to move about.  Lewis Carroll was precisely the same.  This, of course, led to a great deal of work and trouble, and made the illustrating of his books more a matter of artistic interest than of professional profit.  I was seven years illustrating his last work, and during that time I had the pleasure of many an interesting meeting with the fascinating author, and I was quite repaid for the trouble I took, not only by his generous appreciation of my efforts, but by the liberal remuneration he gave for the work, and also by the charm of having intercourse with the interesting, if somewhat erratic genius.

A book very different in character from “Sylvie and Bruno,” but under the same well-known pseudonym, appeared about the same time.  I refer to “Pillow Problems,” the second part of the series entitled “Curiosa Mathematica.”

“Pillow Problems thought out during wakeful hours” is a collection of mathematical problems, which Mr. Dodgson solved while lying awake at night.  A few there are to which the title is not strictly applicable, but all alike were worked out mentally before any diagram or word of the solution was committed to paper.

The author says that his usual practice was to write down the answer first of all, and afterwards the question and its solution.  His motive, he says, for publishing these problems was not from any desire to display his powers of mental calculation.  Those who knew him will readily believe this, though they will hardly be inclined to accept his own modest estimate of those powers.

Still the book was intended, not for the select few who can scale the mountain heights of advanced mathematics, but for the much larger class of ordinary mathematicians, and they at least will be able to appreciate the gifted author, and to wonder how he could follow so clearly in his head the mental diagrams and intricate calculations involved in some of these “Pillow Problems.”

His chief motive in publishing the book was to show how, by a little determination, the mind “can be made to concentrate itself on some intellectual subject (not necessarily mathematics), and thus banish those petty troubles and vexations which most people experience, and which—­unless the mind be otherwise occupied—­will persist in invading the hours of night.”  And this remedy, as he shows, serves a higher purpose still.  In a paragraph which deserves quoting at length, as it gives us a momentary glimpse of his refined and beautiful character, he says:—­

Perhaps I may venture for a moment to use a more serious tone, and to point out that there are mental troubles, much worse than mere worry, for which an absorbing object of thought may serve as a remedy.  There are sceptical thoughts, which seem for the moment to uproot the firmest faith:  there are blasphemous thoughts, which dart unbidden into the most reverent souls:  there are unholy thoughts, which torture with