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Especial attention should, perhaps Post Date: Tue, 12 Aug 2008 11:16:59 +0000 Every teacher of geometry knows the disastrous consequences which follow when a student fails to keep together in thought the letter and the figure to which it properly relates. Words are in a measure open to a like objection that the student sometimes goes astray because word and meaning get separated ; but a letter, for the reasons given, is much more in danger of losing its connection. Especial attention should, perhaps, be devoted to the type of idea which the student is called upon to develop in con- nection with the definition of a point as a purely geometri- cal conception. Autor of the post: Undefined The definition illustrates very fully Post Date: Tue, 12 Aug 2008 10:58:38 +0000 " A point," he is told, " has no dimensions, but denotes position simply." This definition of a point is an effort, as is the definition of a line, to aid the student in getting rid of the direct sensory content which is always present in any real point which he observes. The definition illustrates very fully the conclusion which was reached in the last chapter, when it was pointed out that space can never be recognized except in terms of some kind of sen- sory content. Autor of the post: Undefined This definition calls attention Post Date: Tue, 12 Aug 2008 10:43:06 +0000 One tries by the substitution of an abstract verbal idea for sensory content to get rid of as much sen- sory content as he can, in order to leave behind the pure spatial elements of experience. Another definition which may be noted as especially sig- nificant for our psychological study is that in which the student is told that any figure is a limited portion of space. This definition calls attention to the fact that the same kind of space extends on all sides of a given figure. Autor of the post: Undefined Any limited portion of space Post Date: Tue, 12 Aug 2008 10:25:47 +0000 There is an implication here which comes out more clearly in later discussions; namely, the implication that any limited portion of space has the fundamental characteristics of all space. Indeed, geometry recognizes that all space is absolutely homogeneous. Any limited portion of space which is se- lected for discussion is a representative of other limited portions of space or of space in general. Autor of the post: Undefined The mental processes required Post Date: Tue, 12 Aug 2008 10:09:27 +0000 We may regard a figure, therefore, as a representative example rather than as a single isolated experience significant in itself. Thus, while our definition of a point attempts to get rid of all the content, the statement that a given figure is a limited portion of space attempts to generalize or extend the conclusions of any particular discussion to the larger field of all space. The mental processes required to understand the defini- tions which have been discussed up to this point are space perception and the higher processes of analysis, abstraction, and interpretation of symbols. Autor of the post: Undefined What can be said about Post Date: Tue, 12 Aug 2008 9:54:55 +0000 We pass now to another phase of geometrical science. Not only are the different parts and aspects of figures to be distinguished, but the properties of figures and geomet- rical elements must be discovered. What can be said about a point differs from the definition of the point itself, in that higher and more complex groups of ideas are involved. Autor of the post: Undefined It is, rather, an effort Post Date: Tue, 12 Aug 2008 9:44:15 +0000 Thus one of the statements which is made about the prop- erties of a point is that " through a point an indefinite number of straight lines may be drawn." This statement, though included among definitions, is not strictly a defini- tion. It is, rather, an effort to get the student to consider carefully some of the facts which are outside of the point, but closely related to it. Autor of the post: Undefined The words of this state- Post Date: Tue, 12 Aug 2008 9:25:36 +0000 The statement is put at the be- ginning of the science because it is obvious. The student can find the truth of this fact if he scrutinizes his experi- ences, and brings to clear consciousness that which he can readily justify, but which he has probably never before recognized with explicit clearness. The words of this state- ment are intended to call up in the students mind a whole series of experiments. Autor of the post: Undefined It is a matter Post Date: Tue, 12 Aug 2008 9:07:55 +0000 He will try to draw different lines through an imaginary point, and will note the characteristic relation between these different lines and the point. The mental process thus induced is relatively direct, but it is by no means a mere matter of space perception. It is a matter of experimentation and a matter of comparison, with a view to discovering and recording explicitly facts which will be of use in later elaborate studies. Autor of the post: Undefined It was a discovery, not Post Date: Tue, 12 Aug 2008 8:52:51 +0000 By guiding students through such simple experiments as these the geometrician prepares them to try all possi- ble combinations and recombinations of figures. One can imagine, for example, that the first student of geometry who explicitly formulated the statement that an indefinite number of lines can be drawn through a single point re- garded the fact as a discovery. It was a discovery, not of an obscure fact, but a discovery in explicit form of a fun- damental fact which all along had been present in experi- ences, but was now brought to attention as of sufficient importance to be treated as a distinct and explicit matter of comment. 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