, r.)
Description. 4, 13.7 x 18.9 cm., the text being 9x15 cm.
6 ff. unnumb. + 24 numb, (in Roman) = 30 ff., 30-35 11. Augs-
burg, 1514.
PRINTED BOOKS
101
&m 1ft eav gcotfwet ift ecb
enbiecbUn
DenangenDdifcbtKern
mw fampt fcer Kegd be Crp/v tt& fecfce regclw &
rejjd joptrnft vfl artbcnt gilteti fr^
gen
FIG. 55. TITLE PAGE OF THE 1514 BOSCHENSTEYN
102 KARA ARITHMETICA
Editions. Kobel's Rechenbuch appeared under such varied
titles and in such different combinations with his other books
that it is difficult to say whether a given edition is a new work
or merely a revision. It will aid the student if he recognizes
in the first place that Kobel wrote three distinct books, (i) the
4 Rechenbuchlein,' (2) 'Mit der Kryden,' (3) the 'Vysierbuch.'
The ' Rechenbuchlein ' first appeared at Augsburg in 1514, 4;
the ' Vysierbuch,' a treatise on gauging, at Oppenheim, s. a.
(1515); and * Mit der Kryden' at Oppenheim in 1520.
When the 1518 edition of the Rechenbuchlein (p. 108) appeared
the title was changed, and a few pages were slightly altered.
The 1531 edition (p. 108), however, shows many changes, cer-
tain chapters being entirely rewritten, and others considerably
expanded. Although bearing a similar title, this might with
some justice be called a different treatise ; and yet it is so mani-
festly a revision of the 1514 work that it may more properly
be classed as a new edition.
The three books were sometimes published as one and some-
times separately. The following list of editions is, therefore,
probably incomplete, and it should be understood that any book
mentioned may have been published with some other one.
Editions of the Rechenbuchlein : Augsburg, 1 5 14, 4 (p. 100) ;
Oppenheim, 1514, 4 (p. 106); ib., s. a. (c. 1515); Augsburg,
1516,4; 1517, 4 ; Oppenheim, 1518, 4 (p. 108; 'zumDritte
male gebeffert,' and hence the third revision, although at least
the sixth edition); two other editions before 1520 (Unger) ;
Oppenheim, 1522, 8; ib., 1525, 12; Frankfort, 1527, 8;
ib., 1531, 8 (p. 108); Oppenheim, 1531 ; ib., 1532 ; ib., 1535 ;
Frankfort, 1 537, 8 (p. no); ib., 1544, 8; ib., 1549; ib., 1564
(p. in); ib., 1573; 1575; Frankfort, 1584, 8.
Editions of ' Mit der Kryde od' Schreibfedern/ durch die zei-
ferzal zu reche // Ein neiiw Rechepiichlein/ den angenden Schu-
lern d' rechnugzu ere getruckt ' : Oppenheim, 1520 ; Frankfort,
!537> 8 (p. no); probably included in various other editions of
the Rechenbuchlein.
PRINTED BOOKS
103
mbiecblitiauf Den lituen
mtr iftecbcit pfenfogcn : oat
clnfev cbchcb3it lemen
mit oren nt>
FIG. 56. TITLE PAGE or KOBEL'S Rechenbiechlin (1514)
1O4
KARA AR1THMETICA
55
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LIX
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75
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77
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LXXVI
LXXVII
LXXVJII
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LXXX
LXXXl
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LXXXIII
.LXXXIIU
85 LXXXV
85LXXXVI
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92
97
98
99
ICO
101
102
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105
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107
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112
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2CCO
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XCI
XCII
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XCIIII
XCV
XCVI
xcvn
XCVIII
XC1X
c
CI
CII
cm
CHH
cv
CVI
CVII
cvni
cix
ex
CXI
CXII
CXIIItc
cc
ccc
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VIP
Vfli
1XG
oco
Jcco
oco
7cco
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I CCCC
ZOOOO
i4co
V*
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IBCCCC
(TD IIIIC
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1600
I<5|2
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vnc
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FIG. 57. A PAGE FROM KOBEL'S Rechenbiechlin (1514)
PRINTED BOOKS 105
bet>eut bif? figur far felbett toy! afoe
I ZtfefTe figur ifl t>it benefit airt (t ertcl Wrt flfoej
fill gflrtsert/fllfo mag mflit fliicfc flirt furtfftail/^$
fe4)ffatl/flw ftbeittflit o&erjwflt fec^fffli'1 2c vttb die
VI 3D$ fern Scd;^ ribtattfcM fern fafcefml 6ey
VIII ad^t aiit g<w8 madpert .
IX ?Dif? ^tgur b ct5 ^ fcjt flttrt new ft &y i! f)ttul dflO fc^fH
XI lyL *tAil/6cr XI flirt 0flrt^ tnfld)crt 4
XX
XXXI ftgC tflil /t)fle feirt
imbrctffigt flirt gflrt^ntfl^ert
li
o
Iei4)tU4) lemert fl!l brSc^/t>ie bit t'n beirtert revert
Commcrt/nue bu bie ver(leert/ flrtfc^retbert vrtrtb flrtf?
ert etlid) Kegdrt/^rflgm/vrtrtb ^mnpel in
^crt ^fllert/wtcmflrt bi
frag frfrmbt/w berert bie crjl sflle gepto^ert ift/ vn&
tie mittel *mb tetfte Qflle gfln^ plcibcrt/ So mufiit bte
bcrt (let iDci- gclcid^cit wiupu bit fctfiQfll fltid)
bert felbert bruct) vrtb gefcbic^t bfld fl
b^ccrjt Vrtrtb Ictp^flle in
wto ^ Orto <rtfo gcfd?ic^(/eo
FIG. 58. A PAGE FROM KOBEL'S Rechenbiechlin (1514)
106 KARA ARITHMETICA
Editions of the ' Vysierbuch ' : Oppenheim, s. a. (1515), 4
(p. 113); Frankfort, 1527; ib., 1531, 8 (p. 108) ; Oppenheim,
1531 ; ib., 1532 ; s. 1., 1584; probably included in various other
editions of the Rechenbuchlein (p. in, 1564).
This is the first edition of this well-known arithmetic. As already
stated, the title was occasionally changed, but the work was essentially
but little altered. It is a purely commercial book, with all of the opera-
tions performed by counters as was still the custom of the time in most
parts of Germany. Kobel treats of the rule of three ('die Gulden Regel,
die von dem Walen de Try genant wirt'), partnership, reduction, inher-
itances, and exchange. The fundamental operations include progres-
sions, and Roman numerals are used except in the section on notation
(Fig. 57). Kobel makes a curious use of the Arabic method of writing
fractions, the terms being written in Roman, as in the case of
mfxx for % <*"*>
Altogether, Kobel was a vigorous writer, and his Gothic style shows him
to have been no more a follower of the Italian arithmeticians than Diirer
and Holbein were of the Italian artists.
Other works 0/1514. Boethius, p. 27, 1488 ; Grammateus, p. 123 ;
Jordanus, p. 65, 1496 ; Stromer, p. 83, 1504 ; Torrentini, p. 76, 1501 ;
Hermannus Buschius, ' Algorithmus linealis Proiectiliu : de itegris perpul-
chris Arithmetice artis regulis : earundemque probationibus claris exor-
natus,' Vienna, 4, 4 ff. (also catalogued as anonymous, and as written
by Johannes Cusanus; see p. 43, c. 1490, for Nicolaus Cusa; see also
p. 86, 1505). There was also published c. 1514 an anonymous work
entitled * Arithmetics practice Tractatus qui dicitur Algorismus, cum
additionibus utiliter adjunctis,' Paris, 4.
JAKOB KOBEL. Ed. pr. 1514. Oppenheim, 1514.
See p. 100.
Title. The title page is missing. Fol. A ij begins : * Dem
Ernueften Dietherichen//Remerer von Wormbsrgenant von Dal-
burgk : // meinen befundern giinftigen lieben Junck-//herzen/
Eubeutich Jacob K6bel/ //dieffer zeit Statfchreiber zu Op-//
penheim/mein willig dinft // allerzeyt bereydt // zu vor.'
Colophon. ' Getruckt zu Oppenheym.// Anno, i c M.CCCCC.
XIIII.' (F. no. XXIIII. See Fig. 59.)
PRINTED BOOKS 107
tocten gelcn/t>n6 tt>$ rm fiber to ey tfbertg bkf br/ 5ae
ilocl?
nm fimfcn $clcn/t>fi n>ae rm ate^an aber mate iiba
ctyetbabet
tx>i5crumb
blabr fc^ictb aud) t>ff/ 60 ^) alfo $wy$ t>olb!ac^c
abcr 6ief iiberbiybat
i.ilbcrtgf/
tiibctt^/ fo tegt vff 6tc luiicn ccx*
ttoc^5oatf5em Qde mtr fUnffttt
iibcc blteben t(? i>rt5 ale mantel? mat ate 5u one
act?r/picpuleim^ernaienoc^ fybm fiberblybm iff/
fo^utaerociflfolrualwe^poj one xy. t>f6ic Uitten
le^m/Bo 5<id alfo t>otnb:ac^r fo rl?u 5 u 5^ 0ale ju
fanwncn An an Sum/T>n5 n>ae 5a rau^ notrt/ ^ouon
joint ale 6t^f^u fanl^unbcrr x>n5 funf^ebcn 6ouort
5 vl^m/tpae ale dan am lef?e u bei-blecb r/ fo md lift fc
e
l?ot/t>n& tj
f cm (^e cfcbeit/un fo id) cmpfm6 mic^ to mir tc^t
nuo gcfc^affr l)abeu/ wd u^ fMfifttgf Uc^ mtr gottce
w5 6cr oc^^elatc t^tlff fc$ merm vtA becm/aud)
atc t^tlff fc$ merm vtA beflcm/aud)
v>rfircit/i>ett>
ftl^cn /pii5 aug fccc ^Infmctttrf \>u6 (Seomccraoffcit
baren.
(S
FIG. 59. LAST PAGE OF KOBEL'S Rechenbiechlin (OPPENHEIM, 1514)
I0 8 RARA ARITHMETICA
Description. 4, 14.1 X 19.6 cm., the text being 9. 2 x 15. 8 cm.
5 ff. unnumb. -I- 24 numb. = 29 ff., 28-34 11. Oppenheim, 1514.
See p. 1 06. It is a curious fact that the first and second editions
should have appeared in the same year at two different places.
JAKOB KOBEL. Ed. pr. 1514. Oppenheim, 1518.
See p. 100.
Title. See Fig. 60.
Colophon. 'Alfo Endet fich Seliglich d' Drit // Truck dif3
Rechenbuchleins/ Zu Oppenheym zufam-//men gefegt/ vnd
volendt vff Dinftag des Hey-//ligen Criig Erhohungs tag.//
Anno dm .1518.' (F. XLVI, r.)
Description. 4, 13.7 X 18.4 cm., the text being 9.2 x 14.8
cm. 4 ff. unnumb. +46 numb. (I to XLVI) = 50 ff., 32-33 11.
Oppenheim, 1518.
See p. 1 06.
JAKOB KOBEL. Ed. pr. 1514. Frankfort, 1531.
See p. 100.
Title. ' Ein new geor-//denet Kiinftlich Rechenbuchlin/
Ja-//cob Kobels/ Stattfchreiber zu Oppen-//heim/ Auff den
Linien vnd Spacien/ mit Rech-//enpfeningen. Den angehn-
den Schulern Rech//nens gantz leichtlich zu lernen. Vnd zu
Kellerei//en Ampten/ Kauffmanfchafften/ vnd Krame-//reien
dienlich vnd brduchlich. Mit vilen erkla-//rungen/ Leren/
Regeln vnd Exempln.//,g@ t3:> Mehr dan vormals ie getruckt/
//gebeffert/ vnd zugefetzt.// Im Jare M. D. XXXI.//Vifir
Buchlin/ //Den Jungen/ angehnden/ // Leijfchen Vifirern gantz
leichtlich zu lernen/ //verftehn vnd Rechnen. Wie man ein //
Vifirrut machen/vnnd damit ein//iedes vafz vifirenn/ folle Zu
//ende difes Rechenb6ch//lins angehenckt.' (F. I, r.)
Colophon. 'C.Getruckt zu Franckenfurt/ am Mein. In Ver//
lag vnnd Gemeynfchaff t des Ernhaffetnn // vnd Furnemen Herrn
Jacoben K6-//bels/ Stattfchreiber zu Oppen-//heim. Bei Chrif-
tian Ege-//nolffen. Im Mertzen/ // Nach der geburt // Chrifti.
//M.D.XXXI. Jar.' (F. CXII, r.)
PRINTED BOOKS
109
new ftecfe
Gp acteit / nut
ffmanfcl?aff t
rccfyc lew? mag* / jum ID: ttte malc
L ebeficrt vn ju
paeer nurarbetr t^ct wnb ffifj
!Da6 cv rut tt>iitb/t>etacl)t x>ewtd?t /
*Dct Kawff &t |5>Bc^ \>ii leg mtt pei'h^^
i>n^ ntercf / tx^ae tc^ yrt t>n&cctt)etg
SDem f cm cr noc^ nttr frctem mSt
FIG. 60. TITLE PAGE OF KOBEL'S RechTepuchlein (1518).
i io RARA ARITHMETICA
Description. 8, 9.7 X 14.9 cm., the text being 6.9 X 11.5 cm.
112 ff. numb, in Roman, 28-29 11. Frankfort, 1531.
Editions. See p. 102.
It is often stated that the ' Jacob's staff ' used by surveyors was first
described by Kobel in this year (1531) and that it received its name in
his honor. The name was old before this time, however, as applied to
some form of surveying instrument, for in the Margarita Philosophica
(1503, Bk. VI, tract. II) is a conversation between a master and his
disciple, in which it is mentioned : ' MAG. baculo que lacob dicunt.
Dis. Qualis is eft baculus?' The master thereupon describes the instru-
ment, and a picture of it is given.
JAKOB KOBEL. Ed. pr. 1514. Frankfort, 1537.
See p. too.
Title. * Zwey rech-//enbuchlin: v ff der // Linien vnd Zipher/
Mit eym angehenck-//ten Vifirbuch/fo verftendtlich fiir//geben/
das iedem hieraufz on//ein lerer wol zulernen.// C.Durch den
Achtbarn vnd wol erfarnen//H. Jacoben Kobel Statfchreiber
//zu Oppenheym.// Franc. Chrift. Egen.' (F. i, r.)
Colophon. ' Ende/Im lar M.D.XXXVII.' (F. numb. 144, r.)
Description. 8, 9.5 X 15 cm., the text being 7.1 X 11.9 cm.
8 ff. unnumb. +9-144 numb. = 144 ff., 30 11. Frankfort, 1537.
Editions. See p. 102.
This is the earliest of Mr. Plimpton's copies containing the three
books written by Kobel, (i) Rechenbuchlein,' (2) 'Mit der Kryde,'
(3) the * Vysierbuch.' The combination of the three in a single volume
formed one of the best books of the time, giving the operations both
with counters and according to algorism. The latter is given in the part
entitled ' Mit der kreiden // odder fchreibfederen/ durch // die ziffer-
zal zu rechen/ Ein new Rechen-//buchlin/ den angehnden fchulern
der//rechnung zu eren getruckt.' (F. 106.) In this work Kobel also
includes the usual business problems of the period and the chapter on
the calendar required by the Church schools. An unusually complete
treatment of gauging is given in the part entitled 'Eyn new Vifir//
Buchlin/ den Leyen/ zu // leichtem vnd begreiflichem verftandt //
verordnet/ Durch H. Jacob //K6bel Stattfchreiber zu // Oppenheym.'
(F. 95, r.) It is much more complete than the 1514 edition of the
' Rechenbuchlein ' (see p. 106), and is substantially the same as the
PRINTED BOOKS
ii i
1531 edition (see p. 108) except that it contains the third part, ' Mit
der Kreiden,' which the latter does not.
The Hindu-Arabic numerals were still considered difficult (' den die
Zifer zal am erften zulernen fchwere,' f. 9, v.), and teachers still felt it
better to begin with the common Roman characters (' wil ich zum erften
die felb Teutfche zal ... hie anzeygen vn erkleren ').
JAKOB KOBEL. Ed.
See p. 100.
Title. ' Rechenbuch/
//Auff Linien vnd Ziffern.
//Mit einem Vifir buchlin/
Klar // vnd verftendtlich
furgeben. // Gerechnet
Buchlin/ auff alle // Wahr
vnd Kauffmanfchafft /
Muntz/ // Gewicht/ Elen/
vnd Mafz/ viler Land //
vnd Stett verglichen.//
Durch H. Jacob Kobel. //
(Woodcut.) Cum Gratia &
Priuilegio.// Franckfurt/
Bei Chr. Egen. Erben //
M. D. LXIIII.' (F. i,r.)
Colophon. ' Getruckt zu
Franckfurt // am Mayn
bey Chriftian E-//genolffs
Erben.// M. D. LXIIII.'
(F. 194, v.)
Description. 8, 9. i x
15.2 cm., the text being
6.3 X 12.2 cm. 12 ff.
unnumb. + 194 numb. =
206 ff., 28-2911. Frank-
fort, 1564. (There were
which this is the sixth.)
pr. 1514.
Frankfort, 1564.
Von vetf tt-tttt frajfon 57
orfrt ft eft/ vims tyro af fo nad^'ompf* /fc
(Tit (a*d;r Juf? f *riieit rcdjncrt/ alics o<ss Di'n'ii
pfenm'njj / f<> fie betiti 0ompt / ftnOet ftc j'm
forblm '
f tCbu/rtfsob^clcrt/ fo fomptDfirjcbm/
il byi'cnbnt fieVmb ctncnpfett*
ni0/ VttMft wolfcyl
i
3 v
FIG. 6 1. FROM KOBEL'S Rechenbuch (1564)
at least eight Frankfort editions, of
ii2 KARA ARITHMETICA
Editions. See p. 102.
The first 165 ff. are the same as in the 1537 edition (see p. 1 10). The
rest of the book is not found in the 1537 edition, or in any other
edition in the Plimpton library. It consists of a description of foreign
and domestic money, with numerous illustrations of coins. This begins
S
bttgctttttt/ w<ctt mi't efoanoe* gen &otit
$^e/t>n&i5qfim'dj?fl8 lt/vn& mo4>t
rfrtwwgtt^t mcbr t>ttmjebtndlettge>
nbcym t)cit (ConQ vonCcc-
bec/^lfo vat 0on ^cynrtc^ Conceit 90.
Hun ifloicfr^cj //it tttt'e Vtt trt^cn Cdt$
xnt> t>tc 5tx?cn jttfamen f o inincn feint? f
5Dt'c V>nt> Derr;Icid)cn fi'itc;rj?it|ru a((b cnfc
fd? cidcit. 3imt crfrcn Ic0 Me mciIcn/Mc Sort
^ann^fc meiln/t>tc Convtit emcn w0 iiKb*
^Hl
FIG. 62. FROM KOBEL'S Rechenbuch (1564)
(f. 165, r.): 'Von Frembden vnd Hie//landifchen Muntzen/ So difer zeit
in//Teutfch vnnd Welfchen landen/inn aller // Kauffmanfchaf? t vnd
Gewerb/ Hindeln/ //viler Land art im brauch/ geng/ gibig oder // ver-
ruffen Muntzen/ . . .' The book closes (f. 184, v.) with a set of tables
and (f. 192, v.) a ' Regifter.'
PRINTED BOOKS 113
JAKOB KOBEL. Ed. pr. 1515. Oppenheim, s. a. (1515).
See p. 100.
Title. ' Eyn New geordet // Vyfirbuch. Helt yfi.//Wie man
vff eins yden Lands Eych // vn Mafz/ ein gerecht Vyfirfit mache
FIG. 63. LAST PAGE OF KOBEL'S Vysirbhch (1515)
// vn do mit ein ygklich onbekant Vafz // vyfieren/ auch feynen
inhalt erlernen // folle. Den anhebenden Schulern Vi//firens
u 4 RARA ARITHMETICA
Leichtlich/ mit Figuren vnnd//Exempeln/zu lernen/angezeigt.
//Angehengt Tafeln.//Die Erften Fyer halten yn gerechet/
//fo eyn Fikter weins kaufft wirt/ vmb // Guide zu XXVI. Od'
XXIIII. AT5. jc.// Was die Ome/das Fyrtel/ vnnd die Mafz
gelten.//Die Andern Tafeln/Zeyge an/ Ver//anderug vn wech-
felug einer Muntz//durch die ander/als p. in Ib. 2c. //Ge-
druckt zu Oppenheym.' (F. i, r.)
Colophon. 'Gedruckt zu Oppenheym.' (F. 32, v. See Fig. 63.)
Description. 4, 14.4 X 19.7 cm., the text being 9.4 x 14.5
cm. 4 ff. unnumb. + 28 numb. = 32 ff., 30-32 11. Oppenheim,
s. a. (1515). There is no date on the title page nor in the colo-
phon, but the prefatory statement closes with the words, * Vol-
nendet vff dornftag noch Letare. Anno & c. 1515.'
Editions. See p. 106.
The work is semiarithmetical, quite as much so as the chapters on
mensuration in our textbooks ; chapters, indeed, which owe their origin
in no small degree to these treatises on gauging so often appended to
the old arithmetics. This work is illustrated with quaint woodcuts show-
ing the use of the ' Vyfirftab ' or gauging measure. (See Fig. 63.)
GIROLAMO AND GIANNANTONIO TAGLIENTE.
Ed. pr. 1515. s. 1. (Venice), 1525.
Venetian arithmeticians of c. 1500.
Title. ' Opera che//insegna//A fare ogni Ragione//de Mer-
catia // et a pertegare le Terre // Con arte giometrical // Intito-
lata Componimeto // di arithmetica // Con grati a & preuilegio //
M. D. XXV.' (F. i, r.)
Description. 8, 10.3 X 15.1 cm., the text being 7.5 x 12.5
cm. 91 ff., 28-33 H- S. 1. (Venice), 1525.
Editions. Venice, 1515, 8 (De Morgan having erred in say-
ing 'apparently before 1500'); ib., 1520; ib., 1523, 8; ib. (s. 1.),
1525, 8 (here described); ib., 1526; ib., 1527; ib., 1528, 4;
s. a. (1530?), 8; Venice, 1541 ; Milan, 1541, 8 (p. 115) ; s. 1.,
1547, 8; Milan, 1548; Venice, 1548 (with probably a second
Venetian edition, 1548, under the title 'Thesoro universale de
abacho,' by ' lucha atonio de Uberti,' 8);ib., 1550; ib., 1554; ib.,
PRINTED BOOKS 115
1557; ib., 1561; ib., 1564; ib., 1567; ib., 1570; Milan, 1570;
ib., 1576; ib., 1579; ib., 1586. Riccardi also mentions eleven
other editions, s.a., and four such appeared in the Boncompagni
sale, and four in the Fisher sale of 1906. The work also ap-
peared, and is frequently catalogued, without the authors' names.
These various editions have been the object of critical study
by E. A. Cicogna and Prince Boncompagni. The former set
forth his results in his Saggio di bibliografia Venesiana, Venice,
1847, p. 218, ascribing the work to Girolamo Tagliente 'con
1'ajuto del suo consanguineo Giannantonio Tagliente.' In the
edition of 1525, here described, only the former name appears,
the text beginning, 'C.A1 benigno lettore//Hieronymo Tagliente.'
Boncompagni's investigations, setting forth the differences in the
various editions, appeared in the Atti delf Accademia Pontif. de
Nuovi Lincei, XVI, 139, 147, 155, 304. See also Riccardi, I,
2, 484, and Boncompagni's Bulletino^ XIII, 247.
There was also a treatise published by the Taglientes entitled
' Regole di mercatura intitolato componimento di arithmetica,'
Venice, 1524, 8, probably another edition of this work. See
also the treatise on bookkeeping mentioned on p. 141, 1525.
The book opens with a brief treatment of notation and finger symbols.
Then follow in order the multiplication table, the proof of sevens, various
methods of multiplication, division by the galley method, addition chiefly
of denominate numbers, subtraction, the operations with fractions in the
same order, exchange, rule of three, and applied problems. There are
numerous interesting woodcuts, and such familiar problems as those of
the couriers, the testament, and the sale of eggs are given with illustra-
tions. In spite of the arrangement of topics, there were few textbooks
so influential as this in shaping the subsequent teaching of arithmetic.
GIROLAMO AND GIANNANTONIO TAGLIENTE.
Ed. pr. 1515. Milan, 1541.
See p. 114.
Title. ' Libro // dabaco che in//segna a fare // ogni ragione
mercadantile, & // pertegare le terre co 1'arte di //la Geometria,
e altre no//bilifsime raginoe ftra-//ordinarie co la Ta-//riffa come
n6 RARA ARITHMETICA
refpon//deno li pefi & // Monede de molte terre del mon-//do con
la inclita citta di Vene-//gia. Elquel Libro fe chiama //Thefauro
vniuerfale.' (F. I, r.)
Colophon. ' Stampato in Milano per lo. Antonio da Borgho.//
Nell' anno del. M. D. XLI.' (F. 80, v.)
Description. 8, 10 X 14.6 cm., the text being 7.7 X 12.3 cm.
80 ff. unnumb. + 7 blank = 87 ff., 23-30 11. Milan, 1541.
Editions. Seep. 114.
FIG. 64. FROM THE 1541 TAGLIENTE
This differs but little from the 1525 edition, the ' Opera che insegna'
(see p. 114), except in having a set of tariff tables at the end: * C.Qui
comenza el terzo Libro di la fruttifera opera // chiamata la Tariffa'
(f. K iiii, v.). There is some slight change in the phraseology, particu-
larly at the beginning of the various sections. For two curious forms of
multiplication see Fig. 64. Such arrangements of the work in multipli-
cation were quite common, particularly in the early Spanish and Italian
arithmetics of the first half of the sixteenth century. That they should
have found place in a popular mercantile treatise is, however, rather
surprising.
PRINTED BOOKS 117
VARIOUS AUTHORS. Ed. pr. 1515. Vienna, 1515.
JOANNES DE MURIS (JEAN DE MEURS, MURS, MURIA)- was born in
Normandy, c. 1310; died after 1360. He wrote on arithmetic, astronomy,
and music.
THOMAS BRADWARDIN. See p. 61.
NICOLAUS HOREM (NICOLAS ORESME) was born at Caen (?) c. 1323;
died at Lisieux, July n, 1382. He taught in the College de Navarre at
Paris, and in 1377 became Bishop of Lisieux. He wrote also an ' Algoris-
mus Proportionum,' in which the idea of fractional exponents first appears.
GEORG VON PEURBACH. See p. 53.
JOANNES DE GMUNDEN (JOHANN VON GMUNDEN, JOHANN WISSBIER ?
NYDEN ? SCHINDEL? JOHANNES DE GAMUNDIA) was born c. 1380, at
Gmunden on the Traunsee, or Gemiind in Lower Austria, or Gemiind in
Swabia; died at Vienna, February 23, 1442. He was educated at Vienna,
and taught there, being the first professor of mathematics alone in Austria.
Title. See Fig. 65.
Colophon. ' Impreffum Vienne per Joannem Singrenium //
Expenfis vero Leonardi i Luce Alantfe // f ratrum Anno domini
.M.ccccc.xv.//Decimono die Maij.' (F. 54, r.)
Description. 4, 14.3 X 18.4 cm., the text being 9.7 X 15.5
cm. (without the marginal references). 54 ff. unnumb., 26 11.
Vienna, 1515.
Editions. There is no other edition of this combination of
works. See pp. 53, 61, 118, for the individual treatises.
This interesting work consists of five parts. The first is the arith-
metic of Joannes de Muris. While it is called an extract from the
arithmetic of Boethius, it is merely suggested by that treatise and is
really the work of Muris. This part of the work begins on f. 2 with
the following title : ' Incipit Arithmetica comunis ex // diui Seuerini
Boetij Arithmetica per M. Joannem // de muris compendiofe excerpta.
// Prohemium.' De Morgan (p. 3) mentions a possibly earlier edition,
s. 1. a., 4. (See also p. 86, 1505.)
The second part of the volume begins on f. 17, v., and is the work
on proportion by Thomas Bradwardin. It is a theoretical treatment of
the subject, and has the following title : ' Tractatus breuis proportion u :
ab-//breuiatus ex libro de Proportionibus. D. Thome // Braguardini
Anglici.'
The third part begins on f. 27, v., and is a treatise by Nicolaus
Horem, with the following title : * Tractatus de Latitudinibus forma-//
rum fcdm doctrina magiftri Nicolai Horem.' This subject attracted
considerable attention in the latter part of the Middle Ages.
u8 RARA ARITHMETICA
The fourth part is the algorism of Peurbach, and has the following
title on f. 37, v. : ' Opufculu Magiftri Georgij //Peurbachij doctifs.' As
already stated (p. 53), this work takes up the four fundamental opera-
tions and progressions, giving merely a theoretical discussion of the
subject.
The fifth part (f. 44, v.) begins : ' Incipit tractatus de Minucijs
phi-//ficis : compofitus Vienne Auftrie per magiftrum //Joannem de
Gmunden.' This is the treatise of Gmunden on sexagesimal fractions,
or, as they were called in the Middle Ages, physical fractions. These
fractions, still used by us in our degrees (or hours), minutes, and seconds,
served the purposes of the later decimal fractions. They were carried
much farther than is now the case, a number like 3 15' 40" 15'" 45 Iv
meaning merely 3 + + + + 4 , or 3 The present
symbolism (, ', ") is relatively modern.
The book is particularly interesting because it combines in one
volume five well-known books by mediaeval writers. In no other
Stttbrncttca coiftitiunfe*
pzopomonca bmics,
>claritiK>imbi)9
ocmiiiucgo pbiftcfe.
FIG. 65. TITLE PAGE OF THE VIENNA WORK OF 1515
single volume could the inadequacy of the mediaeval treatment of
mathematics be better seen. Indeed, a manuscript of 1515, found by
Gerhardt in the Wolfenbuttler Bibliothek, expressly states that the
lectures on arithmetic given in the universities of that period were
based on the above works of Muris, Bradwardin, Peurbach, and Joannes
de Gmunden. (Monatsberichte der K. P. Akad. d. Wissensch. zu Berlin,
1867, p. 43.)
PRINTED BOOKS 119
JOANNES DE MURIS. Ed. pr. 1515. Mainz, 1538.
See p. 117.
