⁂ The letter P. followed by a number, refers to the
page of Dr. Peacock’s treatise in which the work or
its author is mentioned. The brackets [ ] enclose
statements which I have taken from others, and
cannot therefore affirm from personal inspection.
For O and N see those letters in the Index.
_Florence_, fourteen-ninety-one. =Philip Calandri.= ‘Philippi Calandri
ad nobilem et studiosum Julianum Laurentii Medicem de Arimethrica
opusculum.’ _Octavo_ (small).
This book, which very few have mentioned at all, and fewer still from
inspection, is a part of the rich bequest of the late Mr. Grenville to
the nation. It begins with a picture of Pythagoras teaching, headed
‘Pictagoras Arithmetrice introductor.’ In the preface is the following
on Leonard of Pisa: ‘Vero e che il modo del notare e numeri con decte
figure dice Lionardo pisano haver nel Mcc. incirca rechato dindia in
Italia: et decti carateri: o vero figure essere indiane: et appresso
deglindi havere imparato la copulatione desse.’ He explains the leading
rules, except division, for integers and for lire, soldi, and denari.
Division he puts down in examples, and appears to have mistaken the
mode of working, or to have had an incorrect printer. His notion of
a divisor is curious. When he divides by 8, he calls his divisor 7;
demanding, as it were, that quotient which, with _seven_ more like
itself, will make the dividend. He also describes the rules for
fractions, and gives some geometrical and other applications. The book
is in black letter, but the numerals are in a thin and small-bodied
type. At the end of the book is, ‘Impresso nella excelsa cipta de
Firenze per Lorenzo de Morgiani et Giovanni Thedesco da Maganza finito
a di primo di Bennaio 1491.’
_Venice_, M.cccc.lxliiii., fourteen-ninety-four. =Lucas Pacioli=, de
Burgo Sancti Sepulcri. ‘Summa de Arithmetica Geometria Proportioni et
Proportionalita.’ _Folio in fours._
_Tusculano_, fifteen-twenty-three. =Lucas Pacioli.= ‘Summa de
Arithmetica geometria. Proportioni: et proportionalita: Novamente
impressa In Toscolano su la riva dil Benacense et unico carpionista
Laco: Amenissimo Sito: de li antique et evidenti ruine di la nobil cita
Benaco ditta illustrato: Cum numerosita de Imperatorii epithaphii di
antique et perfette littere sculpiti dotato: et continens[8] finissimi
et mirabil colone marmorei: inumeri fragmenti di alabastro porphidi et
serpentini. Cose certo lettor mio diletto oculata fide miratu digne
sotterra se ritrovano.’ _Folio in fours_ (ON).
[8] Cns in the title.
These are the full titles of the two editions of this celebrated
work. Both editions are by one printer, Paganino of Brescia, and are
beautifully printed: the type of the second being a good instance
of the black letter in its state of approach to what is now called
Roman letter. The work itself has been described by Hutton, Montucla,
Peacock, Libri, &c.; but it would yet require a volume of description
to do it justice. It is sometimes called the first work on arithmetic
printed; but Calandri, Peter Borgo, three already mentioned in the
Introduction, and perhaps more, take precedence. But it is certainly
the first printed on algebra, and probably the first on book-keeping.
P. 414, 424, 429 &c., 432 &c., 451, 460, 462; Hutton, _Tracts_, vol.
ii. p. 201.
On comparing my copy of the first edition with that in the British
Museum, I found one of those phenomena which so frequently occur in
very old printed books. The first leaves of the two copies, to the
number of about thirty, and the first leaf of the geometry, are not
from the same setting up. The endings of the pages, and the ornaments
of the capital letters, are different in the two. Nor does either of
them agree with the second edition. A part of the first impression
may have been lost, so that a second setting of types was required to
replace that part. Several mathematical bibliographers of note enter
Pacioli as Lucas di Borgo, and even as Borgodi, Lucas, and some do not
mention the first edition.
The Latin _Paciolus_ is usually spelt in Italian Paccioli, but Libri
spells it Pacioli. I believe that the various assertions that Pacioli
wrote Euclid in _Italian_, arise from the geometry of the work above
described being Italian: but it is not Euclid, though of course founded
on him.
There is another old book of which I have reserved the mention till
now, because its author has been confounded with Pacioli. Brunet
gives it as ‘La Nobel opera di Arithmetica ... compilata per B.
[should be P.] Borgi,’ _Venice_, fourteen-eighty-four, _quarto_:
and Hain has ‘Borgo (Pietro) Venet. Aritmetica,’ _Venice_,
fourteen-eighty-two, _quarto_. This edition Hain had not seen: but he
had seen the next (the one mentioned by Brunet), which begins ‘Qui
comenza la nobel opera di arithmetica ... per Piero Borgi,’ _Venice_,
fourteen-eighty-four, _quarto_. And he gives two other editions,
_Venice_, fourteen-eighty-eight and ninety-one, _quarto_. Accordingly,
Mattaire sets down the first edition of _Lucas di Borgo_ as of 1484,
and Dr. Peacock adopts this date. It is quite certain that Dr. Peacock
must have had one of the editions of Pacioli before him when he wrote;
and we are therefore to suppose that, in stating the date of the first
edition, he followed Mattaire, and presumed an edition earlier than any
he had seen. Tartaglia refers to Peter Borgo. P. 458.
_No place_, no date. =John Muris.= ‘Arithmetices Compendium ex
Boetii Libris per Johannem Muris excellentis ingenii virum Accurate
congestum.’ _Quarto._
This is a different edition from the one presently mentioned. John
Muris seems to have escaped the notice of bibliographers as an
arithmetician. I think I remember that there is in Hawkins’s History
of Music some discussion of his ideas on the musical scale, [on which
was published his work, _Leipsic_, fourteen-ninety-six, _folio_]. Muris
lived before the invention of printing, but when I cannot ascertain.
[His astronomical tables are preserved in manuscript.] The present
tract has twelve leaves unpaged, and is certainly of the very earliest
part of the sixteenth century, if not of the fifteenth.
_Venice_, fifteen-one. =Geo. Valla.= ‘De expetendis et fugiendis rebus
opus.’ _Folio in fours._
The first part of this is ‘De Arithmetica libri iii, ubi quædam a
Boetio pretermissa tractantur’ (O). Certainly he supplies some of
the omissions of Boethius; the four rules for instance. The work was
published by the son, Joh. Pet. Valla. I have seen an earlier edition
mentioned, but I cannot find any trustworthy account of such a thing.
_Cologne_, fifteen-one. =John Huswirt.= ‘Enchiridion Novus Algorismi
summopere visus De integris. Minutiis vulgaribus Projectilibus Et
regulis mercatorum sine figurarum (more ytalorum) deletione percommode
tractans,’ &c. _Quarto, twos and threes_ (O).
A short treatise, apparently one of the earliest printed in Germany
on the Arabic system. The _projectilia_ are counters. The rules are
verified by casting out the nines. The words ‘sine figurarum deletione’
are not made good in the rule of division. On this work see the
_Companion to the Almanac_ for 1844, p. 3.
_Paris_, fifteen-three. =Joh. Faber Stapulensis=, =Jodocus
Chlichtoveus=, and =Carolus Bovillus=. ‘In hoc libro contenta Epitome
compendiosaque introductio in libros Arithmeticos divi Severini
Boethii,’ &c. &c. _Folio in fours_.
This book contains an epitome of Boethius by Faber of Étaples; with a
commentary and an arithmetical collection of rules by Chlichtoveus,
his pupil; a compendium of geometry, a book on the quadrature of the
circle, and _cubication_ of the sphere, and a book on perspective, by
Charles Bovillus; with an astronomical compendium by Faber already
mentioned. This book is one of the earliest printed by Henry Stephens
and his partner Wolfgang Hopilius. It is the first edition, I have no
doubt, of Faber’s epitome of Boethius, though Heilbronner and Murhard
assert the contrary, perhaps misled by Faber’s edition of Jordanus. As
to the contents, the Arithmetic of Boethius was the classical work of
the middle ages. It consists of statements of the commonest properties
of numbers, under a great many classifications, to each of which a
name is given. The second work, of arithmetical rules, shews the very
low state of the art. It takes pages upon pages to explain the simple
rules, though no examples are ventured on which have more than three
figures. Montucla says that the quadrature of Bovillus was only saved
from the laughter of geometers by its obscurity. But a work printed by
Henry Stephens, and containing Boetius and Faber, must have been very
far from obscure in its day. The historians of mathematics confine
themselves to works the reputation of which has lasted: but they ought
not to make the state of their own minds, with respect to the rest, a
criterion of that of the contemporary readers.
_Basle_, fifteen-eight. =Gregorius Reisch.= ‘Margarita Philosophica,
cum additionibus novis: ab auctore suo studiosissima revisione tertio
superadditis.’ _Quarto in fours._
According to Kloss (from his own copies) the first edition of this
curious book is _Friberg_ fifteen-three, the second and third both
_Strasburg_ fifteen-four (one printed by Gruninger, which I have seen,
and one by Schott), and he calls the one before me the fourth. But
Hain marks as the first edition one which appears to be printed at
_Heidelberg_, fourteen-ninety-six, _quarto_. The one before me is not
the third; it is evident that a former title-page has been reprinted:
for though this edition is printed (as appears from the colophon) by
Furter and Scotus of Basle, the title-page bears ‘Jo. Schottus Argent.
lectori S.’ Which agrees with Kloss’s statement that the previous
edition to this one was printed by Schott of Strasburg. The Arithmetic
(O), which is a part of the system of philosophy here laid down, has a
frontispiece representing Boethius at one table with Arabic numerals
before him; and Pythagoras at another with counters. Pythagoras among
the Greeks, Apuleius and Boethius among the Romans, were often made the
inventors of arithmetic. The arithmetic is divided into speculative
and practical. The former is a summary of Boethius, often in the words
of John de Muris. The latter is a short treatise on _Algorithm_, as it
was called, or the rules of computation by the Arabic numerals. There
is also computation by counters, fractions common and sexagesimal, and
the rule of three. Many works of fifty years later do no more difficult
questions. That Boethius was the author of the Arabic numerals was a
common notion at the time, revived in our own day. I have seen another
edition of _Strasburg_, fifteen-twelve; and there is said to be another
edition by Orontius Fineus, _Paris_ (?) fifteen-twenty. There is also
an Italian translation, _Venice_, fifteen-ninety-nine, _quarto_, with
the additions of Orontius, translated by Giovan Paolo Galucci.
If the number were sufficient of those who wish to take their notions
of liberal education in Europe at the time immediately preceding the
Reformation from original sources, and not from the reports of others,
a reprint of the _Margarita Philosophica_ would be made. The diversity
of the matters which it treats, and the largeness of its circulation,
stamp it as the best book for such a purpose.
The _Margarita Philosophica_ is the earliest work I have found in which
mention is made of that peculiar system of measures which was current
among the mathematicians of the sixteenth century, and which has caused
no little confusion among writers on metrology. I have already given
some account of this ill-understood system, and shall here endeavour to
present the whole case with some additional evidence.
The Roman foot (of 11·62 inches English) was of course established
throughout the empire: and with it the Roman pace[9] of five feet, or
58·1 inches or 4·84 feet English. The natural pace of a man in our
day is as nearly as possible five English feet: Paucton’s experiments
on the walking paces of individuals gave him 59·7 inches English,
or 4·98 feet. In the British army the step, both what is called the
ordinary step and the quick step, is, by regulation, thirty inches:
making a pace of five feet. The Roman pace, by which distances were
actually measured, was that of a soldier on the march: and, as might
be expected, the weight of his arms and other equipage seems to
have shortened his pace a little. But so near were these measures,
as actually used by the Romans, to the natural ones from which they
derived their names, that it was customary, not only to recur to legal
standards, but, in the absence of ready access to them, to make use
of the natural foot and pace. And we also know from Roman writers,
that a somewhat fanciful relation, though not very far from the truth,
was established between the breadth of the hand across the middle of
the fingers, the _palm_, and the length of the _foot_. It was taken
that four palms made a foot: and a palm was made of course to consist
of four (average) finger-_breadths_ or digits. This division into
palms and digits was the most recognised division of the Roman foot:
that into inches, or _unciæ_, is well known not to belong to the foot
merely, but to any thing else. Whatever magnitude was called unity, the
_uncia_ was its twelfth part. Accordingly, _all_ the Roman foot-rules
which have been found in ruins or excavations have the digital
division, to which _some_ (most, I believe) have the uncial division
superadded. All this I take to be too well established to require the
citation of any authorities.
[9] The pace is derived from the double step, being the distance from
the extremity of the heel at the place from which it is removed in
walking to the same at the place in which it is set down again.
It is quite out of my power, or, as far as I know, of that of any
one else, to trace the gradual alteration of the foot in different
countries. It does not appear that any means were taken to institute
comparisons of various measures, to be preserved as public records.
Such means could have been found: that which was then the common church
of Christendom might have easily regulated the weights and measures
of Europe, even without appearing to do so. But in all probability,
the extent of the variations was not well known until it was too late.
We know, however, as a fact, that the geometers were successful in
establishing a measure among themselves, and communicating it through
Europe on paper. This measure, I have no doubt, they believed to be the
true Roman foot: for they divide it in Roman denominations, make use
of it in their quotations from Roman authors, and never hint at their
having any other notion of a Roman foot. And moreover, the writers who,
in the sixteenth century, recovered the true Roman foot, never mention
any peculiar geometrical foot in use among mathematicians, or in any
way distinguish the latter from the _wrong_ Roman foot which they were
correcting. That the geometers believed the Roman foot, that is, their
own foot, to be the _human_ foot, might be easily proved. And, with
such belief, they would make their so-called Roman foot too short.
From a hundred measures of the feet of adult men, furnished to me by
a boot-maker, and taken as they came in his books, I find the average
length of the Englishman’s foot to be 10·26 inches, in our day: or an
inch and a third shorter than the Roman lineal measure.
This _geometrical foot_ of the mathematicians is, I make no doubt, the
geometrical foot to which writers of the seventeenth century refer, or
mean to refer. But, not long after the true restoration of the ancient
measures, there arose a disposition among those who inquired into the
subject to seek a mystical origin of weights and measures, on the
supposition of some body of exact science once existing, but now only
seen in its vestiges: a disposition which is not yet entirely extinct.
