_Newcastle_, eighteen-twenty-eight. =William Tinwell= (edited by James
Charlton). ‘Treatise on Practical Arithmetic, with Book-keeping, by
single and double entry.’ Twelfth edition. _Duodecimo in threes._
_Dublin_, eighteen-twenty-eight. =John Garrett.= ‘An Essay on
Proportion.’ _Octavo._
Mostly arithmetical, with something on the connexion of number and
magnitude.
_London_, —— =George Peacock.= ‘Arithmetic.’
This is from the _Encyclopædia Metropolitana_. As part of the first
volume of the _Pure Sciences_, the date is eighteen-twenty-nine: but
it was separately published, in the parts, in eighteen-twenty-five
or twenty-six. This is the article mentioned in the Introduction.
I subjoin a few remarks, either in correction of some slips of the
pen, or in addition to what has been said. The paging is that of the
_Encyclopædia Metropolitana_.
P. 402, note. A sexagesimal table is sometimes pasted into a work to
which it does not belong, by some old owner.
P. 404. For ‘Regiomontanus, in his Opus _Palatinum_ de Triangulis,’
read ‘in his work _de Triangulis_; and for ’as we learn from the
relation of Valentine Otho, _in his preface to that work_,’ read ‘in
his preface to the _Opus Palatinum_.’ But further observe, that though
Regiomontanus did change the usual sexagesimal radius into a decimal
one, no decimal tables of his were _published_ until some time after
Apian had published decimal tables of his own. At least, after much
research, I can find none even mentioned. It is a mistake (a universal
one) to say that decimal sines were published with the work _de
Triangulis_, in 1533. It arises from such tables being in the _second_
edition of that work, in 1561.
P. 411. Add that Chaucer, in his work on the _Astrolabe_, makes
_augrime_ figures to be exclusively the Arabic numerals.
P. 414. The work of _Pacioli_ (not Paccioli, though very often so
spelt) was published in 1494 (not 1484). Canacci, not Caracci.
P. 419 (note). Besides the _Fasciculus Temporum_ (of which, by the way,
there are editions dated 1474, and two or three without date, probably
earlier, according to Hain), there are the _Almanac_ and _Ephemerides_
of Regiomontanus, described in the _Companion to the Almanac_ for 1846,
with extensive masses of Arabic numerals.
P. 425. Note that there is an express commentary by Jamlichus
on Nicomachus, and that the former probably did not write the
_Theologumena_ at all.
P. 434. I have no doubt that Dr. Peacock has authority for saying that
the old English Arithmeticians called the then common mode of dividing
the _scratch_ way (as indeed it was); but I have never met with the
phrase.
P. 438 (note). It was quite common, even before the invention of
printing, to speak of _Algus_ as the inventor of decimal notation, or
_Algorithm_: and several of the ornamented title-pages of Simon de
Colines, the successor of Henry Stephens, have figures of Ptolemy,
Orpheus, Euclid, and _Algus_, on one side, opposite to the Muses of
Astronomy, Music, Geometry, and _Arithmetic_, on the other: as if
paired for a dance.
P. 440. See what I have said on the _Disme_ of Stevinus (1585); and
note that Simon of Bruges is Stevinus himself.
P. 442. There is nothing about decimal fractions in Hunt’s Handmaid.
P. 444. I suggest as the derivation of _furlong_, a corruption of
_forty-long_. The ordinary derivation, _furrow-long_, can hardly have a
foundation in fact; for the length of a furrow depends upon that of a
field.
P. 454. For ‘Wingrave’ read ‘Wingate.’ As to the note, I differ
greatly (as appears elsewhere) from Dr. Peacock’s opinion of Cocker.
With respect to the last sentence: ‘It may be worth while observing
that this modest and useful book is not honoured with poetical
recommendations;’ it would seem that the copy consulted wanted the
frontispiece, on which, under the portrait of Cocker, are these lines:
“Ingenious COCKER! (now to Rest thou’rt Gone)
Noe Art can Show thee fully but thine own,
Thy rare _Arithmetick_ alone can show
Th’ vast _Sums_ of Thanks wee for thy Laboure owe.”