Title. <Arithme//ticae specvlativae // Libri duo loannis de
Muris ab in-//numeris erroribus quibus hacte-//nus corrupt!, &
uetuftate fer//me perierant diligen-//ter emendati,// Pvlcherri-
mis qvoqve exemplis, Formisq; nouis declarati & in // ufum
ftudiofae iuuentutis Mogun-//tinae iam reccens ex-//cufi.// Mo-
gvntiae excvdebat // Ivo Scoffer anno.//M. D. XXXVIII.'
(F. i, r.)
Colophon. ' Mogvntiae excvdebat // Ivo Scheffer anno.//
M. D. XXXVIII.' (F. 90, v.)
Description. 8, 9.6 X 14.5 cm., the text being 6.6 X 1 1.9 cm.
4 pp. unnumb. + 3-88 numb. + 5 blank -f- I with woodcut = 96
pp. Mainz, 1538.
Editions. This is the second dated edition of the arithmetic
of this popular mediaeval teacher. (See p. 1 17.) It is more com-
plete than the one of 1515, but it does not, like the latter, give
the marginal references to Boethius, upon which it is so largely
based. (See also Boethius, p. 27.)
See p. 117.
JOANNES FCENISECA. Ed. pr. 1515. Augsburg, 1515.
An Augsburg teacher of c. 1500.
Title. See Fig. 66.
Colophon. * Impreffa Augufte Vindelicorum/ communibus im-
penfis Io/ //annis Miller atq3 loannis fcenifece. Anno a natiuita
//te domini. M.D. XV. ad. IIII. Cal.' Maias.' (F. 20, r.)
Description. 4, 15.1 X 21 cm., the text being n X 13.8 cm.
20 ff. unnumb., 7-39 11. Augsburg, 1515.
Editions. There was no other edition.
This is an extract from a larger volume, for the folios have been
numbered by hand 40-59, and the register begins with 'aa i.' Only
two pages (aa ii, v., and aa iii, r.) are devoted to ' Arithmetica,' and these
relate only to the Boethian system. The rest of the book is devoted
chiefly to geometric figures, the mediaeval astronomy, and music. Such
120
RARA ARITHMETICA
a book shows the superficiality and general emptiness of the work of
the schools that were supposed to stand for culture in the period of the
early Renaissance.
Opera loannis Fceni&c* Augn.
htciniehabent.
Quadratum iapfenri^rconrinens in fefeptem
a r ccs liberties vettrum*
Circulos biblif .iiii.in quibus mctaphyfica
mofaica.
Commentaria horum.
Ad Hcc/libri rubricai nfmiis
fignati:nece(Tarii fu nu
f^Grammatica
terino Logica
Monaftica
Oeconomica
.i.mos
math cacica
plulofbpliia
(^ Medicina fabaltcrrm
thcologia Metaphyfica
Alcarithmiis fiibalrcrnus
Arirfimctfca
Geomccrta
Pcr(pc<ftjiia fiibalrmia
Mulica
Aftronomia
don.aIex.gua.Ia/ca.Pocri
nouus
bocrius
boecius
pcTUsiacobi
boerfus
boccius cphcmerid
pcolcm^us Hiftonc.)
JfofcoricJes
bibiiUmtripTcx
aai.
FIG. 66. TITLE PAGE OF FCENISECA'S Opera (1515)
ANONYMOUS. Ed. pr. c. 1515. Leipzig, s. a. (c. 1515).
Title. ' Melchiar Lotthervs Ivnior candido lectori salutem.'
(Line i.) ( Articularis Bedae prefbyteri numeroru computatio.'
(Line 24.)
Description. One sheet, 28 x 36.4 cm., the text being 21.6 x
26.7 cm. 42 11. Printed on one side of a single sheet.
PRINTED BOOKS 121
Editions. There was no other edition, and only this copy is
known of this one.
This broadside was published by Letter c. 1515, and is a brief state-
ment of the numerical finger-symbolism of the ancients, particularly
as described by the Venerable Bede. The symbolism is practically the
same as that described by such writers as Paciuolo and Aventinus (see
pp. 57 and 136.)
CASPAR LAX. Ed. pr. 1515. Paris, 1515.
Born at Sarinena, Spain, c. 1487 ; died at Saragossa, February 23, 1560. He
taught at Paris and Saragossa. His only works are the two here described.
Title. See Plate III. ' Proportiones magiftri Gafparis // lax
aragonenfis de farinyena . . .' (F. 101, r. Separately cata-
logued, see below.)
Colophon. ' Explicit Arithmetica fpeculatiua Magiftri Gafparis
Lax Aragonenfis de Sarinyena duode-//cim libris demonftrata.
Impreffa Parifius opera ac characteribus Magiftri Nicolai de la
barre. //Expends honefti viri Hemundi le feure Bibliopole Pari-
fius in vico diui Jacobi fub figno Crefce//tis albi vitam degentis.
Anno Domini .1515. Die vero.i3. Menfis Decembris.' (F. ioo,v.)
Description. Fol., 19.8 x 26.5 cm., the text being 14.1 X 21.8
cm. 100 ff. unnumb., 5961 11. Paris, 1515.
Editions. There was no other edition.
A very prolix treatment of theoretical arithmetic, based on Boethius
and his mediaeval successors. As the title shows, Lax was a Spanish
teacher, one of several from the southern peninsula who taught in the
University of Paris in the fifteenth century. Among the others were
Rollandus (originally from Lisbon, mentioned later in connection with
the manuscripts) and Ciruelo (p. 58). All of the contributions of these
scholars were of this general theoretical character. De Morgan face-
tiously remarks, For anything that appears the author (Lax) could not
count as far as 100.'
CASPAR LAX. Ed. pr. 1515. Paris, 1515.
See above.
Title. ' Proportiones magiftri Gafparis //lax aragonenfis de fa-
rinyena.// Venundatur Parifius In vico diui //Jacobi ab Emundo
le feure fub fi-//gno crefcentis albi vitam degente.' (F. 101, r.)
122 KARA ARITHMETICA
Colophon. ' Expliciunt proportiones Ma//giftri Gafparis Lax
Aragonen//fis de Sarinyena impreffe Pa-//rifius opera Magiftri
Nicolai // de la barre pro Emundo le feure // Anno dfii M. d. xv.
die vo vi. me//fis Octobris.' (F. 26, 126 of the whole book, r.)
Description. Fol., 19.5 X 26.6cm., printed in double columns,
each being 7 x 21.5 cm. 26 ff. unnumb., 66 11. Paris, 1515.
Editions. There was no other edition.
Bound with the ' Arithmetica Speculativa' (p. 121). This is a prolix
treatment of mediaeval ratios after the Boethian manner, and as such it
ranks with works like those of Bradwardin (p. 61), Jordanus (p. 62),
and Faber Stapulensis (p. 82).
Other works 0/1515. Boethius, p. 27, 1488; Bradwardin, p. 61,
1495; Kobel, p. 102, 1514; Lanzut, pp.83, 97, 1504, 1513; Licht,
p. 70, 1500 ; Ortega, p. 93, 1512; Peurbach, p. 53, 1492 ; Torrentini,
p. 76, 1501 ; Juan Andre's, ' Sumario breve de la practica de la arithme-
tica,' Valencia (from the book it appears that it was written in Saragossa
in 1514; it was reprinted at Seville in 1537) ; V. Rodulphus Spoletanus,
* De proportione proportionvm dispvtatio,' Rome, 4.
Works 0/1516. Johann Boschensteyn, p. 100, 1514; Capella, p. 66,
1499; Ciruelo, p. 60, 1495; Kobel, p. 102, 1514; Stromer, p. 83,
1504; Widman, p. 36, c. 1488.
PIETRO MARIA BONINI. Ed. pr. 1517. Florence, 1517.
A Florentine writer of the first half of the sixteenth century.
Title. ' Lvcidario darithmetica.' (Large woodcut. F. 3, r.
Fig. 67.)
Colophon. * Cjmpreffo nella excelfa cipta di Firenze per //
Gianftephano di Carlo da Pauia//adi 7 di Gennaio.' (F. 18, r.)
Description. 8, 10.9 x 15.9 cm., the text being 8 x 12.7 cm.
19 ff. unnumb., 24 11. Florence, 1517.
Editions. There was no other edition. This interesting
volume was known to De Morgan only by hearsay when he
wrote his Arithmetical Books. It came into his possession,
however, after that work was published, as is shown by his
autograph on the title page (see Fig. 67). It is not often men-
tioned by bibliographers, and is one of the rare books of the
century. In the Boncompagni sale (no. 1441) there is mentioned
a fteeulatiuama
ansi
fcefimnpen
oe^^
PLATE III. TITLE PAGE OF LAX
PRINTED BOOKS
123
ffLVCIDARIO DARITHMETICA.
an edition of 1 547 ; but this is a misprint for 1 5 1 7, as appears from
Riccardi, vol. I, col. 1534. Riccardi mentions only three copies
known to him or to Boncompagni,
but there was one in the Fisher sale
(London, 1906).
The book is a small octavo, the first
two-thirds being given to mercantile prob-
lems on exchange and the reduction of
money. The last part treats exclusively of
mensuration : ' Speculationi geometriche
di piu sorte : & prima laquadratura del
triangolo.'
Other works of 1517. Anonymous,
'Algorithmus linealis' (see Widman), p. 36,
c. 1488; Borghi, p. 16, 1484; Feliciano,
p. 146, 1526; Kobel, p. 102, 1514; Reisch,
p. 82, 1503; Sacrobosco, p. 32, 1488; FIG. 67. TITLE PAGE OF
Widman, p. 36, c. 1488. BONINI'S Lvcidario
HENRICUS GRAMMATEUS.
Ed. pr. 1518.
Frankfort, 1535.
HEINRICH SCHREIBER; HENRICUS SCRIPTOR ; Latinized Greek, GRAM-
MATEUS. Born at Erfurt, at least as early as 1496. He describes him-
self as ' Henrich Grammateus // von Erffurt/ der fiben f reien kiinf ten Meyf ter.'
He was a student at Cracow and at the University of Vienna (1507). The
dates of his birth and death are unknown, but a record at Vienna reads :
' Anno domini millesimo quingentesimo septimo in festo sanctorum Tibureii
et Valeriani martirum . . . Henricus Scriptoris de Erfordia.' He also
taught at Vienna.
Title. See Fig. 68.
Description. 8, 9.2 x 15.2 cm., the text being 6.7 X 11.3 cm.
96 ff. unnumb., 31 11. Frankfort, 1535.
Editions. Vienna, 15 18, 8; Frankfort, 1535 (here described);
s. 1. (Frankfort), 1544; Frankfort, 1572.
Grammateus also published an ' Algorithmus proportionum
una cum monochordi generis Dyatonici compositione . . .',
Cracow, 1514, 4; ' Libellus de compositione regularum pro
vasorum mensuratione,' Vienna, 1518; ' Behend unnd khunst-
lich Rechnung nach der Regel und welhisch practic,' Nurnberg,
I24
RARA ARITHMETICA
1521, 8, an extract from the work here described ; ' Algorismus
de integris Regula de tri cum exemplis/ Erfurt, 1523; ' Eyn
(BemeincnKegdnbetre.
ttelfcbcn mactic. Kcgcln falfr.
'
J frioflu
/ 0eIpfci
L ^rcfce
zna I/ B -pd vn&
fllDificr rutcn ju m^^ert Dur6 *><**
t> :a t / pnb ^rtangel/mtt an o e r n Iufti0m
(hic^en ber (0eomerrei.
FIG. 68. TITLE PAGE OF THE 1535 GRAMMATEUS
kurtz newe Rechenn unnd Visyrbuechleynn gemacht durch
Heinricum Schreyber,' Erfurt, 1523.
That the book was written at Vienna in 1518 appears by the dedi-
cation to ' Dem Edlen fiirfichtigen weifen Johan//fen Tfchertte einer
PRINTED BOOKS 125
des Senats zu Wien,' which ends: * Gebe//zu Wi//en in O-//fterreich
im jar//nach der geburt vn-//fers Seligmachers. M. D. XViij.' In the
chapter on bookkeeping is the date 1535, so that probably the work
was revised for this edition.
C&Mtfe,
31&te fern 30 abbiren bie qua witet einea n a*
tnene/ald tfl. mit H: pzi m arm i t prima/fecunba
mit fe cuba/tf rtia mit tertt'a K."X>nb man b:au*
cjjet folcfcetticfcen aid -f- iff mefj:/t>nb /rum
fcer/m welder fein 50 merrfen b:ei Ke cjel.
ODann ein quan titct f>a r an beybcn o if en -f-
ober fo fol man n fol^e quan ti tet abbirn ^ in
50 0efa$t bad $ey d^e n
old 9ptr-f-?
p:u-4-y^T> s
3ffl in ber &bcrn quantitet -]- x?nb fn ber tm
br rn / vnb (- ubertriff t / fo fol bic rnber
quant i te t von ber dbern fubt r a f) r t treirben/vn
$ubemub:trtcn fe^ (-Soabcrbic rnber qua
tiret iff 0roffer/fo fubtra j)ir bie Hn'nern uo ber
0r5(|ern/pfi 5U bem b6 bo bleitenb i(l/fecje
ale
So mberob0cfaRtenqtintitet iptfrr fanbe
unb in ^cr vnbern +/pnb ubertriff t +/
fo |ubfra6tretn0 von bcm anbern/vnbjum ti*
b:igcn fcb:cib 3f? es fiber/bae bie r nbcr qua
tittttibcrrriffl bre Sbern/fojie^eemdvoi) bem
anbcrn/x>nb 50 bf m cr fien fe tje -j- ab
FIG. 69. FROM THE 1535 GRAMMATEUS
The work is for the most part a mercantile arithmetic, the opera-
tions being given according to both the abacists (with counters) and
the algorists (by the Hindu- Arabic numerals), and a chapter on book-
keeping being appended. Grammateus gives, however, some considera-
tion to the theory of numbers, the rules of the Coss (algebra), music,
126 RARA ARITHMETICA
bookkeeping, and gauging. In the treatment of the ' Regula falsi,' or
rule of false position, the signs + and are first found in this connec-
tion (f. Eiij). (See Widman, p. 40, 1489, who uses them for another
purpose, and Vander Hoecke, p. 183, 1537.) Grammateus also uses
these signs in writing algebraic binomials, as shown in Fig. 69. It is
interesting to know that Rudolff (p. 150) learned algebra from Gram-
mateus, as he states in the following words : ' Ich hab von meister
Heinrichen so Grammateus genennt/der Cofs anfengklichen bericht
emphangen. Sag im darumb danck.'
Other works 0/1318. Boschensteyn, p. 100, 1514 ; Calandri, p. 48,
1491; Feliciano, p. 145, 1526; Kobel, p. 102, 1514; Riese, p. 139,
1522 ; Torrentini, p. 76, 1501 ; Perez de Oliva, ' Dialogus in laudem
Arithmetics, ' Paris.
ANONYMOUS. Ed. pr. 1519. Venice, 1519.
Title. See Fig. 70.
Colophon. ' Venetijs in Edibus Petri Liechtenftein // Anno
virginei partus .1519.' (F. 12, v.)
Description. 4, 15.6 X 20.4 cm., the text being 10. 1 X 16.3
cm. 12 ff. (i blank), 33-38 11. Venice, 1519.
fcrcailrononnefundainc/itumpulcbcrrnnunicoB/
tinens, Clerico non minus vrilis $ necelV
farms: cnmaddiriombus quibuft
damnouiterapzeflis*
* *
SSnno. i 5 1 9 .Scncti/sin jdibus
pern Kiccbtenftcin*
FIG. 70. TITLE PAGE OF THE Coputus nouus
Editions. There was no other edition, so far as I know,
although there are several anonymous computi, and some may
be the same as this.
This is one of the rare works setting forth the computus as it was
taught in the Church schools of the Middle Ages. (See p. 7.)
PRINTED BOOKS 127
The verses on the calendar mentioned on p. 33 (Anianus, 1488)
here appear as follows :
'Aprilis: Junius: September atqj Nouember
Hij trigenta habent vnum reliqui fuperaddunt
Februarius vigint infuper octoqj dies.' (F. 9.)
Other works 0/1519. Feliciano, p. 146, 1526 ; Blasius, p. 97, 1514 ;
VVidman, p. 37, 1489.
JOHANNES FRANCISCUS PICUS MIRANDULA.
Ed. pr. 1520. S. 1., 1520.
A nephew of Pico de Mirandola, and biographer of his uncle. He was
murdered in 1 533. Like his uncle he was a savant of reputation.
Title. ' loannis Francisci Pici Mirandvlae domini, et // Con-
cordiae comitis, examen vanitatis do//ctrinae gentivm, et veri-
tatis Chri-//stianae disciplinae,//distinctvm in libros sex, qvorvm
tres // omnem philosophorvm sectam vni-//versim, reliqvi Aris-
toteleam //et Aristoteleis armis // particvlatim im-//pvgnant.//
vbicvnqve avtem Christiana et // asseritvr et celebratvr // dis-
ciplina.' (F. I, r.)
Description. Fol, 20.6 X 30.6 cm., the text being 16.9 X 24 cm.
6 ff . unnumb. + 208 numb. + I blank = 2 1 5 ff . , 44 11. S. 1. , 1 5 20.
Editions. There was no other edition. The dedication (f. 2, v.)
bears date M. D. XX, and the privilege M. D. XIX.
The book hardly deserves place in a list of this kind. It has, how-
ever, been included because of the following brief chapters on the
nature of arithmetic : ' Quod fuper mathematicis artibus arithmetica
& geometria, fuperqj mediis Aftrologia & mufica, gentium philofophi
non conueniunt. Cap. vii ' ; ' De opinione pythagoricoru, & de ratione
et philolai & poftidonii . . . Cap. ix ' ; * Quid aduerfus arithmetica
facultate pyrrhonii difputauerint. Cap. vii ' (of liber III).
ANONYMOUS. S. 1. et a. (c. 1520).
Title. ' Von dem Rachnen auff den Linien.' (Running
headline.)
Thirteen fragments of proof sheets of an unknown German arith-
metic, three duplicates. The date is purely conjectural. The work was
of at least 46 pages, since the folios were numbered and part of f. 23
i 2 8 KARA ARITHMETICA
is among the fragments. The title of the book probably appears in the
running headline above given, although this may be the title of only
part of the work. There was at least one illustration of line reckoning
in the book. The lines are 6.4 cm. in length, but there is no complete
page among the fragments. Several anonymous works have already
been mentioned, with some such title as 'Algorismus linealis,' and
possibly this is one of them. It would probably be possible to identify
it if one should examine the types and compare the fragments with
possible originals.
ESTIENNE DE LA ROCHE, Villefranche.
Ed. pr. 1520. Lyons, 1520.
Born at Lyons, c. 1480.
Title. See Fig. 71.
Colophon. ' C.Cy finift larifmetique de maiftre Eftienne de la
roche diet ville franche natif de Lyon // fus le rofne. Imprimee
par Maiftre guillaume huyon. Pour Conftantin fradin mar-//chant
: libraire du diet Lyon. Et fut acheuee Ian . 1 520. le 2 e . de Juing.'
(F. 234, r.).
Description. Fol., 17.3 X 25.5 cm., the text being 13x21 cm.
I f. blank + 4 ff . unnumb. + 230 numb. = 235 ff., 49 11. Lyons,
1520.
Editions. Lyons, 1520, fol. (here described); ib., 1538, fol.
(seep. 130).
This is the best of the early French arithmetics. Since it is semi-
mercantile in character, it was naturally printed at Lyons, then the com-
mercial center of France, the theoretical books being usually printed at
Paris under the influence of the Sorbonne. De la Roche gives a very
complete treatment of the operations with integers, fractions, and com-
pound numbers, and a large number of business applications. Perhaps
no arithmetic published in France in the sixteenth century gives a more
comprehensive view of the science and art of arithmetic and of the
applications of the subject. Unfortunately, however, de la Roche took
much of his work bodily from a manuscript of his master, Chuquet,
which he had in his possession, and which has since been published.
Other works 0/1520. Kobel, p. 102, 1514 ; Peurbach, p. 53, 1492 ;
Raggius, p. 98, 1514; Stromer, p. 83, 1504; Suiseth, p. io,tc. 1480;
Tagliente, p. 114, 1515 ; Anonymous (Tagliente ?), * Libro de Abaco,'
PRINTED BOOKS
129
fus ie TRof ue oiuifec en oeujc parties pont I A p in t ere tracts
DCS r>p:teres gfecn'80 rt regies oe Ufcicte fcie:come U no*
toe entierae ndbze routaa regletrov5 : )La regie ouiic
faulfepofition^eoeuyfauirespofitifeioappofmon em !
moti5n>e la regie oe mediatid entre le plus et Ie mofeioe la
Kg4eoe!acbofe:etDelaquatiteice < pgrem60et^poi60. ;
CxafecScUtractewla p:actiqu6WceHeapplicqiieeeii fait
ce mono? es:en touted mar cbadifee c6iue wapperie : efpi \
cerie:mercerieetentoute0aultre0inarcbadifequifevewi
dent a mefurc au poi5 ou au nobxtcn copaignies et en ttw
que0:eecb5ge0 etmeritesrenfinoo: et wrgent etenlava^ ]
leur oiceu>:,n arget le rev et en fincarget owe. &* oai&? ]
raufc adages et eflat5.tantt>eloiqueoelarget./gten geo^
metric applicqueeauF are mecWique0c5meau):malTon0
dwpetieroetatousaultreeMongnfeaiart&mefurr.
etmt
FIG. 71. TITLE PAGE OF THE FIRST EDITION OF DE LA ROCHE
1 30 KARA ARITHMETICA
Venice, 8; Anonymous, 'Libretto de Abaco,' s. 1., 8; Anonymous,
4 Algorismus novus de integris, de minutiis vulgaribus, de minutiis
physicis,' Augsburg. To an edition of Sacrobosco's Sphere the com-
mentator, Johannes Guyion, prefixed a treatise on arithmetic, 'De
quantitate discreta, 1 Avignon, s. a., c. 1520. There was published,
s. 1. a., possibly at Lyons in this year, ' Le liure des gectz grande-
ment profitable pour messeigneurs les marchans et aultres,' 4, a work
on counter reckoning (see p. 7).
ESTIENNE DE LA ROCHE, Villefranche.
Ed. pr. 1520. Lyons, 1538.
See p. 128.
Title. ' Larifmetique & Geometrie de maiftre // Eftienne de
la Roche diet Ville Fran//che, Nouuellement Imprimee & des
faultes corrigee,// a la qvelle font adiouftees les Tables de diuers
comptes, auec leurs Ca-//nons, calculees par Gilles Huguetan
natif de Lyon, Par lefquelles on pourra facil-//lement trouuer les
comptes tous faictz, tant des achatz que uentes de toutes mar-//
chandifes. Et principalement des marchandifes que fe uendent,
ou achetent a la // mefure, come a Laulne, a la Canne, a la Toyfe,
a la Palme, au Pied, & aultres fem-//blables. Au poix, come a la
Liure, au Quintal, au Millier, a la Charge, au Marc,// & a Lonce,
a la Piece, au Nobre, a la Douzaine, a la Groffe, au cent, & au
Millier.// Auec deux Tables feruantz aulx Librayres uendeurs
& acheteurs de papier. En-//femble une Table de defpence, a
fcauoir a tant pour iour, combien on defped Lan // & le Moys,
& a tant le moys, combien reuient Ian & le iour, & a tant pour
an, co-//bien on defpend tous les moys, & a combien reuient
pour chafcun iour.// Davantaige, les Tables du fin dor & dargent,
pour fcauoir (fcelon que le Marc de billon tiendre//daloy, ou de
fin) combien il uauldra de poix de fin or, ou dargent fin.// On
les uend . . . lenfeigne de la Sphaere,//cheulx Gilles & Jacques
Huguetan freres.// 1538.' (F. i, r.)
Colophon. *C.Cy finift Larifmetique i Geometrie de maiftre
Eftienne de la Roche diet Villefranche // Imprime a Lyon par
maiftre Jacques myt Lan. 1538.' (F. 160, r.)
PRINTED BOOKS 131
Description. Fol., 21.1 X 33 cm., the text being 16.3 X 30.6
cm. 2 ff. unnumb. + 158 numb. = 160 ff., 5 9-60 11. Lyons, 1538.
Editions. See p. 128.
The Huguetan referred to is the one mentioned on p. 188.
Ed. pr. 1521. Basel, 1563.
' The Venerable Bede ' was born in England, probably near Wearmouth,
Durham, c. 673, and died in 735. He was the most distinguished scholar of
his time, and his works cover all the branches of learning then known.
Title. ' Opera // Bedae // Venerabi-//lis Presbyte-//ri, Anglo-
saxonis : vi-//ii in divinis atqve hv-//manis literis exercitatifsimi :
omnia in octo to-//mos diftincta,prout ftatim poft Praefa-//tionem
fuo Elencho enu-//merantur.//AdditoRerum & Verborum Indice
// copiofifsimo.// Cum Caefareae Maieftatis gratia & priuile-//gio,
Regisque Galliarum ad //decennium.//Basileae, per loannem //
Heruagium, Anno M. D. LXIII.' (Surrounded by an elaborate
woodcut with inscriptions.) (F. i, r.)
Description. Fol., 24.2 x 38 cm., the text being 17.7 x 28.6
cm., printed in double columns, each 8.5 cm. wide, 61 11. 152
pp. unnumb. + 271 numbered by columns (i.e., 2 numbers to a
page) = 423 pp. in vol. I. 8 vols. bound in 4. Basel, 1563.
Only the first volume, containing the arithmetic, is described here.
Editions. Some of his arithmetical work is said to have been
published in 1521, fol. ; 1 5 2 5 (in part, see p. 140) 51529 (in part,
seep. 159); Paris, 154445 (first edition of the Opera) ; ib.,1554;
Basel, 1563, fol. (here described). See also c. 1515, anonymous.
His ' Historia Ecclesiastica' appeared as early as 1473.
The first volume contains the ' De Arithmeticis nvmeris liber ' (cols.
98-1 1 6), with little save an elaborate multiplication table and a dialogue
on number, names, and symbols ; ' De Arithmeticis proportionibvs ' (cols.
133-146), with the ' Propofitiones ad acuendos iuuenes' often attributed
to Alcuin, but certainly not Baeda's ; 'De ratione calcvli ' (cols. 147-
158), chiefly multiplication tables of Roman money; ' De nvmerorvm
divisione ' (cols. 159-163) ; ' De loqvela per gestvm digitorvm, et tem-
porvm ratione' (cols. 164-181), or, as the headline states it, 'De indigi-
tatione,' giving us almost our only knowledge of the finger reckoning or
symbolism of the Middle Ages in western Europe, and possibly spurious ;
i 3 2 KARA ARITHMETICA
' De ratione vnciarvm ' (cols. 182-184), a treatise on Roman fractions ;
an extensive treatment of the calendar and the computus, with a descrip-
tion of the astrolabe.
FRANCESCO GHALIGAI. Ed. pr. 1521. Florence, 1552.
A Florentine arithmetician of the first part of the sixteenth century. He
died February 10, 1536.
Title. ' Practica // d'Arithmetica. //di // Francesco Ghaligai
// Fiorentino.// Nuouamente Riuifta, & con fomma // Diligenza
Riftampata.// In Firenze // Appreffo i Givnti // M. D. LII.'
(F. i,r.)
Colophon. ' In Firenze // Appreffo i Givnti // M. D. LII.'
(F. 114, r.)
Description. 4, 15.2 x 20.3 cm., the text being 12 x 17.3 cm.
2 ff. unnumb. + 112 numb. = H4ff., 37-38 11. Florence, 1552.
Editions. Florence, 1521, 4 (see Boncompagni's Bulletino,
VII, 486; XIII, 249) ; ib., 1 548, 4; ib., 1 5 52, 4 (here described).
The 1521 edition is entitled ' Summa De Arithmetica,' but it
is the same as the 1552 edition here described. Some bibli-
ographers mention other editions, as of 1540, 1551, 1562, 1572,
1582, 1591, but Boncompagni's careful investigation, supported
by Riccardi's, throws doubt upon all these.
The book is written in the general style of the Italian works of the
sixteenth century, more or less resembling Borghi (p. 16). It was
intended for the use of merchants, and contains a large number of
practical problems showing the conditions of trade at the time of its
publication. Books 10 to 13 relate to algebra, and their chief interest
attaches to the symbolism employed.
Other works of 1521. Boethius, p. 31, 1488 ; Grammateus, p. 124,
1518; Paxi,p. 77, 1503 ; Sacrobosco, p. 32, 1488.
CUTHBERT TONSTALL. Ed. pr. 1522. London, 1522.
TUNSTALL. Born at Hackforth, Yorkshire, in 1474; died November 18,
1559. He was educated at Oxford, Cambridge, and Padua, was a man of
great learning and energy, and held important positions in the Church and
State. He was bishop of London, and later of Durham.
Title. ' De arte svppvtandi // libri qvattvor // Cvtheberti //
Tonstalli.' (Surrounded by a woodcut. F. i, r. See Fig. 72.)
PRINTED BOOKS
133
SVPPVTAND
BRI QVATTVOR
CVTHEBERTI
TONSTALLI.
FIG. 72. TITLE PAGE OF THE FIRST EDITION OF TONSTALL
i 3 4 KARA ARITHMETICA
Colophon. ' Impress. Londini in aedibvs Ri-//chardi Pynsoni.
Anno ver-//bi incarnati .M.D.XXII.// Pridie idvs octo-//t>ris.
Cvm pri-//vilegio a//rege in-//dvl-//to.' (p. 2 O2, r.)
Description. 4, 15.7 X 20.9 cm., the text being 10.5 x 16.4
cm. 202 ff. unnumb., 29 11. London, 1522.
Editions. London, 1522, 4 (here described); Paris, 1529,
4 (p. 135); ib., 1535; ib., 1538, 4 (P- 135); Strasburg, 1543;
ib. f 1544, 8 (p. 136); ib., 1548; ib., 1551.
This is the first edition of the first book wholly on arithmetic that
was printed in England. (But see p. 10, 1480.) In the dedicatory
epistle Tonstall states that in his dealing with certain goldsmiths he
suspected that their accounts were incorrect, and he therefore renewed
his study of arithmetic so as to check their figures. On his appoint-
ment to the See of London he bade farewell to the sciences by printing
this book in order that others might have the benefit of a work which
he had prepared for his own use. The treatise is in Latin, and, although
it was written for the purpose of supplying a practical handbook, is
very prolix and was not suited to the needs of the mercantile class.
