The extensive study of geometry in Lancashire and the northern counties
generally is a fact which has forced itself upon the attention of several
observers; but none of these have attempted to assign any reasons for so
singular an occurrence. Indeed, the origin and progress of the study of any
particular branch of science, notwithstanding their attractive features,
have but rarely engaged the attention of those best qualified for the
undertaking. Fully satisfied with pursuing their ordinary courses of
investigation, they have scarcely ever stopped to inquire _who_ first
started the subject of their contemplations; nor have they evinced much
more assiduity to ascertain the _how_, the _when_, or in _what_ favoured
locality he had his existence: and hence the innumerable misappropriations
of particular discoveries, the unconscious traversing of already exhausted
fields of research, and many of the bickerings which have taken place
amongst the rival claimants for the honour of priority.
Mr. Halliwell's _Letters on the Progress of Science_ sufficiently show that
the study of geometry was almost a nonentity in England previously to the
commencement of the eighteenth century. Before this period Dr. Dee, the
celebrated author of the preliminary discourse to Billingsley's _Euclid_,
had indeed resided at Manchester (1595), but his residence here could
effect little in flavour of geometry, seeing, as is observed by a writer in
the _Penny Cyclopædia_--
"The character of the lectures on Euclid was in those days extremely
different from that of our own time ... the propositions of Euclid
being then taken as so many pegs to hang a speech upon."
Similar remarks evidently apply to Horrocks and Crabtree (1641); for
although _both_ were natives of Lancashire, and the latter a resident in
the vicinity of Manchester, their early death would prevent the exertion of
any considerable influence; nor does it appear that they ever paid any
attention to the study of the ancient geometry. Richard Towneley, Esq., of
Towneley (1671), is known to have been an ardent cultivator of science, but
his residence was principally in London. It may, however, be mentioned to
his honour, _that he was the first to discover what is usually known as
"Marriotte's Law"_ for the expansion of gases. At a later period
(1728-1763), the name of "John Hampson, of Leigh, in Lancashire," appears
as a correspondent to the _Lady's Diary_; but since he mostly confined his
speculations to subjects relating to the Diophantine Analysis, he cannot be
considered as the originator of the revival in that branch of study now
under consideration. Such being the case, we are led to conclude that the
"Oldham Mathematical Society" was really the great promoter of the study of
the ancient geometry in Lancashire; for during the latter half of the last
century, and almost up to the present date, it has numbered amongst its
members several of the most distinguished geometers of modern times. A
cursory glance at some of the mathematical periodicals of that date will
readily furnish the names of Ainsworth, whose elegant productions in pure
geometry adorn the pages of the _Gentleman's_ and _Burrow's Diaries_;
Taylor, the distinguished tutor of Wolfenden; Fletcher, whose
investigations in the _Gentleman's Diary_ and the _Mathematical Companion_
entitle him to the highest praise; Wolfenden, acknowledged by all as one of
the most profound mathematicians of the last century; Hilton, afterwards
the talented editor of that "work of rare merit" the _Liverpool Student_;
and last, though not least, the distinguished Butterworth, whose elegant
and extensive correspondence occupies so conspicuous a place in the
_Student_, the _Mathematical Repository_, the _Companion_, the _Enquirer_,
the _Leeds Correspondent_, and the _York Courant_. Besides these, we find
the names of Mabbot, Wood, Holt (Mancuniensis), Clarke (Salfordoniiensis),
as then resident at Manchester and in constant communication with, if not
actually members of the society; nor can it be doubted from the evidence of
existing documents that the predilection for the study of the ancient
geometry evinced by various members of this Lancashire School, exercised
considerable influence upon the minds of such distinguished proficients as
Cunliffe, Campbell, Lowry, Whitley, and Swale.
Hence it would seem that _many_, and by no means improbable, reasons may be
assigned for "the very remarkable circumstance of the geometrical analysis
of the ancients having been cultivated with eminent success in the northern
counties of England, and particularly in Lancashire." Mr. Harvey, at the
York meeting of the British Association in 1831, eloquently announced "that
when Playfair, in one of his admirable papers in the _Edinburgh Review_,
expressed a fear that the increasing taste for analytical science would at
length drive the {58} ancient geometry from its favoured retreat in the
British Isles; the Professor seemed not to be aware that there existed a
devoted band of men in the north, resolutely bound to the pure and ancient
forms of geometry, who in the midst of the tumult of steam engines,
cultivated it with unyielding ardour, preserving the sacred fire under
circumstances which would seem from their nature most calculated to
extinguish it." Mr. Harvey, however, admitted his inability clearly to
trace the "true cause of this remarkable phenomenon," but at the same time
suggested that "a taste for pure geometry, something like that for
entomology among the weavers of Spitalfields, may have been transmitted
from father to son; but who was the distinguished individual _first_ to
create it, in the peculiar race of men here adverted to, seems not to be
known." However, as "the two great restorers of ancient geometry, Matthew
Stewart and Robert Simson, it may be observed, lived in Scotland," he asks
the important questions:--"Did their proximity encourage the growth of this
spirit? Or were their writings cultivated by some teacher of a village
school, who communicated by a method, which genius of a transcendental
order knows so well how to employ, a taste for these sublime inquiries, so
that at length they gradually worked their way to the anvil and the loom?"
