On these two different objects of physical investigation, the co-existing elements of bodies, and their successions of changes, it may be of advantage to dwell a little more fully in elucidation of the method which we have to pursue in our own department of physical research; for, though it may perhaps at first appear to you, that to treat of the principles of inquiry, in the physics of _matter_, is to wander from the intellectual and moral speculations which peculiarly concern us; it is in truth only as they are illustrative of inquiries which we are to pursue in the _physiology of the mind_, that I am led to make these general remarks. The principles of philosophic investigation are, as I have already said, common to all the sciences. By acquiring more precise notions of the objects of any one of them, we can scarcely fail to acquire, in some degree, more precision in our notions of every other, and each science may thus be said to profit indirectly by every additional light that is thrown upon each. It is by this diffusive tendency of its spirit, almost as much as by its own sublime truths, and the important applications of these to general physics, that the study of geometry has been of such inestimable advantage to science. Those _precise definitions_ which insure to every word the same exact signification, in the mind of every one who hears it pronounced, and that lucid progress in the developement of truth after truth, which gives, even to ordinary powers, almost the same facility of comprehension with the highest genius, are unquestionably of the utmost benefit to the mathematical student, while he is prosecuting his particular study, without any contemplation of other advantages to be reaped from them. But there can be no doubt that they are, at the same time, preparing his mind for excellence in other inquiries, of which he has then no conception; that he will ever after be less ready to employ, and be more quicksighted than he would otherwise have been in detecting vague and indefinite phraseology, and loose and incoherent reasoning; and that a general spirit of exactness and perspicuity may thus at length be diffused in society, which will extend its influence, not to the sciences merely, but, in some faint degree, also to works of elegant literature, and even to the still lighter graces of conversation itself. “The spirit of geometrical inquiry,” says Fontenelle, “is not so exclusively attached to geometry, as to be incapable of being applied to other branches of knowledge. A work of morals, of politics, of criticism, or even of eloquence, will, if all other circumstances have been the same, be the more beautiful, for having come from the hand of a geometrician. The order, the clearness, the precision, which, for a considerable time, have distinguished works of excellence on every subject, have most probably had their origin in that mathematical turn of thought, which