Some speculated on the pyramids of Egypt, and tried to establish that
the intention of building those great masses was that a record of
measures founded on the most exact principles might exist for ever. But
more turned their attention to the measurement of the earth, and, by
assuming nothing more difficult than that a degree of the meridian a
thousand times more accurate than that of Eratosthenes was in existence
hundreds, if not thousands, of years before him, it was easy enough
to make out that the whole system of Greek, Roman, Asiatic, Egyptian,
&c. measures was a tradition from, or a corruption of, this venerable
piece of lost geodesy. There runs through all these national systems a
certain resemblance in the measures of length: if a bundle of faggots
were made of foot-rules, one from every nation ancient and modern,
there would not be any very unreasonable difference in the lengths of
the sticks.
The metrologists who treat this subject handle it according to their
several theories. Those who have none in particular either neglect
it altogether, or speak of its length as uncertain, or define, with
Dr. Bernard, the geometrical pace as being five feet of its own kind,
without saying what this kind is. Those who have the notion of the
old measure of the meridian accommodate it to their supposed ancient
measure; but at the same time, those of most research and note make it
_less_ than their Roman foot. Thus Paucton makes it nine-tenths of the
Roman foot, which, with his version of that measure, is 10·9 inches
English, and with the true one, 10·5 of the same. Similarly, Romé de
L’Isle makes it more than half an inch (French) less than _his_ Roman
foot. As they do not refer to the geometers of the middle ages, I
cannot guess whence they get their notion, otherwise than from their
theory.
I now proceed to demonstrate the existence of this geometrical foot,
which I believe to have been the effort of mathematicians to perpetuate
and make common what they took to be the Roman foot, on the supposition
that it was nothing but the average length of the _human foot_.
Passing over the general expressions of writers who refer to the use
of the parts of the body in measurement, and who sometimes distinctly
state that the determination of the human foot is necessarily that
of the Roman measure, I take first the statement of Clavius, whose
term of active life was the latter half of the sixteenth century and
who says[10] very distinctly that the mathematicians, to avoid the
diversity of national measures, had laid down a system for themselves.
The table of measures which he gives (and dozens of other writers
before him) is as follows:
[10] ‘Enumerandæ sunt mensuræ quibus mathematici, maxime geometræ,
utuntur. Mathematici enim, ne confusio oriretur ob diversitatem
mensurarum in variis regionibus (quælibet namque regio proprias habet
propemodum mensuras), utiliter excogitarunt quasdam mensuras, quæ certæ
ac ratæ apud omnes nationes haberentur.’—_Comm. in Sacroboscum._
1 breadth of a barleycorn.
4 = 1 digit.
16 = 4 = 1 palm (across the middle of the fingers).
64 = 16 = 4 = 1 foot.
96 = 24 = 6 = 1½ = 1 cubit.
160 = 40 = 10 = 2½ = 1⅔ = 1 step (gressus).
320 = 80 = 20 = 5 = 3⅓ = 2 = 1 pace.
640 = 160 = 40 = 10 = 6⅔ = 4 = 2 = 1 perch (pertica).
125 paces 1 Italic stadium.
8 stadia 1 Italic mile.
4 Italic miles a German mile.
5 Italic miles a Swiss mile.
The constant reference to this barleycorn measure (which is seldom,
if ever, omitted) induced me to try what it would really make. There
is some difference between the breadths of barleycorns. A certain
statement of Thevenot (cited in the _History of Astronomy, Lib. Usef.
Kn._) makes the breadths of 144 grains of oriental barley give 1½
French feet; at which rate 64 would only give 8·53 inches English. A
sample from a London shop gave me (when the largest grains were picked
out) 33 to just more than five inches. Some other samples, procured
from two different parts of England as the finest which could be got,
gave 33 to 5 inches, 33 to 5·1 inches, and 33 to 5·1 inches. The
average of these is 9·8 inches to 64 grain-breadths: a result which
coincides more nearly than could have been expected with the following
determinations.
There is a chain of writers who have studied to perpetuate their
geometrical foot by causing a line to be laid down on the page,
representing the digit, the palm, or the foot. Sometimes the palm or
foot is divided into digits: and of course I rely most on those whose
subdivisions are the best. As paper is apt to shrink as it becomes old,
the foot deduced from these will be somewhat too small, and it may be
afterwards discussed how much it should be lengthened. Leaving this for
the present, I give the measurements from different authors.
_Margarita Philosophica_, above described. In the Strasburg edition
of fifteen-four, the length of the geometrical palm is less than 2½
inches by from half to three fourths of the 24th of an inch. Taking it
half way between these, the four-palm foot is 9·9 inches English. In
the Basle edition of fifteen-eight, in which the woodcuts are of much
rougher execution, the palm is 2·64 inches, giving a foot of 10·56
inches. The palm only is given in both cases.
_Oppenheim_, fifteen-twenty-four. Stöffler. ‘Elucidatio Fabricæ Ususque
Astrolabii.’ _Folio in threes_ (quarto size).
The digit, palm, and foot, are separately given; the foot is divided
into palms, and all agree excellently well with one another. The foot
is exactly 9·75 inches English.
_Paris_, fifteen-twenty-six. John Fernel. ‘Monalosphærium.’ _Folio in
threes._
_Paris_, fifteen-twenty-eight. John Fernel. ‘Cosmotheoria.’ _Folio in
threes._
The historical mistake arising out of these works is the most
remarkable circumstance attending the loss of the geometrical foot.
While Fernel was publishing the first work, he was meditating (or
perhaps executing) his famous measurement of a degree of the meridian.
In this first work he lays his geometrical foot down the page, with
great care, as he says (_omni molimine_). In two copies of this work
which I have examined, the length of the foot is within a sixtieth
of an inch of nine inches and two-thirds, giving 9·65 inches. In the
second work, in which he announces his measure of the degree, he states
that five of his paces and those of men of ordinary stature make six
_geometrical_ paces; which, he adds, is agreeable to the opinion of
Campanus and others (at least a century and a half before), who made
the mile of 1000 common paces to be 1200 geometrical paces. Allowing
60 inches (English) to a common pace, which is rather over than under
the truth, this gives a geometrical pace of 50 inches, and a foot of
_ten_ inches. This is enough to shew that Fernel was, in the second
instance, speaking in rough terms of the foot which he printed _omni
molimine_ in the first. The geometrical pace being forgotten, and the
_monalosphærium_ also, the modern historians have assumed that Fernel
used the Paris foot: by which he is made to appear to come very near
the real degree, whereas he is fifteen miles wrong.
_Paris_, fifteen-fifty-two. Jac. Koebelius. ‘Astrolabii Declaratio.’
_Octavo._
This worthy astrologer, after referring to the perfect notoriety
of the system of measures, gives a digit and a palm. The digit is
nine-sixteenths of an inch (English), giving a foot of nine inches. The
palm is 2½ inches and one-sixteenth, giving 10 inches and a quarter to
the foot. The book is small, and the palm incorrectly subdivided.
_Frankfort_, sixteen-twenty-one. Peter Ryff. ‘Questiones Geometricae.’
_Quarto._
Four very accordant palms are given, indifferently subdivided into
digits. Each palm is 2¼ inches and three-sixteenths, giving a foot of
9·75 English inches.
From these different sources, good and bad, we have for the geometrical
foot 9·8, 9·9, 10·56, 9·75, 9·65, 10, 9, 10·25, 9·75 inches: the mean
is 9·85. But much the best authorities are Fernel and Stöffler, because
they are the greatest names, have given the whole foot, and have taken
the greatest pains with the subdivisions. Their results are 9·75 and
9·65, with a mean of 9·7.
Taking this as the foot on paper, it remains to ask how much it must be
lengthened to allow for the shrinking of the paper. At first, relying
on the plate in Dr. Bernard’s work on ancient weights and measures, in
which the English foot appears to have shrunk by its 42nd part, I was
disposed to lengthen the above in the ratio of 41 to 42. But observing
that an older English foot, figured in the ‘Pathway to Knowledge,’
1596, has shrunk only by its sixtieth part, I am rather inclined to
consider the shrinking of Bernard’s as an extreme case. And moreover,
the two copies of the Monalosphærium give the same foot within
one-hundredth of an inch certainly, and less: and it is very unlikely
that if the paper had shrunk much, it should have shrunk so equally
in two different copies. But, taking one-fiftieth as the outside, it
follows that the geometrical foot is any thing the reader pleases
between 9·7 and 9·9 English inches. The result from modern barley[11]
gives 9·8, as above shewn.
[11] Those who have tried to make the _lengths_ of three barleycorns
into an inch will probably think little of this mode of judging. But I
observed that in samples of barley of very different apparent fineness,
the difference was in the length of the corns, the breadths hardly
varying at all.
It is remarkable how completely the English writers are in ignorance of
the existence and use of the geometrical foot among their continental
neighbours. Blundeville takes Stöffler, in the work above mentioned, to
be speaking of a _German_ foot, which, says he, Stöffler makes to be 2½
inches less than the English foot.
_Paris_, fifteen-fourteen. =Boethius=, =Jordanus Nemorarius=,
=Shirewode= (?), =J. Faber= (Stapulensis). ‘In hoc opere contenta
Arithmetica ... Musica ... epitome ... Boethii: rithmimachie ludus qui
et pugna numerorum appellatur.’ _Folio in fours._ Second edition.
The first arithmetic is by Jordanus, the second by Boethius, both
edited and commented by Faber. The music (a tract on which was in
those days little but a tract on fractions under musical names)
is also by Faber. The _Rithmimachia_ I suppose to be the work of
Shirewode’s mentioned in the Introduction, but no author is named. It
is a short triple dialogue on the properties of numbers, with some
sort of numerical game. Libri attributes it to Faber, probably from
its appearance in this edition without an author’s name, and headed
by an address from Faber, the editor. This second edition was printed
by H. Stephens [the first, _Paris_, fourteen-ninety-six, _folio_, was
printed by Hopilius, whose partner H. Stephens afterwards was, under
the superintendence of David _Lauxius_,[12] of Edinburgh].
[12] _Capiat qui capere potest_ is the only general principle on which
names like this can be read back into the vernacular. I once found
James Hume, well known as a mathematician at Paris in the beginning of
the seventeenth century, described as _Scotus Theagrius_. What part
of the land of cakes this might be, would probably have eluded all
the books in existence: but a Scottish friend, to whom I mentioned my
difficulty, solved it at once, by telling me that he must have been one
of the Humes of _Godscroft_, a place in which it seems certain Humes
are, or were, lords of the soil.
_Paris_, fifteen-fourteen. =Nicolas Cusa.= ‘Hæc accurata recognitio
trium voluminum operum Clariss. P. Nicolai Cusæ Card. &c.’ In three
volumes, _folio in fours_.
Cardinal Cusa is put down in several arithmetical lists because one of
his _opuscula_ is entitled _de Arithmeticis Complementis_. But it is
not a work on arithmetic, nor does it even proceed by arithmetic. Or
it maybe that John Cusa, next mentioned, may have been confounded with
Nicolas. For Cusa, see the _Companion to the Almanac_ for 1846, p. 14,
and _Penny Cyclopædia_, ‘Motion of the Earth.’
_Vienna_, fifteen-fourteen. =Joh. Cusanus.= ‘Algorithmus linearis
projectilium, de integris perpulchris arithmetrice artis regulis,’ &c.
_Quarto size._
A tract of seven pages on counters.
_Augsburg_, fifteen-fourteen. =Jacob Kobel= (printer). ‘Ain Nerv
geordnet Rechen biechlin auf den linien mit Rechen pfeningen:’ &c.
_Quarto, twos and threes_ (duodecimo size).
Computation by counters and Roman numerals: the Arabic numerals are
explained, but not used. In the frontispiece is a cut representing the
mistress settling accounts with her maid-servant by an abacus with
counters. This book is said by Kloss to have been also printed by Kobel
himself at Oppenheym in the same year.
_Oppenheym_, fifteen-fifteen. =Jacob Kobel.= ‘Eyn New geordnet
Vysirbüch.’ _Quarto in threes._
A work on gauging, in which the Arabic numerals are used.
_Vienna_, fifteen-fifteen. =Boethius=, =J. de Muris=, =Thos.
Bradwardin=, =Nic. Horem=, =Peurbach=, =de Gmunden=. ‘Contenta in hoc
libello. Arithmetica communis, Proportiones breves, de latitudinibus
formarum, Algorithmus D. Georgii Peurbachii in integris, Algorithmus
Magistri Joh. de Gmunden de minuciis physicis.’ _Quarto._
The name of George Tannstetter as editor of this collection is a fair
guarantee for the genuineness of the several works.
The first is an abridgment of Boethius, by John de Muris, of whom
elsewhere.
The second is on proportion by Thomas Bradwardin (of the time of Edward
III.), who is Bradowardinus, Bragvardinus (as in the work before
me), Bragadinus (a confusion perhaps between him and Bragadini), &c.
according to the fancy of the speller: Tanner says the book was printed
at Paris in 1495. Bradwardine is better known among theologians for
his book against Pelagius, than among arithmeticians. He begins thus:
‘Proportion is either that which is commonly so called, or properly so
called. Proportion commonly so called is the mutual habitude of two
things. Proportion properly so called, is the mutual habitude of two
things _of the same kind_.’ Perhaps the authority of an Archbishop of
Canterbury (for he was or was to be nothing less) may induce those
who teach the rule of three to remodel their plan according to the
archiepiscopal dictum, which is also that of common sense. [Other
mathematical works of Bradwardine were printed.]
The third book is on the areas of figures (though nothing to any
purpose is done), with the leading idea that the difficulty arises from
variation of breadth. Peurbach’s work is a summary of operations like
that of Sacrobosco; but though written to explain the Arabic notation,
it obviously takes for granted that that notation is generally known.
John de Gmunden’s work is on sexagesimal fractions (_minuciæ physicæ_).
