86 THE WORLD AS IDEA.
case of certain complicated mathematical propositions at which we only arrive by chains of syllogisms; for example, the calculation of the chords and tangents to all arcs by deduction from the proposition of Pythagoras. But even such a truth as this cannot essentially and solely rest upon abstract principles, and the space-relations which lie at its foundation also must be capable of being so presented a priori in pure intuition or perception that the truth of their abstract expression is directly established. But of mathematical demonstration we shall speak more fully shortly.
It is true we often hear men speak in a lofty strain of sciences which rest entirely upon correct conclusions drawn from sure premises, and which are consequently unassailable. But through pure logical reasoning, however true the premises may be, we shall never receive more than an articulate expression and exposition of what lies already complete in the premises; thus we shall only explicitly expound what was already implicitly understood. The esteemed sciences referred to are, however, specially the mathematical sciences, particularly astronomy. But the certainty of astronomy arises from the fact that it has for its basis the intuition or perception of space, which is given a priori, and is therefore infallible. All space-relations, however, follow from each other with a necessity (ground of being) which affords a priori certainty, and they can therefore be safely deduced from each other. To these mathematical properties we have only to add one force of nature, gravity, which acts precisely in relation to the masses and the square of the distance; and, lastly, the law of inertia, which follows from the law of causality and is therefore true a priori, and with it the empirical datum of the motion impressed, once for all, upon each of these masses. This is the whole material of astronomy, which both by its simplicity and its certainty leads to definite results, which are highly interesting on account of the
