at its incongruity, that I was going to take the child Polly with me to the theatre, to see the grown-up Polly act!  Both pictures—­Polly as a child, and Polly as a woman, are, I suppose, equally clear in my ordinary waking memory:  and it seems that in sleep I had contrived to give the two pictures separate individualities.

Of all the mathematical books which Mr. Dodgson wrote, by far the most elaborate, if not the most original, was “Euclid and His Modern Rivals.”  The first edition was issued in 1879, and a supplement, afterwards incorporated into the second edition, appeared in 1885.

This book, as the author says, has for its object

to furnish evidence (1) that it is essential for the purposes of teaching or examining in Elementary Geometry to employ one text-book only; (2) that there are strong a priori reasons for retaining in all its main features, and especially in its sequence and numbering of Propositions, and in its treatment of Parallels, the Manual of Euclid; and (3) that no sufficient reasons have yet been shown for abandoning it in favour of any one of the modern Manuals which have been offered as substitutes.

The book is written in dramatic form, and relieved throughout by many touches in the author’s happiest vein, which make it delightful not only to the scientific reader, but also to any one of average intelligence with the slightest sense of humour.

Whether the conclusions are accepted in their entirety or not, it is certain that the arguments are far more effective than if the writer had presented them in the form of an essay.  Mr. Dodgson had a wide experience as a teacher and examiner, so that he knew well what he was writing about, and undoubtedly the appearance of this book has done very much to stay the hand of the innovator.

The scene opens in a College study—­time, midnight.  Minos, an examiner, is discovered seated between two immense piles of manuscripts.  He is driven almost to distraction in his efforts to mark fairly the papers sent up, by reason of the confusion caused through the candidates offering various substitutes for Euclid.  Rhadamanthus, another equally distracted examiner, comes to his room.

The two men consult together for a time, and then Rhadamanthus retires, and Minos falls asleep.  Hereupon the Ghost of Euclid appears, and discusses with Minos the reasons for retaining his Manual as a whole, in its present order and arrangement.  As they are mainly concerned with the wants of beginners, their attention is confined to Books I. and II.

We must be content with one short extract from the dialogue:—­

Euclid.—­It is, I think, a friend of yours who has amused himself by tabulating the various Theorems which might be enunciated on the single subject of Pairs of Lines.  How many did he make them out to be?

    Minos.—­About two hundred and fifty, I believe.