In writing the present work, the author has been in a great measure prompted by the desire of supplying a really elementary text-book by which "the Royal Road to Geometry" might be opened up to young students of both sexes; so elementary indeed as to place it within the reach of even children, as a means of teaching them to think. The author refers deliberately to both sexes, for, in his humble opinion, woman stands even in greater need than man of mathematical training as a practical course of logic, to compensate the great excitability of her nervous system, and her inherent propensity to yield to feeling and passion rather than to strict reasoning. Hitherto, mathematics in general, and Geometry in particular, have been a sealed book for "the million"; it is only of late years that the tendency to keep the profanum vulgus at a respectful distance, has somewhat subsided among geometers. The author, so far from wishing to keep the masses in ignorance, would rather throw the doors of Geometry wide open to them, and make this science part and parcel of elementary instruction for all classes of society. Let that veil be removed, which can no longer have any other object but to "wrench awe from fools, and tie the wiser souls by its false seeming";

let "the case and habit of place and form," to use Shakspeare's own words, become generally known, and a vast stride will have been made towards that popularisation of knowledge which our age is now endeavouring to achieve.

But the author is not blind to the fact that rendering Geometry, and indeed all sciences, easier to learn, will not improve greatly the instruction given in schools, unless the teachers simplify it still further by a rational and judicious method of teaching. In using this book, the teacher who means to do really efficient work, will have to explain thoroughly the definitions contained in each chapter, illustrating each point with numerous diagrams, before the theorems are read; he will have to supply, at every stage of the demonstration of each proposition, supplementary diagrams for the complete mastering of each difficulty. Whenever it is possible, he should give other demonstrations than those contained in the book, and make his pupils in their turn try to form others for themselves, never omitting to discuss their merits and demerits at full length. As every-day exercises, he should require from them the demonstration of each corollary, or else give it himself; for, on no account must any theorem be taken up before the demonstration of all the corollaries of the

preceding one be gone through. Also, he should constantly test the reasoning and inventive powers of his pupils, by proposing them new theorems to be demonstrated, and problems to be solved and discussed, by means of the theorems and corollaries previously read. And last, not least, he should assist the students in finding out the whole of the inverses and converses of each proposition, and prove their truth or falseness as the case may be. In a word, the teacher must apply himself to the spirit, and not merely to the letter of the book. The way in which a text-book of Geometry should be used is forcibly expressed in the following passage from a speech of Dr. HIRST, President of the Association for the Improvement of Geometrical Teaching:

:

"The text-book indeed, to be properly used, should always be subordinated to the teaching; but to do this it is absolutely essential that the teacher should, by his own study, have risen not merely up to, but above the level of the text-book he employs. Until he has so mastered the subject that it has become plastic in his hands, his teaching must necessarily remain defective; for geometrical truth, it must be remembered, has, like all other truth, many sides, and no text-book can present all or necessarily the one which, to individual pupils, is the

most accessible.

Alternative methods of demonstration,

inquiries into the interdependence of propositions, judicious variation of data, and just discrimination between the contingent and necessary properties of figures ;-these and numerous other matters, all essential to geometrical culture, can only be properly supplied by the teacher; no text-book could be weighted with them. Above all it is to him that we must mainly look for the cultivation of that scientific method of inquiry under whose guidance solely problem-solving can be raised in character above what has been termed 'exalted conundrum guessing,' and acquire its full educational value."

The author takes this opportunity to acknowledge the great help he has derived from J. M. C. DUHAMEL'S book: On Methods in Argumentative Sciences, and to express his deep regret at not having been able to avail himself of J. DELBŒUF's valuable treatise: Philosophic Prolegomena of Geometry, and of J. UEBERWEG's dissertation: Scientific Exposition of the Principles of Geometry, at an earlier stage of his labours; these last two works not having come to his hands until one half of The Elements had already gone through the press.

In conclusion, the author begs to dedicate his work, which is the result of long persistent labours, to all

Teachers and Students who appreciate the sincerity of his intentions, and who sympathize with his efforts. The book lays no claim to perfection. Many perhaps will find numerous defects in it, whilst the most benevolent of readers will probably hold that it is open to considerable improvement. To all the author begs to say that he courts criticism in the interest both of his work and of the cause it is designed to subserve; and he will look upon all those as personal friends who, by pointing out to him such imperfections and shortcomings as they may discover in his book, shall afford him the means of improving it in the future.