character; also remains of plants, saurians, turtles, &c. It is remarked that the eocene and miocene of the Atlantic states contain in all their numerous organic remains only about 2 or 3 species to the hundred in common; a fact indicative of the lapse of a considerable period between the completion of the one and the commencement of the other, during which no strata were deposited in that region. With the progress to later members of the tertiary, the similarity of the animal vestiges, especially of the testacea, to living forms rapidly increases. Of the larger land quadrupeds the species and many of the genera even, as of the mastodon, were lost before the present epoch. In some of the comparatively modern beds of this group found in Georgia between the Altamaha and Turtle rivers, containing shells of the same species with those inhabiting the neighboring waters, are found remains of the extinct species of the horse, hippopotamus, bison, elephant, mastodon, megatherium, and mylodon. Remains of the same elephant and mastodon are also found in New Jersey and New York beneath the beds of peat lying in the upper portion of the pleistocene drift. The genera of fossil shells peculiar to the principal geological divisions are thus given by Woodward in his "Manual of the Mollusca." Some of the genera named have a wider range, but are found most abundant in the division with which they are included in the table. Such is the case with the belemnites. The names in italics are of existing genera. Divisions. PLIOCENE. MIOCENE. EOCENE. UPPER CRETACEOUS. Beloptera, lychnus, megaspira, Glandina, typhis volutilithes, clavella, pseudoliva, seraphs, rimella, conorbis, strepsidura, globulus, phorus, velates, chilostoma, vol varia, lithocardium, teredina. Belemnitella, conoteuthis, turrilites, ptychoceras, hamites, scaphites, pterodonta, cinulia, tylostoma, Acteonella, globiconcha, trigonosemus, magas, lyra, Neithea, inoceramus, hippurites, caprina, caprotina. LOWER CRE- Crioceras, toxoceras, hamulina, baculina, requienia, caprinella, sphæra, thetis. Coccoteuthis, acanthoteuthis, leptoteuthis, nautilus, spinigera, purpurina, nerinæa, neritoma, pteroperna, trichites, hypotrema, diceras, trigoma, pachyrisma, Sowerbia, Tan TACEOUS. UPPER JUBASSIC. LOWER JUKASSIC. TRIAS. PERMIAN. CARBONIFEROUS. DETONIAN. credia. SILURIAN. CAMBRIAN. By turning Actinoceras, phragmoceras, trochoceras, ascoceras, theca, holopella, Murchisonia, atrypa, Retzia, cardiola, clidophorus, grammysia, goniophorus. Camaroceras, endoceras, gonioceras, pterotheca, Maclurea, raphistoma, holopea, platyceras, orthisina, platystrophia, porambonites, pseudo-crania, ambonychia, modiolopsis, lyrodesma. back to the groups of strata and the vestiges of organic beings which they successively introduce, one cannot fail to perceive a marked progress in the condition of the surface of the earth, adapting it for higher organizations of animal life, and at the same time the appearance of these forms, as it seems by successive acts of creation. The broad seas of the lower silurian epoch, stretching over large portions of the globe, contain their peculiar types of marine life, crustacea, cephalopoda, and algæ, all perfect of their kind, and as admirably constructed as their representatives found in modern seas. To these invertebrata next appeared in the deposits left by the waters of the devonian period the first vertebrate animals in the form of fishes. Aquatic reptiles succeed; and when in the luxuriant vegetation of the carboniferous epoch land appears, it is productive only of those forms of animals adapted to moist tropical districts scarcely elevated above the sea level. No vestiges of mammalia or of birds yet appear. The reptiles first attain their highest development in the lias and the oolite periods; and at last in the tertiary epoch, distinguished for its broad areas of dry land, the mammalia are found in numbers and variety which continually increase up to the introduction of their highest representative, man. This general scheme of the creation is established by most careful geological investigations conducted in various parts of the world. New discoveries may somewhat modify its minor features, but where uniformity in the operations of nature is found to obtain so generally, and no exception is detected, it is unphilosophical to refuse to accept the evidence because the whole has not yet been submitted to our examination.