_No place_, date, nor author’s nor printer’s name. ‘Algorithmus
Integrorum Cum Probis annexis.’ _Quarto in threes._
I think this is the oldest book in my list. The type is of that
manuscript appearance which is so common in books printed before 1480:
and this one can hardly be later than 1475, and may be ten years
earlier. The matter is that of a writer prior to the school of Borgi
and Pacioli. Though the nine digits are given, they are not used. The
rules only are given, and they are Sacrobosco’s set; for which see p.
15.
_No place_ nor date. ‘Algorismus novus de integris compendiose sine
figurarum (more Italorum) deletione compilatus. artem numerandi
omnemque viam calculandi enucleatim brevissimè edocens. una cum
algorismis de minuciis vulgaribus videlicet et phisicalibus. Addita
regula proportionum tam de integris quam fractis que vulgo mercatorum
regula dicitur. Quibus habitis qui vis modica adhibita diligentia omnem
calculandi modum facillime adipisci potest.’ _Quarto in fours._
Three different editions of this work are mentioned by bibliographers,
all without dates. Numeration is called _prima species_; addition,
_secunda species_; and so on. There are slight examples, except upon
the square root, in which only results are given. It is something
between such a work as the last, with rules only, and that of Borgi, to
which I now come, in which there is decided force of examples.
_Venice_, fourteen-eighty-eight. =Piero Borgi.= Work on Arithmetic
without title, as follows. _Quarto in fours._
This is one of the editions alluded to in page 2, and for the power of
mentioning it from inspection (as well as the works of Ghaligai and
Texeda following) I am indebted to Dr. Peacock. It begins with the
following verses:
Chi de arte matematiche ha piacere
Che tengon di certeza el primo grado
Avanti che di quelle tenti el vado
Vogli la presente opera vedere
Per questa lui potra certo sapere
Se error sara nel calculo notado
Per questa esser potra certificado
A formar conti di tutto [_sic_] maniere
A merchadanti molta utilitade
Fara la presente opera e afatori
Dara in far conti gran facilitade
Per questa vederan tutti li errori
Ede iquaterni soi la veritade
Danari acquisterano e grandi honori
In la patria e de fuori
Sapran far le rason de tutte gente
Per le figure che son qui depente.
After a page of description of symbols, comes ‘Qui comenza la nobel
opera de arithmeticha ne laqual se tracta tute cosse amercantia
pertinente facta et compilata per Piero borgi da Veniesia.’ At the end
is ‘Stampito in Veniexia per zovanne de Hallis 1488.’
The author begins by saying that there are plenty of sufficiently
good masters, and not less abundance of most excellent authors. His
arrangement of the five _acts_ of arithmetic is strange; they are
numeration, multiplication, partition, summation, and subtraction.
He uses the word _million_, and his numeration goes to millions of
millions of millions. He then proceeds to multiplication, and forms a
table of the nine multiples of 1, 2, ... 10, with those of 12, 16, 20,
24, 32, and 36. He then points out how to prove processes by casting
out sevens and nines; in doing which he shews how to divide by 7, and
uses the sum of the digits to find the remainder to 9. Then follow
multiplications done each in one line, even when the multiplier has
three or four figures. After this follows the common method, in which
the final process is addition, done without any formal rule (in fact,
addition is not made a formal rule any where). Division is said, at the
head of a chapter, to be done in three ways; but only two are given:
that _per cholona_, where the whole is done in one line, and that _per
batelo_, which is what I have always designated by (O). Subtraction is
then explained; and after some applications, the rules for fractions
(which are expressed in the same way as now) are introduced. The rule
of three (_riegola del tre_) is given, and a large number of money
applications.
