can be arranged in an hexameter line.
As Schott here describes the digit of his geometrical pace as being
forty poppy-seeds placed side by side, I thought it possible I might
get a fellow to the barley measure (see page 8). But it seems that he
has not played fair, or has thought it necessary to bring the degrees
of glory of the Virgin up to the 256th power of 2, at any sacrifice.
For I find that forty poppy-seeds, side by side, measure an inch and a
half (English); so that the geometrical foot of sixteen digits would be
two feet English.
_Leyden_, sixteen-sixty. =Vincent Leotaud.= ‘Institutionum
arithmeticarum libri quatuor, in quibus omnia quæ ad numeros simplices,
fractos, radicales, et proportionales pertinent precepta clarissimis
demon strationibus tum arithmeticis tum geometricis illustrata
traduntur.’ _Quarto_ (O).
This is a clear but heavy work, running to 700 pages.
_Herbipoli_ (_Wurtzburg_), sixteen-sixty-one. =Gaspar Schott.= ‘Cursus
Mathematicus.’ _Folio in threes._
It has a shabby arithmetic, which, considering the magnitude of the
course, reminds us of Falstaff’s halfpennyworth of bread. In the
frontispiece, a lion and a bear draw a car surmounted by an armillary
sphere, and having for its wheels (or rather castors) the celestial
and terrestrial globes, along an amphitheatre the pavement of which
is studded with diagrams. These diagrams contain a sly little bit
of freemasonry. Schott, as a good Jesuit, adheres to the Ptolemaic
system, grudgingly, and because his church had condemned the other,
as appears from his words. But in the diagrams on the pavement of his
frontispiece, he draws the Copernican system, and no other. Dr. Peacock
(p. 408) refers to the ‘Arithmetica Practica’ of Schott [_Herb._
sixteen-sixty-two——].
_London_, sixteen-sixty-four. =James Hodder.= ‘Hodder’s Arithmetick:
or, That necessary Art made most easie.’ The third edition, much
enlarged. _Duodecimo_ (O).
Had this work given the new mode of division, it must have stood
in the place of Cocker. The ninth edition was published in
sixteen-seventy-two. [The first edition is said to have had the date
1661.] In Davis’s sale-catalogue (1686), there are marked as by Hodder,
a Vulgar Arithmetic, 1681, and a Decimal Arithmetic, 1671. There are no
decimal fractions in the book before us.
_Deventer_, sixteen-sixty-seven. =Joachim Camerarius.= ‘Explicatio
in duos libros Nicomachi Geraseni ... et notæ =Samuelis Tennulii= in
Arithmeticam Jamblichi....’ _Quarto._ Also, (one publication, though
different in place and date.)
_Arnhem_, sixteen-sixty-eight. =Jamblichus.= ‘In Nicomachi Geraseni
Arithmeticam introductionem et de Fato.’ _Quarto_ (Gr. Lat.).
These comments of Jamblichus and Camerarius are the easiest accesses to
a knowledge of the matter of the work of Nicomachus, of whom no Latin
version exists, of which I can find any mention.
_London_, sixteen-sixty-eight. =John Newton.= ‘The Scale of Interest:
Or the Use of Decimal Fractions.’ _Octavo._
This book is further described in ‘Tables,’ _Penny Cyclop. Suppl._: it
was expressly intended for a school-book, though it is a strange one
for the time.
_London_, sixteen-sixty-eight. =William Leybourn.=
A
Platform { } Purchasers
Guide { for } Builders
Mate { } Measurers.
_Octavo_ (O).
The first book is on interest, the second and third on building and
mensuration of work.
_Toulouse_, sixteen-seventy. =Diophantus.= ‘Arithmeticorum libri sex.’
_Folio in twos._
This is perhaps the best edition of Diophantus. It has the commentaries
of Bachet and of Fermat. The character of the work of Diophantus on
the properties of numbers, treated algebraically, is too well known to
need description. I do not consider him as belonging to the genuine
Greek school. His date is always spoken of as very uncertain, though
the preponderance of opinion seems to place him in the second century.
But I think I have given sufficient reason for supposing him to have
written as late as the beginning of the seventh century. (See Dr.
Smith’s Dictionary of Biography, article _Hypsicles_.)