_Paris_, fifteen-fifteen. =Gaspar Lax.= ‘Arithmetica speculativa
magistri Gasparis Lax Aragonensis de sarinyena duodecim libris
demonstrata.’ Also, same date and place, ‘Proportiones magistri
Gasparis,’ &c. _Folio in threes._
This is a very diffuse and extensive work, in small black letter. It
treats only of the simple properties of numbers; though it is evident
that to a knowledge of Boethius, Lax added that of the arithmetical
books of Euclid. The thing which appears most surprising in this and
other works of the same kind, is the apparent difficulty of dealing
with numbers. Here are upwards of 250 small black letter pages in
folio, filled with propositions on the simplest properties of numbers,
and the reader looks in vain for any number so high as 100 used by way
of exemplification. For any thing that appears the author could not
count as far as 100. Montucla says that Lax was afterwards Pope; but I
am assured by a learned Catholic bishop that there never was any Pope
of the name.
_Lyons_, fifteen-fifteen. =John de Lortse= (Dominican). ‘Oeuvre
tressubtile et profitable de lart et science de aristmeticque: et
geometrie translate nouvellement despaignol en francoys Auquel est
demonstre par figure evidemment: tant le nombre entier: nombre rompu:
regie de compaignies: soubde fin: que toutes aultres choses qui par
geometrie et aristmeticque peuvent estre comprises: comme appert par la
table cy apres mise.’ _Quarto_ (O) (166 folios).
This is perhaps the first book in French on commercial arithmetic. I
have not found mention of it in any catalogue. It is a good Italian
importation, through Spain: and must have made the readers of the
Boethian school stare to see what arithmetic really could do. We have
thus two Spanish books published at Paris in one year: another (see
Siliceus presently mentioned) had been published there the year before.
_Venice_, fifteen-twenty. =R. Suiseth.= ‘Calculator. Subtilissimi
Ricardi Suiseth Anglici Calculationes noviter emendate atque revise.’
_Folio in threes._
This man is Richard at the beginning of the book, Raymund at the end,
while Tanner [and Gesner] call him Roger. His name seems to have been
Swinshead, latinised into Swincetus, Suicetus, Suineshevedus, &c. The
title of his book, _Calculator_, has given rise to the fiction that he
was a great arithmetician, and even algebraist; which, as he lived in
the fourteenth century, would have made him a very remarkable man. But
nothing can be further from the truth: the book is full of philosophy
about intension and remission, and siccity, and calidity and frigidity,
and other _quisquiliæ Suiceticæ_, as some afterwards called them. That
number and magnitude are used occasionally, is all that can be said.
Brucker, according to Enfield, Hallam, &c., says this book is as scarce
as a white raven. But probably this refers to one of the previous
editions. Hain gives three of the fifteenth century, marked ‘Padue
(no date),’ ‘Papiæ, 1488,’ and ‘Papie, 1498.’ All these Suissets are
_Richards_. (See Tanner _in verb._ ‘Swincet;’ Wallis _Op._ t. iii. pp.
673, 675, 680, 685.)
_Lyons_, fifteen-twenty. =Stephen de la Roche Villefranche.=
‘Larismethique nouvellement composee divisee en deux parties dont
la premiere tracte des proprietes parfections et regles de la dicte
Science,’ &c. _Folio in fours_ (O).
A large treatise, very full on commercial arithmetic, and containing
also geometry. I suppose we should find it largely indebted to the
Spanish books above named.
_Paris_, fifteen-twenty-one. =Boethius.= ‘Divi Severini Boetii
Arithmetica duobus discreta libris: adjecto commentario, mysticam
numerorum applicationem perstringente, declarata.’ _Folio in threes._
The commentary is by Girardus Ruffus, and for absurdity and dulness,
it ought to rank high among the mystic commentaries: even to the
justification of Cornelius Agrippa, when he asserts that arithmetic is
not less superstitious than vain, adding, like a philosopher of his
day, that it is only valuable to merchants for the low and mean benefit
of keeping their accounts.
_London_, fifteen-twenty-two. =Tonstall= (Bp. of London). ‘De Arte
supputandi libri quatuor Cutheberti Tonstalli.’ _Quarto._ (Printed by
Pynson.)
_Paris_, fifteen-twenty-nine. The same title. _Quarto in fours._
(Printed by Rob. Stephens.)
_Strasburg_, fifteen-forty-four. ‘De Arte ... Tonstalli, hactenus in
Germania nusquam ita impressi.’ _Octavo._
[There is said to have been an earlier Strasburg edition, but I have
not seen it: there are several other editions, but no English reprint.]
For the life of Tonstall see Anth. Wood, _Ath. Oxon. in verb_. This
book was a farewell to the sciences on the author’s appointment to
the see of London (see the preface): it was published (that is,
the colophon is dated) on the 14th of October, and on the 19th the
consecration took place. This book is decidedly the most classical
which ever was written on the subject in Latin, both in purity of style
and goodness of matter. The author had read every thing on the subject,
in every language which he knew, as he avers in his dedicatory letter
to Thomas More, and had spent much time, he says, _ad ursi exemplum_,
in licking what he found into shape. The wonder is, that after this
book had been reproduced in other countries, and had become generally
known throughout Europe, the trifling speculations of the Boethian
school should have excited any further attention. For plain common
sense, well expressed, and learning most visible in the habits it had
formed, Tonstall’s book has been rarely surpassed, and never in the
subject of which it treats. It seems to have been very little known to
succeeding English writers. P. 419, 426, 427, 439.
_Venice_, fifteen-twenty-three. =Sacrobosco.= ‘Algorismus domini Joh.
de Sacro Busco noviter impressum.’ _Quarto._
The edition of this work mentioned by Watt, as by Cirvellus,
fourteen-ninety-eight, is a mistake: [it was the work on the sphere
which Cirvellus edited in that year] (Hain). Dr. Peacock (p. 416)
thinks that this work is attributed to Sacrobosco without sufficient
reason, and mentions it only as a manuscript. Mr. Halliwell reprinted
it in the _Rara Mathematica_, evidently under the impression that it
had never been printed. If it be the work of Sacrobosco, it establishes
beyond doubt that he had the Arabic numerals and the method of local
value: being nothing but a summary of rules for that arithmetic. The
words _noviter impressum_ are ambiguous: they may either imply a first
impression or any succeeding one, though I have commonly found them
used in the latter sense.
I should pause before I rejected as spurious a work which is attributed
to Sacrobosco in many manuscript copies extant in different countries,
which was printed under his name as early as 1523, and is often cited
as his. Dr. Peacock lays stress upon there being no mention of the
Arabic system in his other works, those on the sphere and on the
calendar. But against the presumption drawn from this circumstance, it
may be urged that it was not likely he would introduce mention of a new
system of arithmetic in works intended for common use, though he might
write a separate work to explain that system. And he would have no
motive for alluding to a method which his readers were not acquainted
with. There is great probability that Sacrobosco was acquainted with
the modern system, which all other evidence goes to shew was introduced
into Italy before his time.
It is of course very possible, or, looking at the progress of other
things, most probable, that isolated individuals had obtained the
Arabic notation from the East, or from Italy, practised it, and written
in it, long before it obtained the smallest general currency.
But there is one circumstance, which seems to me to lend more than
presumption to a still wider supposition, namely, that Arabic
notation and rules were known beyond the bounds of Italy, in the time
of Sacrobosco, to _more_ than a few isolated individuals. Except
by importations from the East, it is impossible to say whence the
philosophers of the thirteenth century could have got any thing
like a set of rules. They certainly had not got any thing Greek,
except from the Arabs: the system of Boethius does not give rules of
computation. Moreover, the name _algorithm_ was in later times so
invariably connected with Arabic arithmetic, that the presumption is
strong it was so from the beginning. Now Roger Bacon, the contemporary
of Sacrobosco, not only used the adjective _algoristicus_ several
times, but recapitulates the names of a set of rules. The theologian,
he says (Jebb, p. 138), should abound in the power of numbering,
that he may know all the algoristic modes, not only for integers,
but for fractions, to numerate, to add, to subtract, to _mediate_
(divide by two), to multiply, to divide, and to extract roots. Now
this is precisely the set of rules given in the treatise attributed
to Sacrobosco, with one omission: this last treatise distinguishes
_duplation_ (multiplication by two) from other cases of multiplication,
which Bacon does not; and _progression_, which is, however, only a
mixture of other rules.
There is a sentence of Bacon’s contemporary, Matthew Paris, quoted
by Mr. Hallam, in which he speaks of what can be done with
Greek notation, as being what cannot be done either in Latin, or
in algorithm, _vel in algorismo_. These words might easily be
interpolated; so instead of using them, as Mr. Hallam does, to give
ground of presumption that Bacon had the Arabic notation, I should
rather use what I have drawn from Bacon to strengthen the genuineness
of the words of Paris; and I think Mr. Hallam will be ready to do the
same.
In ‘Rara Mathematica,’ _London_, eighteen-forty-one, _octavo_, second
edition, Mr. Halliwell has inserted this treatise of Sacrobosco,
together with the poem _de algorismo_ attributed to him in some of the
manuscripts, but probably enough without foundation. The seven rules
above mentioned, are thus given:
Septem sunt partes, non plures, istius artis;
Addere, subtrahere, duplareque dimidiare
Sextaque dividere est, sed quinta est multiplicare
Radicem extrahere pars septima dicitur esse:
on which Mr. Halliwell quotes a manuscript of perhaps nearly as early a
date as Sacrobosco, which describes the seven rules and the necessity
of the _letters called figures_, as follows:
En argorismo devon prendre Et de radix enstracion
Vii especes ... A chez vii especes savoir
Adision Subtracion Doit chasevn en memoire avoir
Doubloison mediacion Letres qui figures sont dites
Monteploie et division Et qui excellens sont ecrites
In the same ‘Rara Mathematica’ are to be found an English manuscript of
the 14th century ‘On the numeration of Algorism,’ and John Norfolk ‘In
artem progressionis summula,’ dated fourteen-forty-five at the end.
_Paris_, fifteen-twenty-six. =John Martin Siliceus.= ‘Arithmetica Joh.
Martini Silicei theoricen praxinque luculenter complexa’ &c. _Folio in
threes._
This is a third edition by Thomas Rhætus; Heilbronner says that the
first was _Paris_, fifteen-fourteen, and that the author, who was a
professor at Salamanca, named _Gujieno_ (Silex), died at Toledo in
fifteen-fifty-seven. Montucla says he was Archbishop of Toledo. This
is a work of considerable extent of subject, and seems to have no
great fault except the usual one of prolixity. It gives a treatise on
theoretical arithmetic (Boethian in character), one on the rules of
computation, one on the mode of calculating by pebbles or the abacus,
and one on fractions. The second edition, fifteen-nineteen, by Orontius
Finæus, is in the Royal Society’s Library.
_Paris_, fifteen-twenty-eight. =John Fernel.= ‘De proportionibus Libri
duo.’ _Folio in threes._
A book on proportion of this date is in great part filled with Boethian
arithmetic. This book deserves attention: its author had a much better
grasp of Euclid than most of his contemporaries.
_Leyden_, fifteen-thirty-one. =Joach. Fortius Ringelbergius.= ‘De
Ratione Studii.’ _Octavo_ (duodecimo size).
At the end, same place, year, and form, with another title-page,
apparently another work, but of this I am not sure, is J. F. R.
‘Compendium de scribendis versibus,’ to which is attached ‘Arithmetica’
(O), the most Lilliputian treatise I know of, giving in 17 pages an
epitome of Boethian terms, algorithmic rules, and counter calculation.
[It is said that the author’s name was Sterk.]
_Venice_, ——. =Giovanni Sfortunati.= ‘Nuovo Lume, Libro di Arithmetica
... Composto per lo acutissimo prescrutatore delle Archimediane, et
Euclidiane dottrine Giovanni Sfortunati da Siena.’ _Quarto in fours._
The end of the work torn out, so I cannot give the date. He mentions L.
de Burgo, Calandri, and P. Borgio, and Tartaglia mentions him: which
gives certain limits. By the manner in which the author is described,
this must be a reprint: it is not likely a work would be originally
printed at Venice, one cause of the production of which is stated to
be the barbarism of the Venetian dialect. I can find no mention of
Sfortunati in catalogues. Cardan, writing in fifteen-thirty-seven,
mentions the work of Fortunatus, the same as the above, no doubt. P.
458, 460.
_Nuremberg_, fifteen-thirty-four. ‘Algorithmus demonstratus.’ _Quarto._
John Schoner, the editor, attributes this to Regiomontanus. It is a
series of demonstrated propositions of numbers connected with the
Arabic notation, involving not only the rules of computation, but
such as that the number of figures in the cube cannot exceed three
times that in the root, &c. J. Schoner, who edited several writings
of Regiomontanus, is likely enough to be right on this one, which is
moreover not unworthy of such an author. It is not in the list which he
published himself, but it seems to be generally recognised. (Weidler
_in verb._)
_Paris_, fifteen-thirty-five. =Orontius Fineus.= ‘Arithmetica practica
libris quatuor absoluta ... Recens ab Authore castigata ... hoc est in
nativum splendorem (quem priorum impressorum amiserat incuria) summa
fidelitate restituta.’ _Folio in fours._
This must be at least the third edition; Leslie says the first was in
fifteen-twenty-five. Orontius died in 1555.
_Strasburg_, fifteen-thirty-six. =Hudalrich= or =Huldrich Regius=.
‘Utriusque Arithmetices Epitome ex variis authoribus concinnata per
Hudalrichum Regium.’ _Octavo_ (duodecimo size).
The first book is on the Boethian arithmetic: the second on the rules
of computation with integers and fractions, and on the use of the
abacus. There are but 96 small folios with large print: so that if I
wished to bring any person into the closest contact with the middle
ages at the least expense of reading, with reference both to their mode
of expression and operation, I should certainly prescribe this book.
