_Recorde._ Division is a partition of a greater summe by a lesser.
_Wingate._ Division is that by which we discover how often one number
is contained in another, or (which is the same) it sheweth how to
divide a number propounded into as many equal parts as you please.
_Johnson._ Apparently considers division not enough of a technical term
to need definition: his first example is, “I would divide 65490 pound
amongst 5 men.”
_Moore._ Division is that part of conjunct Numeration, whereby one
Number is subtracted from another, as often as it is contained in it,
and by that meanes it is found how many of the one is contained in the
other.
_Bridges._ Division is that by which we discover how often one number
is contained in another.
_Hodder._ Division is that by which we know how many times a lesser sum
is contained in a greater.
_Cocker._ Division is the Separation, or Parting of any Number, or
Quantity given, into any parts assigned; Or to find how often one
Number is Contained in another; Or from any two Numbers given to find
a third that shall consist of so many Units, as the one of those two
given Numbers is Comprehended or contained in the other.
The six predecessors of Cocker whom I have chosen, stop when they think
enough has been said. But the illustrious discoverer, or at least the
first general propagator, of the fact that two and two make four (for
his current reputation amounts to this) must have had more in view.
He seems to be laying every offence against accuracy in different
ways, so that the unfortunate schoolboy who commits it may be sure
of a flogging under one count or another of his definition. And the
vice of confounding abstract and concrete number, which leads him to
imply that five shillings can be multiplied by five shillings, runs
through his whole book: as does also the tendency to prolixity and
reduplication of things which confuse each other. As to the general
notion of what Arithmetic is, Cocker tells his beginners that it
is either Natural, Artificial, Analytical, Algebraical, Lineal, or
Instrumental. The natural is “that which is performed by the Numbers
themselves; and this is either Positive or Negative. Positive, which
is wrought by certain infallible numbers propounded, and this either
Single or Comparative; Single, which considereth the nature of numbers
simply by themselves; and Comparative, which is wrought by numbers as
they have Relation one to another. And the Negative part relates to
the Rule of False.” Artificial Arithmetic is performed by artificial
or borrowed numbers invented for that purpose, called Logarithms.
Analytical Arithmetic “is that which shews from a thing unknown to find
truly that which is sought; always keeping the Species without Change.”
Algebraical Arithmetic “is an obscure and hidden Art of accompting by
numbers in resolving of hard Questions.” Lineal Arithmetic “is that
which is performed by lines, fitted to proportions, as also Geometrical
projections.” Instrumental Arithmetic “is that which is Performed by
Instruments, fitted with Circular and Right lines of proportions, by
the motion of an index or otherwise.” So much for Cocker (or Hawkins)
as an explainer. As to the actual modes of operation, they are neither
better nor worse than those pointed out before by Wingate, Moore, and
Bridges. The famous book looks like a patchwork collection, and, I
believe, is nothing more. The reason of its reputation I take to be
the intrinsic goodness of the processes, in which the book has nothing
original; and the systematic puffing with which it was introduced. The
long-promised book of the great Mr. Cocker, with Collins and fifteen
other teachers to recommend it, pushed aside better productions. I am
of opinion that a very great deterioriation in elementary works on
arithmetic is to be traced from the time at which the book called after
Cocker began to prevail. This same Edward Cocker must have had great
reputation, since a bad book under his name pushed out the good ones.
_London_, seventeen-hundred. =Christopher Sturmius.= ‘Mathesis
Enucleata, or the Elements of the Mathematicks.... Made English by J.
R. M. and R. S. S.’ _Octavo._
Arithmetic and Algebra are here very nearly separated.
_Edinburgh_, seventeen-one. =George Brown.= ‘A Compendious, but a
Compleat System of Decimal Arithmetick, Containing more Exact Rules
for ordering Infinites, than any hitherto extant.... First Course.’
_Quarto._
The author’s knowledge of what was then extant, seems far from complete.
_London_, seventeen-three. =John Parsons= and =Thos. Wastell=. ‘Clavis
Arithmeticæ.’ _Octavo_ (small) (ON).
The old system of division is rather recommended. There is a very neat
work on algebra at the end.
_Amsterdam_, seventeen-four. =Andrew Tacquet.= ‘Arithmeticæ Theoria et
Praxis.’ _Octavo_ (small) (ON).
There had been several preceding editions. The theory consists in a
version of the seventh, eighth, and ninth books of Euclid: the practice
in an ordinary treatise. Both methods of division are given: the
Italian as best, but least used.
_London_, seventeen-six. =William Jones.= ‘Synopsis Palmariorum
Matheseos; or a New Introduction to the Mathematics.’ _Octavo in twos_
(N).
Jones is well known among the contemporaries of Newton, and was the
father of his celebrated namesake, the Indian Judge. The book, as its
name imports, is a kind of syllabus.
_London_, seventeen-seven. ****, Art. Bac. Trin. Col. Dub. ‘Arithmetica
absque Algebra aut Euclide demonstrate. Cui accesserunt, Cogitata
nonnulla de Radicibus Surdis, de Æstu Aeris, de Ludo Algebraico, &c.’
_Octavo in twos_ (NO).
There is no doubt that the author was the celebrated Bishop Berkeley,
then a youth under twenty. The object of the arithmetic is stated in
the title; and at that time the effort was much wanted. The algebraical
game defies brief explanation.
_London_, seventeen-seven. =John Smart.= ‘Tables of simple interest and
discount.’ _Octavo in twos_ (small).
This is the first edition of these celebrated tables, and is little
known. The second edition, _London_, seventeen-twenty-six, _quarto_,
is the best having Smart’s name. But in reality the Tables in Francis
Baily’s ‘Doctrine of Interest and Annuities,’ _London_, eighteen-eight,
_quarto_, are Smart’s, and do not profess to be any thing else. Mr.
Baily (who, by the way, did not know this first edition) says, ‘I have
neither time nor inclination to calculate them anew; and therefore I
give them to the world with all their imperfections on their head. I am
happy, however, to observe, that after many years experience, I have
not met with any errors but such as might be discovered on inspection.’
Brand’s edition of seventeen-eighty is said by Mr. Baily to have a good
many errors.
There is yet another edition, _London_, seventeen-thirty-six. John
Smart. ‘Tables of Interest, &c. Abridged for the Use of Schools, in
Order to Instruct Young Gentlemen in Decimal Fractions.’ _Octavo._
Here is another instance of what I have before remarked, that compound
interest has always been considered _the_ application of decimal
fractions, by those whose arithmetic has been commercial.
_Manuscript_ (in my possession) no place, after seventeen-ten.
I insert this here, as proving that, so late as the date above
mentioned, there were French schools in which the decimal point was not
introduced, the old method of division was employed, and the Ptolemaic
system was taught. It is the collection of notes of lectures made by a
young Englishman educated in France, and was sold a few years ago by
his descendant.
_London_, seventeen-fourteen. =Samuel Cunn.= ‘A New and Compleat
Treatise of the Doctrine of Fractions, Vulgar and Decimal,’ &c. _Octavo
in twos_ (small) (N).
Prefixed is a testimonial from Halley, vouching for the goodness of the
work and the novelty of some of its rules.
_London_, seventeen-fourteen. =Edward Wells.= ‘The young Gentleman’s
Course of Mathematicks.’ Three volumes. _Octavo._
These volumes (in my copy) exhibit what I have more often found at
the beginning of the eighteenth century than at any other time;
namely, volumes of different editions in one set. The Arithmetic
title-page of the first volume, which follows the general title-page,
has ‘seventeen-twenty-three,’ and ‘second edition.’ The other volumes
have seventeen-fourteen and seventeen-eighteen. This Wells is the same
whose work I have mentioned above; he qualifies himself D.D. and rector
of Cotesbach in Leicestershire. When I quoted his diatribe against
butter and cheese, ginger and pepper (which I did before I had seen
this work), I sympathised with him, thinking he meant that liberal
education had its wants as well as professional. But I was mistaken:
it is _gentlemanly_ education, as opposed to that of “the meaner part
of mankind,” that he wants to provide for. Every page is headed on one
side ‘The young gentleman’s,’ on the other ‘arithmetic,’ ‘geometry,’
or ‘mechanics,’ as the case may be. The gentlemen are those whom God
has relieved from the necessity of working, for which he expects they
should exercise the faculties of their minds to his greater glory. But
they must not ‘be so Brisk and Airy, as to think, that the knowing how
to cast Accompt is requisite only for such Underlings as Shopkeepers
or Tradesmen;’ and, for the sake of taking care of themselves, ‘no
Gentleman ought to think Arithmetick below Him, that do’s not think
an Estate below Him.’ This Wells might be made as useful now as the
Spartans used to make their slaves. The Arithmetic is an abridged
version of the work of sixteen-ninety-eight above described.