Diophantus is usually placed in the second century, because Suidas
says that the celebrated Hypatia wrote a comment on the astronomical
table of _Diophantus_. This was a common name: Fabricius has collected
upwards of twenty writers or philosophers who bore it; and the
arithmetician of whom I am speaking shews no appearance[15] of having
attended to astronomy. Diophantus mentions Hypsicles, a mathematician:
now (unless this mention make a second) there is only one Hypsicles in
Greek literature, namely, the author of the two last books of Euclid’s
elements. Of Hypsicles, Suidas says that he was a pupil of “Isidore the
philosopher.” But in another place (and this passage, occurring at the
word _Syrianus_, has been overlooked till now) Suidas quotes Damascius
for his information upon this Isidore. Now it is certain that the
Isidore, whose life John Damascius wrote, must have been a contemporary
of the Emperor Justinian, at the earliest: and his pupil, Hypsicles,
can hardly have written before the middle of the sixth century.
Diophantus, who makes mention of Hypsicles, may have flourished
towards the end of the sixth, or the beginning of the seventh century.
The silence of Proclus, Pappus, and Theon, as to both Hypsicles and
Diophantus, is a strong presumption of the latter two having come after
the former three. When Dr. Peacock (p. 397) says that Theon was well
acquainted with the writings of Diophantus, I suppose it is upon the
presumption that the father and teacher could not have been ignorant of
the author on whom the daughter and pupil was said to have published
a commentary. But, this apart, I cannot imagine any reason to suppose
that Theon had ever seen the work of Diophantus. P. 404.
[15] For this reason Fabricius wants to alter the words “a commentary
upon of Diophantus the astronomical table,” into “a commentary upon
Diophantus and an astronomical table.”
_London_, sixteen-seventy-three. =Sam. Morland.= ‘The Description and
Use of two Arithmetick Instruments, together With a Short Treatise
explaining and Demonstrating the Ordinary Operations of Arithmetick,
As likewise A Perpetual Almanac And several Useful Tables.’ _Octavo_
(duodecimo size).
There is also the following second title-page (first in order of time)
which is all that some copies have: _London_, sixteen-seventy-two.
‘A New, and most useful Instrument for Addition, and Subtraction of
Pounds, Shillings, Pence, and Farthings ... By S. Morland.’ A very
miscellaneous book, embodying computation (N), some of Euclid, tables
for Easter, description of the calculating machine, &c.
_London_, sixteen-seventy-three. =John Kersey.= ‘Mʳ Wingate’s
Arithmetick, containing a plain and familiar method,’ &c. _Octavo._
This is called the sixth edition of Wingate (N), one of the very best
of the old writers. This edition has an appendix in augmentation of the
commercial part, by Kersey, who says the first edition (which I have
never seen) was published about sixteen-twenty-nine.
_Amsterdam_, sixteen-seventy-three. =William Bartjen.= ‘Verniende
cyfferinge....’ _Octavo_ (small) (O).
A book of a decidedly commercial character, and with good force of
examples.
_London_, sixteen-seventy-four. =John Mayne.= ‘Socius Mercatoris: or
the merchant’s companion, in three parts.’ _Octavo_ (N).
The three parts are on arithmetic, vulgar and decimal, on interest,
and on solids and cask-gauging: the second and third have two separate
title-pages, dated sixteen-seventy-three. There is a little algebra.
_London_, sixteen-seventy-four. =John Collins.= ‘An Introduction to
Merchants-Accompts.’ _Folio in twos._
Book-keeping by double entry. Collins says that his first edition was
in 1652, and his second (which was nearly all destroyed by the fire) in
1664-5.
_London_, sixteen-seventy-six. =A. Forbes.= ‘The Whole Body of
Arithmetick Made Easie.’ _Duodecimo_ (O).
Not a very easy system.
_London_, sixteen-seventy-seven. =Michael Dary.= ‘Interest epitomised,
both compound and simple ... whereunto is added, a Short Appendix
For the Solution of Adfected Equations in Numbers by Approachment:
Performed by Logarithms.’ _Quarto_ (octavo size) (N).
Of Dary, who was a gunner, then a tobacco-cutter, then a teacher, &c.,
and who was the correspondent of Collins, J. Gregory, Newton, &c., a
good deal is to be found in the Macclesfield correspondence. This tract
contains the distinct announcement and use of a principle which is now
well known, and of which Newton’s (or Briggs’s) method of approximation
is only a particular case. If an equation be reduced to the form _x_ =
φ_x_, φ_x_ being a known function of _x_, successive formation of φ_a_
= _b_, φ_b_ = _c_, φ_c_ = _d_, &c., gives an approximation to a root
of the equation, whenever the results do not increase without limit.
Dary saw and applied this, expediting the process occasionally by using
the mean of the two last results, instead of the last result alone,
to produce the next result. The book itself is quite out of notice;
and neither the ingenuity nor the importance of the principle was
appreciated by Dary’s contemporaries.
