Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members. On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the student by a point moving in accordance to this law, is the parabola.
If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates,—reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology … bears to anatomy.

It has hitherto been a serious impediment to the progress of knowledge, that is in investigating the origin or causes of natural productions, recourse has generally been had to the examination, both by experiment and reasoning, of what might be rather than what is. The laws or processes of nature we have every reason to believe invariable. Their results from time to time vary, according to the combinations of influential circumstances; but the process remains the same. Like the poet or the painter, the chemist may, and no doubt often' does, create combinations which nature never produced; and the possibility of such and such processes giving rise to such and such results, is no proof whatever that they were ever in natural operation.

Nature has but one plan of operation, invariably the same in the smallest things as well as in the largest, and so often do we see the smallest masses selected for use in Nature, that even enormous ones are built up solely by fitting these together. Indeed, all Nature’s efforts are devoted to uniting the smallest parts of our bodies in such a way that all things whatsoever, however diverse they may be, which coalesce in the structure of living things construct the parts by means of a sort of compendium.

The Law of Causation, the recognition of which is the main pillar of inductive science, is but the familiar truth that invariability of succession is found by observation to obtain between every fact in nature and some other fact which has preceded it.

The universe is what it is and can’t be changed by jiggery-pokery. It works by exact rules, like a machine. … Natural law never takes a holiday. The invariability of natural law is the cornerstone of science.

There is nothing which Nature so clearly reveals, and upon which science so strongly insists, as the universal reign of law, absolute, universal, invariable law... Not one jot or tittle of the laws of Nature are unfulfilled. I do not believe it is possible to state this fact too strongly... Everything happens according to law, and, since law is the expression of Divine will, everything happens according to Divine will, i.e. is in some sense ordained, decreed.