At the end of this work will be found an Index containing, besides the
names of the authors mentioned in the Catalogue, who may be known by
the paginal figures opposite, every name which I have found _any where_
as belonging to a writer on arithmetic before 1800. Since that period
I have not swelled the list from mere _catalogues_, but have contented
myself with what I had either seen, or learnt from some other source.
For the existence of such writers, or of their works, of course I do
not vouch, except in those cases in which the work is found in the
body of the Catalogue. Of those who stand unpaged in the index, all I
can say is, that they are taken from many sources, and that there is
in each case plenty of reason to make inquiry for their writings, and
strong presumption that their works are still to be found.
A word with an asterisk before it is the name of a writer who is
referred to by Dr. Peacock. The great apparent preponderance of German
names arises partly from the number of works written on the subject
in Germany being very considerable, partly from the bibliographical
catalogues of that country being more full than those of any other,
and partly from the names which seem to be German, really including
Danes, Dutch, Belgians, and some Swiss. I am far from thinking that
this list contains even the third part of the names of those who have
really written on the subject. I have not been particular in searching
for any thing after 1750, though I have not refused what came in my
way. I might have very much increased the list of recent Germans, from
Rogg’s _Bibliotheca Selecta_ (Tubingen, 1830). Looking at the various
countries which enjoyed the art of printing from 1500 downwards, I
have an impression, from all that I have gathered, which would lead me
to suppose that the number of works on arithmetic published in Latin,
French, German, Dutch, Italian, Spanish, and English, up to the middle
of last century, cannot be less than three thousand, which gives to
each language less than an average of one a year. Few of these would
seem to fall within the province of the historian. Dr. Peacock refers
to about a hundred and fifty. Unfortunately, history must of necessity
be written mostly upon those works which, by being in advance of their
age, have therefore become well known. It ought to be otherwise, but it
cannot be, without better preservation and classification of the minor
works which people actually use, and from which the great mass of those
who study take their habits and opinions. The _Principia_ of Newton
is, if we believe the title-page, a work of the seventeenth century:
but the account of the effect which it produced on science belongs to
the eighteenth. It was not till many years after the publication of
the _Principia_, that its predecessor in doctrine, the great work of
Copernicus, produced its full effect upon general thought and habit.
Nor have we any reason to suppose that it could have been otherwise.
The great exceptions will always bear, perhaps, as large a ratio to the
average power as ever they did: it is as likely as not, that if the
intelligence of the sixteenth century had been sufficient to verify
and receive the opinion of Copernicus at once, some predecessor of his
might have been Copernicus, and he or another of his day might have
been Newton.
It is then essential to true history, that the minor and secondary
phenomena of the progress of mind should be more carefully examined
than they have been. We must distinguish between the progress of
possibilities and that of actual occurrences. Our written annals shew
us too much what might have been, and too little what was: they give
some words to the slow reception of an improvement, and more sentences
to the account of the one man who was able to make it before the world
at large could appreciate it.
The public is beginning to demand that civil history shall contain
something more than an account of how great generals fought, great
orators spoke, and great kings rewarded both for serving their turn.
The progress of nations might well be described, for the most part,
with much less mention of any of the three: but the parallel does not
hold of knowledge. Copernicus and Newton would fill a large space,
though the history were written down to that of every individual who
ever opened a book: but it seems to me that they, and their peers,
are made to fill _all the space_. Nor will it be otherwise until the
historian has at his command a readier access to second and third rate
works in large numbers; so that he may write upon effects as well as
causes.
This list contains upwards of fifteen hundred names, of which a few
may be duplicates of others, arising from the wrong spelling of my
authorities.[2] But these must be much more than counterbalanced by the
number of names which belong to two or more authors, but which only
appear once in my list. Thus there are two Digges’s, three Riese’s,
two named Le Gendre, Walker, Newton, Wallis, &c., and several Taylors,
Butlers, &c., though each name is only mentioned once. There are
several cases in which I have not ventured to strike out one of two
names, though there is every reason to suppose one is a mistake for
the other; as Caraldus and Cataldus, Cappaus and Cuppaus, Döhren and
Dühren. But I have often been deceived in this way; and have more than
once been obliged to re-insert names which I had struck out, supposing
them to be only different spellings of names already in the list. There
are several whom I have seen in more places than one, who are clearly
Germans with names metamorphosed by wrong reading of the black letter,
or use of the genitive for a nominative, or both. Thus Petzoldt has
become Pekoldts, Seckgerwitz has been made Sectgerwik, Schultze has
been made Schulken, and so on. These obvious mistakes of course have
not been admitted. Moreover, several persons are, I suppose, down in
the list under two names by which they are known. Thus, Fortius may be
Ringelbergius (whose real name was Sterk), Blasius may be Pelacanus.
But as I cannot undertake to assert that there is no Fortius except
Ringelbergius, &c., I have let both names stand in the list. Again,
next to Buckley comes Budæus. Many foreign writers, Heilbronner
among the rest, have turned _Buclæus_ into _Budæus_, so that in all
probability the second of these names is a mistake for the first.
And yet there may be also some Budæus who has written on arithmetic;
though, having excluded writers on weights and measures only, I have
not put down the author of the work _De Asse_.
[2] It is necessary, for instance, to keep close watch upon a writer
who introduces among his English authors _Gul. de Cavendy dux de
Xeucathle_.
I have had, in one or two instances, to throw away German _authors_,
for a very obvious reason. The reader will not find the works of
_Anleitung_, or _Grundriss_, or _Rechenbuch_ in my list, which is more
than can be said of every one which has preceded it.
I have not attempted to translate the names of those who wrote in
Latin, at a time when that language was the universal medium of
communication. In every such case I consider that the Latin name is
that which the author has left to posterity; and that the practice
of retaining it is convenient, as marking, to a certain extent, the
epoch of his writings, and as being the appellation by which his
contemporaries and successors cite him. It is well to know that
Copernicus, Dasypodius, Xylander, Regiomontanus, and Clavius were
Zepernik, Rauchfuss, Holtzmann, Muller, and Schlüssel. But as the
butchers’ bills of these eminent men are all lost, and their writings
only remain, it is best to designate them by the name which they bear
on the latter, rather than the former. In some cases, as in that of
Regiomontanus, both names are frequently used: in others, where the
Latin consists in a termination only, as Tonstallus for Tonstall,
or Paciolus for Pacioli, it matters nothing which is used. It may
happen that errors are introduced by returning to the vernacular in a
wrong way. I should like to know how it is shewn that Orontius Fineus
was Oronce Finée: or in what respect this reading is more finically
correct than Horonce Phine, which has great antiquity in its favour?
