inner rationality, rising up into the light of day, into the pure form of the Idea, which is the essence of every science and, in philosophy as the absolute science, is this pure Idea itself; of that peculiar and yet free scientific formation of a science, geometry gives a brilliant example, envied by the other sciences. Just as little is it the case that all reality must be denied to sciences constituted like those named above, on the ground that they are properly empirical; for just as every part or every side of philosophy is capable of being an independent science, so each is thereby immediately also an independent and complete image, and can be taken up and presented in the shape of an image drawn from an intuition which purely and happily keeps itself free from contamination by fixed concepts.

The completion of science, however, requires that intuition and the image be united with the logical and taken up into the purely ideal, just as it requires that the separated, though true, science be relieved of its singularity, and its principle be cognized according to its higher connection and necessity, and precisely thereby itself be completely liberated. This alone also makes it possible to cognize the boundaries of the science, boundaries of which, without this, it must remain in ignorance, because otherwise it would have to stand above itself and cognize the nature of its principle according to its determinacy in the absolute form; for from this cognition there would immediately follow for it the cognition and certainty of the extent of the equality of its various determinacies; but as it is, it can only relate to its boundaries empirically, and must now make false attempts to overstep them, now suppose them narrower than they are, and therefore undergo quite unexpected extensions; just as geometry too — which, for example, knows how to demonstrate the incommensurability of the diameter and the side of the square, but not that of the diameter and the circumference of a circle ¹ — gives an example, and still

1. Fichte prides himself somewhat (in the Introduction to his Natural Right) on the simplicity of his insight into the ground of this latter incommensurability, namely, that the curved is, in earnest, not straight. The superficiality of this ground is self-evident, and is also immediately refuted by the first incommensurability, that of the diameter and the side of the square — both of which are straight — as well as by the quadrature of the parabola. As for the aid that, in the same place, is sought from sound common sense against mathematical infinity — namely, that a polygon of infinitely many sides cannot, precisely because it is a polygon of infinitely many sides, be measured — this same aid would, on the one hand, have to be available also against the infinite progress in which the absolute Idea is supposed to realize itself; and on the other hand, nothing at all is thereby determined about the main point — positive infinity, which is not an infinite quantity but identity — namely, whether this is to be posited; which amounts to saying that nothing is determined about commensurability or incommensurability at all. ↩︎

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