_Paris_, fifteen-thirty-eight. =Nicomachus= of Gerasa. ‘Νικομαχου
Γερασινου Ἀριθμητικης βιβλια δυο Nicomachi Gerasini Arithmeticæ Libri
duo. Nunc primum typis excusi, in lucem eduntur. Parisiis in ofiicina
Christiani Wecheli.’ _Quarto._
Nicomachus, who lived about the time of Tiberius, was a Pythagorean,
and of course devoted to arithmetic. He was in his day, as Lucian
intimates, a Cocker, a person whose name passed as an allusion to
reckoning and numbering. The work of Boethius is wholly drawn from
him: some people call it a mere translation of, others a comment on,
Nicomachus. Perhaps a free rendering might be correct. This edition is
Greek without Latin: [there is another, also Greek without Latin, at
the end of the _Theologumena Arithmeticæ_ attributed to Jamblichus,
_Leipsic_, eighteen-seventeen, _octavo_ (?)]. It is then ultimately
to Nicomachus being a Pythagorean that we owe those once never-ending
denominations of numbers and ratios, of which it might now be difficult
to trace any vestige in modern language, except the common words
_multiple_ and _sub-multiple_ and the _sesquialter_ stop of an organ.
_Milan_, fifteen-thirty-nine. =Jerome Cardan.= ‘Practica Arithmetice et
mensurandi singularis.’ _Octavo_ (O).
This was reprinted, under the title of ‘Practica Arithmetica,’ in the
fourth volume of his collected works, _Leyden_, sixteen-sixty-three,
ten volumes, _folio in twos_. On the arithmetic there is no remark to
make, except that, as might be expected from an Italian of that day,
Cardan shews more power of computation than the French and German
writers. There is a chapter recapitulating the numbers which have
mystic properties as he calls them, one use of which is in foretelling
future events. These are mostly the numbers mentioned in the Old and
New Testaments, but not altogether: 400, for instance, makes its
appearance, because it was (but it was not) the number of bishops at
the Nicene Council. In the tenth volume of the same collection is an
unfinished treatise by Cardan, headed ‘Artis Arithmeticæ Tractatus
de Integris,’ which seems to be the commencement of a more extensive
treatise than the former one. It may be cited from it that decidedly,
in Cardan’s opinion, it was Leonard of Pisa who first introduced the
Arabic numerals into Europe.
_London._ ‘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.
Newlye invented and compyled by =Syr Rycharde Benese=, Chanon of Marton
Abbay besyde London. Prynted in Southwarke in Saynt Thomas hospitall,
by me James Nicolson.’ _Quarto._
Again—_London_. ‘The Boke of measuryng of Lande as well of Woodland as
Plowland, and pasture in the feelde: and to compt the true nombre of
Acres of the same. Newly corrected and compiled by =Sir Richarde de
Benese=. ¶ Impynted at London, by Thomas Colwell.’
The approximate date of the first we must guess at from the fact
that James Nicolson’s dated works are from fifteen-thirty-six to
thirty-eight, and from the presumption that a printer’s undated works
come before the dated ones. Thomas Colwell’s dated works are from
fifteen-fifty-eight to seventy-five. The acre is four roods, each rood
is ten _daye-workes_, each daye-worke four perches. So the acre being
40 daye-workes of 4 perches each, and the mark 40 groats of 4 pence
each, the aristocracy of money and that of land understood each other
easily.[13] The reprint of the above work wants a set of tables which
were in the original, and which I take to be the earliest mathematical
tables published in England, except the smaller tables of the same kind
which appeared in some acts of parliament. They are of double entry,
the entries being made by perches in length and breadth. Thus under 79
opposite 9 we find
4·1
7·3
meaning that 79 perches by 9 is 4 acres 1 rood, 7 day-works and 3
square perches. Perhaps this sort of table became common: I find it,
without the slightest deviation of form, in Arthur Hopton’s ‘Baculum
Geodæticum sive Viaticum, or the Geodeticall Staffe,’ _London_,
sixteen-ten, _quarto_.
[13] More easily than they do at the time when this is written. In the
old day, when the aristocracy of money was a Jew, the aristocracy of
land used to shut him up, and draw a tooth a day until he protected
agriculture to the amount demanded. When Christians became wealthy, a
tax upon the use of the teeth seems to have been substituted for their
forcible removal. The debates on the abolition of this tax are now
proceeding. But, speaking of weights and measures, there is something
else which tells a tale about the feasts and Sundays of old England.
The sack of wool was 13 tods of 28 pounds each, or 364 pounds. This
was arranged, that the tasks of those who spun, &c. might be easily
calculated; a pound a day being a tod a month and a sack a year. But
where are the Sundays and holidays? Most likely they were made up on
the other days.
_Paris_, fifteen-thirty-nine. =Johannes Noviomagus.= ‘De Numeris libri
duo.’ _Octavo_ (small) (O).
Professedly on computation, but interspersed with many curious
historical remarks. It contains the first hints I have found of the
most probable explanation of the origin of the Roman numerals.
_Paris_, fifteen-forty-four. ‘=Orontius Finæus.= Arithmetica Practica
... multisque accessionibus locupletata.’ _Octavo_ (O).
_Paris_, fifteen-fifty-five. =Orontius Finæus.= ‘De Arithmetica
Practica libri quatuor.’ _Quarto_ (O).
An augmented edition. Had this book been simply headed as arithmetic, I
might have let it alone. But as it is called _practical_ arithmetic, it
must stand as an index of the great advance of the English over their
neighbours in computation, as shewn by Tonstall and Recorde. P. 428,
436.
_Nuremberg_, fifteen-forty-four. =Michael Stifel.= ‘Arithmetica
integra. Authore Michaele Stifelio. Cum præfatione Philippi
Melancthonis.’ _Quarto._ (Hutton’s Tracts, v. ii. p. 237.)
The two first books are on the properties of numbers, (O), on surds
and incommensurables, learnedly treated, and with a full knowledge of
what Euclid had done on the subject. The third book is on algebra,
and passes for the introduction of algebra into Germany. But Stifel
himself, in his preface, acknowledges his obligations to Adam Risen,
and professes to have taken all his examples from Christopher Rudolph.
Of the latter I can find nothing except a statement that his work
was subsequently published: but Kastner (v. i. p. 108) gives some
information on the works of Adam and Isaac Riese. P. 424, 436, 474, 450.
Stifel is supposed to have been the inventor of the signs + and - to
denote addition and subtraction: and the manner in which he speaks of
these signs seems to imply that they were of his own introduction (p.
110). It is “_we place_ this sign,” &c. and “_we say_ that the addition
is thus completed;” and so on. The sign +, in the hands of Stifel’s
printer, has the vertical bar much shorter than the other: and when it
is introduced in the woodcuts by the engraver, the disproportion is
greater still. One would (supposing the wood-engraver to have imitated
the handwriting) have supposed that - was first used, and that + was
derived from it by putting a small cross-bar for a distinction. This
form is rather against the late learned Professor Rigaud’s idea (see
Mr. Davies’ _Solutions of Hutton’s Mathematics_, p. 11) that + is a
corruption of P, the initial of _plus_, and also against Mr. Davies’
ingenious conjecture that it is a corruption of _et_ or &. Stifel does
not call the signs _plus_ and _minus_, but _signum additorum_ and
_signum subtractorum_. In no instance do I find him using the first
pair of appellatives at all.
I had written the above, and become fully impressed, by repeated
examination, with the suspicion that the vertical bar in + was, in
Stifel’s invention, a small distinction superadded on the sign -, when
I happened, in the immediate preparation of these sheets for press, to
remember Colebrooke’s statement that the Hindoos (our original teachers
in algebra) use a dot for subtraction, and nothing but the absence of
the dot for addition. It may be likely then, that in the first instance
the Hindoo dot was elongated into a bar, to signify subtraction,
addition having no sign: and that the first who found it convenient to
introduce a sign for addition, merely adopted the sign for subtraction
with a difference.
Against the above conjecture it may be argued, that the European
algebra comes from India through the Mahometan writers, who use no
signs at all. On this I can only remark at present, that I have long
suspected, from many circumstances, that there was a more direct
communication with India in the introduction of algebra than any one
now believes.
Dr. Ritchie suggested that perhaps + was used to denote addition as
being two marks joined together, and made significative of two numbers
joined together: and that - denoted subtraction, as indicating what is
left after one of the marks of the junction is removed. This suggestion
is more like what _ought to have been_ the derivation than any I have
seen: but there is little reason to suppose that any such attempt at
significant symbols would have been made in the sixteenth century.
M. Libri attributes the invention of + and - to Leonardo Da Vinci, in
whose manuscripts he says he had seen them. But as it happened to me to
discover, in a manuscript of that justly celebrated man (if it be not
more correct to say _unjustly_ celebrated, as not celebrated enough for
his merits) now in the British Museum, the sign + used for the figure
4, I think the question must rest open until citations which establish
the point without mistake can be produced.
_No place_, nor date. =Wm. Buckley.= ‘Arithmetica memorativa sive
compendaria Arithmeticæ tractatio.’ Apparently _octavo_.
This book has been described both by Leslie and Peacock, who have
severally quoted some of the verses of which it consists. They say
it was first printed in fifteen-fifty, and afterwards at the end of
Seton’s Logic in sixteen-thirty-one. I should probably not have seen it
to this day (for after Leslie’s full citations, I should hardly have
made active search for it), had not an anonymous correspondent (whom I
have only this way of thanking) obligingly forwarded to me an imperfect
copy which he happened to find among his papers. It is, I suppose,
the extract from Seton’s Logic: the signature in the title-page is
Q 2, after a _libellus ad lectorem_, ‘Quisquis Arithmeticam ... qui
meminisse dedit:’ and there is a preface signed T. H. It is worth
noting that Buckley’s Latin name, _Buclæus_, has been frequently
written _Budæus_ by the continental bibliographers, so that in all
probability the learned author of the book _De Asse_ has been taken for
the author of these verses. Hylles calls him _Buckle_. P. 437.
[According to Watt, there were editions of Seton’s Logic, with ‘Huic
accessit ob artium ingenuarum inter se cognationem Arithmetica
Gulielmi Buclæi,’ _London_, fifteen-seventy-two, seventy-four, and
seventy-seven, _octavo_.]
_Basle_, fifteen-fifty. =John Scheubel.= ‘Euclidis Megarensis ...
Algebræ porro Regulæ ... his libris præmissæ sunt....’ _Folio in twos._
The algebra was reprinted, _Paris_, fifteen-fifty-two. _Quarto._
‘Algebræ compendiosa facilisque descriptio,’ &c. For a description of
the work see Hutton’s Tracts, vol. ii. p. 241.
_Leyden_ (_Lugduni_, without further description), fifteen-fifty.
=Henry Cornelius Agrippa.= ‘De Occulta Philosophia Libri Tres.’
_Octavo_ (Italic letter).
As the preface is dated _Mechlin_, fifteen-thirty-one, and [an edition
appeared there in fifteen-thirty-three], this latter must be the
first. This book has as much right in my list as that of Jordano Bruno
presently noticed, treating of numbers in much the same way. It is a
property of seven that there are seven angels who stand before God;
of twelve, that there are twelve apostles, twelve permutations of the
letters of the tetragrammaton, &c. A good many of the more formal works
on arithmetic of this period abound in enumerations of the same kind,
evidently considered as appertaining to the higher uses of the science.
_Augsburgh_, fifteen-fifty-four. =Psellus.= ‘De quatuor scientiis
mathematicis.’ Edited by Xylander. Gr. Lat. _Octavo_ (duodecimo size).
Psellus lived in the ninth century. The part relating to Arithmetic is
a mere compendium of arithmetical terms, perhaps from Boethius.
_Venice_ (the first two parts fifteen-fifty-six, the rest
fifteen-sixty). =Nicolas Tartaglia.= ‘La prima parte dei numeri e
misure aritmetica geometria et algebra.’ _Folio in threes_ (O).
Of this enormous book I may say, as of that of Pacioli, that it
wants a volume to describe it. P. 430, 450. It was partly translated
into French as follows: _Paris_, sixteen-thirteen. ‘L’Arithmetique
de Nicolas Tartaglia, Brescian, grand mathematicien et prince de
praticiens, recueillie et traduite d’Italien en Francois par Guillaume
Gosselin de Caen.’ _Octavo._
_London_, fifteen-fifty-seven. =Robert Recorde.= ‘The whetstone
of witte, whiche is the seconde parte of Arithmetike: containyng
thextraction of Rootes: The Cossike practise, with the rule of
Equation: and the woorkes of Surde Numbers.’ _Quarto._
On this see the _Companion to the Almanac_ for 1837, p. 36; Hutton’s
Tracts, v. ii. p. 243; P. 369. It is rarely remembered that the old
name of Algebra, the _cossic_ art (from _cosa_, thing), gave this first
English work on algebra its punning title the _whetstone_ of wit, _Cos_
ingenii.
_Lione_, fifteen-fifty-eight. =Giov. Franc. Peverone.= ‘Due brevi e
facili trattati. Il primo d’Arithmetica: l’altro di Geometria,’ &c.
_Quarto._
A book of the very simplest examples of computation.
_Leyden_, fifteen-fifty-nine. =Joh. Buteo.= ‘Joan. Buteonis Logistica
quæ et Arithmetica vulgo dicitur.’ _Quarto in fours._ (O).
In the same year was published, by the same author, his _Opera
Geometrica_, the first of which is a plan of Noah’s ark, with
calculations as to its sufficiency for holding all the animals and
their provender. P. 437.
_Paris_, fifteen-sixty. =James Pelletier.= ‘Jacobi Peletarii Cenomani,
de occulta parte numerorum, quam algebram vocant, libri duo.’ _Quarto._
[First edition, fifty-eight.]
This is said to be the first French work on algebra. There is little
purely arithmetical in it: at the end is a table of squares and cubes
up to those of 140. See Hutton’s Tracts, vol. ii. p. 245.
_London_, fifteen-sixty-one. =Robert Recorde.= ‘The Grounde of artes:
Teaching the worke and practise of Arithmetike, both in whole numbres
and Fractions, after a more easyer and exacter sorte than any like
hath hitherto been sette forthe: Made by M. Robert Recorde, Doctor of
Physik, and now of late ouerseen and augmented with new and necessarie
Additions.
I[ohn] D[ee].
All youth and Elde that reasons Lore
Within your breastes will plant to trade
Of Numbers might the endles store
Fyrst vnderstand, than farther wade’
_Octavo_ (duodecimo size) (O).