_Paris_, fourteen-ninety-six. ‘In hoc opere contenta. Arithmetica decem
libris demonstrata Musica libris demonstrata quatuor Epitome in libros
arithmeticos divi Severini Boetii. Rithmimachie ludus qui et pugna
numerorum dicitur.’ _Folio in fours._
This is the first edition of the work mentioned in page 10: so that I
was wrong in page 4. I was misled by bibliographers mentioning this
as an edition of Jordanus only. The copy before me comes to me in two
different parts, though the register at the end (which has the first
words of all the sheets) proves that these parts belong to one work.
It would be odd if it should have been customary to divide it in this
manner, so as to lead to the above imperfect description. The corrector
of the press was David Lauxius, and the printers John Higman and
Wolfgang Hopilius.
_Venice_, no date, but apparently before fifteen-hundred. ‘Dabaco che
insegna a fare ogni ragione mercadantile: et a pertegare le terre con
larte di la geometria: et altre nobilissime ragione straordinarie con
la Tariffa come respondeno li pesi et monede de molte terre del mondo
con la inclita citta de Vinegia. El qual Libbro se chiama Thesauro
universale.’ _Octavo._
The order of the operations is as in P. Borgi. The book is very full of
florid ornaments and woodcuts.
* * * * *
Among books of the fifteenth century which I have had no opportunity of
seeing are the following, described by Hain and others:
Without date. ‘Arithmeticæ Textus communis, qui pro Magisterio fere
cunctis in gymnasiis ordinarie solet legi, correctus corroboratusque
perlucida quadam ac prius non habita commentatione a Conrado Norico.
Lipsiæ per Martinum Herbipolensem.’ _Folio._
Without date, _quarto_, with the mark of Martin of Wurtzburg
(Herbipolis). ‘Algorithmus linealis,’ beginning ‘Ad evitandum
multiplices Mercatorum errores &c.’
_Treviso_, fourteen-seventy-eight, _quarto_. ‘L’arte del Abbacho,
Practica molto bona et utile a chiachaduno qui vuole uxare l’arte
merchandantia.’
Watt mentions an edition, _Rome_, fourteen-eighty-two, _quarto_, of the
‘Ludus Arithmomachiæ,’ attributed to John (he should have said William)
Shirewood, bishop of Durham. Hain does not mention this.
_Babenberg_, fourteen-eighty-three, _duodecimo_. ‘Rechnungsbüchlein.’
_Cologne_, fourteen-eighty-five, _quarto_. ‘Ars numerandi, seu
compendiosus tractatus de dictionibus numerabilibus.’
_Strasburg_, fourteen-eighty-eight, _quarto_. Anianus. ‘Compotus
manualis magistri aniani. metricus cum commento Et algorismus.’
_Leipzig_, fourteen-eighty-nine, _octavo_. ‘Rechnung auf alle
Kaufmannschaft.’ Murhard and Kloss (who cites Panzer) give this book to
Joh. Widman, whose name appears in an address to the reader; but Hain,
though naming Widman, puts the work down as anonymous.
Two collections of the works of Boethius, with the Arithmetic among
them, _Venice_, fourteen-ninety-one and ninety-seven, _folio_.
Deventer (Daventrie), fourteen-ninety-nine, _quarto_. ‘Enchiridiom
Algorismi sive tractatus de numeris integris.’
Without date, but with a letter from Balth. Licht dated
fifteen-hundred, _Leipzig_, printed by Melchiar Lotter, _quarto_.
‘Algorithmus linealis cum pulchris conditionibus Regule detri: septem
fractionum: regulis socialibus,’ &c. And another, seemingly of the same
date, by the same printer, ‘Algoritmus linealis.’
No doubt the first English print on Arithmetic is cap. x. of ‘The
Mirrour of the World or Thymage of the same,’ headed ‘And after
of Arsmetrike and whereof it proceedeth,’ printed by Caxton in
fourteen-eighty. (Peacock, p. 419, and Ames.)
_Basle_, fifteen-three. =Nicolas de Orbelli.= ‘Cursus librorum
philosophie naturalis venerabilis magistri Nicolai de Orbelli ordinis
minorum secundum viam doctoris subtilis Scoti.’ _Quarto in fours._
This course of natural philosophy requires two pages of explanation of
arithmetical, and, in particular of Boethian, terms, and nothing more.