_London_, seventeen-fourteen. =Joh. Ayres.= ‘Arithmetick Made Easie For
the Use and Benefit of Trades-Men.’ _Duodecimo._ Twelfth edition; with
an Appendix on Book-keeping by Chas. Snell.
A work of the immediate school of Cocker.
_London_, seventeen-fifteen. =John Hawkins.= ‘Cocker’s English
Dictionary.’ Second edition. _Octavo in twos_ (small).
I have entered this book here only because Hawkins asserts that it was
the work of the celebrated arithmetician: which I do not believe, for
the reason above given. [The first edition is said to have the date
seventeen-four.]
_London_, seventeen-seventeen. =Wm. Hawney.= ‘The Compleat
Measurer....’ _Duodecimo in threes._
A full treatise on decimal arithmetic.
{seventeen}
_No place marked_, seventeen-{eighteen }. =George Brown.=
‘Arithmetica Infinita, or the Accurate Accomptant’s Best Companion.’
_Octavo in twos_ (small, oblong).
This is not a work on arithmetic, but a set of tables, which will
certainly be reprinted as soon as the decimals of a pound gain their
proper footing. The main part of it is the first nine multiples (and
the 365th) of the decimals which express each farthing of the pound.
Thus under 4_s._ 1¼_d._ are given the multiples of ·20520833..... The
whole work is copperplate engraving from beginning to end. From several
indications, I gather that Geo. Brown of this work is also the author
mentioned under seventeen-one.
_London_, seventeen-seventeen. =Roger Rea.= ‘The Sector and Plain
Scale, Compared ... Unto which is annexed, So much of Decimal
Arithmatick and the Extraction of the square Root, as is necessary for
the Working of Arithmetical Trigonometry.’ _Octavo_ (N).
The treatise of an illiterate and confused person. Nothing has been
more common than for those who write on application to consider it
advisable not to trust the books to teach, nor the readers to know,
decimal fractions, and to supply a fresh treatise. Rea says he uses
the Italian mode of division (N) as being that which is _most commonly
used_: nothing more than this, even in 1717.
_London_, seventeen-eighteen. =Wm. Bridges.= ‘An Essay to facilitate
Vulgar Fractions; After a New Method, and to make Arithmetical
Operations Very Concise &c. _Duodecimo._’
_London_, seventeen-nineteen. =Good.= ‘Measuring made Easy.’ _Octavo_
(duodecimo size).
A description of Coggeshall’s sliding rule, corrected and enlarged by
James Atkinson.
_London_, seventeen-nineteen. =John Ward.= ‘The Young Mathematician’s
Guide.’ The third edition. _Octavo in twos_ (N).
This useful course, which commences with arithmetic [was first
published about seventeen-six]. It is recommended by Raphson and
Ditton. The sixth edition was in seventeen-thirty-four; the eighth
edition was in seventeen-forty-seven.
_London_, seventeen-twenty-one. =Wm. Beverege= (Bp. of St. Asaph).
‘Institutionum Chronologicarum Libri Duo. Unà cum totidem Arithmetices
Chronologicæ Libellis.’ Third edition. _Octavo in fours._
The date of the preface is sixteen-sixty-eight. The arithmetical part
is a treatise on the numerals of different nations, learned, but not
always judicious, according to modern views of the history of symbols.
It is followed by a brief elementary treatise on arithmetic, with
chronological examples.
_London_, seventeen-twenty-six. =E. Hatton.= ‘The Merchant’s Magazine:
or, Trades-Man’s Treasury.’ _Quarto._
This is the eighth edition of a work of some celebrity, but which must
not be confounded with Hatton’s edition of Recorde. The only guide to
the date of the first edition which I have is the statement of the
eighth that it was reviewed in the _Ouvrages des Savans_[17] for 1695.
[17] Or else the _Acta Eruditorum_. This comes of translating. The
phrase is “works of the learned.”
It is interspersed with copperplate pages of flourished writing,
containing examples and definitions. There is somewhat more of reason
given for rules than was very common, and a vast quantity of mercantile
terms, usages, &c. are explained.
_Witemberg_, seventeen-twenty-seven. =Joh. Fr. Weidler.= ‘De
Characteribus Numerorum Vulgaribus et eorum Ætatibus ... Dissertatio
Critico-mathematica.’ _Quarto._
_London_, seventeen-twenty-eight. =E. Hatton.= ‘A Mathematical Manual:
or Delightful Associate.’ _Octavo._
Mostly on the use of the globes, but containing some “mysterious
curiosities in numbers.”
_London_, seventeen-thirty. =Alexander Malcolm.= ‘A New System of
Arithmetick, Theorical and Practical.’ _Quarto._
One of the most extensive and erudite books of the last century,
having 640 heavy quarto pages of small type; “wherein,” to go on with
the title-page, “the science of numbers is demonstrated in a regular
course from its first principles through all the parts and branches
thereof, either known to the ancients or owing to the improvements
of the moderns; the practice and application to the affairs of life
and commerce being also fully explained: so as to make the whole a
complete system of theory, for the purposes of men of science; and of
practice for men of business.” I quote this lengthy title as a true
description of the work, at the date of publication. Probably the union
of such masses of scientific and commercial arithmetic made the book
unusable for either purpose.
_London_, seventeen-thirty-one. =Edw. Hatton.= ‘An Intire System of
Arithmetic ... containing, I. Vulgar. II. Decimal. III. Duodecimal.
IV. Sexagesimal. V. Political. VI. Logarithmical. VII. Lineal. VIII.
Instrumental. IX. Algebraical. With the Arithmetic of Negatives and
Approximation or [sic] Converging Series,’ &c. Second edition. _Quarto._
A sound, elaborate, unreadable work, of 500 pages, of the same
character as Malcolm’s.
_London_, seventeen-thirty-one. =Wm. Hodgkin.= ‘A Short New and Easy
Method of Working the Rule of Practice in Arithmetick.’ _Octavo in
twos._
An author who chooses his own examples can write a short method on any
rule: but the first example taken at hazard will probably defy the
abbreviations.
_London_, seventeen-thirty-two. =Joseph Champion.= ‘Practical
Arithmetick compleat.’ _Octavo in twos._
_London_, seventeen-thirty-five. =John Kirkby.= ‘Arithmetical
Institutions, containing a Compleat System of Arithmetic, Natural,
Logarithmical, and Algebraical.’ _Quarto in ones._
A system of arithmetic is mixed up with algebra. In the extraction of
roots, Halley’s formula is applied in such a manner as to make the
operation seem continuous, though it is just as difficult as before.
_London_, seventeen-thirty-five. =James Lostau.= ‘The Manual
Mercantile, Second Book: Concerning Decimal Arithmetic....’ _Quarto._
The first book was never published. This work contains a slight
treatise on Arithmetic, but the body of it consists of all the various
integers and fractions that may be useful in commerce, with the first
nine multiples of each. It is 452 pages entirely of copperplate, the
figures being rudely worked in, apparently by the author’s own hand. It
is a posthumous work, and the editor says it took 17 years. This mode
of stereotyping was adopted in several instances in the first half of
the last century. And it must be observed, that if decimal arithmetic
have not thriven in commercial affairs, it has not been for want of a
great many attempts to facilitate the use of it, by publishing books of
multiples.
_London_, seventeen-thirty-five. =Benjamin Martin.= ‘A new Compleat and
Universal System or Body of Decimal Arithmetick.’ _Octavo in twos._
A very full system of decimal arithmetic, applied to all parts of
commercial arithmetic.
_London_, seventeen-thirty-six. =Thomas Weston.= ‘A New and Compendious
Treatise of Arithmetick.’ _Quarto._ Second edition.
A simple and useful treatise.
_Edinburgh_, seventeen-thirty-six. ——. ‘Arithmeticæ et Algebræ
Compendium.’ _Octavo in twos_ (N).
There is a small treatise on arithmetic. The publishers are Thos. and
Wal. Ruddiman.
_London_, seventeen-thirty-eight. =William Pardon.= ‘A New and
Compendious System of Practical Arithmetick.’ _Octavo in twos._
Four hundred full octavo pages is not a very compendious book on
arithmetic, or would not be so now: but it looked small by the side of
Hatton, Malcolm, and Kirkby.
_London_, seventeen-thirty-eight. =Tho. Everard.= ‘Stereometry ...
by the Help of a Sliding-Rule.’ Edited by Leadbetter, tenth edition.
_Duodecimo._
A book which once had a great reputation among excise collectors.
