characteristics which are uniformly found in all objects so named.  This, however, is not the case.[1] We speak of many things which we cannot represent:  names do not always stand for ideas.  The definition of the word triangle as a three-sided figure bounded by straight lines, makes demands upon us which our faculties of imagination are never fully able to meet; for the triangle that we represent to ourselves is always either right-angled or oblique-angled, and not—­as we must demand from the abstract conception of the figure—­both and neither at once.  The name “man” includes men and women, children and the aged, but we are never able to represent a man except as an individual of a definite age and sex.  Nevertheless we are in a position to make a safe use of these non-presentative but useful abbreviations, and by means of a particular idea to develop truths of wider application.  This takes place when, in the demonstration, those qualities are not considered which distinguish the idea from others with a like name.  In this case the given idea stands for all others which are known by the same name; the representative idea is not universal, but serves as such.  Thus when I have demonstrated the proposition, the sum of all the angles of a triangle is equal to two right angles, for a given triangle, I do not need to prove it for every triangle thereafter.  For not only the color and size of the triangle are indifferent, but its other peculiarities as well; the question whether it is right-angled or obtuse-angled, whether it has equal sides, whether it has equal or unequal angles, is not mentioned in the demonstration, and has no influence upon it. Abstracta exist only in this sense.  In considering the individual Paul I can attend exclusively to those characteristics which he has in common with all men or with all living beings, but it is impossible for me to represent this complex of common qualities apart from his individual peculiarities.  Self-observation shows that we have no general concepts; reason, that we can have none, for the combination of opposite elements in one idea would be a contradiction in terms.  Motion in general, neither swift nor slow, extension in general, at once great and small, abstract matter without sensuous determinations—­these can neither exist nor be perceived.

[Footnote 1:  Against the Berkeleyan denial of abstract notions the popular philosopher, Joh.  Jak.  Engel, directed an essay, Ueber die Realitaet allgemeiner Begriffe (Engel’s Schriften, vol. x.), to which attention has been called by O. Liebmann, Analysis tier Wirklichkeit, 2d ed., p. 473.]