P. 464. For ‘James Peele’ read ‘John Mellis,’ and alter the date to
1588, as above. See on Stevinus what I have said above.
P. 471. The _leuca_ was originally a measure of the Gauls; and it may
be much doubted whether the measure was ever out of France since the
time of Cæsar. See the article _League_ in the _Penny Cyclopædia_.
Surely Dr. Peacock must have written ‘France’ meaning to have written
‘England.’ This alteration of one word makes every thing right.
The small number of inaccuracies noted above are all I have been able
to lay my hands on, in going through Dr. Peacock’s most valuable
article, with the materials for this list about me. I may add, that
in two, or perhaps three instances, I remember to have found an
edition mentioned as the earliest which is not so. But to this every
writer is subject who has the courage to attempt history upon such
bibliographical guides to the sources as now exist.
_London_, [eighteen-twenty-nine]. =James Parker.= ‘Arithmetic and
Algebra.’ _Octavo._
An early work of the Society for the Diffusion of Useful Knowledge.
Arithmetical and algebraical demonstrations are connected.
_London_, eighteen-thirty. =Joh. White.= ‘An Elucidation of the Tutor’s
Expeditious Assistant.’ _Duodecimo._
The author claims as a new discovery in arithmetic, the notion
of forming examples in which the figures of the answer have some
perceptible relation, unknown to the pupil, but known to the teacher,
who can thereby see whether it is right or wrong. All the questions
set, for instance, in multiplication have digits which descend by twos,
as in 86420, 20864, 75319, &c. So that it is gravely proposed that the
pupil shall be trained upon selected questions in which the answers
have all the figures odd, or all even. A pupil so trained would be in
danger of being led to think that 3728 could not be the product of any
numbers.
_Edinburgh_, eighteen-thirty. =Adam Anderson.= ‘Arithmetic.’ _Quarto._
This is the article so headed in Brewster’s Cyclopædia, vol. ii. part
1. It has a fair amount of arithmetical history, though hardly enough
for the extent of the work of which it forms a part.
_London_, eighteen-thirty-one. =Frederic Rosen.= ‘The Algebra of
=Mohammed Ben Musa=.’ _Octavo in twos._ English and Persian.
This algebra is very arithmetical. It was published by the Oriental
Translation Fund. M. Libri, in the work presently mentioned (vol. i.
pp. 253-297), has given a long extract from old Latin translations now
in manuscript in the Royal Library at Paris. This book passes for the
one by means of which Leonard of Pisa was the first European who learnt
algebra, or at least who wrote upon it.
_Derby_, eighteen-thirty-three. =Samuel Young.= ‘A System of Practical
Arithmetic.’ _Duodecimo._
A work with great force of rules, and examples in machinery,
manufactures, &c.
_Springfield (U. S.)_, eighteen-thirty-three. =Zerah Colburn.= ‘A
Memoir of Zerah Colburn, written by himself ... with his peculiar
methods of calculation.’ _Duodecimo in threes._
A great many will remember that in 1812 and 1815 two young boys, Zerah
Colburn and George Bidder, astonished every one by a power of rapid
mental calculation to which the most practised arithmeticians could not
make the least approach. Mr. Colburn was, in 1833, a minister among the
Methodists in the United States; Mr. Bidder, there is little occasion
to say, is a civil engineer in England. The peculiarity of Colburn
was, that he could extract roots and find the factors of numbers, to
an extent which the mathematician himself had no organised rule for
doing. Speaking in the third person, Mr. Colburn says: “Some time in
1818, Zerah was invited to a certain place, where he found a number
of persons questioning the Devonshire boy (Mr. Bidder). He displayed
great strength and power of mind in the higher branches of arithmetic;
he could answer some questions that the American would not like to
undertake; but he was unable to extract the roots and find the factors
of numbers.” This treatise contains an account of Colburn’s methods.
_London_, eighteen-thirty-seven. =Daniel Harrison.= ‘A New System
of Mental Arithmetic, by the acquirement of which all numerical
questions may be promptly answered without recourse to pen or pencil
... an entirely novel method of reducing the largest sums of money
... to their lowest denominations by means of original quadrantal
rationale’.... _Duodecimo in threes._ Second edition.