An attentive consideration of these questions in all their bearings has
produced in the mind of the writer a full conviction that we must look to
other sources for the revival of the study of the ancient geometry than
either the writings of Stewart or Simson. It has been well observed by the
most eminent geometer of our own times, Professor Davies--whose signature
of PEN-AND-INK (Vol. ii., p. 8.) affords but a flimsy disguise for his
well-known _propria persona_--that "it was a great mistake for these
authors to have written their principal works in the Latin language, as it
has done more than anything else to prevent their study among the only
geometers of the eighteenth century who were competent to understand and
value them;" and it is no less singular than true, as the same writer
elsewhere observes, "that whilst Dr. Stewart's writings were of a kind
calculated to render them peculiarly attractive to the non-academic school
of English geometers, they remain to this day less generally known than the
writings of any geometer of these kingdoms." The same remarks, in a
slightly qualified form, may be applied to most of the writings of Simson;
for although his edition of Euclid is now the almost universally adopted
text-book of geometry in England, at the time of its first appearance in
1756 it did not differ so much from existing translations as to attract
particular attention by the novelty of its contents. Moreover, at this time
the impulse had already been given and was silently exerting its influence
upon a class of students of whose existence Dr. Simson appears to have been
completely ignorant. In one of his letters to Nourse (_Phil. Mag._, Sept.
1848, p. 204.) he regrets that "the taste for the ancient geometry, or
indeed any geometry, seems to be quite worn out;" but had he instituted an
examination of those contemporary periodicals either wholly or partially
devoted to mathematics, he would have been furnished with ample reasons for
entertaining a different opinion.
We have every reason to believe that the publication of Newton's
_Principia_ had a powerful effect in diffusing a semi-geometrical taste
amongst the academical class of students in this country, and it is equally
certain that this diffusion became much more general, when Motte, in 1729,
published his translation of that admirable work. The nature of the
contents of the _Principia_, however, precluded the possibility of its
being adapted to form the taste of novices in the study of geometry; it
served rather to exhibit the _ne plus ultra_ of the science, and produced
its effect by inducing the student to master the rudimentary treatises
thoroughly, in order to qualify himself for understanding its
demonstrations, rather than by providing a series of models for his
imitation. A powerful inducement to the study of pure geometry was
therefore created by the publication of Motte's translation: ordinary
students had here a desirable object to obtain by its careful cultivation,
which hitherto had not existed, and hence when Professor Simpson, of
Woolwich, published his _Algebra_ and the _Elements of Geometry_ in 1745
and 1747, a select reading public had been formed which hailed these
excellent works as valuable accessions to the then scanty means of study.
Nor must the labours of Simpson's talented associates, Rollinson and
Turner, be forgotten when sketching the progress of this revival. The pages
of the _Ladies' Diary_, the _Mathematician_, and the _Mathematical
Exercises_, of which these gentlemen were severally editors and
contributors, soon began to exhibit a goodly array of geometrical
exercises, whilst their lists of correspondents evince a gradual increase
in numbers and ability. The publication of Stewart's _General Theorems_ and
Simson's edition of _Euclid_, in 1746 and 1756, probably to some extent
assisted the movement; but the most active elements at work were
undoubtedly the mathematical periodicals of the time, aided by such
powerful auxiliaries as Simpson's _Select Exercises_ (1752) and his other
treatises previously mentioned. It may further be observed that up to this
period the mere English reader had few, if any means of obtaining access to
the elegant remains of the ancient geometers. Dr. Halley had indeed given
his restoration of Apollonius's _De Sectione Rationis_ and _Sectione
Spatii_ in 1706. Dr. Simson had also issued his edition of the _Locis
Planis_ in 1749; but unfortunately the very language in which these
valuable works were written, precluded the possibility of {59} these
unlettered students being able to derive any material advantages from their
publication: and hence arises another weighty reason why Simpson's writings
were so eagerly studied, seeing they contained the leading propositions of
some of the most interesting researches of the Alexandrian School.
