acteristics were not universal. With the introduction of the life record it was found possible to define periods of geological history with more definiteness, often placing their boundaries at unconformities which marked a break in the preservation of the life record, thus making a good dividing line. This study has led to the necessity for the introduction of new names and the abandonment of some of the old ones. Very commonly the new names are geographic-Devonian, from Devonshire, England, and Permian, from Perm, Russia, for example-being adopted from the region where the study necessitating the new name was made. The use of fossils has also made it possible to subdivide the larger divisions of geologic history, and the names thus introduced are usually geographical and of local significance. Thus, those of Texas differ from those of New York, California, India, or England. But the large divisions are of world-wide application. The following table gives the names commonly in use in America for the main divisions: Archæan.... Permian. Huronian (Algonkian, United States In a given region a broad statement of the stratigraphic geology would start with the oldest rocks, perhaps the Archæan, and continue down to the present. It would treat of the fossils, their characteristics, variations, and associations, and it would include a study of the structure, position, and relations of the rocks themselves. These studies would be applied to an interpretation of the history of the region, both in general and in detail: The evolution of life; the climate and its variations; the relation of sea and land, and their variations in relation; the nature of sedimentation and the conditions accompanying it; the geographic conditions and the changes in past geography, with causes; periods of volcanic activity and their effects; the growth of mountains and their reduction; in a word, all the many and complex changes and interactions and interrelations of conditions which have helped to make the geological history. It is such a complicated subject that no adequate abstract is possible in an article of this scope. In fact, stratigraphic geology, being a history of the past, differs for each locality, and can be properly discussed only in treatises on geology. Much on stratigraphic geology is, however, given in various articles on specific topics. See PALEONTOLOGY; PALEOBOTANY; ARCHÆAN SYSTEM; CAMPRIAN SYSTEM; SILURIAN SYSTEM; etc. GLACIAL GEOLOGY. One of the last great episodes in geological history was the advent of great ice sheets from northern lands, invading and overwhelming Northeastern America and Northwestern Europe. Because of its recency (in the Pleistocene periIt od), the record of this invasion is clear." lowered the hills, deepened the valleys, scoured, grooved, and polished the rocks, and transported soil and boulders in its onward march, leaving them in complex deposits when it melted back. These deposits clogged the valleys, turning streams aside and causing them to carve new valleys, which are now gorges with rapids and falls, and by making dams across the streams many lakes were ponded back in the stream valleys. In its advance the ice sheet drove out both animal and plant life, and many interest ing effects on life were produced. A study of these records, and an interpretation of the events which they record, is the province of glacial geology. The time of coming, the length of duration of the ice invasion, and the length of time since its withdrawal, are not known in years. From 5000 to 30,000 years is the estimated time since the withdrawal of the ice-probably nearer the former than the latter. The duration of the ice invasion was many times the length of the PostGlacial period, and was great enough for a large amount of work to be performed. The beginning and the end of the Glacial period are included in the Pleistocene, so that even the time of coming is a recent geological event, being post-Tertiary. There is increasing evidence that the Glacial period was complex, consisting of several ice advances, with intermediate periods of deglaciation, or interglacial epochs. Much discussion has arisen on the question of the cause of the Glacial period, without, how ever, arriving at definite results. That the land in the glaciated regions at the beginning of the Glacial period was higher than now is demon strated; and it seems certain that, could the land be once more raised to that elevation, glaciation would again set in. The land is now rising again in the glaciated region, and the query may well be raised: Are we living in an interglacial period? See GLACIAL PERIOD. ECONOMIC GEOLOGY. A great number of geological products have economic value, and our industrial development of the present time is dependent upon these prod ucts. The investigation of these from the standpoint of their occurrence, origin, and uses be longs to the economic geologist. Of the topics of economic geology, undoubtedly the most impertant is the soil. Its origin, distribution, variations in texture and chemical composition, and the means of bettering it and of properly utiliz ing it, are questions of high importance. Building products-the building-stones, cement materials, and clays-form a second