It purports to have been intended for those who were studying theology.
The _Margarita Philosophica_ (page 5,) seems to give the highest
arithmetical limit in liberal education, and the work before us the
lowest.
* * * * *
See page 12. John de Lortse, &c. I was perfectly justified in writing
Lortse, for the name is so spelt in my copy in the clearest black
letter. Nevertheless, it is John de L’Ortie, and the book is a French
translation of the work of Juan de l’Ortega, mentioned by Dr. Peacock,
pp. 426 and 436. Seeing de L’Ortie [sic] mentioned by Meynier, and
the word thus written being nothing but the translation of L’Ortega
(nettle), I compared the French work with Dr. Peacock’s description of
the Spanish, and found that the two must necessarily be the same. On
looking very narrowly at the ſ which caused the mistake, I found in it
a small hook at the bottom, which the letter ſ never has elsewhere in
the work, and the letter i always has. The ſ then in Lortse is an i in
which the dot (or rather small line, for so it is in the black letter)
has been accidentally battered into the top of an ſ. But the alteration
is so perfect that every one reads it for an ſ. This is the second
paragraph which one battered letter has caused. In the new edition of
Brunet it is noted that Heber’s catalogue spells the word Lort_s_e. My
copy is the one which was in Heber’s library.
This French translation gives Juan de l’Ortega an earlier date than the
Spanish bibliographers, or those who have devoted themselves to the
Dominicans, are aware of. Antonio, in his Spanish lists, and Quétif,
in his Dominican one, both give Ortega no earlier date than 1534. But
it appears that he was translated into French in 1515. Panzer (xi.
465, according to Brunet) mentions a work of L’Ortega, _Messina_,
fifteen-twenty-two, beginning ‘Sequitur la quarta opera di arithmetica
et geometria.’
_Florence_ (?), no date. =Francisco di Lionardo Ghaligaio.= ‘Summa De
Arithmetica.’ _Quarto in threes_ (N).
_Florence_, fifteen-sixty-two. =Francesco di Lionardo Ghaligaio.=
‘Pratica D’Arithmetica ... Nuovamente Rivista.’ _Quarto in fours._
Two editions of the same book. The first must have been published
before fifteen-twenty-three, the year in which Julius de Medicis, to
whom it is dedicated as Cardinal, would have been addressed as Pope. It
is a complete and advanced treatise of arithmetic, in thirteen books.
The tenth book begins algebra, or _Arcibra_, the introduction of which
from the Arabic seems to be attributed to Guglielmo de Lunis; Leonard
of Pisa and Giovanni del Sodo are also mentioned as writers.
_Venice_, fifteen-twenty-six. =Joh. Fr. Dal Sole.= ‘Libretto di Abaco
novamente Stampato: Composto per lo excellente maestro Joanne Francisco
dal sole Ingegnero: utilissimo a cadauno per imparare per se stesso
senza maestro. Dimandato breve introductione.’ _Octavo in twos_ (O).
A very elementary work, from the school of Borgi and Pacioli,
proceeding up to the square root.
——— fifteen-forty-five. =Gaspar de’ Texeda.= ‘Suma De Arithmetica
pratica y de’ todas Mercaderias Con la 5 orden de’ contadores. Hecho
por Gaspar de’ Texeda, Con Privilegio Imperial.’ _Quarto in fours_ (ON).
The colophon, which probably contains the place, is torn out. In
numeration the places are distinguished throughout the first part of
the book. Thus, 950777200000 is 950║777q⁰200║000, where ║ (the two
lines should be joined at the bottom) stands for _mill_, and q⁰ for
_cuento_, a million. The proof by sevens and nines appears, as in
Borgi. The work is rather more advanced than that of the latter, and
proceeds to the extraction of roots. The greater part is commercial.
Rules for fractions are given, not complete. The rule of three opens
with an invocation of the Trinity: Sfortunati goes further, for he
connects the rule with the doctrine.
