MODERN GEOMETRY OF POINT, RAY, AND CIRCLE BY WILLIAM BENJAMIN SMITH, A.M., PH.D. (Goett.) PROFESSOR OF MATHEMATICS AND ASTRONOMY IVERS TY LIBRARY 26*217 COPYRIGHT, 1892, BY MACMILLAN AND CO. TYPOGRAPHY BY J. S. CUSHING & Co., BOSTON, U.S.A. PRESSWORK BY BERWICK & SMITH, BOSTON, U.S.A. PREFACE. THIS book has been written for a very practical purpose, namely, to present in simple and intelligible form a body of geometric doctrine, acquaintance with which may fairly be demanded of candidates for the Freshman class, and is in fact demanded at this University of the State of Missouri. This purpose has regulated both the amount and the character of the matter introduced. The former might have been made larger, the latter more uniform and scientific, but only — so at least it seemed to the author—at a sacrifice of usefulness under existing conditions. Much more than one year's study can hardly be given to Plane Geometry in the majority of High Schools and Academies - a fact that sets rather narrow limits to practicable treatment of the subject. In such a course the Apollonian problem would seem to present itself as a natural and rightful goal. Besides, in its solution the logical play, while direct and simple, is yet highly instructive and even artistic, — all the concepts of the foregoing sections are summoned up and marshalled and brought to bear upon a single point. But if any such goal is to be attained in such a time, the path pursued must not be tortuous, and there will be little leisure for lateral excursions. In the Exercises, however, the view of the student is considerably widened so as to embrace most of the more familiar theorems omitted from the V text. These Exercises have, in fact, been chosen with especial reference not so much to their merely disciplinary as to their didactic value, the author being persuaded that quite as good exercise may be found in going somewhither as in walking round the square. The problems proposed for solution will not merely drill the student in what he already knows, but will greatly extend his knowledge, in particular, of projection and perspective, guiding him nearly as far as he can conveniently go without the help of the Cross Ratio - - a notion which the narrow scope of the work as a mere introduction seemed to exclude from employment. It is believed that advocates of the heuristic method may find in these problems ample playroom for the ingenuity of their pupils. As regards both the matter and the arrangement of this part of the book, the author would lay little claim to originality, but would rather acknowledge indebtedness to his predecessors in the attempt to modernize geometrical teaching, particularly to the valuable and indeed admirable works of Dupuis, Halsted, Henrici, Newcomb, Frischauf, Henrici and Treutlein, and Mueller. Up to the Taction-Problem the notion of Form has dominated the whole discussion, but in the following sections certain metric relations of great importance receive due consideration. With respect to the methods employed and the point of view assumed, a preface is no place for apology. With such as approve the resolution of the 31st Assembly of German edu cators: "Im Unterricht der Elementargeometrie an Realschulen und Gymnasien bleibt die Euclidische Geometrie dem System nach bestehen, wird aber im Geiste der neueren Geometrie reformiert," argument would be needless; with others it might be useless. The case stands in a measure as with the Gospel saying, made |