71 SECTION II. NOTES ON EUCLID. WITH SPECIAL REFERENCE TO THE SOLUTION OF I HAVE often wondered that among the various attempts to correct the obvious defects--for educational purposes-of Euclid regarded as a text-book of mathematics, nothing should have yet been done to remodel the book itself. Various writers have published books of geometry, each fondly hoping that his book will not only displace Euclid, but dispose of all rivals. The result has proved rather confusing. A boy who has been at a public school where one of these books has been used, goes perhaps afterwards to a private tutor who prefers another text-book of geometry; thence, perhaps, to a London college, where yet another book is employed; and finally to one of the Universities, where he finds Euclid still holding the place of honour. Now, if Euclid simplified could be put into boys' hands at school, and all other text-books diligently eschewed, there would be a common system at all schools, and no trouble when the old-fashioned Euclid was taken up, at whatever stage of mathematical progress. There can be no doubt Euclid perplexes many boys, and disgusts not a few. For my own part, though I was introduced to Euclid in the absurdest of ways, I loved him from the beginning. I had been counted rather a dullard at Geometrical Exercises, which I disliked, because there were Rules without Reasoning. Just as the foolish arithmetics of those days told us in hard words what to do, but never showed us why, so the Geometrical books told us to rule this line and describe that circle, in order to bisect lines, set up perpendiculars, and so forth, with no proof that the methods were sound. Besides, while mapping (a most instructive exercise), I had intuitively invented such methods for myself; so that rules had not even the charm of novelty. At this stage of my progress— or want of it—a preposterous under-master pitchforked me into a higher class where Euclid was read, and where, as it chanced, the 16th Proposition of the First Book was in hand. It was a new thing to me to find reasoning about matters geometrical. Theoretically, the folly of putting me at the 16th Proposition first ought to have made Euclid hateful to me; but, as a matter of fact, the case proved otherwise: I loved him from the first. I read alone the Definitions, Axioms, and preceding propositions; then went along with propositions ahead; till, before very long, I was in the Spider's Web of the last proposition but one of Book XII. Yet I was by no means a sound, only an eager, student of Geometry; for I remember devising a new construction for that most delightful of all propositions the 10th of Book IV., which, though much shorter and easier than the original, laboured under the trifling defect of being incorrect. Still, I think my case was exceptional. Most boys do not take kindly to Euclid, and certainly there is much in his outer appearance which is not inviting. In particular, the method of first giving the abstract proposition, and then describing a particular case, tests somewhat painfully the young student's power of attention. It is so much harder to make out what the enunciation means than it would be if each part were explained as in the opening words of the proposition, that we cannot wonder if boys are bewildered and wearied. To the more advanced it is pleasing to note how each enunciation has the qualities of a good definition in precisely indicating the abstract idea without any reference to a special case. But Euclid did not write for boys. Again, there is a charm in the skill with which Euclid, having adopted a certain method, gets over the difficulties involved in applying that method to particular cases. The famous Pons Asinorum is a case in point. Euclid's plan will not allow him to use the bisector of the angle B A C, because he has not yet shown how that bisector can be drawn. Nor can he allow himself to suppose his initial figure, repeated line for line, and then applied, after being turned over, to the original figure, after the manner already employed in Proposition IV., because he has not yet shown how the 'copy' is to be made. Either method would have given him a very simple proof, and as it is certain that there must be a line bisecting the angle BA C, and again that another figure precisely like that already drawn is conceivable (in the same sense that a straight line or a circle is conceivable from its definition), he was, logically, free to employ either plan. But he had assigned himself certain limits, and he makes out his proof within those limits very ingeniously and prettily-though confusingly to many boys. The First Book of Euclid treats chiefly of the properties of triangles and parallelograms. An examination of the book suffices to show that Euclid had proposed a definite line of treatment leading up to certain important propositions. Hence many useful properties are left untouched in this book. It is surprising, however, how many valuable propositions Euclid has succeeded-by a judicious method of treatment—in introducing into his plan without marring its symmetry. In attacking deductions either immediately depending on the First Book of Euclid or involving it in part only, it is necessary that the student should have at his fingers' ends, so to speak, all the most useful properties established in the First Book, and also several important properties deducible from this book. We proceed to examine the most valuable of these. In the first place, let us run through the First Book and notice whether there are any properties whose converse theorems, though not proved in Euclid, may be readily established. Euclid has proved the converse of Prop. 5 in Prop. 6, of Prop. 13 in Prop. 14, of Prop. 18 in Prop. 19, of Prop. 24 in Prop. 25, of Props. 27 and 28 in Prop. 29, of Prop. 37 in Prop. 39, of Prop. 38 in Prop. 40, and of Prop. 47 in Prop. 48. The other propositions which admit of a converse are the following: C E D FIG. 40. Euc. I., 15, of which the converse is, 'If two straight lines CE, DE (Fig. 40), on opposite sides of a line AB, make equal angles CEA, DEB with A B; then CE and ED are in the same straight line. This is obviously true, since if CE produced fell in some other direction, as EF, we should have the angle BEF equal to the vertical angle C E A, and therefore to the angle BED, which is absurd. We may refer to this proposition as Euc. Book I., Prop. 15, Conv. Euc. I., 17.-The converse of Prop. 17 is Axiom 12. We touch here on the great defect of Book I., |