_London_, sixteen-seventy-eight. =William Leybourn.= ‘Arithmetick
Vulgar, Decimal, Instrumental, Algebraical.’ _Octavo._ Fourth edition.
(The four parts have separate titles and paging.) (ON).
I have not met with any earlier edition. This work seems to have been
substantially inserted in the course of mathematics by Leybourn,
presently mentioned.
_London_, sixteen-eighty. =Thomas Lawson.= ‘A Mite into the Treasury,
being a Word to Artists, Especially to Heptatechnists.’ _Quarto._
This book I have several times met with in lists, as one of arithmetic,
so that I here insert a true account of it. It is written against
human learning, and arithmetic among the rest, as being of no use to
“open the seals of the book,” or to interpret the Bible. It reminds
me much, difference of age and manners apart, of some books I have
lately seen, of arithmetical examples, in which no name is introduced
except from the Bible: so that Judas Iscariot and the devil may be
mentioned, but Socrates or Milton may not. Of arithmetic T. Lawson
says: “Herein any member of Italian Babilon with his Mass-Book, Mass
for the Dead, Fabulous Legend: any Mahumetan with his dreggy Alcoran;
any Flint-hearted Jew with his Talmud, a mingle-mangle of Jewish Divine
and Humane matters; any Dead, Dry, Unfruitful Formalist, may grow
Profound, Exquisit, Nimble, yea, and though involved in the intricate
windings of Degeneration, out of the Royal state of Regeneration and
Heavenly Transformation may apprehend the Feates, Termes and Parts of
this Natural Art, as Digits, Articles mixt Numbers, Cyphers, Terniries,
Golden Rule direct, Golden rule reverse, a Cube, Phythagoras’s Table,
Algorism, &c. yet be Strangers to the Divine Exercise which leads to
Christ, the Lion of the Tribe of Judah, who alone opens the Seals of
the Book.” Attention is then directed to the number of the Beast.
_London_, sixteen-eighty-one. =Jonas Moore.= ‘A new Systeme of the
Mathematicks.’ Two volumes. _Quarto._
The first volume contains arithmetic, much abridged from Moore’s larger
work above noted. This course was written for Christ’s Hospital.
_London_, sixteen-eighty-two. =Gilbert Clark.= ‘Oughtredus explicatus,
sive Commentarius in Ejus Clavem Mathematicam.’ _Octavo._
This commentary is now of no use. A wise commentator will endeavour
to fill his book as much as he can with such little matters as will
be historically interesting a couple of centuries after his own
time; and this will in many cases cause him to outlive his author.
I suppose this commentator is the Gilb. _Clerke_ who published, in
sixteen-eighty-seven, on what he called the _Spot-dial_.
_London_, sixteen-eighty-three. =Peter Galtruchius.= ‘Mathematicæ
totius ... Clara, Brevis et accurata Institutio.’ _Octavo._
There is a short treatise on arithmetic (O), and logarithms are
mentioned as a recent invention, without any examples of their use.
_London_, sixteen-eighty-four. =Rich. Dafforne.= ‘The Merchant’s
Mirrour.’ _Folio in twos._
A book on book-keeping, translated from the Dutch.
_London_, sixteen-eighty-four. =Abr. Liset.= ‘Amphithalami or the
Accountants closet.’ _Folio._
Book-keeping by double entry. This and the two last books on the
subject (Dafforne and Collins) I have bound together with the _Lex
Mercatoria_ of Gerard Malynes and other books. Mr. M’Culloch has
precisely the same collection, and gives them all as one publication
(_Liter. Polit. Econ._ p. 129-30). But they have different printers,
different dates, and different signatures: most being in twos, and the
rest simple folios; and from Collins’s preface it seems that his book
was certainly a separate publication. I merely mention this (having
seen the like in other cases) to advert to the probability of it not
having been unusual for booksellers to bind parcels of books on the
same subject together for sale.
_London_, sixteen-eighty-four. =Wm. Leybourn.= ‘The line of Proportion
or [of?] Numbers, Commonly called Gunter’s Line made easie,’ &c.
_Duodecimo._
On the sliding rule, which invention is attributed to Seth Partridge.
But he says that Milbourn (see Sherburne’s Manilius, or _Penny
Cycl._ ‘Horrocks’) ‘disposed it in a Serpentine or Spiral Line;’ the
logarithmic spiral, of course (_Penn. Cycl._ ‘Slide Rule’).
_London_, sixteen-eighty-four. =Thos. Baker.= ‘The Geometrical Key: or
the Gate of Equations unlocked:’ &c. _Quarto._
Polyglott: Latin and English. An attempt to do without arithmetic
in the solution of equations, by means of construction. See the
correspondence of Baker and Collins in the Macclesfield correspondence,
beginning of vol. ii.