important group; mineral fuels, including coal, natural gas, and petroleum, a third group; and metallic GEOLOGY. products, including both the precious and baser Besides these, metals, form a fourth group. there are many lesser products-the precious stones, abrasive materials, salt, gypsum, fertilizers, etc. The number of industries dependent upon this varied list of geological products, and the vital relation of several of them to modern civilization, show the value of a thorough and scientific knowledge of the nature and cause of It is the importance of this their occurrence. economic aspect of geology that has led governments, both State and national, to support expensive geological surveys. For a scientific study of economic geology, other aspects of geology must also be considered; consequently the whole field of geology has profited from the need of study of the economic aspect. See ORE DEPOSITS; MINING. THE HISTORY OF GEOLOGY. Catastro 571 Geology ranks as one of the youngest of the sciences. In the latter part of the eighteenth century the discussion was being waged with warmth by Hutton and his followers on the one hand, and Werner and his followers on the other hand, as to whether any but the most recent igneous rocks were to be ascribed to other than aqueous agencies, as Werner affirmed. phism was rampant, and articles on that phase of natural philosophy which dealt with the earth history were mainly philosophical polemics deThe clergy took a fending some hypothesis. share in the discussions, opposing any theory of earth history which seemed at variance with It had the then existing dogmas of theology. not yet come to be the custom in the natural sciences to gather facts patiently, weigh them carefully, and endeavor to draw logical conclusions from them. Rather it seems to have been the custom to take such facts as appeared, philosophize upon them, and defend the conclusions with vigor against all comers and all fact. James Hutton, in 1785, sounded the first note of the new geology when he said that he saw "no traces of a beginning, no prospect of an a foundation end." This generalization, now was based stone of the geological structure, upon a wide and thoughtful study and upon In Playfair's many carefully gathered facts. Illustrations of the Huttonian Theory are to be found many of the principles of modern geology. A second great epoch, in the history of geology was the work of William Smith, at the close of the As has been stated above, eighteenth century. his work made possible the division of the geobased upon scientific logical record into ages principles. His work, therefore, stands as the foundation of stratigraphic geology. The work of Hutton and Smith made it possible for others to follow, and quickly facts began to accumulate and conclusions to be drawn which gave to geology the right to be considered as a separate science. Sir Charles Lyell, sometimes called the founder of modern geology, gathered these results and added to them his own, putting them together as a system in his Principles of GeoloHe vigorously gy, still a geological classic. promulgated his system, and was, without doubt, the greatest and most effective of geological teachers. In these earliest days of geology as a science Americans had but little share; but before the middle of the century James Hall, James D. VOL. VIII.-37. or Dana, and others were vigorously at work on the Out of the old natural philosophy have come GENERAL WORKS: Playfair, BIBLIOGRAPHY. Illustrations of the Huttonian Theory of the Earth (Edinburgh, 1802); Lyell, Principles of Geology (2d ed., London, 1875); A. Geikie, Textbook of Geology (London, 1893); Dana, Manual of Geology (4th ed., New York, 1895): Le Conte, Elements of Geology (New York, 1891); Prestwich, Geology, Chemical, Physical, and Stratigraphical (Oxford, 1886-88): A. Heilprin, Principles of Geology (1890); Bischof. Chemical and Physical Geology (London, 1854-59); Scott, Elementary Geology (New York, 1883); Tarr, Elementary Geology (New York, 1897); JukesBrowne, Handbook of Physical Geology (New York, 1893); Winchell, Geological Studies (New York, 1886); Geikie, Outlines of Field Geology (3d ed., London, 1883).-COSMICAL GEOLOGY: Croll, Climate and Time (Edinburgh, 1885); Fisher, Physics of the Earth's Crust (London, 1881); Ball, The Earth's Beginning (London, 1901); Green, Vestiges of a Molten Globe (London, 1887).-—MINERALOGY AND PETROGRAPHY: Dana, Manual of Mineralogy (3d ed., New York, 1878); Moses and Parsons, Mineralogy, Crystallography and Blowpipe Analysis (New York, 1895); Rosenbusch, Mikroskopische Physiographie der Mineralien und Gesteine (Stuttgart, 188587); Zirkel, Lehrbuch der Petrographie (Leipzig, 1893-95).-DYNAMIC AND STRUCTURAL GEOLOGY: Suess, Das Antlitz der Erde (Prague, 1883-88); Daubrée, Etudes synthétiques de géologie expérimentale (Paris, 1879); Merrill, Rocks, RockWeathering and Soils (New York, 1897); Geikie, Earth Sculpture (London, 1898); Shaler, Aspects of the Earth (New York, 1890); Reade, The Origin of Mountains (London, 1886); Dana, Characteristics of Volcanoes (New York, 1890); Bonney, Volcanoes (London, 1898); Russell, Volcanoes of North America (New York, 1897); Geikie, Ancient Volcanoes of Great Britain (London, 1897); Hull, Volcanoes, Past and Present (London, 1892); Judd, Volcanoes (New York, 1881); Milne, Earthquakes (New York, 1886); Powell, Canyons of the Colorado (Meadville, Pa., 1895); Russell, Rivers of North America (New York, 1898); Russell, Lakes of North America (Boston, 1894); Hovey, Celebrated American Caverns (Cincinnati, 1882); Darwin, Coral Reefs (London, 1891); Dana, Corals and Coral Islands (New York, 1890).