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The term " angle " Post Date: Tue, 12 Aug 2008 6:05:26 +0000 The angle comes to be a part of a general spatial scheme. The relation between all angles is suggested at the same time the properties of a single angle are exhibited. The term " angle " thus comes to have a very much broader connotation than it does in the first defini- tion. Autor of the post: Undefined Thus the student is introduced Post Date: Tue, 12 Aug 2008 5:49:45 +0000 Or one may say that a principle of construction is substituted for the figure. The chapter on definitions next explains certain simple forms of comparison which are not spatial at all but are relational in one of the higher senses referred to in the dis- cussions a few pages back. Thus the student is introduced to methods of superposition. Autor of the post: Undefined When we say that likeness Post Date: Tue, 12 Aug 2008 5:31:24 +0000 When two lines are superim- posed upon each other or two angles are superimposed on each other, they are compared with a view to discovering their likenesses and differences. The discovery of likenesses among lines is comparable to the discovery of likenesses in economic values or the discovery of likenesses in rhetorical forms. When we say that likeness or equality is a general characteristic discoverable in many different mental experi- ences, we must amplify this statement so as to bring out the fact that the method of reaching the judgment of likeness is different in different cases according to the material com- pared. Autor of the post: Undefined These examples show Post Date: Tue, 12 Aug 2008 5:12:58 +0000 If one compares economic values, for example, he does not superpose one value upon another. If one com- pares two musical notes, he cannot superimpose them to discover whether they have the sajne pitch. These examples show that the comparison of two lines is a special problem which requires the student to know not only how to recog- nize the one line and the second line but also how to make the comparison hi a definite and well-ordered fashion. Autor of the post: Undefined The single figure Post Date: Tue, 12 Aug 2008 4:59:14 +0000 The geometrician finds it necessary to direct his student, there- fore, in methods of making comparisons. Comparisons are necessaiy in order to bring out the remoter characteristics of figures. The single figure is no longer the subject of recognition and discussion ; the single figure must be understood by finding its proper place in science, where likeness and unlikeness are quite as significant as distance and form. Autor of the post: Undefined Geometry is rather the science Post Date: Tue, 12 Aug 2008 4:42:52 +0000 Indeed, for purposes of further mental develop- ment and for purposes of application, comparisons are indispensable. It is thus made clear that it is a mistake to assume that geometry is the art of seeing space. Geometry is rather the science of finding out all that can be discovered with regard to the characteristics of spatial figures. Autor of the post: Undefined We select a few easily Post Date: Tue, 12 Aug 2008 4:31:35 +0000 The scientific in- vestigation of the characteristics of figures leads us into the most elaborate logical processes of comparison and in- ference. Indeed, so large a part of geometry is made up of comparisons that the mere recognition of figures is likely to drop in the background. We select a few easily acces- sible samples of angles and compare these with each other. Autor of the post: Undefined He then general- izes Post Date: Tue, 12 Aug 2008 4:19:17 +0000 We select a few examples of triangles and compare these with each other. The student learns how to make the com- parisons in these relatively simple cases. He then general- izes his findings in the form of a principle, which principle is not a matter of perception at all. Autor of the post: Undefined To be sure, the analytical Post Date: Tue, 12 Aug 2008 4:06:49 +0000 We may digress for a moment from our study of the textbook to call attention to the fact that the science of geometry may contribute little or nothing to the students perception of space. Space, as it confronts the student in architectural forms, in the shapes of animals and trees, in the perspective of the street, and in the shadows of solid objects, may be entirely overlooked while the student buries himself in the pages of definitions and comparisons which go to make up his book on the science of geometry. To be sure, the analytical studies of figures might make the stu- dent more intelligent in the observation of form if he would apply his analysis to objects about him. Autor of the post: Undefined Thestudgnt must be taught geometrical Post Date: Tue, 12 Aug 2008 3:50:12 +0000 It has been one of the mistakes of the modern school that it has assumed that the student can readily carry over his textbook studies into ordinary life. What is needed is more attention to the ob- servation and analysis of all forms. Thestudgnt must be taught geometrical analysis, and, at the same time, he must be taught to apply this analysis to figures outsida of the textbook. 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