After the death of Simpson, the Rev. John Lawson, who appears to have
inherited no small portion of the spirit of his predecessors, began to take
the lead in geometrical speculations; and having himself carefully studied
the principal writings of the ancient geometers, now formed the happy
project of unfolding these treasures of antiquity to the general reader, by
presenting him with English translations of most of these valuable remains.
With this view he published a translation of Vieta's restoration of
_Apollonius on Tangencies_, in 1764, and to this, in the second edition of
1771, was added the _Treatise on Spherical Tangencies_, by Fermat, which
has since been reprinted in the _Appendix to the Ladies' Diary_ for 1840.
In 1767 appeared Emerson's _Treatise on Conic Sections_; a work which,
notwithstanding its manifest defects, contributed not a little to aid the
student in his approaches to the higher geometry, but whose publication
would probably have been rendered unnecessary, had Dr. Simson so far
loosened himself from the trammels of the age, as to have written his own
admirable treatise in the English language. The frequency, however, with
which Mr. Emerson's treatise has been quoted, almost up to the present
date, would appear to justify the propriety of including _it_ amongst the
means by which the study of geometry was promoted during the last
generation. The success which attended Mr. Lawson's first experiment
induced him to proceed in his career of usefulness by the publication, in
1772, of the _Treatise on Determinate Section_; to which was appended an
amended restoration of the same work by Mr. William Wales, the well-known
geometer, who attended Captain Cook as astronomer, in one of his earlier
voyages. In 1773 appeared the _Synopsis of Data for the Construction of
Triangles_, which was followed in 1774 by his valuable _Dissertations on
the Geometrical Analysis of the Ancients_; and although the author used an
unjustifiable freedom with the writings of others, Dr. Stewart's more
especially, it is nevertheless a work which probably did more to advance
the study of the ancient geometry than any other separate treatise which
could be named. As these publications became distributed amongst
mathematicians, the _Magazines_, the _Diaries_, and various other
periodicals, began to show the results of the activity which had thus been
created; geometrical questions became much more abundant, and a numerous
list of contributions appeared which afford ample proof that their able
authors had entered deeply into the spirit of the ancient geometry. During
the year 1777 Mr. Lawson issued the first portion of Dr. Simson's
restoration of _Euclid's Porisms_, translated from the _Opera Reliqua_ of
that distinguished geometer; and though the work was not continued,
sufficient had already been done to furnish the generality of students with
a clue to the real nature of this celebrated enigma of antiquity. The last
of these worthy benefactors to the non-academic geometers of the last
century was Mr. Reuben Burrow, who by publishing in 1779 his _Restitution
of Apollonius Pergæus on Inclinations_ gave publicity to a valuable relic
which would otherwise have remained buried in the Latin obscurity of Dr.
Horsley's more elaborate production.
During the greater portion of the time just reviewed, Mr. Jeremiah
Ainsworth was resident in the neighbourhood of Manchester, and so early as
1761 was in correspondence with the editors of the _Mathematical Magazine_.
He subsequently associated with Mr. George Taylor, a gentleman of kindred
habits, then resident in the immediate vicinity, and these worthy veterans
of science, as time wore on, collected around them a goodly array of pupils
and admirers, and hence may truly be said not only to have laid the
foundation of the "Oldham Society," but also to have been the fathers of
the Lancashire school of geometers. Such then was the state of affairs in
the mathematical world at the period of which we are speaking; all the
works just enumerated were attracting the attention of all classes of
students by their novelty or elegance; Dr. Hutton and the Rev. Charles
Wildbore had the management of the _Diaries_, each vieing with the other in
offering inducements for geometrical research; whilst both, in this
respect, for a time, had to contend against the successful competition of
Reuben Burrow, the talented editor of Carnan's _Diary_: correspondents
consequently became numerous and widely extended, each collecting around
him his own select circle of ardent inquirers; and thus it was, to use the
words of Mr. Harvey, and answer the questions proposed, that inquiries
which had hitherto been "locked up in the deep, and to them unapproachable
recesses of Plato, Pappus, Apollonius and Euclid * * porisms and loci,
sections of ratio and of space, inclinations and tangencies,--subjects
confined among the ancients to the very greatest minds, (became) familiar
to men whose condition in life was, to say the least, most unpropitious for
the successful prosecution of such elevated and profound pursuits."
The preceding sketch is respectfully submitted as an attempt to answer the
queries of PEN-AND-INK, so far as Lancashire is concerned. It is not
improbable that other reasons, equally cogent, or perhaps corrective of
several of the preceding, may be advanced by some of your more learned
correspondents, whose experience and means of reference are superior to my
own. Should any such {60} be induced to offer additions or corrections to
what is here attempted, and to extend the inquiry into other localities,
your pages will afford a most desirable medium through which to compare
_notes_ on a very imperfectly understood but most important subject of
inquiry.