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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 Rara 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. 


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 TuUy 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- 
og^phy 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- 
th^ue nationale (unfortunately only just begun), and through 
bibliographies like those of Graesse and Hain and Coppinger. It 


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 unk]ue 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 
practkally 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. 


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 


c^ jcfttions relating to the science or art of numbers. No effort 
has been made to add to the supplementary lists books that 
arc not purely arithmetical, such as treatises on the ancient 
measures, although those that are in Mr. Plimpton*s library, and 
arc of value in the study of the history of arithmetic, have been 

The arrangement is chronological by first editions, but the Index • 
alUmrs for alphabetical and geographical reference. Although the 
nature of the work is usually discussed very briefly, an examina- 
ti«io of the Index will show that a fairly complete history of 
Renaissance arithmetic has been included in the work — a his- 
t»ry 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 ^miliar with it, while our popular units would be unknown to 
most others. These measures, like the number of lines to a page, 
«if 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- 
;^^^lhical needs, although many have a marked historical interest. 

In copying the titles it has been the intention to follow the 

• »n;;inal as closely as possible, without attempting to imitate par- 
*.p.ular forms of t)!^ or to use capitals except as initials. In the 
oases of misspelled words, omitted capitals, and peculiar punctua- 
t>io, the errors have been copied as faithfully as possible. At 
the nme time mistakes must have been made in transcribing, 
although it is hoped that they are not of a serious nature. 




I*: 41 K I. From a manuscript of boethius, c. 1294, show- 
ing riGL'RATE NUMBERS Frontispiece 

T: %T» II. From the Margarita philosophica 82 

A. Geometry B. Arithmetic 

I I %TC III. Title page oe lax 122 

pATf IV. From the campanus manuscript ok euclid, 

c- 1260 433 

I L %TE V*. From a manuscript of euclid, c. 1294 .... 436 

y %TE VI. From a manuscript of boethius, c. 1300 . . 440 

! . %TE VII. From a manuscript of giovanni da firenze, 

Maa 446 

;':%TE VIII. From a manuscript of sacrobosuo, c. 1442 . 450 
?: %rc IX From an anonymous manuscript, c. 1460 • . 462 


c, circa^ about 

cm., centimeters 

ed. pr., edito princepSy first edition 

f., f!., 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 locoy 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 



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. Deceby : : .1478.* (F. 62, r. 
See Fig. 4.) 

Description. 4®, 14.6x20.5 cm., the text being 7.3 X 12.8 
tm. 62 if. 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, 
hb results appearing in the Atti delP Accademia Pontificia de' Nuovi 
Uncei^ vol. 16. This particular copy was in the Pinelli collection, and 
vassold on February 6, 1790, to a^Mr. WodhuU. 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. 



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 

^imoiiiiniicii MM p urt k tt mohoboiM ct wwi 
totfidMoBocbi vookvpttlartrwla iiicrd;a# 

gAn [iri'j^mirinrlimwbbtecfco* ^ 

Tt^ f f g CO pot r piB voicr «a •Bnini 
I ^ouani t mi mobo ajkcoflum : K 
UtMit f innulnMiioavQiirrvokr 
IfArr La mrrcbadtmwtdsrprrlosa 

iTflinotc mepitcffflirftlfiuli^Kine « 

..a |»ocJ)6c« ttfrgJA en ftfioo^iMldx fendiinte 
cnvt bm V tnAnrtnnwisisiiMn volstnminr 
l«bted)0.CIn4e io ronftrrtto |srr Mios « bio: ft 
ntfditf id vfiiitavi nici cbi pmciuUno qorlltilir 
ffondob iNKob mtdlisrntia xA msfsnoimotbo 
drUmft>reiionaiaitaUiipanrniiir faofFart c 
losoAdo cbr bio viniioa wfidmi vok froao re 
oofif po4cMow3b nomr ti w odoncte t fosiio 
Bcrpsimoiiiiocl «iw> vtlgoiAiiocBlltKldo. 
t' €liraKVr0olbcfccwb|)ifimoii8iDe 
btiio Mbico rtodiicniifnmr rttonr V 
mmifro fono fhi fbrmadr.^ cdi come f<v 
lioibam>M fir cosnoToitif -prroiir b cosmnoiie 
tctttir b co^f^llrfta pMcOM f iwcrfTarb • ^pr 
dncrtriid,|)pofttomioq)rmio f«|:t bctDif^fce qn/ 
fob«l|»opDefcoiiolho:ODmrioctiit molmis* 
aine coiMrrsa n onrro mfembiMb P0 molK viii« 
tftdr.rr tl mrnoui vo viitttar.comf f ^dqoolr 

c bpHmermmoir mimnoxi)elirfnioiia.l««# 
pMrrqadbcofaitMbqnfiboan coCifivaia 
«MJ9r80d8fiofirtxfcr fe tnioiM mmim were 
WMWJgft^^yftcHimaiwimyoli wfteJM 
uoiuiiiiciviHiMib«^icnofr "^ 




Fig. I. First page of the treviso arithmetic 

Boethius (p. 25), who WTote in the beginning of the sixth century, fol- 
lowing the Greek models of Nicomachus (p. 186) and Euclid (p. 11). 
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 


<-.j< h men as Bradwardin (p. 6i), Albert of Saxony (p. 9), and 
i'nianus (p. 62). 

3. Tlu algorisms (a^orismi). These were practical arithmetics, 

t na ki^ lk u t ftr CMcbterodi <)itab' bffaro al fhUH 
o OKKiiKnnhlo h cjEcmpb' foi (obiivtMt n fosfiM* 

fit.9 1 4.rnott wnrloperliqaaiiionKHif comr 


f » }^4. 


o/lo / o /I 
/9\ /s[/4-\ i 

» 95 ^ 9 • 


Fii*. 1. From the treviso ARiTHMFnc, SHo^^^N(; variois forms 





*Jix^tx^g^^ fir 
fi^lP99grfifo^^ p pop p p\ 31 t6k 

m t compita dnoK r vi nuto to Atip9eniifro.5or il 
mio tmnjoiMdoh^brfeltrc 14* 1 6 <M5e. 9 
Mj« 5£JL wl»ff<f <I«ca^903 sup i^^y^ok 

, , .5 _i. 4iXo2r64 

(be Itre4000«e $ Mbraimodiicaaa^o.g.^* 
bquabTonovnonanow tiioouaio«jBicbt ^ 
rt^onerqnefte ftanoregurammie bene* 
Bmfzndo texhe qnando banerai m fare qoakbe 
rayone «atmpo2tanoa:rdbem«ubiii:iionfoui 
|»uouaiT piu r<fguraniett:cbt vob^re U 104 rapo# 
ne .al inodo cbe bai vifto ne It trr n%de pseditte. 
Undrprrqitefterprrleabreniicone pxedinetb 
qaalc woo in tato naineroqaindti^nM puo iium 

Fio. 3. From the treviso arithmetic, showing multipication 




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, knowni by the name of aL- 
Khowarazmi (from his birthplace, Khwarazm), and from the Latinized 
form of his name came the word algorism, 

3. Thf 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 jacerc^ to throw), or counters^ from 
which English form we have expressions like 
*cast an account.' The table on which the 
calculi were cast is still called a counter in our 
shops, but, like the sand tables used for com- 
puting, it was in early times called an abacus. 
Just at the opening of the Renaissance the con- 
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 pj^ Last page of 
k)ng before this was done in other countries. ^^^g trkviso arith- 
In Germany the arithmetics frequently have metic 

in their titles the expression ' auff der Linien 

und Fedem' (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 (* ziifer ') was written. 

4. TTu computi. The computus or compotus was a treatise upon 
the Giurch 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. 




ISIDORUS OF SEVILLE. Ed. pr. 1472. Venice, 1483. 

Bom, 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. *€.Incipit epiftola Ifidori iunioris hifpalenfis epi-//fcopi 
ad Braulionem cefar auguftanu epifcopu.' (F. i, r.) 

With this is : * Cin chrifti nomine incipit liber primus fiancti // 
Ifidori hifpalenfis epifcopi de fumo bono.// Qd deus fumus 2 
incomutabilis fit CCap. I.* (F. i, r., following the first f. 105.) 

Colophon. * ftFinit liber etymologiarum // Clfidori hifpalenfis 
epifcopi.' (F. loi, v.) 

Colophon of the second part. * CFinit liber tertius : vltim^ de 
sumo bono // fancti Ifidori hyfpalenfis epi : Impreffus // Venetijs 
per Pctru 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. I29ff. numb, -f- 4 unnumb. = 133 ff., 
58 11. Venice, 1483. 

Editions. Augsburg, 1472; Venice, 1483 (here described); 
ib., 1485 (.?); ib., 1493, fol-J Basel, 1577 (mentioned below). Also 
two editions s. 1. a. (Strasburg, 1470 ?), and one s. 1. a. (Cologne, 

147^78 ?). 

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) : 'Clncipit liber tertius // CDe vocabulo arithmetice 
// difciplinf CCap. 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 Pemam. ' 
(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. 


ALBERT OF SAXONY. Ed. pr.c. 1478. Venice (?),c. 1478. 

Born r. 1330. He lectured in the University of Paris, was Rector at 
Vienna in 1305, and was Bishop of Halberstadt from 1366 to 1390. He 
wrote »everal SM.ientitfic works. 

TitU. • EcxelietifTimi magistri albcrti de // faxonia tractatus 
I •-•rtumum incipit feliciter.* (F. i, r.) 

L cU>pham, * Explitiunt pi)ortiones magiftri //alberti de faxonia.' 

F 9. r ) 

Ik-scripthn, Fol., 19.7 x 28.1 cm., in double columns, each 
Vin;; 5.6 X 16.7 cm. 9 ff. unnumb., 39 11. S. 1. a. (Venice?, 
. I4;8> 

EMtions, S. 1. a. (Venice .^ c. 1478, here described) ; Padua, 
14S2 (12 ff.); ib., 1484 (12 ff.); ib., s. a., but before 1487; 
Venice, M.cccc.xxxlvii for 1487 (10 ff.); ib., 1494 (10 ff.); ib. 

.«::h another work), 1496; ib., 1496 (but no copies extant ?) ; 
B-I'«>;na (with another work), 1502; ib., 1506 (commentary by 
\*:::«»ri>; Paris, s. a. An epitome by Padre Isidoro Isolani 
M'.ldncae was published in Pavia in 1513, and again in 1522. 
I*rincc Boncompagni, in an elaborate discussion of the various 
o'.tHms {BulUtimo, IV, 498), mentions this rare first edition, of 
«hich be knew but one other copy, that in the Biblioteca 
Ambnjsiana at Milan. He was of the opinion that it was 
; nntcd in Venice, in a type used before 1480. 

\hhoiigh the date is uncertain, this Tractatus may contest with the 
lirti^» arithmetic (p. 3) and the Ars Numerandi (p. 23) the honor of 
* ' ng the first printed work devoted wholly to some phase of arithmetic. 
I* treats of ratios according to the cumbersome method of Boethius as 
! .'•'.vwcd during the Middle Ages. It is purely theoretical and repre- 
'irr.t% the university treatment of scientific arithmetic in that period. 

Otkrr u*orks 0/1472-1480, The mathematical activity in Italy dur- 
-jC this period was xtry considerable. From 1472 to 1480 there were 
;'^ mathematical works printed in the country. In the next decade 
••<Te mtrc 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 
••-la thirty years. 

Brf->re the Treviso book there were printed at least three works which 
' •.ched briefly upon arithmetic. These were (1) * I^ re militari libri 


XII,' by Robertus 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 (VincentiusBellovacensis), 
the greatest mediaeval encyclopaedia, printed at Strasburg by Mentelin, 
i469(?)-i473, 10 vols., fol. The second part of this work contains one 
book (no. 16) 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 nanw 
possibly Roger or Raymund), entitled 'Opus aureum calculationum pei 
Johanem de Cipro emendatum et explicit.', with subsequent editions a 
Pavia in 1488(7), 1497, 1498, fol., at Venice in 1505, 1520, and a 
Salamanca in 1520. I have seen an edition of this work, s. a., assigne< 
to c. 1477. For Nicolaus Cusa, see p. 42. 

GIORGIO CHIARINL Ed. pr. 1481. Florence, 1481 

A Florentine arithmetician of the fifteenth century. 

Title, * Qvesto e ellibro che //tracta di mercatantie //et vsar 
ze de paesi.' (F. i, r. See Fig. 5.) 

Colophon, * Finite ellibro di tvcti // ichostvmi : cambi : mon 
//te : pesi : misvre : & vsanze // di lectere di cambi : & ter// mir 
di decte lectere che//nepaesi sicostvmaet in //diverse terr< 
Per me France //fco di Dino di lacopo Kartolaio Fiore//tin 
Adi X di Dicembre MCCCCLXXXI. In Firenze Apreffo 
almusiftero di Fuligno. * (F. 102, r.) 


Description. 8°, 13.2 X 21.2 cm,, the text being 7.1 X 14. i cm. 
3 ff. blank + 6 unnumb. +96 numb. (Roman numerals) = 105 ff., 
24 11. Florence, 148 1. 

Editions. Florence, 148 1, 8° (here described); s. a. (1498), 
8**. The undated edition was not the first ; Copbinger has 
shown that it was printed in 1498. 

Wliile 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- 


Fig. 5. TrTLE of chiarini's work 

able in showing the nature of the practical problems of the time. Copies 
<rf 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/ f urgefchriben vnd // dargethon ift.// Aufs Griechifcher 
fprach in die Teiitfch gebracht/ aigene-//tlich erklirt/ Auch mit 
verftentlichen Exempeln/ griindV/lichen Figuren/vnd allerlaj 
den nutz furangen ftellen-//den Anhangen geziert/ Dermaf fen 
vormals //in Teiitfcher fprach nie gefehen // worden.// Alles zu 
lieb vnd gebrauch den Kunftliebenden Teiitfchen/fo fych der 


Geo-//metrj vnd Rechenkunft anmaffen/mit vilfdltiger miihc 
vnd arbait //zum trewlichften eramet/vnd in Truckh ge-//geben/ 
Durch// Wilhelm Holtzman/genant Xylander/ //von Augfpurg. 
//Getruckt zfl Bafel.' (P. i.) 

Colophon. * VoUendet durch Jacob Kiindig/ zu Bafel/ in // 
Joanns Sporini koften/im jar is62.//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. -H 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 EucUd (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 

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 0/1482, In 1482 appeared the first German arithmetic, 
if we except the *Ars Numerandi' (p. 23). It was written by L'lrich 
Wagner, a Ntimberg Rechenmeister, and was printed by HeinricK Pet- 
zensteiner at Bamberg. Only nine small pieces of parchment proof sheets 
remain. They contain the following colophon: *Anno dni . . • 148a 
kri6. lunij p. Henr. peczensteiner Babenberge : finit Ulrich wagnei 
Rechemeister zu Ntimberg.' * Ludus Arithmomachiae,* by John Sher- 
wood (Shirewode), was published at Rome in the same year. (See alsc 
p. 63, and on Albert of Saxony see p. 9, c. 1478.) 


Ed. pr. 1483. Padua, 1483. 

PROSDOCIMO DE Beldamandt was bom 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, niusic, and astronomy. 

Johannes de Liverius (Liveriis, Lineriis) was a Sicilian writer on 
astronomy who flourished c. 1300- 1350. 

Ti(h. * Profdocimi de beldamandis algo-//rifmi tractatus 
perutilis j neceffarius // foeliciter incipit. qui de generibus cal-// 
culationum fpecie preteri.t nulla5.q falte // neceffaria ad h^ art9 
:gnit65 fuerat.' (F. i, r. See Fig. 6.) 

Colophon. * Algorifmus. Profdocimi de beldamadis // vna cum 
minuciis. Johanis de liueriis. hic//felicite'finit Impreff^ padue. 
Anno . //die zz. februarii.' (F. 21, v.) 

Description, 4®, 15 X 20.8 cm., the text being 9 x 14.7 cm. 
2^ S. 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 tam vulgaribus qu^m phyficis magi- 
Vftri loannis de Liuerijs Siculi.' The • vulgar minutes ' were the com- 
mon fractions, and the 'physical' were the sexagesimal fractions. 
Toirards 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 fuflficiant.' 
(F. D 5, v.) The work of Prosdocimo contains the first reference that 
I have seen to a slate. ' Indigebat etiam calculator semper aliquo /apide, 
^■el sibi conformi super quo scribere atque faciliter delere posset figuras 
<nm 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), 


niim traoitiw pcniiilid t Mceiriniia 


|md«fcfi4)|rfcap< kk ai6<Mii<tia^piCT 
iptf opmtion^plMfoce.'v^boiie focraif adncjtinit cii 
amifliinodiJDimfirittttofi: <|^fiinaliq;c8:oiIo8nroloKo<nci 
to q^ mot fine adbiK (tfia 4>p^Di» cjriltcM» t boc 4)pc fi0ii 
ftgarag eSflbofwabttfa calaitoii o > jUybccoiafana fai 
dio(49tq;bDoaoMinibin&(iiiit»l^iolaiMKUBcda< mqno 
aoinipoienLSciu til (|r in bocUbcDo pontf a»iM&lo mil <• 
4 sd cflladu iicccfr«m finUlb ^aioUelibfiBpi^^ 

(r€uiicr0oia)diii0frkdeiiiiiMneira(incb)a<iififinii6nc ^^A**^ 
NQCriipmid)oaRaob.CITIamaii0(rgo^l£iidide^?.fu< drvu^^JW 
0Minclrfc>tf*J3obttt(up?fiKsriihteftt(k(iif^^ ^oMtih 
c«iwtoBbiigtggrcgmJgtd[onfti6iniidsq?iiiMqpc<p^ Vttitoi&jA 
<ilf mM. '^«bJC0T<kffinil5jiumbkpoftaqno? oniigeno 

So:fiinKn*idcpXi> ofeo <{Bo fcaiiqiifi nikmrcpofl^^^ 
(oT«8»?<a^pirwkT»wjxdroboiiiBlibdli. mq? <n& tniiaf 

Fig. 6. First page ok prosdocimo de beldamandi 

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 BulUtino, vol. XII. 


' 'ther works of i48j, Valturius (p* lo). In this year the second 
< 'rrnian anthmetic was printed at Bamberg, only one (incomplete) copy 
•» nj: known. It was possibly written by Ulrich Wagner (p. 12), and, 
k.r the 1482 work, it was printed by Petzensteiner, as appears from 
■ • '• folkiwing colophon : • In zale Xpi .1483. kl* .1 7. des Meyen Rech- 
r 'uf m mani herley weys in Babenberg durch henr^ // petzenlteiner 
• .TT.tfcn : %olendet.* 

Ed. pr. 1483. Venice, 1540. 

////*•. • Algorismvs de ln//tegris Magistri Prosdoci-//mi 
iKlicldamaiKlis Pataui fimul cu algorifmo de dc-//niinutiis (eu 
*'a«. ti«>nibiis magiftri loanis de Liuerij // ficuli. Reintegratus ab 
rrninbus comiflis a fcri-//ptoribus, a me Federico Delphino ar- 
•. m. & /medicine doctore, mathematicarum difci//plinaru in 
. •. .iberrimo gymnafio Pata//uino publico profeffore, additis ali// 
, .Siis verbis, in aliquibus locis,//pro maiori claritate. Et da-// 
: s impreflfioni ad infta-//tiam meoB fcholariu // nunc algorifmu 
rr.axime dc-//ridcran-//tium.// Venelijs. M. D. XXXX/ 
F i.r) 

C ^i^h^m. • Venetijs per loannem Antonium de Vulpinis dc 
Ca (trogiufredo. Anno domini .M. D. XXXX. //die octauo 
r.cnfi» Aprilis.' (F. 52, v.) 

Inscription, 8^ 9.6 X 14.5 cm., the printed part being 7.1 x 
1 2 5 cm. 44 ff. unnumb., 30 11. Venice, 1 540. (This particular 
py has 8 ff. more, the sheet F appearing in duplicate.) 
xe p. 13- 


Ed. pr. 1484. S. 1, a. (Florence, c. 15 16). 

R %rr AFiK FKwrRjMM. A Florentine philosopher of the latter part of 
•He fifteenth century. 

T::U. • Vcrificatio Vniversalis in // regulas Ariftotelis de 
- • - mm rcccV/dens a cdmuni MathemaV/tico^i dtHTtrina.* 
F I.r Sec Fig. 7.) 


Description. 4°, 13 x 19.9 cm., the text being 9 X 16.7 cm. 

8 ff . unnumb., 4 1-42 11. S. 1. a. (Florence, B. Zucchetta, c. 1 5 16). 

Editions. Pisa, 1484, 8"; this edition, s. 1. a. (Florence, c. 

1 5 16). 

This is a brief treatment of proportion, hardly worthy of ranking as 
an arithmetic. The applications relate to problems of Aristotle. 


itgulas Ariftorelis de mom non recc^ 

dcQsa comuni Marhema^ 


Fig. 7. Title of the verificatio of franciscus 
PIETRO BORGHI. Ed. pr. 1484. Venice, 1484. 

Pi ERG BoRGi. A Venetian arithmetician ; died after 1494. 

Title. * Qui comenza la nobel opera de // arithmethica ne la 
qual fe tracta // tute coffe amercantia pertinente // facta 2 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.//augufU) 
.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. 4- 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**(seep. 19); ib., 1491, ^ (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 Maittain 
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 secort 
commercial arithmetic printed in Italy and long thought to be the firsi 
This particular copy belonged to Count Paolo Vimercati-Sozzi. 


The text of the first edition is closely followed in that of 1488 (see 
p. 19), except for the index, * Tauola de li capitoli ptegnudi i qila 
opa' (f. 118, r.). This does not appear in the second edition, at least 
in Mr. PHmpton'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 
L 118, v.y in connection with a set of verses beginning as follows : 


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 limprefTor de augufta Errardo experto 

dl 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 milhon,' and making no mention whatever of the 
Roman numerals. In the same spirit he eliminates all of the medieval 
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 
Tolcn-//do hora tratar de quelle chof//fe che f olo amerchadati apertien : 
pero tratado folo de quelli //che ale chofTe merchadatefche fono necef- 
iarij io laffero ogni //altra maniera de numeri.' (F. numb, i, v.) 

The sequence is now peculiar, for multiplication is the first operation 
treated. (*CChe cofla 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 10 & 90,' the products of 12, 16, 20, 24, 32, 36, 
by 3 ... 10, are given, these having been necessary on account of the 
moDetaiy tables of the time. The author then gives the method of 


checking by casting out 7's and 9*5. (*CDela 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, * CX>e\ multiplicar p 
cholona/ f. numb. 9, r.), with its checks by 7 and 9, and • per crocetta ' 
(our * cross multiplication/ * CDel multiplicar per chroxeta,* f . numb. 
13, v.), showing that these were the common methods in Venice. Divi- 
£"9^ . sion is then explained by the gal- 

SlSSiVS^d^S^ leyformCCComofediepartirp 

JSogmUttfUfeaUopcn^tdcre ^°' ''-)> °^ present method, then 

OcrqacftalmlMtraoatObpat known as the method of giWng, 

SearotfimindCfltallOIIOCadO 'a danda,' and described bv 

amercbadMtimoltavriUrad€ f^ ^^X not being '"en loned. 

firdlflprcrcnteopmcafiltMi Then follow addition (although 

mnifnf0IXj:»r:'gratTfilcflJt9dr this was used in multiplication), 

prrquefta VCdcratt tUttiU ttVCti subtraction, denominate num- 

&CiqMt€nnMlBt€riUdt ^^ bers, common fractions (also 

*«nan«aimncrojOC^^ beginning with multiplication), 

0$ptMfBtlcraMtl€tUit€Smc "^^^ ^^ ^^ (; .^^ riegola del 
tytrltdSWCCbcronvriOtpattC -3- ). partnership, barter, aUiga- 

FiG. 8. First page of the first edi- ^^on,^^^ false position. The rule 

TION OF BORGHI'S ARTTHMETIC ^^ ^^^^ ^^ ^^^ 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. 

OtArr works 0/1484. Albert of Saxony, p. 9, c. 1478; Vincent 
de Beauvais, p. 10. 


PIETRO BORGHI. Ed. pr. 1484. Venice, 1488. 

See p. 16. 

TitU, * Chi de arte matematiche ha piacere // Che tengon di 
certera el primo grado // Auanti che di quelle tenti el vado // 
Vogli la prefente opera vedere//Per quefta lui potra certo 

IT^CmopA fiMiMtw ifiillen 

_= 1 mmm d J^itd^a ft 5fc tcttmr per fcttc fi^tirt 
"^ t'ncjtidtp mo6o . loooooo .pmbc Uffpauu 

fi^lM ikn fnuogaDCnimra D<:Miijra;:cpcrcbc ••OOOPO 
milkm^^fj fanoTJio million ^ctdfcwdoujciud 
lud^o Ufi^r*i cbf fipHttni^ ww p<rro t^cnc 
DJrtovTio miltc^,^^ iqueftpmo4o , 1 100000 iaa-wi- 

0fmVii0miUorT <: cento miliaipcrdxclrnd ' ••PO^i 
fd^n^ luo^o t>^ ^cnmr3 ra ^c m ij ra :fono lan^wr a cb< Hpi< 
iaitivii0i^£t^&cncdiEo vno miiUon ictmo mi\\^ .^a (ti . . m^^^ 
w^modQ^ * t loocfo.mna vnc^milioruccnfo*: dicicmilii ^^^ 

wnb<9(cridiii(bcii canto rE^tlij ;m (uoi^oock tȣrciic&c^ 
murfbnolflffgtira c5e ripitxmiJ^ vno fi cbe bcjicdito vivo 
m^t3?ne£aifO<dtexcnitJu.^j [iiatjcflornodo, i 1 1 \qooxh' | | | |^o^ 

ill rnmtliofi c<iiwe vifdcii: riifl^a per ibc c\tY^ cinuljon ccnta 
cdfai miiiji w biogioDtf nurtpcrioctnJjrfi^niolartgura ibc np 
tisavtto:fld>«lxneMto\>nmiI:c]KcntocvndcfC[nitu.^a 1 1 1 1 loa 
At^omc4o. 1 1 M loa.oirUTnmtli^ticcittof vndctcui^^ 
mtwm-petcbcohn c\m\M0n raito c vmlctf mtUii: mluop 
afdiinplKcantcti.irrono l^f^^ufiicbc n'piacutd ^rnoifubc 
taeidbvnmiltort cokj t ^\\^cun%i\Uiui\u\lS>^ mqucflo 
B0d»; M 1 1 1 lO.Dirta vnm((CLm cento t xn&itcmiii^ cento 1 1 1 1 1 1 P 
cHoilKTtb^ e*Itr*i cfmtk'oti coitocvitdae n;i7iac cento ;i 
ltadf»eck rimpiiL*; Dcjrcncfcno Jafiijora cbe npjcrcnta vno, 
I^hii;ucf1omodl0. 1 1 1 « U imriaYtiniittorucrirociPiide 1 1 1 1 1 1 1 
urnmaniot cbe flncbc m fuo^jo ocle riiiq>Uce 
twiu -folia Uftguracbef/pjacntair no Jubc bfnc cdirpvif 
>c vtidacmifia ccrjtoc tfidc^exccbofi p^occdcdo 

wpiiamiwc liMdiracrf cumoQCOic ocoiKwmxbc ri» 

icrcttmi.«IMflobg(to cctcba loomoaimitioiroocl 

jrJtacbcteMiDtoiittipoite ptoccdcr joa cbomvmi 

3<Mnl ton npffoKcio MiUtfrfr q^ootopo^ 

Fig. 9. From borghi's arithmetic, 1488 edition 

fapere//Se error fara nel calculo notado//Per quefta effer 
potra ccrtificado // A formar conti di tutto maniere // A mer- 
thadanti molta vtilitade // Fara la prefente opera e afatori // 
Dara in far conti gran facilitade // Per quefta vederan tutti li 
errori//Ede iquatemi foi la veritade // Danari acquifterano e 


grandi honori//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 2 compilata 
per Piero // borgfi da Veniefia.' (F. 2, r.) 

Colophon, * Stampito in Veniexia per zouane de Hair 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. 16. 

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*, 15.4 X 20.8 cm., the text being 11.7 x 16.8 
cm. 100 ff. unnumb., 40-43 11. Venice, 1491. 

See p. 16. This is the fourth edition. 

PIETRO BORGHI. Ed. pr. 1484. Venice, 1517. 

See p. 16. 

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 21.6cm., the text being 13.3 x 17 cm. 
2 ff. blank + 100 numb. = 102 ff., 41 11. Venice, 15 17. 

See p. 1 6, 


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. 'CStampato in Venetia per Fracefco Bindoni, & 
Mapheo // Pafyni compagni. Nel anno .M.D.XXVIII.// Adi 
.XVIII. Del mefe di Zenaro.' (F. icx), v.) 

Description, 8*, 15. i x 20.5 cm., the text being 13.4 x 17.1 
cm. 100 CF. numb., 41 11. Venice, 1528. 

See p. 16. 

PIETRO BORGHI. Ed. pr. 1484. Venice, 1534. 

Seep. 16. 

Title. This is substantially the same as that of the 1484 
edition already described. 

Colophon. * CStampato in Vinegia per Fracefco Bindoni, & 
Mapheo // Pafini compagni. Nel anno .M.D.XXXIIII.// Adi 
.25. Del mefe di Settembre.' (F. 100, v.) 

Description. 8**, 154 X 20.9 cm., the text being 13.5 x 17. i 
cm. 100 ff. numb., 41 11. Venice, 1534. 

See p. 16. The various editions changed but little. 

PIETRO BORGHI. Ed. pr. 1484. Venice, 1540. 

Seep. 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 
vedcre.// 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 i f attori // Dara in far conti gran f elicitade 
// Per quefta uederan tutti gli errori // E delli quatemi fuoi la 
ueritade // Danari acquiftaranno, e grandi honori.// In la patria e 
di f uori // Sapran far le ragg^on de tutte gente // Per le figure che 
fon qui depente.// Auenga che alquanto per me f u promefTo aiTai 


fufficientemente // alia promeffa fatisfaceffe, niente dimancho 
per fatisfar alle pre-//giere di qualch'uno, e maflime di alcuni 
Impreffori, iquali era-//no per ftampar la prefente Opera, bo 
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 Topera // infi. Et pero pro nunc mi paffo 
con alcune cofette aggionte nel//ligar de metalli, lequal princi- 
piano a carte ,tt, & anchora in fvaf/A^ Topera 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 mente di quelli, iquali me//hanno pregato 
non fuffino k fuo modo fatisfatte, prego quelli //me habbino 
per ifcufato/ (F. i, r.) 

Colophon. * Stampato in Venetia per Bernardino de Bindoni.// 
Ne I'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. -h 98 numb. = 100 ff., 39-40 11. Venice, 1540. 

See p. 16. 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 1 540 edition already described. It bears the date, * Anno 
Domini M. D. L.' 

Colophon. *€LStampato in Vinegia per Francefco Bindoni, 
&//Mapheo Pafini. 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. 16. This is the twelfth edition, and at least five subsequent 
editions appeared in the sixteenth century. 


ANONYMOUS. Ed. pr. c. 1485. S. 1. a. (Cologne ?, c. 1485). 

TitU. * Ars numerandi. Incipit copendiofus tractatul^ quin 
//tupliciu dcionii numealiu in quo docet'//luculet' quo ordlant' 
variatur oponuty/et abinuicem deriuatur dictones nume// rales.* 
(F. 2, r. ; see Fig. 10.) Without abbreviations this would appear 
as follows : ' Incipit compendiosus tractatulus quintuplicium die* 
tionum numeralium in quo docetur luculenter quomodo ordinan- 
tur variantur componuntur et ab invicem derivantur dictiones 

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., 
Princifia Typographical London, 1858, vol. Ill) fails to throw 
any light upon the date. Riccardi attributes it to Zell, c. 147 1, 
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 worhs 0/148^-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 Nicol6 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. 
00 arithmetic. See Isidonis, p. 8, 1483. 



Ard tiumcttin^i • 

Inolpit iDDpmbiofiid tmdatnl^qtiiit 
tupltdii tmonif ntimcaltSm (^ 
Ittcuteif quo o^hnfvmmr ofonth^ 
ftabintiicein tmuStur bidoncB mime 

9 . tur catbhtaU9«^ta {S^Im^ qta ^i():rt 

Imtitttf fiu« bi^tifie^qt&M^UiM^a qtft 

fiiullipUcatitt^ ftuc a^itbialw*^ notft 
tou a^ nitmm ^iuerfia ^i Aionib? fi^eft^ a cp 
imtui fi|ffiM.mati«a« cottabut mtf ic bSntia ct ^Ttaaw^iM 
MMtati utpatsb^ tPabiaaut^dottib^pot tiumnalca 
^biS «(%biclbiir«<St pito te ^v^tattb^q biaitur ^cd« ^ieiif 
id cactotalW'' qi ficat ofVa ^tit &a catbtnc^ce cac»ttiaUa# 
<nit(tif di ita ^^nw attc nilaUa ^tiitiit a vo^ 
|»bdlttlt to i(Ha "Vd ^ioif catbhtaUs 2)1! pit 
dpate ' <|ib0Dfi» ah« it^Uaub iftia bnt oti 
gtiiSVel 9t oi^ qfi pitdpalM' qi pAdpafk ligitfc 
iiA| <Ct fciidu 9^ t>upli0W mt ^cdm oardma iPupHdaflf 
iwf q^a^cScrcCa ei ducda nbtVUtt &>< cScrett carbbiatta^ 
aut catDfaiatifasqx te ipia afic tniuan^inio i 
btfinqtu)><6nwlicjp<adutnumccSdo Vnu0 
bttai>twa<|iiattiiM«qftc9«fir)>fcpt8«o<9DmattS Jp o^tuMia 
be&iniMi'buotai'tixtttiK|uattuatt8d^ tioia nuaUa 

iton trts bitsri^tbM Audio tt Afintbuct^cDcT 

tJbtto^a fie u(b; ab tngmtaftio modo^ ^ti^m 
ta«tvi^tattti9'^trigtnta'bttO« a fie (uo mode tt 
aliia:«aXttad«aamta iCtutquagmta^Scpiglta 
<0n<Jlufi^'eMiduo« n ftc fUo mo ufcjf buixtpa^ 
t>uc3ti«m«ta«buailiun7'buc?Cidtto« 4^ vltmi? 

Fig. io. First page of the anonymous ars numerandi 




Ed. pr. 1488. Augsburg, 1488. 

Borrius. 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. 

TitU. • Arithmetica boetij.' (F. i, r.) 

Colophon. *Finit arithmetica Boetij bene re//uifa ac fideli 
(tudio emendata Im//preffa per Erhardu ratdolt viri fo-// 
iertiflimi eximia iduftria t mira im-//primedi arte : qua nup 

I ci^pxcipiicinccr 

I fsdUlK dwIIMIlf 


I nccflb bocjriM 

flft HbflriliM ilfiBrfCtiiiiKniii:ncc 
*flofii«q6 taimWiiitailiiol£ 



iinlUti8i|ibi64ttit0t8ntiicftfUecte i^ ^^. mLT^I 
iitoM5v0:vtiidaimitquo(»« •^'f/^ ^-f^ 
liu8&iuilicilMMI6cm.114[itod9 ^9£Uvr4 n^JL 
gtoiM^fiMrniotefUniteuBSiil^ JAMJi-rtri^ 
o^cn^(lllcUeUtK>tdbtotoll< ivV,,.J^ 
^(illM0M0ratioalctcillMffttfb -«*^^»^'^ 
CM 11MI1110 poUii opoit iriioi Qipc 
cMt.jte ptoBrc ro«a)i»t^bqlc<5 




<t«if crrMrt opatondfl Uam 

lN»Coibnnn0ai&«itiarciii] . ^ 

nollriUboiitfdWbluiiol^a^fo' . . ^^ ,/^ ^ 
dUoi^ciirriccttctmii.Cociiilbbw f^ra^ mUh^ 

^ Wonrniiat^^ciQOfiihaini 

nuuit^lUpiemoopcriinipoiiMzin IfJ^jL-^tJi 

indidamttUt flrtflmptobcrciKiit 
potot (0tair ttcetqiMmi iwM fn 
botiMcaiUbtfcraiUbciliilm te 
ffii0QDerdCttf memiatdodcaf 05/ 
Kcnt>ac.ti|( idling mactojoMito 
iii0 a^caqiif IbmcaligmflNwtoi 


fielilnHMWfs mradonitc^cpcn 
MondtaMcfHRifl^ftMc Poi/ 

Mcfl liiara(|iicnm.£um wjMrmi 
temiionteciverribtM €(mm^ 

Fig. II. First PAGE of the 1488 boethius 






? — 
































5 — 

^ - 









































^ — 






















£d^.27« fecii^*r€^lnbtltarioliletla^atn(tji/ 

3igifbito^Utte> fcmari'binariii.ceititeriuirBms: 

ra^plHHttf fottttttif ji nlbu»i?fiemariu.qitit8qrtu Jinr/ 

fadof Jguimab vno narijittimeroiitateawiroen&iC:vc 

ab* io«. io.jpcebm 8 jjtcriwriu.*TeeJbediaifec|ii£cii 

ii«rdpiciif:tbiafitb jM^mituKtspmralicmepieterMnb 

certamotMes copa /^tmototrangukiisalHcia^ab 

im:qtn fcUicet a*4*aii0ii(timitidpi 9.ind)0fo logimbine lactaMnAt 

entesnn vigencm tenninu pontmr: trioetiie altrinfixus niiiiiarte]ct£Mt: 

X>\x^\sx\t>f&^p&xaV^^ ctb<ccii^latimbuieetlo^^i/ 

ckitiaoflcnbJtiiniiaTtpiinniapti^ itccdpaKtiifstriplQct'^cteNnfei^ 

Fig. 12. MuLTiPucATioN table, 1488 boethius 


Ncnciijs nuc // auguftf excellet nominatiflimus. // Anno diii 
M cccc. Ixxxviij. Men-//fis maij die vigefima.' (F. 48, r. Fig. 14.) 

I>escripti<m. 4®, 1 5. 1 X 20.8 cm., printed in double columns, 
c.ix h being 5.1 x 14.7 cm. 47 ff. unnumb. -f 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); 
Tari!*, 1496, 4® ; Venice, 1497 ; ib., 1499 (bearing also the date 
I4v7>, fol. ; s. 1. a. (the Compendium of Muris, 
V 1 500) ; Paris, 1 SOI, fol. ; ib., 1 503 (see p. 29) ; 
/:. . 1507. 4**; ib., 1510 (see p. 30), fol.; ib., 
151 1. 4''; ib., 1514, fol.; Vienna, 1515 (the 
Miiris * Compendium,' in Tannstetter's works); 
Paris, 1521 (sec p. 31), fol.; ib., 1522, fol.; 
1528, fol.; Paris, 1530, 8"", * De differentiis 
:■ picis libri quatuor'; Basel, 1536, 8"*; Basel, 
1546. • Opera,' fol. ; ib. and Paris, 1549; Basel, !•««• >3. Fi«t'R- 
1553, -ajcctis explic. per J. Scheubelium,' 8°; ^''^^SboethS 
Paris, 1553; Basel, 1570; Venice, 1570, 'Opera,' 
: I There are undoubtedly various other editions of the Arith- 
mciic, or the Epitome by Faber Stapulensis, these works often 
*^-ing bound with such treatises as the Arithmetic of Jordanus 
Ncmorarius Murhard (I, 160) and Rogg (p. 137) mention an 
c<ii:Mm of Fabcr's 'Compendium,' s. 1., 1480, but it is not given 
': \ other bibliographers. (See Hain, I, 468 ; Brunet, Man., I, 
1059; Gracsse, Tr/sor, I, 464; Riccardi, I, i, 159; Boncom- 
l^inii. BulUiino, XII, 148.) 

The text is practically that followed by Friedlein in his 
•standard edition of the 'Opera' of Boethius (Leipzig, 1867), 
•-\ccpt as to numerals. Here, as in the later manuscripts, the 
Arabic characters have replaced the Roman of the original 
Tcit <Sce Fig. 12.) 

The arithmetic of Boethius was based upon the Greek work of 
N 4 'Hnachas (fl. c. 100 a.d.), and related only to the theory of numbers, 
*-tf^ 'A^t^fTur^, as distinguished from the practical calculations, the 


AoyurrtKif, 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.) llie work 
was the standard in the Church schools throughout the Middle Ages. 

itii<^iimiiiiMiwfiiM>Fti ^^ — - *^ ^^ -^ ' 

■&-^HL I av- ti^ -_ ^^_ 



ttMHHHtTHBi^M^^^^i^ ■ > MBMM VHP 

Fig. 14. Last two pages of the 1488 boethius 


Ed. pr. of the Arithmetic, 1488. Venice, 1491-92. 

See p. 25. 

Title, See Fig. 15. 

Colophon, * Impreflis venetijs per Joanne de Forli-//uio 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 ^ Gre//goriu de gregorijs fratres f jlici exitu ad fine vfqj 
pductu//accurati(rimeq5 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 flf. unnumb. + 345 numb. + i blank 
= 349 ff., 66-70 11. Venice, 1491-92 (see the colophons). 


Editions. This is the editio pHnceps of the works of Boethius. 
For other editions sec p. 27. 


Fig. 15. TriLE page, 1491-92 boethil^ 

BOETHIUS. Ed. pr. of the Arithmetic, 1488. Paris, 1503. 
Seep. 15. 
Ti/le. * In hoc libro contenta.// Epitome/ compendiofaqj in- 
Trriductio in libros // Arithmeticos diui Seuerini Boetij: adiccto 
fa- ' 'miliari commentario dilucidata.// Praxis numerandi certis 
-^ .ibufdam rcgulis//conftricta.//Introductio in Geometriam 


breuiufculis an-//notationibus explanata. fex libris diftincta.// 
Primus de magnitudinibus et earu circuftan-//tiis.// Sc'dus dc 
cofequetibus/ cotiguis/ & c6tinuis.//Tertiusdepunctis.//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: * Geometrici introductorij:/, 
fexti / etvltimi libri finis.// Editi anno domini // millefimo quin- 
//getefimo pri//mo : vicefi-//nia quin//ta no-//uem//bris.* And 
on f . Ill, v., the following: * Id opus imprefferut Volphgangus 
//hopilius et Henricus f tephanus // ea in arte focii in Almo 
pari-//fiorum ftudio Anno Chri//fti Celorum totiufqs // nature 
coditoris.// 1503. Die vice//fimafepti-//nia Iu-//nij/ 

Description. 8®, 20 X 26.8 cm., the text being 15.6 x 22.3 
cm. 48 flf. 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. 

Colophon, * CAbfolutum in almo Parifiorum ftudio/ // Anno 
domini qui numero definiuit // omnia 1503. Et emidum ex offi- 
//cina Henrici ftephani Anno // Christi faluatoris // omnium 
1 5 10 de-//cima 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., 46-56 11. Paris, 15 10. 

This is one of the editions containing the commentary of Jodocus 
Clichtoveus on the < Epitome' of Boethius by Jacobus Faber Stapulensis. 
(Seep. 27.) 


BOETHIUS. Ed. pr. of the Arithmetic, 1488. Paris, 1521. 

Seep. 25. 

7i//r . * Divi Severi-//ni Boetii Arithmetica,// dvobvs discreta 
i: Sris ; adic-/A'to commentario, mysticam nvnie-//rorum applica- 
:: •ncm perftringente, declarata.// 
(\\*<Kxlcut with initials of the 
printer: S.D.C.). Vaenundatur 
ai'ud Simoncm Coli-Z/naevm, e 
rt-^ionc fcholae Decretorum.* 
'F. I, r.) 

C oU>phoH. * Excudebat Simon 


C • '.inaeus. ninrns, Anno 

' 1521 

O.iinto , Iduslulias.' (F. 139, v.) 

I^^scription. Fol., 1 9.9 X 28. 2 

. v\ . the text being 13.9X 26 

I -- 4 ff . unnumb. + 1 36 numb. 

^ 140 ff, 40-52 II. Paris, 1 52 1. 

lliis \s the first edition of Boethius 
•A :S tbc commentary of Girardus 
K .rtus: 'C^'irardi Rvffi, in duos 
^z thmrticx Boetii libros, commen- 
'A-.\< (F. 5, r.) This commentary 
^-'ratly exceetis the text in extent, 

^'••! ^ to ponderosity it leaves little to be desired. As a piece of 
:*. j->griphy, however, this is one of the best editions of Boethius. 


Ed. pr. 1488. Strasburg, 1488. 

A^iA^scs was a fifteenth <entury astronomer and poet, of Strasburg. 


lloiTBi SH. H<>LYWAU)E, lIoLYFAX. HALIFAX) was bom at Halifax (Holy. 
wood). Yorkshire; died at Paris in 1244 or 1256. He studied at Oxford and 
>ctmrHi at Paris. He wrote on astronomy and algorism. 

TitU. ' Copotus manua/,1is magri aniani. // metricus cu 
•^cto 'Et algorifmus.' (F. i, r. Fig. 17.) F. 45 begins, 
* Int.ipft textus algorifmi.* 

Fig. 16. Printer's device, 
edition ok boethiis 



Colophon, * Impreffum Argn. per Johamem "^vj^JI Anno do- 
mini .{488 .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 U. Strasburg, 

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., 15 10; Vienna, 15 17; Cracow, 


Fig. 17. TriLE, first EDmoN of anianus 

1504, 1509, 1 52 1, 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), V\ 
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, I549'' 
Wittenberg, 1550; ib., 1568; Antwerp, s. a. (c. 1558); ib., 
1559. There are probably others s. 1. a., and some appear 


under the name of Sacrobosco. Of this first edition Boncom- 
pagni knew only the Munich copy. {Bulktino^ 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 fupadditur 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 
Bcdc under the title * Verfus Prifciani, de Astronomia,' as * Hinc Aries, 
Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Arcitenens, Capri- 
comus, & Vma, 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 neceflariis tarn in 
fuis locis q5 in fine libri pofitis.// Incipit feliciter.' (F. i, r.) 

Description. 4", 1 3. i X 19.4 cm., the text being 9.3 X 1 3. i 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 


contains a number of illustrations showing the use of the hand 
and fingers in assisting in calendar reckoning, the title of * Com- 

ubcroDCConvoM fnkMmt^momn^ etma/ 

tg|]cmtaeftfu(toWdIimfta3(ftaver rJ^^^.i 

bapoifamoiiplidterconflcloim.piiiiiopiit '^^^j ' 
Mdoeoeo 4cftbi)tvcr0*ioeoMcd>ac oautd^^ / 
Uitomdt<iKlo*i&oeilbiliicc oiaY 9^ba.(* 
£«tlq:tcra<i lUomtooiiiiiebontfii^veiiie/ 
t^ftibmtc inttitdA*0edidopoitiC oerdouia* 
£n eft lidda*9n<^bii0 verbis adc5me^ 
irfccrcanaisair.lSrftito eihtait^efclWealtinidoiKedoraebac 
qa<)dwdrUa.0ccui«loUrsttiidogloa<)fapcr bocgfCWtoi/ 

i«oiteco<«rio.»«il6e(ic(ltodcilTalidil aptcdoramqiKMi p^,?!^^ 
ttimialtdotimpalteiolMtvaUdumtiKiK^ ^boAif (/' f2* m«<n QtmanOScfta. ntfnatoeda^ 
ddoeabiicUcoiata&iin^pioruecreaticnre^ tan^tata 
brara(nqpaii<baDcpiaidt.oqrfiigAtU0tJirdaM9« vfr^ 
bu0.1^itin!imfiio^aiicm^ boedit raiDefiipeolosoaitf _ 
ineMa^SSS^tlmi^qaeterafiiittcAii^^ 2. 

|lraq5€om§beiUTonevaaad9«Sad5e ficnUnddtiaii^ fumS 
boiiaquodbabalai^cii20lodofamfdaebinoueriraj^^ ^^ 
todtvera.imno{;pbanre^nfdoitird2de4i^ ' 

<idi babtos aie ronaUe no tonama fed aoHTme oim buanay K 
nimtndaeatrii: ^todudbomaiievlttgAcniatrfic/cpftlaflcba/ 
Utile po*qilttadtaBquode]diteii0(naIa*fedomiieaiudq^ 
dttoaiaaiir eUtaieaiitpotaiaaiitpaflio.Crboctdfaitarifto* 
fn fccOdo eMcott* (p fda no ii c palio p^ qe pafTtonea font 1^^ 
bmtaierdano 2bm5t.cr0oxc.(r no ficporcaap(5.qt ^^bs po/ 
Mrqrn$ mpo(aida»9flcb&ii8a!eronaU9pr5Bpdiaa.9a(ic 

a tj 

Fig. 18, Beginning of anianus, EDmoN 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. 


AXIANUS. Ed. pr. 1488. Lyons, 1504. 

Se« p. 31. 
TitU, •Corapotus cu//comniento/ (F. i, r.) 
Colophon, * CLiber compoti cum c5mento finit feliciter // Im- 

jrciTus Lugduni perClaudiu nour//ri. Annodomini .M.ccccc.iiij. 

. , die .iiij. Orlobris/ (F. 32, r.) 

Inscription. 8*, 14.5 X 20 cm., the text being 8.9 x 14. i cm. 

\i ff. unnumb., 34-40 II. Lyons, 1504. 

See p. 33. In this edition there is a curious misprint in the calendar 
\rxst given on p. 25 (f. 13, r.). It here begins * Julius (instead of 
t.nius) aprilis feptembre & ipfe nouember.' Like the edition of c. 1495 
V- 11"^^ ^i' ^ ^"^11 illustrated. It differs from the 1488 edition in the 
^.'trs of the commentator, but is practically identical with that of 
i. 1495- 

JOHANNES SACROBOSCO. Ed. pr. 1488. Venice, 1501. 

See p. 31. 

TitU, 'Algorifmus Domini Ioa-//nis De Sacro Bufco// 
N«»uitcr ImprefTu J.// Cum Gratia Et Priuilegio.' (F. i, r.) 

Colophon. • ImpreiTum Venetijs per Bemardinum Venetum // 
\>c Vitalibus: Anno Diii .M.CCCCC.I.//Die Tertio Meii. 
Fcbruarij.* (F. 8, r.) 

I>escription, 4*, 14. 1 X 19.6 cm., the text being 10.5 x 16.8 
m 8 fir. unnumb., 39 11. Venice, 1501. 

See p. 32. 


Ed. pr. 1488. S. I. a. (Venice, 1523). 

See p. 31. 
Tu/r, ' Algorifmus Domini Joannis // de Sacro Bufco noui- 
tcr impreflum.' (F. i, r.) 

Cclophm. •Imprcffum Venetiis per Melchiorem Seffam & 
Pctnifn//dc Rauanis Socios. Anno domini .M.D.XXIII.// 
z XXIIII. Octobris.' (F. 8, r.) 

Inscription. 4*, 14.8 X 18.6 cm., the text being 10.3 X 16.8 
8 ff. unnumb., 40 11. S. 1. a. (Venice, 1523). 


JOHANN WIDMAN (?). Ed. pr. c. 1488. Leipzig, c. 1488. 

Bom 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 i486. He evidently 
received the doctor's deg^e about the same time, for he wrote a medkal 
work in 1497 with the title, * Tractatus clarissimi medicina ? 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, * Algorithmus Linealis.' (F. i, r.) * Ad euitadum mul- 
tipli//ces Mercatorum txxQ'lIxt,?^ et alteri^ . . .' (F. 2, r.) 

Description. 4®, 14 X 19.5 cm., the text being 7.8 X 14.5 cm. 
I4ff. umiumb., 31-34II. Initials in red, by hand. Leipzig,c. 148S. 

Editions, Leipzig, s. a. (c. 1488, here described); ib., possi- 
bly 1490, 1493; 1516; 15 17, 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 Wfirzborg 
(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, I>e Duplatione, De Mediatione, De Multi- 
plicatione, De Diuifione, De Progreflione, De radicum extractiooe, 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 0/1488. Suiseth (p. 10, c. 1480) ; Borghi (p. 16, 1484 >. 

JOHANN WIDMAN. Ed. pr. 1489. Pforzheim, 1500. 

See above. 

Title. 'Behennd vnd hupfch // Rechnung vff alien //kaufT- 
manfchafften.' (Woodcut of a schoolroom.) (F. i, r.) 

Colophon. * Gedruckt zfl Pf ortzheim von Thoman // antzhelm 
Im Jubel Jar als man zalt i5CX)//Got fey lob.' (F. 163, v.) 



I>€Scription, i6', lo X 13.2 cm., the text being 6.5 X 10.4 cm. 
1 1 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, 15 19, 8* (p. 40); 


glii^ ilortl infii^ bef4^ltf flVn 4^ bw tAffd 
^nlqoA^r«;tl^fI4>e Ntnn^ic aitbcr 0efc^ i(l 
Ale Mnn bit bnrruc^ irtU<^ ^rt tr fdbfl fcniti 

£nrnwoltnir|ini bMemnuTcuT^mtt 

%ir <U« rci^no nj^ {f^mctn 


P.M<5 l'8=<|=4'=- 

Fig. 19. From the 1500 widman 

Aagmburg, 1526, 8* (p. 40). The title of the first edition was 
A> foDows : • Behede vnd hubsche // Rechnung auff alien // 
•.aufFmanschafTt/ and the colophon (f. 236, r.), *Gedruckt In 



der Furstlichen Stath // Leipezick durch Conradu Kacheloffen 
// Im 1489 lare.' 


iu^^te^d\0httmm^kM^ fine 
©WW 424 15 4 

505 »J 

iloree f^ Z04 f 5 ^& 9 ii6iff 

ICit^rc^Mt izz 



40 7 

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 
* Intomo ad un Trattato d' Aritmetica di Giovanni Widmann di 
Eger* in the Bulletino, IX, 188, the best discussion of this 
arithmetic that has appeared. 


W'idinmn's arithmetic was the first great German textbook on the 
«'.: irct, although minor works had abeady appeared before 1489. It 
.% in the main a practical treatise, with good problems, and it set the 
«tan<iard for (^rmany much as Borghi's book did for Italy. Among its 
r.< *'e»x»rthy features is the use of the plus and minus signs for the first 
: :.>c in a printed work. (See Fig. 21.) These are not used, however, 
A^ signs of operation, but as symbols of excess or deficiency in ware- 
'.• -'ISC measures. The book is illustrated (see Fig. 20) with pictures 
^*.<rming mercantile customs, and with geometric diagrams. Widman 
at knowkxlges his indebtedness to men like Sacrobosco, * ab da lert 
I« annes delacrobufto vn ander mer,' although his work shows no depeod- 
rr.i e upon the ' Algorismus ' named. 

After the * Inhalt difz buchs in einer gemein,' Widman devotes 2^ 
; alters to * Numeratio,' 2| to 'Additio,' 2^ to * Subtrahiren ' (including 
•:cr.<miinate numbers in these topics), i to * Dupliren,' 2 to * Medieren ' 
I I.e. multiplying and dividing by 2), 11^ to * MuUipliciren,' 5 to divi- 
N 'H, 2} to progressions, and 14 to roots (4 referring to cube root). He 
'^tn takes up fractions in the same order, this work being followed by 
<>/7r.{ound numbers and proportion. He then gives a large number 
-f t\pe firoblems, reguiae as he calls them, although they are not stated 
'. the form of rules as we now know them. These include the • Regula 
■ !' \r\ k rule of three, treated as distinct from proportion), and the regulae 
* .^tu detri con versa, positionis, equalitatis, legis, augmenti, plurima, 
V r tcm iarum, suppositionis, residui, excessus, coUectionis, quadrata, 
.-/':< a, rectprocationis, lucri, pagamenti, and alligationis. 

Otkrr works 0/ I48g. Boethius, p. 27, 1488 ; Anianus, p. 32, 1488. 

JOHANN WIDMAN. Ed. pr 1489. Pforzheim, 1508. 

Seep. j6. 

Titif, • Behend vnd hupfch // Rechnung vff alien // Kauff- 
ranfihafftcn.' (F. i, r.) 

Colophon. 'CGedruck zfl Pfhorzheim von Thoman//Anfzhelm 
It. tar als man zalt 1508/ (F. 161.) 

Description. 8**, 9.7 X 13 cm., the text being 6.5 x lO cm. 
7 rf. blank + 161 numb. = 168 ff., 26 11. Pforzheim. 1508. 

Editions, See p. 37. This is the third edition of this famous 
ir:thmclic, and is by the same publisher as the second (1500), 
^ .:: is from different type. It is about as rare as the first 

': i.TKJfl. 



JOHANN WIDMAN Ed. pr. 1489. Hagenau, 15 19. 

See p. 36. 

Title, * Behend vnd hupfch // Rechnung vfT alien // Kauff- 
manfchafFten/ (F. i, r.) 

Colophon, *Getruckt zfl Hagenaw durch Thoman // Anshelm. 
Im iar als man zalt // isi9^* (F. 1 5 1 , r.) 

Description, 8°, 9.7 X 14.3 cm., the text being 6.8 x 11 cm. 
I f. unnumb. -1-151 numb. = 152 ff., 20-29 11. Hagenau, 15 19. 

Editions. See p. 37. This is the fourth edition of Widroan'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. 


See p. 36. 

Ed. pr. 1489. 

Augsburg, 1526. 

Title. * Behennde vnnd // hiibfche 
Rechnug auff alien // KaufFmanfchaflF- 
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 11 cm. 2 flf. unnumb. 4-190 
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. 

4 + f DMkbotMwrP 
4 ' *P Wtyiitf ittymb 

y ^T* ^T "• VW ^^l^^^^^M 

tx^Wr con 4? J o*T£>rt^ eifrbcn 4 K * 
fr.nim rpiid:) 1 00 tf Pj* i(] fin jtFirncf 

Fig. 2 1 . 

8JWf« , 

From the 




PETRUS DE ALLIACO. Ed. pr. 1490. Augsburg, 1490. 

Alyaco, Hbliaco, d* a illy. Bom in Compi^gne in 1350; chancellor of 
the Univeraity of Paris, Bishop of Cambray, and Cardinal. He died in 1420. 

TUk, • Cocordatia aftronomie cu theologia // Cocordatia af- 
tronomie cu hyftorica // narratione. Et elucidariu duory pre- 
//cedentium : diii Petri de Aliaco car//dinalis Cameracenfis.' 
(F. I, r.) 

Colophon. * Explicit tractatus de cocordia aftronomice veri- 
tatis 2 narrationis hi(torice//a dno Petro cardinali Cameracen. 
completus in ciuitate Bafilien. anno xpi // 1414 : mentis. Maij die 
decima.' (F. 33, r.) ' Opus concordantie aftronomie cum theo- 
logia necnon hyftorice verita: nar//ratione explicit feliciter. 
Magiftri Joannis angeli viri peritiflfimi diligeti cor//rectione. 
Erhardiq5 Ratdolt mira imprimendi arte: qua nuper Venetijs 
nuc // Augufte vindelicorum excellit nominatiilimus. 4. nonas 
Januarij, 1490.' (F. 55, v.) 

Description. 4^ 15.6 X 20.4 cm., the text being ii.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 7 ar//tes liberales : 
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 
Fliilofophia x ar//tes liberales. Es impreffo enla infigne y muy 
leal ciudad // de Seuilla en cafa de Juan Cr5berger.//Afto de 
.M.d.xxxviij.' (F. Ixxij, r.) 

Description. Fol., 19 X 26.9 cm., the text being 15.3 x 24.3 
cm. 72 ff. numb., 42 11. Seville, 1538. 

Editions. Seville, 1489, fol.; ib., 1538, fol. (here described). 



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. 



i^iTando j>fl 1 fltrauelTando cfte fendcro : rmirron atama Dd 

Qig g|nictf :fl D0fecome\aiifl vn marauillalocflrniiia: el ^t lot guiD 

rJ^^cn pnlugaroecafaeap3li3aojmu2rvngalffrfi: lalapumioe 





Fig. 22. From the 1538 alonso delatore 

NICOLAUS CUSA. Ed. pr. c. 1490. Strasburg.?, c. 1490. 

NicoLAUs CusANUs, NicoLAUS Chrypffs OR Krebs. Bom at Kucs on 
the Mosel in 1401 ; died at Todi, Umbria, August 11, 1464. He held peti- 
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 

Tit/e, Usually known as the *Opuscula.' The work begins: 
* Prohemium.// [I]n hoc volumine ptinentur certi tractatus 2 libri 
altiflime X5templato//nis et doctrine: a preclare memorie pre- 
ftantiflimo doctiffimoqs viro // Nicolao de Cufa.' (F. i, v.) 

Description. Fol., 17.5 X 25.4 cm., the text being 1 1.7 x 18 cm. 
163 ff. unnumb. -f 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.) 


Editions. Cusa's *Opuscula varia' first appeared s. 1. a. (Stras- 
burg?, c. 1490 or earlier). His * Opera' appeared at Paris in 
151 1 and 15 14 (see below), and again at Basel in 1565. 

Thiscon tains fifteen of Cusa's tractati, including ' Reparatio kalendarij,' 
*De Apice theorie' (4 if.), 'De mathematicis complementis/ '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, 15 14. 

See p. 42. 
TitU. • Hec in hoc fecudo vo//lumine contenta.// 

Dialogus de ignoto. 2. De tranfmutationibus 
Dialogus de annuncia- geometricis. 33. 

tione. 3. De Arithmeticis com- 
Excitationu libri X. 7. plementis. 54. 

Coniectura de nouifli- De mathematicis com- 

mis diebus.// i. plementis. 59. 

Septem epiftolae. 3. Complementum theo- 
Reparatio Caledarii. 22. logicum. 92. 

Correctio Tabulanim De prefectione mathe- 

Alphonfi. 29. matica. in.' 

(Woodcut of printing-press, with the words: *Prelu Afcefianu.') 
• Venudantur cum cgteZ/ris eius operibus in Aedibus Afcenfi- 
anis.' (F. I, r.) 

Coicpfum. * Emissvm est hoc librorvm Cvsae opvs // egregivm 
Parisiis: ex ofllicina Ascen//siana anno Christi pientissimi om// 
nivm Redemptoris MDXIIII, octa//va Assvmptionis semper 
San//ctae semperqveVirginis// Christi Deiqve Matris//Mariae. 
qva patroci//nante apvd Filivm //portvm salv-//tis spera-//mvs 
et//veniae.' (F. CXVI, r.) 

Description. Fol., 20x29 cm., the text being 12.7x26.8 
cm. 114 £F. numb. (Roman numerals) + 2 unnumb. = 1 16 fF., 
46 11. Paris, 1 5 14. 

Editions. See above. 


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, • Algorithmus Integro?//Cum Probis annexis.' (F. i,r. 
See Fig. 23.) 

Description, 8°, 14.9 X 20.6 cm., the text being T.% 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 {Abhandlungeny V, 158) believes 
that Widman wrote this work, the *Algorithmus linealis' (p. 36), 
and also the * Algorithmus Minutiarum Phisicarum.' 

The work opens with a quotation from Boethius, the same indeed 

as the opening sentence of Sacrobosco's Algorismus. After treating 

, ^ ^ • • «■ ®^ numeration, addi- 

//MflOari mittg ntttettCOtt tion. subtraction, du 

' 5' ^SZ%W«4l*>*i»S tt^ P''^*'""' mediation. 

t ttm g^OWg ^WVPSk *> multiplication, and 

^ division, the author 

Fig. 23. TrrLE of the algorithmus integrorum takes up progressions, 

roots, and the proofs 
of the various processes. There are no applications in the book. 

Other works of 1 4^0. Anianus, p- 32, 1488 ; Boethius, p. 27, 14SS. 
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 propor//tionuni 
vel Mercatorum regula/ (F. 10, r.) 

Description, 4*, 13.7 X 19.4 cm., the text being 8.6 X 15.6 cm. 
10 ff. unnumb., 34-36 11. S. 1. a. (c. 1491). 


Editions. This is not the edition described by De Morgan 
p ()()). because the title has 'addita etiam regula/ instead of 
•ai-iita regula.' The word 'regula* instead of 'regla' at the 

^)[ tgo:ifiimg nouudpe 



Fh^ 24. Trrij: pa(;e of the alu)RIsml's ok c. 1491 

• '. : «hows it to be different from the one described by Brunet 

'. .::i<in of i860). It is probably no. 827 of Main, and the inter- 

■' evidence makes 1491 the probable year of the composition 



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. 

piicrAgottid aritbiiKtttceititMMitictor 

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 callexi. (• De regub 
proportionum // Sive aliter Regula Mercatorum dicta.') It is not as 
practical as the elaborate title would seem to indicate. 


Florence, 1491. 

nULIPPI CALANDRI. Ed. pr. 1491. 

A Florentine aritbmeticiaa of the fifteenth century. 
TiiU. ' Pictagoras arithmetrice introductor.' (Woodcut of 

^ 11 



'Pvd f|4^>per 8] 

n^9> — 


'Ptm I e ^o 'P»eti n>i vt c^ 

M a 


'pvd (o e I 'Pud > e I 

Co— J > — ^ 

♦«• LL. ?^/^ 

■Jotne I Co UieiiK o ff 



Fm;. 26. From calandri*s arithmetic 

r^thai^oras.) (F. i, v. Fig. 25.) * Philippi Calandri ad nobi- 
■ -: ct ftudiofu5 Julia//num Laurentii Mcdice dc arimethrica 
• :. ulu.' (F. 2, r.) 



do e ptcna non merren 
doflttotatoto fiuotae 
beladcctafbnteini f 
uom Id fontc 7 meoen 
do ilcontiocto t fhira 
dotluomoio inqtuti 
di fara ptenab deaa 


fbndodufi poyy che e 
Qolendooftire fiiorao 
crdtpot lanoctefcmi: 
-5 dibrardo : uo Capet 
inquatidt fara fuora 



Saraptcnain C >di 






>J (^j. 

i • f 

farafiioni ir^f di 


Fig. 27. From calandri's arithmetic 

Colophon, * Impreffo nella excelfa cipta di Firenze per ///, 
Lorenzo de Morgiani et Giouanni // Thedefco da Maganza fiy; 

nito a di primo di//Genaio 
1491/ (F. 104, V. Fig. 30 > 
Description, 8*, g.SxiS- 
cm., printed in double col 
umns, each 3.3 X 10.7 cm. to 
1 1.5 cm. 104 ff. unnumb, 

"^ '^r~^ ^"^ \ ^,w^.^ 9-26 11. Florence, 1491. 

FiG. 28. From calandri's arithmetic ^^ ^ ,. . ^, 

Edttums, Florence, 149^ 

8* (here described) ; ib., 1 5 18, printed by Bernardo Zuchetta, 4' 



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 

alM 4 obricctoT dap 
cbcctargbo )o broc 

ftaddfuunc Tilted 




loocaiiaLiriM ddfut 
braccw fcncnippct 



iee| f-oo 
HtiMTcrito iC brjc 

Fk;. 29. From calandri*s arfthmetic 

the first printed Italian arithmetic with illustrations accompanying prob- 
lems, and the first to give long division in the modem form (Fig. 26) 
known to the Italian writers by the name 
• a danda.' Indeed Calandri gives only the 
'a danda' method, omitting the galley 

-' — "- ' 9 41 

Fig. 30. Colophon or 


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 
mell, the length of the hypotenuse, and the broken tree were familiar 
in Calandri's time. 