_London_, fifteen-forty-six. ‘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 oure Englyshe tonge by the which all maner of difficile
questions may easely be dissolved and assoyled Anno. 1546.’ _Octavo._
At the end is: ‘Imprynted at London in Aldersgate Strete by Jhon
Herford.’
_London_, fifteen-seventy-four (printed by John Awdeley). ‘An
introduction of Algorisme, to learn to recken wyth the Pen or wyth
the Counters, in whole numbers or in broken. Newly overseen and
corrected....’ _Octavo_ (small) (O).
These are, no doubt, reprints of the work of nearly the same title,
mentioned by Dr. Peacock (p. 419) as having been printed at St. Albans
in 1537. It was a good predecessor of Recorde’s _Grounde of Artes_, but
it makes only a poor follower.
See page 19. In speaking of Rudolph, I had overlooked that Dr. Peacock
(p. 424, note) describes his _Algebra_ from inspection, as having been
edited by Stifel himself in 1571.
_Rome_, fifteen-eighty-three. =Christ. Clavius.= ‘Epitome Arithmeticæ
Practicæ.’ _Octavo_ (small) (O). See page 33.
_Witeberg_, sixteen-four. =Gemma Frisius.= ‘Arithmeticæ Practicæ
Methodus Facilis.’ _Octavo_ (small). See page 25.
See page 26. Macpherson (Annals of Commerce, eighteen-five, quarto,
vol. i. p. 146) gives (from Anderson, i. 409) a French edition of
Stevinus on book-keeping, _Leyden_, sixteen-two, _folio_, as quoted
by Anderson from a copy in his own possession. The title he gives is
‘Livre de compte de prince à la manière d’ltalie en domaine et finance
ordinaire: contenant ce en quoi s‘exerce le très illustre et très
excellent Prince et Seigneur Maurice Prince d’Orange, &c.’
_Frankfort_, sixteen-thirteen. =Joh. Faulhaber.= ‘Ansa inauditæ et
mirabilis novæ artis....’ _Quarto._ Same place, author, and form,
sixteen-fourteen. ‘Numerus figuratus ...’ and sixteen-fifteen,
‘Mysterium Arithmeticum....’
These singular medleys of arithmetic, algebra, prophecy, and nonsense,
seem to be all one publication. The date of the first is JVDICIVM,
written at the bottom of the title-page. Alter the order of the letters
into M.DC.VVJII., and it becomes 1613.
_London_, sixteen-fourteen. =James Dowson.= ‘De Numerorum Figuratorum
Resolutione.’ _Octavo._
This work gives Pascal’s table.
_No place_ nor date. =Joh. Harpur.= ‘The Jewell of Arithmetick: or, the
explanation of a new invented arithmeticall Table....’ _Quarto._
Perhaps this work is the _Merchant’s Jewel_ mentioned by Bridges (p.
44) as of 1628, a date which, from the character of the work, I should
be disposed to give it. I can find no mention nor use of decimal
fractions in it. The operations are performed on a kind of abacus.
Johnson, presently mentioned, describes a ‘most excellent instrument
invented by Mr. William Pratt, called, The Jewell of Arithmatick;’
which is perhaps the one described in the work above, though with a
different inventor’s name.
_London_, sixteen-thirty-three. =Joh. Johnson= (Survaighor). ‘Johnson’s
Arithmatick In 2 Bookes. The first, of vulgare Arithma: with divers
Briefe and Easye rules: to worke all the first 4. partes of Arithmatick
in whole numbers and fractions by the Author newly Invented. The
Second, of Decimall Arithmatick wherby all fractionall operations are
wrought, in whole Numbers, in Marchants accomptes without reduction;
with Interest, and Annuityes.’ Second edition. _Duodecimo_ (O).
In his decimal fractions, Johnson has the rudest form of notation; for
he generally writes the places of decimals over the figures, thus—
1.2.3.4.5.