It is confessedly based upon Italian models, and it is apparent that
Tonstall must have known, from his residence in Padua and his vari-
ous visits to Italy, the works of the leading Italian writers. The book
includes many business applications of the day, such as partnership,
profit and loss, and exchange. It also includes the rule of false, the rule
of three, and numerous applications of these and other rules. It is, how-
ever, the work of a scholar and a classicist rather than a business man.
The word ' supputandi,' in the title, was not uncommon at that time.
Indeed there was some tendency to use the name ' supputation ' for
arithmetic and to speak of calculations as ' supputations.'
Tonstall dedicates the work to his friend Sir Thomas More, whose
talented daughter Erasmus addressed as ' Margareta Ropera Britanniae
tuse decus,' ornament of thine England. More speaks of Tonstall
in the opening lines of his Utopia : ' I was colleague and companion
to that incomparable man Cuthbert Tonstal, whom the king with such
universal applause lately made Master of the Rolls ; but of whom I will
say nothing ; not because I fear that the testimony of a friend will be
suspected, but rather because his learning and virtues are too great
for me to do them justice, and so well known, that they need not my
commendation unless I would, according to the proverb, " Show the sun
with a lanthorn."' The Utopia was first printed in 1516, so this sono-
rous praise was written some years before Tonstall's arithmetic appeared.
PRINTED BOOKS 135
Some idea of the prolixity of the treatise may be obtained from the
number of closely-printed quarto pages assigned to certain topics. The
chapter * De Numeratione ' fills n pages, ' De Additione ' 14 pages,
' De Svbdvctione ' 15 pages, ' De mvltiplicatione ' 14 pages, ' De parti-
tione' 27 pages (the old galley method being used exclusively), and so
on for the other subjects. Some 66 pages, for example, are given to the
theory of ratio and proportion.
The title page was engraved by Holbein, and was evidently printed
after the book was completed, because in this copy the errata appear
on the reverse of the first folio. The work was printed by Richard
Pynson, the successor to Caxton.
Following the arithmetic is an appendix : Appendix ex Bvdaei libro
de as-//fe excerpta : in qua prifca Latinorum et Grecoru // fupputatio,
ad aeftimationem pecunie, turn Gallicae,// turn Angli-//cae reuocatur.'
Tonstall also published a work in 1518: 'In Lavdem matrimonii
oratio,' second edition in 1519, now very rare.
CUTHBERT TONSTALL. Ed. pr. 1522. Paris, 1529.
See p. 132.
Title. 'De arte svppvtandi libri qvatvor//Cvthberti Tonstalli.
(Picture of a tree from which is falling a broken branch, and the
words: Nolialtum fa//pere, fed time.) Parisiis ex officina Roberti
Stephani//M.D.XXIX.' (P. i.)
Colophon. ' Parisiis //excvdebat Robert vs Stepha-//nvs. Ann.
M.D.XXIX. Prid. id. ivn.' (P. 279.)
Description. 4, 13 X 18.8 cm., the text being 9.4 x 15.9 cm.
271 pp. numb. + 8 unnumb. + 2 blank = 281 pp., 36-38 11. Paris,
1529.
Editions. See p. 134.
The text is the same as in the first edition of 1522. There has
been added, however, a second appendix with the following title :
' Gvlielmi Bvdaei Parisiensis,// secretarii regii, breviari-//vm de asse.'
CUTHBERT TONSTALL. Ed. pr. 1522. Paris, 1538.
See p. 132.
Title. ' De arte svp-//putandi libri qua-//tuor, Cutheberti //
Tonftalli.// (Large woodcut.) Parisiis.// Ex officina Roberti Ste-
phani.//M.D.XXXVIII.' (P. i.)
136 KARA ARITHMETICA
Colophon. * Excvdebat Robertvs Stephanvs Parisiis, // ann.
M. D. XXXVIII.//xvi. cal. novemb.' (P. 259.)
Description. 4, 14.4 X 20.1 cm., the text being 9.5 x 16 cm.
259 pp. numb., 39 11. Paris, 1538.
See p. 134.
CUTHBERT TONSTALL. Ed. pr. 1522. Strasburg, 1544.
See p. 132.
Title. ' De arte //svppvtan//di, libri qvatvor//Cvthberti Ton-
stalli,//hactenus in Germania nus-//quam ita impreffi.// loan.
Stvrmivs.//Arithmeticam Cvthbertvs // Tonftallus prae caeteris
dilucide & pure tradidit : atq3 // ita tradidit, ut ars ipfa dum hie
author extat, con-//tenta fcriptore, doctorem non maximopere
aliquem // requirat. Non nego, poffe ex alijs quoque difci : // fed
hie docet erudite, perfpicue latine, id quod non fa//ciunt caeteri:
nee abest longe a perfectione, qui eius // praecepta intelligit.//
Argentorati, ex offi.// Knobloch. per Georg. Machaerop.' (P. I.)
Colophon. * Argentorati, ex officina//Knoblochiana, per Ge-//
orgivm Machaero-//poevm, mense //febrvario//anno,//M. D.
XLIIII.' (P. 478.)
Description. 8, 9.5 x 15 cm., the text being 6.8 x 11.4 cm.
25 pp. blank + 453 numb. = 478 pp., 26 11. Strasburg, 1544.
This copy is bound with the arithmetic of Victorius Strigelius.
Editions. See p. 134. This is the same as the edition of 1 529,
having the second appendix there mentioned. It is the second
Strasburg edition. It is interesting to see that the classical
influence on the Continent was such that seven out of the eight
editions appeared in Paris or Strasburg.
JOHANNES AVENTINUS.
Ed. pr. 1522. Regensburg, 1532.
THURNMAYER. Born at Abensberg, Bavaria, July 4, 1477 ; died at Regens-
burg, January 9, 1534. He wrote on history.
Title. See Fig. 73.
Colophon. ' Ratifpon^ apud loannem Khol // Anno. MD-
XXXII.' (Large woodcut and date, 'Io. Kol 1532.') (F. 12, r.)
PRINTED BOOKS 137
Description. 4, 14.2 x 19.5 cm., the text being 9.9 x 14 cm.
12 ff. unnumb., 26 11. Regensburg, 1532.
Editions. Niirnberg, 1522; Regensburg, 1532, 4 (here
described).
The book is primarily a treatise on numerical finger symbolism, and
contains the most complete explanation of that subject extant. It gives
QVE VETVSTISSIMA, VETERVM
latinorumperdigitos manufcpnume*
randi(quinetiam loquendi) cofue*
tudo^Exbedacupiduris & ima*
ginibus, inucta reginoburgij
flue ractobonae r ,in biblio#
thcca diui hacmerani,
Atcp hoc conuc*
tu auguftali
Rcucrendi
Atcp do&iflimi Domini Lucac bonfi] de#
cani patauini fccretarfj Rcucrcndifli^
miCardinalis Laurcntrj Campcgf]
zc^ Aufpkijs AIo* Aucn/
tino Edita*
Gcrmania IlluflrancJa*
FIG. 73. TITLE PAGE OF THE 1532 AVENTINUS
illustrations showing the representation of the numbers up to one million
by means of the fingers and arms (see Fig. 74). This finger symbolism
is found in the works of Baeda, it was practical in both the East and
138
KARA ARITHMETICA
West during the Middle Ages, and it is mentioned by several sixteenth-
century arithmeticians. (Compare Fig. 74 with Fig. 35, p. 57.)
r
b jj
FIG. 74. FINGER SYMBOLISM FROM AVENTINUS
ADAM RIESE. Ed. pr. 1522.
Leipzig, 1538.
RYSE, Ris, RIES. Born at Staffelstein, near Bamberg, c. 1489; died at
Annaberg, March 30, 1559. One of the most celebrated Rechenmeisters
of the sixteenth century, and the most influential of all the Germans in
replacing the counter reckoning (' auff der Linien ') by the written computa-
tions (' auff Federn ').
Title. ' Rechnung auff // der Linien vnd Federn //Auff aller-
ley handtirung ge-//macht/ durch Adam Rifen.// (Woodcut of
counting house, with reckoning on the line abacus.) Item auffs
new vberfehen vnd //an viel orten gebeffert.// M.DXXXVIII.'
(F. i,r.)
Colophon. ' Gedruckt zu Leiptzigk // durch Valentinum //
Schumann. 1538.' (F. 63, v.)
Description. 8, 9.7 X 15.1 cm., the text being 7.2 x 1 1.5 cm.
63 ff. unnumb. + 2 blank (with manuscript notes) = 65 ff., 28-
33 11. Leipzig, 1538.
Editions. In order to make clear the different editions of
Riese's work, it is necessary to distinguish between the four
arithmetics which he published. These were as follows :
PRINTED BOOKS 139
1 . * Rechnung auff der linihen gemacht durch Adam Riesen
vonn Staffelsteyn // in massen man es pflegt tzu lern in alien
rechenschulen gruntlich begriffen anno 1518.' (Graesse, followed
by Unger, p. 50, who knew of no extant copy.) A second edi-
tion appeared in 1525 (' Getruckt tzu Erffordt durch Mathes
Maler M. CCCCCxxv Jar/ 8, 43 ff.), and a third in 1527. This
was embodied in his second arithmetic which is here described.
The work is rare.
2. 'Rechnung auff //der Linien vnd Federn/ Riese's best-
known work. The title of the first (i 522) edition was as follows:
' Rechenung auff der linihen vnd federn in zal/mafs/vnd gewicht
auff allerley handierung/ gemacht vnd zusamen gelesen durch
Adam Riesen von Staffelstein Rechenmeister zu Erffurdt im
1522 Jar. Itzt vff sant Annabergk durch in fleyssig vbersehen/
vnd alle gebrechen eygentlich gerechtf ertigt/ vnd zum letzten
eine hiibsche vnderrichtung angehengt.' The following editions
of this work are known to me: Erfurt, 1522; ib., 1525, 8;
Niirnberg, 1527; 1528, 8; Erfurt, 1529, 8; ib., 1530; Leip-
zig* I 533> 8 (first edition containing Helm's Visirbuch ; see p.
142); Frankfort, 1535, 12; Annaberg, 1535, 8 (p. 141); 1536,
12; Leipzig, 1538, 8 (here described); 1541; Frankfort, 1544,
8; Leipzig, 1544; s. L, 1548, 8; Leipzig, 1548,8; ib., 1550;
Breslau, 1550; Frankfort, 1552, 8; Leipzig, 1554; 1556, 12;
Frankfort, 1558, 8 (p. 141); Leipzig, 1562, 12; Frankfort,
1564, 8; Frankfort, 1565, 8 (p. 142); Frankfort (a. Oder ?),
1568, 8; Stettin (Frankfort?), 1570, 8; Leipzig, 1571, 8
(p. 142); Frankfort, 1574; Magdeburg, 1579, 8; Frankfort,
1581, 12; ib., 1585, 8; Leipzig, 1586; Frankfort, 1586;
Wittenberg, 1587; Ntirnberg, 1592, 8; Frankfort, 1592, 8
(p. 143); Leipzig, 1598, 12. There were several editions after
1600. It is possible that some of the editions here mentioned
may be of Riese's fourth book, the titles being much alike and
printers varying them from time to time.
3. ' Ein Gerechent Blichlein,' first published at Leipzig in
1533; second edition in 1536. See p. 171.
i 4 o RARA ARITHMETICA
4. ' Rechnung nach der lenge/auff den Linihen vnd Feder,'
first published in 1550, 4. See p. 250.
Kuckuck's remark, that over twenty-six editions of Riese's
arithmetics appeared, greatly underestimates the number. More
than forty appeared in the sixteenth century alone, and several
were published in the seventeenth century.
This was probably the most popular commercial arithmetic of the
sixteenth century. So firmly did it impress itself upon the schools that
' nach Adam Riese ' is a common expression in Germany to-day, nearly
four hundred years after the first of his books appeared. It was to
Germany what Borghi's book was to Italy and Recorde's to England.
It differed from Riese's first book in that it emphasized computation
by the aid of the Hindu-Arabic numerals instead of the counters.
There is no other book that gives as good a picture of the sixteenth-
century mercantile problems of Germany, and of the methods of
solving them.
Other works of .1522. Albert of Saxony, p. 9, c. 1478 ; Boethius,
p. 27, 1488; Budaeus, p. 99, 1514; Kobel, p. 102, 1514; Ortega, p.
93, 1512 ; Sacrobosco, p. 32, 1488; Francisco Pelacani, 'Arithmetica
prattica,' Florence ; Ludovico Vincento (Vincentino) degl' Arrighi, ' La
operina ... da ... bellissima Ragione di Abbacho,' Rome, 4, with
editions at Venice in 1532, 1533, chiefly on chirography.
Works 0/1523. Borriglione, p. 86, 1506; Ciruelo, p. 60, 1495;
Grammateus, p. 124, 1518; Paciuolo, p. 54, 1494; Reisch, p. 82,
1503; Sacrobosco, p. 35, 1488; Tagliente, p. 114, 1515. Rodrigo Fer-
nandez de Santaella (or Valencia, see p. 269, 1555), ' Ars coputandi,'
Saragossa, fol. There was also written in this year, but published s. 1. a.,
a work by Vincenzo Barziza entitled ' Operetta nouamente composta,'
8, 39 ft 7 ., containing a few mercantile rules and tables.
Works of 1524. Feliciano, p. 145, 1526; Tagliente, p. 115, 1515.
Works 0/1525. Herodianus, p. 60, 1495 ; Kobel, p. 102, 1514;
Riese, p. 139, 1522 ; Rudolff, p. 151, 1526; Tagliente, p. 114, 1515 ;
Bede et al., ' Valetius Probus et Petrus Diaconus de notis Romanorum,
Demettius Alabaldus de minutiis, ponderibus et mensuris, Ven. Beda de
computo per gestum digitorum,' etc., Venice, 4; Angelus Mutinens
(i.e. of Modena), 'Thesavro de Scrittori opera artificiosa le quale con
grandissima arte, si per pratica come per geometria insegna . . .,' s. 1.,
M.D.XXxv, has four folios at the end relating to arithmetic, with the
note 'Angelus Mutinens composuit,' and there seems to have been
an edition in 1525, and another s. a. published at Rome; Giovanni
PRINTED BOOKS 141
Tagliente published two editions of a work chiefly on bookkeeping, 4,
24 ff., Venice, beginning, ' Considerando io loanni Taiente quanto e
necessaria cosa ali nostri magnifici getilhomeni & ad altri mercatanti.'
ADAM RIESE. Ed. pr. 1522. Annaberg, 1535.
See p. 138.
Title. The title page is missing.
Colophon. 'CNach difer vnderrichtung kanftu auffs be//hen-
deft alle Exempel in der Falfi machen : W61//left folch Buch-
lin vnnd kurtze erklerung ietzt/ // welches ich zum andern mal
laffe aufzge-//hen/ zu danck an nemen/ wil ich ver-//dienen/
vnd dir auffs eheft ich // mag die Practica nach al-//lem fleifz
heraufzftrei//chen. Datum //auff fanct // Annaberg/ // Dinftag
nach // Martini. Im Iar//M. D. XXV.' (F. 55, r.) On f. 69,
v., is the following colophon: ' Alfo ift ktirtzlich // befchriben
vnd // begriffen // die Confection // der Vifier riiten mit // Irer
iibung vnd gebrauch.//LEnd:// An. M. D. XXXV.// Im Christ-
monat.' This latter is evidently the date of printing of the entire
book.
Description. 8, 8.8 x 13.2 cm., the text being 6.8 x 1 1.6 cm.
69 ff. unnumb., 31 11. Annaberg, 1535.
Editions. See p. 139.
This contains the ' Vifirbuchlin ' of Erhart Helm, as in the 1533
edition, but it does not give his name. See the 1565 edition (p. 142).
ADAM RIESE. Ed. pr. 1522. Frankfort, 1558.
Seep. 138.
Title. The title page is missing. Page numbered 2 begins as
follows : ' Vorrede in difl Rechen-//h>uch/ Adam Rifen.'
Colophon. 'C.End.//Zu Franckfurt bei Chr. Egeb. erben///
Anno 1558.' (F. numb. 87, r.)
Description. 8, 9.4 X 15 cm., the text being 6.5 X 12.2 cm.
87 ff. numb. + i blank = 88 ff., 31 11. Frankfort, 1558.
Editions. See p. 139.
This contains the * Vifirbuchlin ' of Erhart Helm, as in the 1533
edition.
i 4 2 RARA ARITHMETICA
ADAM RIESE. Ed. pr. 1522. t Frankfort, 1565.
See p. 138.
Title. ' Rechenbuch/ Vff Lini//en vnnd Ziphren/ In allerley
// Handtierung/ Gefchefften vnd Kauff-//mannfchafft. Durch
Adam // Rifen.// Mit new en kunftlichen Regeln vnd Ex-//
emplen gemehrt/ Innhalt furge-//ftelten Regifters.//Vifier vnd
Wechfelruten kunftlich vnd // gerecht zumachen/ autf dem Qua-
drat/ // Durch die Arithmetic vnd Geometri. // Von Erhart
Helm/ Mathema//tico zu Franckfurt/ be // fchriben.// Alles von
newem jetzund widerumb erfe-//hen vnd corrigirt.//( Woodcut of
counting house) Franck, Bei Chr. Egen. Erben. 1565.' (F.i,r.)
Colophon. 'M. D. LXV.' (F. 113, r.)
Description. 8, 9.3 X 15.5 cm., the text being 6.5 X 1 1.9 cm.
105 ff. numb. 4- 8 unnumb. = 1 13 ff., 28 11. Frankfort, 1565.
Editions. See p. 139.
See p. 140. The Visirbuch of Helm begins, with no separate title
page, on f. 7 7, v. : * Vifirbuchlin // Hernach folget der ware // Procefz/
vnnd kurtzeft weg/ wie mann Vifir riilhen machen fol/ aufz dem //
Quadraten/ Auff alle Eich.' The name of the author, 'Erhart Helmen,'
appears in the headlines of each folio recto. This part of the book is
strictly speaking not an arithmetic, but it includes a few explanations of
those processes that are necessary in gauging. It also includes a table
of square roots to the equivalent of three decimal places, and a brief
explanation of roots. It was published in separate form in 1529.
ADAM RIESE. Ed. pr. 1522. Leipzig, 1571.
See p. 138.
Title. ' Rechnung auff // der Linien vnd Federn/ // auff
allerley Handtierung/ //Gemacht durch//Adam Rifen. //(Wood-
cut of Adam Riese, with motto : 'Anno 1550 Adam Ries Seins
Alters Im LVIII.') Auffs newe durchelefen/ vnd // zu recht
bracht.// 1571.' (F. i, r.)
Colophon. 'Zu Leipzig druckts // Hans Rhambaw/ //Im Jar//
1571-' (F. 94, v.)
Description. 8, 9.7 x 1 5.6 cm., the text being 6.9 X 1 1.8 cm.
94 ff. unnumb., 24 11. Leipzig, 1571.
PRINTED BOOKS 143
Editions. See p. 139.
See p. 140. This is substantially identical with the 1538 edition
except as to the title page.
/
ADAM RIESE. Ed. pr. 1522. Frankfort, 1592.
See p. 138.
Title. See Fig. 75.
Description. 8, 9.5 X 15.5 cm., the text being 6. 7 X n.8 cm.
1 f. unnumb. + 79 numb. = 80 ff., 26-27 11. Frankfort, 1592.
Editions. See p. 1 39. Bound with this is Helm's work of 1 592
(described later).
FRANCESCO DAL SOLE. Ed. pr. 1526. Ferrara, 1546.
A French arithmetician, born c. 1490, and living in Ferrara at the time of
writing his books.
Title. See Fig. 76.
Colophon. ' In Ferrara Nella Stampa di .M. Giouanni de bugl-
hat &. M. Antonio // Hucher Compagni, Ad Inftantia de .M.
Rinaldo, cuoco dello Illuftriffi-//rno fignor Duca, nel mefe di
zenaro 1546.' (F. 42, v.)
Description. 4, 14 X 19 cm., the text being 12.1 x 16.5 cm.
2 ff. unnumb. +40 numb. =42 ff., 31 11. Ferrara, 1546.
Editions. Sole published a ' Libretto di Abaco ' in Venice in
1526, 8, and this is merely a revision of that work. A third
edition appeared in 1564 (see p. 146).
This is little more than a primer of arithmetic. It contains the funda-
mental operations, a few of the more important applications, eight pages
of products and roots, and several pages on astrology. The part on astrol-
ogy includes some theory of the calendar, as may be seen by the title :
' Incominciano le regoline daftrologia, p ritrouare ha quati di et minute
fa la luna, la Ira dnicale, et infinite getileffe, Belli circuli, elementi, et
natura, del monde.' The most distinctive feature of the arithmetic is
the combination of number and space concepts. For example, in addi-
tion the author considers not only abstract but compound numbers as
well as geometric magnitudes. (' Regola dellae addition! in generalita,
tan to geometrica, quanto arithmetica. Ca. 6.') The same idea is carried
out in the other fundamental operations.
144
KARA ARITHMETICA
rtu manfcDrtfft / mtt ncnwtn ffinfU
FIG. 75. TITLE PAGE OF THE 1592 RIESE
PRINTED BOOKS
'45
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LIBRETTI NVOVI CON LE RE*
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tionejdella fanajefifle mobile et ir
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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,
dadano dela muy infigney co
ronada Ciudad de
J vVvfvwVVVt
^ Con Priuilcgio de fu Magcflad^
por ticmpo dc diez anos
EN r
En ofa dc loan de Me?, Handrof
FIG. 121. TITLE PAGE OF AUREL
2 5 6
KARA ARITHMETICA
LILII GREGORII
GYRALDI FERRARIEN. SVA*
R V M Q.V ARVNDAM ANNOc
tationum DialogifmiXXX. ad Ants
plifi. Cftrd* Sduiitum.
Item Lturcntij frizzolij Solitncnfls Dialogifinw unicm
de ipflm Lilij uitct cr opmbus.
ApudGutltcrumScottum
M D L I I I.
j
FIG. 122. TITLE PAGE OF THE 1553 GYRALDUS
PRINTED BOOKS 257
This set of dialogues is of interest in the history of mathematics in
that the second and third parts deal with notation and finger symbolism.
' Dialogismvs secvndvs de manus & digitorum nominibus de"q; nume-
randi per eos antiquorum ratione ' (p. 10) is an almost unknown sketch of
finger symbolism. ' Dialogismvs tertivs ad Baptistam Lucarinum FR.
filium optimse fpei ac indolis puerum, de notis & finguris numerorum,
quibus Latini ac Graeci utebantur ' (p. 20) is an equally interesting
sketch of the Greek and Latin numerals. Gyraldus also published a
'Brevis instructio de Grecor^z numerali supputatione ' in 1513, fol.
Other works 0/1552. Agricola, p. 171, 1533; Gemma, p. 203,
1540; Ghaligai, p. 132, 1521; Herman Giilfferich, < Ein new kurtz
Rechenbiichlein,' Frankfort, 8, with editions ib., 1555, 12, 1568, 8
(see also p. 244, 1546, and p. 292, 1559); Ortega, p. 94, 1512 ; Peletier,
p. 245, 1549 ; Recorde, p. 214, c. 1542 ; Riese, p. 139, 1522 ; Joachim
Camerarius, ' Arithmologia,' 1 2 (there was also an edition at Basel, s. a.,
1 6); Dunkel, ' Arithmetical Leipzig, 8; Christopher Falconius, ' Re-
chenbuch,' Konigsberg in Preuften, 4 (Murhard cites as a different
work of the same date ' Rechenbuch auff die Preusche miintz mass und
Gewicht,' ib., 4).
DOMENICO MANZONI. Ed. pr. 1553. S. 1. (Venice), 1553.
A sixteenth-century arithmetician, born at Oderzo.
Title. See Fig. 123.
Description. 8, 10.5 x 15.8 cm., the text being 8.1 x 13. 4 cm.
1 6 ff. unnumb., 26 11. except where arranged in sections. S. 1.
(Venice), 1553.
Editions. There was no other edition of this work. Manzoni
had already published a textbook entitled ' Libretto molto utile
per imparar a leggere, scrivere et Abaco, con alcuni Fondamenti
della Dottrina Christiana,' Venice, 1546, 8, 31 pp. of which are
devoted to arithmetic. He also published in 1553 a more elabo-
rate treatise, < La Brieve Risolvtione di Aritmetica universale
in qualsi voglia negotio, doue interuenga numero, peso, & misura,'
Venice, 1553, 8, 246 ff.
A beautifully printed little manual of the fundamental operations of
arithmetic. The Abbreviatvre delle monete, Pefi, & mifure, chi fi ufano
in Vinegia' (f. 2) is helpful to students of the history of sixteenth-
century arithmetic and to all who are interested in the metrology of the
countries with which Venice traded.
258
KARA ARITHMETICA
ABACHETTO NVOV
MOLTO COPIOSO , ET AKTI-
ficiofamente ordinaco a modo
di tariffa d'alcune moltipli*
cationi <5 partimenti di
numcri rapprefentantt
diuerfe nature, $C
qualita di mo*
nete,peil,ec
mifurc,
OPERETTA A CIASCVNO
Vtilisfima , & uniuerfalmente in
tutce le Citta & Prouintie
molco neceiTaru .
FIG. 123. TITLE PAGE OF MANZONI
MICHAEL STIFEL. Ed. pr. 1553. Konigsberg i. Pr., 1553.
See p. 223.
Title. See Fig. 124.
Colophon. 'Gedruckt zu K6nigs-//berg in Preufse durch Alex-
andrum // Behm von Luthomifl/ Voll // endet am dritten tag
PRINTED BOOKS 259
defs Herbft-//monats/ Als mann zalt nach // der geburt vnfers
lieben // herrn Jefu Chrifti.// 1554.' (P. 505.)
Woff*
far Ccf*
Stfcl
f r r ?
FIG. 124. TITLE PAGE OF STIFEL'S RUDOLFF
Description. 4, 14.7 X 19.6 cm., the text being 9.9 X 16 cm.
507 pp. (4 blank, 12 unnumb.), 2 1-26 11. Konigsberg in Preufien,
1553-
2 6o KARA ARITHMETICA
Editions. Konigsberg in Preufien, 1553, 4 (here described);
1571. See also p. 226.
Although the book is nominally an algebra (the word ' cofs ' coming
from the Italian cosa, ' thing,' meaning the unknown quantity), the first
part of the book is devoted entirely to arithmetic. This part is intended
as an introduction to the algebra that comes later, and gives the funda-
mental operations as treated by Rudolff, with Stifel's commentary. It
is entirely theoretical, and it naturally leads into the theory of irrational
numbers, which constitutes the first part of the algebra.
Other works 0/1533. Albert, p. 180, 1534 ; Boethius, p. 27, 1488 ;
Gemma, p. 204, 1540 : Gyraldus, p. 254, 1552 ; Mariani, p. 181, 1535 ;
Morsianus, p. 182, 1536; Ringhieri, p. 253, 155 i ; Rudolff, p. 152, 1526 ;
Bernardu Wojewodki, ' Algorithm, to jest nauka liczby, po polsku na
linijoch uczyniony,' Cracow.
CLAUDE DE BOISSIERE. Ed. pr. 1554. Paris, 1554.
CLAUDIUS BUXERIUS. Born in the diocese of Grenoble, probably c. 1500.
He also wrote on poetry, music, and astronomy. For his Rythmomachia see
p. 271.
Title. See Fig. 125.
Colophon. 'Acheue d'Imprimer le xiij. // iour d'Octobre,//
1554.' (F. 74, i.)
Description. 4, 10 x 14.8 cm., the text being 7.2 x 12.9 cm.
2 ff. unnumb. + 73 numb. = 75 ff., 32 11. Paris, 1554.
Editions. Paris, 1554, 8 (here described); ib., 1563, 8 (p. 262).
This is a theoretical work in two ' livres,' probably written for
students in the University of Paris. It is the second printed work which
I have noticed that carries the system of numeration as high as thousands
of quintillions, ' Mille de Quintillions,' although Chuquet, in his manu-
script of 1484, carries it to ' nonyllions.' De la Roche (p. 128) followed
Chuquet in this as in other respects. Boissiere's plan starts with the
names of millions, bimillions (for million millions), trimillions, etc. He
then says that to avoid ambiguity, as of bimillions for two millions, these
names are abridged to billion, trillion, etc. He moreover numerates in
periods of six figures each, as in England at present, and not in periods
of three figures as is now the custom in France and America.
He first treats of the classes of number according to the old plan,
distinguishing between digits (1-9), articles (multiples of 10), and com-
posites (articles -f- digits), of which names we still use the digit. Instead
of extending the fundamental operations to six, or even nine, as his
PRINTED BOOKS 261
contemporaries so often did, he limits them to four, as we do. His work
in addition and multiplication is substantially like ours, but his ' sub-
straction ' is as follows (1563 edition) :
Refte 9399739 01 C.
Debte ou fomme 1840006503 A.
Paye ou a Subftraire 900032602 B.
I/ART
D'ARYTHME-
TICLVE CONTENANT
TOVTE DIMENTION, TRES-
SINGVLIER ET COMMODE,
tant pout 1'art militaire o,ue
autres calculations.
Auec priuilege
du Roy.
fmprlme'a parK,p<tr^nncttnere,A I'cnfci
jclalian , rue des Porfa .
FIG. 125. TITLE PAGE OF BOISSIERE
262 KARA ARITHMETICA
In division he uses the galley method only. Boissiere is one of the first
writers, I believe the first in France, to invert the divisor in the division
of fractions, as Stifel had done a few years before in Germany. Book I
closes with a somewhat extended treatment of the rule of three.
The second book relates largely to mensuration, figurate numbers,
roots, and the mediaeval proportions.
Some of the rules are in verse. Of military matters, referred to in
the title page, there is little mention.
CLAUDE DE BOISSIERE. Ed. pr. 1554. Paris, 1563.
See p. 260.
Title. This is practically identical with that of the 1554 edi-
tion, except for the following : ' Reueu & augmente par Lucas
Trembley Parifien,// profeffeur des Mathematiques.// A Paris,//
Pour Guillaume Cauellat, a 1'enfeigne de la Poulle//graffe, deuant
le college de Cambray.// 1563.' (F. I, r.)
Description. 8, 10.2 x 1 6. 8 cm., the text being 6. 8 X 13 cm. 3ff.
blank + I unnumb. -f 71 numb. = 75 ff., 25-30 11. Paris, 1563.
See p. 260. Two other books are bound with this : La Declara//
tion et Vsage de // L'inftrument nomm Canomettre,// Par G. des
Bordes, Gentilho-// me bordelois, profeffeur ez // Mathematiques.// . . .
