answers to most of the questions in 1768, but none of them were printed. He also says the principal portion of Ainsworth's writings appear in Burrow's Diary, a periodical of great excellence, which flourished from 1776 to 1778, twelve years. His first contribution was to the Mathematical Magazine, 1761, a paper of which only five numbers were published. He would then be eighteen years of age. In 1771, he sent some mathematical papers to the Manchester Journal, in which paper he got into something less than friendly rivalry with Dr. Henry Clarke, of Salford. He proved too clever for the Doctor, who soon left the field to his opponent. Dr. Clarke evidently long felt the bitterness of his defeat, for in 1777 we find him writing to the Rev. Mr. Lawson, who had sent Ainsworth some of his works to be revised, a letter, strongly impeaching the character of Ainsworth as a gentleman, a letter worthy only of a tittle-tattle, scandalmongering old vixen. Our learned member, Mr. John E. Bailey, has printed this letter, at pp. 62-3 of his Memoir of Dr. Clarke, published this year. Ainsworth, who, Mr. Bailey tells us, "had been the main support of Burrow's Diary, died in 1784, and Dr. Clarke took his place. I think it may be easily and reasonably suspected why Clarke had kept aloof till this time. Mr. Ainsworth also contributed some papers to the London Magazine, in the years 1777-82. His solution to question 481 in the Gentleman's Diary, for 1782, is a fine specimen of neatness, directness, and lucidity in geometrical demonstration. The prize question for the same year, an astronomical one of exceedingly great difficulty, was also answered by Mr. Ainsworth. The fact that Mr. Lawson sent his publications to Mr. Ainsworth for revision is proof of the high esteem that great geometer had for Mr. Ainsworth's abilities. With characteristic garrulousness and self-conceit, Mr. Wilkinson is never weary of telling his readers that Mr. Lawson's Theorems were not all his own, as if nobody had found this out besides himself. The fact is, Mr. Lawson never pretended they were original, any more than Euclid did -the Elements. Ainsworth was a complete mathematician in , every respect. He was, as most of you know, the grandfather of the novelist, Mr. William Harrison Ainsworth. It is said he resided a short time in the neighbourhood of Hollinwood, and was the tutor of Wolfenden. It is certain he was most

of his life a schoolmaster, and for a few years before his death had a school in Hanging Ditch or Long Millgate. There is a memoir of the Ainsworth family, by Blanchard, prefixed to the romance of Rookwood, one of Mr. Harrison Ainsworth's works. From this it would seem that Thomas, the son of Jeremiah, was a more remarkable man in his way than either the mathematician or the novelist. He was a Nonconformist minister, and made £60,000 by land speculations. He certainly took considerable thought for this world, however he might regard the next. Mr. Jeremiah Ainsworth was born in 1743 and died in 1784. He got his early education in the Manchester Grammar School.

Dr. Henry Clarke created a lively sensation amongst his own particular species, the race of pedagogues, about a century ago. He was born in Salford, in 1743, and as a schoolmaster here in the years from 1766 to 1790, he spent the most active and useful years of his chequered life. Like most schoolmasters he had a smattering of every kind of knowledge, and professed a great proficiency in all. It is difficult, and I am not called upon to say, in which he was most accomplished. Mr. John E. Bailey, in his biography of him, published in the beginning of this year, with his kindly good nature, ever ready to throw the mantle of protection over human weaknesses, especially those of parsons and schoolmasters, has done more than justice to his multifarious merits. Never was mere schoolmaster so honoured. About the merits of Clarke as a schoolmaster, a mathematician, and an author, there may be differences of opinion; about his biography there can be but one of commendation for the fulness, the care, and great ability with which it has been executed. If I were asked my opinion of the so-called bagatelle, School Candidates, which Mr. Bailey has printed and bound up with his memoir, I should say it is one of the arrantest pieces of inflated doggerel ever printed. Mr. Bailey gives full and particular details of his works upon mathematical subjects, including his contributions to the periodicals of his time, and quotes the opinions of Bishop Horsley and the late Mr. Wilkinson in testimony of his great powers as a mathematician. The bishop, in the heat of partisanship and severe personal controversy with Sir Joseph Banks, calls him an "inventor in Mathematics," and Mr. Wilkinson, with bouncing phraseology, "one of the most distinguished ornaments of the Lancashire School." Judging by