In the case of Vieta, Viète is certainly wrong: he was called by his
contemporaries Viet (which I suspect to be derived from the Latin form)
and de Viette, never Viète.
It may also be asked, why the unlatinizing process should, for
confusion’s sake, be practised by the learned only; when it is pretty
certain that the world at large will never reconvert Melancthon into
Schwartzerd, or Confucius into Kaen-foo-tzee. Neither will they restore
to the Popes and other priests of the Roman Catholic Church the names
under which they were born and educated.
In many cases it would be impossible to recover these last. For myself,
I am well content with the name under which an author was known in
literature to his contemporaries, and has been handed down to us, his
successors. I know of no canon under which it is imperative to speak
of a writer rather as his personal acquaintance than as his reader:
and, so far as feeling of congruity is concerned, I think _Alexander
ab Alexandro_ looks better at the head of a Latin preface than
_Saunders Saunderson_. Those who really wish to catch the tone of the
middle ages, and shew themselves quite at home, should dwell on the
_Christian_ name, and make the surname a secondary distinction; should
learn to think of _Nicolas_ (who happened to be called Copernicus, or
Zepernik, it should matter little which) and _Christopher_, to whom the
calendar was entrusted.
That this list must be very imperfect I am well aware, for I have been
able to add many names to it which I never found in any catalogue. But
it will not be useless. It may furnish a reason for preserving any
work whatsoever which comes in the way of the reader of this book.
If the name be in the list, a book should not be destroyed which has
been somewhere catalogued and recognised as a portion of the existing
materials for the history of science. But if the name be not in the
list, it is obvious that there is some curiosity about a writer whose
name is not in the most numerous catalogue of arithmetical authors that
has ever been collected in any one place. And it is undeniable that
every name must be in one or the other predicament.
I now come to the Catalogue which forms the body of this work. The
defects, which every one who has examined the lists that are extant,
knows to prevail, arise, in great measure, from the titles and
descriptions of books being copied by those who had never seen the
books themselves. This is not the worst; for a true copy of a true
copy is a true copy: many of these accounts have finally become formal
representations of informal titles, which, though not originally
intended for more than sufficient indication of the book mentioned,
have been used as if they had been full and accurate descriptions.
There is a large number of works, not much distinguished in the history
of science, each of which has, nevertheless, done its part in its day.
The minor points of the same history depend much upon these books,
which, being neither of typographical curiosity, nor of literary fame,
are gradually finding their way either to the waste paper warehouse,
or the public library. These two depositories are almost equally
unfavourable to works of no note assuming their place in the annals of
the knowledge to the progress of which they have contributed. Take the
library of the British Museum, for instance, valuable and useful and
accessible as it is: what chance has a work of being known to be there,
merely because it is there? If it be wanted, it can be asked for; but
to be wanted, it must be known. Nobody can _rummage_ the library,
except those officially employed there, who will only now and then
have leisure to turn their opportunities to account in any independent
literary undertaking. And it would perhaps be difficult to make any
regulation, under which persons not belonging to the institution might
have access to see what is there. Nor will the publication of the
catalogue do much towards supplying the defect. Titles of books are
but vague indicators of their contents; and a catalogue[3] of half a
million of entries, even if its contents could be guessed at by the
titles of the books, is not made to be read through.
[3] When the great catalogue of the Museum is published, those who can
give house-room to forty or fifty volumes, and time enough to their
examination, may have some of the advantage which they would derive
from actual access to the books themselves. And those known to be
engaged in research will derive a still larger portion of the same
advantage from the readiness with which the officers of the Museum will
go out of the usual routine of duty to help them.
It would be something towards a complete collection of mathematical
bibliography, if those who have occasion to examine old works, and
take pleasure in doing it, would add each his quotum, in the shape of
description of such works as he has actually seen, without any attempt
to appear more learned than his opportunities have made him. This is
what I have done in the descriptive catalogue of works on arithmetic,
without selection, or other arrangement than order of date. The only
reason for a work being in this list, is that it has come in my way:
the only reason for one being out of it is, that it has not come in my
way, at least at the time of compilation. Whenever any statement is
made which is taken from any other writer, it will be put in brackets
[], except when the statement itself cites an authority: though I have
sometimes put the brackets even in the latter case. The only other
mode of proceeding would be, to collect lists from authority, naming
the sources of information. But, having found so many errors in these
sources, when my opportunities have enabled me to bring them to the
test, I did not feel inclined to be the tenth transmitter of inaccurate
copies. My mistakes shall be of my own making, and it would not be easy
to invent one which should want high precedent for its species.
The description of the books in the Catalogue is uniformly as follows.
There are given:
1. The place at which the work was printed, in Italics, and generally
in English.
2. The date of the title-page, or colophon, or, when both are wanting,
of the preface, in words; but not quite at length. Thus instead of
fifteen-hundred-and-eighty-eight will be found fifteen-eighty-eight.
I am sure that dates will never be given correctly until this plan
is universally adopted: for two reasons. First, the chance of error
in printing is very much diminished: particularly the risk of a
transposition of figures _at the press_. Secondly, the writer, who
is, on the whole, and one date with another, more to be feared than
the printer, has time to be accurate. A glance at four numerals and
four strokes with the pen, is too rapid a process for certainty: and
those who think they can rely upon themselves to stop _every time_,
and look at what they have done, will frequently find reason to wish
they had been less confident. But _Incidit in Scyllam_, &c.: I had
very nearly announced an edition of Cocker as being unmistakably the
seventeen-hundred-and-twentieth (see page 56).
3. The author’s name. When an initial only is given, it is because the
author has left no more, either in title or preface.