_London_, sixteen-eighty-five. =Seth Partridge.= ‘The description and
use of an instrument called the double scale of proportion.’ _Octavo._
On the sliding rule.
_London_, sixteen-eighty-five. =John Collins.= ‘The doctrine of decimal
arithmetick, simple interest, as also of compound interest ... made
publick by T. D.’ _Octavo_ (small).
A clear and short work, posthumous.
_Naples_, sixteen-eighty-seven. =Ægid. Franc. de Gottignies.=
‘Logistica Universalis.’ _Folio in twos._
The author is the man mentioned by Montucla (ii. 643) as having claimed
some of the discoveries of Cassini. He was a Jesuit, and was professor
at Rome. This work is the last of several, of which it seems to contain
the substance, amplified (Heilbronner, _in verb_.). I have never met
with this book till very lately; and, though I have had no time as yet
to examine it fully, I have been much surprised to find it unnoticed
by mathematical historians. In his views of algebra, the author seems
much before his time. His recognition of its existence at that time as
an art only; of its containing principles and definitions which are
not arithmetic, but perfectly distinct; its definition as a science in
which subtraction is universalised by express convention; the avowed
enlargement of meaning of + and - &c. &c.—shew that the author had
got more than a glimpse of what was coming. The book is a folio of
400 pages, with much prolixity of expression, but vigorous efforts of
thought.
_London_, sixteen-eighty-seven. =William Hunt.= ‘The Gauger’s Magazine.’
A treatise on decimal fractions and mensuration, once well known. There
is a table of squares in it up to that of 10,000.
_London_, sixteen-eighty-seven. =D. Abercromby.= ‘Academia
Scientiarum..... Being a Short & Easie Introduction to the Knowledge Of
the Liberal Arts and Sciences.’ _Octavo_ (small).
A real smatterer’s book, published contemporaneously with the
_Principia_ of Newton. It gives names and general notions, with the
names of a few celebrated authors; just enough to set up a man about
town. There is another book, which may go with this, out of its place,
being the earliest I have met with of the kind.
_London_, sixteen-forty-eight. =Sir Balthazar Gerbier.= ‘The
interpreter of the Academie for forrain languages, and all noble
sciences, and exercises.’ _Quarto._
A polyglott, English and French, announcing the intention of setting
up a College in London. It gives general ideas on the sciences in
dialogue; and that which relates to arithmetic passes between the tutor
and a page of the court, who begins thus: “If you would favour me now
with some particular discourses on all the necessarie sciences (which
would make me capable of the Theoricall part), I should esteeme myselfe
very much obliged unto you for that it would make me passe for a
gallant wit in good companies of men that love to discourse pertinently
of things.” With this view, my predecessor (in intention) gives two
pages and a half of arithmetical terms, and then the young gentleman
says: “Ther’s enough for my Theory.” This book was dated from Paris the
day before the funeral of Charles I. of England: but it was dedicated
to the Duke of York, with a hope that his turn for literature might
increase the comfort of his great parents.
_London_, sixteen-eighty-eight. =Jonas Moore.= ‘Moore’s Arithmetick: in
Four Books.’ The third edition. _Octavo._
A complete book of Arithmetic (N), with book III. on Logarithms, and
book IV. on algebra, both dated sixteen-eighty-seven. This book was
edited by John Hawkins, Cocker’s editor, who has put his name to the
last three parts. See more of this presently, under _Cocker_.
_London_, sixteen-eighty-eight. =Peter Halliman.= ‘The Square and Cube
Root compleated and made easie.’ _Octavo_ (second edition) (O).
That this book should have reached a second edition shews how great
an object it was to save a little calculation. The author simply
approximates to the fractional part of the root by first interpolation.
Thus his general formula would be:
____________ _b_
ⁿ√(_a_ⁿ + _b_) = _a_ + —————————————————
(_a_ + 1)ⁿ - _a_ⁿ
to help which he has a small table of squares and cubes, with first
differences. But he evidently seems to think he has got the _exact_
square and cube roots: and he says:
Now Logarithms lowre your sail,
And Algebra give place,
For here is found, that ne’er doth fail,
A nearer way, to your disgrace.
_London_, sixteen-ninety. =Henry Coggeshall.= ‘The Art of Practical
Measuring Easily Performed, By a Two Foot Rule Which slides to a Foot,’
&c. _Duodecimo in threes._
On the sliding rule and the use of logarithms.