-In what manner new species have been introduced is a question that has occupied the closest study of many philosophers, but which remains in the same obscurity as the origin of individual life. Some, adopting the views of Geoffroy St. Hilaire, contend that the ancient species were the progenitors of all that succeeded and that now live. This view was supported by De Lamarck, and ingeniously sustained by the authors of the "Vestiges of Creation." But the arguments adduced to support the doctrine of the so called transmutation of species, which if established destroys the real existence of species in nature, were based on assumed facts, or carried out to an extent unwarranted by any of those within our experience. The changes of form to which species are subject in successive generations by change of circumstances are very limited, and appear to be restrained within certain bounds by constant laws. No instance has ever been established of one species being succeeded after many generations by another even of the same genus now exhibit. It is shown that even the climate or family; in the immense collections of fossil of any part of the world may be materially remains which have been most carefully studied modified by the varying conditions of portions no evidence of a passage of this character has of its surface, whether covered by water or been proved. On the other hand, instances are raised high above the ocean's level. The existevery where encountered in which new types of ence of elevated continents in polar latitudes, life appear, not through any gradation, but so and at the same time of seas spreading over the far as we can judge by a sudden step, the strange equatorial regions, would necessarily produce a being having none of its genus, order, or even cold climate over large portions of the earth; class to point back to as its possible progenitor. while, the conditions being reversed, a tropical Moreover, the fossil remains present no evidence climate might reach into high latitudes, and of a progress in the succession of species to those plants of kindred character to those that make of higher organization leading to higher types up the ancient coal beds might again thrive of life. On the large scale a progress to higher where the forest remains of the carboniferous orders is perceived, as already observed; but the epoch now lie buried beneath perpetual snow. earliest silurian trilobite exhibited in the struc- It is not perhaps extravagant to suppose that ture of that most delicate organ, the eye, the same such conditions of the surface may in former complexity and wonderful perfection that is per- times have obtained; for the rocks themselves ceived in the eye of the fly and butterfly of the testify, in the marine fossils they contain, that present day, and which is perhaps unequalled in almost all parts of the earth's surface have been living crustacea; while the fishes possessed as beneath the level of the sea; and changes of perfect an organization as that belonging to any level, similar to those which have been observed of the later members of the same families.-In by man, either suddenly taking place, as in the the sketch thus far presented of the develop- rising at different times of the coast of Chili, ment of geological science no special attempt or by a constant imperceptible motion like that has been made to distinguish between the sub- which has continued for centuries to raise the jects relating to descriptive and those belonging coast of Sweden, and that again which in like to the department of physical geology. As the manner carries down the coast of Greenland phenomena introduced suggested the causes of their production, these have been incidentally considered, and thus it is that some of the grandest subjects of physical geology, especially those of a speculative nature, have escaped notice. In both departments important omissions must be observed, which in part at least may be attributed to the necessary limits imposed upon the present article, and to the treatment of their subjects under separate heads. Thus the phenomena of some of the mighty forces which disturb the arrangement and modify the structure of the strata are partially treated in the articles CENTRAL HEAT, Earthquake, ETNA, and VOLCANO; of those which distributed some of the comparatively recent deposits, in ALLUVIUM, DILUVIUM, and GLACIER; and the mode of production of many calcareous beds is explained in ATOLL, and CORAL. The subject of MINERAL VEINS is treated under its own head, as also in the articles COPPER, GOLD, IRON, LEAD, &c. But for a satisfactory elucidation of these and other topics reference must be made to works specially devoted to their treatment. In Lyell's "Principles of Geology" may be most profitably studied the subject of geological dynamics. The work is expressly an attempt to explain the former changes of the earth's surface by the operation of causes now in action, and consequently comprehends a minute investigation of the changes taking place both in inorganic and organic nature. The movements going on upon the surface and coming under the observation of man are considered sufficient, if extended through indefinite periods of time, to account for all the phenomena presented in the strata of the earth, without the necessity of attributing to the forces any greater intensity of action than they such as these, if continued during the long geological periods, would suffice to produce the