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GEOMETRY is the science of form, position, and magnitude, the subject of which it treats consisting of figures drawn according to some definite law, while the aim of the science is the determination of relations of position and magnitude necessarily holding good between different parts of the figured system, though not expressly mentioned in the rule by which the latter is originally defined in the apprehension of the student. Thus, for example, the simplest kinds of figure are the straight line and the plane, and accordingly rectilineal figures, or figures constructed of straight lines and plane surfaces (and primarily the triangle as the rectilineal figure of fewest sides), form the earliest subject of geometrical investigation. Now the form of a triangle may be varied at pleasure, by changing the proportion between the sides, without necessarily raising the question, whether there be any corresponding variation in the proportion of the angles. We may imagine a triangle
geometry, enabling us to di the axioms, as with all ex al has always been regarded as the system.
As the only effective test of ment of the end which has so view, the system proposed is a ing pages to the geometry of of Euc simply in the
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I assumes the form of a necessary truth, or of re truism, in case the thing signified by the defined (as in the foregoing example) is of a nature that it cannot be made the object ntemplation without the distinct recognition e analysis enounced in the definition; and if remises in our systems of geometry had been Dosed exclusively of propositions owing their ›rity to such a principle, the necessity of the usions would have been involved in none of mystery which has been so fertile a source of lation. Hitherto, however, geometers have ucceeded in laying an adequate foundation of cience in definitions alone. It has always been d necessary, either openly or covertly, to call e aid of axioms, or propositions, the truth of h we find ourselves compelled, after more or reflection, to admit, although we may be unto explain the intellectual process by which assent is extorted.
n justification of the appeal to an authority of ha nature, the axioms are commonly spoken as self-evident truths, to which appellation their im has not been very clearly expounded. A 'f-evident proposition ought to carry conviction the face of it irresistible to all who rightly unrstand the terms of the proposition, and this n only be the case when the correct conception f the subject (as in definitions) involves the re
cognition of the features constituting the predicate of the proposition. To perceive the necessary truth of the proposition, that "if two straight lines meeting a third, make the two internal angles less than two right angles, the two straight lines shall meet if produced far enough" (the axiom of Euclid relating to parallel lines), requires an effort of the understanding essentially differing from the mere comprehension of the meaning of the proposition; and the axiom is probably at the outset accepted by a large proportion of students on the authority of the teacher without any clear apprehension of the evidence of the assertion. Before the geometer is contented to rest his system upon principles of whose authority he is able to render so little account, he ought to be thoroughly satisfied that he has exhausted the resources of definition, that his premises exhibit the ultimate analysis of the conceptions concerning which he proposes to reason, or their original construction out of the elementary materials of thought.
It requires little consideration to show, that such a limit is far from being attained in the ordinary system of geometry. It is a sufficient proof of shortcoming, that it contains no effective definition of a straight line. The assertion, that a straight line is "a line lying evenly between its extreme points," amounts to no more than this, that it is a line lying straight between its extreme
points; and as a proposition so manifestly identical can lead to no real advance in reasoning, the definition is never afterwards referred to, and forms no part of the real premises of the system.
The definitions of parallel straight lines, and of a plane surface, are as follows:
Parallel straight lines are such as are in the same plane, and being produced ever so far both ways do not meet.
A plane surface is that in which any two points being taken, the straight line between them lies wholly within such surface.
In neither of these cases does the definition exhibit a simple analysis of the essential meaning of the term defined. We can distinctly imagine a pair of parallel straight lines, or a plane surface, without a thought in our minds of the indefinite prolongation of the lines in the one case, or of the system of straight lines joining every separate pair of points in the plane, in the other case. We apprehend the planeness of a surface by passing our hand over it in a track, of which it is possible, that no portion may consist of a single straight line. The geometrical figure is in neither case defined by the relations of its own essential elements, but by conditions involving a reference to some external system, the notion of which necessarily presupposes the distinct conception of the figure under definition. We must plainly be