Kitabı oku: «Notes and Queries, Number 34, June 22, 1850», sayfa 4

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ON THE ORIGIN AND PROGRESS OF THE STUDY OF GEOMETRY IN LANCASHIRE

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 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 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."