species or orders of polygons is equal to the number of the prime numbers, underm or (m-1), m representing the number of points. Thus, for five points, we have two orders; for seven points, three orders; for nine points, only two orders: but, for eleven points, we have five, and so on. Also, the sum of all the inward angles of any polygon is for the first order 2 (m-2) right angles; for the second order = 2 (m-4) right angles; for the third order = 2 (m-6) right angles, &c. where m is the number of points, or number of sides, in the original polygon. Having laid down some of these general principles and definitions, the author proceeds to shew the analogy which these figures bear to the common polygons of the same denomina tion; after which he investigates, in a similar manner, the several orders of polyhedrons; in both cases deducing many very neat and curious results, which our limits will not allow us to exhibit. The analytical investigation of the figures, however, we are informed, is much more interesting than their geometrical properties, which from the nature of the subject we can readily conceive to be the case; and we shall be glad of the opportunity, as M. POINSOT proposes to continue his researches, of giving our readers, in a subsequent number, a sketch of his analytical processes. It may be that the discussion is devoid of any great útility, but it is certainly curious and interesting; and we can see no reason for making mathematicians the only persons who are to be denied the privilege of amusing themselves at times in their own way, with light and easy reading. ART. XV. Mémoires de la Classe des Sciences, &c.; i. e. Memoirs of the National Institute of France, Vols. VII.-X. THE [Article concluded from the last Appendix, p. 544-] HE Memoirs of the year 1809 are, as usual, preceded by an analysis of the labours of the class, divided into the Mathematical and Physical departments, the former drawn up by M. DELAMBRE, the latter by M. CUVIER. PHYSICAL PART. Vol. X. The reporter commences with the discoveries that have taken place in chemistry during the year, and occupies several pages with a sketch of the experiments of Sir H. Davy and MM. GayLussac and Thenard on the decomposition of the alkalies, and other subjects in which the action of the Galvanic pile is employed to produce chemical decomposition. The account is drawn up with M. CUVIER's usual accuracy and clearness, but, in consequence of the additional information that has been received since it may be supposed to have been written, it is in a considerable degree superseded; so that we cannot be certain how far it gives a correct representation of the opinions of the French at the present period. The writer next refers to M. Gay-Lussac's important experiments on the proportion in which the different gases combine; which is now admitted to be in the simple ratio of their bulk, and without any relation to their weight. The remaining subjects are not individually of great importance, although proper to be noticed in their places. In the department of Mineralogy, is an account of M. Guyton's observations of a new form of the crystallization of the diamond, with his remarks on the diminution which lead experiences in its specific gravity by the operation of hardening; and we learn that M. Sage has discovered that the chrysolite of volcanoes, reduced to powder, may be substituted for emery.In Geology, our attention is occupied by the detail of the interesting discoveries of M. Cuvier, who has completed the mineralogical geography of the environs of Paris, and has exa'mined many parts of the shores of the Mediterranean, in pursuit of his favourite object, the fossil remains which are found in them. A rock, similar to that of Gibraltar, is discerned in other "parts, composed of a red cement, which unites together, in a confused manner, numerous fragments of bones and the wreck of calcareous strata. The bones contained in these rocks all belong to herbivorous animals, the greatest part of the kinds of which are known, and still exist in the same places: but the rocks are probably of considerable antiquity, because some of them include bones of animals at present unknown. We have also an account of other kinds of strata and alluvial soils, with the remains found in them, many of which belong to species no longer seen. Among the notices of the progress of Botanical science, we find that M. Jussieu, who is said to be the legislator of bo'tanical methods,' has established a new order of plants under the name of Monimea, which is placed immediately before the 'family of the Urticea. M. Palisot- Beauvois is said to have studied the organs of fructification in the Graminea more accurately than they had been before examined. M. Labillardiere has described a new Palm, under the name of Ptychosperma, a native of New Ireland. M. Lamoureux has paid particular attention to marine plants, and describes their texture and parts of fructification : but it does not appear that he has advanced any new opinion on these points. M. Mirbel is said to be still continuing his researches on vegetable physiology; and he brings forwards some experiments in order to prove more decisively than others had done - done before, that the albumen of the seed nourishes the young plant after germination: as also some new researches on the germination of the nelumbo. Lastly, M. Poiteau has been occupied with examining to what part of the seed of the gramince the name of cotyledon is applicable. — In Zoology, we have an account of some memoirs by MM. Cuvier and Geoffroy, on particular classes of animals, which are principally important as tending to establish their distinguishing characters with increased precision. Under the section of Physiology, we have a sketch of a set of experiments by MM. Humboldt and Provençal, on the respiration of fish; and also of another set by the latter on the respiration of the mammifera after the division of the eighth pair of nerves, and on the air-vessel of fishes, a subject which has likewise attracted the attention of M. Cuvier.MM. Magendie and Delisle have performed a number of experiments on the poison with which the natives of Java and Borneo infect their weapons, extracted from the Upas tree; and the juice or extract from this tree is found to act as a very subtile poison, although it is generally understood that the stories respecting its vapor or exhalation are entirely fabulous. The report concludes with a few notices on Medicine and Surgery, and on Agriculture. MEMOIRS. On the Tenacity of ductile Metals, and Observations on the Changes of Density in Lead by the Processes in which it is hardened, and