The pretensions of this work are manifestly exaggerated; but the
methods are ingenious. Books of mental arithmetic choose their own
examples, and thus make their own rules work well. The _quadrantal_
method (why so called I cannot guess) is making use of a rule for
turning pounds, &c. into farthings, which requires the learning of a
table; and then making such applications as the following:—A pound troy
contains six times as many grains as there are farthings in the pound;
therefore the grains in 4 pounds troy are the farthings in six times
as many pounds sterling. Some of these rules would really become very
effective, if the method of decimalising the parts of a pound were
used. Multiply the pounds by a thousand, and subtract 4 per cent, and
the result is the number of farthings. Thus:
£1763 . 17 . 9¾ is 1763·89062
1763890·62
4 per cent 70555·62
——————————
Subtract 1693335 = No. of farthings in £1763 . 17 . 9¾
_Berwick-upon-Tweed_, eighteen-thirty-eight. =James Gray= (edited by
Wm. Rutherford). ‘An Introduction to Arithmetic.’ _Duodecimo in threes._
Mr. Rutherford says this neat little work went through more than forty
editions in the half century preceding this publication. How many works
on arithmetic there must be which I have never seen! I never met with
any one of the forty.
_Paris_, eighteen-thirty-eight, thirty-eight, forty, forty-one.
=Guillaume Libri.= ‘Histoire des Sciences Mathématiques en Italie,
depuis la renaissance des lettres jusqu’à la fin du xviiᵉ siècle.’
_Octavo._
The first volume of this work was printed _Paris_,
eighteen-thirty-five, _octavo_, but it was never sold; the whole
impression (except a few copies distributed as presents) was burnt.
The work is yet unfinished; four volumes, dated as above, being all
that have appeared. It is a valuable history as concerns arithmetic,
both in text and notes; the latter contain—the _liber augmenti et
diminutionis_ compiled by Abraham (supposed to be Aben Ezra) _secundum
librum qui Indorum dictus est_ (vol. i. pp. 304-376)—Account of
an extract from the _Liber Abbaci_ of Leonard of Pisa, written in
twelve-hundred-and-two (vol. ii. pp. 287-304)—the _practica Geometriæ_
of the same author, being the whole of his algebra, written in
twelve-hundred-and-twenty (vol. ii. pp. 305-476)—extracts from Pacioli
(vol. iii. pp. 277-294): besides many other matters not relating to
arithmetic. As this work stands, it can be little used for want of an
index.
_London_, eighteen-thirty-eight. =Thos. Keith.= ‘The Complete Practical
Arithmetician.’ _Duodecimo._
The editor (Mr. Maynard, the mathematical bookseller, who ought to
know) calls this the 12th edition: it has a deservedly high character
among books of rules only, for precision and completeness. Keith says
that the bare names of those who have written on arithmetic in England,
from the time of Wingate, would fill a moderate volume. I suspect not:
after having examined every source within my reach, and got only 1500
names, out of all times and countries, I should think it impossible
that Keith could have known of 300 Englishmen, within the limits
mentioned.
_London_, eighteen-forty. =Thomas Stephens Davies.= ‘Solutions of the
Principal Questions of Dr. Hutton’s Course of Mathematics.’ _Octavo._
This work is also a running comment on the original, and it has a
specific right to be in this list from its abounding with instances
of Mr. Horner’s various methods of manipulating arithmetical and more
particularly algebraic questions, independently of his celebrated
method of solving equations. In eighteen-forty-one the same author
published an edition of Hutton’s Course itself.
_London_, eighteen-forty-two. ‘Encyclopædia Britannica.’ Vol. III.
_Quarto._
The article on arithmetic was written, I suppose, by Leslie; or if not,
by a follower of his views. It is meagre in its history.
_London_, eighteen-forty-three. =Alfr. Crowquill= (as he calls
himself). ‘The Tutors Assistant, or Comic Figures of Arithmetic.’
