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philosophers. Was there ever seen one number of any of the three periodicals mentioned on the table of any library or institution in the empire, not in a university town? Thus a Lancashire inquirer, living in the densest and wealthiest population of the globe, not in a seat of government, who may conceive the reasonable desire to look at the memoir of Mr. Cayley, to which reference has been made, or to obtain a glimpse of what the foremost mathematical investigators are doing, must, before he can attain his object, travel hundreds of miles !

It may, perhaps, not be utterly unworthy of the attention of the next man of science who may undertake to draw up a report of progress for the British Association, to endeavour to estimate the number, without reference to attainments or reputation, of those who pursue or have pursued mathematical knowledge in this country, from a genuine love of it; excluding those who merely undergo the interesting process of cramming for examinations, and distinguishing, on the one hand, those actual cultivators of the science who may be supposed to have derived their impulse, as competitors in any way for honour and reward, from our ancient seats of learning, and from the coun tenance of our Government; and, on the other, those who have`drunk in their inspiration chiefly from the perennial fountain of the Diary. I confess it to be my relief, from a limited observation of graduates and non-graduates, that when the difference between the prizes awarded by the authorities on either side is considered, an incomparably greater share of the glory of kindling and cherishing a pure and lasting love of mathematical science in men as well as boys must be attributed to the immortal Ladies' Diary than to all the universities and colleges of these kingdoms put together, to all our Lyceums, Athenæums, and Philosophical Societies, to all our Imperial Boards of Peace and War.

Mr. Kirkman speaks of the "imperishable Diary," but it has disappeared, with thirty-three other mathematical periodicals which have for longer or shorter periods flourished contemporaneously with it. In the year 1800 there were seven high-class mathematical periodicals in circulation. The Ladies' Diary, the oldest, and nearly always in point of excellence deservedly at the head of them, was the last to give way. It is a somewhat curious circumstance that as the great manufacturing and commercial progress, upon which we pride ourselves so much, commenced and sped its way to its present state of repletion, the support accorded to these journals began gradually to decline. Of the one hundred and seventy contributors to the Diary in 1842 there were only sixty in 1871. About five pounds a year was as much as the great English public could afford to pay for purely scientific periodical literature. Mechanics' institutions, co-operative societies, and their bastard progeny, limited liability companies, have been the death of true

self-culture, and the pursuit of knowledge for its own sake. The grandchildren of Wolfenden, Butterworth, Kay, and Mellor (the botanist) are committee-men of co-operative societies, or directors of limited companies; and the clerical successors of Wildbore, Crakelt, and Lawson busy 'their little brains over price and share lists. Forty years ago Mr. Holyoake wrote and published a useful, beautiful little book Mathematics no Mystery - a popular commentary upon Euclid. If he asked his children, the co-operators, to study such a book to-day they would question his sanity.

In my former paper I stated that no Lancashire man had ever distinguished himself by the creation of any original, high-class, standard work. The work of the Lancashire Mathematicians is to be found exclusively in the Diaries and similar publications, complete copies of which are now very scarce. It is doubtful if there be more than two or three complete sets of the Diaries in the whole country. However, I have made out a list of all the Lancashire contributors to these and other mathematical journals, which I will read over to you, with such remarks upon them and their work, as may occur to me as I go on. I know not if by this means their names can be preserved for any great length of time, but certainly the names, if not the work connected with them, are fast disappearing with the publications in which they were first written. Dr. Hutton, in the preface to a collection of the Diaries published in 1775, says it was his intention to have given biographical notices of the editors and principal contributors up to that time, but found it impossible to collect the necessary information. The same difficulty exists to-day with reference to the contributors since that time; hence in most cases I can only give the name and residence of the correspondent, with a reference to the publication in which his communications appear, and the date of them. In 1817 Professor Leybourn published an edition of the Ladies' Diaries for 113 years, in which he gives the names of all those correspondents whose questions and answers are printed. Out of nine hundred and twenty-eight of a total, I find thirty-three from Lancashire-a fair proportion in number as compared with other counties; but when I come to look at the actual amount of work done it seems miserably deficient. Out of one thousand two hundred and ninety-eight questions proposed,

Lancashire supplied thirty-nine; and of three thousand answers nearly, it gave only ninety-six.. Why, one correspondent, the Rev. Charles Wildbore, of Nottingham, alone proposed thirtyfour and answered one hundred and forty-two questions, all of a high-class character, in the course of forty years, during more than twenty of which he was also editor of the Gentleman's Diary. I ought to mention, however, that the most famous Lancashire mathematicians, Griffith Jones, of Liverpool, Wolfenden, Butterworth, and Kay, did not become contributors to the Ladies' Diary till 1816 and afterwards.