_London_, sixteen-ninety. =William Leybourn.= ‘Cursus Mathematicus ...
in nine books.’ _Folio in twos and fours._
The arithmetic (ON) Vulgar, Decimal, Instrumental, and Algebraical,
is stated to be the first book: but algebra is afterwards promoted to
follow geometry. Leybourn gives both methods of division, but thinks
the old one not inferior to the new.
_Leyden_, sixteen-ninety. =Claude Fr. Milliet Dechales.= ‘Cursus seu
Mundus Mathematicus.’ _Folio in twos._ Four volumes.
The first volume contains a course of Arithmetic (O), on which I may
remark, as just now in the case of Schott, that it is very scanty. It
is, perhaps, only in a collected course of Mathematics, that we see how
difficult a thing computation must have been considered. And I begin to
see perfectly, by instance after instance, that it was not the Greeks
only who fled to geometry to escape from arithmetic.
_Amsterdam_, sixteen-ninety-two. =Bernard Lamy.= ‘Elemens des
Mathematiques,’ &c. _Duodecimo._ Third edition (O).
The arithmetic is very poor: p. 410, a friend sends him the mode of
using the _carriage_ in subtraction; he having previously _borrowed_
from the upper line: he prints this as a novelty.
_London_, sixteen-ninety-three. =John Wing.= ‘Heptarchia Mathematica
... Arithmatick....’ _Octavo_ (N).
A plain book, containing, among other things, arithmetic, and a larger
collection than usual of tables useful in mensuration. It has the
licenser’s permission dated after the Revolution (Sept. 14, 1689).
_Paris_, sixteen-ninety-three. =Nicolas Frenicle.= In the ‘Divers
Ouvrages de Mathématique et de Physique,’ _folio_, are contained the
following tracts of Frenicle, namely, ‘Methode pour trouver la solution
des Problemes par les exclusions—Abregé des combinaisons—Des Quarrez
magiques—Table generale des Quarrez magiques de quatre de costé.’ And
in another collection of Frenicle’s tracts only, _quarto_, no date
nor place, with only fly title-pages (pp. 374, register A-Yy), there
are the preceding tracts and also ‘Traité des Triangles Rectangles en
nombres,’ in two parts, [of which last there is said to be an edition
_Paris_, sixteen-seventy-six, _octavo_, which may be the one just
mentioned, for aught I know].
Frenicle (who died in 1675) was one of those _useless_ men, as many
appliers of mathematics have called them, who work subjects into
shape before the time is come for applying them. In this point we
have degenerated from our ancestors in the soundness of our views
as to what knowledge is, and how it comes. For instance, at the end
of the seventeenth century, Picard, an eminently useful (as well as
_useful_) applier of mathematics, who never speculated on any subject,
was the careful preserver of Frenicle’s manuscript combinations, and
exclusions, and magic squares. A century after, Condorcet, a useful
(but _useless_) speculator, who never handled any so-called _practical_
subject, finds it necessary, in his _éloge_ of Frenicle, to apologise
for the _useless_ character of his writings, though the doctrine
of probabilities,—our power over which depends on the knowledge of
combinations, &c., in which Frenicle was a successful workman—was,
when the _éloge_ was written, almost as far advanced as the theory
of probable errors and the method of least squares, on which every
observer who knows his business now relies, and to which astronomy in
particular is much indebted for its modern accuracy. Condorcet had
caught a slang which was current in his day, and has been very popular
among us. The time is coming when really learned men will again be
ashamed of not seeing the value of all the uses of mind: when nothing
but thoughtlessness or impudence, mercurial brain or brazen forehead,
will aver that no knowledge is practical, except that which ends in the
use of material instruments.
_Paris_, sixteen-ninety-four. =Jean Prestet.= ‘Nouveaux Elemens de
Mathematiques.’ _Quarto_ (two volumes, third edition) (ON).
This work is a complete body of arithmetic and algebra, treated
with depth and clearness, and embracing a great extent of subject.
The author is much more impressed with the necessity of strict and
sustained reasoning in algebra than most of his contemporaries; and
in particular, his demonstrations of the processes of arithmetic are
ample; a very uncommon thing in his day. Montucla, who does not appear
to have known this third edition (for he mentions only the two former
ones as of sixteen-seventy-five and eighty-nine), dismisses it with an
opinion that it is _ouvrage très estimable_. He speaks of many inferior
works with much higher praise.