-PHYSIOGRAPHIC GEOLOGY: Gilbert, Geology of the Henry Mountains (Washington, 1877); Davis, Physical Geography (Boston, 1900); Tarr, Elementary Physical Geography (New York, 1895); Geikie, The Scenery of Scotland (London, 1887); Avebury (Lubbock), Scenery of Switzerland (London, 1896).-STRATIGRAPHICAL GEOLOGY AND PALEONTOLOGY: ZittelEastman, Text-book of Paleontology (London, 1900); Keyser-Lake, Text-book of Comparative Geology (Stuttgart, 1893); Nicholson and Ly dekker, Manual of Paleontology (London, 1889). -GLACIAL GEOLOGY: Geikie, The Great Ice Age (New York, 1895); Wright, The Ice Age in North America (New York, 1890); Russell, Glaciers of North America (Boston, 1897); Bonney, Ice Work (New York, 1896).-ECONOMIC GEOLOGY: Phillips, Treatise on Ore Deposits (London, 1896) Kemp, Ore Deposits of the United States and Canada (3d ed., New York, 1901); Tarr, Economic Geology of the United States (New York, 1894).-HISTORY OF GEOLOGY: Geikie. The Founders of Geology (London, 1897). GEOLOGICAL REPORTS AND PERIODICALS: The gov ernments in both America and Europe have geological bureaus which are actively engaged in the investigation of geological problems. In the United States this bureau, known as the United States Geological Survey, publishes reports, bulletins, and monographs of great value. There are also geological surveys in operation in the different States. Among the leading geological journals in America may be mentioned: Journal of Geology (Chicago); American Geologist (Minneapolis); American Journal of Science (New Haven); Bulletin of the Geological Society of America (Rochester). In England the leading journals are the Geologist (London), and Quar terly Journal of the Geological Society (London). The leading German periodicals are: Neues Jahrbuch für Geologie, Mineralogie und Paläontologie (Stuttgart), and Zeitschrift für praktische Geologie (Stuttgart). GE'OMAN'CY. See SUPERSTITION. GEOMETRIC MEAN. If three quantities, a, b, c, are in geometric progression, b is called the geometric mean between a and o; e.g. 2. 4. 8 are three such numbers, 2 being the rate, and 4 is the geometric mean. From the nature of the series b a b с or b2 = ac, and b = Vac. = The positive value of the square root is usually, but not necessarily, taken as the geometric mean when a and c are positive, the negative value being taken when a and c are negative-e.g. the geometric mean between 2, 8 is v 16 + 4, but between -2, -8, it is V16-4. The several terms of a geometric series which lie between two numbers, as a, l, are called the geometricmeans between a, l. The geometric mean of n positive real quantities is the positive value of the nth root of their product-e.g. the geometric mean of 8, 27, 64 is ₫ 8 · 27 · 64 = 24. GEOMETRIC PROGRESSION. See SERIES GEOM/ETRID MOTH. A moth of the family Geometrida, whose caterpillars are called inchworms, loopers, measuring worms, or spanworms. The family name ('ground-measurer') refers to the peculiar method of locomotion of these caterpillars. The geometrids form one of the largest divisions of the Lepidoptera, scattered over all parts of the world, characterized by slenderness of body and the absence of tufts or crests on the thorax; the wings are usually lightcolored, with intricately variegated markings, but some species are distinctly marked with bright colors. They are of medium or small size. For an account of the caterpillars, see MEASUR ING WORM. The caterpillars of the geometrids are among the most destructive pests of tree foliage and garden plants, as is described in the accounts of such species as the canker-worm, gooseberryworm, raspberry-worm, etc. A magnificent monograph of the American geometrids, by Dr. A. S. Packard, was issued as volume x. of its quarto publications by the United States Geological Survey in 1876; it contains thirteen colored plates and gives all American species then known. GEOMETRY (Lat. geometria, from Gk. yewμerpla, geometria, from yewμérpns, geometris, geometer, from yn, ge, earth + μérpov, metron, measure). The science of form. Geometric concepts arise from the consideration of forms of actual objects, just as numerical concepts arise from the consideration of collections of objects E.g. the idea of a cube results from observing that the corresponding physical object, as a die occupies a certain part of space. the first geometric assumption, viz. that space is divisible. In this case it is divided into two parts, that within the cube and that outside of it. Geometry considers only the former, the space occupied by a substance. This space is This implies GEOMETRY. 