_London_, seventeen-thirty-nine. =Christian Wolff.= ‘A treatise of
Algebra,’ translated from the Latin by J. H. M. A. _Octavo._
There is very little of arithmetic, and that mostly on the properties
of numbers.
_Cambridge_, seventeen-forty. =Nicholas Saunderson.= ‘The Elements of
Algebra, in ten books.’ _Quarto_, two volumes.
This is a posthumous work of the well-known Professor Saunderson, the
blind lecturer on optics. The first volume contains a synopsis of
Arithmetic, and the editor’s account of the calculating board, by which
Saunderson supplied the want of sight.
_London_, seventeen-forty. =Wm. Webster.= ‘Arithmetick in Epitome.’
Sixth edition. _Duodecimo_ (ON).
The eighth edition of the author’s book-keeping is _London_,
seventeen-forty-_four_, _octavo in twos_ (small); and the eighth
edition of his ‘Attempt towards rendering the Education of Youth more
easy and effectual,’ is _London_, seventeen-forty-_three_ (with paging
continued from that of the last). He says, “When a Man has tried all
Shifts, and still failed, if he can but scratch out any thing like a
fair _Character_, tho’ never so stiff and unnatural, and has got but
_Arithmetick_ enough in his Head to compute the Minutes in a Year,
or the Inches in a Mile, he makes his last Recourse to a Garret,
and, with the Painter’s Help, sets up for a Teacher of _Writing_ and
_Arithmetick_; where, by the Bait of low Prices, he perhaps gathers a
Number of Scholars.”
_London_, seventeen-forty. ——. ‘A small Treatise of the Square and
Cube....’ Second edition. _Quarto_ (O). Also,
_London_, seventeen-forty. ‘A Supplement to the Square and Cube....’
_Quarto in twos._
The author of this treatise (the last, I think, in which I have seen
the old method of extracting the square root) is a copier of Peter
Halliman, or some similar authority (see sixteen-eighty-eight), only
his denominator is less by a unit.
_London_, seventeen-forty. =Rob. Shirtcliffe.= ‘The Theory and Practice
of Gauging,’ &c.
A work once held in high estimation by the revenue-officers.
_Edinburgh_, seventeen-forty-one. =John Wilson.= ‘An introduction to
Arithmetick.’ _Octavo in twos._
A good demonstrative book, in a large type; very full on the _complete_
operations with circulating decimals, the _ignes fatui_ which have led
many an arithmetical writer astray.
_London_, seventeen-forty-two. =John Marsh.= ‘Decimal Arithmetic made
perfect.’ _Quarto._
Almost entirely on infinite or circulating decimals. The predecessors
whom he cites in his history of the subject are Wallis; Jones, 1706;
Ward; Brown, 1708 or 1709 (he has not the work, but it is above at
seventeen-one); Malcolm; Cunn; Wright, 1734; Martin, 1735; and Pardon.
This subject of circulating decimals was at one time suffered to
embarrass books of practical arithmetic, which need have no more to do
with them than books on mensuration with the complete quadrature of the
circle.
_Leipzig_, seventeen-forty-two. =Jo. Christoph. Heilbronner.= ‘Historia
Matheseos Universæ.’ _Quarto._
Though called a history of mathematics, and really a bibliography
_raisonnée_, yet it is peculiarly devoted to arithmetic, the authors on
which have a separate list. There are also dissertations on numerals,
on their history, &c. The index of this book is of rare goodness.
_Geneva_, seventeen-forty-three, forty-six, forty-seven, forty-nine,
forty-one. =Christian Wolff.= ‘Elementa Matheseos Universæ.’ _Quarto._
A second or later edition (N). Five volumes.
The first of the five volumes contains a short treatise on arithmetic.
Here, as happens so often in works of this period, a set is made up out
of different editions. The first four volumes have _editio novissima_,
the fifth has only _nova_. Wolff’s course would be better known if it
were scarcer. The ordinary reader passes it by as an old book; the
collector as one which is very common. But it is replete with pieces of
information, which are historical references and suggestions. As far as
I can remember, Wolff is much the most learned historian of those who
have written extensive courses.
_London_, seventeen-forty-five. =John Hill.= ‘Arithmetick, Both in the
Theory and Practice.’ _Octavo in twos._
This is the seventh edition of a work of much celebrity. It seems to
have owed its fame partly to a recommendation of Humphrey Ditton,
prefixed to the first edition (about 1712), praising it in the
strongest terms. Perhaps at this time the only things which would catch
the eye are the table of logarithms at the end, and the powers of 2 up
to the 144th, very useful for laying up grains of corn on the squares
of a chess-board, ruining people by horseshoe bargains, and other
approved problems.
_London_, seventeen-forty-eight. =Charles Leadbetter.= ‘The Young
Mathematician’s Companion.’ _Duodecimo in threes._
Second edition. Begins with an ordinary treatise on arithmetic.
_London_, seventeen-forty-eight. =William Halfpenny.= ‘Arithmetick and
Measurement, Improved by Examples.’ _Octavo in twos_ (N).
This is a surveyor’s and artisan’s book of application: but it contains
decimal fractions.
_London_, seventeen-forty-nine; second part, seventeen-forty-eight.
=Solomon Lowe.= ‘Arithmetic in two parts.’ _Duodecimo in threes._
This is a work both learned and foolish: but with the learning and
folly so distinct that they can be used separately. The folly consists
mostly in an attempt to give the rules of Arithmetic in English
hexameters and in alphabetical order; I give a couple of instances.
_Barter._
Barter, exchange of commodities: the rule to
proportion ’em as follows: What’s to be changd,
Value: then, see what That will purchase
of T’other. If an advanc’d price of one, a
proportionable find for the other.
_Casting out of Nines._
Prove by a careful review: ’tis the safest: the
readiest, as follows: SUB.] right;
when -hend and remainder (together) make up the
compound. ADD MULT DIV] add the digits
together and cast-out the nines: then _Right_;
if remainder of Facits agrees with remainder of
factors, Multiplied in Mul: -sor and quotient in
Div; to which add the remainder.
The learning consists in a great knowledge of former writers, and a
copious account of weights, measures, and coins: together with a list
of authors, which I have copied into my own, so far as I could not find
the names elsewhere.
_London_, seventeen-fifty. =James Dodson.= ‘The Accountant, or the
method of Book-keeping Deduced from Clear Principles.’ _Quarto._
As far as I can find, this is the first book in which double entry is
applied to retail trade. James Dodson (my great-grand-father) is best
known to mathematicians in general by his _Antilogarithmic Canon_,
_London_, seventeen-forty-two, _folio_ (_see Penny Cycl._ ‘Tables’).
_London_, seventeen-fifty. =Daniel Fenning.= ‘The Young Algebraist’s
Companion.’ _Duodecimo in threes._
There is a system of fractional arithmetic in this book, which is
written in dialogue. The author thought it impossible to understand
algebra without some better works on arithmetical fractions than
then existed. As it is, says he, it is impossible to understand the
_Algorithm_ much less the _Algorism_, which he explains by saying
that the former means the first principles, and the latter their
practice. In this curious confusion of terms we see at its commencement
an instance of a process which is always going on (though in this
instance it has been arrested), the attachment of different meanings
to different spellings of the same word. My curiosity led me to take a
little trouble to trace Fenning to his authorities. And I find that of
two writers who must have been in his hands, Saunderson and Kirkby, the
first uses _Algorithm_ for first principles, and the second _Algorism_
for practical rules. I think I remember having seen a comparatively
recent edition of this work.
_London_, ——. =R. T. Heath=, assisted by =W. Davidson=. ‘The Practical
Arithmetician: or Art of Numbers improved.’ _Duodecimo._ (Revised by J.
Bettesworth.)
Robert Heath was a person who made noise in his day, and in so doing
established a claim to be considered a worthless vagabond. He was
editor of the _Ladies’ Diary_ from 1746 to 1753, when the Stationers’
Company found it absolutely necessary to strike out some of his
scurrility, and dismiss him; appointing Thomas Simpson in his place.
But before this, in 1749, Heath had commenced the _Palladium_, an
annual publication resembling the _Ladies’ Diary_, the very first
mathematical question of which is so expressed as to convey an
indecent double-meaning, in a manner obviously intended. From 1750
or thereabouts, he began to write against Thomas Simpson. One of his
publications against the latter is headed, “Miss Billingsgate in a
salivation for a black eye:” and in a letter published in a newspaper,
in 1751, he remarks of Simpson and another, that “the best writer
against both is one who shall sign the warrant for their execution.”
_London_, seventeen-fifty-one. =T. Smith.= ‘Compendious Division.
