Chapter XI. Polygons and symmetry. Varieties of polygons; symmetry with respect to a line; symmetry with respect XII. A frustum of a pyramid. Construction; description . XIII. A truncated pyramid. Construction; description XIV. Curved surfaces and lines. The circle; railroad curves; three ways of drawing a circumference XV. A cylinder. Construction; description; length of a cir- XVII. Solids of revolution. The sphere; description; area of the surface; map drawing; volume XVIII. Figures for practice. General questions; a truncated triangular prism; two quadrangular prisms; the regular Quadrilaterals. Construction of various kinds. Polygons. Description; sum of the angles; pentagons; XXIII. Circles. Arrangements of two circles; chords; arcs; Chapter XXV. Constructions. Straight lines; bisecting straight lines; perpendiculars; arcs of given size; angles of given size; bisecting arcs and angles; circumscribed and inscribed circles; miscellaneous problems XXVIII. Ratio and proportion. Ratios between two lines; pro- portions among four lines; means and extremes; to com- INTRODUCTION IN the works of nature and of man Geometry plays a most important rôle. The rays of light from the sun suggest the straight line; the surface of still water, the plane; the faces of crystals, a variety of elementary plane figures bounded by straight lines; while the crystals themselves suggest the most common figures bounded by planes. Moreover, the myriad other forms in the animal, the vegetable, and the mineral kingdoms furnish unending variety of symmetrical and complex geometric forms, while the creations of the artist and the architect, and the problems of the engineer and the astronomer, all have their basis in Geometry. The practice of training pupils early in observing the simple geometric forms and relations of the objects which come under their every-day notice, of teaching them the use of the simplest tools of geometric construction, and of making them familiar with a variety of means of finding lengths, areas, and volumes, is a most natural and potent means of training their powers of observation, and at the same time of cultivating habits of concentrated and continuous attention. The old arithmetics with their puzzling problems furnished a powerful means for the cultivation of the powers of analysis, but they did not furnish in any adequate sense the careful training of the child's faculties of observation. It is true that many of their problems were of a practical nature, and were invaluable as a means of familiarizing the student with some of the simple rules of mensuration, and of creating an interest in the methods of making measurements for obtaining the data for problems to which these rules might be applied—problems in finding the contents of bins and boxes, and of calculating the amount of lumber used in their construction; problems in finding the areas of various shaped fields; problems in finding the heights of trees from their shadows, etc. Such geometric problems often awaken an intense interest, and a desire to know the reasons for the rules employed in their solution, and so create an appetite for the study of Formal Geometry. The want, however, of careful and systematic development of the subject as a means of cultivating the faculties of observation caused a revolt against the arithmetic problems, and resulted in the substitution of nature studies to a considerable extent in the schools for the drill in such problems. But nature studies, which are taught mainly to direct attention to plant and animal life, and to the mere observation of form, fail to give that sharpness to the mental faculties, and that severe training in vigorous thinking, which the consideration of mathematical problems alone can give. The Observational Geometry combines the training of the nature studies, so far as these educate the eye to keen and intelligent perception, with the training which the more valuable problems of the old arithmetics furnish, and so gives a mental discipline at once rigorous and entirely free from that one-sidedness which either of these systems fosters when taken alone. It gives the hand dexterity and skill in making drawings and models of geometrical figures. It trains the eye to estimate with accuracy forms and distances. It teaches an ap |