O/A^r works of i4gi, Anianus, p. 32, 1488 ; Boethius, p. 27, 1488 ; 
Borgbi, p. 16, 1484. 

.< Uc^o 



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 nominat5 C6pendi6//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 copendi5 de abaco per mei/ //ftro Nicolo 
benedeti he meiftro Jacobino fuigo de fancto ger//mano. Ke\ 
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. (Bninet, IV. 
475; Graesse, Tresor^ V, 100; Riccardi, I, 2, 256.) The only go»i 
description of the work is that given by Boncompagni in the Att: 
deir Accad, Pontif. de" nuovi Linceiy XVI, 161, 332, evidently ahei 
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, t>c 
rule of three, certain rules relating to weights, time, money, and other 
measures, and such topics as partnership, barter, interest, alloy's, a&i 
the rule of false position, single and double. He closes the work wit*: 
a chapter on mensuration, or as he calls it, * De la art de ieumentra * 
(* ieumetria ' in the title), and gives a number of interesting woodcut^. 
The chief interest of the book attaches, however, to the fact that 
Pellos came very near the invention of decimal fractions, and th:: 
he actually used the decimal point as is shown in the iUustiatii " 
(Fig. 32). It cannot be said, however, that he had any concept* " 
of the real value of the decimal fraction as such, the first bo- ^ 
devoted to this subject being * La Disme* of Stevin (1585), hereafti: 
described. Pellos simply uses the decimal point to indicate di\-is;c<: 
by some power of ten, writing a common fraction in the quotie'*. 
Thus, to divide 425 by 70, Pellos would divide 42.5 by 7, writing t^ 
result 67^0- 



-4?^i;«:&iy&>i* i •'^j.'^t 

Fig. 31. Title pace of pellos 


dNnferpcr 2 ol 

» T 

s o 

Ct>AiA'P<r > • • 

^ 9 7 < y ♦« 7 . 

CPMrtfrpcr 4^ 0! 
7 9 9 6 $ .7 t 

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o 9 

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a T < c 8 .7 o I 

dtiodeM I p » I 9 — 


^Li&ciidqftftmiitfcrtpodce Arrc jbeioiw rcmblwparitoa 

Fig. 3a. From the pellos arithmetic "» 



Ed. pr. 1492. Wittenberg, 1534. 

PuRBACH, Pbuerbach. Bom at Feuerbach, 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 Mitller, of Konigsberg) was his pupiL His interests were almost 
entirely in astronomy. 

Title. * Elemen//ta Arithmetices // Algorithmvs de // nu- 
meris integris auctore // Georgio Feurbachio.// De Nvmeris 
Yx2s:\S&Jl Regulis communibus & // Proporcionibus.// Cum prae- 
fatione Philippi // Melanchthonis.// M. D. XXXIIIL' (F. i, r.) 

Colophon, 'Impressvm Vitebergae//per losephvm C\vgJ/ 
Anno M. D. XXXIIIL' (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, 


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., 15 10 and ib., 1511; Vienna, s. a. but ear- 
Uer than 151 1, 4"*; ib., 1511,4*; ib., 1512; Niimberg, 1 5 1 3 ; 
Vienna, 15 15, 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- 
rithm!/ ' 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 
fur 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 // fingulari pru- 
dentia coUecta k lodne Vo-//gelin profeffore Mathematico in 


//fchola Viennenfi.//Arithmeticae 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.// Algorithm vs de nv-//nieris 
integris, f ractis, Regulis // communibus, & de Pro-//portionibus. 
// Authore Georgio Peurbachio.// Omnia recens in lucem aedita 
fide & // diligentia fingulari.// Cum praefatione PhiHp. Melanth.' 

Colophon. *Venetijs loan. Anto. de Nicolinis de Sabio.// 
Sumptu uero D. Melchioris Seffae. 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 I4g2, Anianus, p. 32, 1488; Boethius, p. 27, 1488. 
IVorks 0/ I4gj. Anianus, p. 32, 1488; Anonymous (see Widman), 
p. 36, c. 1490; Isidorus, p. 8, 1483; Vincent de Beauvais, p. 10. 

Ed. pr. 1494. Venice, 1494. 

Paciolus, Patiulus, Pacioli. Bom in Borgo San Sepolcro, Tuscany, c. 
1 44 5-1 450; 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*, finis decima nouembris ipofitus f uerit : huic tamen 
pti : die vigefi//ma eiufdem ipofitus fuit. M?cccc.lxliiij. Per eof- 
dem correctorem 2 imprefforem vt i fine Geo*, hr.' (F. 232, 
numb. 224, V.) There is also the following date on f. i, v., 
* M?ccc.?lxliiij?xx'. Nouembris.' 

Description. Fol., 21.5 X 30.5 cm., the text being 19 x 24.2 
cm. 8 ff. unnumb. 4- 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). 


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dtTO)tofin^(fc^ ^ 

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Hnc wn iX|iniiiQiii» 
C moht otoccoCt o {TJiidiHbiif pfMCri^lhitto cS/ 

Fic. 35- Title page or the 1494 paciuolo 



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 

«X 7X «X 

f/ •/ s/ 
/* /^ /* 

9 7 4 §69 

Fig. 34. Gelosia multiplication, 1494 paciuolo 

the rule of three, the rule of false (*E1 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 proportione,' which was pub- 
lished at Venice in 1509 (p. 87). He also published an edition of 
Euclid at Venice in 1509. 

Other works of I4g4. Albert of Saxony, p. 9, 1478; Anianus, p. 
32» 1488. 




35. Finger symbolism, 1494 paciuou) 


Ed. pr. 1494. Toscolano, 1523. 

Title, * Summa de // Arithmetica geo//metria. Proportioni : 
et proportionalita : // Nouamente impresfa In Tofcolano fu la 
riua dil Benacenfe et // vnico carpionifta Laco : Amenisfimo 
Sito : de li antique 2 // euidenti ruine di la nobil cita Benaco ditta 
illuftra-//to : Cum numerofita de Impatorij epithaphij //di antique 
2 perfette littere fculpiti do-//tato : x CU5 finisfimi 2 mirabil co-// 
lone marmorei: inumeri // f ragmenti di alaba-//ftro porphidi 1 
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 : * €.Et fi fequeti pti pnci- 
pali Geo*, finis decima nouembris Ipofitus f uerit : huic tamen pti 
die vigefima // eiusde impofitus fuit .M?ccccc.xxiij. Per eosdem 
correctors impresforem vt in fine Geo', hetur.' (F. 232, v.). 

Description. Fol., 21. i X 30.4 cm., the text being 18.2 x 23.5 
cm. 9 ff. unnumb. -f 223 numb. = 232 ff. Toscolano, 1523. 

Bound with this is the * Tractatus Geometria. Pars fecunda 
principalis huius operis 2, 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. 

Ed. pr. 1495. Paris, 1505. 

Bom at Daroca in Aragon, c. 1470 ; died at Salamanca in 1 560. One of the 
most learned men of his time. He was professor of philosophy at AkaU. 

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.) 



Fig. 36. First pack of text, 1523 pacivolo 


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 ff. unnumb., 35-39 11. Paris, 1505. 

Editions. Paris, 1495,4"*; ib., 1505, 4* (here described); ib., 
1509, 4**; ib., 1 5 13, 4* (see below). Ciruelo also wrote a 
*Cursus quattuor mathematicarvm artiu liberaliu,' Paris, 15 16; 
ib., 1523; ib., 1526; ib., 1528; Alcald, 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 of I4.g§. Anianus, c. 1495, p. 32, 1488. There also 
appeared in 1495 ^^^ 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, 15 13. 

See p. 58. 

Title. * Tractatus arithmetice // Practice qui dicitur algorif- 
mus.// (A large woodcut with the name Jehanlambert.) Venuiv 
dantur Parrhifijs a Johaue La//berto eiufdem ciuitatis biblio- 
pola in fte-//niate diui claudij manente iuxta gymna-//fium 
coquereti.' (F. i, r.) 

Colophon. * CArithmetice practice feu Algorifmi tractatus a 
Petro fan//che Ciruelo nouiter compilatus Explicit Impreffus 
Parifius //per Anthonium Auffourt. Pro Johane Lamberto 
eiufdem ci-//uitatis bibliopola in ftemmata diui claudij manente 
iuxta gym-//nafium coquereti. Anno diii .1513. Dievero .21. 
menfis Martij.' (F. 12, r.) 


Description. 4*, 13.1 X 19 cm., the text being 9.6 x 14.8 cm. 
12 ff. unnumb.y 43-46 11. Paris, 15 13. 

Editions. This is the fourth edition (see p. 60), and is an 
exact reprint of the 1 505 copy, save as to spelling and pagina- 
tion. As a piece of typography it is much better than its 

THOMAS BRADWARDIN. Ed. pr. 1495. Paris, c. 1510. 

Bragwardine, Brandnardinus, Bredwardyn, Bradwardyn, dk 
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. 

TitU. 'Arithmetica thome brauardini.// Olivier Senant// 
V'enum exponuntur ab Oliuiario fenant in vico diui Jacobi fub 
si-//gno beate Barbare fedente.' (F. i, r.) 

Cohphon. * Explicit arithmetica fpeculatiua thoe brauardini 
bn T^'f/uiiaL et 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 
propcrtionibus ' appeared in several editions, as follows : Paris, 
1495; Venice, 1505, fol.; Vienna, 151 5 (p. 117); and a com- 
mentary by Vittori appeared at Bologna in 1 506. 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 
Baeda and Alctiin. 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. 


STAPULENSIS. Ed. pr. 1496. Paris, 1496. 

For the biography of Boethius see p. 25. 

JoRDANUs Nemorarius (Jordanus de Saxonia) was bom at Borgent 
reich, in the diocese of Paderbom, and died in 1236. He studied at Pans 
and was the greatest mathematician of his time save Leonardo Fibonacci 
of Pisa. 

Jacobus Faber Stapulensis (Jacques le FfevRE d*£staplbs) was born 
at Estaples, near Amiens, in I455> ^^^ ^^^^ ^^ N^rac 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 Fran9ois 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., 60-63 H- Paris, 1496. 

Editions. The arithmetic of Jordanus went through varioiLs 
editions as follows: Paris, 1496, fol. (here described); ib., 1503, 
fol.; ib., 1 507, fol. (p. 65); ib., 15 10, fol.; ib., 15 14 (p. 65). Rog^; 
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 o: 
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 Fa^^er 
Stapulensis. The work of Jordanus is similar to that of Boethius, and > 
concerned only with theory of numbers. In particular, the Greek the^-jry 
of ratios, as elaborated during the Middle Ages, is extensi\'ely treated. 

The second part consists of the work of Jacobus Faber Stapulensis or. 
music, in four books. 

The third part is the Epitome of the Arithmetic of Boethiu> 
* CJacobi Fabri Stapulenfis Epitome in duos libros Arithmeticos / . ' o . 
Seuerini Boetij ad Magnificum dnum Joannem Stephanum // Feneriura 
Epifcopum Verfellenfem.* 


The fourth part, consisting of four and a half pages, is a description 
of the arithmetical game of Rithmimachia, possibly by Shirewode (John 
Shinrood, Bishop of Durham, who died in 1494), but usually ascribed 
to Faber Stapulensis. An edition appeared at Erfurt in 1577, 4^. 



hnDeBriibiiicita0 1 Attliee»» 
ICdRpoic tatn iDttlio Dodc kiiiicivroioic9i 

AiiM Mftict coiiite9 (taoo niiicnift liM* 
TCcpboDcicmepoft tcmpoiaimiloirciittmb 

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Fig. 37. Ttfle page of the 1496 boethius 

Other works 0/1406. Albert of Saxony, p. 9, c. 1478 ; Bradwardin, 
p. 6 1 » 1 495 ; Z. Lillius, ' De origine et laudibus scientiarum,' Florence, 
4^ (one page on arithmetic). There was also published at Paris, s. a. 



(c. 1496), an anonymous treatise entitled ' De arte numerskli copediu 
putile icipit feliciter. (Q)uoniam rogatus a plurib5 compedium artis 
numerandi ac breue tractalutu . . .* 

Works of I4gj. Boethius, p. 27, 1488 (the colophon of the 1497 


edition has the incorrect date Mcccclxxxvii) ; Suiseth, 1497 and 1498. 
p. 10, c. 1480. 

IVorks 0/ I4q8. Anianus, p. 32, 1488; Bradwardin, p. 61, 1495 ; 
Chiarini, p. 11, 1481 ; Anonymous, 'Enchiridion sive tractatus de 
numeris integris, fractis,' etc., 4® (doubtless the work published at 
Deventer in 1499, p. 67). 


STAPULENSIS. Ed. pr. 1496. Paris, 1507. 

S«c p. 62. 

TUU, • In hoc opere contenta.// Arithmetica decern libris 
licmcmftrata // Mufica libris demonftrata quattuor // Epitome I 
l:hros arithmeticos diui Seuerini Boetij // Rithmimachie ludus 
, : pugna nuero^ appellat.' (F. i, r.) 

Colophon. * ClmprcfTum Parifij in officina Henrici ftephani 
c regione Schole decretorum fita.//Anno Chrifti fiderum con- 
iii*>ris 1 507. Decimo die Noucmbris.* (F. 78, r.) 

Dtscriptiom. 8^ 20x26.6 cm., the text being 18.1x21.5 
m. 78 flF. unnumb. + i blank = 79 ff., 60 11. Paris, 1507. 

Ediiums. See pp. 2Ty 62, for the editions of Boethius and 

like mo!»t of the works from the press of Stephanus, this book is 
''«-^utifuUy printed. It contains the * Klementa Arithmetica* of Jorda- 
r.'>, with the demonstrations of Faber Stapulensis, the Epitome of Boe- 
thius by Faber Stapulensis, the • Rithmimachia,' the commentary on 
Nu robosro's astronomy by Faber Stapulensis, and the first four books 
f Koclid ' a Boetio in latinum tranflate.' 

STAPULENSIS. Ed. pr. 1496. Paris, 15 14, 

Sec p. 63. 

7i//r. • In hoc opere contenta // Arithmetica decern libris // 

onHmftrata,// Muflca libris demoftrata // quatuor.// Epitome 

-; libros Arith-//nieticos diui Seuerini //Boetij.// Rithmima- 

•..e ludus qui//et pugna numeroru ap-//pellatur.V/ (Sur- 

• -'incied by an elaborate wcxxlcut with the following wording : 
Hacisccvndariacs//tctcastigat//issimaexofficina//aemissio* // ) 

• CHi-c fecundaria fuperioru operum seditio // venalis habetur 
r z*- ■":)'»:/ 'in officina Henrici Stephani e regicme fchole Decre- 

• r:m/ ^F I, r.) 

L'^lophen. * CHas duas Quadriuij partes et artium libcralium 

• ''.M(Hias atqj duces cu quibufdam ammini-//cularijs adicctis: 
. -.'jujt ex fecunda recognitione vna formulis cmcdatilTime 


mandari ad ftudiorum // vtilitatem Henricus Stephanus fuo 
grauiffimo labore et fumptu Parhifijs Anno falutis domini:// 
qui omnia in numero atq5 harmonia formauit 15 14. abfolutumq5 
reddidit code anno: die feptima // Septembris/ fuum laborem 
vbicunq5 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. 


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 neq5 cum Commentatiis: neq5 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 1 2.1 X 22.3 
cm. I24(f. 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, 1577, fol. (p. 68); Leyden, 1592, 8° (p. 68) ; ib., 1599, %"" ; 
and later. An Italian translation was published at Mantua in 
1578. For bibliography, see Boncompagni's BulUtino^ XV, 506. 

This work is a medley of prose and verse, and forms a kind of enc>- 
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. 

E:- :r 

j::' - 

Ji.-uzr _::•.-.-=■ 


tx3- U»r Ir-^z* 



di .: ■■>;■*■ '■■■■■ '— 

Anno SaJuui X ' ' 

Bcrtboc-uEi. T ::»:. " 

Descrtfmim F ■- 3^ ' 2*- - *~ j 
On f. 69. V . tat par. n a-nrairr.. i-_ 
100 ff. in tbe c^-t y>Jt- umcni.^- -i 

*- -^ 




Ed. pr. 1499. Basel, 1577. 

See p. 66. 

Title. * Isidori // Hispalensis // Episcopi // Originum libri 
viginti // ex antiquitate eruti.// Et // Martiani Capellae // 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 Pemam.' (F. i, r.) 

Descriptiofi, FoL, 20.5 X 31 cm., printed in double columns, 
each being 8.2 x 24.4 cm., 7 ff . -f 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. 


Ed. pr. 1499. Leyden, 1592. 

See p. 66. 

Title. * M. Capella.// Martiani // Minei Capellae // Carthagi- 
nensis // de Nvptiis Philolo-//giae, & 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 I2.b 
cm. 4 pp. blank + 18 unnumb. + 396 numb. =418 pp., 30 11 
Leyden, 1592. 

Editiofis. See p. 66. 


HcBtiDftDltg l^oiiiber0di.flritbiiietn^ot0 Doccf 
jDllozatnHiiiio Undid lipocnrinDeedit^IRo 

An«t fbocinccepttcat tib(r<80lt Qttrf 

Sfbdi8Di^tMqiiopi»flito t»tfoa^nomia8 
~~ ' rviMbmtiMtttibi'CDnaiaKj 
indnoltmn cotnnUt db boni 


Fig. 40. Title page of 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. 15, v.) 

Description. 4**, 14 X 19.9 cm., the text being 9.3 X 16.2 era. 
15 ff. unnumb., 22-37 11- Leipzig, s. a. (1500). 

Editions. Leipzig, s. a. (1500), here described; ib., 1509; 
15 1 3; Leipzig, 1515, 4^ which may be the *Algorithmus 
linealis,' s. a. and 1505, by Letter, referred to by De Morgan 
(p. loi). 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 15 12. 
(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 Lyptzeii Anno 1500/ 
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). 


Ed. pr. c. 15CX). 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. 1 500, here described) ; 
Florence, 1514 (?) ; Basel, 1520, 4°; ib., 1530, 8^ 

A work on ancient measures, using the Roman numerals throughouu 
except in the index. Such treatises are of value in studying the histor> 
of arithmetic, but are not, in general, included in the bibliographic^ 
lists of this work. 


Other works of c, 1^00. Anianus, c. 1500, p. 32, 1488; Boethius, 
c. 1500, p. 27, 1488 ; Bradwardiuy 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 
'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 ' {Abhandlungeriy I, 
24 ; Bninet, Man.y 6 (i), 458). 







Fig. 41. TnxE page of leonardus fortius 
GEORGIUS VALLA. Ed. pr. 1501. Venice, 1501. 

Bom 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 f vgiendis // rebvs opvs, in qvo haec // continentvr.// De arith- 
metica libri .iii. ubi quaedam a Boetio praetermifTa 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 
(piritalibus, Catoptricis, ac Opticis, deq; // quadrato circuli habe- 
tur tractatus.// De tota Aftrologia libri .iiii. in qua fabrica,ufus<^; 
aftrolabi exaratur, & quae fi-Z/gnorum in exhibendis medica- 
minibus fit habenda obferuatio.// De Phyflologia libri .iiii. ubi & 


Metaphyfices qda lectu q dignifs. utiliflimaq;.//De Medicina 
libri .vii. ubi de fimplicium natura per ordinem litterarum.//Pro- 
blematum liber unus.// De Grammatica libri .iii.// De Dialectica 
libri .iii.//De Poetica liber unus.//De Rhethorica libri .ii.//De 
Morali Philofophia liber unus.// De Oeconomia, fiue adminiftra- 
tione domus libri .iii. in quibus de Architectu-//ra, re4; 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-//nia, Secudus de corpore, Tertius uero de 
urinis ex Hippocrate, ac Paulo aegi-//neta, deq; Galeni quaeftioni- 
*bus in Hippocratem.// De Rebus extemis liber unus, ac ultimus, 
ubi de Gloria, Amplitudine, & caete-//ris huiufmodi.// Haec fum- 
matim, fed infunt, & alia plurima,quae legedo licet cognofcere/ 
(F. I, r.) 

Colophon. * Venetiis, in aedibus Aldi RomaV/ni impensa, ac 
studio Ioan-//nis Petri Vallae filii pi-//entiss. mense Decern-//' 
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 -f 308 unnumb. = 31 1 ff., 55 11. Venice, 1 501. 

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 o{)era- 
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 octa\x> 
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. 



Ed. pr. 1 50 1. Venice, 1501. 

Arnauld DC ViLLKNEUVE, Arnald Bachuone. Bom in 1248, at Villa 
Nova (Catalonia), or possibly Villeneuve, near Montpellier; died in 1314, 
fthipwrecked 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. 

£cmpixmB £ccleiufticit079ftrotia^ 

micm £dime a atf^agiftro 3iu 


titter 3mp:drtiiiu 

£ttmi6:dtfa ^t K^riuaeglo^ 

Fm.. 42. Title pace of arnaldo de villa nova 


Title. * Computus Ecclefiafticus \ Aftrono-//micus Editus a 
Magiftro Ar-//naldo de villa Noua No-//uiter Impreffum.// 
Cum Gratia Et Priuilegio.' (F. i, r. See Fig. 42.) 

Colophon. *Clmpreffum Venetijs per Bemardinu Venetu 
deVitalibus.//Anno Dni.M.CCCCCJ.Die .xvij.Men.Februarij.' 
(F. II, 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. 


Ed. pr. 1 50 1. Cologne, 1501. 

A German arithmetician of c. 1500. The name Sanensis suggests hb 
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 ftudioqj p 
Johane hufwirt sanefez // elaboratus. caracteri p^^mifTus Colo- 
nie In oflBcina felicis memorie ho//nefti viri Henrici Quentell. 
Anno repatoris humane feruitut5 Mccccci.' (F. 20, r.) 

Description. 4**, 14.3 X 20.6 cm., the text being 9 x 15.4 cm. 
20 fT. unnumb., 25-47 11. Cologne, 1501. 

Editions. Cologne, 1501, 4** (here described); ib., 1503; ib., 
1504, ^ (p. T7)\ 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 


miscellaneous rules ('Tractatus quartus de regulis mercato:fi' etc.). 
In the algoristic treatment of integers Huswirt places 'duplatio 
(doubling) after multiplication, and * mediatio ' (halving) after division ; 


@[it Atntiiott m 


boyf^wHtrtlflff fttTflwfeiBfroiftHfc 


tniucle neftttesfluiguS compcToe Girmtem 
Nsc nimiu nbiats gamuus did l^bm. 

/Uic peieiaiatcascfupaisincc^itus) vmbtai 

A^ illkpodusfitcs* S^iutjgja roilcc 


Fig. 43. Title page of the 1501 huswirt 

tiut 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 
.'\ges and well into the sixteenth century. 



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 Vo theca. circul? cifra. 
Hue 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 


fXymolfiplkifiioae OpGnoMrtni 

_ ^ 


lepore fugiente,* f. i6, r.), 
although it has nothing to 
do with the hound and hare, 
but relates to a traveler going 
from Cologne to Rome, and 
followed five days later by 
another traveler who in 
due time overtakes him. 
(* CAmbulat qdam de Colo- 
nia verfus roma et ambulat 
4tidie 9 miliaria, alius aut 
pfequit' ipfum poft 5 dies,* 

Other works of i^Oi. 
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, 
4**; Strasburg, 1505, 4°; 
Hagenau, 1507, 4"*; ib., 
15 10, 4^*; ib., 15 12, 4**; Strasburg, 15 14, 4**; 15 15, 4**; Strasburg, 
1518, 4^ Paris, 1530, 8**; ib., 1535, 8^ Cologne, 1536, 8*; Paris, 
1550, 8^ 

Works 0/1302. Anianus, p. 32, 1488; Albert of Saxony, p. 9, c. 
1478 ; Bradwardin, p. 6, 1495 \ Nicolo Calvino, a work on arithnoetic 
and geometry, Milan, of which no extant copy is known (see Riccanii, 
part I, col. 213). 

{^r{^{L0nJnrLl^ Q4irfnadrrrMa<n(ntiirtf4ritfw.kiarci \ ffvikn foo^ipafff 

true (nu[ripts[ jnoncniiMi^irutum lut^t U fctjr^^^Bpi* ta(iA bvatfC^h^A 

rrfjm {rori«,q4a0 \tm^it fmi£iipti«.crp:odiicivfltti'!enM Bl^JHM# 

PvVKcrpfoitnHctGiimM.vipJicfuifigDri £fuiipiiW-fcPB»'* 

Ene^ifl* ft % imTcifnn* ^utm^mxmifi^tsm 

£iLg^u« A 4 tdl^Rsrii wiftfJbict4^*>ft 

43 nrlui « 

p:tWp3i3iCty(T0PoOp«ficrffjfiT eb (^ptre.!""" 

f^npiitifi ffl.pjcipoi'iruj ifj^u &u^UB ni^ncrli |^MW} i#«ww « . ^^ 
[i I pf FH?iifl rmilrspl li jjt o zocrabuar mtmcm* i*a »rf (i**fr^ fern*** 

rcmft^ijrjmBt rr>e*r^ rtffniiJrjriT Aj|uiri!N^.<r paawc Mq ^" "' ?f* ^^|?? 

iniCu?nmimcra][|iTi rm iTKiJf^ drnpt Nrhff -i^gam ( 
erir fjj'rfra.^i]a«rtm?'uibn.pTkitum ^rum fimliJi 

itfimpLuni f 7 ** 

J^ 4 


Fig. 44. Complementary multipucation, 
HUSWIRT (1501) 



Ed. pr. 1 501. Cologne, 1504? 

See p. 74. 

TitU, 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 1 501. 


Ed. pr. 1503. Venice, 1503. 

Pasi. a Venetian writer of the fifteenth and sixteenth centuries. 

TitU. '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- 
ccllcfe 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, 


Editions. Venice, 1503, 4* (here described); ib., c. 15 10, 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. 




A VBRemo aduc]p prima a dechiaraie a 
uoftte exoellctk nitte lerobe che fe ueda 
no 1 Venecia a pexo groflb & quelle che 
Ic uedanoapexo Ibtileedelordine di pe 
xi de U arzenti:6£ de le coditionc di pan^ 
n j de lane francefche fatce in Venetia:6C 
eciaiti dele conditione di pannj de firda: 
e panni doro:6C del ordine del uedextc de 
leipecie;&deleruetai:e:8;deIoidine di 
pcjd dele farine e biichorit &del ordine 
de le mifurc dU uini:& del otdine del ucn 
'der del oiotdc in diemodo e pexo (e ucn 
deno igcadiiOt del oidine di fratti che (e ucndeno a nome di ftcr: e 
die pexo hano cadaun ftcr:e come refponde el pexo groflb con tu 
ta la lalta:8f tuto el leuante 6C ponente:& etiam come ipexj fubti^ 
li e pexi grofli lefpondeno con molte tene de Italia : de dalmatia c 
de Ieuante:& come tefpondeno le mefure di panni de lana con ru^ 
ta ]calia,*e con turo el kuante;& caam come refpondeno k mdii^ 
re di panni de fedaie panni doro e darzento con tuta ltalta:6C co ta 
to d leuante 8C ponentcSC come refpondeno le mefure dii^ terra 
2oe la uirga da londra,*e de la taripha dalixadria;6; etiam qtiella de 
damafcho & la taripha da leppo.e come torna la (porta dalixadria 
con molte terte de leuante e de ponete e de la ltaUa:6c comee! can 
ter forfori dalixandria refponde con alchune altre terre de leuante 
de ponente e de Italia«'6£ etiam come re(ix>nde el cantec 2eroui con 
aldiune terre del leuante del ponente e de Italia;^; come ctfpSde el 
cento de le mene con alchune terre del leuante emoltede Italia 6t 
del ponente:8f come refpondeno tun icantera de leuante e del p<v 
nente con el pexo fubrile da Venetia: 8C come reipode el canter da 
napoli de reame con molte terre del leuante del ponete e con moP 
te de ltalia:8; come refponde el canter de Conftatinopoli con mol^ 
teteixcdeleuante&etiamdeltaUa:6rconierclpondeel canter de 

Fig. 45. The beginning of the 1503 paxi ait 



Ed. pr. 1503. S. I a. (Venice ?, c. 15 10). 

See p. 77. 
7V//r. 'Tariffa De Pefi e mefure cor-//refpondenti dal Le 
uantc al Ponete : da // vna terra a laltra : e a tutte le parte del // 
mofido :con la noticia delle robe//che fe trazeno da vno Paefe 
per laltro. Nouainente//c5 diligentia Ri-/Atampata .'. H^' 
F i,r.) 

Coloplum, •CFinilTe il prohemio de // Miffer Bartholomio di // 
I*aii da Vencfia.// Finis/ (F. 218. v.) 

Description, 8*, I0.2 X 1 5. 1 cm., printed in double columns, 
coi h being 4. 1 x 12.5 cm. 218 if. numb., 30 11. S. 1. a. (Venice ?, 
t 1510.) 

Sec p. 77* This is the second edition of this popular ' Tariffa.' Since 
:t was one of the first books of its kind to appear in Venice, its five 
oiitioQS are easily explained. 


Ed. pr. 1503. Venice, 1540, 

Seep. 77. 

TiiU, • Tariffa // de i pcsi, e misvre // corrifpondenti dal Le- 

.juitc al Poncnte : //e da una terra, e luogo allaltro, qua fi p// 

t ;tte le parti dil Mondo : con la dichiaV/ratione, e notificatione 

•:; tutte le robbe://che fi tragono di uno paefe per laltro.// 

C* »mpnsta per M. Bartholomeo di // Pafi da Vinetia. Con la fua 

Tauola copiofifHrna, e faciV/lif^ma a trouare ogni cofa // per 

r<iinc, nuouam€te // fatta : e con fomma // diligeza reuista,//e 

<^tapata. In Vinetia. M.D. XL.* (F. i, r.) 

Colophon. * In Vinegia. Nelli cafe di Pietro di Nicolini da 
>Libbso._'Ne glianni dilla falutifera Circoncifione dil no-//ftro 
>:,n»rc. M. D. XL.// Dil mefe di Genaio.' (F. 212, r.) 

DrscripHon. 8^ 1 0.3 X 14.7 cm., printed in double columns, 
c.A. h being 3.5 x 12.3 cm. 11 ff. unnumb. + 1 blank + 200 
r. :mb. » 212 flf., 30 U. Venice, 1540. 

Nce p. 77- 



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 PaoloGherardo. 
//M. D. LVII/ (F. i,r.) 

Colophon, * In Vinegia per // Comin da Trino.// M. D. LVIL* 
(F. numbered 200, r.) 

Description. 8**, 10.6 X 15 cm., printed in double columns, 
each being 3.7 X 1 2.3 cm. 1 1 ff . unnumb. -h 200 numb. = 211 ff., 
30 11. Venice, 1557 

See p. 77. 

STAPULENSIS. Ed. pr. 1503. Paris, 1503. 

For the biographies of Boethius and Faber Stapulensis see pp. 25, 62. 
JoDocus CLICHTOVEUS was bom at Nieuport, Flanders; died at Chartres, 
September 22, 1543. He was educated at the Sorbonne, and was canon 
of Saint- Jean, at Chart res. Like Faber Stapulensis, he was known chiefly 
as a commentator. 

Title, * In hoc libro contenta // Epitome/ copendiofaqs // 
introductio in libros // Arithmeticos diui Seuerini Boetij : adie// 
cto familiari cometario dilucidata.// Praxis numerandi certis 
quibufdam re-//gulis c5ftricta.// Introductio igeometria : ex 
libris diftlcta // Prim^ de magnitudinib^ & earu // circuftantiis.// 
Secudus de cofequentibus/ conti-//guis/ & cotinuis.// (Sur- 
rounded by an elaborate woodcut.) CTertius de puctis. CQuar- 
tus de lineis. CQuItus de fuperficieb9.//CSextus de corporibus. 
CLiber de quadratura circuli. CLiber de cubica//tione fphere. 
CPerfpectiua introductio. CInfuper aftronomicon,' (F. i, r.) 

Colophon, * Id opus imprefferut Volphgangus // hopilius et 
Henricus ftephanus // ea in arte focii in Almo pari-//fioruin 
ftudio Anno Chri//fti Celorum totiufqs // nature coditoris./. 
1503. Die vice//finiafepti-//nia Iu-//nij/ (F. cxi, v.) 

Description, Fol., 19x26.6 cm., the text being 16x22.6 
cm. 112 ff. numb., 47-54 11. Paris, 1503. 


Editions. This is the first edition of this combination of 
works, the second (somewhat changed) appearing at Paris c. 
1507 (see below), and the third ib., 1 5 10. The epitome appeared 
as a 'Compendium arithmetices Boethii/ s. 1., 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. 15 15, 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 
1 5 10, fol. 

The copy here described is bound with the 1509-10 edition of 
BoviUus (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. 

STAPULENSIS. Ed. pr. 1503. Paris, c. 1507. 

See p. 80. 

Title. 'Introductio/Zjacobi fabri Stapulefis in Arithme//cam 
Diui Seuerini Boetij pariter 2 Jordani // Ars fupputadi tam per 
calcu-//los 95 notas arithmeticas fuis quidem regulis elegater 
expreffa// Judoci Clichtouei Neoportuenfis.// Queftio baud 
indigna de numeroru//et pdigitos t particulos finita pgrelTione 
ex Aurelio Auguftino // CEpitome rerum geometricaru5 ex 
Geometrico introductorio // Caroli Bouilli.// €.De 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. 


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. 

Bom at Balingen, Wiirttemberg ; 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. ^i-f/zs Margarita phy- 
losophica tractans // de omni genera 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 // Mathi^ 
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 -f 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 GuUucci, 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 


B. lie 
Pmik II. From the M.\K(;.\KnA phiiosophica 


•>n arithmetic (see p. 80), and ' Questione di S. Agostina della 
progressione dei nutneri per li digiti» et per li articoli/ Hart- 
f elder (Ztitsch, 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 modem encyclopedia to appear in print. It contains 
A compendiuni of the trivium, the quadrivium, and the natural and moral 
Kicoces. It is made up of twelve hooks, 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 
< U-ssification 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 
roou. 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 
ami the rule of three. The illustrations are particularly interesting. 
Nre Plate II.) 

Oiker works 0/1503* Boethius, p. 27, 1488; Bradwardin, p. 61, 
M95 ; Faber Stapulensis, p. 6a, 1496; Huswirt, p. 74, 1501 ; Jor- 
duius, p. 6at 1496; Orbellis, p. 23, 1485; Peurbach, p. 53, 1492; 
>acrobo900, p. 32, 1488 ; Torrentini, p. 76, 1501 ; Anonymous, 
' lextns arithmetics comunis, cum Conradi Norici commentatione,' 
Ir:png, fol. 

W^rks of 1504. Anianus, p. 35, 1488 ; Bradwardin, p. 61, 1495 ; 
) -hannes Carolus (see Landshut, below) ; Huswirt, p. 74, 1501 ; Reisch, 
p. 82, 1503; Saoobosco, p. 32, 1488; Johann Karl von I^andshut 
I I^aiKQt), *.Algorithmus integrorum,' Leipzig (see also p. 97, 15 13, 
1515); Henricus Stromer, ' Algorithmus linealis cum Regula de Tri,* 
I^-.;izig« 4*, with other editions in 1510; 1512, 4^; 1514 ; Leipzig, 
1516,4*; ib., 1517,4*; 1520. 

GREGORIUS REISCH. Ed. pr. 1503. Basel, 1508. 

Sc« p. 82. 
7V//r. * Margarita philofophica // cu additionibus nouis : ab 
u ,K turc fuo // ftudiofiflfima reuifide tertio fup additis. // Jo. 
5* h«->ttus Argeii. lectori. S.// Hanc emo/ non prefTam mendaci 
.'•..rniate/ Lector:// Pluribus aft auctam perlegc idoctus ens.// 
Kk\\\c^. 1508 ' (F. I, r.) 


Colophon. * CTertio induftria complicu Micha//elis Furterij/ 
et Joanis Scoti//ftudiofiiffime preffa. Ba-//file§.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 if. unnumb., 42 11. In this edition the leading initials 
are inserted by hand and the illustrations are colored. Basel, 

See p. 83. 

GREGORIUS REISCH. Ed. pr. 1503. Basel, 1535. 

See p. 82. 

Title. * Marga-//rita philosophica, rati-//onalis, Moralis phi- 
lofophiae princi-//pia, doudecim libris dialogice c6ple-//ctens, 
olim ab ipfo autore recognita : // nuper aut ab Orontio Fineo 
Delphi//nate caftigata & aucta, un4 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. 4- 1498 numb. + 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 elajxv 
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 arithmel 

Title, See Fig. 46. 

A German arithmetician of c. 1500, from Monte Gaudio (MongavensisV, 


C0l€fkan, 'CAlgorithmi. qui ars dicitur numerandi. de integris 
, , per figuranim (more Alemanorum) deletiones. Nec//n6 de 
pfK>rtionib;» ingeniofi Pythagorifte Theodo//rici Tzwyuel. poft 
pluhina praxin iam tande in hoc // opendiu3 reducti finis adeft. 
qiKxl et puplica ob vti//litatem in magiftrali artis impreffori e 
tabeina ingcV/nuorum liberorum Quentell iterato dideminari pro 


opiilhtlatmo Cdrotioriciomi 

I>vt$catehtiii]slibeffi Tetiafticlioa 
joanoteMuniidlij Runmi^ 

PyiliagorenuiiMlmdil^ ^ujs amac 

DcfbpliBii huiK iiiodicii in coi n paKt^ libni 

Fig. 46. TrrLE page of tzwivel 

«. urauit. Anno a natali dominico Millefimo quin/Zgenetcfimo- 
:VpiinK)/ (F. 9, v. Sec Fig. 47.) 

Description, 4*, 13.9 X 20 cm., the text being 8.8 X 14.5 cm. 
10 flf. (I blank), 46 11. Cologne, 1507. 

Editions. Giinther mentions a Mtinster edition of 1505, but 
I have not seen it ; there was a Cologne edition in the same 
\ car ; Cologne, 1 507, 4* (here described). 

The work is divided into two parts, the first beginning as follows : 
* C^Vlgorithmus de integris p figurarii (more alemanof) deleti-/ one 


arte numeradi enucleatim ppendioreq5 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 : < CAlgorithm^ 
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 alemanor^) 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 i^os^ 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- 

^SS^^l^vSSSS^S!^ «J?»^'' P- 76. 1501 ; Anonymous. 
ndMpomonft9ingcmoflt>yi|bfl0oiilleZb(Mlo 'Tractatus perutilis m arithme- 
%&^&:XS:iT^1^ tica speculativa.' Paris; HennaB- 
latinminiiii0ifbaliamaAnpidronctaRfiitNi0C^ nus Buschius, 'Enchiridion no\iis 
owMit. aiuio • ucali OomimcD tOMm qate Algonsmi,' Cologne, s. a. ( 1 504), 4 
floiidiinorcpttaitf. (see also p. 106, 15 14); Nicdaus 

Fig. 47. Colophon OF TzwrvEL Horem, 'Tractatus proportionu/ 

Venice, in a volume with Bradwar- 
din and Suiseth (see also p. 116, 1515);* Algorithmus linealis Baccalariii 
Wolfgangum Monacenseni,' Leipzig, 4^ ; Georg Leunbach, an arithmetic. 
Works of 1306, Albert of Saxony, p. 9, c. 1478 ; Anonjmious, 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. * Cimpreffum 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 153 
cm. 28 ff. unnumb. -f i blank — 29 fF., 28-34 11. Leipzig, 1507. 


Ediiions. Leipzig, 1 507 (here described) ; ib., 1 509, 4"*; s. 1. a. 
(Numberg?, c. 15 10). 

This resembles several of the works on algorism appearing about this 
time, such as \Vidman*s (?), Licht's, and Huswirt's. It contains a very 
X^Mci treatment of the fundamental operations, including duplation and 
n^ediatioD. In division, only a single example is given, that of 1456 + 
I J. After a similarly brief treatment of fractions, the Boethian propor- 

tittftw ttmiMH iiyft aff uttnt 

99W^vy^ ^9WW9^gw^ww^Fww^ www jg^^^^ww 


utln InlualllllS 


Fig. 48. TrTLE of the 1507 A/gorithmus 

;ioos (ratios) are taken up : ' Sequitur Algorithmus proportioniL' Then 
fuUow the Rule of Three and several other rules now entirely forgotten, 
sQch as * Regula legis,' * Regula augmenti/ * Regula plurima,' ' Regula 
pulchnu' and * Regula falfi.' Such * regulae ' were not stated like modem 
rules but consisted of groups of similar problems. 

OiJUr works of 1507* Boethius, p. 27, 1488 ; Faber Stapulensis, c. 
I507, p. 81, 1503 ; Huswirt, p. 74, 1501 ; Peurbach, p. 53, 149a ; Tor- 
rrntmi, p. 76, 1501. 

Works of i^oH. Anianus, p. 32, 1488 ; Reisch, p. 82, 1503 ; Widman, 
?'• 39» 14^9 ; Hieronimus de Hangest, * Liber proportionum,' Paris, 4^ 

LLXA PACIUOLO. Ed. pr. 1509. Venice, 1509. 

Seep. 54. 

TitU. After a vocabtUary and index the work begins on f. 
I, r., as follows : * Excellentiflimo principi Ludouico mariae Sfor. 
An^lo Mediolanen/Zfium duci: pacis et belli omamento fratris 
Lucae pacioli ex Burgo fancti // Sepulchri ordinis Minorum: Sa- 
c T3t theologiae pfefToris. De diuina pro//portione epiftola.' 




Fig. 49. From the 1507 Aigorithmus 


Colophon. *€.Venetiis 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 impreffum fub quouis //colore in publicum ducat 
fub penis in dicto priuilegio contentis. Anno Re//demptionis 
noftre.M.D. Villi. Klen. lunii. Leonardo Lauretano Ve.//Rem.Pu. 
Gubemante. Pontificatus lulii.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 vras 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 
gi^-en by ' Platone e Ariflo. e yfidoro i le fue ethimologie. El feuerin 
Hoetio in fua arithmetical (F. 5, r.) Most of the treatise is, however, 
dewted to geometry. 

CAROLUS BOVILLUS. Ed. pr. 1509-10. Paris, 1509-10. 

BouvELLES, BouELLEs, BouiLLES, BouvEL. Bom 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-//nientis Finis 
Anno Salutis Humane // 1 509 lanuarij Die Deci//maoctaua // 
CEditum est vniversvm hoc volvmen Ambianis in edibvs Re// 
uerendi in Christo Patris Francifci De Hallewin Eiufdem Loci 
Pontificis .*. // Et emiiTum ex officina Henrici ftephani. Impenfis 
eiufdem et loannis parui in chalcotypa//arte fociorum Anno 
ChristiSaluatoris omnium 1 5 10. 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. Fontis has described (Toulouse Mem. (9) VI, 155- 
167, for 1894) a quarto of 15 10, published by Stephanus of 

AwUjf mifmif ^i^jSwi. 

Clnrup tnadimatfcuopot qiiadnpartit6(rDeNariier»*Pcr£e^4[^ 
Mathcmadds Rods fj^De Geomectids Corporibtts 
([De Gcomctrids Supplementui 

Fig. 50. Title page of bovillus 


Paris, entitled, ' Caroli Bovilli liber de numeris perfectis/ This 
Un>k 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 L 1 48, v., and ends on f. 171, r., with the colophon : * Libri dvodecim 
nvmerorvm finis : editi // in domo- R« P. Francifci de HaUewin/ Pon* 
tifiiis Ambi//anenfis. Anno ab autore numerorum Incamato.// 1510: 
Maij derimafexta.' It relates solely to the mystery of numbers and to 
the (treek theory. The part on perfect numbers, numbers which equal 
i^c phis the sum of their factors, begins on f. 172, r., and ends on f, 
1 H<>, r., with the following colophon : * CLiber perfectorvm nvmerorvm 
fin.s. 7 Plerfecto/ trinoqj deo laudes in gentes. Anno domi-//ni/ 1509 : 
bntiarij 4/ The dates of the several parts vary from October 25,1 509, 
to January 18, 1509 (1510 N. S.). 

OtJk^ works of ijog. Anianus, p. 32, 1488; Anonymous, p. 87, 
1 507 ; Borghi, p. 16, 1484 ; Ciruelo, p. 60, 1495 ; Licht, p. 70, 1500 ; 
.Nit robosco, p. 32, 1488. 

OiJUr works 0/ i^io. Anonymous, p. 87, 1507 ; Anonymous, c. 
>5 io« P* 46, 1491 : Boethius, p. 30, 1488 ; Bovillus, p. 89, 1 509 ; Brad- 
mirilm, c. 1510, p. 61, 1495 ; Clichtoveus, p. 81, 1503 ; Faber Stapu- 
W-a^is p« 62, 1496 ; Jordanus, p. 62, 1496 ; Peurbach, p. 53, 1492 ; 
Mrrobosco, p. 32, 1488 ; Stromer, p. ^^^ 1504 ; Pftxi, c. 1510, p. 79, 
150; ; Torrentini, p. 76, 1501. 

H'orks of ts't. Anianus, p. 32, 1488; Boethius, p. 27, 1488; 
MiiTei (Maphaeus), p. 86, 1506 ; Peurbach, p. 53, 1492 ; Simon Eisen- 
-:..Lnn, * Enchiridion arithmetica,' Ijeipzig, fol. 

JTAN DE ORTEGA. Ed. pr. 1512. Rome, 1515. 

JoHJt PB LoRTxa. A Spankh priest of the Dominican order, from 
Anfoo. He was ttiU living in 1567. 

TiiU. ' Svma // de Arithmetica : Geometria // Pratica vtilifli- 

- a: cirdina//ta per Johane de Or//tega Spagnolo//Palentino.// 

C;m Priuilegio.' (Surrounded by an elaborate woodcut. F. i, r.) 

CalpfAcm, * Impreffo in Roma per Maftro Stephano Guilleri 

c Lorena//anno del noftro Signore 1515 adi 10 de Noudbre 

Kr^cnante Leo//ne Papadeciroo in fuo Anno tertio.* (F. 1 16, r.) 

DfscripHam. Fol., 20.6 X 30.2 cm., the text being 1 3. i X 22.7 

■ Ti 2 ff. unnurob. + 114 numb. » 116 £F., 32-38 11. Rome, 



Fig. 51. From the 15 15 ortega, introduction to barter 


Editions, Barcelona, 15 12; Lyons, 15 12, 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 1 5 1 2 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 1 5 1 5 
belonged to .Prince Boncompagni and has his collation on the 
cover. It js beautifully printed and is one of the best examples 
of the early Italian mathematical typography. 

This b 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. 

Oth^r works of i§i2. Bradwardin, p. 61, 1495 J 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. 15 12. Seville, 1542. 

See p. 9t. 

Title. ' Tratado // fubtiliffimo de Arifmeti-//ca y de Geome- 
tria : c6-//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 imprefTo 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 
SeuiUa:a cinco dias//de deziembre de. M.d. y//quareta y dos 
aiioz.' (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. 


JUAN DE ORTEGA. Ed. pr. 1512. Seville, 1552. 

See p. 91. 

Title. * Tractado // Subtiliffimo 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 Gon^alo Bufto d'muchos//errores 
que aula en algunas im/Zpreffiones pafladas.// CVan aiiadidas 
en efta impreffion las//prueuas defde reduzir hafta partir que- 
//brados. Y en las mas de las figuras de //geometria fus pnie- 
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.) 

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 2 muy leal // ciudad de Seuilla por Jua canalla, enla // 
coUacion 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// 2 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. 15 13. Paris, 15 13. 

See p. 80. 

Title. See Fig. 52. 

Colophon, * CExpletum eft hoc opufculum & ex officina emif- 
//fum/ in alma Parifiorum academia : ano domi-//ni (qui omnia 
numero definiuit) decimoter//tio fupra millefimu & quingente- 
fimu/ // decimafexta die Decembris. Per // Henricu ftephanii/' 
artis excu//fori5 libroru fedulu & indu//ftriu opifice/ e rcgione 
//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, -h 3 unnumb. + 2 blank = 46 ff ., 42 11. Paris, 1 5 1 3. 


EditioHS. There was no other edition. 

Tbu 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 
.n hu 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 

CDemyflkaminicroratnfigntficatioiie opu^ 

iciiIiiin:eoniin prffotun qd ia faois liu 


mm dcfenarionrm (iicdn* 



Fig. 52. Tmx page of the 15 13 cuchtoveus 

inr Itided in a list of arithmetics, for» while there is nothing of compu- 
utum in the work, it is not unrelated to the number theories of the 
niriiueval writers and even of the Pythagoreans. 

Clichtoveus discusses, as is usual among such writers, the religious 
•^ .niAcance of one (' Quid vnitas/ numerorum fons et origio defignat. 
t a|t. I.' ) and the numbers of the first decade. He also mentions several 
Li'^jeer numbers which were supposed to have some scriptural significance, 
r • -t forgetting, of course, 666, * the number of the beast.' 

There b also in this work a chapter, generally unrecognized by writers 
-1 the history of the subject, on finger- reckoning : * Quomodo antiqui: 
- :*neru% omnes per certas digitoru & manuum figurationes/ fignificare 
'.t liniii.' Cap. XXVIII. 

;« >ANNES MARTINUS BLASIUS, Villagarciensis. 

Ed. pr. 1513. Paris, 15 13. 

A SpuiUh astrologer and arithmetician of c. 1 500. In this edition the 
aathor's name appears as * loannes Martinus Blafius diocefis Paccri«/ and 
ni I be 1519 edition as 'loannes Martinus Silecevs (and Sdliceus) Uiocesis 

TiiU^ Sec Fig. 53. 

C4>Jopkam. 'Explicit liber Arithme//tices practice magri Joan- 
' •. Martini Blafij VilV/lagarciefis : PariAjs edit*» in honcftifTima 
IV : ia.- ' corii paleftra : irapredus vero a calcographorum ex-// 
:-.-rtiiIinio Thoma Kees: Vvcfalienfe expenfis pro//bi(rim<)rum 



Fig. 53. Title page of the 1513 blasius 


\ir«>rum : Joannis Parui et J oannis // Lambert. Anno domini. 
1513. in \igilia diui JoVAnnis baptifte.' (F. 26, r.) 

Drscrifitiim. Fol., 19.5 X 28. 1 cm., printed in double columns, 
each being 6.5 X 21.3 cm. 26ff. unnumb.,64-6611. Paris, 1513. 

Eiiitians. Paris, isiji fol. (here described); ib., 15 14, large 
8*; ib., 1519 (sec below); ib., 1526, fol. 

.Mthough an algorism, the work is mediaeval in character. The author 
Ar%t discusses the fundamental operations with integers, including series 
and roots as was the custom, but not considering duplation and media- 
ti«>a as distinct topics. He is one of the earliest writers to adopt the 
spelling subsirtutio^ 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 
vipputatorii.* The ' tertius tractatus ' is devoted to common fractions, 

* fractiooes vulgares * ; the ' quartus tractatus ' to sexagesimal fractions, 

* fractiooes phisicae ' ; and the * quintus tractatus ' to the rule of three 

* * Pnma regula 2 fudametalis qua detri dicunt.*) There are no practical 
ap(»luations of any ^-alue. 

ThiA first edition differs greatly from the third (15 19) described 
bciow. It includes only the algorism, while the latter consists of two 
lorts, the first being on Boethian arithmetic, and the second being 
vulvoaotially identical with the 15 13 edition. 

Otk^r works of lyij. Albert of Saxony, p. 9, c. 1478; Ciruelo, 
p. 60. 1495 i C;yraldus, p. 254, 1553 ; Licht, p. 70, 1500 ; Peurbach, p. 
5 ;. 1492 ; johann Karl von landshut (I^nzut), * Algorithmus linealis,' 
Oaf'ow, with editions, ib., 1515, 15 19, 4* (see p. 83, 1504). 


Ed. pr. 1513. Paris, 15 19. 

See p. 05. 

<Ut>!rrti s FixAKis, editor. Sec p. 160. 

Ttih. ' Arithmctica // loannis '/Martini. Scili-//cei, in the-^en, ct praxim // fciffa, nupcr ab Orontio Fine, Del//phinate, 
fumma diligentia caftigata. Ion-/ 'geq5 caftigatius q prius, ipfo 

-ra- tc imprefTa : omni hominu //conditioni pcrq // vtilis,// 
.V net- ctTaria. ' ' Vircfcit vulnere virtus/ Surrounded by an elab- 

'.Atc miwidcut, with the following on four sides: ' Emissa ex 


officina Henrici Stephani, e regione //scholae Decretorvm Com- 
morantis,// vbi et vaenalis reperitvr.// Parisiis anno Christi. 
1 5 19.') (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, 15 19. 

Editions, See p. 97. 

The first half of this rare work, not found in the 15 13 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 Mathematicaru 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. 

Clnhocopafciilohcccontiocntiit* ^'^^- ^^ ^^^ 

CQofdfitpiopoctfo&qnocdttS Ipdcs r^^l^YlmlZi 

. - ^ . . .. ti. . Bemarduro Zuc- 

acdptatur&inafcop^^ ^*^^"^ // Anno. 

^ M.D.XX. lanua- 

CQ«««ttiorniihociict«oportiod^ ^J- ^^' (^- ^ ^'^^) 

Description, 4, 

iCQaidpaiplnqiiftas&rcinotlo 134x20.1 cm.. 

CCoofutaaoncsaigBmmtonimdlcnlatorfe X15 cm. n ff. 

_, ^ unnumb. , 34-37 11. 

Fig. 54. Title page of racxsius ^, ^^ -^^ 

Florence, 1520. 

Editions. Florence, 15 14, 4"; ib., 1520, 4** (here described). 


This work consists of a theoretical treatment of proportion. While 
[tartly arithmetical, this treatment relates to the fundamental theory, 
Ann\ is equally applicable to geometry. The book is dedicated to the 
:'.!<:>tnous Giovanni Salviati, uncle of Cosimo I, Grand Duke of Tuscany. 

GUILLIELMUS BUDAEUS. Ed. pr. 1514. Florence, 1562. 

Gi'iLLACME Bvvt. Bom at Paris in 1467 ; died at Parts, August 23, 1540. 
Scm of jean Bud^, grand aodiencer of France. He became secretary to 
Louis XII, master of requests to Franfots I, royal librarian, and ambsusa- 
dor to lieo X. He was a man of great erudition, and was instrumental in 
founding the College de France. 

Tir/r, •Trattato//delle Monete//e Valvta loro,// Ridotte 
ilal coftume antico, allVfo mo-//demo, Di M. Guglielmo//Bvdeo. 
, . Tradotto per M. Giouan Bernardo // Gualandi Fiorentino.// 
In Fiorenza ; // Apresso I Givnti // MDLXII.// Con licenza, & 
Priuilcgio.' (P. 1.) 

Colophon. * In Fiorenza aprefTo gli heredi di // Bernardo 
Giunti/7 1562.* (P. 318.) 

Description. 8*, 10.5 X 17.2 cm., the text being 6.8 X 13.1 
cm. 8 pp. unnumb. + 3 blank + 309 numb. = 320 pp., uncut, 
2« 11. Florence, 1562. 

Editions, The dedicatory epistle is dated ' Da. Viterbo il xxx. 
<rAg(»fto. MDLXI,* so that this is the first (as it is the only) 
ctlition of Gualandi's translation of Budaeus. I have not com- 
pared it with the ' Libri de asse et partibus ejus,' Paris, 1514 ; 
Mxond edition, Venice, Aldus, 1522. Although on the title 
l*a;;e it is called a translation, the various books, six in number, 
'^K";;in 'Trattato delle//Monetc// di M. Gio. Bernardo /y^ Gvalandi 
Cittadino// Fiorentino,' leading to the belief that it may have 
h-ccn rewritten by Gualandi. 

Thi^ treatise is ptirely historical, describing in a prolix manner the 
an' jcsiX measures, a subject of interest to arithmeticians in the sixteenth 
f rntury on account of the great number of tables of denominate numbers 
.r. -:« in Italy, France, and Germany. 

There was also a work by Budaeus entitled ' Minervae Aragonise Assis 
y :tfirani siipfmtatio compendiaia ad monetam i)onderaque et mensuras 
>l:spaoie nostre,' etc., published at Saragossa in 1536, S""* 



Ed. pr. 1 5 14. Augsburg, 15 14. 


SENSTEIN, PoscHENSTEiN, Besentinus, etc. Bom at Esslingen, Swabia, in 
1472; died in 1532. He taught Hebrew at the universities of Ingolstadt 
and Heidelberg, and also at Antwerp and Niimberg. Luther and Melanch- 
thon were among his pupils. 

Title. See Fig. 55. 

Colopfion, *Getruckt in der Kayferlichen ftat Augfpurg durch 
//Erhart 6glin Anno 15 14 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, 1514. 

Editions. Augsburg, 15 14, 4* (here described); ib., 15 16; 
ib., 1 5 18. Boschensteyn is also said to have published at 
Augsburg in 15 14 * 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 se>'en 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 * ReguU 
Fufti ' (where he gives his rule in verse). 

JAKOB KOBEL. Ed. pr. 15 14. Augsburg, 15 14. 

KoBEL, KoBELius, KoBiLiNUs. Bom at Heidelberg in 1470; died at 
Oppenheim, January 31, 1533. He studied at Cracow, where Copemicu> 
was his fellow-student. He was a man of varied attainments, meeting with 
success as a Rechenmeister, printer, engraver, woodcarver, poet, and public 

Title. See Fig. 56. 

Colophon. 'Getruckt tzu Augfpurg durch Erhart oglin,// Anno 

Description. 4*, 13.7 X 18.9 cm., the text being 9x15 cm. 
6 ff. unnumb. 4- 24 numb, (in Roman) = 30 flf., 30-35 11. Augs- 
burg, 1 5 14. 



cnbtecblin mit tmnff^m 

Octtdti0aiodircMikni jAwtf'S^ 

iMiycp^Hip 00000 wio 

Fic. 5$. Trruc pack or thk 1514 ihnchknsievn 


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 
* Rechenbiichlein/ (2) *Mit der Kryden,' (3) the *Vysierbuch.' 
The * Rechenbiichlein * first appeared at Augsburg in 15 14, 4**; 
the *Vysierbuch/ a treatise on gauging, at Oppenheim, s. a. 
(15 1 5); and *Mit der Kryden' at Oppenheim in 1520. 

When the 1 5 18 edition of the Rechenblichlein (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 Rechenbiichlein : Augsburg, 1 5 14, 4* (p. 100) ; 
Oppenheim, 15 14, 4** (p. 106); ib., s. a. (c. 15 15); Augsburg, 
1 5 16, 4* ; 1 5 1 7, 4° ; Oppenheim, 1 5 1 8, 4** (p. 108 ; * zum Dritte 
male gebeflert,' 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, 1537, S**^^- no); ib., 1544, 8*; ib., 1549; ib., 1564 
(p. HI); ib., 1573; 157s; Frankfort, 1584, 8*. 

Editions of * Mit der Kryde od' Schreibfedem/ durch die zei- 
f erzal zu reche // Ein neiiw Rechepiichlein/ den angenden Schu- 
lem d' rechnug zu ere getruckt ' : Oppenheim, 1520 ; Frankfort, 
IS37» 8* (p. 1 10); probably included in various other editions of 
the Rechenbiichlein. 



But Uterp 0eo2Oti ct iRccb 
oibtecblmauf oen limeit 

mfc iWctaipfoiingtneoat 

€tel <y<bc|M )3ttl<rn<« 

' T 

Fwi. 56. TiTLB PAGE or kobel's Rectunbiechlin (1514) 



58 LVJa 
99 UX 
6t LXI 

61 Lxni 



69 LXIX 


71 LXXI 

7? Lxxra 

74 LXXIin 

75 LXXV 


81 LXXXt 































































1 500 












m mic 




mviiic , 

Fig. 57. A page from kobel's Rechenbuchlin (15 14) 


nU samm/alf^ mag mm tm^ «btfit^fiaa/im 

XX M^tebeeaic^$»9at9^4illtm^ny^ 

Illi<^LX Urt«n^rc(^JBll46igMietiMC^* 

3M 6€ti 6b0(|i|>Hb€ii ^(pirat/vfi 0^Mi^>4if nM|^k 
l(ft^lUc^kntftt4Ubl0(^^Mr6t^c<ftm recall f9r# 


^ mA(4 wtMeifie Qole 9m9 pbiboi/ 60 mtifiit tte 
€i|{8ale«K^brcM/ifibmMc^toi>eyir 0ef%r(# 

^ydMylMWM bte# vmtd ItmnOe in h^MfUUS 

04cM^(«6i(mr/'imitbr«O«e4l(^arf4^/6« f^U 

Fw. 58. A PAGE FROM kobel's ReckefibiechHn (15 14) 


Editions of the * Vysierbuch ' : Oppenheim, s. a. (15 15), 4"* 
(p. 113); Frankfort, 1527; ib., 1531, 8' (p. 108); Oppenheim, 
1 53 1 ; ib., 1532 ; s. 1., 1584; probably included in various other 
editions of the Rechenbiichlein (p. iii, 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 Regpel, 
die von dem WaJen 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 
TTC 200 

imcTx *°' ^ (see Fig. 58). 

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 of i§i4. 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. 15 14 an anonymous work 
entitled * Arithmeticae practicae Tractatus qui dicitur Algorismus, cum 
additionibus utiliter adjunctis,' Paris, 4^ 

JAKOB KOBEL. Ed. pr. 15 14. Oppenheim, 15 14. 

See p. 100. 

Title, The title page is missing. Fol. A ij begins: *Dem 
Emueften Dietherichen//Remerer von Wormbsigenant von Dal- 
burgk : // meinen befundem giinftigen lieben Junck-//her2en/ 
Eubeiitich Jacob K6bel/ //dieffer zeit Statfchreiber zu Op-'/ 
penheim/mein willig dinf t // allerzeyt bereydt//zfl vor.' 

Colophon. * Getriickt zu Oppenheym.// Anno. 2 c M.CCCCC. 
XIIII.' (F. no. XXIIII. See Fig. 59.) 


^)^ftirr«0m/im6f<l^tab<»<r8ciiflK^Mt(f/ VMf 



ID w i KtM ^g y 9^ ^<|<l^>€m mtf mir fUiiffai 

«(9rMem4ai^0ar9«lc»o<^ fyboi flb<H>lfbai ifi/ 



)f^mApa*«l*ME«mkfilObcrblci>r/jom(ti|i 5<f 

SUfo iidl ic^ dteffem ikcM^ 

b0cM<tt/nK6icm vcc^it0C9/0a>ff90<rt^oii/wi5 
fcm(Bii6c0i^/jpftf»ti^<mp^ ^m^^ 

~ 5al>MM<itf^ffbinMraiwi»bc(rcni/au<^ 

CMC dwffMkjroDfi t 

fc9c» /Vll^4Hl0^<r 2lnfmmuf Mi5 ^com'amaffm 

. 50. Lvn* PACK OF kobel's Rcchcnhiechlin (oppenheim, 15 14) 


Descriptiofi. 4*, 14. i X 19.6 cm., the text being 9.2 x 15.80111. 
5 ff. unnumb. 4- 24 numb. = 29 ff., 28-34 11. Oppenheim, 15 14. 

See p. 106. 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. 15 14. Oppenheim, 15 18. 

See p. icx>. 

Title. See Fig. 60. 

Colophon. *Alfo Endet fich Seliglich d' Drit// Truck dif5 
Rechenbiichleins/ Zu Oppenheym zufam-//nien gefegt/ vnd 
volendt vff Dinftag des Hey-//ligen Criig Erhfihungs tag.// 
Anno diii .1518.' (F. XLVI, r.) 

Description. 4®, 13.7 X 18.4 cm., the text being 9.2 X 14.8 
cm. 4 ff. unnumb. -f-46 numb. (I to XLVI) = 50 ff., 32-33 11. 
Oppenheim, 15 18. 

See p. 106. 

JAKOB KOBEL. Ed. pr. 1514. Frankfort, 1531. 

See p. 100. 

Title. * Ein new geor-//denet Kiinftlich Rechenbfichlin/ 
Ja-//cob K6bels/ Stattfchreiber zu Oppen-//heim/ Auff den 
Linien vnd Spacien/ mit Rech-//enpfeningen. Den angehn* 
den Schulern Rech//nens gantz leichtlich zfl lemen. Vnd zu 
Kellerei//en Ampten/ Kauffmanfchaff ten/ vnd Krdme-//reien 
dienlich vnd brduchlich. Mit vilen erklS-//rungen/ Leren/ 
Regeln vnd Exempln.//fl®*Mehr daii vormals ie getruckt,' 
//gebeffert/ vnd zugefetzt.// Im Jare M. D. XXXL// Vifir 
Bflchlin/ //Den Jungen/ angehnden/ // Leijfchen Vifirem gantz 
leichtlich zu lernen/ //verftehn vnd Rechnen. Wie man ein /.' 
Vifirrut machen/ vnnd damit ein // iedes vafz vifirenn/ folle Zu 
//ende difes Rechenbflch//lins angehenckt.* (F. i, r.) 

Colophon. *CGetruckt zu Franckenfurt/am Mein. In Ver/ 
lag vnnd Gemeynfchaff t des Ernhaffetnn // vnd Fiimemen Herm 
Jacoben K6-//bels/ Stattfchreiber zu Oppen-//heim. Bei Chrif- 
tian Ege-//nolffen. Im Mertzen/ // Nach der geburt // Chrifti. 
//M.D.XXXL Jar.' (F. CXII, r.) 




Me new ^tdjc 

ptf ctjUm XO u mann uff ^cn 

I Linitn tmnft Spactat / imt 

;2\ccbfpf«inmffc/ ^auffmanfcljatff 

|vn5 'JTcfflic^c l)«»ibdMrtge/lficbirlict> 

, r«l>flancmaflrc/5um!&Jtctcmalc/ 

I i 1 8 ^^^ 


^TOicjnari Kfctjcn fot/ttfi rouffgcrn/ 


M cr ntrarbctr r^ct vtnb fiifi 

;m<»npr/t>n5 BcllcnVtJ^rtc^t/ 

i^aft or tiir wurj)/»cracbt pcrnicht/ 

per Bauff dtl^^uct; vn leg mtr ^ci^^ 

^^^ incrcf / wflo ict) yn t>n5ci;ttici(§/ 

iin Fom a ncib mtr frctcm mflr 

'cljijrvn6gof/S«l/JL'fU>/»n gSf 




Fii>. 60. TiiLE FAGi: OK kobel's RechepiichUin (15 18). 


Description, 8**, 9.7 X 14.9 cm., the text being 6.9 X 1 1.5 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 ef\ baculus?' The master thereupon describes the instru- 
ment, and a picture of it is given. 

JAKOB KOBEL. Ed. pr. 15 14. Frankfort, 1537. 