required conditions. The geological formations in the phenomena of their deposition demand these periods, perhaps fully as long as could be required on the supposition that no other and no more energetic causes than those now in action had influenced their production. But those who oppose this theory maintain, that although periods of time of illimitable duration must be admitted, it is unphilosophical to make the intensity of the forces in action, as observed in the short period of man's existence, a measure of the intensity of the same forces in all past epochs. In particular the elevation of mountain chains, it seems to many, must have exhibited more energetic action than has ever been displayed in the experience of man. Several so called theories of elevation have been applied to explain these phenomena. Von Buch and Elie de Beaumont regarded many mountain masses as "craters of elevation," produced by the expansive force of elastic vapors beneath, lifting up and breaking open the horizontal strata, and leaving them arranged in conical form, or sloping on each side from a central line. Thus Etna and Vesuvius, they contended, were lifted up by a single movement; and the same force of elastic vapors might in like manner have been the agent which suddenly uplifted the mountain chains. (See ELIE DE Beaumont.) These views are opposed by many serious objections. The theory of the professors Rogers is referred to in the article EARTHQUAKE.-The views of Prof. James Hall upon this subject have not been before published. They are the result of extended observations, continued for many years, of the strata in their most dis turbed condition in their eastern belt, and of the same formations in the western states, where they lie nearly horizontally, and are filled with organic remains. Toward the west the thickness of the sedimentary deposits is known to gradually decrease; over the great western plateau there is no evidence in the formations of metamorphic action, and trap dikes are of very rare occurrence. Approaching the Appalachian range, the strata become more variously folded and contorted, and the structure of the rocks more compact, in some instances crystalline, and their cleavage more irregular. The introduction of crystalline minerals, and a thicker bedding of the limestones, also bear evidence of progressive metamorphism, which as traced eastward is accompanied by a more broken topography, and ridges and valleys of more marked outlines, corresponding in direction to the synclinal and anticlinal axes. Neither in New England, from the sections across this portion of the range made by Prof. Hall in 1843 and 1844, nor in Pennsylvania and Virginia, according to the observations of other geologists, is there an appearance of any great central axis of older date than the rest, or of igneous rock, theoretically regarded as the necessary nucleus of a mountain chain. On the contrary, many of the highest ridges are composed of strata lying in synclinal axes, the newest, and not the oldest, forming the summits. Seeing that these phenomena could not be explained by reference to any central axis or other theory of elevation, Prof. Hall was led to adopt entirely new views of the origin of mountain chains, after submitting the results of his own widely extended observations and those of others to careful study. The strata uniformly increasing in thickness toward the mountain ranges was an effect evidently not due to their folding. The Appalachian chain, composed of palæozoic strata, coinciding in direction with the original line of greatest accumulation of deposits of that period, the currents which produced them must have had the same direction; and on each side the sediments must have gradually thinned away in finer materials. The anticlinal and synclinal axes having also this same general direction, and the action of metamorphism traceable along the same lines increasing with the increasing thickness of the rocks, and with the frequency of the plications of the strata, Mr. Hall infers that the mountain ranges result from the original accumulation of sediments, their height depending on the quantity of this original deposit, and that the elevation of mountains can nowhere be so great as the original thickness of the sediments. By subsequent action their surface has been abraded and their forms modified; but the greatest portion of the mass still lies below the horizon of our observations. This conclusion is quite true respecting the Appalachian chain, where the sedimentary deposits present a thickness varying from 30,000 to 50,000 feet, while the summits range only from 4,000 to 6,000 feet in height. The same views, sustained by a large accumulation of facts, he applies to the Rocky mountain range, where the sediments of later periods increase the elevation, and the same principles are believed by the author of the theory to be applicable to all mountain ranges, the elevations being dependent