to earn a little money beyond what I had to give my mother for board wages. I had one offer of a situation as clerk in a small office; but sitting at a desk was never very congenial, and the muscular exercise of weaving seemed preferable.

The weeks of my commencing work at the loom saw the Reform agitation at its crisis; being the summer of 1831. I was caught in the fury of the movement, and attended the public demonstrations on the links; going in procession with youths of my own age, or thereby. I could not help contracting Reform sympathies, which, in the shape of liberalism in general politics, stuck to me through life.

I was now thirteen. In another year, or little more, I had taken to scientific studies. Before that, however, I must have trifled away my time with mere diversions, of which, Church music was one. It was long a hobby, little gifted as I was with a musical ear. I used to attend evening classes for singing practice; and, when I was free, I ran after churches where there chanced to be a good precentor-the Methodist Chapel for one.

Another memorable incident was my becoming intimate with two brothers, named Stewart, sons of a blacksmith, whose smithy was an agreeable lounge, especially in winter evenings. The father

was a remarkable man for his station, and had collected a library, partly of theology, like that of my father, but much more wide-ranging, and partly of works of science and even metaphysics,—all which the sons turned to account. The eldest may have been five years my senior. The brothers were already self-taught mathematicians, like myself, and were also well up in elementary Physics and Astronomy. They were no longer at school, being at work with their father, and thus always very accessible; while, on Saturday evenings in particular, I used to spend hours with them in their home. In all my scientific studies, I had them henceforth as companions and prompters. They also introduced me to the Library of the Mechanics' Institution, then in existence about ten years, and at a somewhat low ebb after a flourishing commencement. The books of the library were almost exclusively scientific, and were, therefore, very helpful, as I cared only for science. The Stewarts joined me in endeavouring to revive the Institution in various ways; and we ultimately all became members of the Managing Committee,-I being for a long time the Secretary.

Reverting to my scientific studies, I must have begun with Algebra, and worked at the higher equations and related parts. My memory furnishes

no cue at this point; but luckily, I have preserved a sort of diary, or daily jottings of work and study, extending over my seventeenth year. When this begins, I am occupied with fluxions, in Hutton's Course of Mathematics (the old edition), and devote about two hours every evening to solving exercises in maxima and minima, drawing tangents, and calculating the lengths of curves. I found myself unable to solve the more difficult examples, and was not sure that my solutions were right when I seemed to succeed. I had got beyond the Stewarts, and had no one else to assist me. At this point, the diary shows a remarkable turn in my mathematical labours.

It was a great surprise, besides being a lesson in the treacherousness of memory, to learn from an authentic source that I had been working at the higher Algebra for an indefinite period without having entered upon Geometry through the portal of Euclid. But I well remember the fact of going through the first book of Euclid, at a two or three hours' sitting. I fully satisfied myself that I understood the demonstration of every proposition, although, of course, I must have conceived many of the propositions loosely. I used to fancy that I was only about fourteen when I performed this feat; but, in point of fact, I was sixteen, and had been engaged in Algebra for probably a

year and a half or two years previously. However, having once begun Euclid, I went on at the same pace. Next day, I did the second book, and part of the third. Two days sufficed for the fourth, and made a beginning in the fifth. Sunday intervened, when I never did any scientific work; but, on Monday, a few more propositions were scored. Nothing further was done till Saturday, when nearly the entire day was devoted to the work, the record being the remainder of the fifth book, the whole of the sixth, and part of the eleventh. On Monday, fourteen propositions of the same book were grappled with. On Tuesday, the book was finished and the twelfth begun. On Wednesday, the twelfth was finished and a few propositions of Data gone through, which is the end of the Euclid record. The Data proved very dry and uninviting, and I doubt if I ever looked at it again.

I followed up Euclid by getting, out of the Mechanics' Library, Hutton's Recreations, and working at the geometrical exercises. This went on for a fortnight, when I exchanged Hutton at the Library for Simpson's Geometry, which I seem to have studied closely for a month (though I ought to have worked at it much longer), and then replaced it for Hutton's Dictionary—

READING IN MATHEMATICS.

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taking out at the same time the Algebraical Geometry of the "Library of Useful Knowledge". The two numbers of this series on the Differential Calculus I bought for myself, and made a beginning of the study of the Calculus. Hutton's Dictionary was a mine of wealth, and I often went back upon it. The historical portions and the lives of the mathematicians, I found intensely interesting. There were also considerable treatises upon all the great mathematical departments, and many smaller articles on special topics. I then got from the Library Vince's Fluxions and was devoted to it for the next week or two, solving problems and making extracts; while I made a purchase of the new edition of Hutton's Course of Mathematics, as remodelled by Professor Ramsay of Glasgow, the most important change being the substitution of the Differential Calculus for the portion on Fluxions. This was about the time when the Calculus took the place of Fluxions in Cambridge. It appears that the first use I made of Hutton was to begin Conic Sections. My next borrowing from the Library was Simpson's Fluxions,―a most satisfactory work, like everything that came from the hands of Thomas Simpson.

At the date now reached, October, 1834, I made an important move by availing myself for the evening hours of a mathematical school, taught by