See p. 100. • 

Title. * Zwey rech-//enbfichlin: vff der // Linien vnd Zipher/ 
Mit eym angehenck-//ten Vifirbuch/fo verftendtlich iur/fgebcn/ 
das iedem hieraufz on // eiii lerer wol zulemen.// CDurch den 
Achtbam vnd wol erf amen // H. Jacoben K6bel Statfchreiber 
// zu Oppenheym.// Franc. Chrift. Egen.* (F. i, r.) 

Colop/ion. ' 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, 1 537. 

Editions. See p. 102. 

This is the earliest of Mr. Plimpton's copies containing the three 
books written by Kobel, (i) • Rechenbtichlein,' (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 fchulem 
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/,' 
Bfichlin/ den Leyen/ zu // leichtem vnd begreiflichem verfbuidt /,' 
verordnet/ Durch H. Jacob // K8bel Stattfchreiber zu // Oppenheym.' 
(F. 95, r.) It is much more complete than the 15 14 edition of th« 
* Rechenbiichlein ' (see p. 106), and is substantially the same as the 



1 53 1 edition (see p. io8) 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 zulemen 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 '). 


See p. loa 

TitU. * Rechenbfich/ 
. /AuflF Linien vnd Ziflfem. 
//Mit dnem Vifir bflchlin/ 
Klar//vnd verftendtlich 
furgeben. // Gerechnet 
Bichlin/ auff aUe // Wahr 
Mid Kauffmanfchafft/ 
Muntz/ // Gewicht/ Elen/ 
v-nd Malz/ viler Land // 
vnd Stett verglichen.// 
Durch H. Jacob Kobel. // 
(Woodcut.) Cum Gratia & 
Priuilegio.// Franckfurt/ 
Bei Chr. Egen. Erben // 
M.D. LXIIIL' (F. i,r.) 

Colophon. *Getrucktzu 
Franckfurt //am Mayn 
bey Chriftian E-//genoIfifs 
Erben.// M. D. LXIIIL' 
(F. 194, V.) 

Description. 8*, 9.1 x 
15.2 cm., the text being 
6.3 X 12.2 cm. 12 ff. 
unnumb. -|- 194 numb. = 
206 flf., 28-29 U- Frank- 
fort, 1564. (There were 
which this is the sixth.) 

pr. 1 5 14. 

Frankfort, 1564. 

VoffvisrfttfBfbigaf. 5-f 

(In \ci4tfxh&i (critcft rcc^fn/ aOca DAsMte 
;fn<m€m boffcit tmD vof ooffcn ffin^OM 

%9m «o9cr(e)«mpcf» 

tm tiutrdfr/ rostffr vberbimpr cTii lUrblfti 

tntt 3l<bncKbynt/^Anitfib gibr (fta^^Qcbm 

pftititfnfl / fo ^xth^bffX fompt/ftiiOct (letoi 

rJrbltti foofi^m vitb ai^r^^r^i'i^'^ 


bcf tCbtt/alflobgdert/ (b rpmptbfrfcbm/ 

3(Ifo vd byrcn bnt f!e>mb cAtm p(itt# 



Fig. 61. From kobel's Rechenbuch (1^64^) 
at least eight Frankfort editions, of 


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 

tytOmTbhcgps (m$ (l>ppettbcfm/fl»i« 

ttmtaemdft m^t Domi^at tmOmsu 

}DffbAU>atgrctt20oii -ityntid) neon totf 

btt/Mfowat^n 'icfntid^ €oti^im9o. 
indl<of&r0dngen/cbc Coneanflc^^^bibAt 

Vfuntjlt^tfras Hn wit vtf tagtn Cdntf: 

JD^et^ttb hix^tidi^cnfcagwit^ a\{6txiit 
fi^tSbm^tiMX crflm leg Me ntctlat/9<c 6(011 
igcyntrib CotiQnif&rgangcntfl/^As Mnd 
9o.tncil(n/9ffDC!Ln3iircbcnban<f/Y)nD n-A 
^annoft mcilen/i>u Conrar nncn c^s inebt 

Fig. 62. From k6bel*s Rechenbuch (1564) 

(f. 165, r.): * Von Frembden vnd Hie//l&ndifchen Mfintzen/ Sodifer ztri 
in//Teutfch vnnd Welfchen landen/inn aller // Kauffmanfchaff t \-nii 
Gewerb/ Hindeln/ //viler Land art im branch/ geng/ gibig oder // ver- 
ruflPen Mfintzen/ . . .* The book closes (f. 184, v.) with a set of Ubks 
and (f. 192, V.) a * Regifter.* 



JAKOB KOBEL. Ed. pr. 1515. Oppenheim, s. a. (1515). 

See p. loa 

7///r. * Eyn New geordet // Vyfirbuch. Helt yn.// Wie man 
Nt! cins ydcn Lands Eych// vn Ma(z/ein gerecht Vyfirflt mach€ 



Fig. 63. Last pack of kobel*s yysirf>f4ch (15 15) 

N n do mit ein ygklich onbekant Vafz // vyficrcn/ auch fcyncn 
:alt cricrncn // folic. Den anhebendcn Schulem Vi//firens 


Leichtlich/ mit Figuren vnnd//Exempeln/zu lemen/angezeigt. 
//Angehengt Tafeln.//Die Erften Fyer halten yii gerechet/ 
// fo eyn Fflder weins kauflFt win/ vmb //Guide zu XXVI. Od* 
XXIIII. Alb. 2C.// Was die Ome/das Fyrtel/ vnnd die Ma£z 
gelten.//Die Andern Tafeln/Zeyge an/ Ver//anderug vii wech- 
felug einer Muntz//durch die ander/als p, in lb. ic. //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. 15 15.' 

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.) 


Ed. pr. 1515. s. 1. (Venice), 1525. 

Venetian arithmeticians of c. 1 500. 

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' & 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 flf., 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. L), 
1525, 8** (here described); ib., 1526; ib., 1527; ib., 1528, 4**; 
s. a. (1530?), 8*; Venice, 1541 ; Milan, 1541, 8** (p. 115); s. L, 
1 547, 8** ; Milan, 1 548 ; Venice, 1 548 (with probably a second 
Venetian edition, 1 548, under the title * Thesoro universale de 
abacho,' by * lucha atonio de Uberti,' 8**); ib., 1550; ib., 1554; ib.. 


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 Veneziana, Venice, 
1847, p. 218, ascribing the work to Girolamo Tagliente *con 
Tajuto 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 deW Accademia Pontif, de' 
Suoin Lincei, XVI, 139, 147, 155, 304. See also Riccardi, I, 
2, 484, and Boncompagni's BulletinOy XIII, 247. 

There was also a treatise published by the Taglientes entitled 
*Rcgolc 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 
numeious 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. 


Ed. pr. 15 1 5. Milan, 1541. 

See p. 114. 

Title. * Libro // dabaco che in//segna a fare // ogni ragione 

roercadantile, & // pertegare le terre c6 I'arte di // la Geometria, 

e altre no//bilifsime raginoe ftra-//ordinarie c6 la Ta-//riffa come 


refpon//deno li pefi & // Monede de molte terre del inon-//docon 
la inclita citta di Vene-//gia. Elquel Libro fe chiama //Thefauro 
vniuerfale.' (F. i, r.) 

Colophon. * Stampato in Milano per lo. Antonio da Borgho.// 
Neiranno del. M. D. XLI/ (F. 80, v.) 

Description, 8**, 10 X 14.6 cm., the text being J.J x 12.3 cm. 
80 ff. unnumb. + 7 blank = 87 ff., 23-30 11. Milan, 1541. 

Editions. See p. 114. 

!-^"^' '^ * 

Fig. 64. From the 1541 tagliente 

This diflfers but little from the 1525 edition, the * Opera che insegna' 
(see p. 114), except in having a set of tariff tables at the end: *€LQui 
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 


VARIOUS AUTHORS. Ed. pr. 15 15. Vienna, 1515. 

Joannes de Muris (Jean de Meurs, Murs, Muria) was bom in 
Normandy, c. 13 lo; died after 1360. He wrote on arithmetic, astronomy, 
and music 

Thomas Bradwardin. See p. 61. 

NicoLAUs HoREM (NicoLAS Oresme) was bom at Caen (?) c. 1323; 
died at Lisieux, July 11, 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. 

Gborg von Peurbach. See p. 53. 

Joannes de Gmunden (Johann von Gmunden, Johann Wissbier? 
NvoEN } Schindel ? Johannes de Gamundia) was bom 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. 

Tt'tU. See Fig. 65. 

Colophon. • Impreffum Vienne per Joannem Singrenium // 
Expenfis vero Leonardi 2 Luce Alantfe//fratrum 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, 15 15. 

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 c5munis ex//diui Seuerini 
Boetij Arithmetica per M. Joannem //de muris compendiofe excerpta. 
// Prohemium.' De Morgan (p. 3) mentions a possibly earlier edition, 
s. I. 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 proportionu : 
ab-//lM^uiatus ex libro de Proportionibus. D. Thome // Braguardini 

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 
coosklerable attention in the latter part of the Middle Ages. 


The fourth part is the algorism of Peurbach, and has the following 
title on f. 37, V. : ' Opufculu Magiftri Georgij //Peurbachij doctiis.' As 
already stated (p. 53), this work takes up the four fundamental opera- 
tions and progressions, giving merely a theoretical discussion of the 

The fifth part (f. 44, v.) begins: 'Incipit tractatus de Minucijs 
phi-//ficis : compofitus Vienne Auftrie per magiibrum//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^ 

meaningmerely3 + i5 + ^, + Ji+ii,or3^^. TTie pitsent 

symbolism (**, ', ") is relatively modem. 

The book is particularly interesting because it combines in one 
volume five well-known books by mediaeval writers. In no other 

Vi0Kft!bno9,0^jCem9 1^olltacb9ftlAlcegris. 

Fig. 65. Tttle page of the vienna wcxtK of 1515 

single volume could the inadequacy of the medisval treatment of 
mathematics be better seen. Indeed, a manuscript of 15 15, found by 
Gerhardt in the Wolfenbtittler 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 dtr K. P. Akad. d. Wissensch. tu BerHn, 
1867, p. 43.) 


JOANNES DE MURIS. Ed. pr. 1515. Mainz, 1538. 

See p. 117. 

Title. *Aiithme//ticae specvlativae // Libri duo loannis de 
Muris ab in-//numeris erroribus quibus hacte-//nus corrupti, & 
uetuftate ter//mh 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 + i with woodcut = 96 
pp. Mainz, 1538. 

Editions. This is the second dated edition of the arithmetic 
of this popular mediaeval teacher. (See p. 117.) It is more com- 
plete than the one of 15 15, 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. • ImprefTa Auguftj Vindelicorum/ communibus im- 
penfis lo/ //annis Miller atq5 loannis foenifec^. Anno a natiuita 
//te domini. M.D. XV. ad. IIII. Cal.' Maias.' (F. 20, r.) 

Description. 4*, 1 5.1 X 21 cm., the text being 11 X 13.8 cm. 
20 ff. unnumb., 7-39 11. Augsburg, 15 15. 

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 



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. 

Open loannis Fcmificff Au^* 

Quadncum iaplenrif :condnens in ftfcptem 

amsUbciaks vetnrum. 

Circulos biblj(.uii.inquibusinaaphyfici 

Cofiuncncaria honim* 

Ad hV/libri rubricaiRftrius 

faino Logica 
' Monafttca 
(6o9 Oeconomica 
•i.mos J^^PoKiica 

P^ Alcarichmus fiibaltcrnus 
mathcanca PcrlpcA jua fiibaltema 


^ Mcdidna /ubalccma 
chcologta MccaphyOca 




bocDus cphcfticrito 
pc^rfcmsus HAoad 


Fig. 66. Tftle page of fceniseca's Of era (15 15) 

ANONYMOUS. Ed. pr. c. 15 15. Leipzig, s. a. (c. 15 15). 

Tit/e, *Melchiar Lotthervs Ivnior candido lectori salutem.* 
(Line i.) * Articularis Bedae prefbyteri numeroni computatic* 
(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. 


Editions. There was no other edition, and only this copy is 
known of this one. 

This broadside was published by Lotter c. 15 15, 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. 1 5 1 s . Paris, 1515. 

Bom at Sarifiena, Spain, c. 1487 ; died at Saragossa, Febraary 23, 1 560. He 
taught at Paris and Saragossa. His only works are the two here described. 

Title. See Plate III. * Proportiones magiftri Gafparis // lax 
aragonenOs de farinyena . . .' (F. 10 1, 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.//Expenfis honefti viri Hemundi le feure Bibliopole Pari- 
fius in vico diui Jacobi fub figno Crefce//tis albi vitam degentis. 
AnnoDomini.iSi5.Dievero.i3.MenfisDecembris.' (F. 100, v.) 

Description. Fol., 19.8 X 26.5 cm., the text being 14. i X 21.8 
cm. 100 ff. unnumb., 59-61 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 Cinielo (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. 15 15. Paris, 15 15. 

See above. 
Title. • Proportiones magiftri Gafparis //lax aragonenfis de fa- 
rinyena.// Venundatur Parifius In vico diui //Jacobi ab Emundo 
le feure fub (i-//gno crefcentis albi vitam degente.' (F. 10 1, r.) 


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 dni M. d. xv. 
die vo vi. me//fis Octobris.' (F. 26, 126 of the whole book, r.) 

Description, FoL, 19.5 X 26.6 cm., 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 of 1515. Boethius, p. 27, 1488 ; Bradwardin, p. 61, 
1495; Kobel, p. 102, 1514; Lanzut, pp. 83, 97, 1504, 1513; lichi, 
p. 70, 1500 ; Ortega, p. 93, 151 2; Peurbach, p. 53, 1492 ; Torrentini, 
p. 76, 150 1 ; Juan Andres, ' Sumario breve de la practica de la arithme- 
tica,' Valencia (from the book it appears that it was written in Saragossa 
in 15 14 ; it was reprinted at Seville in 1537) ; V. Rodulphus Spoletanus, 
* De proportione proportionvm dispvtatio,' Rome, 4". 

Works of J s 16, Johann Boschensteyn, p. 100, 15 14 ; Capella, p. 66, 
1499; Ciruelo, p. 60, 1495; Kobel, p. 102, 15 14; 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. *€LlmprelTo 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. 144 1) there is mentioned 


« « « 1^ « 


ttnata. » ^^ « « 


WW-*- I 

rTmunlianir|d.tnrm« <nWcoWu( 
J[.i:ot)iab€mimt>o Ic tf iire W^ fi* 



— ;; ' ■ m 




IM MK III. Tim k w.v hk l.vx 




an edition of 1 547 ; but this is a misprint for 1 5 1 7, as appears from 
Rki ardi, vol. I, col. 153-4. Riccardi mentions only three copies 
Kr.Hwn to him or to Boncompagni, 
hut there ^as one in the Fisher sale 
L>ndon, 1906). 

The book is a small octavo, the first 
:«<> thirds being given to mercantile prob- 
icins on exchange and the reduction of 
m. .ney. The last part treats exclusively of 
mr nsuration : ' Speculationi geometriche 
<1: I'iu torte: & prima laquadratiira del 

Othrr works of tsij. Anonymous, 
'Al^'onihmus linealis* (see Widman), p. 36, 
r. 1488 ; Borghi, p. 16, 1484 ; Feliciano, 
p. 146, 1516; Kobel, p. 102, 1 5 14; Reisch, 
\^ 8i, 1503; Sacrobosco, p. 32, 1488; 
W 'irnan, p. 36, c. 1488. 

Fig. 67. Title pace of 
BONiNi's Lvcidario 

Ed. pr. 1 5 18. 

Frankfort, 1535. 

Hkihkich Scurkiber; Hrnricus Scriptor ; latinized Greek, Gram- 
M4TRr!L Bom at Erfurt, at least as early as 1496. He describes him- 
«etf as* II enrich Grammateus// von EHTurt/derfiben freien kiinften Meyfter.' 
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 septiroo in festo sanctorum Tibureii 
cf Valeriani martirum . . . Henricus Scriptoris de Erfordia.* He also 
Uofht at Vienna. 

TitU. Sec Fig. 68. 

I^€ serif fwH, 8*, 9.2 X 15.2 cm., the text being 6,7 x 1 1 .3 cm. 
/■» :!. unnurob., 31 11. Frankfort, 1535. 

Editions. Vienna, 15 18, 8*; Frankfort, 1535 (here described); 
^ I /Frankfort), 1544; Frankfort, 1572. 

Graromateus also published an ' Algorithmus proportionum 
rw cum monochordi generis Dyatonici compositione . . .*, 
<'r.iiow, 1 5 14, 4*; * Libellus de compositione regularum pro 
. -•-•rum mensuratione,' Vienna, 15 18; * Behend unnd khunst- 

h Rcchnung nach der Regel und welhisch practic,* Niimberg, 



1 521, 8°, an extract from the work here described ; ♦ Algorismus 
de integris Regula de tri cum exemplis,' Erfurt, 1523; 'Ep 

I (Brme^ncitKe0rInberrt« 

imacitc* AC 
I lltoaog{ttf[)eylcnfttonoc^l^9/(Dl 


/ 0clpftffFbi/9nbanbir3ntfrainft/ 
I. ^ar9Crjftnbttn0pirl^a0O2e* 
CBfic^^licn ^urcb^M gotnal/Sape vn5 

f[Di(terr&cn|0 itt4((efi Minb bctt (Diit# 
tfur/ pub Crf angcl/mii attocrn Io(h0cii 
(faicf en bar (Promerrd. 

f'iG. 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 15 18 appears by the dedi- 
cation to * Dem Edlen ftirfichtigen weifen Johan//fen Tfchertte einer 


des Senats zu Wien,' which ends : * Gebe // zu Wi//en in 0-//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. 

mam/oiBVLmitVkpUmzmtt pttiM/ftmnba 

bn/tn votUlfit ftin $0 merrfen btti KrgcL 
|[eie£rfi Kegel. 
tDannetn quantittt ^tanbtyhtn onen-f* 
#ber ^fofol monn fplc^eqaanrfret abbirn^ifi 
lie 9 pn*-j-7W« s psi.— 4l^» 

s pti#-4«y WL • pii.— ioCT# 

3li in ber &bern qnamittt -4- vnb fn ber vtt 
brrn— /pnb*f-tjbertriffr— /fofolbfevnber 
qBOittifet vonber 5bern fobtraft ti iverben/pn 
S0befiifibnaeiirf0-|*6oaber bfc vnber quS 
filer t(l 0r5(rcr/ro fubera^ir ))te tldnerit v5 bee 
0r&ffern/vfi ju bent bae bo blettenb i^/fene— 
ele tfpti.H-d'N: 4p»^H-*N« 
lapgL— 4 N* gprir— gN« 
tSptu— |-2N. lopn,— 4N« 
CdieMiu Kegel. 
60 in ber ob0cfar;rcti qitanttrer vrUrr fonbS 
— pfib in ^cr vndern -^/vtib — oberrrifft +/ 
fo lubrrabir etne ron bctti anbern/vnbsum ci^ 
bsgrn fd)2ctb— 3(ie6aber/baebie rnber qua 
tuetobcrrnff ( bic bbern/fo jte^ eine von bent 
anbertt/9nb 5a btm er (ien fcge *f- aid 

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, 


bookkeeping, and gauging. In the treatment of the ' Regula folsi,' or 
rule of false position, the signs -f and — are first found in this connec- 
tion (f. £ iij). (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 anfengldichen bericht 
emphangen. Sag im darumb danck.' 

Other works 0/1518. Boschensteyn, p. 100, 1514 ; Calandri, p. 48, 
1491 ; Feliciano, p. 145, 1526; Kobel, p. 102, 1514; Riese, p. 139, 
1522 ; Torrentini, p. 76, 1501 ; Pdrez de Oliva, 'Dialogus in laudem 
Arithmeticae,' Paris. 

ANONYMOUS. Ed. pr. 1519. Venice, 1519 

Title, See Fig. 70. 

Colophon. * Venetijs in Edibus Petri Liechtenftein // Ann<» 
virginei partus .1519.' (F. 12, v.) 

Description. 4**, 15.6 X 20.4 cm., the text being 10. i x 16.3 
cm. 12 ff. (i blank), 33-38 11. Venice, 1519. 

fcrcSllrononile fund amentniti palcberrimum con/ 

titicns. iClertcononmiim8V(flto$nec€f' 

r0ilU9:aiinaddtnombii9 qtubuO 


* ^ * 

8ntio.i 5 1 9 *€laitcf/6fn £da>ii« 
pari ItccbtetiftcUu 

Fig. 70. Title page of the Cdputus nouus 

Editions. There was no other edition, so far as I kmw. 
although there are several anonymous computi, and some niu\ 
be the same as this. 

This is one of the rare works setting forth the computus as it «s> 
taught in the Church schools of the Middle Ages. (See p. 7.) 


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 luperaddunt 
Februarius vigint infuper octoq) dies/ (F. 9.) 

Other works of i§ig, Feliciano, p. 146, 1526 ; Blasius, p. 97, 1514 ; 
Widman, p. 37, 1489. 


Ed. pr. 1520. S. 1., 1520. 

A nephew of Pico de Mirandola, and biographer of his uncle. He was 
murdered in 1533. 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. 
6ff. unnumb. + 208 numb. + i blank = 215 ff., 44 11. S. 1., 1520. 

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, fuperqs mediis Ailrologia & 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). 

TitU, •Von dem Rdchnen auff den Linien.* (Running 

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 


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 iiith 
possible originals. 

ESTIENNE DE LA ROCHE, Villefranche. 

Ed. pr. 1520. Lyons, 1520. 

Bom at Lyons, c. 148a 

Title. See Fig. 71. 

Colophon, * CCy finift larifmetique de maiftre Eftienne de la 
roche diet ville tranche natif de Lyon // fus le rofne, Imprimee 
par Maiftre guillaume huyon. Pour Conftantin f radin mar-//chant 
2 libraire du diet Lyon. Et fut acheuee Ian . 1 520. le 2«. de Juing.* 
(F. 234, r.). 

Description, FoL, 17.3 X 25.5 cm., the text being 13x21 cm. 
I f. blank -f 4 ff. unnumb. + 230 numb. = 235 ff., 49 11. Lyons, 

Editions, Lyons, 1520, fol. (here described); lb., 1538, fol 
(see p. 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 sbcteenth 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, Chuqutt, 
which he had in his possession, and which has since been publishei. 

Other works 0/1520, Kobel, p. 102, 1514 ; Peurbach, p. 53, 1402 . 
Raggius, p. 98, 15 14; Stromer, p. ^^^ 1504; Suiseth, p. io,tc. 14S0; 
Tagliente, p. 114, 1515 ; Anonymous (Tagliente ?), * Libro de Abaco/ 




-, ...--„. -^ PC U rotbt wa iUiWiitbf njtif Ex xr^ [ 

I telflMlltHMilkcnocai: p«ftl»oantU pmtaf Uiat 1 
I 00 «ttn»^n^ (t rtglc* tx Umac raia;c^mek n^ 

J BMlA:^ U rt^U IX mfduti6 mot U pliitf rt V moi#:ot l« 
I nncimucm tt en ttf*ii(»*nitTCi m*«b A*itf{# qui fc tciw j 

rli mfcqaci iu[ Jr» ifictt4^ii«# a^lTu jm: tiuf^i* 

^i xr2:rTT 





^ sins 'i- 

F>«.. 71. Title page of the first «)itic)n of de la r(k^he 


Venice, 8**; Anonymous, 'Libretto de Abaco,* s. 1., 8**; Anonymous, 
'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,* 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 & Geometric 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 Toy fe, 
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 lan & le iour, & a tant pour 
an, c6-//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. *CCy finift Larifmetique 2 Geometric de maiftre 
Eftienne de la Roche diet Villefranche // Imprime a Lyon par 
maiftre Jacques myt Lan. 1538.' (F. 160, r.) 


Description. FoL, 21. i X 33 cm., the text being 16.3 X 30:6 
cm. 2flF. unnumb. + isSnumb. = i6ofF., 59-60 11. Lyons, 1538. 
Editions. See p. 128. 
The Huguetan referred to is the one mentioned on p. 1 88. 

BiEDA. Ed. pr. 1521. Basel, 1563. 

* The Venerable Bede ' was bom 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-//ri in divinis atqve hv-//inanis Uteris exercitatifsimi : 
omnia in octo to-//nios diftincta,prout ftatim poft Praefa-//tionem 
(uo Elencho enu-//nierantur.// AdditoRerum & Verbomm Indice 
//copiofifsimo.// Cum Caefareae Maieftatis gratia & priuile-//gio, 
Regisque Galliarum ad//decennium.//Basilese, per loannem// 
Heruagium, Anno M. D. LXIIL' (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. 4- 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, containingthe arithmetic, is described here. 

Editions. Some of his arithmetical work is said to have been 
published in 1521, fol. ; 1525 (in part, see p. 140); 1529 (in part, 
see p. 1 59) ; Paris, 1 544-45 (first edition of the Opera) ; ib. , 1 5 54 ; 
Basel, 1563, fol. (here described). See also c. 15 1 5, anonymous. 
His * Historia Ecclesiastica' appeared as early as 1473. 

The first volume contains the < De Arithmeticis nvmeris hber ' (cols. 
98-1 16), 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 Basda's; 'De ratione calcvli' (cols. 147- 
158), chiefly multiplication tables of Roman money ; < De nvmerorvm 
divistone' (cols. 159-163) ; * De loqvela per gestvm digitorvm, et tem- 
porvm ratione' (cols. 1 64-1 81), or, as the headline states it, <De indigi- 
utione,' giving us almost our only knowledge of the finger reckoning or 
symbolism of the Middle Ages in western Eiuope, and possibly spurious ; 


' De ratione vnciarvm * (cols. 182-184), ^ 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 // Appreflo 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. = 1 14 ff., 37-38 11. Florence, 1552. 

Editions. Florence, 1521, 4' (see Boncompagni's Bulletino, 
VII, 486; XIII, 249) ; ib., 1 548, 4**; ib., 1 552, 4*" (here described). 
The 1 521 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, 1 591, 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 /J"-?/. Boethius, p. 31, 1488 ; Grammateus, p. 1 24, 
1518 ; Paxi, p. 77, 1503 ; Sacrobosco, p. 32, 1488. 

CUTHBERT TONSTALL. Ed. pr. 1522. London, 1522. 

TuNSTALL. Bom at Hackforth, Yorkshire, in 1474; died November i!^ 
1559. He was educated at Oxford, Cambridge, and Padua, was a main o: 
g^eat learning and energy, and held important positions in the Churdi 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.) 



Fic. 7t. TnxE PACE or the hrst edition of tonstall 


Colophon, * Impress. Londini in aedibvs Ri-//chardi Pynsoni. 
Anno v^X'/fhi incamati .M.D.XXII.//Pridie idvs octo-//bris. 
Cvm pri-//vilegio a//rege in-//dvl-//to/ (F. 202, r.) 

Description. 4*, 15.7 X 20.9 cm., the text being 10.5 x 16.4 
cm. 202 if. unnumb., 29 11. London, 1522. 

Editions, London^ 1522, 4* (here described); Paris, 1529, 
4"* (p. 13s); ib., 1535; ib., 1538, 4* (p. 13s); Strasburg, 1543; 
ib., 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 
tuae 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 Rolb ; 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 son 
with a lanthom."' The Utopia was first printed in 15 16, so this sono- 
rous praise was written some years before Tonstall's arithmetic appeared. 


5>ome idea of the prolixity of the treatise may be obtained from the 
nimbrr of closely- printed quarto pages assigned to certain topics. The 
< hapter ' I>e Numenttione ' fills 1 1 pages, * De Additione ' 14 pages, 
' I >e Svbdvctione ' 15 pages, * I>e mvlti plica tione ' 14 pages, ' De parti- 
(Mine * 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 
IVnsoo, the successor to Caxton. 

Following the arithmetic is an appendix : ' Appendix ex Bvdaei libro 
iitr as- 'V^ excerpta : in qua prifca Latinorum et Gr^corii // fupputatio, 
jil arftimationem pecunif, tum Gallicae,// tum Angli-//cx reuocatur.' 

Tonsull also published a work in 15 18: 'In Lavdem matrimonii 
(Tatio,* second edition in 15 19, now very rare. 

CUTHBERT TONSTALL. Ed. pr. 1522. Paris, 1529. 

Sec p. 132. 

Ttf/t. *De arte svppvtandi libri qvatvor//Cvthberti Tonstalli. 
Picture of a tree from which is falling a broken branch, and the 
u«>rds: Noli altum fa//pere, fed time.) Parisiis ex officina Roberti 
Stcphani/ZM.D.XXIX/ (P. i.) 

CoUyphan, * Parisiis //excvdebat Robert vs Stepha-//nvs. Ann. 
M.D XXIX.' (P. 279.) 

Description, 4*, 13 X 18.8 cm., the text being 9.4 X 15.9 cm. 
2' I pp. numb. + 8 unnumb. + 2 blank » 281 pp., 36-38 11. Paris, 

liJttions. See p. 134. 

I'ne text is the same as in the first edition of 1522. There has 
' »-t-n added, however, a second appendix with the following title : 

• i A bclmi B\daei Parisiensis,// secretarii regii, breviari-//vm de asse.* 

CUTHBERT TONSTALL. Ed. pr. 1522. Paris, 1538. 

ht^ p. 13J. 

TitU. • Dc arte svp-//putandi libri qua-//tuor, Cutheberti // 
T nfialli. '' ' (Large woodcut.) Parisiis.// Ex officina Roberti Ste- 

• rani / M.D.XXXVIII.* (P. I.) 


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 / / swip^wtain/ /A\y libri qvatvor//Cvthberti Ton- 
stalli,//hactenus in Germania nus-//quam ita impreffi.// loan. 
Stvrmivs.//Arithmeticam Cvthbertvs // Tonftallus prae caeteris 
dilucide & pure tradidit : atq5 // ita tradidit, ut ars ipfa dum hie 
author extat, con-//tenta fcriptore, doctorera 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 k 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 4- 4S 3 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. 


Ed. pr. 1522. Regensburg, 1532. 

Thurnmayer. Bom at Abensberg, Bavaria, July 4, 1477 ; died at Rcgens< 
burg, January 9, 1534. He wrote on history. 

Title, See Fig. 73. 

Colophon. * Ratifpong apud loannem Khol // Anno. MD- 
XXXII.* (Large woodcut and date, *Io. Kol 1532.') (F. 12, r.) 


Description, 4*, 14.2 X 19.5 cm., the text being 9.9 X 14 cm. 
12 IT. unnumb., 26 U. Regensburg, 1532. 

EJitions. Numberg, 1522; Regensburg, 1532, 4* (here 

IIm; book IS primarily a treatise on numerical finger symbolism, and 
cunuins the most complete explanation of that subject extant. It gives 



latinonimpord^tos manufcpnutne^ 

randiCquiiKdani lo^endi) corue# 

cado,Exbcdacupiduris ZC ima^ 

gjnibas, mu&ar^moburgti 

fiue nerobofiar,mbibIio# 

cfacca diui haemcrani, 



Aic^do^lifliini Domini Lucae bonfi) dc# 

anipacauini (earetarij Rcucrcndifli^ 

miCardinalis Laurcncij Campc^i) 

zc» Aiiipiajs AIo. Aucm* 

tino Edita* 

Gcrmania lUuibranda* 

Fig. 73. Tm-E pace of the 1532 aventims 

. i»tratioos showing the representation of the numl>en> tip to one million 
■ y means of the fingen and anns (see Fig. 74). This finger symliuliMn 
> '.• rtoA in the works of Beda, it was practical in both the East an<i 


West during the Middle Ages, and it is mentioned by several sixteenth- 
century arithmeticians. (Compare Fig. 74 with Fig. 35» p« 57*) 

«fKr»«f^«f»^ ** '^Zf^tlUM 

Fig. 74. Finger symbolism from aventinus 
ADAM RIESE. Ed. pr. 1522. Leipzig, 1538. 

Ryse, Ris, Ries. Bom at Staff elstein, near Bamberg, c. 14S9; 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 compuu- 
tions (* auff Fedem *). 

Title. * Rechnung auff // der Linien vnd Fedem // AmS. aller- 
ley handtirung ge-//niacht/ durch Adam Rifen.// (Woodcut of 
counting house, with reckoning on the line abacus.) Item auffs 
new vberfehen vnd // an viel 6rten gebefTert.//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 flf. unnumb. + 2 blank (with manuscript notes) = 65 flf., 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 : 


1. 'Rechnung auff der linihen gemacht durch Adam Riesen 
vonn Staff elsteyn // in massen man es pflegt tzu lern in alien 
rechenschulen gruntlich begriffen anno 1 5 1 8.' (Graesse, followed 
by linger, 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 1 527. This 
was embodied in his second arithmetic which is here described. 
The work is rare. 

2. 'Rechnung auff //der Linien vnd Fedem/ Riese's best- 
known work. The title of the first (1522) edition was as follows : 
* Rechenung auff der linihen vnd federn in zal/mafs/ vnd gewicht 
auif allerley handierung/ gemacht vnd zusamen gelesen durch 
Adam Riesen von Staffelstein Rechenmeister zu Erffurdt im 
1522 Jar. Itzt vffsant 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*",- 
Niimberg, 1527; 1528, 8**; Erfurt, 1529, 8*; ib., 1530; Leip- 
^^Zy IS33» 8* (first edition containing Helm's Visirbuch; see p. 
142); Frankfort, 1535, 12**; Annaberg, i53S,8*(p. 141); 1536, 
12'; Leipzig, 1538, 8* (here described); 1541; Frankfort, 1544, 
8'; Leipzig. 1544; s. I., 1548, 8*^; Leipzig, 1548, 8**; ib., 1550; 
Breslau, 15 50; Frankfort, 1552, 8*; Leipzig, 1554; 1556, 1 2*; 
Frankfort, 1558, 8* (p. 141); Leipzig, 1562, 12*; Frankfort, 
1564, 8*; Fxankfort, 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; Nurnberg, 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 Biichlein,' first published at Leipzig in 
'533; second edition in 1536. See p. 171. 


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 Recorders 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 0/1522, Albert of Saxony, p. 9, c. 1478; Boethius, 
p. 27, 1488; Budaeus, p. 99, 15 14; Kobel, p. 102, 15 14; 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, 15 18; Paciuolo, p. 54, 1494; Reisch, p. 82, 
1503; Sacrobosco, p. 35, 1488; Tagliente, p. 114, 15 15. Rodrigo Fer- 
nandez de Santaella (or Valencia, see p. 269, 1555), * Ars cdputandi,' 
Saragossa, fol. There was also written in this year, but published s. L a., 
a work by Vincenzo Barziza entitled * Operetta nouam^nte composta,' 
8^, 39 ff., containing a few mercantile rules and tables. 

Works 0/1524. 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 Romanonini. 
Demettius Alabaldus de minutiis, ponderibus et mensuris, Ven. Beda dc 
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. L, 
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; Giovanr* 


UclMrnte |niblished two editions of a work chiefly on bookkeeping, 4^ 
24 tf.. Venue, beginning, * Considerando io loanni Taiente quanto e 
Tit • t•^>ma cosa ali nostri magnifici getilhonieni & ad altri roercatanti.' 

ADAM RIESE. Ed. pr. 1522. Annaberg, 1535. 

See p. 1 3S. 

7 ;///-. The title page is missing. 

Ci'lophon, •CNach difer vnderrichtung kanftu auffs bc//hen- 

.:'ii alle Exempel in der Falfi machen: W61//left folch BiSch- 
: n vnnd kurtze erklerung ietzt/ // welches ich zum andem mal 
■.r:c aufzge-/,'hen/ zu danck an nemen/ wil ich ver-//dienen/ 
\r..i dir auffs eheft ich //mag die Practica nach al-//lem fleife 
*:.craufzftrei//chen. Datum// auff fanct // Annaberg/ // Dinftag 
r.t. h , Martini. Im Iar//M. D. XXV/ (F. 55, r.) On f. 69. 
\ . IS the following colophon: *Alfo ift kiirtzlich //befchriben 
•.'.'!,. begriffen/'/die Confection // der Vifier riiten mit //Irer 

:. .ng vnd gebniuch.//eEnd:// An. M. D. XXXV.// Im Christ- 
•T. nat.* This latter is evidently the date of printing of the entire 

PfScriptiofL 8*, 8.8 X 1 3.2 cm., the text being 6.8 X 1 1.6 cm. 
'-/ rf unnumb., 31 11. Annaberg, 1535. 
Iltiitions. See p. 139. 

I\:^ conuins the * Vifirbfichlin ' of Krhart Helm, as in the 1533 
:. -n, btit it does not give his name. See the 1565 edition (p. 142). 

ADAM RIESE. Ed. pr. 1522. Frankfort. 1558. 

See p. 1 58. 

TiTli', The title page is missing. Page numbered 2 begins as 
: \%^ : • Vorretle in dil* RechenV/biich/ Adam Rifen.* 

1 1 i\>ph(m, 'CEnd.//Zu Franckfurt bei Chr. Egeb. erben/// 
\--» 1558/ (F. numb. 87, r.) 

Iit's^nption. 8*, 9.4 X 15 cm., the text being 6.5 X 12.2 cm. 
•7 rf numb + I blank = 88 ff., 31 11. Frankfort. 1558. 

r.Jitions, See p. 1 39. 

r-. H c<mtains the ' VifirbGchlin ' of Krhart flelm. as in the 1533 


ADAM RIESE. Ed. pr. 1522. Frankfort. 1565, 

See p. 138. 

Title, * Rechenbuch/ Vff Lini//en vnnd Ziphren/ In allerley 
// Handtierung/ Gefcheff ten vnd Kauflf-Z/mannfchafft. Durch 
Adam // Rifen.// Mit new en kflnftlichen Regeln vnd Ex-// 
emplen gemehrt/ Innhalt f iSrge-//ftelten Regifters.//Vifier vnd 
Wechfelrfiten kunftlich vnd // gerecht zumachen/ auP 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 ot 
counting house) Franck, Bei Chr. Egen. Erben. 1565.' (F.i.rj 

Colophon, * M. D. LXV/ (F. 113, r.) 

Description, 8*, 9.3 X 15.5 cm., the text being 6.5 X 1 1.9cm 
105 ff. numb, -h 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. 77, v. : * Vifirbfichlin // Hemach folget der ware //Procefz, 
vnnd kurtzeft weg/ wie mann Vifir rfllhen machen fol/ aufe 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 i< 
strictly speaking not an arithmetic, but it includes a few explanations of 
those processes that are necessary in gauging. It also includes a tab!-. 
of square roots to the equivalent of three decimal places, and a bnc: 
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 Fedem/ //zM 
allerley Handtierung/ //Gemacht durch// Adam Rifen.// (Wood- 
cut of Adam Riese, with motto : * Anno 1550 Adam Ries Sein> 
Alters Im LVIII.') Auffs newe durchelefen/vnd //zu recht 
bracht.//iS7i.' (F. i, r.) 

Colophon, *Zu Leipzig druckts // Hans Rhambaw/ //Ini Jar. 
1 571.* (F. 94, V.) 

Description, 8^ 9.7 X 1 5.6 cm., the text being 6.9 X 11 .8 cr. 
94 ff. unnumb., 24 11. Leipzig, 1571. 


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 1 1.8 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, bom c. 1490, and living in Ferrara at the time of 
writing his books. 

Titk. 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-//mo 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. -h 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). 

Thb 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 daArologia, p ritrouare ha quati di et minute 
fa la luna, la Ira diiicale, et infinite getilefle, Delli 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 additioni in generalita, 
tanto geometrica, quanto arithmetica. Ca. 6.') The same idea is carried 
out in the other fundamental operations. 





Fig. 75. TrrLE page of the 1592 riese 








iMii|bri^ GiOwmn, Ardbttdton^ Arttfc*&rt/(i^ 7 l ij:>l6 
ijhv^ pjorm^i iejkjfucjckw^e, ft mt ifrr* 

$so^j^U hmM^lejtflf inoUf ft im§ 

uilffgio Dhr^i Per am 
cgci ^^ ATS ft €kkn€ 


Fig. 76. Title page op the 1546 Francesco dal sole 


FRANCESCO DAL SOLE. Ed. pr. 1526. Ferrara, 1564. 

See p. 143. 

Title, * Instrvtioni // et Regvle // di Francesco // dal ?^t>J I 
Francese.//Cittadino di Ferrara, Sopra il fon-//damento delle 
alme fcientie d'Abbac-//co, Arithmetica, Geometria, Col-/ /mo- 
grafia, & Mathematica, No-//uamente riftampate, & con//par- 
ticolare addittioni di//effo Authore,//aggionte.//In Ferrara, 
Aprefso Francefco di Rossi //da Valenza.//M. D. LXIIL' (F. i , r.) 

Description. 4*, 14.8 X 20 cm., the text being 11. i X 15.5 cm. 
4 pp. unnumb. -1-71 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 * ^he Sun), a play upon the author's name, which 
appeared in the first edition, gives place to some Latin lines in this one« 


Ed. pr. 1526. Venice, 1526. 

Bom at Lazisa, near Verona ; he was living in 1 563. 

Title, See 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 fignore, 1527. Del mefe // di Zenaro. Regnante il // Sere- 
nifTimo Princi-//pe meffer An-//drea Gritti.// A B C D E F G 
HIKLMNOPQRST V.//Tutti fono duemi.' (F. 80. r.) 

Description, 4^ 15.1 X 20.8 cm., the text being 13.4 X 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 1 5 18), 8*; ib., 15 19, 
8*; ib., 1 524, 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, 1 526,4** (here described) ; ib., 1 527, 4*; 
ib., 1536, 4* (p. 148); ib., 154s, 4* (p. 149); ib., 1550, 4* (P- I49); 



iibzoDiBritbmcria^ -^comctria 

fpcculartua i ps^tfcak : jCompoffo per macftro 

fraiKdcordidano M Xajifio Scroncfc 

^rfmloto Ocflfa gramalddlu 

VIouamcntc ftamparo* 

Chi vo! aprirvm firragliafbrre 
] ncttna d' vni Tocrc oucr CiftcUo 
fiifogna haucr U Scih iccio di quelld 
Giongcr fi pofTa al!c ferrate portc 

Aprcftofa meftirr Fhucjm Tcco pof tc 
CNdu hiuendo la Chiaue del fugcUa^ 
Vn frrrodicapclbtoc grimalcfcllo 
Stnon d'aprir Pimprcfc fiic ficn carce^ 

Ta]quelh) mio libretto moftra come 
S^gkr fipofla allaltc fcrratiux 
fctquellcaprirfaira fanca^auc» 

Infinite ragionjponti^fic raiturc 
Qycflochiirircccon modofuauc 
Cm fcila egrimalddlo c tl proprio nonie. 
Ma ncn lodar mie come 

Col taoKciD biarm^Fjchcl biafmo nafcc 
Sol dc cut ncl mal dir B^alegra e pafcc. 

M. D. XXVit 

£on gratia t priuilcgio- 

Fig. 77. Title page of feliciano 


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 modem 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. 


Ed. pr. 1526. Venice, 1536. 

See p. 145. 

Title. * Libro di Arithmetica \ Geometria // fpeculatiua 2 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, aprelTo 
// fanto Moyfe nelle cafe nuoue luftiniane : Per // Fracefco di 
AlefTandro Bindoni, & Ma-//pheo Pafini, compagni. Nelli anni / ; 
del fignore. 1536. Del mefe //di Zenaro. Regnanteil//SereniiTi- 
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 flf. unnumb., 41 11. Venice, 1536. 

See above. 



Ed. pr. 1526. Venice, 1545. 

S«« p. 146. 

TitU, 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 1 536 edition. Venice, 1 545. 

Editions. Sec p. 146. 


Ed. pr. 1526. Venice, 1550. 

See p. 146. 

TitU. This is substantially the same as in the 1526 edition. 

Coiop/ton, *Stampato nella inclita Citta di Vinegia, per// 
Franccfco Bindoni, & Mapheo Pafmi,// Nclli anni del noftro 
Si^Tiore.//M. D. L.// Registro.// A B CDEFGHIKLM 
N O // P Q R S T V.//Tutti fonno duemi.' (F. 80. r.) 

Description. 4*, 15.6 X 20.8 cm., the te.xt being 13.4 x 17.2 
<r\. 80 flf. unnumb., 41 11. Venice, 1550. 

Etiitions. See p. 146. 


Ed. pr. 1526. Venice, 1560-61. 

Se« p. 146. 

Tit/c. This is substantially the same as in the 1526 editicm 
147;, but bears the date M.D LX. 
CoUphon. *Stampato nella Inclita Citta di Vincgia, Per Fran- 
ce o dc Lcno. Neiranno del N. Signore.// M. D. LXL* 

F. 7a r.) 
Description. 4*, 1 5.9 X 21. 1 cm., the text being 14.3 x 17.4 

T. 79 ff. unnumb., 41 11. Venice, 1560 (colophon 1561). 
Editions. See p. 146. Riccardi mentions two identical cditicms 

• :ms year, one of them, here described, with the colophon date 



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 flf. unnumb. (part III missing). Verona, 1563. 
Editions, See p. 146. 

The third part, on geometry, is missing in this copy. 


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 Citt^.// Con le gionta della Regola 
del Catain del medefimo.// In Verona, Apreffo Dionigi Filiberi. 
CIDIDC II.' (P. I.) 

Colophon, *In Verona,// Nella 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 -f 7 unnumb. -f 276 numb. = 288 pp., 32 11. 
Verona, 1602. 


Ed. pr. 1526. Pkdua, 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. 1629.// Con licenza de' Superiori.' (F. i, r. ) 

Description. 4*, 15 X 20.5 cm., the text being 11.4 x 16 cm. 
4 pp. unnumb. -I- 276 numb. = 280 pp., 32-34 11. Padua, 1629. 

Editions, See p. 146. 



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 in Tre Libri. . . . Di nuouo riftampato, e da 
molti errori corretto, & accrefciuto di molte cofe da // M. Filippo 
Macario Veronefe Rafonato publico della Magnifica // Citti. Con 
Taggionta della Regola del Catain del medefimo.// Al Molt' Illuf- 
tre Signor, e Padron Colendifs. il Signor Gio: Battista Sorer.// 
Venetia, MDCLXIX.// Preffo Gio: Giacomo Hertz/ (F. i, r.) 

Description, 4*, 16. i X 22 cm., the text being 11.8 X 16 cm. 
6 pp. blank 4- 6 unnumb. -f- 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. Numberg, 1534. 

Bom 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 vn Stet //gewicht/ Elnmas//Muntz ec. Alles durch // 
Chriftoffen Rudolff zu/ Wein verfertiger.// 1534.' (F. i, r.) 

Colophon. *Getrukt zu Nfirmberg bey//Johan Petreio//im 
iar nach // der geburt Chrifti // 1 1 1 1 .' (F, 1 20, r.) 

Description. 8^ 10.2 x 14.9 cm., the text being 6.7 X 11.8 
cm. 1 19 ff. unnumb. -f i blank = 1 20 flf ., 3 1 11. Numberg, 1 534. 

Editions. Rudolff published three books as follows: (i) the 
Coss, an algebra, in 1525 (see p. 258 for the Stifel edition of 
1553); (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 (Numberg ?), 1526, 8*; Numberg, 1532, 8**; ib., 


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) Grundbtichlein, the fundamental operations with 
abstract and concrete numbers, integers, and fractions, with and without 
the abacus ; (2) Regelblichlein, the rule of three (* Kegel de Tri *) ax^ 
Welsch practice ('WeUisch rechnung*) ; (3) ExempelbOchlein, 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 ntltzlichfte Kegel, dadurch unxeli- 
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 gefchwioder 
aufzug in der Kegel de Tri gegrUndet, wirt fie auch derhalben practica 

O/A^r works 0/1526. Ciruelo, p. 60, 1495 J Blasius, p. 97, 1513 ; 
Tagliente, p. 114, 15 15; Widman, p. 37, 1489. Sterner mentions an 
anonymous Kechenbtichlein as printed this year at NUmberg. 


Ed. pr. 1526. Niimberg, 1540 

See p. 151. 

Title, This is substantially the same as in the 1534 edition, 
but bears the date 1540. (F. i, r.) 

Colophon, * CGetruckt zu Nflrmberg bey Johaii // 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, * Kfinftlich rech-//nung rait 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/ Mfintz \c, auff //etlich Landt vnd Stett. 
//Gemehrt mit 293 Exempeln/von man-//cherleyKaufFhendeln/ 


mit crklerung/ wie // die fclben zu machen vnd in die//Regel 
zu (eUen fein.// Auffs new widerumb fleiiTig vberfehen/ //vnd 
an Wl orten gebeffert.// AUes durch Chriftoifen Rudolff zu // 
Wicn verfertiget.// 1553/ (F. i, r.) 

CohphoH. • Gcdruckt zu Nurmberg/ durch //Gabriel Hayn.// 
1553/ (F. 206, V.) 

• Drscriptim. 8*, lo X 157 cm., the text being 6.8 x 1 1.8 cm. 
206 ff. unnumb. + 2 blank » 208 ff., 24-26 11. Numberg, 1553. 

Editions. See p. 152. 

As the title states, this is a revision of the 1526 book, with some 
A(l<led matter and a considerable number of new examples. The new 
nutter begins on f. T 8. The book closes with a list of gauger*s char- 
j' ters, 'die vifier ziffer/ not found in the 1534 edition, the integers 
U'lng represented by what are practically the mediaeval numerals, and 
the fractions being generally unit fractions. 

CHRISTOFF RUDOLFF. Ed. pr. 1526. Numberg, 1557. 

See p. 151. 

Description, This edition of RudolflTs arithmetic is substan- 
ti-illy verbatim with that of 1553 (p. 152). 8*, 9.5 X 15 cm., the 
text being 7X 1 1.8 cm. 206 ff. unnumb., 26 11. Numberg, 1557. 

ANONYMOUS. Ed. pr. 1527. Cologne, 1527. 

TttU. • Compendia-//ria artis nvmerandi ratio, et // expeditif- 
:* ma practicandi uia, figuris Arithme//ticis omnes numeroru f or- 
-r.-las c6prchcn//dens, additis quibufdam, ut raris, ita // utilibus 
rc^'.:iis.//Radicis Cvbicae Extractio.// 

13886 151 
C /k:* 653534 29952 

Ka'iix 40 28 

12120. 6 

9648 o 

3886 2144.* (F. 1, r.) 

Cclophon, • Coloniae apvd Mclchiorem // Nouefienfem Anno 
M D XXVII/VMcnfe maic' (F. 29. r.) 


Description, 4*, 13.4 X 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, bom c. 1500. 

Title. * Rvdi//menta Arithmetices // Authore lohanne 
Vuolphio // Herfbrugienfe.// Elemen//tale Geometricvm, ex // 
Euclidis Geometria, 4 loanne Voegelin,// 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.XXXIIIL' (F. 56, r.) 

Description. 8^ 9.7 X 15 cm., the text being 6.8 x 11.7 cm. 
56 ff. unnumb., 24-29 11. (The arithmetic occupies only 27 ff.) 
Frankfort, 1534. 

Editions. Nurnberg, 1527; Frankfort, 1534, 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. 1 548 

See above. 
Title. * Rvdi-Z/menta Arithmeti-//ces, Autore loanne Vuol- 
phio Herfbru-//gienfe.// Elementa-//le Geometricvm,// Ex 


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 1 5.3 cm., the text being 8.2 X 1 1.8 cm. 

60 flf. unnumb., 22-27 H- (The arithmetic occupies only 28 fF.) 

Frankfort, 1548. 

PETRUS APIANUS. Ed. pr. 1527. Ingolstadt, 1527. 

Peter Bienewitz, or Bennewitz. Bom at Leisnig, in 1495 : <^^®<^ ^^ 
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. *Gedruckt vnd volendt zfl Ingolftadt//durch Ge- 
orgium Apianum von Ley^//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. + 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 
Astronamen^ Mathematiker^ und Physiker der Diocese Eichstdtt. 

Apianus follows RudolfT 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 coimters 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 fedem 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 



before this one, although the arrangement had doubtless long been 
more or less familiar to mathematicians. 


jDntoerwevfunot aKcr Ba uftrnfitif? Kcdi 
tnuitg lit mcfehH&.etwmit fchoncn ^c 
gcirt vn fnttjflucf en Se^nffrii . tSuriOc r« 
fidi vcas foirl vjinU 0^«nDtgt<iir in t>cr 

^revHtft/ Oesglevd^ett fiVmiAlp mOrr in 

&ot;fct;cr noch m ltXMfd)tr fpiadt nvc 

ge&incf c. ^urd) pctriitti 2iptanu 

VOit lfV;Tm'cF/$ 2(|?ronomci 



Fig. 78. Title page of the first edition of apianus 

0/A^r works 0/ ij2^, Feliciano, p. 146, 1526 ; Riese, p. 139, 1522. 
the first NUmberg edition so far as I know; Tagliente, p. 114, 1515 ; 
W. Peer, * Ain new guet Rechenbuchlein,' Nurnberg, 8®. 


PETRUS APIANUS. Ed. pr. 1527. Frankfort, 1537. 

Seep. 155. 

Title, * Ein newe vnd wolgeV/griindte vnderweifung aller // 
Kauff mans Rechnung in dreien Bfi//chern/ mit fch6nen Regeln 
vnd fragftiicken be-//griffen. Sunderlich was fortel vnnd behen- 
digVA^it in der Welfchen Practica vnnd Tolle-/Aen gebraucht 
wurt/des gleichen vor//mals weder inn Teutfcher noch in// 
Welifcher Spraach nie getruckt.// CDurch Petrum Apianum 
von Leylznick der // Aftronomei zfl Ingolftatt Ordinarium.// 
(Woodcut of merchants using counters.) Franc. Chri. Egen.' 
(F. I, r.) 

Cohphon, * Zu Franckf urt/ bei Chriftian Egenolfif/ // Anno 
Domini. M. D. xxxvij.//Im Herbftmon.* (F. 183, v.) 

Description, 8^ 9.8 X 15.3 cm., the text being 7 x 1 1.5 cm. 
183 flF. unnumb. + i blank = 184 ff., 25-28 11. Frankfort, 1537. 

See p. 155. 
JOHANNES FERNELIUS. Ed. pr. 1528. Paris, 1528. 

Jean Fern el. Bom 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-//n^erorum tum fimplicium tum // fractorum rationes 
edocet.// Pofterior, ipfas proportiones coV/parat : earumqs 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. -f 24 numb. = 28 ff., 44-45 U. Paris, 1528. 

Editions. There was no other edition. 

This is one of the best of the sixteenth-century treatises on the 
medieval proportion. It follows the Boethian treatment, as seen also 
in the work of Bradwardin. 



Title page of fernelius 


OdUrwarks 0/ i^^S. Borght, p. i6, 1484 ; Cassiodorus, p. 2 1 1, 1540 ; 
('iruelo» p. 60, 1495; Riese, p. 139. 1522; Tagliente, p. 1 1 4, 1515; 
( hhstierDtts Torchillus Morsianus, ' Arithmetica brevis et dilucida in 
qtiiiMioe partes digesta/ Cologne, 8^ (but see p. 182, 1536). 

H'^rks 0/ ijjg. Anianus, p. 32, 1488 ; Riese, p. 139, 1522 ; Tonstall, 
P* iJS* i5'> > Bscda, ' I)e natura rerum et temporum ratione libri duo/ 
Easel, cap. I being ' De computu vel loquela digitorum'; there was an 
niition by Novioniagus, Cologne, 1537 (see also pp. 131, 140,263, 1521, 
>5'5t «554)- 

CHRISTOFF RUDOLFF. Ed. pr. 1530. Augsburg, 1530. 

.Sc«p. 151. 

TiiU, Sec Fig. 80. 

Hccl^nang belatigcnd«d^r; 

bey/cm DttBitd^Xintctfon/tP^UiV 

cbcr gcflolbe Me»crgkrchnM/Oera%tnM(| 
Oorcfo bettSfrcM/bcr pf mtt Onrd) thxet0 

JDorcb mtdxctm ^er fTUmf titrd» 

BongiMffceitrrmtvcrtc/gejjcn tin 

Mbctiaetlcwen tmttpenUfvf» 

^fi XXHen in 0^enrf<b/btttdi 

niitgcn pcffofiQi 
fwi twr # 


f ■ . Ho. TrtLE i*a«;e of the first edition of kiikilff's K.\rmf^i 



Colophon, *Getruckt in der loblichen Reychftat Aug-/Apurg/ 
durch Heynrichen Stayner/ // Volendet am 3 1 May im jar//M. D. 
XXX/ (F. 7S,v.) 

Description, 8®, 9.9 X 15.1 cm., the text being 7.4 x 12.2 cm. 
75 ff. unnumb. 4- 2 blank = yj ff., 28 11. Augsburg, 1530. 

Editions, Augsburg, 1530, 8* (here described); Numberg, 
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 Fin6. Born at Brian9on 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. -j 
cm. 9 ff. unnumb. -f 208 numb. = 217 ff., 48 11. Paris, 1 530-32. 

The title page of the geometry appears on f. 49, r., with the 
date M. D. XXX; the cosmography on f. lOi, r., with the date 
M. D. XXX ; the horography on f. 157, r., with the date M, D. 

Editions, Paris, 1530-32, 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 Parifioram 
Calendis lanuarij 1531,' or 1532 new style. The part on arithmetic is 



Fig. 8i. Title page or the first edition of finaeus 


n^iiuti nzn 3nir >:trifis n*-i.inii: -^t^^^^zzr^^^' -wzir ^ae^prs* common frac- 
liiiTtn jj^^i^j^snTii. Tarrri.mrfc imr ^rr-rtm — ittt. T S aw i F ire no applications 

S= si* 


Fy:,.S2. Fr* M the rnt>T trcroN or r>ir:.7> 

worthy the n^rive, ar>i the work has liitje to comnygpi h. Sctoe inter 
estirg illustrarlons showing the use of the mediJCT^ «>;Vittinzs in? sb:*" 
in Fig. Si. 


ORONTIUS FINAEUS. Ed. pr. 1530-32. Paris, 1542. 

See p. 160. 

Title. * Orontii // Finei Delphin. Re-//gii Mathematicarvm // 
Professoris://arithmetica//practica, libris qva-//tuor abfoluta, 
omnibus qui Ma-//thematicas ipfas tractare volunt // perutilis, 
admodumqueneceffa-//ria: Ex nouiffima authoris reco-//gnitione, 
amplior, ac emenda-//tior facta.// iEdito tertia.// Parisiis.// Ex 
officinaSimonisColinaei.// 1 542.//Cum gratia &priuilegio Chri-// 
ftianiflimi Francorum Regis/ (F. i, r.) 

Description, FoL, 20.9 X 30.1 cm., the text being 16 X 28.2 
cm. 2 ff. unnumb. 4- 66 numb. = 68 ff., 40 11. Paris, 1542. 

See p. 160. 

ORONTIUS FINAEUS. Ed. pr. 1530-32. Paris, 1544. 

See p. 160. 

Title, * Orontii // Finaei Delphi-//natis, Regij Mathe-//niati- 
caru Lutetiae //Profefforis,// Arithmetica//Practica, in com- 
pendia per Authorem // ipfum redacta, multifqs acceffionibus 
//locupletata: Ijs qui ad liberam quauis,//nedu Mathematica 
adfpirant philofo-//phia perutilis, admodumq3 neceffaria.// Lvte- 
tiae Parisiorvm // Apud Simonem Colinaeum.// I544.//Virefcit 
vulnere virtus.' (F. i, r.) 

Description, 8**, 11 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. i6o. 

ORONTIUS FINAEUS. Ed. pr. 1530-32. Paris, 1555. 

See p. 160. 

Title, * Orontii Finaei // Delphinatis, Regii // Mathematicarum 
Lutetiae// profefforis, //de arithmetica practi-//ca libri quator: 
Ab ipfoauthore uigi-//lanter recogniti, multffque //accefsionibus 
recins // locupletati.// Lvtetiae Parisiorvm,// apud Michaelem 
Vafcofanum,// 1555.// Ex privilegio regis.// Virefcit uulnere 
uirtus.' (F. I, r.) 

Description. 4*, 15.4 X 20.7 cm., the text being 1 1.6 X 17.2 
cm. 4 ff. unnumb. + 72 numb. = 76 ff., 34-35 11. Paris, 1555. 


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, 
// Tradotte // 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. i X 16.8 
cm. 8 fF. unnumb. + 81 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 
deir hauere ftampato Topera d'Orontio nella noftra Tofcana 
lingua,' with the date * Di Venetia, il di 7. di Luglio, 1587.' 

See p. 160. 
ANDREAS ALCIATUS. Ed. pr. 1530. Hagenau, 1530. 

Bom at Alzano, near Milan, May 8, 1492; died there June 12, 155a He 
was an Italian jurist. 

Titk, See Fig. 83. 

Description, 8*, 10.3 X 15.4 cm., the text being 7.8X iii 
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 . F5. 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 of 1530, Boethius, p. 27, 1488; Bradwardin, p. 61, 
1495; Clatovenus, p. 292, 1558; Maffei, p. 86, 1506; Riese, p. i39» 
1522; Tagliente, p. 114, 1515; Torrentini, p. 76, 1501 ; Johann 
Kolross, a primer entitled ' Enchiridion : das ist HandbCichlin tlitscher 


Orthographi . . . Auch wie mann die Cifer vnd ttidtsche zaal verston 
vl,' BaseU with another edition, ib., 1534, 8^ There was also an 
Anunymous work entitled ' La vraye mani^re pour apprendre k chiffrer 
ct compter/ published at Lyons, s. a., I3^ c. 1530. 





PhUppi MeUitcbiben,4eifimM 


Amo H. o. XXX* 


Fi»:. 83. Title page of alciati's 


Ed. pr. 1531. Lcyden, 1531. 

joAc-HiM Stckck Rimgblbf.rch. Bont at Antwerp, c. I4i)<): died c. 
1 5j6. lie taught philosophy and matheuMtics in various cities o( Gemuuiy 
and France. 

Tt/U. * loachimi // Fortii Ringel-//bergij Andouerpiani opera, 
' 'q:-jt proxima pagiiia//enumerantur.// Virtvte dvce //comite 
F .rtviu.//Apvd Gryphivm // Lvgdvni, //anno// M. D. XXXI.' 
}• I) 

1 66 


Colophon. ' Lvgdvni apvd // Seb. Giypivm,// 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. 

s« 10. FORTH RING. 
Vytaniim nmed hoc paOo diffrmhr. 









Fic. 84. From the first edition of ringelberuius 

Editions. Leyden, 1531,8° (here described) ; ib. (at least the 
arithmetic part), 1539,8°; Basel, 1541, 8° (see p. 167); Leyden, 


1556, 8* (see p. 168). The * E4)istola ad Lectorem' is dated 

* Louanij Idib. Augusti, 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 (bidij ' 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 17 pages (about 10 pages 
excluding the illustrations), the author treats of the Boethian ratios, 
6gurate 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. 

Othtr works 0/1531, Kobel, p. 101,1514; Juan Gutierrez de Gualda, 

* Arte breue y muy prouechos de cuenta castellana y arismetica,' Toledo, 
4^*; Jb., 1539 ; Saragossa, 1557, 1564; Alcali, 1570 ; H. C. Agrippa, * De 
occulta philosophia libri tres,' s. L, 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. 


Ed. pr. 1531. Basel, 1541. 

Se« p. 165. 

Title, ' loachimi // Fortii Ringelber//gii Andoverpiani 1 vc vbra- 
//tiones, uel potius abfolutifsima KVKKoiralhihcB : nem-//pe liber 
de Ratione ftudij, utriusq; linguae, Gramatice,// Dialectice, Rhe- 
torice, Mathematice, & fublimioris // Philofophiae multa. Quorii 
cX€7xo9 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 11.8 
cm. 796 pp. numb, -f 2 unnumb. = 798 pp., 25 11. Basel, 1541- 

See p. 166. Although the title differs from that of 1531, the work is 
the same. 



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.) 

Cohphan, ' Lvgdvni,// ex officina ty pogra-//phica Michaelis // 
Sylvii,//M. D. LVL' (P. 663.) 

Description, 8^ 10.5 X 15.2 cm., the text being gx 13 cm. 
4 pp. unnumb. + 4 blank + 660 numb. = 668 pp., 26-31 IL 
Leyden, 1556. 

See p. 166. Although the title differs from that of 1531, the work is 
the same. 

MICHAEL PSELLUS. Ed. pr. 1532. Venice, 1532 

Called the Younger, to distingubh him from a philosopher of the same 
name who lived about 870 a.d. Bom at Constantinople in 1020; died in a 
cloister in mo. He studied at Athens and taught philosophy at Con- 

Title, See Fig. 85. 

Description, 8^ 9.9 X 14.9 cm., the text being 7.4 x 11.4 cm. 
I f . blank+ 104 unnumb. = 105 ff., 24-25 11. Greek, Venice, 1532. 

Editions, Venice, 1532,8**, Greek (here described); Paris, 1 5 38, 
Greek, 4**; Paris, 1545, Greek and Latin; Augsburg, 1554, 8*, 
Greek and Latin; Wittenberg, 1556, Latin; Basel, 1554 and 
1556, 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 00 
arithmetic that attracted attention in the Renaissance period. The arith- 
metic is merely a primer for the study of Nicomachus. 


Other works 0/ 1 jij 2. Aventinus, p. 136, 1522; Capella,p. 66, 1499; 
Feliciano, p. 146, 1526; Kobel, p. 102, 15 14; Rudolif, p. 151, 1526; 
Stifel, p. 223, 1544; Vincento, p. 140, 1522 ; Johann Brandt, 'Kunst- 

fms^MtiS 9 hgidfomnUji , Maim/ 

Zi/ro4^iff«i:0iffina)Ai3/«S MVflw» 
Adjoins Atm^ffiH^ As(oif^/«ff * 


]i opus diluddum in quattuor Ma; 

tnematicas dliTdplinas, Aiith/ 


triam^K AftroDomiam# 

Nnmcronim hfccontraAiof explicatiOt 
Elaboratiim Mofices Compendium* 
Copendiii riiifas Geometric lationii* 
Aftronomss coadio peiipiciia^ 


C Ciimgfatia» 

Fig. 85. Title page of the first edition of psellus 

lichc Rechnung mit der Zyffem vnd Pfennigen, Auff allerley handt- 
tierung,' Cologne, 8^, 39 fi, ; Georg Reichelstain, ' Kauffmans hand- 
b(k:hlin. AUer Rechennschafft behendigkeyt, auff Linien vnd Ziffern,' 
Frankfort, sm. 8^ with another edition in 1534. 


MICHAEL PSELLUS. Ed. pr. 1532. Paris, 1557 

Scep. 168. 

TitU. * Michael // Psellvs de // Arithmetica,// M vsica, Geo- 
metria://& Proclus de // Sphaeia,// Elia Vineto Santone inter- 
prete.// (Woodcut with motto: Inpingvi Gallina.) Parisiis,// 
Apud Gulielmum Cauellat, in pingui gallina,// ex aduerfo coUegij 
Cameracenfis. //1557.' (F. i,r.) 