_London_, sixteen-ninety-four. =Wm. Leybourn.= ‘Pleasure with Profit:
Consisting of Recreations of Divers Kinds, viz. Numerical, &c.’ _Folio
in twos._
The recreations on arithmetic contain a great many of those which are
still common. But some have gone out of vogue. I do not remember ever
having had the pleasure or profit of the following:—To put together
five odd numbers to make 20. Answered as follows:—Three nines turned
upside down, and two units. Honest Leybourn thinks this answer is a
_Falacy_; in which I differ from him: I think the question more than
answered, viz. in _very_ odd numbers.
_London_ (no date, but about sixteen-ninety-five). =Venterus Mandey.=
‘Synopsis Mathematica Universalis: or the Universal Mathematical
Synopsis of John James Heinlin, Prelate of Bebenhusan.’ _Octavo._
Translated from the third edition, Latin, _Tubingen_, sixteen-ninety.
Heinlin’s system of arithmetic (O) is somewhat theological. ‘In Unity
all numbers are virtually, although they are infinite. So all things
are in God, in him they live and move. Seven is a Sacred Number,
chiefly used in Holy Scripture. It seems to have its Original from
the Inscrutable Unity of Divine Essence, and Sextuple in respect of
the Divine Persons among themselves: whence also the Description of a
Septangled form is impossible, and cannot be known by Human Minds.’
_London_, sixteen-ninety-six. =Samuel Jeake.= ‘Λογιστικηλογία, or
Arithmetick Surveighed and Reviewed.’ _Folio_ (ON).
To see the size and weight of this book, one would have thought
arithmetic had been a branch of controversial divinity. But it is now
very valuable, from the variety of the information which it contains,
particularly on weights, measures, and coins; and from the goodness of
the index. There is a good deal of algebra in it, many quaint names,
and stories of the same kind. Thus, in stating the famous story of the
Delian oracle telling the inquirers to double a cube, in order to rid
themselves of the plague, it is stated that the meaning was, that the
best method to deliver realms from such contagion, was to abate of
their voluptuousness, and apply themselves to literature. But for all
this, those who know the value of a large book with a good index, will
pick this one up when they can. P. 442.
_Paris_, sixteen-ninety-seven. =De Lagny.= ‘Nouveaux Elemens
d’Arithmetique et d’Algebre.’ _Octavo_ (duodecimo size).
He has the new method of division, but is obliged to write down the
divisor afresh every time he wants to use it. As an instance of
the effect of change of times and methods, the following trivial
circumstance may be worth repeating. De Lagny says that it would take
an ordinary computer a month to find the integers in the cube root of
696536483318640035073641037. To shew what would be thought of such an
assertion now, I had put down the work for all the integers and a few
decimals, for insertion in the _Companion to the Almanac_. When the
time came for making up the manuscript, the slip of paper on which the
above was written found itself (as the French say, which was more than
I could do) in a heap of unarranged papers: and it was better worth my
while to repeat De Lagny’s month’s work, than to sort the papers and
pick it out.
_Oxford_, sixteen-ninety-eight. =E. Wells.= ‘Elementa arithmeticæ
numerosæ et speciosæ.’ _Quarto_ (octavo size) (N).
This is an elegant work, in which arithmetic and algebra are exhibited
to the university student in a simple form, accompanied by the
historical learning which such a student ought to have. The works on
arithmetic were taking too commercial a turn to be all-sufficient for
the purposes of liberal education: and Wells, without by any means
rejecting commercial questions, writes in order that the student may
not of necessity be driven to the works in which, as he says, _exempla
non aliunde petuntur quam a Butyro et Caseo, Zingibere et Pipere,
aliisque consimilibus_.
_London_, no date (but printed for W. and J. Marshal). =Andrew
Tacquet.= ‘The Elements of Arithmetick In Three Books, The Seventh,
Eighth and Ninth of Euclid: With the Practical Arithmetic In Two
Books.... Translated into English by an Eminent Hand.’ _Octavo._
Dr. Peacock mentions Tacquet as a late instance of the old method
of division. Both methods are given in this translation, “whereof
the first,” meaning the Italian method, “is least used but is the
best; the other most used, but the difficultest.” [According to Dr.
Peacock, the original is “Arithmeticæ Theoria et Praxis,” _Antwerp_,
sixteen-fifty-six ——.] I have described an edition of Tacquet elsewhere
(see the Index).
_London_, seventeen-hundred. =Edward Cocker.= ‘Cocker’s Arithmetick:
Being A plain and familiar Method, suitable to the meanest Capacity for
the full understanding of that Incomparable Art, as it is now taught
by the ablest SchoolMasters in City and Country. Composed By Edward
Cocker, late Practitioner in the Arts of Writing Arithmetick, and
Engraving. Being that so long since promised to the World. Perused and
Published by John Hawkins Writing-Master near St. Georges Church in
Southwark, by the Authors correct Copy, and commended to the World by
many eminent Mathematicians and Writing-Masters in and near London.’