578 called a geometric solid or simply a solid. The boundary between the space and that outside of it is a surface. A surface, being itself an element of space, is also divisible, and the boundary between two parts of it is called a line. A line, in turn, is divisible by a point. The number, comparative size, and position of these elements unite to make the concept cube. With accurate ideas of point, line, surface, solid, it is easy to imagine a world of geometric figures formed by their combinations. It is then only necessary to add concise definitions and axioms (q.v.) to found a system of geometry. But the validity of these assumed premises must determine the validity and scope of the resulting science-a fact forcibly exemplified in the case of Euclidean geometry. Geometry was developed by the ancients, espe cially by the Greeks, to a high degree. But their constructions and solutions in elementary geome try were generally effected by the use only of the straight edge and compasses (instruments corresponding to the geometric elements, straight line and circle). Their achievements were, there fore, limited, and such problems as the trisection of an angle, the duplication of a cube, and all those which cannot be expressed by equations of the first or second degree, remained unsolved until the introduction of other instruments. The word geometry' signifies land-measure, and Herodotus attributes the origin of this science to the necessity of resurveying the Egyptian fields following each inundation of the Nile. He refers to the plan of taxation enforced by Sesostris (Rameses II.), which required a survey of the land. Proclus also confirms the Egyptian origin of geometry by saying that Thales introduced this art from that country into Greece. The greatest among the disciples of Thales was Pythagoras, who formulated deductive geometry, and discovered many important propositions. Among the illustrious successors of Pythagoras were Anaxagoras, Enopides, Bryson, Antiphon, Hippocrates of Chios (who duplicated the cube, but not by elementary geometry), Zenodorus, Democritus, and Theodorus. To this list should be added the name of Plato, who introduced a new epoch in the science by formulating the method of geometric analysis, and emphasizing the necessity of accurate definition. Menæchmus, a contemporary of Plato, discovered the conic sections. Among those who studied at the Academy of Plato were Eudoxus, the inventor of proportion, exhaustions, and many theorems found in Euclid's Elements, and Aristotle, who improved many geometric definitions. The name of Euclid marks another epoch in the history of geometry. Euclid's work is remarkable not for its originality, but for its simplicity and perfection as a logical system, based as it was on the discoveries of his predecessors. This work of fifteen books, called the Elements, has for over two thousand years formed the basis of ele mentary instruction in geometry wherever the For the development science has been taught. of the geometry of conic sections we are indebted to Apollonius of Perga, and to Archimedes. The later Greeks also cultivated geometry enthusi astically, as is attested by Nicomedes and Hip parchus, and in the Christian Era by Ptolemy and Pappus. The elementary plane geometry ordinarily studied in our schools is based directly, or indirectly through the work of Legendre, upon Euclid's Elements. Of this classic work, the first as a text-book in the schools of England and her The basis of ancient geometry as set forth in the Elements went practically unchallenged until the nineteenth century. The renewed interest in the science, growing out of the Renaissance, inspired the investigation of Euclid's assumptions, and led mathematicians to seek to demonstrate the fifth postulate or twelfth axiom (given by Brill as the eleventh), viz. that two unlimited straight lines intersect on that side of a transversal on which the sum of the interior angles is less than a straight angle. Among the eminent mathematicians who sought to show the dependence of this proposition upon those preceding it were Legendre and Gauss. Lobatchevsky and Bolyai were the first to construct a geometry innon-Euclidean geometry. dependent of Euclid's assumption, and thus to found the so-called Then at once followed a great advance toward exploring the new field, and from the researches of Riemann, Helmholtz, and Beltrami, it is concluded that ten of the Euclidean assumptions are valid for all geometry, but that the one just mentioned and "two straight lines (or, more generally, two geodetic lines) include no space," are limited to the properties of particular space. Riemann and Helmholtz formulated assumptions for a geometry in space of n-ply manifoldness and with constant curvature, and observed that on the sphere, whose curvature is constant and positive, the sum of the angles of a triangle is less than a straight angle, this characterizing the space of the geometry of Bolyai and Lobatchevsky. Klein has designated these three geometries respectively, the elliptic, parabolic, and hyperbolic Starting with this broader view, many of the leading mathematicians of the last quarter of a century, including Cayley, Lie, Klein, Pasch, Killing, Fiedler, and Mansion, have given much attention and made valuable contributions to the subject of geometry. Without questioning the validity of Euclidean geometry, there have grown out of it in modern times two great systems-an analytic, or coördinate (see ANALYTIC GEOMETRY), and a synthetic, or 'modern' geometry. The latter embraces descriptive and projective geometry, although systems of coördinates have been introduced also in the latter. DESCRIPTIVE GEOMETRY. This has for its object the representation of solids upon two planes at right angles to each other, these planes then being, for convenience, flattened out into a single plane. This may be done in a variety of ways, but the original method is that of parallel rays perpendicular to the planes, and known as the or thographic or orthogonal projection. These projections are commonly made, one on a horizontal plane (called the plane of the figure), and one on a vertical plane (called the elevation); e.g. take H P a circle as the given figure, and let H V be the planes of projection intersecting in X X'. Draw P P2, P' P', perpendicular to H, P P1, P' P', perpendicular to V. The rays from P determine the plane perpendicular to X X' at P', and those from P' determine a plane 1 to X X' at P'. Continuing in this way, the circle is projected into an ellipse on H, and into an ellipse on V. The plane V may now be revolved about X X' through 90°, causing the projection P2 P', to form P, P', and thus representing two projections of the circle in the same plane. This process is entirely reversible, from which it is clear that a figure may be constructed from its projections. Descriptive geometry is a powerful agent in solving the prob lems of mechanics and the constructive arts; e.g. in the planning of machinery, arches, and conduits. 2 the axial pencil (planes with a common axis) corresponds to the pencil of rays (lines with a common point). If two ranges of points, as A, B, C,.... and A', B', C'...., or as A, B, C.... and A", B", C".... in the accompanying figure, are such that the lines which join corresponding points concur, as at S, the two ranges are said to be in perspective; but A, B, C and A", B", C" are said to be projective. The anharmonic ratio (see ANHARMONIC RATIO) of projective ranges is constant, i.e. (A" B" C" D") = (ABCD). This property forms the basis of the general definition of projective plane figures, which may be stated thus: Any two plane figures in which for every point of the one there is a point in the other, and for every line in the one there is a line in the other, and so related that the anharmonic ratio of any corresponding ranges of four points or corresponding pencils of four lines are equal, are said to be projective. transforming a plane figure into a plane figure by means of projective pencils, or three-dimensional figures into three-dimensional figures by means of a sheaf of rays. In the broader sense. projective geometry also includes the study of the corresponding forms of various dimensions; e.g. HYPER-GEOMETRY. Generalization has led geometers to imagine other spaces than that in which we live, and to seek the properties of figures existing in space of more than three dimensions. The result has been the building up of a geometry of hyper-space or of n dimensions. Reasoning in this geometry is possible only by the use of symbols. Since a line segment, i.e. a figure of one dimension, is represented by an algebraic quantity of degree 1, such as a; since a square having two dimensions is represented by the algebraic expression a2; and, finally, since a cube, having three dimensions, is represented by the algebraic expression a3-the idea naturally sug gests itself that some figure of four dimensions corresponds to the symbol a*, and that, in general, some figure of n dimensions corresponds to the symbol an. The fact that four dimensions cannot be represented in the three-dimensional space in which we live has little bearing upon the idea itself; a three-dimensional figure (a solid) cannot completely be represented on a plane, and yet mathematical thought involving the concept of three-dimensional space would remain logical and useful even if all actual figures were only two-dimensional. The idea of the fourth dimension thrusts itself upon the mind even more prominently in studying rectangular coördinates in analytic geometry; ax = b represents a point, one axis being necessary; ax + by =c represents a line, two axes being necessary; and ax + by + c = d represents a plane, three axes being necessary. This suggests that ax + by + cz + dwe may represent a three-dimensional figure in a fourdimensional space. It is evident that just as we can draw in a plane the nets of the five regular bodies, we ought to be able, by analogy, to model in three-dimensional space the solid nets of all the six structures of four-dimensional space corresponding to the five regular bodies. This has been done by Schlegel, the models being made by Brill of Darmstadt. The figure corresponding to the square and cube may be described as follows: It is bounded by 8 cubes, just as the cube is bounded by 6 squares; it has 16 corners, 24 squares, and 32 edges, so that from every corner 4 edges, 6 squares, and 4 cubes proceed, and from every edge 3 squares and 3 cubes. Thus, reasoning by analogies, mathematicians have gradually developed higher geometric systems, and have succeeded in greatly extending the scope of geometry The idea of higher dimensions has been brought |