Containing, A Great Variety of Curious and Easy Contractions of
Division.’ _Octavo in twos._
_London_, seventeen-fifty-three. =Sam. Stonehouse.= ‘A Compendious
Treatise of Arithmetic, By Way of Question and Answer.’ _Octavo._ Third
edition.
Question and answer is well when the difficulty is in the question and
the solution in the answer: but to turn “Decimals are divided like
integers” into “Are decimals divided like integers? The division is
exactly the same,” is trifling. This book gives the abbreviated rule
for decimals of a pound, which very few books have given.
_London_, seventeen-fifty-six. =John Playford.= ‘Vade Mecum, or the
Necessary Pocket Companion.’ Nineteenth edition. _Quarto_ (long and
narrow size).
This book is a ready reckoner, with miscellaneous tables. I have no
information about the origin of these books, which I think are not so
ancient as many may suppose. I could almost think from the preface
(but such deductions are very deceptive) that the earliest of the
books which are now called ready reckoners, meaning those which have
totals at given prices ready cast up, was the following: _London_,
sixteen-ninety-three. Wm. Leybourn. ‘Panarithmologia; Being A Mirror
For Merchants, A Breviate For Bankers, A Treasure For Tradesmen, A
Mate For Mechanicks, And A Sure Guide for Purchasers, Sellers, Or
Mortgagers of Land, Leases, Annuities, Rents, Pensions, &c. In present
Possession or Reversion. And A Constant Concomitant Fitted for All
Men’s Occasions.’ _Octavo._
_London_, seventeen-fifty-eight. =Benjamin Donn.= ‘A new Introduction
to the Mathematicks, being Essays on Vulgar and Decimal Arithmetick.
Containing, Not only the practical Rules, but also the Reasons and
Demonstrations of them.’ _Octavo_ (ON).
This good book fulfils to a great extent the profession of the
title-page as to demonstration. Donn had a good deal of miscellaneous
information. He was, if I remember right, one of Humphrey Davy’s early
teachers.
_London_, seventeen-fifty-nine and sixty-four. =Benjamin Martin.=
‘A New and Comprehensive System of Mathematical Institutions.’ Two
volumes. _Octavo in twos._
Old _Ben Martin_ (as his admirers called him) was an able, and in this
instance a concise writer. He wrote on every mathematical subject (and
never otherwise than well, I believe, except on biography), and a
complete set of his works is rarely seen. He was a bookseller.
_London_, no date, but about seventeen-sixty. =William Weston.=
‘Specimens of Abbreviated Numbers.’ _Octavo in twos._
Some supposed new rules for formation and use of decimals.
_London_, seventeen-sixty. =Jacob Welsh.= ‘The Schoolmaster’s General
Assistant.’ In two volumes. _Octavo._
The author claims a hundred curious discoveries; what they are, I
cannot find.
_London_, seventeen-sixty. =James Dodson.= ‘A Plain and Familiar
Method for Attaining the Knowledge and Practice of Common Arithmetic.’
_Octavo._
An edition of Wingate’s Arithmetic (N) called the nineteenth. But
Wingate, who first published it about 1629, would not have known his
own book, after the various dressings it received from Kersey, Shelley,
and Dodson. One of George Shelley’s editions of John Kersey the son’s
edition of John Kersey the father’s edition of Wingate, called the
fourteenth, is _London_, seventeen-twenty, _octavo in twos_.
_Amsterdam_, seventeen-sixty-one. =Isaac Newton.= ‘Arithmetica
Universalis.’ _Quarto._
This is the edition published by Castiglione, in two volumes, and is
the best. The original book, published by William Whiston, consisted of
the records laid up in the University Archives of the lectures which
Newton delivered as Lucasian professor. [S’ Gravesande, in the preface
to his edition of _Leyden_, seventeen-thirty-two, says it was published
without the author’s knowledge, and much to his displeasure: but
Whiston, in his Memoirs, says it was with Newton’s consent: most likely
it was both with his consent, and to his displeasure. Though properly a
book on Algebra, and its application to Geometry, yet it does contain
a system of Arithmetic. Various alterations, both of matter and
arrangement, were made in the (so called) second edition [By Machin,
1722], which are supposed to have been approved, if not furnished, by
Newton himself.]
This information comes from Castiglione. The other editions I have
seen are the original Latin of Whiston, _Cambridge_, seventeen-seven,
_octavo_. _London_, seventeen-twenty. ‘Universal Arithmetic, ...
translated ... by the late Mr. Raphson, and revised and corrected
by Mr. Cunn.’ _Octavo in twos._ The same reprinted, _London_,
seventeen-twenty-eight, _octavo in twos_, called second edition; it
is advertised as carefully compared with the correct edition that was
published in seventeen-twenty-two. There is also Wilder’s edition of
Raphson’s edition, _London_, seventeen-sixty-nine, _octavo_.
_London_, seventeen-sixty-five. =J. Randall.= ‘An Introduction To
so much of the Arts and Sciences, More immediately concerned in an
Excellent Education for Trade In its lower Scenes and more genteel
Professions,’ &c. _Duodecimo in threes._
Mr. Randall was a quaint man, but his book is well done. It contains
arithmetic, mensuration, and geography; and ends with a dialogue
between the heavenly bodies, upon their mutual arrangements, in which
the earth insists upon being allowed to stand still, and quotes
Scripture like an anti-Copernican, but is brought to reason by the
arguments of the others. This is almost the only writer I have met
with who has given the student a few hints upon habits of computation.
Thus he will not let him say, three and four are seven, seven and five
are twelve, &c.; but only three, seven, twelve, &c. For, says he, (the
example being the addition of some rents) “as you have this pretty
Income, you must talk like a Gentleman to your Figures.”
_London_, seventeen-sixty-six. =W. Cockin.= ‘A Rational and Practical
Treatise of Arithmetic.’ _Octavo in twos._
_Dublin_, seventeen-sixty-eight. =Dan. Dowling.= ‘Mercantile
Arithmetic.’ _Octavo in fours._
Mostly on exchanges. The author has given the rule for the
instantaneous formation of the fourth and fifth and succeeding
places of the decimal of a pound, which I never saw till now in
any book but my own. The third edition of this book is _Dublin_,
seventeen-ninety-five, _duodecimo in threes_. There is another author
of the same name presently mentioned.
_London_, seventeen-seventy-one. =Wm. Rivet.= ‘An Attempt to illustrate
the Usefulness of Decimal Arithmetic.’ Second edition. _Octavo_ (small).
This book contains what I thought no one had given before myself, a
_complete_ head-rule for turning fractions of 1_l._ into decimals to
any number of places. This method, which is much wanted in commercial
arithmetic, is here lost amidst attempts to compute with interminable
fractions; things on which real business will never waste a thought.
_Amsterdam_, seventeen-seventy-three. =Nicolas Barreme.=
‘Comptes-faits, ou Tarif général des Monnoies.’ _Duodecimo in threes._
This is a Dutch reprint of the work of a man who has given his name
to ordinary mercantile computation in France, even more than Cocker
in England. That he was a real person appears from the _privilège_
copied into this book, dated Jan. 26, 1760, whereby Louis XV. grants
the usual rights over his book to Nicolas Barreme. I have also seen
in a catalogue an edition marked seventeen-forty-four. The name
became an institution of France, which even the Revolution did not
destroy. The Citoyen Blavier published a ‘Nouveau Barême,’ _Paris_,
seventeen-ninety-eight, _octavo_, which he called a _Barême décimal_,
in which there is a well-marked distinction between Barême the person
and Barême the thing.
_London_, seventeen-seventy-three. =Thomas Dilworth.= ‘Miscellaneous
Arithmetic.’ In seven parts. _Duodecimo._
This work is but little known. Its contents are on the calendar; on
logarithms; on the rule of three, when the first term is 1_l._ and all
the terms are money; on the weather; a collection of riddles, answered,
in the midst of which are seriously set forth Bacon’s paradoxes on
the characteristics of a Christian, and an essay on the education of
children. Dilworth had made his name a selling one, and was determined
to make use of it.
_Shrewsbury_, seventeen-seventy-three. =Thomas Sadler.= ‘A Complete
System of Practical Arithmetic ... on an entire new plan.’ _Duodecimo
in threes._
The newness of plan seems to consist in putting the rules into
unintelligible verse, and beating even the older rule-mongers in
puzzling plain questions. Thus, a cargo consists of 84, 61, and 35
tons, of which three-fifths is lost; what must each bear of the loss?
This is done by first taking ⅗ of 180, namely 108; then 108 is divided
by 180, producing ·6; then each of the parcels is multiplied by ·6.
_London_, seventeen-seventy-four. =Anth.= and =Joh. Birks=.