_Duodecimo._
This is called “The _twentieth_ edition carefully corrected, with
additions:” but I find that it had become usual, when any one
edition was augmented, to make the assertion with reference to all
succeeding editions. The first edition of this famous work was in
sixteen-seventy-seven (I have seen one copy, which appeared in a sale
a few years ago), the fourth in sixteen-eighty-two, the twentieth
as above, the thirty-third in seventeen-fifteen, the thirty-fifth
in seventeen-eighteen, the thirty-seventh in seventeen-twenty. The
earliest edition I ever possessed is one of sixteen-eighty-five;
what edition it is, is not stated. But there is confusion among
the title-pages. For though the above is unmistakeably marked
seventeen-hundred, and twentieth edition, I have also compared
together two editions, both of seventeen-_twelve_, one called the
_fourteenth_, and the other the _thirtieth_: all three by one printer,
Eben. Tracy, at the three Bibles on London Bridge. How long it went
on in England I do not know: but there was an edition at Edinburgh in
seventeen-sixty-five, and another at Glasgow in seventeen-seventy-one,
both edited by John Mair, whose preface is dated seventeen-fifty-one.
Some account of Cocker is given in the _Penny Cyclopædia_ (art.
‘Cocker’). Not much more is known of him than this, that he was a
skilful writing-master and engraver. At the beginning of the Arithmetic
is a recommendation signed “John Collens” (no doubt the famous John
Collins, or intended to pass for him) certifying that the deceased
author was “knowing and studious in the Mysteries of Numbers and
Algebra, of which he had some choice Manuscripts and a great collection
of Printed Authors in several Languages.” Collins doubts not “but
he hath writ his Arithmetick suitable to his own Preface and worthy
acceptation:” which means that Collins or Collens had only seen the
preface of the forthcoming work at most. Then follows the attestation
of fifteen teachers to the merits of the work. All this looks odd;
because, according to the editor, the book was that which had been
long promised to the world by a celebrated writer. All attestation was
unnecessary; and the certificate of a celebrated name, wrong spelt, to
the effect that he had no doubt the work, then printed, would be good,
may now excite a little curiosity, if not suspicion.
I am perfectly satisfied that Cocker’s Arithmetic is a forgery of
Hawkins, with some assistance, it may be, from Cocker’s papers: that is
to say, there has certainly been more or less of forgery, without any
evidence being left as to whether it was more or less. I could easily
believe that all was forged; and my reasons are as follows:
In both the editions of Hodder which I have seen (1664 and 1672) is
the following advertisement: “There is newly printed Mr. Cocker’s book
called the Tutor to Writing and Arithmetic.” It appears then that
during his lifetime he had published a book on Arithmetic, which I
suspect to have been what would now be called an arithmetical copybook,
with engraved questions and space left for the work. But neither the
posthumous work, nor its preface signed by Cocker himself, make the
least allusion either to the previous work, or to the promise of
another. On the contrary, the language of Cocker’s own preface implies
that it is the first work he has published on arithmetic, and agrees
with many other prefaces (which are usually written last) in speaking
of the work _as already published_. To establish these and other
contradictions, I first give Hawkins’s account in _his_ preface (with
my own Italics). “I Having the Happiness of an Intimate Acquaintance
with Mr. Cocker in his Life-time often solicited him to remember his
Promise to the World, of Publishing his Arithmetick, _but_ (for Reasons
best known to himself) _he refused it_; and (after his Death) the Copy
falling _accidentally_ into my hands, I thought it not convenient to
smother a work of so considerable a moment,” &c. But Cocker himself
writes, or is made to write, as follows: “By the sacred Influence of
Divine Providence, I have been Instrumental to the benefit of many;
by vertue of those useful Arts, Writing and Engraving: and do _now_
with the same _wonted alacrity_ cast this my Arithmetical Mite into
the Publick Treasury.... For you the pretended Numerists[16] ... was
this Book composed and published....” This is an odd preface for a book
which the author never meant to publish, and refused to publish, though
pressed to do so. Of course it is possible that though he wrote with an
intention of publishing, he afterwards changed his mind. This is one
explanation; that Hawkins forged clumsily is another: and which is the
most probable must be gathered, from a review of all the circumstances.
[16] _Numerist_ is a word which Hawkins uses in his own professed
writings: and it was by no means a common word.