Description. 8*, 10.6 X 16.2 cm., the text being 6.6 X 13.} 
cm. 2 ff. unnumb. -f 76 numb. -f 3 blank = 81 flF. (18 ff. on 
arithmetic), 22 11. Paris, 1557. 

Editions. See p. 168. This is one of the Latin editions. 

See p. 168. 

MICHAEL PSELLUS. Ed. pr. 1532. Leipzig, 1590. 

See p. 168. 

Title. * Pselli // Philofophi & Mathemati-//ci clarifsimi / 
Arithmetica// Edita ftudio//M. Chriftophori Meureri,//Malhe- 
matum Profefforis publici // in Academia // Lipfienfi.// 1 590. 
Plato interrogatus, cur homo fit a-//nimal fapientifsimum : on 
apiO'/Zfuip iirlqarai^ refpondit.// Lipsiae.' (F. i, r.) 

Colophon. * Lipfiae, imprimebat Michafil // 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 natura 1 590* 
The text is in Greek and Latin. 

See p. 168. 

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.// 1616.// Lipsiae / 
Typis Abrahami Lambergi.* (F. i, r.) 


DfuripHon. 8*, 9 x 14.4 cm., the text being 6.6 x 12.1 cm. 
30 pp., 1-8 unnurab., then numb. 1-31; 24-25II. Leipzig, 1616. 

sw p. 16S. 
ADAM RIESE. Ed. pr. 1533. Leipzig, 1536. 

Seep. 138. 

Title. See Fig. 86. 

PescriptioH. 4*, 1 4.6 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 iT., 
17-24 II. in the Introduction (3 ff.), the rest of the book con- 
^:Ntin^ of tables. Leipzig, 1536. 

Editions, Leipzig, 1533; ib., 1536, 4* (here described). 

'Ilii!» is a set of mercantile tables for the multiplication and di\nsion 
•f ilcnominate numbers. 

(lEORGE AGRICOI^. Ed. pr. 1533. Paris, 1533. 

Bom March 24^ 1490, at Glauchau, Saxony; died November 21, 1555, ^^ 
<*heiimit£« He was rector of a school at Zwickau (151S-1522), and in later 
iiie a physician. He wrote a number of scientific works. 

Tali'. See Fig. 87, 

lU'scriptioH. 8', I0.2 X 15 cm., the text being 6.8 x 12.4 cm. 
7 j»p. unnumb. -I- 3-261 numb. +6 blank -h i with woodcut = 273 
:;. . 28 II. Paris, 1533. 

Tdttions, Paris, 1533, 8* (here described); Venice, 1533, 8"*; 

• ^535. f<>'-; Basel, 1549, 8"; ib., 1550, fol. There was also 
a-. • Epitome omnium Georgii Agricolae de mensuris ct ponderibus 
>-T G. Philandrum' published at Lyons in 1552, 8^ 

I he work can hardly be called an arithmetic, but, like a few others 

-' vittl in this list, it is a valuable book of reference on the history of 

XT. irni measures. It consists of five books as follows: * Liber primus, 

- nvcnfuris Romanis* (p. 9); * Liber secundus de Mensuris Cirsecis* 

. 75 K * IJbcr tertius, de Pondere renim quas metimur * (p. 144) ; 

.•itrr quartus, de Ponderibus Romanis' (p. 188): • Lil)er quintus, 

> P«»oderibus Graecis* (p. 219). The book is also valuable to the 

*• .i'-ni kA Roman and (ireek numerals, and of the various symlx)ls 

• measures. Such works explain the origin of certain systems of 




Ht> 0fttnbegcft)t<gt/ JO c^*^ 



tneld^tot; lotter. 


Fig. 86. Title page of the 1536 riese 




r:.-i»iirw employed before the metric system was developed, and of 
y.' h »}-mb<)ls as are still used by apothecaries. 


futfieie Umftm$ er PoMMtw^jiri* 

FORT I O p<nw mjt m imf i 

V A R I S I I S» 

lM,MH»B«/2ern/r. Aim 

Fk;. 87. TrTLE pace or the first EnrrK)N ok aciriojla 

< ."s^r worts 0/ /^jj. Agrippa, p. 167, 1531 ; Apianus, p. 6j : Jorda- 
• .;. 62,1496; Riese, p. 139, 1522 ; Schonerus(editor),p. 178, 1534; p. 140, 1522; Anonymous, 'Libretto de Abaco,' Venice; 

-inxxi* (sometimes attributed to Kegiomontanus), p. 178, 1534- 


GIOVANNI MANENTI. Ed. pr. 1534. Venice, 1534 

ZUAN Manbnti. a Venetian mathematician of the sixteenth century. 

Title, See Fig. 88. 

Colophon, * In Vinegia per Giouan' Antonio di Ni-//coIini 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 XXXIIIL' (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 v-as 
intended for the use of Venetian bankers and merchants. 

GIOVANNI SFORTUNATI. Ed. pr. 1534. Venice, 1534 

JoHANN ES Infortunatus. An Italian arithmetician, bom at Siena c. 1 50c. 

Title, See Fig. 89. 

Colophoft, *Stampata in Vinegia per Nicolo di Ariftotiic; 
detto Zoppino.//M.D. XXXIIIL' (F. 129, r.) 

Description, 4°, 15.4 X 21 cm., the text being 13.2 x 16.9001. 
129 ff., 40-41 11. Venice, 1534. 

Editions, Venice, 1534, 4* (here described); ib., 1543, 4°; 
ib., 1544 (colophon 1545, see p. I77),4^ ib-, iS45,4*(p. I77)- 
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 fento 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. 





C6poOa r<rZulM3ficiitl. 

Ccartinlef^iodcl tflalLSeflatovcacto 
ch*aliricheZJi«anctf infra anni X M 
fir M fir Mpat no la poCi, fc*to le PC 
■c cfiCHiateai quellaM D X XXIUL 

Fig. 88. Tttle pace of manknti 





rnntuliio: KaouoLttnw fmpoche moJte 4>pofino 

ni cH€ f>tT aim aiiEon:fono fatfanictt rdclufcj in 

quclVo fi nn^danoj& caitigano:con chiatet la 

adaSc tipcn« dim ojlraiioniin^olto bent di 

fniflcA'VcntillaTcCfi ?nobmietTartaro 

di Gconicirii:pef quaVo a vnopratico 

Agrinicnfotc fi conucgai con ra^ 

uoft da JTcmporrc Ic cordada mi 

furat^U miuu di ctafchuiij 


ml diqocUc* 

Compofto p«t To acuufTioio ptcfcmtatortc^Jenc 

Atchimeduoc&Eiididiane doctmK 








Fig. 89. Title page of the hrst edition of sfortunati 



Ed. pr. 1534. Venice, 1 544-45- 

S«« p. 174. 

TitU, The title page is practically identical with that of 1534, 
except for the date: M. D. XLIIII. 

CdophoH, * Stampata in Vinegia per Bernardino dc Bindoni 
//Milancfe Anno domini. M. D. XLV/ (F. 129, n) 

DescripHoH, 4*, 14.8 x 20 cm., the text being 12.2 x 16.7 cm. 
2 flF. 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. 

Seep. 174. 

GIOVANNI SFORTUNATL Ed. pr. 1534. Venice, 1545. 

Sc«p. 174. 

TiiU. The title page is practically identical with that of the 
1534 edition. 

CohphoH, 'In Vinegia per Giouan* Antonio & Pietro fratel-//li 
de Nicolint da Sabio. Ad inftantia di // Giacomo da Coneano 
hbraro a £an//Fantin. M: D. XLV.* (F. 129, r.) 

DtscripHon. 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 

See p. 174. 

GIOVANNI SFORTUNATL Ed. pr. 1534. Venice, 1561. 

Seep. 174. 

TttU. The title page is practically identical with that of the 
1 5 54 edition. 

Cd^M' • In Venetia per Francefco del Leno,// M D LXL' 

iy€scription. 4*, 15.4X 21.6 cm., the text being 12.9 X 17.3 
r 11. 5 ff. blank + 129 numb. « 134 ff., 41-43 H- Venice, 1561. 
^«^e p. 1 74- 


ANONYMOUS. (Schonerus editor.) 

Ed. pr. 1534. Nuraberg, 1534. 

Johannes Schonerus (Schoner) was bom at Karlstadt, near Wtirzburg, 
lanuaiy 16, 1477* and died at Niirnberg January 16, 1547. He was z 
preacher at Bamberg, and later (i 526-1 546) a teacher of mathematics in the 
Aegidiengymnasium at Niirnberg, in which Melanchthon took such interest 

TifU. See Fig. 90. 

cV/.//i«w. * Norimbergae apud lo. Petreium,// Anno M. D. 
XXXIIII.' (F. 32, r.) 

Ptscn/^tiim. 4*, 15 X 20.9 cm., the text being 1 1.6 X 15.6 cm. 
3J ff. unnumb., 29-43 11. Niirnberg, 1534. 

EJifiiyfis, This is merely an edition of the anonymous medi- 
XN-al ' Algorithmus Demonstratus,' with notes by Schonerus. As 
might be expected, therefore, it is purely theoretical, being a 
late \-ariation of the Boethian works. In the preface Schonerus 
$|^K:aks of it as the ' Algorithmus Demonftratus incerti autoris.' 
Dc 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 
giH?s back at least to the fourteenth century. There is said to 
have been an edition published at Niirnberg in 1533, attributed 
lo Regiomontanus, but I have not seen it. 

JOHANN ALBERT. Ed. pr. 1534. Wittenberg, 156 1. 

.\ Wittenberg Rechenmeister of c. 1500-1565. 

Tif/t\ ' Rechenbfichlin // Auff der Federn/ Gantz // leicht/ 
aus rechtem Grund/ In // Gantzen vnd Gebrochen/ Neben // an- 
gchcflftem vnlangft ausgelaffnem Bijch-//lin/ Avff den Linien/ 
l)cm einfel-//tigen gemeinem Man/ vnd anhe-//benden der Arith- 
mctica//zu g^t.//Durch Johann. Albert/ //Rechenmeifter zu 
\Vittenibergk/zufamen//bracht. Auff s new/ mit allemvleis vber- 
■ fchon/ gemehrt vnd gebeffert/ //zum dritten mal.// Wittem- 
borg. '/ 1561/ (F. I, r.) 

ColifhopL * Gedruckt zu // Wittemberg/ //durch Geor-//gen 
l^iwcn // Erben.// 1 561.* (F. 1 18, v.) 



90. Tttle PAt;E OK THE 1 534 Alf^oritkmvs Demonslratts 


Description, 8*, 9.4 X 15.1 cm., the text being 6.7 X 10.8 cm. 
120 ff. unnumb. (two blank), 7-24 U. 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 unff-/Aundert/ 
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 
Fedem*) 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 Rechenbflchlein/ 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 of i§j4. Borghi, p. 16, 1484; Bradwardin, p. 61, 
1495 > Jordanus, p. 62, 1496; Kolross, p. 164, 1530; Ortega, p. 93, 
15 12; Peurbach, p. 53, 1492; Reichelstain, p. 169, 1532; RudolflF, 
p. 151, 1526 ; Wolphius, p. 154, 1527 ; Rabbi Elias Misrachi, PZN^ 
nCDDH or nODDH IDD (perhaps 1533? 1532?), Constantinople, 4** 
(another edition, with a Latin translation by Schreckfuchs, and a com- 
mentary entitled nCDDH nrN^D IVap by Mtinster, Basel, 1546, 4*). 

GIOVANNI MARIANI. Ed. pr. 1535. Venice, 1580. 

ZuANE Mariani. a Venetian arithmetician of the sixteenth century. 

Titk, * Tariff a perpetva // Con le ragion fatte per fcontro di 
//qualunque Mercadante fi voglia,//che dimoftra quanto roonta 
ogni // quantiti de cadauna mercantia ad // ogni pretio, si a pefo 
come a nume//ro. Buona per og^iuno in Venetia,// Dalmatia, & 
altri luoghi; nelli quali // fi ragiona, & fi fpende a moneda//Vene- 
tiana. Et h buona per Verona,// Breffa, Bergamo, Milan, Cre- 
mona,// Mantoua, & altri luoghi d6ue fi // ragiona, & fi fi>cnde 
a moneda Im-//periale, & Breffana : Con la reduti6//di raoneda 
Venetiana in mone-//da Imperiale, & della imperiale in / / 


Venetiana. Et h buona a ridurre // ogni forte de on 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 1 3.3 cm. 279 ff . numb. + 20 unnumb. = 299 ff ., 
32 U. Venice, 1580. 

Editions. This is apparently the first of two books by Mariani, 
the various editions appearing as follows : Venice, 1 5 3 S ; ib. , 1 5 5 3 ; 
ib, 1559; ib., 1564, 8*; ib., 1567; ib., 1569, 16**; ib., 1572; 
^^ 1575 ; ib., 1579; ib., 1580, 12** (this edition); ib., 1591, I6^ 
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, 
aod 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, 1522 ; Ton- 
stall, p. 134, 1522 ; Torrentini, p. 76, 1501 ; Pedro Melero, * Compendio 
de los numeros 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 ii.i cm. 
I f. unnumb. + 103 numb. — 104 ff., 17-22 11. Freiburg, 1550. 


Editions, Strasburg, 1536, 8**; Freiburg, 1543,8"; ib., 1550. 
8" (here described). 

This work was intended for the Latin schook. 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 0/1536. Boethius, p. 2 7 , 1 488 ; Bradwardin, p. 6 1 , 1 495 ; 
Budaeus, p. 99, 15 14; Feliciano, p. 148, 1526; Ortega, p. 93, 15 12; 
Peurbach, p. 53, 1492 ; Riese, p. 139, 1522 ; Torrentini, p. 76, 1501 ; 
Christiemus Torchillus Morsianus, ' Arithmetica practica,' Basel, 8**, with 
subsequent editions, ib., 1538, 1553, 8** (but see p. 159, 1528) ; Georg 
W&lckl, * Die W&lfch practica/ gezoge auf5 der kunfl der Proportion,' 
Strasburg (Ntirnberg ?), 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 soUen,' Ntirnberg, 8*; ib., 1537. 

ABRAHAM BOSCHENSTEYN. Ed. pr 1536. S. 1,1536. 

The son of Johann Boschensteyn (see p. 99). 

Title. * Ein niitzlich // Rechenbfich-//lin der Zyffer/ darauP 
ein // yeder/ durch fein aygen fleyfz mit // kleyner hiiff / lemen 
mag anfeng//klich rechenen/ Aufzgange durch // Abraham 
B6fchenfteyn/ Vnnd //yetzo zfim dritten mal mit fleyfz //vber- 
fehen vnnd Corrigiert/ mit // erlichen zflgethanen Exem//plen/ 
Durch Johann // B6fchenfteyn/ den alte.// M. D. XXXVI.' 
(F. I, r.) 

Description, 8*, 9.5 X 14.5 cm., the text being 6.8 x 11.9 cm. 
40 fT. unnumb., 20-23 U- 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 19.// tag Aprilis/ An-//no jc. im 30 //jar d* min// 
derenn //zal.' The work is very rare. 


In the epistle the author mentions his father's work (p. 99) : « Wie- 
wol meyn Herr vatter/ herr Johan B6fchenfteyn vor 17. jaren auch der 
gleych zfi Augfpurg inn den Truck milgetheyh hat/ vnnd zflm drittenn 
mal getruckt worden.' Seventeen years before 1530 was 15 13, when 
Johann's book was probably written, since it was published in 15 14. 

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 
ens Heeren ; M.CCCCC. ende // XXXVII. den. ix. dach Febru- 
ari].' (F. 180, V.) 

Description. 8*, 9.4 X 14.7 cm., the text being 7.6 x 12.8 
cm. 5 ff. unnumb. -I- 176 numb. = 181 £f., 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 15 14 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. {S^^Bibliotheca 
Mathematical 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, 15 18, who used them in a similar way.) 


imiaini^fiincvii|Cf ivnnpiifniiviim iinuuiif 
^^te R0iil(iMn btpin ftit0^^cIcfiiiiit(Mtt8lili 

MWMtW/ dftyiHff/ ft wtTflffw/ l^^^lfffilf BIiw ^H MWIMtf 
lidir4Han(ftipVMi^^^(|[tisiiliii€oi. ^ 

TCfBW<BtCOODWl8 wMli W >1MWllllll V fiM <Bf fl5 WI 

mcmncaHniK ocROopt in p w Mi^ y t/ irowy fnponwy 
QL^fe n0idf ttM^tcftlfnp/tnccMu^ 

Q^f(K0llk1MttfhtttllMChn0( oft (Mil (fcf IttC 

iiocr AiKoiitfaii09Ctstta itniuaiciiiu 
«^^K0iiKiNin iiwigciiii0i/iMii0||iifiii'nn 

_^{iM(COKiMnmtmfiicfKl«e^^ (rfMtlii 
tcnalfpiotf lMtoiibC0an0cl9ciBltiftttilMi tMta/off 

Sit cfrto i ltcrt cliocyfaciwtinftflfootiyi tt ^ ^ 

Fig. 91. Title page of vander hoecks 


CJtfmwfiWaftrffhcttl^ If iwntt: |J fort 
fcmtbcpbtftcqiwawmtcoft I 7~ batrnaniuW 
ptt(mtD^«taiouai{;tacc iiicmtt Aniwvncofmti Dtt 
irmiMtii 7^2 rtii r^ noMl^Wcrafio -J 

CX>lrliTactottt!c|cat i.iet — ofl met+ttacr 6m 
«lfcli»cBib(ntucr;li(/al0tricftttk \ ^^'"^^^ \ 

CttanmidiaftrcclimiV: ^ Mnttc ^ rc/It^ ^ 

CltoitifVttt'l'Of^— <iMn — ^ibtraDcmo^e-i- 
iMit — * of^ — *uan + amtm ixdnfufaaxtit fo ucr« 
It «to a&Omltr ftin oft fttlitralimHf. 

C iVtedt^liraoc mtrnt ^ iMnraftoitalm. 
jjrroaoinutltipKffltn jndmi^focnicct iAtf/hp 
^^inottfMbnaluOcnoinmcreDan cciiOfi iiatu 
italoB: (rimiMiriicntn met fftnpclai tiomiturfec 
mocf ohiOniiUNnmcrmuttipbcmn ttac&tquahtrvt 
fimii|r< midt^^cit 4 iitbacr fcbiot cocim 1^ 10 1:0 
nuiSipHcrcrtotmtJO (0fti44lH(rttic trcttBiroft 
fe^Utimiitiplitcat m(t4 nemtn, aIooo;at. 
lWittiinitltvlifcrti^«*8imt;fo noittviicccrtlot 

Fia 92. From vander hoecke's Arithmetica 


Other works of 1^37- Apianus, p. 155, 1527 ; Baeda, p. 159, 1529; 
Benese, p. 182, 1536; Kobel, p. no, 1514; Rudolff, p. 152, 1526; 
Wolphius, p. 154, 1527 ; Andr^, 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 leme 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. 

Bom 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 wods 
of his are extant, this treatise and a * Harmonices Manuale.' 

Title, See Fig. 93. 

Description, 4^ 15.7 X 23.1 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 1 5 54 (p. 263). For the commentar)' 
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. 188) 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 a^ 
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 mediae^iil 
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 inductile 
instead of deductive in treatment. It is also a matter of interest that the 
first multiplication table, the * mensa Pythagorica ' of mediaeval writere. 
to be found in any treatise appears here, although Hilprecht found therr. 
on the Babylonian cylinders of about 2000 B.C. The best edition of the 
works of Nicomachus is that of Hoche (Leipzig, 1866). 



Zl'Nor A>l«MHTIKirZ' BIBAI'a Ai'o. 





In offidna Chriniani Wechdi. 
M. D. XXXVlIi 

Fic. 93. Title pace of Nia)MACHus 


GILLES HUGUETAN. Ed. pr. 1538. Lyons, 1538. 

A Lyons arithmetician, bom c. 1500. 

Title. See Fig. 94. 

Coloplwn, * Icy finiffent las tables des comptes compofees et 
calculees // par Gilles huguetan : Et imprimees cheux // ledict 
Gilles et Jacques huguetan // freres. Lan//i538.' (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., 59-64 11. Lyons, 

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. 

lAMBLICHUS. Ed. pr. 1538. Amheim-Deventer, 1668. 

Bom at Chalcis, in Coele-Syria, c. 283 ; died at Alexandria, c. 33a 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 
S 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 centur\- 
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 iSjS. De la Torre, p. 41, 1489 ; Glareanus, p. 191 . 
1539; Mariani,p. 181, 1535 ; Morsianus,p. 182, 1536; Peurbach, p. 55, 




alailccsparGlLLES HVGVETAN.fmiTdcLyom 
Pv Itfandks on pourra&cilcfiicm troiniar let Compics coin fai^ 
a h Ch«sc, ou au Nocnbffc. 

£^Tdbkfaiiffi diiBn Dor &Dargcm, pour louiofr^rcclon que k Marc dc billon 
I ck fm^ou <ialoy,combkn il uauldradcpoixck finOr,ouDarscm £in. 

kiur,ioofBbknondc(pcfidlan &kMo>*t,&any(bnduMoyscocnbicnrcukiicpour 
M ac pov dia<amuMir«&a ca» pour Ancoiiil>kn ondcfpc^ 

La innkfc dc Aiiali»,ooRcduyre par kclkiTabki lootaMoninrcMa Ifc^ 

t L ART 8ckknotd€N<Mtbar,AiknRafimAnl^^ 
ScPrntt^fm k ooo^pce dctGccts> 

Onksooid aLyofvaknIeigiiedchSpieat,cIicax 
G3k»,iC laques HuguetaD,fiperca, 


Fk;. 94. Title page of hlguftan 


1492 ; Psellus, p. 168, 1532 ; Riese, p. 138, 1522 J Roche, p. 130, 1520; 
Rudolff, p. 160, 1530 ; TonstaU, p. 135, 1522 ; Tonus Klos, • Algoritmus: 



Ex Coele- Syria 


Atithmeticam incroda^onem, 

B T D£ 

F A T O. 

J\(mcprimume(littu,mLatimmfermonem coavfr/iu, 

attk ferfttnit iUt^tem 




Explicatio m duos Ltbros Nicomtchi) 
cutD Indice rerum & verborum locuplea(fiino. 

A R N H E M I i£* 

Proftantapud jutt. FRIDERICUM HAGIUMi 
Davcntrixtypisdefcripfit WiLHELMUf Wil«, 

Cl9 l9 C LXVIII. 

Fig. 95. Title page of the 1668 iamblichus 

to iesth nauka Liczby, Polska rzecza wydana : Prrez Ksiedza Tomasza 
Klosa/ Cracow (reprinted at Cracow in 1889) ; Eysenhut, *Ein ktinstbch 
Rechenbuch,* Augsburg, 8®. 



Ed. pr. 1539. Paris, 1543. 

Ixiftm, LoRETf. Born at Mollis, canton of Glanis, Switierland, in June, 
1 4H.S ; died at Freiburg, Breisgau, May 28, 1 563. He was professor of mathe- 
nvitim and philosophy at Basel (1515-1521), and professor in the College 
de France, Paris (1 521-1524), and later Uught at Basel and Freiburg. He 
wn>te on arithmetic, music, and geometry. 

TttU. Sec Fig, 96. 





Ex ogiaw Ucoki CaztBiJkb in- 

najtt CaMeracenjis, 
> 5 45 

Fk;. 96. Title page of the 1543 clareanus 


Description, 8**, 10.6 X 17 cm., the text being 6.7 X 12.9 cm. 
2 ff. unnumb. H- 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, 
1554, 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, I5S8»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 c^ 
the elementary operations with integers ; then, briefly, of progressions 
and proportion. There is nothing in the little book to commend it. 


Ed. pr. 1539. Freiburg, 1550 

See p. 191. 

Title, * De. V I . Ari//thmeticae // Practicae Speciebvs // Henrin 
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. TJ^ 

Description. 8*, 10 X 15.8 cm., the text being 6.4 X 1 1.7 cm 
2 pp. unnumb. + 2 blank + 74 numb. = 78 pp., 22 11. Freiburg:, 



Ed. pr. 1539. Paris, 1551. 

See p. 191. 

Title, 'Dc fex Arith-//nieticae Practi-//cac Speciebvs,//Hen- 
Hii Glarcani // Epitome.// Parisiis,//Apud Gulielmum Caucl- 
lat, in pingui Gallina,// ^ rcgione coUeij Cameracenfis.// ISS^.* 
(F. i,r.) 

Colophon, * Excudebat Lutctiae Parifiorum Benedictus // Pre- 
i;.»tius Ty|K)graphus in vico Fre-//mentello, fub Stella Aurea:// 
in. Xon. lanuarij,// 1551/ (F. 23, r.) 

Ikscriptiopt, 8*, 10.4 X 16. 1 cm., the text being 6.4 X 11.9 
cm. 2 ff. unnumb. + 21 numb. H- i blank = 24 ff., 29 II. Paris, 

Sec p. 192. 


Ed- pr- >539 Milan, 1539. 

OmoMMo or GiRoi.AMO Cardano, Jerome Carda.n. Horn at Pavia, 
September 24, 1501 ; died at Rome, September 21, 1576. lie was a phyM- 
iiAn and professor of mathematics at Milan (i 534-1 5S9) ^nd profe!v«ior of 
mcduine at Pavia and (1562-1570) Bologna, l^ter he wa.H a papal pen- 
sioner at Rome. He was one of the most acute mathematicians of hU 
century, and wrote numerous treatises on mathematics and natural science. 

Tt(U. See Fig. 97. 

Colophon, *Anno a Virgineo partu.// M. D. XXXIX. // 
I- • Antcmins Caftellioneus Me//diolani Imprimebat Im-//penfis 
iVrmardini VCalufci.* (Printer's mark, with * B.C.') (F. 304. v.) 

r^scription, 8*, 9.9 X 1 5.3 cm., the text being 8.2 x 12.7 cm, 
304 ff. unnumb., 33 11. Milan, 1539. 

Editions, Milan, 1539, 8* (here described); Ntimberg, 1541; 
:•> , 1542. See also p. 338, 1570. A 1537 edition is given by 
V.II'.cus, and I have seen it mentioned in a dealer's catalogue; 
'•' .: I think the date a misprint, or that some one has taken the 
:a:c of the preface instead of looking at the colophon. 

This is one of the most pretentious arithmetic s of the sixtc-cnth cen- 
f-nr. and it did much to influence the advanced teaching of the subject. 
It . f in no sense a practical book, having been written by a mathcmati< ian 


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 



metice^SL Menriirandi{bgcilaris.laqaa 

^€ pieccr Mz^cddamottjftiSi 


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 ancit^nt 
theory. The work then runs into algebra, combining this with arithmetic. 


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-knovni * Ars Magna' (algebra) appeared in 1545. 

JOHANN NOVldMAGUS. Ed. pr. 1539 Paris, 1539. 

Neomagus, Jan Bronckhorst. Bom 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 Baeda and Ptolemy. 

Titk. 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 ^nger notation as found 
in the works of Baeda, the astrological numerals of the Middle Ages, and 
the Boethian theory of numbers. 

Other works of 1^39^ Capella, p. 66, 1499 > Peurbach, p. 53, 1492 ; 
Ringelbergius, p. 166, 1531 ; Vogelin (see Peurbach, p. 53, 1492); Wol- 
phius, p. 15411527; Anonymous, ' Abacho novo con il quale ogni persona 
puol impaiar Abacho senza che alcuno li insegni,' Venice ; Anon3anous, 
* An introduction to algorisme, to leame to reckon with the penne,* Ten- 
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. QvoV/rum prior Logifticen, 
& ueterum nu//merandi confuetudinem : pofterior//Theoremata 
numerorum complecti-//tur, autore loan. Nouiomago. // Nunc 
recins ab ipfo autore recogniti.// (Woodcut with motto: *Discite 
Ivsticiam moniti.') Coloniae loan. Gymnicus excudebat,// Anno 
M.D.XLIIII.' (F. I, r.) 




Lodfticen 8c vecerum niimerandi oonluenidi* 

ncffl , polteribr Thcorcmata numcronun com- 

ple£tjnir,ad do&ifiimum'virum Aadro^ 

am E^erdem profeflbrem 


Nunc rccjns in lacem eaiM authora 
loanne Notuomago. 


Exofnana Chriftiani Wccheli^iub fcilH 

Bafilicnfi^in vico Iacobaeo:& (iib 

Fega(b,in vico BelloiuccniL 


Fig. 98. Title page of noviomagus 


Inscription, 8*, lO X 14.8 cm., the text being 6.7 X 1 1.5 cm- 
5»i tf., 25 II. Cologne, 1544. 
Editions, Sec p. 195. 
Ihis U merely a reprint of the first edition (p. 195, 1539). 

JOHANN NOVIOMAGUS. Ed. pr. 1539. Deventer, 1551. 

See p. 195. 

Title, * Dc nvmc//ris libri II. qvo-//rvm prior logisticen, et 

ucicrum numerandi confuetudinem : poftc-//rior Theoremata 
niimerorum com//plectitur, Autore Ioan.//Nouiomago.//Nvnc 
rc^cns ab ipso //autore recogniti.// (Woodcut, and 'T. B. Fons 
L»\i>.*) Daventriae,// Theodoricus Bomius 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 tf. unnumb.. 25 11. Deventer, 1551. 

Editions, Sec p. 195. 

1 ike the 1544 edition, this is a reprint of that of 1539. 

JODOCUS VVILLICHIUS. Ed. pr. 1540. Strasburg, 1540. 

Wii.KE, WiU'KR, Wii.n. Bom at Kesel. Kast Pni.^Ma: died at l^bus 
No\cinl)er 12, 1 552. He was profes5»or of (f reek (i 540), and then of medicine, 
in the university of Frankfort an derOder. 

Title. See Fig. 99. 

Colophon, 'Argentorati ex oflficina // Cratonis Mylii,// mcnse 
S-t /anno//M. D. XL.* (P. 125.) 

Inscription, 8*, I0.2 X 1 5.4 cm., the text being 8.2 x 1 1 .4 cm. 
2 : j». unnumb. + 123 numb. = 125 pp., 26 11. Strasburg, 1540. 

Editions. Strasburg, 1540, 8* (here described) ; ib., 1545. 

!*:.:'» is a book intended for the classical schools. It is written chiefly 

". I^tin, hut contains numerous extracts from the Cireek. It is l>ascd 

• •■:. drwk models, and contains several quotations from Nicomachus. 

1 --*• jLin of treatment is catechetical (see Fig. 100), and it is interesting 

• r.« tc that this work ap[)eared in the same year in which Rerorde may 

■ .-kr p<:M>he<l his *(»round of Artes* in England (see p. 213^, a boi)k 

'. »S* h the author also adopted the catechism form. It is manifestly 

-.-: .r»ii hv H<)ethius, ami is hair-splitting in theory and uselos in prac- 

: ••- \\ iliu hius begins in a grand ilo(|uent style, • I)e Arithmeti<\T, <iikt 


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 Algebrx 





'lit D. XL. 

Fig. 99. TrrLE page of wiluchius 

explicatur.' P. 19.) Willichius follows the ancient Greek plan of diNid- 
ing arithmetic into two parts, the first being the practical, the k)gistici 
of the classical civilization (*vna irpaicruc^, qua fupputatio domeftica 
fit . . . apud ueteres k ratiocinando Xoyt^i^c^ dicta eft,' p. 19), and the 
second being theoretical, the ancient arithmetica. (* Eft dcMpTrur^, qw 
velut fanctiora myfteria continentur, & haec fola intelligentia aniroi 


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 
calUng one the father and two the mother of numbers. ('Proinde apte 
Zaiathas Pythagorae preceptor dixit, fMvaSa efle numerorum patrem, 8va&i 



QVi$^iitcrn$mrmpar{ I VST. Vft 
fmterimpar^iparim mf^jOnOf 

t^* ^990iJEfl4uim9eimfrimmiidimihtrjmxfit 

indudpUlkpm 14* «<• ^** NICOL; 

tmnugiftrEiidiit$i I VST. SkiifTijhui 

fft. NICOU CutyiwmimiWiwtamAf 

CwrmcdKt pmifi^fiaupcrimparm mdtiplk^iioncM: it 
fari€rimfir Ustir9faurimi$ybi$qim^diiur^ 

VermfimJtimcMemfimbbttyimidtm m^ 

ndmeMfftptwni^m^ bituoio 

txBoiMocaBigiitwriAUadium BudiiiilfcMUt 

tttr. mcOJ^.SmtmbukiiOloJifiotAto 

SytMdex rautaf I VST. QsUnif VmBSc/r .Simto 

Fig. 100. 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. 100. It has at least the merit of 
having been written for beginners. 


GEMMA FRISIUS. Ed. pr. 1540. Wittenberg, 1542. 

More properly Gemma Rainer or Regnier, the Frisian. Bom at Dockum, 
in East Friesland December 8, 1508; died at Louvain, May 25, 1555. He 
received the degree of Doctor of Medicine in 1 541, when he abandoned HLs 
mathematical studies. He wrote upon astronomy and arithmetic, and his 
son Cornells was a contributor to the former science. 

Title, See Fig. 10 1. 

Colophon, * Impreffum Vitebergae apud // Georgiura Rhau.// 
Anno M.D.XLIL' (F. 80, r.) 

Description. 8**, 10 X 15.3 cm., the text being 7.8 x 15.3 era. 
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, S'* (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 {Abhandlungeny 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. 



The editions of Gemma varied but little until Peletarius, 
Ncvcral years before the former's death, added his notes. These 
amplified the text, but they made no changes of importance. 

l*his was the most popular arithmetic of the sixteenth century, at 
IcA^t among those intended for the Latin schools. It combined the older 



per Gemmam.Frifium Medi^ 
cum ac MathematJcum, 


Fi«;. loi. Title page of the 1542 ckmma frkii's 


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, 1*544 

See p. 200. 

Title, This edition is substantially the same as that of 1 542 
with the following exceptions : F. i, r., title page : * Vitebergx 
Anno M,D,XLIIII.' 

Colophon. *AnnoM,D,XLIIII.' (F. 88, r.) 
Description. 8*, 9.5 X 15 cm., the text being 7.9 x 11.3 cm 
88 flf. unnumb., 23 11. Wittenberg, 1544. 

See p. 201. 
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 ipio 
//authore emendata & multis in lo-//cis infigniter aucta.// Hvc 
accesservnt lacobi Pe-//letarii Cenomani annotationes : Hufdem 
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.') Parisiis.// Apud Gulielmum CaueIIat,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 flf., 26 U. Paris, 1549. 


Editions, This is the ninth edition (see p. 200), and the first 
which I have seen with the notes of Peletarius, although these 
noics bear the date 1545 (f. Tj^ r.) They are here given in an 
appendix (*Iacobvs Peletarius Lectori/ f. JT, 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. 

TitU, Except for the date (1550) the title page is the same 
a.s that of the 1 549 edition (p. 202). 

Dfscriptian. 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, 15 50. 

See p. 301. 

GEMMA FRISIUS. Ed. pr. 1540. Wittenberg, 155 1. 

See p. 200. 

TitU. • Arithme//ticae practicae // methodvs facilis, per // 
(}emmam Frifium Medi-//cum ac Mathe-//niaticum.// (Wood- 
' ut 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 1 1.2 cm. 
•V ff. unnumb. + i blank = 88 flf., 23 11. Wittenberg. 1551. 

Set: p. 201. 

GEMMA FRISIUS. Ed. pr. 1540. Antwerp. 1552. 

See p. 200. 

TttU. • Arithme-//ticae practicae metho-//dus facilis, per Gem- 
r-iam Frifium Me-//dicum ac Mathematicum,iam re-//ccns ab ipfo 
i jitorc emcn-//data, & multis in locis // infigniter aucta.// Cor. 
Graphcvs.//Si numerandi artem, cunctis ex artibus illam//Vel 
: •-rmain, exacts difcere lector amas,// Hanc gemma, ingenio fum- 
*-. ;s quam Gemma libello//Hoc paruo includit, carpito, doctus 
^•'i^ '/ Sunt conati alij prolixis tradere chartis // Hanc artem, at 


multis non placet ille labor.// Porr6 hie Gemma fuam gemraam 
fic temperat, ipfa // Vt placeat cunctis commoditate breui.// 
CAntuerpiae, apud Gregorium Bontium // k Caef. Maieft. libra- 
rium admiffum.// Cum gratia & priuilegio.// 1552/ (F. i, r.) 

Description, 8*, 9.5 X 13.9 cm., the text being 7.4 x 1 1.8 cm. 
4 ff. unnumb. -f 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 1 549 edition (p. 202). 

Description, 8**, 9.8 X 14.8 cm., the text being 6.8 x 12.7 cm. 
I f. unnumb. -f 95 numb. = 96 ff., 26 11. Paris, 1553. 

See p. 201. 

GEMMA FRISIUS. Ed. pr. 1540. S. 1. (Leipzig), 155^^ 

See p. 200. 

Title, * Arithme-//ticae practicae // methodvs facilis, per ;, 
Gemmam Frifium 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 11 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 Gcnv 
I Imox^ Frisivm Medi-//cum ac Mathematicum.// (Woodcut a> 
in the 1542 edition.) Vvitebergae // ex officina Haeredvm / 
Georgii Rhavv//M.D.LXL' (F. i, r.) 

Colophon, * Vvitebergae // ex officina Haeredvm // Georgii 
Rhavv//M.D.LXI.' (F. 88, v.) 


Description, 8*, 9.8 X 14.9 cm,, the text being 7,7 X 1 1.2 cm. 
^9^ ff. umiumb., 23 11. Wittenberg, 1561. 
Sec p. 201, 

GEMMA FRISIUS. Ed. pr. 1540. Paris, 1563. 

S«c p. 200. 

Tit/i\ The title page is substantially the same as that of 
1549 (p. 202), with the following addition: loth line, 'Quibus 
ticnuim ab eodem Peletario additae funt Radicis // vtriufque 

Description. 8*, 10.4 X 16.3 cm., the text being 7.1 x 12.5 
«.m. 2 ff. unnumb. -f 102 numb. = 104 ff., 26 11. Paris, 1563. 

Sec p. 201. 

GEMMA FRISIUS. Ed. pr. 1540. Wittenberg, 1563. 

See p. 200. 

Title. • Arithme-//ticae practical me-//thodvs facilis, per// 
Gcmmam Frifium Medicum ac // Mathematicum.// VVitebergae 

M.D.LXIII.' (F. i.r.) 

Colophon, * VVitebergae // ex officina Haercdvm.// Georgii 
khavv./, M.D.LXIII.* (F. 88, r.) 

Description, 8', 10.5 X 15.6 cm., the text being 8.1 X 1 1.2 cm. 
SS ff. unnumb., 23 II. Wittenberg, 1563. 

liiiitions. See p. 2<X>. 

GEMMA FRISIUS. Ed. pr. 1540. Venice. 1567. 

See p. 200. 

Tttlc, • Aritmelica//prattica facilissima,//composta da Gem- 
r..i Frisio Medico,// et Matematico ; // Con Taggiunta dell* Ab- 
'•'euiamcnto de i Rotti Aftronomici di // Giacomo Pelletario; 
A: del conofccre k mente le Calende,//gr Idi, Ic None, le Fefte 
M. V'lii. il luoco del Sole, & della Luna nel // Zodiaco; & la dimo- 
iT.itionc dcUa Radice Cubica : lequali tutte //cofc dal latino, ha 
n cjucfta lingua ridotte Oratio Tosca-//nella della famiglia di 
.\!.icftro Luca Fiorentino; & halle de-//dicate // alio illvstre 


fignore, il fignor//Ettore Podocataro.// (Woodcut.) In Venetia, 
Apreffo Giouanni Bariletto.// MDLXVII.' (F. i, r.) 

Colophon. ' Registro.// ♦ABCDEFGHIKL M.// Tutti 
fono Quademi ; eccetto M,//che h Quintemo //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., 40-42 11. Venice, 1567. 

Editions. See p. 200. This is the only Italian edition of the 
sixteenth century. 

See p. 201. The < Maeftro Luca Fiorentino' mentioned in the titk 
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-//cins ab 
ipfo authore emendata, & multis//in locis infigniter aucta.// 
Hvc accesservnt lacobi Pe-//letari] Cenomani annotadones : 
Eiufdem item de // Fractionibus Aftronomicis compendium: Et 
// de cognofcendis per memoriam Calendis,// Idibus, Nonis, 
Feftis mobilibus, // & loco Solis & Lunae in//Zodiaco.//Nunc 
ver6 a loanne Stein recognita, & no-//uis aucta additionibus.;'/ 
(Woodcut with motto: Benedices // Coronas Anni//Benignitatis 
//Tvae. Psal. 64//) Coloniae,// Apud Matemum 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. i 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, 'ArithmeV/ticae practicae // methodvs facilis,// per / ' 
Gemmam Frisivm // Medicum ac Mathematicum.// (Woodcut 
representing a counting house, with line reckoning.) Lipsiae ' ' 
lohannes Rhamba excudebat.// M. D. LXXV.' (F. i, r.) 


Dfuripticn, 8*, 9.5 X 15.3 cm., the text being 7.8 x 11 cm. 
S; ff. unnumb., 23-24 11. Leipzig, 1575. 

See p* 201. The book has been rebound with added pages which 
ha\e been used for manuscript. These notes are in I^tin, in a Ger- 
nan sixteen th*century hand, and include both work on the fundamental 
<t{ierations and numerous commercial problems. 

GEMMA FRISIUS. Ed. pr. 1540. Paris, 1578. 

Sec 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 
\*iia.//Parisiis,//Apud Hieronymum de Mamef, & viduam// 
(iilicl. Cauellat, fub Pelicano monte D. Hilarij.// 1 578.* (F. i , r) 

Description, 8*, 10.9 X 17.3 cm., the text being 7,4 x 14. i cm. 
3 ff. unnumb. + 93 numb. = 96 ff., 29 11. Paris, 1578. 

See p. 201. 
GEMMA FRISIUS. Ed. pr. 1540. Antwerp, 1581. 

Sec p. 200. 

TitU, ' Arithraeticae/Zpracticae method vs fa-//cilis, per Gem- Frisivm, Medi-//cum, ac Mathematicum confcripta: iam 
:c«.ens ab Auctore // pluribus locis aucta & recognita.//In ean- 
\:rci Inannis Steinii & Iacobi//Peletarii Annotationes.//Eiufdem 
.•_• Fractionibus Aftronomicis Compendium,// & de cognofcendis 
>:r memoria Kalendis, Idibus, No-//nis, Feftis mobilibus, loco'qj 
> lis & Lunf in Zodiaco. // (Woodcut of astronomer.) // Ant- 
•'•'piac.// Apud loannem Bellerum ad infigne Aquilae // aureac. 
A- no 1 581.* (P. I.) 

Description, 8*, 9.7 X 15.7 cm., the text being 6.5 X 13 cm. 
f p:* unnumb. + 178 numb.+ i blank = 184 pp., 31 11. Antwerp, 
!5'^l. With this copy is bound: *Ivl. Pacii//a Beriga // Insti- 
• . t:. »ncs // Logicae,// in vsvm scholarvm//Bemensivm // cditae, 

Bc-TOc. M.D.C 

v^ p. JOI. 


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 Welacij. 
//M. D. LXXXIIL' (F. i,r.) | 

Description, 8®, 9.5 X 15 cm., the text being 7.9 X 11.2 era. 
87 ff. unnumb., 23 11. Wittenberg, 1583. 

See p. 201. 

GEMMA FRISIUS. Ed. pr. 1540. Leipzig, 1588. 

See p. 200. 

Title. * Arithme-//ticae practicae // method vs 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, 159- 

See p. 200. 

Title, * Arithme//ticae Practicae // Methodvs facilis,// per 
Gemmam Frifium Medi-//cum ac Mathematicum.// (Woodcut a> 
in the 1 542 edition.) Lipsiae,// Anno // M. D. XCIL' (F. i, r . 

Colophon. (Woodcut of Pegasus.) * Lipsiae,//ex officina tj^xv 
graphia Abrahami// Lambergi.//Anno//M.D.XCII.' (F.8S,r 

Description, 8^ 9.2 x 15.5 cm., the text being 8. i X 1 1.4 ^^^ 

88 ff. unnumb., 24 11. Leipzig, 1592. 
See p. 201. 

HENRICUS URANIUS. Ed. pr. 1540. Solingen, 154c 

A German classicist, bom at Reesz, Prussia. He lived at Emmehcb \l '"- 
rich) when he wrote this work. 

Title. See Fig. 102. 


Dfscription, 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. 

lliis is a semi-historical discussion of the various measures which the 
sixteenth century received from the Roman civilization, or which were 


deribusEpttomecx Budso^Porck^Aldaio 

CrGeorgio Agrkolacondimam 

per Hcimaim Vranium 



ZoflrMapn^pff^iMMfeicfM Toiittfoidip 


f ALINGMCI, loamm Sour 
cxcudcbar. iUI.ILD.ZU 

Fig. 102. TrrLE page of iranivs 

•^ rniioned in the most commonly read classics of the Renaissance |x?riod. 
N M Arithmetical ofierations are given in the book. It begins with the 
f". t.oruil parts of the <j/, the twelfth being called the uncia (the Troy 
• .r« c K the sixth the sextans, the fourth the quadrans, and so on. 



PHILIP MELANCHTHON. Ed. pr. 1540. Leyden, 1540. 

Bom 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» 




G I A B B N C O- 




Phstnomenaloachiini Camcrarijt 
cl<ganti(simo cannuie 








r i^^^^O^ 


w /-iiraR fl^P'^ 



VS^ ^w*t^^^ 




^Tr i^ J 











I V G D V N I, 


Fig. 103. ' 

Title page of melanchthon 


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 
On-naeus, dated ' Vitebergae, mefe Augufto. M. D. XXXf ' ; ( 2 ) to Johann 
ktiffenstein, dated * Menfe Augusto. Anno M. D. XXXVI * ; (3) to Johann 
S( honerus (see p. 178), dated ' Vuittebergae, Mefe Augusto, Anno M. D. 
.XXXVI/ These epistles are all upon the value and the nature of mathe- 
nutical thought, and are replete with classical and religious references. 
As mathematics they have no value. 

The verses of Camerarius, 'loachimi Camerarii Phaenomena ad 
clarissimum iwenem Danielum Stibarum,' relate chiefly to astronomy. 


Ed. pr. 1540. Paris, 1550. 

rAssioiH)KUTS. Bom at Scylaceum, c. 470 ; died, pro)>ably at Rome, c. 564. 
A Roman statesman and historian. 

TitU, 'Aurelij Cafsiodori Se-//natoris Cos.qve Romani//de 
•^•latuor Mathematicis difciplinis//Compendium.//Parisiis//Apud 
\'afco(anum, uia lacobaea ad infigne Fontis.// M. D. L.* (F. i, r.) 

Inscription, 4*, 1 5.9 X 20.8 cm., the text being 97 X 17.2 cm. 
I f. unnumb. + 7 numb. = 8 ff., 25-29 11. Paris, 1550. 

Editions, Paris, 154O; ib., 1550, 4* (here described); ib., 1580. 
The 'Opera* appeared in Paris in 1598 (and 1584.?). TheCompen- 
'\\\\TCi isembodied in the ' Disciplinarum Uberalium orbis,cx P. Con- 
sent ioet Magno Aurelio Cassiodoro,' published at Basel in 1528. 

lliis brief treatise on the nature of arithmetic, music, geometry, and 
i^trrmomy, the four mathematical disciplines, was held in high esteem 
•r. the Middle Ages. 

Otiur works 0/1^40. Anianus, p. 32, 1488 ; Bekiamandi, p. 1 5, 1483 ; 
}*,p. 182, 1536; Borghi, p. 21, 1484 ; Ortega, p. 93, 1512 ; Paxi, 
;•- T9- *5o3 ; RudoW, p. 152, 1526 ; p. 160, 1530; Scheubel, p. 233, 
1S45 ; Wolphius, p. 154, 1527; Anonymous (Ortega?, p. 91, 1512), 
• « \ u\-TC tres subtile y profitable de Tart y science de arismeticque y 
t' metrie transbte nouvellement d'Espaignol en Francoys,* Paris, 8*. 

U\*rksof iS4i. Agrippa, p. 167, 1531 ; Albert, p. 180, 1534 ; Cardan, 
7- «'J3. «539; Riesc, p. 139, 1522 ; Ringelbergius, p. 167, 1531 ; Ta- 
.: -ntc, p. 1 15, 1515 ; Georg Rheticus, Arithmetic, Strasburg. 


ANONYMOUS. Various authors. 

Ed. pr. 1542. Cologne, 1542 

Title. See Fig. 104. 



Stio ex uarifs authoribus cow 

fV '/'/'•) 





ColonUeexcudebat loannesGymnicm 
^mo M. D. XLIL 

^Q a^xy^U *t^^V£- -A^^V, r^ru. vt/^ . ^ 

Fig. 104. Title page of the 1542 anonymous work 

Description, 8^ 97 X 15.1 cm., the text being 6.9 X 12.3 cm. 
20 ff. unnumb., 25-29 11. Cologne, 1542. 


Editions, Cologne, 1542, 8"* (here described); ib.,1546; Dort- 
mund, 1 549, 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 6 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 6.1 x 12.3 cm. 
48 pp. unnumb. (i blank), 25 11. Dortmund, 1549. 

See above. 

ROBERT RECORDE. Ed. pr. c. 1542. London, 1558. 

Bom at Tenby, Pembroke, c. 1510; died in Southwark prison, probably 
soon after June 28, 1 558 (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. LVIIL* (F. 205, v.) 

Description, 8*, 8.9 X 13. i cm., the text being 8 x 11.8 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 Recorders. It appeared, however, 
between 1 540 and 1 542. 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 


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., 1561, 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); 


A R r B s: 

7e0cb(ngtt)etboo2lif Atib pzacttTc of 

arftbinetaie> liott) in Vbholcnumbies 

Kito jTraccJons > after a moit eafper 

anD cractec fozce, tbeit anpe Ifyt 

l)attj Dprbetro bee ne 


utve ueib fiD» 


flt£bebpiE8|.a o b er tb 

R E C O R D E 

SDocto) of )dt)prilie« 

Fig. 105. Tptle page of the 1558 recorde 

ib., 1586, 8°; ib., 1590, 8* (the Dee and Mellis edition); ib. 
1594, 8^ (p. 207); ib., I596,8«(p.2i9);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. 



none otlicr cramplcB fo; to Irarnett^ mime 
vanon ef ttfi§ foiroc. 

l^ut fijit (ball f ott marlie,tliaf as f on tpn 
to tt)( otDn fcrni) of flrUlyincttfi (ct a pncKr 
totDcplACCo of tDouCsn^s, mttifotboetlie 
70U 0)aU Ctt a ftarrcao f on Trr bcfoic* 

S.St^oi? pcrccauc ^umcranen : tut 3 
ptepe f ott, liott fl^aU 3 do in tt)to art to aUt 
'ftooo nimmcf o) moit togttOcr? 



«ff ^eaOcftttMifiiittHoartr,f9toaeie 

^'*- tat ttoo fttmmcs at oneo togrtftct: 

^otti be iu rou mare aDftc me^e^as 3 toil rrl 

ronanoiie* tDcvcfbtc tbijcnne poo topllc 

tMc ttoo aimitico,fouQall fpiftr ftt OotoiK 

one of tlKm^tt fntttti not tomcat, aod ttftn 

bp (t OiOto a Ipne croOie t^ ort^cr If tR8.ano 

aftcttoacoc (ittteDoitiKebtort^ruinmr > To 
tfiactOat Ipncmape 

bcb(ttociittDcm;a0 ^^.^ 
if rott tooulbe aottt *•^•' 
Us9 to iJis , ?oa »»». 

naft(ietfoocfAm;o | 

Ufwtec bcrc» -••♦»•*.- 
aob HCM If poa ^^_J 
Iplt. po*j miipc aoa; 
tbc 01c tot!)C otber in tlir (smt pl&te. 02 els 
rou map a(br tb .in borbr roOttUrrin a iirto 
puut : tobict) toAr,brc2ufc ic !8 inoa pipitcft 




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 Recorders, because in the preface 
to the latter the author sajrs : * And if any man obiect, that other books 
haue bene written of Arithmetike alreadie fo fufficiently, 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 iatisfie 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 Englifh 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 betv^een 
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^ 

Recorders works were the most influential English mathematical pub- 
lications of the sixteenth century. Thomas Willsford, in his 1 662 edition 
of the * Ground of Artes,* was able to say with much truth that this book 
was * entaird upon the People, ratified and fign'd by the approbation of 

Other works 0/1^42, 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**; Giambatti^u 
Verini, * Spechio del mercatanti,* Milan, 8® (Brunet sa)rs, * Libro de 
Abaco e gioco de memorie,' Milan, sm. 8**); Han vanderWehn, *Exctn- 
pelrechenschaf t der Regel de Tri, die man nennt die Kaufmanns gtildene 
Regel ganz und gebrochen,' s. 1., 8^. 


ROBERT RECORUE. Ed. pr. c. 1342. London, 1579. 

See p. 213. 

Title. ' The // Grounde of Artes :// teaching the work and 
pra- /ctifeof Arithmetikc, bothe in whole numbers // and Frac- 
ijons, after a more eafyer and //exacter forte than any like hath 
hithcr-//to bin fet foorthe :// Made by Mayfter Roberte Re-// 
(ord, Doctor in Phyfike, and now of late// diligently ouerfeene 
ami augmented // with new and neceffarie //Additions.// 

I. D. 

That which my freende halh well bei;onnf , 
For vcr)' louc to common vveale, 
Neede not all whole to be new done, 
But new encreafe I do reueale. 

Some thyng heeretn, I once red re ft, 
And novve agayne for thy behoofe, 
Of zeale 1 doe, and at requef t, 
Both mend and adde, fitte for all proof e. 

Of Numbers vfe, the endleffe might, 
No witte nor language can exprefle, 
Applie and Trie, both day and night, 
And then this truth thou wilt confede. 

I'r^ntcd at London by H. Bynneman.// Anno Domini. 1579/ 
F. i,r.) 

Colophon. ' Imprinted at London by Henry // Binneman, and 
j hn Harifon.//Anno Domini M.D.LXXVIL' (F. 261, r.) 

Description. 8*, 8.7 x 1 3.9 cm., the text being 6.5 x 1 1.7 cm. 
^Oi ff. unnumb., 27 11. London, 1579. 

It mil) be noticed that the colophon is dated 1577. This is therefore one of 
* - • A«cs where a large edition wax printed, and a new title page was added from 
. "At 10 year as necessary. The edition is more rare than its date would suggest. 

ROBERT RECORDE. Ed. pr. c. 1542. London, 1594, 

See p. 213. 

TttU. • The // Grovnd of // Artes, teaching // the perfect 
•A "ke and practife //of Arithmeticke, both in whole numbers // 
t' •! Fractions, after a more eafie and exact //fort, than hitherto 


hath been fet foorthe.// Made by M. Robert Record,// D. in 
PhifickeV/And now lately diligently corrected and beautiV/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/// (P. i.) 

Colophon. * Imprinted at London by Thomas Dawfon, for 
lohn // Harrifon, dwelling in Paules // Churchyard, at the figne 
of // the Greyhound,// 1 594/ (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 + 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 impreflions 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 1 596 

See p. 216. The first two parts, covering 404 pp. of this book, are 
substantially identical with the 1558 edition (p. 213). The third pan 
(pp. 405-557), 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, digrefl- 
ing fomething from M. Recordes,' * LoiTe 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 r61e until the latter 
half of the nineteenth century. Those in this treatise related to number 
guessing, the rules being easily developed by pur present algebra, but 
rather mysterious by sixteenth -century arithmetic. 


ROBERT RECORDE. Ed. pr. c. 1542. London, 1596. 

Seep. 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-, 30-31 U- 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 1 594 (p. 2 1 7). 

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 fmce 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 


M, F- for John Harifon, and are to be fold fay ' 'Geo: WTiitting- 
ton, and Xath: Brooks^ at the f:gn of , , the An^ell in Com-hiU. 
1646/ iV. I.; 

Description. 8*, lO X 15.5 cm., the text being 8.6 x 13.7 cm. 
27 pp. unnumb. -r 629 numb. =656 pp., 31 U. London, 1646. 

Editions. Sec p. 213. 

Like the 1594 edidoo 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 Urstisiiis. 

ROBERT RECORDE. Ed. pr. c. 1542. London, 1646 (?) 

Seep. 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 
Arithroetick' (l.i) are here omitted ; the last lines read ' Printed 
by M. F. for John Harifon, and are to be II 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. -f 629 numb. = 656 pp., 32 11. London, s. a. 

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. 4-536 numb. = 5 58 pp., 33 11. London, 1662. 

See p. 216. 


ROBERT RECORDE. Ed. pr. c. 1542. London, i668. 

See p. 313. 

TitU, 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 
Bi^hopsgate-//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. -h 536 numb. = 558 pp., 33 11. London, 1668. 

JOHANN PREY. Ed. pr. 1543. Nurnberg, s. a. (iS43). 

A Nurnberg gauger of the middle of the sixteenth century. 

TitU, Sec Fig. 107. 

Colophon, • Gedruckt zfl Nurnberg durch //Georg Wachtcr.* 
F. 36, r.) 

Dtscriptiofi. 8*, 9.5 X 15.5 cm., the text being 7. i x 10.9 cm. 
36 ff. unnumb., 21-26 11. Nurnberg, s. a. (1543). 

Editions. There was no other edition. The book bears no 
.iatc except in the dedicatory epistle to the reader, which closes 
T*ith the words, 'Anno 1543.' This epistle gives, also, the only 
reference to the author, who there speaks of himself as *ich 
Johan Prey/ burger zu Nurmberg.* 

The subject of gauging occupied a great deal of attention on the 
(art of German writers on arithmetic in the sixteenth century, and 
(if cAsionally, as in this instance, separate books were prepared. It was 
Dot so common in England as on the Continent, not appearing, for 
'-xamjjlc, in as extensive a work as the Dee and Mellis edition of Recorde 
mcntioDed above. It had, however, some standing in the early Amer* 
V an arithmetics, and is found as a separate chapter as late as the mid- 
Ik: of the nineteenth century. The title page gives some idea of the 
«'>rk of the gauger before the days of stsindardization of casks. The 
Vrncri<-an rule for gauging a cask was substantially as follows : Add to 
:-^^ head diameter 0.7, 0.65, 0.6, or 0.55 of the difference between the 
r* ad and bung diameters (according to the degree of curvature of the 
<:a\cs), and multiply the square of this sum by the length ; divide by 
;?9 for ale or beer gallons, and by 294 for wine. Thus a cask with 
' \T.^ diameter 36 in., head diameter 30 in., and length 48 in., contiins 
f 5 ;.65 ak gallons. 



This particular work is interesting because, although it was printed 
^ i543> 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 

VifUtm/vn^ fcidfteimlfdkttttnf 
nen f^VQbtffkmp gelkficrt 

^ ^ H H -rn Mi ^^[>, 


1 ^^'^^ 





1 .'-^3-^'^'^^ 


fe ^3((fr ^ 



Fig. 107. TrrLE page of frev 

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. 


ANONYMOUS. Ed. pr. 1543. Paris, 1543. 

The author of the Theologoumena is unknown. He lived after Anatolius 
(Bbhop 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. i cm. 
7 pp. unnumb. +61 numb. (S-65) + 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, 18 17. 

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 1^43' 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 Arithmeticae practicae,' s. 1., 8* (with subsequent 
editions: Wittenberg, 1550; ib., 1558; Leipzig, 1556; Weissenfels, 
1564, 8**). 

MICHAEL STIFEL. Ed. pr. 1544. Nurnberg, 1544. 

Stiefel, Stypbl. Bom 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. Niimberg, 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. 


roJTMENA XAZ A'pieMHTIxfix. 

Habcs hie o ftudio 


culum antehac nufquam excurum,inquoi'ca 

Niinierorum ratio explicatur,vt non fit obfcu- 

rum intelligere hanc arithmeticaad fntehore 

ilia de pnilofophia di(putationem,quam 

Tneologiam veteres vocabant, 

comerre plurimum. 

Apud Chriftianum Nrechelum Tub fcuto Bafi- 
lien(t,in vico Iacobxo:& Tub Pega{b,in vi- 
coBellouacenfi. M. D. XLIII. 

Fig. 108. Title page of the Theologoumena 





Aiithore Midiacle Stifclio* 

Norimbcrgse apud lohan.Pctreiunu 
AnnoChiifli m*x>.xliiii« 

Cum gratia & priuilcgio Cvfarco 
ac^ Rcgio ad Scxcnnium. 

Fk;. 109. Title page of stifei.'s second arithmetic 


2. * Arithmetica Integra,' Nurnberg, 1 544, 4* (here described); 
ib., IS4S, 4^ 1546; Nurnberg, 1548; ib., 1586, 4^ 

3. * Deutsche Arithmetica' (p. 231), Niimberg, 1545, 4*. 

4. * Rechenbuch von der Welfchen vnd Deutfchen Practick/ 
auff allerley vorteyl vnd behendigkeit/ mit erklerung viler Exera- 
peln/ . . /, Nurnberg, 1546. 

5. » Die Coss', Konigsberg, 1553-54, 4** (p. 259); 1571 ; 1615. 

The dedication and Melanchthon's preface, both of which have been 
removed from this copy, are dated * Vuitebergae 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 wort. 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 'i 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 wa^ 
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. 

Bom 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 13.2X 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 ^^Va/i/iirtf^, De harenae numero,' is the 
sixth work in the collection. Basel, 1 544. 



f MP fV^nMCy wwWStt 



cekntiflimi Opera , quxquidcm fxunr>omno/nuld$ iam (ccuUsdcG^ 

dcraot^acf a quam pauciffimishaAcnus uilajnunc^ 


can cdita. 

QlioniinCaialogiimucrla pagbiarepcrkt. 

K^iicSd <juo^funt 


bfOtConuneotanaiiccfii Gnrct &Lattnf, 
nunquam ancea cxcuCi. 

Cum C^J' Maiefl.gratia ^primU^io 

E%^ SI LE %UE. 

lodtmts titrudpim cxcuiifkiL 

All.M o XL 1 I 1 1. 

Fk;. no. Title pa(;e ok the 1544 Am himedi'Is 


Editions, This seems to be the first edition of the works of 
Archimedes to contain the * De axenae 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 1 543. It appeared in the elaborate editions 
of Barrow (1675), Torelli (1792), Peyrard (1807-8), and Hei- 
berg (1880). Riccardi says that it *fu illustrato dal Clavio ne' 
suoi commenti alia sfera del Sacrobosco/ 

There are several works of Archimedes extant. The *De arenae 
oumero,' included in this edition, is not an arithmetic, but it treats oi 
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™ • x** = x™ "*" °. 

ARCHIMEDES. Ed. pr. 1544. Paris, 1557. 

See p. 226. 

Title. *Paschasii//Hamellii Regii // Mathematici//Commen- 
tarius // in // Archimedis // Syracufani prseclari Mathematici 
libru // de numero arenae, multis locis per // eundem Hamellium 
//emendatum.// Lvtetiae // Apud Gulielmura Cauellat, fub pin- 
gui Gallina,//ex aduerfo coUegij Cameracenfis.// 1557.' (F. i , r. ) 

Description. 8**, 1 1 X 17. 1 cm., the text being 7.1 x 12.6 cm. 
48 ff. numb., 23-30 11. Paris, 1557. 

Editiofis. See above. 

This commentary of Hamellius is quite as satisfactory as any of the 
earlier ones. 

ARCHIMEDES. Ed. pr. 1544. Venice, 155S 

See p. 226. 

Title. ' Archimedis //opera non nvlla //k Federico Comman- 
dino // Vrbinate // nvper in Latinvm conversa, // et commcnta- 
riis//illvstrata.// Quorum nomina in fequenti pagina leguntur. 


, Cvm privilegioin annos X.// Venetiis,//apud Paulum Manu- 
lium, 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- 
mcdis./V Venetiis,// apud Paulum Manutium, Aldi F.// M D 
LVIII/ (F. 60, r.) It contains the book entitled 'Archimedis// 
IJbcr de Arenae//nvmero.' 

Sec p. 228. 

JUAN SARAVIA, de la calle Beronese. 

£d. pr. 1544. Medina, 1544. 

A Spanish arithmetician, of Medina, of the middle of the sixteenth century. 

Title. See Fig. 1 1 1 . 

Colophon, 'CFue impreffa la prefente:// obrallamada Inftrucio 
'ic mercaderes enla muy // noble villa de medina d'l campo por 
Tc'lro de // Caftro IprelTor. Acofta d'Antoiio de vruena // mer- 
i aclcr d'libros. Acabofe atreynta dias // del mes de Julio. Ano 
(!c mil 2 quinie-//tos i quarenta 2 quatro anos.* (F. cvij, i.) 

Description, 8', 14x20.1 cm., the text being 12, i X 16.9 
tm, 2 ff. unnumb. + 105 numb. (Roman numerals) = 107 ff., 
50 II. Medina, 1544. 

Editions. Medina, 1544, 4* (here described); ib.. 1547, 4*; 
Venice, 1 56 1, 8** (p. 231.) 

This rare work was evidently written about 1542 (see f. xcviij, r.). 
I: :s not a textbook on arithmetic, but it relates to commercial problems, 
r.c topics being ciuiously interspersed with biblical illustrations to show 
trtr jistice of the customs involved. 

Other works 0/1^44. Apianus, p. 155, 1527 ; Reda, p. 13U 1521 ; 
\ .'.aetis. p. 163, 1530-32 ; (lemma, p. 200, 1540 ; Grammateus, p. 123, 
1 5 I A . KolieU p. 102, 15 14 ; Noviomagus, p. 195, 1539 ; Peurbach, p. 53, 
149- • Riesc, p. 139, 1522 ; Sfortunati, p. 177, 1534*. Tonstall, p. 136, 
1-22 : Vander Hoecke, p. 183, 1537 ; H. Bock, * Kin new Rechenbuch- 
'.• r,/ NiaTibcrg, 8*; Johannes Bogardus, a work on finger reckoning 
avr*! on Aventinus (p. 136), Psu^is ; Ix^onard Hegelin. 'Kin ki'msllich 
.^-cf henl>ik:hlin aulf Zyffer >'nnd andern hUpschen Kegeln/ l'lm(?); 

Jl^l%uiJf.lm^i^ C^OJ^h 




mnrp20iiecbor^,i£nIa qual fe cnren9como6aien Xon 
tar (as vfiirasbe todoo foetractoe bcventas? com 
conipaina:)^otro8muc&08Contrato8» l^artjcnlsr 
inenrercb:ibta5et tracto {>e(a8lana8» ZTambicnar 
cambJ08 licit08 Krepxniadod* 'f^ncuamentccotn- 

Fig. III. Title page of saravia 


Antonio Martin, • Tractado de Arithm^tica y Geonietria/ Alcald, 4* ; 
H. Vuelpius, * De minutiis phyficis et practicis aftronomicae arithmeticae 
nrgulis/ Cologne, 4^ and ' Libellus de communibus et ufitatis arithme- 
ticae practicae regulis/ ib., 4^ 

JUAN SARAVIA. Ed. pr. 1544. Venice, 156 1. 