See the _Companion to the Almanac_ for 1837, p. 32. That this book
was published about fifteen-forty there is internal evidence: and
Tanner gives it that date. Dr. Peacock has fifteen-forty-two. But I
have never seen any edition earlier than this. John Dee published it
again, _London_, fifteen-seventy-three, _octavo_, with other verses in
the title-page, and the original dedication to Edward VI. restored.
The preceding edition, though published under Elizabeth, was probably
arranged in the life of her sister, which may account for the omission
of the dedication. There have been many editions. Hartwell’s edition of
Mellis’s edition was published thrice at least, in sixteen-forty-eight
and sixteen-fifty-four, and once more with the date torn out in our
copy. Mellis added a third part, on practice and other things: in
Roman letter, the sacred text of Recorde having always its old black
letter. The last edition I know of it is by Edw. Hatton, _London_,
sixteen-ninety-nine, _quarto_; it has an additional book called
‘Decimals made easie.’ P. 408-10, 434, 454, 476.
_Vienna_, fifteen-sixty-one. =Christoffer Rudolff.= ‘Kunstliche
rechnung mit der ziffer und mit der zalpfenningen,’ &c. _Octavo_
(small).
This is the man to whom Stifel refers as his instructor in algebra.
But there is nothing like either of the signs + or - in his book: so
that Stifel does not appear to have got them from him. This book is not
algebra: and whether any thing of Rudolph’s deserving of that name was
published is more than I can now settle.
_Antwerp_, fifteen-sixty-five. =Valentine Menher de Kempten.=
‘Practicque pour brievement apprendre à Ciffrer, et tenir livre de
Comptes, avec la regle de Coss, et Geometrie.’ _Octavo._ [Date at the
end, sixty-four, July.]
Here are books on Arithmetic, Book-keeping, Algebra, Geometry, and
Trigonometry: it appears that part of the arithmetic had been published
in fifteen-fifty-six, of the geometry in fifteen-sixty-three, of the
algebra in fifteen-fifty-six. The arithmetic is an early specimen of
the book of mere rules. In this and other French books of the same
date, and even much later, the words _septante_ and _nonante_ are
used for seventy and ninety. The book-keeping is a short treatise, or
rather example, of double entry. The algebra is more extensive, on the
model of Stifelius, with many questions leading to simple equations,
each embellished with an illustrative picture in wood. The geometry
is mensuration. The spherical trigonometry is from Regiomontanus, and
refers to no other tables. This book has some pretension to be called a
course of mathematics. Menher is often quoted in England for his tables
of rebate or discount.
_Antwerp_, fifteen-sixty-seven. =Pedro Nunez.= ‘Libro de algebra en
arithmetica y geometria.’ _Octavo_ (small).
This book is wholly algebraical. See Hutton’s Tracts, vol. ii. p. 250.
_London_, fifteen-sixty-eight. =Humfrey Baker.= ‘The Well Spryng
of Sciences, which teacheth the perfecte woorke and practise
of Arithmeticke ... beautified with moste necessary rules and
questions....’ _Octavo_ (very small size) (O).
The reprints will be presently mentioned.
_Paris_, fifteen-seventy-one. =Alex. Vandenbussche.= ‘Arithmetique
militaire.’ _Quarto._
A short work, mostly on the arithmetic of the arrangement of troops,
with a large number of military maxims and observations. The first
book ends with a question described as _non moins plaisante que
belle_, as follows. A _soldat désespéré_ asked of an _arithméticien
fantastique_ how far it was from Milan to hell. The mathematician,
placing it at the centre of the earth, made the distance 1145¼ leagues.
This, and _Je prie à Dieu nous garder d’y aller_, is the whole of the
joke.
_London_, fifteen-seventy-one. =Leonard Digges.= ‘A Geometricall
Practise, named Pantometria ... framed by Leonard Digges,
Gentleman, lately finished by Thomas Digges his sonne. Who hath
also thereunto adjoyned a Mathematicall Treatise of the five
regulare Platonicall bodies,’ &c. _Quarto._ Reprinted as follows:
_London_, fifteen-ninety-one. ‘A Geometrical practical treatize named
Pantometria.... First published by Thomas Digges Esquire ... lately
reviewed by the author himself,’ &c. _Quarto._
Almost the whole is application of Arithmetic to Geometry, in
measurement both of planes and solids.
In this book I find the earliest printed mention I ever met with of
the _theodelite_: a word the derivation of which has long been a
puzzle. It is known to be an exclusively English word in all time
preceding the middle of the last century. Digges calls it ‘the circle
called _theodelitus_.’ A moveable radius travelling round a circle,
for the measurement of angles, had been long denoted by the Arabic
word _alhidada_ (whence the French word _alidade_). _Theodelite_
seems unlikely to be a corruption of this, at first: nor should I
have suspected such a thing, if I had not found in Bourne’s ‘Treasure
for travailers,’ _London_, fifteen-seventy-eight, _quarto_, the
intermediate formation, _athelida_, used for the same thing. I
hold it then pretty certain that this is the true origin of the
word _theodelite_, which was so spelt, not _theodolite_. (See the
_Philosophical Magazine_, April, 1846.)
_Venice_, fifteen-seventy-five. =Franc. Maurolycus.= ‘Arithmeticorum
libri duo.’ _Quarto in fours._
On the properties of numbers and the doctrine of incommensurables;
a superior work to the mass of those which then treated of similar
subjects.
_Basle_, fifteen-seventy-nine. =Christian Urstisius.= ‘Elementa
Arithmeticæ, logicis legibus deducta.’ _Octavo_ (duodecimo size) (O).
It is said that Urstisius was a follower of Copernicus and a teacher
of Galileo. This Arithmetic was translated by T. Hood, _London_,
fifteen-ninety-six, _octavo_ (small). ‘The Elements of Arithmeticke
most methodically delivered,’ &c.
_Brescia_, fifteen-eighty-one. =Ant. Maria Vicecomes= (Visconti).
‘Practica numerorum et mensurarum,’ &c. _Quarto._
A book of mixed arithmetic and algebra, in which the modern method of
division is used. It would, I think, repay an examination, particularly
on the part which relates to the extraction of roots.
_Antwerp_, fifteen-eighty-one. =Gemma Frisius.= ‘Arithmeticæ practicæ
methodus facilis, in eandem J. Steinii et J. Peletarii annotationes,’
&c. _Octavo_ (small size) (O).
This is a reprint, much augmented, of a compendious work of Frisius,
the ‘Ar. pract. meth. fac.’ _Witteberg_, fifteen-sixty-three, _octavo_
(small). But there is an earlier edition of Gemma Frisius, also by
Peletarius, being the earliest I have met with, as follows:
_Paris_, fifteen-sixty-one. ‘Arithmeticæ Practicæ Methodus facilis,
per Gemmam Frisium ... jam recens ab ipso authore emendata, et multis
in locis insigniter aucta. Huc accesserunt Jacobi Peletarii Cenomani
annotationes ... Quibus demum ab eodem Peletario additæ sunt Radicis
utriusque demonstrationes.’ _Octavo_ (small) (O).
These demonstrations of the rules for the square and cube roots
Peletarius treats as a great boon to his reader: and states that he was
very near contenting himself with a reference to his book on _occult_
properties of numbers (his Algebra of 1560, before mentioned).
_London_, fifteen-eighty-three. =H. Baker=, ‘The Well spring of
sciences. Which teacheth the perfect worke and practise of Arithmeticke
... set forthe by Humfrey Baker, Londoner, 1562. And nowe once agayne
perused augmented and amended in all the three partes, by the sayde
Aucthour.’ _Octavo_ (duodecimo size).
Reprinted (same place and size) in sixteen-fifty-five, with a reference
at the end to an edition of sixteen-forty-six. It is one of the books
which break the fall from the ‘grounde of Artes’ to the commercial
arithmetics of the next century. There are some short rules for
particular cases, and great attention to the rule of practice. Among
the peculiarities of the book is a notion, apparently, that none but
fractions should deal with fractions: for Baker will not double ⅗, for
instance, by multiplying by 2, but only by dividing by ½. P. 452.
_London_, fifteen-eighty-four. =John Blagrave.= ‘The mathematical
jewel.’ _Folio in twos._
See the _Companion to the Almanac_ for 1837, p. 41. This book is
curious, because the woodcuts are all done by the author’s own hand,
and Walter Venge, the printer, is not known to have printed any other
work. [For Blagrave’s legacy to the three parishes of Reading, see
Ashmole’s Berkshire, v. iii. p. 372: for the appearance of his ghost,
“credibly reported by many honest and discreet persons,” see the _Annus
Mirabilis_, 1662, p. 49.] This book is not arithmetical, but the
contrary: it is an avowed attempt to drive computation out of astronomy
by the introduction of an instrument called the _Jewel_, which is a
projection of the sphere. Of the tables of sines he says, ‘But now
there are in like maner an infinite number of intricate questions more
harder far than any yet propounded, which by Regiomont. Copernicus and
others doctrine, grow to great toile with their Synes, calculations,
and proportions, wherein first they hunt about for one Syne, which they
call _inventum primum_, then for another, which they call _inventum
secundum_, and commonly _inventum tertium_, and perhaps _quartum_, and
so foorth. After all these are found, then they multiplie and devide
them, and compare their proportions: and when al is done, that they
have founde the Seyne sought for: they yet are faine to goe to their
tables for the arch correspondent.’ There was a great disposition
at this time among the English to try to substitute instruments for
computation. Blagrave’s Jewel stood in high estimation; and as late as
1658, John Palmer, who published a description of it under the name of
the _Catholic Planisphere_, says it was a subject of frequent inquiry
where the instrument and the description were to be met with.
_Leyden_, fifteen-eighty-five. =Simon Stevinus.= ‘L’Arithmetique
de Simon Stevin de Bruges: Contenant les computations des nombres
arithmetiques ou vulgaires: Aussi l’algebre, avec les equations de
cinc quantitez. Ensemble les quatre premiers livres d’Algebre de
Diophante d’Alexandrie maintenant premierement traduicts in François.
Encore un livre particulier de la Pratique d’Arithmetique, contenant
entre autres, Les Tables d’Interest, La Disme; Et un traicté des
Incommensurables grandeurs: Avec l’Explication du Dixiesme livre
d’Euclide.’ _Octavo._
_Leyden_, sixteen-thirty-four. =Albert Girard.= ‘Les Oeuvres
Mathematiques de Simon Stevin de Bruges,’ &c. _Folio in threes._
See Hutton, Tracts, vol. ii. p. 257, and Peacock, Enc. Metr.
Arithmetic, p. 440. My copy of the first work is imperfect, stopping at
the end of the Diophantus; but as Alb. Girard gives every thing in the
collected works, just in the order announced in the above title-page,
I suppose there can be no doubt that every thing there announced was
published in 1585. The _Disme_ is the first announcement of the use
of decimal fractions. Dr. Peacock supposes it was first published in
Dutch about 1590: we here see that it was published in French in 1585.
Hutton speaks of a Dutch edition in 1605. I do not know where either
date[14] was got from. On looking at the verses with which Stevinus’s
friends (according to custom) have eulogised him, a part of a book to
which I am now and then under obligation, I find that the _Disme_, &c.,
was attached to the edition of 1585, with a new paging. Jagius Tornus,
_philomathes_, be he whom he may, quotes and writes as follows, with a
side-note:
[14] Perhaps 1605 is a mistake for 1626, at which date [R. Soc. Libr.
Cat.] De Thiende, leerende alle rekeninghen, &c. was published at Gouda.
Non fumum ex fulgore sed ex fumo dare lucem
Cogitat, ut speciosa dehinc miracula promat.
Sume unum è multis. quid non Decarithmia præstat Decarithmia
Divinum scriptoris opus? cui non ego si vel La Disme
Aurea mî vox sit, centum linguæ, oráque centum pag. 132.
Omni ætate queam laudes persolvere dignas.
Accordingly, assuming Girard to have faithfully copied his original in
his headings (which he has done as far as my means of comparison go),
the method of decimal fractions was announced before 1585, in Dutch.
The characteristic of Stevinus is originality, accompanied by a great
want of the respect for authority which prevailed in his time. For
example, great names had made the point in geometry to correspond with
the unit in arithmetic: Stevinus tells them that 0, and not 1, is the
representative of the point. And those who cannot see this, he adds,
may the Author of nature have pity upon their unfortunate eyes; for the
fault is not in the thing, but in the sight which we are not able to
give them. P. 426, 440, 460.
[Dr. Peacock (p. 440) mentions an English translation of this tract
by Richard Norton, sixteen-eight, under the title ‘Disme, the arte
of tenths, or decimal Arithmetike ... invented by the excellent
mathematician, Simon Stevin.’]
Since writing the above I have examined the account of Stevinus
given by M. Quetelet, presently referred to. It is stated that
the first work of Stevinus was his table of interest, _Antwerp_,
fifteen-eighty-four——; this was reprinted the next year in the
Arithmetic. In the Supplement of the _Penny Cyclopædia_, ‘Tables,’
I have noted, from the contents of this table and of the _Disme_,
the great probability that the table of compound interest _suggested
decimal fractions_: the only doubt was, which was written first, the
table or the Disme. The statement of M. Quetelet clears this up: and I
hold it now next to certain that the same convenience which has always
dictated the decimal form for tables of compound interest was the
origin of decimal fractions themselves.
_London_, fifteen-eighty-eight. =John Mellis.= ‘A briefe instruction
and maner hovv to keepe bookes of Accompts after the order of Debitor
and Creditor, and as well for proper Accompts partible, &c. By the
three bookes named the Memoriall Journall and Leager, and of other
necessaries appertaining to a good and diligent marchant. The which
of all other reckoninges is most lawdable: for this treatise well and
sufficiently knowen, all other wayes and maners may be the easier and
sooner discerned, learned and knowen. Newely augmented and set forth
by John Mellis Scholemaister, 1588. Imprinted at London by John Windet,
dwelling at the signe of the white Beare, nigh Baynards Castle. 1588.’
_Octavo._
This is the earliest English book on book-keeping by double entry which
has ever been produced. At the end of the book-keeping is a short
treatise on arithmetic (O). But Mellis says: ‘Truely I am but the
renuer and reviver of an auncient old copie printed here in London the
14 of August, 1543. Then collected, published, made and set forth by
one Hugh Oldcastle Scholemaster, who, as appeareth by his treatise then
taught Arithmetike and this booke, in Saint Ollaves parish in Marke
Lane.’
