INTRODUCTION.

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

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of three equal sides, or a triangle in which one of the sides is much greater than either of the others, without, in the first place, considering whether the angles will be equal or unequal, or which of them will be the greater; but geometry teaches us, that if two sides of a triangle are equal, the angles opposite to the equal sides are also equal to each other; and if the sides are unequal, the angle which is opposite to the greater side will be greater than the angle which is opposite to the less.

The principles of most obvious authority in reasoning are the propositions laying down the sense in which the terms of the demonstration are to be understood. The student who uses a certain term to signify the conception expressed in detail in a given expression, will perceive, that every actual example of the thing signified must necessarily be possessed of the characteristics mentioned in the defining expression, because it is only by the possession of those characteristics that an actual object can earn a title to the designation in question. If I use the word triangle to signify a rectilineal figure of three sides, I can only recognise a particular figure as a triangle by the apprehension of the three straight lines of which it is composed; and, accordingly, I perceive that every triangle must necessarily be bounded by three straight lines.

Thus it is, that every definition rightly under

stood assumes the form of a necessary truth, or of a mere truism, in case the thing signified by the term defined (as in the foregoing example) is of such a nature that it cannot be made the object of contemplation without the distinct recognition of the analysis enounced in the definition; and if the premises in our systems of geometry had been composed exclusively of propositions owing their authority to such a principle, the necessity of the conclusions would have been involved in none of that mystery which has been so fertile a source of speculation. Hitherto, however, geometers have not succeeded in laying an adequate foundation of the science in definitions alone. It has always been found necessary, either openly or covertly, to call in the aid of axioms, or propositions, the truth of which we find ourselves compelled, after more or less reflection, to admit, although we may be unable to explain the intellectual process by which our assent is extorted.

In justification of the appeal to an authority of such a nature, the axioms are commonly spoken of as self-evident truths, to which appellation their claim has not been very clearly expounded. A self-evident proposition ought to carry conviction on the face of it irresistible to all who rightly understand the terms of the proposition, and this can only be the case when the correct conception of 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