Sec p. 329. 

TitU, *Institvtione//de* Mercanti//che tratta del comparare 
Vet vendere,//et della vsvra chepvo //occorrere nella Mercan- 
tia//iiisieme con vn trattato // de* Cambi.// Et in somma fi ra- 
pona di // tutto quello che al Mercante Christiano // fi conuienc. 
. / Coroposta per il Dottor Sarava,// & nuoamente tradotta di lin- 
pia spagnuola // dal S. Alfonso D'VUoa.// Cvm Privilegio.// In 
Venctia/ZApreffo Bolognino Zaltieri.// M D LXI.' (F. i, r.) 

Colophon. 'In Venetia apreffo Bolognino Zaltieri // 1561/ 
(F, 135. V.) 

Description. 8*, I0.2X 15. i cm., the text being 7,6 X 12.2 
cm, 135 ff. (3 unnumb.), 29 II. Venice, 1561. 

Editions. See p. 229. 

This is an Italian translation of the Spanish edition of 1544 (p. 229). 
rhe book was evidently written in 1542 as already stated, for the author 
^'fieaks (f. 109) *del capo quei\o anno M D XLII.' 

MICHAEL STIFEL. Ed. pr. 1545. Numbcrg, 1545. 

See p. 223. 

TitU, Sec Fig. 1 1 2. 

Colophon, • Zu Nurnberg Truckts Johan Pctreius.// 1545/ 
F. 96.r.) 
Description. 4', 15.5 X 20.7 cm., the text being 10.8 x 15.5 
• r.i. 4 ff. unnumb. + 92 numb. =96 ff., 32-34 II. Number^, 


Eiiitums. See p. 226. 

I be • Deutsche Arithmetical is divided into three parts. In the 

' H^rifsrechnung ' there are ta chapters, all relating to the arithmetic 

' the common people, there being * no household so narrow and |)oor 

•At common arithmetic is not both serviceable and necessary to its 

**- •arc.* rEs ilt kein Haufzhalttmg/ niendart fo gering vnnd fi hlcrht/ 


iSSUin Utitt itftt/^ttOi Mm Me (iof$(wi^ifttinStiutp 
Hdimn^^^cmttmTttit^tmkt) U§^arNn<^)atffc^/int(vd 
(WmbDrn wottm/vcmtenat vnt ocrMmb/rd^tpcr ^dctMfm/5o 
lotrt fu ite mit new crfunMtcm vou^ctt vnnO Otcddn / fc^t Idd^ 
»n&f ur$ ^(rj^fttad^t vnt)0dc§tt/ vn( mtt gntm <Qcmrd|m^ 
f ontltc^m fDoMm vA ^jrmtpebi ariixpftt. .^W oitMr (b|iinii 
Adrrt wtrt vpn^dr^an^d^nitn^vtib^d^cratAtmns/bmisf 
fdnm^m'd^f dnttdfom ntjt rid».3tflc»Mird& ♦Omgftcfe«l6<»« 
(d/att ffdw^cfo n&flt mwe vttO (ctc^te m» drjfcKcf. 


Fig. 112. Title page of stifel's Deutfcht Arithmetica 


das jr die Haufzrechnung oder gemeine rechnung/ nicht ntitzlich 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 (* Von der 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. 

ScHEVBL, ScHEUBELius. Bom at Kirchheim, Wurttemberg, August iS, 
1494; died February 20, 1570. Professor of Mathematics in the University 
of Tiibingen, to which institution he bequeathed most of his manuscripts. 
lie wrote on arithmetic and algebra, and edited part of Euclid. 

Title. See Fig. 1 14. 

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. 4- 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 
* 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 



-f- »nb -r . VIL 

von tHlm }ctc|m -f- vnb ^/fOdl (eUidtt vttitMt 
ttt9/ dttm:o(ar0um:X o^orf^.tT. ^Skr^ctc^ 

titttid Nr )<ifon.^a tc^ nu rcDc von d(a'<^ |(t((2/ 
foffu c» va|te§n von •+• vnh -r- / oMr von — vnb — . Sffo 
<mc|/n>at'cl^vbnvnd((<tl^m)(<cj^fc»(/(bvafcfie(o/«on -h 
i»nb — . 

® ^cn nit feile imp io'c^m -f-vnb — / cin fcnbalUfyn 
QUgottt^nmm/mUi^Ufi ^it fuatn vOIaitff 4 tXcscfn. 'Qcnn o 
de|^t((|ttmt30dot<t|mot(r vndar(d^nct(nta(cn iv^dttivoOfc< 
|m tvirfr/vnbairco nw votfitn d(r<>d(tfl t>on Mfhtnomcn fum: 
ritm:X tf. Doo dcg^KC oOco |i( §ar / <Uo vnf cr on dnidoi OUao* 

«ni)6tt6(ra§Arm. VIIL 
IX>(y0kic^Sctc^/mad^eit cbm bat fd' 
btg$ad[Ktt/im2(66ireni>ti GabtrA^^/ 
^Italian (<> bttimfitbiral>irenbtesal/bic 

s Cum: -f- 7. 
i*dwi: -f- II. 

zo&um: -f- 18, 

« ®iim: — It. 
3 dun: — 6^ 

*» &um: — »4. 


Fig. 1 1 3. From sufel's Deutfchf Arithmetiea 


to the ordinary classical school. A great deal of attention is given to 
e\( hange, 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, llie nature of the work can be somewhat understood 



leu r^Iis computacionum opulculum, 

a loanne Scheubdto compoGtunu 

Noniblum ad ufum qucndam uul 

gatcfn,(ed ctutm a^nu'donem 

6C (bcndam exqumtioran 

ztiduatticx aao« 


^^r jf/tfj^'f/fAutm^A^'VA. 

M. D. XLV. 

Fir.. 114. Title pa(;k ok schhbkl 

*' m the titles of the five ' tractati * : i. * I>e numeris integris * ; 2. * I)e 
: r'^ji^^^rtionilMis, proportionalitatibus & alijs'; 3. M)e minutijs uulgari- 
• -» ' ; 4. • I)c minutijs phyficalibus ' ; 5. * Aliquot reg\ilae.' 

U'hile Scheubel is not much apprec iateil to-day, he was really ahead 
< f h ^ linie. He tried to Ixinish the expression • rule of three * and to 
? S-titutc 'rule of projwrtion.' His explanation of stjuare root is in 


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-//goCom- 
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. i X 15.5 cm., the text being 6.5 x 10.8 
cm. 1 34 ff . unnumb. -f- 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.) 



DeicriptioH. 8*, 9.8 X 15.1 cm., the text being 67 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 
Ixiuks of Euclid that relate to some part of the theory of arith- 

I qviNDici 









Cmp^9frbiUif0id S. N. S. Pmi9Tn^, 

Fi<;. 115. TrrLE page of the 1545 eiclid 

Tctic. Of these separate books, this is the rare first Italian 
is *;:i<in. Among others may be mentioned those of Wittenberg, 
i54^>; 1549; Paris, 155 1» 4* (P- 238); ib., 1554. 4** (p. 238); 
1355 rhy Scheubel) ; Paris, 1557, 8** (p. 240) ; Wittenberg, 1 564, 


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, lliere is a Greek 
copy in the British Museum. A. De Morgan, Feb. 29, 185 2." 

Other works of 1^45 - 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. II. 

Title. See Fig. 116. (Note the signatureofGiulianode Medici.) 
Description, 4**, 16.3 X 23.4 cm., the text being 9.5 x 17.3 cm. 
162 flf. (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 S 
cm. 18 ff. numb., 29 11. Paris, 1554. 

Editions. See p. 237. 

This work is made up of certain extracts from the various books o: 
Euclid relating to Arithmetic. It is in Greek with a Latin translation 
following each definition or theorem. It consists of such standard oM 
definitions as ' Numerus autem, ex unitatibus composita multitudo,* ani 
such theorems as ' Omnis primus numerus, ad omnem numerum qucm 
non metitur, primus est.* There are no discussions, illustrations^ or 
proofs of the propositions. 


E uclidis clcmcncorum 

Montaiueo intaprete. 

Adloannem BdlaiiimCardinalcm. 


M. D. LI. 

Fio. 116. Title pace of the 155 i Eici-in 


EUCLID. Ed. pr. (arith. books) 1545. Paris, 1557. 

See p. II. 

Title, * Evclidis // Elementorvm // Libri XV. Grae-//ci & 
Latin^,// Quibus, cum ad omnem Mathematicae fcientiae // par- 
tem, turn ad quamlibet Geometriae tra-//ctationem, facilis com- 
paratur aditus.// E7ri7pa/x/ui TraXatoV.// S;^i;fuira irein-c nXaTc»- 
1/09, a Tlv0ay6pa^ a-o-Z/ifm €vp€,// IlvdaySpa^ ao^m rf/JC, Tlkdrmv 
B* apiSrfX* iBi'//Ba^€v // EivkXeiBrj^ iirl rolai xXeo^ ircpucdXXi^ 
€T€v^€v, Lvtetiae,//Apud Gulielmum Cauellat, in pingul 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. 
16 ff. unnumb. -h 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.// Ali4s in 
ordine reliquorum // Septimvs : Qui citra // praecedentium Sex 
librorum // Geometricorum opem erudit^//perfequitur, cum reli- 
quis duobus // fequentibus, uera principia ac // folidiora funda- 
menta Logi-//ftices,id eft,ut uocant,//ArithmeticesPra-//ctic2e- 
// Per //loan. Sthen. Luneb.//In scholarvm vsvm Kara to art 
tractatus ipcorrffJuiTiKm, difquifitione nimi-//rum Dialectica qua- 
Dialogorum // est propria.// 1564.* (F. i, r.) 

Colophon, * VVittenbergae.// Anno // 1 564.* (F. 106, v. ) 

Description. 8®, 9.2 X 14.8 cm., the text being 6.3 x 1 1.5 cm. 
107 ff. unnumb., 22-24 11- 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' Arithmetic, u 
En la muy noble //y felice villa de Valladolid (Pincia // otro 
tiempollamada) En la offici-//na de Francifco Fernandez ' de 



Fig. 117. Title pace of texeoa 


cordoua/ junto alas // efcuelas ma//yores // Acabofe a quatro 
dias del mes // de Henero defte aiio del // fenor de mill i quini* 
//entos 2 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, 
Me numerar en caflellano y en guarifmo.' For example, his two methods 
of writing 160,462,009,621, are : 
c. Ix. U462 ^s. . ix U621 
160 U462 4s 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 
XLVL' (F. 64, r.) 

Description, 4*, 15.3 x 20.8 cm., the text being 12 x 16 cm 
I f. unnumb. + 63 numb. =64 flf., 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, A^ (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 • bin- 
cvocolo ' for the common form of multiplication, and gives the *a danda * 
division before the galley form, recommending it as * molto neceiTano/ 
The applications, while not numerous, are practical, and throw some 
light upon the business customs of Siena and Venice. Cataneo was ix>t« 
however, an original writer. His arithmetic is composed quite largel> 


(»f 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. 






invbhetia m p xlvi 

Fig. 118. Tm-B page or the 1546 cataned 

Otlur works of 1^46' Anonymous, p. 313, 1542; Boethius, p. 37, 
t4-'^^: Euclid, p. 237, 1545; Glareanus, p. 193, 1539; lielmrcich, 
;- joj. 'S*' ; Manzoni, p. 257, 1553; Misrachi, p. 180, 1534; 


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. S^ ; Anony- 
mous, • Ein new kurtz Rechenbtichlein auff der Linien und Federn/ 
Frankfort, 8", possibly by Gtiliferich (p. 269, 1555); Alfonzo Ldpezde 
Corella, * Secretos de las cuatro mathemdticas ciencias,* Valladolid. 

Works 0/ i^4J. Gemma, p. 200, 1540; Saravia, p- 229, 1544: 
Tagliente, p. 114, 15 15. 

Works 0/1^48. 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 d ve prime // Matematiche // di 
Pietro Cataneo // con la aggionta,// libro d'Albaco e Geotne- 
tria con il // pratico e uero modo di mifurar la Terra.// Non pi^ 
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 11.4X 15 i 
cm. I f. unnumb. -f 82 numb. + i blank = 84 ff., 32-35 U 
Venice, 1559. 

See p. 242. 

PIETRO CATANEO. Ed. pr. 1546. Venice, 156; 

See p. 242. 

Title. * Le // pratiche //delle dve prime //Matematiche /. ^'. 
Pietro Cataneo Senese,// ricorrette, & meglio ordinate, con ale;:- 
ne ag-//giontioni de lo steffo Autore.// Diuife in libri quattro. 


(WcKxlcut of griffin, with motto: *Virtvte dvce,//comite For- 
tvna.') In Venetia, apreffo Giouanni Griffio,//M D LXVIL* 
(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. 1., 1607. 

Bom at Mans in 151 7 ; 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 T^J Y, 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. 


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 Arithmeticae.' (Seethe 1 560 edition.) 

Colopfion. * Basileae, per lacobvm // Parcvm, expensis // loan- 
nis Oporini,// Anno 1549.' (F. 87, r.) 

Description, Sm. 8*, 87 X 13.7 cm., the text being 6.3 X 1 1.8 
cm. 87 fF. unnumb., 27 11. Basel, 1549. 

Editions. Basel, 1 549, sm. 8** (here described) ; ib., 1 560, 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 
Scheubel's *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-//ticaB Artis, vt bre-//uif- 
sum ita long6 utilifsimum eru//diendis tyronibus, non folum pro- 
I I^X^r ordinem, quo paucis perftrin-//guntur omnia huius artis 
capita : fed // etiam caufa perfpicuitatis, quae plu-//rimiim de- 
lectat & iuuat difcentes,// fummoperi expetedum : per loan-// 
nem Scheubeliu adomatum // & confcriptum.// lam denu6 ab ipfo 
autore recognitum // & emandatum.// Continent autem utrunqj 
hoc Compendiu,// numerorum fcilicet & calculorum, feu//pro- 
iectilium (ut uocant) ra-//tiocincationem.// Bafilae, anno 1 560/ 
(P. I.) 

Colophon, * Basiliae // excudebat lacobus Parous,// 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. -h 191 numb. (3-193) + i blank = 206 pp., 23- 
24 11. Basel, 1 560. 

See above. 



Ed. pr. 1549. Stettin, s. a. (1565 ?). 

A Gennan Rechenmeister of the second half of the sixteenth century. 

Title. * Ein kurtz Rechenbuchlein fi5r die anfahende Schuler 
geroacht //Durch Johann Fifcher. Gedruckt zu Alten Stettin// 
in Johan Eichorns Druckerey/ (F. i, r.) 

Description, 8", 9.8 X 1 5.4 cm., the text being 6.7 x 1 1.2 cm. 
16 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. IS59> 1582, 1592, 1598, and Wittenberg, 1592, all 8^ 
Of the Gennan translation there were editions as follows : Stettin, 
s. a, (1 565 ?), 8* (here described) ; Frankfort an der Oder, 1 566 ; 
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, bom at Durango in 1525. On f. 3, r., 
is a large portrait with the inscription * loannes de Yciar xtatis sve anno xxv.' 
He Uved in Saragossa, and was well known as a calligrapher. 

Title. See Fig. 119. 

Colophon. * CFue impreffo el prefente libro en la muy noble 
//y leal ciudadZ/d'^arago^a 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. 6i, v.) 

Description. Fol., 19 x 28.9 cm., the text being 14.3 X 21.5 
cm. 4 ff. unnumb. -h 56 numb. + 3 blank = 63 if., 27-38 11. 
Saragossa, 1549. 



Fig. 119. Title page of yciar 


Editions. Saragossa, 1549, fol. (here described) ; ib. 1555,4**; 
ib., 1564, 4®. The book probably had several other editions, for 
Heredia (I, 1 54) says that this is * une des plus anciennes Edi- 
tions de ce trait^/ 

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 
cumto is used for million as was their general custom. (See p. 60, 
Cinielo.) The treatment of the fundamental operations is followed by 
progressions, compound numbers, roots, mensuration, and such common 
applications as ' las compaiiias fm tieropo ' and ' con tiempo,' and * las 
reglas de teftamentos.' 

Oih^r works of 1^49* 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, 15 14 ; Recorde, p. 214, 
c. 1542 ; Hans Bock, < Ein new Rechenbtkhlein auif der Linien und 
Fedem,' Niimberg (probably a second edition of the 1 5 44 book, p. 2 29) ; 
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 ?) 


Ed. pr. 1550. Antwerp, 1565. 

Mennhkr. A Geitnan-Dutch arithmetician of the sixteenth century. See 
also p. 281, 1556. 

Title. * Practicqve // pour brievement // apprendre i Ciff rer, 
& tenir Liure //de Comptes, auec la Regie de // Cofs, & Geome- 
tric. //Par M. V. Menher Alleman.// (Woodcut of counting 
house.) A Anvers, Tan 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. *Imprim6 en Anuers par iEgidius Dieft,//rAn 
de noftre Seigneur lefu Chrift.//M. D. LXV.// 19. lanuarij.' 
(F. ii3,v.) 


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 sph^riques.' Antwerp, 1565. 

Editions. Menher wrote three or four arithmetics, as follows : 

1. * Practique briesve pour cyfrer et tenir Livres de compte/ 
Antwerp, 1 5 50, 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 reWsionof 

* 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 

* 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, produccni 
some very practical textbooks. 

ADAM RIESE. Ed. pr. 1550. Leipzig, I550 

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. -f 196 numb. = 200 ff ., 29-3 1 11. Leipzig, 1 55^ 



• Dttb Scbcr. 

©ttrc^ 9f bom 9(i!cf<tu <^ 


Cum gratia & priuilcgCo 

Fig. 1 20. Title pace or the 1550 ri^ise 


Editions. See p. 140. This is the fourth of Riese's books (p. 
1 39). Bound with it is Isaac Riese's arithmetic of 1 580 (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 dcr 
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 of ij^O. Agricola, p. 171, 1533; Agrippa, p. 167, 
1531 ; Anianus, p. 32, 1488 ; Borghi, p. 22, 1484 ; Cassiodorus, p. 21 1, 
1540; Feliciano, p. 149, 1526; Gemma, p. 200, 1540; Glareanus, 
p. 192, 1539; Lonicerus, p. 253, 1551; Medlcrus, 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 c5sueto di tutti li lochi della Italia al modo mercantile,' s. 1. a., with 
some mercantile arithmetic ; William Buckley, ' Arithmetica memorati^a 
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/Zprofeffore in academia Tubingenfi. 
//Parisiis,//Apud Gulielmum Cauellat, in Pingui Gallina,//ex 
aduerfoCollegiiCameracenfis.//i5Si.//Cvmprivilegio." (F. i,t ^ 

Colophon. * Excudebat Lutetiae Parifiorum, Benedictus Preuiv 
tius Typo-//graphus,invico Frementello, fub infigni ftellae aureae. 
//1551/ (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 tV.e 
extracting of roots by the galley method, and therefore shoi^s the }>er- 
sistence of this mediaeval plan. 


ROBERT RECORDE. Ed. pr. 1551. London, 1596. 

Seep. 213. 

Title, 'The Castle // of // Knowledge.// To Knowledge is 
this Castle fet,// All Learnings friends wil it fupport,// So 
(hall their name great honour get,// And gaine great fame with 
g<KKi report. // Though (pitefull Fortune tum'd her wheele,// 
To ftay the Sphere of Vranie,// Yet doth the fphere refift that 
\^hcele,// 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, 
afbigned //by Bonham Norton.// 1596.' (P. i.) 

Colophon. • Imprinted at London by Valentine // Simmcs. 
1596/ (P. 236.) 

Description, 4', 14.4 X 19.2 cm., the text being 12.5 x 17.5 
cm. 3 pp. unnumb. + 1 blank + 232 numb. = 236 pp., 41 11. 
London, 1596. 

Editions, London, 1551 ; ib., 1556, fol.; ib., 1596, 4* (here 

Recorders name does not appear on the title page, but he signs the 
It tier of dedication to * Princesse Marie,* • Robert Record Phyficion.* 
Jh<f work is on astronomy, and is of interest in the history of arith- 
metic only in the operations involving sexagesimal fractions. Division 
-> jierforroed by the galley method, and there are no symbols for 
•i'trmrs, minutes, and seconds. The arithmetical part includes the rule 
•'f three. Like the author's other works (pp. 213, 286), this is in the 
I atc< hism form. 

Othrr works of ij^i, Borghi, p. i6, 1484 ; Euclid, p. 238, 1545 ; 
<»^nfima, p. 200, 1540; Glareanus, p. 193, 1539 ; Noriomagus, p. 197, 
15 ;9; Pelcticr, p. 245, 1549; Recorde, p. 214, c. 1542; Tonstall, 
F'- »34. >522; Adam Ijonicerus, • Arithmcticae brevis Introductio,* 
Frankfort, 8*, with subsequent editions, ib., 1568, 8"; 1570, 12"; 1581; 
I 5^5 : 1600, 8* (Tropfkc puts the first edition as 1550) ; Andres (larcla 
•:r Ix)vas, •Tratado del c6mputo,* 5^alamanca, 8* ; Innocenzo Ringhieri, 

• (Vnti gi\-ochi liberali ... in dieci libri descritti,' Bologna, 4*, with 

* .^»vr<juent editions, Venice, 1553, 4*; Bologna, 1580; Lyons (French 
translation), i555t 4* (contains some number games). 


MARCO AUREL. Ed. pr. 1552. Valencia, 1552. 

Aurel was, as he states, a German. lie lived, however, for several yean 
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 16.6 cm. 
4 ff. unnumb. -H 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, b*Jt 
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 nanie. 
Division is performed entirely by the galley method, but the figures art 
not canceled as is generally the case. * Proporcion ' is used for ratio. 
and * proporcionalidad ' for proportion, as was generally the custom :n 
the early arithmetics of all Latin countries, a custom derived from ir.c 
Boethian books. The * proporciones ' (ratios) are treated at cons?:-:'- 
able length after the fashion set by the mediaeval writers. Theappl: a- 
tions are almost entirely under the ' Regla de tres * (rule of three •. 

In the part devoted to algebra, surd numbers are first treated, re 
root symbols showing the German influence. The plus and minus >:ps 
are also used as extensively as in the works of writers like Stifei xr.'i 
Scheubel, and the symbols for the various powers of the uakr-«/»- 
quantity are such as are found in the works of contemporary wnters .= 
other countries. 


Ed. pr. 1552. Venice. 15 55 

A philosopher of Ferrara, of the middle of the sixteentii cennrr 

Title, See Fig. 122. 

Descriptiofi, 8^ 9.3 X 15.5 cm., the text being c i x i^acr- 
184 pp. (2 blank, 16 unnumb.), 29 11. Venice, 1553 

Editions, Venice, 1552, 8'; ib., 1553- S* .here* 
This treatise also appeared in his * Opera,* BaseL 15S0. <= - - 
and Leyden, 1696. 



»4lgehattca,emi4ikdfe twmm el arte Meramtmolp 

con otTM miiehas i(t£asdilartemeiur^U^I(egla dei 

JtffirM,>idgfvmmtt1Lmuida Jrtemifftr,o\^Jt 



wnen Gt§metria:€omfneJh,itrdeikuU,y becbo Imfri* 

wur p» Af4iv» Aurel, natural ^temoH : IntitnlaJo, Ve$ 

^ertad» deingpiht.f^adirigdo almMf wufgA* 


J aJamM a wuffia/hiejf 


^ Con Prioilcgio de fu Mageftad^ 
por ticmpo de dicz anos. 

Ett a& dc loto dcMcf . HanM 
Ano-' IS S <• 

Fi<;. III. Title pace ok alkel 







UmUmat^Tricoal^Sotunailts Didogftmwikm 


VBHETIIA ApHdamOttrmSttUm. 
H D L I I I. 

Fig. 122. Title page of the 1553 cyraldus 


I'his set of dialogues is of interest in the history of mathematics in 
that the second and third parts deal with notation and finger symbolism* 
' I >talogismvs secvndvs de manus & digitorum nominibus ddq; nume- 
randi f)er eos antiquorum ratione ' (p. 10) is an almost unknown sketch of 
finder symbolism. * Dialogismvs tertivs ad Baptistam Lucarinum FR. 
Milium optimae fpei ac indolis puerum, de notis & finguris numerorum, 
•; ::l>us Ijitini ac (rraeci utebantur' (p. 20) is an equally interesting 
>Let( h of the (ireek and Ijitin numerals. Gyraldus also published a 
' Hrevis instructio de Grecor^ numerali supputatione 'in 15 13, fol. 

Otktr works 0/1^52. Agricola, p. 171, 1533; Gemma, p. 203, 
1540; (ihaligai, p. 132, 152 1 ; Herman (vtilfferich, *Ein new kurtz 
K<:< henbiichlein/ Frankfort, 8*, with editions ib., 1555, i2', 1568, 8' 
iNcc also p. 244, 1546, and p. 292,1559); Ortega, p. 94, 1512 ; Peletier, 
V' 245« >549; Recorde, p. 214, c. 1542 ; Riese, p. 139, 1522 ; Joachim 
C amerahus, ' Arithmologia/ 1 2° (there was also an edition at Basel, s. a., 
16*); I>unkel, ' Arithmetica,' I^eipzig, 8'; Christopher Falconius, • Re- 
chcnlmch,* Konigsberg in Preufien, 4* (Murhard cites as a different 
m<>rk of the same date ' Rechenbuch auff die Preusche miintz mass und 
( k-wit ht,' ib., 4*). 

DOMENICO MANZONI. Ed. pr. 1553. S. 1. (Venice), 1553. 

A stxteenth<entury arithmetician, bom at Oderzo. 

Title, See Fig. 123. 

Description. 8*, 10.5 X 15.8 cm., the text being 8.1 x 13.4 cm. 
16 fT. unnumb., 26 11. except where arranged in sections. S. 1. 
(Venice), 1553. 

Editions. There was no other edition of this work. Manzoni 
h.i<l already published a textbook entitled * Libretto molto utile 
f Krr imparar a leggere, scrivere et Abaco, con alcuni Fondamenti 
*:e!!a Dottrina Christiana,* Venice, 1546, 8*, 31 pp. of which are 
• icvotcd to arithmetic. He also published in 1553 a more elabo- 
nitc treatise, • La Brieve Risolvtione di Aritmetica universale 
:n i\ .ralsi voglia negotio, doue interuenga numero, peso, & misura,* 
W-nue, 1553, 8*, 246 flf. 

A beautifully printed little manual of the fundamental operations of 

1' 'hmetic. The • Abbreviatvre delle monete, Pefi, & mifure, chi fi uiano 

r. \ inegia* (f. 2) is helpful to students of the history of sixleenth- 

i '-ntMry arithmetic and to all who are interested in the metrology of the 

. -.-intncH with which Venice traded. 



Fig. 123. Title page of manzoni 
MICHAEL STIFEL. Ed. pr. 1553. Konigsberg i. Pr., I55J 

See p. 223. 
Title. See Fig. 124. 

Colophon, *Gednlckt zu Kdnigs-//berg in Preufce durch Alcx- 
andrum // Behm von Luthomifl/ Voll//endet am dritten tag 


dcfs Herbft-//monats/Als mann zalt nach // der geburt vnfcrs 
licbcn // herrn Jefu Chrifti.// 1554* (P. 505.) 

$11 l^fti^^flfsiitpmil&ti 

mMdttJ bnrd^ 2a««mbntm 

I T t ?• 

Fui. 134. Title pai;e of stikel's Rt'ix)LFF 

Pescriptum. 4*, 147 X 19.6 cm., the text being 9.9 X 16 cm. 
507 pp. (4 blank, 12 unnumb.), 21-26II. Konigsberg in Preufien. 



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/ IJSJ' Albert, p. 180, 1534 ; Boethius, p. 27, 148S; 
Gemma, p. 204, 1540; Gyraldus, p. 254, 1552 ; Mariani, p. 181, 1535; 
Morsianus, p. 182, 1536 ; Ringhieri, p. 253, 155 1 ; Rudolif, p. 152, 1526 ; 
Bemardu Wojewddki, * Algorithm, to jest nauka liczby, po polsku lu 
linijoch uczyniony,' Cracow. 

CLAUDE DE BOISSIERE. Ed. pr. 1554. Paris, 1554 

Claudius Buxerius. Bom in the diocese of Grenoble, probably c. 150c. 
He also wrote on poetry, music, and astronomy. For his Rythmcmackta see 
p. 271. 

Title, See Fig. 125. 

Colophon, *Acheu6 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 fo: 
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 thousand^ 
of quintillions, *■ Mille de Quintillions,' although Chuquet, in his manu- 
script of 1484, carries it to * nonyllions/ De la Roche (p. 128) follo»«i 
Chuquet in this as in other respects. Boissibre*s plan starts with the 
names of millions, bimillions (for million millions), trimiUions, 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 pUn. 
distinguishing between digits (1-9), articles (multiples of 10), and com- 
posites (articles + digits), of which names we still use the digit. Insteai 
of extending the fundamental operations to six, or even nine, as h:< 


contemponries 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 93997390' C. 

Debte ou fomme 1840006503 A. 

Paye ou k Subftraire 900032602 B. 





tant pour fart milltaire que 
aucres calculations. 

Auec priuilege 
du Koy . 

Jiiafildm , rue da P$rw . 

X J 5 4- 

Fig. 125. TfTLE pace of boissi^rk 


In division he uses the galley method only. Boissiere is one of the fint 
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 & augment^ par Lucas 
Trembley Parifien,// profeffeur des Mathematiques.// A Paris,// 
Pour Guillaume Cauellat, 4 Tenfeigne de la Poulle//graffe, deuani 
le college de Cambray.// 1563.' (F. i, r.) 

Description. 8*, 10.2 X 16.8 cm., the text being 6.8 x 13 cm. 3flF. 
blank -f i unnumb. -h 71 numb. = 75 if., 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, Gentilhd-// me bordelofs, 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 

Bom at Bamberg, April 12, I5cx>; died at Leipzig, April 17, 1574. The 
office of chamberlain (Kammermeister) to the Prince-Bishop of Bamber; 
being hereditary in the family of Liebhard, he took the Latin name o\ 
Camerarius. He was a distinguished classicist, a friend of Melanchthon. 
and a professor at Tiibingen and Leipzig. Of the various commentaries c r. 
Nicomachus, his was the most important of the Renaissance. 

Title. * Explicatio // loachimi Camerarii // Papebergenfis/, ' in 
d vos libros // Nicomachi Geraseni // Py thagorei // Deductionis 
Ad Scientiam Numerorum.// Et Notae //Samuelis Tennulii / /in 
//Arithmeticam // Jamblichi Chalcidensis.//DaventriaB,//Tj*|Ks 
Wilhelmi Wier, CID IDCLXVII.' (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 wiih 


the commentary of lamblicbus on Nicomachus (p. 188, 1538). 
Dcventer, 1667. 

Editions, The fir^t edition was Augsburg, 1554, 8^ There 
was also an edition published in 1 569. 

Caroerarius also wrote a work entitled * De logistica/ pubhshed 
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). 

Oihir works 0/1^^4. Albert, p. 180, 1554; Baeda, p. 131, 1521; 
Buteo, p. 292, 1559 ; Euclid, p. 238, 1545 ; Finaeus, p. 160, 1530-3* ; 
timber (Piscator) p. 247, 1549; Glareanus, p. 192, 1539; Huswirt, 
p. 74« 1501 ; Nicomachus, p. 186, 1538 ; Peletier, p. 245, 1549 ; Psellus, 
p. 168, 1532 ; Riese, p. 139, 1522 ; Stigelius, p. 249, 1549 ; Tagliente, 
p. 114, 1515 ; Barth. Barchi, 'TarifTe della valuta di tutte le monete,' 
\fantaa« 4* (hardly an arithmetic) ; Claudio Bertholio, ' De numerandi 
raiione aphorismi,' Paris, 8^ ; Ian Gentil (Vander Schuere, in his 1634 
c^ittion« f. 201 — see p. 424 — refers to his arithmetic as published at 
Paris, 1554); Caspar Hdtzler, ' Eyn behende und Kunstrik Rekens- 
lioeck op. der Linien und Tziferen,* Liibeck, 8^ 

PETRUS RAMUS, Ed. pr. 1555. Paris. 1555. 

PiERRK DE LA RamAe. Bom at Cuth, Vennandois, 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. 

Titir, Sec Fig. 126. 

Colophon. • P. Rami Eloqventise et Phi-//losophiae profcssoris 
' Regij Arithmeticae,// Finis.' (P. in.) 

Desaription. 4*, 13.8 X 19.1 cm., the text being 9.5 x 17.2 cm. 
128 pp. (2 blank, 16 unnumb.), 26-27 l'- Paris, 1555. 

Editions. Paris, 1555, 4* (here described); ib.,1557, 8*; Basel, 
1567; Paris, 1584. See also Gleitsman, 1600 (p. 427), and the 
• Ubri 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 Oreek 
writers, a little work on the fundamental operations, and the medieval 
tr^rory of ratios. Ramus had not the faculty of putting together a text- 
u>A that should be a rival to that of C^mma Frisius. 


P. Rami, eloquentias 



mecicx libri tres, 
Carolum Lotharmfftm Cardifialem, 


Apud Andrcam Wcchclum,(ub Pegalb«ia 
vicoBellouaco^ ^nno S^lMOt, 


Cum priuilegio Regis. 

Fig. 126. Title page of ramus 


JACOBUS MICYLLUS. Ed. pr. 1555. Basel, 1555. 

MoLTCKR. Bom at Strasburg, April 6, 1 503 ; died at Heidelbetg (?), January 
28, 155K. He was a well-known classical scholar. 

Titlf. See Fig. 127. 



duo^cx diuerfis cius am's fcnV 

pcoribus collet* ^ & exemplis 

plurimis, Tfdettfcjf uxSHSt. 


pia lAcoBVM ai# 

Gun g^tia ft priuflfgio ImpcruU 



Fig. 127. Title pa(;e ok micylli*s 

( ' lifhan. • Basileae, ex officina//IoannisOporini, Anno Salutis 
-.anae//M. D. LV. Menfc Martio/ (P. 319.) 
Discription. 8*, 9,3 X 15.2 cm., the text being 7.8 X 1 1.3 cm. 
-o j.p. (3 blank, 23 unnumb.), 25-27 11. Basel, 1555. 


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 fuppnta- 
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, quem Com- 
putum Ecclefiafticum, barbaro nomine, uocant.' (P. 269.) The folk>w- 
ing is an example of his applied problems : * Scribit Plinius, Alexandriam 
k Rhodo diftare 583000 pafsuum. Cupio autem fcire, quot ftadia i}dem 
pafsus efficiant. Diuido igitur 583000 per 125.' 

MICHAEL NEANDER. Ed. pr. 1555. Basel, 1555. 

Bom in the Joachimsthal, April 3, 1529; died at Jena, October 2j» 15S1. 
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, -h IS unnumb. -f i blank =120 pp., 25-29 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, 2jA 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 Romas 
abbreviations. The text is mostly in Latin, but eight pages of ' FaX^tvY 
irtpi MtTptav #cai Sratf/Aoiv * are in Greek. 



one menfiirabiUiim (ecundum Romanoff 

^ntR$nti&imiM auchoribut huiut generis 

c6aada,opera MiCHAELXf nban^ 

ORi txVaHeUidhimkMp 

Anno M D Liiiu 


tbikmniCHAELis NBAIfc 

Item term ftuerbonmifofiiioiiiiiibai 
mcmonUUuni Lufob 


8ASILEAE, PEa i o A N^ 

Fio. laS. Title pace or neander 



ANTOINE CATHALAN (?). Ed. pr. 1555. 

Paris, 1556. 

A French arithmetician. The author's name does not ^ypear in this 
edition. A woriL with the same title is assigned to Cathalan, Lyons, 1555. 

Title. See Fig. 129. 




liSffftK €H nomhfi miiir ^ r&m^ 

fUffufiUadfprmJfi^fjy trtp^ 

& corrigee. 

Cjr. extmfUspoUTfme UfatnafUii 
f^*l^»&fl»s Ugttti 4 €9m* 

A P A H I S. 

Par Iclian RucIIc,Jcmourant en la Rue 

faind^ Iacques,a Teofcigoc de 

Jaijucuede Regard. 

Fig. 129. Tttle page of the 1556 cathalan 

Description. Small 8**, 7.3 x 11.4 cm., the text being 5.3 ^ 
9.5 cm. 79 ff. numb, -f i unnumb. = 80 ff., 27 11. Paris, 1556 


Editions, Lyons, 1555, i6*; Paris, 1556, sm. 8' (here 


This is a very good little primer of algorism for the time. Concern- 
ing this work the author says : * Lequel art trouua premierement vn 
{ihilofophe d'Arabie, nomm^ Algus. Dont cefte fcience prent Ton nom 
(1 Algortfme ' (f. 3). In multiplication the gelosia arrangement is given 
a> a third method, under the name ' Mvltiplication per quarreaux.' In 
il:\):iion only the galley form appears. The * premier liure* covers the 
f< wx fundamental operations with integers, and a brief treatment of pro- 
trn-vsions. The ' fecond liure' relates to the use of counters ('gectz *), 
w Ku h the author esteems so highly as to say : ' Ht note que cefte efpece 
«ic addition eft plus vtile & facile aux gectz que aux chifres ; ' ' il 
c :t facile par les gectz, & difficile par les chifres.' The last part of the 
'■> 'k is de^-oted to such standard problems as the testament, the pipes 
h\\:\^^ the cistern, and the broken eggs, and to applications like ex- 
i range and pasturage. 

LODOICO BAitZA. Ed. pr 1555. Paris, 1555. 

A Spamish scholar of the sixteenth century. 

Tit/c, See Fig. 130. 

L cli>phon, • Excudebat Benedictus Preuotius, via // Fremen- 
tciia. ad infigne ftellac aureae.* (F. 68, r.) 

Description, 8*, 10.3 X 15.6 cm., the text being 7.8 X 13 cm. 
2 tf. unnumb. -f-66 numb. =68 ff., 26-27 '•• Paris, 1555. 

Editions. This rare first edition of Baeza's arithmetic is sel- 
■i-rn found in libraries. The second edition (1556), 8*, is, how- 
r'.cr, the same impression with the title page changed. 

I htr book is entirely theoretic al, making much of the classification 
* n-jmlicrs and of the ancient ratio systems. It is in I^tin with nu- 
'z.'T***i% (ffeek quotations. 

Otkrr works 0/ JSSS- Euclid, p. 237, 1545; Finaeus, p. 163, 
i:;^>-32; (^mma, p. 200, 1540; (ilareanus, p. 192, 1539; Herman 
«. ^Vrwh, p. 257, 1552 ; Mariani, p. 181. 1535 ; Ringhieri, p. 253, 
: : : I ; Valturius, p. 10, 1472 ; Yciar, p. 249, 1549 ; Jacob Cuno, 'Arith- 
-'t.«a/ Wittenberg, 8*; Taddeo Duni, •Liber de arithmetica/ Basol, 
4* . MekhiorCioldammer, •Arithmetica Prat ica,' Wittenberg, 8*; Jer6- 
r. n..» de Valencia, •Arte de computo/ in Santaella*s •Vocabularium 
r toiasticum,* a separate edition appearing at Saragossa in 1601. (See 
.:-i p. 140, 1523.) 

^-^ ^-^ *i n-METICA 



•*^"^iA»A aiTHOOO 

X. TT«TI Af; 

i '.i '% -* .» »*..-% 



Fic. 132. From BoissikRE's Rythmomathia 





mus ludusPythagorcus Cqui Rythmo 
machiA Dominatur) in vriltuccm & re- 
laxationcm ftudiofbrum comparacus 
ad reram & facilem proprictatcm & 
racioncm numcrorum aflequendam, 
nunc radem per Claudium Buxcrium 
Ddphinarcm iUuifa-atus. 

L V T B T I AB, 

^fudGuMmum CdueBatifubpin^mGdl^ 
lina,exadHerfo coUegij Camcracenfis. 

abacus (^ calculi ytmunt in Pdailo, 
apud loanncm Gcntil. 



Fig. 131. Title page of the 1556 Boissiii^ 




fcl^$nct:!Kcge(it avtffJLiniin 



ineyf!ertnnX)Im/ ^bm^ 
elm ^(fc^nben* 

Fig. 133. Title page of spanun 


DOMENICO DELFINO. Ed. pr. 1556. Venice, 1565. 

A sixteenth-century Venetian. He was bom of a noble family, which had 
produced a doge, a general of the Camaldoli, and other men of prominence, 
and which wa» later to produce a cardinal and a captain-general of the 
Venetian fleet. 

TitU. 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, S**; ib., 1584, 8*, and later. The title page bears the 
tiatc 1565 as stated, and the dedicatory letter *A1 Mag.«<> et 
Ecc."*° Sig. Nicol6 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 * AU'Illvstris- 
simo et Reverendissimo Signor, II S. Cristoforo Madrvccio, 
Cardinal di Trento, Govemator Meritissimo di Milano,' signed 
h\ F. Nicolo Croce, and dated * Di Vinetia il di primo di Settem- 
brc. M D L VI.' There is no colophon to fix the date more 
ac curately, 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 // 
:r. icntori, utilitiU modo, & altri fecre // ti Cap. III.' The author is highly 
li'ided in Croce's letter as of noble family, * Illuftre per lo splendor 
i' 1 Cangue, non dimeno uia piu per Tomamento e per la gloria della 
i ttrina.' The Delfioo family was at that period very prominent in 
'• t.riue. 

MCOL6 TARTAGLIA. Ed. pr. 1556. Venice. 1556. 

Tart A LEA. Bom at Brescia, c. 1506; died at Venice in 1559. He was 
one of the best mathematicians of his time, and was the 6 rat to give a gen- 
eral M>lution of the cubic equation. 

TitU. See Fig. 135. 

Inscription, Fol., 21. 1 X 29. 1 cm., the text being 13.8 X 23.6 
v) 285 ff. (3 blank, 6 unnumb.), 52-56 11. Venice, 1556. 

Editions. Tartaglia's works include the * Nova Scientia,* 
' :crly algebraic, editions of Euclid, Archimedes, and Jordanus, 



D r rv n r tele 

S C I E N Z E. 



Domenico DeQim , Bofe/g ymtiano. 



moI[e"tSc app^tnen cna al uiuere hunuu ioT" 
St illi cognTnon di D to. 

C H. T K I V I L E G I 0. 


IN YXHimA^ p riR E s s (v^Tiraicm : 

M n I T V 

Fig. 134. TnxE page or delfino 







aaifc in ciioa Parte ntgptUri^Sc mtraunditjooi anchor in ognialm 

VtfAinc^oucr ditdpika, douc inccnicnghi il qIo4o» 



Ik VtiegU far Curth Tr<dam ieiNaai. 
M D Ll^J. 

Fig. 135. Title page of tartaglia 


the ' Quesiti et inventioni ' and * Regola generale/ 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 Nicol6 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 Lice 
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 gi^'es 
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 fuyuantc.. . 
Recueillie, & traduite d'ltalien en Francois, par // Gvillavme 
Gosselin de Caen.//Auec toutes les demonftrations Mathcma- 
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.//i57^^ 
// 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. 


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. 

Sec p. 275. 

Titk, * Tvtte Topere // d'arithmetica // del famosissimo // 
Nicol6 Tartaglia.// Nelle qvale in XVII. libri con varie prove,// 
& ragioni, moftrafl 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, £conti,//giuochi, traffico di 
compagnie, compre, vendite, portar mercantie da un paefe // 
all'altro, conuertir monete, congiungimento di metalli, & opere 
de'2ecchieri.//Sopra le qualcofe tutte, formanfi belliffimi que- 
fiti, & fi fciolgono le diflfi-//co'ti, con vgual chiarezza, & dili- 
genza, per vtile rileuato de i mercanti, & te-//forieri, i Capitani, 
e Matematici, & Aftrologhi, &c. // Parte Prima.// Con Privilegio. 
// In Venetia,// Air Infegna del Leone. M. D. XCII.' (F. i, r.) 

Description. 4**, 13.6 X 18.8 cm., the text being 1 1.5 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 b 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. 6ff. unnumb. + 136 numb. = 142 flf., 15-34 11. Paris, 1556. 





De rebus mathcmacicis, 
hadcnus dcfidcratis, 

Libri nil. 

^ Quibus inter attcrz. Circuit quadiacuia Centum 

modisjSc Cu^tz, per cundem Qrontium 

reenter czcogitatisjdcmonlhatur. 

Anno Chrifti Seruatoris, 
M. D. LVI. 

EzofficinaMichaelis Vafcofani^uia lacolNta 

Cum Priuilcgio Regis. 

Fig. 136. Title page of the 1556 finaeus 


Editions. There was no other edition. For his arithmetic, 
sec p. 160. 

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 NABOD. Ed. pr. 1556. Cologne, 1556. 

Naibod, Naiboda. Born at Cologne; died at Padua, March 3, 1593. He 
vas for a time professor of mathematics at Cologne. He wrote on astronomy 
as weU as arithmetic. 

TiiU. 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. = lOO ff., 28 11. Cologne, 1556. 
Editions. There was no other edition. 

Ihis is a latin work, with occasional Greek passages, and was written 
f< if the cbssical schools of (lermany. Although assuming to be a prac- 
t:« al 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, 
i:ke other latin writers of his time, assigns the Arab-Hindu numerals to 
the Pythagoreans. 


Ed. pr. 1556. Antwerp, 1556. 

See p. 249- 

Titlt. See Fig. 138. 

Colophon. • Imprim^ en Anuers par Ian Loe //TAn de noftre 
Sc-:,.n»eur 1556 //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. 
1S4 ff. unnumb., 27-28 11. Antwerp, 1556. 

/editions. See p. 250. 

This is usually mentioned as the second of Menher*s arithmetics. It 
•k. however, the same as the work described on p. 249, 1550, save for a 
:«:- « minor changes. 





incronim4ue natmaSe^o- 





Maieftatis Confiliariiim. 


Apud hxredes Amoldi Birck- 
maiml 1556. 

Fig. 137. Title page of nabod 







Auec ffut 8c priuilege de l*£in- 
pcreur po ur qiucte ans. 

Fig. 138. Title pare of menher 


PIERRE FORCADEL. Ed. pr. 1556-57- Paris, 1556-57. 

Bom at B^ziers ; died at Paris in 1 574. He was ( 1 560) professor of mathe- 
matics in the College Royale, Paris. He also wrote on astronomy, and trans- 
lated the works of seversd 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 T Algebre.// Le tout nouuellement 
inuent^ par Tauteur.// (Woodcut with motto : In Pingvi Gallina.) 
A Paris,// Chez Guillaume Cauellat, k I'enfeigne de la//poulIe 
graffe, deuant le college de Cambray.// 1556.// Avec Privilege.' 

(F. 94.) 

The title page of Book III reads as follows : 

* Le //Troysieme Li-//vre de TArithmetiqve //de P. Forca- 
del de Beziers.// Avqvel sont traictees les de-//monftrations 
de toutes fortes de racines, auecques Tentiere pra-//ctique de 
I'extraction d'icelles, enfemble plufieurs queftions, reigles,//& 
demonftrations Mathematiques, auecques le propre fubiect de 
TAlgebre.// Le tout de Tinuention de rAutheur.//A Paris,,'.' 
Chez Guillaume Cauellat, a Tenfeigne de la // PouUe graffe, de- 
uant le college de Cambray.// 1557.// 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, I ID, 4+ 1 1 1, in the above three books); 25-31 U 
Paris 1556-57. 

Editions. Forcadel wrote four arithmetics, as follows : 

1. * L'arithmetiqve,' Paris, 1556-57, here described. 

2. * L'Arithm^tique par les gects,' Paris, 1558, 8*. 

3. * Arithmetiqve entiere et abregee,* Paris, 1565 (p. 316). 

4. * Arithm^tique demonftrde,' 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, O: 




Toicad^dcBczicfx. / §ss£ - ^•r ^ '«'< 

OTTi FAiidiaicticqaecoiiffilellfanoircogBoL 
|llitdis^iiitt,<mcanfieftf »prapret giiidoiif Je 
__ JcoitteccmipittadonMidieiiiidcqiic^ dc£ 

ian^ttugfx Kioo Icitf|N'opc'C otcovt* 


Fk;. 139. First page ok text of the 1556-57 forcadel 


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 Bulktino, 
II, 424. 

Other works of iss6, Baeza, 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* ; Gasp. Pauerus, • Logis- 
tice astronomica,' Wittenberg, 8® ; Joannes Pierius Valerianus, * Hiero- 
glyphica sive de Sacris Aeg3rptiorvm Uteris 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 Aritm^tica,' Mexico (the first arithmetic printed in 

ROBERT RECORDE. Ed. pr. 1557. London, 1557 

See p. 213. 

Title, See Fig. 140. 

Colophon. * Clmprinted 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 
Coflike 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 tbt 
f aied diametralle nomber.* The treatise on * Cossike nombers ' begins on 
f. S I, and *The rule of equation, commonly called Algebers Rnfe* 00 
f . Gg 4. The sign of equality, * a paire of paralleles, or Gemowe lines d 
one lengthe, thus :=, bicaufe noe .2. thynges, can be nooare equalk,' 
is found for the first time in print on f. Ff i (see Fig. 141). 



tftnafttootiK WlttCtfUffuMb, 


nt Doonkofaites Sdhfdfikijhmr 
limtf^mtlfiymr%i$hti$ "fAHH^ 


CflAeft SfidM arf ti bie IUte>tt 
Mie WftoMOjT or iMtito^ 

Fig. 140. Title page of recorde's Whctj'tonc of witte 



M tfidr fDo;tei Me ertenue ) to MSi'iKte (f omit Atta 
tUioo partem* atbemftbcfirfteto, ifbewmmmiirtt 

flMoatciliiaicngcontomiiebert ttatpoacfMci 
sonrnombem 9 totdeirleafie umomfnctfoiUt Mt 
flmlblb fii)iittijM'o}t soo pioce He snp fiutber* 

flno ogamyfrsont MMLifjM befoclicy ttttnpegf^ 
tdIeMioimnat(onCipi(r» bcfofneotoaiippaiteorf 
componti^nomberyfoit ftalltenntettro^ttatflpe 
nomberortbegteaMte Cgnealotiey maiefiamieis 

aiidtiiuiiiaUttatiuatetbtobetaagUe, amcevf 

fiolobeftyfti^eaflcaltenittt orejM»if«f.3| MUpjo^ 
ponnoe a fttue er<ple0,bfcaaft tlie ertnution ort|e<c 
roatci,matetbemo;ea|itfebeefD;ong|ite. anttsa' 
uo<de tbe teMouft repetitton of tiieTe ioooilM : iMv 
qnallc to : jfliiOl fette aasi ooe often in tDooAetife»a 
pa(ceorparaUeIeo,o;<IDemoiDe Uneo of one lengttt» 
tftttflt'^ y bfeanfe noe.2* tb;nge0,can be moaio 
evtalle* anonotomarbetbcftnombero. 

I. 1 4»sg,*-4— •! J*f ~^ i.f 

:• 2 o.jg^. ^•i & y ■ ^ 1 o2*f • 

?• 26.5-H— I o Jg^— 9.5*— I o:;^— H-2 1 ;.f; 

4. 1 9*2t~*~^ 92.f «— ri o§*- I l osy — 19;^ 

6. J45. i2Sg^-««4o2g^-+— 48of — 9.5* 

I. 2n tbc firftc tbcre appcamb. 2 • nombcri , tbat u 


Fio. 141. From recorders Whetftone of witte 


LUCAS LOSSIUS. 1557. Frankfort a. d. Oder, 1557. 

A German arithmetician of the sixteenth century, bom at Liineberg. 
Title. Sec Fig. 142. 






huiut vtiliisifnae ants, & Regub»quam 

vocant, Detn'tbremter oC per- 

fjpkai traduDtur* 


hem km teem €iU4» 






Fi<;. 143. Title pa(;e or lovsiis 

Deicriptiom. 8*, 9.2 X 14 cm., the text being 6.8 x 1 1.9 cm. 
4 tT. unnumb. + 27 numb. = 31 ff., 22-24 ••• Frankfort. For 
Jaie see f. 4, v.: ' Datae Lunebar, Anno 1557.// 7. Februarij.' 


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 Ariih- 
metica? Est ben^ & artificioi^ 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 J 3 57. 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 Linkn 
end Cyfferen na allerley Hantieringe,' Deventer (second edition 15771: 
Sixtus Medicus, *De Latinis numerorum notis,' Venice, 4* (colophon 
date, 1556; hardly an arithmetic). 


Ed. pr. 1558. Lyons, 1581. 

An Italian arithmetician of the sixteenth century, bom at Cnneo, in 

Title, See Fig. 143. 

Description, 4*, 15.3 X 21.7 cm., the text being 10.2 x 176 
cm. 136 pp. (60 on arithmetic), 35-37 11. The dedicator)' 
epistle is dated 1556. Bound with this is *I1 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 :: 
includes the * gelosia' multiplication of the early Venetian writers. The 
second book relates to fractions, the thutl to business operations^ and 
the fourth to roots. The work is in no sense a scholarly production. 

Fig. 143. Title page of peverone 


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. i 
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 of ijjS, 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 Ntimberg, 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. Bom in Dauphin^, c. 1485-1489; died in a 
cloister in 1560 or 1564. He belonged to the order of St. Anthony, ani 
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 ot 
position, the third of algebra, the fourth of arithmetical problems, acci 
the fifth of algebraic problems. The problems are not of practical 
value, and hence the arithmetic never attained any popularity. 

Other works of 1559* Albert, p. 180, 1534 ; Anianus, p. 32, 14SS 
Cataneo, p. 244, 1546; Gemma, p. 200, 1540; Mariani, p. 181, 1535 
Johann Fischer (Piscator), p. 247, 1549 ; Herman Giilfferich, • Ein ve^ 
Rechenbtichlin auff der Linien und Fedem' (perhaps the work men- 
tioned on p. 257, 1552); Pedro Juan Monz6 (Monzoni), ^Elementa 
Arithmeticae,' Valencia, 8*, and s. 1., 1566, 1569. 



Fig. 144. Title page of casanova 


B V T E O N I S 


& Aricbinetica vulg6 dicitur in Ii« 

bros quinque mgefta:quo> 

turn index fummatim 



^d loeHMVitruuif cvrm^tKm TestitmU^m^ 

dtfnftninse Ufidmn tnmttidiirminiibiiU^ 

firamtHtLibn Decm$» 


Im t<*». 


H ^ 

^ nlwl'8^ 


> /^ 



H 1^ 






Z ^ 



•" Oi 


''P CyLlZLMVM EoriLttrMi 



Ji. D. LiX. 

Cmm fmlU^toSfgiS' 

Fig. 145. 

Title page of buteo 


FRANCISCUS BAROCIUS. Ed. pr. 1560. Pavia, 1560. 

FRANCRiiC'u Barozzi. Bom at Venice, c. 1 538 ; died after 1 587. He wrote 
on cosmography, and edited ProcluA. 

TitU. *Francisci Barocii//patritii Vencti//opvscvlvm,//in 
quo vnoaOratio, & duaeQufftiones:// altera de certitudine, & altera 
//de medietate // Mathematicarum continentur.// Ad Reueren- 
(lisfimum Danielem Barbarum Patriarcham// Aquileienfcm defig- 
natum Virum Clariff.// Patavii, E. G. P.//M. D. LX/ (F. i. n) 

Description, 4^ 14. i x 19.6 cm., the text being 10. i 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 
v>me reference to the old Boethian arithmetic. No arithmetical pro- 
c esses are discussed. 

OLIVIERO FONDULI. Ed. pr. 1560. Bologna, 1560. 

A Bolognese teacher of the middle of the sixteenth century. 

TitU. See Fig. 146. 

Dcscriptiofi, 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. Septcmb. 1560.' 
B^ilogna, 1560. 

Editions, There was no other edition. 

11) is rare little handbook of commercial customs, from the press of 
I'c*legrino Bonardo of Bologna, is hardly an arithmetic, although it 
r\ plains certain arithmetical processes, and gives a considerable number 
of business problems. 

SIMON JACOB. Ed. pr. 1560. Frankfort, 1565. 

Bom at Coburg ; died at Frankfort am Main, June 24, 1 364. He was one 
of the best-known Rechenmeisters of his time. 

TitU. See Fig. 147. 

Co/iyfhon, 'Getruckt zu Franck-//furt am Main/ boy Gcorg 
Rabcn///in verlegung Sigmund Feyerabends/ // vnd Sim<m 
! I liters. Sigmund Feyerabent // Simon Hutter // M. D. LX V.' 
'F. 363. V.) 





vtOilsiine a ciaicheduna per(bna ) di mandore a 

memorialehreuelnaetioni &bdchace(bpn 

il Valutarclepefi, &ini{ure^ &alcrem 

ftrutdoni necceflarie da fapere . 

Ecattdiona^oadnce Muiaelic » Tafleili , & Copcrd^' 

TttMk> Fieno , at Legoe.-Coiiladecdiiantioiie^ae 

Excoipli loro come kgcndo intendcnu. 

Fig. 146. Title page of fonduu 



jmptUt^lSk^limftamff^ tor 

H^i^faiiMiCDKiiMiAlldilflfiHMMiiaitMMi/ tab Mftt 

Fic. 147. Title page of jacx)b 


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 Ziffem/ Frank- 
fort, 1557, 12^ 1574, 12^^ 1589; 1590; 1599. 

2. * Ein new vnd wolgegrundt Rechenbuch,' 1 560 ; 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, * Aritm^tica,' 
a rare work, s. 1. a., which appeared in Spain about this time. 

SIMON JACOB. Ed. pr. 1560. Frankfort, 1600. 

See p. 295. 

Titk. This is practically the same as in the 1565 edition 
described above. 

Colophon, *Getruckt zu Franckfurt am Mayn/ bey // Matthes 
Beckem/In Verlegung Chris-//ftian Egenolphs Erben.//Anno 
1600.' (F. 360, v.) 

Description, 4*, 15.5 X 19.2 cm., the text being 10.5x14^ 
cm. 1 1 ff . unnumb. + 349 numb. = 360 ff ., 28 11. Frankfort, 1600. 

See above. 

JOSEPHUS UNICORNUS. Ed. pr. 1561. Venice, 1561 

Giuseppe Unicorn o. A Bergomese arithmetician, bom in 1523; died in 

Title. See Fig. 148. 

Description. 8^ 9.4 X 13.9 cm., the text being 7.4 x 127 cm 

I f. unnumb. -h 78 numb. = 79 ff., 30 11. Venice, 1561. 





R V. M A Jl T I V «, 

VtfioantUdeLtfiarmdtd quJem fdtuperiiiaa* 
4* , 4imr4tme4igM^ cr bmaio tfid ' 
wtttfftru ptfsimr^trm. 

VtMITlIS, A^I>)MnK)M^NicoiMr, 
AffwDMWM, H O L X I. 

Fig. 148. Title pace of unic»rnts 


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. Numberg. 1561. 

A Numberg Rechenmeister, bom c. 1 520. 

Title, See Fig. 149. 

Colophon. 'Gedriickt zu Niimberg/ //durch Johann vom Berg/ 
//vnd Ulrich Newber.' (F. 156, r.) 

Descriptiofu 4**, 14.2 X 19.3 cm., the text being 10. i 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, soh^ 
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 %2y how much will 7 books cost? This method was ^^ery 
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. 


Ed. pr. 1561. Diisseldorf, 1561. 

A Gemtian 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. 




gcoMucbbetongr/mit fotibmn votteyl/vnb b^ 
N^fltett mifritMflc^ei:tey (trr praiffac^ 

vttbfonRmrnigi!Iid)en$tt miQtfi 


Fig. 149. Title page of werner 



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 O- 



fut4nii drttm trdicmitm UUm^qHtiMk 

num€torum,qu4mqu£ climcdculit^ 



crfm pofiHs^ 


JuSlore lo. Monhemm 


Aimo 1 5 <5 I- 

Fig. 150. Tftle 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 cxplanatk n 
of counter reckoning. 


BENEDICTUS HERBESTUS. Ed. pr. 1561. Cracow, 1577. 

Born at Novomiaati, 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). 

\ I^tin book intended for the Church schools, and revised, as the 
' lYoxmivm auctoris ' states, ' Mariaeburgi in Pnifsia, fub finem Anni 
homini 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.*), 
ar;<l one on the rule of three (• De tribvs nvmeris integris. Capvt VIIL'). 

ANDREAS HELMREICH. Ed. pr. 1561. Eislebcn. 1561. 

A Halle Rechenmeister of the latter half of the sixteenth century. The 
dedication of the 1561 edition is signed by * Andreas Hebnreich Rechen- 
meifter vnd Vifierer zu Halle/ and the 158S edition by * Andreas Helmreich 
von Kifzfeldt/ Notarius publicus/ Rechenmeifter vnd Vifierer zu Halle.* 

TitU, See Fig. 152. 

Colophon *Gedruckt zu Eisleben/ boy // Vrban Ganbifch.* 
<F. Ill, r.) 

Inscription, 4*, 14.2 X 19.3 cm., the text being 9.3 X 14.9 cm. 
I 1 1 ff . unnumb. + 1 blank ^ 1 12 ff., 27-33 11. Eisleben, 1561. 
Kditions, This is probably the first edition of Helmreich's arith- 
rictic, Che dedicatory epistle being dated 'zu Halle inSachffen den 
S»niag Oculi/ welcher war der 9. tag des Monats Marcij/nach 
C h ri f ti Ihefu vnfers lieben Herm vnd Seligmachers Geburt 1 56 1 .* 
M ij rhard (I, i S S) mentions, however, an edition appearing at I lalle 
m I 546, 4*. The book was republished at Leipzig in 1588 (p 
3<V>», and again in 1595, 4* (p. 306), and possibly ib., 1596, 4*. 

71m work is an unsuccessful attempt to combine the old and new 
^-:thnietic$. The author begins with a semi-Boethian treatment of 
•'^t.'X, assuming a knowledge of the fundamental operations with 
•' teeen, and then introduces a course in mercantile arithmetic. He 
, .-.'•^crs with a considerable amount of work on mensuration, ini hiding 




dot adiunda 





lESV Trejhyteroy 

cumfacultate Superiorum. 


In Officitta Mattbai Skheucy 
Cher : t^niio DONM 


Fig. 151. Title page of the 1577 herbestijs 



fcfcttacg^er©e(fc§m ^tat^a/tok frw 

fampt geipfff rr raft be()f nber <trt/nuuicttffry t>fnfr# 
rotten pn& 0ctndTr auff aIUOl7meniCi£fcb 
5a maftKn/jOfamen bracb^^of<b 



gauging (Visierrechnung). It is somewhat surprising that such a weari- 
some treatise ever went through three editions. 

Other works of ij6i, Albert, p. 178, 1534 ; Borghi, p. 16, 1484; 
Feliciano, p. 148, 1526; Gemma, p. 204, 1540; Recorde, p. 214, 
c. 1542 ; RudolfT, p. 152, 1526 ; Sfortunati, p. 177, 1534 ; Tagliente, 
p. 115, 15 15; Wolphius, p. 154, 1527; A. Citolini, * Tipocosmia,* 
Venice, 8° (a synopsis of science, incUiding arithmetic) ; Ognibene de 
Castellano, * Opera,' Venice, 4* (principally on geometry, but including 
some arithmetic ; see also p. 37511582); 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-/Aeit/ 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 // Cube, auflF alle Ohme vnd Eiche zu 
machen/ alles mit war-/Aafftigen perfpectiuifchen/ Georoctri- 
fchen vnd Arithmetifchen demon-//ftrationen, au£z den Bdchem 
Euclidis fundirt/mit fleifz//zufammen gebracht vnd in Druck 
aufzgangen///Durch// Andream Helmreich/Rechenmeiftcr x-nd 
Vifierer // zu Hall in Sachffen.// (Cut showing dimensions of a 
cask.) Imjahris88.' (P. 3.) 

Colophon, *Gedruckt zu Leipzig/ //Durch Abraham Lamberg: 
//Anno//M. D. LXXXVIIL' (P. 336.) 

Description, 4^ 1 5.5 X 19.2 cm., the text being 10. i X 14.5 cm. 
9 pp. blank -f 16 unnumb. + 317 numb. = 342 pp., 31 11. Leipzig, 

See p. 303. 

ANDREAS HELMREICH. Ed. pr. 1561. Leipzig. 1595. 

See p. 303. 

Title, * Rechenbuch/ // Erftlich/ Von // Vortheil vnnd Bchen- 
//digkeit/ nach der Welfchen Practica/ mit ih-//ren vnterfchied- 
lichen Proportionibus, Grund vnd vr-//fach der Regel Detri oder 
proportionum.//n. Von zubereitung mancherley Vifier Ruthen/ 


/ vnnd Schnfiren/damit alle V4hffer vnnd andere Corpora nach 

ilcm // Quadrat vnd Cubo auff alle Ohme vnd Eiche-Vifiere wer- 

ilcn.// III. Wie man kflnftlich das Feld vnd Erdrcich/ auflf man- 

, chcrley art/ mit gewiffcr MeOruthen vnd Schnuren/ nach eines 

jcdcn Landcs//oder Stadt gebrauch/ Geometrifcher vnd Jdio- 

iiU her weiiy vnd was hier-//innen der Vnterfcheid fey/ recht 

mclTcn foL// IHI. De Diftantijs Locorum, das ift/ Wie man 

\\ undcrbar-//Hcher Geographifcher vnd Cofmographifcher/ nach 

Geometrifcher weife/ durch // Arithmeticam, in der gantzen 

Wclt/zweyer Stcdte oder Wonung Diftantiam, oder//wie weit 

the vcm einander gelegen/ nach ihren Longitudinibus, vnd Lati- 

ludinibus, // fol Rechncn vnd finden/ Sampt einer LandtafTel 

(iaraus zu // machen vnd zu befchreiben.// V. Vnnd wie man die 

Funff Horologia communia, oder geV/meine Sonnen Vhrn/ als 

Horizontale, Meridionale, Septentrionale, Ori-//entale, vnd Occi- 

iicntale, auff einen Cubum, aufzwendig/ oder fonften an die Wen- 

de/ Mawer vnd ebenen/ Vnd das Fundamentum Horologiorum 

abrciffen. Item/ //der Sonnen vnd des Monds am Himmel ge- 

K hwinden Lauff rechnen. Vnd auch eine fonderliche gar kunft- 

. . hche Sonnen Vhr/der Cylinder genandt/ darinne alle Stunde 

nach der Sonnen fchein deff Tages grund-//lich zu erfehen vnd 

/ii be finden/ machen/ vnd fampt andem mehr fch6nen Kfinften 

\n«l VbunV/Rcn/ nfitzlich gebrauchen fol.// AUes mit warhaff- 

t:-:cn Demonftrationibus vnd Figuren/ aus den Biichem Euclidis 

\ nd andem fundirt. Durch // Andream Helmreich von Eil'feldt/ 

Kcchenmeiftervnd// Vifierer zu Halle in Sachfen an der Sahle. 