on its alteration in Water. By M. GUYTONMORVEAU.The principal objects of attention in this paper are two facts respecting lead, both of which appear to be anomalous; Ist, that lead becomes of less specific gravity, when it is hardened by hammering or by being drawn into wires; 2dly, that distilled water acts more readily on this metal than common river-water. The increase of density in lead, by the process of hardening, seems to have been originally noticed by Musschenbroeck, but was simply stated as a fact, without any attempt at explanation. The first object of the present author was to verify it, and to examine it under all its different relations. Having obtained specimens of very pure lead, he subjected them to blows of the hammer, and examined the effect which was produced; he then passed the lead through a rolling press, and afterward had it drawn into wire, and also strongly compressed by a stamp, while inclosed in a die. When the lead was submitted to strong pressure, but so confined that its bulk was not increased, the specific gravity was augmented, as is the case with the other metals. It was in the course of these experiments that the author was induced to notice the action of distilled water on lead: which action, he observed, was not exercised when common spring or river 薯 water was employed; and it was found to depend on small portions of some of the neutral salts that exist in this latter fluid. The substance formed appears to be a hydrate of lead. Sketch of the present State of vegetable Anatomy and Physiology. By M. MIRBEL.This gentleman, who is justly celebrated for the acumen which he has exercised on the subject of the present paper, traces in a few words the history of the science, from Theophrastus, through Grew and Malpighi, to our own times. He then describes the essential difference between the monocotyledonous and the dicotyledonous plants, with respect to their mode of germination and the structure of the future stem; remarks on the vegetable tissue, as consisting of a system of membranous cells; and speaks of the tubes with which plants are furnished, and the manner in which the fluids are carried through them. He then inquires by what force the fluids are attracted by the vessels from the earth and the air. He supposes that a vital power is inherent in the vegetable tissue, which operates to produce this attraction: but respecting its mode of acting we are left without any certain information. This incomprehensible power, as he terms it, resides in the soft and delicate liber, or inner bark, of which the buds, leaves, young roots, &c. are said to be the expansion. The essential difference between annual and perennial plants, the causes of the decay and death of vegetables, and the progressive changes which operate for their reproduction, as well as for the gradual developement of the different organs of the embryo, are then traced, according to the opinions which were adopted on the continent when this paper was written. Many alterations, however, have taken place in this country during the last six years, in consequence of the ingenious hypotheses of Mr. A. Knight; which, together with the interesting microscopical discoveries of Mrs. Ibbetson, have produced a considerable revolution in our theories of vegetable physiology, espe cially on some of those points to which M. MIRBEL had more particularly directed his researches. MATHEMATICS and ASTRONOMY, &c. The History in this division is neither very long nor very important. It relates principally to the Memoirs in this and the preceding volume, which we either have noticed or shall have occasion to notice in the present article. Much of it, also, is occupied in discussing the great generality and utility of Lagrange's papers on the variation of arbitrary constant quantities; and of a memoir by Poisson, published in the "Journal de l'Ecole Polytechnique," on the new problem which Lagrange had had announced in his last paper, relative to the uniformity of rotation in the earth and other planets; and respecting which we have not room now to enter into particulars. From the memoir, which is not given in the volume, the author has drawn the following remarkable conclusion, viz. "That the perturbations of the motion of rotation of solid bodies of any figure, and arising from any attractive forces, depend on the same equa tions with the perturbations in the motion of a point attracted towards any fixed centre;" so that the precession of the equinoxes, and the nutation of the terrestrial axis, will be expressed by the same formula which give the variations in the elliptic elements of the planets. MEMOIRS. Researches on the extraordinary Refractions which are observed very near the horizon. By M. BIOT.In this very long and interesting memoir, the author commences by an historical sketch of the labours of different philosophers relative to this singular phænomenon; detailing many of their observations, and examining several of their theories, particularly those of Professor Vince, Dr. Wollaston, and M. Monge. M. BIOT's object is to establish a mathematical theory of the extraordinary refractions observed very near the horizon. He had previously turned his attention to this interesting subject, when he was sent to Dunkirk with M. Mathieu to ascertain the latitude of that northern extremity of the French arc; which situation offered a fine opportunity for making experiments on this terrestrial refraction. In company with his colleague, he observed the depression of the horizon from different known heights, and under various barometrical pressures and degrees of temperature; and he has here detailed, and represented by plates, all the varieties and singularities that were noticed; which these gentlemen followed with the greatest care and perseverance, and measured with the utmost accuracy by means of a repeating circle. They have also detailed with precision the different degrees of temperature, the depression of the apparent horizon and of the objects, their distances, their configur ations, and the relation of their real form to that of their reflected images. Finally, they have determined the nature of the trajectories, and those of the caustics which limit them, from direct experiment; and they have ultimately reduced, to one general and comprehensive mathematical theory, phænomena which, from their variety and transitory nature, seemed to bid defiance to the powers even of the modern analysis. Memoir on the Motion of Light in diaphanous Mediums. By M. LA PLACE.-We have here a remarkable application of the principle of least action to the phænomena of ordinary and extraordinary 8 |