RELATION (Mathematics). What we here mean by this word could have been explained in the article EQUATION, if we had confined urselves to the explanation of arithmetical algebra; but having in the Hides ALGEBRA and OPERATION endeavoured to give higher views, we re induced to insert the present article by remembering that the difli ulties of such a subject are of very different kinds to different persons, nsomuch that any point of view may be usefully taken with reference ,0 some minds, and any detail upon a fundamental notion may remove nisapprehension in one quarter or another.
All reasoning is the discovery of relations which are not evident from lose which are ; or rather, since the proposed result is sometimes svideut in itself, reasoning is the establishment of one relation as a necessary consequence of others. The term relation would be difficult o define in a manner satisfactory to all; it is enough for our present aurpose to say a relation exists between any two objects, whether, of sense or intellect, whenever they have anything in common ; that is to say, the common point, whatever it may be, may be made the means of referring one to the other, or bringing our thoughts from one to the Aber, so as to think of both at the same time, and to compare the two. All the manifold senses of the word may be derived from this one: the relationship of blood implies a common ancestry; the relationship of office, common duties. In mathematics, the relation of greater, equal, or less, implies that one of the magnitudes is the same as to quantity with part or all of the other, and so on. Sameness In every respect would constitute identity ; sameness in one or more respects, relation. The triangles in Euclid, i. 4, are by hypothesis related in a given manner in three particulars : a change of place shows that they can be made identical; that is, their difference before the change of place was difference of position only, not at all of form; in all that can distinguish one triangle from another, except its position in space, they are iden tical We do not quarrel with the phrase that they arc the same triangles differently placed, because sameness is understood with a reservation, and the preceding means that they are the same except in difference Of place.
In an algebraical expression we may have to consider its meaning, form, magnitude, source, mode of derivation, and properties. The meaning
depends upon the fundamental definitions which are employed and the form ; the form, upon the arrangement of the symbols ; the magni tude, when magnitude is signified, upon the form and the particular values given to the symbols; so that these various sources of relation arc closely connected with one another. The fundamental meaning of the sign = implies equality of quantity or magnitude, and some insist that it shall always retain this meaning. There can be no objection to any one insisting on this point for himself ; but the learner—who, if he be wise, will learn all languages with the majority, even though lie should afterwards teach with tho minority—must make himself accus tomed to various uses of this sign, as follows : 1. The sign = means that on one side we have an operation to be performed, and on the other side the result of performing that opera tion by general rules, as in (x + a) (x — a) = — as Whenever the resulting form is intelligible both in form and magni tude, the resulting relation is equality of magnitude under difference of form, independently of the particular values of the symbols ; but when the result is unintelligible, as is the second of the preceding results when first obtained, this relation no longer exists : the proceSs described in INTERPRETATION makes it exist. In all such cases the relation is that of sameness of value and properties, sameness in fact of everything but form; and the relation is independent of the magnitude of the algebraical symbols.
But did no relation exist in a', until we had interpreted the then unknown symbol ce to mean unity ? We answer, that a relation did exist—namely, sameness of properties. The value of the first side is unity ; the unknown symbol of the second side would be found on trial to have all the properties of the unit, when common algebraical rules are applied. If we were to refuse the interpretation, and consider e as a self-contradictory symbol, we could not deprive it of the properties of a unit, or rather, we could not deprive ourselves of the knowledge that the algebraical use of it would produce the same results as the algebraical use of a unit.