the hedge row, whilst Asplenium Lanceolatum seems to form an intermediate link. Are not the differences in the formation of these three plants probably due to the physical differences of the places of their growth ?-those differences so long established that it is now imposible to grow the one from the seed of the other. In the same way, the difference between Asplenium Trichomanes and A. Viride is very slight and may easily be explained by the different character of their ordinary habitats; the one stronger and hardier, growing on rocks and walls in the lower country, where at times it is liable to suffer from drought; the other, generally high up on the fells, only in places where there is plenty of moisture, and where the colour of its frond may have grown delicate and green from the influence of frequent mists. The two Hymenophyllums resemble each other very closely; yet, who has seen the more robust Wilsoni assume the more spreading frond of the Tunbridgense, or the more pointed seed vessel of the former become jagged at its mouth like the latter? The connection between the oak and the beech fern is very near, and the connection between the former and the limestone polypody still nearer; so much so, that we might look for a common origin for these beautiful ferns. Yet no one has So much for the

succeeded in growing one from the other. permanence of species, the characteristics of which have been developed by a long course of growth in certain localities. It may be said there are ferns the form of which we can change and which will produce most curious and beautiful varieties under cultivation. The Hart's Tongue, for instance, will produce a divided and a subdivided and, indeed, a perfectly branched frond; the edge will become frilled and curled, or jagged, or narrow like a strap, or broad, almost circular, in shape. In the same way the Northern Hard Fern, the Lady Fern, the Male Fern, all change almost under our eyes. These changes, however, are evidently owing to accidents of soil and

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aspect. They are rarely persistent and such plants readily go back to the parent type. Cunning gardeners will very likely produce more remarkable varieties than any we now have.

In these few pages I have attempted to put down some observations on the growth and dwelling place of this most graceful and beautiful tribe of plants, and having done so let me apologize for ever having undertaken the task. Very loud are the lamentations of every lover of nature on the wholesale devastations which, year by year, are being made on the rarer specimens of ferns, and many doubts are felt whether any will soon be left by the road side or by the mountain stream. The author of the "Ferns of Derbyshire" grows very indignant against a certain bearded dealer, who brings his packages from Westmoreland to sell or to exchange for the natives of the former county. Lest similar accusations should be brought against me I have veiled most of my localities in the widest terms, so that the fern fancier will not find them without a certain amount of labour and careful observation. That labour amid the finest scenery our island affords is most healthgiving and delightful; those observations cannot but be the source of instruction and enjoyment. I can only wish that whoever undertakes this branch of field botany may be rewarded by as many successful mountain rambles as have fallen to my lot.

AN ACCOUNT OF THE LIFE AND WRITINGS

OF THE LATE HENRY BUCKLEY.

By T. T. Wilkinson, F.R.A.S. &c.

(READ 5TH FEBRUARY, 1863.)

For more than a hundred years the operatives of Lancashire and Yorkshire have been peculiarly distinguished for their love of pure geometry. Most of these were purely self-taught men, and owed their ultimate fame to ability only. Located in groups, which spread over the two counties, and even extended into Cheshire and Northumberland, they pursued their studies at the loom, or in the mine, undeterred by difficulties; their diagrams being either sketched on slates which they hung upon their weaving-frames, or mentally depicted so as to be seen only by the "mind's eye." Yet, whether in darkness or in seclusion, they laboured on, diffusing their influence around, until at length they succeeded in giving a tone even to the studies of our universities. A wandering journeyman hatter (Fletcher) in process of time connected. them with some kindred spirits resident in the metropolis— the journals of the day began to include some of their best speculations and, ultimately, pure geometry reigned supreme in the hands of Wolfenden, Ryley, Butterworth, Whitley, Swale, Hilton, Lawson, Gompertz, Campbell, Smith, Cunliffe, Holt, Rutherford and Kay. These able geometers, as well as many others whose names I have not space to mention, took the ancient Greek forms of Geometry for their models and at times well nigh exhausted the subjects they undertook to investigate. But, in general, it may be said that they

followed no settled plan in their courses of study; and hence their isolated solutions require some previous systematic arrangement before their real value can be perceived. When this has been done the Greek geometry will be found to owe several most important extensions to their skilful handling; and some of our ablest modern writers on these subjects will be seen to have been indebted to the pages of our mathematical periodicals for many of their best examples. On several former occasions* I have directed attention to the labours of some of these able men and endeavoured to systematize a portion of their writings. In what follows I propose to add one more to the number of those worthies, and to offer his example to those self-taught aspiring students who are not yet extinct in the counties of Lancaster and York.

Henry Buckley was born at Turney Bank, in Saddleworth, on the 11th March, 1811. In early youth, he was sent to a dame-school in the village, but the instruction there given did not extend beyond the merest rudiments. On arriving at manhood he was scarcely able to read or write but, having become acquainted with some young men who were studying geometry, he began to feel himself degraded in their company and determined to imitate their example. Elementary reading and writing were soon mastered. Arithmetic and Mensuration followed in their wake, and he then mustered courage sufficient to attack the "Elements of Euclid." By diligence and perseverance he mastered the eight books of "Thomas

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Simpson's Geometry" in a very short period, and then began to apply their principles to the solution of problems and theorems. In this he was assisted by the late Mr. John Butterworth, of Haggate, near Oldham and, in process of

See my paper" on the Lancashire Geometers, and their Writings," volume xi. of the "Manchester Memoirs;" also, "The Life and Writings of Swale;" and "Liverpool Mathematicians," in former volumes of these "Transactions."

time, Mr. Buckley became a worthy disciple of that worthy man and able geometer.

Although Turney Bank is at a considerable distance from Haggate, Mr. Buckley was not deterred either by this or the state of the weather from visiting his tutor several times a week; and on the occasion of one of my last visits to Woodhouse, he pointed out with allowable pride, not only the bleak road towards Haggate, over the Saddleworth moors, but also the quiet nook in one of the fields adjoining where he used to pursue his studies undisturbed by the bustle of the world. It was here that he conceived some of his best processes; and it was here also that he thought out the analyses of many of those difficult problems which occasionally puzzled both his tutor and their common friends, the late Mr. John Whitley, and Professor Davies. As may be expected from his associates, his taste was almost wholly geometrical. Geometry was ever his favourite subject of thought, although his papers contain sufficient to prove that algebra, trigonometry, series and the calculi had not escaped his attention. His highest ambition, however, was to follow in the steps of Mr. Butterworth, and in this he succeeded in an eminent degree. Of his numerous solutions of problems and theorems, now in my possession, the great majority relate to the construction of triangles from given data—to sections of ratio and of space-to inclinations, porisms &c., of which a small portion has been contributed to the "Diaries," the "York Courant" and the "Educational Times." From the numbering of some of his collections it would appear that he intended to form them into separate essays; but the most extended are necessarily very incomplete. In what follows it is proposed to point out the nature of some of these investigations, and to add an occasional specimen of his mode of treating the subjects under consideration. The problems of Inclinations, combined with those of Tangencies, give rise to