1570.' and ' Vsage // Dv // Compas // de // Proportion.// Par D. Hen-
rion, Mathem.// . . . M. DC. XVIII.'
JOACHIM CAMERARIUS. Ed. pr. 1554. Deventer, 1667.
Born at Bamberg, April 12, 1500; died at Leipzig, April 17, 1574. The
office of chamberlain (Kammermeister) to the Prince-Bishop of Bamberg
being hereditary in the family of Liebhard, he took the Latin name of
Camerarius. He was a distinguished classicist, a friend of Melanchthon,
and a professor at Tubingen and Leipzig. Of the various commentaries on
Nicomachus, his was the most important of the Renaissance.
Title. ' Explicatio // loachimi Camerarii // Papebergenfis// in
dvos libros//Nicomachi Geraseni//Pythagorei//Deductionis//
Ad Scientiam Numerorum.// Et Notae // Samuelis Tennulii // in
//Arithmeticam // Jamblichi Chalcidensis.//Daventriae.//Typis
Wilhelmi Wier, CI3 IOCLXVII.' (P. i.)
Description. 4, 14.7 x 19.8 cm., the text being 10.3 X 14
cm. 2 pp. unnumb. + 239 numb. = 241 pp., 34 11. Bound with
PRINTED BOOKS 263
the commentary of lamblichus on Nicomachus (p. 188, 1538).
Deventer, 1667.
Editions. The first edition was Augsburg, 1554, 8. There
was also an edition published in 1569.
Camerarius also wrote a work entitled ' De logistica,' published
at Augsburg in 1554; ib., 1557, 8; Leipzig (?), 1569, 8. See
also p. 257, 1552.
A commentary on the theoretical work of Nicomachus (p. 186).
Other works 0/1554. Albert, p. 180, 1534; Baeda, p. 131, 1521;
Buteo, p. 292, 1559 ; Euclid, p. 238, 1545 ; Finaeus, p. 160, 1530-32 ;
Fischer (Piscator) p. 247, 1549; Glareanus, p. 192, 1539; Huswirt,
p. 74, 1501 ; Nicomachus, p. 1 86, 1538 ; Peletier, p. 245, 1549 ; Psellus,
p. 168, 1532 ; Riese, p. 139, 1522 ; Stigelius, p. 249, 1549 ; Tagliente,
p. 114, 1515 ; Earth. Barchi, 'Tariffe della valuta di tutte le monete,'
Mantua, 4 (hardly an arithmetic) ; Claudio Bertholio, ' De numerandi
ratione aphorismi,' Paris, 8 ; Ian Gentil (Vander Schuere, in his 1634
edition, f. 201 see p. 424 refers to his arithmetic as published at
Paris, 1554); Caspar Hiitzler, < Eyn behende und Kunstrik Rekens-
boeck op. der Linien und Tziferen,' Liibeck, 8.
PETRUS RAMUS. Ed. pr. 1555. Paris, 1555.
PIERRE DE LA RAMEE. Born at Cuth, Vermandois, 1515; killed at Paris
the night of August 24-25, 1572, in the Massacre of St. Bartholomew. He
lectured on philosophy at Paris, and for a short time at Heidelberg. He
wrote on arithmetic, geometry, optics, and mathematics in general.
Title. See Fig. 126.
Colophon. ' P. Rami Eloqventiae et Phi-//losophiae professoris
//Regij Arithmeticae,// Finis.' (P. in.)
Description. 4, 13.8 X 19.1 cm., the text being 9.5 x 17.2 cm.
128 pp. (2 blank, 16 unnumb.), 26-27 11. Paris, 1555.
Editions. Paris, 1555, 4 (here described); ib.,1557, 8; Basel,
1567; Paris, 1584. See also Gleitsman, 1600 (p. 427), and the
' Libri Duo' of Ramus, 1569 (p. 330).
This arithmetic was popular in the Latin schools for half a century.
It is theoretical, consisting largely of definitions, extracts from the Greek
writers, a little work on the fundamental operations, and the mediaeval
theory of ratios. Ramus had not the faculty of putting together a text-
book that should be a rival to that of Gemma Frisius.
264 KARA ARITHMETICA
P. Kami, eloquentix
ET PHILOSOPHIC PROFES-
SORIS REGII,ARITH-
metics libri tres,
Carolum Lotbaringum Cardinatem,
PARTSIIS,
Apud Andream Wechelum,fub Pcgafo,in
vicoBellouaco, ^4nno Salfttif*
1 5 5 5-
Cum priuilegio Regis.
FIG. 126. TITLE PAGE OF RAMUS
PRINTED BOOKS 265
JACOBUS MICYLLUS. Ed. pr. 1555. Basel, 1555.
MOLTZER. Born at Strasburg, April 6, 1 503 ; died at Heidelberg (?), January
28, 1558. He was a well-known classical scholar.
Title. See Fig. 127.
; Arithmetics
LOGISTIC AE DIBRI
duo , ex diuerfis dus artis fcnV
ptoribus colle&ij & exemplis
plurimis, ijfdem'cp utilifs,
nuperilluftrati;
IACOBVM
Cum gratia &priuilfgio Imperiali
ad deceniuum.
BASILEAE, PER I O A N*
FIG. 127. TITLE PAGE OF MICYLLUS
Colophon. s Basileae, ex officina//IoannisOporini, Anno Salutis
humanse//M. D. LV. Menfe Martio.' (P. 319.)
Description. 8, 9.3 x 15.2 cm., the text being 7.8 X 11.3 cm.
320 pp. (3 blank, 23 unnumb.), 25-27 11. Basel, 1555.
266 KARA ARITHMETICA
Editions. There was no other edition. Treutlein's statement
(Abhandlungen, I, 15) that the book was published in Heidelberg
is incorrect. It probably came from the fact that the ' Epistola
Nvncvpatoria' is dated ' Heidelbergae, 10 Calend. Nouemb. Anno
Domini 1553.'
The book was written for the classical schools. Although Micyllus
gives the fundamental operations in a practical manner, the latter part
of his book is theoretical, presenting some of the ancient arithmetic in
the Greek language. The work includes an unusually complete treat-
ment of sexagesimals, ' De partibvs Aftronomicis, & earum fupputa-
tione ' (p. 201), and a chapter on the computus (see p. 7), ' De
Temporvm svppvtatione, qva Ecclefiaftici utuntur.' The latter shows
that the name ' Computus ' had become unpopular, for it begins with the
following statement : ' Est & temporum fupputatio quaedam, quern Com-
putum Ecclefiafticum, barbaro nomine, uocant.' (P. 269.) The follow-
ing is an example of his applied problems : ' Scribit Plinius, Alexandriam
a Rhodo diftare 583000 pafsuum. Cupio autem fcire, quot ftadia ijdem
pafsus efficiant. Diuido igitur 583000 per 125.'
MICHAEL NEANDER. Ed. pr. 1555. Basel, 1555.
Born in the Joachimsthal, April 3, 1529; died at Jena, October 23, 1581.
He was professor of mathematics, Greek, and medicine, in the university of
Jena. He wrote on physics and cosmography.
Title. See Fig. 128.
Colophon. ' Basileae, ex Officina // loannis Oporini, Anno //
Salutis humanae M. D. LV. Men-//fe lanuario.' (P. 119.)
Description. 4, 13.9 X 19.5 cm., the text being 9.1 x 15.7 cm.
104 pp. numb. + 15 unnumb. -f I blank =120 pp., 2529 11.
Basel, 1555.
Editions. There was no other edition. The * Epistola Nvncv-
patoria ' is dated * Die Pafchalis, Anno LIIII,' but the book was
printed, as the colophon shows, in 1555.
This is a historical treatise on Greek, Egyptian, Roman, Arabic,
and mediaeval European weights and measures, and is particularly inter-
esting because of the symbolism which it contains. The origin of our
present apothecary's symbols of measure is seen in the ancient Roman
abbreviations. The text is mostly in Latin, but eight pages of '
Merpcov KCU ^TaO/jiotv ' are in Greek.
2YNO
MENSVRARVM ET PON-
DER VM, PONDERATIONIS'*
que menfurabilium fecundum Romanos,
Athcnienfes , y^f^ov^, H9u iTmot^cvsy ex
pracftantifsimis authoribus huius generis
cotraAaopera MICHAELIS
DRi ex Vallc
Anno M D
ACCE55ERVNT ETIAM, Q^VAE APVD
plenum hdttemif cxtdbdntdcponderumzrmenfurarumrdtionente
bcmcntcr dcpr<mtfd,mmc Greece cr UW m/^o corr^wrrf,
Item rcrum uerborum in his omnibus
mcmorabilium Index,
LEVITICI XIX.
BASILEAE, PER 10
w^w Oporinum.
FIG. 128. TITLE PAGE OF NEANDER
268 KARA ARITHMETICA
ANTOINE CATHALAN (?). Ed. pr. 1555.
Paris, 1556.
A French arithmetician. The author's name does not appear in this
edition. A work with the same title is assigned to Cathalan, Lyons, 1555.
Title. See Fig. 129.
L'ARITHMETIQJE
ET MANIERB DE APPKBNDIIE
a Chijfrfr & comfter far la flume & fat
lesgeftz en nombre entier & yam*
t>u facile a appeudre& tref-
utileatoutesgenSi
Denouueau reueuc
&corrigee.
^4 tyuelle font adiouftees fhfottrs tjue
&. txftnptes pour f air e lafcienrtfltt*
facile t &- flus Ifgttn a com-
frendre.
A PARIS.
Par leJian Ruclle,Jemournnt en la Rue
fainft Iacquts,a TeDfcignc de
Ja queue de Regard.
1 T 5 *
FIG. 129. TITLE PAGE OF THE 1556 CATHALAN
Description. Small 8, 7.3 x 11.4 cm., the text being 5.3 X
9.5 cm. 79 ff. numb. + i unnumb. = 80 ff., 27 11. Paris, 1556.
PRINTED BOOKS 269
Editions. Lyons, 1555, 16; Paris, 1556, sm. 8 (here
described).
This is a very good little primer of algorism for the time. Concern-
ing this work the author says : ' Lequel art trouua premierement vn
philofophe d'Arabie, nomme Algus. Dont cefte fcience prent fon nom
d'Algorifme ' (f. 3). In multiplication the gelosia arrangement is given
as a third method, under the name ' Mvltiplication per quarreaux.' In
division only the galley form appears. The ' premier liure ' covers the
four fundamental operations with integers, and a brief treatment of pro-
gressions. The * fecond liure' relates to the use of counters ('gectz'),
which the author esteems so highly as to say : ' Et note que cefte efpece
de addition eft plus vtile & facile aux gectz que aux chifres ; ' ' il
eft facile par les gectz, & difficile par les chifres.' The last part of the
book is devoted to such standard problems as the testament, the pipes
filling the cistern, and the broken eggs, and to applications like ex-
change and pasturage.
LODOICO BAEZA. Ed. pr. 1555. Paris, 1555.
A Spanish scholar of the sixteenth century.
Title. See Fig. 130.
Colophon. ' Excudebat Benedictus Preuotius, via // Fremen-
tella, ad infigne ftellae aureae.' (F. 68, r.)
Description. 8, 10.3 X 15.6 cm., the text being 7.8 X 13 cm.
2 ff. unnumb. + 66 numb. = 68 ff., 26-27 N- Paris, 1555.
Editions. This rare first edition of Baeza's arithmetic is sel-
dom found in libraries. The second edition (1556), 8, is, how-
ever, the same impression with the title page changed.
The book is entirely theoretical, making much of the classification
of numbers and of the ancient ratio systems. It is in Latin with nu-
merous Greek quotations.
Other works of 1555. Euclid, p. 237, 1545 ; Finaeus, p. 163,
1530-32; Gemma, p. 200, 1540; Glareanus, p. 192, 1539; Herman
Giilfferich, p. 257, 1552; Mariani, p. 181, 1535; Ringhieri, p. 253,
1551 j Valturius, p. 10, 1472 ; Yciar, p. 249, 1549 ; Jacob Cuno, 'Arith-
metica,' Wittenberg, 8 ; Taddeo Duni, ' Liber de arithmetical Basel,
4 ; Melchior Goldammer, 'Arithmetica Pratica,' Wittenberg, 8 ; Jero-
nimo de Valencia, 'Arte de compute,' in Santaella's 'Vocabularium
ecclesiasticum,' a separate edition appearing at Saragossain 1601. (See
also p. 140, 1523.)
2 7
RARA ARITHMETICA
N V M E R A N-
DI DOCTRINA
PR AE CLARA METHODO
expofita, in qua brcuitcr continents,
cxponuntur aperte ea 3 qu& ex yn
rithmetieajuntadyfitm potiora.
futhorc Lodoico B<te%4*
L V T E T I AF,
liclmum Cauetlat 9 /ub pingui Galli-
n<t,ex aJuerfe cottegij Cameracenfis.
1 5 5 5-
FIG. 130. TITLE PAGE OF BAEZA
PRINTED BOOKS 271
CLAUDE DE BOISSIERE. Ed. pr. 1556. Paris, 1556.
See p. 260.
Title. See Fig. 131.
Description, 8, 10.4 x 16.6 cm., the text being 6.7 X 13.1
cm. i f. unnumb. + 52 numb. + i blank = 54 ff., 26-27 ^-
Paris, 1556.
Editions. Paris, 1556, 8 (here described); French edition,
ib., 1556, 8.
Of the three standard treatises on the ancient number game of
Rythmomachia mentioned in this list, the others being the one of 1496
of uncertain authorship and Barozzi's work of 1572, this is the clearest.
It describes very carefully the checkerboard on which the game is
played, the nature of the calculi used, and the general mode of procedure.
Moreover, it is profusely illustrated (Fig. 132), which adds much to the
value of the book. The game was connected with the mediaeval number
classifications and ratios, and could never have been understood by any
save those who were well educated in the ancient theoretical arithmetic.
GALLUS SPANLIN. Ed. pr. 1556 (?). Niirnberg, 1566.
An Ulm Rechenmeister, as he describes himself on the title page.
Title. See Fig. 133.
Colophon. 'Gedruckt zu Nurmberg//durch Chriftoff // Heufz-
ler.' (P. 382.)
Description. 8, 9.2 x 14.7 cm., the text being 7.1 x 1 1.5 cm.
15 pp. unnumb. + 4 blank + 365 numb. = 384 pp., 21-25 U-
Niirnberg, 1566.
Editions. The dedication to ' Den Edlen Ehrnveften/ Furfich-
tigen/ Erfamen vnd Weyfen Herren/Eltern Burgermeiftern vnd
Rath/des heyligen Reichs Stadt Ulm' is dated < 3 Julij/Anno
Chrifti/ 1556,' so that this is possibly the date of the first edi-
tion. I know of no other edition except this of 1566.
The author devotes twenty-one pages to ' Rechnung auff der Linien,'
this counter reckoning being still the popular method at the time he
wrote. He then (p. 24) takes up the processes with Arabic numerals,
at first using a few abstract numbers, but soon, as was the custom,
introducing many practical applications. On the whole the book may
be said to be a rather good exponent of Riese's school.
272
KARA ARITHMETICA
NOBILISSI-
MVS ET ANTIQVISSI-
mus ludusPythagoreus Cqui Rythmo
machia nominator) in vtilitatem & re-
laxationem ftudioforum comparatus
ad veram & facilem proprietatem &
rationem numcrorum affcquendam,
nunc tadem per Claudium Buxcrium
Delphinatcm illuftratus.
L V T B T I AB,
ngut Gti-
ge x'adtterfo collegij Cameracfnfis
in
apttd lottnncm Gentil.
M 5 6
CVM PRIVILEGIO REGIS.
FIG. 131. TITLE PAGE OF THE 1556 BOISSIERE
PRINTED BOOKS
273
T8
V
V
FIG. 132. FROM BOISSIERE'S Rythmomachia
274
KARA ARITHMETICA
miff Jltirii
alf urn epnK
'
FIG. 133. TITLE PAGE OF SPA'NLIN
PRINTED BOOKS 275
DOMENICO DELFINO. Ed. pr. 1556. Venice, 1565.
A sixteenth-century Venetian. He was born of a noble family, which had
produced a doge, a general of the Camaldoli, and other men of prominence,
and which was later to produce a cardinal and a captain-general of the
Venetian fleet.
Title. See Fig. 134.
Description. 8, 9.8 x 14.4 cm., the text being 8. 5 x 12.7 cm.
416 pp. (56 unnumb.), 26 11. Venice, 1565.
Editions. Venice, 1556, 4; ib., 1565, 8 (here described);
ib., 1568, 8; ib., 1584, 8, and later. The title page bears the
date 1565 as stated, and the dedicatory letter ' Al Mag. co et
Ecc. mo Sig. Nicolo Crasso,' signed by Ludouico Dolce, is dated
' In Venegia a VII. di Febraio. M D LXIIII,' 1565 new style.
The author's preface, ' Intentione dell' avtore,' which follows,
bears no date, but succeeding this there is a letter ' AH'Illvstris-
simo et Reverendissimo Signor, II S. Cristoforo Madrvccio,
Cardinal di Trento, Governator Meritissimo di Milano,' signed
by F. Nicolo Croce, and dated ' Di Vinetia il di primo di Settem-
bre. M D L VI.' There is no colophon to fix the date more
accurately, unless it appeared on some page after 360, with which
this copy terminates.
The book, a ' summary of all the sciences,' devotes a brief chapter
to the Boethian arithmetic (pp. 43-48) : ' Dell' Aritmetica, de' svio //
inuentori, utilita, modo, & altri fecre // ti Cap. III.' The author is highly
lauded in Croce 's letter as of noble family, ' Illuftre per lo splendor
del fangue, non dimeno uia piu per Pornamento e per la gloria della
dottrina.' The Delfino family was at that period very prominent in
Venice.
NICOLO TARTAGLIA. Ed. pr. 1556. Venice, 1556.
TARTALEA. Born at Brescia, c. 1506; died at Venice in 1559. He was
one of the best mathematicians of his time, and was the first to give a gen-
eral solution of the cubic equation.
Title. See Fig. 135.
Description. Fol., 21.1 X 29. 1 cm., the text being 13.8 X 23.6
cm. 285 ff. (3 blank, 6 unnumb.), 52-56 11. Venice, 1556.
Editions. Tartaglia's works include the ' Nova Scientia,'
chiefly algebraic, editions of Euclid, Archimedes, and Jordanus,
276
KARA ARITHMETICA
S C I E N Z E,
DEL M^f G TV I F I CO M E S S E 1
Domenico Deljino y nobile Vmitia.no>
DALQVALE SI POSSONO IMPARAB
molte colc_appartenenti ai umere humano ,
& alia cognmon di D i o
Ccn U "Tauola, & le FojMe deiie cofe ptu notattili,
IN VlN^GIA_jjP PRESS (T~GATB 1C ~L
G10LITO DF FERRARI.
M n f T v
FIG. 134. TITLE PAGE OF DELFINO
7
LA PRIMA PARTE DEL
GENERAL TRATTATO DI N V*
MERI, ET HISVRE DI NICOLO TARTAGLIA,
NELLACLVALE IN DIECISETTE
LZBRI SI DICHIARA TVTTI GLI ATTI OPEI^ATIV/,
PRATICHE, ET REGOLE NECESSARIE NON SOL
uicntc in tucta 1'arte ncgor ian'a,& mcrcan tile,ma anchor in ognialtra
attc/dcnciajOucr difcip!ina,doue intcrucngliiil calculo.
WALIGNITA*
CON LI SVOI PRIVILEGII.
In l^intgid per Cttrtio Troia.no deiNauo,
M D Ll^I.
FIG. 135. TITLE PAGE OF TARTAGLIA
278 KARA ARITHMETICA
the * Quesiti et invention! ' and ' Regola generate,' chiefly phys-
ical, and the ' General Trattato ' containing the arithmetic. Of
the arithmetic there were editions in Venice, 1556 and 1560,
4 (?) ; ib., 1592-93, 4 (p. 279) ; Paris, 1578, 8 (below) ; Ant-
werp, 1578, 8. There was also a ' Scelta di Abbaco ridotto dal
famosissimo Nicolo Tartaglia,' published at Venice in 1596.
The entire work consists of six volumes, bound usually in three, and
the publication extended over a period of five years, from 1556 to 1560
inclusive. This volume contains the arithmetic, the most scholarly con-
tribution to the subject that appeared in the sixteenth century. It is
more elaborate than the treatise of Paciuolo (p. 54, 1494), and like
that work it enters into the various minute details of the operations
and commercial rules of the Italian arithmeticians. For example, Tar-
taglia gives seven methods for the multiplication of integers, and four
for division ; he enters very fully into the discussion of denominate
numbers and exchange ; and his treatment of such rules as ' Regula de
tri ' is unusually elaborate. Indeed, there is no other treatise that gives
as much information concerning the arithmetic of the sixteenth cen-
tury, either as to theory or application. The life of the people, the
customs of the merchants, the struggles to improve arithmetic, are all
set forth by Tartaglia in an extended but interesting fashion.
NICOLO TARTAGLIA. Ed. pr. 1556. Paris, 1578.
See p. 275.
Title. L' Arithmetiqve // de Nicolas // Tartaglia Brescian,//
grand mathematicien,// et prince des practiciens.// Diuifee en
deux parties.// La .declaration fe verra en la page fuyuante.//
Recueillie, & traduite d'ltalien en Francois, par // Gvillavme
Gosselin de Caen.//Auec toutes les demonftrations Mathema-
tiques : & plufieurs in-//uentions dudit Gosselin, efparfes chacune
//en fon lieu. //A tres-Illustre & Vertueufe Princeffe Mar-//
gverite de France, Royne de Nauarre.// Premiere Partie.//A
Paris, //Chez Gilles Beys, rue S. lacques, au Lis blanc.// 1578.
//Avec Privilege dv Roy.' (F. I, r.)
Description. 8, 10.6 X 16.6 cm., the text being 7.5 x 13.4
cm. 286 ff. (152 in part I, 134 in part II, 28 being unnumb.),
27-32 11. Paris, 1578.
PRINTED BOOKS 279
See p. 278. The * Seconde Partie ' follows f. numb. 136 of the first
part, and is of the same date. The Privilege follows f. numb. 122 of
the second part, and is dated 'Paris le 17. Septembre. 1577.'
This is merely a French translation of the work already described
with notes by Gosselin. These notes are, rather naively, printed in
more prominent type than the original text, but are generally of little
value. They cover such points as multiplying by or dividing numbers
ending in zero, they amplify certain demonstrations (as in the division
of fractions), and they adapt the commercial chapters to French usage.
NICOL6 TARTAGLIA. Ed. pr. 1556. Venice, 1592.
See p. 275.
Title. ' Tvtte Fopere // d'arithmetica // del famosissimo //
Nicolo Tartaglia.// Nelle qvale in XVII. libri con varie prove,//
& ragioni, moftrafi ogni prattica naturale, & artificiale; i modo, &
le regole da//gli Antichi, & Moderni vfate nell' arte mercantile;
& oue interuiene calcolo,//pefi, denari, tariffe, calmeri, baratti,
cambi di banchieri, e di fiere, faldi, fconti,//giuochi, traffico di
compagnie, compre, vendite, portar mercantie da un paefe //
all'altro, conuertir monete, congiungimento di metalli, & opere
de'zecchieri.//Sopra le qual cofe tutte, formanfi belliffimi que-
fiti, & fi fciolgono le diffi-//colta, CO n vgual chiarezza, & dili-
genza, per vtile rileuato de i mercanti, & te-//forieri, a Capitani,
e Matematici, & Aftrologhi, &c. // Parte Prima.// Con Privilegio.
// In Venetia,//All'Infegna del Leone. M. D. XCII.' (F. i, r.)
Description. 4, 13.6 X 18.8 cm., the text being 11.5 X 18 cm.
479 ff. (204 in part I, 275 in part II, 8 unnumb.), 40 11. Venice,
1592. Riccardi says that it was not completed until 1593.
Editions. See p. 278.
This is substantially the same as the first volume of the ' General
Trattato ' of 1556 (p. 275), except that it is in quarto instead of folio.
ORONTIUS FINAEUS. Ed. pr. 1556. Paris, 1556.
See p. 160.
Title. See Fig. 136.
Description. Fol., 19.6 X 28.5 cm., the text being 13 X 21.4
cm. 6 ff. unnumb. + 1 36 numb. = 142 ff., 1 5-34 11. Paris, 1 5 56.
ORONTIIFINAEI,
DELPHINATIS.RE-
GII MATHEMATI-
CARVM PRO-
FESSORIS,
De rebus mathematicis,
ha<5tenus defiderans,
Libri IIII.
^ Qwbus inter cetera, Circuli quadratura Centum
modis,& fupra, per cundcm Orontium
rcccntcr excogitatis,dcmonftratur.
LVTETIAE PARISIORVM,
Anno Chrifti Scruatoris,
M. D. LVI.
ExofficinaMichaelis Vafco(ani,uia lacob^a
adinfigncFontis.
Cum Pnuilegio Regis.
FIG. 136. TITLE PAGE OF THE 1556 FINAEUS
PRINTED BOOKS 281
Editions. There was no other edition. For his arithmetic,
see p. 1 60.
Although chiefly on geometry, this work is included in the list of
arithmetics because its treatment of proportion is more arithmetical
than that of Euclid and his followers. (Ff. 25-29.) There was, how-
ever, nothing original in the work of Finaeus.
VALENTIN NABOB. Ed. pr. 1556. Cologne, 1556.
NAIBOD, NAIBODA. Born at Cologne; died at Padua, March 3, 1593. He
was for a time professor of mathematics at Cologne. He wrote on astronomy
as well as arithmetic.
Title. See Fig. 137.
Colophon. ' Coloniae, Typis Iohannis//Bathenij.' (F. 100, r.)
Description. 8, 9.5 X 14.9 cm., the text being 6.3 x 1 1.9 cm.
8 ff. unnumb. + 92 numb. = 100 ff., 28 11. Cologne, 1556.
Editions. There was no other edition.
This is a Latin work, with occasional Greek passages, and was written
for the classical schools of Germany. Although assuming to be a prac-
tical treatise on calculating, it so lacks the merit of brevity as to be
unfitted for use as a textbook. It has few problems, and these are of
no commercial value. In his desire to exalt the classical learning, Nabod,
like other Latin writers of his time, assigns the Arab-Hindu numerals to
the Pythagoreans.
VALENTIN MENHER de Kempten.
Ed. pr. 1556. Antwerp, 1556.
See p. 249.
Title. See Fig. 138.
Colophon. ' Imprime en Anuers par Ian Loe //1'An de noftre
Seigneur i556//le 20 iour d'Auril.' (F. 184, v.)
Description. 8, 9.8 x 14.9 cm., the text being 7.8 x 13.2 cm.
184 ff. unnumb., 27-28 11. Antwerp, 1556.
Editions. See p. 250.
This is usually mentioned as the second of Menher's arithmetics. It
is, however, the same as the work described on p. 249, 1550, save for a
few minor changes.
282 KARA ARITHMETICA
VALENTI-
NI NABOD I D E
CALCVLATORIA NV-
merorumque natura Seftio-
nesquatuor.
CLARIS SIMVM Vl^fU
GASPARVM DOVCIVM
JJLORENTINVM, CAESAREAE
Maieftatis Coniiliarium.
COLONIAfi AGRIPPINAE,
Apud hseredes Arnold! Birck-
manni 1556.
FIG. 137. TITLE PAGE OF NABOB
PRINTED BOOKS
283
ARITHMETI
QJ/E SECONDE PAR
~M. VALENTIN MENNHER
deKempten.
Auec grace & priuilege de PEm-
pereur pour quaere ans.
FIG. 138. TITLE PAGE OF MENHER
284 KARA ARITHMETICA
PIERRE FORCADEL. Ed. pr. 1556-57. Paris, 1556-57.
Born at Beziers ; died at Paris in 1574. He was (1560) professor of mathe-
matics in the College Royale, Paris. He also wrote on astronomy, and trans-
lated the works of several Greek mathematicians.
Title. The title page of this copy of Book I is missing. The
title page of Book II reads as follows :
' Le // Second Livre de // T Arithmetiqve de P. For-//cadel de
Beziers.// Av qvel seront declarees les// fractions vulgaires, auec
leurs demonftrations, par les // quantitez continues, & premieres
caufes des egaliffemens // de F Algebre.// Le tout nouuellement
inuente par 1'auteur.// (Woodcut with motto : In Pingvi Gallina.)
A Paris,// Chez Guillaume Cauellat, a 1'enfeigne de la // poulle
graffe, deuant le college de Cambray.// i556.//Avec Privilege.'
(F- 94-)
The title page of Book III reads as follows :
* Le // Troysieme Li-//vre de 1' Arithmetiqve // de P. Forca-
del de Beziers.// Avqvel sont traictees les de-//rnonftrations
de toutes fortes de racines, auecques 1'entiere pra-//ctique de
1'extraction d'icelles, enfemble plufieurs queftions, reigles,//&
demonftrations Mathematiques, auecques le propre fubiect de
1' Algebre.// Le tout de 1'inuention de l'Autheur.//A Paris,//
Chez Guillaume Cauellat, a 1'enfeigne de la // Poulle graffe, de-
uant le college de Cambray.// i $$?.// Avec Privilege.' (F. i , r.,
of Part III.)
Description. 4, 13.2 x 19.2 cm., the text being 10. i x 16 cm.
323 ff. (5+93, 1 10, 4+ 1 1 1, in the above three books); 25-31 11.
Paris 1556-57.
Editions. Forcadel wrote four arithmetics, as follows :
1. * L'arithmetiqve,' Paris, 1556 57, here described.
2. ' L'Arithmetique par les gects,' Paris, 1558, 8.
3. 'Arithmetiqve entiere et abregee,' Paris, 1565 (p. 316).
4. ' Arithmetique demonftree,' Paris, 1570,4.
This is perhaps the most elaborate French treatise on arithmetic
published in the sixteenth century. Its three books of about one hun-
dred pages each form a work of the nature of the great contemporary
Italian arithmetic of Tartaglia, or rather of Tonstall's Latin treatise. Of
PRINTED BOOKS
285
L'ARITHMETICQVE DE P
Forcadel -deBczicn. /
Vela cognoijfince & dftofiicndfs figures ><Mtrcment de*
nombresjimplft.
o v T E 1'A rithmcticque confiftc a fjauoir cognoi-
ftredix figures, oucarafteres,propres guidons de
toutc computation Mathematicque;qui font cy dcf
(bus rcngcz felon leur propre nature.
