tratado de geometría

JKSfl [...]... is, it de- clares that a certain property belongs to a certain thing; or it is practical, that is, it declares that something can be done Propositions are either demonstrable, that is, they may be established by the aid of reason; or they are indemon- simple and evident that they can not more so by any course of reasoning strable, that is, so be made A Theorem A Problem An Axiom is is is a demonstrable,... itself a magnitude 39 By the idea of motion, one magnitude taentally applied to another, and their may be form and extent compared This is called the method of superposition, and is the the methods of demon- most simple and useful of all stration used in geometry The student will meet with many examples EQUALITY 40 When two equal magnitudes are compared, it is found that they may coincide; that is, each... the distances its extent — to — The points of a magnitude may he have from each other any directions whatever, thus Postulate of Form made giving the magnitude any conceivable form These two are all the postulates of geometry rest in the very ideas of space, form, 3T Magnitudes which have they differ in extent, are called Any the They and magnitude same form while Similar point, line, or surface in... enounced in general first usually followed by a particufact, referring to a diagram In the latter part follows the demonstration or solution of the work The these steps ^re frequently shortened student is advised to conclude every demonstration with the general proposition which he has proved The student meeting a reference, should be certain that he can and apply the state principle referred to GENEKAL AXIOMS... Theorem A Problem An Axiom is is is a demonstrable, theoretical proposition a demonstrable, practical proposition an indemonstrable, theoretical propo- sition A Postulate is an indemonstrable, practical propo- sition A ing, proposition which flows, without additional reason- from previous principles, This term is also the demonstration of which is called a is frequently applied to is Corollary propositions,... extent and form Magnitude Geometry is is called a the science of magnitude Geometry is used whenever the size, shape, or position of any thing is investigated It establishes the principles upon which all measurements are made It is the basis of Surveying, Navigation, and Astronomy In addition to these uses of Geometry, the study cultivated for the purpose of training the student's ers of language,... yond the which represents we may conceive them of such extent as to The astronomer knows that the universe sun and his planes extend be- These ideas are expressed in the fol- lowing Postulate of Extent A magnitude may he made to have any extent whatever 36 Magnitudes may, in our minds, have any form, from the most simple, such as a straight line, to that ATo may of the most complicated piece of machinery... applied to is Corollary propositions, very brief and simple 4 The reasoning by which a proposition called the Demonstration The explanation how a thing is is proved done constitutes the Solution of a problem A Direct Demonstration proceeds from the premises by a regular deduction An Indirect Demonstration attains its object by showing that any other hypothesis or supposition than the one advanced would... surface in a similar figure, Homologous Magnitudes which have the same extent, while they differ in form, are called Equivalent are called MOTION AND SUPERPOSITION 38 The postulates are of constant use in geomet- rical reasoning Since the parts of a magnitude may have any posithey may change position By this idea of mo- tion, FIGURES tion, the 21 mutual derivation of points, surfaces, lines, and solids... Geometry consists of definitions, proposiA work on tions, demonstrations, and solutions, with introductory or explanatory remarks Such remarks sometimes have the name of scholia GENERAL AXIOMS —The student should 5 Eemark to state separately the He condition, if any whether it is learn each proposition, so as and the conclusion, also the hypothesis should also learn, at each demonstration, direct . by the aid of reason; or they are indemon- strable, that is, so simple and evident that they can not be made more so by any course of reasoning. A Theorem is a demonstrable, theoretical proposition. A Problem is a demonstrable, practical proposition. An Axiom. given in uninterrupted connection. No attempt is made to exclude any method of demonstration, but rather to present examples of all. The books most freely used are, "Cours de gfeomfetrie el6mentaire, par A as 1825, defined parallel lines as lines having the same direc- tion. Euclid's definitions of a straight line, of an angle, and of a plane, were based on the idea of direction, which is, indeed,