[Watt and others (but not Ames) mention a _quarto_ treatise having this
date, fifteen-forty-three, and entitled ‘A profitable Treatyce called
the Instrument or Boke to learne to knowe the good order of the kepyng
of the famouse reconynge, called in Latyn, Dare and Habere, and in
Englyshe, Debitor and Creditor.’ But no author’s name is given.]
Beckman quotes Anderson as to a book of James Peele, _London_,
fifteen-sixty-nine, _folio_, on book-keeping: but it may be doubted
whether this was not a book on single entry.
_London_, fifteen-ninety. =Cyprian Lucar.= ‘A treatise named
Lucarsolace,’ &c. _Quarto._
A work on mensuration, interspersed with arithmetical rules. It
contains the description of a fire-engine. We find from it that the
surveyors used, not black-lead, but ‘fine-pointed coles or keelers,’ of
which three or four might be bought of any painter for a penny. Lucar
also translated part of Tartaglia on Artillery, with a long appendix.
_London_, fifteen-eighty-eight. _Folio in threes._
_London_, fifteen-ninety. =Thomas Digges.= ‘An Arithmetical warlike
treatise named Stratioticos ... first published by Thomas Digges
Esquire, anno Salutis, 1579,’ &c. _Quarto_ (O).
There is here a brief and good treatise on arithmetic, and some algebra
of the school of Recorde and Scheubel: but the greater part of the work
is on military matters.
_Helmstadt_, fifteen-ninety. =Heizo Buscher.= ‘Arithmeticæ libri duo.’
_Octavo_ (small) (O).
A short and very backward work.
_Bergomi_, fifteen-ninety-one. =Peter Bungus.= ‘Numerorum Mysteria.’
Second edition. _Quarto in fours._
Dr. Peacock gives some description of this fantastical work, page 424.
_Frankfort_, fifteen-ninety-one. =Jordano Bruno.= ‘De Monade numero
et Figura liber Consequens Quinque de Minimo magno et Mensura. Item de
Innumerabilibus, immenso, et Infigurabili; seu De Universo et Mundis
libri octo.’ _Octavo_ (duodecimo size).
On Jordano Bruno see Bayle, Drinkwater-Bethune’s Life of Galileo, p. 8,
and Libri, vol. iv. p. 141, &c. who makes him a vorticist before Des
Cartes, an optimist before Leibnitz, a Copernican before Galileo. The
end of it was that he was roasted alive at Rome, February 17, 1600,
at the age of fifty: ostensibly for heresy (he had been a Dominican),
but most probably in revenge for his satirical writings. He does not
seem, however, to have been a Protestant martyr; as his opinions would
probably have procured his death in almost any state in Europe, as
matters were at that time. Bruno, says M. Libri, seems to have become
a Copernican by a sort of intuition, for he was any thing rather than
a mathematician. His book on monad and number fully confirms this last
statement: it is a collection of dissertations on individual numbers.
But it has the advantage over Pacioli and Bungus (P. 424), that the
Latin is better, and a great part of it is in verse. In the _triads_,
for instance:
Efficiunt totum Casus, Natura, Voluntas,
Dat triplicem mundum Deitas, Natura, Mathesis,
Hinc tria principia emanant, Lux, Spiritus, Unda,
Est animus triplex Vitâ, Sensu, Ratione.
M. Libri has quoted at length (vol. iv. note x.) the Copernican chapter
of the work _de Immenso_.
_Frankfort_, fifteen-ninety-two. =Peter Ramus= and =Lazarus Schoner=.
‘Petri Rami Arithmetices Libri Duo, et Algebræ totidem: a Lazaro
Schonero emendati et explicati. Ejusdem Schoneri libri duo: alter, De
Numeris figuratis; alter, De Logistica Sexagenaria.’ _Octavo_ (O).
The first edition of Ramus is said to be of fifteen-eighty-four. P. 427.
_London_, fifteen-ninety-two. =Thomas Masterson=, ‘his first booke of
Arithmeticke.’
_London_, fifteen-ninety-two. =Thomas Masterson=, ‘his second booke of
Arithmeticke.’
_London_, fifteen-ninety-four. =Thomas Masterson=, ‘his Addition to his
first booke of Arithmetick.’
_London_, fifteen-ninety-five. =Thomas Masterson=, ‘his thirde booke of
Arithmeticke.’ All _Quarto_ (O).
The first book and the addition are on abstract numbers and fractions:
the second book is on commercial arithmetic: the third book is on the
extraction of roots, on surds, and on cossic numbers, or algebra.
Masterson must have been a valuable help to the student: though he
would have been more so if he had used the modern method of division.
_London_, fifteen-ninety-four. =Blundevile.= ‘M. Blundevile His
Exercises, containing sixe Treatises,’ &c. _Quarto in fours._
He was the first introducer of a complete trigonometrical canon into
English, the first announcer of Wright’s discovery of meridional parts,
and he exercised much influence on the studies of the first part of
the seventeenth century. Nor is he altogether out of date yet; for
I lately observed (_Mechanic’s Magazine_, vol. xliv. p. 477) that a
patent for an improvement in horseshoes was upset in the Court of
Chancery, on proof that Blundevile had described it in one of his books
on horses, to which he refers several times in the present work. The
first of these exercises is a treatise on arithmetic, in the form of
question and answer, written for Elizabeth Bacon, the sister of the
great philosopher. It is a dogmatical treatise, sufficiently clear. The
seventh edition of these exercises was published in sixteen-thirty-six,
and I believe there was none later.
_London_, fifteen-ninety-six. ‘The Pathway to Knowledge. Conteyning
certaine briefe Tables of English waights, and Measures ... With the
Rules of Cossicke, Surd, Binomicall, and Residuall Numbers, and the
Rule of Equation, or of Algebere ... And lastly the order of keeping of
a Marchants booke, after the Italian manner, by Debitor and Creditor
... Written in Dutch, and translated into English, by W. P.’ _Quarto_
(O).
This work has escaped the notice of Ames, and was perhaps confounded
with the geometrical work of the same name by Recorde. The English
translator’s preface, giving an account of the existing weights
and measures, is the most complete thing there is of the day. The
arithmetic is not equal to that of Recorde or Masterson, nor the
book-keeping to that of Mellis, and the algebra is far behind
Recorde’s. The translator gives the following verses, the first of
which are now well known. I suspect he is the author of them, having
never seen them at an earlier date: Mr. Halliwell, who is more likely
than myself to have found them if they existed very early, names no
version of them earlier than 1635:
Thirtie daies hath September, Aprill, June, and November.
Febuarie eight and twentie alone, all the rest thirtie and one.
Looke how many pence each day thou shalt gaine,
Just so many pounds, halfe pounds and groates:
with as many pence in a yeare certaine,
Thou gettest and takest, as each wise man notes.
Looke how many farthings in the weeke doe amount.
In the yeare like shillings, and pence thou shalt count.
To complete this subject: Mr. Davies [Key to Hutton’s Course,p. 17]
quotes the following from a manuscript of the date 1570, or near it:
Multiplication is mie vexation,
And Division is quite as bad,
The Golden Rule is mie stumbling stule,
And Practice drives me mad.
_Alcmaar_ (preface dated from Amsterdam), fifteen-ninety-six. =Nicolaus
Petri.= ‘Practicque omte Leeren Rekenen Cypheren ende Boeckhouwen met
die Reghel Coss....’ _Octavo_ (small) (O).
Contains arithmetic, algebra, geometry, and an example of book-keeping.
_Venice_, fifteen-ninety-nine. =Joh. Bapt. Benedictus.= ‘Speculationum
Liber.’ _Folio in twos._
The first speculation is entitled ‘Theoremata Arithmetica,’ and is
a laboured explanation of the principles of arithmetical rules by
reference to geometry, beginning with the old difficulty of the product
of two fractions being less than either of the factors.
_London_, sixteen-hundred. =Thomas Hylles.= ‘The arte of vulgar
arithmeticke, both in integers and fractions, devided into two bookes
... Nomodidactus Numerorum ... Portus Proportionum ... whereunto is
added a third book entituled Musa Mercatorum, comprehending all rules
used in the most necessarie and profitable trade of merchandise ...
Newly collected, digested, and in some part devised, by a wel willer to
the Mathematicals.’ _Quarto_ (O).
This is in dialogue; but whenever any rule or theorem is delivered,
it is in verse. It is a big book, heavy with mercantile lore. The
following are specimens of the verses:
Number is first divided as you see
For number abstract, and number contract
And numbers abstract are such as stand free
From every substance so cleare and exact
That they have no sirname or demonstration
Save the pure units of their numeration.
All primes together have no common measure
Exceeding an ace which is all their treasure.
Addition of fractions and likewise subtraction
Requireth that first they all have like basses
Which by reduction is brought to perfection
And being once done as ought in like cases,
Then adde or subtract their tops and no more
Subscribing the basse made common before.
_The partition of a shilling into his aliquot parts._
A farthing first findes fortie eight
An halfepeny hopes for twentie foure
Three farthings seekes out 16 streight
A peny puls a dozen lower.
Dicke dandiprat drewe 8 out deade
Twopence tooke 6 and went his way
Tom trip and goe with 4 is fled
But goodman grote on 3 doth stay
A testerne only 2 doth take
Moe parts a shilling cannot make.
In spite of all this trifling, Hylles was a man of learning. He cites
both _Lucas_ de Burgo and _Peter_ de Burgo.
_Paris_, sixteen-hundred. =John Chambers.= ‘Barlaami Monachi Logistica
nunc primum Latiné reddita et scholiis illustrata.’ _Quarto._
The Greek is given at the end. Barlaam lived in the fourteenth century,
and his work is mostly on fractions and on proportions. See the _Penny
Cyclopædia_, ‘Barlaam,’ for information on the author and on this
edition.
_Leyden_, sixteen-three. =C. Dibuadius.= ‘In Geometriam Euclidis
prioribus sex Elementorum libris comprehensam Demonstratio Numeralis.’
_Quarto._
The six books of Euclid are, for the most part, here verified by
arithmetical or trigonometrical instances. The usual mode, or _linear_
demonstration, is made to follow in another book.
_Leyden_, sixteen-five. =Simon Stevinus.= ‘Tomus secundus
Mathematicorum hypomnematum de Geometriæ praxi.’ _Folio in threes._
(The printing delayed beyond the title-date.)
This work contains the celebrated Latin treatise on book-keeping
(Peacock, _Encyl. Metrop._ ‘Arithmetic,’ p. 464), in which the
difficulty of finding Latin renderings of the technical terms is well
got over. All writers attribute this translation to the celebrated
Willebrord Snell, who passed from fourteen to eighteen years of age
while the work was printing. Now the fact is, that not only does
Stevinus date his own preface to this treatise on book-keeping in
August 1607, but at the end of the volume he excuses himself from
giving several matters which had been announced at the beginning,
because the printer was tired of waiting, and he had not satisfied
himself about them. And in this same final notice he mentions _Will.
Sn._ (Snell) as a person who had written a letter for him: styling him
_mathematum neque ignarus neque expers_; much too poor a compliment for
a youth who, at such an age, had translated, and therefore understood,
all the writings of Stevinus. The story about the translation is from
Gerard Vossius, who was afterwards Snell’s colleague, and who made a
mistake and confusion of persons which was copied by Bayle, Beckmann,
&c. M. Quetelet has recently published a notice of Stevinus, which
states that he was born in 1548, and died in 1620. P. 464.
_Augsburgh_, sixteen-nine. =Geo. Henischius.= ‘Arithmetica perfecta et
demonstrata.’ _Quarto_ (ON).
A laboured work in seven books. Its algebra is that of a former day;
its power of computation very fair. Dr. Peacock refers several times to
another work of Henischius, ‘De numeratione multiplici,’ sixteen-five.
_Mayence_, sixteen-eleven. =Christopher Clavius.= ‘Epitome Arithmetices
Practicæ.’ _Folio in threes_ (O).
This is in the second volume of Clavius’s collected works. [It is said
to have been first published in fifteen-eighty-three.] Perhaps there
are more extensive examples of the square root worked by the old method
in this treatise than would easily be found elsewhere. P. 426.
_Bologna_, sixteen-thirteen. =Pietro Antonio Cataldi.= ‘Trattato del
modo brevissimo di trovare la Radice quadra delli numeri.’ _Folio._
The rule for the square root is exhibited in the modern form, and
Cataldi shews himself a most intrepid calculator. But the greatest
novelty of the book is the introduction of continued fractions, then,
it seems, for the first time presented to the world. Here, with great
labour, but still with success, Cataldi reduces the square roots of
even numbers to continued fractions of the form
_m_ _m_
_a_+ ———— ———— &c.
_n_+ _n_+
He then uses these fractions in approximation, but without the
assistance of the modern rule by which each approximation is educed
from the preceding two. Thus he reduces, among other examples, the
square root of 78 to
3073763825935885490683681
8 —————————————————————————
3695489834122985502126240
_London_, sixteen-thirteen. =John Tap.= ‘The Pathway to Knowledge,’ &c.
_Octavo_ (O).
This professes to be a reprint of the work published under that name
(1596), and above described. But it is in fact the substance of the
above work thrown into the form of dialogue, with the book-keeping
reprinted, and some algebra taken from V. Menher above mentioned.
_London_, sixteen-thirteen. =Richard Witt.= ‘Arithmeticall questions,
touching the Buying or Exchange of Annuities; Taking of Leases for
Fines, or yearly Rent; Purchase of Fee-Simples; Dealing for present or
future Possessions; and other Bargaines and Accounts, wherein allowance
for disbursing or forbeareance of money is intended; Briefly resolved
by means of certain Breviats.’ _Quarto_ (octavo size) (O).
There is also in the title-page ‘Examined also and corrected at the
Presse, by the Author himselfe.’ A great many circumstances induce me
to think that the general fashion of correcting the press by the author
came in with the seventeenth century, or thereabouts.
