Euclid .
By JOHN Purr "F'N' .4 EUCLID of Alexandria, a famous: geometrician; whose book of Elements, revised and improved, still holds its place as an English school-book, although superseded as such in America and on the Continent." That is an extract from a popular Encyclopaedia published a few years back. In one respect it is already out of date. The process of superiession hai been carried a stage further ; England has followed the example of her neighbours, and no longer requires:her sons and daughters to draw their geometrical inspiration from' the time- honoured Alexandrian fount. " Euclid " has disappeared froth the academic curriculum and troubles the schoolboy no more.
-Those Of Us who were -brought up in the-old-tradition may or may not regret the change. That is a matter of individual preference or individual experience. There were some to whom Euclid was all plane sailing, who swallowed his axioms and theorems with unfailing relish ; to others he spelt pain and vexation which no passage of time can ever wholly obliterate froth- the 'memory. But at least we were all agreed in taking him for granted; He was part of.the school routine, like Chapel or Latin prose or compulsory football; We might like it, or we might not ; but go through with it we must. Why art ancient Greek pedant, dead some two thousand years, should be allowed to impose his dialectics upon successive generations of modern 'English boys, was a question that it never occurred to us to pose.
Now that is all changed. Euclid has lost his position Of privilege ; henceforward he mint stand, if he is to stand at all, on' his own intrinsic merits. Will- our: old friend (or old enemy, as you will) emerge successfully from the ordeal ? Will he retain, in open competition- with his fellow-authors, any of the immortality so long conferred upon him by a conspiracy of pedagogues ? One circumstance, not without significance in this * The Elements of Euclid. Edited. by Todhunter with an Intro- duction by Sir Thomas Heath. (J. M. Rent. 28.) connexion, deserves to be noted. The Elements have appeared quite recently in a new milieu ; they have won a place in the popular series of volumes known as " Every- man's Library." If they look a shade embarrassed in their strange surroundings, that is only what was to be expected ; few people are wholly at their ease on first admission to unwonted company. 'Euclid at any rate is fortunate in his social sponsor. Sir Thomas Heath, who contributes an introduction to the new volume, speaks up bravely for his protégé. " Qualified readers," he tells us, will find the book fascinating, "a book to be read in bed or on a holiday, a book as difficult as any detective story to lay down when once begun." 'Here is a novel role for the -pons asinorum and the squares on the of a right-angled triangle. True, even in the old days, we sometimes studied these masterpieces in bed ; but it was in no holiday mood and with feelings in which fascination played no part. Desperation would be nearer the mark ; for a desperate business it was, with the grim examiner awaiting us in the grey dawn and but one brief matutinal hour still left -in which to repair the wastage of a term's idleness or procrastination.
Nevertheless, Euclid is entitled to fair' play. Higher mathematicians may sneer at him—they sneer at every- body, one another included—but the " qualified reader " will be disposed to give him his chance. Now that he has doffed his official livery, we are better able to judge him dispassionately, to test from an angle of detachment his qualities of head and heart. Let us see how he responds to the treatment. 'Take the first axiom, for example :- "Things-which are' equal to the same thing are equal to one another." That sounds plausible enough, but its' practical application is not quite so easy. It assumes premisses which might be constant in a logically appointed universe, but which, under the makeshift conditions of actual life, betray a disconcerting element of fluidity. Turn to that most respectable of institutions, the League of Nations. Each of its components (that is one of its' fundamental.. principles) is equal. to the, same thing, namely, the abstraction known as a " State Member of the League ". ; but are they all equal to one another ? Is Liberia the equal of France, or Nicaragua of the British Empire ? -Again, " two straight lines cannot enclose a space " ; that, too, has a certain superficial air of veracity. But consider two lines of latitude—it will hardly be objected that they are not straight lines ; they are as straight as anything is capable of being on a globular planet. Follow their full circuit round the world, and then—are you prepared to lay your hand on your heart and assert that they do not enclose-a- space ?. • " In -the same way it may be- shown," and so. forth. Q.E.D., in fact. As to the whole being greater- than the part, that hoary fallacy has been exposed a thousand times.
But if Euclid's head is sometimes at fault, his heart is always in the right place. He is on the side of the angels, as well as the angles. He dealt in " problems," it is true : but on his lips the word was innocent almost to bloodlessness. The dubious associations that it was to acquire on the modern stage were undreamed of in his simple philosophy. A problem to him was just a problem, and nothing more ; that is to say, it was—well, it was something that wasn't a theorem and ended with Q.E.F. instead of Q.E.D. That's plain enough, isn't it ? So to) with his triangles ; they had nothing " eternal " about them ; they produced no social complications ; they were just three-sided objects with sharp corners, whose full identity—such was their modesty—was hidden under mere initials such as A.B.C. or D.E.F. No : on the score of propriety and sound moral tone, the Elements are above all reproach ; they are characterized, like Mr., Podsnap's ideal Englishman, by " an absence of anything calculated . to calla blush into the cheek of a young person." What praise could be higher ? And let no one be frightened by the strange long words : rest assured that they are all perfectly harmless. Some of them are even rather attractive ; hypotenuse, for instance, has a touch of romance about it, a faint suggestion of roses and cham- pagne. Rhomboid is frankly prosaic ; homologous and equimultipk sound like• post-prandial tests of sobriety. Presumably they all mean something, but it is certainly nothing that need cause a moment's embarrassment to the most immature " young person " that ever stepped out of the pages-of a Victorian novel.
The merits of Euclid, regarded as-light holiday reading, are another matter. On this point Sir Thomas Heath may or may not carry the public with him, though it appears that there is at least one undergraduate at Cambridge (Sir Thomas does not tell us his name) who has embraced the new doctrine. There must be many a reader who is jaded with detective stories and would welcome a new literary sensation with enthusiasm. Here is a chance for him to make an experiment. Let him slip the book into his suit-case when he next goes away on a holiday and try the effect of reading it in bed. It may prove a sovereign specific, better even than the old trick of counting imaginary sheep.
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