DA x. a. 3. 4. 5. 6. 7. 8. $>. o. zero.
3. 4. 5. tf. 7. 8. p. o. i. premiere f.
4. IT. 7. 8. p. o. j.^2. (econdes.
5. 6. 7. 8. $. o. i. a. 3. tierces.
6. 7. 8. p. o. i. a. 3. 4. quartcs.
7. 8. 9. o. i. a. 3. 4. j. quintes.
7. 8. p. o. i. i. 3. 4. 5. 6. fixiefmes.
p, o. i, i. 3, 4. 5. (T. 7. feptiefmes.
o. i. 2. 3. 4. 5. 6. 7. 8. huiticfmes.
'a^jnu no ojaz o. i. 2. 3. 4.5. 6.7.8. 9, ncufielmcs,
3 *3 ? '* 3* S*^ S"
FIG. 139. FIRST PAGE OF TEXT OF THE 1556-57 FORCADEL
286 KARA ARITHMETICA
practical problems it has very few, and it is equally deficient in the
theories of the ancient arithmeticians. It is simply a ponderous work
on the theory of arithmetical calculations and rules, valuable for a
scholar but useless as a practical textbook.
For a biographical study of Forcadel see Boncompagni's Bullet ino,
II, 424.
Other works of 1556. Bae'za, p. 269, 1555 ; Cathalan (Anonymous),
p. 269, 1555 ; Gemma, p. 200, 1540 ; Medlerus, p. 223, 1543 ; Medicus,
p. 290, 1557 ; Psellus, p. 168, 1532 ; Recorde, p. 214, c. 1542 ; Riese,
p. 139, 1522 ; Ringelbergius, p. 168, 1531 ; Xylander, p. 356, 1577 ;
Anonymous, ' Arithmetices Epitome,' Freiburg, 12; A. Lottini, ' Calculi
e conti per quelli che hanno denari,' Lyons, 8 ; Casp. Pauerus, ' Logis-
tice astronomica,' Wittenberg, 8 ; Joannes Pierius Valerianus, ' Hiero-
glyphica sive de Sacris Aegyptiorvm literis Commentarii ' (ancient finger
reckoning), Basel, fol., with editions ib., 1567, fol. ; ib., 1575, fol. ; Lug-
duni, 1579, fol. ; Juan Diaz Freyle, ' Sumario compendioso . . . con algunas
reglas tocantes al Aritmetica,' Mexico (the first arithmetic printed in
America).
ROBERT RECORDE. Ed. pr. 1557. London, 1557.
See p. 213.
Title. See Fig. 140.
Colophon. * Cjmprinted at London, //by Jhon Kyngfton.//
Anno domini. 1557.' (F. 164, v.)
Description. 4, 13 X 18 cm., the text being 8.1 x 14.7 cm.
164 ff. unnumb., 36 11. London, 1557.
Editions. There was no other edition.
Recorde speaks of this work as ' The feconde parte of Arithmetike,
containyng the extraction of Rootes in diuerfe kindes, with the Arte of
Coffike nombers, and of Surdes nombers also, in fondrie fortes.' It is
not, however, purely algebraic, for the first half of the book is a treatise
on Boethian arithmetic. For example, the following is one of the defi-
nitions : ' A Diametralle nomber, is fuche a number as hath twoo partes
of that nature : that if thei bee multiplied together, thei will make the
faied diametralle nomber.' The treatise on ' Cossike nombers ' begins on
f. S i, and 'The rule of equation, commonly called Algebers Rule' on
f. Gg 4. The sign of equality, ' a paire of paralleles, or Gemowe lines of
one lengthe, thus ; , bicaufe noe .2. thynges, can be moare equalle,'
is found for the first time in print on f. Ff i (see Fig. 141).
PRINTED BOOKS 287
Cljctoljetftone
oftoftte,
to&tc&e is tl)e fcconor parts of
Arithmetiketconf flfnpn
tfon of ttocte*: HS&c&J?/
fottlj tfje rule of ? fuffo
T^oag^ way/tones doe learegrtate price,
The ^hctftone is for eterfce
jfs neadefuttjmdm Tworkj atftrattnge:
Quite t&nges andkardeiftoillfo dating
jfndmdy tbem/karfaio right good* ft:
Mlartefintn 1&oW,tbei can mt cbufe,
$ut *fe bit belpeyet at men fee,
Hoefbarptneffefemetb in it to bee.
TAegroundeofartes dtdlrede tbttftone:
His \>fi isgreatfjtmlmoare then one,
Here 1/1 ou lift your %ittts to Ttbttte,
toulle toitte thereby doegrettely mtnde,
Sbarpe Tutte* are fined to tbeirfulle tvde.
jfndtojottrfel/be not +nl(inde
CM cfe 250ofte* are to bee folDe, at
tie OTette ooo;e of pouiw,
FIG. 140. TITLE PAGE OF RECORDE'S Whetftone of witte
288 KARA ARITHMETICA
as tf)ct r too;fer0 Doe ctfcnfic ) to Diaincte it onety into
tfooo parte*. SflJfcereof tbc firfte is, v^f OB* nomberu
cqutllc vnto one other. 3nO ttje feconDe is , *k onr now;
if r u compared as e^aalle Vnto.^otbcr nombers*
ai toaics lulling ^ou to rcmclicr, tljat f o u reduce
pournombew, totljeirlcaffc Denominations, anu
Onallelte fo;me0^efo;e ?ou p^oceDe anp farther*
0nD again, if pour equation be for tjc, tbat tljc grca-
tefte Denomination G/?/^ ? be i oineD to an? parte of 4
compounDc nomfaer , T?OU Qall ton rnc it fo , ti>at tlje
nombcrof tljegreatcftc 0gne alone , inaicttanDcaa
cquallctotticrcttc.
anD tins is all tljat neaDetb to be taugljte , conccc-
npngtbustooojUe*
!i^o tubcit, fo^ caOfc alteratio of eftutiont.% tuf II p^o^
pounDe a fetoe craples, bicaufc t^e extraction of tbeic
rootes,maie toe mo^e aptlp bee tojo ugtjte. anD to a-
uotDe tlje teDioufe repetition of tljefe tuoojoe* : is c^
rjuallc to : 31 foill fette as 31 Doe often m luo o:Uc ijfc, a
paire of paralleles,o^ >eniotne lines of one lengtbc,
rqualle. 3nD noto marUe tljcfc nombers*
1. 1 4.t
2. 2 o, t
v 26. > I
4.
T 1 8.2^ f -2 4*
6. 545- -- I2^=4o2e > ~-f 48of 9,5-
i. 3in tbc firftc there nppearctlj, 2 , nomfaf rs , tbat ts
14-%.
FIG. 141. FROM RECORDE'S Whetftone of witte
PRINTED BOOKS 289
LUCAS LOSSIUS. Ed. pr. 1557. Frankfort a. d. Oder, 1557.
A German arithmetician of the sixteenth century, born at Liineberg.
Title. See Fig. 142.
ARITHME'
TICES EROTE*
MAT A PVE-
R I L I A.
IN QVIBVS SEX SPECIES
huius vtilifsimae artis, & Regula>quam
vocant, Detrijbreuiter & per-
fpicue traduntun
IN GRATtAM ET VSVM SCHOLAa
rum puerilim Ldtmnvm coUettt , cr m
lucem iani recent cdita*
- A. ; -
LFCA LOSS 10
Luneburgenfi.
FRANCOFORDI^ AD ODERAM
IN OFFICINA IOHANNIS
EICHORNL
FIG. 142. TITLE PAGE OF LOSSIUS
Description. 8, 9.2 x 14 cm., the text being 6.8 x 11.9 cm.
4 ff. unnumb. + 27 numb. = 31 if., 22-24 U- Frankfort. For
date see f. 4, v. : ' Datae Lunebar, Anno 1557.7/7. Februarij.'
2 9 o KARA ARITHMETICA
Editions. Frankfort an der Oder, 1557, 8 (here described);
s. 1., 1562, 8; Leipzig, 1568, 8; Frankfort, 1569; Magdeburg,
1585, 8.
A small book, intended, as the title states, for beginners in the Latin
schools. It is arranged on the catechism plan, a feature not common
with arithmetics printed in Germany at this time, although extensively
used by Recorde in England (see p. 210). It begins : * Quid eft Arith-
metica? Est bene & artificiofe numerandi & computandi fcientia.'
(F. 5.) The ' species ' are treated in a practical way, but the problems
are all traditional, part of them being based upon biblical incidents.
Other works of 1557. Archimedes, p. 228, 1544; Camerarius,
p. 263, 1554; Gemma, p. 200, 1540; Gutierrez de Gualda, p. 167,
1531 ; Jacob, p. 298, 1560; Paxi, p. 80, 1503; Psellus, p. 170, 1532 ;
Ramus, p. 263, 1555; Rudolff, p. 153, 1526; Tagliente, p. 115, 1515;
Martinus Carolus Cressfelt, Arithmetica, Reeckeninge op den Linien
end Cyfferen na allerley Hantieringe,' Deventer (second edition 1577) ;
Sixtus Medicus, ' De Latinis numerorum notis,' Venice, 4 (colophon
date, 1556; hardly an arithmetic).
GIOVANNI FRANCESCO PEVERONE.
Ed. pr. 1558. Lyons, 1581.
An Italian arithmetician of the sixteenth century, bom at Cuneo, in
Piedmont.
Title. See Fig. 143.
Description. 4, 15.3 x 21.7 cm., the text being 10.2 x 17.6
cm. 136 pp. (60 on arithmetic), 35-37 11. The dedicatory
epistle is dated 1556. Bound with this is 'II breve Trattato//
di Geometria.' Lyons, 1581.
Editions. Although the dedicatory epistle is dated 1556, and
the portrait of the author bears the date 1550, I know of no
edition before 1558, Lyons, 8. This edition of 1581 seems to
have been the second.
The first part of the work treats of the operations with integers. It
is not a particularly progressive textbook, as is seen in the fact that it
includes the ' gelosia' multiplication of the early Venetian writers. The
second book relates to fractions, the third to business operations, and
the fourth to roots. The work is in no sense a scholarly production.
ARITHMETICA
E GEOMETRIA
DEL SIC.*
Gio. Francefco Peuero/te di Canto.
IN LTONE*
P1K CJO* DI TORNES.
M. D. LXXXI.
Con Priuilcgio del Re.
FIG. 143. TITLE PAGE OF PEVERONE
292 KARA ARITHMETICA
ALVISE CASANOVA. Ed. pr. 1558. Venice, 1558.
A Venetian teacher of the sixteenth century.
Title. See Fig. 144.
Description. 4, 15.5 X 20.8 cm., the text being 12.5 x 16. 1
cm. 136 ff. (partly numbered), 33-38 11. Venice, 1558.
Editions. There was no other edition.
Although this rare work is usually classed as a commercial arithmetic,
it is rather a collection of bookkeeping problems. As such it was one
of the most prominent of the century, and it gives an idea of the busi-
ness questions of the Venetian merchants of its time.
Other works 0/1558. Anianus, p. 32, 1488; Archimedes, p. 228,
1544 ; Forcadel, p. 284, 1556-57 ; Gemma, p. 204, 1540 ; Glareanus,
p. 192, 1539 ; Mariani, p. 181, 1535 ; Medlerus, p. 223, 1543 ; Recorde,
p. 213, c. 1542 ; Riese, p. 141, 1522; Andreas Clatovenus, 'Arith-
metica Bohemice,' Prag, 8 (Wydra, in his Historia Matheseos in
Bohemia, p. 18, mentions it as printed in Niirnberg, 1530). There were
also two arithmetics published s. a., but c. 1558, viz. : Benese, p. 182,
c. 1536 ; Gemma, p. 200, 1540.
JOHANNES BUTEO. Ed. pr. 1559. Lyons, 1559.
BOTEO, BUTTON, BATEON. Born in Dauphine, c. 1485-1489; died in a
cloister in 1560 or 1564. He belonged to the order of St. Anthony, and
wrote chiefly on geometry, exposing the pretenses of Finaeus.
Title. See Fig. 145.
Description. 8, 10.4 X 16.9 cm., the text being 7.2 X 13.2 cm.
400 pp., 23-27 11. Lyons, 1559.
Editions. Lyons, 1559, 8 (here described); ib., 1560,8. An
edition of his 'Opera' appeared in 1554.
The work is divided into five books, the first treating of the funda-
mental operations with integers, the second of fractions and the rule of
position, the third of algebra, the fourth of arithmetical problems, and
the fifth of algebraic problems. The problems are not of practical
value, and hence the arithmetic never attained any popularity.
Other works 0/1559. Albert, p. 180, 1534 ; Anianus, p. 32, 1488;
Cataneo, p. 244, 1546 ; Gemma, p. 200, 1540 ; Mariani, p. 181, 1535 ;
Johann Fischer (Piscator), p. 247, 1549 ; Herman Giilfferich, ' Ein new
Rechenbuchlin auff der Linien und Federn' (perhaps the work men-
tioned on p. 257, 1552); Pedro Juan Monzd (Monzoni), ' Elementa
Arithmeticae,' Valencia, 8, and s. 1., 1566, 1569.
PRINTED BOOKS
293
SPECCHIO LVCIDJSSIMO,
nelqjuale.fiucdeno. effere dlfFmito _
tutti i modi , & ordini dc
fcrittura, che fi.dcuc
inenare nelli nego-
tianienti della
Mcrean-
J(EC./MB!1
It loro corriftondeBtie > difearba-
gliando , jjr illuminaitdo
Citttettetto a nego-
tianti .
OPERA NON piv VEDVTA,
compolta per Aluifc Cafanoua ,
Cittadiil.Venetiano . Annc
Dommi.
O ON O R A TT A ET
FIG. 144. TITLE PAGE OF CASANOVA
294
KARA ARITHMETICA
B V TE O N IS
LOG IS TIC A, QV &
& Arithmetica vulgo dicitur in h-
bros quinqne digefla:quo-
rum index fummatim
habetur inter go.
E 1 V S 7> E M ,
i/td locumUitruuii corruftum '&stitutio,qui eft
dcfroponme lafidum mitten Jorum ad baliftt
Dccimo,
I V G D V N 1,
JIPVD
SVB SCVTO VENETO.
JA. 7). LIX.
Cum
FIG. 145. TITLE PAGE OF BUTEO
PRINTED BOOKS 295
FRANCISCUS BAROCIUS. Ed. pr. 1560. Pavia, 1560.
FRANCESCO BAROZZI. Born at Venice, c. 1538 ; died after 1587. He wrote
on cosmography, and edited Proclus.
Title. ' Francisci Barocii // patritii Veneti // opvscvlvm,// in
quo vna Oratio, & cluaeQueftiones://altera de certitudine, & altera
// de medietate // Mathematicarum continentur.// Ad Reueren-
disfimum Danielem Barbarum Patriarcham//Aquileienfem defig-
natum Virum Clariff.// Patavii, E. G. P.//M. D. LX.' (F. I, r.)
Description. 4, 14.1 X 19.6 cm., the text being 10.1 x 15.3
cm. 40 ff. numb., 26 11. Pavia, 1560.
Editions. There was no other edition.
This philosophical discussion has been mentioned because it includes
some reference to the old Boethian arithmetic. No arithmetical pro-
cesses are discussed.
OLIVIERO FONDULI. Ed. pr. 1560. Bologna, 1560.
A Bolognese teacher of the middle of the sixteenth century.
Title. See Fig. 146.
Description. 8, 9.9 X 14.6 cm., the text being 6.9 X I i.i cm.
23 ff. unnumb. (possibly one missing at the end), 24 11. The
date is given on f. I, v.: 'Dat. Bonoiae . die xxv. Septemb. 1560.'
Bologna, 1560.
Editions. There was no other edition.
This rare little handbook of commercial customs, from the press of
Pellegrino Bonardo of Bologna, is hardly an arithmetic, although it
explains certain arithmetical processes, and gives a considerable number
of business problems.
SIMON JACOB. Ed. pr. 1560. Frankfort, 1565.
Born at Coburg; died at Frankfort am Main, June 24, 1564. He was one
of the best-known Rechenmeisters of his time.
Title. Sec Fig. 147.
Colophon. ' Getruckt zu Franck-//furt am Main/ bey Georg
Raben///in verlegung Sigmund Feyerabends/ //vnd Simon
Huters. Sigmund Feyerabent // Simon Hutter // M. D. LXV.'
(F. 363, v.)
296
KARA ARITHMETICA
PRATICHE
DE PIORETTI MERCHANTILI.
vtilifsime a ciafcheduna perfbna 3 di mandare a
memorialebreuelnuetioni fabrichate fopr*
al Valutar de pefi 5 & mifiire, & altre in
ftruttioni necceflarie da fapere .
Et inchora 3 quadrare Muraglie , Taflelli , & Coperti *-
Tuade , Fieno , & Legne.-Con la decchiaratrone, &
Exempli loro come legendo intenderai.
FIG. 146. TITLE PAGE OF FONDULI
PRINTED BOOKS
297
m&mfttenVwtnvfiMem/
extraaian Radicum,t>it t> on ton <propom'onm/mtf tnfen fuftigm Sra^nvft
^tcic^en noc^ nie an tag fommm. CDii6 &ann i?on t> Ocomct via/ tvic man raaitc^ erle;? ^eU
iD e&itf /ancf) atterlep (Totpota?7vegufarla bnb 3rr(0ulatrta/meffen/2(ream finteen t>fi reefy*
ttcnfo( 2iflc8t>ur^0imoit3aco& t>0n$o(urav^ih-acri>ut>3fccJ)enmcifteritt
FIG. 147. TITLE PAGE OF JACOB
29 8 KARA ARITHMETICA
Description. 4, 15.2 x 18.6 cm., the text being 10.3 x 15 cm.
354 ff. (349 numb.), 30 11. Frankfort, 1565.
Editions. Jacob published two works, as follows :
1. ' Rechenbiichlein auf den Linien und mit Ziffern,' Frank-
fort, 1557, 12; 1574, 12; 1589; 1590; 1599.
2. ' Ein new vnd wolgegrundt Rechenbuch,' 1560 ; Frankfort,
1565, 4 (here described); 1569; 1600, 4 (below), and in the
seventeenth century.
Jacob's arithmetics followed the general plan of the popular books
of Riese, and were deservedly well received in the second half of the
century. They were commercial textbooks, and although they do not
show any mathematical advance they are historically valuable for their
applied problems. The title page (fig. 147) is interesting because of the
variety of mathematical instruments illustrated.
Other works of 1560. Borghi, p. 16, 1484 ; Buteo, p. 292, 1559;
Feliciano, p. 149, 1526 ; Psellus, p. 168, 1532 ; Scheubel, p. 246, 1549 ;
Tartaglia, p. 278, 1556 ; Gasparo Rizzo, ' Abbaco nvovo molto copioso
et artificiosamente ordinato,' Venice, 8 ; Juan Ventallol, * Aritme'tica,'
a rare work, s. 1. a., which appeared in Spain about this time.
SIMON JACOB. Ed. pr. 1560. Frankfort, 1600.
See p. 295.
Title. This is practically the same as in the 1565 edition
described above.
Colophon. ' Getruckt zu Franckf urt am Mayn/ bey // Matthes
Beckern/ In Verlegung Chris-//ftian Egenolphs Erben.// Anno
1600.' (F. 360, v.)
Description. 4, 15.5 x 19.2 cm., the text being 10.5 XI4.8
cm. 1 1 ff. unnumb. + 349 numb. = 360 ff., 28 11. Frankfort, 1600.
See above.
JOSEPHUS UNICORNUS. Ed. pr. 1561. Venice, 1561.
GIUSEPPE UNICORNO. A Bergomese arithmetician, born in 1523; died in
1610.
Title. See Fig. 148.
Description. 8, 9.4 x 13.9 cm., the text being 7.4 x 12.7 cm.
i f. unnumb. + 78 numb. = 79 ff., 30 11. Venice, 1561.
PRINTED BOOKS
299
10SEPHI VNICORNI
BERGOMATIS
LIBER
DE VTILITATE MATHEMATICAL
R V. M A JR T I V flU
In quo candide Ltftor mulu quidem fcituperiucw*
d<t , admiratione digna, cr bmano ufid *
neceffirit pafsim reptriet.
V E K s T 1 1 s , ApudDominicumde Nicolinis,
AnnoDomim, M D L x I-
FIG. 148. TITLE PAGE OF UNICORNUS
300 KARA ARITHMETICA
Editions. There was no other edition.
This work is not an arithmetic, Unicorn's textbook on this subject
appearing much later (p. 412, 1598). It is a prolix dissertation on the
uses of mathematics, and of arithmetic in particular, with extracts from
the ancient and mediaeval writers. Like Agrippa's ' De vanitate scien-
tiarum,' it is interesting but profitless.
NICOLAUS WERNER. Ed. pr. 1561. Niirnberg, 1561.
A Niirnberg Rechenmeister, born c. 1520.
Title. See Fig. 149.
Colophon. ' Gedriickt zu Niirnberg/ //durch Johann vom Berg/
//vnd Ulrich Newber.' (F. 156, r.)
Description. 4, 14.2 X 19.3 cm., the text being 10.1 X 15.5
cm. 156 ff. unnumb., 23-31 11. Niirnberg, 1561.
Editions. There was no other edition.
This is a purely mercantile arithmetic, and like many of the Italian
textbooks it assumes some preliminary knowledge of the fundamental
operations. It begins with a treatment of Welsch (Italian) practice,
and is composed almost entirely of practical problems of the day, solved
by this method. Welsch practice differs from the rule of three only in
having, ordinarily, unity for the first term, as in the following example :
If i book cost $2, how much will 7 books cost? This method was very
popular with all German arithmeticians of the sixteenth century, and
survived in the chapter on Practice still to be found in English arith-
metics. It appeared also in the early American textbooks. The book
gives a very good idea of the business requirements of the second half
of the sixteenth century. The value placed upon exchange and * profit
and loss ' may be seen in the fact that twenty-six pages are devoted to
the former and forty-seven to the latter, while partnership also has
twenty-six pages for its share.
JOHANNES MONHEMIUS.
Ed. pr. 1561. Diisseldorf, 1561.
A German teacher of the second half of the sixteenth century.
Title. See Fig. 1 50.
Description. 8, 9.4 X 15 cm., the text being 6.5 x 12 cm.
22 ff. unnumb., 25-27 11. Diisseldorf, 1561.
Editions. There was no other edition.
PRINTED BOOKS
301
nxlot
fottbetlidj foui'e! beeHfcitbei;0ifcl)ettS.<mb9 mt/twb
gcbjtiucbbdangr/mi't fonbetm vwteyl/nib be=
igfci f mjff Jncl;criey art pradtce
$Kicol(ttim
vubfoiiflinenigHiebenju in8 in
FIG. 149. TITLE PAGE OF WERNER
3 02
KARA ARITHMETICA
This work is almost unknown to students of arithmetic. It seems to
be the only edition, although the introduction bears the date 1542. It
is a small work and is based somewhat on the mediaeval texts. It gives
M E T H
COMPVTATORlAE,OMNEM
puttnti Arttm trddcm:t41tl4m,<i*mti$
numcroruni,cjtuni qii olimcalculis,
nunc numwisfuper <eque diftdn
tcslinets (itrfurn <tc dc*
orfurtt poff/w,
per fie it ur.
Auttore lo. Mbnbemioi
DVSSELDORPIl
Anno t | * i-
FIG. 150. TITLE PAGE OF MONHEMIUS
the fundamental operations with integers and a brief treatment of the
rules of three, partnership, and false. There is also a brief explanation
of counter reckoning.
PRINTED BOOKS 303
BENEDICTUS HERBESTUS. Ed. pr. 1561. Cracow, 1577.
Born at Novomiasti, Poland, in 1531 ; died at Jaroslaw, March 4, 1593.
He was a Jesuit priest at Cracow.
Title. See Fig. 151.
Description. 8, 9.5 X 15.3 cm., the text being 8 X 11.5 cm.
42 ff. unnumb., 27-28 11. Cracow, 1577.
Editions. 1561; Cracow, 1564,8; ib., 1566; ib., 1569; ib.,
1577, 8 (here described).
A Latin book intended for the Church schools, and revised, as the
' Proaemivm auctoris ' states, ' Mariaeburgi in Prufsia, fub finem Anni
Domini 1576.' It consists of two parts, the first being devoted to the
arithmetical operations, chiefly with counters (and this as late as 1577),
and the second to the calendar. Besides the operations, the first part
also includes a chapter on progressions (' De Progressione. Capvt VII.'),
and one on the rule of three (' De tribvs nvmeris integris. Capvt VIII.').
ANDREAS HELMREICH. Ed. pr. 1561. Eisleben, 1561.
A Halle Rechenmeister of the latter half of the sixteenth century. The
dedication of the 1561 edition is signed by 'Andreas Helmreich Rechen-
meifter vnd Vifierer zu Halle,' and the 1588 edition by ' Andreas Helmreich
von Eifzfeldt/ Notarius publicus/ Rechenmeifter vnd Vifierer zu Halle.'
Title. See Fig. 152.
Colophon, ' Gedruckt zu Eisleben/ bey // Vrban Ganbifch.'
(F. in,r.)
Description. 4, 14.2 x 19.3 cm., the text being 9.3 x 14.9 cm.
in ff. unnumb. + I blank = 112 ff., 27-33 H- Eisleben, 1561.
Editions. This is probably the first edition of Helmreich' s arith-
metic, the dedicatory epistle being dated 'zu Halle inSachffen den
Sontag Oculi/ welcher war der 9. tag des Monats Marcij/ nach
Chrifti Ihefu vnfers lieben Herrn vnd Seligmachers Geburt 1561.'
Murhard (1, 1 5 5) mentions, however, an edition appearing at Halle
in 1546, 4. The book was republished at Leipzig in 1588 (p.
306), and again in 1595, 4 (p. 306), and possibly ib., 1596, 4.
The work is an unsuccessful attempt to combine the old and new
arithmetics. The author begins with a semi-Boethian treatment of
ratios, assuming a knowledge of the fundamental operations with
integers, and then introduces a course in mercantile arithmetic. He
closes with a considerable amount of work on mensuration, including
3 o 4 KARA ARITHMETICA
ARITHM
TICA LINEA*
RIS,
eicj adiun&a
FIGVR.ATA,
cumquibufdamex
COMPVTO
necdTarijs;
IESV Trejbytero,
cum facilitate Superiorum.
CVH GRATIA ET PRtoifego S. R. M.
In Officina Mattbai Sicktuy
J 5 7 7.
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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,
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Xucfc imt JJefrraifdjcr/ ridjifdjrr/ wib Otfmifcfcrr $j?an$f/r*
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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/
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SDoraus rtt fleffctar &fer / ct
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aaff i?ter ma(
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Irictt/ f wv fic|) felbp / bie ^inien Ovec^nuitg Ictnen Ian/
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FIG. 207. TITLE PAGE OF SCHLEUPNER
L'ARITHMETICA
VNIVERSALE
. DEL SIG.
IOSEPPO VNICORNO.
MATHEMATICO
ECCELLENTISSIMO.
PartePrima:
Ndlaquale fi contiene non folo la Theorica di tu tti i Hum eri.
ma ancoralaPrattica appartcticntc a
tuttti negocij humani.
Trattata , d* amplifcatacon fimma erudition* ,econ
ou'h & tfyui/tti modi di chiare%$4*
CON PRIVILEGIO.
FIG. 208. TITLE PAGE OF UNICORNUS
PRINTED BOOKS 415
Editions. There was no other edition.
This is one of the most elaborate treatises on arithmetic published
in Italy in the sixteenth century. It consists of six books, the first four
making up Part I. The first book treats in a detailed fashion of the
fundamental operations. Unicornus, for example, gives six methods of
multiplication, a treatment that recalls those of Paciuolo and Tarta-
glia. There is a good discussion of the two general methods of dividing,
the downward ('a danda') method having as much attention as the
galley plan. Fractions are also treated in Book I. Book II deals with
the theory of numbers after the Boethian method. Book III treats of
roots, surds, and proportion ; Book IV, of the rules of three and false ;
and Book V, of business arithmetic, including exchange, interest, and
alligation.
The work was too theoretical to be popular, but it is an excellent
source for the study of the development of elementary mathematics.
Unicornus gives a number of interesting historical references.
Other works 0/1598. Cassiodorus, p. 2 1 1 , 1 5 40 ; Fischer (Piscator),
p. 247, 1549 ; Moya, p. 310, 1562 ; Paolini, p. 393, 1589 ; Raymundus
Lullius, * Opera ea quae ad adinventam ab ipso artem universalem . . .
pertinent cum diversorum commentariis,' Strasburg, 1598, 8 (see also
p. 457); Fabricio Mordente, ' Le propositioni di Mordenti,' Rome, 4
(geometric, but with a little arithmetic).
ANTHON NEWDORFFER. Ed. pr. 1599. Nurnberg, 1599-
One of the famous Newdorffer family of Niirnberg Rechenmeisters.
Title. ' Ktinftliche vnd Ordentliche An//weyfzung der gantzen
Practic vff // den Jetzigen fchlag vnd derfelbenn // herlichen
gefchwinden Exempel // vff s kiirtzt zufammen getzogen &c.//
Meinen lieben Difcipeln zu fon-//derlichem Nutzen geftelt.
Durch // mich Anthonium Newdorffer //Rechenmaifter vnd Mo-
dift der//Statt Nurnberg// Anno .M.D.IC.' (F. 2, r.) On f.
i, r., is an engraved frontispiece with figures of Euclid and Py-
thagoras, and a triangular multiplication table. On f. i, v., are
14 11. of verse ' Ad Stvdiosos Artis Nvmerandi.'
Colophon. ' Gedruckt zu Nurnberg/ // durch Paulum Kauff-
mann.//M. D. XCIX.' (F. 65, r.)
Description. 4, 15 x 19.3 cm., the text being 11.4 x 15. 5 cm.
65 ff. unnumb., 22-32 11. Nurnberg, 1599.
416 RARA ARITHMETICA
Editions. There was no other edition.
This is an excellent illustration of the work of the celebrated Niirn-
berg Rechenmeisters. It is divided into twelve books, of which Das erfte
Buchlein handelt von der Venetianifchen oder Kauffmennifchen Practic,'
usually called Welsch practice by the German writers. It is composed
chiefly of problems, the rules and explanations being left for the master.
