on his readers so patiently and helpfully that not one shall be left behind. It is the slowest and weakest that he especially wishes to take by the hand. The very difficulty they have in taking in Euclid's meaning shows that they most need Euclid. Such is the writer's object, and his first thought, in putting into print the substance of his conversational method of teaching Euclid, has had reference to his old Eton pupils. He wishes, though no longer able to speak to them face to face, thus to offer to them the helping-hand of which they used to take a friendly grip. But he looks to others also with whom he has had no personal connection. The many hearty and very friendly letters which he received from teachers both in England and America when, some years ago, he gave in a tract1 a sketch of this mode of teaching Euclid, leads him to hope that they may find this treatise of use in their schools. He trusts, too, that those who are struggling with Euclid, without any one at hand to explain it, and to remove difficulties, may find the treatise a help to them. And he will be glad if some, who look askance, with a half-smile and half-shudder at the idea of reading Euclid, shall be led on by these familiar explanations to a study that will be of incalculable benefit to them. Another set of readers whom the writer has in his mind are those classes of women who, having tried this or that fragment of a scientific subject, have 1 A Narrative Essay on a Liberal Education, Hamilton & Adams. come to the sound conclusion that they must begin at the beginning, and lay a foundation,' and who have, most wisely, chosen Euclid for that foundation." To these classes he hopes that his treatise may be useful. But as they will be in earnest, and assiduous, a half-apology is due to them for the homely and humorous illustrations introduced to arrest attention and awaken interest. Let them kindly remember that the following explanations were given to, and are written for, boys, who, high-spirited and thoughtless, are not fully alive to the importance of what they are learning, or bent, as they are, on self-improvement. It has been already said that these explanations are not written for quick and able boys, but for the great mass of those of average intellect, who have not naturally the accuracy of thought, and the powers of reasoning with which some few are gifted. Yet even an able boy may start at an advantage by a rapid perusal of these familiar explanations. Possibly he may gain a fuller view of Euclid's reasoning than he would have had without them, and pick up thoughts that he might otherwise have missed. One word more: a body of men, able mathematicians, are now working together with a view of improving our methods of geometrical teaching. The writer deprecates the charge of antagonism with them. They are to a great degree entering on that profession which he is leaving. Euclid was king in the writer's time, was still seated on the throne which he has occupied for more than two thousand years. If the common consent of the leading mathematicians of England, in the next generation, decide that he must be dethroned, he must go. But geometrical reasoning will not go. And as the aim of this treatise is to make learners appreciate geometrical reasoning, the writer counts on the sympathy of all who are desirous of improving our geometrical teaching and are working towards that end. A WORD of ADVICE to the LEARNER. PROVIDE YOURSELF with a pair of compasses having a pen and pencil leg, a small flat ruler, and a hard pencil (and keep its point fine). Also it would be well to have by you a little box-wood triangle, called by instrument makers a 'set square.' In default of this, double down and cut off the corner of a sheet of cardboard; it will answer the same purpose. While you are reading this treatise, have your pen much in your hand, and often try if you understand, and remember what you have read, by writing it out, not copying it. Remember Bacon's apophthegm:— Much reading makes a full man, |