146·03817 would be 1 4 6|0 3 8 1 7
Otherwise, his system is tolerably complete.
_London_, sixteen-forty-nine. =R. B.= (Mr of Arts). ‘Arithmetick
Symbolical In one Book. In which the Mystery of Numeration by Symbols
is revealed.’ _Octavo._
A short and easy treatise on Algebra.
_London_, sixteen-fifty-five. =R. B.= ‘An Idea of Arithmetick at first
Designed for the use of the Free-Schoole at Thurlow in Suffolk. By R.
B. Schoolmaster there.’ _Octavo_ (O).
Decimals and algebra, all on the plan of Oughtred.
_Antwerp_, sixteen-sixty-two. =Jean Raeymaker.= ‘Traicte d’Arithmetique
Contenant les quatres especes, avec la regle de trois, et la
Practique.’ _Octavo._
Not a first edition. Nothing but questions and their answers, as in
John Speidell’s work of sixteen-twenty-eight.
_London_, sixteen-seventy-five. =C. H.= ‘The Golden Rule made Plain and
Easie.’ _Octavo._ Second edition.
_No date_ nor place (I think about sixteen-eighty). =J. W.= (of
Brandon). ‘A progression in Arithmetical Progression ... with Some
things very remarkable in that Mystical Number 666.’ _Octavo in twos._
The author shews how to form squares and cubes by differences. He is
very charitable, for his day, to the Papists: for, setting down twelve
as the divine number, and that of the apostles, he seems to imply that
the Roman church, as having a number made up of sixes, may be half
Christian and half apostolic.
_London_, sixteen-eighty-four. —— ‘Enneades Arithmeticæ; the Numbering
Nines. Or, Pythagoras His Table extended to All Whole Numbers under
10000.’ _Octavo in twos._
The plan is, to have 99 rods instead of nine, in Napier’s set; so as to
have the first nine multiples of all numbers short of 100.
_London_, sixteen-eighty-five. =John Wallis.= ‘A treatise of Algebra,
both Historical and Practical.’ _Folio in twos._
_London_, sixteen-ninety-three. =John Wallis.= ‘De Algebra Tractatus;
Historicus et Practicus. Anno 1685 Anglice editus; Nunc Auctus Latine.’
_Folio in twos._
The Latin, or augmented edition, is the second volume of Wallis’s
collected works. See the Introduction to this Catalogue.
_London_, sixteen-ninety-seven ——. ‘Computatio Universalis, seu
Logica Rerum. Being an Essay attempting in a Geometrical Method, to
Demonstrate an Universal Standard, whereby one may judge of the true
Value of every thing in the World, relatively to the Person.’ _Octavo._
The author, cutting down life to its half, to allow for childhood,
sickness, &c., proceeds to a calculation how the true value of
every thing is to be estimated numerically. This, and Craig’s more
famous book (see Useful Knowledge Library, _Probability_,[21]) are
consequences of the Principia. It is curious that as soon as _force_
was widely made known as an object of numerical measurement, some
people began to fancy prudence, pleasure, and pain could be submitted
to the same process.
[21] I have seen this book sold with my name upon the cover as the
author. To this I have no objection, except my knowledge of the
fact that it is the joint production of Sir John Lubbock and Mr.
Drinkwater-Bethune.
_Breslau_, seventeen-seventy-nine. =Joh. Ephr. Scheibel.= ‘Einleitung
zur mathematischen Bücherkentnis. Eilftes Stück.’ _Octavo_ (small).
See the Introduction. This part contains the arithmetical bibliography;
but the paging does not begin after that of the ‘Zehntes Stück.’ In
fact, the bibliography of this book of bibliography is a study of
itself.
_Göttingen_, seventeen-ninety-six, ninety-seven, ninety-nine,
eighteen-hundred. =Abr. Gotthelf Kastner.= ‘Geschichte der
Mathematik....’ Four volumes _octavo_.
See the Introduction, as to Kästner.