4. As much of the title as will certainly identify the book. In
spelling, initial capitals, &c., I have generally followed the author
closely. When there is any defect in this respect, I suppose it will be
in those works which I had not any opportunity of re-examining while
the sheets were in proof. Imitation of type I have not attempted. To
have given the fall titles would have swelled my book too much: at
certain periods the authors of elementary works were much given to
write out descriptive chapters in their title-pages; scores of them
would each have filled two or three pages of even the smaller type used
in the Catalogue.
5. The form in which the work is printed; a matter which will require
some explanation.
A folio, quarto, octavo, duodecimo, or smaller work, is now generally
known by its size, though not always. In the folio the sheet of paper
makes two leaves or four pages, in the quarto four leaves, in the
octavo eight, in the duodecimo twelve, and so on. But even a publisher
thinks more of size than of the folding of the sheet when he talks
about octavo or quarto; and accordingly, when he folds a sheet of
paper into _six_ leaves, making what ought to be a _sexto_ book, he
calls it a duodecimo printed in half sheets, because such printing is
always done with half-sized paper, or with half sheets, so as to give
a duodecimo size. From a very early period it has been universal to
distinguish the sheets by different letters called _signatures_. In the
book now before the reader, which is a half-duodecimo (or what I call
a _duodecimo in threes_), the first sheet which follows the prefatory
matter, B, has B on the first leaf, and B 2 on the third; which is
enough for the folder’s purpose. But in former times the signatures
were generally carried on through half the sheet, and sometimes through
the whole. Again, in modern times, no sheet ever goes into and forms
part of another; that is, no leaf of any one sheet ever lies between
two leaves of another. But in the sixteenth century, and even later
in Italy, it was common enough to print in _quire-fashion_. Imagine a
common copybook, written through straightforward, and the string then
cut: and suppose it then separates into four double leaves besides the
cover. It would then have sixteen pages, the separate double leaves
containing severally pages 1, 2, 15, 16; 3, 4, 13, 14; 5, 6, 11, 12;
7, 8, 9, 10. If a book were printed in this way, it would certainly
be a folio, if the four double leaves of any one quire or gathering
were each a separate sheet: and if the sheet were the usual size, it
would give the common folio size. But if each gathering had the same
letter on all its sheets, if the above for instance were marked A₁ on
page 1, A₂ on page 3, A₃ on page 5, and A₄ on page 7; the book, when
made up, would have all the appearance of a more recent octavo in its
signatures. In order to give the size of a book, and at the same time
to give the means of identifying the edition by its signatures, I have
adopted the plan which gives the following rules:
(a) The words _folio_, _quarto_, _octavo_, _duodecimo_,
_decimo-octavo_, refer entirely to size, as completely as in a modern
sale-catalogue, the maker of which never looks at the inside of a book
to tell its form. All the very modern distinctions of _imperial_,
_royal_, _crown_, _atlas_, _demy_, &c. &c. &c. I have relinquished to
paper-makers and publishers, who alone are able to understand them. But
in old books, the reader must expect to see the several sizes, each one
of them, smaller than in the modern books. When the work is decidedly
small of its name, I have noted it by the word ‘small.’
(b) When the single word occurs, without any thing more, the signatures
are as in the genuine meaning of the word. Thus, as to signatures,
_folio_ has two leaves to one letter of signature, _quarto_ four
leaves, _octavo_ eight, _duodecimo_ twelve; and of course double the
number of pages.
(c) When the modern word occurs with the addition of _in twos_, or _in
threes_, &c. the addition expresses the number of _double leaves_ which
belong to one letter of signature: and which I believe would be found,
if the books were taken to pieces, to be in each quire or gathering.
Thus, _folio in ones_, or _quarto in twos_, or _octavo in fours_, or
_duodecimo in sixes_, would in each case be unnecessary repetition; for
the first word, when alone, is intended to express the gathering in the
third. But _folio in twos_ would mean the folio size with two double
leaves in one quire, _folio in fours_ with four double leaves. Thus,
a book of the octavo size, with the quarto signatures, is _octavo in
twos_: had it been larger, I should have called it _quarto_.
By this means there is something as to size, and something as to
signatures, in every description. But whether any book which I call
_octavo in twos_, for instance, really was printed on whole sheets,
or on half sheets, that is, really was a small quarto, or a divided
octavo, is more than I can in any case undertake to say. All I know is,
that with these rules, the reader has two indications in every case, to
guide him in determining whether the book he has in his hand be the one
I describe or not.
This is no unnecessary excess of description. For so frequently, in
the sixteenth and seventeenth centuries, were there issues of the same
impression under different titles, that all I have done will in many
cases only give a presumption as to whether a book in hand is or is not
the edition I have described. Were I to begin this work again, I would
in every instance make a reference to some battered letter, or defect
of lineation, or something which would be pretty certain not to recur
in any real _reprint_. Ordinary errata would not be conclusive: for
these might be reprinted for want of perceiving the error.
Rules are given for determining the form of printing by the waterlines
of the paper, and by the catchwords. It is supposed that the latter
are always at the end of the sheet, and also that the waterlines are
perpendicular in folio, octavo, and decimo-octavo books, and horizontal
in quarto and duodecimo. But in the first place, a great many old books
have catchwords at the bottom of every page, many have none at all; and
as to the rule of waterlines, I have found exceptions to every case of
it. Pacioli’s Euclid, _Venice_, fifteen-nine, _folio in fours_, has
horizontal waterlines: the Hypomnemata of Stevinus, an undoubted folio,
has thick waterlines _both ways_.
6. In the smaller type are entered such remarks as suggested themselves
on the manner or matter of the work, or on any point arising out of it.
In some cases these have extended themselves to short dissertations,
such as, on the geometrical foot, page 5,—on Sacrobosco’s knowledge
of the Arabic numerals, page 14,—on the invention of + and -, page
19,—on the age of Diophantus, page 47,—on the genuineness of Cocker’s
Arithmetic, page 56. But for the most part I have contrived to keep
within a very moderate compass as to what is said under each work.
