PREFACE TO SECOND EDITION.

THE principal changes, made in this Second Edition of "Pillow-Problems", are as follows:—

(1) After the numeral, which precedes each Question, Answer, or Solution, references are given to the pages at which the corresponding matter may be found.

(2) Some of the Solutions have been re-arranged, and duplicate-diagrams have been inserted, in order that every portion of text may have its illustrative diagram visible along with it, and that the reader may thus be saved the trouble, and the strain on his temper, involved in turning a leaf backwards and forwards while referring from the one to the other.

(3) In the title of the book, the words "sleepless nights" have been replaced by "wakeful hours".

This last change has been made in order to allay the anxiety of kind friends, who have written to me to express their sympathy in my broken-down state of health, believing that I am a sufferer from chronic "insomnia", and that it is as a remedy for that exhausting malady that I have recommended mathematical calculation.

The title was not, I fear, wisely chosen; and it certainly was liable to suggest a meaning I did not intend to convey, viz. that my "nights" are very often wholly "sleepless". This is by no means the case: I have never suffered from "insomnia": and the over-wakeful hours, that I have had to spend at night, have often been simply the result of the over-sleepy hours I have spent during the preceding evening! Nor is it as a remedy for wakefulness that I have suggested mathematical calculation; but as a remedy for the harassing thoughts that are apt to invade a wholly-unoccupied mind. I hope the new title will express my meaning more lucidly.

To state the matter logically, the dilemma which my friends suppose me to be in has, for its two horns, the endurance of a sleepless night, and the adoption of some recipe for inducing sleep. Now, so far as my experience goes, no such recipe has any effect, unless when you are sleepy: and mathematical calculation would be more likely to delay, than to hasten, the advent of sleep.

The real dilemma, which I have had to face, is this: given that the brain is in so wakeful a condition that, do what I will, I am certain to remain awake for the next hour or so, I must choose between two courses, viz. either to submit to the fruitless self-torture of going through some worrying topic, over and over again, or else to dictate to myself some topic sufficiently absorbing to keep the worry at bay. A mathematical problem is, for me, such a topic; and is a benefit, even if it lengthens the wakeful period a little. I believe that an hour of calculation is much better for me than half-an-hour of worry.

The reader will, I think, be interested to see a curiously illogical solution which has been proposed, by a correspondent of the Educational Times, for Problem 61, viz. "Prove that, if any 3 Numbers be taken, which cannot be arranged in , and whose sum is a multiple of 3, the sum of their squares is also the sum of another set of 3 squares, the 2 sets having no common term."

The proposed solution is as follows:—

"Let , , be the three Numbers; then Also

"

Now, if we denote, by '', the property "which cannot be arranged in , and whose sum is a multiple of 3," and, by '', the property "the sum of whose squares is also the sum of another set of 3 squares, the 2 sets having no common term," we see that all, that this writer has succeeded in proving, is that certain selected Numbers, which have property '', have also property '': but this does not prove my Theorem, viz. that any Numbers whatever, which have property '', have also property ''. If his argument were arranged in a syllogistic form, it would be found to assume a quite untenable Major Premiss, viz. "that, which is true of certain selected Numbers which have property '', is true of any Numbers whatever which have property ''."

C. L. D.

CH. CH., OXFORD. September, 1893.