Next, at the end of the work, Hawkins gives a hint of a book on
decimals which would be forthcoming in time. Accordingly, we have:
_London_, sixteen-eighty-five. ‘Cocker’s Decimal Arithmetick ...
Whereunto is added his Artificial Arithmetick ... also his Algebraical
Arithmetick ... according to the Method used by Mr. John Kersey in his
Incomparable Treatise of Algebra. Composed by Edw. Cocker ... Perused,
Corrected, and Published by John Hawkins....’ _Octavo._
This book came to its third edition in seventeen-two. The artificial
(or logarithmic) arithmetic, and the algebra, have separate
title-pages, dated sixteen-eighty-four. Cocker gives no preface here,
but Hawkins does, stating that he had in the preceding work given “an
account of the speedy publication of his Decimal, Logarithmical, and
Algebraical Arithmetick.” He has here mended his hand: for, except the
words “such Questions being more applicable to Decimals are omitted
till we come to acquaint the Learner therewith,” the first treatise
does not give a hint of the second. Again, Kersey’s Algebra, on which
part of Cocker’s second work is founded, was published in 1673, and
the latter had been dead some time before the manuscript of the first
work of 1677 “accidentally” fell into Hawkins’s hands. This is again
singular, on any supposition but that of the forgery. Moreover, at
the end of the preface, Hawkins writes a letter to his friend John
Perkes, _in cipher_ (_Penny Cycl._ ‘Cocker’) in which he says, “If you
please to bestow some of your spare houres in perusing the following
tereatise, you will then be the better able to judg how I have spent
mine.” This looks like a confession of authorship. And in 1704, as
presently noted, appeared _Cocker’s_ English Dictionary by _John
Hawkins_, who would perhaps, had he lived, have found Cocker’s Complete
Dancing-Master and Cookery-book among the papers of the deceased.
The famous book itself I take to be a compilation or close imitation
in all its parts. Even the Frontispiece, &c. is fashioned upon
Hodder. Thus, Hodder begins with his own portrait, and verses of
exaggerated praise under it; and so does Cocker. The former begins
his title with _Hodder’s Arithmetick_, the latter with _Cocker’s
Arithmetic_. The former speaks of that _necessary art_, the latter
of that _incomparable Art_. The former has it ‘explained in a way
_familiar_ to the _capacity_,’ the latter a ‘_familiar_ method suitable
to the meanest _capacity_:’ the two words being then by no means so
common in the senses put upon them as they are now. Turning over the
title-page we find that each of them ‘humbly dedicateth this Manual
(manuel in Hodder) of Arithmetick,’ the first to a ‘most worthily
honoured friend,’ the second to ‘much honoured friends;’ the first ‘in
token of true gratitude for _unmerited_ kindnesses,’ the second ‘as
an acknowledgment of _unmerited_ favours.’ There are too many small
coincidences here. And it must be remembered that every resemblance to
a work so well known as Hodder’s (it is one of the few English works of
the century which have found their way into Heilbronner’s list) would
help the sale.
From all these circumstances I was tolerably sure that there was
no dependence to be placed on the famous Cocker being any body but
Hawkins, so far as this book is concerned: though I must say I hardly
expected to find such confirmation as would arise from catching Hawkins
at a similar trick in another quarter. But on looking at the work above
described as the third edition of Jonas Moore’s Arithmetick, my eye
was caught by the following sentence: “You may likewise prove Division
by Division, as I have shewed at large in the 7. chap. Page 100, 101,
102. of Mr. Cocker’s Arithmetic, printed in the year 1685.” Now Jonas
Moore was dead before 1685; and moreover, could have shewn nothing in
Cocker’s Arithmetic: and on looking farther to see who it is that thus
speaks in the first person, I find the name of John Hawkins to the
second part of the work, as editor. And on looking farther, I find that
a good deal from Moore’s _own_ editions has been introduced _verbatim_
into Cocker. For instance, this sentence is in both: “Notation teacheth
how to describe any number by certain notes and [or] characters, and
to declare the value thereof being so described.” And throughout the
book, paragraphs are frequently introduced from Moore, with alterations
of phrase here and there. So that we have Hawkins arbitrarily altering
and adding, in the first person, to the text of a book which had been
for thirty years before the world under Moore’s name. What are we
to suppose he would do with Cocker’s papers, if indeed he had any?
Moreover, we find in Cocker sentences which had been previously written
by Moore.
To see whether much was gained by Cocker’s Arithmetic, as well as
for the interest of the comparison itself, I will write down the
definitions of addition, subtraction, multiplication, and division,
from Recorde, Wingate, Johnson (see Additions to this work), Moore,
Bridges, Hodder, and Cocker.