In so simple a figure as the circle lies the key to the solution of a multitude of problems, each of which would demand various appliances; whereas the solution results of itself, as it were, as one of the infinite number of elegant properties of this figure. Are we, for example, asked to construct a triangle, being given the base and vertical angle? The problem is indeterminate, _i.e._ it can be solved in an infinite number of ways. But the circle embraces them altogether as the geometrical locus of the vertices of triangles satisfying the given conditions. Again, suppose that two lines are to cut one another so that the rectangle under the segments of the one should be equal to the rectangle under the segments of the other; the solution of the problem from this point of view presents much difficulty. But all chords intersecting inside a circle divide one another in this _proportion_. Other curved lines suggest other purposive solutions of which nothing was thought in the rule that furnished their construction. All conic sections in themselves and when compared with one another are fruitful in principles for the solution of a number of possible problems, however simple is the definition which determines their concept.--It is a true joy to see the zeal with which the old geometers investigated the properties of lines of this class, without allowing themselves to be led astray by the questions of narrow-minded persons, as to what use this knowledge would be. Thus they worked out the properties of the parabola without knowing the law of gravitation, which would have suggested to them its application to the trajectory of heavy bodies (for the motion of a heavy body can be seen to be parallel to the curve of a parabola). Again, they found out the properties of an ellipse without surmising that any of the heavenly bodies had weight, and without knowing the law of force at different distances from the point of attraction, which causes it to describe this curve in free motion. While they thus unconsciously worked for the science of the future, they delighted themselves with a purposiveness in the [essential] being of things which yet they were able to present completely _a priori_ in its necessity. _Plato_, himself master of this science, hinted at such an original constitution of things in the discovery of which we can dispense with all experience, and at the power of the mind to produce from its supersensible principle the harmony of beings (where the properties of number come in, with which the mind plays in music). This [he touches upon] in the inspiration that raised him above the concepts of experience to Ideas, which seem to him to be explicable only through an intellectual affinity with the origin of all beings. No wonder that he banished from his school the man who was ignorant of geometry, since he thought he could derive from pure intuition, which has its home in the human spirit, that