on the amount of original accumulation of sediments; where these deposits consist of the aggregate of several geological periods, all the higher will be the mountain chains. The mountains of the northern United States and Canada, capped by rocks of the Laurentian age, never reach the elevation of those of the Appalachian, composed of the palæozoic formations; and the height of these is surpassed by the summits of the central and western portions of the continent, capped by strata of still later age. In the Alps and other ranges of great elevation the summits are sometimes composed of still more modern formations. These views would restore to some extent the opinions of the older geologists, who always regarded mountain chains as part of the continental elevation. The hitherto unexplained feature of the plication of the strata in numerous nearly parallel axes are accounted for by Mr. Hall thus. It is well understood that the ocean of these ancient deposits must have been always shallow, notwithstanding the immense thickness they attained; and that its bed must have been more or less continually subsiding. This movement would involve a lateral pressure, which would result either in ruptures below or a sliding upon each other of the middle strata, or else cause the upper portions of the mass to become folded in lines parallel to the line of subsidence, which would be the original line of accumulation. The same cause that raised the more elevated portions uplifted the thinner and less disturbed deposits, the whole being a continuous elevation. The present Gulf stream of the Atlantic and other great oceanic currents are collecting ranges of sediments, which if uplifted with the whole oceanic area would no doubt present features similar to those ofexisting mountain chains from which the direction and force of the currents might be traced. GEOMETRY (Gr. yn, the earth, and μerpew, to measure), the science which treats of order and proportion in space. One of the oldest and simplest of sciences, it is nevertheless variously defined, and its fundamental definitions are variously announced, according to the philosophy of the writer. Thus it has been defined, in accordance with the etymology of the word, as the science of measurement, as though it dealt with material things. A portion of space such as might be filled by a solid body is itself called in geometry a solid. The boundaries of this geometrial solid are called surfaces. But geometrical surfaces may be conceived as not bounding solids, but as simply separating space from space. They constitute one form of the zero of space, having length and breadth without thickness; they are a zero of solidity, but have a magnitude of their own, called super ficial area. If a surface be limited in extent, the boundary on any side is a line. A line is the 2d form of the zero in space, being zero not only in solidity, but in superficial area, and having a magnitude in length alone. A line may be conceived as not bounding a superficies, but as being simply a fixed zero of the 2d form. If the line be limited in extent, its extremities are points. A point may also be conceived as a zero in space, of the 3d order, a zero not only in solidity and in superficies, but in length also -having no magnitude or proportion, and retaining only order, or position, as the sole clement of its existence. In the position of points, the difference in the direction of a first and seccond point from a third is called an angle, and the natural unit for measuring angles is oppositeness of direction. We have here all the elemental conceptions of geometry, viz.: a point, a line, a surface, a solid, and an angle, that is, order of position; and magnitude, or proportion of distances. From these definitions as data a vast amount of geometrical science may be deduced by the laws of logic; and geometry has by some been regarded as a purely logistic science, which merely develops the truths implied in its definitions. Geometers themselves, however, usually regard these definitions as descriptions of realities independent of matter; to them a sphere is more real than a globe, and their science is daily discovering sublime truths of the relations of space, not implied in these elemental conceptions, but only connected with them by indissoluble ties of mutual relationship. The history of geometry is divided by Chasles, in his valuable Aperçu historique des méthodes en géométrie, into 5 periods. The 1st is that of the Greek geometry, lasting about 1,000 years, and ending about A. D. 550. Then after a pause of 1,000 years, the 2d period began in the revival of ancient geometry about 1550. A 3d period or epoch was marked in the beginning of the 17th century by Descartes' coordinates. The 4th period was inaugurated in 1684 by the "sublime invention" of the differential calculus. The 5th era is marked in our own century by Monge's "Descriptive Geometry," by which he developed the idea