Directions are given, however, in the case of fractions. Book II relates
to denominate numbers and the ' Regula de Tribus.' Book III treats of
the weighing of commercial products. Book IV ' Handelt von der Rech-
nung eines Caffierers,' including exchange. Book V is entitled ' lornates,
Das ift/ Rechnung von allerley Handtierung,' and consists of practical
problems relating to the purchase of goods. Book VI ' Tractiert von der
Regel Converfa vnd Quinque,' inverse and compound proportion. The
rest of the twelve books are devoted to practical business questions of
the day.
OBERTO CANTONE. Ed. pr. 1599. Naples, 1599.
A Genoese arithmetician of the sixteenth century, residing in Naples.
Title. See Fig. 209.
Description, Fol., 14.5 x 20.1 cm., the text being 10.4 X 15.7
cm. 304 pp. (292 numb.), 34~37 n - Naples, 1599.
Editions. This is the first edition of this work, three other
editions appearing in the seventeenth century.
Arithmetics written by Genoese masters in the sixteenth century
are rare, even in manuscript form, and this is probably the only such
work printed in Naples. Genoa was a mercantile center, but its dialect
was not conducive to the success of a textbook, as is seen in the case
of Zucchetti's treatise (p. 425). Naples was too far from the path of
international commerce to produce many mercantile works. But
although Oberto Cantone was a Genoese, he was a ' professor delle dis-
cipline matematiche ' in Naples, and he dates his dedicatory epistle
' Di Napoli li 15. luglio 1599.' Naples, however, had a commerce of
its own, even if not as extensive as that of Venice or Florence, and by
the close of the century it was natural to expect works of this kind.
The book is mercantile and is based upon Borghi and similar writers
of the North. It gives our present method of multiplication, but makes
no use of the Venetian or Florentine names, with the exception of * per
colonna.' Two or three short methods are given, but no such extended
treatment of varied forms appears as in Tartaglia, or his great prede-
cessor Paciuolo. Division, except in simple cases, is postponed to p. 142,
L VSO PRATTICO
DELL'ARITMETICA
OBERTO CANTONE
DA GENOVA>
PROFESSOR DELLE DISCIPLINE
Maecmatiche .
QTALE CON NVQVA INVENTIONS
sinfegna in mat ma di conti , I'vfo tanto delta Regia Camera delta
Sommaria, quanta di Negotiant^ Mercadanti&Artegiani.
come Nafolicamby, f recambijin ciafcuna piazza.
CON PRIVILEGIO.
o
w
P
TARQJTINIO LONG HO. 2^. D.
Si vendono dal ir.edcilir6 Aucorc a Banchi nuoui.
FIG. 209. TITLE PAGE OF CANTONE
418 KARA ARITHMETICA
and is there treated in the modern form, ' a danda.' Most of the applied
problems are in exchange, although a few other types are given. The
book is poorly constructed, being too prolix in the treatment of the
operations, and too narrow in its applications.
JOANNES MARIANA. Ed. pr. 1599. Toledo, 1599.
Born at Talaver^ de la Reina, in 1536; died at Toledo, February 17,
1624. He was a Spanish Jesuit and a famous historian.
Title. See Fig. 210.
Colophon. 'Toleti, Apud Thomam Guf-//manium, Anno . 1 599.'
(P. 206.)
Description. 4, 1 4 x 1 8.6 cm. , the text being 9.3 X 1 5 .6 cm. 6 pp.
blank + 8 unnumb. + 192 numb. = 206 pp., 26 11. Toledo, 1599.
Editions. There was no other edition.
The author is not the same as the Giovanni Mariani mentioned on
p. 1 80, but a Spanish Jesuit of some fifty years later. The work is on
the history of the weights and measures used in Spain in the sixteenth
century. It traces these measures from the Roman, Greek, and Hebrew
sources, and is valuable for the study of the history of the subject. It
closes with a table of comparative measures.
MARTIN WENCESLAUS. Ed. pr. 1599. Middelburg, 1599.
A Dutch arithmetician of the sixteenth century.
Title. See Fig. 211.
Colophon. ' Eynde defes onfes eerften Vo-//lumens ; Ghedruct
int Jaer ons Hee-//ren 1599. ^ n defe vermaerde Coop-//ftadt
van Middleburgh in // Zeelandt.'// Also in French : ' Fin de ce
noftre premier Volu-//irien: Imprime 1'An de noftre Seig-//neur
1 599. A la tref-renommee // ville Marchande de Middel-//b>ourg
en Zeelande.' (F. 157, r.)
Description. 4, 15.2 X 20.1 cm., printed in double columns,
one in Dutch and the other in French, the Dutch being 6.7
X 15.7 cm., the French 5 X 15 cm. 4 ff. blank + 18 unnumb.
+ 137 numb. = 159 ff. Dutch: 30-41 11.; French: 16-26 11.
Middelburg, 1599.
Editions. This is evidently the first edition, since the dedica-
tion is dated 'Den 10. dach van Decem. ftijlo nouo. Anno Chrifti.
PRINTED BOOKS
419
JO ^KKIS
MARIANAE
Hifpani,
E SOCIE* IESV,
DE P O N<DE Rl%rs ET
menfaris.
Anno
CVM PRIVILEGIO.
Volet i, lAfudThornxm Gufmanium.
FIG. 210. TITLE PAGE OF MARIANA
420 KARA ARITHMETICA
T'FONDAMENT
Ueicnfc^e ^actijcb / mtDtfgaDeta bailee
nbotturnatrtjftc (luchm tian&cnUrgljdtian
met r tDdicUe ow r tcnft cmmmglje of re
Conto^Dantiett*
MART1NVM VVENCESLAVM,
AQVISGRANENSEM.
LE FONDEMENT
DE DARITHMETIQVE AVEC
LAPRACTICQVE ITAL1ENNE ENSEM,
ble Ics pieces , les plus necefsairesde la BsCglc
d'Imercft.
ENSEMBLE EN BAS ALLEMAN ET EN
!ran$ois >avec raifonnable Harmonie on
Concordance.
tl TOVT
Par <JfA
MIDDELBVRGH.
Cn^e men tomtfe te c oope/ bp Adriaen uan be Vivere . 25oer febercooper/
UiooncuDc bp be nieutot ^urf^/inDen bergulom 23plicl,
FIG. 211. TITLE PAGE OF WENCESLAUS
PRINTED BOOKS 421
1598,' 'le 10. lour de Decembre ftylo nouo. Anno. Christi- 1598.'
The long and stupid preface is dated November 30 of the same
year. Wenceslaus had already published two works before this
one, the ' Proportionale ghesolveerde Tafelen van Interest,' 1 594,
8, and the ' Boukhoudens Instruction,' 1595.
The book is interesting, both because of its arrangement of the Dutch
and French texts in parallel columns, thus serving a purpose in lan-
guage teaching in a bilingual country, and because of the mercantile
problems which reveal, as is particularly true of the Dutch books, the
contemporary life of the people.
Other works of 1599. St. Augustine (see Reisch), p. 82, 1503 ;
Barlaamo, p. 343, 1572; Benedetti, p. 364, 1580; Boethius (see
Reisch), p. 82, 1503; Bungus, p. 384, 1583-84; Capella, p. 66,
1499 ; Clichtoveus (see Reisch), p. 82, 1503; Faber Stapulensis (see
Reisch), p. 82, 1503; Finaeus (see Reisch), p. 82, 1503; Gallucci
(see Reisch), p. 82, 1503; Jacob, p. 298, 1560; Jordanus (see
Reisch), p. 82, 1503; Ramus, p. 330, 1569; Reisch, p. 82, 1503;
Schleupner, p. 412, 1598; Striibe, p. 391, 1588; Johann Heere,
Rechenbuchlein von allerhand gebraiichlichen Fragen,' Niirnberg,
8 ; Andreas Reinhard, Drey Regifter Arithmetifcher Anfang zur
Practik,' Leipzig, 8, with a second edition in 1600.
JACOB VANDER SCHUERE.
Ed. pr. 1600. Haarlem, 1600.
A Dutch arithmetician of Meenen, c. 1550-1620.
Title. See Fig. 212.
Description. 8, 9.4 X 14.4 cm., the text being 7.2 x 12.6 cm.
2 ff. unnumb. + 202 numb. = 204 ff., 22-23 11. Haarlem, 1600.
Editions. That the date of the first edition is 1600 appears
in the Voor-Reden of the 1625 edition, where his son, Denys,
says that the book was published by the father ' eerst in't laer
1600. ende dit is al de vierde mael dat het gedrukt is.' There
were various editions in the seventeenth century, including
the following: Haarlem, 1611 ; Rotterdam-Schiedam, 1624, 8
(p. 423) ; Haarlem, 1625 ; c. 1630, 8 (p. 423) ; Gouda, 1634, 8
(p. 424); Amsterdam, 1643, 8 (p. 424) I Rotterdam, 1653,
8 (p. 425); Amsterdam, 1675.
422 KARA ARITHMETICA
ARITHMETICA,
l^eten tonff/
toed fcljoone
C'jrempelen/feer nut boo? alle Coop*
Uefen/ f arteurjs/ Caflter^ / <8ntfan-
fitters/etc.
IAQVES VAN DER SCHVERE
VAN MEENEN.
Ku ter tijJtFrancoyfche School-meefter
tot HAERLIM.
FIG. 212. TITLE PAGE OF VANDER SCHUERE
PRINTED BOOKS 423
Vander Schuere's work on bookkeeping is mentioned under
the later editions (p. 424), although it was not published in the
sixteenth century. No effort has here been made to complete
the list of seventeenth-century editions.
This is one of the many practical arithmetics that appeared in
Holland about this time. It takes up the fundamental processes, rule
of three, fractions, the rule of practice, partnership, commissions, in-
heritance problems, profit and loss, interest, exchange, barter, alliga-
tion, and the various other rules in use at that time. It was one of the
most successful Dutch textbooks.
JACOB VANDER SCHUERE.
Ed. pr. 1600. Rotterdam-Schiedam, 1624.
See p. 421.
Title. ' Arithmetica//Oft// Reken-konft;//Verciert met veel
fchoone // Exempelen/ zeer nut voor alle vlijtighe // Oeffenaers
ende leer-ghierighe Aenvanghers// defer Konft/ etc. Ghemaeckt
door // lacob Vander Schuere, Meenenaer,// Nu ter tijdt Fran-
foyfche. School-meefter//tot Haerlem.//Ende nu int herdrucken
overfien ende ghebetert. // (Woodcut of Vander Schuere with
motto: Door siet den Grond.)//Tot Rotterdam,// Voor Pieter
van Waefberghe/ //Anno 1642.' (F. i, r.)
Colophon. 'Tot Schiedam, //Ghedruckt by Adriaen van Delf.//
Anno 1624.' (F. 208, r.)
Description. 8, 8.7 X 14.7 cm., the text being 7.6 x 12.9
cm. 2 ff. unnumb. + 2o6 numb. = 208 ff., 22-27 H- Rotterdam-
Schiedam, 1624.
See above.
JACOB VANDER SCHUERE.
Ed. pr. 1600. S. 1. a. (?), c. 1630.
See p. 421.
Title. ( Arithmetica oft Reken-konst.'
This edition is without date, the title page being missing, but
is probably c. 1630. F. I is missing. 9 X 1 5.6 cm., the text being
7.3 X 12.2 cm. 5 ff. unnumb. + 252 numb. = 257 ff., 35 11.
4 2 4 KARA ARITHMETICA
JACOB VANDER SCHUERE. Ed. pr. 1600. Gouda, 1634.
See p. 421.
Title. ' Arithmetica // oft // Reken-konf t . // Door lakob vander
Schuere, Meenenaer // Eertyts Frangoysche Schoolmeefter//tot
Haerlem,//En in't herdrucken by hem ouerfien // verbetert en
vermeerdert, en noch by-//geuoecht een kort onderricht van //
'tltaliaens Boeck-houden // ter Goude//By Pieter Rammefeyn,
Boeck-verkooper // inde Korte Groenen-dal, in't Vergult A, B, C.
//a 1634. W Akerfl : fee.' (The title page is elaborately en-
graved, with a portrait of the author surrounded by the following:
' Doorseit den Grond lakob Vander Schvere aet 50.') (F. i, r.)
Description. 8, 8.8 x 14.2 cm., the text being 7 x 12.7 cm.
8 ff. unnumb. + 208 numb. = 216 ff. Bound with this is ' Kort
onderricht //over het // Italiaens // Boek-houden ; // Nu int licht
ghebracht // Door lakob vander Schvere,' etc., with I f. unnumb.
+ 37 numb. = 38 ff., making a total of 254 ff. in the book,
28-33 11- Gouda, 1634.
See p. 423.
JACOB VANDER SCHUERE.
Ed. pr. 1600. Amsterdam, 1643.
See p. 421.
Title. 'Arithmetica //oft // Reken-konf t.// En een kort onder-
richt van't Italiaens Boeckhoude // Door lacob vander Schvere
Meenenaer. // By den Autheur overfien, verbeetert en vermeer-
dert. //Tot Amsterdam, //voor Michiel de Groot Boeckverkooper
// op de nieuwendyk inde Bieftkens Bybel. 1643.' (Engraving of
author with the following wording : ' lacob vander Schvere Mee-
nenaer. Out67.Iaer. Doorsiet denGrondt. Anno 1643.') (F. i,r.)
Description. 8, 9.5 X 14.7 cm., the text being 7.2 X 12.5 cm.
272 ff. (216 numb -f 8 unnumb. in the arithmetic), 26-31 11.
Amsterdam, 1643. Bound with this is ' Kort onder-richt //Over
het //Italiaens // Boeck-houden.// In't Licht gebracht // door //
Jacob van der Schuere. //t'Amsteldam,// By Michiel de Groot,
PRINTED BOOKS 425
Boeckverkooper // op de Nieuwendijck/tuffchen de twee Haer-//
lemmer Sluyfen. A. 1675.' (F. i, r., of the bookkeeping.)
See p. 423.
JACOB VANDER SCHUERE.
Ed pr. 1600. Rotterdam, 1653.
See p. 421.
Title. * Arithmetica // ofte // Reken-konft, // En een kort
onder-richt van't Italiaens // Boeck-houden.// Door // lacob van
der Schuere Meenenaer.//By den Autheur over-fien, verbetert
en // vermeerdert.//Tot Rotterdam, //Gedruckt by Pieter Waef-
berge, woo-//nende op't Steyger/ in de gekroonde // Leeuw/
Anno 1653.' (F. J > r -)
Description. 8, 8.7 x 14.1 cm., the text being 7 x 12.1 cm.
275 pp. (219 numb. + 8 unnumb. in the arithmetic), 26-33 H-
Rotterdam, 1653. Bound with this is 'Kort onder-richt // over
het // Italiaens Boeck-houden. // In't Licht gebracht // door //
Jacob van der Schuere.// Tot Rotterdam, //Gedruckt by Pieter
van Waefberge,//Ordinaris Drucker/ woonende op't Steyger//
in de gekroonde Leeuw/ Anno 1653.' (F- J > r -> f tne book-
keeping.)
See p. 423.
GIOVANNI BATTISTA ZUCHETTA.
Ed. pr. 1600. Brescia, 1600.
Born April 21, 1550. A Genoese arithmetician.
Title. See Fig. 213. The privilege is dated ' In Genoua nel
di 30. di Genaro. M. CCCCCC.'
Description. Fol., 24.2 x 33.4 cm., the text being 17.1 X 26
cm. 444 pp. (412 numb.), 33-37 11. Brescia, 1600.
Editions. There was no other edition. Brunet says this ' Prima
Parte ' is the only one that appeared.
The work has several interesting features, not the least one being the
apology ' Al generoso lettore,' in which the author speaks of the criticism
liable to be directed against a Genoese author on account of his pro-
vincial Italian. The ' Prologo ' is a curious dissertation on the Arti,
426
KARA ARITHMETICA
ato che rifolue qualunque qneCto bifogaofo
a ZecchierijOrefici,* ArgentarL
Cvi
CO 7^
FIG. 213. TITLE PAGE OF ZUCHETTA
PRINTED BOOKS 427
Scienze, & altro,' with some ninety-eight arguments to show the need
for arithmetic on the part of all classes of humanity. The farmer, the
musician, the thief, the cook, the prelate, all are shown to have need of
number ; and Nature, Intelligence, and even God himself make use of it.
The book presupposes a knowledge of the arithmetic of integers,
and opens with a treatment of fractions. The rule of three, in all of
its forms, and with most unbusinesslike numbers, is then discussed at
great length, and this is followed by various complications of the Regola
del Cattaino, ' cosi detta da gli Arabi inuentori di quello, ch' in lingua
nostra fignifica falfa pofizione.' The latter part of the book treats of
such topics as partnership, barter, and alligation. The work was not of
a nature to have any influence on Italian arithmetic.
WILHELM SCHEY. Ed. pr. 1600. Basel, 1600.
A German Rechenmeister at Solothurn. Born c. 1560.
Title. See Fig. 214.
Description. 4, 15.1 X 19.4 cm., the text being 10 x 15 cm.
486 pp. (470 numb.), 36-37 11. Basel, 1600.
Editions. That this is the first edition appears from the dedi-
cation : 'Datum den 12. Septembris. Anno Domini 1600.'
The book is an attempt at a complete commercial arithmetic, but is
not well constructed. The author likes to arrange his computations in
a bizarre fashion, for effect. He extends his explanations unduly, and
for so large a book the mercantile information is not as complete as it
should be. There is a curious arrangement of the figures
in the proofs by casting out nines, as here shown, although
whether this is a typographical matter or a notion of Schey's
there is nothing to indicate. In the ' Regula Falsi ' there is
a rather early use of -f- for the minus sign. In general the
book is reactionary, giving only the galley division (with
much attempt at effect), and mentioning, although not treating,
' Duplatio ' and ' Mediatio.'
Other works 0/1600. Barlaamo, p. 343, 1572 ; Buscher, p. 393,
1590; Chambers (see Barlaamo), p. 343, 1572; Herodianus, p. 60,
1495 ; Hylles, p. 396, 1592 ; Jacob, p. 298, 1560 ; Lonicerus, p. 253,
1551 ; Ramus, p. 263, 1555 ; Reisch, p. 82, 1503 ; Reinhard, p. 421,
1599; Schulze, p. 383, 1584; M. van den Dycke, Chyfer-Boeck '
(second edition, I do not know the first), Antwerp, 8, and ' La vraye
reigle d'Arithmetique ' (a translation of the other?), ib., 8; Georg
Gleitsmann, Kiinftliches Rechenbuch fowohl auf Linien als mit Ziffern
428
KARA ARITHMETICA
ARITMETICA
Att^ ttd^ttttt titttb ^itt^di
*>&cr izoo. (^jrcmpl<i
ri&cit/bermafrm/l>a8Ctn
^M ^ W Wuntttic^en ferid^ t/bon im fcl&tj/ WPl
ncn anfafxntm ^cfyuUrn/imft cimm K^n ^cgimg
Anno M DC
FIG. 214. TITLE PAGE OF SCHEY
PRINTED BOOKS 429
nach defs Rami Arithmetica geftellt,' Frankfort, 8 ; M. Johann Taf,
' Schones neues . . . Rechenbuch,' Cologne, 4. There was also pub-
lished, s. a., but c. 1600, a work by Vincenzo da Bergamo, ' Arith-
metiche instruttioni.'
There were many other arithmetics published in the sixteenth cen-
tury without date, including the following : Anonymous, ' Art et Science
de Arismetique,' Paris, 12, 96 ff. (Boncompagni sale); Anonymous,
* Livre des gects,' s. 1., 4 (see p. 130, 1520) ; Anonymous, ' Abbaco di
succincte dimostrazioni,' s. 1. (Milan?) (Brunet) ; Anonymous, ' (Q)ui
apresso e inaci col nome di dio intedo di tractare e scriuere alquatimo
di e regole sopra larte del numero altrimeti chiamato algurismo,' s. 1.,
33 ff. (Riccardi) ; Joachim Ammonius, ' Isagoge Arithmetices . . . cum
praefatione P. Melanchthoni,' Wittenberg ; Angelus Mutinens, p. 140,
1525 ; Sarafino da Campora, * Delia ragione dell' Abbaco,' and a work
on the calendar (Messina, 1559, and Rome, 1560) (Riccardi); Lauro
Quirini, ' De mysterio numerorum ; ' Matteo Ricci, ' T'ung-wen suan-
chih ' (Practical arithmetic in 1 1 books, the work of a Jesuit missionary
in China, b. 1552, d. 1610; possibly not published during his lifetime).
PART II
MANUSCRIPTS
^tmar cnic Uncc tatt futt;"ttcttlt
PLATE IV. FROM THE CAMPANUS MANUSCRIPF OF EUCLID, c. 1260
MANUSCRIPTS
EUCLID. Latin MS., c. 1260.
See p. n.
Title. ' In hoc libro otinet' geomet'a euclidis // cQ omento
magri campani.' (F. i, v.)
Colophon. ' C.Explicit geometria euclidis cu comen//to magiftri
campani.' (F. 165, v.)
Description. Fol., 17.8 x 25.8 cm., the text being 8.5 x 17.2
cm. besides the marginal figures. 165 ff., 23-44 11. Written on
vellum c. 1260.
Editions. This work being primarily a geometry, I have not
given a list of the editions. For the arithmetical books, which
(except for Book V, on proportion) are not included in this
manuscript, see p. 237.
This is a Latin manuscript of Euclid, with the commentary (proofs)
of Campanus, written on vellum about 1260. The translation of the
theorems is that made by Adelard (^thelhard) of Bath, c. 1 120, but in
the early printed editions of the ' Elements ' it is generally referred to
as that of Campanus.
Of Campanus himself not very much is known. His first name was
probably Johannes, and he is known to have prepared a set of planet-
ary tables, and to have been chaplain to Urban IV. (See Boncompagni's
Bulletino, I, 5, and XIX, 591.)
This manuscript has been studied by Mr. C. S. Peirce (Science, N.S.,
XIII, 809), who believes it to be the copy given by Campanus himself
to Jacques Pantaleon when the latter was Patriarch of Jerusalem, hence
before August 29, 1261, when he became Pope Urban IV. He bases
his belief on a sentence written in a cursive hand just below the colo-
phon, containing the words ' Jacobus Dei gratia Patriarcha Jerusalemi-
torum.' The complete sentence is : 'In noTe dm ame Jacobus dei g ia
433
434 KARA ARITHMETICA
patriarcha Jerufalemitar omibus xpi fidelibus falutem defiderium,' ' In
the name of the Lord, amen; Jacob by the grace of God patriarch
of the Jerusalemites to all the faithful of Christ, greeting and love.'
While this seems more of a blessing or quasi imprimatur than a mark
of ownership, it is equally valuable in serving to fix the date. On f . i , r.,
is the inscription in a fourteenth-century hand, ' mgri adolphi di Werda,'
and a statement that the manuscript belonged to the Phillips collection,
no. 4633. On f. 165, v., is an inscription in an English hand of c. 1400,
' lib 6 ifte fuit Di armachani,' ' This book belonged to Dominus Arma-
chanus.' (For a page of this MS. see Plate IV.)
ANICIUS MANLIUS SEVERINUS BOETHIUS.
Latin MS., c. 1294.
See p. 25.
Title. The manuscript begins : ' Incipit prolog in arithmetics
boetii.' (F. i, v.)
Colophon. ' Explic* arifme ca // boetij ad fymacu^ patriciu c .'
(F. 28, r.)
Description. Fol., 19.8 X 27 cm., in double columns, each being
7 x 20. i cm. 28 ff., 41 11. Written on vellum c. 1294. Bound
with the Euclid described below but in a different hand. It is
beautifully written and illuminated. The contemporary pigskin
binding has the inscription, 'lib' arifmetice boecij.' The text
is practically that followed by the Friedlein edition (Leipzig,
1867). (See Plate I.)
Editions. For printed editions see p. 27.
See p. 27.
EUCLID. Latin MS., c. 1294.
See p. ii.
Title. The work begins without title : ' Punct^ eft cui9 ps He.'
(F. 29, v.)
Colophon. ' Explicit geometri euclidis cum comto campani.'
(F. in, v.)
Description. Fol., 19.8 x 27 cm., the text being 17.5 X 19.5
cm. in ff. (the first 28 ff. being the arithmetic of Boethius
MANUSCRIPTS 435
described on p. 434), 44 11. Written on vellum, c. 1294. It has
beautifully executed figures and is a fine specimen of the work of
the mediaeval scribe. It is written in a different hand from that
of the Boethius with which it is bound. The cover, which seems
contemporary with the manuscript, has the number cclxxxxiiij,
possibly for the date 1 294, the M being omitted as is often the
case. (See Fig. 215 and Plate V.)
ANICIUS MANLIUS SEVERINUS BOETHIUS.
Latin MS., c. 1300.
See p: 25.
Title. The manuscript begins: Tncipit libr arismetice art . . .'
Description. Fol., 13 x 18.6 cm., the text being 8.9 x 14.1
cm. 37 ff., 32-36 11. Beautifully written on vellum. The work
is complete, and, like the manuscript described on p. 434, this
shows a text very similar to that followed by the Friedlein
edition. At the end of the manuscript are two folios of com-
mentary, closing with two almost illegible lines containing the
words ' Com . . . campani . . . ,' referring to the commentary of
Campanus. Roman numerals are used throughout the text,
which was not always the case in manuscripts of this date. (See
Plate VI.) The commentary, which seems to have been added
about a century later, has some Hindu-Arabic numerals.
PAOLO DAGOMARI. Italian MS., c. 1339.
PAOLO DELL'ABACO, PAOLO ASTROLOGO, PAOLO GEOMETRA, PAOLO ARIS-
METRA, PAUL OF THE ABACUS. Bom in Prato, c. 1281 ; died at Florence
in 1374, or, according to some writers, in 1365. He was a celebrated Flor-
entine arithmetician, 'geometra grandissimo, e peritissimo aritmetico . . .
diligentissimo osservatore delle Stelle, e del movimento de'cieli,' as Villani
(Le Vite cT Uomini ilustri Fiorentini) calls him.
Title. ' Trattato d' Abbaco, d' Astrono-//mia, e di segreti natu-
rali //e medicinali.' (F. i, r.)
Description. Fol., 21.7 X 29.3 cm., the text being 15 X 21.4
cm. 138 ff. (7 blank), 32-35 11., clearly written on paper, c. 1339
(possibly copied later).
436 RARA ARITHMETICA
"
UJTO mCOn Hat eft' ^a
biojor^ cw > ftt^mtiftfo.l lm"7
Vim
aranim
^V o <f Ht
<n 4 fovlar-
m Jnmft vV awufl f&rquoufip nu
jiff \^tt*
x
? ! {~. *~nt - ' " ^w 11 * nn mcMctaa c i yar
/ V * i -1 ' * t wyd.*
A ? l>-w <
FIG. 215. FROM THE 1294 MANUSCRIPT OF EUCLID
Bernardino Baldi testifies to the esteem in which the author of this
work was held, in the following words : ' Di patria Fiorentino fu Pauolo ;
il quale, per 1' eccellenza ch' egli hebbe ne le Matematiche, lasciato il
proprio cognome, fu chiamato da tutti il Geometra. Come apunto fra
gli antichi auenne ad Apollonio Pergeo.'
This manuscript is primarily a treatise on arithmetic. The writer,
however, left a number of blank pages at the end, and these have been
filled in from time to time by various owners.
\vdWf-i?.iu ft-'wruifiTT -Ouf .
U.TiuiijT 4 - P" cqleui ne.< f.lun-nuicr >M
,leiHnc.< f.UinTiiug-^m O.j,^. .^ihniOTnu/intJJ
aatmwnn.mouqaw)umiU'Utq7^'-<:nrq,-^Y lvts -^ 1|! lt > t I Mni<qbpa.vq
^rtnt^tnll)^<^irtm^.^f-tq^ui4.nuyAdt|.v't^l 1 an(f.i.ap.5v5.bujlciiKi
Ct sf'fr u'innlF-7nirli3io co^ithfmi n^nni(.l>f'j|hntnjtF\'>tt^'f8^'i
icUoawji rocfctucf U^tnu
'^* V/ Ki ^, fpaUflDyu.-!^ tqdrthtaau; ""Sileto
Pw^W^-lSoSKSfah^^Sa^ttS^
PLATE V. FROM A MANUSCRIPT OF EUCLID, c. 1294
MANUSCRIPTS 437
The arithmetic begins (f . i , r.) : ' El nome sia di Dio et a reverentia
//della fuo potentia et della fanta trinitade. Et dello fuo madre//
uirgino fempre fanta maria Et del beato Sco Giouani batifto // . . . Al
chominiciamento del noftro trattato . . .' On f. i, r., is a table of
V
T
FIG. 216. FROM THE DAGOMARI MANUSCRIPT
money. F. 2, r., f. 3, v., is a table of contents : ' Quefti sono echapitoli
del noftro tractato,' and this, states that the work includes the * Re-
gholvzze del Maestro Pagolo Astrologo.' (ff. 121, r., 131, r.). This
part of the volume is an ordinary commercial
arithmetic such as the Florentine teachers pro-
duced in the fourteenth and fifteenth centuries.
The writing and the forms of the numerals indi-
%'
1 I
-\ 9
cate this period, and are not unlike that of a
fourteenth-century computus (see p. 445) in this
library. This may, however, be a fifteenth-century copy. The exam-
ples in division are unique, since they follow neither the galley nor the
'a danda' method, as is here shown in the case of 49289^-23 =2143.
The peculiar position of the remainders should be noted. The o indi-
cates no remainder, and the 2,2, and i are excesses of y's in the proof.
438
KARA ARITHMETICA
That the author considers this the ' a danda ' plan (the forerunner of
our present long division) appears in his use of this name immediately
after. Another odd feature is the placing of the divisor second in the
FIG. 217. FROM THE DAGOMARI MANUSCRIPT
division of fractions, it usually appearing first in that period. The
problems are of the usual Florentine mercantile character. The first
few are followed by a treatise on 'Nvmeri perfetti ' (f. 67, r.), and this
MANUSCRIPTS
(f. 73, r.) by further business problems. The work
curious illustrations (Figs. 217, 218).
The most interesting feature of the treatise is,
however, the internal evidence as to its date. It is
usually possible to determine quite accurately the
date of a Florentine manuscript on arithmetic by
the examples in equation of payments, a favorite
application with the Tuscan arithmeticians, and one
requiring the year to be stated. This is the case here,
where the dates in these examples are all, save two,
1339. The writer has also used approximately these
dates in other examples, in part as follows: 1329-
1332 (a problem on the calendar, f. 27, r.) ; 1310-
1404 (an astronomical table, which would naturally
extend well into the future, f . 79, r.) ; 1 330 (f. 1 1 6, r.),
and 1339 frequently. A table of the ig-year cycle
(f. 123, v.) begins with 1337, as is easily computed
from a marginal note in a different hand bearing the
dates 1394 and 1412. This part of the work ends
on f. 131, r., and there can be no doubt that it was
written about 1339, the date so frequently used in
the problems.