Cum Priuile^io.' (P. i.) 

L olophon. *Gedruckt zu Leipzig/durch Zacha-//riam Berwakl. 
Im lahr/ // M. D. XCV/ (R 645.) 

I^fscriptum. 4*, 14.5 x 18.7 cm., the text being 10.9 X 15.4 
< m 17 pp. unnumb. + 3 blank + 627 numb. =647 pp.. 26-34 11. 
I^-:pzig, 1595. 

Editions, See p. 303. 

'1 hi> edition contains the Rechenbuch of 1561, togithtr wuh much 
: !it:anal matter as summarized on the title i>agt% \\i, lHX)k!> III, IV 


(pp. 306-631). This includes a considerable amount of work on gaug- 
ing, surveying, cosmography, and the * Sphere.' The first 304 page^ 
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. * Vx2jz\Sq.'^I I Mercantile// Modema.// Di Guglielmo Pa- 
gini // Lucche£e.//In Lucca per il Bus-//dragho. MDLXII/ 

(P. 5.) 

Description. 4*, 14.5 x 20.6 cm., the text being 9.1 x 15.3 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 ^-ith 
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 piccoU/ or 
short division ; then the ' Modo di partir per ripiegho,' ' Modo di 
partire per colonna,' and finally the ' Modo di partire, k danda per 
altro nome partire grande.' Following division is the work in multipli- 
cation, including * regoletto,' * biricuocholo,' • crocetta,' • per col<Mina/ 
and * per ripiegho.' Subtraction or * traction ' (* Modo di trare, o ibttrare ' i 
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 !os> 
(* Gvadagni e perdite *), partnership (• Compagnie *), barter, interest aiwl 
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 

Bom in San Stefano (Santisteban del Puerto), in the Siena Mofena. tr. 
the first third of the sixteenth century. He studied at Alcala and Sala nv a n c^ 
He was canon at Granada in the latter half of the century. 

Title. See Fig. 153. 

Colophon. * En Salamanca.//Por Mathias GAli.//Aiio de 1 562/ 

(P. 765.) 




latiua delBachiller luan 
Perez dcMoya. 


Agon nneuamente coiregidt > y afiadulai 
por el mifino author nuchas colat,coii 
otiof dos libRM^y vna Tabia muy copio 
lade ht cofasmasooubles de codo lo 
^ae en efte Ubro fe conaenc; 

Va £riffda dnuiy akoy mm^ptim* 


Per Haihiat Gift* 

Fk:. 153. Title PAr.E of the 1562 mova 


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 matemiticas,' Alcali, 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 etymobgical 
meaning, adding the rest of the ancient title, * almucabala *), practical 
geometry (< Trata algvnas reglas de Geometria pratica neceflarias 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 ven- 
fully, giving the Greek, Roman, Hebrew, digital, and astrological systems 
together with a brief mention of other systems. Altogether it is ihe 
most noteworthy book on mathematics published in Spain in the 
sixteenth century. 


Ed. pr. 1562. Barcelona, 1705. 

See p. 308. 

Title. * Arithmetica//Practica,//y Especvlativa,// del Ba- 
chiller // Jvan Perez //de Moya.//Aora nvevamente corregi<ii. 
//y aftadidas por el mifmo Autor mu-//chas cofas.// Con otr - 
dos Libros,//y vna Tabla muy copiofa de las cofas mas //nota- 
bles de todo lo que en efte Libro//fe contiene.// Afto 170? 
Con Licencia: En Barcelona, en la Imprenta//de Rafael Fi- 
guer6.' (F. i, r.) 


Description, 8*, 15 X 20 cm., the text being 10.7 X 17.9 cm. 
3(>6 pp. (380 numb.), 39-41 11. Barcelona, 1703. 
Editions. See p. 310. 

See p. 310. It shoe's the popularity of this Spanish treatise that this 
ctiition should appear 141 years after the first one, and that another 
olition was published as late as 1761. 

.\\ONYMOUS. 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 
I m. 98 pp. numb., 24-30 11. Paris, 1562. 

Editions, There was no other edition. 

'l*he work is largely theoretical, but it contains a few applied prob- 
Irms. It was without merit. 

Other works 0/1^62. Benese, p. 182, 1536 ; Budaeus, p. 99, 1514 ; 
I ut lid, p. II, 1482; Gemma, p. 200, 1540; Ghaligai, p. 132, 1521 ; 
I.r,siu$, p. 290, 1557; Ramus, p. 330, 1569; Riese, p. 139, 1522; 
XyUnder, p. 356, 1577 ; Francesco Spinola, •De Intercalandi ratione 
I ifrrigenda, & tabellis quadratorum numerorum, it Pythagoreis disposi- Venice, 8". 


Kd. pr. 1563. I^ipzig. 1563- 

Striokl. Bom at Kaufbeum, December 26, 1524: died at |{eideU>erg. 
June 26, 15^)9. lie was profes5or of theology at Jena, I^ipzig. and Heidel- 
berg, and, aside from the work here mentioned, wrote entirely on that subject. 

litle. 5iec Fig. 155. 

Description, 8*, 9. 1 X 14.9 cm., the text being 6.3 X 1 1.8 cm. 
s 5 ff. unnumb. + i blank = 84 ff., 21-27 11. Leipzig, 1563. 

EJitions, There was no other edition. It seems by the pref- 
f. c that the work was written in 1551. See p. 249, 1549. For 
•'-.c date of the printing of this work, 1563, see f. 6, v., of 
\:.c preface. F. 10, v., gives the year in which it was written, 

'fhis comparatively unknown work presents the subject of arithmetic 
*r.*m the classical standpoint. The author speaks of the di^mity of 
ir.thmetic (' De dignitate arithmeticae '), and follows this discusMon by 



A R I T H M E- 

TI C A. 

» A B. I S I I S» 

Apud AndreatnNV^echelam. 

I 5 ^ a. 

Cum privilegio Regis. 

Fig. 154. Title pace of the anommocs 1562 AwnnKnic 



^ Arithmeticus 




Vi<^orino Strigelio. 


Lipfa, ^^<> 

/N o p r I C I N u lit UN ^. 

Fig. 155. Title pace of striuelius 



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, bom at Avignon c. 1525. 

Title, * L' Arithmetiqve // de Pierre Savonne,// diet Talon, 
natif d' Auignon // comt6 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 :// neceffaire pour tous Maistres de mon- 
noyes, Orfeures & // Changeurs, avec le fait & maniement des 
Changes & Ban-//ques qui fe font iournellement 4 Lyon, & par 
les places ac-//coustumees : comme Flandres, Angleterre, Hef- 
pagne,// Italic, & autres lieux.// A Lyon, par Benoist Rigavd. 
//M. D. LXXL' (F. I, r.) 

Colophon. * A Lyon, de rimprimerie 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, S* 
(here described) ; ib., 1585; ib., 1588,8*. 

This is one of the early French commercial arithmetics, weU 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. Baeda, p. 131, 152 1 ; Boissiere, p. 262, 1554 : 
Feliciano, p. 148, 1526; Gemma, p. 205, 1540; Moya, p. 310, 1562; 
Ortega, p. 93, 151 2; Peletier, p. 245, 1549; Wolifgang Hobel, • Ein 
niitzlich Rechenbuchlein mit viel fchonen Regeln und Fragftiickcn,* 
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". 


COSIMO BARTOLI. Ed. pr. 1564. Venice, 1589. 

A Florentine geometer, bom in 1503; died in 1572. He also translated the 
works of Ftnaeus (see pp. 160, 164). 

Title. • Cosimo Bartoli //gentir hvomo, et // Accadcmico Fio- 
rcnlino,// Del Modo di Misvrare // le diftantie, le fuperficie, i 
corpi, le //piante, le prouincie, le profpcttiue, & // tutte le altre 
c<»fe terrene, che po(To-//no occorrere a gli huomini,// Secondo le 
vere rcgole d* Euclide, & de gli altri//piu lodato fcrittori.// (Elab- 
orate woodcut.) In Venetia, Per Francefco Francefchi Sancfe. 
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. 1 564, 4*; ib., 1 589, 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 * lihro sesto ' (f. 
130, r.) is upon square and cul)e root. The galley method is used, and 
the common sixteenth -century device of annexing 2 n ciphers in s<}uare 
nK>t and di\iding the root by 10" (and similarly for cube root) is cm- 
pl<;yed. Bartoli also gives a table of s<)uares to 663^ llie chapter on 
r«>ots is followed by one on the * Regola delle tre cofe, ouero quattro 

Works of 1564. Albert, p. 180, 1534; Apiamis, p. 155, 1527; 
Bcne^e, p. 182, 1536; Euclid, p. 240, 1545; Gutierrez de Gualda, 
p. 167, 1 53 1 ; Herbestus, p. 303, 1561 ; Kolx^l, p. iii, 1514 ; Mariani, 
p. 181, 1535 •» Mwllerus, p. 223, 1543 i Riese, p. 139, 1522; Sole, 
y>. 146, 1526; Tagliente, p. 115, 15 15; lliierfelder, p. 391, 1587; 
I'iman, p. 391, 1587 (Thierfelder) ; Yciar, p. 249, 1549; Barlaamo, 

* Arithmetica demonstratio eorum (]uae in secundo lihro Klementorum 
^F'lirlidis) sunt,' Strasburg (see also p. 343, 1572) ; (iio\-anni Camilla, 

• Knthosiasmo/ Venice, 8* (containing a little work on arithmetic); 
>f antje! Femindez Lagasa, ' Lihro de (]uentas/ Salamanca, 4^ with a 
l:ttlc work on arithmetic ; Petrus Nonius (NuiWz), ' I.ibro de Algebra 
^n Arithmetica y (ieomctrla,* with two editions at Antwerp in 1567, 8" ; 
the * Opera' of Nonius appeared at Basel in 1592. 


PIERRE FORCADEL. Ed. pr. 1565. Paris, 1565. 

See p. 284. 

Title. See Fig. 1 56. 

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. Hb bst 
includes the names of Feliciano, Faber Stapulensis, Buteo» ScheabeU 
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 kist 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 follovring title : * Compendio 





Cha Chades Perier, cue S Jen 4e BcuMfa^ 
m BcUaDohoo* 

I 5 tf J. 

Fk;. 156. TrixE pace of the 1565 kokcadel 


y breue // inftruction 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 


ca por Antich Rocha 

de Gcrona compucfta , y dc varios 
\u£(;ores recopilada:prduedu 
para todos eftados de gentes. 

Vaanadido vn Compendio^para ttner y ro. 
gir los libros de Cuentaitraduzido de len. ' 
g«a Fiancda en Romance CaflcUano. 


En cafa de Claudio Bomat, a la AgoiU fiicnc 

t i 6 s. 

Con priuilegiopor diez aiios. 

Fig. 157. Title page of rocha 

Claudio Bomat//al Aguila fuerte. 1565.// Con priuilegio por di^' 
afios.' (F. 286.) This relates, as the title suggests, entirely to bookkeei>- 
ing, and is one of the earliest treatises upon the subject in the Span:« 


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 Aritm^tica by Juan Ventallol, of which we do 
not know the date, but which must have appeared before 1565 (p. 29H). 

ERHART HELM. Ed. pr. 1565. Frankfort, 1592. 

A Frankfort arithmetician and gauge r of the middle of the sixteenth 

Title. •1592. Erhart Helm/ // Mathematicus // zu Franck- 
furt am Mayn/ // von // Geomctrifcher Abmcdung der Erden.// 
Item :// Kunftliche Vificr vnnd // Wechfelruthen/aul^ dem Qua- 
drat/ //durch die Arithmeticam vnnd Geometri-//am/gerecht 
/umachcn/fampt einer luftigen behen-//digkeit in Weinrech- 
niing/ //alles durch obgedachten // Authorem bcfchrieben/ vnd 
jctzt von neuwem // widerumb fleiflig 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 II. Bound with Adam Ricse's arithmetic of 1592 
q). 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 sei^arate edition before this one of 1592. 

Iliis is a brief treatise on mensuration, and in particular on gauging, 
S :t it contains some explanation of arithmetical processes, including 
:r.e extraction of roots, and a table of square roots to the etiuivalent of 
thrive decimal places. 

Othrr works 0/1^65. Agrip|>a, p. 167, 1531 ; Apianus, p. 62, 1496 
\ »rdaniis) ; Belli, p. 343, 1573 ; Cusa, p. 43, c. 1490 ; IXMfino, p. 275, 
1556 ; Fischer (I*iscator), p. 247, 1549 ; (lemma, p. 200, 1540 ; Hobel, 
p- 314. '563; Jacob, p. 295, 1560; Jordanus, p. 62, 1496; Menher, 
p. 249, 1550 ; PadoNanius, p. 389, 1587 ; Riese, p. 142, 1522 ; Savonne, 
p. .;i4* 1563; Priscian, Rhemnius, Fanius, Ba^da, Mctianus, M.iberde 
'■. inimis, ponderibus, mensuris, numeris, eorumque notis, et de wtere 
< .jmputandi ratione, ab Elia Vinelo emendati,* Paris, 8**. (Priscian's 
• I »e figuris et nominibus numerorum * had already api)eared in his 
■*«>rk* published at Venice in 1470, fol., with later olitions in 1488, 
UO^t >495» MQ^t »5<9. 1525. Such books, of which Mr. Plimpton 
h iH ^\*eral, are not generally included in this list.) 


HIERONYMUS MUNYOS. Ed. pr. 1566. Valencia, 1566. 

Bom 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 Mufioz. 

Title. See Fig. 158. 

Description. 4**, 15 X 20.4 cm., the text being 9.3 x 14.6 cm. 
4 ff. unnumb. -f 77 numb, -f i blank = 82 ff., 27-31 11. Valencia, 

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, bom c. 1 525. 

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 & augmentde pour la 
quatri^me edition,// de plufieurs regies & articles,// par TAuthcur. 
// A Lyon,// par Michel love,// et lean Pillehotte.// k Tenfeignc 
du Iefus.//i578.//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, -f 5 unnumb. = 380 pp., 30-31 11. Lyons, 1578 

Editions. Lyons, 1566; ib., 1571 ; ib., iS78,8*(heredescribed). 
The dedication of the 1578 edition bears the date • De Lyon ce 
9. de luillet 1 571,* 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, i6io, 1632, and 1643 

Trenchant was one of the best of the sixteenth-century textbook- 
makers of commercial arithmetic in France. His work is divided in\o 






Machcmacicas facultatct ncaflariaB; 
4f§ V4leniin$fuUic$ 



Ex CTpompbia loannii Mcjr. 

Fig. 158. Title page of munyos 


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 bom 
at MonopoU, a town in the province of Bari, on the Adriatic. He «a$ a 
priest, and wrote only the work here described. 

• Title, See Fig. 159. 

Colophon, * In Napoli // AprefTo Mattio Cancer. M.D.LXIX. 
//Con Priuilegio per anni diece.//Marius Carrafa Archiepifcopu> 
Neapolitanus.' (P. 262.) 

Description, 4**, 14. i X 21 cm., the text being 10.8 X 16.9 cm 
13 pp. unnumb. -f 250 numb. = 263 pp., 29 11. Naples, 1569. 

Editions. Naples, 1566, I^tin edition with the title 'Defam: 
liarite arithmeticae et geometriae,' this date appearing also in the 
privilege, * Datum Neapoli die vltimo lulii M.D.LXVI ; ' ib., 1 566 
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 st^'ic. 
Lapazzaia begins with the fundamental operations with integers, v^ 
then treats of ratio, fractions, and progressions. He considers also */- 
rule of three, the rule of fw^y interest, exchange, partnership, alligat ' 
rule of false, and the extraction of roots. The last part of the work -- 
on mensuration. 

Other works of 1566. Belli, p. 343, 1573 ; Fischer (Piscator), p- 2-* 
1549 ; Gemma, p. 200, 1540; Herbestus, p. 303, 1561 ; Monr<$, p.2-- 
1559 ; Georg Meyer, * Rechenbiichlein dels Silberkauffs und gemacb: 
Arbeit,* Augsburg, 16°; Mathaus Nessen (Nesse), *Zwei neue RecK' 
biicher,* Breslau, 8°; Johannes de Segura, ' Mathematicae quae*:-' 
selectae propositiones,' AlcalA, 4**, and * Compendium Arithmciic^ • 
Geographiae partis,' Alcala, 4®. 





Fig. 159. Title page of the 1569 lapazzaia 


GEORGIO LAPAZZAIA. Ed. pr. 1566. Naples, 1590 

See p. 322. 

Title, ' Opera // terza // de Aritmetica // et Geometria./; 
Deirabbate Georgio Lapazaia//da Monopoli.//IntitoIatail 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. -t- 1 blank -h 1 74 numb. = 1 76 pp., 29 11. Naples, 

Editiofts, 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, // delFAbbate 
Giorgio Lapazzaia // Canonico Monopolitano, e Protonotano 
Apoftolico.// Nouamente in queft'vltima imprefsione efpurgato 
da molti errori, & arric-//chite d'vna Prattica d'Abbaco, non 
meno vtilifsima, che neceffaria.//Al Signor Diego d'Aldana, 
Prefidente della Regia Camera della Summ. per Sua Maiefta. 
(Large woodcut, coat of arms.) In Napoli,// AprefTo Tarquini^ 
Longo. MDCI.' (P. i.) 

Colophon, *In Napoli,// Apreffo Tarquinio Longo. MDCr 
(P. 215.) 

Description, 4", 14.9 x 20 cm., the text being 10.7 X 15.8 irr. 
216 pp. (3 blank, 8 unnumb.), 30-38 11. Naples, 1601. 

See p. 322. 


NICOLAUS PETRI. Ed. pr. 1567. Amsterdam, 1635. 

Bom at I)e venter. He taught at Amsterdam from 1567 to 1588. He abo 
« rote on algebra and astronomy. 

Title, 'Practicque/ZOm te leeren//Reeckenen/Cypheren// 
cndc Boeckhouwen/ met die regel Cofs/ //en^^ Geometrie/ fecr 
profijtelijcken voor alien // koop-luyden. Van nieus gecorrigeert 
//cnde vermeerdert///Deur Nicolaum Petri Daventrienfem.// 
L'homme propofe, Et dieu dispofe. A° 1603.// (Woodcut of 
author.) t*Amstclrcdam,//Voor Hendrick Laurentfz. Boeckver- 
ox>pcr 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 • 1 596. // loumael-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 
trii'lry form of division, and its chief value to the student of history lies 
in it^ business problems. 

Other works 0/1567. Agrippa, p. 167, 1531 ; Borghi, p. 16, 1484 ; 
(^ataneo, p. 244, 1546 ; Gemma, p. 205, 1540 ; Mariani, p. 181, 1567 ; 
.Nonius (Nuftez), p. 315, 1564; Ramus, p. 263, 1555; Taglienle, 
V- I iSt 15' 5 ; Conradus Dasypodius, *Logistica,* Strasburg, 8^; Guill. 
<ie 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, '' 
Nf . D. LXVIII.' (F. 34, r.) 

T>rscription. 4*, 1 3.5 X 19.5 cm., the text being 10. 1 x 16.5 
m. 1 1 flF, unnumb. -f 33 numb. =44 ff., 37 11. Brescia. 1568. 
Hdiitions. There was no other edition. 

'ITic dedicatory epistle shows that this work was written at Brescia 
.-, < K tober, 1568. It is composed almost entirely of tables for the use 








s^^^ >^.-^-^^ J^J^.' ^..^ .j^. .crszz: 

T A V O L E 


A K I T II M E- 


Cora iequali ckfcun^ a^euatifiifn^- 




Fig. i6o. Title page of ghebeuno 


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 1 587. 

Title, *The Well fpring of // Sciences.// Which teacheth the 
perfect // worke and practife of Arith-//nieticke, both in whole 
NumV/bers and Fractions : fet // forthe 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, 1 568, 8** (written in 1 562, 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 
Recorders * 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 
* 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 iame.' He complains of the criticism of foreigners that 
English arithmetic is not as advanced as that on the continent : * For 
w'hen I perceyued the importunitie of certayne Araungers 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, 


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 lame effecte 
they haue of late paynted the comers 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 
Pailime.' 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. i74» 
1534; Mauritius Steinmetz, 'Aritmeticae praecepta in quaeftiooes 
raiacta,* lieipzig, sm. 8®. 

HUMPHREY BAKER. Ed. pr. 1568. London. 1659 

See p. 327. 

Title. * The // Wel-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.) 


Description. 8°, 8.4 X 13.6 cm., the text being 6.3 X 1 1.2 cm. 
366 pp. (54 unnumb.), 26-27 11. 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 Impreifion of 1659, the whole Book hath been revifed, every 
Queftion therein examined, the Faults that were committed in former 
Impreilions, 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 
difcemed) 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, i68§.//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 -f 10 unnumb. -f 228 numb. = 246 pp., 32-39 11. Lon- 
don, 1687. 


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-//nieticae 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** (Stegerus edition) ; ib., 1 592, 8*" (Schonerus) ; ib., 1 596, 8* (the 
Snellius and Schonerus * Explicationes,' p. 333); ib., 1599, M 
(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 Arithmctickc 
in whole numbers and fractions ... by P. Ramus . . . translated 
... by William Kempe,* London, 1 592, 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. 


PETRUS RAMUS. Ed. pr. 1569. Paris, 1577. 

Sec p. 263. 

TitU. • Petri Rami // Professoris Regii,// Arithmeticae // li- 
bri dvo.// Parisiis,// Apud Dionyfium 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 II. Paris, 1577. 

See p. 330. 

PETRUS RAMUS. Ed. pr. 1569. Basel, 1580. 

See p. 263. 

TitU. * P. Rami // Arithmeticae // libro dvo // Geometriae // 
scptem et viginti.// (Woodcut.) Basileae, per Evsebivm // Epifco- 
pium, & Nicolai fratris haBredes.//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 11- Basel, 1580. 

See p. 330. 


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., 26-32 11. Frankfort, 

Bound with this is * P. Rami, Regii // Eloquentiae et // Philo- 
S4 3phiae Pro-//fefforis, liber de moribus//veterum Gallorum,// ad 
. '. ' Carolum Lotharingum // Cardinalem.// Parisiis,// Apud An- 
fiream Wechelum.// 1562.// 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. 






toddcmtaLAZARo Schonb&o 

Ev^dem Schok eki libri tb$0: ^tr, Di 
7(smerisfgirMi$i alter » De Lofffiicd 

Apud heredes Andreae Wcchcli, 


Fig. i6i. Title page of the 1586 ramus 


The first of these works is one of several commentaries on the theo- 
rt'tical arithmetic of Ramus (p. 263). It is more practical than that of 
SncUius (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 * I^azari Schoneri De logistica 

scxagenaria liber.' llie former is, as the title suggests, a treatise on 

the Oreek theory of numbers, and the second is on the sexagesimal 

fractions used by the astronomers. Schonerus writes his sexagesimals 

. llae lae o 1 H . ^ , , , . 20 ,40 

thus: for 3-6o* + 39'6o -f 40 -f 7- -f 7^. 

3. 39. 40. 20. 40., ^ ^7 ^ ^^2 

Ihi.s is one of the early approaches to our symbols ^ ', ". 


Ed. pr. 1569. Frankfort, 1596. 

See p. 263. Snrlmi's, bom at Oudewater, October 8, 1546; died at 
l^eyden, March 3, 1613. 

7i//<-. 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 
Gcome-Z/triam Praele-Z/ctiones,* and * Rudolph! // Snellii in 
Sphxram Cor-//nelii Valerii // pnelectioncs,' both of 1596. 

Editions, See p. 330. The first edition of Snell's commentar)-. 

I ike the arithmetic of Ramus, this work is theoretical rather than 
Y^M ti< al. Only the prominence of Ramus could have justified such 
rti »rts as these of Snellius, Salignacus, and Urstisius. 

I'KTRUS RAMUS. Ed. pr. 1569. Frankfort, 1599. 

See p. 263. 

TitU, ' Petri Rami // Arithmeticae // Hbri dvo: Geometriae // 
>cptcm et viginti.// A Lazaro Schonero recogniti & aucti.// Fran- 
t oivrti.// Apud Andrea Wecheli heredes.// Claudium Mamium, 
& I'lannem Aubrium.// M. D. XCIX.' (P. 3.) 

Dtscription. Fol., 17 x 22,$ cm., the te.xt being 11.9X 17.6 
« • 1 244 pp. in the arithmetic. 184 pp. in the geometry, 39 II. 
Frankfort, 1 599. Bound with this is the geometrj' of Ramus. 






Explicationes le^ifimae: 



inpenfiihcndom Petri Fi&hctL 


Fig. 162. Title page of the 1596 ramus and snelui's 


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 humanae//M. D. LXIX.' (P. 190 
of the geometry, bound with the above, or 534 of the entire book.) 

Description, 4^ 17.5 X 23.3 cm., the text being 12.3 x 18.2 
cm. 16 pp. unnumb. -f 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. 


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 1 1.5 x 16 cm. 
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 


Af«»»r i^rmm f^tupm^ S<»» ^fms Mi^A^m^ 





Fig. 163. Title page of the Ltdri vnvs et trigtnta of R.011? 



y tratantes difodidoty decenmiuidos,por 

dPadre Prefentado Fray Thomas 



Conlicenday priuilegio reaL 

EfUtafladoea cineofcalei. 
Fi<;. 164. 'IrriK pack ok mkkcaimi 


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 

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; Monz6, p. 292, 1559; Urstisius, p. 361, 1579; Jacob Frey, 

* Exempelbiichlein allerley Kaufmannshandel,' Niimberg (there was also 
an Augsburg edition of 1 603, 16®) ; Adriaen van der Gucht, * Cyferbouck,' 
Bruges, 4® ; James Peele, * The pathewaye to perfectnes in th' accomptes 
of debitour and creditour,' London, fol. (second edition). 


Ed. pr. 1570. Basel, 1570. 

See p. 193. 

Title. See Fig. 165. 

Colophon, * BasileaB,//ex oflicina Henricpetrina, Anno//Salv- 
tis M. D. LXX. Mense // Martio.' (P. in 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, algebraicae logifticae fuae . . .,* I20pp. 
(Ill numb.), 41 11. Basel, 1570. 

Editions, There was no other separate edition of the * Opus 

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 proportioa 
less geometric than that given in Euclid. See also p. 193, 1539. 

Other works 0/1570. Belli, p. 343, 1573; Boethius, p. 27, 14SS; 
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 arithm^tique fort facile k comprendre • ; Johann Weber, 

* Gerechnet Rechenbiichlein auf Erfurtischen Wein- und Tranks-Kaiift. 
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, i555 • 





Mjiihcmatici clorifsfmi, 





fiinirurandafum,iion foliim Gcomctrico more ftabilicum/cd etiain 

iianjs cxpnrimcnds & obfcruationibusrcnini innacura.folcru 

dcnion(lradoncinu{faranim,ad muldplicrs ufus ac# 

coininodaniin,& in V libros digeAuin. 




dC iocxhaiilliit plane codos Anchmetf cat ibcfauiuS|ib 

audiore recent niu!ds ia lodt rccogoi' 


1 T n M. 


lopSkksr rujr,numcTOS rrcondica nunsenndi fubttli utr,firaindum Gco^ 

mntiQ% quanricatcf /llqlltr(nri^• , ncciEiru Ccwoni*, 

nunc dciuuni in luccin c4iu. 


CumCarf:Maicft.Gratia& Priuilcgio. 

B A S I L 13 ;C. 



Trenchant, p. 320, 1566; Anon3anous, * Les principaux fondemecs 
d'arithm^tique * ; Nicolaus Eschenburg, * Arithmetica logiftica,' Frank- 
fort ; Alex. Vandenbussche, * Arithm^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 flF. unnumb. + 23 numb. = 26 ff., 17-36 11. Venice, 1572. 

Edition. There was no other edition. A German translator 
\vas published in Leipzig in 16 16. 

This is an attempt to popularize the mediaeval number game c^ 
Rithmomachia (Rithmimachia, Rythmomachia), set forth in I^tin pc'^- 
sibly by Shirwode or by Faber Stapulensis, in an edition of Boethius ji 
1496 (see p. 63), and afterwards amplified by Claude Boissiere (s^t 
p. 271, 1556). The game was often, with no authority, attributed ii> 
Px-thagoras. Barocius (or Barozzi) amplified the treatment attribute: 
to Faber Stapulensis, and his discussion of the subject is clearer tha?. 
that of the latter, although hardly equal to that of Boissiere already 
described. He had already published a philosophical discussion ct 
arithmetic as stated on p. 295. 


Ed. pr. 1572. London, 1579 

Leonard Digges came of an ancient family whose scat was Digges Courr. 
Barham, Kent. He studied at Oxford, and was an expert mathematician u 
the time. He died c. 1571. 

Thomas was a son of Leonard, and was bom in Kent. He was educate »I 
at Oxford, and died in London, August 24, 1595. 

Title. See Fig. 167. 

Colophon. * Imprinted at Lon-//don, by Henrie Bynnenur. 
dwel-//^ing 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 ^^* 
16 pp. unnumb. -f 191 numb. = 207 pp., and one plan (p. 17^ • 
35 11. London, 1579. 

Editions. LQndon, 1572, 4*; ib., 1579, 4* (here describcti : 
ib., 1585, 4''; ib., 1590, 4*. The 1579 edition was a ren^ion. 







Mknudtt fer vtiiiti,6'fiU^ dettiStitMt^ 

Etal preiente per Franccfco Barozzi Gentil'huofflo 

Veaetiano in lingua volgare in mododj 

Paraphraii compofto. 

jiffreffoCratiofo Tfrchacim* > 57^ ' 

Fu;. 1 66. Title pa(;e ok barozzi 



c An Arithmedcall Milicarc Treati/e,named 

Compendioujly teachingthe Science of U\^ubers, 

as well in Fracflions as Integers, and fo much ofdie Ru- 
les and i£quations Algcbraicall and Arte of Numbers 
Cofsicall^as are requifice for die Profefsion of a Soldiour^ 

Together with the Modcme Militart Di(apUne^Officcs,Lawc$ and 

Ducties in eucr}' wxl couemcd Giuipc aiid Annie to be obfcnicd : 

LongGnceatteptedbyLBo N ARO D i o c b s Genrlcnun, 

Att§nientc(l,€ligeftc€l, and lately (inifhcd, by 

Thoua» DicGBs,hu Sonne. 

^y her if he bath nlfi dMeptedcen^ve QuelHent ef great OrdhutiKe^ 

refrluediPi his eeher Tre^tz^ of Py lycccbny andffresit 

ArttUerie^ hereafter tehee fuhbjhed. 

VivBT Post Fvnbra Virtvs. 


Printed by Hemic By nnemaik 

Fig. 167. Title page of the 1579 digges 


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 
[>e Morgan includes, and which is in Mr. Plimpton's library, but which 
1 have omitted because it is in no sense an arithmetic. 

Other works of I§J2. Buckley, p. 253, 1550 ; Gemma, p. 300, 1540 ; 
Grammateus, p. 1 23, 1 5 1 8 ; Mariani, p. 1 8 1 , 1 535 ; Barlaamo, ' Aoyurrun;, 
sive arithmetiae, algebraicae 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. 

Bom at Vincenza. He died in 1575. He was an architect at Rome and 
Fenrarm, and wrote on practical geometry. 

Title. See Fig. 168. 

Description. 4*, 15 X 20.6 cm., the text being 9X 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 Profx^rtione * in 1 595 (Venice) 
to form the ' Quattro libri geometric!. ' 

This work is included in this list because of the treatment of medix\'al 
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 
i:xt\. 8 pp. unnumb +93 numb.+ i blank =s 102 pp., 50 11. Venice, 




V I C E N T I N O 
T> E L,l, jd 


Communi Paflioni del Quanto . 


Zftilif (^ necejfarij alia <vera , f^ fuiU inteUigenUa^ 

deW Arithmetical, della Geometric, O* di 

tutte lefiientie ^ arti, 

Al Magnanimo Aleffandro Farncfo Card. 

(^0 N n>KiyiLEGlO. 

In WtviriAfjIfprelfoFrancefiodetrMctfibiSMefu. if7}- 

Fig. 1 68. Title page of belli 










MtrsDBM ruMt^AMra* AtcrtoirrM ttamm rnrs 










Fig. 169. Title pace of pakiis 


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 cont^ns 
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 1S.5 
cm. 144 pp. (127 numb.), 38 11. Venice, 1573. 

See above. 

VALENTIN MENHER. Ed. pr. 1573. Antwerp, IS73 

See p. 249. 

Edited by Michiel Cognet, born c. 1 549 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 11. 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 lime. 
It shows, better than most works of the kind, the state of commerce ic 
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 Netherkods 
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 of 1573. Kobel, p. 102, 15 14 ; Moya, p. 310, 156^: 
Recorde, p. 214, c. 1542; Peter Beausard, * Arithmetices jMraxis,' 
Louvain, 8® ; Simon Kopfer, * Grundbiichlein der Regel Dctri,* NCm- 
berg ; Bartolomeo Piccini, * Trattato de' Cambi,' Florence, 4*. 


Horks of 1SJ4. Anonymous, p. 244, 1546; Baker, p. 327, 1568; 
Buckley, p. 252, c. 1550; Jacob, p. 298, 1560; Riese, p. 139, 1522; 
Kudolff, p. 152, 1526 ; Seton (see Buckley, p. 252, c. 1550) ; Lorenzo 


contcnanc pliificurs belles que(lions&dc« 
mandes^propres & vtiles a tous ccux tjui 

hantencla Trafiquede Marchandifc. 
Cwipofiparftu ydkiUin^ennh€rutil€mdnd: 
rciuu,corrig(l,& aumentc mftuficurs endrMS 

par Michiel Cognec* 


Fne dmfledi'cUmiionfurUfMdcs Changes. 


Vnum difctmrs di bien erdmSmem^fomter^mic 
L Sotutiimjur Suerfis ofmonsy frofojtes. 


La Solution des queftions Matfaemafiques 

par la (uppiuationde Sinus^illuftrees &am« 

plificcs paries demonftradons Geome* 

criques neceflaires i icellcs. 



CUi^ ItM xvMsiergieydhfaideFiMdres, 


Fic. 170. Title page of the 1573 mekher 

IV>fiocchio, • Breve et universale risolutione d*aritmetica,' Brest ia (Ven- 
H e ?>• 4* ; ib., 1 597 ; Milles de Norry, • Arithm^ti<]ue,* Paris, 4* ; johann 
>t:kt:t:Twiu, • Rerhenbiichlein auff allerley Handthierung/ Breslau. 


DIOPHANTUS. Ed. pr. 1575. Toulouse, 1670. 

A Greek mathematician, c. 300 a.d. He was the first great writer upon 

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 // Acceffit Doctrinae Analyticae inucntum 
nouum, collectum // ex varijs eiufdem D. de Fermat Epiftolis.// 
(Engraving ' Rabault Facit/ with motto : * Obloqvitvr nvmeris 
septem discrimina vocvm.*) Tolosae, // Excudebat Bemardvs 
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 4- 6 unnumb. + 341 numb. + 48 of notes = 400 
pp., 50-55 11. Toulouse, 1670. 

Editions. Basel, 1575. There was no other sixteenth-centun 

Athough entitled an arithmetic this is really a treatise on algelva, 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,' * Uber 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 publishtd. 


Ed. pr. 1575. Venice, 1575 

Francesco Maurolico. Bom at Messina, September i6> 1494: died nejr 
there, July 21, 1575. He entered the priesthood and later became profe>>oT 
of mathematics at Messina. He wrote chiefly on astronomy, and editcti 
several works of the Greek mathematicians. 

Tit/e. See Fig. 171. 

Colophon. *In monafterio S. Maria //i parte 19. lulij die,. 
5. II. Indictionisy/i553.* (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. 




Opufciila Mathemattca \ 

7^(wtc frimitm in lucem itdim , mm rerum omnium 
mmtu dtgnarnm^. 


coram Cataloguseft. 


Vcnedjs, Apud Fnaciicum Francifcium Senenfcm . 
M D L X X V. 

Kh;. 171. Title page ok the Opuscula of MAikoLvas 


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) ^ 
* Compvtvs ecclesiasticvs in svmmam coUectvs.* 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). 


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. Ciim // Meffanas 
cum multo pontis & arcus // apparatu expectaretur lo. Cerda,// 
Methynenfium Dux,//Prorex. Indict. 15.//M. D. LVII.// Venc- 
tiis, M D LXXV.//Apud Francifcum Francifcium Senenfem.* 
(P. 183.) 

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 oi 
numbers. It was one of the last of the extensive sixteenth-centur>- 
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 BRUCiEUS. Ed. pr. 1575. Rostock, 1575 

Bom at Alost, Flanders, c. 1531 ; died at Rostock, December 31, 150^ 
He was professor of mathematics at Rome, and later professor of medkire 
at Rostock. 

Title. See Fig. 173. 

Description. 8*, 9.5 X 15.1 cm., the text being 6.5 X 1 1.6 cm. 

76 flf. (I blank), 25 11. Rostock, 1575. 



Mathematici celebcrrimi . 



(^um nrum omnium mmhilmmj . 

Vencd)s> Apud FrancifimmFiandfcium Senenfem. 
M D LX X V» 

Fig. 172. Title page of the Libri duo of maurolycus 


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 georoetn', 






axcvoiBAT lAcosrt tvcava 


Fig. 173. Title page of BRUCiCUS 

and some attention being given to surd numbers, roots, and the nik of 

Other works of isjs. Gemma, p. 206, 1540; Kobel, p. 102, iSM^ 
I^pazzaia, p. 322, 1566; Mariani, p. 181, 1535; Salignac, p. 359' 


1577: Xylander, p. 356, 1577: Mauricius Steinmetz Gersbach, * Arith- 
nuti< cs praecepta/ l^ipzig, 8*. 

Works of 1576, Gemma, p. 200, 1540; Petri, p. 325, 1567 ; Ta- 
j;iicnle, p. 115, 15 15 ; Joseph I^nge, * Arithmctica,* Co|)enhagcn, 8". 


Ed. pr. 1577. Czemy, 1577. 

A l*o)Uh Rechenroeister of the latter half of the sixteenth centur}*. 

Title, Sec Fig. 174. 

Colophon, *Wytin"lenow // Starem Heftc Pra5 kim// v Girijka 
gAemcho.//Lctha Pane///M. D. LXXVII/ (F. XC, r.) 

Description, 8^ 9.5 X 15.9 cm., the text being 7.2 X 1 1.7 cm. 
9 a imnumb. + 89 numb. (Roman) = 98 ff., 21 11. Czerny, 1577. 

Editions, There was no other edition. 

This is one of the few arithmetics in the Polish language published 
I'l the sixteenth century, and is very rare, 'inhere was a copy in the 
K inc om|>agni sale, but the book is seldom mentioned by bibliographers. 
\\ consists of ii\'e parts, the first dealing chiefly with the fundamental 
«*'f rations with counters, the second with written operations, the 
t^ni with fractions, the fourth with business arithmetic, and the fifth the rule of false and allied topics. 

DIONIS GRAY. Ed. pr. 1577. London. 1577. 

A Ix»ndon goldsmith of the second half of the sixteenth century. 
Title Sec Fig. 175. 

Description, Sm. 8*, 8.8 x 13.8 cm., the text being 6.5 x 12 
1 m 126 ff. (8 unnumb,). London, 1577. 

Editions, London, 1577, 8* (here described); ib., 1586, 8*. 

Thi^ i^ a practical arithmetic, consisting of four parts. 'The firft 

• 'r.taineth foundrie partes of Arithmeti(|ue, that is to fay/ the ftmda- 
TT.mtal 0|jerations with integers, including progressions. * The feconde 

• irtf , containeth the faid partes feniyng for praclife of broken numl)ers 
■r f-arcions.* 'The third part containeth the fondrie Rules of pro- 

:-ri;on, furthcretl by vfe of the foreHiid partes,* and includes alligation 
.' i the nik? of false. ^'Ihe fourth |)arte containeth fomlrie Rules of 
■ '•-sctie/ or short processes. It is one of the earliest Knglish arith- 

• ••:.'» to contain niles and definitions in rhyme. For example, in 



fti-L <lJLJ^^^ 

* ■ •t:^s- 

^ ■*'*■' ' 




°|(5ttiP4 (C^r(4 $cKHf|lcTft4 

ProlfKitt/ « dMmt9^ ^iv^iw 



Fig. 174. Title page of girjka g<mila z oorlssteyna 



>|K'aking of aildition Gray says : ' And for to ainplifie the effecte, take 
here a fewe lines in verfe : 

V^The ftore-lioufe of 

ridicaicdkc 9 comaifiyng afvvcll the ^fpj 
f0tmiri€farus0fibtSa€9KemnhQU ^ 

«ittb;0ktniiitsnbcr0,tiiicbt(eiltt< t^ 

lM0rp}(pott<ra,fiirt(Krettop)efr i^ 

taUenre : 9f alfo fiimrrie rules of '&i 

9}nittirofiiio}ke»ofrarc,plr«« Ptn 
faunte. aim rdmmot jeu« 


flmprktcdat IMpitfarirtllLtm 


Fig. 175. Title pa(;e ok gray 

• ^Of fondrie fommes perticulars, one totall for to frame. 

Set them doune right orderly, as worke doeth l)eft require : 
What place ye giue to any one, the reft let haue the f.ime. 
So maie you well per forme the 'ffccte, of what you doe dciirr ' 

The rule is then continued in a series of versus. 



Ed. pr. 1577. Heidelberg, 1577. 

WiLHELM HoLTZMANN. Bom at Augsburg, December 26, 1532; died at 
Heidelberg, February 10, 1 576. He was professor of Greek at Heidelberg. 

Title. See Fig. 176. 

Colop/ion. * Excudebat lacobus Mylius, impenfis // Matthaj 
Harnifch.//M. D. LXXVII/ (F. 52, v.) 

Description, 4**, 14.9 X 19.2 cm., the text being 10.5 X 16 i 
cm. 2 ff. unnumb. -f 50 numb. = 52 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, 1 575, 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 firsi 
and most extensive (ff. 1-22) relates to astronomy. The second !> 
purely arithmetical (ff. 22-36), and treats of common fractions, ginr.g 
the usual operations and the rule of three. The arrangement of thiN 
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 fractiurii 
having a common denominator, but he merely follows the usual nik of 
cross multiplication (f. 29, v.). The third section is * De svTdis, q^^^ 
vocant, nvmeriis iis, qvi a qvadratis primo nafcuntur, Inftitutio d> 
cendo explicanda,' a chapter now conventionally placed in our altie- 
bras. The fourth section relates to the celestial globe and the astrolai* 
(ff. 46-50). 


Ed. pr. 1577. Bologna, 1577 

Cataldi, Cataldo. Bom at Bologna in 1548 ; died at Bologna, Febrja'^ 
1 1, 1626. Professor of mathematics and astronomy at Florence (1563). P*'- 
gia (1572), and Bologna (1584). He wrote several mathematical works, itj 
to him is due the beginning of the theory of continued fractions (1613) 

Title. Sec Fig. 177. 

Description. 4°, 14.3 X 19.6 cm., the text being 10.9 x 16; 
cm. 8 ff. unnumb., 40 11. Bologna, 1577. 





GKtLiBLMi xrLjin9\t 


VtSwrUnmHamimmmumm &tnll^ukm Utr» 
DtyfoGhH&TUmifkar^ trsOam, 


EzcadebatlacobBsMyUas . impeofia 

M. D. LXXYlt 

Fig. 176. Title pa(;k of xylandkr 






M DslJegno d$ Perugta, 


Signori Academici , il Signer Caualicr Padotto^ 

& il Signer Caualicro Anal^gi . 

In Bolocna. 

PerGiouanni RoGi mdlxxvi f. 

CmlkmiAde Sufenm, 

Frc. 177. Title page of cattau)i 


Editions, There was no other edition. Cattaldi wrote, under 
ihc pseudonym Pcrito Annotio, a • Prima Parte della Pratica 
Aritmetica' (Bologna, 1602) and a *Trattato dei numeri per- 
fctti' (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 
ha\*e 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 
Ux>k is a typographical one ; for, the printer not being able to set such 
numliers, all of the fractions have been written in by hand. 

Other xvorks o/i^jj, Borghi, p. i6, 1484 ; Buckley, p. 252, c. 1550 ; 
Cai)clla, p. 68, 1499 ; Cressfelt, p. 290, 1557 ; Herbestus, p. 303, 1561 ; 
Hubel, p. 314, 1563 ; Ramus, p. 331, 1569 ; Seton, p. 252 (see Buckley, 
c. 1 552); Anonymous, * Arithmetica,' Debreczin ; Miguel Berenguer, • I)e 
n'lnierorum antiquorum notis,' Saragossa ; Georg Gehrl, ' Ein nutzlich 
und kunftlich Rechenbuch auff der Fedem,' Prag, 8* ; Johann Jung, an 
arithmetic, LObeck, c. 1557 (no copy extant?); Bemhard Salignacus, 

* 1 ractatus de Arithmetica Partium et AUigationis,' Frankfort (Peacock 
^^^y** »575)^ 2ilso • Regula veri,* Heidelberg, 1578, and * Arithmeticae 
Libri II, et Algebrae totidem,' Frankfort, 1580, 4^; ib., 1593» 4^ 


Ed. pr. 1578. Paris, 1578. 

Profe^or of mathematics in the University of Paris, in the second half of 
the sixteenth century. 

TitU. See Fi^. 178. 

Piscription. 8^ 10. 1 X 17 cm., the text being 7.5 X 14 cm. 
9 pf>. unnumb. + 383 numb. = 392 pp., 28-29 J'- 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- 
Wx of his applied problems from army life, llje arrangement of the 
U>ok is not peculiar, but the great array of military problems is unitiue, 
a:;d, having been prepared es{)ecially for this work, would form an 






nois^ProfelleUr ^s Mathematiques ca 

rVniuerfite de Paris^diuifees en 

quaere partiesrauec vn pe« 

ocTraid^ des fra&ions 



ChtL Hierofme de Mamef,au mont 
S.Hiiaire^a Tenfeigne du Pelican* 



Fig. 178. Title page of chauvet champenois 


interesting source for the study of army conditions in France in the 
sixttrnih century. 

Other works of j^yS. Capelia, p. 66, 1499 » Ciemnaa, p. 200, 1540 ; 
Ramus, p. 355, 1569; Salignacus, p. 359, 157?; Tartaglia, p. 278, 
1556 ; Trenchant, p. 320, 1566. 

JOHANN OTTO. Ed. pr. 1579. Leipzig, 1579. 

A Freiburg Rechenmeister, bom c. 1529. The dedicatory epistle U dated 
1 57(>. and is signed * Johan Otto. ifltati» fuae 50/ which approximately fixes 
the date of Otto's birth. 

Title. See Fig. 179. 

Colophon, •Getruckt zu Leipzig/ bey Johan // Rhambawes 
Scligcrs hindcrlaf-//nen Erben/ 1579.' (P. 423.) 

Description, 4*, 14.5 X 19.1 cm., the text being 11.7 X 17.5 
(m. 424 pp. (381 numb.), 28-38 II. Leipzig, 1579. 

Editions, There was no other edition. 

l*he book is composed almost entirely of tables, although the first 
tt'w Images give a brief treatment of counter reckoning. 

CHRISTIAN URSTISIUS. Ed. pr. 1579. Basel, 1579. 

ALiJ^ssiHERrs. Alu^ssisi DERI'S, WuRSTEisEN« Urstis. Bom at Basel in 
1 544 ; died at Basel March 3a 1 588. He was educated at Basel and became 
pr<>fev<K>r of mathematics (1565) and afterwards (1585) of theology in that 

Title, See Fig. 180. 

Colophon, ' Basileae Hclve-//tiorvm,//per Sebastianvm Hen- 

rupctri. An. hvmanitatis // Filii Dei, CID. ID. LXXIX.// 
.Mcnfc Augufto.* (P. 192.) 

Description, 8*, 10.3 X 16.6 cm., the text being 6.4 x 1 1.2 cm. 
i'>j pp.. 25-28 11. Basel, 1579. 

liditions. On p. 192 is a woodcut with the date 1569, evi- 
■:«-ntly used from some earlier work of the printer's. Murhard 
■ .''L I, p. 173) mentions a work by Urstisius, * Zwey Biichcr von 
»!• r Rechenkunft, defsglcichen in der deutfchen Sprache nie aus- 
^'f^-angen,* Basel, 1569, 4*, in which this woodcut may have 
U.*cn used. He also mentions an edition at Basel in 1595. An 
hn;;]ish translation by T. Hood appeared in 1596. 


hU^ futallc/ foAnthmcticam lUb t^M 
3iifotit)erbrtt ahtt fur ^aiiflfTcttt / STrnprtperfwiw/ 

O<rroitffl) ftft*6ed £tat«iffflt»»nM«r6wifo» juAtfpJto gdm^fta} 
fitlti'Xatttantttnettbm 0!ri*df(tte6iwpit4/ote fu^ranrffertoitcr 

tcuff auff etntiiagttt/ vntonlwr fftmtOTH*m»^™»«[»*^^^ 
ma ffarbeboift 0«*Miii"pl»arm*«*O''»^"^»"'**"""'*l;f 'tSSS 
fpecies fctTAddition otmrettmiflirwa/ wftaB* '»**nS!"f*?"S!r 


Pfalm.90. , ^ . 


I 5 ? »• 

Fig. 179. Title page of otto 


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 
Ikxrthian arithmetic must give way to the practical treatment demanded 


A R I T H- 




In ufttin Academiae BaiiL 

Cbristiani VasTisir^ 



Fig. 180. TrrLE page of ursfbius 

I y modem conditions. As a result it is a somewhat heavy treatment of 
{*art of the old theory of numbers, together with the ftmdamenul oper- 
ations, roots, and certain commercial applications of the rule of three, 
of (partnership, and of alligation. Urstisius expresses his indebtedness 


chiefly to Euclid, Ramus, Salignacus, Gemma Frisius, and Scheubel, not 
a very extensive list for 1579. 

Other works of i§'/g. Digges, p. 340, 1572 ; Gemma, p. 200, 1540; 
Mariani, p. 181, 1535; Riese, p. 139, 1522; Tagliente, p. 115, 1515: 
Anonymous, * Nouvelle et facile m^thode d'arithm^tique,* Lyons, I6^ 
Miguel de Eleyzalde, * Guia de contadores,' Madrid, 4**. 


Ed. pr. 1580. Turin, 1585. 

Johannes Baptista Benedictus. Bom at Venice^ August 14, 1530; died 
at Turin, January 20, 1 590. Philosopher and mathematician of the Duke of 

Title, * lo. Baptistae // Benedicti // Patritij Veneti Philofophi. 
//Diversarvm Specvlationvm // Mathematicarum, & Phyficanim 
// Liber.// Quarum feriem fequens pagina indicabit.// Ad Sere- 
nissimvm Carolvm Emanvelem // AUobrogvm, et Svbalpinor\m 
// D vcem In vie tissimvm.// (Large woodcut.) Tavrini , Apud Hacre- 
dem Nicolai Beuilaquae, MDLXXX V.//Superioribus permifl"um/ 

(F. 3.) 

Description, Fol., 20.7 X 30 cm., the text being 14.5 x 23 cm. 
425 pp. numb, -f 10 unnumb. 4- 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 enlitleti 
* Theoremata Arithmetical The other five parts relate respectively to 
perspective, mechanics, Aristotle, Euclid's book on proportion, an«i 
physics. The arithmetic is a scientific consideration of various matteT> 
of theory, and is best illustrated by the following theorems, which, as :> 
usually the case, are stated in the form of questions : *Theorema IIIL Cn 
multiplicaturi fractos cum integris, recte multiplicent numcranterr. 
fracti per numerum integrorum, partianturque productum per denoroi- 
nante fracti, ex quo numerus quaefitus coUigitur.* 'Theorema XXIX. 
Qvid caufae eft, cur fubtracto duplo producti duorum numeronim a»l 
inuicem multi plica toru ex fumma fuorum quadratorum, fempcr quc»; 
fuper eft duorum numerorum quadratum differentise fit?* All syi^ 
questions are answered by the aid of diagrams, quite as Euclid wocivi 
have done. The graphic treatment is even applied to such problems x< 
that of the couriers (* Theorema CXIIII'). There is an intercst:r.4 


* Ap|A.*ndix de specvlatione regvlae falsi/ which closes the arithmetic, 
and sets forth an elaborate explanation, with graphic aids, of the rule of 
t.ilM.* which was then so common. 

As a siiecimen of graphic arithmetic, combining the Euclidean the- 
ory with the Renaibsance practice, Benedetti's book is worthy of more 
.mention than it has received. It may be inferred from some of his 
sutements that, although purely a theorist himself, he recognized the 
obsolete nattire of much that the practical arithmetics had to offer, 
it is |)ussit)le that the common partnership problems were already con- 
^i<lcre<i 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 Kie»e (p. 138). 

TttU, See Fig. 181. 

Colophon, *Gedruckt zu Leipzig; // ben Hans Rhambaw/ //im D. LXXX.' (P. 402.) 

Description, 4', 14. 1 X 17.9 cm., the text being 1 1 x 16.5 cm. 
36 pp. unnumb. -f 366 numb. = 402 pp. Leipzig. 1580. 

liditions. There was no other sixteenth-century edition. 

Although this book contains a brief treatment of the operations, it 
:s largely made up of tables for the use of merchants. It is therefore 
n- »t a textbook, and I believe it went through only one other edition 
»i.cip/ig, 1619). 

WILLEM RAETS. Ed. pr. 1580. Antweqi, 1580. 

A Dutch arithmetician of Maastricht. The privilege is dated May 22, 15761 
and mentions only the name of Kaets. Coignet (Cognet) in his preface, how- 
ever, speaks of his particular friend ('mijn zoderlinge goet vrient*) Raets as 
dc jd (* meynen ouerledc vrient *), so that he \ ery likely died between 1 576 and 
15^0. Coignet was also the editor of Menher's arithmetic (see p. 34()). 

Title. Sec Fig. 182, 

Description. 8*. 9.8 X 14.7 cm., the text being 7.4 x 1 1.9 cm. 
SS rf. unnumb., 27-31 11. Antwerp, 1580. 

Editions, There was no other edition. 

'I his work is one of several elementary business arithmetics ap|>earing 
•r. the IjOw Countries al)out this time. It is a small lxx)k of no siiec ial 
'i.fTit save as it shows the style of (ommercial j)r<)blenis of the peritxl. 
I ;h: similarity of the title page to that of Menher's 'Arilhmcti<ivc 
vci onde' of 1556 (p. 283) is interesting. 



mxmt> ^funHgtwi<^t/alba itt £mmtv 

filr r 1 o /pfiniDt gdtMrPfgf (/ Dflrinnm Mc ^r^iUmge 
indnfaufffn t>ni ipcrpauffcn/ one fonBae itt^dfufftiac Stcc^nuna 
jar bf&cnD ju pnbcn : 3tuc^ wn atlcrf^i;) 3Sa^ / tf (f n /wU ®c* 
m(c^t fauff- €ampt mc^r angc^tngtcn .^affdn auff Wr SOja* 

©fitic^f girid&m. ^ml<icf}m tim ^ffl^fcl SXrcl^nuiig duff 
aNfiin?Mt)@o(Of/T?- :aUcrt&aufflcuim/ S)cnUmf 

gfaac S^terc«/35utacr »tt& ©teffwc 

Jfti^tit In jf ten ^4rcii nlf^r tutj^jufinxftit. 
Fig. i8i. Title pace of isaac riese 




Other works 0/1580. Apianus, p. 155, 1527 ; Baker, p. 327, 1568 ; 
(assiodorus, p. 211, 1540; (kmma, p. 200, 1540; Gyraldus, p. 354, 
1552 ; Mariani, p. 180, 1535 ; Maurolycus, p. 350, 1575 ; Ramus, p. 331, 
1569; Ringhicri, p. 253, 1551; Salignacus, p. 359, 1577; J. Ammo- 


€ctt fwm ctifrctboecft/.bmt 

iMkiti 0M9/ f 9Ml9}M fixt% 

TVaer In die FotKhmenten ieer srond^lifdc verebert 
eBroecvecIfchooneqoeftien gheiniilmen?vor« 
den.cot nut ^nde oorbter van alk CoopUedl 
code liefbebbender feloer Cooficn. 

mt mA am Trs&Sit tdndi VVlfftlrHde^ ma Am^ 
Idtitit vgcnn^ditgr Uuhul C^ignii 


tdvUocmr* s 5 > o. 

Met Pniil^ie tia tlucn laeren* 

Fig. 182. Title pa(;f. of raets 

t;::h, * Isagoge Arithmetica,* Wittenberg, 8**; H. Fliiker, • Arithmcti< cs 
intrcjductio/ Cologne, 8", and 'Compendium calculorum, seu projectili- 
',ni ratiocinationis,' ib., 8*; C. Zuccantini, * Libro d'Albaco,* Siena. 1 2*. 



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 XXXfc//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 155 1 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 1^8 1, Anonymous, P- 195 » 1539 ; Fischer (Piscator u 
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, * Arithmetica,' Regensburg (referred to in the care- 
lessly prepared Boncompagni sale catalogue, but probably the Kaudler 
book of 1 591). There was also *A short Introduction to Arithmetic ' 
published anonymously in London c. 1581-90, 8^ 


Ed. pr. 1582. Frankfort, 1678. 

A German-Italian teacher of the sixteenth century. 
Title. *Julii Caesaris // von Padua // Arithmetifche // Prac- 
tick/ // Welche in alien Lindern // fehr niStzlich kan gebraucht 
// werden/ bey Kauff- und Ver-/Aauffung allerley Wahren 
auch//die groffen M&ntz-Sorten in//kleine/und die kleine m 
grofTe//zu verwandeln :// Samt der Erkldrung/ wor-//inn ein 
Jedweder/der nur die//Ziffem keftet/ alfobald fehen kan/, 
wie diO Bikhlein zu verftehen ift.// Nebenft Morgen- und Abend- 
//Gebehtenund Gef4ngen/den//reyfenden Perfonen gar bequem 
// bey fich zu fflhren.// Und dann letzlich/ eine kurtze Be- 
fchreibung/ der denckwurdigften//Sachen/fo von Anfang dcr 
Welt/ bio //zu diefer Zeit/vorgangen.//Franckfurt am Mayn 
//Druckts Blafius JWner/im Jahr 1678.' (P. i.) 






Pradica Noaieconiin, 8c Meofnnrain, ac Allttuioois pamcioaen* 
iaocft^aadi . 8c rt ia lodice requeoti . 


B R 1 X 1 AE. 

4lc Twlinis Fratrcs . i j 8 < • 

Fig. 183. Title pa(;e ok visconti 


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 U. 
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.- 
LXXXIL' (F. 65, V.) 

Description. FoL, 21 X 30.2 cm., the text being 12.7 x 195 
cm. 65 flF., 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 k 
monete,' and is a historical treatise on money and coinage, toaching 
slightly on exchange. 

GASPARO SCARUFFI. Ed. pr. 1582. Reggio, 1582. 

See above. 

Title, * Breve Instrvttione // sopra il discorso// fatto dal Mag. 
M.//Gasparo Scarvflfi,// per regolare le cose delli//danaTi. 
(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 





Fig. 184. Title page of scarifti 


Editions. There was no other edition. 

A verbose commentary on certain parts of Scaruffi's work. See 
p. 370. 


Ed. pr. 1582. Basel, 1582. 

A Swiss educator of the sixteenth century. 

Title, * loan. Thomae // Freigii I. V. D.// Paedagogvs.// Hoc 
est, libellvs//ostendens, qva ratio-//ne prima artivm ini-//tia 
pueris quim izciWi-/ /mh tradi pof-//fint.// H^XexJf 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. i X 15.8 cm., the text being 6.8 X 12.4 
cm. 17 pp. unnumb. + i blank + 366 numb. = 384 pp., 30-31 U. 
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 0oannes 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 ' Awea 
Regula . . . uulg6 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 // f tudiofi> 
gratificaturus. Anno Chrifto nato CI3. ID. LXXXL* (On map 
at end.) 

Description, 8^ 10 X 14.6 cm., the text being 7.6 X 1 3.3 cm. 
61 pp. numb. -f 3 blank = 64 pp., 20-32 11. Antwerp, 1582 
(colophon 1 581). 

Editions. There was no other edition. 


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<}2^c foot coafi4ciuda 1b Mmcial 


Q^atmfiietaBt aocslLonit 
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C. too. 

CX9- CD. CI 3* tooo. X&fff. MiBf. 

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Fic 185. From the Pudago^^s of freiuius 


D E 




3idtthM HoSlo auciorc» 

Ex officina Chriftoplion Plantini. 

M. O. L X X X I I. 

Fig. i86. Title page of hostus 


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 quibuidam vfitatis,' 
a set of mediaeval astrological numerals also fully described by 

MAFFEO POVEIANO. Ed. pr. 1582. Bergamo, 1582. 

A Veronese arithmetician of the sixteenth century. 

TitU. 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 0/1^82. Gemma, p. 200, 1540; Moya, p. 310, 1562 ; 
sarrobosco, 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, 
( he Euclide & altri Eccellentiss. Mathematici \ik trattato oscuramente/ 
X'incenza, 8^ (contains some theory of numbers ; see also p. 306, i S^O ; 
Mellema, ' Arithm^tique compost de plusieurs inventions et probl^mes 
nouveaux,' Antwerp, 2 vol., 1582 and 1586. 

CHRISTOPHER CLAVIUS. Ed. pr. 1583. Rome, 1583. 

Chkistoph Klai*. Bom at Bamberg in 1537; died at Rome. Kebruar)- 6, 
i6i3. 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. i88. 

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, Rome, 1583, 8* (here described) ; Cologne, 1584; 
Rome, 1585, 8' (p. 378); Cologne, 1592. There were also edi- 
ii<»ns after 1600 (sec 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 161 2, fol. 








Opera noua, & vtili(fima> 

DtUepikbrtia,&genertlifiratttebe , ebe »farftpcffa9, 
utttfpaie ad ogn'mn . 

Conliccnza dc Supcriori. 

M D L X.X X I I. 

Per Comin Ventura , Sumpatorcincflaa'td. 
Fig. 187. Title page of poveiano 








t^OHUE E»Typtpapb'uDmimciBjfd,i$ii, 

Fig. 188. Tttle page of clavtus 


Clavius was an excellent teacher of mathematics, and his textboob 
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, bat 
it was influential in the classical schools. 

CHRISTOPHER CLAVIUS. Ed. pr. 1583. Rome, 1585. 

See p. 375. 

Titk, * Christophori // Clavii // Bambergensis // e Societate 
// lesv // Epitome Arithmeticae // Practicae nunc denuo ab ipfo 
auctore//recognita.// (Woodcut with I. H. S. in center.) Permis- 
sv Svperiorvm // Romae Ex Typographia Dominici Balae. 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 //Reucz. 
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 dei Svperiori., , 
(Woodcut with I H S in center.) In Roma,// Nella Stamperia di 
Domenico Bafa.//M. D. LXXXVL' (P. i.) 

Description, 8*, 10.3 X 1 5.6 cm., the text being 8.2 x 1 3.3 cm. 
302 pp. (27s 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 mercantik 
schools of Italy, although several editions appeared after 1600. 

CHRISTOPHER CLAVIUS. Ed. pr. 1583. Rome, 1602. 

See p. 375. 
Title, *Arithmetica//Prattica//compostadal Molto //Reuer. 
Padre Chriftoforo Clauio // Bambergenfe della Com-//pagnia di 


lESV.// Et tradotta da Latino in Italiano dal Signor Lorenzo// 
Caftellano Patritio Romano.// Reuifta dal medemo Padre Clauio 
// con alcune aggiunte.//Con Licentia de i Svperiori.//In Roma, 
// (woodcut with L H. S. in center,) Per li Heredi di Nicol6 
Mutij//M. DC. n/ (P. I.) 

Colophon. • In Roma,// Per li Heredi di Nicol6 Mutij M D C 

11/ (P 3«2.) 

Description, 4*, 10.4 X 1 5.7 cm., the text being 8.3 x 13.3 cm. 
312 pp. (281 numb.), 39 11. Rome, 1602. 

Editions, This is the second Italian edition (p. 375). 

Sec p. 378. 

NICOLAUS REYMERS. Ed. pr. 1583. Leipzig, 1583. 

A Gennan surveyor, bom at *Henttede in Dietmarachen.* The 'Be- 
schluss* » dated *xu Hattftede in Diethmarchen,' September 14, 1583. 

Title. * Geodaesia // Ranzoviana.// Landt Rechnen/ // vnd Feld- 
mcffen/ fampt meffen aller-//hand grflffe. AUes auff eine leichte/ 
bchende/ // vnd vormals vnbekandte newe art/ kfinft-//lich/ 
^rfindlich vnd deutlich // befchrieben/ // Zu Ehren // Dem 
Edlen/ Beftrengen // vnd Ehrnuehften Herm/ Heinrichen // 
Kantzouen/ Herm Johans feligen Sohnc/ der // K6n. Mayft. 
zu Dennemarcken/ etc. In den // Ffirftenthumben Schlefewick/ 
Holftein/vnd Diethmar-//fchen/Stadthaltem/Rhat vnd Ambt- 
man auff // Segeberge/ Erbgefeffen zum // Breitenbergc/ etc.// 
Durch//Nicolaum Reymers/von Henftcde///in Dietmarfchcn. 
//Cvm Privilegio.' (F. i, r.) 

Colophon. * Gedruckt zu Leipzig bey // Georg Defner/ // Im 
Uhr//M. D. LXXXIIL* (F. 44, v.) 

Description. 4*, 15.1 X 18.6 cm., the text being 9.7 X 14.5 cm. 
44 ff. unnumb., 25 11. Leipzig, 1583. 

Editions. There was no other edition. 

.\lthough nominally a book on surveying, this work may properly rank 
» an arithmetic, the • Erfte Buch * being entirely devoted to that sub- 
wi L Of this book the first chapter is entitled • von zahlen ' ; the second, 
Mon brilchen'; the third, • von fummieren * ; the fourth, 'von viel- 
frltigen,* multiplication thus directly following addition ; the fifth, * von 


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 manuab, it shows 
no insight into educational problems, and the treatment is very unsatis- 

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 weils,' Erfurt, 8^ ; Johnn 
Weber, * Ein new Kunstlich Rechenbuch auff den linien und ziffem.' 
Leipzig, 4* (see also p. 338, 1570). 

PETRUS BUNGUS. Ed. pr. 1583-84. Bergamo, 1584-85 

Bom at Bergamo ; died September 24, 160 1, at Bergamo. He was a canon 
of the cathedral in that city. He wrote only on the mystery of nnmben. 

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 // facultatis Scriptonim, 
ita exacte, dilucide, & accurate differi-//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 Philoft>- 
phis, Mathematicis, atque alijs ftu-//diofis omnibus, tam vtile, 
quam iucundum fit futurum.//De Svperiorvm licentia.//Bcrg»^ 
mi CID ID XXCIV.//Typis Commini Venturae, eiufdem Vrbis 

Description, Fol., 20.6 X 30.4 cm., the text being 17 x 28.9 
cm. 276 pp. (24s 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 eili- 
tion of the first part was evidently exhausted before that of the 







Opus maxiinaroin teram « & planmanim do^'iuL, flu* 
vaxaK, ,cofix,t & uarieate lefercum,. 