of reducing problems of solid geometry to problems in a plane. Since Chasles' Aperçu historique was published, a 6th period has been introduced by the publication in 1853 of Hamilton's "Quaternions." Greek geometry began, it is said, with Thales and Pythagoras, who obtained their first ideas from the Egyptians and from India. The Pythagorean school demonstrated the incommensurability of the diagonal of a square with its sides, and investigated the 5 regular solids. They had some knowledge of triangles and circles, and possibly were acquainted, as it is affirmed they were, with the fact that the circle and the sphere are the largest figures of the same perimeter or the same surface. About a century after Pythagoras, the great Plato and his disciples commenced a course of rapid and astonishing discoveries, through the study of the analytic method, conic sections, and geometric loci. The ancient analytic mode consisted in assuming the truth of the theorem to be proved, and then showing that this implied the truth only of those propositions which were already known to be true. In modern days the algebraic method, since it allows the introduction of unknown quantities as data for reasoning, has usurped the name of analytic. Conic sections embrace, as is well known, the study of the curves generated by intersecting a cone by a plane surface. Within 150 years after Plato's time this study had been pushed by Apollonius and others to a degree which has scarcely been surpassed by any subsequent geometer, and his works, embracing his predecessors' discoveries as well as his own, proved 19 centuries afterward the foundation of a new system of astronomy and mathematics. Geometrical loci are lines or surfaces defined by the fact that every point in the line or the surface fulfils one and the same condition of position. The investigation of such loci has been from Plato's day to the present one of the most fruitful of all sources of geometrical knowledge. Just before the time of Apollonius, Euclid introduced into geometry a device of reasoning which was exceedingly useful in cases where neither synthesis (i. e. direct proof) nor the analytic mode is readily applicable; it consists in assuming the contrary of your proposition to be true, and then showing that this implies the truth of what is known to be false. Contemporary with Apollonius was Archimedes (287-212 B. C.), who introduced into geometry the fruitful idea of exhaustion. By calculating circumscribed and inscribed polygons about a curve, and increasing the number of sides until the difference between the external and internal polygons becomes exceedingly small, it is evident that the difference between the curve and either polygon will be less than that between the polygons themselves; and this method may be continued by increasing the number of sides, until the difference between the curve and the polygon is as small as we please. Hipparchus in the 2d century before Christ, and Ptolemy in the 2d century after Christ, applied mathematics to astronomy; at the date of the latter writer the doctrine of both plane and spherical triangles had been well discussed by Theodosius and Menelaus. Vieta (A. D. 1540-1603), to whom we principally owe the invention of algebra, enlarged Plato's analytic method by applying algebra to geometry. To him also is due the merit of discovering the ratio of the increase of angles and the increase of their sizes. Kepler (1571-1630) introduced into geometry the idea of the infinitesimal, thus perfecting the Archimedean exhaustion; and also first made the important remark which leads to the solution of questions of maxima, that when a quantity is at its highest point its rise becomes zero. To Kepler we owe also one of the first examples of a problem of descriptive geometry, in the graphic solution of an eclipse of the sun. Soon after Kepler, Cavalieri published (1635) his Geometria Indivisibilibus, a further step in the road from Archimedes' exhaustions to Newton's fluxions. Roberval gave a method of drawing tangents identical in its philosophy with fluxions. Fermat (who shares with Pascal the credit of inventing the calculus of probabilities) introduced the infinitesimal into algebraical calculation, and applied it with great success to geometrical questions. Pascal, applying his wonderful genius to the conic sections, anticipated some of the latest inventions by his famous theorem concerning the relation of 6 points arbitrarily chosen in a conic section. But most wonderful of all the geometrical inventions of the 17th century was that of Descartes, published in 1637; it consisted simply in considering every line as the locus of a point whose position is determined by a relation between its distances from 2 fixed lines at right angles to each other. The relation between these distances, being expressed in algebraical language, constitutes the equation of the curve. By later geometers this method has been generalized