necessitated to cognise this if we have only to do _a priori_ with the formal purposiveness of our representations; but the fact that we are compelled to look out beyond it inspires at the same time an admiration for the object that impels us thereto. We are accustomed to speak of the above-mentioned properties of geometrical figures or of numbers as _beautiful_, on account of a certain _a priori_ purposiveness they have for all kinds of cognitive uses, this purposiveness being quite unexpected on account of the simplicity of the construction. We speak, _e.g._ of this or that _beautiful_ property of the circle, which was discovered in this or that way. But there is no aesthetical act of judgement through which we find it purposive, no act of judgement without a concept which renders noticeable a mere _subjective_ purposiveness in the free play of our cognitive faculties; but an intellectual act according to concepts which enables us clearly to cognise an objective purposiveness, _i.e._ availableness for all kinds of (infinitely manifold) purposes. We must rather call this _relative perfection_ than a beauty of the mathematical figure. To speak thus of an _intellectual beauty_ cannot in general be permissible; for otherwise the word beauty would lose all determinate significance, or the intellectual satisfaction all superiority over the sensible. We should rather call a _demonstration_ of such properties beautiful, because through it the Understanding as the faculty of concepts, and the Imagination as the faculty of presenting them, feel themselves strengthened _a priori_. (This, when viewed in connexion with the precision introduced by Reason, is spoken of as elegant.) Here, however, the satisfaction, although it is based on concepts, is subjective; while perfection brings with itself an objective satisfaction. § 63. _Of the relative, as distinguished from the inner, purposiveness of nature_ Experience leads our Judgement to the concept of an objective and material purposiveness, _i.e._ to the concept of a purpose of nature, only when[105] we have to judge of a relation of cause to effect which we find ourselves able to apprehend as legitimate only by presupposing the Idea of the effect of the causality of the cause as the fundamental condition, in the cause, of the possibility of the effect. This can take place in two ways. We may regard the effect directly as an art product, or only as material for the art of other possible natural beings; in other words, either as a purpose or as a means towards the purposive employment of other causes. This latter purposiveness is called utility (for man) or mere advantage (for other creatures), and is merely relative; while the former is an inner purposiveness of the natural being.