The principal point on which I have distinguished one work from
another, is as to the use of the _old_ or _new_ method of performing
division, which, more than any other single point, decides the
character of a work. The letters O, N, or ON, will tell whether the
work uses the old, the new, or both. Not that the verbal distinction is
here very correct; for neither method is older than the other; and both
appear in Pacioli. A description of the now disused method is given by
Dr. Peacock, p. 433.
With regard to the works themselves, I have made no selection, as
before noticed. No book that I _have_ seen during the compilation has
been held too bad to appear; no book that I _have not_ seen too good
to be left out. I have had but one discretion to exercise, namely,
to determine the extent to which algebra should be considered as
arithmetic. In the earlier day, the distinction was slight: I doubt
whether I have not overrated it; but it is not an easy line to draw.
The history of Arithmetic, as the simple art of computation, has found
little notice from the historians of mathematics in general. They
shew themselves deficient in the knowledge of its progress, and of
the connexion of that progress with the rest of their subject. The
writers whom I can name as having attempted—some more and some less—to
supply this defect, are Wallis, Dechâles, Heilbronner, Scheibel,
Kästner, Leslie, Delambre, Peacock, and Libri. I speak of the progress
of arithmetical writings as works on science, independently of
bibliography properly so called, and biography.
_Wallis_ (p. 44 and Additions) was one of those writers whose works
remain the standard of the erudition of their day. His _algebra_, so
called, is rather the history, theory, and practice of both arithmetic
and algebra. Its miscellaneous and badly indexed form prevents any from
knowing what is in it, except those who make a study of it, which none
of our day will do, unless they intend to go rather deeply into the
history of the exact sciences. But many and many a page by which the
writer intended to gain the credit of research, will be found to be a
transcript from Wallis. As a connected history, however, it is nothing;
and as a bibliography, less. For example, that Regiomontanus used
decimal _fractions_—a very common story—is the consequence of Wallis’s
confused method of stating that he introduced the _decimal_ instead
of the _sexagesimal_ radius into trigonometry; the confusion arising
from his not having a clear knowledge of what had been published of
Regiomontanus. And again, we find him, though the editor of an edition
of Oughtred, balancing as to which was written first, Oughtred’s
_Clavis_, or Harriot’s _Praxis_ (which were published in the same
year), apparently ignorant that the great method as to which the two
authors were chiefly to be compared, did not appear in Oughtred’s first
edition at all. But for all this, if Wallis be cautiously watched as to
books and dates, his works are most valuable magazines of historical
suggestion. From them might be collected a much better _scientific_
history of arithmetic than existed in his time, or, indeed, in any time
preceding the publication of Dr. Peacock’s article on the subject.
_Claude Francis Milliet Dechâles_ was a Jesuit, who published, in
four very large folios, a complete course of mathematics, including
architecture, carpentery, fireworks, and all that was then held to
belong to the exact sciences (see page 53). The first volume opens
with about a hundred pages (large folio, double column) _de progressu
matheseos_, consisting entirely of description of books, in order of
date. The part relating to arithmetic fills nine of these pages. The
whole is done with much care, and is, for the mode of describing books
current at the time, very accurate; and the opinions given on the books
shew that Dechâles had read them. But he is strongly addicted to the
very common mistake of judging the books according to what ought to
have been said of them, if they had been published in his own time.
For example, he finds that there is not sufficient demonstration in
Tonstall; which is true, absolutely speaking; but Tonstall is a very
Euclid by the side of his contemporaries.
_Heilbronner’s_ Historia, &c. (page 69), though it professes only to
give writers up to the beginning of the sixteenth century, makes a
particular exception in favour of arithmetic. Up to the year 1740,
about 170 authors are recorded, a great many of whom he had not seen.
There is also historical dissertation on points of arithmetic. This
is a work of great value to the inquirer: he must not rest upon its
statements; but he will find more than usual materials for further
research.
_Scheibel_ (Additions) may be considered as partly repetition, partly
extension, of Heilbronner. He is one of those bibliographers who
collect from various sources the names and dates of more editions than
those who know catalogues will readily believe in.
_Kästner_ (Additions) falls under the following censure from Dr.
Peacock: “The meagre sketch which Kästner has given of some insulated
works on the subject, generally contrives to omit almost every
particular which is essentially connected with the history of the
progress of the science.” This is well merited, inasmuch as the
author chose to call his work a history, instead of a _bibliography_;
and as the former, nothing can be more incomplete. I have almost
as good a right to call my work a history, as Kästner his. But, as
a bibliography, it may be urged in defence, that he gave fuller
descriptions of books than his predecessors. Scheibel, Heilbronner, and
Dechâles sink into title-writers (and bad ones) before Kästner.
The late Professor _Leslie_ (page 89) was one of those men, the
strength and asperity of whose opinions would make it fair to deal with
them as they dealt with others. In his _Philosophy of Arithmetic_ he
has entered incidentally into much of its history. He was, by taste, a
searcher of old books; and various dates, &c. occur, which shew that
he had more knowledge of books than can be got from catalogues. A few
words will sometimes shew this to a person who has compared the books
with the accounts of them. But, writing in a popular manner, he does
not give references to his authorities, which is a serious diminution
of the value of his work. Of Leslie, as an historian on controverted
points, one principal thing to be cautious of is, his almost monomaniac
antipathy to every thing Hindoo—a most unfortunate turn for an
arithmetical investigator. Those who inquire into this subject will see
what he is in Colebrooke’s hands; those who do not, may compare his
description of the _Lilivati_, “a very poor performance, containing
merely a few scanty precepts,” with the summary of the contents of that
work in the _Penny Cyclopædia_, article _Viga Ganita_. Leslie also
generalises most fearfully every now and then. He informs us that it
was the _practice throughout Europe_ to reduce the rules of arithmetic
to memorial verses, and that Buckley’s _Arithmetica Memorativa_ appears
at one period to have _gained possession of the schools and colleges of
England_. Now the truth is, that the _verses_ attributed to Sacrobosco
had never even been printed when Leslie wrote; and Buckley, so far as
is known, was printed only once alone, and two or three times as an
appendix to a work on logic. Dr. Peacock expresses the truth in saying
that, _before the invention of printing_, the practice of _writing_
memorial verses was common, as appears by manuscript libraries. It is
needless to say that, had the practice of _using them_ been common, the
presses of the fifteenth and sixteenth centuries would have given them
forth in great numbers. But I cannot learn that any metrical work was
printed in the fifteenth century, except the _Compotus_ of Anianus, and
that only once.