so that the distances may be measured from any fixed point or line, and measured in a straight line or in a given curved line; or instead of some of the distances, directions or angles may be introduced. For a majority of the most important cases, Descartes' coördinates are, however, still the best. Huyghens, whose treatise on the pendulum is ranked by Chasles with Newton's Principia, making a glorious combination of Descartes' methods with those of his predecessors, added to geometry the beautiful theory of evolutes, which are the curves formed by the intersection of straight lines at right angles to a given curve; and he applied it not only to the pendulum, but to the theory of optics. Soon after (1686) Tschirnhausen published a wider conception of the generation of curves by straight lines. His famous caustics were made by the intersection of reflected or refracted rays of light; and he proposed other curves made by a pencil point stretching a thread whose ends were fastened, and which also wrapped and unwrapped from given curves. About the same time also De la Hire and Le Poivre invented, independently of each other, modes of transforming one plane curve into another, by making the given curve a peculiar basis for the locus of a new curve. They thus transformed the circle into all the conic sections, without any reference to a cone. The great Newton also invented a means to the same end, so that the consideration of the ellipse and parabola became independent of that of any solid. Thus these methods, especially that of Le Poivre, anticipated descriptive geometry, and perhaps prepared the way for it. In 1700 Parent generalized the method of Descartes from representing a line to representing a curve surface by an equation between the distances of a point in the surface from 3 given planes, at right angles to each other; but this was not methodically arranged, and it was left for Clairaut, in 1781, to finish this great step. Meanwhile Newton's fluxions and Leibnitz's differential calculus had come into use, and Newton, Maclaurin, and Cotes had made the most exhaustive investigation into curves of the 3d degree, and many fine discoveries in regard to curves in general. The enthusiasm which Newton's example aroused in England and Scotland for pure geometry was followed by a lull of about a century, when Monge by his "Descriptive Geometry" gave the whole study new life. The essence of descriptive geometry lies in the transmutation of figures, the reduction of geometry of. 3 dimensions to geometry in a plane. One beautiful example of this branch of science may be found in linear perspective, which simply projects the points of a solid upon a plane, by straight lines of light from the eye. Carnot, at the beginning of this century, in his "Geometry of Position" and "Theory of Transversals," also introduced valuable methods; in the first showing how to indicate the direction of lines more exactly by the use of positive and negative signs, and how to use the idea of motion in a more effective manner than before in geometry; in the second introducing that general form of the theory of transversals, i. e., of the intersections of a system of lines by one not belonging to the system, which Chasles employs so happily in his Géométrie supérieure (1852). This writer develops two principles in the correspondence of figures: one, the principle of duality, by which for a given figure a second is found such that points, planes, and straight lines in one correspond to planes, points, and lines in the other; the second, the principle of homography, by which for any figure a second is drawn such that points, planes, and lines in one correspond to points, planes, and lines in the other; the utility of each being to transfer the demonstrations of truth in one figure to the problems of another figure. In our own time many new ideas have been published, which cannot yet be properly appreciated, because their actual fruitfulness has not been tested. Sir William Rowan Hamilton of Dublin, in attempting to develop a science of time, discovered it to be a pure algebra; and pushing his inquiries further, formed an algebraic notation for questions of space, which promises to become the most valuable geometrical engine yet invented. His "Lectures on Quaternions" (1853) develop the mode of using algebraic notations with a geometrical meaning, as follows. A line, perfectly expressed, has direction as well as length. The sum of 2 sides of a triangle is therefore equal to the 3d side; in other words, you arrive at the same point whether you go through 2 sides in succession, or only through the 3d side. Quotient is the expression of ratio, and the ratio of 2 lines if perfectly expressed must give their directions as well as length. The quotient of one side of a triangle divided by another must therefore be a quaternion, that is, embrace 4 numbers, one to express their ratio of length, one their angle with each |