figure; in the same way, the idea of infinitely great is acquired by denying form to a figure. In denying the idea of a terminal point to a line, the mind conceives a line infinitely great; also, an infinite surface is that to which the idea of a terminal line is denied; and an infinite solid is that to which the idea of a terminal surface is denied. Inasmuch as, according to the preceding, the extension of a point added to any line, or the breadth of a line added to any surface, or the thickness of a surface added to any solid, does not alter the magnitude of the line, the surface or the solid respectively-the number of points contained in a line, or of lines contained in a surface, or of surfaces contained in a solid, is infinite: it is evident that a quantity infinitely small repeated a finite number of times cannot produce a finite quantity. It is obvious, also, that an infinite line cannot be contained in a finite surface or solid, nor an infinite surface or solid in a finite solid. From what has just been said it will be admitted that the magnitude of a geometrical figure is essentially distinct from its form, since magnitude is common to all figures, and is consequently immaterial in the study of an isolated figure. 17. The description of the form, magnitude, and place, of a particular figure is called its construction. If, in the construction of a figure, the magnitude be not stated, then the form only is determinate, and the figure may be of any magnitude. From the homogeneity of space it follows that a determinate portion, and in fact any portion of it, may differ from another of the same form by its magnitude, but not by its properties. Several figures may therefore have the same form and the same magnitude, or the same form only, or the same magnitude only. 18. Figures which have the same form and the same magnitude, are equal to each other, or simply equal. (Eucl. Ax. 8.) That is, figures are equal when they may be taken indiscriminately one for the other. From the homogeneity of space, it follows immediately that two or more equal figures may coincide in all their parts, and also that equal figures which coincide in all their parts, form but one figure and are not distinct from one another: they have consequently the same properties.1 It is therefore evident that, in the study of Geometry, any figure may always be replaced by its equal, and also that if two or more figures be proved to coincide in all their parts, they are equal. Again, two figures, each of which is equal to a third, are obviously equal to each other, for they may both be made to coincide with the third, and consequently with each other. 19. Figures which have the same form, but whose magnitude is indeterminate, are similar to each other, or simply similar. Similar figures which have the same magnitude are necessarily equal to one another; and two figures, each of which is similar to a third, are similar to each other, for they have all the same form. 20. Figures which are equal, or similar, each to each, to the same number of other figures placed in the same order, are said to be placed similarly to the latter. 21. Figures which have the same magnitude, but which may have any form, are equivalent to each other, or simply equivaleni. Equivalen; figures have obviously the same value, and conversely, figures which have the same value are equivalent to each other; also, it is evident that there are no equal magnitudes but those of equivalent figures, and that, when geometrical figures are compared to one another in point of magnitude, equal magnitudes may be interchanged with each other, although they may not belong to equal figures. Two equivalent similar figures are equal to each other; also two equal figures are necessarily similar and equivalent, but two similar, or equivalent, figures need not be equal. In accordance with the preceding definitions, it is always possible to change the magnitude of a figure and yet preserve its form, and to change the form of a figure without changing its magnitude. 1 When two or more figures are spoken of, distinct figures only are meant, unless the contrary is expressly stated. 22. Expanding a figure is increasing its magnitude without altering its form; reducing a figure is diminishing its magnitude without altering its form. Reducing two figures to the same magnitude is reducing the greater, or expanding the lesser, of them until they become equivalent.1 Two figures are necessarily similar to each other if, by reduction, they become equal to each other; also, it is evident that, if two similar figures be reduced to the same magnitude, they will be equal to each other. 23. Transforming a figure is changing its form without altering its magnitude. Geometry, like other sciences, has necessarily a descriptive part, which consists in the definition and classification of figures and their properties; the science itself studies these properties by means of Reasoning.2 1 For the sake of brevity, the term reducing is used in The Elements for both expanding and reducing: that is, altering the magnitude of a figure without altering its form; unless the sense indicates clearly that expanding is not meant. The short digression on Reasoning which follows forms no part of Geometry, but is given here as a help to the student who has not been previously acquainted with Logic. The terms not defined here are those used by logicians, and are supposed to be known. II. ON REASONING. The principle of geometrical reasoning is deduction, and the method for the whole science is synthesis. Although the general method of Geometry is synthetical, geometers often make use of analysis in particular cases, as a help to synthesis. Every new thing spoken of in a science must have some relation to a thing already known; the clear statement of such relation, made to distinguish the new thing from any other, is called a definition. The enunciation of a truth, or of what is supposed to be true, is called a proposition. Two propositions are said to be contradictory to each other, or simply contradictory, when the one affirms what the other depies; so that if the first be true, the second is false, and, vice versâ, if the first be false, the second is true. An axiom is a self-evident truth. Axioms are the foundation of reasoning; the following only are ́used in Geometry : 1. The whole is greater than its part. (Eucl. Ax. 9.) 2. The whole is equal to all its parts taken together. The name of postulate, or demand, is given to truths less evident than axioms, but the admission of which may be demanded without proof. As a consequence of the homogeneity of space, let the following general postulate be granted :— I. In the reduction of a figure, every part thereof is reduced. A theorem is a truth which is not evident of itself, but which becomes so by reasoning. The enunciation of every theorem is composed of two parts: the hypothesis, which is the supposition made on the subject of the theorem, and on which the reasoning must be founded; and the conclusion, which is the inference deduced from the hypothesis, and must be the aim of the reasoning. The hypothesis and conclusion are said to be complex when there is more than one supposition made on the subject, or more than one inference drawn from the hypothesis. The inverse theorem, or simply inverse of a theorem expresses that if the conclusion does not exist as true, the hypothesis does not exist either; the truth or falseness of the inverse of a theorem is directly deduced from that of the theorem itself. When the hypothesis, or the conclusion, is complex, there are as many inverses to the theorem as there are suppositions made on the subject or inferences drawn from the hypothesis. There are as many con- The name of converse theorem, or simply converse of a theorem, is given to a new theorem in which the conclusion is taken as a hypothesis, and the hypothesis as a conclusion. verses to a theorem as there are inverses. theorem expresses a fact more general than that which is expressed by the theorem itself, it often happens that the converse is false; in which case its inverse is equally false. The opposite theorem, or simply opposite of a theorem, is the inverse of its converse. When the opposite of a theorem already proved, is proved also, the theorem acquires, if possible, a greater appearance of truth. The reasoning by which the truth of a theorem is proved is called demonstration. There are several kinds of demonstrations : The direct, or synthetical, demonstration consists in stating propositions admitted to be true, deducing others as necessary consequences of them, then new ones from these, and so on until the proposed theorem is reached, and shown to be equally true. The only difficulty is to choose the starting propositions, and to choose among their consequences those leading to the proposed theorem. |