Besides the leading features of the work which we have thus briefly described, it contains many new and beautiful demonstrations of general principles in Geometry, to which the author was lead by his new methods of investigation. Among these we may mention one, viz., "The square of the hypothenuse of a right-angled triangle equals four times the area of the triangle, plus the square of the difference of the other two sides." This principle has been known to mathematicians by means of arithmetic and algebra, but has never before, we believe, been reduced to a geometrical demonstration. The demonstration of this principle by Mr. Smith is one of the clearest, simplest, and most beautiful in Geometry. The work is divided into three parts, I. The Philosophy of Geometry, II. Demonstrations in Geometry, and III. Harmonies of Geometry. The demonstrative character of it is occasionally enlivened by philosophical and historical observations, which will add much to its interest with the general reader. We have too little skill in studies of this sort to be altogether confident in our opinion, but certainly it strikes us from an examination of the larger and more important portion of Mr. Smith's essay, that it is an admirable specimen of statement and demonstration, and that it must secure to its author immediately a very high rank in mathematical science. We shall await with much interest the judgments of the professors. It makes a handsome octavo of some 200 pages. * * * * * M. FLANDIN, an eminent dilettante and designer attached to the French embassy in Persia, has published in the last number of the _Revue des Deux Mondes_ an interesting memoir of the ruins of Persepolis, under the title of "An Archaiological Journey in Persia." On his route to the ruins he witnessed melancholy evidence, in the condition of the surface and population, of the improvidence and noxiousness of Oriental despotism. He tells us that the remains of the magnificent palace of Darius are dispersed over an immense _plateau_, which looks down on the plain of Merdacht. "Assuredly, they are not much, compared with what they must have been in the time of the last Prince who sheltered himself under the royal roof. Nevertheless, what is now found of them still excites astonishment, and inspires a sentiment of religious admiration for a civilization that could create monuments so stupendous; impress on them a character of so much grandeur; and give them a solidity which has prereserved the most important parts until our days, through twenty-two centuries, and all the revolutions by which Persia has been devastated. The pillars are covered with European names deeply cut in the stone. English are far the most numerous. Very few, however, are of celebrated travelers. We observed, with satisfaction, those of Sir John Malcolm and Mr. Morier, both of whom have so successfully treated Persian subjects." * * * * *