387-389. "Nor can we fail to notice how frequently the law which men have invented proves to have been already known and used in nature. The mathematician devises a geometric locus or an algebraic formula from _a priori_ considerations, and afterward discovers that he has been unwittingly solving a mechanical problem, or explaining the form of a real phenomenon. Thus, for example, in Peirce's 'Integral Calculus,' published in 1843, is a problem invented and solved purely in the enthusiasm of following the analytic symbols; but in 1863 it proved to be a complete prophetic discussion and solution of the problem of two pendulums suspended from one horizontal cord. Thus also Galileo's discussion of the cycloid proved, long afterward, to be a key to problems concerning the pendulum, falling bodies, and resistance to transverse pressure. Four centuries before Christ, Plato and his scholars were occupied upon the eclipse as a purely geometric speculation, and Socrates seemed inclined to reprove them for their waste of time. But in the seventeenth century after Christ, Kepler discovers that the Architect of the heavens had given us magnificent diagrams of the eclipse in the starry heavens; and, since that time, all the navigation and architecture and engineering of the nineteenth century have been built on these speculations of Plato. Equally remarkable is the history of the idea of extreme and mean ratio. Before the Christian era geometers had invented a process for dividing a line in this ratio, that they might use it in an equally abstract and useless problem--the inscribing a regular pentagon in a circle. But it was not until the middle of the present century that it was discovered that this idea is embodied in nature. It is hinted at in some animal forms, it is very thoroughly and accurately expressed in the angles at which the leaves of plants diverge as they grow from the stem, and it is embodied approximately in the revolutions of the planets about the sun.... Now, in all these cases of the embodiment in nature of an idea which men have developed, not by a study of the embodiment, but by an _a priori_ speculation, there seems to us demonstrative evidence that man is made in the image of his Creator; that the thoughts and knowledge of God contain and embrace all possible _a priori_ speculations of men. It is true that God's knowledge is infinite, and beyond our utmost power of conception. But how can we compare the reasonings of Euclid upon extreme and mean ratio with the arrangement of leaves about the stem, and the revolutions of planets around the sun, and not feel that these phenomena of creation express Euclid's idea as exactly as diagrams or Arabic digits could do; and that this idea was, in some form, present in the creation?"--The Natural Foundations of Theology. By T. Hill, D.D., LL.D.