‘Arithmetical Collections and Improvements.’ Second edition. _Octavo._
_London_, seventeen-seventy-four. =Nich. Salomon.= ‘The Expeditious
Accountant; or, Cyphering rendered _so short_, That Half the
Trouble.... A VERY CURIOUS WORK, Totally different from all that have
preceded it.’ _Octavo in twos._
There is something new, says the author, in almost every rule: but I
cannot find it. The head-rule for the decimals of a pound is introduced.
_London_, seventeen-seventy-seven. =James Hardy.= ‘The Elements or
Theory of Arithmetic.’ _Duodecimo in threes._
The author was a teacher at Eton, where, according to common notions,
there could have been no such thing at the date above as a teacher
of arithmetic. It is true there is an ambiguous comma in “Teacher of
Mathematics, and writing-master at Eton College.” The book is a very
creditable one, of great extent, including logarithms, &c.
_London_, seventeen-eighty. =John Bonnycastle.= ‘The Scholar’s Guide to
Arithmetic.’ _Duodecimo._
The first edition of this well-known work. It had from the beginning
algebraical demonstrations attached.
_Birmingham_, seventeen-eighty-three. =Wm. Taylor.= ‘A Complete System
of Practical Arithmetic.’ _Octavo in twos._
An enormous book of 600 pages, with arithmetic, mensuration, geography,
astronomy, algebra, book-keeping, &c., in the above order.
_London_, seventeen-eighty-four. =G=[=eorge=] =Anderson.= ‘The
Arenarius of Archimedes ... from the Greek ... the dissertation of
Christopher Clavius on the same subject....’ _Octavo._
_Oxford_, eighteen-thirty-seven. =Steph. Pet. Rigaud.= ‘On the
Arenarius of Archimedes.’ _Octavo._
I choose this edition to introduce the only purely arithmetical work
of Archimedes. Its object is to shew that any number, even a universe
full of grains of sand, can be easily expressed: a thing by no means
likely to be self-evident to a Greek, whose numerical notation was,
till Archimedes and Apollonius shewed how it might be extended, far
from sufficient for such a purpose. Professor Rigaud, one of the most
learned and accurate of our modern inquirers into mathematical history,
has given an account of Anderson, and many valuable remarks on his
translation, as well as on the subject of it.
_London_, seventeen-eighty-four. =Thomas Dilworth.= ‘The Schoolmaster’s
Assistant.’ _Duodecimo._
This is the twenty-second edition. By the dates of the commendations
prefixed, it would seem that the first edition was published in
seventeen-forty-four or forty-five.
_Great-Yarmouth_, seventeen-eighty-five. =Thos. Sutton.= ‘The
Measurer’s Best Companion; or, Duodecimals brought to perfection.’
_Octavo._
In truth this is the most elaborate system of duodecimals I have met
with.
_London_, seventeen-eighty-six. =George Atwood.= ‘An Essay on the
Arithmetic of Factors applied to various Computations which occur in
the Practice of Numbers.’ _Quarto._
_Edinburgh_, seventeen-eighty-six. =John Mair.= ‘Arithmetic, Rational
and Practical.’ _Octavo._ Fourth edition.
The name of Mair, who was rector of the Perth Academy, is highly
respected in Scotland. His ‘Book-keeping moderniz’d,’ of which the
_ninth_ edition is _Edinburgh_, eighteen-seven, _octavo_, had a
great run. Completeness of subjects, and copiousness of examples,
characterise both works, which extend to six hundred pages of small
print each.
_London_, seventeen-eighty-seven. =C. G. A. Baselli.= ‘An Essay
on Mathematical Language; or, an Introduction to the Mathematical
Sciences.’ _Octavo in twos._
It cannot be both, the reader will say: and in truth it is only the
second, for arithmetic and algebra, with some good points about it.
_London_, seventeen-eighty-eight. ‘Clavis Campanologia, or a Key to the
Art of Ringing.’ _Duodecimo in threes._
As large a list as the present ought to have one book at least on
bell-ringing, the whole theory of which is arithmetical. No art has had
greater enthusiasts for it. The authors of the present treatise, Wm.
Jones, Joh. Reeves, and Thos. Blakemore, are of the number. They record
several names of inventors to whom they give words of praise which
might apply to Newton or Euler; among them is Hardham, who is known
to this day by the snuff mixture which he invented and sold in Fleet
Street, where his name still remains. I should think that few of those
whose noses he has tickled are aware that he may have done the same for
their ears.
_Oxford_, seventeen-eighty-one (one volume); _London_,
seventeen-eighty-eight (the other). =James Williamson.= ‘The Elements
of =Euclid=, with dissertations....’ _Quarto._
The arithmetical books of Euclid are here with the rest. I have chosen
this edition by which to introduce the name, because it is the only
modern translation of _Euclid_. All the works which go by that name
are versions thickly scattered with the views of the editors as to
what Euclid _ought to have been_, instead of the rendering of _what he
was_. For these “many tamperings with his text,” a countryman[18] of
Robert Simson has been the first to call them the “perfidious editors
and translators of Euclid;” a name which, in a sense, they richly
deserve. Williamson was a real disciple of Euclid; and he translated
so closely, that such words as, not being in the Greek, English idiom
renders necessary, are put in Italics. For many editions of Euclid, the
reader may consult my article on that name, in Dr. Smith’s Biographical
Dictionary: and, for the contents of the tenth book, the article
_Irrational Quantities_ in the _Penny Cyclopædia_.
[18] Sir William Hamilton of Edinburgh, in his notes to Reid, p. 765.
If the demonstrative system of Euclid had taken as great a hold in
arithmetic as in geometry, we should not have had to complain of one of
the best exercises of thought being employed for no other purpose than
to make machines.
_London_, seventeen-eighty-eight. =Thomas Keith.= ‘The complete
practical Arithmetician.’ First edition. _Duodecimo in threes._
_London_, seventeen-ninety. =Thomas Keith.= ‘A Key to the complete
practical Arithmetician.’ First edition. _Duodecimo in threes._
_London_, no date; about seventeen-ninety (?). =John Duncombe.= ‘A new
Arithmetical Dictionary.’ _Octavo in twos._
The rules and terms of arithmetic, in alphabetical order.
_Calcutta_, seventeen-ninety. =Joh. Thos. Hope.= ‘A Compendium of
Practical Arithmetick.’ _Octavo in twos._
A clear, prolix book, for the Orphan School at Calcutta.
_London_, seventeen-ninety-one. =William Emerson.= ‘Cyclomathesis,
or an easy Introduction to the several branches of the Mathematics.’
_Octavo._
These are Emerson’s works, collected (not reprinted) in thirteen
volumes, with new title-pages. The arithmetic, which is in the first
volume, is said by the editor to be of seventeen-sixty-three. Emerson
was the writer of many works, which had considerable celebrity: but he
was as much overrated as Thomas Simpson was underrated. There is a most
amusing life of him prefixed to the collection.
_London_, seventeen-ninety-one. =Thos. Keith.= ‘The New Schoolmaster’s
Assistant.’ _Duodecimo in threes._
An abridgment of the larger work above mentioned.
_Berlin_, seventeen-ninety-two. =Leonard Euler.= ‘L’Arithmétique
raisonnée et démontrée, oeuvres posthumes de Léonard Euler, traduite en
Francois par Bernoulli, Directeur de l’Observatoire de Berlin, &c. &c.’
_Octavo in twos_ (ON).
The editor calls this the first work of Euler in his preface, and
posthumous in the title-page. It is, I suppose, a translation of the
work which is set down in Fuss’s list as ‘Anleitung zur Arithmetic, 2
Th. Petersb. 1738. 8,’ the _third_ of Euler’s separate works. It is
mostly on commercial arithmetic, and shews that Euler did not, in 1738,
consider the old method of division quite exploded.
_London_, seventeen-ninety-four. =Rich. Carlile.= ‘A Collection of one
hundred and twenty ... Arithmetical, Mathematical, Algebraical and
Paradoxical Questions.’ _Octavo._
What _Paradox_ is, as a science, I do not know: but the other
distinctions are well known. All who know much of the country schools
remember that _mathematics_ meant geometry, as opposed to arithmetic
and algebra. And it was right it should have been so: for neither the
schoolboy’s arithmetic nor his algebra were _disciplines_.
_London_, seventeen-ninety-four. =Henry Clarke.= ‘The Rationale of
Circulating Numbers.’ A new edition. _Octavo._
Another tract on repeating decimals, with some additions on other
subjects.
