small."'"2 In a special theorem he emphasizes the actual character of elements so defined, exactly distinguishing them from the "indefinitely small" elements: the latter still remain in the process of diminution and belong to the series of finite magnitudes; but it would be meaningless to seek the infinitely small segments in the series." Bernoulli already believed the actual existence of infinitely small elements: "Sic omnes termini hujus progressionis, 1⁄2, 4, %, 18,.... actu existunt, ergo existit infinitesimus....si decem sunt termini existit utique decimus, si centum sunt termini, existit utique centesimus...ergo si numero infinito sunt termini, existit infinitesimus." But Bernoulli still seems to believe that the terminus infinitesimus is given in the series itself; of course, were this true, Veronese says, the concept of the infinitely small would involve a contradiction, because all the terms of this series according to the definition are finite. But it is logically possible to regard the series as defining a certain element beyond itself, which does not belong to the class of numbers given in the series, being always smaller than every term of it; and nevertheless this element may have some exactly determined and definable properties. In a certain circle of problems the assumption of such element may even be inevitable. Suppose we hypothetically accept the assumption of such a system, where "every finite segment, variable as to length and becoming indefinitely small, contains an element, which is different from its terminal points"; in the first place this presupposition is logically possible and implies the definition of infinitely small elements given above; in the second place it is obvious that certain systems (for instance the system of the spatial points) satisfy this hypothetically accepted condition. From what has just been said it seems to follow that 32 Veronese, Grundzüge der Geometrie von mehreren Dimensionen, p. 116. 38 Ibid., p. 141. 34 the infinitely small may have mathematical existence by itself. But still it remains true that all the determinations ascribed to this element are, properly speaking, of a derivative nature. What Veronese might have had in mind is only a convenient way of expressing certain properties and proportions in a certain class of systems called continuous. He explains himself clearly in this way. If the distance between the two foci of the ellipse remains only indefinitely small, without any suggestion of an element of a different order transcending the potential series of such indefinitely small distances, the circle might never be considered as a particular case of ellipse. The "actual existence" of an infinitely small distance in this case means nothing but the possibility of passing from the formula given for the ellipse to that for the circle. The "actual existence" has here no meaning beyond the methodical value of certain operations; and this methodical value of the "infinitely small" element consists in what it is doing in the system, rather than in what it is. The correlation between the finite distance and the infinitely small element defined by the process of its continuous determination, is no relation between quantities, but in our present case, the relation of affinity between two different laws (circle-ellipse). Their "truth" consists in what stands behind their formal definition, in the methodological back-ground of their "existence." Still in terms of our present example, we may say that the point (as infinitely small distance) can be regarded as a “part” of the line only because a certain class of analytic forms (circles) can be regarded as a "part" of another more general class of forms (ellipses). It is obvious that the problem here again harks back to the problem of qualitative infinity. Résumé. From what has been said it follows that infinity in all the cases of its application has a purely method84 Ibid., p. 144. ical value. It is not a "thing in itself," not something ready given and self-existent independent from science; it is not a "thing" of whatever sort; it is a method, rather a methodical aspect of reality than reality itself. I don't want to allege that it is a pure product of our mind, unless we understand this term "mind" in a purely logical sense, as a system of methodical presuppositions of science, action or art. Then and only then, in this exactly restricted sense of "our mind," it may be logically created by it, i. e., every instance of infinity may be and, as a matter of fact, is a result of certain presuppositions. The reproach of artificiality does not affect our position in any way; in this broad and vague sense everything may be called artificial; I don't see any reason why any finite magnitude or any limited field of experience is less artificial than a transfinite number. We are too much inclined to forget, that a long period first of biological adaptation and then of logical and mathematical reasoning were required to perceive the limits of the real objects and to conceive the meaning of the "end." It is an old truth that all the boundaries in this world are artificial, i. e., they are based upon a long system of presuppositions. But since these presuppositions are not artificial at all, since they have their meaning and purpose, the result of their logical activity loses its artificial character also. Thus to persist in the thesis that everything in our world of experience is limited, is in itself a logical limitedness: It must be considered as a modern positivistic extreme, as a reaction against the metaphysical exaggeration of the value of infinity. The opposition of finite and infinite is not a contraposition of the different realms or worlds separated from each other; it is only a cooperation of two different methods one of which is quite as justified as the other. HENRY LANZ. PALO ALTO, CALIFORNIA. R IMAGINATION. SERVANT OR MASTER. EASON'S eye is calm and steady, Gazing ever straight ahead, Seeing clearly every object In its level vision spread. But Imagination cries: "Look upward! Never master had a servant Who could give him such delight, But 'tis well that Reason watch her, See her safely home at night. The scholar struggles slowly Through the records of the past, Pondering o'er their meaning vast. Breaks from Reason's curbing rein He who walks beside the river Silent, motionless, completed, Like the changeless truth of God. There's a pathway up the mountain, Steep, laborious, and slow, Lighted only by the witch-fire Of Imagination's glow. That lone path which thought has traveled Since the Reason's earliest youth, Struggling upward toward the cloud-cap That still veils the Greater Truth. Not for fame and not for riches Of revealing hidden lights. There's no joy for human nature Like the mind's exultant thrill When the new-born thought leaps living, Bringing that ecstatic chill |