F. 1 2 1, r., begins ' Regholvzze del Maestro Pagolo
Astrologo,' Dagomari being referred to in the table
of contents as Paul of the Abacus, ' Regholuze del
Maeftro pagholo delabacho . . .' The ' Regholvzze '
was first published in Libri's Histoire, vol. Ill, p. 296.
Dagomari is included in the list of Bernardino
Baldi's (1589) biographies, published in the Bon-
compagni Bulletino, XIX, 600.
Ff. 131, v., and 132, r., are in a different hand.
These and the following leaves were originally blank,
and after ff. 132, v., 133, v., had been written
upon, some owner used these two blank pages. He
has also left his dates, viz. 1400, 1402, 1406, 1412,
1435, the 1402 being used several times. This was
therefore written about 1400-1435. In this occurs
the first per-cent sign I have met, other than p. i oo,
which is in the earlier part of this volume. This
unknown writer of about 1425 uses a symbol which,
by natural stages, developed into our present %.
439
contains several
FIG. 218. FROM
THE DAGOMARI
MANUSCRIPT
440 KARA ARITHMETICA
Instead of writing 'per ioo/ 'p 100,' or 'p cento,' as had commonly
been done before him, he wrote ' p <_? ' for ' p c,' just as the Italians
wrote !,,... and i, 2, . . . for primo, secundo, etc. In the manu-
scripts which I have examined the evolution is easily traced, the <_?
becoming g about 1650, the original meaning having even then been
FlG. 219. FROM THE ADDITION (c. 1400-1435) TO DAGOMARI
lost. Of late the ' per ' has been dropped, leaving only % or %. See
Figs. 219, 220.
Ff. 132, v., and 132, r., are in a different hand. They refer to the
calendar, and contain the dates 1380 and 1382, each twice. F. 133,
v., is in a still different hand, although also on the calendar. The date
of this part is fixed by the expression, ' et i ifto prefeti aho. f. 1447 finit
circuluf anni.'
EUCLID. Latin MS, c. 1350.
See p. ii.
Title. None. Fragments of the ' Elements.'
Description. 4, 16.5 x 21.4 cm., the text being 10.5 x 15.8
cm, with marginal drawings. 6 ff. unnumb, 32-41 11. A Latin
manuscript, written on vellum in a hand of c. 1350.
The manuscript is a fragment of 6 folios, and includes part of Books
III and IV, and a list of propositions evidently based upon, but not
identical with, Euclid's sequence.
^viiKTc.1 tucdxof cvpUcjr ; -^ ^^ ^r
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Ana m-ou -
utniT. $$r.H.ru\fcnr cv ur?a tif w.tn-.-u.u .o-Ktf
ir tcnitl JwW . ] b ucrwr t'.u : fi tutl avf
c^i.w f x \ M /^ntriwfllbtf 4tn*Sli'*utti f.]:il -.f i- ^.Mnc
uicmr.ft..r.ab.viu.t .^ ici
k . C -r cf XWMC fxib mmtracib 4 idmi . c. ct ^oi ftr ?r
cy^lty^ttcf tint*'ca[tui-v. (^ ^ fjf ^i- <j<ocf . tunico').
N 4 * I r "0n C4 . 1 itnc^t^T 1
-- u ^, t ,,;;t^;-!!
Au
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tit
ii , f . " !
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j^^j^^^r^TTsrrdSH ^tesc^sSIvJ
, ^ ^ " ltnrfcl ^.JT^^^^feAnd^wcA . .w.?^R ^t^mril.^.^tH^S^.
V7 m, w Aii ' m U ^;U x ^|^u r i iftu.^.- muck cvunnacvf tndi..
i
PLATE VI. FROM A MANUSCRIPT OF BOETHIUS, c. 1300
MANUSCRIPTS 44I
Ji ^
fl
<n
Ortfl
*s
610
\4~*
i
>?
I
iSi. ^
FIG. 220. FROM A 1684 MANUSCRIPT, NOT HERE CATALOGUED
442 KARA ARITHMETICA
ALBERTUS MAGNUS. Latin MS., c. 1350.
ALBERTUS TEUTONICUS, DE COLONIA, or RATISBONENSIS. Born at Lau-
ingen, Swabia, c. 1 193 ; died at Cologne, November 25, 1280. He was Count
of Bollstadt, a Dominican priest, and Bishop of Regensburg. He studied at
Padua and taught at Bologna, Strasburg, Freiburg, Cologne, and Paris. He
was so prominent as a philosopher that he was known as ' Doctor Universalis.'
Title. 'De C^lo et Mundo.' (F. i, r.)
Description. 22.4 x 32 cm., the text being 14 x 21.3 cm. with-
out the marginal notes, and arranged in double columns, each 6.5
cm. wide. 90 ff., 50 11.
This is a beautifully written manuscript, in a fourteenth-century
hand, with fine initials in red, black, and blue. The first folio contains
part of the calendar, and a few random memoranda, including an old
price mark of three ducats. The twenty-ninth folio is blank except for
some crude circles, and one folio has been cut out. There is a break
here in the manuscript, f. 28, v., closing with these words from the fourth
tractatus of liber I : * Rei a' gnate e ultim i finis in tpr aut actu aut po.
quia si e corrupta tuc ht actu fine 2 si e adhuc corrupt ' (* Rei autem
generatae est ultimum et finis in tempore aut actu aut potentia : quia
si est corrupta, tune habet actu finem : et si est adhuc corruptabilis ' -
the rest of the sentence being * habet finem potentia.') After the
blank folio, f. 31, r., opens with these words from tractatus I of liber II :
' que eft f b'a sepata ' (' quse est substantia separata,' the preceding miss-
ing words of the sentence being ' quse tamen non limitant operationem
formae ejus ') Aside from this the manuscript is complete. F. 89 closes
in col. i with the words ' Explicit liber de celo et mudo fratris alberti deo
agamus gratias.' F. 90, r., is blank, but 90, v., has the zodiac and planets.
There is a brief reference to Pythagorean arithmetic in liber I,
tractatus I, caput II.
The best edition of this work is in ' B. Alberti Magni // Ratisbonensis
Episcopi, ordinis Prsedicatorum,// opera omnia,// . . . cura ac labore //
Augusti Borgnet.// Volumen quartum // Parisiis . . . MDCCCXC.' The
first edition of the Opera Omnia appeared in Leyden in 1651.
EUCLID. Latin MS, 1375.
See p. 1 1.
Title. The work begins : ' Punctus eft cui9 ps no e".'
Description. Fol, 20.9 X 30.1 cm., the text being 11.9 x 19.5
cm. 39 ff. unnumb. + i blank = 40 ff, 44-47 11. A Latin man-
uscript, written on paper, c. 1375.
MANUSCRIPTS 443
This manuscript includes the first five books of Euclid. This is fol-
lowed by a treatise upon astronomy and mensuration, also in Latin, in a
different but contemporary hand. The mensuration includes some work
on areas and volumes.
ANONYMOUS. Latin MS., 1384.
Title. None. A Computus manualis.
Description. 4, 13.3 x 18.8 cm., the text being 9 x 13.3 cm.
33 ff. (3 blank), 40-44 11., written on paper.
The first written folio has been torn, and the opening lines are miss-
ing. The work is that particular kind of mediaeval computus (see p. 7)
in which a finger mnemonic system is used (see p. 34). It is a copy of
an older treatise, the text of which is here written in Gothic characters,
the copyist's notes appearing in a smaller hand. As in most computi,
the numerals are generally in Roman, both in the text and in the
commentary, but the date is twice given, as follows (f. 19, r.) : 'ano
dnj 1000.300.80.4'; 'anno dnj 1000.300.80.4' (see Fig. 221). The
text is iri Latin, except for one page which is in French. A later owner
has written the date of his ownership, 1600 (f. 2, r.).
ANONYMOUS. Italian MS., 1393.
Title. ' CQui chomincia ilpologho d-1 conpoto //d-lcorfo d-1
folo i d-lla luno// Prolagho.' (F. 3, r., Fig. 222.)
Description. Fol., 22 x 29.7 cm., written in double columns,
each being 7.5 x 20.5 cm. 70 ff. (6 blank), 35 11.
This is an excellent example of a computus (see p. 7). The verse
' Thirty days hath September,' the only relic of the old computi now
familiar to most people, appears in this manuscript in the following
Italian form: * Trenta di a noumbre apile // giugno & fettenbre di
uentotto ( ) vno tutti glialtrj fono trentuno.' (F. 12, v.) (See
also p. 33.)
GIOVANNI, the son of Luca da Firenze. Italian MS., 1422.
The son of Maestro Luca, a celebrated Florentine arithmetician, mentioned
on p. 468.
Title. ' Trattato di aritmetica.' (F. I, r.)
Description. 4, 14.9 X 2 1.9 cm., the text being 9. 3 X 16.3 cm.
145 ff. 27-30 11.
444
KARA ARITHMETICA
.*
*o a* **- -ok *~x* frov~S> -|H * ^^r r 4 "*~ t * ^^T"
Jjftna *%3*Sg!^sfesi3
5JJT^ S, r.-^u^*- c^^*-^ 5 ?*- <^1;
"A fi^V'^ j*- ^1 \C<^ &*~n
biW^ril^S^^P
gS^-^J? ^.JHSZJ F-T ? ia^
itlii tftu*)
'qii^J w
tit Ufg?t0 flmuV; Vmntt
FIG. 221. FROM THE 1384 COMPUTUS
MANUSCRIPTS
445
This is one of the best examples of an early fifteenth-century com-
mercial arithmetic known. Florence was at this time an important
financial center, and the arithmeticians of the city were highly esteemed.
Several of the applications found in arithmetics for the next three cen-
turies had their origin here. Subjects like equation of payments and
partnership involving time, customs like ' days of grace ' in exchange,
and forms like time drafts in sets of two or three can be studied to good
advantage in the arithmetical manuscripts of this period.
ft* Srtttu rrc<lv*m ^ftx. prft 7 p*
FIG. 222. FIRST PAGE OF THE 1393 COMPUTUS
446 KARA ARITHMETICA
On f . i , v., the writer says he proposes to treat of arithmetic or the
abacus : ' arifmetricho volgharemente e chiamata abacho.' The word
1 abacus ' had come at this time, in Italy, to mean simply arithmetic,
the original meaning having been lost. Following a custom of Floren-
tine arithmeticians, Giovanni gives an extensive multiplication table, for
purposes of reference, and then begins at once with examples in com-
pound numbers and fractions, thus presupposing a knowledge of the
fundamental operations. These examples are of a mercantile character
and constitute the entire portion devoted to arithmetic. The examples
in the equation of payments serve, as usual, to fix the date of the manu-
script. They all refer to the years 1418-1426 (ff. 113-122). The
date is, however, fixed exactly by the closing lines of a folio near the
end : ' quefto libro Ifcriffe Giouannj del maeftro lucho dellabacho e
finillo quefto d dottobre 1422 - ' (f. 136, v.). In the first page is the
date 'ad 28 dottobre 1422.'
The book also contains a section on mensuration and the calendar,
with curious illustrations of the months (PL VII). The last five folios,
originally blank, seem to have been written by a different hand about
the same period.
ROLLANDUS. Latin MS., 1424.
A native of Lisbon, canon of Sainte-Chapelle, Paris, c. 1425.
Title. l Scientia de numero ac virtute numeri.' (F. 3, r.)
Description. Fol., 21 X 29.8 cm., the text being 15.5 X 21.5
cm. 168 ff., 28-31 11.
This is an exceedingly interesting manuscript, written in the year
1424. It was prepared at the command of John of Lancaster, Duke of
Bedford, son of Henry IV of England, at one time Protector of England
and Regent of France. To him Rollandus dedicates the treatise : ' Illuf-
triffimo ac fereniffimo principi metuendiffimo domino domino lohanni
patruo domini noftri regis ffrancie et anglie regent! Regnum ffrancie.
duci bethfordie Rollandus fcriptoris veftre celfitudinis phyficus vlex-
bonenfis fe ipfum ex debito iuramenti.' In 1423 Lancaster issued an
ordinance for the restoration of studies in the University of Paris, and
it was probably as a result of this that this textbook was written by a
Portuguese physician, Rollandus, who was then a canon of Sainte-
Chapelle in Paris. The dedication sets forth Lancaster's interest in
learning in France and the status of mathematics at that time. Rol-
landus covers all of theoretical arithmetic as then known, but takes up
no practical problems. He also treats of irrational numbers, a topic
MANUSCRIPTS
447
which is now considered part of algebra. It is doubtful if there is a
manuscript extant which throws more light upon the nature of French
university mathematics at the time this was written. Rollandus also
wrote a work on surgery and one on physiognomy. He may possibly
be the Rolland who in 1410 was rector of the University. Since this
manuscript is evidently a copy, others must have existed, but I have
found no reference to them. Fig. 223 shows
the forms of the numerals used at this time.
It should be noticed that the numerals that
have changed materially in form since the
twelfth century are 4, 5, and 7. These are
shown opposite the letters b, /, and /, re-
spectively, in Fig. 223. The changes in
the other forms have been more evidently
due to the fashion in handwriting. All of
the forms are, however, quite different from
the primitive ones found in the cave inscrip-
tions of India.
ANONYMOUS.
Italian and Latin MSS., c. 1430
and 1478.
Title. The first folio is missing.
Description. Fol., 17 X 22.1 cm., the
text being 11.2X14.4 cm. 183 ff.,
25-28 11.
This volume consists of two Florentine
manuscripts, one on commercial arithmetic
and the other on the computus. The por-
tion on arithmetic was probably written
between 1420 and 1444, these being the
extremes of the dates in the examples in
equation of payments. It is not unlike the
other contemporary Florentine arithmetics
described on pp. 443 and 464. The author
assumes the student's ability to perform the fundamental operations
with integers, although, after numerous multiplication tables, he gives
(see Fig. 224) an example under Multiplicha p modo de barichocholo '
(the Florentine name for our present method), and one under ' Multi-
plicha p modo de Quadrate' (the 'gelosia' method of the Venetians).
FIG. 223. FROM THE
1424 ROLLANDUS
448
KARA ARITHMETICA
FIG. 224. FROM THE ANONYMOUS MANUSCRIPT OF c. 1430
MANUSCRIPTS 449
In the latter part of the work, as if it were a new topic, the author has
introduced (f. 126, v.) a chapter entitled ' Partire a danda.' He closes
with several folios on mensuration.
The second part (beginning on f. 154, v.) is in a different hand. It
is a computus, and from the dates it would seem that it was written in
1475. There is a third part (beginning on f. 172, r.) consisting of a set
of religious verses, in a still different hand, bearing the date 1478. The
language is Italian, excepting for the computus which is Latin.
ANONYMOUS. Latin MS., c. 1435.
Title. ' De tempore de compositione quadratis et compositione
astrolabii plani,' etc.
Description. Bound with the 'Computus cyrometralis ' of 1476.
Written on paper, in a German hand. Some dates which it con-
tains indicate that the book was written about 1435. It is related
to the history of arithmetic only in the forms of numerals used.
JOANNES DE GMUNDEN. Latin MS., c. 1439.
See p. 117.
Title. None. A treatise on the computus.
Colophon. On f. 17, v., are the words, ' Explicit kalendariu mgri
Joh'is gmund.'
Description. Fol., 19. i X 25.7 cm., the written part being about
14. IX 18.5 cm. (The tables vary and are larger.) 20 ff., 32-38 11.
Latin manuscript. Written on vellum, in red and black, about
1439, as shown by the dates on f. 16.
Although entitled a Kalendarium,' it is so much like the mediaeval
computi as to have a place in this list. It is really a semiarithmetical
treatise on the Church calendar.
ANONYMOUS. Latin MS., c. 1441.
Title. None. A computus.
Description. Fol., 21.5 X 28.5 cm. (varies), 10 ff. (i blank);
the number of lines to each folio varies considerably. Latin manu,
script, c. 1441.
The first part of this manuscript is a computus, not very extended.
It relates rather to the calendar itself than to the computations upon
450 RARA ARITHMETICA
which it is founded. The last folio is in a later hand, and from the
tables which it contains it seems to have been written c. 1524.
ANONYMOUS. Latin MS., c. 1442.
Title. None. A treatise on the planets.
Description. Bound with the ' Computus cyrometralis ' of 1476.
Written on paper, in a German hand. It contains the date 1442
in two places (f. 102). Its value in connection with arithmetic
is confined to the forms of the numerals used.
JOHANNES SACROBOSCO. Latin MS., c. 1442.
See p. 31.
Title. An algorismus, beginning with the words, ' Omnia que
a primeua rerum origine processerunt.' (F. 3. r.)
Description. 4, 14.1 X 20 cm., the text being 10.5 X 17.6 cm.
21 ff., 25-3911.
The first folio has a picture of an astronomer with a celestial sphere,
and the name of the student who copied the MS., ' hainricus muglinchk '
(f, i, r.). On the next page (f. i, v.) Mugling's name again appears,
together with three dates: 'Item hainric^ mugling aftronimV 1442,
1443, 1444. The ' Algorismus ' begins on folio 2, r. : ' Omnia que a pri-
meua //reru orignj,' and ends on f. 10, v. : ' Explicat algorafmus.' Then
follows a picture (PL VIII) in colors, representing a master teaching his
pupil the Hindu numerals from a kind of large hornbook, with a motto :
'Ich pin algorifm9 genant//Das ... (?)... hau ich in mein//nes
hant.' A table explaining place value is given on f. n, r. Beginning
on f. n, v., is another treatise on arithmetic, giving the fundamen-
tal operations and some work in progression, and ending with a multi-
plication table (ff. 1 8, v. ; 19, r.). A later hand has added three pages
on progressions, rule of three, partnership, and interest. (Ff. 19, v. ; 20.)
Sacrobosco's algorismus was the first arithmetic, based on the new
numerals, written by an English scholar. It consists of eleven chapters,
viz. Numeratio, Additio, Subtractio, Mediatio, Duplatio, Multiplicatio,
Divisio, Progressio, Perambulum ad radicum extractionem, Extractio
radicum in cubicis. For the various editions of this work see p. 32.
The title of the Paris edition of 15 10 is ' Opvscvlvm de praxi numerorum
quod algorifmum vocant,' and the work consists merely of four folios
containing the chapter ' De arte numerandi.' There was also published
at Antwerp in 1547 (with later editions, Paris 1550, Venice 1564,
MANUSCRIPTS 451
Wittenberg 1578), Sacrobosco's * Libellvs, de anni ratione: seu vt voca-
tur vulgo, Computus ecclesiasticus.' His work on the Sphere was pub-
lished in 1488 at Venice. For a discussion of the authenticity of the
Algorismus, see De Morgan, p. 13.
As stated on p. 31, the date of the death of Sacrobosco is uncertain.
It is either 1244 or 1256, according as we interpret certain lines on his
tomb:
' M. Christi bis C quarto deno quater anno
De Sacro Bosco discrevit tempora ramus,
Gratia cui dederat nomen divina Johannes.'
MARO ANTONIO ROZINO. Latin MS., 1447.
Title. ' Qories Marci Antonij rozoni artiu // doctoris sacre the-
ologie magn et papie phiam legends.' (F. i, r.)
Description. Fol., 20.3 X 28.1 cm., the text being 13.7 X 24
cm. 99 ff., 32-33 11.
This is not an arithmetic, and it has been included in this list only
because it is semi-mathematical and is bound with Bradwardin's treatise
on the theory of proportion. It is a Latin treatise on the theory of per-
spective. It is written in a clear Italian hand, and was part of a volume
numbered 493 in the Boncompagni sale, containing four manuscripts.
Some dealer has removed the first of these manuscripts, the Perspectiva
communis ' of John Peckham, archbishop of Canterbury. A memo-
randum in the Boncompagni catalogue, probably from a leaf removed
with the first treatise, shows that the manuscript was copied in 1447 :
' scripte per me antonium confaronesium ut (vocatur?) de lavilata Anno
drii M"ccccxlvij.' The other manuscripts bound with this are described
below and on p. 452.
THOMAS BRADWARDIN. Latin MS., 1447.
See p. 61.
Title. None. A treatise on proportion.
Colophon. ' Expliciiit propor nes thomi brardi //scripte p me.'
(F. 115, r.)
Description. Fol., 20.3 X 28.1 cm., the text being 13.7 X 22.8
cm. 15 ff., 33 11.
This forms part of the volume last mentioned, and is written in the
same hand. Bradwardin's treatise on proportion was published in
Paris, 1495 (p. 61).
452 KARA ARITHMETlCA
ANONYMOUS. Latin MS, 1447.
Title. A treatise on lenses.
Description. Fol., 20.3 X 28.1 cm., written in two columns,
each being 6.6 X 22.9 cm. 7 ff, 24-33 H-
This rather early work on optics forms part of the volume mentioned
on p. 451, and is written in the same hand.
ST. BERNARD OF SIENA. Latin MS., c. 1450.
Born at Massa, Tuscany, in 1383; died at Aquila in 1444. He was a
zealous founder of monasteries, and wrote various religious treatises.
Title. This book of sermons begins as follows: ' S mo .33.9 .de
reftitutioe.// Dorii .q. in quadrageffla ordo dicendo^ p feptimana
fequete3 I li de x'ana re//ligione .a. R do p. S. B. no de fenif ordonif
mino^ edito.' (F. i, r.)
Colophon. ( Explicit tra//ctatuf de uf3if 2 pctib3 303 Bntu3 //
Bnardim'3 (?) de fenif. ordif minor^.' (F. 129, r.)
Description. 4, 14.8 x 19.8 cm., the text being 10 x 13.8 cm.
129 ff., part vellum and part paper. After f. 60 the pages are
arranged in double columns, each being 5 x 14.5 cm, 44-46 11.
This beautifully written manuscript of c. 1450 is included in this
list because it contains several sermons bearing upon the mercantile
customs of the time, including ' De usuris,' * De cambiis,' ' De con-
tractibus,' and ' De mercatoribus.'
Part of this work was translated into Italian in the fifteenth century,
as appears from a codex in the Biblioteca Comunale at Siena. One of
the sermons was published by Riccomanni, in the Scelta di Curiosita Let-
terarie inedite o rare (no. 13) of Romagnoli, Bologna, 1862. This ser-
mon, ' Sulle Soccite di Bestiami,' contains considerable information as
to the business problems of monastic institutions of the fifteenth cen-
tury. A copy of several of these sermons, made by one Eustachio da
Feltre in 1469, is mentioned on p. 466.
ANONYMOUS. Latin MS, c. 1450.
Title. None. On the Quadrivium.
Description. 4, 14.7 x 21 cm, the text being 14.5 x 17 cm,
47 pp., 18-40 11.
This is a Latin manuscript on the Quadrivium. It is written on
paper, in a German hand of about 1450. It includes a brief treatment
MANUSCRIPTS
453
of arithmetic (5 ff. + i blank, the text occupying ff. 18 - 22, v.), and a
brief treatment of the calendar and the zodiac (ff. 15, r . 1 7, v.). The
geometry begins with modifications of Euclid's definitions : ' Pvnct9 eft c9
pars no eft/Linea e longido fn latitu"' 2 pfuditate.' It closes : C.Explicat
prim9 liber euclidis cu <ometo capani.'
Latin MS., c. 1450.
JOHANNES SACROBOSCO.
See p. 31.
Title. ' Spera mundi secundum Joha//nnem de facrobofco '
(F. i, r.)
Colophon. < Explicit tractat9 de fpa fm // Johem de facrobofco
am/ (F. 35, v.)
FIG. 225. FROM SACROBOSCO'S Spera, c. 1450
Description. Fol., 35 ff., written on vellum.
Printed works of this kind have not been included in this catalogue.
This manuscript, however, shows the state of the numerals about 1450,
and therefore is mentioned.
The treatise begins (f. i, r.) : ' De fpera in quatuor capitula diftin-
gui9 // dicentes p'mo quid fit fpera/ quid eius // centrum/ quid axis fpere
/et quid fit po//lus.'
The first four folios have a marginal commentary, closely written in
a later hand. The figures are carefully drawn throughout (see Fig. 225).
454 KARA ARITHMETICA
There is nothing to show the exact date of the MS., but the handwriting
and numerals indicate the last half of the fifteenth century.
Sacrobosco's * Sphere ' was published in 1488, and often thereafter. It
was the great mediaeval work on astronomy.
ANONYMOUS. Latin MS., c. 1450.
Title. ' Incipiut floref Arifmetrice.' (F. i, r.)
Description. 4, 15 x 21.8 cm., the text being 8.4 X 15.5 cm.
70 ff. unnumb. + 2 blank = 72 ff., 45 11. Latin MS., in a Ger-
man hand, c. 1450.
This is a theoretical treatise on arithmetic and algebra, written
respectively on the Boethian and Al-Khowarazmian models. Only a
little elementary treatment of the fundamental operations (chiefly mul-
tiplication) is given, the writer devoting most of his attention in the
first part of the work to subjects like progressions, ratios, and propor-
tions. The latter part of the book is algebraic and may prove to be a
copy of some mediaeval work of importance. It resembles in some
places the work of al-Khowarazmi, a manuscript of which follows this
one in the same volume.
MOHAMMED IBN MUSA AL-KHOWARAZMI.
Latin MS., 1456.
ABU 'ABD ALLAH MOHAMMED IBN MUSA AL-KHUWARIZMI. Born in
the province of Khwarazm (whence his name), died c. 831. The most cele-
brated algebraist of his time, and the first to write a book bearing the title
Algebra. From his name comes the word algorism (see p. 7).
Title. ' Liber mabometi de Algebra et almuchabilax comparif
et oppof.' (See Fig. 226.)
Colophon. There is none, but f. numb. 85 bears the date 1456,
and the forms of the numerals and letters are of that period.
Description. 4, 15 x 21.8 cm., the text being 12 x 17.2 cm.
23 ff. unnumb. + I blank = 24 ff., 44-48 11. Latin MS. in Ger-
man hand, 1456.
This interesting manuscript of the first book bearing the name alge-
bra is more complete than the one found by Libri in the Bibliotheque
Nationale (Histoire des sciences mathematiques, I, note XII). It more
nearly resembles the one which I found in the Columbia University
Library in 1904, and showed to be in the handwriting of Scheubel.
The two deserve to be edited and compared with the Rosen translation
MANUSCRIPTS
455
FIG. 226. FIRST PAGE OF THE 1456 AL-KHOWARAZMI
456 KARA ARITHMETICA
(London, 1831 ). This manuscript is particularly valuable because, unlike
the one published by Libri, it has the Arabic numerals and the mediaeval
algebraic symbols. The et used for plus so closely resembles the + as to
leave little doubt that the latter was derived from this Latin word. Like
Regiomontanus, the writer uses ^> for minus.
This is followed (ff. numb. 97) by a brief treatise on rhetoric, and
by three manuscripts on mathematics of 1501, c. 1475, and c. 1550,
described on pp. 480, 468, 486.
ANONYMOUS. Italian MS., c. 1450.
Title. A treatise on mensuration, roots, and algorism.
Description. Fol., 21.6 X 30.3 cm., the text being 10.5 x 15.8
cm. 177 pp. (2 blank), 20-25 H- Written on paper, in a Flor-
entine hand, c. 1450.
This is a general treatise on mathematics, with divisions as follows :
A 'praticha de Geometria' (ff. 2-11); fractions (ff. 13-16); square
root (ff. 17-19, r.) ; cube root (ff. 19, v., 21); ratios (ff. 22-26);
algebra (ff. 26-30); mensuration (ff. 31-46); algorism (ff. 47-177).
The algorism is evidently the work of some Florentine teacher, and
the handwriting is that of the middle of the fifteenth century. It
includes the usual applications of the period, such as profit and loss,
partnership, and interest, and it makes prominent the rule of three and
the rule of false position.
PETER PAUL VERGERIUS. Latin MS., c. 1450.
Born at Capo d'Istria, Venice, c. 1350; died in Hungary c. 1420, or pos-
sibly in 1444. Bishop of Capo d'Istria.
Title. ' De Ingeniis Moribus et liberalibus Studiis. Francis-
cvs//fenior auus tuus cui9 ut extant //plurime res magnifice gefte
ita//et multa paffim ab eo fapieter i// dicta memorantur. . . .'
(F. i,r.)
Colophon. ' Petrj pauli uergerij de uiftinopoli ad uber-//tinu3
Cararienfem de ingenijs moribs i / / liberalibs ftudijs adolefcentie
liber felici-//ter. Explicit.// Amen.' (F. 23, v.)
Description. 4, 14 X 20.3 cm., the text being 8.8 X 13.6 cm.
23 ff. (10 vellum, 13 paper), 26-36 11. The first 10 (vellum)
folios are in Gothic script; the last 14 are plainly written in
cursive characters.
MANUSCRIPTS 457
The treatise refers to ' arifmeticha ' and * geumetria '(.15, r.) among
the liberal arts. It is bound with several other manuscripts.
RAYMUNDUS LULLIUS. Latin MS., c. 1450.
RAYMOND LULLY, RAMON LULL. Born in Palma, Majorca, c. 1234; died
in 1315. He was a Catalan alchemist, philosopher, and missionary, and was
known as ' Doctor illuminatus.'
Title. 'Ars Brevis.' A note by a somewhat later hand, written
on f. i, r., reads : 'Ars breuis Raymundi Luli quam scripsit Pisis
//in monasterio Sancti Dominici anno 1307. a. 0.23.' The Ars
brevis begins on f. 5, r. : 'Deus cu tua gra fapia et amore Incipit
ars breuis que eft //ymago artis gnalis. Na ifta fata ab intelectu
fb'tili et fun//dato ipe pot fcire gnale arte.'
Colophon. 'Finiuit Raymundus arte breuen pifis in monaft'io
fac//ti dominici Anno ab incarnatione dni. 1 307. // Explicit feli-
citer.' (F. 23.)
Description. 4, 15.4x20.8 cm., the text being 9.9 X 13.2
cm. 24 ff., 28 11.
This is the second part of a manuscript of 59 folios, of which
the first is the Sensuale of Lullius. This and several other man-
uscripts are bound with the Vergerius already described (p. 456).
The Ars brevis was originally written in 1307 ; this copy was made
about 1450. Although not an arithmetic, the work contains several
mathematical definitions.
JOHANNES ROS. Latin MS., 1450.