_London_, seventeen-ninety-four. =Thomas Molineux.= ‘The Scholar’s
Question-book, or an introduction to Practical Arithmetic. Part the
second. For the use of Macclesfield School.’ _Duodecimo._
An ordinary school-book on fractions and commercial arithmetic. I
never saw the first part. The school-seal, which is engraved on the
title-page, gives the learner to understand the mode adopted of
explaining difficulties; it displays a pedagogue with a birch-rod in
his right hand and a book in his left; illustrative of primary and
secondary method. The fourth edition of this second part is _London_,
eighteen-twenty-two, _duodecimo_. As may be supposed from the date, the
little hint does not appear.
_Paris_, seventeen-ninety-five. =Agricol de Fortia.= ‘Traité des
Progressions ... précédé par un Discours sur la nécessité d’un Nouveau
Systême de Calcul.’ Third edition. _Octavo._
The new system of calculation is a proposal to annex to addition,
multiplication, and involution, the next step, as the author takes
it to be, in the chain of operations. But he is wrong in his use of
the analogy. It appears that he was the author of several works on
arithmetic.
_Dublin_, seventeen-ninety-six. =John Gough= (edited by his son).
‘Practical Arithmetick in four books.’ _Duodecimo._
This work, I am told, had such extensive currency in Ireland (where
it was first published in seventeen-fifty-eight) that the name of the
author became almost synonymous with arithmetic; insomuch that when
Professor Thomson’s Arithmetic was first published in that country,
it went by the name of ‘Thomson’s Gough.’ The second edition appears
to have been an augmented and octavo work, afterwards reduced again
for schools. It is a book of the ordinary character, and abounds in
examples for practice. The last edition I have seen is _Dublin_,
eighteen-thirty-one, _duodecimo in threes_, edited by M. Trotter.
_New York_, seventeen-ninety-seven. =William Milns.= ‘The American
Accountant.’ _Octavo in twos._
The author seems to have been an emigrant from St. Mary Hall, Oxford.
His book has the peculiarity of giving in lieu of answers, the
remainders to nine of the answers, for guide in the proof by casting
out nines.
_Paris_, an VI. (seventeen-ninety-seven or ninety-eight). =Condillac.=
‘La Langue des Calculs.’ _Octavo._
A posthumous work. The views of a clear-headed mathematician and
metaphysician upon the foundation of arithmetic and the formation of
its language.
_London_, seventeen-ninety-eight. =Rich. Chappell.= ‘The Universal
Arithmetic.’ _Octavo in twos_ (small).
This book deserves notice for the author’s attempt to introduce
the practice of subtracting in division, without writing down the
subtrahend. He versifies his tables, _ex. gr._
So 5 times 8 were 40 Scots
Who came from Aberdeen,
And 5 times 9 were 45,[19]
Which gave them all the spleen.
[19] The North Briton, No. 45.
_London_, seventeen-ninety-eight. =Francis Walkingame.= ‘The
Tutor’s Assistant; being a Compendium of Arithmetic, and a complete
question-book.’ _Duodecimo in threes._
This is the twenty-eighth edition; when the first was published I
do not know, any more than what edition is now current. I should be
thankful to any one who would tell me who Walkingame was, and when the
first edition was published: for this book is by far the most used of
all the school-books, and deserves to stand high among them. I have
before me John Fraser’s ‘Walkingame modernized and improved,’ _London_,
eighteen-thirty-one, called _seventy-first edition_; John Little’s
edition, _London_, eighteen-thirty-nine; William Birkin’s edition,
_Derby_, eighteen-forty-three, called the _fifty-first_, and bearing
proof that at least seven _Birkins_ had appeared; and Samuel Maynard’s
edition of F. Crosby’s edition, _London_, eighteen-forty-four. All
these are _duodecimo_. When editors do not agree within twenty as to
the number of editions of their author which have been published, that
author is surely a man of note.
_London_, seventeen-ninety-eight. =Wm. Playfair.= ‘Lineal Arithmetic,
applied to shew the Progress of the Commerce and Revenue of England
during the present century.’ _Octavo in fours._
Not arithmetic, but plates arranging the several matters in curves, in
the manner now much more familiar than it was then.
_London_, seventeen-ninety-nine. =Charles Vyse.= ‘The Tutor’s Guide,
being a complete system of Arithmetic.’ _Duodecimo in sixes._ (Tenth
edition, edited by J. Warburton; eleventh in eighteen-one.)
In the same place, year, form, and by the same editor, was published a
new edition of the Key. It appears that the first edition was reviewed
in the _Monthly Review_ for 1771, so that it is probably of the year
before. Vyse is one of the most celebrated of the illustrious band who
used to adorn the shelves of a country schoolmaster at the beginning
of this century; Vyse, Dilworth, Walkingame, Keith, Joyce, Hutton,
Bonnycastle (with Cocker for a lost Pleiad). He is also the poet of the
lot: and some of his examples have gone through many other books. The
following specimen of the muse of arithmetic should be preserved, as
the best known in its day, and the most classical of its kind:
When first the Marriage-Knot was tied
Between my Wife and me,
My Age did her’s as far exceed
As three Times three does three;
But when ten Years, and Half ten Years,
We Man and Wife had been,
Her Age came up as near to mine
As eight is to sixteen.
Now, tell me, I pray,
What were our Ages on the Wedding Day?
The book (this tenth edition at least) is crowded with examples, which
circumstance makes the Key very large. On the execution there is no
remark to make. If a new edition were published, some of the examples
must be omitted, as rather opposed to modern ideas of decency.
_Paris_, eighteen-hundred. ‘Séances des Ecoles Normales, recueillies
par des sténographes, et revues par les Professeurs. Nouvelle édition.’
Thirteen volumes. _Octavo._
When the Normal School was founded at Paris, in 1794, the professors
engaged “pledged themselves to the representatives of the people and to
each other” neither to read nor to repeat from memory. Their lectures
were taken down in short-hand, and these volumes contain some of them.
The professors of mathematics were Lagrange and Laplace: and few
persons are aware that the mode in which the two first mathematicians
in Europe taught the humblest elements of arithmetic and algebra can
thus be judged of. The contents are, so far as these subjects are
concerned; vol. I. p. 16, programme and Laplace, arithmetic; 268,
Laplace, arithmetic; 381, Laplace, algebra: II. 116, Laplace, algebra;
302, do. do.: III. 24, Laplace, algebra; 227, Lagrange, arithmetic;
276, Lagrange, algebra; 463, do. do.: IV. 41, Laplace, geometry; 223,
Laplace, algebraic geometry; 401, Lagrange, algebraic geometry: V. 201,
Laplace, new system of measures: VI. 32, Laplace, probability: VII. 1,
Biot, account of the ‘Mécanique Céleste.’
The last three volumes contain the _debates_, or conferences, between
the teachers and their pupils, of which there are three in the first of
them, on arithmetic and algebra, not at all worth reading.
_Taunton_, (no date, perhaps before eighteen-hundred). =William
Wallis.= ‘An Essay on Arithmetic ... Briefly, Shewing, First, The
Usefulness; Secondly, It’s extensiveness; Thirdly, The Methods of it.’
This is a remonstrance by the author, a teacher at Bridgwater in
Somersetshire, against the prevailing modes of teaching arithmetic.
The following is an extract: “And I have seen a _Fair-Book_ (as ’tis
call’d) of a young Man’s, about 17 _Years_ of Age, who had been 6
_Years_ at _School_, but never went through that Rule [of three]: In
the same Book I found 132 Questions in _Reduction_, in the working
of them were 2680 Figures, which might have been better done in 500,
so that there were 2180 _superfluous Ones_. In another Rule I saw an
Example, in which were 174 Figures, but might have been done in 23; and
one of 80 that might have been done in 12: In general, I have found
in the Boys Books, 3, or 4 Times as many Figures as need be. These
_Methods_ have so far hindred their Advances in Learning, that amongst
30 Scholars, since I came hither, I have not found one that understood
a Rule beyond _Division_, tho’ some of them were 14 or 15 _Years_ of
Age, and had been kept at School, ever since they were capable of being
taught.”
_Paris_, eighteen-hundred. =Condorcet.= ‘Moyen d’apprendre à compter
surement et avec facilité.’ Second edition. _Octavo_ (duodecimo size).
One of the simplest explanations of the most elementary arithmetic
which has ever appeared. It was written in the last days of the author,
while hiding from the fate which he only finally avoided by suicide:
and the last sheet was hardly finished when his retreat was discovered.
_Madrid_, eighteen-one. ‘Aritmetica y Geometría práctica de la real
Academia de San Fernando.’ _Octavo._
A very clearly written and printed work.
_Buckingham_, eighteen-two; second edition, eighteen-eight. =John
King.= ‘An Essay intended to establish a new universal System of
Arithmetic....’ _Octavo._
The title of the second edition is more modest: it is ‘An Essay, or
attempt towards establishing....’ The system is the _octonary_ system,
in which 10 means eight, 100 is sixty-four, &c.