_Breviats_ are tables. As far as I know, this is the first English
book of tables of compound interest. And there are _real_ tables
of half-yearly and quarterly compound interest. Again, decimal
fractions are really used: the tables being constructed for ten
million of pounds, seven figures have to be cut off, and the reduction
to shillings and pence with a _temporary_ decimal separation, is
introduced when wanted. For instance, when the quarterly table of
amounts of interest at ten per cent is used for three years, the
principal being 100_l_. (page 99), in the table stands 137266429, which
multiplied by 100 and seven places cut off gives the first line of the
following citation:
‘The Worke
{l 1372 | 66429
_Facit_ {sh 13 | 2858
{d 3 | 4296.’
Giving 1372_l._ 13_s._ 3_d._ for the answer. And the tables are
expressly stated to consist of _numerators_, with 100 ... for a
denominator.
[A reprint of this work, by T. Fisher, _London_, sixteen-thirty-four,
_duodecimo_, is in the Royal Society’s Library.]
_Paris_, sixteen-fourteen. =Artabasda.= ‘Nic. Smyrnæi Artabasdæ Græci
Mathematici ΕΚΦΡΑΣΙϹ Numerorum Notationis per gestum digitorum. Item
Venerab. =Bedæ= de Indigitatione et manuali loquelæ lib.’ _Octavo in
twos_ (edited by F. Morell, the first Gr. Lat., the second Lat.).
These tracts are on nothing but the mode of representing numerals
by the fingers. Bede’s tract is not a separate work, but a chapter
from his treatise _de natura rerum_, in which it will be found in
the ‘Bedæ ... opuscula plura,’ edited by Noviomagus, _Cologne_,
fifteen-thirty-seven, _folio in twos_. But the first edition is,
_Venice_, fifteen-twenty-five, Joh. Tacuinus (editor and printer), ‘hoc
in volumine hæc continentur M. Val. Probus ... Ven. Beda de Computo per
gestum digitorum ... nunc primum edita.’ _Quarto in fours._
_Paris_, sixteen-fourteen. =Honorat Meynier.= ‘L’Arithmetique ...
enrichie de ce que les plus doctes mathematiciens ont inventés ...
tant aux formes que nos anciens ont practiquees comme en celles qui se
practiquent aujourd’huy en France, en Holande, en Allemagne, en Espagne
et autres nations.’ _Quarto_ (O).
There is a great deal of matter in this book, which runs to 664 pages;
and it might be historically useful. Part of it is an attack on
Stevinus, who, by the way, is represented as then alive. The following
is one of the examples in subtraction: ‘L’an 1535, Jean Calvin composa
son labirinthe abominable (puisque dans iceluy s’est perdu un grand
nombre d’hommes de bien) ... je demande combien s’est passé d’années
depuis qu’il composa le dit Dedale.’
_London_, sixteen-fourteen. =Thomas Bedwell.= ‘De Numeris Geometricis.
Of the nature and properties of geometrical numbers, first written by
Lazarus Schonerus, and now Englished,’ &c. _Quarto._
On figurate, square, &c. numbers, with applications to mensuration.
_Bologna_, sixteen-sixteen. =P. A. Cataldi.= ‘Quarta parte della
pratica Aritmetica Dove si tratta della principalissima, et
necessariissima Regola chiamata comunemente del Tre,’ &c. _Folio_
(quarto size).
A long and wordy treatise on the rule of three, but venturing on
examples of a much higher order of difficulty of computation than had
been previously attempted.
_Edinburgh_, sixteen-seventeen. =John Napier.= ‘Rabdologiæ seu
Numerationis per Virgulas Libri duo: cum Appendice de expeditissimo
Multiplicationis Promptuario, Quibus accessit et Arithmeticæ Localis
Liber unus. Authore et inventore Johanne Nepero,’ &c. _Duodecimo_ (O).
A posthumous work. Napier’s rods are well known, and this book is the
_descriptio princeps_ of them, with applications. The use of decimal
fractions, expressly attributed to Stevinus, renders it remarkable. It
is stated (P. 441) that Napier invented the decimal point, but this is
not correct: 1993·273 is written by him 19932′7″3‴.
_Oppenheym_, sixteen-seventeen and nineteen. =Robert Fludd.= ‘Utriusque
Cosmi Majoris scilicet et minoris metaphysica, physica atque technica
historia.’ Two volumes. _Folio in twos._
The second volume has another title-page, purporting that it gives
the supernatural, natural, preternatural, and contranatural history.
Of Robert Fludd (though he was great enough in his day to engage the
attention of Descartes) I shall here say nothing, except that he gave
his mixture of mysticism and science two dedications, one on each side
of a leaf. The first, signed _Ego, homo_, was addressed to his Creator:
the second, signed ‘Robert Fludd’ to James I. of England. The first
volume contains a treatise on Arithmetic, and on _Cossic_ Arithmetic,
or Algebra. The arithmetic is rich in the description of numbers, the
Boethian divisions of ratios, the musical system, and all that has any
connexion with the numerical mysteries of the sixteenth century. The
algebra is of the four rules only, referring for equations and other
things to Stifel and Recorde. The signs of addition and subtraction are
P and M with strokes drawn through them. The notation for powers is
the _q_, _c_, _qq_, &c. series of symbols. Perhaps the most remarkable
thing about the algebra is, that Fludd, who wrote it in France for the
instruction of a Duke of Guise, should have known nothing of Vieta. The
second volume is strong upon the hidden theological force of numbers.
_London_, sixteen-nineteen. =Henry Lyte.= ‘The art of tens, or decimall
arithmeticke.’ _Octavo_ (duodecimo size).
One of the earliest English users of decimal fractions. P. 440.
_Hamburgi_, sixteen-twenty. =Francis Brasser= and =Otto Wesellow=.
‘Arithmetica Francisci Brasseri ... ab Ottone Wesellow ex Germanico in
Latinum sermonem versa, atque in lucem edita.’ _Octavo_ (O).
Brasser seems to have been a celebrated teacher: it is not clear
whether he published or not. The book itself is a mixture of arithmetic
and algebra in Scheubel’s form, with much of algebraical application to
commercial questions. There is extraction of roots as far as the fourth.
_Hamburg_, sixteen-twenty-one. =Peter Lauremberg= of Rostoch.
‘Institutiones Arithmeticæ.’ _Octavo_ (small).
A demonstrative book.
_Rostoch_, sixteen-twenty-one. =John Lauremberg.= ‘Organum analogicum,
sive instrumentum proportionum.’ _Quarto._
This instrument is the sector.
_Rostoch_, sixteen-twenty-three. =John Lauremberg.= ‘Virgularum
numeratricum et promptuarii arithmetici descriptio, figuræ et usus.’
_Quarto._
A tract of fourteen pages on Napier’s rods.
_London_, sixteen-twenty-four. =William Ingpen.= ‘The Secrets of
numbers, according to theologicall, arithmeticall, geometricall, and
Harmonicall computation.’ _Quarto._
A worthy follower of Peter Bungus. P. 425.
_London_, sixteen-twenty-four. =Thomas Clay.= ‘Briefe easie and
necessary tables of interest and rents forborne,’ &c. _Octavo_ (very
small size).
There is no hint here of decimal fractions. Attached is a
‘chorologicall discourse’ on the management of estates.
_London_, sixteen-twenty-eight. =John Speidell.= ‘An Arithmeticall
Extraction or Collection of divers questions with their answers.’
_Octavo_ (small).
For John Speidell, see the article _Tables_ in the Supplement of the
_Penny Cyclopædia_. The book is literally what it professes to be,
questions with their _facits_ or answers, and nothing else, mostly in
reduction, the rule of three, and practice.
_Amsterdam_, sixteen-twenty-nine. =Albert Girard.= ‘Invention nouvelle
en l’algebre,’ &c. _Quarto._
In this celebrated work there is a slight treatise on arithmetic, the
most remarkable part of which is, that the author gives no examples
in division by more than one figure, and seems to decline them as too
difficult for his readers: he gives some results only. P. 426, 442. On
one occasion he uses the decimal point.
=Oughtred’s= _Clavis_ is a book of arithmetic as well as of algebra,
and one of great celebrity. The editions that have fallen in my way are:
_London_, sixteen-thirty-one. ‘Arithmeticæ in numeris et speciebus
institutio: quæ tum logisticæ, tum analyticæ, atque adeo totius
mathematicæ, quasi clavis est.’ Signed ‘Guilelmus Oughtred,’ at the end
of the dedication. _Octavo._
_London_, sixteen-forty-seven. ‘The Key of the Mathematicks New Forged
and Filed: Together with A Treatise of the Resolution of all kinde of
Affected Æquations in Numbers. With the Rule of Compound Usury; And
demonstration of the Rule of false Position. And a most easie Art of
delineating all manner of Plaine Sun-Dyalls. Geometrically taught by
Will. Oughtred.’ _Octavo._
_Oxford_, sixteen-fifty-two. ‘Guilelmi Oughtred ... Clavis Mathematicæ
denuo limata, sive potius fabricata ... Editio tertia auctior et
emendatior.’ _Octavo._
_Oxford_, sixteen-sixty-seven. ‘Do. do. ... Editio quarta auctior et
emendatior.’ _Octavo._
_Oxford_, sixteen-ninety-three. ‘Do. do. ... Editio quinta auctior et
emendatior.’ _Octavo._
_Oxford_, sixteen-ninety-eight. ‘Do. do. Editio quinta auctior et
emendatior. Ex recognitione D. Johannis Wallis, S.T.D. Geometriæ
Professoris Saviliani.’ _Octavo._
_London_, sixteen-ninety-four (again in seventeen-two). ‘Mr. William
Oughtred’s Key of the Mathematics. Newly translated ... with Notes ...
Recommended by Mr. E. Halley.’ _Octavo._
The third edition was an extended re-translation into Latin; and the
preface of it, which was copied into succeeding editions, shews that
it was carefully performed under Oughtred’s own eye. The treatise on
dialling was translated into Latin by a young man of sixteen, of Wadham
College, whom Oughtred describes as already an inventor in Astronomy,
Gnomonics, Statics, and Mechanics, and of whose future fame he augurs
great things: his name was Christopher Wren. There are two fifth
editions in the list, the second of them revised, it appears, by Dr.
Wallis, who ought, one would think, to have known better the history
of the book he edited. But on examining them, I find that they are the
same impressions with different title-pages: so that it would seem as
if Wallis had allowed his eminent name to appear as a guarantee for a
book he had never revised. Wrong again, however: for in the preface to
the third edition it appears that Wallis had given all the help of an
editor at that stage of the work; so that his name should rather have
appeared before.
The editions which follow the first, besides other additions, have
the solution of adfected Equations, or Vieta’s method (see the Index,
‘Horner’). Throughout, the old, or _scratch_ method of division,
is retained. Dr. Peacock observes of this method that it lasted
nearly to the end of the seventeenth century: but it thus appears
that it even got into the eighteenth, and that with Halley’s name as
a recommendation. Throughout all the editions, Oughtred’s ancient
algebraical notation is retained, as also his way of writing decimal
fractions
(12|3456 for our 12·3456).
——————
I cannot tell why Dr. Peacock always spells his name Oughtred_e_ (p.
441, &c.).
_London_, sixteen-thirty-two. =William Oughtred.= ‘The Circles of
Proportion and the Horizontal Instrument. Both invented and the uses
of both Written in Latine by Mr. W. O. Translated into English: and
set forth for the publique benefit by William Forster.’ Published with
it, _London_, sixteen-thirty-three, ‘An addition unto the use of the
instrument ... For the Working of Nauticall Questions ... Hereunto is
also annexed the excellent Use of two Rulers for Calculation. And it is
to follow after the 111 Page of the first Part.’ _Quarto._
The second edition, _Oxford_, sixteen-sixty, _octavo_, is ‘by the
Author’s consent Revised, Corrected ... by A. H. Gent.’ [The editor was
Arthur Haughton.] The addition above named gives the first description
of the _sliding rule_. The circles of proportion are what would now
be called circular sliding rules; and the two rulers are the common
sliding rule, the rulers bring kept together by the hand. Various
errors are afloat about the invention of the sliding rule. Some give it
to Wingate, some to Gunter, some to Partridge; the official book of the
Excise Office gives it to an excise officer whose name I forget. But
the truth is that Oughtred invented it. (See Penny Cycl. ‘Slide Rule.’)
_London_, sixteen-thirty-three. =Robert Butler.= ‘The Scale of
Interest; or Proportionall Tables and Breviats ..... Together with the
valuation of Annuities,’ &c. _Octavo._
These tables are for discount or present value; and as far as
annuities, &c. are concerned, that is, in compound interest questions,
resemble Witt’s. But decimals are really used in the applications, the
separating notation being as in 12|3456.
_London_, sixteen-thirty-three. =Nicholas Hunt, M.A.= ‘The Hand-Maid
to Arithmetick refined: Shewing the variety and facility of working
all Rules in whole Numbers and Fractions, after most pleasant and
profitable waies. Abounding with Tables above 150. for Monies, Measures
and Weights, tale and number of things here and in forraigne parts;
verie usefull for all Gentlemen, Captaines, Gunners, Shopkeepers,
Artificers, and Negotiators of all sorts: Rules for Commutation and
Exchanges for Merchants and their Factors. A Table from 1l. to 100
thousand, for proportionall expences, and to reserve for Purchases.’
_Octavo_ (ON). P. 442.
The first thing that strikes any one in this book is the slavishness
of the dedication to an Earl, and the grotesque appearance which the
use of the ambiguous word _rare_ gives it. “When I reflect on ancient
Nobility, earth’s glory, it being found in the way of vertue; this so
rare a thing transports my soule with thoughts of a glorious eternitie.
And for as much as the Lord hath said (_Dixit dii estis_) that you are
titular and tutelar terrestriall gods, powerfully protecting the meaner
shrubbes under the spreading branches of your tall Cedars: I wonder
not to see almost every Pamphlet to crouch and deject it selfe in all
humility and lowlinesse, seeking patronage from persons of eminency.”
He afterwards calls on his patron to “imitate the propitiousnesse
of the divine essence,” which will excite him to “a voluntarie
prostitution of his humble service,” if the patron will protect him
from “that squint-ey’d foole, that antipathie to vertue (the envious
man).”