_Delambre_ (page 84) wrote on the history of Greek Arithmetic in such
a manner as to set that part of the subject fairly going. I doubt if
any one since the time of Wallis, at least of their order of note, has
exhibited so much of the union of the _scholar_ and the _computer_.
Of the obligations under which the history of astronomy lies to him,
it is unnecessary to speak here, or any where: no one man was ever so
closely connected with that science, with its past, its present, and
its future, by history, observations, and tables.
Of my opinion of Dr. _Peacock’s_ work (page 91) I need hardly say any
more, after the pains which I have taken to give reasons on every point
in which I differ from its author, and to correct every little error
which I have found. This I judge to be the most undoubted compliment
which can be paid to any work. Having looked carefully over it, with
a great many of the works mentioned in it in my hands, it would be
a strong evidence in its favour, were it needed, that I have found
no more to set right than is there noted, in matters of dates and
circumstances. It is much to be wished that this treatise should be
published separately:[4] those who can obtain it will find that it
gives life and spirit to the catalogue of books which forms the main
part of this work. Up to this moment it is the only work which can be
called a history of arithmetic.
[4] It was once intended to publish these treatises separately. Nine
years ago, the proprietors of the _Encyclopædia Metropolitana_ so
fully intended to publish separately, that they considered themselves
aggrieved because I, who had written the _mathematical_ article on
probabilities, wrote a _popular_ work with that word in the title-page,
which they alleged, through their agent, was in effect a republication
of the former work, &c. Not being able to get them either to litigation
or arbitration, I was obliged to write a pamphlet to prove that the
charge was frivolous. The pamphlet is unanswered, and all the treatises
unpublished (separately) to this day. The latter I regret, for the sake
of science. It is a great pity that Sir John Herschel’s treatises on
light and sound, Dr. Peacock’s arithmetic, Mr. Airy’s tides, &c. are
thus locked up.
M. _Libri’s_ history of science in Italy (page 95) is the work of a
man who, to the character of a mathematician, adds that of a man who
is well versed in literature, and a successful collector of the rarest
works in that of his own country. There is much which is interesting in
its early history of Italian Arithmetic. Unfortunately, it is not yet
finished; and, I am told, is not likely to be speedily resumed.
The history of most of the sciences resembles a river which sinks
underground at a certain part of its course, and emerges again at a
distant spot, swelled by certain tributaries, which have joined it in
the tunnel. Of arithmetic in particular, we note the disappearance in
the seventh century of all that was good in the Greek system; and we
see the rise, at the invention of printing, of the most trivial part of
it, combined with the additions which are now well known to have been
received from Eastern sources.
The manuscript literature of the middle ages will prove no very
productive source of information, to judge by all that has been made of
it hitherto. But then it is to be remembered how seldom, if ever, it
has happened, that the investigator of it has united the character of a
sufficient mathematician with that of an industrious and well-trained
palæographist. Neither the one nor the other can proceed alone: the
former has not only to learn how to read, but what to read; for the
actual habitat of the manuscripts, and how to get preliminary knowledge
of their existence, is a study of itself. The latter is apt to know no
difference between what is sound, and what is worthless.
When Euclid, Ptolemy, &c. are first seen to reappear, they come in as
Arabic writers. They may be called Greek, but the first translations
are from the Arabic; and their effect upon the literature of Europe is,
in the first instance, just what it might have been if the authors had
been Persians or Saracens. How the communication took place has been
considered as a point of small curiosity compared with the importation
into Europe of the Arabic numerals.
This subject, both generally, and with reference to the several
countries, has been long on the anvil: not the authorship of the
letters of Junius has tasked research and ingenuity more than the
introduction of the nine digits and the cipher. I suppose nobody would
listen to the hypothesis that the former wrote themselves: but I am
much inclined to suspect that the latter introduced themselves. The
_endosmose_ constantly going on between nations connected by war and
commerce would not merely explain so easy a matter, but would render
it very difficult to explain how it did not happen, if it had not
happened. That the Venetian merchants should not know the system of
accounts of those with whom they traded, is incredible: that, at the
beginning of the thirteenth century, many priests and some soldiers,
returned from their crusades, should not bring back with them the
account of so very elementary a difference between themselves and those
with whom they had treated, and whom they had in many instances served
as slaves taken in war, is unlikely. That Leonard of Pisa, as he is
called, was the first who wrote on or in the new system, is pretty
generally affirmed by the Italians of the fifteenth and sixteenth
centuries: and there can be no doubt of it. But because we trace the
first formal user or expounder to Italy, it by no means follows that
the other parts of Europe owe the system to that country in the first
instance, however certain it may be that they owe much assistance in
the course of the general establishment. It was so common in England
in the thirteenth century, that Roger Bacon recommends it (page 14)
as a study, of course as a practicable study, for the clergy. Had it
not been commonly known, at least as to what it _was_, he would have
done more than give the name of the science, and the names of its
rules; he would have added some description. Those who have searched
into the matter, merely with a view to Arabic _numerals_, and without
any collateral thoughts about _arithmetic_ in their heads, may have
passed over much valuable evidence. They did not know, perhaps,
that the organised rules of computation always went with the Arabic
system, and never with the Roman or Boethian: that if ever they came
to the connected mention of addition, subtraction, multiplication,
and division, it ought to have been a sign that they were reading on
_algorism_, as distinguished from _arithmetic_. This passage of Roger
Bacon has been neglected; though the mere occurrence of the word
_algorism_, in a most interpolable clause of one sentence, occurring in
Matthew Paris, has received notice and discussion from Mr. Hallam, from
a learned writer in the Archæologia, and probably from others.
The rejection of the work attributed to Sacrobosco may be accounted
for from the general ignorance of its having been printed under his
name at an early period. Though this does not prove the genuineness of
the work, it very much adds to the evidence for it. As early as 1523,
the learned world was invited to dissent from the assertion that the
treatise in question was written by Sacrobosco, and did not do it. It
was often cited afterwards, and never, as far as I have seen, with
any doubt. But the only writer of this or the last century that I can
find, who describes the _printed_ edition, is the Italian editor of
Fabricius, _Bibliotheca Latina_.