_Duodecimo in sixes._
Walkingame’s arithmetic, with comic woodcuts: but the subject will not
furnish good materials. Under ‘Subtraction of time,’ for instance, is
represented the stealing of a watch; as a heading for _Troy weight_,
the _wooden horse_; and for _measure of capacity_, a phrenologised
head. The worst of it is, that the joke will remain on hand too long
for the learner: a picture of a gamekeeper producing a hare from a
poacher’s pocket must stare him in the face, as an illustration of
‘proof,’ through twenty-two mortal questions of addition. A comic
arithmetic, with the cuts in illustration of the examples for exercise,
would give the artist much fairer play.
_Liège_, eighteen-forty-four. =H. Forir.= ‘Essai d’un cours de
Mathématiques à l’usage des élèves du collège communal de Liège ...
Arithmétique.’ Eighth edition. _Octavo in fours._
A good and well-printed treatise, with many examples.
_London_, eighteen-forty-five. =Thomas Tate.= ‘A Treatise on the First
Principles of Arithmetic.’ _Duodecimo._
Two pages at the end, on the use of the properties of nine in
constructing questions for pupils, are well worthy the attention of
teachers.
_London_, eighteen-forty-six. =A. De Morgan.= ‘The Elements of
Arithmetic.’ Fifth edition, with Appendixes. _Duodecimo in threes._
The previous editions were eighteen-thirty, thirty-two, and
thirty-five, _duodecimo_, and eighteen-forty _in threes_.
Books of bibliography last longer than elementary works; so that I have
a chance of standing in a list to be made two centuries hence, which
the book itself would certainly not procure me.
The following are some additional works more briefly described. Here 2
+ 2, 4 + 4, 6 + 6, mean quarto in ones, octavo in twos, duodecimo in
threes.
+--------------+------------------+----------------------+
| Date. | Place. | Author. |
+--------------+------------------+----------------------+
| SEVENTEEN- | | |
| | | |
|-thirty-four | London | ALEX. WRIGHT |
|-fifty-four | Exon. | JOSEPH THORPE |
|-fifty-seven | London | R. GADESBY |
|-sixty-one | London | JOHN DEAN |
|-sixty-two | London | RICH. RAMSBOTTOM |
|-sixty-six | Sheffield | JOHN EADON |
|-sixty-six | Birmingham | WM. CRUMPTON |
|-seventy | Exeter | G. DYER |
|-seventy-one | London | WM. SCOTT |
|-eighty-eight | Birmingham | WM. TAYLOR |
|-eighty-nine | Edinburgh | WM. GORDON |
| | | |
| EIGHTEEN- | | |
| | | |
|-four | Bingham | W. BUTTERMAN |
|-five | Shrewsbury | GEO. BAGLEY |
|-eight | Paris | EDM. DEGRANGE |
|-eleven | London | JOH. HARRIS WICKS |
|-eleven | Birmingham | J. RICHARDS |
|-twelve | Paris | J. CL. OUVRIER DELILE|
|-sixteen | Edinburgh | R. HAY |
|-seventeen | London | JAMES MORRISON |
|-eighteen | London | JOH. MATHESON |
|-eighteen | Dundee | JAMES NICOLSON |
|-twenty | Glasgow | C. MORRISON |
|-twenty | Paris | J. B. JUVIGNY |
|-twenty-two | Bedford | W. H. WHITE |
|-twenty-three | Canterbury {| H. MARLEEN and} |
| | {| W. SMEETH } |
|-twenty-three | Lausanne | EM. DEVELEY |
|-twenty-four | Cork | JOH. PENROSE |
|-twenty-six | London | ANTH. PEACOCK |
|-twenty-six | London | J. W. HOAR |
|-twenty-seven | London | WM. PHILLIPS |
|-twenty-eight | London | S. P. REYNOLDS |
|-twenty-eight | London | ROB. FRAITER |
|-twenty-eight | [London] | R. W. WOODWARD |
|-twenty-eight}| London | ANONYMOUS |
| [_circa_] }| | |
|-twenty-nine | Paris | H. L. D. RIVAIL |