In deference to the genius of two young men who first shed lustre upon the scientific annals of Lancashire, I commence with the names of Horrox and Crabtree. Although not mathematicians in the sense we understand the term, as investigators of abstract truths, they must have had an acquaintance with the Elements of Euclid and the Conics of Apollonius. The cases of Copernicus, Galileo, Kepler, Gassendi, Newton, Dr. Halley, and Laplace; and of Herschel, Adams, and Leverrier, in our own day, prove that a profound acquaintance with Mathematics is essential to the highest. walks in Astronomy; but the fact that these young men, with many others, especially James Ferguson, of Banff, as great a prodigy as Horrox himself, possessed themselves of high attainments in the science, with but a very moderate and sometimes no knowledge of geometry, shows that proficiency in the latter science is not a necessity to the pursuit of the former. The case of Ferguson is so peculiar that I may be excused if I relate it. On one of his lecturing tours he stayed with Dr. Hutton, then a schoolmaster at Newcastle, and one evening the conversation turned upon the division of a grindstone into a given number of ⚫ equal concentric areas. Dr. Hutton, of course, soon resolved the question in figures. Ferguson said little and soon went to bed. The following morning he came down stairs highly delighted, and showing the doctor some rings of cardboard, told him he had solved the question. The astronomer out of his bright, mechanical genius, with a pair of compasses only, had cut out of the board concentric rings of equal areas, and had verified their correctness by weighing them. He confessed to the astonished doctor he never could comprehend a proposition of Euclid in his life.

It is remarkable that although both Horrox and Crabtree felt

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the need of a decimal notation, and that one had been invented, as well as logarithms, long before their time, they never seem to have availed themselves of either in their calculations. Mr. Whatton, in his life of Horrox, published in 1859, says that neither Mathematics nor Physical Science were taught at Cambridge whilst Horrox was there; but in this he must be mistaken, as I find that Henry Briggs, the great improver of logarithms, was Examiner and Lecturer there in the Faculty of Mathematics in 1592. He afterwards went to Gresham College, London; but no doubt he had a successor at Cambridge. In all probability Horrox's friend Foster, afterwards Professor of Astronomy at Gresham College, London, occupied the position of Examiner and Lecturer in Mathematics at Cambridge at the time Horrox was there. Foster went to Cambridge in 1616 and left for London in 1636, four years after Horrox joined the university. Mr. Whatton, in his ardour to make out Horrox a greater prodigy than he really was, also alludes to his want of books, as well as tutorial instruction, in the university, and goes on to say there was at that time no public observatory either in England or France, but forgets to tell his readers that there had been one in existence at Nuremberg one hundred and sixty years, the observations made in which, over seventy years, had been published by Walther in 1544, and which Horrox had, together with the works of Copernicus, Kepler, Tycho Brahe, and many others, as may be seen in a list of his books given at page eleven of Whatton's Memoir. Then Whatton says Horrox was the first to predict the transit of Venus; but that is not so. Kepler, who died in 1630, nine years before the transit, foretold it—certainly making an error of a day, which undoubtedly he would have corrected had he lived a few years longer. He had predicted a transit of Mercury, which Gassendi witnessed in Paris in 1631. But really the reputation of Horrox as a scientific man needed no embellishments of this kind. His memory, with which that of Crabtree is imperishably entwined, has been many times crowned with graceful little chaplets from the pens of many great men. Horrox is the one Lancashire name Hutton gives a place to in his Dictionary. The two men, if not gifted with mathematical genius, possessed the finest powers of observation, and their philosophy was founded upon the unerring induction of facts. No cobwebs

of theology or scholasticism ever interrupted their clear vision. So charming, so fascinating is the picture of these young men at their studies that one never wearies of looking at it. It is highly creditable to the character of the Rev. Mr. Brickell that he has caused a suitable memorial of Horrox to be erected at Hoole. Perhaps a marble tablet in memory of Crabtree will be found upon the walls of the cathedral of Manchester some day.

Very early in the last century (1718) a Mathematical Society was formed in Manchester. In the Chetham Library is a printed copy of two lectures (or rather of one and part of another, for the whole of the second is not there) delivered to this society by "the late ingenious Mr. John Jackson," as the title-page reads, and whom the late Mr. T. T. Wilkinson somewhat fulsomely calls "the father of the Lancashire School of Geometers." There is an account of these lectures in No. 103 of the First Series of Notes and Queries, by Mr. Crossley, the learned president of the Chetham Society. They are remarkable for earnestness and the quaintness with which the excellencies and claims of mathematical studies are set forth. They were printed in 1719. Whether Mr. Jackson wrote anything upon the elements of the science I do not know; his name does not appear in the Ladies' Diary which had then been in circulation fifteen years.

The first Lancashire contributor to the Diary is Mr. John Hampson, of Leigh. He first appears with an answer to the prize question in 1728, and his last contribution is in 1764, in answer to a question in spherical trigonometry. In these thirty-six years he proposes five questions, four of which are Diophantine, and one a geometrical problem. He answers seven questions, four Diophantine, and three geometrical problems. This gentleman was of more than average ability, and shows a considerable mastery over figures for his time.

Mr. Jeremiah Ainsworth, Manchester, one of the cleverest mathematicians Lancashire has produced up to our time, only appears once in Leybourn's edition of the Ladies' Diary, viz., in 1768, with a Diophantine problem of no great difficulty. Mr. Wilkinson, who in a paper on Lancashire Geometers, in the eleventh volume, second series, of the Memoirs of the Manchester Literary and Philosophical Society, 1852, and in the Mechanic's Magazine, Vol. 61, 1854, gives an account of him, says he sent

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