Tbttlofft ,TbilofiflM , MMthemMkut Mfie sl^sfiudiofis 




T^fpis Comiai Vennux , & Sodj . 

Fig. 189. Title page ok the 1585 


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., 1 591, 4** (which De Morgan incorrectly calls 
the second edition); ib., 1614 (p. 384). It appeared under the 


CO " » 





00 oj 









130 00 



DO «»«> 

I03 •••. 


Q3 ^ ^''^ '***• 
00 00 €o» 

Fig. 190. From the 16 14 EorrioN of bungus 

title *Numerorum mysteria,' Bergamo, 1599, 4"* (P- 384)- ^ *^^^'^ 
an edition under this title, published at Paris in 1618 (1617 
in the colophon), 4'. For the 'Praecipuae numerorum notac ci 
earum valor,' Parma, 1689, see p. 384. 

This is a mass of erudition, prolix and unscientific, relating to tbc 
mystery of numbers. It was written by a priest for the use of preacbcn. 


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 
albo of much value in showing the nature of the Roman numerals in use 
in the sixteenth century. (See Figs. 190, 191.) 


00 <«» 











C0I03 00 



CCIOD »• •• 




CCI33 00 o> 




Fm;. 191. From the 16 14 EDmoN of BtN<a-s 

Oihfr works 0/1584, Buckley, p. 252, 1550; Cassiodorus, p. 211, 
1540; Clavius, p. 375» "5^3; I>elfino, p. 275, 1556; Kol)cl, p. 102, 
I 5 14 ; Ramus, p. 263, 1555 ; Anton Schulze, * Arithmctica oder Rech- 
. '/.LK hicin,* ( * Arithmetica oder Rechenl)uch nel)en ciner dienlirhen 
\nl<fitung zum Buchhalten*?) s. L, with subsequent editions at Liegniu, 
t^*<M^, 4^ and Frankfort, 1600, 4^ 


PETRUS BUNGUS. Ed. pr. 1583-84. Bergamo, 1599. 

See p. 3S0. 

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 approbationc.// 
Bergomi, Typis Comini Venturae, eiufdem vrbis Typographi.// 
00 ID XCIX.' (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 : Inquo mirus inprimis, 
idemq; perpe-//tuusArithmeticaePythagoric?cum Diuinae Paginx 
Nu-//nieris confenfus, multiplici ratione probatur.// Poftrema hac 
editione ab Auctore ipfo copiofo Indice, & ingenti // Appendice 
avctvm.// lUuftriffimo viro, Virtutum omnium, ac difciplinanim 
//genere omatiffimo //Ranvtio Gambarae // Comiti Virolae. & c. 
// Bergomi, Typis Comini Venturae. 1614.' (P. i.) 

Descriptiofi, 4', 17.5 X 23.4 cm., the text being 12.6X 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 /fHoXstJ/ Et earum valor,.' ' 
Secundum // Petrum // Bungum.//Parmae, Ex Typographia Du- 
cali.//CLO. IDC. LXXXIX.' (P. i.) 



Description, 4*, 1 1. 1 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 Mysteria.' 

The treatment of Roman numerals by Bungus is the most elaborate 
and interesting to be found in any of the works of Renaissance writers. 

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 x9.8 cm., 
printed on vellum in double col- 
umns, each 1.8 X 9.5 cm. 144 pp. 
(icx)numb.),27-2811. Lyons,i585. 

Editions, There was no other 
edition. The privilege is dated 
August 6, 1 581, 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 
looo. 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 n6bre de quarante roille.' 
This is dedicated 'A pvissant et illustre 
Seigneur le Baron de Mont-clar.' 


admirab!e,poiir (afac ioqm 
forte dc c6pce • tant de mac< 
ehondire » comoie dc ckSger 
dinerfc maniere^poQc vendre 
& Ackeccr^laqaelle fctiiif a en 
coat le monde ancc grandc 
facilicftfans gecosaephune. 

fir « U/?» Ic ifNgfm dc Mff frv i« 
nrercic m WuiJh^i* am 

U cottc iMMioellemf DC compofl^ 9c 
mif cnliimteiCMr le Moon He- 
gal Pirdmoniob . ProfciTcot dc 
tftfhenuciqm en TValiietfitidc 

A L Y O H 

U Ic Tend co me Mcreiere i 

I'cnrei'gne dc la Sphere* 

1 f • f • 
Auecff'iKslcge da Ttny. 


192. Title page of monte 
regal piedmontois 


SIMON STEVIN. Ed. pr. 1585. Leyden, 1585. 

Bom at Bruges in 1548; died at the Hague in 1620. He was a merchant, 
a soldier, and an officer in the dvil 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. + 1 blank + 642 numb. -I- (203 numb, -f 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, 
1 582. His arithmetic first appeared in Flemish, at Leyden, 1 58$, 
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 Li\Tes 
d'Algebre de Diophante d'Alexandrie,' translated by Steviu, apparently 
from Xyknder's text ; (3) * La Pratiqve d'Arithmetiqve,' an attempt ai 
a practical textbook, but too scholarly for its purposes. The P^atiq^% 
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 conuertk 
en Frangois,' 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). 


Ed. pr. of the arithmetic, 1585. Leyden, 1634 

See ahove. 

Title. 'Les//OEuvres//Mathematiques//de//Simon Ste\in, 
// Augment^s // Par Albert Girard.' (P. i .) 

Page 3 reads : Les // CEuvres // Mathematiques //de Simon 
Stevin de Bruges.// Ou font inferos les // Memoires Mathema- 
tiqves,// Efquelles s'eft exerci le Tres-haut & Tres-illuftrc Prince 




Contcnanc les computations des nombrcs 

ArithroctiqucsoQ vnlgaires : 
Jafi tM?eif:, SMclisefumiuiimc fMHteb. 
Enlcmbic Icsquatrc premiers liures cfAlgebce 
dc Diophante d' Alcxaiidric»maintenant pre- 
miercment tradui^ en Fran9ois« 

Immt m kmfmkMif it U Prmiqiu i^Arhbmaifie^ 
ci«fniA;f mri Mres,Us Tdkjd' Inter eOM D^ini 
It rn tTAtcle dci ImmmenfmjbUs fjrmdcms : 
MutlExiUistmdHDixiifm Uwred'Eudidi. 

A L £ T D E» 

De rimprimcricdcChriftophlcPlaniin* 
do. Id. lxxxv. 

Fi<;. 193. TiTij: pa(;e ok thk 1585 mkmn 




K A T X O N« 

ESmt dcnnez, nmbfa it Difmii^pn IYmMt far 

ExplkdtiihduJhnni Uyatxoisotdresdenonibies^ 

Di(inc> defquelsle premier 17 ®8®4®7®Je^lnix^ 

ixf^tmdunq^. flnousfiut ^^j^^^ 

crouverlettrfomme. anfimBmk ®®®® 

On mema ks xiombres doonex X 7 S 4 7 

en ordrc comme djoigntiiCt les m^^^Z^ 

aiouftant felon la vulgdre maniere > 7 5 7 • ^ 

d*atoaftcrnombresentiers»enctfie 941 $04 

Donne (bmme (par te i probleme dt lAtkhmed* 

Sue) 941304, aai (one fee cpe dcmonftrent les figpes^ 
eflus les nombies ) 94X © J o @ 4®- ^ di » oue^ 
les mefines Com la fomme reoaifc. JDeMMffir/fiM. ^ Les 
27 (6) 8 ® 4 (3) 7 (D donnex, fMt (par b j« defininod^ 

*7;;V- tJo^ if-^ enfemble 17 iV^> ^i^*^ 
raifonlesj7@^®7@5®v^lcnM7p^t **» 
8.7 J ® 7 ® 8 ® 4 ® fcront 87; ,V^* fcfjjjls mris 

nombte$.commc 17 ^*^; 37 ij^i ??J l^osR^ 
cnfcmble Cpar le io« problcme de 1 Ancb.; 941 -J^, 
mais autancTautaufli la fomme 941 ® }(D © ® 4(!i 


Fig. 194. From the 1634 edition of stevin 


Maurice //de Nassau, Prince d'Aurenge, Gouvemeur des Pro- 
vinces des // PaTs-bas unis, General par Mcr & par Terre, &c.// 
Lc tout reveu, corrig<5, & augment^ // Par Albert Girard Sarnie- 
Inis, Mathematicien.// A Leyde // Chez Bonaventurc & Abraham 
KUcvier, Imprimeurs ordinaires //de rUniverfit^, Anno do Id c 

Description. Fol., 21.7 x 34 cm., printed in double columns, 
each being 8. 1 X 28 cm. 910 pp. (232 on arithmetic, 10 unnumb.), 
f)3 11. Leyden, 1634. 

Editions, See p. 386. 

Othfr works of 158s ' Benedetti, p. 364, 1580; Hungiis, p. 382, 
15S3 ; Caesar, p. 370, 1582 ; Claviiis, p. 378, 1583 ; Digges, p. 340, 
1572: (xemma, p. 200, 1540; Lonicerus, p. 253, 1551 ; I^ossius, 
j>. ^S9, 1557 ; Psellus, p. 168, 1532 ; Riese, p. 139, 1522 ; Savonne, 
I'- 3Mf 1563 ; lo. Frans. Fulconis, ' Cisterna Fulconica, libro d^abaco in 
i.r.g'ia pro\*enzale,' s. 1. (Lyons?), 8*; Thilman Ofenlach, * Rechcn- 
1 ui hk'in mit der Ziffer und auf den Linien mit Zahlpfennigen,' Basel, 
^' . Johann Schreckenberger, * Rechenbuchlein auf! den Linien und der 
I t^iem/ Strasburg, 8*; Samuel Kisenmenger (pseud. Siderotrates), 

* (/v< lopaedia Paracelsica Christiana,* s. 1., with a section on arithmetic. 

Works of is86. Clavius, p. 378, 1583; Gray, p. 353, 1577; 
Nft-Ilcma, p. 375, 1582; Ramus, p. 331, 1569; Riese, p. 139, 1522; 
s« honerus, p. 331 (Ramus, 1569); Sterin, p. 386, 1585; Slifel, 
p. 226, 1544 ; Tagliente, p. 115, 1515 ; Anonymous (Sterner mentions 
a Kechenb(k:hlein without title page, Magdeburg) ; (leorg Hiiflein, 

• Rc< henbiichlein mit der Ziflfer und mit den Zahlpfennigen auf der 
/ initr/ Strasburg, 8^; Paulus Alexandrinus, * Rudimenta/ (ireek and 
I^tin, Wittenberg, 4^ (with some notes on Jewish arithmetic). 

JOHANNES PADOVANIUS. Ed. pr. 1587. Verona, 1587. 

Giovanni Paim>vani. A Veronese mathematician of the second half of 
the fiiiteenth century. 

Titir, See Fig. 193. 

Description. 4*, 14.4x20.4 cm., the te.\t beinj; 9.6 x 15.7 
i m. 7 pp. unnurob. + 73 numb. = 80 pp., 29-32 11. Verona, 

ILditions. Padovani published a work in Venice in 1 565 con- 
>: sting of six parts, the fourth of which was * De Arithmetica.* 





folum omnis generis numerandi ars tarn Latino 

rermoDe,quam Grxco pcrdifcitur: aerum 

ctiam i^uicquid ad quafcunque ra« 

tiocinaciofics peninet ^ faciii 

do^ina aperitur» 

OmibMSf &itt frmsuifiAlkdlnirafU fiitdiofis permceffgnm. 


ExTypographiaScbaftiani a Donnis. ij87' 
Delicentia Stiperiorum. 

Fig. 195. Title page of padovavius 


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. 


Ed. pr. 1587. Niirnberg, 1587. 

A Gennan Rechenmeister of Steyer, bom c. 1525. 

Title, See Fig. 196. 

Colophon, ' Gedruckt zu Nfimberg/ // durch Leonhardt // 
Heufzler.' (P. 369.) 

Description. 8*, 9.4 X 1 5.2 cm., the text being 7x13.3 cm. 30 
pp. unnumb. -f 369 numb. = 399 pp., 25-26 11. Niirnberg, 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 Rudolf! 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 J587. 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 scienzc,' Rome, 4"* (with a 
brief treatment of arithmetic) ; Conr. Poeppingius, * Neues Rechen- 
biichlein auf Linien und Fedem,' Braunschweig, 8^ (four editions after 

Works 0/1588, Gemma, p. 208, 1540; Helmreich, p. 306, 1561 ; 
Rudolif, 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., 15991 I2^ 







Hcfebffc^t((6cft/fnn gra3wi& 
^W*^ Cur* i^i^Ssi;^'' ^ 



Fig. 196. Title page of thierfelder 


Ed. pr. 1589. Dordrecht, 1609 

A French schoolmaster at Dordrecht (Dort), in the second half of the six 
teenth century. 

Tit/e. * Een corte ende een-//vuldige Inftructie/om lichtclij- 
eke // en by hemfelve/ fonder eenige meefter oft onden^-ij-/ (er 


le leeren cijfferen. Seer nut eii profijtelijcken //alle menfchen/ 
die in dc Confte van Arith-//n^etica heel f techt/ onervaren // 
cndc ccnvuldich zijn.// Geftelt ende by een vergadert, door 
Bemaert // Stockmans Ismiz, Fran^oyfche Schoolmeefter//inde 
vermaerde Coop-ftadt Dordrecht.// Hier zijn ooc bygevoecht de 
Different ien van de co-//ren-Mate der voomaefte ftede in Hol- 
lant/ t'Sticht/ // Zeelant/ Brabant/ Vlaenderen/ Gelderlant/ 
En-//gelant/ Vrancrijck/ eii Ooftlant/ tot dienft // van alien 
Gracn-ccx)pers.// Van nieus overfien ende verbetert door C. P. 
B<>eye.// Item/noch van nieus bygevoecht een clare onder-// 
wyfinge om de tafelen van Intereft te leeren maken/ //alles tot 
tlicnft vanden onervarenen.// Sapiente 11. verf. 22.// Maer ghy 
hcbbet alles in Mate/ Tellen/ // ende Ghewichte gheordineert.// 
Tot Dordrecht // By my Pieter Verhagen/ woonende inde Drue 
, kcrije/ teghen over de Wijnbrugge. 1609/ (F. i, r.) 

Colophon. 'Tot Dordrecht.// By Adrien lanfz Bot.//Anno 
1^09.* (F. 214, V.) 

Description, 8*, 9.1 X 14 cm., the text being 7.4 x 1 19 cm. 
4 fT. blank + 211 unnumb. = 215 fT., 31-33 11. Dordrecht, 1609. 

Editions. This book was first printed in 1589, and the dedi- 
1 atorj' 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 commcr- 
< lal work, and like Vander Schuere's book it gives an excellent view of 
the mercantile life of Holland in this period. 

Other u*orks of ij8g. Jacob, p. 298, 1560; Moya, p. 310, 1562; 
F ahio Paolini (Paulinas), ' Hebdomades, sive septem de septenario 
i.!>ri/ Venice, 4^ containing a little arithmetic, and ib., 1598 (?). 

li'orks of isgo. Bungus, p. 382, 1583; Digges, p. 340, 1572; 
fjiiob, p. 298, 1560; Lapazzaia, p. 324, 1566; Psellus, p. 170, 1532; 
Ke^orde, p. 214, c. 1542; Heizo Buscher (Boscherus), •Arithmetics 
i l»ri duo/ Helmstadt, sm. 8% with editions, ib., i59«. 8**; Hamburg, 
f 592, 8'; ib., 1597, 8**; Frankfort, 1600, 8*, and later; Franciscus 
I Crasser, an arithmetic, Liibeck, with later editions : Cyprian Lucar, • A 
trcratise named Lucarsolace,* Ix)ndon, 4**; Bartolom^ Solorzano, • IJbrode 
4^^va V Manual de cuentas de Mercaderes,' Madrid. 


BERNAERT STOCKMANS. Ed. pr. 1589. Gouda, 1644 

See p. 392. 

Title, * Aritmetica,// Door Bemardus Stockma // eertyts 
Franfoyfche fchool-//meefter inde vermaerde coopf tadt // Dor- 
derecht nu van nieu' curieus //gecorigert ende verbetert noch is 
// hier by gevoecht een taf elken om te // rabatteeren op fulcken 
tyt ofte intereft // men begeert door //Abel. W. Waesenaer,, 
Rekenmeefter tot Vtrecht // (Portrait of author.) Gedruckt tot 
Vtrecht by//Efdras Willemfsen Snellaert.//boeckvercooper,y 
Anno. 1637.' (P. I.) 

Colophon. *Ter Govde,// Gedruckt by Pieter Rammazevn, 
BoQc\i'// drucker in't vergulde A B C.// Door Efdras WUlemfz 
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 II. 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 Bagnacivalk?. 

Title. See Fig. 197. 

Description. 4*, 14.2 X 17.9 cm., the text being lo.i 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 Bor^ model 
(see p. 16) which served for so many of the be^t Italian writen. It 
was written, as the dedicatory epistle states, at Bagnacavallo, and is ovt 
of the few books on mathematics published in the sixteenth ceotuiy at 
Ferrara. It has no merit save as its applied problems give a riev of 
the business life of the time. In its numerical work it is reactioiian. 
making, for example, a strong plea for the galley as opposed to the 
' a danda ' or modem method of division. 





Oa M. Francesco Pagani da Bagnacauallo, 

J^IU quMle/icontiene iivero, ^ficili 
modo di confeggiarf^. 

Con molti Q^edti important! ,& Deceflarij 

i Ragionieriysl Mcrcantii<S: ad ogni 

perfona^in tutti i Paefi. 


(TU^t^^-d njT^^fj^^^!^ (/^-rt**"^***/ 

7^ FEIf^KAKA, 

Appreffb Vittorio Baldini 

Con licenza dc' Supcrioti. m. d. xci. 

Fi<-.. 197. Title page ok pagani 

-. r. 1 591. Cologne, 1591. 

J : 1? ..m., printed in double columns, 

::. ;S pp. unnumb. -I- 798 numb. = 

. -J r vji :n the volume, and if they were 

r=^ zcv must have constituted a section by 

J-.uij^ne, 1591. 
o :j other edition. 

*:sL::ie on the history of monetary measures con- 

.ci Bndelios), director of the Bavarian mint, 

. .cr* oy the following writers : Albertus Brunus 

^.uis of France, and ambassador ; Johannes 

. •tv'gjer Stoffler who died in 1531 ; Bilibaklus 

. L «:eiebrated humanist; Martinus Garatus 

-. . lapter ' De monetis ' ; Franciscus Curtius, 

. - : \vignon, who write on the same topic; 

- , 5 'J o ; : Didacus Covarrubias (151 2-1577 », 

rienhcus Mameranus, a Belgian printer: 

-ancibcus de Aretio (1418-1483), a cele- 
.xjijs E\"erardus (i 473-1 532), a celebrated 
.. . »irg: Jacobus Menochius (i 531-1607), an 
^> jihers. It is a monumental work, and is 
.1 }t the history of monetary tables. 
. .>iker. p. 327, 1568 ; Bungus, p. 382, 1583 
x.mBau p. 200, 1540; Mariani, p. 181, 1535 
>:ilus» p. 168, 1532 ; Ramus, p. 331, 1$^ 
^ . :o. 1473; Jordanus Bruno, 'De Monade 
. . ->anki6rt, 8* ; Johann Kaudler, * Arithmetica 
oiea und mit Ziffem,' Regensburg, 8^ ; Hans 
. on dem Rechnen mit den Zahlpfennigen oder 

S. Ed. pr. 1592. London, 1600 

. ,iaMfn-mn of the latter part of the sixteenth centviy. 

^2 X 19.6 cm., the text being 10.8 x 169 
^ - >70 numb. = 285 ff ., with 2 charts at the 









clone um veterum»qukmN€otericoraoi Authoruiii,t|al 
deMonctts»ean2ndetn4uevalore,ti«, ponderer 
ac fimiltbusrcripreninc 

Qtmwmtmiam CMdtipmf^^gmddaUiawumiUM. 



rcQdirs.atqQe IlIuftrirs.Principisac D.Dommi Ernefti Elcftoris 

Colonien BauaricDuc.&c. Monetarum^tam Rhc 

nenfittm,qu2^m Vvcftphaticarum Archie. 

pircopalium Pntfcdo. 
Cm Stmmariis et Indici CoPioaa 





Fig. 198. Title pa(;e of bidel et al. 



The Arte of vulgar arith- 

mcticke , both in Integers and Fradtions, 
{ieutdedinto mo ^oohsijphereofthe firfl is emitted 

Nomodiddcfm Ntmenmm^mciiht (econd Psrtm Pr9fmii$mum. wiih 

ceitcine Demonftnuions^ieduccd into fo plainc and pcricd Mc- 

iW/f/ ihiHf bmhmthttherf heimfmhbfitedm ingk^ • Wktitmmt 

isaddd a third Bookf ^cncituled c^i^ il/rrr ii«rMi : Cooi- 

frcbcndtngall tlie moft nece(&ric and profiabkUct 

In an which thrceBookcs : thcRuIeS)Precq>cs^nKiMaziiiis,atiioc 

^f$cly eimfpfidmmeeterfir the tetter retsim^g $ftbemsm mammm^ 

hex, alTo the opcrrions. cxamplcs.dciiioiillratioDs« aod qodhoM^ 

mem mMe4i[Uwijeen9imJkdm»A€XfUmd^mth€fmwm 

of a Oul-tguctfot flic kcjdcfs mocc ckac ? cdcf fUr Sa%. 

aiul£ldtcrs« pmtablefor Mocham^andgjcncnO/ 

Newly coUcAcdjdisfftcd, and in ibmeMtt 

EcdcBtliicni* cip* 1^ 
Le arning vmo foolabgfcttcn on ihewfecic and Uaicki^ 
thctrnebthaiid:biittodiciHfcicisaIcwcIlof| " 
aBocdetvpoohii right atmc* 

Boctratlibr. i« Amb* rif. u 
OwmU^futemuimspimMMM mttmrstmfirtiEiMjpmi^mKmmmm mim 
rmime/mma^^HHiMMfiiafrmcsfdUmmkm ft i 


Ki<i. 199. TriLE pac;e or the 1600 M\i.ii> 


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. 


Ed. pr. 1592. Venice, 1592. 

Giovanni Antonio Magini. Bom at Padua, June 13, 1555; died at 
Bologna, February 11, 161 7. He was professor of astronomy and mathe- 
matics at Bologna, and wrote numerous works on these subjects. 

Title. * lo.// 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 valdi vtile Geometris, Aftronomis, Geographis, Mechanicis, 
Ar-//chitectis, Militibus, Agrorum menforibus, & denique // om- 
nibus Mathematicarum profefforibus.// Cvm Privilegio.// Vene- 
tiis, apvd Robertvm Meiettvm.// M. D. XCIL* (F. i, r.) 

Cohpfton, 'Bononiae,// apud lannem Baptiftam Ciottum,// 
Typis Victorij Benacij,//Anno Domini, M. D. XCII.// Superi- 
orura permiffu.' (F. 114, v., and 132, v.) 

Description. 4*, 16 X 21.6cm., the text being 1 1.8 X 18.2 cm. 
;/' 4 if. unnumb. -H 1 10 numb, (in the first book) + 124 numb. 4- 4 



unnumb. (in the second book) = 242 fF., 34-38 U. 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 ino»t 
extensive table of squares and roots that had appeared up to 1592. 


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. i cm., the text being 10. i x 15.6 
cm. 255 pp. (230 numb.), 34-35 11. London, 1592-95. Bot)k 
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 1 594- 

The author says in his dedicatory epistle, dated 'London this 
20 Augufl Anno. 1592,' that he has < vnder taken to write and publifhe 
fixe bookes of the Art of Arithmeticke : with this order and methods, 
that the firft, third, and fift bookes, fhalbe as a fummarie and groond, 
teaching, the true ingenious, inuentions, and the perfect iiguratiue and 
caractericall operations of the fame Art . . . Then the feconde, fourth, 
and fixt bookes, fhalbe of Arithmeticall quedions and demaundes, with 
the application of the definitions, common fenteces and inftnictiors 
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 fractiooN 
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 centurr 
(see Fig. 201). Book III would now be classed as part of algebra (5<?t 
Fig. 202), since it refers chiefly to irrational numbers. 

Other works 0/ i^g2, Buscher, p. 393, 1590 ; Capella^ p. 68, 1400 • 
Clavius, p. 375, 1583; Fischer (Piscator), p^ 247, 1549; Gemma. 






Shewing the iiigeiiM)miiiieDdoas9iid 

tkms,bywliicnioakuljKcbemie CblotioDoraolwcccs 

of Arkhmcdcallaiicftioiis: aficra moreper&ft, pl^uo^ 

bncfe» weUordeccd Anthmcdcadl way , then any 

ocbcr hcRcdbrcpabliflied : vcdc 



iDpcinced at London by Richard FicId^dvrcOii^ in 

ihcBbckemcnncareLudgate. >. 


Fic aoo. Title page of masierson 



p. 200, 1540; Nonius, p. 3i5» 1564; Psellus, p. 168, 1532; Ramus, 
p. 33i» 1569 ; Riese, p. 139, 1522 ; Tartaglia, p. 279, 1556 ; Rutilio 
Cosentino Benicansa, * Corona di tutte le scientie de Abaco,' Naples, 


5 . • 200 • • 9 Fmi xio 
soo « • 4 • « 80.0 
1 00 
300 • • 4 • • lao.o 

1 5 o • • 4. • * <oo 

So • • a6oo « • ISO 
FMii 3900. 
40O.i * S • « aooo 

KfrxrBcookeoot iiSlpooncL 
{33 TiivoiiurdianCsiiiadcacoinpaoicApacin30opoimdrar^ 
inoocthcs, and then fmttcth yet m 100 poand , and 6nioiieihei 
aRv that taketh out aoo poiid,and wim the reft ranaineth ▼*• 
tilltbeycarcscndfpatm xoo poondfor onenioDeth,andcfaoi 
putt^yet in7oo pooiul 9 and tf iiK>ncthcs afm diat takcdi <»t 
accrtaine fiinunc of money , and widi the reft rcniainet faninfl 
cheyeato endjind then findc to hane gained together 400 poM 
wheneofB mud haue 80 pound more then A • the<pielboii» 
hfn much money B tookc out of thccompani^ without lecfco- 
aiDg intereft vpcm intcreft. 

300 •• 2 • • 600 400 
100 So 

400. • 6 • .3400 320 
2 00 1^0 

Too . .4, • ^of 

x6o • « 3800 • • 240 

t 00 • •.! » • 100 

700 __j 

Too. • ^ ^.4800 K iriarBtookcout^4op^*"'°' 
160 y 

Fig. 201. From masi'erson 

8** ; Julius Caesar of Padua, * Gewiffe Erinnerung einer allgemeinen 
arithmetifchen Practic,' Cologne, 16** (see also p. 368, 1582); Johana 
Krafft, * Ein neues vnd wohlgegriindtes Rechenbuch,' Ulm, 



haoii« here fhcir munbtf written Oder dieiiL 
I. 1. 1. ♦ !• *• 7- *• 9. 10. II. II. 

ThefirftcanAer written ehus ^, doeth figpiifie anymmi. 

bo before whkb it u written, tobe the firft number giiicn. 

talecn^ or iaugined;and u called r^iur, or rooce.lbr chat all the 

otlwfCJraQerthaiK rbeiroriaoaUocof-^ringof ir« The(e« 

cond written cbm ^t is called My^ or fqoarc, and doth ligoifie 

aojrimnbcrbeCocc whidiit is written, to be die prodoA of dio 

firftmaliiplicacionoftheroote bjritrclfe: thatisofthcrootc 

two times taken and nrakiplicd. The chird written thascC,ts 

called ^#, and dothfigmfie the number following the fiime 

robe theproduaofthefecondnmltiplicatiooof the rootc»chrce 

eimestakedandmultiplied, that bofthe ^multiplied by the 

i^.Thefoorth is called «a^««i^» and doth fignibe the num • 


carion^that is the piodn A of the %. feure times taken and fflul * 

tipiied Tbefiftiscalled/Svyeiyr,anddoth fignifietfaenamber 

MiowingcbeiameitobetiK produft of the tonrtfa nmltiprica> 

cioo* Tbefixt is caUcd ^(tiiyStfim « and doth fim 

felkming the finic»to be dicpcodnft of the m mnltipUcatioa 


feoifie tbe number folk> wing the fimc^to be the prod u A of the 

«t mnktplioMioo.Thedght it called <#i^«A^^«#i^»& dotb 

fignifie the number foUowing the &ine,to be the produA of the 

ISmntfa mulciplicatioo. The nmth is called raiAfC4irAr , and dod^ 

%Oifi^ the number following the Cun^to be theptoduA of the 

e^ multiplicatioii.The tctth is called M^gfurfiUi, &dodi 

ficiifie the number foUowti^ the iame, tobetMpfoduA&c 

lie daienth, is called i/br>9£fr/)r third rurfolMk.!^ 

bcdled ui^tjmfKmie. The durteenth is called dfrnrfrUk^ or 

fioith fivfolide. You majr prooeede fiuther at your pkafurc if 

you markethat the fiftcanAer is the firft furfbUde, thefcttenrh 

the Iccood firiblide* and thenexrvnoompound number iol- 

bwin^rfiencitfivfidid^fo infinicdycucr the next vncom- 

Fig. 202, FkuM ma2>tlkm>n, shouim; au.khkvic mmh<u«s 


SIGISMUNDUS SUEVUS. Ed. pr. 1593. Breslau, 1593. 

A German priest, living at Breslau. Bom c 1550. 

Title, See Fig. 203. 

Colopfion, 'Gedruckt zu Brefslaw/ durch // Georgium Baw- 
mann/J. Jn Mitvorle-//gung Andreae Wolcken. Jin 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 Gematm 
used to foretell the famine of Poland represent the applications. 


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 being6.3 X 13.2cm. 
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 a« k- 
ward monetary systems of the time. See Paxi, p. 77 ; Mariani, p. iS.-. 

Other works of IS93* Gemma, p. 200, 1540; Tartaglia, p. r7*j. 
1556 ; Horatio Galasso, * Giochi di Carte,' Venice, 8*, with an editi. r. 
at Verona in 1597, 12^ and a French translation in 1603 (containir.^ 
several * giochi d'abbaco '). 


Ed. pr. 1594 Madrid. i6a5 

A Spanish merchant and arithmetician of the second half of the sixte« - 
century, bom at Valencia. He lived at Seville. 

Title. * Libro de // Arithmetica //especvlativa, y pra-/. tici. 
intitvlado, el Dorado // Contador, contiene la fineza y reglas ce 




iHPii ni ^^ipVfiiii VHP uii|civmiii pcvcHiiiiii 

MA OoPiC^- SdfitMtti, 


Fkj. 203. Title pa«e ok sievus 




N V O V A^ 


dellt Cecchini , d^ Lire 

died I efotdi vno^ 

Fino a / 12 / 8. Laqual feme 

anco per ogni (brtcdi 


Alcflandro AlamagoL 


Apprcflb Gio. Am. Racnp azctto. 1 5 95. 
Fig. 204. Title page of alamagki 


contar // oro y plata, y los Aneages de // Flandes.// Por moder- 
noy com-//pendiofo eftilo.//Compvesto por Migvel Gero-//niino 
lie Santa-Cruz, natural de la Ciudad, y //Reino de Valencia, y 
vezino de // Seuilla.// Al L<^ Don Pero de Baraiz,// Tenience 
mayor de Corregidor de la infigne Villa de// Madrid, Corte del 
Rci nueftro Seftor.//Aflo i643//Con licencia.// En Madrid, Por 
Fnincifco Martinez.// A coftade luan BautiftaTabino, Mercader 
dc 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 Tab^no 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- 
(\antile arithmetics of the sixteenth century. It is not, however, a very 
practical book, the author having been too much influenced by the 
thforctical side of works like Tartaglia's. 

Othfr works of IS94» 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., 
> ^97• 4% Ai^d others appeared after i6ot) ; Smiraldo Borghetti, 'Opera 
d'ablx&co,' Venice, 8* ; Jerdnimo Cort^, * Arithmetica practica,* Valencia. 

Works of JS95* Belli, p. 343, 1573; Helmreich, p. 306, 1561 ; 
Ma>terson, p. 400, 1592 ; Urstisius, p. 361, 1579. 

JOANNES BILSTENIUS. Ed, pr, 1596. Basel. 1596. 

A sixtecnth<entury German educator. 

TitU\ Syntagma // PhilippoV/rameum // Artium Li-//bera- 
hum, Methodo brevi ac per-//fpicua concinnatum //^tcr // loan. 
Kilstcnium // Marfbergianum.// In // Gratiam Tyronum partim 
ditlicilibus // vocibus Germanica adiecta // est explicatio.// 
Basilcx.//Typis Conr. Waldkirch.//CID ID XCVL* (F. i. r.) 

Description. 8*, lO X 16. 1 cm., the text being 6.8 X 12 cm. 
1 6 pp. unnumb. + 586 numb. = 602 pp., 26-28 11. Basel, 1 596. 

Editions, There was no other edition. 


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 i^gd, Helmreich, p. 303, 1561 ; Hood (translator ; 
see Urstisius), p. 361, 1579; Petri, p. 325, 1567; Ramus, p. 330, 
1569 ; Recorde, p. 219, c. 1542 ; SneUius (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.XCVIL//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 11.8 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 numerab 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 of 1^97* 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 Arithmetiqiie/ 
London, 8®. 


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- 
Beamois.' The dedicatory epistle is dated at * Kimpercorentin.* 

Title, * L* Henry-metre,// instrvment royal, et // vniversel, 
avcc sa theoriqve,// vsage, et pratiqve demonstrce par // les Pro- 
(Mifitions Elementaires d*Euclide, & regies familicres //d'Arith- 
mctique : & aufli fans Arithmetique : Lequel prend toutes 
mcfurcs Geometriques, & Aftronomiques, qui luy // font circu- 
lairemcnt oppoftes tant au Ciel, qu'en la Terre,//svr vne sevle 
station, par vn // feul triangle Orthogone, fans le bouger de la 
place, ny aller // mefurer aucune diftance de ftation, ainfi qu'on 
eft con-//trainct de faire auec les autres Inftruments Geome- 
triques.// DeTinuention/ZD'Henry de Suberuille Breton, Cha- 
noine en TEglife Cathedrale S. Pierre //de Xaintes : & Aduocat 
en la Cour de Parlement de Bourdeaux.// Item,// Vn petit traict6 
fur la Theorique, & Pratique que de T Extraction des racines 
<iuarr&s,// pour dreffer les Scadrons, & Bataillons quarrfe.// 
Dcdies au Roy.// Diev a difpofc^ toutes chofes en Poids, Nombre, 
& // Mesvre. Sap. 1 1. 21.// A Paris,// Chez, Adrien Perier, rue 
fainct laques en la //boutique de Plantin au Compas.// 1598.// 
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, 1 598. 
Editions. There was no other edition. 

Although this is primarily a treatise on mensuration by the use of a 

tr:gi>nometric instrument somewhat resembling a quadrans, it is properly 

1 IK luded in this list because it contains several chapters on fractions 

.md denominate numbers. These are of no special merit, and are 

- r.troduced as a preliminary to the calculations involved in the use of the 

* f Icnry-metre * which the author invented. The book is an interesting 

t rtort to perpetuate an inventor's name by a work of no special scholar- 

- ' :p dcfscribing an instrument of no particular value. There is, however, 

a \alue in the general study of all of these early instruments. Many of 

r r.em are easily constructed, the quadrans for example, and their more 

«_- \ tensive use in the teaching of trigonometry would be very helpful. 




Ed. pr. 1598. Freiburg in Uchtland, 1598. 

A Freiburg Rechenmeister of the latter half of the sixteenth centoiy. 

Titk. See Fig. 205. 


m onfobrti^m Aritbmetkif {nacbrott^ 

UM^fire VVdhetmi M*f$. Anno 


Fig. 205. Title page of lautenschlager 



Description. 8*, 9.4 x 14.7 cm., the text being 6.9 X 12.3 cm. 
4 pp. unmimb. + 58 mimb. =62 pp , 21-30 11. Freiburg in 
Uchtland, 1598. 


<5mM«t MM iftvtWMwtm 

99^ll( ^iKicr die km^ 3aM 

€0?ii Oft nKclmn afbmiltl 
CIcil spra^ifrf mit tcmvMbcmtf 1 

ep IMM Oir M« 5^9 Mi» Soff 9^ • 

10MCtfUi«9(hii»ai iff »a»(iMi 

e^am Wttj^ in 91c MMini srM' 
;Olr mirlcr Sa^lratill ffommnfcO/ 
;iUaifrc fk/foM>0 Smoffco w4t 


9 I 3Mm 

Fl<;. 206. From LALTENSCHLAtitR 

liditians. There was no other edition. 

This is the first arithmetic that I have seen comj)osed entirely in 
. \ 'lie. Various others had already contained verses, but thib is 


substantially all in meter. Like all sach educational vagaries, it is vrik 
in subject matter, the most difficult example in division being tLi: .: 
65443 by 6. The illustration (Fig. 206) shows the rule of three z 

CASPAR SCHLEUPNER. Ed. pr. 1598 Leipzig, i;:i> 

A Breslau Rechenmeister, bom at Niimbeig c. 1535. 

7V//(f. See Fig. 207. 

Colophon. * Gedruckt zu // Leipzig/bey Frantz // Schnellb. It: 
//Tyipxs haeredum Beyeri.// (Woodcut.) Im lahr// M. D. XC\i. 
(F. 99, V.) 

Description. 8*, 9.3 X 14.7 cm., the text being 6.9 x 12.2 cz. 
99 ff. unnumb., 22-26 11. Leipzig, 1598. 

Editions. Leipzig, 1 598, 8* (here described) ; Breslau, 1 599. S' 

Schleupner was one of the last serious advocates of the oki * L-^e 
reckoning' with counters. He was a disciple of the Newdorffer sch:- . 
of Niimberg Rechenmeisters, as he states in his preface, and had him- 
self taught at * Neyfe* and * Breizlaw ' for many years when he decyic^i 
to write this book. He makes an attempt at an easy method oi p^- 
senting the fundamental operations, hoping, as he says, to set forth tht 
doctrines of Adam Riese and Johan Seckerwitz (another Niimbcrr 
Rechenmeister) in simple fashion. The work is made up of a series .t 
impossible questions and answers between a father and his son. Tx 
latter always begins with 'Geliebter Vater,' and the father's replxs. 
while always clear, are often very extended, the entire 99 pages covef- 
ing little more than the four operations, together with a few insijT*'r- 
cant problems. As a simple presentation of line reckoning, however, 
the book has few equals. 

JOSEPHUS UNICORNUS. Ed. pr. 1598. Venice. 150^^ 

See p. 298. 

Title. See Fig. 208. 

* Parte Seconda,' of the same date, follows f . 204, the two bein^ 
bound in one, and the pages being numbered consecutively. 

Colophon. * In Venetia,// AprefTo Francefco de* Fiancefch: 
Senefe.//M DXCVIIL' (F. 413, v.) 

Description. 4^ 15.5 X 2 1.6 cm., the text being 10.4X 176 cm. 
1 2 fif. unnumb. -f 395 numb. = 407 fT., 29-39 U- Venice, I59^ 



t>ni> ^7(§r fTan^r/ fc§rtiii^ul^|ii 

?)araitf ifti (idMjcr £cfcr/ od- 

dxr dOdnc fdn (£m nuil^ntf (t| aufftkc omI 

!Qcnii(/aufittai»^ fow/ 061K tknate 'yMaitit^ Uf 

M(t/ fte fi4 firtt^/ Wt»Hm'XtaiumittaMt0u/ 

iCOd^misnVtt/ wUbtt jumt R^ 

Ccro0ldc^ (Knior/ lii.<C)aiC(<^ 

6(rpr4toei0 deficit/ 

©ur* ffofpor 6cf )laipncr/ 

yon t^ilrmBfrg /S)cui>|<l&m ec^irfmcw 

p<riii^rc^lan>. £0?cnifi9li(l)(n/vn» 

btun itt l\th€n "ivittib/ ya mm 


tttklMm.Wktf.ttUtf. StnMu 

3n vorlesuns Tinitti QBoUf <n ^tbtn /)Qb 

Fig. 207. TrrLE page or schleupnt.k 






Parte Prima: 


ott ancoralaPcatdca appartencnte a 



Fig. 208. Title page or unicornus 


Editions, There vfzs no other edition. 

lliis is one of the most elaborate treatises on arithmetic pubhshed 
in luly in the sixteenth century. It consists of six books, the first four 
making up Part 1. llie 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 ha\nng 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 
r(X)ts, surds, and proportion ; Book IV, of the rules of three and false ; 
and Book V, of business arithmetic, including exchange, interest, and 

The work was too theoretical to be popular, but it is an excellent 
source for the study of the development of elementary mathenuitics. 
rnicomus gives a number of interesting historical references. 

Oiktr works o/i^gS. Cassiodorus, p. 2 1 1 , 1 5 40 ; Fischer ( Piscator), 
f». 247, 1549 ; Nfoya, p. 310, 1562 ; Paolini, p. 393, 1589 ; Raymundus 
I nil ins, « ()f)era ea quae ad adinventam ab ipso artem universalera . . . 
ficrtincnt 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. Numberg. 1599 

One of the famous Newdorffer famUy of Ntirnberg Rechentncisters. 

Title. • Kiinftliche vnd Ordentliche An//weyfzung dcr gantzen 
Practic vfF // den Jetzigen fchlag vnd derfelbenn // herlichen 
^cfchwinden Exempel // vff s kiirtzt zufammen getzogen &c.// 
Mcincn lieben Difcipcln zu fon-//derlichem Nutzen geftelt. 
Oiirch // mich Anthonium Newd6rfFer//Rechenmaifter vnd Mo- 
.jjft dcr//Statt Numberg// Anno .M.D.IC (F, 2, r.) On f. 
I , r., is an eng^naved frontispiece with figures of Euclid and Py- 
thagoras, and a triangular multiplication table. On f. i, v.,are 
1 4. 11. of verse • Ad Stvdiosos Artis Nvmerandi.' 

Colophon. 'Gedruckt zu Numberg/ //durch Paulum Kauff- 
mann./'M. D. XCIX.' (F. 65. r.) 

Description, 4*, 1 5 X 19.3 cm,, the text being 1 1.4 x 15.5 cm. 
#05 ff. unnumb., 22-32 11. Numberg, 1599. 


Editions, There was no other edition. 

This is an excellent illustration of the work of the celebrated Niim- 
berg Rechenmeisters. It is divided into twelve books, of which * Das erfte 
Bfichlein 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 * lomates. 
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 U. 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 ca^ 
of Zucchetti*s treatise (p. 425). Naples was too far from the path of 
international commerce to produce many mercantile worls. But 
although Oberto Cantone was a Genoese, he was a ' professor deUe 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 bv 
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, bat makes 
no use of the Venetian or Florentine names, with the exception of * per 
I olonna.' Two or three short methods are given, but no such cxtende^i 
treatment of \'aried forms appears as in Tartaglia, or his great prede- 
cessor Paciuolo. Division, except in simple cases, is postponed to p. 1 4* • 








s mjigna $n materia di conti , tvfi tanto itlla Htgia Camera dtJU 

Sommaria, qttanto di Nt^otiantit Mtrcadantit^Arttgiam, 

9 ewu Ni^BtMobift erttambyin eiafiunafiaiza, 







' K H.^ V o L ip 
n^pfrelf^TAKQyxsio Long ho. Af. O. TC. 

Si f cndooo dal tLcdc&Lo Aucorc a Baochi ououi. 
Fig. 209. Title page ok cantone 


and is there treated in the modem 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. 

Bom at Talavera de la Reina, in 1536; died at Toledo, Febraaiy 17, 
1624. He was a Spanish Jesuit and a famous historian. 

Titk, See Fig. 210. 

Colopfton. * Toleti, Apud Thomam Guf-//manium, Anno . 1 599/ 
(P. 206.) 

Descriptiofi, 4**, 14 X 18.6 cm., the text being 9.3 XI 5.6 cm. 6pp. 
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. 180, 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 Hebcew 
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,i599. 

A Dutch arithmetician of the sixteenth century. 

Title, See Fig. 211. 

Colophon, * Eynde defes onfes eerften Vo-//lumens : Ghedrxxrt 
int Jaer ons Hee-//ren 1599. In defe vermaerde Coop-//ftadt 
van Middleburgh in // Zeelandt.'// Also in French : * Fin dc ce 
noftre premier Volu-//nien: Imprim^ TAn de noftre Seig-//ncur 
1 599. A la tref-renommfe // ville Marchande de Middel-//b<Hi^ 
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 5x15 cm. 4 ff. blank -h 18 unnumb. 
4-137 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. 









'^ ^W* 

Toleti» ^AfuiThttmum Cufrndmum. 

Fig. 210. Trru: page of makuna 



bamfcl^e ]^}aeti)cb / mttitniaDetia D'aOec 
ie»epd(jBiin ji^toetDuptieienDeittiPtanc||oij8/ 






ble Ics pieces, Ics plus nccdsaiiode la B^e 




«a^c mmtifittff ce coove/ bp Adriaen nan ^ Viyere, jeoni 
UMMinendchp IM nteutof »utr</tnDm luqpdDciiaBi^ 

Fig. 211. Title page of wenceslaus 


1 598/ 'le 10. lour dc Decembrc ftylo nouo. Anno. Christi- 1 598.' 
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 IS99' St. Augustine (see Reisch), p. 82, 1503 ; 
Harlaamo, p. 343, 1572; Benedetti, p. 364, 1580; Boethius (see 
kcisch), p. 82, 1503; Bungus, p. 384, 1583-84; Capella, p. 66, 
1499; Clichtovcus (see Reisch), p. 82, 1503; Faber Stapulensis (see 
Rtisch), p. 82, 1503; Finaeus (see Reisch), p. 82, 1503; Gallucci 
(S4.% Reisch), p. 82, 1503; Jacob, p. 298, 1560; Jordanus (see 
Rrisch), p. 82, 1503; Ramus, p. 330, 1569; Reisch, p. 82, 1503; 
Shieupner, p. 412, 1598; Strttbe, p. 391, 1588; Johann Hcere, 
* Rechenbijchlein von allerhand gebraiichlichen Fragen,' Niirnlierg, 
«• ; Andreas Reinhard, * Drey Regifter Arithmetifi her Anfang zur 
I'ractik,* Leipzig, 8*, with a second edition in 1600. 


Ed. pr. 1600. Haarlem, 1600. 

A Dutch arithmetician of Meenen, c 1550-1620. 

7//A-. Sec Fig. 212. 

Description. 8*, 9.4 X 14.4 cm., the text being 7.2 x 12.6 cm. 

2 ff. unnumb. -I- 202 numb. = 204 ff , 22-23 ''• Haarlem, 1600. 

Jiiiitiiyns. That the date of the first edition is 1600 appears 

ij\ the V<K)r-Rcden of the 1625 edition, where his son, Denys, 

^.i\ s that the book was published by the father 'eerst in*t laer 

I /%oo. ende dit is al de vierde mael dat het gedrukt is.' There 

v.t-rc: various editions in the seventeenth century, including 

•:io following: Haarlem, 161 1; Rotterdam-Schiedam, 1624.8* 

; ► ^23) ; Haarlem, 1625 ; c. 1630, 8° (p 423) ; Gouda, 1634, 8* 

:•. -4.24); Amsterdam, 1643, 8* (p. 424); Rotterdam, 1653, 

K-* (\>. 425); Amsterdam. 1675. 



iSft ^tkmw&l 

Btx^tttt met beel rcHootie 



KttttctiidtFiaiicoyfche School-mceflet 
toe Hab&iim. 

Tot Habrlbm, ^ ^ ^ 

S0|^ «fflWtemiim»oecfth}«lttr/fitfei9Mr 

Fig. 212. Title page of vander schuere 


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. 


Ed. pr. 1600. Rotterdam-Schiedam, 1624. 

See p. 421. 

Title, • Arithmetica//Oft// Reken-konft;//Vcrcicrt met vccl 
fchoone// Exempelen/zeer nut voor alle vlijtighe // Oeffcnaers 
cnde leer-ghierigheAenvanghers //defer Konft/ etc. Ghemaeckt 
d<K)r// Jacob Vander Schuere, Meenenaer,//Nu ter tijdt Fran- 
f<»yfche. School-mecfter//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 Dell.// 
Anno 1624.* (F. 208, r.) 

Description. 8*, 8.7 X 14.7 cm., the text being 7.6 x 12.9 
cm. 2 flf. unnumb. + 2o6 numb.s208 fif., 22-27 'l- Rotterdam- 
Schiedam, 1624. 

Sec above. 


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 
i s probably c. 1630. F. i is missing. 9 X 1 5.6 cm., the text being 
y 3 X 12.2 cm. 5 ff. unnumb. -f 252 numb. = 257 ff,, 35 11. 


JACOB VANDER SCHUERE. Ed. pr. 1600. Gouda, 1634. 

See p. 421. 

Title. * Arithmetica// oft// Reken-konft.// Door lakob vander 
Schuere, Meenenaer // Eertyts Fran5oysche 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 RammefejTi, 
Boeck-v^rkooper // 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. -f 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. 
-h 37 numb. = 38 ff., making a total of 254 ff. in the book, 
28-33 11. Gouda, 1634. 

See p. 423. 


Ed. pr. 1600. Amsterdam, 1643. 

See p. 421. 

Title. * Arithmetica//oft // Reken-konft.// Eneen kort onder- 
richt van't Italiaens Boeckhoude // Door lacob vander Schvere 
Meenenaer. // By den Autheur overfien, verbeetert en verroeer- 
dert.//Tot Amsterdam,// voor Michiel de Groot Boeckverkoopcr 
// op de nieuwendyk inde Bieftkens Bybel. 1643.* (Engraving of 
author with the following wording : * lacob vander Schvere Mee- 
nenaer. Out 67. laer. DoorsietdenGrondt. Anno 1643.') (F. i,r.) 

Description. 8°, 9.5 X 14.7 cm., the text being 7.2 X 12.5 cm 
272 flf. (216 numb + 8 unnumb. in the arithmetic), 26-31 11. 
Amsterdam, 1643. Bound with this is * Kort onder-richt //0\*er 
het //Italiaens // Boeck-houden.// In*t Licht gebracht //door 
Jacob van der Schuere.// t'Amsteldam,// By Michiel de Groot, 


Boeckverkooper // op de Nieuwendijck/tuffchen de twee Haer-// 
Icminer Sluyfen. A°. 1675/ (F. i, r, of the bookkeeping.) 
See p. 423. 


Ed pr. 1600. Rotterdam, 1653. 

See p. 431. 

Title, • Arithmetica // ofte // Reken-konft, // En een kort 
onder-richt van't Italiaens // Boeck-houden.// Door // lacob van 
dcr Schuere Meenenaer.// By den Autheur over-fien, verbetert 
en // venneerdert.//Tot Rotterdam,// Gedruckt by Picter Waef- 
berge, woo-//nende op't Steygcr/ in de gekroonde // Leeuw/ 
Anno 1653/ (F. i, r.) 

Description, 8^ 8.7 X 14,1 cm., the text being 7 x 12.1 cm. 
275 PP- (219 numb. -h 8 unnumb. in the arithmetic), 26-33 "- 
Rotterdam, 1653. Bound with this is 'Kort onder-richt // over 
he I // Italiaens Boeck-houden. // In't Licht gebracht // door // 
Jacob van der Schuere.// Tot Rotterdam,// Gedruckt by Pietcr 
van Waefberge,//Ordinaris Drucker/ woonende op*t Steyger// 
in de gekroonde Leeuw/ Anno 1653.* (F. i, r., of the book- 
See p. 423. 


Ed. pr. 1600. Brescia, 1600. 

Bom April 21, 1550. A Genoese arithmetician. 

Tit/e. See Fig. 213. The privilege is dated • In Genoua nel 
t!i 30. di Genaro. M. CCCCCC 

Description, Fol., 24.2 X 33.4 cm., the text being 17. i x 26 
< m. 444 pp. (412 numb.), 33-37 11. Brescia, 1600. 

Editions, There was no other edition. Brunet says this * Prima 
I*arte * is the only one that appeared. 

The work has several interesting features, not the least one being the 
.1 : « >]<>gy * Al generoso lettore,* in which the author speaks of the criticism 
! a Me to be directed against a (lenoese author on account of his pro- 
% n( ial Italian. The 'Prologo' is a curious dissertation on the * Arti, 




CiTtADnvo G«|lOVItf. 

Fig. 213. Title page of zuchetta 


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 Gennan Rechenmeister at Solothum. Bom c. 156a 

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 \ 3 / 
whether this is a typographical matter or a notion of Schey's \y/ 
there is nothing to indicate. In the * Regula Falsi ' there is ?\. 
a rather early use of -i- for the minus sign. In general the / 2 \ 
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'Arithm^tique ' (a translation of the other?), ib., 8**; Georg 
Gleitsmann, * Kiinflliches Rechenbuch fowohl auf Linien als mit Ziffem 




M fc^txtti 5^cd<ltt «tt(f alletUi* Km 

dn$ nt^itm @irunt» gdn^ fmitb tmlgoim^aatc 

tartmun »fr<t izoo> ^ctti)?U/9<n|flMrimrtKT|ttotfi(fe| 

\fStn @d^"* *'"«^ iKw^mmaflflr ju @oto 

beg TtM^MM/mit J?Uff bcgTUlmci^flSM 90«I»M(> 

Anno M. DC 

Fig. 314. Title page of schev 


nach deb Rami Arithmetica geftellt,' Frankfort, 8* ; M. Johann Taf, 
' Schones neues • « . Rechenbuch/ Cologne, 4^ There was also pub- 
Hbhed, s. a., but c« 1600, a work by Vincenzo da Bergamo, *Arith- 
aietiche instruttioni.' 

Inhere were many other arithmetics published in the sixteenth cen- 
tury without date, including the following : Anonymous, * Art et Science 
de Arismeti<iue,' Paris, li**, 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 ak]uatimo 
di e regole sopra larte del numero altrimeti chiamato algurismo,' s. L, 
^j ff. (Riccardi) ; Joachim Ammonius, * Isagoge Arithmetices . • . cum 
praefatione P. MeUnchthoni,' Wittenberg ; Angelus Mutinens, p. 140, 
1 535 ; Sarafino da Campora, ' Delia ragione dell' Abbaco,' and a work 
on the calendar (Messina, 1559, and Rome, 1560) (Riccardi) ; Lauro 
C^irini, • I)e 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). 



KIX'LID. I^itin MS., c. 1260. 

See p. II. 

TttU. *In h(K libro otinct* geomet'a eudidis // cu ..mcnto 
maj^rl camixini.* (F. i, v.) 

Colophon, *CExplicit geomctria cuclidis cu c6mcn//to magiftri 
camiKini.* (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 '•• Written on 
vellum c. 1260. 

Editions, This work being primarily a geometr)', I have not 
;,'iven a list of the editions. For the arithmetical b(X)ks, which 
(except for Book V, on proportion) are not included in this 
manuscript, see p. 237. 

'Vh'xs is a latin manuscript of Euclid, with the commentary (proofs) 
<if C'ampanus, written on vellum about 1260. The translation of the 
thc«>rems is that made by Adelard (.-f^thelhard) of Bath, c. 1 1 20, but in 
rhe early printed editions of the ' Elements * it is generally referred to 
as that of Cam})anus. 

Of Campanus himself not very much is knoH-n. His first name was 
].ri»l»ahly Johannes, and he is known to have prepareti a set of planet- 
.iTV tiibles, and to have been chaplain to Urban IV. (See Boncom]>agni's 
A////W//f<7, 1, 5, and XIX, 591.) 

'I 'his manuscript has been studied by Mr. C. S. Peirce (.SV/V//<y, n.s., 
XIII* 809), who believes it to be the copy given by Cam|)anus himself 
t.# Jacipics Pantalton when the latter was Patriarch of Jerusalem, hence 
|^-f«>rc Aug\ist 29, 1 26 1, when he became Poj>e Url)an IV. He l>ases 
r s 1 relief on a sentence written in a cursive hand just 1k*1ow the colo- 
• 'r\<9rk^ containing the words •Jacobus Dei gratia Patriarcha Jerusalemi- 
t. .'uni-* The complete sentence is : 'In noie dni ame Jacobus dei g* 



patriarcha Jerufalemitar omibus xpi fidelibus ialutem 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® ifte fuit Di armachani,' — • This book belonged to Dominus Arma- 
chanus/ (For a page of this MS. see Plate IV.) 


Latin MS., c. 1294, 

See p. 25. 

Title. The manuscript begins : * Incipit prolog in arithmetica 
boetii.' (F. i, v.) 

Colophofi, * Explic* arifme^* // boetij ad fymacu^ patriciu^.* 
(F. 28, r.) 

Description. Fol., 19.8 X 27 cm., in double columns, each being 
7 X 20.1 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 L) 

Editiofts. 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. Ill, V.) 

Description. Fol., 19.8 X 27 cm., the text being 17.5 X 19.5 
cm. 1 1 1 ff . (the first 28 ff. being the arithmetic of Boethius 


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, 
|xissibly for the date 1294, the M being omitted as is often the 
case. (See Fig. 2 1 5 and Plate V.) 


Latin MS., c. 1300. 

See p. 25. 

Titk. The manuscript begins: 'Incipit libf arismetice art . . .* 
Description. Fol., 13 X 18.6 cm., the text being 8.9 x 14. i 
cm. 37 ff., 32-36 11. Beautifully written on vellum. The work 
i.s 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 
I'late VI.) The commentary, which seems to have been added 
aix>iit a century later, has some Hindu-Arabic numerals. 

I'AOLO DAGOMARI. Italian MS., c. 1339. 

r\<>U) deix'Araco, Paolo Astrolot.o, Paolo GsoMETRAt Paolo Aris- 
MCTRA, PaI'L of THE Abacis. Bom in Prato, c. 1281; died at Florence 
in 1374. or, according to some writern, in 1365. He was a celebrated Flor- 
entine arithmetician. *geometra grandissimo, e peritissimo aritmetico . . . 
diltgentiftsimo osservatore delle Stelle« e del movimento de'cieli/ as Villani 
(A/ Viit d^Comtni dustri Ftorenttni) calls him. 

Tide. ' Trattato d* Abbaco, d* Astrono-//mia, e di segreti natu- 
re. 1 1 ^'e medicinali.* (F. i, r.) 

Description. Fol., 21.7 X 29.3 cm., the text being 15 X 21.4 
ni. 138 ff. (7 blank), 32-35 11., clearly written on paper, c. 1339 
: -* >ssibly copied later). 



cdinar* <>■ > ** ^ .»*»-^***y«>***»*j 


2? ^-^ '^-^^'tSir;5r"i^& 


Mrt0 onintni* «iiii«M»«jiq»rfi4H*fir^«d 
^ i-r-* ' cii?iii*te«#«t-|imr€i- ^^"^^^Sll^S^ 

Fig. 215. From the 1294 manuscript of eucud 

Bernardino Baldi testifies to the esteem in which the authw of this 
work was held, in the following words : * Di patria Fiorentino fu P^uiok> . 
il quale, per V 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 ha^'e been 
filled in from time to time by various owners. 


The arithmetic begins (f . i , r.) : ' Kl nome sia di Dio et a reverentia 
//della fuo potentia et della ianta trinitade. Et dello fuo madrc// 
uirgino fempre (anta maria Et del beato Sco Giouaiii batifto // . . . Al 
chominiciamento del noibx> trattato . . .' On f. i, r., is a table of 

^ ••4^.1- fi#e-i.#4t < y^ ^^r 'T^^'^'^^^T^^^rr^t^-^^- 

T)wiwiit y u>»i ^ W V mr "y j^ ^* ^ * iii»tn['y*uiit«yf<'g-/l^,^^M#^-' 
#»«•# Uii i M ii»cnc Aifii '««|^'{^«»MaA t,L|»iU f ^ <l* " <i' i^Jr t ^ 't* 
«i<^mf|fr^«i»r-5^'fMn«m« Y*^n0»tm nm U ^^^ *^* ^^^^^^^ ^^ 

Fig. 2i6. From the dagomari manuscript 

money. F. 2, r., — f . 3, %•., is a table of contents : * Quefti sono echapitoli 

del noftro tractato/ and this states that the work includes the * Re- 

^holvzze del Maestro Pagolo Astrologo.* (ff. 121, r., — 131, r.). Tliis 

p*art of the volume is an ordinary commercial 

arithmetic such as the Florentine teachers pro- 

4 ! need in the fourteenth and fifteenth centuries. 

The writing and the forms of the numerals indi- 

€ Me this period, and are not unlike that of a 

f « Mirteenth-century computus (see p. 445) in this 

Iilirary. This may, however, be a fifteenth-century copy. The exam- 

j lc5 in division are unique, since they follow neither the galley nor the 

* a danda * method, as is here shown in the case of 49289 -h 23 = 2 1 43. 

The pe< uliar position of the remainders should be noted. The o indi- 

< ates no remainder, and the 2, 2, and i are excesses of 7*s in the proof. 

I 4" 



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 


yr|»^1<iii p » ; |lf»-f^ (^*«***) Pt^ y*ow»« f^f^U o V 'prW ^ I ^ 1 1 i j fmm twutr 

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 



(f. 73, r.) by further business problems. The work contains several 
curious illustrations (Figs. 217, 218). 

'llie 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, 
v^here the dates in these examples are all, save two, 
^^^9• I'hc writer has also used approximately these 
<lates in other examples, in part as follows: 1329- 
133' (^ 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 19-year cycle 
I f. 123, v.) begins with 1337, as is easily computed 
trom a marginal note in a different hand bearing the 
«lates 1394 and 14 12. This part of the work ends 
on f. 131, r., and there can be no doubt that it was 
uraten about 1339, the date so frequently used in 
the problems. 

F. 121, r., begins • Regholvzze del Maestro Pagolo 
Astrologo,' Ilagomari being referred to in the table 
< »f contents as Paul of the Abacus, * Regholuze del 
NLicftro pagholo delabacho . . .' The * Regholvzze* 
v^as first published in Libri's Histoire^ vol. Ill, p. 296. 
Dagomari is included in the list of Bernardino 
ilaUli's (1589) biographies, published in the Bon- 
c onifiagni BulUHno^ XIX, 600. 

Ff. i3if v., and 132, r., are in a different hand. 

J hese and the following leaves were originally blank, 

.-r.«l after ff. 132, v.,— 133, v., had been written 

•.;K>n, some oi^-ner used these two blank pages. He 

; also left his dates, viz. 1400, 1402, 1406, 14 12, 

I 435. the 1402 being used several times. This was 

:riertrf<>re uTitten about 1400-1435. In this occurs 

tr e fir^i per-cent sign I have met, other than p. 100, 

^* h .« h Ls in the earlier part of this volume. This 

• .r known writer of about 1425 uses a symbol which, 

' \ riatural stages, developed into our present 

\K\, 218. From 




Instead of writing *per 100/ *p 100/ or *p cento,* as had commonly 
been done before him, he wrote * p c^* for • p d,' just as the Italians 
wrote f , 5, . . . and i®, 2", ... for primo, secundo, etc. In the manu- 
scripts which I have examined the evolution is easily traced, the c-^ 
becoming g about 1650, the original meaning having even then been 

"'jfer -7-- ^ « 

■Hr.|^^fvwwr ^[M^^v^ *tW^^Gii» t^ - _ j 

From the addition (c. 1400- 143 5) to dacx>mari 

^g^y^^WT^p^nf^- ^- '•//• t^^n ^^t**— ^^iT 

lost. Of late the ' per' has been dropped, leaving only g 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. i33» 
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 ano. f. 1447 finii 
circuluf anni.' 

EUCLID. Latin MS., c. 1350 

See p. II. 

Tit/e. None. Fragments of the * Elements.' 

Description, 4**, 16.5 X 21.4 cm., the text being 10.5 X 15-S 
cm., with marginal drawings. 6ff. 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. 


to ' 




y a<»< 

-^- tf"73 


Kh;. 2JO. From a 1684 xnsistKin, n<ii muK i\TM.(*.nu 


ALBERTUS MAGNUS. Latin MS., c. 1350. 

Albertus Teutonicus, de Colonia, or Ratisbonensis. Bom at Lau- 
ingen, Swabia, c. 1 193 ; died at Cologne, November 25, 1 280. 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 Caelo et Mundo/ (F. i, r.) 

Description. 22.4 X 32 cm., the text being 14x21.3 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-centun* 
hand, with fine initials in red, black, and blue. The first folio contains 
part of the calendar, and a few random memoranda, including an oki 
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' gTIate e ultim ) finis in tpr aut actu aut po. 
quia si e comipta 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, tunc habet actu finem : et si est adhuc comiptabilis ' — 
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 ' (* quae est substantia separata,* the preceding miss- 
ing words of the sentence being * quae 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 planeu. 

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 // Ratisbonenst^ 
Episcopi, ordinis Prxdicatorum,// opera omnia,// . . . cura ac labore 
Augusti Borgnet.// Volumen quartum // Parisiis . . . MDCCCXC* The 
first edition of the Opera Omnia appeared in Leyden in 165 1. 

EUCLID. Latin MS., 1375. 

See p. II. 

Title. The work begins : * Functus eft cui^ ps no e~.* 
Description. Fol., 20.9 X 30. 1 cm., the text being 1 1.9 x 19 ; 
cm. 39 ff . unnumb. -f i blank = 40 fF., 44-47 11. A Latin man- 
uscript, written on paper, c. 1375. 


This manuscript includes the first five books of Euclid. This is fol- 
lowed by a treatise upon astronomy and mensuration, also in I^tin, in a 
different but contemporary hand. The mensuration includes some work 
on areas and volumes. 

ANONYMOUS. Latin MS.. 1384. 

TitU. 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 II., 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 (vothic 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 
dhj 1000.300.80.J * ; 'anno dnj 1000.300.80.4' (see Fig. 221). The 
text is in I^tin, 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. 

Tith\ 'CQui chomincia ilpologho dl conpoto//dlcorfo dl 
folo J dlla 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, ap[)ears in this manuscript in the following 
Italian form: 'Trenta di a noumbre apile // giugno & fcttcnhrc di 
ucntotto( ) vno tutti gUaltrj fono trentuno.' (F. 12, v.) (See 

GIOVANNI, the son of Luca da F'ircnze. Italian MS., 1422. 

The son of Maestro Luca, a celebrated Florentine arithmetiv lan, meniitmrd 
on p. 468. 

Title. 'Trattato di aritmetica/ (F. i. r,) 
Description. 4*, 14.9 X 21.9 cm., the te.xt bcinggj x 16.3 cm. 
145 ff. 27-30 11. 