Come how they might, however, we find ourselves, at the invention of
printing, in possession of the knowledge that two distinct systems of
arithmetic were current. The streams which had united in one bed had
not mingled their waters; and for nearly a century two distinct classes
of writings present themselves. Their only common point is the use of
the Arabic or Indian numerals and method of notation.
The first, or _algoristic_ system, as we may call it, proceeded by
systematic rules to the performance of questions of computation. It
embraced mercantile arithmetic, and what was then called algebra:
and laborious and prolix efforts were made to combine the two; that
is, to represent in such a form as we should now call algebraic,
as distinguished from arithmetical, the solutions of questions
of commerce, or of what might become such, if the connexion were
diligently fostered. I cannot suppose that the multifarious problems of
exchange of different kinds which abound in Pacioli and his immediate
followers, were actually useful to the merchant: but there is an
obvious leaning to the idea that they ought to be so. Geometry was also
an application of this arithmetic; and in a manner which strongly marks
its eastern character, as will appear to those who compare the work of
Pacioli with the Indian books.
The second, or _Boethian_ system, as I shall call it, because the
work of Boethius was its great text-book, did not give any rules of
calculation, nor apply itself to any application. The study was the
properties of numbers, and particularly of their ratios. There was no
art about it; and we have no means of telling whether the philosophers
of this school reckoned on their fingers, or used an abacus, or put pen
to paper for the performance of some organised method of computation.
To judge by the smallness of the numbers used in the instances adduced,
we must suppose that the writers left it to their readers to do as they
liked best.
For some specimens of the laborious manner by which the Pythagorean
Greeks, in the first instance, and afterwards Boethius in Latin,
had endeavoured to systematise the expression of numerical ratios,
I may refer the reader to the article “Numbers, old appellations
of,” in the Supplement to the _Penny Cyclopædia_. If I were to give
any account of the whole system, on a scale commensurate with the
magnitude of the works written on it, the reader’s patience would not
be _subquatuordecupla subsuperbipartiens septimas_—or, as we should now
say, _seven per cent_—of what he would find wanted for the occasion.
Not that the books I am speaking of get quite so far as this. I am
hardly prepared to say exactly what number under fifty ought to be
named as the terminus of the Boethian store of numbers; but certainly
we rarely find them choose their instances from numbers above it. The
system broke down under its own phraseology, as did that of the Yancos
mentioned by Dr. Peacock, who could not get further than _three_,
because they could not express this idea by any thing more simple than
_Poettarrarorincoaroac_.
The Italian school of _algorists_, with Pacioli at their head,
found followers in Germany, England, France, and Spain; and in all
but England, the Boethian school also. I cannot discover a single
English work which pays any detailed attention to the latter class
of arithmeticians. Tonstall, and still more Recorde, for the former
appears to have been little known, were the preservers; and before
the end of the sixteenth century the ordinary style of commercial
arithmetic, which has prevailed among us ever since, was in course of
establishment.
This gradual formation of the English school of commercial works will
be apparent enough in my list. From the time of Recorde, we always were
conspicuous in numerical skill as applied to money. The questions of
the English books are harder, involve more figures in the data, and
are more skilfully solved. It is possible that I might, if my French,
German, and Italian list were more complete, produce exceptions to this
rule. But nothing, I think, could arise to alter my conviction that the
efforts which were made in this country towards the completion of the
logarithmic tables in the seventeenth century, and the instantaneous
appreciation of the value of the discovery of logarithms, were the
result of that superiority in calculation which I assert to have been
formed in the sixteenth. And yet this last-named century produced one
man in France, one in Germany, and one in Italy, with either of whom
no one English calculator could compare in extent of operations: I
allude to Vieta, Rheticus, and Cataldi. There was no opportunity to
compete with these men; for the subjects on which they worked were not
introduced here. It was only towards the _end_ of the sixteenth century
that what were then the higher parts of the mathematical sciences began
to be disseminated with effect in Britain.
To the commercial school of arithmeticians above noted we owe the
destruction of demonstrative arithmetic in this country, or rather the
prevention of its growth. It never was much the habit of arithmeticians
to prove their rules: and the very word _proof_, in that science, never
came to mean more[5] than a test of the correctness of a particular
operation, by reversing the process, casting out the nines, or the
like. As soon as attention was fairly diverted to arithmetic for
commercial purposes alone, such rational explanation as had been handed
down from the writers of the sixteenth century began to disappear,
and was finally extinct in the work of Cocker, or Hawkins, as I think
I have shewn reason for supposing it should be called. From this
time began the finished school of teachers, whose pupils ask, when a
question is given, what rule it is in, and run away, when they grow up,
from any numerical statement, with the declaration that any thing may
be proved by figures—as it may, to them. Any thing may be unanswerably
propounded, by means of figures, to those who cannot think upon number.
Towards the end of the last century, we see a succession of works,
arising one after the other, all complaining of the state into which
arithmetic had fallen, all professing to give rational explanation, and
hardly one making a single step in advance of its predecessors.
[5] At first I had written “degenerated into nothing more;” but this is
incorrect. The original meaning of the word _proof_, in our language,
is testing by trial.
It may very well be doubted whether the earlier arithmeticians could
have given general demonstrations of their processes. It is an
unquestionable fact of observation, that the application of elementary
principles to their apparently most natural deductions, without
drawing upon subsequent, or what ought to be subsequent, combinations,
seldom takes place at the commencement of any branch of science. It
is the work of advanced thought. But the earlier arithmeticians and
algebraists had another difficulty to contend with: their fear of
their own half-understood conclusions, and the caution with which it
obliged them to proceed in extending their half-formed language. It was
not merely by oversight, I suspect, that Oughtred so often calculates
_ab_ + _ac_ by two multiplications, instead of using _a_(_b_ + _c_):
but rather from that same general fear of abbreviation, and suspicion
that error may lurk in it, which possesses men of business who dare
not multiply by 10 by the annexation of a cipher, but proceed with
each figure, and carriage, as they would do if the multiplier were
8 or 7. I have seen this often enough; and things nearly as strange
times without number. But what shall we say to the following; a most
sufficient recommendation of the study of old works to the teacher, as
shewing that the difficulties which it is now (I speak to the _teacher_
not the _rule-driller_) his business to make smooth to the youngest
learners, are precisely those which formerly stood in the way of the
greatest minds, and sometimes effectually stopped their progress.
