limits," then the reply is that in such case, not the line itself is dealt with geometrically, but certain definite lines artificially put in quasi-definite relations to it, the indefinite becomes cognizable only through the medium of the hypothetically definite. Turning to the second illustration, the rejoinder to be made is that in so far as it concerns the relations between the ideal and the real, the analogy drawn does not shake but strengthens my argument. For whether considered under its geometrical or under its dynamical aspect, and whether considered in the necessary order of its development or in the order historically displayed, Astronomy shows us throughout that truths respecting simple, theoretically-exact relations, must be ascertained before truths respecting the complex and practically inexact relations that actually exist can be ascertained. As applied to the interpretation of planetary movements, we see that the theory of cycles and epicycles was based on pre-existing knowledge of the circle: the properties of an ideal curve having been learned, a power was acquired of giving some expression to the celestial motions. We see that the Copernican interpretation expressed the facts in terms of circular movements otherwise distributed and combined. We see that Kepler's advance from the conception of circular movements to the conception of elliptic movements was made possible by comparing the facts as they are with the facts as they would be were the movements circular. We see that the subsequently-learned deviations from elliptic movements were learned only through the presupposition that the movements are elliptical. And we see, lastly, that even now predictions concerning the exact positions of planets, after taking account of perturbations, imply constant references to ellipses that are regarded as their normal or average orbits for the time being. Thus, ascertainment of the actual truths has been made possible only by preascertainment of certain ideal truths. To be convinced that by no other course could the actual truths have been ascertained, it needs only to suppose any one saying that it did not concern him, as an astronomer, to know anything about the properties of circles and ellipses, but that he had to deal with the Solar System as it exists, to which end it was his business to observe and tabulate positions and directions and to be guided by the facts as he found them.