The book itself is very full on weights and measures, and commercial
matters generally. It does not treat of decimal fractions: what the
author calls ‘decimall Arithmeticke’ is the division of a pound into
10 primes of two shillings each; each shilling into _six_ primes
of two pence each. There are some verses, perhaps suggested by the
republication of Buckley, headed: ‘Arithmeticke-Rithmeticall, or
the Handmaid’s Song of Numbers;’ of which the rules of addition and
subtraction will give a sufficient specimen:
Adde thou upright, reserving every tenne,
And write the Digits downe all with thy pen,
The proofe (for truth I say,)
Is to cast nine away.
From the particular summes, and severall
Reject the Nines; likewise from the totall
When figures like in both chance to remaine
Clear light of working right shal be your gain;
Subtract the lesser from the great, noting the rest,
Or ten to borrow, you are ever prest,
To pay what borrowed was thinke it no paine,
But honesty redounding to your gaine.
_Paris_, sixteen-thirty-four (with another title-page,
sixteen-forty-four, but it is all one impression). =Peter Herigone.=
‘Cursus Mathematici tomus secundus.’ _Octavo_ (O).⁗
The volumes of this course, which is said to be the first complete
course published, have different dates. It is a polyglott book, Latin
and French. This second volume contains practical arithmetic. It
introduces the decimal fractions of Stevinus, having a chapter ‘des
nombres de la dixme.’ The mark of the decimal is made by marking the
place in which the last figure comes. Thus when 137 livres 16 sous is
to be taken for 23 years 7 months, the product of 1378′ and 23583‴
is found to be 32497374⁗ or 3249 liv. 14 sous, 8 deniers. There is
a _memoria technica_ for numbers in this book, which I subjoin,
that those who know the old system of Grey may compare the two. The
consonants and vowels which stand for numbers are:
P B C D T F G L M N
1 2 3 4 5 6 7 8 9 0
a e i o u ar er ir or ur }
ra re ri ro ru }
So that _r_ signifies _five_ except before or after _u_, when it
destroys the meaning altogether. Grey’s system is:
B D T F L S P K N Z
1 2 3 4 5 6 7 8 9 0
a e i o u au oi ei ou y
The vowel-systems of the two might be combined.
_London_, sixteen-thirty-four. =William Webster.= ‘Webster’s tables for
simple interest direct .... Also his Tables for Compound Interest ...’
_Octavo_ (small). Third edition.
This work treats decimal arithmetic as a thing known: but all the
tables are not decimals. Neither is there yet any recognised decimal
point; only a partition-line to be used on occasion. In this book I
find the first head-rule for turning a decimal fraction of a pound into
shillings, pence, and farthings: though not so perfect a one as was
afterwards found. I have seen the second edition, place, title, and
form as above, sixteen-twenty-nine. The copy of each edition which I
have examined is bound up with King James’s list of customs, ‘The Rates
of Merchandizes’ &c. promulgated in 1605.
_London_, sixteen-thirty-seven (again in forty-seven). =Leonard
Digges.= ‘A book named Tectonicon .... Published by Leonard Digges,
Gentleman, in the yeere of our Lord 1556.’ _Quarto._
A book of mensuration: one of the instances in which a book was
republished for the name of its author, long after its methods were
obsolete. This occurred so frequently in the seventeenth century that
we must take for granted the existence of a class of mathematicians to
whom the light of Napier, &c. had not penetrated.
_London_, sixteen-thirty-eight; reprinted in seventeen-forty-eight.
=John Penkethman.= ‘Artachthos, or A New Book declaring The Assise or
Weight of Bread by Troy and Avoirdupois Weights. Containing divers
Orders and Articles made and set forth by the Right Honorable the Lords
and Others of his Majesty’s most Honorable Privy Council.... Published
by their Lordships Orders.’ _Quarto._
This does not look like a book of arithmetic; but Penkethman has
thought it necessary to prefix instruction in numeration, both Roman
and Arabic.
‘Note that IV signifies IIII, as IX signifies nine which takes as it
were by stealth, or pulls back one from foure and ten.’ So that, in
fact, I stands behind X and picks his pocket.
[_Leyden_, sixteen-forty.] =Adrian Metius.= ‘Arithmetica Practica.’
_Quarto_ (N).
There is no prolixity about this book. Sexagesimal fractions are
taught, but not decimal ones. The title is torn out in my copy.
_Herbornæ Nassoviorum_, sixteen-forty-one. =Joh. Henr. Alsted.=
‘Methodus Admirandorum Mathematicorum Novem libris exhibens universam
Mathesin.’ _Duodecimo_ (N).
This has a slight treatise on arithmetic, with a few words on algebra:
nothing on decimal fractions.
_Sora_, sixteen-forty-three. =John Lauremberg.= ‘Arithmetica ...
exemplis historicis illustrata, itidem algebræ principia.’ _Quarto_ (N).
A book which uses the Italian method of division is a rarity among
German books of this period. The examples are drawn from historical
matters, and from the military art.
_Paris_, sixteen-forty-four. =Theon= of Smyrna. ‘Τῶν κατὰ μαθηματικὴν
χρησίμων εἰς τὴν τοῦ Πλάτωνος ἀνάγνωσιν.’ _Quarto._ Edited by Ismael
Bullialdus (Bouillaud).
_Leyden_, eighteen-twenty-seven. =Theon= of Smyrna. The same title.
_Octavo in twos._ Edited by J. J. de Gelder.
These are the only editions of Theon of Smyrna, neither having the
complete work, if indeed any complete manuscript exist. The first
has the arithmetic and music; the second, with various readings,
the arithmetic only. The arithmetic is nothing more than that
classification and nomenclature of numbers which is also given by
Nicomachus (the two were contemporaries), and followed by Boethius. The
music, as in other cases, is only discussion of fractions under names
derived from the scale. Theon of Smyrna must not be confounded with his
greater namesake of Alexandria.
_London_, sixteen-forty-five. =Edmund Wingate.= ‘Use of the Rule of
Proportion in Arithmetique and Geometric.’ _Duodecimo._ (A translation
from Wingate’s own French.)
This book is on the use of Gunter’s logarithmic scale, not the sliding
rule, which Wingate never wrote upon.
_Leyden_, sixteen-forty-six. =Francis Vieta.= ‘Opera Mathematica.’
_Folio in twos._
In this collection is the ‘De numerosa potestatum purarum atque
adfectarum ad exegesin resolutione tractatus,’ which was first
published, _Paris_, sixteen-hundred, _folio_, and is the first
exposition of the method by which the general evolution of the roots of
equations is connected in operation and principle with the extraction
of the square root and common division. See the references presently
given under the name of Horner. The first departure from Vieta’s
form of solution, which amounts to incorporating with it the method
afterwards known by the name of Newton, was made by Henry Briggs in the
‘Trigonometria Britannica,’ _Gouda_, sixteen-thirty-three, _folio in
twos and threes_. Accordingly, Briggs has a fair claim to be the first
user of that which is called Newton’s method of approximation, though
he does not explain it, but leaves the explanation to be collected
from his examples. The work which introduced Vieta’s method into
England was the ‘Artis Analyticæ Praxis’ of Thomas Harriot, _London_,
sixteen-thirty-one, _folio_. All these writers would have called
their process algebraical, but the progress of the art of computation
will make their works essential parts of the history of arithmetic. I
believe the most complete account of Vieta is that which I have given
in the _Penny Cyclopædia_: with which may be read the account of his
works in Hutton’s _Tracts_, vol. ii. pp. 260-274.
_London_, sixteen-forty-eight. =Seth Partridge.= ‘Rabdologia, or the
Art of numbring by Rods.’ _Duodecimo_ (O).
The use of Napier’s rods or bones explained.
Napier’s _Rabdologia_ brought the well-known _Napier’s rods_ into vogue
for half a century (P. 411). And the reason seems to have been that
this contrivance really was found useful, in the state of the habit of
computation as it then existed. P. 432, 440.
_Leyden_, sixteen-forty-nine. =Joh. Hen. Alsted.= ‘Scientiarum omnium
Encyclopædia.’ Two volumes. _Folio in threes._
The first edition is said to be sixteen-thirty, and the preface of
this one is dated sixteen-twenty-nine, but it refers to a course of
philosophy published in sixteen-twenty, of which it seems to be an
enlargement. As far as I can learn, this is the first of the works to
which moderns attach the idea of a _Cyclopædia_. For though Ringelberg
and Martinius had published under that name, and though the _Margarita
Philosophica_ above mentioned and other works ran through the whole
circle of elementary knowledge—yet I cannot establish that any work
before Alsted’s was so large and comprehensive as to be a Cyclopædia
in the sense in which Chamber’s Dictionary was afterwards so called.
The course of arithmetic (O) in the second volume is rather scanty, but
seems less so when the bareness of the professed mathematical courses
is considered. There is a table of the squares and cubes of all numbers
up to 1000. Logarithms are not mentioned, but Napier’s rods are.
Moreover, by the date of the first edition, we must call this a course
of mathematics prior to Herigone’s, usually called the first.
_London_, sixteen-fifty. =John Wybard.= ‘Tactometria. seu,
Tetagmenometria. Or, The Geometry of Regulars practically proposed ...
by J. W.’ _Octavo in twos._
An excellent book of mensuration of solids, full of remarkable
information on the subject of weights and measures.
_London_, sixteen-fifty. =Jonas Moore.= ‘Moore’s Arithmetick:
Discovering the secrets of that Art, in Numbers and Species.’ _Octavo_
(N).
This is Jonas Moore’s first work, and is a very good one. It is
very complete in decimals, giving the contracted multiplication and
division. It has the use of logarithms (in which the operations with
the negative characteristic are fully given), algebra from Oughtred,
and squares and cubes of all numbers up to 1000, and the fourth, fifth,
and sixth powers up to 200. The second edition is:
_London_, sixteen-sixty. ‘Moor’s Arithmetick in two books.’ _Octavo._
It is worthy of remark that the common arithmetic of this edition is in
black letter, the second book, or Algebra, being Roman letter, as all
the previous edition was. Perhaps Recorde, &c. had given common readers
a prejudice for black letter arithmetics. P. 442.
_Dantzig_, sixteen-fifty-two. =Joh. Broscius.= ‘Apologia pro Aristotele
et Euclide.... Additæ sunt duæ disceptationes de numeris perfectis.’
_Quarto._
A perfect number is one which is equal to the sum of its less factors:
thus 28, being the sum of 1, 2, 4, 7, and 14, is a perfect number. And
a number was called defective or redundant, according as the sum of its
factors fell short of or exceeded the number. Thus 8 is defective, and
12 is redundant.
_London_, sixteen-fifty-three. =Noah Bridges.= ‘Vulgar Arithmetique,
explayning the Secrets of that Art, after a more exact and easie way
than ever.’ _Octavo_ (N).
_London_, sixteen-sixty-one. =Noah Bridges.= ‘Lux Mercatoria,
Arithmetick Natural and Decimal,’ &c. _Octavo_ (N).
Dr. Peacock (p. 452) has given the first book notoriety by his
citations from the eulogistic poetry at the beginning, which he has by
no means exhausted. The poem of Geo. Wharton, the royalist astrologer,
on the existing state of things, is a political satire. Another
commends the plainness of the work, hints at decimals and logarithms
becoming too common, and pronounces that a merchant
‘may fetch home the Indies, and not know
What Napier could, or what Oughtred can doe.’
Dr. Peacock speaks lightly of the work, in mentioning the excessive
praises with which it was announced. But a very short treatise,
explicit upon the modern mode of division, which those who praised it
had perhaps never seen before, and upon the use of practice, was rather
an exception to the rule at the time at which this book was published.
The second work is more learned, and has an appendix on decimals. The
author disapproves of the use which some would make of decimals, and
avers that the rule of practice is more convenient in many cases; which
is perfectly true. Bridges seems to have had a very clear view of the
capabilities of arithmetic without formal fractions. He mentions some
preceding authors, and his list is ‘The Merchant’s Jewel,’ 1628; ‘The
Handmaid to Arithmetick,’ 1633; ‘The Map of Commerce,’ 1638; Masterman,
Johnson, Hill, Leybourn, Recorde, Baker, Val. Menher, Boteler.
_London_, [sixteen-fifty-four]. =E=[=dmund=] =W=[=ingate=]. ‘Ludus
Mathematicus. Or, The mathematical Game: Explain Explaining the
description, construction, and use of the Numericall Table of
Proportion.’ _Duodecimo._ (Again in eighty-one.)
This is the description of a logarithmic instrument, which it would
be impossible to give a notion of without the instrument itself, or a
drawing. The date is cut out in my copy.
_London_, sixteen-fifty-five. =Thos. Gibson.= ‘Syntaxis Mathematica.’
_Octavo._
This work is filled with the construction of equations from Des Cartes;
but it gives the arithmetical solution, or Vieta’s Exegesis, from
Harriot: also interest-tables, partly from _Simon Stevens_.
_Oxonii_, sixteen-fifty-seven. =John Wallis.= ‘Mathesis Universalis:
sive Arithmeticum Opus Integrum,’ &c. _Quarto._
This and other works, with their separate title-pages, are collected
(apparently so published) as _Opera Mathematica_, and were afterwards
republished, in substance, in the folio collection of Wallis’s works.
The Arithmetic gives both modes of division: but the old notation, as
12|345,
————
is used for decimal fractions. Wallis afterwards adopted the decimal
point in his Algebra.
_Wurtzburg_, sixteen-fifty-seven-and-eight. =Gaspar Schott.= ‘Magia
Universalis Naturæ et Artis.’ _Quarto_ (four parts, variously bound in
volumes).
This work, and others of a similar character by the same author,
are the precursors of the works of mathematical amusement compiled
by Ozanam, Montucla, Hutton, &c. In the fourth part he treats of
arithmetic, mostly on the wonders of combinations. The gem of the book
is his accurate calculation of the degrees of grace and glory of the
Virgin Mary, which are exactly
1157920892373161954235709850086879078532
69984665640564039457584007913129639936
not one more nor less. This is the 256th power of 2, and is repeated
three times, printed in the same way. Others solved the same problem,
and found out the number of stars at the same time, by writing down
every possible way in which the words of