Perhaps no man of his day had so much power over mathematical language
as Wallis. But the following extract of a letter from him to Collins
(Macclesfield Collection, vol. ii. p. 579), in 1673, shews that he once
had doubts whether he might dare write down the square root of 12 as
being twice the square root of 3, however certain he might be that it
is so; because no one had so written it. Speaking of the square root
of a negative quantity, he says, “Only, though I had from the first a
good mind to it, I durst not without a precedent, when I was so young
an algebraist as in the history my late letter reports, take upon me
to introduce a new way of notation, which I did not know of any to
have used before me. And it was not without some diffidence that I
ventured on 2√3 instead of √12 prefixed to a surd root; but I found
it so expedient, not only for the discovering the root of a binomial,
whether quadratic, cubic, or others, but for the adding and subducting
of commensurable surds, that I resolved to use it for my own occasions,
before I knew whether others would approve of it or no; especially
having found in the first edition of Oughtred’s Clavis (for I had not
then seen the second) one instance or two for it, to justify myself if
it should be questioned. But since that time it is grown more common,
and I perhaps have somewhat contributed thereunto.”
If I were to take such things from old and unknown writers, in which
they abound, I should be thought to waste the time of my readers by
giving an undue importance to the difficulties of very inferior minds.
But those who do not want such a confession as the above from a Wallis,
to help to persuade them that the difficulties of progress are of the
same character in all ages, will find in old works plentiful and useful
supplies of thought for a modern teacher.
Another instance of the halting progress of language is the mode of
introduction of the decimal point. Stevinus, the undoubted introducer
of the decimal fraction (though others may have seen very clearly the
great use of 10, 100, 1000, &c. as divisors), had no such thing, but
only a distinct mode of denoting the meaning of each decimal place. Dr.
Peacock mentions Napier as being the person to whom the introduction is
unquestionably due: a position which I must dispute, upon additional
evidence.
The inventor of the single decimal distinction, be it point or line, as
in 123·456 or 123|456, is the person who first made this distinction a
permanent language: not using it _merely as a rest in a process_, to be
useful in pointing out afterwards how another process is to come on,
or language is to be applied; but making it his final and permanent
indication as well of the way of pointing out where the integers end
and the fractions begin, as of the manner in which that distinction
modifies operations. Now, first, I submit that Napier did not do this;
secondly, that if he did do it, Richard Witt did it before him (page
34).
I have not seen Wright’s summary of logarithms, of 1616, to which
Dr. Peacock refers for some indications of the decimal point. But I
take with confidence his assertion that the first distinct notice of
the thing in question, by or by means of Napier, is contained in the
_Rabdologia_ (page 35).
In this famous tract there are but two instances to the purpose: a
scanty number, be it noticed, for a person who had seized the idea of
completing the decimal scale. The first is an instance of division
which Dr. Peacock cites. Here the quotient is, as we should now say,
1993·273. Napier uses a comma in his quotient, as a rest, and writes
1993,273, and then presents his answer in the form 19932′7″3‴, as
Stevinus would have done. Dr. Peacock states this, and notes it as
a “partial conformity to the practice of Stevinus.” Unfortunately
Napier has not given us instances enough to tell us whether he had a
_practice_ of his own: and in the only other case, which Dr. Peacock
also cites (I have verified both citations), he adheres still more
closely to the method of Stevinus. For though, in a question of
addition, he has drawn a line down the interval where our decimal
points would be, which line would be as distinctive in his total as in
his _addenda_, he places the exponents of Stevinus in this total, which
thus has both distinctions, and stands 1994|9′1″6‴0⁗.
I cannot trace the decimal point in this: but if required to do so,
I can see it more distinctly in Witt (page 33), who published four
years before Napier. But I can hardly admit him to have arrived at
the notation of the decimal point. For though his tables are most
distinctly stated to contain only numerators, the denominator of which
is always unity followed by ciphers; and though he has arrived at a
complete and permanent command of the decimal _separator_ (which with
him is a vertical line) in every operation, as is proved by many scores
of instances; and though he never thinks of multiplying or dividing
by a power of 10 in any other way than by altering the place of this
decimal separator; yet I cannot see any reason to suppose that he gave
a meaning to the quantity with its separator inserted. I apprehend
that if asked what his 123|456 was, he would have answered: It
_gives_ 123-456/1000, not it _is_ 123-456/1000. This is a wire-drawn
distinction: but what mathematician is there who does not know the
great difference which so slight a change of idea has often led to?
The person who first distinctly saw that the answer -7 always implies
that the problem requires 7 things of the kind diametrically opposed
to those which were assumed in the reasoning, made a great step in
algebra. But some other stepped over his head, who first proposed to
let -7 _stand for_ 7 such diametrically opposite things.
Who, then, was the real inventor of the decimal separator, to give both
interpretation and operative form? This is a question I will not answer
positively: nor attempt to answer till I have pointed out how the
case stands with several possible claimants. And first, as to Briggs.
He does not indeed arrive at the simple decimal _point_ (which is a
strong presumption against such a thing being suggested by Napier; for
who would have learnt it from him, if not Briggs?)—but he omits the
denominator, and draws a line under the decimals. And it is a further
presumption against any such idea of abbreviation being common, that
Briggs explains his notation as quite peculiar to himself. In the
preface, or _Ad Lectorem_, of the _Arithmetica Logarithmica, London_,
sixteen-twenty-four, _folio in twos_, he commences thus:
“Ut obscuritatis crimen, quantum in me situm fuerit effugiam, paucula
hæc humanissime Lector, te admonendum censui. 1. Si numerus occurrat
cui linea subscribitur, notæ illæ quæ supra lineam sunt descriptæ,
Numeratorem partium constituunt, quarum Denominator semper intelligitur
unitas, cum tot cyphris, quot sunt notæ superius positæ. ut 75
designant ⁷⁵/₁₀₀ vel ¾. sic 5 9321 designant
9321
5 —————— .”