|-twenty-nine | Paris | A. SAVARY |
|-twenty-nine | London | ANONYMOUS |
|-twenty-nine | London | DANIEL DOWLING |
|-twenty-nine | London | JAMES PERRY |
|-thirty | London | W. PUTSEY |
|-thirty-one | London | GEO. HUTTON |
|-thirty-two | Edinburgh | ROB. CUNNINGHAM |
|-thirty-four | London | RICH. CHAMBERS |
|-thirty-four | Newcastle-on-Tyne| THOS. THOMPSON |
|-thirty-four | London | CHRIST. KNOWL. SOC. |
|-thirty-six | London | RICH. MOSLEY |
|-thirty-six | Edinburgh | JOH. DAVIDSON |
|-thirty-seven | Boston (U. S.) | JOS. BARTRUM |
|-thirty-nine | London | J. FELTON |
|-forty | Cambridge | W. C. HOTSON |
|-forty-three | London | MERCATOR |
|-forty-five | London | W. WATSON |
|Without date | London | ADAM TAYLOR |
| —— | London | HENRY AULT |
| —— | Dartmouth | WM. JARVIS |
+--------------+------------------+----------------------+
+--------------+-----------------------------+---------------+
| Date. | Leading Word of Title. | Form. |
+--------------+-----------------------------+---------------+
| SEVENTEEN- | | |
| | | |
|-thirty-four | Fractions | 6 + 6. |
|-fifty-four | Treatise | 4 + 4. |
|-fifty-seven | Decimal | 4 + 4. |
|-sixty-one | Rule of Practice | 8 (2d ed.) |
|-sixty-two | Fractions anatomised | 6 + 6. |
|-sixty-six | Guide | 6 + 6. |
|-sixty-six | Decimals and Mensuration | 6 + 6. |
|-seventy | School-Assistant | 12. |
|-seventy-one | New System | 8. |
|-eighty-eight | Guide | 6 + 6. |
|-eighty-nine | Institutes | 4 + 4. |
| | | |
| EIGHTEEN- | | |
| | | |
|-four | Dialogue | 6 + 6. |
|-five | Assistant | 2 + 2. |
|-eight | Pratique | 8. |
|-eleven | Merchant’s Companion | 6 + 6. |
|-eleven | Practical | 6 + 6. |
|-twelve | Appliquée au Commerce | 8. |
|-sixteen | Beauties | 8. |
|-seventeen | Concise System | 12. |
|-eighteen | Theory and Practice | 12. |
|-eighteen | Modern Accountant | 6 + 6. |
|-twenty | Young Lady’s Guide | 6 + 6. |
|-twenty | Application au Commerce | 8. |
|-twenty-two | Complete Course | 6 + 6. |
|-twenty-three | Assistant | 6 + 6. |
| | | |
|-twenty-three | D’Emile | 8. |
|-twenty-four | Treatise | 6 + 6. |
|-twenty-six | Mental | 12. |
|-twenty-six | Examinations | 18. |
|-twenty-seven | New and concise System | 12. |
|-twenty-eight | Practical | 4 + 4(small). |
|-twenty-eight | Short System | 6 + 6(2d ed.) |
|-twenty-eight | Guide | 12. |
|-twenty-eight}| Intellectual (Pestalozzian) | 6 + 6. |
| [_circa_] }| | |
|-twenty-nine | D’après Pestalozzi | 12. |
|-twenty-nine | Arithmétique | 8. |
|-twenty-nine | Concise Arithmetician | 6 + 6. |
|-twenty-nine | Improved System | 12. |
|-twenty-nine | Middle Stage and Key | 6 + 6. |
|-thirty | Practical | 6 + 6. |
|-thirty-one | Theory and Practice | 12. |
|-thirty-two | Text-Book | 6 + 6. |
|-thirty-four | Introduction | 6 + 6. |
|-thirty-four | Business Book | 4 + 4. |
|-thirty-four | Taught by Questions | 8 (square). |
|-thirty-six | Elements | 12. |
|-thirty-six | Guide | 6 + 6. |
|-thirty-seven | Arithmetic | 6 + 6. |
|-thirty-nine | Improved Method | 6 + 6. |
|-forty | Principles | 6 + 6(2d ed.) |
|-forty-three | Expeditious Calculation | 4 + 4. |
|-forty-five | School | 6 + 6(4th ed.)|
|Without date | [Useful Arithmetic] & Sequel| 12. |
| —— | Requisites of Business | 6 + 6(pt. 1). |
| —— | Catechism | 8 (small). |
+--------------+-----------------------------+---------------+