_London_, eighteen-three. =Rob. Goodacre.= ‘Arithmetic adapted to
different classes of learners.’ _Duodecimo in sixes._
The ninth edition of this work, by Samuel Maynard, is _London_,
eighteen-thirty-nine, _duodecimo_.
_Dublin_, eighteen-four. =P. Deighan.= ‘A Complete Treatise on
Arithmetic, rational and practical.’ Two volumes. _Octavo in twos._
This treatise has a list of a thousand subscribers, and has amused me
very much. The old notions of the style of a book were, it seems, not
extinct in Ireland a hundred years after they had been exploded in
England. The author, who handles his subject ably, puts _philomath_
after his name, and is perhaps the last of those who rejoiced in a
title which, though self-conferred, its owners would not have changed
for F.R.S. It is dedicated “to all those who think that a knowledge
of accounts is useful to mankind, from the king on the throne to the
lowest subject.” It has the praises of the author’s friends in prose
and poetry, duly prefixed. I quote a few lines from one of the poets,
desiring the reader to observe where sophs come from, unsought, and
whence Irish authors got their stationery.
“How many sophs, to sense and science blind,
Range through the realms of nonsense unconfin’d,
Unaw’d by shame, and unrestrain’d by law,
Their labour chaff, and their reward a straw;
Neglected and despis’d, they sink in shame
To that _oblivion_ whence, _unsought_, they came.
The muse, indignant, oft with grief has seen
An author led by ignorance and spleen,
With snail-paced speed, but unremitting toil,
In attic chamber waste the midnight oil,
With waste of paper, loss of ink combined,
_And pens from public offices purloined_.[20]—
But DEIGHAN of a more enlightened mind,
More innate genius, talents more refined,” &c.
[20] Really I am afraid that there must have been some truth in this.
Mr. Thomas Moore gives a translation of the _Pennis non homini datis_
of Horace, which shews that he had heard of the thing at least. Of
course he can clear himself: at any rate Lalla Rookh has not much the
air of having been written with a pen from a public office.
_London_, eighteen-five. =Christ. Dubost.= ‘Commercial Arithmetic, with
an Appendix upon Algebraical Equations.’ _Duodecimo in threes._
Of all works I know professing to be strictly commercial, this has the
fullest explanations in words of the rules and processes.
_Madras_, eighteen-six. =James Brown.= ‘A course of Military and
Commercial Arithmetic.’ _Small octavo size_: no signatures.
As might be expected, this is full on Indian exchanges, weights, and
measures.
_London_, eighteen-six. =William Frend.= ‘Tangible Arithmetic; or the
Art of Numbering made easy, by means of an Arithmetical Toy which will
express every number up to 16,666,665.’ _Octavo._ Second edition.
The toy is the Chinese instrument or abacus, called the _Schwanpan_,
for a description of which see Peacock, p. 408.
_Paris_, eighteen-seven. =J. B. V.= ‘L’Arithmétique enseignée par des
moyens clairs et simples.’ _Octavo._
This is in dialogue between a mother and her boy: the author assures us
that they are from real life; and it must have been so; for the _ancien
officier du génie_, as he calls himself, could no more have written
these dialogues than the mother and child could have constructed a sap.
The lady has the awkward name of Madame Épinogy; and as the object of
the dialogues is to make the child invent, I can find no origin for
this name, except the supposition that it is a blundering derivative
from ἐπίνοια. Nevertheless the dialogues are exceedingly good.
_Paris_, eighteen-eight. =F. Peyrard.= ‘Oeuvres d’Archimède, traduites
littéralement, avec un commentaire.’ Second edition, two volumes.
_Octavo._
I mention this work, not only for the _Arenarius_ already noticed,
but for the disquisition by Delambre on the arithmetic of the Greeks,
which afterwards appeared in the ‘Histoire de l’Astronomie Ancienne,’
_Paris_, eighteen-seventeen, _quarto_, two volumes. Delambre was a
real reader of the works he cites. He collected his materials from
Nicomachus, the _Theologumena_, Barlaam, the two Theons, Ptolemy,
Eutocius, Pappus, and Archimedes. I may as well say here what I have to
say on those of the above who have not been mentioned elsewhere.
The best edition of Ptolemy is that of Halma, which is a collection
of Ptolemy and his commentators, published at different times, and
separately,—the whole making distinct works, as well as a set. Of this
I know only four volumes, of which the two to the present purpose
are _Paris_, eighteen-thirteen and sixteen. ‘Κλαυδιου Πτολεμαιου
μαθηματικη Συνταξις....’ (Gr. Fr.) _quarto_, two volumes. Brunet says
that the commentaries of Theon, and the Κανονες Προχειροι of Ptolemy,
are in five more volumes, _Paris_, eighteen-twenty-one, twenty-two,
twenty-two, twenty-three, twenty-five, _quarto_.
The commentaries of Eutocius on the works of Archimedes are to be found
in several editions, but best in that of Joseph Torelli, _Oxford_,
seventeen-ninety-two, ‘Ἀρχιμηδους τα σωζομενα μετα των Εὐτοκιου
Ἀσκαλωνιτου ὑπομνηματων’ (Gr. Lat.). _Folio._
The fragment of the second book of Pappus (the only part of the first
two books published as yet, if, indeed, any more exist) is to be found
(Gr. Lat.) in the third volume of John Wallis’s _Opera Mathematica_,
Oxford, sixteen-ninety-nine, _folio in twos_.
_London_, eighteen-ten. =W. Tate.= ‘A System of Commercial Arithmetic.’
_Duodecimo._
A work approximating more nearly to modern business than most of those
then in use, in its _additions_; but, like most attempts to improve
real commercial arithmetic, wanting the corresponding _omissions_.
_Hawick_, eighteen-eleven. =Chas. Hutton=, edited by Alex. Ingram. ‘A
Complete Treatise on Practical Arithmetic and Book-keeping.’ _Duodecimo
in threes._
According to Hutton’s Catalogue, the fifth edition was in
seventeen-seventy-eight, and the twelfth in eighteen-six: and at his
death he possessed no edition previous to the fifth. The late Dr.
Olinthus Gregory published what he called the eighteenth edition,
enlarged, &c. _London_, eighteen-thirty-four, _duodecimo_: and a new
edition of Ingram’s Hutton, by James Trotter, appeared _Edinburgh_,
eighteen-thirty-seven, _duodecimo_.
_London_, eighteen-twelve. =Thomas Clark.= ‘A New System of Arithmetic;
including Specimens of a Method by which most Arithmetical Operations
may be performed without a Knowledge of the Rule of Three; and followed
by Strictures on the Nature of the Elementary Instruction contained in
English Treatises on that Science.’ _Octavo._
This is an able attempt to draw public attention to the state of
instruction in arithmetic. The author asserts, 1. There is not in the
English language, a work of any repute whatever, employed in school
education, in which the four fundamental rules of arithmetic are
clearly and comprehensively laid down. 2. Not one in which the rules
laid down are accompanied by examples so detailed as to remove the
difficulties which these rules must present to beginners. 3. None in
which the rules and examples for abstract and concrete numbers are kept
distinct from each other. 4. There is not a work of this description
in which ordinary and decimal fractions are properly arranged. 5. Or
in which the rationale of arithmetical operations seems of sufficient
importance to the instructor to induce him to incorporate it with
his work. 6. Or in which the principles and algebraical signs used
in arithmetic are given and explained at the time when the science
requires their introduction.
_Dublin_, eighteen-twelve. =John Walker.= ‘The Philosophy of Arithmetic
... and the Elements of Algebra.’ _Octavo._
Mr. Walker was a good scholar, an excellent mathematician, and a
most original thinker. Both this work and that which he published on
geometry shew great power.
_Sheffield_, eighteen-thirteen. =Joseph Youle.= ‘The Arithmetical
Preceptor ... to which is added a Treatise on Magic Squares.’
_Duodecimo in threes._
_London_, eighteen-thirteen. =Edward Strachey.= ‘Biga Ganita; or the
Algebra of the Hindus.’ _Quarto._
_Bombay_, eighteen-sixteen. =John Taylor.= ‘Lilawati; or a Treatise
on Arithmetic and Geometry by =Bhascara Acharya=.’ _Quarto size_ (no
signatures).
_London_, eighteen-seventeen. =Henry Thomas Colebrooke.= ‘Algebra,
with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and
Bháscara.’ _Quarto._
The first work has notes by S. Davis, and is from a Persian version
of Bhascara’s Sanscrit. The second work is also from the Persian. The
third, which contains not only the two works of Bhascara, but also
an arithmetical chapter from Brahmegupta, is all from the Sanscrit.
