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