10000
Oughtred adopted both the vertical and sub-horizontal separators, thus
shutting up the numerator in a semi-rectangular outline. In the mean
time, Gunter, after adopting Briggs’s notation in the first instance,
gradually dropt it, and substituted the decimal point. The history of
this symbol in his hands is rather curious, on account of its having
been so completely overlooked, though all the circumstances were to be
seen in widely circulated works.
Gunter published in 1620 his _Canon Logarithmorum_, sines and tangents
on Briggs’s system. To this table there is no explanation. But an
explanation was written by him, and, though I cannot make out that it
was ever published separately, yet it appears in the collection of his
works. I have what is called the second edition of this collection,
_London_, sixteen-thirty-six, _quarto_, which, for various reasons,
I suspect to be the first, thinking that the announcement of second
edition merely refers to the fact of its having been previously
published during Gunter’s life. After the _finis_, with new paging
and signatures, comes ‘The Generall use of the Canon and Table of
Logarithmes.’ This I suspect to have then made its first appearance:
but from the very manner in which the decimal fractions are treated,
it is clear that it was written before or with the work on the sector,
&c., which first[6] came out in sixteen-twenty-three; and this, though
it makes reference to that work, or at least to its matter. At the
beginning of this ‘Generall use, &c.’ common fractions are used, even
when the denominators are decimal. At page 13, Briggs’s notation
appears, without explanation: and 116 04 is the third proportional
to 100 and 108; this continues through page 14. In page 15, a dot is
added to Briggs’s notation in one instance: 100_l._ in 20 years at
8 per cent becomes 466.095_l._ At the bottom of this page Briggs’s
notation disappears thus, “It appeareth before, that 100_l._ due at
the yeares end is worth but 92 592 in ready money: If it be due at
the end of 2 yeares, the present worth is 85_l._ 733: then adding
these two together, wee have 178_l._ 326 for the present worth of 100.
pound annuity for 2 yeeres and so forward.” After this change, thus
made without warning[7] in the middle of a sentence, Briggs’ notation
occurs no more in the part which relates to numbers. But in the
following chapter, which is trigonometrical (and which may have been
written first, for Gunter’s _own_ logarithms are only trigonometrical),
it reappears, sometimes with and sometimes without the dot. In the
previous work on the sector, &c., the simple point is always _used_:
but in explanation the fraction is not thus written, but described
as parts. Thus, 32.81 feet in operation is 32 feet 81 parts in the
description of the question or answer.
[6] A printed title-page has 1623: an additional engraved and
ornamented title-page has 1624.
[7] As far as I can find, the words _cosine_ and _cotangent_ (which
Gunter introduced) appear in the same manner, without warning or
explanation.
It was long before the simple decimal point was fully recognised in all
its uses, in England at least: and on the continent the writers were
rather behind ours in this matter. As long as Oughtred was widely used,
that is, till the end of the seventeenth century, there must always
have been a large school of those who were trained to the notation
123|456. To the first quarter of the eighteenth century, then, we must
refer, not only the complete and final victory of the decimal point,
but also that of the now universal method of performing the operation
of division and extraction of the square root.
This evident slowness in the admission of such important improvements
will not appear singular to those who observe what is now taking place
with reference to Horner’s method for the solution of equations. It
will be the business of some successor of mine, a century or two hence,
to distinguish in his list, by some appropriate mark, the works of our
period which adopt this method from those which do not: just as I have
done with the two methods of division. I hope, if I live to publish a
second edition, that I may be able to add to my list the name of the
_first_ work, written for students in the University of Cambridge,
which shall contain this most fundamental addition to the processes of
pure arithmetic: which, though it has found its way into the public
examinations, is not yet in the books which prepare students for them.
This has been a long digression, on what is perhaps the most
interesting part of the subject, its language. We owe every thing,
almost, to the simplicity of certain modes of expression. Nothing is
more clear than that the Greek geometers, with all the acuteness and
perseverance necessary to carry results of arithmetical application
to a high pitch, and all the taste for numbers which would turn
their thoughts that way, were stopped by their insufficient system
of numeration, and the tediousness of their processes. The history
of language, then, is of the highest order of interest, as well as
utility: its suggestions are the best lessons for the future which a
reflecting mind can have.
In looking over the list of books, it will be observed, that during
the eighteenth and nineteenth centuries there is a great preponderance
of English writers. The law I imposed on myself, of entering no books
which I could not see, necessarily brought this about. During the
last two centuries, the elementary works of the different countries
in Europe have not gained great general circulation. They have ceased
to be written by the greatest names, for the most part; and they have
rarely been cited by the historians of science. Copies which have found
their way to this country have probably soon been destroyed. As it is
my intention to endeavour to extend this list, whether I publish the
extensions or not, and in any case to provide for the preservation of
the materials I collect, it will be worth the while of any one who is
able, to furnish me with information on works which I have not seen.
I think it probable that any one who has had the curiosity to rescue
three books of arithmetic from a stall, will find that one of them
is not in this list. From any such person I shall thankfully receive
the full title of the neglected work, with the form in which it is
printed. Nor need it by any means be presumed that because a book is
wholly unknown, it proves nothing in the history of the science. A
book so thoroughly lost as that of Witt, contains a nearer approach
to the decimal point than was made by Napier. Horner found a close
approximation to his own method, for the case of the cube root, in
an obscure compendium of arithmetic. I might also instance Dary’s
discovery mentioned in page 48; and other things of the same kind. The
history of hints given before the time at which they were (perhaps
could be) made to bear fruit, would be a very curious one: and the
progress of science will never be well understood until some little
account can, in each case, be given of the reason why a notion should
be so productive at a particular period, which was so barren at a
previous one.
* * * * *
On looking at the fourth edition of Gunter’s work (sixteen-sixty-two),
I find that some liberties have been taken with the original text:
among others, Briggs’s notation is restored in several places, though
not entirely. It should also be noticed that the point which generally
occurs after the first figure of a logarithm is not a fractional
separator, but only divides one integer from the rest.