A Valencian priest.
Title. ' Artificium artis arithmeticae.' (F. i, r.) '[DJEus qui
es unus . . . Incip artifi" 1 ? atis alfmet'ce De alphabeto.' (F. 24, r.)
Colophon. * Ad laudem // oipotentis dei et uirginis marie . . .
fi//niuit frat' Johanis ros de Valencia puincie aragonii hoc //
artificiu arifmetice pad ... I loco fratrum rnino^ de ofpitali //
1450 die .5. Januarij I uigilia epiphanie domini.' (F. 14 = 37, r.)
Description. 4, 15.1 X 19 cm., the text being 9.2 X 13.7 cm.
36 ff., 28 11. Bound with the Lullius and Vergerius described
above and on p. 456.
458 KARA ARITHMETICA
This rather early Spanish monastic treatise on algorism is the third
of three manuscripts in the same hand, written c. 1450, of which the
first two are mentioned on pp. 456, 457. It begins on f. 24, r., and
ends on f. 37, r. After some definitions, tables, and computus figures,
the author takes up the ' nine subjects ' : ' Noue f'biecta ponut I arif-
metica ' (f. 3 = 26, v.). These are treated very briefly. This is followed
by a religious work, in the same hand (to f. 57).
ANONYMOUS. Latin MS., 1442.
Title. ' De ymagine mundi.' (F. 2, r.)
Colophon. * <Ldeo grat. anj. (amen ?) Ano. dnj. 1442 . . . C.Ex-
plicit lib. 9 de ymagine mundi deo grat . .' .'
Description. 8, 14.5 x 21.5 cm., the text being 11.5 x 19 cm.
1 58 ff. in the entire manuscript, n ff. in this portion. The other
portions are described on pp. 477, 478.
This is the first manuscript in a collection of 156 folios, and begins
with the lines ' I. H. S.//C.de ymagine Mundi ' (f . 2, r.). It is written
in Latin and treats of physical and descriptive geography. It has some
interest in the history of arithmetic through the Roman numerals which
are generally employed except in the case of large numbers. This manu-
script and the others bound with it are described in Narducci's catalogue
of the Boncompagni manuscripts (Rome, 1862, no. 81 ). As there stated,
at one time it belonged to Alessandro Padovani, a celebrated collector
of the sixteenth century.
ANONYMOUS. Italian MS., c. 1456.
Title. The title is missing.
Description. 4, 14.6 x 21 cm., the text being n x 15.6 cm.
in ff., (12 with drawings), 30 11. No. 168 in the Boncompagni
sale catalogue.
This is a business arithmetic, written in northern Italy, and com-
pleted, as appears from a note, July 15, 1456. As in most of the busi-
ness arithmetics of the period, the first pages contain a set of tables,
these being followed by a discussion of the fundamental operations with
denominate numbers. Although written in 1456 it is probably a copy
of an earlier work of about 1420, for the examples in the equation of
payments involve dates from 1418 to 1425 (ff. 78-84). The work is
also interesting because it contains the early form of the sign % (see
Fig. 227) already mentioned on p. 439. The column tables used by
MANUSCRIPTS 459
merchants in their multiplication < per colonna,' and common in the
Italian manuscripts of this nature, are shown in Fig. 228.
FIG. 227. FROM AN ITALIAN ARITHMETIC OF c. 1456
ANONYMOUS. Italian MS., c. 1460.
Possibly by Raffaele Canacci, a Florentine mathematician.
Title. There is none given, but the work is a general treatise
on mathematics.
Description. Fol., 28.2 x 39 cm., the text being 16.5 x 28 cm.
2 blank + 322 numbered ff. = 324 ff., 5 1 11. Italian manuscript,
c. 1460.
This is an Italian manuscript, beautifully written on vellum, with
finely executed initials in colors and gold at the beginning of each of
its sixteen books. It belonged at one time to Libri, and later to Bon-
compagni. Narducci describes it in the catalogue of the latter's man-
uscripts (no. 14). The author begins (f. i) with a description of the
work : Come e in che modo eldetto trattato e diuifo/ cioe cio che
lopa cotiene.' The successive chapters are as follows :
460 KARA ARITHMETICA
r: " - <M - i..
i -*.n ** i 7 *
41 . >o .
-, - *"' *
> 4- I -*>S> *" 1 *
^ . 4-| |^ <M
| -f I
1 - *-| | -UJ, ^. ^ . ^ o
x's <e> " K
11~+ 4/ ^ - 4-
. ^l^J e "V O
,00
r
?
FIG. 228. FROM AN ITALIAN ARITHMETIC OF c. 1456
MANUSCRIPTS
461
I. ' Qui chomincia el pimo libro del detto trattato, & pima pone la
diuifione del detto primo libro/ laquale e achapitoli/ cioe e diuifo i .4.
capitolj.' (F. i, r.) The four ' capitoli ' are as follows :
1 . 'El primo capitolo del pimo libro/ doue fimoftra lordine e modo
del numerare le fighure chefufano afcriuere enumeri.' (F. i, v.) In this
are explained the Hindu-Arabic notation, the nine ' figure fignifichatiue,'
and the 0, ' che i arabia fidice cero.'
2. The title of chapter 2 is wanting, f. 3 having for some reason been
left blank. It related to the addition of integers and compound numbers.
3. 'Qui chomincia el terzco capitolo del pimo libro/ doue fitratta
del modo & hordine del trarre el numero minore del numero maggiore.'
(F. 5 , r.) The method is that
of borrowing and repaying.
4. ' El quarto capitolo
del primo libro di quefto
trattato/ Doue fitratta del
modo & hordine del @chare
e numerj.' (F. 6, v.) The
column form of the multipli-
cation table is first given,
XlX
K>-
FIG.
229. MULTIPLICATION
QUADRATO,' C. 1460
PER
and is followed by various
methods of multiplying. The
names ' El berichuocholo ' (f .
10, v.) and <p quadrato'
(f . u, r.) show the work to
be Florentine rather than
Venetian, and the handwrit-
ing and the numerous refer-
ences to Florence confirm
this fact. The method * per quadrato ' is shown in Fig. 229. Five pages
of multiplication tables are given, such an elaborate treatment having
been rather common in the arithmetics of that city. Division is not
treated in Book I.
II. 'El fecondo libro del detto trattato. Nelquale fi contiene la
natura & proprieta De numerj. & prima come e diuifo & aprj lontel-
letto.' (F. 17, r.) This is divided into two chapters :
1. 'El pimo capitolo del fjo libro/ Nelqual fitratta la natura che a
ife el nuo con diffinitionj fapute.' (F. 1 7, r.) This relates to such ancient
classifications of number as odd and even, prime and composite.
2. 'El fecondo capitolo del fjo libro/ Doue fitratta de numerj nomi-
nati per nomj apropiati alle fighure geometre.' (Figurate numbers.)
462
KARA ARITHMETICA
III. * El terzco libro della praticha darifmetricha. Nelquale fitratta el
modo di partire p nuj. & pima I che forma e modo el detto libro e diuifo.'
(F. 23, r.) The subject of division is now taken up in three chapters :
i. By the use of the table, el modo di partire per gli numerj fcripti
I fulle librettinj.' (F. 23, v.)
2. The ' ripieghi ' method (f. 28, v.), by the
successive factors of the divisor.
3. Long division. (F. 32, r.) The author
gives not only the galley method, but an early
description of the method ' a danda,' substan-
tially our present plan (see Fig. 230).
IV. This treats of common fractions. (F.
33. v -)
V. Ratios, ' quantita proportional].' (F.
46, v.)
VI. Mercantile arithmetic (f. 58, v.), with
numerous references to the customs of Floren-
tine merchants. Sixty-six large folios (132 pp.)
are given to this book.
VII. The rule of false position : ' El feptimo
libro diquefto trattato nelquale fitratta del
modo delafoluere de chafi p lofemplice modo
delchatain che p moltj fidicono principij del
chatain.' (F. 124, v.)
VIII. Simple and compound interest (f.
134, v.), with 2 pages (4 incomplete) of tables
giving the interest on ^100 for 1-21 yrs. at
rates varying from ' 5 p TOO. lano ' (f. 140, v.) to ' 40 p 100. lano ' (f.
150, v.). Equation of payments is also presented in this book (f. 152,
v.), the dates ranging from 1458 to 1464.
IX. False position as treated by Leonardo of Pisa. (F. 170, v.)
Leonardo is mentioned in PI. IX.
X. Miscellaneous problems. (F. 176, r.) These are largely tradi-
tional examples, and include the hare and hound, finding numbers satis-
fying given conditions, problems about eating, the jealous husbands, and
the testament complication.
XI. Proportion, based on Euclid V. (F. 225, r.)
XII. Algebra, 'della regola della algebra.' (F. 233, v.) Unfortunately
few of the figures for this book were drawn. The treatment is rhetor-
ical, practically no symbolism being used.
FIG. 230. FROM AN
ITALIAN MANUSCRIPT
OF C. 1460
^^rcu^^u^^.^
^"X.mpUxovu^o ^
CUA '
\;sh to -5SftS,Sit;s
J^ll ^ P( Vv ' Pn, crto dbbtno ^Ub(lKft
/ - >^ -NT - ^^ T3TIVI .
S^s^^^-^^^^^
^Sffi^asS
PLATE IX. FROM AN ANONYMOUS MANUSCRIPT, c. 1460
MANUSCRIPTS 463
XIII. Algebra continued, < la regola de Algebra amucabale.' (F. 279,
r.) This is a very interesting treatment of the subject fecondo ghugliel-
mo de lunis ' and ' Lionardo pifano,' and it throws some light upon alge-
bra as studied in the fifteenth century. (See Plate IX.)
XIV. Algebra continued (f. 295, v.), according to Master Biagio
(' certj cafi che ferine m? biagio nel fuo trattato di pratica') of 1340,
Master Gratia de Castellani (' fecodo che fcriue m? gratia de caftellani),
and Leonardo of Pisa.
XV. Algebra continued (f.- 312, r.), according to certain 'maeftrj
antichi,' viz. : ' Maeftro paolo,' ' m? Antonio,' ' m? giouanj,' Leonardo
pifano,' * m? biagio che circha al. 1340. anj morj,' m? paolo fiorj che
circha al. 1360. duro,' ' m? michele padre di m? mariano,' ' m? lucha,'
and ' un altro m biagio.'
XVI. This is missing, the manuscript ending with Book XV not
quite completed. On f. i, r., there is mentioned the ' fedecimo e ultimo'
chapter, and this might have contained the name of the author had the
work been finished.
EUCLID. Latin MS., c. 1460.
See p. n.
Title. None. The first book of the Elements.
Colophon. ' CExplicat prim^ liber euclidis cu ometo capani.'
(F. 14, v.)
Description. 4, 14.7 x 2 1.2 cm., the text being 9.9 X 1 5.8 cm.
22 ff., 18-40 11.
This manuscript of the first book of Euclid, with the commentary by
Campanus, has been included on account of the forms of the numer-
als used. It is written in a German hand of c. 1460, and is bound
(ff. 1-14) with the two manuscripts next described.
ANONYMOUS. Latin MS., c. 1460.
Title. None. A treatise on the calendar.
Description. See the preceding manuscript. This is ff. 1 5-17
of the volume.
This manuscript, bound with the Euclid just described, was probably
written by the same hand. It is a brief treatise on the calendar, and
was intended, as usual, for the Church schools. In it occurs the date
1460.
464 KARA ARITHMETICA
ANONYMOUS. Latin MS, c. 1460.
Title. None. On the Quadrivium.
Description. See p. 463. This is ff. 18-22 of the volume.
This is a general treatise on the quadrivium, and therefore contains
a chapter on arithmetic.
BENEDETTO DA FIRENZE. Italian MS, c. 1460.
A Florentine arithmetician of the first half of the fifteenth century.
Title. ' Inchomincia el trattato darifme//tricha efpelialmete
quella pte // che e fotto pofta alia mercatatia // e comminciando
alnome didio.' (F. n, r.)
q.q.1
7
t 7
f T*-9 78<?-
T.-H;
400
9
^ 9
u 9
90
FIG. 231. TABLES FROM BENEDETTO DA FIRENZE
Description. 8, 8.3 x 11.7 cm, the text measurement vary-
ing. 2 ff. missing + 348 ff. unnumb. = 350 ff, 20 11. Manuscript
on parchment.
This is a parchment codex written about the middle of the fifteenth
century. It lacks the first two folios and possibly also a few at the end.
It is one of the best examples of the mercantile arithmetics in Italy
MANUSCRIPTS
465
preceding the printed works. The author begins, as usual, with several
pages of tables (see Fig. 231), the multiplication table including the
prime numbers below fifty. There is also a table of squares and one for
the multiplication of compound numbers. Benedetto presupposes that
the reader is able to perform the fundamental operations with integers,
and he begins at once with operations on compound numbers and frac-
tions. The applications include exchange, partnership, and equation of
P'lG. 232. FROM BENEDETTO DA FIRENZE
payments, the dates in the problems including the years from 1460 to
1464. The latter part of the book contains a number of such traditional
problems as those of the grains of wheat on a chess-board, the couriers,
the hare and hound (see Fig. 232), the jealous husbands, and the testa-
ment of the dying man. The author closes with a brief treatment of
mensuration.
ALBERT OF SAXONY. Latin MS., 1462.
See p. 9.
Title. ' Tractatus proportionum. IncipiGt pportioef copo te a
dno albertutio.// [ ] Roportio coitr da e ducxp .operito^ in a co t'rio
uni co //ad inuice hltudo.' (F. i, v.)
Colophon. ' Explicat pportioef 2po te // p reuedo mro alberto
defifona // Finis //' (F. 12, v.)
4 66 KARA ARITHMETICA
Description. 4, 14.5 X 21.8 cm., the text being n.8 x 13.8
cm. (varies). 19 ff., 23 11. Bound with the Vergerius and Lullius
mentioned on pp. 456, 457.
This is the treatise on proportion that was printed without date, prob-
ably at Venice c. 1478 (p. 9). It is followed by another treatise in the
same hand, 'De latitudinibus formarum,' which bears the date 1462.
ST. BERNARD OF SIENA. Latin MS., 1469.
See p. 452.
Title. None. Sermons of St. Bernard (Bernardinus) of Siena.
Description. 4, 15 X 20.6 cm., the text being 11.5 x 14 cm.
130 ff. (12 blank), 28 11.
Editions. The works of St. Bernard were first printed in
Venice in 1591, 4, and again in Paris in 1636, 2 vol., fol. Some
of his sermons were, however, printed in Florence, and some in
Venice, in 1495.
This manuscript, containing several of the sermons of St. Bernard,
was copied by one Eustachio de Feltre in 1469. Two dates are given
showing the completion of parts of the work, September 22, 1469, and
October 6, 1469. For the reasons for including this manuscript, see
p. 452.
ANONYMOUS. Italian MS., 1473.
Title. 'Trattato della Arithmetica.' (F. i, r.)
Description. Fol., 16.7 x 23.7 cm., the text being 10 x 16.2
cm. 190 ff., 35 11.
This treatise was composed, as the first folio states (Fig. 233), in the
year 1473. I n tne examples in partial payments the dates given are
about 1490, and this particular manuscript may have been copied about
that time. There is, however, one example with the date 1392, which
would seem to indicate that at least part of the book was copied from
some earlier writer. There is also (f. 183) a brief treatment of the
calendar with two dates 1443. On f. 180 there are also the dates 1452
and 1453. In the margins some sixteenth-century dates have been added
by a later owner.
The work is beautifully written on vellum. The first part consists
entirely of arithmetic, treated from the mercantile standpoint. That it
is a copy of an earlier work also appears from the fact that a few pages
MANUSCRIPTS
467
A-emo pvtnetpio Alnornc Jt du> <^d
riQ/rxiTno
Jel Jo
_ __ Sro CrtcE.ntio.>Tv<ano
_ Jovtofa wazHrc. n-u^ S CD LrittL-ertHo cc
ov
ccitftuale. cot-hz. cltjocivcidifo c^ndli daxtoHximATTtcrtfu ^J>
mo ttJjbnn>o m^o^finc. acD7anidhio giv*X dihmac-
i (jiuJi* naoai tB feno dllovo |aticl<z.Q^ve.xte.renj
oni-a (ui. cUUmtX
f \dt a.^didi'cB.Tntrudt di S'cu" GorUiani
s
'on^tt miriifi
_ _ _ nclx p/ente.'^ihv. atmo c-ltx^q^fD JT(O j
to ctcJA. ndlui'Xi-at'tiotci/c ifri'ibuivc u
atlino (tcwz.
irii. oikc. at fUttut^tlc. cozfo
mo
rvactt V)in-ViA filLtcu, nie.nt. cttmeno (aiiioiiiYt\.^p^. at cjitGilj
e.-pnv Additive, audio ^/^ lovo J^4o dA UnAhirA coaJu
UJA wlonmA. t>Oti2jvhX-*(W<|udL. fcrnvji-G. i-i"a)ra_iajiitzllx 'pv^
<^ fwivi lorm'o
jL ^loiictT'hL ciAi* jYyyp^TJx <><-"* >^^
FIG. 233. FIRST PAGE OF THE 1473 Trattato
468 KARA ARITHMETICA
are left blank for subsequent insertions, the scribe having written : ' Hie
dificiunt quatuo // chartse q non funt//i exemplario ' (f. 65, v.) The
second part of the work consists of practical geometry and mensuration
(ff. 103 to 128). The third part (f. 129) relates to irrational numbers.
On f. 135 begins the ' Regole de la Arcibra,' a chapter on algebra.
There is nothing to indicate the name of the author or the copyist.
ANONYMOUS. Latin MS., c. 1475.
Title. <De duplirj Arti Viforir.' (F. I, r.)
Description. 4, 15 x 21.8 cm., the text being 10 x 17 cm.
(varies). 1 1 ff . + I blank = 12 ff., 40-45 11. Written in a Ger-
man hand of c. 1450-1475. Bound with the manuscript of
Mohammed ibn Musa (p. 454).
This manuscript gives some account of gauging, and closes with a
brief treatment of trigonometry. The mediaeval numeral forms are used
throughout. A copy also appears in the 1501 manuscript described on
p. 480.
LUCA DA FIRENZE. Italian MS., c. 1475.
Maestro Luca da Firenze lived in the fifteenth century, and was the son
of the celebrated Florentine arithmetician Matteo, who was born in the
fourteenth century.
Title. ' Inprencipio darte dabaco.' (F. i, r.) F. 2 begins:
' Inprencipio darte dabacho fecondo loftile dinfegniare del ma-//
eftro luca di Matteo da fflrenc.e.' (Fig. 234.)
Description. 4, 16.8 x 23.3 cm., the text being 11.7 x 13.8
cm. 46 ff., 29 11.
Although Fabbroni's Storia delF Universita di Pisa (I, 97) says that
Luca's son Giovanni went with Lorenzo dei Medici to the University
of Pisa in 1515, thus putting his birth about 1495 and Luca's birth
perhaps about 1450, I feel that either this is incorrect, or it is to some
other Giovanni and Luca that he refers. One of the best evidences of
the date of an arithmetic is found in the dates given in its problems.
Authors usually mention years that are not remote from the time when
they write, and in the examples in equation of payments (f. 29, v.) Luca
uses dates from 1410 to 1441. I therefore think that either he copied
a problem from his father (Matteo), or, what is more probable, he him-
self wrote about that time, say c. 1425, and that in either case this Gio-
vanni was a descendant but not a son.
MANUSCRIPTS
469
The arithmetic resembles numerous others written in Florence about
this time, such as those described on pp. 443 and 464. The author
begins with the fundamental operations and follows these by a treatment
zengwg.me.ta. fVlH
dvc
fWuc-ra fufte -ZCTqKmi'
firtc cfjo-n-ptjvrtc. ft-cttf&zaTno pi-mn^
le >Mio ''---. -'
FIG. 234. FIRST PAGE OF LUCA DA FIRENZE
of fractions and denominate numbers, closing with a series of applica-
tions to the business problems of the time. Subjects like equation
of payments, partnership, and exchange are given the most attention.
The handwriting indicates that the manuscript is a copy made about
470 KARA ARITHMETICA
1475-1500. The symbols in Fig. 235 may also throw some light on the
disputed origin of our symbol for dollars, $. No one seriously considers
such fanciful theories as the combination of U and S, or the Spanish
banner about the Pillars of Hercules. The symbol first appears in print
in The American Accomptant, by Chauncey Lee (Lansingburgh, 1797),
but in a very different form from that now used. It was common
among merchants for some time before it was cast in type form, for
a note in one of the early American arithmetics says that the symbol
was in use, but that there was no type for it. The third edition of
Pike's arithmetic (1798) uses m., c., d., D., and E., for mills, cents,
dimes, dollars, and eagles, but Daboll's School maffer's Afflftant (4th ed.,
1799, p. 20) gives the symbol $ very nearly in its present form. Now
whence come symbols like this ? If they are invented de novo it is
usually easy to find their first appearance, as in the case of symbols like
TT, e, and / (for V i). But mercantile symbols usually develop slowly,
like from libra t /- from the old s (/) from soldi or solidi, and d from
denarii. So it is probable that $ was simply developed from some
earlier symbol of value, such as that for pounds or reales. Now the
symbol for pounds (libra, lire) has various forms, appearing in England
as or lb., but generally in Italy as L, or Ib. with two bars across. The
former is seen in Fig. 243, from a manuscript of 1545, and the latter in
Fig. 216 from Dagomari's work. The latter form, in the fifteenth and
sixteenth century manuscripts, appears as practically our dollar sign, as
shown in Figs. 235, 237, 246, and it is not improbable that our early
American merchants used it for the new unit of value, the dollar, just as
the /- for shilling is still used in many parts of our country for \2\ c.,
although the original meaning is entirely lost. The symbol came into
general use in printed books between 1800 and 1825.
ANONYMOUS. Latin MS., 1476.
Title. ' Computus cyrometralis.'
Description. 4, 15.5 x 20.9 cm. A volume of 180 ff., con-
sisting of four treatises, the other three being elsewhere described.
The first of the four manuscripts is a computus of the usual fifteenth-
century type. (See Fig. 236.) In it the Latin original of the verses
beginning ' Thirty days hath September ' appears (f . 6) as follows :
' Ja. mar
ma. iul. aug. oc. de. deca-
trib9. et. vno. alij. trigeta.
f$. februus. octo. viginte.'
MANUSCRIPTS
18
13
'?
n-O
iO v
r
rana V 7 T
*/rto-rx- / fo
|o/>n.vvi'n4-/Jp t
Sn*5\|;?
>&/* 4*" 11$'
'</$*
*<v
FIG. 235. TABLES FROM LUCA DA FIRENZE
472
KARA ARITHMETICA
JWKC *S. WMk<Mr*.-MM* n
f^HM-aft <j* 4pfe* v
BMtCgfe. ..aaMap^a'^ayttgi
'jr*~&***nr***r* 1
" ' i"^~^ ' T~- -. -- -i . -.
&**+*>+&.& $4 *
^^9-teI^^
Bi&^-^
^^>^^^^tfi&j^ <-~TPCH
^iv^eS^f^fiBVp^^ ^J^^^i
*?<3a ^r pr f^.te^ a^fe t*r As^^Si
01 -.^ % I ' f^V L ._ C _^ _ -v xv* . .-**
FIG. 236. FROM THE 1476 Computus cyrometralis
MANUSCRIPTS 473
(See also p. 33.) It does not start here, however, but goes back at
least as far as Sacrobosco (i3th cent., see p. 31), in whose Computus
(see p. 45 1 ) it appears as follows :
' Sep. No. lun. Ap. triginta dato, reliquis magis uno.
Ni fit Biffextus, Februus minor efto duobus.'
(1545 edn., f. K 6.)
JACOBUS OBERNHEYM. Latin MS., 1476.
A Nurnberg computist of c. 1431.
Title. ' Computus norembergensis.'
Colophon. ' Anno dm 1476 complete. '
Description. See the preceding manuscript, with which it is
bound.
This is a German computus written by the same hand as the preced-
ing. It is evidently a copy of a work written in 1431, for the following
statement appears on f. 65 : 'Anno dm mccccxxxi quarta feria p9
judica finitg e liber ifte per me Jacobu Obernheym.'
ANONYMOUS. Latin MS., 1477.
Title. None. A calendarium. The first folio begins 'Albeto
belo caro decagolo ca9 nobilis.'
Description. 4, 15.4 x 20.4 cm., the text being 10.5 x 16.4
cm. 14 ff., 17 11. Bound with the Vergerius and Lullius de-
scribed on pp. 456, 457.
This is a manuscript of 14 folios on the calendar, and has two dates,
M.cccc.lviij and M.cccc.lxxvij, on the first folio. It also has some
lunar tables for 1364-1381, the treatise having probably been copied
about 1458-1477 from another one of a century earlier.
NICOLO DE ORBELLI. Latin MS., 1478.
Title. 'Incipit op9 fratris (?) dorbelli fup . . .' (F. I, r.). ' In-
cipit. Matematica' (F. 132, r.).
Colophon. On f . 1 26, v. : ' Explicunt Scripta f fs nicolai de orbel-
lis // doctoris eximini fup ... // Deo gratias et xpo ihu Amen
//M? 4 78.'
Description. 6, 10.5 X 15.6 cm., the text being 9.2 X 1 1.9 cm.
270 ff. (6 blank), 41 11. Latin MS. on paper, 1478.
474 KARA ARITHMETICA
The first part of the book is devoted to dialectics and logic. The
mathematics begins on f. 132, r., and consists of two folios on the theory
of numbers and three on geometry. On f. 140, r., begins a treatise on
philosophy, containing (f. 184, v.) a drawing of 'Johs fcotus docts
fubtilis.' On f. 193, r., begins a treatise on astronomy: ' Hie // Incipit
liber de // celo & mundo.' The rest of the work relates to science in
general. (See p. 23, 1485.)
LEONARDUS MAYNARDUS. Latin MS., c. 1488.
A fifteenth-century mathematician of Cremona, Italy.
Title. ' Leonardi Cremonenfis artis metri//ce pratice compi-
latio. Primus tractatus.' (F. i, r.)
Description. 4, 17.2 x 24.4 cm., the text being 7 x 10.8 cm.
24 ff., 25 11., figures on the margins.
This is a treatise on trigonometry, and has been included in this
catalogue because of the arithmetical nature of some of the problems.
Favaro's careful investigation of the time when Maynard lived still
leaves the matter in doubt. He may have lived in the latter part of the
fifteenth century, or he may possibly have been the 'Leonardus de
Antoniis de Cremona, ordinis minorum, bacalarius ' who lived early in
the fifteenth century.
In the catalogue of the Boncompagni sale another Latin manuscript
of Leonardo is given as 'in pelle di 33 carte membranacie del secolo
XIV.' If this is correct, which is doubtful, Leonardo must have lived
before 1400. See also Enestrom in Bibliotheca Mathematica IV (3),
p. 290, and Favaro in the same journal IV (3), p. 334. The latter gives
a bibliography, with some quotations.
In 1902 M. Curtze translated and edited one of the three manu-
scripts of this work known to him. This is in the Venetian dialect,
and belongs to the University library at Gottingen. He also consulted
the two Latin manuscripts formerly belonging to the Boncompagni
library, of which this is the later by a few years. This manuscript
belonged to the advocate Cav. Carlo Morbio, in Milan, before Bon-
compagni secured it. A note on f . 15, r., shows that in 1655 it belonged
to Bonifacio or Joseffo Aliprandi. Another note, on f . i , refers to a pas-
sage in a work by Franciscus Arisius, printed at Parma in 1702. This
passage is as follows: ' LXXXVIII (i.e. 1488) Leonardus Maynardus
Insignis Astronomus, Physicus et Mathematicus, cuius opusculum M. S.
Mediolani servatur, mihi indicatum ab eruditissimo Viro Lazaro Augstino
Cotta I. C. amico meo nequaquam satis laudato, cui est initium : (Here
MANUSCRIPTS 475
follow the first few lines of this work, so nearly identical with the
manuscript as to show that Arisius probably had this very one at hand.)
. . . Fuit ante Blasium Leonardos Maynardus, qui suo tempore non tan-
turn inter nos, sed etiam inter omnes in iis studiis tenuit principatum.'
A similar passage appears in the manuscript Biografia Cremonese of
Vincenzo Lancetti, now in the civic library at Cremona, from which we
may infer that he lived before Battista Piasio, of a noble Cremonese
family, a philosopher, physician, and astrologer, who flourished about
1500. This manuscript of Maynardus is described (no. 254) in Nar-
ducci's catalogue of the Boncompagni manuscripts (Rome, 1862).
ANONYMOUS. Italian MS., c. 1490.
Title. None. A manuscript on elementary mathematics.
'Choncio fia chofa che fono noue fighure nellaba//cho.' (F. i, r.)
Description. 4, 14.5 X 21.5 cm., the text being 9 x 15.2 cm.
158 ff. (5 blank), 26 11.
The arithmetic is of the ordinary commercial type, and includes the
fundamental operations together with a considerable range of business
applications. It follows the general style of the Florentine arithmetics
and uses the Tuscan berichuocholo instead of the Venetian scachero to
designate our present multiplication : * Volendo multiprichare .2. numeri
p berichuocolo ' (f. 17). The dates in the problems in equation of pay-
ments indicate that it was written about 1490 (ff. 83-86-129). The
per-cent symbol % here appears as p c, p c, p c- , as well as p 100
(ff. 84, 86, 134, v. See Fig. 237).
The second part of the work (f. 91) relates to algebra : ' Qui aprffo
fcriuerio lareghola dellarabre (dell' algebra). mochabiln (e muqabala).'
The work is rhetorical, there being practically.no symbolism employed.
This part of the work closes with the words, 'Voglia hora fare fine
enondire piu fopa a que//fta reghola delagebremghabile . . . (f. 109, r.).
The geometry begins on f. 109, r., and is confined to simple men-
suration. The work closes (ff. 123-153) with a series of miscellaneous
problems.
ANDREA DI GIOVANNI BATTISTA LANFREDUCCI.
Italian MS., c. 1490.
An officer of the Republic of Pisa in 1 505.
Title. None. A treatise on arithmetic. 'Choncio fia chofa
che fono noue figure nellabacho.' (F. 3, r.)
476 KARA ARITHMETICA
tl
ea
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it too cmnfraU'Uria <nlrncfc etUi u*4i
cntcfl
Sucllc
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C tU Ud mClC C ^ at A- |<3Jfr C CdC-A- T-&VA& t(t1C
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.^
^^
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
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