^tiiTii/fcttt ht ^fliw H^iti^ Vw«» 

, ,1„^ i«^,„f-«^ ^f(A?^ <^f^ S-*?--* 
Fig. 221. From the 1384 comfutus 




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. 

9^ fUf^m^ 9^^ »^«rt 

M9»r»iK -jW«»»«^Ct. 4 fti »» n> t^fViN 
ft iniiii > ii» H»f»H y*»»^ » »* f » w f f i^ 

j^m^ ^Sf^C^ ^ i*.||rjim»ii#fc 

^mi ^m^ ptn^ f *M;. *^ 


f^flW ^r^'^^ 

' » 

Fig. 222. First pace of the 1393 computi's 


On f . I , v., the writer says he proposes to treat of arithmetic or the 
abacus : ' arifmetricho volgharemente e chiamata abacho.' The word 
< 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, far 
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 14 18-142 6 (flF. 113-122). The 
date is, however, fixed exactly by the closing lines of a folio near the 
end : * quefto libro IfcrifTe 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 (PI. VII). The last five fobos, 
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. 

Tit/e. * 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 : ' IQcf- 
triflimo ac fereniifimo principi metuendiffimo domino domino lohanni 
patruo domini noftri regis ffrancie et anglie regent! Regnum ffrancie. 
duci bethfordie Rollandus fcriptoris veftre celfitudinis phyficus vfcx- 
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 op 
no practical problems. He also treats of irrational numbers, a topk 


which is now considered part of algebra. It is doubtful if there is a 
manubcript extant which throws more light upon the nature of French 
uni>i!rsity mathematics at the time this was written. Kollandus 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. 2 23 shows 

the forms of the numerals used at this time. /\| «^—|OOQl|/| ITT 
It should be noticed that the numerals that ^ — ^/gl tfl( . 
have changed materially in form since the ^^ r^ 0^ 

twelfth century are 4, 5, and 7. These arc P^ -2€iZI£^, 
shown opposite the letters b^ /, and /, re- ^•— — ^i}^ ' * 
si^Ttively, in Fig. 223. The changes in ii — ^ ^7^?^* 
the other forms have been more e\idently Vy — 7^^ • 

due to the fashion in handwriting. All of T 

the forms are, however, quite different from O ~ ••> ^ ' 
the primiti\'e ones found in the cave inscrip- M ■ $ I QJL' 
turns of India. '^f-'^^^^QO^^ 

ANONYMOUS. k ^| ^g[ 

Italian and Latin MSS., c. 1430 j ^ 6ii 

and 1478. r» ^tC^ 

Title, The first folio is missing. xJ 

Description, Fol., 17 X 22.I cm., the ^^^T^ '^^ • 

text being 11.2x14.4 cm. 183 fT., ^-p""""^*^- 

25-28 11. ^ ^^ 

Q - \^ 
This volume consists of two Florentine ^ ^ ' 

manuscripts, one on commercial arithmetic ^ ""** ^ • 

and the other on the computus. The fK)r- P "* "* jC • 

tion on arithmetic was prolmbly written ^t^«.« 1^ 

t)etween 1420 and 1444, these l)eing the ^ «^,*«rf- 

extremes of the dates in the examples in VWH.KTTi. 

expiation of payments. It is not unlike the Fu;. 223. Fk<>m thk 
other contem|)orary Florentine anthmctirs >4^4 R'»iiamm s 

ciescril)ed on pp. 443 and 464. The author 

aNsumes the student's ability to jicrform the fumlanienta! o|H.Tations 
with integers, although, after numerous multipli* ation lahlt^s, he gives 
(see Fig. 224) an example under •Miihiph* ha p nvulo <ie lurit h<M h<»]o' 
(the Florentine name for our jjrescnl niethiHl), ami one umler • Mulii- 
plicha p modo de (Juadrato* (the •gelosia* methoil of the Venetians). 





,00 »f^- fits 

^1 I I f o< 


KlHp:<^ii^m«(j« Vt ^.tMtJt, 

Fig. 224. From the anonymous manuscript or c. 1430 


In the latter part of the work, as if it were a new topic, the author has 
introduced (f. 1 26, v.) a chapter entitled * Partire a danda.* He closes 
with several folios on mensuration. 

llie 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 ct compositione 
astrolabii plant,' 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 
J<)h*is gmiind.' 

Description, Fol., 19.1 X 25.7 cm., the written part being about 
14. 1x18.5 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 medise\*al 
romputi 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 manuscri{)t is a computus, not very extended. 
It relates rather to the calendar itself than to the computations ui>on 


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. 


See p. 31. 

Title, An algorismus, beginning with the words, * Omnia que 
a primeua rerum origine processerunt.' (F. 3. r.) 

Description, 4^ 14. i X 20 cm., the text being 10.5 X 17.6 cm. 
21 ff., 25-39 11- 

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 a^^iears, 
together with three dates : * Item hainric9 mugling albronim9,' 1442, 
1443, 1444. The * Algorismus ' begins on folio 2, r. : * Omnia que a pri- 
meua //rerii orignj,* and ends on f. 10, v. : • Explicat algorafmns.' Then 
follows a picture (PI. 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. 11, r. B^inning 
on f. II, v., is another treatise on arithmetic, giving the fundamen- 
tal operations and some work in progression, and ending with a multi- 
plication table (ff. 18, 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^ Extrictio 
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 numerorQin 
quod algorifmum vocant,' and the work consists merely of four folkc* 
containing the chapter < De arte numerandi.' There was also published 
at Antwerp in 1547 (with later editions, Paris 1550, Venice 1564, 


Wittenberg 1578), Sacrobosco's * Libellvs, de anni ratione: seu vt voca- 
tur vulgo, Computus ecclesiasticus.' His work on the Sphere was pub- 
hshed in 1488 at Venice. For a discussion of the authenticity of the 
Algorismus, see De Morgan, p. 13. 

As stated on p. 3 1 , the date of the death of Sacrobosco is uncertain. 
It is either 1244 or 1256, according as we interpret certain lines on his 

• M. Chrtsti bis C quarto deno quater anno 
I)e Sacro Bosco discrevit tempera ramus, 
Gratia cui dcderat nomen divina Johannes/ 


Ti/U. • Qoiies Marci Antonij rozoni artiu // doctoris sacre the- 
(tlogie magn et papi; phiam legentis.' (F. i, r.) 

Description, Fol., 20.3 X 28.1 cm., the text being 13.7 X 24 
cm. 99 flf.. 32-33 11. 

This is not an arithmetic, and it has been included in this list only 
lic<*ause it is semi-mathematical and is bound with Bradwardin's treatise 
4>n the theory of proportion. It is a Latin treatise on the theory of per- 
>{»ecti\%. 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 
c f immunis ' of John Peckham, archbishop of Canterbury. A memo- 
randum in the Boncompagni catalogue, probably from a leaf removed 
vh ith the first treatise, shows that the manuscript was copied in 1447 : 
*■ scripte per me antonium confaronesium ut (vocatur?) de lavilata Anno 
<lni M"ccccxlvij.' The other manuscripts bound with this are described 
\»^\om and on p. 452. 


See p. 61. 

Titk, None. A on proportion. 

Colophon, • Explicint proper"** thomi brardi //scripte p mc.* 
iF'. 115. r.) 

Description, Fol., 20.3 X 28.1 cm., the text being 13.7 x 22.8 
cm. 15 ff., 33 "• 

This forms \aj\ of the volume last mentioned, and is written in the 
,^^nne hand. Bradwardin's treatise on proportion was published in 
1*3JTS, 1495 (P-6i). 


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 volmne mentioned 
on p. 451, and is written in the same hand. 

ST. BERNARD OF SIENA. Latin MS., c. 1450. 

Bom 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: * ST .33.' -de 
reftituti5e.// Dotii .q. in quadragefTia ordo dicendo^ p feptimana 
fequete5 1 li** de x'ana re//ligione .a. R*!" p. S. B.n<> de fenif ordonif 
mino^edito.' (F. i, r.) 

Colophon, * Explicit tra//ctatuf de uf5if 2 5ctib5 f$u3 Bntuj 
Bnardini5 (?) de fenif. ordif minory.' (F. 129, r.) 

Description, 4^ 14.8 X 19.8 cm., the text being 10 x 13.8 cm. 
129 if., part vellum and part paper. After f. 60 the pages are 
arranged in double columns, each being 5 x 14.5 cm., 44-46 U 

This beautifully written manuscript of c. 1450 is included in th:.^ 
list because it contains several sermons bearing upon the mercanti'r 
customs of the time, including * De usuris,' * De cambiis,' *De cod- 
tractibus,' and * De mercatoribus.' 

Part of this work was translated into Italian in the fifteenth centun . 
as appears from a codex in the Biblioteca Comunale at Siena. One f 
the sermons was published by Riccomanni, in the Scelia di Curiosita U: 
terarie inedite o rare (no. 13) of Romagnoli, Bologna, 1862. This scr- , 
mon, ' SuUe Soccite di Bestiami,' contains considerable information :^ | 
to the business problems of monastic institutions of the fifteenth ctr- | 
tury. A copy of several of these sermons, made by one Eustachio ili 
Feltre in 1469, is mentioned on p. 466. 

ANONYMOUS. Latin MS., c. 14.-^ 

Title, None. On the Quadrivium. 

Description. 4*", 14.7 X 21 cm., the text being 14.5 x 17 cc 
47 pp., 18-40 11. 

This is a Latin manuscript on the Quadrivium. It is written '' 
paper, in a German hand of about 1450. It includes a brief treatmrz 



of arithmetic (5 ff. + i blank, the text occupying ff. 18 — 32, 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 : ' Pvnct<) eft c9 
()arsD5eft/Lineaelongi<'^fn latitu*^)pfCiditate.' It closes: 'CExplicat 
priro^ liber euclidis cu ometo capani.' 


See p. 31. 

TitU. * Spent mundi secundum Joha//nncm de facrobofco/ 
(F. I, r.) 

Colophon, • Explicit tractat** de fpa ^m 1 1 ]ciCitTti dc facrobofco 
am; (F. 35, V.) 

■in ^M l ^pt^Mfi, 

Fio. 225. From sacrob(sco's Sprra^ c. 1450 

Description, Fol., 35 flf., written on vellum. 

I*rinted works of this kind have not been included in this catalogue. 
I h:s manuscript, however, shows the state of the numerals about 1450, 
in<l therefore is mentioned. 

'Fhe treatise begins (f. i, r.) : * I)e fpera in quatuor capitula diftin- 

. 1 : *• // dicentes p'mo quid fit fpera/ quid eius // centrum/ quid axis fpere 

et quid fit po//lds.' 

Ilw first four folios have a marginal commentary, closely written in 

L La cer haiMi. The figures are carefully drawn throughout (see Fig. 225). 


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 1 488, and often thereafter. It 
was the great mediaeval work on astronomy. 

ANONYMOUS. Latin MS., c. 1450. 

Title, * Incipiut fioref Arifmetrice.' (F. i, r.) 
Description. 4°, 15 X 21.8 cm., the text being 8.4 x 15.5 cm. 
70 fF. unnumb. -f 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 pro>'e to be a 
copy of some mediaeval work of importance. It resembles in scMne 
places the work of al-Khowarazmi, a manuscript of which follows this 
one in the same volume. 


Latin MS., 1456. 

Abu <Abd Allah Mohammed ibn Musa ai^Khuwarizml Bora in 
the province of Khwarazm (whence his name), died c. 831. The most cele- 
hrated algebraist of his time, and the first to write a book bearing the title 
Algebra. From his name comes the word algarism (see p. 7). 

Title. ' Liber mahometi de Algebra et almuchabila x compafif 
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 BibKotheqiie 
Nationale {Histoire des sciences mathematigues, I, note XII). It more 
nearly resembles the one which I found in the Columbia Universiiv 
Library in 1904, and showed to be in the handwriting of ScheuUrl. 
The two deserve to be edited and compared with the Rosen translation 




Kk;. 226. FiRsi pa«;e ok tiik 1456 ai.-kh<>**k*/mi 


(London, 1 831 ). 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 1 501, 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 11. 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. 2-1 1); 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. 1430. 

60m 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 cui^ ut extant //plurime res magnifice gefte 
ita//et multa pafTim ab eo fapieter i// dicta memorantur. . . .* 
(F. I, r.) 

Colophon. * Petrj pauli uergerij de uiftinopoli ad uber-//tinu$ 
Cararienfem de ingenijs moribs x 1 1 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. 


The treatise refers to ' arifmeticha ' and ' geumetria ' (f. 1 5, r.) among 
e liberal arts. It is bound with several other manuscripts. 

AYMUNDUS LULLIUS. Latin MS., c. 1450. 

Raymond Lully, Ramon Lull. Bom 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 

•n f. I, r., reads : * Ars breuis Raymundi Luli quam scripsit Pisis 

/in monasterio Sancti Dominici anno 1307. ^. c.23.' The Ars 

f reins begins on f. 5, r. : *Deus cu tua gra fapla et amore Incipit 

irs breuis que eft //ymago artis gnalis. Na ifta fata ab intelectu 

ib'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*, 15.4X 20.8 cm., the text being 9.9 X 13.2 
cm. 24 if., 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. 


A Valendan priest. 

Title. 'Artificium artis arithmeticae.' (F. i,r.) *[D]Eus qui 
es unus . . . Incip artifi^r^ atis alfmet'ce De alphabet©.' (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 f ratrum mino^ 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. 


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 ' : * None fbiecta 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, * Cdeo grat. anj. (amen ?) Ano. diij. 1442 . . . CEx- 
plicit lib.9 de ymagine mundi deo grat . . .' 

Description, 8^ 14.5 X 21.5 cm., the text being 1 1.5 x 19 cm. 
1 58 ff . in the entire manuscript, 1 1 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.//Cde 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 catal<^;ue 
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 1 1 x 15,6 cm. 
Ill 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 equatioiD of 
payments involve dates from 1418 to 1425 (flF. 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 bv 


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 aiothmetic of c. 1456 
ANONYMOUS. Italian MS., c. 1460. 

Possibly by Raifaele Canacci, a Florentine mathematician. 

Titk. 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 fJ., 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 : 


"»- • 



4^ • 

!»• • 


1 ' 

4- • 

4-1 . 


5. • 


• "'-♦f 

4-1 « 


. v»>» 

r . 

♦ I 


♦-/ . 


. -..^/T. 

> • 

4- 1 

• -^«> 

4-1 • 

/ • 

• "^•r* 

% • 


• 1^» 

t-l . 

4- • 

. ,r^ 

1 ' 



4-1 • 

■> • 

. .^}. 



4-1 • 

•%. • 

• %•«.• 

^ • 

*■? • 


+1 • 

!»• . 


1 • 


^l ' 


9- • 


4-1 • 

5« • 



4-1 • 

■»-i r 

4-> - 



V . 


4-1 • 



/•• . 


> • 

♦1 * 


4-i • 

f • 




^1 • 
4-1 • 


. 1 >>-• 


Fig. 228. From an itauam arithmetic of c. 1456 

t • 


1 ' 


#- • 


< ■ 


c • 


> • 


s • 


7 • 

♦ > 

i» • 


4-> • 

!•• • 4->»» 

4-> . 

?• • ■»"^V 

4-> • 

%• • l>r» 

1-> • 

7.. IX^o 

4-> • 

<• • •«.%•».• 

♦■> • 

f . -i-lf* 

4-> • 

^» . 1 %%• 

+•> • 

j» • t4-t • 

4-> • 

•»- • • 54-- 



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 figni^chatiue/ 
and the 0, ' che i arabia fidice ^ero.' 

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 terzgo capitolo del pimo libro/ doue fitratta 
del modo & hordine del trarre el numero minore del numero maggiore.' 
(F.s.r.) The method is that . ^^w wy 
of borrowing and repaying. \^ * '^-.a'2^^% 

4. *E1 quarto capitolo 
del primo libro di queilo 
trattato/ Doue fitratta del 
modo & hordine del @chare 
e numerj.' (F. 6, v.) The 
column form of the multipli- 
cation table is first given, 
and is followed by various 
methods of multiplying. The 
names ' El berichuocholo ' (f . 
10, V.) and 'p quadrato' 
(f. II, 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 quadra to ' 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. *E1 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 difiinitionj fapute.' (F. 17, 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.) 

229. Multiplication 
quadrato,* c. i460 



III. * El terz^o libro della praticha darifmetricha. Nelquale (itratta 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 ieptirao 
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 jCtoo for i— 21 yrs. at 

rates varying from '5 p 100. laiio * (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 




* ^„. 

fete j*?^ *^ * f^*" ^^ ~ '^ •w 

I p^t%m Uwm «-£» ft — p W« tf» *M li #Lf*TiBjyn^ f»U«a»« V* 

k^t-Hfelld <*^ •UrfteffV (*«■ f i l l i y<l« > iU*to •«tt4 ^»U^Mir\« 

1*1. MF !\. 1 KmM \N WmWM.Ms MWlMKiri. v. ||rM> 


XIII. Algebra continue^!, * la rcgola de Algebra amucabale/ (F. 279, 
r.) 'lliis is a very interesting treatment of the subject ' fee ondo ghuglieU 
mo de lunis* and ' Lionardo pilano/ and it throws some light upon alge- 
bra as studied in the fifteenth century. (See Plate IX.) 

XIV. Algebra continue<i (f. 295, v.), according to Master Biagio 
(*certj cafi che fcriue m* biagio nel fuo trattato di pratica*) of 1340, 
Master Gratia de Castellani (* fccodo che fcriue m:* gratia de caftellani;, 
and Ijeonardo of Pisa. 

XV. Algebra continued (f. 312, r.), according to certain 'maeftrj 
antic hi/ viz. : ♦ Maeftro jkioIo/ * m* Antonio/ * m " giouanj/ * leonardo 
piiano/ 'nv biagio che circha al. 1340. anj morj/ 'm" fxiolo fiorj che 
circha al. 1360. duro/ ♦m^ michele padre di m? mariano,' 'm" Imha/ 
and ' un altro m" biagio.* 

XVI. This is missing, the manuscript ending with Book XV not 
c)uite 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. 1 1. 

Ti'/U. None. The first book of the Elements. 

Colophon, 'CExplicat prim*> liber euclidis cu ^;mcto capani/ 
(F. 14, V.) 

Description. 4*, 14.7 X 2 1.2 cm., the text being 9.9 x 1 5.8 cm. 
22 flf., 18-40 II. 

This manuscript of the first book of Euc lid, with the commentary by 
Campanus, has been included on account of the forms of the numer- 
als used. It is written in a (terman hand of c. 1460, and is bound 
(fT. 1-14) with the two manuscripts next described. 

ANONYMOUS. Latin MS., c. 1460. 

Title, None. A treatise on the calendar. 

Description, Sec the preceding manuscript. This is flf. 15-17 
of the volume 

This manuscript, lx)und with the Euclid just described, was prolubly 
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 



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 poftaalla mercatatia // e comminciando 
alnome didio.' (F. 11, r.) 






1 -». f 





1 1 

t 1 

1 •«. 
1 -v 

1 1 
1 ^ 

1 ^ 

1 ^ 






1 ^ 

1 ^ 

1 7 
1 7 

1 t 
1 t 

1 ^ 

-1^ • 






n. 1 






-1. ^ 

-V 1 
•V t 






Fig. 231. Tables from Benedetto da firenze 

Descriptim, 8*, 8.3 X 1 1.7 cm., the text measurement \-ar)- 
ing. 2 ff . missing -f 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 



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 


aiw^ A^^' oVii^i <U*^ *ta^ 

Fig. 232. From Benedetto da firenze 

payments, the dates in the problems including the years from 1460 to 
1 464. 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 

ALBERT OF SAXONY. Latin MS., 1462. 

See p. 9. 

Tit/c. * Tractatus proportionum. Incipiut pportioef copo^« a 
dno albertutio.// [ ] Roportio coitr da e duoy ;Dperitoy in a^° t'rio 
um<^**//ad inuice hitudo.' (F. i, v.) 

Colophon, * Explicat pportioef opo^® // p reuedo mfo alberto 
defifona// Finis//' (F. 12, v.) 


Description. 4^ 14.5 X 21.8 cm., the text being 11.8 X 13.8 
cm. (varies). 19 ff., 23 11. Bound with the Vergerius and Lullius 
mentioned on pp. 456, 457. 

This is the treatise oh 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 gi«n 
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. In the 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. ITiere 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 pagers 


.Axvmo rnindino 4lnom( cli cho ^a* UfiRfprimx itt>^» 

riO<9fiTno pnapcdcli «ifpoftoti A^ic6 pteh-o 

_ diortbA ntAzHw. tv^ Sc? JLtturenno ^9^^ gtotto 

ScS GtJufio^ mi^ ${? Gorm> clf4tMl[tcre:«(^cnci«ilm^ 
P9U yccM 9rU- 4 uiulo. 7 ian^^'ntSL reoarcTvliK Ji Kilte t^^ 
ccUrtuilt. cmrh octpdmclt/o cftullt cUutoHxtiriAnuttlu^^bs. 
JBO AtmI^cii^io vncjo^Htna nomciuimo ^vitix dtt^neic- 
hitti <nutDi tnocll dy ficno •lUovo Uiidit ^rcuevenj^Smf^T* 
^>4ibonora aSfiluht cLtLiTntX aniTTixnC Aino cfiic A4 oa^l^vW 
^Jt "^f diiiaaii^ilai ax ^!u Govuiyxrxi^'>oA'n^i<^^ 

CWia.(uiai^L^4utti.luintvn4lt K4hoiviIt Uiinih 

Id cUU rwi4iifX:_Jiiiolci:«. lihibuirtf uunoVt^w^ 

*j7 ^1 J»«*^i ^ttia(u-di Uo£ J(ilt (ftiali alliTio noft 

wno'Vtmre «iifinc. dt Waliniairu. alJ>tuu d/ 

mjIa tfrno eon/titiuno lUiumpo Loro-.9^^(h» ^larnlmii'o 
nttcUi piti*^ AlUck mcntft Jimuno Lntofttmisfai «/t citulli 
€; pur cC(cimura. aticllo ^/vioro cufo JilUTumii'A cdcvJu 

ctt^dottofpcinUtncnhL ^ (piti lomio dcLtUTtVi^no €iA9ohe:ii^ 

Fio. 233. First page of the 1473 Trattato 


are left blank for subsequent insertions, the scribe having written : ' Hie 
dificiunt quatuo // chartae 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 algelMa. 
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. 147S 

Maestro Luca da Firenze lived in the fifteenth century, and was the son 
of the celebrated Florentine arithmetician Matteo, who was bora 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 fflren^e.' (Fig. 234.) 

Description, 4*, 16.8 x 23.3 cm., the text being 11.7 x 138 
cm. 46 flf., 29 11. 

Although Fabbroni's Storia delP University di Pisa (I, 97) says that 
Luca*s son Giovanni went with Lorenzo dei Medici to the University 
of Pisa in 15 15, 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 pa3anents (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. 



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 

— ' — *^' -^i%m/mm m l m^tmnJ ^ wmwk* 


»<><»>o p /kfhim 

Kv^* MiU zimilm^ Mg»yn» <^%»^»i»' Z.f«b«- <|«*iiO« •i^A- 

•l-n- ^V 


9|^<««i^^(k < w J U ( > fnii> ft yi(Mi»> W'Ul^ 
fM «brfii^ pWWiii. fljuvi. f.fW ii-1 p 

Fig. 234. FikhT PAtiE of \x\\ da firen/e 

of fractions and denominate numbers, closing with a scries of applica- 
tions to the business problems of the time. Subjects like equation 
of payments, partnership, and exchange are given the most a lien lion. 
The handwriting indicates that the manuscript is a copy matie alx)ut 


1 4 75-1 500. 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 I^€ American Accomptanty 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 £., for mills, cents, 
dimes, dollars, and eagles, but DabolPs Schoolmaftet^ s Affiftant (4th cd., 
'799> P* 20) gives the symbol t 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 
w, ^, and i (for V— i). But mercantile symbols usually develop slowly, 
like J[^ from libra^ /- from the old s (J) from soldi or soiidi^ 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 lb. 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 12^ 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 if., 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 folk>ws : 

•Ja. mar 
ma. iul. aug. oc. de. deca- 
trib9. et. vno. alij. trigSta. 
f $. februus. octo. viginte.' 


47 > 





















C\ ^ 

'^t^i^t^^'** •7./~ 

Fig. 235. Tabi.ks from u'( a da kirkn/k 



::fTC^ ♦*) 

«.po*A7 *«.» <»i ««4« o«-*«J^ iffij^^^ 


Pig. 236. From the 1476 Computus cyro me traits 


(See also p. 33.) It does not surt here, however, but goes back at 
least as far as Sacrobosco (13th cent., see p. 31), in whose Computus 
^!ice p. 45 1 ) it appears as follows : 

•Sep. No. lun. Ap. triginta dato, reliquis magis uno. 
Ni tit Bidextus, Februus minor cfto duohus.' 

(1545 edn., f. K 6.) 


A Nurnberg computist of c. 1431. 

Title. • Computus norcmbergensis.* 
Colophon. 'Anno dni 1476 complete' 

Description, See the preceding manuscript, with which it is 

lliis is a (jerman computus written by the same hand as the pre< ed- 
ing. It is evidently a copy of a work i^ritten in 1431, for the following 
sutement appears on f. 65 : ' Anno dhi m"c( cc'xxxi" quarta feria p*« 
judica finite e liber ifte per me Jacobu Obemheym/ 

ANONYMOUS. Latin MS. 1477. 

Title. None. A calendarium. The first folio begins *Albclo 
bclo caro dccagolo ca*» 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 Vergcrius and Lullius de- 
scribed on pp. 456, 457. 

This is a manus< ript of 14 folios on the calendar, ami has two dates, 
M.cccc'.Knij" and M.cccc^.lxxvij, on the first folio. It also has !K>me 
I'lnar tables for 1364-1 381, the treatise having proluibly been ro]»ietl 
al«>ut 1458-1477 from another one of a century earlier. 

NICOL6 DE ORBELLI. I^itin MS.. 147S. 

Title. *Incipit op*» fratris (?) dorbelli fup . . .* (F. i. r.). * In- 
cipit. Matematica' (F. 132, r.). 

Colophoti. On f. 1 26, V. : • Explicunt Scripta f rJ5 niiolai dc orlx.'l- 
Iis//doctoris eximini fup . . . //Deo gratias et xpo ihu Amen 


Description. 6^ 10.5 X 15.6 cm., the text being 9.2 X 1 1.9 cm. 
270 fT. (6 blank). 41 II. Latin MS. on pajwr, 1478. 


The first part of the book is devoted to dialectics and logic The 
mathematics begins on f. 132, r., and consists of two folios an the theory 
of numbers and three on geometry. On f. 140, r., b^ins a treatise on 
philosophy, containing (f. 184, v.) a drawing of 'Johs fcotos docts 
fubtilis.' On f. 193, r., begins a treatise on astronomy: ' Hie // Indpit 
liber de // celo & mundo.' The rest of the work relates to science in 
general. (See p. 23, 1485.) 


A fifteenth-century mathematician of Cremona, Italy. 

Title, * Leonardi Cremonenfis artis inetri//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 if., 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 ctf the 
fifteenth century, or he may possibly have been the *Leonardns 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 U\^ 
before 1400. See also Enestrom in Bibliotheca Mathematica IV (31. 
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 1 702. This 
passage is as follows : * LXXXVIII (i.e. 1488) Leonardus Maynardus 
Insignis Astronomus, Physicus et Mathematicus, cuius opuscuhim M. S. 
Mediolani servatur, mihi indicatum ab eruditissimo Viro Lazaro Augstino 
Cotta I. C. amico meo nequaquam satis laudato, cui est initium : (Here 


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 Leonardus Maynardus, qui suo tempore non tan- 
tum 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 fiachofa 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 2, p c% p ^^ 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 aprlTo 
fcriuerio lareghola dellarabre (deir algebra). m°ochabiln" (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//fla reghola delagebrem'^ghabile ... (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 


Italian MS., c. 1490. 

An officer of the Republic of Pisa in 1 505. 
TitU. None. A treatise on arithmetic. *Choncio fia chofa 
che fono noue figure nellabacho.' (F. 3, r.) 


licit ^^Itiictiw \\^ujiJiap\i0 aa{^.^c 
caiiciio Jxncixicnc ^vtv^uadAfnia dJb 

dnio cavimU-UriB etlitiefc etiat uedftdiiZiV 

eavvello fzmyo Jrcncicic-wWc (oyva JdUpxmA 
yixhttx cauailo Jutffn <M% xacnc ^xcJluitv- U 
^iiiclcttci tuctifn AuticliC. hsrA iW>ucmo - U 
Jt^Hnisa^ne acwiamo ucattc actAtirv fem^ 

Ce wcni^ v'^ 4f iP^f" ^uoA^^tAnc in/ 
weftf c.^ Jit ^4to^if^JixJii6i^S^rc,a\AAn^ 

hemw <raall4]^i9nA Mtfita oALa t^4 C0e daJi 

nc( 4d ' «^i^ ' 5' *^' V^ aLom a^ mcttmte • ^t. dtr* 
l^*^ •'^ f tncfi -^ Ji ^\opcc»C A'p^^^t- 

ueacteiiu4tihp femp^ eciAU4i» vtn4 j!iivtttn.A(U 

clvniAi^o YiclU^^ ecu' '^ yn^x c- ^-di^hna^i 

da) \AcJictc au^tihTe JalUjnnwi ttirtifn ^iU^ 

lu^id -net. ^d*eu4.ti mervticli mraieiftwtt 

Fig. 237. From the anonymous manuscript of c. 1490 


CoUyphon. • Qveflo Libro. afcritto. difua. popia ajano. andrea. 
1-anfrcduccj.' (F. 96, v.) 

Description, 4*, 16.8 X 23.4 cm., the text being 12.1 X 16.5 
tm. 96 ff. (15 blank), 29 II. 

lliis book is an Italian commercial arithmetic of the Florentine type 
written alK>ut 1490. After a brief treatment of notation the author 
takes up the ' Librettine minore/ or the smaller multiplication table to 
to X 10, following this by several examples. He then gives (f. 9, r.) 
* le libretine maggiore ' or larger table, with multiplication * pcjuailrato ' 
an<l • p berichuoc hulo,* these names showing the work to be Tuscan. 
I >i\ision is followed by a large number of problems, per cent (il c^ and 
p too) playing a large part. The problems in e<]uation of |)ayments show 
the book to have been written between 1489 and 1491 (fT. 72-74). 


A Niirnbcrg RechentncLster, c. 1450-1500. 

TiiU, •HannsDimpfel //(Multiplication table) //Johann New- 
dorflcr RechenV/maiftcr vnd Modift zu N.// 1492.' (F. i, r.) 

Description, 8*, 10.5 X 13.9 cm. 11 fif., 20 11. Written on 

This is a beautifully written primer apiarently done under the guid- 
ance of one of the celebrated Rechenmeisiers of the Newtliirffer family 
of Nikmberg. It was written in 1492 and is evidently the work of a 
Uginner in commercial arithmetic. It opens with the addition and 
multiplication of comi)Ound numl)ers, 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 Newdurflfer. As to the latter, it is not 
improliable that he was the father of the celebrated St hreil>- and Rechen- 
mtisler johann Newdorffer, the founder of the (German calligraphy, 
w ho was bom at Niimberg in 1497, and died there Novemlwr 12, 1 563, 
aii<l whose son Johann (b. February 22, 1543, d. Octolwr 28, 1581) 
A aN also a well-known Rechenmeister. 

ANONYMOUS. Latin MS., c. 1500. 

Title, • De comutata ppornc.* (F*. 14.) 

Description. Fol., 14.5 x 21.5 cm , the le.xt beinj; 1 1.5 x 19cm 
This is the second part of the 1442 manuscript, flf. 14-32 (p. 458). 



This is a Latin treatise on arithmetic, apparently written about i45»>- 
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 ifcritabilej. 
apud doctof ppli* olim. f. fabiie.//pure cernut oculi vere demTz/fhikm 
niic i fine feculi ' (f. 30, v.) ; * quadratura circuli ' (f. 32, r.). The 
table in Fig. 238 shows some work in series such as is common in ariih- 




1 •^•~| 



^ - 


lo. 1 























Fig. 238. From De comutata pporlu^ 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 fi. 
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 ai 
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 iS. 
1490, mentioning his coronation as king of Bohemia on August 25, 147 1. 
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 pai>er. It occupies flf. 
39-44 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 




M^ v)'^^p|»«r*'/S'ipp(vMT mulh flint* ffr^' <i ^^Jn^ti^i /yJiia'**^ 

Fk;. 239. From Z^r coHiutata fporike, c. 1500 


The fifth portion of the volume (ff. 45-52) consists of a set of Lftin, 
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 V^erge- 
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 
* 1 501 adj feptembr In Niimberg.' 

Description. 4*, 15 x 21.8 cm., the text being 9.8 x 17.1 cm. 
89 ff. unnumb. -h 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. 

MERCATELLO. Italian MS., Mercatello, c. 1522 

An Italian teacher, bom at Mercatello, and living there in 1522. He was 
a pupil of Paciuolo. 

Title, 'Svmme//Arismetice.* (F. i, r.) * Stephano. D.B.Del- 
list pha//ni.damercatello. at vtti.qve//ili.liqvali. in arte.mercan 
tile.exercitare.sidilectano.' (F. i, v.) 

Description, 8°, 13.5 x 20.7 cm., the written part beinj: 
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. 


This is a manuscript on commercial arithmetic, unusually complete 
m its applications, and also unusually well written. It is of the general 
Florentine type, but, as appears from a date on f. loi, r., was written 
at Merc atello, a town south of Ferrara and east of Florence, llie 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^^* 
( Mercatello), in the latter case as • Stefano di Bap»* Stefanj f j.' The 
cxam|>les in equation of payments are dated 1371-72, showing that 
thcM! were copied from some earlier work, and in fact they were taken 
from Paciuolo (' Diftinctio nona, tractatus quintus'), who in turn 
U)rrowed them from some predecessor. 

Stephano states in his de<licatory epistle that he was a pupil of 
Pariuolo, and that he is chiefly indebted to him for his material : ' et 
niaxime dal mio K'^\ et ex" af!me & pceptor'. M. Luca dal Borgo.* 
I his is <]uite evident on comparing certain passages ; for, while Stephano 
(i(>cs not usually copy his master verbatim, there is often a great simi- 
brity )>etween them, and sometimes (as in the chapter * De le. 3. falfe 
{Mifitioni ') 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 
I >f 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 numl)ers. A little 
work in mensuration and the calendar is given at the end of the lxx>k. 

BARTOLOMEO ZAMBERTO, editor. Latin MS., c. 1525 

A Venetian scholar of c. 1500. He was bom c. 1473. 

7///r. * Evclidis // Megarenfis graeci philofo-Z/phi ex Theonc 
;;ra'co com-Z/metare Interprete Zabcr-//to veneto triplex prlci- 
pioru // jjcnus pr{mu diffinitioncs: // Sign^^Z/est cuius |)ars nulla: 

Linca vero, lon-//gitudo ilia tabilis.// Linex autem limitcs, funt 
f:;;na.// Recta linea, est quae ex xqua-//1i ftia interiacet fij^na. 
Siif)erfi-//cies, est quae longitudinem latitu//dinemqj tantum 
habct. Super-//^ficiei extrema, funt lineae. Pla-//na fuperficies, 
cs»l qux ex at*qua-/.^i« fuas interiacet lineas.* (F. 2, r.) 

Description. I2^ 8x 11.7 cm. 152 ff. (2 blank), 16-20 11. 
Written on parchment. 

This beautifully written manus<Tipt has l^een included in this cat- 
alogue of arithmetics without much justification, since it docs rot 


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 
1 5 13. The complete translation was first printed in Venice in 15 13, 
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 15 16, and in Basel in iS37» 1546, and 1558. See 
Riccardi, II, i, 644; Weissenbom, Die Ueberseizungen 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*, 16.6 X 23.3 cm., the text being 1 1 .8 x 16.3 cm, 
81 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 aU 
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 

Tit/e. * Trattato d' Aritmetica, e del Misure.' (F. i , r.) 
Description. Fol., 18x23.8 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, 6S 
show, about 1535. It is a commercial treatise, beginning, as was fre- 
quently the case, with the fundamental operations with compoand 
numbers. The method of division 'a danda * is preferred to that *per 
galea,* although both are given : ' II partir a galea e molto l^iadro fct 



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.) 

/ J •«.# 

f-' • 

Fig. 240. From the anomymocs IVattato ov c. 1535 

%f uch 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. 341.) 


pM/si9 cu'^utA ^^^* 

Fig. »4i. 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). 


A sUteenth-century student. 
Title. '.M .D. XLV.// Ludouichoalt de Salifpurga.' (F. i. r > 
Description. 8', 12.6 X 17.8 cm. 92 ff. 


This is a business arithmetic written by some student in 1545. The 
atithor first treats of the fundamental operations with denominate num- 
bers, following this by a treatment of fractions. (Fig. 242.) The last })art 

/iOwfi- oUi j>t^^ 

V I'"' J \ J?"V , 


Fk ;. 242. Fr< )M 1 H 1: lA* IX )V IC« > A I T 

€»f the work is a 'praticha' (f. 48), and conNiilcrs the ordinary l»iisinfss 
arithmetic of the time. The author use** almost the same form of ihi- j>cr- 
cent symlwl as the i*Titcr of 1456 referred to on p. 458. \i/. p c^ (see 
Fig. 243). The manuMrijit isdescril)e<i m Nanlutt i's tatal«»^"ie, p. 152. 





Fig. 243. From the ludovico alt 

ANONYMOUS. Latin MS., c. 155a 

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 U- Written in a Ger- 
man hand, c. 1550. 



This manuscript is the last one in the volume containing the algebra 
of Mohammed ibn Musa (p. 454). It consists of introductions, book by 
lxx>k, to books I~XV of Euclid. The symbob -h, -•- (for — ), V . and 
V/ (for </') are used. 

HONORATUS. I-atin MS., c, 1 550-1600. 

A Venetian monk of the sixteenth century. 

Ti'fle, 'Opus Arithmetica D. Honorati vene-/Ai monachj 
coenobij S. Lauretij.* (F. i.) 


Description. 8', 10.9 X 16 cm., the text being about 9.5 x 
15.8 cm. 1 II fT. 

This manuscript was written by a pupil of a Venetian monk namt^d 
Honoratus evidently about 1550-1600. It is a practical arithmeticp 


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 
c^ (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. 166 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. Sni. 4", 1 1.9 x 16.8 cm., 279 ff., 20-25 11. 

It 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", 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' 


Vno hi rictuto d'innrtM) ^ c9^.i^ 
jpew i r%pone di^y ^j>t, too.^' domaiu 
dA dun^ue qiutuo fix U fondo. 

XO000-- >•-— 

14 >: 

— «>— 4. 


(Tooo — — 

xoo — - 
9 — ^Ca» 

31 /ixi^o:^ > 

9J^ . 

37 • 
OX -^ 


X^ X.0 

Fm;. J45. From an Italian maniscriit ok r. 1560 


(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' Torala algise* scritto per mi // Camillo 
sachetto Tanno 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 grofre,//ebene, oder 
nidere, in die lenge, hohe, breite vnd//tiefte als graben Ciftemen, 
vnd Brunnen.' (F. i, r.) 

Description. I2^ 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. 



— ^ 






Fiti. 246. From an Italian manl-scrift of c. 1565 


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. 


Italian MS., 1579. 

A Venetian student of 1 579. 

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. 1 579. 

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 inflti- 
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 bom November 14, 1522, at Florence. 

Title, None. 

Colophon, *Di Lione adi 13. di Maggio 1579: Francesco 

Description, 4*, 15.5 X 21.8 cm., 32 ff. (2 blank), 26-30 U. 
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. 


JOHANNES KLUMPIUS. Latin MS., 1598-99. 

TitU. * Varij Tractatus Mathematices // k Joanne Klumpio 
philixV/ftudiofo excpt Ingol-//ftadij Anno// 1598 el fequcntj.' 
(P. X) 

Colophon. At the end of the • Arithmetica practica * arc the 
words • Abfoluim9 1 5 Januarij // Anno 1 599/ 

Description. 4*, 15.3 X 20 cm., the text being 10.5 x 16.5 cm. 
(varj'ing). 70 pp. blank + 442 pp. written = 512 pp., 19-21 II. 
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 
cif two distinct parts, probably the result of two courses of lectures, 
llie first treats of the four fundamental operations with integers and 
< ommon fractions ; the second of practii al 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 Oennan student of c. 1600. 

Title, * Das Funfft Capitel.// Item Im Funfften Capitel ift 
von dem Algorifmo//oder Cofs.* (F. 4, r.) 

Colophon. Not entirely legible. It contains the date, August 4, 


Description. 4", 18.8 x 24.6 cm., the text being 13.5 x 19.8 
cm. 321 flf. (13 blank), the lines varying. 

This is a (lerman manuscript containing the solutions of the problems 
in StifePs edition of RudolfT's Coss, beginning with Chapter 5. (See 
p. 358.) lliis manuscript was written, as the colophon shows, in 1599. 
The name of the writer is not entirely legible, but it seems to lie Brite- 
nils 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. Ncme. 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. 


This is a Dutch manuscript on gauging and general mensuration. 
The gauger's tables are as clearly written as in the best Florentine 

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. (xTiries). 
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 tabulais et formularis,' clearly written on vellum, cod> 
taining multiplication (or < area ') tables of primes, of * oddly even ' and 
' evenly odd ' numbers, of * solid numbers,' and the like. 

ANONYMOUS. German MS., c. 1600. 

Title, * Nutzlicher Gebrauch // Der Weldt Kugel.' (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. 


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 : 

Pace i6. The 1561 edition of Borghi has been acquired. 

Pa<;e 23. The 1485 edition of Nicol6 de Orbelli has been acquired. 
See also page 473. 

Pace 36. There has been acquired a work on the calendar by 
Regiomontanus : 'Almanach magif^ri Johanis//de monteregio ad anos. 
xviij.Z/acuratiifime 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-//l>ri 
Stapv-//leii i Artiii diuifione itroductio . . .' Colophon : • Cracouif . . . 
'534»* 4*» *3-8 X 'S.7 cm., the text being 10.2 x 16 cm. 

PA<iE 82. The first (1503) edition of the Margarita Philosophira has 
been acquired. The colophon is as follows : ' Chalchographatum primi- 
ciali hac // peffura/ Friburgi p loanne Scho//ttu Argen. citra fcftu 
Margarethf //anno gratise MiCCCCC.IlI.' 

Pace 114. An edition of Taglicnte's * Libro //de abaco,' • Vinegia . . . 
M.DJCLIIII ' has been acquired. Also the Venice e<lition of 1570, and 
an edition s. a. 

Pa(;e 115. The 1570 Milan edition of Tagliente has been acquired : 

• Libro //'de Abbaco che inse-//gna a fare ogni ragione // mercadantile 
. . . Milano 1570.' 

Pac;e 139. The 1525 edition of Riese's first book has been ac(]uired : 

• Rechnung auff der linihen //gemacht durch Adam Riefen vonn Staffcl* 
//fteyn/in mailen man es pflegt tzu lern in alien //rec hen frhulen 


gruntlich begriffen anno i5i8.//vleyrigklich vberlefen/ vnd zum andeni 
mall //in tnigk vorfertiget.//CGetruckt zu Erffordt 2Cum//Schwart2cn 
Horn.// 1^25.' 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 
10.7 cm. 44 ff., 17-20 11. 

Page 139. Another Erfurt edition of Riese's second work has been 
acquired: *Rechnung auff//der Lynihen vn Fedem/ // Auff allerley 
handthirung/gemacht durch//Adam Ryfen.// Zum andem 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. 

4 + 79 ff. 

Page 167. 'Ulrich Kern von Freysing Eyn new Kunftlichs wolge-// 
grtlndts Vifierbuch/ gar grvifl vnnd behend//aulz rechter art der 
Geometria/ . . . M.D JCXXI.' FoL, 18 x 27.8 cm., the text being 1 2.7 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 Witem- // bergae, per 
Vi-//tum Creutzer.' 8% 9 x14.8 cm., the text being 7 x11.2 cm. 
7ff., 27 11. 

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//dearitmeti-//ca et geo-//metna. 
// Deir Abbate Georgio Lapazaia // da Monopoli. // Intitolata il Rama- 
glietto.//In NapohV/Apreflo Mattio Cancer. M.D.LXXV.' 14.2 x 20 
cm., the text being 11 x 16 cm. 4 + 169 pp., 29 11. 

Page 322. The 1566 edition of Lapazzaia has been acquired : ' Famt- 
liarita // d'arithmetica, e geometria // con Tvsitata prattica Napo-// 
litana, Compofla & ordinata per Abbate Geor-//gio Lapizzaya Canonico 
Monopolitano // Nuouamente con fomma diligentia // Riflampata, 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. 


Page 340. * L*Arithmetiqve//militaire d'Alexandre//Vandenbu88che 
Flandrois//departie en deux liurcs.//. . . A Paris . . .' (s. a.). 8*, 14.8 x 
20.8 cm., the text being 9.8 x 16 cm. 35 ff., 34 IL Paris 1571. The 
Proeine is dated *' De la grode 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: 

* IJvre // d*Arithme-//tique contenant plufieurs belles que-//rtions & 
demandes, bien propres & // vtiles k tous Nfarchans, // Par M . Valentin 
Mennher // de Kempten. // Reueu par M. Melchior van Elftaer.// . . •' 

Page 375. The 1584 Cologne edition of Clavius has been acquired. 

Pa(;e 383. • Arithmetica // oder // Kechenbuch // • . . durch // An- 
thonium Schultzen/ . • . Zur Liegnitz. .. 1600/ FoL, 13.6 x 18.5 cm., 
the text being 10.3 x 13.5 cm. 4 + 259/7., 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 Cosmogiaphia, at Venice. 
It contains 5 pp. on arithmetic 

Pac;e 391. The work of Aurelio Marinati has been acquired: M^ 
prima parte // della // sommadi tvtte // le scienze // nella qvale si tratta 
delle // sette arti liberali // . • . Roma ... 1 587.* 4*, 1 6 x a 1 .5 cm., the 
text being 11 x 17.7 cm. 8 + 1 56 pp. The chapter ' Dell' antmetica ' 
begins on p. 99 and ends on p. 11 4. 

Page 408. Vila's arithmetic has been acquired: ' Reglas // bre^-s 
de Arith-//metica . . . per Bemat // Vila, • • . Barcelona . . • Any. M.I).- 
I-XXXXVI.* 8\9.8x 14.3 cm., the text being 7.5 x 12cm. 8+136!!. 

Page 418. The 161 1 edition of Mariana's work has been acquired. 

* loannis // Marianae // Hifpani //, c socie. lesv, // I)e Ponderibvs // et 
Mens\-ris. //Typis Wecheliants. //Anno M.DC.Xl.' 

Pa(;e 435. a manuscript of Zuchetta's work, copied in 1692, has 
been acquired. 

Pa(;e 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 f!., on paper, in I^tin, in a ("terman harnl. 

Page 480. A manuscript of Bede, ' De Scientia romputandi,' written 
in Latin^ c. 1520, has been acquired. FoL, 22.7 x 33 cm., the text 


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 Gcr- 
bert begins * Incipit liber geometri^ artis aeditvs a Dn5 // 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., the 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. Fortius. There was also an edition, Rome, 1524, 4*. 

Page 81. Boethius, Clichtoveus, and Stapulensis. There was also an 
edition, Paris, 15 14, fol. 

Page 97. There is an anonymous *Algorithmus linealis,* 15 13 at 
Gottingen, — probably Licht. 

Page 106. Kobel. There was an edition of the * Vysierbuch,' Oppcn- 
heim, 15 19. 

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. Glilfferich. 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 
prgecipuorum . . .,' 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 soW (for 
example, see page 122), most of them were purchased by Lord Over- 
stone and presented to the University of London, where they niay now 
be seen. 





8, 9. 10 


• •4* 122 


245* 249 




122 1 


249. 353 




133, 133 


252. 253 


II, 13 


123, 126 


254. 257 




136, 137 


357. 260 


15. «8 


127. 128 


2fja 263 




«3i* "32 


263, 2fi9 




132. «40 


271. 2W> 






386. 3(>0 






2f^ 2i)2 










•43. «52 


2<^5. 2<t& 




•53. '56 


2ilS. 306 




•57. 159 












159. 164 






165. 167 


3«6. 3»9 






320, 322 










174. «8o 


325. 328 




180, 181 


330. 338 

150 J 



181, 183 






183, 186 






186, 188 


340. 343 




191. 195 






197.211 1 












313. 316 






221, 333 


353- 350 




333. 329 


350. .^»i 




331. 338 


3<>i. 3^*4 




240, 343 


364, 367 


94. 97 






98. 106 









375» 380 


392. 393 




380. 383 




407, 408 


385. 389 

1 591 











389. 391 








404. 407 


425. 427 



433 1 












435 1 


454. 458 










440 1 






442 ] 


















439 > 












446 1 






























AbaitiA, 7, 70. Ste Countera 

Adriaen van der Gucht, jjS 

Agncola, 171 

AUmagni, 404 

Albert, J., 178 

Albert of Saxony, 9, y 465 

Albertus Magnus, 443 

Akali, 60, 167. 231. 310,32a 

Akiatus, 164 

Alexandrinus, 3S9 

Algebra, 56, 125, 132, J36, 233, 252, 

260, 286. 400. 403, 454 
Algebra (facsimile), 455 
Algebraic symbols (facsimile), 403 
Algorism, 5, 7, 10, 1 3, 1 5, 28, ^^, 74, 

A 1- Khowaraxmi, 7, 454 
Alkmaar, 325 
Alligation. 18 
Almadiano, 23 
Alon%o Delatore, 41 
America, 286 
Ammonius, 367, 429 
Amsterdam, 325. 424 
Andrea Lanfreducci, 475 
Andres, 132 

Andres Garcia de Ix>vas, 353 
Angelas Mutinens, 140 
Anianus, 31. 7 
Annaberg, 139 
Anonymous, 3, 23, 64, 67, 71, 83, 

106, 120, 126, 127, 128, 130, 153. 

178, 186, 195, 31 1, 313, 323, 243. 
352, 386, 31 1. 338, 340, 359, 368, 

408. 4^9. 443« 447. 449» 45°' 452. 
45^ 45*« 459. 463. 464. 466. 468, 
47> 475. 477. 478.480. 482, 486. 

Antonius de Barreft, 238 

Antwerp, 32, 183, 200, 249, 250, 278, 2hi, 

3'5' 345' 365* 372, 375. 386. 4*7 
Apian us, 1 55, 62 
Aquila, 396 

Arabic numerals, 1 1 1, 447 
Arabic numerals (facsimiles). 447, 1*1. 

VIII, et passim 
Archimedes 226 
Aristotle, 15 
254, Arithmomachia. 12. Stf Kithmomai hia 
Amheim, 18.S 
Astrolabe, 72 

Astrological numeraK, 375 
106, Astronomical fractions. Sre Sexa^r%i- 

Augsburg, 8, 10, 25, 37, 37. 41, too. 102, 

119, 130, 152, 160, 168, 186, 190, 238, 

263. 32*. 338 
Augustine, 83 
Aurel, 254 

Austria. Set Cracow, Vienna, eu. 
Autorif, 250 
Aventinus, 136, 229 
Avignon, 130 

Bachet, 348 

Baeda, 131, 140, 159 

Haeza, 2^19 

Baker. 327 
> 86, Bamberg, 12, 15 
173. Barcelona. 93. 31a 316, 375. 408 
244, Barchi, 263 
389, Barlaamo. 315, 343 
454, Barocius (Baro//i). 295. 340 
470, Barres, 238 
488, Barter. 18, 3.;9 

Bartoli, 315 




Barziza, 140 

Basel, 8, II, 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 Baeda 
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, 16 
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 
Bnicaeus, 350 
Bruges, 338 
Bruno, 396 
Brunus, 396 
Buckley, 252 
Budaeus, 99, 35, 164 
Budapesth, 67 
Bud^. See Budaeus 

Budel, 39^ 
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, 2ii 

Camilla, 315 

Campanus, 433 

Canacci, 459 

Cantone, 416 

Capella, 66 

Cardanus, 193, 338 

Cardanus (portrait), 194 

Casanova, 292 

Cassiodonis, 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, 37S 

Clement, 375 

Clichtoveus, 30, 80, 82, 94 

Cognet, 346, 36s 

Coignet. See Cognet 

Cologne, 23. 27, 46. 74, 76. 85. 153. 154. 
159, 167, 169, 180, 195. 200. 212, 231, 
281, 367, 375, 396. 402, 429 

Computus, 7, 31, 41, 73, 126, 44> 440- 
463, 473 

Computus (facsimiles), 444, 445, 47' 

Constantinople, 180 

Copenhagen, 353 

Corella, 244 



Cortes, 407 

Com. 1 25, 226, 233. 260, 286. S£g Algebra 
Ounter*, 7. 3^>. 155. 271, 412, 482 
Counters (facsimileit), 69, 103, 156, 166, 

201. 214, 215 
Covamibian, 396 
C racow, 32, V7, 123, 190, 260, 303 
Cuenio, 60, 249 
Culman, 182 
Cuno, 269 
Curtiui, 3g6 

C*u*a, J., 106. See aha Nicolauft Cusa 
Ciem^ 353 

Dagoinari, 435 

Dcbrecxin, 35(> 

I>e<.imal point (facMiniles), 52, 388 

t>ecuiiab, 50, 329, 330, 386, 388 

I>ee« 214 

l>e U Roche, 128 

l>el6no, 275 

I>enmark. See Copenhagen 

I)e SubervUle, 409 

I >« venter, 64, 67, 76, 188, 195, 262 

IhAling, 480 

Dicgo el Castillo. 46 

l>iKge*t 340 

Diophantus, 348 

l>ivi«ion« 18, 49, 449 

IHvtsion (fac«iinile)i) — 
A danda, 47. 462, 489 
Galley. 6» ioo» 483, 486. 487 
Of fractions, 485 

IH^tftion of fractions. 233, 262 

IK>llar sign, 470 

I dollar sign (facsimiles), 471. 491 

I >(>iiienicho de Valsugana, 492 

Dordrecht (l)ort), 392 

Dortmnnd, 213 

l>uni, 269 

Huplation, 75 

Ihtsseldorf, 300 

I>)cke, 427 

Eclipse (facMmile), 453 

Ki««niiiann, 91 

Kt^eniiienger, 3S9 

Ki«leben, 303 

Kita« Mt^rachi, 180 

England, 1 34* 2 1 6. 32 7, 4 50. See London 

Equality sign (facsimile), 288 

Equation of payments, 439. 446 

Erfurt, 124, 139, 338 

Eschenbuig, 340 

Espinosa, 238 

Euclid. II. 236, 4. 56, 43> 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 

Femandex de Santaella, 140, 26(> 

Fernelius. 157 

Ferrara, 143. 146. 3114 

Figurate numbers (facsimiles), 27, PL I 

Finaeus, 160, 279, 82, 97, 106 

Finger symbols, 34, 57, ^5, 115, 121, 
J3«. «37. 257 

Fmger symbols (facsimiles). 57. 1 38 

Fischer. 247 

Flicker, 367 

Florence. 10, 15, 16. 47. 63, 70. 98. 9.^, 
122. 132, 140. 346 

Fceniseca, 119 

Fonduli, 295 

Fonduli (portrait), 296 

Forcadel, 284, 316 

Fractions. 106. See I)e<.imak(, Sexa- 
gesimals, Division 

France, 284. .SV^ Tarln, Lyons. Avignon 

Francesco dal Sole. 1 43 

Franciscus, 15 

Franciscus de Oreti(». 396 

Frankfort a, M., 32. ^^^ 102, 106, 123, 

»3<). «S4. »55« ^^*^ ^^^ -00. -44. a5> 
269, 2<>o. 2g5, 2.^S. 3ig. 330. 331. 340. 

359. Z(^* 3**^3. 3"^. Z'^^ 40.V 4^0 
Frankfort a. d. Oder, ly^ 247. 2'*^t> 
Frascada. 490 

Freiburg, 82. 181, 1 82, ic)2, 301, 410 
Freigius, 372 
Frey, Jacob, 338 
Frey, J oh an n, 221 
Freyle. 2S6 


FulconU, 389 
Furet, 93 


Galasso, 404 

Gauging, 114, 142, 221, 468 

Gauging (facsimiles) 113, 222 

Gehrl, 359 

Gematria, 404 

Gemma Frisius, 2cx> 

Gempelius, 391 

Genoa, 416, 425 

Gentil, 263 

George of Hungary, 67 

Gennany. See Augsburg, Bamberg, 

Heidelberg, etc. 
Gersbach, 353 
Getons. See Counters 
Ghaligai, 132 
Ghebelino, 325 
Giovanni da Firenze, 443 
Girard, 386 

Girjka 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 
Gray, 353 
Gualda, 167 
Gucht, 338 
Giilfferich, 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 

Hommannus, 396 

Horoscope, 478, 492 

Hostus, 372 

Huguetan, 188, 131 

Hungary. See Budapesth, Debreciiii 

Huswirt, 74 

Hiitzler, 263 

Hylles, 396 

lamblichus, 188, t86 

Ingolstadt, 155 

Instruments (facsimiles), 124, 283, 297, 


Isidorus, 8, 3, 10 

Italy. See Florence, Venice, etc 

Jacob, 295 
Jacob's staff, 1 10 
JenSnimo de Valencia, 269 
Joannes de Gmunden, 117, 449 
Joannes de Muris, 1 17 
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 

Landshut, 83, 97 



Lanfredncci, 475 
lange, 353 

lanxut. S€t londshut 
lapauaia, ^21 
LaiMuzaui (portrait), 323 
Laudensis, 396 
Lauteiuch lag«r, 4 1 o 

l^JU 131 

iaxesto, 146 

I^ipxig. 27, 36» 37, 44, 53, 70, 83, 86. 91, 
I50, 138, 139, 155. 168, 171. 200, 223, 
^Ih «47. 249* 250. 263. 290. 303' 3"» 
33 «. 340. 353» 361. 365. 379. 412, 42 « 

l^mgo, 330 

l^eonardut Maynardus, 474 

I^eunbach. 86 

Iveyden, 66, 165, 166, 210, 254, 386. Sri 

licht, 70 

IJegnitx, 383 

I jllius, 63 

Liverius, 13 
^ l^oodon, 10. 132, 134, 182, 195. 213, 214, 

244, 252, 253, 286, 327. 330, 338, 340» 
353- 3<^. 39«« 393. 396. 400. 407, 408 

l^nkerus 253 
lx>pe< de Corella, 244 
l^ritus, 191 
Lortie, 91 

IXMftiUA, 389 

lx>tter, 70 

liottini, 286 

Ijouvaio, 238, 346 

I»vaa, 253 

Lubcck, 263, 393 

Luca da Firenie, 468 

Luca de Borgo. 54, 87 

l.ucar. 3<)3 

Ludovico Alt, 484 

Lttgdoni (Lyons 0r Leyden), 167, 300, 

Luniut,4i5, 457 
I.uyck, 408 
Lyona, ^2^ 93, 128, 130, 165, 171, iq3, 

245, 253. iCn), 286. 290, 292, 314, 320, 
385. 389. S^e Lugduni 

Madrid, vo» 393' 404 

Maffei, 86 

Magdeburg, xyy, 180, 290, 389 

Maginus, 399 

Mainz, 23, it8 

Mameranus, 396 

Mancnti, 174 

Mantua, 66, 263 

Manzoni, 357 

Mariana, 418 

Mariani, 180 

Mannati, 391 

Martin, 231 

Mastenon, 400 

Mathematics, vllue of, 72, 127, 231, 300 

Maurolycus, 348, 350 

Maynardus, 474 

Measures, 70, 99, 164, 171, 209, 266. 319 

Mediato, 75 

Medina, 229 

Medlerus, 223 

Melanchthon, 210, 53, 164, 326, 236 

Melero, t8i 

Mellema, 375 

Mellis, 218, 214, 391 

Menher, 249, 281, 346 

Menochius, 396 

Mensuration, 478 

Mercado, 335 

Mercatello, 480 

Meres, 40S 

Messina, 93, 429 

Mewrer, 39<> 

Mexico, 2S6 

Meyer, 322 

M icy 11 us, 265 

Middelburg, 418 

Milan, 76, 114, 115, 193, 216 

Military arithmetic, 343. 359, 3H5 

Million, 17 

Million (facsimile). 19 

Mirandula, 127 

Misrachi, iSo 

Modena, 66 

Mohammed ibn Musa, 7, 454 

Molinaeus. 3<)6 

Monhemius, 300 

Monte kepal Piedmont ois, 3S5 

Mon/o, 3g2 

Mordente, 41 5 

Morsianus, I5<>, 182 

Moya, 30S 

Mulling, 450 



Multiplication, 56, 115, 447 

Multiplication (facsimiles) — 
Complementary, 76 
Fanciful, 116 
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 

Miinster, 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 

Nicol6 de Orbelli, 23, 473 

Nicomachus, 186, 4, 27 

Nonius, 315 

Norico, 44 

Norry, 347 

Noviomagus, 195, 159 

Numerals. See Arabic, Astrological, 

Numeration, 260 
Nuftez, 315 
Numberg, 53, 62, 86, 123, 137, 139, 151, 

1521 154. 156. 160, 178, 182, 193, 221, 

223, 226, 231, 249, 271, 292,300, 314, 

338,346, 391,412, 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. 106, 121, 

126, 131, 134, 157, 160, 167. 168,171. 

186, 191, 192, 195, 200, 211, 223, 228. 

229, 233, 237, 245. 252, 260, 263, 26S> 

269, 271, 278. 279, 284, 31 1. 314. 316. 

330f 340, 343» 347, 359, 38^, 409, 446^ 

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 
Pecle, 338 
Peer, 156 
Pelacani, 140 
Peletier, 245 
Pellos, 50 

Per-ccnt sign, 439, 441, 458 
Per-cent sign (facsimiles), 437, 440, 441. 

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 
Piricheymer, 396 
Pisa, 16 
Piscator, 247 

Plus and minus, 40, 126, 1S3, 254, 456 
Plus and minus (facsimiles), 40» 125. 

i85» 234 
Poeppingius, 391 
Poitiers, 245 



Poland, 353. Se* Crmcow 
Fortius, 70 
Fo\eiano, 375 
Prag. 29a, 359 
Friscian, 319 
Froblema (facsimilet) — 

Cistern, 48 

Couriers, 113 

Kallen tree, 49 

Fish, 438 

Gauging, 1 1 3, zti 

Hare and hound, 465 

Horseshoe nails, 484 

Market women, 1 1 1 

Partnership, 402 

Snake and well (tree), 48, 439 

Thief, 438 
Proportion, 46, 89, 99, 235 
Proportion (facsimile), 28 
Prosdocimo, 13 
Psellus, 168 
Pythagoras (portrait), 46 

Quadrans (facsimile), 162 
(^uirini, 429 

Racu, 365 

Rainer, 300 
Ramus, 263, 330, 335 
Raphael Franciscus, 15 
Raymundus Lullius, 457 
Recordc, 213, 253, 286 
Recreations, 391 
Regensburg. 136. 137. 396, 368 

Regpo. 370 
Regiomontanus, 62 
Regius, 181 
Regnaudus, 396 
Regnier, 200 
Reichelstain, 169 
Reinhard, 421 
Reisch, 82 
Reymere, 379 
Rheticos, 211 
Ricci, 429 

Riese, A., 138, 171, 250. 7 
Riese, A. (portrait), 251 
Riese, U 252. 365 
Ringclbcrgius, 165 

Ringhieri, 253 

Rithmomachia (Rithmimachia, Ryth* 

momachia), 12, 63, 271, 340 
Rithmomachia (facsimiles), 64, 273 
Riuo, 298 
Rocha, 316 
Roche, 128 

Rodolphus Spoletanus, 122 
Rodriguei, 408 
Rollandus, 446 
Roman numerals, 104, 106, 249, 382, 383, 


Roman numerals (facsimiles), 104, 10$, 

107. 37> 582, 383 
Rome, 32, 86, 91, 93, 122, 140. 236, 37$, 

378. 39«. 4i5.429 
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, 2>S, 253, 308. 315, 

Salignacus, 359 
Salispurgo, 4H4 
Santa Crui, 404 
Santaella, 140, 269 
Sarafino da Campora, 429 
Saragossa, 9*). 122, 140. 167. iSi, 247. 

24Q, 2i^ 35c> 

Sarmvia, 239 

Savon ne. 314 

Scaruffi, 370 

Scheubel 233. 246, 252, 454 

Schcy. 4^7 

Sihtcdam, 423 

Si-hleupncr, 41a 

Schonerus, J., 178 

Schonerus. K, ^O' 333 

Sihrevkenb^rpcr, ;Vi 

Schrevkenfmh^iu* (Sihrrt.kfui hM* i^^ 

Sihreilwr. Srt Orammateu^ 

Schuere* 424 



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 Alcali, 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. 

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- 
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, 168 

Trenchant, 320 

Trevisano, 408 

Treviso, 3 

Trigonometry, 474 

Tiibingen, 74 

Turin, 50, 86, 364 

Turkey. See Constantinople 

Tzwivel, 84 

Uberti, 114 
Ulm, 229, 402 
Ulman, 391 
Unicomus, 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- 
71. 73» 77i 82, 87. 99. 114, 115, 130, 
140, 141, 146, 148, i6o, 168, 171, i7> 
174, 180, 181, 195. 200, 229, 242, 253. 
254, 257, 275, 278, 292, 298, 315. 34a 
343' 346, 347» 348. 350* 364. 38*' 389. 
393. 399. 404. 407. 408, 4*2 

Ventallol, 298, 319 

Vergerius, 456 



Verint, 216 

Verona, 10, 148, 389, 404 

Venc, 399, 41 li 100, 262 

Verte (factimile), 411 

Vienna, 27, 32. 53, 61,66, 106. 117. 123, 

151. 152 
ViU. 408 

Vincent de Beauvais, 10 
Vtnccnto, 140 
Vincenza, 66, 37 5 
Vincenzo da Bergamo, 429 
Viscontl, 368 
Vittori, 61 
Vogelin. 195 
Von Szily, 67 
Vuelpius, 231 

Wagner, 12, 15 
Wllck). 182 
Weber, Johann, 338 
Weights. St* MeaAures 
WeissenfeU, 223 

WeUch ptactice, 152, 416 

Wenceslaua, 418 

MTemer, 300 

Widman, 36. 40, 44 

Willichius, 197 

Willaford, 216 

Wittenberg, 32, 53, 61, 139, 168. 178, 

180, 200, 223, 236, 237, 247, 269, 286. 

167. 389, 429 
Wojewodki, 260 
Wolphiun, 154 

Xy lander, 356 

Vciar. 247 

Zamberto. 481 
Zero, 76 
Zuccantint, 367 
Zuchetta, 425 
Zuchetta (portrait), 426 
Zurich, 391, 396 


3 1 

3 2044 024 343 386 


Harvard Collaga WManar Ubrary 
CambrMga, MA02138 1617