It is also pretty copious in selections from the Commentators, and
has a large body of dissertation by Colebrooke himself. But it does
not entirely supersede the former two, which have likewise valuable
annotations.
_Edinburgh_, eighteen-thirteen. =Elias Johnston.= ‘A sure and easy
Method of learning to Calculate.’ _Duodecimo in sixes._
This is a translation of the work of Condorcet mentioned under the date
1800.
_Paris_, eighteen-thirteen. =F. Peyrard.= ‘Les Principes fondamentaux
de l’Arithmétique.’ _Octavo in twos._
An elegant mixture of arithmetic and algebra, by the editor and
translator of Euclid, and the translator of Archimedes.
_Lille_, eighteen-fourteen. ——. ‘Manuel d’Arithmétique ancienne et
décimale.’ _Duodecimo in threes._
A small book, in question and answer: a transition book from the
old system to the new, containing both, and intended for commercial
purposes. At the end are some forms for letters of ceremony and
business, and for petitions: and it seems rather strange to English
eyes to see that a petition for a son condemned to death for homicide,
ranks among the matters which are considered near enough to the
ordinary course of business to find a place; and a place which, when
opened, gives the option of reading the way of turning francs into
roubles.
_London_, eighteen-fourteen. =S. F. Lacroix.= ‘Traité Élémentaire
d’Arithmétique à l’usage de l’école centrale des quatre-nations.’ Tenth
edition. _Octavo._
A well-known work, by one of the most systematic and most widely
circulated of elementary writers. The sixteenth edition was _Paris_,
eighteen-twenty-three, _octavo_. There was an English translation,
_London_, eighteen-twenty-three, _octavo_, anonymous,—an attempt to
introduce demonstrative arithmetic into our schools.
The third edition of an American translation, by John Farrar, appeared
_Cambridge (U.S.)_, eighteen-twenty-five, _octavo in twos_.
_Dublin_, eighteen-fourteen. =R. F. Purdon.= ‘Theory of some of the
Elementary Operations in Arithmetic and Algebra.’ _Octavo._
_Oxford_, eighteen-fourteen. =Charles Butler.= ‘An Easy Introduction to
the Mathematics.’ _Octavo._
This book fulfils the promise of the title-page well, and has been
frequently cited for the historical introductions to the several
subjects, which are very good, and, as parts of a learner’s course,
unexampled.
_London_, eighteen-fifteen. =J. Carver.= ‘The Master’s and Pupil’s
Assistant.’ _Duodecimo in threes._
The author of this work, dependent as the sale of it was on teachers,
has had the sense and courage to say, that questions with answers are
_for the benefit of the masters_ and the _injury of the pupils_. It
is dated from _Belgrave House, Pimlico_,—a name and site which might
puzzle an antiquary a century hence.
_Paris_, eighteen-sixteen. =Bezout.= ‘Traité d’Arithmétique.’ Eighth
edition. _Octavo._
A work of a somewhat older stamp than those of Lacroix and Bourdon. The
eleventh edition was _Paris_, eighteen-twenty-three, _octavo_, edited
by A. A. L. Reynaud, with notes and a table of logarithms; the notes
a separate work, with another title-page. The next year appeared the
twelfth edition of Reynaud’s own work, _Paris_, eighteen-twenty-four,
‘Traité d’Arithmétique,’ _octavo_, also augmented by a table of
logarithms. This work enters rather more on the theory of numbers.
_London_, eighteen-sixteen. =Thos. Taylor.= ‘Theoretic Arithmetic, in
three books; containing the substance of all that has been written on
this subject by Theo of Smyrna, Nicomachus, Jamblichus, and Boetius.
Together with some remarkable particulars respecting perfect, amicable,
and other numbers, which are not to be found in the writings of any
ancient or modern mathematicians. Likewise, a specimen of the manner in
which the Pythagoreans philosophised about numbers; and a development
of their mystical and theological arithmetic.’ _Octavo._
_Edinburgh_, _circa_ eighteen-sixteen. =A. Melrose= (edited by A.
Ingram). ‘A Concise System of Practical Arithmetic.’ Second edition.
_Duodecimo in threes._
Horner refers to this book as containing what is nearly an anticipation
of his method, in the case of the simple cube root.
_London_, eighteen-seventeen. =Thos. Preston.= ‘A New System of
Commercial Arithmetic ... a perfect, a permanent and universal ready
reckoner.’ _Duodecimo in threes._
Perhaps the plan of this book is partly taken from one published by
Girtanner in seventeen-ninety-four (_Penny Cycl. Suppl._ Tables), in
which the logarithms of numbers and certain intermediate fractions are
given. But, in the main, it is an application of the same principle
as that which has long been used in astronomical logarithms, namely,
giving the logarithms of integers, with a column in which those
integers, considered as seconds, are turned into degrees, minutes, and
seconds. In this way are given the logarithms of pence up to 130_l._;
of pounds up to 7 cwt. 16lbs.; of twelfths up to 333⅓; of sixteenths
up to 312½; of sixteenths of gallons, considered as fractions of a tun
of 236 gallons, for seed-oils; the same for a tun of 256 gallons, for
fish-oils and wines; of pounds of 120 to the cwt. up to five tons; of
grains, up to 25 oz. troy; and two for days and for pounds at 5 per
cent, by which one operation gives the interest for days on any number
of pounds up to 2600_l._
_Leipsic_, eighteen-seventeen. =Fred. Astius.= ‘Theologumena
Arithmeticæ.... Accedit =Nicomachi= Geraseni Institutio Arithmetica.’
_Octavo._ (See p. 17.)
These θεολογούμενα have been attributed to Jamblichus and to
Nicomachus: but they seem rather to consist of extracts from
Nicomachus, Anatolius, and others. They are explanations of the
Pythagorean and Platonic opinions on numbers; and form a very good
accompaniment for the works of Nicomachus. The notes are full and good.
_London_, eighteen-eighteen. =George G. Carey.= ‘A Complete System of
Theoretical and Mercantile Arithmetic.’ _Octavo._
A commercial book with a table of logarithms in it is rare in the
nineteenth century.
_Edinburgh_, eighteen-eighteen. =William Ritchie.= ‘A System of
Arithmetic.... and a Course of Book-keeping.’ _Duodecimo in sixes._
A book of much greater merit than could be guessed from its pretensions
or its notoriety: and, for its size, one of the most comprehensive I
have met with. Its author was a little (about twenty years) in advance
of his age, and the greater part of the edition was sold as waste paper.
_London_, eighteen-nineteen. ‘Philosophical Transactions.’ _Quarto._
=W. G. Horner.= ‘A New Method of Solving Numerical Equations of all
orders by Continuous Approximation.’
_London_, eighteen-thirty. =Thos. Leybourn.= ‘New Series of the
Mathematical Repository.’ Volume five. _Octavo in twos._ =W. G.
Horner.= ‘Horæ Arithmeticæ.’
The first-mentioned paper contains the most remarkable addition
made to arithmetic in modern times, the value of which is gradually
becoming known. On this subject I may refer to Mr. Horner‘s paper
on Algebraic Transformation in the Mathematician (vols. i. and ii.
various numbers); to J. R. Young, ‘An elementary treatise on Algebra,’
_London_, eighteen-twenty-six, _octavo_, as the first elementary
writer who saw the value of Horner’s method; J. R. Young, ‘Theory and
solution of Algebraical Equations of the higher orders,’ _London_,
eighteen-forty-three, _octavo_; Thos. Stephens Davies, ‘A Course
of Mathematics ... by Charles Hutton,’ twelfth edition, _London_,
eighteen-forty-three, _octavo_; T. S. Davies, ‘Solutions of the
principal questions in Dr. Hutton’s Course of Mathematics,’ _London_,
eighteen-forty, _octavo_; Peter Gray, four papers in the Mechanic’s
Magazine for March, eighteen-forty-four; A. De Morgan, ‘Notices of
the progress of the problem of Evolution,’ in the _Companion to the
Almanac_ for eighteen-thirty-nine, with the two articles headed
‘Involution and Evolution’ in the _Penny Cyclopædia_, and in the
Supplement (eighteen-thirty-eight and forty-five), and a Letter to the
Editor of the Mechanic’s Magazine, published in that work for February,
eighteen-forty-six.
In connexion with this subject I ought to mention Mr. Thomas Weddle’s
‘New simple and general method of solving numerical equations of all
orders,’ _London_, eighteen-forty-two, _quarto_. This is an organised
process, in which the principle of each step is the correction of the
preceding result by multiplication, not by addition.
_Edinburgh_, eighteen-twenty. =John Leslie.= ‘The Philosophy of
Arithmetic.’ _Octavo._