his performances, I should discount these praises very considerably, and at the best would say with the sober and clear-headed Dr. Hutton, Clarke was "an ingenious author" simply. It is evident the work he himself thought the most highly of was his translation of Lorgna on Infinite Series, and of this the opinion of a most competent authority, Mr. John Landen, was, that it was a piece of gross and ignorant impertinence to place it before English mathematicians. His work on the Rationale of Circulating Numbers, a copy of which is in the Chetham Library, is one of but average ability. Dr. Hutton, in his dictionary, gives the sum and substance of it in half-a-dozen lines. Dr. Clarke's contributions to the Ladies' Diary, from 1772 to 1784, are perhaps the cream of his mathematical works. In these thirteen years there are printed six questions and thirteen answers to questions from him. Of the first, two are questions in mensuration, both as I think, but especially the second one, clumsily concocted. I will read it

QUESTION 706, BY MR. HENRY CLARKE:

A gentleman having in his garden a circular grass plot, which is exactly level, and upon the area a tall fir tree, and in the mound of it six oaks at equal distances from each other, was desirous of having a tumulus terreus raised on it, with an obelisk, seventeen yards high, on the top or vertex of it. He agreed with the workmen at id. a solid yard; but now, the work being done, they are at a loss to determine the solidity and value. From the observations they have made it appears that the tumulus is an equilateral hyperboloyd, with its vertex exactly over the centre of the base, and its semi-transverse equal to the height of the obelisk; also that the present foot of the fir is twenty yards below the level of the foot of the obelisk, and the two oaks are equally distant from it; and moreover the gentleman himself remembers that the continual product of the distances of the fir from the six oaks is 16,883,942,000 yards. From hence the workmen hope, by the assistance of the Diarians, to know what is due to them.

Is it possible to imagine anything more stupidly conceived than this? Fancy a gentleman remembering the product of six numbers into 16,883,942,000! The next is a neat little geometrical problem, of which he gives an equally neat solution. The other three questions involve the summation of series, all founded upon forms given by Lorgna, and therefore not original. None of them can be said to be of more than ordinary difficulty in solution. One of them is done by a correspondent in half a dozen lines. His thirteen answers to questions comprise two in common algebra,

two in the summation of series, one in common mensuration, an answer to a curious question concerning the hands of a clock, and the remaining seven are solutions to geometrical problems. None of them, with one exception, marked with the precision and clearness which is the most striking feature in all the best geometers, both ancient and modern; and certainly they discover no traces of that systematic investigation for which Wilkinson gives him credit, as "among the first of those who in their geometrical studies manifested any system at all.”

Skipping over a pretty long list I have here,* of which I can do no more than give the names, residences, the number and character of the questions they proposed and answered, and where they are to be found, I come to three of the six or seven Lancashire mathematicians worthy of the name. I refer to Wolfenden, Butterworth, and Kay, all of them born at or in the neighbourhood of Royton. Humble, lowly, quiet, unambitious men, destitute alike of vanity, pride, and selfishness; great souls, spotless and pure as driven snow, whose ever active and only aspirations were the pursuit and discovery of truth, and commensurate with these aspirations gifted with fine intellects and great reasoning powers, capable of seeing and correlating the sublime truths of the highest and deepest natural philosophy, I must confess myself utterly unable to fully appreciate their great faculties for mathematical inquiries. Mr. Wilkinson has given brief accounts of them in the paper I have referred to before in the eleventh • volume of the Memoirs of the Manchester Literary and Philosophical Society, where he rightly describes them as the best of Lancashire geometers. Wilkinson has also given details of their lives in Fielding's Historical Gleanings in South Lancashire, a book published by Abel Heywood in 1852. With somewhat questionable taste, in my opinion, he tells how sorely Wolfenden and Butterworth were pinched by poverty and want in their old days; how "kind" it was of one of Mr. Wilkinson's friends to give Butterworth £2 for books which were worth £10, with a promise to "lend" the old man one now and then, and this by a person who, whatever reputation he had for learning, was solely indebted to Butterworth for it. Some Manchester gentlemen, whose names

*See list at the end of this Paper.