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the magnitude of the angle made by those lines (which is measured by the quantity of plane surface intercepted between them abutting on the point of intersection) will continually increase as the direction of the moveable arm approaches to the relation of transverseness, or as distance in the direction C P includes a larger proportion of distance in the direction C B, and a less in the direction CA. Thus the magnitude of the angle between two straight lines may be used to define their relative direction; and as the magnitude of any angular segment may always be expressed by a direct numerical ratio to that of the rectangular segment between straight lines in transverse directions (which itself constitutes one quarter of the whole superficial expanse round any point in a plane), it affords a more simple and often more convenient measure of the difference in direction than the fundamental distinction, by reference to the decomposition of distance in the direction of one arm of the angle in certain proportions into distance in the direction of the other arm, and in a direction transverse to it respectively.
The principles which govern the relations of magnitude, Equal, Greater, and Less, are laid down in the nine first axioms of Euclid, the whole of which may conveniently be replaced by a single proposition enouncing the conditions by which the equality of given dimensions, or the superior magnitude of one or the other of them, is ultimately to be determined.
The meaning of equality, and of greater and less, is identity or excess, on one side or the other, in respect of the quantity of space occupied by the dimensions compared ; and thus the relative magnitude of particular dimensions must ultimately be tried by bringing them both actually to occupy the same space, or making one of them to occupy a certain space which is wholly filled by a part of the other. And such, in fact, is the end to which all processes of measurement are directed, either by bringing the dimensions compared into actual coincidence with each other (taking them to pieces when necessary) or with a third magnitude which can readily be compared with both the original dimensions. By such means, it is made evident to sense whether they occupy the same space, or on which side is the excess. In like manner, in order to demonstrate or make evident to the understanding the relative magnitude of dimensions which are the object of that faculty, as the dimensions occupying certain positions in definite species of figure, it will suffice to shew, from the construction of the figures in which they lie, that the magnitudes compared, or their component parts, may be applied to each other, so as wholly to coincide or occupy the same space, or to coincide to the whole extent of one of the magnitudes, leaving
a portion of the other unoccupied. The result in the former case, will be to shew that dimensions of the kind in question are necessarily equal to each other; in the latter, that the dimension which wholly coincides with a part of the other, is the less of the two. Hence the proposition which is to replace the axioms of Euclid relating to equality and inequality.
Two equal magnitudes are such, that one of the two (or the sum of all its parts) may be made to coincide with the other or the sum of all its parts. But if one of the two magnitudes, or the sum of all its parts, coincide with a portion of the other, leaving a portion of the second magnitude unoccupied, the including magnitude is said to be the greater, the included the less.
The process by which it is shewn that one figure may be applied to, or made to coincide with, another, for the purpose of ascertaining their relative magnitude, is called the proof by superposition, and is, in truth, the process upon which all demonstration of equality or excess must fundamentally rest.
The nature of the reasoning is, however, not unfrequently misapprehended by beginners, and the basis of the demonstration looked on with suspicion, as if the certainty of the conclusion would thus be left to depend upon the exactitude of particular measurements. But it must be observed, that the process does not consist in actual measurement of the figures by which the reasoning is illustrated, but in showing from the conditions of the case, that figures constructed according to a certain plan, when properly applied to each other, must necessarily either wholly coincide, or partially coincide, leaving a clear excess on the one side or the other.
Whatever may have been the reason which gave Euclid a prejudice against this necessary part of the proof, it is certain that he keeps it as much as possible in the background, and frequently resorts to circuitous reasoning and ex absurdo proofs, in order to demonstrate propositions, which might be proved at once by direct superposition. But nothing is added to the cogency of demonstration by the length of the deductive process; and it would surely be more philosophical in all cases to lead the student, by the shortest path, to the basis on which his conviction must ultimately rest.
The definition of proportion, in the fifth book of Euclid, is often the source of difficulty to the student, who is conscious that the very complex relation described in that proposition, is not what he understands under the name of proportion, and is embarrassed by the real inconsequence of applying the results of Euclid's demonstrations to proportional quantities in his own sense of the term. The truth appears to be, that Proportion is
an elementary relation, the discernment of which between distances observed on different occasions, originally gives rise to the conception of magnitude, as a quality of which the distances compared exhibit different phases; in the same way that the discrimination of blue and white and red gives rise to the idea of colour, as the quality embracing the whole of these phenomena. If colour consisted exclusively of white and black, one tint would differ from another only in the degree of illumination, and the case would be precisely analogous with that of magnitude, which differs only in degrees of more or less. A greater magnitude is then, fundamentally, a dimension which bears a greater proportion than a second to some common standard ; or conversely, a dimension to which the common standard bears a less proportion than to a second dimension; while the second dimension is, in either case, determined by the same fundamental conditions as the less of the two magni.. tudes.
Thus an increase of the antecedent of two dimensions, or a diminution of the consequent, corresponds to an increase of the proportion between them, and conversely a diminution of the antecedent, or an increase of the consequent to a dimi, nution of the proportion. It is evident then, that by making both members of a proportion greater or smaller in the same proportion, that is, by